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    Fig. 1.

    Flow diagram of an example of a two-stage coupling. The full set of coupling variables is given in Table 1

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    Fig. 2.

    Time series of sensible heat flux for all eight tiles in TESSEL for a point in Siberia with (top) the old coupling and (bottom) the new implicit scheme. (66.7°N, 155°E) This is from a forecast at T159 resolution with a 1-h time step (20 min in the vertical diffusion scheme because of sub– time stepping). The fractional cover of the different tiles for this point is 0.00, 0.00, 0.53, 0.04, 0.00, 0.37, 0.00, and 0.06 for tiles 1–8, respectively

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A Proposed Structure for Coupling Tiled Surfaces with the Planetary Boundary Layer

M. J. BestMet Office, and Joint Centre for Hydro-Meteorological Research, Wallingford, United Kingdom

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A. BeljaarsEuropean Centre for Medium-Range Weather Forecasts, Reading, United Kingdom

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J. PolcherLaboratoire de Météorologie Dynamique, IPSL CNRS, Paris, France

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P. ViterboEuropean Centre for Medium-Range Weather Forecasts, Reading, United Kingdom

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Abstract

A generalized coupling is proposed between atmospheric models and surface schemes (land and ocean). A set of input and output variables is defined for this purpose in such a way that it can be used by many current and future models, including mosaic or tile schemes. The basic concept is to pass atmospheric variables from the lowest model level and their relation to corresponding fluxes to the surface scheme. The surface scheme returns the fluxes. In this framework, there is no need for the atmospheric model to have detailed information about the surface. Only the result of the surface computations is needed; namely, the fluxes, which are applied as a boundary condition. The equations for fully implicit coupling are derived, and the relevance for numerical stability is demonstrated. It is also shown that fully implicit coupling in a tile scheme leads to more robust results than partially implicit coupling.

Corresponding author address: Anton Beljaars, European Centre for Medium-Range Weather Forecasts, Shinfield Park, Reading RG2 9AX, United Kingdom. Email: anton.beljaars@ecmwf.int

Abstract

A generalized coupling is proposed between atmospheric models and surface schemes (land and ocean). A set of input and output variables is defined for this purpose in such a way that it can be used by many current and future models, including mosaic or tile schemes. The basic concept is to pass atmospheric variables from the lowest model level and their relation to corresponding fluxes to the surface scheme. The surface scheme returns the fluxes. In this framework, there is no need for the atmospheric model to have detailed information about the surface. Only the result of the surface computations is needed; namely, the fluxes, which are applied as a boundary condition. The equations for fully implicit coupling are derived, and the relevance for numerical stability is demonstrated. It is also shown that fully implicit coupling in a tile scheme leads to more robust results than partially implicit coupling.

Corresponding author address: Anton Beljaars, European Centre for Medium-Range Weather Forecasts, Shinfield Park, Reading RG2 9AX, United Kingdom. Email: anton.beljaars@ecmwf.int

1. Introduction

At present there is no generic structure in the way that land surface schemes are coupled to boundary layer schemes. This means that it would be difficult to interchange land surface schemes within any given general circulation model (GCM). To help us to understand the different behaviors of various land surface schemes and ultimately to further our understanding of the complex exchange between the surface and the atmosphere, it is desirable to be able to use different land surface schemes within the same GCM. This will allow the attribution of differences in the model as a whole to differences in land surface schemes. We therefore need to design a consistent “plug compatible” structure for the interface between these land surface schemes and the boundary layer schemes. The interface needs not only to cover all the physical processes that are currently treated at the interface, but it should also be compatible with all existing numerical schemes.

There are several ways in which a surface scheme can be coupled to a GCM depending upon how implicit or explicit this coupling should be. Polcher et al. (1998) discuss how the different options may be achieved. Based on findings in the Project for Intercomparison of Land-surface Parameterization Schemes, they propose a standard interface for the coupling between the surface and the atmosphere. However, recent developments in both surface schemes and boundary layer schemes mean that we need to readdress the work that was presented in that paper to make the coupling more general.

Recent land surface models have a simple representation of surface heterogeneity by coupling the atmosphere to a series of subsurfaces in a mosaic or tile approach (Koster and Suarez 1992; Ducoudré et al. 1993; van den Hurk et al. 2000; Essery et al. 2003). The idea is that the internal boundary layers from subsurfaces have merged at or below the lowest model level (blending height; see, e.g., Claussen 1995), and therefore the different tiles can be coupled to a single homogenized boundary layer. This is only possible if the subsurfaces have limited horizontal scales (of the order of 10–100 times the height of the lowest model level). For subsurfaces with large horizontal scales it would be necessary to maintain the surface heterogeneity throughout the boundary layer. How adequate and accurate the different representations are is still a matter of active research (e.g., Doran and Zhong 2002). For a universal coupler it is sufficient to return tile-averaged fluxes to the atmospheric model, because atmospheric models that resolve the tile structure will need coupling tile by tile, and so it is similar to a high-resolution model that resolves the heterogeneity.

The time scale of the processes involved is often short in comparison with the model time step, and therefore a fully implicit coupling between atmosphere and land surface may be required. It means that vertical exchange processes in the boundary layer and its coupling to the top layer of the surface are formulated in terms of variables at the new time level. The implication is that the new time-level variables cannot be computed level by level in an explicit way, but that a tridiagonal matrix system has to be solved that involves the top layer of the land surface scheme, including the tiles (Polcher et al. 1998). The benefit brought by an implicit coupling to the surface–atmosphere interaction was recently shown by Schultz et al. (2001). A generalized coupler should thus be able to accommodate for this option.

The purpose of this paper is to define the variables that need to be passed between the atmospheric model and the land surface scheme and to show how a fully implicit coupling can be achieved with a tiled land surface model. It is also demonstrated that a fully implicit coupling is more stable and less noisy than a partially implicit coupling. The more general problem of a coupler that accounts for changes in resolution and/or grid structure between atmosphere and surface adds considerable numerical complication and is beyond the scope of this paper (e.g., Jones 1999). Having a coupling strategy that allows for a tiled surface should help to handle this problem, but it is not clear at this stage whether the interpolation procedures would interfere with numerical stability.

2. Definition of the atmosphere–surface interface

Deciding how to define the interface between the atmosphere and the land surface is by no means trivial. There are basically two options for coupling: (i) at the surface, by providing boundary conditions for temperature, moisture, and wind to the atmosphere, that is, Dirichlet-type boundary conditions, or (ii) at the lowest model level, by providing the atmospheric model with surface fluxes, that is, Neumann-type boundary conditions. In the following subsections, advantages and disadvantages of the two options are discussed.

a. Heat and moisture fluxes

Coupling with surface variables is used in many models but usually leads to an intimate coupling of atmospheric and land surface code. The reason is that many land surface–related parameters are necessary in the atmospheric model, for example, roughness lengths, moisture availability, vegetation characteristics, and skin temperature. In the case of a tiled land surface, all these parameters are tile dependent. It is therefore more convenient to couple at the lowest model level by passing its variables to the land surface scheme, which returns the surface fluxes to the atmosphere. Fluxes can be tile averaged for most models, because the atmosphere does not distinguish between tiles, and so it is sufficient to return a single flux. Models that maintain the tile structure in the boundary layer may need fluxes from individual tiles, but this is not a fundamental problem. The disadvantage is that the turbulent exchange between the lowest model level and the surface becomes part of the land surface module. Turbulent exchange is traditionally part of the atmospheric vertical diffusion scheme, and ideally the surface layer (between the lowest model level and the surface) should be consistent with the parameterization of the outer layer (above the surface layer). However, many models have different formulations for turbulence in the surface layer and the outer layer. For instance, the European Centre for Medium-Range Weather Forecasts (ECMWF) model has observed Monin–Obukhov similarity functions in the surface layer, but empirically tuned stability functions in the outer part of the stable boundary layer (Viterbo et al. 1999). Also, the numerical formulation of turbulent transport in the surface layer is fundamentally different from the one in the outer layer. In the surface layer, bulk transfer formulations are used, whereas in the outer layer central differencing is applied with evaluation of diffusion coefficients on flux levels defined between model levels. Furthermore, the aerodynamic coupling with the surface needs surface characteristics. Often the surface moisture driving the fluxes is taken to be the saturated value at the surface temperature, but this is by no means universal, and alternatives are possible (Milly 1992). It is therefore better to have all the complexity of the surface coupling in the surface scheme and to exchange well defined and “measurable” quantities like heat and moisture fluxes with the atmosphere.

So for tiled schemes, there is a clear advantage in coupling at the lowest model level with the surface scheme providing fluxes to the atmospheric model. Also for the ocean, it is not a limitation to couple in this way, and in the future there may be additional advantages. It will rationalize the coupling to wave models, simplify the coupling when fast skin variables are introduced at the ocean surface (cool skin, warm layer; e.g., Beljaars 1999), and allow land and water tiles at coastlines. The surface model needs to apply to all surfaces, that is, land and ocean, including sea ice. The atmosphere only sees fluxes and does not need to know about typical land surface variables like skin temperatures, stomatal resistances, and roughness lengths.

b. Momentum fluxes

The momentum boundary condition is usually simpler, that is, a no-slip condition at the surface, although ocean currents may be considered in the future (Janssen et al. 2003). The main difficulty is again over tiled surfaces.

To enable complete plug compatibility, the atmospheric model should not need to know any details about how the land surface scheme solves its equations. This means that setting the no-slip condition in the atmospheric model is not an option, because then the surface code needs to pass information on the roughness length, which can be an aggregated one (“effective roughness length”) or roughness information on the individual tiles.

It is better to leave the momentum flux computation to the land surface code, because it can do a proper weighting of drag over all the tiles to produce a gridbox value.

To enable the coupling of the land surface scheme and the atmospheric model to remain as general as possible, we need to be able to predict the type of changes that may be made to the parameterizations within the near future. One such change is likely to be the way in which the orographic drag is computed within the model. Many models have an orographic contribution to the aerodynamic roughness length that is independent of the tiles and some compensation in the roughness length for heat and moisture transfer (Mason 1985; Taylor et al. 1989; Hewer and Wood 1998). However, new boundary layer schemes are being developed that directly put a distributed drag profile within the boundary layer (Wood et al. 2001; Beljaars et al. 2004). This means that the orographic drag needs to be in the atmospheric code and not in the land surface scheme. Models that use the effective-roughness-length concept should isolate the drag due to subgrid orography and apply it as a separate drag term to the lowest model level of the atmospheric model. In this way the land surface scheme can have its own parameterization of roughness lengths independent of orographic form drag.

c. Explicit and implicit solutions

To maintain generality, both implicit and explicit coupling should be an option. Implicit coupling is often necessary for stability because the time scales of the processes can be shorter than the time step of the model. In this case the land surface computations should be done between the downward and upward sweeps of the tridiagonal matrix equation for updating the boundary layer state variables, that is, the boundary layer scheme straddles the land surface scheme (see Fig. 1). However, some boundary layer schemes need surface fluxes in the parameterization of the diffusion coefficients (e.g., Troen and Mahrt 1986; Beljaars and Viterbo 1998). The diffusion coefficients are always evaluated using profiles and/or fluxes from the previous time level. It is possible to keep the fluxes from the previous time level, but for technical reasons it is sometimes preferable to recompute them just before evaluation of the diffusion coefficients. It is also the only option for the first time step. Thus, an explicit coupling must be an option of the interface. In this case the lowest-model-level variables are specified explicitly, and the land surface scheme must return the fluxes. We call this the “diagnostic mode” because it is used to estimate fluxes, for example, to set diffusion coefficients before doing time integration. With the diagnostic mode called before the implicit computation, a two-stage coupling is necessary, as illustrated in Fig. 1.

To achieve fully implicit coupling, it is necessary to pass a linear relation between the lowest model level variables for wind, dry static energy, and specific humidity (Unl, Vnl, Snl, qnl) and the corresponding fluxes (τx, τy, H, E)1:
i1525-7541-5-6-1271-e1
where the superscript n indicates the time level. The time index n can be the new time level t + 1 as in an implicit scheme, or a time level extrapolated in time (overimplicit) to avoid nonlinear instability (Kalnay and Kanamitsu 1988; Beljaars 1992). With implicit coupling, the As and Bs are the result of the downward elimination of the tridiagonal matrix of the vertical diffusion equation. The scheme is implicit because Unl, Vnl, Snl, and qnl are only known after the computation of fluxes (τx, τy, H, E) by the surface scheme. We call this the “time-stepping mode,” because the atmospheric model as well as the surface scheme do one integration step in time.

In numerical models the vertical diffusion scheme has to be implicit, and so the coupling is through the As and Bs as the result of the tridiagonal elimination. However, in stand-alone simulations, the lowest-model-level variables are replaced by an imposed time series of parameters, which can be from observations or atmospheric simulations. This can be done by using the B coefficients and by setting the As to zero.

3. The surface scheme

The surface scheme needs to provide the atmospheric model with the information on the surface boundary condition. The different input and output variables are described in Table 1. The fluxes, as suggested in the table, are tile-aggregated ones, which assumes that the tile structure only exists in the surface model. As has been described above, two modes of coupling are required: a diagnostic mode to diagnose preliminary fluxes to support the parameterization and a time-stepping mode with full implicit coupling during the time integration. Also, a third mode (“enquiry mode”) is needed to make surface quantities such as albedo and emissivity available to the atmospheric model. The diagnostic and time-stepping modes are described in the following two subsections.

a. Explicit coupling (diagnostic mode)

For the explicit coupling used in the diagnostic mode of the land surface model, the atmospheric conditions are prescribed, and it is sufficient to express the exchange of heat and moisture with the help of the transfer coefficients CM, CH, and CQ into the differences between the lowest-model-level and surface variables:
i1525-7541-5-6-1271-e2
where τx, τy, H, and E are the momentum fluxes, the heat flux, and the moisture flux; α and β are moisture availability coefficients; Sl and Ss (=cpTs) are dry static energy at the lowest model level and at the surface; ql is specific humidity at the lowest model level; qsat(Ts) is the saturation specific humidity at surface temperature Ts; cp is heat capacity at constant pressure; and ρ is density. Superscript t indicates the old time level. The surface scheme uses the surface temperature Tts and the moisture availability coefficients α and β from the old time level, and the fluxes can simply be obtained from Eqs. (2). So there is no time stepping involved in this mode.

b. Implicit computation using the surface energy balance equation (time-stepping mode)

In this section we consider a homogeneous surface, which can be the surface cover in a full grid box, but also a single tile as part of a tiled surface. Heat and moisture fluxes are discussed only, because the momentum fluxes are relatively simple because of the no-slip boundary condition. The stresses can simply be obtained by replacing time level t of the linear part of Eqs. (2) by the new (implicit) time level n and combining it with Eqs. (1).

The derivation of heat and moisture fluxes follows a Penman–Monteith approach in which the equations are linearized and the surface temperature is eliminated. First the fluxes of heat and moisture are rewritten in implicit form, that is, with surface variables at time level n:
i1525-7541-5-6-1271-e3
In Eqs. (3) the saturated humidity at the surface is linearized with respect to the surface temperature at the previous time level Tts, and the equations are written as a linear expression in Snl, qnl, and Sns:
i1525-7541-5-6-1271-e4
with coefficients
i1525-7541-5-6-1271-e5
The energy balance equation for the top soil layer with heat capacity C and temperature Ts reads
i1525-7541-5-6-1271-e6
where RSW and RLW represent net shortwave and net longwave radiation at the surface, L is the latent heat of evaporation, and Λ(TsTd) is the heat flux to a deeper layer with temperature Td, using a conductivity Λ. Equation (6) can also be used for a skin layer with zero heat capacity (C = 0) as in the ECMWF model (Viterbo and Beljaars 1995). In that case, Td represents the temperature of the top soil layer and Λ is the skin layer conductivity. Combining this equation with Eq. (4), it can be rewritten in the following form (dropping the coefficients that are zero):
i1525-7541-5-6-1271-e7
In the longwave radiation, the effect of the surface temperature on the outgoing radiation has been included by linearizing with respect to Tts. It is usually assumed that the downward longwave radiation is independent of surface temperature so that dRLW/dTs = −4εsσT3s, with εs being the surface emissivity and σ being the Stefan– Boltzmann constant. If the radiation code is not called every time step to save computer time, the previous radiation time level should be used as reference. Equation (7) is linear in Sns (or Tns) and can be written in the following form:
i1525-7541-5-6-1271-e8
The surface dry static energy Sns can be eliminated from Eqs. (4), and the fluxes can be rewritten as
i1525-7541-5-6-1271-e9
with
i1525-7541-5-6-1271-e10
With Eq. (9), a linear expression of fluxes in terms of lowest-model-level variables is available. Combined with Eq. (1), it gives a fully implicit solution in both atmosphere and surface variables. In applications for which the lowest model variables are known (e.g., in stand-alone simulations forced by an observed time series), Eq. (9) provides the fluxes from the land surface scheme. The surface scheme is still implicit because the surface temperature responds implicitly to the forcing.

c. Implicit coupling of a tile scheme

With tile (or mosaic) schemes, the surface fluxes are calculated for several different types of surface and then these fluxes are averaged according to the fraction of each type of surface in a grid box. In other words, a weighted averaged of the fluxes in Eq. (9) is required.

Since the lowest-model-level variables are independent of the tiles, the weighted average of the fluxes over the tiles can be obtained by taking a weighted average of the coefficients in Eqs. (10). Superscript i is used to identify the ith tile, and νi is the fractional surface area of the ith tile:
i1525-7541-5-6-1271-e11
The overbar indicates the gridbox average of the fluxes. The next step is to write Eqs. (1) with mean fluxes
i1525-7541-5-6-1271-e12
The A and B coefficients are the result of the downward elimination of the tridiagonal matrix of the vertical diffusion equations. It is straightforward now to solve H, E, Snl, and qnl from Eqs. (11) and (12).

4. Results from the ECMWF model

In this section, two versions of the ECMWF land surface scheme [called the Tiled ECMWF Scheme for Surface Exchanges over Land (TESSEL); van den Hurk et al. 2000] are compared for a point with noisy tile fluxes. The difference between the two versions is in the implicitness of the tile coupling. The results presented here are comparable to those of Schultz et al. (2001) except that in this paper the complexity of the subgrid variability is included.

In the TESSEL scheme, the skin level of each tile (open water, frozen water, wet interception reservoir, low vegetation, exposed snow, high vegetation, sheltered snow, and bare soil) is connected to the lowest model level by means of an aerodynamic resistance (inverse of transfer coefficient times wind speed). For moisture, there can be an additional canopy resistance to control evaporation/transpiration from the vegetation or bare soil. The skin level has no heat capacity and responds instantaneously to the energy forcing. The scheme can be represented by the equations presented in the previous sections.

In the original implementation, the tile coupling was solved in a way that was not fully implicit. Instead of Eqs. (1), the tridiagonal solver provided a linear relation between the lowest-model-level dry static energy and moisture and the skin values of temperature and moisture. These relations were used together with the surface energy equation for each tile to solve for the skin temperature. In this procedure it is assumed that the tile for which the skin temperature is solved covers 100% of the grid box. After the skin temperatures are found, they are kept constant and used as a fixed boundary condition. The contribution to the fluxes from the tiles is weighted by the fraction of the respective tile to solve for the lowest-model-level values of S and q.

This procedure is only fully implicit for a tile that covers 100% of the gridbox area. When small fractions are present, the lowest model level is mainly affected by the tiles that cover substantial fractions, and the procedure is not implicit, because the lowest-model-level changes caused by other tiles have no influence on the tile with a small fraction. The consequence in the ECMWF implementation is that the results can be noisy and that even blowups occur in some experimental model versions.

The new coupling as described in the previous sections has also been coded in the ECMWF model and compared with the old coupling. The code has been verified by artificially setting the tile fraction to 100% successively for the eight tiles in the old as well as the new scheme. For this situation, the old code and the new code give identical results for the tile under consideration.

To see the impact of the new surface coupling, a global forecast was run at spectral T159 resolution (about 110 km) for 12 h with the old and the new coupling. The initial condition is the ECMWF operational analysis of 1200 UTC 15 December 2002 interpolated to this resolution. The time step is 1 h. This case was selected, because it is known to be close to instability at one particular point, although the global flux fields simulated with the two schemes appear virtually identical.

The point close to instability is shown in Fig. 2. Note that the design of the code is such that tiles with zero or negligible fraction are still part of the computation, although they do not contribute to the averaged fluxes. This choice was made for simplicity because it avoids selecting a threshold and conditional code inside all the loops over the tiles. The disadvantage is that instabilities in nonrelevant tiles can cause blowups. In this particular case, the heat flux for some of the tiles goes to very extreme values, particularly for the snow tiles 5 and 7 and, to a lesser extent, for the high-vegetation tile 6. The new scheme does not have this problem, because it is fully implicit and therefore much more stable. So it can be concluded that the new way of coupling is more robust and stable, as can be expected from an implicit scheme. The new code is also much simpler and easier to understand.

5. Concluding remarks

In this paper we have discussed a general method for coupling land surface schemes to the atmosphere. The method described employs an implicit coupling, which increases the stability of the coupling and maintains consistency between the atmosphere and the land surface.

Although the general principles of this coupling were first presented by Polcher et al. (1998), we have extended the coupling method to ensure that it is more general and, therefore, more applicable to future land surface and boundary layer schemes. The details of how a tiled (or mosaic) land surface scheme can also be coupled using this method have been given in the paper.

Different coupling methods, such as using fluxes or state variables, have been discussed. It has been argued that coupling through fluxes is the most logical option. Furthermore it has been demonstrated that it allows for a straightforward implicit coupling for tile schemes. So it is recommended in this paper that all surface fluxes (including the momentum flux from vegetative roughness elements) be calculated within the land surface scheme.

Acknowledgments

The authors thank the anonymous reviewers for their detailed and constructive comments. Their input helped considerably to improve the manuscript.

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Fig. 1.
Fig. 1.

Flow diagram of an example of a two-stage coupling. The full set of coupling variables is given in Table 1

Citation: Journal of Hydrometeorology 5, 6; 10.1175/JHM-382.1

Fig. 2.
Fig. 2.

Time series of sensible heat flux for all eight tiles in TESSEL for a point in Siberia with (top) the old coupling and (bottom) the new implicit scheme. (66.7°N, 155°E) This is from a forecast at T159 resolution with a 1-h time step (20 min in the vertical diffusion scheme because of sub– time stepping). The fractional cover of the different tiles for this point is 0.00, 0.00, 0.53, 0.04, 0.00, 0.37, 0.00, and 0.06 for tiles 1–8, respectively

Citation: Journal of Hydrometeorology 5, 6; 10.1175/JHM-382.1

Table 1.

Variables to be passed within the coupling scheme. This table does not include the domain-describing variables such as geographical coordinates, time, or height of atmospheric levels. The surface scheme should respond with output dependent on a mode flag (enquiry mode, explicit mode, or time-stepping mode). The output variables are all tile averaged

Table 1.

1

Another way of achieving implicit coupling is by specifying a relation between lowest-model-level variables and surface variables as described by Polcher et al. (1998). However, it will be shown that there are clear advantages to using the form of Eqs. (1) for tiled surfaces.

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