1. The Solar Interior NSO Solar Physics Summer School Tamara Rogers, HAO June 14, 2007 trogers@ucar.edu

4. Equations of Stellar Structure Hydrostatic Equilibrium Mass Conservation Energy Generation Energy Transport SSM assumes spherical symmetry and neglects rotation and magnetism SSM uses MLT to describe energy transfer in the convection zone. These equations are solved for radial structure of density, temperature, pressure, mass structure etc. in the solar interior

5. Standard Solar Model (SSM) Solve the previous equations with an equation of state (EOS) and opacity Start with zero age main sequence (ZAMS) and evolve the equations forward to present day, where we can compare the Mass, Radius, Luminosity and composition to observed values. ADJUSTABLE PARAMETERS: Helium abundance, heavy element abundance and mixing length parameter Difference between SSM and helioseismology VERY GOOD AGREEMENT OVERALL, but clear issues At the base of the convection zone Pre-Asplund et al. Pijpers, Houdek et al. Model S Z = 0.015 *

6. The Core (Energy Source) PP PP 4 1 H P P nn 1 4 He + ENERGY Once generated, the energy must escape The core is that region of the Sun that is hot and dense enough for nuclear reactions to take place, it includes approximately to 10% of the solar mass

7. Energy Transfer b b s Adiabactic displacement The process by which energy is transferred from the core depends on the density/ temperature gradient (and to some extent the composition gradient). r1r1 r2r2 if @ r 2 blob continues to move upward => convectively unstable Sub-adiabatic (stable, radiative) Super-adiabatic (unstable, convective)

8. The Radiation Zone In the radiation zone Can calculate a mean free path for the generated photons to interact with the matter in the radiation zone ~0.5cm, so the photons random walk out of the radiative interior taking ~ 30000 years!!

9. The Convection Zone At some point the opacity increases substantially, temperature gradient increases and convection sets in. Unlike radiation, heat transfer by convection is very complicated and inherently 3 dimensional ==> 1D SSM use Mixing Length Theory (MLT) The treatment of convection remains one of the major uncertainties in modern SSM Convective element travels a “mixing length”, written as a fraction of the pressure scale height, before diffusing and sharing its excess heat with the surroundings is the fraction and is the adjustable “mixing length” parameter The excess heat of the blob combined with its velocity can give you an Estimate of the amount of energy transferred ==> convective flux *

10. Flows in Solar Interior La Palma Granulation * Differential Rotation Meridional Circulation

11. The (Magneto-) Hydrodynamic Equations Mass Conservation Momentum Conservation Energy Conservation Magnetic Induction *

12. The problem with solving these equations The Sun is highly turbulent Resolving length scales from the radius of the sun down to (say) a sunspot would require ~10 10 grid points resolved for timescales of at least several rotation periods (to understand rotation) and 22 years (to understand dynamo) *for 10 year resolution, ~ 10 6 -10 7 processor hours!! Convection (and dynamo) are inherently 3D Cant get a dynamo in 2D (Cowlings Theorem - next lecture) 3D convection significantly different than 2D Equations are highly nonlinear Velocity depends on magnetic field and density, which both depend on velocity…Equations must be solved as a coupled system (7 equations + eos) Nevertheless, people try… Numerical simulations are always carried out at lower Re (not so turbulent) out of computational necessity, with the hope that once in a turbulent regime qualitative behavior is the same

13. SOHO La Palma LOCAL MODELS OF COMPRESSIBLE MHD OBSERVATIONAL DATA Numerical Modeling HelioseismologySimulation GLOBAL MODELS OF ANELASTIC MHD *

14. Cattaneo & Emonet *

15. Stein et al. *

16. Local Simulations Generally done in cartesian coordinates representing some region in convection zone, sometimes with rotation and magnetic fields Granulation (observed) Simulation Cattaneo *

17. Global Simulations (circa 1985) 3D spherical shell simulations of convection zone in anelastic approximation * * Anelastic approximation filters sound waves, good approx. when v c << c s Simulations showed “banana cell” structure reminiscent of Taylor-Proudman constraint At surface

18. Taylor Proudman Taylor-Proudman columns occur when system is in geostrophic balance Pressure gradients balance Coriolis force taking the curl one gets (assuming incompressible) Fluid velocity is uniform along lines parallel to

19. Along came helioseismology… Differential rotation observed at surface persists through CZ- ANGULAR VELOCITY CONSTANT ON RADIAL LINES *NOT* ON COLUMNS *

20. HelioseismologySimulation Global Simulations (circa 2000) Solve full nonlinear 3D equations in the convection zone under the Anelastic approximation Unfortunately….still get angular velocity constant on cylinders 3D simulation (M. Miesch) *

21. Global Simulations (circa 2000)

22. Model for Differential Rotation (Rempel 2005) If there is a latitudinal entropy gradient in the tachocline (or at base of solar convection zone) can break Taylor-Proudman balance ==> Thermal Wind In steady state, incompressible, neglecting viscosity Negative latitudinal entropy gradient leads to negative vertical rotation gradient

23. Rempel (2005) Solve axisymmetric MEAN FIELD equations with a (parametrized) model for angular momentum transport and no convection *

24. Revised 3D numerical simulations If 3D simulations impose a latitudinal entropy gradient as bottom boundary condition 3D simulation (M. Miesch) *

25. Why Latitudinal Entropy gradient? Its just most likely culprit for balancing the differential rotation… its not cleary how Reynolds stresses/Magnetic Stresses affect this balance… hasn’t been studied, interested? This leads to the obvious question as to what causes the strong differential rotation in the tachocline (radial and latitudinal) ==> Ultimately, why is the interior rotating uniformly? Magnetic Field confined to the radiative interior enforces uniform rotation via Ferraro’s isorotation law *

26. Ferraro’s Isorotation Law For a steady state, axisymmetric, poloidal field angular velocity must be constant along field lines (axisymmetry) (poloidal field) (steady state)

27. Magnetic Model for Uniform Rotation MacGregor & Charbonneau, Gough & McIntyre 1998, Garaud & Rogers, etc. *Currently favored model Field lines open to convection zone Field lines are confined to Radiation zone When field lines are confined to radiative interior can enforce uniform rotation (as expected from Ferraro)…HOWEVER, if the field lines open to convection zone --> no uniform rotation: HOW TO CONFINE THE FIELD

28. Gough & McIntyre (1998) Meridional Circulation at BCZ confines the field However, its not clear that the MC at the BCZ is strong enough to confine the field, simulations seem to indicate its not More recent results indicate that convective overshoot is able to confine the field, at the moment it is still not 100% clear

29. *

30. Understanding the Internal Rotation profile is a key ingredient to understanding the solar dynamo….the source of all magnetic activity

31. Convection Zone (observed): Convection Differential Rotation Meridional Circulation Differential rotation and meridional circulation observed using p-modes From observations, know there are pressure waves, large scale meridional flow, azimuthal flow and small scale convection

32. Radiation Zone (observed): Differential Rotation Torsional Oscillations What we expect: Internal gravity waves, meridional circulation, small scale turbulence

33. Gravity Wave Model for Uniform Rotation Talon, Kumar & Zahn 2002 1. Wave-Mean Flow oscillation in the solar tachocline (analogous to QBO) 2. Prograde shear layer has larger amplitude than retrograde layer due to magnetic spin down ==> filters prograde waves & allows through only retrograde waves (negative angular momentum) 3. The deposition of negative angular momentum brings about uniform rotation