Interface Instabilities

Instabilities at Astrophysical Fluid Interfaces

Jonathan Dursi

CITA|ICAT UTK, 26 Feb 2008

Fluids: Almost Everything

• 99% of the visible matter in the Universe is in the form of fluids

• Most of the astrophysical systems we don’t fully understand, it’s the fluid dynamics tripping us up M42 - Orion Nebula

Credit: NASA, ESA, M. Robberto (STScI/ESA) and the Hubble Space Telescope Orion Treasury Project Team

http://antwrp.gsfc.nasa.gov/apod/ap060119.html



Astrophysical Fluids

• Typically ionized plasmas

• Often can use MHD → hydro + magnetic fields

• No surface tension; viscosity negligible (Re ~ 1015 not unusual)

• Eqn of state can be complex

• Often highly compressible Gravitational Instabilityin a cold disk

Opportunity!• Many astrophysical systems

depend on (nearly)-familiar fluid behaviours

• But with added physics, or in different regime

• Interesting variations on familiar problems

• Interesting consequences

Cat’s Eye NebulaNASA, ESA, HEIC, and The Hubble Heritage Team (STScI/AURA)




Today: Three Variations on Familiar Problems

• Work I’ve been involved in recently

• Involves familiar behaviours in less familiar circumstances

• Interesting generalizations of problems

• Real importance to astrophysical systems

Mixing in Classical Novae• Nova explosions happen on

surface of white dwarf star

• Light incoming material needs to ‘dredge up’ heavy material

• Variation on Miles (1957) explanation for wind-water mixing

Nova Cygni 1992 Credit: NASA, ESA, HST, F. Paresce, R. Jedrzejewski (STScI)

http://apod.nasa.gov/apod/ap951227.html



Draping of Magnetic Fields in Galaxy Clusters

• Projectile in a magnetized medium can ‘sweep up’ magnetic field

• Builds a strong thin magnetized layer - but do shear instabilities destroy it?

• Kelvin-Helmholtz with thin magnetized layer

Flames in Type Ia Supernovae

• Complete incineration of white dwarf star

• Burning begins as flame

• Flame instabilities greatly increase burning

• Instability properties of these flames

Supernova 1994D (STScI)

Classical Novae: Resonant Driving of Gravity Waves



Alexakis et al (2004)



Classical Novae• White Dwarf orbited by a

companion

• Companion expands, material (mostly H/He) accretes onto WD

• This layer is hot; when it builds up to high enough densities, can ignite

Accretion: NASAhttp://commons.wikimedia.org/wiki/Image:Accretion_disk.jpg

http://commons.wikimedia.org/wiki/Image:Accretion_disk.jpg

http://commons.wikimedia.org/wiki/Image:Accretion_disk.jpg


http://apod.nasa.gov/apod/ap951227.html Nova Persei 1901 Credit: NOAO/AURA/NSF

http://jumk.de/astronomie/special-stars/nova-persei.shtml





Why is Explosion so Energetic?

• Burning of H,He is actually pretty slow; hard to have explosion

• Lots of C,O would help; could catalyze burning (and is seen in ejecta)

• Can’t come in amount needed from donor star

WD(C,O)

H/Heenvelope

Kelvin-Helmholtz can’t

help us• Accretion shear (or shear from

early convection stages) might do the mixing

• It can’t in time needed; density ratio high enough to significantly reduce mixing

C,Oρ~11

H,Heρ ~ 1

But there is a way to do this..

• Wind does drive waves

• Air/Water-1:1000 density ratio

• Also generates higher moisture content in air over water

Waves in Nanaimo, BChttp://flickr.com/photos/druclimb/530928245/

http://flickr.com/photos/druclimb/530928245/

http://flickr.com/photos/druclimb/530928245/

Miles (1957)• Resonantly drive gravity waves

• Assume existing boundary flow

• If ∃ gravity wave with velocity

equal to U(y) for some y, can drive that wave.

U(y)

Linear Theory• Assume incompressible flow,

sinusoidal surface wave, deep ‘water’.

• ϕ is related to stream function; k is wave number; Im(c k) is growth rate

• Max unstable mode (k) ~ ρ2/ρ1 times larger than KH

!!! !!

k2 +U !!

U ! c

"! = 0

k2c2 ! "1

"2

#c2k !!|0 + c U !|0

$! gk

!1! "1

"2

"= 0

Linear Theory• Crucial parameter: Froude

number

• Growth rate of instability ~

• Can only examine modest range of shear velocities

!!! !!

k2 +U !!

U ! c

"! = 0

k2c2 ! "1

"2

#c2k !!|0 + c U !|0

$! gk

!1! "1

"2

"= 0F = U/

!g!

e!4.9At/F 2

Weakly Nonlinear Analysis

• Vortices form between the crests of the waves

• In phase with wave motion, drives them

• Indications for how mixing occurs

Nonlinear Growth

• What we really want is nonlinear effects - mixing

• Among first ever full nonlinear simulations of Miles (1957)mechanism

• Impossibly slow for air/water, tractable in this regime

An important tool: The Flash Code

Cellular detonation

Compressed turbulence

Helium burning on neutron stars

Richtmyer-Meshkov instability

Laser-driven shock instabilitiesNova outbursts on white dwarfs Rayleigh-Taylor instability

Flame-vortex interactions

Gravitational collapse/Jeans instability

Wave breaking on white dwarfs

Shortly: Relativistic accretion onto NS

Orzag/Tang MHDvortex

Type Ia Supernova

Intracluster interactions

MagneticRayleigh-Taylor

AMR is valuable for interface

problems• Adaptivity in mesh allows

resolution where necessary - interface

• Allows for higher resolution of interface for same resources

http://www.astro.sunysb.edu/mzingale/flame_vortex/flames_FLASH.html



Some subtleties

• Had to modify hydrodynamic solver to do hydrostatic equillibrium over long timescales well

• Handful of techniques for modifing these sorts of problems in Godunov codes (Zingale, Dursi, et al)

Mixing Rates: • Series of FLASH simulations

varying Froude number, initial conditions

• If KH can play a role, mixing is through cusp instabilities

• Otherwise overturning

• Mixing occurs until thickened layer prevents more

Use as input model for

larger scales• Once a scaling can be

determined, can be placed into larger scale model

• Can look at effects of early-stage simmering...

And long term evolution of

novae• Can accretion wind over long

period of time drive enough mixing to make novae energetic enough?

• Under right conditions, yes!

ApJ, Physics of Fluids, and

• Other prestigious publications

‘Bubble Wrap for Bullets’: draped magnetic layers

Dursi (2007)

Abel 1689Credit: NASA, N. Benitez (JHU), T.

Broadhurst (The Hebrew University), H. Ford (JHU), M. Clampin(STScI), G. Hartig (STScI), G. Illingworth (UCO/Lick Observatory), the ACS Science

Team and ESA

Abel 2029

Virgo:R. White (UA; optical), S. Snowden, R. Mushotzky (NASA/GSFC; X-ray)

ArchesNASA/CXC/Northwestern/F.Zadeh et al.

Hydra ANASA/CXC/SAO

http://chandra.harvard.edu/photo/1999/0087/



Bubbles in Galaxy Clusters

•Radio Bubbles (radii 6-20 kpc), seen as voids in X-rays

•Thought to be inflated by high-energy jets from active central galaxies

•Seen to have very sharp interfaces

•Conduction should dissipate these in ~108 years

NASA/IoA/A.Fabian et al.

Perseus:A. Fabian (IoA Cambridge) et al., NASA




Cluster Bubbles

• Bubble’s existence at a distance from inflation point is a puzzle

• Purely hydrodynamic bubble will rip itself to a smoke ring in one crossing time

Robinson, Dursi et al (2004)

Does it matter?

• Can such a thin layer have interesting dynamic effects?

• Linear theory analysis

• Three layers; velocity +/- U, magnetized layer of some thickness/strength

Does it matter?

• Localized B-fields make linear analysis more tedious, but remains doable

• Eigenvalue problem; boundary conditions at interface

0.02 0.05 0.1 0.2 0.5 1 2l!

1

1.5

2

3

5

7

10

kvA2 g stable

0.02 0.05 0.1 0.2 0.5 1 2l!

1

1.5

2

3

5

7

10

v A2 U2

stable

Rayleigh-Taylor Kelvin-Helmholtz

If VA is a few times relevant velocity, can stabilize againstwavelengths an order of magnitude longer than thickness of layer

layer thickness layer thickness

stable

(Alfv

én S

peed

/Gra

v Sp

eed)

2

(Alfv

én S

peed

/She

ar S

peed

)2

stable

VA = 0.2 U VA = 1.25 U

Run with v3.0 of the Athena code

U

U

!x of kinematic solution

-2 0 226

28

30

32

34

36

38!y of kinematic solution

-2 0 226

28

30

32

34

36

38!z of kinematic solution

-2 0 226

28

30

32

34

36

38

!x around draped projectile

-2 0 226

28

30

32

34

36

38!y around draped projectile

-2 0 226

28

30

32

34

36

38!z around draped projectile

-2 0 226

28

30

32

34

36

38

!x / u

-0.07 -0.03 0.00 0.03 0.07

!y / u

-0.4 -0.2 0.0 0.2 0.4

!z / u

-1.0 -0.7 -0.3 0.1 0.4

Potentialflow

around solid

sphere

3D AMR results

vx vy vz

Supernovae: Flame Instabilities

Supernovae Ia

Supernova 1994D (STScI)

• Very bright events• Few x 1051 ergs (1028

megatons TNT)• ~28 day rise time• No H in spectrum

• Can outshine host galaxy• Can be seen at great distances• Leave behind no remnant• Cosmologically interesting

Supernova Ia Mechanism

accretion ignition flame propagationsimmering

detonation?instabilities

Flame vs Detonation

• Flame/Deflagration

• Subsonic

• Heat propagates by conduction

• Detonation

• Supersonic

• Shock-driven heating

Landau-Darrieus Instability

Clanet & Searby (1998)

• Planar flame front • Initial wrinkle grows in time• Driven by density jump across

moving interface• Grows fastest at small scales

• More wrinkling → more surface area → more burning

Effect of Curvature, Strain

• Why does this flame remain so flat?

Effect of Curvature, Strain• Geometry affects heat transport

• Negative curvature focuses transport, speeding flame

• Positive curvature dilutes transport, slowing flame

• Flame resists wrinkling on small scales.

• Chemical flames have species diffusion which counteracts this

Physical Setup• Inward/outward propagating 1d

spherical flame• Ignite w/ top-hat hot ash region

beside cold fuel• Flame ignites, propagates

• At different radii, strain+curvature varies

• Can read off flame velocity vs. strain+curvature

• This is something you can still do with an explicit code -- just 1d flames.

Results

• Lack of species diffusion means that astrophysical flames act against strain/geometry to flatten

• Effect depends on composition, weakly on density (degeneracy)• Quantified effect can be put into (eg) level-set method to

improve models

• Given predictions for turbulence, can build increasingly accurate models for burning in turbulent velocity field.

Predictions

• Flame Stability• Stable to LD on different scales, depending on

composition•~10 flame thicknesses (pure carbon)•~50 flame thicknesses (50% carbon)

Limits of Explicit Codes

• Anything beyond 1d flame evolution is too expensive with an explicit code

• Mach numbers ~10-4 - 10-2

• 100 - 10,000 CFL timesteps before flame moves a single grid cell.

Need specialized methods

• Can’t use analastic, boussinesq approximations:

• Significant density jumps across flame

Bell et al. (2004)• Generalization of previous gas chemical flame

work to arbitrary EOS

• Mach expansion:

• Ignoring linear accoustics (p1) and in a box where thermodynamic pressure p0 is constant,

Bell et al (2004)

• New velocity divergence constraint becomes

Predictions confirmed:

Box size < 50 flame widths

~ 50 widths

> 50 widths

• UCSC, LBL CCSE • Fully resolved multidimensional

simulations of LD

• Various densities, 50% Carbon/Oxygen flames

• Instabilities completely suppressed if box < 50 flame widths; partially suppressed at 50; unsuppressed at larger sizes

Effect of Magnetic Fields

• Landau-Darrieus: Purely hydrodynamic• Significant Magnetic Fields on some white dwarfs:

• Surface fields: 108 – 109 G

• Interior fields: ?? but potentially much higher

• Naively, magnetic field lines will provide tension against wrinkling

Magnetic Landau-Darrieus

• Consider flame as thin interface propagating parallel to or perpendicular to magnetic field lines

• Perform linear theory analysis

Field Transverse to Propagation

• No competition between Alfvén speed and flow speed

• Only one case to consider

Field Transverse to Propagation

• Little effect until Alfvén speed ~ flame speed

• After that, significant suppression

Field Along Propagation • Perturbations

generate waves that travel up/downstream

• Super-Alfvénic:

• No Alfvén waves can travel upstream

• Sub-Alfvénic:• Only one Alfvén wave

downstream

• Trans-Alfvénic:

• Sub-Alfvénic in fuel, super-Alfénic in ash

Field Along Propagation: Stability Regions

• Super-Alfvénic

• Unstable for large density jump

• Sub-Alfvénic

• Unstable for large destabilizing gravity and sufficiently small jump

• Trans-Alfvénic• Nonevolutionary

Opportunity!• Many astrophysical systems

depend on (nearly)-familiar fluid behaviours

• But with added physics, or in different regime

• Interesting variations on familiar problems

• Interesting consequences

Cat’s Eye NebulaNASA, ESA, HEIC, and The Hubble Heritage Team (STScI/AURA)




Documents

Interface Instabilities