Connecting solar radiance variability to the
solar dynamo using the virial theorem
Oskar Steiner
Kiepenheuer-Institut fur Sonnenphysik, Freiburg i.Br.
http://www.uni-freiburg.de/˜steiner/
toc ref
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.
toc Oskar Steinerref
What can the virial theorem tell us? (cont.)
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.
toc Oskar Steinerref
What can the virial theorem tell us? (cont.)
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]
toc Oskar Steinerref
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)
toc Oskar Steinerref
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 .
toc Oskar Steinerref
Estimates of the various virials (cont.)
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.
toc Oskar Steinerref
Estimates of the various virials (cont.)
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.
toc Oskar Steinerref
3. Magneto-convective energy transport
core
convection zone
Magnetic flux tube at the bottom of the
convection zone (dashed circle),
toc Oskar Steinerref
3. Magneto-convective energy transport
core
convection zone
Magnetic flux tube at the bottom of the
convection zone (dashed circle), rising
through the convection zone to the
surface (solid red circle),
toc Oskar Steinerref
3. Magneto-convective energy transport
core
convection zone
Magnetic flux tube at the bottom of the
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.
toc Oskar Steinerref
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.
toc Oskar Steinerref
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.
expz
At the critical height zcrit = zexp, the
magnetic pressure must vanish and the flux
tube “explodes” into a cloud of weak field.
toc Oskar Steinerref
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.
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
toc Oskar Steinerref
Magneto-convective energy transport (cont.)
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”.
toc Oskar Steinerref
Magneto-convective energy transport (cont.)
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.
toc Oskar Steinerref
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
toc Oskar Steinerref
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.
toc Oskar Steinerref
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.
toc Oskar Steinerref
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
toc Oskar Steinerref
'&
$%
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
3. Magneto-convective energy transport
4. The solar cycle in terms of the virial theorem
5. Conclusion
6. References
toc Oskar Steinerref
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
toc Oskar Steinerref
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