INTRODUCTION TO THE MAGNETOHYDRODYNAMICS

1

• INTRODUCTION TO THE MAGNETOHYDRODYNAMICS
• BOUNDARY PHENOMENA
• Isak Beilis
• Tel-Aviv University
• Electrical Discharge Plasma Laboratory

2
The Main Issues

• The subject. Plasma as fluid flow
• Applications
• The main factors determined the MHD
• The base system of equations
• Tensor of electrical conductivity
• Equation of magnetic field (MF) induction.
  Diffusion and frozen
• MF-plasma interaction. Magnetic Pressure
• Adiabatic invariant of magnetic moment. Plasma
  confinement
• Near wall electrical phenomena

3
The Subject

• The plasma is a conductive fluid flow
• Current and magnetic field induction
• Magneto-hydrodynamics (MHD) hydrodynamics
  flowelectro-magnetic phenomena
• Application Electromagnetic body acceleration,
  MHD- power generation, Plasma accelerators,
  thrusters,

4
MHD and the Applications

• Electromagnetic solid body acceleration
  (Railgun)
• The electromagnetic force generates in the
  current loop
• ILindU/dt, IB
• When the current increases about to 1 MA the arc
  plasma and the body (1g) accelerated and reach
  velocity V10 km/s

Accelerated Plasma Arc
V
I
E
Accelerated Solid Body
5
MHD and the Applications

• IN MAGNETIC FIELD
• MHD-Generation of Electrical Power
• GIVEN plasma flow V
• OBTAIN E-electrical field and the current
• MHD-Plasma Accelerator
• GIVEN E-electrical field
• OBTAIN plasma flow V

Magnetic field
X
j
Plasma flux
R
6
The Main Factors in MHD

• Electromagnetic (Lorenz) force
• FB jxB
• Volume energy dissipation
• qqEqB j(EvxB)
• Electron energy transfer
• qe - 5jkTe/2e
• In the E ? B fields the electrons and ions are
  drift in one direction and the plasma moves with
  drift velocity vE/B
• B?H D?E
• ?- magnetic permeability, ?- permittivity

v
B
E
7
MHD Equations

• The mathematical description of MHD flow is based
  on the solution of Boltzmann system of equations
  determined the velocity distribution function
  f(t,r,v) of different plasma particles. The
  particle density ?, velocity v and temperature T
  are obtained from integrals
• ?? f(t,v.r) mdv
• v? f(t,v.r) mvdv
• T? f(t,v.r) (mv2/3)dv
• taking in account the volume electromagnetic
  force and energy dissipation

8
The Base System of Equations

• The plasma (e,i,a) flow as continuous media
• Continuity
• ??/?t div(?v) 0, ?-mass density e,i,a
• Momentum
• ?dv/dt - ?P ?eE jxB, Pressure PNkT
• Energy
• ?d(? v2/2)/dt - divq jE
• Operator
• d/dt ?/ ?t vx?/?t vy?/?t vz?/?t

9
Equations of Electromagnetic Field

• Maxwell Equations
• rot H (4?j ?E/?t)/c
• rot E - (?H/?t)/c
• div E 4??e
• div H 0
• Ohm Law
• J ?EvxH/cgradPe/cnejxH/cene

10
Tensor of Conductivity

• ?e2.107H(Gs) ?i104ZiH(Gs)/Ai
• ?e4.104Te3/2eV/Ncm -3
• when ?e gt1
• the plasma is magnetized
• E?E grad Pe/cne
• J ?E? -?(jxB)/B
• Jx ?Ex?- ?e(jyBz-jzBy)/B
• Jy ?Ey?- ?e(jzBx-jxBz)/B
• Jz ?Ez?- ?e(jxBe-jeBx)/B

?e,ieH/(cme,i) ?e ?e,i?e,i
z
x
B
y
11
Tensor of Conductivity

• Jx ?Ex?
• Jy ?(Ey?- ?eEz?)/(1?e2)
• Jz ?(- ?eEy? Ez?)/(1?e2)

12
Tensor of Conductivity

• Ji ? ?ikEk? ,
• ix,y,z k1,2,3
• (1?e2) 0 0
• ?ik ?/(1?e2)0 1 - ?e
• 0 - ?e 1

13
Hall Current

• Ex? Ey? 0 EEz?
• Jx 0
• Jy - ??eE/(1?e2) Hall current
• Jz ?E/(1?e2) Magnetic Field (MF) decreases
  the conductivity in E direction.

14
Equation of MF Induction

• rot E - ?B/?t
• j ?EvxB E in above Eq.
• ?B/?t rot (vxB) j/ ?, using from Max Eq.
• j (rot B)/?
• Equation of magnetic induction
• ?B/?t rot (vxB) rot (??)-1rotB

15
MF Diffusion and Frozen

• ?B/?t rot (vxB) ?m ?2B
• ?m (??)-1- magnetic viscosity
• ?? ? ?B/?t rot (vxB)
• MF is frozen in the plasma and they move together
• v? 0 ?B/?t ?m ?2B
• MF diffusion through the plasma

16
Physics of the Plasma-MF Interaction

• When external Bo changes in time the drift
  current j is created
• The j larger when ? is larger and the plasma
  inducted MF Bi also is large
• by ? ? ? must be j ? ? that IMPOSSIBLE !
• This means that the magnetic field does not move
  relative to plasma,
• i.e. Plasma velocityField velocity
• (FROZEN!)

Bo
?
j
Bi
17
MF Pressure

• MF frozen and MF pressure
• It is known MF changes j appears and their
  interaction is due to electromagnetic force F
  j?H/c.
• We consider it as a force of magnetic pressure
• Using dH/dy 4?j/c the force in a unit volume is
• F(HdH/dy)/4? d(H2/8?)/dy or
• PMF H2/8?

z
H
x
j
y
dH/dy
18
MF Pressure

• When the plasma is an ideal conductor (j??) the
  magnetic pressure action is similar to gas
  pressure interaction with solid wall.
• This is the frozen case.
• For an ideal insulator (j? 0) no interaction with
  magnetic pressure.
• This is the case transparent to MF.
• In the real case the plasma conductivity is a
  finite value and therefore the MF penetration is
  by diffusion mechanism

19
Skin Layer and Time

• The Diffusion coefficient is
• Dm ?m (??)-1 or Dm c/4??
• The skin layer is
• L (Dmt)1/2 t/??
• The skin time is
• t L2/Dm ??L2
• In case of laboratory low temperature plasma this
  time is relatively small and in case of space
  plasma it can be very large due to difference of L

20
Frozen Effect and High-strongly Magnetic Field

• When the plasma compression time due to explosion
  is much smaller than the MF penetration time the
  enhanced MF can be obtained from
• BoSo BS as
• kB/BoSo/S
• Bo, So are the values before explosion

Explosion
So
Explosion
21
Magnetic Moment Invariant

• Magnetic moment for circular (area S) current
• MI.S
• Adiabatic invariant of magnetic moment is
• M mv?2/2B const
• v? - perpendicular ion velocity
• v?v.sin ? v?? v.cos ?
• The v? increases and v?? decreases with B
  increasing

v??
v?
Plasma Reflection
22
Plasma Confinement

• Magnetic trap
• Trapped Particle if
• sin? gt (B/Bm)
• Particles penetrated if
• sin? lt (B/Bm)

23
The Main Criterions in MHD

• The MHD criterions are obtained using the
  dimensionless variables in the system of equation
• Equation of MF induction
• (Bo/to)?B/?trot(vxB)(voBo/L)rot(?m
  rotB)(?moBo/L2)
• By division Bo/to, and to L/vo
• ?B/?trot(vxB)rot(?m rot B)Rem-1
• Rem voL/?mo voL??o - Magnetic Reynolds
• Large L, ? -frozen Rem gtgt1 space plasma
• Small L, ? - diffusion Rem ltlt 1 lab plasma

24
Main Criterions in MHD

• Parameter of MHD interaction
• Nm ?H2L/?c2v
• Parameter of electrical loading
• K cE/vH
• Alfen Number Aa ca/v ? relation between
  electromagnetic and inertial forces
• caH/(4?v2?) ? velocity of Alfen wave
• Aa characterize also the relation between the
  energy of magnetic field to the plasma kinetic
  energy

25
Near Wall Electrical Phenomena

• For MHD equation solution electrical boundary
  condition must be given!
• j0 insulator,
• ? ?w electrode that ? ?p
• ?? ?p - ?w - ?
• jp - ?
• qp - ?

?p
Plasma
qp
?
?w
jp
?
?
?
?
?
?
?
?
?
Wall
26
Near Wall Electrical Phenomena

• Sheath (space charge)
• div E 4?(?e- ?i), ?eN
• ?e Boltzmann distribution
• ?e Function of j and ?
• Problem Instable sheath potential
• Presheath (quasineutral plasma)
• Equations of ion and electron motion with
  electric, magnetic and electron pressure forces.
• Solution Ion acceleration toward the sheath and
  smooth sheath-presheath potential distribution

Presheath
Sheath
?
Ion acceleration
Ech
Eo
? Ni
Ne
Wall
Esh gtgt Eo
27
Magnetized Sheath
.
e

• Magnetized electrons
• Non-magnetized ions
• Sheath with closed electron drift
• Ion acceleration in the sheath electrical field

jy
H
i
Ex
Accelerated Ions
i
i
e
e


Plasma


Drift Electron Current jy enecEx/H Ion
energy Wi eExdsh
e
e
i
i
Electron Drift Current
28
Magnetized Presheath
Presheath
Sheath

• Smooth transition
• Bohm Vo/Cs ? 1
• Cs(Te/mi)1/2
• Magnetized Solution
• ?rd/ri
• rd Debye radius
• ri vi/?i
• ?i ?i?i
• ? je/ji

Vo
B
Wall
Cs
Eo
y
x
?(x)
z
zz
?
?
?
?
29
Near-Wall Plasma Volume

• Ambipolar Diffusion
• je -eNbeE - De?N
• ji eNbeE - Di?N
• je jiDA?N, begtgtbi
• From je ji be,ieDe,i /Te,i
• DA (be Di bi De)/ (be bi)
• DA (Te Ti)Di
• In MF DA ? DA/(1 (?e?e)2)

?N
E
je
?
jp
?
Plasma
Wall
30
Exercises

• 1. Determine the electron and Cu-ion cyclotron
  frequencies for MF of 1 T.
• 2. Calculate the Hall current density for plasma
  density N1014 cm-3 and 1016 cm-3 , when H104
  Gs, ? 10 Ohm-1 cm-1, Te2eV, Ez10 V/cm (Ex
  Ey0)
• 3. The plasma conductivity is ? 104 Ohm-1m-1.
  The plasma moves with velocity 1km/s through
  magnetic field in channel of length 0.1m. What
  is the skin layer? Is the MF frozen?
• 4. The plasma flux with velocity 1 km/s flows in
  transverse MF of 1T. How much is the inducted
  electric field?
• 5. What is the magnetic pressure when the B1 T?