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1 molod 1.9 \subsection{Fizhi: High-end Atmospheric Physics}
2 edhill 1.7 \label{sec:pkg:fizhi}
3     \begin{rawhtml}
4     <!-- CMIREDIR:package_fizhi: -->
5     \end{rawhtml}
6 molod 1.3 \input{texinputs/epsf.tex}
7 molod 1.1
8 molod 1.9 \subsubsection{Introduction}
9 molod 1.1 The fizhi (high-end atmospheric physics) package includes a collection of state-of-the-art
10     physical parameterizations for atmospheric radiation, cumulus convection, atmospheric
11 molod 1.11 boundary layer turbulence, and land surface processes. The collection of atmospheric
12     physics parameterizations were originally used together as part of the GEOS-3
13     (Goddard Earth Observing System-3) GCM developed at the NASA/Goddard Global Modelling
14     and Assimilation Office (GMAO).
15 molod 1.1
16     % *************************************************************************
17     % *************************************************************************
18    
19 molod 1.9 \subsubsection{Equations}
20 molod 1.1
21 molod 1.9 Moist Convective Processes:
22 molod 1.1
23 molod 1.5 \paragraph{Sub-grid and Large-scale Convection}
24 molod 1.1 \label{sec:fizhi:mc}
25    
26     Sub-grid scale cumulus convection is parameterized using the Relaxed Arakawa
27 molod 1.10 Schubert (RAS) scheme of \cite{moorsz:92}, which is a linearized Arakawa Schubert
28 molod 1.1 type scheme. RAS predicts the mass flux from an ensemble of clouds. Each subensemble is identified
29     by its entrainment rate and level of neutral bouyancy which are determined by the grid-scale properties.
30    
31     The thermodynamic variables that are used in RAS to describe the grid scale vertical profile are
32     the dry static energy, $s=c_pT +gz$, and the moist static energy, $h=c_p T + gz + Lq$.
33     The conceptual model behind RAS depicts each subensemble as a rising plume cloud, entraining
34     mass from the environment during ascent, and detraining all cloud air at the level of neutral
35     buoyancy. RAS assumes that the normalized cloud mass flux, $\eta$, normalized by the cloud base
36     mass flux, is a linear function of height, expressed as:
37     \[
38     \pp{\eta(z)}{z} = \lambda \hspace{0.4cm}or\hspace{0.4cm} \pp{\eta(P^{\kappa})}{P^{\kappa}} =
39 jmc 1.19 -\frac{c_p}{g}\theta\lambda
40 molod 1.1 \]
41     where we have used the hydrostatic equation written in the form:
42     \[
43 jmc 1.19 \pp{z}{P^{\kappa}} = -\frac{c_p}{g}\theta
44 molod 1.1 \]
45    
46     The entrainment parameter, $\lambda$, characterizes a particular subensemble based on its
47     detrainment level, and is obtained by assuming that the level of detrainment is the level of neutral
48     buoyancy, ie., the level at which the moist static energy of the cloud, $h_c$, is equal
49 molod 1.10 to the saturation moist static energy of the environment, $h^*$. Following \cite{moorsz:92},
50 molod 1.1 $\lambda$ may be written as
51     \[
52 jmc 1.19 \lambda = \frac{h_B - h^*_D}{ \frac{c_p}{g} \int_{P_D}^{P_B}\theta(h^*_D-h)dP^{\kappa}},
53 molod 1.1 \]
54    
55     where the subscript $B$ refers to cloud base, and the subscript $D$ refers to the detrainment level.
56    
57    
58     The convective instability is measured in terms of the cloud work function $A$, defined as the
59     rate of change of cumulus kinetic energy. The cloud work function is
60     related to the buoyancy, or the difference
61     between the moist static energy in the cloud and in the environment:
62     \[
63 jmc 1.19 A = \int_{P_D}^{P_B} \frac{\eta}{1 + \gamma}
64     \left[ \frac{h_c-h^*}{P^{\kappa}} \right] dP^{\kappa}
65 molod 1.1 \]
66    
67 jmc 1.19 where $\gamma$ is $\frac{L}{c_p}\pp{q^*}{T}$ obtained from the Claussius Clapeyron equation,
68 molod 1.1 and the subscript $c$ refers to the value inside the cloud.
69    
70    
71     To determine the cloud base mass flux, the rate of change of $A$ in time {\em due to dissipation by
72     the clouds} is assumed to approximately balance the rate of change of $A$ {\em due to the generation
73     by the large scale}. This is the quasi-equilibrium assumption, and results in an expression for $m_B$:
74     \[
75 jmc 1.19 m_B = \frac{- \left. \frac{dA}{dt} \right|_{ls}}{K}
76 molod 1.1 \]
77    
78     where $K$ is the cloud kernel, defined as the rate of change of the cloud work function per
79     unit cloud base mass flux, and is currently obtained by analytically differentiating the
80     expression for $A$ in time.
81     The rate of change of $A$ due to the generation by the large scale can be written as the
82     difference between the current $A(t+\Delta t)$ and its equillibrated value after the previous
83     convective time step
84     $A(t)$, divided by the time step. $A(t)$ is approximated as some critical $A_{crit}$,
85     computed by Lord (1982) from $in situ$ observations.
86    
87    
88     The predicted convective mass fluxes are used to solve grid-scale temperature
89     and moisture budget equations to determine the impact of convection on the large scale fields of
90     temperature (through latent heating and compensating subsidence) and moisture (through
91     precipitation and detrainment):
92     \[
93 jmc 1.19 \left.{\pp{\theta}{t}}\right|_{c} = \alpha \frac{ m_B}{c_p P^{\kappa}} \eta \pp{s}{p}
94 molod 1.1 \]
95     and
96     \[
97 jmc 1.19 \left.{\pp{q}{t}}\right|_{c} = \alpha \frac{ m_B}{L} \eta (\pp{h}{p}-\pp{s}{p})
98 molod 1.1 \]
99 jmc 1.19 where $\theta = \frac{T}{P^{\kappa}}$, $P = (p/p_0)$, and $\alpha$ is the relaxation parameter.
100 molod 1.1
101     As an approximation to a full interaction between the different allowable subensembles,
102     many clouds are simulated frequently, each modifying the large scale environment some fraction
103     $\alpha$ of the total adjustment. The parameterization thereby ``relaxes'' the large scale environment
104     towards equillibrium.
105    
106     In addition to the RAS cumulus convection scheme, the fizhi package employs a
107 molod 1.10 Kessler-type scheme for the re-evaporation of falling rain (\cite{sudm:88}), which
108 molod 1.1 correspondingly adjusts the temperature assuming $h$ is conserved. RAS in its current
109     formulation assumes that all cloud water is deposited into the detrainment level as rain.
110     All of the rain is available for re-evaporation, which begins in the level below detrainment.
111     The scheme accounts for some microphysics such as
112     the rainfall intensity, the drop size distribution, as well as the temperature,
113     pressure and relative humidity of the surrounding air. The fraction of the moisture deficit
114     in any model layer into which the rain may re-evaporate is controlled by a free parameter,
115     which allows for a relatively efficient re-evaporation of liquid precipitate and larger rainout
116     for frozen precipitation.
117    
118     Due to the increased vertical resolution near the surface, the lowest model
119     layers are averaged to provide a 50 mb thick sub-cloud layer for RAS. Each time RAS is
120     invoked (every ten simulated minutes),
121     a number of randomly chosen subensembles are checked for the possibility
122     of convection, from just above cloud base to 10 mb.
123    
124     Supersaturation or large-scale precipitation is initiated in the fizhi package whenever
125     the relative humidity in any grid-box exceeds a critical value, currently 100 \%.
126     The large-scale precipitation re-evaporates during descent to partially saturate
127     lower layers in a process identical to the re-evaporation of convective rain.
128    
129    
130 molod 1.5 \paragraph{Cloud Formation}
131 molod 1.1 \label{sec:fizhi:clouds}
132    
133     Convective and large-scale cloud fractons which are used for cloud-radiative interactions are determined
134     diagnostically as part of the cumulus and large-scale parameterizations.
135     Convective cloud fractions produced by RAS are proportional to the
136     detrained liquid water amount given by
137    
138     \[
139 jmc 1.19 F_{RAS} = \min\left[ \frac{l_{RAS}}{l_c}, 1.0 \right]
140 molod 1.1 \]
141    
142     where $l_c$ is an assigned critical value equal to $1.25$ g/kg.
143     A memory is associated with convective clouds defined by:
144    
145     \[
146 jmc 1.19 F_{RAS}^n = \min\left[ F_{RAS} + (1-\frac{\Delta t_{RAS}}{\tau})F_{RAS}^{n-1}, 1.0 \right]
147 molod 1.1 \]
148    
149     where $F_{RAS}$ is the instantanious cloud fraction and $F_{RAS}^{n-1}$ is the cloud fraction
150     from the previous RAS timestep. The memory coefficient is computed using a RAS cloud timescale,
151     $\tau$, equal to 1 hour. RAS cloud fractions are cleared when they fall below 5 \%.
152    
153     Large-scale cloudiness is defined, following Slingo and Ritter (1985), as a function of relative
154     humidity:
155    
156     \[
157 jmc 1.19 F_{LS} = \min\left[ { \left( \frac{RH-RH_c}{1-RH_c} \right) }^2, 1.0 \right]
158 molod 1.1 \]
159    
160     where
161    
162     \bqa
163     RH_c & = & 1-s(1-s)(2-\sqrt{3}+2\sqrt{3} \, s)r \nonumber \\
164     s & = & p/p_{surf} \nonumber \\
165 jmc 1.19 r & = & \left( \frac{1.0-RH_{min}}{\alpha} \right) \nonumber \\
166 molod 1.1 RH_{min} & = & 0.75 \nonumber \\
167     \alpha & = & 0.573285 \nonumber .
168     \eqa
169    
170     These cloud fractions are suppressed, however, in regions where the convective
171     sub-cloud layer is conditionally unstable. The functional form of $RH_c$ is shown in
172 molod 1.13 Figure (\ref{fig.rhcrit}).
173 molod 1.1
174     \begin{figure*}[htbp]
175     \vspace{0.4in}
176 jmc 1.18 \centerline{ \epsfysize=4.0in \epsfbox{s_phys_pkgs/figs/rhcrit.ps}}
177 molod 1.1 \vspace{0.4in}
178 molod 1.13 \caption [Critical Relative Humidity for Clouds.]
179     {Critical Relative Humidity for Clouds.}
180     \label{fig.rhcrit}
181 molod 1.1 \end{figure*}
182    
183     The total cloud fraction in a grid box is determined by the larger of the two cloud fractions:
184    
185     \[
186     F_{CLD} = \max \left[ F_{RAS},F_{LS} \right] .
187     \]
188    
189     Finally, cloud fractions are time-averaged between calls to the radiation packages.
190    
191    
192 molod 1.9 Radiation:
193 molod 1.1
194     The parameterization of radiative heating in the fizhi package includes effects
195     from both shortwave and longwave processes.
196     Radiative fluxes are calculated at each
197     model edge-level in both up and down directions.
198     The heating rates/cooling rates are then obtained
199     from the vertical divergence of the net radiative fluxes.
200    
201     The net flux is
202     \[
203     F = F^\uparrow - F^\downarrow
204     \]
205     where $F$ is the net flux, $F^\uparrow$ is the upward flux and $F^\downarrow$ is
206     the downward flux.
207    
208     The heating rate due to the divergence of the radiative flux is given by
209     \[
210     \pp{\rho c_p T}{t} = - \pp{F}{z}
211     \]
212     or
213     \[
214     \pp{T}{t} = \frac{g}{c_p \pi} \pp{F}{\sigma}
215     \]
216     where $g$ is the accelation due to gravity
217     and $c_p$ is the heat capacity of air at constant pressure.
218    
219     The time tendency for Longwave
220     Radiation is updated every 3 hours. The time tendency for Shortwave Radiation is updated once
221     every three hours assuming a normalized incident solar radiation, and subsequently modified at
222     every model time step by the true incident radiation.
223     The solar constant value used in the package is equal to 1365 $W/m^2$
224     and a $CO_2$ mixing ratio of 330 ppm.
225     For the ozone mixing ratio, monthly mean zonally averaged
226     climatological values specified as a function
227 molod 1.10 of latitude and height (\cite{rosen:87}) are linearly interpolated to the current time.
228 molod 1.1
229    
230 molod 1.5 \paragraph{Shortwave Radiation}
231 molod 1.1
232     The shortwave radiation package used in the package computes solar radiative
233     heating due to the absoption by water vapor, ozone, carbon dioxide, oxygen,
234     clouds, and aerosols and due to the
235     scattering by clouds, aerosols, and gases.
236     The shortwave radiative processes are described by
237 molod 1.10 \cite{chou:90,chou:92}. This shortwave package
238 molod 1.1 uses the Delta-Eddington approximation to compute the
239     bulk scattering properties of a single layer following King and Harshvardhan (JAS, 1986).
240     The transmittance and reflectance of diffuse radiation
241 molod 1.10 follow the procedures of Sagan and Pollock (JGR, 1967) and \cite{lhans:74}.
242 molod 1.1
243     Highly accurate heating rate calculations are obtained through the use
244     of an optimal grouping strategy of spectral bands. By grouping the UV and visible regions
245     as indicated in Table \ref{tab:fizhi:solar2}, the Rayleigh scattering and the ozone absorption of solar radiation
246     can be accurately computed in the ultraviolet region and the photosynthetically
247     active radiation (PAR) region.
248     The computation of solar flux in the infrared region is performed with a broadband
249     parameterization using the spectrum regions shown in Table \ref{tab:fizhi:solar1}.
250     The solar radiation algorithm used in the fizhi package can be applied not only for climate studies but
251     also for studies on the photolysis in the upper atmosphere and the photosynthesis in the biosphere.
252    
253     \begin{table}[htb]
254     \begin{center}
255     {\bf UV and Visible Spectral Regions} \\
256     \vspace{0.1in}
257     \begin{tabular}{|c|c|c|}
258     \hline
259     Region & Band & Wavelength (micron) \\ \hline
260     \hline
261     UV-C & 1. & .175 - .225 \\
262     & 2. & .225 - .245 \\
263     & & .260 - .280 \\
264     & 3. & .245 - .260 \\ \hline
265     UV-B & 4. & .280 - .295 \\
266     & 5. & .295 - .310 \\
267     & 6. & .310 - .320 \\ \hline
268     UV-A & 7. & .320 - .400 \\ \hline
269     PAR & 8. & .400 - .700 \\
270     \hline
271     \end{tabular}
272     \end{center}
273     \caption{UV and Visible Spectral Regions used in shortwave radiation package.}
274     \label{tab:fizhi:solar2}
275     \end{table}
276    
277     \begin{table}[htb]
278     \begin{center}
279     {\bf Infrared Spectral Regions} \\
280     \vspace{0.1in}
281     \begin{tabular}{|c|c|c|}
282     \hline
283     Band & Wavenumber(cm$^{-1}$) & Wavelength (micron) \\ \hline
284     \hline
285     1 & 1000-4400 & 2.27-10.0 \\
286     2 & 4400-8200 & 1.22-2.27 \\
287     3 & 8200-14300 & 0.70-1.22 \\
288     \hline
289     \end{tabular}
290     \end{center}
291     \caption{Infrared Spectral Regions used in shortwave radiation package.}
292     \label{tab:fizhi:solar1}
293     \end{table}
294    
295     Within the shortwave radiation package,
296     both ice and liquid cloud particles are allowed to co-exist in any of the model layers.
297     Two sets of cloud parameters are used, one for ice paticles and the other for liquid particles.
298     Cloud parameters are defined as the cloud optical thickness and the effective cloud particle size.
299     In the fizhi package, the effective radius for water droplets is given as 10 microns,
300     while 65 microns is used for ice particles. The absorption due to aerosols is currently
301     set to zero.
302    
303     To simplify calculations in a cloudy atmosphere, clouds are
304     grouped into low ($p>700$ mb), middle (700 mb $\ge p > 400$ mb), and high ($p < 400$ mb) cloud regions.
305     Within each of the three regions, clouds are assumed maximally
306     overlapped, and the cloud cover of the group is the maximum
307     cloud cover of all the layers in the group. The optical thickness
308     of a given layer is then scaled for both the direct (as a function of the
309     solar zenith angle) and diffuse beam radiation
310     so that the grouped layer reflectance is the same as the original reflectance.
311 molod 1.13 The solar flux is computed for each of eight cloud realizations possible within this
312 molod 1.1 low/middle/high classification, and appropriately averaged to produce the net solar flux.
313    
314 molod 1.5 \paragraph{Longwave Radiation}
315 molod 1.1
316 molod 1.10 The longwave radiation package used in the fizhi package is thoroughly described by \cite{chsz:94}.
317 molod 1.1 As described in that document, IR fluxes are computed due to absorption by water vapor, carbon
318     dioxide, and ozone. The spectral bands together with their absorbers and parameterization methods,
319     configured for the fizhi package, are shown in Table \ref{tab:fizhi:longwave}.
320    
321    
322     \begin{table}[htb]
323     \begin{center}
324     {\bf IR Spectral Bands} \\
325     \vspace{0.1in}
326     \begin{tabular}{|c|c|l|c| }
327     \hline
328     Band & Spectral Range (cm$^{-1}$) & Absorber & Method \\ \hline
329     \hline
330     1 & 0-340 & H$_2$O line & T \\ \hline
331     2 & 340-540 & H$_2$O line & T \\ \hline
332     3a & 540-620 & H$_2$O line & K \\
333     3b & 620-720 & H$_2$O continuum & S \\
334     3b & 720-800 & CO$_2$ & T \\ \hline
335     4 & 800-980 & H$_2$O line & K \\
336     & & H$_2$O continuum & S \\ \hline
337     & & H$_2$O line & K \\
338     5 & 980-1100 & H$_2$O continuum & S \\
339     & & O$_3$ & T \\ \hline
340     6 & 1100-1380 & H$_2$O line & K \\
341     & & H$_2$O continuum & S \\ \hline
342     7 & 1380-1900 & H$_2$O line & T \\ \hline
343     8 & 1900-3000 & H$_2$O line & K \\ \hline
344     \hline
345     \multicolumn{4}{|l|}{ \quad K: {\em k}-distribution method with linear pressure scaling } \\
346     \multicolumn{4}{|l|}{ \quad T: Table look-up with temperature and pressure scaling } \\
347     \multicolumn{4}{|l|}{ \quad S: One-parameter temperature scaling } \\
348     \hline
349     \end{tabular}
350     \end{center}
351     \vspace{0.1in}
352 molod 1.12 \caption{IR Spectral Bands, Absorbers, and Parameterization Method (from \cite{chsz:94})}
353 molod 1.1 \label{tab:fizhi:longwave}
354     \end{table}
355    
356    
357     The longwave radiation package accurately computes cooling rates for the middle and
358     lower atmosphere from 0.01 mb to the surface. Errors are $<$ 0.4 C day$^{-1}$ in cooling
359     rates and $<$ 1\% in fluxes. From Chou and Suarez, it is estimated that the total effect of
360     neglecting all minor absorption bands and the effects of minor infrared absorbers such as
361     nitrous oxide (N$_2$O), methane (CH$_4$), and the chlorofluorocarbons (CFCs), is an underestimate
362     of $\approx$ 5 W/m$^2$ in the downward flux at the surface and an overestimate of $\approx$ 3 W/m$^2$
363     in the upward flux at the top of the atmosphere.
364    
365     Similar to the procedure used in the shortwave radiation package, clouds are grouped into
366     three regions catagorized as low/middle/high.
367     The net clear line-of-site probability $(P)$ between any two levels, $p_1$ and $p_2 \quad (p_2 > p_1)$,
368     assuming randomly overlapped cloud groups, is simply the product of the probabilities within each group:
369    
370     \[ P_{net} = P_{low} \times P_{mid} \times P_{hi} . \]
371    
372     Since all clouds within a group are assumed maximally overlapped, the clear line-of-site probability within
373     a group is given by:
374    
375     \[ P_{group} = 1 - F_{max} , \]
376    
377     where $F_{max}$ is the maximum cloud fraction encountered between $p_1$ and $p_2$ within that group.
378     For groups and/or levels outside the range of $p_1$ and $p_2$, a clear line-of-site probability equal to 1 is
379     assigned.
380    
381    
382 molod 1.5 \paragraph{Cloud-Radiation Interaction}
383 molod 1.1 \label{sec:fizhi:radcloud}
384    
385     The cloud fractions and diagnosed cloud liquid water produced by moist processes
386     within the fizhi package are used in the radiation packages to produce cloud-radiative forcing.
387     The cloud optical thickness associated with large-scale cloudiness is made
388     proportional to the diagnosed large-scale liquid water, $\ell$, detrained due to super-saturation.
389     Two values are used corresponding to cloud ice particles and water droplets.
390     The range of optical thickness for these clouds is given as
391    
392     \[ 0.0002 \le \tau_{ice} (mb^{-1}) \le 0.002 \quad\mbox{for}\quad 0 \le \ell \le 2 \quad\mbox{mg/kg} , \]
393     \[ 0.02 \le \tau_{h_2o} (mb^{-1}) \le 0.2 \quad\mbox{for}\quad 0 \le \ell \le 10 \quad\mbox{mg/kg} . \]
394    
395     The partitioning, $\alpha$, between ice particles and water droplets is achieved through a linear scaling
396     in temperature:
397    
398     \[ 0 \le \alpha \le 1 \quad\mbox{for}\quad 233.15 \le T \le 253.15 . \]
399    
400     The resulting optical depth associated with large-scale cloudiness is given as
401    
402     \[ \tau_{LS} = \alpha \tau_{h_2o} + (1-\alpha)\tau_{ice} . \]
403    
404     The optical thickness associated with sub-grid scale convective clouds produced by RAS is given as
405    
406     \[ \tau_{RAS} = 0.16 \quad mb^{-1} . \]
407    
408     The total optical depth in a given model layer is computed as a weighted average between
409     the large-scale and sub-grid scale optical depths, normalized by the total cloud fraction in the
410     layer:
411    
412 jmc 1.19 \[ \tau = \left( \frac{F_{RAS} \,\,\, \tau_{RAS} + F_{LS} \,\,\, \tau_{LS} }{ F_{RAS}+F_{LS} } \right) \Delta p, \]
413 molod 1.1
414     where $F_{RAS}$ and $F_{LS}$ are the time-averaged cloud fractions associated with RAS and large-scale
415     processes described in Section \ref{sec:fizhi:clouds}.
416     The optical thickness for the longwave radiative feedback is assumed to be 75 $\%$ of these values.
417    
418     The entire Moist Convective Processes Module is called with a frequency of 10 minutes.
419     The cloud fraction values are time-averaged over the period between Radiation calls (every 3
420     hours). Therefore, in a time-averaged sense, both convective and large-scale
421     cloudiness can exist in a given grid-box.
422    
423 molod 1.12 \paragraph{Turbulence}:
424 molod 1.9
425 molod 1.1 Turbulence is parameterized in the fizhi package to account for its contribution to the
426     vertical exchange of heat, moisture, and momentum.
427     The turbulence scheme is invoked every 30 minutes, and employs a backward-implicit iterative
428     time scheme with an internal time step of 5 minutes.
429     The tendencies of atmospheric state variables due to turbulent diffusion are calculated using
430     the diffusion equations:
431    
432     \[
433     {\pp{u}{t}}_{turb} = {\pp{}{z} }{(- \overline{u^{\prime}w^{\prime}})}
434     = {\pp{}{z} }{(K_m \pp{u}{z})}
435     \]
436     \[
437     {\pp{v}{t}}_{turb} = {\pp{}{z} }{(- \overline{v^{\prime}w^{\prime}})}
438     = {\pp{}{z} }{(K_m \pp{v}{z})}
439     \]
440     \[
441     {\pp{T}{t}} = P^{\kappa}{\pp{\theta}{t}}_{turb} =
442     P^{\kappa}{\pp{}{z} }{(- \overline{w^{\prime}\theta^{\prime}})}
443     = P^{\kappa}{\pp{}{z} }{(K_h \pp{\theta_v}{z})}
444     \]
445     \[
446     {\pp{q}{t}}_{turb} = {\pp{}{z} }{(- \overline{w^{\prime}q^{\prime}})}
447     = {\pp{}{z} }{(K_h \pp{q}{z})}
448     \]
449    
450     Within the atmosphere, the time evolution
451     of second turbulent moments is explicitly modeled by representing the third moments in terms of
452     the first and second moments. This approach is known as a second-order closure modeling.
453     To simplify and streamline the computation of the second moments, the level 2.5 assumption
454 molod 1.10 of Mellor and Yamada (1974) and \cite{yam:77} is employed, in which only the turbulent
455 molod 1.1 kinetic energy (TKE),
456    
457     \[ {\h}{q^2}={\overline{{u^{\prime}}^2}}+{\overline{{v^{\prime}}^2}}+{\overline{{w^{\prime}}^2}}, \]
458    
459     is solved prognostically and the other second moments are solved diagnostically.
460     The prognostic equation for TKE allows the scheme to simulate
461     some of the transient and diffusive effects in the turbulence. The TKE budget equation
462     is solved numerically using an implicit backward computation of the terms linear in $q^2$
463     and is written:
464    
465     \[
466 jmc 1.19 {\dd{}{t} ({{\h} q^2})} - { \pp{}{z} ({ \frac{5}{3} {{\lambda}_1} q { \pp {}{z}
467 molod 1.1 ({\h}q^2)} })} =
468     {- \overline{{u^{\prime}}{w^{\prime}}} { \pp{U}{z} }} - {\overline{{v^{\prime}}{w^{\prime}}}
469 jmc 1.19 { \pp{V}{z} }} + {\frac{g}{\Theta_0}{\overline{{w^{\prime}}{{{\theta}_v}^{\prime}}}}
470     - \frac{ q^3}{{\Lambda}_1} }
471 molod 1.1 \]
472    
473     where $q$ is the turbulent velocity, ${u^{\prime}}$, ${v^{\prime}}$, ${w^{\prime}}$ and
474     ${{\theta}^{\prime}}$ are the fluctuating parts of the velocity components and potential
475     temperature, $U$ and $V$ are the mean velocity components, ${\Theta_0}^{-1}$ is the
476     coefficient of thermal expansion, and ${{\lambda}_1}$ and ${{\Lambda} _1}$ are constant
477     multiples of the master length scale, $\ell$, which is designed to be a characteristic measure
478     of the vertical structure of the turbulent layers.
479    
480     The first term on the left-hand side represents the time rate of change of TKE, and
481     the second term is a representation of the triple correlation, or turbulent
482     transport term. The first three terms on the right-hand side represent the sources of
483     TKE due to shear and bouyancy, and the last term on the right hand side is the dissipation
484     of TKE.
485    
486     In the level 2.5 approach, the vertical fluxes of the scalars $\theta_v$ and $q$ and the
487     wind components $u$ and $v$ are expressed in terms of the diffusion coefficients $K_h$ and
488 molod 1.10 $K_m$, respectively. In the statisically realizable level 2.5 turbulence scheme of
489     \cite{helflab:88}, these diffusion coefficients are expressed as
490 molod 1.1
491     \[
492     K_h
493     = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell \, S_H(G_M,G_H) \, & \mbox{decaying turbulence}
494 jmc 1.19 \\ \frac{ q^2 }{ q_e } \, \ell \, S_{H}(G_{M_e},G_{H_e}) \, & \mbox{growing turbulence} \end{array} \right.
495 molod 1.1 \]
496    
497     and
498    
499     \[
500     K_m
501     = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell \, S_M(G_M,G_H) \, & \mbox{decaying turbulence}
502 jmc 1.19 \\ \frac{ q^2 }{ q_e } \, \ell \, S_{M}(G_{M_e},G_{H_e}) \, & \mbox{growing turbulence} \end{array} \right.
503 molod 1.1 \]
504    
505     where the subscript $e$ refers to the value under conditions of local equillibrium
506     (obtained from the Level 2.0 Model), $\ell$ is the master length scale related to the
507     vertical structure of the atmosphere,
508     and $S_M$ and $S_H$ are functions of $G_H$ and $G_M$, the dimensionless buoyancy and
509     wind shear parameters, respectively.
510     Both $G_H$ and $G_M$, and their equilibrium values $G_{H_e}$ and $G_{M_e}$,
511     are functions of the Richardson number:
512    
513     \[
514 jmc 1.19 {\bf RI} = \frac{ \frac{g}{\theta_v} \pp{\theta_v}{z} }{ (\pp{u}{z})^2 + (\pp{v}{z})^2 }
515     = \frac{c_p \pp{\theta_v}{z} \pp{P^ \kappa}{z} }{ (\pp{u}{z})^2 + (\pp{v}{z})^2 } .
516 molod 1.1 \]
517    
518     Negative values indicate unstable buoyancy and shear, small positive values ($<0.2$)
519     indicate dominantly unstable shear, and large positive values indicate dominantly stable
520     stratification.
521    
522 jmc 1.19 Turbulent eddy diffusion coefficients of momentum, heat and moisture in the
523     surface layer, which corresponds to the lowest GCM level
524     (see {\it --- missing table ---}%\ref{tab:fizhi:sigma}
525     ),
526 molod 1.1 are calculated using stability-dependant functions based on Monin-Obukhov theory:
527     \[
528     {K_m} (surface) = C_u \times u_* = C_D W_s
529     \]
530     and
531     \[
532     {K_h} (surface) = C_t \times u_* = C_H W_s
533     \]
534     where $u_*=C_uW_s$ is the surface friction velocity,
535     $C_D$ is termed the surface drag coefficient, $C_H$ the heat transfer coefficient,
536     and $W_s$ is the magnitude of the surface layer wind.
537    
538     $C_u$ is the dimensionless exchange coefficient for momentum from the surface layer
539     similarity functions:
540     \[
541 jmc 1.19 {C_u} = \frac{u_* }{ W_s} = \frac{ k }{ \psi_{m} }
542 molod 1.1 \]
543     where k is the Von Karman constant and $\psi_m$ is the surface layer non-dimensional
544     wind shear given by
545     \[
546 jmc 1.19 \psi_{m} = {\int_{\zeta_{0}}^{\zeta} \frac{\phi_{m} }{ \zeta} d \zeta} .
547 molod 1.1 \]
548     Here $\zeta$ is the non-dimensional stability parameter, and
549     $\phi_m$ is the similarity function of $\zeta$ which expresses the stability dependance of
550     the momentum gradient. The functional form of $\phi_m$ is specified differently for stable and unstable
551     layers.
552    
553     $C_t$ is the dimensionless exchange coefficient for heat and
554     moisture from the surface layer similarity functions:
555     \[
556 jmc 1.19 {C_t} = -\frac{( \overline{w^{\prime}\theta^{\prime}}) }{ u_* \Delta \theta } =
557     -\frac{( \overline{w^{\prime}q^{\prime}}) }{ u_* \Delta q } =
558     \frac{ k }{ (\psi_{h} + \psi_{g}) }
559 molod 1.1 \]
560     where $\psi_h$ is the surface layer non-dimensional temperature gradient given by
561     \[
562 jmc 1.19 \psi_{h} = {\int_{\zeta_{0}}^{\zeta} \frac{\phi_{h} }{ \zeta} d \zeta} .
563 molod 1.1 \]
564     Here $\phi_h$ is the similarity function of $\zeta$, which expresses the stability dependance of
565     the temperature and moisture gradients, and is specified differently for stable and unstable
566 molod 1.10 layers according to \cite{helfschu:95}.
567 molod 1.1
568     $\psi_g$ is the non-dimensional temperature or moisture gradient in the viscous sublayer,
569     which is the mosstly laminar region between the surface and the tops of the roughness
570     elements, in which temperature and moisture gradients can be quite large.
571 molod 1.10 Based on \cite{yagkad:74}:
572 molod 1.1 \[
573 jmc 1.19 \psi_{g} = \frac{ 0.55 (Pr^{2/3} - 0.2) }{ \nu^{1/2} }
574 molod 1.1 (h_{0}u_{*} - h_{0_{ref}}u_{*_{ref}})^{1/2}
575     \]
576     where Pr is the Prandtl number for air, $\nu$ is the molecular viscosity, $z_{0}$ is the
577     surface roughness length, and the subscript {\em ref} refers to a reference value.
578     $h_{0} = 30z_{0}$ with a maximum value over land of 0.01
579    
580     The surface roughness length over oceans is is a function of the surface-stress velocity,
581     \[
582 jmc 1.19 {z_0} = c_1u^3_* + c_2u^2_* + c_3u_* + c_4 + \frac{c_5 }{ u_*}
583 molod 1.1 \]
584     where the constants are chosen to interpolate between the reciprocal relation of
585 molod 1.10 \cite{kondo:75} for weak winds, and the piecewise linear relation of \cite{larpond:81}
586 molod 1.1 for moderate to large winds. Roughness lengths over land are specified
587 molod 1.10 from the climatology of \cite{dorsell:89}.
588 molod 1.1
589     For an unstable surface layer, the stability functions, chosen to interpolate between the
590     condition of small values of $\beta$ and the convective limit, are the KEYPS function
591 molod 1.10 (\cite{pano:73}) for momentum, and its generalization for heat and moisture:
592 molod 1.1 \[
593     {\phi_m}^4 - 18 \zeta {\phi_m}^3 = 1 \hspace{1cm} ; \hspace{1cm}
594     {\phi_h}^2 - 18 \zeta {\phi_h}^3 = 1 \hspace{1cm} .
595     \]
596     The function for heat and moisture assures non-vanishing heat and moisture fluxes as the wind
597     speed approaches zero.
598    
599     For a stable surface layer, the stability functions are the observationally
600 molod 1.10 based functions of \cite{clarke:70}, slightly modified for
601 molod 1.1 the momemtum flux:
602     \[
603 jmc 1.19 {\phi_m} = \frac{ 1 + 5 {{\zeta}_1} }{ 1 + 0.00794 {\zeta}_1
604     (1+ 5 {\zeta}_1) } \hspace{1cm} ; \hspace{1cm}
605     {\phi_h} = \frac{ 1 + 5 {{\zeta}_1} }{ 1 + 0.00794 {\zeta}
606     (1+ 5 {{\zeta}_1}) } .
607 molod 1.1 \]
608     The moisture flux also depends on a specified evapotranspiration
609     coefficient, set to unity over oceans and dependant on the climatological ground wetness over
610     land.
611    
612     Once all the diffusion coefficients are calculated, the diffusion equations are solved numerically
613     using an implicit backward operator.
614    
615 molod 1.5 \paragraph{Atmospheric Boundary Layer}
616 molod 1.1
617     The depth of the atmospheric boundary layer (ABL) is diagnosed by the parameterization as the
618     level at which the turbulent kinetic energy is reduced to a tenth of its maximum near surface value.
619     The vertical structure of the ABL is explicitly resolved by the lowest few (3-8) model layers.
620    
621 molod 1.5 \paragraph{Surface Energy Budget}
622 molod 1.1
623     The ground temperature equation is solved as part of the turbulence package
624     using a backward implicit time differencing scheme:
625     \[
626     C_g\pp{T_g}{t} = R_{sw} - R_{lw} + Q_{ice} - H - LE
627     \]
628     where $R_{sw}$ is the net surface downward shortwave radiative flux and $R_{lw}$ is the
629     net surface upward longwave radiative flux.
630    
631     $H$ is the upward sensible heat flux, given by:
632     \[
633     {H} = P^{\kappa}\rho c_{p} C_{H} W_s (\theta_{surface} - \theta_{NLAY})
634     \hspace{1cm}where: \hspace{.2cm}C_H = C_u C_t
635     \]
636     where $\rho$ = the atmospheric density at the surface, $c_{p}$ is the specific
637     heat of air at constant pressure, and $\theta$ represents the potential temperature
638     of the surface and of the lowest $\sigma$-level, respectively.
639    
640     The upward latent heat flux, $LE$, is given by
641     \[
642     {LE} = \rho \beta L C_{H} W_s (q_{surface} - q_{NLAY})
643     \hspace{1cm}where: \hspace{.2cm}C_H = C_u C_t
644     \]
645     where $\beta$ is the fraction of the potential evapotranspiration actually evaporated,
646     L is the latent heat of evaporation, and $q_{surface}$ and $q_{NLAY}$ are the specific
647     humidity of the surface and of the lowest $\sigma$-level, respectively.
648    
649     The heat conduction through sea ice, $Q_{ice}$, is given by
650     \[
651 jmc 1.19 {Q_{ice}} = \frac{C_{ti} }{ H_i} (T_i-T_g)
652 molod 1.1 \]
653     where $C_{ti}$ is the thermal conductivity of ice, $H_i$ is the ice thickness, assumed to
654     be $3 \hspace{.1cm} m$ where sea ice is present, $T_i$ is 273 degrees Kelvin, and $T_g$ is the
655     surface temperature of the ice.
656    
657     $C_g$ is the total heat capacity of the ground, obtained by solving a heat diffusion equation
658 molod 1.10 for the penetration of the diurnal cycle into the ground (\cite{black:77}), and is given by:
659 molod 1.1 \[
660 jmc 1.19 C_g = \sqrt{ \frac{\lambda C_s }{ 2\omega} } = \sqrt{(0.386 + 0.536W + 0.15W^2)2\times10^{-3}
661     \frac{86400}{2\pi} } \, \, .
662 molod 1.1 \]
663 jmc 1.19 Here, the thermal conductivity, $\lambda$, is equal to $2\times10^{-3}$ $\frac{ly}{sec}
664     \frac{cm}{K}$,
665 molod 1.1 the angular velocity of the earth, $\omega$, is written as $86400$ $sec/day$ divided
666     by $2 \pi$ $radians/
667     day$, and the expression for $C_s$, the heat capacity per unit volume at the surface,
668     is a function of the ground wetness, $W$.
669    
670 molod 1.9 Land Surface Processes:
671 molod 1.1
672 molod 1.5 \paragraph{Surface Type}
673 molod 1.10 The fizhi package surface Types are designated using the Koster-Suarez (\cite{ks:91,ks:92})
674     Land Surface Model (LSM) mosaic philosophy which allows multiple ``tiles'', or multiple surface
675     types, in any one grid cell. The Koster-Suarez LSM surface type classifications
676 molod 1.1 are shown in Table \ref{tab:fizhi:surftype}. The surface types and the percent of the grid
677     cell occupied by any surface type were derived from the surface classification of
678 molod 1.10 \cite{deftow:94}, and information about the location of permanent
679     ice was obtained from the classifications of \cite{dorsell:89}.
680 molod 1.13 The surface type map for a $1^\circ$ grid is shown in Figure \ref{fig:fizhi:surftype}.
681 molod 1.1 The determination of the land or sea category of surface type was made from NCAR's
682     10 minute by 10 minute Navy topography
683     dataset, which includes information about the percentage of water-cover at any point.
684 molod 1.13 The data were averaged to the model's grid resolutions,
685 molod 1.1 and any grid-box whose averaged water percentage was $\geq 60 \%$ was
686 molod 1.13 defined as a water point. The Land-Water designation was further modified
687 molod 1.1 subjectively to ensure sufficient representation from small but isolated land and water regions.
688    
689     \begin{table}
690     \begin{center}
691     {\bf Surface Type Designation} \\
692     \vspace{0.1in}
693     \begin{tabular}{ |c|l| }
694     \hline
695     Type & Vegetation Designation \\ \hline
696     \hline
697     1 & Broadleaf Evergreen Trees \\ \hline
698     2 & Broadleaf Deciduous Trees \\ \hline
699     3 & Needleleaf Trees \\ \hline
700     4 & Ground Cover \\ \hline
701     5 & Broadleaf Shrubs \\ \hline
702     6 & Dwarf Trees (Tundra) \\ \hline
703     7 & Bare Soil \\ \hline
704     8 & Desert (Bright) \\ \hline
705     9 & Glacier \\ \hline
706     10 & Desert (Dark) \\ \hline
707     100 & Ocean \\ \hline
708     \end{tabular}
709     \end{center}
710 molod 1.17 \caption{Surface type designations.}
711 molod 1.1 \label{tab:fizhi:surftype}
712     \end{table}
713    
714     \begin{figure*}[htbp]
715 jmc 1.18 \centerline{ \epsfysize=4.0in \epsfbox{s_phys_pkgs/figs/surftype.eps}}
716 molod 1.13 \vspace{0.2in}
717 molod 1.17 \caption {Surface Type Combinations.}
718 molod 1.1 \label{fig:fizhi:surftype}
719     \end{figure*}
720    
721 jmc 1.18 % \rotatebox{270}{\centerline{ \epsfysize=4in \epsfbox{s_phys_pkgs/figs/surftypes.eps}}}
722     % \rotatebox{270}{\centerline{ \epsfysize=4in \epsfbox{s_phys_pkgs/figs/surftypes.descrip.eps}}}
723 molod 1.13 %\begin{figure*}[htbp]
724 jmc 1.18 % \centerline{ \epsfysize=4in \epsfbox{s_phys_pkgs/figs/surftypes.descrip.ps}}
725 molod 1.13 % \vspace{0.3in}
726     % \caption {Surface Type Descriptions.}
727     % \label{fig:fizhi:surftype.desc}
728     %\end{figure*}
729 molod 1.1
730    
731 molod 1.5 \paragraph{Surface Roughness}
732 molod 1.1 The surface roughness length over oceans is computed iteratively with the wind
733 molod 1.10 stress by the surface layer parameterization (\cite{helfschu:95}).
734     It employs an interpolation between the functions of \cite{larpond:81}
735     for high winds and of \cite{kondo:75} for weak winds.
736 molod 1.1
737    
738 molod 1.5 \paragraph{Albedo}
739 molod 1.10 The surface albedo computation, described in \cite{ks:91},
740 molod 1.1 employs the ``two stream'' approximation used in Sellers' (1987) Simple Biosphere (SiB)
741     Model which distinguishes between the direct and diffuse albedos in the visible
742     and in the near infra-red spectral ranges. The albedos are functions of the observed
743     leaf area index (a description of the relative orientation of the leaves to the
744     sun), the greenness fraction, the vegetation type, and the solar zenith angle.
745     Modifications are made to account for the presence of snow, and its depth relative
746     to the height of the vegetation elements.
747    
748 edhill 1.16 \paragraph{Gravity Wave Drag}
749 molod 1.9
750 molod 1.12 The fizhi package employs the gravity wave drag scheme of \cite{zhouetal:95}).
751 molod 1.1 This scheme is a modified version of Vernekar et al. (1992),
752     which was based on Alpert et al. (1988) and Helfand et al. (1987).
753     In this version, the gravity wave stress at the surface is
754     based on that derived by Pierrehumbert (1986) and is given by:
755    
756     \bq
757 jmc 1.19 |\vec{\tau}_{sfc}| = \frac{\rho U^3}{N \ell^*} \left( \frac{F_r^2}{1+F_r^2}\right) \, \, ,
758 molod 1.1 \eq
759    
760     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
761     surface wind speed, $h$ is the standard deviation of the sub-grid scale orography,
762     and $\ell^*$ is the wavelength of the monochromatic gravity wave in the direction of the low-level wind.
763     A modification introduced by Zhou et al. allows for the momentum flux to
764     escape through the top of the model, although this effect is small for the current 70-level model.
765     The subgrid scale standard deviation is defined by $h$, and is not allowed to exceed 400 m.
766    
767 molod 1.10 The effects of using this scheme within a GCM are shown in \cite{taksz:96}.
768 molod 1.1 Experiments using the gravity wave drag parameterization yielded significant and
769     beneficial impacts on both the time-mean flow and the transient statistics of the
770     a GCM climatology, and have eliminated most of the worst dynamically driven biases
771     in the a GCM simulation.
772     An examination of the angular momentum budget during climate runs indicates that the
773     resulting gravity wave torque is similar to the data-driven torque produced by a data
774     assimilation which was performed without gravity
775     wave drag. It was shown that the inclusion of gravity wave drag results in
776     large changes in both the mean flow and in eddy fluxes.
777     The result is a more
778     accurate simulation of surface stress (through a reduction in the surface wind strength),
779     of mountain torque (through a redistribution of mean sea-level pressure), and of momentum
780     convergence (through a reduction in the flux of westerly momentum by transient flow eddies).
781    
782    
783 molod 1.9 Boundary Conditions and other Input Data:
784 molod 1.1
785     Required fields which are not explicitly predicted or diagnosed during model execution must
786     either be prescribed internally or obtained from external data sets. In the fizhi package these
787     fields include: sea surface temperature, sea ice estent, surface geopotential variance,
788     vegetation index, and the radiation-related background levels of: ozone, carbon dioxide,
789     and stratospheric moisture.
790    
791 molod 1.13 Boundary condition data sets are available at the model's
792 molod 1.1 resolutions for either climatological or yearly varying conditions.
793     Any frequency of boundary condition data can be used in the fizhi package;
794     however, the current selection of data is summarized in Table \ref{tab:fizhi:bcdata}\@.
795     The time mean values are interpolated during each model timestep to the
796 molod 1.13 current time.
797 molod 1.1
798     \begin{table}[htb]
799     \begin{center}
800     {\bf Fizhi Input Datasets} \\
801     \vspace{0.1in}
802     \begin{tabular}{|l|c|r|} \hline
803     \multicolumn{1}{|c}{Variable} & \multicolumn{1}{|c}{Frequency} & \multicolumn{1}{|c|}{Years} \\ \hline\hline
804     Sea Ice Extent & monthly & 1979-current, climatology \\ \hline
805     Sea Ice Extent & weekly & 1982-current, climatology \\ \hline
806     Sea Surface Temperature & monthly & 1979-current, climatology \\ \hline
807     Sea Surface Temperature & weekly & 1982-current, climatology \\ \hline
808     Zonally Averaged Upper-Level Moisture & monthly & climatology \\ \hline
809     Zonally Averaged Ozone Concentration & monthly & climatology \\ \hline
810     \end{tabular}
811     \end{center}
812     \caption{Boundary conditions and other input data used in the fizhi package. Also noted are the
813     current years and frequencies available.}
814     \label{tab:fizhi:bcdata}
815     \end{table}
816    
817    
818 molod 1.5 \paragraph{Topography and Topography Variance}
819 molod 1.1
820     Surface geopotential heights are provided from an averaging of the Navy 10 minute
821     by 10 minute dataset supplied by the National Center for Atmospheric Research (NCAR) to the
822     model's grid resolution. The original topography is first rotated to the proper grid-orientation
823 molod 1.10 which is being run, and then averages the data to the model resolution.
824 molod 1.1
825 molod 1.10 The standard deviation of the subgrid-scale topography is computed by interpolating the 10 minute
826     data to the model's resolution and re-interpolating back to the 10 minute by 10 minute resolution.
827 molod 1.1 The sub-grid scale variance is constructed based on this smoothed dataset.
828    
829    
830 molod 1.5 \paragraph{Upper Level Moisture}
831 molod 1.1 The fizhi package uses climatological water vapor data above 100 mb from the Stratospheric Aerosol and Gas
832     Experiment (SAGE) as input into the model's radiation packages. The SAGE data is archived
833 edhill 1.15 as monthly zonal means at $5^\circ$ latitudinal resolution. The data is interpolated to the
834 molod 1.1 model's grid location and current time, and blended with the GCM's moisture data. Below 300 mb,
835     the model's moisture data is used. Above 100 mb, the SAGE data is used. Between 100 and 300 mb,
836     a linear interpolation (in pressure) is performed using the data from SAGE and the GCM.
837    
838 molod 1.8
839 molod 1.9 \subsubsection{Fizhi Diagnostics}
840 molod 1.8
841 molod 1.9 Fizhi Diagnostic Menu:
842 molod 1.14 \label{sec:pkg:fizhi:diagnostics}
843 molod 1.8
844     \begin{tabular}{llll}
845     \hline\hline
846     NAME & UNITS & LEVELS & DESCRIPTION \\
847     \hline
848    
849     &\\
850     UFLUX & $Newton/m^2$ & 1
851     &\begin{minipage}[t]{3in}
852     {Surface U-Wind Stress on the atmosphere}
853     \end{minipage}\\
854     VFLUX & $Newton/m^2$ & 1
855     &\begin{minipage}[t]{3in}
856     {Surface V-Wind Stress on the atmosphere}
857     \end{minipage}\\
858     HFLUX & $Watts/m^2$ & 1
859     &\begin{minipage}[t]{3in}
860     {Surface Flux of Sensible Heat}
861     \end{minipage}\\
862     EFLUX & $Watts/m^2$ & 1
863     &\begin{minipage}[t]{3in}
864     {Surface Flux of Latent Heat}
865     \end{minipage}\\
866     QICE & $Watts/m^2$ & 1
867     &\begin{minipage}[t]{3in}
868     {Heat Conduction through Sea-Ice}
869     \end{minipage}\\
870     RADLWG & $Watts/m^2$ & 1
871     &\begin{minipage}[t]{3in}
872     {Net upward LW flux at the ground}
873     \end{minipage}\\
874     RADSWG & $Watts/m^2$ & 1
875     &\begin{minipage}[t]{3in}
876     {Net downward SW flux at the ground}
877     \end{minipage}\\
878     RI & $dimensionless$ & Nrphys
879     &\begin{minipage}[t]{3in}
880     {Richardson Number}
881     \end{minipage}\\
882     CT & $dimensionless$ & 1
883     &\begin{minipage}[t]{3in}
884     {Surface Drag coefficient for T and Q}
885     \end{minipage}\\
886     CU & $dimensionless$ & 1
887     &\begin{minipage}[t]{3in}
888     {Surface Drag coefficient for U and V}
889     \end{minipage}\\
890     ET & $m^2/sec$ & Nrphys
891     &\begin{minipage}[t]{3in}
892     {Diffusivity coefficient for T and Q}
893     \end{minipage}\\
894     EU & $m^2/sec$ & Nrphys
895     &\begin{minipage}[t]{3in}
896     {Diffusivity coefficient for U and V}
897     \end{minipage}\\
898     TURBU & $m/sec/day$ & Nrphys
899     &\begin{minipage}[t]{3in}
900     {U-Momentum Changes due to Turbulence}
901     \end{minipage}\\
902     TURBV & $m/sec/day$ & Nrphys
903     &\begin{minipage}[t]{3in}
904     {V-Momentum Changes due to Turbulence}
905     \end{minipage}\\
906     TURBT & $deg/day$ & Nrphys
907     &\begin{minipage}[t]{3in}
908     {Temperature Changes due to Turbulence}
909     \end{minipage}\\
910     TURBQ & $g/kg/day$ & Nrphys
911     &\begin{minipage}[t]{3in}
912     {Specific Humidity Changes due to Turbulence}
913     \end{minipage}\\
914     MOISTT & $deg/day$ & Nrphys
915     &\begin{minipage}[t]{3in}
916     {Temperature Changes due to Moist Processes}
917     \end{minipage}\\
918     MOISTQ & $g/kg/day$ & Nrphys
919     &\begin{minipage}[t]{3in}
920     {Specific Humidity Changes due to Moist Processes}
921     \end{minipage}\\
922     RADLW & $deg/day$ & Nrphys
923     &\begin{minipage}[t]{3in}
924     {Net Longwave heating rate for each level}
925     \end{minipage}\\
926     RADSW & $deg/day$ & Nrphys
927     &\begin{minipage}[t]{3in}
928     {Net Shortwave heating rate for each level}
929     \end{minipage}\\
930     PREACC & $mm/day$ & 1
931     &\begin{minipage}[t]{3in}
932     {Total Precipitation}
933     \end{minipage}\\
934     PRECON & $mm/day$ & 1
935     &\begin{minipage}[t]{3in}
936     {Convective Precipitation}
937     \end{minipage}\\
938     TUFLUX & $Newton/m^2$ & Nrphys
939     &\begin{minipage}[t]{3in}
940     {Turbulent Flux of U-Momentum}
941     \end{minipage}\\
942     TVFLUX & $Newton/m^2$ & Nrphys
943     &\begin{minipage}[t]{3in}
944     {Turbulent Flux of V-Momentum}
945     \end{minipage}\\
946     TTFLUX & $Watts/m^2$ & Nrphys
947     &\begin{minipage}[t]{3in}
948     {Turbulent Flux of Sensible Heat}
949     \end{minipage}\\
950     \end{tabular}
951    
952     \newpage
953     \vspace*{\fill}
954     \begin{tabular}{llll}
955     \hline\hline
956     NAME & UNITS & LEVELS & DESCRIPTION \\
957     \hline
958    
959     &\\
960     TQFLUX & $Watts/m^2$ & Nrphys
961     &\begin{minipage}[t]{3in}
962     {Turbulent Flux of Latent Heat}
963     \end{minipage}\\
964     CN & $dimensionless$ & 1
965     &\begin{minipage}[t]{3in}
966     {Neutral Drag Coefficient}
967     \end{minipage}\\
968     WINDS & $m/sec$ & 1
969     &\begin{minipage}[t]{3in}
970     {Surface Wind Speed}
971     \end{minipage}\\
972     DTSRF & $deg$ & 1
973     &\begin{minipage}[t]{3in}
974     {Air/Surface virtual temperature difference}
975     \end{minipage}\\
976     TG & $deg$ & 1
977     &\begin{minipage}[t]{3in}
978     {Ground temperature}
979     \end{minipage}\\
980     TS & $deg$ & 1
981     &\begin{minipage}[t]{3in}
982     {Surface air temperature (Adiabatic from lowest model layer)}
983     \end{minipage}\\
984     DTG & $deg$ & 1
985     &\begin{minipage}[t]{3in}
986     {Ground temperature adjustment}
987     \end{minipage}\\
988    
989     QG & $g/kg$ & 1
990     &\begin{minipage}[t]{3in}
991     {Ground specific humidity}
992     \end{minipage}\\
993     QS & $g/kg$ & 1
994     &\begin{minipage}[t]{3in}
995     {Saturation surface specific humidity}
996     \end{minipage}\\
997     TGRLW & $deg$ & 1
998     &\begin{minipage}[t]{3in}
999     {Instantaneous ground temperature used as input to the
1000     Longwave radiation subroutine}
1001     \end{minipage}\\
1002     ST4 & $Watts/m^2$ & 1
1003     &\begin{minipage}[t]{3in}
1004     {Upward Longwave flux at the ground ($\sigma T^4$)}
1005     \end{minipage}\\
1006     OLR & $Watts/m^2$ & 1
1007     &\begin{minipage}[t]{3in}
1008     {Net upward Longwave flux at the top of the model}
1009     \end{minipage}\\
1010     OLRCLR & $Watts/m^2$ & 1
1011     &\begin{minipage}[t]{3in}
1012     {Net upward clearsky Longwave flux at the top of the model}
1013     \end{minipage}\\
1014     LWGCLR & $Watts/m^2$ & 1
1015     &\begin{minipage}[t]{3in}
1016     {Net upward clearsky Longwave flux at the ground}
1017     \end{minipage}\\
1018     LWCLR & $deg/day$ & Nrphys
1019     &\begin{minipage}[t]{3in}
1020     {Net clearsky Longwave heating rate for each level}
1021     \end{minipage}\\
1022     TLW & $deg$ & Nrphys
1023     &\begin{minipage}[t]{3in}
1024     {Instantaneous temperature used as input to the Longwave radiation
1025     subroutine}
1026     \end{minipage}\\
1027     SHLW & $g/g$ & Nrphys
1028     &\begin{minipage}[t]{3in}
1029     {Instantaneous specific humidity used as input to the Longwave radiation
1030     subroutine}
1031     \end{minipage}\\
1032     OZLW & $g/g$ & Nrphys
1033     &\begin{minipage}[t]{3in}
1034     {Instantaneous ozone used as input to the Longwave radiation
1035     subroutine}
1036     \end{minipage}\\
1037     CLMOLW & $0-1$ & Nrphys
1038     &\begin{minipage}[t]{3in}
1039     {Maximum overlap cloud fraction used in the Longwave radiation
1040     subroutine}
1041     \end{minipage}\\
1042     CLDTOT & $0-1$ & Nrphys
1043     &\begin{minipage}[t]{3in}
1044     {Total cloud fraction used in the Longwave and Shortwave radiation
1045     subroutines}
1046     \end{minipage}\\
1047     LWGDOWN & $Watts/m^2$ & 1
1048     &\begin{minipage}[t]{3in}
1049     {Downwelling Longwave radiation at the ground}
1050     \end{minipage}\\
1051     GWDT & $deg/day$ & Nrphys
1052     &\begin{minipage}[t]{3in}
1053     {Temperature tendency due to Gravity Wave Drag}
1054     \end{minipage}\\
1055     RADSWT & $Watts/m^2$ & 1
1056     &\begin{minipage}[t]{3in}
1057     {Incident Shortwave radiation at the top of the atmosphere}
1058     \end{minipage}\\
1059     TAUCLD & $per 100 mb$ & Nrphys
1060     &\begin{minipage}[t]{3in}
1061     {Counted Cloud Optical Depth (non-dimensional) per 100 mb}
1062     \end{minipage}\\
1063     TAUCLDC & $Number$ & Nrphys
1064     &\begin{minipage}[t]{3in}
1065     {Cloud Optical Depth Counter}
1066     \end{minipage}\\
1067     \end{tabular}
1068     \vfill
1069    
1070     \newpage
1071     \vspace*{\fill}
1072     \begin{tabular}{llll}
1073     \hline\hline
1074     NAME & UNITS & LEVELS & DESCRIPTION \\
1075     \hline
1076    
1077     &\\
1078     CLDLOW & $0-1$ & Nrphys
1079     &\begin{minipage}[t]{3in}
1080     {Low-Level ( 1000-700 hPa) Cloud Fraction (0-1)}
1081     \end{minipage}\\
1082     EVAP & $mm/day$ & 1
1083     &\begin{minipage}[t]{3in}
1084     {Surface evaporation}
1085     \end{minipage}\\
1086     DPDT & $hPa/day$ & 1
1087     &\begin{minipage}[t]{3in}
1088     {Surface Pressure tendency}
1089     \end{minipage}\\
1090     UAVE & $m/sec$ & Nrphys
1091     &\begin{minipage}[t]{3in}
1092     {Average U-Wind}
1093     \end{minipage}\\
1094     VAVE & $m/sec$ & Nrphys
1095     &\begin{minipage}[t]{3in}
1096     {Average V-Wind}
1097     \end{minipage}\\
1098     TAVE & $deg$ & Nrphys
1099     &\begin{minipage}[t]{3in}
1100     {Average Temperature}
1101     \end{minipage}\\
1102     QAVE & $g/kg$ & Nrphys
1103     &\begin{minipage}[t]{3in}
1104     {Average Specific Humidity}
1105     \end{minipage}\\
1106     OMEGA & $hPa/day$ & Nrphys
1107     &\begin{minipage}[t]{3in}
1108     {Vertical Velocity}
1109     \end{minipage}\\
1110     DUDT & $m/sec/day$ & Nrphys
1111     &\begin{minipage}[t]{3in}
1112     {Total U-Wind tendency}
1113     \end{minipage}\\
1114     DVDT & $m/sec/day$ & Nrphys
1115     &\begin{minipage}[t]{3in}
1116     {Total V-Wind tendency}
1117     \end{minipage}\\
1118     DTDT & $deg/day$ & Nrphys
1119     &\begin{minipage}[t]{3in}
1120     {Total Temperature tendency}
1121     \end{minipage}\\
1122     DQDT & $g/kg/day$ & Nrphys
1123     &\begin{minipage}[t]{3in}
1124     {Total Specific Humidity tendency}
1125     \end{minipage}\\
1126     VORT & $10^{-4}/sec$ & Nrphys
1127     &\begin{minipage}[t]{3in}
1128     {Relative Vorticity}
1129     \end{minipage}\\
1130     DTLS & $deg/day$ & Nrphys
1131     &\begin{minipage}[t]{3in}
1132     {Temperature tendency due to Stratiform Cloud Formation}
1133     \end{minipage}\\
1134     DQLS & $g/kg/day$ & Nrphys
1135     &\begin{minipage}[t]{3in}
1136     {Specific Humidity tendency due to Stratiform Cloud Formation}
1137     \end{minipage}\\
1138     USTAR & $m/sec$ & 1
1139     &\begin{minipage}[t]{3in}
1140     {Surface USTAR wind}
1141     \end{minipage}\\
1142     Z0 & $m$ & 1
1143     &\begin{minipage}[t]{3in}
1144     {Surface roughness}
1145     \end{minipage}\\
1146     FRQTRB & $0-1$ & Nrphys-1
1147     &\begin{minipage}[t]{3in}
1148     {Frequency of Turbulence}
1149     \end{minipage}\\
1150     PBL & $mb$ & 1
1151     &\begin{minipage}[t]{3in}
1152     {Planetary Boundary Layer depth}
1153     \end{minipage}\\
1154     SWCLR & $deg/day$ & Nrphys
1155     &\begin{minipage}[t]{3in}
1156     {Net clearsky Shortwave heating rate for each level}
1157     \end{minipage}\\
1158     OSR & $Watts/m^2$ & 1
1159     &\begin{minipage}[t]{3in}
1160     {Net downward Shortwave flux at the top of the model}
1161     \end{minipage}\\
1162     OSRCLR & $Watts/m^2$ & 1
1163     &\begin{minipage}[t]{3in}
1164     {Net downward clearsky Shortwave flux at the top of the model}
1165     \end{minipage}\\
1166     CLDMAS & $kg / m^2$ & Nrphys
1167     &\begin{minipage}[t]{3in}
1168     {Convective cloud mass flux}
1169     \end{minipage}\\
1170     UAVE & $m/sec$ & Nrphys
1171     &\begin{minipage}[t]{3in}
1172     {Time-averaged $u-Wind$}
1173     \end{minipage}\\
1174     \end{tabular}
1175     \vfill
1176    
1177     \newpage
1178     \vspace*{\fill}
1179     \begin{tabular}{llll}
1180     \hline\hline
1181     NAME & UNITS & LEVELS & DESCRIPTION \\
1182     \hline
1183    
1184     &\\
1185     VAVE & $m/sec$ & Nrphys
1186     &\begin{minipage}[t]{3in}
1187     {Time-averaged $v-Wind$}
1188     \end{minipage}\\
1189     TAVE & $deg$ & Nrphys
1190     &\begin{minipage}[t]{3in}
1191     {Time-averaged $Temperature$}
1192     \end{minipage}\\
1193     QAVE & $g/g$ & Nrphys
1194     &\begin{minipage}[t]{3in}
1195     {Time-averaged $Specific \, \, Humidity$}
1196     \end{minipage}\\
1197     RFT & $deg/day$ & Nrphys
1198     &\begin{minipage}[t]{3in}
1199     {Temperature tendency due Rayleigh Friction}
1200     \end{minipage}\\
1201     PS & $mb$ & 1
1202     &\begin{minipage}[t]{3in}
1203     {Surface Pressure}
1204     \end{minipage}\\
1205     QQAVE & $(m/sec)^2$ & Nrphys
1206     &\begin{minipage}[t]{3in}
1207     {Time-averaged $Turbulent Kinetic Energy$}
1208     \end{minipage}\\
1209     SWGCLR & $Watts/m^2$ & 1
1210     &\begin{minipage}[t]{3in}
1211     {Net downward clearsky Shortwave flux at the ground}
1212     \end{minipage}\\
1213     PAVE & $mb$ & 1
1214     &\begin{minipage}[t]{3in}
1215     {Time-averaged Surface Pressure}
1216     \end{minipage}\\
1217     DIABU & $m/sec/day$ & Nrphys
1218     &\begin{minipage}[t]{3in}
1219     {Total Diabatic forcing on $u-Wind$}
1220     \end{minipage}\\
1221     DIABV & $m/sec/day$ & Nrphys
1222     &\begin{minipage}[t]{3in}
1223     {Total Diabatic forcing on $v-Wind$}
1224     \end{minipage}\\
1225     DIABT & $deg/day$ & Nrphys
1226     &\begin{minipage}[t]{3in}
1227     {Total Diabatic forcing on $Temperature$}
1228     \end{minipage}\\
1229     DIABQ & $g/kg/day$ & Nrphys
1230     &\begin{minipage}[t]{3in}
1231     {Total Diabatic forcing on $Specific \, \, Humidity$}
1232     \end{minipage}\\
1233     RFU & $m/sec/day$ & Nrphys
1234     &\begin{minipage}[t]{3in}
1235     {U-Wind tendency due to Rayleigh Friction}
1236     \end{minipage}\\
1237     RFV & $m/sec/day$ & Nrphys
1238     &\begin{minipage}[t]{3in}
1239     {V-Wind tendency due to Rayleigh Friction}
1240     \end{minipage}\\
1241     GWDU & $m/sec/day$ & Nrphys
1242     &\begin{minipage}[t]{3in}
1243     {U-Wind tendency due to Gravity Wave Drag}
1244     \end{minipage}\\
1245     GWDU & $m/sec/day$ & Nrphys
1246     &\begin{minipage}[t]{3in}
1247     {V-Wind tendency due to Gravity Wave Drag}
1248     \end{minipage}\\
1249     GWDUS & $N/m^2$ & 1
1250     &\begin{minipage}[t]{3in}
1251     {U-Wind Gravity Wave Drag Stress at Surface}
1252     \end{minipage}\\
1253     GWDVS & $N/m^2$ & 1
1254     &\begin{minipage}[t]{3in}
1255     {V-Wind Gravity Wave Drag Stress at Surface}
1256     \end{minipage}\\
1257     GWDUT & $N/m^2$ & 1
1258     &\begin{minipage}[t]{3in}
1259     {U-Wind Gravity Wave Drag Stress at Top}
1260     \end{minipage}\\
1261     GWDVT & $N/m^2$ & 1
1262     &\begin{minipage}[t]{3in}
1263     {V-Wind Gravity Wave Drag Stress at Top}
1264     \end{minipage}\\
1265     LZRAD & $mg/kg$ & Nrphys
1266     &\begin{minipage}[t]{3in}
1267     {Estimated Cloud Liquid Water used in Radiation}
1268     \end{minipage}\\
1269     \end{tabular}
1270     \vfill
1271    
1272     \newpage
1273     \vspace*{\fill}
1274     \begin{tabular}{llll}
1275     \hline\hline
1276     NAME & UNITS & LEVELS & DESCRIPTION \\
1277     \hline
1278    
1279     &\\
1280     SLP & $mb$ & 1
1281     &\begin{minipage}[t]{3in}
1282     {Time-averaged Sea-level Pressure}
1283     \end{minipage}\\
1284     CLDFRC & $0-1$ & 1
1285     &\begin{minipage}[t]{3in}
1286     {Total Cloud Fraction}
1287     \end{minipage}\\
1288     TPW & $gm/cm^2$ & 1
1289     &\begin{minipage}[t]{3in}
1290     {Precipitable water}
1291     \end{minipage}\\
1292     U2M & $m/sec$ & 1
1293     &\begin{minipage}[t]{3in}
1294     {U-Wind at 2 meters}
1295     \end{minipage}\\
1296     V2M & $m/sec$ & 1
1297     &\begin{minipage}[t]{3in}
1298     {V-Wind at 2 meters}
1299     \end{minipage}\\
1300     T2M & $deg$ & 1
1301     &\begin{minipage}[t]{3in}
1302     {Temperature at 2 meters}
1303     \end{minipage}\\
1304     Q2M & $g/kg$ & 1
1305     &\begin{minipage}[t]{3in}
1306     {Specific Humidity at 2 meters}
1307     \end{minipage}\\
1308     U10M & $m/sec$ & 1
1309     &\begin{minipage}[t]{3in}
1310     {U-Wind at 10 meters}
1311     \end{minipage}\\
1312     V10M & $m/sec$ & 1
1313     &\begin{minipage}[t]{3in}
1314     {V-Wind at 10 meters}
1315     \end{minipage}\\
1316     T10M & $deg$ & 1
1317     &\begin{minipage}[t]{3in}
1318     {Temperature at 10 meters}
1319     \end{minipage}\\
1320     Q10M & $g/kg$ & 1
1321     &\begin{minipage}[t]{3in}
1322     {Specific Humidity at 10 meters}
1323     \end{minipage}\\
1324     DTRAIN & $kg/m^2$ & Nrphys
1325     &\begin{minipage}[t]{3in}
1326     {Detrainment Cloud Mass Flux}
1327     \end{minipage}\\
1328     QFILL & $g/kg/day$ & Nrphys
1329     &\begin{minipage}[t]{3in}
1330     {Filling of negative specific humidity}
1331     \end{minipage}\\
1332     \end{tabular}
1333     \vspace{1.5in}
1334     \vfill
1335    
1336     \newpage
1337     \vspace*{\fill}
1338     \begin{tabular}{llll}
1339     \hline\hline
1340     NAME & UNITS & LEVELS & DESCRIPTION \\
1341     \hline
1342    
1343     &\\
1344     DTCONV & $deg/sec$ & Nr
1345     &\begin{minipage}[t]{3in}
1346     {Temp Change due to Convection}
1347     \end{minipage}\\
1348     DQCONV & $g/kg/sec$ & Nr
1349     &\begin{minipage}[t]{3in}
1350     {Specific Humidity Change due to Convection}
1351     \end{minipage}\\
1352     RELHUM & $percent$ & Nr
1353     &\begin{minipage}[t]{3in}
1354     {Relative Humidity}
1355     \end{minipage}\\
1356     PRECLS & $g/m^2/sec$ & 1
1357     &\begin{minipage}[t]{3in}
1358     {Large Scale Precipitation}
1359     \end{minipage}\\
1360     ENPREC & $J/g$ & 1
1361     &\begin{minipage}[t]{3in}
1362     {Energy of Precipitation (snow, rain Temp)}
1363     \end{minipage}\\
1364     \end{tabular}
1365     \vspace{1.5in}
1366     \vfill
1367    
1368     \newpage
1369    
1370 molod 1.9 Fizhi Diagnostic Description:
1371 molod 1.8
1372     In this section we list and describe the diagnostic quantities available within the
1373     GCM. The diagnostics are listed in the order that they appear in the
1374 molod 1.14 Diagnostic Menu, Section \ref{sec:pkg:fizhi:diagnostics}.
1375 molod 1.8 In all cases, each diagnostic as currently archived on the output datasets
1376     is time-averaged over its diagnostic output frequency:
1377    
1378     \[
1379 jmc 1.19 {\bf DIAGNOSTIC} = \frac{1}{TTOT} \sum_{t=1}^{t=TTOT} diag(t)
1380 molod 1.8 \]
1381 jmc 1.19 where $TTOT = \frac{ {\bf NQDIAG} }{\Delta t}$, {\bf NQDIAG} is the
1382 molod 1.8 output frequency of the diagnostic, and $\Delta t$ is
1383     the timestep over which the diagnostic is updated.
1384    
1385     { \underline {UFLUX} Surface Zonal Wind Stress on the Atmosphere ($Newton/m^2$) }
1386    
1387     The zonal wind stress is the turbulent flux of zonal momentum from
1388     the surface.
1389     \[
1390     {\bf UFLUX} = - \rho C_D W_s u \hspace{1cm}where: \hspace{.2cm}C_D = C^2_u
1391     \]
1392     where $\rho$ = the atmospheric density at the surface, $C_{D}$ is the surface
1393     drag coefficient, $C_u$ is the dimensionless surface exchange coefficient for momentum
1394     (see diagnostic number 10), $W_s$ is the magnitude of the surface layer wind, and $u$ is
1395     the zonal wind in the lowest model layer.
1396     \\
1397    
1398    
1399     { \underline {VFLUX} Surface Meridional Wind Stress on the Atmosphere ($Newton/m^2$) }
1400    
1401     The meridional wind stress is the turbulent flux of meridional momentum from
1402     the surface.
1403     \[
1404     {\bf VFLUX} = - \rho C_D W_s v \hspace{1cm}where: \hspace{.2cm}C_D = C^2_u
1405     \]
1406     where $\rho$ = the atmospheric density at the surface, $C_{D}$ is the surface
1407     drag coefficient, $C_u$ is the dimensionless surface exchange coefficient for momentum
1408     (see diagnostic number 10), $W_s$ is the magnitude of the surface layer wind, and $v$ is
1409     the meridional wind in the lowest model layer.
1410     \\
1411    
1412     { \underline {HFLUX} Surface Flux of Sensible Heat ($Watts/m^2$) }
1413    
1414     The turbulent flux of sensible heat from the surface to the atmosphere is a function of the
1415     gradient of virtual potential temperature and the eddy exchange coefficient:
1416     \[
1417     {\bf HFLUX} = P^{\kappa}\rho c_{p} C_{H} W_s (\theta_{surface} - \theta_{Nrphys})
1418     \hspace{1cm}where: \hspace{.2cm}C_H = C_u C_t
1419     \]
1420     where $\rho$ = the atmospheric density at the surface, $c_{p}$ is the specific
1421     heat of air, $C_{H}$ is the dimensionless surface heat transfer coefficient, $W_s$ is the
1422     magnitude of the surface layer wind, $C_u$ is the dimensionless surface exchange coefficient
1423     for momentum (see diagnostic number 10), $C_t$ is the dimensionless surface exchange coefficient
1424     for heat and moisture (see diagnostic number 9), and $\theta$ is the potential temperature
1425     at the surface and at the bottom model level.
1426     \\
1427    
1428    
1429     { \underline {EFLUX} Surface Flux of Latent Heat ($Watts/m^2$) }
1430    
1431     The turbulent flux of latent heat from the surface to the atmosphere is a function of the
1432     gradient of moisture, the potential evapotranspiration fraction and the eddy exchange coefficient:
1433     \[
1434     {\bf EFLUX} = \rho \beta L C_{H} W_s (q_{surface} - q_{Nrphys})
1435     \hspace{1cm}where: \hspace{.2cm}C_H = C_u C_t
1436     \]
1437     where $\rho$ = the atmospheric density at the surface, $\beta$ is the fraction of
1438     the potential evapotranspiration actually evaporated, L is the latent
1439     heat of evaporation, $C_{H}$ is the dimensionless surface heat transfer coefficient, $W_s$ is the
1440     magnitude of the surface layer wind, $C_u$ is the dimensionless surface exchange coefficient
1441     for momentum (see diagnostic number 10), $C_t$ is the dimensionless surface exchange coefficient
1442     for heat and moisture (see diagnostic number 9), and $q_{surface}$ and $q_{Nrphys}$ are the specific
1443     humidity at the surface and at the bottom model level, respectively.
1444     \\
1445    
1446     { \underline {QICE} Heat Conduction Through Sea Ice ($Watts/m^2$) }
1447    
1448     Over sea ice there is an additional source of energy at the surface due to the heat
1449     conduction from the relatively warm ocean through the sea ice. The heat conduction
1450     through sea ice represents an additional energy source term for the ground temperature equation.
1451    
1452     \[
1453 jmc 1.19 {\bf QICE} = \frac{C_{ti}}{H_i} (T_i-T_g)
1454 molod 1.8 \]
1455    
1456     where $C_{ti}$ is the thermal conductivity of ice, $H_i$ is the ice thickness, assumed to
1457     be $3 \hspace{.1cm} m$ where sea ice is present, $T_i$ is 273 degrees Kelvin, and
1458     $T_g$ is the temperature of the sea ice.
1459    
1460     NOTE: QICE is not available through model version 5.3, but is available in subsequent versions.
1461     \\
1462    
1463    
1464     { \underline {RADLWG} Net upward Longwave Flux at the surface ($Watts/m^2$)}
1465    
1466     \begin{eqnarray*}
1467     {\bf RADLWG} & = & F_{LW,Nrphys+1}^{Net} \\
1468     & = & F_{LW,Nrphys+1}^\uparrow - F_{LW,Nrphys+1}^\downarrow
1469     \end{eqnarray*}
1470     \\
1471     where Nrphys+1 indicates the lowest model edge-level, or $p = p_{surf}$.
1472     $F_{LW}^\uparrow$ is
1473     the upward Longwave flux and $F_{LW}^\downarrow$ is the downward Longwave flux.
1474     \\
1475    
1476     { \underline {RADSWG} Net downard shortwave Flux at the surface ($Watts/m^2$)}
1477    
1478     \begin{eqnarray*}
1479     {\bf RADSWG} & = & F_{SW,Nrphys+1}^{Net} \\
1480     & = & F_{SW,Nrphys+1}^\downarrow - F_{SW,Nrphys+1}^\uparrow
1481     \end{eqnarray*}
1482     \\
1483     where Nrphys+1 indicates the lowest model edge-level, or $p = p_{surf}$.
1484     $F_{SW}^\downarrow$ is
1485     the downward Shortwave flux and $F_{SW}^\uparrow$ is the upward Shortwave flux.
1486     \\
1487    
1488    
1489     \noindent
1490     { \underline {RI} Richardson Number} ($dimensionless$)
1491    
1492     \noindent
1493     The non-dimensional stability indicator is the ratio of the buoyancy to the shear:
1494     \[
1495 jmc 1.19 {\bf RI} = \frac{ \frac{g}{\theta_v} \pp {\theta_v}{z} }{ (\pp{u}{z})^2 + (\pp{v}{z})^2 }
1496     = \frac{c_p \pp{\theta_v}{z} \pp{P^ \kappa}{z} }{ (\pp{u}{z})^2 + (\pp{v}{z})^2 }
1497 molod 1.8 \]
1498     \\
1499     where we used the hydrostatic equation:
1500     \[
1501     {\pp{\Phi}{P^ \kappa}} = c_p \theta_v
1502     \]
1503     Negative values indicate unstable buoyancy {\bf{AND}} shear, small positive values ($<0.4$)
1504     indicate dominantly unstable shear, and large positive values indicate dominantly stable
1505     stratification.
1506     \\
1507    
1508     \noindent
1509     { \underline {CT} Surface Exchange Coefficient for Temperature and Moisture ($dimensionless$) }
1510    
1511     \noindent
1512     The surface exchange coefficient is obtained from the similarity functions for the stability
1513     dependant flux profile relationships:
1514     \[
1515 jmc 1.19 {\bf CT} = -\frac{( \overline{w^{\prime}\theta^{\prime}} ) }{ u_* \Delta \theta } =
1516     -\frac{( \overline{w^{\prime}q^{\prime}} ) }{ u_* \Delta q } =
1517     \frac{ k }{ (\psi_{h} + \psi_{g}) }
1518 molod 1.8 \]
1519     where $\psi_h$ is the surface layer non-dimensional temperature change and $\psi_g$ is the
1520     viscous sublayer non-dimensional temperature or moisture change:
1521     \[
1522 jmc 1.19 \psi_{h} = \int_{\zeta_{0}}^{\zeta} \frac{\phi_{h} }{ \zeta} d \zeta \hspace{1cm} and
1523     \hspace{1cm} \psi_{g} = \frac{ 0.55 (Pr^{2/3} - 0.2) }{ \nu^{1/2} }
1524 molod 1.8 (h_{0}u_{*} - h_{0_{ref}}u_{*_{ref}})^{1/2}
1525     \]
1526     and:
1527     $h_{0} = 30z_{0}$ with a maximum value over land of 0.01
1528    
1529     \noindent
1530     $\phi_h$ is the similarity function of $\zeta$, which expresses the stability dependance of
1531     the temperature and moisture gradients, specified differently for stable and unstable
1532 molod 1.10 layers according to \cite{helfschu:95}. k is the Von Karman constant, $\zeta$ is the
1533 molod 1.8 non-dimensional stability parameter, Pr is the Prandtl number for air, $\nu$ is the molecular
1534     viscosity, $z_{0}$ is the surface roughness length, $u_*$ is the surface stress velocity
1535     (see diagnostic number 67), and the subscript ref refers to a reference value.
1536     \\
1537    
1538     \noindent
1539     { \underline {CU} Surface Exchange Coefficient for Momentum ($dimensionless$) }
1540    
1541     \noindent
1542     The surface exchange coefficient is obtained from the similarity functions for the stability
1543     dependant flux profile relationships:
1544     \[
1545 jmc 1.19 {\bf CU} = \frac{u_* }{ W_s} = \frac{ k }{ \psi_{m} }
1546 molod 1.8 \]
1547     where $\psi_m$ is the surface layer non-dimensional wind shear:
1548     \[
1549 jmc 1.19 \psi_{m} = {\int_{\zeta_{0}}^{\zeta} \frac{\phi_{m} }{ \zeta} d \zeta}
1550 molod 1.8 \]
1551     \noindent
1552     $\phi_m$ is the similarity function of $\zeta$, which expresses the stability dependance of
1553     the temperature and moisture gradients, specified differently for stable and unstable layers
1554 molod 1.10 according to \cite{helfschu:95}. k is the Von Karman constant, $\zeta$ is the
1555 molod 1.8 non-dimensional stability parameter, $u_*$ is the surface stress velocity
1556     (see diagnostic number 67), and $W_s$ is the magnitude of the surface layer wind.
1557     \\
1558    
1559     \noindent
1560     { \underline {ET} Diffusivity Coefficient for Temperature and Moisture ($m^2/sec$) }
1561    
1562     \noindent
1563     In the level 2.5 version of the Mellor-Yamada (1974) hierarchy, the turbulent heat or
1564     moisture flux for the atmosphere above the surface layer can be expressed as a turbulent
1565     diffusion coefficient $K_h$ times the negative of the gradient of potential temperature
1566 molod 1.10 or moisture. In the \cite{helflab:88} adaptation of this closure, $K_h$
1567 molod 1.8 takes the form:
1568     \[
1569 jmc 1.19 {\bf ET} = K_h = -\frac{( \overline{w^{\prime}\theta_v^{\prime}}) }{ \pp{\theta_v}{z} }
1570 molod 1.8 = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell \, S_H(G_M,G_H) & \mbox{decaying turbulence}
1571 jmc 1.19 \\ \frac{ q^2 }{ q_e } \, \ell \, S_{H}(G_{M_e},G_{H_e}) & \mbox{growing turbulence} \end{array} \right.
1572 molod 1.8 \]
1573     where $q$ is the turbulent velocity, or $\sqrt{2*turbulent \hspace{.2cm} kinetic \hspace{.2cm}
1574     energy}$, $q_e$ is the turbulence velocity derived from the more simple level 2.0 model,
1575     which describes equilibrium turbulence, $\ell$ is the master length scale related to the layer
1576     depth,
1577     $S_H$ is a function of $G_H$ and $G_M$, the dimensionless buoyancy and
1578     wind shear parameters, respectively, or a function of $G_{H_e}$ and $G_{M_e}$, the equilibrium
1579     dimensionless buoyancy and wind shear
1580     parameters. Both $G_H$ and $G_M$, and their equilibrium values $G_{H_e}$ and $G_{M_e}$,
1581     are functions of the Richardson number.
1582    
1583     \noindent
1584     For the detailed equations and derivations of the modified level 2.5 closure scheme,
1585 molod 1.10 see \cite{helflab:88}.
1586 molod 1.8
1587     \noindent
1588     In the surface layer, ${\bf {ET}}$ is the exchange coefficient for heat and moisture,
1589     in units of $m/sec$, given by:
1590     \[
1591     {\bf ET_{Nrphys}} = C_t * u_* = C_H W_s
1592     \]
1593     \noindent
1594     where $C_t$ is the dimensionless exchange coefficient for heat and moisture from the
1595     surface layer similarity functions (see diagnostic number 9), $u_*$ is the surface
1596     friction velocity (see diagnostic number 67), $C_H$ is the heat transfer coefficient,
1597     and $W_s$ is the magnitude of the surface layer wind.
1598     \\
1599    
1600     \noindent
1601     { \underline {EU} Diffusivity Coefficient for Momentum ($m^2/sec$) }
1602    
1603     \noindent
1604     In the level 2.5 version of the Mellor-Yamada (1974) hierarchy, the turbulent heat
1605     momentum flux for the atmosphere above the surface layer can be expressed as a turbulent
1606     diffusion coefficient $K_m$ times the negative of the gradient of the u-wind.
1607 molod 1.10 In the \cite{helflab:88} adaptation of this closure, $K_m$
1608 molod 1.8 takes the form:
1609     \[
1610 jmc 1.19 {\bf EU} = K_m = -\frac{( \overline{u^{\prime}w^{\prime}} ) }{ \pp{U}{z} }
1611 molod 1.8 = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell \, S_M(G_M,G_H) & \mbox{decaying turbulence}
1612 jmc 1.19 \\ \frac{ q^2 }{ q_e } \, \ell \, S_{M}(G_{M_e},G_{H_e}) & \mbox{growing turbulence} \end{array} \right.
1613 molod 1.8 \]
1614     \noindent
1615     where $q$ is the turbulent velocity, or $\sqrt{2*turbulent \hspace{.2cm} kinetic \hspace{.2cm}
1616     energy}$, $q_e$ is the turbulence velocity derived from the more simple level 2.0 model,
1617     which describes equilibrium turbulence, $\ell$ is the master length scale related to the layer
1618     depth,
1619     $S_M$ is a function of $G_H$ and $G_M$, the dimensionless buoyancy and
1620     wind shear parameters, respectively, or a function of $G_{H_e}$ and $G_{M_e}$, the equilibrium
1621     dimensionless buoyancy and wind shear
1622     parameters. Both $G_H$ and $G_M$, and their equilibrium values $G_{H_e}$ and $G_{M_e}$,
1623     are functions of the Richardson number.
1624    
1625     \noindent
1626     For the detailed equations and derivations of the modified level 2.5 closure scheme,
1627 molod 1.10 see \cite{helflab:88}.
1628 molod 1.8
1629     \noindent
1630     In the surface layer, ${\bf {EU}}$ is the exchange coefficient for momentum,
1631     in units of $m/sec$, given by:
1632     \[
1633     {\bf EU_{Nrphys}} = C_u * u_* = C_D W_s
1634     \]
1635     \noindent
1636     where $C_u$ is the dimensionless exchange coefficient for momentum from the surface layer
1637     similarity functions (see diagnostic number 10), $u_*$ is the surface friction velocity
1638     (see diagnostic number 67), $C_D$ is the surface drag coefficient, and $W_s$ is the
1639     magnitude of the surface layer wind.
1640     \\
1641    
1642     \noindent
1643     { \underline {TURBU} Zonal U-Momentum changes due to Turbulence ($m/sec/day$) }
1644    
1645     \noindent
1646     The tendency of U-Momentum due to turbulence is written:
1647     \[
1648     {\bf TURBU} = {\pp{u}{t}}_{turb} = {\pp{}{z} }{(- \overline{u^{\prime}w^{\prime}})}
1649     = {\pp{}{z} }{(K_m \pp{u}{z})}
1650     \]
1651    
1652     \noindent
1653     The Helfand and Labraga level 2.5 scheme models the turbulent
1654     flux of u-momentum in terms of $K_m$, and the equation has the form of a diffusion
1655     equation.
1656    
1657     \noindent
1658     { \underline {TURBV} Meridional V-Momentum changes due to Turbulence ($m/sec/day$) }
1659    
1660     \noindent
1661     The tendency of V-Momentum due to turbulence is written:
1662     \[
1663     {\bf TURBV} = {\pp{v}{t}}_{turb} = {\pp{}{z} }{(- \overline{v^{\prime}w^{\prime}})}
1664     = {\pp{}{z} }{(K_m \pp{v}{z})}
1665     \]
1666    
1667     \noindent
1668     The Helfand and Labraga level 2.5 scheme models the turbulent
1669     flux of v-momentum in terms of $K_m$, and the equation has the form of a diffusion
1670     equation.
1671     \\
1672    
1673     \noindent
1674     { \underline {TURBT} Temperature changes due to Turbulence ($deg/day$) }
1675    
1676     \noindent
1677     The tendency of temperature due to turbulence is written:
1678     \[
1679     {\bf TURBT} = {\pp{T}{t}} = P^{\kappa}{\pp{\theta}{t}}_{turb} =
1680     P^{\kappa}{\pp{}{z} }{(- \overline{w^{\prime}\theta^{\prime}})}
1681     = P^{\kappa}{\pp{}{z} }{(K_h \pp{\theta_v}{z})}
1682     \]
1683    
1684     \noindent
1685     The Helfand and Labraga level 2.5 scheme models the turbulent
1686     flux of temperature in terms of $K_h$, and the equation has the form of a diffusion
1687     equation.
1688     \\
1689    
1690     \noindent
1691     { \underline {TURBQ} Specific Humidity changes due to Turbulence ($g/kg/day$) }
1692    
1693     \noindent
1694     The tendency of specific humidity due to turbulence is written:
1695     \[
1696     {\bf TURBQ} = {\pp{q}{t}}_{turb} = {\pp{}{z} }{(- \overline{w^{\prime}q^{\prime}})}
1697     = {\pp{}{z} }{(K_h \pp{q}{z})}
1698     \]
1699    
1700     \noindent
1701     The Helfand and Labraga level 2.5 scheme models the turbulent
1702     flux of temperature in terms of $K_h$, and the equation has the form of a diffusion
1703     equation.
1704     \\
1705    
1706     \noindent
1707     { \underline {MOISTT} Temperature Changes Due to Moist Processes ($deg/day$) }
1708    
1709     \noindent
1710     \[
1711     {\bf MOISTT} = \left. {\pp{T}{t}}\right|_{c} + \left. {\pp{T}{t}} \right|_{ls}
1712     \]
1713     where:
1714     \[
1715 jmc 1.19 \left.{\pp{T}{t}}\right|_{c} = R \sum_i \left( \alpha \frac{m_B}{c_p} \Gamma_s \right)_i
1716 molod 1.8 \hspace{.4cm} and
1717 jmc 1.19 \hspace{.4cm} \left.{\pp{T}{t}}\right|_{ls} = \frac{L}{c_p} (q^*-q)
1718 molod 1.8 \]
1719     and
1720     \[
1721     \Gamma_s = g \eta \pp{s}{p}
1722     \]
1723    
1724     \noindent
1725     The subscript $c$ refers to convective processes, while the subscript $ls$ refers to large scale
1726     precipitation processes, or supersaturation rain.
1727     The summation refers to contributions from each cloud type called by RAS.
1728     The dry static energy is given
1729     as $s$, the convective cloud base mass flux is given as $m_B$, and the cloud entrainment is
1730     given as $\eta$, which are explicitly defined in Section \ref{sec:fizhi:mc},
1731     the description of the convective parameterization. The fractional adjustment, or relaxation
1732     parameter, for each cloud type is given as $\alpha$, while
1733     $R$ is the rain re-evaporation adjustment.
1734     \\
1735    
1736     \noindent
1737     { \underline {MOISTQ} Specific Humidity Changes Due to Moist Processes ($g/kg/day$) }
1738    
1739     \noindent
1740     \[
1741     {\bf MOISTQ} = \left. {\pp{q}{t}}\right|_{c} + \left. {\pp{q}{t}} \right|_{ls}
1742     \]
1743     where:
1744     \[
1745 jmc 1.19 \left.{\pp{q}{t}}\right|_{c} = R \sum_i \left( \alpha \frac{m_B}{L}(\Gamma_h-\Gamma_s) \right)_i
1746 molod 1.8 \hspace{.4cm} and
1747     \hspace{.4cm} \left.{\pp{q}{t}}\right|_{ls} = (q^*-q)
1748     \]
1749     and
1750     \[
1751     \Gamma_s = g \eta \pp{s}{p}\hspace{.4cm} and \hspace{.4cm}\Gamma_h = g \eta \pp{h}{p}
1752     \]
1753     \noindent
1754     The subscript $c$ refers to convective processes, while the subscript $ls$ refers to large scale
1755     precipitation processes, or supersaturation rain.
1756     The summation refers to contributions from each cloud type called by RAS.
1757     The dry static energy is given as $s$,
1758     the moist static energy is given as $h$,
1759     the convective cloud base mass flux is given as $m_B$, and the cloud entrainment is
1760     given as $\eta$, which are explicitly defined in Section \ref{sec:fizhi:mc},
1761     the description of the convective parameterization. The fractional adjustment, or relaxation
1762     parameter, for each cloud type is given as $\alpha$, while
1763     $R$ is the rain re-evaporation adjustment.
1764     \\
1765    
1766     \noindent
1767     { \underline {RADLW} Heating Rate due to Longwave Radiation ($deg/day$) }
1768    
1769     \noindent
1770     The net longwave heating rate is calculated as the vertical divergence of the
1771     net terrestrial radiative fluxes.
1772     Both the clear-sky and cloudy-sky longwave fluxes are computed within the
1773     longwave routine.
1774     The subroutine calculates the clear-sky flux, $F^{clearsky}_{LW}$, first.
1775     For a given cloud fraction,
1776     the clear line-of-sight probability $C(p,p^{\prime})$ is computed from the current level pressure $p$
1777     to the model top pressure, $p^{\prime} = p_{top}$, and the model surface pressure, $p^{\prime} = p_{surf}$,
1778     for the upward and downward radiative fluxes.
1779     (see Section \ref{sec:fizhi:radcloud}).
1780     The cloudy-sky flux is then obtained as:
1781    
1782     \noindent
1783     \[
1784     F_{LW} = C(p,p') \cdot F^{clearsky}_{LW},
1785     \]
1786    
1787     \noindent
1788     Finally, the net longwave heating rate is calculated as the vertical divergence of the
1789     net terrestrial radiative fluxes:
1790     \[
1791 jmc 1.19 \pp{\rho c_p T}{t} = - \p{z} F_{LW}^{NET} ,
1792 molod 1.8 \]
1793     or
1794     \[
1795 jmc 1.19 {\bf RADLW} = \frac{g}{c_p \pi} \p{\sigma} F_{LW}^{NET} .
1796 molod 1.8 \]
1797    
1798     \noindent
1799     where $g$ is the accelation due to gravity,
1800     $c_p$ is the heat capacity of air at constant pressure,
1801     and
1802     \[
1803     F_{LW}^{NET} = F_{LW}^\uparrow - F_{LW}^\downarrow
1804     \]
1805     \\
1806    
1807    
1808     \noindent
1809     { \underline {RADSW} Heating Rate due to Shortwave Radiation ($deg/day$) }
1810    
1811     \noindent
1812     The net Shortwave heating rate is calculated as the vertical divergence of the
1813     net solar radiative fluxes.
1814     The clear-sky and cloudy-sky shortwave fluxes are calculated separately.
1815     For the clear-sky case, the shortwave fluxes and heating rates are computed with
1816     both CLMO (maximum overlap cloud fraction) and
1817     CLRO (random overlap cloud fraction) set to zero (see Section \ref{sec:fizhi:radcloud}).
1818     The shortwave routine is then called a second time, for the cloudy-sky case, with the
1819     true time-averaged cloud fractions CLMO
1820     and CLRO being used. In all cases, a normalized incident shortwave flux is used as
1821     input at the top of the atmosphere.
1822    
1823     \noindent
1824     The heating rate due to Shortwave Radiation under cloudy skies is defined as:
1825     \[
1826 jmc 1.19 \pp{\rho c_p T}{t} = - \p{z} F(cloudy)_{SW}^{NET} \cdot {\rm RADSWT},
1827 molod 1.8 \]
1828     or
1829     \[
1830 jmc 1.19 {\bf RADSW} = \frac{g}{c_p \pi} \p{\sigma} F(cloudy)_{SW}^{NET}\cdot {\rm RADSWT} .
1831 molod 1.8 \]
1832    
1833     \noindent
1834     where $g$ is the accelation due to gravity,
1835     $c_p$ is the heat capacity of air at constant pressure, RADSWT is the true incident
1836     shortwave radiation at the top of the atmosphere (See Diagnostic \#48), and
1837     \[
1838     F(cloudy)_{SW}^{Net} = F(cloudy)_{SW}^\uparrow - F(cloudy)_{SW}^\downarrow
1839     \]
1840     \\
1841    
1842     \noindent
1843     { \underline {PREACC} Total (Large-scale + Convective) Accumulated Precipition ($mm/day$) }
1844    
1845     \noindent
1846     For a change in specific humidity due to moist processes, $\Delta q_{moist}$,
1847     the vertical integral or total precipitable amount is given by:
1848     \[
1849     {\bf PREACC} = \int_{surf}^{top} \rho \Delta q_{moist} dz = - \int_{surf}^{top} \Delta q_{moist}
1850 jmc 1.19 \frac{dp}{g} = \frac{1}{g} \int_0^1 \Delta q_{moist} dp
1851 molod 1.8 \]
1852     \\
1853    
1854     \noindent
1855     A precipitation rate is defined as the vertically integrated moisture adjustment per Moist Processes
1856     time step, scaled to $mm/day$.
1857     \\
1858    
1859     \noindent
1860     { \underline {PRECON} Convective Precipition ($mm/day$) }
1861    
1862     \noindent
1863     For a change in specific humidity due to sub-grid scale cumulus convective processes, $\Delta q_{cum}$,
1864     the vertical integral or total precipitable amount is given by:
1865     \[
1866     {\bf PRECON} = \int_{surf}^{top} \rho \Delta q_{cum} dz = - \int_{surf}^{top} \Delta q_{cum}
1867 jmc 1.19 \frac{dp}{g} = \frac{1}{g} \int_0^1 \Delta q_{cum} dp
1868 molod 1.8 \]
1869     \\
1870    
1871     \noindent
1872     A precipitation rate is defined as the vertically integrated moisture adjustment per Moist Processes
1873     time step, scaled to $mm/day$.
1874     \\
1875    
1876     \noindent
1877     { \underline {TUFLUX} Turbulent Flux of U-Momentum ($Newton/m^2$) }
1878    
1879     \noindent
1880     The turbulent flux of u-momentum is calculated for $diagnostic \hspace{.2cm} purposes
1881     \hspace{.2cm} only$ from the eddy coefficient for momentum:
1882    
1883     \[
1884     {\bf TUFLUX} = {\rho } {(\overline{u^{\prime}w^{\prime}})} =
1885     {\rho } {(- K_m \pp{U}{z})}
1886     \]
1887    
1888     \noindent
1889     where $\rho$ is the air density, and $K_m$ is the eddy coefficient.
1890     \\
1891    
1892     \noindent
1893     { \underline {TVFLUX} Turbulent Flux of V-Momentum ($Newton/m^2$) }
1894    
1895     \noindent
1896     The turbulent flux of v-momentum is calculated for $diagnostic \hspace{.2cm} purposes
1897     \hspace{.2cm} only$ from the eddy coefficient for momentum:
1898    
1899     \[
1900     {\bf TVFLUX} = {\rho } {(\overline{v^{\prime}w^{\prime}})} =
1901     {\rho } {(- K_m \pp{V}{z})}
1902     \]
1903    
1904     \noindent
1905     where $\rho$ is the air density, and $K_m$ is the eddy coefficient.
1906     \\
1907    
1908    
1909     \noindent
1910     { \underline {TTFLUX} Turbulent Flux of Sensible Heat ($Watts/m^2$) }
1911    
1912     \noindent
1913     The turbulent flux of sensible heat is calculated for $diagnostic \hspace{.2cm} purposes
1914     \hspace{.2cm} only$ from the eddy coefficient for heat and moisture:
1915    
1916     \noindent
1917     \[
1918     {\bf TTFLUX} = c_p {\rho }
1919     P^{\kappa}{(\overline{w^{\prime}\theta^{\prime}})}
1920     = c_p {\rho } P^{\kappa}{(- K_h \pp{\theta_v}{z})}
1921     \]
1922    
1923     \noindent
1924     where $\rho$ is the air density, and $K_h$ is the eddy coefficient.
1925     \\
1926    
1927    
1928     \noindent
1929     { \underline {TQFLUX} Turbulent Flux of Latent Heat ($Watts/m^2$) }
1930    
1931     \noindent
1932     The turbulent flux of latent heat is calculated for $diagnostic \hspace{.2cm} purposes
1933     \hspace{.2cm} only$ from the eddy coefficient for heat and moisture:
1934    
1935     \noindent
1936     \[
1937     {\bf TQFLUX} = {L {\rho } (\overline{w^{\prime}q^{\prime}})} =
1938     {L {\rho }(- K_h \pp{q}{z})}
1939     \]
1940    
1941     \noindent
1942     where $\rho$ is the air density, and $K_h$ is the eddy coefficient.
1943     \\
1944    
1945    
1946     \noindent
1947     { \underline {CN} Neutral Drag Coefficient ($dimensionless$) }
1948    
1949     \noindent
1950     The drag coefficient for momentum obtained by assuming a neutrally stable surface layer:
1951     \[
1952 jmc 1.19 {\bf CN} = \frac{ k }{ \ln(\frac{h }{z_0}) }
1953 molod 1.8 \]
1954    
1955     \noindent
1956     where $k$ is the Von Karman constant, $h$ is the height of the surface layer, and
1957     $z_0$ is the surface roughness.
1958    
1959     \noindent
1960     NOTE: CN is not available through model version 5.3, but is available in subsequent
1961     versions.
1962     \\
1963    
1964     \noindent
1965     { \underline {WINDS} Surface Wind Speed ($meter/sec$) }
1966    
1967     \noindent
1968     The surface wind speed is calculated for the last internal turbulence time step:
1969     \[
1970     {\bf WINDS} = \sqrt{u_{Nrphys}^2 + v_{Nrphys}^2}
1971     \]
1972    
1973     \noindent
1974     where the subscript $Nrphys$ refers to the lowest model level.
1975     \\
1976    
1977     \noindent
1978     { \underline {DTSRF} Air/Surface Virtual Temperature Difference ($deg \hspace{.1cm} K$) }
1979    
1980     \noindent
1981     The air/surface virtual temperature difference measures the stability of the surface layer:
1982     \[
1983     {\bf DTSRF} = (\theta_{v{Nrphys+1}} - \theta{v_{Nrphys}}) P^{\kappa}_{surf}
1984     \]
1985     \noindent
1986     where
1987     \[
1988 jmc 1.19 \theta_{v{Nrphys+1}} = \frac{ T_g }{ P^{\kappa}_{surf} } (1 + .609 q_{Nrphys+1}) \hspace{1cm}
1989 molod 1.8 and \hspace{1cm} q_{Nrphys+1} = q_{Nrphys} + \beta(q^*(T_g,P_s) - q_{Nrphys})
1990     \]
1991    
1992     \noindent
1993     $\beta$ is the surface potential evapotranspiration coefficient ($\beta=1$ over oceans),
1994     $q^*(T_g,P_s)$ is the saturation specific humidity at the ground temperature
1995     and surface pressure, level $Nrphys$ refers to the lowest model level and level $Nrphys+1$
1996     refers to the surface.
1997     \\
1998    
1999    
2000     \noindent
2001     { \underline {TG} Ground Temperature ($deg \hspace{.1cm} K$) }
2002    
2003     \noindent
2004     The ground temperature equation is solved as part of the turbulence package
2005     using a backward implicit time differencing scheme:
2006     \[
2007     {\bf TG} \hspace{.1cm} is \hspace{.1cm} obtained \hspace{.1cm} from: \hspace{.1cm}
2008     C_g\pp{T_g}{t} = R_{sw} - R_{lw} + Q_{ice} - H - LE
2009     \]
2010    
2011     \noindent
2012     where $R_{sw}$ is the net surface downward shortwave radiative flux, $R_{lw}$ is the
2013     net surface upward longwave radiative flux, $Q_{ice}$ is the heat conduction through
2014     sea ice, $H$ is the upward sensible heat flux, $LE$ is the upward latent heat
2015     flux, and $C_g$ is the total heat capacity of the ground.
2016     $C_g$ is obtained by solving a heat diffusion equation
2017 molod 1.10 for the penetration of the diurnal cycle into the ground (\cite{black:77}), and is given by:
2018 molod 1.8 \[
2019 jmc 1.19 C_g = \sqrt{ \frac{\lambda C_s }{ 2 \omega } } = \sqrt{(0.386 + 0.536W + 0.15W^2)2x10^{-3}
2020     \frac{86400.}{2\pi} } \, \, .
2021 molod 1.8 \]
2022     \noindent
2023 jmc 1.19 Here, the thermal conductivity, $\lambda$, is equal to $2x10^{-3}$ $\frac{ly}{sec}
2024     \frac{cm}{K}$,
2025 molod 1.8 the angular velocity of the earth, $\omega$, is written as $86400$ $sec/day$ divided
2026     by $2 \pi$ $radians/
2027     day$, and the expression for $C_s$, the heat capacity per unit volume at the surface,
2028     is a function of the ground wetness, $W$.
2029     \\
2030    
2031     \noindent
2032     { \underline {TS} Surface Temperature ($deg \hspace{.1cm} K$) }
2033    
2034     \noindent
2035     The surface temperature estimate is made by assuming that the model's lowest
2036     layer is well-mixed, and therefore that $\theta$ is constant in that layer.
2037     The surface temperature is therefore:
2038     \[
2039     {\bf TS} = \theta_{Nrphys} P^{\kappa}_{surf}
2040     \]
2041     \\
2042    
2043     \noindent
2044     { \underline {DTG} Surface Temperature Adjustment ($deg \hspace{.1cm} K$) }
2045    
2046     \noindent
2047     The change in surface temperature from one turbulence time step to the next, solved
2048     using the Ground Temperature Equation (see diagnostic number 30) is calculated:
2049     \[
2050     {\bf DTG} = {T_g}^{n} - {T_g}^{n-1}
2051     \]
2052    
2053     \noindent
2054     where superscript $n$ refers to the new, updated time level, and the superscript $n-1$
2055     refers to the value at the previous turbulence time level.
2056     \\
2057    
2058     \noindent
2059     { \underline {QG} Ground Specific Humidity ($g/kg$) }
2060    
2061     \noindent
2062     The ground specific humidity is obtained by interpolating between the specific
2063     humidity at the lowest model level and the specific humidity of a saturated ground.
2064     The interpolation is performed using the potential evapotranspiration function:
2065     \[
2066     {\bf QG} = q_{Nrphys+1} = q_{Nrphys} + \beta(q^*(T_g,P_s) - q_{Nrphys})
2067     \]
2068    
2069     \noindent
2070     where $\beta$ is the surface potential evapotranspiration coefficient ($\beta=1$ over oceans),
2071     and $q^*(T_g,P_s)$ is the saturation specific humidity at the ground temperature and surface
2072     pressure.
2073     \\
2074    
2075     \noindent
2076     { \underline {QS} Saturation Surface Specific Humidity ($g/kg$) }
2077    
2078     \noindent
2079     The surface saturation specific humidity is the saturation specific humidity at
2080     the ground temprature and surface pressure:
2081     \[
2082     {\bf QS} = q^*(T_g,P_s)
2083     \]
2084     \\
2085    
2086     \noindent
2087     { \underline {TGRLW} Instantaneous ground temperature used as input to the Longwave
2088     radiation subroutine (deg)}
2089     \[
2090     {\bf TGRLW} = T_g(\lambda , \phi ,n)
2091     \]
2092     \noindent
2093     where $T_g$ is the model ground temperature at the current time step $n$.
2094     \\
2095    
2096    
2097     \noindent
2098     { \underline {ST4} Upward Longwave flux at the surface ($Watts/m^2$) }
2099     \[
2100     {\bf ST4} = \sigma T^4
2101     \]
2102     \noindent
2103     where $\sigma$ is the Stefan-Boltzmann constant and T is the temperature.
2104     \\
2105    
2106     \noindent
2107     { \underline {OLR} Net upward Longwave flux at $p=p_{top}$ ($Watts/m^2$) }
2108     \[
2109     {\bf OLR} = F_{LW,top}^{NET}
2110     \]
2111     \noindent
2112     where top indicates the top of the first model layer.
2113     In the GCM, $p_{top}$ = 0.0 mb.
2114     \\
2115    
2116    
2117     \noindent
2118     { \underline {OLRCLR} Net upward clearsky Longwave flux at $p=p_{top}$ ($Watts/m^2$) }
2119     \[
2120     {\bf OLRCLR} = F(clearsky)_{LW,top}^{NET}
2121     \]
2122     \noindent
2123     where top indicates the top of the first model layer.
2124     In the GCM, $p_{top}$ = 0.0 mb.
2125     \\
2126    
2127     \noindent
2128     { \underline {LWGCLR} Net upward clearsky Longwave flux at the surface ($Watts/m^2$) }
2129    
2130     \noindent
2131     \begin{eqnarray*}
2132     {\bf LWGCLR} & = & F(clearsky)_{LW,Nrphys+1}^{Net} \\
2133     & = & F(clearsky)_{LW,Nrphys+1}^\uparrow - F(clearsky)_{LW,Nrphys+1}^\downarrow
2134     \end{eqnarray*}
2135     where Nrphys+1 indicates the lowest model edge-level, or $p = p_{surf}$.
2136     $F(clearsky)_{LW}^\uparrow$ is
2137     the upward clearsky Longwave flux and the $F(clearsky)_{LW}^\downarrow$ is the downward clearsky Longwave flux.
2138     \\
2139    
2140     \noindent
2141     { \underline {LWCLR} Heating Rate due to Clearsky Longwave Radiation ($deg/day$) }
2142    
2143     \noindent
2144     The net longwave heating rate is calculated as the vertical divergence of the
2145     net terrestrial radiative fluxes.
2146     Both the clear-sky and cloudy-sky longwave fluxes are computed within the
2147     longwave routine.
2148     The subroutine calculates the clear-sky flux, $F^{clearsky}_{LW}$, first.
2149     For a given cloud fraction,
2150     the clear line-of-sight probability $C(p,p^{\prime})$ is computed from the current level pressure $p$
2151     to the model top pressure, $p^{\prime} = p_{top}$, and the model surface pressure, $p^{\prime} = p_{surf}$,
2152     for the upward and downward radiative fluxes.
2153     (see Section \ref{sec:fizhi:radcloud}).
2154     The cloudy-sky flux is then obtained as:
2155    
2156     \noindent
2157     \[
2158     F_{LW} = C(p,p') \cdot F^{clearsky}_{LW},
2159     \]
2160    
2161     \noindent
2162     Thus, {\bf LWCLR} is defined as the net longwave heating rate due to the
2163     vertical divergence of the
2164     clear-sky longwave radiative flux:
2165     \[
2166 jmc 1.19 \pp{\rho c_p T}{t}_{clearsky} = - \p{z} F(clearsky)_{LW}^{NET} ,
2167 molod 1.8 \]
2168     or
2169     \[
2170 jmc 1.19 {\bf LWCLR} = \frac{g}{c_p \pi} \p{\sigma} F(clearsky)_{LW}^{NET} .
2171 molod 1.8 \]
2172    
2173     \noindent
2174     where $g$ is the accelation due to gravity,
2175     $c_p$ is the heat capacity of air at constant pressure,
2176     and
2177     \[
2178     F(clearsky)_{LW}^{Net} = F(clearsky)_{LW}^\uparrow - F(clearsky)_{LW}^\downarrow
2179     \]
2180     \\
2181    
2182    
2183     \noindent
2184     { \underline {TLW} Instantaneous temperature used as input to the Longwave
2185     radiation subroutine (deg)}
2186     \[
2187     {\bf TLW} = T(\lambda , \phi ,level, n)
2188     \]
2189     \noindent
2190     where $T$ is the model temperature at the current time step $n$.
2191     \\
2192    
2193    
2194     \noindent
2195     { \underline {SHLW} Instantaneous specific humidity used as input to
2196     the Longwave radiation subroutine (kg/kg)}
2197     \[
2198     {\bf SHLW} = q(\lambda , \phi , level , n)
2199     \]
2200     \noindent
2201     where $q$ is the model specific humidity at the current time step $n$.
2202     \\
2203    
2204    
2205     \noindent
2206     { \underline {OZLW} Instantaneous ozone used as input to
2207     the Longwave radiation subroutine (kg/kg)}
2208     \[
2209     {\bf OZLW} = {\rm OZ}(\lambda , \phi , level , n)
2210     \]
2211     \noindent
2212     where $\rm OZ$ is the interpolated ozone data set from the climatological monthly
2213     mean zonally averaged ozone data set.
2214     \\
2215    
2216    
2217     \noindent
2218     { \underline {CLMOLW} Maximum Overlap cloud fraction used in LW Radiation ($0-1$) }
2219    
2220     \noindent
2221     {\bf CLMOLW} is the time-averaged maximum overlap cloud fraction that has been filled by the Relaxed
2222     Arakawa/Schubert Convection scheme and will be used in the Longwave Radiation algorithm. These are
2223     convective clouds whose radiative characteristics are assumed to be correlated in the vertical.
2224     For a complete description of cloud/radiative interactions, see Section \ref{sec:fizhi:radcloud}.
2225     \[
2226     {\bf CLMOLW} = CLMO_{RAS,LW}(\lambda, \phi, level )
2227     \]
2228     \\
2229    
2230    
2231     { \underline {CLDTOT} Total cloud fraction used in LW and SW Radiation ($0-1$) }
2232    
2233     {\bf CLDTOT} is the time-averaged total cloud fraction that has been filled by the Relaxed
2234     Arakawa/Schubert and Large-scale Convection schemes and will be used in the Longwave and Shortwave
2235     Radiation packages.
2236     For a complete description of cloud/radiative interactions, see Section \ref{sec:fizhi:radcloud}.
2237     \[
2238     {\bf CLDTOT} = F_{RAS} + F_{LS}
2239     \]
2240     \\
2241     where $F_{RAS}$ is the time-averaged cloud fraction due to sub-grid scale convection, and $F_{LS}$ is the
2242     time-averaged cloud fraction due to precipitating and non-precipitating large-scale moist processes.
2243     \\
2244    
2245    
2246     \noindent
2247     { \underline {CLMOSW} Maximum Overlap cloud fraction used in SW Radiation ($0-1$) }
2248    
2249     \noindent
2250     {\bf CLMOSW} is the time-averaged maximum overlap cloud fraction that has been filled by the Relaxed
2251     Arakawa/Schubert Convection scheme and will be used in the Shortwave Radiation algorithm. These are
2252     convective clouds whose radiative characteristics are assumed to be correlated in the vertical.
2253     For a complete description of cloud/radiative interactions, see Section \ref{sec:fizhi:radcloud}.
2254     \[
2255     {\bf CLMOSW} = CLMO_{RAS,SW}(\lambda, \phi, level )
2256     \]
2257     \\
2258    
2259     \noindent
2260     { \underline {CLROSW} Random Overlap cloud fraction used in SW Radiation ($0-1$) }
2261    
2262     \noindent
2263     {\bf CLROSW} is the time-averaged random overlap cloud fraction that has been filled by the Relaxed
2264     Arakawa/Schubert and Large-scale Convection schemes and will be used in the Shortwave
2265     Radiation algorithm. These are
2266     convective and large-scale clouds whose radiative characteristics are not
2267     assumed to be correlated in the vertical.
2268     For a complete description of cloud/radiative interactions, see Section \ref{sec:fizhi:radcloud}.
2269     \[
2270     {\bf CLROSW} = CLRO_{RAS,Large Scale,SW}(\lambda, \phi, level )
2271     \]
2272     \\
2273    
2274     \noindent
2275     { \underline {RADSWT} Incident Shortwave radiation at the top of the atmosphere ($Watts/m^2$) }
2276     \[
2277     {\bf RADSWT} = {\frac{S_0}{R_a^2}} \cdot cos \phi_z
2278     \]
2279     \noindent
2280     where $S_0$, is the extra-terrestial solar contant,
2281     $R_a$ is the earth-sun distance in Astronomical Units,
2282     and $cos \phi_z$ is the cosine of the zenith angle.
2283     It should be noted that {\bf RADSWT}, as well as
2284     {\bf OSR} and {\bf OSRCLR},
2285     are calculated at the top of the atmosphere (p=0 mb). However, the
2286     {\bf OLR} and {\bf OLRCLR} diagnostics are currently
2287     calculated at $p= p_{top}$ (0.0 mb for the GCM).
2288     \\
2289    
2290     \noindent
2291     { \underline {EVAP} Surface Evaporation ($mm/day$) }
2292    
2293     \noindent
2294     The surface evaporation is a function of the gradient of moisture, the potential
2295     evapotranspiration fraction and the eddy exchange coefficient:
2296     \[
2297     {\bf EVAP} = \rho \beta K_{h} (q_{surface} - q_{Nrphys})
2298     \]
2299     where $\rho$ = the atmospheric density at the surface, $\beta$ is the fraction of
2300     the potential evapotranspiration actually evaporated ($\beta=1$ over oceans), $K_{h}$ is the
2301     turbulent eddy exchange coefficient for heat and moisture at the surface in $m/sec$ and
2302     $q{surface}$ and $q_{Nrphys}$ are the specific humidity at the surface (see diagnostic
2303     number 34) and at the bottom model level, respectively.
2304     \\
2305    
2306     \noindent
2307     { \underline {DUDT} Total Zonal U-Wind Tendency ($m/sec/day$) }
2308    
2309     \noindent
2310     {\bf DUDT} is the total time-tendency of the Zonal U-Wind due to Hydrodynamic, Diabatic,
2311     and Analysis forcing.
2312     \[
2313     {\bf DUDT} = \pp{u}{t}_{Dynamics} + \pp{u}{t}_{Moist} + \pp{u}{t}_{Turbulence} + \pp{u}{t}_{Analysis}
2314     \]
2315     \\
2316    
2317     \noindent
2318     { \underline {DVDT} Total Zonal V-Wind Tendency ($m/sec/day$) }
2319    
2320     \noindent
2321     {\bf DVDT} is the total time-tendency of the Meridional V-Wind due to Hydrodynamic, Diabatic,
2322     and Analysis forcing.
2323     \[
2324     {\bf DVDT} = \pp{v}{t}_{Dynamics} + \pp{v}{t}_{Moist} + \pp{v}{t}_{Turbulence} + \pp{v}{t}_{Analysis}
2325     \]
2326     \\
2327    
2328     \noindent
2329     { \underline {DTDT} Total Temperature Tendency ($deg/day$) }
2330    
2331     \noindent
2332     {\bf DTDT} is the total time-tendency of Temperature due to Hydrodynamic, Diabatic,
2333     and Analysis forcing.
2334     \begin{eqnarray*}
2335     {\bf DTDT} & = & \pp{T}{t}_{Dynamics} + \pp{T}{t}_{Moist Processes} + \pp{T}{t}_{Shortwave Radiation} \\
2336     & + & \pp{T}{t}_{Longwave Radiation} + \pp{T}{t}_{Turbulence} + \pp{T}{t}_{Analysis}
2337     \end{eqnarray*}
2338     \\
2339    
2340     \noindent
2341     { \underline {DQDT} Total Specific Humidity Tendency ($g/kg/day$) }
2342    
2343     \noindent
2344     {\bf DQDT} is the total time-tendency of Specific Humidity due to Hydrodynamic, Diabatic,
2345     and Analysis forcing.
2346     \[
2347     {\bf DQDT} = \pp{q}{t}_{Dynamics} + \pp{q}{t}_{Moist Processes}
2348     + \pp{q}{t}_{Turbulence} + \pp{q}{t}_{Analysis}
2349     \]
2350     \\
2351    
2352     \noindent
2353     { \underline {USTAR} Surface-Stress Velocity ($m/sec$) }
2354    
2355     \noindent
2356     The surface stress velocity, or the friction velocity, is the wind speed at
2357     the surface layer top impeded by the surface drag:
2358     \[
2359     {\bf USTAR} = C_uW_s \hspace{1cm}where: \hspace{.2cm}
2360 jmc 1.19 C_u = \frac{k}{\psi_m}
2361 molod 1.8 \]
2362    
2363     \noindent
2364     $C_u$ is the non-dimensional surface drag coefficient (see diagnostic
2365     number 10), and $W_s$ is the surface wind speed (see diagnostic number 28).
2366    
2367     \noindent
2368     { \underline {Z0} Surface Roughness Length ($m$) }
2369    
2370     \noindent
2371     Over the land surface, the surface roughness length is interpolated to the local
2372 molod 1.10 time from the monthly mean data of \cite{dorsell:89}. Over the ocean,
2373 molod 1.8 the roughness length is a function of the surface-stress velocity, $u_*$.
2374     \[
2375 jmc 1.19 {\bf Z0} = c_1u^3_* + c_2u^2_* + c_3u_* + c_4 + {c_5}{u_*}
2376 molod 1.8 \]
2377    
2378     \noindent
2379     where the constants are chosen to interpolate between the reciprocal relation of
2380 molod 1.10 \cite{kondo:75} for weak winds, and the piecewise linear relation of \cite{larpond:81}
2381 molod 1.8 for moderate to large winds.
2382     \\
2383    
2384     \noindent
2385     { \underline {FRQTRB} Frequency of Turbulence ($0-1$) }
2386    
2387     \noindent
2388     The fraction of time when turbulence is present is defined as the fraction of
2389     time when the turbulent kinetic energy exceeds some minimum value, defined here
2390     to be $0.005 \hspace{.1cm}m^2/sec^2$. When this criterion is met, a counter is
2391     incremented. The fraction over the averaging interval is reported.
2392     \\
2393    
2394     \noindent
2395     { \underline {PBL} Planetary Boundary Layer Depth ($mb$) }
2396    
2397     \noindent
2398     The depth of the PBL is defined by the turbulence parameterization to be the
2399     depth at which the turbulent kinetic energy reduces to ten percent of its surface
2400     value.
2401    
2402     \[
2403     {\bf PBL} = P_{PBL} - P_{surface}
2404     \]
2405    
2406     \noindent
2407     where $P_{PBL}$ is the pressure in $mb$ at which the turbulent kinetic energy
2408     reaches one tenth of its surface value, and $P_s$ is the surface pressure.
2409     \\
2410    
2411     \noindent
2412     { \underline {SWCLR} Clear sky Heating Rate due to Shortwave Radiation ($deg/day$) }
2413    
2414     \noindent
2415     The net Shortwave heating rate is calculated as the vertical divergence of the
2416     net solar radiative fluxes.
2417     The clear-sky and cloudy-sky shortwave fluxes are calculated separately.
2418     For the clear-sky case, the shortwave fluxes and heating rates are computed with
2419     both CLMO (maximum overlap cloud fraction) and
2420     CLRO (random overlap cloud fraction) set to zero (see Section \ref{sec:fizhi:radcloud}).
2421     The shortwave routine is then called a second time, for the cloudy-sky case, with the
2422     true time-averaged cloud fractions CLMO
2423     and CLRO being used. In all cases, a normalized incident shortwave flux is used as
2424     input at the top of the atmosphere.
2425    
2426     \noindent
2427     The heating rate due to Shortwave Radiation under clear skies is defined as:
2428     \[
2429 jmc 1.19 \pp{\rho c_p T}{t} = - \p{z} F(clear)_{SW}^{NET} \cdot {\rm RADSWT},
2430 molod 1.8 \]
2431     or
2432     \[
2433 jmc 1.19 {\bf SWCLR} = \frac{g}{c_p } \p{p} F(clear)_{SW}^{NET}\cdot {\rm RADSWT} .
2434 molod 1.8 \]
2435    
2436     \noindent
2437     where $g$ is the accelation due to gravity,
2438     $c_p$ is the heat capacity of air at constant pressure, RADSWT is the true incident
2439     shortwave radiation at the top of the atmosphere (See Diagnostic \#48), and
2440     \[
2441     F(clear)_{SW}^{Net} = F(clear)_{SW}^\uparrow - F(clear)_{SW}^\downarrow
2442     \]
2443     \\
2444    
2445     \noindent
2446     { \underline {OSR} Net upward Shortwave flux at the top of the model ($Watts/m^2$) }
2447     \[
2448     {\bf OSR} = F_{SW,top}^{NET}
2449     \]
2450     \noindent
2451     where top indicates the top of the first model layer used in the shortwave radiation
2452     routine.
2453     In the GCM, $p_{SW_{top}}$ = 0 mb.
2454     \\
2455    
2456     \noindent
2457     { \underline {OSRCLR} Net upward clearsky Shortwave flux at the top of the model ($Watts/m^2$) }
2458     \[
2459     {\bf OSRCLR} = F(clearsky)_{SW,top}^{NET}
2460     \]
2461     \noindent
2462     where top indicates the top of the first model layer used in the shortwave radiation
2463     routine.
2464     In the GCM, $p_{SW_{top}}$ = 0 mb.
2465     \\
2466    
2467    
2468     \noindent
2469     { \underline {CLDMAS} Convective Cloud Mass Flux ($kg/m^2$) }
2470    
2471     \noindent
2472     The amount of cloud mass moved per RAS timestep from all convective clouds is written:
2473     \[
2474     {\bf CLDMAS} = \eta m_B
2475     \]
2476     where $\eta$ is the entrainment, normalized by the cloud base mass flux, and $m_B$ is
2477     the cloud base mass flux. $m_B$ and $\eta$ are defined explicitly in Section \ref{sec:fizhi:mc}, the
2478     description of the convective parameterization.
2479     \\
2480    
2481    
2482    
2483     \noindent
2484     { \underline {UAVE} Time-Averaged Zonal U-Wind ($m/sec$) }
2485    
2486     \noindent
2487     The diagnostic {\bf UAVE} is simply the time-averaged Zonal U-Wind over
2488     the {\bf NUAVE} output frequency. This is contrasted to the instantaneous
2489     Zonal U-Wind which is archived on the Prognostic Output data stream.
2490     \[
2491     {\bf UAVE} = u(\lambda, \phi, level , t)
2492     \]
2493     \\
2494     Note, {\bf UAVE} is computed and stored on the staggered C-grid.
2495     \\
2496    
2497     \noindent
2498     { \underline {VAVE} Time-Averaged Meridional V-Wind ($m/sec$) }
2499    
2500     \noindent
2501     The diagnostic {\bf VAVE} is simply the time-averaged Meridional V-Wind over
2502     the {\bf NVAVE} output frequency. This is contrasted to the instantaneous
2503     Meridional V-Wind which is archived on the Prognostic Output data stream.
2504     \[
2505     {\bf VAVE} = v(\lambda, \phi, level , t)
2506     \]
2507     \\
2508     Note, {\bf VAVE} is computed and stored on the staggered C-grid.
2509     \\
2510    
2511     \noindent
2512     { \underline {TAVE} Time-Averaged Temperature ($Kelvin$) }
2513    
2514     \noindent
2515     The diagnostic {\bf TAVE} is simply the time-averaged Temperature over
2516     the {\bf NTAVE} output frequency. This is contrasted to the instantaneous
2517     Temperature which is archived on the Prognostic Output data stream.
2518     \[
2519     {\bf TAVE} = T(\lambda, \phi, level , t)
2520     \]
2521     \\
2522    
2523     \noindent
2524     { \underline {QAVE} Time-Averaged Specific Humidity ($g/kg$) }
2525    
2526     \noindent
2527     The diagnostic {\bf QAVE} is simply the time-averaged Specific Humidity over
2528     the {\bf NQAVE} output frequency. This is contrasted to the instantaneous
2529     Specific Humidity which is archived on the Prognostic Output data stream.
2530     \[
2531     {\bf QAVE} = q(\lambda, \phi, level , t)
2532     \]
2533     \\
2534    
2535     \noindent
2536     { \underline {PAVE} Time-Averaged Surface Pressure - PTOP ($mb$) }
2537    
2538     \noindent
2539     The diagnostic {\bf PAVE} is simply the time-averaged Surface Pressure - PTOP over
2540     the {\bf NPAVE} output frequency. This is contrasted to the instantaneous
2541     Surface Pressure - PTOP which is archived on the Prognostic Output data stream.
2542     \begin{eqnarray*}
2543     {\bf PAVE} & = & \pi(\lambda, \phi, level , t) \\
2544     & = & p_s(\lambda, \phi, level , t) - p_T
2545     \end{eqnarray*}
2546     \\
2547    
2548    
2549     \noindent
2550     { \underline {QQAVE} Time-Averaged Turbulent Kinetic Energy $(m/sec)^2$ }
2551    
2552     \noindent
2553     The diagnostic {\bf QQAVE} is simply the time-averaged prognostic Turbulent Kinetic Energy
2554     produced by the GCM Turbulence parameterization over
2555     the {\bf NQQAVE} output frequency. This is contrasted to the instantaneous
2556     Turbulent Kinetic Energy which is archived on the Prognostic Output data stream.
2557     \[
2558     {\bf QQAVE} = qq(\lambda, \phi, level , t)
2559     \]
2560     \\
2561     Note, {\bf QQAVE} is computed and stored at the ``mass-point'' locations on the staggered C-grid.
2562     \\
2563    
2564     \noindent
2565     { \underline {SWGCLR} Net downward clearsky Shortwave flux at the surface ($Watts/m^2$) }
2566    
2567     \noindent
2568     \begin{eqnarray*}
2569     {\bf SWGCLR} & = & F(clearsky)_{SW,Nrphys+1}^{Net} \\
2570     & = & F(clearsky)_{SW,Nrphys+1}^\downarrow - F(clearsky)_{SW,Nrphys+1}^\uparrow
2571     \end{eqnarray*}
2572     \noindent
2573     \\
2574     where Nrphys+1 indicates the lowest model edge-level, or $p = p_{surf}$.
2575     $F(clearsky){SW}^\downarrow$ is
2576     the downward clearsky Shortwave flux and $F(clearsky)_{SW}^\uparrow$ is
2577     the upward clearsky Shortwave flux.
2578     \\
2579    
2580     \noindent
2581     { \underline {DIABU} Total Diabatic Zonal U-Wind Tendency ($m/sec/day$) }
2582    
2583     \noindent
2584     {\bf DIABU} is the total time-tendency of the Zonal U-Wind due to Diabatic processes
2585     and the Analysis forcing.
2586     \[
2587     {\bf DIABU} = \pp{u}{t}_{Moist} + \pp{u}{t}_{Turbulence} + \pp{u}{t}_{Analysis}
2588     \]
2589     \\
2590    
2591     \noindent
2592     { \underline {DIABV} Total Diabatic Meridional V-Wind Tendency ($m/sec/day$) }
2593    
2594     \noindent
2595     {\bf DIABV} is the total time-tendency of the Meridional V-Wind due to Diabatic processes
2596     and the Analysis forcing.
2597     \[
2598     {\bf DIABV} = \pp{v}{t}_{Moist} + \pp{v}{t}_{Turbulence} + \pp{v}{t}_{Analysis}
2599     \]
2600     \\
2601    
2602     \noindent
2603     { \underline {DIABT} Total Diabatic Temperature Tendency ($deg/day$) }
2604    
2605     \noindent
2606     {\bf DIABT} is the total time-tendency of Temperature due to Diabatic processes
2607     and the Analysis forcing.
2608     \begin{eqnarray*}
2609     {\bf DIABT} & = & \pp{T}{t}_{Moist Processes} + \pp{T}{t}_{Shortwave Radiation} \\
2610     & + & \pp{T}{t}_{Longwave Radiation} + \pp{T}{t}_{Turbulence} + \pp{T}{t}_{Analysis}
2611     \end{eqnarray*}
2612     \\
2613     If we define the time-tendency of Temperature due to Diabatic processes as
2614     \begin{eqnarray*}
2615     \pp{T}{t}_{Diabatic} & = & \pp{T}{t}_{Moist Processes} + \pp{T}{t}_{Shortwave Radiation} \\
2616     & + & \pp{T}{t}_{Longwave Radiation} + \pp{T}{t}_{Turbulence}
2617     \end{eqnarray*}
2618     then, since there are no surface pressure changes due to Diabatic processes, we may write
2619     \[
2620 jmc 1.19 \pp{T}{t}_{Diabatic} = \frac{p^\kappa}{\pi}\pp{\pi \theta}{t}_{Diabatic}
2621 molod 1.8 \]
2622     where $\theta = T/p^\kappa$. Thus, {\bf DIABT} may be written as
2623     \[
2624 jmc 1.19 {\bf DIABT} = \frac{p^\kappa}{\pi} \left( \pp{\pi \theta}{t}_{Diabatic} + \pp{\pi \theta}{t}_{Analysis} \right)
2625 molod 1.8 \]
2626     \\
2627    
2628     \noindent
2629     { \underline {DIABQ} Total Diabatic Specific Humidity Tendency ($g/kg/day$) }
2630    
2631     \noindent
2632     {\bf DIABQ} is the total time-tendency of Specific Humidity due to Diabatic processes
2633     and the Analysis forcing.
2634     \[
2635     {\bf DIABQ} = \pp{q}{t}_{Moist Processes} + \pp{q}{t}_{Turbulence} + \pp{q}{t}_{Analysis}
2636     \]
2637     If we define the time-tendency of Specific Humidity due to Diabatic processes as
2638     \[
2639     \pp{q}{t}_{Diabatic} = \pp{q}{t}_{Moist Processes} + \pp{q}{t}_{Turbulence}
2640     \]
2641     then, since there are no surface pressure changes due to Diabatic processes, we may write
2642     \[
2643 jmc 1.19 \pp{q}{t}_{Diabatic} = \frac{1}{\pi}\pp{\pi q}{t}_{Diabatic}
2644 molod 1.8 \]
2645     Thus, {\bf DIABQ} may be written as
2646     \[
2647 jmc 1.19 {\bf DIABQ} = \frac{1}{\pi} \left( \pp{\pi q}{t}_{Diabatic} + \pp{\pi q}{t}_{Analysis} \right)
2648 molod 1.8 \]
2649     \\
2650    
2651     \noindent
2652     { \underline {VINTUQ} Vertically Integrated Moisture Flux ($m/sec \cdot g/kg$) }
2653    
2654     \noindent
2655     The vertically integrated moisture flux due to the zonal u-wind is obtained by integrating
2656     $u q$ over the depth of the atmosphere at each model timestep,
2657     and dividing by the total mass of the column.
2658     \[
2659     {\bf VINTUQ} = \frac{ \int_{surf}^{top} u q \rho dz } { \int_{surf}^{top} \rho dz }
2660     \]
2661 jmc 1.19 Using $\rho \delta z = -\frac{\delta p}{g} = - \frac{1}{g} \delta p$, we have
2662 molod 1.8 \[
2663     {\bf VINTUQ} = { \int_0^1 u q dp }
2664     \]
2665     \\
2666    
2667    
2668     \noindent
2669     { \underline {VINTVQ} Vertically Integrated Moisture Flux ($m/sec \cdot g/kg$) }
2670    
2671     \noindent
2672     The vertically integrated moisture flux due to the meridional v-wind is obtained by integrating
2673     $v q$ over the depth of the atmosphere at each model timestep,
2674     and dividing by the total mass of the column.
2675     \[
2676     {\bf VINTVQ} = \frac{ \int_{surf}^{top} v q \rho dz } { \int_{surf}^{top} \rho dz }
2677     \]
2678 jmc 1.19 Using $\rho \delta z = -\frac{\delta p}{g} = - \frac{1}{g} \delta p$, we have
2679 molod 1.8 \[
2680     {\bf VINTVQ} = { \int_0^1 v q dp }
2681     \]
2682     \\
2683    
2684    
2685     \noindent
2686     { \underline {VINTUT} Vertically Integrated Heat Flux ($m/sec \cdot deg$) }
2687    
2688     \noindent
2689     The vertically integrated heat flux due to the zonal u-wind is obtained by integrating
2690     $u T$ over the depth of the atmosphere at each model timestep,
2691     and dividing by the total mass of the column.
2692     \[
2693     {\bf VINTUT} = \frac{ \int_{surf}^{top} u T \rho dz } { \int_{surf}^{top} \rho dz }
2694     \]
2695     Or,
2696     \[
2697     {\bf VINTUT} = { \int_0^1 u T dp }
2698     \]
2699     \\
2700    
2701     \noindent
2702     { \underline {VINTVT} Vertically Integrated Heat Flux ($m/sec \cdot deg$) }
2703    
2704     \noindent
2705     The vertically integrated heat flux due to the meridional v-wind is obtained by integrating
2706     $v T$ over the depth of the atmosphere at each model timestep,
2707     and dividing by the total mass of the column.
2708     \[
2709     {\bf VINTVT} = \frac{ \int_{surf}^{top} v T \rho dz } { \int_{surf}^{top} \rho dz }
2710     \]
2711 jmc 1.19 Using $\rho \delta z = -\frac{\delta p}{g} $, we have
2712 molod 1.8 \[
2713     {\bf VINTVT} = { \int_0^1 v T dp }
2714     \]
2715     \\
2716    
2717     \noindent
2718     { \underline {CLDFRC} Total 2-Dimensional Cloud Fracton ($0-1$) }
2719    
2720     If we define the
2721     time-averaged random and maximum overlapped cloudiness as CLRO and
2722     CLMO respectively, then the probability of clear sky associated
2723     with random overlapped clouds at any level is (1-CLRO) while the probability of
2724     clear sky associated with maximum overlapped clouds at any level is (1-CLMO).
2725     The total clear sky probability is given by (1-CLRO)*(1-CLMO), thus
2726     the total cloud fraction at each level may be obtained by
2727     1-(1-CLRO)*(1-CLMO).
2728    
2729     At any given level, we may define the clear line-of-site probability by
2730     appropriately accounting for the maximum and random overlap
2731     cloudiness. The clear line-of-site probability is defined to be
2732     equal to the product of the clear line-of-site probabilities
2733     associated with random and maximum overlap cloudiness. The clear
2734     line-of-site probability $C(p,p^{\prime})$ associated with maximum overlap clouds,
2735     from the current pressure $p$
2736     to the model top pressure, $p^{\prime} = p_{top}$, or the model surface pressure, $p^{\prime} = p_{surf}$,
2737     is simply 1.0 minus the largest maximum overlap cloud value along the
2738     line-of-site, ie.
2739    
2740     $$1-MAX_p^{p^{\prime}} \left( CLMO_p \right)$$
2741    
2742     Thus, even in the time-averaged sense it is assumed that the
2743     maximum overlap clouds are correlated in the vertical. The clear
2744     line-of-site probability associated with random overlap clouds is
2745     defined to be the product of the clear sky probabilities at each
2746     level along the line-of-site, ie.
2747    
2748     $$\prod_{p}^{p^{\prime}} \left( 1-CLRO_p \right)$$
2749    
2750     The total cloud fraction at a given level associated with a line-
2751     of-site calculation is given by
2752    
2753     $$1-\left( 1-MAX_p^{p^{\prime}} \left[ CLMO_p \right] \right)
2754     \prod_p^{p^{\prime}} \left( 1-CLRO_p \right)$$
2755    
2756    
2757     \noindent
2758     The 2-dimensional net cloud fraction as seen from the top of the
2759     atmosphere is given by
2760     \[
2761     {\bf CLDFRC} = 1-\left( 1-MAX_{l=l_1}^{Nrphys} \left[ CLMO_l \right] \right)
2762     \prod_{l=l_1}^{Nrphys} \left( 1-CLRO_l \right)
2763     \]
2764     \\
2765     For a complete description of cloud/radiative interactions, see Section \ref{sec:fizhi:radcloud}.
2766    
2767    
2768     \noindent
2769     { \underline {QINT} Total Precipitable Water ($gm/cm^2$) }
2770    
2771     \noindent
2772     The Total Precipitable Water is defined as the vertical integral of the specific humidity,
2773     given by:
2774     \begin{eqnarray*}
2775     {\bf QINT} & = & \int_{surf}^{top} \rho q dz \\
2776 jmc 1.19 & = & \frac{\pi}{g} \int_0^1 q dp
2777 molod 1.8 \end{eqnarray*}
2778     where we have used the hydrostatic relation
2779 jmc 1.19 $\rho \delta z = -\frac{\delta p}{g} $.
2780 molod 1.8 \\
2781    
2782    
2783     \noindent
2784     { \underline {U2M} Zonal U-Wind at 2 Meter Depth ($m/sec$) }
2785    
2786     \noindent
2787     The u-wind at the 2-meter depth is determined from the similarity theory:
2788     \[
2789 jmc 1.19 {\bf U2M} = \frac{u_*}{k} \psi_{m_{2m}} \frac{u_{sl}}{W_s} =
2790     \frac{ \psi_{m_{2m}} }{ \psi_{m_{sl}} }u_{sl}
2791 molod 1.8 \]
2792    
2793     \noindent
2794     where $\psi_m(2m)$ is the non-dimensional wind shear at two meters, and the subscript
2795     $sl$ refers to the height of the top of the surface layer. If the roughness height
2796     is above two meters, ${\bf U2M}$ is undefined.
2797     \\
2798    
2799     \noindent
2800     { \underline {V2M} Meridional V-Wind at 2 Meter Depth ($m/sec$) }
2801    
2802     \noindent
2803     The v-wind at the 2-meter depth is a determined from the similarity theory:
2804     \[
2805 jmc 1.19 {\bf V2M} = \frac{u_*}{k} \psi_{m_{2m}} \frac{v_{sl}}{W_s} =
2806     \frac{ \psi_{m_{2m}} }{ \psi_{m_{sl}} }v_{sl}
2807 molod 1.8 \]
2808    
2809     \noindent
2810     where $\psi_m(2m)$ is the non-dimensional wind shear at two meters, and the subscript
2811     $sl$ refers to the height of the top of the surface layer. If the roughness height
2812     is above two meters, ${\bf V2M}$ is undefined.
2813     \\
2814    
2815     \noindent
2816     { \underline {T2M} Temperature at 2 Meter Depth ($deg \hspace{.1cm} K$) }
2817    
2818     \noindent
2819     The temperature at the 2-meter depth is a determined from the similarity theory:
2820     \[
2821 jmc 1.19 {\bf T2M} = P^{\kappa} (\frac{\theta*}{k} ({\psi_{h_{2m}}+\psi_g}) + \theta_{surf} ) =
2822     P^{\kappa}(\theta_{surf} + \frac{ \psi_{h_{2m}}+\psi_g }{ \psi_{h_{sl}}+\psi_g }
2823     (\theta_{sl} - \theta_{surf}) )
2824 molod 1.8 \]
2825     where:
2826     \[
2827 jmc 1.19 \theta_* = - \frac{ (\overline{w^{\prime}\theta^{\prime}}) }{ u_* }
2828 molod 1.8 \]
2829    
2830     \noindent
2831     where $\psi_h(2m)$ is the non-dimensional temperature gradient at two meters, $\psi_g$ is
2832     the non-dimensional temperature gradient in the viscous sublayer, and the subscript
2833     $sl$ refers to the height of the top of the surface layer. If the roughness height
2834     is above two meters, ${\bf T2M}$ is undefined.
2835     \\
2836    
2837     \noindent
2838     { \underline {Q2M} Specific Humidity at 2 Meter Depth ($g/kg$) }
2839    
2840     \noindent
2841     The specific humidity at the 2-meter depth is determined from the similarity theory:
2842     \[
2843 jmc 1.19 {\bf Q2M} = P^{\kappa} \frac({q_*}{k} ({\psi_{h_{2m}}+\psi_g}) + q_{surf} ) =
2844     P^{\kappa}(q_{surf} + \frac{ \psi_{h_{2m}}+\psi_g }{ \psi_{h_{sl}}+\psi_g }
2845 molod 1.8 (q_{sl} - q_{surf}))
2846     \]
2847     where:
2848     \[
2849 jmc 1.19 q_* = - \frac{ (\overline{w^{\prime}q^{\prime}}) }{ u_* }
2850 molod 1.8 \]
2851    
2852     \noindent
2853     where $\psi_h(2m)$ is the non-dimensional temperature gradient at two meters, $\psi_g$ is
2854     the non-dimensional temperature gradient in the viscous sublayer, and the subscript
2855     $sl$ refers to the height of the top of the surface layer. If the roughness height
2856     is above two meters, ${\bf Q2M}$ is undefined.
2857     \\
2858    
2859     \noindent
2860     { \underline {U10M} Zonal U-Wind at 10 Meter Depth ($m/sec$) }
2861    
2862     \noindent
2863     The u-wind at the 10-meter depth is an interpolation between the surface wind
2864     and the model lowest level wind using the ratio of the non-dimensional wind shear
2865     at the two levels:
2866     \[
2867 jmc 1.19 {\bf U10M} = \frac{u_*}{k} \psi_{m_{10m}} \frac{u_{sl}}{W_s} =
2868     \frac{ \psi_{m_{10m}} }{ \psi_{m_{sl}} }u_{sl}
2869 molod 1.8 \]
2870    
2871     \noindent
2872     where $\psi_m(10m)$ is the non-dimensional wind shear at ten meters, and the subscript
2873     $sl$ refers to the height of the top of the surface layer.
2874     \\
2875    
2876     \noindent
2877     { \underline {V10M} Meridional V-Wind at 10 Meter Depth ($m/sec$) }
2878    
2879     \noindent
2880     The v-wind at the 10-meter depth is an interpolation between the surface wind
2881     and the model lowest level wind using the ratio of the non-dimensional wind shear
2882     at the two levels:
2883     \[
2884 jmc 1.19 {\bf V10M} = \frac{u_*}{k} \psi_{m_{10m}} \frac{v_{sl}}{W_s} =
2885     \frac{ \psi_{m_{10m}} }{ \psi_{m_{sl}} }v_{sl}
2886 molod 1.8 \]
2887    
2888     \noindent
2889     where $\psi_m(10m)$ is the non-dimensional wind shear at ten meters, and the subscript
2890     $sl$ refers to the height of the top of the surface layer.
2891     \\
2892    
2893     \noindent
2894     { \underline {T10M} Temperature at 10 Meter Depth ($deg \hspace{.1cm} K$) }
2895    
2896     \noindent
2897     The temperature at the 10-meter depth is an interpolation between the surface potential
2898     temperature and the model lowest level potential temperature using the ratio of the
2899     non-dimensional temperature gradient at the two levels:
2900     \[
2901 jmc 1.19 {\bf T10M} = P^{\kappa} (\frac{\theta*}{k} ({\psi_{h_{10m}}+\psi_g}) + \theta_{surf} ) =
2902     P^{\kappa}(\theta_{surf} + \frac{\psi_{h_{10m}}+\psi_g}{\psi_{h_{sl}}+\psi_g}
2903 molod 1.8 (\theta_{sl} - \theta_{surf}))
2904     \]
2905     where:
2906     \[
2907 jmc 1.19 \theta_* = - \frac{ (\overline{w^{\prime}\theta^{\prime}}) }{ u_* }
2908 molod 1.8 \]
2909    
2910     \noindent
2911     where $\psi_h(10m)$ is the non-dimensional temperature gradient at two meters, $\psi_g$ is
2912     the non-dimensional temperature gradient in the viscous sublayer, and the subscript
2913     $sl$ refers to the height of the top of the surface layer.
2914     \\
2915    
2916     \noindent
2917     { \underline {Q10M} Specific Humidity at 10 Meter Depth ($g/kg$) }
2918    
2919     \noindent
2920     The specific humidity at the 10-meter depth is an interpolation between the surface specific
2921     humidity and the model lowest level specific humidity using the ratio of the
2922     non-dimensional temperature gradient at the two levels:
2923     \[
2924 jmc 1.19 {\bf Q10M} = P^{\kappa} (\frac{q_*}{k} ({\psi_{h_{10m}}+\psi_g}) + q_{surf} ) =
2925     P^{\kappa}(q_{surf} + \frac{\psi_{h_{10m}}+\psi_g}{\psi_{h_{sl}}+\psi_g}
2926 molod 1.8 (q_{sl} - q_{surf}))
2927     \]
2928     where:
2929     \[
2930 jmc 1.19 q_* = - \frac{ (\overline{w^{\prime}q^{\prime}}) }{ u_* }
2931 molod 1.8 \]
2932    
2933     \noindent
2934     where $\psi_h(10m)$ is the non-dimensional temperature gradient at two meters, $\psi_g$ is
2935     the non-dimensional temperature gradient in the viscous sublayer, and the subscript
2936     $sl$ refers to the height of the top of the surface layer.
2937     \\
2938    
2939     \noindent
2940     { \underline {DTRAIN} Cloud Detrainment Mass Flux ($kg/m^2$) }
2941    
2942     The amount of cloud mass moved per RAS timestep at the cloud detrainment level is written:
2943     \[
2944     {\bf DTRAIN} = \eta_{r_D}m_B
2945     \]
2946     \noindent
2947     where $r_D$ is the detrainment level,
2948     $m_B$ is the cloud base mass flux, and $\eta$
2949     is the entrainment, defined in Section \ref{sec:fizhi:mc}.
2950     \\
2951    
2952     \noindent
2953     { \underline {QFILL} Filling of negative Specific Humidity ($g/kg/day$) }
2954    
2955     \noindent
2956     Due to computational errors associated with the numerical scheme used for
2957     the advection of moisture, negative values of specific humidity may be generated. The
2958     specific humidity is checked for negative values after every dynamics timestep. If negative
2959     values have been produced, a filling algorithm is invoked which redistributes moisture from
2960     below. Diagnostic {\bf QFILL} is equal to the net filling needed
2961     to eliminate negative specific humidity, scaled to a per-day rate:
2962     \[
2963     {\bf QFILL} = q^{n+1}_{final} - q^{n+1}_{initial}
2964     \]
2965     where
2966     \[
2967     q^{n+1} = (\pi q)^{n+1} / \pi^{n+1}
2968     \]
2969    
2970    
2971 molod 1.9 \subsubsection{Key subroutines, parameters and files}
2972 molod 1.6
2973 molod 1.9 \subsubsection{Dos and donts}
2974 molod 1.6
2975 molod 1.9 \subsubsection{Fizhi Reference}
2976 molod 1.17
2977     \subsubsection{Experiments and tutorials that use fizhi}
2978     \label{sec:pkg:fizhi:experiments}
2979    
2980     \begin{itemize}
2981     \item{Global atmosphere experiment with realistic SST and topography in fizhi-cs-32x32x10 verification directory. }
2982     \item{Global atmosphere aqua planet experiment in fizhi-cs-aqualev20 verification directory. }
2983     \end{itemize}
2984    

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