Error estimation algorithms for ocean models

Errors due to discretization are inherent to the numerical solutions of Ocean General Circulation Models (OGCM). However, obtaining reliable estimates for these errors is a difficult undertaking. In this thesis, we develop a stochastic dual-weighted error estimator for the estimation of the discretization error in physical quantities of interest (goals) that is applicable to ocean models. Towards this aim, we extend the dual-weighted error estimation technique by a stochastic process with memory. In this, we extend previous work on memory-less stochastic dual-weighted methods for two-dimensional wave-type flows. The introduction of memory effects is the key new element of our extension and is shown to be crucial in the estimation of goal errors in an ocean model setting. The memory governs the temporal evolution of the stochastic process. We interpret the memory as a stochastic representation of physical constraints on the time-evolution of the essential building block of our stochastic dual-weighted error estimator – the local truncation error. The memory of the stochastic process is represented by temporal correlation coefficients directly or by their upper bound. The temporal correlation coefficients and other required stochastic quantities of the stochastic process are estimated from high-resolution model information at near-initial times. Our resulting stochastic dual-weighted approach is equivalent to a linearized stochastic-physics ensemble, but in contrast to the ensemble it only requires a single model integration and a single adjoint integration. In order to study the applicability of our stochastic dual-weighted error estimator for OGCMs, we focus on important oceanic features: the presence of lateral boundaries with their associated boundary currents, and the phenomenon of baroclinic instability within a stratified ocean. Both phenomena are studied by means of idealized experiments, the Munk gyre and the flow against an island for lateral boundaries, and the spherical channel experiment for baroclinic instabilities. For flows with boundaries, we find that our stochastic error estimator provides meaningful error bounds for a range of physically relevant goals. For the eddying flow regime due to baroclinic instabilities, the stochastic process of our error estimator is modeled as a compound of a horizontal, a vertical, and a temporal structure. To be applicable in our stochastic dual-weighted error estimation framework, we propose a generalization of its temporal structure to include additional, possibly negative time-correlations.

[1]  W. Large,et al.  Oceanic vertical mixing: a review and a model with a nonlocal boundary layer parameterization , 1994 .

[2]  A. Stuart,et al.  Extracting macroscopic dynamics: model problems and algorithms , 2004 .

[3]  M. Giorgetta,et al.  Icosahedral Shallow Water Model (ICOSWM): results of shallow water test cases and sensitivity to model parameters , 2009 .

[4]  P. Roache.  QUANTIFICATION OF UNCERTAINTY IN COMPUTATIONAL FLUID DYNAMICS , 1997 .

[5]  Christopher K. Wikle,et al.  Atmospheric Modeling, Data Assimilation, and Predictability , 2005, Technometrics.

[6]  Andreas Griewank,et al.  Automatic Differentiation of Algorithms: From Simulation to Optimization , 2000, Springer New York.

[7]  Andrew T. Wittenberg,et al.  Impacts on Ocean Heat from Transient Mesoscale Eddies in a Hierarchy of Climate Models , 2015 .

[8]  D. E. Harrison,et al.  On the wind-driven ocean circulation , 2019 .

[9]  Timothy G. Trucano,et al.  Verification and Validation in Computational Fluid Dynamics , 2002 .

[10]  D. Darmofal,et al.  Review of Output-Based Error Estimation and Mesh Adaptation in Computational Fluid Dynamics , 2011 .

[11]  Rolf Rannacher,et al.  An optimal control approach to a posteriori error estimation in finite element methods , 2001, Acta Numerica.

[12]  D. Venditti,et al.  Adjoint error estimation and grid adaptation for functional outputs: application to quasi-one-dimensional flow , 2000 .

[13]  Jan Riehme,et al.  On the use of discrete adjoints in goal error estimation for shallow water equations , 2010, ICCS.

[14]  M. Maltrud,et al.  Vertical heat transport in eddying ocean models , 2008 .

[15]  Sergey Danilov,et al.  Two finite-volume unstructured mesh models for large-scale ocean modeling , 2012 .

[16]  Paul D. Williams,et al.  Stochastic Parameterization: Towards a new view of Weather and Climate Models , 2015, 1510.08682.

[17]  T. Ringler,et al.  Analysis of Discrete Shallow-Water Models on Geodesic Delaunay Grids with C-Type Staggering , 2005 .

[18]  Carl Wunsch,et al.  Practical global oceanic state estimation , 2007 .

[19]  Beth A. Wingate,et al.  Lateral Mixing in the Eddying Regime and a New Broad‐Ranging Formulation , 2013 .

[20]  H. Mori.  Transport, Collective Motion, and Brownian Motion , 1965 .

[21]  Serge Prudhomme,et al.  On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors , 1999 .

[22]  Armin Köhl,et al.  Evaluation of the GECCO2 ocean synthesis: transports of volume, heat and freshwater in the Atlantic , 2015 .

[23]  W. Collins,et al.  Evaluation of climate models , 2013 .

[24]  I. Babuska,et al.  A‐posteriori error estimates for the finite element method , 1978 .

[25]  Andreas Griewank,et al.  Evaluating derivatives - principles and techniques of algorithmic differentiation, Second Edition , 2000, Frontiers in applied mathematics.

[26]  P. Heimbach,et al.  Ocean Data Assimilation in Support of Climate Applications: Status and Perspectives. , 2016, Annual review of marine science.

[27]  J. McWilliams.  Fundamentals of Geophysical Fluid Dynamics , 2011 .

[28]  M. Balmaseda,et al.  Evaluation of the ECMWF ocean reanalysis system ORAS4 , 2013 .

[29]  Alexandre J. Chorin,et al.  Estimating the uncertainty in underresolved nonlinear dynamics , 2013, 1306.2981.

[30]  J. Tinsley Oden,et al.  Estimation of modeling error in computational mechanics , 2002 .

[31]  M. Giles,et al.  Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality , 2002, Acta Numerica.

[32]  K. Bathe,et al.  Review: A posteriori error estimation techniques in practical finite element analysis , 2005 .

[33]  Alistair Adcroft,et al.  Formulating the Equations of Ocean Models , 2013 .

[34]  M. Giles,et al.  Adjoint Error Correction for Integral Outputs , 2003 .

[35]  Jochem Marotzke,et al.  The Oceanic Eddy Heat Transport , 2002 .

[36]  M. Cox,et al.  Baroclinic Instability in Ocean Currents , 1973 .

[37]  Christopher J. Roy,et al.  Review of Discretization Error Estimators in Scientific Computing , 2010 .

[38]  M. Giles.  On adjoint equations for error analysis and optimal grid adaptation in CFD , 1997 .

[39]  Martin Leutbecher,et al.  A Spectral Stochastic Kinetic Energy Backscatter Scheme and Its Impact on Flow-Dependent Predictability in the ECMWF Ensemble Prediction System , 2009 .

[40]  J. McWilliams.  The Nature and Consequences of Oceanic Eddies , 2013 .

[41]  P. Gent,et al.  Isopycnal mixing in ocean circulation models , 1990 .

[42]  Peter Korn,et al.  Ensemble-type numerical uncertainty information from single model integrations , 2015, J. Comput. Phys..

[43]  M. Redi.  Oceanic Isopycnal Mixing by Coordinate Rotation , 1982 .

[44]  Hazime Mori,et al.  A New Expansion of the Master Equation , 1974 .

[45]  Eusebio Valero,et al.  The estimation of truncation error by τ-estimation revisited , 2012, J. Comput. Phys..

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

[47]  R. Pacanowski,et al.  Parameterization of Vertical Mixing in Numerical Models of Tropical Oceans , 1981 .

[48]  Geoffrey K. Vallis,et al.  Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation , 2017 .

[49]  Thomas Richter,et al.  A Posteriori Error Estimation for the Fractional Step Theta Discretization of the Incompressible Navier-Stokes Equations , 2015 .