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Keywords:

  • diffusion;
  • exchange coefficient;
  • weather and climate models

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Fractional time-stepping schemes for the nonlinear diffusion equation
  5. 3. Tuning the parameter K
  6. 4. Situation-dependent adjustments of the decentring and the exchange coefficient
  7. 5. Discussion and conclusions
  8. Acknowledgements
  9. Appendix A. Proof that ρ(γ) ≤ 1
  10. Appendix B. Proof that γopt is linearly stable for P = 1 when η ≥ 1/2.
  11. References

This article addresses the role of the tuning of the parameters of the subgrid parametrizations of diffusive processes in numerical models. To this end, it examines the stability and accuracy of a set of fractional time-split integration schemes for the nonlinear diffusion equation. The best values of the time-decentring parameters are established. The dependency of the diffusion on the exchange coefficient is tuned to get the most accurate simulation of the steady-state solution and this is then tested in a non-steady-state model integration. The following questions are then addressed: (i) what is the relative role of parameter ‘tuning’ compared to the choice of the numerical scheme? and (ii) what is the effect of using a parametrization in one model that has been tuned in another model?

The presented tests suggest that the best results are obtained by controlling the details of the numerics while taking the physical value of the exchange coefficient, instead of tuning the physics parametrization. This suggests that it is beneficial to consider ‘tuning’ the physics parametrization schemes via the freedom in the numerics as an alternative to tuning the parameters of the physical processes. Copyright © 2012 Royal Meteorological Society


1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Fractional time-stepping schemes for the nonlinear diffusion equation
  5. 3. Tuning the parameter K
  6. 4. Situation-dependent adjustments of the decentring and the exchange coefficient
  7. 5. Discussion and conclusions
  8. Acknowledgements
  9. Appendix A. Proof that ρ(γ) ≤ 1
  10. Appendix B. Proof that γopt is linearly stable for P = 1 when η ≥ 1/2.
  11. References

Numerical atmospheric models consist of a dynamical core that integrates a set of equations (hydrostatic primitive equations, non-hydrostatic inviscid Euler equations, or variations thereof) that govern the grid-scale-resolved fields and a set of physics parametrizations that estimate the exchanges of mass, energy and momentum by the unresolved processes. These are typically all the relevant subgrid processes such as turbulence, deep convection, and subgrid orographic interactions. The equations for the radiative energy fluxes are generally too complex to be solved in operational weather and climate models without serious simplifications, so, in practice, radiation schemes also have the character of a parametrization. Many of the subgrid parametrizations add some diffusivity to the system.

Physics parametrizations in general do not solve any theoretical equations but rather use empirical relations to make approximate estimates of their impact on the resolved-scale dynamics via exchanges in the budgets of energy, momentum and the masses of the different water species. Consequently, they contain many parameters that have no absolute physical equivalent in nature and which are subject to a certain degree of arbitrariness. This induces considerable errors in the models, on top of those originating from the discretisations in the numerical schemes of the dynamical core. In practice, this freedom of choice is used to ‘tune’ these parameters to achieve better model performance. For instance, in numerical weather prediction (NWP), such tunings are carried out by minimising model verification scores with respect to observations. In climate models, the parametrizations are tuned to obtain the best model climatology over a long period in the past.

Besides discretisation errors of the dynamics, models also have numerical errors originating from the way in which the physics are coupled to the dynamics, e.g. Staniforth et al. (2002). In atmospheric models, such parametrizations are computed by distinct numerical packages. They change the model state, and are then aggregated and added to the atmospheric state. However there is a certain degree of freedom in how such results can be combined. For instance, one may compute all the contributions separately and then add the tendencies of the variables. This is called parallel coupling. On the other hand, one may establish a computation order for the different processes and provide the result of each process to the next one. This is called sequential coupling. As shown in Termonia and Hamdi (2007), this has an impact on the numerical properties of the entire scheme of the model.

Tuning of parametrizations represents a substantial part of the process of model building. Some attempts have been made to automate this by using data assimilation techniques and ensemble methods, e.g. Annan et al. (2005). But most often, tuning is considered an art rather than a science, i.e. there is no systematic understanding nor a science-based methodology for the process of tuning. The main goal of the present article is an attempt to advance this understanding by addressing the question: What is the impact of parameter tuning on model errors compared with errors originating from the numerical techniques? In order to make this a tractable problem, this question will be limited to the study of diffusive processes, isolated from the interaction with the dynamics.

Diffusive processes often take the form of a differential operator expressed in terms of exchange coefficients. The form of these exchange coefficients can vary, e.g. Louis (1979), Louis et al. (1981), Cuxart et al. (2000) and Redelsperger et al. (2001). Such schemes rely on some closure assumptions such that their actual behaviour is only an approximation of reality. In practice, the tunings of these schemes often appear in the choice of the situation dependency on parameters like the Richardson number (Geleyn, 1987) and the mixing length (Teixeira and Cheinet, 2004).

Besides diffusive physics processes, models should be supplemented with an amount of numerical diffusion in the dynamical core in order to prevent an accumulation of energy at the spatial scales near the model resolution. Váňa et al. (2008) developed a quite sophisticated horizontal diffusion scheme, the so-called semi-Lagrangian horizontal diffusion (SLHD). As has been shown by Barkmeijer (2010), the tuning of such a scheme can also have a substantial impact on the behaviour of deep convective systems in convection-permitting models.

The exchange coefficients in turbulence schemes are nonlinear functions of the meteorological variables. As a consequence, the numerical schemes may start to exhibit nonlinear behaviour, which may result in spurious oscillations (Kalnay and Kanamitsu, 1988, denoted as KK88 henceforth). As shown in that article, this can be improved by introducing or increasing the time decentring of the numerical scheme that solves the diffusion equation. Interestingly, Bénard et al. (2000) proposed a scheme where this decentring is situation-dependent. This raises the following question: Is it better to tune physical parametrizations such as, for instance, the dependence of the turbulent exchange coefficients on the Richardson number and the mixing length, as mentioned above, or could such practice of adapting physical quantities be replaced by a ‘tuning’ of the details of the numerics?

The latter case is much more attractive since the discretisation of the equations is a free choice anyway, whereas notions such as Richardson numbers and mixing lengths have a physical interpretation and the formulation of the turbulence exchange processes should be based as much as possible on empirical knowledge (even if our understanding of turbulence is still subject to considerable uncertainties).

In the European modelling community, it is becoming increasingly important for physics parametrizations that have been tuned for one NWP model to be used in another model. For instance, within the European ALADIN and HIRLAM consortia*, it has been decided to exchange parametrizations developed in different models to be plugged into a dynamical core that is commonly shared between the two consortia. Specifically for surface models, Best et al. (2004) proposed a general interface to make such an exchange possible. By contrast, plugging physics schemes into another model has turned out to be less straightforward than expected. The reasons may be diverse, but one of the unknown factors in this could be the tuning of the parametrizations. For instance, the work in Termonia and Hamdi (2007) shows that the time-step organisation of the ALADIN model (ALADIN International Team, 1997) should be expected from a purely paper analysis to be inferior to that of the Integrated Forecast System (IFS) of the European Centre for Medium-Range Weather Forecasts (ECMWF). The fact that both these models exhibit a state-of-the-art performance suggests that the past interplay of model development and tuning (not studied in that article) have played a major role in this. Besides the first question mentioned above, a second question will be addressed in the present article: What is the effect of using a parametrization in one model that has been tuned in another model?

The main problem of plugging parametrizations into other models is the potential existence of so-called compensating errors, where errors in one of the parametrizations may be counteracted by errors in another parametrization. The investigation of the above question in a full three-dimensional atmospheric model is thus problematic from a methodological point of view. Therefore, clean tests can only be carried out in an idealized set-up, where all aspects of the diffusion are numerically controlled. The aim of the present article is to study a simplified form of the diffusion equation, with a specific idealised experimental set-up. The simplifications here will be determined by

  • (i)
    the specific choice of the initial state,
  • (ii)
    the restriction of the equation to one single variable,
  • (iii)
    some simplified dependence of the diffusion coefficients on the remaining variable, and
  • (iv)
    an idealized form of the forcing of the diffusion equation.

The advantage of this approach is that the solution of the steady state can be analytically derived. The steady state can be seen as an intrinsic property of the scheme depending on the forcing. The latter is provided to the scheme via the input variables of the routines in the model. In the presented tests in this article, it will be shown that tuning the scheme to get a good steady state leads to a better behaviour when the scheme is coupled to an external forcing, which makes the tuning more independent from the rest of the model into which the scheme is plugged, and this may help avoiding compensating errors. The implications of the present numerical tests for realistic three-dimensional models will be discussed at the end of this article.

The article is organised as follows. In section 2, we look at the stability and accuracy of fractional time-stepping schemes for the nonlinear diffusion equation. We derive some analytical results and compare with numerical experiments. In section 3, we consider the problem of tuning of the parameter of the diffusion equation to improve the accuracy of fractional time-stepping schemes, and compare its accuracy with that obtained by an optimal choice of the numerical scheme. In section 4, we introduce a situation-dependent choice of the decentring of the numerical scheme, following the work of Bénard et al. (2000). Section 5 presents a discussion and the conclusions.

2. Fractional time-stepping schemes for the nonlinear diffusion equation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Fractional time-stepping schemes for the nonlinear diffusion equation
  5. 3. Tuning the parameter K
  6. 4. Situation-dependent adjustments of the decentring and the exchange coefficient
  7. 5. Discussion and conclusions
  8. Acknowledgements
  9. Appendix A. Proof that ρ(γ) ≤ 1
  10. Appendix B. Proof that γopt is linearly stable for P = 1 when η ≥ 1/2.
  11. References

Diffusion in atmospheric models has the general form

  • equation image(1)

where Ψ is a shorthand notation for the meteorological variables, such as wind, temperature, pressure and the quantities of different water species. The diffusion coefficient depends on the meteorological variables in a way that is determined by the details of the parametrization. In Eq. (1), ℱ represents a forcing term which can represent, for instance, a simplified version of a radiative forcing.

Many methods exist to discretise the vertical derivatives in three-dimensional models; for instance, finite-difference methods (with staggering) and vertical finite elements. Here we shall limit ourselves to studying solutions of the form

  • equation image(2)

with X real. We choose a forcing of the form ℱ = S sin(mz) and we study a ‘parametrization’ of the form ν = k|X(t)|P, with k > 0 and P ≥ 0. This can be seen as a leading-order term in a Taylor expansion of the diffusion coefficient around some predefined state. Also, Eq. (2) can be seen as a single mode in the Fourier decomposition and for such a mode the derivatives can be computed exactly. The fact that the derivatives are exact ensures that we are not adding any spurious numerical diffusion to the system. Solutions of the form of Eq. (2) exist and Eq. (1) is then reduced to

  • equation image(3)

with the coefficient K = km2 > 0, and there is no dependency on z. The aim here is to discretise the time derivative of this equation and it can be easily seen that, also in this case, the solution will stay on the mth mode, irrespective of the chosen numerical scheme. As will be shown below, this idealized set-up has the advantage that the steady state can be analytically derived. Equation (3) will be referred to below as the nonlinear damping equation, or more briefly the damping equation.

Equation (3) was also used in KK88 to analytically study the properties of a set of numerical schemes of the exchange of sensible and latent energy in atmospheric models between the lowest model level and the ground.

In the exchange coefficient (K|X|P), the constant K determines the degree of stiffness and the exponent P determines the degree of nonlinearity. The term S is usually taken to be an oscillatory forcing term. The results of this section are only strictly valid for a constant forcing. It can be assumed that the results remain valid for a slowly varying forcing. We will find this to be the case later, when we treat an oscillatory forcing in our numerical experiments.

Equation (3) can be rewritten conceptually as

  • equation image(4)

where D and S are the two physics terms. Different time-stepping schemes can be used to solve this equation. KK88 compared the stability of various two-time-level schemes, each treating the two right-hand-side terms S and D in a concurrent manner. We will make a further comparison with parallel and sequential schemes. In these cases, the updating for the two physics terms is performed by fractional time-stepping, where

  • the parallel computation is carried out as

    • equation image(5)
    • equation image(6)
    • equation image(7)

    and

  • the sequential as

    • equation image(8)
    • equation image(9)
    • equation image(10)

A concrete example of D(X1,X) and D(X∗∗,X) will follow later, in Eq. (11).

Note that η = 0 is equivalent to the parallel case (X = X, X∗∗ = X1), due to the fact that the forcing is independent of X. Henceforth, it will be studied as a specific case of Eqs (8)–(10). Also, if the sequential scheme is run without any additional forcings besides S, the step (8) always directly follows step (10), except for the first and the last time step of the integration. However, following Caya et al. (1998), the properties of these schemes will be determined with respect to their steady-state solution and, as will be shown below, these will be different for different values of η. Due to the nonlinearity and issues of stability, the updating of the damping is usually performed at least in a partly implicit way. The present study is limited to the set of schemes in Eqs (8)–(10). For instance, iterative methods (e.g. Diamantakis et al., 2006; Wood et al., 2007) are not considered.

We will discretise Eq. (4) by means of the two-time-level schemes denoted as explicit coefficient, extrapolated temperature schemes in KK88. For these schemes, a popular choice is

  • equation image(11)

where, for simplicity, we have assumed X > 0, which we will assume in our derivations below. The decentring parameter γ determines the amount of implicitness of the numerical scheme. The subcases γ = 0, γ = 1, γ = 0.5 correspond to explicit, implicit and (second-order) Crank–Nicholson schemes respectively.

The concurrent scheme studied in KK88 is then given by:

  • equation image(12)

Performing a linear stability analysis around the stationary solution equation image, KK88 found the following linear stability criterion:

  • equation image(13)

where equation image. If γ is chosen so that γ > 0.5(P + 1), the scheme is linearly stable for any value of the forcing. In practice, a value γ = 1.5 is often used, for instance in the IFS of ECMWF.

For schemes with fractional time-splitting, the result is expected to be different. Also, the issue of correct representation of the stationary solution, first touched upon by Caya et al. (1998), must be taken into account. In the work of Dubal et al. (2004), parallel and sequential schemes were compared for the linear damping equation with constant forcing. In our notation:

  • equation image(14)

with σK in the case P = 0. The steady-state solution is given by Xss = S/σ. This is correctly reproduced by the concurrent scheme. It is found that the parallel scheme results in an incorrect steady-state solution:

  • equation image(15)

The error becomes large for large time steps, unless an explicit scheme (γ = 0) is used. This becomes relevant in NWP due to the large time steps allowed by semi-Lagrangian time schemes. However, the sequential scheme can represent the correct solution, provided that the implicitness and η are appropriately chosen. In the case that S is treated symmetrised around the damping σX, one finds (Dubal et al., 2004) the solution

  • equation image(16)

The correct Xss can be obtained by choosing γ = η. The stability criterion of KK88 can now be considered as well, but note that the perturbation δX should be taken with respect to the numerical steady state (16), i.e. possibly the ‘incorrect’ one. One finds the same criterion as above in (13), essentially due to the linearity of the equation considered here.

Let us see how these results hold up in the nonlinear case. We will repeat the analysis of Dubal et al. (2004) for the nonlinear damping equation with σ(X) = KXP. Then we perform a stability analysis in the fashion of KK88. We consider the sequential scheme with parameters γ, η, and the choice of forcing treated as symmetrised around the damping. As mentioned above, this includes the parallel scheme as a subcase with η = 0.

2.1. Best decentring for reproducing the stationary state Xss

We will first be concerned with the choice of the decentring parameter γ for the diffusion. To choose it, the most straightforward approach is to take the smallest value that still guarantees stability, both in the linear sense and in the sense that no spurious oscillations are generated, as observed in KK88. However, choosing the decentring larger than is strictly needed for stability reasons usually leads to a loss of accuracy. Therefore some schemes, like the one discussed by Bénard et al. (2000), adapt the value of the decentring to the state of the actual forcing per time step. Here we will derive the optimal choice of γ given a forcing S in order to best reproduce the steady state. This will then be used below as a reference.

The sequential split scheme leads, after using Eq. (11), to the equation

  • equation image(17)

with

  • equation image(18)

because the damping is computed after the addition of the first part of the forcing in step (8). Thus, it follows that the ‘numerical steady state’ is given implicitly by:

  • equation image(19)

Requesting that the numerical steady state equals the analytic steady state equation image, leads to a condition relating γ and η:

  • equation image(20)

Defining sSΔt, kKΔt, this can also be written as:

  • equation image(21)

For the linear-damping case P = 0, this simplifies to γ = η, in accordance with the results of Dubal et al. (2004). For the quadratic case P = 1, we get:

  • equation image(22)

Let us call γ defined by Eq. (20) the optimal choice denoted by γopt. In general, for P > 0, γopt depends on s and k. In the limits Δt [RIGHTWARDS ARROW] 0 and Δt [RIGHTWARDS ARROW] ∞, γopt approaches (P + 1)η and η respectively. From Eq. (20), one can show that γopt is a monotonically decreasing function of Δt.

From the previous analysis, one should keep in mind that:

  • The stationary state cannot be correctly reproduced if γ is outside the range [η,(P + 1)η]. Specifically, the parallel scheme (η = 0) cannot reproduce Xss.

  • For any s and k, a correct reproduction of the steady state can always be obtained by choosing γ appropriately in the range [η,(P + 1)η].

However, in practical models, the decentring is not tuned towards obtaining the correct stationary solution, but is determined by stability criteria (e.g. Bénard et al., 2000), which we study below.

2.2. Stability to perturbations around Xss, P = 1

We start with an analytical treatment of the case P = 1. This is a mild nonlinearity, but all interesting features are already present. The results for general P will then be discussed without giving the explicit details of their derivation. The sequential split scheme corresponds to Eqs (17) and (18), with P = 1. The analytical steady state is given by equation image, while the ‘numerical steady state’ is given implicitly by Eq. (19). Let us consider a perturbation around the numerical steady state as follows:

  • equation image(23)

following the analysis in KK88. Inserting this into Eq. (17) and keeping only terms of first order in δX gives

  • equation image(24)

The first five terms on the right-hand side add up to zero, when the form of the stationary solution Xss is imposed using Eq. (19). The remaining terms are the first-order perturbation terms. Let us define αKXssΔt and βKSΔt2. The expression can now be rewritten as

  • equation image(25)

The terms with α are also present in KK88, while the extra ‘β terms’ come from the fractional time-stepping. For the amplification factor ρ = δX+/δX, one must have |ρ| < 1 to avoid (linear) instabilities§. From Eq. (25), we can see immediately that ρ < 1 for 0 ≤ γ ≤ 2η. In fact, we can show that ρ < 1 for γ ≥ 0 (Appendix A). The condition on ρ thus becomes ρ > −1. After some manipulation, this gives a condition for the decentring:

  • equation image(26)

For example, when equation image,

  • equation image(27)

Note that, without the β terms, condition (26) would be given by

  • equation image(28)

which corresponds to the condition derived in KK88 for the concurrent scheme for P = 1.

Note that the criterion (26) can be easier to satisfy due to the β terms being present. As an example, let us consider the case of very large time steps. In the limit Δt [RIGHTWARDS ARROW] ∞, we have Xss [RIGHTWARDS ARROW] (γη)SΔt, so that α [RIGHTWARDS ARROW] (γη)β. It is easy to see that the stability criterion (26) reduces to γ > 1/2 in this limit. This means that less decentring is needed than in the case of a concurrent treatment of forcing and diffusion (where one would require γ > 1). As discussed further in the next section, this turns out to be a general result.

As a further specific example, let us choose γ as in the previous section to recover the correct steady state, using Eq. (22). We have equation image, so that equation image. The stability condition then gives the following stable range:

  • equation image(29)

A plot of F(α) is shown in Figure 1 for η = 0.5.

thumbnail image

Figure 1. F(α) for η = 0.5.

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The decentring γopt is given by

  • equation image(30)

One can check that this is in the stable range for all values of α: γopt > F(α).

In general, one can prove that the optimal γ given by Eq. (22) is always in the stable range for η ≥ 0.5. Appendix B gives a proof.

Another example is given by the parallel scheme (η = 0); in this case we have

  • equation image(31)

The stationary state cannot be reproduced, unless γ = 0 (explicit scheme). For stability, we find also in this case that γ > 1/2 suffices for stability in the limit Δt [RIGHTWARDS ARROW] ∞.

2.3. Stability to perturbations around Xss, general P

As in the previous section, we look at perturbations around the stationary state Xss. The sequential-split scheme leads to Eq. (17), with σ(X) = K(X + ηs)P. It is not feasible to derive an explicit analytic expression for the amplification factor ρ, because the stationary state itself is given implicitly as the root of an order P + 1 polynomial. However, we can derive a useful implicit expression.

Let us define YX + ηs, so that σ(Y) = KYP. The stationary solution is then given (implicitly) by

  • equation image(32)

An analysis along the lines of the previous section gives the following amplification factor:

  • equation image(33)

It can be shown that ρ < 1 for γ ≥ 0 (Appendix A). Note that the stability does not depend explicitly on η. For the purpose of stability, it does not matter how the forcing is split around the damping. The third term in the numerator comes from the combined effect of the nonlinearity and the use of a sequential scheme. The typical result is that a smaller value of γ suffices to assure stability, compared with the concurrent scheme. We demonstrate this in the next section by plotting ρ as function of γ. This can be done by solving Eq. (32) numerically with a root-finding algorithm (we used Brent's method), and substituting in Eq. (33).

2.4. Numerical experiments

We consider Eq. (3) with a periodic forcing

  • equation image(34)

as in KK88. The discretisation is performed with fractional time-stepping; the forcing Sn at time step tn is given by

  • equation image(35)

In general, one would expect the analytical results derived above to hold for slowly varying forcing (i.e. ΔtT = 20).

We considered the cases P = 1, 2, 3, 4 and K = 10, 100, 1000. To create an accurate reference solution for each case, we used the concurrent scheme in Eq. (12) with γ = 0.5 and very small time step Δt = 0.0001. We compared the reference solutions with solutions obtained with fractional time-stepping schemes with η = 0, 0.5, 1 for time steps Δt = 1/2j (j = 0,...,7). The solutions were always started from the same initial condition, X(t = 0) = 0.6.

We optimise the γ of the sequential scheme using Eq. (20). Since S is now a function of time, it is now not clear which value for S we should use to calculate γopt. While we could use a different γ every time step (section 4), in actual NWP models calculating γopt every time step might become too computationally expensive. We therefore opted to calculate γopt only at the beginning of the integration, taking the average value of S, S = 1 = avg{S(t)}, which in practice could be obtained by using a climatological value. The experiments clearly show that optimizing γ, i.e. using γopt, indeed improves the results of the sequential scheme significantly (Figure 2, which shows the case K = 10, P = 1 with η = 1 and Δt = 1/4). The improvement holds up in the other studied cases as well. As noted above, optimizing γ in this way is not possible when using the parallel scheme, since γopt = 0, which leads to numerical instability (unless Δt is small enough). This makes the sequential scheme more attractive than the parallel scheme.

thumbnail image

Figure 2. Sequential scheme with optimised γ (= γopt(S = 1) ≈ 1.56; dash-dotted line) and unoptimised γ (=1; dashed). The solid line is the reference solution. In all cases, K = 10, P = 1, η = 1 and Δt = 1/4.

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Let us now illustrate the difference in optimal decentring between concurrent and sequential schemes. Consider a numerical example with S = 1, K = 100, P = 2, Δt = 1, η = 0.5. The optimal γ can be computed from Eq. (20) yielding γopt ≈ 0.70. For S, we take the amplitude of the oscillatory forcing. In Figure 3, 50 time steps with γ = γopt are computed, for the concurrent and sequential schemes. It is clear that the concurrent scheme suffers from spurious oscillations, because it needs a larger decentring parameter. The sequential scheme is stable for γ = 0.70. Note that, because of the large oscillations in the concurrent scheme, X becomes negative at some time steps. This is not a problem, but one should ensure that the exchange coefficient is always positive in the numerical calculation, i.e. the exchange coefficient is given by K|X|P in general.

thumbnail image

Figure 3. Sequential (dash-dotted line) versus concurrent (dashed line) scheme, with small decentring parameter γ = γopt ≈ 0.70, and the other parameters K = 100, P = 2, Δt = 1, η = 0.5. The solid line is the reference solution.

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As a further illustration, we compare ρ(γ) as computed from Eq. (33), using Brent's root-finding method to compute Yss. In Figure 4, we present ρ(γ) for the concurrent and sequential schemes, for K = 100, P = 2, η = 1, Δt = 1, S = 1. It can be seen that the concurrent scheme needs a larger decentring parameter to ensure stability. Note that, if ρ < 0, the sign of the perturbation δX alternates per time step. In practice, however, we have noticed that in the complete solution such alternations only rarely occur, due to the nonlinearity of the original equation. Nevertheless, such negative values should be handled carefully or avoided when possible.

thumbnail image

Figure 4. ρ(γ) for a sequential scheme with η = 1 (circles) and the concurrent scheme (crosses) for K = 100, P = 2 and Δt = 1. The stable region is above the horizontal line (ρ = −1).

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Finally, we determine numerically the minimum value of γ necessary for linear stability, for K = 10, 100, 1000 and P = 0, 1, 2, 3, 4 (with S = 1). We find that a value around γ = 0.8 is sufficient for (linear) stability, (Figure 5). Numerical calculations show that, for the cases we studied, γopt is always in the (linearly) stable range when η = 1/2 or η = 1. We suspect this is a general feature of γopt as long as η ≥ 1/2. For the case P = 1, we have a proof (Appendix B). The condition η ≥ 1/2 can be understood by looking at the limit Δt [RIGHTWARDS ARROW] ∞, where we have γopt [RIGHTWARDS ARROW] η and ρ [RIGHTWARDS ARROW] 1 − (1).

thumbnail image

Figure 5. Minimum value of γ for linear stability (i.e. ρ(γ) ≥ −1) with S = 1 and different vlues of P and K.

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3. Tuning the parameter K

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Fractional time-stepping schemes for the nonlinear diffusion equation
  5. 3. Tuning the parameter K
  6. 4. Situation-dependent adjustments of the decentring and the exchange coefficient
  7. 5. Discussion and conclusions
  8. Acknowledgements
  9. Appendix A. Proof that ρ(γ) ≤ 1
  10. Appendix B. Proof that γopt is linearly stable for P = 1 when η ≥ 1/2.
  11. References

We have seen that schemes with fractional time-stepping do not correctly reproduce the stationary state, unless some sort of fine tuning is performed. One could also try to improve model behaviour by replacing Kphys by Kt:

  • equation image(36)

which amounts to inserting some extra tunable degrees of freedom in the physics term by inserting the general and dimensionless function F in the equation. An alternative way to see this here is that the ‘physical’ coefficient (i.e. the value of the coefficient as given by the theory) Kphys is replaced by a different ‘tuned’ value Kt in the numerical scheme. One could then invent some form for F and choose its parameters to get the best model behaviour, i.e. the best scores.

Since the model studied here is simple enough, we can actually derive analytically the best choice for F to obtain the best representation of the stationary state of the analytic equation

  • equation image(37)

which is given by

  • equation image(38)

To derive an expression for Kt, we refer to Eqs (17)–(19). Instead of using Kphys in Eq. (18), we insert Kt, taken such that the numerical steady state (Eq. (19) with K = Kt) equals the physical steady state in Eq. (38). This leads to the choice

  • equation image(39)

where Xss is the stationary state (38) and from which the form of the function F can be identified:

  • equation image(40)

In practice, one does not know the true stationary state. Tuning consists of modifying some of the parameters of the parametrization of the exchange coefficient to obtain satisfactory model results. This is done by trial and error while comparing model output with observations. In the present case, we consider a tuning by means of the adapted value of the stiffness factor K. The value of Kt derived above represents the best value for the present system, which a modeller is trying to find by trial and error, but nevertheless we will call it the tuned value.

Kt can be computed numerically for given values of η and γ. Note that Kt can become negative if η < γ. To avoid this, one can take γη, but large enough to satisfy stability. Note that, in the parallel scheme (η = 0), negative Kt cannot be avoided in some cases. A negative Kt can be taken as a sign that the time step is too large, and that a lower time step should be considered. However, if one insists on using this large time step, our previous results show that one should consider tuning the decentring parameter γ, instead of K. However, since this is not possible for the parallel scheme, and negative Kt cannot be avoided in the parallel scheme when Δt is large, we find again that the sequential scheme is more attractive, as we did before in section 2.4.

As an example, consider the case η = 1 (sequential physics) and γ = 1 (this guarantees stability, see above). For P > 0, Kt must be taken smaller than the physical K. For large Δt and P = 3 or P = 4, Kt can differ from Kphys by several orders of magnitude.

Numerical tests corroborate Eq. (39) in the sense that the computed value for Kt gives the best results. However, the accuracy of the solutions is poor compared to the case where γ is optimised instead. Figure 6 illustrates this with a specific case. Moreover, this general conclusion was confirmed by computing root-mean-square errors (RMSEs) for larger P values, showing a clear superiority of optimising γ; Figure 7 shows an example.

thumbnail image

Figure 6. Solution with K tuned (γ = 1 and Kt ≈ 2.40; dashed line) versus reference solution (solid) and solution with γ optimised (γopt ≈ 1.24, Kphys = 10; dash-dotted). In all cases, P = 1, K = 10, S = 1, Δt = 1, η = 1.

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thumbnail image

Figure 7. Accuracy plots of the logarithmic root-mean-square error (RMSE) of both K tuned (dashed) versus the solution with γ optimised (dash-dotted) as a function of the time step Δt = (1/2)j represented by j, for P = 2, K = 10, S = 1, η = 1. The RMSE was computed with respect to the reference and averaged over the time interval t = 51 to t = 100.

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From studying numerical examples with P = 1, 2, 3, 4 and K = 1, 10, 100, 1000, we obtain the following conclusions:

  • Tuning K to obtain the correct stationary state is possible, but gives less accurate results than when optimising γ.

  • Kt can become negative if γ > η. This leads to instabilities, as can be seen explicitly in the numerical experiments.

  • For the sequential case with η = 1, it seems best to take γ = 1 for stability, and then use Kt according to Eq. (39). Remember that γ = 1 gives the correct stationary state in the linear case (P = 0), and is stable, so it seems like a good starting ‘guideline’. The value of Kt depends on P and different situations of boundary-layer stability can correspond to quite different values of P (Appendix of KK88). This indicates that, in order to get adequate results in realistic NWP models, it might be necessary to tune K differently in different situations of boundary-layer stability.

As a final test, we carry out the following numerical experiments:

  • We tune the exchange coefficient factor K in the parallel scheme (η = 0) with the steady state of the parallel scheme to obtain equation image, and use it for K in a sequential scheme (η≠0), while leaving the forcing S and the time step Δt unchanged;

  • Vice versa: we tune the exchange coefficient in the sequential scheme, obtaining equation image, and use its value in the parallel scheme.

This mimics, although admittedly in a very crude way, what is currently being tried in full NWP models. For instance, Best et al. (2004) proposed a solution to externalise surface schemes from the atmospheric models by a standardized interface, which may create the possibility of running surface schemes developed for one model in another. Both the Tiled ECMWF Scheme for Surface Exchanges over Land (TESSEL) of the IFS model of ECMWF and the SURface EXternalisée (SURFEX, Le Moigne et al., 2009) of Météo-France currently have such an interface which, in principle, could technically be tested in the other models. Of course, these models were developed and tuned within their original atmospheric model and it may be expected that they should be retuned when applied in another model.

An example of the second experiment is presented in Figure 8. K is tuned for the sequential scheme (with η = γ = 1), giving equation image. We then plug equation image into the parallel scheme (with η = 0 and γ = 1). The amplitude of the solution with ‘wrong tuning’ is twice the one of the reference solution, which is well approximated when the correct tuning is performed.

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Figure 8. Tuned sequential scheme (η = 1, equation image; dashed), wrongly tuned parallel scheme (η = 0, equation image; dash-dotted) and reference solution (solid line), all with P = 1, K = 10, S = 1, Δt = 1/4, γ = 1.

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4. Situation-dependent adjustments of the decentring and the exchange coefficient

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Fractional time-stepping schemes for the nonlinear diffusion equation
  5. 3. Tuning the parameter K
  6. 4. Situation-dependent adjustments of the decentring and the exchange coefficient
  7. 5. Discussion and conclusions
  8. Acknowledgements
  9. Appendix A. Proof that ρ(γ) ≤ 1
  10. Appendix B. Proof that γopt is linearly stable for P = 1 when η ≥ 1/2.
  11. References

In the ALADIN model, the decentring is chosen in a situation-dependent manner, as explained in Bénard et al. (2000), but their aim was to control the stability and not the accuracy. Here the dependency is defined by solving Eq. (20) or Eq. (39) at each time step, where the forcing S evolves according to Eq. (35). This is primarily aimed at improving the accuracy of the scheme.

Figure 9 shows four different numerical solutions for the case P = 1, K = 10. The exact (reference) solution is shown, and also three solutions computed with the sequential scheme (η = 1). Two of them use the optimised γ from Eq. (20). In the first case, the equation is used just once as before, to compute γopt(S = 1) ≈ 1.24. In the second case, Eq. (20) is used at every time step to compute γopt(S) for the next time step. Finally, the third case makes use of Eq. (39) to compute Kt at every time step. Due to the large time step, there is still a substantial error when optimizing γ just once. It is clear that the time-dependent adjustment works quite well to reduce this error.

thumbnail image

Figure 9. Comparison of the situation-dependent adjustment of γ and K, for P = 1, K = 10, Δt = 1: reference solution (solid line), γ optimised (long dash), γ optimised at every time step (dotted), and K tuned at every time step (short dash).

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In conclusion, for the tests of the runs with non-steady-state solutions:

  • Situation-dependent exchange coefficients and decentring substantially improve the performance of scheme compared to a prefixed optimal choice of γ and a prefixed tuning of K.

  • The improvements obtained by situation-dependent γ on the one hand and situation-dependent K on the other hand lead to comparable improvements.

It is important to stress here that a γ that is optimal for representing the steady state is also optimal for a general run away from the steady state.

5. Discussion and conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Fractional time-stepping schemes for the nonlinear diffusion equation
  5. 3. Tuning the parameter K
  6. 4. Situation-dependent adjustments of the decentring and the exchange coefficient
  7. 5. Discussion and conclusions
  8. Acknowledgements
  9. Appendix A. Proof that ρ(γ) ≤ 1
  10. Appendix B. Proof that γopt is linearly stable for P = 1 when η ≥ 1/2.
  11. References

Although tests presented here are simplified compared to the numerical parametrizations in NWP and general circulation models, they capture the essence of the form of the diffusive processes, and a few important conclusions can be drawn from the analysis in the previous sections.

  • To guarantee numerical stability, one can get away with smaller values of the decentring parameter γ when using a sequential-split scheme as opposed to the concurrent scheme. A value γ = 0.8 suffices for stability, while concurrent schemes, for instance the IFS in which γ = 1.5, can still be unstable for higher P.

  • The type of splitting, determined by the parameter η, does not influence the stability properties.

  • Accuracy can be greatly reduced in a sequential-split scheme at large time steps, due to the fact that the stationary solution is not correctly reproduced. It is possible to tune the parameters γ, η, K to overcome this, but not for a parallel scheme. The problem is that the tuning depends on stiffness and nonlinearity, so that a compromise must be found.

On a general level, the present study suggests the following conclusions. Plugging a diffusive scheme that has been tuned in one model into another model leads to wrong magnitudes of the signal, and thus one may a priori expect to be forced to retune the physics parametrization. In fact, they can be coupled to the dynamics and other physics in a sequential or parallel way (or more commonly some mixture of both), and it is rarely done the same way in two models. Our results suggest that this may lead to problems when a physical package that is developed and tuned in one model is naively introduced into another model. The question then becomes: is there a better way to improve model behaviour than tuning or a retuning of the exchange coefficient?

From Figures 6 and 7, it can be seen that, with a fixed tuning of the exchange coefficient (dashed line) and a fixed optimal choice of the decentring (dash-dotted), a (fixed) optimal choice of γ, tuned to have the best steady state, gives the best model performance away from the steady state. Moreover, the optimal ‘tunings’ of the exchange coefficient in this article can differ by orders of magnitude from the ‘true’ (physical) exchange coefficient for a large time step and can sometimes even become negative, which can then not be used because this would lead to an unstable numerical scheme. Using less than optimal tunings would then lead to less accurate descriptions of the steady state. As explained, this is not a problem for the parameters of the numerical scheme, at least in the sequential case. Additionally, from Figure 9 it can be seen that both a situation-dependent exchange coefficient and a situation-dependent optimal choice of the decentring parameter give comparable results. However, in practice, carrying out tunings where the functions are situation-dependent complicates the model building. By contrast, the approach in Bénard et al. (2000) shows that an implementation of a situation-dependent decentring choice, based on the given steady state, is a rather straightforward exercise in numerical analysis. In conclusion, these tests for the sequential scheme suggest that it may pay off more to make an extra effort in controlling the details of the numerics, based on the ad hoc steady state of the solution, rather than tuning the parametrization.

The novelty here is that such situation-dependent numerics not only deal with the nonlinear instabilities, as was originally the aim of Bénard et al. (2000), but the results here suggest that it is also useful to improve the accuracy. Steady states are an intrinsic property of the schemes, and as such may be good candidates to make the schemes model-independent, rather than relying on a retuning of the parameters of the physics parametrization.

In the present article, the steady state of the numerical scheme is computed in Eq. (19), which is solved by a root-finding algorithm for a given S. In practice, this may turn out to be more difficult. To our knowledge, there is at least one case where this is done in a realistic model set-up: the decentring is computed based on the steady state in Bénard et al. (2000). It might be feasible to extend this to simple surface schemes, e.g. that of Noilhan and Mahfouf (1996) which has been implemented in an operational model (Giard and Bazile, 2000).

It should be stressed that this study is still very restricted with regard to realistic applications. For instance, it did not treat any compensating errors between different parametrizations or feedbacks, nor the fact that the steady state of the scheme is not the correct steady state of reality. As a consequence, parameter tuning will always be necessary. Nevertheless, the present results advocate that (a) steady states should be considered more when developing physics packages and (b), by optimising the details of the numerics, one can obtain better results than by a tuning of the physics parametrization. The discretisation of surface schemes would seem to us to be a good application to test this claim.

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Fractional time-stepping schemes for the nonlinear diffusion equation
  5. 3. Tuning the parameter K
  6. 4. Situation-dependent adjustments of the decentring and the exchange coefficient
  7. 5. Discussion and conclusions
  8. Acknowledgements
  9. Appendix A. Proof that ρ(γ) ≤ 1
  10. Appendix B. Proof that γopt is linearly stable for P = 1 when η ≥ 1/2.
  11. References

The authors thank D. Degrauwe for useful discussions and the two anonymous reviewers for their many helpful and pertinent comments on the manuscript. This work was financially supported by the Belgian Science Policy (BELSPO), project no. MO/34/013.

Appendix A. Proof that ρ(γ) ≤ 1

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Fractional time-stepping schemes for the nonlinear diffusion equation
  5. 3. Tuning the parameter K
  6. 4. Situation-dependent adjustments of the decentring and the exchange coefficient
  7. 5. Discussion and conclusions
  8. Acknowledgements
  9. Appendix A. Proof that ρ(γ) ≤ 1
  10. Appendix B. Proof that γopt is linearly stable for P = 1 when η ≥ 1/2.
  11. References

From Eq. (32) and S > 0 it follows that

  • equation image(A1)

which simplifies to Yss > γs, assuming K > 0 and X > 0. Equation (33) can be written as ρ = 1 − Z(Yss,s,k,γ), with Z having the form

  • equation image(A2)

for some denominator N > 0. Using the inequality for Yss above, this leads to

  • equation image(A3)

proving that ρ ≤ 1.

Appendix B. Proof that γopt is linearly stable for P = 1 when η ≥ 1/2.

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Fractional time-stepping schemes for the nonlinear diffusion equation
  5. 3. Tuning the parameter K
  6. 4. Situation-dependent adjustments of the decentring and the exchange coefficient
  7. 5. Discussion and conclusions
  8. Acknowledgements
  9. Appendix A. Proof that ρ(γ) ≤ 1
  10. Appendix B. Proof that γopt is linearly stable for P = 1 when η ≥ 1/2.
  11. References

For P = 1, we have

  • equation image(B1)

(Eq. (22)). Requiring that γopt is linearly stable leads to the condition

  • equation image(B2)

(Eq. (29)). Substituting Eq. (B1) into the inequality (B2) gives the condition

  • equation image(46)

with Q(α,η) a third-degree polynomial in α with coefficients depending on η. One can easily check that all these coefficients are positive when η ≥ 1/2. Since α > 0, this proves that γopt is linearly stable when η ≥ 1/2.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Fractional time-stepping schemes for the nonlinear diffusion equation
  5. 3. Tuning the parameter K
  6. 4. Situation-dependent adjustments of the decentring and the exchange coefficient
  7. 5. Discussion and conclusions
  8. Acknowledgements
  9. Appendix A. Proof that ρ(γ) ≤ 1
  10. Appendix B. Proof that γopt is linearly stable for P = 1 when η ≥ 1/2.
  11. References
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  • *

    ALADIN: Aire Limitée Adaptation Dynamique Développement INternational. HIRLAM: HIgh-Resolution Limited-Area Model.

  • This is the terminology of Dubal et al., 2004. It does not necessarily imply η = 0.5; η is kept variable.

  • In fact, in some models an option to specify the degree of decentring may even not be available.

  • §

    i.e. large oscillations when ρ < −1, convergence to a wrong solution when ρ > 1 (KK88).

  • In the case S(t) = 0, the following expressions are used: γopt(t) = (P + 1)η and Kt(t) = Kphys