8.3 Propagation of a wave in the atmosphere - Line shapes

A plane wave propagating along the z direction in the atmosphere can be represented by its electric field vector ${\bf E}$:

$${\bf E}(z,t)= {\bf E}_m e^{-j2\pi\nu(t-n_c \frac{z}{c})}$$ (8.20)

where n_c= n + jn_i is the complex refraction index.

The real part, the true refraction index, n= c/v_p (where v_p is the phase velocity of the wave), is often expressed in terms of the refractivity,

$$N = 10^6(n-1) \simeq 0.223\rho_a + 1760\rho_w/T$$ (8.21)

The right hand sum, which was empirically derived, is known as the Smith-Weintraub equation. It separates the contribution of the dry air component (first term, where $\rho_a$ is the air density, expressed in gm^-3) from that of water vapor (second term, where $\rho_w$ is the density of water vapor).

The imaginary part of the refraction index, n_j causes an exponential attenuation of the wave amplitude, and is related to the power absorption coefficient $\kappa_\nu$ by:

$$n_j= {\frac{c\kappa_\nu}{4\pi\nu}}$$ (8.22)

At a frequency $\nu$ close to a rotational transition $\nu_{lu}= (E_u-E_l)/h$, the optical depth is:

$$\tau_\nu = \int \kappa_\nu{(T, P)} dh$$ (8.23)

$\kappa_{lu}$, the absorption coefficient integrated over the transition lu is given by the standard asymmetric top formula (see e.g. [Townes & Schawlow 1975] p.102):

$$\kappa_{lu}= \frac{8\pi^2 h^{1.5}}{3c(kT)^{2.5}} (\frac{\rho_... ...-2.5} (1-\frac{h\nu}{kT}) e^{E_l/kT} \nu_{lu}^2\Phi(\nu-\nu_o)$$ (8.24)

Replacing A, B, C, the rotational constants (here in Hz), and $\mu = 1.85$ Debye  $= 1.85\,10^{-18}$ esu.cm, the electric dipole moment, by their values for H₂O, and setting $(1-\frac{h\nu}{kT})=1$,

$$\kappa_{lu}^w [cm^{-1}]= 5.7 10^{-24}(\rho_w/m_w)g_IS_{lu}(T/273)^{-2.5} e^{E_l/kT}\nu_{lu}^2\Phi(\nu-\nu_o)$$ (8.25)

S_lu is the transition intrinsic strength, g_I= 3/2 or 1/2 is the nuclear relative statistical weight of the ortho and para levels (see below), and $\Phi(\nu-\nu_o)$ the line profile. $\nu_{lu}$ is now in GHz.

The line profile is given to a good approximation by the well known Van Vleck and Weisskopf collisional profile (here, multiplied by $\pi$, [Townes & Schawlow 1975] p.342):

$$\Phi(\nu-\nu_o)= \frac{\Delta\nu}{(\nu-\nu_o)^2 + (\Delta\nu)^2} + \frac{\Delta\nu} {(\nu+\nu_o)^2 + (\Delta\nu)^2}$$ (8.26)

with $\Delta\nu = \frac{1}{2\pi\tau}$, where $\Delta\nu$ is the line width and $\tau$ the mean time between molecular collisions.

At the centre of the line, the second term of Eq.8.26 becomes negligible, and

$$\Phi(\nu-\nu_o) = 1/\Delta\nu = 2\pi\tau \sim \frac{2\pi}{\frac{\rho_a}{m_a} v\sigma^2}$$ (8.27)

$v\sim \sqrt{T}$ is the molecular velocity and $\sigma$ the collisional cross section.

Note that the density in Eq.8.24 is $\rho_w$, whereas that in Eq.8.27 is $\rho_a$: the absorption coefficient at the centre of the line is proportional to $\rho_w/\rho_a$. It is independent of the total air pressure, as long as this ratio (hence RH) stays constant. This is not the case, of course, away from the centre, since $\Delta\nu \sim \rho$: as the density drops, the lines become narrower and narrower. In the far wings of the line, the second term in Eq.8.26 and the contribution from the wings of other lines cannot anymore be neglected. In fact, a better fit to the water emission in the far wings is reached if Eq.8.26 is replaced by another collisional line shape, called the ``kinetic profile''

$$\Phi(\nu-\nu_0) = \frac{4 \nu \nu_0 \Delta \nu}{(\nu^2-\nu_0^2)^2 + 4 \nu^2 \Delta\nu^2}$$ (8.28)

The opacity and width of the main H₂¹⁶O lines are large: $\tau_o= 60$ and $\Delta\nu\simeq 20$GHz, for the fundamental line in normal conditions of p and T, and for $\rho_w = 1$g.m^-3 and w_w= 1 mm (dry weather). These lines dominate most of the millimetre and submillimetre atmospheric attenuation; deviations from theoretical line shapes in the far wings (typically 1/10th of intensity) are accounted for by an empirical continuum. The rare isotopomer H₂¹⁸O, a few hundred times less abundant than H₂¹⁶O, makes a negligible contribution (see the discussion by [Waters 1976])