Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 333 / 3

At the displaced free surface, constant atmospheric pressure and the hydrostatic approximation give the dynamic conditionThe linearized kinematic condition is $w=D_g\eta /Dt$. Combining these relations withgives

Linearization about rest replaces the material derivative by ${\partial }_t$, so the quasi-geostrophic potential-vorticity equation becomesInsert the separation of variables $\psi =\varphi (x,y,t)P(z)$ and imposeThe horizontal amplitude then satisfiesThe Linear Rossby-wave equation therefore governs each vertical normal mode, whose Rossby deformation radius is $c/f_0$.

The rigid lower boundary has $w=0$. For a time-dependent normal mode this givesThe free-surface condition from part i givesWriting $q=N/c$, the lower condition selectsThe upper condition therefore requiresEquivalently, with $\mu =NH/c$ and $ϵ=N^{2}H/g$,Each positive root ${\mu }_n$ determines the barotropic mode or a baroclinic mode through $c_n=NH/{\mu }_n$.

Suppose $ϵ=N^{2}H/g\ll 1$. The smallest root satisfiesso the leading external gravity wave eigenfunction is depth-independent:For $n\ge 1$, the roots lie just above $n\pi$:and henceIn particular, up to an arbitrary normalization and sign,The hierarchy $c_0>c_1>c_2>\cdots$ separates the fast, nearly barotropic free-surface mode from the internal baroclinic modes.

Expand both the forcing and response in the orthogonal vertical normal modes. Projection onto $P_n$ givesIf the horizontal forcing scale obeys $L_h\gg c_n/f_0$, the relative vorticity term ${\nabla }_h^2{\varphi }_n$ is small compared with the stretching term. The resulting long-wave equation isThus each mode communicates the forcing westward at its long Rossby wave speed $C_n$, with the barotropic mode fastest because $c_0$ is largest.

The method of characteristics shows that a point is affected only after a westward characteristic from the forcing region reaches it. At a western observation point $X_1$, during the intervalthe fast barotropic signal has arrived but the first and higher baroclinic signals have not. Consequentlywhich is nearly independent of depth. At the eastern point $X_2$, no westward Rossby-wave characteristic arrives from the forcing region, so the disturbance remains zero under the stated initial and radiation conditions.

The propagation speed derived in part v is $C_n=\beta c_n^2/f_0^2$. Thus the question's notation $c_{0}t>L_X>c_{1}t$ must be read as a comparison with these modal Rossby signal speeds; using the gravity-wave speeds $c_n$ literally would not describe the long-wave equation derived above.

In a steady state the time derivative of quasi-geostrophic potential vorticity vanishes, leaving the local balanceChoose the undisturbed state immediately east of the compact forcing as the integration condition. The solution is thenThe discontinuity in this ideal expression reflects the discontinuous prescribed forcing at $z=-D$; a vertically smooth forcing gives the corresponding smooth vertical profile.

The adjustment begins with the rapidly propagating barotropic mode and is therefore initially almost depth-independent. Successively slower baroclinic modes then arrive from the forcing region. Their Fourier series in the vertical normal modes progressively reconstructs the forcing's upper-layer profile, tending to the stated steady solution while points to the east remain unaltered.