% 表題  2 次元非弾性流体モデル deepconv-mars 解説: 離散モデル編
%       第 6 節  地面モデルの離散化
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% 履歴  2001/11/01  小高正嗣
%       2007/04/26  小高正嗣: 英語版作成
%
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\section{\DselJ{地表面熱収支モデル}{Ground surface}} \Dseclab{離散化地面モデル}

\DselJ{
地中の温度変化を計算する熱伝導方程式の離散化は時間方向には Clank-Nicolson 
法, 空間方向には中心差分を用いて行う. 温度と格子間隔を整数格子点, 熱フ
ラックスを半整数格子点で評価する. 鉛直方向の格子点数は $J'$ とし, 最下
層から $j=1,2,...J'$ とする. 最上層の温度 $T_{i,J'}$ が地表面温度 
$T_{sfc,i}$ である.}
{
The time integration of 1D thermal conduction equation of grand
    surface (equation (55) in Part I is performed by
    the Crank-Nicolson scheme. 
The space differencing is evaluated by the second
    order centered scheme.
The grand temperature and depth are evaluated on the grid point and
    the heat flux is evaluated on the half grid point.
The number of vertical grid point is $J'$ and the suffix
    of the lowest grid point is $j = 1$.
The $T_{i,J'}$ is assumed to the surface temperature $T_{sfc,i}$.
The finite difference 1D thermal conduction equation is represented as follows.
}

\begin{eqnarray}
  \frac{T_{i,j}^{n+1}-T_{i,j}^{n}}{\Delta t}
  &=& \frac{\kappa}{4\Delta z_{j}}
  \left( \frac{T_{i,j+1}^{n+1} - T_{i,j}^{n+1}}
              {\Delta z_{j+1}+\Delta z_{j}} -
         \frac{T_{i,j}^{n+1} - T_{i,j-1}^{n+1}}
              {\Delta z_{j}+\Delta z_{j-1}}
  \right. \nonumber \\
  && + 
  \left. \frac{T_{i,j+1}^{n} - T_{i,j+1}^{n}}
              {\Delta z_{j+1}+\Delta z_{j}} -
         \frac{T_{i,j}^{n} - T_{i,j-1}^{n}}
              {\Delta z_{j}+\Delta z_{j-1}}
  \right).
\end{eqnarray}

\DselJ{
ここで $\kappa = k_{g}/\rho _{g}c_{p,g}$ である. 時刻 $(n+1)\Delta t$ の
項を左辺に, 時刻 $n\Delta t$ の項を右辺にまとめると,
}
{
or,
}
\begin{eqnarray}
&& 
- \frac{\kappa \Delta t}{\Delta z_{j}}
  \frac{T_{i,j-1}^{n+1}}
       {\overline{\Delta z}_{i}}
+ \left[
  4 + \frac{\kappa \Delta t}{\Delta z_{i,j}}
  \left(
  \frac{1}{\overline{\Delta z}_{j+\frac{1}{2}}}+
  \frac{1}{\overline{\Delta z}_{j-\frac{1}{2}}}
  \right)
  \right]T_{i,j}^{n+1}
- \frac{\kappa \Delta t}{\Delta z_{j}}
  \frac{T_{i,j+1}^{n+1}}
       {\overline{\Delta z}_{j+\frac{1}{2}}} \nonumber \\
&=& 
+ \frac{\kappa \Delta t}{\Delta z_{j}}
  \frac{T_{i,j-1}^{n}}
       {\overline{\Delta z}_{j-\frac{1}{2}}}
+ \left[
  4 - \frac{\kappa \Delta t}{\Delta z_{j}}
  \left(
  \frac{1}{\overline{\Delta z}_{j+\frac{1}{2}}}+
  \frac{1}{\overline{\Delta z}_{j-\frac{1}{2}}}
  \right)
  \right]T_{i,j}^{n}
+ \frac{\kappa \Delta t}{\Delta z_{j}}
  \frac{T_{i,j+1}^{n+1}}
       {\overline{\Delta z}_{j+\frac{1}{2}}}.
\end{eqnarray}

\DselJ{
ここで $\overline{\Delta z}_{j+\frac{1}{2}}=(\Delta z_{j+1}+\Delta
z_{j})/2$ とした. $\Dvect{T}^{n}=(..., T_{i,j}^{n}, T_{i,j+1}^{n},
T_{i,j+2}^{n}, ...)^{T}$ とすると, 行列式の形で
}
{
where $\kappa = k_{g}/\rho _{g}c_{p,g}$.
This equation can be represented in matrix form as follows.
}

\begin{equation}
  \Dvect{A}\cdot \Dvect{T}^{n+1} = \Dvect{B}\cdot \Dvect{T}^{n},
  \Deqlab{熱伝導方程式の行列表現}
\end{equation}

\DselJ{
と表すことができる. ここで $\Dvect{A},\Dvect{B}$ はそれぞれ,
}
{
where 
  $\Dvect{T}^{n}=(..., T_{i,j}^{n}, T_{i,j+1}^{n}, T_{i,j+2}^{n}, ...)^{T}$.
The elements of $\Dvect{A}$, $\Dvect{B}$
  are represented as follows.
}
\begin{eqnarray*}
&&
 A_{jj}=4 + \frac{\kappa \Delta t}{\Delta z_{j}}
  \left( \frac{1}{\overline{\Delta z}_{j+\frac{1}{2}}}+
         \frac{1}{\overline{\Delta z}_{j-\frac{1}{2}}}
  \right),
 A_{j,j+1}= - \frac{\kappa \Delta t}{\Delta z_{i}}
  \frac{1}{\overline{\Delta z}_{i+\frac{1}{2}}},
 A_{j,j-1}= - \frac{\kappa \Delta t}{\Delta z_{i}}
  \frac{1}{\overline{\Delta z}_{i-\frac{1}{2}}}, \\
&&
 B_{jj}=4 - \frac{\kappa \Delta t}{\Delta z_{j}}
  \left( \frac{1}{\overline{\Delta z}_{i+\frac{1}{2}}}+
         \frac{1}{\overline{\Delta z}_{i-\frac{1}{2}}}
  \right),
 B_{j,j+1}= + \frac{\kappa \Delta t}{\Delta z_{i}}
  \frac{1}{\overline{\Delta z}_{i+\frac{1}{2}}},
 B_{j,j-1}= + \frac{\kappa \Delta t}{\Delta z_{i}}
  \frac{1}{\overline{\Delta z}_{i-\frac{1}{2}}},
\end{eqnarray*}

\DselJ{
を要素に持つ $J'\times J'$ 行列である. 
}{}

\DselJ{
上端での境界条件と下端で断熱境界条件を考慮すると,
\Deqref{熱伝導方程式の行列表現}は
}
{
Considering the boundary condition of upper and lower boundaries,
  \Deqref{熱伝導方程式の行列表現} is modified as follows.
}
\begin{equation}
  \Dvect{A}\cdot \Dvect{T}^{n+1} = \Dvect{B}\cdot \Dvect{T}^{n} + 
                                   \Dvect{S}
\end{equation}
\DselJ{
となる. したがって
}
{
Therefore, the grand temperature is given by the solution of the
   following matrix equation.
}
\begin{equation}
  \Dvect{T}^{n+1} = \Dvect{A}^{-1} \cdot 
                    (\Dvect{B}\cdot \Dvect{T}^{n} + \Dvect{S}),
\end{equation}

\DselJ{
を解くことになる.
ここで係数行列 $\Dvect{A}, \Dvect{B}$ の第 1 行および第 $J'$ 行の対角要素は,
}
{
where the elements of $\Dvect{A}$ and  $\Dvect{B}$ are modified as follows.
}
\begin{eqnarray*}
&&
 A_{11}=4 + \frac{\kappa \Delta t}{\Delta z_{1}}
  \left( \frac{1}{\overline{\Delta z}_{\frac{3}{2}}} \right),
 B_{11}=4 - \frac{\kappa \Delta t}{\Delta z_{1}}
  \left( \frac{1}{\overline{\Delta z}_{\frac{3}{2}}} \right), \\
&&
 A_{J'J'}=4 + \frac{\kappa \Delta t}{\Delta z_{J'}}
  \left( \frac{1}{\overline{\Delta z}_{J'-\frac{1}{2}}} \right),
 B_{J'J'}=4 - \frac{\kappa \Delta t}{\Delta z_{J'}}
  \left( \frac{1}{\overline{\Delta z}_{J'-\frac{1}{2}}} \right),
\end{eqnarray*}

\DselJ{
$\Dvect{S}$ は $J'$ 行列ベクトルで, 
}
{
$\Dvect{S}$ 
   is a column vector whose dimension is
   $J'$ are represented as follows.
}
\[
 S_{j} = \left\{
 \begin{array}{ll}
   \frac{\Delta t}{\rho _{g}c_{p,g}\Delta z_{J'}}
        [- F_{s}(1-A) + F_{IR,net} + H], & j=J' \\
  0, & j\neq J'
 \end{array}
\right.
\]
\DselJ{
と表される.
}{}
