Magnetic Buoyancy Instabilities in the Solar Tachocline

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Magnetic Buoyancy Instabilities in the Solar
Tachocline

• David Hughes
• Department of Applied Mathematics
• University of Leeds

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Possible Instability Mechanisms in the Tachocline

• Dynamic instabilities driven by differential
  rotation (e.g. Rayleigh instability).

2. Baroclinic instability.
3. Shear instabilities possibly stabilised
by magnetic field (Rayleigh criterion,
Richardson criterion, semi-circle theorems).

• Diffusive instabilities modification to shear
  instabilities GSF instability.

5. Joint instabilities driven by a
combination of latitudinal differential rotation
and a toroidal field.
6. Magnetic buoyancy instabilities.
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If the solar magnetic field is maintained by some
sort of a?-dynamo then helioseismology
measurements have pinned down the site of the
?-effect.

• Site of a-effect not determined via
  helioseismology. Various possibilities
• Surface a-effect (Babcock, Leighton).
• Distributed convection-zone a-effect.

Consensus that the bulk of the toroidal field is
generated (and stored) at, or just below, the
base of the convection zone regardless of the
nature of the dynamo mechanism.
Actually quite an old idea, based on other
physical considerations magnetic buoyancy
(Parker), magnetoconvection (Spiegel Weiss
Golub et al).
Magnetic field escapes from the tachocline via
magnetic buoyancy instability. It is the only
instability in the tachocline whose consequences
we can observe directly.
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What is meant by Magnetic Buoyancy?

• A means of causing the rise of isolated flux
  tubes (Parker 1955)

This is a non-equilibrium phenomenon not
an instability.
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2. An instability mechanism of continuously
stratified fields
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3. An instability mechanism of isolated flux tubes
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Instability can only occur for superadiabatic
atmospheres. Furthermore, increasing the field
strength decreasing ß is stabilising.
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Instability mechanisms (2) and (3) are clearly
very different.
Which is the more appropriate for the
tachocline? Is the field in the form of (i)
flux tubes, in mechanical equilibrium?
(ii) a diffuse
field?
Neither, really. The field is probably a
complicated, tangled mess, pulled out by the
shearing flow, buffeted by convective plumes from
above. It will have both toroidal and poloidal
components, though one might expect the toroidal
field to dominate.
However, to a rough first approximation, I
consider it more appropriate to regard the field
as diffuse and stratified, and not as existing as
isolated tubes in a field-free environment.
For an alternative view, see the works of
Schüssler and his co-workers.
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Basic mechanism of magnetic buoyancy instability
(2), for field B(z)x (no velocity shear for the
moment). A stratified horizontal magnetic field
that increase with depth supports more gas than
would be possible in its absence. Atmosphere is,
to some extent, top-heavy. Release of
gravitational potential energy can result in
instability.
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Stabilising effect of subadiabatic gradient can
be diminished (possibly strongly) by double
diffusive effects.
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The Influence of Rotation
Some effects of rotation can be captured by local
analysis though not the effects of shear.
For uniform rotation in the régime VA2 ltlt O2H2 ltlt
c2, instability if
(Acheson Schmitt Rosner)
i.e. instability driven by decrease with height
of B/?, rather than B.
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How should we interpret the diffusivity ratio ?/??
In a turbulent regime maybe ?T ? ?T. If
diffusivities assume molecular values then ?/? ltlt
1 though we do not expect these to be
isotropic. Possible therefore that this ratio
varies throughout the solar cycle as field is
amplified.
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Nonlinear Evolution
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Fully three-dimensional evolution
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Higher resolution simulation to examine
undulations.
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The Scale of the Emerging Flux
How does the instability manage to produce
large-scale field structures?
1. Via strong subcriticality. Conceivable if
field is held down by overshooting convection.
2. Through modulation of the instability
mechanism.
e.g. instability of layer of field with B
(Bx(z), By(z), 0) to motions independent of y.
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Effect of Velocity Shear
Consider the linear, ideal MHD stability of
magnetic buoyancy instability influenced by a
shear flow (Tobias Hughes 2004).
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Now we consider the complementary problem of
investigating, numerically, the unstable
modes. Consider the effect on a basic state,
unstable in the absence of shear, of two
different shear flows. Basic state chosen is
unstable to undulatory modes, but stable to
interchanges. Basic unsheared states have a
linear field profile B B0(1 ?z) and are also
taken to be isothermal. Without shear, two
other parameters define basic state plasma ß and
?gd/VA2.
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Flow (i) U d(z3 / 6 z2 / 4 13 z / 12 )
Flow (ii) U U0 tanh a (z zs)
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• Comparison of role of shear for
• moderate and large l, flow 1, ß 10,
• k 0.32.
• l 4 (dashed)
• l 8 (solid)

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Eigenfunctions flow 1, d 10.
Eigenfunctions no shear.
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Effect of flow 1. Shear is stabilising. No effect
on k 0 modes, which here are stable.
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Growth rate as a function of location of the
shear (zs) for flow 2. U0 0.7 a 1, 5,
10. Optimal stabilising effect at zs 0.75, at
which the eigenfunctions of the unsheared
instability are peaked.
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Eigenfunctions of u and Bz for a 10 and zs 0,
0.25, 0.5, 0.75.
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Contour plot of growth rate for flow 2, with a
10 and zs 0.75.
Growth rates in the absence of shear.
Effect of shear is again stabilising.
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Things to Think About
Regarding the magnetic buoyancy instability

• What is the role of diffusion on the
  buoyancy/shear instability?

2. How are the diffusion coefficients and
hence the instability influenced by a
(weak) magnetic field?
3. Nearly all instability studies have been
performed with a purely toroidal
field. What is the role of the poloidal field?

• What is the competition between the magnetic
  buoyancy instability and the
• overshooting convection?

5. What is the extent of the downward
influence of the instability?

• Can magnetic buoyancy instability play a
  regenerative role in the dynamo process?
• Rotationally-influenced instability
  leads to a mean e.m.f. (and hence an a-effect),
  which
• can act to transform BT to BP.

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Things to Think About
More generally

• Is the solar cycle field generated and/or stored
  in the tachocline?

2. If so, how much of the tachocline does it
occupy?
3. Is this compatible with a tachocline with
circulation in an essentially field-free region?
i.e. a two-layer tachocline