The Solar Interior NSO Solar Physics Summer School

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

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

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. Pijpers, Houdek et ADJUSTABLE PARAMETERS Helium abundance, heavy element al. abundance and mixing length parameter Pre-Asplund et al. Z= 0. 015 Difference between SSM and helioseismology VERY GOOD AGREEMENT OVERALL, but clear issues Model S At the base of the convection zone

The Core (Energy Source) 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 P P 4 1 H n Pn P + ENERGY 1 4 He Once generated, the energy must escape

Energy Transfer The process by which energy is transferred from the core depends on the density/ temperature gradient (and to some extent the composition gradient). r 2 b Adiabactic displacement s r 1 b if @ r 2 Sub-adiabatic (stable, radiative) Super-adiabatic (unstable, convective) blob continues to move upward => convectively unstable

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. 5 cm, so the photons random walk out of the radiative interior taking ~ 30000 years!!

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 ==> 1 D SSM use Mixing Length Theory (MLT) 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 The treatment of convection remains one of the major uncertainties in modern SSM *

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

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

The problem with solving these equations 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) Convection (and dynamo) are inherently 3 D Cant get a dynamo in 2 D (Cowlings Theorem - next lecture) 3 D convection significantly different than 2 D The Sun is highly turbulent Resolving length scales from the radius of the sun down to (say) a sunspot would require ~1010 grid points resolved for timescales of at least several rotation periods (to understand rotation) and 22 years (to understand dynamo) *for 10 year resolution, ~106 -107 processor hours!! 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

* Numerical Modeling OBSERVATIONAL DATA SOHO LOCAL MODELS OF COMPRESSIBLE MHD La Palma GLOBAL MODELS OF ANELASTIC MHD Helioseismology Simulation

* Cattaneo & Emonet

Stein et al. *

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

Global Simulations (circa 1985) 3 D spherical shell simulations of convection zone in anelastic approximation* Simulations showed “ba reminiscent of Taylor-Pr At surface * Anelastic approximation filters sound waves, good approx. when vc << cs

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

Along came helioseismology… * Differential rotation observed at surface persists through CZANGULAR VELOCITY CONSTANT ON RADIAL LINES *NOT* ON COLUMNS

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

Global Simulations (circa 2000)

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

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

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

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 *

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

Magnetic Model for Uniform Rotation *Currently favored Mac. Gregor & Charbonneau, Gough & Mc. Intyre 1998, Garaud model & Rogers, etc. 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

Gough & Mc. Intyre (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

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

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

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

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