Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 77 (Fundamentals of Atmosphere--ocean Dynamics)

Let ${\rho }_0$ be a constant reference density and define total buoyancy acceleration by $\sigma =-g(\rho -{\rho }_0)/{\rho }_0$. Absorb the reference hydrostatic pressure into a kinematic pressure $p$. The ideal nonrotating Boussinesq equations areFor a resting density profile ${\rho }_b(z)$, the background satisfies $p_b^{\prime }(z)={\sigma }_b(z)$ and the squared buoyancy frequency isStable stratification has $N^2>0$. The Boussinesq approximation requires fractional density variations to be small over the domain and parcel excursions, so density can be replaced by ${\rho }_0$ in inertia while its small variation is retained in the gravitational force. For an approximately uniform density gradient over height $H$, this requires $N^{2}H/g\ll 1$. The stipulated incompressibility alone does not justify neglecting density variation in inertia.

Put $b=\sigma -{\sigma }_b$ and $p^{\prime }=p-p_b$. To linear order the two-dimensional internal gravity wave equations areThe exact impermeability condition on $z=\eta (x,t)$ is the kinematic boundary conditionTaylor expansion about $z=0$ permits $w(x,0,t)={\eta }_t$ at leading order if the boundary slope and its displacement relative to the vertical wave scale are small: $|kϵ|\ll 1$ and $|mϵ|\ll 1$. The second condition controls both the omitted evaluation correction $\eta w_z$ and the product $u{\eta }_x$, since continuity gives $u\sim (m/k)w$. Hence for the harmonic forcing,The propagating calculation assumes $k\ne 0$ and $0<|\omega |<N$; a time-independent boundary does no oscillatory work on fluid initially at rest. If $k=0$ but $\omega \ne 0$, incompressibility forces the vertical velocity to be independent of height, so the unbounded column has no finite-energy radiating internal-wave solution. The propagating wave-generation problem therefore needs the stated nonzero horizontal wavenumber.

For a plane mode with phase $\theta =kx+mz-\omega t$, let $U,W,P,B$ be its complex amplitudes. The linear equations giveSubstituting into $-i\omega W=-imP+B$ gives the dispersion relationNo waves enter from above. The appropriate radiation condition therefore selects upward vertical group velocity, not upward phase propagation:The boundary-forced internal gravity wave satisfying the boundary condition isThe displacement is $ϵ\sin \theta$, consistent with $b=-N^2$ times the displacement. For a right-moving boundary choose $k>0$, $\omega >0$ without loss of generality. Then $m<0$, and the constant-phase lines of an internal gravity wave have slope $dz/dx=-k/m>0$: the crests rise toward the right. Their acute angle with the horizontal isThe group velocity is along these crests in the upward-right direction, whereas the wavevector and phase propagation point downward-right.

The physical pressure perturbation is ${\rho }_0{p}^{\prime }$. Pressure work at the lower boundary, to second order in amplitude, is its period-mean pressure times vertical boundary velocity. Static hydrostatic terms average to zero over a cycle. ThereforeAs an independent energy check, the wave's mean kinetic-plus-buoyancy energy density is ${\rho }_0{ϵ}^2{N}^2/2$, and multiplication by $c_{gz}$ gives exactly this upward energy flux.

For the exact horizontal mean-momentum equation, incompressibility writes the horizontal momentum equation asAverage over a horizontal period. Require the pressure and velocity fluxes to be periodic, or more generally their horizontal derivative averages to vanish; in particular there must be no imposed mean pressure gradient. ThenHorizontal averaging also makes ${\partial }_z\overline{w}=0$; the zero mean boundary displacement rate gives $\overline{w}=0$. These conditions exclude an independent mean mass flux or horizontal force. The mean equation applies in the fluid, with flattening of the small moving boundary understood at the order used above.

For slow switch-on, define $A(z,t)=ϵ(t-z/c_{gz})$, using $ϵ(s)=0$ for $s<0$. The outgoing internal-wave envelope radiation condition is $A_t+c_{gz}{A}_z=0$. Replace $ϵ$ in the oscillatory fields by $A$. A modulation time $T$ must satisfy $|\omega |T\gg 1$ and $|m|c_{gz}T\gg 1$, with a frequency bandwidth sufficiently narrow to linearize the dispersion relation. The exact boundary derivative also contains $\dot{ϵ}\sin (kx-\omega t)$; it is smaller by the slow-modulation parameter and is beyond this leading envelope approximation.

To leading slow-modulation order, the wave momentum flux isSince ${\partial }_z{A}^2=-c_{gz}^{-1}{\partial }_t{A}^2$, integrating the mean-momentum equation from rest gives the internal-wave momentum depositionThis is second order in wave amplitude, with smaller relative slow-modulation corrections. Mean-flow feedback on the leading wave is higher order. The mean current is created as the wave envelope arrives; an already uniform wavetrain has no local flux divergence.

Finally move horizontally with speed $c_p=\omega /k$. The initial uniform relative flow is $-c_p$, with infinite total energy in the unbounded column. Its finite mean-energy change is insteadAt finite time the envelope is absent sufficiently far above, so integrating the momentum equation givesThis is the boundary work and mean-flow energy of an internal wave relation. In the moving frame the boundary pattern is stationary to leading modulation order. It extracts energy from the incident mean flow and converts it to outgoing internal-wave energy; in the laboratory frame the same energy is supplied as boundary work.

Here write $\sigma$ for the buoyancy perturbation about the resting constant-$N$ stratification; the total buoyancy also includes its background gradient $N^2$. Use kinematic perturbation pressure $p$. The weak-forcing Linearized Boussinesq equations areWeakness means that perturbation advection is small compared with the retained linear terms. With the specified streamfunction convention, the transverse vorticity isDifferentiate the horizontal momentum equation in $z$ and subtract the $x$ derivative of vertical momentum. The pressure derivatives cancel, givingThe second equation uses ${\sigma }_t=N^2{\psi }_x$ and is the forced internal-wave equation.

For free waves at frequency $|\omega |\ll N$, the internal gravity wave dispersion relation gives $|k|/|m|\simeq |\omega |/N\ll 1$. Thus the horizontal response scale is much larger than the vertical scale. Vertical acceleration is small in the vertical momentum equation, leaving the hydrostatic approximation $p_z=\sigma$. Correspondingly the transverse vorticity is predominantly $u_z$: neglect $-w_x={\psi }_{xx}$ relative to ${\psi }_{zz}$ in the inertial vorticity term. The restoring term $N^2{\psi }_{xx}$ must still be retained, since it balances the small time derivatives.

For $m\ne 0$ put $\psi =e^{imz}P(x,t)$, $\sigma =e^{imz}B(x,t)$ and interpret physical fields as real parts. The reduced equation isOne can solve it while obtaining the buoyancy simultaneously. Define $U=imP$ and $H=iB/(cm)$. The first-order hydrostatic equations becomeHence $R_{+}=U+H$ and $R_{-}=U-H$ obeyApply the method of characteristics with zero fields in the past. Withthe solution is $R_{\pm }=I_{\pm }$. Consequently the hydrostatic response to a localized horizontal force isThese are exactly the required two characteristic integrals, derived with their relative sign fixed by the buoyancy equation. Quiet-past initial conditions exclude additional freely propagating solutions.

For the consistency check, set $\tau =s-t$, so both integrals run over the fixed range $(-\infty ,0)$. Differentiation at fixed $x,z$ now givesThis change of variable is important: it includes the time dependence of both the sampled forcing and the moving integration geometry without omitting a term. Each straight path intersects $|x|\le a$ for a time interval of length at most $2a/c$. If $L=\sup |f_{xt}|$, thenThe characteristic size of the hydrostatic buoyancy and pressure-gradient balance is $|m|a{f}_{\max }$, as follows from the analogous bound on $B$ in its integral formula. Thus the omitted acceleration has relative scale at most a constant timesTogether with the low-frequency assumption, this verifies the requested hydrostatic consistency. It is a comparison of characteristic magnitudes, rather than division by a buoyancy value at a node where both terms may vanish. The capital $N$ here is the buoyancy frequency, as printed in the PDF.

For $t\ge 0$, only $0\le s\le t$ contributes. If $|x|>ct+a$, every point $x\mp c(s-t)$ lies outside the source interval, so both integrals vanish:This finite propagation cone belongs to the reduced hydrostatic wave equation.

The hypotheses do not require an even forcing, so no single symmetric graph follows for every permitted $f$. For an illustrative smooth even nonnegative profile, $U$ is even and $H$ is odd. At late time and fixed $x$, if $A_f={\int }_{-a}^{a}f(r,\infty )dr$, thenThus the streamfunction develops a broad nearly level interior response with outgoing fronts, while the buoyancy changes sign across the source and approaches opposite interior plateaus to its left and right. Both vanish beyond the fronts. For $m>0$, a vertical phase with $\sin (mz)=1$ gives real fields $\psi =U/m$ and $\sigma =cmH$. The figure plots those profiles up to positive normalization factors, for an even compact forcing increasing slowly from zero.

Choose a fixed vertical datum so the physical free surface is at $H+\zeta$, with $H$ constant; its depth above the bottom is $h=H+\zeta -b$. The gradient of the constant datum does not enter the pressure force. With $D_H={\partial }_t+u{\partial }_x+v{\partial }_y$, the rotating shallow water equations areThe fluid is hydrostatic with depth-independent horizontal velocity at this order. The pressure-gradient force uses the free-surface slope, not the layer-depth slope: $\nabla \zeta =\nabla h+\nabla b$.

Let $q=v_x-u_y$ be relative vertical vorticity, $q_a=f+q$ the absolute vorticity, and $\delta =u_x+v_y$. Taking the horizontal curl of the two momentum equations and expanding the advective derivatives givesMass conservation is $D_{H}h=-h\delta$. The two stretching factors cancel, proving the exact shallow-water potential vorticity conservation lawThe denominator is $h$, the actual column thickness that changes under horizontal convergence; an arbitrary surface elevation $\zeta$ does not obey that column mass equation. There is no $w{\partial }_z$ term because the reduced layer quantities have no vertical dependence. The vertical motion has already supplied the layer-thickness equation, rather than remaining an independent coordinate of this model.

Now take $H=h_{00}$ and assume $|q|\ll |f|$, $|\zeta |,|b|\ll H$. To first order,For a small Rossby number, leading geostrophic flow satisfies $fv=g{\zeta }_x$ and $fu=-g{\zeta }_y$. ThusFor $f\ne 0$, the barotropic deformation radius is $L_R=\sqrt{gH}/|f|$. Multiplication of the first-order PV by $H$ givesThis is the required shallow-water quasi-geostrophic potential vorticity, including the bottom contribution.

Write $S=fϵ/H$ and $L=L_R$. The initial state has $\psi =0$, so ${\mathcal{Q}}_i=f+(f/H)b$. The specified homogenization sets ${\mathcal{Q}}_f=f$ in $|y|<a$ and leaves it unchanged outside. Hence the requested change isFor $f>0$ this is a descending straight segment from $+Sa$ to $-Sa$, with zero exterior values and jumps at the two edges. Endpoint values do not affect the inversion. It is the anomaly, not the full final PV profile, that appears as the source below.

For the zonal mean, $v=0$, $u=-{\psi }^{\prime }(y)$, and potential-vorticity inversion requireswith decaying disturbance at both infinities. The odd source makes the unique decaying solution odd. A particular interior solution is $S{L}^{2}y$, so writeBoth $\psi$ and ${\psi }^{\prime }$ must be continuous at $y=\pm a$: a jump would introduce a delta function or its derivative into a source that has only finite jumps. Matching at $a$ givesand thereforeDifferentiate the interior streamfunction to obtainThe exterior profiles needed for the sketches are $u=(D/L)e^{-(|y|-a)/L}$ and $\zeta =(f/g)\mathrm{sgn}(y)D{e}^{-(|y|-a)/L}$.

For $L\gg a$, put $A=a/L$, $Y=y/L$, so $|Y|\le A$. Uniform Taylor expansion givesConsequentlyFor $f>0$ the leading interior current is westward and approximately parabolic.

For $L\ll a$ and away from the edges, $u\simeq -S{L}^2$ and $\zeta =(f/g)\psi \simeq ϵy$: the surface follows the bottom slope centrally. Near $|y|=a$, the exponentially thin corrections give positive edge jets of width $L$ and peak scale $SLa/2$ when $f>0$. Those jets decay in the exterior. The elevation is odd, has opposite-sign extrema just inside the edges, and decays to zero outside; at the positive edge $\zeta (a)\simeq ϵ(a-L)/2$. The following sketch uses the full matched solution with $L/a=0.08$. For negative $f$, the current and PV-change signs reverse, while this elevation profile does not.

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.