--- manual/s_phys_pkgs/text/fizhi.tex 2004/10/12 18:16:03 1.7 +++ manual/s_phys_pkgs/text/fizhi.tex 2005/07/14 19:18:02 1.8 @@ -892,6 +892,2139 @@ the model's moisture data is used. Above 100 mb, the SAGE data is used. Between 100 and 300 mb, a linear interpolation (in pressure) is performed using the data from SAGE and the GCM. + +\subsection{Fizhi Diagnostics} + +\subsubsection{Fizhi Diagnostic Menu} +\label{sec:fizhi-diagnostics:menu} + +\begin{tabular}{llll} +\hline\hline + NAME & UNITS & LEVELS & DESCRIPTION \\ +\hline + +&\\ + UFLUX & $Newton/m^2$ & 1 + &\begin{minipage}[t]{3in} + {Surface U-Wind Stress on the atmosphere} + \end{minipage}\\ + VFLUX & $Newton/m^2$ & 1 + &\begin{minipage}[t]{3in} + {Surface V-Wind Stress on the atmosphere} + \end{minipage}\\ + HFLUX & $Watts/m^2$ & 1 + &\begin{minipage}[t]{3in} + {Surface Flux of Sensible Heat} + \end{minipage}\\ + EFLUX & $Watts/m^2$ & 1 + &\begin{minipage}[t]{3in} + {Surface Flux of Latent Heat} + \end{minipage}\\ + QICE & $Watts/m^2$ & 1 + &\begin{minipage}[t]{3in} + {Heat Conduction through Sea-Ice} + \end{minipage}\\ + RADLWG & $Watts/m^2$ & 1 + &\begin{minipage}[t]{3in} + {Net upward LW flux at the ground} + \end{minipage}\\ + RADSWG & $Watts/m^2$ & 1 + &\begin{minipage}[t]{3in} + {Net downward SW flux at the ground} + \end{minipage}\\ + RI & $dimensionless$ & Nrphys + &\begin{minipage}[t]{3in} + {Richardson Number} + \end{minipage}\\ + CT & $dimensionless$ & 1 + &\begin{minipage}[t]{3in} + {Surface Drag coefficient for T and Q} + \end{minipage}\\ + CU & $dimensionless$ & 1 + &\begin{minipage}[t]{3in} + {Surface Drag coefficient for U and V} + \end{minipage}\\ + ET & $m^2/sec$ & Nrphys + &\begin{minipage}[t]{3in} + {Diffusivity coefficient for T and Q} + \end{minipage}\\ + EU & $m^2/sec$ & Nrphys + &\begin{minipage}[t]{3in} + {Diffusivity coefficient for U and V} + \end{minipage}\\ + TURBU & $m/sec/day$ & Nrphys + &\begin{minipage}[t]{3in} + {U-Momentum Changes due to Turbulence} + \end{minipage}\\ + TURBV & $m/sec/day$ & Nrphys + &\begin{minipage}[t]{3in} + {V-Momentum Changes due to Turbulence} + \end{minipage}\\ + TURBT & $deg/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Temperature Changes due to Turbulence} + \end{minipage}\\ + TURBQ & $g/kg/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Specific Humidity Changes due to Turbulence} + \end{minipage}\\ + MOISTT & $deg/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Temperature Changes due to Moist Processes} + \end{minipage}\\ + MOISTQ & $g/kg/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Specific Humidity Changes due to Moist Processes} + \end{minipage}\\ + RADLW & $deg/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Net Longwave heating rate for each level} + \end{minipage}\\ + RADSW & $deg/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Net Shortwave heating rate for each level} + \end{minipage}\\ + PREACC & $mm/day$ & 1 + &\begin{minipage}[t]{3in} + {Total Precipitation} + \end{minipage}\\ + PRECON & $mm/day$ & 1 + &\begin{minipage}[t]{3in} + {Convective Precipitation} + \end{minipage}\\ + TUFLUX & $Newton/m^2$ & Nrphys + &\begin{minipage}[t]{3in} + {Turbulent Flux of U-Momentum} + \end{minipage}\\ + TVFLUX & $Newton/m^2$ & Nrphys + &\begin{minipage}[t]{3in} + {Turbulent Flux of V-Momentum} + \end{minipage}\\ + TTFLUX & $Watts/m^2$ & Nrphys + &\begin{minipage}[t]{3in} + {Turbulent Flux of Sensible Heat} + \end{minipage}\\ +\end{tabular} + +\newpage +\vspace*{\fill} +\begin{tabular}{llll} +\hline\hline + NAME & UNITS & LEVELS & DESCRIPTION \\ +\hline + +&\\ + TQFLUX & $Watts/m^2$ & Nrphys + &\begin{minipage}[t]{3in} + {Turbulent Flux of Latent Heat} + \end{minipage}\\ + CN & $dimensionless$ & 1 + &\begin{minipage}[t]{3in} + {Neutral Drag Coefficient} + \end{minipage}\\ + WINDS & $m/sec$ & 1 + &\begin{minipage}[t]{3in} + {Surface Wind Speed} + \end{minipage}\\ + DTSRF & $deg$ & 1 + &\begin{minipage}[t]{3in} + {Air/Surface virtual temperature difference} + \end{minipage}\\ + TG & $deg$ & 1 + &\begin{minipage}[t]{3in} + {Ground temperature} + \end{minipage}\\ + TS & $deg$ & 1 + &\begin{minipage}[t]{3in} + {Surface air temperature (Adiabatic from lowest model layer)} + \end{minipage}\\ + DTG & $deg$ & 1 + &\begin{minipage}[t]{3in} + {Ground temperature adjustment} + \end{minipage}\\ + + QG & $g/kg$ & 1 + &\begin{minipage}[t]{3in} + {Ground specific humidity} + \end{minipage}\\ + QS & $g/kg$ & 1 + &\begin{minipage}[t]{3in} + {Saturation surface specific humidity} + \end{minipage}\\ + TGRLW & $deg$ & 1 + &\begin{minipage}[t]{3in} + {Instantaneous ground temperature used as input to the + Longwave radiation subroutine} + \end{minipage}\\ + ST4 & $Watts/m^2$ & 1 + &\begin{minipage}[t]{3in} + {Upward Longwave flux at the ground ($\sigma T^4$)} + \end{minipage}\\ + OLR & $Watts/m^2$ & 1 + &\begin{minipage}[t]{3in} + {Net upward Longwave flux at the top of the model} + \end{minipage}\\ + OLRCLR & $Watts/m^2$ & 1 + &\begin{minipage}[t]{3in} + {Net upward clearsky Longwave flux at the top of the model} + \end{minipage}\\ + LWGCLR & $Watts/m^2$ & 1 + &\begin{minipage}[t]{3in} + {Net upward clearsky Longwave flux at the ground} + \end{minipage}\\ + LWCLR & $deg/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Net clearsky Longwave heating rate for each level} + \end{minipage}\\ + TLW & $deg$ & Nrphys + &\begin{minipage}[t]{3in} + {Instantaneous temperature used as input to the Longwave radiation + subroutine} + \end{minipage}\\ + SHLW & $g/g$ & Nrphys + &\begin{minipage}[t]{3in} + {Instantaneous specific humidity used as input to the Longwave radiation + subroutine} + \end{minipage}\\ + OZLW & $g/g$ & Nrphys + &\begin{minipage}[t]{3in} + {Instantaneous ozone used as input to the Longwave radiation + subroutine} + \end{minipage}\\ + CLMOLW & $0-1$ & Nrphys + &\begin{minipage}[t]{3in} + {Maximum overlap cloud fraction used in the Longwave radiation + subroutine} + \end{minipage}\\ + CLDTOT & $0-1$ & Nrphys + &\begin{minipage}[t]{3in} + {Total cloud fraction used in the Longwave and Shortwave radiation + subroutines} + \end{minipage}\\ + LWGDOWN & $Watts/m^2$ & 1 + &\begin{minipage}[t]{3in} + {Downwelling Longwave radiation at the ground} + \end{minipage}\\ + GWDT & $deg/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Temperature tendency due to Gravity Wave Drag} + \end{minipage}\\ + RADSWT & $Watts/m^2$ & 1 + &\begin{minipage}[t]{3in} + {Incident Shortwave radiation at the top of the atmosphere} + \end{minipage}\\ + TAUCLD & $per 100 mb$ & Nrphys + &\begin{minipage}[t]{3in} + {Counted Cloud Optical Depth (non-dimensional) per 100 mb} + \end{minipage}\\ + TAUCLDC & $Number$ & Nrphys + &\begin{minipage}[t]{3in} + {Cloud Optical Depth Counter} + \end{minipage}\\ +\end{tabular} +\vfill + +\newpage +\vspace*{\fill} +\begin{tabular}{llll} +\hline\hline + NAME & UNITS & LEVELS & DESCRIPTION \\ +\hline + +&\\ + CLDLOW & $0-1$ & Nrphys + &\begin{minipage}[t]{3in} + {Low-Level ( 1000-700 hPa) Cloud Fraction (0-1)} + \end{minipage}\\ + EVAP & $mm/day$ & 1 + &\begin{minipage}[t]{3in} + {Surface evaporation} + \end{minipage}\\ + DPDT & $hPa/day$ & 1 + &\begin{minipage}[t]{3in} + {Surface Pressure tendency} + \end{minipage}\\ + UAVE & $m/sec$ & Nrphys + &\begin{minipage}[t]{3in} + {Average U-Wind} + \end{minipage}\\ + VAVE & $m/sec$ & Nrphys + &\begin{minipage}[t]{3in} + {Average V-Wind} + \end{minipage}\\ + TAVE & $deg$ & Nrphys + &\begin{minipage}[t]{3in} + {Average Temperature} + \end{minipage}\\ + QAVE & $g/kg$ & Nrphys + &\begin{minipage}[t]{3in} + {Average Specific Humidity} + \end{minipage}\\ + OMEGA & $hPa/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Vertical Velocity} + \end{minipage}\\ + DUDT & $m/sec/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Total U-Wind tendency} + \end{minipage}\\ + DVDT & $m/sec/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Total V-Wind tendency} + \end{minipage}\\ + DTDT & $deg/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Total Temperature tendency} + \end{minipage}\\ + DQDT & $g/kg/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Total Specific Humidity tendency} + \end{minipage}\\ + VORT & $10^{-4}/sec$ & Nrphys + &\begin{minipage}[t]{3in} + {Relative Vorticity} + \end{minipage}\\ + DTLS & $deg/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Temperature tendency due to Stratiform Cloud Formation} + \end{minipage}\\ + DQLS & $g/kg/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Specific Humidity tendency due to Stratiform Cloud Formation} + \end{minipage}\\ + USTAR & $m/sec$ & 1 + &\begin{minipage}[t]{3in} + {Surface USTAR wind} + \end{minipage}\\ + Z0 & $m$ & 1 + &\begin{minipage}[t]{3in} + {Surface roughness} + \end{minipage}\\ + FRQTRB & $0-1$ & Nrphys-1 + &\begin{minipage}[t]{3in} + {Frequency of Turbulence} + \end{minipage}\\ + PBL & $mb$ & 1 + &\begin{minipage}[t]{3in} + {Planetary Boundary Layer depth} + \end{minipage}\\ + SWCLR & $deg/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Net clearsky Shortwave heating rate for each level} + \end{minipage}\\ + OSR & $Watts/m^2$ & 1 + &\begin{minipage}[t]{3in} + {Net downward Shortwave flux at the top of the model} + \end{minipage}\\ + OSRCLR & $Watts/m^2$ & 1 + &\begin{minipage}[t]{3in} + {Net downward clearsky Shortwave flux at the top of the model} + \end{minipage}\\ + CLDMAS & $kg / m^2$ & Nrphys + &\begin{minipage}[t]{3in} + {Convective cloud mass flux} + \end{minipage}\\ + UAVE & $m/sec$ & Nrphys + &\begin{minipage}[t]{3in} + {Time-averaged $u-Wind$} + \end{minipage}\\ +\end{tabular} +\vfill + +\newpage +\vspace*{\fill} +\begin{tabular}{llll} +\hline\hline + NAME & UNITS & LEVELS & DESCRIPTION \\ +\hline + +&\\ + VAVE & $m/sec$ & Nrphys + &\begin{minipage}[t]{3in} + {Time-averaged $v-Wind$} + \end{minipage}\\ + TAVE & $deg$ & Nrphys + &\begin{minipage}[t]{3in} + {Time-averaged $Temperature$} + \end{minipage}\\ + QAVE & $g/g$ & Nrphys + &\begin{minipage}[t]{3in} + {Time-averaged $Specific \, \, Humidity$} + \end{minipage}\\ + RFT & $deg/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Temperature tendency due Rayleigh Friction} + \end{minipage}\\ + PS & $mb$ & 1 + &\begin{minipage}[t]{3in} + {Surface Pressure} + \end{minipage}\\ + QQAVE & $(m/sec)^2$ & Nrphys + &\begin{minipage}[t]{3in} + {Time-averaged $Turbulent Kinetic Energy$} + \end{minipage}\\ + SWGCLR & $Watts/m^2$ & 1 + &\begin{minipage}[t]{3in} + {Net downward clearsky Shortwave flux at the ground} + \end{minipage}\\ + PAVE & $mb$ & 1 + &\begin{minipage}[t]{3in} + {Time-averaged Surface Pressure} + \end{minipage}\\ + DIABU & $m/sec/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Total Diabatic forcing on $u-Wind$} + \end{minipage}\\ + DIABV & $m/sec/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Total Diabatic forcing on $v-Wind$} + \end{minipage}\\ + DIABT & $deg/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Total Diabatic forcing on $Temperature$} + \end{minipage}\\ + DIABQ & $g/kg/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Total Diabatic forcing on $Specific \, \, Humidity$} + \end{minipage}\\ + RFU & $m/sec/day$ & Nrphys + &\begin{minipage}[t]{3in} + {U-Wind tendency due to Rayleigh Friction} + \end{minipage}\\ + RFV & $m/sec/day$ & Nrphys + &\begin{minipage}[t]{3in} + {V-Wind tendency due to Rayleigh Friction} + \end{minipage}\\ + GWDU & $m/sec/day$ & Nrphys + &\begin{minipage}[t]{3in} + {U-Wind tendency due to Gravity Wave Drag} + \end{minipage}\\ + GWDU & $m/sec/day$ & Nrphys + &\begin{minipage}[t]{3in} + {V-Wind tendency due to Gravity Wave Drag} + \end{minipage}\\ + GWDUS & $N/m^2$ & 1 + &\begin{minipage}[t]{3in} + {U-Wind Gravity Wave Drag Stress at Surface} + \end{minipage}\\ + GWDVS & $N/m^2$ & 1 + &\begin{minipage}[t]{3in} + {V-Wind Gravity Wave Drag Stress at Surface} + \end{minipage}\\ + GWDUT & $N/m^2$ & 1 + &\begin{minipage}[t]{3in} + {U-Wind Gravity Wave Drag Stress at Top} + \end{minipage}\\ + GWDVT & $N/m^2$ & 1 + &\begin{minipage}[t]{3in} + {V-Wind Gravity Wave Drag Stress at Top} + \end{minipage}\\ + LZRAD & $mg/kg$ & Nrphys + &\begin{minipage}[t]{3in} + {Estimated Cloud Liquid Water used in Radiation} + \end{minipage}\\ +\end{tabular} +\vfill + +\newpage +\vspace*{\fill} +\begin{tabular}{llll} +\hline\hline + NAME & UNITS & LEVELS & DESCRIPTION \\ +\hline + +&\\ + SLP & $mb$ & 1 + &\begin{minipage}[t]{3in} + {Time-averaged Sea-level Pressure} + \end{minipage}\\ + CLDFRC & $0-1$ & 1 + &\begin{minipage}[t]{3in} + {Total Cloud Fraction} + \end{minipage}\\ + TPW & $gm/cm^2$ & 1 + &\begin{minipage}[t]{3in} + {Precipitable water} + \end{minipage}\\ + U2M & $m/sec$ & 1 + &\begin{minipage}[t]{3in} + {U-Wind at 2 meters} + \end{minipage}\\ + V2M & $m/sec$ & 1 + &\begin{minipage}[t]{3in} + {V-Wind at 2 meters} + \end{minipage}\\ + T2M & $deg$ & 1 + &\begin{minipage}[t]{3in} + {Temperature at 2 meters} + \end{minipage}\\ + Q2M & $g/kg$ & 1 + &\begin{minipage}[t]{3in} + {Specific Humidity at 2 meters} + \end{minipage}\\ + U10M & $m/sec$ & 1 + &\begin{minipage}[t]{3in} + {U-Wind at 10 meters} + \end{minipage}\\ + V10M & $m/sec$ & 1 + &\begin{minipage}[t]{3in} + {V-Wind at 10 meters} + \end{minipage}\\ + T10M & $deg$ & 1 + &\begin{minipage}[t]{3in} + {Temperature at 10 meters} + \end{minipage}\\ + Q10M & $g/kg$ & 1 + &\begin{minipage}[t]{3in} + {Specific Humidity at 10 meters} + \end{minipage}\\ + DTRAIN & $kg/m^2$ & Nrphys + &\begin{minipage}[t]{3in} + {Detrainment Cloud Mass Flux} + \end{minipage}\\ + QFILL & $g/kg/day$ & Nrphys + &\begin{minipage}[t]{3in} + {Filling of negative specific humidity} + \end{minipage}\\ +\end{tabular} +\vspace{1.5in} +\vfill + +\newpage +\vspace*{\fill} +\begin{tabular}{llll} +\hline\hline + NAME & UNITS & LEVELS & DESCRIPTION \\ +\hline + +&\\ + DTCONV & $deg/sec$ & Nr + &\begin{minipage}[t]{3in} + {Temp Change due to Convection} + \end{minipage}\\ + DQCONV & $g/kg/sec$ & Nr + &\begin{minipage}[t]{3in} + {Specific Humidity Change due to Convection} + \end{minipage}\\ + RELHUM & $percent$ & Nr + &\begin{minipage}[t]{3in} + {Relative Humidity} + \end{minipage}\\ + PRECLS & $g/m^2/sec$ & 1 + &\begin{minipage}[t]{3in} + {Large Scale Precipitation} + \end{minipage}\\ + ENPREC & $J/g$ & 1 + &\begin{minipage}[t]{3in} + {Energy of Precipitation (snow, rain Temp)} + \end{minipage}\\ +\end{tabular} +\vspace{1.5in} +\vfill + +\newpage + +\subsubsection{Fizhi Diagnostic Description} + +In this section we list and describe the diagnostic quantities available within the +GCM. The diagnostics are listed in the order that they appear in the +Diagnostic Menu, Section \ref{sec:fizhi-diagnostics:menu}. +In all cases, each diagnostic as currently archived on the output datasets +is time-averaged over its diagnostic output frequency: + +\[ +{\bf DIAGNOSTIC} = {1 \over TTOT} \sum_{t=1}^{t=TTOT} diag(t) +\] +where $TTOT = {{\bf NQDIAG} \over \Delta t}$, {\bf NQDIAG} is the +output frequency of the diagnostic, and $\Delta t$ is +the timestep over which the diagnostic is updated. + +{ \underline {UFLUX} Surface Zonal Wind Stress on the Atmosphere ($Newton/m^2$) } + +The zonal wind stress is the turbulent flux of zonal momentum from +the surface. +\[ +{\bf UFLUX} = - \rho C_D W_s u \hspace{1cm}where: \hspace{.2cm}C_D = C^2_u +\] +where $\rho$ = the atmospheric density at the surface, $C_{D}$ is the surface +drag coefficient, $C_u$ is the dimensionless surface exchange coefficient for momentum +(see diagnostic number 10), $W_s$ is the magnitude of the surface layer wind, and $u$ is +the zonal wind in the lowest model layer. +\\ + + +{ \underline {VFLUX} Surface Meridional Wind Stress on the Atmosphere ($Newton/m^2$) } + +The meridional wind stress is the turbulent flux of meridional momentum from +the surface. +\[ +{\bf VFLUX} = - \rho C_D W_s v \hspace{1cm}where: \hspace{.2cm}C_D = C^2_u +\] +where $\rho$ = the atmospheric density at the surface, $C_{D}$ is the surface +drag coefficient, $C_u$ is the dimensionless surface exchange coefficient for momentum +(see diagnostic number 10), $W_s$ is the magnitude of the surface layer wind, and $v$ is +the meridional wind in the lowest model layer. +\\ + +{ \underline {HFLUX} Surface Flux of Sensible Heat ($Watts/m^2$) } + +The turbulent flux of sensible heat from the surface to the atmosphere is a function of the +gradient of virtual potential temperature and the eddy exchange coefficient: +\[ +{\bf HFLUX} = P^{\kappa}\rho c_{p} C_{H} W_s (\theta_{surface} - \theta_{Nrphys}) +\hspace{1cm}where: \hspace{.2cm}C_H = C_u C_t +\] +where $\rho$ = the atmospheric density at the surface, $c_{p}$ is the specific +heat of air, $C_{H}$ is the dimensionless surface heat transfer coefficient, $W_s$ is the +magnitude of the surface layer wind, $C_u$ is the dimensionless surface exchange coefficient +for momentum (see diagnostic number 10), $C_t$ is the dimensionless surface exchange coefficient +for heat and moisture (see diagnostic number 9), and $\theta$ is the potential temperature +at the surface and at the bottom model level. +\\ + + +{ \underline {EFLUX} Surface Flux of Latent Heat ($Watts/m^2$) } + +The turbulent flux of latent heat from the surface to the atmosphere is a function of the +gradient of moisture, the potential evapotranspiration fraction and the eddy exchange coefficient: +\[ +{\bf EFLUX} = \rho \beta L C_{H} W_s (q_{surface} - q_{Nrphys}) +\hspace{1cm}where: \hspace{.2cm}C_H = C_u C_t +\] +where $\rho$ = the atmospheric density at the surface, $\beta$ is the fraction of +the potential evapotranspiration actually evaporated, L is the latent +heat of evaporation, $C_{H}$ is the dimensionless surface heat transfer coefficient, $W_s$ is the +magnitude of the surface layer wind, $C_u$ is the dimensionless surface exchange coefficient +for momentum (see diagnostic number 10), $C_t$ is the dimensionless surface exchange coefficient +for heat and moisture (see diagnostic number 9), and $q_{surface}$ and $q_{Nrphys}$ are the specific +humidity at the surface and at the bottom model level, respectively. +\\ + +{ \underline {QICE} Heat Conduction Through Sea Ice ($Watts/m^2$) } + +Over sea ice there is an additional source of energy at the surface due to the heat +conduction from the relatively warm ocean through the sea ice. The heat conduction +through sea ice represents an additional energy source term for the ground temperature equation. + +\[ +{\bf QICE} = {C_{ti} \over {H_i}} (T_i-T_g) +\] + +where $C_{ti}$ is the thermal conductivity of ice, $H_i$ is the ice thickness, assumed to +be $3 \hspace{.1cm} m$ where sea ice is present, $T_i$ is 273 degrees Kelvin, and +$T_g$ is the temperature of the sea ice. + +NOTE: QICE is not available through model version 5.3, but is available in subsequent versions. +\\ + + +{ \underline {RADLWG} Net upward Longwave Flux at the surface ($Watts/m^2$)} + +\begin{eqnarray*} +{\bf RADLWG} & = & F_{LW,Nrphys+1}^{Net} \\ + & = & F_{LW,Nrphys+1}^\uparrow - F_{LW,Nrphys+1}^\downarrow +\end{eqnarray*} +\\ +where Nrphys+1 indicates the lowest model edge-level, or $p = p_{surf}$. +$F_{LW}^\uparrow$ is +the upward Longwave flux and $F_{LW}^\downarrow$ is the downward Longwave flux. +\\ + +{ \underline {RADSWG} Net downard shortwave Flux at the surface ($Watts/m^2$)} + +\begin{eqnarray*} +{\bf RADSWG} & = & F_{SW,Nrphys+1}^{Net} \\ + & = & F_{SW,Nrphys+1}^\downarrow - F_{SW,Nrphys+1}^\uparrow +\end{eqnarray*} +\\ +where Nrphys+1 indicates the lowest model edge-level, or $p = p_{surf}$. +$F_{SW}^\downarrow$ is +the downward Shortwave flux and $F_{SW}^\uparrow$ is the upward Shortwave flux. +\\ + + +\noindent +{ \underline {RI} Richardson Number} ($dimensionless$) + +\noindent +The non-dimensional stability indicator is the ratio of the buoyancy to the shear: +\[ +{\bf RI} = { { {g \over \theta_v} \pp {\theta_v}{z} } \over { (\pp{u}{z})^2 + (\pp{v}{z})^2 } } + = { {c_p \pp{\theta_v}{z} \pp{P^ \kappa}{z} } \over { (\pp{u}{z})^2 + (\pp{v}{z})^2 } } +\] +\\ +where we used the hydrostatic equation: +\[ +{\pp{\Phi}{P^ \kappa}} = c_p \theta_v +\] +Negative values indicate unstable buoyancy {\bf{AND}} shear, small positive values ($<0.4$) +indicate dominantly unstable shear, and large positive values indicate dominantly stable +stratification. +\\ + +\noindent +{ \underline {CT} Surface Exchange Coefficient for Temperature and Moisture ($dimensionless$) } + +\noindent +The surface exchange coefficient is obtained from the similarity functions for the stability + dependant flux profile relationships: +\[ +{\bf CT} = -{( {\overline{w^{\prime}\theta^{\prime}}}) \over {u_* \Delta \theta }} = +-{( {\overline{w^{\prime}q^{\prime}}}) \over {u_* \Delta q }} = +{ k \over { (\psi_{h} + \psi_{g}) } } +\] +where $\psi_h$ is the surface layer non-dimensional temperature change and $\psi_g$ is the +viscous sublayer non-dimensional temperature or moisture change: +\[ +\psi_{h} = {\int_{\zeta_{0}}^{\zeta} {\phi_{h} \over \zeta} d \zeta} \hspace{1cm} and +\hspace{1cm} \psi_{g} = { 0.55 (Pr^{2/3} - 0.2) \over \nu^{1/2} } +(h_{0}u_{*} - h_{0_{ref}}u_{*_{ref}})^{1/2} +\] +and: +$h_{0} = 30z_{0}$ with a maximum value over land of 0.01 + +\noindent +$\phi_h$ is the similarity function of $\zeta$, which expresses the stability dependance of +the temperature and moisture gradients, specified differently for stable and unstable +layers according to Helfand and Schubert, 1993. k is the Von Karman constant, $\zeta$ is the +non-dimensional stability parameter, Pr is the Prandtl number for air, $\nu$ is the molecular +viscosity, $z_{0}$ is the surface roughness length, $u_*$ is the surface stress velocity +(see diagnostic number 67), and the subscript ref refers to a reference value. +\\ + +\noindent +{ \underline {CU} Surface Exchange Coefficient for Momentum ($dimensionless$) } + +\noindent +The surface exchange coefficient is obtained from the similarity functions for the stability + dependant flux profile relationships: +\[ +{\bf CU} = {u_* \over W_s} = { k \over \psi_{m} } +\] +where $\psi_m$ is the surface layer non-dimensional wind shear: +\[ +\psi_{m} = {\int_{\zeta_{0}}^{\zeta} {\phi_{m} \over \zeta} d \zeta} +\] +\noindent +$\phi_m$ is the similarity function of $\zeta$, which expresses the stability dependance of +the temperature and moisture gradients, specified differently for stable and unstable layers +according to Helfand and Schubert, 1993. k is the Von Karman constant, $\zeta$ is the +non-dimensional stability parameter, $u_*$ is the surface stress velocity +(see diagnostic number 67), and $W_s$ is the magnitude of the surface layer wind. +\\ + +\noindent +{ \underline {ET} Diffusivity Coefficient for Temperature and Moisture ($m^2/sec$) } + +\noindent +In the level 2.5 version of the Mellor-Yamada (1974) hierarchy, the turbulent heat or +moisture flux for the atmosphere above the surface layer can be expressed as a turbulent +diffusion coefficient $K_h$ times the negative of the gradient of potential temperature +or moisture. In the Helfand and Labraga (1988) adaptation of this closure, $K_h$ +takes the form: +\[ +{\bf ET} = K_h = -{( {\overline{w^{\prime}\theta_v^{\prime}}}) \over {\pp{\theta_v}{z}} } + = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell \, S_H(G_M,G_H) & \mbox{decaying turbulence} +\\ { q^2 \over {q_e} } \, \ell \, S_{H}(G_{M_e},G_{H_e}) & \mbox{growing turbulence} \end{array} \right. +\] +where $q$ is the turbulent velocity, or $\sqrt{2*turbulent \hspace{.2cm} kinetic \hspace{.2cm} +energy}$, $q_e$ is the turbulence velocity derived from the more simple level 2.0 model, +which describes equilibrium turbulence, $\ell$ is the master length scale related to the layer +depth, +$S_H$ is a function of $G_H$ and $G_M$, the dimensionless buoyancy and +wind shear parameters, respectively, or a function of $G_{H_e}$ and $G_{M_e}$, the equilibrium +dimensionless buoyancy and wind shear +parameters. Both $G_H$ and $G_M$, and their equilibrium values $G_{H_e}$ and $G_{M_e}$, +are functions of the Richardson number. + +\noindent +For the detailed equations and derivations of the modified level 2.5 closure scheme, +see Helfand and Labraga, 1988. + +\noindent +In the surface layer, ${\bf {ET}}$ is the exchange coefficient for heat and moisture, +in units of $m/sec$, given by: +\[ +{\bf ET_{Nrphys}} = C_t * u_* = C_H W_s +\] +\noindent +where $C_t$ is the dimensionless exchange coefficient for heat and moisture from the +surface layer similarity functions (see diagnostic number 9), $u_*$ is the surface +friction velocity (see diagnostic number 67), $C_H$ is the heat transfer coefficient, +and $W_s$ is the magnitude of the surface layer wind. +\\ + +\noindent +{ \underline {EU} Diffusivity Coefficient for Momentum ($m^2/sec$) } + +\noindent +In the level 2.5 version of the Mellor-Yamada (1974) hierarchy, the turbulent heat +momentum flux for the atmosphere above the surface layer can be expressed as a turbulent +diffusion coefficient $K_m$ times the negative of the gradient of the u-wind. +In the Helfand and Labraga (1988) adaptation of this closure, $K_m$ +takes the form: +\[ +{\bf EU} = K_m = -{( {\overline{u^{\prime}w^{\prime}}}) \over {\pp{U}{z}} } + = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell \, S_M(G_M,G_H) & \mbox{decaying turbulence} +\\ { q^2 \over {q_e} } \, \ell \, S_{M}(G_{M_e},G_{H_e}) & \mbox{growing turbulence} \end{array} \right. +\] +\noindent +where $q$ is the turbulent velocity, or $\sqrt{2*turbulent \hspace{.2cm} kinetic \hspace{.2cm} +energy}$, $q_e$ is the turbulence velocity derived from the more simple level 2.0 model, +which describes equilibrium turbulence, $\ell$ is the master length scale related to the layer +depth, +$S_M$ is a function of $G_H$ and $G_M$, the dimensionless buoyancy and +wind shear parameters, respectively, or a function of $G_{H_e}$ and $G_{M_e}$, the equilibrium +dimensionless buoyancy and wind shear +parameters. Both $G_H$ and $G_M$, and their equilibrium values $G_{H_e}$ and $G_{M_e}$, +are functions of the Richardson number. + +\noindent +For the detailed equations and derivations of the modified level 2.5 closure scheme, +see Helfand and Labraga, 1988. + +\noindent +In the surface layer, ${\bf {EU}}$ is the exchange coefficient for momentum, +in units of $m/sec$, given by: +\[ +{\bf EU_{Nrphys}} = C_u * u_* = C_D W_s +\] +\noindent +where $C_u$ is the dimensionless exchange coefficient for momentum from the surface layer +similarity functions (see diagnostic number 10), $u_*$ is the surface friction velocity +(see diagnostic number 67), $C_D$ is the surface drag coefficient, and $W_s$ is the +magnitude of the surface layer wind. +\\ + +\noindent +{ \underline {TURBU} Zonal U-Momentum changes due to Turbulence ($m/sec/day$) } + +\noindent +The tendency of U-Momentum due to turbulence is written: +\[ +{\bf TURBU} = {\pp{u}{t}}_{turb} = {\pp{}{z} }{(- \overline{u^{\prime}w^{\prime}})} + = {\pp{}{z} }{(K_m \pp{u}{z})} +\] + +\noindent +The Helfand and Labraga level 2.5 scheme models the turbulent +flux of u-momentum in terms of $K_m$, and the equation has the form of a diffusion +equation. + +\noindent +{ \underline {TURBV} Meridional V-Momentum changes due to Turbulence ($m/sec/day$) } + +\noindent +The tendency of V-Momentum due to turbulence is written: +\[ +{\bf TURBV} = {\pp{v}{t}}_{turb} = {\pp{}{z} }{(- \overline{v^{\prime}w^{\prime}})} + = {\pp{}{z} }{(K_m \pp{v}{z})} +\] + +\noindent +The Helfand and Labraga level 2.5 scheme models the turbulent +flux of v-momentum in terms of $K_m$, and the equation has the form of a diffusion +equation. +\\ + +\noindent +{ \underline {TURBT} Temperature changes due to Turbulence ($deg/day$) } + +\noindent +The tendency of temperature due to turbulence is written: +\[ +{\bf TURBT} = {\pp{T}{t}} = P^{\kappa}{\pp{\theta}{t}}_{turb} = +P^{\kappa}{\pp{}{z} }{(- \overline{w^{\prime}\theta^{\prime}})} + = P^{\kappa}{\pp{}{z} }{(K_h \pp{\theta_v}{z})} +\] + +\noindent +The Helfand and Labraga level 2.5 scheme models the turbulent +flux of temperature in terms of $K_h$, and the equation has the form of a diffusion +equation. +\\ + +\noindent +{ \underline {TURBQ} Specific Humidity changes due to Turbulence ($g/kg/day$) } + +\noindent +The tendency of specific humidity due to turbulence is written: +\[ +{\bf TURBQ} = {\pp{q}{t}}_{turb} = {\pp{}{z} }{(- \overline{w^{\prime}q^{\prime}})} + = {\pp{}{z} }{(K_h \pp{q}{z})} +\] + +\noindent +The Helfand and Labraga level 2.5 scheme models the turbulent +flux of temperature in terms of $K_h$, and the equation has the form of a diffusion +equation. +\\ + +\noindent +{ \underline {MOISTT} Temperature Changes Due to Moist Processes ($deg/day$) } + +\noindent +\[ +{\bf MOISTT} = \left. {\pp{T}{t}}\right|_{c} + \left. {\pp{T}{t}} \right|_{ls} +\] +where: +\[ +\left.{\pp{T}{t}}\right|_{c} = R \sum_i \left( \alpha { m_B \over c_p} \Gamma_s \right)_i +\hspace{.4cm} and +\hspace{.4cm} \left.{\pp{T}{t}}\right|_{ls} = {L \over c_p } (q^*-q) +\] +and +\[ +\Gamma_s = g \eta \pp{s}{p} +\] + +\noindent +The subscript $c$ refers to convective processes, while the subscript $ls$ refers to large scale +precipitation processes, or supersaturation rain. +The summation refers to contributions from each cloud type called by RAS. +The dry static energy is given +as $s$, the convective cloud base mass flux is given as $m_B$, and the cloud entrainment is +given as $\eta$, which are explicitly defined in Section \ref{sec:fizhi:mc}, +the description of the convective parameterization. The fractional adjustment, or relaxation +parameter, for each cloud type is given as $\alpha$, while +$R$ is the rain re-evaporation adjustment. +\\ + +\noindent +{ \underline {MOISTQ} Specific Humidity Changes Due to Moist Processes ($g/kg/day$) } + +\noindent +\[ +{\bf MOISTQ} = \left. {\pp{q}{t}}\right|_{c} + \left. {\pp{q}{t}} \right|_{ls} +\] +where: +\[ +\left.{\pp{q}{t}}\right|_{c} = R \sum_i \left( \alpha { m_B \over {L}}(\Gamma_h-\Gamma_s) \right)_i +\hspace{.4cm} and +\hspace{.4cm} \left.{\pp{q}{t}}\right|_{ls} = (q^*-q) +\] +and +\[ +\Gamma_s = g \eta \pp{s}{p}\hspace{.4cm} and \hspace{.4cm}\Gamma_h = g \eta \pp{h}{p} +\] +\noindent +The subscript $c$ refers to convective processes, while the subscript $ls$ refers to large scale +precipitation processes, or supersaturation rain. +The summation refers to contributions from each cloud type called by RAS. +The dry static energy is given as $s$, +the moist static energy is given as $h$, +the convective cloud base mass flux is given as $m_B$, and the cloud entrainment is +given as $\eta$, which are explicitly defined in Section \ref{sec:fizhi:mc}, +the description of the convective parameterization. The fractional adjustment, or relaxation +parameter, for each cloud type is given as $\alpha$, while +$R$ is the rain re-evaporation adjustment. +\\ + +\noindent +{ \underline {RADLW} Heating Rate due to Longwave Radiation ($deg/day$) } + +\noindent +The net longwave heating rate is calculated as the vertical divergence of the +net terrestrial radiative fluxes. +Both the clear-sky and cloudy-sky longwave fluxes are computed within the +longwave routine. +The subroutine calculates the clear-sky flux, $F^{clearsky}_{LW}$, first. +For a given cloud fraction, +the clear line-of-sight probability $C(p,p^{\prime})$ is computed from the current level pressure $p$ +to the model top pressure, $p^{\prime} = p_{top}$, and the model surface pressure, $p^{\prime} = p_{surf}$, +for the upward and downward radiative fluxes. +(see Section \ref{sec:fizhi:radcloud}). +The cloudy-sky flux is then obtained as: + +\noindent +\[ +F_{LW} = C(p,p') \cdot F^{clearsky}_{LW}, +\] + +\noindent +Finally, the net longwave heating rate is calculated as the vertical divergence of the +net terrestrial radiative fluxes: +\[ +\pp{\rho c_p T}{t} = - {\partial \over \partial z} F_{LW}^{NET} , +\] +or +\[ +{\bf RADLW} = \frac{g}{c_p \pi} {\partial \over \partial \sigma} F_{LW}^{NET} . +\] + +\noindent +where $g$ is the accelation due to gravity, +$c_p$ is the heat capacity of air at constant pressure, +and +\[ +F_{LW}^{NET} = F_{LW}^\uparrow - F_{LW}^\downarrow +\] +\\ + + +\noindent +{ \underline {RADSW} Heating Rate due to Shortwave Radiation ($deg/day$) } + +\noindent +The net Shortwave heating rate is calculated as the vertical divergence of the +net solar radiative fluxes. +The clear-sky and cloudy-sky shortwave fluxes are calculated separately. +For the clear-sky case, the shortwave fluxes and heating rates are computed with +both CLMO (maximum overlap cloud fraction) and +CLRO (random overlap cloud fraction) set to zero (see Section \ref{sec:fizhi:radcloud}). +The shortwave routine is then called a second time, for the cloudy-sky case, with the +true time-averaged cloud fractions CLMO +and CLRO being used. In all cases, a normalized incident shortwave flux is used as +input at the top of the atmosphere. + +\noindent +The heating rate due to Shortwave Radiation under cloudy skies is defined as: +\[ +\pp{\rho c_p T}{t} = - {\partial \over \partial z} F(cloudy)_{SW}^{NET} \cdot {\rm RADSWT}, +\] +or +\[ +{\bf RADSW} = \frac{g}{c_p \pi} {\partial \over \partial \sigma} F(cloudy)_{SW}^{NET}\cdot {\rm RADSWT} . +\] + +\noindent +where $g$ is the accelation due to gravity, +$c_p$ is the heat capacity of air at constant pressure, RADSWT is the true incident +shortwave radiation at the top of the atmosphere (See Diagnostic \#48), and +\[ +F(cloudy)_{SW}^{Net} = F(cloudy)_{SW}^\uparrow - F(cloudy)_{SW}^\downarrow +\] +\\ + +\noindent +{ \underline {PREACC} Total (Large-scale + Convective) Accumulated Precipition ($mm/day$) } + +\noindent +For a change in specific humidity due to moist processes, $\Delta q_{moist}$, +the vertical integral or total precipitable amount is given by: +\[ +{\bf PREACC} = \int_{surf}^{top} \rho \Delta q_{moist} dz = - \int_{surf}^{top} \Delta q_{moist} +{dp \over g} = {1 \over g} \int_0^1 \Delta q_{moist} dp +\] +\\ + +\noindent +A precipitation rate is defined as the vertically integrated moisture adjustment per Moist Processes +time step, scaled to $mm/day$. +\\ + +\noindent +{ \underline {PRECON} Convective Precipition ($mm/day$) } + +\noindent +For a change in specific humidity due to sub-grid scale cumulus convective processes, $\Delta q_{cum}$, +the vertical integral or total precipitable amount is given by: +\[ +{\bf PRECON} = \int_{surf}^{top} \rho \Delta q_{cum} dz = - \int_{surf}^{top} \Delta q_{cum} +{dp \over g} = {1 \over g} \int_0^1 \Delta q_{cum} dp +\] +\\ + +\noindent +A precipitation rate is defined as the vertically integrated moisture adjustment per Moist Processes +time step, scaled to $mm/day$. +\\ + +\noindent +{ \underline {TUFLUX} Turbulent Flux of U-Momentum ($Newton/m^2$) } + +\noindent +The turbulent flux of u-momentum is calculated for $diagnostic \hspace{.2cm} purposes + \hspace{.2cm} only$ from the eddy coefficient for momentum: + +\[ +{\bf TUFLUX} = {\rho } {(\overline{u^{\prime}w^{\prime}})} = +{\rho } {(- K_m \pp{U}{z})} +\] + +\noindent +where $\rho$ is the air density, and $K_m$ is the eddy coefficient. +\\ + +\noindent +{ \underline {TVFLUX} Turbulent Flux of V-Momentum ($Newton/m^2$) } + +\noindent +The turbulent flux of v-momentum is calculated for $diagnostic \hspace{.2cm} purposes +\hspace{.2cm} only$ from the eddy coefficient for momentum: + +\[ +{\bf TVFLUX} = {\rho } {(\overline{v^{\prime}w^{\prime}})} = + {\rho } {(- K_m \pp{V}{z})} +\] + +\noindent +where $\rho$ is the air density, and $K_m$ is the eddy coefficient. +\\ + + +\noindent +{ \underline {TTFLUX} Turbulent Flux of Sensible Heat ($Watts/m^2$) } + +\noindent +The turbulent flux of sensible heat is calculated for $diagnostic \hspace{.2cm} purposes +\hspace{.2cm} only$ from the eddy coefficient for heat and moisture: + +\noindent +\[ +{\bf TTFLUX} = c_p {\rho } +P^{\kappa}{(\overline{w^{\prime}\theta^{\prime}})} + = c_p {\rho } P^{\kappa}{(- K_h \pp{\theta_v}{z})} +\] + +\noindent +where $\rho$ is the air density, and $K_h$ is the eddy coefficient. +\\ + + +\noindent +{ \underline {TQFLUX} Turbulent Flux of Latent Heat ($Watts/m^2$) } + +\noindent +The turbulent flux of latent heat is calculated for $diagnostic \hspace{.2cm} purposes +\hspace{.2cm} only$ from the eddy coefficient for heat and moisture: + +\noindent +\[ +{\bf TQFLUX} = {L {\rho } (\overline{w^{\prime}q^{\prime}})} = +{L {\rho }(- K_h \pp{q}{z})} +\] + +\noindent +where $\rho$ is the air density, and $K_h$ is the eddy coefficient. +\\ + + +\noindent +{ \underline {CN} Neutral Drag Coefficient ($dimensionless$) } + +\noindent +The drag coefficient for momentum obtained by assuming a neutrally stable surface layer: +\[ +{\bf CN} = { k \over { \ln({h \over {z_0}})} } +\] + +\noindent +where $k$ is the Von Karman constant, $h$ is the height of the surface layer, and +$z_0$ is the surface roughness. + +\noindent +NOTE: CN is not available through model version 5.3, but is available in subsequent +versions. +\\ + +\noindent +{ \underline {WINDS} Surface Wind Speed ($meter/sec$) } + +\noindent +The surface wind speed is calculated for the last internal turbulence time step: +\[ +{\bf WINDS} = \sqrt{u_{Nrphys}^2 + v_{Nrphys}^2} +\] + +\noindent +where the subscript $Nrphys$ refers to the lowest model level. +\\ + +\noindent +{ \underline {DTSRF} Air/Surface Virtual Temperature Difference ($deg \hspace{.1cm} K$) } + +\noindent +The air/surface virtual temperature difference measures the stability of the surface layer: +\[ +{\bf DTSRF} = (\theta_{v{Nrphys+1}} - \theta{v_{Nrphys}}) P^{\kappa}_{surf} +\] +\noindent +where +\[ +\theta_{v{Nrphys+1}} = { T_g \over {P^{\kappa}_{surf}} } (1 + .609 q_{Nrphys+1}) \hspace{1cm} +and \hspace{1cm} q_{Nrphys+1} = q_{Nrphys} + \beta(q^*(T_g,P_s) - q_{Nrphys}) +\] + +\noindent +$\beta$ is the surface potential evapotranspiration coefficient ($\beta=1$ over oceans), +$q^*(T_g,P_s)$ is the saturation specific humidity at the ground temperature +and surface pressure, level $Nrphys$ refers to the lowest model level and level $Nrphys+1$ +refers to the surface. +\\ + + +\noindent +{ \underline {TG} Ground Temperature ($deg \hspace{.1cm} K$) } + +\noindent +The ground temperature equation is solved as part of the turbulence package +using a backward implicit time differencing scheme: +\[ +{\bf TG} \hspace{.1cm} is \hspace{.1cm} obtained \hspace{.1cm} from: \hspace{.1cm} +C_g\pp{T_g}{t} = R_{sw} - R_{lw} + Q_{ice} - H - LE +\] + +\noindent +where $R_{sw}$ is the net surface downward shortwave radiative flux, $R_{lw}$ is the +net surface upward longwave radiative flux, $Q_{ice}$ is the heat conduction through +sea ice, $H$ is the upward sensible heat flux, $LE$ is the upward latent heat +flux, and $C_g$ is the total heat capacity of the ground. +$C_g$ is obtained by solving a heat diffusion equation +for the penetration of the diurnal cycle into the ground (Blackadar, 1977), and is given by: +\[ +C_g = \sqrt{ {\lambda C_s \over {2 \omega} } } = \sqrt{(0.386 + 0.536W + 0.15W^2)2x10^{-3} +{ 86400. \over {2 \pi} } } \, \, . +\] +\noindent +Here, the thermal conductivity, $\lambda$, is equal to $2x10^{-3}$ ${ly\over{ sec}} +{cm \over {^oK}}$, +the angular velocity of the earth, $\omega$, is written as $86400$ $sec/day$ divided +by $2 \pi$ $radians/ +day$, and the expression for $C_s$, the heat capacity per unit volume at the surface, +is a function of the ground wetness, $W$. +\\ + +\noindent +{ \underline {TS} Surface Temperature ($deg \hspace{.1cm} K$) } + +\noindent +The surface temperature estimate is made by assuming that the model's lowest +layer is well-mixed, and therefore that $\theta$ is constant in that layer. +The surface temperature is therefore: +\[ +{\bf TS} = \theta_{Nrphys} P^{\kappa}_{surf} +\] +\\ + +\noindent +{ \underline {DTG} Surface Temperature Adjustment ($deg \hspace{.1cm} K$) } + +\noindent +The change in surface temperature from one turbulence time step to the next, solved +using the Ground Temperature Equation (see diagnostic number 30) is calculated: +\[ +{\bf DTG} = {T_g}^{n} - {T_g}^{n-1} +\] + +\noindent +where superscript $n$ refers to the new, updated time level, and the superscript $n-1$ +refers to the value at the previous turbulence time level. +\\ + +\noindent +{ \underline {QG} Ground Specific Humidity ($g/kg$) } + +\noindent +The ground specific humidity is obtained by interpolating between the specific +humidity at the lowest model level and the specific humidity of a saturated ground. +The interpolation is performed using the potential evapotranspiration function: +\[ +{\bf QG} = q_{Nrphys+1} = q_{Nrphys} + \beta(q^*(T_g,P_s) - q_{Nrphys}) +\] + +\noindent +where $\beta$ is the surface potential evapotranspiration coefficient ($\beta=1$ over oceans), +and $q^*(T_g,P_s)$ is the saturation specific humidity at the ground temperature and surface +pressure. +\\ + +\noindent +{ \underline {QS} Saturation Surface Specific Humidity ($g/kg$) } + +\noindent +The surface saturation specific humidity is the saturation specific humidity at +the ground temprature and surface pressure: +\[ +{\bf QS} = q^*(T_g,P_s) +\] +\\ + +\noindent +{ \underline {TGRLW} Instantaneous ground temperature used as input to the Longwave + radiation subroutine (deg)} +\[ +{\bf TGRLW} = T_g(\lambda , \phi ,n) +\] +\noindent +where $T_g$ is the model ground temperature at the current time step $n$. +\\ + + +\noindent +{ \underline {ST4} Upward Longwave flux at the surface ($Watts/m^2$) } +\[ +{\bf ST4} = \sigma T^4 +\] +\noindent +where $\sigma$ is the Stefan-Boltzmann constant and T is the temperature. +\\ + +\noindent +{ \underline {OLR} Net upward Longwave flux at $p=p_{top}$ ($Watts/m^2$) } +\[ +{\bf OLR} = F_{LW,top}^{NET} +\] +\noindent +where top indicates the top of the first model layer. +In the GCM, $p_{top}$ = 0.0 mb. +\\ + + +\noindent +{ \underline {OLRCLR} Net upward clearsky Longwave flux at $p=p_{top}$ ($Watts/m^2$) } +\[ +{\bf OLRCLR} = F(clearsky)_{LW,top}^{NET} +\] +\noindent +where top indicates the top of the first model layer. +In the GCM, $p_{top}$ = 0.0 mb. +\\ + +\noindent +{ \underline {LWGCLR} Net upward clearsky Longwave flux at the surface ($Watts/m^2$) } + +\noindent +\begin{eqnarray*} +{\bf LWGCLR} & = & F(clearsky)_{LW,Nrphys+1}^{Net} \\ + & = & F(clearsky)_{LW,Nrphys+1}^\uparrow - F(clearsky)_{LW,Nrphys+1}^\downarrow +\end{eqnarray*} +where Nrphys+1 indicates the lowest model edge-level, or $p = p_{surf}$. +$F(clearsky)_{LW}^\uparrow$ is +the upward clearsky Longwave flux and the $F(clearsky)_{LW}^\downarrow$ is the downward clearsky Longwave flux. +\\ + +\noindent +{ \underline {LWCLR} Heating Rate due to Clearsky Longwave Radiation ($deg/day$) } + +\noindent +The net longwave heating rate is calculated as the vertical divergence of the +net terrestrial radiative fluxes. +Both the clear-sky and cloudy-sky longwave fluxes are computed within the +longwave routine. +The subroutine calculates the clear-sky flux, $F^{clearsky}_{LW}$, first. +For a given cloud fraction, +the clear line-of-sight probability $C(p,p^{\prime})$ is computed from the current level pressure $p$ +to the model top pressure, $p^{\prime} = p_{top}$, and the model surface pressure, $p^{\prime} = p_{surf}$, +for the upward and downward radiative fluxes. +(see Section \ref{sec:fizhi:radcloud}). +The cloudy-sky flux is then obtained as: + +\noindent +\[ +F_{LW} = C(p,p') \cdot F^{clearsky}_{LW}, +\] + +\noindent +Thus, {\bf LWCLR} is defined as the net longwave heating rate due to the +vertical divergence of the +clear-sky longwave radiative flux: +\[ +\pp{\rho c_p T}{t}_{clearsky} = - {\partial \over \partial z} F(clearsky)_{LW}^{NET} , +\] +or +\[ +{\bf LWCLR} = \frac{g}{c_p \pi} {\partial \over \partial \sigma} F(clearsky)_{LW}^{NET} . +\] + +\noindent +where $g$ is the accelation due to gravity, +$c_p$ is the heat capacity of air at constant pressure, +and +\[ +F(clearsky)_{LW}^{Net} = F(clearsky)_{LW}^\uparrow - F(clearsky)_{LW}^\downarrow +\] +\\ + + +\noindent +{ \underline {TLW} Instantaneous temperature used as input to the Longwave + radiation subroutine (deg)} +\[ +{\bf TLW} = T(\lambda , \phi ,level, n) +\] +\noindent +where $T$ is the model temperature at the current time step $n$. +\\ + + +\noindent +{ \underline {SHLW} Instantaneous specific humidity used as input to + the Longwave radiation subroutine (kg/kg)} +\[ +{\bf SHLW} = q(\lambda , \phi , level , n) +\] +\noindent +where $q$ is the model specific humidity at the current time step $n$. +\\ + + +\noindent +{ \underline {OZLW} Instantaneous ozone used as input to + the Longwave radiation subroutine (kg/kg)} +\[ +{\bf OZLW} = {\rm OZ}(\lambda , \phi , level , n) +\] +\noindent +where $\rm OZ$ is the interpolated ozone data set from the climatological monthly +mean zonally averaged ozone data set. +\\ + + +\noindent +{ \underline {CLMOLW} Maximum Overlap cloud fraction used in LW Radiation ($0-1$) } + +\noindent +{\bf CLMOLW} is the time-averaged maximum overlap cloud fraction that has been filled by the Relaxed +Arakawa/Schubert Convection scheme and will be used in the Longwave Radiation algorithm. These are +convective clouds whose radiative characteristics are assumed to be correlated in the vertical. +For a complete description of cloud/radiative interactions, see Section \ref{sec:fizhi:radcloud}. +\[ +{\bf CLMOLW} = CLMO_{RAS,LW}(\lambda, \phi, level ) +\] +\\ + + +{ \underline {CLDTOT} Total cloud fraction used in LW and SW Radiation ($0-1$) } + +{\bf CLDTOT} is the time-averaged total cloud fraction that has been filled by the Relaxed +Arakawa/Schubert and Large-scale Convection schemes and will be used in the Longwave and Shortwave +Radiation packages. +For a complete description of cloud/radiative interactions, see Section \ref{sec:fizhi:radcloud}. +\[ +{\bf CLDTOT} = F_{RAS} + F_{LS} +\] +\\ +where $F_{RAS}$ is the time-averaged cloud fraction due to sub-grid scale convection, and $F_{LS}$ is the +time-averaged cloud fraction due to precipitating and non-precipitating large-scale moist processes. +\\ + + +\noindent +{ \underline {CLMOSW} Maximum Overlap cloud fraction used in SW Radiation ($0-1$) } + +\noindent +{\bf CLMOSW} is the time-averaged maximum overlap cloud fraction that has been filled by the Relaxed +Arakawa/Schubert Convection scheme and will be used in the Shortwave Radiation algorithm. These are +convective clouds whose radiative characteristics are assumed to be correlated in the vertical. +For a complete description of cloud/radiative interactions, see Section \ref{sec:fizhi:radcloud}. +\[ +{\bf CLMOSW} = CLMO_{RAS,SW}(\lambda, \phi, level ) +\] +\\ + +\noindent +{ \underline {CLROSW} Random Overlap cloud fraction used in SW Radiation ($0-1$) } + +\noindent +{\bf CLROSW} is the time-averaged random overlap cloud fraction that has been filled by the Relaxed +Arakawa/Schubert and Large-scale Convection schemes and will be used in the Shortwave +Radiation algorithm. These are +convective and large-scale clouds whose radiative characteristics are not +assumed to be correlated in the vertical. +For a complete description of cloud/radiative interactions, see Section \ref{sec:fizhi:radcloud}. +\[ +{\bf CLROSW} = CLRO_{RAS,Large Scale,SW}(\lambda, \phi, level ) +\] +\\ + +\noindent +{ \underline {RADSWT} Incident Shortwave radiation at the top of the atmosphere ($Watts/m^2$) } +\[ +{\bf RADSWT} = {\frac{S_0}{R_a^2}} \cdot cos \phi_z +\] +\noindent +where $S_0$, is the extra-terrestial solar contant, +$R_a$ is the earth-sun distance in Astronomical Units, +and $cos \phi_z$ is the cosine of the zenith angle. +It should be noted that {\bf RADSWT}, as well as +{\bf OSR} and {\bf OSRCLR}, +are calculated at the top of the atmosphere (p=0 mb). However, the +{\bf OLR} and {\bf OLRCLR} diagnostics are currently +calculated at $p= p_{top}$ (0.0 mb for the GCM). +\\ + +\noindent +{ \underline {EVAP} Surface Evaporation ($mm/day$) } + +\noindent +The surface evaporation is a function of the gradient of moisture, the potential +evapotranspiration fraction and the eddy exchange coefficient: +\[ +{\bf EVAP} = \rho \beta K_{h} (q_{surface} - q_{Nrphys}) +\] +where $\rho$ = the atmospheric density at the surface, $\beta$ is the fraction of +the potential evapotranspiration actually evaporated ($\beta=1$ over oceans), $K_{h}$ is the +turbulent eddy exchange coefficient for heat and moisture at the surface in $m/sec$ and +$q{surface}$ and $q_{Nrphys}$ are the specific humidity at the surface (see diagnostic +number 34) and at the bottom model level, respectively. +\\ + +\noindent +{ \underline {DUDT} Total Zonal U-Wind Tendency ($m/sec/day$) } + +\noindent +{\bf DUDT} is the total time-tendency of the Zonal U-Wind due to Hydrodynamic, Diabatic, +and Analysis forcing. +\[ +{\bf DUDT} = \pp{u}{t}_{Dynamics} + \pp{u}{t}_{Moist} + \pp{u}{t}_{Turbulence} + \pp{u}{t}_{Analysis} +\] +\\ + +\noindent +{ \underline {DVDT} Total Zonal V-Wind Tendency ($m/sec/day$) } + +\noindent +{\bf DVDT} is the total time-tendency of the Meridional V-Wind due to Hydrodynamic, Diabatic, +and Analysis forcing. +\[ +{\bf DVDT} = \pp{v}{t}_{Dynamics} + \pp{v}{t}_{Moist} + \pp{v}{t}_{Turbulence} + \pp{v}{t}_{Analysis} +\] +\\ + +\noindent +{ \underline {DTDT} Total Temperature Tendency ($deg/day$) } + +\noindent +{\bf DTDT} is the total time-tendency of Temperature due to Hydrodynamic, Diabatic, +and Analysis forcing. +\begin{eqnarray*} +{\bf DTDT} & = & \pp{T}{t}_{Dynamics} + \pp{T}{t}_{Moist Processes} + \pp{T}{t}_{Shortwave Radiation} \\ + & + & \pp{T}{t}_{Longwave Radiation} + \pp{T}{t}_{Turbulence} + \pp{T}{t}_{Analysis} +\end{eqnarray*} +\\ + +\noindent +{ \underline {DQDT} Total Specific Humidity Tendency ($g/kg/day$) } + +\noindent +{\bf DQDT} is the total time-tendency of Specific Humidity due to Hydrodynamic, Diabatic, +and Analysis forcing. +\[ +{\bf DQDT} = \pp{q}{t}_{Dynamics} + \pp{q}{t}_{Moist Processes} ++ \pp{q}{t}_{Turbulence} + \pp{q}{t}_{Analysis} +\] +\\ + +\noindent +{ \underline {USTAR} Surface-Stress Velocity ($m/sec$) } + +\noindent +The surface stress velocity, or the friction velocity, is the wind speed at +the surface layer top impeded by the surface drag: +\[ +{\bf USTAR} = C_uW_s \hspace{1cm}where: \hspace{.2cm} +C_u = {k \over {\psi_m} } +\] + +\noindent +$C_u$ is the non-dimensional surface drag coefficient (see diagnostic +number 10), and $W_s$ is the surface wind speed (see diagnostic number 28). + +\noindent +{ \underline {Z0} Surface Roughness Length ($m$) } + +\noindent +Over the land surface, the surface roughness length is interpolated to the local +time from the monthly mean data of Dorman and Sellers (1989). Over the ocean, +the roughness length is a function of the surface-stress velocity, $u_*$. +\[ +{\bf Z0} = c_1u^3_* + c_2u^2_* + c_3u_* + c_4 + {c_5 \over {u_*}} +\] + +\noindent +where the constants are chosen to interpolate between the reciprocal relation of +Kondo(1975) for weak winds, and the piecewise linear relation of Large and Pond(1981) +for moderate to large winds. +\\ + +\noindent +{ \underline {FRQTRB} Frequency of Turbulence ($0-1$) } + +\noindent +The fraction of time when turbulence is present is defined as the fraction of +time when the turbulent kinetic energy exceeds some minimum value, defined here +to be $0.005 \hspace{.1cm}m^2/sec^2$. When this criterion is met, a counter is +incremented. The fraction over the averaging interval is reported. +\\ + +\noindent +{ \underline {PBL} Planetary Boundary Layer Depth ($mb$) } + +\noindent +The depth of the PBL is defined by the turbulence parameterization to be the +depth at which the turbulent kinetic energy reduces to ten percent of its surface +value. + +\[ +{\bf PBL} = P_{PBL} - P_{surface} +\] + +\noindent +where $P_{PBL}$ is the pressure in $mb$ at which the turbulent kinetic energy +reaches one tenth of its surface value, and $P_s$ is the surface pressure. +\\ + +\noindent +{ \underline {SWCLR} Clear sky Heating Rate due to Shortwave Radiation ($deg/day$) } + +\noindent +The net Shortwave heating rate is calculated as the vertical divergence of the +net solar radiative fluxes. +The clear-sky and cloudy-sky shortwave fluxes are calculated separately. +For the clear-sky case, the shortwave fluxes and heating rates are computed with +both CLMO (maximum overlap cloud fraction) and +CLRO (random overlap cloud fraction) set to zero (see Section \ref{sec:fizhi:radcloud}). +The shortwave routine is then called a second time, for the cloudy-sky case, with the +true time-averaged cloud fractions CLMO +and CLRO being used. In all cases, a normalized incident shortwave flux is used as +input at the top of the atmosphere. + +\noindent +The heating rate due to Shortwave Radiation under clear skies is defined as: +\[ +\pp{\rho c_p T}{t} = - {\partial \over \partial z} F(clear)_{SW}^{NET} \cdot {\rm RADSWT}, +\] +or +\[ +{\bf SWCLR} = \frac{g}{c_p } {\partial \over \partial p} F(clear)_{SW}^{NET}\cdot {\rm RADSWT} . +\] + +\noindent +where $g$ is the accelation due to gravity, +$c_p$ is the heat capacity of air at constant pressure, RADSWT is the true incident +shortwave radiation at the top of the atmosphere (See Diagnostic \#48), and +\[ +F(clear)_{SW}^{Net} = F(clear)_{SW}^\uparrow - F(clear)_{SW}^\downarrow +\] +\\ + +\noindent +{ \underline {OSR} Net upward Shortwave flux at the top of the model ($Watts/m^2$) } +\[ +{\bf OSR} = F_{SW,top}^{NET} +\] +\noindent +where top indicates the top of the first model layer used in the shortwave radiation +routine. +In the GCM, $p_{SW_{top}}$ = 0 mb. +\\ + +\noindent +{ \underline {OSRCLR} Net upward clearsky Shortwave flux at the top of the model ($Watts/m^2$) } +\[ +{\bf OSRCLR} = F(clearsky)_{SW,top}^{NET} +\] +\noindent +where top indicates the top of the first model layer used in the shortwave radiation +routine. +In the GCM, $p_{SW_{top}}$ = 0 mb. +\\ + + +\noindent +{ \underline {CLDMAS} Convective Cloud Mass Flux ($kg/m^2$) } + +\noindent +The amount of cloud mass moved per RAS timestep from all convective clouds is written: +\[ +{\bf CLDMAS} = \eta m_B +\] +where $\eta$ is the entrainment, normalized by the cloud base mass flux, and $m_B$ is +the cloud base mass flux. $m_B$ and $\eta$ are defined explicitly in Section \ref{sec:fizhi:mc}, the +description of the convective parameterization. +\\ + + + +\noindent +{ \underline {UAVE} Time-Averaged Zonal U-Wind ($m/sec$) } + +\noindent +The diagnostic {\bf UAVE} is simply the time-averaged Zonal U-Wind over +the {\bf NUAVE} output frequency. This is contrasted to the instantaneous +Zonal U-Wind which is archived on the Prognostic Output data stream. +\[ +{\bf UAVE} = u(\lambda, \phi, level , t) +\] +\\ +Note, {\bf UAVE} is computed and stored on the staggered C-grid. +\\ + +\noindent +{ \underline {VAVE} Time-Averaged Meridional V-Wind ($m/sec$) } + +\noindent +The diagnostic {\bf VAVE} is simply the time-averaged Meridional V-Wind over +the {\bf NVAVE} output frequency. This is contrasted to the instantaneous +Meridional V-Wind which is archived on the Prognostic Output data stream. +\[ +{\bf VAVE} = v(\lambda, \phi, level , t) +\] +\\ +Note, {\bf VAVE} is computed and stored on the staggered C-grid. +\\ + +\noindent +{ \underline {TAVE} Time-Averaged Temperature ($Kelvin$) } + +\noindent +The diagnostic {\bf TAVE} is simply the time-averaged Temperature over +the {\bf NTAVE} output frequency. This is contrasted to the instantaneous +Temperature which is archived on the Prognostic Output data stream. +\[ +{\bf TAVE} = T(\lambda, \phi, level , t) +\] +\\ + +\noindent +{ \underline {QAVE} Time-Averaged Specific Humidity ($g/kg$) } + +\noindent +The diagnostic {\bf QAVE} is simply the time-averaged Specific Humidity over +the {\bf NQAVE} output frequency. This is contrasted to the instantaneous +Specific Humidity which is archived on the Prognostic Output data stream. +\[ +{\bf QAVE} = q(\lambda, \phi, level , t) +\] +\\ + +\noindent +{ \underline {PAVE} Time-Averaged Surface Pressure - PTOP ($mb$) } + +\noindent +The diagnostic {\bf PAVE} is simply the time-averaged Surface Pressure - PTOP over +the {\bf NPAVE} output frequency. This is contrasted to the instantaneous +Surface Pressure - PTOP which is archived on the Prognostic Output data stream. +\begin{eqnarray*} +{\bf PAVE} & = & \pi(\lambda, \phi, level , t) \\ + & = & p_s(\lambda, \phi, level , t) - p_T +\end{eqnarray*} +\\ + + +\noindent +{ \underline {QQAVE} Time-Averaged Turbulent Kinetic Energy $(m/sec)^2$ } + +\noindent +The diagnostic {\bf QQAVE} is simply the time-averaged prognostic Turbulent Kinetic Energy +produced by the GCM Turbulence parameterization over +the {\bf NQQAVE} output frequency. This is contrasted to the instantaneous +Turbulent Kinetic Energy which is archived on the Prognostic Output data stream. +\[ +{\bf QQAVE} = qq(\lambda, \phi, level , t) +\] +\\ +Note, {\bf QQAVE} is computed and stored at the ``mass-point'' locations on the staggered C-grid. +\\ + +\noindent +{ \underline {SWGCLR} Net downward clearsky Shortwave flux at the surface ($Watts/m^2$) } + +\noindent +\begin{eqnarray*} +{\bf SWGCLR} & = & F(clearsky)_{SW,Nrphys+1}^{Net} \\ + & = & F(clearsky)_{SW,Nrphys+1}^\downarrow - F(clearsky)_{SW,Nrphys+1}^\uparrow +\end{eqnarray*} +\noindent +\\ +where Nrphys+1 indicates the lowest model edge-level, or $p = p_{surf}$. +$F(clearsky){SW}^\downarrow$ is +the downward clearsky Shortwave flux and $F(clearsky)_{SW}^\uparrow$ is +the upward clearsky Shortwave flux. +\\ + +\noindent +{ \underline {DIABU} Total Diabatic Zonal U-Wind Tendency ($m/sec/day$) } + +\noindent +{\bf DIABU} is the total time-tendency of the Zonal U-Wind due to Diabatic processes +and the Analysis forcing. +\[ +{\bf DIABU} = \pp{u}{t}_{Moist} + \pp{u}{t}_{Turbulence} + \pp{u}{t}_{Analysis} +\] +\\ + +\noindent +{ \underline {DIABV} Total Diabatic Meridional V-Wind Tendency ($m/sec/day$) } + +\noindent +{\bf DIABV} is the total time-tendency of the Meridional V-Wind due to Diabatic processes +and the Analysis forcing. +\[ +{\bf DIABV} = \pp{v}{t}_{Moist} + \pp{v}{t}_{Turbulence} + \pp{v}{t}_{Analysis} +\] +\\ + +\noindent +{ \underline {DIABT} Total Diabatic Temperature Tendency ($deg/day$) } + +\noindent +{\bf DIABT} is the total time-tendency of Temperature due to Diabatic processes +and the Analysis forcing. +\begin{eqnarray*} +{\bf DIABT} & = & \pp{T}{t}_{Moist Processes} + \pp{T}{t}_{Shortwave Radiation} \\ + & + & \pp{T}{t}_{Longwave Radiation} + \pp{T}{t}_{Turbulence} + \pp{T}{t}_{Analysis} +\end{eqnarray*} +\\ +If we define the time-tendency of Temperature due to Diabatic processes as +\begin{eqnarray*} +\pp{T}{t}_{Diabatic} & = & \pp{T}{t}_{Moist Processes} + \pp{T}{t}_{Shortwave Radiation} \\ + & + & \pp{T}{t}_{Longwave Radiation} + \pp{T}{t}_{Turbulence} +\end{eqnarray*} +then, since there are no surface pressure changes due to Diabatic processes, we may write +\[ +\pp{T}{t}_{Diabatic} = {p^\kappa \over \pi }\pp{\pi \theta}{t}_{Diabatic} +\] +where $\theta = T/p^\kappa$. Thus, {\bf DIABT} may be written as +\[ +{\bf DIABT} = {p^\kappa \over \pi } \left( \pp{\pi \theta}{t}_{Diabatic} + \pp{\pi \theta}{t}_{Analysis} \right) +\] +\\ + +\noindent +{ \underline {DIABQ} Total Diabatic Specific Humidity Tendency ($g/kg/day$) } + +\noindent +{\bf DIABQ} is the total time-tendency of Specific Humidity due to Diabatic processes +and the Analysis forcing. +\[ +{\bf DIABQ} = \pp{q}{t}_{Moist Processes} + \pp{q}{t}_{Turbulence} + \pp{q}{t}_{Analysis} +\] +If we define the time-tendency of Specific Humidity due to Diabatic processes as +\[ +\pp{q}{t}_{Diabatic} = \pp{q}{t}_{Moist Processes} + \pp{q}{t}_{Turbulence} +\] +then, since there are no surface pressure changes due to Diabatic processes, we may write +\[ +\pp{q}{t}_{Diabatic} = {1 \over \pi }\pp{\pi q}{t}_{Diabatic} +\] +Thus, {\bf DIABQ} may be written as +\[ +{\bf DIABQ} = {1 \over \pi } \left( \pp{\pi q}{t}_{Diabatic} + \pp{\pi q}{t}_{Analysis} \right) +\] +\\ + +\noindent +{ \underline {VINTUQ} Vertically Integrated Moisture Flux ($m/sec \cdot g/kg$) } + +\noindent +The vertically integrated moisture flux due to the zonal u-wind is obtained by integrating +$u q$ over the depth of the atmosphere at each model timestep, +and dividing by the total mass of the column. +\[ +{\bf VINTUQ} = \frac{ \int_{surf}^{top} u q \rho dz } { \int_{surf}^{top} \rho dz } +\] +Using $\rho \delta z = -{\delta p \over g} = - {1 \over g} \delta p$, we have +\[ +{\bf VINTUQ} = { \int_0^1 u q dp } +\] +\\ + + +\noindent +{ \underline {VINTVQ} Vertically Integrated Moisture Flux ($m/sec \cdot g/kg$) } + +\noindent +The vertically integrated moisture flux due to the meridional v-wind is obtained by integrating +$v q$ over the depth of the atmosphere at each model timestep, +and dividing by the total mass of the column. +\[ +{\bf VINTVQ} = \frac{ \int_{surf}^{top} v q \rho dz } { \int_{surf}^{top} \rho dz } +\] +Using $\rho \delta z = -{\delta p \over g} = - {1 \over g} \delta p$, we have +\[ +{\bf VINTVQ} = { \int_0^1 v q dp } +\] +\\ + + +\noindent +{ \underline {VINTUT} Vertically Integrated Heat Flux ($m/sec \cdot deg$) } + +\noindent +The vertically integrated heat flux due to the zonal u-wind is obtained by integrating +$u T$ over the depth of the atmosphere at each model timestep, +and dividing by the total mass of the column. +\[ +{\bf VINTUT} = \frac{ \int_{surf}^{top} u T \rho dz } { \int_{surf}^{top} \rho dz } +\] +Or, +\[ +{\bf VINTUT} = { \int_0^1 u T dp } +\] +\\ + +\noindent +{ \underline {VINTVT} Vertically Integrated Heat Flux ($m/sec \cdot deg$) } + +\noindent +The vertically integrated heat flux due to the meridional v-wind is obtained by integrating +$v T$ over the depth of the atmosphere at each model timestep, +and dividing by the total mass of the column. +\[ +{\bf VINTVT} = \frac{ \int_{surf}^{top} v T \rho dz } { \int_{surf}^{top} \rho dz } +\] +Using $\rho \delta z = -{\delta p \over g} $, we have +\[ +{\bf VINTVT} = { \int_0^1 v T dp } +\] +\\ + +\noindent +{ \underline {CLDFRC} Total 2-Dimensional Cloud Fracton ($0-1$) } + +If we define the +time-averaged random and maximum overlapped cloudiness as CLRO and +CLMO respectively, then the probability of clear sky associated +with random overlapped clouds at any level is (1-CLRO) while the probability of +clear sky associated with maximum overlapped clouds at any level is (1-CLMO). +The total clear sky probability is given by (1-CLRO)*(1-CLMO), thus +the total cloud fraction at each level may be obtained by +1-(1-CLRO)*(1-CLMO). + +At any given level, we may define the clear line-of-site probability by +appropriately accounting for the maximum and random overlap +cloudiness. The clear line-of-site probability is defined to be +equal to the product of the clear line-of-site probabilities +associated with random and maximum overlap cloudiness. The clear +line-of-site probability $C(p,p^{\prime})$ associated with maximum overlap clouds, +from the current pressure $p$ +to the model top pressure, $p^{\prime} = p_{top}$, or the model surface pressure, $p^{\prime} = p_{surf}$, +is simply 1.0 minus the largest maximum overlap cloud value along the +line-of-site, ie. + +$$1-MAX_p^{p^{\prime}} \left( CLMO_p \right)$$ + +Thus, even in the time-averaged sense it is assumed that the +maximum overlap clouds are correlated in the vertical. The clear +line-of-site probability associated with random overlap clouds is +defined to be the product of the clear sky probabilities at each +level along the line-of-site, ie. + +$$\prod_{p}^{p^{\prime}} \left( 1-CLRO_p \right)$$ + +The total cloud fraction at a given level associated with a line- +of-site calculation is given by + +$$1-\left( 1-MAX_p^{p^{\prime}} \left[ CLMO_p \right] \right) + \prod_p^{p^{\prime}} \left( 1-CLRO_p \right)$$ + + +\noindent +The 2-dimensional net cloud fraction as seen from the top of the +atmosphere is given by +\[ +{\bf CLDFRC} = 1-\left( 1-MAX_{l=l_1}^{Nrphys} \left[ CLMO_l \right] \right) + \prod_{l=l_1}^{Nrphys} \left( 1-CLRO_l \right) +\] +\\ +For a complete description of cloud/radiative interactions, see Section \ref{sec:fizhi:radcloud}. + + +\noindent +{ \underline {QINT} Total Precipitable Water ($gm/cm^2$) } + +\noindent +The Total Precipitable Water is defined as the vertical integral of the specific humidity, +given by: +\begin{eqnarray*} +{\bf QINT} & = & \int_{surf}^{top} \rho q dz \\ + & = & {\pi \over g} \int_0^1 q dp +\end{eqnarray*} +where we have used the hydrostatic relation +$\rho \delta z = -{\delta p \over g} $. +\\ + + +\noindent +{ \underline {U2M} Zonal U-Wind at 2 Meter Depth ($m/sec$) } + +\noindent +The u-wind at the 2-meter depth is determined from the similarity theory: +\[ +{\bf U2M} = {u_* \over k} \psi_{m_{2m}} {u_{sl} \over {W_s}} = +{ \psi_{m_{2m}} \over {\psi_{m_{sl}} }}u_{sl} +\] + +\noindent +where $\psi_m(2m)$ is the non-dimensional wind shear at two meters, and the subscript +$sl$ refers to the height of the top of the surface layer. If the roughness height +is above two meters, ${\bf U2M}$ is undefined. +\\ + +\noindent +{ \underline {V2M} Meridional V-Wind at 2 Meter Depth ($m/sec$) } + +\noindent +The v-wind at the 2-meter depth is a determined from the similarity theory: +\[ +{\bf V2M} = {u_* \over k} \psi_{m_{2m}} {v_{sl} \over {W_s}} = +{ \psi_{m_{2m}} \over {\psi_{m_{sl}} }}v_{sl} +\] + +\noindent +where $\psi_m(2m)$ is the non-dimensional wind shear at two meters, and the subscript +$sl$ refers to the height of the top of the surface layer. If the roughness height +is above two meters, ${\bf V2M}$ is undefined. +\\ + +\noindent +{ \underline {T2M} Temperature at 2 Meter Depth ($deg \hspace{.1cm} K$) } + +\noindent +The temperature at the 2-meter depth is a determined from the similarity theory: +\[ +{\bf T2M} = P^{\kappa} ({\theta* \over k} ({\psi_{h_{2m}}+\psi_g}) + \theta_{surf} ) = +P^{\kappa}(\theta_{surf} + { {\psi_{h_{2m}}+\psi_g} \over {{\psi_{h_{sl}}+\psi_g}} } +(\theta_{sl} - \theta_{surf})) +\] +where: +\[ +\theta_* = - { (\overline{w^{\prime}\theta^{\prime}}) \over {u_*} } +\] + +\noindent +where $\psi_h(2m)$ is the non-dimensional temperature gradient at two meters, $\psi_g$ is +the non-dimensional temperature gradient in the viscous sublayer, and the subscript +$sl$ refers to the height of the top of the surface layer. If the roughness height +is above two meters, ${\bf T2M}$ is undefined. +\\ + +\noindent +{ \underline {Q2M} Specific Humidity at 2 Meter Depth ($g/kg$) } + +\noindent +The specific humidity at the 2-meter depth is determined from the similarity theory: +\[ +{\bf Q2M} = P^{\kappa} ({q_* \over k} ({\psi_{h_{2m}}+\psi_g}) + q_{surf} ) = +P^{\kappa}(q_{surf} + { {\psi_{h_{2m}}+\psi_g} \over {{\psi_{h_{sl}}+\psi_g}} } +(q_{sl} - q_{surf})) +\] +where: +\[ +q_* = - { (\overline{w^{\prime}q^{\prime}}) \over {u_*} } +\] + +\noindent +where $\psi_h(2m)$ is the non-dimensional temperature gradient at two meters, $\psi_g$ is +the non-dimensional temperature gradient in the viscous sublayer, and the subscript +$sl$ refers to the height of the top of the surface layer. If the roughness height +is above two meters, ${\bf Q2M}$ is undefined. +\\ + +\noindent +{ \underline {U10M} Zonal U-Wind at 10 Meter Depth ($m/sec$) } + +\noindent +The u-wind at the 10-meter depth is an interpolation between the surface wind +and the model lowest level wind using the ratio of the non-dimensional wind shear +at the two levels: +\[ +{\bf U10M} = {u_* \over k} \psi_{m_{10m}} {u_{sl} \over {W_s}} = +{ \psi_{m_{10m}} \over {\psi_{m_{sl}} }}u_{sl} +\] + +\noindent +where $\psi_m(10m)$ is the non-dimensional wind shear at ten meters, and the subscript +$sl$ refers to the height of the top of the surface layer. +\\ + +\noindent +{ \underline {V10M} Meridional V-Wind at 10 Meter Depth ($m/sec$) } + +\noindent +The v-wind at the 10-meter depth is an interpolation between the surface wind +and the model lowest level wind using the ratio of the non-dimensional wind shear +at the two levels: +\[ +{\bf V10M} = {u_* \over k} \psi_{m_{10m}} {v_{sl} \over {W_s}} = +{ \psi_{m_{10m}} \over {\psi_{m_{sl}} }}v_{sl} +\] + +\noindent +where $\psi_m(10m)$ is the non-dimensional wind shear at ten meters, and the subscript +$sl$ refers to the height of the top of the surface layer. +\\ + +\noindent +{ \underline {T10M} Temperature at 10 Meter Depth ($deg \hspace{.1cm} K$) } + +\noindent +The temperature at the 10-meter depth is an interpolation between the surface potential +temperature and the model lowest level potential temperature using the ratio of the +non-dimensional temperature gradient at the two levels: +\[ +{\bf T10M} = P^{\kappa} ({\theta* \over k} ({\psi_{h_{10m}}+\psi_g}) + \theta_{surf} ) = +P^{\kappa}(\theta_{surf} + { {\psi_{h_{10m}}+\psi_g} \over {{\psi_{h_{sl}}+\psi_g}} } +(\theta_{sl} - \theta_{surf})) +\] +where: +\[ +\theta_* = - { (\overline{w^{\prime}\theta^{\prime}}) \over {u_*} } +\] + +\noindent +where $\psi_h(10m)$ is the non-dimensional temperature gradient at two meters, $\psi_g$ is +the non-dimensional temperature gradient in the viscous sublayer, and the subscript +$sl$ refers to the height of the top of the surface layer. +\\ + +\noindent +{ \underline {Q10M} Specific Humidity at 10 Meter Depth ($g/kg$) } + +\noindent +The specific humidity at the 10-meter depth is an interpolation between the surface specific +humidity and the model lowest level specific humidity using the ratio of the +non-dimensional temperature gradient at the two levels: +\[ +{\bf Q10M} = P^{\kappa} ({q_* \over k} ({\psi_{h_{10m}}+\psi_g}) + q_{surf} ) = +P^{\kappa}(q_{surf} + { {\psi_{h_{10m}}+\psi_g} \over {{\psi_{h_{sl}}+\psi_g}} } +(q_{sl} - q_{surf})) +\] +where: +\[ +q_* = - { (\overline{w^{\prime}q^{\prime}}) \over {u_*} } +\] + +\noindent +where $\psi_h(10m)$ is the non-dimensional temperature gradient at two meters, $\psi_g$ is +the non-dimensional temperature gradient in the viscous sublayer, and the subscript +$sl$ refers to the height of the top of the surface layer. +\\ + +\noindent +{ \underline {DTRAIN} Cloud Detrainment Mass Flux ($kg/m^2$) } + +The amount of cloud mass moved per RAS timestep at the cloud detrainment level is written: +\[ +{\bf DTRAIN} = \eta_{r_D}m_B +\] +\noindent +where $r_D$ is the detrainment level, +$m_B$ is the cloud base mass flux, and $\eta$ +is the entrainment, defined in Section \ref{sec:fizhi:mc}. +\\ + +\noindent +{ \underline {QFILL} Filling of negative Specific Humidity ($g/kg/day$) } + +\noindent +Due to computational errors associated with the numerical scheme used for +the advection of moisture, negative values of specific humidity may be generated. The +specific humidity is checked for negative values after every dynamics timestep. If negative +values have been produced, a filling algorithm is invoked which redistributes moisture from +below. Diagnostic {\bf QFILL} is equal to the net filling needed +to eliminate negative specific humidity, scaled to a per-day rate: +\[ +{\bf QFILL} = q^{n+1}_{final} - q^{n+1}_{initial} +\] +where +\[ +q^{n+1} = (\pi q)^{n+1} / \pi^{n+1} +\] + + \subsection{Key subroutines, parameters and files} \subsection{Dos and donts}