GMD Geoscientific Model Development GMD Geosci. Model Dev. 1991-9603 Copernicus Publications Göttingen, Germany 10.5194/gmd-6-901-2013 A test of numerical instability and stiffness in the parametrizations of the ARPÉGE and ALADIN models Tudor M. https://orcid.org/0000-0002-2683-2652 1 Croatian Meteorological and Hydrological Service, Zagreb, Grič 3, Croatia 05 07 2013 6 4 901 913 Copyright: © 2013 M. Tudor 2013 This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/ This article is available from https://gmd.copernicus.org/articles/6/901/2013/gmd-6-901-2013.html The full text article is available as a PDF file from https://gmd.copernicus.org/articles/6/901/2013/gmd-6-901-2013.pdf

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 Kessler-type scheme) as a stiff problem, particularly the combination of terms arising due to the evaporation of snow.

 References Beljaars, A. C. M.: Numerical schemes for parametrizations, ECMWF Seminar proceedings 9–13 September 1991 Numerical methods in atmospheric models, Vol II, 1–42, 1991. Beljaars, A. C. M., Bechtold, P., Kohler, M., Morcrette, J.-J., Tompkins, A., Viterbo, P., and Wedi, N. P.: The numerics of physical parametrization, ECMWF Seminar proceedings 6–10 September 2004 Recent developements in numerical methods for atmosphere and ocean modelling, 113–134, 2004. Bénard, P., Marki, A., Neytchev, P. N., and Prtenjak, M. T.: Stabilization of nonlinear vertical diffusion schemes in the context of NWP models, Mon. Weather Rev., 128, 1937–1948, 2000. Brown, P. S.: Clark's partially implicit method for efficient solution of the condensation equations, J. Atmos. Sci., 34, 1142–1144, 1977. Brown, P. S. and Árnason, G.: Efficient numerical integration of the equations governing droplet growth by condensation, J. Atmos. Sci., 30, 245–248, 1973. Brown, P. and Pandolfo, J. P.: A numerical predictability problem in solution of the nonlinear diffusion equation, Mon. Weather Rev., 110, 1214–1223, 1982. Courtier, P. and Geleyn, J.-F.: A global spectral model with variable resolution application to the shallow water equations, Q. J. Roy. Meteor. Soc., 114, 1321–1346, 1988. Courtier, P., Freydier, C., Geleyn, J.-F., Rabier, F., and Rochas, M.: The ARPÉGE project at METEO-FRANCE, Proceedings from the ECMWF workshop on numerical methods in atmospheric models, 9–13 September 1991, 2, 193–231, 1991. Davies, H. C.: The stability of some planetary boundary layer diffusion equation, Mon. Weather Rev., 111, 2140–2143, 1983. Epperson, J. F.: An introduction to numerical methods and analysis, John Wiley & sons, 576 pp., 2001. Geleyn, J.-F.: Use of a modified Richardson number for parameterizing the effect of shallow convection, in: Short and medium range weather prediction, edited by: Matsuno, Z., special volume, J. Meteor. Soc. Japan, 141–149, 1987. Geleyn, J.-F. and Hollingsworth, A.: An economical analytical method for computation of the interaction between scattering and line absorption of radiation, Contr. Atmos. Phys., 52, 1–16, 1979. Geleyn, J.-F., Girard, C., and Louis, J.-F.: A simple parametrization of moist convection for large-scale atmospheric models, Beitr. Phys. Atmos., 55, 325–334, 1982. Giard, D., and Bazile, E.: Implementation of a new assimilation scheme for soil and surface variables in a global NWP model, Mon. Weather Rev., 128, 997–1015, 2000. Girard, C. and Delage, Y.: Stable schemes for nonlinear vertical diffusion in atmospheric circulation models, Mon. Weather Rev., 118, 737–745, 1990. Higham, D. J. and Trefethen, L. N.: Stiffness of ODEs, BIT, 33, 285–303, 1993. Hortal, M.: The developement and testing of a new two-time-level semi-Lagrangian scheme (SETTLS) in the ECMWF forecast model, Q. J. Roy. Meteor. Soc., 128, 1671–1687, 2002. Ivatek-Šahdan, S. and Tudor, M.: Use of high-resolution dznamical adaptation in operational suite and research impact studies, Meteorol. Z., 13, 1–10, 2004. Jacobson, M. Z.: Fundamentals od atmospheric modelling, Cambridge university press, 656 pp., 2000. Kalnay, E. and Kanamitsu, M.: Time schemes for strongly nonlinear damping equations, Mon. Weather Rev., 118, 1945–1958, 1988. Kessler, E.: On distribution and continuity of water substance in atmospheric circulations, Met. Mon. Am. Met. Soc., 10, 84 pp., 1969. Louis, J.-F., Tiedke, M., and Geleyn, J.-F.: A short history of PBL parametrization at ECMWF, Proceedings from the ECMWF workshop on planetary boundary layer parametrization, 25–27 November 1981, 59–79, 1982. Machenhauer, B. and Haugen, J. E.: Test of a spectral limited area shallow water model with time-dependent lateral boundary conditions and combined normal mode/semi-Lagrangian time integration schemes, Proceedings from the ECMWF workshop on techniques for horizontal discretization in numerical weather prediction models, 2–4 November 1987, 361–377, 1987. Pielke, R. A.: Mesoscale meteorological modeling, Academic press, 676 pp., 2002. Ritter, B. and Geleyn, J.-F.: A comprehensive radiation scheme for numerical weather prediction models with potential applications in climate simulations, Mon. Weather Rev., 120, 303–325, 1992. Robert, A.: A semi-lagrangian and semi-implicit numerical integration scheme for primitive meteorological equations, J. Met. Soc. Japan, 60, 319–324, 1982. Ropp, D. L., Shadid, J. N., and Ober, C. C.: Studies of the accuracy of time integration methods for reaction-diffusion equations, J. Comp. Phys., 194, 544–574, 2004. Schlegel, M., Knoth, O., Arnold, M., and Wolke, R.: Implementation of multirate time integration methods for air pollution modelling, Geosci. Model Dev., 5, 1395–1405, <a href="http://dx.doi.org/10.5194/gmd-5-1395-2012">https://doi.org/10.5194/gmd-5-1395-2012</a>, 2012. Sednev, I. and Menon, S.: Analyzing numerics of bulk microphysics schemes in community models: warm rain processes, Geosci. Model Dev., 5, 975–987, <a href="http://dx.doi.org/10.5194/gmd-5-975-2012">https://doi.org/10.5194/gmd-5-975-2012</a>, 2012. Simmons, A. J. and Burridge, D. M.: An energy and angular momentum conserving vertical finite-difference scheme and hybrid vertical coordinate, Mon. Weather Rev., 109, 758–766, 1981. Stull, R. B.: An Introduction to Boundary Layer Meteorology, Kluwer, 666 pp., 1988. Teixeira, J.: Boundary layer clouds in large scale atmospheric models: cloud schemes and numerical aspects, PhD Thesis, ECMWF, Reading, UK, 190 pp., 2000. Telišman Prtenjak, M., Marki, A., and Bénard, P.: Refinement of the vertical diffusion scheme in the ARPÉGE/ALADIN model, Geofizika, 15, 27–51, 1998. Wolfram, S.: Mathematica, Wolfram media & Cambridge university press, 3rd Edn., 1403 pp., 1996. Wood, N., Diamantakis, M., and Staniforth, A.: A monotonically-damping second-order-accurate unconditionally-stable numerical scheme for diffusion, Q. J. Roy. Meteor. Soc., 133, 1559–1573, 2007.