## 1. Introduction

The location of the sinking of cooled water in polar regions is one of the fundamental issues that needs clarification for the theory of the ocean’s overturning circulation. Recent work on that sinking (e.g., Pedlosky and Spall 2005) has emphasized the enhancement of the sinking in the vicinity of lateral boundaries of the basin where the vorticity produced by stretching can be dissipated by friction. There have been many other studies of the process (e.g., Pedlosky 1968; LaCasce 2004). However, in these earlier studies, the sinking was supposed to occur in boundary regions with significant vertical stratification. The importance of boundary mixing for the meridional overturning circulation has been emphasized by Marotzke (1997) and Marotzke and Scott (1999). In each of these studies the zones of upwelling or sinking have been substantially stratified.

In a recent paper (Spall 2008), the downwelling induced by buoyancy loss in a boundary current was studied in an attempt to describe the process by which cooled water in polar regions sinks. As previous studies have shown, the tendency is for the sinking to take place adjacent to boundaries where the vorticity induced by the stretching of vortex columns by the sinking fluid can be dissipated by friction. The calculation in Spall’s study used the full Massachusetts Institute of Technology (MIT) general circulation model (Marshall et al. 1997). A current was introduced at the entrance to a channel and cooling, uniform in the down-channel direction, produced an evolution of the current in that direction such that an along-channel pressure gradient in geostrophic balance drove fluid to the right-hand boundary of the channel where it underwent strong sinking. In contrast to earlier work, the model develops a mixed layer of very weak but nonzero vertical stratification in which the sinking occurs but the lateral temperature gradients in the layer drive a geostrophic flow forcing the downwelling. This contributes to making the sinking region extremely narrow and the narrowness of the sinking region is such that only a few grid points in the calculation represent the boundary layer, so its spatial resolution is marginal. Although it is not thought that this affects the overall strength of the downwelled fluid, it appears conceptually important to resolve the structure of the dynamics with a simple analytical model: that is the goal of the present study.

One of the curious features of the numerical results is the nonmonotonic behavior of the along-channel flow near the boundary. The numerical model has a double boundary layer structure in which a broad Prandtl-type boundary layer appears to act to satisfy the no-slip condition on the along-channel flow: yet, as the boundary is approached within this layer, *u*, the along-channel flow, *increases* before finally being brought to zero in a very narrow region within the no-slip layer. In the discussion that follows a very simple linear model of a weakly stratified fluid, cooled at the upper surface, is employed to discuss, in particular, the inner region of the boundary layer where the overshoot of *u* occurs and where the strong sinking is found. The use of this linear model is suggested by the relative insensitivity in the numerical model of Spall (2008) to the degree of nonlinearity. Indeed, Spall suggested that the layer was a modified form of the nonhydrostatic Stewartson layer (Stewartson 1957) found in the theory of homogeneous rotating fluids. There is much that is unrealistic in the analytic model and yet its ability to reproduce salient features of the full numerical model implies that those features are robust and not dependent on the nonlinear nature of the original calculation.

Section 2 describes the basic model. Section 3 outlines the equations for the interior flow outside the Stewartson layer, while section 4 describes the Stewartson layer and the matching condition of the layer to the interior, setting a boundary condition on the interior flow. Section 5 presents the main results for a simple example of the theory. In section 6 some final remarks are made on the overall nature of the problem and its dependence on stratification (or its lack).

## 2. The model

*L*and depth

*D*. The fluid in the channel is cooled at the surface at a rate

*H*such that at the upper surface,

*z = D*,

*T*is the temperature

*anomaly*above a weak background vertical gradient, that is,

*κ*is the thermal diffusivity in the vertical direction, and

_{ν}*x*and

*y*are the long-channel and cross-channel coordinates, respectively.

*α*is the coefficient of thermal expansion,

*g*is the acceleration due to gravity,

*ρ*is the constant reference density, and

_{o}*f*is the constant Coriolis parameter. The Rossby number of the flow,

*S*, the Burger number, is

*f*plane are

The thermal conditions represent a nonuniform heating at the upper boundary and a fixed temperature at the lower boundary, which is our substitute in this simple model for a fairly passive fluid layer beneath. The sidewalls are thermally insulated.

*z*= 0 and 1 are understood to be at the edge of the Ekman layer.

*S*is small but will insist that the temperature anomaly is small enough to maintain a stable stratification consistent with our linearization. Our interest will be focused on the region between the Ekman layers and, in particular, on the boundary layers on the sides of the channel where we anticipate the major vertical motion will occur. We will consider the parameter limit of weak stratification expressed by [see (4.4) below]

## 3. The interior

*I*,

*z*. From the boundary conditions (2.12) it follows that

*w*is

_{I}*O*(

*E*) at

_{υ}*z*=0,1 and so must be of that order for all

*z*. If

*S*≪ 1, and assuming that

*σ*is

_{υ}*O*(1), it follows that the vertical advection of temperature is negligible in the thermal equation (2.8e), which then becomes

*z*independent) constant of integration from the thermal wind relation. The vertical velocity in the interior is very weak, of order

*E*,

_{H}*E*(which for simplicity we will assume are of the same order). However, it is not possible to determine the solution of (3.2) until boundary conditions are specified on the sidewalls. At the horizontal boundaries the interior temperature must satisfy the conditions (2.11a,b). To find the appropriate boundary conditions for (3.2) and to determine the barotropic component of the interior flow, it is necessary to examine the boundary layers at y = 0 and 1.

_{υ}## 4. The sidewall boundary layer

*D*/

*L*. Fundamentally though the basic ideas are not significantly altered. In the original theory there are two possible boundary layers; an outer layer with thickness that depends on the quarter power of the friction and which act to satisfy the no-slip condition. That

*E*

_{H}^{1/4}layer depends on vorticity dissipation in Ekman layers on solid horizontal surfaces bounding the fluid on at least one horizontal boundary. We have chosen to examine a layer satisfying a no-stress condition on

*z*= 0 and 1: It is easy to show, with the application of (2.12), that this outer layer is no longer possible. That will present a strong constraint on the interior flow. The inner boundary layer, in our notation, has a thickness,

*dimensional*units,

*L*.

*y*= 0 as

*U*is an unknown scaling constant for all variables and

*υ*is the geostrophically balanced part of the correction to

_{g}*υ*in the boundary layer while

*υ*is the ageostrophic part. All correction variables are functions of the stretched

_{a}*y*variable,

*η*=

*y*/

*δ*and must vanish for large

_{b}*η*. To lowest order in the small parameter

*σ*/

_{H}S*E*

_{H}^{2/3}(

*D*/

*L*)

^{2/3}, the correction functions satisfy

*ND*/

*f*, multiplied by

*σ*

_{H}^{1/2}, the square root of the Prandtl number.

*w*

*z*= 0,1, from which it follows that

*= 0 at those points. This implies that the solution for*p

_{z}*p*

*p*

With the solutions to the boundary layer equations, we are now in a position to carry out the matching procedure at *y* = 0. A similar process will occur at *y* = 1 but those details can be skipped.

*y*= 0,

*u*,

_{I}*υ*, and

_{I}*T*are

_{I}*O*(1) and

*w*is

_{I}*O(E*

_{H}). In the classical Stewartson layer problems involving a homogeneous fluid, the interior velocity normal to the boundary is zero or, if there is a geostrophic, order one

*υ*, then the geostrophically balanced

_{I}*υ*must satisfy the zero conditions on its own. This, however, cannot be the case here. Since the parameter,

_{I}*interior temperature gradient*must, to lowest order, satisfy the

*insulating condition*on

*y*= 0. It is then impossible for the interior to satisfy

*that*condition

*and*the condition on

*υ*. We are forced to the conclusion, then, that we must choose

*U*, the scale for the boundary layer correction to achieve that balance; that is,

*S*goes to zero. Then, the thermal equation is decoupled from the vertical advection and the temperature satisfies a form of Laplace’s equation. It is obvious in this limit that the temperature field must directly satisfy the insulating condition directly with the interior variables.

*A*

_{1j},

*A*

_{2j}, and

*A*

_{3j}. Note that this implies that the flow in the boundary layer is forced by the impinging

*geostrophic*flow in the interior at the wall, a conclusion that Spall (2008) has found in his numerical study. Thus, the sinking in the boundary layer is forced indirectly by the cooling as it generates a down-channel pressure gradient and a geostrophic flux toward the boundary where the sinking takes place.

*T*= 0 at

_{Iy}*y*= 0,1. That determines the baroclinic flow in the interior. The barotropic component of the interior flow, for which

*T*and

*w*are zero, satisfies

*b*denotes a barotropic pressure field independent of

*z.*The boundary conditions for this flow are that the geostrophic, barotropic velocities must cancel the vertical average of the baroclinic solution obtained from (3.2) so that

## 5. An example

*y*= 0, where the cooling will take place (for specified values of

*x*). The form is chosen to make the satisfaction of the thermal conditions on the sidewalls,

*T*= 0, very simple. The solution to (3.2) that accomplishes that is

_{Iy}*y*= 0,1 the boundary conditions for the barotropic velocities are

*A, B, E*, and

*F*are determined by applying (5.4). The result is shown in appendix B.

Figure 1 shows the contours of the velocity in the cross-channel direction at a position, *kx* = 1.25*π*, where the flow in the upper part of the water column is being driven toward the boundary *y* = 0. Half the channel width is shown: The solution for *υ* is antisymmetric about *y* = 0.5. In 0 ≤ *y* ≤ 0.5 flow is being driven toward the boundary *y* = 0 where it sinks in a boundary layer of width *δ _{b}* = 0.037. The resulting velocity profile of the zonal velocity shown in the half channel (the zonal velocity is

*symmetric*across the channel) at

*z*= 0.8 in Fig. 2. Note the monotonic decrease of the zonal velocity as

*y*= 0 is approached by the interior solution but, as we approach the boundary layer, the structure of the Stewartson layer produces a local enhancement of the down-channel velocity—just as found in the numerical model of Spall (2008). We can see here that this is due entirely to the damped oscillatory behavior of the layer’s structure and the relatively large amplitude of the correction driven by the need of the Stewartson layer to bring the cross-channel velocity to rest. Figure 3 shows the profile of the vertical velocity, again at

*z*= 0.8. The vertical velocity is entirely limited to the sidewall boundary layer since the interior velocity, frictionally driven, is extremely small, that is,

*O*(

*E*,

_{H}*E*). As in the numerical model, the sinking is limited to the boundary region where the vorticity production due to vortex tube stretching can be balanced by viscous dissipation.

_{v}*υ*in the boundary layer, which satisfies

*y*is evident. Note that the ageostrophic velocity in the boundary layer is largely driven by the

*geostrophic*velocity in the interior.

## 6. Discussion

A simple linear model has been used to discuss the response of a weakly stratified layer to a nonuniform cooling of the surface. The nonuniformity in the downstream direction is imposed to mimic the downstream variation that would appear naturally if the cooling were uniform but if, as in the nonlinear model of Spall (2008), nonlinear advection is included. In particular, the cooling has been chosen so that the geostrophic velocity in the cross-channel direction, forced by the cooling, drives an amplified response in a narrow sidewall boundary layer that is, for all intents and purposes, the same as the boundary layer introduced by Stewartson (1957). The principal difference in the treatment here is that for the layer to bring the *geostrophic* cross-channel velocity to rest at the boundary requires an amplitude for the boundary layer correction that is much larger than in the classical theory, where the interior flow is two-dimensional and the cross-channel velocity in the interior is weak and ageostrophic.

The analysis confirms the interpretation of Spall (2008) that the narrow zone to which strong vertical motion is limited in the numerical model that he investigated is essentially the same as the linear Stewartson layer. This is somewhat intriguing since the numerical model is strongly nonlinear, but it does seem to imply that the basic process determining the region of sinking by cooling is robustly governed by the balance between vorticity generated by vortex stretching and the dissipation of that vorticity in narrow regions near the boundary. In particular, the net downwelling at the boundary is set by the magnitude of the geostrophic flow in the interior, driven to the boundary by the along-channel pressure gradient produced by the cooling. The local boundary enhancement of the along-channel velocity is also well described by this simple analytical model and is, again, a fundamental feature of the Stewartson *E _{H}*

^{1/3}layer when that layer is forced by a geostrophic interior impinging flow.

If the stratification were increased enough to reverse the inequality (4.4) so that *σ _{H}S* ≫

*E*

_{H}^{2/3}(

*D*/

*L*)

^{2/3}(so that the deformation radius exceeds the width of the Stewartson layer), then the Stewartson layer would split (Barcilon and Pedlosky 1967) into a hydrostatic layer of thickness (

*σ*)

_{H}S^{1/2}and a very narrow buoyancy layer whose thickness is (

*E*/

_{H}D*L*)

^{1/2}/(

*σ*)

_{H}S^{1/4}; however, with insulating sidewalls this second layer is essentially absent. The vertical velocity is much reduced in the stratified hydrostatic layer, and the constraint on the interior velocity eliminates the forcing of the boundary layer by the geostrophic cross-channel flow. Thus, as might be expected, the strong downwelling seen in the model described here depends essentially on the weakness of the stratification, which in the numerical model of Spall is self-generated as the cooling-forced mixed layer.

## Acknowledgments

I am grateful to M. Spall for sharing with me his unpublished numerical calculations, which suggested the analysis of the present paper and for many helpful conversations concerning this work. This research was supported in part by NSF Grant OCE 0451086.

## REFERENCES

Barcilon, V., and J. Pedlosky, 1967: A unified theory of homogeneous and stratified rotating fluids.

,*J. Fluid Mech.***29****,**609–621.Greenspan, H. P., 1968:

*The Theory of Rotating Fluids*. Cambridge University Press, 327 pp.LaCasce, J. H., 2004: Diffusivity and viscosity dependence in the linear thermocline.

,*J. Mar. Res.***62****,**743–769.Marotzke, J., 1997: Boundary mixing and the dynamics of the three-dimensional thermohaline circulations.

,*J. Phys. Oceanogr.***27****,**1713–1728.Marotzke, J., and J. R. Scott, 1999: Convective mixing and thermohaline circulation.

,*J. Phys. Oceanogr.***29****,**2962–2970.Marshall, J., C. Hill, L. Perlman, and A. Adcroft, 1997: Hydrostatic, quasi-hydrostatic and nonhydrostatic ocean modeling.

,*J. Geophys. Res.***102****,**5733–5752.Pedlosky, J., 1968: Linear theory of the circulation of a stratified ocean.

,*J. Fluid Mech.***35****,**185–205.Pedlosky, J., and M. Spall, 2005: Boundary intensification of vertical velocity in a

*β*-plane basin.,*J. Phys. Oceanogr.***35****,**2487–2500.Spall, M. A., 2008: Buoyancy-forced downwelling in boundary currents.

,*J. Phys. Oceanogr.***38****,**2704–2721.Stewartson, K., 1957: On almost rigid rotations.

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## APPENDIX A

## APPENDIX B

The profile of the down-channel velocity *u* at *z* = 0.8 for the same parameters as in Fig. 1.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3996.1

The profile of the down-channel velocity *u* at *z* = 0.8 for the same parameters as in Fig. 1.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3996.1

The profile of the down-channel velocity *u* at *z* = 0.8 for the same parameters as in Fig. 1.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3996.1

The vertical velocity profile at *z* = 0.8. Parameters as in Fig. 1.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3996.1

The vertical velocity profile at *z* = 0.8. Parameters as in Fig. 1.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3996.1

The vertical velocity profile at *z* = 0.8. Parameters as in Fig. 1.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3996.1

The circulation streamfunction of the ageostrophic velocity in the *y*–*z* plane. All parameters are as in Fig. 1. The streamfunction contours are scaled with its numerical maximum = 0.007 63.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3996.1

The circulation streamfunction of the ageostrophic velocity in the *y*–*z* plane. All parameters are as in Fig. 1. The streamfunction contours are scaled with its numerical maximum = 0.007 63.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3996.1

The circulation streamfunction of the ageostrophic velocity in the *y*–*z* plane. All parameters are as in Fig. 1. The streamfunction contours are scaled with its numerical maximum = 0.007 63.

Citation: Journal of Physical Oceanography 39, 4; 10.1175/2008JPO3996.1