Connecting solar radiance variability to the solar … solar radiance variability to the solar dynamo using the virial theorem Oskar Steiner Kiepenheuer-Institut fu¨r Sonnenphysik,

Connecting solar radiance variability to the

solar dynamo using the virial theorem

Oskar Steiner

Kiepenheuer-Institut fur Sonnenphysik, Freiburg i.Br.

[email protected]

http://www.uni-freiburg.de/˜steiner/

toc ref

http://www.kis.uni-freiburg.de

http://www.kis.uni-freiburg.de

1. What can the virial theorem tell us?

Assuming that the global solar luminosity varies approximately like solar irradiance,

then the virial theorem unavoidable implies structural changes in the Sun or in partial

layers of it on that same time scale. What kind of structural changes might it be?

Composite of the total solar irra-

diance. From C. Frohlich under

www.pmodwrc.ch → more

toc Oskar Steinerref

What can the virial theorem tell us? (cont.)

A general form of the virial theorem, including the magnetic field, is given by

Chandrasekhar & Fermi (1953):

1

2

d2J

dt2= 2K + Ω + M + 3

Z

R

P dV + S ,

J(t) =

Z

R

ρr2dV ,

K(t) =1

2

Z

R

ρv2dV ,

Ω(t) =1

2

Z

R

ρΨdV ,

M(t) =

Z

R

B2

2µ0

dV ,

3

Z

R

P dV = 3(γ − 1)U(t) ,

S(t) = −

I

∂R

PtotdS +1

µ0

I

∂R

(r · B)B · dS .

where J is the moment of inertia, K the kinetic energy of mass motion, Ω the

gravitational binding energy of the star with Ψ being the gravitational potential, M the

magnetic energy, U the internal energy, and S a surface integral over the boundary

∂R of the region R occupied by the star.



1

2

d2J

dt2= 2K + Ω + M + 3

∫R

P dV + S

A variation of the total internal energy (as a consequence of luminosity

variation) goes along with corresponding changes in one or several of the other

virials, one of which being surely the total magnetic energy.



The region R encompasses the convection zone, where the lower boundary lies

beneath the overshoot region in the radiative zone with pg = const and B = 0. At

the upper boundary Pg = 0.

R1

R2

R

gp = const

core

convection zone

S(t) = −

I

∂R

PtotdS +1

µ0

I

∂R

(r · B)B · dS

= 4πR31pg(R1)

| z

1.4×1040[J]

+4πR32f

B2(R2)

2µ0| z

1.5×1028[J]

+1

µ0

4πR32fB2

⊥(R2)

| z

3×1028[J]


2. Estimates of the various virials

The magnetic virial: M(t) =∫R

B2/(8π)dV . One solar cycle generates a

magnetic flux of Φ ≈ 1016 Wb. If this flux resides in the overshoot layer in the

form of flux tubes with a strength of 10 T, it has a total magnetic energy of

M ≈ 5 × 1032 J

The inertial virial: d2J/dt2 ≈ J/τ2cyc ≈

∫R

ρr2dV/τ2cyc. ⇒

d2J/dt2 ≈ 10−3 × M

The thermal virial: The equivalent variation in thermal energy over a solar cycle

due to the solar luminosity variation is δL ≈ 0.001 · L⊙ so that

∆Urad ≈ 1032 J

“Coincidence” of equal magnetic and thermal energy change (Schussler, 1996)


Estimates of the various virials (cont.)

The kinetic virial: The kinetic energy of convective motion in the convection

zone is Kcon = (1/2)∫R

ρv2

condV = 7.8 × 1031 [J] = 0.16 × M .

In the tachocline, where magnetic intensification presumably takes place, the

available convective kinetic energy reduces to Kcon ≈ 10−4 × M .

Also, because Fconv ∝ v2conv <

!10−3, convective motion cannot be tapped

in sufficient amounts. Since turbulent, convective motion is unlike to produce

coherent large scale magnetic flux ropes in the overshoot layer, we assume the

variation of Kcon with the solar cycle to be negligible, ∆Kcon = 0.

The kinetic energy of rotation in the convection zone is

Krot = (1/2)∫R

ρ(Ω(θ, r) × r)2dV = 5.5 × 1034 [J] = 110 × M .

However, the difference to the kinetic energy of rigid rotation under the

condition of equal rotational momentum, i.e., the available kinetic energy from

differential rotation is ∆Krot = 1 × 1033 [J] = 2 × M .



The available kinetic energy from differential rotation in the tachocline alone

amounts to ∆Krot = 1 × 830 [J] = 0.015 × M .

Thus, differential rotation in the magnetic layer would have to be very efficiently

replenished from layers atop for feeding the energy required for field

intensification. In lack of such a mechanism we have

K(t) = (1/2)∫R

ρv2dV = const

“Torsional oscillations” are due to a geostrophic flow driven by temperature

variations at the surface according to Spruit (2003), not to Lorentz forces.



Considering the variations over a solar cycle, only three significant virias of the

full virial equation, (1/2)d2J

dt2= 2K + Ω + M + 3

∫R

P dV + S, remain:

Ω + M + 2U = 0 .

Note that for the total convection zone

Ω = 2 × 1040[J] and

U = 6 × 1039[J]

A possible scenario that relates the three terms Ω, M , and U over a solar

cycle is provided by the flux-tube dynamo.


3. Magneto-convective energy transport

core

convection zone

Magnetic flux tube at the bottom of the

convection zone (dashed circle),



core

convection zone


convection zone (dashed circle), rising

through the convection zone to the

surface (solid red circle),



core

convection zone


convection zone (dashed circle), rising

through the convection zone to the

surface (solid red circle), where it forms

a sunspot pair

- Schussler, Caligari, Ferriz-Mas, Moreno Insertis et al.

- Fisher, Fan, Longcope, Linton et al.

- D’Silva, Choudhuri et al.


Magneto-convective energy transport (cont.)

0zcrit

pp0e

ln

20B 2

ln

ei

µ

Gas pressure decrease more rapidly in

the superadiabatic external

atmosphere, pe, than in the adiabatic

flux-tube atmosphere, pi.



0zcrit

pp0e

ln

20B 2

ln

ei

µ





expz

At the critical height zcrit = zexp, the

magnetic pressure must vanish and the flux

tube “explodes” into a cloud of weak field.



0zcrit

pp0e

ln

20B 2

ln

ei

µ





expz

At the critical height zcrit = zexp, the

magnetic pressure must vanish and the flux

tube “explodes” into a cloud of weak field.

⇒ Flux intensification by “flux-tube explosion”:

Moreno Insertis, Caligari, & Schussler (1995), Rempel & Schussler (2001) → more



Simulation of the flux-tube explosion by Matthias Rempel

Evolution of the entropy and the magnetic field strength in the course of a

“flux-tube explosion”.



Thermal and flow structure beneath a sunspot. From the SOI/MDI home page by

Kosovichev, et al.

Left: The sound speed beneath a sunspot. Redder colors are faster (hotter) and bluer colors are

slower (cooler). Right: Motion beneath a spot. Material flows at the depth of 0 — 3 Mm (a) and at

6 — 9 Mm (b). The outline of the sunspot umbra and penumbra is shown. The colors indicate

vertical motion with red corresponding to downflow. The arrows show horizontal motion. The

longest arrows correspond to 1.0 km/s.


http://soi.Stanford.EDU/results/2001_MDI_sr_review/

4. The solar cycle in terms of the virial theorem

m1 Weak toroidal field of B ≈ 1 T. Ω1, M1, and U1 initial virials.m2 This field, located in a sheet near the bottom of the convection zone, gets

amplified to 10 T by the process of “flux-tube” explosion.

∆M = M2 − M1 ≈ M ≈ 1032 J > 0 , ∆Ω = Ω2 − Ω1 ≥?

0 ,

so that, according to the virial theorem,

∆U = U2 − U1 = −1

2(∆Ω + M) ≈ −1032 J < 0 ,

leaving an internal energy deficiency of another 1032 J, which must be gained

by a reduction in radiation loss from the Sun in fulfillment of

Etot 1 = U1 + Ω1 + M1 = Etot 2 = U2 + Ω2 + M2 + Rր


The solar cycle in terms of the virial theorem (cont.)m3 Magnetic field gets ejected or annihilated, clearing stage for the next solar

cycle with opposite magnetic sign. Time of magnetic activity at the solar

surface.

∆M = M3 − M2 ≈ M ≈ −1032 J < 0 , ∆Ω = Ω3 − Ω2 ≤?

0 ,

so that, according to the virial theorem,

∆U = U3 − U2 = −1

2(∆Ω + M) ≈ 1032 J > 0 ,

leaving an internal energy excess of 1032 J, which must be lost by an

excess of radiation from the Sun.


The solar cycle in terms of the virial theorem (cont.)

The thermodynamic cycle:

1st stage 2nd stage

conv

ectio

nzo

ne

extra entropy−richupflow

conv

ectio

nzo

ne

throttledenergy flux

extraradiation

extra entropy−deficientdownflow

These processes work on a hydrodynamic time-scale.


5. Conclusion

According to the present ideas

- Solar radiance variability is an expression of the thermodynamic cycle

associated with the magnetic solar cycle

- The virial theorem provides an explanation for the hitherto “coincidence” of

the magnetic energy generated over one solar cycle equating the solar

luminosity variation integrated over the same period

- Observational consequences might bear the predicted

- variation of the superadiabaticity, δ, with the solar cycle,

- correlation between stellar luminosity variation and the magnetic energy

generated over a stellar cycle


'&

$%

Hydrodynamic vs. magnetohydrodynamic mixing-length convection

∆s

mixT∆

s

T

∆ Tmc mix

Ftot = (1 − ϕ)Fc + ϕFmc δFtot = ϕ(Fmc − Fc)

→ backto § 3HH

'&

$%

The excess energy flux caused by the flux-tube explosion is according to Rempel (2001)

δL = m∆sT

= mTcp

Hp

(∇−∇ad)| z

δ

∆r = mg

∇ad

δ ∆r ,

where m is the mass per cycle period to be discharged from the initial magnetic sheet

at the bottom of the convection zone:

d Rc

A = πR 2c8

m = ρ0Ad/τcyc ,

d =Φ

12πRcB0

= 1.2 · 107m .

δL = 0.002 · L⊙

»δ

10−6

–

·

»∆r

108m

–

→ backto § 3HH

'&

$%

Abstract flux intensification process

convection zone

core

dweak flux sheet strong flux sheetdense flux sheetsubadiabatic

superadiabatic

Mass in a weak magnetic

layer is bottled and ...

... returned on top.

→ Strong flux sheet

buoyant below a slightly

lifted atmosphere.

Upward drift in

subadiabatic overshoot

layer reestablishes

mechanical equilibrium.

U1, Ω1, M1 ≈ 0 U2 < U1, Ω2 =?

Ω1,

M2 > 0

→ backto § 3

'&

$%

Facular regions as seen in the ecliptic plane and perpendicular to it

ecliptic planeview perpendicular toview in ecliptic plane

→ backto § 1

Table of content

1. What can the virial theorem tell us?

2. Estimates of the various virials


4. The solar cycle in terms of the virial theorem

5. Conclusion

6. References


References

Chandrasekhar, S. and Fermi, E.: 1953, ApJ 118, 116

Frohlich C. and Lean J., 1998, in New Eyes to see inside the Sun and Stars, F.L.

Deubner (ed.), IAU Symp. 185, Kluwer, Dordrecht

Moreno Insertis, F., Caligari, P., and Schussler, M.: 1995, ApJ 452, 894

Rempel, M.: 2001, Struktur und Ursprung starker Magnetfelder am Boden der solaren

Konvektionszone, PhD-Thesis, Georg-August-Universitat Gottingen

Rempel, M. and Schussler, M.: 2001, ApJ 552, L171

Schussler, M.: 1996, in Solar and Astrophysical Magnetohydrodynamic Flow, NATO ASI

Series Vol. 481, K.C. Tsiganos (ed), Kluwer, Dordrecht, p. 17

Steiner, O.: 2003, Astronomische Nachrichten, 324, 106

Steiner, O.: 2004, in Multi-Wavelength Investigations of Solar Activity, IAU Symp. 223,

A. V. Stepanov, E.E. Benevolenskaya, & A.G. Kosovichev (eds.), in press


Connecting solar radiance variability to the solar … solar radiance variability to the solar dynamo using the virial theorem Oskar Steiner Kiepenheuer-Institut fu¨r Sonnenphysik,

Documents