A criterion for the existence of an inertial boundary current in a baroclinic ocean is derived. It is shown that the effect of topographical variations can play a decisive role in determining whether an inertial boundary current can exist, even if the velocity field is highly baroclinic

Pedlosky, Joseph

English

Yale University

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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. https://creativecommons.org/licenses/by-nc-sa/4.0/   A Necessary Condition far the Existence of an Inertial Boundary Layer in a Baroclinic Ocean' Joseph Pedlosky Department of M athematics M assachusetts Institute of Technology ABSTRACT A :,criterion for the existence of an inertial boundary current in a baroclinic ocean is deri ved . It is shown that the effect of topographical variations can play a decisive role in determining whether an inertial boundary current can exist, even if the velocity field 1s highly baroclinic. Introduction. The question of the existence of inertial boundary layers plays a central role (Charney 1955) in most theories of the steady general oceanic circulation. Greenspan (1962) showed that a necessary condition for the ex-istence of an inertial boundary layer in a homogeneous ocean could be found in the form -+ _,. _,. a (U1 · n) - (f/h) < o. as Here U1 is the fluid velocity of the interior flow at the edge of the boundary layer, f is the Coriolis parameter, h is the ocean depth, and-; is a unit normal vector directed toward land at the ocean boundary. The scalar s measures arc length (counterclockwise) along the perimeter of the basin. The criterion ( 1) suggests that bottom topography could play a decisive role in determining whether an inertial boundary current can exist. One important weakness of this theorem exists in the assumption that the ocean is essentially vertically homogeneous in both density and velocity. If, on the other hand, the velocities at great depths are considerably smaller than the surface currents, as in a baroclinic ocean, one might expect the influence r. This research was supported by the U.S. Air Force, Grant AF-AFOSR-492-64. Accepted for publication and submitted to press I 8 March 1965. 69 J ournal of M arine Research [23,2 of topography to be correspondingly reduced. T o investi gate this question further it is necessary to consider the condition fo r the existence of an inertial boundary current in a baroclini c ocean. The Baroclinic Crit erion. For the sake of convenience only, consider that the relevant boundary runs south to north, along which they axis is ali gned; let x measure positi ve-eastward into the ocean, and z measure positi ve-upward from the mean ocean bottom. A scalin g analysis can show that for geophysicall y relevant phenomena in middle latitudes the appropriate equation governing the boundary-layer motion in the asymptotic region near the edge of the boundary layer is the quasigeostrophic potential vorti city equation. The density is assumed to be a conservative property for any fluid element. In addition, the scale height of the density fi eld is assumed large compared to the depth of the fluid. In the asymptoti c region, the potential vorti city equation can be line-arized about the interi or moti on, and one obtains as the relevant asymptoti c equation governing the boundary-layer motion: (2a) In (2a), U1(y, z) is the interi or eastward fl ow velocity at the edge of the boundary layer and is a function of latitude and depth, v is the northward velocity of the boundary current, and y is the positi ve rati o, -/2 /g(fJ l neJ/fJz). The quantity (! I is the density fi eld of the interi or fl ow at the edge of the boundary layer. The appropriate boundary conditions fo r (2a) express the vanishing of the vertical velocity at the top of the ocean, z = H, while at the ocean's bottom a possible south-north variation of bottom topography, 'Y/y , will produce a vertical velocity, V'f)y - The east-west variati on of topography is ignored fo r purposes of simpli city. Unless the topographical variati on is very large in the x di recti on, this may be done without loss of generalit y (see G reen-span 1962). If the equation expressing the conservation of density is utili zed to evaluate the verti cal velocity, it can be shown that the boundary conditions may be expressed as foll ows: at z = H , at z = 'Y/· (2 b) (2c) Two assumptions are now made. First, it is assumed that the tangent of the angle made by the hori zontal-velocity vector to the shore is always finit e i. e. there is always some component of the boundary-layer fl ow parallel t~ the coast. This may be expressed in the asymptotic region as (3) Pedlosky: An Inertial Boundary Layer Second, it is assumed that, in the asymptotic region, Vxx - > O. V 71 This insures a boundary-layer character for the flow in the asymptotic region. The equation for v/ U1 can be shown to be (yu; (v/ U1)z)z + (v/ U1) (u1 df) + (:!!.-) u; = o. dy U1 xx (5) Multiply (5) by the nonsingular quantity (v/U1)- 1 and then integrate from z = 'YJ to z = H to obtain H H H U2 (v/ U1): d: u 2 Vxx d: u df d: u ( )f J --- y Z + I - Z + I - Z + I 'YJ 'YJy = 0 . (v/ U1)2 v dy (6) n n n Since the first two terms in (6) are positive, one must have, as a necessary condition for the existence of a baroclinic inertial boundary current, H df~ - U1dz + JU1 ('Y/) 'YJy < o, dy (7) n or H (H - 'YJ)2 U1 ('Y/) !!_ (_j- ) + df \ [U1(z) - U1 (rJ)] dz< o. dy H-rJ dy J (8) n If U1 is independent of z, i.e. if the ocean is barotropic, (8) reduces to (1)-the criterion derived by Greenspan for the homogeneous ocean. Define the vertically averaged flux out of the boundary layer, M, as H M=~U1dz . n Then the criterion (8) can be written (9) (1 o) Thus, if the bottom of the ocean is flat, ( 1 o) states that the flux, M, must be negative. The vertically integrated interior velocity must be directed toward the coast. On the other hand, if drJ/dy is not zero and if the bottom velocity, Ui('Y/), is not zero, the effect of topography may completely alter this criterion 72 · J ournal of Marine Research on the transport. If dr;/dy is 0(10- 2) and Hi s 0(105 cm), which are reasonable Gulf Stream values, then, if U1 (r;)H/M > 10-2, the effect of topography will be at least as important as the variation of the Coriolis parameter. Assume that the interior Row, U1 , takes on its maximum value at the surface, z = H; then, if the ratio of the bottom velocity to the surface velocity is as large as one to one hundred, the topographical effect will play a decisive role, even in such a highly baroclinic ocean, in determining whether an inertial boundary current may exist. Thus, such a phenomenon as the separation of the Gulf Stream at a point where (10) is violated, may, in fact, be completely determined by topographical effects, as Greenspan originally argued on the basis of his treatment of the homogeneous-ocean problem. Finally, I feel that assumptions (3) and (4) are probably not independent and that only one need be assumed. However, such a speculation has not as yet been proven. · REFERENCES CHARNEY, J. G. 1955. The Gulf Stream as an inertial boundary layer. Proc. nat. A cad. Sci., JI: 73 1-740. GREENSP AN, H. P. 1962. A criterion for the existence of inertial boundary layers in oceanic circulation. Proc. nat. Acad. Sci., 48: 2034-2039. 

A necessary condition for the existence of an inertial boundary layer in a baroclinic ocean

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A necessary condition for the existence of an inertial boundary layer in a baroclinic ocean

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