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Tidal vorticity over isolated topographic features
ContinentalShelf Research, Vol. 14, No. 13/14, pp. 1583-1599, 1994
~ )
Pergamon
Copyright © 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0278-4343/94 $7.00 + 0.00
Tidal vorticity over isolated topographic features M . - J . P A R K * t a n d DONG-PING W A N G *
(Received 28 February 1991; accepted 23 April 1991) Abstract--The transient and residual transport vorticity of tidal flow over a bump was examined using a vertically-integrated model. Results indicate that two counter-rotating transport vorticities are generated over the bump and the advection of these two vorticities leads to an anticyclonic residual transport vorticity at the center, and two smaller cyclonic residual transport vorticities at the perimeter of the bump. Effects of the residual vorticity on sediment transport were investigated by computing the divergence of bed load. Results indicate that a circular bump tends to deform into an elliptical shape with the major axis inclining cyclonically relative to the tidal stream axis. Results for a bump are compared with those for a hollow of the same shape but with an opposite bottom slope.
1. I N T R O D U C T I O N
IN many coastal areas, the tidal motion is one of the dominant physical processes. It affects the local flow persistently, and as a result, the tidal flow plays an important role in the long term distribution of materials. The tidal residual flow in open sea is induced from the nonlinear advection of the instantaneous vorticity produced by column stretching or shrinking and bottom stress curl over an irregular sea bed (e.g. ZIMMERMAN,1981 ; ROBINSON, 1983). When the tidal flow is studied with a vertically-integrated two-dimensional model, the depth-averaged velocity is obtained from the volume transport divided by a water depth. The instantaneous vorticity is often studied with the vorticity of the depth-averaged velocity, which, however, is neither the depth-mean vorticity integrated over a water column (ZIMMERMAN, 1986), nor the depth-mean vorticity of a water column (volume transport) (PARK and WANG, 1991). This suggests that although the depth-averaged velocity may represent the depth-mean velocity of a water column, the vorticity of the depth-averaged velocity may not properly represent the depth-mean vorticity of a water column. PARK and WANG (1991) investigated the instantaneous and residual vorticity productions by the rectilinear tidal flow over a circular hollow in a straight channel. By comparing the instantaneous (transient) vorticity of the volume transport over a circular hollow with that of the depth-averaged velocity, they clearly indicated the disparity between two vorticities. The transient vorticity of the volume transport showed a cyclonic rotation of a counter-rotating vorticity pair over the hollow, whereas that of the depth-
*Marine Sciences Research Center, State University of New York, Stony Brook, NY 11794, U.S.A. -~'Present address: Department of Oceanography, Chungnam National University, Taejon, Korea, 3/)5-764. 1583
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M.-J. PARKand DONG-PINGWANG
averaged velocity showed a non-rotating vorticity pair. However, after averaging over a tidal cycle, the residual vorticity patterns from two different approaches became similar. The vorticity production over a sand band was studied previously (e.g. ZIMMERMAN, 1981), but the detailed process how the instantaneous vorticity leads to the residual vorticity has not been clearly shown. This may be partly due to the fact that the vorticity of the depth-averaged velocity is insufficient to represent the instantaneous vorticity of a water column. We extend the previous study of PARK and WANG (1991) and examine the transient and residual vorticity productions over a circular bump and compare the results with those for a circular hollow. ZIMMERMAN (1981) reviewed the natural sand bank development in tidal areas and indicated that the sand ridges are of the order of the tidal excursion length and oriented cyclonically (8-15 °) to the tidal stream axis. HUTHNANCE (1982a,b) suggested that a small isolated bump tends to extend across the tidal stream first and an elliptical bank tends to rotate to be cyclonically inclined to the tidal stream axis. However, he assumed steady state, and neglected Coriolis term. PARK and WANe (1991) used the approach of HUTnNANCE (1982a,b) for calculation of sediment transport over a hollow, and suggested that a circular hollow tends to be elliptical with the major axis of the ellipse oriented cyclonically to the tidal stream axis. We extend the previous study to examine how a circular bank changes and discuss the results with those of previous studies. 2. MODEL The generation of transient and residual transport vorticities by tidal flows over a bump is investigated in a straight channel with earth rotation included. The channel is oriented parallel to the x-axis and bounded by two rigid walls. It is 30 km wide, 50 km long and 20 m deep, and opens at both left and right ends. The tidal wave in the channel propagates from left to right (i.e. in the positive x-direction) with the amplitude of 1 m and the velocity of about 0.6 m s -1. The model uses the vertically-integrated equations of motion and the continuity equation, following PRITCHARD (1971). --
OvU
Oif_U+ O u U + _ _ _ f V = at Ox Oy
-gH
O~
U ( U 2 q- V2) 1/2
+ Dr
(1)
OV+ OuV+ a v V + f u = _ g H O ~ _ k V ( U 2 + V2) 1/2 at Ox Oy ay H2 + D~,
(2)
Ox
-k
H2
+ o _u+ o v = o
at
Ox
(3)
ay
where
Dx= A . ~x 2H ~x +
Dy= AH[~-}\
oy[
H
\ay
+
7x
'
ay] a~x[ kay ax]jj'
u = U/H, v = V/H, H = h + ~, U and V are the volume transports in x- and y- directions, is the sea surface elevation, h is the water depth, H is the total water depth (H = h + ~), D X and D s are the horizontal momentum diffusions in x- and y-directions, k is frictional
Tidal vorticityover isolated topographicfeatures
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coefficient (= 0.0025), A/_/is horizontal eddy coefficient (= 100 m 2 s-l), f i s the Coriolis parameter (9.5 × 10 -5 s -1) and g is the gravity acceleration. At the open boundaries, sea levels and currents are specified based on a frictionally damped progressive Kelvin wave solution (see Appendix). At the closed boundaries, the normal velocity is zero. Equations (1), (2) and (3) are solved numerically using a finite element method (WANG, 1975) with a horizontal resolution of 2 km. 3. TIDAL FLOW OVER A LOW BUMP Numerical experiment was first conducted for a low bump. The bump is 22 km wide and has a circular Gaussian shape with a height of 4.9 m. The size of bump is about 2.6 times the tidal excursion length [E = (uT#r) - 8.5 km]. The Rossby [Ro = (u/fL)] and the Ekman [Ek = (ku/fh)] numbers for this experiment are about 0.3 and 0.9, respectively. In other words, the flow is marginally non-linear, but highly dissipative. 3.1. Transient vorticity dynamics Cross-differentiating equations (1) and (2), we obtain the transport vorticity equation, 0~
+u'Va-2
f~ dH
0-T A
B
d-7C
V(
1
dH/xu-f D
=-gJ(H,~)-kVx E
F
+VxD G
(4)
H
where f~ = (OV/Ox - OU/Oy) and J is the Jacobian operator. In equation (4), term A is the local acceleration of transport vorticity, term B is the vorticity advection, term C represents the scattering of vorticity over the topography, term D arises from the column stretching gradient, term E is the vorticity from sea surface divergence, term Fis the topographic vorticity tendency, term G is the friction and term H is the vorticity diffusion. The topographic vorticity tendency represents the vorticity generation when tidal flow crosses the isobath. Expansion of the friction term yields
- k Vx U-~2I - - k t-~ ~ + - ~ Ux V{UI - k lUI ux G
G-1
G-2
(5)
G-3
The first term on the right hand side of equation (5) is the vorticity dissipation by bottom friction, whereas the second and the third terms, the frictional torque, are the vorticity generations due to transport shear and depth gradient, respectively. Equation (4) states that the instantaneous transport vorticity is generated by the topographic vorticity tendency (term F), transport shear (term G-2) and depth gradient (term G-3), and dissipated by bottom friction (term G-I) and horizontal diffusion (term/-/), The instantaneous transport vorticity shows a pair of counter-rotating tidal vorticities rotating cyclonically around the bump (Fig. 1). The vorticity pair is confined to the bump and the maximum vorticity is found at maximum flood and maximum ebb. The vorticity amplitude is significantly reduced at about two hours before either the maximum flood or the maximum ebb [Fig. l(c) and (f)]. The instantaneous transport vorticity is generated by the topographic vorticity tendency and the frictional torque; the Ekman number of 0,9
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M.-J. PARK and DONG-PING WANG
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"F v
'.... ........ • -
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2'0
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o
~
(kin)
2'o
30
Fig. I. Transient transport vorticlty over a low bump; (a)-(f) are in 1/6 tidal cycle interval starting at the time of the maximum ebb current. The solid lines are for the positive and the chain-dotted lines are for zero and the negative values. The contour interval is 10 x IW 5 m s 1. The dotted line is the perimeter of the bump.
suggests that these two terms are comparable for the vorticity generation. To further analyze the vorticity generation, the transient transport vorticity balance is examined at two locations over the bump. Figure 2 shows the topographic vorticity tendency, the vorticity advective term [sum of terms B, C and D of equation (4)], the friction and the
1587
Tidal vorticity over isolated topographic features
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:
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8
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Fig. 2. Transient vorticity balance at the northwest corner (a) and at the center (b) of a low bump. The time starts from about 0.3 h before the maximum ebb. The solid line is for the vorticity advective term, the chain-dotted line is for the topographic vorticity tendency, the dashed line is for the friction, and the dotted line is for the vorticity diffusion. The vorticity generation by the frictional torque at the northwest corner (c) and at the center (d) of a low bump. The solid line is for the friction term [sum of terms G-I, G-2 and G-3 of equation (5)], the chain-dotted line is for the vorticity dissipation by bottom friction [term G-1 of equation (5)], the dashed line is for the vorticity generation by transport shear [term G-2 of equation (5)] and the dotted line is for the vorticity generation by depth gradient [term G-3 of equation (5)].
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M.-J. PARK and DONG-PING WANG
vorticity diffusion over a tidal cycle at the northwest corner (second quadrant) and at the center. The frictional torque associated with transport shear and depth gradient and the vorticity dissipation due to bottom friction are also shown in Fig. 2. At the northwest corner of the bump, the topographic vorticity tendency and the vorticity advective term are small, but are comparable. Also, the frictional torque is much smaller than the topographic vorticity tendency. At the center of the bump, the topographic vorticity tendency is the dominant term, but it is largely linear. The frictional torque due to depth gradient also is large. However, since the frictional torque due to depth gradient is counterbalanced by the frictional torque due to transport shear, the instantaneous transport vorticity over the low bump is generated mainly by the topographic vorticity tendency. The frictional torque tends to reinforce the topographic vorticity tendency at the times of maximum flood and ebb. This implies that the vorticity advection is important at the times of maximum flood and ebb (Fig. 2).
3.2. Residual vorticity dynamics Averaging instantaneous transport vorticity over a tidal cycle, we obtain the residual transport vorticity [Fig. 3(a)]. The pattern shows a negative (anticyclonic) vorticity over the top of the bump and two positive (cyclonic) vorticities at the perimeter. The magnitude of the positive vorticity is about 60% of the negative vorticity, and the major axis of the negative vorticity ellipse inclines cyclonically relative to the tidal stream axis (i.e. x-axis). The amplitude of the residual transport vorticity, that is, difference between peak positive vorticity and peak negative vorticity, is about 16% of the amplitude of the instantaneous vorticity. PARKand WAN~ (1991) showed that the pattern of the residual transport vorticity over a hollow is similar to that of the residual velocity vorticity. The pattern of the residual transport vorticity over a bump is also similar to that of the residual velocity vorticity. On the other hand, the residual transport is deflected anticyclonically as the flow passes over the bump [Fig. 3(b)], whereas the residual current forms an anticyclonic eddy at the top of the bump and two smaller cyclonic eddies at the perimeter [Fig. 3(c)]. The difference between residual transport and residual current is due to the presence of strong Stokes transport. In order to understand how the residual transport vorticity arises, we analyze the residual transport vorticity balance. Averaging equation (4) over a tidal cycle, we obtain the residual transport vorticity equation. This consists of residual vorticity advective term, residual topographic vorticity tendency, residual friction, and residual vorticity diffusion u. Vf~ - 2 1) d_HH_ V H dt H~-
xU + ( g J ( H , ~)) = - k Vx
+ (VxD)
(6)
where ( ) indicates the time-averaging over a semi-diurnal tidal cycle. The residual vorticity advective term is mainly positive at the second and fourth quadrants of the bump and negative at the first and third quadrants; the maximum negative value occurs at the center of the bump [Fig. 4(a)]. We can also compare the instantaneous vorticity balance (Fig. 2) with the residual vorticity balance. For example, at the second quadrant of the bump, the vorticity advective term is largely positive during a tidal cycle and the residual vorticity advective term becomes positive. On the other hand, the vorticity advective term at the center of the bump is mostly negative during a tidal cycle, which results in negative residual
Tidal vorticity over isolated topographic features
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8
r
o.
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i
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(km)
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Fig. 3. (a) Residual transport vorticity over a low bump. The solid lines are for the positive and the chain-dotted lines are for zero and the negative values. The contour interval is 1.5 x 10,5 m s- 1. The dotted line is the perimeter of the bump. (b) Residual transport. The maximum value is 0.34 m; s 1. (c) Residual currents. The maximum velocity is 0.9 cm s -1.
v o r t i c i t y a d v e c t i v e t e r m . T h e r e l a t i v e l y large r e s i d u a l v o r t i c i t y a d v e c t i v e t e r m at t h e s e c o n d a n d f o u r t h q u a d r a n t s o f t h e b u m p reflects t h e i m p o r t a n c e o f v o r t i c i t y a d v e c t i o n d u r i n g t h e m a x i m u m flood a n d e b b . T h e r e s i d u a l t r a n s p o r t v o r t i c i t y b a l a n c e is m a i n l y b e t w e e n t h e r e s i d u a l v o r t i c i t y a d v e c t i v e t e r m [Fig. 4(a)] a n d t h e r e s i d u a l f r i c t i o n [Fig. 4(c)].
1590
M.-J. PARK and DONG-PING WANG (b)
(o) . . . . ......J \ < . \
E o
.?
...... I0
2'0
30
0
10
(kin)
(km)
(c)
(d)
20
30
20
30
,b
.............S.
v o
..?J/. 10
(kin)
2~0
30
0
10
(kin)
Fig. 4. Residualtransport vorticity balance over a low bump. The solid lines are for the positivc and the chain-dotted lines are for zero and the negative values. The contour interval is 10 x 10- J0 m s 2. The dottedlineistheperimeterofthebump. (a)Residualvorticityadvectiveterm;(b)rcsidual topographic vorticity tendency; (c) residual friction; and (d) residual vorticity diffusion.
4. COMPARISON BETWEEN THE LOW BUMP AND THE HIGH BUMP CASES Numerical experiment was also conducted for tidal flow over a high bump. The high b u m p is 22 km wide and has a Gaussian shape with the m a x i m u m height of 9.8 m. Thus, the bottom slope of the high b u m p is twice that of the low bump. The Rossby and E k m a n numbers are about 0.3 and 1.2, respectively. The instantaneous transport vorticity has a pair of counter-rotating vorticity rotating cyclonically over the high b u m p , which is similar to the low b u m p case. The vorticity amplitude over the high b u m p is about 2.4 times the vorticity amplitude over the low bump. The instantaneous vorticity is controlled mainly by the topographic vorticity tendency. The frictional torque reinforces the topographic vorticity tendency at the m a x i m u m flood and ebb, as in the case of a low bump. H o w e v e r , since water is shallower over the high b u m p , the frictional torque becomes m o r e important over the high b u m p than over the low bump. Consequently, the instantaneous transport vorticity over the high b u m p is intensified at the m a x i m u m flood and ebb, and it is significantly reduced at the slack water. This means that the vorticity advection at the m a x i m u m flood and ebb are more important over the high b u m p than over the low bump. By averaging the instan-
Tidal vorticity over isolated topographic features
1591
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(k,~) Fig. 5. (a) Residual transport vorticity over a high bump. The solid lines are for the positive and the chain-dotted lines are for zero and the negative values. The contour interval is 3.0 x 10--5 m s t. The dotted line is the perimeter of the bump. (b) Residual transport. The maximum value is 0.55 m2s-l. (c) Residual currents. The maximum velocity is 2.3 cm s t. t a n e o u s v o r t i c i t y o v e r a tidal cycle, the r e s i d u a l t r a n s p o r t vorticity was o b t a i n e d [Fig. 5(a)]. T h e p o s i t i v e (cyclonic) v o r t i c i t y is c o n f i n e d within the s e c o n d a n d f o u r t h q u a d r a n t s while the n e g a t i v e ( a n t i c y c l o n i c ) v o r t i c i t y f o r m s a b a n d e x t e n d i n g f r o m t h e first to the third q u a d r a n t s with t h e m a x i m u m at the c e n t e r of the b u m p . T h e m a j o r axis of the a n t i c y c l o n i c v o r t i c i t y ellipse inclines c y c l o n i c a l l y r e l a t i v e to t h e tidal s t r e a m axis. T h e a m p l i t u d e o f the r e s i d u a l t r a n s p o r t v o r t i c i t y is a b o u t 14% o f t h e a m p l i t u d e of the i n s t a n t a n e o u s vorticity.
1592
M.-J. PARKand DONG-PINGWANG
The vorticity pattern over the high bump is similar to that over the low bump [Fig. 3(a)], but the magnitude of the anticyclonic vorticity over the high bump is about twice that over the low bump. In addition, the ratio of the positive vorticity at the perimeter to that of the negative vorticity at the center increases from 60% over the low bump to 75% over the high bump. The residual transport is shown in Fig. 5(b); because of the stronger vorticity, the residual transport is larger and it bends more sharply over the high bump than over the low bump. The residual vorticity of the depth-averaged velocity over the high bump is similar to that over the low bump. The anticyclonic vorticity over the high bump is about 2.5 times larger than that over the low bump, and consequently, the anticyclonic eddy is much stronger over the high bump [Fig. 5(c)]. The residual currents show three eddies. It is noted that the increase of residual vorticity and residual currents from the low bump case to the high bump case is comparable to the increase in the bottom slope. The residual transport vorticity balance is between the residual vorticity advective term and the residual friction, as in the low bump case. 5. COMPARISON BETWEEN THE BUMP AND THE HOLLOW CASES Since the bottom slope of a hollow is opposite to that of a bump, it would be useful to compare the results for a bump with those for a hollow. The dynamics of tidal vorticities over a hollow was previously examined by PARK and WANG (1991). The hollow is 22 km wide and has a Gaussian shape with the maximum depth of 38 m. The model conditions other than the hollow geometry are the same as those for the bump; the bottom slope of the hollow is about twice that of the high bump. A counter-rotating transient transport vorticity pair exists both over a bump and over a hollow. As the bottom slope changes sign from bump to hollow, the signs of the transient transport vorticity also change. The transient transport vorticity over a bump is much stronger at the maximum flood and ebb than near the slack water due to the increased frictional torque; in contrast, the transient vorticity over a hollow does not change appreciably through a tidal cycle. Nevertheless, the topographic vorticity tendency is the major source for the transient vorticity generation over the bump and the hollow cases. The residual transport vorticity in both cases is largely controlled by the net vorticity advection. Because of the sign change in the bottom slope, the residual transport vorticity is anticyclonic at the top of a bump, whereas it is cyclonic at the center of a hollow [Fig. 6(a)]. On the other hand, because the frictional torque reinforces the topographic vorticity tendency over the bump, the cyclonic vorticity at the perimeter of the bump becomes comparable to the anticyclonic vorticity at the center. The cyclonic vorticity at the center of the hollow is much stronger than the anticyclonic vorticity at the perimeter, since the vorticity advection is mainly contributed by the topographic vorticity tendency. Also, the major axis of the anticyclonic vorticity ellipse over the bump inclines cyclonically relative to the tidal stream axis, whereas the major axis of the cyclonic vorticity ellipse over the hollow is perpendicular to the tidal stream axis. The pattern of the residual vorticity for both the bump and the hollow cases is similar to that of the residual transport vorticity. The residual transport flows anticycionically over the bump, while it bends cyclonically over the hollow [Fig. 6(b)]. The three residual eddies over the bump are comparable, but the cyclonic eddy at the center of the hollow is much stronger than the anticyclonic eddies at the perimeter [Fig. 6(c)].
Tidal vorticity over isolated topographic features
1593
(a)
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.
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(kin) Fig. 6. (a) Residual transport vorticity over a hollow (from PARK and WANG, 1991). The solid lines are for the positive and the chain-dotted lines are for zero and the negative values. The contour interval is 5 × 10 -5 m s -1. The dotted line is the perimeter of the hollow. (b) Residual transport. The maximum value is 0.57 m 2 s- n. (c) Residual currents. The maximum velocity is 1.7 cm s -] .
6. SEDIMENT DYNAMICS Previous studies (e.g. ZIMMERMAN,1981) found a definite relationship between residual eddy and sand bank. We examine effects of residual eddy on the sediment transport over a bump. In this study, our primary interest is to obtain a qualitative pattern of sand
1594
M.-J. PARKand DON6-PINGWAN6
transport. Following essentially the approach in HUTHNANCE(1982a,b), we assume that the bed load transport rate Qb (volume/unit width/time) is related to [ul 3,
Qb ~ lul2(u + 21ulVh)
(7)
where 2 is k/tan(friction angle). There are some uncertainties in equation (7) (HUTHNANCE, 1982a,b), and quantitative results cannot be expected. However, the qualitative results obtained using equation (7) were found to be fairly robust to the changes in the transport formula (HUTHNANCE,1982a, b). The instantaneous sand transport pattern over a low bump was examined by computing the divergence of bed load using equation (7). The enhancement of the down-slope transport was not considered since the slope of the bump is small (O(10-4)). The instantaneous bed load divergence pattern indicates that the erosion occurs as currents flow over the bump and the deposition occurs as currents pass the bump. This pattern is related to the variation of tidal currents with the water depth that tidal velocity increases toward the top of the bump. The erosion and deposition are of equal magnitude and are mostly confined within the bump. The residual bed load divergence, obtained from averaging the instantaneous value over a tidal cycle, indicates that the deposition occurs mostly at the first and third quadrants of the bump while the erosion occurs at the second and fourth quadrants [Fig. 7(a)]. This will cause a circular bump to change to an elliptical one with the major axis of the ellipse inclining cyclonically relative to the tidal stream axis. HUTHNANCE (1982b) indicated that for a uniform flow, the residual bed load divergence pattern over a small bump is similar to the pattern of the gradient of the residual current along tidal stream axis, O(u)/Ox. Since in our model the far-field tidal current, ut, increases toward the right-hand wall, facing the direction of the tidal wave propagation, the equivalent form is 2(u2)(O(u)/Ox). Indeed, this pattern [Fig. 7(b)] approximates the residual bed load divergence pattern. Also, although the residual bed load divergence pattern is somewhat different from the residual transport vorticity pattern, the deposition tends to occur over the area of negative vorticity and the erosion over the area of positive vorticity. The erosion-deposition pattern over the high bump is similar to that over the low bump. The residual bed load divergence over the high bump shows that the deposition occurs at the first and third quadrants, whereas the erosion occurs at the second and fourth quadrants. However, the major axis of the deposition ellipse over the high bump tends to incline more toward the tidal stream axis due probably to the increased effect of bottom friction. The magnitude of deposition over the high bump increases about three times that over the low bump suggesting the non-linearity of the sediment transport. The deposition and erosion will also cause the circular bank to contract laterally and become elliptic with the major axis of the ellipse inclining cyclonically relative to the tidal stream axis. This result agrees with the observation that the natural sand banks are linear and they are inclined cyclonically to the tidal stream axis (ZIMMERMAN,1981). It also supports the result of HUTHNANCE(1982b) on sand bank formation; a sand bank tends to contract laterally (unless very narrow) and tends to rotate to be cyclonically inclined toward the tidal stream axis. For the hollow case (PARKand WANG, 1991), because of the sign change in bottom slope, the transient and residual bed load divergence have opposite signs to those over a bump. Both circular bump and hollow tend to contract laterally and become elliptic with the major axis of the ellipse inclining cyclonically relative to the tidal stream axis. This
Tidal vorticity over isolated topographic features
1595
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10
30
(km) Fig. 7. (a) Residual bed load divergence over a low bump. (b) The pattern of 2(~)(0(u)/0x). The solid lines are for the positive and the chain-dotted lines are for zero and the negative values. The contour interval is 1.0 × 10-7 mas-3. The dotted line is the perimeter of the bump. The u1 is the far-field tidal current and the ( ) denotes time-averaging over a semi-diurnal tidal period. suggests that b o t h circular b u m p and circular hollow are very unstable bed form configurations and m a y not be f o r m e d in the tidal areas. 7. D I S C U S S I O N A vertically-integrated, finite-element m o d e l was used to study the t o p o g r a p h i c rectification of tidal currents over low and high bumps. T h e model d o m a i n was designed to have typical current velocity - O(1 m s-1), water depth - O(10 m) and size of the b u m p - O(10 km) in the shallow tidal areas (HUTHNANCE, 1982b; ROBINSON, 1983). T h e m a x i m u m tidal residual vorticity is expected w h e n the t o p o g r a p h i c length scale is o r d e r of the tidal excursion length (ZIMMERMAN, 1981; ROBINSON, 1983). The size of our low and high b u m p s and the hollow ( - 2 2 km) is a b o u t 2.6 times the tidal excursion length ( - 8 . 5 km), and this ratio is suitable for the m a x i m u m response. Field observations of the residual vorticity over a sand b a n k are scarce, but the m o d e l result m a y be c o m p a r e d to the analytic solution. T h e m a x i m u m tidal residual vorticity of the d e p t h - a v e r a g e d velocity over the high b u m p m a y be estimated f r o m ROBINSON (1983); WO - - 0 . 4 5 ( f u / o h ) ( O h / O s )
--
0.8 X 10 -5 f r o m the c o l u m n stretching
1596
M.-J. PARKand DONG-PINGWANG
~Oo ~ 0 . 4 ( k / o ) ( O ( u / h ) / O n ) ~ 0.6 x 10 .5 from the frictional torque
where cr is the tidal frequency, s is the distance measured along the tidal stream and n is the normal. The residual vorticity from our model (not shown; PARK, 1990) is about 1.0 x 10 .5 , which agrees reasonably well with the analytic solution. The results on the low bank and the hollow also showed similar reasonable agreements with the analytic solutions. The tidal current asymmetry and mean anticyclonic circulation around the elliptic sand bank were noticed in the shallow tidal seas (CAsTON and Sa'RIDE, 1970). Our results show that the anticyclonic residual eddy develops over a circular bank, but it is elliptic rather than circular. The results also indicate that as the height of b u m p increases, the strength of transient and residual transport vorticities increases in proportion to the increasing bottom slope. Also, as the water depth becomes shallower, the relative importance of the frictional torque to the topographic vorticity tendency becomes greater. Consequently, the cyclonic residual transport vorticity at the perimeter of the high b u m p becomes comparable to the anticyclonic residual vorticity at the center. Comparison between the b u m p and the hollow case indicates that apart from a sign change, the transient and residual transport vorticities look similar. However, due to the effect of frictional torque, the transient transport vorticity over a b u m p is significantly stronger at the m a x i m u m flood and ebb c o m p a r e d to other parts of the tidal cycle. On the other hand, the strength of the transient transport vorticity over a hollow does not change appreciably through a tidal cycle. Also, unlike the b u m p cases, the residual transport vorticity at the center of the hollow is much stronger than that at the perimeter. Due to the effect of the increased frictional torque the major axis of the anticyclonic residual vorticity over a b u m p inclines cyclonically relative to the tidal stream axis, whereas the major axis of the cyclonic residual vorticity over a hollow is perpendicular to the tidal stream axis. Unlike the hollow case, over a bump, the deposition tends to be associated with the anticyclonic residual vorticity and the erosion with the cyclonic residual vorticity (HurHNANCE, 1982b). The present model uses a depth-averaged velocity to calculate sediment transport. Thus, the E k m a n veering effect (MCCAVE, 1979) can not be precisely represented, but the effect of b o t t o m friction is incorporated in the depth-averaged velocity. O t h e r possible effects from the secondary currents (HEATHERSHAWand HAMMOND, 1980) and transport lag (KENNEDY, 1969) may be less important than the primary distortion of the depth-averaged current (HUTHNANCE, 1982a, b) and were not considered in this study. Although the present study deals with the rectilinear tidal currents, the results on sand bank evolution by the elliptical tidal currents may be similar (HuTI4NANCE, 1982a). Acknowledgements--The computation of this study was supported by the Cornell National Supercomputer
Facility. The first author wishes to thank the partial support from Chungnam National University, Korea.
REFERENCES CASTONV. N. D. and A. H. STRIDE(1970) Tidal sand movement between some linear banks in the North Sea off northeast Norfolk. Marine Geology, 9, M38-M42. HEATHERSHAWA. D. and F. D. C. HAMMOND(1980) Secondary circulation near sand banks and in coastal embayments. Deutsche Hydrographische Zeitschrift, 33,135-151. HUTHNANCEJ. M. (1982a) On one mechanism forming linear sand banks. Estuarine, Coastal and Shelf Science, 14, 79-89.
Tidal vorticity over isolated topographic features
1597
HUTHNANCEJ. M. (1982b) On the formation of sand banks of finite extent. Estuarine, Coastal and Shelf Science, 15, 277-299. KENNEDY J. F. (1969) The formation of sediment ripples, dunes and antidunes. Annual Reviews of Fluid Mechanics, l, 147-168. McCAVE I. N. (1979) Tidal currents at the North Hinder Lightship, southern North Sea; flow directions and turbulence in relation to maintenance of sand banks. Marine Geology, 31,101-114. PARKM.-J. (1990) Transient tidal vorticity in coastal seas. Ph.D. Thesis, Marine Sciences Research Center. State University of New York, Stony Brook, 105 pp. ])ARK M.-J. and D.-P. WANG (1991) Transient tidal vorticity over a hollow. In: Tidal hydrodynamics, B. B. PARKER, editor, John Wiley and Sons, New York, pp. 419-434. PR1TCHARDD. W. (1971) Hydrodynamic Models. In: Estuarine modelling: an assessment, G. H. WARD,JR and W. H. ESPEV, JR, editors, Water Quality Office, U. S. Environmental Protection Agency, pp. 5-33. ROBINSON I. S. (1983) Tidally induced residual flows. In: Physical oceanography o f coastal and shelf seas, B. JOHNS, editor, Elsevier, Amsterdam, pp. 321-356. WANG H. P. (1975) Modelling an ocean pond; a two-dimensional finite element hydrodynamic model of Ninigret Pond, Charlestown, Rhode Island. University of Rhode Island Marine Technical Report 40, 60 pp. ZIMMERMAN Z. T. F. (1981) Dynamics, diffusion and geomorphological significance of tidal residual eddies. Nature, 290, 549-555. Z1MMERMANZ. T. F. (1986) Principal differences between 2D- and vertically averaged 3D- models of topographic tidal rectification. In: Physics of shallow estuaries and bays, J. VAN DE KREEKE, editor, Springer-Verlag, New York, pp. 120-129.
APPENDIX
Open boundary condition 1. Analytical solution. For the solution of a frictionally-damped progressive Kelvin wave in a rotating channel of uniform depth and width, we consider the linearized depth-averaged equations of motion and the continuity equation. The channel is oriented parallel to the x-axis and is bounded by two rigid walls in the y-axis. The wave propagates in the positive x-direction and the normal velocity vanishes at the rigid walls. Assuming ~ << h, thc equations are Ou Ot -
O~ g ~x
Fu
fu = - g Uyy Ou ax
l O~ h at
(8)
(9)
(10)
where F is a linearized friction coefficient. We look for a progressive wave solution periodic in x and t, and we let
= Re{ ~oa(x)fl(y) e i(k ..... )}
( I 1)
where ~o is the initial wave amplitude before entering the channel, a(x) is a functmn to account for the frictional damping of the wave along the channel, fl(y) is a function to account for the cross-channel variation of the wave due to the Earth's rotation, k is the wave number and cr is the wave frequency. Manipulation of equations (8) and (10) yields a wave equation,
O2¢+FO¢ Ot2
202~-- 0
0 - 7 - C° OX ~ --
w'here C o = "v/~. Substituting equation (11) into equation (12), we get
(121
1598
M.-J. PARK a n d DONG-PING WANG
O2a 4- (k~ -- k2)a = 0
(13)
2 k a O_a_~+ F k 2 a = 0 Ox
(14)
07
where ko is the initial wave n u m b e r ( = a/Co). The solutions for a in equations (13) and (14) are a(x) = e -&x,
k I = X/k2~
(15)
a ( x ) = e -iqx,
~:1 = Fk2°" 2ka
(16)
From equations (15) and (16), we obtain an equation for k 402k 4 - 4 O 2 k o 2k2 - F 2k4o = 0 .
(17)
k = _ ~ ( . X ~ l 4- (14/0) 2 4- 1) 1/2.
(18)
The solution for k in equation (17) is
Using equation (18) in equation (15), we get
k 1 = -~-~(X/14- (F/a) 2 - 1) 1/2.
(L9)
Thus, the wave solution we look for is = ~_,,e - k , x f l ( y ) COS ( k x - a t )
(20)
where k = ~(VI~(F~
d 4- 1) 1/2
and /,kl = ~22(X/1-t- (F/o) 2 - 1) I/2. Substituting ~ of equation (20) in equation (10), we get
GC,k,, e u =
k'~fi(Y).cos ( k x - at - ~ )
hV, k 2 + k 2
(2l)
,
where q~ = tan -1 ( k l / k ) . Using the solutions of ¢ [equation (20)] and u [equation (21)] in equation (9) and assuming that qb << I, we obtain fl(y) = e-/Ky/c,,
ko
K = ,/k2-T~ 1
(22)
W h e n the bottom friction is negligible, i.e. F - 0, then k 1 - 0, and K - i. Thus, we get fl(y) = eo-.ty/( , which is the cross-channel variation function of the Kelvin wave. Using fl(y) in equation (22), we find the solutions for a frictionally-damped Kelvin wave in a straight rotating channel, ¢ = ¢o e kl* e /Ky/c cos ( k x - ot
)
u = ; ~,,CoK e -k,x e-fKylC,, cos ( k x - at -- ~ ) h
v=0 where
(23) (24)
(25)
Tidal vorticity over isolated topographic features
(a)
1599
o .
.
.
:-.....
.
,, ,.. . . . . . . . . 0
10
,.. ;
30
20
40
,50
(km) 0
(b)
"::
0N '
E
........
: : - - , k , 4 , , - - - " • k k 4 # 4 4 4 4
~
"""""
v
~
~
V ~ I 4
0
0
~ A A & 4 A ~ 4 4 4 4
: ~--~ k 4
~1 4
~
110
, 4 . 4 , , d 4 4 4
4
d
4
4
4
~
2~0
4
~
3'0
4
4
4
4
4
~ 4 4 4 4
410
5O
(km) Fig. 8. Residual currents (a) when the uniform sea level is applied at the open boundaries and (b) when the analytical solution is applied with a slight adjustment. The background residual current velocities arc 0 (10) cm s I for (a) and 0 (0.1) cm s -I for (b).
k = (~/1X/2 k° + (FI0)2)2+ 1)L/2,
k] = V~2 (X/f + (F/a) 2 - 1) ]/2,
K- X/~'k°
and qb = tan I (kl/k)" 2. Open boundary condition. When the model is driven by a uniform sea level of 1 m amplitude at both ends of the channel, it produces large noise around open boundaries. With the dynamically consistent frictionallydamped Kelvin wave solution [equations (23)-(25)], the model rcmoves most of the large noise, but small noisc is still present near the open boundary. The noise is eliminated with the slight adjustment of the analytical solution. In the case of using constant sea level at the open boundaries, large half-circle residual eddies are generated near the boundaries. Thcsc eddies are completely removed when thc analytical solution is applied (Fig. g).
~ )
Pergamon
Copyright © 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0278-4343/94 $7.00 + 0.00
Tidal vorticity over isolated topographic features M . - J . P A R K * t a n d DONG-PING W A N G *
(Received 28 February 1991; accepted 23 April 1991) Abstract--The transient and residual transport vorticity of tidal flow over a bump was examined using a vertically-integrated model. Results indicate that two counter-rotating transport vorticities are generated over the bump and the advection of these two vorticities leads to an anticyclonic residual transport vorticity at the center, and two smaller cyclonic residual transport vorticities at the perimeter of the bump. Effects of the residual vorticity on sediment transport were investigated by computing the divergence of bed load. Results indicate that a circular bump tends to deform into an elliptical shape with the major axis inclining cyclonically relative to the tidal stream axis. Results for a bump are compared with those for a hollow of the same shape but with an opposite bottom slope.
1. I N T R O D U C T I O N
IN many coastal areas, the tidal motion is one of the dominant physical processes. It affects the local flow persistently, and as a result, the tidal flow plays an important role in the long term distribution of materials. The tidal residual flow in open sea is induced from the nonlinear advection of the instantaneous vorticity produced by column stretching or shrinking and bottom stress curl over an irregular sea bed (e.g. ZIMMERMAN,1981 ; ROBINSON, 1983). When the tidal flow is studied with a vertically-integrated two-dimensional model, the depth-averaged velocity is obtained from the volume transport divided by a water depth. The instantaneous vorticity is often studied with the vorticity of the depth-averaged velocity, which, however, is neither the depth-mean vorticity integrated over a water column (ZIMMERMAN, 1986), nor the depth-mean vorticity of a water column (volume transport) (PARK and WANG, 1991). This suggests that although the depth-averaged velocity may represent the depth-mean velocity of a water column, the vorticity of the depth-averaged velocity may not properly represent the depth-mean vorticity of a water column. PARK and WANG (1991) investigated the instantaneous and residual vorticity productions by the rectilinear tidal flow over a circular hollow in a straight channel. By comparing the instantaneous (transient) vorticity of the volume transport over a circular hollow with that of the depth-averaged velocity, they clearly indicated the disparity between two vorticities. The transient vorticity of the volume transport showed a cyclonic rotation of a counter-rotating vorticity pair over the hollow, whereas that of the depth-
*Marine Sciences Research Center, State University of New York, Stony Brook, NY 11794, U.S.A. -~'Present address: Department of Oceanography, Chungnam National University, Taejon, Korea, 3/)5-764. 1583
1584
M.-J. PARKand DONG-PINGWANG
averaged velocity showed a non-rotating vorticity pair. However, after averaging over a tidal cycle, the residual vorticity patterns from two different approaches became similar. The vorticity production over a sand band was studied previously (e.g. ZIMMERMAN, 1981), but the detailed process how the instantaneous vorticity leads to the residual vorticity has not been clearly shown. This may be partly due to the fact that the vorticity of the depth-averaged velocity is insufficient to represent the instantaneous vorticity of a water column. We extend the previous study of PARK and WANG (1991) and examine the transient and residual vorticity productions over a circular bump and compare the results with those for a circular hollow. ZIMMERMAN (1981) reviewed the natural sand bank development in tidal areas and indicated that the sand ridges are of the order of the tidal excursion length and oriented cyclonically (8-15 °) to the tidal stream axis. HUTHNANCE (1982a,b) suggested that a small isolated bump tends to extend across the tidal stream first and an elliptical bank tends to rotate to be cyclonically inclined to the tidal stream axis. However, he assumed steady state, and neglected Coriolis term. PARK and WANe (1991) used the approach of HUTnNANCE (1982a,b) for calculation of sediment transport over a hollow, and suggested that a circular hollow tends to be elliptical with the major axis of the ellipse oriented cyclonically to the tidal stream axis. We extend the previous study to examine how a circular bank changes and discuss the results with those of previous studies. 2. MODEL The generation of transient and residual transport vorticities by tidal flows over a bump is investigated in a straight channel with earth rotation included. The channel is oriented parallel to the x-axis and bounded by two rigid walls. It is 30 km wide, 50 km long and 20 m deep, and opens at both left and right ends. The tidal wave in the channel propagates from left to right (i.e. in the positive x-direction) with the amplitude of 1 m and the velocity of about 0.6 m s -1. The model uses the vertically-integrated equations of motion and the continuity equation, following PRITCHARD (1971). --
OvU
Oif_U+ O u U + _ _ _ f V = at Ox Oy
-gH
O~
U ( U 2 q- V2) 1/2
+ Dr
(1)
OV+ OuV+ a v V + f u = _ g H O ~ _ k V ( U 2 + V2) 1/2 at Ox Oy ay H2 + D~,
(2)
Ox
-k
H2
+ o _u+ o v = o
at
Ox
(3)
ay
where
Dx= A . ~x 2H ~x +
Dy= AH[~-}\
oy[
H
\ay
+
7x
'
ay] a~x[ kay ax]jj'
u = U/H, v = V/H, H = h + ~, U and V are the volume transports in x- and y- directions, is the sea surface elevation, h is the water depth, H is the total water depth (H = h + ~), D X and D s are the horizontal momentum diffusions in x- and y-directions, k is frictional
Tidal vorticityover isolated topographicfeatures
1585
coefficient (= 0.0025), A/_/is horizontal eddy coefficient (= 100 m 2 s-l), f i s the Coriolis parameter (9.5 × 10 -5 s -1) and g is the gravity acceleration. At the open boundaries, sea levels and currents are specified based on a frictionally damped progressive Kelvin wave solution (see Appendix). At the closed boundaries, the normal velocity is zero. Equations (1), (2) and (3) are solved numerically using a finite element method (WANG, 1975) with a horizontal resolution of 2 km. 3. TIDAL FLOW OVER A LOW BUMP Numerical experiment was first conducted for a low bump. The bump is 22 km wide and has a circular Gaussian shape with a height of 4.9 m. The size of bump is about 2.6 times the tidal excursion length [E = (uT#r) - 8.5 km]. The Rossby [Ro = (u/fL)] and the Ekman [Ek = (ku/fh)] numbers for this experiment are about 0.3 and 0.9, respectively. In other words, the flow is marginally non-linear, but highly dissipative. 3.1. Transient vorticity dynamics Cross-differentiating equations (1) and (2), we obtain the transport vorticity equation, 0~
+u'Va-2
f~ dH
0-T A
B
d-7C
V(
1
dH/xu-f D
=-gJ(H,~)-kVx E
F
+VxD G
(4)
H
where f~ = (OV/Ox - OU/Oy) and J is the Jacobian operator. In equation (4), term A is the local acceleration of transport vorticity, term B is the vorticity advection, term C represents the scattering of vorticity over the topography, term D arises from the column stretching gradient, term E is the vorticity from sea surface divergence, term Fis the topographic vorticity tendency, term G is the friction and term H is the vorticity diffusion. The topographic vorticity tendency represents the vorticity generation when tidal flow crosses the isobath. Expansion of the friction term yields
- k Vx U-~2I - - k t-~ ~ + - ~ Ux V{UI - k lUI ux G
G-1
G-2
(5)
G-3
The first term on the right hand side of equation (5) is the vorticity dissipation by bottom friction, whereas the second and the third terms, the frictional torque, are the vorticity generations due to transport shear and depth gradient, respectively. Equation (4) states that the instantaneous transport vorticity is generated by the topographic vorticity tendency (term F), transport shear (term G-2) and depth gradient (term G-3), and dissipated by bottom friction (term G-I) and horizontal diffusion (term/-/), The instantaneous transport vorticity shows a pair of counter-rotating tidal vorticities rotating cyclonically around the bump (Fig. 1). The vorticity pair is confined to the bump and the maximum vorticity is found at maximum flood and maximum ebb. The vorticity amplitude is significantly reduced at about two hours before either the maximum flood or the maximum ebb [Fig. l(c) and (f)]. The instantaneous transport vorticity is generated by the topographic vorticity tendency and the frictional torque; the Ekman number of 0,9
1586
M.-J. PARK and DONG-PING WANG
(b)
(o)
E
•
. _,-=--:
/:
o
'
~
:
1C)
2'0
10
30
(k,-,-,)
(kin)
(c)
(d)
i
20
30
2~0
30
o
o 0
10
(kin)
2'0
o
3O
i 10
(kin) (f)
(,)
t,~ °-
"F v
'.... ........ • -
10
'
(kin)
2'0
3o
o
~
(kin)
2'o
30
Fig. I. Transient transport vorticlty over a low bump; (a)-(f) are in 1/6 tidal cycle interval starting at the time of the maximum ebb current. The solid lines are for the positive and the chain-dotted lines are for zero and the negative values. The contour interval is 10 x IW 5 m s 1. The dotted line is the perimeter of the bump.
suggests that these two terms are comparable for the vorticity generation. To further analyze the vorticity generation, the transient transport vorticity balance is examined at two locations over the bump. Figure 2 shows the topographic vorticity tendency, the vorticity advective term [sum of terms B, C and D of equation (4)], the friction and the
1587
Tidal vorticity over isolated topographic features
100 i
50 E X
:
:
0 -50 -100
0
2
4
6 Time (hour)
8
;(}
loo ~ 5o . . . . . . . .
i2
III ~. . . . . . . . . . .
...........
/
:
......
E
-100 ] 0
4
100 rq
E
6 Time (hour)
8
1'0
12
8
1'0
12
!
50-
0-50" -100
4
6 Time (hour)
1oo 50 ~'~::~:.~-. ~: . . . . . . . . . . . . . . . . . . . i. . . . . . . . . . . . . . . . . . . . . . f
-5o
-100
.......
~
2
4
,
6 Time (hour)
.
---...
: 8
1'0
12
Fig. 2. Transient vorticity balance at the northwest corner (a) and at the center (b) of a low bump. The time starts from about 0.3 h before the maximum ebb. The solid line is for the vorticity advective term, the chain-dotted line is for the topographic vorticity tendency, the dashed line is for the friction, and the dotted line is for the vorticity diffusion. The vorticity generation by the frictional torque at the northwest corner (c) and at the center (d) of a low bump. The solid line is for the friction term [sum of terms G-I, G-2 and G-3 of equation (5)], the chain-dotted line is for the vorticity dissipation by bottom friction [term G-1 of equation (5)], the dashed line is for the vorticity generation by transport shear [term G-2 of equation (5)] and the dotted line is for the vorticity generation by depth gradient [term G-3 of equation (5)].
1588
M.-J. PARK and DONG-PING WANG
vorticity diffusion over a tidal cycle at the northwest corner (second quadrant) and at the center. The frictional torque associated with transport shear and depth gradient and the vorticity dissipation due to bottom friction are also shown in Fig. 2. At the northwest corner of the bump, the topographic vorticity tendency and the vorticity advective term are small, but are comparable. Also, the frictional torque is much smaller than the topographic vorticity tendency. At the center of the bump, the topographic vorticity tendency is the dominant term, but it is largely linear. The frictional torque due to depth gradient also is large. However, since the frictional torque due to depth gradient is counterbalanced by the frictional torque due to transport shear, the instantaneous transport vorticity over the low bump is generated mainly by the topographic vorticity tendency. The frictional torque tends to reinforce the topographic vorticity tendency at the times of maximum flood and ebb. This implies that the vorticity advection is important at the times of maximum flood and ebb (Fig. 2).
3.2. Residual vorticity dynamics Averaging instantaneous transport vorticity over a tidal cycle, we obtain the residual transport vorticity [Fig. 3(a)]. The pattern shows a negative (anticyclonic) vorticity over the top of the bump and two positive (cyclonic) vorticities at the perimeter. The magnitude of the positive vorticity is about 60% of the negative vorticity, and the major axis of the negative vorticity ellipse inclines cyclonically relative to the tidal stream axis (i.e. x-axis). The amplitude of the residual transport vorticity, that is, difference between peak positive vorticity and peak negative vorticity, is about 16% of the amplitude of the instantaneous vorticity. PARKand WAN~ (1991) showed that the pattern of the residual transport vorticity over a hollow is similar to that of the residual velocity vorticity. The pattern of the residual transport vorticity over a bump is also similar to that of the residual velocity vorticity. On the other hand, the residual transport is deflected anticyclonically as the flow passes over the bump [Fig. 3(b)], whereas the residual current forms an anticyclonic eddy at the top of the bump and two smaller cyclonic eddies at the perimeter [Fig. 3(c)]. The difference between residual transport and residual current is due to the presence of strong Stokes transport. In order to understand how the residual transport vorticity arises, we analyze the residual transport vorticity balance. Averaging equation (4) over a tidal cycle, we obtain the residual transport vorticity equation. This consists of residual vorticity advective term, residual topographic vorticity tendency, residual friction, and residual vorticity diffusion u. Vf~ - 2 1) d_HH_ V H dt H~-
xU + ( g J ( H , ~)) = - k Vx
+ (VxD)
(6)
where ( ) indicates the time-averaging over a semi-diurnal tidal cycle. The residual vorticity advective term is mainly positive at the second and fourth quadrants of the bump and negative at the first and third quadrants; the maximum negative value occurs at the center of the bump [Fig. 4(a)]. We can also compare the instantaneous vorticity balance (Fig. 2) with the residual vorticity balance. For example, at the second quadrant of the bump, the vorticity advective term is largely positive during a tidal cycle and the residual vorticity advective term becomes positive. On the other hand, the vorticity advective term at the center of the bump is mostly negative during a tidal cycle, which results in negative residual
Tidal vorticity over isolated topographic features
] 589
(a)
v
£-
o I ~3
2~0
30
2fO
50
(kin)
(b)
o.
o
i
10
(kin)
(c)
v
~
8
r
o.
".
i
•
"
i
10
(km)
210
30
Fig. 3. (a) Residual transport vorticity over a low bump. The solid lines are for the positive and the chain-dotted lines are for zero and the negative values. The contour interval is 1.5 x 10,5 m s- 1. The dotted line is the perimeter of the bump. (b) Residual transport. The maximum value is 0.34 m; s 1. (c) Residual currents. The maximum velocity is 0.9 cm s -1.
v o r t i c i t y a d v e c t i v e t e r m . T h e r e l a t i v e l y large r e s i d u a l v o r t i c i t y a d v e c t i v e t e r m at t h e s e c o n d a n d f o u r t h q u a d r a n t s o f t h e b u m p reflects t h e i m p o r t a n c e o f v o r t i c i t y a d v e c t i o n d u r i n g t h e m a x i m u m flood a n d e b b . T h e r e s i d u a l t r a n s p o r t v o r t i c i t y b a l a n c e is m a i n l y b e t w e e n t h e r e s i d u a l v o r t i c i t y a d v e c t i v e t e r m [Fig. 4(a)] a n d t h e r e s i d u a l f r i c t i o n [Fig. 4(c)].
1590
M.-J. PARK and DONG-PING WANG (b)
(o) . . . . ......J \ < . \
E o
.?
...... I0
2'0
30
0
10
(kin)
(km)
(c)
(d)
20
30
20
30
,b
.............S.
v o
..?J/. 10
(kin)
2~0
30
0
10
(kin)
Fig. 4. Residualtransport vorticity balance over a low bump. The solid lines are for the positivc and the chain-dotted lines are for zero and the negative values. The contour interval is 10 x 10- J0 m s 2. The dottedlineistheperimeterofthebump. (a)Residualvorticityadvectiveterm;(b)rcsidual topographic vorticity tendency; (c) residual friction; and (d) residual vorticity diffusion.
4. COMPARISON BETWEEN THE LOW BUMP AND THE HIGH BUMP CASES Numerical experiment was also conducted for tidal flow over a high bump. The high b u m p is 22 km wide and has a Gaussian shape with the m a x i m u m height of 9.8 m. Thus, the bottom slope of the high b u m p is twice that of the low bump. The Rossby and E k m a n numbers are about 0.3 and 1.2, respectively. The instantaneous transport vorticity has a pair of counter-rotating vorticity rotating cyclonically over the high b u m p , which is similar to the low b u m p case. The vorticity amplitude over the high b u m p is about 2.4 times the vorticity amplitude over the low bump. The instantaneous vorticity is controlled mainly by the topographic vorticity tendency. The frictional torque reinforces the topographic vorticity tendency at the m a x i m u m flood and ebb, as in the case of a low bump. H o w e v e r , since water is shallower over the high b u m p , the frictional torque becomes m o r e important over the high b u m p than over the low bump. Consequently, the instantaneous transport vorticity over the high b u m p is intensified at the m a x i m u m flood and ebb, and it is significantly reduced at the slack water. This means that the vorticity advection at the m a x i m u m flood and ebb are more important over the high b u m p than over the low bump. By averaging the instan-
Tidal vorticity over isolated topographic features
1591
(a)
v £-
J i
o
10
20
50
20
30
(b)
o
i
10
0
(k~) (c)
:':.,)
•
,,r
•
q
v
•
•
q
v
f
•
4
•
E £-
; •
,i A
o
ttl,
~
~,~
~.~
1()
210
30
(k,~) Fig. 5. (a) Residual transport vorticity over a high bump. The solid lines are for the positive and the chain-dotted lines are for zero and the negative values. The contour interval is 3.0 x 10--5 m s t. The dotted line is the perimeter of the bump. (b) Residual transport. The maximum value is 0.55 m2s-l. (c) Residual currents. The maximum velocity is 2.3 cm s t. t a n e o u s v o r t i c i t y o v e r a tidal cycle, the r e s i d u a l t r a n s p o r t vorticity was o b t a i n e d [Fig. 5(a)]. T h e p o s i t i v e (cyclonic) v o r t i c i t y is c o n f i n e d within the s e c o n d a n d f o u r t h q u a d r a n t s while the n e g a t i v e ( a n t i c y c l o n i c ) v o r t i c i t y f o r m s a b a n d e x t e n d i n g f r o m t h e first to the third q u a d r a n t s with t h e m a x i m u m at the c e n t e r of the b u m p . T h e m a j o r axis of the a n t i c y c l o n i c v o r t i c i t y ellipse inclines c y c l o n i c a l l y r e l a t i v e to t h e tidal s t r e a m axis. T h e a m p l i t u d e o f the r e s i d u a l t r a n s p o r t v o r t i c i t y is a b o u t 14% o f t h e a m p l i t u d e of the i n s t a n t a n e o u s vorticity.
1592
M.-J. PARKand DONG-PINGWANG
The vorticity pattern over the high bump is similar to that over the low bump [Fig. 3(a)], but the magnitude of the anticyclonic vorticity over the high bump is about twice that over the low bump. In addition, the ratio of the positive vorticity at the perimeter to that of the negative vorticity at the center increases from 60% over the low bump to 75% over the high bump. The residual transport is shown in Fig. 5(b); because of the stronger vorticity, the residual transport is larger and it bends more sharply over the high bump than over the low bump. The residual vorticity of the depth-averaged velocity over the high bump is similar to that over the low bump. The anticyclonic vorticity over the high bump is about 2.5 times larger than that over the low bump, and consequently, the anticyclonic eddy is much stronger over the high bump [Fig. 5(c)]. The residual currents show three eddies. It is noted that the increase of residual vorticity and residual currents from the low bump case to the high bump case is comparable to the increase in the bottom slope. The residual transport vorticity balance is between the residual vorticity advective term and the residual friction, as in the low bump case. 5. COMPARISON BETWEEN THE BUMP AND THE HOLLOW CASES Since the bottom slope of a hollow is opposite to that of a bump, it would be useful to compare the results for a bump with those for a hollow. The dynamics of tidal vorticities over a hollow was previously examined by PARK and WANG (1991). The hollow is 22 km wide and has a Gaussian shape with the maximum depth of 38 m. The model conditions other than the hollow geometry are the same as those for the bump; the bottom slope of the hollow is about twice that of the high bump. A counter-rotating transient transport vorticity pair exists both over a bump and over a hollow. As the bottom slope changes sign from bump to hollow, the signs of the transient transport vorticity also change. The transient transport vorticity over a bump is much stronger at the maximum flood and ebb than near the slack water due to the increased frictional torque; in contrast, the transient vorticity over a hollow does not change appreciably through a tidal cycle. Nevertheless, the topographic vorticity tendency is the major source for the transient vorticity generation over the bump and the hollow cases. The residual transport vorticity in both cases is largely controlled by the net vorticity advection. Because of the sign change in the bottom slope, the residual transport vorticity is anticyclonic at the top of a bump, whereas it is cyclonic at the center of a hollow [Fig. 6(a)]. On the other hand, because the frictional torque reinforces the topographic vorticity tendency over the bump, the cyclonic vorticity at the perimeter of the bump becomes comparable to the anticyclonic vorticity at the center. The cyclonic vorticity at the center of the hollow is much stronger than the anticyclonic vorticity at the perimeter, since the vorticity advection is mainly contributed by the topographic vorticity tendency. Also, the major axis of the anticyclonic vorticity ellipse over the bump inclines cyclonically relative to the tidal stream axis, whereas the major axis of the cyclonic vorticity ellipse over the hollow is perpendicular to the tidal stream axis. The pattern of the residual vorticity for both the bump and the hollow cases is similar to that of the residual transport vorticity. The residual transport flows anticycionically over the bump, while it bends cyclonically over the hollow [Fig. 6(b)]. The three residual eddies over the bump are comparable, but the cyclonic eddy at the center of the hollow is much stronger than the anticyclonic eddies at the perimeter [Fig. 6(c)].
Tidal vorticity over isolated topographic features
1593
(a)
/ 0 0
10
0
10
2'0
50
2PO
30
(b)
Y
9
I
0
(k~)
(c) b o.
d •
"C v o.
q
,
.
,:
2]0
10
.30
(kin) Fig. 6. (a) Residual transport vorticity over a hollow (from PARK and WANG, 1991). The solid lines are for the positive and the chain-dotted lines are for zero and the negative values. The contour interval is 5 × 10 -5 m s -1. The dotted line is the perimeter of the hollow. (b) Residual transport. The maximum value is 0.57 m 2 s- n. (c) Residual currents. The maximum velocity is 1.7 cm s -] .
6. SEDIMENT DYNAMICS Previous studies (e.g. ZIMMERMAN,1981) found a definite relationship between residual eddy and sand bank. We examine effects of residual eddy on the sediment transport over a bump. In this study, our primary interest is to obtain a qualitative pattern of sand
1594
M.-J. PARKand DON6-PINGWAN6
transport. Following essentially the approach in HUTHNANCE(1982a,b), we assume that the bed load transport rate Qb (volume/unit width/time) is related to [ul 3,
Qb ~ lul2(u + 21ulVh)
(7)
where 2 is k/tan(friction angle). There are some uncertainties in equation (7) (HUTHNANCE, 1982a,b), and quantitative results cannot be expected. However, the qualitative results obtained using equation (7) were found to be fairly robust to the changes in the transport formula (HUTHNANCE,1982a, b). The instantaneous sand transport pattern over a low bump was examined by computing the divergence of bed load using equation (7). The enhancement of the down-slope transport was not considered since the slope of the bump is small (O(10-4)). The instantaneous bed load divergence pattern indicates that the erosion occurs as currents flow over the bump and the deposition occurs as currents pass the bump. This pattern is related to the variation of tidal currents with the water depth that tidal velocity increases toward the top of the bump. The erosion and deposition are of equal magnitude and are mostly confined within the bump. The residual bed load divergence, obtained from averaging the instantaneous value over a tidal cycle, indicates that the deposition occurs mostly at the first and third quadrants of the bump while the erosion occurs at the second and fourth quadrants [Fig. 7(a)]. This will cause a circular bump to change to an elliptical one with the major axis of the ellipse inclining cyclonically relative to the tidal stream axis. HUTHNANCE (1982b) indicated that for a uniform flow, the residual bed load divergence pattern over a small bump is similar to the pattern of the gradient of the residual current along tidal stream axis, O(u)/Ox. Since in our model the far-field tidal current, ut, increases toward the right-hand wall, facing the direction of the tidal wave propagation, the equivalent form is 2(u2)(O(u)/Ox). Indeed, this pattern [Fig. 7(b)] approximates the residual bed load divergence pattern. Also, although the residual bed load divergence pattern is somewhat different from the residual transport vorticity pattern, the deposition tends to occur over the area of negative vorticity and the erosion over the area of positive vorticity. The erosion-deposition pattern over the high bump is similar to that over the low bump. The residual bed load divergence over the high bump shows that the deposition occurs at the first and third quadrants, whereas the erosion occurs at the second and fourth quadrants. However, the major axis of the deposition ellipse over the high bump tends to incline more toward the tidal stream axis due probably to the increased effect of bottom friction. The magnitude of deposition over the high bump increases about three times that over the low bump suggesting the non-linearity of the sediment transport. The deposition and erosion will also cause the circular bank to contract laterally and become elliptic with the major axis of the ellipse inclining cyclonically relative to the tidal stream axis. This result agrees with the observation that the natural sand banks are linear and they are inclined cyclonically to the tidal stream axis (ZIMMERMAN,1981). It also supports the result of HUTHNANCE(1982b) on sand bank formation; a sand bank tends to contract laterally (unless very narrow) and tends to rotate to be cyclonically inclined toward the tidal stream axis. For the hollow case (PARKand WANG, 1991), because of the sign change in bottom slope, the transient and residual bed load divergence have opposite signs to those over a bump. Both circular bump and hollow tend to contract laterally and become elliptic with the major axis of the ellipse inclining cyclonically relative to the tidal stream axis. This
Tidal vorticity over isolated topographic features
1595
(a)
il-.
-_ ..i;"i 0
10
20
30
(kin)
(b)
/
/ '
~
o.
o
20
10
30
(km) Fig. 7. (a) Residual bed load divergence over a low bump. (b) The pattern of 2(~)(0(u)/0x). The solid lines are for the positive and the chain-dotted lines are for zero and the negative values. The contour interval is 1.0 × 10-7 mas-3. The dotted line is the perimeter of the bump. The u1 is the far-field tidal current and the ( ) denotes time-averaging over a semi-diurnal tidal period. suggests that b o t h circular b u m p and circular hollow are very unstable bed form configurations and m a y not be f o r m e d in the tidal areas. 7. D I S C U S S I O N A vertically-integrated, finite-element m o d e l was used to study the t o p o g r a p h i c rectification of tidal currents over low and high bumps. T h e model d o m a i n was designed to have typical current velocity - O(1 m s-1), water depth - O(10 m) and size of the b u m p - O(10 km) in the shallow tidal areas (HUTHNANCE, 1982b; ROBINSON, 1983). T h e m a x i m u m tidal residual vorticity is expected w h e n the t o p o g r a p h i c length scale is o r d e r of the tidal excursion length (ZIMMERMAN, 1981; ROBINSON, 1983). The size of our low and high b u m p s and the hollow ( - 2 2 km) is a b o u t 2.6 times the tidal excursion length ( - 8 . 5 km), and this ratio is suitable for the m a x i m u m response. Field observations of the residual vorticity over a sand b a n k are scarce, but the m o d e l result m a y be c o m p a r e d to the analytic solution. T h e m a x i m u m tidal residual vorticity of the d e p t h - a v e r a g e d velocity over the high b u m p m a y be estimated f r o m ROBINSON (1983); WO - - 0 . 4 5 ( f u / o h ) ( O h / O s )
--
0.8 X 10 -5 f r o m the c o l u m n stretching
1596
M.-J. PARKand DONG-PINGWANG
~Oo ~ 0 . 4 ( k / o ) ( O ( u / h ) / O n ) ~ 0.6 x 10 .5 from the frictional torque
where cr is the tidal frequency, s is the distance measured along the tidal stream and n is the normal. The residual vorticity from our model (not shown; PARK, 1990) is about 1.0 x 10 .5 , which agrees reasonably well with the analytic solution. The results on the low bank and the hollow also showed similar reasonable agreements with the analytic solutions. The tidal current asymmetry and mean anticyclonic circulation around the elliptic sand bank were noticed in the shallow tidal seas (CAsTON and Sa'RIDE, 1970). Our results show that the anticyclonic residual eddy develops over a circular bank, but it is elliptic rather than circular. The results also indicate that as the height of b u m p increases, the strength of transient and residual transport vorticities increases in proportion to the increasing bottom slope. Also, as the water depth becomes shallower, the relative importance of the frictional torque to the topographic vorticity tendency becomes greater. Consequently, the cyclonic residual transport vorticity at the perimeter of the high b u m p becomes comparable to the anticyclonic residual vorticity at the center. Comparison between the b u m p and the hollow case indicates that apart from a sign change, the transient and residual transport vorticities look similar. However, due to the effect of frictional torque, the transient transport vorticity over a b u m p is significantly stronger at the m a x i m u m flood and ebb c o m p a r e d to other parts of the tidal cycle. On the other hand, the strength of the transient transport vorticity over a hollow does not change appreciably through a tidal cycle. Also, unlike the b u m p cases, the residual transport vorticity at the center of the hollow is much stronger than that at the perimeter. Due to the effect of the increased frictional torque the major axis of the anticyclonic residual vorticity over a b u m p inclines cyclonically relative to the tidal stream axis, whereas the major axis of the cyclonic residual vorticity over a hollow is perpendicular to the tidal stream axis. Unlike the hollow case, over a bump, the deposition tends to be associated with the anticyclonic residual vorticity and the erosion with the cyclonic residual vorticity (HurHNANCE, 1982b). The present model uses a depth-averaged velocity to calculate sediment transport. Thus, the E k m a n veering effect (MCCAVE, 1979) can not be precisely represented, but the effect of b o t t o m friction is incorporated in the depth-averaged velocity. O t h e r possible effects from the secondary currents (HEATHERSHAWand HAMMOND, 1980) and transport lag (KENNEDY, 1969) may be less important than the primary distortion of the depth-averaged current (HUTHNANCE, 1982a, b) and were not considered in this study. Although the present study deals with the rectilinear tidal currents, the results on sand bank evolution by the elliptical tidal currents may be similar (HuTI4NANCE, 1982a). Acknowledgements--The computation of this study was supported by the Cornell National Supercomputer
Facility. The first author wishes to thank the partial support from Chungnam National University, Korea.
REFERENCES CASTONV. N. D. and A. H. STRIDE(1970) Tidal sand movement between some linear banks in the North Sea off northeast Norfolk. Marine Geology, 9, M38-M42. HEATHERSHAWA. D. and F. D. C. HAMMOND(1980) Secondary circulation near sand banks and in coastal embayments. Deutsche Hydrographische Zeitschrift, 33,135-151. HUTHNANCEJ. M. (1982a) On one mechanism forming linear sand banks. Estuarine, Coastal and Shelf Science, 14, 79-89.
Tidal vorticity over isolated topographic features
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HUTHNANCEJ. M. (1982b) On the formation of sand banks of finite extent. Estuarine, Coastal and Shelf Science, 15, 277-299. KENNEDY J. F. (1969) The formation of sediment ripples, dunes and antidunes. Annual Reviews of Fluid Mechanics, l, 147-168. McCAVE I. N. (1979) Tidal currents at the North Hinder Lightship, southern North Sea; flow directions and turbulence in relation to maintenance of sand banks. Marine Geology, 31,101-114. PARKM.-J. (1990) Transient tidal vorticity in coastal seas. Ph.D. Thesis, Marine Sciences Research Center. State University of New York, Stony Brook, 105 pp. ])ARK M.-J. and D.-P. WANG (1991) Transient tidal vorticity over a hollow. In: Tidal hydrodynamics, B. B. PARKER, editor, John Wiley and Sons, New York, pp. 419-434. PR1TCHARDD. W. (1971) Hydrodynamic Models. In: Estuarine modelling: an assessment, G. H. WARD,JR and W. H. ESPEV, JR, editors, Water Quality Office, U. S. Environmental Protection Agency, pp. 5-33. ROBINSON I. S. (1983) Tidally induced residual flows. In: Physical oceanography o f coastal and shelf seas, B. JOHNS, editor, Elsevier, Amsterdam, pp. 321-356. WANG H. P. (1975) Modelling an ocean pond; a two-dimensional finite element hydrodynamic model of Ninigret Pond, Charlestown, Rhode Island. University of Rhode Island Marine Technical Report 40, 60 pp. ZIMMERMAN Z. T. F. (1981) Dynamics, diffusion and geomorphological significance of tidal residual eddies. Nature, 290, 549-555. Z1MMERMANZ. T. F. (1986) Principal differences between 2D- and vertically averaged 3D- models of topographic tidal rectification. In: Physics of shallow estuaries and bays, J. VAN DE KREEKE, editor, Springer-Verlag, New York, pp. 120-129.
APPENDIX
Open boundary condition 1. Analytical solution. For the solution of a frictionally-damped progressive Kelvin wave in a rotating channel of uniform depth and width, we consider the linearized depth-averaged equations of motion and the continuity equation. The channel is oriented parallel to the x-axis and is bounded by two rigid walls in the y-axis. The wave propagates in the positive x-direction and the normal velocity vanishes at the rigid walls. Assuming ~ << h, thc equations are Ou Ot -
O~ g ~x
Fu
fu = - g Uyy Ou ax
l O~ h at
(8)
(9)
(10)
where F is a linearized friction coefficient. We look for a progressive wave solution periodic in x and t, and we let
= Re{ ~oa(x)fl(y) e i(k ..... )}
( I 1)
where ~o is the initial wave amplitude before entering the channel, a(x) is a functmn to account for the frictional damping of the wave along the channel, fl(y) is a function to account for the cross-channel variation of the wave due to the Earth's rotation, k is the wave number and cr is the wave frequency. Manipulation of equations (8) and (10) yields a wave equation,
O2¢+FO¢ Ot2
202~-- 0
0 - 7 - C° OX ~ --
w'here C o = "v/~. Substituting equation (11) into equation (12), we get
(121
1598
M.-J. PARK a n d DONG-PING WANG
O2a 4- (k~ -- k2)a = 0
(13)
2 k a O_a_~+ F k 2 a = 0 Ox
(14)
07
where ko is the initial wave n u m b e r ( = a/Co). The solutions for a in equations (13) and (14) are a(x) = e -&x,
k I = X/k2~
(15)
a ( x ) = e -iqx,
~:1 = Fk2°" 2ka
(16)
From equations (15) and (16), we obtain an equation for k 402k 4 - 4 O 2 k o 2k2 - F 2k4o = 0 .
(17)
k = _ ~ ( . X ~ l 4- (14/0) 2 4- 1) 1/2.
(18)
The solution for k in equation (17) is
Using equation (18) in equation (15), we get
k 1 = -~-~(X/14- (F/a) 2 - 1) 1/2.
(L9)
Thus, the wave solution we look for is = ~_,,e - k , x f l ( y ) COS ( k x - a t )
(20)
where k = ~(VI~(F~
d 4- 1) 1/2
and /,kl = ~22(X/1-t- (F/o) 2 - 1) I/2. Substituting ~ of equation (20) in equation (10), we get
GC,k,, e u =
k'~fi(Y).cos ( k x - at - ~ )
hV, k 2 + k 2
(2l)
,
where q~ = tan -1 ( k l / k ) . Using the solutions of ¢ [equation (20)] and u [equation (21)] in equation (9) and assuming that qb << I, we obtain fl(y) = e-/Ky/c,,
ko
K = ,/k2-T~ 1
(22)
W h e n the bottom friction is negligible, i.e. F - 0, then k 1 - 0, and K - i. Thus, we get fl(y) = eo-.ty/( , which is the cross-channel variation function of the Kelvin wave. Using fl(y) in equation (22), we find the solutions for a frictionally-damped Kelvin wave in a straight rotating channel, ¢ = ¢o e kl* e /Ky/c cos ( k x - ot
)
u = ; ~,,CoK e -k,x e-fKylC,, cos ( k x - at -- ~ ) h
v=0 where
(23) (24)
(25)
Tidal vorticity over isolated topographic features
(a)
1599
o .
.
.
:-.....
.
,, ,.. . . . . . . . . 0
10
,.. ;
30
20
40
,50
(km) 0
(b)
"::
0N '
E
........
: : - - , k , 4 , , - - - " • k k 4 # 4 4 4 4
~
"""""
v
~
~
V ~ I 4
0
0
~ A A & 4 A ~ 4 4 4 4
: ~--~ k 4
~1 4
~
110
, 4 . 4 , , d 4 4 4
4
d
4
4
4
~
2~0
4
~
3'0
4
4
4
4
4
~ 4 4 4 4
410
5O
(km) Fig. 8. Residual currents (a) when the uniform sea level is applied at the open boundaries and (b) when the analytical solution is applied with a slight adjustment. The background residual current velocities arc 0 (10) cm s I for (a) and 0 (0.1) cm s -I for (b).
k = (~/1X/2 k° + (FI0)2)2+ 1)L/2,
k] = V~2 (X/f + (F/a) 2 - 1) ]/2,
K- X/~'k°
and qb = tan I (kl/k)" 2. Open boundary condition. When the model is driven by a uniform sea level of 1 m amplitude at both ends of the channel, it produces large noise around open boundaries. With the dynamically consistent frictionallydamped Kelvin wave solution [equations (23)-(25)], the model rcmoves most of the large noise, but small noisc is still present near the open boundary. The noise is eliminated with the slight adjustment of the analytical solution. In the case of using constant sea level at the open boundaries, large half-circle residual eddies are generated near the boundaries. Thcsc eddies are completely removed when thc analytical solution is applied (Fig. g).