Journal of Oceanography, Vol. 53, pp. 207 to 229. 1997

Review

On the Theories that Underlie Our Understanding of Continental Shelf Circulation

G. T. CSANADY

Old Dominion University, Norfolk, VA 23529, U.S.A.

(Received 10 January 1996; accepted 1 October 1996)

Our present understanding of continental shelf circulation rests on a few conceptual Keywords: models which yield highly idealized mathematical representations of shelf topography ⋅ Continental shelf circulation, and dynamics. This review examines several of such models in the light of experimental ⋅ evidence accumulated within the past two decades or so: what are the successes of the coastal oceanogra- models, and what overidealizations underlie their shortcomings. The coastal jet model, phy, ⋅ coastal upwelling, coastally trapped wave models, the dynamic height model, Joint Effect of Baroclinity and ⋅ coastally trapped Relief (JEBAR) models, self-advection of density and front models are discussed in turn. waves, The final section on the interaction of shelf and boundary currents draws attention to the ⋅ thermohaline lack of satisfactory models for this important aspect of shelf dynamics. coastal currents.

1. Introduction it turns out that the problem of the open boundary condition “Coastal” and “Deepwater” physical oceanography resurfaces in the different theories and is the root cause of have developed separately on account of disparate vertical their principal weaknesses. It is not too far-fetched to con- and horizontal scales, and the consequent dissimilar role of clude that these theories in one way or another fail to account such forcing effects as wind stress or freshwater runoff. for observation, because they neglect interactions with deeper Waters over continental shelves abut waters of the deep waters offshore. A study of such interactions promises to be ocean, however, so that separate treatment of their movements a rewarding research activity for the future. requires a decision on where the one ends and the other begins. Furthermore, such treatment implies a priori that 2. Coastal Upwelling continental shelf circulation can be understood in isolation, Owing to its ecologic and economic importance, coastal that any interactions with the deep ocean can be suitably upwelling is the most discussed and researched phenomenon represented in shelf circulation models. Although all our in all of coastal oceanography. In a recent review, Huyer theories of shelf circulation implicitly assume this, condi- (1990) gives a thorough and very readable summary of the tions to be imposed at the “open” boundary separating the physical processes involved in wind-induced upwelling, continental shelf from the deep ocean remain the Achilles with emphasis on the major upwelling regions off the west heel of shelf circulation models, whether analytical or nu- coast of North and South America, distinguished “by their merical. high productivity and their cool and foggy weather”. High The separation of continental shelf and deep ocean is productivity comes from nutrients brought to the surface by already drastic surgery. Irregular topography, water mass upwelling, cool and foggy weather from the associated cold distribution, wind fields force further idealization. Theories water temperatures. Although of lesser economic importance, of shelf circulation have employed different levels of ide- transient, wind-induced inner shelf upwelling (and its obvious alization in pursuing two different objectives: one, to simulate counterpart, downwelling) is commonplace on most other complex reality as faithfully as possible, two, to understand shelves (Pettigrew and Murray, 1986) and along the coasts prominent observed phenomena in terms as simple as pos- of inland seas such as the Baltic (Walin, 1972) or the Great sible. The latter approach rests on “conceptual models”, and Lakes (Mortimer, 1963). builds simple theories to explain the essential physics of A great deal of very detailed information has come to phenomena. The purpose of this article is to review several light on coastal upwelling since the early 60-s. Reviews of such “minimally complex” theories, and to adjudge how the major field programs by Smith (1981) and Brink et al. well and how completely they account for the observed (1983), and of upwelling-related phenomena in continental phenomena they are designed to explain. Not surprisingly, shelf circulation by Huyer (1990), summarize the many

207 Copyright  The Oceanographic Society of Japan. Fig. 1. Temperature and velocity cross-shelf sections in the CODE II experiment: (a) Top row, two days after the start of upwelling- favorable wind; (b) Bottom row, three days later. From Lentz (1987). interesting findings. With a view to relating observations to waters come from, to replace those moving offshore in the theory, we briefly sketch here occurrences in a prototype surface layer, but Lentz’s analysis of mass balance shows it upwelling event, the “spring transition” along the U.S. west to be two-dimensional in the cross-shore plane, on the coast, particularly well documented in the course of the average during the five days of the spring transition. In CODE experiment off northern California (Lentz, 1987, and 130 m depth the flow is offshore in the top 45 m or so, other papers in the same issue). onshore below, the depth-integrated transports balancing. Temperature gradients in the top 45 m are significant even 2.1 Prototype coastal upwelling at the beginning of the upwelling event. The onshore flow is Figure 1(a) shows the distribution of cross-shore and drawn from 200Ð300 m depth over the continental slope, alongshore velocity and temperature less than two days after with little influence on waters deeper than 400 m. Lentz the arrival of strong upwelling favorable wind in the central (1987) gives a wealth of other detail. transect of CODE, from Lentz’s paper. The well-known This relatively straightforward picture gets more signatures of coastal upwelling are all present: upward complicated even before the first relaxation of the upwelling- sloping isotherms, a strong coastal jet, rapid offshore mo- favorable wind: 3 days later the sections look different, Fig. tion in the surface layer with strong divergence close to 1(b). The cross-shelf velocity has a stagnation point a few shore. The displacement of the isotherms is large: the km from shore, and shoreward from there the alongshore 11.5°C isotherm has moved in the preceeding 24 hours from velocity is now barotropic and poleward, opposite to the a more or less constant depth of 15 m to the surface, wind. Lentz (1987) and Send et al. (1987) demonstrate that intersecting it now at 7 km from shore. During the following this flow component is pressure-gradient driven. Later in the 24 hours the coastal jet moves offshore by some 10 km while upwelling season, when upwelling and relaxation cycles the 11.0°C isotherm also moves to the surface and offshore. alternate, alongshore pressure gradient driven flow comes to Cross-shelf advection of both temperature and momentum supply much of the upwelling fluid mass which is perma- thus plays a major role. The figures do not show where the nently removed seaward (Send et al., 1987; Winant et al.,

208 G. T. Csanady 1987). On this longer time scale the mass balance thus becomes three-dimensional, the convergence of the along- shore transport balancing the surface layer divergence. How well does existing theory account for the diverse phenomena observed on occasion of wind-driven coastal upwelling, and more broadly, in the course of upwelling- downwelling cycles caused by variable wind?

2.2 Two layer coastal jet model The classical, minimally complex model of transient coastal upwelling is due to Charney (1955), the model originally meant to simulate the Gulf Stream. Transplanted to coastal oceanography in the sixties, it represents the coastal zone as a constant depth sea with a vertical coast, a light layer overlying a heavy one, extending from the coast Fig. 2. Coastal upwelling and coastal jet generation by alongshore to “infinity”. The wind stress is suddenly applied, the wind wind, according to the two-layer Charney model. From Csanady (1977). force distributed evenly over the light surface layer. Ev- erything is supposed the same in every cross-section inde- pendently of alongshore distance, including in particular the pressure distribution. Postulating time-independent cross- direct consequence of postulated alongshore uniformity. shore motion, the model portrays events in coastal upwelling Dynamically, linear superposition of barotropic and as illustrated in Fig. 2. Alongshore wind accelerates the baroclinic solutions is responsible for the entire scenario. surface layer: with interface friction absent, the surface fluid One weakness of this model is the postulate of time- speeds up, as earth rotation slowly deflects the motion independent cross-shore motion: this cannot just material- seaward. The coast, however, suppresses cross-shore motion ize as the wind begins to blow. The full analytical solution over an e-folding distance of c/f, the baroclinic radius of of the same problem, impulsively applied wind to a two- deformation (c is the phase speed of internal waves, f layer frictionless fluid bounded by a vertical wall (Crépon, Coriolis parameter). Farther out cross-shore Ekman trans- 1967), shows essentially the same behavior, with some gaps port in the surface layer comes to balance the wind stress. filled in: the cross-shore motion is set up in the wake of an Within the range of the coast’s influence wind stress re- internal gravity wave propagating offshore, in a period of mains partly unbalanced and steadily accelerates an along- order fÐ1, Ekman transport offshore arises in the same period, shore “coastal jet” in the surface layer, with highest velocity and is accompanied by inertial oscillations. at the coast. The near-shore divergence of the cross-shore Both Charney’s and Crépon’s model are linearized, motion in the surface layer raises the interface under the valid only for small interface displacements (compared to coastal jet, causing convergence in the lower layer, and either layer depth). A minimally complex extension supposes inducing onshore motion farther seaward. In the surface again two layers and constant total depth, but allows a large layer, the cross-shore pressure gradient associated with the wind stress force to be exerted impulsively in a short time raised interface balances the Coriolis force acting on the (compared to fÐ1, Csanady, 1977). Following the impulse, coastal jet. In the lower layer, the Coriolis force acting on the the two layers adjust to geostrophic equilibrium, conserving onshore motion accelerates the fluid in the wind direction. potential vorticity. A large enough wind impulse causes the Offshore from the coastal jet, but at distances much shorter thermocline to rise to the surface, and to move offshore. The than the barotropic radius of deformation, approximate two- displacement from the coast of the surfaced thermocline is dimensional mass balance prevails, so that the depth-inte- proportional to the wind stress impulse. The “coastal jet” is grated alongshore Coriolis force (nearly) vanishes. The now shifted offshore, and becomes a “frontal jet”, associ- water column here gains net windward momentum, the gain ated with the surface “front” (sharp horizontal density (nearly) equaling the wind stress impulse. Much farther gradient) of the upwelled interface. from the coast, at distances large compared to the barotropic It is at once evident that the extended coastal jet model radius of deformation, the Coriolis force of cross-shore captures key features of the CODE spring transition in its Ekman transport in the surface layer still balances the wind early phase, portrayed in Fig. 1(a). The surface layer coastal stress, but the lower layer is stationary. The water mass jet is prominent, its depth-integrated momentum comparable leaving the shelf via Ekman transport comes from the to the wind impulse initially, while later the jet moves continuous lowering of sea level over an e-folding distance offshore. Isotherms rise to the surface and also move offshore. equal to the large barotropic radius of deformation. Mass Mass balance is approximately two-dimensional, so that the balance is thus two-dimensional in the cross-shore plane, a acceleration of the lower layers may be ascribed to the

On the Theories that Underlie Our Understanding of Continental Shelf Circulation 209 Coriolis force acting on the onshore flow. The divergence of layer, as in the two-layer model, the series can be summed, the surface flow is confined to about 10 km from the shore, and reveal a response in every way similar to the two-layer a distance of order c/f, given reasonable assumptions on model’s (Csanady, 1982a, pp. 67–83). In particular, the surface layer depth and density defect. (Alternatively, c may surface mixed layer carries the full Ekman transport offshore, be estimated from the phase speed of the internal Kelvin while the return flow is evenly distributed over all the wave, Huyer et al., 1987). stratified layers, regardless of the density distribution. On The principal shortcoming of the two layer coastal jet the other hand, if stratification extends to the surface, and if model is that it says nothing about the depth range of the the wind-force affects a surface stratified layer, the onshore upwelling circulation, 200Ð300 m according to Lentz (1987) flow under the coastal jet comes from layers immediately in the spring transition study. In the completely different below the surface (wind-driven) layer, according to the physical setting of the New York Bight, summer upwelling results of Yoon and Philander (1982). and associated offshore Ekman transport are accompanied by much shallower onshore flow, causing a salinity maxi- 2.4 Summary: flaws of the extended coastal jet model mum layer in the 30Ð70 m range to intrude shoreward over The remarkable success of the extended coastal jet the shelf (Gordon et al., 1976). In the case of the Peru shelf model in simulating the initial phase of upwelling is marred the “apparent source depth of upwelling waters” is 50–100 by the uncertainty of source water depth. Deepwater models m, according to a thorough analysis by Huyer et al. (1987). fail to resolve the uncertainty: in the discrete layer model the Moreover, the source depth remains the same in an El Niño source water depth is arbitrary; in the Yoon-Philander model episode as in normal times, in spite of a greatly depressed it derives from the summation of internal modes and pre- thermocline. The “source” is of course located offshore in sumably depends on the details of the stratification, but the deep water, and the uncertainty affecting its depth comes calculations do not reveal in what manner. Even if they did, from disregarding any interactions between the shelf and the the connection between the deepwater model and the shelf- deep ocean. slope region would remain unclear. We have seen in Fig. 1(b) that the response of shelf 2.3 The source depth of upwelling waters waters to wind forcing changes qualitatively in one or two Deepwater models give some idea of where the source days, when alongshore variability and pressure gradients waters may come from, although not a satisfactory answer. become important. The (qualified) validity of the extended A three-layer extension of the finite amplitude adjustment coastal jet model is thus limited to times not much greater model under large impulsive wind stress, with total depth than fÐ1. still constant (Csanady, 1982b) is one such model, if only As emphasized before, mass balance is two-dimensional bottom layer is taken to be much deeper than the top two. by hypothesis in the cross-shore plane in all versions of the The second (middle) layer of this model represents the coastal jet model, lowering of the sea level (and of interior “thermocline”, the waters of which, in the case of full isopycnal surfaces) on the distance scale of the barotropic upwelling, come to occupy the coastal zone. This is so radius of deformation supplying fluid for offshore Ekman because “the wind peels off the surface layer over which it transport. There is no direct evidence for this, and the acts, bringing only the next lighter layer to the surface, implied distance scale depends on the total depth of the two- without causing the second (or third or fourth) interface to layer model, so that it is arbitrary. Again this problem arises move upward by more than a fraction of its initial depth”. from failure to connect the shelf model to the deep ocean. The question remains, of course, what the “next lighter layer” is in a given (continuously) stratified body of water. 3. Coastally Trapped Waves Somewhat more useful is a constant depth, linearized A phenomenon rivaling coastal upwelling in the at- model with continuous stratification. Analytical solutions tention it has received from oceanographers is the sponta- portraying upwelling response to suddenly applied wind, in neous alongshore propagation of sea level changes and a constant depth stratified ocean, have been discussed by associated flow patterns, collectively known as coastally several authors, including Yoshida (1967), Csanady (1982a) trapped waves. In contrast to the two-dimensional coastal jet and Yoon and Philander (1982). The results appear as sums models, alongshore variation is an essential element in all over one barotropic mode and an infinite series of baroclinic coastally trapped wave models, as is cross-shore depth modes. To gain a true picture of the response, the series have variation in some of them. The simplest such model is the to be summed, or at least many modes have to be added up: barotropic Kelvin wave in a constant depth sea, familiar Yoon and Philander summed 192 modes. An important from tidal wave theory (Taylor, 1920). Long, slowly difference arises from the choice of stratification versus propagating coastally trapped waves (long compared to wind-force distribution. If an unstratified surface mixed shelf width, slow compared to gravity waves) share the layer is taken to overlie the stratified layers, as observation property of the Kelvin wave of propagating in the suggests, and if the wind force is distributed evenly over that “rightbounded” direction in the northern hemisphere,

210 G. T. Csanady leftbounded south of the equator. First we briefly review was signal propagation related to the movement of weather what observations tell us about these remarkable features of systems or of any other forcing. shelf circulation. In many more investigations, in diverse locations, although individual propagating events were not identified, 3.1 Observations of propagating events there was statistical evidence for signal propagation, or at The pioneering study to show slow alongshore propa- least for the downwave displacement of response to forcing. gation of pressure signals, independently of any translating One recent example is a study of Schwing (1989), from the forcing, was Hamon’s (1962) analysis of coastal sea level on rugged east coast of Canada, where bottom topography and the east coast of Australia, revealing a northward propagating friction immensely complicate shelf circulation. Strub et al. signal of several days’ period. Another classic of the same (1987) discussed the possible influence of alongshore period is Mortimer’s (1963) demonstration (from water propagation on the CODE spring transition observations, intake temperatures) that temperature signals propagate but only found such indirect evidence as a maximum response cyclonically around Lake Michigan at speeds conforming to in sea level “several hundred kilometers” north of the an internal Kelvin wave model. Early observations off the maximum of southward wind stress (their figures show Oregon coast also showed sea level changes slowly propa- about 200 km). Winant et al. (1987) reported, in the same gating northward (Mooers and Smith, 1968). These were vein, that alongshore volume transport is best correlated later found to be accompanied by changes in the alongshore with wind stress about 30 km to the south. Brown et al. (1987) velocity, more or less constant with depth, in geostrophic concluded, on the other hand, that the pressure field is very balance with cross-shore pressure gradients (Smith, 1974; well correlated with the local wind, and, as one would Kundu and Allen, 1976; Allen, 1980). Off the Peru coast, sea expect, propagates slowly with it. They confirmed that level changes propagate southward coupled with tempera- alongshore velocity changes occur in geostrophic balance ture as well as velocity signals (Smith, 1978). In Lake with cross-shore pressure gradients, as many other studies Ontario, where the absence of tides and sustained boundary had shown, and also propagate. In other words, direct currents makes the clear identification of individual propa- forcing by transient winds was prominently present, gating events possible, the two kinds of propagating flow alongshore propagation less so. structure, with and without accompanying upwelling of The most ambitious observational study aimed spe- isotherms, could be separately documented (Csanady and cifically at coastally trapped waves was the recent Austra- Scott, 1974; Csanady, 1976). In none of the quoted cases lian Coastal Experiment (ACE), fittingly located along the

Fig. 3. Contours of equal time-lagged correlation of coastal sea level in the time-lag—distance plane. The dotted line connects local maxima, indicating northward signal propagation at a speed of 3.34 m sÐ1. From Freeland et al. (1986).

On the Theories that Underlie Our Understanding of Continental Shelf Circulation 211 NSW coast, where Hamon first found them (Freeland et al., shelf wave, accounts for alongshore propagation through 1986; Church et al., 1986a, b). An illustration from Freeland the well-known vortex-stretching effect of the sloping sea- et al. (1986), Fig. 3 here, shows alongshore propagation of floor, see e.g. Gill (1982). This model has an extensive coastal sea level at a speed of 3.34 m sÐ1, revealed by maxima literature: milestones in the development of the theory are of lagged correlation. The associated alongshore velocity papers of Robinson (1964), Mysak (1966, 1980), Buchwald was recorded on moorings along three cross-shore transects. and Adams (1968), Adams and Buchwald (1969), Gill and Inspection of the data at the inshore moorings suggests some Schumann (1974), Huthnance (1975); a recent review is northward propagation of barotropic velocity changes, al- Brink (1991). Water depth is taken to increase monotonically though most of the observed motion cannot be ascribed to from the coast to a finite asymptotic depth far offshore, propagating flow events, certainly not to free waves independent of alongshore distance: h = h(x), dh/dx > 0, h ⇒ propagating across the entire array, as Freeland et al. (1986) H (x ⇒ ∞). Continental shelf waves are a subset of wave emphasize. In other words, free waves were certainly present, motions possible over such topography. They are by hy- albeit in a complex combination of motions locally forced pothesis “long”, so that the “boundary layer approximation” by weather systems as well as induced by meanders and may be invoked (alongshore scales much longer than cross- eddies of the East Australia Current. shore ones), of a frequency much lower than inertial, and Observations in various places established that the nondivergent. most efficient generators of propagating flow events are These idealizations reduce the vorticity equation to a transient alongshore winds affecting a limited portion of the simple balance between vorticity tendency and vortex coast. Any such forcing affects shelf circulation in the stretching. The boundary condition at the coast is vanishing “downwave” direction (right-bounded in the northern hemi- cross-shore mass transport. The boundary condition at the sphere, leftbounded in the southern), or from a different open boundary (“infinity” in idealized models) has been a perspective, local circulation responds to remote forcing source of difficulty, however. The original choice, made by from the upwave direction, as well as to local winds. The one Robinson, Mysak, Huthnance and others, was vanishing major exception to universal forcing by alongshore wind is pressure (or sea surface elevation ζ) far offshore. The un- the case of Peru, where coherent pressure, velocity and derlying physical idea is that the pressure field generated in temperature signals arrive from far upwave, perhaps from as shallow coastal waters by the wind or other forcing should far as the equatorial wave guide (Brink et al., 1983). The ACE not extend to deep water, the tail should not wag the dog. experiment attempted to pin down the exact manner and With pressure gradients absent, the wind stress would then location of upwave forcing responsible for the observed be balanced by Ekman transport far offshore. Under these propagating signals, but the results were inconclusive in this conditions the vorticity tendency vanishes together with the respect, much sophisticated statistical analysis notwith- vortex stretching term (see the Appendix for a précis of standing (Griffin and Middleton, 1991; McIntosh and relevant vorticity arguments). Schahinger, 1994; Schahinger and Church, 1994). An alternative boundary condition at infinity is van- To sum up what is empirically established and what is ishing mass-transport streamfunction, ψ. Buchwald and not, coastally trapped waves, barotropic and internal Kelvin Adams (1968) introduced this, presumably having found it wave-like, are found on many continental shelves. They are more convenient to work in terms of ψ. For free waves the more obviously present in some locations than others, but two conditions are equivalent, but forcing by alongshore nowhere are they in any sense dominant phenomena. They wind leads to a difficulty: a vanishing streamfunction ex- are generated mainly by transient alongshore winds, but cludes cross-shore Ekman transport. When Adams and questions remain on their exact generation mechanism. Buchwald (1969) calculate the response of the shelf to suddenly imposed alongshore wind, they find an alongshore 3.2 Coastally trapped wave models pressure gradient balancing the wind stress in deep water. In How well do theoretical models account for this evi- inverting their Fourier transform, they speak of the dence? The classical such model is the internal Kelvin wave, “troublesome pole” at κ = 0 (κ is the transform variable), and in a constant depth, two-layer sea. Proudman (1953) discusses remark incorrectly that it “has no real physical meaning”. As this model in detail, attributing it to Defant. Being a constant Mysak (1980) points out, the corresponding directly forced depth model, it shares some weaknesses of the coastal jet (non-wavelike) part of the solution implies onshore model, two-layer or continuously stratified. It does, never- geostrophic flow and consequent vortex stretching, and is theless, account for the baroclinic type of event propagation, the proximate cause of wave generation in this version of the that involving vertical movement of isotherms or isopycnals, forced continental shelf wave model. about as well as the coastal jet model accounts for upwelling. Gill and Schumann (1974) adopted the ψ = 0 offshore This is well covered in the literature, see e.g. Clarke (1977) boundary condition, and went on to replace it by a condition or Csanady (1982a). of vanishing cross-shore pressure gradient, ∂ζ/∂x = 0, ap- Another conceptual model, the barotropic continental proximately equivalent to ψ = 0, as Buchwald and Adams

212 G. T. Csanady already noted. The approximation derives from matching flow on the shelf to a constant depth abyss, where the length scale of the motion is large compared to shelf width (the barotropic radius of deformation, similarly to the costal jet model). The approximate condition is applied where the shelf-slope region joins the constant depth abyss. Physically this means vanishing alongshore geostrophic velocity at shelf-slope edge, while allowing significant cross-shore geostrophic velocity and relatively large cross-shore geostrophic mass transport (Allen, 1976). What do the different offshore boundary conditions imply for wind-forced shelf circulation? The idealizations Ð1 of the long continental shelf wave model reduce the vorticity Fig. 4. Streamlines of volume transport at time t = 5f after equation and the coastal boundary condition to: sudden start of alongshore wind over a sloping semi-infinite shelf. Ordinate points offshore, abscissa is alongshore downwave distance in arbitrary units. From Csanady (1995). ∂ 2  ∂ζ  dh ∂ζ h  + f = 0, ∂t∂x  ∂x  dx ∂y ()1 case, offshore Ekman transport at shelf-edge fed by inflow ∂ 2ζ ∂ζ fG from downwave, “closed” cells in the second case, h + fh = ()x = 0 . streamlines on the shelf being closed by offshore loops ∂t∂x ∂y g extending far into deep water. Mathematically, a directly forced solution takes care of the mass balance, as Adams and As remarked before, the vorticity equation contains Buchwald (1969) have shown for the ψ = 0 boundary con- only vorticity tendency and vortex stretching terms, the dition, Csanady (1995) for the zero pressure boundary forcing term, the alongshore wind stress G, appearing in the condition, illustrated here in Fig. 4. The full solution con- boundary condition (Mysak, 1980). tains an infinite series of continental shelf waves in addition Taking the offshore boundary condition to be: to the directly forced flow pattern, in both cases. Shelf width largely determines the properties of shelf ζ = 0, x ⇒∞, h finite waves: this is common to many different shelf-slope ge- ometries. For any depth profile monotonically increasing to the vorticity equation and the coastal boundary condition a finite asymptotic depth offshore the properties of free shelf imply the integral constraint: waves can be rigorously calculated. The directly forced solution, however, makes wind-induced shelf response ∞ ∂ζ ∞ G dramatically different from a sum of shelf waves. That a ∫ h dx = ∫ dy.2() particular or directly forced solution has to be part of system 0 ∂x y g response, is common to many problems in applied mechanics. Which offshore boundary condition is “right” for forced This calls for accomodating Ekman transport crossing waves? Although a pressure gradient in deep water is the shelf edge within an alongshore geostrophic coastal unappealing, a motionless deep ocean is also an current. The inflow/outflow is g/f times the right hand side, overidealization. The CODE observations we discussed occurring upwave, i.e. in the half-space from which conti- showed two-dimensional mass balance in the xz plane in the nental shelf waves propagate to the local section. The local initial period, albeit confined to the top few hundred meters, transport by an alongshore geostrophic current is g/f times and without any indication that the inflow was accompanied ψ the integral on the left. By contrast, the = 0 boundary by an alongshore pressure gradient opposing the wind. A condition at infinity implies: few days later alongshore flow supplied the fluid for offshore Ekman transport, as in the shelf wave model with vanishing ∞ ∂ζ ∫ h dx = 03() pressure offshore. The long-wave approximation can le- 0 ∂x gitimately be applied only in this later period, at a time long compared to fÐ1 after the imposition of the wind. Mass or vanishing net alongshore transport. The two boundary balance in the xy plane must also have its limitation in time, conditions are equivalent for free waves (G = 0), but for the because the fluid that leaves the shelf must somehow be wind-forced problem they imply different flow patterns: replaced. what might be called “open” circulation cells in the first Returning to the fundamental idea of vorticity genera-

On the Theories that Underlie Our Understanding of Continental Shelf Circulation 213 tion through vortex stretching, the question is, how far from and others. It rests on the further assumption of (very) small shore vorticity tendency becomes evanescent, where Ekman vertical displacements of the constant density surfaces. transport fully balances wind stress. How far offshore this According to Huthnance’s comprehensive analysis, the free happens, that is at what exact depth the zero pressure waves have properties intermediate between barotropic shelf boundary condition may apply, is not clear. One suspects waves and internal Kelvin waves. Published studies of that the stratification of the deep ocean has much to do with forced, friction-affected waves invoke “thin” surface and it, perhaps also the depth profile. The problem of shelf wave bottom Ekman layers to transmit the effects of surface and generation by wind thus remains incompletely understood: bottom stress to the interior (e.g. Clarke and Brink, 1985). the problem of the offshore boundary condition remains to As mentioned above in connection with the Yoon and bedevil the theory. By contrast, the properties and behavior Philander (1982) study, a crucial aspect of forcing by wind of free inviscid shelf waves are well understood, and described seems to be the “mapping” of the wind stress force onto the in the copious literature of the subject. density distribution. This is ignored in the Ekman dynamics approach, an unnecessary oversimplification: the equations 3.3 Extensions of the continental shelf wave model of Clarke and Brink could be solved without it, a step that Frictionless solutions are unrealistic in nearshore might throw added light on the “source water” problem. shallow waters where bottom friction sets a limit to the Further to complicate this issue, in stratified fluid the force acceleration of the coastal current. A highly idealized repre- of gravity affects the bottom boundary layer over the sloping sentation of bottom shear stress is Ðrv, r a friction coefficient seafloor (Lentz and Trowbridge, 1991; Trowbridge and of the dimension of velocity, v alongshore velocity com- Lentz, 1991; Garrett et al., 1993), with the result that ponent, the stress acting in the alongshore direction. When “Ekman pumping” becomes an altogether dubious model. this is introduced into the linearized equations, both the Nor does the surface boundary layer conform to the Ekman vorticity equation and the coastal boundary condition acquire dynamics model (Lentz, 1992). Close to the coast, the the corresponding terms: overturning circulation rapidly modifies the density distri- bution and invalidates the theory. Nevertheless, further development of this extension of the shelf wave model may ∂ 2  ∂ζ  ∂ 2ζ dh ∂ζ h  + r + f = 0, well repay the effort. Of particular interest would be a ∂ ∂  ∂  ∂ 2 ∂ t x x x dx y calculation of the forced, frictionless response of stratified ()4 waters to suddenly imposed wind, including any directly forced solution, and a sum of all shelf wave modes, or at ∂ 2ζ ∂ζ ∂ζ fG h + r + fh = ()x = 0 . least of as many as required to describe the full linear theory ∂ ∂ ∂ ∂ t x x y g behavior of the isopycnal surfaces. The shelf wave model, anywhere in the continuum With the zero pressure offshore boundary condition the between the frictionless and the arrested case, accounts for integral mass-transport constraint remains valid. Time-in- the establishment of coastally trapped pressure fields in dependent solutions under forcing sinusoidal in the along- response to time- and space-variable winds. The fields shore direction are called “arrested topographic waves” extend to the downwave half-space from the forcing region, (Csanady, 1978a). The total response to such forcing, sud- through slow propagation and frictional control of faster denly imposed, consists of the steady solution plus some currents. These are very important insights, but one must not frictional shelf waves (Thompson, 1987). The steady solu- forget that they have been gained at the cost of a whole host tion satisfies the mass transport constraint, which becomes of approximations and idealizations. They certainly do not the dominant feature of the streamline pattern. Forcing justify overly sanguine notions to the effect that all observed sinusoidal both alongshore and in time raises forced frictional shelf circulation could be represented by an appropriate sum waves. Power et al. (1989, 1990) have explored both free of shelf waves, an idea at least implicit in a considerable and forced frictional wave-like solutions, for different shelf- literature (e.g. Clarke and Van Gorder, 1986). An example slope topographies. The forced solutions are similar to from the CODE experiment is Chapman’s (1987) hindcast arrested topographic waves induced by the same wind, of bottom pressure and alongshore velocity at the central while free waves decay fairly rapidly (typically in less than transect from early April to late July, based on a short sum a wavelength). Solutions of this kind are useful in interpreting of forced, frictional shelf waves. The hindcast generally quasi-steady pressure fields arising in response to wind tracks observations as to sign, but the amplitude is too small stress (e.g. Hickey and Pola, 1983; Hickey, 1989). by about half. Furthermore, the calculated modal structures The barotropic continental shelf wave model has been are nearly barotropic, inherently incapable of representing extended to continuous stratification, over depth monotoni- the surface intensified coastal jet, the overturning circula- cally increasing with distance from shore, by Wang and tion, the temperature structure or cross-shelf currents. As a Mooers (1976), Huthnance (1978), Clarke and Brink (1985), general principle, such an approach to hindcasting should be

214 G. T. Csanady avoided merely on the grounds that conceptual models velocities result from the combination of cross-shore den- intentionally do not aim at simulating complex reality. sity gradients and alongshore wind: near Galveston, observed westward alongshore velocities are often of order 1 m sÐ1. 4. Baroclinic Currents The Alaska Coastal Current is another fast, though Density variations arise in coastal waters from many more massive thermohaline current. It flows along the edge causes, freshwater runoff, differential heating, evaporation of an inner shelf front arising from distributed runoff into the and precipitation, or simply advection of lighter or heavier Gulf of Alaska (Royer, 1983), at speeds similar to those seen fluid. Horizontal density gradients created by such processes over the Louisiana-Texas shelf. Easterly winds enhance the entail pressure gradients and act as prime movers of “ther- speed of the frontal current, as off the Texas coast, and are mohaline”, or density-driven circulation. A complication is thought to be responsible for holding the plume, and the that wind-forced motions may also be accompanied by associated coastal current, within a short distance of the horizontal density gradients through the geostrophic ad- coast for hundreds of kilometers. To this extent, the two justment of the pressure field, as in coastal upwelling, and it situations are similar. The shoreface of this coast is excep- is not always easy to separate wind-driven from thermohaline tionally steep, however, the water depth much greater than circulation. the surface plume of freshened water, already very close to Regardless of their origin, once horizontal density the coast. Therefore the isopycnal surfaces underlying the gradients are of significant magnitude, or equivalently, once plume are quasi-horizontal, except at the outer edge of the isopycnal surfaces are noticeably inclined, they influence Alaska Current where the speeds are highest. A celebrated circulation by sustaining baroclinic currents. We first briefly feat of this coastal current was the rapid conveyance of the review observations on such currents. Valdez oil spill out of Prince William Sound (Royer et al., 1990). 4.1 Density fronts and related flows Similar to the Alaska Coastal Current, the shelf edge River plumes provide good examples of density-driven frontal currents off the east coast of North America, Labrador flow. Münchow and Garvine (1993a, b) have explored the to New England, carry a water mass distinct from the Delaware river plume close to its entrance onto the shelf; adjacent ocean and overhang the continental slope to a Fig. 5 shows the salinity distribution at times of low and high considerable distance. Both the depth and the width of the discharge. Low discharge generates a diffuse front with overhang diminish southward, but the amount of shelfwater nearly vertical isopleths already fairly close to the estuary, over the slope per unit length of the coast remains compa- while a tight, quasi-horizontal front is seen at the same rable to what lies over the shelf. The overhang is, moreover, location, when water discharge is 14 times greater. In the variable and episodically extends by more than a shelf- latter case, baroclinic currents with velocities of order 20 width offshore. One consequence of the overhang is that any cm sÐ1 accompany the density gradients. One remarkable upper slope current contributes to the alongshore transport aspect of such plumes is that they transport shelfwater along of shelfwater. At the outer edge of the overhang the baroclinic the coast (usually in the “downwave” direction of coastally frontal current also carries substantial transport, much as in trapped waves) at a rate many times greater than the fresh- the Alaska Coastal Current, although in lesser amounts. water outflow from the estuary. Strong wind opposing an established frontal current displaces the plume offshore, 4.2 Dynamic height Fig. 6, resulting in a “lens-shaped” halocline in cross- What conceptual models and theories elucidate the section, resembling the similarly shaped spring thermocline physics of baroclinic currents? In steady state, much is made in the Great Lakes, also occurring in response to opposing clear by what is perhaps the oldest conceptual model in wind (Csanady, 1971). The Mississippi-Atchafalaya plume oceanography, the distribution of “dynamic height”, also behaves analogously on the Louisiana-Texas shelf, moving known as “steric height” or (multiplied by the acceleration offshore and eastward under winter northwest winds, of gravity) “geopotential anomaly”. In deep water, dynamic alongshore and westward under summer easterlies. The height is calculated relative to a deep reference level, on influence of this plume reaches far westward in summer, which horizontal pressure gradients are weak, in which case affecting the flow along this very flat and wide shelf for geostrophic velocities calculated at lesser depths approxi- hundreds of kilometers. Figure 7 (from Jochens and Nowlin, mate observable ones. We may extrapolate the same cal- 1994) shows the density distribution observed about 300 km culations into coastal waters, but should have a clear un- west of the outflow of the Atchafalaya: nearly vertical derstanding of what exactly the results mean. isopleths extend more than 60 km from shore, almost to a Dynamic height is defined by an integral of the hy- haline front with a surface expression centered at 80 km. drostatic equation: Vertical isopleths also characterize the top 12 m of the water column farther offshore, clearly a signature of surface stir- z p = p − ∫ ρgdz ()5 ring by wind and convection. Some very high alongshore h −h

On the Theories that Underlie Our Understanding of Continental Shelf Circulation 215 Fig. 5. The Delaware plume at low (top) and high (bottom river discharge, surface salinity (left) and cross section at arrow (right). From Münchow and Garvine (1993a).

Fig. 6. As previous figure, under high opposing wind. From Münchow and Garvine (1993b).

216 G. T. Csanady Fig. 7. Density distribution off the Texas coast in late spring. From Jochens and Nowlin (1994).

where ph is pressure at z = Ðh. In deep water the reference Note here that the horizontal gradients ∇H of pressure depth is a constant h, and the calculation of the geostrophic and density are taken at constant z, including especially velocity is facilitated by the introduction of the dynamic ∇Hph, the gradient of pressure at the seafloor, but not along height D: the seafloor. On the other hand, variations of the density- defect σb are those occurring along the seafloor. In analogy with the deepwater calculation of D, one would now expect = 1 ×∇ = g ×∇ () ug k H p k H D 6 to define the dynamic height gradient as the sum of the last fρ f two terms, relegating the bottom pressure gradient to a separate component of the pressure field. Integrating along where: some chosen path between two points A and B, we thus tentatively define the difference of dynamic heights at z D = ∫ σdz, constant level z by: −h

z z B σ = − ρ ρ D − D = ∫ σ()B dz − ∫ σ()A dz + ∫ σ ∇ h.ds ()8 1 / 0 B A −h −h A b H ρ σ with 0 a reference density, the small nondimensional where ds is a directed element of the integration path from density defect (gσ is buoyancy). In a coastal domain the A to B. For different integration paths the above definition of depth varies from zero to, say, 100 m, and no depth qualifies dynamic height difference is unique only if the last integral as a suitable constant reference level for calculating D. The is independent of the path. This is the case if and only if the bottom depth is the only plausible choice for the lower limit line integral over any closed circuit containing A and B of the integration, above which the hydrostatic equation vanishes, ∫ σ ∇ h.ds = 0. In that case, according to Stokes’ yields the pressure gradient (as a “kinematic” quantity, i.e. b H σ ∇ divided by the reference density): theorem, the curl of the vector b Hh vanishes everywhere:

∂σ ∂ ∂σ ∂ z b h − b h ≡ ()σ = () ∇ p =∇ p + g∫ ∇ σdz + gσ ∇ h.7() J b ,h 09 H H h −h H b H ∂x ∂y ∂y ∂x

On the Theories that Underlie Our Understanding of Continental Shelf Circulation 217 where J(σb, h) is the Jacobian of σb and h, the vanishing of which means that bottom density is constant along lines of constant depth. Our tentative definition of dynamic height differences therefore applies if and only if σb = func(h). Fortunately, density fields observed in coastal domains often satisfy this condition to a reasonable approximation. Where the condition is satisfied, the dynamic height fields of deep ocean and shelf match, if they are calculated on the shelf according Eq. (8). The last integral is evaluated along the bottom, over whatever path, from the intersection of the deep reference level with the seafloor, to points on the shelf (point B). Integrals from the bottom up in the water column to any depth z, at point B and off the shelf at point A, complete the calculation. Figure 8 illustrates the integration path (Csanady, 1979). The integrand on the seafloor is Fig. 8. Extension of dynamic height calculations to coastal actually everywhere σ dz, because ∇ h.ds equals the rise of waters: where the deep reference level runs into the continen- b H tal slope, density is integrated upward along any path (if the seafloor dz over an element ds of the integration path. bottom density is constant along isobaths) or along “fall lines” The calculated geostrophic velocities blend smoothly be- of the seafloor (otherwise), to points on the shelf or slope. tween the two domains, if σb = func(h). They define a baroclinic component of the velocity field, independent of any barotropic motion. In the contrary case, when bottom density varies along isobaths, we return to Eq. (7) and calculate the vertical cruises. In May, westward wind stress and the Mississippi- component of the curl for each of the four vectors in the Atchafalaya outflow combine to generate strong baroclinic coastal currents along the Texas coast. In August, the annual equation. The vectors ∇Hp and ∇Hσ are differentials at constant z and their curl vanishes. The other two vectors shift in the wind regime disrupts this circulation pattern, follow the seafloor, and have nonzero curl, the sum of the causing the river plume to move offshore and then eastward. two curls adding up to zero, however. Together with our last In May constant bottom density contours closely follow result this leads to: isobaths and the dynamic height map faithfully reflects the observed circulation pattern. By contrast, on account of along-isobath density gradients, the August map shows  ∂  ∂p   ∂  ∂p  cross-shore geostrophic velocities in the western Gulf, in k.∇ ×∇ p ≡   b  −   b  b H h  ∂x  ∂y   ∂y ∂x  conflict with observation. =−gJ()σ ,h .10() b 4.3 JEBAR on the shelf To put JEBAR in perspective, consider the idealized This density-related curl of the bottom pressure gradient case of vertically well mixed water columns, with density forces motion just as the curl of wind-stress or of bottom varying in the alongshore direction only, over much of the stress does. The phenomenon is known as JEBAR, joint width of a continental shelf, the alongshore buoyancy gra- effect of baroclinity and relief, and is discussed further in the dient constant shoreward of an outer shelf isobath, σ = γy, next section. γ = constant, x < l. Something very similar exists over the JEBAR notwithstanding, it is possible to extend the Mid Atlantic Bight in winter, when the waters off the Gulf dynamic height concept to arbitrary bottom density distri- of Maine are considerably colder than waters off Virginia. bution, by simply choosing specific integration paths along As a typical density gradient we take δσ = 10Ð3 over a dis- the seafloor, and following the same recipe as in the case of tance of 1000 km, or γ = Ð10Ð9 mÐ1. The bottom slope shall constant density along isobaths. A plausible choice is inte- also be constant, typically s = dh/dx = 10Ð3, over a shelf of gration perpendicular to the isobaths, along what are known width l = 100 km, to the 100 m isobath. as “fall lines”. This method again defines a single-valued Fluid columns being heavier in the north, the along- dynamic height field. The associated geostrophic velocities shore bottom pressure gradient in this situation pushes are, however, divergent, and cannot exist in steady state southward, and is greater in deep water than in shallow, without some kind of compensating effect. exerting an anticyclonic torque on the shelf water mass, Figure 9 shows dynamic heights so determined, aver- according to Eq. (10). The torque either generates vorticity, aged over nine May cruises of the LATEX experiment or forces steady circulation of such character as to compensate (Jochens and Nowlin, 1994), Fig. 10 over eight JulyÐAugust for JEBAR.

218 G. T. Csanady Fig. 9. Dynamic heights extrapolated to the Lousiana-Texas shelf in May, when bottom density contours approximately coincide with isobaths. From Jochens and Nowlin (1994).

Fig. 10. As previous figure, for August, when bottom density contours cut across isobaths. From Jochens and Nowlin (1994).

On the Theories that Underlie Our Understanding of Continental Shelf Circulation 219 Suppose first that the curl of the bottom pressure g ∂σ gradient locally generates vorticity. Substituting the u =− ()z + h . f ∂y Jacobian of bottom density and depth from Eq. (9) we have: With σ decreasing northward, geostrophic mass ∂ω ∂σ =−gs .11() transport is offshore directed and divergent. That is not the ∂t ∂y total cross-shore mass transport, however: the alongshore momentum balance contains the bottom stress, which must How long does it take for the vorticity to reach a be balanced by onshore directed Ekman transport, τb/f = rv/ magnitude compatible with observation? The alongshore f = rgsσ/f2. The total cross-shore mass transport is then: velocity at the shelf edge is typically 0.1 m sÐ1, zero at the coast, for a typical ω of 10Ð6 sÐ1. We calculate that the gh2 ∂σ rgsσ alongshore density gradient spins up the fluid to the typical U =− − .16() ∂ 2 vorticity in 105 s, or about a day. f y f Because the winterÐspring density distribution lasts for a season, frictional equilibrium is a more realistic model. The transports satisfy mass balance: We seek therefore a steady-state pressure distribution complementing the dynamic height field so as to sustain ∂()∂ vh + U = steady motion against bottom pressure torque. The rules for 0. ∂y ∂x calculating dynamic height yield in this example: D = σ(sl + z). For the total pressure we write gζ, ζ = D + ζ , dynamic t The divergences balance exactly: offshore mass trans- height plus a barotropic component to be determined. port by the baroclinic flow is divergent (remember that ∂σ/ As in the frictional shelf wave model, we write τ = rv b ∂y is negative), the alongshore barotropic flow convergent, for the y-component of the bottom stress, and neglect the x- as northward flow diminishes northward. JEBAR draws in component. The vorticity equation then reduces to a balance fluid from the south (the downwave direction or negative-y between bottom pressure torque and bottom stress curl: domain) to supply the offshore geostrophic transport driven by density contrasts. This offshore transport must be supposed ∂v ∂ζ r + gs = 0 ()z =−h .12() to be accomodated in some way, perhaps in a shelf-edge ∂x ∂y frontal current. The particular solution does not satisfy the boundary Substituting ζ = D + ζt into this equation we obtain: condition at the coast, so that a solution of the homogeneous version of Eq. (13), ζh, must be added, to cancel the con- tribution of the particular solution to ∂ζ /∂x at the coast: r ∂ 2ζ ∂ζ ∂σ t t + t =−s ()l − x .13() ∂ 2 ∂ ∂ fs x y y ∂ζ h =−sσ ()x = 0 .17() ∂x A particular solution is found by inspection: This is a “boundary flux” condition in the heat analogy. ζ =− σ()− t s l x . The effects of the boundary flux spread out from every length element of the coast (acting as “sources”), seaward Adding the dynamic height component we find the and toward negative y, over a coastal boundary layer of simple result for the total pressure field: δ ()′− ′ width = 2 ry y / fs . The influence of all elements dy upwave (toward positive y) of a given location add up to a ζ = D + ζ = γysx()+ z .14() t solution. Outside the boundary layer, on the outer shelf, the particular solution only is significant. Within the boundary The sea surface is here a tilted plane, sloping down layer northward mass transport is suppressed, and the di- toward the coast and to the north, bottom pressure vanish- vergence of total alongshore transport equals the Ekman ing. The alongshore geostrophic velocity is depth-inde- transport arriving from the outer shelf. pendent: At the southern limit of our region, where σ = 10Ð3, the alongshore velocity outside the coastal boundary layer in v = gsσ / f ()15 this model is 0.1 m sÐ1, maximum offshore velocity a factor of ten smaller. while the cross-shore geostrophic velocity vanishes at the While this conceptual model of JEBAR-influenced seafloor:

220 G. T. Csanady shelf circulation is overidealized, the results should indicate the kinds of effects to expect from alongshore density gradients over a sloping seafloor. Referring to the LATEX dynamic heights in Figs. 9 and 10, we would suspect little influence from JEBAR in the first case, rather more in the second, where the dynamic heights in the western Gulf cross the shelf much as in our simple example. Keep in mind, however, that all these calculations are diagnostic, i.e. they do not allow for advection of density by the calculated velocities. One must suppose that the density gradients in our example are maintained by surface heating as the water penetrates to lower latitude. A further point to remember is Fig. 11. Self-advection of density perturbation ρ′ on a mid-shelf that a “skyhook” was needed at the shelf edge to absorb the isobath, following uniform surface cooling for a limited period fluid leaving the shelf: again an open boundary problem! (scaled time t = 0 to 1), over a limited alongshore distance (scaled y = 0 to Ð1). The distance scale is 100 km, the time scale 4.4 “Self-advection” of density 47 days, the density perturbation scale 0.24 kg mÐ3. From Shaw Localized cooling occurs frequently on continental and Csanady (1983). shelves under cold air outbreaks, and result in fairly sharp along-isobath density gradients. Through JEBAR, these give rise to a regional circulation pattern which modifies the Figure 11 from Shaw and Csanady (1983) shows a calculated density field by advection. The scalar product of velocity case, the density anomaly distribution at the end of a cooling and temperature gradient in the heat balance equation quan- event, and the time history of a section along a mid-shelf tifies this effect. The resulting nonlinear coupling of the isobath. The initial condition was uniform cooling for a time flow and density fields acts as feedback, sharpens density period of 2fÐ1, over 0 < y < ÐY. Shaw (1982) has discussed gradients, and leads to spontaneous motion, a kind of self- several other examples. He was further able to relate the advection of the density field. simple model to the establishment of the cold pool in the A minimally complex model of this phenomenon is a Mid-Atlantic Bight, through late winter cooling in the Gulf shelf with parallel isobaths, the water column impulsively of Maine. cooled at t = 0 to an excess density uniform across-shelf and Huthnance (1984) discussed other hypothetical cases over the depth from the coast to the shelf edge, along a of JEBAR forcing, Csanady (1984) a diagnostic model of segment of the coast of length Y (Shaw and Csanady, 1983). river-plume circulation, with results similar in principle to Outside the coastal boundary layer the JEBAR-induced what we found in the previous section, i.e. showing fluid alongshore velocity is as we calculated in the previous drawn from the “forward” portion of the shelf (negative y) section, Eq. (15), except that now σ is density anomaly of the to compensate for the divergence of the baroclinic veloci- well-mixed water column, variable in time and alongshore ties. distance. The advection-diffusion equation of density anomaly simplifies to (Shaw and Csanady, 1983): 4.5 Front models “Fronts”, or boundaries between different water masses ∂σ ∂σ ∂ 2σ are ubiquitous features of the coastal ocean. The Bering Sea + V = K ()18 ∂t ∂y ∂y2 shelf provides several examples, see Fig. 12 from Coach- man et al. (1980). The change of a fluid property (tem- where K is a horizontal eddy mixing coefficient. Substitut- perature, salinity, etc.) in a relatively short horizontal dis- ing for v, and transforming the alongshore coordinate into tance earmarks a front. Usually this also means density η = Ðfy/gs we arrive at Burgers’ equation for the density change, and has then dynamical implications: lighter water anomaly: tends to intrude over heavier fluid, the latter to underflow the former. Earth rotation deflects the intruding fluid, until a quasi-steady state of geostrophic balance is reached, with ∂σ ∂σ ∂ 2σ + σ = λ ()19 the front in an inclined position supplying the pressure ∂t ∂η ∂η2 gradient which balances the Coriolis force of an alongfront jet. Friction and mixing slowly destroy this balance, how- with λ = f2K/g2s2, both η and λ having the physical dimen- ever, unless some other process continuously restores it. sion of time. Burgers (1948) proposed this equation as a Moreover, the alongfront jet is hydrodynamically unstable, simple model of turbulence: its solutions show formation develops meanders and eddies, which become agents of and propagation of fronts and their gradual dissolution. cross-front mixing.

On the Theories that Underlie Our Understanding of Continental Shelf Circulation 221 force, the vortex stretching and tilting terms vanish, and the vorticity tendency equation for η becomes (Kao et al., 1978):

∂η ∂b ∂v = + f + K ∇2η ()20 ∂t ∂x ∂z m

where Km is eddy viscosity, the term with Km representing diffusion of alongfront vorticity. The total derivative on the left includes advection of η in the (xz) plane. Apart from advection and diffusion, two “source terms” appear in Eq. (17), ∂b/∂x and f∂v/∂z. When the sum of these vanishes, the thermal wind equation is satisfied, geostrophic equilibrium prevails, and there is nothing to generate circulation in the cross-front plane. Departure from geostrophic equilibrium forces cross-front circulation: this is the simple physical content of the η-equation. The accompanying tendency equations for buoyancy and alongfront velocity show that, in the absence of cross-front circulation, both b and v-gra- Fig. 12. Different flow and mixing domains on the Bering Sea dients decay under the influence of mixing and friction, continental shelf. From Coachman et al. (1980). destroying thermal wind balance. The balance is restored by the cross-front circulation much as in the adjustment mod- els. Numerical studies revealed further details, and effects of different boundary conditions (see e.g. Kao et al., 1978; Minimally complex models of front formation start Wang, 1984; Chapman and Lentz, 1994). with a state of disequilibrium (for example a rigid vertical A minimally complex model of a front maintained by membrane separating two fluids of different density), and freshwater runoff from land (Stommel and Leetmaa, 1972) allow the fluid to adjust to a balance between Coriolis force is two-dimensional in the xz plane, and prescribes the and pressure gradient (upon withdrawal of the membrane). horizontal depth-integrated buoyancy flux B (resulting from Neglecting friction, mixing and time-dependent motions, the runoff) at the coast. Vertical buoyancy fluxes at the surface equilibrium shape of the boundary between the fluid masses and bottom are supposed to vanish. Depth is taken to be then follows from potential vorticity conservation, as in the constant, the width of the frontal zone much greater than case of “full” upwelling. The Coriolis force associated with depth, vertical velocities much smaller than horizontal ones, 2 the cross-front adjustment motion generates the jet. The |w| << |u|. Alongfront wind stress τy = ρu* is prescribed at impulse of that force totals fd, where d is cross-front dis- the surface, at the bottom vanishing velocities u and v or a placement from the initial state, the impulse equal to the jet suitable drag law. Linearized versions of the η and v equations velocity. are supposed to apply, plus the b equation with the hori- The initial state of disequilibrium may include mo- zontal advection term retained, all without time dependence. mentum, representing impulsive acceleration, due to wind The flux conditions imply that the depth-integrated stress, for example. This is how a lens-shaped pycnocline buoyancy flux is independent of distance from the coast: may arise: a fast moving light fluid mass near the coast, initially fenced in by a membrane, is released. As it moves  ∂  = 0 b + = () bodily offshore, the alongshore Coriolis force cancels out B ∫  Kb ub dz constant. 21 −h  ∂x  initial momentum and the fluid then spreads out on the surface, to a lens shape with opposing jets at its fronts (Csanady, 1978b). Although these models resemble observed A solution of the form b = b0(x) + b1(z) satisfies the b- fronts, they are much less steep, jet velocities much smaller equation and boundary conditions. The resulting depth- than calculated, on account of less extreme initial gradients, integrated buoyancy flux is: as well as friction and mixing in the course of adjustment. Fluid intruding along the bottom, in particular, does not get ∂b B = ()K + K h 0 ()22 very far. b s ∂x The balance of alongfront horizontal vorticity, η = ∂u/∂z Ð ∂w/∂x, throws some light on how a front is maintained where Ks is “shear diffusivity”, a concept due to Taylor against friction and mixing. Supposing two-dimensional (1954): cross-front circulation, with buoyancy (b = Ðgσ) the driving

222 G. T. Csanady 1 0 tions. In this final section we examine what observations say K = ∫ U 2dz s −h about such interactions. Kbh 5.1 Outer shelves and upper slope currents 0 with U = ∫ udz′ . The result follows directly from the Many outer shelves are under the influence of a bound- −h boundary conditions, using only the b-equation, the advec- ary current or upper slope current, the more so, the greater the tion-diffusion equation for buoyancy. The remaining equa- instability of that current. The best explored example is the tions constitute a familiar Ekman layer problem with ther- outer shelf of the South Atlantic Bight, where eddies and mal wind, allowing the determination of the velocity distri- meanders of the Gulf Stream are important players in the bution. The solution yields the velocities u and v as functions mean alongshore momentum balance, in addition to a strong of depth z, from which U may be calculated. The prescribed alongshore sea level gradient of Gulf Stream origin (Lee et integrated buoyancy flux B then determines the buoyancy al., 1984). The resulting mean flow is northward, both on the gradient db0/dx, and the full solution. outer and mid-shelf. According to studies of Sturges (1974), The alongfront transport, in particular, is: Blaha (1984) and Noble and Gelfenbaum (1992), sea level changes on this long coast are impressed by the Gulf Stream, and fluctuate with the latter’s transport. The mean alongshore 0 ∂b ∫ Vdz = ()h2 /2f 0 .23() sea level gradient is about 2 × 10Ð7, the same as along the −h ∂ x inshore edge of the Gulf Stream, and apparently arises from

Ð6 Ð2 Ð4 Ð1 the Stream’s dynamics. If db0/dx = 10 s , h = 50 m, f = 10 s , the transport Mass exchange between shelf and Gulf Stream is also is 12.5 m2sÐ1 per unit cross-shore distance, or 1.25 × 105 3 Ð1 vigorous: a classic illustration of Lee et al. (1991) portrays m s over a 10 km broad frontal zone, a significant portion the mechanism, Fig. 13. Hydrodynamic instability causes of the typical alongshore transport on a shelf. the Gulf Stream surface front to contort into a wavy shape, The Stommel-Leetmaa model represents a front subject the wave crests breaking backward, to enclose some to intense mixing and internal friction, something of an shelfwater in a cyclonic eddy. The decay of these eddies extreme case, and yet it demonstrates that sustained buoy- deposits on the shelf nutrient-rich waters upwelled from the ancy flux from the coast generates persistent shelf circula- Gulf Stream thermocline, in exchange for shelfwater tion of significant amplitude. The primary effect of the transported away. The figure also indicates the different buoyancy flux is to sustain a buoyancy gradient cross-shore. domains on the South Atlantic Bight shelf, an inner shelf Cross-shore velocities advecting buoyancy help sustain the freshened by runoff, a mid-shelf where wind-driven currents flux, and also, through their alongshore Coriolis force, coexist with the alongshore pressure gradient impressed by maintain the alongfront current nearly in geostrophic equi- the Gulf Stream, and the outer shelf, with its zoo of eddies. librium. Even where the unstable boundary current lies some To sum up conceptual models of thermohaline circu- distance offshore, outer shelf circulation may be affected by lation briefly, the dynamic height model is certainly an cast-off eddies, as in the Mid-Atlantic Bight or on the important diagnostic tool, but it offers no insight as to how Louisiana-Texas shelf, by warm core rings spun off the Gulf density differences drive circulation. JEBAR models help Stream and the Loop Current respectively. On the Louisi- somewhat in this regard, but in their present state of devel- ana-Texas outer shelf the impinging rings generate residual opment they barely scratch the surface. Front models are eastward flow, in the Mid-Atlantic Bight they merely inten- overidealized and two-dimensional, inherently incapable of sify mass exchange. generating alongshore gradients of bottom pressure, and Along the U.S. west coast, the summer “coastal jet” of therefore portraying only baroclinic flow, much as dynamic the Washington and Oregon outer shelf now appears to be height. part of the boundary current system (Huyer et al., 1991), veering off the coast of Northern California to become the 5. Interaction of Shelf and Boundary Currents core of the California Current. The density distribution off In earlier sections we have repeatedly encountered Oregon indeed strongly suggests the interpretation that the difficulties with the postulate that shelf circulation can be lighter isopycnic layers have little to do with shelf circula- understood in isolation from the “deep ocean”. This was not tion, even if their outcropping shuttles back and forth cross- meant in the sense that there would be no interaction, only shore under the influence of wind events. On the other hand, that any such interaction could be suitably quantified and the poleward undercurrent (centered on the upper slope) used in shelf models, without the need of building coupled exerts a dominant influence on flow in the lower layers of the models. The difficulties of the shelf models suggest that the outer shelf, and indeed affects the middle and inner shelves conditions to be imposed at the open boundary should be as well. On the Peru shelf the undercurrent encompasses based on a better understanding of shelf-deep water interac- most of the water column, excepting only a 20 m or so

On the Theories that Underlie Our Understanding of Continental Shelf Circulation 223 Fig. 13. Gulf Stream meanders and eddies on the outer shelf of the South Atlantic Bight. The shaded zone shows nutrient-rich water upwelling from the Gulf Stream thermocline. From Lee et al. (1991).

surface layer, with the result that the return leg of the illustration is that the West Greenland and Labrador bound- upwelling circulation carries the waters (and the alongshore ary currents, as well as an upper slope current farther south, momentum opposing the wind) of the undercurrent waters flow alongside the shelf currents to central Nova Scotia. onshore (Brink et al., 1983). Moreover, the upper slope current off the Mid Atlantic The extreme example of a dominant poleward bound- Bight, intermittently transporting some 3 × 106 m3sÐ1 ary current on an eastern ocean boundary is the Leeuwin southwestward (Csanady and Hamilton, 1988), was missed Current off the west coast of Australia: this in not an by the artist. In an earlier detailed study of the Mid Atlantic undercurrent, but a surface one, apparently displacing sea- Bight circulation, Chapman et al. (1986) concluded that the ward the usual eastern boundary equatorward current. The bulk of the water on the shelf as far south as Virginia is of adjacent outer shelf domain also carries poleward flow, even Scotia shelf origin, and that it flows southwestward, parallel in southern summer, when the winds are strong and to the isobaths of the shelf all along its 1000 km long stretch. equatorward (Smith et al., 1991). A strong alongshore The alongshelf flow is here independent of, and in most pressure gradient, of deep ocean origin, provides the driving places in opposition to, the mean wind. Southwestward flow force, overcoming both wind and bottom stress. occupies the entire water column whether stratified or well The most striking observed property of long-term mean mixed, and requires a long-term mean alongshelf pressure shelf circulation is its very long range. Chapman and gradient to drive the flow against bottom friction. The Beardsley (1989) constructed a circulation scheme for the physical origin of this pressure gradient, and of the apparently shelves of northeastern North America, shown in Fig. 14. very long scale of shelf circulation, has been the subject of Coastal waters as far south as Virginia originate, according much speculation. Diagnostic studies of circulation showed to this scheme, from the West Greenland Current, in the an alongshore sea level gradient of order 10Ð7 off the Mid- sense that they contain some water of that origin, by the Atlantic Bight (e.g. Scott and Csanady, 1976). According to evidence of conservative tracers. A suggestive feature of the Smith and Schwing (1990) the shelves of eastern Canada

224 G. T. Csanady Fig. 14. Coastal currents off northeastern North America. From Chapman and Beardsley (1989).

On the Theories that Underlie Our Understanding of Continental Shelf Circulation 225 behave much as the Mid-Atlantic Bight, so that it is reason- Appendix: Vorticity Balance in Frictionless Barotropic able to suppose that their mean southwestward flow is also Flow over Sloping Bottom sustained by alongshore pressure gradients. Let the small density differences one finds in the coastal Chapman and Beardsley (1989) attribute the long- ocean be disregarded, on the hypothesis that they do not range east coast mean circulation to freshwater runoff and actively influence a certain class of motions. Upon integration similarly acting ice melt. Conceptual models of thermoha- over depth, the equations of motion and continuity yield line circulation (e.g. Stommel and Leetmaa, 1972, see above) relationships between the components of volume transport, suggest that baroclinic currents may indeed transport waters or depth-mean velocity, and pressure, or surface elevation, of this shelf southwestward, but in lesser amounts than also containing surface and bottom stress. From these, a observed. The models also do not account for the alongshore potential vorticity equation may be derived, of the following pressure gradients. It seems that the upper slope currents form (Huthnance, 1981): adjacent to the northeast shelves at least partly drive shelf circulation by imposing the alongshore pressure gradient,  ∂  ω + f  1  τ − τ  much as the Gulf Stream apparently does.  + u.∇ =− k.∇× s b .A.1()  ∂t  h  h  h  5.2 Any models? Several studies of coastal circulation have addressed Here u is depth-averaged velocity vector, with com- ω ∂ ∂ ∂ ∂ the problem of forcing by oceanic alongshore pressure ponents (u,v), = v/ x Ð u/ y the vertical component of gradients (e.g. Middleton, 1987; Chapman and Brink, 1987; depth-averaged vorticity, f is Coriolis parameter, supposed τ τ Kelly and Chapman, 1988). The models are of the coastally constant, h is water depth, s, b surface and bottom stress in trapped wave type, arrested or otherwise, with a pressure kinematic units (divided by density). Horizontal momentum ′ ′ gradient applied at the shelf edge, supposedly of oceanic transport u v by turbulence or by departures ud, vd from the origin. These models do not elucidate how such gradients depth-averaged velocities u, v has been neglected. It is im- arise, what circulation feature in the deep ocean is respon- portant to keep in mind that any deductions from the above sible for them, and to what extent, if any, the shelf itself is equation relate to the depth-average velocities, so that any involved in their sustenance. overturning circulation such as occurs in coastal upwelling Models of the arrested topographic wave type also is outside the scope of the theory, and that the effects of show that the steep continental slope has an “insulating” overturning on the depth-average motion have been ne- effect, restricting shelf-deep ocean interaction to upper glected. slope currents (e.g. Csanady and Shaw, 1983). A model of The physical content of Eq. (A.1) is that the time-rate an upper slope current with bottom friction due to Hill of change of the potential vorticity (ω + f)/h, following the (1995) illustrates how such a current generates alongshore fluid motion, equals the curl of the wind stress- less bottom pressure gradients on the adjacent shelf. stress-force, divided by depth. Supposing vorticity advec- The above are a few tentative steps in elucidating shelf- tion, wind stress and bottom stress negligible yields the slope interactions, inadequate at least in respect of portraying linearized equation: what must be mutual influences from slope to shelf as well as in reverse, requiring a two-sided approach, models of shelf ∂ω f = u.∇h ()A.2 and slope matched at their interface. A recent paper by ∂t h Condie (1995) modeling coastal currents interacting with the East Australian Current (EAC) is a first step in this the physical content of which is that vortex stretching by direction. Condie applies a geostrophic adjustment model to cross-isobath flow locally generates vorticity. This vorticity describe the shoreward penetration of the EAC front. He balance characterizes the (free) continental shelf wave model. suggests that a similar model may also apply to the Gulf Wind-forcing of shelf waves (neglecting bottom friction) Stream front in the South Atlantic Bight. requires addition of the wind stress curl: As earlier sections have demonstrated, the same un- solved problem of shelf-slope interaction undermines the ∂ω f  τ  success of such shelf circulation models as we have, and = u.∇h − k.∇× s .A.3() ∂   forces us to resort various skyhooks, debatable boundary t h h conditions offshore. If uniform wind stress acts parallel to isobaths, the Acknowledgements driving force τs/h is large over shallow depth, small over Support by the Office of Naval Research is gratefully deep, generating cyclonic or anticyclonic vorticity accord- acknowledged. ing to the direction of the wind and of the depth gradient. A three-way balance between wind-force, vortex stretching

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On the Theories that Underlie Our Understanding of Continental Shelf Circulation 229