AR8210: Using Data to Drive a Model Corona over Photospheric Timescales

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AR8210 Using Data to Drive a Model Corona over
Photospheric Time-scales

• W.P. Abbett
• SSL UC Berkeley
• RHESSI Sonoma Workshop, Dec. 2004

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A Quick Synopsis of whats needed

• An MHD model active region requires both
• A means of specifying the photospheric boundary
  in a way consistent with the observed evolution
  of the vector magnetic field in the photosphere.
• E- (from u and B) and the fluxes into the volume
  (?uz, pxuz, pyuz, pzuz, euz) at z0 for all t
• An initial atmosphere throughout the
  computational volume, which must encompass the
  ß1 plasma of the photosphere along with the
  low-ß coronal plasma.
• Bx, By, Bz, px, py, pz, ?, e at t 0 for all
  x, y, and z
• A means of specifying the other external
  boundaries.

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What we Have

• With new means of specifying physically
  consistent photospheric flows (ILCT, MEF, MSR),
  we have u and B (and thus E-) at z0 for all t.
• Using the Wheatland technique, we can generate a
  divergence-free, force-free magnetic field
  throughout the volume that matches the imposed
  boundary conditions at t 0.

E- , ?uz, pxuz, pyuz, pzuz, euz at z 0 for all
t Bx, By, Bz, px, py, pz, ?, e at t 0 for
all x, y, z
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What we still need

• Inversion techniques like ILCT and MEF do not
  provide the vertical gradients of u (since a
  single vector magnetogram gives no information
  about the vertical gradients of B) nor do they
  say anything about the evolution of the
  transverse components of the magnetic field.
• Yet higher than first-order accurate numerical
  schemes require information from cells below the
  boundary layer to properly calculate the fluxes
  at the boundary face necessary to update the
  solution.

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The Strategy
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The Active Boundary

• Treat as a code-coupling exercise --- we now have
  two distinct 3D regions
• An MHD model with a domain that encompasses the
  layers above the photosphere and extends out to
  the low corona, and
• A 3D dynamic boundary layer, with its lower
  boundary at the photosphere, and its upper
  boundary at the base of the MHD model corona

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The Active Boundary

• Make a physical assumption Coronal forces do not
  affect photospheric motions.
• Then in the boundary layer, we can assume that
  ILCT or MEF flows permeate the entire layer, and
  implicitly solve (using the ADI method) the
  induction equation given the prescribed flow
  field.

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Coupling the codes

• Use a modified version of this ADI boundary
  code to implicitly solve the induction,
  continuity, and (a simple) energy equation given
  the ILCT or MEF prescribed flow field
• The upper boundary of the ADI code extends into
  the lower active zones of the MHD model, and the
  ghost cells of the MHD model are specified by the
  upper active zones of the ADI code.

E- , ?uz, pxuz, pyuz, pzuz, euz at z 0 for all
t

• The cadence is determined by the (explicit) MHD
  code.

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• Timestep severely restricted by CFL condition up
  in the corona --- the characteristic flow speed
  in the corona far exceeds that of the
  photosphere.
• Simple scaling of e.g. the update cadence, field
  strength, or size scale of AR will not suffice,
  if one desires to maintain the physics of the
  transition layers

• ?B/?t ? x (v x B)

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Principal Challenges

• Extreme separation of time scales
• CFL condition ?t lt ?z/c
• Sound speeds large in the low corona
• Characteristic flow speeds along loops can be
  100s of kms per second
• Extreme separation of spatial scales
• Evolution of the ARs magnetic field evolution
  is not independent of the global magnetic field
  (8210 connected by a trans-equatorial loop to
  another AR complex!)
• Convective granulation pattern small compared to
  the size scale of AR at the photosphere

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Methods of Attack

• Explicit temporal differencing
• Solve for qn1 directly from the state of the
  atmosphere at qn
• Fully implicit differencing
• Define qn1 implicitly in terms of qn ---
  requires the inversion of a large, sparse matrix
• Semi-implicit techniques

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Advantages/Disadvantages of each approach

• Explicit methods Accurate, but numerically
  stable only if CFL condition is satisfied
• Semi-implicit methods efficient when stiffness
  comes from linearities.
• Coronal plasma
• Fully-implicit methods efficient when
  stiffness comes from non-linearities
• Timestep restricted by the dynamic timescale of
  interest

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Fully-implicit technique NR

• Widely used in complex non-linear 1D problems
  (e.g., non-LTE radiation hydro)
• Basic idea Multi-dimensional Taylor expansion of
  F(q) about the current state vector qk (?, px,
  py, pz, e, Bx, By, Bz)
• F(qk1) F(qk) (dF/dq)k (qk1 - qk)
• Then for 2nd order corrections solve
• (dF/dq)k dqk - F(qk) then update qk
• qk1 qk dqk

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Why is this not in general use for 3D MHD
problems?

• The calculation and storage of J (dF/dq)k is
  computationally prohibitive!
• J is an N X N matrix where N neqnxnynz
• But there is a solution to this problem!!
• Jacobian-free Newton-Krylov methods (new
  technique used in fluid dynamics, Fokker-Planck
  codes, and even 2D Hall MHD)

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The Basic Idea

• Make an initial guess for the correction vector
  dq and form a linear residual r0
• Solve for dq using a Krylov-based GMRES
  technique
• where ßi is given by the minimization of
• in a least squares sense.
• The Important Point We need only to calculate a
  matrix vector product Jv which we approximate
  using

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Caveats

• Cant get something for nothing! The GMRES
  convergence rate slows as the timestep is
  increased
• Solution pre-condition dq before Krylov
  iteration.
• More info about technique see Knoll Keyes JcP
  2004 193,57.

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Initial Tests

• IMHD (Implicit MHD)
• F95 3D resistive finite-volume MHD code
• Not operator split
• Set up to interface with PARAMESH (to address the
  problem of disparate spatial scales)
• Efficient use of memory (can run on a laptop)
• Can generate static solutions to improve initial
  configuration
• In its infancy! First light only a few weeks
  ago..

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Some tests..