Solitary wave propagation in solar flux tubesRobert Erdélyi and Viktor Fedun Citation: Phys. Plasmas 13, 032902 (2006); doi: 10.1063/1.2176599 View online: http://dx.doi.org/10.1063/1.2176599 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v13/i3 Published by the American Institute of Physics. Related ArticlesEffect of ion temperature on ion-acoustic solitary waves in a plasma with a q-nonextensive electron velocitydistribution Phys. Plasmas 19, 104502 (2012) A study of solitary wave trains generated by injection of a blob into plasmas Phys. Plasmas 19, 102903 (2012) Quantum ring solitons and nonlocal effects in plasma wake field excitations Phys. Plasmas 19, 102106 (2012) Dust-ion-acoustic Gardner solitons in a dusty plasma with bi-Maxwellian electrons Phys. Plasmas 19, 103706 (2012) The collision effect between dust grains and ions to the dust ion acoustic waves in a dusty plasma Phys. Plasmas 19, 103705 (2012) Additional information on Phys. PlasmasJournal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors

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PHYSICS OF PLASMAS 13, 032902 �2006�

Solitary wave propagation in solar flux tubesRobert Erdélyia� and Viktor FedunSolar Physics and Upper-Atmosphere Research Group, Department of Applied Mathematics,University of Sheffield, Hounsfield Road, Hicks Building, Sheffield S3 7RH, United Kingdom

�Received 14 October 2005; accepted 23 January 2006; published online 10 March 2006�

The aim of the present work is to investigate the excitation, time-dependent dynamic evolution, andinteraction of nonlinear propagating �i.e., solitary� waves on vertical cylindrical magnetic flux tubesin compressible solar atmospheric plasma. The axisymmetric flux tube has a field strength of1000 G at its footpoint, which is typical for photospheric regions. Nonlinear waves that develop intosolitary waves are excited by a footpoint driver. The propagation of the nonlinear signal isinvestigated by solving numerically a set of fully nonlinear 2.0D magnetohydrodynamic �MHD�equations in cylindrical coordinates. For the initial conditions, axisymmetric solutions of the lineardispersion relation for wave modes in a magnetic flux tube are applied. In the present case, we focuson the sausage mode only. The dispersion relation is solved numerically for a range of plasmaparameters. The equilibrium state is perturbed by a Gaussian at the flux tube footpoint. Two solitarysolutions are found by solving the full nonlinear MHD equations. First, the nonlinear wavepropagation with external sound speed is investigated. Next, the solitary wave propagating close tothe tube speed, also found in the numerical solution, is studied. In contrast to previous analytical andnumerical works, here no approximations were made to find the solitary solutions. A naturalapplication of the present study may be spicule formation in the low chromosphere. Future possibleimprovements in modeling and the relevance of the photospheric chromospheric transition regioncoupling by spicules is suggested. © 2006 American Institute of Physics.�DOI: 10.1063/1.2176599�

I. INTRODUCTION

Observational data clearly show that the solar atmo-sphere is heavily structured. Recent magnetohydrodynamic�MHD� wave studies in magnetic structures �e.g., flux tubesin the solar atmosphere� have given a boost by TRACE andSOHO observations since the MHD waves became directlyobservable by these satellites �see the reviews in Ref. 1�.Observations show that the MHD waves are mainly linear bythe time they appear at coronal temperatures. On the otherhand, there is little progress made on how these wavespropagate to coronal heights; where do they come from;what is their connection to the transition region �TR� or toeven lower parts of the atmosphere; or is there a connectionat all? Very recent observations of the transition region, inparticular spicules and moss oscillations, detected byTRACE and by SUMER on board SOHO, may bring uscloser to the origin of the running �propagating� waves ofcoronal loops. The correlations on arcsecond scales betweenchromospheric and transition region emission in active re-gions were studied in Ref. 2. The discovery of active regionmoss, i.e, dynamic and bright upper transition region emis-sion at transition region heights above active region �AR�plage, provides a powerful diagnostic tool to probe the struc-ture, dynamics, energetics, and coupling of the magnetizedsolar chromosphere and transition region. In Ref. 2, the in-teraction of the chromosphere with the upper TR, by study-ing correlations �or the lack thereof� between emission atvarying temperatures, was carried out in great detail: from

a�

Electronic address: Robertus@sheffield.ac.uk

1070-664X/2006/13�3�/032902/9/$23.00 13, 03290

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the low chromosphere �Ca II K-line�, to the middle and up-per chromosphere �H��, to the low transition region �C IV41550 Å at 0.1 MK�, and the upper transition region �Fe IX/X171 Å at 1 MK and Fe XII 195 Å at 1.5 MK�. The highcadence �24–42 s� data sets obtained with the SwedishVacuum Solar Telescope �SVST, La Palma� and TRACE al-lowed us to find a relation between upper transition regionoscillations and low-lying photospheric oscillations. The cor-relation analysis gave some partial answers to the question ofhow the heating mechanisms of the chromosphere are relatedand whether the spatial and temporal variability of moss �andspicules� can be used as diagnostics for coronal heating.Next, in Ref. 3, the intensity oscillations in the upper TRabove AR plage were analyzed. A possible role of a photo-spheric driver in the appearance of moss �and spicule� oscil-lations was suggested. Wavelet analysis of the observations�by TRACE� verifies strong ��5–15% � intensity oscilla-tions in the upper TR footpoints of hot coronal loops. Arange of periods from 200 to 600 s, typically persisting forfour to seven cycles, was found. A preliminary comparisonof photospheric vertical velocities �using the Michelson Dop-pler Imager on board SOHO� revealed that some upper TRoscillations show a correlation with solar global acoustic pmodes in the photosphere. In addition, the majority of theupper TR oscillations are directly associated with upperchromospheric oscillations observed in H�, i.e., periodicflows in spicular structures. The presence of such strong os-cillations at low heights �of order 3000 km� provides an idealopportunity to study the propagation of oscillations from

photosphere and chromosphere into the TR and corona �see,

© 2006 American Institute of Physics2-1

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032902-2 R. Erdélyi and V. Fedun Phys. Plasmas 13, 032902 �2006�

for example, Ref. 4�. It can also help us to understand themagnetic connectivity in the chromosphere and TR, and shedlight on the source of chromospheric mass flows such asspicules. Especially this latter aspect gave us the motivationto study how a signal �linear or weakly nonlinear� excited�periodically or solitary� at the photospheric level would bemanifested higher up in a stratified magnetic solar atmo-sphere �see also Ref. 5�.

In the present work, we report on some initial resultsobtained by studying the propagation of linear and nonlinearwaves �waves of the solitary type� that are excited by a foot-point driver. The propagation of the signal is investigated bysolving numerically a set of fully nonlinear 2.0D MHD equa-tions. First, we investigate the time-dependent evolution ofthe fast and slow sausage surface waves in a magnetic fluxtube. Next, we compare our results with asymptotic analyti-cal solutions as given in Ref. 6 that were obtained by usingthe method of multiple scale expansion. The nonlinear stageof the present numerical simulations is compared with simu-lations carried out in Ref. 7, where only the approximateLeibovich-Roberts �LR� evolution equation

�f

�t+ cT

�f

�z+ �f

�f

�z+ �

�3

�z3�−�

+� f�s,t���2 + �z − s�2�1/2ds = 0

was solved numerically as opposed to the current full MHDsolution. Here f denotes the velocity perturbation along theflux tube. The parameters � ,�, etc. are given in Ref. 8,where this equation was first analytically derived. We alsoshow that a solution similar to the asymptotic solution of theLR equation found in Ref. 6 in a more general case indeedcan be found in the framework of a full MHD description.

II. THE BASIC EQUATIONS AND ASSUMPTIONS

The analysis of the linear or nonlinear excitation andwave propagation in a magnetic flux tube is of fundamentalimportance. Pioneering analytical investigations of the linearproblem in slab and tube geometry were carried out by, e.g.,Refs. 9–14. The study of nonlinear waves in flux tubes wasinvestigated by, e.g., Ref. 8 and 15–20 and many others. Themain issue is not just to derive the governing equation forweakly nonlinear perturbations, which has been more or lesssuccessful, but to find solutions to this governing equation.The latter, one may say, was less successful unless approxi-mation theories �long wavelength or thin flux tube; methodof multiple scale expansion� were used. Alternatively, theanalytically derived approximation governing equations weresolved by numerical schemes. However, to the best of ourknowledge, no attempt was made to solve, even numerically,the full nonlinear MHD equations in the nonlinear regime fora vertical homogeneous flux tube.

In the present work we consider, numerically, the wavepropagation in a cylindrical magnetic flux tube in a nonstrati-fied solar atmosphere. The set of the full MHD equationsreads as follows:

�V+ �V · ��V = −

�p+

1�� � B� � B , �1�

�t � ��

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��

�t+ � · ��V� = 0, �2�

�B

�t= � � �V � B� , �3�

� · B = 0, �4�

p = p0� �

�0��

, �5�

where � is the density, p the pressure, V the velocity, Bmagnetic induction, and ��=5/3� is the adiabatic index.These equations are solved in cylindrical coordinate systemr, , z. We denote the tube radius by r0, and all dependentvariables inside the tube have no index, while outside thetube they are denoted with index e. We are interested inwaves that are much longer than the lateral dimensions of themagnetic field. The coordinate system describing the equilib-rium model is shown in Fig. 1. Because we assume cylindri-cal symmetry, the azimuthal components of the velocity andmagnetic field are set equal to zero. �0 and p0 are the undis-turbed density and the pressure inside the flux tube; �e0 andpe0 are the undisturbed density and the pressure outside theflux tube; there is no undisturbed velocity, i.e., V0=0. Ne-glecting gravity, the static state of equilibrium is defined by

p0 + B02/2� = pe0, �6�

where B0= �0,0 ,B0� is the undisturbed magnetic field insidethe flux tube. Let us now consider small velocity perturba-tions V= �u�r ,0 ,z� ,0 ,w�r ,0 ,z��, pressure p, density �, andmagnetic field b= �br�r ,0 ,z� ,0 ,bz�r ,0 ,z�� inside and outsidethe magnetic tube. The dynamics of wave modes may be

FIG. 1. �Color online� The geometry of the problem.

described by the well-known equation as given in Ref. 21,

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032902-3 Solitary wave propagation in solar flux tubes Phys. Plasmas 13, 032902 �2006�

�2

�t2 �2

�t2 − �VA2 + C0

2��2 + VA2C0

2 �2

�z2�2 = 0, �7�

where =� ·u. Assuming that =R�r�ei��t−n−kz� and substi-tution into �7� yields

1

r

�

�r�r

�R

�r� + �k0

2 −n2

r2 �R = 0, �8�

as obtained in Ref. 14, where

k02 =

��2 − k2VA2���2 − k2C0

2��VA

2 + C02���2 − k2CT

2�. �9�

Here VA=B0 /���0 is the Alfvén speed, C0= ��p0 /�0�1/2 isthe sound speed inside the tube, and CT=C0VA /��C0

2+VA2�

denotes the tube speed. The state outside of the tube is de-scribed by a similar equation �all dependent variables withindex e�. In the present work, we consider the outside regionto be nonmagnetic. Therefore,

ke2 =

�2 − k2Ce2

Ce2 , �10�

where Ce= ��p0e /�0e�1/2 is the sound speed outside the tube.Equation �8� corresponds to Bessel’s equation with variablek0r. The general solution of �8� has the form, providedk0

2 ,ke2�0,

R�r� = An0Jn�k0r� + Bn0Yn�k0r� , r r0,

AneJn�ker� + BneYn�ker� , r � r0, �11�

where Jn and Yn are the Bessel functions of the first andsecond kind, respectively. These solutions correspond to spa-tially oscillating waves. However, for k0

2=−m02 0 and

ke2=−me

2 0 we obtain solutions describing surface waves,

R�r� = Cn0In�m0r� + Dn0Kn�m0r� , r r0,

CneIn�mer� + DneKn�mer� , r � r0, �12�

where In and Kn are the modified Bessel functions of the firstand second kind. The amplitude of the perturbations at theorigin and at infinity �r→�� must be finite, so the constantsDn0, Bn0, and Cne are equal to zero. By substituting thesesolutions back into Eqs. �1�–�5� for surface waves only, weobtain, for the perturbations inside the flux tube �see, e.g.,Ref. 13�,

� = iCn0�01

�In�m0r� ,

u = Cn0�2 − k2C0

2

m02�2

d

drIn�m0r� ,

w = − Cn0C0

2

�2 ikIn�m0r� ,

�13�

br =k

�uB0,

bz = iCn0�2 − k2C0

2

3 B0In�m0r� ,

�

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p = iCn0�0C0

2

�In�m0r� .

Similar expressions hold for the external region,

�e = iDne�0e1

�Kn�mer� ,

ue = Dne

�2 − k2Ce2

m02�2

d

drKn�mer� ,

we = − Dne

Ce2

�2 ikKn�mer� ,

�14�bre = 0,

bze = 0,

pe = iDne

�eCe2

�Kn�mer� .

The conditions for matching the inside and outside solutionsat r=r0 are

ue�r0� = u�r0� , �15�

pe = p +1

�B0bz. �16�

Eigenfunctions of the surface mode, in a thin flux tube, areshown in Fig. 2. The amplitude of the radial velocity insidethe tube increases linearly, while outside the tube it exponen-tially decreases �Fig. 2�a��. The full pressure is constantacross the tube �see Fig. 2�c��. The longitudinal velocity atthe boundary of the tube has discontinuity �Fig. 2�d��. Elimi-nating the velocity amplitudes and applying the boundaryconditions yields the general dispersion equations; see, e.g.,Refs. 9–11, 13, and 14. In the case of body waves that are

FIG. 2. The eigenfunctions of radial and longitudinal components of veloci-ties �a, d� and magnetic field �b, e�, pressure �c�, density �f� for �kr0=0.3�and Vph=5183 m/s. The amplitude of the radial component u=0.2.

spatially oscillating inside the waveguide but evanescent out-

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032902-4 R. Erdélyi and V. Fedun Phys. Plasmas 13, 032902 �2006�

side �m02 0,me

2�0�, the cylindrically symmetric mode �i.e.,n=0, sausage mode� is the solution of

�2�ek0J1�r0k0�J0�r0k0�

= �0�VA2k2 − �2�me

K1�r0me�K0�r0me�

. �17�

In the case of surface waves �m02�0,me

2�0� the disper-sion relation for the cylindrically symmetric mode is as fol-lows:

�2�em0I1�r0m0�I0�r0m0�

= �0�VA2k2 − �2�me

K1�r0me�K0�r0me�

. �18�

Numerical solutions of the dispersion relations for sur-face and body waves are shown in Figs. 3 and 4 �see alsoRef. 14�. The numerical procedure is to select a value for kr0,and then determine those values of the phase speedVph=� /k for which the dispersion relation is satisfied. In thecase of body waves, of course, we find many solutions be-cause Jn is oscillatory. For the numerical calculations, weapplied typical photospheric conditions �see Table I�. Thetypical diameter of an intense flux tube is �100–300 km.

In the thin-flux-tube approximation �kr0�1�, Eq. �18�has the approximate solution for the slow surface sausagemode as given in Ref. 6,

FIG. 3. �Color online� The phase speed �km/s� of modes under photosphericconditions VA�Ce�C0�CT. Many slow body waves are shown.

FIG. 4. �Color online� Solutions of the dispersion relations for the case

Ce�VA�C0�CT.

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� = CTk + 2�k3�ln� k

2+ 0.577� + O�k5 ln k � , �19�

where

� =�e0CT

5

8�0VA4 r0

2, �2 = �1 −CT

2

Ce2�r0

2.

Figure 5 shows the phase speed Vph as a function of thedimensionless wave number �kr0�. We see that for kr0�1,the exact numerical solution of �18� �solid line� coincideswith the approximate solution for the long-wavelength limit�19� �dot-dashed line�. The dashed line indicates the tubespeed.

The boundary of the flux tube can be found by using theequation valid at r=r0+��t ,z� �see, e.g., Ref. 6�

u =��

�t+ w

��

�z. �20�

III. NUMERICAL CALCULATIONS

To solve numerically the full MHD problem of the time-dependent evolution of nonlinear surface sausage waves onvertical cylindrical magnetic flux tubes, the Versatile Advec-tion Code �VAC� is applied �see, e.g., Ref. 22�. All calcula-tions were carried out by using the combinations of the totalvariations diminishing �TVD� and the TVD Lax-Friedrich�TVDLF� methods. We apply the free boundary conditions atthe domain boundaries in order to achieve reasonable reduc-

TABLE I. The typical photospheric conditions.

B0

�G�VA

�m/s�C0

�m/s�Ce

�m/s�

tube which is coolerthan its surroundings

1000 9�103 �6.4�103 �11�103

intense cool tube 1000 9�103 4.5�103 �7.8�103

FIG. 5. The phase speed �m/s� of a slow surface mode for a tube that iscooler than its surroundings. The exactly �solid line� and approximately

�dot-dashed line� solutions.

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032902-5 Solitary wave propagation in solar flux tubes Phys. Plasmas 13, 032902 �2006�

tion of reflection. For the numerical calculations, the set ofthe full MHD equations, that are actually solved, reads asfollows:

��V

�t+ � · �V�V − BB� + �ptot = − �� · B�B , �21�

��

�t+ � · ��V� = 0, �22�

�e

�t+ � · �Ve − BB · V + Vptot� = − ��� · B�B · V , �23�

�B

�t+ � · �VB − BV� = �� · B�V , �24�

p = �� − 1��e − �V2/2 − B2/2� ,

ptot = p + B2/2, �25�

with conservative variable density �, momentum density �V,total energy density e, and magnetic field B. The magneticfield is rescaled as B=B /��. The terms that are proportionalto � ·B are necessary in order to eliminate numerical prob-lems related to the divergence of the magnetic field. Formore details on VAC, see, e.g., Ref. 22.

At the boundary of the flux tube, the full pressure bal-ance �17� should be satisfied after starting the calculations.Numerically we studied two distinct cases: �i� linear and �ii�nonlinear stage of wave propagation along the tube. For thelinear MHD wave simulations, �i� we use a cylindricallysymmetric domain �0,80r0� �400 grid points� in the z direc-tion and �0,2r0� �140 grid points� in the r direction. Initially,the vertical magnetic field is perturbed along the tube, seeFig. 6.

After starting the linear calculations, the boundary pro-file monotonically increases toward a sin profile �see Figs. 7and 8� for various snapshots. These profiles are symmetricabout the z axis. Figure 8 shows the vertical perturbations forall variables. Observe that the full pressure is constant across

FIG. 6. �Color online� The initial magnetic fields profile �0.2B0�.

the whole domain. It corresponds to the condition of pressure

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balance �16� achieved properly in the numerical experiment.Next, in Fig. 9 we plot a snapshot of the radial velocity

at the same time when Fig. 8 is shown. The perturbationprofile in the r direction has the constant shape which corre-sponds perfectly to the analytical solutions �13�, �14�, andFig. 2.

In what follows, we plot the numerical solutions of thefull set of the MHD equations for the nonlinear case �ii�. Forthe latter, we employ the symmetric domain �0,320r0� �1600grid points� in the z direction and �0,2r0� �150 grid points� inthe r direction. Initially, the magnetic field is perturbed at thefootpoint. The perturbation has a Gaussian spatial distribu-tion,

bz = B0�1 + CI0�m0r�I0�m0r0�

�e−�zk − 16r0k�2, �26�

where C=0.15.After the initial perturbation, the evolution of any vari-

ables with time can be observed throughout the whole com-putational domain. As an example, we present results of thesimulations for the u and w components of the velocity. Wefound actually two distinct solutions already predicted in the

FIG. 7. The time-dependent evolution �in s� of the flux tube boundary forthe initial perturbations of magnetic fields 0.2B0.

FIG. 8. �Color online� 2D snapshots of perturbations for case �i�.

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032902-6 R. Erdélyi and V. Fedun Phys. Plasmas 13, 032902 �2006�

asymptotic analytical studies: �a� a fast nonlinear sausagesurface wave, which propagates with a velocity approxi-mately equal to Ce, and �b� a slow nonlinear sausage surfacewave, which propagates with CT.

A. Nonlinear solution close to Ce

The full time necessary for the formation and propaga-tion of a fully developed fast nonlinear sausage surface waveis, formally, divided into four stages. First, linear propaga-tions are shown in Fig. 10. During this period of time, theinitial perturbations propagate with small �linear� amplitudes.However, this period of time is rather short. The second stageis where dispersion dominates. At this stage, the major role

FIG. 9. �Color online� 2D snapshot of the radial velocity perturbation u forthe linear case �i�. The velocity is measured in m/s. x corresponds to theradial direction �in meters�, and z is the direction along the tube �in meters�.

FIG. 10. The time-dependent evolution of the initial signal. The vertical line

indicates the footpoint.

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on the propagation of the initial signal is the dispersive ef-fect. This solution is nonlinear in the sense that the amplitudeis increasing with the propagation speed. This behavior istypical for cnoidal waves; see, e.g., Ref. 23.

The third stage is nonlinear �as shown in Fig. 11�. Thewave amplitude of the oscillatory solution grows and steep-ens further, and is balanced by the tendency for dispersiveeffects to spread the wave. The dispersion prevents waveoverturning and shock development. Finally, the fourth stageis solitary �Fig. 12�. At this stage, we see the formation of

FIG. 11. The time-dependent evolution of the initial signal at the third stage.The vertical line denotes the footpoint.

FIG. 12. The time-dependent evolution of the initial signal at the last �the

fourth� stage. Again, the vertical line here indicates the footpoint region.

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032902-7 Solitary wave propagation in solar flux tubes Phys. Plasmas 13, 032902 �2006�

solitary waves �nonlinear waves of the solitary type� whichare propagating without any permanent change with constantspeed of propagation ��Ce�. This behavior is typical for soli-tary waves. Analytically the propagation of the nonlinear fastsausage surface waves in a magnetic slab embedded in amagnetic-free environment was considered by Ref. 24 in thelimit of small amplitude.

In Fig. 13, we plot a snapshot of the radial velocity at thesame time as that of Fig. 12. The perturbation profile in the rdirection inside the tube is linear, as expected, and corre-sponds approximately to the Bessel function of the first kind.Outside the tube, the profile is proportional to the Besselfunction of the second kind �this is a typical for behaviorsurface waves�. In order to establish whether this wave isindeed a solitary wave, we have to analyze their interactionwith each other. In Fig. 14, the temporal evolution of solitarywave interaction for type �a� is displayed. Two consecutivewaves are launched. Since their initial amplitudes are differ-ent, they will catch up with each other. After their collision,the profiles show only very small deviations from the initialprofiles. In the case presented here, the amplitude of thepropagating signal is decreased by ��5–10% �. This effectcan be explained by radiation in the form of small oscilla-tions on the tails of signals after the interaction �see, e.g.,

FIG. 13. �Color online� 2D snapshot of the radial velocity perturbation u forthe nonlinear case �ii�. The units are the same as in Fig. 9.

FIG. 14. The nonlinear wave interaction of waves traveling with the exter-

nal sound speed.

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Ref. 23�. Therefore, we may conclude that these waves arenot solitary waves in the very strict sense of this word. How-ever, these waves can be labeled as nonlinear waves of thesolitary type. A reasonable check for this statement would beto plot the width, d, of the wave as a function of its ampli-tude, A. In Fig. 15, the solid line represents the dependenceof solitary wave width d on amplitude of wave A. We seethat the numerical solution for the nonlinear wave whichpropagates with Ce speed is close to the Korteweg–de Vriessolution.

B. Nonlinear solution close to CT

Next, we study slow nonlinear sausage surface waves.After a time of about t=180s, the initial Gaussian profilechanges. Sharp peaks develop in the front that cause profileasymmetry. These large gradients develop in the front as aresult of the wave breaking effect. The amplitude of the pro-file is monotonically increasing �Fig. 16�.

Perhaps the decay of this nonlinear wave will proceeduntil the profile reaches the critical amplitude. The criticalamplitude for the numerical solution of the LR equation wasfound by Ref. 7 in an asymptotic way. The numerical solu-tion found in the current case, in the form of a slow solitarywave, shows very similar properties to the analytical solution

FIG. 15. The dependence of solitary wave width d on amplitude A for thenumerical solution close to Ce �the solid line�. The dashed line �KdV� cor-responds to a width-amplitude dependence of the Korteweg–de Vries equa-tion, the chain-dotted line �BO� to the Benjamin-Ono equation, and thedotted line �MR� to the Molotovshchikov and Ruderman �1987� approxi-mate result.

of the LR equation found by Ref. 6 in the general form,

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032902-8 R. Erdélyi and V. Fedun Phys. Plasmas 13, 032902 �2006�

�f

�t+ cT

�f

�z+ bf

�f

�z+

�

�

�3

alz3�−�

+�

F��−1�z − z���f�z��dz�

= 0,

where all parameters are explained in details in Ref. 6.

IV. CONCLUSION

In the present paper, we study the linear and nonlinearpropagation of an initial Gaussian perturbation in a straightnonstratified magnetic flux tube. The solutions of the fullMHD equations are obtained numerically in the form of twosets of nonlinear waves. First, we found nonlinear surfacewaves �of solitary type� that develop and propagate in a fluxtube. These waves travel with the speed which approxi-mately equals the sound speed outside the flux tube �fastsausage mode�. Next, we also found a wave that propagateswith the tube speed �slow sausage mode�. The behavior ofthis second solution to the full nonlinear MHD equations issimilar �at their early stage of formation� to those shown byRef. 7. Interactions of two solitary waves with externalsound speed were also investigated. We found that the sig-nals kept their identity after the interaction, however theyhave shown no phase shifts during their interaction �Fig. 17�.

FIG. 16. Time evolution of an initial Gaussian profile of the longitudinalvelocity perturbation w velocity. Each panel corresponds to different snap-shots. Observe also that two nonlinear waves propagate with different ve-locities; the solitary wave with external sound speed Ce preceding the non-linear signal propagating with tube speed CT.

This feature is typical for, e.g., Benjamin-Ono-type solitary

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waves that were analytically found for slab geometry usingthe method of matched asymptotic approximations. Thewidth-amplitude dependence of these waves, however,shows a resemblance to those of KdV type solitary waves.This behavior is also speculated in Ref. 12. The interactionof the nonlinear waves with tube speed was more complex.Although the initial shapes were preserved after their colli-sion, we also found minor amplitude decreases. It is veryhard to estimate whether this energy loss is due to a nonlin-ear phase or due to numerical viscosity. Further detailed in-vestigations of the propagation and interaction of these typesof solitary waves are necessary in order to evaluate the slownonlinear sausage surface wave formation.

ACKNOWLEDGMENTS

The authors thank Professor M. Ruderman for a numberof useful discussions. R.E. acknowledges M. Kéray for pa-tient encouragement.

The authors are also grateful to NSF, Hungary �OTKA,Ref. No. TO43741� and to the University of Sheffield for thefinancial support they received.

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FIG. 17. �Color online� Solitary wave trajectories on a space-time diagram.The diagram shows no phase shifts of solitary waves with external soundspeed after the interactions �which occurred approximately at t=1050 s�.Also, observe the propagation of the fast nonlinear sausage surface waves�stage three�.

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