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). Stochastic parameterization: Toward a new view of weather and climate models. Bull. Am. Meteorol. Soc., 98(3):565-588. [9] Cooper, F. C. and Zanna, L. (2015). Optimisation of an idealised ocean model, stochastic parameterisation of sub-grid eddies. Ocean Modell., 88:38-53. [10] Crommelin, D. and Vanden-Eijnden, E. (2008). Subgrid-scale parameterization with conditional Markov chains. J. Atmos. Sci., 65(8):2661-2675. [11] Dijkstra, H. A. (2005). Nonlinear physical oceanography: a dynamical systems approach to the large scale ocean circulation and El Nino, volume 28

Abstract

New avenues are explored for the numerical study of the two dimensional inviscid hydrostatic primitive equations of the atmosphere with humidity and saturation, in presence of topography and subject to physically plausible boundary conditions for the system of equations. Flows above a mountain are classically treated by the so-called method of terrain following coordinate system. We avoid this discretization method which induces errors in the discretization of tangential derivatives near the topography. Instead we implement a first order finite volume method for the spatial discretization using the initial coordinates x and p. A compatibility condition similar to that related to the condition of incompressibility for the Navier- Stokes equations, is introduced. In that respect, a version of the projection method is considered to enforce the compatibility condition on the horizontal velocity field, which comes from the boundary conditions. For the spatial discretization, a modified Godunov type method that exploits the discrete finite-volume derivatives by using the so-called Taylor Series Expansion Scheme (TSES), is then designed to solve the equations. We report on numerical experiments using realistic parameters. Finally, the effects of a random small-scale forcing on the velocity equation is numerically investigated.

,D. Crom- melin, S. I. Dolaptchiev, et al., Stochastic parameterization: Toward a new view of weather and climate models, Bulletin of the American Meteorological Society, 98 (2017), pp. 565-588. [5] T. Berry and J. Harlim, Linear theory for filtering nonlinear multiscale systems with model error, Proc. Roy. Soc. London, 470A (2014). [6] M. Bonavita, L. Isaksen, and E. Holm, On the use of EDA background error variances in the ECMWF 4D-Var, Quarterly Journal of the Royal Meteorological Society, 138 (2012), pp. 1540-1559. [7] M. Bonavita, L. Raynaud, and L. Isaksen

. Juricke, V. Kitsios, F. Lott, V. Lucarini, S. Mahajan, T. Palmer, C. Penland, M. Sakradzija, J. von Storch, A. Weisheimer, M. Weniger, P. Williams, and J. Yano. “Stochastic parameterization: Toward a new view of weather and climate models.” In: Bull. Amer. Meteorol. Soc. 98.3 (2017), pp. 565–588. [7] R. Buizza, M. Miller, and T. Palmer. “Stochastic representation of model uncertainties in the ECMWF ensemble prediction system.” In: Quart. J. R. Meteorol. Soc. 125 (1999), pp. 2887–2908. [8] G. F. Carnevale. “Statistical features of the evolution of two

subgrid forces (log-linear scales) for the DNS model (blue – scheme E, red – scheme INMCM, yellow – scheme Z). Green line corresponds to stochastic parameterization and shown to indicate its small magnitude. From Fig. 2a we see that for the all three schemes the full DNS advection tendency ( J ( ψ , ω )) could be well reproduced by its coarse-grained part ([ J ( ψ , ω )] h ) in large scales. However at the small scales subgrid part of advection is important (note the difference at the right end of the spectrum between black and colored curves in Fig. 2a ). Apart

. Crommelin, J. Biello, and S. Böing. A data-driven multi-cloud model for stochastic parametrization of deep convection. Phil. Trans. of the Roy. Soc. A, 371(1991), 2013. [28] R. Plant and G. C. Craig. A stochastic parameterization for deep convection based on equilibrium statistics. J. Atmos. Sci, 65(1):87–105, 2008. [29] M. Sakradzija, A. Seifert, and T. Heus. Fluctuations in a quasi-stationary shallowcumulus cloud ensemble. Nonlin. Processes Geophys., 1:1223–1282, 2014. [30] G. C. Craig and B. G. Cohen. Fluctuations in an equilibrium convective ensemble. Part I

model and nullifies the checkerboard grid noise ((−1) i + j , i and j are indices along the x - and y - directions) if a second order approximation is used. Zero Dirichlet boundary conditions are applied. The physical reasoning for stochastic parameterization with lateral white noise is again taken from [ 23 ], where subgrid turbulence was shown to produce stochastic forcing in small resolved scales. In contrast to [ 19 ], we generate random streamfunction, instead of random Reynolds stress components. Streamfunction approach gives analogous wavenumber spectrum

, Convective forcing fluctuations in a cloud-resolving model: Relevance to the stochastic parameterization problem. J. Climate 20 (2007), No. 2, 187–202. 10.1175/JCLI3954.1 Shutts G. Palmer T. Convective forcing fluctuations in a cloud-resolving model: Relevance to the stochastic parameterization problem J. Climate 20 2007 2 187 202 [37] G. Shutts, A stochastic convective backscatter scheme for use in ensemble prediction systems. Q. J. R. Meteorol. Soc . 141 (2015), No. 692, 2602–2616. 10.1002/qj.2547 Shutts G. A stochastic convective backscatter scheme for use in

. Duan, A stochastic approach for parameterizing unresolved scales in a system with memory, J. Alg. Comput. Tech. 3 (2009), 319–405. [11] J. Duan and B. Nadiga, Stochastic parameterization for large eddy simulation of geophysical flows, Proc. American Math. Soc. 135 (2007), 1187–1196. [12] P. A. Durbin and B. A. Petterson Reif, Statistical Theory and Modeling for Turbulent Flows, 2nd ed., Wiley, Chichester, 2010. [13] R. M. Errico, R. Langland and D. P. Baumhefner, The workshop on atmospheric predictability, Bull. Amer. Meteor. Soc. 74 (2002), 1341–1343. [14] Y. T

N -dimensional state of the system, g ( φ, t ) is the system nonlinear operator, W ˙ $\dot{W} $ is a Gaussian white noise with its amplitude γ being a state- and time-independent vector with covariance matrix 〈 γγ T 〉 = 2 Γ . The non-linear operator of the system is assumed to be time-periodic with period Δ ( g ( φt ) = g ( φ, t + Δ ) ) reflecting the annual cycle in solar forcing. The noise term represents the unresolved small scale processes or stochastic parameterizations in the model [ 37 , 45 ]. The probability density of system ( 1.1 ) can be