We consider a system formed by an infinite viscous liquid layer with a constant horizontal temperature gradient, and a basic nonlinear bulk velocity profile. In the limit of long-wavelength and large nondimensional surface tension, we show that hydrotherma
bythecoexistenceofcoherentspatialstructureswithtemporalchaos[14].Inparticular,thesearchforabetterunderstandingofthelongwavelengthpropertiesoftheKSequationhasbeenintensivelypursuedlately[15].Anditispreciselyinthelong-wavelimitthatwehaveobtainedtheKSequationasgoverninghydrothermalsurface-waveinstabilitiesinathinviscouslayerof uidwithhorizontaltemperatureandvelocitygradients,andinthelimitoflargenondimensionalsurface-tension.Thisresultcouldpossiblybeconnectedtoarecentexperiment[9]whichhasbeenperformedtostudythefeaturesofhydrothermalwavesappearingwhenacylindricalliquidlayerislaterallyheated.Theexistenceofhydrothermalwaveinstabilitieshasbeenobservedexperimentally,andtheresultswerefoundtopartiallyagreewiththeprevioustheoreticalpredictionsofSmithandDavis[1].Ifwebelievethat,despitethedi erenceinthegeometrytheresultspresentedinRef.[9]cansomehowbeconsideredasanapproximationtothetheoreticalpredictions,thenthelong-wavedynamicsofthefreesurfacedescribedbytheKSequationshouldplayaroleintheexplanationofthepropagativepatternsobservedinthiskindofexperiment.
Atlast,wewouldliketospeculateonapossibleconnectionbetweentheresultsobtainedinthisletterandhot-wireexperiments[10,11].Theseexperimentsconsistofahotwireplacedhorizontallyjustbelowafreesurfaceofa uid.Aboveacertaincriticalvalueoftheelectricalpowersuppliedtothehotwire,propagatingpatternsareobserved.Theexistenceofbothhorizontaltemperaturegradientsandabasicvelocity ow,asthe uidisconvecting,makescontactwiththetheoreticalsettinginwhichhydrothermalwavesshowup.Ofcourse,thisisnotenoughtoensurethattheaboveresultscanhaveanimmediateapplicationtothehot-wireexperiments.However,sincethesystemunderconsiderationherepresentssimilaritieswiththeabovementionedexperiments,theycouldhelptoelucidatethephysicalmechanismresponsiblefortheobservedpropagativepatterns.Theideatomimicfeaturesofaconvecting uidbyatwo-dimensionalsystemwithavelocitygradienthasbeenexaminedbyRoz`e[16],basedondataofanexperimentwithamuchshorterwire[17].However,thisapproach,whichusedalinear owpro leandlinearizedequations,turnedouttoberatherdisappointing.Infact,asshowninRef.[1],thereisnolong-waveinstabilityinthiscase.