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Battalgazi Yildrim Thesis.Pdf Topics in Numerical Ocean Simulation by B. Gazi YILDIRIM M.Sc., Applied Mathematics, Brown University; Providence, RI, 2008 M.Sc., Aerospace Engineering, Mississippi State University; Starkville, MS, 2003 B.Sc., Aeronautical Engineering, Istanbul Technical University; Turkey, 2000 A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Division of Engineering at Brown University PROVIDENCE, RHODE ISLAND May 2012 c Copyright 2012 by B. Gazi YILDIRIM This dissertation by B. Gazi YILDIRIM is accepted in its present form by The Division of Engineering as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date George Em Karniadakis, Ph.D., Advisor Recommended to the Graduate Council Date Martin Maxey, Ph.D., Reader Date Kenneth Breuer, Ph.D., Reader Approved by the Graduate Council Date Peter M. Weber, Dean of the Graduate School iii Acknowledgements I would like to thank to my advisor Professor George Em Karniadakis for giving me opportunity to work with his group and for fixing many wrong decisions of my life during this study. I also thank to my Ph.D. dissertation readers Professor Martin Maxey and Kenny Breuer for useful feedbacks and discussions. Professor Changsheng Chen provided their numerical simulation output and in- troduced action balance equation to me. I am grateful to him for his contribution to this dissertation work. Dr. Nico Booij helped me to understand numerical ocean wave better and correct my misunderstandings. I am indebted to him for his help. I also want to thank to Prof. Rainald Lohner for supporting my decision to study at Brown University. Martin Hunt helped me to use PUQ for the third part of my thesis. I truly thank to him. I thank to my friends Cagatay Demiralp, Hatice Sahinoglu and Momotazur (Shantu) and Mila Rahmnan for their true friendship. I worked on my disserta- tion and lived in West Lafayette, IN for last three years. I thank to Selma and Turkay Yolcu, Derya and Rafet Maltas for their support during my work at Purdue university. I also thank to Francesca Randazzo for her friendship. I would like to thank to my parents Zeynep and Celil Yildirim for their uncondi- tional support and love. I dedicated this dissertation to my parents. I thank to my sister Mehtap Yildirim. My wife Yesim Yildirim was patient with me and kept me iv going on, without her support I would not be able to finish this dissertation work. I have baby daughter Elif Ecrin born in September 2010, she tried all tricks to stop my dissertation. In spite of this, this thesis is dedicated to her along with my parents. I also thank to parents of my wife, especially for Fatma Serin who was taking care of Elif and us during the last part of my study. v Abstract of “ Topics in Numerical Ocean Simulation ” by B. Gazi YILDIRIM, Ph.D., Brown University, May 2012 We investigate two primary aspects of numerical simulations for the oceans. The first is the degree of spatial complexity of the dynamics in regional ocean models. Using simulation results from three different regional ocean models (HOPS, ROMS and FVCOM) we show that just a few spatio-temporal POD (proper orthogonal decomposition) modes are sufficient to describe the most energetic ocean dynamics. In particular, we demonstrate this with the simulated ocean dynamics for the New Jersey coast, Massachusetts Bay and Gulf of Maine. Moreover, the extrema of the POD spatial modes are very good locations for sensor placement and accurate field reconstruction. We employ a modified POD theory to incorporate a limited number of measurements in reconstructing the velocity and temperature fields, and we study systematically the corresponding reconstruction errors as a function of the sensor location, number of sensors, and number of POD modes. This new approach is quite accurate in short-term simulation, and hence it has the potential of accelerating the use of real-time adaptive sampling in data assimilation for ocean forecasting. The second theme is the accurate prediction of surface gravity waves in the ocean. We develop a new, high-order, hybrid discretization of the phased-averaged (ac- tion balance) equation to simulate ocean waves. We employ Discontinuous Galerkin (DG) discretization on an unstructured grid of the geophysical space and Fourier- collocation for the wave number direction and frequency coordinates. The original action balance equation is modified to facilitate absorbing boundary conditions in the frequency domain. This modification enforces periodicity at the frequency bound- aries so that the fast convergence of Fourier-collocation still holds. In addition, a mapping of the wave number directional coordinate is introduced to cluster the col- location points about the sharply defined, directional spectra. Time-discretization vi is accomplished by a TVD Runge-Kutta scheme. The overall convergence of the scheme is exponential (spectral). We successfully verified and validated the method against several analytical solutions, observational data, and experimental results. The last part of this study uses a generalized polynomial chaos (gPC) framework to quantify the uncertainty of the ocean wave simulations. To this end, we adopted the collocation stochastic method. The collocation points in the random space were chosen from the Smolyak grid. This assured us substantial reduction in the number of samples for the large random variables. We considered two random inputs: (1) the activated source term parameters and (2) the current field provided from the exper- iment. We introduced a random perturbation current field from a Karhunen-Loeve expansion of the Gauss correlation function. The uniform distribution probability function was chosen for all random inputs. We finally quantified the uncertainty first for each individual contribution (from either the random variables associated with source term parametrizations or the random process of the current field) and then evaluated the uncertainty with both random sources. Contents Acknowledgments iv 1 Motivation and Objective 1 2 Data simulation and POD 13 2.1 Ocean Circulation Models: ROMS, HOPS, and FVCOM . 14 2.2 PODFormulation............................. 16 2.3 POD-MethodofSnapshot . 19 2.4 SimulationsandLow-dimensionality. ... 21 2.4.1 NewJerseyandMiddleAtlanticBight . 22 2.4.1.1 TheSnapshotLengthEffect . 24 2.4.2 MassachusettsBay . 28 2.4.3 GulfofMaine ........................... 30 2.4.4 The “numerical” ocean is low-dimensional . 31 2.5 POD-basedReconstruction. 35 2.5.1 Unsteadyflowpastacylinder . 41 2.6 Massachusetts Bay: Results and Discussion . ... 46 2.6.1 Totalvelocity ........................... 48 2.6.2 Temperature............................ 50 2.7 SummaryandDiscussion. 52 3 Numerical Ocean Wave 58 3.1 ActionBalanceEquation . 67 3.2 ModelingtheSourceTerms . 71 3.2.1 Source generation by wind (Sin)................. 72 3.2.2 Dissipation by white-capping (Swc) ............... 73 3.2.3 Nonlinear wave-wave interactions (Snl) ............. 75 3.2.3.1 Quadruplet wave-wave interactions (Snl4)....... 77 vii 3.2.3.2 Triad wave-wave interactions (Snl3).......... 80 3.2.4 Depth-induced breaking (Sbr) .................. 83 3.2.5 Bottom friction (Sbfr)....................... 86 4 Hybrid Spectral/DG Method for Phase-averaged Equation 89 4.1 SpaceDiscretizations . 90 4.1.1 TimeDiscretization. 102 4.1.2 ForwardandBackwardTransformations . 104 4.1.3 MappingintheSpectralDirection. 106 4.1.4 Absorbing Boundary Treatment in Frequency Direction . 111 5 Verification and Validation 115 5.1 Verification of Temporal and Spatial Accuracy . 116 5.1.1 TemporalAccuracyVerification . 117 5.1.2 SpatialAccuracyVerification . 117 5.2 VerificationandValidationCases . 121 5.2.1 Case I: Numerical diffusion in deep water (S =0) ....... 122 5.2.2 Case II: Current-induced shoaling and retractions in deep wa- ter (S =0) ............................126 5.2.3 Case III: Depth-induced shoaling and refractions in shallow water (S =0)...........................130 5.2.4 Case IV: Duration-limited growth in deep water (S = Sin + Snl4 + Swc) ............................133 5.2.5 Case V: Fetch-limited growth in deep water (S = Sin +Snl4 +Swc)136 5.2.6 Case VI: Depth-induced wave breaking (S = Sbr) .......138 5.2.7 Case VII: Flume test case (S = Sbr + Snl3)...........141 5.2.8 Case VIII: HISWA Experiment (S = Sb,fr + Sbr + Snl3) .... 144 5.3 Resultsanddiscussion . 152 6 Uncertainty Quantification for Numerical Ocean Wave Prediction 156 6.1 Uncertaintyinsourceterms . 159 6.1.1 Sensitivity analysis of source term parameters . 159 6.1.2 Resultsanddiscussion . 164 6.2 Uncertaintyincurrentfield . 171 6.2.1 Resultsanddiscussion . 174 6.3 Uncertaintyofsourceandcurrentfield . 179 6.3.1 Resultsanddiscussion . 181 6.4 SummaryandConclusions . 187 7 Summary and Future Works 190 A Spectral Methods for Discontinous and Sharp-Gradient Problems 195 A.1 Introduction................................ 196 A.2 ChebyshevCollocationMethods . 197 A.2.1 Filtering.............................. 199 viii A.2.2 CoordinateTransformation . 202 A.2.3 TimeIntegration . 205 A.3 Numericalresults ............................. 206 A.3.1 The one-dimensional viscous Burger equation . 206 A.3.2 The two-dimensional inviscid and viscous Burger Equations . 210 A.3.2.1 Spectralconvergencetestcase . 210 A.3.2.2 Circularshocktestcase . 212 A.3.3 Euler equations test case (Shock tube problem) . 214 A.4 Summary .................................217 B Sea Spectrums 224 B.1 Pierson-MoskowitzSpectrum. 225 B.2 JONSWAP (Joint North Sea Wave Project) Spectrum.
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