Introduction to SimulationsIntroduction to SimulationsIntroduction to MHD SimulationIntroduction to MHD Simulation

LecturesLectures 9 & 109 & 10

Categories of Partial Differential Equations• Partial Differential Equations are usually classified into three categories

based on the characteristics or curves of information propagation.

– Hyperbolic equations – e.g. a wave equation where v=const. is the velocity of

propagation.

– Parabolic equations- e.g. a diffusion equation where D is the diffusion coefficient.

– Elliptic equations – e.g. Poisson equation where ρ is the source term. For ρ=0 we have Laplace's equation.

• The first two examples are initial value problems – they tell how a system

evolves in time and are solved by integrating forward in time.

• The third is a boundary value problem - there the system must give the

solution everywhere simultaneously. (You can’t integrate in from the

boundary like you integrate forward in time).

• In global MHD modeling we interested in solving a set of hyperbolic

equations at the Sun, heliosphere and magnetosphere and an elliptic

equation in the ionosphere.

2

2

2

2

2

x

uv

t

u

∂∂

=∂∂

∂∂

∂∂

−=∂∂

x

uD

xt

u

( )yxy

u

x

u,

2

2

2

2

ρ=∂∂

+∂∂

Boundary Value Problems• Stable solutions to boundary value problems are usually easy to achieve.

Most effort is on making them efficient.

• These become the solution of a number of simultaneous algebraic

equations.

• Consider a solution to

• Represent u(x,y) by its values at specific points where ∆ is the grid spacing

• Substitute into the Poisson equation and let uj,l=u(xj.yl)

( )yxy

u

x

u,

2

2

2

2

ρ=∂∂

+∂∂

Lllyy

Jjjxx

l

j

,...1,0

,...1,0

0

0

=∆+=

=∆+=

lj

ljljljljljlj uuuuuu,2

1,,1,

2

,1,,1 22ρ=

∆+−

+∆

+− −+−+

ljljljljlj uuuu ,

2

,1,,1,1 4 ρ∆=−++ −−+

Solutions to Boundary Value Problems

• This can be converted into a matrix equation following

the prescription in Press (1986) page 618 and the resulting matrix can be inverted.

• Define for j=0,1…J, and l=0,1,….,L. In this way i increases most rapidly along the columns

representing y values.

• The equation now becomes

• The points j=0,j=J, l=0, and l=L are the boundaries

where either the function or its derivative are specified.

• This gives an equation of the form and can be

solved by a number of inversion techniques.

( ) lLji ++≡ 1

( ) iiiiLiLi uuuuu ρ2

1111 4 ∆=−+++ −++−++

buA =⋅

A Note on Conservation Laws• Consider a quantity that can be moved from place to

place.

• Let be the flux of this quantity – i.e. if we have an element of area then is the amount of the

quantity passing the area element per unit time.

• Consider a volume V of space, bounded by a surface

S.

• If σ is the density of the substance then the total

amount in the volume is

• The rate at which material is lost through the surface

is

fr

Ar

δ Afrr

δ⋅

∫V

dVσ

∫ ⋅S

Adfrr

∫ ∫ ⋅−=V S

AdfdVdt

d rrσ

Flux Conservation Continued

– Use Gauss’ theorem

– An equation of the preceding form means that the

quantity whose density is σ is conserved.

∫ ∫ ⋅−=V S

AdfdVdt

d rrσ

0=

⋅∇+

∂∂

∫ dVft

V

rσ

ft

r⋅−∇=

∂∂σ

Flux-Conservative Initial Value Problems• Many initial value problems can be written in flux-conservative form where u

and F are vectors. F is called the conservative flux.

• The simplest general flux-conservative equation is the advection equation

with a constant velocity v

• The solution of this equation is a wave propagating in the positive x-

direction where f is some function.

• Let's try the most straightforward approach and select equal points in x and

t

( )x

uF

t

u

∂∂

−=∂∂

rrr

x

uv

t

u

∂∂

−=∂∂

( )vtxfu −=

Nntntt

Jjxjxx

n

j

,...1,0

,...1,0

0

0

=∆+=

=∆+=

Flux-Conservative Initial Value Problems –The Advection Problem 2

• Forward Euler differencing – First order accurate in time- can calculate

quantities at n+1 knowing only previous quantities.

• Second order approximation in space using only quantities know at time n.

• The resulting approximation in the advective equation called Forward Time

Centered Space (FTCS) becomes

• Too bad it doesn't work!

)(

1

,

tOt

uu

t

un

j

n

j

nj

∆+∆−

=∂∂ +

)(2

211

,

xOx

uu

x

un

j

n

j

nj

∆+∆−

=∂∂ −+

∆−

−=∆− −+

+

x

uuv

t

uun

j

n

j

n

j

n

j

2

11

1

Flux Conservative Initial Value Problems: Stability

• The FTSC approach is an explicit scheme – i.e. ujn+1 can be calculated

from things already known.

• Assume that the coefficients of the difference equations are slowly varying.

In that case the independent solutions (eigenmodes) of the difference

equations have the form where k is a real spatial wave number

and is a complex number.

• The difference equations will have an exponentially growing mode if

• The number ξ is called the amplification factor at a given wave number k.

• Substitute the equation for ξ into the FTCS equation and get

• FTCS is unconditionally unstable because the absolute value of ξ is always

greater than one.

xikjnn

j eu∆= ξ

( )kξξ =

( ) 1>kξ

( ) xkx

tvik ∆

∆∆

−= sin1ξ

Flux Conservative Initial Value Problems: The Courant Condition

( )n

j

n

j

n

j uuu 112

1−+ +→• Let and substitute into the advection

equation to give

(Lax Method)

•The amplification factor becomes

•The stability requirement gives . This is called the

Courant-Friedrichs-Lewy stability criterion. Information propagates

with a maximum velocity of v. If the differencing scheme requires

information from too far away to propagate to a given point it will

be unstable. Therefore ∆t must not be too large.

( ) ( )n

j

n

j

n

j

n

j

n

j uux

tvuuu 1111

1

2

1−+−+

+ −∆∆

−+=

xkx

tvixk ∆

∆∆

−∆= sincosξ

1≤∆∆x

tv

Flux-Conservative Initial Value Problems: The Courant Condition 2

• How does replacing ujn in the time derivative by its average lead to stable

solutions?

• Rewrite the equations for the Lax method as

• This is the FTCS representation for the equation

• The extra term is a diffusion term – we call this adding numerical dissipation

or numerical viscosity.

• From the amplification factor unless and the wave amplitude

decreases. This is better than spurious increases.

• Finite-difference schemes also can exhibit dispersion or phase errors. Even

if the amplification factor is . At each time step the modes

get multiplied by different phase factors depending on their value of k.

Eventually the wave packet disperses.

• The Lax method is first order in time. Other methods have second order or

higher in both time and space. Higher order in time allows larger time steps.

∆+−

+

∆−

−=∆− −+−+

+

t

uuu

x

uuv

t

uun

j

n

j

n

j

n

j

n

j

n

j

n

j 1111

12

2

1

2

( )t

x

x

uv

t

u

∆∆

+∂∂

−=∂∂

2

2

xtv ∆=∆ 1<ξ

xtv ∆=∆ xike ∆−=ξ

Flux-Conservative Initial Value Problems: Shocks and Upwind Differencing

• Numerical viscosity tends to control the formation of spurious shocks

• In space physics shocks are physically real.

• For wave equations propagation (amplitude or phase) errors are usually the most troublesome but for advection transport errors are important.

• For example in the Lax scheme j propagates to j+1 and j-1 in the next time step. If v is positive the propagation is only in the plus direction.

• One way to deal with transport errors is to use upwind differencing. This uses the fact that depending on the sign of v an advectedquantity propagates in one direction.

• This is useful where advected variables pass through a shock.

<

∆−

−=∆−

>

∆−

−=∆−

++

−+

0,

0,

1

1

1

1

n

j

n

j

n

jn

j

n

j

n

j

n

j

n

j

n

jn

j

n

j

n

j

vx

uuv

t

uu

vx

uuv

t

uu

An Example of an Implicit Scheme

• Consider the parabolic diffusion equation in one spatial dimension

• This can be written in flux-conservative form if

• Let D be a constant.

∂∂

∂∂

=∂∂

x

uD

xt

u

x

uDF

∂∂

−=

2

2

x

uD

t

u

∂∂

=∂∂

An Implicit Scheme 2

• Difference it explicitly

• Note that the right hand side has a second derivative that has been differenced.

• This time the amplification factor is

and stability is achieved for

• The maximum time step is related to the time it takes to

diffuse across the width of a cell or

( )

∆

+−= −+

∆−+

2

11 21

x

uuuD

n

j

n

j

n

j

t

uu nj

nj

( )

∆−=

∆

∆

2sin1

24

2

xkx

tDξ

12

2≤

∆∆

x

tD

D

2λτ ≈

An Implicit Scheme 3

• Since we usually are interested in physical phenomena

on scales much larger than ∆x this would require us to take too many time steps.

• We need to take larger time steps but larger time steps

will not work for small scale phenomena.

• We might be ok if we could figure out a way to do something that works on large scales at the expense of

small scales.

• Consider

• This is same as before except that everything on the

right is at time n+1.

• This is an implicit scheme in which you must solve a

set of simultaneous equations at each timestep for ujn+1

( )

∆

+−=

∆− +

−++

++

2

1

1

11

1

12

x

uuuD

t

uu n

j

n

j

n

j

n

j

n

j

An Implicit Scheme 4

• At large scales the amplification factor becomes

• This is unconditionally stable at any time step size!

• The details of the small-scale evolution of the initial conditions are inaccurate.

• This is first order accurate but higher order schemes are possible.

• The biggest drawback of this approach is that you

frequently have to invert very big matrices to solve the linear equations.

• Some modern approaches combine implicit and explicit approaches. (The solar code used for the flux

cancellation model is semi-implicit.)

• The two MHD codes we will use in this class are explicit.

∆+=

2sin41

1

2 xkαξ

The Magnetohydrodynamic Equations

• Macroscopic plasma properties are governed by basic conservation laws for mass, momentum and energy in a fluid.

Bj

jjBvE

B

EB

Ejv

BjIvvv

v

rr

rrrr

r

rr

rrr

rrtrrr

r

×∇=

=+×−=

=⋅∇

×−∇=

−+=⋅+⋅−∇=

×+⋅−∇=

⋅−∇=

0

2

2

1

0

t

1v

2

1+])[(

t

+)(t

)(t

µ

αηη

∂∂

γρ

∂∂

ρ∂

∂ρ

ρ∂

ρ∂

pepe

e

p

Mass

Momentum

Energy*

Faraday’s Law

Gauss’ Law

Ohm’s Law

Ampere’s Law

* Some solve for a pressure equation ( )3

51 =⋅−+⋅∇−=

∇⋅+∂∂ γγγ Ejvv

rrrrpp

t

p

Properties of MHD

• In ideal MHD the field is frozen-in ( ) provided η is small.

• MHD is useful for a wide variety of problems because basically it is a statement of mass, momentum and

energy conservation.

• Information is propaged by three MHD wave modes.

• MHD equations allow us to capture shock waves (important in the supersonic solar wind).

• The inclusion of resistivity (η) allows us to break the frozen-in approximation and study magnetic

reconnection.

2B

BEv

rrr ×

=

MHD wave modes

• Alfvén speed:

• Sound speed:

• Intermediate (Alfvén) wave

• Fast and slow waves:

2

0

a

Bc

µ ρ=

s

Pc

γρ

=

cosph Av c θ=

2 2 2 2 2 2 2 2 21( ) ( ) 4 cos

2ph s A s A s Av c c c c c c θ = + ± + −

/k B kBθ = ⋅v v

Reconnection– As long as frozen in flux holds plasmas can mix along flux tubes

but not across them.

When two plasma regimes interact a thin boundary will separate

the plasma

The magnetic field on either side of the boundary will be tangential

to the boundary (e.g. a current sheet forms).

– If the conductivity is finite and there is no flow Faraday’s law and Ampere’s law give a diffusion equation

Magnetic field diffuses down the field gradient toward the central

plane where it annihilates with oppositely directed flux diffusing

from the other side.

This reduces the field gradient and the whole process stops but not

until magnetic field energy has been converted into heat via Joule

heating (the resulting pressure increase is what is needed to

balance the decrease in magnetic field pressure).

2

21

0 z

Bt

B x

∂∂=∂

∂σµ

r

– For the process to continue flow must transport magnetic flux toward the boundary at the rate at

which it is being annihilated.

An electric field in the Ey ( ) direction will provide this in flow.

In the center of the current sheet B=0 and Ohm’s law gives

If the current sheet has a thickness 2l Ampere’s law gives

00 xzy BuE −=

σyy jE =

lBj xy 00µ=

2 l JY

EY

EY

Reconnection 2

Equating the EY expressions

Thus the current sheet thickness adjusts to produce a balance between diffusion and convection. This

means we have very thin current sheets.

There is no way for the plasma to escape this system. However, if the diffusion is limited in extent then

flows can move the plasma out through the sides.

( )00

1 zul σµ−=

Reconnection 3

– When the diffusion is limited in space annihilation is replaced by reconnection Field lines flow into the diffusion region from the top and

bottom

Instead of being annihilated the field lines move out the sides.

In the process they are “cut” and “reconnected” to different partners.

Plasma originally on different flux tubes, coming from different places finds itself on a single flux tube in violation of frozen in flux.

The boundary which originally had Bx only now has Bz as well.

– Reconnection allows previously unconnected regions to exchange plasma and hence mass, energy and momentum. Although MHD breaks down in the diffusion region, plasma

is accelerated in the convection region where MHD is still valid.

Reconnection 4

X

Z

Toward a Self-Consistent Solution

Solving the MHD Equations – Magnetospheric Boundary Conditions

• The MHD equations are solved as an

initial value problem.

• Solar wind parameters enter the

upstream edge of the simulation and

interact with fields and plasmas in the

simulation box.

• Boundary conditions at the

downstream edge, the north and south

edges and the east and west edges are

set to approximate infinity.

–Downstream- where

ψ represents the parameters in the MHD equations.

– The bow shock frequently passes

through the side and top and

bottom boundaries. Here setting

the derivative approximately

parallel to the shock to zero works

well.

0=∂∂ xψ

• Map field aligned currents

( ) from the inner boundary to the

ionosphere.

• Using current continuity the relationship

between the ionospheric potential and the

parallel current is

• Use mapped FAC and conductivity model

to solve for the ionospheric potential Φ.– For a scalar conductance (Σ) the potential

becomes

– This is Poisson's equation and can be

solved as a boundary value problem.

• Map the potential back to inner boundary

and use it to determine boundary condition

for the perpendicular flow.

Solving the MHD Equations: The Inner (Ionosphere) Boundary

IJ sin||−=Φ∇⋅∑⋅∇

||JMagnetosphere

Ionosphere

Inner

Boundary

||J

B

Φ∇−

2B

BV

×Φ∇−=⊥

||J

Σ=Φ∇ j2

Ionospheric conductance• The dense regions of the ionosphere (the D, E and F regions)

contain concentrations of free electrons and ions. These mobile charges make the ionosphere highly conducting.

• Electrical currents can be generated in the ionosphere.

• The ionosphere is collisional. Assume that it has an electric field but for now no magnetic field. The ion and electron equations of motion will be

where is the ion neutral collision frequency and is the electron neutral collision frequency.

– For this simple case the current will be related to the electricfield by

where is a scalar conductivity.

• If there is a magnetic field there are magnetic field terms in the momentum equation. In a coordinate system with along the z-axis the conductivity becomes a tensor.

eene

iini

umEe

umEqrr

rr

ν

ν

=−

=

−==

e

ei

n

uueEj

rrrr

0σ

inνenν

0σ

−=

000

0

0

σσσσσ

σ PH

HP

Br

• Specific conductivity – along the magnetic field

• Pedersen conductivity – in the direction of the applied electric field

• Hall conductivity – in the direction perpendicular to the applied field

where and are the total electron and ion momentum transfer collision frequencies and and are the electron and ion gyrofrequencies.

• The Hall conductivity is important only in the D and E regions.

• The specific conductivity is very important for magnetosphere and ionosphere physics. If all field lines would be equipotentials.

– Electric fields generated in the ionosphere (magnetosphere) would map along magnetic field lines into the magnetosphere (ionosphere)

+=

iiee

emm

neνν

σ 112

0

Ω++

Ω+=

iii

i

eee

eeP

mmne

112222

2

νν

ννσ

Ω+Ω

+Ω+

Ω−=

iii

i

eee

eeH

mmne

112222

2

ννσ

eν iνeΩ iΩ

∞→0σ

• Assume a generalized Ohm’s law of the form and that

• The total current density in the ionosphere is

where and refer to perpendicular and parallel to the magnetic field.

• Space plasmas are quasi-neutral so

where

• The current continuity equation can be written where is along the magnetic field.

• Integrate along the magnetic field line from the bottom of the ionosphere to infinity. Since the field lines are nearly equipotentials we can write where the perpendicular height integrated conductivity tensor is

( )BuEjrrrr

×+⋅= σ0=u

r

B

EBEEj HP

rrrrr ×

++= ⊥ σσσ 0

⊥

( ) ( ) 00 =⋅∇+⋅∇=⋅∇ ⊥⊥⊥ EEJrrr

σσ

−=⊥

PH

HP

σσσσ

σ

( )s

jE

∂

∂−=⋅∇ ⊥⊥⊥

rσ

s∂

( ) ( )⊥⊥⊥⊥∞ ⊥ ∑∫ ⋅∇=

⋅∇=∞ EEdsj

s rr0 σ

−=

∑∑∑∑∑ ⊥

PH

HP

Ionospheric Conductance Model• Solar EUV ionization

– Empirical model- [Moen and Brekke, 1993]

• Diffuse auroral precipitation

– Strong pitch angle scattering at the inner boundary of simulation-

[Kennel and Petschek, 1966]

• Electron precipitation associated with upward field-aligned currents

(Knight, [1972] relationship-

where FE is the energy flux to the ionosphere, ∆Φis the parallel potential difference, and j is the parallel current density.)

jFE ∆Φ=

• Ionospheric Pedersen

conductance viewed from dusk.

• Note the large day-

night asymmetry.

This results from ionization by solar

EUV radiation.

Pedersen Conductance• Ionospheric Hall and Pedersen conductance

from a simulation of the magnetosphere during a

prolonged period with southward IMF.

• The white lines show the

ionospheric convection

pattern.

• Precipitation from the magnetosphere

enhances both the Hall and Pedersen

conductances at night.

Hall Conductance

Solving MHD Equations: Grids• A wide variety of grids are

used.

–The grid left is from the model Jimmy Raeder and Mostafa El Alaoui used a non-uniform Cartesian grid.

–The BATSRUS code from the University of Michigan uses a Cartesian mesh which is run time adaptive.

–Ogino uses a uniform Cartesian grid with 108 cells.

–Linker uses a non-uniform spherical grid.

–Lyon uses an unstructured spherical grid.

Solving the MHD Equations: Resistivity• The ideal MHD equations are non-

dissipative.

• Numerical resistivity occurs in all codes

caused by the averaging errors inherent in

the numerical process.

• Most of the MHD modes add dissipation

terms to the ideal MHD equations. Some

form of dissipation is necessary if

phenomena like reconnection, in which ideal

MHD breaks down, are to be included in the

simulation.

•The resistivity models can depend on the

physical parameters. For instance in the

Raeder-El Alaoui global MHD code the

resistivity has a threshold in current density

and then is proportional to the currrent

density squared ( ).2

Jαη =

Solving the MHD Equations: Spatial Differencing for Gas Dynamics

• A conservative finite difference

method to solve the gas dynamics

part of the MHD equations. This is the

procedure used in the Raeder-El

Alaoui code.

• The flux-limiting procedure applies

diffusion only in the regions where it

is necessary to balance dispersive

effects.

)F(Ut

U ⋅−∇=∂

∂

y

UfUf

x

UfUf

t

U jijijiji

∆

−−

∆

−−=

∂∂ −+−+

)()()()(2

1,

2

1,,

2

1,

2

1

))((

))()((

iiiiii

iii

UUccvv

UFUFf

−+++−

+=

+++

++

111

1

2

1

4

1

2

1

jif

,21−

Solving the MHD Equations: Time Stepping

• An explicit conservative predictor-

corrector scheme for time

stepping (second order accurate):

• Stability criterion (Courant-

Friedrichs-Lewy, CFL):

• This must be satisfied everywhere

in the simulation domain.

• Since characteristic velocity is the

Alfvén velocity the Courant

condition is important near the

Earth where the Alfvén becomes

large and the time step becomes

small.

)F(Ut

U ⋅−∇=∂

∂

)F(.

)F(.2

1

2

1

1

2

1

++

+

∇∆−=

∇∆−=

nnn

nnn

UtUU

UtUU

msvv

zyxt

+∆∆∆≤∆ ),,min(

max δ

Solving the MHD Equations: The Field Equations

• Place the magnetic field

components on the center of the

cell faces

• Place the electric field

components on the centers of the

cell edges

21

21

21 ,,,,,,

)(;)(;)( +++ kjizkjiykjix BBB

kjizkjiykjix EEE,,,,,,

21

21

21

21

21

21 )(;)(;)( ++++++

( ) ( ) ( )[ ]( ) ( )[ ] yEE

zEEBt

kjizkjiz

kjiykjiykjix

∆−−

∆−=∂∂

−+++

−++++

,,,,

,,,,,,

21

21

21

21

21

21

21

21

21

Divergence of B Control

• 8-wave scheme – adds source terms to the MHD equations so that magnetic monopoles are advected with

the velocity of the plasma.

• Diffusive control – adds the gradient of to the

induction equations, so that the error in is diffused away.

• Hyperbolic correction adds extra scalar equation to

propagate error in with preset speed.

• Projection scheme eliminates the errors after each

time step by solving a Poisson equation and projecting to a divergence free solution

• Constrained transport use a special discretization that the conserves to round off.

Br

⋅∇Br

⋅∇

Br

⋅∇Br

⋅∇Br

⋅∇=∇ φ2

φ∇−=′ BBrr

Br

⋅∇

Some Examples of Global MHD Simulations: The Earth's Magnetosphere

• A simulation of the Earth's magnetosphere during a large

magnetospheric substorm

using the Raeder- El Alaouicode.

• The thermal pressure in the

noon-midnight meridian plane.

•This snapshot was taken at the time of the substorm

onset.

Some Examples of Global MHD Simulations: Planetary Magnetospheres

• The temperature of the

plasma in Jupiter's

magnetosphere during an

interval in which the IMF

was northward.

• This simulation used the

Ogino-Walker MHD code.

For this calculation the

MHD equations were

advanced on a uniform

Cartesian grid with ~108

cells.

• At Jupiter the model must

include atmospherically

driven corotation and allow

plasma from Io's

volcanoes to populate the

magnetosphere.

Some Examples of Global MHD Simulations: The Heliosphere

• The temperature (color spectrogram) and

magnetic field lines of an

expanding coronal mass ejection.

• This calculation was done

using the BATSRUS code developed at the

University of Michigan. This code uses an

adaptive grid.

Some Examples of Global MHD Models: Modeling the Sun

• Magnetic field lines in the

solar corona.

• This simulations uses a semi-implicit code

developed by Jon Linker and colleagues at SAIC.

• The simulation was run on

a spherical grid system.