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A Bayesian nonparametric test for conditional independence
Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model
Department of Mathematics, Imperial College London, Huxley Building, 180 Queen's Gate, London, SW7 2AZ, UK |
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.
References:
[1] |
R. Abramov,
A simple stochastic parameterization for reduced models of multiscale dynamics, Fluids, 1 (2016), 1-18.
|
[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.
|
[3] |
P. S. Berloff,
Random-forcing model of the mesoscale oceanic eddies, J. Fluid Mech., 529 (2005), 71-95.
doi: 10.1017/S0022112005003393. |
[4] |
P. Berloff and I. Kamenkovich,
On spectral analysis of mesoscale eddies. Part Ⅰ: Linear analysis, J. Phys. Oceanogr., 43 (2013), 2505-2527.
|
[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. |
[6] |
P. Berloff and J. McWilliams,
Material transport in oceanic gyres. Part Ⅱ: Hierarchy of stochastic models, J. Phys. Oceanogr., 32 (2002), 797-830.
|
[7] |
P. Berloff and J. McWilliams,
Material transport in oceanic gyres. Part Ⅲ: Randomized stochastic models, J. Phys. Oceanogr., 33 (2003), 1416-1445.
|
[8] |
J. Bröcker,
Assessing the reliability of ensemble forecasting systems under serial dependence, Q. J. R. Meteorol. Soc., 144 (2018), 2666-2675.
|
[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. |
[10] |
F. Cooper and L. Zanna,
Optimization of an idealised ocean model, stochastic parameterisation of sub-grid eddies, Ocean Model., 88 (2015), 38-53.
|
[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. |
[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. |
[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. |
[14] |
J. Elsner and A. Tsonis, Singular Spectrum Analysis: A New Tool in Time Series Analysis, Plenum Press, New York, 1996.
![]() |
[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. |
[16] |
J. Frederiksen, T. OKane and M. Zidikheri, Stochastic subgrid parameterizations for atmospheric and oceanic flows, Phys. Scr., 85 (2012), 068202. |
[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. |
[18] |
P. Gent,
The Gent–McWilliams parameterization: 20/20 hindsight, Ocean Model., 39 (2011), 2-9.
|
[19] |
P. Gent and J. Mcwilliams,
Isopycnal mixing in ocean circulation models, J. Phys. Oceanogr., 20 (1990), 150-155.
|
[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.
|
[21] |
E. Hairer, S. Nørsett and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems, Springer-Verlag, Berlin, 1993. |
[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.
|
[23] |
D. D. Holm, Variational principles for stochastic fluid dynamics, Proc. A., 471 (2015), 20140963, 19 pp.
doi: 10.1098/rspa.2014.0963. |
[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. |
[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.
|
[26] |
S. Karabasov, P. Berloff and V. Goloviznin,
CABARETin the ocean gyres, Ocean Model., 2–3 (2009), 155-168.
|
[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. |
[28] |
R. Kraichnan,
Small-scale structure of a randomly advected passive scalar, Phys. Rev. Lett., 11 (1968), 945-963.
|
[29] |
C. Leith,
Stochastic backscatter in a subgrid–scale model: Plane shear mixing layer, Phys. Fluids A: Fluid Dynamics, 2 (1990), 297-299.
|
[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.
|
[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. |
[32] |
P. Mana and L. Zanna,
Toward a stochastic parameterization of ocean mesoscale eddies, Ocean Model., 79 (2014), 1-20.
|
[33] |
J. McWilliams,
A note on a consistent quasigeostrophic model in a multiply connected domain, Dynam. Atmos. Ocean, 5 (1977), 427-441.
|
[34] |
E. Mémin,
Fluid flow dynamics under location uncertainty, Geophys. Astrophys. Fluid Dyn., 108 (2014), 119-146.
doi: 10.1080/03091929.2013.836190. |
[35] |
J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. |
[36] |
R. W. Preisendorfer, Principal Component Analysis in Meteorology and Oceanography, Elsevier, Amsterdam, 1988. |
[37] |
P. J. Schmid,
Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656 (2010), 5-28.
doi: 10.1017/S0022112010001217. |
[38] |
I. Shevchenko and P. Berloff,
Multi-layer quasi-geostrophic ocean dynamics in eddy-resolving regimes, Ocean Modell., 94 (2015), 1-14.
|
[39] |
I. Shevchenko and P. Berloff,
Eddy backscatter and counter-rotating gyre anomalies of midlatitude ocean dynamics, Fluids, 1 (2016), 1-16.
|
[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. |
[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. |
[42] |
G. Vallis, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation, Cambridge University Press, Cambridge, UK. |
[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. |
[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. |
[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. |
[46] |
J. Wouters and V. Lucarini, Disentangling multi-level systems: Averaging, correlations and memory, Journal of Statistical Mechanics: Theory and Experiment, (2012), P03003. |
show all references
References:
[1] |
R. Abramov,
A simple stochastic parameterization for reduced models of multiscale dynamics, Fluids, 1 (2016), 1-18.
|
[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.
|
[3] |
P. S. Berloff,
Random-forcing model of the mesoscale oceanic eddies, J. Fluid Mech., 529 (2005), 71-95.
doi: 10.1017/S0022112005003393. |
[4] |
P. Berloff and I. Kamenkovich,
On spectral analysis of mesoscale eddies. Part Ⅰ: Linear analysis, J. Phys. Oceanogr., 43 (2013), 2505-2527.
|
[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. |
[6] |
P. Berloff and J. McWilliams,
Material transport in oceanic gyres. Part Ⅱ: Hierarchy of stochastic models, J. Phys. Oceanogr., 32 (2002), 797-830.
|
[7] |
P. Berloff and J. McWilliams,
Material transport in oceanic gyres. Part Ⅲ: Randomized stochastic models, J. Phys. Oceanogr., 33 (2003), 1416-1445.
|
[8] |
J. Bröcker,
Assessing the reliability of ensemble forecasting systems under serial dependence, Q. J. R. Meteorol. Soc., 144 (2018), 2666-2675.
|
[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. |
[10] |
F. Cooper and L. Zanna,
Optimization of an idealised ocean model, stochastic parameterisation of sub-grid eddies, Ocean Model., 88 (2015), 38-53.
|
[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. |
[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. |
[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. |
[14] |
J. Elsner and A. Tsonis, Singular Spectrum Analysis: A New Tool in Time Series Analysis, Plenum Press, New York, 1996.
![]() |
[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. |
[16] |
J. Frederiksen, T. OKane and M. Zidikheri, Stochastic subgrid parameterizations for atmospheric and oceanic flows, Phys. Scr., 85 (2012), 068202. |
[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. |
[18] |
P. Gent,
The Gent–McWilliams parameterization: 20/20 hindsight, Ocean Model., 39 (2011), 2-9.
|
[19] |
P. Gent and J. Mcwilliams,
Isopycnal mixing in ocean circulation models, J. Phys. Oceanogr., 20 (1990), 150-155.
|
[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.
|
[21] |
E. Hairer, S. Nørsett and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems, Springer-Verlag, Berlin, 1993. |
[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.
|
[23] |
D. D. Holm, Variational principles for stochastic fluid dynamics, Proc. A., 471 (2015), 20140963, 19 pp.
doi: 10.1098/rspa.2014.0963. |
[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. |
[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.
|
[26] |
S. Karabasov, P. Berloff and V. Goloviznin,
CABARETin the ocean gyres, Ocean Model., 2–3 (2009), 155-168.
|
[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. |
[28] |
R. Kraichnan,
Small-scale structure of a randomly advected passive scalar, Phys. Rev. Lett., 11 (1968), 945-963.
|
[29] |
C. Leith,
Stochastic backscatter in a subgrid–scale model: Plane shear mixing layer, Phys. Fluids A: Fluid Dynamics, 2 (1990), 297-299.
|
[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.
|
[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. |
[32] |
P. Mana and L. Zanna,
Toward a stochastic parameterization of ocean mesoscale eddies, Ocean Model., 79 (2014), 1-20.
|
[33] |
J. McWilliams,
A note on a consistent quasigeostrophic model in a multiply connected domain, Dynam. Atmos. Ocean, 5 (1977), 427-441.
|
[34] |
E. Mémin,
Fluid flow dynamics under location uncertainty, Geophys. Astrophys. Fluid Dyn., 108 (2014), 119-146.
doi: 10.1080/03091929.2013.836190. |
[35] |
J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. |
[36] |
R. W. Preisendorfer, Principal Component Analysis in Meteorology and Oceanography, Elsevier, Amsterdam, 1988. |
[37] |
P. J. Schmid,
Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656 (2010), 5-28.
doi: 10.1017/S0022112010001217. |
[38] |
I. Shevchenko and P. Berloff,
Multi-layer quasi-geostrophic ocean dynamics in eddy-resolving regimes, Ocean Modell., 94 (2015), 1-14.
|
[39] |
I. Shevchenko and P. Berloff,
Eddy backscatter and counter-rotating gyre anomalies of midlatitude ocean dynamics, Fluids, 1 (2016), 1-16.
|
[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. |
[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. |
[42] |
G. Vallis, Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation, Cambridge University Press, Cambridge, UK. |
[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. |
[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. |
[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. |
[46] |
J. Wouters and V. Lucarini, Disentangling multi-level systems: Averaging, correlations and memory, Journal of Statistical Mechanics: Theory and Experiment, (2012), P03003. |




















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