We will see in Section 4.4 that for most atmospheric gases k, and hence the optical thickness, has an intricate dependence on wavenumber. This considerably complicates the solution of the radiative transfer equations, since the fluxes must be solved for individually on a very dense grid of wavenumbers, and then the results integrated to yield the net atmospheric heating, which is the quantity of primary interest. The development of shortcuts that can improve on a brute-force integration is an involved business, which in some respects is as much art as science, and leads to equations whose behavior can be difficult to fathom. The radiative transfer equations become much simpler if the optical thickness is independent of wavenumber. This is known as the grey gas approximation. For grey gases, the Schwartzschild equations can be integrated over wavenumber, yielding a single differential equation for the net upward and downward flux. More specifically, we shall assume only that the optical thickness is independent of wavenumber within the infrared spectrum, and that the temperature of the planet and its atmosphere is such that essentially all the emission of radiation lies in the infrared. Instead of integrating over all wavenumbers, we integrate only over the infrared range, thus obtaining a set of equations for the net infrared flux. Because of the assumption regarding the emission spectrum, the integrals of the Planck function nB(v, T) over the infrared range can be well approximated by aT4.
With the exception of clouds of strongly absorbing condensed substances like water, the grey gas model yields a poor representation of radiative transfer in real atmospheres, for which the absorption is typically strongly dependent on wavenumber. Nonetheless, a thorough understanding of the grey gas model provides the starting point for any deeper inquiry into atmospheric radiation. Here, we can find many of the fundamental phenomena laid bare, because one can get much farther before resorting to detailed numerical computations. Further, grey gas radiation has proved valuable as a placeholder radiation scheme in theoretical studies involving the coupling of radiation to fluid dynamics, when one wants to focus on dynamical phenomena without the complexity and computational expense of real gas radiative transfer. Sometimes, a simple scheme which is easy to understand is better than an accurate scheme which defies comprehension.
The grey gas versions of the two-stream Schwarzschild equations are obtained by making tv independent of frequency and integrating the resulting equations over all frequencies. The result is
Grey gas versions of the solutions given in the previous section can similarly be obtained by integrating the relations over all frequencies, taking into account that t is now independent of v. The expressions have precisely the same form as before, except that 1+ and I_ now represent total flux integrated over all longwave frequencies, and every occurence of nB is replaced by aT4. To avoid unnecessary proliferation of notation, when the context allows little possibility of confusion we will use the same symbols I+ and I_ to represent the longwave-integrated flux as we used earlier to represent the frequency-dependent flux spectrum. When we need to emphasize that a flux is a frequency dependent spectrum, we will include the dependence explicitly (as in "I+ (v)" or "I+ (v,p)"; when we need to emphasize that a flux represents the longwave-integrated net flux, we will use an overbar (as in I+).
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