Aspects of Mode
Conversion
Paul Cally
School of Mathematical Sciences
Monash University
Where is Monash
University?
!
• “Main” campus situated in
Melbourne SE suburb of
Clayton
•4 other campuses in and
State of Victoria
(size of GB)
around Melbourne
•2 international campuses
(Malaysia and South Africa)
•Total student enrolment
63,000 (incl 22,000
international); 56.4% female
•29,000 at Clayton
2
Two Illustrative
Problems
Exact Solutions in 2D isothermal stratified
atmosphere with uniform inclined field
(with Shelley C Hansen)
Multiple Scattering from collections of thin
flux tubes (with Chris S Hanson)
I will not discuss 3D fast/Alfvén
conversion (described in Monash leg of the
workshop program; talk to me later)
3
Mode Conversion in 2D Atmosphere
with Uniform Inclined Magnetic Field
— with Shelley Hansen
4
Preliminary Concepts:
fast-slow conversion
Generalized Ray Theory
(Cally 2006; Schunker &
Cally 2006) emphasises the
importance of the a=c
equipartition level and the
attack angle !
Transmission (simplified
version)
T = exp[ ⇡|k|hs sin2 ↵]a=c
Maximal transmission near
minimal attack angle (though
not exactly 1 in full version)
5
Region I: Propagating
acoustic waves
Region II: Propagating
gravity waves
Regions III and IV:
Evanescent waves
Dimensionless variables:
" = kxH wavenumber;
# = $H/c frequency;
n = NH/c B-V frequency
acoustic cutoff freq #c=½
6
Introduce Uniform
Inclined Magnetic Field
Inclined angle % to the vertical in x-z plane; wave
propagation in that plane only.
Isothermal
&→0 at top; &→∞ at bottom. Couple the regimes.
4th order ODE
Exact solutions Zhugzhda & Dzhalilov (1984)
[Meijer G-functions], or Cally (2001)
[hypergeometric functions]
7
Exact solutions connect
top and bottom asymptotics
The exact solutions
each split into 4
asymptotic forms at
both top and bottom
Connection formulae
provide the link
In this case (unusually)
C is a reflected
component (different
from ray theory usage).
8
General Solution
in terms of 2F3 hypergeometric functions, where
( = $H/a
0 as z ∞, and (→∞ as z -∞.
Note: 2F3(a;b;x)
1 as x 0,
so
u ∼ C1(-2" + C2(2" + C3(1-2i"0+2i"
tan%
+ C3(1+2i"0+2i"
tan%
as (→0 (z ∞).
[ "0 = √(#2sec2%-¼) is real above ramp frequency # > ½ cos%]
9
On the other hand …
at the bottom
Can instead write u = c1U1 + … + c4U4
where the Ui are “pure” fast or slow
waves at the bottom (different basis)
c = AC where c=(c1, c2, c3, c4)T,
C = (C1, C2, C3, C4)T, and A=(aij) with
the aij known exactly.
10
11
Wave-Energy Flux
F = CH C = cH c, where
0
0
B i
B
=B ⌫
@ 0
0
0
B
B
=B
@
sec ✓
⌫
0
0
0
i
⌫
1
0
C
0
C
2 C
34 U ( 0 )A
2
44 U (0 )
0
0
2
33 U (0 )
⇤
2
34 U ( 0 )
0
0
0
0
sec ✓
⌫
0
0
1
0
C
0
C
2 C
34 U ( z )A
2
44 U (z )
0
0
2
33 U (z )
⇤
2
34 U ( z )
We have analytic expressions for the coefficients.
!
So now we can take ratios of fluxes out to fluxes in, and hence
calculate Reflection, Transmission, and Conversion coefficients R, T, C
12
Fast Wave Injected from Below
R, T, C against
" and # for
various field
inclinations %.
Notice attack
angle effect!
13
Slow Wave Injected from Above (flare?)
14
Fast Wave Injected from Above
Fast wave at top is
evanescent in all Regions
Can carry energy iff both
growing and decaying
solutions are present:
tunnelling
(diagonalization)
Includes both injected and
reflected
Physical for source at
finite height (e.g. flare?)
15
Fast Wave Injected from Above
16
"=0.55, #=0.7, %=30°
Movies: strong and
weak transmission
cases for fast wave
from below in
Region I.
!
"=0.55, #=0.7, %=-30°
In z<0, the short
wavelength are
slow waves and
long wavelengths
are fast.
17
Slow wave from
above
"=0.6, #=0.8, %=30°
!
• Strong conversion
to slow waves in
z<0
!
"=0.1, #=0.8, %=0°
!
•
Strong
transmission as
fast wave in z<0
18
Fast wave from
above
"=0.25, #=0.8, %=30° (Region I)
!
•Strong conversion
to fast waves in
z<0
!
"=0.1, #=0.3, %=30° (Region IV)
!
•Strong
transmission as
slow waves in z<0
19
Discussion
Isothermal mode conversion
Mode conversion can be studied using
hypergeometric connection formulae
In more complex cases without exact solutions, the
asymptotics must be connected numerically, but
the concept is the same
The concept of “attack angle” is fully verified
In 3D, the equations are 6th order. No exact
solutions available. Brings in Alfvén wave (Cally &
Goossens 2008; Cally & Hansen 2011).
20
Multiple Scattering
from Collections of Thin Flux Tubes
— with Chris Hanson (now at MPS)
21
Thin Tubes in
Stratified Atmosphere
Thin tubes support kink and
sausage modes only
Acoustic waves scatter
between them
Slow waves take energy “down
the plughole”
Near-field (acoustic jacket)
and far-field (propagating
waves)
Near-field induces collective
behaviour if tubes are tightly
packed
22
Why the Acoustic
Jacket?
2
2
2
$ = c (kx +
2
kz )
with kz large because of “wicked
wiggles” (slow waves,
2
kz
2
2
≈ $ /a where a << c)
2
Hence kx < 0; horizontal evanescence (acoustic
jacket). Gets tighter as we go deeper.
Continuous spectrum of slow waves/jacket modes
(open bottom boundary)
Removes energy from the system
A form of mode conversion
23
Nonmagnetic
Model Atmosphere
Truncated polytrope, z<-z0<0, index mp
Velocity potential
!
inc (x, t)
=
np
1
X
X
im Jm (knp r)
p
i(m✓ !t)
m (n ; s)e
n=0 m= 1
where s=-z/z0 is dimensionless depth and
!
)m expressed using Whittaker functions
n=0 is f-mode; truncate p-modes at np
24
Coupling to Tube
Waves
Constant
flux tubes
Sausage
!
!
Kink
!
!

2gs @ 2
g(mp + 1) @
! (2mp + (mp + 1)) +
+
⇠k
z0 @s2
z0
@s
@ inc
= ! 2 (mp + 1)( + 1)
@s
2

2gs
@2
g
@
! z0 +
+
⇠?
(1 + 2 )(mp + 1) @s2
1 + 2 @s
2(1 + ) 2 @ inc
=
! z0
1+2
@x
2
25
Scattered Wave
Field travelling
S
i (ri , ✓i , s)
=
1
X
m= 1
+
" np
X
i
Smn
n=0
N
X
n=np
p
(1) p
im✓i
n (n ; s)Hm (kn ri )e
3
i
Smn
⇣n (jn ; s)Km (knj ri )eim✓i 5 ,
jacket
Scattered field from tube at origin. Scattering
coefficients S derive from sausage and kink
matching of *⊥.
26
Scattered Wave
Field
S
i
=
X
ATin
S
in
n
Using the matrix notation of Kagemoto and Yue (1986),
a single tube’s scatter
diffraction transfer matrix
0
Ai = Bi @ai +
N
X
l=1,l6=i
1
TTli Al A
T = transformation matrix; relates incident and scattered
wave field; derives from Graf’s addition formula for
Bessel functions
S
in
= Tnil
27
I
ln
Various tube
Configurations
Regular (filled) or
random (open)
28
Scattering from
Multiple Tubes
2
4I
N
X
i6=l,l=1
3
0
Bl TTil Bi TTli 5 Al = Bl @al +
29
N
X
i6=l,l=1
1
TTil Bi ai A
Results
Hanson & Cally, ApJ 2014a,b
Incident f-mode wavelength + (other
p possible, but biggest effect for f)
Absorption ! and phase shift ,
Various configurations of multiple
tubes (identical here but not nec.)
With and without T matrix
30
3 tubes closely and widely packed
31
6 or 7 regularly arranged tubes packed closely
32
7 randomly arranged close tubes
All within 33
jacket reach
Multiple Scattering
Conclusions
A powerful calculus that allows us to do many
tubes and many cases
Multiple scattering and acoustic jacket have large
effect
Good agreement with simulations (Felipe et al
2013)
Adds to Hanasoge & Cally (2009): sausage modes;
multiple (>2) tubes/scattering; correction of
symmetry error
34
35