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Computational MHD: A Model Problem for Widely Separated Time and Space Scales

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### 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

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

Weakly Nonlinear

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

Types of MHD ProblemsDifferent Problems Require Different Approaches

- Magneticallydominated 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

- Flow dominated plasmas
- Stellar interiors, convection, dynamo
- Magnetic forces are weak (b >> 1)
- Collisional
- Fluid equations good at all scales
- Closures: characterize sub-grid scale turbulence

Can adapt hydrodynamic 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?

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

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

- Equivalent to two-fluid model

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!

What a Difference B Makes!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

- Lundquist number:

- Explicit time step impractical:

Requires implicit methods

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:

d/L ~ S-a << 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

Inaccuracies can lead to “spectral pollution” and anomalous perpendicular transport

MHD Boundary Conditions

- Hydrodynamics:
- Conservation laws in flux-divergence form:

- Specify normal flux at boundary:
- Depends on inflow/outflow, super/sub-sonic

- Magnetic flux in MHD:

- 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:

- 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

Centroid: r, p, rV

Vertex: A, E

x

Face: B, V

z

Finite Volumes for MHD- Primary and dual grids (one possibility)

Conserves mass, momentum, magnetic flux

- 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

Minimize global error by expansion in basis functions

Galerkin Discrete Approximation

- 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:

- 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

Solenoidal Constraint

- Faraday:

- Depends on

- Cannot be guaranteed in discrete model

- Modified wave system
- Projection
- Diffusion
- Grid properties

Building a Discrete Model:

Temporal Discretization

Temporal Discretization

- Explicit methods
- Solve directly for values at n+1 in terms of values at time step n

- Computationally efficient, but…..

Time step limited by condition for numerical stability:

Temporal Discretization

- Implicit methods
- Values at n+1 in defined implicitly terms of values at time step n

- 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)

- Treat only “fast” part of operator implicitly to avoid time step restriction

- Precise decomposition of W 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 W are known, but an expression for S is difficult to achieve
- W : full MHD operator
- F: linearized MHD operator
- Use operator splitting:

- Expression for S not needed

Semi-Implicit Method

- Recognize that the operator F is completely arbitrary!!

- G can be chosen for accuracy and ease of inversion
- G should be easier to invert than F (or W!)
- 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

Old explicit code

How SI Works

Old explicit code

How SI Works

Old explicit code

How SI Works

- Stabilizes by dispersion
- Slows down unstable waves
- k-dependent inertia

Old explicit code

Semi-implicit term

Semi-Implicit Operator for MHD

- Linearized, ideal MHD wave operator
- Wide spectrum of normal modes
- Highly anisotropic spatial operator
- Basis of many implicit formulations
- Not a simple Laplacian
- Requires specialized pre-conditioners

- Challenge: find optimum algorithm for inverting this operator with CFL ~ 104

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

- Newton’s method
- Evaluation of Jacobian
- “Jacobian-free” methods
- “Newton-Krylov” methods

Example:

MHD Relaxation and the Laboratory “Dynamo”

Nonlinear

Relaxation

Dynamo

Diffusion

Instabilities

Plasma Relaxation- System attempts to minimize energy subject to constraints:

- 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

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

- 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

Self-reversal

- Perturb by 1 part in 106
- Dynamo is nonlinear and chaotic

Example:

Tokamak Disruption

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

Toroidal Fusion Plasmas: TokamakExperimental Tokamak Disruption

- Time dependence at disruption onset
- Growing 3-D magnetic perturbation (n = 1)
- Nonlinear evolution?
- Effect on confinement?
- Can this be predicted?

- Increase in neutral beam power
- Plasma pressure increases
- Sudden termination (disruption)

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 msec.
- Power ~ 5 GW

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

The Heliosphere During Whole Sun Month

August – September 1996

Solar Corona, Solar Wind, and HeliosphereLaunch 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

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
- Require new SI operators (4th order)
- Under development

Limits on Modeling

Balance of algorithm performance and problem requirements with available cycles

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)
- Dt - time step (seconds)

Constraints:

- P - peak hardware performance (Tflop/sec)
- e - hardware efficiency
- eP - delivered sustained performance
- T - problem time duration (seconds)
- C - # of cases / year
- 1 case / week ==> C ~ 50

Implications: Need for Better Closures

Assumptions:

- Performance is delivered
- Implicit algorithm
- Q ind. of Dt (!!)

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

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