Understanding the Connection Between Magnetic Fields in the Solar Interior and Magnetic Activity in the Corona W.P. Abbett and G.H. Fisher, B.T. Welsch,

Understanding the Connection Between Magnetic Fields in the

Solar Interior and Magnetic Activity in the Corona

W.P. Abbettand

G.H. Fisher, B.T. Welsch, D.J. Bercik

SSL Colloquium 10-15-04SolarMURI / CISM

Overview

Co-conspirators:S.A. Ledvina, L. Lundquist,J.M. McTiernan, J. Allred,Y. Fan, S.L. Hawley, A. Nordlund, S. Regnier,R.F. Stein

The Workhorse:24 node 48 processor x86 class Beowulf cluster

Outline:• Connecting observations of magnetic field in the solar photosphere to

numerical models of the solar corona

• Active region magnetic fields in the convective envelope below the photosphere

• Quiet Sun magnetic fields, the turbulent dynamo, and X-ray emission in main sequence stars

ILCT: Inductive Local Correlation Tracking

Goal: • To drive a 3D model corona with a time-series of photospheric magnetograms

AR-8210 4hr MDI AR-8210 15min IVM

ILCT: Inductive Local Correlation Tracking

Challenge:

• MHD codes require the specification of electric fields in the boundary volume to advance the solution.

• Thus, we need a flow field physically consistent with the observed evolution of the magnetic field.

That is, we require flows that satisfy the ideal MHD induction equation:

∂B/∂t = x (v x B).

Re-casting the vertical component of the induction equation into a less common, but very useful form, we have:

∂Bz/∂t + · (vBz − vz B) = 0 Under-determined system!

Vertical gradients are unspecified!

ILCT: Inductive Local Correlation Tracking

• In general, transverse motions of magnetic structures, u, are not identical to transverse flows of magnetized plasma, v. The geometric relation of Demoulin & Berger (2003) can relate the two, however:

A velocity field so obtained is not physical

One approach:

• First obtain the approximate transverse motion of magnetic structures, u

(LCT), by maximizing a cross correlation function between successive magnetograms (LCT applied to magnetic elements).

u v − (B/Bz)vz

ILCT: Inductive Local Correlation Tracking

Since the LHS is known, we have a Poisson equation for φthat can be easily solved.

Then the vertical component of the ideal MHD induction equation canbe expressed as • ∂Bz/∂t + · (Bz u) = 0

Now, let’s define scalar quantities φ and ψ in the following way: • Bzu

−φ + x (ψ z)

Then it immediately follows that

∂Bz/∂t = 2 φ

Note that the vertical componentof the induction equation can be satisfied without specifying the transverse components of the magnetic field!

ILCT: Inductive Local Correlation Tracking

Again, we have a Poisson equation

provided that we stipulate that u(LCT)

accurately represents u .

If we now take the curl of both sides of • Bzu

−φ + x (ψ z) ,

we obtain an expression for ψ:

− (x Bzu)·z = 2 ψ

ILCT: Inductive Local Correlation Tracking

With both ψ and φ specified, we now have the uILCT necessary

to compute a photospheric velocity field v that advances Bz in a way that • Matches the observed time evolution of Bz , and • Satisfies the vertical component of the induction equation.

We need only to algebraically solve the following system:

uILCT = v − (B/Bz)vz

Still under-determined

However, if we have a time-series of vector magnetograms, B is known.

Further, the flow of magnetized plasma along B is not constrained by the ideal induction equation --- thus, we have the freedom to close theabove system with a choice of v · B = 0.

ILCT: Inductive Local Correlation Tracking

Thus, we obtain a solution of the form:

v = uILCT − (u

ILCT · B) B/ B2

vz = −(uILCT · B) Bz / B2

EILCT = −(v x B) / c

Note that:

• The induction equation is a linear system.

• With additional information (Doppler signal, simulated flows), it is possible to express a unique solution (e.g. with non-vanishing v · B) as a linear combination of the above velocity field and a particular solution obtained using the additional information.

• ILCT flows that satisfy the resistive induction equation can be obtained using a similar formalism.

ILCT applied to NOAA AR-8210

Beyond ILCT: Data-driven simulationsWhat we have so far:

• The electric field necessary to evolve Bz on the “plane” of the photosphere in a way that matches observations.

We still need:

• Vertical gradients of B and v that evolve B at the model photosphere in such a way as to match observations.

• An initial atmosphere with a magnetic topology that matches that inferred from observations of the corona.

Beyond ILCT: The Initial Atmosphere

We use the Wheatland et al. (2000) approach to construct a non-linear force-free extrapolation based on the first vector magnetogram of thetime series.

We still need:

• Vertical gradients of v that evolve B at the model photosphere in such a way as to match observations.

Beyond ILCT: The Initial Atmosphere

Potential Non-linear FFF MHD

Chr

omos

pher

eP

hoto

sphe

re

Toward a Data-driven Simulation of AR-8210

We still need:• Vertical gradients of v

that evolve B at the model photosphere in such a way as to match observations.

The Plan:• Simultaneously evolve two fully-coupled 3D codes --- a simplified

kinematic boundary code, and a 3D MHD model corona

Directly measuredDirectly measured Calculated by Calculated by boundary codeboundary code

The Evolution of the Transverse Field in the Boundary Layers

Derived Derived by ILCTby ILCT

Initially from Initially from NLFFFNLFFFextrapolationextrapolation

at photosphere, z = 0 above photosphere, z > 0

)vv()vv(

)vv()vv(

yzyzyxyxy

xzxzxyxyx

BBz

BBxt

B

BBz

BByt

B

Toward a Data-driven Simulation of AR-8210

The Emergence and Decay of Active Region Magnetic Fields Below the Photosphere

The dangers of over-interpreting 2D results:

How much field line twist does a flux

tube need to prevent its fragmentation?

The Emergence and Decay of Active Region Magnetic Fields in the Convection Zone

The effects of convective turbulence on active region-scale magnetic fieldsbelow the surface:

What field strength is necessary for a

flux rope to retain its cohesion?

The Transport of Magnetic FluxThe effects of convective turbulence on initially “neutrally-buoyant” active region-scale magnetic structures with field strengths near the cohesion limit:

How rapidly is magnetic flux transported to the base of the convection zone?

Over the lifetime of an active

region-scale magnetic structure,

we find no systematic tendency

for a net transport of magnetic flux

into either the upper or lower half

of the model convection zone.

The Transport of Magnetic Flux

This is a surprising result --- the

conventional wisdom would

suggest that, on average, the

magnetic flux should be transported

downward in the presence of the

asymmetric vertical flows of stratified

convection.

Turbulent PumpingThe efficient, downward advective transport of magnetic flux --- relevant to:

• Theories of the global solar dynamo• Theories of active region formation and evolution• Penumbral structure in sunspots• The flux storage problem

How efficiently is magnetic flux

transported to the base of the

convection zone in the absence of

an adjoining stable (or nearly stable)

layer?

Turbulent Pumping

A more appropriate experiment:

• Impose a relatively weak, domain-filling magnetic field of the form B=B0x (where B0 is assumed constant) on a statistically relaxed convection zone.

• Allow the simulation to progress, keeping track of the average distribution of signed and unsigned magnetic flux along the way, via:

),,,(1

)(2/

000

tzyxBdzdydxL

t x

LLL

x

zyx

where

zyx LLL

xtzyxBzdzdy

txdx

Ltz

000

),,,(),(

11)(

zy LL

tzyxBdzdytx00

),,,(),(

and

Turbulent PumpingThere are two important time scales to keep in mind:

• The convective time scale Hr / vc

• The “flux expulsion” timescale --- i.e., the amount of time necessary for the field to reach its equilibrium distribution

On a convective time scale

• There is no evidence of a net transport of signed flux over multiple turnover times (t < 5) in the absence of a stable layer (the distribution of unsigned flux is affected by field amplification and the turbulent dynamo).

From Tobias et al 2001. Here, The units of t are expressed in termsof an isothermal sound crossing time

Turbulent Pumping

BvB

t

lBvAB dddt

d

A

To qualitatively understand the initial behavior of the simulations, let’s neglect

the effects of Lorentz forces and magnetic diffusion, and again consider the

ideal MHD induction equation:

2

0

z

y

Lz

L

xzzx BvBvdydt

d

Applying Stoke’s theorem gives:

Since we are interested how signed flux is redistributed through the domain,

let’s consider a closed circuit encompassing the lower half of a single

vertical slice. Our horizontal boundaries are periodic, and vz and Bz are

assumed anti-symmetric across the lower boundary. Thus, the line integral

becomes:

Turbulent Pumping

2

00

0 z

yx

Lz

L

z

L

xvdydx

L

B

dt

d

The initial horizontal magnetic field is constant (of the form B = B0x); thus,

the only way the total amount of magnetic flux above or below the mid-plane

of a vertical slice can change, is by the interaction of vertical flows with the

horizontal layer of flux. Then the average time rate of change of signed

magnetic flux in the lower half of the domain can initially be expressed as:

If there are no bulk flows, or net vertical pulsations in the domain (as is the

case in our dynamically relaxed model convection zone), then

0dt

d

And we should expect no initial tendency for a horizontal flux layer to be

preferentially transported in one direction over the other, solely as a result of

the presence of an asymmetric vertical flow field.

Turbulent Pumping

2

00

1 z

yx

Lz

L

xzzx

L

xBvBvdydx

Ldt

d

Now, let’s extend our analysis well beyond the initial stages of our simulations.

The change in the amount of flux in the lower half of the domain as a function

of time is:

Note that in the MHD approximation, the integrand is –cEy; thus, this equation

simply states that a net transport of magnetic flux into the lower half of the

domain occurs if there is a net component of the electric field (in the y-direction)

along the midplane. This occurs in the marked interval below:

Turbulent Pumping

The net component of the electric field Ey along the midplane is a quantity

sensitive to the distribution of the vertical magnetic field along that plane.

• As flux is expelled into inter-granular regions, field is stretched and amplified, and we find that the strongest vertical magnetic fields tend to be associated with strong, high-vorticity downflows.

Turbulent Pumping

2

00

1 z

yx

Lz

L

xzzx

L

xBvBvdydx

Ldt

d

As vertical fields are amplified and become concentrated in and around

localized, high vorticity downflows, an equilibrium distribution is reached,

and the net transport of flux ceases.

This effect can be understood in

a simple (heuristic) way:

Turbulent Pumping

Thus, our simulations differ from those of penetrative convection:

• Only after ~5 Hr/vc do we see any sign of a net transport of magnetic flux into the lower half of the domain

• Our pumping mechanism is weak, occurs only after the field distribution becomes significantly non-uniform, and ceases once the equilibrium distribution is reached (over a longer, “flux expulsion and amplification” time scale of ~25 Hr / vc)

We therefore conclude that:

• The strong pumping mechanism evident in simulations of penetrative convection (t < Hr / vc ) is primarily the result of the presence of the overshoot layer --- flux entrained in the strong down drafts penetrates into the stable region where it remains for time scales far exceeding that of convective turnover.

• The net transport of flux is uncorrelated to the degree of flow asymmetry in the domain

The Convective DynamoNow let’s introduce a dynamically unimportant seed field into a non-rotating

turbulent model convection zone:

The Convective Dynamo

Exponential growth phase Saturation phase

The Convective Dynamo

• This type of model should not be confused with a global dynamo model --- in our case, there is no need for an interface layer, rotation (differential or otherwise), or prescribed flows.

• Our work differs from that of Cattaneo (1999) in that our domain is highly stratified --- but like his Boussinesq calculation, we find that our small seed field grows exponentially until it saturates at ~7% of the total kinetic energy of the computational domain

The Convective Dynamo and X-ray Flux in Stars

• We use simple stellar structure theory (MLT) to scale our simulations to the outer layers of main sequence stars (in the F0 to M0 spectral range) --- this allows us to estimate the unsigned flux on the surface of “non-rotating” reference stars.

• With these estimates, we use the empirical relationship of Pevtsov et al. (2003) to estimate the amount of X-ray emission resulting from such a turbulent dynamo.

The Convective Dynamo and X-ray Flux in Stars

• Our results compare well with the observed lower limits of X-ray flux (and when scaled to chromospheric levels), compare well with the observed lower limits of Mg II flux

• Thus, we suggest that dynamo action of a non-rotating convective envelope can provide a viable alternative to acoustic heating as a mechanism for the observed “basal” emission level seen in the chromopsheric and coronal emission of main sequence stars

Summary of Recent Results

• We have developed a new means of determining magnetized flows, consistent with both observations of the photospheric magnetic field and the MHD induction equation. Electric fields derived from these flows can be used to drive 3D MHD models of the solar corona.

(B.T. Welsch, G.H. Fisher, W.P. Abbett, and S. Regnier, ApJ 2004, 610 1148)

• We have performed a large-scale parameter space survey of the sub-surface evolution of active region-scale magnetic fields in turbulent model convection zones, and have use these models to understand the evolution of sub-surface structures, drive model coronae, and to better understand the theory of turbulent pumping.

(W.P. Abbett, G.H. Fisher, Y. Fan, and D.J. Bercik, ApJ 2004, 612 557)

• We have demonstrated how convection in a non-rotating, stratified medium can drive a turbulent dynamo, and suggest that this dynamo action can provide a viable alternative to acoustic heating as a mechanism for the observed basal levels of X-ray and Mg II flux in main sequence stars.

(D.J. Bercik, G.H. Fisher, C.M. Johns-Krull, and W.P. Abbett, ApJ 2004, submitted)