Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2022 / 2

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.

It is convenient to use the optical depth above a level,Thermal radiation escapes mainly from levels with $t$ of order unity: emission much deeper is attenuated, while the column far above contains little emitting material. Hence, in an optically thick atmosphere,For a particular viewing direction, $t\sim \mu$ instead gives $p_{\mathrm{rad}}\sim p_s\sqrt{\mu /{\tau }_{\infty }}$. These are characteristic levels, not sharp emitting surfaces; the contribution function depends on the temperature exponent $a$. If ${\tau }_{\infty }\lesssim 1$, the formal level lies below the surface and the surface itself contributes strongly. A useful estimate then clips $p_{\mathrm{rad}}$ at $p_s$.

Write ${\tau }_0={\tau }_{\infty }$, $n=2a$, and $t={\tau }_0(p/p_s{)}^2$. The integrated Planck function is $B(T)=\sigma T^4/\pi$ by the Stefan–Boltzmann law, andThe formal radiative transfer equation solution for upward radiance at the top isThe first term is transmitted surface emission; the second is emission from the atmosphere. Integrating radiance over the upward hemisphere gives the outgoing radiant flux per unit area,Here $E_j$ is an exponential integral. For fixed $a$ and ${\tau }_0\to \infty$, surface transmission is negligible and the upper limit can be extended to infinity. Positivity permits interchanging the integrals; putting $t=\mu x$ givesConsequently the exact hemispheric optically thick result isThis is a dimensional scale times a dimensionless definite integral depending only on $a$. In the commonly used vertical-ray approximation, define $F_{\mathrm{v}}=\pi I(0,1)$. ThenA vertical-ray approximation therefore has the same scaling but omits the angular factor $1/(a+1)$. Stating which quantity is used prevents a purely angular normalization discrepancy.

At fixed $T_s$ and $a>0$, the outgoing radiant flux scales as $F\propto {\tau }_{\infty }^{-2a}$. Increasing the optical depth shifts the effective emitting region to lower pressure and lower temperature. A surface temperature cannot be inferred from emission alone: specify the absorbed stellar radiant flux $F_0$ and impose radiative equilibrium, $F=F_0$. If the no-atmosphere blackbody has $T_0=(F_0/\sigma {)}^{1/4}$, thenThe vertical-ray approximation omits $a+1$ from this prefactor. For fixed $a>0$ and sufficiently large optical depth, the surface is warmer than $T_0$: this is the thermal greenhouse effect within the prescribed temperature profile. It does not by itself establish a self-consistent vertical heating or convection balance.

For $a=0,\frac{1}{2},1,\frac{3}{2},2,\dots$, put $n=2a=0,1,2,3,4,\dots$. Repeated integration by parts gives $\Gamma (n+1)=n!$. The hemispheric thick-limit results areThe second row assumes fixed surface temperature; the third assumes fixed absorbed radiant flux. In the vertical-ray approximation the first of these rows becomes $1,1/{\tau }_0,2/{\tau }_0^2,6/{\tau }_0^3,24/{\tau }_0^4$, and the surface temperature factors are the inverse fourth roots.

Thus increasing $a$ makes the atmosphere colder at every fixed $0<p<p_s$, decreases outgoing radiation at fixed $T_s$, and requires a warmer surface at fixed absorbed radiant flux. The case $a=0$ is isothermal and gives $F=\sigma T_s^4$ exactly, including finite optical depth.

There is an essential approximation limit. The extension of the integral to infinity is valid for fixed $a$ as ${\tau }_0\to \infty$, and is not uniform for arbitrarily large $a$ at fixed ${\tau }_0$. The vertical integrand peaks at $t=2a$, so one needs that peak well above the surface in optical depth, with a negligible tail beyond ${\tau }_0$. For example, at ${\tau }_0=10$ the thick formula predicts $F/(\sigma T_s^4)=20!/(1110^{20})\simeq 0.00221$ for $a=10$ but approximately $0.04773$ for $a=12$. This artificial increase contradicts the exact monotonicity: in the finite-optical depth expression every atmospheric emission factor $(t/{\tau }_0{)}^{2a}$ decreases with $a$. At fixed finite ${\tau }_0$, the exact limiting outgoing radiant flux is the transmitted surface term $2\sigma T_s^4{E}_3({\tau }_0)$, rather than factorial divergence.