Coupling the atmosphere with the underlying surface presents numerical stability challenges in cost-effective model integrations used for operational weather prediction or climate simulations. These are due to the choice of large integration time steps compared to the physical timescale of the problem, aiming at reducing computational burden, and to an explicit flux coupling formulation, often preferred for its simplicity and modularity. Atmospheric models therefore use the surface-layer temperatures (representative of the uppermost soil, snow, ice, water, etc.) at the previous integration time step in all surface–atmosphere heat-flux calculations and prescribe fluxes to be used in the surface model integrations. Although both models may use implicit formulations for the time steps, the explicit flux coupling can still lead to instabilities.

In this study, idealized simulations with a fully coupled implicit system are performed to derive an empirical relation between surface heat flux and surface temperature at the new time level. Such a relation mimics the fully implicit formulation by allowing one to estimate the surface temperature at the new time level without solving the surface heat diffusion problem. It is based on similarity reasoning and applies to any medium with constant heat diffusion and heat capacity parameters. The advantage is that modularity of the code is maintained and that the heat flux can be computed in the atmospheric model in such a way that instabilities in the snow or ice code are avoided. Applicability to snow–ice–soil models with variable density is discussed, and the loss of accuracy turns out to be small. A formal stability analysis confirms that the parametrized implicit-flux coupling is unconditionally stable.

Coupling atmospheric models to the underlying surface model involves both
scientific and technical issues. Models of the atmospheric circulation tend to
be computer intensive and therefore often employ long time steps (up to 1 h), which is a challenge for stability and accuracy

The ideal solution for stability is to combine the boundary
layer heat diffusion and, e.g.,
the snow or ice layer diffusion in a single implicit solver. This has been
demonstrated in a series of papers describing developments in
the ORCHIDEE model

Ongoing work at ECMWF on snow modelling raised similar issues. The existing single layer
snow model

In this paper, we propose a solution, that has the simplicity and modularity of the explicit flux coupling, but still has the stability of the fully implicit system. To derive simple solutions, the fully implicit coupled system is used as a reference. It is shown that the tri-diagonal set of equations corresponding to the discretized diffusion equation (for snow, ice, or soil) can be converted to a relation between temperature and heat flux at the surface. The coefficients in this relation are then parametrized depending on properties of the medium, time step, and vertical discretization. The coefficients are put in dimensionless form, which makes the empirical coefficients universal and applicable to any medium and any discretization.

The experimental environment in this paper is a simple model of a near-surface air layer coupled to a snowpack by turbulent exchange. The atmosphere (e.g. at a height of 10 m, typical for atmospheric models) is assumed to have a diurnal cycle, and the response of temperature in the snowpack is considered. Although the following sections refer to snow only, the dimensionless framework ensures that the outcome is valid for any medium.

The following two sections (2 and 3) describe the equations for the
discretized snow layer and the turbulent coupling between atmosphere and
snow. Sections 4, 5, and 6 describe the numerical solution for an idealized
diurnal cycle, the parametrization of the coefficients that relate heat flux
and top layer snow temperature and the testing of the proposed scheme.
Finally, the results and their applicability are briefly discussed in the
concluding section. Also the implications of non-uniform snow density are
discussed. The numerical solver and a formal stability analysis are described
in Appendices

The numerical grid is defined by the position of the half levels,
i.e. the thickness of the layers. The full levels are in the middle of the
layers, i.e.

We consider the diffusion equation for temperature in snow

In case

To focus on stability of the atmosphere surface coupling, it is assumed that
the evolution of the near atmospheric temperature is known, e.g. as in
stand-alone simulations of the land surface. However, this is not a limitation
in full three-dimensional (3-D) models that typically use an implicit solver for the turbulent
diffusion. In that case the atmospheric model will perform the downward
elimination process (the same way as described in Appendix

With a prescribed air temperature, the heat flux into the snow layer can be
related to the air–surface temperature difference in the following way:

The coupling through a transfer coefficient is standard and represents
the integral profile function according to Monin–Obukhov (MO) similarity

List of parameters used in the idealized simulation of a snow layer.

In the vertically discretized snow (see Fig.

Explicit flux coupling.
This is the traditional approach where the expression for the surface flux
uses the previous time level of the surface temperature leading to the
following discretization of Eq. (

Implicit flux coupling.
The discretization of Eq. (

In this section, solutions are considered for a 1 m thick snow layer with
constant heat capacity and heat diffusion coefficients. Idealized temperature
forcing from the atmosphere is prescribed as a sinusoidal diurnal cycle. The
choice of constants is documented in Table

Diurnal cycle time series of snow skin temperature (left column) and surface heat flux (right column). The simulations were made with 0.2, 0.02, and 0.002 m vertical resolution (top, middle, and bottom panels). The blue curves refer to the fully implicit solution (IMPL); the red curves indicate the solutions with explicit flux coupling (EXPFLX). The solid curves are with a time step of 3600 s and the dashed curves with 100 s.

The first thing to note is that amplitude and phase of the skin temperature
diurnal cycle only have a small dependence on vertical resolution. This is
surprising because the amplitude of diurnal cycle of layer 1 with

Although it is impossible to draw general conclusions about accuracy from
limited experimentation, we note that the fully implicit solution with

Finally, the explicit coupling turns out to be unstable for very thin snow
layers (see lower panels in Fig.

Because of the good stability and accuracy characteristics, we develop in the
next section a parametric form of

As suggested above, it is desirable to have all the flux formulations (also
for the atmosphere–surface exchange) at the new time level

For that purpose, we make use of similarity theory for the diffusion equation
with constant coefficients. If we think of an infinite medium (thick snow
layer) with uniform temperature

Similarly, we can apply an external forcing by suddenly applying a heat
flux

Dimensionless function

The scaling arguments above apply to the continuous system. For the
discretized system, the scaling behaviour of

For small ratios of

Empirical estimates of parameter

Diurnal cycle series of skin temperature
(left columns and) and surface heat flux (right columns). The simulations
were made with 0.2, 0.02, and 0.002 m resolution (top, middle, and bottom
panels). The blue curve refers to the fully implicit solution (IMPL); the
black solid curve is the solution with parametrized

The second parameter for which an empirical formulation is needed is

From Figs.

With the empirical formulations for

Finally, the scheme was further simplified by using the parametric form for

Numerical stability is a critical issue for atmospheric models that are coupled to a fast responding surface, e.g., through a thin snow or ice layer. Very thin snow layers can occur in early winter after the first snow fall and during melt in spring. A fine discretization may also be desirable to allow for a fast response of the surface temperature to changes in radiation. Formal stability analysis confirms that unconditional stability can be achieved by a fully implicit coupling between atmosphere and surface.

Fully implicit coupling leads to a tri-diagonal problem in which atmosphere and surface are solved simultaneously. In practice, often explicit flux coupling is applied; the atmospheric model uses the surface temperature of the previous time level to compute the surface heat flux, which is used later as boundary condition for the heat diffusion in the surface. Explicit surface coupling puts stability limits on the thickness of the top snow layer and on the time step. Explicit flux coupling is often applied, because existing codes do not necessarily have sufficient modularity to support fully implicit coupling.

Although the atmosphere–surface heat diffusion leads to a single tri-diagonal matrix problem, one can also break it up in different steps. It is shown that the elimination part of the solver of the snow heat diffusion problem leads to a linear relation between surface temperature and surface heat flux. This relation can be used together with the atmosphere–surface interaction formulation to solve for the surface heat flux.

Dimensionless

A simple method has been developed to approximate the coefficients in this linear relation. The coefficients are scaled with the characteristic scales of the diffusion equation. This makes the result universal and applicable to an arbitrary medium, e.g. snow, ice, or soil. The depth scale that characterizes the penetration of a perturbation over a time step, turns out to play a crucial role. In this paper the relevant empirical function is “measured” by solving the diffusion equation for a range of vertical resolutions and time steps.

Finally, the empirical functions are used to solve for the coupled diffusion problem and compared with the fully implicit computations. The results are very close. The advantage of the method is that the surface fluxes can be computed without calling any surface code, and behaves like explicit flux coupling. The only difference is that the surface heat-flux expression has a damping term depending on the time step. This damping term is the result of the change of surface temperature related to the heat flux, and stabilizes the result.

Dimensionless

The scaling argument used above only applies for a diffusion equation with
constant properties of the medium. However, in reality there may be a profile
of, e.g., snow density as snow becomes more and more compact in deeper layers,
or vertical resolution may be variable. The latter is numerically equivalent
to a variable diffusion coefficient

In fact the aerodynamic coupling between atmosphere and snow can be interpreted as a big jump in the properties of the medium

. As a simple test, a case was selected where the profile of density is 150 kg mWe conclude that making an estimate of the relation between heat flux and
surface temperature is a practical solution to support explicit flux coupling
and to combine numerical stability for long time steps with a modular code
structure. A formal stability analysis in Appendix

The data that are used in this paper has been produced with a dedicated stand-alone Fortran program. ECMWF's data policy does not allow open access to software. However, the code can be obtained from the first author, subject to license. The license implies non-commercial use, i.e. for research and education only.

The set of equations discussed in Sect.

If the surface heat flux is not known, the first line of the matrix equation
contains a linear relation between

In this section we present the stability properties of the three coupling
methods introduced in the present study, namely the explicit flux coupling
(EXPFLX), the implicit flux coupling (IMPFLX), and the parametrized implicit
flux coupling (IMPPAR). Since the numerical stability is expected to be
greatly influenced by the numerical treatment of the surface boundary
condition, a classical von Neumann stability analysis, which assumes periodic
boundary conditions, would not be adequate. For this reason our study is
based on a matrix stability analysis

Values of

Spectral radius of the matrix

Without loss of generality, we consider in the following that

Explicit flux coupling: the surface temperature involved
in the computation of

Implicit flux coupling: the surface temperature
at time level

Parametrized implicit flux coupling: the temperature
at time level

As shown in Appendix

For the
special cases

Numerical results are obtained for NL

Maximum value of

Empirically, it can be found that the stability condition for the explicit
flux coupling roughly behaves like

The authors declare that they have no conflict of interest.

The authors would like to thank Alex West and Jan Polcher for their
comprehensive reviews and for suggesting numerous improvements to the
manuscript. F. Lemarié acknowledges the support of the French National
Research Agency (ANR) through contract ANR-16-CE01-0007. We also thank Erland
Källén and Nils Wedi for careful reading of an early version of the
manuscript.Edited by: S. Valcke
Reviewed by: A. E. West, F. Lemarié

F. Lemarié reviewed the discussion paper and was added as a co-author during the revision.

, and J. Polcher