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General circulation models

Basic Structure

Atmospheric GCMs solve the three-dimensional, time-dependent equations for the rates of change of surface pressure, wind components, temperature, and moisture content, taking account of sources and sinks of heat, moisture, and momentum. These equations are usually expressed either in finite-difference form for a three-dimensional array of points or in spectral form in each layer of the model. Whichever form is used, the source and sink terms, representing the physical parametrizations, are calculated at grid points. As the land surface processes, which are central to this review, are part of these, the distinction between spectral and finite-difference models need not concern us further. The basic equations with the source and sink terms on the right-hand side may be written:

(1)

Here t , H. E are the vertical fluxes (positive upward) of momentum, heat, and moisture due to vertical subgrid-scale eddies (turbulence and convection), R. is the net downward radiative flux, P is the precipitation or net sink of moisture, and a is the vertical coordinate, being p/r _*, or pressure normalized by surface pressure r _*. Other notation is conventional as given in the list of symbols.

Fluxes at the Land-Atmosphere Interface

The land surface affects the atmosphere through the fluxes R_N, t , H. and E at the bottom boundary (s = 1) of the atmospheric model. The treatment of these fluxes in GCMs has been reviewed by Carson (1982). Though an attempt to emulate Carson's review would be too lengthy for the present purpose and an unnecessary duplication, a summary of the present state of the art may be useful, especially as some information not available to Carson can be included. In this section, we shall first present and discuss the basic formulation for each of the fluxes and then summarize the parametrizations in the main GCMs in current use, as known to the reviewer. These models are listed in table 1. Some models for which no new information is available are omitted here. These include the Australian Numerical Meteorology Research Centre model, now adopted, with some changes, as the NCAR (National Centre for Atmospheric Research) "Community Climate Model," and the Oregon State University (OSU), Siberian Academy of Sciences Computing Centre, and Main Geophysical Observatory models. These have only low vertical resolution and cannot be expected to give a realistic representation of the near-surface processes important in assessing the response to surface perturbations. However, brief descriptions of the OSU model and also the ECMWF (European Centre for Medium Range Weather Forecasting) forecast model are at the end of this paper.

Radiative Fluxes. (A) Basic formulations. The net downward radiative flux at the land surface

(2)

where is the emissivity or absorptivity of the surface for radiation of wavelength and and B(l ) represent the downward and black body fluxes at wavelength lambda. However, in atmospheric radiative transfer, we are concerned mainly with two principal streams of radiation, solar radiation at l = 0.2µm-4µm and terrestrial radiation at l = 3µm-100µm (these wavebands contain over 99% of the radiation from each source). It is therefore convenient to make the twostream approximation and write:

(3)

where s _s is Stefan's constant and T₀ the surface temperature (degrees Kelvin). R_s(0) is the downward solar radiation and a _* R_s(0) is the reflected solar flux. Note that a _*, the albedo, is not a constant for a given surface but depends on the spectral distribution of the incoming solar radiation, which will depend on atmospheric constituents, including cloud, and on the zenith angle of the sun. Thus, formally

(4)

Similarly, the emissivities for terrestrial radiation are means over the appropriate range of wavelengths weighted by the intensity at each wavelength. For the downward flux, this is again dependent on the atmospheric structure (temperature, humidity, and cloud), whilst for the upward radiation it is, for given surface conditions, dependent only on the surface temperature because of the Planck function's temperature dependence.

TABLE 1. GCM models

^aRandall describes two versions of the GLAS model, referred to in this paper as GLAS 1980 and GLAS 1982.

(B) Surface albedo in GCMs. We shall discuss here only the snow-free albedo; snow cover is allowed to increase the albedo in all the GCMs except that the LMD model prescribes a geographical variation taking account of the climatological snow cover. For snow-free albedo, many models, including the AES, GFDL, GLAS, and NCAR models, use the geographical distribution specified by Posey and Clapp (1964), who specified a low albedo (0.07) for tropical forests. As discussed by Rowntree (1975) and Dickinson (1980), this is due to the use of a visible albedo, but there is a strong spectral variation with much larger values for infrared wavelengths, and the mean value for the whole spectrum is about 0.125. Other geographical distributions, used in the LMD model due to Bartman (1981) and the ECMWF model due to Preuss and Geleyn (1980), using satellite data, also have values below 0.1 over parts of the tropical continents. Probably the most satisfactory treatment is that in the GISS model, with a detailed specification of albedos for eight land surface types as classified by Matthews (1983). It allows for seasonal variation and separate values for the visible and nearinfrared. The desert albedo has a moisture dependence that can halve the albedo to 0.175 for a moist surface. Relatively simple formulations are used in the EERM model, with a _* = 0.31-0.17 h_u where h_u is a non-linear function of soil moisture content, and in the MO model, with a. = 0.2, though a vegetation-dependent variation due to M. F. Wilson (personal communication) has recently been tested in the latter.

(C) Surface emissivity in GCMs. Generally a value of 1.0 has been used for emissivity in GCMs. However, the model II of GISS (Hansen et al. 1983) uses realistic emissivities for deserts, snow, and ice. The NCAR model uses values less than unity at 812 µm; the EERM model uses 0.95.

Surface Turbulent Fluxes. (A) Basic formulations. The turbulent fluxes at the surface may be formally written

(5)

Here the formal definition of the flux of a quantity X is given first as , where the primes (') indicate deviations from a time-mean, then the formulation commonly used in GCMs , with C_x being a turbulent transfer coefficient depending on surface roughness and atmospheric stability; ro and are near-surface density and windspeed and OX is an estimate of the excess of a surface value X₀ over the nearsurface atmospheric value is generally taken as zero.

The right-hand terms in equation 5 for heat and moisture transfer are alternative expressions for the fluxes in resistance form. For heat this combines an atmospheric resistance r_H and the temperature gradient d q ; setting q ₀ at the surface temperature is observed to give reasonable estimates. With a wet surface such as a lake, the same is true of the moisture flux with the same, purely atmospheric, resistance r_E and with q₀ = q_sat (T₀), the saturation specific humidity at the surface temperature. However, for moisture transfer over land there is often an additional surface resistance due to the vegetation's stomata! resistance to transfer from moist surfaces within the leaf. This resistance is very large in arid regions and also at certain times in other regions (e.g. at night).

(B) GCM specifications of transfer coefficients. As discussed by Carson (1982), there is a wide range of complexity in the specifications of transfer coefficients. The GFDL and AES models use very simple forms (C_D = C_H = C_E = 2 and 3 x 10^-3 respectively) and do not distinguish between land and sea; similarly simple roughness and stability dependent formulations are used in the EERM, GISS, and MO models. Intermediate, relatively simple forms that depend only on surface type (land or sea) and windspeed are used in the NCAR and LMD models. The roughness lengths (z₀) used in the MO and EERM models over land are fixed (10 and 16 cm respectively), whilst in the GISS model they depend not only on the vegetation type but also on the orographic roughness.

(C) GCM specifications of near-surface variables. The models may be divided into two groups by their specifications of the near-surface values of V_s. q _s, q_s. The simplest approach of taking the values for the lowest model level is used in the GFDL, NCAR, and MO models; for all of these this level is at 70-100 metres above the surface (s = .987-.991).

Values for the EERM model's lowest level (s = .95) are also used directly, except that the wind is turned through an angle dependent on thermal stability, wind speed, and Coriolis parameter. Some such turning is probably beneficial to the accuracy of the surface stress computation even with the lowest level at nearly 100 m. Hansen et al. (1983) found that with realistic cross-isobar angles of surface flow, the ITCZ was sharpened, with decreased rainfall over the southern Sahara and increased Hadley cell mass flux.

Apart from the Randall (GLAS 1982) version of the GLAS model (see Randall 1982), the other models (GLAS 1980, AES, GISS) derive the near-surface values of q_s, and tetas by assuming that the surface flux (eqn. 5) equals a diffusive flux, which, for specific humidity q_s is of the form , with K a stability dependent diffusion coefficient and expressed as the gradient between q_s and the value for the lowest level. One might expect the form for temperature to be , but for the AES model it is with c = 5° C km^-1, whilst for the GISS model it is said to be . In the AES model V_s is also obtained by flux continuity, while in the GLAS 1980 and earlier GISS models, downward extrapolation is used. In the latest GISS model, an Ekman formulation is used with explicit allowance for turning of the wind as discussed above. In middle latitudes this formulation gave increased eddy kinetic energies with warming of high latitudes.

(D) GCM specifications of surface variables. The surface temperature T₀ derived from the subsurface thermal parametrization discussed in the next section, is used for q _c in all the models. The limitation of evaporation in arid conditions is usually allowed for by calculating a potential evaporation E_p from equation (5c) with q₀ = q_sat(T₀) and obtaining the actual evaporation E from

(6)

with b a function of W the normalized soil moisture content W = (m/m_crit), where m is the soil moisture content for the top soil layer, and mcrit the lowest m for which E = E_p, and b = 1 for W>1. The calculation of m is discussed under (B) in Subsurface Processes. In the GFDL, MO, LMD, GISS, and AES models b = W, whilst in the EERM model, a weighted combination of two calculations of E is used, the weights depending on the vegetative cover such that with full vegetation cover b = h_u = 0.9 W² (3-2W). In the NCAR model, b is set to a constant value (0.25). The linear formulation (b = W) has been criticized by Mintz (personal communication) because in arid conditions it gives an excessive value for E_p. To appreciate the problem, it is instructive to consider an alternative formulation of evaporation, that using the PenmanMonteith equation (e.g. Monteith 1973). Equation is written in the resistance form

(7)

Here r_aE is the atmospheric resistance and r_s is the surface resistance (for water vapour).

Because of the difficulty of observing T₀, it is desirable to eliminate it from equation (7). In the Penman-Monteith approach, this is achieved by using the surface energy balance:

(8)

where d T = T₀ - T(z_e), G is downward heat flux into the soil, and r_aH is an atmospheric heat resistance analogous to r_aE.

By defining

so that d T (and so T₀) can be eliminated to give the Penman-Monteith equation:

Priestley and Taylor (1972) analysed observations of drying surfaces using equation (9) and obtained a formulation like (6). However, it differs from (6) because their estimate of the potential evaporation in

(10)

depends only weakly on T₀ and hence on soil moisture. In contrast, in GCMs T₀, and hence qsat (T₀) and so also E_ps, all increase rapidly as the soil dries, so (6), as used in GCMs, is not consistent with observations. One solution to this problem, as proposed by Mintz and Serafini (1981) and used in the 1982 version of the GLAS model, is to use a separate wet surface energy balance to compute the surface temperature needed for the calculation of E_p. Randall (1983) used b = 1 - exp ( - 6.8 W) with this formulation. Randall reports that this formulation gave a considerable reduction in evaporation over subtropical deserts. In the GCM experiments with the 1982 GLAS model, the ground wetness data were based on climatology instead of depending on modelled precipitation and evaporation.

An alternative solution to Mintz and Serafini's may be to introduce the surface resistance r_s explicitly in equation (5c). In one practical application of this approach with observed data, Thompson, Barrie, and Ayles (1982) allow r_s to depend on the minimum stomatal resistance, leaf area index, and available water capacity of different types of vegetation as well as on soil moisture. Evaporation of precipitation intercepted by vegetation is included. An approach of this kind could be used in a GCM. It would allow incorporation of geographical distributions of vegetation characteristics and soil types and the use of multilayer soil models.

Subsurface Processes. (A) Basic formulations. To specify the surface temperature and soil moisture variables needed for calculating the surface turbulent fluxes, some representation of subsurface processes is needed. For a layer of ground between depths z and z + d z, neglecting horizontal subsurface transfers of heat and water, and defining z, G, and the water flux M as positive downward

(11)

(12)

where m is the soil water content, Q_g and N are source and sink terms. For the heat budget (11), the only significant heat sources are due to moisture phase changes. In the moisture budget (12), provided we consider m (and M) to refer to the sum of water vapour and liquid water contents (and flux), there are no sources and sinks except those associated with melting and freezing. With this definition, the "surface" is strictly the interface between vegetation and air and it is there that the surface boundary condition

(13)

applies. The terms in parentheses represent the net contribution of surface and atmospheric processes (rainfall P_R and snowmelt M_S less surface runoff Y(0)) to the downward water flux in the soil, whilst E(0) is the evapotranspiration, which may be partly a sink at the soil surface but in the presence of vegetation also takes water out of the soil throughout the root zone and transfers it to the atmosphere throughout the canopy. For GCMs, with a lowest layer of order 100 m in depth, it is probably unnecessary to apply the boundary condition for the atmospheric model so as to allow for this. However, the distribution of the sink in the soil needs to be taken into account by allowing direct transfer of water to the atmosphere from layers throughout the root zone.

With heat fluxes the heat capacity of the vegetation is probably small enough to justify use of

(14)

as the surface boundary condition.

The treatment of the subsurface moisture fluxes is discussed in the paper by Dooge chap 7). The subsurface heat flux G(z) can be represented by a diffusive term of form

(15)

Typical values of the heat capacity C and conductivity lambdag are given by Geiger (1965); both depend markedly on moisture content so that changes in moisture content due to changes in vegetation or climate can affect G(0). In the tropics G(0) is generally small on seasonal time scales but its diurnal variation can be large for dry surfaces.

(B) GCM treatments of subsurface hydrological processes. Most models represent subsurface water content, generally called soil moisture, by a single variable that is updated according to equation (13), with surface runoff generally represented by limiting the soil moisture to a maximum value (field capacity), generally 100 to 200 kg m^-2. Exceptions to this are the NCAR model (no surface hydrology) and the GLAS and GISS models, with more elaborate parametrizations.

The GISS model has two layers whose water capacities depend on the vegetation characteristics, with diffusion between the layers. During development of the model the upward diffusion coefficient was made infinite during the mid-latitude growing season and all the year in lower latitudes (except deserts) to represent the ready accessibility of water in the deeper soil to vegetation. Together with more rapid downward diffusion of water to the lower layer, this change delayed the limitation of evaporation due to soil dryness by about two months. Runoff is allowed before field capacity is reached, with runoff taken as 0.5 WP_R. The model was quite sensitive to variations in the coefficient; doubling it to unity reduced summer continental temperatures by 7-8 K and increased global mean runoff by 70 mm y^-1.

The 1980 version of the GLAS model treats soil moisture in a way similar to other models but allows runoff of a fraction of the precipitation, this fraction being a nonlinear function of the soil moisture deficit (see Carson 1982).

(C) GCM treatments of subsurface thermal processes. The treatment of subsurface thermal processes in GCMs may be conveniently described in terms of a thermal capacity C. equal to p_gCD as obtained by vertical integration of equation (11) through a depth D. For the GFDL and NCAR models, C_* = 0, so that the substantial diurnal variation of heat storage cannot be represented. Most other models (see Carson 1982 for more details) use a value of C_* roughly appropriate for the diurnal variation of surface temperature by taking , where is the e-folding depth of the diurnal temperature wave (v ₀ = 2p per day). In the MO model, D = d is used, which, with a typical value of C_* gives C_* ~ 21 J cm^-2 K^-1. The LMD model uses C_* ~ 18 J cm^-2 K^-1, whilst in the AES model, l _g and C are specified functions of soil moisture.

Two of the models (EERM and GISS) use two layer treatments with thermal capacity (and also conductivity in the GISS model) dependent on soil moisture. These two-layer treatments allow representation of the seasonal cycle of soil heat flux, though this is unlikely to affect simulations significantly in the tropics.

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