Computational MHD:
A Model Problem for Widely Separated
Time and Space Scales
Dalton D. Schnack
Center for Energy and Space Science
Center for Magnetic Self-Organization in Laboratory and
Space Plasmas
Science Applications International Corp.
San Diego, CA 92121 USA
Computational MHD
• Computational MHD is challenging because of:
–
–
–
–
Widely separated spatial scales
Widely separated time scales
Extreme anisotopy
Closure uncertainties
• Different types of MHD problems
– Kinetic energy dominant (convection, dynamo)
– Magnetic energy dominant (corona, lab plasmas)
– Require different computational approaches
• Will review approaches that have been successful
of simulating highly magnetized plasmas
Types of MHD Problems
Weakly Nonlinear
Strongly Nonlinear
•
•
KE ~ ME
Flow is important
–
–
–
–
–
•
•
•
•
Shocks
Turbulence
Dynamos
Convection
Jets
Properties of “statistical” state with
all wavelengths represented
Relatively simple BCs and
geometry
Important stiffness comes from
nonlinearity
Advection is important
•
•
KE << ME
Dominated by magnetic field
– Coronal loops
– Magnetic fusion
•
Slow departures from equilibrium
(instabilities)
– Culmination in disruptive behavior
•
•
Complex BCs and geometry
Important stiffness comes from
linear properties
– Resistive layers, current sheets
– Slowly growing instabilities
– Parasitic modes
•
Advection not so important
Different Problems Require Different
Approaches
•
Flow dominated plasmas
– Stellar interiors, convection,
dynamo
– Magnetic forces are weak (b
>> 1)
– Collisional
– Fluid equations good at all
scales
– Closures: characterize subgrid scale turbulence
Can adapt hydrodynamic
algorithms
•
Magnetically dominated
plasmas
– Coronal loops, laboratory
fusion plasmas
– Magnetic forces are
strong (b ~ 1 in corona, b
<< 1 in fusion plasmas)
– Low collisionality
– Fluid equations not valid
at small scales
– Closures: characterize
non-local kinetic effects
(collective particle
dynamics)
Need to develop new
algorithms
“Sub-grid Scale” Models
•
•
Computational resources limit the number of degrees of freedom in discrete
model
Important physical processes occur below the scale of the grid
– Large Reynolds’ number turbulence
– Kinetic effects in low collisionality plasmas
•
Must be captured in a sub-grid scale model
– Turbulence
• Averaging - characterize effect of small scales on large scales (new dependent variables) u
= <u> + w, <<u>> = <u>, <w> = 0, <wiwj> ≠ 0
• Closure - expressions or equations for new variables, <ui∂iuj> => -n∂2<uj>
• Maintain consistency - Stokes’ theorem, flux conservation,….
– Kinetic effects
• Characterize non-local kinetic effects as local stress tensor, heat flux, …..
• Chapman-Enskog-like closures (integration along field lines)
• Subcycling kinetic/particle
•
•
We still don’t know exactly what equations to solve!
Extensive validation against experiment and observation is required
To come……
•
•
•
•
•
•
•
•
Building discrete models of physical systems
Physical basis for MHD
MHD boundary conditions
Spatial approximations
Temporal approximations (mostly implicit)
Examples
Extensions to MHD
Where can we go from here?
Finite Dimensional Models
Discrete Models
• Physics described by PDEs on the continuum
– “Infinite” degrees of freedom (eigenmodes)
• Build discrete model with finite (N) degrees of freedom:
PDEs ==> N algebraic equations
– Need convergence: Discrete solution => PDE solution
as N  
• Requirements for convergence:
– Consistency: discrete equations => differential
equations as N   (including boundary conditions!)
– Stability: small errors cannot grow without bound
• Lax’s Theorem: Stability implies convergence for well
posed consistent approximations
Guides for Building Discrete Models
• No unique, consistent, stable discrete analog
• Guide: Discrete system should have as many
properties of physical system as possible
–
–
–
–
Conservation laws
Normal modes (eigenfunctions and eigenvalues)
Self-adjointness, symmetries, anti-symmetries
Boundary conditions
• Not a purely mathematical exercise
• Requires experience and intuition
The MHD Model
Physics Problem:
Modeling Magnetized Plasmas
• Dynamics of electrical conducting fluids in
the presence of a magnetic field
• Must solve material dynamics and Maxwell
simultaneously
• Simplest model is MHD, but…..
Not just hydrodynamics with Lorentz force!
What Equations to Solve?
• Complete description => kinetic equation
– 6 dimensions + time
– Useless in practice
• Take successive velocity moments of the kinetic equation
==> fluid model (3 dimensions + time)
– Each moment equation contains the next higher order moment
– Must truncate, or close, the system
– Closure assumption: highest order moment can be written in terms
of lower order moments
• Assume V2/c2 << 1
– Ignore displacement current
– Implies quasi-neutrality
“Single-fluid” form (me=0, ne=ni=n)
• Combine equations for different species
n
   nV 
t
dV
  p i  p e   J  B    
dt
1
E   V  B  J  B  p e  
J
Ideal MHD ne
Resistive MHD
min
Two- fluid effects
(Hall + diamagnetic)
 Equations for p i and p e , + closures
• Equivalent to two-fluid model

Possible Fluid Models

Model
Vi 1,2
General


,
mn
,

i
L
 1 ,
Vthi 2 
b    
VA  
Momentum
Ohm’s law
Whistlers4
KAW5
GV6
dV
 p  J  B
dt
E  V  B
1
 J  B  pe 
ne
Yes
Yes
Yes
E  V  B
1
 JB
ne
Yes
No
No
E  V  B
No
No
No
E  VE  B
No
Yes
Yes

   ||     gv
i
i
Vthi /   ci
 Vthi

 ci



Drift
Vthi
MHD 7
 2 ci


V
 0
Vthi
3

Hall
MHD

Ideal
MHD


 b    nc
i
mn
dV
 J  B   ||
dt
 b    nc
i
mn
dV
 p  J  B
dt

   ||  b    nc
i

dVE
mn
 mnV* i  VE
dt
 mnV||i  VE    ||
i
 (I  bb) (b    nc
i )
   p1    J  B



1
|| pe
ne
What a Difference B Makes!
Hydrodynamics
• Cannot support shearing
displacements
• Linear:
– Sound wave
• Nonlinear (Riemann
problem):
– 3 waves
• Shock
• Expansion
• Contact discontinuity
– 4 possible combinations
What a Difference B Makes!
Hydrodynamics
• Cannot support shearing
displacements
• Linear:
– Sound wave
• Nonlinear (Riemann
problem):
– 3 waves
• Shock
• Expansion
• Contact discontinuity
– 4 possible combinations
MHD
• Can support shearing
displacements
• Linear:
– Sound wave
– Compressional Alfvén wave
– Shear Alfvén wave
• Nonlinear (Riemann
problem):
– 7 waves
• 3 shocks
• 3 contact discontinuities
• Expansion
– 648 possible combinations!
The Challenges of
Computational MHD
Challenges for Computational MHD
• Strongly magnetized plasmas exhibit:
– Extreme separation of time scales
– Extreme separation of spatial scales
– Extreme anisotropy
• Each has implications for algorithms
1. Extreme Separation of Time Scales
A
S

Alfvén transit time
 evol

Sound transit time
R

MHD evolution time
Resistive diffusion time
• Lundquist number:
R
S 
A
~ 10812  1
• Explicit time step impractical:
t


x
A
L

A
N

Requires implicit methods

 evol
2. Extreme Separation of Spatial Scales
• Important dynamics occurs in internal boundary
layers
– Structure is determined by plasma resistivity or other
dissipation
– Small dissipation cannot be ignored
• Long wavelength along magnetic field
• Extremely localized across magnetic field:
-a
 /L ~ S << 1 for S >> 1
• It is these long, thin structures that evolve
nonlinearly on the slow time scale
• Requires specialized gridding
3. Extreme Anisotropy
• Magnetic field locally defines special direction in
space
• Important dynamics are extended along field
direction, very narrow across it
• Propagation of normal modes (waves) depends
strongly on local field direction
• Transport (heat and momentum flux) is also highly
anisotropic
• Requires accurate treatment of B  
Inaccuracies can lead to “spectral pollution” and
anomalous perpendicular transport

MHD Boundary Conditions
• Hydrodynamics:
– Conservation laws in flux-divergence form:
U
   F
t
– Specify normal flux at boundary:
• Depends on inflow/outflow, super/sub-sonic

• Magnetic flux in MHD:
B
   E
t
• Must also specify Etan, and nothing more!
If algorithm requires more than this, it is inconsistent!!

Building a Discrete Model:
Spatial Discretization
Types of Spatial Discretization
• 2 general types of approximation
– Local (minimize error locally)
• Finite differences
– Taylor series expansion
• Finite volumes
– Local integral formulation
– Global/Galerkin (minimize error globally)
• Based on expansion in basis functions
– Spectral methods
» Basis functions defined globally
– Finite element methods
» Basis functions defined locally
• All are used in computational MHD
Spatial Approximation
• Replace:
• With:
dU
 A(U)
dt
dUi N
  AijU j
dt 
j1
 A(Ui )  F(x)



 i  1,2,...,N ,  xi , i  1,N; x  xi  xi1

Grid


Error
Discrtization

• Cannot separate boundary conditions from
approximation and/or grid
– Discrete problem should not require more BCs
Finite Volumes
• An “algorithm” for generating
finite difference formulas
– Physically motivated
– Integrate conservation laws of
small “finite volume” defined by
grid
– Exactly conservative
– Consistent BCs
• MHD also requires “finite area”
– Magnetic flux conservation
– Consistent BCs

Ftop
atop
Fleft
aleft

aright
V


Fright
abottom


Fbottom



E x top
ltop
Ey
left 
lleft



A
lright
lbottom

E xbottom


Ey
right
Finite Volumes for MHD
• Primary and dual grids (one possibility)
y
Centroid: , p, V
Vertex: A,E
x
Face: B,V
Conserves mass, momentum,
magnetic flux
z
• Faces and edges on physical boundary to preserve BCs
Galerkin (Non-local) Methods
• Finite differences and finite volumes
minimize error locally
– Based on Taylor series expansion
• Galerkin methods minimize error globally
– Based on expansion in basis functions
• Solve “weak form” of problem
u(x,t)
 Lu(x,t)
t


u
 v  Ludx  0
t

Minimize global error by expansion in basis functions

Galerkin Discrete Approximation
Mij
du j
dt
 Lij u j
Mij   dVbia j
Lij   dVbi La j
Mass Matrix
Response Matrix
• Solution generally requires inverting the mass
matrix


• Different basis functions give different methods
– Usually: bi = ai
• ai=exp(ikx) => Fourier spectral methods
• ai=localized polynomial => finite element methods
Finite Elements
• Project onto basis of locally
defined polynomials of degree p,
e.g.: p = 1
• Integrate by parts:
 x  x j1
,

x

x
j
j1

 x
x
a j (x)   j1
,
x j1  x j
0


u j
 2a j
a j
ai a j
Mij
  dVai
u


dV
u

dS
a
uj


j
j
i
t
x 
x

x
x 2
S
Boundary term
•
•
•
•
•
•
Polynomials of degree p converge as hp+1
Natural
implementation of boundary conditions

Automatically preserves self-adjointness
Excellent for smooth functions
Works well with arbitrary grid shapes
Now widely used in MHD
x j1  x  x j
x j  x  x j1
otherwise
Solenoidal Constraint
• Faraday:
B
   E
t
• Depends on


B0
t
    0
• Cannotbe guaranteed in discrete model
U

– Modified wave system t    F  R  B
– Projection
– Diffusion
B B  
 2    B
B 

   E    B ,
  B      B
t
t

– Grid properties


a  b   0
Building a Discrete Model:
Temporal Discretization
Temporal Discretization
t n 1
u
 u
t
t
tn

• Explicit methods

u n 1

un


– Solve directly for values at n+1 in terms of values at
time step n
u n1  u n
 u n  O(t)
t

u n1  Fu n
• Computationally efficient, but…..
Time steplimited by condition for numerical
stability:
 t  t 1
Temporal Discretization
• Implicit methods
– Values at n+1 in defined implicitly terms of values at
time step n
u n1  u n
 u n1  O(t)
t

Aun1  Bun , or u n1  A1Bun
A  I  t
• Requires inversion of operator:
• More work than explicit method, but…
Unconditionally stable for any time
step…..BUT: Must followtime scale of
interest!
Multiple Time Scales
(Parasitic Waves)
• MHD operator contains widely separated time
scales (eigenvalues)
u

u

t Full MHD operator

Fu

Su
Fast time scales:
Slow time scales:
Alfvén waves, soundwaves, etc Resistive instabilities, island evolution,
(parasitic waves)
(interesting physics)
• Treat only “fast” part of operator implicitly to
avoid time step restriction
u n1  u n
 Fu n1  Su n
t
• Precise decomposition of  for complex nonlinear
system
is often difficult or impractical to achieve

Dealing with Parasitic Waves
• Original idea from André Robert (1971)
• In MHD, F and  are known, but an
expression for S is difficult to achieve
–  : full MHD operator
– F: linearized MHD operator
• Use operator splitting:
 F  S
 S   F
u n1  u n 
u n1  u n
n1
n
n
 Fu
 (  F)u  u  tF 

t
 t


• Expression for S not needed

Semi-Implicit Method
u n1  u n 
u n1  u n
n1
n
n
 Fu
 (  F)u  u  tF 

t
t


• Recognize that the operator F is completely arbitrary!!

(I  tG )u n1  (I  t)u n  tGu n
S. I.
operator
Explicit
S. I.
operator
• G can be chosen for accuracy and ease of inversion
– G should be easier to invert than F (or !)
– Gshould approximate F for “modes of interest”
– Some choices are better than others!
• The semi-implicit method originated decades ago in climate
modeling
• Has proven to be very useful for resistive and extended MHD
How SI Works
U n
V n
   a 
t 
x 
n 1/ 2
2
V n 1/ 2
U n 1/ 2
2  V 
 
 b 
 at
 
2 t 
t 
x 
x

How SI Works
U n
V n
   a 
t 
x 
n 1/ 2
2
V n 1/ 2
U n 1/ 2
2  V 
 
 b 
 at
 
2 t 
t 
x 
x

Old explicit code
How SI Works
U n
V n
   a 
t 
x 
n 1/ 2
2
V n 1/ 2
U n 1/ 2
2  V 
 
 b 
 at
 
2 t 
t 
x 
x
Old explicit code

c 2 k 2 t 2  4

How SI Works
U n
V n
   a 
t 
x 
n 1/ 2
2
V n 1/ 2
U n 1/ 2
2  V 
 
 b 
 at
 
2 t 
t 
x 
x
Old explicit code

c 2 k 2 t 2  4

How SI Works
U n
V n
   a 
t 
x 
n 1/ 2
2
V n 1/ 2
U n 1/ 2
2  V 
 
 b 
 at
 
2 t 
t 
x 
x
Old explicit code

c 2 k 2 t 2  4

Semi-implicit term
How SI Works
U n
V n
   a 
t 
x 
n 1/ 2
2
V n 1/ 2
U n 1/ 2
2  V 
 
 b 
 at
 
2 t 
t 
x 
x
Old explicit code

Semi-implicit term
c(k) 2 k 2 t 2  4,
c 2 k 2 t 2  4


c(k) 2  c 2 /(1 ak 2 t 2 )
How SI Works
U n
V n
   a 
t 
x 
n 1/ 2
2
V n 1/ 2
U n 1/ 2
2  V 
 
 b 
 at
 
2 t 
t 
x 
x
Old explicit code

Semi-implicit term
c(k) 2 k 2 t 2  4,
c 2 k 2 t 2  4
c(k) 2  c 2 /(1 ak 2 t 2 )
ak 2 t 2  c 2 k 2 t 2 / 4 1



How SI Works
U n
V n
   a 
t 
x 
n 1/ 2
2
V n 1/ 2
U n 1/ 2
2  V 
 
 b 
 at
 
2 t 
t 
x 
x

Old explicit code
c 2 k 2 t 2  4
Semi-implicit term
c(k) 2 k 2 t 2  4,
c(k) 2  c 2 /(1 ak 2 t 2 )
ak 2 t 2  c 2 k 2 t 2 / 4 1

• Stabilizes by dispersion

• Slows down unstable waves
• k-dependent inertia
Semi-Implicit Operator for MHD
• Linearized, ideal MHD
wave operator
p
    pV    1 p  V
• Wide spectrum of normal
t
V
modes
I  atG   V  V  p  J  B
t
• Highly anisotropic spatial
 0G      V  B 0   B 0    V
Sound waves
operator
Alfvén waves
• Basis of many implicit
Challenge: find
formulations
optimum algorithm for
• Not a simple Laplacian
inverting this operator
• Requires specialized prewith CFL ~ 104
conditioners
B
   V  B
t
Fully Implicit Approach
• Semi-implicit method is efficient when stiffness comes
from linearities
– Can split linear terms from full operator
– Fusion and coronal plasmas
• Fully implicit approach required when stiffness comes from
non-linearities
– Turbulence, etc.
– Must solve non-linear discrete equations
A(U)U  B ,
F(U)  A(U)U  B  0
F l l1
  (U  U l )  F l
U 

•
•
•
•
Newton’s method
Evaluation of Jacobian
“Jacobian-free” methods
“Newton-Krylov” methods
Example:
MHD Relaxation and the
Laboratory “Dynamo”
Plasma Relaxation
• System attempts to minimize energy subject to constraints:
 p
B 2 
W   

dV

1
2


0 
W  K  0

,
K   A  BdV
W /K  minimum
• Leads to preferred (“relaxed”) configurations (Taylor)
• Occursin a variety of situations
– Reversed-field pinch (“dynamo”)
– Tokamak (disruption)
– Solar corona (CMEs?)
• Underlying dynamics are MHD-like
– Turbulence?
– Subset of long wavelength modes?
• Cyclic process
Relaxed State
Nonlinear
Relaxation
Dynamo
Instabilities
Diffusion
Laboratory Dynamo
• Toroidal Z-pinch (RFP)
• Positive toroidal field in
center
• Negative toroidal field at
edge
• Cyclic relaxation
Lab Dynamo Mechanism
•
Laboratory “dynamo” is a nonlinear driven system
•
Plasma is resistive
•
Poloidal flux (toroidal current) sustained by applied voltage
•
Toroidal flux is sustained by a “dynamo”
•
Mediating dynamics
– Nonlinear behavior of long wavelength MHD instabilities
– Driven by resistive evolution of mean fields
– Nonlinear evolution converts poloidal flux => toroidal flux
– Like differential rotation
•
Finite resistivity makes flux conversion irreversible
•
No conversion of toroidal flux => poloidal flux
– Poloidal flux supplied by external circuit
Is it really a dynamo??
One-way
flux conversion
Examples of Dynamo Results
Self-reversal
•
•
•
Begins far from relaxed
state
Time dependence of mode
energy and field at wall
Dynamo sustains discharge
after t ~ 0.07
•
Taylor relaxation during
sustainment
•
•
Perturb by 1
part in 106
Dynamo is
nonlinear and
chaotic
Example:
Tokamak Disruption
Toroidal Fusion Plasmas: Tokamak
Magnetic flux
surfaces,  const.
Toroidal angle 
Integrated Control
Magnetic axis
Current Profile
Control
Optimized Resistive
Wall Mode Control
separatrix
Disruption Detection,
Correction, Mitigation
Poloidal angle 
•
•
•
•
Confine hot plasma to extract fusion energy
Most promising concepts are toroidal - tokamak
Magnetic field lines trace out flux surfaces
Geometry is axisymmetric, dynamics are 3-D
Experimental Tokamak Disruption
• Increase in neutral beam power
• Plasma pressure increases
• Sudden termination (disruption)
• Time dependence at disruption
onset
• Growing 3-D magnetic
perturbation (n = 1)
• Nonlinear evolution?
• Effect on confinement?
• Can this be predicted?
Simulation of Disruption
• Resistive MHD simulation of disruption
• Experimentally measured initial conditions
• Anisotropic heat conduction
• Vacuum region
• Ideal modes grows with finite resistivity (S = 105)
• Magnetic field becomes stochastic
• Non-symmetric heat load on wall and divertor
• Time for crash ~ 200 sec.
• Power ~ 5 GW
Disruption Dynamics
QuickTime™ and a
YUV420 codec decompressor
are needed to see this picture.
Example:
Dynamics of the Solar Corona
Solar Coronal Dynamics
• Coronal magnetic field energized by stresses
(motions) at the photosphere
• Approximately force-free current slowly builds
• Sudden disruption
– Violent relaxation?
• Coronal mass ejection (CME)
• Structure of heliosphere
– Use observed photospheric fields
• Space weather
Solar Corona, Solar Wind, and
Heliosphere
The Heliosphere During Whole Sun Month
August – September 1996
Launch and Propagation of CME
• Disruptions of corona result in CMEs; propagates into
heliosphere
• Coronal dynamics (< 20 Rs) is an implicit problem
– Large variation of Alfvén and sound speeds
– Subsonic to transonic flows
– Wave dominated
• Inner heliosphere (> 20 Rs) is an explicit problem
–
–
–
–
Supersonic and super-Alfvénic flows
Advection dominated
Steep gradients
Shocks
• Can couple implicit (Mikic & Linker) and explicit
(Odstrcil) codes beyond sonic and Alfvénic points to
model launch and propagation of CME into heliosphere
CME Launch and Propagation Using
Coupled Codes
QuickTime™ and a
Animation decompressor
are needed to see this picture.
The Next Steps
Beyond MHD
• MHD is not a great model for hot magnetized
plasmas
– Collisionality is low but not zero
– Current layers width comparable to ion Larmor radius
– Separate ion and electron dynamics important
• Need lowest order FLR corrections
– Full Ohm’s law (Hall + diamagnetic)
– Gyro-viscosity (non-dissipative momentum transport)
• “Extended” MHD
• Implications for algorithms
Extended MHD Algorithms
• FLR terms admit new waves
– Hall term (electrons, Ohm’s law)
• Whistlers (correction to shear branch)
– Gyro-viscosity (ions, momentum
equation)
• Corrections to compressional branch
• These new waves are dispersive:
– Cannot be treated explicitly   Ck 2
– Require new SI operators (4th order)
– Under development


t  Cx 2
Limits on Modeling
Balance of algorithm performance and problem requirements with available cycles
Na Q
t
 3 10 7
Algorithms
Algorithms:
• N - # of meshpoints for each
dimension 
• a- # of dimensions
– 1.5 - transport
– 3 (spatial) fluid
– 5-6 kinetic (spatial +
velocity)
• Q - code-algorithm
requirements (Tflop /
meshpoint / timestep)
• t - time step (seconds)
P
CT
Constraints
Constraints:
• P - peak hardware performance
(Tflop/sec)
•  - hardware efficiency
– P - delivered sustained
performance
• T - problem time duration
(seconds)
• C - # of cases / year
– 1 case / week ==> C ~ 50
Implications: Need for Better Closures
????
Na Q
t
 3 10 7
Algorithms
P
CT
Constraints
Assumptions:
• Performance is delivered
• Implicit algorithm
• Q ind. of t (!!)
Requirements:
• At least 3-D physics required
• Required problem time: 1 msec 1 sec
Conclusions:
• 3-D (i.e., fluid) calculations for
times of ~ 10 msec within reach
• Longer times require next
generation computers (or better
algorithms)
• Higher dimensional (kinetic)
long time calculations unrealistic
• Integrated kinetic effects must
come through low dimensionality
fluid closures
Outlook
• Interesting fluid systems have many degrees of freedom
– Kolmogorov => ~ Re per spatial dimension
– “Realistic” discrete model => N ~ Re3
– Small scale (kinetic or turbulent) physics affects large scale
dynamics
• The computers will never be big enough or fast enough for
“realistic” direct simulation!
• Need more work on closures; partnerships with theory;
validation with experiment
• Need a better idea?
– Abandon traditional initial value/time marching approach?
• “Equation-free multiscale” methods (Kevrikidis & Gear)
• “Project” microscale dynamics onto macroscales
• Perhaps these will lead to economical, realistic
(predictive?) modeling