computational magneto-fluid dynamicsu0016541/mhd_sheets_pdf/cmfdles1.uu09.pdf · computational...

Computational Magneto-Fluid Dynamics CMFD-1

Computational Magneto-Fluid Dynamics

Rony KeppensCentre for Plasma-Astrophysics, K.U.Leuven (Belgium)

& FOM-Institute for Plasma Physics ‘Rijnhuizen’

& Astronomical Institute, Utrecht University

Guest lectures at Utrecht University, May-June 2009

With material based on PRINCIPLES OF MAGNETOHYDRODYNAMICS

by J.P. Goedbloed & S. Poedts (Cambridge University Press, 2004)

and on ADVANCED MAGNETOHYDRODYNAMICS

by J.P. Goedbloed R.Keppens & S. Poedts (CUP, to appear 2009-2010)


Computational Magneto-Fluid dynamics

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Overview: may 11

• Numerical MHD in brief: governing MHD equations; example numerical applica-tions; focus of this course.

• Linear advection equation: discretizations, stability, diffusion, dispersion, order ofaccuracy.

• Linear hyperbolic systems and nonlinear scalar equations: Riemann problem,Burgers equation, shocks and rarefactions.


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Overview: may 18

• Finite Volume discretization: integral versus differential form, explicit time integra-tion, CFL condition, TVD concept and TVDLF method.

• Euler equations: gas dynamics in 1D, solution of the Riemann problem. TVDLFsimulations.

• Multi-D hydro applications: Rayleigh-Taylor and Kelvin-Helmholtz instability devel-opment, demonstrative shock-dominated astrophysical problems.


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Overview: may 25

• Preliminary aspects for MHD simulations: conservative and primitive formulations,generalized Riemann invariants. Compound waves.

• 1.5D isothermal MHD: with TVDLF simulations.

• Multi-D MHD: MHD wave anisotropies; ∇ · B = 0 for shock-capturing schemes


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Overview: june 8

• Demonstrative multi-D MHD applications: nonlinear evolution of linearly unstable,idealized configurations (from planar shear layers to 3D jet simulations, revisitingKelvin-Helmholtz instabilities, modeling kink instabilities).

• Transonic MHD in astrophysical simulations: transmagnetosonic winds, astro-physical jet launching, accretion funnels onto magnetized stars


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Magnetohydrodynamic model:

• macroscopic dynamics of perfectly conducting plasma

⇒ ideal MagnetoHydroDynamic – MHD – description

⇒ continuum, single fluid description of plasma in terms of ρ, v, p, B

⇒ conservation of mass, momentum, energy, and magnetic flux

• magnetic field B introduces Lorentz force J × B

⇒ perpendicular to field lines and current J

⇒ attractive/repulsive forces between parallel current-carrying wires

• no magnetic sources or ‘monopoles’ hence ∇ · B = 0

⇒ contrast to electric charges (sources of electric field)

⇒ magnetic field lines (tangent to B) have no beginning or end

⇒ always form closed loops


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Governing equations:

• 8 non-linear PDE for density ρ, velocity v, temperature T , B

∂ρ

∂t+ ∇ · (vρ) = 0

ρ

(∂v

∂t+ v · ∇v

)

+ ∇p− (∇× B) × B = ρg

ρ

(∂T

∂t+ v · ∇T

)

+ (γ − 1)p∇ · v = Sq

∂B

∂t−∇× (v × B) = SB

• pressure p = ρT (ideal gas only), external gravity g

⇒ Euler for gas dynamics + pre-Maxwell equations

⇒ use EM units where µ0 = 1

• Add ∇ · B = 0 ⇒ no magnetic monopoles

• Ideal MHD : no heat source/sink Sq, no source term SB


• conservation of mass: local density value can alter in 2 ways

∂ρ

∂t= − ρ∇ · v

︸︷︷︸local compressions

− v · ∇ρ︸︷︷︸

advected density gradients

⇒ total mass is conserved (no sinks/sources)

• momentum equation (Newton’s law)

ρ

(∂v

∂t+ v · ∇v

)

+ ∇p− (∇× B)︸︷︷︸

J

×B = ρg

⇒ inertial effects, pressure gradients, Lorentz force, exte rnal gravity

⇒ current found from B directly: Ampere’s law

J = ∇× B

⇒ neglects displacement current term in Maxwell equation

1

c2∂E

∂t+ µ0J = ∇× B

⇒ non-relativistic plasma flows v ≪ light speed c


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The induction equation:

• evolutionary equation for B in ideal MHD: Faraday’s law

∂B

∂t= ∇× (v × B)

︸︷︷︸−E

⇒ field lines are frozen in plasma

⇒ unimpeded flow along B, flow ⊥ B displaces field line

⇒ analytically: if ∇ · B = 0 initially, then always

• electric field in co-moving frame for perfectly conducting fluid

E′ = E + v × B = 0


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Ideal versus resistive MHD: kinetic modeling

• kinetic ‘MHD’: induction equation in multi-D MHD + ∇ · B = 0

⇒ prescribed velocity field, no back-reaction of B on flow (ignore Lorentz force)

⇒ assume B2/2 ≪ ρv2/2: insignificant magnetic energy

• Weiss (1966!) numerical simulations with resistivity

⇒ expulsion of magnetic flux by eddies

• consider medium with constant resistivity η, Ohm’s law

E′ = E + v × B = η j .

⇒ induction equation then given by

∂B

∂t= ∇× (v × B) + η∇2B


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Weiss kinetic simulations

• kinematic modeling: velocity field v time-invariant

⇒ set v = [sin(2πx) sin(2πy − π/2), sin(2πx + π/2) sin(2πy)]

⇒ 2D incompressible flow ∇ · v = 0

⇒ models 4 convection cells on unit square [0, 1]2

• magnetic field evolution from induction equation with resistivity

• ensure ∇ · B = 0: use vector potential

B = ∇× A ⇒ ∇ · (∇× A) = 0

⇒ 2D case: just involves z-component of vector potential Az ≡ A

B =

(∂A

∂y,−∂A

∂x, 0

)


• solve numerically induction equation in 2D

⇒ consider cases η = 0.1, η = 0.01, η = 0.001

⇒ from resistive towards ideal MHD case

⇒ η = 0: frozen in limit (only numerical diffusion)

• perform 302 simulations, initial B = (0, 1)

⇒ evolution of magnetic energy: 3 phases

⇒ field amplification, resistive diffusion, steady state

⇒ turnover: convective term comparable to resistivity


• steady-state configuration:

⇒ magnetic field/flux expelled from centre of eddies as η → 0

⇒ flux concentrates at edges of convective cells

⇒ returns in modern full 2D/3D MHD magnetoconvection models


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Ideal MHD and flux conservation:

• magnetic flux through surface intersecting B lines Ψ ≡∫ ∫

S

B · ndS

S1

S2

⇒ identical for any surface S along ‘flux tube’

⇒ easily found from Gauss theorem:∫ ∫ ∫

V

∇ · B dV =

∫ ∫

σ

B · ndσ = −∫ ∫

S1

B1 · n1dS1 +

∫ ∫

S2

B2 · n2dS2 = 0

• conservation of magnetic flux: basic law of ideal MHD

⇒ flux through surface element moving with fluid will remain con stant:

Ψ =

∫ ∫

C

B · ndσ = constant

⇒ for closed contour C moving with plasma


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Ideal MHD and conservation laws:

• equivalently: conservation laws for density ρ, momentum density m = ρv, H and B

∂ρ

∂t+ ∇ · (vρ) = Sρ

∂m

∂t+ ∇ · (vρv − BB) + ∇ptot = Sρv

∂H∂t

+ ∇ · (vH + vptot − BB · v) = Se

∂B

∂t+ ∇ · (vB − Bv) = SB

• ptot ≡ thermal pressure + magnetic pressure

• total energy density H has 3 contributions

H =p

γ − 1︸︷︷︸internal

+ρv2

2︸︷︷︸kinetic

+1

2B2

︸︷︷︸magnetic

• Sources (Sinks) of conserved quantities in right hand side


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Scale invariance:

• units of length, mass, time (plus µ0 = 1)

⇒ trivially scale out of equations

⇒ take lengthscale l0, field strength B0, density ρ0

⇒ speed v0 = B0/√ρ0 timescale from t0 = l0/v0

• pure MHD signal at Alfv en speed B0/√ρ0

⇒ field line wiggles: magnetic tension as restoring force

x

z

k

y

B0

v1+B0 B1

• MHD can be applied to laboratory, solar, galactic dimensions alike!

⇒ macroscopic dynamics of plasmas in dimensionless form


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Example applications:

• consider STATIC MHD equilibria v = 0, leaves only{−∇p + (∇× B) × B + ρg = 0

and ∇ · B = 0

⇒ governing equations for stratified, magnetostatic equilibria

• Prominences in solarcorona

• translational symme-try

⇒ 2D in cross-section

• Tesla strong B intokamaks

• neglect g, axisym-metry

⇒ 2D in cross-section


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Computing prominence equilibria:

• 2D Problem in cross-section governed by second order elliptic PDE

⇒ in terms of flux function ψ(y, z), field lines on isosurfaces

⇒ along poloidal flux contour: pressure gradient balances gravity

• use 2D Finite Element discretization, Picard iterate to solution

⇒ use local expansion functions , of given polynomial form (here bicubic)

⇒ isothermal, double prominence (left); non-isothermal three-part structure (right)

(from Petrie et al, ApJ, 2007)


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Tokamak equilibria:

• Controlled thermonuclear fusion: magnetically caged (Tesla fields), hot plasmas

⇒ balance of ∇p = J × B: three orthogonal vectors

• 3D visualization: ρ-isosurfaces, p in cross-section, grid impression

⇒ B vectors and selected fieldlines: variation of winding with radius


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Geodynamo simulations:

• Earth’s magnetic field is currently mostly dipolar:

– evidence from magnetized rocks that orientation reverses every few 100000years, taking a few 1000 years for full reversal.

• Earth consist of inner core, outer core, mantle, crust:

– liquid iron outer core (1300 < R < 3400 km) must maintain field

– rotation and convection in moving conducting fluid described by Ohm’s law

E + v × B = η j .

– Inhomogeneity of magnetic field decays in time τD determined by resistivity ηand length scale l0 ∼ ∇−1 of inhomogeneity:

τD = µ0l20/η = l20/η .

– resistive diffusion time scale τD ∼ 5 × 1011 s = 16 000 years

– need sustained B generation by molten iron motion

– convection driven by heat from radioactive decay in inner solid core


• Full 3D MHD simulations by Glatzmaier & Roberts (1995):

– simulated several 100000 years of geodynamo activity

– inner core mediates reversals: its B changes on diffusion time

– captured reversal event, changed dipole orientation in 1000 years:

http://www.es.ucsc.edu/ glatz/(website Gary Glatzmaier)

• spectral method : all variables written in expansion exploiting global functions ,in particular here: Chebyshev polynomials (radial variation) and spherical harmonics(both angular variations)

http://www.es.ucsc.edu/~glatz/


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Eruptive event studies:

• simulating 3D kink-unstable loop evolution

⇒ Torok & Kliem, ApJ 2005, 630, L97

• Further ejection and CME initiation

• Finite difference/volume – Lax-Wendroff method : update local function valuesthrough fluxes computed from neighboring grid points. Here for zero-beta p = 0conditions, neglect g, stabilized by added viscosity terms and ‘artificial smoothing’


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Numerical discretizations and MHD:

• 3 examples given meant to demonstrate

⇒ diversity of MHD problems : static, dynamic, long term slow evolution versussudden events, different geometries, . . .

⇒ diversity of employed discretizations (FEM, spectral, finite difference, finitevolume) and numerical algorithms

• This course can NOT treat all of these in detail

⇒ I will focus on modern, shock-capturing schemes

⇒ pay attention to nonlinear ideal MHD in particular

⇒ give state-of-the-art examples, for Finite Volume approaches


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Reference material:

• Throughout this course, I use material based on

⇒ Principles of Magnetohydrodynamics , Goedbloed & Poedts , CUP 2004

⇒ Advanced Magnetohydrodynamics , Goedbloed, Keppens & Poedts , CUPto appear 2009-2010

• I also recommend (mostly HD):

⇒ P. Wesseling, Principles of Computational Fluid dynamics

⇒ E. F. Toro, Riemann Solvers and Numerical Methods for Fluid dynamics. Apractical Introduction (2nd Edition)

⇒ R. J. LeVeque, Numerical Methods for Conservation Laws

⇒ R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems

⇒ R. J. LeVeque, D. Mihalas, E. A. Dorfi and E. Muller, Computational Methodsfor Astrophysical Fluid Flow


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The advection equation

Trivial problem: linear advection equation

⇒ ∂tρ + v∂xρ = 0

⇒ constant given velocity v

⇒ initial density pulse ρ(x, t = 0) = ρ0(x)

Trivial solution: ρ(x, t) = ρ0(x− vt)

⇒ analytically: done!

⇒ numerically?


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Focus on Riemann Problem

• initial data ρ(x, 0) =

{ρl x < 0ρr x > 0

⇒ two constant states separated by discontinuity

⇒ solution still trivial, graphically:

1 x-t plane

t

x

Linear advection (v>0)

vt

t=0

t=t1ρ

ρ

l

r

x= 0

x= 0

rρl ρ

x= 0

ρl

ρr


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Numerical discretizations

• First Attempt:

⇒ temporal discretization: forward Euler

⇒ spatial discretization: Centered differencing

⇒ ρn+1

i−ρn

i

∆t + vρn

i+1−ρn

i−1

2∆x = 0

• numerically solve Riemann Problem

⇒ manifestation of a numerical instability

⇒ Von Neumann stability analysis: insert ρ(x, tn) = Gnρoeikx

⇒ amplification ρn+1 = Gn+1

Gn ρn ≡ Gρn

⇒ | G |=√

1 +(v∆t

∆xsink∆x

)2> 1 for all k

⇒ unconditionally unstable!


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Upwind method

• Second Attempt:

⇒ temporal discretization: forward Euler

⇒ spatial discretization: upwind, for v > 0:

⇒ ρn+1

i−ρn

i

∆t+ v

ρn

i−ρn

i−1

∆x= 0

• numerically solve Riemann Problem

⇒ first order upwind method (here indentical to TVDLF1)

⇒ note: diffusion off the ‘contact discontinuity’

⇒ Von Neumann stability analysis: amplification

| G |=√

1 + 2v∆t

∆x

(∆t

∆xv − 1

)

(1 − cos k∆x)

⇒ stable for ∆t < ∆x/v


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MacCormack method

• mixture of (spatially) forward and backward difference

⇒ ρ∗i = ρni − ∆t∆xv

(ρni+1 − ρni

)

⇒ ρ∗∗i = ρni − ∆t∆xv

(ρ∗i − ρ∗i−1

)

⇒ ρn+1i = ρ∗

i+ρ∗∗

i

2

• rewritten:

ρn+1i = ρni − ∆t

∆xv(ρn

i+1−ρn

i−1

2

)

+ (∆t)2

(∆x)2v2(ρn

i+1−2ρn

i+ρn

i−1

2

)

⇒ second order accurate

⇒ Von Neumann stability:

| G |=√

1 +(∆t)2

(∆x)2v2 (cos2 k∆x− 2 cos k∆x + 1)

[(∆t)2

(∆x)2v2 − 1

]

⇒ stable | G |≤ 1 for ∆t ≤ ∆x/v


• MacCormack method: dispersive

⇒ manifests Gibbs phenomenon

⇒ non-monotonicity preserving: monotone ρ(x, 0) develops extrema


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Order of accuracy

• ‘Local Truncation Error’: insert exact solution in discretization formula

• consider upwind method

ρn+1i − ρni

∆t+ v

ρni − ρni−1

∆x= 0

⇒ LTE from LUP∆t = 1

∆t [ρ(x, t + ∆t) − ρ(x, t)] + v∆x [ρ(x, t) − ρ(x− ∆x, t)]

⇒ Taylor expand (assuming smooth solutions), use ∂tρ + v∂xρ = 0

⇒ LUP∆t = ∆t

2v

(v − ∆x

∆t

)ρxx + O

(∆t2,∆x2

)

⇒ goes to zero like ∆t for ∆t→ 0: first order method

• order of accuracy of MacCormack method:

⇒ LMC∆t = ∆t2

6v

(∆x2

∆t2− v2

)

ρxxx + O(∆t3,∆x3

)

⇒ MC is second order accurate


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Lax-Friedrichs

• Lax-Friedrichs discretization

ρn+1i =

1

2(ρni+1 + ρni−1) −

∆t

2∆xv(ρni+1 − ρni−1)

⇒ Von Neumann stability analysis: ρ(x, tn) = Gnρ0 exp(ikx)

⇒ amplification ρn+1 = Gn+1

Gn ρn ≡ Gρn

⇒ | G |=| cos(k∆x) − iv∆t∆x

sin(k∆x) | ≤ 1 if ∆t ≤ ∆x/v

⇒ first order since LLF∆t = ∆t

2

(

v2 − ∆x2

∆t2

)

ρxx + O(∆t2,∆x2

)


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Consistency

• Method with LTE → 0 for ∆t→ 0 is consistent

• smooth solution: LTE and global error are same order for stable method

• Lax equivalence theorem: for consistent method:

⇒ stability is necessary and sufficient for convergence!


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Stencil

• Stencil of a method: graphically

Time

Space

First order Lax-Friedrichs

Second order MacCormack

First Order Upwind


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Convergence

• Convergence of a method in accord with order of accuracy

• periodically advect a Gaussian Bell profile (one cycle)

⇒ error as true difference with t = 0 pulse

⇒ compare Upwind with MacCormack solution at N = [50, 100, 200, 400]


• comparison of two second order methods

⇒ periodic advection of Gaussian Bell and Square pulse


• MacCormack versus TVDLF (defined later)

⇒ smooth versus discontinuous initial profile

• both second order accurate methods render 1st order convergence!


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Monotonicity

• Monotonicity preserving method:

⇒ monotone initial data remains monotone

⇒ MC: not monotonicity preserving; TVDLF: monotonicity preserving

⇒ for Riemann problem: no oscillations will appear

• Godunov theorem: linear monotonicity preserving method

⇒ at most 1st order accurate

• second order accuracy + monotonicity preserving:

⇒ TVDLF method depends nonlinearly on data ρni


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TVD concept

• Numerical Total Variation

TVn ≡∑

i| ∆ρni+1/2 |

⇒ summed differences ∆ρi+1/2 = ρi+1 − ρi

⇒ TVD concept: TV diminishes with time:

TVn+1 ≤ TVn

• TVD → monotonicity preserving

⇒ TVD methods degenerate to 1st order accuracy at extrema


• TVDLF method: TVD hence monotonicity preserving

⇒ diffuses the discontinuity

⇒ monotone ρ(x, 0) develops no extrema!

⇒ first order at jump


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Linear Hyperbolic Systems

• constant coefficient linear system

~qt + A~qx = 0

⇒ with ~q(x, t) ∈ ℜm and matrix A ∈ ℜm×m

• hyperbolic when matrix A is diagonalizable with real eigenvalues

⇒ strictly hyperbolic when distinct

⇒ m eigenvectors + m real eigenvalues

A~rp = λp~rp with p : 1, . . . , m


• write as

[A] [~r1 | ~r2 | . . . | ~rm] = [~r1 | ~r2 | . . . | ~rm]

λ1

λ2

. . .λm

⇒ or shorthand AR = RΛ with diagonal matrix Λ

⇒ matrix of right eigenvectors as columns R

• The solution to system ~qt + A~qx = 0 is equivalent to:

⇒ pre-multiply with R−1 or:

(R−1~q)t + R−1(RΛR−1)~qx = 0

⇒ redefine ~v ≡ R−1~q to get

~vt + Λ~vx = 0

⇒ m independent constant coefficient linear advection equatio ns!


• Each advection equation has trivial analytic solution:

vp(x, t) = vp(x− λpt, 0)

⇒ solution to the full linear hyperbolic system is then

⇒ ~q(x, t) =∑m

p=1vp(x− λpt, 0)~rp

⇒ depends on initial data at m discrete points

• nomenclature: ~v are ‘characteristic variables’

⇒ curves x = xo + λpt are “p-characteristics”


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NonLinear scalar equation

• general equation form qt + f(q)x = 0

⇒ with nonlinear flux function f(q): Burgers equation f(q) = q2/2

⇒ with q ≡ ρ ‘advection’ speed increasing with density

• Demonstrates wave steepening (area conservation) and shock formation:

⇒ advect triangular pulse with MacCormack and TVDLF scheme


• Riemann problem has two cases:

⇒ ρl > ρr: shock traveling at speed s = ρl+ρr

2

⇒ ρl < ρr: rarefaction wave ρ(x, t) = ρ(x/t) =

ρl x < ρltx/t ρlt < x < ρrtρr x > ρrt

⇒ Numerically with TVDLF scheme:


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End of day 1: summary and voluntary take-home assignment

• MHD equations intro

⇒ Explain difference between ideal-resistive MHD using induction equation alone

⇒ give (modern) examples where MHD simulations occur in astrophysical contexts

• Numerical algorithms and associated jargon:

⇒ derive stability constraints for mentioned schemes using von Neumann analysis

⇒ explain connection advection equation - linear hyperbolic system - Burgers

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