2 Atmospheric model

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

$$\DD{u}{t} -fv = -c_{p}\Theta _{0}\DP{\pi }{x} + D(u),$$ (1)

$$\DD{v}{t} + fu = D(v),$$ (2)

$$\DD{w}{t} = -c_{p}\Theta _{0}\DP{\pi }{z} + g\frac{\theta }{\Theta _{0}} + D(w),$$ (3)

$$\DP{(\rho _{0}u)}{x} + \DP{(\rho _{0}w)}{z} = 0,$$ (4)

$$\DD{\theta }{t} + w\DP{\Theta _{0}}{z} = \frac{\Theta _{0}}{T_{0}}(Q_{rad}+Q_{dis}) + D(\theta + \Theta _{0}) ,$$ (5)

$$\DD{}{t} = \DP{}{t} + u\DP{}{x} + w\DP{}{z}.$$

(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.

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

$$\DD{P_{0}}{z} = - \rho _{0}g,$$ (7)

$$P_{0} = \rho _{0}RT_{0}.$$ (8)

The deviation of nondimensional pressure function is calculated by using the following equation which is derived from (1)$\sim$(4).

$$c_{p}\Theta_{0} \left[ \rho _{0}\DP[2]{\pi }{x} + \DP{}{z} \left( \rho _{0}\DP{\pi }{z} \right) \right] = \frac{g}{\Theta_{0} }\DP{(\rho _{0}\theta )}{z}$$

$$- \DP{}{x}\left[ \rho _{0}\left ( u\DP{u}{x} + w\DP{u}{z} - D(u) \right) \right]$$

$$- \DP{}{z}\left[ \rho _{0}\left ( u\DP{w}{x} + w\DP{w}{z} - D(w) \right) \right].$$ (9)

Boundary conditions

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

Parameters

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${}_{2}$