Stable Schemes for Nonlinear Vertical Diffusion in Atmospheric Circulation Models

Girard, Claude; Delage, Yves

doi:10.1175/1520-0493(1990)118<0737:ssfnvd>2.0.co;2

Cited by 37 publications

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“…The air pollution and chemistry transport modeling community, as well as various numerical weather forecast centers, have paid substantial attention to the use of numerical techniques in complex models (e.g., Girard and Delage, 1990;Beljaars, 1991;Teixeira, 2000;Benard et al, 2000;Jacobson, 2002;Beljaars et al, 2004;Wood et al, 2007;Zaveri et al, 2008;Schlegel et al, 2012;Tudor, 2012), while climate modelers have focused less on this issue. In this paper we present an example from the global aerosol-climate model ECHAM-HAM Zhang et al, 2012) to demonstrate that numerical errors associated with stiff systems can lead to significant systematic biases in simulations at typical spatial and temporal scales considered in climate research.…”

Section: Introductionmentioning

confidence: 99%

Numerical issues associated with compensating and competing processes in climate models: an example from ECHAM-HAM

et al. 2013

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Abstract. The purpose of this paper is to draw attention to the need for appropriate numerical techniques to represent process interactions in climate models. In two versions of the ECHAM-HAM model, different time integration methods are used to solve the sulfuric acid (H 2 SO 4 ) gas evolution equation, which lead to substantially different results in the H 2 SO 4 gas concentration and the aerosol nucleation rate. Using convergence tests and sensitivity simulations performed with various time stepping schemes, it is confirmed that numerical errors in the second model version are significantly smaller than those in version one. The use of sequential operator splitting in combination with a long time step is identified as the main reason for the large systematic biases in the old model. The remaining errors of nucleation rate in version two, related to the competition between condensation and nucleation, have a clear impact on the simulated concentration of cloud condensation nuclei (CCN) in the lower troposphere. These errors can be significantly reduced by employing solvers that handle production, condensation and nucleation at the same time. Lessons learned in this work underline the need for more caution when treating multi-timescale problems involving compensating and competing processes, a common occurrence in current climate models.

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Section: Introductionmentioning

confidence: 99%

Numerical issues associated with compensating and competing processes in climate models: an example from ECHAM-HAM

et al. 2013

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“…A further consequence of the stiffness of the problem (Kalnay and Kanamitsu, 1988;Girard and Delage, 1990;Bénard et al, 2000) arises when the diffusion coefficient is a function of the variables on which the diffusion operator is itself acting -as is the case for planetaryboundary-layer schemes. The resulting nonlinearity can then lead to numerical instability.…”

Section: Introductionmentioning

confidence: 99%

A monotonically‐damping second‐order‐accurate unconditionally‐stable numerical scheme for diffusion

Wood

Diamantakis

Staniforth

2007

Quart J Royal Meteoro Soc

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ABSTRACT:We present a new two-step temporal discretization of the diffusion equation, which is formally secondorder-accurate and unconditionally stable. A novel aspect of the scheme is that it is monotonically damping: the damping rate is a monotonically-increasing function of the diffusion coefficient, independent of the size of the time step, when the diffusion coefficient is independent of the variable being diffused. Furthermore, the damping rate increases without bound as the diffusion coefficient similarly increases. We discuss the nonlinear behaviour of the scheme when the diffusion coefficient is a function of the diffused variable. The scheme is designed to maintain any steady-state solution. We present examples of the performance of the scheme.  Crown

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“…Girard and Delage (1990) apply the stabilization scheme for vertical diffusion in the Canadian spectral weather forecast model, Bénard et al (2000) in ARPÉGE and ALADIN models, and Wood et al (2007) develop a stable scheme for the Unified Model. Teixeira (2000) describes noisy solutions in the IFS model that are a consequence of the interaction between the radiation, vertical diffusion and cloud schemes.…”

Section: Discussionmentioning

confidence: 99%

“…A solution has been proposed (Girard and Delage, 1990) to vary the implicitness of the scheme locally (in space and time). The scheme is over-implicit only where, when and as much as needed.…”

Section: Tudor: Numerical Instability and Stiffness Of The Arpége mentioning

confidence: 99%

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A test of numerical instability and stiffness in the parametrizations of the ARPÉGE and ALADIN models

Tudor

2013

Geosci. Model Dev.

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Abstract. Meteorological numerical weather prediction (NWP) models solve a system of partial differential equations in time and space. Semi-lagrangian advection schemes allow for long time steps. These longer time steps can result in instabilities occurring in the model physics. A system of differential equations in which some solution components decay more rapidly than others is stiff. In this case it is stability rather than accuracy that restricts the time step. The vertical diffusion parametrization can cause fast non-meteorological oscillations around the slowly evolving true solution (fibrillations). These are treated with an anti-fibrillation scheme, but small oscillations remain in operational weather forecasts using ARPÉGE and ALADIN models. In this paper, a simple test is designed to reveal if the formulation of particular a physical parametrization is a stiff problem or potentially numerically unstable in combination with any other part of the model. When the test is applied to a stable scheme, the solution remains stable. However, applying the test to a potentially unstable scheme yields a solution with fibrillations of substantial amplitude. The parametrizations of the NWP model ARPÉGE were tested one by one to see which one may be the source of unstable model behaviour. The test identified the set of equations in the stratiform precipitation scheme (a diagnostic Kesslertype scheme) as a stiff problem, particularly the combination of terms arising due to the evaporation of snow.

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Stable Schemes for Nonlinear Vertical Diffusion in Atmospheric Circulation Models

Cited by 37 publications

References 0 publications

Numerical issues associated with compensating and competing processes in climate models: an example from ECHAM-HAM

Numerical issues associated with compensating and competing processes in climate models: an example from ECHAM-HAM

A monotonically‐damping second‐order‐accurate unconditionally‐stable numerical scheme for diffusion

A test of numerical instability and stiffness in the parametrizations of the ARPÉGE and ALADIN models

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