WV GROWTH

WAVE GROWTH PATTERNS IN A NON-LINEAR DISPERSIVE SYSTEM WITH INSTABILITY AND DISSIPATION

DINESHGERA, MRIDUL GAUTAMandHOTAV. S. GANGARAO

Department

ofMechanical

& Aerospace

Engineering

CFC, Department

^ofCivilEngineering

West

Virginia University

Morgantown WV

26506

U.S.A

(Received

November 30,

1995)

ABSTRACT. A

simpleone-dimensionalnon linearequation including effects of instability, dissipation,and dispersionisexaminednumerically.Periodic solutionofa non lineardispersive equationis

presented

for different valuesofct,

13,

and y characterizing theconstants for instability, dissipation, and dispersion respectively.

In

thispaper, the

growth

pattern for thewaveatdifferent time intervals is discussed. Various

equilibrium

stateswith different initial

configuration

have been observed depending oninitial conditions.

KEY WORDS

AND

PHRASES:

Nonlineardispersivewaves,nonlinearinstability and dissipation.

1991AMS

SUBJECT

CLASSIlCICATION

CODES:

76E30.

1.0

INTRODUCTION

One

of theintrinsicpropertiesattributed todispersionin anonlinearsystemwithinstabilityand dissipation has been pointedoutbyanumerical initialvalue

problem

concerningasimpleone-dimensional model equation

[1],

given by:

Dispersion worksas aneffectiveimpedancein nonlinearmode coupling

processes

and resultsin saturation athigher amplitudes for strong dispersion leadingtoa non linearequilibrium, i.e.,a rowofsaturated soliton like

pulses. However,

the wave evolutions are chaotic in theabsence ofsufficientdispersion

[2,3].

Itisinterestingtonoticefromthe lineardispersion relation, f2 txk yk +

il3

^k

,

which is obtained bysubstitution ofU exp(ikx +

Qt)

intothelinear versionofEquation-I,smallamplitudesinusoidal waves withlong wavelengthsarelinearly unstable.Thussmall amplitudesinusoidal wavesgrowordamp according towhether

Re

f>O or

Re

f<0. The maximumgrowthrateisalways givenatthewavenumber

l=(a/2, [4].

Itisanticipatedthat theseunstablewavesmay destroyasteadyrow ofpulseswhen the distancebetween adjacentsoliton likepulsesbecometoolong

[5].

One

ofthesimplestnon-lineareffectswhich iscapable of saturatingthegrowth ofalinearlyunstable wave or spectrumofwavesis resonantmodecoupling

[6].

The existenceof both instabilityanddispersion

600 D. GERA, M. GAUTAM AND H. ^V. S. GANGARAO

indicatesthepossibilityofasteadystate, because theenergyinfluxduetotheself excitation is transferred throughmodecouplingtoshortwavelengthand isexpectedtobe balancedby dampingduetothe fourth orderdissipationterm.Whenthe rateof

energy

influxfromthe linearinstabilityisbalanced bytherateof outflow from mode coupling, the steadystateis achieved.

Cohenetal.

[7]

appliedthe similar one dimensionalpartialdifferentialequationas equation

(1)

in analyzingthe non-linear saturationofthedissipativetrappedionmode Thedissipativetrappedionmode is alow frequency,electrostatic drift wavepropagatingintheelectrondiamagneticdirection. The wave is destabilizedbyioncollisionaldampingandLandaudampingduetobothcirculatingandtrappedions

[8].

Equation

(1)

alsorepresentsthe modified K-dVequationwhich includes theenergy dissipationterms

Ott

andSudan

[9]

observed asolitarywavepattern forasmallchangeintheamountofsuchdissipation Ott and Sudan

[9]

modified equation

(1)

andinvestigated three cases, viz,

(a)

magnetic wavesdamped by dectron-ion collisions,

(b)

ion-sound waves withelectron Landaudamping

[8],

and

(c)

shallowwaterwaves damped by viscosity.

A

non linearevolutionequationfor thefreeinterfacedisplacementfromplanar shape^isfoundto possessclassical form as

Eq. (1),

^withinterracial viscositiessupportingthe existence ofadispersiveterm 10-

12]. In

suchasystem, u willrepresenttheperturbedinterface.Therefore,it isimportanttounderstand the role ofdispersionin non linear systems,includingbothgrowthanddampingmechanisms in relationtoinstability waves in fluidsystems.

2.0

NUMERICAL ANALYSES

Equation-1withperiodic boundaryconditions on the interval

[0,L]

isintegrated numericallybyafinite differencemethod inspaceand time. Fivepoint,central differenceapproximationswereusedforspatial derivativesso thatpossible leadingerrors wereof an order less than the fourthpowerof thespatialmesh size Spatialmeshpointsweretakentobe 150 in theperiodicity lengthL=2,withtheperiodic boundarycondition atx=0and x=L.Initial conditionsassignedwere

(a) -cos(x), (b) -sin(rx)-cos(rrx), (c)

^{stepwavewitha unit} amplitude,and

(d)

uniformlydistributedrandomnumbers.

Forthe case when

13u---UUx

^>>tu--y

.u,o

then theEq.(1) describing slow changesof theamplitude of the

Koteweg

de Vries solutionmaybe derivedbymeans oftwo-time asymptoteequation expansionwith slow time scale definedby T=et

(where

^eis asmallperturbation

parameter)

Thesteadysolutionofequation

(1)

isgiven asymptoticallytobe

[5].

u=Asech2Bx

^{[_ 6} tanhBx

ln(codaBx)l

(2)

51/1

J

where

A=(21

^tfS/5y)andB=(7t/20y)

t2.

Temporalevolution of the solution ofeq.

(1)

isalso derivedbyToh and Kawahara

[5].

Theasymptotic solution(Eq.

(2))

^{isused laterforthe}qualitative comparisonof our numericalalgorithm.

3.0

RESULTS AND DISCUSSION

Theresults of numericalintegration ofEquation-1withthe initial condition

(a)

ispresentedinFigure- Values of a,

3,

andy forthe differentcases

I,II,

^and

III

aregiveninTable-1.

Table-1 Instability, DissipationandDispersioncoefficientsfor three cases

0.01 0.1 0.001

2.0e-03 5.0e-04 5.0e-04

III

5.1 e-06 50e-04 5.0e-04

It

is evidentfrom Figure-1 that

case-I

growsatthefastestrateas

compared

tothe othertwocases, this is attributed tothe fact thatithad the

largest

valueofa.Figure

(b)

depictsthemoderate growth ofthe wave,andFigure

(c)

shows a stable waveform. Thestable waveformispredicted becauseofverysmall valueofa.Itisseenfrom Figure-

(a)

thatthetemporalevolutiondevelopsto asolitarywaveform after 2

’secondsfor

case-I,

whilethe case-IIisunderdampedandcase-IIIhas astableamplitude. As decreases

(refer

^toFigure-l(a)through

l(c))

stabilityof thewaveincreases. This isalso qualitativelyobservedbyToh and Kawahara

[5],

who used thesteadysolution asgiven by equation2. Theother case with theperiodic

bounda

^conditionof

(-sin(x) cos(nx)

ispresentedinFigure-2

(the , 13,

andyvaluesforthis caseand therestoftheotherboundarycondition cases are the same as

I).

It growstothe wavesofconstantamplitude after 1.2 seconds. Similar trends were also observedby Oronand Edwards

10]

and Cohenetal.

[7].

Itis interestingtonoticefrom Figure-3,if the step wave is takentoasthe initialdisturbance,thenafter t=0 4 seconds,italsoconvergestothe cyclic wave Lastcase was triedby takingrandom variables asaninitial condition. Itis inferredfrom Figure-4that italsogrowstoastablewaveformafter anelapseof initial 20 seconds

Awavewith sufficiently smallamplitudeandlarge^widthgrowsbecausethegrowthterm a

U,,x

^ismore important than thedampingtermy

U

^{for smallwavenumbers.Meanwhile,}

th,e

^{dispersioncaninhibitmode}

couplingandresult in saturationathigher amplitudes forsufficientdispersion; forwhich thegrowth just balances thedampingand alsothedispersion balancesthe nonlinearity.

Computer

solutions

[5,7,10,12]

indicatethatthe

growth

ofaninitial perturbationis followed by formation of a rowofsolitons forthestrongly dispersive case Also, it isinterestingto notice that the existence of adispersiveeffect canbringaboutakindof organizationin the system that exhibits a turbulent like behaviorif theeffect ofdispersioniscompletely neglected.Numericalresultsrevealed,thatthe number of pulsesincreases astheamplitudeof the initial disturbance increases

Thus,

theequilibriumstateisinitial condition

dependent.

The intervalsbetweenpulsesarenotregularatthe stage when the saturatedpulsesare firstdevelopedbutbecomeregularatlater time.

Equation

(1)

^with

13=0

^{reducestothat}equation describingthe chemical reactions which exhibit a turbulent like behavior

13] A

time evolutionof

eq.(1)

with

13=0

for arbitrary initial disturbances waspursued

602 D. GERA, M. GAUTAM AND H. V. S. GANGARA0

01 9.2 0. 1,6 0.I! 1,2 1.1i 1,6 1.It

S(c)

Figure Temporalevolutionof

U

forinitial condition

-cos(x)

for

(a) Case-I’, (b) Case-H*,

and

(c) Case-HI*.

Figure2 Temporalevolutionof U for initial condition

-cos(rrx)-sin(rrx)

at

(a)

t--.0

see, (b)

t---0.4see(C)w0.8 see

RefertoTable

t-O.|

U

3(a)

⁴

(b)

t--l.2

U

3(b)

⁴

(c)

Figure3 Temporalevolutionof

U

forinitial ,-2.,,,c

condition ofstepwaveat

(a)

0.0sec, and

(b)

0.4sec

U t’O

(a)

0.0sec,

(b)

^0.4sec,

(c)

1.2sec, and

(d)

2.8 sec

4

(a)

604 D. GERA, M. GAUTAM AND H. V. S. GANGARAO

numericallybyYamada

&

Kuramoto 13

].

Itwasconcluded,atanearlystage ofevolution,that aspatially periodicstructuredeterminedbythe maximumgrowthratedevelopsbut itfinallybreaks intoaturbulentstate

4.0

CONCLUSIONS

Short wavelength componentsdue to initialuniformlydistributed randomvariablesassigned at individualmeshpoints quickly

damp

outandsoon generate a waveform. It shouldbeemphasizedthatnot alloftheyounghumps generatedatthe initialstagegrow uptosaturatedsoliton-likepulses. It shouldbe noted here that the numberofsolitons whichemerge fromagiveninitial condition in thepurely dispersivecase has no direct relationtothe final numberofsoliton likepulsesin the case with non zerotandy.

[]

[e]

[3]

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