1. David Hughes Department of Applied Mathematics University of Leeds
The Solar Interior
David Hughes
Department of Applied Mathematics
University of Leeds

2. The solar interior is both very well understood, and not very well understood at all.
Static, one-dimensional, non-magnetic Sun.
Not very well understood
Dynamic, three-dimensional, magnetic Sun.

3. Solar Structure Solar Interior Visible Sun
Internal solar structure determined by solution of differential equations
governing pressure balance, energy input etc.
Solar Interior
Core
Radiative Interior
(Tachocline)
Convection Zone
Visible Sun
Photosphere
Chromosphere
Transition Region
Corona
(Solar Wind)

4. Solar Core Central 25% (175,000 km)
Temperature at centre 1.5 x 107 K Temperature at edge 7 x 106 K
Density at centre g cm Density at edge g cm-3
Temperature in core high enough for nuclear reactions. ENERGY
p-p chain: 3 step process (above) leads to production of He4 and neutrinos (n).
Missing neutrinos (not as many detected as thought). Neutrino mass

5. The Radiative Zone Extends from 25% to ~70% of the solar radius.
Energy produced in core carried by photon radiation.
Density drops from 20 g cm-3 to 0.2 g cm-3.
Temperature drops from 7 x 106 K to 2 x 106 K.

6. The Convection Zone
Extends from 70% of the solar radius to visible surface.
Radiation less efficient as heavier ions not fully ionised (e.g. C, N, O, Ca, Fe).
Fluid becomes unstable to convection. Motions highly turbulent. Motion on large range of scales. Fluid mixed to be adiabatically stratified.
Temperature drops from 2 x 106 K to 5,800 K.
Density drops exponentially to 2 x 10-7 g cm-3
Convection visible at the surface (photosphere) as granules and supergranules.

7. The Photosphere: Granules
Convection at solar surface can
be seen on many scales.
Smallest is granulation.
Granules ~ 1000 km across
Rising fluid in middle.
Sinking fluid at edge
(strong downwards plumes)
Lifetime approx 20 mins.
Supersonic flows (~7 kms-1) .

8. The Photosphere: Supergranules
Can also see larger structures
in convection patterns
(Mesogranules) and Supergranules
Seen in measurements of Doppler
frequency.
Cover entire Sun
Lifetime: 1-2 days
Flow speeds: ~ 0.5kms-1

9. The Sun’s Global Magnetic field
Ca II emission Extreme ultra-violet

10. Solar Rotation
Differential rotation with latitude observed at the solar surface.
Equator rotates in approx 25 days, the poles in approx 40 days.
What about the internal rotation rate?

11. The static one-dimensional Sun omits all the interesting dynamic effects
resulting from convective motions, differential rotation and magnetic fields.
It is though worth pointing out that the rotational and magnetic energies are,
globally, extremely small.

13. We know the correct equations.
What might we expect, from theoretical considerations, for the
internal rotation and magnetic field?
Why is this a difficult problem?
We know the correct equations.
But, we cannot solve them in the appropriate regime.

14. Basics for the Sun
Dynamics in the solar interior is governed by the following equations of MHD:
INDUCTION
MOMENTUM
MASS CONSERVATION
ENERGY
GAS LAW
Or, in dimensionless form:
etc.

15. Basics for the Sun
BASE OF CZ
PHOTOSPHERE
1020
1013
1010
10-7
105
10-3
10-4
0.1-1
1016
1012
106
10-7
10-6 1
(Ossendrijver 2003)

16. Modelling: physical parametersRe Rm
Pm =1
Stars
Liquid metal experiments
simulations IM
Pm=1
102
103
107

17. For Kolmogorov turbulence, dissipation length
Number of grid points, in one direction
Since
where ε is energy generation rate, then
Thus, ratio of largest to smallest physically important scales is:
Current simulations can deal with a dynamic range of O(103).
So if true viscous scale is to be resolved (O(1mm)) then largest outer
scale is of the order of a few metres.
Alternatively, one can simulate the entire Sun, but at Reynolds numbers O(103).

18. Numerical simulations of rotating convection
Anelastic
Differential rotation and meridional circulation
Consistent with Taylor-Proudman theorem.
Brun, Miesch, Toomre

19. Rotating Convection Experiment

21. Rotating Convection Experiment
Increasing the radial heating 
Slow rotation

22. The Sun’s Internal Rotation Rate
Internal solar rotation rate.
Angular velocity constant on radial lines
in the convection zone. Radiative interior
in solid body rotation.
Note the thin layer of strong radial shear at the base of the convection zone – known as the tachocline.

23. Why is there a tachocline?
What is the mechanism that leads to the formation of this thin shear layer?
Take the differential rotation of the convection zone as given (transport of angular momentum by stresses of turbulence)
Spiegel & Zahn (1992): a radiation driven meridional circulation transports angular momentum from the convection zone into the interior along cylinders  no tachocline.

24. Governing equations (thin layer approximation)
hydrostatic equilibrium
geostrophic balance
meridional motions - anelastic approximation
transport of heat
conservation of angular momentum
variables separate:
radiative spreading

25. Radiative spreading at solar age  (Elliott 1997)
boundary conditions (top of radiation zone)
initial conditions
(Elliott 1997)
at solar age 

26. Spiegel & Zahn (1992) postulated that there would be two-dimensional turbulence
in the stably stratified envelope immediately beneath the convective envelope.
They argued that this would provide little stress to transport angular momentum radially.

27. Is there a linear instability leading to turbulence?
Is this turbulence 2D?
If this turbulence is 2D does it really act so as to transport angular momentum towards a state of constant differential rotation?—i.e does the turbulence act as anisotropic viscosity?
Analogy with atmospheric models says that (hydrodynamic) 2D turbulence of this type acts so as to mix PV and drive the system away from solid body rotation (Gough & McIntyre 1998, McIntyre 2003)
Anti-friction
(though the addition of a magnetic field can change angular momentum transport – cf MRIs)
(also atmosphere n/k ~ 0.7, tachocline n/k ~10-6)

28. Magnetic models
A relic field in the interior can keep the interior rotating as a solid body (Mestel & Weiss – 10-2 G)
But if magnetic coupling spins down the radiative interior, why doesn’t angular velocity propagate in along field lines?
MacGregor & Charbonneau (1999) suggested that all the field lines must be contained in the radiative interior (no magnetic coupling)
Hence if hydrodynamic and isotropic,
differential rotation propagates in along cylinders.
If magnetic, then differential rotation propagates in
along field lines

29. Gough & McIntyre Model
Meridional circulation due to gyroscopic pumping  2 cells with upwelling at mid-latitudes.
Field held down where shear is large, brought up where no shear.
Coupling still there, but no differential rotation brought in.
Delicate balance, dependent on Elsasser number
Gough & McIntyre (Nature 1998)

30. Current Thinking… NOMINAL BASE
OVERSHOOTING (AND MAYBE PENETRATIVE) CONVECTION
TURBULENCE IS 3D
MAGNETIC
STRONG MEAN DYNAMO FIELD…
BUOYANCY INSTABILITIES
END OF
OVER-
SHOOTING
MHD TURBULENCE DRIVEN FROM ABOVE
VERY STABLY STRATIFIED – 2D
WEAK MERIDIONAL FLOW
LATITUDINAL ANGULAR MOMENTUM TRANSPORT
WEAK MEAN FIELD, BUT MHD IMPORTANT…
BASE OF
TACHOCLINE

31. Stability of the Tachocline
“Chicken & Egg”
Hydrodynamic instabilities
Radial shear instability (K-H type)
(radial motions suppressed, diffusion can change picture… hydro statement)
Latitudinal differential rotation instability (2D, (q,f))
Fjortoft (1950) stable if
Watson (1981)
Hydrodynamically stable if equator to pole difference < 29 %
(in reality ~ 12%)
Lots of others (shallow water hydro etc) similar conclusions – at best marginally stable.

32. MHD & joint instabilities
Of course the tachocline has a magnetic field.
Field has 2 major effects
Magnetic buoyancy
Imparts tension to plasma
Gilman & Fox (1997)
Joint instability of toroidal flow and toroidal field
cf MRI (Balbus & Hawley 1992, Ogilvie & Pringle 1996)
m=1 mode.
Maxwell stresses in the nonlinear regime would act so as to transport angular momentum towards pole (opposite to Reynolds stresses)

33. Magnetic Buoyancy Instabilities
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.
(Matthews, Hughes & Proctor 1995)