Many interesting mixing problems involve instabilities at the interface
between the two fluids. I have recently considered shear driven mixing
- both resonant wind-wave driven instabilities and the more familiar
Kelvin-Helmholtz type in less familiar circumstances (thin magnetized
layer, or highly compressible with cooling) for various applications,
and this work continues. Also of interest is if the interface itself
is dynamic, such as is the case with flames or detonations, where the
resulting corrugations can greatly increase or decrease burning rates.
Novae are explosions on the surface of white dwarfs, are responsible for
enriching their environment with heavy elements, and may be progenitors of
Type Ia supernovae. It is known that a significant amount of material
must be dredged up from the white dwarf itself and mixed into the
atmosphere to `catalyze' the burning reactions. A robust mechanism
for this mixing remains
enigmatic; it is hard to use light material to dig up heavier material.
Wind-driven mixing through the resonant driving of a gravity wave, from Alexakis et al., 2004.
I have recently been part of a project that using simulations to test a
new mechanism for this dredge-up — the resonant driving of winds on the
surface of the white dwarf, which then break and mix material into the
surface. This mechanism, which can be significantly more efficient than
the usual Kelvin-Helmholtz wind-driven-mixing, is partly responsible for
the humidity in air above bodies of water, where the density difference
is much greater (about a factor of 1000) than in this case. Nonetheless,
the real similarity between terrestrial waves and this much more extreme
environment was enough to get this work noted in both
USA Today and the prestigious publication `Surfing Journal'.
One result from this computational project was that the results from an
extensive parameter study in 2D and 3D were then used to build a model
which could then be incorporated into global models of novae scenario.
Kelvin Helmholtz disrupting a layer with VA < U; for
VA > U, the layer is stabilized. See
Dursi 2007.
The Kelvin-Helmholtz shear instability is extremely well-known and
studied (and can be usefully used as a problem comparing experiment
and computation, eg,
Dimonte et al. 2004,
Calder et al. 2002), but even so adding new dynamics can provide surprising results.
For the magnetic draping work described above, it was necessary to know
if the resulting thin magnetic layer could stabilize interestingly
large modes. If not, then the same shear which gives rise to the
magnetized layer would disrupt it, and nothing interesting could happen.
One's intuition from considering effects such as diffusion or gradual
density gradients leads one to suspect that a thin layer could only
stabilize modes on its own scale.
However, as I showed in Dursi 2007,
using linear theory confirmed with simple 2D simulations, the fact that
a magnetic field can carry signals along the field with Alfvén
speed VA (proportional to field strength) means that as
long as VA is greater than the half-shear speed U, modes
an order of magnitude or more larger than the thickness of the layer
can be stabilized, as the Alfvén waves can communicate the
restoring force over a large distance. This is also true for the
case of Rayleigh-Taylor instability; if VA is sufficiently
greater than a characteristic velocity due to gravitational acceleration,
~ (At g k)1/2, the mode is stabilized.
This work is to be generalized to 3D and more complex geometries.
A hot supersonic wind mixing with the surface of a disk.
Another shear-driven interface instability project involves wind-disk
interactions (eg, the figure to the left). If a central object emits
a wind, then on the surface of a disk orbiting the object a turbulent
boundary layer will be set up, determined by the incoming hot wind and
the cooling rate in the disk material. The turbulent boundary layer
mediates the ablation of the disk by the wind, and determines the
vertical boundary conditions of the disk. I am performing a study of
such disk-wind interactions with Chris Thompson (CITA) to understand the
interplay of cooling, Kelvin-Helmholtz, and rotation in the formation
of the turbulent boundary layer, using extremely large simulations.
Mixing properties even in adiabatic supersonic Kelvin-Helmholtz flows
are surprisingly poorly understood, and adding even fairly simple cooling
physics provides extremely rich dynamics.
Instabilities of Dynamic Interfaces: Deflagrations and Detonations
The instability of interfaces becomes more complex if the interfaces
themselves are propagating with their own dynamics, such as the case of
flame fronts or detonations.
Thermonuclear flames, such as in the case of Type Ia supernovae, have
undergone relatively little careful study, even though the propagation of
flames under degenerate conditions determines how these events unfold,
and the flame propagation itself contains a great deal of interesting
dynamics. Flame instabilities are essential for speeding up burning, as
corrugations in the flame increase the surface being burned at any given
moment. Similarly, detonations are extremely sensitive to curvature,
and even modest enforced curvatures can completely disrupt them.
Flame-turbulence interaction in Type Ia. Note that
the flame suppresses small-scale wrinkling at the front, while material
behind the front is greatly wrinkled; we have measured the Markstein
number of these flames in Dursi et al. (2003).
Much of our understanding of these flame instabilities
comes from the extensive work done in the terrestrial
combustion community. However, as we showed in Dursi et al. (2003), astrophysical flames in degenerate material have some
different properties than terrestrial gas flames. Because of the electron
degeneracy in a white dwarf, there is very little species diffusivity
(compared to thermal conductivity). As a result, the flame reacts strongly
to curvature; the modification due to changes in heat transport cannot
be counteracted by fuel transport. One important consequence is that
astrophysical flames are extremely stable to perturbations on scales less
than about 50 flame thicknesses (much larger than for typical terrestrial
flames); a flame wrinkled on smaller scales tends to `flatten out'
(, see Fig. ). Even when the flame is unstable (or propagating
through a turbulent medium), curvature can significantly reduce the
local speed of burning in regions of high strain, and thus affect the
total turbulent flame speed. This work also establishes the scales on
which shear can affect local burning.
The differences between these flames and the more familiar terrestrial
gas flames are due to the properties of material being burned; in
this case, viscosity and species diffusion are negligible (in terms of
dimensionless numbers, Pr << 1 and Le >> 1); further,
despite the enormous energy release of these flames or detonations,
the resulting change in fluid densities are quite modest (δρ/ρ ~ 0.2)
Because these are well outside of the more usual
terrestrial case, relatively little work in this regime has been done,
meaning there is much interesting new dynamics; this also makes burning-fluid
experiments necessary for comparison.
I have also examined the effect of magnetic fields on flame instabilities
(in Dursi 2003).
Magnetic fields can have only limited roles in
terrestrial flames, where the relatively high magnetic resistivity mean
that no local variations in magnetic field strengths can persist for
dynamically interesting periods of time. For flames in astrophysical
plasmas, however, a strong enough magnetic field - one for which the
Alfvén speed exceeds the flame speed - can completely suppress flame
instability. This can easily occur in other systems, but is less likely
to play a global role in the evolution of a Type Ia supernovae. However,
another interesting regime was uncovered in this work; when the flame
is trans-Alfvénic, the flame becomes non-evolutionary (as is the
case with trans-Alfvénic shocks), and flame-generated MHD turbulence
becomes possible.
A. Alexakis, A. C. Calder, L. J. Dursi, R. Rosner, J. W. Truran, B. Fryxell, M. Zingale, F. X. Timmes, P. Ricker, and K. Olson.
On the Nonlinear Evolution of Wind-Driven Gravity Waves,
Physics of Fluids, 16:3256-3268, Sept 2004.
A. Alexakis, A. C. Calder, A. Heger, E. F. Brown, L. J. Dursi, J. W. Truran, R. Rosner, D. Q. Lamb, F. X. Timmes, B. Fryxell, M. Zingale, P. M. Ricker, and K. Olson.
On Heavy Element Enrichment in Classical Novae.
ApJ, 602:931-937, February 2004.
F. X. Timmes, M. Zingale, K. Olson, B. Fryxell, P. Ricker, A. C.
Calder, L. J. Dursi, H. Tufo, P. MacNeice, J. W. Truran, and
R. Rosner.
On the Cellular Structure of Carbon Detonations.
ApJ , 543:938-954, November 2000.
Sweeping up a Magnetic Sheath: Magnetic Draping over Moving Cores and Bubbles in Galaxy Clusters Accretion and Explosion: the Astrophysics of Degenerate Stars, KITP, May 17 2007 [PDF][Keynote]
A talk given while I was in residence at the KITP program
on supernovae in May 2007.
The talk is on work examining `magnetic draping' in galaxy clusters, including discussing the
linear theory magnetized-thin-layer-KH work. Video or audio of the talk, as well as the slides, are available
at the KITP website.
Simulating Astrophysical Combustion with the FLASH code CAIMS/MITACS Joint Annual Conference, June 2006 [PDF][OpenDocument]
An invited talk to the Scientific
Computing session of the annual Canadian applied mathematics
conference discussing simulating combustion in an astrophysical context,
and the computational choices that this regime suggests.
Towards Understanding some Astrophysical Flows using Multiscale Approaches with the FLASH code HPCS 2006, May 2006 [PDF][OpenDocument]
A discussion of some of our attempts to simulate turbulent astrophysical flows
with interesting physics on disparate scales using simple multiscale approaches; includes
discussion of the wind-disk problem.
Grungy Gastrophysics and Cosmology St. Mary's University, Nov 2005 [PDF][OpenDocument]
This discusses the Type Ia supernovae problem, including discussion of the flame instabilities.
High Performance Reactive Fluid Flow Simulations Using Adaptive Mesh Refinement on Thousands of Processors Supercomputing 2000, Nov 2000 [PDF]
This was the talk I gave (on behalf of the whole FLASH team) for the Gordon Bell Award, discussing
the instabilities of detonation waves.
