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 the following exploratory analyses, I investigate R&R’s mathematical analysis to try to understand what if any assumptions are involved.

Table of Contents

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*}

Initial Attempt

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

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?

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

Second Attempt

I have found a source which derives an equation similar to the one offered by Raval and Ramanathan: Liou, K. N.. An Introduction to Atmospheric Radiation (International Geophysics, Volume 84) (Section 4.2). Elsevier Science. Kindle Edition. In particular, equation (4.2.10a) in that book looks vaguely related. However, there is a clear error in the notation of the equation. So, I’ll need to work through the derivation to see how the correctly notated equation should read.

Following the derivation of K. N. Liu in Section 4.2, I see that a notational problem arises (in the Kindle version of the book) at equation 4.2.5a. The problem is that the text uses the symbol \tau for both \tau (optical depth relative to TOA, at height z) and \tau_s (optical depth of the surface relative to TOA). (Figure 4.4 suggests Liu might have intended to use \tau_\star for what I’m denoting \tau_s.) I believe equation 4.2.5a ought to read:

(20)   \begin{equation*} I^\uparrow_\nu(\tau,\mu) = B_\nu(\tau_s) \, e^{-(\tau_s-\tau)/\mu} + \int_{\tau}^{\tau_s} B_\nu(\tau^\prime)\,e^{-(\tau^\prime-\tau)/\mu}\,\frac{\dd\tau^\prime}{\mu} \end{equation*}

Following along in the derivation, and applying the definition T_\nu(\tau/\mu) = \exp(-\tau/\mu), the correctly-notated version of equation 4.2.7a becomes:

(21)   \begin{equation*} I^\uparrow_\nu(\tau,\mu) = B_\nu(\tau) \, T_\nu[(\tau_s-\tau)/\mu] + \int_{\tau}^{\tau_s} B_\nu(\tau^\prime)\,T_\nu[(\tau^\prime-\tau)/\mu]\,\frac{\dd\tau^\prime}{\mu} \end{equation*}

Liu’s defines upward flux density F_\nu^\uparrow and slab/diffuse transmittance T_\nu^f via:

(22)   \begin{equation*} F_\nu^\uparrow(\tau) = 2\pi \int_0^1 I_\nu^\uparrow(\tau,\mu)\,\mu\,\dd\mu \end{equation*}

(23)   \begin{equation*} T_\nu^f(\tau) = 2 \int_0^1 T_\nu(\tau/\mu)\,\mu\,\dd\mu \end{equation*}

These lead to equation 4.2.10a (with its notation fixed):

(24)   \begin{equation*} F_\nu^\uparrow(\tau) = \pi B_\nu(\tau_s) \, T_\nu^f(\tau_s-\tau) - \int_\tau^{\tau_s} \pi B_\nu(\tau^\prime) \frac{\dd}{\dd \tau^\prime} \, T_\nu^f(\tau^\prime-\tau) \, \dd\tau^\prime \end{equation*}

Now, to try to shift towards something more like the R&R formula, let’s try integrating by parts:

(25)   \begin{equation*} F_\nu^\uparrow(\tau) = \pi B_\nu(\tau) + \int_\tau^{\tau_s} \pi \, T_\nu^f(\tau^\prime-\tau) \, \frac{\dd}{\dd \tau^\prime} B_\nu(\tau^\prime)\, \dd\tau^\prime \end{equation*}

Let’s define a relevant absorbance by A_\nu^f = 1 - T_\nu^f. Then, our equation for upward flux becomes

(26)   \begin{equation*} F_\nu^\uparrow(\tau) = \pi B_\nu(\tau_s) - \int_\tau^{\tau_s} \pi \, A_\nu^f(\tau^\prime-\tau) \, \frac{\dd}{\dd \tau^\prime} B_\nu(\tau^\prime)\, \dd\tau^\prime \end{equation*}

The form of this seems promising in its similarity to the form of R&R’s equation. We will need to change to the normalized pressure coordinate, x = p/p_s, to get to R&R’s formulation.

The equation for upward flux can be written in terms of the variable x as

(27)   \begin{equation*} F_\nu^\uparrow(x) = \pi B_\nu(1) - \int_x^{1} \pi \, A_\nu^f(x, x^\prime) \, \frac{\dd}{\dd x^\prime} B_\nu(x^\prime)\, \dd x^\prime \end{equation*}

where

(28)   \begin{equation*} A_\nu^f(x, x^\prime) = A_\nu^f\left(\tau(x^\prime) - \tau(x)\right) \end{equation*}

Setting x=0 in the above equation and integrating over the frequency \nu would produce R&R’s equation (up to a factor of \pi, which can perhaps be folded into R&R’s definition of B(T)), but only if A_\nu^f(x, x^\prime) is independent of frequency.

So, this analysis again confirms the suspicion that R&R’s analysis relies on assuming a “grey” atmosphere with absorption independent of frequency (or wavelength).

R&R’s Reference

R&R cite Manabe(1967) as the source of their radiative transfer equation. However, it seems possible to me that they should have referenced Manabe(1964), since I find more radiative transfer equations in the latter than in the former. However, neither reference seems to present a formulation of the radiative transfer equation that closely matches the one offered by R&R. Nonetheless, the analyses here support the correctness of R&R’s equation, given the assumption of frequency-independent absorption.