--- manual/s_phys_pkgs/text/fizhi.tex	2006/06/28 15:35:07	1.17
+++ manual/s_phys_pkgs/text/fizhi.tex	2010/08/30 23:09:21	1.19
@@ -36,11 +36,11 @@
 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
@@ -49,7 +49,7 @@
 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.
@@ -60,11 +60,11 @@
 related to the buoyancy, or the difference
 between the moist static energy in the cloud and in the environment:
 \[
-A = \int_{P_D}^{P_B} { {\eta \over {1 + \gamma} } 
-\left[ {{h_c-h^*} \over {P^{\kappa}}} \right] dP^{\kappa}}
+A = \int_{P_D}^{P_B} \frac{\eta}{1 + \gamma}
+\left[ \frac{h_c-h^*}{P^{\kappa}} \right] dP^{\kappa}
 \]
 
-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.
 
 
@@ -72,7 +72,7 @@
 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
@@ -90,13 +90,13 @@
 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
@@ -136,14 +136,14 @@
 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
@@ -154,7 +154,7 @@
 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
@@ -162,7 +162,7 @@
 \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
@@ -173,7 +173,7 @@
 
 \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.}
@@ -409,7 +409,7 @@
 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}.
@@ -463,11 +463,11 @@
 and is written:
 
 \[
-{\dd{}{t} ({{\h} q^2})} - { \pp{}{z} ({ {5 \over 3} {{\lambda}_1} q { \pp {}{z} 
+{\dd{}{t} ({{\h} q^2})} - { \pp{}{z} ({ \frac{5}{3} {{\lambda}_1} q { \pp {}{z} 
 ({\h}q^2)} })} =
 {- \overline{{u^{\prime}}{w^{\prime}}} { \pp{U}{z} }} - {\overline{{v^{\prime}}{w^{\prime}}} 
-{ \pp{V}{z} }} + {{g \over {\Theta_0}}{\overline{{w^{\prime}}{{{\theta}_v}^{\prime}}}} } 
-- { q^3 \over {{\Lambda} _1} }
+{ \pp{V}{z} }} + {\frac{g}{\Theta_0}{\overline{{w^{\prime}}{{{\theta}_v}^{\prime}}}}
+- \frac{ q^3}{{\Lambda}_1} }
 \]
 
 where $q$ is the turbulent velocity, ${u^{\prime}}$, ${v^{\prime}}$, ${w^{\prime}}$ and 
@@ -491,7 +491,7 @@
 \[
 K_h 
  = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell \, S_H(G_M,G_H) \, & \mbox{decaying turbulence}
-\\ { q^2 \over {q_e} } \, \ell \, S_{H}(G_{M_e},G_{H_e}) \, & \mbox{growing turbulence} \end{array} \right.
+\\ \frac{ q^2 }{ q_e } \, \ell \, S_{H}(G_{M_e},G_{H_e}) \, & \mbox{growing turbulence} \end{array} \right.
 \]
 
 and
@@ -499,7 +499,7 @@
 \[
 K_m
  = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell \, S_M(G_M,G_H) \, & \mbox{decaying turbulence}                
-\\ { q^2 \over {q_e} } \, \ell \, S_{M}(G_{M_e},G_{H_e}) \, & \mbox{growing turbulence} \end{array} \right.
+\\ \frac{ q^2 }{ q_e } \, \ell \, S_{M}(G_{M_e},G_{H_e}) \, & \mbox{growing turbulence} \end{array} \right.
 \]
 
 where the subscript $e$ refers to the value under conditions of local equillibrium
@@ -511,16 +511,18 @@
 are functions of the Richardson number:
 
 \[
-{\bf RI} = { { {g \over \theta_v} \pp {\theta_v}{z} } \over { (\pp{u}{z})^2 + (\pp{v}{z})^2 } }
- =  {  {c_p \pp{\theta_v}{z} \pp{P^ \kappa}{z} } \over { (\pp{u}{z})^2 + (\pp{v}{z})^2 } } .
+{\bf RI} = \frac{ \frac{g}{\theta_v} \pp{\theta_v}{z} }{ (\pp{u}{z})^2 + (\pp{v}{z})^2 }
+ =  \frac{c_p \pp{\theta_v}{z} \pp{P^ \kappa}{z} }{ (\pp{u}{z})^2 + (\pp{v}{z})^2 } .
 \]
 
 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,
-which corresponds to the lowest GCM level (see \ref{tab:fizhi:sigma}),
+Turbulent eddy diffusion coefficients of momentum, heat and moisture in the 
+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
@@ -536,12 +538,12 @@
 $C_u$ 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
@@ -551,13 +553,13 @@
 $C_t$ is the dimensionless exchange coefficient for heat and 
 moisture from the surface layer similarity functions:
 \[
-{C_t} = -{( {\overline{w^{\prime}\theta^{\prime}}}) \over {u_* \Delta \theta }} =
--{( {\overline{w^{\prime}q^{\prime}}}) \over {u_* \Delta q }} =
-{ k \over { (\psi_{h} + \psi_{g}) } }
+{C_t} = -\frac{( \overline{w^{\prime}\theta^{\prime}}) }{ u_* \Delta \theta } =
+-\frac{( \overline{w^{\prime}q^{\prime}}) }{ u_* \Delta q } =
+\frac{ k }{ (\psi_{h} + \psi_{g}) }
 \]
 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
@@ -568,7 +570,7 @@
 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 
@@ -577,7 +579,7 @@
  
 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}
@@ -598,10 +600,10 @@
 based functions of \cite{clarke:70},  slightly modified for
 the momemtum flux:  
 \[
-{\phi_m} = { { 1 + 5 {{\zeta}_1} } \over { 1 + 0.00794 {{\zeta}_1}
-(1+ 5 {{\zeta}_1}) } } \hspace{1cm} ; \hspace{1cm}
-{\phi_h} = { { 1 + 5 {{\zeta}_1} } \over { 1 + 0.00794 {\zeta}
-(1+ 5 {{\zeta}_1}) } } .
+{\phi_m} = \frac{ 1 + 5 {{\zeta}_1} }{ 1 + 0.00794 {\zeta}_1
+(1+ 5 {\zeta}_1) } \hspace{1cm} ; \hspace{1cm}
+{\phi_h} = \frac{ 1 + 5 {{\zeta}_1} }{ 1 + 0.00794 {\zeta}
+(1+ 5 {{\zeta}_1}) } .
 \]
 The moisture flux also depends on a specified evapotranspiration
 coefficient, set to unity over oceans and dependant on the climatological ground wetness over
@@ -646,7 +648,7 @@
 
 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 
@@ -655,11 +657,11 @@
 $C_g$ 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:
 \[
-C_g = \sqrt{ {\lambda C_s \over 2\omega} } = \sqrt{(0.386 + 0.536W + 0.15W^2)2\times10^{-3}
-{86400 \over 2 \pi} } \, \, .
+C_g = \sqrt{ \frac{\lambda C_s }{ 2\omega} } = \sqrt{(0.386 + 0.536W + 0.15W^2)2\times10^{-3}
+\frac{86400}{2\pi} } \, \, .
 \]
-Here, the thermal conductivity, $\lambda$, is equal to $2\times10^{-3}$ ${ly\over{ sec}}
-{cm \over {^oK}}$,    
+Here, the thermal conductivity, $\lambda$, is equal to $2\times10^{-3}$ $\frac{ly}{sec}
+\frac{cm}{K}$,
 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,
@@ -710,16 +712,16 @@
 \end{table}
  
 \begin{figure*}[htbp]
-  \centerline{  \epsfysize=4.0in  \epsfbox{part6/surftype.eps}}
+  \centerline{  \epsfysize=4.0in  \epsfbox{s_phys_pkgs/figs/surftype.eps}}
   \vspace{0.2in}
   \caption  {Surface Type Combinations.}
   \label{fig:fizhi:surftype}
 \end{figure*}
 
-% \rotatebox{270}{\centerline{  \epsfysize=4in  \epsfbox{part6/surftypes.eps}}}
-% \rotatebox{270}{\centerline{  \epsfysize=4in  \epsfbox{part6/surftypes.descrip.eps}}}
+% \rotatebox{270}{\centerline{  \epsfysize=4in  \epsfbox{s_phys_pkgs/figs/surftypes.eps}}}
+% \rotatebox{270}{\centerline{  \epsfysize=4in  \epsfbox{s_phys_pkgs/figs/surftypes.descrip.eps}}}
 %\begin{figure*}[htbp]
-%  \centerline{  \epsfysize=4in  \epsfbox{part6/surftypes.descrip.ps}}
+%  \centerline{  \epsfysize=4in  \epsfbox{s_phys_pkgs/figs/surftypes.descrip.ps}}
 %  \vspace{0.3in}
 %  \caption  {Surface Type Descriptions.}
 %  \label{fig:fizhi:surftype.desc}
@@ -752,7 +754,7 @@
 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 
@@ -1374,9 +1376,9 @@
 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.  
 
@@ -1448,7 +1450,7 @@
 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
@@ -1490,8 +1492,8 @@
 \noindent
 The non-dimensional stability indicator is the ratio of the buoyancy to the shear:
 \[
-{\bf RI} = { { {g \over \theta_v} \pp {\theta_v}{z} } \over { (\pp{u}{z})^2 + (\pp{v}{z})^2 } }
- =  {  {c_p \pp{\theta_v}{z} \pp{P^ \kappa}{z} } \over { (\pp{u}{z})^2 + (\pp{v}{z})^2 } }
+{\bf RI} = \frac{ \frac{g}{\theta_v} \pp {\theta_v}{z} }{ (\pp{u}{z})^2 + (\pp{v}{z})^2 }
+ =  \frac{c_p \pp{\theta_v}{z} \pp{P^ \kappa}{z} }{ (\pp{u}{z})^2 + (\pp{v}{z})^2 }
 \]
 \\
 where we used the hydrostatic equation: 
@@ -1510,15 +1512,15 @@
 The surface exchange coefficient is obtained from the similarity functions for the stability
  dependant flux profile relationships:
 \[
-{\bf CT} = -{( {\overline{w^{\prime}\theta^{\prime}}}) \over {u_* \Delta \theta }} = 
--{( {\overline{w^{\prime}q^{\prime}}}) \over {u_* \Delta q }} = 
-{ k \over { (\psi_{h} + \psi_{g}) } } 
+{\bf CT} = -\frac{( \overline{w^{\prime}\theta^{\prime}} ) }{ u_* \Delta \theta } = 
+-\frac{( \overline{w^{\prime}q^{\prime}} ) }{ u_* \Delta q } = 
+\frac{ k }{ (\psi_{h} + \psi_{g}) } 
 \]
 where $\psi_h$ is the surface layer non-dimensional temperature change and $\psi_g$ is the
 viscous sublayer non-dimensional temperature or moisture change:
 \[
-\psi_{h} = {\int_{\zeta_{0}}^{\zeta} {\phi_{h} \over \zeta} d \zeta} \hspace{1cm} and 
-\hspace{1cm} \psi_{g} = { 0.55 (Pr^{2/3} - 0.2) \over \nu^{1/2} } 
+\psi_{h} = \int_{\zeta_{0}}^{\zeta} \frac{\phi_{h} }{ \zeta} d \zeta \hspace{1cm} and 
+\hspace{1cm} \psi_{g} = \frac{ 0.55 (Pr^{2/3} - 0.2) }{ \nu^{1/2} } 
 (h_{0}u_{*} - h_{0_{ref}}u_{*_{ref}})^{1/2}
 \]
 and:
@@ -1540,11 +1542,11 @@
 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
@@ -1564,9 +1566,9 @@
 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, 
@@ -1605,9 +1607,9 @@
 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}
@@ -1710,9 +1712,9 @@
 \]
 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
 \[
@@ -1740,7 +1742,7 @@
 \]
 where:
 \[
-\left.{\pp{q}{t}}\right|_{c} = R \sum_i \left( \alpha { m_B \over {L}}(\Gamma_h-\Gamma_s) \right)_i 
+\left.{\pp{q}{t}}\right|_{c} = R \sum_i \left( \alpha \frac{m_B}{L}(\Gamma_h-\Gamma_s) \right)_i 
 \hspace{.4cm} and 
 \hspace{.4cm} \left.{\pp{q}{t}}\right|_{ls} = (q^*-q)
 \]
@@ -1786,11 +1788,11 @@
 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
@@ -1821,11 +1823,11 @@
 \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
@@ -1845,7 +1847,7 @@
 the vertical integral or total precipitable amount is given by:   
 \[
 {\bf PREACC} = \int_{surf}^{top} \rho \Delta q_{moist} dz = - \int_{surf}^{top} \Delta  q_{moist}
-{dp \over g} = {1 \over g} \int_0^1 \Delta q_{moist} dp
+\frac{dp}{g} = \frac{1}{g} \int_0^1 \Delta q_{moist} dp
 \]
 \\
 
@@ -1862,7 +1864,7 @@
 the vertical integral or total precipitable amount is given by:   
 \[
 {\bf PRECON} = \int_{surf}^{top} \rho \Delta q_{cum} dz = - \int_{surf}^{top} \Delta  q_{cum}
-{dp \over g} = {1 \over g} \int_0^1 \Delta q_{cum} dp
+\frac{dp}{g} = \frac{1}{g} \int_0^1 \Delta q_{cum} dp
 \]
 \\
 
@@ -1947,7 +1949,7 @@
 \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
@@ -1983,7 +1985,7 @@
 \noindent
 where
 \[
-\theta_{v{Nrphys+1}} = { T_g \over {P^{\kappa}_{surf}} } (1 + .609 q_{Nrphys+1}) \hspace{1cm}
+\theta_{v{Nrphys+1}} = \frac{ T_g }{ P^{\kappa}_{surf} } (1 + .609 q_{Nrphys+1}) \hspace{1cm}
 and \hspace{1cm} q_{Nrphys+1} = q_{Nrphys} + \beta(q^*(T_g,P_s) - q_{Nrphys})
 \]
 
@@ -2014,12 +2016,12 @@
 $C_g$ 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:
 \[
-C_g = \sqrt{ {\lambda C_s \over {2 \omega} } } = \sqrt{(0.386 + 0.536W + 0.15W^2)2x10^{-3}
-{ 86400. \over {2 \pi} } } \, \, .
+C_g = \sqrt{ \frac{\lambda C_s }{ 2 \omega } } = \sqrt{(0.386 + 0.536W + 0.15W^2)2x10^{-3}
+\frac{86400.}{2\pi} } \, \, .
 \]
 \noindent
-Here, the thermal conductivity, $\lambda$, is equal to $2x10^{-3}$ ${ly\over{ sec}} 
-{cm \over {^oK}}$, 
+Here, the thermal conductivity, $\lambda$, is equal to $2x10^{-3}$ $\frac{ly}{sec} 
+\frac{cm}{K}$, 
 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, 
@@ -2161,11 +2163,11 @@
 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
@@ -2355,7 +2357,7 @@
 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
@@ -2370,7 +2372,7 @@
 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
@@ -2424,11 +2426,11 @@
 \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
@@ -2615,11 +2617,11 @@
 \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
 \[
-{\bf DIABT} = {p^\kappa \over \pi } \left( \pp{\pi \theta}{t}_{Diabatic} + \pp{\pi \theta}{t}_{Analysis} \right)
+{\bf DIABT} = \frac{p^\kappa}{\pi} \left( \pp{\pi \theta}{t}_{Diabatic} + \pp{\pi \theta}{t}_{Analysis} \right)
 \]
 \\
 
@@ -2638,11 +2640,11 @@
 \]
 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
 \[
-{\bf DIABQ} = {1 \over \pi } \left( \pp{\pi q}{t}_{Diabatic} + \pp{\pi q}{t}_{Analysis} \right)
+{\bf DIABQ} = \frac{1}{\pi} \left( \pp{\pi q}{t}_{Diabatic} + \pp{\pi q}{t}_{Analysis} \right)
 \]
 \\
 
@@ -2656,7 +2658,7 @@
 \[
 {\bf VINTUQ} = \frac{ \int_{surf}^{top} u q \rho dz  } { \int_{surf}^{top} \rho dz  }
 \]
-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 
 \[
 {\bf VINTUQ} = { \int_0^1 u q dp  }
 \]
@@ -2673,7 +2675,7 @@
 \[
 {\bf VINTVQ} = \frac{ \int_{surf}^{top} v q \rho dz  } { \int_{surf}^{top} \rho dz  }
 \]
-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 
 \[
 {\bf VINTVQ} = { \int_0^1 v q dp  }
 \]
@@ -2706,7 +2708,7 @@
 \[
 {\bf VINTVT} = \frac{ \int_{surf}^{top} v T \rho dz  } { \int_{surf}^{top} \rho dz  }
 \]
-Using $\rho \delta z = -{\delta p \over g} $, we have 
+Using $\rho \delta z = -\frac{\delta p}{g} $, we have 
 \[
 {\bf VINTVT} = { \int_0^1 v T dp  }
 \]
@@ -2771,10 +2773,10 @@
 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} $.
 \\
 
 
@@ -2784,8 +2786,8 @@
 \noindent
 The u-wind at the 2-meter depth is determined from the similarity theory:
 \[
-{\bf U2M} = {u_* \over k} \psi_{m_{2m}} {u_{sl} \over {W_s}} =
-{ \psi_{m_{2m}} \over {\psi_{m_{sl}} }}u_{sl}
+{\bf U2M} = \frac{u_*}{k} \psi_{m_{2m}} \frac{u_{sl}}{W_s} =
+\frac{ \psi_{m_{2m}} }{ \psi_{m_{sl}} }u_{sl}
 \]
 
 \noindent
@@ -2800,8 +2802,8 @@
 \noindent
 The v-wind at the 2-meter depth is a determined from the similarity theory:
 \[
-{\bf V2M} = {u_* \over k} \psi_{m_{2m}} {v_{sl} \over {W_s}} =
-{ \psi_{m_{2m}} \over {\psi_{m_{sl}} }}v_{sl}
+{\bf V2M} = \frac{u_*}{k} \psi_{m_{2m}} \frac{v_{sl}}{W_s} =
+\frac{ \psi_{m_{2m}} }{ \psi_{m_{sl}} }v_{sl}
 \]
 
 \noindent
@@ -2816,13 +2818,13 @@
 \noindent
 The temperature at the 2-meter depth is a determined from the similarity theory:
 \[
-{\bf T2M} = P^{\kappa} ({\theta* \over k} ({\psi_{h_{2m}}+\psi_g}) + \theta_{surf} ) = 
-P^{\kappa}(\theta_{surf} + { {\psi_{h_{2m}}+\psi_g} \over {{\psi_{h_{sl}}+\psi_g}} }
-(\theta_{sl} - \theta_{surf})) 
+{\bf T2M} = P^{\kappa} (\frac{\theta*}{k} ({\psi_{h_{2m}}+\psi_g}) + \theta_{surf} ) = 
+P^{\kappa}(\theta_{surf} + \frac{ \psi_{h_{2m}}+\psi_g }{ \psi_{h_{sl}}+\psi_g }
+(\theta_{sl} - \theta_{surf}) ) 
 \]
 where:
 \[
-\theta_* = - { (\overline{w^{\prime}\theta^{\prime}}) \over {u_*} }
+\theta_* = - \frac{ (\overline{w^{\prime}\theta^{\prime}}) }{ u_* }
 \]
 
 \noindent
@@ -2838,13 +2840,13 @@
 \noindent
 The specific humidity at the 2-meter depth is determined from the similarity theory:
 \[
-{\bf Q2M} = P^{\kappa} ({q_* \over k} ({\psi_{h_{2m}}+\psi_g}) + q_{surf} ) = 
-P^{\kappa}(q_{surf} + { {\psi_{h_{2m}}+\psi_g} \over {{\psi_{h_{sl}}+\psi_g}} }
+{\bf Q2M} = P^{\kappa} \frac({q_*}{k} ({\psi_{h_{2m}}+\psi_g}) + q_{surf} ) = 
+P^{\kappa}(q_{surf} + \frac{ \psi_{h_{2m}}+\psi_g }{ \psi_{h_{sl}}+\psi_g }
 (q_{sl} - q_{surf})) 
 \]
 where:
 \[
-q_* = - { (\overline{w^{\prime}q^{\prime}}) \over {u_*} }
+q_* = - \frac{ (\overline{w^{\prime}q^{\prime}}) }{ u_* }
 \]
 
 \noindent
@@ -2862,8 +2864,8 @@
 and the model lowest level wind using the ratio of the non-dimensional wind shear
 at the two levels:
 \[
-{\bf U10M} = {u_* \over k} \psi_{m_{10m}} {u_{sl} \over {W_s}} =
-{ \psi_{m_{10m}} \over {\psi_{m_{sl}} }}u_{sl}
+{\bf U10M} = \frac{u_*}{k} \psi_{m_{10m}} \frac{u_{sl}}{W_s} =
+\frac{ \psi_{m_{10m}} }{ \psi_{m_{sl}} }u_{sl}
 \]
 
 \noindent
@@ -2879,8 +2881,8 @@
 and the model lowest level wind using the ratio of the non-dimensional wind shear
 at the two levels:
 \[
-{\bf V10M} = {u_* \over k} \psi_{m_{10m}} {v_{sl} \over {W_s}} =
-{ \psi_{m_{10m}} \over {\psi_{m_{sl}} }}v_{sl}
+{\bf V10M} = \frac{u_*}{k} \psi_{m_{10m}} \frac{v_{sl}}{W_s} =
+\frac{ \psi_{m_{10m}} }{ \psi_{m_{sl}} }v_{sl}
 \]
 
 \noindent
@@ -2896,13 +2898,13 @@
 temperature and the model lowest level potential temperature using the ratio of the 
 non-dimensional temperature gradient at the two levels:
 \[
-{\bf T10M} = P^{\kappa} ({\theta* \over k} ({\psi_{h_{10m}}+\psi_g}) + \theta_{surf} ) = 
-P^{\kappa}(\theta_{surf} + { {\psi_{h_{10m}}+\psi_g} \over {{\psi_{h_{sl}}+\psi_g}} }
+{\bf T10M} = P^{\kappa} (\frac{\theta*}{k} ({\psi_{h_{10m}}+\psi_g}) + \theta_{surf} ) = 
+P^{\kappa}(\theta_{surf} + \frac{\psi_{h_{10m}}+\psi_g}{\psi_{h_{sl}}+\psi_g}
 (\theta_{sl} - \theta_{surf})) 
 \]
 where:
 \[
-\theta_* = - { (\overline{w^{\prime}\theta^{\prime}}) \over {u_*} }
+\theta_* = - \frac{ (\overline{w^{\prime}\theta^{\prime}}) }{ u_* }
 \]
 
 \noindent
@@ -2919,13 +2921,13 @@
 humidity and the model lowest level specific humidity using the ratio of the 
 non-dimensional temperature gradient at the two levels:
 \[
-{\bf Q10M} = P^{\kappa} ({q_* \over k} ({\psi_{h_{10m}}+\psi_g}) + q_{surf} ) = 
-P^{\kappa}(q_{surf} + { {\psi_{h_{10m}}+\psi_g} \over {{\psi_{h_{sl}}+\psi_g}} }
+{\bf Q10M} = P^{\kappa} (\frac{q_*}{k} ({\psi_{h_{10m}}+\psi_g}) + q_{surf} ) = 
+P^{\kappa}(q_{surf} + \frac{\psi_{h_{10m}}+\psi_g}{\psi_{h_{sl}}+\psi_g}
 (q_{sl} - q_{surf})) 
 \]
 where:
 \[
-q_* =  - { (\overline{w^{\prime}q^{\prime}}) \over {u_*} }
+q_* =  - \frac{ (\overline{w^{\prime}q^{\prime}}) }{ u_* }
 \]
 
 \noindent
