Toward More Realistic 3D MHD Simulations of Magnetic Flux Emergence

(and Decay) in Active Regions

W. P. AbbettSpace Sciences Laboratory

University of California, Berkeley

Challenges:• Computational domain encompasses

magnetized plasma of differing physical characteristics:

– The convecting high-beta super-adiabatically stratified plasma of the sub-surface layers

– The beta~1 surface layers: includes the energetic transition from the optically thick, cool chromosphere to the magnetically-heated, optically thin sparse corona

– The low-beta magnetically dominated corona: energetics of the magnetically-confined coronal plasma dominated by thermal conduction and oprically thin radiative losses

• Extreme disparity in both spatial and temporal scales:

– Active regions on the photosphere evolve over time scales of days, weeks, and months, yet simulations that encompass both photospheric and coronal plasma are often Courant limited by e.g. fast moving magneto-acoustic waves in coronal loops. Resolving these waves can introduce unnecessary numerical diffusion at and below the surface.

– Conductive and radiative timescales further restrict explicit calculations

– Small scale structures can affect the evolution of large scale fields (e.g. the effect of convective turbulence on the emergence and decay of active region fields)

This has motivated a new numerical approach(or more precisely, a combination of existing techniques combined together to form a new code designed specifically for this set of challenges):

• First, the obligatory list of the system of equations that must be solved:

)1(

42

0

00

2

P

QPt

t

BP

t

t

uu

BBuuBB

gΠBB

I v vv

v

• The energy source terms are of critical importance in order to couple a turbulent model convection zone with a low-density, magnetized model corona

TQconv 25.6 Tr

)(22

|| TTB

Qcor

BB

2

5

0|| Tk TT )(

Source terms in the corona:

Source term in the lower layers of the convection zone:

(anisotropic thermal conduction and optically-thin radiative cooling)

(radiative heating in the diffusion approximation with the radiative conductivity approximated using Kramer’s opacity law)

21 QQQQQ corconv

||)(1

),,()(1

)(1

22

2

01

11

BzQ

zyxTzTzR

Qxy

Artificial source terms acting to mimic optically-thick chromospheric cooling, and an average coronal heating mechanism consistent with the empirical relationship of Pevtsov et al. 2005:

However, in practice, each term is multiplied by an envelope function (either depth, temperature or density dependent, depending on the individual term) to ensure that the heating or cooling term is smoothly shut off as the atmosphere transitions to a regime where a particular treatment is inapplicable.

Of course, energy terms due to viscous and resistive heating are included as well.

Each individual source term is included in the MHD energy equation

The Components of the Method:

kk

k

kk qqqF

qFqF

11 )()(

1. A “Jacobian-free” Newton-Krylov fully implicit solver

• The basic idea: a multi-dimensional Taylor expansion of the system F(q) about the current state vector q

)( kk

k

qFqqF

kkk qqq 1

Then for successive second-order corrections to an initial guess for q, solve

and update q via

• A problem with multi-dimensional Newton-Raphson alone is that the matrix J = (dF/dq)|k takes up an excessive amount of memory:

– J is an N X N matrix where N = neq∙nx∙ny∙nz

• The Newton-Krylov approach gets around this by forming a linear residual and solving for the correction vector using a standard Krylov-based technique called “GMRES”:

where β is determined via the minimization of in a least-squares sense.

• The main point of all this, is that all one needs in the above sum to solve for the Newton correction is a matrix-vector product (of dimension N), which can be approximated by the “Frechet” derivative for any vector v, thereby eliminating the need for the explicit formation and storage of the matrix J.

00 )( qJqFr

0

1

00 )( rJqq i

j

ii

/))()(( qFvqFJv

2||)(|| qFqJ j

)(max,,2/1 uuf

a x

kji

),( ,,2/1,,1,,2/1 tP kjikjikji uu

),( ,,2/1,,,,2/1 tP kjikjikji uu

2. We need a numerical representation of the flux divergence terms in the MHD system --- there are a number of techniques available, but for simplicity in incorporating the scheme into the adaptive mesh domain-decomposition framework we employ the semi-discrete central scheme of Kurganov and Levy 2000.

Ignoring the source terms, we have the system:

Which can be represented numerically via:

with flux functions F given by:

0)(

ufut

z

tt

y

tt

x

tt

tkji

zkji

zkji

ykji

ykji

xkji

xkji

)()()()()()( 2/1,,2/1,,,2/1,,2/1,,,2/1,,2/1,, FFFFFFu

)()(2

)(

2

))(())(()( ,,2/1,,2/1

,,2/1,,2/1,,2/1

,,2/1 ttttt

t kjikjikji

xkjikjix

kji

uuaufuf

F

where

and

Here, Pi,j,k represents a third order CWENO (central weighted essentially non-oscillatory) polynomial interpolation of the state vector u to the cell faces, and λl are the eigenvalues of the system.

3. The MPI and structured AMR (Adaptive Mesh Refinement) framework is that provided by PARAMESH v3.0 (MacNeice et al. 2000). Parallelization is achieved through domain decomposition

– the current numerical scheme inherently minimizes inter-processor communication (e.g., block boundary data is exchanged relatively infrequently).

Of course, the disadvantage of any central non-staggered scheme is the need to ensure that the magnetic field is divergence-free. Currently, we incorporate the “8-wave” formulation of Powell et al. 1999 (since our particular domain of interest generally does not exhibit regions of stagnation); though we intend to incorporate a Constrained Transport scheme in the near future.

Putting it all together

A simple semi-implicit temporal differencing scheme:

Where here, e indicates the Newton-Raphson error vector (iterated to convergence), F represents the high-order, non-operator split numerical representation of the flux divergence vector (the quantities in parenthesis in the MHD system of conservation equations), and g denotes the source terms of the system that we wish to treat implicitly(here, we choose that the system be Courant-limited by the terms in f ).

Alternatively, we can define the error vector as

where here, the source terms are included in f. In the context of this particular differencing scheme, F n (obtained from the Kurganov-Levy explicit solution) can be viewed as a “physics-based pre-conditioner” for the Newton-Krylov system --- this greatly reduces the number of implicit iterations necessary for convergence.

nnnnn dtdtdt ggFqqe )1(11

nnnn dtdt ffqqe )1(11

Initial Results:• First, the initial state must be generated --- the most challenging

aspect of a calculation of this type. The state must include a super-adiabatically stratified model convection zone (at least extending ~ 4 Mm below the visible surface), a beta~1 cool model photosphere- chromosphere, along with the low-beta, low density, hot corona.

Left: Photospheric surface layer (background shading denotes temperature perturbations above the mean state; density perturbations are shown by the overlaid contours). A constant radiative flux is imposed at the closed lower boundary, horizontal boundaries of this sub-domain are periodic, and the vertical boundary is assumed stress-free and closed.This state is not yet fully relaxed.

• The transition layers should be reasonably resolved, and the total domain must be able to realistically encompass a small active region

Left: Temperature perturbations along a vertical slice of this initial (un-relaxed) sub-domain (background shading denotes vertical velocity perturbations above the mean state; unsigned horizontal velocities are indicated by the overlaid contours).

Once a dynamically relaxed state is achieved in this initial sub-domain, the periodic slices will be inserted in the MPI, domain-decomposed version of the code, allowing for a much larger, active region-scale domain in which to introduce strong, twisted flux ropes.

Sub-domain solutions at different depths (temperature perturbations at each height shown) from ~4 Mm below the visible surface to a 30000 K transition layer (above the photosphere, but below the corona at the top of the model “overshoot layer”).

Progress on the MPI-AMR version of the code:

Above: Orszag-Tang vortex MHD test. Shown is the x-component of the magnetic field along horizontal slices through the 3D domain. Left: A horizontal slice through a 3D MHD blast test calculation --- shown are contours of the magnetic field strength. The calculation was performed on a Beowulf cluster. Block boundaries are shown; there are 32 cells per block and four levels of refinement.

:

• Once a dynamically relaxed state is achieved, an active region-scale buoyant flux rope can be inserted below the surface (or introduced into the lower boundary of the model convection zone), and magnetic structures can self-consistently emerge, evolve, and decay throughout the model atmosphere (similar to what was done in Abbett et al. 2004 shown below for flux tubes in the deep convection zone)

References and Contact Information:

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

Kurganov, A., and Levy, D., 2000, SIAM J. Sci. Comput., 22, 1461.

MacNeice, P., Olson, K.M., Mobarry, C., deFainchtein, R., and Packer, C., 2000, Computer Physics Communications, 126, 330.

Pevtsov, A.A., Fisher, G.H., Acton, L.W., Longcope, D.W., Johns-Krull, C.M., Kankelborg, C.C., and Metcalf, T.R., 2003, 598, 1387.

Powell, K.G., Roe, P.L., Linde, T.J., Gombosi, T.I., De Zeeuw, D.L, JCP, 1999 154, 284.

Author email: abbett@ssl.berkeley.edu