Table Of ContentMathematics of Planet Earth 10
Bertrand Chapron · Dan Crisan ·
Darryl Holm · Etienne Mémin ·
Anna Radomska Editors
Stochastic
Transport in
Upper Ocean
Dynamics
STUOD 2021 Workshop, London, UK,
September 20–23
Mathematics of Planet Earth
Volume 10
SeriesEditors
DanCrisan,ImperialCollegeLondon,London,UK
KenGolden,UniversityofUtah,SaltLakeCity,UT,USA
DarrylD.Holm,ImperialCollegeLondon,London,UK
MarkLewis,UniversityofAlberta,Edmonton,AB,Canada
YasumasaNishiura,TohokuUniversity,Sendai,Miyagi,Japan
JosephTribbia,NationalCenterforAtmosphericResearch,Boulder,CO,USA
JorgePassamaniZubelli,InstitutodeMatemáticaPuraeAplicada,RiodeJaneiro,
Brazil
This series provides a variety of well-written books of a variety of levels and
styles,highlightingthefundamentalroleplayedbymathematicsinahugerangeof
planetarycontextsonaglobalscale.Climate,ecology,sustainability,publichealth,
diseases and epidemics, management of resources and risk analysis are important
elements. The mathematical sciences play a key role in these and many other
processes relevant to Planet Earth, both as a fundamental discipline and as a key
componentofcross-disciplinaryresearch.Thiscreatestheneed,bothineducation
andresearch,forbooksthatareintroductorytoandabreastofthesedevelopments.
Springer’s MoPE series will provide a variety of such books, including mono-
graphs,textbooks,contributedvolumesandbriefssuitableforusersofmathematics,
mathematicians doing research in related applications, and students interested
in how mathematics interacts with the world around us. The series welcomes
submissions on any topic of current relevance to the international Mathematics
of Planet Earth effort, and particularly encourages surveys, tutorials and shorter
communicationsinalivelytutorialstyle,offeringaclearexpositionofbroadappeal.
ResponsibleEditors:
MartinPeters,Heidelberg([email protected])
RobinsondosSantos,SãoPaulo([email protected])
AdditionalEditorialContacts:
DonnaChernyk,NewYork([email protected])
MasayukiNakamura,Tokyo([email protected])
Bertrand Chapron (cid:129) Dan Crisan (cid:129) Darryl Holm (cid:129)
Etienne Mémin (cid:129) Anna Radomska
Editors
Stochastic Transport in
Upper Ocean Dynamics
STUOD 2021 Workshop, London, UK,
September 20–23
Editors
BertrandChapron DanCrisan
Ifremer–InstitutFrançaisdeRecherche ImperialCollegeLondon
pourl’ExploitationdelaMer London,UK
Plouzané,France
DarrylHolm EtienneMémin
ImperialCollegeLondon CampusUniversitairedeBeaulieu
London,UK Inria–InstitutNationaldeRechercheen
SciencesetTechnologiesduNumérique
AnnaRadomska Rennes,France
ImperialCollegeLondon
London,UK
ThisworkwassupportedbyHorizon2020FrameworkProgramme(856408)
ISSN2524-4264 ISSN2524-4272 (electronic)
MathematicsofPlanetEarth
ISBN978-3-031-18987-6 ISBN978-3-031-18988-3 (eBook)
https://doi.org/10.1007/978-3-031-18988-3
Mathematics Subject Classification: 60Hxx, 60H17, 70L10, 35R60, 37M05, 37-11, 35Qxx, 65Pxx,
00B25
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Preface
This volume contains the Proceedings of the 2nd Stochastic Transport in Upper
OceanDynamicsWorkshopheldon20–23September2021.Afterthesuccessofthe
first workshop, the STUOD Principal Investigators: Prof. Dan Crisan (ICL), Prof.
BertrandChapron(IFREMER),Prof.DarrylHolm(ICL)andProf.EtienneMémin
(INRIA)weredelightedtobebackwithanothereducationalandinspirationalevent.
“Stochastic Transport in Upper Ocean Dynamics” (STUOD) project is supported
by an ERC Synergy Grant, led by Imperial College London, National Institute
forResearchinDigitalScienceandTechnology(INRIA)andtheFrenchResearch
Institute for Exploitation of the Sea (IFREMER). The project aims to deliver new
capabilities for assessing variability and uncertainty in upper ocean dynamics and
providedecisionmakersameansofquantifyingtheeffectsoflocalpatternsofsea
levelrise,heatuptake,carbonstorageandchangeofoxygencontentandpHinthe
ocean.Theprojectwillmakeuseofmultimodaldataandwillenhancethescientific
understandingofmarinedebristransport,trackingofoilspillsandaccumulationof
plasticinthesea.
As in the previous year, the 2nd STUOD Annual Workshop 2021 focused on a
rangeoffundamentaltopicalareas,including:
1. Observations at high resolution of upper ocean properties such as temperature,
salinity,topography,wind,wavesandvelocity
2. Large-scalenumericalsimulations
3. Data-basedstochasticequationsforupperoceandynamicsthatquantifysimula-
tionerror
4. Stochasticdataassimilationtoreduceuncertainty
Each chapter in the present volume illustrates one or several of these topical
areas. Many chapters offer new mathematical frameworks that are intended to
enhancefutureresearchintheSTUODproject.
Theeventbroughttogether65participantsfrom11countries:UK28,France22,
USA 1, Canada 1, Australia 1, Czech Republic 1, Germany 4, Italy 4, Ireland 1,
South Africa 1 and Switzerland 1. Moreover, the workshop was well attended by
early-career academics, post-graduate students, industry representatives (Watson-
v
vi Preface
MarlowFluidTechnologyGroup,OceanScope),seniormembersofthecommunity
andinvitedguests.
Thescientificprogramofthis4-dayhybrideventincludedinvitedpresentations
bySTUODAdvisoryBoardMembers:ProfAlbertoCarrassi(UniversityofRead-
ing,NCEO),ProfFrancoFlandoli(ScuolaNormaleSuperiore)andProfSebastian
Reich (University of Potsdam), Dr Eniko Székely (École Polytechnique Fédérale
deLausanne,SwissDataScienceCenter),individualpresentationsbytheSTUOD
Principal Investigators and post-doctoral Researchers, snapshot presentations and
demos.Thespeakersincludedleadingmid-careerandseniorresearchersaswellas
early-careerresearchers.Moreover,theforumyieldedopportunitiesforinvestigators
at an early stage of their career to have discussions with established scientist,
fostering potential future research collaborations, networking as well as inclusion
andtrainingofthenextgenerationofresearchers.
The photograph above shows some participants attending the event in person
duringabreakbetweenlectures.
Most of the lectures were video-recorded and may be viewed on the
STUODYouTubechannel.
The following is a brief description of the 19 contributions included in the
proceedings:
The submitted manuscripts include the paper by Dan Crisan and Prince
Romeo Mensah, entitled “Blow-up of Strong Solutions of the Thermal Quasi-
GeostrophicEquation”.Thispaperconcernsthesystemofcoupledequationsthat
Preface vii
governstheevolutionofthebuoyancyandpotentialvorticityofafluid.Thissystem
has been shown in recent work of the authors and their collaborators to possess
a local in time solution. In this paper, the authors give a characterization of the
blow-upofsolutionsofthesysteminthespiritoftheclassicalBeale–Kato–Majda
blow-upcriterionforthesolutionoftheEulerequation.
ThecontributionofArnaudDebussche,BerengerHug,andEtienneMémin,
entitled “Modelling Under Location Uncertainty: A Convergent Large-Scale
RepresentationoftheNavier-StokesEquations”,introducesmartingalesolutions
for2Dand3DstochasticNavier-Stokesequationsintheframeworkofthemodelling
under location uncertainty (LU). Such solutions are unique when the spatial
dimension is 2D. The authors also prove that, if the noise intensity goes to zero,
thesesolutionsconvergetoasolutionofthedeterministicNavier-Stokesequation.
Evgueni Dinvay considers in the paper “A Stochastic Benjamin-Bona-
Mahony Type Equation” a particular nonlinear dispersive stochastic equation
recently introduced as a model describing surface water waves under location
uncertainty. The corresponding noise term is introduced through a Hamiltonian
formulation, which guarantees the energy conservation of the flow. The author
showsthattheinitial-valueproblemhasauniquesolution.
Benjamin Dufée, Etienne Mémin, and Dan Crisan investigate in the paper
“Observation-BasedNoiseCalibration:AnEfficientDynamicsfortheEnsem-
ble Kalman Filter” the calibration of the stochastic noise in order to guide its
realizations towards the observational data used for the assimilation. This is done
inthecontextofthestochasticparametrizationunderlocationuncertainty(LU)and
dataassimilation.Thenewmethodologyismathematicallyjustifiedbytheuseofthe
Girsanov theorem and yields significant improvements in the experiments carried
out on the surface quasi-geostrophic (SQG) model, when applied to ensemble
Kalmanfilters.Thetestcasestudiedinthepapershowsimprovementsofthepeak
MSEfrom85%to93%.
ThepaperbyCamillaFiorini,Pierre-MarieBoulvard,LongLi,andEtienne
Mémin, entitled “A Two-Step Numerical Scheme in Time for Surface Quasi
Geostrophic Equations Under Location Uncertainty”, considers the surface
quasi-geostrophic (SQG) system under location uncertainty (LU) and proposes
a Milstein-type scheme for these equations, which is then used in a multi-step
method.TheSQGsystemconsideredinthepaperconsistsofonestochasticpartial
differentialequation,whichmodelsthestochastictransportofthebuoyancy,anda
linearoperatorlinkingthevelocityandthebuoyancy.IntheLUsetting,theEuler-
Maruyamaschemeconvergeswithweakorder1andstrongorder0.5.Theauthors
develophigherorderschemesintime,basedonaMilstein-typeschemeinamulti-
step framework. They compare different kinds of Milstein schemes. The scheme
with the best performance is then included in the two-step scheme. Finally, they
showhowtheirtwo-stepschemedecreasestheerrorincomparisontoothermulti-
stepschemes.
ThecontributionofFrancoFlandoliandEliseoLuongo,entitled“TheDissipa-
tionPropertiesofTransportNoise”,presentsinacompactwaythelatestresults
about the dissipation properties of transport noise in fluid mechanics. Motivated
viii Preface
by the fact that transport noise is natural in a passive scalar equation for the
heat diffusion and transport, the authors introduce several results about enhanced
dissipation due to the noise. Rigorous statements are matched with numerical
experiments to understand that the sufficient conditions stated are not yet optimal
butgiveafirstusefulindication.
DanielGoodairpresentsinthepaper“ExistenceandUniquenessofMaximal
Solutions to a 3D Navier-Stokes Equation with Stochastic Lie Transport” a
criterion for showing that an abstract SPDE possesses a unique maximal strong
solution. This is then applied to a 3D stochastic Navier-Stokes equation. Inspired
by the classical work of Kato and Lai, the author provides a comparable result
in the stochastic case applicable to a variety of noise structures such as additive,
multiplicative and transport. In particular, the criterion is designed to fit viscous
fluiddynamicsmodelswithstochasticadvectionbylietransport.Itsapplicationto
the incompressible Navier-Stokes equation matches the existence and uniqueness
resultofthedeterministictheory.
Darryl D. Holm, Ruiao Hu, and Oliver D. Street present in “Coupling of
Waves to Sea Surface Currents Via Horizontal Density Gradients” a set of
mathematicalmodelsandnumericalsimulationsmotivatedbysatelliteobservations
of horizontal sea surface fluid motions that show the close coordination between
thermal fronts and the vertical motion of waves or, after an approximation, the
slowlyvaryingenvelopeoftherapidlyoscillatingwaves.Thiscoordinationoffluid
movementswithwaveenvelopesoccursmostdramaticallywhenstronghorizontal
buoyancy gradients are present, e.g., at thermal fronts. The nonlinear models of
thiscoordinatedmovementpresentedinthepapermayprovidefutureopportunities
for the optimal design of satellite imagery that could simultaneously capture the
dynamics of both waves and currents directly. The models derived in the paper
appear first in their un-approximated form, then again with a slowly varying
envelope(SVE)approximationusingtheWKBapproach.TheWKBwave-current-
buoyancyinteractionmodelderivedbytheauthorsforafreesurfacewithhorizontal
buoyancy gradients indicates that the mechanism for these correlations is the
ponderomotive force of the slowly varying envelope of rapidly oscillating waves
acting on the surface currents via the horizontal buoyancy gradient. In this model,
the buoyancy gradient appears explicitly in the WKB wave momentum, which in
turngeneratesdensity-weightedpotentialvorticitywheneverthebuoyancygradient
isnotalignedwiththewave-envelopegradient.
The contribution of Ruiao Hu and Stuart Patching, entitled “Variational
Stochastic Parameterisations and Their Applications to Primitive Equation
Models”,presentsanumericalinvestigationintothestochasticparameterizationsof
theprimitiveequations(PE)usingthestochasticadvectionbylietransport(SALT)
andstochasticforcingbylietransport(SFLT)frameworks.Theseframeworkswere
chosenduetotheirstructure-preservingintroductionofstochasticity,whichdecom-
poses the transport velocity and fluid momentum into their drift and stochastic
parts,respectively.Inthispaper,theauthorsdevelopanewcalibrationmethodology
to implement the momentum decomposition of SFLT, and they compare this
methodology with the Lagrangian path methodology implemented for SALT. The
Preface ix
resulting stochastic primitive equations are then integrated numerically using a
modificationoftheFESOM2code.Forcertainchoicesofthestochasticparameters,
the authors show that SALT causes an increase in the eddy kinetic energy field
and an improvement in the spatial spectrum. SFLT also shows improvements in
these areas, though to a lesser extent. The SALT approach, however, produces
an excessive downwards diffusion of temperature, compared to high-resolution
deterministicsimulations.
ThepaperbyOanaLangandWeiPan,entitled“APathwiseParameterisation
for Stochastic Transport”, sets the stage for a new probabilistic approach to
effectively calibrate in a pathwise manner a general class of stochastic nonlinear
fluiddynamicsmodels.Theauthorsfocusona2DEulerSALTequation,showing
thatthedrivingstochasticparametercanbecalibratedinanoptimalwaytomatcha
setofgivendata.Moreover,theyshowthatthismodelisrobustwithrespecttothe
stochasticparameters.
TheworkbyLongLi,EtienneMémin,andGillesTissot,entitled“Stochastic
Parameterization with Dynamic Mode Decomposition”, considers a physical
stochasticparameterizationtoaccountfortheeffectsoftheunresolvedsmallscale
on the large-scale flow dynamics. This random model is based on a stochastic
transport principle, which ensures a strong energy conservation. The dynamic
mode decomposition (DMD) is performed on high-resolution data to learn a basis
of the unresolved velocity field, on which the stochastic transport velocity is
expressed. Time-harmonic property of DMD modes allows the authors toperform
a clean separation between time-differentiable and time-decorrelated components.
Thecorrespondingrandomschemeisassessedonaquasi-geostrophic(QG)model.
ThepaperbyAlexanderLobbe,entitled“DeepLearningfortheBenesFilter”,
concerns the filtering problem, in other words, the optimal estimation of a hidden
state given partial and noisy observations. Filtering is extensively studied in the
theoretical and applied mathematical literature. One of the central challenges in
filtering today is the numerical approximation of the optimal filter. The author
presents a brief study of a new numerical method based on the mesh-free neural
network representation of the density of the solution of the filtering problem
achieved by deep learning. Based on the classical SPDE splitting method, the
algorithm introduced includes a recursive normalization procedure to recover the
normalizedconditionaldistributionofthesignalprocess.Thepresentworkusesthe
Benesmodelasabenchmark:withintheanalyticallytractablesettingoftheBenes
filter, the author discusses the role of nonlinearity in the filtering model equations
for the choice of the domain of the neural network. Further, he presents the first
studyoftheneuralnetworkmethodwithanadaptivedomainfortheBenesmodel.
Data assimilation techniques are the state-of-the-art approaches in the recon-
structionofaspatio-temporalgeophysicalstatesuchastheatmosphereortheocean.
These methods rely on a numerical model that fills the spatial and temporal gaps
in the observational network. Unfortunately, limitations regarding the uncertainty
of the state estimate may arise when considering the restriction of the data
assimilationproblemstoasmallsubsetofobservations,asencounteredforinstance
in ocean surface reconstruction. These limitations motivated the exploration of
