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