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Title: A dynamic closure modeling framework for model order reduction of geophysical flows

Abstract

Here in this paper, a dynamic closure modeling approach has been derived to stabilize the projection-based reduced order models in the long-term evolution of forced-dissipative dynamical systems. To simplify our derivation without losing generalizability, the proposed reduced order modeling (ROM) framework is first constructed by Galerkin projection of the single-layer quasigeostrophic equation, a standard prototype of large-scale general circulation models, onto a set of dominant proper orthogonal decomposition modes. We then propose an eddy viscosity closure approach to stabilize the resulting surrogate model considering the analogy between large eddy simulation (LES) and truncated modal projection. Our efforts, in particular, include the translation of the dynamic subgrid-scale model into our ROM setting by defining a test truncation similar to the test filtering in LES. Finally, the a posteriori analysis shows that our approach is remarkably accurate, allowing us to integrate simulations over long time intervals at a nominally small computational overhead.

Authors:
 [1];  [1];  [1]
  1. Oklahoma State Univ., Stillwater, OK (United States). School of Mechanical and Aerospace Engineering
Publication Date:
Research Org.:
Oklahoma State Univ., Stillwater, OK (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)
OSTI Identifier:
1593561
Alternate Identifier(s):
OSTI ID: 1509520
Grant/Contract Number:  
SC0019290
Resource Type:
Accepted Manuscript
Journal Name:
Physics of Fluids
Additional Journal Information:
Journal Volume: 31; Journal Issue: 4; Journal ID: ISSN 1070-6631
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
42 ENGINEERING; Turbulence modeling; Geophysical flows; Reduced order modeling; Proper orthogonal decomposition; Galerkin projection; Dynamic eddy viscosity closure

Citation Formats

Rahman, Sk. M., Ahmed, S. E., and San, O. A dynamic closure modeling framework for model order reduction of geophysical flows. United States: N. p., 2019. Web. doi:10.1063/1.5093355.
Rahman, Sk. M., Ahmed, S. E., & San, O. A dynamic closure modeling framework for model order reduction of geophysical flows. United States. doi:10.1063/1.5093355.
Rahman, Sk. M., Ahmed, S. E., and San, O. Fri . "A dynamic closure modeling framework for model order reduction of geophysical flows". United States. doi:10.1063/1.5093355.
@article{osti_1593561,
title = {A dynamic closure modeling framework for model order reduction of geophysical flows},
author = {Rahman, Sk. M. and Ahmed, S. E. and San, O.},
abstractNote = {Here in this paper, a dynamic closure modeling approach has been derived to stabilize the projection-based reduced order models in the long-term evolution of forced-dissipative dynamical systems. To simplify our derivation without losing generalizability, the proposed reduced order modeling (ROM) framework is first constructed by Galerkin projection of the single-layer quasigeostrophic equation, a standard prototype of large-scale general circulation models, onto a set of dominant proper orthogonal decomposition modes. We then propose an eddy viscosity closure approach to stabilize the resulting surrogate model considering the analogy between large eddy simulation (LES) and truncated modal projection. Our efforts, in particular, include the translation of the dynamic subgrid-scale model into our ROM setting by defining a test truncation similar to the test filtering in LES. Finally, the a posteriori analysis shows that our approach is remarkably accurate, allowing us to integrate simulations over long time intervals at a nominally small computational overhead.},
doi = {10.1063/1.5093355},
journal = {Physics of Fluids},
number = 4,
volume = 31,
place = {United States},
year = {2019},
month = {4}
}

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Works referenced in this record:

A Reduced-Order Method for Simulation and Control of Fluid Flows
journal, July 1998


Modeling Mesoscale Turbulence in the Barotropic Double-Gyre Circulation
journal, November 2003


An eddy-viscosity subgrid-scale model for turbulent shear flow: Algebraic theory and applications
journal, October 2004


A proposed modification of the Germano subgrid‐scale closure method
journal, March 1992

  • Lilly, D. K.
  • Physics of Fluids A: Fluid Dynamics, Vol. 4, Issue 3
  • DOI: 10.1063/1.858280

Intermodal energy transfers in a proper orthogonal decomposition–Galerkin representation of a turbulent separated flow
journal, September 2003


Dispersive–Dissipative Eddy Parameterization in a Barotropic Model
journal, August 2001


Reduced-order subscales for POD models
journal, July 2015

  • Baiges, Joan; Codina, Ramon; Idelsohn, Sergio
  • Computer Methods in Applied Mechanics and Engineering, Vol. 291
  • DOI: 10.1016/j.cma.2015.03.020

The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows
journal, January 1993


Artificial viscosity proper orthogonal decomposition
journal, January 2011

  • Borggaard, Jeff; Iliescu, Traian; Wang, Zhu
  • Mathematical and Computer Modelling, Vol. 53, Issue 1-2
  • DOI: 10.1016/j.mcm.2010.08.015

A Short Review on Model Order Reduction Based on Proper Generalized Decomposition
journal, October 2011

  • Chinesta, Francisco; Ladeveze, Pierre; Cueto, Elías
  • Archives of Computational Methods in Engineering, Vol. 18, Issue 4
  • DOI: 10.1007/s11831-011-9064-7

Numerical Simulations of a Two-Layer Quasi-Geostrophic Equation of the Ocean
journal, January 2000


Identification strategies for model-based control
journal, July 2013


A dynamic subgrid‐scale eddy viscosity model
journal, July 1991

  • Germano, Massimo; Piomelli, Ugo; Moin, Parviz
  • Physics of Fluids A: Fluid Dynamics, Vol. 3, Issue 7
  • DOI: 10.1063/1.857955

On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far-field boundary treatment
journal, August 2010

  • Kalashnikova, I.; Barone, M. F.
  • International Journal for Numerical Methods in Engineering, Vol. 83, Issue 10
  • DOI: 10.1002/nme.2867

Efficiency of a POD-based reduced second-order adjoint model in 4D-Var data assimilation
journal, January 2007

  • Daescu, D. N.; Navon, I. M.
  • International Journal for Numerical Methods in Fluids, Vol. 53, Issue 6
  • DOI: 10.1002/fld.1316

Model Reduction for Flow Analysis and Control
journal, January 2017


Approximate deconvolution reduced order modeling
journal, January 2017

  • Xie, Xuping; Wells, David; Wang, Zhu
  • Computer Methods in Applied Mechanics and Engineering, Vol. 313
  • DOI: 10.1016/j.cma.2016.10.005

On Low-Dimensional Galerkin Models for Fluid Flow
journal, June 2000


Modal Analysis of Fluid Flows: An Overview
journal, December 2017

  • Taira, Kunihiko; Brunton, Steven L.; Dawson, Scott T. M.
  • AIAA Journal, Vol. 55, Issue 12
  • DOI: 10.2514/1.j056060

A New Closure Strategy for Proper Orthogonal Decomposition Reduced-Order Models
journal, April 2012

  • Akhtar, Imran; Wang, Zhu; Borggaard, Jeff
  • Journal of Computational and Nonlinear Dynamics, Vol. 7, Issue 3
  • DOI: 10.1115/1.4005928

pyMOR -- Generic Algorithms and Interfaces for Model Order Reduction
journal, January 2016

  • Milk, René; Rave, Stephan; Schindler, Felix
  • SIAM Journal on Scientific Computing, Vol. 38, Issue 5
  • DOI: 10.1137/15m1026614

Evaluating Data Assimilation Algorithms
journal, November 2012


Sparse reduced-order modelling: sensor-based dynamics to full-state estimation
journal, April 2018

  • Loiseau, Jean-Christophe; Noack, Bernd R.; Brunton, Steven L.
  • Journal of Fluid Mechanics, Vol. 844
  • DOI: 10.1017/jfm.2018.147

Dynamics of three-dimensional coherent structures in a flat-plate boundary layer
journal, September 1994


An evolve-then-filter regularized reduced order model for convection-dominated flows: EVOLVE-THEN-FILTER REGULARIZED REDUCED ORDER MODEL
journal, February 2017

  • Wells, D.; Wang, Z.; Xie, X.
  • International Journal for Numerical Methods in Fluids, Vol. 84, Issue 10
  • DOI: 10.1002/fld.4363

A reduced-order approach to four-dimensional variational data assimilation using proper orthogonal decomposition
journal, January 2007

  • Cao, Yanhua; Zhu, Jiang; Navon, I. M.
  • International Journal for Numerical Methods in Fluids, Vol. 53, Issue 10
  • DOI: 10.1002/fld.1365

A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems
journal, January 2015

  • Benner, Peter; Gugercin, Serkan; Willcox, Karen
  • SIAM Review, Vol. 57, Issue 4
  • DOI: 10.1137/130932715

A spectral viscosity method for correcting the long-term behavior of POD models
journal, February 2004


Spectral proper orthogonal decomposition
journal, March 2016

  • Sieber, Moritz; Paschereit, C. Oliver; Oberleithner, Kilian
  • Journal of Fluid Mechanics, Vol. 792
  • DOI: 10.1017/jfm.2016.103

Observing the Ocean in the 1990s
journal, October 1982

  • Munk, W.; Wunsch, C.
  • Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 307, Issue 1499
  • DOI: 10.1098/rsta.1982.0120

Proper orthogonal decomposition and low-dimensional models for driven cavity flows
journal, July 1998

  • Cazemier, W.; Verstappen, R. W. C. P.; Veldman, A. E. P.
  • Physics of Fluids, Vol. 10, Issue 7
  • DOI: 10.1063/1.869686

pyROM: A computational framework for reduced order modeling
journal, January 2019


Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison
journal, September 2012

  • Wang, Zhu; Akhtar, Imran; Borggaard, Jeff
  • Computer Methods in Applied Mechanics and Engineering, Vol. 237-240
  • DOI: 10.1016/j.cma.2012.04.015

Stabilization of projection-based reduced-order models: STABILIZATION OF PROJECTION-BASED REDUCED-ORDER MODELS
journal, February 2012

  • Amsallem, David; Farhat, Charbel
  • International Journal for Numerical Methods in Engineering, Vol. 91, Issue 4
  • DOI: 10.1002/nme.4274

Wind-driven ocean circulation and equilibrium statistical mechanics
journal, August 1989


Numerical analysis of the Leray reduced order model
journal, January 2018

  • Xie, Xuping; Wells, David; Wang, Zhu
  • Journal of Computational and Applied Mathematics, Vol. 328
  • DOI: 10.1016/j.cam.2017.06.026

Extreme learning machine for reduced order modeling of turbulent geophysical flows
journal, April 2018


Stabilization of projection-based reduced order models of the Navier–Stokes
journal, August 2012


Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier–Stokes equation
journal, July 2013

  • Balajewicz, Maciej J.; Dowell, Earl H.; Noack, Bernd R.
  • Journal of Fluid Mechanics, Vol. 729
  • DOI: 10.1017/jfm.2013.278

On the hidden beauty of the proper orthogonal decomposition
journal, August 1991

  • Aubry, Nadine
  • Theoretical and Computational Fluid Dynamics, Vol. 2, Issue 5-6
  • DOI: 10.1007/bf00271473

Two-level discretizations of nonlinear closure models for proper orthogonal decomposition
journal, January 2011


Evolution of three-dimensional coherent structures in a flat-plate boundary layer
journal, February 1994


The Quantum Limit to Moore's Law
journal, August 2008


Data-Driven Filtered Reduced Order Modeling of Fluid Flows
journal, January 2018

  • Xie, X.; Mohebujjaman, M.; Rebholz, L. G.
  • SIAM Journal on Scientific Computing, Vol. 40, Issue 3
  • DOI: 10.1137/17m1145136

Physically constrained data‐driven correction for reduced‐order modeling of fluid flows
journal, October 2018

  • Mohebujjaman, M.; Rebholz, L. G.; Iliescu, T.
  • International Journal for Numerical Methods in Fluids, Vol. 89, Issue 3
  • DOI: 10.1002/fld.4684

POD-Galerkin approximations in PDE-constrained optimization
journal, October 2010


On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high-Reynolds-number flow over an Ahmed body
journal, April 2014

  • Östh, Jan; Noack, Bernd R.; Krajnović, Siniša
  • Journal of Fluid Mechanics, Vol. 747
  • DOI: 10.1017/jfm.2014.168

A coarse-grid projection method for accelerating incompressible flow computations
journal, January 2013


Approximate deconvolution large eddy simulation of a barotropic ocean circulation model
journal, January 2011


Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations
journal, October 2010

  • Carlberg, Kevin; Bou-Mosleh, Charbel; Farhat, Charbel
  • International Journal for Numerical Methods in Engineering, Vol. 86, Issue 2
  • DOI: 10.1002/nme.3050

Model reduction for compressible flows using POD and Galerkin projection
journal, February 2004

  • Rowley, Clarence W.; Colonius, Tim; Murray, Richard M.
  • Physica D: Nonlinear Phenomena, Vol. 189, Issue 1-2
  • DOI: 10.1016/j.physd.2003.03.001

Four-Gyre Circulation in a Barotropic Model with Double-Gyre Wind Forcing
journal, June 2000


The chips are down for Moore’s law
journal, February 2016


An intrinsic stabilization scheme for proper orthogonal decomposition based low-dimensional models
journal, May 2007

  • Kalb, Virginia L.; Deane, Anil E.
  • Physics of Fluids, Vol. 19, Issue 5
  • DOI: 10.1063/1.2723149

Wind-Driven Currents in a Baroclinic Ocean; with Application to the Equatorial Currents of the Eastern Pacific
journal, November 1947

  • Sverdrup, H. U.
  • Proceedings of the National Academy of Sciences, Vol. 33, Issue 11
  • DOI: 10.1073/pnas.33.11.318

A Hybrid Approach for Model Order Reduction of Barotropic Quasi-Geostrophic Turbulence
journal, October 2018


Variants of Dynamic Mode Decomposition: Boundary Condition, Koopman, and Fourier Analyses
journal, April 2012

  • Chen, Kevin K.; Tu, Jonathan H.; Rowley, Clarence W.
  • Journal of Nonlinear Science, Vol. 22, Issue 6, p. 887-915
  • DOI: 10.1007/s00332-012-9130-9

Enablers for robust POD models
journal, February 2009

  • Bergmann, M.; Bruneau, C. -H.; Iollo, A.
  • Journal of Computational Physics, Vol. 228, Issue 2
  • DOI: 10.1016/j.jcp.2008.09.024

Stability Properties of POD-Galerkin Approximations for the Compressible Navier-Stokes Equations
journal, March 2000

  • Iollo, A.; Lanteri, S.; Désidéri, J. -A.
  • Theoretical and Computational Fluid Dynamics, Vol. 13, Issue 6
  • DOI: 10.1007/s001620050119

A stabilized proper orthogonal decomposition reduced-order model for large scale quasigeostrophic ocean circulation
journal, May 2015


Optimal nonlinear eddy viscosity in Galerkin models of turbulent flows
journal, February 2015

  • Protas, Bartosz; Noack, Bernd R.; Östh, Jan
  • Journal of Fluid Mechanics, Vol. 766
  • DOI: 10.1017/jfm.2015.14

Turbulence and the dynamics of coherent structures. I. Coherent structures
journal, January 1987

  • Sirovich, Lawrence
  • Quarterly of Applied Mathematics, Vol. 45, Issue 3
  • DOI: 10.1090/qam/910462

An Example of Eddy-Induced Ocean Circulation
journal, July 1980


Inertial gyres in decaying and forced geostrophic turbulence
journal, November 1992