5 Radiation

The radiative heating rate is calculated from the radiative flux by using the second order centered scheme.

$$Q_{rad,i,j}^{n} = Q_{rad,IR,i,j}^{n} + Q_{rad,NIR,i,j}^{n} + Q_{rad,dust,SR,i,j}^{n} + Q_{rad,dust,IR,i,j}^{n},$$ (51)

$$Q_{rad,*,i,j} = - \frac{g}{c_{p}} \frac{F_{*,net,i,j+\frac{1}{2}} - F_{*,net,i,j-\frac{1}{2}}} {\Delta P _{0,j}},$$ (52)

$$F_{*,net,i,j+\frac{1}{2}} = F_{IR,i,j+\frac{1}{2}}^{\uparr... ...d \Delta P_{0,j} = P_{0,j+\frac{1}{2}} - P_{0,j-\frac{1}{2}}.$$

The radiative flux is evaluated on the half grid point The subscript $m$ shows grid point in the wave number. In the following sections, the superscript which shows time step is omitted.

5.1 Radiative transfer of atmospheric CO${}_{2}$

The finite difference form of the infrared radiative flux and the Plank function are represented as follows.

$$F_{IR,i,j+\frac{1}{2}}^{\uparrow} = \sum _{m}\Delta \nu _{m}\left\{ \pi B_{\nu _{i},T_{sfc,i}}{\cal T}_{i}(0,z_{j+\frac{1}{2}}) \right.$$

$$\left. + \sum _{j'=1}^{j} \pi B_{\nu _{m},T_{i,j'}} \frac{ {\cal ... ...cal T }_{m}(z_{j+\frac{1}{2}},z_{j'-\frac{1}{2}}) } { \Delta z_{j'} } \right\},$$ (53)

$$F_{IR,i,j+\frac{1}{2}}^{\downarrow} = \sum _{m}\Delta \nu _{m}\left\{ \sum _{j'=j}^{J}\pi B_{\nu _{m},T... ...al T }_{m}(z_{j+\frac{1}{2}},z_{j'+\frac{1}{2}}) } { \Delta z_{j'} } \right\} ,$$ (54)

$$B_{\nu _{m},T_{i,j}} = \frac{1.19\times 10^{-8}\nu _{m}^{3}} {e^{1.4387\nu_{i}/T_{i,j}}-1},$$ (55)

$$T_{i,j} = \Pi _{0,j}(\theta _{i,j}+\Theta _{0,j})$$ (56)

where the averaged transmission function, the equivalent width, and the effective path length are represented as follows.

$${\cal T}_{m}(z_{j+\frac{1}{2}},z_{j'+\frac{1}{2}}) = \exp... ...{*}(z_{j+\frac{1}{2}},z_{j'+\frac{1}{2}})/ \alpha ^{*}_{m}}},$$

$$u(z_{j+\frac{1}{2}},z_{j'+\frac{1}{2}}) = 1.67 g \vert P_... ...\alpha ^{*}_{m}=\alpha _{m} \frac{\overline{p}_{jj'}}{p_{0}},$$

$$P_{0,j+\frac{1}{2}} = 0.5(P_{0,j}+P_{0,j+1}), \quad \over... ...\frac{1}{2}}\vert} {u(z_{j+\frac{1}{2}},z_{j'+\frac{1}{2}})},$$

The finite form of the near infrared radiative flux and the effective path length are represented as follows.

$$F_{NIR,i,j+\frac{1}{2}}^{\downarrow} = \sum _{m}\Delta \nu ... ...{i}(z_{j+\frac{1}{2}},z_{J+\frac{3}{2}})\cos \zeta \right\}.$$ (57)

$$u(z_{j+\frac{1}{2}},z_{J+\frac{3}{2}}) = \frac{1.67 g \ve... ...{0,J+\frac{3}{2}} \vert} {\mbox{MAX}(\cos \zeta, \epsilon )}.$$

where $\epsilon$ is a small parameter to ensure $u = 0$ when $\cos \zeta =0$.

5.2 Radiative transfer of dust

The finite difference form of the solar radiative transfer equation of dust are represented as follows.

$$F_{dif,\nu_{m},i,j+\frac{1}{2}}^{\uparrow} = F_{dif,\nu_{m},i,j-\frac{1}{2}}^{\uparrow} - \Delta \tau _{\nu_{m... ...frac{1}{2}}^{\uparrow} + F_{dif,\nu_{m},i,j-\frac{1}{2}}^{\uparrow}}{2} \right.$$

$$- \left. \gamma _{2,\nu_{m}}\frac{F_{dif,\nu_{m},i,j+\frac{1}{2}}... ...ga}_{\nu_{m}}^{*} S_{0}e^{-\tau _{\nu_{m},j+\frac{1}{2}}^{*} /\mu_{0}} \right],$$ (58)

$$F_{dif,\nu_{m},i,j-\frac{1}{2}}^{\downarrow} = F_{dif,\nu_{m},i,j+\frac{1}{2}}^{\downarrow} + \Delta \tau _{\nu_... ...frac{1}{2}}^{\uparrow} + F_{dif,\nu_{m},i,j-\frac{1}{2}}^{\uparrow}}{2} \right.$$

$$- \left. \gamma _{1,\nu_{m}}\frac{F_{dif,\nu_{m},i,j+\frac{1}{2}}... ...ga}_{\nu_{m}}^{*} S_{0}e^{-\tau _{\nu_{m},j+\frac{1}{2}} ^{*}/\mu_{0}} \right],$$ (59)

$$\Delta \tau _{\nu_{m}, j}= \frac{\overline{Q}_{e,\nu_{m}}}{... ...+\frac{1}{2}} = \sum _{j'=j+1}^{J+1}\Delta \tau _{\nu_{m},j'}.$$

(58) and (59) can be represented in matrix form as follows.

$$\Dvect{F}_{dif,\nu_{m}}^{\uparrow} = \Dvect{A}\Dvect{F}_{dif,\nu_{m}}^{\uparrow} + \Dvect{B}\Dvect{F}_{dif,\nu_{m}}^{\downarrow} + \Dvect{R},$$ (60)

$$\Dvect{F}_{dif,\nu_{m}}^{\downarrow} = \Dvect{C}\Dvect{F}_{dif,\nu_{m}}^{\uparrow} + \Dvect{D}\Dvect{F}_{dif,\nu_{m}}^{\downarrow} + \Dvect{S}.$$ (61)

where $\Dvect{F}_{dif,\nu_{m}}^{\uparrow}=( F_{dif,\nu_{m},i,\frac{1}{2}}^{\uparrow}, ... ...,i,\frac{3}{2}}^{\uparrow},..., F_{dif,\nu_{m},i,J+\frac{1}{2}}^{\uparrow})^{T}$ and so on. (60) and (61) are solved by iteration method. The elements of the matrixes in (60) and (61) are represented as follows.

$$\begin{eqnarray*} A_{jk} &=& \left\{ \begin{array}{cl} - \frac{1}{2}\Delta \t... ...}}^{*} /\mu_{0}} & j\leq J-1 \\ 0 & j=J \end{array} \right. \end{eqnarray*}$$

The finite difference form of the solar radiative transfer equation of dust are represented as follows.

$$F_{IR,\nu_{m},i,j+\frac{1}{2}}^{\uparrow} = F_{IR,\nu_{m},i,j-\frac{1}{2}}^{\uparrow} - \Delta \tau _{\nu_{m}... ...\frac{1}{2}}^{\uparrow} + F_{IR,\nu_{m},i,j-\frac{1}{2}}^{\uparrow}}{2} \right.$$

$$- \left. \gamma _{2,\nu_{m}}\frac{F_{IR,\nu_{m},i,j+\frac{1}{2}}^... ...wnarrow}}{2} -2\pi (1-\tilde{\omega}_{\nu_{m}}^{*})B_{\nu_{m},T_{i,j}} \right],$$ (62)

$$F_{IR,\nu_{m},i,j-\frac{1}{2}}^{\downarrow} = F_{IR,\nu_{m},i,j+\frac{1}{2}}^{\downarrow} + \Delta \tau _{\nu_{... ...\frac{1}{2}}^{\uparrow} + F_{IR,\nu_{m},i,j-\frac{1}{2}}^{\uparrow}}{2} \right.$$

$$- \left. \gamma _{1,\nu_{m}}\frac{F_{IR,\nu_{m},i,j+\frac{1}{2}}^... ...wnarrow}}{2} +2\pi (1-\tilde{\omega}_{\nu_{m}}^{*})B_{\nu_{m},T_{i,j}} \right].$$ (63)

(62) and (63) can be represented in matrix form as follows.

$$\Dvect{F}_{IR,\nu_{m}}^{\uparrow} = \Dvect{A}\Dvect{F}_{IR,\nu_{m}}^{\uparrow} + \Dvect{B}'\Dvect{F}_{IR,\nu_{m}}^{\downarrow} + \Dvect{R}',$$ (64)

$$\Dvect{F}_{IR,\nu_{m}}^{\downarrow} = \Dvect{C}\Dvect{F}_{IR,\nu_{m}}^{\uparrow} + \Dvect{D}\Dvect{F}_{IR,\nu_{m}}^{\downarrow} + \Dvect{S}'.$$ (65)

where $\Dvect{F}_{IR,\nu_{m}}^{\uparrow}=(F_{IR,\nu_{m},i,\frac{1}{2}}^{\uparrow}, F_{... ...},i,\frac{3}{2}}^{\uparrow},..., F_{IR,\nu_{m},i,J+\frac{1}{2}}^{\uparrow})^{T}$ and so on. (64) and 行列形式のダストの下向き赤外放射 are solved by iteration method. The elements of the matrixes in (64) and 行列形式のダストの下向き赤外放射 are represented as follows.

$$\begin{eqnarray*} B'_{jk} &=& \left\{ \begin{array}{cl} 0 & j=k=1 \\ B_{jk}... ...\nu_{m}, T_{i,j}} & j\leq J-1 \\ 0 & i=N \end{array} \right. \end{eqnarray*}$$