Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP, Radboud Universiteit

A way to describe fully ionized plasmas using a fluid approximation;

Uses a simplified set of equations for electrodynamic processes

Can be formulated entirely in terms of density, velocity, pressure and magnetic field

A quick recap of Hydrodynamics

• 1. Equation of motion for a fluid:

Total comoving derivative

Pressure force

Gravitational force

2. Continuity equation and equation of state (ideal gas):

Mass density ρ must adjust to flow in order to conserve mass!

Compression raises pressure and temperature, expansion lowers it. Ideal mono-atomic gas has γ = 5/3

3. Conservative form of the equations:

Mass Conservation

Momentum Conservation

Energy Conservation

Details can be found at:

http://astro.ru.nl/~achterb/Gasdynamica_2016/

Cosmic plasmas and magnetic fields

• Ionized gases (plasmas) can carry free electric charges, leading to:

- Electrostatic effects - Electromagnetic effects

• In the astrophysical context one is always dealing with highly conducting plasma’s: - Currents and the magnetic fields they generate are

long-lived: the realm of Magnetohydrodynamics

How can you detect Cosmic magnetic fields?

• In stars: Zeeman splitting of some spectral lines;

• In diffuse gas: synchrotron radiation;

• In diffuse gas with dust: polarization of scattered starlight.

• In diffuse gas: Faraday rotation and Rotation measure of

EM radiation (both are index of refraction effects)

Sun in UV light (TRACE satellite)

Synchrotron sources:

Supernova Remnants AGNs/Quasars

Typical field strengths

Object Field strength in Gauss

Earth’s dipole field at N/S Pole 0.6

Solar Dipole field at N/S Pole 10

Sunspot 2500

Solar Wind at 1 AU 2 10-6

Magnetic A Stars 10,000

Pulsar (Neutron Star) 1011-1013

Supernova Remnant 10-4-10-5

Interstellar Medium 3 10-6

Intra-Cluster Gas 10-5-10-6

Intergalactic Medium (“Voids”) < 10-9 for random component

< 10-11 for uniform component

Introduction to plasma physics

Simplest model: two-fluid model consisting of ions and electrons

Extra physics: the electromagnetic coupling between charges (ions and electrons) through the Lorentz Force

Extra equations: Maxwell’s equations that describe how: - free charges generate electric fields - currents generate magnetic fields

Step 1: Equation of motion for a charged fluid

( )

Equation of motion for a charge .

Step 1: replace mass by mass density ( number density charges)

Step 2: re-interpret time derivative:

Step 3: replace ch

dm q qdt c

m nm n

ddt t

× = +

=

∂⇒ + •

∂

v v BE

V

( )

arge by a charge density

Step 4: add a pressure force and (where needed) a gravitational force density

q nq

P nm

nm P nmt c

ρ

ρ

≡

−

∂ × + • = − + + + ∂

g

V BV V V E g

Step 1: Equation of motion for a charged fluid

( )

Equation of motion for a charge .

Step 1: replace mass by mass density ( number density charges)

Step 2: re-interpret time derivative:

Step 3: replace ch

dm q qdt c

m nm n

ddt t

× = +

=

∂⇒ + •

∂

v v BE

V

( )

arge by a charge density

Step 4: add a pressure force and (where needed) a gravitational force density

q nq

P nm

nm P nmt c

ρ

ρ

≡

−

∂ × + • = − + + + ∂

g

V BV V V E g

Step 1: Equation of motion for a charged fluid

( )

Equation of motion for a charge .

Step 1: replace mass by mass density ( number density charges)

Step 2: re-interpret time derivative:

Step 3: replace ch

dm q qdt c

m nm n

ddt t

× = +

=

∂⇒ + •

∂

v v BE

V

( )

arge by a charge density

Step 4: add a pressure force and (where needed) a gravitational force density

q nq

P nm

nm P nmt c

ρ

ρ

≡

−

∂ × + • = − + + + ∂

g

V BV V V E g

Step 1: Equation of motion for a charged fluid

( )

Equation of motion for a charge .

Step 1: replace mass by mass density ( number density charges)

Step 2: re-interpret time derivative:

Step 3: replace ch

dm q qdt c

m nm n

ddt t

× = +

=

∂⇒ + •

∂

v v BE

V

( )

arge by a charge density

Step 4: add a pressure force and (where needed) a gravitational force density

q nq

P nm

nm P nmt c

ρ

ρ

≡

−

∂ × + • = − + + + ∂

g

V BV V V E g

NOTATION CHANGE!!!

Step 1: Equation of motion for a charged fluid

( )

Equation of motion for a charge .

Step 1: replace mass by mass density ( number density charges)

Step 2: re-interpret time derivative:

Step 3: replace ch

dm q qdt c

m nm n

ddt t

× = +

=

∂⇒ + •

∂

v v BE

V

( )

arge by a charge density

Step 4: add a pressure force and (where needed) a gravitational force density

q nq

P nm

nm P nmt c

ρ

ρ

≡

−

∂ × + • = − + + + ∂

g

V BV V V E g

Step 1: Equation of motion for a charged fluid

( )

Equation of motion for a charge .

Step 1: replace mass by mass density ( number density charges)

Step 2: re-interpret time derivative:

Step 3: replace ch

dm q qdt c

m nm n

ddt t

× = +

=

∂⇒ + •

∂

v v BE

V

( )

arge by a charge density

Step 4: add a pressure force and (where needed) a grav

itational force density

nm P nmt

q nq

P nm

cρ

ρ

∂ × + • = − + + + ∂

≡

−

V V BV E g

g

V

Glorious outcome!!

Two-fluid equations for a hydrogen plasma

( )

( )

Ions are protons with mass and charge :

Electrons: mass and charge :

p

i ii p i i i i i p

e

e ee e e e e e e e

m e

n m P n e n mt c

m e

n m P n e n mt c

+

∂ × + • = − + + + ∂

−

∂ × + • = − − + + ∂

V V BV V E g

V V BV V E g

Both species feel the same electromagnetic fields, which they generate together!

The two fluids together generate the EM fields:

( )

( )

( )2

4 4

1

0

4 4 4 4

If self-gravity is important equation for gravitational potential :

4

i p e

i e

i i e e

e

c t

c c t c c

e

t

G n m n

n n

e n

m

n

π π

π π π π

π

ρ −

−

• = =

∂× = −

∂

• =

∂ ∂× = =

∂ ∂

Φ

∇ Φ = +

J

E

BE

B

E V+ +V EB

There are additional equations needed

( )

= i

00

e,

1. Particle number conservation: 0;

2. Ideal gas law:

3. Maxwell's Equations:

4 4 (Coulomb's law)

1 (Faraday's Equation)

0

b

n nt

nP nk T P

t

q n

n

c

γ

α αα

π πρ

∂+ • =

∂

= =

• = =

∂× = −

∂

• =

∑

V

E

BE

B

=e,i

(No magnetic monopoles!)

4 4 4 4

(Ampere's equation)c c t c c t

q nα α αα

π π π π∂ ∂× = =

∂ ∂∑E EB + +J V

Charge density ρ and current density J are determined by both the ions and the electrons

Conductivity for static fields: Ohms Law

Free charge carriers in a plasma conduct current:

electrostatic force frictional force due toon the electrons electr

Simple model: stationary ions and moving electrons:

Balance between electric force and electron-ion friction:

ee e e e e ei e

dn m en n mdt

ν= − −

V E V

on-ion collisions

2

0

Solve for electron velocity in STATIC case:

Associated current density:

=

ee ei

ee e e

e ei

em

e nenm

ν

σν

=

= −

− = ≡

EV

EJ V E

Current density J and driving electric field E are proportional!

Scalar and tensor conductivity

Our simple calculation gives a scalar conductivity:

eσ=J E

More complicated case: a tensor conductivity

(in component form: )i ij jJ Eσ= • =J E

High school version of Ohms Law is the same!

current resistance

current density area

electric field strenght length

V I R

I J

V EL

= × = ×

= × = ×

= = ×

Basic relations:

High school version of Ohms Law is the same!

current resistance

current density area

electric field strenght length

V I R

I J

V EL

= × = ×

= × = ×

= = ×

Basic relations:

Combine:

( = / )I V ELJ E L RR R

σ σ= = = ≡

The MHD condition

• Valid for plasma’s with a very high conductivity!

'c

σ σ × = = +

V BJ E E

Ohm’s law in a moving plasma:

In terms of the resistivity η=1/σ:

cη

σ×

+ = =V B JE J

Limit of infinite conductivity (zero resistivity):

0

c

c

ησ

×+ = = ↓

⇔

×

V B JE J

V BE = -

Alternative interpretation in terms of Lorentz force:

0 if L qc× = =

v BF E + v = V

Lorentz force vanishes for any charge that moves with the plasma!

Basic equations: 1. dynamics

( )

( )

Ions are protons with mass and charge :

Electrons: mass and charge :

p

i ii p i i i i i p

e

e ee e e e e e e e

m e

n m P n e n mt c

m e

n m P n e n mt c

+

∂ × + • = − + + + ∂

−

∂ × + • = − − + + ∂

V V BV V E g

V V BV V E g

Basic assumptions: 1) electrons and protons almost move together: 2) electron and ion densities are the same:

e i ≡V V V

e in n n= ≡

Add the two equations of motion:

( )

( )

i ii p i i i i i p

e ee e e e e e e e e

n m P n e n mt c

n m P n e n mt c

∂ × + • = − + + + ∂ ∂ × + • = − − + + ∂

V V BV V E g

V V BV V E g

( ) ( ) ( ) ( ) ( )0 p e i e i e i i e e

Total plasma Charge density Total cTotal mass densitypressure

1n m m P P n e n e n e n et c

∂ + + • = − + + − + − ∂

+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -V V V E V V ( )0 p e

urrent density

n m m× +

B + g

MHD approximation: neglect charge density, define:

( )( )

0 p e e i

0 i e

mass density , pressure

current density

n m m P P P

n e

ρ = + = +

= −J V V

MHD equation of motion:

( ) Pt c

ρ ρ∂ × + • = − + ∂ V J BV V + g

( ) ( ) ( ) ( ) ( ) ( )0 p e i e i e i i e e 0 p e

Total plasma Charge density Total current densityTotal mass densitypressure

1n m m P P n e n e n e n e n m mt c

∂ + + • = − + + − + − × + ∂

V V V E V V B + g

Maxwell’s equations:

1. We do not need the equations for the electric field! 2. Ampere's equation simplifies considerably:

4 1c c tπ

∂= +×

∂B J E

MHD describes slow phenomena: neglect the displacement current!

Still missing: a dynamical equation for the magnetic field!

Faraday’s equation + Ampere’s equation + Ohms Law:

( )

( )2

2

4

4

ct

cc t

c

σ πσ

π

∂ = − × ∂ × ∂= − + ⇒ = × × + ∇ ∂= ×

B E

V B J BE V B B

J B

The Induction Equation

Summary: basic MHD equations

( )

( )

( )2

2

0

4

4

0

Pt c

t

ct

c

c

ρ ρ

ρ ρ

πσ

π

σ

∂ × + • = − + ∂

∂+ • =

∂

∂= × × + ∇

∂

× =

• =

×= − +

V J BV V + g

V

B V B B

B J

B

V B JE

Force Balance Mass conservation Induction Equation Ampères law No Monopoles!! MHD Condition

Additional relations:

00

and TP P Pγ

ρ ρµ ρ

= =

2 (and 4 for self-gravity)Gπ ρ= − Φ ∇ Φ =g

Ideal gas law and the Equation Of State

Governing equations for the gravitational field

The Lorentz Force in the MHD Approximation

( ) Pt c

ρ ρ∂ × + • = − + ∂ V J BV V + g

( )L

L 44

c

c

ππ

× = × ×

⇒ =× =

J BfB B

f

B J

The Lorentz Force in the MHD Approximation

( ) Pt c

ρ ρ∂ × + • = − + ∂ V J BV V + g

( )L

L 44

c

c

ππ

× = × ×

⇒ =× =

J BfB B

f

B J

( ) ( )4

Pt

ρ ρπ

× ×∂ + • = − + ∂

B BV V V + g

In the end, only the magnetic field appears in the eqn. of motion!