on the influences of nonlinear bottom friction on the topographic rectification of tidal currents

Geophys. Asfrophys. Fluid Dynamics, Vol. 42. pp. 227-245 Reprints available directly lrom the publisher Photocopying permitted by license only 0 1988 Gordon and Breach Science Publishers, Inc. Printed in Great Britain

On the Influences of Nonlinear Bottom Friction on the Topographic Rectification of Tidal Currents DANIEL G. WRIGHT and JOHN W. LODER Department of Fisheries and Oceans, Physical and Chemical Sciences, Bedford Institute of Oceanography, P. 0. Box 1006, Dartmouth, N.S., Canada B2Y 4A2

(Received 10 August 1987)

The influences of nonlinear bottom friction on the along-isobath mean current associated with the topographic rectification of tidal currents are examined in the limit of depth-independence (no vertical structure in the horizontal currents), weak friction and weak nonlinearity. These influences are discussed in terms of the effects of spatial gradients and temporal variations in the effective friction coefficient relating bottom stress to velocity. For typical parameter values, spatial gradients in the coefficient enhance the mean current and temporal variations reduce it. The first influence is the source of Loder’s (1980) suggestion that the mean current is enhanced by nonlinear friction. The second influence accounts for the mean current reduction upon the inclusion of nonlinear friction in the example considered by Huthnance (1981). However, it is shown that each of these influences can have the opposite effect for extreme parameter values. Additionally, the inclusion of mean current contributions to the friction coefficient can significantly increase the effective mean friction coefficient and hence also reduce the mean current. The results confirm that the form of bottom friction is important in models for the topographic rectification of tidal currents.

KEY WORDS: Topographic rectification, bottom friction, tidal currents.

1. INTRODUCTION

Observations indicate that the mean (time-averaged over a tidal period) circulation in tidally-energetic shallow seas is maintained in

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228 D. G . WRIGHT AND J. W. LODER

part by the rectification of tidal currents over submarine topographic features (e.g. Huthnance, 1973; Magnell et al., 1980; Butman et al., 1983; Smith, 1983; Howarth and Huthnance, 1984). Numerical modelling and theoretical studies have complemented these observations by identifying the mechanisms through which momentum and vorticity are transferred from the tide to the mean state (e.g. Huthnance, 1973; Zimmerman, 1978, 1980; Loder, 1980; Huthnance, 1981; Robinson, 1981; Greenberg, 1983; Tee, 1985; Wright and Loder, 1985; Maas et al., 1987).

One result from the theoretical studies is that the representation of bottom friction has a significant influence on the mean circulation. However, the nature of this dependence is not fully understood. For example, Loder (1980) suggested that nonlinearity in the bottom stress law results in an increase in the along-isobath mean current because of enhanced spatial gradients in the bottom stress. In contrast, Huthnance (1981) showed, based on an integral constraint on the fluid motion, that the rectification of a rectilinear tidal current perpendicular to isobaths results in a significantly weaker mean current when a quadratic stress law is used instead of a linear relation between bottom stress and velocity. The primary purpose of this study is to clarify the influences of the commonly-used quadratic stress law on the along-isobath mean current in depth-independent (i.e. with vertical structure in the horizontal currents neglected) models of topographic tidal rectification and in so doing resolve the apparent inconsistency between Loder’s (1980) and Huthnance’s (1981) results.

2. MODEL FORMULATION AND BASIC RESULTS

We consider the rectification of tidal currents in homogeneous water over a topographic feature of depth H = H ( x ) . Beginning with the nonlinear, two-dimensional shallow water equations, and making the assumptions (Le /LH)2 < 1 and LH/L,<l (Le, LH and L, are the tidal excursion, topographic length-scale and tidal length-scale, respectively), we show in the Appendix that the depth-averaged tidal equations may be approximated by


RECTIFICATION OF TIDAL CURRENTS 229

ut - f v = -gz, - ( p H ) - 'R , (2.2)

r: + f u = -gz, - ( p H ) - 's, (2.3)

where U , V are the x-, y- (cross-isobath, along-isobath) components of tidal velocity, Z is the tidal surface elevation, (R ,S ) is the bottom stress at the tidal frequency, p is the water density, f is the Coriolis parameter, g is the acceleration due to gravity, and subscripts x, y, t, indicate partial differentiation. Each of the variables in (2.1)-(2.3) varies only slowly (i.e. on the tidal scale, LT) in the along-isobath direction and hence may be approximated as y-independent over the topographic scale, L,, of interest here (see the Appendix for a more rigorous justification). Further, the derivation presented in the Appendix shows that Z,, = 0 to the same approximation [see (A.6)], a fact which we use below.

The corresponding depth-averaged mean equations (again see the Appendix) are

H - ' ( H E ) , - f v = -g[, - (pH) - ' r, (2.5)

where (U, 17) is the mean horizontal current, r, is the cross-isobath component of mean surface elevation gradient, (F,F) is the mean bottom stress, and the overbar indicates an average over the tidal period.

Differentiating (2.3) with respect to x and using Z,,xO [or cross- differentiating (2.2) and (2.3) and repeatedly using d/dy 2501, then integrating with respect to time, we obtain

t t

V,= -fS U,dt ' -J(S/pH) ,d t ' ,

which is a tidal vorticity equation showing the production of tidal vorticity by planetary vortex stretching and differential bottom friction. Note that, in the absence of friction, the cross-isobath changes in V are in quadrature with U C(2.1) implies that U,=


230 D. G. WRIGHT AND J. W. LODER

( - H , / H ) U ] so that (HUT/),=O and, from (2.6), there is no along- isobath mean bottom stress; thus, friction is essential for the rectification mechanism examined here. Further, using continuity again to eliminate U , from (2.7) and substituting the resulting expression for V, in (2.6) yields

or

t t

S/H= Uf(S/H),dt'= -f U d t ' ( S / H ) ,

s= - H q ( S / H ) , , (2.9)

where r] = f U dt' is the cross-isobath tidal displacement of a water column. Equation (2.9) is a simple relation expressing the mean bottom stress in terms of local tidal quantities.

Before discussing the implications of (2.9) it is instructive to relate this result to Huthnance's (1981) results. For small-amplitude (Ro= U*/fL,G 1, where U* is a typical tidal current amplitude) barotropic wave motion over arbitrary bottom topography, and weak friction ( S * / p f H U * G 1, where S* is a typical amplitude of the tidal bottom stress), Huthnance (1981) showed that the circulation adjusts so that the line integral of the bottom stress divided by the depth following the Lagrangian fluid circuit averages to zero in time; i.e.

where a,, is the total bottom stress (mean plus tidal) and dl is a line element along the fluid circuit. When the tidal displacement r] is small compared to the horizontal scale on which a,/H varies, a truncated Taylor series expansion of ab/H is valid and (2.10) gives

H ~ ' ( S + $ + [ H - ' ( S + S)],q 7zo (2.1 1)

or

s z = H q ( S / H ) , , (2.9')

which is precisely (2.9). The equivalence of these two formulae


RECTIFICATION OF TIDAL CURRENTS 23 1

indicates that the difference in results obtained by Loder [using the idealized model from which (2.9) is derived] and Huthnance [using the approach leading to (2.9)] is due entirely to the choice of tidal current parameters and the parameterizations of S and S.

To identify the stress parameterization characteristics that influence the mean current, consider the general forms

F=k(x)C, S=[K(x,t)V]" (2.12,2.13)

where [*Iw represents the contribution from the enclosed quantity at the tidal frequency, o. The consideration of these forms is based on the premise that the effects of nonlinear friction can be viewed as occurring through spatial gradients and time-dependence in the effective coefficients k and K . Noting that contributions to S from frequencies other than w do not contribute to (2.9), and hence substituting S = K ( x , t ) V and S= k(x)C in (2.9) and expanding, we have

(2.14)

Further, for weak friction,

from (2.7, which allows (2.14) to be rewritten as - _ _ -

C = k - ' { - f ( H , / H ) K q 2 +(H,/H)KVq -K ,Vq} . (2.16)

(a) (b) (4 This expression confirms that the mean current arising from topographic tidal rectification is influenced by spatial gradients (through the third term) and time variations (through all three terms) in the effective friction coefficient in the tidal equation, as well as by the relative magnitudes of this coefficient and its counterpart in the mean equation.

Frequently, k and K are assumed equal and approximated by a constant. Then (2.16) reduces to



which suggests that 6 is independent of the (weak) friction. It is important to realize that this apparent insensitivity to bottom stress is potentially misleading. Equation (2.16) clearly demonstrates that the form of the friction coefficient (reflected in the amplitudes and phases of both K / k and K,/k), as well as the local topography, are critical in the determination of 17, so that the assumption of constant friction coefficients is generally inappropriate. We also note that the limit of no friction is singular in that the mean current is strongest for friction tending to zero (Zimmerman, 1980) yet, as discussed above, it vanishes identically if friction is neglected from the outset.

Finally, a significant Stokes drift can be associated with the rectification of tidal currents. In the present case, the Stokes velocity can be estimated from the tidal velocity using the linearized expression (Longuet-Higgins, 1969)

where (2.1) and (2.15) have been used in the last step. Note that us is independent of the bottom stress and that for K / k = 1 (locally), the first term in (2.16) is just the negative of the Stokes velocity and hence does not contribute to the Lagrangian velocity.

3. EFFECTS OF THE NONLINEAR STRESS LAW

We now consider some examples which illustrate the importance of the influences apparent from (2.16). In each case we assume that the total bottom stress is of the form

where u is the total (mean plus tidal) velocity and r is a friction coefficient whose form varies from case to case, and which may be a function of both space and time. When a quadratic stress law is assumed, r=cDlul where cD is a drag coefficient taken to be 2.5 x The successive cases presented below are intended to



approximate this form of r with increasing levels of sophistication, starting with the simplest case of spatial gradients only (3a) and proceeding to include time-dependence (3b) and finally the fully nonlinear stress law (3c). For each form of r considered, solutions for some typical parameter values are given in Table 1 and numerical solutions for the special case R [ = amp (V)/amp ( U ) ] = f/w = 0.7 (typical of semi diurnal tides at mid-latitudes in shallow water away from coastlines) are presented in Figure 1. Estimates of the results for other parameter values may be determined from Table 1, using Figure 1 as a guide for interpolation in 4.

3a. The influence of spatial gradients in r

We initially consider (Figure la; Table 1: case 1) a time- independent friction coefficient of the general form r = k,(H,/H)" where the subscript 0 indicates evaluation at the local position and CI is a constant which determines the rate of change of r with H . [As a first approximation to the quadratic stress law we could take k,=c,Q, where Qo is the local mean tidal current speed, as in case (1) of Table I]. For this general form of r we have

k = K = k,(H,/H)", ( 3 4 and (2.16) gives

v= (H,/H)[ - fr'+ (1 + C I ) i q ] . (3.3)

Note that (3.3) is independent of k , but depends on the spatial gradients in Y through a. Taking a=O corresponds to a purely linear stress law while the contribution of U [see (2.1)] to spatial gradients in r can be represented with CI= 1.

Now, consider local tidal currents of the form

(UO, V,) = u, [cos(wt), 08 cos(wt + 41, (3.4)

where U , is the local amplitude of the cross-isobath tidal current and 4 is the phase lead of V relative to U. Equation (3.3) may then be rewritten as



Figure 1 Model prediction of the nondimensional along-isobath mean Eulerian current for R = f/o=0.69 with five different parameterizations of the bottom stress: (a) r = k , (......), r=koHo/H ( - - - - - - ) , r=cDQ (.---.); (b) r=cDQo(t)H,/H (---), r = cDQ (-), r =cDq (-). The currents are nondimensionalized by the velocity scale -+(f/o) ( H , / H ) (U,&o)U, which has an approximate value of 3.Ocms-' at Station D on the northwestern side of Georges Bank.


Table 1 C

omparison of the along-isobath m

ean Eulerian currents for the examples discussed in the text. A

ll quantities are normalized by

-f(flo)(H,/H

)(UJm

)U,. W

ith this normalization the Stokes current, &, is

-1. The three term

s in each case correspond to the three term

s in (2.16). The parenthesized factors in the third terms for q5= kn/2 correspond to the effective value of a as defined in the text.

4= fn/2 +=O,n R

-arbitrary R=O

.O

R=0.5

R=

1.0 R

=1.5

(4 1

1 1

1 1

(4 ~

-

f 0.5aw

/ f f aw

/f f 1.5aolf

___ f 0.5

4 f

* wlf

f 1.5w

/ f (1) r

=c

D~

O(H

dH

)’ (b)

-

1 1

1 1

-

f0.5wlf

f d

f f 1.504 f

N

(2) (4

1 r =

cDQ

(x) (b)

-

(3) r=

c~Q

o(t)(Ho/W

’ (b) -

r = cD

Q(x, t)

(b) -

-

f 0.42w

/ f f 4

f f 1.654 f

(4)

r = cD

q(x, t) (b)

-

f0.3

34

f f 0.670/ f

f w

lf (5)

W

ul

(4 ~

-

+0.5(0.74 f0.52flo)wlf

f(0.5 k 0.5 f /w)co/f

f 1.5(0.34f0.43flo)w

/f

(a) 0.67

0.67 0.84

1 1.10

(4 -

-

f 0.42aw

/ f f aolf

f 1.65a0.4 f

(a) 0.67

0.67 0.84

1 1.10

(c) 0.67R

2/(1 + Rz)

-

+0.42(0.5f0.98f/w)w

/ f f(0.25 fO

.75flw)w

lf f 1.65(0.15f0.57f/o)o/J

(a) 0.67( 1 + R

2)/( 1 + 2R2) 0.67

0.66 0.67

0.67

(c) 0.67R

2/(1+2R2)

~ f 0.33(0.5 f

0.98 f/o)w

/f f0.67(0.25 f0.75 f/w

)w/ f

& l(0.15 k0.57 f/cu)o/ f

-

0.420/ f f w

lf f 1.6501 f

-



In a study of tidal rectification over the sides of Georges Bank, Loder (1980) took r = cDU,HO/H, corresponding to c1 = 1. For 4 % 4 2 and R z f / o (typical of that area) spatial gradients in r then increase the along-isobath mean Eulerian current by about 50 % (e.g. Figure la) and the Lagrangian current by 100% [as anticipated, the first term in (3.5) is just the negative of the Stokes velocity and hence does not contribute to the Lagrangian current]. However, for R=O [the example considered by Huthnance (1981) following his equation (5.4)], (3.5) is independent of a. Hence spatial gradients in r cannot explain the reduction in V by a factor of 5 which Huthnance found for R=O when linear friction ( r=cDQo) is replaced by fully nonlinear friction ( r = ~ ~ 1 ~ 1 ) .

It is of interest to consider the more realistic, but still time- independent, parameterization r=c ,Q(x) where Q = ( U z + VZ)'/', as in case (2) of Table 1. Then

k = K = cDQ(x) (3.6)

K , = - K(H, /H)CQ-1(U2 - fvq) l /Q, (3.7)

and

where we have used (2.1) and (2.15). Thus the local spatial variations in K can be described by writing

= c D Q O ( H O / H ) " , (3.8a)

where c1 is a constant given by

c1= CQ - '( U 2 - f vr)l/Q 10. (3.8b)

The latter expression gives the local value of c1 determined by the tidal dynamics, and can be substituted in (3.5) to evaluate V. For the examples presented in Table 1, this value appears as the parenthesized factor in the third term (c) of the solutions for 4= &z/2. The first term in (3.8b) represents spatial gradients in U determined by (2.1). It is less than or equal to 1 and decreases with increasing R since U 2 / Q < Q for RZO. The second term, which can be written as ( f /o) R sin 4(sinz wt/Q)/Q, changes sign following sin 4 but is only weakly dependent on R for 4= fn /2 [see the parenthesized part of the third term for case (2) in Table 11. Thus this term increases c1

near 4=n/2 but equally decreases it near $= -n/2. This occurs



because, as U increases towards shallow water, the associated increments in V lead U by approximately n/2 for weak friction. For 4=n/2 locally, these increments reinforce V while, for 4= -n/2 locally, they oppose 1/: This is the origin of the asymmetry about 4 = 0 in the results in Figure la for r = c,Q(x).

The primary conclusions from these results for time-independent r are that (1) spatial gradients in r significantly affect the mean current associated with tidal rectification, (2) assuming r inversely propor- tional to H is a good approximation to these gradients for some but not all parameter values and (3) the influence of spatial gradients does not account for the reduction of 6 in Huthnance's (1981) example.

3b. The influence of time-dependence in r

We next consider a friction coefficient of the form r =c,Q,(t)(H,/H)" which allows realistic temporal variations and specified spatial dependence (Figure lb; Table 1, case 3). The appropriate tidal and mean friction coefficients are then

and

k =Y( V + g/V= C D Q O ( H O / H ) " , (3.9b)

with the latter expression following because r does not have a component at the tidal frequency so v = 0 . In this case, k is equal to the mean value of K. Substitution of (3.9) in (2.16) now yields

(3.10)

which is directly comparable with (3.3). The results in Figure 1 for r=c ,QoHo/H and r=cDQo(t)HO/H

suggest that time-dependence tends to reduce the magnitude of the along-isobath mean current at least for 4= $-n/2. Comparison of cases (1) and (3) in Table 1 indicates that this conclusion is valid for rectilinear currents (4=0,7t or R=O) for all R, but for 4= &n/2 the time-dependence reduces 1151 for R < 1 and increases it for Iw > 1. For R= 1, Qo is constant so r is time-independent for both parameteriza-



tions. The two-thirds reduction of 117 for rectilinear currents (Table 1) agrees with Huthnance’s (1981) example and reflects a reduced contribution from the first term in (3.10) compared to the corresponding term in (3.3); the second term in both (3.3) and (3.10) vanishes identically for rectilinear currents. The reduction of the first term for the assumed time-dependence is due to q being small when Qo (and hence r and K ) is large so that the effective friction acting on the tides is reduced. For 4 = &n/2 an analogous argument leads to the conclusion that the effective friction acting on the tides is reduced for R < 1 but increased for R> 1 due to the assumed time- dependence.

As in the previous case of time-independent r, it is of interest to consider the spatial dependence determined by the tidal dynamics. Consider then, r =c,Q(x, t) (Figure 1 b; Table 1, case 4), for which the appropriate tidal and mean friction coefficients are

K=cDQ(x, t), k=cDQ(x). (3.11,3.12)

Using (2.1) and (2.15) we find that

Substituting (3.11)-(3.13) in (2.16) then yields

which can be compared with the earlier expressions (3.3) and (3.10). We first note that, in contrast to all of the previous examples, the

third term in (3.14) which arises from term (c) in (2.16) gives a non- zero contribution for rectilinear currents if RZO (see Table 1). This is a reflection of the fact that K , is not generally in phase with K for this particular parameterization of bottom stress. The contribution from this term is relatively small for R<OS but for larger R it can be significant and always increases 6 (at R = 1.5 it increases 17 by 69 %).

For non-rectilinear currents we can write (3.14) in the form (3.10) with an effective a given by



again evaluated locally. Comparison of the results for cases (2) and (4) in Table 1 (recall that a is the parenthesized term in the examples 4 = k 4 2 ) indicates that the inclusion of realistic time-dependence in K and K , generally reduces the contribution from the first term in (3.15) but increases the contribution from the second term. The net result depends on the particular tidal parameter values considered. However, we note that the tendency for smaller values of a corresponding to 4= -n/2 compared to 4=n/2 remains and the explanation given in Section 3a for the case r=c,Q(x) applies here as well.

The primary conclusions of this section are that ( 1 ) the inclusion of time-dependence in r significantly alters the along-isobath mean current arising from tidal rectification, with the current magnitude typically being reduced for lR< 1 and increased for lR> 1; and (2) this influence is the origin of the mean current reduction by a factor of two-thirds in Huthnance’s (1981) example, where the influence has its largest possible effect.

3c. The fully nonlinear stress law

In this section we consider the most realistic example of a fully nonlinear stress law with r=c,q where q=IuI is the current speed including both mean and tidal contributions (Figure lb; Table 1, case 5).

Under the assumption that the mean current is sufficiently weak that its influence on the tides can be neglected, the tidal friction coefficient is again given by (3.11). Thus, differences between this example and the case r = cDQ, which includes only tidal contributions to the current speed, are due solely to changes in k-the mean friction coefficient-which can be determined following Heaps (1978):

For 6’ g U 2 + V z , this becomes

where the result that Q has no component at the tidal frequency has

G.A.F.D.-C



again been used. Substituting (3.11), (3.13) and (3.17) in (2.16) we obtain

which is precisely equation (5.4) of Huthnance (1981), in that case derived directly from the present (2.10).

We emphasize that the only difference between this case and the case r=cDQ(x , t ) is the enhanced k resulting from the proper linearization of the fully nonlinear stress law and manifested as the difference between (3.12) and (3.17). The consequent reduction in V is clearly seen upon comparison of cases (4) and ( 5 ) in Table 1 as well as in Figure lb. For both rectilinear ( ~ = O , X , or R=O) and circular (4 = k 4 2 , R = 1) tidal currents this reduction is precisely (1 + R2)/ (1 +2R2), and for the other examples in Table 1 (4 = f 4 2 , R =0.5,1.5) this expression is a good approximation of the effect. Although the influence discussed in this section has no effect for R=O (and hence is irrelevant to the example considered by Huthnance), it is note- worthy that with it included the mean current is reduced by the factor two-thirds for all rectilinear tidal currents when linear friction ( r = c,Q,) is replaced by fully nonlinear friction ( r = c,q).

The primary conclusion of this section is then that the inclusion of mean current contributions in the friction coefficient generally tends to increase the effective mean friction coefficient and hence reduce the along-isobath mean current. The reduction is maximum (50%) for large R and decreases to zero for R = 0.

4. SUMMARY

Under the assumptions of depth-independence, uniformity along isobaths and weak nonlinearity, a simple equation (2.9) relating the mean bottom stress associated with topographic tidal rectification to local tidal and topographic parameters has been derived. It may be interpreted as stating that the depth-distributed stress (f/H) acting on the local along-isobath mean current is just the negative of the time-averaged, depth-distributed along-isobath stress acting on the tidal flow following the fluid motion. The equation is equivalent to


RECTIFICATION OF TIDAL CURRENTS 24 1

the general formula (2.10) derived by Huthnance C1981, equation

Assuming that the bottom stress is parallel to the local velocity, a further expression (2.16) has been obtained which shows that both spatial gradients and time-dependence in the effective friction coefficients can influence the mean current. Also, differences between the mean values of the friction coefficient for the tidal and mean flows can influence the mean current strength. A number of examples have been presented, indicating that these influences of the nonlinearity of bottom friction do indeed have significant effects on the mean current arising from topographic tidal rectification.

For the typical case of clockwise (counterclockwise) rotary tidal currents in the Northern (Southern) Hemisphere, the results confirm Loder’s (1980) suggestion that spatial gradients in the tidal friction coefficient enhance the anticyclonic mean circulation around shallow topographic features and show that this effect is still present when the influences of time-dependence in the coefficient are added. On the other hand for counterclockwise (clockwise) rotary tidal currents in the Northern (Southern) Hemisphere, the effect of spatial gradients is generally weaker and typically reduces (i.e. makes a cyclonic contribution to) the mean current.

Time-dependence in the tidal friction coefficient typically reduces the mean current strength, but when the along-isobath tidal current amplitude exceeds the cross-isobath amplitude, the tendency can be reversed if the current is not rectilinear. The former, more typical influence of time-dependence accounts for the mean current reduction in Huthnance’s (1981) example when linear bottom friction is replaced by a quadratic law. The apparent inconsistency between Loder’s (1980) and Huthnance’s (1981) results thus arises from Loder (1980) considering only one of the influences of nonlinear friction (i.e. spatial gradients in the friction coefficient) and Huthnance (1981) presenting an unrepresentative example (in which the spatial- gradient influence disappears).

When a fully nonlinear stress law is used (i.e. when mean current contributions to the mean friction coefficient are included) the mean friction coefficient is increased and the along-isobath mean current is consequently reduced. The reduction is a maximum (50%) for relatively large along-isobath tidal currents, but vanishes when the local amplitude of this current component is zero.

(3.811.



The general conclusion is that the form of friction has significant quantitative effects on the mean circulation associated with topographic tidal rectification. An important unresolved question is the magnitude of the effects of nonlinear friction when vertical structure in the horizontal currents is included, in which case both bottom and internal friction influence the mean circulation (Wright and Loder, 1985). Preliminary results obtained by Wright and Loder (1987) using a crude representation of nonlinear friction in a depth-dependent model indicate that the influences identified above remain present, but the magnitude of the effects on the mean current is altered and there is somewhat reduced sensitivity to the parameterization of bottom friction.

Acknowledgements

We would like to thank the two reviewers for their constructive criticisms and comments.

References

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RECTIFICATION OF TIDAL CURRENTS 243 Maas, L. R. M., Zimmerman, J. T. F. and Temme, N. M., “On the exact shape of the

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Appendix

DERIVATION OF SIMPLIFIED EQUATIONS OF MOTION

We begin with the shallow water equations for a homogeneous fluid on an f-plane

au/at+u - Vu+f xu+ru/(H+c)= -gVc, (‘4.2)

where the notation is as detailed in the text except that we use vector notation with f = f k where k is a unit vector in the positive z-direction, u is the depth-averaged horizontal velocity and c is the sea surface elevation. We now introduce the following nondimensionalizations based on the tidal flow: H = H*H’, f = of’ , u= U*u’, c=(L$L,) H*C, where H* is a typical depth, o is the tidal frequency, U* is a typical tidal current speed, L,( E U * / o ) is the tidal excursion, and LT[ E ( g H * ) 1 ’ 2 / o ] is a typical horizontal scale on which tidal quantities vary. In addition, motivated by the work of Maas et al. (1987) who considered the more restrictive case of small-



amplitude topography, we explicitly recognize the multiple-scale nature of the problem by writing

where x' defines the short space-scale on which the depth varies and X' defines the long space-scale of tidal variations. As typically done in multiple-scale problems, we proceed formally treating x' and X as independent variables.

Introducing the above nondimensionalizations into (A. 1) and (A.2), defining E = LH/LT, 6 = L,/LH, F = r/wH*, and dropping the primes on nondimensional quantities, we obtain

E aqat + E ~ U au/ax + 6,611 . V,U + ~f x u = -i aC/ax

- ~ V 2 5 - EU/( H + ESC), (A.5)

where V, = [i(a/aX), j(a/aY)] and i, j are unit vectors in the x, y- directions, respectively. It is immediately apparent that if E 4 1 (i.e. relatively abrupt topography, as we assume), then Hu and 5 do not vary on the fast scale LH and hence, to lowest order in E, may be taken as constant over this scale. Indeed, expanding 5 and u in powers of E (e.g. (= lo + 85' + .), substituting in (A.4) and (A.5), and equating powers of E we obtain

auo/at + 6u0 auo/ax + f x UO = - v,i0 - i ap/ax - H - FUO. ( ~ . 9 )

Note that (A.9) implicitly assumes that both 6 and F are larger than E, but they are not otherwise restricted. If 6 4 1, then (2.1)-(2.3) follow immediately from (A.7)-(A.9). Indeed, since motions at frequencies other than w are [to O(E)] driven solely by the advective term in (A.9), we conclude that the amplitudes of both the means and the higher harmonics are at most of order 6 times the amplitudes of



tidal variations. Thus, the contribution from the nonlinear term in (A.9) at the tidal frequency is at most of order 6', and (2.1)-(2.3) are in fact valid under the weaker assumption 6' + 1.

Time-averaging (A.7) over a tidal period and using U = O away from the topographic variation, we obtain Uo =O [i.e. (2.4)], and time-averaging (A.9) we obtain

As noted above, it is apparent that mean and harmonic variations have amplitudes at most of order 6 times the amplitudes of tidal quantities, so neglecting terms of order 6', we may approximate uo and uo in the advective terms in (A.10), (A. l l ) by their tidal contributions. In addition, we note that the advective terms, and hence also 6' and V,co, vanish in the region away (by a tidal excursion) from the topography (e.g. Maas et al., 1987). Finally, (A.6) shows that lo does not vary on the topographic scale, so V,co vanishes over the entire region of interest. Equations (2.5), (2.6) now follow from (A.lO) and (A. 11).


on the influences of nonlinear bottom friction on the topographic rectification of tidal currents

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