a frequency domain finite element model for tidal

A FREQUENCY DOMAIN FINITE ELEMENT MODELFOR TIDAL CIRCULATION

by

Joannes J. Westerink, Keith D. Stolzenbach,and Jerome J. Conner

Energy Laboratory Report No. MIT-EL 85-006

January 1985

A FREQUENCY DOMAIN FINITE ELEMENT MODEL

FOR TIDAL CIRCULATION

by

Joannes J. Westerink

Keith D. Stolzenbach

Jerome J. Conner

Energy Laboratory

and

R. M. Parsons Laboratory forWater Resources and Hydrodynamics

Department of Civil EngineeringMassachusetts Institute of Technology

Cambridge, Massachusetts 02139

Sponsored by

Northeast Utilities Service Company

and

New England Power Service Company

under the

MIT Energy Laboratory Electric Utility Program

and by

The Sea Grant Office of NOAAU.S. Department of Commerce

Energy Laboratory Report No. MIT-EL 85-006

January 1985

ABSTRACT

A highly efficient finite element model has been developed for the

numerical prediction of depth average circulation within small scale

embayments which are often characterized by irregular boundaries and

bottom topography.

Traditional finite element models use time-stepping and have been

plagued with requirements for high eddy viscosity coefficients and small

time steps necessary to insure numerical stability, making application

to small bays infeasible. These problems are overcome by operating in

the frequency domain, an intrinsically more natural solution procedure

for a highly periodic process such as tidal forcing. In order to handle

non-linearities, an iterative scheme which updates non-linearities as

right hand side force loadings must be implemented.

Pioneering efforts with the harmonic approach have had shortcomings

in either not modeling all physically relevant terms and/or in not

gearing towards application to small scale regions. Small embayments

are often quite shallow and have rapidly varying depth, making the

nonlinear terms in the governing hydrodynamic equations much more

significant. This requires that more frequencies be used in order to

resolve the tide and account for the greater nonlinear coupling due to

bottom friction, convective acceleration and finite amplitude effects.

In order to make the process of handling this wide range of frequencies

manageable, a hybrid frequency-time domain approach is applied. The

iterative scheme revolves around a highly efficient linear core code

which can handle a wide range of frequencies. Furthermore, instead of

Fourier expanding the nonlinear terms, an efficient least squares error

minimization algorithm is used for the discrete spectral analysis of the

iteratively updated psuedo-force time history generated by the non-

linearities.

With this highly efficient scheme it is now possible to efficiently

study both short period and long term residual circulation within small

scale embayments.

-3-

ACKNOWLEDGMENTS

This report is part of a research program to develop more efficient

and accurate ciculation and dispersion models for coastal waters. The

report describes a two-dimensional non-linear frequency domain model

(named TEA-NL) for the analysis of circulation in tidal embayments. The

work is an extension of a linear model (TEA) described in

Westerink, J. J., Connor, J. J., Stolzenbach K. D., Adams,

E. E., and Baptista, A. M., "TEA: A Linear Frequency Domain

Finite Element Model for Tidal Embayment Analysis," Energy

Laboratory Report No. MIT-EL 84-012, February 1984

Also developed as part of this reserch program is a two-dimensional

transport model (ELA) which combines Eulerian and Lagrangian techniques

and is described in

Baptista, A. M., Adams, E. E., and Stolzenbach, K. D.,

"Eulerian-Lagrangian Analysis of Pollutant Transport in Shallow

Water," Energy Laboratory Report No. MIT-EL 84-008, June 1984

(Also published as Technical Report No. 296, R. M. Parsons

Laboratory for Water Resources and Hydrodynamics, M.I.T.)

Support for this research was provided in part by the Sea Grant

Office of NOAA, Department of Commerce, Washington, D.C., and in part by

Northeast Utilities Service Company and New England Power Company

through the M.I.T. Energy Laboratory Electric Utility Program.

-4-

TABLE OF CONTENTS

Page

ABSTRACT ................... .......... 3

ACKNOWLEDGMENTS ................... ....... 4

TABLE OF CONTENTS . .................. ...... 5

LIST OF TABLES ... . .. .... . .. .... .. . . . .. . . . 6

LIST OF FIGURES ................... .. ..... 8

1. INTRODUCTION ................... ....... 13

2. DESCRIPTION OF TIDES IN SHALLOW EMBAYMENTS . ......... . 24

2.1 Governing Equations . .... .............. . . 242.2 Harmonic Tidal Components in Estuaries . ......... 29

3. NUMERICAL FORMULATION .................... .. 43

3.1 Weighted Residual Formulation . ....... ...... 43

3.2 Finite Element Method Formulation. . ............ . 47

3.3 Frequency Domain Formulation . . . . . . . . . . . . . . . 56

4. LINEAR CORE MODEL ................... .... 61

5. NONLINEAR MODEL ............. ... .. ...... . 79

5.1 Harmonic Analysis of Non Linear Pseudo Forcings . . .. . . 79

5.2 Iterative Convergence . .................. 97

6. APPLICATION ................... ....... 104

6.1 Description of Bight of Abaco and Its Tides . ....... 104

6.2 Overtide Computations for the Bight of Abaco .. .... . 124

6.3 Compound Tide Computations for the Bight of Abaco ..... 1706.4 Discussion ................... ..... 193

7. CONCLUSIONS . ...... ............ ....... . 202

REFERENCES .... . . . .... ........ .. ........ 205

-5-

LIST OF TABLES

Table 1.1

Table 2.1

Table 2.2

Table 2.3

Table 2.4a

Table 2.4b

Table 2.5a

Table 2.5b

Table 2.6

Table 4.1

Table 4.2

Table 4.3

Table 5.1

Table 5.2a

Table 5.2b

Page

Wavelengths of a 12.4 Tide in Various Water Depths .. 22

Astronomical Tides of Importance . ......... . 30

Major Overtides. . .................. . 31

Major Compound Tides . ................ 31

Response-Forcing Table for Overtides as Generated byFinite Amplitude Term at Cycle No. 2 of Iteration. . . 35

Response-Forcing Table for Overtides as Generated byFinite Amplitude Term at Cycle No. 3 of Iteration. . . 35

Response-Forcing Table for Compound Tides as Generated

by Finite Amplitude Term at Cycle No. 1 of Iteration . 37

Response-Forcing Table for Compound Tides as Generated

by Finite Amplitude Term at Cycle No. 2 of Iteration . 37

Tides of Interest (High Freq. End) . ......... 40

Sizes and Ranks of Various Matrices. . ........ . 64

Variation of Convergence with cs . . . . . . . . . . . 69

Comparison of Analytical and Numerical Elevations and

Velocities for Example Channel Case at VariousLocations

(a) Linearized Friction Factor X = 0.0000. . ..... . 74

(b) Linearized Friction Factor X = 0.0010. . ..... . 75

(c) Linearized Friction Factor X = 0.0100. . ..... . 76

LSQ Analysis Results Showing Effects of Variation of

Number of Frequencies and Time Sampling Points;Example Simulating Overtide Type Frequencies . .... 89


Number of Frequencies and Time Sampling Points;

Example Simulating Closely Spaced Compound Tide

Frequencies. . ............ . . . . . . . . 92


Number of Frequencies and Time Sampling Points;

Example Simulating Closely Spaced Compound Tide

Frequencies . . . . . . . . . . . . . . . . . . . . . 93

-6-

Table 6.1

Table 6.2

Table 6.3a

Table 6.3b

Table 6.4

Table 6.5a

Table 6.5b

Table 6.6

Table 6.7a

Table 6.7b

Reflection and Transmission Coefficients for a LongWave Passing Over a Step from Depth hi to Depth h2for Various Depth Ratios. . . . . . . . . . . . . . .

Summary of Measured Astronomical Tides Along theOpen Ocean Boundary .. . ... .. . ................

Measurement Error for Each Frequency in Terms ofProportional Variance, Vm . . ..........

J

Measurement Error for Each Frequency in Terms ofProportional Standard Deviation, S ..........

J

Values for Friction Factor cf in Terms of Depth andBottom Roughness (from Wang and Connor, 1975) . .

Overtide Computation Errors Expressed as ErrorBetween Measurements and TEA Predictions in Terms ofProportional Variance, VP .. . . . . ............. .

J

Overtide Computation Errors Expressed as ErrorBetween Measurements and TEA Predictions in Terms ofProportional Standard Deviation, SP . . .......

J

Tides of Possible Interest for M 2 and N2 Interaction.

Compound Tide Computation Errors Expressed as ErrorBetween Measurements and TEA Predictions in Terms ofProportional Variance, V. . . . . . . .............

Compound Tide Computation Errors Expressed as ErrorBetween Measurements and TEA Predictions in Terms ofProportional Standard Deviation, SP .........

i

-7-

Page

. 107

* 111

* 122

* 123

* 136

* 165

* 166

. 171

* 190

* 191

LIST OF FIGURES

Page

Figure 2.1

Figure

Figure

Figure

2.2

3.1

4.1

Figure 4.2

Figure 5.1

Figure 5.2

Figure 5.3

Figure

Figure

Figure

6.1

6.2

6.3

Figure 6.4

Figure 6.5

Definition Sketch Showing Typical Elevation Prescribed

(FT) and flux prescribed (Q ) Boundaries . . . . . . . 28

Schematic of Major Astronomical and Shallow Water Tides 41

Schematic of Iterative Non Linear Scheme. . ...... . 60

Definition Sketch of Depth Varying Channel Which

Illustrates Convergence Problems of Iterative Linear

Scheme. . ........... . .. ............. 68

Finite Element Grid Discretization for Closed Ended

Channel Example Case. . ........ ......... . . 73

Linear Equation Generated by Least Squares Analysis

Procedure . . . . . . . . . . . . . . . . . . . . . . . 83

Effects of Variation in Frequency and Time Sampling

Rates for Typical Overtide Frequency Distribution . . . 88

Effects of Variation in Frequency and Time Sampling

Rates for Typical Compound Tide Frequency Distribution

(maximum period is T = 12.4 hours and maximum synodic

period is TS = 27 days) . .... . . . . . . . . . . . 91

Geography of Bight of Abaco, Bahamas. . ...... .. . 105

Bathymetry of the Bight of Abaco, Bahamas . ...... 109

Field Data for M 2 Astronomical Constituent (after

Filloux & Snyder, 1979)

(a) Amplitude in centimeters. . ... . . . . . . . . . 112

(b) Phase lag in radians. . ......... . . . . . 113

Field Data for M4 Overtide Constituent (after Filloux

& Snyder, 1979)

(a) Amplitude in centimeters. . ...... . . ...... . 114

(b) Phase lag in radians. . ...... . . . . . . . . 115

Field Data for M 6 Overtide Constituent (after Filloux

& Snyder, 1979)

(a) Amplitude in centimeters. . .. . . . . . . . . . . 116

(b) Phase lag in radians. . .... . . . . . . . . . . 117

-8-

Figure 6.6

Figure 6.7

Figure 6.8

Figure 6.9

Figure 6.10

Figure 6.11a

Figure 6.11b

Figure 6.12

Figure 6.13

Figure 6.14

Field Data for N2 Astronomical Constituent (afterFilloux & Snyder, 1979)

(a) Amplitude in centimeters . . . . . . . . . . . . .

(b) Phase lag in radians . . . . .. .............. .

Finite Element Grid Discretization for Bight of Abaco,Bahamas. . . . . . . . . . . . . . . . . . . . . . . ..

Results of TEA with Full Non Linear Friction Effects(cf = 0.003) for M2 Astronomical Constituent


(b) Phase lag in radians . . . . . . . . . . . . . . .


(a) Amplitude in centimeters . . . ... . . . . . ....





Trajectory along which M 2 Elevation Amplitudes areCompared for Varying Friction Factor in Figure 6.11b

Comparison of M 2 Elevation Amplitudes for VaryingFriction Factor, cf, along Trajectory S . . . . . . .

Results of TEA with Full Non Linear Friction Effects(cf = 0.009) for Steady State Constituent. .. ......

Results of TEA with Full Non Linear Friction Effects(cf = 0.009) for M4 Overtide Constituent

(a) Amplitude in centimeters .. . ... ............

(b) Phase lag in radians . . . . ................

Results of TEA with Full Non Linear Friction Effects(cf = 0.009) for M6 Overtide Constituent

(a) Amplitude in centimeters . ............. .

(b) Phase lag in radians ................

-9-

118

119

125

127

128

129

130

131

132

133

134

137

138

139

140

141

Figure 6.15

Figure 6.16

Figure 6.17

Figure 6.18

Figure 6.19

Figure 6.20

Figure 6.21

Figure 6.22

Results of TEA with Full Non Linear Friction Effects

(cf = 0.009) and Finite Amplitude Effects for M2Astronomical Constituent




(c = 0.009) and Finite Amplitude Effects for

Steady State Constituent . . . . . . . . . . . . . . .


(cf = 0.009) and Finite Amplitude Effects for

M4 Constituent




(c = 0.009) and Finite Amplitude Effects forM 6 Constituent




(cf = 0.009), Finite Amplitude Effects and Convective

Acceleration Effects for M 2 Astronomical Constituent





Acceleration Effects for Steady State Constituent. . .


(cf = 0.009), Finite Amplitude Effects and ConvectiveAcceleration Effects for M4 Overtide Constituent





Acceleration Effects for M6 Overtide Constituent


(b) Phase lag in radians . . . . . . . . . . . . . . .-10-

142

143

144

145

146

147

148

150

151

152

153

154

155

156

Figure 6.23a Velocity Results of TEA with Full Non Linear FrictionEffects (cf = 0.009), Finite Amplitude Effects andConvective Acceleration Effects for Steady StateComponent . . . . . . . . . ..................... .

Figure 6.23b Velocity Results of TEA with Full Non Linear FrictionEffects (cf = 0.009), Finite Amplitude Effects andConvective Acceleration Effects for M2 Component atTime of Maximum Ebb for the M 2 Component Relative tothe Ocean Boundary. . . . . . . . . . . . . . . . . .

Figure 6.23c Velocity Results of TEA with Full Non Linear FrictionEffects (cf = 0.009), Finite Amplitude Effects andConvective Acceleration Effects for M4 Component atTime of Maximum Ebb for the M2 Component Relative tothe Ocean Boundary. . . . . . . . ................. .

Figure 6.23d Velocity Results of TEA with Full Non Linear FrictionEffects (c = 0.009), Finite Amplitude Effects andConvective Acceleration Effects for M6 Component atTime of Maximum Ebb for the M2 Component Relative tothe Ocean Boundary. . . . . . . . .................

Figure 6.24a Continuity Equation Pseudo Forcing Vector Ratios Dueto M 2 - N2 Interaction. . . . . . . . . . . . . . . .

Figure 6.24b

Figure 6.25

Figure 6.26

Figure 6.27

Figure 6.28

Momentum Equation Pseudo Forcing Vector Ratios Due toM2 - N2 Interaction . . . . . . . . . . . . .......

Results of TEA with M2-N2 Interaction and with FullNon Linear Friction (cf = 0.009) and Finite AmplitudeEffects for M 2 Astronomical Constituent

(a) Amplitude in centimeters. . . . . . ............. .

(b) Phase lag in radians. . . . . . . . . . . . .....

Results of TEA with M 2-N2 Interaction and with FullNon Linear Friction (cf = 0.009) and Finite AmplitudeEffects for N2 Astronomical Constituent

(a) Amplitude in centimeters. . . . . . ............. .

(b) Phase lag in radians. . . . . . . . . . . ......

Results of TEA with M 2-N2 Interaction and with FullNon Linear Friction (cf = 0.009) and Finite AmplitudeEffects for Steady State Constituent. . . . . . . . ...

Results of TEA with M 2-N2 Interaction and with FullNon Linear Friction (cf = 0.009) and Finite AmplitudeEffects for MN Compound Constituent . . . . . . . . . .

161

173

174

176

177

178

179

180

181

158

159

160

Figure 6.29

Figure 6.30

Figure 6.31

Figure 6.32

Results of TEA with M2-N 2 Interaction and with FullNon Linear Friction (cf = 0.009) and Finite AmplitudeEffects for MN 4 Compound Constituent


(b) Phase lag in radians . . . . . . . . . . . . . ..

Results of TEA with M 2-N2 Interaction and with FullNon Linear Friction (cf = 0.009) and Finite AmplitudeEffects for M4 Overtide Constituent



Results of TEA with M 2-N2 Interaction and with FullNon Linear Friction (cf = 0.009) and Finite AmplitudeEffects for 2MN6 Compound Constituent

(a) Amplitude in centimeters . . . . . . . . . ....


Results of TEA with M 2-N2 Interaction and with FullNon Linear Friction (cf = 0.009) and Finite AmplitudeEffects for M6 Overtide Constituent

(a) Amplitude in centimeters . . . .. ............ .

(b) Phase lag in radians . . . . . . . . . . . ....

-12-

182

183

184

185

186

187

188

189

CHAPTER 1. INTRODUCTION

Recent years have seen the development of numerous coastal

circulation models which apply the finite element method. The principal

advantage of finite element methods over the more traditional finite

difference methods is the greater versatility allowed in grid

discretization which is especially important for small scale coastal

embayments. This feature permits the convenient fitting of the often

irregular boundaries and allows refinement of the grid in such critical

areas as high flow and bottom depth gradient regions and/or the narrow

mouths connecting these embayments to the open ocean.

These circulation models share as a general starting point the well

established shallow water equations which are derived by depth averaging

the conservation of mass and momentum equations with the application of

the hydrostatic and Boussinesq assumptions. Therefore the equations that

are applied are based on first principles and require empirical support

only for the turbulent exchanges and surface and bottom stresses. Major

differences between these models lie in features such as the type of

localized expansions used to resolve the spatial dependence of the

variables and more importantly the way in which they discretize the time

dependence. Traditionally time marching schemes have been applied which

are either explicit or implicit.

Early time domain models [Grotkop, 1973; Taylor and Davis, 1975;

Wang and Connor, 1975; Kawahara, 1978; King et al., 1974] have had

severe problems relating to the economy and accuracy of the schemes

developed. Economic constraints stem both from the large amount of

numerical manipulation required for the schemes and a maximum allowable

-13-

time step required for the accuracy and/or stability of the

computation. For explicit schemes a Courant stability constraint

necessitates the maximum time step to decrease along with element size,

making it especially infeasible to apply these models to small scale

geometries/elements. Furthermore, these early models have been plagued

with accuracy problems which relate to short wave length artificial

oscillations in elevation and velocity produced by the finite

discretization of the domain [Gray, 1980; Sani, et al., 1980]. These

models either require artificially high eddy viscosity to damp out this

short wave length noise or the numerical scheme is such that it is

inherently overdamped. The accuracy problem, however, arises from the

fact that not only the numerical noise is damped, but the longer

physical waves being simulated are also damped. This aspect of

overdamping has drawn strong criticism as to the ability of these models

to adequately simulate the physical problem described by the shallow

water equations. Efforts to overcome these shortcomings in accuracy and

efficiency have been numerous and have had varying degrees of success.

Alternatives which have been investigated include different time

integration schemes [Gray and Lynch, 1977, 1979; Niemeyer, 1979], mass

lumping schemes [Kawahara, 1982] and the examination of the effects of

mixed interpolation of elevation and velocity [Walters and Cheng, 1980;

Walters and Carey, 1983; Platzman, 1981; Williams and Zienkiewicz,

1981]. Certain investigators paid special attention to the numerically

troublesome convective terms by either applying a Petrov-Galerkin

weighting scheme (equivalent to upwinding) [Nakazawa, et al., 1980] or

by using the method of characteristics in conjunction with finite

elements [Benque, et al., 1981]. One of the more promising schemes

-14-

developed is the use of a wave type equation in conjunction with the

fundamental momentum equation as the basis of the finite element

formulation [Lynch and Gray, 1979]. Although some of these alternative

schemes have been successful at eliminating short wave length noise

without damping the longer physical waves and are more efficient than

earlier models, all of the above methods still have maximum allowable

time steps making them economically unattractive for either long term

simulations and/or small scale embayments.

A very attractive alternative to time stepping schemes which has

recently been employed is the use of harmonic analysis in conjunction

with finite elements. Because of the periodic nature of the tidal

phenomenon, the harmonic method is an intrinsically more natural

solution procedure and was one of the traditional methods for analysis

before the advent of finite difference and finite element methods

[Dronkers, 1964]. There are no time stepping limitations since this

procedure generates a set of quasi-steady (or time independent)

equations. In addition, truncation errors and the associated stability

problems caused by time stepping are precluded. Furthermore,

eliminating the time dependence from the governing equations reduces

them from equations of the difficult and time consuming hyperbolic type

to that of the elliptic type which are much more readily solved by

finite element methods. The harmonic method also offers the potential

of economically performing realistic long term simulations in tidal

embayments (e.g., up to 30 days) and calculating the associated residual

circulation.

A possible drawback of the harmonic method is the increased number

of frequencies which would be required to model a non-periodic

-15-

phenomenon such as wind driven circulation. However some winds (steady

winds and sea breezes) are periodic and furthermore any wind spectrum

can readily be harmonically decomposed. Time domain schemes become

economically more attractive when the number of frequencies required to

adequately represent the wind spectrum becomes excessive. A more

significant difficulty which arises in implementing this frequency

domain technique is that non linear terms generate additional responses

at frequencies other than the base forcing frequency.

Strategies to handle this harmonic coupling produced by the

non linear terms (finite amplitude, convective acceleration and bottom

friction terms) have consisted of either iterative procedures [Pearson

and Winter, 1977; Kawahara, 1978] or some type of perturbation analysis

[Askar and Cakmak, 1978; Kawahara et al., 1977; Le Provost and Poncet,

1978; Le Provost et al., 1981]. The iterative scheme applied by Pearson

generates a finite spectral series representing the pseudo-forcings due

to all the non linear components of the shallow water equations. A

linear solution is then used to evaluate the elevation response due to

each pseudo forcing component (at pre-determined frequencies). Pseudo

forcings are then updated using the updated responses in elevation and

velocity until convergence in the elevation is achieved. The full non

linear model has only been tested on geometries with a relatively small

number of elements while a simplified linear version has been applied to

a large grid of a deep bay [Jamart and Winter, 1980].

Kawahara [1977], on the other hand, applied a perturbation analysis

which allows grouping of terms in the expanded (by a power series)

shallow water equations in order to generate several sets of linearized

equations of varying order. In a later paper, Kawahara [1978] applied

-16-

the periodic Galerkin method for the time dependence which produces a

coupled system of non linear simultaneous equations which are then

solved iteratively. An important limitation with Kawahara's schemes is

that both bottom friction and Coriolis are omitted from the governing

equations while keeping the much less important eddy viscosity terms.

A further weak point of both Pearson's and Kawahara's work is that

they have used a Fourier series expansion in terms of integer values of

a base frequency (the M2 tide) to represent the variables. This

precludes the possibility of investigating the interaction between the

majority of the tidal components (see Table 2.1 for major tidal

components). Therefore, such effects as monthly (spring/neap) variation

(caused to a large extent by beating effects of closely packed tidal

components) can not be looked at and only a major base tide (e.g., M2)

and its harmonics may be studied.

Le Provost [1981] applies an expansion which considers the

interaction of the major closely spaced forcing components (M2 , S2 , N2,

K1 ). However, the perturbation analysis and the quasi-linearization for

bottom friction which are used only account for the non linear coupling

between the major forcing component (M2 ) and its first harmonic (M4).

The remaining astronomic constituents of the tide generating potential

(S2 , N2, KI) are treated linearly. The resulting computer code has been

applied to the English Channel and Le Provost states that the method is

constrained in its application to very shallow waterbodies.

Finally, Lynch [1981] has developed a linear harmonic model based

on the same re-arrangement of the fundamental equations as an earlier

time stepping scheme [Lynch and Gray, 1979] which was found to minimize

short wave length noise while retaining the computational accuracy of

-17-

the longer physical waves. The equations which he uses as the starting

point of his finite element scheme are the wave equation (found by

substituting the momentum equation into the continuity equation) and the

momentum equation. He found that when using similar interpolation

orders for elevation and velocity, this scheme will give accurate

solutions without noise.

As previously mentioned the harmonic method naturally lends itself

to the calculation of long term variations and residual circulations.

Residual currents may be thought of as a complicated combination of

steady and slowly varying currents due to the effects of non linear

interactions of both the main and various other astonomic tidal

components [Ianniello, 1977]. Due to infeasibility of running long term

simulation with time stepping programs, investigators have commonly

computed residual circulations either by time averaging the governing

equations [Walters and Cheng, 1980; Bonnefille, 1978; Tee, 1981] or by

time averaging the results produced by a time domain model over one or

several tidal cycles. The former technique places the inability to

perform long term computations in a set of additional unknown time

averaged terms (tidal-stress terms). The latter technique has problems

relating to both expense and accuracy. In order to capture the effects

of the variations in tidal forcing (which cause the non linear

interaction between tides and produce residual/long term circulations)

with sufficient accuracy, the simulation would have to be run over

extended periods of time. It would not only be expensive to run a time

stepping model for long periods of time but round off errors would also

propagate through the solution. Hence it is doubtful that these

previous efforts are able to model the effects that drive residual

-18-

circulation since they are not able to capture the actual physics of the

residual currents.

As is discussed in Chapter 2, the major variation in the currents

of coastal embayments generated by astronomic tidal forcings are

well described by considering a one month period. Responses then occur

at these astronomical forcing frequencies and their associated higher

harmonics (and certain lower frequencies due to the closeness of certain

of the components). Hence we have a limited number of frequencies which

need to be considered.

Therefore it now becomes even clearer that applying the harmonic

method is extremely well suited to assess low period fluctuations and

residual circulation due to tidal forcing components and their non

linear interaction. It is not only computationally convenient to do so

but also allows the calculation to be done in a manner which is based on

the same first principles with which we presently perform short term

circulation computations and furthermore allows all the significant

effects to be caught.

In summarizing the many advantages of the harmonic method, we note

that:

(i) no time stepping constraints due to small element sizes are

required;

(ii) it is well suited for the highly periodic tidal computations

in estuaries;

(iii) the results may be stored in a much more economical and

convenient form for applications with a transport model;

(iv) it allows computations of long term residual currents; and

(v) there are no cold start problems which time domain approaches

often have.

-19-

In spite of its many unique features, the harmonic method is far from

being exploited to its full potential. Pioneering efforts which have

applied this technique have had shortcomings in not modeling all

physically relevant terms, by not allowing for full interaction between

the major tidal components and in not gearing towards application of the

method to residual circulation computation and to small scale

geometries. Even though application of the method to small scale

estuaries increases the computational effort due to the often shallower

depths which increases the significance of non linearities, it is

nonetheless in these small scale regions that the effort associated with

time domain approaches becomes entirely insurmountable due to the

exessively small time steps needed.

We conclude that there is a definite need for the development of

improved strategies for computing tidally induced circulation within

coastal embayments. The present research addresses this issue with the

development of a general harmonic finite element model which allows the

in-depth study of the many complex non linear interactions which occur

in shallow waterbodies. This includes not only the investigation of the

coupling occurring between a given astronomical tide and the harmonics

it generates through the non linear terms in the governing equations but

also the complicated interactions between the various astronomical tides

and their associated harmonics. Among the nonlinear harmonic responses

to be investigated are steady and other very long period residual

circulations which are generated.

A direct iteration scheme will be used to handle the non linear

terms in the governing equations. However, inherent to any solution

scheme which iteratively updates the non linearities as right hand side

-20-

loadings is the limitation that the relative magnitude of the right hand

side non linear terms must be small compared to the left hand side

linear terms (Ketter and Prawel, 1969). The implication of this is that

the non linear solution must be a perturbed linear solution. The

significance of the non linear friction term far exceeds that of the

other non linear terms for the case of tidal estuaries. In order to

minimize the importance of non linear friction as a right hand side

term, a close approximation of the linear part of the friction term (a

major part of the friction term is linear as is shown in Chapter 2) will

be included on the left hand side of the equations. This greatly

enhances iterative stability and allows computations to be performed for

very shallow estuaries.

Furthermore there may be theoretical limitations for iterative

schemes relating to the relative size (with respect to wavelength) of

the estuary. Lamb (1932) shows that when solving for the case of an

open ended canal, the solution obtained by treating the finite amplitude

term by successive approximation will be unstable if 2n () (. ) is not

small, where x = the size of the canal and X = wavelength. Even though

the same difficulty does not necessarily occur for the case of a closed

ended canal, the criterion may be viewed as being indicative of

potential instabilities. Table 1.1 shows the wavelengths associated

with a 12.4 hour tide in various water depths without considering bottom

friction. Since it is the intent to apply the method to estuaries and

bays and not to coastal seas, the stability criterion will be small and

this potential instability will not be a limitation of the method.

In the following chapters the solution strategies applied are

discussed in detail. Chapter 2 describes the governing equations used

-21-

Table 1.1 Wavelengths of a 12.4 Tide in Various Water Depths

-22-

h x(m) (km)

2 200

10 442

100 1400

and furthermore describes important tides and the types of interactions

that may occur. In Chapter 3 the governing partial differential

equations are discretized such that numerous sets of linear algebraic

equations are generated. These sets of equations are in the frequency

domain and are coupled through the non linearities. Chapter 4 examines

various strategies to optimally solve each of these linear sets of

equations and Chapter 5 discusses the details of the implementation of

the fully non linear scheme. In Chapter 6, the program TEA-NL (Non

Linear Tidal Embayment Analysis), developed with the methods described,

is applied to an example case.

-23-

CHAPTER 2. DESCRIPTION OF TIDES IN SHALLOW EMBAYMENTS

2.1 Governing Equations

The equations which are used to describe tidal wave propagation may

be readily developed by depth averaging the Navier-Stokes and continuity

equations with the assumptions of hydrostatic pressure distribution,

constant density fluid, constant pressure at the air-water interface and

negligible momentum dispersion (or eddy viscosity). The resulting

equations are (Dronkers, 1964):

n9t + [u(h + n)],x + [v(h + n)] y 0 (2.1)

u + gn - fv - C /p(h + r) + Tb/p(h + n) + (uu + vu ) = 0 (2.2a)gt ,x x x ,x ,y

v + g~ + fu - /p(h + n) + Tb/p(h + n) + (uv + vv ) = 0 (2.2b),t ,y y y ,x ,y .

where:

u(x,y,t) components of water velocity in x and y directions,v(x,y,t) respectively

r(x,y,t) = surface elevation relative to mean sea level (MSL)

t = time

h = depth to MSL

g = acceleration due to gravity

p = water density

f = the Coriolis factor

x , = the applied surface stressesx y

b bT ,' = the bottom stresses

-24-

Bottom stresses are quantified as

xb /0 c (ux f

[Daily and

2 1/2

b 2 +v2) 1/2S/0 = C +v) v

Cf = friction factor =

1/8 fDW

g/c2

2ng

h

Harleman, 1966]:

(2.3a)

(2.3b)

Darcy-We isbach

Chezy

Manning

(2.4)

The fully non linear friction terms may be approximated by linearized

friction terms as follows:

[h/(h+n)] b/P blin /p= X

b b,lin[h/(h+n)] b/ = ln/p = XV

y y

X = cfU = linearized friction coefficient

(2.5a)

(2.5b)

(2.6)

U = representative flow velocity

For tidal flow with only one frequency present and the linearization

being performed on an equivalent work over a tidal cycle basis, X has

the following form [Ippen, 1966]:

8max 3n f

(2.7)

where

U max representative maximum velocity during a cyclemax

-25-

where

where

I~-----------

Although the linear friction term does not characterize the fully non

linear one in spreading energy to other frequencies, it can approximate

the magnitude of the actual non linear friction term quite reasonably.

Linearized friction is helpful in the fully non linear scheme as an

iterative stabilizer.

Wind stresses may be approximated by the empirical formulas [Van

Dorn, 1953; Wu, 1969]:

S= air U 2 cos 0 (2.8a)x air D 10 w

s = p ir U2 sin e (2.8b)y air D 10 w

where: U10 = wind speed

pair = density of air

cD = wind drag coefficient

S = wind approach anglew

The applied wind stress term in the governing equations may be

simplified by assuming that finite amplitude effects will be unimportant

and approximating 1/(h+n) as 1/h, a very reasonable assumption when

considering the empirical nature of the formula used to represent this

term and the inherent limitations of a depth averaged model in

simulating wind-driven circulation.

Finally, the Coriolis parameter is calculated as:

f = 20 sin6 (2.9)

where 4 = degrees latitude and 0 = radian frequency of the Earth.

The boundary conditions associated with the governing equations are

elevation prescribed and normal flux (or velocity) prescribed conditions

which are respectively expressed as:

-26-

(x,y,t) = n (x,y,t) on P (2.10)

and

Qn(x,y,t) = Qn(x,y,t) on TQ (2.11)

where

P = elevation prescribed boundary

r = flux prescribed boundary

As is shown in Figure 2.1, elevation prescribed boundaries are usually

associated with open ocean boundaries and flux prescribed boundaries are

usually associated with land (including rivers) boundaries.

Since the governing equations (2.1 and 2.2) are based on first

principles (with the exception of surface stresses), their ability to

model the circulation in estuaries depends on whether the assumptions

made in their derivation are satisfied. The depth averaging of the

equations precludes the accurate modeling of strongly stratified

estuaries and/or wind induced circulation in deep waterbodies. The

constant density assumption does not allow density driven currents to be

simulated and the hydrostatic pressure assumption rules out the modeling

of short or intermediate length waves. The importance of eddy viscosity

has been deemed as negligible in most estuaries [Dronkers, 1964] and

hence only empirical support is required for the surface stress terms.

Since we have confidence that Eqs. 2.1 and 2.2 accurately describe tidal

circulation in well mixed estuaries, the task at hand is to implement a

numerical scheme which allows these complicated partial differential

equations to be solved accurately for arbitrarily shaped estuaries

over extended periods, such that the effects of the variations

-27-

RiverInflow

Q

TIDALEMBAYMENT

r

OCEAN

Figure 2.1 Definition Sketch Showing Typical Elevation Prescribed (P )and flux prescribed (rQ) Boundaries

-28-

in tidal forcing can be included. As we have seen, the harmonic method

in conjunction with finite elements is excellently suited for this

purpose.

2.2 Harmonic Tidal Components in Estuaries

Prior to proposing a general harmonic finite element scheme which

allows the treatment of the more cumbersome non linear terms in the

shallow water equations, it will be beneficial to discuss in more detail

the nature of the non linear responses which arise in shallow estuaries.

Tides are the result of complex gravitational interactions between

the moon, sun and oceans. Tides in the open ocean may be described by

the superpositioning of a series of harmonic components which are

unaffected by the non linear terms in the governing equations. In

shallow water, however, non linear effects such as the finite amplitude

of the tide (compared to the overall depth), bottom friction and

convective acceleration become significant, and as a consequence the

astronomical tides present (see Table 2.1) generate shallow water

tides which cannot be ignored [Dronkers, 1964].

Shallow water tides may be classified as either overtides or

compound tides. Overtides correspond to the generation of a tidal

response through the non linearities by one astronomical component. The

frequencies associated with the overtides are exact multiples of the

frequencies of the astronomical tidal components which generate them

(see Table 2.2 for overtides of importance). Compound tides are the

result of the non linear interaction between two astronomical

constituents, and their associated frequencies correspond to sums and

differences of frequencies of the astronomical tide (see Table 2.3 for

-29-

Astronomical Tides of Importance

-30-

-1Tide f(hr - ) T(hr) P /P (%)

1M

LOW PERIOD

Mm 0.00131 763 = 32 days 1.2

MSf 0.00151 661 = 27.5 days 9.1

Mf 0.00305 328 = 13.6 days 17.2

SEMI-DIURNALM 2 0.08054 12 hr 25 min 100

S2 0.08333 12 hr 0 min 46

N2 0.079051 12 hr 39 min 19

K2 0.083449 11 hr 59 min 13

L 2 0.082080 12 hr 11 min 4

DIURNAL

K 1 0.04178 23 hr 56 min 58

01 0.04198 23 hr 49 min 41

P1 0.04158 24 hr 03 min 19

Table 2.1

Table 2.2 Major Overtides

Tide Freq. Comp. f(hr- 1 ) T(hr)

M4 2wM2 0.161074 6.21

2M6 3wM 0.241611 4.14

M8 4n 2 0.322148 3.10

S4 20 0.166667 6.00

S6 302 0.250000 4.00

Table 2.3 Major Compound Tides

Tide Freq. Comp. f(hr-1) T(hr)

MS4 M2 + wS2 0.163870 6.10

2MS6 2M2 + 2 0.244407 4.09

2SM 6 2 +S2 + 2 0.247204 4.05

2MS 2 2 - W S2 0.077740 12.86

MN wM2 + wN2 0.159588 6.27

2MN6 2w + N2 0.240125 4.16

22MN 2

2MN 2 2o2 - ON2 0.082022 12.19

2. . ..

-31-

_~I~_

compound tides of importance). Residual circulation corresponds to both

steady state (zero frequency) circulation which results from an overtide

type interaction or very low period compound tides produced by

frequencies of two major closely spaced tides.

Let us examine how the non linearities interact with the

astronomical forcing tide to produce overtides and compound tides. We

shall do this by following an iterative type procedure which is

indicative of the overall non linear iterative numerical scheme that

has been implemented. The non linearities are treated as right hand

side force loadings of sets of equations produced by harmonically

separating the (time domain) governing equations. Details concerning

this harmonic decomposition of the governing equations are given in

Chapter 3. For the present analysis, we shall simply state that a non

linear forcing at a given frequency will generate a response at

additional frequencies.

In general the variables describing tidal motion may be expressed

as a harmonic series of the form (for a one-dimensional tidal wave):

(x,y,t) = _ J(x,y) cos(wjt + * ) (2.12)

u(x,y,t) = uj(x,y) cos(w t + j) (2.13)

where: wj = frequency of the jth harmonic component

'n = amplitude of elevation of the jth harmonic

uj = amplitude of velocity of the jth harmonic

j = phase shift for elevation of the jth harmonic

*j = phase shift for velocity of the jth harmonic

By definition these series representations are exact for the case of

linear deep water tides where only astronomical species exist. They are

-32-

approximate for the case of non linear shallow water tides and the

accuracy with which they represent the tides depends on how many terms

in the series are considered.

The general type of terms produced by substituting our series

representations into the non linear finite amplitude term may be studied

by only considering a two term series. This yields:

(un),x f {[u cos(wt + ,) + um cos(&mt + ,m)1

[; cos(wit + ) + r m cos(Wmt + m) 9x (2.14)

This may readily be expanded to:

S(u),x (Uln),x cos(W t + I) cos(W t + c 9)

+ (u Im),x cos( Lt + * ) cos( mt + m )

+ (Um1 ,x cos(wmt + Jm) cos(t + )

+ (um ) ,x cos(mt + Vm ) cos(Wmt + m) (2.15)

We note that Eq. 2.15 contains both terms which involve only one

frequency and mixed terms which involve two frequencies. The former

describes the effect of finite amplitude produced by a single tidal

component while the latter describes the interaction of two tidal

components. The mixed term is the representative term in Eq. 2.15 and

may be further expanded through a trigonometric identity as follows:

(U1m),x cos(Wt + ) co(Wmt )

=2 (Ufm) ,x cos[( - ~,)t + (' - m)

+ cos[w + Wm)t + (0 + rm) ] l (2.16)+ COE I it

-33-

-LU~-- EDIIYLil~ ~W~CIX~_ II -~I~ ~--~. UI*E*?6~(1LYL

We conclude that the non linear finite amplitude term may be represented

by a harmonic expansion with associated frequencies equal to the sums

and differences of all the possible combinations of frequencies present

in the expansion for responses.

We now apply the previous expansions to examine how overtides are

generated through the finite amplitude term. We start out our iterative

investigation by assuming that the non linearities are non-existent and

that only one astronomical species of frequency wl exists in our

estuary. If we treat the finite amplitude term as a forcing term in the

continuity equation, we may deduce from Eqs. 2.14 - 2.16 that a tide at

frequency wl produces a forcing at both steady state (or zero

frequency) and at 2wl. Both these forcings in turn generate responses

at their associated frequencies. Hence after one cycle of our iteration

we have responses at frequencies 0, wl and 2wl. Each of these

harmonic responses will again generate two forcings equal to the sum and

difference of the associated frequency. At Cycle 2 this generates

forcings at 0, 201 and 4wl. Furthermore the various harmonic

repsonses will interact to generate mixed forcings at frequencies equal

to the sums and differences of the mixed frequencies. For Cycle 2 this

generates forcings at wl, 201 and 3wI. Table 2.4a summarizes the

response-forcing interaction which are produced at Cycle 2 by the finite

amplitude term when only one astronomical tidal forcing frequency is

present. Table 2.4b shows the interaction occuring at Cycle 3 and the

process is repeated until a sufficient number of frequencies has been

found. These integer multiples of the base frequency wl then are the

overtides associated with the astronomical tide at frequency wl. For

the tidal problem, the energy transferred to each successive harmonic

-34-

Table 2.4a Response-Forcing Table for OvertidesFinite Amplitude Term at Cycle No. 2

as Generatedof Iteration

Table 2.4b Response-Forcing Table for Overtides as Generated byFinite Amplitude Term at Cycle No. 3 of Iteration

ResponseFrequency 1 1 3

0 0 2"1 3w1

2wl 3w1 41w1 0 . 1 2w1

4w1 15w

201 0 .1

6w1301 0

-35-

ResponseFrequency 211

0 0 1 2w1

2w1 3ww1 0 l

4w11 0

__IIILILU l__ylLI_1Ls~.~llY--..C^~___ __ I L -1 11^11.~

will decrease substantially. Hence only a limited number of overtide

frequencies need be considered in order to accurately model the non

linear interactions occurring. In fact, for the finite amplitude term,

the most significant harmonics are the zero frequency and the 2w~,

although an entire series of overtides (3wl, 4wl, etc.) exist. We

note that each of these response-forcing tables is indicative of the

amount of energy being transferred to the various harmonics, in that the

more cycles it takes for a non linear frequency to show up, the less

important it is.

The convective acceleration terms will produce the same overtide

frequencies as the finite amplitude terms, even though the energy will

be distributed somewhat differently to each of the various harmonics due

to there being different phase shifts for the terms.

When two or more astronomical forcing tides exist we can again

deduce from our previous expansion that the finite amplitude and

convective terms will produce not only the harmonic overtides associated

with each individual astronomical tide, but also tidal species with

frequencies equal to the sums and differences of combinations of the

astronomical frequencies present and their associated compound tides and

overtides. Table 2.5a and b illustrate the frequencies present at the

first and second cycle. We note that the cycle at which a frequency

shows up is indicative of its relative importance. Furthermore we note

that all frequencies which appear at a particular cycle are not

necessarily equally important as can be seen by examining the individual

expansions.

The non linear friction term differs somewhat from the finite

amplitude and convective terms in that all frequencies generated by the

-36-

Table 2.5a Response-Forcing Table for Compound Tides as Generated by

Finite Amplitude Term at Cycle No. 1 of Iteration

Table 2.5b Response-Forcing Table for Compound Tides as Generated by

Finite Amplitude Term at Cycle No. 2 of Iteration

-37-

LILI___LL1IILU__Y___LI~-XXI

non linearity are present at the very first cycle. Subsequent cycles

only update the magnitude of the forcings and the associated responses.

Let us examine the bottom friction term for a one dimensional case.

bh x h (2.17)

7 P S-q,cf u u

This equation may be approximated by Taylor series expanding the finite

depth ratio term to get:

bh x u u (2.18)h+ cfpUnU cf1 uI

Assuming only one astronomical tide exists we have at the first cycle of

the iteration a response of

u = u cos(w,t) (2.19)

Substituting into the dominating term of our approximation for friction

and performing a Fourier expansion we find that the non linear forcing

will be:

2

cffulu - cfu (0.8488 cos olt + 0.1698 cos 3w1t - 0.0242 cos 5 1t + ... )

(2.20)

Hence there are forcing terms at all the odd harmonics of the base

frequency wl. We note that the major forcing component remains at the

astronomical frequency, whereas both the finite amplitude and convective

acceleration terms distributed forcing to frequencies other than that of

the responses that were generating the forcing. Hence after the first

cycle we expect responses at frequencies wl, 3w1, 5w1, etc. It is

-38-

quite simple to demonstrate that at the second and subsequent cycles,

when all these odd harmonics are included in the expansion for velocity,

the term cfulul will still only produce forcings at the same odd

harmonics as the first cycle. The precise coefficients of the

approximating Fourier series will be case dependent. However since the

major forcing is at the base frequency, the major reponse generated

should be at the same base frequency. For this case of dominant

response frequency remaining the same, the Fourier series expansion

representing all the interacting frequencies should be quite similar

(for the first few loading terms at least) to the very first expansion

equation (2.20). It is also the case for compound tides that when one

frequency component dominates, the Fourier series approximating cfu uI

will be similar to the one generated if only the dominant terms were

considered. Finally we note that the (n/h)cflulu term will generate

even harmonics (including zero frequency) in the same way that cf uIu

generates odd harmonics. It is obvious that the forcings and hence

responses due to this second order term are of smaller magnitude than

that of the main term.

Table 2.6 shows a summary of the astronomical, overtide and

compound tidal constituents (at the high frequency end) which could be

important in shallow estuaries. Figure 2.2 shows the information from

Table 2.6 in graphical form. It is emphasized that these constituents

are case dependent and are affected by such factors as the relative

importance of the various astronomical tides for the bay under

consideration and/or which non linear terms in the equations are most

important. We note that all these frequencies modulate over a period of

up to 208 days or possibly longer if other components are considered.

-39-

Table 2.6 Tides of Interest (High Freq. End)

f = frequency of tideT = period of tideAf = freq. increment between consecutive tidesT = 1/Af = synodic period (length of record required to distinquish

between tidal constituents)

-40-

S 6

S42SM6 **- "*.* -* .*..__ ...-M4

MN4 * **** * * ** * *

K2

2MN2 & L2 .......

N2 2MS 2

U1 -

p,

MSfsteady--- ---- --

So'C isl I I I

-41-

H,

0II

I

.-

|I I

.-~ --L-LI-L-^LL I~Xr -~~L-1_.ICI__ ~_

,--(I

M2

It is clear that it is not possible to include in a general sense

all of these tidal components with either a perturbation analysis (in

which the modeler determines a priori which terms and interactions are

to be investigated) or a scheme which uses a Fourier expansion in terms

of integer values of a base frequency (allowing only overtides of one

major frequency to be studied).

The most general and computationally convenient scheme which allows

investigation of any number of these closely spaced tidal components is

an iterative scheme in which the non linear interactions are treated as

force loadings which are somehow harmonically analyzed. The following

chapters discuss in detail the numerical techniques applied in the

iterative frequency domain scheme used to develop the numerical code.

-42-

CHAPTER 3. NUMERICAL FORMULATION

3.1 Weighted Residual Formulation

The spatial dependence of our governing equations will be resolved

through the use of the finite element method. In order to apply the

finite element method a weighted residual formulation [Connor and

Brebbia, 1976] must be established which shall be used as the basis of

our finite element formulation. As is shown in Chapter 4, in order to

avoid matrix rank problems the algebraic matrix equations produced with

our finite element scheme must be solved by substituting the finite

element momentum equation into the finite element continuity equation.

The resulting formulation is similar to that of Lynch [1980] with the

exception that the finite element discretization is applied before the

continuity and momentum equations are merged. These manipulations,

however, dictate that the continuity equation be used in order to

establish the symmetrical weak weighted residual form. It may readily

be shown that formulating the fundamental weak form in this fashion

leads to elevation prescribed boundary conditions, j*, being essential

and normal flux prescribed boundary conditions, Qn*, being natural.

Specifically applying Galerkin's method to establish the fundamental

weak form, the error in the continuity equation is weighted by the

variation in elevation, 6r, and is integrated over the interior domain,

Q. Furthermore the natural boundary error must be accounted for by

weighting it with 8n and integrating it over the natural boundary

SN . It is required that the combined integrated and properly

-43-

_ ~__I~ __~TI_*_____*CC3_1___*_A~-.

weighted interior and natural boundary errors vanish and the following

expression results:

ff ft) + [u(h+n)]0

+ [v(h+-)] 18n dO + f T-Q + Q* l i d = 0

Applying Gauss' theorem in order to eliminate the derivatives on the

flux terms yields:

ff {, 8r - u(h+n)(6r) - v(h+)(p) I do9 ,x

+ f fu(h+n)a + v(h+j)a 1 6n dV + (-Q + Q n ) 5 dP = 0nx ny n n

r r

where anx, any are the direction cosines on the boundary.

However, the normal flux may be expressed as:

Q = anx Q + any Qyn nx x ny y

(3.2)

(3.3)

where

Q = u(h+)

Q = v(h+)

Using the previous relationships for flux simplifies Eq. 3.2 to:

ff In I - u(h+n)(8n) - v(h+r)(8 ) I dOy

+ f Qn 6 dr + f (-Qn+ n)' dP = 0r r.,

(3.4a)

(3.4b)

(3.5)

Furthermore the entire boundary, r, is divided into an essential

boundary, r E , and a natural boundary r N . This then leads to:

-44-

(3.1)

ff {rn 8n - u(h+n)(8r) - v(h+n)(8n) } dQ

+ f QnSndr + f Q 8dr - Qn 6dr+ ( Q&n 6dr- o (3.6)r r nrN E N N

The natural boundary integrals of normal flux cancel and the essential

boundary integral of normal flux vanishes due to the selection of

elevation as an essential boundary condition. This implies that "* is

exactly satisfied on PE and therefore the variation, An, is by

definition zero on rE . With these simplifications, Eq. 3.6 leads to

the symmetrical weighted residual weak form:

ff {n tn - u(h+r)(~i) - v(h+n)( n) IdQ + f Q n dTr - 0 (3.7)

N

We now proceed with the weighted residual method by establishing

the weighted residual form of the momentum equations. In order to allow

for the possibility of establishing symmetrical final system matrices

each of the momentum equations is multiplied through by depth, h. The

weighted residual forms for this modified form of the momentum equations

is then obtained by weighting the associated errors with residual

velocities and integrating over the interior domain, resulting in:

s 2 2)1/2f fhu + gh x - fhv - x/p + (h/h+n) cf(u2+ u

+ h(uu + vu )I 6u dQ = 0 (3.8a),x ,y

ff fhv + gh, - fhu -T/o + (h/h+n) cf(u2+ 2 )/2

+ h(uv + vv )1 6v dQ = 0 (3.8b),x ,y

-45-

II1L_1LIIII____Yli I___WIIPI__ _e I___ - .ii._

The weighted residual equations which will be the basis of the

finite element formulation have now been established. The final

weighted residual equations include first order spatial derivatives of

all the variables, n, u, and v. Hence the associated functional

continuity requireumen imposed on these variables is that they be

continuous over the domain. Furthermore there are first order

derivatives taken of the weighting function 6T, again requiring

continuity over the domain, while no derivatives are taken of the

weighting functions 6u and 6v, requiring only that they be finite over

the domain.

Since the non-linearities will be treated in an iterative manner

Eqs. (3.7) and (3.8) are re-arranged such that non linear terms appear

on the right hand side where they can be conveniently updated as pseudo

force loadings to a linear problem with each iteration. All boundary

and non-variable loadings are also placed on the right-hand sides.

Finally, in order to enhance iterative stabilty, a linearized friction

term is included on both sides of the momentum equation. These

modifications result in the following equations:

ff {n &r - uh(6n) - vh(6) } dQ =

fQ 6n dr + ff {un(6) + vyn() } dQ (3.9)N

ff { hu + ghj - fhv + Xu } 6u dQ =,t ,x

ff{Ts/p + (X - (h/h+)cf(u + v2)/2u - h(uu - vu )}6u dQ (3.10a)x ,x ,y

-46-

ff { hvt + ghy - fhu + Xv } 6v dQ -

0 x f,9x ,y

3.2 Finite Element Method Formulation

In order to generate a system of algebraic equations from integral

equations, the finite element method is applied to the final form of the

weighted residual equations. This involves dividing the global domain,

Q, into element subdomains, Qe, and representing the variables within

each element by polynomial expansions. Contributions from all elements

are summed and inter-element functional continuity requirements are

taken into account in order to generate a global system of equations.

To satisfy the minimum functional continuity requirements on the

variables linear or higher order expansions must be used for the finite

element approximations. For the development of this method it was felt

that the simplicity of linear expansions outweighed the improved

accuracy achieved (for the same number of total nodes) of higher order

elements. Therefore the simplest possible element, the linear triangle,

was selected and the variables are expressed within each element as:

S= tl1nI1 + 2"2 + 4 3 13 = (n (3.11)

u = 1u1 + 2 u2 + + 3 u3 = u(n) (3.12a)

(n)v = (1V1 + 2v2 + 3v3 = ()v (3.12b)

where:

-47-

(n) 2 1 (3.13)

are the nodal elevation values for element n.

U1 v 1

u(n ) u2 and v(n) v 2 (3.14a,b)

u3 v3

are the nodal velocity values in each coordinate direction for element

n. Finally,

-- - [ l 2 62] (3.15)

are the normalized element coordinates for each element. The same

linear expansions are used for the weighting functions which is

expressed as:

6 = 4 6 )(n) (3.16)

6u = , 6 u(n) (3.17a)

6v = , 6v ( n ) (3.17b)

Furthermore, linear element expansions were selected for mean water

level depths h. In this manner elevations, velocities and depths are

-48-

all defined at the nodes such that inter-element fluxes are both

continuous and cleary defined. Hence:

h = 4 h(n) (3.18)

Finally, for reasons discussed in Chapter 5 it is desirable to have

nodal values for both the linearized friction factor X and our friction

factor cf, again requiring linear expansions:

X = X (n) (3.19)

and cf = _ (n) (3.20)

Substituting the finite element expansions into the final weighted

residual form of the continuity equation (Eq. 3.7) for each element

sub-domain, Pe, and summing over all elements within the domain, Q,

leads to:

T (n) 8(n) (n) (n) (n) (n) (n) (n)S- 4 u h n) v d, h 4 dQel - ,t -- - -,x ,

e

* 6 (n)dr +(n) (n) (n) (n) (n) (n)JQ _ n dr + ff u 4r 4 Sn +4)v 4)9 4 ( lpdQl

. ..n -,x- - - - ,9rTN ee

(3.21)

Re-arranging somewhat gives:

T T (n) (n)

S6 [(n[) T dQ)n(n) T 4 h(n)4 dQ)u(n) T 6 h(n) d )v(n)el 9 0f X1

e e e

T T (n) (n) T (n) (n)=(-fQ4 dr) + ( tx 4) I 4u +4 4) 4 v }dQ] (3.22)

TN ee

-49-

Letting (n) LU Eu(n)

v(n)

(n) , 6u(n)and 6U

Eq. 3.22 may therefore (n)be expressed

as:

Eq. 3.22 may therefore be expressed as:

(n) (n)-D Uel ) (n) (n)

el

Sp(n),lin +-

where the element sub-matrices are defined as:

S T d

e

-E fT 4 h

e

P(n),linT)

P(n),nl-)

SdQ ff j _ h(n ) d_ -,y

f 4T Q drrN nN e

ff (4e

(n)-G -

(n) T4u +4S- - -,y

(n)6 r

v(n))dQ4)v )dQ

Loading the element sub-matrices into global matrices leads to the

following global equations:

6 T [M , - D U=T) - t = -

lin nl1- -P + P ]

T) -T

-50-

(3.23)

(3.24)

p(n) ,nlIn)

(3.25)

(3.26)

(3.27)

(3.28)

(3.29)

(3.30)

'n

U

11

D

lin

pnl-71

= global

= global

= global

= global

= global

= global

elevation vector (1 elevation per node)

velocity vector (2 velocities per node)

continuity equation coefficient matrix

derivative matrix

load vector for flux prescribed boundaries

load vector for finite amplitude effects

Allowing for an arbitrary variation in 68q leads to a final set of

non linear algebraic equations which minimize the error incurred in the

continuity equation due to the finite representation of the spatial

variables:

(3.31)lin n1M r,t - DU = - Plin + Pn

Substituting the finite element expansion into the weighted

residual forms of the momentum equation (Eqs. 3.10a,b) for each element

sub-domain, Qe, and summing over all elements within the domain, Q,

leads to:

_ (n) (n) u (n) h(n) (n) ( h(n)(n) ( (n) (n)9 [ff{sh u + X $u _f4d h n) v + 0 h _9 dQ

elQ --e

s

ff {- 4 6u(n)} d Q + ff I{A(n) 46 u(n)ldQ9 Q fric-u

Sh) (n) (n) (n) (n)d , u(n) (n)ff{ h 4u 4, u + h v }Su dQ]Q . - -,x- -.. .. ,y- --

(3.32a)

for the x-momentum equation and

-51-

where

Sf f L (n) v(n) n ) (n) (n)+ g (n) (n) (n) _6v(n) dOS[ff{$ h v + 4 v -f h ()un+ _ ( n ) (n

el 9e

s

f {-~. _ 6v(n)}do + ff { ) A-v(n)}dQp fric-v

e e

(n) () (n) (n) () (n) (n)

- ff ( h 6 u( v + h v(n) v },v dQ (3.32b)

Qe

for the y-momentum equation. The right-hand side load terms

representing the difference between linear and non linear friction are

now denoted as:

(n) = h cf(u 2 v2 1/2A c (u 2+ v ) u1 (3.33a)fric-u h+ f

(n)

fric-v = c (u 2 + 2)1/2 v (3.33b)frn) =[ch+n fv

and have not been expanded in terms of the finite element approximations

used since they require a numerical integration scheme.

Re-arranging Eqs. 3-32a,b leads to:

Su(n)[ ,T h(n) i, dQ)u(n) + (fT, X(n)_ dQ)u(n)- -U - -±- -, t -

el _ Qe e

+(_-ff T h(n)_ dQ)v(n) + g(f4T h(n) dQ)n(n)

Sp - ,x

e e

s

T f x(n) ST d)(ff±T -=dQ) + (ffrA() Td)

SP fric-u de e

-( ffT h(n)_ u(n) (n) T h(n) (n)}do)] (3.34a)

ee

-52-

6v(n) Tfr h(n)f dC)v(n) + (-t 9-- - -,tel 9e

+(ffT h(n) d()u(e

(ff T

ep

+ g(ffT (n)+ g(ffh P)

e

+ ( (n4) T dQ)+ (J cfric-v -

e

-(ff{ T h(n)

e

(n) (n) T h(n) (n)}dQ)S,x- ..- - - ,y -

Adding together expressions 3.34a and 3.34b (which does not change these

equations due to the arbitraryness of 6u(n) and 6v(n)) and recalling

expressions 3.23, 3.24 and 3.27 we have:

[M(n) u(n) + n) (n) + g D(n)+ ! U (

+ p(n) ,nlS fric

_ (n),nl-cony

where the element sub-matrices are defined as:

eff Te

f T

Sh(n)4) dQ 0

0 f T h(n)

I QI e

(n)X dQ

f- - )f

-53-

and

dP) l(n)

(3.34b)

e _ ( n )

el

(n)i =

(3.35)

M(n)--U

M(n)=F

(3.36)

(3.37)h(n)X dQ

___IIILYIYY__JYI__~_L-I~~~IWICiYIZI

-f.T h (n)4 dQ)v (n )

jf±T h(n)Qe

(n)C

.(n)

(n) ,nl-A-fric

_fffT

e

dQ C

S

x dQ

s- d- dQ

p

T (n)

ff 4T n) dQ- fric-v

e

dQ

)(3.38)

(3.39)

(3.40)

u(n))dQii )do

ff(4T h(n) 4 u (n) u(n)+- - - - - -,x-

e

T (n) (n) (n) (n) (n) ( y v (n))dQ( T h n 4) u 4) v + 44h 4 v v )d, x- , y -

(3.41)

Again loading element sub-matrices into global matrices leads to the

following set of global equations:

6UT [MU + M U + M-U ,t -F -C

U + g DTnl nl

-W A-fri c cony

-54-

ffe

Sffe

p(n),nl-conv

(3.42)

T h(n) v ( n )

where

_ = global momentum equation mass matrix

M = global linearized frictional distribution matrix

M = global Coriolis matrix-C

P = global wind stress loading vector

nlP - = global load vector containing difference between

linearized friction and full nonlinear friction

nlP = global load vector containing convective-cony acceleration effects (non linear)

Allowing for arbitrary variation in 6U results in the following final

set of non linear algebraic equations which minimize the error incurred

in the momentum equation due to the finite representation of the

spatial variables:

TM U + M U + M U + g D = P + P - P (3.43)-U ,t -F -C - W A-fric cony

Eqs. 3.31 and 3.43 are still differentially time dependent which will be

resolved by applying the harmonic analysis procedure presented in the

next section. With the exception of the non-linear frictional

(n),nldifference load vector, P-f , all the previous element matrices*-d-fric

and vectors may be readily developed in an analytical fashion. The

procedure is very straight-forward and the resulting element matrices

and vectors are presented in Appendix A.

The matrices M , M and M are all symmetric while the Coriolis

matrix, MC, is skew symmetric. In the development of the prescribed

flux load vector (calculated only on natural boundaries TN) a linear

varying normal flux is assumed. For the wind stress load vector PM

wind shear stress is considered constant over each element.

-55-

Since it is not possible to analytically evaluate the integrals in

the friction difference load vector, a numerical integration procedure

is applied. Allowing for the fact that velocity, elevation and depth

all vary linearly, at least a cubic integration formula is required for

the non-linear friction contribution.

3.3 Frequency Domain Formulation

The finite element technique has been used to resolve the spatial

dependence in the governing equations and has thus reduced the non

linear partial differential form of the governing equations to the

following set of non linear differentially time dependent algebraic

equations:

lin nlM - DU = Pin + (3.44)-n - t --n -71

M U +M U + M U + gD P + P (3.45)tUF -tC - _FU

where

nl - (3.46)

-U -A-fric --conv

Both the variables r and U and the loadings P lin Pn P

and P are time dependent vectors. In addition all terms are-U

linear with the exception of the right hand side pseudo forcing terms

pnland Pnl which contain non linear combinations of the variablesn -U

velocity and elevation.

The differential time dependence in Eqs. 3.44 and 3.45 will be

resolved by a scheme which reduces them to sets of harmonic equations

-56-

which are coupled through the non linear terms. The non linear coupling

will be treated by an iterative updating scheme which shall be discussed

in more detail later. The reduction of Eqs. 3.44 and 3.45 from the time

domain to the frequency domain assumes that the responses of the system

in elevation and velocity may be expressed as a harmonic series of the

form:

N

(t) = Re e (3.47)J=1

Nf i t

U(t) = Re{ 7 U. e ) (3.48)j=1 -

It is noted that the complex quantities denoted by ^ include both a

magnitude and a phase shift. Furthermore it is assumed that both the

linear and the non linear load vectors may be represented by similar

harmonic series:

Nf A io tP(t) = Re{ Y P e (3.49)

j=1

Substituting the harmonic series representations of the variables into

Eqs. 3.44 and 3.45 and taking appropriate time derivatives leads to:

Nf A iW t Nf ^ i jtM ( Y iw e ) - D( 7 Ue ) == J= 1= J=1

N Nf lin ijt f ,n iWtj

- y P e + y P e (3.50)j=1 -r j J=1 -J

-57-

and

N f A i t N f iWJ t N f ij t

( 7Ji U e + 7 U e )+ ! ( e )

J=1 j=1 J=l

N N Nf it it f nl ijt

+ gDT( .e j) PW e J+ pUnl e (3.51)

= j=1 =1 j=1 j

These equations may be re-arranged as:

Nf A A Ain An i& t

(iw M - DUj + P P ei 0 (3.52)j=1 j j - - -

fA A A T ^ A nl iWj t(1w !U U + M U + M U + gD -P -P )e = 0 (3.53)

=-J -i _ + F Uj -J - WJ -U

Due to the orthogonality properties of sinusoidal functions (and hence

complex exponentials), each of the expressions within the brackets must

equal zero. This leads to Nf sets of time independent linear equation

of the following type:

A A Anlin 4nliW M - D U. Pi P (3.54)= = - - -i

A A A T A ^ ^nlij 5 + + + g D + P (3.55)

i !M U +M! U +M! U + = W +

Note that the natural flux prescribed boundary conditions are

Alinincorporated in the load vectors Pj . The essential boundary

condition may also be expressed as a harmonic series and hence

Nf A iwj t . ij t

j 1 e r Nj e (3.56)

j=1 N

-58-

which leads to a set of boundary conditions associated with each of the

set of equations (3.54-3.55) of the form:

l N = 'n (3.57)

The iterative solution strategy starts out with the assumption that

the non linear loadings are zero. Each of these Nf sets of equations

are then solved for the boundary loadings imposed. Time histories may

then be generated for velocity and elevation with Eqs. 3.47 and 3.48.

This in turn allows time histories of the non linear psuedo loading

vectors P nl(t) and P nl(t) to be produced. As was assumed earlier,n --U

these time domain pseudo forcings may now be approximated as harmonic

series. Hence the total non linear loadings for continuity and momentum

are distributed to all or some of the Nf sets of frequency domain

equations. Now each of these Nf sets of equations is solved again and

the entire procedure is repeated until convergence is reached. A

schematic of the iterative scheme is shown in Figure 3.1. Strategies of

how to optimally harmonically decompose the nonlinear load vectors and

details such as the number of frequencies required for the harmonic

series representation used will be discussed in Chapter 5.

Each of the Nf sets of equations are linear at each cycle of the

iteration although they are coupled through the non linear loadings. As

is clear from Figure 3.1 solving each set of linear equations is the

heart of our overall non linear solution scheme. In the next chapter we

shall examine methods to solve these linear sets of equations in an

optimal manner.

-59-

GENERATE TIME DOMAIN RESPONSE

HISTORY BY SUPERPOSITIONING

OF SOLUTIONS Uj AND nj

U(t) AND n(t)

GENERATE TIME HISTORY LOADING

P n(t) USING TIME

HISTORY RESPONSE

HARMONIC ANALYSIS OF Pnl(t)

TO GENERATE

FREQ. DOMAIN LOADING P

J = 1,M

SELECT NEXT

SELECT SET OF w j's FOR

WHICH LINEAR CORE IS RUN

Figure 3.1. Schematic of Iterative Non Linear Scheme

-60-

BOUNDARY CONDITIONS

n AND Q

FOR ALL FREQUENCIES wj

j = 1,M

LINEAR SOLUTION AT GIVEN WjA lin ^* ^* An

SP l ,(Q )+

RESULTS IN U AND 'j

I

I: ,-

CHECK CONVERGENCE

CHAPTER 4. LINEAR CORE MODEL

In Chapter 3 the finite element method was applied in order to

resolve the spatial dependence of the governing partial differential

equations and reduce them to a set of non linear algebraic equations in

space with the differential time dependence left unresolved. The

assumption of harmonic forcings on the system and harmonically

decomposible pseudo forcings due to the non linearities allowed the

elimination of the time dependence from this set of equations and thus

produced numerous sets of equations in the frequency domain.

Furthermore the concurrent assumption of an iterative type solution

scheme which updates the non linear pseudo forcings yielded sets of

linear algebraic equations of the following type for each required

frequency:

A A A

is M r - D U = P (4.1)

iw M U + M U + M U + g D n P (4.2)

A

The continuity loading vector P can include both contributions from a

flux loading at frequency w and from the component at frequency W of a

harmonically decomposed finite amplitude pseudo forcing. Similarly the

momentum loading vector PU can include contributions from a harmonic

type wind loading at frequency w and the components of the non linear

pseudo loadings at frequency w due to convective acceleration and the

difference between a linearized friction and the full non linear

-61-

friction. As was noted in Chapter 3 these equations form the core of

our fully non linear scheme.

As was discussed in Chapter 2, numerous frequencies are needed to

accurately simulate shallow water tides with a frequency domain

approach. The required frequencies represent either the astronomical

tides of importance which are present or their associated non linear

over and compound tides. Hence the linear core equations need to be

solved for each of these frequencies. In addition, the fully non linear

model is iterative, which means that all (or at least most) of these

frequencies need to be solved for at each cycle of the iteration until

convergence is reached. Therefore it is important that the linear core

solution strategy be not only accurate and free of spurious oscillations

but also very efficient.

The method selected to solve Eqs. 4.1 and 4.2 should also take into

account the wide range of frequencies that may be required, from the

zero frequency and low frequency astronomical tides and residuals

generated up to the very high frequency harmonics. Finally, the method

applied should take into account the physical characteristics of

typical tidal embayments and the nature of the tidal flows within these

embayments. Let us now examine some of the various possible ways in

which Eqs. 4.1 and 4.2 may be solved.

One possible solution method is to substitute for elevation into

the momentum equation and obtain a final equation with velocity as the

basic variable. Solving for elevation with the continuity equation

yields:

A 1 -1 A A

S= M (P + D U) (4.3)i w T-n =

-62-

Substituting Eq. 4.3 into Eq. 4.2 and re-arranging leads to:

2 T-1 U DT -1(w M + iM + iM+ g DM D) U g D M P (4.4)

U -- = _ -= -- -

Hence Eq. 4.4 is solved for velocity U which may then be back-

substituted into Eq. 4.3 in order to obtain elevation. This strategy,

however, fails due to a rank problem with the total left-hand side

system matrix generated in Eq. 4.4. Due to the w2 factor multiplying

the momentum mass matrix, , and the w factors multiplying the

linearized friction and Coriolis matrices, M and MC, in Eq. 4.4, the

contribution of the gD TM D matrix dominates the sum of these

matrices. For higher harmonics, the effect of &2M could disappear;;U

entirely due to the round off accuracy of the computer. Table 4.1 shows

that the rank of the g DTM1 D matrix is N, making the rank of the total= =T) =

system matrix also N. Clearly we can not solve for 2N velocities with a

system of 2N equations of rank N. This then indicates that the

appropriate manner in which to solve Eqs. 2.1 and 2.2 is to substitute

for velocity in the continuity equation which produces a wave-like

equation with n as the basic variable.

Let us now go into some of the various ways that Eqs. 2.1 and 2.2

can be solved if velocity is substituted into the continuity equation.

The first technique examined involves solving for velocity with the

momentum equation such that only the mass matrix M need be inverted.

Note that if the mass lumping procedures are applied, the matrix M= U

simplifies to a diagonal matrix which makes the required inversion

extremely economical. More details on the mechanics and implications of

lumping procedures are discussed later in this chapter. Hence solving

Eq. 4.2 for U as indicated:

-63-

Table 4.1 Sizes and Ranks of Various MatricesN = Number of Node Points

Matrix Size Rank

U ' M 2N x 2N 2N

M-1 NxN N

D M- 1 D 2N x 2N N

n M- 1 nT N xN N

-64-

^ 1 -1 A A T^U - M P - (M +M ) U - g D T1 (4.5)

- iw U -U =C

The above procedure bars zero frequency cases from being solved due to

the 11w term. This problem may readily be corrected by adding the

term c U to both sides of Eq. 4.2, where cs is an arbitrary non-zero

constant. Therefore Eq. 4.2 is now solved for U as:

^ 1 -1 A T AU 1 - (M + M - c M )U - g D (4.6)TU (i + c s t-U -F -C U

such that the zero frequency case is not excluded from investigation.

Substituting for U using Eq. 4.6 into the continuity equation (4.1) and

re-arranging produces:

2 -1(0 - iwc)M - gD DT =

-1 A A A

-DM P - ( + M - _)UI1 - (i, + cs)P (4.7)_- -U (F C -cn

-1 TNow both the dominating component, gD M U D , and the additional

component, (w2 - iwc )M , of the left hand size matrix are of rank N.s---n

Hence the rank of the left-hand side matrix is sufficient to solve for

N unknown elevations.

However, since Eq. 4.7 contains the variable U on the right hand

side of the equation an iterative scheme is required to solve Eqs. 4.6

and 4.7. This linear type iteration is distinct from the iteration

scheme discussed in previous chapters which updated the non linear terms

in the governing equations. However both types of iterations could

proceed simultaneously.

Placing the very small (w - iWC )M term in Eq. 4.7 on the right-

hand side, leads to the following attractive iterative linear core

-65-

solution scheme:

-1 T ^k+l 2 ^k ^gDM D ( - ic) n + (i + c s -TP

+DM P - (M + -c M ) (4.8)=U -U F +-C sU U

Ak+l 1 -1 k T ^k+1U -( + c M )U -gD n 1 (4.9)- (iW + c ) JU - (F Cs -U

The superscript k refers to the cycle of the iteration. The linear

iterative solution procedure is started by initially evaluating the

^o o ^ 1right-hand side of Eq. 4.8 using r = 0 and U 0, solving for I ,

^1then evaluating U using Eq. 4.9 and continuing the iteration until

convergence is achieved.

As previously mentioned this linear iterative scheme has some very

-1 Tdesirable features. First of all, the system matrix, gD 1 D , that

needs to be solved is both real and symmetric thus saving in

computational expense and storage. Furthermore, the system matrix does

not have frequency, w, embedded in it which eliminates the need to

re-set and re-solve a system matrix for each of the many frequencies

required in the fully non linear scheme.

The advantages of this linear iterative scheme, however, are offset

by the severe iterative stability problems which can occur, either

leading to divergence or extremely slow convergence. The convergence

criterion for this linear core model is:

^k+1 ^k 2 2 cRonv k ^k- = gh (1 + )- (c --- f)(a + c (4.10)^k ^k- gh s h s (4.10)-- ^ +

-66-

where Rconv = local convergence criterion and I = element dimension.

The actual convergence rate may be checked by examining

log(l/Rconv) which indicates how many decimal places the solution is

improving every cycle. For example, Rconv equals 0.1 when the

solution is improving by one decimal place every cycle of the

iteration. The smaller Rconv the more rapid the convergence while for

Rconv > 1 the iteration diverges. It is stressed that Rconv is a

local convergence criterion and must be satisfied everywhere within the

domain.

Typical iterative stability problems can be demonstrated by

examining a zero frequency flow in a depth varying channel with a

constant linear friction factor and no Coriolis effects (see Figure

4.1). For this case the convergence criterion reduces to:

R 1= -. (4.11)cony he

Note that convergence is dependent on both the selection of a value for

c s and the location in the channel. Table 4.2 shows values of Rconv

for various selections of cs at different locations in the channel in

terms of v, the ratio of maximum to minimum depth in the channel. Only

by setting c s = X/hmin will convergence be guaranteed everywhere in

the channel regardless of the depth variation. For example, for a depth

variation of y = 10, the maximum value of Rconv obtained at any point

in the channel by using cs = X/hmin, cs = X/havg and cs =

X/hmax are respectively 0.9, 4.5 and 9. Even though the optimal

selection of cs = X/hmin produces a stable convergent scheme, it is

very slow due to the value of Rconv being close to 1.

-67-

M.S.L.

max min

havg

hmax

depth ratio = y - > 1h. -min

Figure 4.1 Definition Sketch of Depth Varying Channel

Which Illustrates Convergence Problems ofIterative Linear Scheme

-68-

Table 4.2 Variation of Convergence with c s

-69-

Rcony

c shallow end average depth deep end

x0hmin

avg 2

1

0h I1 1+1max -

Y

The convergence criterion shows that for cases in which the depth

or the linear friction factor, X, vary substantially within the domain,

stability problems will arise with this particular scheme. Even though

for some cases it is possible to adjust the global factor c s such that

convergence is guaranLeed everywhere within the domain, it is a

cumbersome procedure to find this value and will result in at best very

slow convergence rates. It is clear from examining either Table 4.2 or

Eq. 4.11, that optimal convergence (i.e. Rconv = 0) is achieved by

selecting local values of c s equal to X/h. What this implies

is that instead of adding c M U using a global value for c s

to the right and left hand sides of Eq. 4.2, local (or element) matrices

S(n) (n) which use the local optimal value for cs should be used to

establish a global matrix which is added to the left and right hand

sides of Eq. 4.2. However, it is noted that this optimal global

matrix is exactly M. What this indicates is that the frictional

distribution matrix should be kept on the left-hand side when solving

Eq. 2.2 for U.

Based on examination of values of Rconv and running computer

simulations which apply the linear iterative scheme discussed above for

a variety of embayments, this iterative scheme proved impractical for

use as the linear core solver. Furthermore, examination of various

other schemes to solve Eq. 2.1 and 2.2 led to the conclusion that a

direct one pass non-iterative solution technique for the linear core was

optimal. As we shall see, this optimal method yields a final system

matrix which is complex and non-symmetric. Furthermore, it has

frequency embedded into it requiring that at each cycle of the non

linear iteration the system matrix must be re-set and re-solved.

-70-

However, the overall amount of computational effort is substantially

less for the one pass linear solution when considering the number of

iterations required for the other linear solution methods. This direct

linear scheme was implemented as the linear core solver for the overall

non linear code.

For the direct solution scheme, the momentum equation is now solved

for U as follows:

A A -1 ^ T A

U - M (P - g D ) (4.12)- TOT -U

where:

TO = (iw M+ MF +) (4.13)

Substituting for U into our continuity equation produces:

^ -1 IT A ^ -1(i M + g D M D ) r - P + DM P (4.14)

= = TOT -n = :TOT -U

As previously mentioned the total left hand system matrix in Eq. 4.14 is

complex, non-symmetric (since the Coriolis matrix M is contained in

MTOT ) and has frequency embedded into it. Due to storage limitations

it is preferable to re-set and resolve the system matrix for each

frequency at each cycle of the iteration rather than storing the matrix

produced for each frequency at the first cycle and using it in

subsequent cycles of the iteration.

Finally in order to make this technique viable, the lumping

procedure has been applied not only to the mass matrix MU, but also to

the linearized frictional distribution matrix M and the Coriolis

matrix, M. This now allows the required inversion of M to be;;c ;O

-71-

performed economically. For the sake of consistency the matrix M was

also lumped. The lumping for the symmetric mass and friction matrices

involves combining all terms on a row onto the diagonal and for the skew

symmetric Coriolis matrix combining rows onto off-diagonal terms (in

order to retain the skew symmetric natural of the matrix). Hence matrix

TOT will be tri-diagonal. The lumping procedure in effect amounts to

a slight re-distribution of mass and the linear friction and Coriolis

forcings between neighboring nodes. Model results have been shown to be

quite insensitive to these lumpings.

The linear core model has been verified against the analytical

solution for a tidal wave entering a rectangular channel closed at

one end both with and without bottom friction damping (Ippen, 1966).

The example channel used for this simulation was 25 km long and 4 km

wide and had a depth of 10 m. The grid representing this channel is

shown in Figure 4.2. A constant element size of 1 km was used yielding

5 nodes across the channel width. A 12.4 hour forcing tide of 1 m in

amplitude at the open (ocean) end was used. The linear core model

was run for a no bottom friction case, a lightly damped case (with a

linearized friction factor, X, equal to 0.001 m/sec) and a heavily

damped case (X = 0.01 m/sec).

Results of the linear core model for all three cases are shown

and compared to the corresponding exact analytical solution (at

various locations) in Table 4.3. Agreement between analytical values

and numerical predictions is excellent for both elevation amplitude

and phase. For the undamped and lightly damped cases, the linear

core model slightly overpredicts elevation amplitudes by an average

-72-

4

i~~.tLL t.S i ;n i A\-.... ~t ~ C-

I V VV V- L//VI LI IVik

x (kin)

Figure 4.2 Finite Element Grid Discretization for Closed Ended Channel Example Case

r I -- , - 1---- -~------,----I

1 '

1 't I . '

/V '1/VV ~ .\~;\; ~-~r \ ~ ~~.'"' \~~1/1/

Table 4.3 Comparison of Analytical and Numerical Elevations and Velocitiesfor Example Channel Case at Various Locations

(a) Linearized Friction Factor X = 0.0000

Elevations

Analytical

Amplitude Phase(m) (rad)

1.00000 0.00000

1.02796 0.00000

1.05113 0.00000

1.06273 0.00000

1.06661 0.00000

Numerical

Amplitude(m)

1.00000

1.02800

1.05119

1.06277

1.06657

Velocities

Analytical

Amplitude(m/sec)

0.36747

0.28178

0.18015

0.08997

0.00000

Phase(rad)

0.00000

0.00000

0.00000

0.00000

Numerical

Amplitude(m/sec)

0.36847

0.28408

0.18181

0.09260

0.00103

-74-

x(m)

0

6000

13000

19000

25000

Phase(rad)

0.00000

0.00000

0.00000

0.00000

0.00000

x(m)

0

6000

13000

19000

25000

Phase(rad)

1.57080

1.57080

1.57080

1.57080

1.57080


(b) Linearized Friction Factor X = 0.0010

Elevations

Analytical

Amplitude Phase

(m) (rad)

1.00000 0.00000

1.02744 -0.02026

1.05039 -0.03637

1.06194 -0.04422

1.06581 -0.04680

Numerical

Amplitude(m)

1.00000

1.02748

1.05044

1.06198

1.06578

Velocities

Analytical

Amplitude

(m/sec)

0.36721

0.28158

0.17915

0.08990

0.00000

Phase

(rad)

1.53906

1.53267

1.52744

1.52485

Numerical

Amplitude(m/sec)

0.36821

0.28388

0.18167

0.09253

0.00103

-75-

x(m)

0

6000

13000

19000

25000

Phase(rad)

0.00000

-0.02029

-0.03641

-0.04425

-0.04678

x(m)

0

6000

13000

19000

25000

Phase(rad)

1.53905

1.53274

1.52749

1.52485

1.52589


(c) Linearized Friction Factor X = 0.0100

Elevations

Analytical

Amplitude Phase

(m) (rad)

1.00000 0.00000

0.98152 -0.18880

0.98420 -0.34690

0.99167 -0.42508

0.99507 -0.45097

Numerical

Amplitude(m)

1.00000

0.98144

0.98409

0.99152

0.99483

Velocities

Analytical

Amplitude Phase(m/sec) (rad)

0.34439 1.27029

0.26328 , 1.20652

0.16729 1.15433

0.08393 1.12844

0.00000

Numerical

Amplitude(m/sec)

0.34530

0.26540

0.16963

0.08637

0.00096

-76-

x(m)

0

6000

13000

19000

25000

Phase(rad)

0.00000

-0.18904

-0.34718

-0.42532

-0.45070

x(m)

0

6000

13000

19000

25000

Phase(rad)

1.27017

1.20734

1.15483

1.12843

1.13888

of about 0.00005 m which equals approximately 0.005 percent of the

amplitude at each point. For the heavily damped case, the numerical

predictions for elevation amplitude are slightly below exact values.

The predictions for velocity amplitude and phase are also excellent

although errors are somewhat larger for velocity than for elevation.

We note that this is consistent with the fact that velocities are

computed as derivatives of elevation. The linear core model consistently

overpredicts velocity amplitudes by about 0.002 m/sec. Although the

absolute error for velocity is about the same throughout the channel,

the relative error (defined as a percentage of the exact velocity

amplitude at a point) becomes large at the closed end of the channel

due to velocities decreasing to zero. We note that this slight

overprediction corresponds to a small amount of leakage through the

closed end of the channel. This is due to the fact that normal flux

is treated as a natural boundary condition and will only be satisfied

exactly (no leakage) in the limit as the grid is refined. However, in

terms of the velocity at the entrance, the error between exact and

predicted velocities is less than 1%. Furthermore we note that the

finite element method is an error minimization procedure. The errors

depend on the degree of spatial discretization and solutions will

improve with increased grid refinement.

The numerically predicted values shown in Table 4.3 are values

for the channel centerline. Node to node oscillations across the width

of the channel were extremely small. The character of the maximum

node to node oscillation remained about the same regardless of the

amount of damping. For elevation, node to node oscillations for this

case were somewhat less than the discrepancy which existed between the

-77-

exact and numerical solution (typical maximum difference in elevation

amplitude across the channel was 0.00002 m which is about 0.002% of

the amplitude at each point). For velocity the node to node

oscillations were slightly greater than the discrepancy between the

exact and numerical solution (typical maximum difference in elevation

amplitude across the channel was 0.004 m/sec). For general two-

dimensional flows these node to node oscillations will increase

somewhat. However it is estimated that even under severe depth and

geometry changes the elevation amplitude oscillations will typically

remain less than 1% with corresponding velocity amplitude oscillations

increasing to several percent.

We conclude that the linear core model accurately simulates the

linearized governing equations at very low node to node oscillation

levels. Now that an accurate linear core model has been developed,

we are ready to proceed with the more complex task of incorporating

the non linear terms in the governing equations into our computations.

-78-

CHAPTER 5. NON LINEAR MODEL

The iterative solution technique used in the development of the

fully non linear model was described in Section 3.3. Chapter 4 examined

strategies for the optimal solution of each of the linear sets of

equations produced with this iterative scheme. Attention will now be

focused on the details regarding the implementation of the fully non

linear scheme. Of primary importance is the selection of a harmonic

analysis procedure for the pseudo loadings generated by the non linear

terms in the governing equations. In addition, issues such as iterative

stability and convergence rates will be addressed in this chapter.

5.1 Harmonic Analysis of Non Linear Pseudo Forcings

The selection of a harmonic analysis procedure for the non linear

pseudo forcing vectors is of vital importance for the efficiency,

accuracy and generality of the fully non linear model. The efficiency

of the model is strongly influenced by the number of time history points

required for the harmonic analysis procedure, since the procedure must

be applied at every node in the grid at each iteration. The accuracy

and generality of the harmonic analysis relate to the type and the

detail of harmonic information extracted from a given time history

record.

A variety of standard Fourier harmonic analysis procedures can be

applied to convert time history loadings to frequency domain loadings.

All standard Fourier procedures operate with integer multiples of some

base frequency. Therefore, Fourier analysis is quite satisfactory when

-79-

examining one major astronomical tide and its overtides. However, as

was noted in Chapter 2, tidal harmonics are not limited to frequencies

which are integer multiples of some base frequency. Tidal energy exists

throughout a wide range of frequencies which may be extremely closely

spaced and are, in general, irregularly distributed. Hence, in order to

obtain sufficient frequency resolution when Fourier analyzing the non

linear pseudo forcing time histories, an extremely small base frequency

step is required. Associated with this very small frequency step is a

very large total number of frequencies being processed, most of which

have no associated tidal energy. In Fourier analysis procedures, time

history record lengths and the total number of time sampling points

increase inversely with respect to the frequency step. Hence, the finer

the desired frequency resolution, the larger the number of time history

data points which need to be generated. Even application of the very

efficient Fast Fourier Transform algorithm would be impractical due to

the excessive amount of numerical operations required to obtain the

frequency resolution needed to separate important tidal components

[Oppenheim and Schafer, 1975; Newland, 1980].

A very attractive alternative to standard Fourier analysis

procedures is the least squares harmonic analysis method. This method

consists simply of a common least squares error minimization procedure

which uses a harmonic series as the fitting function. This harmonic

series only contains frequencies which are known to exist in the time

history record. The method is able to extract extremely closely and

irregularly spaced frequency information, yet it only requires a number

of time history sampling points equal to twice the number of frequencies

contained in the time history record. The almost infinite frequency

-80-

resolution and the extremely low number of required time history

sampling points make the least squares method the optimal choice for

the analysis of tidal records. Furthermore, since there are no set

requirements for record length and time sampling intervals, this method

is ideally suited for the analysis of field data [Munk and Hasselman,

1964; Filloux and Snyder, 1979a; Speer, 1984]. The method is even

better qualified for the analysis of analytically generated harmonic

response and non-linear pseudo forcing histories since these signals are

guaranteed to contain only the exact predictable harmonics associated

with a given set of astronomical forcing frequencies (i.e., there is no

energy due to non-tidal forcings).

In order to harmonically decompose a time history record with

values f(ti) at time sampling points ti; i - 1,N and with known

frequency content wj; J = 1,M, the following harmonic series is used

for the least squares procedure:

Mg(t) = f aj cosw t + b sinj ti (5.1)

j=1

where aj, bj are the unknown harmonics coefficients. The squared

error between the sampling points and the fitting function is:

NE = If(ti) - g(ti ) 2 (5.2)

Hence:

N M 2E = S r 7 (a cos t + b sinwt i ) - f(t (5.3)

i=1 j=1

-81-

The error minimization is accomplished by setting equal to zero the

partial derivatives of E with respect to each of the coefficients aj

and bj. Hence:

N M

E = [ (a coswjLi + b sinw t) -_ f(ti)]cosw t = 0 j=1,N (5.4a)j i=1 J=1

N M

8 = T [ 7 (a cosW ti + b sinw ti) - f(t )]sinj t = 0 j=1,N (5.4b)

j i=1 J=1

These equations lead to a complete set of 2M simultaneous linear

equations:

LSQ a = SLSQ (5.5)

where:

M is the least squares (LSQ) matrix-LSQ

a is the vector of unknown coefficients

S is the LSQ signal vector-LSQ

Equation 5.5 is shown in expanded form in Figure 5.1.

Steady state components in the signal being analyzed simply

correspond to a frequency equal to zero in the harmonic analysis series

(Eq. 5.1). When setting up the least squares matrix, the zero frequency

component will generate a row and a column of zeros. These should

obviously be eliminated when generating the LSQ matrix. This then

results in a 2M-1 x 2M-1 matrix when zero frequency is included as an

analysis frequency.

The harmonic least squares method does not have the matrix ill

conditioning problems often associated with the least squares procedure

-82-

i

cos 2 w ti

Ycosltisinl 1t i

Icoswlticos t i

Ycosw tisinMt i

sinw t Icosw t Tcos m cosw t

Isin2 wltii

sinblticos $t it

YcoswMtisinwl t

2

Tcos tiI

Tsinwtcosoltii

sinwMtisin lti

,sinMt cosw Mt ii

Tsin2 Ntii

Ysinw ltsin NMt ... YsinwMticoswMtii i

rf(ti)cosltii

Yf(ti)sinw lt

f( t )coswMti

ff(ti)sinwMti

Figure 5.1 Linear equation generated by least squares analysis procedure

applied with polynomial fitting functions. As is seen by examining

Figure 5.1, the LSQ matrix is diagonally dominant due to the squaring of

the diagonal summation terms, while terms in the matrix are still of the

same order of magnitude due to the nature of the sine and cosine

functions.

The LSQ matrix, MLSQ, need only be generated once in order to

analyze any of a number of time history signals with the same frequency

content. Hence, SQ is set and LDLT decomposed only once before

the cycling begins in the overall non linear iterative scheme. Upon

each cycle of the iteration the right hand side of Eq. 5.5, SLSQ

which contains the information on the actual time history signal being

harmonically analyzed, is re-set. The vector a, which contains the

harmonic coefficients being sought, may then be solved for by a forward

and backward substitution procedure. Each pair of coefficients ai,

bi, contained in vector a for every analysis frequency, are readily

converted to the complex form required for the frequency domain pseudo

loading vectors.

Setting up the signal vector, SLSQ, involves approximately 8MN

operations and solving the vector a requires roughly 4M2 operations

since the matrix ~SQ is already in decomposed form. These operations

must be performed for every nodal point at every cycle of the

iteration. In addition, the nodal time history values, f(ti), must be

generated at every node for each of the N sampling points. The effort

required for this is dependent on the node to element ratio of the grid

and on which non linear terms are included in the overall non linear

analysis.

-84-

For a time history signal for which the entire frequency content is

both known and used in the harmonic fitting series, the number of time

sampling points required to find the precise signal equals twice the

number of frequencies in the signal. Together with the orthogonality

properties of sinusoidal functions, this requirement insures that all

the equations in the system of equations (5.5) will be linearly

independent. The fact that the reproduction of the signal is precise

may be inferred by examining Figure 5.1 which shows that if f(ti) is

substituted by the original harmonic generating signal, this equation

becomes an identity. We note that although there is no noise in the

signal, there is still an inherent round-off accuracy for the elements

in the LSQ matrix and signal vector which places requirements on the

time sampling procedures. Let us now examine some of the sampling

criteria which allow the LSQ method to be optimally applied.

In order to avoid duplicity of information in the elements of the

LSQ system of equations, time sampling points should be contained within

the overall period of the signal being sampled. However, to allow

maximum dissimilarity in each equation, sampling points should be spread

throughout the period of modulation of the signal. The period of

modulation of a signal is obtained by examining its frequency content

and selecting the maximum value of all periods or synodic periods of the

harmonic components contained in the signal. The synodic period

describes the period of beating of two closely spaced frequencies and is

calculated as:

T 2 (5.6)S 1 2

-85-

~1M~___ ___I__

where wl, w2 = adjacent frequencies in radians/sec. It is

especially important to sample throughout the modulation period of the

signal when analyzing records with extremely closely spaced frequencies

in order to avoid round-off accuracy problems which could lead to

singularity of the matrix M SQ . Hence, when sampling a signal which

contains only two very closely spaced frequencies, the four sampling

points should be spread over the synodic period which will be much

greater than the periods of the individual components. Unlike field

sampling procedures, the numerical generation of time sampling points is

only affected by the number of points generated and is unaffected by the

period of time over which they are generated. If the procedures

discussed are adhered to, harmonic signals generated with a typical

tidal frequency content (e.g. frequencies of Table 2.6) can be exactly

(to within machine accuracy) recuperated while using only twice as many

time sampling points as frequencies contained in the signal.

The pseudo forcing signals generated by the non linear terms are

such that energy is transferred indefinitely to over and compound tidal

frequencies. The amount or order of the energy transfer to each

frequency is roughly described by the various levels of response-forcing

tables in Chapter 2. The number of frequencies appearing, as each

subseqent table is generated, increases exponentially. However, the

amount of energy transferred to new frequencies becomes increasingly

insignificant as they appear. Since it is numerically impossible (and

meaningless) to consider all the frequencies that energy is spread to,

the harmonic series representing the pseudo forcing must be truncated,

which establishes an order of accuracy for the harmonic analysis. This

-86-

accuracy should be compared with the order of spatial accuracy being

achieved by the finite element method.

Truncating the series has the effect that no interaction is allowed

between the truncated harmonics and those being considered. This will

therefore affect the response at the harmonic being analyzed. However,

if we have consistently selected as analysis frequencies all those which

correspond to a harmonic non linear pseudo forcing above some given

threshold, the effect of there being no feedback from the frequencies

not considered into the analysis frequencies introduces no more error

into the overall computation than the neglect of these frequencies in

the first place.

We shall now consider the effects the truncations have on the LSQ

analysis procedure. Let us examine some results of numerical

experiments which are illustrative of the behavior of the LSQ procedure

when fewer frequencies are present in the LSQ analysis series than are

present in the signal being analyzed.

Figure 5.2a shows an input signal with seven equally spaced

harmonics with the same input amplitude at each harmonic. When

analyzing this input signal with all seven frequencies present in the

manner previously prescribed (i.e., 14 time sampling points spread over

the maximum modulation), we find that the input signal is exactly

recuperated (to accuracy of 14 digits) as is shown in Case 1 of Table

5.1. However, if only four sampling frequencies are included in the

analysis series, and 8 time sampling points are used (while still

sampling over the entire modulation period), severe aliasing occurs at

the higher sampling frequencies. Increasing the number of time sampling

points by only 2 corrects the aliasing problem and accurate amplitudes

-87-

_ ~_~__~4~~

-1111110 2 3

F 1 quencyFrequency

1 i4w 5W 6w

(a) Input Signal

0 WF 2eI 3wFrequency

(b) Results for LSQ Analysis with 4 Frequenciesand 8 Time Sampling Points

0 W 2wI 3w1

Frequency

(c) Results for LSQ Analysis withand 10 Time Sampling Points

4 Frequencies

Figure 5.2 Effects of Variation in Frequency and Time Sampling Ratesfor Typical Overtide Frequency Distribution

-88-

2.0

1.0.

2.0.

1.0

1.0-

G I~-I~

I I X .

Table 5.1 LSQ Analysis Results Showing Effects of Variation of Number of Frequencies andTime Sampling Points; Example Simulating Overtide Type Frequencies

Analysis Resulting Coefficients for FrequenciesCase Frequencies N* At**

(hrs) 1 2 3 4 5 6 7

1 1 - 7 14 5.57 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.0000000.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

2 1 - 4 8 9.75 1.000980 1.002718 2.001712 1.998599 - -0.000000 -0.000161 0.005177 0.004993

3 1 - 4 10 7.80 1.000048 1.000252 1.000820 1.002468 - -0.000000 -0.000410 -0.000818 -0.001220

4 1 - 4 14 5.57 0.999265 0.998617 0.998885 0.999371 - - -

0.000000 -0.000639 -0.001276 -0.001911

5 1 - 4 56 1.395 1.003836 1.007666 1.007645 1.007611 - -0.000000 0.000410 0.000821 0.001232

6 1 - 4 224 0.3482 0.999219 0.998439 0.998439 0.9984400.000000 -0.000023 -0.000046 -0.000070

* N = number of time history sampling points**At = time step between consecutive time history sampling points

and phases are recuperated from the signal for all the frequencies

included in the sampling series. However, increasing the number of time

sampling points further does not increase the accuracy obtained

whatsoever. These effects are summarized in both Figure 5.2 and Table

5.1. The extent of aliasing depends on how many frequencies are

neglected and their associated energy level (i.e., neglecting higher

harmonics with small associated amplitudes produces less aliasing).

Finally, we note that in the example considered, which is representative

of an overtide frequency distribution, there was an increase in economy

in neglecting higher harmonics in the LSQ analysis since both the number

of frequencies and the number of time sampling points decreased. Care

must be taken that responses are not calculated for frequencies which

have aliased pseudo forcing amplitudes since this can lead to

instabilities in the overall iterative process.

Let us now consider an input signal which includes closely spaced

frequencies and hence is more representative of compound tidal

frequencies. Figure 5.3a shows an input signal with four groups of

clustered frequencies. Again, if we sample all 10 frequencies present

and adhere to the prescribed sampling procedures (20 time sampling

points distributed over the full modulation), we can exactly recuperate

the input signal, as is shown in Case 1 of Table 5.2a. However, if only

the first seven frequencies are used in the LSQ analysis series (and the

last cluster is neglected), the number of time sampling points must be

increased by more than 4 times over what it was when all ten frequencies

were present. Table 5.2a shows that severe errors occur (e.g., more

than 1000 fold increases in amplitude) if the time sampling point

density is not sufficiently high. This type of error leads to

-90-

0!0 0.5 10 .5 2 0Frequency (hr- 1 )

(a) Input Signal

2.0

1.01

0.0 0.5 1.0 1.5 2.0Frequency (hr

- 1)

(b) Results for LSQ Analysis with 7 Frequencies(one entire cluster neglected) and 80 TimeSampling Points

2.0.

1.0

0.0 0.5 1.0 1.5 2.0

Frequency (hr-1 )

(c) Results for LSQ Analysis with 9 Frequencies(one frequency neglected from within cluster)and 160 Time Sampling Points

Figure 5.3 Effects of Variation in Frequency and Time Sampling Ratesfor Typical Compound Tide Frequency Distribution (maximumperiod is T = 12.4 hours and maximum synodic period is

TS = 27 days) -91-

~.~u~'----- -CI-?~--~-?iliryL"1~- -LLsrr

Table 5.2a LSQ Analysis Results Showing Effects of Variation of Number of Frequencies andTime Sampling Points; Example Simulating Closely Spaced Compound Tide Frequencies

Analysis Resulting Coefficients for FrequenciesCase Frequencies N* At**

(hrs) 1 2 3 4 5 6 7 8 9 10

1 1 - 10 20 38.4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.00000.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

2 1 - 7 14 54.9 3.5800 1.6209 0.9163 1.9452 1.3661 1.9080 -1.3900 - - -0.0000 -0.0109 0.1232 -0.3419 0.0769 -0.2909 -1.0710

3 1 - 7 16 48.0 1.0543 1626. 3.0844 3644. 0.5364 -1623. -361. - - -0.0000 -94. 0.0855 -137. -0.3611 89. 1383.

4 1 - 7 40 19.2 1.0589 3.9483 -0.5780 0.8362 0.9887 1.0152 1.0601 - - -0.0000 -2.0861 1.4183 0.5070 0.0312 0.0746 0.1003

5 1 - 7 80 9.6 1.0271 1.0903 1.0743 1.0735 0.9905 1.0097 1.0490 - - -0.0000 -0.0547 -0.0425 -0.0632 0.0158 0.0606 0.0922

6 1 - 7 160 4.8 1.0120 1.0207 1.0193 1.0320 0.9925 0.9813 1.0056 - - -0.0000 0.0072 0.0068 0.0004 0.0176 0.0603 0.1110


I # 0I I , 4 . I I P

Table 5.2b LSQ Analysis Results Showing Effects of Variation of Number of Frequencies andTime Sampling Points; Example Simulating Closely Spaced Compound Tide Frequencies

Analysis Resulting Coefficients for FrequenciesCase Frequencies N* At** '

(hrs) 1 2 3 4 5 6 7 8 9 10

7 1-6, 8-10 18 48.0 0.1908 0.1101 0.9891 1.0170 0.1001 0.0996 - 0.1064 0.1601 0.02590.0000 0.1948 0.0448 -0.1295 0.0092 0.0068 -0.0155 -0.1720 0.1286

8 1-6, 8-10 40 19.2 0.9809 1.4910 0.7510 0.9872 1.0762 1.0618 - 1.0134 1.0146 1.00200.0000 -0.3460 0.2290 0.0623 0.0157 -0.0147 0.0216 0.0235 0.0293

9 1-6, 8-10 42 18.3 1.9852 1.0468 1.0015 0.9923 0.9977 0.9961 - 0.9993 0.9661 1.00190.0000 0.0843 0.0066 -0.0471 0.0006 0.0042 -0.0036 -0.0886 0.0499

10 1-6, 8-10 80 9.6 1.0028 1.0035 1.0037 1.0044 1.0644 1.0548 - 1.0046 1.0122 1.02410.0000 -0.0157 -0.0149 -0.0048 0.0219 -0.0059 0.0239 0.0305 0.0356

11 1-6, 8-10 82 9.4 0.9973 1.0561 0.8064 1.1660 1.7550 0.5231 - 1.1610 0.8487 0.99250.0000 -0.0213 -0.3497 0.2095 -0.3962 0.3497 0.3483 -0.2594 -0.0128

12 1-6, 8-10 160 4.8 1.0039 1.0053 1.0050 0.9992 1.0498 1.0562 - 0.9954 0.9989 1.01280.0000 -0.0126 -0.0118 -0.0063 0.0529 0.0231 0.0121 0.0252 0.0374

13 1-6, 8-10 162 4.7 1.0039 1.0061 1.0058 0.9994 1.0499 1.0572 - 0.9982 0.9987 1.00620.0000 -0.0133 -0.0124 -0.0073 0.0535 0.0239 0.0059 0.0150 0.0228


exponential growth of response in the full non linear iterative scheme

and causes complete iterative instability in a matter of several

cycles. Again the accuracy of the LSQ analysis doesn't increase with

further increases in the number of time sampling points beyond a certain

number.

Comparing the computational economy of using all the signal

frequencies versus using less sampling frequencies and more time

sampling points, shows that for this case it is far better to include

all the signal frequencies. When using all 10 signal frequencies and 20

time points, 2000 operations are required, while when using only 7

frequencies and the 80 time sampling points, 4676 operations are

required. These numbers are only indicative of the computational effort

required for the actual LSQ analysis per nodal point (i.e., the

conversion of a time history signal to a set of amplitudes in the

frequency domain for the pseudo forcing at a node) and do not include

the effort required to generate the time history sampling points. This

effort could very well be even greater than the increased effort

associated with the LSQ analysis.

In general, signals with very closely spaced frequencies are much

more sensitive to exclusion of frequencies in the LSQ analysis series

than signals with a widely spaced frequency content. The net increase

in the number of required time sampling points is very case specific and

depends on the closeness of frequencies in the signal, the number of

dropped frequencies and their associated magnitude. Often we may want

to increase the number of sampling frequencies, even though they have

magnitude which fall outside of the range of interest, simply in order

to decrease the number of time sampling points. There will be an

-94-

optimal balance between the number of sampling frequencies and the

number of time sampling points for every case. We note that storage

requirements are also effected by this balance between number of time

sampling points and number of sampling frequencies and should be taken

into consideration.

Finally the effect of excluding only one frequency contained in a

given frequency cluster from the analysis series is shown in Figure 5.3c

and Table 5.2b. We note that frequencies within the cluster from which

the frequency is excluded are most sensitive to the exclusion. Although

not illustrated in Table 5.2b, it can be shown that if the requirement

of spreading time sampling points throughout the modulation period is

not met, energy from the missing frequency will be lumped to the other

frequencies in the cluster in a constructive or destructive manner,

depending on where in the modulation period the time sampling points

lie.

Table 5.2b also illustrates a convenient methodology for

determining whether or not a certain number of time sampling points is

sufficient for the number of frequencies and characteristics of the

signal being analyzed. By simply adding two time sampling points and

accordingly adjusting the sampling time step (such that all time points

are evenly spaced over the modulation period) a totally different set of

time sampling points is generated. If the results of the LSQ analysis

procedure are the same, then the number of time sampling points is

sufficient.

As has become apparent in the preceding paragraphs, two categories

of frequencies may be defined. The first category consists of those

frequencies which have sufficiently large associated forcing/response

-95-

--L~- -UI~~-IYYIYIOIF~-Lq~

amplitudes that they should be included in the full non linear analysis

in order for the computation to be consistent to a given order of

accuracy. At these full non linear analysis frequencies, responses are

calculated (and hence the linear core model is run) allowing full

interaction between all frequencies in this category. The second

category of frequencies consists of frequencies which do not have

significant enough levels of forcing/response amplitudes to be included

as full non linear analysis frequencies, but are used to allow the LSQ

procedure to more efficiently extract accurate information at the full

non linear analysis frequencies. These frequencies do not affect the

interaction occuring between the full non linear analysis frequencies.

As seen earlier, the number of time sampling points required is

very case dependent. Furthermore a re-analysis of the signal with a

slightly higher number of time sampling points allows us to assess if

the time sampling density is sufficiently high for the LSQ procedure to

accurately analyze the signal. This allowed the convenient

implementation of an automatic time step selection feature in the

computer code TEA-NL (Non Linear Tidal Embayment Analysis), which

chooses the minimum time sampling rate to achieve a specified level of

accuracy for the non linear frequency domain pseudo forcing amplitudes

at each of the LSQ analysis frequencies. This time step selection

process need only be applied at a few representative nodes in the grid

at each cycle. If the number of time sampling points required exceeds a

user specified maximum number (the number at which it is more efficient

to include more frequencies for the LSQ analysis), TEA-NL stops

execution and allows the user to input additional LSQ frequencies

thereby premitting the LSQ analysis to be performed efficiently.

-96-

The number of frequencies generated for a typical tidal problem

with several significant astronomical forcing frequencies is quite

large. Although the tabular method of Chapter 2 gives a rough idea of

the importance of frequencies, we won't a priori know precisely which

frequencies have sufficiently large forcing/response amplitudes to be

included in the overall non linear analysis. TEA-NL has been set up to

select those frequencies which should be included as full non linear

analysis frequencies based on a user specified non linear pseudo forcing

threshold. The threshold is a percentage of the maximum spatially

averaged non linear forcings of all the frequencies. This automatic

selection procedure of full non linear analysis frequencies allows the

convenient and economical use of TEA-NL while ensuring that the order of

accuracy of the harmonic analysis (and hence the non linear interaction)

is consistent providing the user does not neglect to input certain

important frequencies. Frequencies with spatially average forcing

amplitudes less than the specified percentage of the maximum will be

excluded as full non linear analysis frequencies and are used only as

LSQ analysis frequencies. This percentage is taken as a fraction of the

expected smallest quantity of interest.

5.2 Iterative Convergence

The iterative stability of every scheme which solves a set of non

linear algebraic equations through direct iteration is dependent on the

magnitude of the right hand side term being smaller than the magnitude

of the linear terms on the left hand side of the equations. Therefore

the success of the fully non linear scheme used in TEA-NL is dependent

on the non linearities being of sufficiently small magnitude with

-97-

respect to the linear terms so that the non linear solution is only a

perturbed linear solution. For most tidal embayments, the non linear

terms in the governing equations do not dominate the linear terms.

However, the magnitude of the non linear friction term may become quite

substantial in shallow embayments with rapid velocities. Fortunately,

as is indicated by Eq. 2.20, for the case of a single astronomical

forcing tide, the major portion of the harmonically decomposed friction

term is distributed to the main astronomical forcing frequency itself.

It can also be shown that for the case of several astronomical forcing

tides, the dominant tidal frequency (usually M2 ) will still have the

largest pseudo forcing contribution from the non linear friction term.

Furthermore the magnitude of the coefficient of the forcing at the

dominant frequency will be close to the case where only the dominant

astronomical forcing exists. We note that this dominant harmonic

friction term may be approximated as a linearized friction term and

incorporated with the other linear terms on the left hand side of the

equations. This led to the use of the linearized friction term on both

sides of the momentum equations in Chapter 3. Hence the iteration now

occurs about a right hand side loading term which equals the difference

between a linearized friction and the fully quadratic friction term. If

the linearized friction factor is properly estimated, this can reduce

the right hand side loadings by an order of magnitude. For the fully

non linear scheme this increases the rate of convergence substantially

and in cases where friction dominates, makes an otherwise divergent

iteration converge.

We note that the value of the non linear friction coefficient,

cf, is dependent on the bottom surface, whereas the linearized

-98-

friction coefficient (see Eq. 2.6) is a property of both the non linear

friction coefficient and the local flow. The effectiveness of including

linear friction in decreasing the magnitude of the right hand side

loading term is dependent on how closely the linearized friction term

approximates the loading term of the harmonic expansion for non linear

friction (in Eq. 2.20). The most convenient scheme to obtain a good

local estimate for linear friction X is to update the user prescribed

value at the beginning of the second cycle of iteration using nodal

values of cf and the results of the first iteration for the nodal

magnitude of velocity for the dominant frequency. Hence, program TEA-NL

only requires that a reasonable global linearized friction factor (based

on some global cf and global estimate for velocity) be specified in

order to start the iterative process. The updated nodal values for X

obtained in the second iteration are not only helpful in speeding

convergence rates, but also allow an improved fully linear solution (if

TEA-NL is run in the linear mode) due to the much improved local values

for X.

Finally we note that iterative convergence rates may be improved by

including the computation of finite amplitude and convective

acceleration effects only beyond the second cycle of iteration. The

reason for this is that friction is usually the dominant term and once

the repsonses in elevation and velocity have adjusted for it, the

effects of the other terms are more accurately assessed. For example,

if the user specifies a linearized friction factor which is too low, the

elevation amplitude in the first cycle would be overestimated. This in

turn results in the over-adjustment for finite amplitude effects in the

second cycle, while the elevation amplitudes being calculated in the

-99-

second cycle would have adjusted better to the actual non-linear

friction. However, including finite amplitude effects only beyond the

third cycle of iteration does not allow this type of overcompensation.

So far we have seen how convergence rates could be improved.

However, the question that remains is at what point we can consider the

solution to have converged. Obviously the computational effort of

TEA-NL is directly related to the total number of iterative cycles that

need to be run. When determining the degree of accuracy which the

iteration process should achieve we should consider the following

points:

(i) There is an order of accuracy associated with the finite

element method which was used for the spatial discretization

of the governing differential equations. The accuracy depends

on the grid size and the types of gradients (relative to the

grid size) for both the flow field and the depth variation.

Furthermore, the linear core solution exhibits a certain

degree of node to node oscillation in the solution calculated

for elevation and velocity. This oscillation may be typically

quantified to the order of several percent of the magnitude of

the solution at a given node. The oscillation is somewhat

greater for velocity than for elevation. The accuracy of the

computation is not improved by carrying the iterative accuracy

beyond the estimated percentage of the node to node

oscillation. This then indicates that achieving a relative

accuracy of several percent at each frequency is sufficient

for iterative convergence.

-100-

(ii) We note that non linear pseudo forcings are generated with

elevations and velocities which contain a certain amount of

node to node oscillation. Hence we expect some deterioration

in the solution achieved at each of the various levels of

frequencies described in Chapter 2 (i.e., more node to node

oscillation). The degree of deterioration depends on the

magnitude of oscillation relative to the overall magnitude of

the forcing (signal to noise ratio). Furthermore, it depends

on which of the non linear terms are included in the analysis

and their relative importance. For example, the finite

amplitude forcing term is due to a gradient of the product of

elevation and velocity. If the gradients in elevation and

velocity are smaller than the relative node to node

oscillation (which depends on the magnitude of the terms), we

would expect a meaningless set of harmonic pseudo forcings to

be generated. In general, the forcing signal to noise ratio

will be such that it will allow the meaningful calculation of

significant response frequencies. For the finite amplitude

and convective acceleration pseudo forcing terms, this level

of deterioration increases as energy is cascaded down to the

various levels of frequencies. For the friction pseudo

forcings this signal/noise effect is less pronounced due to

the fact that the forcing term does not include any

derivatives and furthermore energy is cascaded from the major

astronomical forcing levels to all the compound and overtide

frequencies at once. We conclude that this noise in the

-101-

pseudo forcings must be considered when determining the

convergence achievable at each of the frequencies.

(iii) There is an order of accuracy associated with the truncation

of the harmonic series used for the time discretization of the

governing equations. Hence the frequencies taken into

consideration are effected to a certain degree by the lack of

interaction with the missing harmonics. As previously

mentioned the overall accuracy of the computation by not

considering this interaction is no worse than that caused by

not considering these terms in the first place. However,

performing a calculation of a given frequency beyond the

estimated percent of the missing non linear interaction would

be meaningless.

(iv) Bottom friction and bottom topography can only be described to

within a certain degree of accuracy. Therefore the iterative

accuracy being sought should also take into account the

variability in response associated with the uncertainty in

parameters.

All these points should be taken into consideration when

determining the level of accuracy which the iteration process should

achieve. This level is case dependent and also varies for each of the

frequencies for which the calculations are being performed.

Program TEA-NL allows the determination of the level of accuracy

achieved and the rate of convergence by calculating the following

-102-

parameters at each cycle and for each freuqency under consideration:

Dn = maximum global difference between magnitude of elevationcalculated at consecutive cycles

DU = maximum global difference between magnitude of velocitycalculated at consecutive cycles

Dr = average global difference between magnitude of elevationcalculated at consecutive cycles

DU = average global difference between magnitude of velocitycalculated at consecutive cycles

n average global value of elevation amplitude

U = average global value of velocity ampltiude

RP = convergence rate for elevation amplitude

RU - convergence rate for velocity amplitude

L = relative convergence level for elevation

LU = relative convergence level for velocity

The relative convergence levels are defined as:

D

L (5.7a)

DULU -- (5.7b)

U

It is these values that should be used to determine convergence in

TEA-NL. Typically they will be on the order of several percent. We

shall discuss values for these parameters in more detail in Chapter 6.

Finally, we note that R, and RU are not only indicative of the

improvement in accuracy with each cycle of iteration, but may also be

indicative of the level of signal to noise in the pseudo forcings.

-103-

~'-L-~'mrrrrrYr~-lir~jYBrr~ a~rr~. -rrc~

CHAPTER 6. APPLICATION

Program TEA-NL is a very flexible computer code which allows the

general calculation of non linear tides. TEA-NL is unique in its

ability to compute compound tides. As an example application, TEA-NL

will be used to investigate tidal circulation within the Bight of Abaco

in the Bahamas. This is an ideal application not only because

significant non linear tides are generated, but also because extensive

field data collection and analysis have been performed for this shallow

semi-enclosed basin (Filloux and Snyder, 1979). Furthermore, the basin

is such that it allows simple boundary conditions to be applied for the

non linear tides.

6.1 Description of Bight of Abaco and Its Tides

The Bight of Abaco (Figure 6.1) is a shallow embayment with land

boundaries consisting of the Island of Abaco along the southern and

eastern parts of the embayment and the islands of Little Abaco and Grand

Bahama along the northern parts. Although the northwestern part of the

embayment does have a number of shallow connections to the open ocean,

data taken by Filloux and Snyder (1979) showed that these openings were

relatively opaque to the tides and could therefore be treated as land

boundaries. They reached this conclusion by comparing the amplitudes

of major astronomical tides at several locations lying within the bight

along the northwestern boundary to those at a nearby location in the

open ocean.

-104-

Figure 6.1 Geography of Bight of Abaco, Bahamas. Dotted line

represents 200 m contour.

-105-

-- I- --~*axmrrr^-- ---~-*r--rrs(ul EICILI^* ~-lrr ---- ~ l-~(~ (lre~~

The western edge of the bight is connected to the open ocean and is

characterized by an extremely sharp discontinuity in depth. The 200 meter

depth contour lies between 1 to 3 kilometers from the 5 meter contour and

the 1000 meter contour lies between 3 to 15 kilometers from the 5 meter

contour. Ultimately depths drop to between 1500 and 2000 meters. Hence

depths on the ocean side of the boundary drop by a factor of between

200 and 800. As a wave passes over a step of this size, it is largely

reflected. This may be shown by considering the reflection and

transmission coefficients of an incoming long wave passing over a step

from depth h1 to depth h2 which are expressed as (Ippen, 1966):

-1-/h I /h - 1

K = (6.1a)r /h 2 + 11 2

K = (6.1b)t /h/h + 11 2

Values for various depth ratios h1/h2 are shown in Table 6.1. These

equations, which were derived for a vertical step, will apply to the

Bight of Abaco case since even the shortest possible overtide wavelength

(M6 at a depth of 2 m has a wavelength of 66 km) is many times greater

than the distance over which the most substantial portion of the depth

drop occurs. For the Bight of Abaco h /h2 has a range of between 0.005

and 0.001 and it is seen that the reflected wave has an amplitude

between 0.87 and 0.94 of the incoming wave and is reflected out of phase

with respect to the incoming wave. The transmitted wave has an amplitude

between 0.13 and 0.06 of the incoming wave and is in phase with the

-106-

Table 6.1 Reflection and Transmission Coefficients for a Long WavePassing Over a Step from Depth h1 to Depth h2 forVarious Depth Ratios

-107-

h1 Kh K

h r t

0.1 -0.52 0.48

0.01 -0.82 0.18

0.005 -0.87 0.13

0.001 -0.94 0.06

0 -1.00 0.00

_ _____1_LI1LCIIYlliI~~ i I~__~IXI -I --Ue~-~I~ -(~~ -^X

incoming wave. When considering the lateral expansion that occurs, the

amount of reflected energy increases and the transmitted wave becomes

even smaller. Hence it is a reasonable approximation in this case to

assume that the wave is totally reflected. This allows the very

convenient treatment of the boundary conditions for the non linear tides.

Since depths on the open ocean side of the boundary are very deep, there

will be essentially no non linear tides generated there. The non linear

tides generated within the shallow bight, however, will be reflected

back into the bight due to this severe depth discontinuity. As a result

no non linear tidal species will exist in the open ocean and their

amplitudes should be specified as zero along the ocean boundary.

Figure 6.2 shows the bathymetry within the bight. In the region

along the open ocean boundary, depths vary between 2 and 5 meters. This

region actually forms a sill since depths increase again in the interior

of the bight. In the northern half of the bight, a 7 - 8 meter

depression is the dominant feature. Depths become very shallow along

the northern edge.

Bottom characteristics also vary somewhat within the bight. The

sill region has a bottom surface characterized by numerous sand waves

with heights between 1 and 3 meters. These sand bores are not

represented in the depth distribution. The northern depression region

contains muddy mounds with a relief of 10 cm and horizontal scale of

several meters. The relatively flat southern portion of the bight has a

bottom surface consisting of a thin sediment cover over rock, punctuated

in patches by sea fans and corals (Filloux and Snyder, 1979; Snyder,

Sidjabat and Filloux, 1979).

-108-

Figure 6.2 Bathymetry of the Bight of Abaco, Bahamas.Depth Contour in Meters.

-109-

Filloux and Snyder (1979) have run a series of three field

experiments, each lasting approximately one month, which measured

elevation at 15 locations within the bight. At each location bottom

mounted tide gauges with a sensitivity of 1 cm collected a time history

of bottom pressure. The bandpass characteristics of the data are such

that steady motion, surface wave motion and turbulence are excluded from

the data records collected. These time history records were then

harmonically decomposed using a least squares analysis procedure which

uses 5 astronomical frequencies (01, K1 , N2 , M2, S2) and two overtide

frequencies (M4 and M6) for the analysis series. A time point sampling

rate of 4 data points per hour was used from the available recorded

40 readings per hour (Filloux, private communication). Atmospheric

pressure records were also collected and harmonically analyzed in the

same manner as the tidal records. This allowed the bottom pressure

amplitudes to be adjusted to reflect only water pressure variation.

Hence surface elevation amplitude and phase data were obtained for the

M2 , N2 , S2, 01 and K1 astronomical tides and the M4 and M6 overtides

at 25 points throughout the bight. The M2 tide is the major component

having an amplitude of 40 cm along the open ocean boundary. The

amplitudes and phases along the open ocean boundary of the five

astronomical components are summarized in Table 6.2. The amplitudes

and phase lags for surface elevation, obtained at the various measurement

points in the bight for the M2, M4 , M6 and N2 tides are shown in

Figures 6.3 through 6.6. These figures only reflect data which Filloux

and Snyder (1979) considered to yield consistent and stable estimates.

-110-

Table 6.2 Summary of Measured Astronomical TidesAlong the Open Ocean Boundary

-111-

Phase LagTide Amplitude (radians relative

(cm) to M2)

K1 9. 3.5

01 7. 3.7

N2 10. 5.9

M2 40. 0.0

S2 6. 0.9

eru~irarrari-pr---,~--uLI rruu,~,~yiyaF1 __~~~~~_~VLR-

+39.2

419.6 + 21.E

16.4+17.2

17.518.5

16.9 15+18.3 +14

\.15

Measured Elevation

Amplitude

37.4+39.0

+29.2+14.615.613.5 1I. 9 26 +38.8

. 3 +

.3 12.2

+ 39.4

.5 27.9+

15.9 +12.714.5 15.8+

14.1

16.7+ 16.715.9 +16.6

+16.3

19.1

+18.1 19.2+17.617.9 18.8

17.1 1 .1

Figure 6.3 Field Data for M2 AstronomicalConstituent (after Filloux & Snyder, 1979)

(a) Amplitude in centimeters

-112-

-0.07+

+0.87 +0.56

0.96+0.89

1.15+0.96

0.87+0.860.82

1.29 1.321.33 +1.10

1.501l.521.52

+ 1.73

Measured ElevationPhase Lag

0.00+0.00

+0.17

0.35+ +0.19

40.94

-0.02

+ 0.070.17+

+1.70

1.83+1.681.80

1.94+

+ 1.85

2.011.92 1.95+

88 1.921.76

Figure 6.3 Field Data for M2 AstronomicalConstituent (after Filloux & Snyder, 1979)

(b) Phase lag in radians (relative to the

M2 tide)

-113-

_1_I1IILI__II~ *L(LI*_IIY~ -_I~--~- I

0F +0.2

+2.1 +0.8

1.21.4

,1.2+ 1.21 +0.90.8

1.2 0.81.1 +0.7

0.50.5+ 00.5

+ 0.6

M

Measured ElevationAmplitude

+0.20.2

+0.3

0.6+ 40.2 +0.3

+0.2

0.3+

0.9 +.2+

0.3 +0.5

+0.50.30.5

+ 0.7

0.5 .0+0.5 + 0.7

7 0.6 0.9+ 0.9

Figure 6.4 Field Data for M4 Overtide Constituent(after Filloux & Snyder, 1979)


-114-

+1.57

+ 1.22 + 0.33

1.261.33

1.331.06

1.74 1.+1. 52 +1i.

+ 1.

+2.


A+

+; +6.02

0.38+ 44.92+2.39

06 +0.3859

+ 1.95

5.65+

+ 3.56 4.05+4.05+3.91

2.22

2.442.88 +2.91

73

2.602.72 2.60++2.56 2.95

2.88 2.90+ 3.04



M2 tide)

-115-

Y1-Ya~ul---rrrcP;~~iIIYTPll)lli- -r*--a~ , I,

r +0.3

+0.1 +0.5

0.20.3

0.8 0

0.80.7

1.6 0.8+1.3 +1.0

+0.70.7


0.20.2

+0.3

0.6 + +0.4 +0.2

+0.7

+0.2

0.3+

+0.30.7 0.4+

0.4

0.2+0.40.4 + 0.3

+0.6

0.5 070.3 + 0.4 0.5+

0.5 0.5+ 0.6

Figure 6.5 Field Data for M Overtide Constituent(after Filloux &6 Snyder, 1979)


-116-

"" +3.80

+0.23 +1.71

4.404.69

+4.404.82 3.7 4

3.793.63

5.10+5.15 44.284.92

4.97+4.76 3.4.94

5.585.295.67

+5.64

6.6.0


+ 5.674.63

+ 3.16

3.32+ +2.37 +5.25

+3.18

+5.952.34+

91+ 1.74

1.90

+0.45

25 0.10 0.3821 ' 0.70+

.28 1.170,45



M2 tide)

-117-

~II-----"II~ -I~L~L~" -T~ yl ~~_ _

+9.7

+ 3.7 -,o

+3.23.4

4.24.1 2.7

3.83.4

4.3 2.64.0 + 2.9

2.2+ 3.83.3

2.22.73.5

+2.4


+10.39.5

+7.6

+ +7.03.3 +10.3

2t 9

+10.0

7.7+

+2.03.6+3.4

+2.3

2.72.1 1.7 +4.34.0 3.6

3.2 +2.3

Figure 6.6 Field Data for N2 AstronomicalConstituent (after Filloux & Snyder, 1979)


-118-

+0 ' +

o061 '.19

+0.610.52

0.70+0.79

0.87 0.+~1.13 +1.

5.78Measured Elevation

Phase Lag

5.85+6.02

4+ 6.00

6.14+ +6.005.85

61 +0.6810

-.0.006.02

0.93 ++1.03 0.98 1.41

+ 1.411.33 1.95+

0.91+ 1.201.69 + 1.22

41.06

1.31

0.98 1.48 1.50+1.43+ 1.85

1.82 +1.50

Figure 6.6 Field Data for N2 AstronomicalConstituent (after Filloux & Snyder, 1979)


M2 tide)

-119-

-- I.aar~ i)iWII~~~PIYILc--arx~ aar~rs- - ~~-X

Filloux and Snyder (1979) predicted the theoretical statistical

error associated with their data analysis procedure. Their actual

errors, determined by the variability in results at a point from

experiment to experiment, generally substantially exceeded the computed

theoretical statistical error. They suggest that this may be the result

of not having included more harmonic components in their least squares

analysis procedure. As we shall see later in this chapter, not including

more harmonic components could very well have led to increased errors

associated with certain of the harmonics which Filloux and Snyder were

able to extract from the time history records.

The variability in the analyzed measurement data may be quantified

by calculating the proportional variance for each harmonic. The

proportional variance is computed by summing the square of the difference

between the elevation amplitude value obtained from experiment to

experiment at a particular location and the average value at that location

and then dividing by the sum of the squares of all the measured values.

For the computation of proportional variance of measurement data only

locations with more than one data point are used. The proportional

variance of measurement data for each harmonic j is expressed as:

' K K 2L IK m 1 k

nj (x, k) - Z n. (x' k)

Vm= £= k=l k k=1 (6.2)

SKL £ 2

Snj(x , k)£=l k=1

m thwhere n. = measured elevation amplitude component for j harmonic

xk = measurement location within the bight

-120-

L = total number of measurement locations with multipledata points

K = total number of measurement data points at location Z

The reduced data presented by Filloux and Snyder (1979) was used to

compute the proportional variances for each of the seven harmonics

used in their data analysis. These proportional measurement variances,

Vm, were computed for the bight as a whole and are presented in Table

6.3a. Furthermore proportional measurement variances are presented

for the three distinct regions (the sill region, the northern bight

and the eastern bight) into which the bight may be divided. Each of

these three regions contain an equal number of multiple data point

measurement locations. The proportional variances are generally

equal or somewhat lower in the sill region and eastern part of the

bight (when compared to values for the entire bight) and are generally

substantially greater in the northern part of the bight. A somewhat

easier way to interpret these proportional variances is to examine the

proportional standard deviation which is equal to the square root of

the proportional variance:

Sm = i (6.3)j j

The proportional standard deviation may be viewed as the standard

deviation in terms of a fraction of a global representative measure of

amplitude. Table 6.3b presents the associated proportional standard

deviations for measurement data. It is noted that the M2, 01 and K1

tides (K1 measurement errors are consistent with M2 and 01 in the

northern and eastern bight and K1 is therefore included) all show very

-121-

Table 6.3a Measurement Error for Each Frequency inTerms of Proportional Variance, Vm

J

mV

Tide Entire Sill Northern EasternBight Region Bight Bight

K1 0.0063 0.0015 0.0154 0.0006

01 0.0008 0.0007 0.0013 0.0005

N2 0.0171 0.0055 0.0450 0.0126

M2 0.0009 0.0012 0.0007 0.0013

S2 0.0205 0.0134 0.0530 0.0009

M4 0.0105 0.0134 0.0285 0.0014

M6 0.0144 0.0054 0.0294 0.0120

-122-

Table 6.3b Measurement Error for Each Frequency inTerms of Proportional Standard Deviation,

mS

Tide Entire Sill Northern Eastern

Bight Region Bight Bight

K1 0.08 0.04 0.12 0.02

01 0.03 0.03 0.04 0.02

N2 0.13 0.07 0.21 0.11

M2 0.03 0.03 0.03 0.04

S2 0.14 0.12 0.23 0.03

M4 0.10 0.12 0.17 0.04

M6 0.12 0.07 0.17 0.11

-123-

~YI~

modest measurement errors while the N2, S2, M4 and M6 tides all exhibit

somewhat higher errors. There is no correlation between average

amplitude in a region and the error incurred in the reduced measurements.

As previously mentioned, we will later show that the N2, S2 , M4 and M6

tides display greater measurement error due to the existence of significant

(relative to each of these tides) closely spaced compound tides which

were not used in the data reduction procedure.

In the following two sections we shall examine the circulation

patterns predicted by TEA-NL for the bight. First, only the M2

astronomical tide and the steady, M4 and M6 overtides it generates

will be considered. Then the much more complex interaction of the

closely spaced M2 and N2 astronomical tides and their associated

compound and overtides will be investigated.

6.2 Overtide Computations for the Bight of Abaco

TEA-NL requires both a description of the geometry of the

embayment, in the form of a finite element grid, and a set of boundary

conditions for each of the various harmonic components being

considered. The finite element grid discretization for the Bight of

Abaco is shown in Figure 6.7. The grid has been refined in the area

surrounding the small island which lies along the ocean boundary and

also in the region adjacent to the Island of Grand Bahama.

In this section we shall examine the circulation resulting from the

main astronomical (boundary forcing) component and its most significant

overtides. Along the ocean boundary the M2 amplitude of 40 cm measured

by Filloux and Snyder (1979) is used as the elevation prescribed boundary

-124-

Figure 6.7 Finite Element Grid Discretization forBight of Abaco, Bahamas.

-125-

~---~~--L-ieu l----- ----

condition. Furthermore for the steady, M4 and M6 overtides, amplitudes

of zero are prescribed along the ocean boundary due to its reflective

nature. Land boundaries are all specified with zero normal flux for

all frequencies.

In order to determine the sensitivity of variations in bottom

friction, three sets of overtide runs were performed with TEA-NL

corresponding to non linear friction factors of 0.003, 0.006 and 0.009.

Results for both elevation amplitude and phase of the M2 component are

shown in Figures 6.8 through 6.10. The effects of the increase in

friction factor cf are illustrated in Figure 6.11b which shows M2

elevation amplitudes along the trajectory defined in Figure 6.11a.

This trajectory passes through three representative areas in the bight;

the sill region, the central bight and the northern bight. The increases

in bottom friction correspond to decreases in elevation amplitude and

increases in phase shifts although the overall pattern of the distributions

remain similar. The sill region, which corresponds to a very shallow

region which has rapid flow, is a high gradient region and shows the

most substantial reduction (damping) in elevation amplitude and the

largest increases in phase lag. Elevation amplitudes are the smallest

in the center part of the basin and increase again somewhat towards the

eastern and northern land boundaries. The increase in amplitude is the

most significant along the northern edge of the bight where depths decrease

very rapidly (from 8 m to 1 m). We note that for a non linear friction

coefficient of cf = 0.009 there is excellent agreement between the M2

elevations predicted by TEA-NL (Figure 6.10) and the measurements by

Filloux and Snyder (1979) (Figure 6.3). This value for cf is consistent

-126-

32-

528

Figure 6.8 Results of TEA with Full Non LinearFriction Effects (cf = 0.003) for M2Astronomical Constituent

(a) Amplitude in centimeters.

-127-

-L--~-------cPm~i~ Irr~. r~.*~.~g(. -u~~

O.N0.o( \

run D2

PHASE

1.0 2

1.5


(b) Phase lag in radians (relative to M2tide

-128-

run el

AMPLITUDE

2'/32

Figure 6.9 Results of TEA with Full Non LinearFriction Effects (cf - 0.006) for M2Astronomical Constituent


-129-

0.0

10 PHASE

1...

1,5


(b) Phase lag in radians (relative to M2tide

-130-

20

2q 2

Figure 6.10 Results of TEA with Full Non Linear

Friction Effects (cf = 0.009) for M2Astronomical Constituent


-131-

1.1

5,43

Figure 6.10 Results of TEA with Full Non LinearFriction Effects (cf - 0.009) for M2Astronomical Constituent

(b) Phase lag in radians (relative to M2tide)

-132-

__;II_~Y _XI~_;~

Figure 6.11a Trajectory along which M ElevationAmplitudes are Compared for VaryingFriction Factor in Figure 6.11b

-133-

i 'II I .

Sill _ Central Northern

50 Region Bight Depression

45

40

35

n MI 30

(cm) 25

20 - c 0.003

15 cf = 0.006

10 Cf = 0.009

5

0" I I I I I " I I I I

S

Figure 6.11b Comparison of M 2 Elevation Amplitudes for Varying Friction

Factor, cf, along Trajectory S

with what would be expected for a water depth of 2 - 7 m with large

bedforms (dunes of 1 m) as is shown in Table 6.4.

Figures 6.12 through 6.14 show overtide (steady, M4 and M6)

elevation amplitudes and phases corresponding to the M2 forcing tide

with a friction factor cf = 0.009. These computations did not include

the effects of finite amplitude in the continuity equation or convective

acceleration in the momentum equation and hence the overtide responses

shown are generated by the non linear friction term. The most

significant overtide is the M6 tide (Figure 6.14) since as was discussed

in Chapter 2, the harmonically decomposed friction pseudo forcing term

is distributed mainly to the M2 and M6 frequencies. Of secondary

importance is the finite amplitude effect in the bottom friction term

which generates pseudo forcings/responses at even harmonics. This is

reflected in the weaker responses at the steady and M4 overtides (as is

shown in Figures 6.12 and 6.13). We note that TEA-NL requires the

specification of a zero reference at some point in the domain with

respect to which all elevations are computed. For this case a point

along the ocean boundary was used for convenience.

Figures 6.15 through 6.18 illustrate the effects of including in

the computation both the finite amplitude term from the continuity

equation and the friction term (again with a friction factor of

cf = 0.009). As was shown in Chapter 2, the most significant pseudo

forcings due to the finite amplitude term in the continuity equation

are distributed to the steady and M4 overtides and are a result of the

responses at M2. This is reflected in the substantial increase in

response for the steady and M4 overtides (compared to computations

-135-

Table 6.4 Values for Friction Factor cf in Terms of Depth and BottomRoughness (from Wang and Connor, 1975)

HBot- H

tom [m] 1 2 5 10 20 30 40 50 100rough-ness

k[] [seck [m]

Stones0.07 0.025 0.0061 0.0049 0.0036 0.0028 0.0023 0.0020 0.0018 0.0017 0.0013

Smallrocks0.20 0.030 0.0088 0.0070 0.0052 0.0041 0.0033 0.0028 0.0026 0.0024 0.0019

0.035

0.040

0.0095 0.0070

0.0092

0.0056

0.0073

0.004410.003910.0035

0.0058

0.0026

0.00340.0051 0.0046

0.0033

0.0043

-136-

7unes0.50

1.10

________L_ (l ___jllIC--~-~L-~-~ --~I~I

"'

Figure 6.12 Results of TEA with Full Non LinearFriction Effects (cf = 0.009) for SteadyState Constituent. Amplitude incentimeters.

-137-

0.4

run e2

AMPLITUDE0.T

0-

OO.LL

0. L

Figure 6.13 Results of TEA with Full Non LinearFriction Effects (cf - 0.009) for M4Overtide Constituent


-138-

I~L __LLI1___~____1~1_~_C--II Y~U.LII ~tl -^-.~I_

i. o M4

/5 run e2

PHASE

3.O


Friction Effects (cf = 0.009) for M4Overtide Constituent


-139-


Friction Effects (cf = 0.009) for M6Overtide Constituent


-140-

I1I~-~-IIY..P~--- -L-*~ II~I~L~i~~^---~~--- -~

Figure 6.14 Results of TEA with Full Non LinearFriction Effects (cf - 0.009) for M6Overtide Constituent


-141-

20'

Figure 6.15 Results of TEA with Full Non LinearFriction Effects (cf = 0.009) and Finite

Amplitude Effects for M2 AstronomicalConstituent


-142-

I

I

0.7L

I.A Ii

1.8

Figure 6.15 Results of TEA with Full Non LinearFriction Effects (cf = 0.009) and FiniteAmplitude Effects for M2 AstronomicalConstituent


-143-

0.0 STEADY

run e3

AMPLITUDE

.2.5

2,25

1 2.o

2.5

2.25

2,0

2.5


Friction Effects (cf = 0.009) and FiniteAmplitude Effects for Steady StateConstituent. Amplitude in centimeters.

-144-

~I=-- IIOL~I~-Y ..

~------~

Figure 6.17 Results of TEA with Full Non LinearFriction Effects (cf = 0.009) and FiniteAmplitude Effects for M4 OvertideConstituent


-145-

run e3

PHASE

1.5

2.5



-146-

..-..m lux.-r~---^--r*llra~- -* IY"eY"Lnar~~.--



-147-

/

Figure 6.18 Results of TEA with Full Non LinearFriction Effects (cf a 0.009) and FiniteAmplitude Effects for M6 OvertideConstituent


-148-

with friction effects alone) as is shown in Figures 6.16 and 6.17.

Contributions of the finite amplitude pseudo forcing to other

frequencies (M2, M6, etc.) are of secondary importance since these

are generated by either overtide responses (which are much smaller than

the M2 response) or through the interaction of an overtide response

with the main M2 tide. This is unlike the friction term for which

all overtides may be directly generated by the astronomical tides.

The limited effect of the finite amplitude term on frequencies other

than steady and M4 is demonstrated in Figures 6.15 and 6.18 which

show that responses at M2 and M6 remain essentially unchanged

(compared to computation with friction effects alone).

The changes resulting from including convective acceleration

in the computation in addition to friction and finite amplitude

may be seen by examining Figures 6.19 through 6.22. As expected

responses at M2 and M 6 remain essentially unchanged while there are

very slight changes for the steady state and M4 elevation amplitude

distributions.

All three overtides calculated have responses of approximately

equal importance. In general, patterns of elevation amplitude vary

somewhat for these overtides. However they all exhibit high gradients

in the sill region. High gradients for both elevation and flux are

also prominent for the M2 astronomical tide in this region. The high

gradients of the main tide, together with higher M2 velocities and

elevations and (furthermore) shallower depths, result in much greater

non linear pseudo forcings in this region relative to the rest of the

-149-

III__IUY__~___I*__CI__LII___,

Figure 6.19 Results of TEA with Full Non LinearFriction Effects (cf = 0.009), FiniteAmplitude Effects and ConvectiveAcceleration Effects for M2 AstronomicalConstituent


-150-

0.6_-O,L)" o.

I~1e 1

Figure 6.19 Results of TEA with Full Non LinearFriction Effects (cf = 0.009), FiniteAmplitude Effects and ConvectiveAcceleration Effects for M2 AstronomicalConstituent


-151-

~-- (irrrrr~-~- i-i 3 -li L-yCyrsUlpl -_- lrrYI*~C'_r-- s~l- ~_qii~L~LBgirCLirY-n~-~ --.-~ ̂.-~X

__

0.0 STEADY

run e4

AMPLITUDE

2.7 5

Z .5

2.25

Figure 6.20 Results of TEA with Full Non LinearFriction Effects (cf = 0.009), FiniteAmplitude Effects and ConvectiveAcceleration Effects for Steady StateConstituent. Amplitude in centimeters.

-152-

LI.00. , - - '

Figure 6.21 Results of TEA with Full Non LinearFriction Effects (cf = 0.009), FiniteAmplitude Effects and ConvectiveAcceleration Effects for M4 OvertideConstituent


-153-

----~'*siPII-YLXlr~i

run e4

PHASE

3.0



-154-

_ _

1.0 o.5

Run e4

AMPLITUDE

2.5,.5

2.5

2.0

I.0 -

I. ~ c



-155-

_III____YII__LYiUI~_CIIP.I.XI~1__X^~.

3.0 Run e4

PHASE4. 5

4-S

so

6.0

Figure 6.22 Results of TEA with Full Non LinearFriction Effects (cf = 0.009), FiniteAmplitude Effects and Convective

Acceleration Effects for M6 OvertideConstituent


-156-

bight. The sill region is therefore responsible for a substantial

portion of the generation of the overtides.

We note that for the steady state component, the high elevation

gradients in the sill region correspond to a seaward flushing current

which is shown in Figure 6.23a. This steady residual current, which is

the result of the finite amplitude term in the continuity equation,

exists only in the sill region since the gradients in elevation are

small in the rest of the bight. It should be stressed that the steady

residual current computed and shown in Figure 6.23a is a residual

velocity current and is equivalent to the time averaged velocity

expressed as:

uR = u(t) = uw= 0 (6.4)

This steady Eulerian residual velocity is associated with the net drift

of a particle traveling with the velocity of the fluid. It is distinct

from the time averaged flux which gives the following residual flux

current:

QR = u(t)(h + n(t)) = uRh + u(t)n(t) (6.5)

We note that mass is not conserved when considering the steady velocity

currents by themselves since, as was noted in Chapter 3, the harmonic

continuity equation for the steady component is coupled with harmonic

continuity equations for other frequency components. For the residual

flux currents, however, mass is conserved for each individual frequency

constituent. This is due to the fact that the continuity equation in

terms of flux is a linear differential equation and therefore leads to

-157-

Figure 6.23a Velocity Results of TEA with Full Non LinearFriction Effects (cf = 0.009), FiniteAmplitude Effects and Convective AccelerationEffects for Steady State Component.

-158-

Figure 6.23b Velocity Results of TEA with Full Non Linear

Friction Effects (c = 0.009), Finite

Amplitude Effects and Convective Acceleration

Effects for M2 Component at Time of Maximum

Ebb for the M2 Component Relative to the

Ocean Boundary

-159-

Figure 6.23c Velocity Results of TEA with Full Non LinearFriction Effects (cf = 0.009), FiniteAmplitude Effects and Convective AccelerationEffects for M4 Component at Time of MaximumEbb for the M2 Component Relative to the

Ocean Boundary

-160-

Figure 6.23d Velocity Results of TEA with Full Non Linear

Friction Effects (cf = 0.009), FiniteAmplitude Effects and Convective Acceleration

Effects for M6 Component at Time of Maximum

Ebb for the M2 Component Relative to the

Ocean Boundary

-161-

uncoupled harmonic continuity equations for all the various frequencies.

Hence residual flux currents indicate net mass flushing patterns. For

our particular case, while net steady velocity currents are quite

significant (maximum velocities in the bight for the steady residual

component are approximately 10% of the maximum velocities in the bight

for the main M2 component at maximum ebb), the net steady flux currents

will be insignificant. This is due to the fact that the residual velocity

current, uR, is generated by the time averaged finite amplitude term,

u(t)n(t). Hence the two terms in Eq. 6.5 balance to yield no net residual

flux, QR. However, in general, net steady flux currents can exist and

will depend on the type of ocean connections and forcings, the depth and

bottom friction factor distributions, as well as the geometry of the

embayment.

Figure 6.23b shows the predicted velocities at maximum ebb (relative

to the ocean boundary) for the main component (M2). We note that the

scaling is greater by a factor of 10 relative to Figure 6.23a. Figures

6.23c and d show the predicted velocities associated with the M4 and M6

overtide components at the time of maximum ebb for the M2 component

(relative to the ocean boundary). We note again that the velocity scaling

varies for these figures. The actual total velocity is obtained by

adding all four components at any one time. Finally we note that at some

locations along land boundaries, non-zero normal velocities exist. These

are especially severe along the northern boundary where very sharp depth

gradients exist. Hence the actual amount of flux leakage is limited due

to the shallow depths in these areas. Furthermore these non-zero

velocities can be eliminated by refinement of the grid in any problem

-162-

areas since normal fluxes are natural boundary conditions and are thus

satisfied exactly in the limit.

So far we have seen that the overtides generated are all of about

equal magnitude. The friction term is the most important in that it is

responsible for the responses of the main astronomical tide (M2) in

addition to the generation of the M6 overtide. The finite amplitude

term in the continuity equation generates for the most part the M4 and

steady tides, while the convective acceleration term has little effect

compared to the other non linear terms. Let us now compare the

overtides computed by TEA-NL, for a friction factor of 0.009 and with

all non linearities included, to those experimentally obtained by

Filloux and Snyder (1979).

As was previously noted, agreement between TEA-NL predictions and

measurements by Filloux and Snyder (1979) for both elevation amplitude

and phase of the M2 tide was excellent. Since the measurements by

Filloux and Snyder do not reflect steady state circulation, no

comparisons can be made. For the M4 overtide, TEA-NL predictions

(Figure 6.21) and measurements (Figure 6.4) show good agreement.

However, when comparing TEA-NL results for the M6 tide (Figure 6.22)

with measurements (Figure 6.5), there is some discrepancy. The

numerical predictions for the M6 amplitude exceed measurements by a

factor of about 2.0. Phase errors are less pronounced although

agreement is not as good as for the M4 tide.

The variability between the TEA-NL overtide predictions and

measurements may be quantified by calculating the proportional variance

for each of the harmonics at which reduced measurement data are

-163-

available. This proportional variance is computed by summing the

square of the difference between each of the experimental elevation

amplitude values available at each measurement location and the value

computed by TEA-NL at that location and then dividing by the sum of

the squares of all the measured values (Snyder, Sidjabat and Filloux,

1979). Hence the proportional variance evaluating the error between

predictions and measurements for each harmonic j is expressed as:

L KE In (x,, k) - n (x) 2

Vp £=l k=l (6.6)L K1z E

£=l k=l

where n~ measured elevation amplitude component for the jth

3 harmonic

TEA-NL predicted elevation amplitude component for

the jth harmonic

L = total number of measurement locations

K = total number of measurement data points at location k

The proportional prediction variances, V3, are calculated for the

entire bight and the three sub-regions previously defined and are

presented in Table 6.5a. It is noted that regional values for V are

about the same or less in the northern and eastern bight compared to

values for the entire bight while in the sill region they are somewhat

higher. These proportional prediction variances V include the

uncertainties in the reduced measurement data Vm. Hence if the average

measured value at each point were correct, the net error between predicted

-164-

Table 6.5a Overtide Computation Errors Expressed asError Between Measurements and TEA Predictionsin Terms of Proportional Variance, V

j

-165-

P



M2 0.0114 0.0142 0.0038 0.0099

M4 0.0907 0.1489 0.0387 0.0767

M6 0.9198 0.9774 1.2368 0.7892

Table 6.5b Overtide Computation Errors Expressed asError Between Measurements and TEA Predictionsin Terms of Proportional Standard Deviation,

sPi

-166-

Sp



M2 0.11 0.12 0.06 0.10

M4 0.30 0.39 0.20 0.28

M6 0.96 0.99 1.11 0.89

and measured elevation amplitudes would be obtained by subtracting VM.3

from V . Values for proportional prediction standard deviations are3

defined by:

S = (6.7)j j

and are shown in Table 6.5b. All values for the predicted M 6 tide

are greater than the measured amplitude. Table 6.5b shows that this

amplitude excess is equal to approximately one. Hence the predicted

M6 values are too high by a factor of 2.0. Snyder, Sidjabat and

Filloux (1979) had the same overprediction problems for the M6 tide

with their numerical model.

The numerical model applied to the Bight of Abaco by Snyder,

Sidjabat and Filloux (1979) is a frequency domain model which uses

finite differences to resolve the spatial dependence of the governing

equations. As for TEA-NL, the non linear harmonic coupling in their

model is handled with an iterative scheme which cycles through the

various sets of harmonic equations. However, their model is based on an

analytical harmonic separation of the governing equations and uses a

number of approximate expansions for the various terms. Furthermore,

their model only performs computations for 5 astronomical tides (K1,

01, N2 , M2 and S2) and two overtides (M4 and M6) and does not

consider any compound type interactions. Snyder, Sidjabat and Filloux

found that their optimal overall solution was obtained at a friction

factor, cf, equal to 0.007 and that this resulted in proportional

prediction variances of VM2 = 0.07, Vp = 0.33 and VM = 0.79. These2 to 4 proportional prediction standard deviation values of6

correspond to proportional prediction standard deviation values of

-167-

S 0.26 S = 0.56 and S = 0.89. Hence TEA-NL predictions areM2 M4 M6

somewhat better than their predictions for the M2 and M4 tides while

the error for the M6 tide is about the same for both models.

Snyder, Sidjabat and Filloux (1979) were able to obtain better

agreement between their numerical model predictions and their field

measurements by deviating from the standard quadratic law by either

including a linear friction component or by allowing for significant

nontidal currents. The first mechanism assumes bottom friction to be

composed of a linear and a quadratic part of the form:

bT = c u + cf2 uIu (6.8)p fl f2

For this two parameter friction law, the quadratic friction coefficient,

cf2 , is substantially reduced from that used for the fully quadratic

one parameter law. The reasons for their improved agreement may be

readily explained as follows. The friction forcing felt by the M2

tidal component is about the same as the one parameter friction law if

the linear friction factor cfl is sufficiently large to compensate for

the reduction in the quadratic friction factor, cf2 (recall that the

largest portion of the quadratic friction term acted as a linear term at

the actual forcing frequency). This then allows the response at M 2 to

be the same as that calculated when the one parameter fully quadratic

law was used. The M4 overtide response will be largely unaffected

since this is generated by the finite amplitude term. Hence, since the

M2 response remains the same, the finite amplitude forcing at M4

will remain the same. The M6 overtide response, however, will be

reduced depending directly on how much the quadratic coefficient cf2

-168-

has been reduced from the one parameter law. This is due to the fact

that the M6 tide is now generated by the reduced quadratic term

cf21ulu (and the dominant M2 response velocities have remained the

same for both friction laws). Snyder, Sidjabat and Filloux found that

with friction factors of cfl = 0.00086 m/sec and cf2 = 0.0033 they were

able to reduce the proportional prediction variance for the M6 tide

to VP = 0.178 (SP = 0.42). The second mechanism by which they wereM6 M6

able to reduce the error for the M6 tide was to include an rms nontidal

current of 0.28 m/sec. This nontidal current is about equal to the

maximum tidal velocity in the bight. The proportional prediction

variance for the M6 tide was reduced to V' = 0.231 (SP = 0.48) inM6 M6

this manner. However the deviations required from the standard

quadratic law for the first mechanism and the large nontidal current

required for the second mechanism in order to significantly improve

the results are not supported by the flow conditions which exist in

the bight.

As this point we note that Filloux and Snyder (1979) and Snyder,

Sidjabat and Filloux (1979) did not consider steady state in either

their analysis of the experimental data or in their numerical model.

However, as was seen from TEA-NL results, the steady state overtide

response was of the same order of magnitude as other velocities

considered making it inconsistent to not consider this steady term.

Furthermore, Snyder, Sidjabat and Filloux (1979) did not consider any

compound tides generated through the non linear interactions between

the various astronomical tides. In the next section we shall establish

the importance of compound tides in the bight by examining the

interaction of the M2 and N2 tides.

-169-

6.3 Compound Tide Computations for the Bight of Abaco

In this section we shall study the compound tidal interactions in

the bight. In particular, we are interested in examining the

significance of compound tides with frequencies in the neighborhood of

the M and M6 tides. From Table 6.2 we note that the N2 tide is

roughly of the same magnitude as the diurnal constituents K1 and

01 . However, the importance of the compound tides generated by these

diurnal constituents which lie close to the M4 and M6 tides is

much less than the compound tides generated by semi-diurnal

constituents. Furthermore, Table 6.2 shows that the N2 forcing tide

is roughly twice as large as the S2 tide. Therefore looking at the

compound tides generated by the M 2 and N2 tides will give us a good

understanding of the compound tidal interaction. In addition it will

allow us to assess the importance of compound tides in the vicinity of

the M4 and M6 overtides.

The frequencies most likely to be of importance for the M2 - N2

interaction are readily obtained by using the response-forcing tables

discussed in Chapter 2. The frequencies produced with this technique

after the second cycle of the procedure are listed in Table 6.6. We

note that these compound tides separate into five frequency clusters.

The first cluster consists of the steady zero frequency response and two

long period (28 days and 14 days) residual tidal components. The

remaining response clusters are grouped around 12, 6, 4 and 3 hours.

The synodic period for these frequencies is 28 days.

The frequencies listed in Table 6.6 were used in the application of

TEA-NL. A time sampling rate of 132 points (spread over 28 days) was

-170-

Table 6.6 Tides of Possible Interest for M2 and N2 Interaction

Tide Freq. Comp.* Freq. -T ~T

(rad/sec- 1 ) (days)____________ ,________ _________I___

steady

MN

2MN

2NM2

N2

M2

2MN2

3NM 4

N4

MN4

M4

3MN 4

N6

2NM6

2MN6

M6

N8

3NM 8

2MN 8

3MN8

M8

28.05 days

14.02 days

0

1 22w -2w2

wl-2w2

"'21 2

2w2

2w1 23 1-3w 2

132+2w221+22w 1

w 2

2 w1+2w2

1

2 +2 2

0.0

2.5927x10-6

-65.1854x106

-41.3538x10-

1.3797x10 4

1.4056x10-4

1.4316x10-4

2.7335x10-

2.7594x10-

-42.7853x10-4

2.8113x10 4

2.8372x10 4

-44.1391x10-

4.1650x10-

4.1909x10 - 4

4.217x10 - 4

-45.5189x10

5.5447x10-4

5.5707x10--4

5.5966x10-

-45.6225X10 3.10 hrs

I I a

28.05

28.05

0.56

28.05

28.05

28.05

0.56

28.05

28.05

28.05

28.05

28.05

0.56

28.05

28.05

28.05

0.56

28.05

28.05

28.05

*wl = WM2W2 = UN 2

-171-

12.89

12.65

12.42

12.19

6.38

6.33

6.27

6.21

6.15

4.22

4.19

4.16

4.14

3.16

3.15

3.13

3.12

hrs

hr s

hrs

hrs

hrs

hrs

hrs

hrs

hr s

hrs

hrs

hrs

hrs

hrs

hrs

hrs

hrs

I--L--L*ni n~~_~ ~.;~~~a*--*l~l~l~i~P---~ -- ~n~m~~-l----- .~~x.,._

required in order to obtain accurate harmonic analysis results up to and

including the 4 hour period cluster. Friction and finite amplitude

effects were considered while convective acceleration was neglected in

the computations due to its limited importance.

The boundary conditions are specified such that the M2 and N2

astronomical constituents have amplitudes and phases set equal to values

measured by Filloux and Snyder (1979). For all overtide and compound

tides, zero elevation is specified along the ocean boundary.

Globally averaged harmonic pseudo forcing amplitude distributions

for the continuity and momentum equations are shown in Figures 6.24a,b.

These figures show the ratios of the harmonic forcing at each frequency

to the maximum harmonic forcing of all frequencies considered.

Furthermore these figures only reflect tides with a pseudo forcing

greater than 1% of the maximum harmonic continuity or momentum pseudo

forcing. We note that the tides with the most prominent forcings

correspond to frequencies in the first two rows of Table 2.5b.

Figure 6.24a shows the finite amplitude pseudo forcings being

distributed mainly to the steady cluster and the M4 cluster. The

forcings are distributed to the cluster in a similar way as for the

overtide case. However, now not only are certain overtides generated

but compound interactions of relative importance also exist. Besides

the steady pseudo forcing, a 28 day period finite amplitude pseudo

forcing exists. Furthermore both an M4 and MN4 pseudo forcing are

now significant. The N2 overtides themselves are not of importance.

Figure 6.24b shows the friction pseudo forcing being distributed

mainly to the astronomical frequencies themselves. Besides the M2

-172-

P

4 .10.09- N2() 08 - M6.07-.06- N 2MN

.04 12MN 2

.03-

.02 -

.010.00 1.00 2.00 3.00 4.00

-4w x 10 (rad/sec)

Figure 6.24a Continuity Equation Pseudo Forcing Vector RatiosDue to M 2 - N 2 Interaction

-173-

-YIYL6~-~I"--"-~-CYrrr-- ~~I~I~

P

M2.10

2MN2 M.08 2 H 6.07 - M4.06 STEADY.05-

.04 MN MN4 2N 6

.03-

.02 -

.01-0.00 1.00 2.00 3.00 4.00 5.00

x 10- 4 (rad/sec)

Figure 6.24b Momentum Equation Pseudo Forcing Vector RatiosDue to M 2 - N2 Interaction

-174-

overtides, MN, 2MN 2, MN4 and 2MN6 compound tides are now of

significance. We note that for both the continuity equation and

momentum equation pseudo loadings, the compound pseudo forcings shown

are typically 40-50% of the magnitude of the adjacent M 2 overtides.

Hence to be consistent in the order of approximation of the analysis we

must take these compound tides into consideration.

Figures 6.25 through 6.32 show the most significant tides

associated with the non linear interaction of the M2 and N2 tides.

The steady state (Figure 6.27), M2 (Figure 6.25) and M4 (Figure

6.30) constituents are essentially the same as for the M2 overtide

computations. The M6 constituent (Figure 6.32) does show some

reduction in amplitudes but predicted values still substantially

exceed measured values. The N2 astronomical constituent (Figure

6.26) shows very good agreement with measurements (Figure 6.6).

The variability between the TEA-NL compound tide predictions

and measurements are again quantified by calculating the proportional

prediction variance, VP. Values for V are shown in Table 6.7a.3 J

Proportional prediction variances V? are again about the same orJ

less in both the northern and eastern bight compared to values for the

entire bight with the exception of the values for V for the N2 tide

in the northern bight. However as was noted from Table 6.3a, the

proportional measurement variances, Vm, in the northern bight were

in general substantially greater than values for Vm for the bight as

a whole. This was especially true for the N2 tide. Hence the net

error between TEA-NL predictions and measurements will be substantially

reduced. We conclude that in general the numerically predicted

-175-

~"----~~--L-"L~~~L-~^-l*l ll~-at~- -TY r~ -L~~--- -i~r~a~*~il_ 4~,,

20'

Figure 6.25 Results of TEA with M2 -N 2 Interactionand with Full Non Linear Friction (cf -

0.009) and Finite Amplitude Effects for

M2 Astronomical Constituent


-176-

_ __

0.O

I.2-

.5 0'(0

Figure 6.25 Results of TEA with M2-N2 Interactionand with Full Non Linear Friction (cf -0.009) and Finite Amplitude Effects forM2 Astronomical Constituent.


-177-

r~~~-- I"-~a-C -r -~nu~rr~ ~9~

Figure 6.26 Results of TEA with M2 -N 2 Interactionand with Full Non Linear Friction (cf =0.009) and Finite Amplitude Effects forN2 Astronomical Constituent.


-178-

Figure 6.26 Results of TEA with M2 -N 2 Interactionand with Full Non Linear Friction (cf -

0.009) and Finite Amplitude Effects forN2 Astronomical Constituent.


-179-

---~I

Figure 6.27 Results of TEA with M2 -N 2 Interactionand with Full Non Linear Friction (cf =0.009) and Finite Amplitude Effects forSteady State Constituent. Amplitude incentimeters.

-180-

Figure 6.28 Results of TEA with M2 -N 2 Interactionand with Full Non Linear Friction (Cf -

0.009) and Finite Amplitude Effects for MNCompound Constituent

Amplitude in centimeters

-181-

MN4

0,run h3

AMPLITUDE

0.2

0.L

Figure 6.29 Results of TEA with M2 -N 2 Interactionand with Full Non Linear Friction (cf =0.009) and Finite Amplitude Effects forMN4 Compound Constituent


-182-

Figure 6.29 Results of TEA with M2-N 2 Interactionand with Full Non Linear Friction (cf =0.009) and Finite Amplitude Effects forMN4 Compound Constituent


-183-

rrl--~i~i-P. -~ly C1~yl~ Clli~iY~-~L~ s

Figure 6.30 Results of TEA with M2 -N 2 Interactionand with Full Non Linear Friction (cf =0.009) and Finite Amplitude Effects forM4 Overtide Constituent


-184-

Figure 6.30 Results of TEA with M2 -N2 Interactionand with Full Non Linear Friction (cf =0.009) and Finite Amplitude Effects forM4 Overtide Constituent


-185-

ILCILLL~mltl IIII111 ~

00.

0*6

6.6

Figure 6.31 Results of TEA with M2 -N 2 Interactionand with Full Non Linear Friction (cf =0.009) and Finite Amplitude Effects for2MN 6 Compound Constituent.


-186-

Figure 6.31 Results of TEA with M2-N 2 Interactionand with Full Non Linear Friction (cf =0.009) and Finite Amplitude Effects for

21iN46 Compound Constituent.

(b) Phase lag in radians (relative to H2tide)

-187-

~ .I-.IXII-UIL ~I~CI*^ L I~---%~IIItl~._ ~.1II___

0.5,run h3

AMPLITUDE

1.0

Figure 6.32 Results of TEA with M2 -N 2 Interactionand with Full Non Linear Friction (cf =0.009) and Finite Amplitude Effects for

M6 Overtide Constituent


-188-

Figure 6.32 Results of TEA with M2-N2 Interactionand with Full Non Linear Friction (cf -

0.009) and Finite Amplitude Effects for

M6 Overtide Constituent


-189-

_I_____LPL__ILLI____~I ~1IY-~-~.-..l.^~ I_

Table 6 .7a Compound Tide Computation Errors Expressedas Error Between Measurements and TEAPredictions in Terms of ProportionalVariance, V

3

-190-

PVjTide

Entire Sill Northern EasternBight Region Bight Bight

N2 0.0253 0.0187 0.0680 0.0305

M2 0.0114 0.0138 0.0066 0.0077

M4 0.0799 0.1800 0.0384 0.0304

M6 0.4693 0.6011 0.5356 0.3964

Table 6.7b Compound Tide Computation Errors Expressedas Error Between Measurements and TEAPredictions in Terms of ProportionalStandard Deviation, SI

3

-191-

distributions are better for the northern bight than for the bight as

a whole. For the sill region proportional prediction variances, VP ,

for the M4 and M6 overtides exceed values for the bight as a whole.

Values for the proportional prediction standard deviation are shown

in Table 6.7b.

Hence agreement between predictions and measurements for both

the M2 and N2 tides overall is excellent (Snyder, Sidjabat and Filloux

(1979) obtained a proportional prediction variance for the N2 tide

of V~ = 0.02). Agreement is good for the M tide and has improvedN2

for the M6 tide when compared to the overtide computations. For the

M2 tide about 75% of the locations have predicted amplitude values

which exceed the average measured values at a location while for the

N2 tide the fraction is only 60%. For the M4 tide only about 50%

of the locations have overpredicted amplitudes while 25% of the

locations have predicted values equal to the average of the measured

values. Finally for the M6 tide all locations have overpredicted

amplitudes with the exception of locations actually on the ocean

boundary. As may be deduced from Table 6.7b, the overprediction

factor for the M6 tide has been reduced to about 1.7 for these

compound tide computations.

As would be expected from our examination of pseudo forcing

values, there are now also significant compound responses. There is

a monthly varying MN compound tide (Figure 6.28), a MN4 compound tide

(Figure 6.29) adjacent to the M4 and a 2MN6 compount tide (Figure 6.31)

adjacent to the M 6 . Although these compound tides are somewhat smaller

than their adjacent overtides (by a factor of approximately 2 to 3),

-192-

they are important to the dynamics of the bight. Furthermore, it is

noted that patterns for both the elevation amplitude and phase shift

distributions of adjacent compound tides and overtides are very

similar.

6.4 Discussion

In the previous section it was ascertained that agreement between

the reduced experimental data and the TEA-NL numerical predictions which

included the full non linear interaction of the M2 and N2 astronomical

tides was excellent for the astronomical tides themselves and good for

the M4 overtide. However M6 predictions exceeded measured values by

a factor of about 1.7. In this section we shall explore some of the

various possibilities that might explain and/or improve the discrepancy

which exists between measurements and predictions for the M 6 overtide.

Let us first determine what effect neglecting compound tides has

had on the measurement data reduction procedure used by Filloux and

Snyder (1979). As was discussed in Chapter 5, the least squares

harmonic analysis procedure is much more sensitive to the neglect of

frequencies of relative importance within a cluster than when a

frequency is dropped outside of a cluster. Hence the procedure may have

trouble resolving a tide if a closely spaced adjacent tide exists and

is not included as an analysis tide. The associated error in the results

will depend on both the relative significance of the two tides and the

time point sampling density. However the error introduced into the

reduced measurement data under consideration should be about the same

for both the M4 and M6 tides since as the results of the compound tide

-193-

. ................. iJ --- i i--- -

computations showed, the MN4 and 2MN6 compound tides are of about the

same relative importance with respect to their adjacent M4 and M6

overtides. Furthermore the spacing between the MN4 and M4 tides and

between the 2MN6 and M6 tides is equal (Ts - 28 days). Table 6.3

confirms that reduced measurement errors are about the same for both

the M4 and M6 tides. Hence we conclude that data errors for the M6

tide are not responsible for the large discrepancy that exists between

predictions and measurements for that tide.

However the grouping of measurement data errors discussed in

Section 6.1 may be readily explained by examining the importance of

closely spaced compound tides which were neglected in the data reduction.

Recall that the K1, 01 and M2 tides generally had very low measurement

errors (S ~ 0.03) while the N2, S2, M4 and M 6 tides all had larger

errors (Sm - 0.10-0.14). The tides with higher measurement errors all

have relatively important compound tides in their vicinity which were

neglected in the measurement data analysis. As was seen in Figures

6.24a and b, the 2MN2 tide is of relative importance with respect to

the N2 tide and will also certainly be important with respect to the

S2 tide. Furthermore as was previously mentioned, the MN4 and 2MN6

tides are proportionally significant with respect to the M4 and M6

tides. However the tides with lower measurement errors do not have

significant closely spaced compound tides. While the 2MN 2 tide is

significant with respect to the N2 tide, it is not significant with

respect to the much larger M2 tide. The 01 and K1 tides will have

no important compound tides located in their vicinity even when all

five major astronomical tides are included in the computations.

-194-

We note from Table 2.6 that the K1 and 01 tides are extremely closely

spaced (Ts - 208 days). However the error associated with the data

reduction procedure is still very low. Hence we conclude that the

proportional measurement variances for the N2, S2, M4 and M6 tides

can be reduced to the levels achieved for the 01, K1 and M2 tides

if select compound tides (2MN2, MN4 , 2MN6 and most likely the 2MS2,

MS4 and 2MS6 tides) are included in the least squares analysis procedure

used to reduce the measured elevation time history records to harmonic

amplitudes. The time sampling rate could be kept about the same as that

used by Filloux and Snyder (1979) which may be deduced from the fact

that the K1 and 01 reduced measurements showed very low error.

Thus far we have seen that experimental data error levels can

not be responsible for the poor fit of the predicted M6 tide. In

fact the measurement error levels calculated in Section 6.1 are in

general quite modest compared to TEA-NL prediction-measurement error

levels. Therefore let us now examine some possible ways in which the

fit for the M6 tide could be improved.

The first issue to be examined is the correctness of the

boundary conditions which were applied with TEA-NL. A good indication

which justifies treating the shallow connections to the open ocean in

the northwestern part of the bight as land boundaries is the low

prediction-measurement errors in the northern bight. In fact as was

seen in the previous section the net prediction-measurement errors

are in general substantially less in the northern part of the

bight when compared to those for the bight as a whole. This confirms

Filloux and Snyders' (1979) conclusion regarding this boundary.

-195-

I__IYL_____lill*I_~ll~Ll I~-~ _._._ ~i ii.ll LI-LI -XLII L^

Let us now assess whether the treatment of the ocean boundary

along the western edge of the bight as being totally reflective for

overtides is justifiable. In the previous section we noted that

prediction-measurement error levels for the M4 and especially the M6

tides were substantially greater than those for the M 2 and N2 tides.

In addition, regional error levels for both the M4 and M6 tides were

the highest in the sill region. We also note that although measured

values along the ocean boundary are small for both the M4 and M 6

tides, they certainly are not negligibly small compared with overall

bight values for these tides as would be the case for a totally

reflective boundary. These facts possibly indicate that either the

assumption of the boundary being totally reflective is not entirely

correct and/or the location of the reflective boundary is incorrect.

As was seen in Section 6.1, the actual reflection coefficient

was only about 0.90. Hence a certain amount of leakage of overtide

energy into the open ocean does occur. In fact if the ocean boundary

were totally reflective as was assumed, then no astronomical tides

would be allowed to enter the bight either. The question is whether

the somewhat inflated reflection coefficient of 1.0 which was used

contributes significantly to overpredicted overtides. For the M 6

tide almost all predicted values are too high. For the M4 tide only

50% of the comparison locations were overpredicted with another 25%

of the comparison locations having equal predicted and measured

values. However it may be shown that the overall contribution to

the proportional variance for the M4 tide from overpredicted points

far outweighs that from underpredicted points not only due to there

-196-

being more overpredicted points but also due to the fact that the

prediction-measurement differences were on the average much greater

for the overpredicted points. This indicates that both M4 and M6

computations suffer mainly from overprediction which is consistent

with a reflection coefficient which is too high. Hence accounting

for the correct degree of reflection will definitely improve

prediction-measurement error levels for both the M4 and M6 overtides.

However since an approximately equal influence of reflection

coefficients would be expected on both the M4 and M6 tides there

will still remain a substantial discrepancy between the M4 and M6

error levels.

The other possible problem with the open ocean boundary condition

applied for the overtides in the computations could be the location

of the reflection boundary. The grid used (Figure 6.7) has the ocean

boundary located at the beginning of the sharp depth drop as if a vertical

step were located there. However it would seem more logical to place

this boundary somewhere in the middle of the range of the most substantial

depth drop in order to better simulate a would be reflective boundary.

We recall that the 1000 meter contour was between 3 and 15 kilometers

from the present boundary. Hence placing the reflective boundary

halfway between the 5 and 1000 meter contours would put it between

1 to 7 kilometers (depending on where along the boundary) away from

the boundary used in Figure 6.7. This adjustment distance can be

significant when compared to the overall scale of the sill region.

We note that defining an ocean boundary in the manner just described

would create difficulties in this case since there are no astronomical

-197-

tide measurements at that location.

We conclude that allouing for some overtide transmission out of

the bight and accounting for the fact that the reflection does not

totally occur at the upper edge of the depth drop will contribute

towards improving the M4 and M6 distributions but are not the

dominating physical mechanisms which explain the much larger

prediction-measurement errors of the M 6 overtide.

Let us now assess whether the use of one equal value for friction

factor, cf, for the entire bight contributed significantly to the

large error level of the M6 tide. The use of spatially dependent

friction factor values is certainly physically well motivated due to

the significant variation in bottom surface characteristics within

the bight. The sill region has a substantially greater bottom

roughness (dunes of 1 to 3 m in a depth of 2 - 5 m) than other areas

in the bight and, as Table 6.4 shows, a value of cf greater or equal

to 0.009 would be expected in the sill region. Table 6.4 also

indicates that a value of cf = 0.009 is somewhat too high in other

regions of the bight. Hence in the sill region cf could be greater

or equal to the value used in the computations while in the remainder

of the bight a lower value should be used. If the value of cf used

in the computations were significantly below the actual value for the

sill region, then the higher than actual values in other parts of the

bight might compensate for this. However this hypothesis which favors

the use of localized friction factors is not supported by the error

distributions for the M 2 and N2 tides. The M 2 tide was dominantly

overpredicted due to both the number of overpredicted points and the

-198-

fact that the average overpredicted differences far exceed under-

predicted differences. For the N2 tide, overprediction only slightly

dominated underprediction.

As has been previously discussed, the sill region is the most

important region in terms of both the effect of friction on the

main tides and the generation of non linear tides. That the most

significant impact of friction on all distributions is in the sill

region is demonstrated by the high gradients in elevation amplitude

and phase which exist there. Furthermore since the non linearities

are the most significant in the sill region (due to high elevation

amplitudes, high velocities and shallow depths relative to other

parts of the bight), the non linear overtides and compound tides

are largely generated there. Hence a spatially varying friction

factor will not drastically effect any of the computed distributions

if values for cf in the sill region are kept the same. The limited

impact of spatially varying friction factor is confirmed by the

findings of Snyder, Sidjabat and Filloux (1979) who performed a

limited number of computations to check for the sensitivity of

this effect.

In Section 6.2 the effects of variations in global friction

factor were checked with intervals of cf equal to 0.003. Given the

general dominance of the overpredictions it is likely that more

refined increases in cf beyond the value of 0.009 will have some

effect in reducing general error levels. As was state earlier an

increase of the value for cf in the sill region would be justifiable

due to the bottom roughness there. Furthermore values for cf in the

-199-

Illlll__l~i ~-- L- *~*~ llb~

remainder of the bight could most likely be reduced to physically

more realistic values without effecting the tidal distributions in

the bight. An increase in cf in the sill region not only causes

decreases in the amplitudes of the main tide distribution but also

generally decreases the M6 amplitudes. This is due to the fact that

for this case the effect of the reduction in M2 velocities due to

increased friction proportionally outweighs the actual increases in

the friction factor itself. This results in a reduced M6 pseudo forcing

and hence a reduced M6 tide. However extrapolating the effects of

the reduction in amplitudes of the M2 and M6 tides for changes in cf

of 0.003, we conclude that this fine tuning process will not have a

major impact in reducing M6 error levels.

Finally let us consider the effects of only including the M2

and N2 astronomical tides and neglecting the 01, K1 and S2 tides in

the computations performed. As stated in Section 6.3, after the M2

and N2 tides, the S2 will probably be the most influential to the

M6 . A very significant improvement in M6 errors was achieved by

including the N2 tide in the computation (recall VP dropped from

0.9198 to 0.4693 and Sp dropped from 0.959 to 0.685). However asM6

was seen from Table 6.2, the S2 tide is only about half as large as

the N2 tide. Assuming that the improvement in the M 6 solution due

to considering the S2 tide (in addition to the M2 and N2) is

proportionally (to the amplitude of the tide) the same as that

achieved when the N2 tide was included, the error for the M6 could

be brought down to about VP = 0.24 (Sp = 0.50). However thisM 6 M6

assumes that the processes are linear, which of course they are not.

-200-

Hence any actual improvement could be more or less than this. We

note that we do not expect any modification to the M4 tide due to

the inclusion of the S2 tide, in the same way that the M4 remained

essentially unchanged when the N2 tide was added.

We have examined a number of possibilities for their effectiveness

in improving the fit between the predicted and measured overtides.

No single mechanism seems capable of reducing overtide error levels

to those of the astronomical tides. However a combination of these

mechanisms may achieve significantly better overtide fits. The

largest reduction in M6 error will most likely be brought about by

the inclusion of the S2 astronomical tide in addition to the M2 and

N2 tides in the computation. As was seen this could very well lead

to reducing M6 error levels close to those presently achieved for

the M4 tide. Improved treatment of the main ocean boundary would

bring about improvements in fit for both the M4 and M6 tides.

Finally, the fine tuning of the friction coefficient, cf, could

produce minor improvements for the error levels of all tides.

-201-

-~I-r_--rr~unr~r*nluBhrl--dO YIPYYt;P~__ - ~- iiu-.ru~,y---

CHAPTER 7. CONCLUSIONS

A computer model, TEA-NL, which computes tidally driven circulation a

in coastal embayments, has been developed. The finite element method

was used to resolve the spatial dependence in the governing equations

while a hybrid time domain - frequency domain approach was used to

resolve the time dependence. With this approach the non linear terms

are iteratively updated in the time domain to produce time histories

which are then harmonically decomposed with the least squares method.

The least squares method was extremely well suited for this purpose

since it allows the resolution of very closely spaced and narrowly

banded energy in an extremely efficient manner. With harmonic forcings

on the system and the harmonically decomposed non linear terms

(pseudo-forcings), the governing equations separated into sets of

linear equations in the frequency domain. This led to the development

of a linear core solver which solved each set of linear equations

at a given frequency in an extremely efficient manner. The linear

core solver yields accurate solutions (not overdamped) while it

shows very low spurious oscillations.

TEA-NL allows the general investigation of the effects of the

non linear interaction between tidal components in shallow estuaries.

Hence not only can overtides be computed but compound tides can also

be assessed. The importance of compound tides was seen in the

application of TEA-NL to the Bight of Abaco where certain of the

compound tides generated through the interaction of the M2 and N2

astronomical tides had responses equal to about 50% of the corresponding

adjacent M2 overtides. We note that these compound tides can be

-202-

especially important in the assessment of long period residual

fluctuations which exist in addition to any steady state residual

currents.

There are several aspects of TEA-NL which could be improved.

The first aspect concerns the fact that at present the TEA-NL user

must specify all frequencies of possible interest. However use of

an FFT at several selected locations (typifying various parts) in

the embayment would allow the identification of all important

frequencies. These frequencies could then be used for the much

more economical least squares harmonic analysis procedure for all

points in the embayment. This would make the model more user

friendly and also ensure that no important frequencies are neglected.

Furthermore this will simplify the simulation of complex wind

histories.

A further aspect which would improve TEA-NL would be the use

of higher order finite elements because of their increased accuracy

per number of total nodes and the convenience of the larger elements

with which the embayment may be discretized. We note that the size

of the largest element in the domain must reflect the size of the

smallest wavelength present (e.g. Mg) (so that wave shape can be

adequately represented). In very shallow embayments wavelengths

decrease while the importance of higher harmonics increases, possibly

requiring a very fine grid. The higher the order of the element,

however, the larger the minimum element size which can be used.

Furthermore higher order elements would be more convenient to

accomodate rapid changes in geometry and depth.

-203-

~_~ ;__)~ I __X~ _I__ j/*~~_ ___ _i 1_~~1

Finally the treatment of flux boundary conditions could be

improved. TEA-NL presently treats them as natural which, unless

boundaries are sufficiently refined, could lead to leakage. The

specific refinement of boundary areas however is often inconvenient.

Hence TEA-NL would be improved by re-formulating such that fluxes

were treated as essential boundary conditions and as such were

more strictly enforced (i.e., no errors were allowed regardless

of element sizes along the boundary).

Despite these minor inconveniences in usage, TEA-NL is an

effective model for simulating both short term (1 day) and long

term (1 month and more) tidally driven circulations in embayments.

Its most important feature is that it allows an accurate assessment

of compound tides which include long term periodically fluctuating

residual circulations.

p

-204-

REFERENCES

Askar, A. and A.S. Cakmak, 1978. Studies on finite amplitude waves in

bounded waterbodies. Advances in Water Resources, 4:229-246.

Benque, J.P., B. Ibler, G. Labadie and B. Latteux, 1981. Finite element

method for incompressible viscous flows. GAMM Conference on Numerical

Methods In Fluid Mechanics. (4th), (ed. Viviand), 10-20.

Bonnefille, R., 1978. Residual phenomena in estuaries, application to

the Gironde Estuary. Hydrodynamics of Estuaries and Fjords, (ed.

J. Nihoul), 187-195.

Connor, J.J. and C.A. Brebbia, 1976. Finite Element Techniques for

Fluid Flow, Newnes-Butterworth, London.

Daily, J.W. and D.R.F. Harleman, 1966. Fluid Dynamics, Addison Wesley.

Defant, A., 1961. Physical Oceanography, Vol. 2, Pergamon Press, Oxford.

Doodson, A.T., 1921. The harmonic development of the tide-generating

potential. Proceedings of the Royal Society of London, A100:305-328.

Dronkers, J.J., 1964. Tidal Computations in Rivers and Coastal Waters,

North Holland Publ. Co., Amsterdam.

Filloux, J.H. and R.L. Snyder, 1979. A study of tides, setup and bottom

friction in a shallow semi-enclosed basin. Part I: Field experiment and

harmonic analysis. Journal of Physical Oceanography, 9:158-169.

Gray, W.G. and D.R. Lynch, 1977. Time-stepping schemes for finite

element tidal model computations. Advances in Water Resources,

1(2):83-95.

Gray, W.G. and D.R. Lynch, 1979. On the control of noise in finite

element tidal computations: a semi-implicit approach. Computers and

Fluids, 7:47-67.

Gray, W.G., 1982. Some inadequacies of finite element models as

simulators of two-dimensional circulation. Advances in Water Resources,

5:171-177.

Grotkop, G., 1973. Finite element analysis of long-period water waves.

Computer Methods in Applied Mechanics and Engineering, 2:147-157.

Hald A., 1952. Statistical Theory with Engineering Applications, John

Wiley and Sons, Inc., London.

Ianniello, J.P., 1977. Tidally induced residual currents in estuaries

of constant breadth and depth. Journal of Marine Research,35(4):755-786.

Ippen, A.T., 1966. Estuary and Coastline Hydrodynamics, McGraw-Hill

Book Co., Inc., New York.

-205-

Jamart, B.M. and D.F. Winter, 1980. Finite element solution of theshallow water wave equations in Fourier space, with application to

Knight Inlet, British Columbia. Proceedings of 3rd InternationalConference on Finite Elements in Flow Problems, Vol. 2, Banff, Alberta, r(ed. D.H. Norrie), 103-112.

Kawahara, M., K. Hasegawa and Y. Kawanago, 1977. Periodic tidal flow

analysis by finite element perturbation method. Computers and Fluids,5:175-189.

Kawahara, M., 1978. Steady and unsteady finite element analysis of

incompressible viscous fluid. Finite Elements in Fluids, Vol. 3, (ed.Gallegher et al.).

Kawahara, M. and K. Hasegawa, 1978. Periodic Galerkin finite elementmethod of tidal flow. International Journal for Numerical Methods inEngineering, 12:115-127.

Kawahara, M., H. Hirano, K. Tsubota and K. Inagaki, 1982. Selective

lumping finite element method for shallow water flow. InternationalJournal for Numerical Methods in Fluids, 2:89-112.

Ketter, R.L. and S.P. Prawel, 1969. Modern Methods of EngineeringComputation, McGraw Hill Book Co., Inc., New York.

King, I.P., W.R. Norton and K.R. Iceman, 1974. A finite element model

for two-dimensional flow. Finite Element Method in Flow Problems, (ed.Oden et al.).

Lamb, H., 1932. Hydrodynamics, 6th edition, Dover Publications, NewYork.

Le Provost, C. and A. Poncet, 1978. Finite element method for spectralmodeling of tides. International Journal for Numerical Methods inEngineering, 12:853-871.

Le Provost, C., G. Rougier and A. Poncet, 1981. Numerical modeling ofthe harmonic constituents of the tides, with application to the EnglishChannel. Journal of Physical Oceanography, 11:1123-1138.

Lynch, D.R. and W.G. Gray, 1979. A wave equation model for finite

element tidal computations. Computers and Fluids, 7(3):207-228.

Lynch, D.R. 1981. Comparison of spectral and time-stepping approachesfor finite element circulation problems. Proc. Oceans '81, IEEE Pub.,810-814.

Lynch, D.R. 1983. Progress in hydrodynamic modeling; review of U.S.contributions, 1979-1982. Reviews of Geophysics and Space Physics,21(3): 741-754.

Munk, W.H. and K. Hasselmann, 1964. Super-resolution of tides. Studiesin Oceanography, University of Washington Press, 339-344.

-206-

Munk, W.H. and D.E. Cartwright, 1966. Tidal spectroscopy andprediction. Phil. Trans. Royal Society of London, 259:533-581.

Nakazawa, S., D.W. Kelly, O.C. Zienkiewicz, I. Christie, and M.Kawahara, 1980. An analysis of explicit finite element approximationsfor the shallow water equations. Proceedings of 3rd InternationalConference on Finite Elements in Flow Problems, Vol. 2, Banff, Alberta,(ed. D.H. Norrie), 1-12.

Newland, D.E., 1980. An Introduction to Random Vibrations and Spectral

Analysis, Longman, London.

Niemeyer, G., 1979. Long wave model independent of stability criteria.

Journal of the Waterway, Port, Coastal and Ocean Division, ASCE,101:WW1:51-65.

Oppenheim, A.V. and R.W. Schafer, 1975. Digital Signal Processing,

Prentice Hall Inc., Englewood Cliffs, N.J.

Pearson, C.E. and D.F. Winter, 1977. On the calculation of tidal

currents in homogeneous estuaries. Journal of Physical Oceanography,7(4):520-531.

Platzman, G.W., 1981. Some response characteristics of finite element

tidal models. Journal of Computational Physics, 40(1):36-63.

Proudman, J., 1953. Dynamical Oceanography, Methuen and Co. Ltd.,Strand WC2.

Sani, R.L., P.M. Gresho and R.L. Lee, 1980. On the spurious pressuresgenerated by certain GFEM solutions of the incompressible Navier-Stokesequations. Proceedings of the 3rd International Conference on FiniteElements in Flow Problems, Vol. 1, Banff, Alberta, (ed. D.H. Norrie).

Snyder, R.L., M. Sidjabat and J.H. Filloux, 1979. A study of tides,setup and bottom friction in a shallow semi-enclosed basin. Part II:Tidal model and comparison with data. Journal of Physical Oceanography,9:170-188.

Speer, P., 1984. Tidal distortion in shallow estuaries. PhD Thesis,Massachusetts Institute of Technology/Woods Hole OceanographicInstitution.

Tee, K.T., 1981. A three-dimensional model for tidal and residualcurrents in bays. in Transport Models for Inland and Coastal Waters,(ed. Fisher), 284-309.

Taylor, C. and J. Davis, 1975. Tidal and long wave propagation - afinite element approach. Computers and Fluids, 3:125-148.

van de Kreeke, J. and A.A. Chiu, 1981. Tide-induced residual flow inshallow bays. Journal of Hydraulic Research, 19(3):231-249.

Van Dorn, W., 1953. Wind stress on an artificial pond. Journal ofMarine Research, 12(3).

-207-

Walters, R.A. and R.T. Cheng, 1980. Calculations of estuarine residual

currents using the finite element method. Proceedings of the 3rd

International Conference on Finite Elements in Flow Problems, Vol. 2,

Banff, Alberta, (ed. D.H. Norrie).

Walters, R.A. and R.T. Cheng, 1980. Accuracy of an estuarine

hydrodynamic model using smooth elements. Water Resources Research

16(1):187-195.

Walters, R.A. and G.F. Carey, 1983. Analysis of spurious oscillation

modes for the shallow water and Navier-Stokes equations. Computers and

Fluids, 11(1):51-68.

Wang, J.D. and J.J. Connor, 1975. Mathematical modeling of near coastal

circulation. Ralph M. Parsons Laboratory for Water Resources and

Hydrodynamics, T.R.# 200, Massachusetts Institute of Technology.

Williams, R.T. and O.C. Zienkiewicz, 1981. Improved finite element

forms for the shallow water wave equations. International Journal for

Numerical Methods in Fluids, 1:81-97.

Wu, J., 1969. Wind stress and surface roughness at air-sea interface.

Journal of Geophysical Research, 74(2).

Zienkiewicz, O.C., 1977. The Finite Element Method, Third Edition,

McGraw-Hill Book Co., Ltd., London.

1

-208-

a frequency domain finite element model for tidal

Documents