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1 Wind-Driven Flows and Bottom Topography 1 Wind-driven flows and bottom topography 1.1 The depth-integrated vorticity equation We start with the planetary geostrophic momentum equation and continuity, @τ k^ × fu = −∇φ + ; (1) @z @w r · u + = 0: (2) @z Taking the curl of (1) gives @ r · fu = r × τ ; (3) @z or, after splitting up the derivatives on the LHS and applying (2), @w @ u · rf − f = r × τ : @z @z Integrating through the water column, from the bottom at z = −H to the sea surface at z = 0 (ignoring the small contribution from the sea surface height) gives U · rf − fws + fwb = r × τ s − r × τ b; where subscripts refer to surface and bottom values, respectively. Here 0 U = u dz ˆ−H is the depth-integrated horizontal transport. We now assume that the kinematic bound- ary conditions are ws = 0 (i.e. a flat sea surface) and wb = ugb · r(−H) = −ugb · rH; where 1 u = k^ × rφ ; gb f jz=−H 1 is the bottom geostrophic velocity. Finally, we assume that the surface stress is given by the wind, τ s = τ w; and that the bottom stress is proportional to the bottom geostrophic flow, τ b = Rugb . This gives U · rf − fugb · rH = r × τ w − r × Rugb: (4) 1.2 The special case considered in Sverdrup-Stommel theory The classical Sverdrup-Stommel model for wind-driven mid-latitude flows can be re- coverd by one of two sets of assumptions: 1. The bottom is flat and the bottom friction is proportional to the depth-integrated flow. 2. The wind-driven flow does not actually reach the bottom (which may then very well be non-flat), but there is still a friction term which is proportional to the depth-integrated flow. In both cases the kinematic boundary condition falls out and the depth-integrated vorticity equation reduces to U · rf = r × τ w − r × rU; (5) where r = R=H: Introducing streamfunction for the depth-integrated flow (in steady- state the depth-integrated flow is divergentless), so that U = k^ × r ; or @ @ U = − ;V = ; @y @x allows the equation to be written 2 J( ; f) = r × τ w − rr ; 2 ^ where J(a; b) = axby − aybx is the Jacobian operator. Finally, since rf = βi;we arrive at the familiar expression @ β = r × τ − rr2 : @x w Stommel assumed that the interior oceans, away from the western boundary, is governed by the “Sverdrup” balance between advection of planetary vorticity and wind stress curl, i.e. @ β I = r × τ : @x w This is a hyperbolic first-order differential equation which may be solved by integrating the equation along characteristics which in this case are (zonal) f contours. Integrating westward from the eastern boundary where we set I = const: = 0 (from the kinematic boundary condition of zero flow into that wall), the interior solution for any y (latitude) becomes xe 1 I (x) = − r × τw dx: (6) ˆx β As shown in standard texts, near the western wall a boundary layer correction B that solves @ β B = −rr2 ; @x B needs to be added to the Sverdrup solution. 1.3 Bottom topography in a constant-density ocean What if the wind-driven flow does reach the bottom and if that bottom is not flat? As it turns out, the problem can now be re-cast in terms of another set of characteristics, namely q = f=H contours which are the more general form of the ambient potential vorticity. We start with the simplest possibility, namely an ocean of constant density. The starting point then becomes the steady and linear depth-averaged momentum equation, and the steady depth-integrated continuity equation, both for a non-stratified 3 fluid (i.e. the shallow-water equations): τ Ru k^ × fu = −grη + w − ; H H r · uH = 0: A PV equation is easily made by taking the curl of the momentum equation, giving τ Ru r · fu = r × s − r × ; H H or, multiplying and dividing by H on the LHS, f τ Ru r · Hu = r × s − r × : H H H In light of the continuity equation, the LHS can now be re-arranged to give f τ Ru Hu · r = r × s − r × : (7) H H H As seen, the expression becomes equivalent to the Sverdrup-Stommel model (5) when H is constant. But in this more general model the ambient potential vorticity field is q = f=H: So the expression is one that relates the forcing and dissipation to cross-f/H flow. 1.4 Bottom topography in a density-stratified ocean When baroclinicity in a stratified ocean is introduced, the problem may be written either in terms of the depth-integrated flow or, as we will see a bit later, the bottom flow. 4 1.4.1 An expression involving the depth-integrated flow We start by splitting the total depth-integrated transport into geostrophic and Ekman parts, U = (Hugb + U tw) + (U Es + U Eb) ; where Hugb is the depth-integrated transport associated with the geostrophic bottom flow (defined above) and 0 U tw = (ug − ugb) dz; ˆ−H τ U = −k^ × w ; E;s f Ru U = k^ × gb ; E;b f are the depth-integrated thermal wind transport (relative to the bottom flow) and the top and bottom Ekman transports, respectively. Rewriting the bottom stretching term in (4) in terms of a depth-integrated transport, f U · rf − Hu · rH = r × τ − r × Ru ; H gb w gb and then adding bottom stretching terms from the other flow components—to both sides of the equation—gives f f U · rf − U · rH = − (U + U + U ) · rH + r × τ − r × Ru ; (8) H H tw Es Eb w gb or, after dividing by H and using the product rule for derivatives, f f 1 f 1 f U·∇ = − U ·∇H+ r × τ − U · rH − r × Ru + U · rH : H H2 tw H w H2 Es H gb H2 Eb 5 Finally, substituting in the expressions for U Es and U Eb, and then again using the product rule for derivatives, results in f 1 τ Ru U · r = fU · r + r × w − r × gb : (9) H tw H H H This is the generalized version of (5) which takes into account effects of a non-flat bottom in a stratified ocean. Indeed, setting H to a constant recovers (5). Note that the same expression could also be reached by following a slightly different procedure, namely by taking the curl of the depth-averaged momentum equations rather than by taking a depth-integral of the vorticity equation (as we have done here). If bottom friction is confined to western boundary layers, the flow in a “generalized Sverdrup interior” can be found by again introducing a stream function for the depth- integrated flow, giving 1 τ J ( ; q) = fU · r + r × w ; I tw H H with q = f=H: The solution procedure then involves defining a coordinate system that follows f/H contours, with unit vectors ^t and n^ tangent and normal to the contours, respectively. The equation then becomes @ @q 1 τ I = fU · r + r × w ; @s @n tw H H and, assuming I = 0 where the q contours runs into the eastern boundary, the solution everywhere (except near the western boundary) is found by integration along any one q contour: se 1 1 τ w I (s) = − fU tw · r + r × ds: (10) ˆs @q=@n H H This is the generalized Sverdrup solution, equivalent to the flat-bottom version in (6). Although (9) is mathematically correct, it is somewhat dubious from a physical point of view since it suggests that the total depth-integrated flow interacts with the bottom (as shown even more explicitly in Eqn. 8). In reality it is of course only the bottom 6 flow ugb that can actually interact kinematically with a sloping bottom. We note that in order to create the final expression we had to add topographic interaction terms from both the thermal wind transport and the two Ekman transports. In particular, the thermal wind transport interaction term (the first term on the RHS of Eqn. 9) has gotten its own name, JEBAR (Joint Effect of Baroclinicity and Relief), and has received a lot of attention in the literature. It is nonetheless an unphysical term since it measures the kinematic interaction between a sloping bottom and thermal wind flow above that bottom. 1.4.2 An expression involving the bottom flow An alternative formulation that circumvents this issue can be made as follows. We start by expanding the total transport on the LHS of (4): (Hugb + U tw + U Es + U Eb) · rf − fugb · rH = r × τ w − r × Rugb; or, rearranging, Hugb ·rf −fugb ·rH = −U tw ·rf +(r × τ w − U Es · rf)−(r × Rugb + U Eb · rf) : Substituting in the expressions for U Es and U Eb, dividing by H and then again using the product rule for derivatives now gives f 1 f τ f Ru Hu · r = − U · rf + r × w − r × gb : (11) gb H H tw H f H f In line with the arguments above, it can be argued that this expression for what causes the bottom flow to cross f=H contours is somewhat more dynamically correct than the equivalent expression for the total depth-integrated flow (Eqn. 9). The generalized Sverdup solution for the bottom geostrophic flow away from a west- ern boundary layer can be found by introducing a geostrophic streamfunction ^ fugb = k × r ; 7 and introducing this into the equation after removing the bottom friction term and multiplying by q = f=H : f f 2 τ J ( ; q) = − U · rf + r × w ; I H2 tw H2 f or, in a coordinate that follows an q contour, @ @q f f 2 τ I = − U · rf + r × w : @s @n H2 tw H2 f Hence, the general procedure of integrating along a q contour, along s; up to the eastern boundary where I = 0 can be followed here as well.
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