1B11 Foundations of Astronomy Concepts of Magneto-HydroDynamics (MHD)

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1B11 Foundations of AstronomyConcepts of
Magneto-HydroDynamics (MHD)

• Silvia Zane, Liz Puchnarewicz
• emp_at_mssl.ucl.ac.uk
• www.ucl.ac.uk/webct
• www.mssl.ucl.ac.uk/

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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

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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

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.

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1B11 The magnetic Reynolds number

If the scale length of the plasma is L, the
gradient term is (approximately)
The ratio RM between the two terms on the right
hand side of the induction equation is
RM Magnetic Reynolds number

1. If RM ltlt 1 then the diffusion term dominates
2. If RM ltlt 1 then the coupling term dominates

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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.

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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)

C(t1)
Ex If the surface is reduced in area we can
infer the magnetic field strength is increased
at t2.
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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.
B-field
S3
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.
V

P1
V
P2
B
B
A
A
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1B11 MHD forces

The Magnetic Force FM in a MHD plasma is FM j x
B. Using the Maxwell equations
And after a bit of lengthy algebra
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.
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1B11 MHD forces two simple components

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.
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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.

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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

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!)
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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
  RMgtgt1) 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.

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1B11 Magnetic reconnection

• These separate plasma cells are partitioned by
  thin current sheets, which support the change in
  magnetic fields across the boundary. Recall

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!
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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)
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1B11 Magnetic reconnection
Vout V Alfven

The resulting reconnected lines are sharply
bent through the current sheet. Magnetic tension
forces associated with these bent lines
accelerate the plasma along the current sheet and
away from the diffusion region on each side.
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.
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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.