Using plasma dynamics to determine the
strength of a prominence's magnetic fields
GCOE Symposium 2013 @ Kyoto University
Andrew Hillier
What is a Quiescent Prominence?
~10 Mm
Image: Quiescent prominence
observed on 2007/10/03 01:56
UT in the Ca II H line (3968.5 Å)
Temperature:
6000K~10,000K
(Tandberg-Hanssen 1995)
Number Density:
1010~1011 cm-3
(eg Labrosse 2010 &
Hirayama 1986)
Magnetic field strength:
3~30 G
(Leroy 1989)
Ionisation fraction:
~0.2 at centre
(Gunar et al 2008)
Prominences and Space Weather
Prominence eruption on August 31, 2012 observed by the Solar
Dynamics Observatory satellite (courtesy of NASA)
How Well Do We Understand Quiescent
Prominences?
Magnetic field strength:
3~30 G (Leroy 1989)
But only ~15 prominences
have had their magnetic field
measured (to my knowledge)
~10 Mm
But we need to know the
field strength to be able to
model prominences, discuss
there dynamics etc
Image: Quiescent prominence observed on
2007/10/03 01:56 UT in the Ca II H line (3968.5 Å)
The Plumes in Prominences
First observed by
Stellmacher &
Wiehr 1973
Rediscovered by
Berger et al 2008
& De Toma et al
2008
Fig: Prominence
observed in Hα on 8th
Aug 2007 using
Hinode SOT
Courtesy of T. Berger
The Plumes Created by the
Magnetic Rayleigh-Taylor Instability
• The plumes (fingers of low
density material rising
through the dense
prominence material)
were hypothesized to be
created by the RayleighTaylor instability by Berger
et al 2008 & 2010
Key Point 1: plumes have an
elliptical head
Image: Quiescent prominence
observed on 2007/10/03 03:30 UT in Key Point 2: Constant rise
velocity (10 – 30 km/s)
the Ca II H line (3968.5 Å)
The Plumes Created by the
Magnetic Rayleigh-Taylor Instability
• Simulations by Hillier et al
(2012) investigated the 3D
mode of the magnetic RayleighTaylor instability in a
prominence model
Key Point 3: Creates filamentary
structure aligned with Magnetic
field
Using The Key Points to Make a Model
Key Point 1: plumes have an
elliptical head (change
coordinates to make a circle)
Key Point 2: Constant rise velocity
(10 – 30 km/s) (Change reference
frame)
Key Point 3: Creates filamentary
structure aligned with Magnetic
field (Makes it like a tube)
Flow around a circular cylinder
This has now reduced to a classic
fluid dynamics problem
Using the assumptions of invisicid,
irrotational and incompressible it is
possible to calculate the potential
flow around a circular cylinder
1

vr  1  2  cos
 r 
1

v  1  2  sin 
 r 
Potential HD flow around a circular
cylinder – Source Wikipedia
Compression at Top of Plumes
Material is
compressed
High total
pressure
drives
material out
the way
Plume rises
For some plumes we see a thick,
bright hat. As the emission of
prominences is mainly scattering, this
is showing higher density regions
Image: Left - Quiescent
prominence observed on
2007/10/03 02:56 UT in the Ca II H
line (3968.5 Å).
Right – Zoomed image of plume
Mathematical model for the
Compression
Image: Compressible MHD flow round a circular
cylinder. Magnetic field into screen
We can use a classic
solution of flow around a
circular cylinder
+ MHD (Horizontal field
only)
+Compressibility correction
to get the density
distribution (van Dyke
1975).

2(  1)  2

M *2  
1

M


 

How can this be used?

2(  1)  2

M *2  
1


M 



p
 2
B / 8
• By modelling the intensity in
terms of density, the
compression at the top of the
plume can be calculated.
• This will allow for the plasma
beta to be solved for.
Estimate of Prominence Plasma Beta –
Calculating Plume Size and Velocity
The dimension of the plume head (needed for normalisation) are
a~900km and b~1700km
The rise velocity is vrise  12.3  0.6 km s1
Estimate of Prominence Plasma Beta –
Fitting Intensity to calculate β
Assuming that the emission is only proportional to the density we can
fit to solve for M*, giving an estimate of the plasma beta of
 ~ 0.47  1.13 for  ~ 1.4  1.7
Conclusions
• We now have a new way to estimate the plasma beta of
quiescent prominences using the Rayleigh-Taylor plumes
• Application to one prominence gives the plasma β as β=0.47 –
1.13 for γ=1.4 – 1.7.
• There are many potential improvements that can be made,
that will improve the accuracy AND the amount of
information we can extract from the prominence
For greater detail, please see:
Hillier, Hillier & Tripathi (2012) ApJ, 761, 106
Setting for Simulations
• Kippenhahn-Schlűter
prominence model (Priest 1982)
• Buoyant tube put in centre of
prominence to make it unstable
and a velocity perturbation in
the y direction to excite
interchange of magnetic field
• Ideal MHD used (grid
90*150*400)
• Length normalised to pressure
scale height   600 km
z/H
z/H
  1.05

p(0)
 0.5
2
2
( Bx 0  Bz ) / 8
     0.7
A

~ 0.6
     1.3
x/H
y/H
Fig: Mass density (colour) and field lines (contour)
of prominence model. A is x-z cut and B is y-z cut
(y boundary is symmetric)
2D Density Slice of simulation
• Swirling, vortex like structures
formed once instability is
initiated
• Reach height of approx 6Mm
• Upflows: ~ 6 km/s
(approximately constant)
• Width of upflows inversely
cascades from ~100 km to
~1Mm
• Makes threads in the
prominence material
z/H
y/H
Movie: Temporal evolution of instability
in x=0 plane. Colour shows density,
arrows show velocity
Evolution shown in 3D
(1) Rise of cavity releases the
magnetic tension, flattening the
field lines. Instability starts on
small scale
(2) Multiple plumes formed, plume
magnetic field begins to move
through the prominence
(3) Magnetic field lines glide
passed each other in an
interchange process
Fig: Temporal evolution of instability in 3D, lines
represent magnetic field with density isosurface
Application to Simulation Results
To check the data, first we revise the axis to give a circular head.
Note there is no density increase at the top of the plume
Application to Simulation Results –
Velocity Around Plume Head
• Velocities along curve shown in previous slide (both simulated –
solid, and predicted -dashed)
Application to Simulation Results –
Matching Density Distribution
Integrating the density along the x- The above figure shows the
axis shows the increase in column simulated density along the slit
density at the head of the plume
and predicted density
Application to Simulation Results –
Calculating χ2
• By calculating the χ2 for fits to the density profile for different
values of plasma β, we can show that the smallest χ2
corresponds to the simulation plasma β of ~0.55
Can we model the Bright Emission?
For some plumes we see a thick,
bright hat. As the emission of
prominences is mainly scattering, this
is showing higher density regions
Image: Left - Quiescent
prominence observed on
2007/10/03 02:56 UT in the Ca II H
line (3968.5 Å).
Right – Zoomed movie of plume
What if the Magnetic Field is Vertical
• If the field is vertical, then
the compression doesn’t
occur at the head of the
plume
• Rarefaction occurs instead
• It is hard to understand the
observations of the plume
if the prominence field is
vertical
Courtesy of Roger Scott, Montana State University
Estimate of Prominence Plasma Beta –
β as a Function of γ
Necessary improvements for the Model
• Deal with projection effects and magnetic field that is not
along the line of sight (use velocity equations combined with
observed Doppler shifts)
• Include shear between the plume magnetic field and the
prominence magnetic field to give direction of the magnetic
field (use most unstable mode of magnetic Rayleigh-Taylor
instability under shear and the with the observed plume
width)
• Improved model for emission (there must be a way to
improve my simple model for the emission – Suggestions
Please!)