/[MITgcm]/manual/s_phys_pkgs/text/fizhi.tex

Diff of /manual/s_phys_pkgs/text/fizhi.tex

Parent Directory | Revision Log | View Revision Graph Revision Graph | View Patch Patch

-revision 1.7 by edhill,
Tue Oct 12 18:16:03 2004 UTC
+revision 1.12 by molod,
Tue Aug  2 21:17:38 2005 UTC
 Line 1
- \section{Fizhi: High-end Atmospheric Physics}
+ \subsection{Fizhi: High-end Atmospheric Physics}
  \label{sec:pkg:fizhi}
  \begin{rawhtml}
  <!-- CMIREDIR:package_fizhi: -->
  \end{rawhtml}
  \input{texinputs/epsf.tex}
- \subsection{Introduction}
+ \subsubsection{Introduction}
  The fizhi (high-end atmospheric physics) package includes a collection of state-of-the-art
  physical parameterizations for atmospheric radiation, cumulus convection, atmospheric
- boundary layer turbulence, and land surface processes.
+ boundary layer turbulence, and land surface processes. The collection of atmospheric
+ physics parameterizations were originally used together as part of the GEOS-3
+ (Goddard Earth Observing System-3) GCM developed at the NASA/Goddard Global Modelling
+ and Assimilation Office (GMAO).
  % *************************************************************************
  % *************************************************************************
- \subsection{Equations}
+ \subsubsection{Equations}
- \subsubsection{Moist Convective Processes}
+ Moist Convective Processes:
  \paragraph{Sub-grid and Large-scale Convection}
  \label{sec:fizhi:mc}
  Sub-grid scale cumulus convection is parameterized using the Relaxed Arakawa
- Schubert (RAS) scheme of Moorthi and Suarez (1992), which is a linearized Arakawa Schubert
+ Schubert (RAS) scheme of \cite{moorsz:92}, which is a linearized Arakawa Schubert
  type scheme.  RAS predicts the mass flux from an ensemble of clouds.  Each subensemble is identified
  by its entrainment rate and level of neutral bouyancy which are determined by the grid-scale properties.
-Line 43 
 where we have used the hydrostatic equat
+Line 46 
 where we have used the hydrostatic equat
  The entrainment parameter, $\lambda$, characterizes a particular subensemble based on its
  detrainment level, and is obtained by assuming that the level of detrainment is the level of neutral
  buoyancy, ie., the level at which the moist static energy of the cloud, $h_c$, is equal
- to the saturation moist static energy of the environment, $h^*$.  Following Moorthi and Suarez (1992),
+ to the saturation moist static energy of the environment, $h^*$.  Following \cite{moorsz:92},
  $\lambda$ may be written as
  \[
  \lambda = { {h_B - h^*_D} \over { {c_p \over g} {\int_{P_D}^{P_B}\theta(h^*_D-h)dP^{\kappa}}} } ,
-Line 101 
 $\alpha$ of the total adjustment. The pa
+Line 104 
 $\alpha$ of the total adjustment. The pa
  towards equillibrium.
  In addition to the RAS cumulus convection scheme, the fizhi package employs a
- Kessler-type scheme for the re-evaporation of falling rain (Sud and Molod, 1988), which
+ Kessler-type scheme for the re-evaporation of falling rain (\cite{sudm:88}), which
  correspondingly adjusts the temperature assuming $h$ is conserved. RAS in its current
  formulation assumes that all cloud water is deposited into the detrainment level as rain.
  All of the rain is available for re-evaporation, which begins in the level below detrainment.
-Line 186 
 F_{CLD} = \max \left[ F_{RAS},F_{LS} \ri
+Line 189 
 F_{CLD} = \max \left[ F_{RAS},F_{LS} \ri
  Finally, cloud fractions are time-averaged between calls to the radiation packages.
- \subsubsection{Radiation}
+ Radiation:
  The parameterization of radiative heating in the fizhi package includes effects
  from both shortwave and longwave processes.
-Line 221 
 The solar constant value used in the pac
+Line 224 
 The solar constant value used in the pac
  and a $CO_2$ mixing ratio of 330 ppm.
  For the ozone mixing ratio, monthly mean zonally averaged
  climatological values specified as a function
- of latitude and height (Rosenfield, et al., 1987) are linearly interpolated to the current time.
+ of latitude and height (\cite{rosen:87}) are linearly interpolated to the current time.
  \paragraph{Shortwave Radiation}
-Line 231 
 heating due to the absoption by water va
+Line 234 
 heating due to the absoption by water va
  clouds, and aerosols and due to the
  scattering by clouds, aerosols, and gases.
  The shortwave radiative processes are described by
- Chou (1990,1992). This shortwave package
+ \cite{chou:90,chou:92}. This shortwave package
  uses the Delta-Eddington approximation to compute the
  bulk scattering properties of a single layer following King and Harshvardhan (JAS, 1986).
  The transmittance and reflectance of diffuse radiation
- follow the procedures of Sagan and Pollock (JGR, 1967) and Lacis and Hansen (JAS, 1974).
+ follow the procedures of Sagan and Pollock (JGR, 1967) and \cite{lhans:74}.
  Highly accurate heating rate calculations are obtained through the use
  of an optimal grouping strategy of spectral bands.  By grouping the UV and visible regions
-Line 321 
 low/middle/high classification, and appr
+Line 324 
 low/middle/high classification, and appr
  \paragraph{Longwave Radiation}
- The longwave radiation package used in the fizhi package is thoroughly described by Chou and Suarez (1994).
+ The longwave radiation package used in the fizhi package is thoroughly described by \cite{chsz:94}.
  As described in that document, IR fluxes are computed due to absorption by water vapor, carbon
  dioxide, and ozone.  The spectral bands together with their absorbers and parameterization methods,
  configured for the fizhi package, are shown in Table \ref{tab:fizhi:longwave}.
-Line 357 
 Band & Spectral Range (cm$^{-1}$) & Abso
+Line 360 
 Band & Spectral Range (cm$^{-1}$) & Abso
  \end{tabular}
  \end{center}
  \vspace{0.1in}
- \caption{IR Spectral Bands, Absorbers, and Parameterization Method (from Chou and Suarez, 1994)}
+ \caption{IR Spectral Bands, Absorbers, and Parameterization Method (from \cite{chsz:94})}
  \label{tab:fizhi:longwave}
  \end{table}
-Line 428 
 The cloud fraction values are time-avera
+Line 431 
 The cloud fraction values are time-avera
  hours).  Therefore, in a time-averaged sense, both convective and large-scale
  cloudiness can exist in a given grid-box.
- \subsubsection{Turbulence}
+ \paragraph{Turbulence}:
  Turbulence is parameterized in the fizhi package to account for its contribution to the
  vertical exchange of heat, moisture, and momentum.
  The turbulence scheme is invoked every 30 minutes, and employs a backward-implicit iterative
-Line 458 
 Within the atmosphere, the time evolutio
+Line 462 
 Within the atmosphere, the time evolutio
  of second turbulent moments is explicitly modeled by representing the third moments in terms of
  the first and second moments.  This approach is known as a second-order closure modeling.
  To simplify and streamline the computation of the second moments, the level 2.5 assumption
- of Mellor and Yamada (1974) and Yamada (1977) is employed, in which only the turbulent
+ of Mellor and Yamada (1974) and \cite{yam:77} is employed, in which only the turbulent
  kinetic energy (TKE),
  \[ {\h}{q^2}={\overline{{u^{\prime}}^2}}+{\overline{{v^{\prime}}^2}}+{\overline{{w^{\prime}}^2}}, \]
-Line 492 
 of TKE.
+Line 496 
 of TKE.
  In the level 2.5 approach, the vertical fluxes of the scalars $\theta_v$ and $q$ and the
  wind components $u$ and $v$ are expressed in terms of the diffusion coefficients $K_h$ and
- $K_m$, respectively.  In the statisically realizable level 2.5 turbulence scheme of Helfand
+ $K_m$, respectively.  In the statisically realizable level 2.5 turbulence scheme of
- and Labraga (1988), these diffusion coefficients are expressed as
+ \cite{helflab:88}, these diffusion coefficients are expressed as
  \[
  K_h
-Line 568 
 where $\psi_h$ is the surface layer non-
+Line 572 
 where $\psi_h$ is the surface layer non-
  \]
  Here $\phi_h$ is the similarity function of $\zeta$, which expresses the stability dependance of
  the temperature and moisture gradients, and is specified differently for stable and unstable
- layers according to Helfand and Schubert, 1995.
+ layers according to \cite{helfschu:95}.
  $\psi_g$ is the non-dimensional temperature or moisture gradient in the viscous sublayer,
  which is the mosstly laminar region between the surface and the tops of the roughness
  elements, in which temperature and moisture gradients can be quite large.
- Based on Yaglom and Kader (1974):
+ Based on \cite{yagkad:74}:
  \[
  \psi_{g} = { 0.55 (Pr^{2/3} - 0.2) \over \nu^{1/2} }
  (h_{0}u_{*} - h_{0_{ref}}u_{*_{ref}})^{1/2}
-Line 587 
 The surface roughness length over oceans
+Line 591 
 The surface roughness length over oceans
  {z_0} = c_1u^3_* + c_2u^2_* + c_3u_* + c_4 + {c_5 \over {u_*}}
  \]
  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)
+ \cite{kondo:75} for weak winds, and the piecewise linear relation of \cite{larpond:81}
  for moderate to large winds.  Roughness lengths over land are specified
- from the climatology of Dorman and Sellers (1989).
+ from the climatology of \cite{dorsell:89}.
  For an unstable surface layer, the stability functions, chosen to interpolate between the
  condition of small values of $\beta$ and the convective limit, are the KEYPS function
- (Panofsky, 1973) for momentum, and its generalization for heat and moisture:
+ (\cite{pano:73}) for momentum, and its generalization for heat and moisture:
  \[
  {\phi_m}^4 - 18 \zeta {\phi_m}^3 = 1 \hspace{1cm} ; \hspace{1cm}
  {\phi_h}^2 - 18 \zeta {\phi_h}^3 = 1 \hspace{1cm} .
-Line 602 
 The function for heat and moisture assur
+Line 606 
 The function for heat and moisture assur
  speed approaches zero.
  For a stable surface layer, the stability functions are the observationally
- based functions of Clarke (1970),  slightly modified for
+ based functions of \cite{clarke:70},  slightly modified for
  the momemtum flux:
  \[
  {\phi_m} = { { 1 + 5 {{\zeta}_1} } \over { 1 + 0.00794 {{\zeta}_1}
-Line 660 
 be $3 \hspace{.1cm} m$ where sea ice is
+Line 664 
 be $3 \hspace{.1cm} m$ where sea ice is
  surface temperature of the ice.
  $C_g$ is the total heat capacity of the ground, obtained by solving a heat diffusion equation
- for the penetration of the diurnal cycle into the ground (Blackadar, 1977), and is given by:
+ for the penetration of the diurnal cycle into the ground (\cite{black:77}), and is given by:
  \[
  C_g = \sqrt{ {\lambda C_s \over 2\omega} } = \sqrt{(0.386 + 0.536W + 0.15W^2)2\times10^{-3}
  {86400 \over 2 \pi} } \, \, .
-Line 672 
 by $2 \pi$ $radians/
+Line 676 
 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, $W$.
- \subsubsection{Land Surface Processes}
+ Land Surface Processes:
  \paragraph{Surface Type}
- The fizhi package surface Types are designated using the Koster-Suarez (1992) mosaic
+ The fizhi package surface Types are designated using the Koster-Suarez (\cite{ks:91,ks:92})
- philosophy which allows multiple ``tiles'', or multiple surface types, in any one
+ Land Surface Model (LSM) mosaic philosophy which allows multiple ``tiles'', or multiple surface
- grid cell. The Koster-Suarez Land Surface Model (LSM) surface type classifications
+ types, in any one grid cell. The Koster-Suarez LSM surface type classifications
  are shown in Table \ref{tab:fizhi:surftype}. The surface types and the percent of the grid
  cell occupied by any surface type were derived from the surface classification of
- Defries and Townshend (1994), and information about the location of permanent
+ \cite{deftow:94}, and information about the location of permanent
- ice was obtained from the classifications of Dorman and Sellers (1989).
+ ice was obtained from the classifications of \cite{dorsell:89}.
  The surface type for the \txt GCM grid is shown in Figure \ref{fig:fizhi:surftype}.
  The determination of the land or sea category of surface type was made from NCAR's
 minute by 10 minute Navy topography
-Line 735 
 and surface albedo.}
+Line 739 
 and surface albedo.}
  \paragraph{Surface Roughness}
  The surface roughness length over oceans is computed iteratively with the wind
- stress by the surface layer parameterization (Helfand and Schubert, 1991).
+ stress by the surface layer parameterization (\cite{helfschu:95}).
- It employs an interpolation between the functions of Large and Pond (1981)
+ It employs an interpolation between the functions of \cite{larpond:81}
- for high winds and of Kondo (1975) for weak winds.
+ for high winds and of \cite{kondo:75} for weak winds.
  \paragraph{Albedo}
- The surface albedo computation, described in Koster and Suarez (1991),
+ The surface albedo computation, described in \cite{ks:91},
  employs the ``two stream'' approximation used in Sellers' (1987) Simple Biosphere (SiB)
  Model which distinguishes between the direct and diffuse albedos in the visible
  and in the near infra-red spectral ranges. The albedos are functions of the observed
-Line 750 
 sun), the greenness fraction, the vegeta
+Line 754 
 sun), the greenness fraction, the vegeta
  Modifications are made to account for the presence of snow, and its depth relative
  to the height of the vegetation elements.
- \subsubsection{Gravity Wave Drag}
+ \paragraph{Gravity Wave Drag}:
- The fizhi package employs the gravity wave drag scheme of Zhou et al. (1996).
+ The fizhi package employs the gravity wave drag scheme of \cite{zhouetal:95}).
  This scheme is a modified version of Vernekar et al. (1992),
  which was based on Alpert et al. (1988) and Helfand et al. (1987).
  In this version, the gravity wave stress at the surface is
-Line 768 
 A modification introduced by Zhou et al.
+Line 773 
 A modification introduced by Zhou et al.
  escape through the top of the model, although this effect is small for the current 70-level model.
  The subgrid scale standard deviation is defined by $h$, and is not allowed to exceed 400 m.
- The effects of using this scheme within a GCM are shown in Takacs and Suarez (1996).
+ The effects of using this scheme within a GCM are shown in \cite{taksz:96}.
  Experiments using the gravity wave drag parameterization yielded significant and
  beneficial impacts on both the time-mean flow and the transient statistics of the
  a GCM climatology, and have eliminated most of the worst dynamically driven biases
-Line 784 
 of mountain torque (through a redistribu
+Line 789 
 of mountain torque (through a redistribu
  convergence (through a reduction in the flux of westerly momentum by transient flow eddies).
- \subsubsection{Boundary Conditions and other Input Data}
+ Boundary Conditions and other Input Data:
  Required fields which are not explicitly predicted or diagnosed during model execution must
  either be prescribed internally or obtained from external data sets.  In the fizhi package these
-Line 825 
 current years and frequencies available.
+Line 830 
 current years and frequencies available.
  Surface geopotential heights are provided from an averaging of the Navy 10 minute
  by 10 minute dataset supplied by the National Center for Atmospheric Research (NCAR) to the
  model's grid resolution. The original topography is first rotated to the proper grid-orientation
- which is being run, and then
+ which is being run, and then  averages the data to the model resolution.
- averages the data to the model resolution.
- The averaged topography is then passed through a Lanczos (1966) filter in both dimensions
- which removes the smallest
- scales while inhibiting Gibbs phenomena.
- In one dimension, we may define a cyclic function in $x$ as:
- \begin{equation}
- f(x) = {a_0 \over 2} + \sum_{k=1}^N \left( a_k \cos(kx) + b_k \sin(kx) \right)
- \label{eq:fizhi:filt}
- \end{equation}
- where $N = { {\rm IM} \over 2 }$ and ${\rm IM}$ is the total number of points in the $x$ direction.
- Defining $\Delta x = { 2 \pi \over {\rm IM}}$, we may define the average of $f(x)$ over a
- $2 \Delta x$ region as:
- \begin{equation}
- \overline {f(x)} = {1 \over {2 \Delta x}} \int_{x-\Delta x}^{x+\Delta x} f(x^{\prime}) dx^{\prime}
- \label{eq:fizhi:fave1}
- \end{equation}
- Using equation (\ref{eq:fizhi:filt}) in equation (\ref{eq:fizhi:fave1}) and integrating, we may write:
- \begin{equation}
- \overline {f(x)} = {a_0 \over 2} + {1 \over {2 \Delta x}}
- \sum_{k=1}^N \left [
- \left. a_k { \sin(kx^{\prime}) \over k } \right /_{x-\Delta x}^{x+\Delta x} -
- \left. b_k { \cos(kx^{\prime}) \over k } \right /_{x-\Delta x}^{x+\Delta x}
- \right]
- \end{equation}
- or
- \begin{equation}
- \overline {f(x)} = {a_0 \over 2} + \sum_{k=1}^N {\sin(k \Delta x) \over {k \Delta x}}
- \left( a_k \cos(kx) + b_k \sin(kx) \right)
- \label{eq:fizhi:fave2}
- \end{equation}
- Thus, the Fourier wave amplitudes are simply modified by the Lanczos filter response
- function ${\sin(k\Delta x) \over {k \Delta x}}$.  This may be compared with an $mth$-order
- Shapiro (1970) filter response function, defined as $1-\sin^m({k \Delta x \over 2})$,
- shown in Figure \ref{fig:fizhi:lanczos}.
- It should be noted that negative values in the topography resulting from
- the filtering procedure are {\em not} filled.
- \begin{figure*}[htbp]
+ The standard deviation of the subgrid-scale topography is computed by interpolating the 10 minute
-   \centerline{  \epsfysize=7.0in  \epsfbox{part6/lanczos.ps}}
+ data to the model's resolution and re-interpolating back to the 10 minute by 10 minute resolution.
-   \caption{ \label{fig:fizhi:lanczos} Comparison between the Lanczos and $mth$-order Shapiro filter
-   response functions for $m$ = 2, 4, and 8. }
- \end{figure*}
- The standard deviation of the subgrid-scale topography
- is computed from a modified version of the the Navy 10 minute by 10 minute dataset.
- The 10 minute by 10 minute topography is passed through a wavelet
- filter in both dimensions which removes the scale smaller than 20 minutes.
- The topography is then averaged to $1^\circ x 1^\circ$ grid resolution, and then
- re-interpolated back to the 10 minute by 10 minute resolution.
  The sub-grid scale variance is constructed based on this smoothed dataset.
-Line 892 
 model's grid location and current time,
+Line 845 
 model's grid location and current time,
  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{Key subroutines, parameters and files}
- \subsection{Dos and donts}
+ \subsubsection{Fizhi Diagnostics}
+ 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
+ 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 \cite{helfschu:95}. 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 \cite{helfschu:95}. 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 \cite{helflab:88} 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 \cite{helflab:88}.
+ \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 \cite{helflab:88} 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 \cite{helflab:88}.
+ \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 (\cite{black:77}), 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 \cite{dorsell:89}. 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
+ \cite{kondo:75} for weak winds, and the piecewise linear relation of \cite{larpond:81}
+ 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-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
+ \[
+ q^{n+1} = (\pi q)^{n+1} / \pi^{n+1}
+ \]
+ \subsubsection{Key subroutines, parameters and files}
+ \subsubsection{Dos and donts}
- \subsection{Fizhi Reference}
+ \subsubsection{Fizhi Reference}

 Legend:



Removed from v.1.7
 


changed lines


 
Added in v.1.12
 Legend:



Removed from v.1.7
 


changed lines


 
Added in v.1.12
-Removed from v.1.7
+Added in v.1.12

	ViewVC Help
Powered by ViewVC 1.1.22