Transcript Talk 1

Wave Generation and Propagation in
the Solar Atmosphere
Zdzislaw Musielak
Physics Department
University of Texas at Arlington (UTA)
OUTLINE

Theory of Wave Generation

Theory of Wave Propagation

Solar Atmospheric Oscillations

Theory of Local Cutoff Frequencies

Applications to the Sun
The H-R Diagram
Solar structure
Model of the Solar Atmosphere
Averett and Loeser (2008)
Energy Input
From the solar photosphere:
acoustic and magnetic waves
Produced in situ:
reconnective processes
From the solar corona:
heat conduction
Generation of Sound
Lighthill (1952)
James M. Lighthill
Acoustic Sources
Monopole
Dipole
Quadrupole
Efficiency of Acoustic Sources
Lighthill Theory of Sound Generation
(Lighthill 1952)
The inhomogeneous wave equation
Lˆ[  ]  Sˆ[ut ]
with

2 2
ˆ
L  2  cs 
t
2
and the source function
Sˆ [ut ]  Squad
Lighthill-Stein Theory of Sound Generation
(Lighthill 1952; Stein 1967)
The inhomogeneous wave equation
LˆS [ p1 ]  SˆS [ut , p0 ]
with
2
2
2








2
2
2
2
2
LˆS  2  cS    S  cS  BV    2  2 
t
 t   x y 
2
and
p1 
p
p0
and the acoustic cutoff frequency
cS
S 
2H
Lighthill-Stein Theory of Sound Generation
The source function is given by
SˆS [ut , p0 ]  Squad  Sdip  Smon
where
Squad  
4
Sdip   
2
and
S mon  
4
S
2
S
Applications of Lighthill-Stein Theory
Generation of acoustic and magnetic flux tube
waves in the solar convection zone
Collaborators: Peter Ulmschneider and Robert Rosner;
also Robert Stein, Peter Gail and Robert Kurucz
Graduate Students: Joachim Theurer, Diaa Fawzy, Aocheng Wang,
Matthew Noble, Towfiq Ahmed, Ping Huang
and Swati Routh
Acoustic Wave Energy Fluxes
log g = 4
Ulmschneider, Theurer & Musielak (1996)
Generation of Magnetic Tube Waves
Fundamental Modes
Generation of Longitudinal Tube Waves I
The wave operator

2 
2 
ˆ
LT [ p2 ]   2  cT 2   D  p2
z
 t

2
with
p2 
p
 0 B0
2
,
cT 
cS c A
cS2  c A2
and the cutoff frequency (Defouw 1976)
2
c
cT 9
1
S  1
D 

 2
H 16 2 cA  2
Generation of Longitudinal Tube Waves II
The source function is given by
2
2


cT 
2
2  2
e
ˆ


ST [ut ] 


ut
BV
2 
2

2  0 B0 c A  t

or it can be written as
2
ˆ
ST [ut ]  S dip
Generation of Transverse Tube Waves
The wave operator

2 
2 
ˆ
LK [v1 ]   2  cK 2   K  v1
z
 t

2
with
v1  vx 
1/ 4
0
,
cK
K 
4H
2
,
cK 
B0
4 ( e   0 )
The source function
1/ 4

e
0
SˆT [u x , u z ] 
0  e
1
1
 u x  2u x    1
u x   
   u x   1  
 2   g
   g 

    u z
t  t 
z  t 
 t  z   H z 
 t
PROCEDURE
Solution of the wave equations:
- Fourier transform in time and space
Wave energy fluxes and spectra:
- Averaging over space and time
- Asymptotic Fourier transforms
- Turbulent velocity correlations
- Evaluation of convolution integrals
Description of Turbulence
The turbulent closure problem:
- spatial turbulent energy spectrum
(modified Kolmogorov)
- temporal turbulent energy spectrum
(modified Gaussian)
(Musielak, Rosner, Stein & Ulmschneider 1994)
Solar Wave Energy Spectra
Wave Energy and Radiative Losses
Current Work
Modifications of the Lighthill and
Lighthill-Stein theories to include
temperature gradients.
Chromospheric Models
 Purely
Theoretical
 Two-Component
 Self-Consistent
 Time-Dependent
Collaborators: Peter Ulmschneider, Diaa Fawzy,
Wolfgang Rammacher, Manfred
Cuntz and Kazik Stepien
Models versus Observations



Base - acoustic waves
Middle - magnetic tube waves
Upper – other waves and / or
non-wave heating
Fawzy et al. (2002a, b, c)
Solar Chromospheric Oscillations

Response of the solar chromosphere to propagating
acoustic waves – 3-min oscillations (Fleck & Schmitz
1991, Kalkofen et al. 1994, Sutmann et al. 1998)

Oscillations of solar magnetic flux tubes (chromospheric
network) – 7 min oscillations (Hasan & Kalkofen 1999,
2003, Musielak & Ulmschneider 2002, 2003)
Chromospheric oscillations
are not cavity modes!
P-modes
Applications of Fleck-Schmitz Theory
Propagation of acoustic and magnetic flux tube
waves in the solar chromosphere
Collaborator: Peter Ulmschneider
Graduate Students: Gerhard Sutmann, Beverly Stark,
Ping Huang, Towfiq Ahmed,
Shilpa
Subramaniam and
Swati Routh
Excitation of Oscillations by Tube Waves I
The wave operator for longitudinal tube waves is

2 
2 
ˆ
LT [ p2 ]   2  cT 2   D  p2
z
 t

2
with
p2 
p
 0 B0
2
,
cT 
cS c A
cS2  c A2
and the cutoff frequency (Defouw 1976)
2
c
cT 9
1
S  1
D 

 2
H 16 2 cA  2
Excitation of Oscillations by Tube Waves II
The wave operator for transverse tube waves is

2 
2 
ˆ
LK [v1 ]   2  cK 2   K  v1
z
 t

2
2
with
v1  vx 
1/ 4
0
,
cK 
B0
4 ( e   0 )
and the cutoff frequency (Spruit 1982)
cK
K 
4H
Initial Value Problems
LˆT  p2   0
and
v1 t , z   0
IC: lim
t 0
BC:
lim v1 t , z   V0 t 
z 0
and
and
Lˆ K v1   0
 v 
lim  1   0
t 0
 t 
lim v1 t , z   0
z 
Laplace transforms and inverse Laplace transforms
Solar Flux Tube Oscillations
Longitudinal tube waves
Transverse tube waves
Theoretical Predictions
Solar Chromosphere:
170 – 190 s (non-magnetic regions)
150 – 230 s (magnetic regions
Maximum amplitudes are 0.3 km/s
Solar Atmospheric Oscillations
 Solar
Chromosphere:
100 – 250 s
 Solar
Transition Region:
200 – 400 s
Corona:
2 – 600 s
 Solar
TRACE and SOHO
Lamb’s Original Approach (1908)
Acoustic wave propagation in a stratified and isothermal medium is
described by the following wave equation
 2u 2  2u cS u
 cS 2 
0
2
t
z
H z
With
u  u1  01/ 2
, one obtains
2
 2u1
2  u1
2

c


S
S u1  0
2
2
t
z
where
S 
cS
2H
Klein-Gordon
equation
is the acoustic cutoff
frequency
A New Method to Determine Cutoffs
General form of acoustic wave equation in a medium with gradients:
   2 2  2 dcs  d 2 cs 
, 2   i  0
 Lˆs  2 , cs 2 ,
z dz z dz 
  t
i = 1, 2, 3
Transformations:
d  dz cs
give
and
~ 
i  i e

with
   cs (~ ) cs (~ )d~
0
  2 2

2
ˆ
,  i ( )   i  0
 La  2 ,
2

  t 
Using the oscillation theorem and Euler’s equation allow finding
the acoustic cutoff frequency!
Musielak, Musielak & Mobashi Phys. Rev. (2006)
The Oscillation Theorem
d 2
 ( x)  0
2
d
Consider
with periodic solutions
d 2
 ( x)  0
2
d
Another
equation
If
 ( x)  ( x)
for all x
then the solutions of the second equation
are also periodic
Euler’s Equation and Its Turning Point
d  CE
 2 0
2
d
4
2
CE  1
Periodic solutions
CE  1
Turning point
CE  1
Evanescent solutions
Applications of the Method
Cutoff frequencies for acoustic and magnetic
flux tube waves propagating in the solar
chromosphere
Collaborator: Reiner Hammer
Graduate Students: Hanna Mobashi, Shilpa
Subramaniam and Swati Routh
Torsional Tube Waves I
Isothermal and ‘wide’ magnetic flux tubes
Introducing
x
v
R
and
y  Rb
, we have
2x 2 2x
 c A ( s) 2  0
2
t
s
and
 2 y  2 y
 [c A ( s ) ]  0
2
t
s
s
x and y are Hollweg’s variables
Torsional Tube Waves II
Using the method, we obtain
 2 x1  2 x1  cA  x1
 2   
0
2
t

 c A  
and
 2 y1  2 y1  cA  y1
 2   
0
2
t

 c A  
where
cA  dc A / d
Torsional Tube Waves III
Eliminating the first derivatives, we obtain Klein-Gordon
equations
 2 x2  2 x2
2



x ( ) x2  0
2
2
t

and
 2 y2  2 y2
2



y ( ) y2  0
2
2
t

where
 2x ( ) 
3  cA 
 
4  cA 
2
1  cA  1  cA 
1  cA  and
2
 y ( )      
  
2  cA  4  cA 
2  cA 
2
Torsional Tube Waves IV
Making Fourier transforms in time, the Klein-Gordon
equations become
d 2 x2
2
2

[



x ( )] x2  0
2
d
and
d 2 y2
2
2

[



y ( )] y2  0
2
d
Using Euler’s equation and the oscillation theorem,
the turning-point frequencies can be determined.
The largest turning-point frequency becomes the
local cutoff frequency.
Torsional Tube Waves V
Exponential models:
c A ( s)  c A0 e s / mH
where m = 1, 2, 3, 4 and 5
The model basis is located at
the solar temperature minimum
Routh, Musielak and Hammer (2007)
Torsional Tube Waves VI
Since
2
3  c  1  c 
 2x ( )   A    A 
4  cA  2  cA 
and
1  c  1  c 
 2y ( )   A    A 
2  cA  4  cA 
2
For isothermal and thin magnetic flux tubes, we have
cA  const , which gives
x   y  0
cutoff-free propagation!
Musielak, Routh and Hammer (2007)
Current Work
 Acoustic
waves in non-isothermal media
 Waves
in “wide” magnetic flux tubes
 Waves
in “wine-glass” flux tubes
 Waves
in inclined magnetic flux tubes
CONCLUSIONS

Lighthill-Stein theory of sound generation was used to calculate
the solar acoustic wave energy fluxes. The fluxes are sufficient
to explain radiative losses observed in non-magnetic regions of
the lower solar chromosphere.

A theory of wave generation in solar magnetic flux tubes was
developed and used to compute the wave energy fluxes. The
obtained fluxes are large enough to account for the enhanced
heating observed in magnetic regions of the solar
chromosphere.

Fleck-Schmitz theory was used to predict frequencies and
amplitudes of the solar atmospheric oscillations. The theory can
account for 3-min oscillations in the lower chromosphere.

A method to obtain local cutoff frequencies was developed. The
method was used to derive the cutoffs for isothermal and “wide”
flux tubes and to show that the propagation of torsional waves
along isothermal and thin magnetic flux tubes is cutoff-free.
Supported by NSF, NASA and The Alexander von Humboldt Foundation