Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model

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June 2020, 2(2): 173-205. doi: 10.3934/fods.2020010

Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model

Colin Cotter , Dan Crisan , Darryl Holm , Wei Pan and Igor Shevchenko ^,

Department of Mathematics, Imperial College London, Huxley Building, 180 Queen's Gate, London, SW7 2AZ, UK

^* Corresponding author: Igor Shevchenko

Published July 2020

The stochastic variational approach for geophysical fluid dynamics was introduced by Holm (Proc Roy Soc A, 2015) as a framework for deriving stochastic parameterisations for unresolved scales. This paper applies the variational stochastic parameterisation in a two-layer quasi-geostrophic model for a $ \beta $-plane channel flow configuration. We present a new method for estimating the stochastic forcing (used in the parameterisation) to approximate unresolved components using data from the high resolution deterministic simulation, and describe a procedure for computing physically-consistent initial conditions for the stochastic model. We also quantify uncertainty of coarse grid simulations relative to the fine grid ones in homogeneous (teamed with small-scale vortices) and heterogeneous (featuring horizontally elongated large-scale jets) flows, and analyse how the spread of stochastic solutions depends on different parameters of the model. The parameterisation is tested by comparing it with the true eddy-resolving solution that has reached some statistical equilibrium and the deterministic solution modelled on a low-resolution grid. The results show that the proposed parameterisation significantly depends on the resolution of the stochastic model and gives good ensemble performance for both homogeneous and heterogeneous flows, and the parameterisation lays solid foundations for data assimilation.

Keywords: Stochastic parameterisations, stochastic transport noise, uncertainty quantification, geophysical fluid dynamics, multi-layer quasi-geostrophic model.

Mathematics Subject Classification: Primary: 60H15; Secondary: 76U60.

Citation: Colin Cotter, Dan Crisan, Darryl Holm, Wei Pan, Igor Shevchenko. Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model. Foundations of Data Science, 2020, 2 (2) : 173-205. doi: 10.3934/fods.2020010

References:

[1]	R. Abramov, A simple stochastic parameterization for reduced models of multiscale dynamics, Fluids, 1 (2016), 1-18. Google Scholar
[2]	A. Arakawa, Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part Ⅰ, J. Comput. Phys., 1 (1966), 119-143. Google Scholar
[3]	P. S. Berloff, Random-forcing model of the mesoscale oceanic eddies, J. Fluid Mech., 529 (2005), 71-95. doi: 10.1017/S0022112005003393. Google Scholar
[4]	P. Berloff and I. Kamenkovich, On spectral analysis of mesoscale eddies. Part Ⅰ: Linear analysis, J. Phys. Oceanogr., 43 (2013), 2505-2527. Google Scholar
[5]	P. Berloff, S. Karabasov, J. T. Farrar and I. Kamenkovich, On latency of multiple zonal jets in the oceans, J. Fluid Mech., 686 (2011), 534-567. doi: 10.1017/jfm.2011.345. Google Scholar
[6]	P. Berloff and J. McWilliams, Material transport in oceanic gyres. Part Ⅱ: Hierarchy of stochastic models, J. Phys. Oceanogr., 32 (2002), 797-830. Google Scholar
[7]	P. Berloff and J. McWilliams, Material transport in oceanic gyres. Part Ⅲ: Randomized stochastic models, J. Phys. Oceanogr., 33 (2003), 1416-1445. Google Scholar
[8]	J. Bröcker, Assessing the reliability of ensemble forecasting systems under serial dependence, Q. J. R. Meteorol. Soc., 144 (2018), 2666-2675. Google Scholar
[9]	K. K. Chen, J. H. Tu and C. W. Rowley, Variants of dynamic mode decomposition: Boundary condition Koopman and Fourier analyses, J. Nonlinear Sci., 22 (2012), 887-915. doi: 10.1007/s00332-012-9130-9. Google Scholar
[10]	F. Cooper and L. Zanna, Optimization of an idealised ocean model, stochastic parameterisation of sub-grid eddies, Ocean Model., 88 (2015), 38-53. Google Scholar
[11]	C. Cotter, D. Crisan, D. Holm, W. Pan and I. Shevchenko, Data assimilation for a quasi-geostrophic model with circulation-preserving stochastic transport noise, J. Stat. Phys., 179 (2020), 1186-1221. doi: 10.1007/s10955-020-02524-0. Google Scholar
[12]	C. J. Cotter, G. A. Gottwald and D. D. Holm, Stochastic partial differential fluid equations as a diffusive limit of deterministic lagrangian multi-time dynamics, Proc. A., 473 (2017), 20170388, 10 pp. doi: 10.1098/rspa.2017.0388. Google Scholar
[13]	J. Duan and B. T. Nadiga, Stochastic parameterization for large eddy simulation of geophysical flows, Proc. Amer. Math. Soc., 135 (2007), 1187-1196. doi: 10.1090/S0002-9939-06-08631-X. Google Scholar
[14]	J. Elsner and A. Tsonis, Singular Spectrum Analysis: A New Tool in Time Series Analysis, Plenum Press, New York, 1996. Google Scholar
[15]	C. Franzke, A. J. Majda and E. Vanden-Eijnden, Low-order stochastic mode reduction for a realistic barotropic model climate, J. Atmospheric Sci., 62 (2005), 1722-1745. doi: 10.1175/JAS3438.1. Google Scholar
[16]	J. Frederiksen, T. OKane and M. Zidikheri, Stochastic subgrid parameterizations for atmospheric and oceanic flows, Phys. Scr., 85 (2012), 068202. Google Scholar
[17]	F. Gay-Balmaz and D. D. Holm, Stochastic geometric models with non-stationary spatial correlations in Lagrangian fluid flows, J. Nonlinear Sci., 28 (2018), 873-904. doi: 10.1007/s00332-017-9431-0. Google Scholar
[18]	P. Gent, The Gent–McWilliams parameterization: 20/20 hindsight, Ocean Model., 39 (2011), 2-9. Google Scholar
[19]	P. Gent and J. Mcwilliams, Isopycnal mixing in ocean circulation models, J. Phys. Oceanogr., 20 (1990), 150-155. Google Scholar
[20]	I. Grooms, A. Majda and K. Smith, Stochastic superparametrization in a quasigeostrophic model of the Antarctic Circumpolar Current, Ocean Model., 85 (2015), 1-15. Google Scholar
[21]	E. Hairer, S. Nørsett and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems, Springer-Verlag, Berlin, 1993. Google Scholar
[22]	A. Hannachi, I. Jolliffe and D. Stephenson, Empirical orthogonal functions and related techniques in atmospheric science: A review, Int. J. Climatol., 27 (2007), 1119-1152. Google Scholar
[23]	D. D. Holm, Variational principles for stochastic fluid dynamics, Proc. A., 471 (2015), 20140963, 19 pp. doi: 10.1098/rspa.2014.0963. Google Scholar
[24]	W. Hundsdorfer, B. Koren, M. van Loon and J. G. Verwer, A positive finite-difference advection scheme, J. Comput. Phys., 117 (1995), 35-46. doi: 10.1006/jcph.1995.1042. Google Scholar
[25]	I. Kamenkovich, P. Berloff and J. Pedlosky, Anisotropic material transport by eddies and eddy-driven currents in a model of the North Atlantic, J. Phys. Oceanogr., 39 (2009), 3162-3175. Google Scholar
[26]	S. Karabasov, P. Berloff and V. Goloviznin, CABARETin the ocean gyres, Ocean Model., 2–3 (2009), 155-168. Google Scholar
[27]	I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, 113. Springer-Verlag, New York. 1991. doi: 10.1007/978-1-4612-0949-2. Google Scholar
[28]	R. Kraichnan, Small-scale structure of a randomly advected passive scalar, Phys. Rev. Lett., 11 (1968), 945-963. Google Scholar
[29]	C. Leith, Stochastic backscatter in a subgrid–scale model: Plane shear mixing layer, Phys. Fluids A: Fluid Dynamics, 2 (1990), 297-299. Google Scholar
[30]	M. Leutbecher, S.-J. Lock, P. Ollinaho, S. Lang, G. Balsamo, P. Bechtold, M. Bonavita, H. Christensen, M. Diamantakis, E. Dutra, S. English, M. Fisher, R. Forbes, J. Goddard, T. Haiden, R. Hogan, S. Juricke, H. Lawrence, D. MacLeod, L. Magnusson, S. Malardel, S. Massart, I. Sandu, P. Smolarkiewicz, A. Subramanian, F. Vitart, N. Wedi and A. Weisheimer, Stochastic representations of model uncertainties at ECMWF: State of the art and future vision, Q. J. R. Meteorol. Soc., 143 (2017), 2315-2339. Google Scholar
[31]	A. J. Majda, I. Timofeyev and E. Vanden Eijnden, A mathematical framework for stochastic climate models, Comm. Pure Appl. Math., 54 (2001), 891-974. doi: 10.1002/cpa.1014. Google Scholar
[32]	P. Mana and L. Zanna, Toward a stochastic parameterization of ocean mesoscale eddies, Ocean Model., 79 (2014), 1-20. Google Scholar
[33]	J. McWilliams, A note on a consistent quasigeostrophic model in a multiply connected domain, Dynam. Atmos. Ocean, 5 (1977), 427-441. Google Scholar
[34]	E. Mémin, Fluid flow dynamics under location uncertainty, Geophys. Astrophys. Fluid Dyn., 108 (2014), 119-146. doi: 10.1080/03091929.2013.836190. Google Scholar
[35]	J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. Google Scholar
[36]	R. W. Preisendorfer, Principal Component Analysis in Meteorology and Oceanography, Elsevier, Amsterdam, 1988. Google Scholar
[37]	P. J. Schmid, Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656 (2010), 5-28. doi: 10.1017/S0022112010001217. Google Scholar
[38]	I. Shevchenko and P. Berloff, Multi-layer quasi-geostrophic ocean dynamics in eddy-resolving regimes, Ocean Modell., 94 (2015), 1-14. Google Scholar
[39]	I. Shevchenko and P. Berloff, Eddy backscatter and counter-rotating gyre anomalies of midlatitude ocean dynamics, Fluids, 1 (2016), 1-16. Google Scholar
[40]	C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), 439-471. doi: 10.1016/0021-9991(88)90177-5. Google Scholar
[41]	A. Siegel, J. Weiss, J. Toomre, J. McWilliams, P. Berloff and I. Yavneh, Eddies and vortices in ocean basin dynamics, Geophys. Res. Lett., 28, 3183–3186. Google Scholar
[42]	G. Vallis, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation, Cambridge University Press, Cambridge, UK. Google Scholar
[43]	S. Vannitsem, Stochastic modelling and predictability: Analysis of a low-order coupled ocean–atmosphere model, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2018), 20130282, 18 pp. doi: 10.1098/rsta.2013.0282. Google Scholar
[44]	A. Weigel, Ensemble forecasts, In Jolliffe, I. and Stephenson, D., editors, Forecast Verification: A Practitioner's Guide in Atmospheric Science, chapter 8, pages 141–166. John Wiley & Sons, Oxford, UK, 2 edition. Google Scholar
[45]	P. Woodward and P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54 (1984), 115-173. doi: 10.1016/0021-9991(84)90142-6. Google Scholar
[46]	J. Wouters and V. Lucarini, Disentangling multi-level systems: Averaging, correlations and memory, Journal of Statistical Mechanics: Theory and Experiment, (2012), P03003. Google Scholar

show all references

References:

[1]	R. Abramov, A simple stochastic parameterization for reduced models of multiscale dynamics, Fluids, 1 (2016), 1-18. Google Scholar
[2]	A. Arakawa, Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part Ⅰ, J. Comput. Phys., 1 (1966), 119-143. Google Scholar
[3]	P. S. Berloff, Random-forcing model of the mesoscale oceanic eddies, J. Fluid Mech., 529 (2005), 71-95. doi: 10.1017/S0022112005003393. Google Scholar
[4]	P. Berloff and I. Kamenkovich, On spectral analysis of mesoscale eddies. Part Ⅰ: Linear analysis, J. Phys. Oceanogr., 43 (2013), 2505-2527. Google Scholar
[5]	P. Berloff, S. Karabasov, J. T. Farrar and I. Kamenkovich, On latency of multiple zonal jets in the oceans, J. Fluid Mech., 686 (2011), 534-567. doi: 10.1017/jfm.2011.345. Google Scholar
[6]	P. Berloff and J. McWilliams, Material transport in oceanic gyres. Part Ⅱ: Hierarchy of stochastic models, J. Phys. Oceanogr., 32 (2002), 797-830. Google Scholar
[7]	P. Berloff and J. McWilliams, Material transport in oceanic gyres. Part Ⅲ: Randomized stochastic models, J. Phys. Oceanogr., 33 (2003), 1416-1445. Google Scholar
[8]	J. Bröcker, Assessing the reliability of ensemble forecasting systems under serial dependence, Q. J. R. Meteorol. Soc., 144 (2018), 2666-2675. Google Scholar
[9]	K. K. Chen, J. H. Tu and C. W. Rowley, Variants of dynamic mode decomposition: Boundary condition Koopman and Fourier analyses, J. Nonlinear Sci., 22 (2012), 887-915. doi: 10.1007/s00332-012-9130-9. Google Scholar
[10]	F. Cooper and L. Zanna, Optimization of an idealised ocean model, stochastic parameterisation of sub-grid eddies, Ocean Model., 88 (2015), 38-53. Google Scholar
[11]	C. Cotter, D. Crisan, D. Holm, W. Pan and I. Shevchenko, Data assimilation for a quasi-geostrophic model with circulation-preserving stochastic transport noise, J. Stat. Phys., 179 (2020), 1186-1221. doi: 10.1007/s10955-020-02524-0. Google Scholar
[12]	C. J. Cotter, G. A. Gottwald and D. D. Holm, Stochastic partial differential fluid equations as a diffusive limit of deterministic lagrangian multi-time dynamics, Proc. A., 473 (2017), 20170388, 10 pp. doi: 10.1098/rspa.2017.0388. Google Scholar
[13]	J. Duan and B. T. Nadiga, Stochastic parameterization for large eddy simulation of geophysical flows, Proc. Amer. Math. Soc., 135 (2007), 1187-1196. doi: 10.1090/S0002-9939-06-08631-X. Google Scholar
[14]	J. Elsner and A. Tsonis, Singular Spectrum Analysis: A New Tool in Time Series Analysis, Plenum Press, New York, 1996. Google Scholar
[15]	C. Franzke, A. J. Majda and E. Vanden-Eijnden, Low-order stochastic mode reduction for a realistic barotropic model climate, J. Atmospheric Sci., 62 (2005), 1722-1745. doi: 10.1175/JAS3438.1. Google Scholar
[16]	J. Frederiksen, T. OKane and M. Zidikheri, Stochastic subgrid parameterizations for atmospheric and oceanic flows, Phys. Scr., 85 (2012), 068202. Google Scholar
[17]	F. Gay-Balmaz and D. D. Holm, Stochastic geometric models with non-stationary spatial correlations in Lagrangian fluid flows, J. Nonlinear Sci., 28 (2018), 873-904. doi: 10.1007/s00332-017-9431-0. Google Scholar
[18]	P. Gent, The Gent–McWilliams parameterization: 20/20 hindsight, Ocean Model., 39 (2011), 2-9. Google Scholar
[19]	P. Gent and J. Mcwilliams, Isopycnal mixing in ocean circulation models, J. Phys. Oceanogr., 20 (1990), 150-155. Google Scholar
[20]	I. Grooms, A. Majda and K. Smith, Stochastic superparametrization in a quasigeostrophic model of the Antarctic Circumpolar Current, Ocean Model., 85 (2015), 1-15. Google Scholar
[21]	E. Hairer, S. Nørsett and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems, Springer-Verlag, Berlin, 1993. Google Scholar
[22]	A. Hannachi, I. Jolliffe and D. Stephenson, Empirical orthogonal functions and related techniques in atmospheric science: A review, Int. J. Climatol., 27 (2007), 1119-1152. Google Scholar
[23]	D. D. Holm, Variational principles for stochastic fluid dynamics, Proc. A., 471 (2015), 20140963, 19 pp. doi: 10.1098/rspa.2014.0963. Google Scholar
[24]	W. Hundsdorfer, B. Koren, M. van Loon and J. G. Verwer, A positive finite-difference advection scheme, J. Comput. Phys., 117 (1995), 35-46. doi: 10.1006/jcph.1995.1042. Google Scholar
[25]	I. Kamenkovich, P. Berloff and J. Pedlosky, Anisotropic material transport by eddies and eddy-driven currents in a model of the North Atlantic, J. Phys. Oceanogr., 39 (2009), 3162-3175. Google Scholar
[26]	S. Karabasov, P. Berloff and V. Goloviznin, CABARETin the ocean gyres, Ocean Model., 2–3 (2009), 155-168. Google Scholar
[27]	I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, 113. Springer-Verlag, New York. 1991. doi: 10.1007/978-1-4612-0949-2. Google Scholar
[28]	R. Kraichnan, Small-scale structure of a randomly advected passive scalar, Phys. Rev. Lett., 11 (1968), 945-963. Google Scholar
[29]	C. Leith, Stochastic backscatter in a subgrid–scale model: Plane shear mixing layer, Phys. Fluids A: Fluid Dynamics, 2 (1990), 297-299. Google Scholar
[30]	M. Leutbecher, S.-J. Lock, P. Ollinaho, S. Lang, G. Balsamo, P. Bechtold, M. Bonavita, H. Christensen, M. Diamantakis, E. Dutra, S. English, M. Fisher, R. Forbes, J. Goddard, T. Haiden, R. Hogan, S. Juricke, H. Lawrence, D. MacLeod, L. Magnusson, S. Malardel, S. Massart, I. Sandu, P. Smolarkiewicz, A. Subramanian, F. Vitart, N. Wedi and A. Weisheimer, Stochastic representations of model uncertainties at ECMWF: State of the art and future vision, Q. J. R. Meteorol. Soc., 143 (2017), 2315-2339. Google Scholar
[31]	A. J. Majda, I. Timofeyev and E. Vanden Eijnden, A mathematical framework for stochastic climate models, Comm. Pure Appl. Math., 54 (2001), 891-974. doi: 10.1002/cpa.1014. Google Scholar
[32]	P. Mana and L. Zanna, Toward a stochastic parameterization of ocean mesoscale eddies, Ocean Model., 79 (2014), 1-20. Google Scholar
[33]	J. McWilliams, A note on a consistent quasigeostrophic model in a multiply connected domain, Dynam. Atmos. Ocean, 5 (1977), 427-441. Google Scholar
[34]	E. Mémin, Fluid flow dynamics under location uncertainty, Geophys. Astrophys. Fluid Dyn., 108 (2014), 119-146. doi: 10.1080/03091929.2013.836190. Google Scholar
[35]	J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. Google Scholar
[36]	R. W. Preisendorfer, Principal Component Analysis in Meteorology and Oceanography, Elsevier, Amsterdam, 1988. Google Scholar
[37]	P. J. Schmid, Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656 (2010), 5-28. doi: 10.1017/S0022112010001217. Google Scholar
[38]	I. Shevchenko and P. Berloff, Multi-layer quasi-geostrophic ocean dynamics in eddy-resolving regimes, Ocean Modell., 94 (2015), 1-14. Google Scholar
[39]	I. Shevchenko and P. Berloff, Eddy backscatter and counter-rotating gyre anomalies of midlatitude ocean dynamics, Fluids, 1 (2016), 1-16. Google Scholar
[40]	C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), 439-471. doi: 10.1016/0021-9991(88)90177-5. Google Scholar
[41]	A. Siegel, J. Weiss, J. Toomre, J. McWilliams, P. Berloff and I. Yavneh, Eddies and vortices in ocean basin dynamics, Geophys. Res. Lett., 28, 3183–3186. Google Scholar
[42]	G. Vallis, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation, Cambridge University Press, Cambridge, UK. Google Scholar
[43]	S. Vannitsem, Stochastic modelling and predictability: Analysis of a low-order coupled ocean–atmosphere model, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2018), 20130282, 18 pp. doi: 10.1098/rsta.2013.0282. Google Scholar
[44]	A. Weigel, Ensemble forecasts, In Jolliffe, I. and Stephenson, D., editors, Forecast Verification: A Practitioner's Guide in Atmospheric Science, chapter 8, pages 141–166. John Wiley & Sons, Oxford, UK, 2 edition. Google Scholar
[45]	P. Woodward and P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54 (1984), 115-173. doi: 10.1016/0021-9991(84)90142-6. Google Scholar
[46]	J. Wouters and V. Lucarini, Disentangling multi-level systems: Averaging, correlations and memory, Journal of Statistical Mechanics: Theory and Experiment, (2012), P03003. Google Scholar

Figure 1. Shown is a schematic of the computational domain $ \Omega $

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Figure 2. The series of snapshots shows the high-resolution solution $ q^f $ computed on the fine grid $ G^f = 2049\times1025 $ ($ dx\approx dy\approx 1.9\, {\rm km} $), the true solution $ q^a $ computed on the coarse grid $ G^c = 257\times129 $ ($ dx\approx dy\approx 15\, {\rm km} $), and the low-resolution solution $ q^m $ also computed on $ G^c $ by simulating the QG model for the low drag $ \boldsymbol{\mu = 4\times10^{-8}\, {\rm s^{-1}}} $ (heterogeneous flow). All the solutions are given in units of $ [s^{-1}f^{-1}_0] $, where $ f_0 = 0.83\times10^{-4}\, {\rm s^{-1}} $ is the Coriolis parameter. In order to visualize the solutions on the same color scale we have multiplied the ones in the second layer by a factor of 5

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Figure 2, but for $ G^c = 129\times65 $ ($ dx\approx dy\approx 299\, {\rm km} $)">Figure 3. The same as in Figure 2, but for $ G^c = 129\times65 $ ($ dx\approx dy\approx 299\, {\rm km} $)

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Figure 4. The series of snapshots shows the dependence of the solution on the resolution for the low drag $ \boldsymbol{\mu = 4\times10^{-8}\, {\rm s^{-1}}} $ (heterogeneous flow). All the solutions are given in units of $ [s^{-1}f^{-1}_0] $, where $ f_0 = 0.83\times10^{-4}\, {\rm s^{-1}} $ is the Coriolis parameter. In order to visualize the solutions on the same color scale we have multiplied the ones in the second layer by a factor of 5

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Figure 5. The series of snapshots in the figure shows the high-resolution solution $ q^f $ computed on the fine grid $ G^f = 2049\times1025 $ ($ dx\approx dy\approx 1.9\, {\rm km} $), the true solution $ q^a $ computed on the coarse grid $ G^c = 129\times65 $ ($ dx\approx dy\approx 299\, {\rm km} $), and the low-resolution solution $ q^m $ computed on the coarse grid $ G^c $ by simulating the QG model for the high drag $ \boldsymbol{\mu = 4\times10^{-7}\, {\rm s^{-1}}} $ (homogeneous flow). All the solutions are given in units of $ [s^{-1}f^{-1}_0] $, where $ f_0 = 0.83\times10^{-4}\, {\rm s^{-1}} $ is the Coriolis parameter. In order to visualize the solutions on the same color scale we have multiplied the ones in the second layer by a factor of 5

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Figure 6. Shown is a typical dependence of the area of the stochastic cloud $ A^c $ on the size of the stochastic ensemble $ \overline{\bf x} $. The left and right column shows the area of the stochastic cloud (marked in grey color) which consists of $ N = 100 $ and $ N = 400 $ ensemble members, respectively. The stochastic ensemble has been computed for the first 64 leading EOFs capturing 96% of the flow variability (top row) and the first 128 leading EOFs capturing 99% of the flow variability (bottom row). The true solution $ \bar{\bf x}^c $ is marked with a black dot. The plot represents a typical part of the computational domain of size $ [10,45]\times[45,65] $ in the first layer, which can be divided into two regions: a fast flow region (the boundary layer along the northern boundary $ [10,45]\times[60,65] $, the jet occupying the domain $ [10,45]\times[45,52] $) and a slow flow region $ [10,45]\times(52,60) $

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Figure 7. Shown are (a) instantaneous and (b) time-averaged normalized velocity fields for the heterogeneous flow

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Figure 8. Shown is the dependence of the averaged area of the stochastic cloud for the slow, $ \overline{A}^c_s $, and fast, $ \overline{A}^c_f $, flow regions on the number of EOFs, $ K $, and the size of the stochastic ensemble $ N $

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Figure 3; $ G^c = 129\times65 $. Using $ K = \{1,2,4,8,16,32,64\} $ leading EOFs allows to capture 23%, 42%, 60%, 77%, 89%, 96%, and 99% of the flow variability, respectively. The initial conditions for the stochastic model have been computed over the spin up period $ T_{\rm spin} = [-8,0] $ hours">Figure 9. Shown is the dependence of $ \widetilde{R}_{\mathcal{S}} $ for the velocity component $ u^a $ ($ v^a $ is not shown, since it behaves qualitatively similar to $ u^a $), stream function $ \psi^a $, and PV anomaly $ q^a $ on (a) the number of EOFs $ K $ ($ N = 100 $ in this case) and (b) size of the stochastic ensemble $ N $ ($ K = 1 $ in this case) over the time period $ T = [0,21] $ days for the heterogeneous flow in Figure 3; $ G^c = 129\times65 $. Using $ K = \{1,2,4,8,16,32,64\} $ leading EOFs allows to capture 23%, 42%, 60%, 77%, 89%, 96%, and 99% of the flow variability, respectively. The initial conditions for the stochastic model have been computed over the spin up period $ T_{\rm spin} = [-8,0] $ hours

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Figure 9, but for $ \widetilde{R}_{\mathcal{S}_{\sigma}} $">Figure 10. The same as in Figure 9, but for $ \widetilde{R}_{\mathcal{S}_{\sigma}} $

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Figure 9, but for $ G^c = 257\times129 $. We show $ v^a $ component of the velocity field, since it behaves differently from $ u^a $. For the higher resolution, using $ K = \{1,2,4,8,16,32,64\} $ leading EOFs allows to capture 39%, 61%, 77%, 90%, 97%, 99%, and 99.8% of the flow variability, respectively">Figure 11. The same as in Figure 9, but for $ G^c = 257\times129 $. We show $ v^a $ component of the velocity field, since it behaves differently from $ u^a $. For the higher resolution, using $ K = \{1,2,4,8,16,32,64\} $ leading EOFs allows to capture 39%, 61%, 77%, 90%, 97%, 99%, and 99.8% of the flow variability, respectively

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Figure 11, but for $ \widetilde{R}_{\mathcal{S}_{\sigma}} $">Figure 12. The same as in Figure 11, but for $ \widetilde{R}_{\mathcal{S}_{\sigma}} $

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Figure 13. Evolution of $ \overline{<\widetilde{T}_{\mathcal{S}_{\sigma}}>} $ for the heterogeneous flow and for different number of EOFs: $ K = 1 $ (solid line), $ K = 2 $ (solid line marked by a cross), $ K = 4 $ (solid line marked by an circle), as well as for different sizes of the stochastic ensemble: $ N = 100 $ (solid line), $ N = 200 $ (solid line marked by an asterisk), $ N = 400 $ (solid line marked by a square)

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Figure 14. Evolution of the root mean square error $ \overline{<RMSE(|\boldsymbol{u}|)>} $ (solid line) and standard deviation $ \overline{<\sigma(|\boldsymbol{u}|)>} $ (dashed line) of the ensemble mean for the heterogeneous flow and different number of EOFs: $ K = 1 $ (solid/dashed line), $ K = 2 $ (solid/dashed line marked by a cross), $ K = 4 $ (solid/dashed line marked by an circle); and for different sizes of ensemble: $ N = 100 $ (solid/dashed line), $ N = 200 $ (solid/dashed line marked by an asterisk), $ N = 400 $ (solid/dashed line marked by a square). Note that the initial ensemble is biased

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Figure 14, but for an unbiased ensemble. The results for the homogeneous flow look very similar">Figure 15. The same as in Figure 14, but for an unbiased ensemble. The results for the homogeneous flow look very similar

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Figure 16. Evolution of the root mean square error, $ \overline{<RMSE(|\boldsymbol{u}|)>} $ (solid line), and standard deviation, $ \overline{<\sigma(|\boldsymbol{u}|)>} $ (dashed line), of the ensemble mean for the heterogeneous flow computed for $ K = 1 $, $ N = 100 $ and different amplitudes of the noise $ dW $: $ |dW| = 1 $ (solid/dashed line), $ |dW| = 5 $ (solid/dashed line marked by a cross), $ |dW| = 10 $ (solid/dashed line marked by an circle). Not that the initial ensemble is unbiased. The results for the homogeneous flow look very similar

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Figure 3; $ G^c = 129\times65 $. The initial conditions for the deterministic model have been computed over the spin up period $ T_{\rm spin} = [-8,0] $ hours. The results for the homogeneous flow look very similar">Figure 17. Shown is the dependence of $ R_{\mathcal{S}} $ (deterministic case) for the velocity component $ u^a $ ($ v^a $ is not shown, since it behaves qualitatively similar to $ u^a $), stream function $ \psi^a $, and PV anomaly $ q^a $ on (a) the number of EOFs $ K $ ($ N = 100 $ in this case) and (b) size of the stochastic ensemble $ N $ ($ K = 1 $ in this case) over the time period $ T = [0,21] $ days for the heterogeneous flow presented in Figure 3; $ G^c = 129\times65 $. The initial conditions for the deterministic model have been computed over the spin up period $ T_{\rm spin} = [-8,0] $ hours. The results for the homogeneous flow look very similar

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Figure 17, but for $ G^c = 257\times129 $. The results for the homogeneous flow look very similar">Figure 18. The same as in Figure 17, but for $ G^c = 257\times129 $. The results for the homogeneous flow look very similar

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Figure 19. Evolution of $ \overline{<\widetilde{T}_{\mathcal{S}_{\sigma}}>} $ (solid line; stochastic case) and $ \overline{<T_{\mathcal{S}_{\sigma}}>} $ (dashed line; deterministic case) for the heterogeneous flow; the number of leading EOFs is $ K = 1 $, and the size of the ensemble is $ N = 100 $. The results for the homogeneous flow look very similar

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Figure 20. Typical rank histograms for heterogeneous flow velocity $ \mathbf{u} = (u,v) $ at different locations (not shown) and resolutions. Note that a pair of histograms at different resolutions shares the same location in the computation domain. Each histogram is based on 1000 forecast-observation pairs generated by solving the stochastic QG model. For simulating the stochastic QG model we use 100 stochastic solutions and 32 leading EOFs. Each stochastic solution for the rank histogram is selected randomly from the ensemble every 4 hours. The rank histograms at the higher resolution are flatter and t hus the forecasts are more reliable, although this resolution is still much less than the reference simulation

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Figure 21. The series of snapshots shows the true deterministic solution $ q^a_1 $ computed on $ G^c = \{129\times65 $, $ 257\times129 \}$, and the low-resolution parameterised solution $ \bar{q}^p_1 $ (averaged over the stochastic ensemble of size $ N = 100 $) computed on the same coarse grids by simulating the stochastic QG model from a randomly perturbed zero initial condition for the heterogeneous flow; the parameterised solution uses 32 leading EOFs. All solutions are given in units of $ [s^{-1}f^{-1}_0] $, where $ f_0 = 0.83\times10^{-4}\, {\rm s^{-1}} $ is the Coriolis parameter. The true solutions correspond to day 1 and presented here just to show the structure of the flow

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Figure 22. The series of snapshots shows the true deterministic solution $ q^a_1 $, modelled solution $ q^m_1 $ computed with the deterministic QG model, and parameterised solution $ \bar{q}^p_1 $ (averaged over the stochastic ensemble of size $ N = 100 $) computed with the stochastic QG model. All solutions were computed on the same grid $ G^c = 129\times65 $, have the same initial condition, and the parameterised solution uses 32 leading EOFs. All fields are given in units of $ [s^{-1}f^{-1}_0] $, where $ f_0 = 0.83\times10^{-4}\, {\rm s^{-1}} $ is the Coriolis parameter

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Figure 22, but for $ G^c = 257\times129 $">Figure 23. The same as in Figure 22, but for $ G^c = 257\times129 $

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Figure 24. Shown is the relative $ l_2 $-norm error between the true deterministic solution, $ {\bf u}^a $, and modelled solution $ {\bf u}^m $ (red line) and parameterised solution $ {\bf u}^p $ (blue line) as a function of time

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