Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2015 (A11048W1, C5)

Source paper: www.whatdotheyknow.com/request/oxford_physics_past_papers_2/response/2340696/attach/3/Physics%20Papers%202014.pdf?cookie_passthrough=1. The dated cover identifies 16 June 2015, irrespective of the bundle's 2014 filename. Questions 1 and 2 survive and are solved below; source pages 5–10 have no content streams, so their unknown question numbering and requests are not reconstructed. Lettered subdivisions organize the surviving marked blocks.

Notation TODO. The angular-velocity glyph in the source's general equations on page 2 is illegible. We use $\mathbf{\Omega }$ for the angular velocity explicitly described by the surviving words; this does not claim to recover the missing printed glyph. The missing illustration for question 2 is discussed in its own solution.

Let $b$ denote buoyancy in a stably stratified Boussinesq approximation. Geostrophic balance and hydrostatic balance imply the thermal wind relation $f_0{U}_z=-{\bar{b}}_y$. Consequently the specified vertical shear requiresFor northern-hemisphere $f_0>0$ and $\Lambda >0$, the mean potential temperature decreases northward and increases upward: ${\bar{\Theta }}_y=-({\Theta }_0/g)f_0\Lambda$, ${\bar{\Theta }}_z=({\Theta }_0/g)N^2$. The corresponding mean isentropes slope upward toward the north, with $dz/dy=f_0\Lambda /N^2$. Stable potential-temperature stratification does not require the actual temperature to increase upward; the dry adiabatic lapse also enters that comparison.

In the rigid-lid constant-shear baroclinic model, perturbation quasi-geostrophic potential vorticity isThe mean interior potential-vorticity gradient vanishes on this constant-$f_0$, constant-$N$, linear-shear basic state. Linear advection of that perturbation therefore becomes $({\partial }_t+\Lambda z{\partial }_x)q^{\prime }=0$, the supplied interior equation. It retains slow balanced motion and omits friction, planetary-vorticity gradients and nonlinear disturbance advection.

Substitution of ${\psi }^{\prime }=\Psi (z)\cos (ly)e^{ik(x-ct)}$ givesA possible branch has zero interior perturbation potential vorticity, soBoth supplied applications have $f_0>0$, so this equals the source's expression. For an unstable mode the nonzero imaginary part of the phase speed prevents a real critical level $\Lambda z=c$; for real speeds the displayed solution remains a valid zero-interior-PV branch away from, and smoothly through, any critical level. Taking the real part gives the physical disturbance.

The rigid lids enforce $w^{\prime }=0$. Linearized adiabatic buoyancy conservation isUse the geostrophic conventions $v^{\prime }={\psi }_x^{\prime }$ and $b^{\prime }=f_0{\psi }_z^{\prime }$. At a lid this gives $({\partial }_t+\Lambda z{\partial }_x){\psi }_z^{\prime }-\Lambda {\psi }_x^{\prime }=0$, henceThese are the no-normal-flow condition expressed through boundary buoyancy advection, not conditions of zero horizontal velocity. Boundary temperature anomalies can interact even though this solution has no interior perturbation PV, allowing baroclinic instability to extract mean available potential energy.

For $l=0$ put $L_D=NH/f_0$, $\eta =k{L}_D$ with $k>0$, and write the discriminant asThe second factor is positive for every $\eta >0$, since $x>\tanh x$. The first changes sign at the positive solution of $x=\coth x$, $x=\eta /2$. Numerically $x_c=1.19968$, so ${\eta }_c=2.39936$. The growing branch exists for $0<k{L}_D<{\eta }_c$, givingEquality is neutral, so this is an infimum wavelength, rather than an attained shortest growing mode.

In the unstable interval,Both phase-speed branches have the same real part there; the imaginary parts have opposite signs. At and beyond the short-wave cutoff the growth rate is zero and the two real speeds areThey split continuously from $1/2$ and approach the boundary-flow speeds zero and $\Lambda H$ at large $\eta$. As $\eta \to 0$, $D=-{\eta }^2/12+O({\eta }^4)$, so $\sigma \to 0$ linearly with $k$, despite a finite limiting $c_i=\Lambda H/\sqrt{12}$. A zonally uniform disturbance has no exponential growth from this mechanism.

The maximum growth minimizes $D$. Its derivative isThe positive interior minimum is at ${\eta }_m\simeq 1.606$, yieldingThe supplied phase-speed formula therefore predicts fastest growth at a wavelength of order four deformation lengths, not at the short-wave cutoff.

Write the vertical complex amplitude as $\Psi (z)=a(z)e^{i\varphi (z)}$. A constant streamfunction phase has $kx-k{c}_{r}t+\varphi (z)=\mathrm{const}$, hence $dx/dz=-{\varphi }_z/k$. For $k>0$, westward tilt with height implies ${\varphi }_z>0$. With $v^{\prime }={\psi }_x^{\prime }$ and $b^{\prime }=f_0{\psi }_z^{\prime }$, a zonal average over the wave givesfor the northern-hemisphere conventions used here. This is the westward tilt and poleward eddy heat flux relation. Warm anomalies move northward and cold anomalies southward, transporting heat down the mean gradient ${\bar{b}}_y=-f_0\Lambda <0$. The tilt describes the relative phases needed for a nonzero flux, not the literal trajectories of fluid particles along a tilted streamfunction crest.

The flux reduces the mean meridional temperature contrast and converts mean available potential energy into eddy available potential energy. In the perturbation buoyancy-variance budget its production term is $-\overline{v^{\prime }{b}^{\prime }}{\bar{b}}_y/N^2>0$. Correlated rising warm anomalies and sinking cold anomalies have positive $\overline{w^{\prime }{b}^{\prime }}$, converting that eddy potential energy into kinetic energy. The flow is therefore energized by the baroclinic reservoir associated with tilted density/potential-temperature surfaces. It is not simply a claim that the disturbance amplifies while leaving the basic state's thermal structure unchanged.

For nonzero meridional wavenumber,At least one growing zonally varying mode can exist only ifThe upper value is a supremum approached as $k\to 0$, where the growth itself tends to zero. Thus not every permitted meridional wavenumber has the same fastest-growing zonal scale; nonzero $l$ restricts the available band of $k$ and weakens the growth factor $k/\sqrt{k^2+l^2}$ at fixed $\eta$.

Using $N=\sqrt{N^2}$, not the tabulated squared value itself, givesThe planetary radii put these into angular context: $l_{\sup }R$ is about 280 for Earth and 149 for Jupiter, corresponding to minimum meridional angular wavelengths about $1.28^{\circ }$ and $2.42^{\circ }$, respectively. They are reciprocal-length wave numbers; the dimensionless angular counts should not be substituted for $l$ in the Cartesian dispersion relation. If a channel admits a half-wave across its width, the corresponding minimum width is $\pi /l_{\sup }$ rather than a full wavelength.

The oceanic scale is mesoscale, while the much larger deformation length makes the Jovian modes thousands of kilometres across. Both are small fractions of the planetary radius, consistent with using a local patch as a first approximation. Real boundaries, planetary-vorticity gradients, varying stratification and jets can change these limits; this simple model alone does not predict the complete observed flow pattern.

Finally, the source's phrase $c^2<0$ must be interpreted in the mean-flow frame. The actual criterion is $D<0$, or $(c-\Lambda H/2{)}^2<0$. In the original frame $c$ has nonzero real part, so $c^2$ is generally complex, not an ordered negative real number. For example, at $\eta =1$ and $\Lambda H=1$, $c=0.5+0.2511i$ and $c^2\simeq 0.1870+0.2511i$. This is an unstable wave even though the literal printed inequality is inapplicable.

Meridional overturning circulation links relatively warm upper-ocean transport to a colder return flow at depth, with exchanges among basins and vertical branches. Surface heat loss and salinity changes alter water density; sufficiently dense water can sink and spread along deep pathways. Freshwater input lowers salinity and can inhibit dense-water formation. Winds, particularly in the Southern Ocean, and energy supplied to mixing are also important to the global return and upwelling pathways. The circulation is not powered by surface density forcing alone in an otherwise closed, unmixed ocean.

The Atlantic meridional overturning circulation carries heat northward, helping regulate ocean-atmosphere heat exchange and regional climate. Overturning also ventilates the deep ocean and redistributes carbon, oxygen and nutrients. Changes in its strength can alter regional sea-surface temperatures, air-sea heat release and atmospheric circulation. They need not eliminate every western-boundary current or every source of European warmth: wind-driven ocean circulation and atmospheric heat transport are distinct, interacting contributions. The two-box model below isolates a density/salinity feedback rather than representing all these mechanisms.

Illustration TODO. The source's two-volume circulation sketch on page 4 is unavailable. Its exact arrows and layout cannot be recovered. The surviving flow law, salinity/temperature definitions and printed steady polynomial allow the calculation below under an explicit freshwater sign convention; no unavailable diagram values are invented.

Use $\Delta T=T_2-T_1>0$, $\Delta S=S_2-S_1$, and let positive $q$ be the thermally driven state with cold box 1 denser than warm box 2. Take positive $F$ to freshen box 1, represented in a fixed-volume salt budget by an equivalent salt removal $F{S}_1$ from that box and addition to the other reservoir. This is the freshwater convention compatible with the printed $+BF$ and positive $B$. It can represent a compensating saline volume transfer or the usual virtual-salt-flux approximation. The exact pipe assignment is not claimed to have been read from the missing illustration.

Exchange in either circulation direction reduces a salinity difference at rate proportional to $|q|\Delta S$. Steady salt balance therefore givesThe supplied linear equation of state givesEliminating $\Delta S$ yields the two-box overturning model equationHere $A$ and $B$ both have transport units when $q,F,k$ do, while $\alpha \Delta T$ and $\beta S_1$ are dimensionless. Although the source calls $q$ a mass flow, its stated Sverdrup units are volume per time; the common reference density cancels in the Boussinesq salt/volume approximation. At $F>0$, $q=0$ cannot be a finite-salinity steady state in this model. At $F=0$, zero flow needs separate treatment instead of division by $|q|$.

The circulation weakens when freshwater freshens the cold reservoir, and the weaker circulation removes the resulting salinity contrast less effectively. This salt-advection feedback is the mechanism behind multiple steady states, not an additional externally imposed switch.

Use all coefficients exactly as printed and use Sv consistently for transports. The present salt contrast and density-driven friction coefficient areThusThe large numerical value of $k$ reflects division by a small dimensionless density contrast, about $2.07\times 10^{-6}$ with the supplied effective expansion coefficients. It is not the actual overturning transport, which remains $15$ Sv.

For $q>0$, the steady equation is a quadratic with rootsThe critical freshwater forcing in a two-box overturning model occurs where the two roots coalesce:There is also a negative-flow branch for every $F>0$:At the present forcing, the three roots are $15.000$, $6.774$ and $-3.950$ Sv. The prescribed present state is therefore the upper positive branch, not an arbitrary choice among the roots.

Stability needs a salinity evolution law, not just the algebraic steady equation. For fixed temperature contrast and reference salinity, the standard salt-contrast dynamics can be writtenwhere $V_{\mathrm{eff}}>0$ summarizes the reservoir volumes. The linear slope at a positive root is $(A-2q)/V_{\mathrm{eff}}$: the upper branch is stable and the lower branch unstable. At a negative root it is $(2q-A)/V_{\mathrm{eff}}<0$, so the reversed branch is stable. The two stable states below the fold have different attraction basins, demonstrating bistability. Their stability is conditional on this fixed-temperature, fixed-reference-salinity closure.

As forcing increases quasistatically from $0.5$ Sv, the normal branch weakens and reaches a saddle-node bifurcation at $F_c$. Beyond it no positive steady circulation remains, so the reduced dynamics move toward the stable reversed branch. The reversal is salinity dominated: the fresh cold reservoir is no longer the denser one. Recovery under a later forcing change can depend on the system's state and feedback history.

Interpret a uniform increase of $0.3\%$ per year from the present value as a constant slope, $F(t)=F_0(1+0.003t)$ with $t$ in years. Under quasistatic tracking of the stable branch, the time to its fold isIf instead the phrase means a compounded fractional increase, $F=F_0{e}^{0.003t}$, the corresponding time is $\ln (F_c/F_0)/0.003=51.3$ yr. These are two different forcing prescriptions, not two different critical freshwater fluxes.

The branch does not remain exactly at $15$ Sv until that date. Differentiating its quadratic giveswhich is $-24.71$ at the present state. The initial decrease is about $0.0371$ Sv per year under the linear ramp. At the fold the flow is only $10.887$ Sv, about 27 percent below its present intensity. Thus maintained close has no unique duration without a tolerance: for example, a ten-percent decrease to $13.5$ Sv occurs atA five-percent criterion gives about $18.4$ yr. The 55-year value estimates persistence of the positive equilibrium branch, not a perfectly flat present-day circulation.

These times are model extrapolations, not forecasts. The reservoirs' finite adjustment times are unspecified; near a fold, weakening restoring rates can invalidate quasistatic tracking and produce a delayed transition. Real ocean temperatures, reference salinity, winds, mixing, ice, geometry, regional freshwater pathways and the density-to-transport relation do not stay fixed. The imposed freshwater trend and the two-box reduction need independent justification before an actual transition time can be inferred.

A strong weakening of northward Atlantic heat transport could cool the North Atlantic and northern Europe relative to a climate with stronger overturning, altering atmospheric circulation and precipitation. It does not imply that all wind-driven transport vanishes or that greenhouse forcing ceases. The defensible implication is sensitivity of regional climate to ocean heat-transport changes, while the toy model supplies a transparent example of feedback and a threshold rather than a reliable dated prediction.

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