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Wave Generation and Propagation Wave Generation and Propagation in the Solar Atmospherein the Solar Atmosphere
Zdzislaw MusielakZdzislaw Musielak
Physics Department Physics Department
University of Texas at Arlington (UTA)University of Texas at Arlington (UTA)
OUTLINEOUTLINE
Theory of Wave Generation Theory of Wave Generation
Theory of Wave Propagation Theory of Wave Propagation
Solar Atmospheric Oscillations Solar Atmospheric Oscillations
Theory of Local Cutoff FrequenciesTheory of Local Cutoff Frequencies
Applications to the SunApplications to the Sun
The H-R DiagramThe H-R Diagram
Solar structureSolar structure
Model of the Solar AtmosphereModel of the Solar Atmosphere
Averett and Loeser (2008)
Energy InputEnergy Input
From the solar photosphere:From the solar photosphere:
acoustic and magnetic wavesacoustic and magnetic waves
Produced in situ:Produced in situ:
reconnective processesreconnective processes
From the solar corona:From the solar corona:
heat conductionheat conduction
Generation of Sound
James M. Lighthill
Lighthill (1952)
Acoustic Sources
Monopole Dipole
Quadrupole
Efficiency of Acoustic Sources
][ˆ][ˆ tuSL 2
2 22
ˆsL c
t
ˆ [ ]t quadS u S
Lighthill Theory of Sound Generation(Lighthill 1952)
The inhomogeneous wave equation
with
and the source function
1 0ˆˆ [ ] [ , ]S S tL p S u p
22 2 22 2 2 2 2
2 2 2ˆS S S S BVL c c
t t x y
1
0
pp
p
2S
S
c
H
Lighthill-Stein Theory of Sound Generation(Lighthill 1952; Stein 1967)
The inhomogeneous wave equation
with
and and the acoustic cutoff frequency
0ˆ [ , ]S t quad dip monS u p S S S
Lighthill-Stein Theory of Sound Generation
The source function is given by
where
and
4quadS
22SdipS
4SmonS
Applications of Lighthill-Stein TheoryApplications of Lighthill-Stein Theory
Generation of acoustic and magnetic flux tubeGeneration of acoustic and magnetic flux tube
waves in the solar convection zonewaves in the solar convection zone
Collaborators:Collaborators: Peter Ulmschneider and Robert Rosner; Peter Ulmschneider and Robert Rosner;
also Robert Stein, Peter Gail and Robert Kurucz also Robert Stein, Peter Gail and Robert Kurucz
Graduate Students:Graduate Students: Joachim Theurer, Diaa Fawzy, Aocheng Joachim Theurer, Diaa Fawzy, Aocheng Wang, Wang, Matthew Noble, Towfiq Ahmed, Ping Huang Matthew Noble, Towfiq Ahmed, Ping Huang
and Swati Routh and Swati Routh
Acoustic Wave Energy Fluxes
log g = 4
Ulmschneider, Theurer & Musielak (1996)
Generation of Magnetic Tube Waves
Fundamental Modes
22
2
22
2
2
2 ][ˆ pz
ct
pL DTT
22AS
AST
cc
ccc
2
0 0
pp
B
2
2 2
9 1 1
16 2ST
DA
cc
H c
Generation of Longitudinal Tube Waves I
The wave operator
with
and the cutoff frequency (Defouw 1976)
,
222
2
2
2
00
2
2][ˆ tBV
A
TetT u
tc
c
BuS
Generation of Longitudinal Tube Waves II
The source function is given by
diptT SuS ][ˆ 2
or it can be written as
zxxxx
e
ezxT u
zHz
u
tg
tz
ug
tt
u
t
uuuS
1
],[ˆ111
2
2
0
4/10
2 22 2
1 12 2ˆ [ ]K K KL v c v
t z
1/ 41 0xv v 0
04 ( )K
e
Bc
4K
K
c
H
Generation of Transverse Tube Waves
The wave operator
The source function
with
, ,
PROCEDUREPROCEDURE
Solution of the wave equations:Solution of the wave equations:
- Fourier transform in time and space- Fourier transform in time and space
Wave energy fluxes and spectra:Wave energy fluxes and spectra:
- Averaging over space and time- Averaging over space and time
- Asymptotic Fourier transforms- Asymptotic Fourier transforms
- Turbulent velocity correlations- Turbulent velocity correlations
- Evaluation of convolution integrals- Evaluation of convolution integrals
Description of TurbulenceDescription of Turbulence
The turbulent closure problem:The turbulent closure problem:
- spatial turbulent energy spectrum- spatial turbulent energy spectrum
(modified Kolmogorov) (modified Kolmogorov)
- temporal turbulent energy spectrum- temporal turbulent energy spectrum
(modified Gaussian) (modified Gaussian)
(Musielak, Rosner, Stein & Ulmschneider 1994)(Musielak, Rosner, Stein & Ulmschneider 1994)
Solar Wave Energy Spectra
Wave Energy and Radiative LossesWave Energy and Radiative Losses
Current WorkCurrent Work
Modifications of the Lighthill and Modifications of the Lighthill and
Lighthill-Stein theories to includeLighthill-Stein theories to include
temperature gradients.temperature gradients.
Chromospheric ModelsChromospheric Models
Purely TheoreticalPurely TheoreticalTwo-ComponentTwo-ComponentSelf-ConsistentSelf-ConsistentTime-DependentTime-Dependent
Collaborators:Collaborators: Peter Ulmschneider, Diaa Fawzy, Peter Ulmschneider, Diaa Fawzy, Wolfgang Rammacher, ManfredWolfgang Rammacher, Manfred Cuntz and Kazik StepienCuntz and Kazik Stepien
Models versus ObservationsModels versus Observations
BaseBase - acoustic waves - acoustic waves MiddleMiddle - magnetic tube waves - magnetic tube waves UpperUpper – other waves and / or – other waves and / or
non-wave heatingnon-wave heating
Fawzy et al. (2002a, b, c)
Solar Chromospheric OscillationsSolar Chromospheric Oscillations
Response of the solar chromosphere to propagating Response of the solar chromosphere to propagating acoustic waves – 3-min oscillations (acoustic waves – 3-min oscillations (Fleck & SchmitzFleck & Schmitz 1991, Kalkofen et al. 1994, Sutmann et al. 1998)1991, Kalkofen et al. 1994, Sutmann et al. 1998)
Oscillations of solar magnetic flux tubes (chromospheric Oscillations of solar magnetic flux tubes (chromospheric network) – 7 min oscillations (Hasan & Kalkofen 1999, network) – 7 min oscillations (Hasan & Kalkofen 1999, 2003, Musielak & Ulmschneider 2002, 2003)2003, Musielak & Ulmschneider 2002, 2003)
Chromospheric oscillations are not cavity modes!
P-modes
Applications of Fleck-Schmitz TheoryApplications of Fleck-Schmitz Theory
Propagation of acoustic and magnetic flux tubePropagation of acoustic and magnetic flux tube
waves in the solar chromospherewaves in the solar chromosphere
Collaborator:Collaborator: Peter Ulmschneider Peter Ulmschneider
Graduate Students:Graduate Students: Gerhard Sutmann, Beverly Stark, Gerhard Sutmann, Beverly Stark, Ping Huang, Towfiq Ping Huang, Towfiq
Ahmed, Shilpa Ahmed, Shilpa Subramaniam Subramaniam and Swati Routhand Swati Routh
22
2
22
2
2
2 ][ˆ pz
ct
pL DTT
22AS
AST
cc
ccc
2
0 0
pp
B
2
2 2
9 1 1
16 2ST
DA
cc
H c
Excitation of Oscillations by Tube Waves I
The wave operator for longitudinal tube waves is
with
and the cutoff frequency (Defouw 1976)
,
2 22 2
1 12 2ˆ [ ]K K KL v c v
t z
1/ 41 0xv v 0
04 ( )K
e
Bc
4K
K
c
H
Excitation of Oscillations by Tube Waves II
The wave operator for transverse tube waves is
with
,
and the cutoff frequency (Spruit 1982)
Initial Value Problems
0ˆ1 vLK 0ˆ
2 pLT and
0lim 1
0
t
vt
tVztvz
010
,lim
IC: 0,lim 10
ztvt
and
BC: and 0,lim 1
ztvz
Laplace transforms and inverse Laplace transforms
Solar Flux Tube Oscillations
Longitudinal tube waves Transverse tube waves
Theoretical Predictions Theoretical Predictions
Solar Chromosphere:Solar Chromosphere:
170 – 190 s (non-magnetic regions)170 – 190 s (non-magnetic regions)
150 – 230 s (magnetic regions150 – 230 s (magnetic regions
Maximum amplitudes are 0.3 km/s
Solar Atmospheric OscillationsSolar Atmospheric Oscillations
Solar Chromosphere: 100 – 250 sSolar Chromosphere: 100 – 250 s
Solar Transition Region: 200 – 400 sSolar Transition Region: 200 – 400 s
Solar Corona: 2 – 600 sSolar Corona: 2 – 600 s
TRACE and SOHOTRACE and SOHO
Lamb’s Original Approach (1908)
Acoustic wave propagation in a stratified and isothermal medium isdescribed by the following wave equation
02
22
2
2
z
u
H
c
z
uc
t
u SS
With
2S
S
c
H
2/101 uu , one obtains
012
21
22
21
2
uz
uc
t
uSS
whereis the acoustic cutofffrequency
Klein-Gordon equation
A New Method to Determine CutoffsA New Method to Determine Cutoffs
0,,,ˆ2
2
2
22
2
2
iss
ss dz
cd
zdz
dc
zc
tL
scdzd eii ~
i = 1, 2, 3
~)~()~(0
dcc ss
0)(,,ˆ 22
2
2
2
iia tL
General form of acoustic wave equation in a medium with gradients:
Transformations:
and with
give
Using the oscillation theorem and Euler’s equation allow finding the acoustic cutoff frequency!
Musielak, Musielak & Mobashi Phys. Rev. (2006)
The Oscillation Theorem
0)(2
2
xd
dConsider
with periodic solutions
0)(2
2
xd
dAnotherequation
)()( xx If for all x
then the solutions of the second equationare also periodic
Euler’s Equation and Its Turning Point
04 22
2
EC
d
d
1EC
1EC
1EC
Periodic solutions
Turning point
Evanescent solutions
Applications of the MethodApplications of the Method
Cutoff frequencies for acoustic and magneticCutoff frequencies for acoustic and magnetic
flux tube waves propagating in the solarflux tube waves propagating in the solar
chromospherechromosphere
Collaborator:Collaborator: Reiner Hammer Reiner Hammer
Graduate Students:Graduate Students: Hanna Mobashi, Shilpa Hanna Mobashi, Shilpa
Subramaniam and Swati Routh Subramaniam and Swati Routh
Torsional Tube Waves I
Introducing Rvx and
Isothermal and ‘wide’ magnetic flux tubes
Rby , we have
0)(2
22
2
2
s
xsc
t
xA
0])([ 22
2
s
ysc
st
yA
and
x and y are Hollweg’s variables
Torsional Tube Waves II
Using the method, we obtain
0121
2
21
2
x
c
cx
t
x
A
A
and
0121
2
21
2
y
c
cy
t
y
A
A
where ddcc AA /
Torsional Tube Waves III
Eliminating the first derivatives, we obtain Klein-Gordon equations
0)( 22
22
2
22
2
xx
t
xx
and
where
0)( 22
22
2
22
2
yy
t
yy
A
A
A
Ax c
c
c
c
2
1
4
3)(
2
2 and 2
2
4
1
2
1)(
A
A
A
Ay c
c
c
c
Torsional Tube Waves IV
Making Fourier transforms in time, the Klein-Gordon equations become
0)]([ 222
22
2
xd
xdx
and
0)]([ 222
22
2
yd
ydy
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:
Routh, Musielak and Hammer (2007)
mHsAA ecsc /0)(
where m = 1, 2, 3, 4 and 5
The model basis is located atthe solar temperature minimum
Torsional Tube Waves VI
Since
and
A
A
A
Ax c
c
c
c
2
1
4
3)(
2
2
2
2
4
1
2
1)(
A
A
A
Ay c
c
c
c
For isothermal and thin magnetic flux tubes, we have
constcA , which gives 0 yx
cutoff-free propagation!
Musielak, Routh and Hammer (2007)
Current WorkCurrent Work
Acoustic waves in non-isothermal mediaAcoustic waves in non-isothermal media
Waves in “wide” magnetic flux tubesWaves in “wide” magnetic flux tubes
Waves in “wine-glass” flux tubesWaves in “wine-glass” flux tubes
Waves in inclined magnetic flux tubesWaves in inclined magnetic flux tubes
CONCLUSIONSCONCLUSIONS Lighthill-Stein theory of sound generation was used to calculate Lighthill-Stein theory of sound generation was used to calculate
the solar acoustic wave energy fluxes. The fluxes are sufficient the solar acoustic wave energy fluxes. The fluxes are sufficient to explain radiative losses observed in non-magnetic regions of to explain radiative losses observed in non-magnetic regions of the lower solar chromosphere.the lower solar chromosphere.
A theory of wave generation in solar magnetic flux tubes was A theory of wave generation in solar magnetic flux tubes was developed and used to compute the wave energy fluxes. The developed and used to compute the wave energy fluxes. The obtained fluxes are large enough to account for the enhanced obtained fluxes are large enough to account for the enhanced heating observed in magnetic regions of the solar heating observed in magnetic regions of the solar chromosphere. chromosphere.
Fleck-Schmitz theory was used to predict frequencies and Fleck-Schmitz theory was used to predict frequencies and amplitudes of the solar atmospheric oscillations. The theory can amplitudes of the solar atmospheric oscillations. The theory can account for 3-min oscillations in the lower chromosphere. account for 3-min oscillations in the lower chromosphere.
A method to obtain local cutoff frequencies was developed. The A method to obtain local cutoff frequencies was developed. The method was used to derive the cutoffs for isothermal and “wide” method was used to derive the cutoffs for isothermal and “wide” flux tubes and to show that the propagation of torsional waves flux tubes and to show that the propagation of torsional waves along isothermal and thin magnetic flux tubes is cutoff-free. along isothermal and thin magnetic flux tubes is cutoff-free.
Supported by NSF, NASA and The Alexander von Humboldt Foundation