Propagation regimes of interfacial solitary waves in a three-layer fluid

16 Pages·2015·4.03 MB·English

Nonlin.ProcessesGeophys.,22,117–132,2015
www.nonlin-processes-geophys.net/22/117/2015/
doi:10.5194/npg-22-117-2015
©Author(s)2015.CCAttribution3.0License.
Propagation regimes of interfacial solitary waves in
a three-layer fluid
O.E.Kurkina1,A.A.Kurkin1,E.A.Rouvinskaya1,andT.Soomere2,3
1NizhnyNovgorodStateTechnicalUniversityn.a.R.Alekseev,NizhnyNovgorod,Russia
2InstituteofCybernetics,TallinnUniversityofTechnology,Akadeemiatee21,12618Tallinn,Estonia
3EstonianAcademyofSciences,Kohtu6,10130Tallinn,Estonia
Correspondenceto:A.A.Kurkin([email protected])
Received:5December2014–PublishedinNonlin.ProcessesGeophys.Discuss.:6January2015
Revised:22January2015–Accepted:23January2015–Published:4March2015
Abstract. Long weakly nonlinear finite-amplitude internal 1 Introduction
waves in a fluid consisting of three inviscid layers of arbi-
trary thickness and constant densities (stable configuration,
Boussinesq approximation) bounded by a horizontal rigid There are many important topics related to internal waves
bottom from below and by a rigid lid at the surface are de- (IWs)intheocean.LargeIWsarehighlysignificantforsed-
scribeduptothesecondorderofperturbationtheoryinsmall iment resuspension and transport (Bogucki and Redekopp,
parametersofnonlinearityanddispersion.First,apairofal- 1999;StastnaandLamb,2008;Reederetal.,2011)andfor
ternatives of appropriate KdV-type equations with the coef- the biology on the continental shelf (Sandstrom and Elliott,
ficientsdependingontheparametersofthefluid(layerposi- 1984). The currents forced by large or breaking IWs cause
tions and thickness, density jumps) are derived for the dis- powerfulforcesonmarineplatformsandsubmersibles.The
placements of both modes of internal waves and for each associated strong distortion of the density field has a severe
interfacebetweenthelayers.Theseequationsareintegrable impact on acoustic signaling (Apel et al., 2007; Chin-Bing
for a very limited set of coefficients and do not allow for et al., 2009; Warn-Varnas et al., 2009; Sridevi et al., 2011).
properdescriptionofseveralnear-criticalcaseswhencertain Their capacity to break and impact the local microstructure
coefficientsvanish.Amorespecificequationallowingfora has major consequences for the understanding of interior
variety of solitonic solutions and capable of resolving most oceanmixing(MullerandBriscoe,2000).Internalwavesare
near-critical situations is derived by means of the introduc- believedtoberesponsibleforsubstantialdamage(Osborne,
tionofanothersmallparameterthatdescribestheproperties 2010).HighwatervelocitiesinintenseIWscancreateenor-
of the medium and rescaling of the ratio of small parame- mous local loads and bending moments and represent a po-
ters.Thisprocedureleadstoapairofimplicitlyinterrelated tential danger to off-shore structures, such as oil platforms,
alternatives of Gardner equations (KdV-type equations with drillrigs,etc.ThedangerfromIWsisconsideredsocritical
combined nonlinearity) for the two interfaces. We present a that, similar to the systems of tsunami warning, the poten-
detailedanalysisoftherelationshipsforthesolutionsforthe tialforautomateddetectionsystemsforlarge-amplitudeIWs
disturbancesatbothinterfacesandvariousregimesoftheap- (internal soliton early warning systems) is being discussed
pearance and propagation properties of soliton solutions to now.Suchsystemswereeventestedtosupportdrillingcam-
these equations depending on the combinations of the pa- paignsandguaranteethesafetyofdrillingplatforms(Stober
rametersofthefluid.Itisshownthatboththequadraticand andMoum,2011).
thecubicnonlineartermsvanishforseveralrealisticconfig- Most of the studies of IWs focus on large-amplitude lo-
urationsofsuchafluid. calized IWs, which propagate in relatively shallow areas of
oceans, shelf regions or semi-sheltered seas. A fascinating
feature of many waves of this kind is that they propagate
for a long time without any significant change in their en-
PublishedbyCopernicusPublicationsonbehalfoftheEuropeanGeosciencesUnion&theAmericanGeophysicalUnion.
118 O.E.Kurkinaetal.:Propagationregimesofinterfacialsolitarywaves
ergyorshapeassolitonsdo(Jeans,1995).Theycanbeinter- propertiesinthree-layerfluidsandflows.Theyare,however,
preted and adequately described as internal solitary waves oftenverycomplex,andtheresultsarenoteasytovisualize.
(ISWs) and are often referred to as internal solitons (Os- The drawback is that the analytical progress is limited and
trovsky and Stepanyants, 1989; Grimshaw, 2001). Observa- numerical computations should be involved to obtain final
tionsoflarge-amplitudeISWsarepresentedinmanyoriginal localizedsolutionsandtoanalyzetheirproperties.
research papers and reviews (Holloway et al., 2001; Jack- Althoughcontemporarynumericalmethodsandfullynon-
son, 2004; Ostrovsky and Stepanyants, 2005; Sabinin and linear approaches such as the above-discussed method of
Serebryany, 2005; Vlasenko et al., 2005; Apel et al., 2007; conjugate flows allow for extensive studies into properties
Grimshawetal.,2007). of highly nonlinear IWs, many specific features can still be
Theconceptoflayeredfluid,widelyusedforboththeoret- recognized,analyzedandunderstoodusingclassicalmethods
icalresearchandapplications,isconvenientforthedescrip- foranalyticalstudiesintoIWsintheweaklynonlinearframe-
tion of IWs because wave motion in such environments is work. Such fully analytical methods make it possible to es-
describedbyequationscontainingasmallnumberofparam- tablish exactly the appearance of disturbances of different
eters and often allowing for analytical studies of the prop- shapesandamplitudes,and,moreimportantly,tounderstand
erties of solutions. These simple models are able to mirror the specific features of the behavior of waves correspond-
thebasicpropertiesoftheactualinternalwavesystems.The ing to the situation where a substantial change in the over-
efficiencyofthetwo-layermodel–thesimplestsystembasi- all regime of wave propagation is possible (Kurkina et al.,
callyrepresentingthekeypropertiesofIWs–hasbeenestab- 2011a).
lishedin numerousanalyticalandnumericalstudies aswell Weaklynonlinearanalyticalmodelsofdifferentlevelsfor
as by means of in situ observations and laboratory experi- IWsinathree-layerfluidhavebeendevelopedforsomespe-
ments(Funakoshi,1985;FunakoshiandOikawa,1986;Mirie cificsituations.Thepropertiesofmode-1IWsinasymmet-
and Pennell, 1989; Choi and Camassa, 1996, 1999; Pullin ricthree-layerfluid(whentheundisturbedstateissymmetric
andGrimshaw,1998;Craigetal.,2004;Guyenne,2006;Za- about the mid-depth) were analyzed up to the second order
hiboetal.,2007;Camassaetal.,2010). innonlinearitybyGrimshawetal.(1997).Anextensionin-
Thetwo-layermodelonlyconditionallyrepresentsthever- volving up to the fourth-order nonlinear terms was derived
ticalstructureofseasandoceans.Itsdirectextension–three- in Kurkina et al. (2011b). The model of Yang et al. (2010)
layerstratification – has proved to be a proper approxima- involved both modes of IWs in a general three-layer ocean,
tion of the sea water density profile in some basins in the the first order in nonlinearity and dispersion (Korteweg–de
world ocean with specific hydrological conditions (Knauss, Vries (KdV) approximation; only mode-2 waves were ana-
1996;LeppärantaandMyrberg,2009;Kurkinaetal.,2011a). lyzed).Theweaklynonlineartheoryforacontinuouslystrat-
Three-layer models are more complex than two-layer sys- ifiedfluidcanbeusedtoderivetheparametersofevolution
tems,becauseoftheincreasednumberofset-upparameters, equations for IWs in a three-layer fluid (Grimshaw et al.,
buttheyrepresentnewdynamicaleffectsandallowformuch 1997; Yang et al., 2010). These parameters for continuous
moreanalyticalprogresscomparedtothearbitrarystratified stratificationaregivenintheformofintegralexpressionsin-
medium.Anotheradvantageofthree-layermodelsincontrast cludingtheverticalmode(whoseshapeisdeterminedbythe
totwo-layeronesistheabilitytodescribetwomodesofIWs, density stratification), its higher-order corrections and their
theirpropertiesandpossibleinteractions. derivatives (Lamb and Yan, 1996; Pelinovskii et al., 2000;
Nonlinear theory of IWs in a three-layer fluid with con- Grimshawetal.,2002).Whenthistechniqueisadaptedtoa
stant density in each layer is presented in the literature layeredmodel(withapiecewisecontinuousdensityprofile),
in the framework of different approaches. Simplified equa- therelevantintegrandscontainverycomplexexpressions(es-
tions for three-layer conjugate flows are derived in Lamb peciallyforhigher-ordercorrectionsandcoefficients)involv-
(2000) and Rusås and Grue (2002), and some analytic re- ing generalized functions, and analytical progress in their
sultsareobtainedtounderstandconjugateflowsand,incon- analysis is not always possible. Therefore, it is preferable
sequence, large-amplitude, flat-centered (table-like) solitary initially to use equations for layered models and to apply
internalwavesofbothmodes.Comparisonwithsolutionsob- asymptotic expansions to them directly. This approach has
tained for continuous stratifications is also given in Lamb beenused,forinstance,inKurkinaetal.(2011b)andKoop
(2000).Afullynonlinearnumericalmethodforthecalcula- andButler(1981).
tionofsolitarywavesinathree-layerfluid(RusåsandGrue, A specific feature of weakly nonlinear equations for a
2002)allowsfortheinvestigationofbothmode-1andmode- three-layer fluid is that some coefficients at the nonlinear
2 waves, including broad flat-centered waves and extreme termsmayvanishforcertainmodesandcertainspecificden-
(overhanging)waves.Thesimilarityofmode-1wavestothe sitystratifications(KakutaniandYamasaki,1978;Grimshaw
interfacial waves in a two-layer fluid is the probable reason et al., 1997; Yang et al., 2010). This feature is common for
why the analysis is mainly performed for situations when several wave classes in stratified environments (Soomere,
oneofthelayersisrelativelythin.Thesemodelshavegreatly 2003). Some coefficients at the nonlinear terms may vanish
contributed to the understanding of nonlinear internal wave eveninatwo-layerfluidwiththesurfacetensionbetweenthe
Nonlin.ProcessesGeophys.,22,117–132,2015 www.nonlin-processes-geophys.net/22/117/2015/
O.E.Kurkinaetal.:Propagationregimesofinterfacialsolitarywaves 119
Figure1.Definitionsketchofthethree-layerfluidandinternalwavesofmode-1(left)andmode-2(right).
layers(Giniyatullinetal.,2014;Kurkinaetal.,2015).Insuch density of the middle layer is ρ =ρ and the densities in
2
cases, it is often necessary to account for higher-order non- the lowermost and uppermost layers are ρ =ρ+(cid:49)ρ and
1 1
lineartermstodescribethemotionadequately.Inthecaseof ρ =ρ−(cid:49)ρ , respectively. We employ the Boussinesq ap-
3 2
IWsinathree-layerfluid,itisnecessarytoproduceasecond- proximationandassumethatdensitiesinthelayersdifferin-
orderweaklynonlineartheorytoresolvesomeofsuchsitua- significantly((cid:49)ρ /ρ(cid:28)1).Inthiscase,theequationsofthe
1,2
tions.Theattemptsinthisdirectionhavesofarbeenlimited motionareLaplaceequationsforthevelocitypotential(cid:56) in
i
tothespecificcaseofsymmetricstratification(Kurkinaetal., eachlayer:
2011b).
WedevelophereananalyticmodelforlongIWsoffinite
amplitudeforthethree-layerfluidwithanarbitrarycombina- ∇2(cid:56)i =0, i=1,2,3. (1)
tionofthelayers’thicknesses.Suchaverticalstructuremuch
better matches the properties of real seas and oceans. The The kinematic boundary conditions at the bottom and at
relevant nonlinear evolutionary equations are obtained first thefixedupperboundaryreducetotheconditionthatvertical
withtheuseofanasymptoticprocedurefromthegoverning velocitiesattheseboundariesvanish:
equationsforthreehomogeneousfluidlayers.Thederivation
of these equations relies on the second order of the pertur-
bation theory for both interfaces and for the waves of each (cid:56)1z=0atz=0, (cid:56)3z=0atz=H3. (2)
mode.Thedevelopedframeworkmakesitpossibletoinves-
Here and further on, the subscripts x, y, z or t denote a
tigate the properties of both modes of long internal gravity
partialderivativealongtherespectivecoordinate(x,y,z)or
waves in a three-layer fluid. The key development is that,
withrespecttotimet.Theclassicalkinematicanddynamic
foreachequation,thecoefficientsatthenonlineartermsand
boundaryconditionsattheinterfacesbetweenthelayersare
terms expressing linear and nonlinear dispersion of waves
asfollows(Kurkinaetal.,2011b):
areexpressedexplicitlyviaparametersofthefluidconfigura-
tion.Thismakesitpossibletoanalyzecomprehensivelythe 
η +(cid:56) η −(cid:56) =0
boStiesefphrhmet.hacFsveiaoioolerfqrastuthautaenectndhilotoinssowiisntgeuniisasssttidopooirafnsdicsdeau,rltslaooscefptodhtaeherafetfisincipcdtueuieldraantetruittosrerb.neramssTtcihwiaonelnheinedntnghefecotioehsrsrepysenaaecrocryafhnonloricvmnradaeseneaedrr-. ηρ=tt1ρ+(cid:16)2(cid:56)(cid:56)(cid:16)1(cid:56)12txx+2ηtxx+12−(∇21(cid:56)((cid:56)∇121zz(cid:56))=22+)02g+ηg(cid:17)η(cid:17)z=H1+η(x,t), (3)
inordertobalancethenonlinearanddispersiontermsinthe
equations. 
ζ +(cid:56) ζ −(cid:56) =0
2As uGsouvaelrinninthgeeaqnuaalytisoisnsof long internal waves, let us con- ζρ=tt2+ρ(cid:16)3(cid:56)(cid:56)(cid:16)2(cid:56)23txx+3ζtxx+12−(∇21(cid:56)((cid:56)∇23zz2(cid:56))=23+)02g+ζg(cid:17)ζ(cid:17)z=H2+ζ(x,t). (4)
sideramodelsituationofirrotationalmotionsinathree-layer
inviscid fluid with undisturbed interface positions z=H Here, the unknown functions η(x,t) and ζ(x,t) denote the
1,2
(H ≥H )andtotalthicknessH (H ≥H )overlyingaflat instantaneous position of the interface between the bottom
2 1 3 3 2
horizontalbottomlocatedatz=0intheapproximationofa andmiddlelayersandbetweentheupperandmiddlelayers,
rigid lid on the surface of the fluid (Fig. 1). The reference respectively.
www.nonlin-processes-geophys.net/22/117/2015/ Nonlin.ProcessesGeophys.,22,117–132,2015
120 O.E.Kurkinaetal.:Propagationregimesofinterfacialsolitarywaves
3 Asymptoticprocedure

ζ +ε(cid:56) ζ −(cid:56) =0,
Tnoonhmetheaenpcophneaairnraacantcreeeraisoltifiscttihsceceanplevrosirpooefnrmtheeevnoptlrumotciaoeynssseeusqbustotaatbnioetinahliflgoyhrdlaiegpphehtneedd- ζρtt+(cid:16)(cid:56)ε(cid:56)23+xxζ1xxε−(cid:0)(cid:56)(cid:56)23zz(cid:1)=2+0,1(cid:0)(cid:56) (cid:1)2
2 2t 2 2x 2 2z z=H +ζ(x,t). (10)
otworadavneeramilvyoezteiadon.nseSqiimnuaiatliaolranlyyeftoroerdKsmfluurakildiln.(aTbuehttisafilgn.oi(ta2el0)n1aa1mtbup)r,alioltluuydriegmolpoalnliegiss- ++g21ζ(cid:0)(cid:56))=3z(cid:1)ρ23+(cid:16)(cid:56)gζ3t(cid:17)+21ε(cid:0)(cid:56)3x(cid:1)2  2
twosmallparameters.Firstly,thetypicalhorizontalscaleof
wavemotionL(equivalently,thetypicalwavelengthforlin- The asymptotic analysis of the resulting equations is
ear or nonlinear wave trains) considerably exceeds the fluid straightforward. As the procedure is fairly cumbersome,
depthH.ThisassertionL(cid:29)H notonlyensuresthatasmall largely follows a well-known approach (Koop and Butler,
parameterH/Lispresentinthesystem,butalsomatchesthe 1981) and provides almost no instructive aspects, we omit
usual properties of environments supporting the IW motion itsdetailsandpresentonlyitsbasicsteps.First,allunknown
in natural water bodies. It also fits the basic assumptions of functions are expanded into Taylor series in the vicinity of
commonderivationsoftheKdVequationanditsgeneraliza- oneoftheinterfaces:
tions. Another small parameter naturally emerges from the
treersftariccetsio’npothsaittiothnesafmropmlittuhdeeirauonfdtihsteurdbisetdurlboacnactieosntoisthsemianl-l f(x,z=H1+η(x,t),t)=Xj∞=0ηjj! ∂∂jzfj (cid:12)(cid:12)(cid:12)(cid:12)z=H1,
cpoamrampaerteedrotofnthoenlflinueidarditeypεth=. Tah/iHs a(cid:28)sse1rt.ion gives rise to the f(x,z=H2+ζ(x,t),t)=X∞ ζjj! ∂∂jzfj (cid:12)(cid:12)(cid:12)(cid:12) .
Our specific interest is to describe long-living solitary j=0 z=H2
waves, for which the dispersion in some sense balances the
All constituents of the resulting series are then expanded
impactofnonlinearity.Aswillbeshownbelow,aconvenient
into power series with respect to powers of the parameter
parameter that characterizes the role of dispersion in wave
propagation is µ=H2/L2. As the existence of long-living ε(cid:28)1:
nonlinearwavemotionsandsolitarywavesusuallypresumes
a specific balance between the terms representing nonlinear (cid:16) (cid:17)
η=ε η +εη +ε2η +... , (11)
anddispersiveeffects,weassumethatε∼µ0. 0 1 2
(cid:16) (cid:17)
These assumptions and introduced parameters make it ζ =ε ζ +εζ +ε2ζ +... .
0 1 2
possibletoperformtheanalysisoftheproblemEqs.(1)–(4)
√
innondimensionalform.Thestandardprocedureofintroduc-
It is easy to show that all the potentials (cid:56) ∼ ε; conse-
i
ingnondimensionalcoordinatesintoEqs.(1)–(4)underthese
quently,thepowerseriesforthepotentialsareasfollows:
assumptionsleadstothefollowingboundaryproblem:
√ (cid:16) (cid:17)
(cid:56) = ε ϕ +εϕ +ε2ϕ +... , i=1,2,3, (12)
(cid:56) +ε(cid:56) =0, 0≤z≤H , (5) i i,0 i,1 i,2
1zz 1xx 1
(cid:56) +ε(cid:56) =0, H ≤z≤H , (6)
2zz 2xx 1 2 Finally,weemploythetechniqueofmultipletemporaland
(cid:56)3zz+ε(cid:56)3xx =0, H2≤z≤H3, (7) spatial scales (Engelbrecht et al., 1988; Nayfeh, 2000). The
(cid:56) =0atz=0, (cid:56) =0atz=H (8) “slow” time and “stretched” reference frame coordinate are
1z 3z 3
introducedasfollows:

ηηρtt++(cid:16)(cid:56)εε(cid:56)(cid:56)12+xxηη1xxε−−(cid:0)(cid:56)(cid:56)(cid:56)12zz(cid:1)2==+001,,(cid:0)(cid:56) (cid:1)2 ξ =Heεr1e/,2(cxd−encott)e,sthτe=(asε3y/e2tt.unknown)phasespeedof(f1re3e)
++1g12η(cid:0)(cid:56))1=2tz(cid:1)ρ222+(cid:16)(cid:56)gη2t(cid:17)1+x 12ε(cid:0)(cid:56)2 2x(cid:1)12z z=H1+η(x,t), (9) lfionleloawr iinngterrenpallawceamveesn.tTofhitshesuobpsetriatutotirosnofispaerqtuiaivladleernivtattoivtehse:
∂ ∂ ∂ ∂ ∂
=ε1/2 , =−ε1/2c +ε3/2 . (14)
∂x ∂ξ ∂t ∂ξ ∂τ
Substitutionofexpansions(Eqs.11–12)anddefinitionsof
modifiedcoordinates(Eqs.13–14)intoEqs.(5)–(10)leadsto
aninfinitesystemofequationswithrespecttotheelementsof
expansions (Eqs. 11–12). Equations of the resulting system
Nonlin.ProcessesGeophys.,22,117–132,2015 www.nonlin-processes-geophys.net/22/117/2015/
O.E.Kurkinaetal.:Propagationregimesofinterfacialsolitarywaves 121
are scaled by one of the multiples ε1/2 or ε3/2, where they (Fig.1).Thecorrespondingexpressionsforthephasespeeds
arise. The obtained system can be solved recursively until are
any desired order. The system of equations obtained in the
leading(lowest)order(∼ε0)fortheelementsofexpansions c±2= 1(cid:49)ρgH (l (1−l )+rl (1−l )±d) (24)
3 1 1 2 2
(Eqs.11–13)hastheform(Ruvinskayaetal.,2010) 2 ρ
1(cid:49)ρ 1
ϕi,0zz =0, i=1,2,3, (15) = 2 ρ gH3q±,
ϕ =0atz=0, ϕ =0atz=H , (16)
1,0z 3,0z 3
where
q
ϕ1,0z =0, ϕ2,0z =0 (cid:27)z=H , (17) d= r2l22(1−l2)2−2rl1(1−l2)(l2−l1(2−l2))+l12(1−l1)2
−(ρ+(cid:49)ρ)cϕ +(cid:49)ρgη =−ρcϕ 1
1,0ξ 0 2,0ξ
andl =H /H .Thesymmetricmodeofinternalwaves
1,2 1,2 3
hasthelargerphasespeedc+,correspondingtothe“+”sign
ϕ =0, ϕ =0 (cid:27) befored inEq.(24).Forthisreason,wecallitfastmode,or
−2ρ,0czϕ +r(cid:49)3,ρ0zgζ =−(ρ−r(cid:49)ρ)cϕ z=H2, (18) mode-1.Theantisymmetricmode,whosephasespeedisc−,
2,0ξ 0 3,0ξ
iscalledslowmode,ormode-2.Belowweshallconsiderthe
where (cid:49)ρ =(cid:49)ρ, (cid:49)ρ =r(cid:49)ρ. It follows from Eqs. (15)– nonlinearmotionsofbothmodes.
1 2
(18) that the functions ϕ ,i=1,2,3 do not depend on the The first-order and second-order corrections to the linear
i,0
coordinatez. solutionsatisfythefollowingequations:
Thesystemofequationsobtainedinthefirstorder(∼ε1)
is ε2: η± +α±η±η± +β±η± =0, (25)
1τ 1 1ξ 1ξξξ
ϕi,1zz+ϕi,0ξξ =0, i=1,2,3, (19) ζ1±τ +eα±ζ1±ζ1±ξ +βe±ζ1±ξξξ =0,
ϕ (z=0)=0, ϕ (z=H )=0,
1,1z 3,1z 3
ε3: η± +α±(cid:0)η±η±(cid:1) +β±η± +α±η±2η± (26)
2τ 1 2 ξ 2ξξξ 1 1 1ξ
(c=ρη0ρ+ξ(cid:16)+(cid:49)ϕ2ρϕ,)01(cid:16)τ,1ϕ−z1=,c0ϕτ02−,,1ξcϕc+η1,0121ξξϕ+2+2,0ϕ12ξ2(cid:17)ϕ,11z,0=ξ(cid:17)0+(cid:49)ρgη1z=H1, (20) ζ2±τ +++eαγ±±β(cid:0)ζ1±ζ±η1±ζ1±ζ,±52±ξ(cid:1)+ξ++γγ1±βe±η±ζ1±ζ±η2±ζ1ξ±ξξ±ξξξ++=eαγ01±2.±ζη1±1±2ξζη1±1±ξξξ+=βe1±0,ζ1±,5ξ
e1 1 1ξξξ e2 1ξ 1ξξ

cρζ(cid:16)0ϕξ2+,0ϕτ2−,1czϕ=2,01ξ,+c21ζϕ022ξ,0+ξ(cid:17)ϕ+3,1(cid:49)z =ρg0ζ1 z=H2. (21) theTAheppceonedffiixc.ieTnhtes fiorfstE-oqrsd.e(r2e5q)uaantidon(s26(E)qa.re25p)reasree,ntaesdexin-
=(ρ−r(cid:49)ρ)(cid:16)ϕ3,0τ −cϕ3,1ξ +12ϕ32,0ξ(cid:17) pmeocttieodn,stohfebwotehll-thkenoiwntnerfKacdeVs.eTqhueatsieocnosndth-oatrddeersecqriubaetiothnes
± ±
(Eq.26)arelinearequationswithrespecttoη andζ .Note
From Eqs. (19)–(21), we obtain the following relation- thatequationsforη±andζ±haveasimilarstr2ucturef2oreach
n n
ships:
ordernanddifferfromeachotheronlybyvaluesoftheirco-
efficients(notealsothatthecoefficientsintheequationsfor
ϕ1,1z =−ϕ1,0ξξz, ϕ3,1z =ϕ3,0ξξ(H3−z), (22) ζ± have tildes). Furthermore, we omit indices “±” where
n
possible, having in mind different expressions for different
modes.
ϕ H =cη , ϕ (H −H )=c(η −ζ ) , (23)
1,0ξξ 1 0ξ 2,0ξξ 1 2 0 0 ξ
ϕ (H −H )=cζ .
3,0ξξ 2 3 0ξ
Equations (22) and (23) together with Eqs. (17) and (18) 4 Equationsforinterfaces
allowonetodefinethephasespeedcofthelinearIWs.The
relevant equation is bi-quadratic and, not unexpectedly, re- Ineachorder,wealsoobtainrelationshipsbetweenthecor-
vealsthattwowavemodesexistinthissystem.Oneofthem respondingtermsinseries(Eqs.11–12).Theserelationships
results in synchronous in-phase movements of both the in- forthefirst-orderandsecond-ordertermsare
terfaces. The other mode is antisymmetric: the motions of
the interfaces are synchronous but have opposite directions ζ =sη , ζ =sη +s η2+s η x, (27)
1 1 2 2 quad 1 disp 1ξξ
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122 O.E.Kurkinaetal.:Propagationregimesofinterfacialsolitarywaves
where complex and given in the Appendix. As the coefficients β ,
1
s=(l2−2ql1(l2−l1))/l1, pβer1o,vγi1d,eeγt1h,eiγr2e,xeγp2liacriet enxoptraenssailoynzsedheinre.thNisotpeapthear,twβ±e d≡oβen±ot,
squad= 2q(l2−l1)(cid:0)(cid:0)α∗+3q(cid:1)l1−3(cid:1), β1±≡βe1±astheyarejustcoefficientsofk3andk5intheTay-
l1H3 lor expansion at small k (long-wave limit) of the dispersion
α∗=αH /c, relation ω(k) for linear IWs in a three-layer fluid. This dis-
3
persion relation is the same for both interfaces for a fixed
(l −l )H2(cid:16) (cid:16) (cid:17) (cid:17)
s = 2 1 3 2ql (l −l )2+12β∗ −l (l +l ) , mode.
disp 1 2 1 2 1 2
6l
1 The coefficients of the equation for the lower interface
(cid:16) (cid:17)
β∗=β/ cH32 . mode-1displacementη+inthelimitsH1→H2orH2→H3
are reduced into known expressions for internal waves in a
Note that s±=±1 for a symmetric stratification (H1= two-layerfluidwithdensityjumps(cid:49)ρ1+(cid:49)ρ2 and(cid:49)ρ1,re-
H3−H2, (cid:49)ρ1=(cid:49)ρ2, or r =1) analyzed in Kurkina et al. spectively. For the mode-1 displacement of the upper inter-
(2011b)formode-1waves. face ζ+, the vanishingof the lowermost or middlelayer (in
Substitutingζ1,ζ2into(Eq.11),weobtain the limiting process H1→0 or H1→H2) leads to the cor-
(cid:16) (cid:17) responding formulations for parameters of IWs in two-fluid
ζ =ε(ζ +εζ )+O ε3
1 2 systemswithdensityjumps(cid:49)ρ or(cid:49)ρ +(cid:49)ρ ,respectively.
2 1 2
(cid:16) (cid:17) (cid:16) (cid:17) Thesimilarityofthestructuresoftheequationsfordistur-
=εs(η +εη )+ε2 s η2+s η +O ε3 (28)
1 2 quad 1 disp 1ξξ bances of mode-1 and mode-2 considerably simplifies fur-
(cid:16) (cid:17)
=sη+s η2+s η +O ε3 . ther analysis; namely, it is sufficient for solving only one
quad disp xx
of those equations (say, η) and then for using the above-
CombiningEqs.(25)and(26)leadstothefollowinggen- derivedrelationshipstoobtainanapproximatesolutiontothe
eralizationsoftheKdVequationfortheinterfaces(presented otherequation.Therelationship(Eq.28)connectinginterfa-
hereintheoriginalcoordinates): cialdisplacementsshouldbeusedwhentheinitialconditions
forEq.(29)arebeingsetup.Asimilarexpressionexistsfor
η +cη +αηη +βη +α η2η +β η (29)
t x x xxx 1 x 1 5x η(ζ).Thisinterdependenceis,however,effectiveonlyduring
+γ1ηηxxx+γ2ηxηxx =0, limited time intervals. The relevant nonlinear and solitary-
ζt+cζx+eαζζx+βeζxxx+eα1ζ2ζx+βe1ζ5x (30) wave solutions have different wave speeds (see Eq. 24) and
therefore will split over longer time intervals (considerably
+γ ζζ +γ ζ ζ =0.
e1 xxx e2 x xx largerthan1/ε).Thisprocessofseparationofinitiallybasi-
Although Eqs. (29) and (30) can be derived separately, cally equivalent solutions for different modes implicitly ex-
they are not independent. Every such derivation results in a pressesthelimitationsoftheapplicabilityoftheusedasymp-
link to the solutions of the other equation. This link is pre- toticprocedure.
sented explicitly in Eq. (28) for the equation for the lower
interfaceη(x,t).Itallowsforadirectcalculationofthedis-
turbancesfortheupperinterfaceζ(x,t)withanaccuracyof 5 Gardnerequationsforinterfacialdisplacements
O(cid:0)ε3(cid:1).Asimilarlink(notshown)existsforthedetermina-
tionofthedisturbancesofthelowerinterfaceusingthesolu- Equations(29)and(30)areintegrableusingtheinversescat-
tionfortheupperinterface.Thesecouplingsdonottransform teringmethodonlyfortwospecificsetsofnontrivialvalues
anequationforoneinterfacetoanequationfortheotherin- of their coefficients (Newell, 1985; Zwillinger, 1997; Weis-
terface:e.g.,substitutingEq.(28)intoEq.(30)doesnotyield stein,2002a,b).Thesesets,however,arenotapplicabletothe
Eq.(29),butthedifferenceincludestermswhicharehigher problem of internal wave motion in stratified environments,
orderthanthoseretainedinEq.(29).Theequationforonein- forwhichthisequationapparentlyremainsnonintegrable.In
terfaceandtheequationobtainedbysubstitutingtheconnec- thiscase,stationary(inaproperlychosenmovingcoordinate
tionbetweenthetwodisplacementsintotherelationshipfor system) solitary waves normally do not interact elastically.
theotherinterfaceareonlyequivalentasymptoticallyasthe Theymaybegeneratedfromasuitablychosenwavesystem
small parameters go to zero. Similar interrelations become and they often radiate their energy during their motion and
evidentforasimilarpairofGardnerequationsbelow. interact with other components of the wave field in a com-
ThecoefficientsofEqs.(29)and(30)accountforthemain plicatedmanner(MarchantandSmyth,1996;Osborneetal.,
properties of the environment supporting the internal wave 1998;Osborne,2010).
motionsuchasthelocationoftheundisturbedinterfaces,the It was recognized already in the first studies of internal
magnitude of the density jumps between the layers and the waves using the KdV equation several decades ago that the
total fluid depth. The main development here is the deriva- coefficientatthe(firstorderintermsoftheasymptoticpro-
tionofexplicitexpressionsforthecoefficientsoftheseequa- cedure,orquadraticintheappearance)nonlinearterminthis
tions.Theexpressionsforα±,eα±,β±,βe±,α1±,eα1±arequite equation may vanish or change its sign (Kakutani and Ya-
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O.E.Kurkinaetal.:Propagationregimesofinterfacialsolitarywaves 123
masaki,1978;Miles,1979;GearandGrimshaw,1983).The
signofthedispersiveterm,however,isalwaysthesame.Im-
portantly,thesignofthenonlineartermgovernsthepolarity
of solitary waves and soliton solutions to the relevant equa-
tion. The event of vanishing of the relevant coefficient or a
change in its sign therefore leads to substantial rearrange-
mentofthewavepropagationinthevicinityofthevanishing
location.Anaturalconsequenceisthatanapproachingsoli-
tonof,say,elevationwillbetransformedintoasolitonofde-
pression, and vice versa (Knickerbocker and Newell, 1980;
Talipovaetal.,1997;Grimshawetal.,1999).
In order to understand how this process functions, it is
necessarytobuildahigher-orderequationthatisabletode-
scribethephenomenaintheareawherethecoefficientatthe
Figure 2. Shapes of soliton solutions to the Gardner equation for
quadraticnonlineartermvanishes.Moreover,suchequations different combinations of the signs of coefficients at its nonlinear
arecrucialforthedescriptionofthewavemotionindomains terms (idea of representation by R. Grimshaw, E. Pelinovsky and
wherethistermismuchsmallerthanothertermsandwhere T.Talipova).
higher-ordertermsgovernthewaveevolution.Thederivation
of such equations is possible through a detailed analysis of
theroleofhigher-ordercontributions(representedbycoeffi- latedGardnerequations:
cientsα ,α inEqs.29and30)ofthepresentedasymptotic
1 e1
procedureintoEqs.(29)and(30)incaseswhenthequadratic η +cη +αηη +α η2η +βη =0, (31)
t x x 1 x xxx
termsvanishoraresmallenoughtobeignored.
ζ +cζ +αζζ +α ζ2ζ +βζ =0.
TherelevantgeneralizationoftheKdVequationiscalled t x e x e1 x xxx
theGardnerequation.Thisequationcontainsacubicnonlin-
Formally,theyonlydifferfromEqs.(29)and(30)bythe
earterm,thecoefficientatwhichmayalsobesign-variable,
absence of terms β η , γ ηη and γ η η . The actual
dependingonthevariationsinthestratification(Talipovaet 1 5x 1 xxx 2 x xx
difference is, however, much deeper, as Eq. (31) expresses
al., 1999; Grimshaw et al., 2002, 2010; Kurkina, 2011a). If
thedynamicsonanotherscale,governedbytwoindependent
this term is negative (which is the case in a two-layer envi-
small parameters. This difference becomes to some extent
ronment),thesolitons’heightsarelimitedand,intheprocess
evident via another relationship between the soliton ampli-
ofreachingthelimitingheight,thesolitonwidensinfinitely
tude and its width. These equations govern the interfacial
intoatable-likedisturbance.Ifthecubictermhasapositive
motion of weakly nonlinear finite-amplitude unidirectional
coefficient, the Gardner equation possesses several types of
waves neglecting mode interactions. Their coefficients are
solitonswithdifferentpolarities.
called environmental parameters, because they account for
IfthequadraticandcubictermsofEqs.(29)and(30)have
the background conditions (configuration of the medium).
thesameorderofmagnitude,theabove-describedasymptotic
These coefficients are functions of the layers’ positions and
procedureisnotapplicable.Apossibilityistomakeexplicit
densityjumpsbetweenthelayers(ortheirratior).
useofthesmallnessofthecoefficientatthequadraticnonlin-
Similarly to Eqs. (29) and (30), Eq. (31) is interrelated
eartermandtoassumethatα∼δ,whereδ(cid:28)1.Asuitable
withintheaccuracyofO(ε3):
procedure may be constructed using asymptotic expansions
similar to Eqs. (10) and (11) but built for two independent
smallparametersµandε(see,e.g.,LambandYan,1996).It ζ =sη+squadη2. (32)
isnownecessarytoaccountforsecond-ordertermswithre-
specttobothparameters.Thiscanbedonebyre-expanding AninverseofEq.(32),η=η(ζ),canbederivedsimilarly
the above series in terms of α∼δ. A meaningful evolution inastraightforwardmanner.
equationcanthenbeobtainedusingtheconditionthatthree The analytical one-soliton solution of the Gardner equa-
terms(quadraticandcubicnonlinearity,andtheleadingterm tioniswellknown(HelfrichandMelville,2006;Pelinovsky
responsible for linear dispersion) are of the same order of etal.,2007):
magnitude.Thisispossibleifδ∼εandµ∼ε2.
A
Theprocedureitselfissimilartotheabove-describedtech- η(x,t)= . (33)
nique, and sorting out the terms with similar powers of ε 1+Bcosh(γ(x−Vt))
leadstothedesiredevolutionequations.Theseequations,at
thelowestorder,forthedisturbancesattheinterfacesofthe ThesolitonvelocityV =c+βγ2isexpressedthroughthe
three-layer environment are equivalent to a pair of interre- inversewidthofthesolitonγ.TheparametersAandB de-
pendonthecoefficientsofEq.(31)anddeterminethesoliton
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124 O.E.Kurkinaetal.:Propagationregimesofinterfacialsolitarywaves
amplitude a as the extreme value of the function η(x,t) in
Eq.(33):
A 6βγ2 6α βγ2
a= , A= , B2=1+ 1 . (34)
1+B α α2
Theparametersofthefamilyofsolutionscanalsobeex-
pressedthroughitsamplitudea:
2α+aα (cid:16) α (cid:17) α
γ2=a 1, A=a 2+a 1 , B=1+a 1. (35)
6β α α
There are different branches of the soliton solutions de-
pending on the signs of coefficients at the nonlinear terms;
see Fig. 2. The asymptotics of the solution (Eq. 33) is de-
scribedforinstanceinPelinovskyetal.(2007).
6 EnvironmentalparametersofIWs
Whilst the mathematical properties of the Gardner equation
anditssolutionsarequitewellknown,thesepropertiesmay
vary radically along the propagation of realistic IWs in nat-
ural environments (Nakoulima et al., 2004). Therefore, it is
important to analyze the behavior of the environmental pa-
rameters for possible ranges of parameters of the medium,
whichgovernthekinematicandnonlinearcharacteristicsof
the IW field. The above-described explicit expressions for
thecoefficientsofEqs.(29)and(30)(seeAppendix)remain
valid also for Eq. (31) and allow the production of the esti- Figure3.VariationinenvironmentalparametersforIWsoffast(left
matesofnonlinearIWshapes,limitingamplitudesandpro- column)andslow(rightcolumn)modesintheparticularcaser=1
jectionsofpossibletransformationsofsolitaryIWsinanin- ((cid:49)ρ1=(cid:49)ρ2).Zerocontoursaregivenbyboldblacklines.
homogeneousmedium(Kurkinaetal.,2011a).
Todescribethebehavioroftheenvironmentalparameters,
we use their graphs on the “phase” plane of scaled undis- Figure3showsthemapsofvaluesofparametersc±,β±,
turbed interface positions (l =H /H , l =H /H ) for the α±, α± as functions of l and l when r =(cid:49)ρ /(cid:49)ρ =1.
1 1 3 2 2 3 e e1 1 2 2 1
fixedvalueoftheratiorofdensityjumps.Alongthediagonal Theparametersc± andβ± arealwayspositiveforthethree-
l =l ,thetwointerfacescoincide,andthislinerepresentsa layer fluid. They tend to zero only when one of the layers
1 2
two-layer fluid. The other diagonal l =1−l corresponds vanishes for mode-2 waves (c−, β−) or when the thick-
1 2
to the symmetric geometric configuration of layers that has nesses of two out of the three layers simultaneously tend
beenaddressedindetailinKurkinaetal.(2011b).Asweal- tozeroformode-1waves(c+,β+).Theparametersc− and
ways require l ≥l , the triangle to the left of the diagonal β− both have one maximum for the symmetric configura-
2 1
l =l representsallpossiblerangesofl andl .Therefore, tionl =1−l =3/4.Theircounterpartsc+andβ+havethe
2 1 1 2 2 1
we use this triangle to analyze the environmental parame- maximaatthepointl =l =1/2,i.e.,whenthethree-layer
1 2
tersfortheupperinterface(coefficientsintheequationsfor fluiddegeneratesintoatwo-layerfluidwithequallayersand
ζ±). For a Boussinesq fluid, there is symmetry: switching withadensityjumpof2(cid:49)ρ.
the depths of the upper and lower layers and interchanging Aspecificfeatureofthecoefficientsatthequadraticnon-
the density jumps will result in exactly the same behavior linear term α±, α± is that they may vanish for some non-
e
withthewavesflippedover.Thecoefficientsfortheequation degenerate three-layer configurations. These parameters for
for η given in terms of l and l should be the same as the mode-1 IWs vanish along the diagonal l =1−l due to
1 2 2 1
equations for ζ given in terms of 1−l and 1−l and re- thesymmetryofthewavemotion.Theyalsohaveotherco-
2 1
placing r with 1/r. When the Boussinesq approximation is inciding zero contours passing through the points (0; 0.5),
involved, no preferred vertical direction exists; therefore, if (0.3;0.7),(0.5;1)ontheplane(l ,l ).Thesignsofα+ and
1 2
the solution of a fixed polarity exists in this framework, its α+ alwayscoincide.Eachofthequadraticnonlinearcoeffi-
e
mirrordisturbanceisalsoasolution. cientsα−andα−formode-2IWshasasmoothzerocontour
e
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O.E.Kurkinaetal.:Propagationregimesofinterfacialsolitarywaves 125
Figure 4 illustrates possible soliton branches of IWs of
both modes for r =(cid:49)ρ /(cid:49)ρ =1. It is clear that all possi-
2 1
bleregimesshowninFig.2canbeimplementedformode-1
waves. Waves of both polarities can exist (waves of eleva-
tion on both interfaces or waves of depression on both in-
terfaces), but properties of such waves can be different for
differentcombinationsoftheparametersofthemedium.The
leftpanelofFig.4alsopresentsinformationaboutallpoints
atwhichboththecoefficientsatthecubicandquadraticterms
are zero. Higher-order extensions of the weakly nonlinear
models have to be produced to provide a balance between
thenonlinearanddispersivetermsandtodescribeIWsinthe
vicinityofsuchpointsproperly.Thesemodelsforthepoints
Figure 4. Map of the appearance of possible soliton solutions to
correspondingtoasymmetricstratifications(andnotlocated
the Gardner equation for a three-layer fluid for fast (left panel)
atthediagonalofthe(l ,l )plane)aresupposedtobesimilar
1 2
and slow (right panel) internal wave modes in the particular case
tothosedescribedinKurkinaetal.(2011b)forasymmetric
r=1((cid:49)ρ1=(cid:49)ρ2). The legend of patterns is given at the top.
situation.
Bold black curves correspond to α=0. Bold grey curves corre-
e
spond toeα1=0. The markers show points at whicheα=eα1=0. The pool of solitonic mode-2 waves (Fig. 4, left panel)
Black squares indicate the combinations (l1, l2)=(9/26; 17/26) consists of solitons with a limited amplitude. As described
≈ (0.3462; 0.6538), and grey circles the combinations (l1, l2)≈ above, these solitons broaden while approaching this limit
(0.1703;0.5717)and(l1,l2)≈(0.4283;0.8297). and form table-like structures. Their polarity on each inter-
faceisdeterminedbythesignofthecoefficientatquadratic
− −
nonlinear terms α and α . Thus, two types of mode-2
1 e1
passing through the points (0; 0), (0.25; 0.75), (1; 1) on the waves can exist in the three-layer fluid: convex waves dis-
plane(l ,l ).Thesignsoftheseparameters,determiningthe placing the upper interface upward and the lower interface
1 2
polarity of solitary IWs in the KdV approximation, are al- downward, and concave waves doing the opposite. Both
waysopposite,reflectingthestructureofmode-2wavesthat typesofmode-2IWswereobservedintheSouthChinaSea
alwayscreateoppositelydirectedinterfacedisplacements. (Yang et al., 2010), but concave waves occur fairly rarely.
± ±
The coefficients at the cubic nonlinear terms α , α be- Convexwaveswerefoundmuchmoreoften.Theywereex-
1 e1
havedifferentlyforIWsofdifferentmodes(Fig.4).Theco- perimentally observed in Mehta et al. (2002) and their na-
+ +
efficientsα andα intheGardnerequationformode-1IWs turecanbeexplainedasanintrusionofwaterintothemiddle
1 e1
have two distinct closed zero contours passing through the layer.
points(0;1)and(9/26;17/26)ontheplane(l ,l ).Thelatter Thefluidatthemid-depthofasymmetricthree-layerfluid
1 2
pointcorrespondstothesymmetricstratification.Thepossi- (with equal density jumps) is not displaced in the case of
bilityofthechangeinthesignofthecubicnonlinearparam- mode-2 waves (Lamb, 2006), so a rigid boundary placed at
eteratthispointforIWsinathree-layerfluidwasfirstmen- themid-depthwillnotbeaffectedbytheflow.Thesolution
tioned in Grimshaw et al. (1997). For such a stratification, intheupperorlowerhalfofthefluidissimplythesolution
bothcoefficientsattheleadingnonlineartermsintheweakly foratwo-layerfluidinvolvinghalfofthemiddlelayer;these
nonlineartheoryformode-1IWsvanishsimultaneously.The “partial”solutionsofcourseonlyexisttogether,andmustbe
wave dynamics near such a regime is described by an ex- synchronized.Themiddlelayerplaystheroleofafluidthat
tended(2+4)KdVequation(Kurkinaetal.,2011b).There bothunderliestheupperlayerandoverliesthelowerlayer.If
exist two other such points (Fig. 4) in each half-plane. The halfofthemiddlelayerisshallowerthanthelowerandup-
− −
parameters α and α of mode-2 waves are always nega- perlayers,ahumpofelevationmayexistattheupperinter-
1 e1
tive. Their minimal absolute values occur for a symmetric face,andahumpofdepressionatthelowerone.Thisfeature
stratification,whenalllayersarerelativelythick.Whenone wasusedtointerpretmode-2solitaryIWsexperimentallyob-
− −
of the layers vanishes, α and α tend to negative infinity, served by Mehta et al. (2002) with the help of the Gardner
1 e1
reflecting the fact that the three-layer mode-2 wave regime modelforatwo-layerfluid.
doesnothaveavalidasymptoticprocesstowardsatwo-layer A summary of the results of some calculations of fully
dynamics. nonlinear and higher-order weakly nonlinear solitary IWs
For both modes, the absolute values of the coefficients at is presented in Fig. 5. These data are in qualitative agree-
thequadraticandcubictermsoftheequationsfortheupper mentwiththepredictionsofthemodelbasedontheGardner
andlowerinterfacesarenotequalexceptiftheyarezeroor equation. Some differences can be explained by the use of
ifthestratificationissymmetric.Thisfeatureexemplifiesthe smoothedthree-layerstratificationsinnumericalcalculations
asymmetry of the interfacial displacements for asymmetric inLamb(2006).
three-layerstratifications.
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126 O.E.Kurkinaetal.:Propagationregimesofinterfacialsolitarywaves
some choice is made as to what the dependent variable in
theweaklynonlinearequationrepresents:itcouldbethedis-
placementofthelowerinterface,theupperinterface,theav-
erageofthetwo,thevelocityatsomelevelbetweenthetwo
interfaces,etc.Thiswasdiscussed,forexample,inLamband
Yan(1996).Whilethefirst-ordercoefficientsarenotaffected
bythischoice,thesecond-ordercoefficientsare.Asthesigns
ofα+ andα+ arealwaysthesameforanyfixedpointofthe
e
plane(l ,l ),bothofthesesolutionsaretakenfromtheleft
1 2
half-plane(α,α )orbothfromtherighthalf-plane.So,dif-
1
ferentchoicesgivedifferentpredictionswhenwearecloseto
critical values where the cubic nonlinear coefficient is zero.
Forthefullynonlineardispersiveequations,wewillonlybe
inoneregime.
Thewavesfrommode-1producedisturbancesoftheinter-
Figure 5. Points corresponding to stratifications for which mode- facesofthesamepolarity.Therefore,ifα >0,thepolarity
1
1 ISWs were considered in Boussinesq approximation with equal of the solution matches the sign of α. The shaded areas in
density jumps in different approaches. Circles: positive polarity Fig.6(theconfigurationswherethedisplacementsofthein-
waves(fullynonlinearnumericalmethodforISWsinathree-layer
terfacesbelongtodifferentbranchesofthesolution;Eq.33)
fluid)(RusåsandGrue,2002).Triangles:wavesofelevationwitha arelocatedfairlyclosetothecontoursα+=0,α+=0.Con-
minimumamplitudegreaterthanzero;squares:wavesofelevation 1 e1
+ +
sequently,theabsolutevaluesofthecoefficientsα andα
withnominimumamplitude(fullynonlinearnumericalmodel,con- 1 e1
are small in these areas, and the corresponding solution ap-
tinuous almost three-layerstratification) (Lamb, 2006). Left vicin-
ityofthediamondonthelinel2=1−l1:wavesofbothpolarities parentlyresemblestheKdVsoliton.Thisregimeisthelim-
broaden and flatten whileapproaching an amplitude limit, (2+4) iting one for the solutions corresponding to both “upper”
KdVmodelforIWsinasymmetricthree-layerfluid(Kurkinaetal., and“lower”familiesofsmall-amplitudesolitonsdepictedin
2011a,2012). Fig.2.Thisfeatureconfirmsthecorrectnessofthepresented
constructionsandtheapplicabilityofthesolutions(η+,ζ+)
tointernalwavesofmode-1,evenintheseareas.
Inthiscase,itshouldbedecidedtoderiveaGardnerequa-
tion for the lower interface or for the upper interface. The
fact that the relationship (Eq. 32) does not convert the sec-
ond of Eq. (31) to the first of Eq. (31) is connected to the
fact that deriving the Gardner equation for the upper inter-
faceamountstoadifferentchoiceandhenceresultsindiffer-
ent coefficients. As for stratifications with the same density
jumpacrosseachinterface,theinterfacefurthesttothemid-
depth is, probably, the best choice for the dependent vari-
able in the Gardner equation. In this case, one would use
the equation for ζ and the relationship connecting interfa-
cialdisplacementsifl <1−l andtheequationforη with
1 2
connection Eq. (32) otherwise. For different density jumps,
theinterfacewiththelargestdensityjumpshouldbeused.
Figure6.Comparisonofcurvesα1+=0andeα1+=0when(cid:49)ρ1= norImntahlleysneaost avnaldido,caenadnst,htehemaasgsneirttuiodnesro=f d(cid:49)eρn2s/it(cid:49)yρju1m=p1s aist
(cid:49)ρ2. The intersection points are the same as marked points in
Fig.5,leftpanel,whereα+=0andα+=0.Theshadedregions different interfaces can be considerably different. To depict
e
represent the configurations where the displacements of the inter- the changes associated with the above “phase diagrams” of
facesbelongtodifferentbranchesofthesolutionofEq.(33)shown varioussolutions,wechosethevaluesr =2andr =1/2to
inFig.2. characterizethevariationsinthecoefficientsinquestionand
the wave propagation regimes. Different values of r signif-
icantly influence the patterns for mode-1 environmental pa-
Thecontoursofα1+=0andeα1+=0inGardnerequations rameters.Themodificationsarethelargestforthevaluesof
forthelowerandtheupperinterface,respectively,areshown the coefficients at the nonlinear terms. These changes ob-
in Fig. 6 for the case (cid:49)ρ1=(cid:49)ρ2. The difference between viously affect the possible regimes for soliton appearance
zerocontoursasshowninFig.6isrelatedtothefactthatthe andpropagation(Figs.7and8).Interestingly,theparameters
second-order coefficients are not uniquely determined until
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