Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 73 / 3

Take a beta-plane approximation, $f=f_0+\beta y$ with $\beta >0$, and define the depth transport $\mathbf{T}=(U,V)={\int }_{-H}^0{\mathbf{u}}_{h}dz$. For steady flow with no net flux through surface and bed, depth-integrated continuity gives $U_x+V_y=0$. Vertical integration of momentum givesIn the interior, small depth-to-horizontal aspect ratio makes vertical stress divergence the leading viscous contribution; horizontal viscous terms are neglected there. Taking the vertical component of curl eliminates pressure. Sinceneglecting bottom stress gives the Sverdrup balance,Thus the wind-stress curl sets the interior meridional depth transport, not the local surface meridional velocity. A constant-$f$ plane would have no $\beta V$ term; the use of $\beta$ in this question requires latitude-dependent $f$. For a transport streamfunction with $(U,V)=(-{\bar{\psi }}_y,{\bar{\psi }}_x)$, the interior relation is $\beta {\bar{\psi }}_x=W/{\rho }_0$.

To close a depth-integrated bottom-drag model, approximate the bottom horizontal velocity by the depth mean, ${\mathbf{u}}_b\simeq \mathbf{T}/H$. This is the barotropic closure needed to express the stated stress solely in terms of the transport. DefineThe coefficient $\gamma$ in a physical stress law is not itself a drag rate; $r$ has units of inverse time. The ocean basin equation with bottom drag and lateral viscosity follows by taking curl of the integrated momentum equation and yields the combined Stommel boundary layer and Munk boundary layer equation,The bottom contribution is negative because the vertically integrated stress difference is surface stress minus bottom stress. Set the constant wall streamfunction to zero. No normal flow and no tangential slip impose $\bar{\psi }=0$ and ${\partial }_n\bar{\psi }=0$ on the walls.

The inviscid interior solution has ${\bar{\psi }}_I=S(y)x/\beta +C_0(y)$. In a thin east/west layer, $x$ derivatives dominate the smaller $y$ derivatives. Holding $y$ as a parameter givesEquivalently the boundary correction $\varphi =\bar{\psi }-{\bar{\psi }}_I$ obeys $\nu {\varphi }^{\prime \prime \prime \prime }-r{\varphi }^{\prime \prime }-\beta {\varphi }^{\prime }=0$. In a stretched inward coordinate $X=x/\delta$ at the west wall and $X=(1-x)/\delta$ at the east wall, this becomes respectivelyThe first-derivative sign reversal distinguishes east from west. Vanishing $W$ and its derivatives at the north/south boundaries makes leading forcing-dependent solutions compatible with those boundaries, avoiding extra leading meridional layers. These are leading boundary-layer equations, not an exact deletion of every meridional derivative in the full basin PDE.

The nonconstant exponential modes of the leading basin equation satisfyThus a leading uniformly valid form is $\bar{\psi }\simeq S(y)x/\beta +C_0(y)+{\sum }_{j=1}^3{C}_j(y)e^{{\lambda }_{j}x}$, with exponentials referenced to the wall where they decay. The coefficients enforce the four east/west no-slip conditions; their values are not needed to identify the scales.

If $r^3\ll {\beta }^2\nu$, horizontal viscosity dominates drag in the boundary layers. The Munk boundary layer scale is ${\delta }_M=(\nu /\beta {)}^{1/3}$, assumed much smaller than basin width. The roots are ${\delta }_M^{-1}$ and $(-1\pm i\sqrt{3})/(2{\delta }_M)$ to leading order, soThe east layer has exponential thickness ${\delta }_M$ and the west oscillatory layer has envelope thickness $2{\delta }_M$, both of order ${\delta }_M$. The west correction supplies the order-one return transport, while the east no-slip correction is typically smaller in amplitude. Setting $r=0$ still leaves enough viscous modes to impose no-slip.

If $r^3\gg {\beta }^2\nu$, linear drag dominates the broad vorticity layer. Definewith ${\delta }_S$ small compared with basin width. The roots are approximately $-{\delta }_S^{-1}$ and $\pm {\delta }_v^{-1}$; the fast roots have smaller correction $\beta /(2r)$. The drag-dominated basin solution with no-slip layers isThere is a broad western Stommel boundary layer of thickness ${\delta }_S$, a thinner western no-slip layer of thickness ${\delta }_v$, and an eastern no-slip layer of thickness ${\delta }_v$. The narrow layers can have small streamfunction amplitude while supplying an order-one change in wall velocity.

If viscosity is set identically to zero, only the particular solution, a constant and the western Stommel exponential remain. That second-order model cannot generally satisfy both streamfunction and derivative conditions at both walls. Small nonzero viscosity must be retained in the wall skins when no-slip is required. In the question's stress notation, the limits compare $[\gamma /({\rho }_{0}H){]}^3$ with ${\beta }^2\nu$. Neither limit applies at the crossover, where all cubic terms contribute. These composite forms are leading asymptotic solutions of the basin problem; meridional derivatives and interior diffusion supply higher-order corrections.