Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 77 / 4

Let ${\mathbf{\omega }}_a=2\mathbf{\Omega }+\nabla \times \mathbf{u}$ be absolute vorticity, where $\mathbf{\Omega }=(0,0,f/2)$ is constant. Let $\rho$ be density and $\alpha$ a materially conserved stratifying scalar, so $D\alpha /Dt=0$. For the incompressible density-stratified ideal fluid one may choose the advected density itself; in a thermodynamic description one may use an entropy-like scalar with $\rho =\rho (p,\alpha )$. The Rossby–Ertel potential vorticity isErtel's theorem states that it is conserved following a parcel under inviscid, adiabatic motion with conservative body force and the stated stratifying-scalar condition.

Start from the rotating momentum equationHere $\Phi$ includes gravity and any conservative centrifugal potential. Taking its curl givesMass conservation is $D\rho /Dt=-\rho \nabla \cdot \mathbf{u}$, specializing to $D\rho /Dt=0$ here. Combining these equations yieldsDifferentiating the material conservation of $\alpha$ gives $D({\partial }_i\alpha )/Dt=-({\partial }_i{u}_j){\partial }_j\alpha$. Dotting these two equations, the velocity-gradient terms cancel after relabelling their indices. ThereforeIf $\alpha =\rho$, the triple product vanishes immediately. More generally, an equation of state $\rho (p,\alpha )$ makes $\nabla \rho$ a linear combination of $\nabla p$ and $\nabla \alpha$, so it also vanishes. HenceThis proves the conservation law and specifies why $\alpha$ must be a suitable stratifying scalar: an arbitrary unrelated passive tracer need not annihilate the baroclinic triple product. No viscosity or scalar diffusion is permitted in this theorem.

For the Boussinesq approximation, subtract the resting background buoyancy $N^{2}z$ and denote its perturbation by $b$. With constant nonzero $f$ and constant $N>0$, the leading geostrophic flow and hydrostatic balance define the quasi-geostrophic streamfunctionThe leading horizontal flow is nondivergent. At the next order, its relative vertical vorticity $q_g={\nabla }_H^2\psi$ is stretched by the small vertical velocity, while the background buoyancy is advected vertically. Thus the curl and buoyancy equations giveThese follow from $D{q}_a/Dt=-q_a{\nabla }_H\cdot \mathbf{u}$ with $q_a\simeq f$ and ${\nabla }_H\cdot {\mathbf{u}}_a=-w_z$, retaining the geostrophic advection of the first-order anomalies. Vertical advection of those anomalies is of higher quasi-geostrophic order.

Differentiate the buoyancy equation in $z$. It is important not to assume without justification that ${\partial }_z$ commutes with $D_g$. Their commutator on $b$ isby hydrostatic balance and thermal wind. Hence $D_g{b}_z=-N^2{w}_z$. Adding $f/N^2$ times this equation to the vorticity equation eliminates $w_z$, yieldingThis is the quasi-geostrophic potential-vorticity equation on an $f$-plane. The additive $f$ makes $Q$ equal to the first-order normalized Ertel PV of total buoyancy $N^{2}z+b$; using a PV anomaly instead removes that constant and does not change the dynamics.

For stretched coordinates for quasi-geostrophic inversion, putThen $(f^2/N^2){\partial }_z^2={\partial }_Z^2$, and the potential-vorticity relation becomesStable stratification makes this operator elliptic. Uniform $N$ and nonzero $f$ are essential to this simple constant-coefficient stretching.

Potential-vorticity inversion is the diagnostic recovery of the balanced flow from the instantaneous PV distribution, together with appropriate boundary or far-field data. In the unbounded constant-$N$ case the mathematical problem is the Poisson equationRequire a regular solution, with interface continuity in a piecewise-smooth case. For a compactly supported bounded anomaly, the unique decaying solution is the Newtonian-potential integralThe sign follows from ${\nabla }_{\ast }^2[-1/(4\pi r)]={\delta }_0$. A nonzero total anomaly gives a $1/r$ far-field streamfunction, which is allowed by the stated decay condition. The measure is in stretched coordinates; converting to physical coordinates introduces their volume Jacobian.

Two regular solutions with these data differ by a harmonic function tending to zero, which must vanish by the maximum principle. This also excludes imposed uniform currents or shear fields encoded by nondecaying harmonic additions and fixes the irrelevant additive pressure gauge. Once $\psi$ is found, ${\mathbf{u}}_g=(-{\psi }_y,{\psi }_x)$, $p^{\prime }=f\psi$ and $b=f{\psi }_z$ follow. The smaller vertical and ageostrophic velocities require the next-order balance equations.

PV is not by itself an inversion variable for arbitrary unbalanced fluid motion: the quasi-geostrophic approximation supplies the diagnostic geostrophic and hydrostatic relations, filtering the independent inertia-gravity-wave motions. In bounded domains, boundary buoyancy or equivalent pressure data are additional indispensable inputs.

Let $Q_0>0$ denote the small excess PV in the spherical region, with $Q_0\ll |f|$ for quasi-geostrophic consistency. Let its stretched radius be $R$ and set $r=(X^2+Y^2+Z^2{)}^{1/2}$. Spherical symmetry of the source and uniqueness of the decaying inversion imply a radial streamfunction. Its equations areRegularity at the origin removes the interior $1/r$ term. Decay removes the exterior constant. Integrating gives ${\psi }_{min}=Q_0{r}^2/6+C_0$ and ${\psi }_{mout}=C_1/r$. Since the source has no surface delta function, $\psi$ and ${\psi }_r$ match at $R$. Their matching fixes $C_1=-Q_0{R}^3/3$ and $C_0=-Q_0{R}^2/2$. Therefore the spherical potential-vorticity anomaly hasIts induced horizontal flow isThis flow is azimuthal about the vertical axis, so it is tangent to the PV level surfaces and advects no part of this axisymmetric distribution across its spherical boundary. The example is therefore steady under the quasi-geostrophic PV evolution. In physical coordinates the stretched sphere is the ellipsoid $x^2+y^2+(N^2/f^2)z^2=R^2$, with vertical semiaxis $R|f|/N$.

Relative vertical vorticity is the horizontal Laplacian of the streamfunction, not its full stretched three-dimensional Laplacian. Inside the spherical potential-vorticity anomaly, the quadratic solution givesOutside, in the equatorial plane $Z=0$, let $s=(x^2+y^2{)}^{1/2}>R$ and $C=Q_0{R}^3/3$. The azimuthal velocity is $v_{\theta }={\psi }_s=C/s^2$, soThe opposite signs thus follow directly from solid-body rotation inside and the exterior velocity's rapid radial decay. The azimuthal velocity itself has the same rotational sense on both sides; its radial derivative makes the exterior relative vorticity negative.

There is no contradiction with the zero exterior PV anomaly. At $Z=0$ the exterior solution has ${\psi }_{ZZ}=C/s^3$, so $q_g+{\psi }_{ZZ}=0$. The stratification/stretching contribution compensates the horizontal relative vorticity. Inside it contributes $Q_0/3$, supplementing $2Q_0/3$ to give the prescribed excess $Q_0$.

Potential-vorticity conservation makes the evolving PV distribution a set of parcel-carried labels. In the ideal model it can be rearranged into filaments and complicated geometry, but it is not created or destroyed by advection. Apparent homogenization, such as that assumed for the shallow-water experiment, describes a coarse-grained mixed distribution rather than pointwise destruction of materially conserved PV.

Within a balanced regime, potential-vorticity inversion turns those labels and the boundary data into a velocity field. This makes understanding a flow a coupled geometric problem: determine how PV is rearranged, invert the resulting distribution, and use the recovered velocity to find the next rearrangement. Inversion is nonlocal, so a compact PV anomaly influences fluid well outside its support, as the spherical example explicitly demonstrates.

The quasi-geostrophic potential-vorticity equation has only one time derivative:An initial PV field and the inversion data determine the balanced evolution; an independent initial acceleration is not needed. Pressure, buoyancy and the leading velocity are diagnosed from that field at each instant. This reduced initial-data requirement reflects the removal of freely oscillating inertia-gravity waves, which in the full fluid equations require additional initial information. The principles are consequently powerful for slow large-scale flow, but do not recover every unbalanced motion or remove the need to state boundary conditions and the balance approximation.