Annu. Rev. Fluid Mech. 1991.23: 389-412 Copyright © 1991 by Annual Reviews [nco All rights reserved

COASTAL-TRAPPED WAVES AND WIND-DRIVEN CURRENTS OVER THE CONTINENTAL SHELF

K. H. Brink

Department of Physical Oceanography, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts 02543

KEY WORDS: ocean-current variability, wind driving, coastal oceanography, alongshore propagation

1. INTRODUCTION

The 1970s saw a remarkable development in the understanding of wind­ driven current variability over the continental shelf, allowing Allen (1980) to summarize many of the keystones of our current understanding of the subject. Since 1980, our understanding of such processes has become a good deal more sophisticated, especially in terms of coastal-trapped-wave theory. In fact, the point has been reached where models and observations are now often compared with a reasonable expectation of quantitative agreement. Along the way, some exciting new physical insights have also Access provided by Old Dominion University on 01/02/18. For personal use only.

Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org been gained. It thus seems timely to revisit the problem of wind-driven variability over the continental shelf, and therefore the present offering is intended to be a continuation of Allen's (1980) original fine review. The following material focuses mainly on variability having time scales longer than a day but not more than a few weeks. In all cases, attempts are made to emphasize the relation between theoretical concepts and observed behavior in the coastal ocean. The approach is to address "water­ column" models in which nonadiabatic effects, while often present, are not of overwhelming importance. Within this classification, deterministic problems expressed in terms of coastal-trapped-wave theory are treated 389 0066-4189/91/0115--0389$02.00 390 BRINK first, followed by consideration of stochastic models that may or may not be centered on wave models. A central focus on coastal-trapped-wave theory is appropriate because it has proven to be a versatile approach, yielding successful predictions as well as physical insight.

2. COASTAL-TRAPPED-WAVE MODELS 2.1 Introduction Many aspects of wind-driven variability over the continental margin can be explained in terms of the properties of coastal-trapped waves. All continental margins allow the existence of such waves, and the sense of alongshore phase propagation is always such that the coast is on the right (left) when looking in the direction of phase propagation in the Northern (Southern) Hemisphere. Such waves generally have periods longer than the inertial period, so that they often play a role in the response of the ocean to atmospheric weather changes, which typically have time scales of a few days. The waves are important because they are readily excited and, through their propagation, spread the ocean's response in the along­ shore direction. Many important wave properties have been summarized by Huthnance et al (1986).

2.2 Formulation Before proceeding to the properties of free coastal-trapped waves, it is useful to consider the mathematical formulation of the problem. The linearized equations of motion for a stratified, Boussinesq ocean are

I I u,-jv = - -Px+ !�, (la) Po -Po I I v,+ju - -Py+ -T�, (lb) = Po Po

Access provided by Old Dominion University on 01/02/18. For personal use only. (Ic) Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org (ld)

and (Ie)

where u, v, and ware the velocity components in the x, y, and z directions, respectively. The Coriolis parameter isj( = 2Q sin 1>,where Q is the Earth's rotation rate and 1> is latitude), the acceleration due to gravity is g, the pressure perturbation is p, and density is written as WIND-DRIVEN CURRENTS OVER THE SHELF 391

Po+ p(z) + p'(x,y, z, t), where Ip'l« p« Po. The horizontal component of turbulent stress is (, X, ,V) and will be needed to represent wind driving and the bottom frictional drag. Subscripts with respect to independent variables repre­ sent partial differentiation. The offshoreand upward directions are taken

to be x and z, respectively. In the absence of turbulent stresses, it is straightforward to reduce the system (1) to the vorticity equation

( 2 02)(p = (2) + zt\ ° . Pxxt+Pyyt+ J ot2 N2h The buoyancy frequency squared is

N2 = - !Lpz. (3) Po The vorticity equation is to be solved subject to no normal flow at the coast, Pxt+Jpy = ° at x = 0, (4a) a free surface,

N2 ° at z = 0, (4b) pz + �P = 9 no flowthrough the bottom

and boundedness far from the coast, Px -+ 0 as x -+ 00. (4d) Access provided by Old Dominion University on 01/02/18. For personal use only.

Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org Free coastal-trapped waves are assumed to have the form exp [i(wt+ly)]F(x, (5) z), p = with the depth here chosen to be a function only of distance offshore, i.e. h h(x). The problem has now been reduced, for a given alongshore = wavenumber I, to an eigenvalue problem for F(x, z) and w. Even with these simplifications, analytical solutions to (2) subject to (4) are very difficult to find. Thus, F is usually found numerically via resonance iter­ ation (Wang & Mooers 1976, Brink 1982) or inverse iteration (Huthnance 1978). Computer programs for obtaining modal structures and related 392 BRINK

properties (described by Brink & Chapman 1987) are freely available and cover a fairly broad range of realistic situations. System (2-4) can be simplified in the "long-wave" limit, where along­

shore scales are large relative to cross-shelf scales (82/8y2 « 82/8x2) and where frequencies are low (02/012 «/2). In this case, the problem becomes separable and is

Pzt 2( ) 0 (6a) Pxx+f = , N2 z

Pxt+fpy = ° at x = 0, (6b) N2 pz + -p ° at z 0, (6c) g = =

 at z -hex), f2( ) +hx(Pxt+fPY) ° = (6d) =

and

Px-40 as X->

The solutions can be written as

p Y,ly, t)F,,(x, z), (7a) = where

(7b)

The eigenvalue of the problem is now em and members of the complete set (Clarke 1977) of eigenfunctions Fn are orthogonal subject to

Access provided by Old Dominion University on 01/02/18. For personal use only. fbnm FnFmdzlx=o+ rooh xFnFmdxlz=-h(x), (8) Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org = 1° -h Jo

where bnm is the Kronecker delta. The particular normalization implied by (8) must be used in order to conserve energy as the wave propagates through a slowly varying medium (Brink 1989). It is straightforward in the long-wave limit to solve for the wind-forced problem (with no bottom stress). Further, it is frequently assumed that forcing is only through an alongshore wind stress that does not vary in the cross-shelf direction [rX 0, rJ' rJ'(y, t)]. In this case, the pressure can = = still be expressed in terms of the eigenfunctions, WIND-DRIVEN CURRENTS OVER THE SHELF 393

where now the amplitude functions obey

(9b)

Here ,� is the alongshore wind stress at the surface, and

(9c)

is the coupling coefficientof mode n to the wind stress if the surface mixed layer is infinitesimally thin (see, for example. Clarke 1977). A scaling of (2) leads to a stratification parameter (Burger number)

where H, No, and L are typical values of depth, buoyancy frequency, and cross-shelf horizontal scale, respectively. This dimensionless number expresses the relative strength of stratification; alternatively, it can be thought of as the ratio of the internal Rossby radius of deformation to the

horizontal length scale. The barotropic (unstratified) limit of (2), i.e. S --+ 0, leads to the lowest order vorticity balance

(10)

that is, that the vertical velocity does not vary with depth. In this barotropic limit, the flowis then described by (10), (4b), and (4c) (e.g. Clarke & Brink 1985). In the following, the properties of the free modes are first treated. The excitation of coastal-trapped waves is then considered, and finally com­ Access provided by Old Dominion University on 01/02/18. For personal use only.

Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org parisons of models with observations are discussed.

2.3 Free-Wave Properties The general properties of free coastal-trapped waves are summarized nicely by Huthnance (1975, 1978) and by Clarke (1977). For arbitrary topog­ raphy and stratification, there is a free-wave mode similar to the baro­ tropic Kelvin wave plus an infinite set of higher mode, more slowly propagating waves. All of these wave modes have phase propagation in the sense described above (Section 2.1), provided the depth increases monotonically with distance from the coast. Slower (higher order) modes 394 BRINK have progressively more complex spatial structures (Figure I). If the depth increase is not monotonic, then an infinitenumber of waves will also exist, supported by the reverse slopes associated, for example, with trenches (Mysak et al 1979, Mysak & Willmott 1981, Brink 1983a) or by banks (Brink 1983b). Increasing the stratification results in an increased fre­ quency (hence phase speed) for a given wave mode. Further, making the rigid lid assumption [neglecting the divergence due to the displacement of the free surface, i.e. using pz = 0 in lieu of (4b)] leads to an increased phase speed. In the barotropic limit (Huthnance 1975), where stratification can be neglected, all free-wave modes exist for all alongshore wavenumbers but, except for the barotropic Kelvin wave, their frequencies are always less

than the inertial frequency (i.e. w

Distance Offshore(km) 100 200 300 Mode 1

1000

2000 Figure I Model structures of alongshore velocity, computed for the northern Access provided by Old Dominion University on 01/02/18. For personal use only.

Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org Oregon coast (after Battisti & Hickey 200 300 1984). Units are arbitrary. Mode 2

1000 >0

2000 WIND-DRIVEN CURRENTS OVER THE SHELF 395

Alongshore Wavenumber, 1

Figure 2 Schematic dispersion curves for an unstratified ocean (solid lines) and for stratified oceanS (dashed lines).

the wave frequency increases until it reaches the inertial frequency (dashed lines in Figure 2). Huthnance (1978) has shown that the wave frequency will reach ! if the maximum value of

(11) A = 17 :�I, evaluated at the bottom, is greater than or equal to unity. Once a dispersion curve rises above f, strictly trapped waves apparently no longer exist. Physically, this would occur because the coastal wave becomes strongly coupled, through the topography, to the continuum of freely radiating oceanic internal waves. Thus, the coastal-trapped waves are apparently "leaky" in the same sense as are edge waves (e.g. Chapman 1982). In the

limit as S -> 00, the continental margin is much thinner than a typical internal Rossby radius of deformation, and the behavior of the wave modes is like that of pure internal Kelvin waves (e.g. Gill & Clarke 1974). Once a dispersion curve reaches f, the wave's group velocity is generally Access provided by Old Dominion University on 01/02/18. For personal use only.

Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org in the same direction as the phase velocity for all wavenumbers where the wave exists. This substantially changes the nature of scattering off of topographic obstacles, since reflection of energy is no longer possible (Wilkin 1988). A useful diagnostic for a given wave mode is the ratio (averaged over a wave period) between kinetic energy K and potential energy P:

K R=- (12) P

(Brink 1982). The potential energy is the sum of two components: One is 396 BRINK associated with the displacement of the free surface (Pr), and the other is associated with the vertical displacement of water parcels in a stratified ocean (PJ. In an unstratified ocean, Ps = 0 and Pr is small compared with K (except for the barotropic Kelvin wave), so R»l (13) for all shelf-wave modes. In fact, in the rigid-lid limit, we have -2 D = gH(fL) -.00, (14a)

R -. 00. (14b) In the opposite limit of Kelvin wave-type behavior (large S), we have R -.1, and the importance of gravity as a restoring force allows the equipartition of wave energy. Once a specific wave mode is calculated, it is straight­ forward to estimate f{, Pr, and Ps and thus estimate the importance of baroclinic effects. The presence of a mean alongshore flow over the continental margin can alter the free-wave properties through Doppler shifting, modification of the background vorticity field, and by allowing the growth of insta­ bilities. The case of barotropic, stable, coastal-trapped waves has been treated by Brooks & Mooers (1977), who were interested in the effect of the energetic Florida current on coastal-trapped-wave propagation. Luther & Bane (1985) considered the linearly unstable modes associated with the Gulf Stream off the United States Carolina coast and found a strong instability growing on the inshore edge of the mean current. They were able to identify this mode convincingly with Gulf Stream "shingles," which have been frequently observed (e.g. Bane et aI1981). When coastal-trapped waves propagate in the opposite direction to that of the mean current, the possibility of critical coastal-trapped waves arises (Gill & Schumann 1979). This is a situation in which the direction of wave propagation is locally Access provided by Old Dominion University on 01/02/18. For personal use only. Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org reversed owing to the strength of the mean flow. Gill & Schumann (1979) first raised this possibility to explain the observed absence of coastal­ trapped-wave propagation off the Natal coast, where the Agulhas current comes particularly close to shore (e.g. Schumann & Brink 1990). It now seems more likely that the absence of propagation is related to enhanced frictional damping associated with the distortion of the wave's modal structure by the mean-flow vorticity field (Brink 1990). It appears that the main mechanism for the decay of coastal-trapped­ wave energy is bottom friction. It is straightforward to include its effects in the long-wave limit, where the decay of wave modes is accompanied by WIND-DRIVEN CURRENTS OVER THE SHELF 397 an onshore-offshorephase shift associated with the differencein frictional efficiency versus depth (Brink & Allen 1978, 1983, Clarke & Van Gorder 1986). A perturbation approach to the damping of coastal-trapped waves, valid even where the modes are not orthogonal but where the decay time is long relative to the wave period, has been presented by Brink (1990). At general frequencies and wavenumbers, the frictional problem has most often been addressed by solving the complex eigenvalue problem associ­ ated with the frictional version of (2)-(4) (Spillane 1980, Webster & Holland 1987). This approach yields very general results at the cost of substantially increased computational complexity. Power et al (1989) used such an approach to show that, in addition to the onshore-offshore phase shift, the free-wave modal structure also adjusts to increasing friction by having the location of maximum fluctuating velocity shift into progressively deeper water. The effect of this migration is to mitigate somewhat the damping rate of the wave by putting more of its structure into deeper water, where damping is relatively less effective.

2.4 The Excitation of Coastal-Trapped Waves The study of tides over the continental shelf has led to an interesting application of free coastal-trapped-wave theory. At latitudes poleward of 30°, the diurnal tide occurs at a frequency below the inertial frequency, so that it is possible for the diurnal tide to excite free coastal-trapped waves. Fitting current and sea-level data to theoretical wave modes has shown, in several locations, that the diurnal tide over the shelf includes a barotropic Kelvin wave (which dominates sea-level variability but produces weak currents) and a lowest mode coastal-trapped (shelf) wave, which is associ­ ated with strong current fluctuations (e.g. Figure 3). The shelf wave is apparently excited by topographic irregularities such as the Strait of Juan de Fuca (Crawford & Thomson 1984, Flather 1988) or the Gulf of Maine (Daifuku & Beardsley 1983). In addition, Crawford & Thomson (1984) showed that including stratification in computing free-wave properties improves the fitto data considerably and provides rationalization for the Access provided by Old Dominion University on 01/02/18. For personal use only.

Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org neglect of the short shelf-wave mode (which appears in the barotropic case but not when stratification is included). Such tidal studies provide some of the most direct evidence for free coastal-trapped-wave propagation. Most work on the excitation of coastal-trapped waves, however, has concentrated on wind-forced motions in the long-wave limit (e.g. Gill & Schumann 1974). Restriction to this limit is superficially quite reason­ able because of the large scales of weather systems and because of the rela­ tively short onshore-offshore scales of the continental margin. Given this assumption, along with the absence of a mean alongshore flow, the free coastal-trapped-wave modes have the convenient orthogonality property 398 BRINK

N em/sec +E 0� 10

Kelvin Stratified Wave Shelf Wave Sum Observed

15m r�

45m r�

81m r�

Figure 3 Results of fitting eoastal-trapped-wave modes to diurnal tide data from the shelf off British Columbia (after Crawford & Thomson 1984). The current contributions from different modeled waves and their sum are shown, as are observed current ellipses.

(8) which reduces the problem, complete with bottom stress, to an infinite set of coupled equations

c; I Ynt- Yny+ LanmYm b;r:�, (ISa) m =

where Yn is the amplitude of wave mode n. The frictional coupling coeffi­ cients are defined as

(ISb)

where a subscript B refers to a quantity evaluated at the bottom Access provided by Old Dominion University on 01/02/18. For personal use only.

Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org [z = -h(x)], and por is a constant of proportionality between bottom stress and bottom velocity, e.g. r� porv. In the limit of weak friction, the = equations given by (15a) become uncoupled at lowest order, and ann can be thought of as an inverse-decay distance for the frce wave. The physical mechanism for the wave forcing is straightforward. An alongshore wind stress causes an onshore-offshore transport within the surface Ekman layer. In order to conserve mass, a compensating offshore­ onshore flow must take place at greater depth. As this flow crosses isobaths, vortex stretching leads to changes in the local relative vorticity. This vorticity is expressed largely in terms of the alongshore velocity and, WIND-DRIVEN CURRENTS OVER THE SHELF 399 \ in the presence of alongshore variability (%y "I 0), gives rise to wave propagation. A number of practical questions arise when solving coastal-trapped­ wave problems (6 and ISa). First, when computing the free-wave modal structures, there is some question about where to place an artificialvertical coastal wall. Such an arbitrary boundary is used to exclude the highly turbulent inner shelf so that the inviscid free-wave equations (6) remain valid. This problem was addressed by Mitchum & Clarke (l986a), who showed that a reasonable choice is to place the "coast" at the isobath where the water's depth is three times the scale thickness of the turbulent (Ekman) boundary layer. In addition, they showed that the wind still excites motions within the nearshore wedge, so that a correction (often small) to the wave result, proportional to the local alongshore wind stress, must be added to the wave solution's "coastal" sea level in order to achieve accuracy. Their results are dependent on a highly idealized formulation of the bottom stress but nonetheless appear to work well in practice (Mitchum & Clarke 1986b). A second question that often arises is, How many modes need to be summed in order to achieve accurate results? [i.e. what is the range of n in (IS)?] Early applications of coastal-trapped-wave theory yielded adequate to very good results with only a single mode (Brink 1982, Battisti & Hickey 1984). Clarke & Van Gorder (1986), on the other hand, argued that seven or more modes typically need to be used in order to obtain accurate results, especially for alongshore velocity. Chapman (1987) addressed this question in the context of hindcasting currents off northern California and con­ cluded that no improvement in the hindcast occurred when more than three modes were used. In realistic stratified situations, the accurate com­ putation of higher modes tends to involve considerable expense, so this issue is of some practical importance. Lopez & Clarke (1989) found a solution to this difficultyby noticing that for higher modes (cn small), (ISa) reduces to Access provided by Old Dominion University on 01/02/18. For personal use only.

Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org C;; I Ynt+ LanmYm � b;r�. (16)

m In other words, alongshore variability (and hence wave propagation) becomes unimportant. Lopez & Clarke (1989) then proposed the following approach. First, solve for pressure (which converges faster than velocity) for the first few wave modes by using (1Sa). This information is then used to prescribe only Py in system (6), which is then solved for total pressure. This method avoids the difficult calculation of higher modes and provides a rational separation between "local" and "remotely forced" fluctuations. The remotely forced information is that which would be obtained from 400 BRINK solving (15) (i.e. directly, using winds from a large spatial domain), and the locally forced information is that which could be obtained from (16) or, equivalently, the residual of the Lopez & Clarke (1989) method. Coastal-trapped-wave theory has often proved quite successful in prac­ tice. For example, Battisti & Hickey (1984) and Chapman (1987) have used it to demonstrate that a large portion of the sea-level and alongshore current variability off the western United States is associated with wind­ forced coastal-trapped waves. Typical results are shown in Figure 4. The wind forcing occurs along the entire coast but is most effective where the winds are strongest, offnorthern California. Other successful applications of the theory have been carried out for the shelf east of Australia (Church et a11986) and the west Florida shelf (Mitchum & Clarke 1986b). It is fair to say that, in any coastal region where evidence of coastal-trapped-wave phenomena has been seriously sought, it has been found. There is, however, a consistent problem. Although the wave theory works quite well for hindcasting pressure and the alongshore component of current, it has little or no skill at hindcasting density or onshore-offshore currents, (e.g. Chapman 1987). Some reasons for this partial failure are discussed in Section 3. The Gulf of California might prove to be an exception, in that preliminary results suggest that a substantial part of the temperature variability there is related to wave propagation (Merrifield & Winant 1989). Most studies of coastal-trapped-wave generation in nature involve the use of a dynamical model. This approach imposes a bias in the study of data, so it is desirable to seek more empirical approaches. One such effort was made by Chapman et al (1988), who showed that for hindcasting sea level given wind data over a spatial domain, a dynamical model based on (I 5) works, within error, as well as an entirely statistical approach for the region offshore of northern California. This, in effect, says that the first­ order wave-equation model works about as well as any possible linear model, given the inputs. A more ambitious, purely empirical test of coastal-trapped-wave theory Access provided by Old Dominion University on 01/02/18. For personal use only.

Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org was undertaken by Davis & Bogden (1989). They took the approach of trying to explain oceanic variability over the shelf offnorthern California by using a large suite of wind and sea-level information as inputs. They found that a large part of the sea-level variation offCalifornia was coherent and in phase, a result that they tentatively identifiedas a barotropic Kelvin wave of unknown origin. The residual sea-level variation behaved largely as would be expected of coastal-trapped waves. They found that currents near 39°N were influenced by this wave activity and by purely local wind forcing. The local forcing contributed not only to alongshore currents, but also to the onshore current component and to temperature variations. b) :� 30

20 C) 10 � -10 a)OlO t � -20 � !\ -30

-40

-50

(\,J 0.0 50 (,,) 0 � r � \ /'Jill Al v·�. Nli \�Il 40 3D

20 it) 10 @ -0.10 -L. � b 0.10 --r- -1..Q ::c � -20

- � 30

-40 C) -50 0.00 N.' zAkN 50 � t1 t\J% � ::c 40

3D �

20 [;j C) 10 -0.10 � --L. � � ::c TIME (days) � -10 (,.) -20 -30 � -40 � -50 � S;;

Access provided by Old Dominion University on 01/02/18. For personal use only. TIME (days) Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org Figure 4 Observed fluctuations (light lines) and modeled fluctuations (heavy lines) for the continental shelf offnorthern California (with permission ;!5 from Chapman 1987). (a) Bottom pressure (in decibars) from the inner shelf (C2) and the midshelf (C3); (b) alongshore velocity (in centimeters per second) from the inner shelf at 20 m (C2), the midshelf at 55 m (C3), and the outer shelf at 90 m (C4). 402 BRINK Overall, the results are consistent'with those that might be expected from coastal-trapped-wave theory, but they have the added benefit of explaining more variance than wave-type models have been able to rationalize. Up to this point, the discussion has focused on the excitation of coastal­ trapped waves by large-scale alongshore winds. Other possibilities cer­ tainly exist. For example, Webster & Holland (1987) developed a fre­ quency-domain analogue to (15) that can be used to solve for wind-forced motions at any frequency. Gordon & Huthnance (1987) paid particular attention to the forced behavior near the zero group velocity of the free barotropic waves and showed how oscillating currents can be generated in response to a wind impulse. Their results explain observed rotary current variations in the North Sea and thus qualitatively account for some of the variability in onshore-offshore flow, a very rare occurrence in terms of coastal-trapped-wave modeling. Other means of forcing coastal-trapped waves do not involve the wind stress directly. For example, Kroll & Niiler (1976), and subsequently others, have explored the possibility that fluctuations in the midlatitude deep ocean drive shelf waves. No convincing observational evidence for sueh excitation has yet been found. The situation at very low latitudes, however, is different.The equator acts as a wave guide, which is interrupted at the coasts. At the eastern edge of the ocean basin, incoming equatorial wave energy can either be reflected or transmitted poleward along the coast in the form of coastal-trapped waves (Moore 1968, Clarke 1983). Enfield et al (1987) used observations to make a convincing case that equatorial Yanai (mixed Rossby-gravity) waves excite the observed (Smith 1978, Cornejo-Rodriguez & Enfield 1987) free coastal-trapped-wave activity off Peru. Straits or estuaries represent another interesting interruption of the coastal wave guide. A good example is provided by the effect of Hudson Bay and the Hudson Strait on flow over the Labrador shelf. Wright et al (1987) demonstrated that changes in air pressure over the bay lead to fluxes out of the strait and onto the Labrador shelf. At the shelf-strait Access provided by Old Dominion University on 01/02/18. For personal use only. Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org junction, the coastal flux becomes analogous to a wind stress (Middleton 1988) in forcing waves over the shelf. Another interesting example is that of Bass Strait, between mainland Australia and Tasmania. It appears that along-strait winds generate waves within the strait, which in turn excite coastal-trapped waves that propagate northward along the eastern coast of Australia (Clarke 1987, Buchwald & Kachoyan 1987). This explanation for the origin of the northward-propagating waves leaves one interesting question, peculiar to this problem, unanswered: Where does the eastward­ propagating energy along the Australian south coast go? Church & Free- WIND-DRIVEN CURRENTS OVER THE SHELF 403 land (1987) made a convincing case that it does not pass around Tasmania, and Clarke (1987) showed that it cannot pass through Bass Strait. It seems possible that the waves are scattered into the open ocean as they pass the sharp and topographically complex southern tip of Tasmania.

2.5 Summary Coastal-trapped-wave theory provides a powerful conceptual mechanism for understanding large-scale flow over the continental shelf. The under­ lying idea of wind-driven variability propagating in the direction of the coastal-trapped-wave phase velocity is conceptually simple and has been verified repeatedly by observations. The theory works best for pressure (sea level) and alongshore velocity, and only rarely shows any useful skill for density or onshore velocity. This lack of skill is perhaps not surprising in light of the tendency for observed onshorc currents to violate the long­ wave assumption by their magnitude and by having short spatial scales relative to the alongshore velocity (e.g. Kundu & Allen 1976).

3. STOCHASTIC MODELS OF WIND-DRIVEN CURRENTS

The models discussed in the previous section all have the common thread of determinism. That is to say, they could be used, given time series of inputs, to produce time series of outputs that can then be compared with observations. Another approach-the stochastic one-is to use the statistics of input functions to hindcast the statistics of the observed varia­ bility. This approach is, in some ways, more compact and offers some physical insights that might not otherwise be so obvious.

3.1 Wave-Based Models Allen & Denbo (1984) took advantage of the fact that the long-wave wind­ forced problem could be simplified to (15) in order to investigate the Access provided by Old Dominion University on 01/02/18. For personal use only.

Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org statistics of the wind-forced responsc. They further assumed that only a single wave mode is needed. While this assumption may now seem a bit naive, it appeared quite reasonable given information available at that time (e.g. Battisti & Hickcy 1984). For the case of a single mode, or where there are multiple, uncoupled modes (i.e. anm 0 if n "# m), the solution = of (15) is

YnCy,t) = cnbn fOexp(-cn annl1)'o[(y+cn1]),(t-1])]dl1 (17) 404 BRINK for an infinite extent of coast. Once this solution is known, it becomes straightforward to express lagged covariances between Yn at different locations, and between Yn and the wind stress. All of these covariances depend only upon the modal coefficients (bm em and ann) and upon the lagged autocovariance function of the wind-stress forcing, which is pre­ sumed known. Typical results of the Allen & Denbo (1984) calculations are shown in Figure 5. The autocorrelation function of wind stress (R

1200

600

-... E: 0 Rpp � � -600

-1200 -4 -2 0 2 4

1200

600 -... � Rrp � 0

Access provided by Old Dominion University on 01/02/18. For personal use only. � Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org -600

-1200 -4 -2 0 2 4 f (days)

Figure 5 Theoretical correlation diagrams using single-mode theory (with permission from Chapman et al 1988). (Top) Correlation of sea level with sea level at other alongshore (YL) and time (IL) lags. (Bottom) Same for correlation of sea level at (0, 0) with wind stress at other space or time lags. WIND-DRIVEN CURRENTS OVER THE SHELF 405

5 shows the predicted autocorrelation function of sea level Rppo (For a single mode, the function for alongshore velocity is identical.) The tilt of the ellipses indicates alongshore propagation of the signal in the sense of free-wave propagation. The slope of the major axis of the elliptical con­ tours represents a value intermediate between the phase speed of the forcing (infinite in this case) and of the free coastal-trapped wave. In contrast, the correlation between wind and pressure RTp (bottom panel) is not so symmetric. The maximum wind-pressure correlation occurs at a location "upstream" with regard to wave propagation. The location of the maximum correlation lies on a straight line passing through the origin and having a slope exactly equal to the free-wave propagation speed. The Allen & Denbo (1984) results are very powerful in that they can readily be compared with observed statistics. Perhaps most interesting from an obser­ vational standpoint is the finding that pressure (or alongshore velocity) should be best correlated with nonlocal winds, an effect that is indeed observed in nature (e.g. Halliwell & Allen 1984, Winant et al 1987). The Allen & Denbo (1984) analysis has been extended to the case with mUltiple modes by Chapman et al (1988). Two substantial changes occur with this generalization. First, the correlation diagrams become functions of location and of variable. This follows because

n

v = j- 1 L YnFn.(X, z),

n

and because the modal functions Fn are differentfor each n. Also, the tilts of the major axes of the R,p correlation diagrams vary so as to represent a weighted (by modal structure) average of the phase speeds of the free modes. Chapman et al (1988) used these results to show that three-mode correlation diagrams yielded appreciably better theoretical results than single-mode diagrams when compared with observations made off north­ ern California. Access provided by Old Dominion University on 01/02/18. For personal use only. Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org 3.2 More Complete Models The Allen & Denbo (1984) stochastic approach, used with many modes, in principle yields correct results for a broad range of problems where the long-wave limit applies and forcing consists of only an alongshore wind stress independent of cross-shelf distance. This approach, while very enlightening, is thus somewhat limited in its applicability. A more general approach to the linear problem is made by relaxing these assumptions. In this case, the approach used is to Fourier transform the equations of 406 BRINK motion (1) with respect to both time and alongshore direction. This leaves a two-dimensional boundary-value problem to be solved for each fre­ quency and alongshore wavenumber (Spillane 1980, Brink et al 1987), yielding a frequency-wavenumber transfer function. The wind forcing enters both at the coast and at the free surface, while bottom friction provides a damping mechanism. The forcing of the model is prescribed through the frequency-wavenumber spectrum and the cross-shelf structure of the wind stress (Figure 6). One obtains statistics relating currents to

winds as a function of (x, z) location and frequency by integrating results over alongshore wavenumber, using the wind-stress spectrum as a weight­ ing function. Some representative transfer functions at a fixed frequency are shown in Figure 7. Pronounced spikes, associated with rapid phase shifts (not shown), appear in the transfer functions for v and p. These represent the contributions from coastal-trapped-wave resonances. Upon integration, these spikes tend to dominate the amplitude of the response, especially since the wind spectrum is "red" (that is, it has greatest amplitude at long wavelengths). The phase shifts associated with these peaks, however, lead to relatively small co-spectra, or relatively low coherence between wind and v or p. This is the mathematical expression of the fact that propagating energy arrives from locations sufficiently distant that remote, incoherent forcing plays a substantial role. The transfer functions for onshore current and density are not so domi­

nated by resonance spikes. Indeed, the u function increases for short scales and the p function is flat for short scales. This is in distinct contrast to

05 +--'------'------'----t-----'------'-----,---'-----+ / Access provided by Old Dominion University on 01/02/18. For personal use only. Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org

00 -l---r--,-----,--+----.-----,----.------1 -2.2 -1.1 0.0 11 22

Wavenumber feyk";i'x f03)

Figure 6 Frequency-alongshore wavenumber spectrum of alongshore wind stress for the West Coast of the United. States during summer 1982. Units are the base 10 log of (dynes cm-2) (cycles km-1)-1 (cycles day-l)-l (with permission from Brink et aI1987). WIND-DRIVEN CURRENTS OVER THE SHELF 407

25

20

....

�15 15 ,<. . � ',"" :;::. � '," .. " � ,�<0 • .. � � � 010 �, 10 10 � � ,. .. "- ' "

-10 -5 0 5 10 -10 -5 0 5 10 JR(cm-' x fO} J.R(cm'x fO)

Figure 7 Amplitudes of wind-stress transfer functions of midshelf oceanic variables for w = 1.08 x 10- 5 s- I for northern California (with permission from Brink et al 1987).

the v or p functions, which decline for short scales (large alongshore

wavenumber I). This tendency for increased contribution to u at large / only occurs in the realistic case of the wind stress varying with distance

offshore. Physically, the trends in the transfer functions imply that u and p are most sensitive to very short alongshore scales in the wind stress­ scales so short as to be unresolved by existing estimates of the wind spectrum. Nonetheless, aircraft-based measurements of the wind field indi­ cate that there is oftensubstantial energy in scales as short as 10 km (e.g. Stuart 1981, Winant et aI1988). The results of Brink et al (1987) show fairly good agreement with observations for alongshore velocity and pressure. This is consistent with the quality of the wind input, which resolved large scales well but small Access provided by Old Dominion University on 01/02/18. For personal use only.

Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org scales poorly. Also, as with many forced-wave calculations (Figure 4b; e.g. Chapman 1987), the amplitude of the alongshore velocity was also

somewhat underpredicted. The model performed poorly for u and p, presumably at least partly because the wind spectrum did not resolve the shortest scales. It is difficult to tell at this time to what extent the linear model physics is also to blame for this inadequacy. The advantages with stochastic models are that they relax some of the restrictive approximations associated with most forced-wave models, and that they allow different physical insights than those of the wave models. The two approaches are complementary. 408 BRINK

4. CRITIQUE

The above sections really address only a limited physical regime: fluc­ tuating motions driven by fluctuatingwinds in a linearized, nearly inviscid context. While this type of model is successful in some regards, it leaves considerable room for improvement. It thus seems reasonable to consider briefly a few topics that could not be discussed fully here. In the ocean, turbulent surface and bottom boundary layers almost always exist. Since these· are typically of 0(10 m) thick, they occupy a substantial part of the shelf water column. Although some of the effects of these boundary layers are accounted for above, their considerable cur­ rent and thermal perturbations are not considered. Turbulent boundary layers over the shelf have dynamics that are at least two-dimensional and hence substantially more complex than in simpler settings. The most obvious effect occurs near the surface, where horizontal temperature gra­ dients associated with upwelling can lead to the generation of variable boundary-layer thickness and of near-surface fronts (de Szoeke & Richman 1984, Rudnick & Davis 1988). Likewise, flowin a bottom boundary layer beneath a stratified interior and above a sloping bottom leads to buoyancy­ induced anomalies in the boundary-layer structure (Weatherly & Martin 1978). In either case, it is clear that turbulent boundary-layer processes over the shelf and slope will be complex and are important components of a complete coastal-circulation model. Fronts are a common phenomenon in the coastal ocean. Once a front is formed its advection by wind-driven currents can complicate the thermal response of the ocean (e.g. Ou 1984). Further, the frontal distortion of the ambient vorticity distribution can modify coastal-trapped waves and allow for the existence of new, frontal-trapped modes (e.g. Luther & Bane 1985). Finally, various types of coastal fronts appear to be hydrodynamically unstable (e.g. Flagg & Beardsley 1978, Narimousa & Maxworthy 1985, Barth 1989a,b, Gawarkiewicz 1989). Instabilities lead to current fluc­ tuations that are not directly related to the wind, and that may prove to Access provided by Old Dominion University on 01/02/18. For personal use only.

Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org be a substantial mechanism for the exchange of water and materials across the continental margin. Addressing the problem of steady, wind-driven currents over the con­ tinental margin at first seems to be as simple as taking the steady limit of (1). This is not true, since the resulting suppression of the vertical velocity (Ie) will then lead to singularities. Thus, it is necessary to add new physics to any steady, stratified model. One approach has been to add horizontal and vertical mixing, a procedure that leads to results that can be quali­ tatively thought of as the steady limit of coastal-trapped-wave generation (McCreary & Chao 1985). Although wind-driven variability, whether remote or local, seems to dominate alongshore current fluctuations over WIND-DRIVEN CURRENTS OVER THE SHELF 409 the shelf, it is not clear that steady winds play a similarly dominant role. Various influences of the deep ocean on the shelf cannot be entirely discounted (Huthnance 1984, McCreary et al 1986, Kelly & Chapman 1988), and estuarine outflows can also drive alongshore currents (e.g. Beardsley & Hart 1978) and coastal-trapped waves (Whitehead & Chap­ man 1986). Further, nonlinear processes can lead to the generation of a mean flowdriven by fluctuating currents (Loder 1980, Haidvogel & Brink 1986, Samelson & Allen 1987). It is, in fact, difficult to use observations to test models of mean currents. This is so, at least partly, because the absence of fluctuations (by definition of "mean") implies only one real­ ization and hence too few degrees of freedom to make any meaningful statistical test.

5. CONCLUSIONS

It would certainly be fair to conclude that we have developed a sophis­ ticated understanding (and predictive capability) for variations in sea level and alongshore current over the continental shelf for typical periods ranging from the inertial period out to about 20 days. Despite the impres­ sive results of existing models (e.g. Figure 4), there are still some trouble­ some problems. First, existing models tend to underpredict the amplitude of observed fluctuations of alongshore current and sea level, typically by about 10-50%. Second, such models tend to underestimate grossly the observed strength of density and cross-shelf current changes. The latter difficulty appears to be at least partly associated with the long-wave assumption, which suppresses the small spatial scales that typify actual cross-shelf currents. Any of these problems could also be related to frontal, nonlinear, or turbulent processes that are not included explicitly in most existing wind-driven models. Our present understanding of currents over the continental shelf, while satisfying in some regards, is clearly inadequate. This is so because many problems of interest, such as oceanic frontogenesis, coastal upwelling, and Access provided by Old Dominion University on 01/02/18. For personal use only.

Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org the dispersal of pollutants, involve the exchange of properties across the continental margin. Understanding transport across the continental margin requires an understanding of onshore-offshore currents. Such an understanding is presently lacking and is likely to be the focus of con­ siderable research in the coming years.

ACKNOWLEDGMENTS David Chapman and Steven Lentz provided valuable comments. Support for this effort was provided by the Office of Naval Research (ONR) through the Coastal Sciences Program, code 1122CS. 410 BRINK

Literature Cited

Allen, J. S. 1980.Models of wind-driven cur­ 87-24, Woods Hole Oceanogr. Inst., rents on the continental shelf. Annu. Rev. Woods Hole, Mass. 119 pp. FluidMech. 12: 389-433 Brink, K. H., Chapman, D. c.. Halliwell. G. Allen, J. S., Denbo, D. W. 1984. Statistical R. Jr. 1987.A stochastic model for wind­ characteristics of the large-scale response driven currents over the continental shelf. of coastal sea level to atmospheric forcing. J. Geophys.Res. 92: 1783-97 J. Phys. Oceanogr.14: 1079-94 Brooks, D. A., Mooers, C. N. K. 1977.Free, Bane, J. M. Jr., Brooks, D. A., Lorenson, stable continental shelf waves in a sheared, K. R. 1981. Synoptic observations of the barotropic boundary current. J. Phys. three-dimensional structure and propa­ Oceanogr. 7: 380-88 gation of Gulf Stream meanders along the Buchwald, V. T., Kachoyan, B. J. 1987. Shelf continental margin. J. Geophys. Res. 86: waves generated by a coastal flux. Aust.J. 6411-25 Mar. Freshwater Res.38: 429-37 Barth, J. I 989a. Stability of a coastal upwell­ Chapman, D. C. 1982. Nearly trapped inter­ ing front, I, Model development and sta­ nal edge waves in a geophysical ocean. bility theorem. J. Geophys. Res.94: 10,844- Deep-Sea Res.29(4A): 525--33 56 Chapman, D. C. 1987. Application of wind­ Barth, J. 1989b. Stability of a coastal upwell­ forced, long, coastal-trapped wave theory ing front, 2, Model results and comparison along the California coast. J. Geophys. with observations. J. Geophys. Res. 94; Res. 92: 1798-1816 10,857-83 Chapman, D. c., Lentz, S. J., Brink, K. H. Battisti, D. S., Hickey, B. M. 1984. Appli­ 1988. A comparison of empirical and cation of remote wind-forced coastal dynamical hindcasts of low-frequency, trapped wave theory to the Oregon and wind-driven motions over a continental Washington coasts. 1. Phys.Oceanogr. 14: shelf. 1. Geophys.Res. 93: 12,409-22 887-903 Church, J. A., Freeland, H. 1. 1987. The Beardsley, R. C., Hart, J. 1978. A simple energy source for the coastal-trapped theoretical model for the flow of an estu­ waves in the Australian Coastal Experi­ ary onto a continental shelf. J. Geophys. ment region. J. Phys. Oceanogr. 17: 289- Res.83: 873-83 300 Brink, K. H. 1982. A comparison of long Church, J. A., White, N. J., Clarke, A. J., coastal trapped wave theory with obser­ Freeland, H. J., Smith, R. L. 1986. Coast­ vations off Peru. J. Phys. Oceanogr. 12: al-trapped waves on the east Australian 897-913 continental shelf. Part II: Model veri­ Brink, K. H. 1983a. Some effects of strati­ fication. J. Phys. Oceanogr. 16: 1945-57 fication on long trench waves. J. Phys. Clarke, A. J. 1977. Observational and Oceanogr.13:496-500 numerical evidence for wind-forced coast­ Brink, K. H. 1983b. Low-frequency free al trapped long waves. J. Phys. Oceanogr. wave and wind-driven motions over a sub­ 7: 231-47 marine bank. J. Phys. Oceanogr.13: 103- Clarke, A. J. 1983. The reflection of equa­ 16 torial waves from oceanic boundaries. J. Brink, K. H. 1989. Energy conservation in Phys. Oceanogr.13: 1193-1207 coastal-trapped wave calculations. J. Clarke, A. J. 1987. Origin of coastally Phys. Oceanogr.19: 1011-16 trapped waves observed during the Aus­ Brink, K. H. 1990. On the damping of free tralian Coastal Experiment. J. Phys. coastal-trapped waves. Phys. Oceanogr. Oceanogr. 17:1847-59

Access provided by Old Dominion University on 01/02/18. For personal use only. J.

Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org 20: 1219-25 Clarke, A. J., Brink, K. H. 1985. The Brink, K. H., Allen, 1. S. 1978. On the effect response of stratified, frictional flow of of bottom friction on barotropic motion shelf and slope waters to fluctuating large­ over the continental shelf. J. Phys. scale low-frequency wind forcing. J. Phys. Oceanogr.8:919-22 Oceanogr.15:439-53 Brink, K. H., Allen, 1. S. 1983. Reply (to Clarke, A. J., Van Gorder, S. 1986. A T. J. Simons' comments on K. H. Brink method for estimating wind-driven fric­ and J. S. Allen, J. Phys. Oceanogr. 8: tional, time-dependent, stratified shelf and 919-22,1978).J. Phys. Oceanogr.13: 149- slope water flow. J. Phys. Oceanogr. 16: 50 1013-28 Brink, K. H., Chapman, D. C. 1987. Pro­ Cornejo-Rodriguez, M., Enfield,D. B. 1987. grams for computing properties of Propagation and forcing of high-fre­ coastal-trapped waves and wind-driven quency sea level variability along the west motions over the continental shelf and coast of South America. J. Geophys. Res. slope (second edition). Rep. No. WHO[- 92:14,323-34 WIND-DRIVEN CURRENTS OVER THE SHELF 411

Crawford, W. R., Thomson, R. E. 1984. Huthnance, J. M. 1978. On coastal trapped Diurnal-period continental shelf waves waves: analysis and numerical calculation along Vancouver Island: a comparison of by inverse iteration. J. Phys. Oceanogr. 8: observations with theoretical models. J. 74-92 Phys. Oceanogr. 14: 1629-46 Huthnance, J. M. 1984. Slope currents and Daifuku, P. R., Beardsley, R. C. 1983. The "JEBAR." J. Phys. Oceanogr. 14: 795-8 10 KI tide on the continental shelf from Huthnance, J. M., Mysak, L. A., Wang, Nova Scotia to Cape Hatteras. J. Phys. D.-P. 1986. Coastal trapped waves. In Oceanogr. 13: 3-17 Baroclinic Processes on Continental Davis, R. E., Bogden, P. S. 1989. Variability Shelves. Coast. Estuar. Sci. 3, ed. C. N. on the California shelffo rced by local and K. Mooers, pp. 1-18. Washington, DC: remote winds during the Coastal Ocean Am. Geophys. Union Dynamics Experiment. J. Geophys. Res. Kelly, K. A., Chapman, D. C. 1988. The 94: 4763-83 response of stratified shelf and slope de Szoeke, R. A., Richman, J. G. 1984. On waters to steady offshore fo rcing. J. Phys. wind-driven mixed layers with strong hori­ Oceanogr. 18: 906-25 zontal gradients-a theory with appli­ Kroll, J., Niiler, P. P. 1976. The transmission cation to coastal upwelling. J. Phys. and decay of barotropic topographic Oceanogr. 14: 364-77 Rossby waves incident on a continental Enfield, D. B., Cornego-Rodriguez, M., shelf. J. Phys. Oceanogr. 6: 432-50 Smith, R. L., Newberger, P. A. 1987. The Kundu, P. K., Allen, J. S. 1976. Some three­ equatorial source of propagating varia­ dimensional characteristics of low-fre­ bility along the Peru coast during the quency current fluctuations near the 1982-1983 EI Nino. J. Geophys. Res. 92: Oregon coast. J. Phys. Oceanogr. 6: 181- 14,335-46 99 Flagg, C. N., Beardsley, R. C. 1978. On the Loder, J. 1980. Topographic rectification of stability of the winter shelf water/slope tidal currents on the side ofGeorges Bank. water front south of New England. J. Geo­ J. Phys. Oceanogr. \0: 1399-1416 phys. Res. 83: 4623-31 Lopez, M., Clarke, A. J. 1989. The wind­ Flather, R. A. 1988. A numerical model driven shelf and slope water flow in terms investigation of tides and diurnal-period of a local and a remote response. J. Phys. continental shelf waves along Vancouver Oceanogr. 19: 1091-1101 Island. J. Phys. Oceanogr. 18: 115-39 Luther, M. E., Bane, J. M. Jr. 1985. Mixed Gawarkiewicz, G. 1989. Linear stability instabilities in the Gulf Stream over the models of shelfbreak fr onts. PhD thesis. continental slope. J. Phys. Oceanogr. 15: Univ. Dei., Newark. 166 pp. 3-23 Gill, A. E., Clarke, A. J. 1974. Wind-induced McCreary, J. P. Jr., Chao, S.-Y. 1985. Three­ upwelling, coastal currents and sea level dimensional shelf circulation along an changes. Deep-Sea Res. 21: 325-45 eastern ocean boundary. J. Mar. Res. 43: Gill, A. E., Schumann, E. H. 1974. The gen­ 13-36 eration of long shelf waves by the wind. J. McCreary, J. P. Jr., Shetye, S. R., Kundu, P. Phys. Oceanogr. 4: 83-90 K. 1986. Thermohaline forcing of eastern Gill, A. E., Schumann, E. H. 1979. Topo­ boundary currents: with application to the graphically induced changes in the struc­ circulation offthe west coast of Australia. ture of an inertial coastal jet: application J. Mar. Res. 44: 71-92 to the Agulhas Current. J. Phys. Ocean­ Merrifield, M. A., Winant, C. D. 1989. Shelf ogr. 9: 975-91 circulation in the Gulf of California: a Gordon, R. L., Huthnance, J. M. 1987. description of the variability. J. Geophys.

Access provided by Old Dominion University on 01/02/18. For personal use only. Storm-driven continental shelf waves over Res. 94: 18,133-60 Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org the Scottish continental shelf. Cont. Shelf Middleton, J. F. 1988. Long shc1fwaves gen­ Res. 7: lOIS -48 erated by a coastal flux. J. Geophys. Res. Haidvogel, D. B., Brink, K. H. 1986. Mean 93: 10,724-30 currents driven by topographic drag over Mitchum, G. T., Clarke, A. J. 1986a. The the continental shelf and slope. J. Phys. frictional nearshore response to forcing by Oceanogr. 16: 2159-71 synoptic scale winds. J. Phys. Oceanogr. Halliwell, G. R., Allen, J. S. 1984. Large­ 16: 934-46 scale sea level response to atmospheric Mitchum, G. T., Clarke, A. J. 1986b. Evalu­ fo rcing along the west coast of North ation of frictional, wind-forced long-wave America, summer 1973. J. Phys. Ocean­ theory on the west Florida shelf. J. Phys. ogr. 14: 864-86 Oceanogr. 16: 1029-37 Huthnance, J. M. 1975. On trapped waves Moore, D. W. 1968. Planetary-gravity waves over a continental shelf. J. Fluid Mech. 69: in an equatorial ocean. PhD thesis. Har­ 689-704 vard Univ., Cambridge, Mass. 412 BRINK

Mysak, L. A., LeBlond, P. H., Emery, W. 1. Up welling, ed. F. A. Richards, pp. 32-38. 1979. Trench waves. J. Phys. Oceanogr. 9: Washington, DC: Am. Geophys. Union 100 1-13 Wang, D.-P., Mooers, C. N. K. 1976. Coast­ Mysak, L. A., Willmott, A. J. 1981. Forced al trapped waves in a continuously strati­ trench waves. J. Phys. Oceanogr. 11: 1481- fied ocean. J. Phys. Oceanogr. 6: 853-63 1502 Weatherly, G. L., Martin, P. J. 1978. On Narimousa, S., Maxworthy, T. 1985. Two­ the structure and dynamics of the oceanic layer model of shear-driven coastal up­ bottom boundary layer. J. Phys. Ocean­ welling in the presence of bottom topog­ ogr. 8: 557-70 raphy. J. Fluid Mech. 159: 503-31 Webster, 1., Holland, D. 1987. A numerical Ou, H. W. 1984. Wind-driven motion near a method for solving the forced baroclinic shelf-slope front. J. Phys. Oceanogr. 14: coastal-trapped wave problem of general 985-93 form. J. Almos. Ocean. Te chnol. 4: 220- Power, S. B., Middleton, J. H., Grimshaw, 26 R. H. J. 1989. Frictionally modified con­ Whitehead, J. A., Chapman, D. C. 1986. tinental shelf waves and the subinertial Laboratory observations of a gravity cur­ response to wind and deep-ocean forcing. rent on a sloping bottom: the generation of shelf waves. Fluid Mech . 172: 373-99 J. Phys. Oceanogr. 19: 1486-1506 J. Rudnick, D. L., Davis, R. E. 1988. Fronto­ Wilkin, J. L. 1988. Scattering of coastal­ genesis in mixed layers. J. Phys. Oceanogr. trapped waves by irregularities in coastline 18:434-57 and topography. PhD thesis. Mass. Inst. Technol.jWoods Hole Oceanogr. Inst., Samelson, R. M., Allen, J. S. 1987. Quasi­ Cambridge/Woods Hole, Mass. (available geostrophic topographically generated as Rep. No. WHO/-88-47, Woods Hole mean flow over the continental margin. J. Oceanogr. Inst., Woods Hole, Mass.) Phys. Oceanogr. 17: 2043-64 Winant, C. D., Beardsley, R. C., Davis, R . . Schumann, E. H., Brink, K. H. 1990. Coast­ E. 1987. Moored wind, temperature and al-trapped waves off the coast of South current observations made during CODE- Africa: generation. propagation and cur­ 1 and CODE-2 over the northern Cali­ rent structures. J. Phys. Oceanogr. In press fo rnia continental shelf and upper slope. Smith, R. L. 1978. Poleward propagating J. Geophys. Res. 92: 1569-1604 perturbations in currents and sea level Winant, C. D., Dorman, C. E., Friehe, C. A., along the Peru coast. J. Geophys. Res. 83: Beardsley, R. C. 1988. The marine layer off 6083-92 northern California: an example of super­ Spillane, M. C. 1980. A model of wind-forced critical channel flow. 1. Almos. Sci. 45: viscous circulation near coastal boundaries. 3588-3605 PhD thesis. Oreg. State Univ., Corvallis. Wright, D. G., Greenberg, D. A., Majaess, 88 pp. F. G. 1987. The influence of bays on Stuart, D. W. 1981. Sea-surface tem­ adjusted sea level over adjacent shelves peratures and winds during Joint II. Part with application to the Labrador Shelf. J. II. :remporal fluctuations. In Coastal Geophys. Res. 92: 14,610-20 Access provided by Old Dominion University on 01/02/18. For personal use only. Annu. Rev. Fluid Mech. 1991.23:389-412. Downloaded from www.annualreviews.org