Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2019 / 1

Let $R_A,R_B$ be the specific specific gas constants and $c_{pA},c_{pB}$ the specific heat capacities of the noncondensible gas and vapour. Write $e_s(T)$ for the saturation vapor pressure, $r$ for vapour mass per unit dry-gas mass, and $ϵ=R_A/R_B$. For a dilute ideal gas mixture,The second equation is the Clausius-Clapeyron equation, neglecting condensed-phase volume and taking constant latent heat $L$. The first law of thermodynamics for saturated ascent, with condensate removed and negligible condensate heat capacity, is to leading orderSince $dr=r[LdT/(R_B{T}^2)-d\ln p]$, the resulting moist adiabat has logarithmic slopeConsequently a sufficient dry adiabat criterion isThe first two conditions justify neglecting the vapour's effects on the gas law and sensible heat capacity; the last two suppress latent heating. With $x=R_A/c_{pA}$ and $\beta =L/(R_{B}T)$, the latent conditions are $\beta e_s/p\ll 1$ and $x{\beta }^2{e}_s/p\ll 1$. Usually $x\beta \gg 1$, so the squared-latent term gives the stronger condition. At sufficiently low surface temperature, the exponential smallness of $e_s(T_s)$ makes these conditions hold andThese are local conditions along the column. They can be checked at the surface when $x{\beta }_s>1+2x$: on this dry profile the strongest latent parameter is proportional to $\exp [(1+2x)\zeta -{\beta }_s(e^{x\zeta }-1)]$ and decreases upwards. Without that restriction a surface-only test need not bound every altitude. If suspended condensate is retained, its own heat capacity and loading must also be negligible; saturation alone does not specify those quantities.

Interpret the requested mass path as the vapour column; no condensed-water loading is prescribed. Hydrostatic equilibrium gives $dp=-\rho gdz$, so if $q_B={\rho }_B/\rho$ is vapour mass fraction,For the mixture $q_B=ϵe_s/[p+(ϵ-1)e_s]$. Under the dilute conditions of part (a), $q_B\simeq ϵe_s/p$. Introduce $\zeta =-\ln (p/p_s)$, so $T=T_s{e}^{-x\zeta }$ and $-dp=pd\zeta$. Integrating the constant-$L$ Clausius-Clapeyron equation relative to the surface givesThereforeThe explicit $p_s$ cancels: decreasing vapour mass fraction in a denser dry column compensates its greater total mass. This cancellation belongs to the dilute, dry-adiabatic approximation, and is not a general property of moist atmospheres. With suspended condensate, an additional and unspecified condensate mass path would have to be added.

For ordinary condensibles, $L/(R_{B}T)$ is typically of order ten or more near low atmospheric temperatures. At the approximate triple-point temperatures, the supplied substances give $\beta$ about $20$ for water and $10$–$12$ for carbon dioxide, methane and nitrogen. This is consistent with the supplied large ratios of latent heat to sensible heating: for molecular gases $c_p/R$ is only a few. The relevant $x=R_A/c_{pA}$ is typically a few tenths.

The integral is concentrated where ${\beta }_{s}x\zeta =O(1)$. There $x\zeta =O({\beta }_s^{-1})\ll 1$, so $e^{x\zeta }-1\simeq x\zeta$ givesThis is an Laplace's method, rather than an assumption that vapour is uniformly mixed. The effective contributing depth in log-pressure is $\Delta \zeta \sim 1/(x{\beta }_s)$, explaining why the near-surface atmosphere dominates.

An exact representation within the same dilute model makes the error explicit. Set $y={\beta }_s{e}^{x\zeta }$ and define the exponential integral $E_1(b)={\int }_b^{\infty }{e}^{-y}dy/y$. ThenThe saturated vapor column on a dry adiabat estimate overstates the exact dilute-model column by roughly $1/{\beta }_s$, about five to ten per cent for those illustrative values. It is not accurate for a warm, nondilute steam atmosphere.