1. Magnetic Buoyancy Instabilities in the Solar Tachocline
David Hughes
Department of Applied Mathematics
University of Leeds

2. Possible Instability Mechanisms in the Tachocline
Dynamic instabilities driven by differential rotation (e.g. Rayleigh instability).
Baroclinic instability.
Shear instabilities – possibly stabilised by magnetic field (Rayleigh criterion,
Richardson criterion, semi-circle theorems).
Diffusive instabilities – modification to shear instabilities; GSF instability.
Joint instabilities driven by a combination of latitudinal differential rotation
and a toroidal field.
Magnetic buoyancy instabilities.

3. If the solar magnetic field is maintained by some sort of αω-dynamo then
helioseismology measurements have pinned down the site of the ω-effect.
Site of α-effect not determined via helioseismology. Various possibilities:
Surface α-effect (Babcock, Leighton).
Distributed convection-zone α-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.

4. 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.
Te, pe, ρe g
B, Ti, pi, ρi
Total pressure equilibrium between the tube and its surroundings implies:
Provided
then
and the tube will rise.

5. 2. An instability mechanism of continuously stratified fieldsg
B, T, etc
B+δB, etc
B+dB, etc
Consider a static equilibrium atmosphere
with a stratified horizontal magnetic field
(dependent only on depth).
Suppose a parcel of fluid (and field) is raised
in pressure equilibrium with its surroundings,
without bending the field lines. No diffusion.
Conservation of mass and flux:
Conservation of specific entropy:
Total pressure equilibrium:

6. Instability if δρ < dρ
Instability if δρ < dρ. On using the equation of magnetohydrostatic equilibrium this gives:
Alternatively:
i.e. instability if B/ρ falls off sufficiently rapidly with height. Modification to the
Schwarzschild criterion.

7. 3. An instability mechanism of isolated flux tubesg
B, pi, etc
pe, etc
B = 0
pe+dp, etc
Consider an isolated flux tube in pressure
and mechanical equilibrium with its
surroundings. Clearly the tube is cooler,
by a very specific amount, than its
surroundings.
Suppose the tube is raised in pressure
equilibrium with its surroundings,
without bending the field lines.
B+δB, etc
Conservation of mass and flux:
Conservation of specific entropy:
Total pressure equilibrium:

8. Instability if δρ < dρ. Using the above equations, this becomes:
where
is positive (negative) if the external
stratification is superadiabatic (subadiabatic),
and where β is the ratio of the gas pressure to magnetic pressure in the tube.
Instability can only occur for superadiabatic atmospheres.
Furthermore, increasing the field strength – decreasing β – is stabilising.
For modes that allow bending of the field lines, instability to long wavelength
perturbations if
so instability possible for subadiabatic atmospheres.
(Spruit & van Ballegooijen 1982)

9. 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.

10. potential energy can result in instability.
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.
In ideal MHD, instability to interchange modes (no bending of the field lines) if:
i.e.
i.e.
Instability to three-dimensional modes (under ideal MHD) if:
(Newcomb 1961)
Most unstable mode has kx  0.

11. by double diffusive effects.
Stabilising effect of subadiabatic gradient can be diminished (possibly strongly)
by double diffusive effects.
Instability to undulatory modes (as kx  0) if:
(Gilman; Acheson)
There is also an instability (overstability) even if the field increases with height.
Criterion for interchange modes (assuming ν, η  κ):

12. 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 << Ω2H2 << c2, instability if
(Acheson; Schmitt & Rosner)
i.e. instability driven by decrease with height of B/ρ, rather than B.

13. How should we interpret the diffusivity ratio η/κ?
A highly controversial issue (at least in dynamo theory) concerns the suppression of transport
coefficients (the α-effect, turbulent diffusivity) by a very weak large-scale field.
In a turbulent regime maybe ηT  κT.
If diffusivities assume molecular values then η/κ << 1 – though we do not expect these
to be isotropic.
Possible therefore that this ratio varies throughout the solar cycle as field is amplified.

14. Nonlinear Evolution Most dramatic form of instability arises from
a discontinuous magnetic layer. Interchange modes
preferred. g
Field free
Uniform field
Cattaneo & Hughes
Final evolution dominated by
vortex interactions.
Instability of top boundary
causes downward motions
below bottom boundary

15. Fully three-dimensional evolution
Sketch of emergence of magnetic field
as bipolar regions (after Parker 1979).
Simulation of 3d nonlinear evolution of
magnetic buoyancy instability of a layer
of magnetic gas.
Arching produced in nonlinear phase through
Crow instability.
Matthews, DWH. & Proctor

16. Higher resolution simulation to examine undulations.
Wissink et al. (2000)
Higher resolution simulation
to examine undulations.

17. 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.
Cattaneo, Chiueh & Hughes
(a)
(b)
Two cases of equal gravitational PE.
Bx weak (3% of magnetic energy).
(a) Resonant surface near top of
magnetic layer.
(b) Resonant surface towards bottom
of magnetic layer.

18. Effect of Velocity Shear
Consider the linear, ideal MHD stability of magnetic buoyancy instability
influenced by a shear flow (Tobias & Hughes 2004).
Equilibrium state:
Stratified ideal atmosphere (i.e. no dissipation) with a horizontal magnetic field
B(z)x and an aligned shear flow U(z)x.
Investigate linear stability through:
Energy principle – gives sufficient conditions for stability (Bernstein et al
1958, Frieman & Rotenberg 1960)
Numerical solution of eigenvalue problem – enables investigation of
unstable modes.
Cannot be done via local analysis.

19. Linearised equation of motion, in terms of the displacement ξ of a fluid element from
its equilibrium trajectory:
where
with Q =  x (ξ x B).
Normal mode solutions, proportional to exp(iωt):
and hence
All of the integrals are real. Therefore ω is real and the system linearly stable provided that

20. Seek normal mode solutions of the form ξx = ξx(z) exp(i(kx + ly)), etc.
From Euler-Lagrange equations, stationary values of δW given by
Hence δW is guaranteed to be positive if
A > 0 and AC  B2 everywhere.
Stability guaranteed if this is satisfied and δW is a minimum.
with
Criterion (i)
Provided that U2 < cT2 everywhere, stability of the equilibrium state is assured if the
following inequality is everywhere satisfied:
In particular, stability to all modes (for any finite wavenumber) is guaranteed if, everywhere,
Extension of Adam’s (1978) result.
(+ Galilean transformations)

21. Alternatively, from integration by parts, we can write δW as
hence δW is guaranteed to be positive provided that δW is a minimum and:
A > 0 and dB /dz < C everywhere.
Criterion (ii)
If U2 < cT2 everywhere, then stability is assured provided that the following inequality
is everywhere satisfied:
For two-dimensional undulatory modes (l = 0) this simplifies considerably to:
(“Richardson criteria”)

22. 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 + ζz) and are also
taken to be isothermal.
Without shear, two other parameters define basic state: plasma β and λ=gd/VA2.

23. Flow (i):
U = δ(z3 / 6 – z2 / z / 12 )
Flow (ii):
U = U0 tanh α (z – zs)

24. Comparison of role of shear for
moderate and large l, flow 1, β = 10,
k = 0.32.
l = 4 (dashed)
l = 8 (solid)

25. Eigenfunctions – flow 1, δ = 10.
Eigenfunctions – no shear.

26. Effect of flow 1. Shear is stabilising
Effect of flow 1. Shear is stabilising. No effect on k = 0 modes, which here are stable.

27. Growth rate as a function of location
of the shear (zs) for flow 2.
U0 = 0.7; α = 1, 5, 10.
Optimal stabilising effect at zs ~ 0.75,
at which the eigenfunctions of the
unsheared instability are peaked.

28. Eigenfunctions of u and Bz for α = 10
and zs = 0, 0.25, 0.5, 0.75.

29. Growth rates in the absence of shear.
Contour plot of growth rate for flow
2, with α = 10 and zs = 0.75.
Effect of shear is again stabilising.

30. Things to Think About Regarding the magnetic buoyancy instability …
What is the role of diffusion on the buoyancy/shear instability?
How are the diffusion coefficients – and hence the instability – influenced
by a (weak) magnetic field?
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?
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 α-effect), which
can act to transform BT to BP.

31. Things to Think About More generally …
Is the solar cycle field generated and/or stored in the tachocline?
If so, how much of the tachocline does it occupy?
Is this compatible with a tachocline with circulation in an essentially field-free region? i.e. a two-layer tachocline