The purpose of this lecture is to explain howboundaries can be taken into account within

singular perturbation problems Bydefinitionce singular problem is an equation ofReform

ct ne fwith EZO where of E is an operator oforderd i with the following propertyif ne à foe f Consone strong

sense then à is a solution of

JE fwhere À is an operator of order Lol

Asa consequence we cannot enforce the samenumber of boundary conditions force and

for at presence of boundary layers within nen es à BL part

Questions1 How can we construct boundary layersSystematic construction

2 Justification ofthe convergence and

validity of the approximation

Remarks D In some cases the BL hason effect on the mean flow2 Unfortunately there is no completelyunified framework to address the two pointsabove Study of two examples one simpleone more complicated

I IntroductionDAtoymodel.im

Enfin usé L x ECO D

ukot.ua O

Existence and uniqueness of a solution n'byLax Milgram theoremA priori estimate x ne exp Ex

SÈME esseulée ÎE Ç nen é fée e fuséeEN

Elway Muette ICul ci init à sel

Ansatzi vieux tube tué f

1Boundary conditions for à

Need to look at BL close to second set

Plug Ansatz into equation

t LITUBL satisfies dzzubtr dzuf O

u.BE temps f 3

vite satisfies 2g une ciel 0

NE 0

Consequence 7 0 0 no BL ondeleftucxt ffsxueu enpf.EE un

watt satisfies up with small remaindersatisfies the Be at xd

x D with exponentiallysmall remainder

consequence EW uahxlpftlli uaq.seRenard EH Bible E Èeenpf 2 de

OLD

Thea energyestimate mes the boundarylayer

2 A bit of methodologymm

Ref Gérard Vaut Paul

Simplifications Amine that of l is a linearoperator with constant coefficients and that1 RI IRM xRIfflat boundaryItis legitimate to look for modal solutionsofcffi.ie we lookfor special solutions ofthe form

UE x UE enpfik.ae dxnk'ERw tangential Fourier moded C Q Re d 0UE c pend

Boundary layer Red I

Plug into equation linear systemoftheformA k d UE o ci

where Alisa did matrix with polynomialcoefficients in k d land E in generalCe symbolof et eThere exists a mon trivial solution iff

det A Kkl D 0 2

Polynomial int with coefficients that are

polynomial in lsland E sayof degree m

This polynomial has ne complex rootsdz du

Complex analysis IggC Q Jj lé such that

Dj E k EH Jj as E 0

NE kD d solution of 2 with Red 0

d dim Vet U polofen de MCE le

d number of boundary conditions thatcourbe lifted by a boundary layerin the case of Dirichlet boundary conditions

Theorem Gérard Vaut Paul If pj is independent of knon degeneracy then WE mm of modal solfurthertechnicalassumptionsGeneralization in an arbitrary domain 1 work withlocal coordinates vs Symbole AEC k DNondegeneracy assumption The gig's are independentof x karcade connected componentofthe boundary

Remark Point of view equivalent to butslightly different from matched asymptoticexpansions

I Example 1 Boundary layers forthe critical

reflection problem ima stratified fluidJoint work with Roberta Bianchini and LaureSaint Raymond following a previous analysisof Thierry Dauxoir and Bill Young

Starting point Bousainesq model

q v SG F v Vp ni Du bez in rB fb is v F b N'vz j.EEf

in r

b buoyancyGN k s N L inthe following tosimplifyr xs xD EN _x singt nz cosy 0

Conditions on 21 7par 0 dub par 0

Purpose ofAndy by Danois Young studyreflection of an incident wave on a sloping

boundary and retrieve some experimentalobservations trapping of reflected wave dosetothe boundary creation of harmonics etc

Idea if f v k Les then linear plane waves

ofthe in viscid system are approximate solutions tantthey do not satisfy the boundary conditions

Creation of a reflected wave BL in

orderto match the boundary condition Thenonlinear interaction between the incident wave andthe BL creates hamrmonics

Today focus on boundary layers fr43only Forget about nonlinear kun 8 0

Look for solutions of B with 5 0 intheforme

j espfilkx ut d e Ê2 X singtzzasz e normal variable

mary seeing tangential variable

Plug into linear system

As Ç 0

As 4x4 complex matrix

A io auIf.sihI r igxa

Nontrivial solution det A 0

Polynomial of order 6 ind

del Agarn kw ico vu ki 5 22k singury

à n'n'y d k'Casey d

Criticality parameter 5 co singstudyof the behavior ofthe roots of det Ajdepending on N K and 5Proposition Amine u s Nek Ikki WILLc det Ay has exactly 3roots with so dalpaitii The rooto.ve inthe following way321 n s 151 Eu aireradjipotold d ouiJ'fuitÊtre pas Lynn 3Mo ER M2 Mfrreflectedwave bz k dz v

dadzvi l'a Superposition of two BtN BD Superpositionof one of size n'k

2dB BL of one of size v s

different sizes

iii the BL can always lift 3 Bc

dim Vert Ypg dee det Asa Remo3

Construction of an approximate solution ira

weakly nonlinear regime1 Take on invisaid linear wave Corwavepacket2 Couture the associated BL correctsthe defect in B c

Approximate solution of linear systemsatisfying exactly the boundary conditions

3 Deal with NL remainder Out ofthe scope of today's lecture

Justification of the approximationEnergy method case verra 13K n'b

HE Vandchlly templeEst NE Vapp trotteCap C d t s vbulb

Remark Both the incident wave packet andthe BL carry an energyof order OCD

Example 2 Ekman layers

starting point rotatingfluid equationwithsmallvertical viscosity

zu Et VT n È ez ru Tpe Dune EdéhEoinerdiva 0

es Yara1 T O D T R 12oz

Reminder L P Lez r Coriolis operatorE projection auto 1K

1K net r divu quziz.co aziz D

Proposition There exists ce bilbertion basis of 1Kdenokd eDeeezsyoz.cousistimgofeigenvedorsofL

LNG ideaNkdes lez T

Kal

Ref Chemin Desjardins Gallagher Grenier 2006Lions Temaru Wang 1994Babin MahaloriNicolaenko 1999

Nea eikh.dk men b as CTkzznz k simCTkzz

Following Lecture 2 it is natural to lookfor we in the four

n'Cttenpf_tel v Ect

In other wordsideaE

yegebellt Nn corrector

Eg for beenProblem Nea pz 0

2 I

from a spectral pointof view consider the operatorAsX H defined by AE f we where

Lu Due fdiva 0

near _0 hïmmpatüThen Al is not a normal operator L and

D associated with Dirichlet BD do notAself adjoint

commute

partSource of difficulty inthe analysisresponsible forthe presence of Ekmanlayers

Back to the general methodologyLookfor a solution offre linear part fin the forme

E

enplika.xa iaotq.dz BÉE to

anticipatethehigh oscillationsthe system

with Re d 0

linear system

A E É 0

where

AE fixe Kat EM

O I o i kaE

det AE polynomial of degree 6in leCI fact polynomial of degree 3in d

dette Ê tl 1kf fin élite Kelly

f ICI

Investigate the roots of I focussing ontheroots that do not belong to A pet X 42

Remark Inthe présent case since the BL is

created by the trace ofNeon at the boundary itis sufficient to assume that de fork Cha lez C7f Yo fou some lez EZ

Regime I ka 0 IL 1

2 roots X of order E X Ëwhere 2

M t 2in I 2 0

n i u ti

clamsical Ekman layers

1 root oforder 1 Xo No

where I we no Ibert ce 2 0

Desmet correspond to a BL

spore

Regime 2 ka 0 1

Abit artificial But corresponds totheasymptotie

Kd1 root of order E

2 X 2in2

2 roots of order E X

whereLiu YI Liu Ikari Ikeda 0

quasi resonant BL much larger than classicalEkman layer

Regime 3 kenzo 101412classical Ekman layersf root in exactly zero

Regime 4 ka 0 lol L1 classical Ekman layer2 roots are exactly zero

In Regime si hourdagou deal withthe not which is OLD

classical construction use dire de no thatuB k BÉ nulle 0 at 0

forget about do

Since UBL is divergence free and When BCis of sine E u etc Ekman pumpingSome fluid exits the BL and enteurs theinterior

Lifted by a couche vlilt OC.es

oscillating at frequency E D

vlilttfezrvh.lt CD

Source term in the eq.fr vefriction terms