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

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-revision 1.11 by molod,
Tue Aug  2 15:50:51 2005 UTC
+revision 1.19 by jmc,
Mon Aug 30 23:09:21 2010 UTC
 Line 36 
 buoyancy. RAS assumes that the normalize
  mass flux, is a linear function of height, expressed as:
  \[
  \pp{\eta(z)}{z} = \lambda \hspace{0.4cm}or\hspace{0.4cm} \pp{\eta(P^{\kappa})}{P^{\kappa}} =
- -{c_p \over {g}}\theta\lambda
+ -\frac{c_p}{g}\theta\lambda
  \]
  where we have used the hydrostatic equation written in the form:
  \[
- \pp{z}{P^{\kappa}} = -{c_p \over {g}}\theta
+ \pp{z}{P^{\kappa}} = -\frac{c_p}{g}\theta
  \]
  The entrainment parameter, $\lambda$, characterizes a particular subensemble based on its
 Line 49 
 buoyancy, ie., the level at which the mo
  to the saturation moist static energy of the environment, $h^*$.  Following \cite{moorsz:92},
  $\lambda$ may be written as
  \[
- \lambda = { {h_B - h^*_D} \over { {c_p \over g} {\int_{P_D}^{P_B}\theta(h^*_D-h)dP^{\kappa}}} } ,
+ \lambda = \frac{h_B - h^*_D}{ \frac{c_p}{g} \int_{P_D}^{P_B}\theta(h^*_D-h)dP^{\kappa}},
  \]
  where the subscript $B$ refers to cloud base, and the subscript $D$ refers to the detrainment level.
 Line 60 
 rate of change of cumulus kinetic energy
  related to the buoyancy, or the difference
  between the moist static energy in the cloud and in the environment:
  \[
- A = \int_{P_D}^{P_B} { {\eta \over {1 + \gamma} }
+ A = \int_{P_D}^{P_B} \frac{\eta}{1 + \gamma}
- \left[ {{h_c-h^*} \over {P^{\kappa}}} \right] dP^{\kappa}}
+ \left[ \frac{h_c-h^*}{P^{\kappa}} \right] dP^{\kappa}
  \]
- where $\gamma$ is ${L \over {c_p}}\pp{q^*}{T}$ obtained from the Claussius Clapeyron equation,
+ where $\gamma$ is $\frac{L}{c_p}\pp{q^*}{T}$ obtained from the Claussius Clapeyron equation,
  and the subscript $c$ refers to the value inside the cloud.
 Line 72 
 To determine the cloud base mass flux, t
  the clouds} is assumed to approximately balance the rate of change of $A$ {\em due to the generation
  by the large scale}. This is the quasi-equilibrium assumption, and results in an expression for $m_B$:
  \[
- m_B = {{- \left.{dA \over dt} \right|_{ls}} \over K}
+ m_B = \frac{- \left. \frac{dA}{dt} \right|_{ls}}{K}
  \]
  where $K$ is the cloud kernel, defined as the rate of change of the cloud work function per
 Line 90 
 and moisture budget equations to determi
  temperature (through latent heating and compensating subsidence) and moisture (through
  precipitation and detrainment):
  \[
- \left.{\pp{\theta}{t}}\right|_{c} = \alpha { m_B \over {c_p P^{\kappa}}} \eta \pp{s}{p}
+ \left.{\pp{\theta}{t}}\right|_{c} = \alpha \frac{ m_B}{c_p P^{\kappa}} \eta \pp{s}{p}
  \]
  and
  \[
- \left.{\pp{q}{t}}\right|_{c} = \alpha { m_B \over {L}} \eta (\pp{h}{p}-\pp{s}{p})
+ \left.{\pp{q}{t}}\right|_{c} = \alpha \frac{ m_B}{L} \eta (\pp{h}{p}-\pp{s}{p})
  \]
- where $\theta = {T \over P^{\kappa}}$, $P = (p/p_0)$, and $\alpha$ is the relaxation parameter.
+ where $\theta = \frac{T}{P^{\kappa}}$, $P = (p/p_0)$, and $\alpha$ is the relaxation parameter.
  As an approximation to a full interaction between the different allowable subensembles,
  many clouds are simulated frequently, each modifying the large scale environment some fraction
 Line 136 
 Convective cloud fractions produced by R
  detrained liquid water amount given by
  \[
- F_{RAS} = \min\left[ {l_{RAS}\over l_c}, 1.0 \right]
+ F_{RAS} = \min\left[ \frac{l_{RAS}}{l_c}, 1.0 \right]
  \]
  where $l_c$ is an assigned critical value equal to $1.25$ g/kg.
  A memory is associated with convective clouds defined by:
  \[
- F_{RAS}^n = \min\left[ F_{RAS} + (1-{\Delta t_{RAS}\over\tau})F_{RAS}^{n-1}, 1.0 \right]
+ F_{RAS}^n = \min\left[ F_{RAS} + (1-\frac{\Delta t_{RAS}}{\tau})F_{RAS}^{n-1}, 1.0 \right]
  \]
  where $F_{RAS}$ is the instantanious cloud fraction and $F_{RAS}^{n-1}$ is the cloud fraction
 Line 154 
 Large-scale cloudiness is defined, follo
  humidity:
  \[
- F_{LS} = \min\left[ { \left( {RH-RH_c \over 1-RH_c} \right) }^2, 1.0 \right]
+ F_{LS} = \min\left[ { \left( \frac{RH-RH_c}{1-RH_c} \right) }^2, 1.0 \right]
  \]
  where
 Line 162 
 where
  \bqa
  RH_c & = & 1-s(1-s)(2-\sqrt{3}+2\sqrt{3} \, s)r \nonumber \\
     s & = & p/p_{surf} \nonumber \\
-    r & = & \left( {1.0-RH_{min} \over \alpha} \right) \nonumber \\
+    r & = & \left( \frac{1.0-RH_{min}}{\alpha} \right) \nonumber \\
  RH_{min} & = & 0.75 \nonumber \\
  \alpha & = & 0.573285 \nonumber  .
  \eqa
  These cloud fractions are suppressed, however, in regions where the convective
  sub-cloud layer is conditionally unstable.  The functional form of $RH_c$ is shown in
- Figure (\ref{fig:fizhi:rhcrit}).
+ Figure (\ref{fig.rhcrit}).
  \begin{figure*}[htbp]
    \vspace{0.4in}
-   \centerline{  \epsfysize=4.0in  \epsfbox{part6/rhcrit.ps}}
+   \centerline{  \epsfysize=4.0in  \epsfbox{s_phys_pkgs/figs/rhcrit.ps}}
    \vspace{0.4in}
    \caption  [Critical Relative Humidity for Clouds.]
              {Critical Relative Humidity for Clouds.}
-   \label{fig:fizhi:rhcrit}
+   \label{fig.rhcrit}
  \end{figure*}
  The total cloud fraction in a grid box is determined by the larger of the two cloud fractions:
 Line 308 
 cloud cover of all the layers in the gro
  of a given layer is then scaled for both the direct (as a function of the
  solar zenith angle) and diffuse beam radiation
  so that the grouped layer reflectance is the same as the original reflectance.
- The solar flux is computed for each of the eight cloud realizations possible
+ The solar flux is computed for each of eight cloud realizations possible within this
- (see Figure \ref{fig:fizhi:cloud}) within this
  low/middle/high classification, and appropriately averaged to produce the net solar flux.
- \begin{figure*}[htbp]
-   \vspace{0.4in}
-   \centerline{  \epsfysize=4.0in  %\epsfbox{part6/rhcrit.ps}
-              }
-   \vspace{0.4in}
-   \caption  {Low-Middle-High Cloud Configurations}
-   \label{fig:fizhi:cloud}
- \end{figure*}
  \paragraph{Longwave Radiation}
  The longwave radiation package used in the fizhi package is thoroughly described by \cite{chsz:94}.
-Line 360 
 Band & Spectral Range (cm$^{-1}$) & Abso
+Line 349 
 Band & Spectral Range (cm$^{-1}$) & Abso
  \end{tabular}
  \end{center}
  \vspace{0.1in}
- \caption{IR Spectral Bands, Absorbers, and Parameterization Method (from \cite{chzs:94})}
+ \caption{IR Spectral Bands, Absorbers, and Parameterization Method (from \cite{chsz:94})}
  \label{tab:fizhi:longwave}
  \end{table}
-Line 420 
 The total optical depth in a given model
+Line 409 
 The total optical depth in a given model
  the large-scale and sub-grid scale optical depths, normalized by the total cloud fraction in the
  layer:
- \[ \tau = \left( {F_{RAS} \,\,\, \tau_{RAS} + F_{LS} \,\,\, \tau_{LS} \over F_{RAS}+F_{LS} } \right) \Delta p, \]
+ \[ \tau = \left( \frac{F_{RAS} \,\,\, \tau_{RAS} + F_{LS} \,\,\, \tau_{LS} }{ F_{RAS}+F_{LS} } \right) \Delta p, \]
  where $F_{RAS}$ and $F_{LS}$ are the time-averaged cloud fractions associated with RAS and large-scale
  processes described in Section \ref{sec:fizhi:clouds}.
-Line 431 
 The cloud fraction values are time-avera
+Line 420 
 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.
- Turbulence:
+ \paragraph{Turbulence}:
  Turbulence is parameterized in the fizhi package to account for its contribution to the
  vertical exchange of heat, moisture, and momentum.
-Line 474 
 is solved numerically using an implicit
+Line 463 
 is solved numerically using an implicit
  and is written:
  \[
- {\dd{}{t} ({{\h} q^2})} - { \pp{}{z} ({ {5 \over 3} {{\lambda}_1} q { \pp {}{z}
+ {\dd{}{t} ({{\h} q^2})} - { \pp{}{z} ({ \frac{5}{3} {{\lambda}_1} q { \pp {}{z}
  ({\h}q^2)} })} =
  {- \overline{{u^{\prime}}{w^{\prime}}} { \pp{U}{z} }} - {\overline{{v^{\prime}}{w^{\prime}}}
- { \pp{V}{z} }} + {{g \over {\Theta_0}}{\overline{{w^{\prime}}{{{\theta}_v}^{\prime}}}} }
+ { \pp{V}{z} }} + {\frac{g}{\Theta_0}{\overline{{w^{\prime}}{{{\theta}_v}^{\prime}}}}
- - { q^3 \over {{\Lambda} _1} }
+ - \frac{ q^3}{{\Lambda}_1} }
  \]
  where $q$ is the turbulent velocity, ${u^{\prime}}$, ${v^{\prime}}$, ${w^{\prime}}$ and
-Line 502 
 $K_m$, respectively.  In the statisicall
+Line 491 
 $K_m$, respectively.  In the statisicall
  \[
  K_h
   = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell \, S_H(G_M,G_H) \, & \mbox{decaying turbulence}
- \\ { q^2 \over {q_e} } \, \ell \, S_{H}(G_{M_e},G_{H_e}) \, & \mbox{growing turbulence} \end{array} \right.
+ \\ \frac{ q^2 }{ q_e } \, \ell \, S_{H}(G_{M_e},G_{H_e}) \, & \mbox{growing turbulence} \end{array} \right.
  \]
  and
-Line 510 
 and
+Line 499 
 and
  \[
  K_m
   = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell \, S_M(G_M,G_H) \, & \mbox{decaying turbulence}
- \\ { q^2 \over {q_e} } \, \ell \, S_{M}(G_{M_e},G_{H_e}) \, & \mbox{growing turbulence} \end{array} \right.
+ \\ \frac{ q^2 }{ q_e } \, \ell \, S_{M}(G_{M_e},G_{H_e}) \, & \mbox{growing turbulence} \end{array} \right.
  \]
  where the subscript $e$ refers to the value under conditions of local equillibrium
-Line 522 
 Both $G_H$ and $G_M$, and their equilibr
+Line 511 
 Both $G_H$ and $G_M$, and their equilibr
  are functions of the Richardson number:
  \[
- {\bf RI} = { { {g \over \theta_v} \pp {\theta_v}{z} } \over { (\pp{u}{z})^2 + (\pp{v}{z})^2 } }
+ {\bf RI} = \frac{ \frac{g}{\theta_v} \pp{\theta_v}{z} }{ (\pp{u}{z})^2 + (\pp{v}{z})^2 }
-  =  {  {c_p \pp{\theta_v}{z} \pp{P^ \kappa}{z} } \over { (\pp{u}{z})^2 + (\pp{v}{z})^2 } } .
+  =  \frac{c_p \pp{\theta_v}{z} \pp{P^ \kappa}{z} }{ (\pp{u}{z})^2 + (\pp{v}{z})^2 } .
  \]
  Negative values indicate unstable buoyancy and shear, small positive values ($<0.2$)
  indicate dominantly unstable shear, and large positive values indicate dominantly stable
  stratification.
- Turbulent eddy diffusion coefficients of momentum, heat and moisture in the surface layer,
+ Turbulent eddy diffusion coefficients of momentum, heat and moisture in the
- which corresponds to the lowest GCM level (see \ref{tab:fizhi:sigma}),
+ surface layer, which corresponds to the lowest GCM level
+ (see {\it --- missing table ---}%\ref{tab:fizhi:sigma}
+ ),
  are calculated using stability-dependant functions based on Monin-Obukhov theory:
  \[
  {K_m} (surface) = C_u \times u_* = C_D W_s
-Line 547 
 and $W_s$ is the magnitude of the surfac
+Line 538 
 and $W_s$ is the magnitude of the surfac
  $C_u$ is the dimensionless exchange coefficient for momentum from the surface layer
  similarity functions:
  \[
- {C_u} = {u_* \over W_s} = { k \over \psi_{m} }
+ {C_u} = \frac{u_* }{ W_s} = \frac{ k }{ \psi_{m} }
  \]
  where k is the Von Karman constant and $\psi_m$ is the surface layer non-dimensional
  wind shear given by
  \[
- \psi_{m} = {\int_{\zeta_{0}}^{\zeta} {\phi_{m} \over \zeta} d \zeta} .
+ \psi_{m} = {\int_{\zeta_{0}}^{\zeta} \frac{\phi_{m} }{ \zeta} d \zeta} .
  \]
  Here $\zeta$ is the non-dimensional stability parameter, and
  $\phi_m$ is the similarity function of $\zeta$ which expresses the stability dependance of
-Line 562 
 layers.
+Line 553 
 layers.
  $C_t$ is the dimensionless exchange coefficient for heat and
  moisture from the surface layer similarity functions:
  \[
- {C_t} = -{( {\overline{w^{\prime}\theta^{\prime}}}) \over {u_* \Delta \theta }} =
+ {C_t} = -\frac{( \overline{w^{\prime}\theta^{\prime}}) }{ u_* \Delta \theta } =
- -{( {\overline{w^{\prime}q^{\prime}}}) \over {u_* \Delta q }} =
+ -\frac{( \overline{w^{\prime}q^{\prime}}) }{ u_* \Delta q } =
- { k \over { (\psi_{h} + \psi_{g}) } }
+ \frac{ k }{ (\psi_{h} + \psi_{g}) }
  \]
  where $\psi_h$ is the surface layer non-dimensional temperature gradient given by
  \[
- \psi_{h} = {\int_{\zeta_{0}}^{\zeta} {\phi_{h} \over \zeta} d \zeta} .
+ \psi_{h} = {\int_{\zeta_{0}}^{\zeta} \frac{\phi_{h} }{ \zeta} d \zeta} .
  \]
  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
-Line 579 
 which is the mosstly laminar region betw
+Line 570 
 which is the mosstly laminar region betw
  elements, in which temperature and moisture gradients can be quite large.
  Based on \cite{yagkad:74}:
  \[
- \psi_{g} = { 0.55 (Pr^{2/3} - 0.2) \over \nu^{1/2} }
+ \psi_{g} = \frac{ 0.55 (Pr^{2/3} - 0.2) }{ \nu^{1/2} }
  (h_{0}u_{*} - h_{0_{ref}}u_{*_{ref}})^{1/2}
  \]
  where Pr is the Prandtl number for air, $\nu$ is the molecular viscosity, $z_{0}$ is the
-Line 588 
 $h_{0} = 30z_{0}$ with a maximum value o
+Line 579 
 $h_{0} = 30z_{0}$ with a maximum value o
  The surface roughness length over oceans is is a function of the surface-stress velocity,
  \[
- {z_0} = c_1u^3_* + c_2u^2_* + c_3u_* + c_4 + {c_5 \over {u_*}}
+ {z_0} = c_1u^3_* + c_2u^2_* + c_3u_* + c_4 + \frac{c_5 }{ u_*}
  \]
  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}
-Line 609 
 For a stable surface layer, the stabilit
+Line 600 
 For a stable surface layer, the stabilit
  based functions of \cite{clarke:70},  slightly modified for
  the momemtum flux:
  \[
- {\phi_m} = { { 1 + 5 {{\zeta}_1} } \over { 1 + 0.00794 {{\zeta}_1}
+ {\phi_m} = \frac{ 1 + 5 {{\zeta}_1} }{ 1 + 0.00794 {\zeta}_1
- (1+ 5 {{\zeta}_1}) } } \hspace{1cm} ; \hspace{1cm}
+ (1+ 5 {\zeta}_1) } \hspace{1cm} ; \hspace{1cm}
- {\phi_h} = { { 1 + 5 {{\zeta}_1} } \over { 1 + 0.00794 {\zeta}
+ {\phi_h} = \frac{ 1 + 5 {{\zeta}_1} }{ 1 + 0.00794 {\zeta}
- (1+ 5 {{\zeta}_1}) } } .
+ (1+ 5 {{\zeta}_1}) } .
  \]
  The moisture flux also depends on a specified evapotranspiration
  coefficient, set to unity over oceans and dependant on the climatological ground wetness over
-Line 657 
 humidity of the surface and of the lowes
+Line 648 
 humidity of the surface and of the lowes
  The heat conduction through sea ice, $Q_{ice}$, is given by
  \[
- {Q_{ice}} = {C_{ti} \over {H_i}} (T_i-T_g)
+ {Q_{ice}} = \frac{C_{ti} }{ H_i} (T_i-T_g)
  \]
  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
-Line 666 
 surface temperature of the ice.
+Line 657 
 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 (\cite{black:77}), and is given by:
  \[
- C_g = \sqrt{ {\lambda C_s \over 2\omega} } = \sqrt{(0.386 + 0.536W + 0.15W^2)2\times10^{-3}
+ C_g = \sqrt{ \frac{\lambda C_s }{ 2\omega} } = \sqrt{(0.386 + 0.536W + 0.15W^2)2\times10^{-3}
- {86400 \over 2 \pi} } \, \, .
+ \frac{86400}{2\pi} } \, \, .
  \]
- Here, the thermal conductivity, $\lambda$, is equal to $2\times10^{-3}$ ${ly\over{ sec}}
+ Here, the thermal conductivity, $\lambda$, is equal to $2\times10^{-3}$ $\frac{ly}{sec}
- {cm \over {^oK}}$,
+ \frac{cm}{K}$,
  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,
-Line 686 
 are shown in Table \ref{tab:fizhi:surfty
+Line 677 
 are shown in Table \ref{tab:fizhi:surfty
  cell occupied by any surface type were derived from the surface classification of
  \cite{deftow:94}, and information about the location of permanent
  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 surface type map for a $1^\circ$ 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
  dataset, which includes information about the percentage of water-cover at any point.
- The data were averaged to the model's \fxf and \txt grid resolutions,
+ The data were averaged to the model's grid resolutions,
  and any grid-box whose averaged water percentage was $\geq 60 \%$ was
- defined as a water point. The \fxf grid Land-Water designation was further modified
+ defined as a water point. The Land-Water designation was further modified
  subjectively to ensure sufficient representation from small but isolated land and water regions.
  \begin{table}
-Line 716 
 Type & Vegetation Designation \\ \hline
+Line 707 
 Type & Vegetation Designation \\ \hline
 & Ocean \\ \hline
  \end{tabular}
  \end{center}
- \caption{Surface type designations used to compute surface roughness (over land)
+ \caption{Surface type designations.}
- and surface albedo.}
  \label{tab:fizhi:surftype}
  \end{table}
  \begin{figure*}[htbp]
-   \centerline{  \epsfysize=7in  \epsfbox{part6/surftypes.ps}}
+   \centerline{  \epsfysize=4.0in  \epsfbox{s_phys_pkgs/figs/surftype.eps}}
-   \vspace{0.3in}
+   \vspace{0.2in}
-   \caption  {Surface Type Compinations at \txt resolution.}
+   \caption  {Surface Type Combinations.}
    \label{fig:fizhi:surftype}
  \end{figure*}
- \begin{figure*}[htbp]
+ % \rotatebox{270}{\centerline{  \epsfysize=4in  \epsfbox{s_phys_pkgs/figs/surftypes.eps}}}
-   \centerline{  \epsfysize=7in  \epsfbox{part6/surftypes.descrip.ps}}
+ % \rotatebox{270}{\centerline{  \epsfysize=4in  \epsfbox{s_phys_pkgs/figs/surftypes.descrip.eps}}}
-   \vspace{0.3in}
+ %\begin{figure*}[htbp]
-   \caption  {Surface Type Descriptions.}
+ %  \centerline{  \epsfysize=4in  \epsfbox{s_phys_pkgs/figs/surftypes.descrip.ps}}
-   \label{fig:fizhi:surftype.desc}
+ %  \vspace{0.3in}
- \end{figure*}
+ %  \caption  {Surface Type Descriptions.}
+ %  \label{fig:fizhi:surftype.desc}
+ %\end{figure*}
  \paragraph{Surface Roughness}
-Line 754 
 sun), the greenness fraction, the vegeta
+Line 745 
 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.
- Gravity Wave Drag:
+ \paragraph{Gravity Wave Drag}
- The fizhi package employs the gravity wave drag scheme of \cite{zhouetal:96}).
+ 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
  based on that derived by Pierrehumbert (1986) and is given by:
  \bq
- |\vec{\tau}_{sfc}| = {\rho U^3\over{N \ell^*}} \left(F_r^2 \over{1+F_r^2}\right) \, \, ,
+ |\vec{\tau}_{sfc}| = \frac{\rho U^3}{N \ell^*} \left( \frac{F_r^2}{1+F_r^2}\right) \, \, ,
  \eq
  where $F_r = N h /U$ is the Froude number, $N$ is the {\em Brunt - V\"{a}is\"{a}l\"{a}} frequency, $U$ is the
-Line 797 
 fields include:  sea surface temperature
+Line 788 
 fields include:  sea surface temperature
  vegetation index, and the radiation-related background levels of: ozone, carbon dioxide,
  and stratospheric moisture.
- Boundary condition data sets are available at the model's \fxf and \txt
+ Boundary condition data sets are available at the model's
  resolutions for either climatological or yearly varying conditions.
  Any frequency of boundary condition data can be used in the fizhi package;
  however, the current selection of data is summarized in Table \ref{tab:fizhi:bcdata}\@.
  The time mean values are interpolated during each model timestep to the
- current time. Future model versions will incorporate boundary conditions at
+ current time.
- higher spatial \mbox{($1^\circ$ x $1^\circ$)} resolutions.
  \begin{table}[htb]
  \begin{center}
-Line 840 
 The sub-grid scale variance is construct
+Line 830 
 The sub-grid scale variance is construct
  \paragraph{Upper Level Moisture}
  The fizhi package uses climatological water vapor data above 100 mb from the Stratospheric Aerosol and Gas
  Experiment (SAGE) as input into the model's radiation packages.  The SAGE data is archived
- as monthly zonal means at 5$^\circ$ latitudinal resolution.  The data is interpolated to the
+ as monthly zonal means at $5^\circ$ latitudinal resolution.  The data is interpolated to the
  model's grid location and current time, and blended with the GCM's moisture data.  Below 300 mb,
  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.
-Line 849 
 a linear interpolation (in pressure) is
+Line 839 
 a linear interpolation (in pressure) is
  \subsubsection{Fizhi Diagnostics}
  Fizhi Diagnostic Menu:
- \label{sec:fizhi-diagnostics:menu}
+ \label{sec:pkg:fizhi:diagnostics}
  \begin{tabular}{llll}
  \hline\hline
-Line 1381 
 Fizhi Diagnostic Description:
+Line 1371 
 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}.
+ Diagnostic Menu, Section \ref{sec:pkg:fizhi:diagnostics}.
  In all cases, each diagnostic as currently archived on the output datasets
  is time-averaged over its diagnostic output frequency:
  \[
- {\bf DIAGNOSTIC} = {1 \over TTOT} \sum_{t=1}^{t=TTOT} diag(t)
+ {\bf DIAGNOSTIC} = \frac{1}{TTOT} \sum_{t=1}^{t=TTOT} diag(t)
  \]
- where $TTOT = {{\bf NQDIAG} \over \Delta t}$, {\bf NQDIAG} is the
+ where $TTOT = \frac{ {\bf NQDIAG} }{\Delta t}$, {\bf NQDIAG} is the
  output frequency of the diagnostic, and $\Delta t$ is
  the timestep over which the diagnostic is updated.
-Line 1460 
 conduction from the relatively warm ocea
+Line 1450 
 conduction from the relatively warm ocea
  through sea ice represents an additional energy source term for the ground temperature equation.
  \[
- {\bf QICE} = {C_{ti} \over {H_i}} (T_i-T_g)
+ {\bf QICE} = \frac{C_{ti}}{H_i} (T_i-T_g)
  \]
  where $C_{ti}$ is the thermal conductivity of ice, $H_i$ is the ice thickness, assumed to
-Line 1502 
 the downward Shortwave flux and $F_{SW}^
+Line 1492 
 the downward Shortwave flux and $F_{SW}^
  \noindent
  The non-dimensional stability indicator is the ratio of the buoyancy to the shear:
  \[
- {\bf RI} = { { {g \over \theta_v} \pp {\theta_v}{z} } \over { (\pp{u}{z})^2 + (\pp{v}{z})^2 } }
+ {\bf RI} = \frac{ \frac{g}{\theta_v} \pp {\theta_v}{z} }{ (\pp{u}{z})^2 + (\pp{v}{z})^2 }
-  =  {  {c_p \pp{\theta_v}{z} \pp{P^ \kappa}{z} } \over { (\pp{u}{z})^2 + (\pp{v}{z})^2 } }
+  =  \frac{c_p \pp{\theta_v}{z} \pp{P^ \kappa}{z} }{ (\pp{u}{z})^2 + (\pp{v}{z})^2 }
  \]
  \\
  where we used the hydrostatic equation:
-Line 1522 
 stratification.
+Line 1512 
 stratification.
  The surface exchange coefficient is obtained from the similarity functions for the stability
   dependant flux profile relationships:
  \[
- {\bf CT} = -{( {\overline{w^{\prime}\theta^{\prime}}}) \over {u_* \Delta \theta }} =
+ {\bf CT} = -\frac{( \overline{w^{\prime}\theta^{\prime}} ) }{ u_* \Delta \theta } =
- -{( {\overline{w^{\prime}q^{\prime}}}) \over {u_* \Delta q }} =
+ -\frac{( \overline{w^{\prime}q^{\prime}} ) }{ u_* \Delta q } =
- { k \over { (\psi_{h} + \psi_{g}) } }
+ \frac{ k }{ (\psi_{h} + \psi_{g}) }
  \]
  where $\psi_h$ is the surface layer non-dimensional temperature change and $\psi_g$ is the
  viscous sublayer non-dimensional temperature or moisture change:
  \[
- \psi_{h} = {\int_{\zeta_{0}}^{\zeta} {\phi_{h} \over \zeta} d \zeta} \hspace{1cm} and
+ \psi_{h} = \int_{\zeta_{0}}^{\zeta} \frac{\phi_{h} }{ \zeta} d \zeta \hspace{1cm} and
- \hspace{1cm} \psi_{g} = { 0.55 (Pr^{2/3} - 0.2) \over \nu^{1/2} }
+ \hspace{1cm} \psi_{g} = \frac{ 0.55 (Pr^{2/3} - 0.2) }{ \nu^{1/2} }
  (h_{0}u_{*} - h_{0_{ref}}u_{*_{ref}})^{1/2}
  \]
  and:
-Line 1552 
 viscosity, $z_{0}$ is the surface roughn
+Line 1542 
 viscosity, $z_{0}$ is the surface roughn
  The surface exchange coefficient is obtained from the similarity functions for the stability
   dependant flux profile relationships:
  \[
- {\bf CU} = {u_* \over W_s} = { k \over \psi_{m} }
+ {\bf CU} = \frac{u_* }{ W_s} = \frac{ k }{ \psi_{m} }
  \]
  where $\psi_m$ is the surface layer non-dimensional wind shear:
  \[
- \psi_{m} = {\int_{\zeta_{0}}^{\zeta} {\phi_{m} \over \zeta} d \zeta}
+ \psi_{m} = {\int_{\zeta_{0}}^{\zeta} \frac{\phi_{m} }{ \zeta} d \zeta}
  \]
  \noindent
  $\phi_m$ is the similarity function of $\zeta$, which expresses the stability dependance of
-Line 1576 
 diffusion coefficient $K_h$ times the ne
+Line 1566 
 diffusion coefficient $K_h$ times the ne
  or moisture. In the \cite{helflab:88} adaptation of this closure, $K_h$
  takes the form:
  \[
- {\bf ET} = K_h = -{( {\overline{w^{\prime}\theta_v^{\prime}}}) \over {\pp{\theta_v}{z}} }
+ {\bf ET} = K_h = -\frac{( \overline{w^{\prime}\theta_v^{\prime}}) }{ \pp{\theta_v}{z} }
   = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell \, S_H(G_M,G_H) & \mbox{decaying turbulence}
- \\ { q^2 \over {q_e} } \, \ell \, S_{H}(G_{M_e},G_{H_e}) & \mbox{growing turbulence} \end{array} \right.
+ \\ \frac{ q^2 }{ q_e } \, \ell \, S_{H}(G_{M_e},G_{H_e}) & \mbox{growing turbulence} \end{array} \right.
  \]
  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,
-Line 1617 
 diffusion coefficient $K_m$ times the ne
+Line 1607 
 diffusion coefficient $K_m$ times the ne
  In the \cite{helflab:88} adaptation of this closure, $K_m$
  takes the form:
  \[
- {\bf EU} = K_m = -{( {\overline{u^{\prime}w^{\prime}}}) \over {\pp{U}{z}} }
+ {\bf EU} = K_m = -\frac{( \overline{u^{\prime}w^{\prime}} ) }{ \pp{U}{z} }
   = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell \, S_M(G_M,G_H) & \mbox{decaying turbulence}
- \\ { q^2 \over {q_e} } \, \ell \, S_{M}(G_{M_e},G_{H_e}) & \mbox{growing turbulence} \end{array} \right.
+ \\ \frac{ q^2 }{ q_e } \, \ell \, S_{M}(G_{M_e},G_{H_e}) & \mbox{growing turbulence} \end{array} \right.
  \]
  \noindent
  where $q$ is the turbulent velocity, or $\sqrt{2*turbulent \hspace{.2cm} kinetic \hspace{.2cm}
-Line 1722 
 equation.
+Line 1712 
 equation.
  \]
  where:
  \[
- \left.{\pp{T}{t}}\right|_{c} = R \sum_i \left( \alpha { m_B \over c_p} \Gamma_s \right)_i
+ \left.{\pp{T}{t}}\right|_{c} = R \sum_i \left( \alpha \frac{m_B}{c_p} \Gamma_s \right)_i
  \hspace{.4cm} and
- \hspace{.4cm} \left.{\pp{T}{t}}\right|_{ls} = {L \over c_p } (q^*-q)
+ \hspace{.4cm} \left.{\pp{T}{t}}\right|_{ls} = \frac{L}{c_p} (q^*-q)
  \]
  and
  \[
-Line 1752 
 $R$ is the rain re-evaporation adjustmen
+Line 1742 
 $R$ is the rain re-evaporation adjustmen
  \]
  where:
  \[
- \left.{\pp{q}{t}}\right|_{c} = R \sum_i \left( \alpha { m_B \over {L}}(\Gamma_h-\Gamma_s) \right)_i
+ \left.{\pp{q}{t}}\right|_{c} = R \sum_i \left( \alpha \frac{m_B}{L}(\Gamma_h-\Gamma_s) \right)_i
  \hspace{.4cm} and
  \hspace{.4cm} \left.{\pp{q}{t}}\right|_{ls} = (q^*-q)
  \]
-Line 1798 
 F_{LW} = C(p,p') \cdot F^{clearsky}_{LW}
+Line 1788 
 F_{LW} = C(p,p') \cdot F^{clearsky}_{LW}
  Finally, the net longwave heating rate is calculated as the vertical divergence of the
  net terrestrial radiative fluxes:
  \[
- \pp{\rho c_p T}{t} = - {\partial \over \partial z} F_{LW}^{NET} ,
+ \pp{\rho c_p T}{t} = - \p{z} F_{LW}^{NET} ,
  \]
  or
  \[
- {\bf RADLW} = \frac{g}{c_p \pi} {\partial \over \partial \sigma} F_{LW}^{NET} .
+ {\bf RADLW} = \frac{g}{c_p \pi} \p{\sigma} F_{LW}^{NET} .
  \]
  \noindent
-Line 1833 
 input at the top of the atmosphere.
+Line 1823 
 input at the top of the atmosphere.
  \noindent
  The heating rate due to Shortwave Radiation under cloudy skies is defined as:
  \[
- \pp{\rho c_p T}{t} = - {\partial \over \partial z} F(cloudy)_{SW}^{NET} \cdot {\rm RADSWT},
+ \pp{\rho c_p T}{t} = - \p{z} F(cloudy)_{SW}^{NET} \cdot {\rm RADSWT},
  \]
  or
  \[
- {\bf RADSW} = \frac{g}{c_p \pi} {\partial \over \partial \sigma} F(cloudy)_{SW}^{NET}\cdot {\rm RADSWT} .
+ {\bf RADSW} = \frac{g}{c_p \pi} \p{\sigma} F(cloudy)_{SW}^{NET}\cdot {\rm RADSWT} .
  \]
  \noindent
-Line 1857 
 For a change in specific humidity due to
+Line 1847 
 For a change in specific humidity due to
  the vertical integral or total precipitable amount is given by:
  \[
  {\bf PREACC} = \int_{surf}^{top} \rho \Delta q_{moist} dz = - \int_{surf}^{top} \Delta  q_{moist}
- {dp \over g} = {1 \over g} \int_0^1 \Delta q_{moist} dp
+ \frac{dp}{g} = \frac{1}{g} \int_0^1 \Delta q_{moist} dp
  \]
  \\
-Line 1874 
 For a change in specific humidity due to
+Line 1864 
 For a change in specific humidity due to
  the vertical integral or total precipitable amount is given by:
  \[
  {\bf PRECON} = \int_{surf}^{top} \rho \Delta q_{cum} dz = - \int_{surf}^{top} \Delta  q_{cum}
- {dp \over g} = {1 \over g} \int_0^1 \Delta q_{cum} dp
+ \frac{dp}{g} = \frac{1}{g} \int_0^1 \Delta q_{cum} dp
  \]
  \\
-Line 1959 
 where $\rho$ is the air density, and $K_
+Line 1949 
 where $\rho$ is the air density, and $K_
  \noindent
  The drag coefficient for momentum obtained by assuming a neutrally stable surface layer:
  \[
- {\bf CN} = { k \over { \ln({h \over {z_0}})} }
+ {\bf CN} = \frac{ k }{ \ln(\frac{h }{z_0}) }
  \]
  \noindent
-Line 1995 
 The air/surface virtual temperature diff
+Line 1985 
 The air/surface virtual temperature diff
  \noindent
  where
  \[
- \theta_{v{Nrphys+1}} = { T_g \over {P^{\kappa}_{surf}} } (1 + .609 q_{Nrphys+1}) \hspace{1cm}
+ \theta_{v{Nrphys+1}} = \frac{ T_g }{ P^{\kappa}_{surf} } (1 + .609 q_{Nrphys+1}) \hspace{1cm}
  and \hspace{1cm} q_{Nrphys+1} = q_{Nrphys} + \beta(q^*(T_g,P_s) - q_{Nrphys})
  \]
-Line 2026 
 flux, and $C_g$ is the total heat capaci
+Line 2016 
 flux, and $C_g$ is the total heat capaci
  $C_g$ 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:
  \[
- C_g = \sqrt{ {\lambda C_s \over {2 \omega} } } = \sqrt{(0.386 + 0.536W + 0.15W^2)2x10^{-3}
+ C_g = \sqrt{ \frac{\lambda C_s }{ 2 \omega } } = \sqrt{(0.386 + 0.536W + 0.15W^2)2x10^{-3}
- { 86400. \over {2 \pi} } } \, \, .
+ \frac{86400.}{2\pi} } \, \, .
  \]
  \noindent
- Here, the thermal conductivity, $\lambda$, is equal to $2x10^{-3}$ ${ly\over{ sec}}
+ Here, the thermal conductivity, $\lambda$, is equal to $2x10^{-3}$ $\frac{ly}{sec}
- {cm \over {^oK}}$,
+ \frac{cm}{K}$,
  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,
-Line 2173 
 Thus, {\bf LWCLR} is defined as the net
+Line 2163 
 Thus, {\bf LWCLR} is defined as the net
  vertical divergence of the
  clear-sky longwave radiative flux:
  \[
- \pp{\rho c_p T}{t}_{clearsky} = - {\partial \over \partial z} F(clearsky)_{LW}^{NET} ,
+ \pp{\rho c_p T}{t}_{clearsky} = - \p{z} F(clearsky)_{LW}^{NET} ,
  \]
  or
  \[
- {\bf LWCLR} = \frac{g}{c_p \pi} {\partial \over \partial \sigma} F(clearsky)_{LW}^{NET} .
+ {\bf LWCLR} = \frac{g}{c_p \pi} \p{\sigma} F(clearsky)_{LW}^{NET} .
  \]
  \noindent
-Line 2367 
 The surface stress velocity, or the fric
+Line 2357 
 The surface stress velocity, or the fric
  the surface layer top impeded by the surface drag:
  \[
  {\bf USTAR} = C_uW_s \hspace{1cm}where: \hspace{.2cm}
- C_u = {k \over {\psi_m} }
+ C_u = \frac{k}{\psi_m}
  \]
  \noindent
-Line 2382 
 Over the land surface, the surface rough
+Line 2372 
 Over the land surface, the surface rough
  time from the monthly mean data of \cite{dorsell:89}. Over the ocean,
  the roughness length is a function of the surface-stress velocity, $u_*$.
  \[
- {\bf Z0} = c_1u^3_* + c_2u^2_* + c_3u_* + c_4 + {c_5 \over {u_*}}
+ {\bf Z0} = c_1u^3_* + c_2u^2_* + c_3u_* + c_4 + {c_5}{u_*}
  \]
  \noindent
-Line 2436 
 input at the top of the atmosphere.
+Line 2426 
 input at the top of the atmosphere.
  \noindent
  The heating rate due to Shortwave Radiation under clear skies is defined as:
  \[
- \pp{\rho c_p T}{t} = - {\partial \over \partial z} F(clear)_{SW}^{NET} \cdot {\rm RADSWT},
+ \pp{\rho c_p T}{t} = - \p{z} F(clear)_{SW}^{NET} \cdot {\rm RADSWT},
  \]
  or
  \[
- {\bf SWCLR} = \frac{g}{c_p } {\partial \over \partial p} F(clear)_{SW}^{NET}\cdot {\rm RADSWT} .
+ {\bf SWCLR} = \frac{g}{c_p } \p{p} F(clear)_{SW}^{NET}\cdot {\rm RADSWT} .
  \]
  \noindent
-Line 2627 
 If we define the time-tendency of Temper
+Line 2617 
 If we define the time-tendency of Temper
  \end{eqnarray*}
  then, since there are no surface pressure changes due to Diabatic processes, we may write
  \[
- \pp{T}{t}_{Diabatic} = {p^\kappa \over \pi }\pp{\pi \theta}{t}_{Diabatic}
+ \pp{T}{t}_{Diabatic} = \frac{p^\kappa}{\pi}\pp{\pi \theta}{t}_{Diabatic}
  \]
  where $\theta = T/p^\kappa$.  Thus, {\bf DIABT} may be written as
  \[
- {\bf DIABT} = {p^\kappa \over \pi } \left( \pp{\pi \theta}{t}_{Diabatic} + \pp{\pi \theta}{t}_{Analysis} \right)
+ {\bf DIABT} = \frac{p^\kappa}{\pi} \left( \pp{\pi \theta}{t}_{Diabatic} + \pp{\pi \theta}{t}_{Analysis} \right)
  \]
  \\
-Line 2650 
 If we define the time-tendency of Specif
+Line 2640 
 If we define the time-tendency of Specif
  \]
  then, since there are no surface pressure changes due to Diabatic processes, we may write
  \[
- \pp{q}{t}_{Diabatic} = {1 \over \pi }\pp{\pi q}{t}_{Diabatic}
+ \pp{q}{t}_{Diabatic} = \frac{1}{\pi}\pp{\pi q}{t}_{Diabatic}
  \]
  Thus, {\bf DIABQ} may be written as
  \[
- {\bf DIABQ} = {1 \over \pi } \left( \pp{\pi q}{t}_{Diabatic} + \pp{\pi q}{t}_{Analysis} \right)
+ {\bf DIABQ} = \frac{1}{\pi} \left( \pp{\pi q}{t}_{Diabatic} + \pp{\pi q}{t}_{Analysis} \right)
  \]
  \\
-Line 2668 
 and dividing by the total mass of the co
+Line 2658 
 and dividing by the total mass of the co
  \[
  {\bf VINTUQ} = \frac{ \int_{surf}^{top} u q \rho dz  } { \int_{surf}^{top} \rho dz  }
  \]
- Using $\rho \delta z = -{\delta p \over g} = - {1 \over g} \delta p$, we have
+ Using $\rho \delta z = -\frac{\delta p}{g} = - \frac{1}{g} \delta p$, we have
  \[
  {\bf VINTUQ} = { \int_0^1 u q dp  }
  \]
-Line 2685 
 and dividing by the total mass of the co
+Line 2675 
 and dividing by the total mass of the co
  \[
  {\bf VINTVQ} = \frac{ \int_{surf}^{top} v q \rho dz  } { \int_{surf}^{top} \rho dz  }
  \]
- Using $\rho \delta z = -{\delta p \over g} = - {1 \over g} \delta p$, we have
+ Using $\rho \delta z = -\frac{\delta p}{g} = - \frac{1}{g} \delta p$, we have
  \[
  {\bf VINTVQ} = { \int_0^1 v q dp  }
  \]
-Line 2718 
 and dividing by the total mass of the co
+Line 2708 
 and dividing by the total mass of the co
  \[
  {\bf VINTVT} = \frac{ \int_{surf}^{top} v T \rho dz  } { \int_{surf}^{top} \rho dz  }
  \]
- Using $\rho \delta z = -{\delta p \over g} $, we have
+ Using $\rho \delta z = -\frac{\delta p}{g} $, we have
  \[
  {\bf VINTVT} = { \int_0^1 v T dp  }
  \]
-Line 2783 
 The Total Precipitable Water is defined
+Line 2773 
 The Total Precipitable Water is defined
  given by:
  \begin{eqnarray*}
  {\bf QINT} & = & \int_{surf}^{top} \rho q dz \\
-            & = & {\pi \over g} \int_0^1 q dp
+            & = & \frac{\pi}{g} \int_0^1 q dp
  \end{eqnarray*}
  where we have used the hydrostatic relation
- $\rho \delta z = -{\delta p \over g} $.
+ $\rho \delta z = -\frac{\delta p}{g} $.
  \\
-Line 2796 
 $\rho \delta z = -{\delta p \over g} $.
+Line 2786 
 $\rho \delta z = -{\delta p \over g} $.
  \noindent
  The u-wind at the 2-meter depth is determined from the similarity theory:
  \[
- {\bf U2M} = {u_* \over k} \psi_{m_{2m}} {u_{sl} \over {W_s}} =
+ {\bf U2M} = \frac{u_*}{k} \psi_{m_{2m}} \frac{u_{sl}}{W_s} =
- { \psi_{m_{2m}} \over {\psi_{m_{sl}} }}u_{sl}
+ \frac{ \psi_{m_{2m}} }{ \psi_{m_{sl}} }u_{sl}
  \]
  \noindent
-Line 2812 
 is above two meters, ${\bf U2M}$ is unde
+Line 2802 
 is above two meters, ${\bf U2M}$ is unde
  \noindent
  The v-wind at the 2-meter depth is a determined from the similarity theory:
  \[
- {\bf V2M} = {u_* \over k} \psi_{m_{2m}} {v_{sl} \over {W_s}} =
+ {\bf V2M} = \frac{u_*}{k} \psi_{m_{2m}} \frac{v_{sl}}{W_s} =
- { \psi_{m_{2m}} \over {\psi_{m_{sl}} }}v_{sl}
+ \frac{ \psi_{m_{2m}} }{ \psi_{m_{sl}} }v_{sl}
  \]
  \noindent
-Line 2828 
 is above two meters, ${\bf V2M}$ is unde
+Line 2818 
 is above two meters, ${\bf V2M}$ is unde
  \noindent
  The temperature at the 2-meter depth is a determined from the similarity theory:
  \[
- {\bf T2M} = P^{\kappa} ({\theta* \over k} ({\psi_{h_{2m}}+\psi_g}) + \theta_{surf} ) =
+ {\bf T2M} = P^{\kappa} (\frac{\theta*}{k} ({\psi_{h_{2m}}+\psi_g}) + \theta_{surf} ) =
- P^{\kappa}(\theta_{surf} + { {\psi_{h_{2m}}+\psi_g} \over {{\psi_{h_{sl}}+\psi_g}} }
+ P^{\kappa}(\theta_{surf} + \frac{ \psi_{h_{2m}}+\psi_g }{ \psi_{h_{sl}}+\psi_g }
- (\theta_{sl} - \theta_{surf}))
+ (\theta_{sl} - \theta_{surf}) )
  \]
  where:
  \[
- \theta_* = - { (\overline{w^{\prime}\theta^{\prime}}) \over {u_*} }
+ \theta_* = - \frac{ (\overline{w^{\prime}\theta^{\prime}}) }{ u_* }
  \]
  \noindent
-Line 2850 
 is above two meters, ${\bf T2M}$ is unde
+Line 2840 
 is above two meters, ${\bf T2M}$ is unde
  \noindent
  The specific humidity at the 2-meter depth is determined from the similarity theory:
  \[
- {\bf Q2M} = P^{\kappa} ({q_* \over k} ({\psi_{h_{2m}}+\psi_g}) + q_{surf} ) =
+ {\bf Q2M} = P^{\kappa} \frac({q_*}{k} ({\psi_{h_{2m}}+\psi_g}) + q_{surf} ) =
- P^{\kappa}(q_{surf} + { {\psi_{h_{2m}}+\psi_g} \over {{\psi_{h_{sl}}+\psi_g}} }
+ P^{\kappa}(q_{surf} + \frac{ \psi_{h_{2m}}+\psi_g }{ \psi_{h_{sl}}+\psi_g }
  (q_{sl} - q_{surf}))
  \]
  where:
  \[
- q_* = - { (\overline{w^{\prime}q^{\prime}}) \over {u_*} }
+ q_* = - \frac{ (\overline{w^{\prime}q^{\prime}}) }{ u_* }
  \]
  \noindent
-Line 2874 
 The u-wind at the 10-meter depth is an i
+Line 2864 
 The u-wind at the 10-meter depth is an i
  and the model lowest level wind using the ratio of the non-dimensional wind shear
  at the two levels:
  \[
- {\bf U10M} = {u_* \over k} \psi_{m_{10m}} {u_{sl} \over {W_s}} =
+ {\bf U10M} = \frac{u_*}{k} \psi_{m_{10m}} \frac{u_{sl}}{W_s} =
- { \psi_{m_{10m}} \over {\psi_{m_{sl}} }}u_{sl}
+ \frac{ \psi_{m_{10m}} }{ \psi_{m_{sl}} }u_{sl}
  \]
  \noindent
-Line 2891 
 The v-wind at the 10-meter depth is an i
+Line 2881 
 The v-wind at the 10-meter depth is an i
  and the model lowest level wind using the ratio of the non-dimensional wind shear
  at the two levels:
  \[
- {\bf V10M} = {u_* \over k} \psi_{m_{10m}} {v_{sl} \over {W_s}} =
+ {\bf V10M} = \frac{u_*}{k} \psi_{m_{10m}} \frac{v_{sl}}{W_s} =
- { \psi_{m_{10m}} \over {\psi_{m_{sl}} }}v_{sl}
+ \frac{ \psi_{m_{10m}} }{ \psi_{m_{sl}} }v_{sl}
  \]
  \noindent
-Line 2908 
 The temperature at the 10-meter depth is
+Line 2898 
 The temperature at the 10-meter depth is
  temperature and the model lowest level potential temperature using the ratio of the
  non-dimensional temperature gradient at the two levels:
  \[
- {\bf T10M} = P^{\kappa} ({\theta* \over k} ({\psi_{h_{10m}}+\psi_g}) + \theta_{surf} ) =
+ {\bf T10M} = P^{\kappa} (\frac{\theta*}{k} ({\psi_{h_{10m}}+\psi_g}) + \theta_{surf} ) =
- P^{\kappa}(\theta_{surf} + { {\psi_{h_{10m}}+\psi_g} \over {{\psi_{h_{sl}}+\psi_g}} }
+ P^{\kappa}(\theta_{surf} + \frac{\psi_{h_{10m}}+\psi_g}{\psi_{h_{sl}}+\psi_g}
  (\theta_{sl} - \theta_{surf}))
  \]
  where:
  \[
- \theta_* = - { (\overline{w^{\prime}\theta^{\prime}}) \over {u_*} }
+ \theta_* = - \frac{ (\overline{w^{\prime}\theta^{\prime}}) }{ u_* }
  \]
  \noindent
-Line 2931 
 The specific humidity at the 10-meter de
+Line 2921 
 The specific humidity at the 10-meter de
  humidity and the model lowest level specific humidity using the ratio of the
  non-dimensional temperature gradient at the two levels:
  \[
- {\bf Q10M} = P^{\kappa} ({q_* \over k} ({\psi_{h_{10m}}+\psi_g}) + q_{surf} ) =
+ {\bf Q10M} = P^{\kappa} (\frac{q_*}{k} ({\psi_{h_{10m}}+\psi_g}) + q_{surf} ) =
- P^{\kappa}(q_{surf} + { {\psi_{h_{10m}}+\psi_g} \over {{\psi_{h_{sl}}+\psi_g}} }
+ P^{\kappa}(q_{surf} + \frac{\psi_{h_{10m}}+\psi_g}{\psi_{h_{sl}}+\psi_g}
  (q_{sl} - q_{surf}))
  \]
  where:
  \[
- q_* =  - { (\overline{w^{\prime}q^{\prime}}) \over {u_*} }
+ q_* =  - \frac{ (\overline{w^{\prime}q^{\prime}}) }{ u_* }
  \]
  \noindent
-Line 2983 
 q^{n+1} = (\pi q)^{n+1} / \pi^{n+1}
+Line 2973 
 q^{n+1} = (\pi q)^{n+1} / \pi^{n+1}
  \subsubsection{Dos and donts}
  \subsubsection{Fizhi Reference}
+ \subsubsection{Experiments and tutorials that use fizhi}
+ \label{sec:pkg:fizhi:experiments}
+ \begin{itemize}
+ \item{Global atmosphere experiment with realistic SST and topography in fizhi-cs-32x32x10 verification directory. }
+ \item{Global atmosphere aqua planet experiment in fizhi-cs-aqualev20 verification directory. }
+ \end{itemize}

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-Removed from v.1.11
+Added in v.1.19

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