Numerical Models of Oceans and Oceanic Processes

Numerical Models of Oceans and Oceanic Processes

This is Volume 66 in the INTERNATIONAL GEOPHYSICS SERIES A series of monographs and textbooks Edited by RENATA DMOWSKA, JAMES R. HOLTON, and H. THOMAS ROSSBY A complete list of books in this series appears at the end of this volume.

Numerical Models of Oceans and Oceanic Processes

LAKSHMI H. KANTHA University of Colorado Boulder, Colorado

CAROL ANNE CLAYSON Purdue University West Lafayette, Indiana

ACADEMIC PRESS A Harcourt Science and Technology Company

San Diego San Francisco New York Boston London Sydney Tokyo

Cover photo: A NOAA AVHRR image of the Gulf of Mexico on March 19, 1995. A large anticyclonic warm core eddy shed by the Loop Current and strong wintertime cooling of the water column on the U.S. continental shelf are depicted. It is a challenge to simulate and accurately forecast the processes responsible for these features in numerical ocean models, the subject matter of this book. (Photo courtesy of Dr. Robert Amone of the Naval Research Laboratory, Stennis Space Center, MS.)

This book is printed on acid-free paper.

Copyright © 2000 by ACADEMIC PRESS All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Requests for permission to make copies of any part of the work should be mailed to: Permissions Department, Harcourt Brace & Company, 6277 Sea Harbor Drive, Orlando, Florida, 32887-6777. Academic Press A Harcourt Science and Technology Company 525 B Street, Suite 1900, San Diego, California 92101-4495, USA http://www.academicpress.com Academic Press Harcourt Place, 32 Jamestown Road, London NWl 7BY, UK http://www.acadeniicpress.com Library of Congress Catalog Card Number: 99-60406 International Standard Book Number: 0-12-434068-7 PRINTED IN THE UNITED STATES OF AMERICA 00 01 02 03 04 05 MM 9 8 7 6

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To The Office of Naval Research on the occasion of its Fiftieth Anniversary and The as. Navy

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Contents

List of Acronyms xv List of Symbols xxiii Foreword xxvii Preface xxx Prologue xxxiii

Chapter 1

Introduction to Ocean Dynamics 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14

Types, Advantages, and Limitations of Ocean Models Recent Examples 21 Governing Equations 28 Vorticity Conservation 46 Nondimensional Numbers and Scales of Motion 48 Geostrophic Flow and Thermal Wind 56 Inertial Motions 59 Ekman Layers 60 Sverdrup Transport 71 Western Boundary Intensification (Stommel Solution) Gyre Scale Circulation (Munk Solution) 80 Barotropic Currents over Topography 90 Baroclinic Transport over Topography 92 Coastal Upwelling and Fronts 96

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Contents 1.15 Mesoscale Eddies and Variability 100 1.16 Thermohaline Circulation and Box (Reservoir) Models 1.17 Numerical Models 121

Chapter 2

Introduction to Numerical Solutions 2.1

Introduction 129 2.1.1 Architecture 130 2.1.2 Computational Errors 132 2.2 Ordinary Differential Equations 137 2.2.1 Runge-Kutta Method 138 2.3 Partial Differential Equations 141 2.3.1 Consistency, Convergence, and Stability 143 2.3.2 Elliptic, Hyperbolic, and Parabolic Systems 149 2.4 Elliptic Equations and Steady-State Problems 151 2.4.1 Direct Solvers 152 2.4.2 Iterative Solvers and Relaxation Methods 157 2.4.3 Preconditioned Conjugate Gradient Method 161 2.4.4 Multigrid Methods 163 2.4.5 Pseudo-transient Method 167 2.5 Time Dependent Problems 168 2.5.1 Advection Equation and Hyperbolic Systems 168 2.5.2 Diffusion Equation and Parabolic Systems 181 2.6 Finite-Difference (Grid Point) Methods 186 2.6.1 Staggered Grids 193 2.6.2 Time Differencing and Filtering 202 2.6.3 Computational Grids 206 2.7 Spectral (Spectral Transform) Methods 211 2.8 Finite-Element Methods 216 2.8.1 Spectral Element Approach 220 2.9 Parameterization of Subgrid Scale Processes 223 2.10 Lateral Open Boundary Conditions 225 2.11 Computationallssues 231 2.12 Examples 233 2.12.1 Inertial Oscillations 233 2.12.2 Thermohaline Circulation 236 2.12.3 Normal Modes 236 2.12.4 Gyre Scale Circulation 237 2.12.5 Advection Problems 240 2.12.6 M.I.T. Nonhydrostatic Global Model 241

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Contents Chapter 3

Equatorial Dynamics and Reduced Gravity Models Solutions 3.1 3.2 3.3

3.4 3.5

Oceanic Dynamical Response to Forcing 248 Governing Equations 253 Equatorial Waves 259 3.3.1 Kelvin Waves 262 3.3.2 Yanai Waves 265 3.3.3 Rossby Waves 266 3.3.4 Inertia-Gravity (Poincare) Waves 269 Equatorial Currents 271 Reduced Gravity Model of Equatorial Processes

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Chapter 4

Midlatitude Dynamics and Quasi-Geostrophic Models 4.1

4.2 4.3 4.4 4.5

Linear Motions 292 4.1.1 Inertia-Gravity (Sverdrup/Poincare) Waves 4.1.2 Kelvin Waves 298 4.1.3 Planetary Rossby Waves 300 4.1.4 Topographic Rossby Waves 308 Gontinuous Stratification 310 Geostrophic Adjustment and Instabilities 4.3.1 Geostrophic Adjustment 313 4.3.2 Instabilities 316 Spinup 324 Quasi-Geostrophic Models 327 4.5.1 Governing Equations 329 4.5.2 Applications 335

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Chapter 5

High-Latitude Dynamics and Sea-Ice Models 5.1 5.2 5.3

Salient Features of Ice Cover 337 Momentum Equations for Sea Ice 346 Constitutive Law for Sea Ice (Ice Rheology) 5.3.1 Viscous-Plastic Ice Rheology 351 5.3.2 Elastic-Viscous-Plastic Ice Rheology

348 357

Contents 5.4 5.5 5.6

Continuity Equations for Sea Ice 360 Response of Sea Ice to Storm Passage 361 Numerics 363 5.6.1 Governing Equations in Orthogonal Curvilinear Coordinates 364 5.6.2 Solution Technique 372

Chapter 6

Tides and Tidal Modeling 6.1 6.2 6.3

6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15

Description of Tides 379 Formulation: Tidal Potential 387 Body, Load, Atmospheric, and Radiational Tides 399 6.3.1 Body (Solid Earth) Tides 399 6.3.2 Load Tides 401 6.3.3 Atmospheric Tides 404 6.3.4 Radiational Tides 408 Dynamical Theory of Tides: Laplace Tidal Equations 410 Equilibrium Theory of Tides 417 Tidal Analysis: Orthotides 418 Tidal Currents 424 Global Tidal Models 429 Regional Tidal Models 440 Geophysicallmplications 441 6.10.1 Tidal Dissipation and LOD 443 6.10.2 Tidal Energetics 448 Changes in Earth's Rotation 457 Baroclinic (Internal) Tides 471 Long-Period Tides 480 Shallow Water Tides and Residual Currents 486 Summary 490

Chapter 7

Coastal Dynamics and Barotropic Models 7.1 7.2 7.3

Wind- and Buoyancy-Driven Currents Tidal Motions 498 Continental Shelf Waves 499

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Contents 7.4 7.5

Modeling Shelf Circulation 512 Barotropic Models 513 7.5.1 Coastal Ocean Response to Wind Forcing 7.5.2 Storm Surges and Storm Surge Modeling 7.5.3 Response to Pressure Forcing 526

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Chapter 8

Data and Data Processing 8.1

8.2

8.3 8.4

In Situ Observational Data 532 8.1.1 XBT, CTD, CM, ADCP, and Drifter Data 533 8.1.2 Historical Hydrographic Data 549 8.1.3 Historical Marine Surface Data 549 Remotely Sensed Data 554 8.2.1 Sea Surface Temperature from IR Sensors 555 8.2.2 Sea Surface Winds from Microwave Sensors 558 8.2.3 Chlorophyll and Optical Clarity from Color Sensors 8.2.4 Sea Surface Height from Satellite Altimetry 561 NWP Products 569 Preprocessing of Observational Data and Postprocessing of Model Output 571 8.4.1 Graphics and Visualization of Model Output 574 8.4.2 Analyses 576

Chapter 9

Sigma-Coordinate Regional and Coastal Models 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9

Introduction 579 Gkiverning Equations 585 Vertical Mixing 591 Boundary Conditions 593 Mode Splitting 596 Numerics 599 9.6.1 Vertical Direction 603 9.6.2 Horizontal Direction 609 Numerical Problems 611 Applications 614 Code Structure 619

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Contents Chapter 10

Multilevel Basin Scale and Global Models 10.1 10.2 10.3 10.4 10.5 10.6 10.7

Introduction 628 Governing Equations 629 Isopycnal Diffusion 640 Architecture and Other Model Features Applications 647 Hybrid s-Coordinate Models 649 Regional z-Level Models 659

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Chapter 11

Layered and Isopycnal Models 11.1 Layered Models 11.2 Isopycnal Models

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Chapter 12

Ice-Ocean Coupled Models 12.1 Sea-Ice Models 676 12.2 Coupled Ice-Ocean Models

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Chapter 13

Ocean-Atmosphere Coupled Models 13.1 Coupling between the Ocean and the Atmosphere 703 13.2 Coupled Ocean-Atmosphere General Circulation Models 708 13.3 Regional Coupled Ocean-Atmosphere Models 719

Chapter 14

Data Assimilation and Nowcasts/ Forecasts 14.1 Introduction 730 14.2 Direct Insertion 734 14.3 Nudging 735

Contents 14.4 Statistical Assimilation Schemes 736 14.4.1 Kalman Filter 738 14.4.2 Reduced State Space Kalman Filters 742 14.4.3 Optimal Interpolation (01) Scheme 744 14.5 Variational Methods 747 14.5.1 Adjoint Models 750 14.6 Predictability of Nonlinear Systems—Low Order Paradigms 753 14.7 Nowcasts/Forecasts in the Gulf of Mexico 769

Appendix A Equations of State A.l Equation of State for the Ocean 777 A.2 Equation of State for the Atmosphere

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Appendix B Wavelet Transforms B.l

Introduction 786 B.1.1 Theory 788 B. 1.2 Continuous Wavelet Transforms (CWT) 794 B. 1.3 Discrete Wavelet Transforms (DWT) 804 B.2 Examples 808 B.3 Wavelet Transforms and Stochastic Processes 812 B.4 Two-Dimensional Wavelet Transforms 813 B.5 Cross Wavelet Transforms (CrWT) 815 B.6 Error Analysis 818

Appendix C Empirical Orthogonal Functions and Empirical Normal Modes C.l Empirical Orthogonal Functions C. 1.1 Complex EOFs 828 C. 1.2 Singular Spectrum Analysis

820 833

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Contents C. 1.3 Extended EOFs 835 C. 1.4 Coupled Pattern Analysis 840 C.2 Empirical Normal Modes 842

Appendix D

Units and Constants D.l

Useful Quantities 851 D.1.1 SI (International System of Units) Units and Conventions 851 D. 1.2 Useful Conversion Factors 852 D. 1.3 Useful Universal Constants 853 D. 1.4 Useful Geodetic Constants 854 D. 1.5 Useful Physical Constants 855 D. 1.6 Useful Dynamical Quantities 857 D.2 Important Scales and Quantities 858 D.2.1 Length Scales 858 D.2.2 Timescales 859 D.2.3 Velocity Scales 859 D.2.4 Nondimensional Quantities 860 D.3 Useful Websites 862

References 865 Biographies 911 Index 915 List of Volumes in the Series

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List of Acronyms AABW AAM AATSR ABL ACC ACCE ADCM ADCP ADEOS ADI AGCM AJDJEX ALACE AMSU AOML APE ATLAS ATSR AUV AVHRR AXBT BL BMO BVP CEAREX CEOF CFC CFD

AntArctic Bottom Water Atmospheric Angular Momentum Advanced Along Track Scanning Radiometer Atmospheric Boundary Layer Antarctic Circumpolar Current Atlantic Circulation and Climate Experiment Acoustic Doppler Current Meter Acoustic Doppler Current Profiler ADvanced Earth Observation Satellite Alternate Direction Implicit method Atmosphere General Circulation Model Arctic Dynamics Joint Experiment Autonomous Lagrangian Circulation Explorer (floats) Advanced Microwave Sounding Unit Atlantic Oceanographic and Meteorological Laboratory Available Potential Energy Automated Temperature Line Acquisition System Along Track Scanning Radiometer Autonomous Underwater Vehicle Advanced Very High Resolution Radiometer Air-deployed expendable BathyThermograph Boundary Layer British Meteorological Office Boundary Value Problem Central Eastern ARctic Experiment Complex Empirical Orthogonal Functions ChloroFluoroCarbon Computational Fluid Dynamics

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Acronyms CFL CGCM CLIVAR CNES CO ADS COAGCM COAMPS CO ARE COD CODAR CODE COIS CORE CPC CPSST CPU CREAMS CrWT CTD CUPOM CWT CZCS DA DFT DMSP DOE DOM DWBC DWT EAC ECM ECMWF EEZ EEOF EIC EKE ENM ENSO EOF EOS EOS-AM

Courant-Friedrichs-Levy Coupled ocean-atmosphere General Circulation Model CLImate VARiability and predictability program Centre National d'Etudes Spatiales Comprehensive Ocean Atmosphere Data Set Coupled Ocean Atmosphere General Circulation Model Coastal Ocean Atmosphere Modeling and Prediction System Coupled Ocean Atmosphere Response Experiment Computational Ocean Dynamics Coastal Ocean Doppler Radar Coastal Ocean Dynamics Experiment Coastal Ocean Imaging Spectrometer Continuous Observation of the Rotation of the Earth Chmate Prediction Center (at NCEP; former Climate Analysis Center) Cross-Product Sea Surface Temperature Central Processing Unit Circulation Research in East Asian Marginal Seas Cross Wavelet Transform Conductivity Temperature Depth (profiler) University of Colorado version of Princeton Ocean Model Continuous Wavelet Transform Coastal Zone Color Scanner Data Assimilation Discrete Fourier Transform Defense Meteorological Satellite Program Department of Energy Dissolved Organic Matter Deep Western Boundary Current Discrete Wavelet Transform Ensemble Average Closure Electromagnetic Current Meter European Center for Medium-Range Weather Forecasting Exclusive Economic Zone Extended Empirical Orthogonal Functions Equatorial Intermediate Current Eddy Kinetic Energy Empirical Normal Modes El Nino-Southern Oscillation Empirical Orthogonal Function Earth Observing System Earth Observing System AM mission

Acronyms ERBE ERICA ERM ERS ESA EUC FCT FEM FFT FNMOC FOCAL FT FTCS GAC GALE GATE GCM GCOS GEOSAT GEOSECS GEWEX GF GFDL GEO GEOSAT GJ GLOSS GMT GOALS GODAE GOES GOOS GPS GTOS GTS GW HIRS HNLC HRPT ICES IDOE lERS

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Earth Radiation Budget Experiment Experiment on Rapidly Intensifying Cyclones over the Atlantic Exact Repeat Mission Earth Resources Satellite European Space Agency Equatorial Undercurrent Flux-Corrected Transport Finite Element Method Fast Fourier Transform Fleet Numerical Meteorology and Oceanography Center Francais Ocean et Climat dans 1'Atlatique equatorial Fourier Transform Forward in Time, Centered in Space Global Area Coverage (4 km resolution AVHRR) data Genesis of Atlantic Lows Experiment Global Atlantic Tropical Experiment General Circulation Model Global CHmate Observing System GEOdetic SATelHte Geochemical Ocean Sections Study Global Energy and Water cycle Experiment Gigaflops Geophysical Fluid Dynamics Laboratory Follow-On Gigajoules GLObal Sea level observing System Greenwich Mean Time Global Ocean Atmosphere Land System Global Ocean Data Assimilation Experiment Geostationary Orbiting Environmental Satellite Global Ocean Observing System Global Positioning System Global Terrestrial Observing System Global Telecommunication System Gigawatts, also gigawords High resolution Infra Red Sounder High Nitrate Low Chlorophyll High Resolution Picture Tube satellite data receivers International Council for Exploration of the Sea International Decade of Ocean Exploration International Earth Rotation Service

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lES IGBP IGOSS IMSL IOC lOP IPCC IR ISCCP ITCZ IVP JEBAR JGOFS JIC U.S. JMA JPL KG KE LAC LATEX LCE LEADEX LES LHS LIDAR LOD LOM LTE LW MABL MBT MCSST MICOM MIZ MIZEX MJO ML MLD M-0 MODE MODIS

Acronyms Inverted Echo Sounder International Geosphere Biosphere Programme Integrated Global Ocean Services System International Math and Science Library Intergovernmental Oceanographic Commission Intensive Observation Period Intergovernmental Panel on Climate Change InfraRed International Satellite Cloud Climatology Project Inter-Tropical Convergence Zone Initial Value Problem Joint Effect of Baroclinicity and bottom Relief Joint Global Ocean Flux Study Navy/NOAA Joint Ice Center Japan Meteorological Office Jet Propulsion Laboratory (at California Institute of Technology) Kantha-Clayson Kinetic Energy Local Area Coverage (1 km resolution AVHRR) data LouisianA-TEXas Loop Current Eddy LEADs Experiment Large Eddy Simulation Left-Hand Side Laser Induced Detection and Ranging Length of Day Length of Month Laplace Tidal Equations Longwave Marine Atmospheric Boundary Layer Mechanical BathyThermograph Multi-Channel Sea Surface Temperature Miami Isopycnic Coordinate Model Marginal Ice Zone Marginal Ice Zone Experiment Madden-Julian Oscillations Mixed Layer Mixed Layer Depth Monin-Obukhoff Mid-Ocean Dynamics Experiment MODerate resolution Imaging Spectroradiometer

Acronyms MOM MOODS MW MY NADW NAO NASA NAVO NCAR NCDC NCEP NDBC NEC NECC NEMO NESDIS NGDC NMC NMFS NOAA NODC NOGAPS NORAPS NOS NRL NSCAT NWP NWS OCXS ODE OGCM 01 OLR OML OSSE OWS PAGES PALACE PAR PEL

Modular Ocean Model Master Oceanographic Observation Data Set Megawatts, also Megawords Mellor-Yamada North Atlantic Deep Water North Atlantic Oscillation National Aeronautics and Space Administration NAVal Oceanographic office National Center for Atmospheric Research National CHmate Data Center National Centers for Environmental Prediction (formerly NMC) NOAA Data Buoy Center North Equatorial Current North Equatorial Counter Current Navy Earth Map Observer National Environmental Satellite Data and Information Service National Geophysical Data Center National Meteorological Center National Marine Fisheries Service National Oceanic and Atmospheric Administration National Oceanographic Data Center Navy Operational Global Atmosphere Prediction System Navy Operational Regional Atmosphere Prediction System National Ocean Service Naval Research Laboratory NASA SCATterometer Numerical Weather Prediction National Weather Service Ocean Color and Temperature Sensor Ordinary Differential Equation Ocean General Circulation Model Optimal Interpolation Outgoing Longwave Radiation Oceanic Mixed Layer Observing System Simulation Experiment Ocean Weather Station Past Global Changes Profiling Autonomous LAgrangian Circulation Explorer float Photosynthetically Available Radiation Planetary Boundary Layer

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PCA PDE PDO PE PIP PIRATA PMEL POD PODAAC POES POM POLYMODE POP PPM PSU PVD QC QG QH RAFOS RCM RCOAM REOF RHS RK RMS ROV RV SAR SAUV SC SCOR SEASAT SeaWiFS SEC SEQUAL SHEB A SLP SMMR SOFAR SOI SOR

Acronyms Principal Component Analysis Partial Differential Equation Pacific Decadal Oscillation Potential Energy, also Primitive Equation Principal Interaction Pattern Pilot Research moored Array in the Tropical Atlantic Pacific Marine Environmental Laboratory Proper Orthogonal Decomposition Physical Oceanography Distributed Active Archive Center Polar Orbiting Environmental Satellite Princeton Ocean Model Joint USSR POLYgon and U.S. MODE experiments Principal Oscillation Pattern; also Parallel Ocean Program Piecewise Parabolic Method Practical Sahnity Units Progressive Vector Diagram QuaHty Control Quasi-geostrophic Quasi-hydrostatic SOFAR spelled backward Rotor-type Current Meter Regional Coupled Ocean Atmosphere Model Rotated Empirical Orthogonal Functions Right-Hand Side Runge-Kutta Root Mean Square Remotely Operated Vehicle Research Vessel Synthetic Aperture Radar Smart Autonomous Underwater Vehicle Successive Correction Scientific Committee on Ocean Research SEA SATellite Sea-viewing Wide Field-of-view Sensor South Equatorial Current SEasonal response of the EQUatorial AtLantic programme Surface HEat Budget of the Arctic experiment Sea Level Pressure Scanning Multichannel Microwave Radiometer SOund Fixing and Ranging floats Southern Oscillation Index Successive Over Relaxation

Acronyms SSA SSH SSM/I SSS SST STD SVD SW TAO TC TF TFO TOGA TOPEX TOS TOYS T/P TRMM TVD UNESCO UTC UV VACM VMCM VOS TW WCRP WD WFT WMO WOCE WT WWB WWW XBT XCP XCTD

Singular Spectrum Analysis Sea Surface Height Special Sensor Microwave Imager Sea Surface Salinity Sea Surface Temperature; Sea Surface Topography in geodesy Salinity Temperature Depth profiler Singular Value Decomposition Shortwave Tropical Atmosphere Ocean array TOGA/COARE Teraflops TOPEX Follow-On Tropical Ocean Global Atmosphere program TOPography Experiment TOGA Observing System TIROS Operational Vertical Sounder TOPEX/Poseidon Tropical Rainfall Measuring Mission Total Variance Diminishing method United Nations Economic Social and Cultural Organization Universal Temps Coordinee (Universal Time Coordinated) Ultraviolet Vector Averaging Current Meter Vector Measuring Current Meter Voluntary Observing Ship Terawatts World Climate Research Program Wavelet Domain Windowed Fourier Transform World Meteorological Organization World Ocean Circulation Experiment Wavelet Transform Westerly Wind Burst World Wide Web, also World Weather Watch expendable BathyThermograph expendable Current Profiler expendable Conductivity Temperature Depth

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List of Symbols

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