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)
http://antwrp.gsfc.nasa.gov/apod/ap070513.html
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
Nova Cygni 1992 Credit: NASA, ESA, HST, F. Paresce, R. Jedrzejewski (STScI)
http://apod.nasa.gov/apod/ap951227.html
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
Nova Cygni 1992 Credit: NASA, ESA, HST, F. Paresce, R. Jedrzejewski (STScI)
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/
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
http://apod.nasa.gov/apod/ap001031.html
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)
http://antwrp.gsfc.nasa.gov/apod/ap070513.html