Documentation of the PMIP models (Bonfils et al. 1998)

PMIP Documentation for LMD4

Laboratoire de Météorologie Dynamique: Model LMCE LMD4.3 (sin(lat)x7.5 L11) 1991

PMIP Representative(s)

for 0fix and 6fix:

Dr. Pascale Braconnot, Laboratoire des Sciences du Climat et de l´Environnement, Bat 709 CEA-DSM, Orme des Merisiers, F 91191 Gif-sur-Yvette cedex, France, Phone: 33 (1) 69 08 77 11; Fax 33 (1) 69 08 77 16;

for 21fix:

Dr. Gilles Ramstein Laboratoire des Sciences du Climat et de l´Environnement, Bat 709 CEA-DSM, Orme des Merisiers, F 91191 Gif-sur-Yvette cedex, France, Phone: 33 (1) 69 08 77 11; Fax 33 (1) 69 08 77 16;


Model Designation

LMD4.3 (sin(lat)x7.5 L11) 1991

Model Identification for PMIP


PMIP run(s)

0fix, 6fix, 21fix, 0cal, 21cal

Number of days in each month: 30 30 30 30 30 30 30 30 30 30 30 30

Model Lineage

The LMD model derives from an earlier version developed for climate studies (cf. Sadourny and Laval 1984) . Subsequent modifications principally include changes in the representation of radiation and horizontal diffusion, Inclusion of parameterizations of gravity-wave drag and prognostic cloud formation and cloud water pronostic equation (Le Treut 1994 and Ramstein 1998).

Model Documentation

Sadourny R., Laval K., 1984, January and July performance of the LMD general circulation model. In New Perspectives in Climate Modelling, A. Berger and C. nicolis (eds), Developments in Atmospheric Science, 16, Elsevier, pp 173-198

To see the improvements between LMD4ter and LMD4 (previous version) :

Le Treut H., Z.X. Li and M. Forichon, 1994 : Sensitivity of the LMD general circulation model to greenhouse forcing associated with two different cloud water parametrizations, J. Climate, Vol. 7, 1827-1841. .

Z.X. Li and Le Treut, H, 1992, Cloud radiation feedbacks in a general circulation model and their dependence on cloud modelling assumptions, Climate Dynamics, Vol. 7, 133-139.

Ramstein, G., Serafini, Y. V., Le Treut, H., 1998, Cloud processes associated with past and future climate changes, Climate Dynamics, in press.

Numerical/Computational Properties

Horizontal Representation

Finite differences on a uniform-area (constant area), staggered C-grid (cf. Arakawa and Lamb 1977) .

The model has a regular grid in longitude and in sine of the latitude (ie. 7.5° for longitude and latitude between 3° (equator) and 13° (poles)).

Horizontal Resolution

There are 36 grid points equally spaced in the sine of latitude and 48 points equally spaced in longitude. (The mesh size is 335 km north-south and 837 km east-west at the equator, and is about 536 x 592 km at 45 degrees latitude. )

dim_longitude*dim_latitude: 48*36

Vertical Domain

Surface to about 4 hPa. For a surface pressure of 1000 hPa, the lowest atmospheric level is at 979 hPa.

Vertical Representation

Finite-difference sigma coordinates.

Vertical Resolution

There are 11 unevenly spaced sigma levels: the model uses a sigma-coordinate in the vertical with the following levels: 0.991, 0.967, 0.914, 0.830, 0.708, 0.566, 0.411, 0.271, 0.150 0.07, 0.01.

For a surface pressure of 1000 hPa, 3 levels are below 800 hPa and 2 levels are above 200 hPa.

Computer/Operating System

The PMIP simulations were performed using a CRAY C94 under the UNICOS operating system.

Computational Performance

For the PMIP experiment, about 1.5 hours Cray C94 computation time per simulated year.


For the PMIP experiment, atmosphere, soil moisture, and snow cover/depth are initialized for 1 January 1979 from a previous model simulation.

For 21k, the initialisation is done with a second year first day run of the control run. Then the physics part is inhibit, only the dynamics part is run for 10 days which is the time needed by the model to dump the waves generated by the ice-sheet LGM. Then the physics part is switch on at the eleventh day. The run lasts 16 years, we keep only the 15 last years.

For 0k computed SSTs, the flux are computed from the fixed SSts control run and prescribed to the computed SSts control run for 30 years, when the equilibrium is reached. Then 15 years are performed to be analysed as control SST computed run.

For 21k comuted SSTs, the flux from the fixed SSTs control run are corrected for the sea level drop. For a first step, 30 years are performed to reach the equilibrium and 15 new years are performed to be analysed as LGM SST computed run.

Time Integration Scheme(s)

The time integration scheme for dynamics combines 4 leapfrog steps with a Matsuno step, each of length 6 minutes. In the model, physics is updated every 30 minutes, except for shortwave/longwave radiative fluxes, which are calculated every 6 hours. For computation of vertical turbulent surface fluxes and diffusion, an implicit backward integration scheme with 30-minute time step is used, but with all coefficients calculated explicitly. See also Surface Fluxes and Diffusion.


Orography is averaged on the model grid (see Orography). At the four latitude points closest to the poles, a Fourier filtering operator from 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 non diffusive scheme) are filled by borrowing moisture from the level below.

Sampling Frequency

For the PMIP simulation, the model history is written once every 24 hours.

Dynamical/Physical Properties

Atmospheric Dynamics

Primitive-equation dynamics are expressed in terms of u and v winds, potential enthalpy, specific humidity, and surface pressure. The advection scheme is designed to conserve potential enstrophy for divergent barotropic flow (cf. Sadourny 1975a , 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 Ñ (Ñ *Ñ *)Ñ , where Ñ denotes a first-order difference on the model grid, while Ñ * 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 (1987) 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 from Smagorinsky et al. (1965) . Estimation of TKE involves calculation of a countergradient term from 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.

Gravity-wave Drag

The formulation of gravity-wave drag closely follows the linear model described by Boer et al. (1984) . The drag at any level is proportional to the vertical divergence of the wave momentum stress, which is formulated as the product of a constant aspect ratio, the local Brunt-Vaisalla frequency, a launching height determined from the orographic variance over the grid box (see Orography), the local wind velocity, and its projection on the wind vector at the lowest model level. The layer where gravity-wave breakdown occurs (due to convective instability) is determined from the local Froude number; in this critical layer the wave stress decreases quadratically to zero as a function of height.

Solar Constant/Cycles

The solar constant is the AMIP-prescribed value of 1365 W/(m2). The orbital parameters and seasonal insolation distribution are calculated from PMIP recommendations. A seasonal, but not a diurnal cycle in solar forcing, is simulated.


Carbon dioxide concentration is prescribed. The values of 345, 280 and 200 ppm for control, 6ka and 21ka, respectively have been chosen for PMIP. A value of 280 ppm (pre-indusrtrial value) has been chosen for the computed SSTs control. Three-dimensional ozone concentration is diagnosed as a function of the 500 hPa geopotential heights following the method of Royer et al. (1988) . Radiative effects of water vapor and cloud water, but not those of aerosols, are also included (see Radiation).


Shortwave radiation is derived from 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 accurate 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 105 m-1 from 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 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).

Cloud water content and cloud fraction are interactively used in the radiative code through the calculation of cloud optical thickness and cloud emissivity (Le Treut 1994).

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 this decrease.

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.


Both convective and large-scale precipitation are linked to cloud LWC (see Cloud Formation). For warm clouds, the precipitation is parameterized using the relationship proposed by Sundqvist (1981) in which Cl = 2 x 104 kg per kg and Ct = 5.5 10-4 s-1. For cold clouds, we use a relationship that takes into account the terminal falling speed of the crystals as described in Starr and Cox (1985).

Planetary Boundary Layer

The PBL is represented by the first 4 levels above the surface (at sigma = 0.979, 0.941, 0.873, and 0.770). The PBL top is prescribed to be at the sigma = 0.770 level; here vertical turbulent eddy fluxes of momentum, heat, and moisture are assumed to vanish. See also Diffusion, Surface Fluxes, and Surface Characteristics.


Raw orography obtained at 10 x 10-minute resolution from the U.S. Navy dataset (cf. Joseph 1980) is area-averaged over the model grid boxes. The orographic variance about the mean value for each grid box is also computed from the same dataset for use in the gravity-wave drag parameterization (see Gravity-wave Drag). The rugosity is also linked to orographic variance.

For LGM, we add the anomaly of Peltier (Peltier(21k) -Peltier(0k)) to the control run.


For control: SSTs and sea ice extent are specified from the climatolgy of the 79-88 Reynold?s data (1988) used in AMIP (10 years mean).

For 6 fix: SSTs and sea-ice prescribed at their present day value, as in the control run.

For 21 fix: SSTs and sea-ice: the change in SSTs (LGM minus present-day) given by CLIMAP (1981), available at NGDC rather than the LGM absolute values in order to avoid differences due to differences in present day climatologies, was used. When points remain ocean, to obtain seasonally varying SSTs and sea ice edge from data for February and August, a simple sinusoidal variations, with extrems in February and August, is used. When points are ocean in summer and ice in winter (for Nothern Hemisphere), a "trapezoidal" function is used with an ad-hoc duration of the sea-ice deduced from present-day temperatures equivalent.

For computed SSTs experiments, the AGCM was coupled to a mixed layer ocean (50m) (Le Treut et al. 1994). The ocean heat transport is daily prescribed using the present-day diagnosed ocean heat transport accounting for the sea level drop (105m.) as described in the PMIP Newsletter (Webb et al. 1997).

Sea Ice

For 0fix and 6fix, sea ice extents are prescribed monthly. The surface temperature of the ice is predicted from the balance of energy fluxes (see Surface Fluxes) that includes conduction heating from the ocean below. This conduction flux is proportional to the difference between the surface temperature and that of melting ice (271.2 K), and is inversely proportional to the ice thickness (prescribed to be a uniform 3 m). Snow that accumulates on sea ice modifies its albedo and thermal properties. See also Snow Cover and Surface Characteristics.

For 21fix, sea ice edge of CLIMAP data is used for February and August.There is not any percent of sea ice for each grid box. When the CLIMAP 21k sea-ice extent is interpolated over the model grid 48x36 if the result is higher than 50%, we consider that the grid box is sea ice. When the result is lower than 50%, the whole box is free of ice.

For computed SSTs experiments, the sea ice appears when the temperature is lower than -2°C and disappear when it is higher.

Snow Cover

If the air temperature at the first level above the surface (at sigma = 0.979) is <0 degrees C, precipitation falls as snow. Prognostic snow mass is determined from a budget equation, with accumulation and melting over both land and sea ice. Snow cover affects the surface albedo and the heat capacity of the surface. Sublimation of snow is calculated as part of the surface evaporative flux, and snowmelt contributes to soil moisture. See also Surface Characteristics, Surface Fluxes, and Land Surface Processes.

Surface Characteristics

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) , which 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 unity (blackbody emission) for all surfaces.

Surface Fluxes

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 Bunker (1976) . 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).

Land Surface Processes

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 November 9, 1998. For further information, contact: Céline Bonfils ( )