Table Of ContentCIMAT Lectures in Mathematical Sciences
Gerardo Hernández-Dueñas
Miguel Angel Moreles
Editors
Mathematical and
Computational
Models of Flows
and Waves in
Geophysics
CIMAT Lectures in Mathematical Sciences
EditorialBoardMembers
OctavioArizmendi,CIMAT,Guanajuato,Mexico
ArturoHernandez,CIMAT,Guanajuato,Mexico
MiguelAngelMoreles,CIMAT,Guanajuato,Mexico
MiguelNakamura,CIMAT,Guanajuato,Mexico
ManagingEditor
RafaelHerreraGuzman,CIMAT,Guanajuato,Mexico
This series, published by Birkhäuser in collaboration with CIMAT, reports recent
developments in mathematics and its applications, including those in computer
science, mathematical statistics, economics, biology, etc., which are close to the
research fields cultivated at CIMAT. The texts to be considered will be sets of
lecture notes on current or new topics with a strong mathematical content, as
well as presentations of classic fields from novel perspectives. The volumes can
betheproductofspecializedlecturecourses,advancedsummer/winterschools,or
collaborativeeffortsbyresearchgroupstopresentvariousfacetsofagivenfieldin
aninsightfulandunifiedmanner.
Gerardo Hernández-Dueñas (cid:129)
Miguel Angel Moreles
Editors
Mathematical and
Computational Models of
Flows and Waves in
Geophysics
Editors
GerardoHernández-Dueñas MiguelAngelMoreles
InstituteofMathematics DepartmentofMathematics
NationalAutonomousUniversityofMexico CIMAT
Juriquilla,Querétaro,Mexico Guanajuato,Mexico
ISSN2731-6521 ISSN2731-653X (electronic)
CIMATLecturesinMathematicalSciences
ISBN978-3-031-12006-0 ISBN978-3-031-12007-7 (eBook)
https://doi.org/10.1007/978-3-031-12007-7
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Preface
Thisvolumeproposesan integralapproachtostudyingthegeophysicsofEarth.It
is motivatedby a varietyof phenomenafrom naturewith deep and directimpacts
inourlives.Sucheventsmayevolveacrossalargerangeofspatialandtimescales
andmaybeobservedintheocean,theatmosphere,thevolcanicsurface,aswellas
underground.
Thephysicallawsdictatingtheevolutionofsuchphenomenaleadtotheunifying
themeofthismanuscript,thatis,themathematicalandcomputationalmodelingof
flowsandwaves.Consequently,theunderlyingmodelsaregivenintermsofpartial
differential equations (PDEs) whose solutions are approximated using numerical
methods,thusprovidingsimulationsoftheaforementionedphenomena,whichare
tobegiventheappropriategeophysicalvalidationandinterpretation.
The outline of the book is as follows. The first chapter considers rapidly
rotating convectively driven flows in extreme parameter regimes, which are of
relevance in geophysical and astrophysical settings. The underlying model is
the rapidly rotating Rayleigh-Bénardconvection (RRRBC) primitive equations. It
is well known that laboratory explorations and direct numerical simulations of
the governing equations are limited. Consequently, alternative methodologies are
required. For instance, the chapter introduces an asymptotic reduction based on
small Rossby (Ro) number. The resulting equations, the non-hydrostaticbalanced
geostrophicequations(NHBGE),formaclosedsysteminwhichthesmallparameter
Ro no longer appears. The asymptotic procedure eliminates the impediments to
using the primitive equations to describe geophysical and astrophysical flows.
In particular, the reduced equations describe four important dynamical regimes:
cellularconvection,convectiveTaylorcolumns,convectiveplumes,andgeostrophic
turbulence. Some of these are relevant to both astrophysical and oceanographic
applications.Thelatterconnectsthechapterthatfollows.
Chapter2dealswithinteractionsoftheoceanandtheatmosphere,withemphasis
onwavetheory.Thechapterisconcernedwithsomeimportantapplicationsrelated
totheinfluenceofoceansurfacewavesonpresentchallengingissues.Oneparticular
issueisthegastransferacrosstheinterfaceanditspotentialimpactonclimateand
its changes.Anotheroneisthe upperoceandynamicsandthe behaviorof surface
v
vi Preface
currents and drift, greatly associated with transport of pollutants and objects on
the sea surface. It is noteworthy that the chapter develops the underlying theory,
complementingitwithdatatoprovideinsightfulinformationontheair-seatransfer
mechanisms. The aim is to obtain the best knowledge in that respect, in order to
be able to predictin the most appropriatefashionthe ocean-atmosphereexchange
processesofgreenhousegasesforinstance,andinduecoursetopredictclimateand
itschangesthroughtheuseofpowerfulnumericalmodels.
As illustrated by the first two chapters, in recent decades, computational
resources have made possible the simulation of complex phenomena. While the
firstchapterdelvesintothemathematicalphysicstounderstandgeophysicalflows,
the second chapterrelieson data and observation.Both use numericalmethodsto
complement their respective theses. It is apparent that an in depth knowledge of
these methods is essential. Thus, Chap.3 provides a state-of-the-art contribution
for one of such methods.Namely,the work proposesa second-orderaccurate and
robustnumericalmethodfor the conservativelevel-setapproach,which is applied
for capturing the interface between two fluids. The first two chapters include
problems of this sort. The exposition will appeal to a numerical specialist, and
specificsofthemethodareincluded.Forinstance,thetimeintegrationisbasedona
methodthatallowstheselectionbetweencompleteexplicitandimplicitfirst-order
timeformulationsora second-orderCrank-Nicolson(implicit)method.Thespace
discretization is based on a finite-volume method on prisms elements consisting
of unstructured triangular grids on the horizontal directions and several layers in
thevertical.Numericalresultsforthree-dimensionalsimulationsrequiresignificant
computationaltimetobecarriedout.Thus,theentirecodeisdevelopedinparallel.
Theparallelizationofthealgorithmisbasedonadomaindecompositionintoseveral
sub-domainsinthehorizontaldirection,oneforeachparallelprocess,andaparallel
solutionofthelinearsystemusingamulti-colorSOR(MSOR)method.
The second part of the book deals with problems associated to solid Earth.
In Chap.4, granular flows in volcanic environments are considered. As a primer,
continuummodelscanbeusedformodelingthisflowphenomenon.However,unlike
the equations governing Newtonian fluids, such as the Navier-Stokes equations,
granularmodelsmayrequireadditionaltermsthattake intoaccountfrictionaland
collisional loss of energy both between particles and between the mediumand its
substrate. Also, they fail to describe phenomenainherentto the granularityof the
medium, such as particle-size segregation and high-speed ejection of individual
particles.Thisfailurecanbeofcrucialimportancewhenassessinghazardmapsfor
locationspronetorockavalanchesandpyroclasticdensitycurrents.Consequently,
the chapter focuses instead on the description of discrete models. These models
take into account the mechanics of individual particles and are used to explain
the behavior of granular flows. This approach uses molecular dynamics (MD)
algorithms, which first calculate the sum of forces experienced by each of the
individualgrains,andafterwardssolvetheequationsofmotionwithanappropriate
integrator. The basics of the method are fully described. An application to active
volcanoesispresented.
Preface vii
Chapter 5 addresses geophysical exploration of underground resources. The
explorationofaregionofinterestrequiresthemeasurementsofseveralgeophysical
datasetswhichneedtobeinterpretedforcharacterization.Aschemeforcooperative
inversionofseismicandgravitymeasurementsispresented.Theseismicmethodhas
beenparticularlysuccessful,becomingthebasisforotherstate-of-the-artmethods
suchasfullwaveforminversion(FWI).FWIisapowerfulseismic-imagingmethod
used to estimate a seismic-velocity model such that the discrepancies between
observed and synthetic seismograms are minimized. The gravimetric inversion
method is an inversion method (GI) for density estimation. This method is well
knownfor estimatingstructureswith horizontalchangesof mass distribution.The
solution is straightforward using Gauss-Newton minimization to obtain a density
model inverting the square matrix on a single step. This method is widely used
by geophysicists because of its fast convergence. However, it is computationally
expensiveandunfeasibleforlarge-scaleproblems.Full-waveforminversion(FWI)
and gravimetric inversion (GI) are carried out in tandem. First, FWI is used to
estimate a seismic-velocity model. Then, using Gardner’s density-velocity rela-
tionship, GI is performed to update the density model. Again, using Gardner’s
velocitydensityrelation,avelocitymodelisobtainedandtheprocessisiterated.For
FWI,minimizationisapproximatedbyagradient-basedalgorithm.Tocomputethe
gradient,theadjointstatemethodisused.Forthereader’sconvenience,gravimetric
andwaveformforwardmodelingareintroduced,aswellasthenumericalmethods
ofsolution.Resultsareillustratedwiththewell-knownMarmousimodel,ahighly
nontrivialbenchmarkproblemintheliterature.
The lastchapterdealsalso with wave phenomena.The motivationis the useof
electromagneticmethodsingeophysicsexplorationtomaptheresistivestructureof
thesubsurfaceusinginstrumentsthatworkatlowinductionnumbers(LINs).These
methods have been successfully applied to archaeological studies, groundwater
characterization,contaminantmigration,andmineralalterationmapping.Tointer-
pretelectromagneticmeasurements,the electricand magneticfieldsare computed
by solving numerically the Maxwell’s equations in the low induction-number
domain.Thegeophysicalapplicationsunderstudyleadtohighscalecomputations.
Consequently, the differential forms of Maxwell’s equations are solved using the
parsimoniousfinite-differencemethod.SuchmethodisimplementedinFortranand
parallelizedwithOpenMP.
Acknowledgments The editors thank the Mexican Research Center in Mathe-
matics for its hospitality. Both editors were partially supported by the grants
UNAM-DGAPA-PAPIITIN112222&ConacytA1-S-17-634.Theco-editorG.H-D
would like to thank the hospitality of NorthWest Research Associates and the
supportofNSFAward2123740,CTICandDGAPAduringhissabbaticalvisit.
Juriquilla,Mexico GerardoHernández-Dueñas
Guanajuato,Mexico MiguelAngelMoreles
Contents
GeostrophicTurbulenceandtheFormationofLargeScaleStructure..... 1
EdgarKnobloch
OceanSurfaceWavesandOcean-AtmosphereInteractions................ 35
FranciscoJ. Ocampo-Torres,PedroOsuna,HéctorGarcía-Nava,and
NicolasG.Rascle
A 3D Two-Phase Conservative Level-Set Method Using an
UnstructuredFinite-VolumeFormulation..................................... 67
MiguelUhZapataandReymundoItzáBalam
ThePhysicsofGranularNaturalFlowsinVolcanicEnvironments........ 103
G. M.Rodríguez-Liñán,R.Torres-Orozco,V. H.Márquez,L.Capra,
andV.Coviello
CooperativeGravityandFullWaveformInversion:ElasticCase.......... 129
RaulU.Silva-Ávalos,JonásD.DeBasabe,andMrinalK.Sen
Modellingthe 3DElectromagneticWaveEquation:Negative
ApparentConductivitiesandPhaseChanges................................. 171
BeatrizValdés-Moreno,MarcoA.Pérez-Flores,andJonásD.DeBasabe
ix
Geostrophic Turbulence and the
Formation of Large Scale Structure
EdgarKnobloch
Abstract ThisChaptersummarizesrecentprogressinunderstandingrapidlyrotat-
ingconvectivelydrivenflows.Theemphasisisonthebehaviorinextremeparameter
regimesof relevancein geophysicaland astrophysicalsettings. These regimesare
described by an asymptotically reduced system of equations valid in the limit of
small Rossby numbers. The equations describe four regimes as a scaled Rayleigh
(cid:2)
number Ra increases: a disordered cellular regime near threshold, a regime of
(cid:2)
weaklyinteractingconvectiveTaylorcolumnsatlargerRa,followedforyetlarger
(cid:2)
Ra by a breakdown of the convective Taylor columns into a disordered plume
regimecharacterizedbyreducedheattransportefficiency,andfinallybyaturbulent
state called geostrophic turbulence. Properties of these states are described and
illustrated using direct numerical simulations of the reduced equations. These
simulationsrevealthatgeostrophicturbulenceisunstabletotheformationoflarge
scale barotropic vortices or jets, via a process known as spectral condensation.
Thedetailsofthisprocessarequantifiedanditsimplicationsexplored.Theresults
are corroborated via direct numerical simulations of the governing Navier-Stokes
equations. In the presence of lateral boundaries robust boundary zonal flows
resemblingtopologicallyprotectededgestatesinchiralsystemsarepresent.
1 Introduction
Typicalflowsofastrophysicalandgeophysicalrelevancerepresentachallengefor
both laboratory explorations and direct numerical simulations of the governing
equations.Inrapidlyrotatingflowsthisisfortworeasons:theverylargeReynolds
numbersgoverningsuchflows,requiringhighspatialresolution,andthelowvalues
of the Rossby number (equivalently,the Ekman number) for such flows implying
the presence of high frequency inertial waves requiring high temporal resolution.
E.Knobloch((cid:2))
DepartmentofPhysics,UniversityofCaliforniaatBerkeley,Berkeley,CA,USA
e-mail:[email protected]
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2022 1
G.Hernández-Dueñas,M.A.Moreles(eds.),MathematicalandComputational
ModelsofFlowsandWavesinGeophysics,CIMATLecturesinMathematical
Sciences,https://doi.org/10.1007/978-3-031-12007-7_1
