Stability of periodic progressive gravity wave solutions of the Whitham equation in the presence of vorticity

HAL Id: hal-02550499

https://hal.archives-ouvertes.fr/hal-02550499

Submitted on 15 May 2020

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Distributed under a Creative Commons

Stability of periodic progressive gravity wave solutions of the Whitham equation in the presence of vorticity

C. Kharif, M. Abid, J.D. Carter, H. Kalisch

To cite this version:

C. Kharif, M. Abid, J.D. Carter, H. Kalisch. Stability of periodic progressive gravity wave solutions

of the Whitham equation in the presence of vorticity. Modern Physics Letters A, World Scientific

Publishing, 2020, 384 (2), pp.126060. �10.1016/j.physleta.2019.126060�. �hal-02550499�

Stability of periodic progressive gravity wave solutions of the Whitham equation in the presence of vorticity

C. Kharif^a,∗, M. Abid^a, J.D. Carter^b, H. Kalisch^c

aAixMarseilleUniversité,CNRS,CentraleMarseille,IRPHEUMR7342,F-13384,Marseille,France bMathematicsDepartment,SeattleUniversity,USA

cDepartmentofMathematics,UniversityofBergen,Norway

The modulational instability of two-dimensional nonlinear traveling-wave solutions of the Whitham equationinthepresenceofconstantvorticityisconsidered.Itisshownthatvorticityhasasignificant effectonthegrowthrateoftheperturbationsandontherangeofunstablewavenumbers.Waveswith khgreaterthanacriticalvalue,wherekisthewavenumberofthesolutionandhisthefluiddepth,are modulationallyunstable.Thiscriticalvaluedecreasesasthevorticityincreases.Additionally,itisfound thatwaveswithlargeenoughamplitudearealwaysunstable,regardlessofwavelength,fluiddepth,and strengthofvorticity.Furthermore,thesenewresultsareinqualitativeagreementwiththoseobtainedby consideringfullynonlinearsolutionsofthewater-waveequations.

1. Introduction

Itis well known that small-amplitude,two-dimensional, peri- odicwavetrainsarestablewithrespecttothemodulationalinsta- bilitywhenthedispersiveparameterkh,wherekisthewavenum- berand his themean fluid depth,is lessthan thecritical value 1.363.Nevertheless,McLean [1] foundthatStokeswavesaremod- ulationallyunstable when kh=^{1 andak}=⁰.29, where a is the wave amplitude. We can conjecture that strongly nonlinear uni- formwave trainsaremodulationally unstablewithrespect toin- finitesimalperturbations in shallow water. To extend the results ofMcLean [1] toshallowerwater,FranciusandKharif [2] investi- gated instabilitiesof periodic gravity wavesin shallowwater us- ingthefullynonlinearpotential Eulerequations.Forsmallvalues of ak, they found that the dominant instabilities are quasi-two- dimensionalwhereasformoderateandlargesteepness,thedomi- nantinstabilitiesarethree-dimensional.

Whitham [3] proposed an extension of the KdV equation by using the full linear dispersion instead of its third-order trun- cated expression. Consequently, the Whitham equation presents an improvement over the KdV equation for shortwaves. In fact, Carter [4] showed that the Whitham equation provides a more accurate modelforexperimental initial wavesof depressionthan does the KdV equation. Similarly, Moldabayev, Kalisch and Du-

* Correspondingauthor.

E-mailaddress:kharif@irphe.univ-mrs.fr(C. Kharif).

tykh [5] showedthatsolutionsoftheWhithamequationstayclose to solutions of the Euler equations, and Klein et al. [6] gave a mathematicalproof thatsolutions oftheEulerequationsare well approximated by solutions ofthe Whitham equation on smallto intermediatetimescales.

EhrnströmandKalisch [7,8] provedrigorouslythattheWhitham equationadmitssmallandlargeamplitude,periodictraveling-wave solutionsandnumericallycomputedtraveling-wavesolutionswith avarietyofamplitudesincludingthoseclosetothehighestwave.

Later on, Kharifand Abid [9] computed steadily propagating pe- riodic waves in the presence of constant vorticity. The method of computing these solutions was developed by Ehrnström and Kalisch [8] andisalsofoundinSanfordet al. [10] andKharifand Abid [9]. Sanford et al. [10] studied the Whitham equation and foundthat two-dimensional,periodicwavetrainswithkh=^{1 are} stable when the wave steepness, ak, is less than approximately 0.142 and are unstable when the wave steepness is larger than thisthreshold.Toacertainextent,thisresultissurprisingbecause theWhitham equationisvalidforweaklynonlinearwaterwaves.

The latter authors numericallycorroborated the stability analysis of Hur andJohnson [11] whofound that small-amplitude waves withkh<1.145 arestableandareunstablewhenkh>1.145.Note thatBenjaminandFeir [12] andWhitham [13] showedthatStokes wavesareunstablewithrespecttolongwavelengthperturbations ifkh>1.363.Lateron,HurandJohnson [14] incorporated inthe Whithamequation theeffectofconstantvorticity whichmodifies thethresholdvalueofthedispersiveparameter.

Following Whitham [3], Kharif et al. [15] and Kharif and Abid [9] proposed a new model derived from the Euler equa- tions for fullynonlinear waterwaves propagating on a vertically sheared currentofconstant vorticity in shallowwaterthat satis- fiesthe unidirectional lineardispersionrelation. Fromthis model they derived, within the framework of weakly nonlinear waves, ageneralization ofthe Whithamequation whichthey namedthe vor-Whitham equation. At the same time, Hur [16] and Bjørnes- tadandKalisch [17] derivedshallowwaterwave equationsinthe presenceofconstantvorticity.

InordertoextendthepreviousstudiesofSanfordet al. [10] and Hur andJohnson [14], we consider the spectral stability of two- dimensional, periodic traveling-wave solutions of the Whitham equation in thepresence of constant vorticity.Ourstudy focuses on the modulational instability. The second aim is to show that theWhitham equation, whichis anapproximate equation thatis easiertoworkwiththanthefullynonlinearwaterwaveequations, mayprovidereliable stabilityresultsthatareinqualitativeagree- mentwiththoseofthefullequations.

In section 2, we present the vor-Whitham equation. In sec- tion 3, we describe how to compute the stationary solutions to thisequation.Additionally, wepresentthestability ofthesesolu- tionswithrespecttoinfinitesimal perturbations,thegrowthrates ofinstabilities, andthe ranges ofunstable Floquet parameters as functionsofvorticity.Section4containsaconclusionofthiswork.

2. Thevor-Whithamequation

Weconsidertwo-dimensionalgravity wavesthatpropagateon thesurfaceofaninviscid,incompressiblefluidwithashearcurrent paralleltothedirectionofwavepropagationthatvarieslinearlyin theverticaldirection. Weassume thatthewavestravel alongthe x−^{axis and that the z}−^{axis is oriented upward with z}=^{0 rep-} resenting the unperturbedfree surface. In order to focus on the effectsduetovorticity,weassumethatthecurrentvelocityiszero atthefreesurface.Inthissituation,thecurrentvorticity,,iscon- stantandthevor-Whithamequationisgivenby

ηt+^c1()ηηx+^K∗ηx=⁰, (1) where η(x,t)representsthe freesurface displacementandt rep- resentstime.Thecoefficientofthenonlineartermis

c₁()= ^3gh+h²² h

4gh+^h22,

where g is thegravitationalconstant of acceleration.The disper- sivetermisgivenbytheconvolutionproduct, K∗ηx,whichisthe inverseFouriertransformoftheproductoftheFouriertransforms ofK(x)and ηx(x,t).Theintegralkernelis

K(x)= ¹ 2π

+∞

−∞

c(k)e^ikxdk,

wheretheunidirectionaldispersionoflinearwavesinthepresence ofvorticityis

c(k)=tanh(kh)

2k +

gtanh(kh)

k +²tanh²(kh) 4k² .

Weconsidertraveling-wavesolutionsofthevor-Whithamequation oftheform η(x,t)= ¯η(x−^c0t)foragivenphasevelocityc0.Sub- stitutingthisansatzintoequation(1) andintegratingonceleadsto theequationthatdefinesc andη¯

−^c0η¯+^c1()η¯²

2 +^K∗ ¯η=^B, (2)

Fig. 1.Plotsofperiodictraveling-wavesolutionstothevor-Whithamequationwith L=2π,waveheight,H(seecaptions),and(a)= −1.0,(b)=1.0.Notethatthe verticalscalesinthetwoplotsaredifferent.

where B is theconstant ofintegration. Wechoose B so that the solutionη¯ haszeromean.

3. Stabilityanalysis 3.1. Steadywaves

Intheframeofreferencemovingwiththetraveling-wavesolu- tion,thevor-Whithamequationisgivenby

η_τ−c₀η_X+c₁()ηη_X+K∗η_X=0, (3) where X =^x−^c0t and τ ₌t. In this frame of reference, the traveling-wavesolution,η¯(X),isstationary(independentof τ)and satisfiesequation (2).Tocomputethesesolutions,we usethenu- merical methodof Ehrnströmand Kalisch [7]. Details of the nu- mericalmethodanditsvalidationarefoundinKharifandAbid [9].

However, herein we add asupplementary equation that fixes the waveamplitudewhenfollowingsolutionsusingc0asthecontinu- ationparameter.

In order to put the equations in dimensionless form, h and h/g are chosen asthe referencelength andreferencetime, re- spectively.Thischoicecorrespondstosettingh=^{1 and g}=^1.We consider2π-periodictraveling-wavesolutionstothevor-Whitham equation.Consequently,thewavenumberofthesesolutionsisk= 1.

Fig. 1 shows profiles of traveling-wave solutions to the vor- Whitham equation for two values of and four values of the wavesteepness.Theseplots,alongwithothersleftoutforbrevity, demonstratethat:(i)solutionscorrespondingtodifferentvaluesof and L are qualitatively similar, (ii) for solutions with a given periodandvorticity,increasing wavespeedincreaseswave height andsteepness,and(iii)thereappearstobeasolutionofmaximal heightforallwaveperiodsandvaluesofvorticity.

Fig. 2 shows profiles of traveling-wave solutions to the vor- Whitham equation withwave height 0.34 andfour values of . These plots, along with others left out for brevity, demonstrate that: (i) for solutions with a given period and wave height, in- creasingvorticitycausesthewidthofthesolutiontoincreaseand (ii) forsolutionswitha givenperiodandwave height,increasing vorticitycausesboththeminimaandmaximaofη¯ todecrease.

Fig.3 displaysthephase velocityasa functionofwave steep- ness for five values of the vorticity. These plots show that the phasevelocitydecreasesasthewavesteepnessincreases.Forfixed

Fig. 2.Plotsofperiodictraveling-wavesolutionstothevor-Whithamequationwith waveheight,H=0.34,L=2π,andfourdifferentvaluesof.

Fig. 3.Phase velocityofthe traveling-wave solutionsas afunctionofthe wave steepnessforseveralvaluesofthevorticity.=0(◦),−0.1(∗),0.1(),−0.2(), 0.2().

valuesofthewavesteepness,c0 increasesasincreases.Thisfea- turewasobservedbyKharifandAbid [9] withintheframeworkof afullynonlinear,generalizedvor-Whitham equation,whereas the profilesshowninFig.1areweaklynonlinear.

3.2.Stabilityofsteadysolutions

Inordertostudythestabilityofthesesolutionswithrespectto infinitesimalperturbations,let

η(X,τ)= ¯η(X)+η(X,τ), |η_{| | ¯}η_|, (4)

where η¯(X) and η(X,τ) correspond to the 2π-periodic unper- turbedsteady solutionand infinitesimal square integrabledistur- bance,respectively.Substitutingequation(4) intoequation(3) and linearizinggivesthefollowingequationwhichgovernsthe(linear) evolutionoftheperturbations

η_τ−^c0η_X+^c1()(ηη¯ _X+ ¯ηXη)+^K∗η_X=⁰. (5)

The Fourier-Floquet-Hill method of Deconinckand Kutz [18] and Johnson [19] establishesthatallboundedsolutionsofthisproblem havetheform

η(X,τ)=^exp(λτ)exp(ip X)

+∞

j=−∞

a_jexp(i j X), (6)

wherepisarealnumberknownastheFloquetparameter.Substi- tutingequation(6) intoequation(5) gives

+∞

−∞

(c₀−^c1()η¯₋c_p+j)i(p+^j)a_jexp(i j X)

−^c1()η¯X +∞

−∞

ajexp(i j X)=λ

+∞

−∞

ajexp(i j X), (7) where

c_p+j=tanh(p+^j) 2(p+^j) +

tanh(p+^j)

(2(p+^j) +²tanh²(p+^j) 4(p+^j)² .

Equation(7) istransformedintoageneralizedeigenvalueproblem forλ,whichafter truncationat M Fouriermodescan be written asfollows

Au=λBu, (8)

whereu= (a_−M,...,a₀,...,a_M)^T isthecorresponding eigenvector.

The 2M+^{1 unknowncoefficients}(a₋M,...,a0,...,aM)are chosen tosatisfy(7) at2M+1 collocationpointsequallydistributedalong one period of the unperturbed solution. We used M=^{100 and} checkedthattheresultsarethesamewithinsevensignificantfig- ureswhendoublingthisvalue.Thecomplex-valuedmatricesAand B depend on the unperturbedwave, η¯,the vorticity, , and the Floquet parameter, p. Oncethe unperturbedtraveling-wave solu- tion has been computedand p fixed, the generalized eigenvalue problem (8) is solved by using a standard numerical eigenvalue solver.

Theeigenvaluespectrumcorrespondingtotheflatsurface(η¯= 0) is

λj=ⁱ(p+^j)c0−ⁱ(p+^j)cp+^j,

wherethephasevelocityoftheflatsurfaceis c0=tanh(1)/2+

tanh(1)+²tanh²(1)/4.

All ofthese eigenvalues lie onthe imaginary axis.Therefore, the flat surface is spectrally stable. The corresponding eigenfunctions are η_j=ajexp(λjτ)exp(i(p+ ^j)X) which representinfinitesimal wavesoffrequency(p+^j)c₀−(p+^j)c_p+jinthemovingframeof referenceandwavenumber p+^j.

Astheamplitudeoftheunperturbedwaveincreasesfromzero, theeigenvalues moveonthe imaginaryaxisandeigenvalue colli- sionsoccur.Anecessary,butnotsufficient,conditionforinstability isthecollision ofeigenvalues.Thecollision ofeigenvalues canbe expressedas

λj1(p)=λj2(p), (9)

wherethe correspondingwavenumbersare k1=^p+^j1 andk2= p+ ^j2. McLean et al. [20] divided the solutions of (9) into two classes. Depending upon whether j1− ^j2 is even or odd, insta- bilities belong to class I or class II, respectively. Without loss of generality,itisconvenienttoassumethat j₂= −^j1 forclassIand j2= −^j1−^{1 for class II. Herein, we focus only on class I with}

Table 1

Growthrateofthemostunstableperturba- tionasafunctionofthebasicwavesteep- nesswithoutvorticity.

ak p Re(λ)max

0.20 0.101 1.67×10⁻³ 0.19 0.005 0.811×¹⁰⁻⁴ 0.18 0.003 4.19×¹⁰⁻⁵

0.17 O(10⁻¹¹)

0.15 O(10⁻¹¹)

0.10 O(¹⁰⁻¹²)

0.05 O(¹⁰⁻¹²)

j₁=^{1 whichcorrespondto}instabilityofmodulationaltype.These assumptionsallowequation(9) toberewrittenas

2c₀=(1+^p)c_1+p+(1−^p)c_1−p, (10) with

2k₀=^k1+^k2, (11)

where k₀=^{1, k}1=¹+^{p and k}2=¹−^{p. The} subharmonic and superharmonic sidebands correspond to k2 and k1, respectively.

Equations(10) and(11) canbeinterpretedastheresonanceoftwo infinitesimalwaves(sidebands)withthebasicwave,i.e. aresonant four-waveinteraction.

3.3. Numericalresults

As a check on our numerical approach, we considered the caseofSanford et al. [10] correspondingtothe2π-periodicsolu- tion shownin their figure 3(b)withc0=⁰.8002.We found that the maximum rate of growth is 0.000356 and the frequency is 0.00751 corresponding to p= ±⁰.04056. These values obtained with=^{0 areveryclosetothoseofSanford}et al. [10].Notethat weusedtheirtransformation η→³η/4 inordertoensurethatour Whithamequationandsolutionwerethesameastheirs.Addition- ally,Sanford et al. [10] showed that thetraveling-wave solutions arestabletothemodulationalinstabilityiftheirwavesteepnessis lessthanapproximately0.142 whichcorresponds toak≈⁰.19 in ourscaling(seeTable1).

We carried out the stability computations for solutions with steepness ak=0.05 and ak=0.10 for a range of values. We found that they are both stablewith andwithout vorticity.Con- sequently, we decided to examine the stability oftraveling-wave solutionsofwavesteepnessak=⁰.20,avaluelargerthanthecrit- icalvalueinthe=^{0 case.}

Figs. 4 and 5 contain plots of Im(λ) andRe(λ) versus p, the wavenumberofthenormalmodeperturbationforthreevaluesof thevorticityandak=⁰.20.Theupperplotsinthesefiguresshow thecollisionsoftwopurelyimaginaryinthevicinityoftheorigin.

These collisions give rise to instabilities corresponding to inter- valsofinstabilityshowninthelowerplots.Theseplotsshowthat fora fixed value of the wave steepness both the rate of growth andthe sizeoftheintervalofinstabilityincrease asdecreases.

Theseplotsalsoshowthattherearenoinstabilitieswiththesame wavenumberastheunperturbedsolution(i.e. p=^{0)foranyofthe} valuesofvorticityweexamined.

Fig.6showsthemagnitudesofthecoefficientsaj forthemost unstable perturbation corresponding to =⁰.10 and ak=⁰.20.

Thetwo dominantcomponents, j= −^{1 and j}=^{1,correspondto} subharmonicandsuperharmonicsidebandstypicalofmodulational instability.Notethatthewavenumbersofthesubharmonicandsu- perharmonic sidebands are 1−^{p and 1}+^p, respectively. Fig. 7 showstheamplitude spectrum ofthe unperturbedwave ofwave steepnessak=⁰.20 perturbedbyitsmostunstablenormalmode.

Fig. 4.Radianfrequency(top)andrate ofgrowth(bottom)ofthenormalmode perturbationagainstitswavenumberfor=^{0 andak}=⁰.20.

The physicalperturbationcorresponds totherealpartoftheper- turbation giveninequation (6). The amplitude ofthe modeshas been normalized so that the fundamental mode, k=^{1, hasunit} amplitude.Themagnitudeofthesuperharmonicmodeoftheper- turbation, |^a1|^{,is onetenth oftheamplitude ofthe}fundamental mode.

Fig. 8 contains plots of the maximumgrowth rate versus the Floquetparameterfortheperturbationforthreevaluesofthevor- ticity.Theseplotsshow thatthevorticityeffectistwofold:(i) the maximalgrowthrateincreasesasdecreasesexceptforverylong unstable perturbations, (ii) the bandwidthof unstable wavenum- bersincreases as decreases. Notethat thesefeatures were ob- served by Thomaset al. [21] and FranciusandKharif [22] within theframeworkofthenonlinearSchrödingerequationandthefully nonlinearEulerequations,respectively.

Hur andJohnson [11] foundthat small-amplitude,2π/k-peri- odictraveling-wavesolutionsoftheWhithamequationaremodu- lationally unstable ifkh>kh_crit≈¹.146.Lateron,Hur andJohn- son [14] foundaformulafortheboundaryinthe(,k)-planethat separatesstableandunstablesmall-amplitude,periodic,traveling- wavesolutionstothevor-Whithamequation.Aplotofthisbound- ary is included in Fig. 9. We numerically corroborated this an- alytic result for a variety of and k values. For example, we considered solutions justbelow andjust above the criticalvalue of (,k)≈(5,0.96). We found that a small-amplitude (wave height of 4.3∗^{10−3) solutionto the}vor-Whitham equation with (,k)=(5,0.9)isspectrallystable,seeFig.10(a)andthatasmall- amplitude(waveheightof3.3∗^{10−3)solutionwith}(,k)=(5,1) is unstable,see Fig. 10(b).Insummary, inthe absence ofvortic- ity we found a critical value inagreement with that ofHur and Johnson [11] and Sanfordet al. [10] andinthepresenceofvortic- ity,wefound criticalvaluesinagreementwiththefindingofHur andJohnson [14]. Themodulationalinstability oflarge-amplitude solutionsinthepresenceofvorticityisanewfinding.

4. Conclusion

The modulationalinstability oftraveling-wavesolutions ofthe Whitham equation in the presence ofvorticity hasbeen investi- gated numerically.The Whitham equation is an extension of the KdVequationwhichtakesintoaccountthefullrangeofdispersion.

We presented a sampling ofour results which show the impor- tant qualitative results related to the modulational instability of

Fig. 5.Radian frequency (top) and rate of growth (bottom) of the normal mode perturbation against its wavenumber forak=0.20 and (a)= −0.1 (b)=0.1.

Fig. 6.Magnitudeofthecoefficientaj ofthemostunstablenormalmodecorre- spondingto=⁰.1 andak=⁰.20.

Fig. 7.Amplitudespectrumoftheunperturbedwaveofwavesteepnessak=⁰.20 perturbedbyitsmostunstablenormalmodefor=⁰.1.Theamplitudesofthe Fouriercomponentshavebeennormalizedbytheamplitudeofthefundamental modek=^{1 ofthe}unperturbedwave.

Fig. 8.Maximumgrowthrateagainstthewavenumberfor=^{0 (solidline),}=

−⁰.1 (dashedline),=⁰.1 (dot-dashedline)andak=⁰.20.

Fig. 9.Stabilitydiagraminthe(,k)-planeforsmall-amplitude,periodic,traveling- wavesolutionstothevor-Whithamequation.