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Numerical Methods for Atmospheric and Oceanic Sciences
Numerical Methods for Atmospheric and Oceanic Sciences deals with various numerical
methods that are applied to fluid systems such as the atmosphere and hydrosphere.
With a detailed and comprehensive overview of the various numerical methods that
are applied to fluid systems in general and the atmospheric and oceanic sciences
in particular, this book will be useful for students of atmospheric and oceanic
sciencesinbothseniorundergraduateandgraduatecourses.Itprovidesdetailsofthe
application of finite difference methods to various problems that involve processes
like advection, barotropic, shallow water, baroclinic oscillation, and decay. The
concepts of consistency, stability, and convergence are also emphasized. The book
provides clear exposition of concepts such as stability, staggered grid, and nonlinear
computational instability. The book also provides broad details and applications of
advancednumericalmethodssuchasthespectralmethod,finiteelementmethod,and
finitevolumemethod.
A. Chandrasekar is Dean (Academics and Continuing Education) and Outstanding
Professor at the Department of Earth and Space Sciences, Indian Institute of Space
ScienceandTechnology,Thiruvananthapuram. Heisaleadingexpertinatmospheric
scienceinIndiaandhaspublishedwidelyontheabovetopicsthroughouthiscareer.
In2010,heauthoredthebookBasicsofAtmosphericScience.
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Numerical Methods for
Atmospheric and Oceanic Sciences
A. Chandrasekar
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Informationonthistitle: www.cambridge.org/9781009100564
© A.Chandrasekar2022
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Firstpublished2022
PrintedinIndia
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LibraryofCongressCataloging-in-PublicationData
Names: Chandrasekar,A.,author.
Title: Numericalmethodsforatmosphericandoceanicsciences/A.
Chandrasekar,DepartmentofEarthandSpaceSciences,IndianInstitute
ofSpaceScience&Technology,Tiruvananthapuram,India.
Description: Cambridge,UnitedKingdom;NewYork,NY:Cambridge
UniversityPress,2022. |Includesbibliographicalreferencesandindex.
Identifiers: LCCN2021050095(print)|LCCN2021050096(ebook)|ISBN
9781009100564(hardback)|ISBN9781009119238(ebook)
Subjects: LCSH:Atmosphericphysics–Mathematicalmodels. |
Oceanography–Mathematicalmodels. |BISAC:SCIENCE/EarthSciences/
Meteorology&Climatology
Classification: LCCQC880.C4182022(print)|LCCQC880(ebook)|DDC
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... tothememoryofProfessorT.N.Krishnamurti
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Contents
ListofFigures xix
Foreword xxvii
Preface xxix
1 PartialDifferentialEquations 1
1.1 Introduction 1
1.2 DiffusionEquation 4
1.3 First-orderEquations 6
1.4 First-orderEquations: MethodofCharacteristics 9
1.5 Second-orderQuasilinearPDEs: ClassificationUsingMethodof
Characteristics 10
1.6 WaveEquation 15
1.7 LinearAdvectionEquation 17
1.8 LaplaceEquation 18
1.9 MethodofSeparationofVariablesfortheOne-dimensionalHeat
Equation 19
1.10 MethodofSeparationofVariablesfortheOne-dimensionalWave
Equation 20
Exercises 21
2 EquationsofFluidMotion 25
2.1 Introduction 25
2.2 LagrangianandEulerianDescriptionofFluidMotion 26
2.2.1 Substantiveortotalderivative 26
2.2.2 Conservationofmassprinciple: Continuityequation 28
2.2.3 Conservationofmomentumprinciple: Momentumequation 29
2.2.4 Euler’sequationofmotionforanidealfluid 31
2.2.5 Conservationofenergyprinciple: Thermodynamicenergyequation 32
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viii Contents
2.3 EquationsGoverningAtmosphericMotion 34
2.3.1 Rotatingframeofreference 34
2.3.2 Conservationofenergy: Thermodynamicenergyequation
foratmosphere 36
2.3.3 Geostrophicbalanceequations 38
2.3.4 Hydrostaticbalanceequation 38
2.3.5 Governingequationsofmotionofatmospherewithpressureasa
verticalcoordinate 39
2.3.6 Quasi-geostrophicequationsofmotionofatmospherewith
pressureasaverticalcoordinate 40
2.3.7 Shallowwaterequations 41
2.3.8 Vorticityequationforincompressiblefluid: Curlofthe
Navier–Stokesequation 42
2.3.9 Vorticityequationforatmosphericandoceanicflows 43
2.3.10 Non-divergentvorticityequationforatmosphericandoceanicflows 43
2.3.11 Boussinesqapproximation 43
2.3.12 Anelesticapproximation 45
2.3.13 Conservationofwatervapourmixingratioequation 45
2.3.14 Meanequationsofturbulentflowintheatmosphere 46
2.3.15 RANS,LES,andDNSapproaches 48
2.3.16 Parameterizationofphysicalprocessesintheatmosphericmodels 51
2.3.17 Parallelcomputing 51
Exercises 53
3 FiniteDifferenceMethod 57
3.1 Introduction 57
3.2 MethodofFiniteDifference 57
3.2.1 Forwarddifferencescheme 59
3.2.2 Backwarddifferencescheme 60
3.2.3 Centraldifferencescheme 60
3.2.4 Centeredfourth-orderdifferencescheme 61
3.2.5 FinitedifferenceschemeforsecondderivativesandLaplacian 61
3.3 TimeIntegrationSchemes 64
3.3.1 Two-timelevelschemes 65
3.3.2 Three-timelevelschemes 67
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Contents ix
Exercises 68
Pythonexamples 70
4 ConsistencyandStabilityAnalysis 73
4.1 ConsistencyandStabilityAnalysis 73
4.2 BasicAspectsofFiniteDifferences 73
4.2.1 Consistency 74
4.2.2 Convergence 74
4.2.3 LaxEquivalenceTheorem 75
4.3 ErrorsandStabilityAnalysis 75
4.3.1 Introduction 75
4.3.2 Discretizationerror 75
4.3.3 Representationofrealnumbersinacomputer: Round-offerror 76
4.3.4 StabilityanalysisofFTCSschemeasappliedtoone-dimensional
heatconductionequation 78
4.3.5 Richardsoncentralintimeandcentralinspace(CTCS)finite
differenceschemeanditsstability 79
4.3.6 DuFort–Frankelfinitedifferenceschemeanditsstability 80
4.3.7 Backwardintimeandcentralinspace(BTCS)schemeandits
stability 81
4.3.8 Crank–NicolsonSchemeanditsstability 82
4.4 Two-dimensionalHeatConductionEquation 84
4.4.1 FTCSschemeanditsstability 84
4.4.2 BTCSschemeanditsstability 85
4.4.3 AlternatingDirectionImplicit(ADI)method 86
4.5 StabilityAnalysisofOne-dimensionalLinearAdvectionEquation 86
4.5.1 Forwardintimeandcentralinspace(FTCS)scheme 87
4.5.2 Centralintimeandcentralinspace(CTCS)schemeanditsstability 87
4.5.3 Upwindmethods 90
4.5.4 Laxfinitedifferenceschemeanditsstability 91
4.5.5 Lax–Wendroffschemeanditsstability 92
4.5.6 Backwardintimeandcentralinspace(BTCS)schemeandits
stability 93
4.5.7 Crank–Nicolsonschemeanditsstability 94
4.6 MatrixMethodofStabilityAnalysis 94
4.6.1 Matrixmethodfortheone-dimensionalheatequation 95
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