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1B11
Foundations of Astronomy
Concepts of MagnetoHydroDynamics (MHD)
Silvia Zane, Liz Puchnarewicz
[email protected]
www.ucl.ac.uk/webct
www.mssl.ucl.ac.uk/
1B11 Introduction
• To understand in detail the various space plasma and solar
activity phenomena it is useful to recap some particular
principles arising from MHD.
• A plasma is a quasi-neutral gas consisting of positively
and negatively charged particles (usually ions and electrons)
which:
1) are subject to electric, magnetic and other forces, and
2) exhibit collective behaviour such as bulk motion,
oscillations and instabilities.
•Frozen-in flux approximation: a central tool to understand
the behaviour of a plasma in presence of a B-field
1B11 The Frozen-In Flux Approximation
We start from the MHD induction equation, describing the
evolution of a magnetic field in a plasma with conductivity 
and permeability 0
B
1
2
   v  B  
 B
t
 0
1) The first term of the right hand side describes the
behaviour (coupling) of the magnetic field with the
plasma moving at velocity v
2) The second term on the right hand side represents
diffusion of the magnetic field through the plasma.
1B11 The magnetic Reynolds number
If the scale length of the plasma is L, the gradient term is
(approximately)  ~ 1/ L
The ratio RM between the two terms on the right hand side
of the induction equation is
RM ~ 0Lv
RM= Magnetic Reynolds number
1) If RM << 1 then the diffusion term dominates
2) If RM << 1 then the coupling term dominates
1B11 B from plasma flow
• In a typical space plasma the conductivity  is very high,
and the scale lengths, L, large
• In the solar wind and the magnetosphere RM ~1011.
Hence the diffusion term is negligible in these contests and
the magnetic field convects exactly with the plasma flow.
Or, the plasma particles are frozen with B and forced to
move along the field lines.
 This is often referred as “ideal MHD limit” or “frozen-in
flux approximation”.
 It is an extremely important concept since it allows us to
study the evolution of the field, and particularly the
topology of the field lines, by looking at the plasma flow.
1B11 Plasma flow from B
Or course the concept can be reversed: if we know the topology
of the magnetic field, then we know the plasma fluid flow.
1) A surface S1 in a plasma bounded by a
closed contour C encloses a given amount
of magnetic flux at time t1.
2) The surface may be subsequently
deformed and/or relocated by motions of
the plasma
3) However, under the frozen-in flux approx.,
the surface will enclose the same amount
of magnetic flux C at a later time t2
B-field
C(t2)
n2
S2
C(t1)
n1
 C   B n1dA   B n2 dA
S1
S2
Ex: If the surface is reduced in area we can infer the magnetic field
strength is increased at t2.
S1
1B11 Magnetic flux tube
We can define a magnetic flux tube: by
taking the closed loop and moving it parallel
to the field it encloses.
The surface, or tube S3, thus created has
zero flux through it and consequently the
fluid elements that form the flux tube at one
moment, form the flux tube at all instants.
Also: if two fluid elements P1 and P2
are originally linked by the field lines
A and B, they will remain connected by
field lines A and B whatever the
individual motions V1 and V2 of the
individual volumes.
B-field
n2
S2
n1
S3
S1
V
P1
V
P2
B
A
B
A
1B11 MHD forces
The Magnetic Force FM in a MHD plasma is FM =j x B.
Using the Maxwell equations:
 B2  1
FM    B  B  
  B  B  .....
0
 2 0   0
And after a bit of lengthy algebra:
1
 B2  B2 R C
FM   

2
2


R
0
C
 0
Rc is the local radius of curvature of the field line. It points
towards the centre of curvature of the field line.
Let us examine the two terms in the latter equation….
1B11 MHD forces: two simple components
 B2  B2 R C
FM   

2
 20  0 RC
The magnetic force can be resolved into two conceptually
simple components:
1) A force perpendicular to the B-field which has the form of a
pressure (it is the gradient of a scalar quantity B2/20), and
2) A force towards the instantaneous centre of curvature that
depends on: i) radius of curvature and ii) field strength B. This
is equivalent to a tension force acting along the field lines.
Thus forcing the field lines together results in an opposing
perpendicular pressure force, while trying to bend the field
lines results in an opposing tension force.
1B11 MHD waves
This magnetic pressure and magnetic tension represent two
kinds of restoring force that arise in a plasma in the presence of
a magnetic field.
They are associated to wave modes: waves and perturbations
that propagate in the plasma.
Most important ones:
1) Alfven waves
2) Magnetosonic wave modes, in which both the
magnetic field strength and the plasma pressure vary.
1B11 MHD waves
The Alfven wave is a very important one. It is entirely due to
the tension force associated with the magnetic field.
1) It is essentially a magnetic wave, as there is no associated
compression of the plasma as in the case of sonic (pressure)
waves.
2) It propagates preferentially along the field direction (and not
across it) with speed:
2
VA   B /
0 
This wave causes magnetic field and plasma velocity
perturbations which are perpendicular to the background
field (and to the wave propagation vector) and thus is
sometimes called transverse wave or shear wave. It is
analogous to waves on a string under tension (guitar strings!)
1B11 Magnetic reconnection
•The concept of magnetic reconnection is key in
understanding the coupling between solar wind and
planetary atmospheres (auroras), as well as acceleration
of particles in space and astrophysical plasmas (solar
flares, etc..)
 The frozen-in flux approximation (in the case of
RM>>1) leads to a picture of space plasmas contained
within separated regions.
 For example, field and plasma from the Sun (slar
winds,..) are frozen-out of the region occupied by a
planetary magnetic field.
1B11 Magnetic reconnection
•These “separate” plasma cells are partitioned by thin
current sheets, which support the change in magnetic
fields across the boundary. Recall:
  B  0 j
However, exactly because of their thinness (small spatial
lengths!), the magnetic Reynolds number within the
current sheet may be relatively small:
 diffusion of the magnetic field through the plasma
start to be important!
1B11 Magnetic reconnection
If there is a strong B-field gradient
and the fields on either side of the
gradient are anti-parallel, then
diffusion of the field at the gradient
can led to a loss of total magnetic flux
this situation is called magnetic
annihilation.
Field lines are convected into the
diffusion region and merge with field
lines with opposite orientation (which
originally where on the other side of
the gradient)
B field
j
E 




Vin
Diffusion
1B11 Magnetic reconnection
Vout ~ V Alfven
The resulting “reconnected lines”
are sharply bent through the
current sheet.
j
Magnetic tension forces
associated with these bent lines
accelerate the plasma along the
current sheet and away from the
diffusion region on each side.

Vin
Diffusion
B
The simplest magnetic reconnection geometry. Anti-parallel field lines are
separated by a thin current sheet (light grey) across which the field reverses. Due to
the small scale lengths, the frozen-in flux approximation breaks down  magnetic
flux diffuse from both sides. The field reconnects to form 2 hairpin like field lines,
which rapidly contract away from the neutral “X”-point. Outflows jets of plasma
are formed, also moving away from the “X”-point, on both sides.
1B11 Magnetic reconnection
Magnetic reconnection is an extremely important process:
 It allows the two sides of the field gradient to be
linked by the newly reconnected line.
 It allows plasmas from either side to flow along the
field and mix with those from other side.
 Also, magnetic energy is continuously liberated in
the process, causing accelerated and heated plasma
flows.