PPARC 2006

The Solar Interior

David Hughes

Department of Applied Mathematics

University of Leeds

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The solar interior is both very well understood, and not very well understood at all.

Well understood

Static, one-dimensional, non-magnetic Sun.

Not very well understood

Dynamic, three-dimensional, magnetic Sun.

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Solar Structure

Internal solar structure determined by solution of differential equationsgoverning pressure balance, energy input etc.

Solar Interior

1. Core2. Radiative Interior3. (Tachocline)4. Convection Zone

Visible Sun

1. Photosphere2. Chromosphere3. Transition Region4. Corona5. (Solar Wind)

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Solar Core

Central 25% (175,000 km)Temperature at centre 1.5 x 107 K Temperature at edge 7 x 106 KDensity at centre 150 g cm-3 Density at edge 20 g cm-3

Temperature in core high enough for nuclear reactions. ENERGYp-p chain: 3 step process (above) leads to production of He4 and neutrinos ().

Missing neutrinos (not as many detected as thought). Neutrino mass

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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.

PPARC 2006The 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.

PPARC 2006The Photosphere: Granules

Convection at solar surface canbe 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) .

PPARC 2006The Photosphere: Supergranules

Can also see larger structuresin convection patterns

(Mesogranules) and Supergranules

Seen in measurements of Dopplerfrequency.

Cover entire Sun

Lifetime: 1-2 days

Flow speeds: ~ 0.5kms-1

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The Sun’s Global Magnetic field

Ca II emission Extreme ultra-violet

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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?

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The static one-dimensional Sun omits all the interesting dynamic effectsresulting from convective motions, differential rotation and magnetic fields.

It is though worth pointing out that the rotational and magnetic energies are,globally, extremely small.

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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.

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Or, in dimensionless form: ,1

)( 2BBuB

Rmtetc.

Basics for the SunDynamics in the solar interior is governed by the following equations of MHD:

.

terms,loss)(

,0).(

,.

),0.()( 2

TRpDt

pD

t

pt

t

otherviscous

u

FFgBjuuu

BBBuB

INDUCTION

MOMENTUM

MASS CONSERVATION

ENERGY

GAS LAW

PPARC 2006Basics for the Sun

PHdgRa

4

ULRe

ULRm

Pr

Pm

LURo 2

scUM

202

Bp

1020

1013

1010

10-7

105

10-3

10-4

0.1-1

1016

1012

106

10-7

10-6

1

110-3-0.4

BASE OF CZ PHOTOSPHERE

(Ossendrijver 2003)

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Modelling: physical parameters

Re

Rm

Pm

=1

Stars

Liquid metal experiments

simulations

IM

Pm=1

102

103

103 107

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lLN /

Lu3

Number of grid points, in one direction

43

ReN

Thus, ratio of largest to smallest physically important scales is: 43

min

max ReLL

For Kolmogorov turbulence, dissipation length 41

3

l

Since where ε is energy generation rate, then

Current simulations can deal with a dynamic range of O(103).

So if true viscous scale is to be resolved (O(1mm)) then largest outerscale is of the order of a few metres.

Alternatively, one can simulate the entire Sun, but at Reynolds numbers O(103).

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Brun, Miesch, Toomre

Anelastic Differential rotation and meridional circulation

Numerical simulations of rotating convection

Consistent with Taylor-Proudman theorem.

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Rotating Convection Experiment

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Rotating Convection Experiment

Increasing the radial heating

Slow rotation

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The Sun’s Internal Rotation Rate

Internal solar rotation rate.

Angular velocity constant on radial linesin the convection zone. Radiative interiorin solid body rotation.

Note the thin layer of strong radial shear at the base of the convection zone – known as the tachocline.

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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.

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Governing equations (thin layer approximation)

transport of heat

meridional motions - anelastic approximation

variables separate:

conservation of angular momentum

radiative spreading

hydrostatic equilibrium

geostrophic balance

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Radiative spreading

(Elliott 1997)

at solar age

boundary conditions (top of radiation zone)

initial conditions

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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.

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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 / ~ 0.7, tachocline / ~10-6)

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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

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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 22

20

r

B

Gough & McIntyre (Nature 1998)

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Current Thinking…

OVERSHOOTING (AND MAYBE PENETRATIVE) CONVECTIONTURBULENCE IS 3DMAGNETICSTRONG MEAN DYNAMO FIELD…BUOYANCY INSTABILITIES

MHD TURBULENCE DRIVEN FROM ABOVEVERY STABLY STRATIFIED – 2DWEAK MERIDIONAL FLOWLATITUDINAL ANGULAR MOMENTUM TRANSPORTWEAK MEAN FIELD, BUT MHD IMPORTANT…

NOMINALBASE

END OFOVER-SHOOTING

BASE OFTACHOCLINE

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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, ())• 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.

0)]()1[( 22

2

dd

411010~ 42

2

2

U

NRi

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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)

cossinab

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Sketch of emergence of magnetic fieldas bipolar regions (after Parker 1979).

Simulation of 3d nonlinear evolution ofmagnetic buoyancy instability of a layerof magnetic gas.

(Matthews, Hughes & Proctor 1995)

Magnetic Buoyancy Instabilities

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