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

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.

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)

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 150 g cm-3 Density at edge 20 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

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.

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.

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) .

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

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

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?

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.

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.

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.

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 10-3-0.4 (Ossendrijver 2003)

Modelling: physical parameters Re Rm Pm =1 Stars Liquid metal experiments simulations IM Pm=1 102 103 107

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).

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

Rotating Convection Experiment

Rotating Convection Experiment Increasing the radial heating  Slow rotation

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.

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.

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

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

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.

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)

Magnetic models A relic field in the interior can keep the interior rotating as a solid body (Mestel & Weiss 1987 10-3 – 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

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)

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

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.

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)

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)