UCB-SSL Plans for Next Year
Joint CCHM/CWMM Workshop, July 2007
W.P. Abbett, G.H. Fisher, and B.T. Welsch
RADMHD: Modeling the combined convection
zone-to-corona system:
The code solves the resistive, fully-compressible MHD system of equations:
u 0
t
u B 2 BB
uu p
4 Π g
I
t 8
B
uB Bu B
t
e
eu p u B Q
2
t 4
Closure relation: a non-ideal equation of state obtained through an inversion
of the OPAL tables (Rogers 2000),
p p ( , e)
Modeling the combined convection zone-to-corona
system in a physically self-consistent way:
The source term Q in the energy equation,
e
eu p u B Q
2
t 4
must include the important physics believed to govern the evolution of the
combined system. In the corona, this includes
• radiative cooling (in the optically thin limit),
• the divergence of the electron heat flux,
• a coronal heating mechanism (if necessary).
In the lower atmosphere at and above the visible surface,
• radiative cooling (optically thick)
Below the surface in the deeper layers of the convective interior
• radiative cooling (in the optically thick diffusion limit)
Modeling the combined convection zone-to-corona system:
We represent the source term Q as follows:
Q Qr Qc QB
In order to extend the spatial domain to active region scales, we choose not
to solve the optically-thick LTE transfer equation to obtain an expression for
surface cooling, Qr . Instead, we choose to approximate this cooling in
a way that successfully reproduces the average stratification and
solar-like convective turbulence of the more realistic simulations of
Bercik (2002) and Stein et al. (2003):
Q1 ne nh (T )
Qr 1Q1 2Q2 3Q3 where Q2 1 e e( , T0 )
Q3 R ( , T )T
and 1 , 2 , and 3 represent dimension-less envelope functions that restrict
each term to the appropriate range of densities or depths in such a way as to
avoid sharp cutoffs. (T ) is obtained from the CHIANTI atomic database.
Modeling the combined convection zone-to-corona system:
The structure of the transition region and corona depend strongly on the remaining
non-radiative terms in Q Qr Qc QB: the divergence of the electron heat flux,
T
Qc B || (T ) B 2
B
and an additional coronal heating rate QB (if necessary). We employ an
empirically-based description of coronal heating consistent with the observed
relationship between unsigned magnetic flux and the power dissipated in the
atmosphere by a coronal heating mechanism, QB dV c .
The RHS of this equation represents the Pevtsov et al. (2003) power law
relationship between X-ray luminosity and unsigned magnetic flux at the
photosphere Lx c . If we choose a simple heating function of the form
QB B / B (consistent with Lundquist et al. 2007), we arrive at an
empirically-based form of coronal heating consistent with Pevtsov’s Law:
c B
QB
BdV
Modeling the combined convection zone-to-corona system:
The calibrated radiative source term Qr in Q Qr Qc QB , coupled with a
constant radiative flux lower boundary condition (on average) maintains the
super-adiabatic stratification necessary to initiate and sustain convection.
The thermodynamic structure of the model is controlled by the energy
source terms, the gravitational acceleration and the applied thermodynamic
boundary conditions. No stratification is imposed a priori.
Numerical techniques and challenges:
A dynamic numerical model extending from below the photosphere out into
the corona must:
• span a ~ 10 - 15 order of magnitude change in gas density and a
thermodynamic transition from the 1 MK corona to the optically thick, cooler
layers of the low atmosphere, visible surface, and below;
• resolve a ~ 100 km photospheric pressure scale height while
simultaneously following large-scale evolution (we use the Mikic et al. 2005
technique to mitigate the need to resolve the 1 km transition region scale
height characteristic of a Spitzer-type conductivity);
• remain highly accurate in the turbulent sub-surface layers, while still
employing an effective shock capture scheme to follow and resolve shock
fronts in the upper atmosphere
• address the extreme temporal disparity of the combined system
RADMHD: Numerical techniques and challenges
• For the Quiet Sun: we use a semi-implicit, operator-split method.
u 0
t
u B 2 BB
uu p I Π g
t 8 4
B
uB Bu B
t
e
eu p u B Q
2
t 4
• Explicit sub-step: We use a 3D extension of the semi-discrete method of
Kurganov & Levy (2000) with the third order-accurate central weighted essentially
non-oscillatory (CWENO) polynomial reconstruction of Levy et al. (2000).
• CWENO interpolation provides an efficient, accurate, simple shock capture
scheme that allows us to resolve shocks in the transition region and corona
without refining the mesh. The solenoidal constraint on B is enforced implicitly.
RADMHD: Numerical techniques and challenges
• For the Quiet Sun: we use a semi-implicit, operator-split method
u 0
t
u B 2 BB
uu p I Π g
t 8 4
B
uB Bu B
t
e
eu p u B Q
2
t 4
• Implicit sub-step: We use a “Jacobian-free” Newton-Krylov (JFNK)
solver (see Knoll & Keyes 2003). The Krylov sub-step employs the generalized
minimum residual (GMRES) technique.
• JFNK provides a memory-efficient means of implicitly solving a non-linear
system, and frees us from the restrictive CFL stability conditions imposed by
e.g., the electron thermal conductivity and radiative cooling.
RADMHD: Numerical techniques and challenges
• The MHD system is solved on an adaptive, domain-decomposed mesh.
Note: With our numerical techniques, AMR is not needed to
simulate the Quiet Sun. However, RADMHD has the capability
of interfacing with the PARAMESH libraries (MacNeice et al. 2000)
to provide an adaptive framework.
• Spatial disparities of the combined convection zone-to-corona system
are addressed via the CWENO explicit scheme, the domain decomposition
strategy, and AMR capability if necessary.
• Temporal disparities of the combined convection zone-to-corona system
are addressed via the JFNK implicit scheme. Pre-conditioning is an
essential requirement if one wishes to rapidly relax atmospheres by
significantly exceeding the CFL limit.
• Boundary conditions of the Quiet Sun simulations: Periodic in the
transverse directions, constant radiative flux in through a closed lower
boundary, open coronal boundary
The Quiet Sun magnetic field in the model chromosphere
Magnetic field generated
through the action of a
convective surface
dynamo.
Fieldlines drawn (in both
directions) from points
located 700 km above the
visible surface.
Grayscale image
represents the vertical
component of the velocity
field at the model
photosphere.
The low-chromosphere
acts as a dynamic, high-β
plasma except along thin
rope-like structures
threading the atmosphere,
connecting strong
photospheric structures to
the transition region-
corona interface.
Plasma-β ~ 1 at the
photosphere only in
localized regions of
concentrated field (near
strong high-vorticity
downdrafts
From Abbett (2007)
Flux submergence in the Quiet Sun and the connectivity between an initially
vertical coronal field and the turbulent convection zone
From Abbett (2007)
Reverse Granulation
• A brightness reversal with height in the atmosphere is a common feature of Ca II H and K
observations of the Quiet Sun chromosphere.
• In the simulations, a temperature (or convective) reversal in the model chromosphere occurs
as a result of the p div u work of converging and diverging flows in the lower-density layers
above the photosphere where radiative cooling is less dominant.
Gas temperature and Bz at ~ 700 km above and below the model photosphere
From Abbett (2007)
Flux cancellation and the effects of resolution:
The Quiet Sun magnetic flux
threading the model
photosphere over a 15
minute interval. Grid
resolution ~ 117 X 117 km
Average unsigned flux per
pixel: 34.5 G
Simulated noise-free
magnetograms reduced to MDI
resolution (high-resolution
mode) by convolving the dataset
with a 2D Gaussian with a
FWHM of 0.62” or 459 km.
Average unsigned flux per pixel
is now: 19.9 G
Simulated noise-free
magnetograms reduced to Kitt
Peak resolution. FWHM of the
Gaussian Kernel is 1.0” or 740
km. Average unsigned flux per
pixel: 15.0 G
Observed unsigned flux per pixel at
Kitt Peak: 5.5 G
log B
log β Bz log B
log J
Characteristics of the Quiet Sun model atmosphere:
Note: Above movie is not a timeseries!
PLANS: A Focus on the Physics of Active Region
Flux Emergence….
Extend the RADMHD quiet Sun simulations to active region spatial scales using
our new allocation on NASA’s Discover cluster at Goddard Space Flight Center
Once the large-scale quiet Sun model has energetically relaxed, introduce active
region strength magnetic fields from below by
(1) introducing a magnetic flux tube directly into the portion of the domain
representing the convection zone (similar to the more idealized simulations of,
e.g., Magara 2004, Manchester 2004, Archontis 2007);
(2) introduce magnetic flux into the domain from below using previous ANMHD
calculations of buoyant Ω-loops in the deep interior (Abbett 2000, 2004);
(3) introduce an interacting pair of twisted flux ropes into the RADMHD domain
below the visible surface, and study the magnetic topology of the corona as one
system emerges into another;
The Physics of Flux Emergence….
(4) study the effects of magnetic flux emergence at small scales (e.g., quiet-
Sun fields generated by surface convection or ephemeral active regions), and
the effect of small scale flux emergence on the large scale magnetic topology
of the model corona;
(5) follow the evolution of the model active region after emergence, study the
decay process and magnetic connectivity as convective turbulence interacts
with loop footpoints;
(6) test and validate our inversion techniques and boundary driving schemes by
driving RADMHD model corona with synthetic magnetograms and comparing
the resultant model coronae against the self-consistent calculations.
RADMHD Flux Emergence: Computational Requirements
The most computationally intensive portion of the project is to relax
the convectively unstable portion of the large-scale domain.
We reduce the computational cost by
(1) relaxing periodic sub-domains on our local cluster,
(2) filling the blocks of the larger grid with these solutions, and
(3) introducing an entropy perturbation throughout the large-scale domain to
break the symmetry.
We find that this process takes fewer processor hours than if we simply perturb
large-scale horizontally-invariant, super-adiabatically stratified background
atmospheres and allow the magnetoconvection to develop from scratch.
This is still an expensive process.
By comparison, the AR emergence timescale is quite rapid, and the emergence
runs are relatively inexpensive (though we choose to remain constrained by the
magnetosonic wavespeed in the corona)!