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-revision 1.18 by jmc,
Fri Aug 27 13:15:37 2010 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
 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 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 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 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 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 536 
 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 551 
 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 568 
 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 577 
 $h_{0} = 30z_{0}$ 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 598 
 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 646 
 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 655 
 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 752 
 In this version, the gravity wave stress
+Line 754 
 In this version, the gravity wave stress
  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 1374 
 In all cases, each diagnostic as current
+Line 1376 
 In all cases, each diagnostic as current
  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 1448 
 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 1490 
 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 1510 
 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 1540 
 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 1564 
 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 1605 
 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 1710 
 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
  \[
-Line 1740 
 $R$ is the rain re-evaporation adjustmen
+Line 1742 
 $R$ is the rain re-evaporation adjustmen
  \]
  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 1786 
 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 1821 
 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 1845 
 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 1862 
 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 1947 
 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 1983 
 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 2014 
 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 2161 
 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 2355 
 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 2370 
 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 2424 
 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 2615 
 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 2638 
 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 2656 
 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 2673 
 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 2706 
 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 2771 
 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 2784 
 $\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 2800 
 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 2816 
 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 2838 
 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 2862 
 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 2879 
 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 2896 
 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 2919 
 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

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