Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2016 (A11048W1, C5)

Source paper: www.whatdotheyknow.com/request/physics_past_exam_papers_2015_16/response/973997/attach/3/Cowell%20Attachment.zip?cookie_passthrough=1. The dated cover identifies the Part C sitting of 14 June 2016. All eight options are treated below, using the native radiometer plot for the graphical estimates.

Optical response at fixed cloud structure. Put $b=1-g$ and write the two-stream cloud albedo as $A=b\tau /(2+b\tau )$. At fixed scattering asymmetry parameter,At zero optical depth the last quotient is understood by its finite limiting value. Optical depth counts extinction along a ray. For an approximately homogeneous, monodisperse cloud,This uses spherical droplets, independent extinction, representative constant $N,r$, and the geometric optics limit in which the droplet radius greatly exceeds the relevant visible wavelength and $Q_{\mathrm{ext}}\simeq 2$. Diffraction supplies part of the extinction cross-section; it is not legitimate to obtain the factor two from geometric interception alone. A size distribution requires its second moment rather than an arbitrary mean radius.

With liquid-water density ${\rho }_w$ and cloud liquid-water content $L$, the volume constraint gives $r^3=3L/(4\pi {\rho }_{w}N)$. ConsequentlyThis provesKeeping $L,h,{\rho }_w,g$ fixed isolates the Twomey effect:It is a partial response, not a claim that cloud water and cloud geometry stay unchanged under every real aerosol perturbation.

Absolute and fractional susceptibility. Choose $N_{\ast }$ so that $A=1/2$ there, and put $n=N/N_{\ast }$. ThenThe cloud albedo susceptibility per added droplet decreases monotonically. Its small-$n$ asymptote is proportional to $n^{-2/3}$, while its large-$n$ asymptote is proportional to $n^{-4/3}$. Thus a fixed absolute droplet-number addition has much more leverage in a clean cloud than in an already droplet-rich cloud. The formal divergence as $N\to 0$ is not a physical infinite response: fixed water content would require indefinitely large drops, eventually invalidating the optical and cloud assumptions.

For a fixed fractional number addition the response is instead $dA/d\ln N=A(1-A)/3$, maximal at $A=1/2$ and small at either very low or nearly saturated albedo. This distinction is essential when comparing interventions specified by an absolute particle addition with those specified by a percentage increase.

Brightening and net radiation. Marine cloud brightening would increase the population of suitable cloud condensation nuclei, for example by producing sea-salt aerosol from seawater and allowing transport into susceptible marine boundary-layer clouds. Activation depends on particle size, hygroscopicity and the cloud's attained supersaturation; adding particles does not guarantee a one-for-one increase of droplets. At fixed cloud water, more activated droplets mean smaller radii, larger total extinction area, larger shortwave optical depth and more reflected sunlight.

Promising physical conditions combine relatively clean, persistent low clouds, appreciable incident sunlight and a dark underlying ocean. The extensive marine stratocumulus regions over the eastern subtropical oceans are examples of this combination. An already polluted, optically saturated cloud has little remaining shortwave leverage. Bright clouds over snow or ice also add relatively little contrast with the surface. Low clouds have temperatures closer to the surface than high clouds, so their extra longwave greenhouse effect is generally less able to offset the shortwave cooling. Sunlit seasons and latitudes matter; no shortwave benefit occurs at night.

Schematically, the fixed-cloud shortwave perturbation is $\delta F_{\mathrm{SW}}\sim -\mathcal{S}{f}_c\delta A$, where $\mathcal{S}$ is the local incident solar flux and $f_c$ the affected cloud fraction. Changes in cloud fraction, precipitation, entrainment, liquid water and longwave emission must also be evaluated before equating the positive derivative above with a guaranteed net cooling. The derivative identifies susceptibility, not a complete climate-response calculation.

Potential temperature and a parcel. For a dry ideal gas of fixed composition, the first law of thermodynamics per unit mass isA reversible adiabatic process has $\delta q=0$. Taking $c_p,R_d$ constant gives $d\ln T=\kappa d\ln p$, where $\kappa =R_d/c_p$. Integrating to a fixed reference pressure $p_0$ defines the potential temperatureIt is the temperature a parcel would have after reversible dry adiabatic process taking it to $p_0$. Equivalently $ds=c_{p}d\ln \Theta$ under these assumptions, making potential temperature a thermodynamic entropy label useful for diagnosing stratification and air-mass motion. Latent heating, radiation, mixing and variable composition require modifying the conservation statement.

An air parcel is a small material volume followed with the flow. The parcel model treats it as internally approximately uniform, large compared with molecular scales and small compared with environmental variation scales. Rapid acoustic wave adjustment makes its pressure approximately equal to the surrounding pressure, while slower heat and mass exchange can be neglected during an idealized displacement. It is not a rigid container or an isolated molecule.

At the parcel's height, environmental hydrostatic pressure supplies an upward pressure force ${\rho }_{e}g$ per unit volume. Dividing by parcel density gives the upward accelerationThe last steps use equal pressure, equal gas constants and the same reference pressure. Moisture-dependent density requires virtual temperature or its potential-temperature counterpart. Nonhydrostatic pressure perturbations and entrainment are excluded from this elementary buoyancy expression.

The convective available potential energy is the positive buoyancy work per unit mass between the level of free convection and the equilibrium level:One sums positive-buoyancy layers when there is more than one. Negative work below free convection contributes convective inhibition rather than positive CAPE. In a lossless parcel estimate, conversion of CAPE to vertical kinetic energy gives $w_{\max }\simeq \sqrt{2\mathrm{CAPE}}$; pressure work beyond this model, mixing and precipitation can change that speed.

Linear stability. Let $\xi$ denote displacement from a neutrally buoyant reference altitude. Dry motion through an adiabatic process retains ${\Theta }_p={\Theta }_e(0)$, while ${\Theta }_e(\xi )={\Theta }_e(0)+\xi {\Theta }_e^{\prime }(0)+O({\xi }^2)$. The linear vertical momentum equation is thereforeFor $N_0^2>0$, $\xi =C\cos N_{0}t+D\sin N_{0}t$: stable buoyancy oscillations with period $2\pi /N_0$. For $N_0^2=0$, $\xi =C+Dt$: neutral stratification with no restoring force. For $N_0^2<0$, growing and decaying exponentials occur, indicating convective instability. The growing solution is generic unless the initial data select only the decaying branch. These are local small-displacement results with an approximately constant environmental gradient.

Lapse-rate estimate. Define $\Gamma =-dT/dz$. Hydrostatic balance and the ideal gas law imply $p^{\prime }/p=-g/(R_{d}T)$, soIn the free-parcel-oscillator interpretation, identifying the ridge encounter interval with the buoyancy period gives $P=10^4/20=500\mathrm{s}$ and $N_0=2\pi /P=0.01257{\mathrm{s}}^{-1}$. With $T=293.15\mathrm{K}$, $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$ and $c_p\simeq 1004\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$,The frequency identification is an extra modeling assumption. A stationary mountain gravity wave generally obeys ${\hat{\omega }}^2=N_0^2{k}_h^2/(k_h^2+m_z^2)$ without rotation, with $\hat{\omega }=U{k}_h$. Ridge spacing fixes $\hat{\omega }$, not the vertical wavenumber $m_z$. For example, $m_z=k_h$ allows the same spacing and wind with $N_0=\sqrt{2}\hat{\omega }$, and a lapse rate about $0.3\mathrm{K}\mathrm{k}{\mathrm{m}}^{-1}$. Thus $5.1$ is the requested parcel-period estimate, not a uniquely identifiable lapse rate for every possible mountain-wave field; a resolved wave field needs its vertical structure as well.

Normalization and angular pattern. A scattering phase function describes the distribution of single-scattered power over outgoing solid angle, for a specified incident direction, wavelength and polarization convention. Use the common normalizationThe scattering asymmetry parameter is the mean cosine of the scattering angle: forward scattering has positive $g$, backward scattering negative $g$, and a forward–backward symmetric pattern has $g=0$.

Let $C=E_0(2\pi /\lambda {)}^4[(n^2-1)/(n^2+2){]}^2{r}^6$. The given small-sphere angular intensity is $I=C(1+{\mu }^2)/2$, where $\mu =\cos \Theta$. Its total scattered power isNormalizing givesDimensionally the supplied $I$ is radiant intensity, power per solid angle. If interpreted as detector irradiance at a distance, the appropriate inverse-square distance factor is also needed; it cancels out of the normalized angular pattern.

The polar plot has equal forward and backward lobes, each with radius $3/2$, and lateral values $3/4$. A small nonabsorbing particle behaves as an induced electric dipole. Radiation transverse to the dipole and the average over two orthogonal incident polarizations produce the $1+{\cos }^2\Theta$ form. Its odd angular moment vanishes:As the size parameter $2\pi r/\lambda$ grows, Rayleigh scattering in the electric-dipole approximation no longer suffices. Higher multipoles and interference produce structure in the angular pattern, and large particles generally develop a strong narrow forward diffraction peak. Their $g$ is normally positive, although detailed values and angular features depend on refractive index, size distribution and shape; they are not fixed by radius alone. Large spheres may show rainbows and backscattering structure, while nonspherical ice introduces further angular features.

Optically thick transmission. In a nonabsorbing two-stream or transport approximation, it is the transport optical depth $(1-g)\tau$ that controls the loss of forward directional memory. Using the same idealized albedo closure as in option 1 gives total forward-hemisphere transmittanceFor the same large but finite extinction optical depth,The respective transport-depth factors are $0.15$, $0.30$ and $1$. The water cloud scatters most strongly forward and returns the least light backward. All transmittances tend to zero in the infinite-depth limit. The strictly unscattered direct-beam transmittance is instead $e^{-\tau }$, identical in the three cases: the ranking concerns total transmitted radiation, including forward-scattered light, not that direct component alone. Full radiative transfer can also depend on higher angular moments beyond $g$.

Third body and steady partition. The third molecule removes excess association energy and momentum, stabilizing newly formed ozone rather than being consumed in the net reaction. Its decreasing abundance with altitude suppresses three-body ozone formation strongly. Write $x=[\mathrm{O}]$, $y=[{\mathrm{O}}_3]$, $a=k_1{n}_2{n}_M$, $j=J_3$ and $b=k_3$, with $n_2,n_M$ held as reservoirs. The kinetic equations areThe fast association/photolysis subsystem conserves odd oxygen, $x+y$, and has eigenvalues $0,-(a+j)$. Its partition-adjustment time is thereforeThis neglects the slower production and mutual-destruction terms while finding that fast mode. On its equilibrium partition, $ax\simeq jy$, and total odd-oxygen equilibrium gives $bxy=J_2{n}_2$. HenceWith fixed oxygen mole fraction and temperature, the atomic-oxygen mixing ratio scales as $x/n_M\propto n_M^{-3/2}$ in this approximation, and thus increases upward as pressure decreases.

Lifetime versus relaxation. The distinction is formalized by Chapman family lifetime versus relaxation time. In the ozone-dominated regime $y\gg x$, the family abundance is approximately $y$, and its gross chemical lifetime isThis recovers the printed ${\tau }_S$ as a family turnover time under its additional partition assumption. It is not generally an e-folding relaxation time. With rapid partitioning, the family equation is $\dot{X}=P-k{X}^2$. Linearizing about equilibrium gives relaxation rate $2k{X}_e$ but turnover rate $k{X}_e$, a factor of two difference. More generally the approximate family lifetime is $(x+y)/(2bxy)$, and its relaxation time is half that quantity. Neither can be replaced by $1/(2bx)$ when atomic oxygen dominates.

For a direct check using the complete two-species equations, the equilibrium Jacobian matrix isThe numerical high-altitude example below gives eigenmode times about $321\mathrm{s}$ and $4.0\times 10^5\mathrm{s}$, whereas the printed ${\tau }_S$ is only $5.15\times 10^3\mathrm{s}$. Thus interpreting both printed expressions as universal eigenmode time constants would be false, even though the fast partition and the ozone-dominated turnover approximations are useful.

Setting $x=y$ in the fast-partition approximation gives $a=j$, henceRetaining mutual destruction gives $a=j+bx$ and changes this estimate by less than one percent. The formal high-altitude limit of ${\tau }_F$ is $1/J_3\simeq 333\mathrm{s}$: ozone photolysis, unlike three-body association, does not slow merely because collider density decreases. This is a partition timescale, not the lifetime of all odd oxygen. Extrapolating the fixed-reservoir approximation indefinitely upward would eventually cease to be self-consistent.

Day and night. At $p=0.01\mathrm{hPa}=1\mathrm{Pa}$ and $230\mathrm{K}$, $n_M=p/(k_{B}T)=3.15\times 10^{20}{\mathrm{m}}^{-3}$ and $n_2=0.2n_M$. The exact equilibrium is conveniently obtained fromThe physical root givesTheir ratio is $156$, consistent with the requested approximate factor $150$; the simpler fast-partition estimate gives approximately $151$. The printed fast time is $331\mathrm{s}$ at this pressure.

For nighttime Chapman ozone, after sunset set both photolysis rates to zero. Ozone initially grows by converting atomic oxygen, but simply putting every initial oxygen atom into one ozone molecule neglects the simultaneous $\mathrm{O}+{\mathrm{O}}_3$ sink. The night equations areWhile sufficient atomic oxygen remains, the ozone balance tends towardThis is a useful long-night sunrise estimate, roughly thirty times the sunset ozone abundance, not an exact result for an unspecified night duration. The precise phase-plane relation isTogether with the night equations it supplies $y(t_{\mathrm{night}})$ for any supplied duration. For illustration only, a twelve-hour dark interval gives ${\chi }_{O_3}\simeq 5.8\times 10^{-5}$; no twelve-hour duration is stated by the paper. Even the infinite-dark-time endpoint is slightly below $a/b$, because the finite initial atomic reservoir is consumed. This conditional estimate avoids inventing a unique sunset-to-sunrise duration.

Detector noise and graphical reading. Noise-equivalent power specifies the incident optical power producing signal-to-noise ratio one, with the electrical bandwidth and averaging convention stated. A spectral NEP has units $\mathrm{W}\mathrm{H}{\mathrm{z}}^{-1/2}$; a band-integrated NEP has units W. Specific detectivity normalizes the latter by detector area and noise bandwidth:Its usual unit, the Jones, is $\mathrm{cm}\mathrm{H}{\mathrm{z}}^{1/2}{\mathrm{W}}^{-1}$. Removing the expected square-root area and bandwidth dependence helps compare different detector sizes and readout bandwidths, provided noise conditions and spectral response are comparable. This does not normalize away every background, temperature or implementation difference.

Reading the native embedded voltage plot gives approximate means $V_C\simeq 0.331\mathrm{V}$, $V_S\simeq 0.406\mathrm{V}$ and $V_H\simeq 0.493\mathrm{V}$. The one-second root mean square scatter is about $0.006\mathrm{V}$; the plotted peak-to-peak excursions are larger and must not be substituted for an root mean square noise. Independent graphical digitization of the blue trace gives means near $0.331,0.406,0.493$ V and scatters about $0.006$–$0.007$ V, consistent with this estimate. These are readings from a drawing, not exact original data.

The supplied Planck approximation gives $B_C\simeq 0.08590$ and $B_H\simeq 0.12836$ in $\mathrm{W}{\mathrm{m}}^{-2}\mathrm{s}{\mathrm{r}}^{-1}(\mathrm{c}{\mathrm{m}}^{-1}{)}^{-1}$. Thus the noise-equivalent radiance of one one-second sample isHere the calibrating means are used to determine the response slope; their own uncertainty is included explicitly in the final scene estimate below.

Optical throughput and detectivity. A paraxial $f/F$ cone has half-angle approximately $1/(2F)$ and projected solid angle approximately $\pi /(4F^2)$. With $F=2$ and $A_d=10^{-6}{\mathrm{m}}^2$,Using the exact cone angle $\tan \theta =1/(2F)$ gives projected etendue $A_d\pi {\sin }^2\theta =1.85\times 10^{-7}{\mathrm{m}}^2\mathrm{sr}$, within the intended approximation. The radiance noise corresponds to detector power noiseThe wavenumber bandwidth is $1\mathrm{c}{\mathrm{m}}^{-1}$ and must match the radiance's per-wavenumber unit; it is not an electrical noise bandwidth. For the common order-of-magnitude assignment $\Delta f\sim 1\mathrm{Hz}$ to a one-second sample,Area in the last expression is $1\mathrm{m}{\mathrm{m}}^2=0.01\mathrm{c}{\mathrm{m}}^2$. For a specified one-second boxcar with white noise, the one-sided equivalent noise bandwidth is instead $1/(2t)=0.5\mathrm{Hz}$. That convention gives spectral NEP $\simeq 2.5\times 10^{-10}\mathrm{W}\mathrm{H}{\mathrm{z}}^{-1/2}$ and $D_{\ast }\simeq 3.9\times 10^8$ Jones. The source does not specify the electrical filter, so the two conventions should not be mixed or assigned spurious precision.

Two-point retrieval and its uncertainty. Assume linear, stable response for the three views. The radiometer two-point calibration isWith target radiances treated as known, differentiating givesNotice $b<0$ for an intermediate scene: raising the hot voltage while keeping the scene voltage fixed reduces the inferred scene radiance. Numerically $a\simeq 0.2621$, $b\simeq -0.1214$ in radiance units per volt.

The scene interpolation fraction is about $0.463$, giving $B_S\simeq 0.1056$ andFor independent white noise in each view, each hundred-sample mean has standard error ${\sigma }_{\bar{V}}=0.006/10=0.0006\mathrm{V}$. The radiance error isUsing the given temperature sensitivity,A scatter estimate closer to $0.007$ V changes this to roughly $0.14$ K. These are random standard errors under the averaging assumptions. Target-temperature/emissivity uncertainty, response drift and atmospheric or surface-emissivity corrections are separate: the retrieved quantity is a brightness temperature, not automatically the true sea-surface temperature.

Potential-vorticity dynamics. The quasi-geostrophic potential-vorticity equation expresses material conservation of a balanced potential vorticity by a horizontally nondivergent geostrophic flow. Its terms represent relative vorticity, planetary vorticity and the layer-thickness/free-surface contribution. The Rossby deformation radius $L_d$ measures the scale at which that thickness contribution competes with the horizontal-vorticity term. The quasi-geostrophic approximation filters fast acoustic waves and gravity waves, making the slower balanced evolution accessible to early digital forecasts. The 1950 barotropic-vorticity forecasting study demonstrated practical twenty-four-hour forecasts from observed mid-tropospheric fields, while identifying discretization and omitted baroclinicity as error sources.^[ref]

Near the north pole put $\delta =\pi /2-\theta$ and $r\simeq a\delta$. ThenUsing projected rather than arc distance gives the same leading polar expansion. It is a north-polar expression; the sign structure is changed consistently at the opposite pole.

In cylindrical coordinates, horizontal advection is $u_r{\partial }_{r}Q+(u_{\varphi }/r){\partial }_{\varphi }Q$. Substituting the specified streamfunction givesThe factors of $1/r$ are required by cylindrical geometry.

An exact traveling eigenfunction. This is a polar gamma-plane Rossby wave. The radial equation implies ${\nabla }_h^2\psi -\psi /L_d^2=-{\alpha }^2\psi$, so $Q=f-{\alpha }^2\psi$. Since $J(\psi ,\psi )=0$, its nonlinear self-advection vanishes exactly, regardless of amplitude. Also $f_r=-\gamma r$, giving $J(\psi ,f)=\gamma {\psi }_{\varphi }$. HenceFor $\psi =F(r)\cos (\omega t-n\varphi )$, this yieldsThe wave is retrograde relative to the balanced mean flow, since $\omega /n<0$ in this convention.

Multiplying the radial equation by $r^2$ givesFor $K^2={\alpha }^2-L_d^{-2}>0$, setting $y=Kr$ gives the standard Bessel equation with derivatives with respect to $y$. ThusRegularity at the pole excludes $Y_n$ for the usual nonnegative integers. If $K^2<0$, the corresponding general solutions are modified Bessel functions; at $K=0$ they are $r^{\pm n}$, with a constant and $\ln r$ for $n=0$. The printed Bessel hint mixes its independent-variable labels; using $y=Kr$ and derivatives in $y$ resolves that notation.

What the hexagon fixes. With the north-polar approximation and the supplied rotation period,The intrinsic phase speed at that radius is $c_{\varphi }=r_0\omega /n=-\gamma r_0/{\alpha }^2$. Therefore the measured $-4\mathrm{m}{\mathrm{s}}^{-1}$ fixesThe polar-wave deformation-radius identifiability issue is that a unique $L_d$ additionally requires a radial wavenumber or boundary condition. The quoted peak-to-peak meander is an amplitude, not a radial wavelength: multiplying the exact eigenfunction by a constant changes its amplitude without changing $K,\alpha$ or $\omega$. Thus it cannot be silently converted into $K=2\pi /\Delta r$.

A transparent order-of-magnitude closure is a slowly varying radial envelope near $r_0$, so $F^{\prime \prime }+F^{\prime }/r$ is small relative to $n^{2}F/r_0^2$. Then $K^2\simeq n^2/r_0^2$ andThis is a conditional estimate, not a uniquely determined observation. The general real-$K$ bound is $L_d\ge {\alpha }^{-1}\simeq 1.7\times 10^6$ m. Different radial structures give different values: for example $K=3.0\times 10^{-7}$ and $5.0\times 10^{-7}{\mathrm{m}}^{-1}$ yield about $2.0\times 10^6$ and $3.2\times 10^6$ m, respectively, at the same measured phase speed. Their amplitudes remain free. The given $5^{\circ }$ meander corresponds to a radial half-amplitude about $2.5\times 10^6$ m and helps characterize the pattern's size, but alone removes none of this eigenvalue ambiguity.

Signed radial balance. Let $r$ increase outward from the vortex centre and $U$ be signed azimuthal velocity, positive anticlockwise. At $45^{\circ }$N, $f=2{\Omega }_{\oplus }\sin 45^{\circ }>0$. The positive speed is $V=|U|$, $f$ is the Coriolis parameter, the density is $\rho$, and $p_r=\partial p/\partial r$. In circular motion the inward centripetal acceleration is $-U^2/r$, while the outward radial acceleration from the Coriolis force is $fU$. The pressure gradient acceleration is $-p_r/\rho$. HenceThe term $U^2/r$ is an acceleration, not an additional physical force in the rotating-frame momentum equation. It can be moved to the force-balance side and depicted as outward centrifugal force in a frame following the circular parcel.

For a cyclone, $U=+V$ and $p_r/\rho =V^2/r+fV>0$: pressure increases outward and the centre is a low. Its inward pressure force balances the outward Coriolis and centrifugal terms. For a conventional anticyclone, $U=-V$ and $G=-p_r/\rho >0$; the balance isThe inward Coriolis force balances the outward pressure and centrifugal terms, and pressure has a maximum at the centre.

Anticyclonic branches and near-centre regularity. The anticyclonic gradient-wind branches follow from the quadratic balance. Solving givesThe minus root is the normal branch connected continuously to geostrophic flow for small forcing. The plus root is an anomalous high-speed branch and must not be discarded algebraically, even though it often fails the stability conditions relevant to large-scale anticyclones. Reality requiresA finite smooth central pressure therefore has zero radial gradient at the origin, with $-p_r/\rho$ no larger than order $r$. For $p=p_c-\rho C{r}^2/2+\cdots$, the limiting curvature obeys $C\le f^2/4$. A pressure maximum with a finite nonzero limiting outward pressure force cannot support the required circular anticyclonic balance arbitrarily close to its centre.

For positive $\mathrm{Ro}=V/(fr)$ and geostrophic magnitudes,The cyclonic gradient wind is subgeostrophic and the normal anticyclonic wind supergeostrophic. On the latter branch $0\le \mathrm{Ro}\le 1/2$. At a point on the eastern side of a cyclone, geostrophic and actual winds point north but the actual arrow is shorter; the ageostrophic difference points south. On the eastern side of a normal anticyclone both geostrophic and actual winds point south, with the actual arrow longer and its ageostrophic addition also south.

At lower latitude, the same $V$ and radius give a larger Rossby number and larger differences from geostrophic wind. The contour topology remains a central low or a conventional central high, but for fixed $V,r$ the required radial pressure gradients change with $f$: the cyclonic gradient is $\rho (V^2/r+fV)$ and the anticyclonic outward pressure force is $fV-V^2/r$. The latter conventional branch cannot be maintained once the reduced Coriolis contribution is too weak. Actual spatial gradients, rather than an arbitrary spacing of contour values, determine how tightly drawn contours should be.

The high-centre conclusion is conditional on the conventional anticyclonic branch. Gradient balance alone also allows a sufficiently rapid clockwise circulation about a low: if $V>fr$, then $p_r/\rho =V^2/r-fV>0$. For example, $V=2fr$ gives $p_r/\rho =2f^{2}r>0$. Such a configuration is outside the usual geostrophic anticyclone regime. Tornado-scale vortices have very large Rossby number, so the dominant balance is nearly cyclostrophic, $p_r/\rho \simeq V^2/r>0$. Their centres are low pressure irrespective of the sign of rotation; low pressure supplies the inward acceleration needed to bend the fast flow.

Balances and density gradients. The first two relations describe horizontal geostrophic balance between pressure gradient and Coriolis force accelerations. The vertical relation is hydrostatic balance between the vertical pressure force and gravity. They describe a slowly varying large-scale interior circulation, excluding the frictional surface/bottom layers and large ageostrophic accelerations.

Differentiate the horizontal balances vertically and use $p_z=-\rho g$, treating $f,{\rho }_0$ as vertically constant:These are the oceanic thermal wind relations. With no motion at $z=-H$, the exact surface relations involve the depth-integrated density gradients. If the measured density is representative of the vertical mean, or if those gradients are approximately uniform with depth,Equivalently these hold exactly for the horizontal gradients of $\bar{\rho }=H^{-1}{\int }_{-H}^0\rho dz$ in this fixed-depth reference model. A single arbitrary surface-density gradient does not otherwise determine a vertical integral. In the Northern Hemisphere the inferred surface current runs along horizontal isopycnals with denser water on its left when the no-motion reference and this vertical-gradient approximation apply.

For a steady density pattern and a ship whose actual track velocity is $v_s(\cos \theta ,\sin \theta )$,If the field changes during the survey an Eulerian time derivative must also be included. Here $H=\Delta z>0$ is the upward separation from reference depth to surface, so its sign must not be reversed.

Inversion and velocity estimate. Let $C=f{\rho }_0{v}_s/(gH)$ and denote the two ship measurements by $y_{\theta },y_0$, with the east-going ship assigned the subscript zero. The observation system and its inverse areIt is invertible for $\sin \theta \ne 0$. For two measurements and two unknowns this inverse is also the stated least-squares estimate; the nearly parallel-track geometry explains its poor conditioning at small angle.

The supplied parameters give $C=5.654\times 10^{-5}$ in units converting velocity to the measured density tendency. With $y_{\theta }=1.9\times 10^{-5}$ and $y_0=1.8\times 10^{-5}\mathrm{kg}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$,The current therefore has eastward and southward components. Signs follow from the positive-upward, east–north coordinate convention and the specified no-motion reference.

Uncertainty and track design. The two-track thermal-wind covariance exposes the geometrical limitation. The stated error already applies to the difference of successive density observations. Dividing by the sampling interval, rather than adding an extra factor of $\sqrt{2}$, gives ${\sigma }_y=7.2\times 10^{-4}/3600=2.0\times 10^{-7}\mathrm{kg}{\mathrm{m}}^{-3}{\mathrm{s}}^{-1}$. Assume independent errors of that size for the two ships, as in the supplied covariance model. ThenAt $5^{\circ }$,The correlation is positive,Both estimates depend on the east-going ship; its positive measurement error lowers $v_g$ and also lowers $u_g$, explaining the sign of the covariance. If the other ship goes north instead, $\theta =90^{\circ }$ gives orthogonal observations: $u_g=y_{90}/C$ and $v_g=-y_0/C$. Both standard errors become $0.00354\mathrm{m}{\mathrm{s}}^{-1}$ and their covariance vanishes. The north-going track improves the poorly constrained zonal component by about a factor of sixteen, while leaving the meridional precision unchanged under the same noise assumptions.