Introduction into MHD (magnetohydrodynamics)
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Last updated: 2018-04-10
Introduction into MHD
(magnetohydrodynamics)
see also scripts by B.D. Scott, S. Guenter, many books:
Goldston, Boyd & Sanderson, Biskamp, Hazeltine&Meiss,...
!
http://www2.ipp.mpg.de/~bds/lectures/mhd-lecture.html
http://www2.ipp.mpg.de/~ppg/2018_SS/
1outline
• reminder: hydrodynamic equations
• MHD equations
• connection to kinetic theory
• limits and flavors of MHD
2• constant N , changing V
¢E = N kB ¢T
uation summary: MHD-equations 2 ¢nelements,
• pressure field exerts force on boundary ¢
¢V inward; app
fluid (hydrodynamic)
summary: equations (no charges)
MHD-equations
¢N = 0
n
=°
IV ¢
±F = °p dS F=° dS p
awgo to infinitesimal volumes S
• change in density with time following flow
co-moving frame:
continuity equation 3 dT
• put in electric force, go to infinitesimaldn
continuity nk + pr · v = 0 volumes
= °nr · v
in addition equation
B
2
dt
dt dv
nm = °rp + nqE
• but we also have the form using fixed volumes dt as a reference
forceequation,
s equations
as pressure equation p = nk•Bnow
T , with general ratio sp.
+
@n heats
forces °;
force equation transform to fixed frame, note it is a Lorentz
@t
= °r transforma
· nv
µ ∂ ≥
@v
nm + v · rv = °rp + nq E
-Γ
@p @t
• we reconcile these by considering them as the same statement i
energy conservation
law:
Ohm s law or + v · rp + ° pr · v = 0
Ohm s law •
@t
† one global frame (lab frame)
main diÆerences to particle motion are pressure, velocity self
† one local frame (near each infinitesimal volume)
in general (non-isotropicincase),
in ∇ · T has to be
addition
addition
added to force balance; T stress tensor
Maxwells sequations
Maxwell equations
off-diagonal elements: viscous stress ν (Δv +
2/3 δij∇ · v) (Navier Stokes equation)
adiabatic law: 3ated by forces straightforward extension: add forces due to
dv electric fields
Plasma and Lorentz force
Electrodynamics
Nm =F
dt
• one set of fluid equations for each charged particle species
on boundary elements, inward; apply Stokes theorem
I I @n
p dS F = ° dS p F=° dV rp @t + r · nv = 0
S V
µ ∂ ≥ ¥
@v v
o infinitesimal volumes nm + v · rv = °rp + nq E + £B
@t c
co-moving frame:
dv @p
nm = °rp + nqE + v · rp + ° pr · v = 0
dt @t
• itplus
ame, note is a Maxwell’s equations for E
Lorentz transformation
µ ∂ ≥ ¥
@v 1 @B v 1 @E 4º
nm + v · rv = °rp + nq E + +£B
r£E = 0 + J = r£B
@t c @t c c @t c
r·B=0 r · E = 4ºΩch
le motion are pressure, velocity self advection
• charge density and current are summed over species
X X
Ωch = nÆ q Æ J= nÆ qÆ vÆ
Æ 4 Æcombining
ø kc
• “low the two
frequencies” means fluids
! ø into one: the electrons
kc
† cannot
c °1
(@/@t)be ‘special’
terms (except @B/@t since r£E is in general (except
hence neglect terms nonzero)@B/@t since r£E is in general no
A Single Fluid
free1.slow dynamics
current† basic result:- divergence
compared freetocurrent
light waves
4º 1 @E
A
rentz force in v equation and velocity divergences Single Fluid
in p equation
4º 1 @E
r£B = J+ r£B = J+
c c @t
at no species (sp. electrons) c v c @t
is “special” with regard to p or
• we have for each fluid, Lorentz force in v equation and velocity divergences in p equat
2. small
r£B =
4ºelectron mass - neglect electron
J hence r·J=04ºinertia
J£B † basic requirement
becomes
ism
thatøno =
species
r£B (sp. J hence
electrons) is “special” with to p· Jor=
regard r
E
ch ! ø c n e e n M
i i c
c
3. current as relative
A Single Fluid drift between Athe
J£B Single
species Fluid
is smaller than fluid
Ωch E ø ne me ø ni Mi
velocity (all species have
rpe the same v) vc
ø vi º v e ø E º ° £B
! • we have for each fluid, ne e J c
Lorentz force in v equation and rp velocity
e divergences
v in p equ
orentz force in v equation and velocity ø vdivergences
i º ve in p equation ø E º ° £B
! ne e ne e c
† basic(easy),
quasineutrality requirement is thatratio
small mass no species
(easy),(sp.
. . . electrons) is “special” with regard to p o
that 4.quasi-neutral plasma,isalso
no species (sp. electrons) dynamically:
“special” with regard to p or v
† required assumptions: quasineutrality (easy), small mass ratio (easy), . . .
ed a! posteriori (not so easy) J£B
Ωch E ø
J£B † last two must be checked ne me ø ni Mi
Ωch E
! ø a posteriori
ne me ø nicMi (not so easy)
mass density; c single charge density is zero
5.pressure
• singleforce smaller
density J than
is summed electric/Lorentz
mass density; single chargeforces
rpe (MHD
density is zero ordering
v
vs drift ordering) rp ø vi º v e v ø E º ° £B
velocity
ø vi º ve• single velocity is the
e
ne e ø E º ° £B ne e c
neExB
e velocity c
total pressure single pressure is summed total 5
pressure
† required assumptions: quasineutrality (easy), small mass ratio (easy), . . .
•with these assumptions:
6y, velocity, pressure,
@Ω and magneticMHD
summary field equation
+ r · Ωv = 0
@t
µ
@Ω
+∂r · Ωv = 0 continuity
@v @t J£B
Ω + v · rv = ° rp
@t
µ ∂ c
@v J£B
Ω@p + v · rv = ° rp force balance
+
@tv · rp + ° pr · v = c 0
@t
@p@B energy
+ v=· r£(v£B)
rp + ° pr · v = 0
@t @t conservation
@B
= r£(v£B) induction law
ere’s law @t
4º
J = r£B
mpere’s law c
4º Ampere’s law
has submerged, being given J = r£B
by the ExB velocity
c
ideal Ohm’s law:
E£B v£B
d hasvsubmerged,
=c being given by the
E = ExB
° implies
velocity that E // small,
B2 c
7 E⊥ can be largekinetic derivation
f is a single particle distribution function:
f(x,v) tells us the number of particles that can be found in
Fluid description of Plasmas
dxdv
Start with kinetic equation:
do not consider the distribution function but moments:
• moments of the distribution function:
• neglect kinetic effects, i.e. the different response of particles with
different velocities to external fields (e.g. Landau damping)
8Moments of the distribution function
density (k=0):
centre-of-mass velocity (k=1):
temperature (k=2):
9Moments of the distribution function
local quantities i.e. fluid description is only possible if the mean free path in
smaller than the scale length of the processes under investigation!
problematic for:
•collision-free plasmas
• small-scale processes
time scale of the processes under investigation has to be much longer
than the collision time :
10continuity equation
0-th order moment of kinetic equation
energy and momentum conservation:
(if only one particle species)
v and r independent phase space variables, thus:
11continuity equation
• E independent of v
-∞
+ =0
similar for force equation and energy equation
closure for energy equation needed: thermodynamics (see above)
12Magnetohydrodynamics (MHD)
single fluid description:
assumption: fluids and fields fluctuate on the same time and
length scales (ions length and time scales)
⇒ all effects that are connected to the electron dynamics are
neglected
non-relativistic description
Te = Ti even higher collision rate necessary than for two fluids
energy exchange between ions and electrons must happen on the time
scale under consideration:
13MHD-equations
(2) force equations: add force equations for ions and electrons (single charged
ions, isotropic pressure)
electron inertia is negligible
⇒
in the static case the pressure gradient is balanced by currents perpendicular to
the magnetic field (and gravitational forces)
14@B+=vr£(v£B)
· rp + ° pr · v = 0
@t
@t = r£(v£B)
@t
summary: MHD-equations
@B
= r£(v£B)
re’s lawlaw
mpere’s
@t
4º4º
continuity
by Ampere’s J J==r£B
law equation r£B
c c 4º
J = r£B
c
ashas
d submerged, being
beinggiven
force equation
submerged, givenby
bythe
theExB
ExB velocity
velocity
ric field has submerged, being given by the ExB velocity
E£BE£B Text v£B
c= cv =
v =v Ohm s law
2
E£B =°
EEE= °v£B
=° c
B 2Bc B 2 c
in addition
Maxwell s equations
adiabatic law:
15formation has? beenA studied with moving-mesh methods
previously20 .
Some physics
sponding dynamical properties of ideal MHD
frequencies follow equations
typical
This paper wavenumbers
is organized of global
as follows. First, o
Newcomb’s
Lagrangian formulation for ideal MHD in Lagrangian
labeling is briefly reviewed. Next, we introduce DEC
2º 2º Lagrangian, and de-
• ‘frozen’ magnetic field lines, flux conservation
to spatially discretize Newcomb’s
kk ª k? ª Then we implement the
rive the variational integrators.
Lk in 2D and show numerical
method ¢? results that artifi-
• linearised equations: sound waves, Alfven waves, coupled
cial reconnection does not take place. In the end, the
strengths and weaknesses of the method will be summa-
by ratio of kinetic and magnetic pressure
rized and discussed.
l ordering is
• typically: Lflavours of MHD
• MHD-equations
ideal/resistive MHD: 7 independent variables: v (3),
(3) generalised Ohm s law
B (2), n, p
• incompressible MHD (∇·v = 0, n=const):
4 independent variables: v (2), B (2)
• reduced MHD (strongly magnetised): 2 variables
• Hall MHD: add jxB in Ohm’s law
• MHD has to be dropped in favour of two fluid
Plasma motion resistivity Hall term
equations if ExB drift is comparable to diamagnetic
perpendicular to B
drift Bx∇p
17summary
• MHD equations are one fluid equations
• all electron inertia terms and parallel dynamics
are neglected
• valid on large scales compared to free mean path
(parallel) and Larmor radius (perpendicular)
• valid on long time scales compared to collision
times
• strong ExB flows assumed compared to
diamagnetic drifts
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