The Solar Interior NSO Solar Physics Summer School

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The Solar InteriorNSO Solar Physics Summer
School
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(No Transcript)
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Equations of Stellar Structure
SSM assumes spherical symmetry and
neglects rotation and magnetism
Hydrostatic Equilibrium
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
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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
Pre-Asplund et al.
Difference between SSM and helioseismology VERY
GOOD AGREEMENT OVERALL, but clear issues At the
base of the convection zone
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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
Once generated, the energy must escape
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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).
r2
Adiabactic displacement
r1
blob continues to move upward gt
convectively unstable
Sub-adiabatic (stable, radiative)
Super-adiabatic (unstable, convective)
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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!!
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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 gt 1D 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 gt convective flux
The treatment of convection remains one of the
major uncertainties in modern SSM
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Flows in Solar Interior
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The (Magneto-) Hydrodynamic Equations
Mass Conservation
Momentum Conservation
Energy Conservation
Magnetic Induction
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The problem with solving these equations
Equations are highly nonlinear Velocity depends
on magnetic field and density, which both depend
on velocityEquations must be solved as a
coupled system (7 equations eos)
Convection (and dynamo) are inherently 3D Cant
get a dynamo in 2D (Cowlings Theorem - next
lecture) 3D convection significantly different
than 2D
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
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SOHO
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Global Simulations (circa 1985)
3D spherical shell simulations of convection zone
in anelastic approximation
Simulations showed banana cell
structure reminiscent of Taylor-Proudman
constraint
At surface
Anelastic approximation filters sound waves,
good approx. when vc ltlt cs
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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
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Along came helioseismology

Differential rotation observed at
surface persists through CZ- ANGULAR
VELOCITY CONSTANT ON RADIAL LINES NOT ON
COLUMNS
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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)
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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 gt Thermal Wind
In steady state, incompressible, neglecting
viscosity
Negative latitudinal entropy gradient leads to
negative vertical rotation gradient
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Rempel (2005)

Solve axisymmetric MEAN FIELD equations with a
(parametrized) model for angular momentum
transport and no convection
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Revised 3D numerical simulations

If 3D simulations impose a latitudinal entropy
gradient as bottom boundary condition
3D simulation (M. Miesch)
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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 hasnt been studied, interested?
This leads to the obvious question as to what
causes the strong differential rotation in the
tachocline (radial and latitudinal)
gt Ultimately, why is the interior rotating
uniformly? Magnetic Field confined to the
radiative interior enforces uniform rotation via
Ferraros isorotation law
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Ferraros Isorotation Law
(axisymmetry)
(poloidal field)
(steady state)
For a steady state, axisymmetric, poloidal field
angular velocity must be constant along field
lines
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Magnetic Model for Uniform Rotation
MacGregor Charbonneau, Gough McIntyre 1998,
Garaud Rogers, etc.
Currently favored model
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 --gt no uniform
rotation HOW TO CONFINE THE FIELD
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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
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Understanding the Internal Rotation profile is a
key ingredient to understanding the solar
dynamo.the source of all magnetic activity
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Convection Zone (observed)
Convection
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
Meridional Circulation
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Radiation Zone (observed)
What we expect Internal gravity waves,
meridional circulation, small scale turbulence
Torsional Oscillations
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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
gt filters prograde waves allows through only
retrograde waves (negative angular momentum)
3. The deposition of negative angular
momentum brings about uniform rotation