- R = C = 0: T(x) = (1 - x)/RC :
- C = 0, A^2 = R/RC : T(x) = exp (-A x)
- RC = small: T(x) ~ exp(-R/C x).
onsdag 9 juni 2010
A Too Simplistic Climate Model
A simplistic climate model including effects of conduction, convection and radiation takes the form
R T + C T_x - RC T_xx = 0 for x in (0,1), - RC T_x(0) = Q, T(1) = small,
where T(x) is temperature at altitude x, T_x is the first and T_xx the second derivative with respect to x, R is a coefficient of net outgoing radiation, C is a convection velocity, RC is a
a combined conduction/radiation coefficient and Q (= 1 say) is a heat source.
Depending on the size of the coefficients R, C and RC, the temperature profile T(x) takes the following main forms:
We see a linearly decaying temperature profile with climate sensitivity T(0) = 1/RC,
or an exponentially decaying profile with T(0) = 1.
In the case of substantial net outgoing radiation with R not small and R/RC and R/C large, the exponential decay is rapid. If R/RC and R/C is small, then T(x) is almost constant.
The linear profile observed in reality is realized only in the first case with RC dominating R and C, and then with a large climate sensitivity if RC is small.
The above model is thus not easy to bend to fit with observation of a constant lapse rate which is not large, and thus seems too simplistic.
What the model is missing is the thermodynamics of an atmosphere under gravitation which
has an equilibrium (isentropic) state with constant not large lapse rate, as shown in posts on
We thus find that radiation/conduction must be combined with thermodynamics to describe the action of the atmopshere as a global air conditioner maintaining a surface temperature of 15 C under permanent heating from the Sun, including the crucial aspect of climate sensitivity.
The above argument gives yet another indication that a simplistic estimate of basic climate sensitivity of 1 C based on radiation only, may not be close to any reality.