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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.

For the single-line greenhouse deficit, use hydrostatic balance, fixed surface pressure $p_s$, constant gravity $g$, and the dry adiabatThe gas constant and specific heat refer to the background gas. Let ${\mu }_d=\cos \tilde{\theta }>0$ be the supplied effective ray cosine. The vertical optical depth and slant transmittance from level $p$ to space areThis permits pressure-dependent line widths. With local thermodynamic equilibrium, negligible infrared scattering, a black thermal surface, and no incident thermal radiation from space, the formal radiative transfer solution isThe weights add to one. Subtract the transparent-surface radiance and put $B(p)=B_{{\nu }_0}(T(p))$, using the narrow-line approximation only in the line deficit. Define its pressure-dependent equivalent widthApproximating each hemispheric flux by $\pi$ times the effective-ray radiance preserves the black-surface flux. The outgoing longwave flux is thereforeIntegration by parts gives the particularly transparent formBoth boundary terms vanish: $E(0,q)=0$ and $B(p_s)-B(p_s)=0$. The dry adiabat and monotonic Planck function give $dB/dp>0$. Atmospheric emission replaces part of the warm surface radiation by emission from cooler levels, reducing emission to space for a specified $T_s$. The inequality is strict when opacity occurs over a nonzero range with temperature contrast; an isothermal absorbing layer at the surface temperature would not produce such a contrast.

For a spherical planet the mean absorbed instellation is $F_{\ast }=(1-\alpha )L_{⊛}/4$. The required global energy balance isConsequently every equilibrium of this model has $T_s\ge T_{\mathrm{bare}}=[(1-\alpha )L_{⊛}/(4{\sigma }_{\mathrm{SB}}){]}^{1/4}$, with strict warming when $D>0$. At fixed temperature profile, increasing $q$ increases $E$ and hence $D$. This argument does not assume that the greenhouse gas changes the stellar-band albedo, surface pressure, or background thermodynamics; those are held fixed. For sufficiently broad or overlapping saturated wings the full frequency-dependent Planck function must replace the narrow-line approximation.

Linearize about a uniform horizontal flow $U$ and a hydrostatic, vertically stratified basic state. The basic-flow material derivative is $D_0={\partial }_t+U{\partial }_x$. The first two equations are the horizontal and vertical perturbation momentum balances: pressure gradients accelerate the disturbance, while the vertical equation retains the buoyancy force $-g{\rho }^{\prime }/{\rho }_0$ after the basic hydrostatic balance is subtracted.

The Boussinesq approximation neglects fractional density variations in inertia and enforces incompressibility, $u_x+w_z=0$, but retains them in buoyancy. Advecting the stable background density gradient gives $D_0{\rho }^{\prime }+wd\bar{\rho }/dz=0$. With $N^2=-(g/{\rho }_0)d\bar{\rho }/dz>0$, this is the fourth supplied equation. A displaced parcel then has a restoring buoyancy tendency.

Small amplitude allows products of perturbations to be discarded; two-dimensional motion removes $y$ dependence and transverse flow. Rotation, viscosity, diffusion, diabatic heating and mean shear are neglected. ${\rho }_0$ is the reference inertial density, whereas the small background density gradient still sets $N$. In a compressible real atmosphere the appropriate stability measure involves potential temperature; this Boussinesq system is a local idealization that filters acoustic waves rather than a full compressible atmospheric model.

For a hydrostatic Boussinesq approximation, introduce reduced pressure $\pi =p^{\prime }/{\rho }_0$ and buoyancy $b=-g{\rho }^{\prime }/{\rho }_0$. The geostrophic balance and perturbation hydrostatic balance equations areCommuting the mixed derivatives gives the height-coordinate thermal wind relationsEquivalently $u_{g,z}=g{\rho }_y^{\prime }/(f{\rho }_0)$ and $v_{g,z}=-g{\rho }_x^{\prime }/(f{\rho }_0)$. In an atmosphere the corresponding ideal-gas form is conveniently derived from geopotential $\Phi$ on pressure surfaces: $f{u}_g=-{\Phi }_y{|}_p$, $f{v}_g={\Phi }_x{|}_p$, and ${\Phi }_{\ln p}=-R_{a}T$. HenceSince $d\ln p/dz=-g/(R_{a}T)$, this gives the vertical shear at a fixed horizontal location,The horizontal temperature derivatives here are on isobars; replacing them by fixed-height derivatives is a small-slope approximation. This qualification avoids mixing compressible and Boussinesq density gradients.

For an illustrative poleward temperature drop $40\mathrm{K}$ across $3000\mathrm{km}$, mean $T\sim 250\mathrm{K}$, $f\sim 10^{-4}{\mathrm{s}}^{-1}$ and depth $10\mathrm{km}$,Adding a modest surface wind gives an upper-level westerly jet of order $50$–$60\mathrm{m}{\mathrm{s}}^{-1}$. The thermal wind fixes vertical shear, not the absolute wind without a boundary or reference value.

The underlying principle is material conservation of potential vorticity for adiabatic, inviscid, rotating, stably stratified motion. In the quasi-geostrophic approximation, the leading horizontal velocity is nondivergent and geostrophic:Here $f_0=2\Omega \sin {\varphi }_0$ is the reference Coriolis parameter, $\beta =(df/dy{)}_0=2\Omega \cos {\varphi }_0/a$ on a spherical planet, and $f=f_0+\beta y$. The positive buoyancy frequency squared is $N^2$; in a Boussinesq fluid it is $-(g/{\rho }_0)d\bar{\rho }/dz$. To quasi-geostrophic order the potential vorticity is the background $f_0+\beta y$ plus the stated perturbation $q^{\prime }$.

Linearizing its material derivative about uniform zonal velocity $U$ giveswhich is the supplied equation. The uniform basic flow contributes no relative-vorticity gradient or vertical shear term.

The approximation needs small Rossby number, time scales long compared with $|f_0{|}^{-1}$, vertical hydrostatic balance, stable stratification, small departures from the reference state and meridional extent small enough for a local beta plane. Buoyancy and rotation must balance at the relevant aspect ratio. Friction, diabatic heating and strong nonlinear advection are neglected here. It applies to balanced midlatitude ocean and atmospheric motions; an atmospheric version also requires a consistent reference-density or pressure-coordinate treatment. It fails in a sufficiently near-equatorial region where $f_0$ vanishes.

Notation. The source uses $p_s$ for both the hypothetical dry surface pressure and the actual surface pressure after evaporation. Distinguish them as $P_d$ and $P$, respectively, and interpret the specified pressure thickness $\Delta p$ as fixed. This conserves the original hydrogen inventory and permits $P$ to vary. Reading both occurrences as one fixed number would be inconsistent with adding water vapor to a closed atmospheric column mass budget.

Let $E(T)$ denote the saturation pressure of water, let $x$ be the mole fraction of water vapor in the mixed layer, and let $M_d,M_v$ be the molar masses of hydrogen and water. Surface phase equilibrium and the ideal gas law giveBecause the layer has constant temperature, its relative humidity isThus every level strictly above the surface is subsaturated: hydrostatic balance makes $p<P$, whereas the saturation pressure stays constant. The result does not require a dilute water vapor approximation.

The Clausius-Clapeyron relation, neglecting the specific volume of liquid water and treating the water vapor as an ideal gas, givesHere $T_t,E_t$ are the triple point values, $L_v$ is the specific latent heat, and $R$ is the molar gas constant. All temperatures in these expressions are absolute. As throughout the question, this is a constant-latent heat, ideal gas model, not an extrapolation valid through the real critical point of water.

For a well-mixed ideal gas mixture, its mean molar mass and water vapor mass fraction areBy hydrostatic balance, the mixed layer has total mass per unit area $\Delta p/g$, of which $w_v\Delta p/g$ is water vapor. The remaining atmosphere contains no water vapor. Consequently conservation of hydrogen mass givesEquivalently the requested quadratic equation isFor each $T_s$, calculate $E$, take the positive root of this quadratic equation, and retain it only if $0\le x<1$ and $P>\Delta p$. The last condition leaves a nonempty dry layer. In particular the dry constituent's surface partial pressure is not generally $P_d$: $P_d$ measures its column weight, and the hydrogen and water vapor have different molar masses.

Before condensation, assume a closed parcel with fixed atmospheric mixing ratio, constant specific heat capacities, and negligible irreversible heating. The first law of thermodynamics for a dry adiabatic process isThe parcel remains in approximate pressure equilibrium with a hydrostatic environment. With the standard approximation that parcel and environmental densities are close, $dp/dz\simeq -\rho g$, henceThis is the dry adiabatic lapse rate, about $9.8\mathrm{K}\mathrm{k}{\mathrm{m}}^{-1}$ for air. More precisely the environmental hydrostatic balance gives $dT/dz=-g{\rho }_{\mathrm{env}}/(c_p{\rho }_{\mathrm{parcel}})$; the familiar constant dry adiabatic lapse rate uses their ratio close to one.

Let $e_v$ be the parcel's water vapor partial pressure. Fixed mole fraction before condensation means $e_v/p$ is constant. Its dew point $T_D$ is defined by $E(T_D)=e_v$, where $E$ is the saturation pressure. The Clausius-Clapeyron relation givesThe last approximation uses hydrostatic balance, the ideal gas law, and an environmental temperature close to the parcel's. Rearrangement yieldsFor a modest initial dew-point depression, $T_D/T\simeq 1$ in absolute temperature, soWith $L_v\simeq 2.5\times 10^6\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}$, $R_d\simeq 287\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$, and $R_v\simeq 461.5\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$, $H\simeq 1.58\times 10^5\mathrm{m}$. This dew point scale is much longer than the atmospheric pressure scale height.

The exponential function is an approximation, not the exact consequence of a dry adiabatic process. Retaining $T=T_s-{\Gamma }_{d}z$ gives $p/p_s=(T/T_s{)}^{c_p/R_d}$ and the useful comparisonIt also shows explicitly that expansion lowers the dew point even before any water vapor condenses.

For the variable-latent-heat saturation curve of a pure saturated water atmosphere, the total pressure equals the saturation pressure. The exact Clausius-Clapeyron relation along the phase coexistence curve iswhere $v_v,v_c$ are the specific volumes of the vapor and condensed thermodynamic phase. Neglecting $v_c$ and using the ideal gas law, $v_v=R_{\mathrm{wv}}T/p$, givesThe triple point provides the integration constant. Below it, the prescribed latent heat is constant, soAbove it, write $K=L_{\ast }/[R_{\mathrm{wv}}(T_c-T_{\ast })]$. Integrating $K(T_c/T^2-1/T)$ givesThese are the requested implicit temperature profiles $T(p)$. Both branches meet at $(T_{\ast },p_{\ast })$, and each is monotone before the limiting liquid-gas critical point. The lower branch describes equilibrium with ice, rather than a stable liquid surface. A stable liquid ocean surface belongs to the upper branch; the colder upper atmosphere may use the lower branch. Hydrostatic balance is needed to convert this relation into a height profile, but not to obtain $T(p)$.

For constant gravitational acceleration, hydrostatic balance gives the atmospheric column mass $m=p_{\mathrm{sfc}}/g$, with negligible pressure at the top. For the same surface temperature $T_{\ast }<T_{\mathrm{sfc}}<T_c$, the variable latent heat gives a smaller integrated $\ln (p_{\mathrm{sfc}}/p_{\ast })$, and henceMultiplying by the same planetary surface area gives the same comparison for total mass. The models agree for a surface at or below the triple point, where they use the same sublimation branch. The comparison assumes the same surface temperature, rather than the same surface pressure.

The pressure-linear grey opacity model interprets $\kappa$ as a mass absorption coefficient. If it is instead a molecular absorption cross section, replace $\kappa$ below by that cross section divided by molecular mass. Assume a plane-parallel atmosphere, hydrostatic balance, local thermodynamic equilibrium, and thermal absorption and emission without scattering of light. With height $z$ increasing upwards,Integrating upwards from the surface givesThusThe vertical optical depth is dimensionless. The atmosphere is optically thick when ${\tau }_{\infty }\gg 1$ and optically thin when ${\tau }_{\infty }\ll 1$. A ray with zenith-angle cosine $\mu$ sees optical depth ${\tau }_{\infty }/\mu$, so even a vertically thin layer can be opaque along a sufficiently grazing ray.

The constants $R_a,R_b,R_d$ are effective reservoir capacities: stored carbon dioxide mass anomaly per unit effective concentration anomaly, in GtCO${}_2$ per ppmv. Only $C_a$ is a literal atmospheric mole fraction; $C_b,C_d$ are scaled reservoir variables chosen to share its equilibrium value. The exchange coefficients $r,s$ are conductances, in GtCO${}_2$ per ppmv per year: $r(C_a-C_b)$ is transfer from the atmosphere to the fast reservoir and $s(C_b-C_d)$ transfer from that reservoir to the deep ocean. They are not by themselves inverse times; division by a reservoir capacity gives an inverse-time rate.

Hydrostatic balance estimates total atmospheric mass as $M_{\mathrm{air}}=4\pi R_E^2\bar{p}/g$. One ppmv of a dilute ideal gas is a mole fraction $10^{-6}$, so, with dry-air molar mass $M_{\mathrm{mol},\mathrm{air}}\simeq 28.97$ g/mol and carbon dioxide molar mass $44.01$ g/mol,The final factor converts kilograms to gigatonnes. This is carbon dioxide mass, not the smaller carbon mass.

Hydrostatic balance is $dP/dr=-GM(r)\rho /r^2$. A free-fall estimate gives $t_{\mathrm{dyn}}\sim \sqrt{R_{\odot }^3/(G{M}_{\odot })}\simeq 1.6\times 10^3$ s; a test particle falling in a fixed point-mass field adds factor $\pi /(2\sqrt{2})$, giving about 30 minutes.

For an ideal nonrelativistic gas with negligible boundary pressure, the virial theorem is $2U+\Omega =0$, so total energy is $\Omega /2$. Since $\Omega =-{\int }_0^{M}Gmdm/r(m)$ and $r\le R$, $|\Omega |\ge G{M}^2/(2R)$. This is a minimum magnitude, not a lower bound on the negative energy: unrestricted central concentration can make the latter arbitrarily negative. A uniform-density estimate gives $\Omega =-3G{M}^2/(5R)$ and available contraction energy $3G{M}^2/(10R)$. Its release at present solar luminosity lasts $\boxed{t\sim 10^7\mathrm{yr}}$ (order $10^7$–$10^8$ years for alternative structural estimates), much less than a nuclear-burning lifetime.

Hydrostatic balance and the ideal gas law give $dp/dz=-pg/(R_{a}T)$, hence approximately $p=p_0{e}^{-z/H}$ with $H=R_{a}T/g$ for weak temperature variation. For well-mixed absorber abundance, ${\rho }_{\mathrm{abs}}\propto p/T$, while pressure broadening gives $k\propto p$. Thus $k{\rho }_{\mathrm{abs}}\propto e^{-2z/H}$ and its integral above height $z$ has the same scale. ThereforeThe factor two depends on both pressure dependence of absorption and constant mixing ratio; it is not the ordinary density scale height.

Write the density and pressure as ${\rho }_0(z)+{\rho }^{\prime }(x,z,t)$ and $p_0(z)+p^{\prime }(x,z,t)$, respectively. The background satisfies hydrostatic balance, $p_0^{\prime }=-g{\rho }_0$, and ${\rho }_0^{\prime }/{\rho }_0=-\alpha$. Linearizing the Euler equations for an inviscid fluid, incompressibility and advection of density givesDifferentiate the horizontal equation in $x$ and use incompressibility to obtain $p_{xx}^{\prime }={\rho }_0{w}_{zt}$. Differentiate the vertical equation in time and twice in $x$:Dividing by ${\rho }_0>0$ gives the equation for internal waves in an exponentially stratified incompressible fluid,The first-derivative term retains the variation of inertial density with height; no Boussinesq approximation has been made.

An atmospheric vertical coordinate using pressure instead of geometric height. Under hydrostatic balance, pressure decreases upward and geopotential obeys ${\partial }_p\Phi =-1/\rho$. With the ideal gas law, ${\partial }_{\ln p}\Phi =-RT$. Horizontal gradients taken at fixed $p$ lie on an isobaric surface. A log-pressure coordinate is a useful related coordinate with the opposite upward sign.

Combining geostrophic balance, $fu=-{\Phi }_y$, $fv={\Phi }_x$, with hydrostatic balance, ${\Phi }_p=-RT/p$, and commuting the horizontal and vertical derivatives proves the displayed equations. Horizontal derivatives hold pressure fixed. This relates vertical shear to horizontal temperature gradients away from the equator; the derivation requires nonzero Coriolis parameter and the validity of geostrophic balance.

A grey atmosphere model with mass absorption coefficient proportional to pressure. Under hydrostatic balance, $\kappa ={\kappa }_{s}p/p_s$ gives downward optical depth $t={\kappa }_s{p}^2/(2g{p}_s)$. For $T=T_s(p/p_s{)}^a$, a non-scattering optically thick column emits hemispheric radiant flux $F\sim \sigma T_s^4{t}_s^{-2a}\Gamma (2a+1)/(a+1)$. The angular factor follows from integrating ray attenuation over the upward hemisphere; a vertical-ray approximation omits it. The extension of the emitting-depth integral to infinity requires fixed $a$ as $t_s\to \infty$.

Prescribe a dry adiabat for a background ideal gas A,where $R_A$ is its specific gas constant, $c_{pA}$ its constant specific heat capacity at fixed pressure, and $T_s,p_s$ the surface temperature and pressure. A dilute saturated vapour B has specific gas constant $R_B$, constant latent heat $L$, and saturation vapor pressure $e_s(T)$. Put $ϵ=R_A/R_B$ and $\beta =L/(R_B{T}_s)$.

The ideal-vapour Clausius-Clapeyron relation integrates toIts dilute vapour mass fraction is approximately $ϵe_s/p$. Under constant-$g$ hydrostatic balance, the vapour mass path is thereforeFor the second equality, set $\zeta =\log (p_s/p)$ and then $y=\beta e^{x\zeta }$. Thus the exponential integral follows from the combination of adiabatic cooling and rapidly falling saturation vapor pressure.

For $\beta \gg 1$, integration by parts gives $e^{\beta }{E}_1(\beta )\sim {\beta }^{-1}(1-{\beta }^{-1}+2{\beta }^{-2}-\cdots )$, henceThe column is weighted toward its warm lower part. Diluteness must hold throughout the column. This calculation prescribes the dry adiabat; if condensation heating changes that profile, a consistent moist adiabat is needed instead.

For a nonrotating self-gravitating star, let $p_1,{\rho }_1,{\varphi }_1$ be Eulerian perturbations and $\Delta p=-\gamma p\nabla \cdot \mathbf{\xi }$ the adiabatic Lagrangian perturbation. Integration of the linear momentum equation against ${\mathbf{\xi }}^{\ast }$, using hydrostatic balance and the perturbed Poisson equation, yields the displayed Rayleigh quotient. The gravitational integral is over all space and the other integrals over the star. The compression denominator is $\gamma p$, not $\gamma \rho$. Surface terms vanish for compactly supported trials or the physical free-surface and exterior potential conditions.