http://www.lmd.ens.fr/ . Subsequent modifications principally include changes in the representation of radiation and horizontal diffusion, and inclusion of parameterizations of gravity-wave drag and prognostic cloud formation. . Other key model documents include publications by Sadourny and Laval (1984) , Laval et al. (1981) , and Le Treut and Li (1991) . Details of computational aspects are described by Butel (1991) . , with points equally spaced in sine of latitude and in longitude. Horizontal advection of moisture is by a semi-upstream advection scheme. See also Horizontal Resolution. Surface Fluxes and Diffusion. Orography). At the four latitude points closest to the poles, a Fourier filtering operator after Arakawa and Mintz (1974)  is applied to the momentum, thermodynamics, continuity, and water vapor tendency equations to slow the longitudinally propagating gravity waves for numerical stability. Negative moisture values (arising from vertical advection by the centered nondiffusive scheme) are filled by borrowing moisture from the level below. , b ). Total energy is also conserved for irrotational flow (cf. Sadourny 1980) . The continuity and thermodynamics equations are expressed in flux form, conserving mass and the space integrals of potential temperature and its square. The water vapor tendency is also expressed in flux form, thereby reducing the probability of spurious negative moisture values (see Smoothing/Filling).
- Linear horizontal diffusion is applied on constant-pressure surfaces to
potential enthalpy, divergence, and rotational wind via a biharmonic operator
del(del*del*)del, where del denotes a first-order difference on the model grid,
while del* is a formal differential operator on a regular grid without
geometrical corrections. Because of the highly diffusive character of the
flux-form water vapor tendency equation (see Atmospheric Dynamics), no
further horizontal diffusion of specific humidity is included. Cf. Michaud
 for further details.
- Second-order vertical diffusion of momentum, heat, and moisture is applied only within the planetary boundary layer (PBL). The diffusion coefficient depends on a diagnostic estimate of the turbulence kinetic energy (TKE) and on the mixing length (which decreases up to the prescribed PBL top) that is estimated after Smagorinsky et al. (1965) . Estimation of TKE involves calculation of a countergradient term after Deardorff (1966)  and comparison of the bulk Richardson number with a critical value. Cf. Sadourny and Laval (1984)  for further details. See also Planetary Boundary Layer and Surface Fluxes.
- Shortwave radiation is modeled after an updated scheme of Fouquart and Bonnel (1980)
. Upward/downward shortwave
irradiance profiles are evaluated in two stages. First, a mean photon optical
path is calculated for a scattering atmosphere including clouds and gases. The
reflectance and transmittance of these elements are calculated by,
respectively, the delta-Eddington method (cf. Joseph et al. 1976) and by a simplified two-stream
approximation. The scheme evaluates upward/downward shortwave fluxes for two reference cases: a conservative atmosphere and a first-guess absorbing
atmosphere; the mean optical path is then computed for each absorbing gas from the logarithm of the ratio of these reference fluxes. In the second stage,
final upward/downward fluxes are computed for two spectral intervals (0.30-0.68 micron and 0.68-4.0 microns) using more exact gas transmittances (Rothman 1981) and with adjustments made for the presence of clouds (see Cloud Formation). For clouds, the asymmetry factor is prescribed, and the optical depth and single-scattering albedo are functions of cloud liquid water content after Stephens (1978)
- Longwave radiation is modeled in six spectral intervals between wavenumbers 0 and 2.82 x 10^5 m^-1 after the method of Morcrette (1990, 1991 ). Absorption by water vapor (in two intervals), by the water vapor continuum (in two intervals in the atmospheric window, following Clough et al. 1980) , by the carbon dioxide and the rotational part of the water vapor spectrum (in one interval), and by ozone (in one interval) is treated. The temperature and pressure dependence of longwave absorption by gases is included. Clouds are treated as graybodies in the longwave, with emissivity depending on cloud liquid water path after Stephens (1978) . Longwave scattering by cloud droplets is neglected, and droplet absorption is modeled by an emissivity formulation from the cloud liquid water path. For purposes of the radiation calculations, all clouds are assumed to overlap randomly in the vertical. See also Cloud Formation.
- When the temperature lapse rate is conditionally unstable, subgrid-scale
convective condensation takes place. If the air is supersaturated, a moist
convective adjustment after Manabe and Strickler (1964)
 is carried out: the temperature profile is
adjusted to the previous estimate of the moist adiabatic lapse rate, with total
moist static energy in the column being held constant. The specific humidity is
then set to a saturated profile for the adjusted temperature lapse, and the
excess moisture is rained out (see Precipitation).
- If the temperature lapse rate is conditionally unstable but the air is
unsaturated, condensation also occurs following the Kuo (1965)
 cumulus convection scheme, provided there is
large-scale moisture convergence. In this case, the lifting condensation level
is assumed to be at the top of the PBL, and the height of the cumulus cloud is
given by the highest level for which the moist static energy is less than that
at the PBL top (see Planetary Boundary Layer). It is assumed that all
the humidity entering each cloudy layer since the last call of the convective
scheme (30 minutes prior) is pumped into this cloud. The environmental humidity
is reduced accordingly, while the environmental temperature is taken as the
grid-scale value; the cloud temperature and humidity profiles are defined to be
those of a moist adiabat.
- The fractional area of the convective cloud is obtained from a suitably
normalized, mass-weighted vertical integral (from cloud bottom to top) of
differences between the humidities and temperatures of the cloud vs those of
the environment. As a result of mixing, the environmental (grid-scale)
temperature and humidity profiles evolve to the moist adiabatic values in
proportion to this fractional cloud area, while the excess of moisture
precipitates (see Precipitation). Mixing of momentum also occurs.
- There is no explicit simulation of shallow convection, but the moist convective adjustment produces similar effects in the moisture field (cf. Le Treut and Li 1991) . See also Cloud Formation.
- Cloud cover is prognostically determined, as described by Le Treut and Li (1991)
. Time-dependent cloud liquid water content (LWC) follows a conservation
equation involving rates of water vapor condensation, evaporation of cloud
droplets, and the transformation of small droplets to large precipitating drops
(see Precipitation). The LWC also determines cloud cover (see below) and
cloud optical properties (see Radiation).
- The fraction of convective cloud in a grid box is unity if moist
convective adjustment is invoked; otherwise, it is given by the surface
fraction of the active cumulus cloud obtained from the Kuo (1965)
 scheme (see
Convection). Cloud forms in those layers where there is a decrease in
water vapor from one call of the convective scheme to the next (every 30
minutes), and the cloud LWC is redistributed in these layers proportional to
- The fraction of stratiform cloud in any layer is determined from the probability that the total cloud water (liquid plus vapor) is above the saturated value. (A uniform probability distribution is assumed with a prescribed standard deviation--cloud typically begins to form when the relative humidity exceeds 83 percent of saturation.) This stochastic approach also crudely simulates the effects of evaporation of cloud droplets. Cf. Le Treut and Li (1991)  for further details. See also Precipitation.
- For each grid box, 8 coexisting land surface types are specified from aggregation of the data of Matthews (1983; 1984): bare soil, desert, tundra, grassland, grassland with shrub cover, grassland with tree cover, deciduous forest, evergreen forest, and rainforest. The fractional areas of each surface type vary according to grid box.
- The surface roughness lengths over the continents are prescribed as a
function of orography and vegetation from data of Baumgartner et al. (1977)
, and their seasonal modulation is inferred
following Dorman and Sellers (1989)
Roughness lengths over ice surfaces are a uniform 1 x 10^-2 m. Over
ocean, the surface drag/transfer coefficients (see Surface Fluxes) are
determined without reference to a roughness length.
- Surface albedos for oceans and snow-free sea ice are prescribed from
monthly data of Bartman (1980)
, and for
snow-free continents from monthly data of Dorman and Sellers (1989)
. When there
is snow cover, the surface albedo is modified according to the parameterization
of Chalita and Le Treut (1994)
takes account of snow age, the eight designated land surface types, and spectral
range (in visible and near-infrared subintervals).
- The longwave emissivity is prescribed as 0.96 for all surfaces.
- The surface solar absorption is determined from surface albedos, and
longwave emission from the Planck equation with prescribed emissivity of 0.96
(see Surface Characteristics).
- In the lowest atmospheric layer, surface turbulent eddy fluxes of
momentum, heat, and moisture are expressed as bulk formulae multiplied by
drag/transfer coefficients that are functions of wind speed, stability, and
(except over ocean) roughness length (see Surface Characteristics). The
transfer coefficient for the surface moisture flux also depends on the vertical
humidity gradient. Over the oceans, the neutral surface drag/transfer is corrected according to the local condition of surface winds. For strong surface winds, the drag/transfer coefficients are determined
(without reference to a roughness length) as functions of surface wind speed
and temperature difference between the ocean and the surface air, following
. For conditions of light surface winds over the oceans, functions of Golitzyn and Grachov (1986) that depend on the surface temperature and humidity gradients are utilized. In the transition region between these wind regimes, surface drag/transfer coefficients are calculated as exponential functions of the surface wind speed.
- In addition, the momentum flux is proportional to the wind vector
extrapolated to the surface. The sensible heat flux is proportional to the
difference between the potential temperature at the ground and that
extrapolated from the atmosphere to the surface. The surface moisture flux is
proportional to the potential evaporation (the difference between the saturated
specific humidity at the surface and the extrapolated atmospheric humidity)
multiplied by an evapotranspiration efficiency beta. Over oceans, snow, and ice, beta is set to unity, while over land it is a function of soil moisture (see Land Surface Processes).
- Above the surface layer, but only within the PBL, turbulent eddy fluxes are represented as diffusive processes (see Diffusion and Planetary Boundary Layer).
- Ground temperature and bulk heat capacity (with differentiation for bare
soil, snow, and ice) are defined as mean quantities over a single layer of
thickness about 0.15 m (over which there is significant diurnal variation of
temperature). The temperature prediction equation, which follows Corby et al. (1976)
, includes as forcing the surface
heat fluxes (see Surface Fluxes) and the heat of fusion of snow and ice.
- Prognostic soil moisture is represented by a single-layer "bucket" model after Budyko (1956) , with uniform field capacity 0.15 m. Soil moisture is increased by both precipitation and snowmelt, and is decreased by surface evaporation, which is determined from the product of the evapotranspiration efficiency beta and the potential evaporation from a surface saturated at the local surface temperature and pressure (see Surface Fluxes). Over land, beta is given by the maximum of unity or twice the ratio of local soil moisture to the constant field capacity. Runoff occurs implicitly if the soil moisture exceeds the field capacity. Cf. Laval et al. (1981)  for further details.
Last update August 13, 1996. For further information, contact: Tom Phillips ( email@example.com)