--- manual/s_phys_pkgs/text/fizhi.tex 2005/08/02 15:50:51 1.11 +++ 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 @@ -162,22 +162,22 @@ \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} + \caption [Critical Relative Humidity for Clouds.] + {Critical Relative Humidity for Clouds.} + \label{fig.rhcrit} \end{figure*} The total cloud fraction in a grid box is determined by the larger of the two cloud fractions: @@ -308,20 +308,9 @@ 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 -(see Figure \ref{fig:fizhi:cloud}) within this +The solar flux is computed for each of eight cloud realizations possible 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}. @@ -360,7 +349,7 @@ \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} @@ -420,7 +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}. @@ -431,7 +420,7 @@ 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. @@ -474,11 +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 @@ -502,7 +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 @@ -510,7 +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 @@ -522,16 +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 @@ -547,12 +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 @@ -562,13 +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 @@ -579,7 +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 @@ -588,7 +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} @@ -609,10 +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 @@ -657,7 +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 @@ -666,11 +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, @@ -686,13 +677,13 @@ 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 10 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} @@ -716,25 +707,25 @@ 100 & Ocean \\ \hline \end{tabular} \end{center} -\caption{Surface type designations used to compute surface roughness (over land) -and surface albedo.} +\caption{Surface type designations.} \label{tab:fizhi:surftype} \end{table} - \begin{figure*}[htbp] - \centerline{ \epsfysize=7in \epsfbox{part6/surftypes.ps}} - \vspace{0.3in} - \caption {Surface Type Compinations at \txt resolution.} + \centerline{ \epsfysize=4.0in \epsfbox{s_phys_pkgs/figs/surftype.eps}} + \vspace{0.2in} + \caption {Surface Type Combinations.} \label{fig:fizhi:surftype} \end{figure*} -\begin{figure*}[htbp] - \centerline{ \epsfysize=7in \epsfbox{part6/surftypes.descrip.ps}} - \vspace{0.3in} - \caption {Surface Type Descriptions.} - \label{fig:fizhi:surftype.desc} -\end{figure*} +% \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{s_phys_pkgs/figs/surftypes.descrip.ps}} +% \vspace{0.3in} +% \caption {Surface Type Descriptions.} +% \label{fig:fizhi:surftype.desc} +%\end{figure*} \paragraph{Surface Roughness} @@ -754,16 +745,16 @@ 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 @@ -797,13 +788,12 @@ 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 -higher spatial \mbox{($1^\circ$ x $1^\circ$)} resolutions. +current time. \begin{table}[htb] \begin{center} @@ -840,7 +830,7 @@ \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. @@ -849,7 +839,7 @@ \subsubsection{Fizhi Diagnostics} Fizhi Diagnostic Menu: -\label{sec:fizhi-diagnostics:menu} +\label{sec:pkg:fizhi:diagnostics} \begin{tabular}{llll} \hline\hline @@ -1381,14 +1371,14 @@ 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. @@ -1460,7 +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 @@ -1502,8 +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: @@ -1522,15 +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: @@ -1552,11 +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 @@ -1576,9 +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, @@ -1617,9 +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} @@ -1722,9 +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 \[ @@ -1752,7 +1742,7 @@ \] where: \[ -\left.{\pp{q}{t}}\right|_{c} = R \sum_i \left( \alpha { m_B \over {L}}(\Gamma_h-\Gamma_s) \right)_i +\left.{\pp{q}{t}}\right|_{c} = R \sum_i \left( \alpha \frac{m_B}{L}(\Gamma_h-\Gamma_s) \right)_i \hspace{.4cm} and \hspace{.4cm} \left.{\pp{q}{t}}\right|_{ls} = (q^*-q) \] @@ -1798,11 +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 @@ -1833,11 +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 @@ -1857,7 +1847,7 @@ the vertical integral or total precipitable amount is given by: \[ {\bf PREACC} = \int_{surf}^{top} \rho \Delta q_{moist} dz = - \int_{surf}^{top} \Delta q_{moist} -{dp \over g} = {1 \over g} \int_0^1 \Delta q_{moist} dp +\frac{dp}{g} = \frac{1}{g} \int_0^1 \Delta q_{moist} dp \] \\ @@ -1874,7 +1864,7 @@ the vertical integral or total precipitable amount is given by: \[ {\bf PRECON} = \int_{surf}^{top} \rho \Delta q_{cum} dz = - \int_{surf}^{top} \Delta q_{cum} -{dp \over g} = {1 \over g} \int_0^1 \Delta q_{cum} dp +\frac{dp}{g} = \frac{1}{g} \int_0^1 \Delta q_{cum} dp \] \\ @@ -1959,7 +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 @@ -1995,7 +1985,7 @@ \noindent where \[ -\theta_{v{Nrphys+1}} = { T_g \over {P^{\kappa}_{surf}} } (1 + .609 q_{Nrphys+1}) \hspace{1cm} +\theta_{v{Nrphys+1}} = \frac{ T_g }{ P^{\kappa}_{surf} } (1 + .609 q_{Nrphys+1}) \hspace{1cm} and \hspace{1cm} q_{Nrphys+1} = q_{Nrphys} + \beta(q^*(T_g,P_s) - q_{Nrphys}) \] @@ -2026,12 +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, @@ -2173,11 +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 @@ -2367,7 +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 @@ -2382,7 +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 @@ -2436,11 +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 @@ -2627,11 +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 \[ -{\bf DIABT} = {p^\kappa \over \pi } \left( \pp{\pi \theta}{t}_{Diabatic} + \pp{\pi \theta}{t}_{Analysis} \right) +{\bf DIABT} = \frac{p^\kappa}{\pi} \left( \pp{\pi \theta}{t}_{Diabatic} + \pp{\pi \theta}{t}_{Analysis} \right) \] \\ @@ -2650,11 +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 \[ -{\bf DIABQ} = {1 \over \pi } \left( \pp{\pi q}{t}_{Diabatic} + \pp{\pi q}{t}_{Analysis} \right) +{\bf DIABQ} = \frac{1}{\pi} \left( \pp{\pi q}{t}_{Diabatic} + \pp{\pi q}{t}_{Analysis} \right) \] \\ @@ -2668,7 +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 \[ {\bf VINTUQ} = { \int_0^1 u q dp } \] @@ -2685,7 +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 \[ {\bf VINTVQ} = { \int_0^1 v q dp } \] @@ -2718,7 +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 \[ {\bf VINTVT} = { \int_0^1 v T dp } \] @@ -2783,10 +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} $. \\ @@ -2796,8 +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 @@ -2812,8 +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 @@ -2828,13 +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 @@ -2850,13 +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 @@ -2874,8 +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 @@ -2891,8 +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 @@ -2908,13 +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 @@ -2931,13 +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 @@ -2983,3 +2973,12 @@ \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} +