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Subsections


6.4.3 Fizhi: High-end Atmospheric Physics

6.4.3.1 Introduction

The fizhi (high-end atmospheric physics) package includes a collection of state-of-the-art physical parameterizations for atmospheric radiation, cumulus convection, atmospheric boundary layer turbulence, and land surface processes. The collection of atmospheric physics parameterizations were originally used together as part of the GEOS-3 (Goddard Earth Observing System-3) GCM developed at the NASA/Goddard Global Modelling and Assimilation Office (GMAO).

6.4.3.2 Equations

Moist Convective Processes:


6.4.3.2.1 Sub-grid and Large-scale Convection

Sub-grid scale cumulus convection is parameterized using the Relaxed Arakawa Schubert (RAS) scheme of Moorthi and Suarez [1992], which is a linearized Arakawa Schubert type scheme. RAS predicts the mass flux from an ensemble of clouds. Each subensemble is identified by its entrainment rate and level of neutral bouyancy which are determined by the grid-scale properties.

The thermodynamic variables that are used in RAS to describe the grid scale vertical profile are the dry static energy, $ s=c_pT +gz$, and the moist static energy, $ h=c_p T + gz + Lq$. The conceptual model behind RAS depicts each subensemble as a rising plume cloud, entraining mass from the environment during ascent, and detraining all cloud air at the level of neutral buoyancy. RAS assumes that the normalized cloud mass flux, $ \eta $, normalized by the cloud base mass flux, is a linear function of height, expressed as:

$\displaystyle {\partial \eta(z) \over {\partial z}} = \lambda \hspace{0.4cm}or\... ...\eta(P^{\kappa}) \over {\partial P^{\kappa}}} = -{c_p \over {g}}\theta\lambda $

where we have used the hydrostatic equation written in the form:

$\displaystyle {\partial z \over {\partial P^{\kappa}}} = -{c_p \over {g}}\theta $

The entrainment parameter, $ \lambda $, characterizes a particular subensemble based on its detrainment level, and is obtained by assuming that the level of detrainment is the level of neutral buoyancy, ie., the level at which the moist static energy of the cloud, $ h_c$, is equal to the saturation moist static energy of the environment, $ h^*$. Following Moorthi and Suarez [1992], $ \lambda $ may be written as

$\displaystyle \lambda = { {h_B - h^*_D} \over { {c_p \over g} {\int_{P_D}^{P_B}\theta(h^*_D-h)dP^{\kappa}}} } , $

where the subscript $ B$ refers to cloud base, and the subscript $ D$ refers to the detrainment level.

The convective instability is measured in terms of the cloud work function $ A$, defined as the rate of change of cumulus kinetic energy. The cloud work function is related to the buoyancy, or the difference between the moist static energy in the cloud and in the environment:

$\displaystyle A = \int_{P_D}^{P_B} { {\eta \over {1 + \gamma} } \left[ {{h_c-h^*} \over {P^{\kappa}}} \right] dP^{\kappa}} $

where $ \gamma$ is $ {L \over {c_p}}{\partial q^* \over {\partial T}}$ obtained from the Claussius Clapeyron equation, and the subscript $ c$ refers to the value inside the cloud.

To determine the cloud base mass flux, the rate of change of $ A$ in time due to dissipation by the clouds is assumed to approximately balance the rate of change of $ A$ due to the generation by the large scale. This is the quasi-equilibrium assumption, and results in an expression for $ m_B$:

$\displaystyle m_B = {{- \left.{dA \over dt} \right\vert _{ls}} \over K} $

where $ K$ is the cloud kernel, defined as the rate of change of the cloud work function per unit cloud base mass flux, and is currently obtained by analytically differentiating the expression for $ A$ in time. The rate of change of $ A$ due to the generation by the large scale can be written as the difference between the current $ A(t+\Delta t)$ and its equillibrated value after the previous convective time step $ A(t)$, divided by the time step. $ A(t)$ is approximated as some critical $ A_{crit}$, computed by Lord (1982) from $ in situ$ observations.

The predicted convective mass fluxes are used to solve grid-scale temperature and moisture budget equations to determine the impact of convection on the large scale fields of temperature (through latent heating and compensating subsidence) and moisture (through precipitation and detrainment):

$\displaystyle \left.{{\partial \theta \over {\partial t}}}\right\vert _{c} = \alpha { m_B \over {c_p P^{\kappa}}} \eta {\partial s \over {\partial p}} $

and

$\displaystyle \left.{{\partial q \over {\partial t}}}\right\vert _{c} = \alpha ... ...er {L}} \eta ({\partial h \over {\partial p}}-{\partial s \over {\partial p}}) $

where $ \theta = {T \over P^{\kappa}}$, $ P = (p/p_0)$, and $ \alpha $ is the relaxation parameter.

As an approximation to a full interaction between the different allowable subensembles, many clouds are simulated frequently, each modifying the large scale environment some fraction $ \alpha $ of the total adjustment. The parameterization thereby ``relaxes'' the large scale environment towards equillibrium.

In addition to the RAS cumulus convection scheme, the fizhi package employs a Kessler-type scheme for the re-evaporation of falling rain (Sud and Molod [1988]), which correspondingly adjusts the temperature assuming $ h$ is conserved. RAS in its current formulation assumes that all cloud water is deposited into the detrainment level as rain. All of the rain is available for re-evaporation, which begins in the level below detrainment. The scheme accounts for some microphysics such as the rainfall intensity, the drop size distribution, as well as the temperature, pressure and relative humidity of the surrounding air. The fraction of the moisture deficit in any model layer into which the rain may re-evaporate is controlled by a free parameter, which allows for a relatively efficient re-evaporation of liquid precipitate and larger rainout for frozen precipitation.

Due to the increased vertical resolution near the surface, the lowest model layers are averaged to provide a 50 mb thick sub-cloud layer for RAS. Each time RAS is invoked (every ten simulated minutes), a number of randomly chosen subensembles are checked for the possibility of convection, from just above cloud base to 10 mb.

Supersaturation or large-scale precipitation is initiated in the fizhi package whenever the relative humidity in any grid-box exceeds a critical value, currently 100 %. The large-scale precipitation re-evaporates during descent to partially saturate lower layers in a process identical to the re-evaporation of convective rain.


6.4.3.2.2 Cloud Formation

Convective and large-scale cloud fractons which are used for cloud-radiative interactions are determined diagnostically as part of the cumulus and large-scale parameterizations. Convective cloud fractions produced by RAS are proportional to the detrained liquid water amount given by

$\displaystyle F_{RAS} = \min\left[ {l_{RAS}\over l_c}, 1.0 \right] $

where $ l_c$ is an assigned critical value equal to $ 1.25$ g/kg. A memory is associated with convective clouds defined by:

$\displaystyle F_{RAS}^n = \min\left[ F_{RAS} + (1-{\Delta t_{RAS}\over\tau})F_{RAS}^{n-1}, 1.0 \right] $

where $ F_{RAS}$ is the instantanious cloud fraction and $ F_{RAS}^{n-1}$ is the cloud fraction from the previous RAS timestep. The memory coefficient is computed using a RAS cloud timescale, $ \tau$, equal to 1 hour. RAS cloud fractions are cleared when they fall below 5 %.

Large-scale cloudiness is defined, following Slingo and Ritter (1985), as a function of relative humidity:

$\displaystyle F_{LS} = \min\left[ { \left( {RH-RH_c \over 1-RH_c} \right) }^2, 1.0 \right] $

where


$\displaystyle RH_c$ $\displaystyle =$ $\displaystyle 1-s(1-s)(2-\sqrt{3}+2\sqrt{3} \, s)r$  
$\displaystyle s$ $\displaystyle =$ $\displaystyle p/p_{surf}$  
$\displaystyle r$ $\displaystyle =$ $\displaystyle \left( {1.0-RH_{min} \over \alpha} \right)$  
$\displaystyle RH_{min}$ $\displaystyle =$ $\displaystyle 0.75$  
$\displaystyle \alpha$ $\displaystyle =$ $\displaystyle 0.573285 .$  

These cloud fractions are suppressed, however, in regions where the convective sub-cloud layer is conditionally unstable. The functional form of $ RH_c$ is shown in Figure (6.9).

\begin{figure*} % latex2html id marker 31551 \vspace{0.4in} \centerline{ \epsf... ...ive Humidity for Clouds.] {Critical Relative Humidity for Clouds.} \end{figure*}

The total cloud fraction in a grid box is determined by the larger of the two cloud fractions:

$\displaystyle F_{CLD} = \max \left[ F_{RAS},F_{LS} \right] . $

Finally, cloud fractions are time-averaged between calls to the radiation packages.

Radiation:

The parameterization of radiative heating in the fizhi package includes effects from both shortwave and longwave processes. Radiative fluxes are calculated at each model edge-level in both up and down directions. The heating rates/cooling rates are then obtained from the vertical divergence of the net radiative fluxes.

The net flux is

$\displaystyle F = F^\uparrow - F^\downarrow $

where $ F$ is the net flux, $ F^\uparrow$ is the upward flux and $ F^\downarrow$ is the downward flux.

The heating rate due to the divergence of the radiative flux is given by

$\displaystyle {\partial \rho c_p T \over {\partial t}} = - {\partial F \over {\partial z}} $

or

$\displaystyle {\partial T \over {\partial t}} = \frac{g}{c_p \pi} {\partial F \over {\partial \sigma}} $

where $ g$ is the accelation due to gravity and $ c_p$ is the heat capacity of air at constant pressure.

The time tendency for Longwave Radiation is updated every 3 hours. The time tendency for Shortwave Radiation is updated once every three hours assuming a normalized incident solar radiation, and subsequently modified at every model time step by the true incident radiation. The solar constant value used in the package is equal to 1365 $ W/m^2$ and a $ CO_2 $ mixing ratio of 330 ppm. For the ozone mixing ratio, monthly mean zonally averaged climatological values specified as a function of latitude and height (Rosenfield et al. [1987]) are linearly interpolated to the current time.

6.4.3.2.3 Shortwave Radiation

The shortwave radiation package used in the package computes solar radiative heating due to the absoption by water vapor, ozone, carbon dioxide, oxygen, clouds, and aerosols and due to the scattering by clouds, aerosols, and gases. The shortwave radiative processes are described by Chou [1990,1992]. This shortwave package uses the Delta-Eddington approximation to compute the bulk scattering properties of a single layer following King and Harshvardhan (JAS, 1986). The transmittance and reflectance of diffuse radiation follow the procedures of Sagan and Pollock (JGR, 1967) and Lacis and Hansen [1974].

Highly accurate heating rate calculations are obtained through the use of an optimal grouping strategy of spectral bands. By grouping the UV and visible regions as indicated in Table 6.10, the Rayleigh scattering and the ozone absorption of solar radiation can be accurately computed in the ultraviolet region and the photosynthetically active radiation (PAR) region. The computation of solar flux in the infrared region is performed with a broadband parameterization using the spectrum regions shown in Table 6.11. The solar radiation algorithm used in the fizhi package can be applied not only for climate studies but also for studies on the photolysis in the upper atmosphere and the photosynthesis in the biosphere.


Table 6.10: UV and Visible Spectral Regions used in shortwave radiation package.
UV and Visible Spectral Regions


Region Band Wavelength (micron)
UV-C 1. .175 - .225
  2. .225 - .245
    .260 - .280
  3. .245 - .260
UV-B 4. .280 - .295
  5. .295 - .310
  6. .310 - .320
UV-A 7. .320 - .400
PAR 8. .400 - .700



Table 6.11: Infrared Spectral Regions used in shortwave radiation package.
Infrared Spectral Regions


Band Wavenumber(cm$ ^{-1}$) Wavelength (micron)
1 1000-4400 2.27-10.0
2 4400-8200 1.22-2.27
3 8200-14300 0.70-1.22


Within the shortwave radiation package, both ice and liquid cloud particles are allowed to co-exist in any of the model layers. Two sets of cloud parameters are used, one for ice paticles and the other for liquid particles. Cloud parameters are defined as the cloud optical thickness and the effective cloud particle size. In the fizhi package, the effective radius for water droplets is given as 10 microns, while 65 microns is used for ice particles. The absorption due to aerosols is currently set to zero.

To simplify calculations in a cloudy atmosphere, clouds are grouped into low ($ p>700$ mb), middle (700 mb $ \ge p > 400$ mb), and high ($ p < 400$ mb) cloud regions. Within each of the three regions, clouds are assumed maximally overlapped, and the cloud cover of the group is the maximum cloud cover of all the layers in the group. The optical thickness of a given layer is then scaled for both the direct (as a function of the solar zenith angle) and diffuse beam radiation so that the grouped layer reflectance is the same as the original reflectance. The solar flux is computed for each of eight cloud realizations possible within this low/middle/high classification, and appropriately averaged to produce the net solar flux.

6.4.3.2.4 Longwave Radiation

The longwave radiation package used in the fizhi package is thoroughly described by Chou and M.J.Suarez [1994]. As described in that document, IR fluxes are computed due to absorption by water vapor, carbon dioxide, and ozone. The spectral bands together with their absorbers and parameterization methods, configured for the fizhi package, are shown in Table 6.12.


Table 6.12: IR Spectral Bands, Absorbers, and Parameterization Method (from Chou and M.J.Suarez [1994])
IR Spectral Bands


Band Spectral Range (cm$ ^{-1}$) Absorber Method
1 0-340 H$ _2$O line T
2 340-540 H$ _2$O line T
3a 540-620 H$ _2$O line K
3b 620-720 H$ _2$O continuum S
3b 720-800 CO$ _2$ T
4 800-980 H$ _2$O line K
    H$ _2$O continuum S
    H$ _2$O line K
5 980-1100 H$ _2$O continuum S
    O$ _3$ T
6 1100-1380 H$ _2$O line K
    H$ _2$O continuum S
7 1380-1900 H$ _2$O line T
8 1900-3000 H$ _2$O line K
    K: k-distribution method with linear pressure scaling
    T: Table look-up with temperature and pressure scaling
    S: One-parameter temperature scaling



The longwave radiation package accurately computes cooling rates for the middle and lower atmosphere from 0.01 mb to the surface. Errors are $ <$ 0.4 C day$ ^{-1}$ in cooling rates and $ <$ 1% in fluxes. From Chou and Suarez, it is estimated that the total effect of neglecting all minor absorption bands and the effects of minor infrared absorbers such as nitrous oxide (N$ _2$O), methane (CH$ _4$), and the chlorofluorocarbons (CFCs), is an underestimate of $ \approx$ 5 W/m$ ^2$ in the downward flux at the surface and an overestimate of $ \approx$ 3 W/m$ ^2$ in the upward flux at the top of the atmosphere.

Similar to the procedure used in the shortwave radiation package, clouds are grouped into three regions catagorized as low/middle/high. The net clear line-of-site probability $ (P)$ between any two levels, $ p_1$ and $ p_2 \quad (p_2 > p_1)$, assuming randomly overlapped cloud groups, is simply the product of the probabilities within each group:

$\displaystyle P_{net} = P_{low} \times P_{mid} \times P_{hi} . $

Since all clouds within a group are assumed maximally overlapped, the clear line-of-site probability within a group is given by:

$\displaystyle P_{group} = 1 - F_{max} , $

where $ F_{max}$ is the maximum cloud fraction encountered between $ p_1$ and $ p_2$ within that group. For groups and/or levels outside the range of $ p_1$ and $ p_2$, a clear line-of-site probability equal to 1 is assigned.


6.4.3.2.5 Cloud-Radiation Interaction

The cloud fractions and diagnosed cloud liquid water produced by moist processes within the fizhi package are used in the radiation packages to produce cloud-radiative forcing. The cloud optical thickness associated with large-scale cloudiness is made proportional to the diagnosed large-scale liquid water, $ \ell$, detrained due to super-saturation. Two values are used corresponding to cloud ice particles and water droplets. The range of optical thickness for these clouds is given as

$\displaystyle 0.0002 \le \tau_{ice} (mb^{-1}) \le 0.002$   for$\displaystyle \quad 0 \le \ell \le 2$   mg/kg$\displaystyle ,$

$\displaystyle 0.02 \le \tau_{h_2o} (mb^{-1}) \le 0.2$   for$\displaystyle \quad 0 \le \ell \le 10$   mg/kg$\displaystyle . $

The partitioning, $ \alpha $, between ice particles and water droplets is achieved through a linear scaling in temperature:

$\displaystyle 0 \le \alpha \le 1$   for$\displaystyle \quad 233.15 \le T \le 253.15 . $

The resulting optical depth associated with large-scale cloudiness is given as

$\displaystyle \tau_{LS} = \alpha \tau_{h_2o} + (1-\alpha)\tau_{ice} . $

The optical thickness associated with sub-grid scale convective clouds produced by RAS is given as

$\displaystyle \tau_{RAS} = 0.16 \quad mb^{-1} . $

The total optical depth in a given model layer is computed as a weighted average between the large-scale and sub-grid scale optical depths, normalized by the total cloud fraction in the layer:

$\displaystyle \tau = \left( {F_{RAS} \,\,\, \tau_{RAS} + F_{LS} \,\,\, \tau_{LS} \over F_{RAS}+F_{LS} } \right) \Delta p, $

where $ F_{RAS}$ and $ F_{LS}$ are the time-averaged cloud fractions associated with RAS and large-scale processes described in Section 6.4.3.2. The optical thickness for the longwave radiative feedback is assumed to be 75 $ \%$ of these values.

The entire Moist Convective Processes Module is called with a frequency of 10 minutes. The cloud fraction values are time-averaged over the period between Radiation calls (every 3 hours). Therefore, in a time-averaged sense, both convective and large-scale cloudiness can exist in a given grid-box.

6.4.3.2.6 Turbulence

:

Turbulence is parameterized in the fizhi package to account for its contribution to the vertical exchange of heat, moisture, and momentum. The turbulence scheme is invoked every 30 minutes, and employs a backward-implicit iterative time scheme with an internal time step of 5 minutes. The tendencies of atmospheric state variables due to turbulent diffusion are calculated using the diffusion equations:

$\displaystyle {{\partial u \over {\partial t}}}_{turb} = {{\partial \over {\par... ...}})} = {{\partial \over {\partial z}} }{(K_m {\partial u \over {\partial z}})} $

$\displaystyle {{\partial v \over {\partial t}}}_{turb} = {{\partial \over {\par... ...}})} = {{\partial \over {\partial z}} }{(K_m {\partial v \over {\partial z}})} $

$\displaystyle {{\partial T \over {\partial t}}} = P^{\kappa}{{\partial \theta \... ...{{\partial \over {\partial z}} }{(K_h {\partial \theta_v \over {\partial z}})} $

$\displaystyle {{\partial q \over {\partial t}}}_{turb} = {{\partial \over {\par... ...}})} = {{\partial \over {\partial z}} }{(K_h {\partial q \over {\partial z}})} $

Within the atmosphere, the time evolution of second turbulent moments is explicitly modeled by representing the third moments in terms of the first and second moments. This approach is known as a second-order closure modeling. To simplify and streamline the computation of the second moments, the level 2.5 assumption of Mellor and Yamada (1974) and Yamada [1977] is employed, in which only the turbulent kinetic energy (TKE),

$\displaystyle { {1\over2} }{q^2}={\overline{{u^{\prime}}^2}}+{\overline{{v^{\prime}}^2}}+{\overline{{w^{\prime}}^2}}, $

is solved prognostically and the other second moments are solved diagnostically. The prognostic equation for TKE allows the scheme to simulate some of the transient and diffusive effects in the turbulence. The TKE budget equation is solved numerically using an implicit backward computation of the terms linear in $ q^2$ and is written:

$\displaystyle {{d \over {d t}} ({{ {1\over2} } q^2})} - { {\partial \over {\par... ...erline{{w^{\prime}}{{{\theta}_v}^{\prime}}}} } - { q^3 \over {{\Lambda} _1} } $

where $ q$ is the turbulent velocity, $ {u^{\prime}}$, $ {v^{\prime}}$, $ {w^{\prime}}$ and $ {{\theta}^{\prime}}$ are the fluctuating parts of the velocity components and potential temperature, $ U$ and $ V$ are the mean velocity components, $ {\Theta_0}^{-1}$ is the coefficient of thermal expansion, and $ {{\lambda}_1}$ and $ {{\Lambda} _1}$ are constant multiples of the master length scale, $ \ell$, which is designed to be a characteristic measure of the vertical structure of the turbulent layers.

The first term on the left-hand side represents the time rate of change of TKE, and the second term is a representation of the triple correlation, or turbulent transport term. The first three terms on the right-hand side represent the sources of TKE due to shear and bouyancy, and the last term on the right hand side is the dissipation of TKE.

In the level 2.5 approach, the vertical fluxes of the scalars $ \theta_v$ and $ q$ and the wind components $ u$ and $ v$ are expressed in terms of the diffusion coefficients $ K_h$ and $ K_m$, respectively. In the statisically realizable level 2.5 turbulence scheme of Helfand and Labraga [1988], these diffusion coefficients are expressed as

$\displaystyle K_h = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell... ...l \, S_{H}(G_{M_e},G_{H_e}) \, & \mbox{growing turbulence} \end{array} \right. $

and

$\displaystyle K_m = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell ... ...l \, S_{M}(G_{M_e},G_{H_e}) \, & \mbox{growing turbulence} \end{array} \right. $

where the subscript $ e$ refers to the value under conditions of local equillibrium (obtained from the Level 2.0 Model), $ \ell$ is the master length scale related to the vertical structure of the atmosphere, and $ S_M$ and $ S_H$ are functions of $ G_H$ and $ G_M$, the dimensionless buoyancy and wind shear parameters, respectively. Both $ G_H$ and $ G_M$, and their equilibrium values $ G_{H_e}$ and $ G_{M_e}$, are functions of the Richardson number:

$\displaystyle {\bf RI} = { { {g \over \theta_v} {\partial \theta_v \over {\part... ...{\partial u \over {\partial z}})^2 + ({\partial v \over {\partial z}})^2 } } . $

Negative values indicate unstable buoyancy and shear, small positive values ($ <0.2$) indicate dominantly unstable shear, and large positive values indicate dominantly stable stratification.

Turbulent eddy diffusion coefficients of momentum, heat and moisture in the surface layer, which corresponds to the lowest GCM level (see [*]), are calculated using stability-dependant functions based on Monin-Obukhov theory:

$\displaystyle {K_m} (surface) = C_u \times u_* = C_D W_s $

and

$\displaystyle {K_h} (surface) = C_t \times u_* = C_H W_s $

where $ u_*=C_uW_s$ is the surface friction velocity, $ C_D$ is termed the surface drag coefficient, $ C_H$ the heat transfer coefficient, and $ W_s$ is the magnitude of the surface layer wind.

$ C_u$ is the dimensionless exchange coefficient for momentum from the surface layer similarity functions:

$\displaystyle {C_u} = {u_* \over W_s} = { k \over \psi_{m} } $

where k is the Von Karman constant and $ \psi_m$ is the surface layer non-dimensional wind shear given by

$\displaystyle \psi_{m} = {\int_{\zeta_{0}}^{\zeta} {\phi_{m} \over \zeta} d \zeta} . $

Here $ \zeta$ is the non-dimensional stability parameter, and $ \phi_m$ is the similarity function of $ \zeta$ which expresses the stability dependance of the momentum gradient. The functional form of $ \phi_m$ is specified differently for stable and unstable layers.

$ C_t$ is the dimensionless exchange coefficient for heat and moisture from the surface layer similarity functions:

$\displaystyle {C_t} = -{( {\overline{w^{\prime}\theta^{\prime}}}) \over {u_* \D... ...e}q^{\prime}}}) \over {u_* \Delta q }} = { k \over { (\psi_{h} + \psi_{g}) } } $

where $ \psi_h$ is the surface layer non-dimensional temperature gradient given by

$\displaystyle \psi_{h} = {\int_{\zeta_{0}}^{\zeta} {\phi_{h} \over \zeta} d \zeta} . $

Here $ \phi_h$ is the similarity function of $ \zeta$, which expresses the stability dependance of the temperature and moisture gradients, and is specified differently for stable and unstable layers according to Helfand and Schubert [1995].

$ \psi_g$ is the non-dimensional temperature or moisture gradient in the viscous sublayer, which is the mosstly laminar region between the surface and the tops of the roughness elements, in which temperature and moisture gradients can be quite large. Based on Yaglom and Kader [1974]:

$\displaystyle \psi_{g} = { 0.55 (Pr^{2/3} - 0.2) \over \nu^{1/2} } (h_{0}u_{*} - h_{0_{ref}}u_{*_{ref}})^{1/2} $

where Pr is the Prandtl number for air, $ \nu$ is the molecular viscosity, $ z_{0}$ is the surface roughness length, and the subscript ref refers to a reference value. $ h_{0} = 30z_{0}$ with a maximum value over land of 0.01

The surface roughness length over oceans is is a function of the surface-stress velocity,

$\displaystyle {z_0} = c_1u^3_* + c_2u^2_* + c_3u_* + c_4 + {c_5 \over {u_*}} $

where the constants are chosen to interpolate between the reciprocal relation of Kondo [1975] for weak winds, and the piecewise linear relation of Large and Pond [1981] for moderate to large winds. Roughness lengths over land are specified from the climatology of Dorman and Sellers [1989].

For an unstable surface layer, the stability functions, chosen to interpolate between the condition of small values of $ \beta$ and the convective limit, are the KEYPS function (Panofsky [1973]) for momentum, and its generalization for heat and moisture:

$\displaystyle {\phi_m}^4 - 18 \zeta {\phi_m}^3 = 1 \hspace{1cm} ; \hspace{1cm} {\phi_h}^2 - 18 \zeta {\phi_h}^3 = 1 \hspace{1cm} . $

The function for heat and moisture assures non-vanishing heat and moisture fluxes as the wind speed approaches zero.

For a stable surface layer, the stability functions are the observationally based functions of Clarke [1970], slightly modified for the momemtum flux:

$\displaystyle {\phi_m} = { { 1 + 5 {{\zeta}_1} } \over { 1 + 0.00794 {{\zeta}_1... ...= { { 1 + 5 {{\zeta}_1} } \over { 1 + 0.00794 {\zeta} (1+ 5 {{\zeta}_1}) } } . $

The moisture flux also depends on a specified evapotranspiration coefficient, set to unity over oceans and dependant on the climatological ground wetness over land.

Once all the diffusion coefficients are calculated, the diffusion equations are solved numerically using an implicit backward operator.

6.4.3.2.7 Atmospheric Boundary Layer

The depth of the atmospheric boundary layer (ABL) is diagnosed by the parameterization as the level at which the turbulent kinetic energy is reduced to a tenth of its maximum near surface value. The vertical structure of the ABL is explicitly resolved by the lowest few (3-8) model layers.

6.4.3.2.8 Surface Energy Budget

The ground temperature equation is solved as part of the turbulence package using a backward implicit time differencing scheme:

$\displaystyle C_g{\partial T_g \over {\partial t}} = R_{sw} - R_{lw} + Q_{ice} - H - LE $

where $ R_{sw}$ is the net surface downward shortwave radiative flux and $ R_{lw}$ is the net surface upward longwave radiative flux.

$ H$ is the upward sensible heat flux, given by:

$\displaystyle {H} = P^{\kappa}\rho c_{p} C_{H} W_s (\theta_{surface} - \theta_{NLAY}) \hspace{1cm}where: \hspace{.2cm}C_H = C_u C_t $

where $ \rho $ = the atmospheric density at the surface, $ c_{p}$ is the specific heat of air at constant pressure, and $ \theta $ represents the potential temperature of the surface and of the lowest $ \sigma $-level, respectively.

The upward latent heat flux, $ LE$, is given by

$\displaystyle {LE} = \rho \beta L C_{H} W_s (q_{surface} - q_{NLAY}) \hspace{1cm}where: \hspace{.2cm}C_H = C_u C_t $

where $ \beta$ is the fraction of the potential evapotranspiration actually evaporated, L is the latent heat of evaporation, and $ q_{surface}$ and $ q_{NLAY}$ are the specific humidity of the surface and of the lowest $ \sigma $-level, respectively.

The heat conduction through sea ice, $ Q_{ice}$, is given by

$\displaystyle {Q_{ice}} = {C_{ti} \over {H_i}} (T_i-T_g) $

where $ C_{ti}$ is the thermal conductivity of ice, $ H_i$ is the ice thickness, assumed to be $ 3 \hspace{.1cm} m$ where sea ice is present, $ T_i$ is 273 degrees Kelvin, and $ T_g$ is the surface temperature of the ice.

$ C_g$ is the total heat capacity of the ground, obtained by solving a heat diffusion equation for the penetration of the diurnal cycle into the ground (Blackadar [1977]), and is given by:

$\displaystyle C_g = \sqrt{ {\lambda C_s \over 2\omega} } = \sqrt{(0.386 + 0.536W + 0.15W^2)2\times10^{-3} {86400 \over 2 \pi} } \, \, . $

Here, the thermal conductivity, $ \lambda $, is equal to $ 2\times10^{-3}$ $ {ly\over{ sec}} {cm \over {^oK}}$, the angular velocity of the earth, $ \omega$, is written as $ 86400$ $ sec/day$ divided by $ 2 \pi$ $ radians/ day$, and the expression for $ C_s$, the heat capacity per unit volume at the surface, is a function of the ground wetness, $ W$.

Land Surface Processes:

6.4.3.2.9 Surface Type

The fizhi package surface Types are designated using the Koster-Suarez (Koster and Suarez [1992,1991]) Land Surface Model (LSM) mosaic philosophy which allows multiple ``tiles'', or multiple surface types, in any one grid cell. The Koster-Suarez LSM surface type classifications are shown in Table 6.13. The surface types and the percent of the grid cell occupied by any surface type were derived from the surface classification of Defries and Townshend [1994], and information about the location of permanent ice was obtained from the classifications of Dorman and Sellers [1989]. The surface type map for a $ 1^\circ$ grid is shown in Figure 6.10. The determination of the land or sea category of surface type was made from NCAR's 10 minute by 10 minute Navy topography dataset, which includes information about the percentage of water-cover at any point. The data were averaged to the model's grid resolutions, and any grid-box whose averaged water percentage was $ \geq 60 \%$ was defined as a water point. The Land-Water designation was further modified subjectively to ensure sufficient representation from small but isolated land and water regions.


Table 6.13: Surface type designations.
Surface Type Designation


Type Vegetation Designation
1 Broadleaf Evergreen Trees
2 Broadleaf Deciduous Trees
3 Needleleaf Trees
4 Ground Cover
5 Broadleaf Shrubs
6 Dwarf Trees (Tundra)
7 Bare Soil
8 Desert (Bright)
9 Glacier
10 Desert (Dark)
100 Ocean


Figure 6.10: Surface Type Combinations.
\begin{figure*}\centerline{ \epsfysize=4.0in \epsfbox{part6/surftype.eps}} \vspace{0.2in} \end{figure*}

6.4.3.2.10 Surface Roughness

The surface roughness length over oceans is computed iteratively with the wind stress by the surface layer parameterization (Helfand and Schubert [1995]). It employs an interpolation between the functions of Large and Pond [1981] for high winds and of Kondo [1975] for weak winds.

6.4.3.2.11 Albedo

The surface albedo computation, described in Koster and Suarez [1991], employs the ``two stream'' approximation used in Sellers' (1987) Simple Biosphere (SiB) Model which distinguishes between the direct and diffuse albedos in the visible and in the near infra-red spectral ranges. The albedos are functions of the observed leaf area index (a description of the relative orientation of the leaves to the sun), the greenness fraction, the vegetation type, and the solar zenith angle. Modifications are made to account for the presence of snow, and its depth relative to the height of the vegetation elements.

6.4.3.2.12 Gravity Wave Drag

The fizhi package employs the gravity wave drag scheme of Zhou et al. [1995]). This scheme is a modified version of Vernekar et al. (1992), which was based on Alpert et al. (1988) and Helfand et al. (1987). In this version, the gravity wave stress at the surface is based on that derived by Pierrehumbert (1986) and is given by:

$\displaystyle \vert\vec{\tau}_{sfc}\vert = {\rho U^3\over{N \ell^*}} \left(F_r^2 \over{1+F_r^2}\right) \, \, ,$ (6.33)

where $ F_r = N h /U$ is the Froude number, $ N $ is the Brunt - Väisälä frequency, $ U$ is the surface wind speed, $ h$ is the standard deviation of the sub-grid scale orography, and $ \ell^*$ is the wavelength of the monochromatic gravity wave in the direction of the low-level wind. A modification introduced by Zhou et al. allows for the momentum flux to escape through the top of the model, although this effect is small for the current 70-level model. The subgrid scale standard deviation is defined by $ h$, and is not allowed to exceed 400 m.

The effects of using this scheme within a GCM are shown in Takacs and Suarez [1996]. Experiments using the gravity wave drag parameterization yielded significant and beneficial impacts on both the time-mean flow and the transient statistics of the a GCM climatology, and have eliminated most of the worst dynamically driven biases in the a GCM simulation. An examination of the angular momentum budget during climate runs indicates that the resulting gravity wave torque is similar to the data-driven torque produced by a data assimilation which was performed without gravity wave drag. It was shown that the inclusion of gravity wave drag results in large changes in both the mean flow and in eddy fluxes. The result is a more accurate simulation of surface stress (through a reduction in the surface wind strength), of mountain torque (through a redistribution of mean sea-level pressure), and of momentum convergence (through a reduction in the flux of westerly momentum by transient flow eddies).

Boundary Conditions and other Input Data:

Required fields which are not explicitly predicted or diagnosed during model execution must either be prescribed internally or obtained from external data sets. In the fizhi package these fields include: sea surface temperature, sea ice estent, surface geopotential variance, vegetation index, and the radiation-related background levels of: ozone, carbon dioxide, and stratospheric moisture.

Boundary condition data sets are available at the model's resolutions for either climatological or yearly varying conditions. Any frequency of boundary condition data can be used in the fizhi package; however, the current selection of data is summarized in Table 6.14. The time mean values are interpolated during each model timestep to the current time.


Table 6.14: Boundary conditions and other input data used in the fizhi package. Also noted are the current years and frequencies available.
Fizhi Input Datasets


Variable Frequency Years
Sea Ice Extent monthly 1979-current, climatology
Sea Ice Extent weekly 1982-current, climatology
Sea Surface Temperature monthly 1979-current, climatology
Sea Surface Temperature weekly 1982-current, climatology
Zonally Averaged Upper-Level Moisture monthly climatology
Zonally Averaged Ozone Concentration monthly climatology


6.4.3.2.13 Topography and Topography Variance

Surface geopotential heights are provided from an averaging of the Navy 10 minute by 10 minute dataset supplied by the National Center for Atmospheric Research (NCAR) to the model's grid resolution. The original topography is first rotated to the proper grid-orientation which is being run, and then averages the data to the model resolution.

The standard deviation of the subgrid-scale topography is computed by interpolating the 10 minute data to the model's resolution and re-interpolating back to the 10 minute by 10 minute resolution. The sub-grid scale variance is constructed based on this smoothed dataset.

6.4.3.2.14 Upper Level Moisture

The fizhi package uses climatological water vapor data above 100 mb from the Stratospheric Aerosol and Gas Experiment (SAGE) as input into the model's radiation packages. The SAGE data is archived as monthly zonal means at $ 5^\circ$ latitudinal resolution. The data is interpolated to the model's grid location and current time, and blended with the GCM's moisture data. Below 300 mb, the model's moisture data is used. Above 100 mb, the SAGE data is used. Between 100 and 300 mb, a linear interpolation (in pressure) is performed using the data from SAGE and the GCM.

6.4.3.3 Fizhi Diagnostics

Fizhi Diagnostic Menu:

NAME UNITS LEVELS DESCRIPTION
       
UFLUX $ Newton/m^2$ 1
Surface U-Wind Stress on the atmosphere
VFLUX $ Newton/m^2$ 1
Surface V-Wind Stress on the atmosphere
HFLUX $ Watts/m^2$ 1
Surface Flux of Sensible Heat
EFLUX $ Watts/m^2$ 1
Surface Flux of Latent Heat
QICE $ Watts/m^2$ 1
Heat Conduction through Sea-Ice
RADLWG $ Watts/m^2$ 1
Net upward LW flux at the ground
RADSWG $ Watts/m^2$ 1
Net downward SW flux at the ground
RI $ dimensionless$ Nrphys
Richardson Number
CT $ dimensionless$ 1
Surface Drag coefficient for T and Q
CU $ dimensionless$ 1
Surface Drag coefficient for U and V
ET $ m^2/sec$ Nrphys
Diffusivity coefficient for T and Q
EU $ m^2/sec$ Nrphys
Diffusivity coefficient for U and V
TURBU $ m/sec/day$ Nrphys
U-Momentum Changes due to Turbulence
TURBV $ m/sec/day$ Nrphys
V-Momentum Changes due to Turbulence
TURBT $ deg/day$ Nrphys
Temperature Changes due to Turbulence
TURBQ $ g/kg/day$ Nrphys
Specific Humidity Changes due to Turbulence
MOISTT $ deg/day$ Nrphys
Temperature Changes due to Moist Processes
MOISTQ $ g/kg/day$ Nrphys
Specific Humidity Changes due to Moist Processes
RADLW $ deg/day$ Nrphys
Net Longwave heating rate for each level
RADSW $ deg/day$ Nrphys
Net Shortwave heating rate for each level
PREACC $ mm/day$ 1
Total Precipitation
PRECON $ mm/day$ 1
Convective Precipitation
TUFLUX $ Newton/m^2$ Nrphys
Turbulent Flux of U-Momentum
TVFLUX $ Newton/m^2$ Nrphys
Turbulent Flux of V-Momentum
TTFLUX $ Watts/m^2$ Nrphys
Turbulent Flux of Sensible Heat


NAME UNITS LEVELS DESCRIPTION
       
TQFLUX $ Watts/m^2$ Nrphys
Turbulent Flux of Latent Heat
CN $ dimensionless$ 1
Neutral Drag Coefficient
WINDS $ m/sec$ 1
Surface Wind Speed
DTSRF $ deg$ 1
Air/Surface virtual temperature difference
TG $ deg$ 1
Ground temperature
TS $ deg$ 1
Surface air temperature (Adiabatic from lowest model layer)
DTG $ deg$ 1
Ground temperature adjustment
QG $ g/kg$ 1
Ground specific humidity
QS $ g/kg$ 1
Saturation surface specific humidity
TGRLW $ deg$ 1
Instantaneous ground temperature used as input to the Longwave radiation subroutine
ST4 $ Watts/m^2$ 1
Upward Longwave flux at the ground ( $ \sigma T^4$)
OLR $ Watts/m^2$ 1
Net upward Longwave flux at the top of the model
OLRCLR $ Watts/m^2$ 1
Net upward clearsky Longwave flux at the top of the model
LWGCLR $ Watts/m^2$ 1
Net upward clearsky Longwave flux at the ground
LWCLR $ deg/day$ Nrphys
Net clearsky Longwave heating rate for each level
TLW $ deg$ Nrphys
Instantaneous temperature used as input to the Longwave radiation subroutine
SHLW $ g/g$ Nrphys
Instantaneous specific humidity used as input to the Longwave radiation subroutine
OZLW $ g/g$ Nrphys
Instantaneous ozone used as input to the Longwave radiation subroutine
CLMOLW $ 0-1$ Nrphys
Maximum overlap cloud fraction used in the Longwave radiation subroutine
CLDTOT $ 0-1$ Nrphys
Total cloud fraction used in the Longwave and Shortwave radiation subroutines
LWGDOWN $ Watts/m^2$ 1
Downwelling Longwave radiation at the ground
GWDT $ deg/day$ Nrphys
Temperature tendency due to Gravity Wave Drag
RADSWT $ Watts/m^2$ 1
Incident Shortwave radiation at the top of the atmosphere
TAUCLD $ per 100 mb$ Nrphys
Counted Cloud Optical Depth (non-dimensional) per 100 mb
TAUCLDC $ Number$ Nrphys
Cloud Optical Depth Counter


NAME UNITS LEVELS DESCRIPTION
       
CLDLOW $ 0-1$ Nrphys
Low-Level ( 1000-700 hPa) Cloud Fraction (0-1)
EVAP $ mm/day$ 1
Surface evaporation
DPDT $ hPa/day$ 1
Surface Pressure tendency
UAVE $ m/sec$ Nrphys
Average U-Wind
VAVE $ m/sec$ Nrphys
Average V-Wind
TAVE $ deg$ Nrphys
Average Temperature
QAVE $ g/kg$ Nrphys
Average Specific Humidity
OMEGA $ hPa/day$ Nrphys
Vertical Velocity
DUDT $ m/sec/day$ Nrphys
Total U-Wind tendency
DVDT $ m/sec/day$ Nrphys
Total V-Wind tendency
DTDT $ deg/day$ Nrphys
Total Temperature tendency
DQDT $ g/kg/day$ Nrphys
Total Specific Humidity tendency
VORT $ 10^{-4}/sec$ Nrphys
Relative Vorticity
DTLS $ deg/day$ Nrphys
Temperature tendency due to Stratiform Cloud Formation
DQLS $ g/kg/day$ Nrphys
Specific Humidity tendency due to Stratiform Cloud Formation
USTAR $ m/sec$ 1
Surface USTAR wind
Z0 $ m$ 1
Surface roughness
FRQTRB $ 0-1$ Nrphys-1
Frequency of Turbulence
PBL $ mb$ 1
Planetary Boundary Layer depth
SWCLR $ deg/day$ Nrphys
Net clearsky Shortwave heating rate for each level
OSR $ Watts/m^2$ 1
Net downward Shortwave flux at the top of the model
OSRCLR $ Watts/m^2$ 1
Net downward clearsky Shortwave flux at the top of the model
CLDMAS $ kg / m^2$ Nrphys
Convective cloud mass flux
UAVE $ m/sec$ Nrphys
Time-averaged $ u-Wind$


NAME UNITS LEVELS DESCRIPTION
       
VAVE $ m/sec$ Nrphys
Time-averaged $ v-Wind$
TAVE $ deg$ Nrphys
Time-averaged $ Temperature$
QAVE $ g/g$ Nrphys
Time-averaged $ Specific \, \, Humidity$
RFT $ deg/day$ Nrphys
Temperature tendency due Rayleigh Friction
PS $ mb$ 1
Surface Pressure
QQAVE $ (m/sec)^2$ Nrphys
Time-averaged $ Turbulent Kinetic Energy$
SWGCLR $ Watts/m^2$ 1
Net downward clearsky Shortwave flux at the ground
PAVE $ mb$ 1
Time-averaged Surface Pressure
DIABU $ m/sec/day$ Nrphys
Total Diabatic forcing on $ u-Wind$
DIABV $ m/sec/day$ Nrphys
Total Diabatic forcing on $ v-Wind$
DIABT $ deg/day$ Nrphys
Total Diabatic forcing on $ Temperature$
DIABQ $ g/kg/day$ Nrphys
Total Diabatic forcing on $ Specific \, \, Humidity$
RFU $ m/sec/day$ Nrphys
U-Wind tendency due to Rayleigh Friction
RFV $ m/sec/day$ Nrphys
V-Wind tendency due to Rayleigh Friction
GWDU $ m/sec/day$ Nrphys
U-Wind tendency due to Gravity Wave Drag
GWDU $ m/sec/day$ Nrphys
V-Wind tendency due to Gravity Wave Drag
GWDUS $ N/m^2$ 1
U-Wind Gravity Wave Drag Stress at Surface
GWDVS $ N/m^2$ 1
V-Wind Gravity Wave Drag Stress at Surface
GWDUT $ N/m^2$ 1
U-Wind Gravity Wave Drag Stress at Top
GWDVT $ N/m^2$ 1
V-Wind Gravity Wave Drag Stress at Top
LZRAD $ mg/kg$ Nrphys
Estimated Cloud Liquid Water used in Radiation


NAME UNITS LEVELS DESCRIPTION
       
SLP $ mb$ 1
Time-averaged Sea-level Pressure
CLDFRC $ 0-1$ 1
Total Cloud Fraction
TPW $ gm/cm^2$ 1
Precipitable water
U2M $ m/sec$ 1
U-Wind at 2 meters
V2M $ m/sec$ 1
V-Wind at 2 meters
T2M $ deg$ 1
Temperature at 2 meters
Q2M $ g/kg$ 1
Specific Humidity at 2 meters
U10M $ m/sec$ 1
U-Wind at 10 meters
V10M $ m/sec$ 1
V-Wind at 10 meters
T10M $ deg$ 1
Temperature at 10 meters
Q10M $ g/kg$ 1
Specific Humidity at 10 meters
DTRAIN $ kg / m^2$ Nrphys
Detrainment Cloud Mass Flux
QFILL $ g/kg/day$ Nrphys
Filling of negative specific humidity











NAME UNITS LEVELS DESCRIPTION
       
DTCONV $ deg/sec$ Nr
Temp Change due to Convection
DQCONV $ g/kg/sec$ Nr
Specific Humidity Change due to Convection
RELHUM $ percent$ Nr
Relative Humidity
PRECLS $ g/m^2/sec$ 1
Large Scale Precipitation
ENPREC $ J/g$ 1
Energy of Precipitation (snow, rain Temp)










Fizhi Diagnostic Description:

In this section we list and describe the diagnostic quantities available within the GCM. The diagnostics are listed in the order that they appear in the Diagnostic Menu, Section 6.4.3.3. In all cases, each diagnostic as currently archived on the output datasets is time-averaged over its diagnostic output frequency:

$\displaystyle {\bf DIAGNOSTIC} = {1 \over TTOT} \sum_{t=1}^{t=TTOT} diag(t) $

where $ TTOT = {{\bf NQDIAG} \over \Delta t}$, NQDIAG is the output frequency of the diagnostic, and $ \Delta t$ is the timestep over which the diagnostic is updated.

UFLUX Surface Zonal Wind Stress on the Atmosphere ( $ Newton/m^2$)

The zonal wind stress is the turbulent flux of zonal momentum from the surface.

$\displaystyle {\bf UFLUX} = - \rho C_D W_s u \hspace{1cm}where: \hspace{.2cm}C_D = C^2_u $

where $ \rho $ = the atmospheric density at the surface, $ C_{D}$ is the surface drag coefficient, $ C_u$ is the dimensionless surface exchange coefficient for momentum (see diagnostic number 10), $ W_s$ is the magnitude of the surface layer wind, and $ u$ is the zonal wind in the lowest model layer.

VFLUX Surface Meridional Wind Stress on the Atmosphere ( $ Newton/m^2$)

The meridional wind stress is the turbulent flux of meridional momentum from the surface.