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G0B28A

### Stefaan Poedts

Centre for Plasma-Astrophysics, KU Leuven (Belgium)

Course G0B28A, KU Leuven

Notes by S. Poedts (and R. Keppens& J.P. Goedbloed) based on PRINCIPLES OF MAGNETOHYDRODYNAMICS

(2)

G0B28A



### •

The ideal MHD equations: postulating the basic equations, scale independence,

what is a physical model?; [ book: Sec. 4.1 ]

### •

Magnetic flux: flux tubes, global magnetic flux conservation; [ book: Sec. 4.2 ]

### •

Conservation laws: conservation form of the equations, global conservation laws, local conservation laws – conservation of magnetic flux; [ book: Sec. 4.3 ]

### •

Discontinuities: shocks and jump conditions, boundary conditions for interface

plas-mas; [ book: Sec. 4.5 ]

### •

Model problems: laboratory models I–III, astrophysical models IV–VI.

(3)

The MHD model: The ideal MHD equations (1) 4-2

 

### Postulating the basic equations

Equations of magnetohydrodynamics can be introduced by

### •

averaging the kinetic equations by moment expansion and closure through transport theory (book: Chaps. 2 and 3);

### •

just posing them as postulates for a hypothetical medium called ‘plasma’ and use physical arguments and mathematical criteria to justify the result (Chaps. 4, . . . ).

[ There is nothing suspicious about posing the basic equations. That is what is actually done with all basic equations in physics. ]

In the second approach, since the MHD equations describe the motion of a conducting fluid interacting with a magnetic field, we need to combine Maxwell’s equations with the equations of gas dynamics and provide equations describing the interaction.

(4)

The MHD model: The ideal MHD equations (2) 4-3

### •

Maxwell’s equations describe evolution of electric field

### E(r, t)

and magnetic field

### B(r, t)

in response to current density

and space charge

0

2

0

0

−1/2

(‘Amp `ere’)

0

(Poisson)

(no monopoles)

### •

Gas dynamics equations describe evolution of density

and pressure

### + ∇ · (ρv) = 0 ,

(mass conservation)

### + v · ∇p + γp∇ · v = 0 ,

(entropy conservation)

where

### + v · ∇

(5)

The MHD model: The ideal MHD equations (3) 4-4

### •

Coupling between system described by

### {E, B}

and system described by

### {ρ, p}

comes about through equations involving the velocity

### v(r, t)

of the fluid:

‘Newton’s’ equation of motion for a fluid element describes the acceleration of a fluid element by pressure gradient, gravity, and electromagnetic contributions,

### = F ≡ −∇p + ρg + j × B + τE ;

(momentum conservation)

### (7)

‘Ohm’s’ law (for a perfectly conducting moving fluid) expresses that the electric field

### E

 in a co-moving frame vanishes,



(‘Ohm’)

### •

Equations (1)–(8) are complete, but inconsistent for non-relativistic velocities:

### (9)

(6)

The MHD model: The ideal MHD equations (4) 4-5  

### Consequences of pre-Maxwell

1. Maxwell’s displacement current negligible [

2

2

### )

] for non-relativistic velocities:

2

2

2

0

0

### l

0 [using Eq. (8)]

### ,

indicating length scales by

### l

0 and time scales by

0, so that

0

0.

### ⇒

Recover original Amp `ere’s law:

0

### (10)

2. Electrostatic acceleration is also negligible [

2

2

]:

2

2

2

0

0

2

0

### l

0 [using Eqs. (3), (8), (10)]

### ⇒

Space charge effects may be ignored and Poisson’s law (3) can be dropped.

3. Electric field then becomes a secondary quantity, determined from Eq. (8):

### (11)

(7)

The MHD model: The ideal MHD equations (5) 4-6

 

### •

Exploiting these approximations, and eliminating

and

### j

through Eqs. (10) and (11), the basic equations of ideal MHD are recovered in their most compact form:

0

### ⇒

Set of eight nonlinear partial differential equations (PDEs) for the eight variables

,

,

, and

.

### •

The magnetic field equation (15)(b) is to be considered as a initial condition: once satisfied, it remains satisfied for all later times by virtue of Eq. (15)(a).

(8)

The MHD model: The ideal MHD equations (6) 4-7  

### •

Alternative thermodynamical variables (replacing

and

### e

– internal energy per unit mass (

temperature

) and

### s

– entropy per unit mass. Defined by the ideal gas relations, with

e

i

:

v

v

i

v

−γ

### •

From Eqs. (12) and (14), we obtain an evolution equation for the internal energy,

### (17)

and an equation expressing that the entropy convected by the fluid is constant (i.e. adi-abatic processes: thermal conduction and heat flow are negligible),

−γ

### (18)

(9)

The MHD model: The ideal MHD equations (7) 4-8  

### •

In many astrophysical systems, the external gravitational field of a compact object (represented by point mass

### ∗

situated at position

### r = r∗

far outside the plasma) is more important than the internal gravitational field. The Poisson equation

2

gr

### (19)

then has a solution with negligible internal gravitational acceleration (2nd term):

gr

3







3

3



### •

Estimate gravitational forces

ex/ing

### ≡ ρg

ex/in compared to Lorentz force

B

### ≡ j × B

:

1) Tokamak (with radius

### a

of the plasma tube and

,

### ∗

referring to the Earth):

B

2

0

6

−2

−2

exg

ex

2

−6

−2

−2

ing

in

2

−24

−2

−2

### .

(10)

The MHD model: The ideal MHD equations (8) 4-9

2) Accretion disk (replace

d, and

### ∗ → R = 0.1R

d ):

R Z 0.1Rd Hd ϕ jet jet

(a) Young stellar object (YSO)

d

d

18

−3 ):

B

−12

exg

−9

ing

−19

### .

(b) Active galactic nucleus (AGN)

d

d

8

12

−3 ):

B

−21

exg

−27

### (23)

(11)

The MHD model: The ideal MHD equations (9) 4-10  

### •

The MHD equations (12)–(15) can be made dimensionless by means of a choice for

the units of length, mass, and time, based on typical magnitudes

### l

0 for length scale,

### ρ

0 for plasma density, and

### B

0 for magnetic field at some representative position. The

unit of time then follows by exploiting the Alfv ´en speed :

0

A,0

0

0

0

0

0

0

### •

By means of this basic triplet

0,

0,

### t

0 (and derived quantities

0 and

### v

0), we create dimensionless independent variables and associated differential operators:

0

0

0

0

### (25)

and dimensionless dependent variables:

0

0

0

02

0

0

02

### •

Barred equations are now identical to unbarred ones (except that

### μ

0 is eliminated).

### ⇒

Ideal MHD equations independent of size of the plasma (

### l

0), magnitude of the magnetic field (

### B

0), and density (

### ρ

0), i.e. time scale (

### t

0).

(12)

The MHD model: The ideal MHD equations (10) 4-11

 

### Scales of actual plasmas

l0(m) B0(T) t0(s)

tokamak 20 3 3× 10−6

magnetosphere Earth 4 × 107 3 × 10−5 6

solar coronal loop 108 3 × 10−2 15

magnetosphere neutron star 106 108 10−2

accretion disc YSO 1.5 × 109 10−4 7 × 105

accretion disc AGN 4 × 1018 10−4 2 × 1012

galactic plasma 1021 10−8 1015

(= 105ly) (= 3 × 107y)

* Some recently discovered pulsars, called magnetars, have record magnetic fields of1011T : the plasma Universe is ever expanding!

Note Tokamak: 1 min ( 60 s ) ⇒ 20 × 106 crossing times ,

(13)

The MHD model: The ideal MHD equations (11) 4-12  

### A crucial question:

Do the MHD equations (12)–(15) provide a complete model for plasma dynamics? Answer: NO!

Two most essential elements of a scientific model are still missing, viz. 1. What is the physical problem we want to solve?

2. How does this translate into conditions on the solutions of the PDEs?

This brings in the space and time constraints of the boundary conditions and initial data. Initial data just amount to prescribing arbitrary functions

i

i

i

i

### (r)

on domain of interest

### (27)

Boundary conditions is a much more involved issue since it implies specification of a

magnetic confinement geometry.

### ⇒

magnetic flux tubes (Sec.4.2), conservation laws (Sec.4.3), discontinuities (Sec.4.4),

(14)

The MHD model: Magnetic flux (1) 4-13 

### •

Magnetic flux tubes are the basic magnetic structures that determine which boundary conditions may be posed on the MHD equations.

b a

### •

Two different kinds of flux tubes:

(a) closed onto itself, like in thermonuclear tokamak confinement machines,

(b) connecting onto a medium of vastly different physical characteristics so that the flux tube may be considered as finite and separated from the other medium by suitable jump conditions, like in coronal flux tubes.

(15)

The MHD model: Magnetic flux (2) 4-14

 

### •

Magnetic fields confining plasmas are essen-tially tubular structures: The magnetic field equation

### (28)

is not compatible with spherical symmetry. Instead, magnetic flux tubes become the es-sential constituents. S1 S2

Gauss’ theorem:

V

S1

1

1

1

S2

2

2

2

### = 0 ,

Magnetic flux of all field lines through surface element

### dσ

1 is the same as through arbitrary other element

### dσ

2 intersecting that field line bundle.

S

is well defined

### (29)

(16)

The MHD model: Magnetic flux (3) 4-15

 

### •

Kinematical concept of flux tube comes from

.

### •

Dynamical concept of magnetic flux conservation comes from the induction equation,

a contraction of Faraday’s law,

### −∇ × E

, and ‘Ohm’s’ law,

:

### •

Example: Global magnetic flux conservation inside toroidal tokamak. ‘Ohm’s’ law at the wall (where the conductivity is perfect because of the plasma in front of it):

w

(A.2)

w

t

w

w

Since

t

### = 0

, and the boundary condition that there

is no flow across the wall,

w

(on

)

### (32)

the other contribution has to vanish as well:

w

(on

)

### ⇒

magnetic field lines do not intersect the wall. ϕ

V

W

(17)

The MHD model: Magnetic flux (4) 4-16

 

### Global magnetic flux conservation (cont’d)

ϕ Btor Spol a Bpol W magn.axis b Stor ϕ d l pol d l tor

Apply induction equation to toroidal flux:

tor

Spol

tor

tor

tor

tor

tor

pol

because

### l

tangential to the wall.

### ⇒

BCs (32) & (33) guarantee

tor

### = const

.

Similarly for poloidal flux:

pol

Stor

pol

pol

pol

### (35)

(18)

The MHD model: Conservation laws (1) 4-17

 

### •

Next step: systematic approach to local conservation properties.

### •

The MHD equations can be brought in conservation form:

### (36)

This yields: conservation laws, jump conditions, and powerful numerical algorithms!

### •

By intricate vector algebra, one obtains the conservation form of the ideal MHD

equa-tions (suppressing gravity):

### ⇓

From now on, putting

0

1 2

2

1 2

2

12

2

12

2

2

### (40)

It remains to analyze the meaning of the different terms.

(19)

The MHD model: Conservation laws (2) 4-18 

### •

Defining – momentum density :

### (41)

– stress tensor :

12

2

### (42)

– total energy density :

12

2

γ−11

12

2

– energy flow :

12

2

γ−1γ

2

– (no name):

yields

### + ∇ · π = 0

(conservation of mass),

### + ∇ · T = 0

(conservation of momentum),

### + ∇ · U = 0

(conservation of energy),

### ∂B

(20)

The MHD model: Conservation laws (3) 4-19



### •

Defining – total mass:

### (50)

– total momentum:

– total energy :

### (52)

– total magnetic flux :

### (53)

gives, by the application of the BCs (32), (33):

Gauss

Gauss

12

2

Gauss

Stokes!

### v × B · dl = 0 . (57)

(21)

The MHD model: Conservation laws (4) 4-20



### •

Kinematic expressions for change in time of line, surface, and volume element:

### (dσ) = −(∇v) · dσ + ∇ · v dσ ,

(crucial for flux conservation!)

### (60)

1 d v(r+d 1) d 2 dσσ v(r) v(r+d 2) b c a r v r+d v(r+d ) d O d 2 d 3 d 1

(22)

The MHD model: Conservation laws (5) 4-21

 

### •

Combine dynamical relations (46)–(48) with kinematics of volume element:

12

2

### ⇒

Mass of co-moving volume element conserved locally,

momentum and energy change due to force and work on the element.

### •

However, dynamics (49) for the flux exploits kinematics of surface element:

### ⇒

Magnetic flux through arbitrary co-moving surface element constant (!) :

### 

C

(23)

The MHD model: Discontinuities (1) 4-22  

### Jump conditions

Extending the MHD model

### •

The BCs for plasmas surrounded by a solid wall:

w

(on

)

### ⇒

no flow accross the wall,

w

(on

)

### ⇒

magnetic field lines do not intersect the wall.

Under these conditions, conservation laws apply and the system is closed.

### •

For many applications (both in the laboratory and in astrophysics) this is not enough. One also needs BCs (jump conditions) for plasmas with an internal boundary where the magnitudes of the plasma variables ‘jump’.

Example: at the photospheric boundary the density changes

−9.

### •

Such a boundary is a special case of a shock, i.e. an irreversible (entropy-increasing) transition. In gas dynamics, the Rankine–Hugoniot relations relate the variables of the subsonic flow downstream the shock with those of the supersonic flow upstream.

We will generalize these relations to MHD, but only to get the right form of the jump conditions, not to analyze transonic flows (subject for a much later chapter).

(24)

The MHD model: Discontinuities (2) 4-23 

### •

Excite sound waves in a 1D compressible gas (HD): the local perturbations travel with the sound speed

.

### ⇒

Trajectories in the

### −t

plane (characteristics):

### •

Now suddenly increase the pressure, so that

### p

changes in a thin layer of width

### δ

:

δ

p

x

2 1

shocked unshocked

### ⇒

‘Converging’ characteristics in the

plane.

### ⇒

Information from different space-time points accumulates, gradients build up until steady state reached where dissipation and nonlinearities balance

### ⇒

shock.

(25)

The MHD model: Discontinuities (3) 4-24

 

### •

Wihout the non-ideal and nonlinear effects, the characteristics would cross (a).

With those effects, in the limit

### → 0

, the characteristics meet at the shock front (b).

t a x t b x c2 shock c1

### ⇒

Moving shock front separates two ideal regions.

### •

Neglecting the thickness of the shock (not the shock itself of course), all there remains is to derive jump relations across the infinitesimal layer.

(26)

The MHD model: Discontinuities (4) 4-25

 

### Procedure to derive the jump conditions

Integrate conservation equations across shock from

1 (undisturbed) to 2 (shocked).

### •

Only contribution from gradient normal to the front:

δ→0

2 1

δ→0

2 1

1

2

### •

In frame moving with the shock at normal speed

:

shock

finite

δ→0

2 1

δ→0

2 1

n u 1 2 v1 v2

### •

Hence, jump conditions follow from the conservation laws by simply substituting

(27)

The MHD model: Discontinuities (5) 4-26  

### •

Conservation of mass,

### •

Conservation of momentum,

1 2

2

12

2

### •

Conservation ot total energy,

1

2

2

12

2

12

2

2

12

2

γ−11

12

2

12

2

γ−1γ

2

### •

Conservation of magnetic flux,

### ∇ · B = 0

(28)

The MHD model: Discontinuities (6) 4-27

 

### •

Simplify jump conditions by transforming to co-moving shock frame, where relative plasma velocity is



### ≡ v − un

, and split vectors in tangential and normal to shock:

n

(mass)

n 2

12

t2

### ]] = 0 ,

(normal momentum)

n

t

n

t

### ]] ,

(tangential momentum)

n

12

n 2

t2

γ−1γ

t2

n

t

t

(energy)

n

(normal flux)

n

t

n

t

### ]] .

(tangential flux)

### ⇒

6 relations for the 6 jumps

,

n

,

t

,

,

n

,

t

.

### •

Do not use entropy conservation law since shock is entropy-increasing transition: not



n

but

−γ

### (79)

(29)

The MHD model: Discontinuities (7) 4-28

 

### Two classes of discontinuities:

(1) Boundary conditions for moving plasma-plasma interfaces, where there is no flow accross the discontinuity (vn

)

### ⇒

will continue with this here.

(2) Jump conditions for shocks (vn

)

### ⇒

leave for advanced MHD lectures. 



When

n

### = 0

, jump conditions (73)–(78) reduce to:

12

t2

### ]] = 0 ,

(normal momentum)

n

t

### ]] = 0 ,

(tangential momentum)

n

t

t

(energy)

n

(normal flux)

n

t

### ]] = 0 .

(tangential flux)

### •

Two possibilities, depending on whether

### B

intersects the interface or not:

(a) Contact discontinuities when

n

### = 0 ,

(30)

The MHD model: Discontinuities (8) 4-29

 

### •

For co-moving interfaces with an intersecting magnetic field,

n

### = 0 ,

the jump conditions (80)–(84) only admit a jump of the density (or temperature, or entropy) whereas all other quantities should be continuous:

– jumping:

– continuous:

n

t

n

t

### ]] = 0 .

Examples: photospheric footpoints of coronal loops where density jumps, ‘divertor’ tokamak plasmas with

### B

intersecting boundary.

### •

These BCs are most typical for astrophysical plasmas, modelling plasmas with very different properties of the different spatial regions involved (e.g. close to a star and far away): difficult! Computing waves in such systems usually requires extreme resolu-tions to follow the disparate time scales in the problem.

(31)

The MHD model: Discontinuities (9) 4-30  

### •

For co-moving interfaces with purely tangential magnetic field,

n

### = 0 ,

the jump

conditions (80)–(84) are much less restrictive:

– jumping:

t

t

– continuous:

n

n

12

t2

### ]] = 0 .

Examples: tokamak plasma separated from wall by tenuous plasma (or ‘vacuum’), dayside magnetosphere where IMF meets Earth’s dipole.

### •

Plasma–plasma interface BCs by transforming back to lab frame,

n

n

:

(

interface)

### n · [[v]] = 0

(normal velocity continuous)

12

2

### ]] = 0

(total pressure continuous)

### •

Jumps tangential components,

t

&

t

### ]]

, due to surface current & surface vorticity:

lim

δ→0, |j|→∞

### ≡

lim

(32)

The MHD model: Model problems (1) 4-31  

### •

We are now prepared to formulate complete models for plasma dynamics

MHD equations

### +

specification of magnetic geometries

appropriate BCs.

### •

For example, recall two generic magnetic structures: (a) tokamak; (b) coronal loop.

b a

### •

Generalize this to six model problems, separated in two classes:

### ⇒

Models I–III (laboratory plasmas) with tangential discontinuities;

(33)

The MHD model: Model problems (2) 4-32

 

### Laboratory plasmas (models I–III)

model I model II (*) model III

ϕ n wall plasma a a n ϕ b n ϕ c b c coil vacuum vacuum plasma pl-vac wall vac. / plasma ( * ) plasma pl-vac

plasma–wall plasma–vac–wall plasma–vac–coil–vac

toroidal

(34)

The MHD model: Model problems (3) 4-33

 

### •

Model I: axisymmetric (2D) plasma contained in a ‘donut’-shaped vessel (tokamak) which confines the magnetic structure to a finite volume. Vessel

### +

external coils need to be firmly fixed to the laboratory floor since magnetic forces are huge.

### ⇒

Plasma–wall, impenetrable wall needs not be conducting (remember why?).

### ⇒

Boundary conditions are

(at the wall)

(at the wall)

### ⇒

just two BCs for 8 variables!

### •

These BCs guarantee conservation of mass, momentum, energy and magnetic flux:

the system is closed off from the outside world.

### •

Most widely used simplification: cylindrical version (1D) with symmetry in

and

### ⇒

Non-trivial problem only in the radial direction, therefore: one-dimensional.

(35)

The MHD model: Model problems (4) 4-34

 

### •

Model II: as I, but plasma separated from wall by vacuum (tokamak with a ‘limiter’).

### ⇒

Plasma–vacuum–wall, wall now perfectly conducting (since vacuum in front).

### •

Vacuum has no density, velocity, current, only

### ˆ

pre-Maxwell dynamics:

### (95)

BC at exterior interface (only on

### ˆ

, consistent with

t

):

### n · ˆB = 0

(at conducting wall)

### •

BCs at interior interface (

### B

not pointing into vacuum and total pressure balance):

### n · B = n · ˆB = 0

(at plasma–vacuum interface)

12

2

### ]] = 0

(at plasma–vacuum interface)

### ⇒

Consequence (not a BC) is jump in

### B

t, i.e. skin current:

### = n × [[B]]

(36)

The MHD model: Model problems (5) 4-35

 

### •

Variant of Model II with vacuum replaced by tenuous plasma (negligible density, with or without current), where again the impenetrable wall needs not be conducting.

### ⇒

Applicable to tokamaks to incorporate effects of outer plasma.

### ⇒

Also for astrophysical plasmas (coronal loops) where ‘wall’ is assumed far away.

### •

BCs at exterior interface for outer plasma:

(at the wall)

(at the wall)

### •

BCs at interior interface for tangential plasma-plasma discontinuity:

### n · B = n · ˆB = 0

(at plasma–plasma interface)

### n · [[v]] = 0

(at plasma–plasma interface)

12

2

### ]] = 0

(at plasma–plasma interface)

### .

Note: Model II obtained by just dropping conditions on

and

### ˆv

.

(37)

The MHD model: Model problems (6) 4-36

 

### •

Model III is an open plasma–vacuum configuration excited by magnetic fields

### ˆ

that are externally created by a coil (antenna) with skin current.

### ⇒

Open system: forced oscillations pump energy into the plasma.

### ⇒

Applications in laboratory and astrophysical plasmas: original creation of the confining magnetic fields and excitation of MHD waves.

### •

BCs at coil surface:

### n · [[ ˆB]] = 0

(at coil surface)

c

### (r, t)

(at coil surface)

where

c

### (r, t)

is the prescribed skin current in the coil.

### •

Magnetic field outside coil subject to exterior BC (96) at wall (possibly moved to

### ∞

), combined with plasma-vacuum interface conditions (97) and (98):

### n · B = n · ˆB = 0

(at plasma–vacuum interface)

### ,

(38)

The MHD model: Model problems (8) 4-37

 

### Astrophysical plasmas (models IV–VI)

model V model IV a θ a pl. line tying plasma line tying b pl. plasma line tying b θ c model VI φ

closed loop open loop

stellar wind 2D

(39)

The MHD model: Model problems (9) 4-38

 

### •

In model IV, the field lines of finite plasma column (coronal loop) are line-tied on both sides to plasma of such high density (photosphere) that it is effectively immobile.

### ⇒

Line-tying boundary conditions:

### v = 0

(at photospheric end planes)

### ⇒

Applies to waves in solar coronal flux tubes, no back-reaction on photosphere:

### •

In this model, loops are straightened out to 2D configuration (depending on

and

### z).

Also neglecting fanning out of field lines

### ⇒

quasi-1D (finite length cylinder).

(40)

The MHD model: Model problems (10) 4-39

 

### •

In model V, the magnetic field lines of a semi-infinite plasma column are line-tied on one side to a massive plasma.

### ⇒

Line-tying boundary condition:

### v = 0

(at photospheric end plane)

### ⇒

Applies to dynamics in coronal holes, where (fast) solar wind escapes freely:

(41)

The MHD model: Model problems (12) 4-40



### •

In model VI, a plasma is ejected from photosphere of a star and accelerated along the open magnetic field lines into outer space.

### ⇒

Combines closed & open loops (models IV & V), line-tied at dense photosphere, but stress on outflow rather than waves (requires more advanced discussion).

### •

Output from an actual simulation with the Versatile Advection code: 2D (axisymm.) magnetized wind with ‘wind’ and ‘dead’ zone. Sun at the center, field lines drawn, veloc-ity vectors, densveloc-ity coloring. Dotted, drawn, dashed: slow, Alfv ´en, fast critical surfaces.

[ Keppens & Goedbloed, Ap. J. 530, 1036 (2000) ]

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