Download - Wave propagation in a non-uniform, magnetised plasma: Finite beta James McLaughlin Leiden March 2005
Wave propagation in a non-uniform, magnetised plasma:
Finite beta
James McLaughlin
Leiden
March 2005
Introduction
• Models of wave motions that should occur in the neighbourhood of 2D X-point. There are three types of wave motions that should occur; slow MA, fast MA and Alfvén. Wave motions recently observed in the corona.
• Research has also illustrated importance of magnetic topology. Extrapolations predict null points; Alfvén speed zero.
• Aim - Bring these areas together. Show how plasma waves behave when they travel into magnetic structures (inhomogeneous medium).
Model
• Perform perturbation about equilibrium magnetic field – simple 2D X-point.
( )0 ,0,B
x za
= -B
Model
• Perform perturbation about equilibrium magnetic field – simple 2D X-point.
( )0 ,0,B
x za
= -B
• Idea is to send in wave from top boundary, and see (and explain!) what happens.
Model II
• Perturbation analysis on ideal MHD equations- linearised perturbation equations
• Take velocity of the form:
101
101
011
1011
0
0.
1
v
v
BvB
BBv
pt
pt
t
pt
yBBBB
Bv ˆvv
0000
0||1 yv
A
Model III
• Equations decouple – fast and slow MA wave & Alfvén wave.
• MA waves:
• Alfvén wave -
2222
22
221
11
1122
42
,
2
2
zx
xz
zx
xz
zx
zx
zz
xx
zxt
p
xt
b
zt
b
z
pz
x
px
t
z
px
x
pz
z
b
x
bzx
t
zx
xz
z
vz
x
vx
t
b
z
bz
x
bx
t
v yyyzxy
,
v~ 0B
||0 v~ B
Fast MA wave I
• Fast wave case reduces to single wave equation:
• Where (non-dimensionalised. Spatially dependent).
• Fast wave solved numerically - two-step Lax-Wendroff scheme. -6 ≤ x ≤ 6 and -6 ≤ z ≤ 6 .
• Boundary conditions
2 2Av x z= +
2
2
2
22
2
2
,zx
zxvt A
0
0,0,0
otherwise0
0sin6,
666
zxx zxx
ttx
Fast MA wave II
• Linear, fast magnetoacoustic wave travels towards the vicinity of the X-point and bends around it.
• Alfvén speed spatially varying - travels faster further away from the origin.
• Wave demonstrates refraction – wraps wave around null
• Key feature of fast wave propagation.
Alfvén wave I
• Alfvén wave solved numerically - two-step Lax-Wendroff scheme.
• 0 ≤ x ≤ 6 and 0 ≤ z ≤ 6 (first quadrant)
• Boundary conditions:
0,0,0
otherwise0
30
03
cos1sin6,
060
6
z
y
x
y
x
y
z
y
y
z
v
x
v
x
v
z
vx
txtxv
Alfvén wave II
• Alfvén wave travels down from top boundary and begins to spread out, following the field lines.
• As the wave approaches the separatrix, thins but keeps its original amplitude. The wave eventually accumulates very near the separatrix (x axis).
Fast and slow MA waves
• How does this simple model change with the inclusion of a finite term?
• Main questions:
1. How does the slow wave couple to the (driven) fast wave?
2. How does the nature of the coupling change with ?
3. What is the effect of pressure at the origin?
Fast and slow I
• Now consider finite case. Recall equations:
• First thing; much more complicated! We now have a coupled system: driving the system with a fast wave will generate a slow wave, which in turn will act back on the system.
• Difficult to think about: pure fast and pure slow wave concepts no longer valid.
2222
22
221
11
1122
42
,
2
2
zx
xz
zx
xz
zx
zx
zz
xx
zxt
p
xt
b
zt
b
z
pz
x
px
t
z
px
x
pz
z
b
x
bzx
t
zx
xz
Fast and slow II
• Also, now concerned that so (even though the fast wave may be able to pass through the origin.
• Fast wave solved numerically - two-step Lax-Wendroff scheme.
-4 ≤ x ≤ 4 and -4 ≤ z ≤ 4 . Boundary conditions
04,,otherwise0
0sin4,
x
ttx
2sound
22Fast cvc A 02 Av
• We find that the fast wave (driven on the upper boundary) travels into the box (across fieldlines) mainly as before, with some small additions:
),
Fast and slow
• Main questions:
1. How does the slow wave couple to the (driven) fast wave?
2. How does the nature of the coupling change with ?
3. What is the effect of pressure at the origin?
How does the slow wave couple to the (driven) fast wave? I
• There is now coupling to the parallel velocity component, such that two waves are produced:
firstly, an aspect travelling at the same speed and frequency as the driven (fast) wave; this wraps around the null in a similar way to the zero beta fast wave.
secondly, an aspect that trails behind the first, travelling at a much slower speed ( ). This is the slow wave.
How does the slow wave couple to the (driven) fast wave? II
slow wavegenerated bythe system
/
homogeneoussolution
(driven) perpendicularwave showing up
in parallelvelocity component
/
inhomogeneoussolution
Parallel velocity component
Fast and slow
• Main questions:
1. How does the slow wave couple to the (driven) fast wave?
2. How does the nature of the coupling change with ?
3. What is the effect of pressure at the origin?
How does the nature of the coupling change with ?
• The size of the used will affect the magnitude of the coupling and determine how much the pressure and parallel velocity feedback to affect the (driven) perpendicular velocity wave.
• In the corona, typical values of are often cited.01.0
Position reached by (homogenous solution) slow wave after time t=1.5 along x=1.5
)
Linear plotGradient = -0.989
Linear plot makes sense if
~SLOWc
Time taken by (homogenous solution) slow waveto reach z=3.0 along x=1.5
Linear plotGradient = 1.223
Linear plot makes sense if
1~time
time
distance~~SLOW
c
Amplitude of (homogenous solution) slow waveat (x, z)=(1.5,3.0)
) e
Gradient = 1.599
Amplitude ~
So perhaps negligibleFor coronal vales of ?
6.1
Amplitude of (inhomogenous solution) waveat (x, z) = (0.5,1.0) at t=1.67
1~time
time
distance~~
sc
Linear plotGradient = 0.8702
Amplitude increases
Also, position of max alwaysoccurs at same point in space,
i.e. position of couplingindependent of
~
Fast and slow III
• Main questions:
1. How does the slow wave couple to the (driven) fast wave?
2. How does the nature of the coupling change with ?
3. What is the effect of pressure at the origin?
What is the effect of pressure at the origin? I
• Idea is that, as opposed to case, when all wave is packed around null, pressure may now critically change the results; i.e. either expel the wave away from the null or in some other way impede its propagation (because of the term).
• However, we do not find that the inclusion of a pressure term significantly changes the system.
• Yes, the pressure is increasing as we approach the null ( at null), and yet fast wave still wraps around the null in a very similar way to before, and so wave will still dissipate close to the origin.
• We also find no evidence of the fast wave travelling through the null (perhaps is still true, but may be tiny (so dissipates).
1p
0
2sound
22Fast cvc A
soundc
Conclusions I – Fast wave
• When fast magnetoacoustic wave propagates near X-type neutral point, wave wraps itself around due to refraction (at least in 2D).
• Large current density accumulation at the null.
Build up exponential in time.
• Refraction of the wave focuses energy of the incident wave towards null point - wave continues to wrap itself around null point, again and again.
Conclusions II – Alfvén wave
• Alfvén wave - wave propagates along field lines, accumulating on separatrix (along separatrices due to symmetry). Wave thins and stretches.
• The current jx increases and accumulates along the separatrix, whilst jz decays away.
Conclusions III – wave coupling
• Perpendicular and parallel velocity components now (inevitably) coupled with inclusion of .
• Driven perpendicular (fast) wave generates waves in parallel velocity component (one of which is slow wave).
• Magnitude of the coupling dependent on , especially on amplitude of parallel velocity waves. Hence in a low corona, effect is very small, i.e. preferential heating still occurs at the origin for the fast MA wave.
• Pressure does have an effect on the system, but only causes a very small impediment; fast wave still refracts around origin, wrapping around again and again.
• Hence, inclusion of finite term does not significantly change the nature of the system.
0