Analysis: Checking Raval and Ramanathan

A seminal paper from Raval and Ramanathan (1989)1The original paper was published in Nature in 1989, but is behind a paywall. Fortunately, it was reprinted in a conference proceeding soon after; that’s the link I’ve offered. defines the normalized greenhouse effect, \nghe, and indicates that, in the absence of lapse rate or water vapor feedbacks, \nghe wouldn’t be expected to change as the surface temperature increases. In this analysis, I investigate their mathematical analysis to try to understand what if any assumptions are involved.

Their key equation

R&R present the following radiative transfer equation (TRE):

(1)   \begin{equation*} \OLR = B(\tsurf) - \int_0^1 A(x)\,\frac{\dd B}{\dd x}\,\dd x \end{equation*}

where x is the pressure divided by surface pressure; the blackbody emission is B(T) = \sigma\,T^4 = \int \tilde B_\lambda\,\dd\lambda; and the effective absorptivity A is the “integral of the monochromatic absorptivity A_\lambda weighted with \dd \tilde B_\lambda, and normalized by \dd B.”2For this, R&R cite Manabe & Wetherald (1967). However, I don’t see the equation in M&W (1967). “Thus A(x) is the absorptivity between the TOA (x=0) and the pressure level x.”

Let’s try transforming the prior integral using integration by parts:

(2)   \begin{equation*} \OLR = B(\tsurf) - \left[ A(1)\,B(T(1)) + A(0)\,B(T(0)) - \int_0^1 \frac{\dd A(x)}{\dd x}\,B(T(x))\,\dd x \right] \end{equation*}

(3)   \begin{equation*} \OLR = \left[1 - A(1)\right]\,B(\tsurf) + \int_0^1 \frac{\dd A(x)}{\dd x}\,B(T(x))\,\dd x \end{equation*}

Schwarzchild’s equation of radiative transfer

Let’s try to see how the above relates to Scharzchild’s equation of radiative transfer (which governs propagation of thermal radiation in the absence of scattering):

(4)   \begin{equation*} \frac{\dd I_\lambda}{\dd s} = \alpha_\lambda\,\left[B_\lambda(T) - I_\lambda \right] \end{equation*}

where s is distance in a particular direction and \alpha_\lambda is the wavelength-dependent absorption coefficient. This equation has the solution

(5)   \begin{equation*} I_\lambda(s) = I_\lambda(0) \, R_\lambda(0,s) + \int_0^s \alpha_\lambda(s^\prime) \, B_\lambda(T(s^\prime)) \, R_\lambda(s^\prime,s) \, \dd s^\prime \end{equation*}

where R_\lambda(s_1, s_2) is the transmittance between s_1 and s_2,

(6)   \begin{equation*} R_\lambda(s_1, s_2) = \exp\left[-\int_{s_1}^{s_2} \alpha_\lambda(s^\prime) \,\dd s^\prime \right] \end{equation*}

If we define U_\lambda(s^\prime,s) = 1-R_\lambda(s^\prime,s), then

(7)   \begin{equation*} \frac{\dd U_\lambda(s^\prime,s)}{\dd s^\prime} = -\alpha_\lambda(s^\prime) \, R_\lambda(s^\prime,s) \end{equation*}

Thus integration by parts leads to:

(8)   \begin{equation*} I_\lambda(s) = I_\lambda(0) \, R_\lambda(0,s) - \left[ U_\lambda(s,s)\,B_\lambda(T(s)) - U_\lambda(0,s)\,B_\lambda(T(0)) -\int_0^s U_\lambda(s^\prime,s) \, \frac{\dd B_\lambda(T(s^\prime))}{\dd s^\prime} \, \dd s^\prime \right] \end{equation*}

Applying the definition of U_\lambda(s^\prime) leads to:

(9)   \begin{equation*} I_\lambda(s) = B_\lambda(T(0)) + \left[I_\lambda(0) - B_\lambda(T(0))\right] \, R_\lambda(0,s) +\int_0^s U_\lambda(s^\prime,s) \, \frac{\dd B_\lambda(T(s^\prime))}{\dd s^\prime} \, \dd s^\prime \end{equation*}

If we assume the surface air temperature matches the surface temperature and the surface is a blackbody, this becomes:

(10)   \begin{equation*} I_\lambda(s) = B_\lambda(T(0)) +\int_0^s U_\lambda(s^\prime,s) \, \frac{\dd B_\lambda(T(s^\prime))}{\dd s^\prime} \, \dd s^\prime \end{equation*}

Now let’s change this to a more useful coordinate, by setting s \,\cos\theta = z. This leads to:

(11)   \begin{equation*} I_\lambda(z) = B_\lambda(T(0)) +\int_0^{z} U_\lambda(\frac{z^\prime}{\cos\theta},\frac{z}{\cos\theta}) \, \frac{\dd B_\lambda(T(z^\prime))}{\dd z^\prime} \frac{\dd z^\prime}{\dd s^\prime} \, \frac{\dd z^\prime}{\cos\theta}   \end{equation*}

(12)   \begin{equation*} I_\lambda(z) = B_\lambda(\tsurf) +\int_0^{z} U_{\lambda,\theta}(z^\prime,z) \, \frac{\dd B_\lambda(T(z^\prime))}{\dd z^\prime} \, \dd z^\prime  \end{equation*}

where

(13)   \begin{equation*} U_{\lambda,\theta}(z_1, z_2) = 1-\exp\left[-\frac{1}{\cos\theta} \int_{z_1}^{z_2} \alpha_\lambda(s^\prime) \,\dd s^\prime \right] \end{equation*}

Now let’s change variables again, to use x = p(z)/p(0) where p is pressure. This leads to:

(14)   \begin{equation*} I_\lambda(x) = B_\lambda(\tsurf) -\int_x^1 V_{\lambda,\mu}(z^\prime,z) \, \frac{\dd B_\lambda(T(x^\prime))}{\dd x^\prime} \, \dd x^\prime  \end{equation*}

where \mu = \cos\theta and

(15)   \begin{equation*} V_{\lambda,\mu}(x_1, x_2) = 1-\exp\left[-\frac{1}{\mu} \int_{x_1}^{x_2} \tilde\alpha_\lambda(x^\prime) \,\dd x^\prime \right] \end{equation*}

(16)   \begin{equation*} \tilde\alpha_\lambda(x^\prime) = \alpha_\lambda(x^\prime) \, \frac{p(0)}{\dd p/\dd z} \end{equation*}

To get from intensity to flux, we need to integrate over all directions (within the upward hemisphere of solid angles). This involves an integration of the form 2\pi\int_0^1 f(\mu)\,\dd\mu where \mu = \cos\theta. This leads to:

(17)   \begin{equation*} F_\lambda(x) = 2\pi\,B_\lambda(\tsurf) - 2\pi \int_x^{1} W_{\lambda}(x^\prime,x) \, \frac{\dd B_\lambda(T(x^\prime))}{\dd x^\prime} \, \dd x^\prime  \end{equation*}

where

(18)   \begin{equation*} W_{\lambda}(x_1,x_2) = \left\{ 1 - \int_0^1 \exp\left[-\frac{1}{\mu} \int_{x_1}^{x_2} \tilde\alpha_\lambda(x^\prime) \,\dd x^\prime \right] \, \dd\mu \right\} \end{equation*}

Correspondence #1

If we define \tilde B_\lambda = 2\pi B_\lambda, A_\lambda(x) = W_{\lambda}(x,1), and compute \OLR_\lambda = F_\lambda(0), and this becomes

(19)   \begin{equation*} \OLR_\lambda(x) = \tilde B_\lambda(\tsurf) - \int_0^{1} A_{\lambda}(x^\prime) \, \frac{\dd \tilde B_\lambda(T(x^\prime))}{\dd x^\prime} \, \dd x^\prime  \end{equation*}

We can integrate this over wavelength, \lambda, and obtain R&R’s version of the RTE, but only if we assume that the absorption coefficient, \tilde\alpha_\lambda(x^) is independent of wavelength, i.e., we have a “gray” atmosphere.

However, R&R indicated that A_\lambda(x) is a “weighted average,” so perhaps we haven’t yet found the correspondence that they intended.

Correspondence #2

TBD

Conclusions

Thus, the result that \nghe will not change as \tsurf varies depends on these assumptions:

  • Negligible scattering (as is implicit in Schwarzchild equation)
  • Fixed lapse rate (as R&R name)
  • Fixed atmospheric composition (including absolute humidity)
  • Temperature varies linearly with altitude and pressure varies exponentially with altitude
  • Surface is a blackbody with a temperature that matches the surface atmospheric temperature
  • The atmospheric absorption/emission is “gray”, i.e., independent of wavelength