277 lines
13 KiB
TeX
277 lines
13 KiB
TeX
\section{Model equations and numerics}
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The core of the model is a set of primitive equations. They describe
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the conservation of momentum, mass, and thermal energy.
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Using spherical coordinates and the sigma system and with
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the aid of the equation of state they can be written in the
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dimensionless form as follows:
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{\bf Conservation of momentum:}
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Vorticity equation
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\begin{equation}
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\pd{(\zeta + f)}{t} = \frac{1}{(1-\mu^2)}\pd{F_v}{\lambda}-\pd{F_u}{\mu}+P_{\zeta}
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\label{vortgl}
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\end{equation}
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Divergence equation
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\begin{equation}
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\pd{D}{t} = \frac{1}{(1-\mu^2)}\pd{F_u}{\lambda}+\pd{F_v}{\mu}-\nabla^2\left(\frac{U^2+V^2}{2(1-\mu^2)}+\Phi+T_0\ln p_s\right)+P_D
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\label{divgl}
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\end{equation}
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Hydrostatic approximation
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\begin{equation}
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\pd{\Phi}{\ln\sigma} = -T
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\label{hydrogl}
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\end{equation}
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{\bf Conservation of mass:}
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Continuity equation
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\begin{equation}
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\pd{\ln p_s}{t} = -\int\limits_0^1Ad\sigma
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\label{kontigl}
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\end{equation}
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{\bf Conservation of energy:}
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First law of thermodynamics
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\begin{equation}
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\pd{T'}{t} = -\frac{1}{(1-\mu^2)}\pd{(UT')}{\lambda}-\pd{(VT')}{\mu}+DT'-\dot{\sigma}\pd{T}{\sigma}+\kappa\frac{T}{p}\omega+\frac{J}{c_p}+P_T,
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\label{thermogl}
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\end{equation}
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with:
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\bfl
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$ \di F_u = V(\zeta+f)-\dot{\sigma}\pd{U}{\sigma}-T'\pd{\ln p_s}{\lambda} $
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\efl
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\bfl
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$ \di F_v = -U(\zeta+f)-\dot{\sigma}\pd{V}{\sigma}-T'(1-\mu^2)\pd{\ln p_s}{\mu} $
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\efl
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\bfl
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$ \di A = D+\vec{V}\cdot\nabla\ln p_s $
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\efl
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and \quad $U = u\cos\phi$, $V = v\cos\phi$.
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Where the variables denote:
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\begin{tabbing}
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xxxxxxxxxxxxxxxxx\=xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx\kill
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$T$ \> temperature \\
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$T_0$ \> reference temperature \\
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$T'=T-T_0$ \> temperature deviation from $T_0$ \\
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$\zeta$ \> relative vorticity \\
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$D$ \> divergence \\
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$p_s$ \> surface pressure \\
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$p$ \> pressure \\
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$\Phi$ \> geopotential \\
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$t$ \> time \\
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$\lambda$, $\phi$ \> longitude, latitude \\
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$\mu=\sin\phi$ \> \\
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$\sigma=p/p_s$ \> sigma vertical coordinate \\
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$\dot{\sigma}=d\sigma/dt$ \> vertical velocity in $\sigma$-system \\
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$\omega=dp/dt$ \> vertical velocity in $p$-system \\
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$u$, $v$ \> zonal, meridional component of horizontal velocity \\
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$\vec{V}$ \> horizontal velocity with components $U$, $V$ \\
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$f$ \> Coriolis parameter \\
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$J$ \> diabatic heating rate \\
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$c_p$ \> specific heat of dry air at constant pressure \\
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$\kappa$ \> adiabatic coefficient \\
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\end{tabbing}
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The set of differential equations consists of the four prognostic
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equations (\ref{vortgl}), (\ref{divgl}), (\ref{kontigl}), and
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(\ref{thermogl}). Vorticity $\zeta$ and divergence $D$ are
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scaled by the angular velocity of the earth $\Omega$, pressures $p$
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and $p_s$ are scaled by the global mean surface pressure $P_s=1011\,hPa$,
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temperatures $T$ and $T_0$ are scaled by $a^2\Omega^2/R$, geopotential
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$\Phi$ is scaled by $a^2\Omega^2/g$, and time $t$ is scaledby $\Omega^{-1}$, where
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$a$ is the radius of the earth, $R$ is the gas constant of dry air,
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and $g$ is the gravitational acceleration.
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For the parameterizations $P_\zeta$, $P_D$ and $P_T$ see section
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\ref{parametrisierungen}. The model can be run with or without orography.
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The horizontal representation of any model variable is given
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by a series of spherical harmonics. If $Q$ is an arbitrary
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model variable, then its spectral representation has the form:
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\begin{equation}
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\label{spektral}
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Q(\lambda,\mu,t) = \sum_\gamma Q_\gamma(t)\,Y_\gamma(\lambda,\mu).
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\end{equation}
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Here, $Y_\gamma$ are the spherical harmonics, and $Q_\gamma$ the
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corresponding complex amplitudes, where $\gamma=(n,m)$ designates
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the spectral modes ($n=1,\,2,\,3,\ldots$: total wave number;
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$m=0,\,\pm1,\,\pm2,\,\pm3,\ldots$: zonal wave number),
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with $|m|\le n$ \citep{holton}. The latter condition follows
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from the triangular truncation in wave number space.
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The truncation is done at the total wave number $n_T$, which
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can be set to $n_T=21,31,42,85,127,170$, i.e. the model can be
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used with the T21,\ldots,T170 spectral resolution. The vertical
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resolution is given by $n_L$ equidistant $\sigma$-levels with the
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standard value $n_L=5$. At the upper ($\sigma=0$) and lower boundary
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($\sigma=1$) of the model domain the vertical velocity is set to zero
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($\dot{\sigma}=0$).
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The linear contributions to the tendencies are calculated in the spectral
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domain, the nonlinear contributions in grid point space. Therefore, at
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every time step, the necessary model variables are transformed from
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spectral to grid point representation by Legendre and Fast Fourier (FFT)
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transformations, and then the calculated tendencies are transformed back
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into the spectral domain where the time step is carried out
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\citep[for the transform method see][]{orszag70, eliasen70}. Because of
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the semi-implicit time integration scheme \citep*{hossim75, simhosburr78}
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the terms due to gravity wave propagation are integrated in time implicitly,
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and the remaining terms are integrated explicitly, the latter with a
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leap-frog time step. In the standard model, a time step of one hour is used.
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A Robert-Asselin time filter \citep*{halwilli} is applied to avoid decoupling
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of the two leap-frog time levels. The contributions to the tendencies due
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to vertical advection are calculated by an energy and angular-momentum
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conserving vertical finite-difference scheme \citep*{simburr81}.
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\section{Parameterizations}
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\label{parametrisierungen}
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\subsection{Friction}
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The dissipative processes in the atmosphere are parameterized using a linear
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approach (Rayleigh friction), which describes the effects of surface drag and
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vertical transport of the horizontal momentum due to small scale turbulence
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in the boundary layer. To achieve this, vorticity $\zeta$ and divergence $D$
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are damped towards the state of rest ($\zeta=0,\,D=0$) with the time scale $\tau_F$.
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The parameterization terms $P_\zeta$ and $P_D$ appear in the model equations
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(\ref{vortgl}) resp. (\ref{divgl}) and have the form:
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\beqa
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P_\zeta & = & \frac{\zeta}{\tau_F}+H_\zeta \label{paraz} \\
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P_D & = & \frac{D}{\tau_F}+H_D. \label{parad}
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\eeqa
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The time scale $(\tau_F)_l$ depends on the $\sigma$-level $l$ ($l=1,\ldots,n_l$).
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Usually, for the upper levels ($l=1,\ldots,n_l-1$) it is set to
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$(\tau_F)_l=\infty$ (no friction) and for the lowest level ($l=n_l$)
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a typical value is $(\tau_F)_l=1\,d$. An explanation of the hyperdiffusion terms
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$H_\zeta$ and $H_D$ follows in section \ref{diffusion}.
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\subsection{Diabatic heating}
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\label{diabatischeheizung}
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All the diabatic processes considered in the model are also parameterized using
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a linear approach (Newtonian cooling). They include the diabatic heating due to
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absorption and emission of short and long wave radiation, as well as latent and
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sensible heat fluxes (convection). The temperature $T$ relaxes towards the
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restoration temperature $T_R$ with the time scale $\tau_R$. The parameterization
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term in the thermal energy equation (\ref{thermogl}) is given by:
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\begin{equation}
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\frac{J}{c_p}+P_T = \frac{T_R-T}{\tau_R}+H_T. \label{parat}
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\end{equation}
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For the hyperdiffusion $H_T$ see section \ref{diffusion}. $\tau_R$ depends on
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the $\sigma$-level $l$, $T_R$ on the latitude $\phi$ and on the vertical
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coordinate $\sigma$. The restoration temperature field has the form:
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\begin{equation}
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\label{glgTr_2d}
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T_R(\phi,\sigma) = T_R(\sigma)+f(\sigma)\,T_R(\phi).
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\end{equation}
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The vertical profile is described by:
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\begin{equation}
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T_R(\sigma) = (T_R)_{tp}+\sqrt{\left[\frac{L}{2}\Big(z_{tp}-z(\sigma)\Big)\right]^2+S^2}\,+\,\frac{L}{2}\Big(z_{tp}-z(\sigma)\Big),
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\end{equation}
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with $\di (T_R)_{tp}=(T_R)_{grd}-L\,z_{tp}$. Here, $z$ denotes the geometric
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height, $z_{tp}$ the global constant height of the tropopause,
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$L=-(\partial T_R)/(\partial z)$ the vertical restoration temperature gradient,
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$(T_R)_{grd}$ and $(T_R)_{tp}$ the restoration temperature at the surface and at
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the global isothermal tropopause, respectively. $S$ provides a smoothing of the
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profile at the tropopause. $z(\sigma)$ is determined by an iterative method.
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The profile is determined by setting the parameters $(T_R)_{grd}$, $z_{tp}$,
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$L$ and $S$. Figure \ref{Tr_z} shows the vertical profile for the standard
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parameter values.
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% 37.4.09: erscheint so im naechsten Abschnitt: \begin{figure}[!!!b]
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\begin{figure}[h]
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\begin{minipage}{.5\linewidth}
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\includegraphics{Pics/Tr_z_Eng}
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%\centering \epsfig{figure=Tr_z.Eng.eps, width=\linewidth}
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\end{minipage}
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\begin{minipage}{.5\linewidth}
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\includegraphics{Pics/Tr_sigma_Eng}
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%\centering \epsfig{figure=Tr_sigma.Eng.eps, width=\linewidth}
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\end{minipage}
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\caption{\footnotesize{Vertical profile of the restoration temperature $T_R$ as
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function of the geometric height $z$ (left) and as function of the dimensionless
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vertical coordinate $\sigma$ (right) for standard parameter values:
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$(T_R)_{grd}=288\,K$; $z_{tp}=12\,km$; $L=6.5\,K/km$; $S=2\,K$.}} \label{Tr_z}
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\end{figure}
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The temperature contrast between low and high latitudes due to the differential
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radiative energy balance, which drives the general circulation, is described by
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the meridional form of the restoration temperature:
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\begin{equation}
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\label{Tr_phi}
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T_R(\phi) = (\Delta T_R)_{NS}\, \frac{\sin\phi}{2}-(\Delta T_R)_{EP}\, \left(\sin^2\phi-\frac{1}{3}\right).
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\end{equation}
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The meridional gradient decreases with height and vanishes at the tropopause:
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\begin{equation}
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f(\sigma) = \left\{\ba{ll}
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\di\sin\left(\frac{\pi}{2}\left(\frac{\sigma-\sigma_{tp}}{1-\sigma_{tp}}\right)\right) &
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\mbox{if}\quad\sigma\ge\sigma_{tp} \\
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0 &\mbox{if}\quad\sigma<\sigma_{tp},
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\ea\right.
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\end{equation}
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with the height of the tropopause
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\begin{equation}
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\sigma_{tp} = \left(\frac{(T_R)_{tp}}{(T_R)_{grd}}\right)^{\frac{g}{LR}}.
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\end{equation}
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In equation (\ref{Tr_phi}), \dtepE represents the constant part of the meridional
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temperature contrast, and \dtns the variable part, corresponds to the annual cycle.
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Figure \ref{Tr_2d} shows the meridional and vertical form of the restoration
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temperature field (see eqn. (\ref{glgTr_2d})).
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\afterpage{
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\begin{figure}[!!!t]
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\includegraphics[width=16cm]{Pics/plot_Tr_aqua}
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%\centering \epsfig{figure=plot_Tr_aqua.eps, angle=270, width=\linewidth}
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\caption{\footnotesize{Restoration temperature field $T_R$ in $\degrees$C as function
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of latitude $\phi$ and the $\sigma$-level $l$ for standard parameter values as in
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figure \ref{Tr_z} and with $(\Delta T_R)_{EP}=70\,K$, $(\Delta T_R)_{NS}=0\,K$.}} \label{Tr_2d}
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\end{figure}
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}
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Usually, for the lower model levels, the time scale $\tau_R$ is set to smaller values
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(stronger diabatic heating) than for the upper levels in order to account for the
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stronger impact of the turbulent heat fluxes near the surface. The standard $\tau_R$ setting
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for $n_l=5$ levels is $(\tau_R)_{l=1,\ldots,3}=30\,d$, $(\tau_R)_{l=4}=10\,d$,
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$(\tau_R)_{l=5}=5\,d$.
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\subsection{Diffusion}
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\label{diffusion}
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The parameterizations (\ref{paraz}), (\ref{parad}) and (\ref{parat}) contain the
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hyperdiffusion terms $H_\zeta$, $H_D$ and $H_T$, respectively. The hyperdiffusion
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parameterizes both the subgrid scale horizontal mixing and the energy cascade into
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these scales and its subsequent dissipation, because the dissipative range of the
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wavenumber-energy-spectrum is not included with the relatively coarse model resolution.
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If $Q$ is one of the model variables $\zeta$, $D$ or $T$, then the hyperdiffusion
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is given by equation (\ref{hypergitter}) for grid point representation and by equation
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(\ref{hyperspektral}) for spectral representation (see also eqn. (\ref{spektral}))
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\beqa
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H & = & -(-1)^hK\,\nabla^{2h}\,Q(\lambda,\mu,t) \label{hypergitter} \\
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& = & -(-1)^hK\,\nabla^{2h}\sum_\gamma Q_\gamma(t)\,Y_\gamma(\lambda,\mu). \label{hyperspektral}
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\eeqa
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The hyperdiffusion for one spectral mode $\gamma$ is then \citep{holton}:
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\beqa
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H_\gamma & = & -(-1)^hK\,\nabla^{2h}\,Q_\gamma(t)\,Y_\gamma(\lambda,\mu) \\
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& = & -K\left(\frac{n(n+1)}{a^2}\right)^hQ_\gamma(t)\,Y_\gamma(\lambda,\mu). \label{hgamma.2}
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\eeqa
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With the condition that the spectral modes with $n=n_T$ are damped with a prescribed
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time scale $\tau_H$:
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\begin{equation}
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H_\gamma = -\frac{1}{\tau_H}\,Q_\gamma(t)\,Y_\gamma(\lambda,\mu)\quad\mbox{}
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\end{equation}
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${if} n=n_T,$ substitution into Equation (\ref{hgamma.2}) yields:
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\begin{equation}
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K = \frac{1}{\tau_H}\left(\frac{a^2}{n_T(n_T+1)}\right)^h.
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\end{equation}
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Thus, from Equation (\ref{hgamma.2}) it follows that:
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\begin{equation}
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H_\gamma = -\frac{1}{\tau_H}\left(\frac{n(n+1)}{n_T(n_T+1)}\right)^hQ_\gamma(t)\,Y_\gamma(\lambda,\mu). \label{hn2}
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\end{equation}
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In the model the hyperdiffusion is applied in the form (\ref{hn2}). For the shortest
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waves ($n=n_T$) the damping is maximal, for the mean ($n=0$) the damping vanishes.
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The integer exponent with the standard value $h=4$ leads to an additional reduction
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of the damping at small wavenumbers. The diffusion time scale is usually set to
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$\tau_H=1/4\,d$.
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