Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2017 / 3

Put the rigid lid at $z=0$, the interface at $z=-h(x,y,t)$ and ${\rho }_1={\rho }_0-\Delta \rho$. Assume hydrostatic vertical balance and a resting deep layer. Its pressure is $p_2=C-{\rho }_{0}gz$, with horizontally uniform $C$. Pressure continuity at the interface then fixes the upper-layer pressure:Hence its horizontal pressure gradient at fixed $z$ is ${\nabla }_h{p}_1=\Delta \rho g{\nabla }_{h}h$, independent of depth. Taking the horizontal momentum equation and a vertically uniform upper-layer velocity givesThe exact density denominator for upper-layer acceleration is ${\rho }_1$; the familiar ${\rho }_0$ denominator is the small-density-contrast Boussinesq approximation. The restoring pressure is weaker than for a free surface because only the density difference contributes.

The horizontal shallow-layer equations use the vertical component of planetary rotation: $f=2\Omega \sin \phi$. Thus the source's vector notation is interpreted as the projected vector $f\mathbf{k}$ after the traditional approximation. Taking the full magnitude $2\Omega$ as a midlatitude Coriolis parameter would discard the latitude dependence. Together with layer-volume conservation, the model has ${\partial }_{t}h+{\nabla }_h\cdot (h\mathbf{u})=0$.

Both descriptions are depth-averaged hydrostatic shallow-layer models. In a one-layer free-surface shallow water equations, a homogeneous layer overlies a rigid bottom and its upper surface moves. Surface elevation changes give a restoring acceleration based on full gravity $g$, with external-wave speed $\sqrt{g{H}_0}$. Barotropic coastal storm surges, long surface gravity waves or a basin seiche are examples.

In the rigid-lid reduced gravity model, the movable boundary is a density interface beneath the active upper layer, while the much deeper lower layer remains at rest. The internal interface mode instead has speed $\sqrt{g^{\prime }{h}_0}$ and represents, for example, thermocline displacements and an upper-ocean baroclinic eddy. The mathematical shallow-water form is retained with $g$ replaced by $g^{\prime }$ and with the appropriate interface-depth convention. A freely moving two-layer surface would support both external and internal modes; the rigid lid filters out the fast external mode, and the resting abyss removes its independent dynamics.

Write $h=h_0+\eta$, with small interface displacement $\eta$ and horizontal variation scale $L$. The geostrophic pressure-Coriolis balance givesSubtracting the uniform reference state and the mass-conserving linear potential-energy term, the perturbation energy densities per unit horizontal area areThereforeThe internal deformation radius $L_R$ is the distance an internal gravity wave travels in an inverse-Coriolis time. Balanced broad disturbances $L\gg L_R$ mainly store available potential energy; balanced fine structures $L\ll L_R$ require relatively large kinetic energy. This scale result presupposes small Rossby number $U/(|f|L)$; reducing the length at fixed displacement eventually violates geostrophy. It does not say an initially resting small-scale height perturbation already has that large kinetic energy: such a perturbation undergoes geostrophic adjustment.

Let $K=(k^2+l^2{)}^{1/2}$ and $c=\sqrt{g^{\prime }{h}_0}$. The three branches areFor nonzero $K$ and $f$, the zero-frequency mode is stationary geostrophic flow, with $f\mathbf{k}\times \mathbf{u}=-g^{\prime }\nabla \eta$ and zero horizontal divergence. It carries the linear potential-vorticity disturbance. There is no Rossby-wave propagation on a constant-$f$ plane; a varying Coriolis parameter would be needed for that.

The other two branches are oppositely phased inertia-gravity waves. Their phase-speed magnitude and radial group velocity are $|\omega |/K=\sqrt{c^2+f^2/K^2}$ and $|\partial \omega /\partial K|=c^{2}K/\sqrt{f^2+c^2{K}^2}$. When $K{L}_R\gg 1$, gravity dominates: $|\omega |\simeq cK$ and the waves are approximately nondispersive, with both speeds approaching $c$. When $K{L}_R\ll 1$, rotation dominates: $|\omega |\simeq |f|+c^2{K}^2/(2|f|)$, the frequency approaches the inertial frequency and the group velocity tends to zero. The large long-wave phase speed is not a large signal speed.

Without rotation the propagating branches become nondispersive gravity waves, $\omega =\pm cK$, and the stationary branch is a nondivergent vortical velocity disturbance rather than a finite-pressure geostrophic balance. At $K=0$, the modes reduce to constant interface height and spatially uniform inertial oscillations. The wave branches have zero linear potential-vorticity perturbation; it is the stationary branch that retains the initial potential vorticity.

A numerical scale estimate requires a reference ocean density and latitude not specified exactly. Taking ${\rho }_0\simeq 1025\mathrm{kg}{\mathrm{m}}^{-3}$, $g\simeq 9.81\mathrm{m}{\mathrm{s}}^{-2}$ and a representative midlatitude $|f|\simeq 10^{-4}{\mathrm{s}}^{-1}$ givesFor a small-amplitude interface displacement of scale $L=1\mathrm{km}\ll L_R$, gravity drives rapid horizontal acceleration and radiates most of the displacement in inertia-gravity waves. The gravity adjustment time is $L/c\sim 8$ minutes, much shorter than $1/|f|\sim 2.8$ hours. A weak balanced remnant remains in an unbounded inviscid adjustment problem; energy is carried away by waves, not dissipated merely by geostrophic adjustment.

For $L=100\mathrm{km}\gg L_R$, rotational adjustment produces a largely retained interface displacement with a geostrophic current. Wave emission and inertial oscillations accompany the adjustment, but the balanced structure is much more important. To quantify that distinction under an initially resting, linear single-Fourier-component assumption, potential-vorticity conservation givesThis is about $0.002$ for the kilometre scale and $0.95$ for the hundred-kilometre scale. These fractions use $L$ as an inverse wavenumber scale, not a full wavelength, and are illustrative rather than unique predictions for an unspecified disturbance or bounded basin.