Considerations in the Calculation of Vertical Velocityin Three-Dimensional Circulation Models

Richard A. Luettich, Jr.University of North Carolina at Chapel Hill

Institute of Marine SciencesMorehead City, NC, USA 28557

Julia C. MuccinoDepartment of Civil and Environmental Engineering

Arizona State UniversityTempe, AZ, USA 85287

Michael G. G. ForemanDepartment of Fisheries and Oceans

Institute of Ocean SciencesSidney, British Columbia, CANADA V8L 4B2

Submitted to: Journal of Atmospheric and Oceanic TechnologyDate: January 8, 2002

1/29

1. AbstractThe vertical velocity, , in three-dimensionalcirculationmodelsis typically computed

from the three-dimensionalcontinuity equation assumingthat the depth-varying horizontal

velocityfield wascalculatedearlierin thesolutionsequence.Computing in thiswayappearsto

requirethesolutionof anover-determinedsystemsincethecontinuityequationis first order, yet

mustsatisfytwo boundaryconditions(oneat the freesurfaceandoneat thebottom).At least

threemethodshave beenpreviously proposedto compute : (i) the “traditional” methodthat

solvesthe continuity equationwith the bottomboundaryconditionandignoresthe free surface

boundarycondition,(ii) a “vertical derivative” methodthat solves the vertical derivative of the

continuityequationusingbothboundaryconditionsand(iii) an“adjoint” approachthatminimizes

acostfunctionalcomprisedof residualsin thecontinuityequationandin bothboundaryconditions.

Thelattersolutionis equivalentto the"traditional"solutionplusacorrectionthatincreaseslinearly

over thedepthandis proportionalto themisfit betweenthe"traditional"solutionat thesurfaceand

the surface boundary condition.

In thispaperweshow thatthe"verticalderivative" methodyieldsinaccurateandphysically

inconsistentresultsif it is discretizedas hasbeenpreviously proposed. However, if properly

discretizedthe"verticalderivative" methodis equivalentto the“adjoint” methodif thecostfunc-

tion is weightedto exactly satisfythe boundaryconditions. Furthermore,if the horizontalflow

field satisfiesthe depth-integratedcontinuity equationlocally, oneof the boundaryconditionsis

redundantand obtainedfrom the"traditional" methodshouldmatchthefreesurfaceboundary

condition. In this case,the “traditional,” “adjoint” andproperlydiscretized“vertical derivative”

approachesyield thesameresultsfor . If thehorizontalflow field is not locally massconserving,

themassconservationerroris transferredinto thesolutionfor . Thisis particularlyimportantfor

models that do not guarantee local mass conservation, such as some finite element models.

w

w

w

w

w

w

w

2/29

2. IntroductionThis paperinvestigatesthemeritsof differentapproachesfor calculatingverticalvelocity

fieldsby solvingthethree-dimensionalcontinuityequation,assumingthehorizontalvelocityfields

are known.

In typical three-dimensionalcirculationmodels[e.g.,FUNDY (Lynch and Werner, 1987),

POM (Blumberg and Mellor, 1987),QUODDY (Lynch and Werner, 1991, Lynch and Naimie,

1993),ROMS(Haidvogel and Beckmann, 1999)]theverticalvelocityisdeterminedfromthethree-

dimensional continuity equation:

(1)

where is the horizontalvelocity with components , and

is the vertical velocity. Here arethe horizontalcoordinates, is the vertical

coordinate(positive upward, at themeanwatersurface), is time and is thehorizontal

gradient operator. The kinematic boundary condition on vertical velocity at the bottom is:

at (2a)

where is the mean water depth. The analogous condition at the surface is:

at (2b)

where is thesurfaceelevation.[Note, for future reference, hasbeendefinedasthe

verticalvelocityatthebottomasdeterminedfrom theboundaryconditionand hasbeendefined

astheverticalvelocity at thesurfaceasdeterminedfrom theboundarycondition.]Equations(1),

(2a),(2b)aresolvedfor assumingthatthehorizontalvelocityandsurfaceelevationsareknown

from a previous part of the modelsolution. The primary difficulty of solving theseequationsis

that, together, they constituteanover-determinedsystemof equationsfor themostgeneralcase;

∂w∂z------- ∇ V•–=

V x y z t, , ,( ) u x y z t, , ,( ) v x y z t, , ,( ),( )

w x y z t, , ,( ) x y,( ) z

z 0= t ∇

w V– ∇h• wb= = z h–=

h x y,( )

w∂η∂t------ V ∇η•+ ws= = z η=

η x y t, ,( ) wb

ws

w

3/29

that is, (1) is a firstorder equation that must be solved subject to twoboundary conditions. As will

be shown, in numerical schemes that are locally mass conserving, one of the boundary conditions

becomes redundant and thus the system is not over-determined.

We will consider here two approaches to solving the over-determined system:

1. Solution of the vertical derivative of the continuity equation:

(3)

where indicates vertical velocity computed using the this approach (henceforth,

VDC). This is a second order differential equation and thus both boundary conditions

can be satisfied (Lynch and Naimie, 1993).

2. Solution of the over-determined system in a “best fit” sense by admitting residuals in

the first order continuity equation and both boundary conditions. An optimal solution

is then sought that minimizes those residuals in a weighted least squared sense. Because

this approach involves the adjoint of the continuity equation, we call it the “adjoint”

approach (henceforth, ADJ). It will be described in more detail in Section 3.

A third approach, called the “traditional” approach (TRAD) simply neglects one of the boundary

conditions; this approach will be shown here to be a component of ADJ.

Muccinoet al. (1997) found that VDC and ADJ provide different vertical velocity fields

regardless of resolution in the vertical or horizontal and that ADJ better approximates the analytic

solution in a simple test problem than VDC. The objectives of this paper are to reconcile the numer-

ical differences between VDC and ADJ and to make overall recommendations for the computation

of vertical velocity in three-dimensional circulation models. Results are provided for tidally forced

∂2wvdc

∂z2

----------------∂∂z----- ∇ V•( )–=

wvdc

4/29

circulation in a quarter annular test case and a wind, density and boundary forced circulation off

the southwest coast of Vancouver Island.

3. A summary of the adjoint approach (ADJ)ADJ admits residuals in the continuity equation (1) and the boundary conditions (2a) and

(2b) at each horizontal node:

(4a)

at (4b)

at (4c)

A cost functional, which is formed from the squares of these residuals is defined:

(5)

where , and are constant weights and is the total

water depth and is included in the denominator of the first term to normalize the vertical integral

and also to maintain dimensional consistency. The weights are defined as the inverses of the cova-

riances of the residuals:

, , (6)

where the covariances are defined as:

, , (7)

and indicates expected value. Thus the dimensions of , and are , and

, respectively. We will assume that . Additionally, given that only the relative

f z( ) ∂w∂z------- ∇ V•+=

ib w V ∇h•+= z h–=

is w∂η∂t------ V ∇η•––= z η=

IW f

H-------- f{ }2

dz

h–

η

∫ W b ib{ }2W s is{ }2

+ +=

W f W b W s H x y t, ,( ) h x y,( ) η x y t, ,( )+=

W f1

C f------= W b

1Cb------= W s

1Cs------=

C f f2⟨ ⟩= Cb ib

2⟨ ⟩= Cs is2⟨ ⟩=

⟨ ⟩ W f W b W s T2

T2/L

2

T2/L

2W b W s=

5/29

values of the weights is significant (not the absolute values of the weights themselves) we set

without further loss of generality. Thus, (5) can be written:

(8)

The adjoint solution, , minimizes and can be shown to be:

, (9)

where is the “traditional solution” to the governing equation (1) and the bottom boundary

condition (2a). [For mathematical details leading to (9) see Muccino (1997); for a similar derivation

using “representers,” see Appendix A of Muccino and Bennett (2001)] Thus, the adjoint solution

is a sum of and a correction that is linear in and proportional to the misfit between the tradi-

tional solution evaluated at the surface and the surface boundary condition. In the limit of

(that is, no weight given to the three-dimensional continuity equation in ), the

correction reduces to:

for (10)

In this case, the correction is zero at the bottom and equal to the surface boundary condition misfit

of the traditional solution at the surface. Consequently, the correction causes the adjoint solution

to satisfy both boundary conditions exactly although it may diminish the mass conserving proper-

ties of the solution. In the limit of , (that is, no weight given to the boundary condi-

tions in ) (9) reduces to:

for (11)

In this case, the correction approaches a constant (half the surface boundary condition misfit of the

W b W s 1= =

IW f

H-------- f{ }2

dz

h–

η

∫ ib{ }2is{ }2

+ +=

wadj I

wadj z( ) wtrad z( ) wc z( )+= wc z( ) ws wtrad η( )–( )W f /H

2h z+( )/H+

2W f /H2

1+-----------------------------------------------=

wtrad

wtrad z

W f /H2

0= I

wc z( ) ws wtrad η( )–( )h z+H

-----------= W f /H2

0=

W f /H2 ∞→

I

wc z( )ws wtrad η( )–

2---------------------------------= W f ∞→

6/29

traditional solution) over the depth. This distributes any error evenly between the surface and

bottom boundary conditions, but has no impact on the mass conserving property of the solution.

Clearly, intermediate values of generate corrections that fall between these limits. These

observations are illustrated in Figure 1, which shows the correction, normalized by the surface

boundary condition misfit:

(12)

for various values of .

W f /H2

wn z( )wc z( )

ws wtrad η( )–---------------------------------

W f /H2

h z+( )/H+

2W f /H2

1+-----------------------------------------------= =

W f /H2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0Figure 5: Parameters as in Fig. 4: Node 401, Adjoint correction

Amplitude (m/s)

Dim

ensi

onle

ss D

epth

L=0 L=1 L=10 L=100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.2 0.4 0.6 0.9

Dim

ensi

onle

ss D

epth

hz

+(

)/H

()

Normalized correction

0.1 0.3 0.5 0.7 0.8

Figure1. Normalized ADJ correctionprofilesfor a rangeof . For , thenormalized correction rangesfr om 0.0 at the bottom to 1.0 at the top. For large ,the normalized misfit approaches a constant value of 0.5.

W f /H2

W f /H2

0=W f /H

2

wn( )1.0

W f /H2

0.1=W f /H

21=

W f /H2

10=W f /H

2100=

W f /H2

0=

00

7/29

4. Numerical ImplementationTwo approaches to determining the vertical velocity have

been described and these will now be tested numerically. A vertical

sequence of three nodes indicated by subscripts , and

will be used, as shown in Figure 2. Superscripts and indi-

cate quantities evaluated over the intervals and

, respectively, (e.g., ).

4.1 Discretization of VDC

Using centered finite difference, the LHS of (3) can be discretized:

(13)

and the RHS of (3) can be discretized:

(14)

Combining (13) and (14) yields:

i 1+

i

i 1–

Figure 2. Notation usedfor vertical discretization

+

–i 1– i i 1+

– +

i 1– i,{ }

i i 1+,{ } ∆z+

zi 1+

zi

–=

∂2wvdc

∂z2

----------------∂∂z-----

∂wvdc

∂z--------------

∂wvdc

∂z--------------

+ ∂wvdc

∂z--------------

-

–

12--- ∆z

+ ∆z-

+( )--------------------------------------------------≈=

wvdci 1+

wvdci

–

∆z+

----------------------------- wvdc

iwvdc

i 1––

∆z-

-----------------------------

–

12--- ∆z

+ ∆z-

+( )----------------------------------------------------------------------------=

2

∆z+ ∆z

-+( )

----------------------------wvdc

i 1+

∆z+

------------ wvdci 1

∆z+

--------- 1

∆z-

--------+ –

wvdci 1–

∆z-

------------+=

∂∂z-----– ∇ V•( ) ∇ V•( )+ ∇ V•( )-

–12--- ∆z

+ ∆z-

+( )------------------------------------------------–≈

2

∆z+ ∆z

-+( )

----------------------------–12--- ∇ V•( )i 1+ ∇ V•( )i

+[ ] ∇ V•( )i ∇ V•( )i 1–+[ ]–{ }=

8/29

(15)

wherethe ‘+’ and ‘-’ have beenaddedto the two termsinside the curly brackets of (15) as a

reminderthat the first groupin squarebrackets is evaluatedover the interval andthe

second group is evaluated over the interval .

The LHS of (15) is tridiagonalandefficiently solved usinga tridiagonalsolver like the

Thomasalgorithm.The RHS of (15) requiresthe evaluationof the horizontalvelocity on level

coordinatesurfaces.Most three-dimensionalcirculationmodelsusestretchedor terrainfollowing

coordinatesin whichtheverticaldimensionis transformedfrom to , where

and are arbitrary constants (e.g., , ):

(16)

Derivativesin theterrainfollowing coordinatesystemarerelatedto derivativesin thelevel coordi-

nate system using the chain rule:

(17a)

(17b)

where indicateshorizontalderivativesonalevel surfaceand indicateshorizontalderivatives

on a stretched surface. Using (17a) and (17b) to expand terms on the RHS of (15) gives:

wvdci 1+

∆z+

------------ wvdci 1

∆z+

--------- 1

∆z-

--------+ –

wvdci 1–

∆z-

------------+ 12---– ∇ V•( )i 1+ ∇ V•( )i

+[ ]+

∇ V•( )i ∇ V•( )i 1–+[ ]

-–{ }=

i i 1+,{ }

i 1– i,{ }

h– z η< < b σ a< < a

b a 1= b 0=

σ aa b–( ) z η–( )

H----------------------------------+=

∇•V ∇σ•Va b–( )

H---------------- σ b–( )

a b–( )-----------------∇η σ a–( )

a b–( )-----------------∇h+ •

∂V∂σ-------–=

∂∂z-----

a b–( )H

---------------- ∂∂σ------=

∇ ∇σ

9/29

(18a)

(18b)

(18c)

(18d)

Substituting (18a) through (18d) and recognizing that yields:

(19)

In practice, the horizontal derivatives and therefore (19) reduces to:

∇•V( )i 1+[ ]+

∇σ•V( )i 1+ a b–( )H

---------------- σi 1+b–

a b–---------------------

∇η σi 1+a–

a b–---------------------

∇h+ •Vi 1+ Vi

–

∆σ+------------------------

–=

∇•V( )i[ ]+

∇σ•V( )i a b–( )H

---------------- σib–

a b–--------------

∇η σia–

a b–--------------

∇h+ •Vi 1+ Vi

–

∆σ+------------------------

–=

∇•V( )i[ ]-

∇σ•V( )i a b–( )H

---------------- σib–

a b–--------------

∇η σia–

a b–--------------

∇h+ •Vi Vi 1–

–

∆σ-------------------------

–=

∇•V( )i 1–[ ]-

∇σ•V( )i 1– a b–( )H

---------------- σi 1–b–

a b–--------------------

∇η σi 1–a–

a b–--------------------

∇h+ •Vi Vi 1–

–

∆σ-------------------------

–=

∆z H∆σ/ a b–( )=

wvdci 1+

∆σ+------------ wvdc

i 1

∆σ+---------- 1

∆σ----------+

–wvdc

i 1–

∆σ-------------+

12---–

Ha b–( )

---------------- ∇σ•V( )i 1+ ∇σ•V( )i+[ ]

+ Ha b–( )

---------------- ∇σ•V( )i ∇σ•V( )i 1–+[ ]

-–

=

σi 1+b–( )

a b–( )--------------------------∇η σi 1+

a–( )a b–( )

--------------------------∇h+ •Vi 1+ Vi

–

∆σ+------------------------

–

σib–( )

a b–( )------------------∇η σi

a–( )a b–( )

------------------∇h+ •Vi 1+ Vi

–

∆σ+------------------------

Vi Vi 1–

–

∆σ-------------------------

––

σi 1–b–( )

a b–( )-------------------------∇η σi 1–

a–( )a b–( )

-------------------------∇h+ •Vi Vi 1–

–

∆σ-------------------------

+

∇σ•V( )i[ ]+

∇σ•V( )i[ ]-

=

10/29

(20)

where no ambiguity is introduced by dropping the and superscripts on the remaining hori-

zontal velocity gradient terms.

An alternative expression for is obtained from (15) if it is assumed that the horizontal

derivatives . Using the chain rule, (15) reduces to:

(21)

It is clear that (20) and (21) are identical only if . Previous investigators (Lynch

and Naimie, 1993; Muccino et al., 1997) have used an expression for that is equivalent to

wvdci 1+

∆σ+------------ wvdc

i 1

∆σ+---------- 1

∆σ----------+

–wvdc

i 1–

∆σ-------------+

12---–

Ha b–( )

---------------- ∇σ•V( )i 1+ ∇σ•V( )i 1––[ ]

=

σi 1+b–( )

a b–( )--------------------------∇η σi 1+

a–( )a b–( )

--------------------------∇h+ •Vi 1+ Vi

–

∆σ+------------------------

–

σib–( )

a b–( )------------------∇η σi

a–( )a b–( )

------------------∇h+ •Vi 1+ Vi

–

∆σ+------------------------

Vi Vi 1–

–

∆σ-------------------------

––

σi 1–b–( )

a b–( )-------------------------∇η σi 1–

a–( )a b–( )

-------------------------∇h+ •Vi Vi 1–

–

∆σ-------------------------

+

+ -

wvdc

∇•V( )i[ ]+

∇•V( )i[ ]-

=

wvdci 1+

∆σ+------------ wvdc

i 1

∆σ+---------- 1

∆σ----------+

–wvdc

i 1–

∆σ-------------+

12---–

Ha b–( )

---------------- ∇σ•V( )i 1+ ∇σ•V( )i 1––[ ]

=

σi 1+b–( )

a b–( )--------------------------∇η σi 1+

a–( )a b–( )

--------------------------∇h+ •Vi 1+ Vi

–

∆σ+------------------------

–

σi 1–b–( )

a b–( )-------------------------∇η σi 1–

a–( )a b–( )

-------------------------∇h+ •Vi Vi 1–

–

∆σ-------------------------

+

∂V∂σ-------

+ ∂V∂σ-------

-=

wvdc

11/29

(21). However, results from Muccino et al. (1997) show that these values of do not agree with

vertical velocities obtained with ADJ. Below, we demonstrate that (20) does yield values that are

essentially identical to ADJ, while (21) yields significantly inferior estimates of vertical velocity.

4.2 Discretization of ADJ

Recall, the adjoint solution, (9), consists of the sum of the traditional solution (the

solution of the governing equation using the bottom boundary condition and neglecting the surface

boundary condition) plus a correction. Thus, the numerical solution proceeds by first finding

by discretizing (1):

(22)

Using the chain rule, this becomes:

(23)

The calculation is initiated using the bottom boundary condition, and then marched up the water

column; no matrix solver is required. Once is known, is easily computed algebraically

and added to to yield the adjoint solution, .

wvdc

wtrad

wtrad

wtradi

wtradi 1–

–

∆z-

-------------------------------12--- ∇ V•( )i ∇ V•( )i 1–

+[ ]–=

wtradi

wtradi 1–

–

∆σ--------------------------------

12--- H

a b–------------ ∇σ•V( )i ∇σ•V( )i 1–

+[ ]

–=

σib–( )

a b–( )------------------∇η σi

a–( )a b–( )

------------------∇h+ •Vi Vi 1–

–

∆σ-------------------------–

σi 1–b–( )

a b–( )-------------------------∇η σi 1–

a–( )a b–( )

-------------------------∇h+ •Vi Vi 1–

–

∆σ-------------------------–

wtrad wc

wtrad wadj

12/29

4.3 Comparison of VDC and ADJ

Consider VDC first. The second order equation should admit, in addition to the solution of

the associated first order equation, terms that are linear and constant in . For example, suppose

the vertical velocity that satisfies the first order equation (1) is:

(24a)

Consider a different vertical velocity field with addition terms, linear and constant in :

(24b)

where and are constants. The second derivatives of both (24a) and (24b) are:

(25)

Thus, the additional terms in the vertical velocity are transparent in VDC. Furthermore, since the

VDC solution is forced to satisfy both boundary conditions, these terms take on a form such that

satisfies both boundary conditions.

Now consider ADJ with . Equation (10) can be written in the form of (24b),

where:

(26)

and therefore:

, . (27)

Thus, VDC and ADJ with should theoretically yield the same vertical velocity solu-

tion. Indeed, this equivalence is found numerically when (20) is used for the VDC solution

z

w z( ) g z( )=

z

w z( ) g z( ) c1z c2+ +=

c1 c2

∂w2

∂z2

--------- ∂g2

∂z2

--------=

w z( )

W f /H2

0=

g z( ) wtrad z( ) ws wtrad η( )–( ) zH---- ws wtrad η( )–( ) h

H----+ +=

c1

ws wtrad η( )–( )H

--------------------------------------= c2 ws wtrad η( )–( ) hH----

=

W f /H2

0=

13/29

although it does not occur when (21) is used for the VDC solution. This is demonstrated below

using two different test problems.

5. Quarter Annular Harbor Test CaseWe begin numerical tests in a quarter annular harbor with quadratic bathymetry and peri-

odic boundary forcing; the analytic vertical velocity for this problem is given in Muccino et al.

(1997) and thus serves as a useful starting point for comparing the accuracy of these approaches.

That solution is repeated in the Appendix for convenience.

The geometry of the quarter annular harbor is as in Muccino et al. (1997) and is shown in

Figure 3. The boundaries at , , and are no-flow boundaries.

The open boundary, located at , is forced by an M2 tide with frequency

and amplitude . The bathymetry of the harbor, as shown in

Figure 3b, is quadratic in and constant in , such that , where .

r r1 4x104m= = θ 0= θ π/2=

r r2 1x105m= =

ω 1.4054–×10 s

1–= η0 0.10 m=

r θ h h0r2= h0 6.259–×10 m

1–=

r1r2

Figure 3b. Side view of quarter annularharbor with quadratic bottom.

Figure 3a. Plan view of quarter annularharbor with opening at r r2=

θ π/2=

r1 4x104m= r2 1x105m=x

y

N

S

D

14/29

The horizontal velocity is calculated using the analytic solution in Lynch and Officer

(1985); the horizontal velocities are calculated analytically, rather than numerically, so that any

deviations of the numeric vertical velocity solution from the analytic vertical velocity solution

owe entirely to the vertical velocity solution technique, not numerical errors in the horizontal

velocity.

However, evaluation of horizontal velocity derivatives is considered to be a component of

the vertical velocity calculation procedure. Thus (20), (21) and (23) are discretized in the horizon-

tal using Galerkin finite elements with linear basis functions. The solution is evaluated using the

grid shown in Figure 3a. The grid has 825 nodes and 1536 elements in the horizontal and 32

evenly spaced sigma layers in the vertical. Results are presented here for the two representative

nodes shown in the figure: Node S is shallow ( ) and Node D is deep

( ).

The results in Muccino et al. (1997) are for two sets of parameters, and are shown at just

one instant in the tidal cycle. Here, we use one set of parameters (consistent with Figure 4 in Muc-

cino et al. (1997): and , see the Appendix for definition of these

parameters), and present the vertical velocity as amplitude and phase in Figure 4. The parameter

is set to zero here; its impact on the ADJ solution is investigated next. In these figures,

the analytic, ADJ and VDC (20) amplitude and phase are coincident, with VDC (21) amplitude

and phase differing from them. Results for different values of and and and at other nodes

are not shown but are qualitatively similar, indicating that the observations regarding Figure 4 do

not depend upon particular choices of parameters or bathymetry, but rather are quite general.

Now we will consider the impact of on the ADJ solution. Recall the normalized

correction profiles for various values of in Figure 1; to obtain the actual correction, these

profiles are scaled by the surface boundary condition misfit of the traditional solution (9). In the

h 11.29 m=

h 56.41 m=

λ 6.627 6.627i+= K 102.1=

W f /H2

λ K h0

W f /H2

W f /H2

15/29

quarter annular harbor, the magnitude of this misfit is two or three orders of magnitude smaller than

. Thus, the ADJ correction is insignificant, regardless of the value of , and ADJ

essentially collapses to the traditional approach.

6. Application to the Pacific Coast of Southwest Vancouver Island.The summer circulation off the western continental margin of Vancouver Island is charac-

terized by a moderately intense upwelling of nutrients that supports high biological productivity

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

x 10−5

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

230 235 240 245 250 255 260 265 270 275−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

230 235 240 245 250 255 260 265 270 275−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

Vertical velocity phase (degrees)235 245 255 265 275

Node D(Phase)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

x 10−5

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Dim

ensi

onle

ss D

epth

zh

+(

)/H

Dim

ensi

onle

ss D

epth

zh

+(

)/H

0

1

0

1

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 235 245 255 265 275

Node S(Amplitude)

Node D(Amplitude)

Node S(Phase)

AnalyticADJ, VDCVDC

(20)(21)

Figure 4. Amplitude and phase of vertical velocity in the quarter annular harbor at Node Sand Node D. Note that the analytic, ADJ and VDC (20) solutions are coincident, butdifferent from the VDC (21) solution.

Vertical velocity amplitude (m/s)5–×10

5–×10

wtrad W f /H2

16/29

and a lucrative commercial fishery. Circulation models (e.g., Foreman et al. 2000) have been devel-

oped to better understand both spatial and temporal variations in this upwelling, and these physical

models are now being coupled to biological models in order to simulate specific components of the

food chain. The accuracy of these models is highly dependent on both their adherence to mass

conservation (so that nutrients are not falsely depleted or created) and the accuracy of the vertical

velocities that move nutrients and biota up the water column. A circulation model for this region is

thus a useful test for the vertical velocity calculations described here.

The model used here is the pseudo-nonlinear FUNDY5 (Lynch and Werner, 1987) that

incorporates root-mean square tidal velocities in the bottom friction and vertical viscosity coeffi-

cients (see Foreman et al.(2000) for further details). This model solves the three-dimensional,

harmonic shallow water equations in the sequential manner described in Section 1 using linear

triangular finite elements. In this test application we consider only the steady state solution

resulting from steady forcing. Combined wind- and buoyancy- driven flows are forced with average

winds measured at a meteorological buoy near the middle of the model domain and a three-dimen-

sional density field that was constructed through Kriging of temperature and conductivity measure-

ments taken in late July, 1998. Boundary conditions for these calculations were computed through

a combination of geostrophic radiation conditions and adjustments to the surface elevations so that

the vertically-integrated flows passed through the boundary without any reflection. Analogous to

the inversion described in Foreman et al.(2000), further boundary condition adjustments were also

made to introduce a California Undercurrent consistent with observations at two current meter

moorings lying along the continental slope.

The computational grid is comprised of 9767 nodes horizontally (Figure 5) and 41 evenly

spaced sigma surfaces vertically. Several nodes are highlighted in the figure for future reference;

17/29

Figure 5. The Southwestern Vancouver Island domain (a) contours of bathymetry(meters); (b) computational grid. Labeled points are discussed further in the text.

A

ED

B

C

A

ED B

C

Barkley Canyon

Nitinat Canyon

(a)

(b)

0100200300400500600700800900100012501500200025002700

18/29

each of these nodes is representative of certain regions in the domain, as detailed in Table 1.

Vertical velocity profiles for Nodes A through E calculated with ADJ and a range of values

and VDC (21) are shown in Figure 6. Since, the VDC (20) solution is coincident with the ADJ solu-

tion when , it is not distinct in Figure 6. Several observations may be made regarding

these figures:

1. In all cases, the VDC (21) solution is substantially different than the ADJ solutions for

any value of .

2. In the deep ocean (Node A), on the continental shelf (Node B) and near the coast (Node

C), the vertical velocities are very small (order ).

3. Along the sides of Barkley Canyon (Nodes D and E), the vertical velocities are one to

two orders of magnitude greater than at Nodes A, B and C. The ability to predict the

patterns of vertical velocity in regions such as Barkley Canyon is important in under-

standing biological productivity (Allen et al., 2001). Of particular concern here is the

fact that VDC (21) and ADJ predict opposite vertical velocity trends. On the south side

W f /H2

W f /H2

0=

Table 1. Description of representative nodes in SouthwesternVancouver Island Domain

Node Region (m)Surface BC misfit

of traditional solution(m/s)

A Deep Ocean 2020

B Continental Shelf 138

C Coast 35

D South Barkley Canyon 475

E North Barkley Canyon 487

H

6.05–×10–

6.75–×10–

1.57–×10–

1.43–×10–

6.23–×10

W f /H2

15–×10 m/s

19/29

−3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1

x 10−3

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0Figure 10e: Barkley Canyon, North (Node 6946), R=6201e−6

−8 −6 −4 −2 0 2 4 6 8

x 10−4

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0Figure 10d: Barkley Canyon, South (Node 6114), R=−2842e−6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

x 10−5

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0Figure 10c: Coast (Node 7794), R=−0.15e−6

−1.5 −1 −0.5 0 0.5 1 1.5

x 10−5

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0Figure 10b: Shelf (Node 4049, R=−68e−6)

−1.5 −1 −0.5 0 0.5 1 1.5 2

x 10−5

−1

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0Figure 10a: Deep (Node 4257, R=478e−6)

Figure 6. Vertical velocity profiles forNodes A through E (as shown in Figure 5)using ADJ with a range of valuesand VDC (20) and (21). In all cases, theVDC (20) solution is coincident with theADJ solution for (see Section4.3).

W f /H2

W f /H2

0=

Node ANode B

Node C Node D

(Deep)(Continental

(Coast) (Barkley

South)

Node E(Barkley Canyon

North)

0 21-15–×10

-1 0 15–×10

5–×10

0 0.6 1.2 1.8 -4 4 84–×10

-3 -1 03–×10

ADJ W f /H2

0=( ) VDC,

ADJ W f /H2

0.1=( )

ADJ W f /H2

1=( )

ADJ W f /H2

10=( )

VDC

Vertical velocity (m/s)

Vertical velocity (m/s)

Dim

ensi

onle

ss D

epth

zh

+(

)/H

Dim

ensi

onle

ss D

epth

zh

+(

)/H

Dim

ensi

onle

ss D

epth

zh

+(

)/H 1

0

1

0

1

0

(21)

(20)

-2 1

Shelf)

Canyon,

-8 0

20/29

of Barkley Canyon, VDC (21) predicts upwelling while ADJ predicts downwelling, and

on the north side of the same canyon, VDC (21) predicts downwelling while ADJ

predicts upwelling. Further insight into the vertical velocity behavior may be gained by

examining the horizontal velocity field in that region at (

at Nodes D and E) as illustrated in Figure 7; this figure shows a California Undercurrent

that generally follows the bathymetry towards the northwest. However, the flow cuts

across the deepest parts of the canyon, suggesting downwelling on the south side of the

canyon and upwelling on the north side of the canyon. This is the vertical velocity

behavior predicted by ADJ for all values of ; VDC (21) predicts the opposite.

4. Unlike results presented for the quarter annular harbor, there is considerable depen-

dence here of the ADJ solution on for all nodes except near the coast at Node

C. This dependence indicates that the surface boundary condition misfit of the tradi-

tional solution is relatively large.

5. As shown in Figure 1, there is not much change in the vertical velocity solution

obtained with values of and , and thus we consider these

values of to be “large” in the sense that larger values will not substantially

change the solution. Likewise, there is little difference in the solution obtained with

values of and , and thus we consider these values of

to be “small.”

7. Effects of local mass conservationThe previous sections have shown that the traditional, adjoint and vertical derivative (20)

solutions for vertical velocity give essentially identical results when the surface boundary condition

misfit of the traditional solution is equal to zero. This misfit is readily identified as owing to errors

z 300 m–= z h+( )/H 0.4≈

W f /H2

W f /H2

W f /H2

10= W f /H2

100=

W f /H2

W f /H2

0= W f /H2

0.1=

W f /H2

21/29

in the horizontal solution. Integrating (1) from the bottom upward and applying the bottom

boundary condition (2a) yields:

(28)

Applying Liebnitz’s Rule to the integral in (28) yields:

(29)

Figure 7. Horizontal velocity field at z=-300 m (velocities on stretched coordinatesurfacesare interpolated onto the level surface 300 m below the surface).The flow istowards the northwest largely following the bathymetry. However, the flow cuts acrossthe deepestpart of the Barkley Canyon; this would indicate a downwelling on the southside of the canyon and an upwelling on the north side of the canyon.

Barkley Canyon

Nitinat Canyon

E

D

01002003004005006007008009001000125015002000

wtrad z( ) ∇•V zd

h–

z

∫– V h–( )•∇h–=

wtrad z( ) ∇• V zd

h–

z

∫– V z( )•∇z+=

22/29

In the interior of the water column, and the vertical velocity is simply the horizontal diver-

gence of the horizontal velocity field integrated up from the bottom. At the free surface, (29)

becomes:

(30)

Using the surface boundary condition (2b) to replace the final term in (30) and rearranging gives:

(31)

The RHS of (31) is the vertically-integrated continuity equation. Consequently, (31) shows that the

misfit between the traditional solution evaluated at the surface and the surface boundary condition

is nonzero anywhere in the domain that the horizontal velocity field does not conserve mass in the

vertically-integrated sense. Since numerical models solve discrete rather than continuous

governing equations, discretization of the vertically-integrated continuity equation in (31) must

match that used to evaluate (1), (2a) and (2b).Thus, local mass conservation, on the same numerical

stencil used to determine , is required if the misfit on the LHS of (31) is to be zero. If mass is not

locally conserved, error is introduced into the computed vertical velocity field as it is integrated up

the water column.

The quarter annular test problem presented in Section 5 uses horizontal velocities obtained

from the analytical solution and thus these velocities satisfy the vertically-integrated continuity

equation exactly; the surface misfit is insignificant and VDC (20) and ADJ with any value of the

parameter give essentially identical results throughout the domain.

∇z 0=

wtrad η( ) ∇• V zd

h–

η

∫– V η( )•∇η+=

ws wtrad η( )– ∂η∂t------ ∇• V zd

h–

η

∫+=

w

W f /H2

23/29

While finite difference models using Arikawa C grids conserve mass on each computa-

tional cell, Galerkin finite element models are guaranteed to conserve mass only globally (Lynch,

1985; Lynch and Holboke, 1997) and therefore allow for nonzero local residuals in the vertically-

integrated continuity equation. Figure 8 illustrates the surface misfits for the Vancouver Island test

problem; by (31), this plot also represents local vertically-integrated continuity residuals. Consid-

erable surface misfits are observed; areas having large misfits (and thus poor vertically-integrated

mass conservation) typically correspond to areas of steep bathymetric gradients. Plots of surface

misfits or vertically-integrated mass error such as this are easy to construct and provide a useful

diagnostic tool for identifying areas where local mass conservation is relatively poor, and therefore

where significant errors are likely to exist in the vertical velocity solution.

As outlined earlier, most oceanic circulation models follow a sequential solution procedure

in which the vertical velocity solution occurs separate from and with minimal feed back to the hori-

zontal velocity solution. Since mass conservation error in the horizontal solution is the cause of the

vertical velocity solution error identified above, it seems inadvisable to sacrifice the boundary

condition information in favor of stricter adherence to the three-dimensional continuity equation

when determining the vertical velocity. Consequently, we suggest a small value of the ADJ

weighting parameter ( ) as preferable to a high value ( ).

8. ConclusionsThe results presented in this paper help reconcile past uncertainty in the vertical velocity

solution in three-dimensional circulation models. Specifically, we have found the following.

1. Most three-dimensional circulation models use a sequential solution procedure to solve

for the free surface elevation and velocity fields. That is, the vertically-integrated conti-

W f /H2

W f /H2

0.1< W f /H2

10>

24/29

nuity and the three-dimensional momentum equations are solved first for the free

surface elevation and the horizontal velocity fields; then the three-dimensional conti-

nuity equation is solved for the vertical velocity field. Solving the three-dimensional

continuity equation (a first order differential equation in the vertical coordinate) for the

vertical velocity would appear to be problematic given the need to satisfy boundary

conditions at both the bottom and at the free surface. However, the “traditional”

Figure 8. Misfit between the traditional solution at the surface and the surfaceboundary condition for the Vancouver Island domain. Filled, colored contoursrepresent misfits, and contour lines represent bathymetry (with the same contourintervals as Figure 5). Residuals are largest in regions of sharply varying bathymetry.

-25.0-5.0-1.0-0.20.21.05.0

25.03–×10

25/29

(TRAD) vertical velocity solution, obtained by integrating the three-dimensional conti-

nuity equation upward using the bottom boundary condition, will match the surface

boundary condition if the elevation and horizontal velocity fields exactly satisfy the

vertically-integrated continuity equation. In this case, the surface boundary condition is

redundant. If the elevation and horizontal velocity fields are not locally mass

conserving, the misfit between the TRAD solution and the surface boundary condition

is equal to the local error in the vertically-integrated continuity equation.

2. The VDC approach proposed by Lynch and Naimie (1993), in which the vertical

velocity is computed from the vertical derivative of the three-dimensional continuity

equation, is equivalent to an optimal, adjoint solution (ADJ) of the three-dimensional

continuity equation (Muccino et al., 1997) in which the boundary conditions are

preserved in lieu of stricter adherence to the continuity equation. VDC requires the

solution of a tri-diagonal matrix problem over the vertical while ADJ requires no matrix

solution. Thus ADJ is more computationally efficient than VDC.

3. Depending upon the numerical discretization applied to VDC, results are obtained that

are less accurate than the other vertical velocity solutions in the quarter annular harbor

test case and that appear to be physically inconsistent in the Vancouver Island test case.

We recommend that if VDC is used, the discretization presented in (20) be used rather

than the discretization presented in (21).

4. The ADJ solution is the solution that minimizes the cost functional (5) which penalizes

misfits to the three-dimensional continuity equation and the misfits of the bottom and

surface boundary conditions. The ADJ solution can be shown to be the sum of the

TRAD solution and a linear correction. In the limit of satisfying both the bottom and

26/29

free surface boundary conditions ( ) the correction is zero at the bottom and

equal to the misfit of the TRAD solution and the free surface boundary condition at the

surface. In the limit of maximizing adherence to the three-dimensional continuity equa-

tion, , the correction is a constant over the entire water column and equal

to one half of the misfit of the TRAD solution and the free surface boundary condition.

5. If there is no misfit between the TRAD solution and the free surface boundary condi-

tion, TRAD, ADJ and VDC (20) give identical solutions for the vertical velocity.

6. Results from models that do not enforce strict local mass conservation, such as finite

element models, will be susceptible to the vertical velocity errors described herein. We

recommend plotting maps of the error in the vertically-integrated continuity equation

as a diagnostic tool for determining areas in the domain that may be subject to signifi-

cant vertical velocity errors. The first choice for improving the computed vertical

velocity is to reduce errors in vertically-integrated mass conservation, either by

improved grid resolution or by smoothing the bathymetry (Oliveira et al., 2000). Mass

conservation may also be improved in Generalized Wave Continuity Equation based

finite element models by increasing the primitive continuity equation weighting param-

eter (known as or ), (Kolar et al., 1992, 1994). If local mass conservation cannot

be achieved, we suggest use of ADJ with the weighting coefficient set to preferentially

favor the surface and bottom boundary conditions ( ).

9. AcknowledgementsThe authors would like to thank Chris Naimie for independently confirming that the results

presented here are consistent with FUNDY5. The authors also thank Rick Thomson, Susan Allen,

Dave Mackas and the crew of the CCGS John P. Tully for collecting the salinity, temperature, and

W f /H2

0=

W f /H2

10>

τ0 G

W f /H2

0.1<

27/29

velocity data that were used in this study. These observations were taken during a GLOBEC

Canada cruise that was co-sponsored by the Natural Sciences and Engineering Research Council

of Canada and the Department of Fisheries and Oceans. R. Luettich would like to acknowledge

partial funding for this work by the Office of Naval Research award N00014-97-C-6010 and J.

Muccino would like to acknowledge partial funding for this work by the National Science Foun-

dation under OCE-9520956.

10. Appendix: Analytic solution quarter annular harborConsider a quarter annular harbor with no-flow boundaries at , , and

and an open boundary at . The open boundary is subject to periodic forcing

. The bathymetry of the harbor is quadratic in and constant in , such that

. The eddy viscosity, , and bottom friction, , vary such that:

and , (32)

are constant. The analytic solution for surface elevation and horizontal velocity (Lynch andOfficer,

1985) are:

(33)

(34)

where: ,

,

r r1= θ 0=

θ π/2= r r 2=

η Re η0eiωt{ }= r θ

h h0r 2= N k

KkhN------= λ iωh2

N------------=

η r t,( ) Re Ars1 Br

s2+( ) iωt( )exp

=

v r σ t, ,( ) Re v0 r( ) 1 δ λσ( )cosh–( )eiωt{ }=

Aη0s2r 1

s2

s2r 2

s1r 1

s2 s1r 1

s1r 2

s2–------------------------------------------= B

η– 0s1r 1

s1

s2r 2

s1r 1

s2 s1r 1

s1r 2

s2–------------------------------------------=

s1 1– 1 β2–+= s2 1– 1 β2––=

β2 ω2 iωτ–( )/ gh0( )=

28/29

is the gravitational constant and . The vertical velocity is (Muccino et al., 1997):

(35)

where:

,

11. References

Allen, S. E., C. Vindeirinho, R. E. Thomson, M. G. G. Foreman and D. L. Mackas, 2001: Physicaland biological processes over a submarine canyon during an upwelling event. Canadian Journalof Fisheries and Aquatic Sciences, 59(4), 671-684.

Blumberg, A. F. and G. L. Mellor, 1987: A description of a three-dimensional coastal ocean circu-lation model. In: C.N.K. Mooers, [ed], Three-dimensional Coastal Ocean Models, Coastal andEstuarine Sciences 4, AGU Press, Washington, DC, 1-16.

Foreman, M. G. G., R. E. Thomson, and C. L. Smith, 2000: Seasonal current simulations for thewestern continental margin of Vancouver Island. J. Geophys. Res., 105(C8), 19,665-19,698.

Haidvogel, D.B. and A. Beckmann. 1999: Three-dimensional ocean models. Chapter 4 in Numer-ical Ocean Circulation Modeling, Imperial College Press, London, England, 121-162.

Kolar, R. L., W. G. Gray, J. J. Westerink and R. A. Luettich, Jr., 1994: Shallow water modeling inspherical coordinates: Equation formulation, numerical implementation and application. J.Hydraul. Res., 32(1), 3-24.

τ Nh2----- λ2 λtanh

λ λ2

K----- 1–

λtanh+

---------------------------------------------=

v0 r( ) giωr--------– s1 Ar

s1 s2Brs2+( )=

g i 1–=

w r σ t, ,( ) Re 2γ α1δ σ λσ( )cosh λ( )cosh+( ) λσ( )sinh λ( )sinh+λ

------------------------------------------------–

=

γ α2 σ 1 δ λσ( ) λ( )sinh+sinh( )λ

--------------------------------------------------------–+ 2γ α1 1 δ λ( )cosh–[ ]+ + e

iωt

σ z/h=

γgh0

iω-------- iωt( )exp=

α1 As1rs1 Bs2rs2+= α2 As12rs1 Bs2

2rs2+=

δ 1

λ( )cosh 1λK---- λ( )tanh+

-----------------------------------------------------------=

29/29

Kolar, R. L., W. G. Gray and J. J. Westerink, 1992: An analysis of the mass conserving propertiesof the generalized wave continuity equation. Computational Methods in Water Resources IX, 2,T. Russell et al., [eds], Computational Mechanics Publications, Southampton, UK, pp. 537-544.

Lynch, D.R. and M.J. Holboke, 1997: Normal flow boundary conditions in 3D circulation models.Int. J. Numer. Meth. Fluids, 25, 1185-1205.

Lynch, D. R. and C. E. Naimie, 1993: The M2 tide and its residual on the outer banks of the Gulfof Maine. J. Phys. Oceanogr., 23, pp. 2222-2253.

Lynch, D. R. and F. E. Werner, 1991: Three-dimensional hydrodynamics on finite elements. Part I:Nonlinear time-stepping model. Int. J. Numer. Meth. Fluids, 12, 507-533.

Lynch, D. R. and F. E. Werner, 1987: Three-dimensional hydrodynamics on finite elements. PartII: Linearized harmonic model. Int. J. Numer. Meth. Fluids, 7, 871-909.

Lynch, D. R. and C. B. Officer, 1985: Analytic test cases for three-dimensional hydrodynamicmodels. Int. J. Numer. Meth. Fluids, 5, 529-543.

Lynch, D. R., 1985: Mass balance in shallow water simulations. Communications in AppliedNumerical Methods, 1, 153-159.

Muccino, J.C. and A. F. Bennett, 2001: Generalized inversion of the Korteweg-de Vries Equation.submitted to Dyn. Atmos. Oceans.

Muccino, J. C., W. G. Gray and M. G. G. Foreman, 1997: Calculation of vertical velocity in three-dimensional, shallow water equation, finite element models. Int. J. Numer. Meth. Fluids, 25,779-802.

Oliveira, A., A. B. Fortunato and A. M. Baptista, 2000: Mass balance in Eulerian-Lagrangian trans-port simulations in estuaries. J. Hydraul. Eng., 126, 605-614.