# C. Details of the Model

Here, the radiation scheme used in the numerical calculations and the characteristics of the scheme will be described. The key is to discretize the radiative flux equation in two ways and then join the two discretized equations, which allows radiation calculations for a gray atmosphere with an arbitrary optical depth profile.

#### Finite-Difference Representation of Radiative Flux

Net upward radiative flux for the case of a gray atmosphere is given by the following equation:

where is optical depth at the top of the atmosphere, thus is . πB is equal to σT4.

Here, the radiative flux equation is discretized in the same way as in Nakajima et al. (1992), and thus

Here, KMAX represents the upper-most layer, and KMAX = 32 in a typical model calculation. The radiative flux is calculated by joining two sets of discretized equations. Specifically, the integrals for the contributions to net upward radiative flux from optically thin layers and optically thick layers are evaluated using the upper and lower options in the equation, respectively. The fraction of the usage of the upper and lower options in the equation changes according to the value of . If a significantly large value is taken for , the integrals are evaluated entirely with the upper option, which results in a radiative scheme identical to that used in Numaguti and Hayashi (1991).

There exists no that makes the upper and lower options identical. If such hypothetically exists, the relationship

needs to hold. Rearranging this condition by letting leads to

Unfortunately, no solutions exist for this equation. More accurately, a discontinuity emerges in the evaluated fluxes at the altitude at which the transition between the two options occurs. However, the actual difference between the evaluated fluxes is almost negligibly small. In fact, no evident visible discontinuities exist in the vertical profiles of the radiative flux. (For more information, see Figure 5 of Characteristics of One-Dimensional Radiative-Convective Equilibrium Solutions.)

The above-mentioned complex procedure is necessary because problems arise if either the upper or lower option in the discretized radiative flux equation alone is used. The reasons that the problem arises will be described below, and the values of that are suitable for 3D calculations will be examined.

#### For the Case of Large CΔτ

 First, the extreme limits are considered in which is set to a significantly large value. This corresponds to using the upper option of the discretized equation for the entire layer, that is, using the radiative scheme of Numaguti and Hayashi (1991). As an example, results will be shown for = 106, Pn0 = 105, and a vertical grid resolution of L1000. The relationship between Tg and OLR from this case is almost identical to that from = 10-1 (not shown). However, the vertical profile of radiative flux at high temperatures is not calculated properly. Figure 1 shows the vertical profile of net upward radiative flux for the case of Tg = 550 K. Unlike in the case of = 10-1, net upward radiative flux increases toward the surface in the layer below the vicinity of log10σ = - 2. Net upward radiative flux decreases from the top to a certain level in the middle layer, and below the level flux increases again. This tendency is attributable to the following reason. If the atmosphere becomes optically thick, only the terms associated with j = k remain in the lower option of the discretized equation, which then leads to for the evaluation of net upward radiative flux. Properly, Fnet should be represented as (a) (b) Figure 1: The results for the case of = 106. For L1000, pn0 = 105, and Ts= 550 K. (a) The vertical profile of net upward radiative flux (W/m2). (b) The vertical profile of radiative heating rate (K/sec).

In this example calculation, the values of σT4 and net upward radiative flux are

 For the case of L32, the above-described phenomenon is more amplified. Net upward radiative flux starts increasing near log10σ = - 3 from higher to lower altitudes, however, it starts decreasing again around log10σ = -1 (Figure 2(a)). This profile results because the net upward radiative flux is determined by σT4k-1 - σT4k below log10σ = -3.0. The decrease in net upward radiative flux from higher to lower altitudes near the surface is attributable to small vertical grid intervals near the surface. The net upward radiative flux values around log10σ = - 1 for the case of = 106 are larger than those for the case of = 10-1 by 400 W/m2. The differences in both upward and downward radiative fluxes in the middle layer between the case of = 106 and the case of = 10-1 are approximately 200 W/m2. On the other hand, the error in the radiative heating profile which is important for time evolution problems in 3D model is relatively small. (Figure 2(b)). (a) (b) Figure 2: The results for the case of = 106. For L32, pn0 = 105 Pa, and Ts = 550 K. (a) The vertical profile of net upward radiative flux (W/m2). (b) The vertical profile of radiative heating rate (K/sec).

In the case of L32, a discontinuity arises in the middle layer of the vertical profile of radiative flux for ≥ 10.0. Though not shown, in the case of L32, the radiative flux can be calculated with few problems as long as ≤ 1.0.