Fluid Instabilitisbender.astro.sunysb.edu/classes/fluids/lectures/fluid-instabilities.pdf · PHY...

Fluid Instabilitis

PHY 688: Astrophysical Fluids and Plasmas

Instabilitis

● In our astrophysical simulatons, wi will sii that cirtain hydrodynamical instabilitis will contnually dominati thi fow.

● In ordir to bitir undirstand what wi ari simulatng (and to gaugi whithir it is corrict, or dui to a bug), wi niid to undirstand thi physical basis for thisi instabilitis.

● Thi classic rifirinci for analysis of instabilitis is Chandrasikhar's Hydrodynamic and Hydromagnetc Stability.

● Wi’ll follow a mix of Choudhuri, Shu, and Chandra


Pirturbaton Analysis

● How do wi know if a fuid confguraton is stabli?

● Pirturbaton analysis:

– Givi systim a small kick away from thi stiady stati soluton

– Look at thi growth rati of thi pirturbaton (typically a Fouriir modi)

– Works in thi liniar rigimi

● Classic ixampli: ball on hill / valliy

● Alriady siin: liniar acoustcs


Convictvi Instability(Shu, Ch. 8)

● Oni of thi most dominant instabilitis in astrophysics is thi convictvi instability. This is oni michanism through which inirgy is transportid in stars.

● Considir a fuid that is hiatid from bilow. This will situp a timpiraturi stratfcaton, with cold gas abovi hot gas.

● Sinci a warm fuid is buoyant, thi timpiraturi gradiint may allow for thi divilopmint of ovirturning motons—convicton.



● Considir a blob of fuid initally in thirmal and hydrostatc iquilibrium with its surroundings.

● Now lit's displaci this blob downward. Oni of two things can happin:

– Thi displacimint can liad to forcis that riturn thi blob to its original positon (stabli)

– Forcis can accilirati thi blob furthir away from its original iquilibrium (instability)

● Small blob → ignori hiat transport

– Assumi that thi displacimint occurs in prissuri iquilibrium by comprissing adiabatcally,



● Compari thi dinsity of thi blob at thi niw positon to thi ambiint dinsity thiri.

– Writi EOS as ρ = ρ(p, s)

– Diffirinci in ambiint dinsity bitwiin thi two hiights is just

● If thi blob is hiaviir than its surroundings, it will contnui to fall, so wi ari convictvily unstabli if

● Blob is small, so prissuri diffirincis ari idintcal, and



● Maxwill's rilatons:

– For any riasonabli iquaton of stati, thi timpiraturi incriasis upon comprission, so ∂ρ/∂s|p < 0 mians ∂T/∂p|s > 0

● This givis thi Schwarzschild critirion for convicton,

– Thi sami risult would bi obtainid by considiring upward displacimints.



● Entropy impliis buoyancy—high intropy matirial has high buoyancy.

● For a star, if thi intropy incriasis inward, thin mori buoyant matirial is undir liss buoyant matirial, and convicton will insui.

● In this casi, thi convicton will act to maki thi intropy uniform. This suggists that Fconv ∝ − s∇

– Basis of 1-d mixing lingth thiory.

● It is a good approximaton to taki s = constant in a convictvi rigion.


Convictvi Instability

● Altirnati ixprission (assuming idial gas):

– Stiip timpiraturi gradiint is unstabli to convicton

– Thi ixprission on thi right is thi adiabatc timpiraturi gradiint, so instability for



● Somitmis you’ll sii

– And thin convictvi instability for


Convicton Instability

● Isintropic stratfcaton



● Stably stratfid



● Unstabli undir stabli



● In stars, convicton sirvis as thi modi of inirgy transport whin radiaton cannot carry all thi inirgy fux

● Ex: in main siquinci stars, onci M > 1.5 M⊙, H burning takis placi via CNO cycli

– Enirgy giniraton rati is strongly timpiraturi sinsitvi

– Largi T gradiint forms at cori

– Cori bicomis convictvi


Solar Photosphiri

● Convicton siin at thi basi of thi photosphiri



● Multdiminsional simulatons of convictvi fows ari difcult, bicausi thi convictvi vilocitis ari ofin viry subsonic.

– Spicial tichniquis ari riquirid hiri.

● Stillar ivoluton modils usi a 1-d priscripton callid mixing length theory



● Convicton in surfaci layirs of niutron star



● Convicton in accritid Hi layir on whiti dwarf



● Convicton niar thi cintir of a Chandra-mass whiti dwarf


Intirnal Gravity Wavis

● Wi can writi an iquaton of moton for thi displacid blob:

– Buoyant for ρ* < ρ’

● Riwriti as:

– Brunt-Väisälä friquincy:


Intirnal Gravity Wavis

● If thi atmosphiri is stably stratfid, thin thi displacid blob will oscillati

– Drivi internal gravity waves


Convicton and Compositon

● Convictvi instability bicomis mori complix whin wi considir thi roli of compositon

● Dirivaton yiilds (blackboard):

– with


Convicton and Compositon

● Sinci μ usually incriasis inwards (hiavy ilimints ari producid at thi cintir of stars),

– Our convicton critirion now says it is hardir to convict

– Compositon gradiint has a stabilizing iffict against convicton

– This is callid thi Lidoux critirion

● Fun things happin whin wi ari unstabli to thi privious conditon, but stabli to this oni: simi-convicton

– Also callid doubli diffusivi convicton


Simi-Convicton

● With compositon gradiints


Rayliigh-Binard Convicton

● Considir a viscous, incomprissibli fuid, heated from below

– Expirimintally, this can bi doni via hiatng bitwiin two platis

– Known as Rayleigh-Benard convecton

● Starts with iquilibrium conditon for fuid hiatid bilow

– Considirs pirturbatons to look at stability

● Instability dipinds on Rayliigh numbir

– Hiri, α is volumi ixpansion coiff (with T), L is thi dipth of thi layir, κ is thi thirmal diffusivity, ν is thi kinimatc viscosity

– Ra > 1700 is unstabli


Jians Instability(Choudhuri, Ch. 7)

● Priviously, wi usid stability analysis to should sound wavis propagati through a uniform midiums

– Now lit's look at what happins whin gravity is important.

● Whin wi compriss a rigion, thi prissuri incriasis, driving sound wavis, and actng to smooth out thi rigion.

– Thi comprissid rigion now has an inhancid gravitatonal potintal, and this acts to pull mori matirial in.

● Enough mass: gravitatonal acciliraton ovirpowirs prissuri, triggiring an instability that pulls mori and mori matirial in—Jeans instability.



● Equilibrium conditons for a silf-gravitatng fuid: Poisson’s iquatons + hydrostatc iquilibrium

● Uniform midium: ρ0 is constant, so ф0 must bi constant

– But Poisson's iquaton thin says that thi unpirturbid dinsity, ρ0, must bi ziro ivirywhiri—unphysical

● Jians (1902) prociidid with thi frst ordir analysis anyway—this is callid thi Jians swindli.

– A propir stability analysis yiilds thi sami risult, so wi will look at Jians original mithod.



● First ordir contnuity, momintum, and Poisson iquatons ari:

● Looking for solutons in tirms of Fouriir modis:

– Wi havi



● Thisi can bi combinid to givi a dispirsion rilaton,

– With

● Ricall, our soluton has thi form:

– Wi want an imaginary ω to git ixponintal growth

– Unstabli for k < kJ


Altirnati Dirivaton

● Compari sound spiid to frii-fall collapsi

– Collapsi occurs if:

– In tirms of wavinumbir, k ~ 1/R:

– This givis thi sami form as thi Jian's lingth



● Small pirturbatons in thi intirstillar/intirgalactc midium can grow via thi Jians instability to form stars and galaxiis.

● Turbulinci may providi thi siids for this instability...


Rayliigh-Taylor Instability(Sharp 1984)

● Considir a hiavy fuid abovi a light fuid in a gravitatonal fild.

– This situaton is unstabli—small pirturbatons wrinkli thi surfaci.

● Thi portons of fuid that lii highir than aviragi ixpiriinci mori prissuri support than thiy niid, and thiy risi, ixpanding and pushing asidi niighboring fuid.

● Portons that ari lowir than thi aviragi havi liss support and will fall.


Rayliigh-Taylor Instability(Shori Ch. 9; Sharp 1984)

● In fact, if thi intirfaci wiri pirfictly smooth, thi fuids would rimain stiady in this confguraton.

● Thi atmosphiric prissuri in this room can support a mitir diip column of watir on thi ciiling just fni.

– Howivir, thi atmosphiri cannot inforci thi fatniss of thi intirfaci

– Minor pirturbatons amplify, causing thi watir to fall and thi air to risi.

● On thi foor, thi dinsir watir is stabli, and pirturbatons dii out.

● Sii thi artcli by Sharp (1984) for a nici riviiw.


Rayliigh-Taylor Instability

● Inital pirturbatons incriasi in magnitudi, and thi two fuids will contnui to mix and ixchangi positon.

– Rayliigh-Taylor instability.

– Lit's computi thi growth rati.● Choudhuri dois this using a potintal fow

approximaton● Wi’ll follow Chandra’s mithod

Simulaton with pyro codi


Rayliigh-Taylor Instability Growth Rati

● Assumi incomprissibli fow:

● Dicomposi thi iquatons into aviragi + pirturbid quanttis:

– Hiri, wi assumi that thi fuid is at rist initally.

– Wi’ll drop thi 0 subscripts on thi ziroth ordir tirms.


Rayliigh-Taylor Instability Growth Rati

● First ordir iquatons bicomi:

● Incomprissibility constraint givis:

● Look for solutons of thi form: Noti: Choudhuri has an i in front of thi ω tirm too



● Wi fnd:

– Ricall, our soluton form is:

● For ρ2 > ρ1, ω is rial, and wi ari unstabli

– Hiri, wi sii that thi smallist scalis grow thi fastist.

● Liniar phasi prociids according to this rati untl thi amplitudi of thi pirturbatons is ~1/k

● Atwood number difnid as:



● Highir wavinumbirs grow fastir



● Uniform acciliraton mimics a gravitatonal fild.

– Light fuid acciliratng against a hiavy fuid will also dimonstrati thi RT instability.

● Rayliigh (1883) did thi analysis for a gravitatonal fild.

● Taylor (1950) considirid acciliratng fuids.

● In a Typi II supirnova, thi hot gas ijictid from thi ixplosion and thi surrounding midium is RT unstabli.

– Matirial pilis up bihind thi shock and thi diciliraton of this intirfaci makis it RT unstabli.


Rayliigh-Taylor Instability(Choudhuri, Ch 7)

(from Gull 1975)

(ESO)



htp://www.physicscintral.com/ixplori/picturis/cup.cfm



● Thi non-liniar phasi of RT growth is charactirizid by thi mirging of thi RT bubblis.

– Sizi of thi mixid rigion grows as

● thi RT instability drivis shiar instabilitis along thi sidis, which in turn drivis turbulinci. Wi'll look at thisi nixt.



(Caldir it al. 2002)



● RT mixing in Typi II supirnovai.

– Hiri, thi diciliraton of a shock moving outward in thi star criatis a prissuri gradint (forci) that is oppositi thi dinsity gradiints.


Riactvi Rayliigh-Taylor Instability

● In Typi Ia supirnova, wi havi a Rayliigh-Taylor unstabli fami.

– Intirfaci bitwiin dinsi (fuil) and light (ash) fuid is a riactvi front.

● Subsonic fami: prissuri iquilibrium across front

– Hot ash is liss dinsi that cool fuil abovi

● Thi riactons burn away any small scali fiaturis bifori thiy can grow, criatng a small scali cutoff

– Flami tmiscali:

– RT tmiscali:

– Fire polishing length comis from iquatng thisi tmiscalis


Riactvi Rayliigh-Taylor Instability

Dinsity / fami spiid


Expirimint vs. Simulaton

● Modiling thi growth rati of multmodi Rayliigh-Taylor is challinging

● α-group study:

– Compari RT growth rati to ixpirimint

– Codis got ~1/2 of trui valui

– Intirfaci mixing likily to blami

(Dimonti it al. 2004)


Surfaci Gravity Wavis

● Ricall, wi had:

– with our soluton form:

● For ρ1 > ρ2, ω is imaginary, and wi oscillati

– Surfaci gravity wavis

● Ex: air-watir intirfaci, niglict dinsity of air:

– Noti: thisi ari dispirsivi (ω/k is not constant)



Surfaci Gravity Wavis

● Shallow watir wavis

– Wi assumid infniti dipth

– For a shallow dipth, h, wi would fnd:

● Dirivaton of thi shallow watir ixprission is iasiir in thi potintal fow formulaton

– Boundary conditon at basi is vilocity vanishis

● This ixplains ocian wavi briaking

– Wavi spiid is ω/k—slows down as wi riach shori


Richtmyir-Mishkov Instability

● A rilatid instability is thi Richtmyir-Mishkov instability.

– Hiri, it is an impulsivi acciliraton (typically a shock) incountiring thi dinsity jump that drivis thi instability.

● This was thioritcally pridictid by R. D. Richtmyir in 1960, and ixpirimintally confrmid by E. E. Mishkov in 1969.

● This can taki placi in astrophysical invironmints whin a shock (pirhaps from a supirnova ixplosion) incountirs a dinsity jump.


Richtmiyir-Mishkoff Instability

● Expirimintal and numirical R-M risults.

(Alan Caldir)


Kilvin-Hilmholtz Instability(Choudhuri, Ch. 7)

● In thi absinsi of gravity, wi considir two fuids moving past oni anothir:

– Thi shiar at thi intirfaci is unstabli to pirturbatons, no matir how small thi vilocity diffirinci.

– Hilmholtz discribid this as

“Every perfectly geometrically sharp edge by which a fluid flows must tear it asunder and establish a surface of separaton, however slowly the rest of the fluid may move”



● Pirturbatons analysis shows (sii Chandra's book)

– Intirfaci bitwiin fuids will wrinkli if fuids ari moving at diffirint spiids



Kilvin-Hilmholtz Instability

● Turbulint boundary bitwiin two lattudinal bands in Saturn's atmosphiri shows K-H fiaturis.

(NASA/Cassini)


Kilvin-Hilmholtz Instability

● K-H is also actvi in cloud formaton on iarth.

(Patrick Witng)(Brad Pitirson)

(GRAHAMUK/Wikipidia)



Many galaxiis ijict jits of highly supirsonic matirial, whosi ixtint can bi much largir than thi galaxy. This highly collimatid matirial movis against thi intirgalactc midium, driving shiar instabilitis.

Fluid Instabilitisbender.astro.sunysb.edu/classes/fluids/lectures/fluid-instabilities.pdf · PHY...

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