Western boundary current theory
Contents
Western boundary current theory¶
Motivation {#motivation .unnumbered}¶
We have seen that the Sverdrup balance ties together the surface Ekman layer and the geostrophic interior (which actually overlaps with the Ekman layer), yielding a prediction of the vertically-integrated meridional velocity as a function of the curl of the wind stress. Using conservation of mass, we can then calculate the zonal velocity component as well, provided we can integrate from a point in the middle of the basin all the way to the eastern boundary of the basin. This leads to two questions: (1) why do we integrate to the eastern boundary, not the western? (2) how do we close the mass budget, that is, ensure that across any basin-wide zonal line the net meridional transport is zero? The answer to both questions requires the concept of a western boundary current, a particular type of boundary layer.
Boundary layers {#boundary-layers .unnumbered}¶
The boundary layer concept occurs often in fluid mechanics. Recall that to make progress in understanding any particular fluid flow, we almost always have to simplify the situation—figure out what are the essential elements in the dynamical balance, and ignore everything else. The essential elements may be different in different parts of the domain, however, and in particular there are often relatively thin layers around the boundaries where the dominant force balance is quite different from what it is in the interior of the domain. Typically, it is friction that becomes important in a boundary layer, whereas the interior can be approximated as inviscid. The Ekman layer is an example of a frictional boundary layer. Sometimes the essential new element in the boundary layer is not friction but nonlinearity, brought about by strong flow and short spatial scales (such as the thickness of the layer).
Why is it just on the western boundary? {#why-is-it-just-on-the-western-boundary .unnumbered}¶
Now, the Sverdrup balance relates the vertically-integrated meridional flow to the curl of the wind stress, which could, for example, be independent of longitude. Then there would be a net Sverdrup transport to the south (given negative curl in the northern hemisphere)—but in a closed basin, this cannot be a complete steady-state solution. For a steady state, the total transport across any parallel of latitude must be zero, so it is impossible for the Sverdrup balance to be valid across the entire basin. There must be a region—a boundary layer—where the Sverdrup balance is violated, so that the interior Sverdrup transport to the south can be returned to the north. One might expect that there would be boundary layers on both the eastern and the western boundaries, but it turns out that the significant boundary layer occurs only on the western side.
This is a robust result. Four separate types of model, or argument, can be used to reach the same conclusion: the return flow is entirely in a western boundary current. Like the Sverdrup balance, all of the arguments depend crucially on the variation of Coriolis parameter with latitude, that is, the “beta effect”. Separately, the arguments involve
simple linear damping of momentum, equivalent to bottom friction in a homogeneous (constant density) model;
lateral friction;
nonlinearity;
or time-dependence, that is, the “spinup” from an initial state of rest to the mean circulation.
Each of these mechanisms modifies the simple, steady, linear vorticity balance in such a way as to permit a strong return flow only on the western boundary. The last argument is the most profound, general, and directly applicable to the ocean. The first two are of great historical importance and nicely isolate the essential role of the beta effect, but they are so idealized that it does not make sense to apply them directly to the ocean. The third mechanism, nonlinearity, is more relevant to real western boundary currents.
Stommel’s model: simple damping {#stommels-model-simple-damping .unnumbered}¶
The first and simplest explanation of “The westward intensification of wind-driven ocean currents” was given by Henry Stommel in 1948, in a paper of that title (in Trans. Amer. Geophys. Union, 29, 202–206). He used a rectangular, flat-bottomed, homogeneous ocean with a simple pattern of wind stress and a steady, linear, vertically-integrated momentum balance. Wind stress and a damping term then appear as body forces. It turns out that this simple linear momentum damping is equivalent to the net effect of a bottom Ekman layer—they look the same in the vorticity balance, causing a linear vorticity damping. With this model, Stommel showed that the circulation in a basin with constant Coriolis parameter (\(f\)) is symmetric in the east-west direction; but when \(f\) varies linearly with latitude, the circulation becomes much weaker and highly asymmetric, with the Sverdrup balance in the interior and with a frictional boundary layer returning the meridional flow on the western side of the basin.
The vorticity balance in this model can be written as $$\frac{\partial \zeta}{\partial t}+ \beta v = \frac{1}{\rho}\frac{1}{H}\mbox{curl}_z(\vec{\tau}_w)
R\zeta, \label{E-dampedvort}$\( where \)R\( is a damping coefficient, \)H\( is the depth of the basin, and both \)\zeta\( and \)v$ are depth-averaged. The time derivative is actually zero, because we are looking for a steady solution; I left it in to show how the damping term would work in a time-dependent situation. Then the rate of change of vorticity would be proportional to minus the vorticity itself, and if there were no other terms in the equation, any initial vorticity pattern would just decay exponentially.
In the interior of the basin, we have the Sverdrup balance: $\(\beta v = \frac{1}{\rho}\frac{1}{H}\mbox{curl}_z(\vec{\tau}_w). \label{E-sver}\)\( Now suppose we have a frictional boundary layer on the eastern or western boundary, in which the primary balance is \)\(\beta v = - R\zeta. \label{E-stommelwbc}\)\( Can this happen on either the eastern or the western boundary? The difference between the two cases is the sign of the vorticity in the layer. If the layer is on the west, there must be a northward velocity (for our northern hemisphere mid-latitude gyre example) increasing to a maximum at the boundary. We have no lateral friction, so the appropriate boundary condition is free-slip, hence the velocity maximum on the boundary. The relative vorticity is then negative in this current. Multiplying by the \)-R\( on the right of ([\[E-stommelwbc\]](#E-stommelwbc){reference-type="ref" reference="E-stommelwbc"}), we get a positive term. On the left, we also have a positive term, because \)v\( is positive, that is, northward. Hence the balance in ([\[E-stommelwbc\]](#E-stommelwbc){reference-type="ref" reference="E-stommelwbc"}) works for the western boundary case. If we tried to put the current on the eastern side, the one thing that would change would be the sign of the relative vorticity (\)\zeta$). The right hand side of the equation would now be negative, but the left would still be positive—hence no balance is possible, and the current can’t exist on the eastern boundary.
Munk’s model: lateral friction {#munks-model-lateral-friction .unnumbered}¶
A somewhat more complex model was published in 1950 by Walter Munk. Instead of Stommel’s linear friction, he used a constant lateral eddy viscosity, but no bottom friction. The boundary condition is then no-slip, that is, both velocity components equal zero on the boundary, and the structure of the resulting boundary layer is more complicated than in Stommel’s model. The details are not important here. The important point is that the conclusion from Munk’s model is exactly the same as from Stommel’s: the frictional boundary layer that carries the poleward return flow to balance the equatorward Sverdrup interior flow can occur only on the western boundary.
Morgan’s and Charney’s models: nonlinearity {#morgans-and-charneys-models-nonlinearity .unnumbered}¶
A few years later, after discussions with Stommel, Morgan and Charney independently developed inertial models, showing that at least in their formation regions, western boundary currents might not depend on friction at all. The main idea is that potential vorticity is conserved following a water column, so a western boundary return flow to the north (that is, \(f\) is increasing) requires some combination of increasing water column height and increasingly negative relative vorticity. To keep the argument as simple as possible, consider a model in a homogeneous ocean with a flat bottom, so that water column height is constant. (The Morgan and Charney models considered the upper layer in a two-layer ocean, so spatially-varying water column height was important, but this does not change the basic argument.) The vorticity balance would then be $\(\mbox{\)\vec{u}\(}\cdot \nabla\zeta + \beta v = 0. \label{E-nlvort}\)\( Just as in the Stommel model, the boundary current will satisfy the free-slip conditions, will be strongest at the boundary, and will therefore have negative relative vorticity if it is on the western boundary. ([\[E-nlvort\]](#E-nlvort){reference-type="ref" reference="E-nlvort"}) then requires the first term---the downstream component of the relative vorticity gradient---to be negative, because \)\beta v\( is positive. This works. The negative vorticity gets stronger downstream as the boundary current gets narrower and/or carries more transport. In contrast, a northward boundary current on the eastward boundary would have to have positive relative vorticity, but the \)\beta$ term would cause this positive vorticity to weaken downstream.
Pedlosky’s Rossby wave argument: spinup {#pedloskys-rossby-wave-argument-spinup .unnumbered}¶
Still later, people developed models not just of the steady circulation, but of how the circulation would respond to change in the wind stress. In 1965, Pedlosky pointed out that the dynamics of the time-dependent case actually provides another good explanation for westward intensification. The argument rests on the characteristics of Rossby waves, which we will study in some detail later. For now we just note a few essentials. Rossby waves are large-scale potential vorticity disturbances that propagate as waves. Their propagation results from the change in Coriolis parameter with latitude, that is, the beta effect. Unlike the sorts of waves with which we are most familiar, Rossby waves propagate in a highly anisotropic manner: their phase propagation always has a westward component, and although they can propagate energy either east or west, the Rossby waves that propagate energy to the west have large scales and relatively high propagation speeds, while those propagating energy to the east have small east-west scales and are so slow that they tend to get dissipated before they get very far. Westward (long) Rossby waves are reflected at the western boundary as (short) eastward Rossby waves. Hence energy (and “information” about the interior of the basin) tends to pile up in small scales on the western boundary, forming a western boundary current.
Pedlosky’s Rossby wave argument provides the most fundamental explanation of why the boundary layer is in the west; it is implicit in any model of the steady western boundary current, including the models discussed above.
Topography {#topography .unnumbered}¶
Recently, attention has been focussed on the importance of ocean bottom topography in the dynamics of western boundary currents. Contrary to all of the steady models discussed above, there is evidence that in most western boundary currents, the change in water column height associated with a downslope (for poleward-going western boundary currents) or upslope (for equatorward) component of flow is of primary importance; potential vorticity is approximately conserved following a water parcel in a western boundary current over much of its path.
Conclusion {#conclusion .unnumbered}¶
The one essential element in the westward intensification of ocean currents is the beta effect, as Stommel showed. Given that, a western boundary current theory can be constructed in many ways. The models and ideas discussed above are not intended to provide a realistic picture of an actual western boundary current, but to illustrate some of the dynamical concepts and balances that might be included in such a picture.
The most fundamental concept to remember is the Pedlosky argument: information about forcing at low frequencies propagates effectively only to the west. This information about vorticity forcing propagates via potential vorticity waves—Rossby waves—which depend on the beta effect. Unlike the eastern boundary, the western boundary “knows” about all the forcing in the interior, and therefore some sort of western boundary layer must form in response to that forcing.
Source
The original source of this content can be found in PDF form from Eric Firing’s course notes.