2379 lines
71 KiB
TeX
2379 lines
71 KiB
TeX
% set global definitions
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%
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\newcommand{\D}{\displaystyle}
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%
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\begin{center}
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{\large PUMA-II Parameterizations} \\
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\end{center}
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\begin{center}
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{\Large Very draft version} \\
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\end{center}
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\begin{center}
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{\LARGE WARNING:} \\
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\end{center}
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\begin{center}
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{\Large This is a very preliminary and unfinished
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documentation and parts of it may be
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inconsistent with the actual code}
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\end{center}
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\newpage
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%\tableofcontents
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\section{Parameterizations}
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\subsection{Surface Fluxes and Vertical Diffusion}
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\subsubsection{Surface Fluxes \label{surflux}}
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The bulk aerodynamic formulas are used to
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parameterize
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surface
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fluxes of zonal and meridional momentum (wind
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stress)
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$F_u$ and
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$F_v$,
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sensible heat $F_T$ and latent heat $L \, F_q$, where
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$F_q$ is the surface flux of moisture and $L$ is the
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latent heat of vaporisation $L_v$, or, depending on
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temperature, the latent heat of sublimation $L_s$:
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\begin{equation}\label{fluxes}
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\begin{array}{rcl}
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\D F_u & = & \D \rho \, C_m \, |\vec{v}| \, u \\
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&& \\
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\D F_v & = & \D \rho \, C_m \, |\vec{v}| \, v \\
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&& \\
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\D F_T & = & \D c_p \, \rho \, C_h \, |\vec{v}| \,
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(\gamma
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T
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- T_S ) \\
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&& \\
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\D L \, F_q & = & \D L\, \rho \, C_h \, C_w \, |\vec{v}|
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\,
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(\delta
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q - q_S )
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\end{array}
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\end{equation}
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All fluxes are positive
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in downward direction. $\rho$ denotes the density,
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$c_p$ is the specific
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heat for moist air at constant pressure ($c_p= c_{pd} \,
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[1+(c_{pv}/c_{pd}-1)\, q]$, where
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$c_{pd}$ and $c_{pv}$ are the specific heats at
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constant pressure for dry air and water vapor,
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respectively). $C_m$ is the drag
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coefficient, $C_h$ is the transfer coefficient for heat,
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$T_S$ is
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the surface temperature, $q_S$ is the surface specific
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humidity
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and $|\vec{v}|$ is the absolute value of the horizontal
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velocity at the lowermost level. The wetness factor
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$C_w$
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accounts
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for different evaporation efficiencies due to surface
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characteristics (Section \ref{hydro}). $u$, $v$,
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$T$ and $q$ are the zonal and meridional wind
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components, the
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temperature and the specific humidity, respectively,
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of the lowermost model level. The factors $\gamma$
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and ${\delta}$ are used to relate the model quantities
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to
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the respective near surface
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values. $\delta$ is set to 1 and $\gamma$ is set to
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give a potential temperature:
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\begin{equation}\label{gamma}
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\gamma = \left(\frac{p_S}{p}\right)^{\frac{R_d}{c_{pd}}}
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\end{equation}
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where p is the pressure of the lowermost
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model level, $p_S$ is the surface pressure and $R_d$
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is the gas constant for dry air.
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While $\gamma$, $\rho$,
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$C_m$, $C_h$,
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$|\vec{v}|$,
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$T_S$ and $q_S$ apply to time level $t - \Delta t$,
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values
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for $u^{t+ \Delta t}$, $v^{t+ \Delta t}$, $T^{t+ \Delta
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t}$
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and $q^{t+ \Delta t}$ are computed implicitly
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from the discretized tendency equations:
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\begin{equation}
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\begin{array}{rcccl}
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\D \frac{u^{t+\Delta t}-u^{t-\Delta t}}{2 \Delta
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t} & = & \D -
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\,
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\frac{1}{\rho \, \Delta z}\, F_u^{t+\Delta t} & = & \D
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- \,
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\frac
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{g \, \rho \, C_m \, |\vec{v}|}{p_S \, \Delta \sigma} \,
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u^{t +
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\Delta t} \\
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&&&& \\
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\D \frac{v^{t+\Delta t}-v^{t-\Delta t}}{2 \Delta t} & =
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&
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\D
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- \,
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\frac{1}{\rho \, \Delta z}\, F_v^{t+\Delta t} & = & \D
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- \,
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\frac
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{g \, \rho \, C_m \, |\vec{v}|}{p_S \, \Delta \sigma} \,
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v^{t +
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\Delta t} \\
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&&&& \\
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\D \frac{T^{t+\Delta t}-T^{t-\Delta t}}{2 \Delta t} &
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= &
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\D
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-\,
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\frac{1}{c_p \, \rho \, \Delta z} \, F_T^{t+\Delta t} &
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= & \D -
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\,
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\frac{g \, \rho \, C_h \, |\vec{v}|} {p_S \, \Delta
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\sigma} \,
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(\gamma T^{t + \Delta t} - T_S) \\
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&&&& \\
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\D \frac{q^{t+\Delta t}-q^{t-\Delta t}}{2 \Delta t} & =
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&
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\D -
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\,
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\frac{1}{\rho \, \Delta z} \, F_q^{t+\Delta t}& = & \D
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- \,
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\frac{g \, \rho \, C_h \, C_w \, |\vec{v}|} {p_S \, \Delta
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\sigma} \,
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(\delta q^{t + \Delta t} - q_S)
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\end{array}
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\end{equation}
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where $g$ is the gravitational acceleration and $\Delta
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\sigma = \Delta p/p_S $ is the thickness of the
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lowermost model layer.
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In addition to the tendencies, the surface fluxes of
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momentum, sensible and latent heat and the
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partial derivative of the sensible and the latent heat flux
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with respect to the surface temperature
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are computed:
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\begin{equation}\label{fluxes2}
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\begin{array}{rcl}
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\D F_u & = & \D \rho \, C_m \, |\vec{v}| \,
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u^{t+\Delta t} \\
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&& \\
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\D F_v & = & \D \rho \, C_m \, |\vec{v}| \, v^{t+
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\Delta t}\\
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&& \\
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\D F_T & = & \D c_p \, \rho \, C_h \, |\vec{v}| \,
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(\gamma
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T^{t + \Delta t}
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- T_S ) \\
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&& \\
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\D L \, F_q & = & \D L\, \rho \, C_h \, C_w \, |\vec{v}|
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\,
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(\delta
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q^{t+\Delta t} - q_S ) \\
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&& \\
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\D \frac{\partial F_T}{\partial T_S} & = & \D - c_p \,
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\rho \, C_h \, |\vec{v}| \\
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&& \\
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\D \frac{\partial (L \, F_q)}{\partial T_S} & = & \D -
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L\, \rho \, C_h \, C_w \, |\vec{v}| \,
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\frac{\partial q_S(T_S)}{\partial T_S}
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\end{array}
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\end{equation}
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The derivatives of the fluxes may be used, for
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examples, for an implicit calculation of the
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surface temperature (see Section \ref{surtemp}).
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\subsubsection*{ Drag and transfer coefficients}
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The calculation of the drag and the transfer
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coefficient $C_m$ and $C_h$ follows the method
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described in Roeckner et al.~(1992) for the ECHAM-3
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model, which bases on the work of Louis (1979) and
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Louis et al.~(1982). A Richardson number dependence
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of
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$C_m$ and $C_h$ in accordance to the
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Monin-Obukhov
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similarity theory is given by
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\begin{equation}
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\begin{array}{rcl}
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C_m & = & \left( \frac{k}{\ln (z/z_0)}\right)^{2} \,
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f_m
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(Ri, z/z_0) \\
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&&\\
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C_h & = & \left( \frac{k}{\ln (z/z_0)}\right)^{2} \, f_h
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(Ri, z/z_0)
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\end{array}
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\end{equation}
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where $k$ is the von Karman constant ($k$ = 0.4) and
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$z_0$ is the roughness length, which depends on the
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surface characteristics (Section~\ref{landsurf} and
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Section~\ref{seasurf}). The Richardson
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number $Ri$ is
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defined as
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\begin{equation}
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Ri=\frac{g\, \Delta z \,(\gamma_E T - \gamma_E T_S)}{\gamma T
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\, |\vec{v}|^2}
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\end{equation}
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with $\gamma$ from Eq.~\ref{gamma} and $\gamma_E$ transfers temperatures to virtual
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potential temperatures to include the effect of moisture.
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\begin{equation}\label{gammaE}
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\gamma_E = \left(1- \left(\frac{R_v}{R_d}-1\right)\, q
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\right)
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\,\left(\frac{p_S}{p}\right)^{\frac{R_d}{c_{pd}}}
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\end{equation}
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where $q$ refers to the respective specific humidities and
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$R_v$ is the gas constant for water
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vapor.
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Different empirical formulas for stable ($Ri \ge 0$)
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and
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unstable ($Ri < 0$) situations are used. For the stable
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case, $f_m$ and $f_h$ are given by
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\begin{equation}\label{fmfh1}
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\begin{array}{rcl}
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\D f_m & = &\D \frac{1}{1+(2\,b\,Ri) /\sqrt{\,1+ d\,
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Ri}}
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\\
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&& \\
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\D f_h & = &\D \frac{1}{1+(3\,b\,Ri) /\sqrt{\,1+ d\,
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Ri}}
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\end{array}
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\end{equation}
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while for the unstable case, $f_m$ and $f_h$ are
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\begin{equation}\label{fmfh2}
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\begin{array}{rcl}
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\D f_m & = & \D 1- \frac{2\,b\,Ri}{1+3\,b\,c\, [
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\frac{k}{\ln(z/z_0+1)}]^2\sqrt{-Ri\, (z/z_0+1)}} \\
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&& \\
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\D f_h & = &\D 1- \frac{3\,b\,Ri}{1+3\,b\,c\, [
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\frac{k}{\ln(z/z_0+1)}]^2\sqrt{-Ri\, (z/z_0+1)}}
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\end{array}
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\end{equation}
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where $b$, $c$, and $d$ are prescribed constants and
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set
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to
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default values of $b$ = 5, $c$ = 5 and $d$ = 5.
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\subsubsection{Vertical Diffusion}
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Vertical diffusion representing the non resolved
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turbulent exchange is applied to the horizontal wind
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components $u$ and $v$, the potential temperature
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$\theta$ ($= T (p_S/p)^{R_d/c_{pd}}$) and the
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specific
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humidity
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$q$. The
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tendencies due to the turbulent transports are given by
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\begin{equation}
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\begin{array}{rcccl}
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\D
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\frac{\partial u}{\partial t} & = & \D \frac
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{1}{\rho}\frac{\partial J_u}{\partial z} & = & \D \frac
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{1}{\rho}\frac{\partial }{\partial z} ( \rho\, K_m \,
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\frac{\partial u}{\partial z}) \\
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&&&& \\
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\D \frac{\partial v}{\partial t} & = & \D \frac
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{1}{\rho}\frac{\partial J_v}{\partial z} & = & \D \frac
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{1}{\rho}\frac{\partial }{\partial z} ( \rho \, K_m \,
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\frac{\partial v}{\partial z} )\\
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&&&& \\
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\D \frac{\partial T}{\partial t} & = &\D \frac
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{1}{\rho}\frac{\partial J_T}{\partial z} & = & \D \frac
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{1}{\rho}\frac{\partial }{\partial z} ( \rho \, K_h \,
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(\frac{p}{p_S})^{R_d/c_{pd}}\,\frac{\partial
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\theta}{\partial
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z})
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\\
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&&&& \\
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\D \frac{\partial q}{\partial t} & = & \D \frac
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{1}{\rho}\frac{\partial J_q}{\partial z} & = & \D \frac
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{1}{\rho}\frac{\partial }{\partial z}( \rho\, K_h \,
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\frac{\partial q}{\partial z} )
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\end{array}
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\end{equation}
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where p is the
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pressure, $p_S$ is the
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surface pressure, $R_d$ is the gas constant
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for dry air and $c_{pd}$ is
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the specific heat for dry air at constant pressure. Here,
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the turbulent
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fluxes (positive downward) of zonal and meridional
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momentum $J_u$ and
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$J_v$,
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heat $c_{pd} \, J_T$
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and moisture $J_q$ are parameterized by a linear
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diffusion along the vertical gradient with the exchange
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coefficients $K_m$ and $K_h$ for momentum and
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heat,
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respectively. $K_m$ and $K_h$ depend on the actual
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state (see below).
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As the effect of the surface fluxes are computed
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separately (Section \ref{surflux}), no flux boundary
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conditions for the vertical diffusion scheme are
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assumed
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at the top and the bottom of the atmosphere but the
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vertical diffusion is computed starting with
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initial values for $u$, $v$, $q$ and $T$ which include
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the tendencies due to the surface fluxes.
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As for the surface fluxes, the equations are formulated
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implicitely with exchange coefficients applying to the
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old time level. This leads to sets of linear equations for
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$u^{t+\Delta t}$, $v^{t+\Delta t}$, $T^{t+\Delta t}$
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and $q^{t+\Delta t}$, which are solved by a back
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substitution method.
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\subsubsection*{Exchange coefficients}
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The calculation of the exchange coefficient $K_m$ and
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$K_h$ follows the mixing length
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approach as an extension of the similarity theory used
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to
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define the drag and transfere
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coefficients (Section \ref{surflux} and Roeckner et
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al.~1992):
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\begin{equation}
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\begin{array}{rcl}
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\D K_m & = & \D l_m^2\, \left|
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\frac{\partial\vec{v}}{\partial
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z}
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\right| \, f_m(Ri) \\
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&&\\
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\D K_h & = & \D l_h^2\, \left|
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\frac{\partial\vec{v}}{\partial
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z}
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\right| \, f_h(Ri)
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\end{array}
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\end{equation}
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where the functional dependencies of $f_m$ and $f_h$
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on
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$Ri$ are the same as for $C_m$ and $C_h$
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(Eq.~\ref{fmfh1} and Eq.~\ref{fmfh2}), except that
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the
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term
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\begin{equation}
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\left[\frac{k}{\ln(z/z_0+1)}\right]^2\sqrt{(z/z_0+1)}
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\end{equation}
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is replaced by
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\begin{equation}
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\frac{l^2}{(\Delta z)^{3/2} \, z^{1/2}}\left[ \left(
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\frac{z+\Delta z}{z}\right)^{1/3} -1 \right]^{3/2}
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\end{equation}
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The Richardson number $Ri$ is defined as
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\begin{equation}
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Ri=\frac{g}{\gamma T} \frac{\partial (\gamma_E
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T)}{\partial z} \left| \frac{\partial \vec{v}}{\partial z}
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\right|^{-2}
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\end{equation}
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with $\gamma$ from Eq.~\ref{gamma} and $\gamma_E$ from Eq.~\ref{gammaE}. According
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to
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Blackadar (1962), the mixing lengths $l_m$ and $l_h$
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are
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given by
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\begin{equation}
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\begin{array}{rcl}
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\D \frac{1}{l_m} & = & \D \frac{1}{k\, z}
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+\frac{1}{\lambda_m} \\
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&&\\
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\D \frac{1}{l_h} & = & \D \frac{1}{k\, z}
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+\frac{1}{\lambda_h}
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\end{array}
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\end{equation}
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with $\lambda_h = \lambda_m\sqrt{(3 d)/2}$. The
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parameters $\lambda_m$ and $d$ are set to default
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values
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of $\lambda_m = 160~m$ and $d= 5$.
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\newpage
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\subsection{Horizontal Diffusion}
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The horizontal diffusion parameterization based on the
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ideas of Laursen and Eliasen (1989),
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which, in the ECHAM-3 model (Roeckner et al.~1992),
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improves the results compared with a
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${\nabla}^k$ horizontal diffusion. The diffusion is
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done in spectral space. The contribution to
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the tendency of a spectral prognostic variable $X_n$ is
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\begin{equation}
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\frac{\partial X_n}{\partial t} = -k_X L_n X_n
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\end{equation}
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where $n$ defines the total wave number. $L_n$ is a
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scale selective function of the total wave
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number and is chosen such that large scales are not
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damped while the damping gets stronger
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with increasing $n$:
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\begin{equation}
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L_n = \left\{ \begin{array}{lcl} (n-n_{\star})^{\alpha}
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& \mbox{for} & n > n_{\star} \\
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&&\\
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0 & \mbox{for} & n
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\le n_{\star} \end{array}
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\right.
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\end{equation}
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where $n_{\star}$ is a cut-off wave number. The
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parameters $n_{\star}$ and $\alpha$ are set
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to default values of $n_{\star}$~=~15 and
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$\alpha$~=~2 similar to the ECHAM-3 model in T21
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resolution (Roeckner et al.~1992). The diffusion
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coefficient $k_X$ defines the timescale of the
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damping and depends on the variable. In the model,
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$k_X$ is computed from prescribed
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damping time scales $\tau_X$ for the smallest waves.
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Default values of
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$\tau_D$~=~0.2~days for divergence,
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$\tau_{\xi}$~=~1.1~days for vorticity and
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$\tau_T$~=~15.6~days for temperature and humidity
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are chosen, which are comparable with
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the respective values in the T21 ECHAM-3 model. In
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contrast to ECHAM-3, however, no level or
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velocity dependent additional damping is applied.
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\newpage
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\subsection{Radiation}
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\subsubsection{Short Wave Radiation}
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The short wave radiation scheme bases
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on the ideas of Lacis and Hansen (1974) for the cloud
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free
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atmosphere. For the cloudy part, either constant
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albedos and
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transmissivities for high- middle- and low-level clouds
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may be prescribed or parameterizations
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following Stephens (1978) and Stephens et al.~(1984)
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may be used.
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The downward radiation flux density
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$F^{\downarrow SW}$ is assumed to be the
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product of the extrateristical solar flux density
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$E_0$ with different transmission factors for various
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processes:
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\begin{equation}
|
|
F^{\downarrow SW}= \mu_0 \, E_0 \cdot {\cal T}_R \cdot {\cal T}_O
|
|
\cdot {\cal T}_W \cdot {\cal T}_D \cdot
|
|
{\cal T}_C \cdot
|
|
{\cal R}_S
|
|
\end{equation}
|
|
|
|
Here, $\mu_0$ refers to the cosine of the solar zenith
|
|
angle and the factor ${\cal R}_S$ incorporates
|
|
different surface
|
|
albedo values. The Indices of the transmissivities ${\cal T}$
|
|
denote Rayleigh scattering ($R$), ozone
|
|
absorption ($O$), water vapor absorption ($W$) and
|
|
absorption and scattering by aerosols
|
|
(dust; $D$) and cloud droplets ($C$), respectively.
|
|
$E_0$ and $\mu_0$ are computed following
|
|
Berger (1978a, 1978b). The algorithm used is valid to
|
|
1,000,000 years past or hence. The numeric to compute
|
|
$E_0$ and
|
|
$\mu_0$ is adopted from the
|
|
CCM3 climate model (Kiehl et al.~1996, coding by E.~Kluzek 1997).
|
|
The
|
|
calculation accounts
|
|
for earths orbital parameters and the earths distance
|
|
to the sun, both depending on the year and the time
|
|
of the year.
|
|
|
|
Following, for example, Stephens (1984) the solar
|
|
spectral range
|
|
is divided into two regions: (1) A visible and
|
|
ultraviolet
|
|
part for wavelengths $\lambda < 0.75$ $\mu$m with
|
|
pure cloud
|
|
scattering, ozone absorption and
|
|
Rayleigh scattering, and without water vapor
|
|
absorption. (2) A
|
|
near infrared part for
|
|
wavelengths $\lambda > 0.75$ $\mu$m with cloud
|
|
scattering and
|
|
absorption and with water vapor absorption. Absorption
|
|
and
|
|
scattering by aerosols is neglected in the present
|
|
scheme. Dividing
|
|
the total solar energy $E_0$ into the two spectral
|
|
regions results in the
|
|
fractions ${E_1}$~=~0.517 and $E_2$~=~0.483 for
|
|
spectral
|
|
ranges 1 and 2, respectively.
|
|
|
|
\subsubsection*{Clear sky}
|
|
|
|
For the clear sky part of the atmospheric column
|
|
parameterizations following Lacis and Hansen
|
|
(1974) are used for Rayleigh scattering, ozone
|
|
absorption and water vapor absorption.
|
|
|
|
{\bf Visible and ultraviolet spectral range ($\lambda <
|
|
0.75$
|
|
$\mu$m)}
|
|
|
|
In the visible and ultraviolet range, Rayleigh
|
|
scattering and ozone absorption are considered for
|
|
the clear sky part. Rayleigh scattering is confined to
|
|
the lowermost atmospheric layer. The
|
|
transmissivity for this layer is given by
|
|
|
|
\begin{equation}
|
|
{\cal T}_{R1}=1 - \frac{0.219}{1+0.816\mu_0}
|
|
\end{equation}
|
|
|
|
for the direct beam, and
|
|
|
|
\begin{equation}
|
|
{\cal T}_{R1}=1 - 0.144
|
|
\end{equation}
|
|
|
|
for the scattered part.
|
|
|
|
Ozone absorption is considered for the Chappuis band
|
|
in the visible ${\cal A}^{vis}$ and for the
|
|
ultraviolet range ${\cal A}^{uv}$. The total transmissivity
|
|
due to ozone is given by
|
|
|
|
\begin{equation}
|
|
{\cal T}_{O1} = 1 - {\cal A}^{vis}_O - {\cal A}^{uv}_O
|
|
\end{equation}
|
|
|
|
with
|
|
|
|
\begin{equation}
|
|
{\cal A}^{vis}_O = \frac{
|
|
0.02118x}{1+0.042x+0.000323x^2}
|
|
\end{equation}
|
|
|
|
and
|
|
|
|
\begin{equation}
|
|
{\cal A}^{uv}_O=\frac{1.082x}{(1+138.6x)^{0.805}}+\frac{
|
|
0.0658x}{1+(103.6x)^3}
|
|
\end{equation}
|
|
|
|
where the ozone amount traversed by the direct solar
|
|
beam, $x$, is
|
|
|
|
\begin{equation}
|
|
x=M \; u_{O_3}
|
|
\end{equation}
|
|
|
|
with $u_{O_3}$ being the ozone amount [cm] in the
|
|
vertical column above the considered
|
|
layer, and $M$ is the magnification factor after
|
|
Rodgers (1967)
|
|
|
|
\begin{equation}
|
|
M= \frac{35}{(1224 {\mu_0}^2 +1)^{\frac{1}{2}}}
|
|
\end{equation}
|
|
|
|
The ozone path traversed by diffuse radiation from
|
|
below is
|
|
|
|
\begin{equation}
|
|
x^{*}=M \; u_{O_3}+\overline{M} \; (u_t -u_{O_3})
|
|
\end{equation}
|
|
|
|
where $u_t$ is the total ozone amount above the main
|
|
reflecting layer and $\overline{M}$=1.9
|
|
is the effective magnification factor for diffusive
|
|
upward radiation.
|
|
|
|
{\bf Near infrared ($\lambda > 0.75$ $\mu$m)}
|
|
|
|
In the near infrared solar region absorption by water
|
|
vapor
|
|
is considered only. The transmissivity is given by
|
|
|
|
\begin{equation}
|
|
{\cal T}_{W2}=1-\frac{2.9 y}{(1+141.5y)^{0.635} +
|
|
5.925y}
|
|
\end{equation}
|
|
|
|
where $y$ is the effective water vapor amount [cm]
|
|
including an approximate correction for the
|
|
pressure and temperature dependence of the absorption
|
|
and the magnification factor $M$. For
|
|
the direct solar beam, $y$ is given by
|
|
|
|
\begin{equation}
|
|
y=\frac{M}{g}
|
|
\int\limits^p_0 0.1 \; q
|
|
\left(\frac{p}{p_0}\right)\left(\frac{T_0}{T}\right)^
|
|
{\frac{1}{2}} dp
|
|
\end{equation}
|
|
|
|
while for the reflected radiation reaching the layer from
|
|
below, $y$ is
|
|
|
|
\begin{equation}
|
|
y=\frac{M}{g}
|
|
\int\limits^{p_S}_0
|
|
0.1 \; q
|
|
\left(\frac{p}{p_0}\right)\left(\frac{T_0}{T}\right)^
|
|
{\frac{1}{2}} dp
|
|
+
|
|
\frac{\beta_d}{g} \int\limits^{p_S}_{p}
|
|
0.1 \; q
|
|
\left(\frac{p}{p_0}\right)\left(\frac{T_0}{T}\right)^
|
|
{\frac{1}{2}} dp
|
|
\end{equation}
|
|
|
|
with the acceleration of gravity $g$, the surface
|
|
pressure $p_S$, a reference pressure
|
|
$p_0$~=~1000~hPa, a reference temperature
|
|
$T_0$~=~273~K, the specific humidity $q$
|
|
[kg/kg] and the magnification factor for diffuse
|
|
radiation $\beta_d$~=~1.66.
|
|
|
|
\subsubsection*{Clouds}
|
|
|
|
Two possibilities for the parameterization of the effect
|
|
of clouds on the short wave radiative fluxes are
|
|
implemented: (1) prescribed cloud properties and (2) a
|
|
parameterization following Stephens (1978) and
|
|
Stephens et al. (1984), which is the default setup.
|
|
|
|
{\bf Prescribed cloud properties}
|
|
|
|
Radiative properties of clouds are prescribed
|
|
depending on
|
|
the cloud level. Albedos ${\cal R}_{C1}$ for cloud
|
|
scattering in
|
|
the visible spectral range ($\lambda < 0.75$ $\mu$m),
|
|
and
|
|
albedos ${\cal R}_{C2}$ for cloud scattering and
|
|
absorptivities
|
|
${\cal A}_{C2}$ for cloud absorption in the near infrared
|
|
part
|
|
($\lambda > 0.75$ $\mu$m) are defined for high,
|
|
middle
|
|
and low level clouds. The default values are listed in Table \ref{tabcl1}.
|
|
|
|
{\protect
|
|
\begin{table}[h]
|
|
\begin{center}
|
|
\begin{tabular}{|c|c|c|c|}\hline
|
|
Cloud & Visible range
|
|
&\multicolumn{2}{c|}{Near
|
|
infrared} \\
|
|
Level & ${\cal R}_{C1}$ & ${\cal R}_{C2}$ &
|
|
${\cal A}_{C1}$ \\
|
|
\hline
|
|
&&& \\
|
|
High & 0.15 & 0.15 & 0.05 \\
|
|
Middle & 0.30 & 0.30 & 0.10 \\
|
|
Low & 0.60 & 0.60 & 0.20 \\
|
|
|
|
\hline
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{\label{tabcl1} Prescribed cloud albedos
|
|
${\cal R}_{C}$
|
|
and absorptivities ${\cal A}_{C}$} for spectral range 1 and 2
|
|
\end{table}
|
|
}
|
|
|
|
{\bf Default: Parameterization according to Stephens
|
|
(1978) and Stephens et al. (1984)}
|
|
|
|
Following Stephens (1978) and Stephens et al. (1984)
|
|
cloud parameters are derived from the cloud liquid
|
|
water path $W_L$ [g/m$^2$] and the cosine of the solar zenith
|
|
angel $\mu_0$. In the visible and ultraviolet range
|
|
cloud scattering is present only while in the near
|
|
infrared both, cloud scattering and absorption, are
|
|
parameterized.
|
|
|
|
|
|
{\bf Visible and ultraviolet spectral range ($\lambda <
|
|
0.75$
|
|
$\mu$m)}
|
|
|
|
For the cloud transmissivity ${\cal T}_{C1}$ Stephens
|
|
parameterization for a non absorbing medium is
|
|
applied:
|
|
|
|
\begin{equation}
|
|
{\cal T}_{C1}=1-
|
|
\frac{\beta_1\tau_{N1}/\mu_0}{1+\beta_1\tau_{N1}
|
|
/\mu_0} = \frac{1}{ 1+\beta_1 \tau_{N1}/\mu_0}
|
|
\end{equation}
|
|
|
|
$\beta_1$ is the backscatter coefficient, which is
|
|
available in tabular form. In order to avoid interpolation
|
|
of tabular values the following interpolation formula is
|
|
used
|
|
|
|
\begin{equation}
|
|
\beta_1 = f_{b1} \; \sqrt{\mu_0}
|
|
\end{equation}
|
|
|
|
where the factor $f_{b1}$ comprises a tuning
|
|
opportunity for the cloud albedo and is set to a default
|
|
value of 0.035.
|
|
|
|
$\tau_{N1}$ is an effective optical depth for which
|
|
Stephens (1979) provided the interpolation formula
|
|
|
|
\begin{equation}
|
|
\tau_{N1}= 1.8336 \; (\log{W_L})^{3.963}
|
|
\end{equation}
|
|
|
|
which is approximated by
|
|
|
|
\begin{equation}
|
|
\tau_{N1}= 2\; (\log{W_L})^{3.9}
|
|
\end{equation}
|
|
|
|
to be used also for the near infrared range (see below).
|
|
|
|
{\bf Near infrared ($\lambda > 0.75$ $\mu$m)}
|
|
|
|
The transmissivity due to scattering and absorption of
|
|
a cloud layer in the near infrared spectral range is
|
|
|
|
\begin{equation}
|
|
{\cal T}_{C2}=\frac{4u}{R}
|
|
\end{equation}
|
|
|
|
where u is given by
|
|
|
|
\begin{equation}
|
|
u^2=\frac{(1-
|
|
\tilde{\omega}_0+2\; \beta_2 \; \tilde{\omega}_0)}{(1-
|
|
\tilde{\omega}_0)}
|
|
\end{equation}
|
|
|
|
|
|
and R by
|
|
|
|
\begin{equation}
|
|
R=(u+1)^2 \exp{(\tau_{eff})}
|
|
-(u-1)^2 \exp{(-\tau_{eff})}
|
|
\end{equation}
|
|
|
|
with
|
|
|
|
\begin{equation}
|
|
\tau_{eff}=\frac{\tau_{N2}}{\mu_0}\sqrt{(1-
|
|
\tilde{\omega}_0)(1-\tilde{\omega}_0 + 2 \; \beta_2 \;
|
|
\tilde{\omega}_0)}
|
|
\end{equation}
|
|
|
|
where the original formulation for the optical depth
|
|
$\tau_{N2}$ by Stephens (1978)
|
|
|
|
\begin{equation}
|
|
\tau_{N2}=2.2346 \; (\log{W_L})^{3.8034}
|
|
\end{equation}
|
|
|
|
is, as for the visible range, approximated by
|
|
|
|
\begin{equation}
|
|
\tau_{N2}= 2 \; (\log{W_L})^{3.9}
|
|
\end{equation}
|
|
|
|
Approximations for the table values of the back
|
|
scattering coefficient $\beta_2$ and the single
|
|
scattering albedo $\tilde{\omega}_0$ are
|
|
|
|
\begin{equation}
|
|
\beta_2=\frac{f_{b2}\; \sqrt{\mu_0}}{\ln{(3+0.1\;
|
|
\tau_{N2})}}
|
|
\end{equation}
|
|
|
|
and
|
|
|
|
\begin{equation}
|
|
\tilde{\omega}_0=1-
|
|
f_{o2}\;\mu_0^2\;\ln{(1000/\tau_{N2})}
|
|
\end{equation}
|
|
|
|
where $f_{b2}$ and $f_{o2}$ provide a tuning of the
|
|
cloud
|
|
properties and are set to default values of $f_{b2}$=0.04
|
|
and $f_{o2}$=0.006.
|
|
|
|
The scattered flux is computed from the cloud albedo
|
|
${\cal R}_{C2}$ which is given by
|
|
|
|
\begin{equation}
|
|
{\cal R}_{C2}=[\exp{(\tau_{eff})}-\exp{(-\tau_{eff})}]
|
|
\; \frac{u^2-
|
|
1}{R}
|
|
\end{equation}
|
|
|
|
|
|
|
|
\subsubsection*{Vertical integration}
|
|
|
|
For the vertical integration, the adding method is used
|
|
(e.g. Lacis and Hansen 1974, Stephens 1984). The
|
|
adding method calculates the reflection ${\cal R}_{ab}$
|
|
and transmission ${\cal T}_{ab}$ functions for a
|
|
composite layer formed by combining two layers one
|
|
(layer $a$) on top of the other (layer $b$). For the
|
|
downward beam ${\cal R}_{ab}$ and ${\cal T}_{ab}$ are given by
|
|
|
|
\begin{eqnarray}\label{LH31}
|
|
{\cal R}_{ab} & = &{\cal R}_{a}+{\cal T}_{a}{\cal R}_b{\cal T}^{*}_a/(1-
|
|
{\cal R}^*_a{\cal R}_b) \nonumber \\
|
|
{\cal T}_{ab} & = &{\cal T}_a{\cal T}_b/(1-{\cal R}^*_a{\cal R}_b)
|
|
\end{eqnarray}
|
|
|
|
where the denominator accounts for multiple
|
|
reflections between the two layers. For illumination
|
|
form below ${\cal R}^*_{ab}$ and ${\cal T}^*_{ab}$ are given by
|
|
|
|
\begin{eqnarray}\label{LH32}
|
|
{\cal R}^*_{ab} & = &{\cal R}^*_b+{\cal T}^*_b{\cal R}^*_a{\cal T}_b/(1-
|
|
{\cal R}^*_a{\cal R}_b) \nonumber \\
|
|
{\cal T}^*_{ab} & = &{\cal T}^*_a{\cal T}_b/(1-{\cal R}^*_a{\cal R}_b)
|
|
\end{eqnarray}
|
|
|
|
The following four steps are carried out to obtain the
|
|
radiative upward and downward fluxes at the boundary
|
|
between two layers from which the total flux and the
|
|
absorption (heating rates) are calculated:
|
|
|
|
1) ${\cal R}_l$ and ${\cal T}_l$, $l=1, L$ are computed for each
|
|
layer and both spectral regions according to the
|
|
parameterizations.
|
|
|
|
2) The layers are added, going down, to obtain
|
|
${\cal R}_{1,l}$ and ${\cal T}_{1,l}$ for $L=2,L+1$ and
|
|
${\cal R}^*_{1,l}$ and ${\cal T}^*_{1,l}$ for $L=2,L$.
|
|
|
|
3) Layers are added one at the time, going up, to obtain
|
|
${\cal R}_{L+1-l,L+1}$, $l=1,L-1$ starting with the ground
|
|
layer, ${\cal R}_{L+1} = {\cal R}_S$ which is the surface albedo and
|
|
${\cal T}_{L+1}$=0.
|
|
|
|
4) The upward $F^{\uparrow SW}_l$ and downward
|
|
$F^{\downarrow SW}_l$ short wave radiative fluxes at
|
|
the interface of layer
|
|
($1,l$) and layer (l+1,L+1) are determined from
|
|
|
|
\begin{eqnarray}
|
|
F^{\uparrow SW}_l & = &{\cal T}_{1,l}\;{\cal R}_{l+1,L+1}/(1-
|
|
{\cal R}^*_{1,l}\;{\cal R}_{l+1,L+1}) \nonumber \\
|
|
F^{\downarrow SW}_l & = &{\cal T}_{1,l}/(1-
|
|
{\cal R}^*_{1,l}\;{\cal R}_{l+1,L+1})
|
|
\end{eqnarray}
|
|
|
|
The net downward flux at
|
|
level
|
|
$l$, $F_l^{\updownarrow SW}$, is given by
|
|
|
|
\begin{equation}
|
|
F_l^{\updownarrow SW}=F_l^{\downarrow SW}-F_l^{\uparrow SW}
|
|
\end{equation}
|
|
|
|
Finally, the temperature tendency for the layer between
|
|
$l$ and $l+1$ is computed:
|
|
|
|
\begin{equation}
|
|
\frac{\Delta T_{l+\frac{1}{2}}}{2\Delta t} = -
|
|
\frac{g}{c_p
|
|
\, p_S}\frac{F_{l+1}^{\updownarrow SW}-F_{l}^{\updownarrow SW}}{\Delta
|
|
\sigma}
|
|
\end{equation}
|
|
|
|
\newpage
|
|
|
|
\subsubsection{Long Wave Radiation}
|
|
|
|
{\bf Clear sky}
|
|
|
|
For the clear sky long wave radiation, the broad band
|
|
emissivity method is employed (see, for example,
|
|
Manabe and M\"oller 1961, Rodgers 1967, Sasamori
|
|
1968, Katayama 1972, Boer et al. 1984).
|
|
Using the broad band transmissivities
|
|
${\cal T}_{(z,z^{\prime})}$ between level $z$ and level
|
|
$z^{\prime}$, the upward and downward fluxes at
|
|
level
|
|
$z$, $F^{\uparrow LW}(z)$ and
|
|
$F^{\downarrow LW}(z)$, are
|
|
|
|
\begin{equation}
|
|
\begin{array}{rcl}
|
|
\D F^{\uparrow LW}(z) & = &\D {\cal A}_S \, B(T_S)
|
|
{\cal T}_{(z,0)} +
|
|
\int\limits_0^z B(T^{\prime}) \frac{\partial
|
|
{\cal T}_{(z,z^{\prime})}}{\partial z^{\prime}} d z^{\prime}
|
|
\\
|
|
& & \\
|
|
\D F^{\downarrow LW}(z) & = & \D \int\limits_{\infty}^z
|
|
B(T^{\prime}) \frac{\partial
|
|
{\cal T}_{(z,z^{\prime})}}{\partial z^{\prime}} d z^{\prime}
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
|
|
where $B(T)$ denotes the black body flux ($ B(T) = \sigma_{SB}
|
|
T^4$) and
|
|
${\cal A}_S$ is the surface emissivity. The effect
|
|
of water vapor, carbon dioxide and ozone is included in the
|
|
calculations of the transmissivities ${\cal T}$
|
|
(with ${\cal T} = 1 - {\cal A}$, where ${\cal A}$ is the
|
|
absoroptivity/emissivity). The transmissivities for water vapor
|
|
${\cal T}_{H_2O}$, carbon dioxide ${\cal T}_{CO_2}$ and
|
|
ozone ${\cal T}_{O_3}$ are taken from Sasamori (1968):
|
|
|
|
\begin{equation}\label{taus}
|
|
\begin{array}{lcl}
|
|
\D {\cal T}_{H_2O}& = & \D 1-0.846\;(u_{H_2O}+3.59 \cdot 10^{-5})^{0.243}
|
|
-6.90\cdot 10^{-2} \\
|
|
&& \\
|
|
\multicolumn{3}{l}{\mbox{for } u_{H_2O} < 0.01\mbox{ g, and }}\\
|
|
&& \\
|
|
\D {\cal T}_{H_2O}& = & \D 1-0.240\log{(u_{H_2O}+0.010)}+0.622 \\
|
|
&& \\
|
|
\multicolumn{3}{l}{\mbox{else.}}\\
|
|
&&\\
|
|
&&\\
|
|
\D {\cal T}_{CO_2}& = & \D 1-0.0825\;u_{CO_2}^{0.456}\\
|
|
&& \\
|
|
\multicolumn{3}{l}{\mbox{for } u_{CO_2} \le 0.5 \mbox{ cm, and }}\\
|
|
&& \\
|
|
\D {\cal T}_{CO_2}& = & \D 1-0.0461\log{(u_{CO_2})}+0.074 \\
|
|
&& \\
|
|
\multicolumn{3}{l}{\mbox{else.}}\\
|
|
&&\\
|
|
&&\\
|
|
\D {\cal T}_{O_3} & = & \D 1-0.0122\log{(u_{O_3}+6.5 \cdot 10^{-4})}+0.0385
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
where $u_{H_2O}$, $u_{CO_2}$ and $u_{O_3}$ are the effective
|
|
amounts of water vapor, carbon
|
|
dioxide and ozone, respectively, which are obtained from:
|
|
|
|
\begin{equation}
|
|
u(p, p^{\prime}) =
|
|
\frac{f}{g}
|
|
\int\limits_{p}^{p^{\prime}} q_X
|
|
\left(\frac{p^{\prime\prime}}{p_0}\right)
|
|
dp^{\prime\prime}
|
|
\end{equation}
|
|
|
|
where $q_X$ denotes the mixing
|
|
ratios [kg/kg] of water vapor, carbon dioxide and ozone,
|
|
respectively, $g$ is the gravitational acceleration, $p$
|
|
is pressure and $p_0$~=~1000 hPa is the reference
|
|
pressure. The factor $f$ is used to transfer the units to g/cm$^2$
|
|
for $u_{H_2O}$ and cm-STP for
|
|
$u_{CO_2}$ and cm-STP for $u_{O_3}$, which are used in Eq.~\ref{taus}.
|
|
|
|
To account for the overlap between
|
|
the water vapor and the carbon dioxide bands near
|
|
15~$\mu$m, the CO$_2$ absorption is
|
|
corrected by a H$_2$O transmission at 15~$\mu$m,
|
|
${\cal T}_{H_2O}^{15\mu m}$, with
|
|
${\cal T}_{H_2O}^{15\mu
|
|
m}$ given by
|
|
|
|
\begin{equation}
|
|
{\cal T}_{H_2O}^{15\mu m} = 1.33-0.832 \, (u_{H_2O}
|
|
+
|
|
0.0286)^{0.26}
|
|
\end{equation}
|
|
|
|
{\bf Clouds}
|
|
|
|
Clouds can be either treated as gray bodies with a prescribed cloud flux emissivity (grayness) or
|
|
the cloud flux emissivity is obtained from the cloud liquid water contend. If the cloud flux
|
|
emissivity (grayness) ${\cal A}^{cl}$ is externally prescribed, the value is
|
|
attributed to each cloud layer. Otherwise, which is the default, ${\cal A}^{cl}$ is calculated
|
|
from the cloud liquid water (e.g. Stephens 1984)
|
|
|
|
\begin{equation}
|
|
{\cal A}^{cl}=1.-\exp{(-\beta_d \; k^{cl} \; W_L)}
|
|
\end{equation}
|
|
|
|
where $\beta_d$~=~1.66 is the diffusivity factor, $k^{cl}$ is the mass absorption coefficent
|
|
(with
|
|
is set to a default value of 0.1~m$^2$/g (Slingo and Slingo 1991)) and $W_L$ is the
|
|
cloud liquid water path.
|
|
|
|
For a single layer
|
|
between $z$ and $z^{\prime}$ with fractional cloud
|
|
cover
|
|
$cc$, the total transmissivity ${\cal T}^*_{(z, z^{\prime})}$
|
|
is
|
|
given by
|
|
|
|
|
|
\begin{equation}
|
|
{\cal T}^*_{(z, z^{\prime})} = {\cal T}_{(z, z^{\prime})} \, (1 - cc \,
|
|
{\cal A}^{cl})
|
|
\end{equation}
|
|
|
|
where ${\cal T}_{(z, z^{\prime})}$ is the clear sky
|
|
transmissivity. When there is more than one cloud
|
|
layer
|
|
with fractional cover, random overlapping of the
|
|
clouds is
|
|
assumed and ${\cal T}^*_{(z, z^{\prime})}$ becomes
|
|
|
|
\begin{equation}
|
|
{\cal T}^*_{(z, z^{\prime})} ={\cal T}_{(z, z^{\prime})} \, \prod_j (1
|
|
- cc_j \, {\cal A}^{cl}_{j})
|
|
\end{equation}
|
|
|
|
where the subscript $j$ denotes the cloud layers.
|
|
|
|
\subsubsection*{Vertical discretization}
|
|
|
|
To compute the temperature tendency for a model
|
|
layer resulting form the divergence of the radiative
|
|
fluxes, the vertical discretization scheme of Chou et al. (2002) is used. The upward and
|
|
downward fluxes, $F_l^{\uparrow LW}$ and $F_l^{\downarrow LW}$, at
|
|
level $l$, which is the interface between two model
|
|
layers, are computed from
|
|
|
|
\begin{equation}\label{rad1}
|
|
\begin{array}{rcllcl}
|
|
\D F_l^{\uparrow LW}& = &\D \sum\limits_{l^\prime =
|
|
l}^{L} B_{l^\prime+\frac{1}{2}} [{\cal T}^*_{(l ,
|
|
l^\prime)}-
|
|
{\cal T}^*_{(l^\prime+1,l)}] & & \D \;\; l=1, \cdots , L\\
|
|
&&&&\\
|
|
&& \D +{\cal T}^*_{(l ,L+1)} \; F_{L+1}^{\uparrow LW} & &\\
|
|
& & & &\\
|
|
&&&&\\
|
|
\D F_l^{\downarrow LW}& = &\D
|
|
\sum\limits_{l^\prime=1}^{l-
|
|
1} B_{l^\prime+\frac{1}{2}} [{\cal T}^*_{(l^\prime+1,l)}-{\cal T}^*_{(l^\prime,l)}] & & \D
|
|
\;\; l=2,
|
|
\cdots , L+1
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
where ${\cal T}^*_{(l,l^{\prime})}$ denotes the
|
|
transmissivity of
|
|
the layer from level $l$ to level $ l^{\prime}$ (see
|
|
above)
|
|
and $B_{l+\frac{1}{2}}$ is the black body flux for
|
|
level
|
|
$l+\frac{1}{2}$. The downward flux at the top of the atmosphere, $F_0^{\downarrow
|
|
LW}$, and the upward flux at the surface ,$F_{L+1}^{\uparrow LW}$, are given by
|
|
|
|
\begin{equation}
|
|
\begin{array}{rcl}
|
|
\D F_0^{\downarrow LW} & = & \D 0 \\
|
|
&&\\
|
|
\D F_{L+1}^{\uparrow LW} & = & \D {\cal A}_S \; B(T_S) + (1-{\cal A}_S) \;
|
|
F_{L+1}^{\downarrow LW}
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
where ${\cal A}_S$ denotes the surface emissivity and $T_S$ is the surface temperature. Note,
|
|
that for a more convenient discription of the scheme,
|
|
$l+\frac{1}{2}$
|
|
denotes a so called full level, where the temperatures
|
|
are
|
|
defined. This may be in contrast to the convention in
|
|
most of the other sections where a full
|
|
level is indicated by $l$.
|
|
|
|
Eqs.~\ref{rad1} can be rearranged to give
|
|
|
|
\begin{equation}\label{vertical1}
|
|
\begin{array}{rclcl}
|
|
\D F_l^{\uparrow LW}& = &\D B_{l+\frac{1}{2}} +
|
|
\sum\limits_{l^\prime=l+1}^{L+1} {\cal T}^*_{(l^\prime ,
|
|
l)} \,
|
|
[B_{l^\prime+\frac{1}{2}} - B_{l^\prime-
|
|
\frac{1}{2}}] & & \D \;\; l=1, \cdots , L\\
|
|
&&&&\\
|
|
& & + {\cal T}^*_{(l,L+1)} \, (1-{\cal A}_S) \; F_{L+1}^{\downarrow LW} & & \\
|
|
& & & & \\
|
|
&&&&\\
|
|
\D F_{l^\prime}^{\downarrow LW}& = & \D
|
|
B_{l^\prime-
|
|
\frac{1}{2}} - \sum\limits_{l=1}^{l^\prime-1}{\cal T}^*_{(l^\prime , l)} \,
|
|
[B_{l+\frac{1}{2}} -
|
|
B_{l-\frac{1}{2}}]
|
|
& & \D \;\; l^\prime=2, \cdots , L+1
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
with the boundary conditions
|
|
|
|
\begin{equation}
|
|
\begin{array}{rcl}
|
|
\D B_{L+\frac{3}{2}}& = & \D {\cal A}_S \,
|
|
B(T_S) \\
|
|
&&\\
|
|
\D B_{\frac{1}{2}} & = & \D 0
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
The net downward flux at
|
|
level
|
|
$l$, $F_l^{\updownarrow LW}$, is given by
|
|
|
|
\begin{equation}
|
|
F_l^{\updownarrow LW}=F_l^{\downarrow LW}-F_l^{\uparrow LW}
|
|
\end{equation}
|
|
|
|
Finally, the temperature tendency for the layer between
|
|
$l$ and $l+1$ is computed:
|
|
|
|
\begin{equation}
|
|
\frac{\Delta T_{l+\frac{1}{2}}}{2\Delta t} = -
|
|
\frac{g}{c_p
|
|
\, p_S}\frac{F_{l+1}^{\updownarrow LW}-F_{l}^{\updownarrow LW}}{\Delta
|
|
\sigma}
|
|
\end{equation}
|
|
|
|
{\bf Emission of a layer}
|
|
|
|
As pointed out by Chou et al.~(2002), the difference between the upward and downward
|
|
emission of a layer will be large, if the layer is rather opaque and the temperature range across
|
|
the layer is large. This, in particular, holds for coarse vertical resolution as in the default version
|
|
of the model. Therefore, the upward and the downward emission of a layer is computed
|
|
separately following the ideas of Chou et al.~(2002):
|
|
|
|
The contribution of the upward flux at level $p$ from the adjecant layer below can be written as
|
|
|
|
\begin{equation} \label{FUPLW}
|
|
\Delta F^{\uparrow LW}(p) = -\int\limits^{p+\Delta p}_{p} B(p^{\prime}) \; \frac{\partial
|
|
{\cal T}_{(p,p^{\prime})}}{\partial p^{\prime}} \; dp^{\prime} = B^u \; (1-{\cal T}_{
|
|
(p+\Delta p, p)})
|
|
\end{equation}
|
|
|
|
where $\Delta p$ is the thickness of the adjacent layer, $B^u$ is the effective Planck flux for the
|
|
adjacent layer, and ${\cal T}_{(p+\Delta p, p)}$ is the flux transmittance between $p$ and $p
|
|
+\Delta p$. Assuming that the Planck function varies linearly with pressure and the
|
|
transmittance decreases exponentially with pressure away from $p$ it follows
|
|
|
|
\begin{equation}
|
|
B(p^{\prime})= B(p) + \frac{(B(p)-B(p+\Delta p)) (p^{\prime} - p)}{\Delta p}
|
|
\end{equation}
|
|
|
|
and
|
|
|
|
\begin{equation}
|
|
{\cal T}_{(p, p^{\prime})} = \exp{(-c\; (p^{\prime}-p))}
|
|
\end{equation}
|
|
|
|
with $c$ ia a constant. From Eq.~\ref{FUPLW} the effective Planck flux for the adjacent layer
|
|
$B^u$ is
|
|
|
|
\begin{equation}
|
|
B^u=\frac{B(p)-B(p+\Delta p)\;{\cal T}_{(p+\Delta p, p)}}{1-{\cal T}_{(p+\Delta p, p)}}
|
|
+\frac{B(p)-
|
|
B(p+\Delta p)}{\ln({\cal T}_{(p+\Delta p, p)})}
|
|
\end{equation}
|
|
|
|
Similarly, for the downward flux at the lower boundary of the layer, the effective Planck
|
|
function of the layer $B^d$ is
|
|
|
|
\begin{equation}
|
|
B^d=\frac{B(p+\Delta p)-B(p)\;{\cal T}_{(p+\Delta p, p)}}{1-{\cal T}_{(p+\Delta p, p)}}
|
|
+\frac{B(p+\Delta
|
|
p)-B(p)}{\ln({\cal T}_{(p+\Delta p, p)})}
|
|
\end{equation}
|
|
|
|
Replacing the respective Planck functions in Eqs.~\ref{vertical1} by $B^u$ and $B^d$ results
|
|
in
|
|
|
|
\begin{equation}\label{vertical2}
|
|
\begin{array}{rclcl}
|
|
\D F_l^{\uparrow LW}& = &\D B^u_{l+\frac{1}{2}} +
|
|
\sum\limits_{l^\prime=l+1}^{L+1} {\cal T}^*_{(l^\prime ,
|
|
l)} \,
|
|
[B^u_{l^\prime+\frac{1}{2}} - B^u_{l^\prime-
|
|
\frac{1}{2}}] & & \D \;\; l=1, \cdots , L\\
|
|
&&&&\\
|
|
& & \D + {\cal T}^*_{(l,L+1)} \, (1-{\cal A}_S) \; F_{L+1}^{\downarrow LW} & & \\
|
|
& & & & \\
|
|
&&&&\\
|
|
\D F_{l^\prime}^{\downarrow LW}& = & \D
|
|
B^d_{l^\prime-
|
|
\frac{1}{2}} - \sum\limits_{l=1}^{l^\prime-1} {\cal T}^*_{(l^\prime , l)} \,
|
|
[B^d_{l+\frac{1}{2}}
|
|
-
|
|
B^d_{l-\frac{1}{2}}]
|
|
& & \D \;\; l^\prime=2, \cdots , L+1
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
where
|
|
|
|
\begin{equation}
|
|
\begin{array}{lcl}
|
|
\D B^d_{l^{\prime}-\frac{1}{2}} & = & \D \frac{B_{l^{\prime}}-B_{l^{\prime}-1} \;
|
|
{\cal T}_{(l^{\prime},l^{\prime}-1)}}{1-{\cal T}_{(l^{\prime},l^{\prime}-1)}} +
|
|
\frac{B_{l^{\prime}} - B_{l^{\prime}-1}}{\ln({\cal T}_{(l^{\prime},l^{\prime}-1)})} \\
|
|
&& \\
|
|
\D B^u_{l^{\prime}-\frac{1}{2}} & = & \D (B_{l^{\prime}} + B_{l^{\prime}-1} ) -
|
|
B^d_{l^{\prime}-\frac{1}{2}}
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
For the calculation of the effective Plank function, the mean transmissivity for a layer partially
|
|
filled with clouds is given by
|
|
|
|
\begin{equation}
|
|
{\cal T}_{(l^{\prime},l^{\prime}-1)} = f_{{\cal T}} \; {\cal
|
|
T}^{cs}_{(l^{\prime},l^{\prime}-1)} \; (1 -
|
|
cc_{(l^{\prime},l^{\prime}-1)}{\cal A}^{cl}_{(l^{\prime},l^{\prime}-1)})
|
|
\end{equation}
|
|
|
|
|
|
with the cloud emissivity ${\cal A}^{cl}$ and the clear sky transmissivity ${\cal T}^{cs}$
|
|
being defined above, and the factor $f_{{\cal T}}$ provides a tuning opportunity.
|
|
|
|
When a model layer spans a region where the temperature lapse rate changes signs, the linearity
|
|
of $B$ with respect to $p$ can not longer be assumed and $B^d$ and $B^u$ are simply
|
|
computed from
|
|
|
|
\begin{equation}
|
|
B^u_{l+\frac{1}{2}}=B^d_{l-\frac{1}{2}}= 0.5 \; B_{l+\frac{1}{2}} + 0.25 \; (B_{l} +
|
|
B_{l^{\prime}})
|
|
\end{equation}
|
|
|
|
|
|
\subsubsection{Ozone}
|
|
|
|
Ozone concentration is prescribed. Either a three dimensional ozone distribution can be
|
|
externally provided or an idealized annual cycle of ozone concentration can be used. The
|
|
idealized distribution bases on the analytic ozone distribution of Green (1964):
|
|
|
|
\begin{equation}
|
|
u_{O_3}(h)=\frac{a+a \; \exp{(-b/c)}}{1+\exp((h-b)/c)}
|
|
\end{equation}
|
|
|
|
where $u_{O_3}(h)$ is the ozone amount [cm-STP] in a vertical column above the altitude $h$,
|
|
$a$ is the total ozone amount in a vertical column above the ground, $b$ the altitude at which
|
|
the ozone concentration has its maximum. While for $a$~=~0.4~cm, $b$~=~20~km and
|
|
$c$~=~5~km
|
|
this distribution fits close to the mid-latitude winter ozone distribution, an annual cycle and a
|
|
latitudinal dependence is introduced by varying $a$ with time and latitude.
|
|
|
|
\subsubsection{Additional Newtonian cooling}
|
|
|
|
For the standard setup with a vertical resolution of five equally spaced sigma-levels, the model
|
|
produces a strong bias in the stratospheric (uppermost level) temperatures. This may be
|
|
attributed to the insufficient representation of the stratosphere and its radiative and dynamical
|
|
processes. The bias also effects the tropospheric circulation leading, for example, to a
|
|
misplacement of the dominant pressure centers. To enable the simulation of a more realistic
|
|
tropospheric climate, a Newtonian cooling can be applied to the uppermost level. Using this
|
|
method, the model temperature $T$ is relaxed towards a externally given distribution of the
|
|
temperature $T_{NC}$ which results in additional temperature tendencies $\dot{T}$ for the
|
|
uppermost model level of
|
|
|
|
\begin{equation}
|
|
\dot{T}=\frac{T_{NC}-T}{\tau_{NC}}
|
|
\end{equation}
|
|
|
|
where $\tau_{NC}$ is the time scale of the relaxation, which has a default value of ten days.
|
|
|
|
\newpage
|
|
|
|
\subsection{Moist Processes and Dry Convection}
|
|
|
|
\subsubsection{Correction of Negative Humidity}
|
|
|
|
Local negative values of specific humidity are an
|
|
artifact of spectral models. In the model, a simple
|
|
procedure corrects these negative values by
|
|
conserving the global amount of water. The correction of negative moisture is performed at the
|
|
beginning of the grid-point
|
|
parameterization scheme. A negative
|
|
value of specific humidity is reset to zero.
|
|
Accumulation of all corrections defines a correction
|
|
factor. A hierarchical scheme of three steps is used. First, the correction is done within an
|
|
atmospheric column only. If there are atmospheric columns without sufficient moisture, a
|
|
second correction step is done using all grid points of the respective latitude. Finally, if there is
|
|
still negative humidity remaining, a global correction is performed.
|
|
|
|
\subsubsection{Saturation Specific Humidity}
|
|
|
|
For parameterizations of moist processes like cumulus
|
|
convection and large scale condensation
|
|
the computation of the saturation specific humidity
|
|
$q_{sat}(T)$ and its derivative with respect
|
|
to temperature $dq_{sat}(T)/dT$ is needed at several
|
|
places. In
|
|
the model, the Tetens formula (Lowe 1977) is used to
|
|
calculate the saturation pressure
|
|
$e_{sat} (T)$ and its derivative with respect to
|
|
temperature $de_{sat}(T)/dT$:
|
|
|
|
\begin{equation}
|
|
\begin{array}{rcl}
|
|
\D e_{sat}(T) & = & \D a_1 \exp{\left(a_2 \,
|
|
\frac{T-T_0}{T-a_3}\right)} \\
|
|
&& \\
|
|
\D \frac{de_{sat}(T)}{dT} & = & \D \frac{a_2 \, (T_0
|
|
- a_3)}{(T-a_3)^2} \, e_{sat}(T)
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
with the constants $a_1$~=~610.78,
|
|
$a_2$~=~17.2693882, $a_3$~=~35.86 and
|
|
$T_0$~=~273.16. The
|
|
saturation specific humidity $q_{sat}(T)$ and its
|
|
derivative $dq_{sat}(T)/dT$ are given by
|
|
|
|
\begin{equation}\label{qsat}
|
|
\begin{array}{rcl}
|
|
\D q_{sat}(T) & = & \D \frac{\epsilon \,
|
|
e_{sat}(T)}{p-(1-\epsilon )\, e_{sat}
|
|
(T)} \\
|
|
&&\\
|
|
\D \frac{dq_{sat}(T)}{dT} & = & \D \frac{p \,
|
|
q_{sat}(T)}{p-(1-\epsilon)\, e_{sat}
|
|
(T)} \frac{de_{sat}(T)}{dT}\\
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
where $p$ is the pressure and $\epsilon$ is the ration
|
|
of the gas constants
|
|
for dry air $R_d$ and water vapor $R_v$ ($\epsilon =
|
|
R_d / R_v$).
|
|
|
|
|
|
\subsubsection{Cumulus Convection}
|
|
|
|
The cumulus convection is parameterized by a
|
|
Kuo-type convection scheme (Kuo 1965, 1974)
|
|
with some modifications to the original Kuo-scheme.
|
|
The Kuo-scheme considers the effect of
|
|
cumulus convection on the large scale flow applying
|
|
the following assumptions. Cumulus
|
|
clouds are forced by mean low level convergence in
|
|
regions of conditionally unstable
|
|
stratification. The production of cloud air is
|
|
proportional to the net amount of moisture
|
|
convergence into one grid box column plus the
|
|
moisture supply by surface evaporation. In a
|
|
modification to the original scheme, the implemented
|
|
scheme also considers clouds which
|
|
originate at upper levels where moisture convergence
|
|
is observed. This type of cloud may occur
|
|
in mid-latitude frontal regions. Therefore, only the
|
|
moisture contribution which takes place in
|
|
the layer between the lifting level and the top of the
|
|
cloud is used instead of the whole column.
|
|
Thus, the total moisture supply $I$ in a period $2
|
|
\Delta t$ is given by
|
|
|
|
\begin{equation}\label{cli}
|
|
I= \frac{2 \Delta t \, p_S}{g}
|
|
\int\limits_{\sigma_{Top}}^{\sigma_{Lift}} A_q \, d
|
|
\sigma
|
|
\end{equation}
|
|
|
|
where $A_q$ is the moisture convergence plus the
|
|
surface evaporation if the lifting level
|
|
$\sigma_{Lift}$ is the lowermost model level.
|
|
$\sigma_{Top}$ is the cloud top level, $p_S$ is
|
|
the surface pressure and $g$ is the gravitational
|
|
acceleration. Lifting level, cloud base and cloud
|
|
top are determined as follows. Starting form the
|
|
lowermost level, the first level with positive
|
|
moisture supply $A_q$ is considered as a lifting level. If the lowermost level $L$ is considered
|
|
to be a lifting level and the surface layer is dry adiabatic unstable ($\theta_S > \theta_L$
|
|
where $\theta$ denotes the potential temperature), the convection starts from the surface.
|
|
Air from the lifting level ($l+1$) is lifted dry
|
|
adiabatically up to the next level ($l$) by keeping its
|
|
specific humidity. A cloud base is
|
|
assumed to coincide with level $l+\frac{1}{2}$ if the
|
|
air is saturated at $l$. Above the cloud
|
|
base the air is lifted moist adiabatically. Distribution of
|
|
temperature $T_{cl}$ and of moisture
|
|
$q_{cl}$ in the cloud is found by first lifting the air
|
|
dry adiabatically
|
|
|
|
\begin{equation}\label{clad}
|
|
\begin{array}{rcl}
|
|
\D (T_{cl})_l^{Ad} & = & \D (T_{cl})_{l+1}
|
|
\left(\frac{\sigma_l}{\sigma_{l+1}}\right)^{\frac{R_
|
|
d}{c_{pd}}} \\
|
|
&&\\
|
|
\D (q_{cl})_l^{Ad} & = & \D (q_{cl})_{l+1}
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
and then by correcting temperature and moisture values
|
|
due to the condensation of water vapor
|
|
|
|
\begin{equation}\label{ccc1}
|
|
\begin{array}{rcl}
|
|
\D (T_{cl})_l & = & \D (T_{cl})_l^{Ad} +
|
|
\frac{L}{c_p} \, \frac{(q_{cl})_l^{Ad} - q_{sat}
|
|
[(T_{cl})_l^{Ad}]}{1+\frac{L}{c_p}\,
|
|
\frac{dq_{sat}[(T_{cl})_l^{Ad}]}{dT}} \\
|
|
&&\\
|
|
\D (q_{cl})_l & = & \D (q_{cl})_l^{Ad}
|
|
-\frac{(q_{cl})_l^{Ad} - q_{sat}[(T_
|
|
{cl})_l^{Ad}]}{1+\frac{L}{c_p}\,
|
|
\frac{dq_{sat}[(T_{cl})_l^{Ad}]}{dT}}
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
where the suturation specific humidity $q_{sat}$ and
|
|
its derivative with respect to temperature
|
|
$dq_{sat}/dT$ are computed from Eqs.~\ref{qsat}.
|
|
$L$ is
|
|
either the latent heat of vapourisation $L_v$ or
|
|
the latent heat of sublimation $L_s$ depending on the
|
|
temperature.
|
|
$c_p$ is the specific
|
|
heat for moist air at constant pressure ($c_p= c_{pd} \,
|
|
[1+(c_{pv}/c_{pd}-1)\, q]$ where
|
|
$c_{pd}$ and $c_{pv}$ are the specific heats at
|
|
constant pressure for dry air and water vapor,
|
|
respectively) and $R_d$ in Eq.~\ref{clad} is the gas
|
|
constant for dry air.
|
|
For reasons of accuracy the calculation (\ref{ccc1}) is
|
|
repeated once where $(T_{cl})^{Ad}$
|
|
and $(q_{cl})^{Ad}$ are now replaced by the results
|
|
of the first iteration.
|
|
|
|
Cumulus clouds are assumed to exist only if the
|
|
environmental air with temperature $T_e$ and
|
|
moisture $q_e$ is unstable stratified with regard to the
|
|
rising cloud parcel:
|
|
|
|
\begin{equation}
|
|
(T_{cl})_l > (T_e)_l
|
|
\end{equation}
|
|
|
|
The top of the cloud $\sigma_{Top}$ is then defined
|
|
as
|
|
|
|
\begin{equation}
|
|
\sigma_{Top}=\sigma_{l+\frac{1}{2}} \; \mbox{if }
|
|
\left\{ \begin{array}{lcll} (T_{cl})_l &
|
|
\le &
|
|
(T_{e})_l & \mbox{and} \\ &&& \\ (T_{cl})_{l+1} &
|
|
> & (T_{e})_{l+1} & \end{array}
|
|
\right.
|
|
\end{equation}
|
|
|
|
Cumulus clouds do exist only if the net moisture
|
|
accession $I$ as given by Eq.~\ref{cli} is
|
|
positive.
|
|
Once this final check has been done, the heating and
|
|
moistening of the environmental air and
|
|
the
|
|
convective rain are computed.
|
|
|
|
In the model either the original scheme proposed by
|
|
Kuo (1968) or the modified scheme with
|
|
the parameter $\beta$ (Kuo 1974) can be chosen,
|
|
where $\beta$ determines the partitioning of
|
|
heating and moistening of the environmental air. In the
|
|
scheme without $\beta$ the surplus $P$
|
|
of total energy of the cloud against the environmental
|
|
air is given by
|
|
|
|
\begin{equation}
|
|
P=\frac{p_s}{g}
|
|
\int\limits_{\sigma_{Top}}^{\sigma_{Base}} (c_p\,
|
|
(T_{cl} -T_{e}) + L\,
|
|
(q_{sat}(T_e)-q_{e})) d\sigma
|
|
\end{equation}
|
|
|
|
|
|
The clouds produced dissolve instantaneously by
|
|
artificial mixing with the environmental air,
|
|
whereby the environment is heated and moistened by
|
|
|
|
\begin{equation}\label{handm}
|
|
\begin{array}{rcl}
|
|
\D (\Delta T)^{cl} & = & \D a \, (T_{cl} -T _e) \\
|
|
&&\\
|
|
\D (\Delta q)^{cl} & = & \D a \, (q_{sat}(T_e) -q _e)
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
where $a$ is the fractional cloud area being produced
|
|
by the moisture supply:
|
|
|
|
\begin{equation}
|
|
a=L\, \frac{I}{P}
|
|
\end{equation}
|
|
|
|
|
|
In the scheme with $\beta$ the fraction 1-$\beta$ of the
|
|
moisture is condensed, while the
|
|
remaining fraction $\beta$ is stored in the atmosphere.
|
|
The parameter $\beta$ depends on the
|
|
mean relative humidity and, in the present scheme, is
|
|
given by
|
|
|
|
\begin{equation}
|
|
\beta = \left( 1 -
|
|
\frac{1}{\sigma_{Base}-\sigma_{Top}}
|
|
\int\limits_{\sigma_{Top}}^{\sigma_{Base}}
|
|
\frac{q_e}{q_{sat}(T_e)} d\sigma \right)^3
|
|
\end{equation}
|
|
|
|
Instead of Eq.~\ref{handm}, the temperature and
|
|
moisture tendencies are now
|
|
|
|
\begin{equation}\label{handm2}
|
|
\begin{array}{rcl}
|
|
\D (\Delta T)^{cl} & = & \D a_T \, (T_{cl} -T _e) \\
|
|
&&\\
|
|
\D (\Delta q)^{cl} & = & \D a_q \, (q_{sat}(T_e) -q
|
|
_e)
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
where $a_T$ and $a_q$ are given by
|
|
|
|
\begin{equation}
|
|
\begin{array}{rcl}
|
|
\D a_T & = & \D \frac{(1-\beta )\, L\, I }{c_p\,
|
|
\frac{p_S}{g}\,
|
|
\int\limits_{\sigma_{Top}}^{\sigma_{Base}} (T_{cl}
|
|
- T_e)\, d\sigma} \\
|
|
&&\\
|
|
\D a_q & = & \D \frac{\beta \, I }{\frac{p_S}{g}\,
|
|
\int\limits_{\sigma_{Top}}^{\sigma_{Base}}
|
|
(q_{sat}(T_e) - q_e) \, d\sigma}
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
The final tendencies for moisture $\partial q / \partial
|
|
t$ and temperature $\partial T / \partial t$
|
|
which enter the diabatic leap frog time step are given
|
|
by
|
|
|
|
\begin{equation}
|
|
\begin{array}{rcl}
|
|
\D \frac{\partial q}{\partial t} & = & \D \frac{(\Delta
|
|
q)^{cl}}{2 \Delta t} - {\delta}^{cl} A_q
|
|
\\
|
|
&& \\
|
|
\D \frac{\partial T}{\partial t} & = & \D \frac{(\Delta
|
|
T)^{cl}}{2 \Delta t}
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
where ${\delta}^{cl}$ is specified by
|
|
|
|
\begin{equation}
|
|
{\delta}^{cl} = \left\{ \begin{array}{ll} 1 & \mbox{if
|
|
} \; \sigma_{Top} \le \sigma \le
|
|
\sigma_{Lift} \\ & \\ 0 & \mbox{otherwise}
|
|
\end{array} \right.
|
|
\end{equation}
|
|
|
|
and $2\Delta t$ is the leap frog time step of the model.
|
|
The convective precipitation rate
|
|
$P_{c}$ [m/s] of each cloud layer is
|
|
|
|
\begin{equation}
|
|
P_{c} = \frac{c_p\, \Delta p}{L\, g \, \rho_{H_2O}}
|
|
\frac{(\Delta T)^{cl}}{2\Delta t}
|
|
\end{equation}
|
|
|
|
where $\Delta p$ is the pressure thickness of the layer
|
|
and $\rho_{H_2O}$ is the density of
|
|
water. $(\Delta T)^{cl}$ is computed from
|
|
Eq.~\ref{handm} or Eq.~\ref{handm2},
|
|
respectively.
|
|
|
|
\subsubsection{Large Scale Precipitation}
|
|
|
|
Large scale condensation occurs if the air is
|
|
supersaturated ($q > q_{sat}(T)$). Condensed water
|
|
falls out
|
|
instantaneously as precipitation. No storage of water
|
|
in clouds is considered. An iterative procedure is used
|
|
to compute final
|
|
values ($T^*$, $q^*$) starting from the
|
|
supersaturated state ($T$, $q$):
|
|
|
|
\begin{equation}\label{lsp1}
|
|
\begin{array}{rcl}
|
|
\D T^* & = & \D T + \frac{L}{c_p} \, \frac{q -
|
|
q_{sat}
|
|
(T)}{1+\frac{L}{c_p}\, \frac{dq_{sat}(T)}{dT}} \\
|
|
&&\\
|
|
\D q^* & = & \D q -\frac{q -
|
|
q_{sat}(T)}{1+\frac{L}{c_p}\,
|
|
\frac{dq_{sat}(T)}{dT}}
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
where the suturation specific humidity $q_{sat}$ and
|
|
its derivative with respect to temperature
|
|
$dq_{sat}/dT$ are computed from Eqs.~\ref{qsat}.
|
|
$L$ is
|
|
either the latent heat of vapourisation or
|
|
the latent heat of sublimation depending on the
|
|
temperature.
|
|
$c_p$ is the specific
|
|
heat for moist air at constant pressure ($c_p= c_{pd} \,
|
|
[1+(c_{pv}/c_{pd}-1)\, q]$ where
|
|
$c_{pd}$ and $c_{pv}$ are the specific heats at
|
|
constant pressure for dry air and water vapor,
|
|
respectively). This calculation is repeated once using
|
|
($T^*$,
|
|
$q^*$) as the new initial state. Finally, The
|
|
temperature
|
|
and moisture tendencies and the precipitation rate
|
|
$P_{l}$ [m/s] are computed:
|
|
|
|
\begin{equation}
|
|
\begin{array}{rcl}
|
|
\D \frac{\partial T}{\partial t} & = &
|
|
\D \frac{T^*-T}{2\Delta t} \\
|
|
&&\\
|
|
\D \frac{\partial q}{\partial t} & = &
|
|
\D \frac{q^*-q}{2\Delta t} \\
|
|
&& \\
|
|
\D P_{l} & = & \D \frac{p_S \, \Delta \sigma}{g \,
|
|
{\rho}_{H_2O}} \frac{ (q-q^*)}{2\Delta t}
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
where $p_S$ is the surface pressure, $\rho_{H_2O}$
|
|
is
|
|
the density of water, $\Delta \sigma$ is the layer
|
|
thickness and $2\Delta t$ is the leap frog time step of
|
|
the model.
|
|
|
|
\subsubsection{Cloud Formation}
|
|
Cloud cover and cloud liquid water content are
|
|
diagnostic quantities. The fractional cloud cover
|
|
of a grid box, $cc$, is parameterized following the ideas of Slingo and Slingo (1991) using the
|
|
relative humidity for the stratiform cloud amount $cc_s$ and the convective precipitation rate
|
|
$P_{c}$ [mm/d] for the convective cloud amount $cc_c$. The latter is given by
|
|
|
|
\begin{equation}
|
|
cc_c= 0.245 + 0.125 \ln{(P_c)}
|
|
\end{equation}
|
|
|
|
where $0.05 \le cc_c \le 0.8$.
|
|
|
|
Before computing the amount of stratiform clouds, the relative humidity $rh$ is multiplied by
|
|
$(1-cc_c)$ to account for the fraction of the grid box covered by convective clouds. If $cc_c \ge
|
|
0.3$ and the cloud top is higher than $\sigma = 0.4$ ($\sigma=p/p_S)$, anvil cirrus is present
|
|
and the cloud amount is
|
|
|
|
\begin{equation}
|
|
cc_s=2\; (cc_c-0.3)
|
|
\end{equation}
|
|
|
|
High-, middle- and low-level stratiform cloud amounts are computed from
|
|
|
|
\begin{equation}
|
|
cc_s=f_{\omega} \left(\frac{rh-rh_c}{1-rh_c}\right)^2
|
|
\end{equation}
|
|
|
|
where $rh_c$ is a level depending critical relative
|
|
humidity. Optionally, a restriction of low-level stratiform cloud amount due to subsidence can
|
|
by introduced by the factor $f_{\omega}$ where $f_{\omega}$ depends on the vertical
|
|
velocity
|
|
$\omega$. In the default version, $f_{\omega}$~=~1.
|
|
|
|
Cloud liquid water content $q_{H_2O}$ [kg/kg] is computed according to Kiehl et al. (1996):
|
|
|
|
\begin{equation}
|
|
q_{H_2O} = \frac{q^0_{H_2O}}{\rho} \exp{(-z/h_l)}
|
|
\end{equation}
|
|
|
|
where the reference value $q^0_{H_2O}$ is $0.21\cdot 10^{-3}$~kg/m$^3$, $\rho$ is the air
|
|
density, $z$ is the height
|
|
and the local cloud water scale height $h_l$~[m] is given by vertically integrated water vapor
|
|
(precipitable water)
|
|
|
|
\begin{equation}
|
|
h_l= 700 \ln{\left(1 + \frac{1}{g} \int\limits^{p_s}_0 q dp \; \right)}
|
|
\end{equation}
|
|
|
|
\subsubsection{Evaporation of Precipitation and Snow
|
|
Fall}
|
|
|
|
Possible phase changes of convective or large scale
|
|
precipitation within the atmosphere or the
|
|
condensational growth of cloud droplets are not
|
|
considered in the model. However, a distinction
|
|
between rain and snow fall at the surface is made. If
|
|
the temperature of the lowermost level
|
|
exceeds the freezing point ($T$~$>$~273.16~K),
|
|
convective and large scale precipitation is
|
|
assumed to be rain, otherwise all precipitation fall out
|
|
as snow.
|
|
|
|
\newpage
|
|
|
|
\subsubsection{Dry Convective Adjustment}
|
|
|
|
Dry convective adjustment is performed for layers which are dry adiabatically unstable, e.g.
|
|
$\partial \theta / \partial p > 0$ where $\theta$ denotes the potential temperature. The adjustment
|
|
is done so that the total sensible heat of the respective column is conserved. Wherever dry
|
|
convection occurs, it is assumed that the moisture is completely mixed by the convective
|
|
process as well. The adjustment is done iteratively. The atmospheric column is scanned for
|
|
unstable regions. A new neutral stable state for the unstable region is computed which consists
|
|
of a potential temperature $\theta_N$ and specific humidity $q_N$:
|
|
|
|
\begin{equation}
|
|
\begin{array}{lcl}
|
|
|
|
\D \theta_N & = & \D \frac{\sum\limits_{l=l_1}^{l_2} T_l \; \Delta\sigma_l}
|
|
{\sum\limits_{l=l_1}^{l_2} \sigma_l^{\kappa} \; \Delta\sigma_l} \\
|
|
&&\\
|
|
\D q_N & = & \D \frac{\sum\limits_{l=l_1}^{l_2} q_l \; \Delta\sigma_l}
|
|
{\sum\limits_{l=l_1}^{l_2} \Delta\sigma_l} \\
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
where $l_1$ and $l_2$ define the unstable region, $\sigma = (p/p_S)$ is the vertical coordinate,
|
|
$T$ and $q$ are temperature and specific humidity, respectively, and $\kappa$ is
|
|
$R_d$/$c_{pd}$ where $R_d$ and $c_{pd}$ are the gas constant and the specific heat for dry
|
|
air,
|
|
respectively.
|
|
|
|
The procedure is repeated starting from the new potential temperatures und moistures until all
|
|
unstable regions are removed. The temperature and moisture tendencies which enter the diabatic
|
|
time steps are then computed from the final $\theta_N$ and $q_N$
|
|
|
|
\begin{equation}
|
|
\begin{array}{lcl}
|
|
\D \frac{T_l^{t+\Delta t}-T_l^{t-\Delta t}}{2\Delta t} & = & \D \frac{\theta_N
|
|
\; \sigma_l^{\kappa} -T_l^{t-\Delta t}}{2\Delta t} \\
|
|
&&\\
|
|
\D \frac{q_l^{t+\Delta t}-q_l^{t-\Delta t}}{2\Delta t} & = & \D \frac{q_N - q_l^{t-\Delta
|
|
t}}{2\Delta t}
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
\newpage
|
|
|
|
\subsection{Land Surface and Soil \label{landmod}}
|
|
|
|
The parameterizations for the land surface and the soil
|
|
include the calculation of temperatures
|
|
for the surface and the soil, a soil hydrology and a river
|
|
transport scheme. In addition, surface
|
|
properties like the albedo, the roughness length or the
|
|
evaporation efficiency are provided.
|
|
As, at the moment, coupling to an extra glacier
|
|
module is not available, glaciers are treated like
|
|
other land points, but with surface and soil properties
|
|
appropriate for ice. Optionally, A simple biome model can be used (see AXEL).
|
|
|
|
\subsubsection{Temperatures}\label{surtemp}
|
|
|
|
The surface temperature $T_S$ is computed from
|
|
the linearized energy balance of the
|
|
uppermost $z_{top}$ meters of the ground:
|
|
|
|
\begin{equation} \label{land.1}
|
|
c_{top} \; z_{top} \; \frac{\Delta T_S}{\Delta t}
|
|
= F_S - G + \Delta T_S \; \frac{\partial (Q_a
|
|
-F_g)}{\partial T_S} - F_m
|
|
\end{equation}
|
|
|
|
$z_{top}$ is a prescribed parameter and set to a default
|
|
value of $z_{top}$~=~0.20~m.
|
|
$Q_a$ denotes the total heat flux from the atmosphere,
|
|
which consists of the sensible heat flux,
|
|
the latent heat flux, the net short wave radiation and the
|
|
net long wave radiation. $Q_g$ is the flux
|
|
into the deep soil. $Q_a$ and $Q_g$ are defined positive
|
|
downwards. $Q_m$ is the
|
|
snow melt heat flux and $c_{top}$ is the
|
|
volumetric heat capacity. Depending on the snow
|
|
pack, $z_{top}$ can partly or totally consist
|
|
of snow or soil solids:
|
|
$z_{top}=z_{snow}+z_{soil}$. Thus, the heat
|
|
capacity $c_{top}$ is a
|
|
combination of snow and soil heat capacities:
|
|
|
|
\begin{equation}
|
|
c_{top} =\frac{ c_{snow} \; c_{soil} \; z_{top}}{c_{snow} \; z_{soil} +
|
|
c_{soil} \; z_{snow}}
|
|
\end{equation}
|
|
|
|
The default value of
|
|
$c_{snow}$ is
|
|
0.6897~$\cdot$~10$^{6}$~J/(kg K) using a snow
|
|
density
|
|
of 330~kg/m$^3$. $c_{soil}$ is set to a default value
|
|
of 2.07~$\cdot$~10$^{6}$~J/(kg K) for
|
|
glaciers and to a value of 2.4~$\cdot$~10$^{6}$~J/(kgK) otherweise.
|
|
|
|
Below $z_{top}$ the soil column is discretized into
|
|
$N$ layers with thickness $\Delta z_i$,
|
|
where layer $1$ is the uppermost of the soil
|
|
layers. The default values for the model are $N$~=~5
|
|
and $\Delta z$~=~(0.4~m, 0.8~m, 1.6~m,
|
|
3.2~m, 6.4~m). The heat flux into layer 1, $Q_g$, is
|
|
given by
|
|
|
|
\begin{equation}
|
|
Q_g=\frac{2 k_{1}}{\Delta z_{1}} (T_S - T_{1})
|
|
\end{equation}
|
|
|
|
where $k_{1}$ and $T_{1}$ are the thermal
|
|
conductivity and the temperature.
|
|
If the snow depth is greater than $z_{top}$, the
|
|
thermal properties of snow are blended with the
|
|
first soil layer to create a snow/soil layer with
|
|
thickness $z_{snow}-z_{top}+\Delta z_1$. The
|
|
thermal conductivity $k_1$ and heat capacity
|
|
$c_1$ of a snow/soil layer are
|
|
|
|
\begin{equation}
|
|
\begin{array}{rcl}
|
|
\D k_1 & = & \D \frac{k_{snow}\; k_{soil}\; (\Delta
|
|
z_1+z_{snow}-z_{top})}{k_{snow}\; \Delta z_1 +
|
|
k_{soil}\; (z_{snow}-z_{top})} \\
|
|
&&\\
|
|
\D c_1 & = & \D \frac{c_{snow}\; c_{soil}\; (\Delta
|
|
z_1+z_{snow}-z_{top})}{c_{snow}\; \Delta z_1 +
|
|
c_{soil}\; (z_{snow}-z_{top})}
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
with default values of $k_{snow}$~=~0.31~W/(m K),
|
|
$k_{soil}$~=~2.03~W/(m K) for
|
|
glaciers and $k_{soil}$~=~7~W/(m K) otherweise.
|
|
|
|
After the surface temperature $T_S$ has been
|
|
calculated
|
|
from Eq.~\ref{land.1}, snow melts when $T_S$ is
|
|
greater than the freezing temperature $T_{melt}$. In this
|
|
case, $T_S$ is set to $T_{melt}$ and a new atmospheric
|
|
heat
|
|
flux $Q_a(T_{melt})$ is calculated from $Q_a$ and
|
|
$\partial Q_a/\partial T_S$. If the energy inbalance is
|
|
positive ($Q_a(T_{melt}) > c_{top} \; z_{top}\; (T_{melt} -
|
|
T_S^{t})/\Delta t$; where $T_S^{t}$ is the surface
|
|
temperature at
|
|
the previous time step), the
|
|
snow melt heat flux $Q_m$ is
|
|
|
|
\begin{equation}\label{melt}
|
|
Q_m=\max(Q_a(T_{melt}) - \frac{c_{top} \; z_{top}}{\Delta
|
|
t} \; (T_{melt} - T_S^{t}), \; \frac{W_{snow} \; L_f}{\Delta t})
|
|
\end{equation}
|
|
|
|
where $W_{snow}$ is the mass of the snow water of
|
|
the
|
|
total snow pack and $L_f$ is the latent heat of fusion.
|
|
Any excess of energy is used to warm the soil.
|
|
|
|
|
|
With the heat flux $F_z$ at depth $z$ of the soil
|
|
|
|
\begin{equation}
|
|
F_z = -k \; \frac{\partial T}{\partial z}
|
|
\end{equation}
|
|
|
|
one dimensional energy conservation requires
|
|
|
|
\begin{equation}
|
|
c\; \frac{\partial T} {\partial t} = - \frac{\partial
|
|
F_z}{\partial
|
|
z}= \frac{\partial}{\partial z} \left [ k \;
|
|
\frac{\partial T}{\partial z} \right]
|
|
\end{equation}
|
|
|
|
where $c$ is the volumetric soil heat capacity, $T$
|
|
is the soil temperature, and $k$ is the
|
|
thermal conductivity.
|
|
|
|
In the model, thermal properties (temperature,
|
|
thermal conductivity, volumetric heat capacity) are
|
|
defined at the center of each layer. Assuming the
|
|
heat flux from $i$ to the interface $i$ and $i+1$
|
|
equals the heat flux from the interface to $i+1$, the
|
|
heat flux $F_i$ from layer $i$ to layer $i+1$
|
|
(positive downwards) is given by
|
|
|
|
\begin{equation}
|
|
F_i= - \frac{2\; k_i\; k_{i+1} (T_i -
|
|
T_{i+1})}{k_{i+1} \; \Delta z_i + k_i\; \Delta z_{i+1}}
|
|
\end{equation}
|
|
|
|
The energy balance for layer $i$ is
|
|
|
|
\begin{equation}
|
|
\frac{c_i\; \Delta z_i}{\Delta t} \; (T_i^{t+\Delta t} -
|
|
T_i ^{t}) = F_i - F_{i-1}
|
|
\end{equation}
|
|
|
|
|
|
The boundary conditions are zero flux at the bottom
|
|
of the soil column and heat flux $F_g$ at the top.
|
|
|
|
This equation is solved implicitly using fluxes
|
|
$F_i$ evaluated at $t+\Delta t$
|
|
|
|
\begin{equation}
|
|
\begin{array}{lclcl}
|
|
\D \frac{c_i \Delta z_i}{\Delta t} (T_i^{t+\Delta t} -
|
|
T_i ^{t}) & = &\D \frac{k_i k_{i+1}
|
|
(T_{i+1}^{t+\Delta
|
|
t} - T_i^{t+\Delta t})}{k_{i+1} \Delta z_i + k_i
|
|
\Delta z_{i+1}} + G & for & \D i = 1 \\
|
|
& & & & \\
|
|
\D \frac{c_i \Delta z_i}{\Delta t} (T_i^{t+\Delta t} -
|
|
T_i ^{t}) & = &\D \frac{k_i k_{i+1}
|
|
(T_{i+1}^{t+\Delta
|
|
t} - T_i^{t+\Delta t})}{k_{i+1} \Delta z_i + k_i
|
|
\Delta z_{i+1}} + \frac{k_i k_{i-1} (T_{i-
|
|
1}^{t+\Delta t} - T_i^{t+\Delta t})}{k_{i-1} \Delta
|
|
z_i + k_i \Delta z_{i-1}} & for & \D 1 < i < N \\
|
|
& & & & \\
|
|
\D \frac{c_i \Delta z_i}{\Delta t} (T_i^{t+\Delta t} -
|
|
T_i ^{t}) & = &\D \frac{k_i k_{i-1} (T_{i-
|
|
1}^{t+\Delta t} - T_i^{t+\Delta t})}{k_{i-1} \Delta
|
|
z_i + k_i \Delta z_{i-1}} & for & \D i = N
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
resulting in a linear system for the $T_i^{t+\Delta
|
|
t}$.
|
|
|
|
\subsubsection{Soil Hydrology}\label{hydro}
|
|
|
|
The parameterization of soil hydrology comprises the
|
|
budgets for snow amount and the soil
|
|
water amount. The water equivalent of the snow layer
|
|
$z_{snow}^{H_2O}$ is computed over
|
|
land and glacier areas from
|
|
|
|
\begin{equation}
|
|
\frac{\partial z_{snow}^{H_2O}}{\partial t} = F_q +
|
|
P_{snow}-M_{snow}
|
|
\end{equation}
|
|
|
|
where $F_q$ is the evaporation rate over snow
|
|
computed from Eq.~\ref{fluxes2}, $P_{snow}$ is the
|
|
snow fall and $M_{snow}$ is the snow
|
|
melt rate (all fluxes are positive downward and in m/s).
|
|
$M_{snow}$ is related to the snow melt
|
|
heat flux $Q_m$ (Eq.~\ref{melt}) by
|
|
|
|
\begin{equation}
|
|
M_{snow}=\frac{Q_m}{\rho_{H_2O}\, L_f}
|
|
\end{equation}
|
|
|
|
where $L_f$ is the latent heat of fusion.
|
|
|
|
The soil water reservoir $W_{soil}$ [m] is
|
|
represented by a single-layer bucket model
|
|
(Manabe 1969). Soil water is increased by precipitation
|
|
$P$ and snow melt $M_{snow}$
|
|
and is depleted by the surface evaporation $F_q$:
|
|
|
|
\begin{equation}
|
|
\frac{\partial W_{soil}}{\partial t} = P + M + F_q
|
|
\end{equation}
|
|
|
|
where all fluxes are defined positive downwards and in
|
|
m/s.
|
|
Soil water is limited by a field capacity $W_{max}$
|
|
which geographical distribution can be prescribed via an external input or is set to a default
|
|
value of
|
|
0.5~m everywhere. If the soil water
|
|
exceeds $W_{max}$ the excessive
|
|
water builds the runoff $R$ and is provided to the river
|
|
transport scheme
|
|
(Section~\ref{runoff}). The ratio of the soil water and
|
|
the field capacity defines the wetness
|
|
factor $C_w$ which is used in Eq.~\ref{fluxes2} to
|
|
compute the surface evaporation:
|
|
|
|
\begin{equation}\label{cwgl}
|
|
C_w=\frac{W_{soil}}{f_{Cw} \; W_{max}}
|
|
\end{equation}
|
|
|
|
where the factor $f_{Cw}$ (with a default value of 0.25) takes into account that maximum
|
|
evaporation will take place even if the bucket is not completely filled. For land points covered
|
|
by glaciers, $C_w$ is set to a
|
|
constant value of 1.
|
|
|
|
\subsubsection{River Transport}\label{runoff}
|
|
|
|
The local runoff is transported to the ocean by a river
|
|
transport scheme with linear advection
|
|
(Sausen et al.~1994). For each grid box (both, land and
|
|
ocean costal points) the river water
|
|
amount $W_{river}$ [m$^3$] is computed from
|
|
|
|
\begin{equation}
|
|
\frac{\partial W_{river}}{\partial t}= ADV + area \; (R - S)
|
|
\end{equation}
|
|
|
|
where $R$ is the local runoff (Section~\ref{hydro}),
|
|
$S$ is the input into the ocean, $ADV$
|
|
is the advection of river water and $area$ is the area of the
|
|
respective grid box. The input into
|
|
the ocean $S$ is given by
|
|
|
|
\begin{equation}
|
|
S=\left\{ \begin{array}{ll} 0 & \mbox{for land points}
|
|
\\
|
|
&\\
|
|
ADV & \mbox{for ocean points} \end{array}
|
|
\right.
|
|
\end{equation}
|
|
|
|
This ensures that $S$ is non-zero only for ocean costal
|
|
points. The advection from grid box
|
|
$(i,j)$ into grid box $(i',j')$, $ADV_{(i,j)\rightarrow
|
|
(i',j')}$, is formulated using an upstream
|
|
scheme:
|
|
|
|
\begin{equation}
|
|
\begin{array}{rcl}
|
|
\D ADV_{(i,j)\rightarrow (i+1,j)} & = & \D \left\{
|
|
\begin{array}{ll} u_{i,j} W_{i,j}, & \mbox{if } \;
|
|
u_{i,j} \ge 0 \\
|
|
& \\
|
|
u_{i,j} W_{i+1,j}, & \mbox{if } \;
|
|
u_{i,j} < 0 \end{array} \right. \\
|
|
&& \\
|
|
\D ADV_{(i,j)\rightarrow (i,j+1)} & = & \D \left\{
|
|
\begin{array}{ll} -v_{i,j} W_{i,j}, & \mbox{if } \;
|
|
v_{i,j} \le 0 \\
|
|
& \\
|
|
-v_{i,j} W_{i,j+1}, & \mbox{if } \;
|
|
v_{i,j} > 0 \end{array} \right. \\
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
where $i$ and $j$ are the zonal and meridional indices
|
|
of the grid box, which are counted
|
|
from the west to the east and from the north to the
|
|
south, respectively. The zonal and
|
|
meridional advection rates $u_{i,j}$ and $v_{i,j}$ are
|
|
defined at the interface of two grid
|
|
boxes and depend on the slope of the orography:
|
|
|
|
\begin{equation}
|
|
\begin{array}{rcl}
|
|
\D u_{i,j} & = & \D \frac{c}{\Delta
|
|
x}\left[\frac{h_{i,j}-h_{i+1,j}}{\Delta
|
|
x}\right]^{\alpha} \\
|
|
&& \\
|
|
\D v_{i,j} & = & \D \frac{c}{\Delta
|
|
y}\left[\frac{h_{i,j+1}-h{i,j}}{\Delta
|
|
y}\right]^{\alpha}
|
|
\end{array}
|
|
\end{equation}
|
|
|
|
where $\Delta x$ and $\Delta y$ are the distances
|
|
between the grid points in the longitudinal
|
|
and the meridional direction. $h$ is the height of the
|
|
orography, which is modified in
|
|
order to omit local minima at land grid points. The
|
|
empirical constants $c$ and $\alpha$ are
|
|
set to the values given by Sausen et al.~(1994) for T21
|
|
resolution ($c = $~4.2~m/s and
|
|
$\alpha =$~0.18).
|
|
|
|
\subsubsection{Other Land Surface
|
|
Parameter}\label{landsurf}
|
|
|
|
Some additional quantities characterizing the land surface of
|
|
each grid box need to be specified for use in the model. The land-sea mask and the orography
|
|
are read from an external file. Optionally, this file may also include other climatological surface
|
|
parameter: the global distribution of the surface roughness length $z_0$, a background albedo
|
|
${\cal R}_S^{clim}$, a glacier mask for permanent ice sheets, the bucked size for the soil water
|
|
$W_{max}$ (see section above) and a climatological annual cycle of the soil wetness
|
|
$C^{clim}_w$ (which may be used instead of the computed $C_w$ from Eq.~\ref{cwgl}. If
|
|
there is no input for the particular field in the file, the parameter is set to be horizontal
|
|
homogeneous with a specific value. The following defaults are used: $z_0$~=~ 2~m,
|
|
${\cal R}_S^{clim}$~=~0.2, no glaciers, $W_{max}$~=~0.5 and $C^{clim}_w$~=~0.25.
|
|
|
|
For snow covered areas, the background albedo is modified to give the actual albedo ${\cal
|
|
R}_S$
|
|
which is used in the radiation scheme. For points, which are not covered by glaciers, ${\cal
|
|
R}_S$ is
|
|
given by
|
|
|
|
\begin{equation}
|
|
{\cal R}_S={\cal R}_S^{clim} + ({\cal R}_S^{snow}-{\cal R}_S^{clim}) \;
|
|
\frac{z_{snow}}{z_{snow} + 0.01}
|
|
\end{equation}
|
|
|
|
where $z_{snow}$ is the snow depth, and the albedo of the snow, ${\cal R}_S^{snow}$,
|
|
depends on
|
|
the surface temperature $T_S$
|
|
|
|
\begin{equation}\label{rsnow}
|
|
{\cal R}_S^{snow}={\cal R}_{max}^{snow} + ({\cal R}_{min}^{snow} - {\cal
|
|
R}_{max}^{snow}) \; \frac{T_S -
|
|
263.16}{10}
|
|
\end{equation}
|
|
|
|
with ${\cal R}_{min}^{snow} \le {\cal R}_S^{snow} \le {\cal R}_{max}^{snow}$ and default
|
|
values
|
|
${\cal R}_{min}^{snow}$~=~0.4 and ${\cal R}_{max}^{snow}$~=~0.8.
|
|
|
|
For glaciers, ${\cal R}_S$ is given by ${\cal R}_S^{snow}$ from Eq.~\ref{rsnow} but with a
|
|
default
|
|
${cal R}_{min}^{snow}$~=~0.6.
|
|
|
|
The surface specific humidity $q_S$ is given by the saturation specific humidity at $T_S$:
|
|
|
|
\begin{equation}
|
|
q_S =q_{sat}(T_S)
|
|
\end{equation}
|
|
|
|
where $q_{sat}(T_S)$ is computed from
|
|
Eq.~\ref{qsat}.
|
|
|
|
\subsection{Sea Surface}\label{seasurf}
|
|
|
|
Sea surface temperatures $T_{sea}$, sea ice distributions
|
|
$c_{ice}$ and surface temperatures over
|
|
sea ice $T_i$ are provided by the ocean and sea
|
|
ice modules (Section HEIKO). From
|
|
these quantities, the following additional parameter are
|
|
computed which enter the atmospheric
|
|
parameterizations. The prescribed surface albedo ${\cal R_S}$
|
|
for open water is set to a default value of
|
|
0.069. For sea ice ${\cal R}_S$ is given as a function of the ice
|
|
surface temperature $T_{i}$:
|
|
|
|
\begin{equation}
|
|
{\cal R}_S=\min{({\cal R}_S^{max}, \, 0.5 + 0.025 \, (273. - T_{i}))}
|
|
\end{equation}
|
|
|
|
where the prescribed maximum sea ice background
|
|
albedo ${\cal R}_S^{max}$ is set to a default value
|
|
of 0.7.
|
|
|
|
The surface
|
|
specific humidity $q_S$ is given by the
|
|
saturation specific humidity at the surface
|
|
temperature $T_S$ which is either $T_{sea}$ or
|
|
$T_{i}$:
|
|
|
|
\begin{equation}
|
|
q_S =q_{sat}(T_S)
|
|
\end{equation}
|
|
|
|
where $q_{sat}(T_S)$ is computed from
|
|
Eq.~\ref{qsat}. The wetness factor $C_w$ which
|
|
enters
|
|
the calculation of the surface evaporation
|
|
(Eq.~\ref{fluxes2}) is set to 1.
|
|
|
|
The roughness length $z_0$ over sea ice is set to a
|
|
constant value of $z_0$~=~0.001~m. Over open
|
|
water,
|
|
$z_0$ is computed from the Charnock (1955) formula:
|
|
|
|
\begin{equation}
|
|
z_0 = C_{char} \frac{u_{*}^{2}}{g}
|
|
\end{equation}
|
|
|
|
with a minimum value of $1.5 \cdot 10^{-5}$~m.
|
|
$C_{char}$ denotes the Charnock constant and is set
|
|
to
|
|
0.018. $g$ is the gravitational acceleration. The
|
|
friction
|
|
velocity $u_{*}$ is calculated from the surface wind
|
|
stress at the previous time level:
|
|
|
|
\begin{equation}
|
|
u_{*}=\sqrt{\frac{|F_u, F_v|}{\rho} }
|
|
\end{equation}
|
|
|
|
where $|F_u, F_v| $ is the absolute value of the surface
|
|
wind stress computed from Eq.~\ref{fluxes2} and
|
|
$\rho$
|
|
is the density.
|
|
|
|
\newpage
|
|
|
|
\subsection{References}
|
|
|
|
Berger, A., 1978a: A simple algorithm to compute
|
|
long-term
|
|
variations of daily insolation. Institute of Astronomy
|
|
and
|
|
Geophysic, Universite Catholique de Louvain, {\bf
|
|
Contribution
|
|
18}, Louvain-la-Neuve, Belgium.
|
|
|
|
Berger, A., 1978b: Long-term variations of daily
|
|
insolation
|
|
and quaternary climatic change. {\it J.~Atmos.~Sci.},
|
|
{\bf 35}, 2362-
|
|
2367.
|
|
|
|
Blackadar, A.~K., 1962: The vertical distributionof
|
|
wind and turbulent exchange in a neutral
|
|
atmosphere. {\it J.~Geophys.~Res.}, {\bf 67},
|
|
3095-3102.
|
|
|
|
Boer, G.~J., N.~A.~McFarlane, R.~Laprise,
|
|
J.~Henderson and J.-P.~Blanchet, 1984: The
|
|
Canadian Climate Centre spectral atmospheric general
|
|
circulation model. {\it
|
|
Atmosphere-Ocean},
|
|
{\bf 22}, 397-429.
|
|
|
|
Charnock, M., 1955: Wind stress on a water surface.
|
|
{\it Q.~J.~R.~Meteorol.~Soc.}, {\bf
|
|
81}, 639-640.
|
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|
|
Chou, M.-D., M.~J.~Suarez, X.-Z.~Liang and M.~M.~H.~Yan, 2001 (Revised 2002):
|
|
A thermal infrared radiation
|
|
parameterization for atmospheric studies. {\it Technical Report Series on Global Modelling and
|
|
Data Assimilation, M.~J.~Suarez Ed.}, {\bf NASA/TM-2001-104606, Vol.
|
|
19}, 55pp.
|
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|
|
Green, A.~E.~S., 1964: Attenuation by ozone and the earth's albedo in the middle ultraviolet.
|
|
{\it AAppl.~Opt.}, {\bf 3}, 203-208.
|
|
|
|
Katayama, A., 1972: A simplified scheme for
|
|
computing
|
|
radiative transfer in the troposphere.
|
|
Department
|
|
of Meteorology, {\it Tech.~Report}, {\bf No. 6},
|
|
University of California, Los Angeles, CA,
|
|
77 pp.
|
|
|
|
Kiehl, J.~T., J.~J.~Hack, G.~B.~Bonan, B.~A.~Boville, B.~P.~Briegleb, D.~L.~Williamson
|
|
and
|
|
P.~J.~Rasch, 1996: Description of the NCAR Community Climate Model (CCM3). {\it NCAR
|
|
Technical Note}, {\bf NCAR/TN-420+STR}, 152pp.
|
|
|
|
Kuo, H.~L., 1965: On formation and intensification of
|
|
tropical cyclones through latent heat
|
|
release by cumulus convection. {\it J.~Atmos.~Sci.},
|
|
{\bf 22}, 40-63.
|
|
|
|
Kuo, H.~L., 1974: Further studies of the
|
|
parameterization of the influence of cumulus
|
|
convection on large-scale flow. {\it J.~Atmos.~Sci.},
|
|
{\bf 31}, 1232-1240.
|
|
|
|
Lacis, A.~A., and J.~E.~Hansen, 1974: A parameterization for the absorption of solar
|
|
radiation in the Earth's atmosphere. {\it J.~Atmos.~Sci.}, {\bf 31}, 118-133.
|
|
|
|
Laurson, L.~and E.~Eliasen, 1989: On the effects of
|
|
the damping mechanisms in an
|
|
atmospheric general circulation model. {\it Tellus},
|
|
{\bf 41A}, 385-400.
|
|
|
|
Louis, J.~F., 1979: A parametric model of vertical
|
|
eddy fluxes in the atmosphere. {\it
|
|
Boundary
|
|
Layer Meteorology}, {\bf 17}, 187-202.
|
|
|
|
Louis, J.~F., M.~Tiedtke and M.~Geleyn, 1982: A
|
|
short history of the PBL parameterisation
|
|
at
|
|
ECMWF. Proceedings, ECMWF workshop on
|
|
planetary boundary layer parameterization,
|
|
Reading, 25-27 Nov. 81, 59-80.
|
|
|
|
Lowe, P.~R., 1977: An approximating polynomial for
|
|
the computation of saturation vapour
|
|
pressure. {\it J.~Appl.~Met.}, {\bf 16}, 100-103.
|
|
|
|
Manabe, S.~and F.~M\"oller, 1961: on the radiative
|
|
equilibrium and heat balance of the
|
|
atmosphere. {\it Mon~Wea.~Rev.}, {\bf 89}, 503-532.
|
|
|
|
Manabe, S., 1969: Climate and ocean circulation. I.
|
|
The atmospheric circulation and the
|
|
hydrology of the earth's surface. {\it
|
|
Mon.~Wea.~Rev.}, {\bf 97}, 739-774.
|
|
|
|
Rodgers, C.~D., 1967: The use of emissivity in the
|
|
atmospheric radiation calculation.
|
|
{\it Q.~J.~R.~Meteorol.~Soc.}, {\bf 93}, 43-54.
|
|
|
|
Roeckner, E., K.~Arpe, L.~Bengtsson, S.~Brinkop,
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