next up previous
: 3 Turbulent parameterization : Two dimensional anelastic model : 1 Outline of the

2 Atmospheric model

The governing equation of model atmospheric dynamics is the 2D anelastic system (Ogura and Phillips, 1962).

    $\displaystyle \DD{u}{t} -fv = -c_{p}\Theta _{0}\DP{\pi }{x} + D(u),$ (1)
    $\displaystyle \DD{v}{t} + fu = D(v),$ (2)
    $\displaystyle \DD{w}{t} = -c_{p}\Theta _{0}\DP{\pi }{z} +
g\frac{\theta }{\Theta _{0}} + D(w),$ (3)
    $\displaystyle \DP{(\rho _{0}u)}{x} + \DP{(\rho _{0}w)}{z} = 0,$ (4)
    $\displaystyle \DD{\theta }{t} + w\DP{\Theta _{0}}{z}
= \frac{\Theta _{0}}{T_{0}}(Q_{rad}+Q_{dis}) + D(\theta + \Theta _{0}) ,$ (5)
    $\displaystyle \DD{}{t} = \DP{}{t} + u\DP{}{x} + w\DP{}{z}.$  

(1), (2), (3) are the horizontal and vertical component of equation of motion, respectively. (4) is the continuity equation and (5) is the thermodynamic equation. $x,y, z,t$ are horizontal, vertical and time coordinate, respectively. $u, v$ are horizontal and vertical velocity, and $\theta, \pi $ are potential temperature and nondimensional pressure function deviation from those of basic state, respectively. $\rho
_{0}, \Theta _{0}, T_{0}$ are density, potential temperature and temperature in basic state. $g$ is gravitational acceleration whose value is equal to 3.72 msec${}^{-2}$. $Q_{rad}$ is radiative heating (cooling) rate per unit mass which is calculated by convergence of the radiative heat flux. $Q_{dis}$ is heating rate per unit mass owing to dissipation of turbulent kinetic energy, which is given by turbulent parameterization.

$D(\cdot )$ term in equation (1)$\sim $(5) represents the turbulent diffusion owing to subgrid scale turbulent mixing.

D(\cdot ) = \DP{}{x}\left[ K\DP{(\cdot )}{x} \right] +
...}{\rho _{0}}\DP{}{z}\left[ \rho _{0}K\DP{(\cdot )}{z} \right].
\end{displaymath} (6)

$K$ is turbulent diffusion coefficient which is calculated by (10) and (11).

The nondimensional pressure function and potential temperature are defined as follows.

\Pi \equiv \left(\frac{p}{P_{00}}\right)^{\kappa } = \Pi _{...
..., \quad
\Pi_{0} = \left(\frac{P_{0}}{P_{00}}\right)^{\kappa }

\Theta \equiv T\Pi^{-1} = \Theta _{0} + \theta, \quad
\Theta _{0} = T_{0}\Pi_{0}^{-1}

where $p$ and $P_{0}$ are pressure and basic state pressure, $P_{00}$ is reference pressure (= 7 hPa), $\kappa = R/c_{p}$, $c_{p}$ is specific heat of constant pressure per unit mass and $R$ is atmospheric gas constant per unit mass. The basic state atmospheric structure is calculated by using the hydrostatic equation as follows.
$\displaystyle \DD{P_{0}}{z}$ $\textstyle =$ $\displaystyle - \rho _{0}g,$ (7)
$\displaystyle P_{0}$ $\textstyle =$ $\displaystyle \rho _{0}RT_{0}.$ (8)

The deviation of nondimensional pressure function is calculated by using the following equation which is derived from (1)$\sim $(4).
$\displaystyle c_{p}\Theta_{0} \left[ \rho _{0}\DP[2]{\pi }{x} + \DP{}{z}
\left( \rho _{0}\DP{\pi }{z} \right) \right]$ $\textstyle =$ $\displaystyle \frac{g}{\Theta_{0} }\DP{(\rho _{0}\theta )}{z}$  
    $\displaystyle - \DP{}{x}\left[ \rho _{0}\left
( u\DP{u}{x} + w\DP{u}{z} - D(u) \right) \right]$  
    $\displaystyle - \DP{}{z}\left[ \rho _{0}\left
( u\DP{w}{x} + w\DP{w}{z} - D(w) \right) \right].$ (9)

Boundary conditions

The model horizontal boundary is cyclic. The vertical wind velocity is set to be 0 at the surface and upper boundary.


表 1: Parameters of atmospheric model
Parameters Standard Values Note
$f$ 0 sec${}^{-1}$  
$g$ 3.72 msec${}^{-2}$  
$P_{00}$ 7 hPa  
$c_{p}$ 734.9 Jkg${}^{-1}$K${}^{-1}$ Value of CO${}_{2}$
$R$ 189.0 Jkg${}^{-1}$K${}^{-1}$ Value of CO${}_{2}$

next up previous
: 3 Turbulent parameterization : Two dimensional anelastic model : 1 Outline of the
Odaka Masatsugu 平成19年4月25日