Stochastic parametrization of subgrid‐scale processes in coupled ocean–atmosphere systems: benefits and limitations of response theory

A stochastic subgrid-scale parameterization based on the Ruelle's response theory and proposed inWouters and Lucarini (2012) is tested in the context of a low-order coupled ocean-atmosphere model for which a part of the atmospheric modes are considered as unresolved. A natural separation of the phase-space into an invariant set and its complement allows for an analytical derivation of the different terms involved in the parameterization, namely the average, the fluctuation and the long memory terms. In this case, the fluctuation term is an additive stochastic noise. Its application to the loworder system reveals that a considerable correction of the low-frequency variability along the invariant subset can be obtained, provided that the coupling is sufficiently weak. This new approach of scale separation opens new avenues of subgrid-scale parameterizations in multiscale systems used for climate forecasts.

[1]  Cohen,et al.  Dynamical Ensembles in Nonequilibrium Statistical Mechanics. , 1994, Physical review letters.

[2]  S. Vannitsem.  Stochastic modelling and predictability: analysis of a low-order coupled ocean–atmosphere model , 2014, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[3]  P. Imkeller,et al.  Reduction of deterministic coupled atmosphere–ocean models to stochastic ocean models: a numerical case study of the Lorenz–Maas system , 2003 .

[4]  Jorgen S. Frederiksen,et al.  Eddy Viscosity and Stochastic Backscatter Parameterizations on the Sphere for Atmospheric Circulation Models , 1997 .

[5]  C. Nicolis,et al.  Stochastic aspects of climatic transitions-Additive fluctuations , 2010 .

[6]  Lai-Sang Young,et al.  What Are SRB Measures, and Which Dynamical Systems Have Them? , 2002 .

[7]  S. Vannitsem,et al.  Low-frequency variability and heat transport in a low-order nonlinear coupled ocean–atmosphere model , 2014, 1412.0621.

[8]  D. Ruelle.  Differentiation of SRB States , 1997 .

[9]  A. Dalcher,et al.  Error growth and predictability in operational ECMWF forecasts , 1987 .

[10]  T. Palmer,et al.  Stochastic representation of model uncertainties in the ECMWF ensemble prediction system , 2007 .

[11]  Zhimin Chen,et al.  Remarks on the Time Dependent Periodic¶Navier–Stokes Flows on a Two-Dimensional Torus , 1999 .

[12]  Rui A. P. Perdigão,et al.  Dynamics of Prediction Errors under the Combined Effect of Initial Condition and Model Errors , 2009 .

[13]  Shaun Lovejoy,et al.  The Weather and Climate: Emergent Laws and Multifractal Cascades , 2013 .

[14]  D. Lawrence,et al.  Weak Land–Atmosphere Coupling Strength in HadAM3: The Role of Soil Moisture Variability , 2005 .

[15]  T. DelSole,et al.  Stochastic Models of Quasigeostrophic Turbulence , 2004 .

[16]  Valerio Lucarini,et al.  Elements of a unified framework for response formulae , 2013, 1310.1747.

[17]  Andrew J. Majda,et al.  A mathematical framework for stochastic climate models , 2001 .

[18]  Rafail V. Abramov.  A Simple Linear Response Closure Approximation for Slow Dynamics of a Multiscale System with Linear Coupling , 2012, Multiscale Model. Simul..

[19]  Michael E. Mann,et al.  Observed and Simulated Multidecadal Variability in the Northern Hemisphere , 1999 .

[20]  G. Shutts.  A kinetic energy backscatter algorithm for use in ensemble prediction systems , 2005 .

[21]  F. Doblas-Reyes,et al.  Stochastic atmospheric perturbations in the EC-Earth3 global coupled model: impact of SPPT on seasonal forecast quality , 2015, Climate Dynamics.

[22]  Mickaël D. Chekroun,et al.  Approximation of Stochastic Invariant Manifolds , 2015 .

[23]  D. Ruelle.  A review of linear response theory for general differentiable dynamical systems , 2009, 0901.0484.

[24]  Sergey Kravtsov,et al.  Stochastic Parameterization Schemes for Use in Realistic Climate Models , 2011 .

[25]  John D. Hunter,et al.  Matplotlib: A 2D Graphics Environment , 2007, Computing in Science & Engineering.

[26]  G. Nicolis,et al.  Stochastic aspects of climatic transitions–Additive fluctuations , 1981 .

[27]  W. Price,et al.  Long-time behavior of Navier-Stokes flow on a two-dimensional torus excited by an external sinusoidal force , 1997 .

[28]  Eric Vanden-Eijnden,et al.  Subgrid-Scale Parameterization with Conditional Markov Chains , 2008 .

[29]  Judith Berner,et al.  Stochastic climate theory and modeling , 2015 .

[30]  C. Nicolis,et al.  Dynamics of Model Error: The Role of the Boundary Conditions , 2007 .

[31]  Valerio Lucarini,et al.  Disentangling multi-level systems: averaging, correlations and memory , 2011, 1110.6113.

[32]  Michael J. Rycroft,et al.  Storms in Space , 2004 .

[33]  K. Hasselmann.  Stochastic climate models Part I. Theory , 1976 .

[34]  M. Ghil,et al.  A highly nonlinear coupled mode of decadal variability in a mid-latitude ocean–atmosphere model , 2007 .

[35]  Rafail V. Abramov,et al.  A Simple Closure Approximation for Slow Dynamics of a Multiscale System: Nonlinear and Multiplicative Coupling , 2012, Multiscale Model. Simul..

[36]  S. Vannitsem.  The role of the ocean mixed layer on the development of the North Atlantic Oscillation: A dynamical system's perspective , 2015 .

[37]  C. E. Leith,et al.  Predictability of climate , 1978, Nature.

[38]  Taikan Oki,et al.  GLACE: The Global Land–Atmosphere Coupling Experiment. Part II: Analysis , 2006, Journal of Hydrometeorology.

[39]  C. Nicolis,et al.  Can error source terms in forecasting models be represented as Gaussian Markov noises? , 2005 .

[40]  J. Frederiksen,et al.  Subgrid-Scale Parameterizations of Eddy-Topographic Force, Eddy Viscosity, and Stochastic Backscatter for Flow over Topography , 1999 .

[41]  I. Moroz,et al.  Stochastic parametrizations and model uncertainty in the Lorenz ’96 system , 2013, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[42]  Valerio Lucarini,et al.  Multi-level Dynamical Systems: Connecting the Ruelle Response Theory and the Mori-Zwanzig Approach , 2012, Journal of Statistical Physics.

[43]  T. Palmer,et al.  Addressing model uncertainty in seasonal and annual dynamical ensemble forecasts , 2009 .

[44]  E. Lorenz.  Atmospheric predictability experiments with a large numerical model , 1982 .

[45]  G. Milstein.  Numerical Integration of Stochastic Differential Equations , 1994 .

[46]  W. Price,et al.  Time dependent periodic Navier-Stokes flows on a two-dimensional torus , 1996 .

[47]  Mickaël D. Chekroun,et al.  Approximation of Stochastic Invariant Manifolds: Stochastic Manifolds for Nonlinear SPDEs I , 2014 .