reviewarticle modeling kelvin–helmholtz instability in soft x-ray...

Review ArticleModeling KelvinndashHelmholtz Instability in Soft X-Ray Solar Jets

Ivan Zhelyazkov1 Ramesh Chandra2 and Abhishek K Srivastava3

1Faculty of Physics Sofia University 1164 Sofia Bulgaria2Department of Physics Kumaun University Nainital 263001 India3Department of Physics Indian Institute of Technology Banaras Hindu University Varanasi 221005 India

Correspondence should be addressed to Ivan Zhelyazkov izhphysuni-sofiabg

Received 8 August 2016 Revised 16 October 2016 Accepted 1 December 2016 Published 8 February 2017

Academic Editor Valery Nakariakov

Copyright copy 2017 Ivan Zhelyazkov et alThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Development of KelvinndashHelmholtz (KH) instability in solar coronal jets can trigger the wave turbulence considered as one of themain mechanisms of coronal heating In this review we have investigated the propagation of normal MHD modes running onthree X-ray jets modeling them as untwisted and slightly twisted moving cylindrical flux tubes The basic physical parameters ofthe jets are temperatures in the range of 52ndash82MK particle number densities of the order of 109 cmminus3 and speeds of 385 437and 532 km sminus1 respectively For small density contrast between the environment and a given jet as well as at ambient coronaltemperature of 20MK and magnetic field around 7G we have obtained that the kink (119898 = 1) mode propagating on movinguntwisted flux tubes can become unstable in the first and second jets at flow speeds of cong348 and 429 km sminus1 respectively The KHinstability onset in the third jet requires a speed of cong826 km sminus1 higher than the observed one The same mode propagating inweakly twisted flux tubes becomes unstable at flow speeds of cong361 km sminus1 for the first and of 443 km sminus1 for the second jet Exceptthe kink mode the twisted moving flux tube supports the propagation of higher (119898 gt 1) MHDmodes that can become unstable ataccessible jetsrsquo speeds

1 Introduction

Jets are considered to be ubiquitous confined plasma ejecta inthe solar atmosphere They have been extensively observedin the solar atmosphere in various wavebands such as H120572[1 2] Ca ii H [3ndash5] EUV [6] and soft X-ray [7] in order tounderstand their multitemperature characteristics X-ray jetswere discovered by the Soft X-Ray Telescope (SXT) on boardYohkoh [8] as transient X-ray energy release and enhance-mentwith apparent collimated ballisticmotions of the plasmaassociated with the flares in X-ray bright points emergingflux regions or active regions (for details see Shibata et al[7]) As it has been pointed out by Shimojo et al [9] jets fromX-ray bright points in active regions most likely appear at thewestern edge of preceding sunspots and exhibit a recurrentplasma propulsion in the solar atmosphere These X-ray jetsare confined plasma dynamics with typical morphologicalproperties for example (1ndash40)times 104 km length and thewidthof 5 times 103ndash105 km Such jets possess apparent velocities of10ndash1000 km sminus1 and lifetime of 100ndash16000 s [9] The electron

densities of the X-ray jets are of the orders of (07ndash4) times109 cmminus3 Their temperatures lie in the range of 3ndash8MK withan average temperature of 56MK [10]

In terms of spatial location jets can be classified as polarjets [11] and active region jets [12] A study of polar jetparameters based on Hinode XRT observations was carriedout by Savcheva et al [13] who showed that jets prefer-ably occur inside the polar coronal holes Culhane et al[11] have found from Hinodersquos Extreme-ultraviolet ImagingSpectrometer (EIS) 4010158401015840 slot observations of a polar coronalhole that jet temperature ranges from 04 to 50MK Thejet velocities had typical values that are mostly less than theSunrsquos escape velocity (618 km sminus1) therefore in consequencemost of the jets fall back in the lower solar atmosphere aftertheir triggering Using the XTR on Hinode Cirtain et al[14] conclude that X-ray jets in polar coronal holes have twodistinct velocities one near the Alfven speed (sim800 km sminus1)and another near the sound speed (200 km sminus1) Moreoverthey were the first to give an evidence for the propagationof Alfven waves in solar X-ray jets Kim et al [15] presented

HindawiAdvances in AstronomyVolume 2017 Article ID 2626495 18 pageshttpsdoiorg10115520172626495

2 Advances in Astronomy

the morphological and kinematic characteristics of threesmall-scale X-rayEUV jets simultaneously observed by theHinode XRT and the Transition Region and Coronal Explorer(TRACE) While observing the coronal jets for two differentwavelength bands they obtain matching characteristics fortheir projected speed (90ndash310 km sminus1) lifetime (100ndash2000 s)and size (11ndash5 times 105 km) Chifor at al [12] have reported2007 January 1516 observations of a recurring jet situatedon the west side of NOAA active region 10938 A strongblue-shifted component and an indication of a weak red-shifted component at the base of the jet were observed around119879e = 16MK in these jets The upflow velocities wereobserved exceeding up to 150 km sminus1 These jets were seenover a range of temperatures between 025 and 25MK whiletheir estimated electron densities lie above 1011 cmminus3 for thehigh-velocity upflow components Yang et al [16] presentedsimultaneous observations of three recurring jets in EUV andsoft X-ray (SXR) which occurred in an active region on 2007June 5 On comparing their morphological and kinematicproperties the authors have found that EUVand SXR jets hadsimilar onset locations directions size and terminal veloci-ties The three observed jets were having maximum Dopplervelocities ranging from 25 to 121 km sminus1 in the Fe xii 120582195 lineand from 115 to 232 km sminus1 in the He ii 120582256 line Extensivemulti-instrument observations obtained simultaneously withthe SUMER spectrometer on board the Solar andHeliosphericObservatory (SoHO) with EIS and XRT on board Hinodeand with the Extreme-ultraviolet imagers (EUVI) of the SunndashEarth Connection Coronal and Heliospheric Investigation(SECCHI) instrument suite on board the Ahead and BehindSTEREO spacecrafts were performed by Madjarska [17] Thedynamic process of X-ray jet formation and evolution hasbeen derived in great detail In particular for the first timethere was found spectroscopically a temperature of 12MK(Fe xxiii 26376 A) and density of 4 times 1010 cmminus3 in the quietSun The author has clearly identified two types of upflowsin which the first one was the collimated upflow along theopen magnetic fields and the second was the formation ofa plasma cloud from the expelled bright point small-scaleloops Chandrashekhar et al [18] studied the dynamics oftwo jets seen in a polar coronal hole with a combination ofEIS and XRTHinode data They found no evidence of helicalmotions in these events but detected a significant shift of thejet position in a direction normal to the jet axis with a driftvelocity of about 27 and 7 km sminus1 respectively

The launch of the Solar Dynamics Observatory (SDO)[19] with the Atmospheric Imaging Assembly (AIA) [20 21]opens a new page in observing the solar jets Moschou etal [22] have reported high cadence observations of solarcoronal jets observed in the extreme-ultraviolet (EUV) 304 Ausing Atmospheric Imaging Assembly (AIA) instrument onboard SDO They registered in fact coronal hole jets withspeeds of 94 to 760 km sminus1 and lifetimes of the order of severaltens of minutes A detailed description of the dynamicalbehavior of a jet in an on-disk coronal hole observed withAIASDOwas presented by Chandrashekhar et al [23]Theirstudy reveals new evidence of plasma flows prior to thejetrsquos initiation along the small-scale loops at the base of the

jet The authors have also found further evidence that flowsalong the jet consisting of multiple quasi-periodic small-scale plasma ejections In addition spectroscopic analysisestimates temperature as Log 589 plusmn 008K and electron den-sities as Log 875 plusmn 005 cmminus3 in the observed jet Measuredproperties of the registered transverse wave have providedevidence that strong damping of the wave occurred as itpropagates along the jet with speeds of sim110 km sminus1 Usingthe magnetoseismological inversion observed plasma andwave parameters the jetrsquos magnetic field is estimated as 119861= 121 plusmn 02G Recently Sterling et al [24] have reportedhigh-resolution X-ray and extreme-ultraviolet observationsof 20 randomly selected X-ray jets that form in coronal holesat the solar polar caps In each jet converse to the widelyaccepted emerging magnetic flux model a miniature versionof the filament eruptions that initiated coronal mass ejectionsdrove the jet producing reconnection process Formation ofa rotating jet during the filament eruption on 2013 April10ndash11 on the base of multiwavelength and multiviewpointobservations with STEREOSECCHIEUVI and SDOAIAwas reported by Filippov et al [25] The confined eruptionof the filament within a null-point topology which is alsoknown as an Eiffel tower magnetic field configuration formsa twisted jet after magnetic reconnection near the null pointThe sign of the helicity in the jet is observed the same as thatof the sign of the helicity in the filament It is noteworthy thatthe untwistingmotion of the reconnectedmagnetic field linesgives rise to the accelerating plasma along the jet

It is well established that themagnetic reconnection at dif-ferent heights in the solar atmosphere plays a key role in trig-gering jet-like events The magnetic reconnection betweenopen and closed fields (standard reconnection scenario) isone of the well-known processes of the jetrsquos occurrence [1526] The jets emerging by this kind of mechanism are knownas standard jets [3 7 27 28] Some other observationaland simulation studies showed that the reconnection at themagnetic null in a fan-spine magnetic topology can alsotrigger jet-like events [29ndash32] Eruptions of small archesfilaments and flux ropes from within this type of magneticfield configurations can be responsible for reconnection andjetsrsquo occurring These types of jets are known as blowoutjets [28 33 34] Moore et al [35] used the full-disk He ii304 A movies from the Atmospheric Imaging Assembly onSDO to study the cool (119879 sim 105 K) component of X-rayjets observed in polar coronal holes by XRT The AIA 304 Amovies revealed that most polar X-ray jets spin as they eruptThe authors examined 54 X-ray jets that were found in polarcoronal holes in XRT movies sporadically taken during thefirst year of continuous operation of AIA (2010 May through2011 April) These 54 jets were big and bright enough in theXRT images to be categorized as a standard jet or as a blowoutjet From the X-ray movies 19 of the 54 jets appeared likestandard jets 32 appeared as blowout jets and three wereambiguous and were not falling in any category Moore et al[36] have studied 14 large-scale solar coronal jets observedin Sunrsquos pole In EUV movies from the SDOAIA each jetwas very similar to most X-ray and EUV jets erupting incoronal holes However each was exceptional in that it went

Advances in Astronomy 3

higher than most of the standard coronal jets They weredetected in the outer corona beyond 22 119877⊙ in images asobserved from the Solar and Heliospheric ObservatoryLargeAngle Spectroscopic Coronagraph (LASCOC2 coronagraph[37])

Schmieder et al [38] proposed a new model between thestandard and blowout models where magnetic reconnectionoccurs in the bald patches around some twisted field linesone of whose foot points is open Pariat et al [39] included themagnetic field inclination and photospheric field distributionand performed another 3D numerical MHD model for thetwo different types of jet events standard and blowout jetsWe note also that stereoscopic studies (multiple points ofview) have been carried out by the EUVISECCHI imagers onboard the twin STEREO spacecraft to estimate the expectedspeed motion andmorphology of polar coronal jets [40] Tosum up the flux emergence [41 42] and flux cancellation [4344] are the two main triggering processes that are known tobe responsible for jetsrsquo occurrence In few observational andprimarily numerical studies the wave-induced reconnectionhas also been suggested as a cause for the onset of jet-likeevents [42 45ndash47]

We consider magnetically structured X-ray solar jets asmoving cylindrical magnetic flux tubes that support theexcitationpropagation of various kind of magnetohydrody-namic (MHD) oscillations and waves While in static solaratmospheric plasma the propagating MHDmodes are stablethe axial motion of the flux tubes engenders a velocity jumpat the tube surface which can trigger a KelvinndashHelmholtz(KH) instability The KH instability arises at the interface oftwo fluid layers that move with different speeds (see egChandrasekhar [48])mdashthen a strong velocity shear arisesnear the interface between these two fluids forming a vortexsheet This vortex sheet becomes unstable to the spiral-like perturbations at small spatial scales [49] In cylindricalgeometry when a magnetic flux tube is axially moving sucha vortex sheet is evolved near tubersquos boundary and it maybecomeunstable against KH instability provided that the tubeaxial velocity exceeds a critical value [50] Further on in thenonlinear stage of KH instability this vortex sheet causes theconversion of the directed flow energy into turbulent energymaking an energy cascade at smaller spatial scales [51]

The KH instability studying in various solar jets overthe past decade arose from the fact that KH vortices wereobserved in solar prominences [52ndash54] in SweetndashParkercurrent sheets [55] in a coronal streamer [56] and in coronalmass ejections [57ndash61] All these observations stimulatedthe modeling of KH instability in moving twisted magneticflux tubes in nonmagnetic environment [62] in magnetictubes of partially ionized plasma [63] in spicules [64ndash66]in photospheric tubes [67] in high-temperature and coolsurges [68 69] in dark mottles [70] at the boundary ofrising coronal mass ejections [60 61 71 72] in rotatingtornado-like magnetized jets [49] and in a chromosphericjet (fast disappearance of rapid red-shifted and blue-shiftedexcursions alongside a larger scale H120572 jet) [73] A moreextensive review on modeling the KH instability in solaratmosphere jets the reader can be seen in Zhelyazkov [74] andreferences therein

The first modeling of KH instability in X-ray jets wascarried out by Vasheghani Farahani et al [75] who exploredtransverse wave propagation along the detected by Cirtainet al [14] coronal hole soft X-ray jets Vasheghani Farahaniet al analyzed analytically in the limit of thin magneticflux tube the dispersion relation of the kink MHD modeand have obtained that this mode is unstable against theKH instability when the critical jet velocity is equal to447VA = 3576 km sminus1 (VA = 800 km sminus1 is the Alfven speedinside the jet) Numerical solving of the same dispersionrelation when considering the jet and its environment ascold magnetized plasmas carried out by Zhelyazkov [7677] yielded a little bit lower critical flow speed for theinstability onset namely 431VA = 3448 km sminus1 The lowestcritical jet speed of 4025VA = 3220 km sminus1 was derived bynumerically solving the wave dispersion relation without anyapproximations that is treating both media as compressibleplasmas But even the latter critical jet speed is still too highfor the KH instability to be detectedobserved in coronal holesoft X-ray jets The reason for obtaining such high criticalspeeds is the circumstance that Vasheghani Farahani et al[75] assumed an electron number density of the order of108 cmminus3 and magnetic field strength of 10G

Here we study the propagation of kink and higher MHDmodes in standard active region soft X-ray jets notablyjets 8 11 and 16 of Shimojo and Shibatarsquos set of sixteenobserved flares and jets [10] and have shown that with oneorder higher electron densities sim109 cmminus3 and moderatemagnetic field sim7G MHD modes in high-speed jets likethese ones can become unstable against the KH instabilityat accessible jets speeds except for jet 16 which requiresa higher flow velocity In the next section we list the basicphysical parameters of jet 11 and the topology of magneticfields inside and outside the moving flux tube modelingjet and also derive the wave dispersion equation for bothuntwisted and twisted tubes The physical parameters ofother two jets (8 and 16) will be provided en route in thenext section The numerical solving MHD wave dispersionrelations and the discussion of the conditions under whichthe KH instability can develop in such moving structuresare presented in Section 3 The last section summarizes themain results obtained in this article and outlooks our futurestudies of KH instability in more complex (rotating) solaratmosphere jets

2 Geometry Magnetic Field Topology andMHD Wave Dispersion Relations

We consider the soft X-ray jet as a straight cylinder of radius119886 and density 120588i embedded in a uniform field environmentwith density 120588e We study the propagation of MHD wavesin two magnetic configurations notably in untwisted andtwisted flux tubes For an untwisted tube magnetic fields inboth media are homogeneous and directed along the 119911-axisof our cylindrical coordinate system (119903 120601 119911) Bi = (0 0 119861i)and Be = (0 0 119861e) respectively (see Figure 1)

Themagnetic field inside the twisted tube is helicoidBi =(0 119861i120601(119903) 119861i119911(119903)) while outside the tube the magnetic field is


Bi

v0

Be

(a)

Bi

v0

Be

(b)

Figure 1 Equilibrium magnetic fields of a soft X-ray solar jet in an untwisted flux tube (a) and in a weakly twisted flux tube (b)

uniform and directed along the tube axisBe = (0 0 119861e) Notethat we assume a magnetic fieldsrsquo equilibrium with uniformtwist for which the magnetic field inside the tube is Bi =(0 119860119903 119861i119911) where119860 and 119861i119911 are constantThe parameter thatcharacterizes the uniform magnetic field twist is the ratio119861i120601(119886)119861i119911 equiv 120576 that is 120576 = 119860119886119861i119911 Our frame of referencefor studying the wave propagation in the jet is attached to thesurrounding magnetoplasmamdashthus k0 = (0 0 V0) representsthe relative jet velocity if there is any flow in the environmentThe jump of the tangential velocity at the tube boundary theninitiates the magnetic KH instability onset when the jumpexceeds a critical value

Before dealing with governing MHD equations it isnecessary to specify what kind of plasma each medium is(the moving tube and its environment) As seen from theEvents List in [10] the electron density in jet 11 is 119899jet equiv119899i = 29 times 109 cmminus3 the temperature is 119879jet equiv 119879i = 55ndash64MK and jet speed (in their notation) is 119881jet = 437 km sminus1Our choice for environment magnetic field electron densityand temperature is 119861e = 67G 119899e = 26 times 109 cmminus3 and119879e = 20MK respectively With 119879i = 55MK the totalpressure balance equation (equality of the sumof thermal andmagnetic pressures in both media) that is

119901i + 1198612i2120583 = 119901e + 1198612e2120583 (1)

where 120583 is the magnetic permeability of vacuum yields thefollowing basic sound and Alfven speeds in the jet and itsenvironment 119888si = 275 km sminus1 and VAi = 111 km sminus1 and 119888se =166 km sminus1 and VAe = 286 km sminus1 respectively Accordinglythe plasma betas in both media are 120573i = 7369 and 120573e =0403mdashthis implies that in principle one can treat the jetas incompressible plasma and its surrounding medium ascool magnetized plasma When studying the MHD wavepropagation in the untwisted magnetic flux tube we shall

use two approaches namely of compressible plasmas in bothmedia and the simplified limit of incompressible and coolplasmasmdasha similarity of dispersion curves patterns obtainedfrom corresponding dispersion relations will eventually jus-tify the usage of the second approach in particular in the caseof twisted tube There are two important input parameters inthe modeling KH instability in moving magnetic flux tubeswhich are the density contrast 120588e120588i equiv 120578 = 0896 andthe ratio of the axial external and internal magnetic fields119861e119861i equiv 119887 = 244 For the twisted tube the second parameterhas the form 119887twist = 119861e119861i119911

In a system of cylindrical coordinates the equilibriumphysical variables (density fluid velocity and pressure) arefunctions of the radial coordinate 119903 only Then their pertur-bations can be Fourier-analyzed putting them proportionalto exp[i(minus120596119905 + 119898120601 + 119896119911119911)] where 120596 is the angular wavefrequency (that in general can be a complex quantity) 119898is the mode number (a positive or negative integer) and 119896119911is the axial wave number We can eliminate all except twoof the perturbations (perturbation 119901tot of the total (thermal+ magnetic) pressure and the radial component 120585119903 of theLagrangian displacement 120585) to get the following governingequations [78]

119863 dd119903 (119903120585119903) = 1198621119903120585119903 minus 1198622119903119901tot119863d119901tot

d119903 = 1198623120585119903 minus 1198621119901tot(2)

The coefficients 119863 1198621 1198622 and 1198623 are functions of theequilibrium variables 1205880 B0 and k0 and of the Doppler-shifted frequencyΩ = 120596minusk sdotk0 and have the following forms

119863 = 1205880 (Ω2 minus 1205962A) 11986241198621 = 21198610120601120583119903 (Ω41198610120601 minus 119898119903 119891B1198624)


1198622 = Ω4 minus (1198962119911 + 11989821199032 )11986241198623 = 1205880119863[Ω2 minus 1205962A + 211986101206011205831205880

dd119903 (1198610120601119903 )]

+ 4Ω4(11986120120601120583119903 )2

minus 12058801198624 4119861201206011205831199032 1205962A

(3)where

1198624 = (1198882s + 1198882A) (Ω2 minus 1205962c) 119891B = 119898119903 1198610120601 + k sdot B01205962A = 1198912B1205831205880 1205962c = 1198882s1198882s + 1198882A1205962A

(4)

Here120596A is the Alfven frequency and120596c is the cusp frequencythe other notation is standard

Eliminating 120585119903 from (2) one obtains the well-knownsecond-order ordinary differential equation [79ndash81]

d2119901totd1199032 + [1198623119903119863 d

d119903 (1199031198631198623 )] d119901totd119903

+ [1198623119903119863 dd119903 (11990311986211198623 ) + 11198632 (11986221198623 minus 11986221)] 119901tot = 0

(5)

By means of the solutions to (5) in both media one can findthe corresponding expressions for 120585119903 and after merging thesesolutions together with those for 119901tot through appropriateboundary conditions at the interface 119903 = 119886 one can derivethe dispersion relation of the normal modes propagating inthe moving magnetic flux tube

21 Dispersion Relation of MHD Modes in an Untwisted FluxTube In an untwisted magnetic flux tube the coefficient1198621 = 0 while 1198623 = 1205880119863(Ω2 minus 1205962A)mdashthen (5) takes the form

d2119901totd1199032 + 1119903 d119901tot

d119903 minus (11989820 + 11989821199032 )119901tot = 0 (6)

where

11989820 = minus(Ω2 minus 11989621199111198882s ) (Ω2 minus 1198962119911V2A)(1198882s + V2A) (Ω2 minus 1205962c) (7)

The cusp frequency 120596c is usually expressed via the so-calledtube speed 119888T notably 120596c = 119896119911119888T where [82]

119888T = 119888sVAradic1198882s + V2A

(8)

The solutions for 119901tot can be written in terms of modifiedBessel functions 119868119898(1198980i119903) inside the jet and 119870119898(1198980e119903) inits surrounding plasma We note that wave attenuationcoefficients 1198980i and 1198980e in both media are calculated

from (7) with replacing the sound and Alfven speeds withthe corresponding values for each medium Recall that inevaluating 1198980e the wave frequency is not Doppler-shiftedmdashit is simply 120596 By expressing the Lagrangian displacements 120585i119903and 120585e119903 in both media via the derivatives of correspondingBessel functions and by applying the boundary conditionsfor continuity of the pressure perturbation 119901tot and 120585119903 acrossthe interface 119903 = 119886 one obtains the dispersion relation ofnormal MHD modes propagating in a flowing compressiblejet surrounded by a static compressible plasma [65 83 84]

120588e120588i (1205962 minus 1198962119911V2Ae)1198980i 1198681015840119898 (1198980i119886)119868119898 (1198980i119886)minus [(120596 minus k sdot v0)2 minus 1198962119911V2Ai]1198980e119870

1015840119898 (1198980e119886)119870119898 (1198980e119886) = 0

(9)

Due to the flowing plasma the wave frequency is Doppler-shifted inside the jetWe recall that for the kinkmode (119898 = 1)one defines the so-called kink speed [82]

119888k = (120588iV2Ai + 120588eV2Ae120588i + 120588e )12 = (1 + 1198612e1198612i1 + 120588e120588i )12 VAi (10)

which as seen is independent of sound speeds and charac-terizes the propagation of transverse perturbations We willshow that notably the kinkmode can becomeunstable againstKH instability

When the jet is considered as incompressible plasma andits environment as a cool one the MHD wave dispersionrelation (9) keeps its form but the two attenuation coefficients1198980ie become much simpler namely

1198980i = 1198961199111198980e = (1198962119911V2Ae minus 1205962)12

VAe (11)

respectively

22 Dispersion Relation of MHD Normal Modes in a TwistedFlux Tube Inside the tube (119903 ⩽ 119886) where 119861i120601 = 119860119903 thequantities 119891B and 120596Ai take the forms

119891B = 119898119860 + 119896119911119861i119911120596Ai = 119898119860 + 119896119911119861i119911radic120583120588i (12)

respectively For incompressible plasma we redefine (withoutthe loss of generality) the coefficients 119863 1198621 1198622 and 1198623 bydividing them by 1198624 to obtain

119863 = 120588 (Ω2 minus 1205962A) 1198621 = minus21198981198611206011205831199032 (119898119903 119861120601 + 119896119911119861119911) 1198622 = minus(11989821199032 + 1198962119911) 1198623 = 1198632 + 1198632119861120601120583 d

d119903 (119861120601119903 ) minus 411986121206011205831199032 1205881205962A

(13)


Radial displacement 120585119903 is expressed through the total pressureperturbation as

120585119903 = 1198631198623d119901totd119903 + 11986211198623119901tot (14)

The solution to this equation obviously depends upon themagnetic field and density profile

With aforementioned coefficients 119863 1198621 1198622 and 1198623evaluated for the jetrsquos medium (5) reduces to the modifiedBessel equation

[ d2

d1199032 + 1119903 dd119903 minus (11989820i + 11989821199032 )]119901tot = 0 (15)

where

11989820i = 1198962119911 [1 minus 411986021205962Ai120583120588i (Ω2 minus 1205962Ai)2] (16)

The solution to (15) bounded at the tube axis is

119901tot (119903 ⩽ 119886) = 120572i119868119898 (1198980i119903) (17)

where 119868119898 is the modified Bessel function of order119898 and 120572i isa constant Lagrangian displacement 120585i119903 by using (14) can bewritten as

120585i119903 = 120572i119903 (Ω2 minus 1205962Ai)1198980i1199031198681015840119898 (1198980i119903)120588i (Ω2 minus 1205962Ai)2 minus 411986021205962Ai120583

minus 2119898119860120596Ai119868119898 (1198980i119903) radic120583120588i120588i (Ω2 minus 1205962Ai)2 minus 411986021205962Ai120583

(18)

where the prime signmeans a differentiation with respect theBessel function argument

For the cool environment with a straight-line magneticfield 119861e119911 = 119861e and homogeneous density 120588e the 1198621minus3 and 119863coefficients take the form

119863 = 120588 (1205962 minus 1205962A) 1198621 = 01198622 = minus[11989821199032 + 1198962119911 (1 minus 12059621205962A)] 1198623 = 1198632

(19)

The total pressure perturbation outside the tube obeys thesame Bessel equation as (15) but 11989820i is replaced by

11989820e = 1198962119911 (1 minus 12059621205962Ae) (20)

which coincides with the attenuation coefficient in the coolenvironment of an untwisted flux tubeThe solution boundedat infinity now is

119901tot (119903 gt 119886) = 120572e119870119898 (1198980e119903) (21)

where 119870119898 is the modified Bessel function of order 119898 and 120572eis a constant

In this case the Lagrangian displacement can be writtenas

120585e119903 = 120572e1199031198980e1199031198701015840119898 (1198980e119903)120588e (1205962 minus 1205962Ae) (22)

and the Alfven frequency is simplified to

120596Ae = 119896119911119861e119911radic120583120588e = 119896119911VAe (23)

Here VAe = 119861eradic120583120588e is the Alfven speed in the surroundingmagnetized plasma

The boundary conditions which merge the solutions ofthe Lagrangian displacement and total pressure perturbationinside and outside the twisted magnetic flux tube have theforms [85]

120585i1199031003816100381610038161003816119903=119886 = 120585e1199031003816100381610038161003816119903=119886 119901tot i minus 1198612i120601120583119886 120585i119903

1003816100381610038161003816100381610038161003816100381610038161003816119903=119886 = 119901tot e1003816100381610038161003816119903=119886 (24)

where total pressure perturbations 119901tot i and 119901tot e are given by(17) and (21) respectively With the help of these boundaryconditions we derive the dispersion relation of the normalMHDmodes propagating along a twisted magnetic flux tubewith axial mass flow k0

(Ω2 minus 1205962Ai) 119865119898 (1198980i119886) minus 2119898119860120596Airadic120583120588i(Ω2 minus 1205962Ai)2 minus 411986021205962Ai120583120588i

= 119875119898 (1198980e119886)(120588e120588i) (1205962 minus 1205962Ae) + 1198602119875119898 (1198980e119886) 120583120588i (25)

where as we alreadymentionedΩ = 120596minusk sdotk0 is the Doppler-shifted wave frequency in the moving medium

119865119898 (1198980i119886) = 1198980i1198861198681015840119898 (1198980i119886)119868119898 (1198980i119886) 119875119898 (1198980e119886) = 1198980e1198861198701015840119898 (1198980e119886)119870119898 (1198980e119886)

(26)

This dispersion equation is similar to the dispersion equationof normal MHDmodes in a twisted flux tube surrounded byincompressible plasma [65]mdashthe only difference is that therein (10) 120581e equiv 1198980e = 1198961199113 Numerical Solutions and Results

Firstly we shall study the dispersion characteristics of thekink (119898 = 1) mode in untwisted moving magnetic flux tube(in two approaches notably (i) considering the jet and itssurrounding plasma as compressible media and (ii) treatingthe jet as an incompressible medium while its environmentis assumed to be cool plasma) and later on explore thesame thing for the kink (119898 = 1) and higher (119898 gt 1)MHDmodes propagating in a twisted moving flux tube Two


dispersion relations (9) and (25) are transcendent equationsin which the wave frequency 120596 is a complex quantityRe(120596) + Im(120596) while the axial wave number 119896119911 is a realvariable The appearance of KH instability (ie Im(120596) gt 0) isdetermined primarily by the jet velocity and in searching for acriticalthreshold value of it wewill gradually change velocitymagnitude V0 from zero to that critical value (and beyond)Our numerical task is to solve the dispersion relation incomplex variables obtaining the real and imaginary parts ofthe wave frequency or as is usually done of the wave phasevelocity Vph = 120596119896119911 as functions of the axial wave number119896119911 at various magnitudes of the velocity shear between thesoft X-ray jet and its environment k0

For numerical solving the wave dispersion relations itis practically to normalize all variables and we do thatnormalizing the wave length 120582 = 2120587119896119911 to the tube radius119886 that implies a dimensionless wave number 119896119911119886 All speedsare normalized with respect to the Alfven speed inside the jetVAi For normalizing the Alfven speed in the ambient coronalplasma VAe we need the density contrast 120578 and the ratio ofthe magnetic fields 119887 = 119861e119861i to find VAeVAi = 119887radic120578 Thenormalization of sound speeds in both media requires thespecification of the reduced plasma betas ie = 1198882sieV2Aie Inthe dimensionless computations the flow speed V0 will bepresented by the Alfven Mach number 119872A = V0VAi31 KelvinndashHelmholtz Instability in Untwisted Flux TubesAmong the various MHD wave spectra that exist in a staticmagnetic field flux tube of compressible plasma surroundedby compressible medium the most interesting for us is thekink-speed wave whose speed for jet 11 according to (10) isequal to

119888k = radic1 + 11988721 + 120578 VAi = 2127 km sminus1

or in dimensionless form 119888kVAi

= 19166(27)

The normal modes propagating in a static homogeneouslymagnetized flux tube can be pure surface waves pseudo-surface (body) waves or leaky waves (see Cally [86]) Thetype of the wave crucially depends on the ordering of thebasic speeds in both media (the flux tube and its surroundingplasma) more specifically of sound and Alfven speeds as wellas corresponding tube speeds In our case the ordering is

VAi lt 119888se lt 119888si lt VAe (28)

which with 119888Ti cong 103 km sminus1 and 119888Te cong 1436 km sminus1 afternormalizing them with respect to the external sound speed119888se yields (in Callyrsquos notation)

119860 = 06687119862 = 16566

119860e = 17229119862T = 062119862Te = 08649

(29)

Since 119860 lt 119862 and 119860 lt 119862Te along with 119860 lt 119862Te lt 119862according to Callyrsquos classification the kink mode in a restflux tube must be pure surface mode type Sminus+ (for detail seeCally [86]) For an Sminus+ type wave 119881 should lie between 119860 and119862 119860 ⩽ 119881 ⩽ 119862 and also 119881 lt 119862Te In our normalizationthe normalized wave velocity should be bracketed between1 and radici = 2478 that is 1 ⩽ 120596(119896119911VAi) ⩽ 2478 Thetypical wave dimensionless wave velocity 120596(119896119911VAi) is thenormalized kink speed 119888kVAi = 19166 which as expectedlies between 1 and 2478 Concerning the relation betweennormalized wave phase velocity and external tube speed inour case we have the opposite inequality that is 119881 gt 119862Tethat in Callyrsquos normalization reads as 12805 gt 08649 andin ours as 19166 gt 12937 The reason for that unexpectedchange in the mutual relations between two velocities is therelatively bigger value of parameter 119887 (=244) that yields alarger magnitude of 119888k than for example in the case when119887 is close to 1mdashthen the inequality 119881 lt 119862Te is satisfied (seeeg Zhelyazkov et al [87])

A reasonable question is how the flow will change thedispersion characteristics of the kink (119898 = 1) modeCalculations show that the flow shifts upwards the kink-speeddispersion curve and splits it into two separate curves [65](A similar duplication is observed for the tube-speed VTidispersion curves too) The evolution of the pair of kink-speed dispersion curves can be seen in Figure 2(a)mdashtheinput parameters for solving the dispersion equation (9) ofthe kink mode (119898 = 1) traveling in a moving flux tubeof compressible plasma surrounded by compressible coronalmedium are 120578 = 0896 119887 = 244 i = 6141 ande = 0336 During calculations the Alfven Mach number119872A was varied from zero (static plasma) to values at whichwe obtained unstable solutions Let us first note that at119872A = 0 we get the kink-speed dispersion curve which at avery small dimensionless wave number 119896119911119886 (=0005) yieldsthe value of the normalized wave phase velocity equal to19168mdashvery close to the previously estimated magnitude of19166 This observation implies that our code for solvingthe wave dispersion relation is correct For small AlfvenMach numbers the pair of kink-speed modes travel withvelocities 119872A ∓ 119888kVAi [65] (in that case their dispersioncurves go practically parallel) At higher 119872A however thebehavior of each curve of the pair 119872A ∓ 119888kVAi turns out tobe completely different As seen from Figure 2(a) for 119872A ⩾3785 both kink curves break and merge forming a familyof semiclosed dispersion curves A further increasing in 119872Aleads to a separation of those curves in opposite directionsThis behavior of the kink (119898 = 1) mode dispersion curvessignals us that we are in a range of the Re(VphVAi)ndash119896119911119886-plane where one can expect the occurrence of KH instabilityA prediction of the Alfven Mach number 119872A at which theinstability will occur can be found from the inequality [88]

|119898|1198722A gt (1 + 1120578) (|119898| 1198872 + 1) (30)

that gives for the kink (119898 = 1) mode 119872A gt 3839 Inour case the KH instability starts at that 119872A for which theleft-hand side semiclosed curve disappearsmdashthis happens at


3943865

3775

3785 38 3825 385

MA = 375

MA = 375

m = 1 120576 = 016

18

2

22

24

26Re

(ph

A

i)

05 1 15 2 25 3 35 40kza

(a)

3893915

MA = 3865

MA = 394

m = 1 120576 = 0

0

01

02

03

04

05

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)

Figure 2 (a) Dispersion curves of stable and unstable kink (119898 = 1) MHD mode propagating in a moving untwisted magnetic flux tube ofcompressible plasma (modeling jet 11) at 120578 = 0896 and 119887 = 244 Unstable dispersion curves located above the middle of the plot have beencalculated for four values of the Alfven Mach number 119872A = 3865 389 3915 and 394 (b) The normalized growth rates of the unstablemode for the same values of 119872A Red curves in both plots correspond to the onset of KH instability

406

3985

3785

38 3845 3885

Incompressible jetcool environment

MA = 375

MA = 375

m = 1 120576 = 0

14

16

18

2

22

24

26

Re(

ph

Ai)

05 1 15 2 25 3 35 40kza

(a)


MA = 406

m = 1 120576 = 0

MA = 3985

0

01

02

03

04

05

06

07

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)

Figure 3 (a) Dispersion curves of stable and unstable kink (119898 = 1) MHD mode propagating in a moving untwisted magnetic flux tube ofincompressible plasma surrounded by cool medium at the same input parameters as in Figure 2 The dispersion curves of unstable modeslocated above the middle of the plot have been calculated for 119872A = 3985 401 4035 and 406 Alfven Mach numbers associated with thenonlabeled black green and purple curves are equal to 3796 3815 and 383 respectively while those of far right green and purple semiclosedcurves have magnitudes of 3925 and 3965 (b) Similar plots as in Figure 2

119872crA = 3865 The growth rates of unstable kink modes are

plotted in Figure 2(b) The red curves in all diagrams denotethe marginal dispersiongrowth rate curves for values ofthe Alfven Mach number smaller than 119872cr

A the kink modeis stable otherwise it becomes unstable and the instabilityis of the KH type With 119872cr

A = 3865 the kink modewill be unstable when the velocity of the moving flux tubeis higher than 429 km sminus1mdasha speed which is below thanthe observationally measured jet speed of 437 km sminus1 Wewould like to underline that all the stable kink modes inthe untwisted (120576 = 0) moving tube modeling jet 11 arepure surface modes while the unstable ones are notmdashthelatter becomes partly surface and partly leaky modes (their

external attenuation coefficients1198980es are complex quantitieswith positive imaginary parts)This circumstance means thatwave energy is radiated outward in the surroundingmediumwhich allows us to claim the KH instability plays a dual roleonce in its nonlinear stage the instability can trigger waveturbulence and simultaneously the propagating KH-mode isradiating its energy outside

When numerically solving (9) for the kink (119898 = 1) modetreating the jetrsquos plasma as incompressible medium and itsenvironment as cool plasma we obtain dispersion curvesrsquoand growth ratesrsquo patterns very similar to those shown inFigure 2 (see Figure 3) Computations show as expected thatthe stable kink mode is a nonleaky surface mode while the


Re(

ph

Ai)

MA = 6625 MA = 6668

MA = 67

MA = 66704

m = 1 120576 = 0

25

3

35

4

45

05 1 15 2 25 3 35 40kza

(a)

MA = 67

MA = 66704

m = 1 120576 = 0

0

01

02

03

04

05

06

07

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)

Figure 4 (a) Dispersion curves of stable and unstable kink (119898 = 1) MHD mode propagating in a moving untwisted magnetic flux tube ofcompressible plasma (modeling jet 8) at 120578 = 0963 and 119887 = 456 Unstable modersquos dispersion curves are pictured in the middle of the plot for119872A = 66704 668 669 and 67 (b) The corresponding normalized growth rates of unstable modes for the same values of 119872A

unstable one possesses a real internal attenuation coefficient1198980i = 119896119911 not changed by the instability but the externalone 1198980e becomes a complex quantity with positive real andnegative imaginary parts Now the threshold Alfven Machnumber is a little bit bigger and yields a critical jet speedfor the instability onset of cong442 km sminus1 being with 5 km sminus1higher than the observationally measured 437 km sminus1 Wenote that as a rule jetrsquos incompressible plasma approximationyields slightly higher threshold Alfven Mach numbers thanthe model of compressible media This ascertainment showshow sensitive to jetrsquos and its environmentrsquos media treatmentthe occurrence of KH instability of the kink mode in our jetis mdashone can expect instability onset at an accessible jet speedonly if bothmedia are considered as compressiblemagnetizedplasmas

The basic physical parameters of jet 8 namely 119899i = 41 times109 cmminus3 and 119879i = 52MK for a low density contrast of 120578 =0963 assuming as before that the temperature of surroundingplasma is119879e = 20MKand the backgroundmagnetic field119861eis equal to 7G yield the following sound and Alfven speeds119888se = 166 km sminus1 119888si = 2675 km sminus1 VAe = 2428 km sminus1 andVAi = 5222 km sminus1 Accordingly plasma betas in both mediaare 120573i = 315 and 120573e = 056 while the magnetic fields ratio119887 = 4563 defines 119861i = 153GThe speed ordering VAi lt 119888se ltVAe lt 119888si according to Callyrsquos [86] classification tells us thatin a rest magnetic flux tube the propagating wave must be apure surface mode of type Sminus+ The numerical computationsconfirm this as well as reproduce the normalized kink speedof 33342 within three places after the decimal pointWith theflow inclusion the behavior of the pair kink-speed dispersioncurves in the region of the instability onset as seen from Fig-ure 4(a) is completely different more specifically the lowersemiclosed kink-speed dispersion curves at 119872A = 6625and 665 (the orange curve) correspond to pseudosurface(body) waves while the higher kink-speed dispersion curvesare associated with surface waves Note that at 119872A = 6668the surface wave dispersion curves brakes in two parts the

right-hand one merges with the lower kink-speed curve at119896119911119886 = 02337 (see the blue line) while its left-hand side formsa narrow semiclosed dispersion curve of pure surface modeThe threshold Alfven Mach number is equal to 66704 (theprediction one according to (30) is 667) which implies thatthe critical flow speed for KH instability onset is equal to3483 km sminus1mdasha value far below the jet speed of 385 km sminus1This circumstance guarantees that even if one considers thejet medium as incompressible plasma and its environment ascool plasma we will get an accessible jetrsquos speed for instabilityonset

In a similar way calculating the sound and Alfven speedsfor jet 16 and its environment (with 119899i = 10 times 109 cmminus3119879i = 74MK 120578 = 095 119879e = 20MK and 119861e = 67G) we get119888se = 166 km sminus1 119888si cong 319 km sminus1 VAe cong 474 km sminus1 VAi =350 km sminus1 and plasma betas 120573i cong 10 and 120573e = 014 Themagnetic fields ratio 119887 is equal to 132 giving 119861i cong 51G Inthis case the speed ordering is 119888se lt 119888si lt VAi lt VAe and thekink (119898 = 1) mode propagating in a rest magnetic fluxtube is a pseudosurface (body) wave of type Bminus+ (see TableI in [86]) The numerical calculations confirm this modetype and yield a normalized value of the kink speed veryclose to the predicted one of 11858 Now the pattern ofstable kink-speed dispersion curves is more complicated(see Figure 5(a))mdashwhile the regular lower kink-speed curvescorrespond to pseudosurface (body) waves the higher kink-speed dispersion curves have the form of narrow and longsemiclosed up to 119872A = 23 loops and are in fact bulk wavesboth attenuation coefficients are purely imaginary numbersAs expected the dispersion curves of unstable kink modepossess complex attenuation coefficients (with positive realand imaginary parts for 1198980i and positive real and negativeimaginary part for1198980e) The threshold Alfven Mach numberbeing equal to 2359305 determines a critical flow velocity of8258 km sminus1 for KH instability onset which is much higherthan the jet 16 speed of 532 km sminus1This consideration showsthat even a relatively high-speed observed soft X-ray jet


23

225

Re(

ph

Ai)

MA = 2325

MA = 2275

MA = 239

MA = 2325

MA = 2359305

MA = 225

m = 1 120576 = 0

08

09

1

11

12

13

14

15

16

17

05 1 15 2 25 3 35 40kza

(a)

MA = 2359305

MA = 238

MA = 239

MA = 237

m = 1 120576 = 0

0

005

01

015

02

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)

Figure 5 (a) Dispersion curves of stable and unstable kink (119898 = 1) MHD mode propagating in a moving untwisted magnetic flux tube ofcompressible plasma (modeling jet 16) at 120578 = 095 and 119887 = 132 Dispersion curves of unstable modes located in the middle of the plot havebeen calculated for 119872A = 2359305 237 238 and 239 (b) Plots of corresponding normalized growth rates

cannot become unstablemdashthe reason for that in this case isthe low density of the jetrsquos plasma

32 KelvinndashHelmholtz Instability in Twisted Flux Tubes Anatural question that immediately raises is how the twist ofthe internal magnetic field Bi will change the conditionsfor instability occurrence and the shape of the dispersioncurves When we model the soft X-ray jet as a movingtwistedmagnetic flux tubes there are two types of instabilitiesthat can develop in the jet namely kink instability due tothe twist of the magnetic field and KH instability owing tothe tangential discontinuity of plasma velocity at the tubeboundary According to Dungey and Loughhead [89] thekink instability will occur provided 119861i120601(119886) gt 2119861i119911 or inour notation as 120576 gt 2 Further on in order to prevent thedeveloping of kink instability wewill consider weakly twistedflux tubes 120576 lt 1 and thus only KH instability can takeplace in such a jet configuration When numerically solvingdispersion equation (25) we need the normalization of thelocal Alfven frequencies 120596Aie The normalization of thesefrequencies is performed by multiplying each of them bythe tube radius 119886 and dividing by the Alfven speed VAi =119861i119911radic120583120588i to get

119886120596AiVAi

= 119898119860119886119861i119911+ 119896119911119886 = 119898120576 + 119896119911119886

119886120596AeVAi

= 119896119911119886119861e119861i119911radic120578 = 119896119911119886119887twistradic120578 (31)

where 119887twist = 119861e119861i119911 = 119887radic1 + 1205762 For small values of 120576 wewill take 119887twist cong 119887

We begin our calculations for the kink (119898 = 1) mode inthe moving twisted tube (modeling jet 11) with 120578 = 0896119887twist cong 119887 = 244 and 120576 = 0025 The results of numericaltask are presented in Figure 6 from which one sees thatthe kink mode dispersion and growth rate curves are verysimilar to those shown in Figure 3 The critical flow velocity

for emerging KH instability now is Vcr0 = 4426 km sminus1calculated by using the ldquoreducedrdquoAlfven speed VAiradic1 + 1205762 =11096 km sminus1 KH instability of the (119898 = minus1) mode willoccur if the jet speed exceeds 442 km sminus1 of which value isbeyond the observationally derived jet speed of 437 km sminus1We note that while the wave attenuation coefficient of the119898 = 1 mode propagating in untwisted moving magnetic fluxtube of incompressible plasma surrounded by a cool mediumis not changed by the flow in the twisted flux tube (in thesame approximations) that attenuation coefficient becomesa complex number with positive imaginary part like theattenuation coefficient in the cool environment

According to the instability criterion (30) the occurrenceof KH instability of higher MHDmodes would require lowerthreshold Alfven Mach numbers (and accordingly lowercritical jet speeds) than those of the kink (119898 = plusmn1) modeOur computations show however that with 120576 = 0025 thethreshold Alfven Mach numbers are generally still high toinitiate the KH instability Their magnitudes for the 119898 =2 3 and 4 modes are 3965 3943 and 3926 respectivelyThe corresponding critical velocities for instability onsetare accordingly equal to 4400 4375 and 4356 km sminus1 Asseen only the 119898 = 4 MHD mode can become unstableagainst KH instability A clutch of unstable dispersion curvesand normalized growth rates are shown in Figure 7 Herewe observe a new phenomenon the KH instability startsat some critical 119896119911119886-number along with the correspondingthreshold Alfven Mach number That critical dimensionlesswave number for the 119898 = 4 MHD mode (look at Figure 7)is equal to 0528 If we assume that jet 11 has a width Δℓ =5 times 103 km then the critical wavelength for a KH instabilityemergence is 120582119898=4cr cong 298Mm

A substantial decrease in the threshold Alfven Machnumber in moving twisted magnetic flux tubes can beobtained by increasing the magnetic field twist parametersay taking for instance 120576 = 04 The dispersion curvesand normalized wave growth rates of the kink (119898 = plusmn1)


Re(

ph

Ai)

MA = 4022

MA = 3989

m = 1 120576 = 0025

21

215

22

225

23

235

24

05 1 15 2 25 3 35 40kza

(a)

MA = 4022

MA = 3989

m = 1 120576 = 0025

05 1 15 2 25 3 35 40kza

0

01

02

03

04

05

06

Im(

ph

Ai)

(b)

Figure 6 (a) Dispersion curves of unstable kink (119898 = 1) MHD mode propagating in a moving twisted flux tube of incompressible plasma(modeling jet 11) at 120578 = 0896 and 120576 = 0025 and for 119872A = 3989 40 (green curve) 4011 (purple curve) and 4022 (b) The normalizedgrowth rates for the same values of 119872A

Re(

ph

Ai)

MA = 395MA = 3926

m = 4 120576 = 0025

206

207

208

209

21

211

212

05 1 15 2 25 3 35 40kza

(a)

MA = 395

MA = 3926

m = 4 120576 = 0025

0

005

01

015

02

025

03

035

04

045Im

(ph

A

i)

05 1 15 2 25 3 35 40kza

(b)

Figure 7 (a) Dispersion curves of unstable (119898 = 4) MHD mode propagating in a moving twisted flux tube of incompressible plasma at thesame parameters as in Figure 6 for 119872A = 3926 3934 (green curve) 3942 (purple curve) and 395 (b) The normalized growth rates for thesame values of 119872Amode are plotted in Figures 8 and 9 One is immediatelyseeing the rather complicated forms of both the dispersioncurves and dimensionless growth rates of the 119898 = minus1 kinkmodemdashsuch a complication was absent at 120576 = 0025 Wenote that computations were performed at 119887twist = 119887radic1 + 1205762and the reference ldquoreducedrdquo Alfven speed inside the jet nowis VAiradic1 + 1205762 = 103 km s minus1 With this Alfven speed thecritical velocities for the instability occurrence are equal to416 km sminus1 (for the 119898 = 1 mode) and cong408 km sminus1 for the119898 = minus1 mode respectively

In solving dispersion equation (25) for the 119898 = 2MHD mode we start with a little bit bigger Alfven Machnumber than that predicted by instability criterion (30) andhave obtained three different kinds of dispersion curves andnormalized growth rates (see Figure 10) notably curves withdistinctive change of the normalized wave phase velocity

in a relatively narrow 119896119911119886-interval (the orange and bluecurves in Figure 10(a)) two piece-wise dispersion curves(the purple and green curves in the same plot) and onealmost linear dispersion curve (the red one which is in factthe marginal dispersion curve obtained for the thresholdAlfven Mach number equal to 4062) Not less interestingare the dimensionless growth rate curves corresponding tothese three kinds of dispersion curves For the thresholdAlfven Mach number of 4062 we obtained one instabilitywindow and instability starts at the critical dimensionlesswave number of 1944 or equivalently at 120582119898=2cr cong 80Mm Forthe slightly superthreshold Alfven Mach numbers of 4075and 41 we got two different instability windows (see the greenand purple curves in Figure 10(b)) and a further increase in119872A leads to merging of those two instability windowsmdashseethe blue and orange curves in the same plot The critical flowvelocity at which KH instability arises is equal to cong418 km sminus1


Re(

ph

Ai)

MA = 404

MA = 413

m = 1 120576 = 04

21

215

22

225

23

235

24

245

05 1 15 2 25 3 35 40kza

(a)

MA = 404 MA = 413

m = 1 120576 = 04

0

01

02

03

04

05

06

07

08

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)

Figure 8 (a) Dispersion curves of unstable (119898 = 1) MHD mode propagating in a moving twisted flux tube of incompressible plasma(modeling jet 11) at 120578 = 0896 and 120576 = 04 and for 119872A = 404 407 (green curve) 41 (purple curve) and 413 (b) The normalized growthrates for the same values of 119872A

Re(

ph

Ai)

MA = 402

MA = 39565

m = minus1 120576 = 04

205

21

215

22

225

23

235

24

05 1 15 2 25 3 35 40kza

(a)

MA = 402

MA = 39565

m = minus1 120576 = 04

0

01

02

03

04

05

06

07Im

(ph

A

i)

05 1 15 2 25 3 35 40kza

(b)

Figure 9 (a) Dispersion curves of unstable (119898 = minus1) MHD mode propagating in a moving twisted flux tube of incompressible plasma(modeling jet 11) at the same parameters as in Figure 8 and for 119872A = 39565 398 (green curve) 40 (purple curve) and 402 (b) Thenormalized growth rates for the same values of 119872A

We have skipped the complicated dispersion and growthrate curves pictured in Figure 10 for the 119898 = 3 and 119898 =4 MHD modesmdashwe have calculated only a few curves forAlfven Mach numbers close to the corresponding thresholdones (see Figures 11 and 12) One can observe that forthe magnetic field twist parameter 120576 = 04 the criticaldimensionless wave numbers are shifted far on the rightmdashtheir values for both modes are equal to 3619 and 4512respectively that yield 120582119898=3cr cong 43Mm and 120582119898=4cr cong 35MmThe critical jet speeds at which KH instability emerges areequal correspondingly to cong422 and 424 km sminus1 We note thatall the threshold Alfven Mach numbers of higher MHDmodes (119898 ⩾ 2) traveling on the moving twisted magneticflux tube are larger than the predicted ones but while for120576 = 0025 they (Alfven Mach numbers) are decreasing withincreasing the mode number at 120576 = 04 we have just the

opposite ordering We would like to notice that finding themarginal dispersion and growth rate curves of the highermodes at 120576 = 04 turns out to be a laborious computationaltask

To finish our survey on KH instability of MHDmodes inmoving twisted magnetic flux tubes we will briefly considerhow a weak twist of the internal magnetic field 120576 = 0025will change the critical jet speeds for the occurrence of KHinstability in jet 8 We are not going to graphically presentthe dispersion curves and growth rates of unstable modesbecause simply there is nothing special in their shapemdashin the incompressible plasma jet approximation and coolenvironment at small magnetic field twist the curves lookmore or less similar Recall that at density contrast of 0963and magnetic fields ratio 119887 = 456 the threshold AlfvenMach number for the kink (119898 = 1) mode is equal to 69047


Re(

ph

Ai)

MA = 42

MA = 4062

m = 2 120576 = 04

215

22

225

23

235

24

245

25

1 15 2 25 3 35 405kza

(a)

MA = 42

MA = 415

MA = 41

MA = 4075

MA = 4062

m = 2 120576 = 04

0

01

02

03

04

05

Im(

ph

Ai)

1 15 2 25 3 35 405kza

(b)

Figure 10 (a) Dispersion curves of unstable (119898 = 2) MHD mode propagating in a moving twisted flux tube of incompressible plasma(modeling jet 11) at the same parameters as in Figure 8 and for119872A = 42 415 41 4075 and 4062 (b)The normalized growth rates for thesame values of 119872A

Re(

ph

Ai)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

2175

218

2185

219

2195

22

2205

26 28 3 32 34 36 38 424kza

(a)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

0

005

01

015

02

025Im

(ph

A

i)

26 28 3 32 34 36 38 424kza

(b)

Figure 11 (a) Dispersion curves of unstable (119898 = 3) MHD mode propagating in a moving twisted flux tube of incompressible plasma(modeling jet 11) at the same parameters as in Figure 8 and for 119872A = 4093 41 4109 and 4125 (b) The normalized growth rates for thesame values of 119872A

which for the reference Alfven speed of 5222 km sminus1 yields acritical flow velocity of 3606 km sminus1 that is lower than the jetspeed of 385 km sminus1 The KH instability of 119898 = 2 3 and 4MHDmodes should occur at flow velocities of cong259 357 and3555 km sminus1 and start at critical dimensionless wave numbers(119896119911119886)cr equal respectively to 0054 0119 and 0205

4 Summary and Conclusion

In this paper we have explored the conditions under whichMHD waves with various mode numbers (119898 = plusmn1 2 3and 4) propagating along untwisted and twisted magneticflux tubes (representing three observed X-ray solar jets) canbecome unstable against the KH instabilityWe have used twomodels of a soft X-ray jet namely considering the jet andits surrounding coronal medium as compressible magnetized

plasmas and another simplified model representing the jetas an incompressible medium and its environment as coolcoronal plasma A comparison of the dispersion curves andnormalized wave growth rates of the kink (119898 = 1) modecalculated on the base of these two models shows that oneobtains similar dispersion curvesrsquo and growth ratesrsquo patterns(see Figures 2 and 3) This circumstance justifies the usageof the second simplified model in studying the propagationcharacteristics of MHD modes in moving twisted magneticflux tubes We must however immediately underline thatthe onset of KH instability in incompressible magnetic fluxtube always requires slightly greater threshold Alfven Machnumbers and correspondingly higher critical jet speeds

Our numerical computations show the following(i) The flow does not change the nature of propagating

stable MHD modes being pure surface waves in a

14 Advances in AstronomyRe

(ph

A

i)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

216

2165

217

2175

218

2185

219

36 38 4 42 44 46 48 534kza

(a)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

0

005

01

015

02

025

Im(

ph

Ai)

36 38 4 42 44 46 48 534kza

(b)

Figure 12 (a) Dispersion curves of unstable (119898 = 4) MHD mode propagating in a moving twisted flux tube of incompressible plasma(modeling jet 11) at the same parameters as in Figure 8 and for119872A = 412 413 414 and 415 (b)The normalized growth rates for the samevalues of 119872A

rest magnetic flux tube but unstable ones due to highenough jet speed become partly surface and partlyleaky wavesThat is why it ismore appropriate to termthem generalized surface waves

(ii) The KH instability of the kink mode (119898 = plusmn1)starts at Alfven Mach numbers very close to thepredicted ones Numerically found threshold119872As forthe higherMHDmodes are in fact greater than theirinitial evaluations Accessible flow velocities for insta-bility onset in jet 11 have been derived for movinguntwisted magnetic flux tube of compressible plasma(429 km sminus1) and for the119898 = 4MHDmode propagat-ing in weakly twisted magnetic flux tube (120576 = 0025)of incompressible plasma surrounded by cool coronalmedium (4356 km sminus1) A nonnegligible decrease ofthe critical jet speed for all consideredmodes runningin twisted moving flux tube is obtained when themagnetic field twist parameter is 120576 = 04mdashthen thecritical velocities of the 119898 = plusmn1 2 3 and 4 modesare equal to 416 cong408 418 cong422 and 424 km sminus1respectively The instability onset of all consideredMHD modes in jet 8 occurs at flow velocities ingeneral lower than the jet speed of 385 km sminus1mdashtheirmagnitudes for the kink (119898 = 1) mode runningon untwisted or slightly twisted magnetic flux tube(with magnetic field twist parameter 120576 = 0025)are equal to 3483 and 3606 km sminus1 respectively Thecorresponding critical flow velocities for the higher(119898 = 2 3 and 4) MHD modes turn out to liebetween aforementioned speeds namely being equalto cong359 357 and 3555 km sminus1 Jet 16 possessingthe highest flow speed of 532 km sminus1 in Shimojo andShibatarsquos Events List [10] is however stable againstthe KH instabilitymdashthe critical flow velocity at whichthe kink (119898 = 1) mode would become unstablepropagating in untwisted magnetic flux tube is equal

to cong826 km sminus1mdashmuch higher than the jet speed Thecritical velocities in twisted tubes are even largerequal to 8524 8483 8448 and 8426 km sminus1 for the119898 = 1ndash4 MHDmodes respectively

(iii) The instability onset for a given mode 119898 is very sen-sitive to the main input parameters density contrast120578 magnetic field ratio 119887 the magnetic field twist 120576and especially the background magnetic field 119861e Ifwe assume that for jet 11 119861e = 10G Alfven speedsin the two media have new values notably VAi =32038 km sminus1 and VAe = 4275 km sminus1 With thesenew Alfven speeds the input parameters for solvingwave dispersion relations (9) and (25) have changedand more specifically i = 07376 e = 01506and 119887 = 12636 Now the critical jet velocity for theinstability onset dramatically increasesmdashit becomes751 km sminus1 for the kink mode in untwisted tube and774 km sminus1 in weakly twisted (120576 = 0025) flux tube

(iv) If we accept the Paraschivrsquos et al [90] suggestion thatplasma density in the region crossed by coronal jetshas to be the sum 119899cor + 119899jet or on average two timeshigher than the plasma in the surrounding corona119899cor then the total pressure balance equation againfor jet 11 can be satisfied at 119861e = 10G and we havethe following set of input parameters 120578 = 0437 VAi =1057 km sminus1 VAe = 4275 km sminus1 i = 67782 e =01506 and 119887 = 3781 With these parameters thecritical flow velocities for KH instability occurrencereduce to 560 km sminus1 for the kink mode in untwistedflux tube and to 575 km sminus1 for the same mode inweakly twisted (120576 = 0025) flux tube Note thatwithin this approach one obtains critical jet speedsthat are approximately with 200 km sminus1 less than thosecalculated for the same background magnetic fieldof 10G Here one raises the big question ldquowhich


approach the standard one or that of Paraschivrsquos etal [90] will be appropriate for studyingmodeling theKH instability in solar X-ray (and other) jetsrdquo Theperfect answer to this question will be obtained whenwe will try to model a really observationally detectedKH instability in soft X-ray jets like that of Foullon etal [57]

It is intriguing to see what critical jet speeds we will obtainfor a numerically modeled solar X-jetmdasha case in point is thestudy of Miyagoshi and Yokoyama [91] These authors havepresented MHD numerical simulations of solar X-ray jetsbased on magnetic reconnection model that includes chro-mospheric evaporation Peculiar to their study is that totalpressure balance equation excludes the magnetic pressureinside the jet saying that it is very weak that is (in theirnotation)

119901jet = 119901cor + 11986122120583 (32)

where 119861 equiv 119861e is the background (coronal) magnetic fieldSupposing that 119899cor = 109 cmminus3 119879cor = 106 K 119861 = 10Gand 119899jet = 45 times 109 cmminus3 they obtain from their totalbalance equation the temperature of the jet 119879jet = 67 times106 Kmdashindeed a reasonable value However to satisfy thestandard total pressure balance (1) (assuming a very smallbut finite value for the magnetic field in the jet 119861i) withthe aforementioned values for both the temperature and jetdensity we were forced to double the coronal density that isto take 119899cor equiv 119899e = 20 times 109 cmminus3 In such a case the soundand Alfven speeds in the jet and its environment are 119888si cong304 km sminus1 119888se = 117 km sminus1 VAi = 4767 km sminus1 and VAe =4875 km sminus1 The ordering of sound and Alfven speeds is thesame as for the 11 X-ray jet in the Shimojo and Shibata paper[10] that implies the propagation of pure surface stablemodesThe magnetic field inside the jet is 119861i cong 147Gmdashhence themagnetic fields ratio is 119887 = 6817 Thus the input parametersfor solving (9) and (25) are 120578 = 0444 i = 405872 and e =00695 and as we already found 119887 = 6817 Note that thisrelatively high value of 119887 would suspect a rather big value ofthe threshold Alfven Mach numbermdashthe instability criterionyields 119872A gt 1242 Numerically found threshold 119872A forinitiating KH instability of the kink (119898 = 1) mode in movinguntwistedmagnetic flux tube of compressible plasma is 12435that yields a critical velocity of 607 km sminus1 For the case ofweakly twisted flux tube (120576 = 0025) in the approximationincompressible jet and cool plasma environment we obtainedfor the same kink mode a critical velocity of 620 km sminus1 Thecritical speeds for the higher (119898= 2ndash4)modes are accordinglyof 615 609 and 606 km sminus1

The three examples of coronal X-jets embedded in abackground magnetic field of 10G show that KH instabilityof basically the kink (119898 = 1) mode can occur if jetsrsquospeeds lie in the range of 500ndash700 km sminus1 Such velocities aregenerally accessible for high-speed jets lower critical speedsbetween 400 and 500 km sminus1 can be observeddetected atmoderate external magnetic fields in the range of 6ndash7GThis

requirement is in agreement with Pucci et al [34] evaluationthat the magnetic field inside a standard X-ray jet wouldbe equal to 28G while for a blowout jet that value shouldbe around 45G To be honest we can say that the most ofsoft X-ray solar jets with speeds below 400ndash450 km sminus1 arestable against the KH instabilitymdashthis instability can developonly in relatively dense high-speed jets Note however thateach soft X-ray jet is a unique eventmdashit requires a separatecareful exploration of the instability conditions Arising KHinstability in turn might trigger wave turbulence consideredas an effective mechanism for coronal heating

Our simplified model of studying the possibilities foremerging of KH instability in fast soft X-ray jets has to beimproved by assuming a radial density gradient or somemagnetic field and flow velocity shears A challenge also is themodeling of the same instability in rotating and oscillatoryswaying coronal hole X-ray jets

Competing InterestsThe authors declare that they have no competing interests

Acknowledgments

This work was supported by the Bulgarian Science Fundand the Department of Science amp Technology Govern-ment of India Fund under Indo-Bulgarian Bilateral ProjectDNTSINDIA 017 IntBulgariaP-212 The authors areindebted to Snezhana Yordanova for drawing one figure

References

[1] J M Beckers ldquoSolar spiculesrdquoAnnual Review of Astronomy andAstrophysics vol 10 no 1 pp 73ndash100 1972

[2] J R Roy ldquoThe magnetic properties of solar surgesrdquo SolarPhysics vol 28 no 1 pp 95ndash114 1973

[3] K Shibata T Nakamura T Matsumoto et al ldquoChromosphericanemone jets as evidence of ubiquitous reconnectionrdquo Sciencevol 318 no 5856 pp 1591ndash1594 2007

[4] C Chifor H Isobe E Mason et al ldquoMagnetic flux cancellationassociated with a recurring solar jet observed with HinodeRHESSI and STEREOEUVIrdquo Astronomy amp Astrophysics vol491 no 1 pp 279ndash288 2008

[5] N Nishizuka M Shimizu T Nakamura et al ldquoGiant chro-mospheric anemone jet observed with Hinode and comparisonwith magnetohydrodynamic simulations Evidence of propa-gating Alfven waves and magnetic reconnectionrdquo The Astro-physical Journal Letters vol 683 no 1 pp L83ndashL86 2008

[6] D Alexander and L Fletcher ldquoHigh-resolution observations ofplasma jets in the solar coronardquo Solar Physics vol 190 no 1-2pp 167ndash184 1999

[7] K Shibata Y Ishido L Acton et al ldquoObservations of X-rayjets with the Yohkoh soft X-ray telescoperdquo Publications of theAstronomical Society of Japan vol 44 no 5 pp L173ndashL179 1992

[8] S Tsuneta L Acton M Bruner et al ldquoThe soft X-ray telescopefor the SOLAR-A missionrdquo Solar Physics vol 136 no 1 pp 37ndash67 1991

[9] M Shimojo S Hashimoto K Shibata T Hirayama H SHudson and L W Acton ldquoStatistical study of solar X-ray jetsobserved with the Yohkoh soft X-ray telescoperdquo Publications of


the Astronomical Society of Japan vol 48 no 1 pp 123ndash1361996

[10] M Shimojo and K Shibata ldquoPhysical parameters of solar X-rayjetsrdquo The Astrophysical Journal vol 542 no 2 pp 1100ndash11082000

[11] L Culhane L K Harra D Baker et al ldquoHinode EUV studyof jets in the Sunrsquos south polar coronardquo Publications of theAstronomical Society of Japan vol 59 supplement 3 pp S751ndashS756 2007

[12] C Chifor P R Young H Isobe et al ldquoAn active region jetobserved with Hinoderdquo Astronomy amp Astrophysics vol 481 no1 pp L57ndashL60 2008

[13] A Savcheva J Cirtain E E DeLuca et al ldquoA study of polar jetparameters based on Hinode XRT observationsrdquo Publicationsof the Astronomical Society of Japan vol 59 supplement 3 ppS771ndashS778 2007

[14] J W Cirtain L Golub L Lundquist et al ldquoEvidence for Alfvenwaves in solar X-ray jetsrdquo Science vol 318 no 5856 pp 1580ndash1582 2007

[15] Y-HKim Y-JMoon Y-D Park et al ldquoSmall-scale X-rayEUVjets seen in Hinode XRT and TRACErdquo Publications of theAstronomical Society of Japan vol 59 supplement 3 pp S763ndashS769 2007

[16] L-H Yang Y-C Jiang J-Y Yang Y Bi R-S Zheng and J-CHong ldquoObservations of EUV and soft X-ray recurring jets in anactive regionrdquo Research in Astronomy and Astrophysics vol 11no 10 pp 1229ndash1242 2011

[17] M S Madjarska ldquoDynamics and plasma properties of an X-ray jet from SUMER EIS XRT and EUVI A amp B simultaneousobservationsrdquo Astronomy amp Astrophysics vol 526 no 2 articleA19 2011

[18] K Chandrashekhar A Bemporad D Banerjee G R Guptaand L Teriaca ldquoCharacteristics of polar coronal hole jetsrdquoAstronomy amp Astrophysics vol 561 article A104 2014

[19] W D Pesnell B JThompson and P C Chamberlin ldquoThe SolarDynamics Observatory (SDO)rdquo Solar Physics vol 275 no 1 pp3ndash15 2012

[20] J R Lemen A M Title D J Akin et al ldquoThe AtmosphericImaging Assembly (AIA) on the Solar Dynamics Observatory(SDO)rdquo Solar Physics vol 275 no 1 pp 17ndash40 2012

[21] P Boerner C Edwards J Lemen et al ldquoInitial calibration oftheAtmospheric Imaging Assembly (AIA) on the Solar DynamicsObservatory (SDO)rdquo Solar Physics vol 275 no 1 pp 41ndash662012

[22] S P Moschou K Tsinganos A Vourlidas and V ArchontisldquoSDO observations of solar jetsrdquo Solar Physics vol 284 no 2pp 427ndash438 2013

[23] K Chandrashekhar R J Morton D Banerjee and G R GuptaldquoThe dynamical behaviour of a jet in an on-disk coronal holeobserved with AIASDOrdquo Astronomy amp Astrophysics vol 562article A98 2014

[24] A C Sterling R L Moore D A Falconer and M AdamsldquoSmall-scale filament eruptions as the driver of X-ray jets insolar coronal holesrdquo Nature vol 523 no 7561 pp 437ndash4402015

[25] B Filippov A K Srivastava B N Dwivedi et al ldquoFormation ofa rotating jet during the filament eruption on 2013 April 10-11rdquoMonthly Notices of the Royal Astronomical Society vol 451 no1 pp 1117ndash1129 2015

[26] M Shimojo K Shibata T Yokoyama and K Hori ldquoOne-dimensional and pseudo-two-dimensional hydrodynamic sim-ulations of solar X-ray jetsrdquo Astrophysical Journal vol 550 no2 pp 1051ndash1063 2001

[27] S Kamio H Hara T Watanabe et al ldquoVelocity structure of jetsin a coronal holerdquo Publications of the Astronomical Society ofJapan vol 59 supplement 3 pp S757ndashS762 2007

[28] R L Moore J W Cirtain A C Sterling and D A FalconerldquoDichotomy of solar coronal jets Standard jets and blowoutjetsrdquoTheAstrophysical Journal vol 720 no 1 pp 757ndash770 2010

[29] E Pariat S K Antiochos and C R DeVore ldquoA model for solarpolar jetsrdquo The Astrophysical Journal vol 691 no 1 pp 61ndash742009

[30] B Filippov L Golub and S Koutchmy ldquoX-ray jet dynamics ina polar coronal hole regionrdquo Solar Physics vol 254 no 2 pp259ndash269 2009

[31] E Pariat S K Antiochos and C R Devore ldquoThree-dimensional modeling of quasi-homologous solar jetsrdquo Astro-physical Journal vol 714 no 2 pp 1762ndash1778 2010

[32] F Jiang J Zhang and S Yang ldquoInteraction between an emerg-ing flux region and a pre-existing fan-spine dome observed byIRIS and SDOrdquoPublications of theAstronomical Society of Japanvol 67 no 4 article 78 2015

[33] A C Sterling L K Harra and R L Moore ldquoFibrillar chromo-spheric spicule-like counterparts to an extreme-ultraviolet andsoft X-ray blowout coronal jetrdquo The Astrophysical Journal vol722 no 2 pp 1644ndash1653 2010

[34] S Pucci G Poletto A C Sterling and M Romoli ldquoPhysicalparameters of standard and blowout jetsrdquoAstrophysical Journalvol 776 article 16 2013

[35] R L Moore A C Sterling D A Falconer and D Robe ldquoThecool component and the dichotomy lateral expansion and axialrotation of solar X-ray jetsrdquoThe Astrophysical Journal vol 769no 2 article 134 2013

[36] R L Moore A C Sterling and D A Falconer ldquoMagneticuntwisting in solar jets that go into the outer corona in polarcoronal holesrdquo Astrophysical Journal vol 806 no 1 article 112015

[37] G E Brueckner R A Howard M J Koomen et al ldquoThe LargeAngle Spectroscopic Coronagraph (LASCO)rdquo Solar Physics vol162 no 1 pp 357ndash402 1995

[38] B Schmieder Y Guo F Moreno-Insertis et al ldquoTwisting solarcoronal jet launched at the boundary of an active regionrdquoAstronomy and Astrophysics vol 559 article A1 2013

[39] E Pariat K Dalmasse C R DeVore S K Antiochos andJ T Karpen ldquoModel for straight and helical solar jets IParametric studies of the magnetic field geometryrdquo Astronomyand Astrophysics vol 573 article 130 2015

[40] S Patsourakos E Pariat A Vourlidas S K Antiochos and JP Wuelser ldquoSTEREO SECCHI stereoscopic observations con-straining the initiation of polar coronal jetsrdquo The AstrophysicalJournal vol 680 no 1 pp L73ndashL76 2008

[41] C Gontikakis V Archontis and K Tsinganos ldquoObservationsand 3D MHD simulations of a solar active region jetrdquo Astron-omy amp Astrophysics vol 506 no 3 pp L45ndashL48 2009

[42] R Chandra G R Gupta S Mulay and D Tripathi ldquoSunspotwaves and triggering of homologous active region jetsrdquoMonthlyNotices of the Royal Astronomical Society vol 446 no 4 pp3741ndash3748 2015

[43] J Chae H Wang C-Y Lee P R Goode and U SchuhleldquoPhotosphericmagnetic field changes associatedwith transition


region explosive eventsrdquo Astrophysical Journal vol 497 no 2pp L109ndashL112 1998

[44] L R B Rubio and C Beck ldquoMagnetic flux cancellation in themoat of sunspots results from simultaneous vector spectropo-larimetry in the visible and the infraredrdquo Astrophysical Journalvol 626 no 2 pp L125ndashL128 2005

[45] L Heggland B De Pontieu andVHHansteen ldquoObservationalsignatures of simulated reconnection events in the solar chro-mosphere and transition regionrdquoTheAstrophysical Journal vol702 no 1 pp 1ndash18 2009

[46] D E Innes R H Cameron and S K Solanki ldquoEUV jets typeIII radio bursts and sunspot waves investigated using SDOAIAobservationsrdquo Astronomy and Astrophysics vol 531 article L132011

[47] A K Srivastava and K Murawski ldquoObservations of a pulse-driven cool polar jet by SDOAIArdquo Astronomy amp Astrophysicsvol 534 article A62 2011

[48] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityThe International Series of Monographs on Physics ClarendonPress Oxford UK 1961

[49] T V Zaqarashvili I Zhelyazkov and L Ofman ldquoStability ofrotating magnetized jets in the solar atmosphere I KelvinndashHelmholtz instabilityrdquo Astrophysical Journal vol 813 no 2article 123 2015

[50] D Ryu T W Jones and A Frank ldquoThemagnetohydrodynamicKelvin-Helmholtz instability A three-dimensional study ofnonlinear evolutionrdquo The Astrophysical Journal vol 545 no 1pp 475ndash493 2000

[51] S A Maslowe ldquoShear flow instabilities and transitionrdquo inHydromagnetic Instabilities and the Transition to Turbulence HL Swinney and J P Gollub Eds pp 181ndash228 Springer BerlinGermany 1985

[52] T E Berger G Slater NHurlburt et al ldquoQuiescent prominencedynamics observed with the Hinode solar optical telescope ITurbulent upflow plumesrdquo The Astrophysical Journal vol 716no 2 pp 1288ndash1307 2010

[53] M Ryutova T Berger Z Frank T Tarbell and A TitleldquoObservation of plasma instabilities in quiescent prominencesrdquoSolar Physics vol 267 no 1 pp 75ndash94 2010

[54] D Martınez-Gomez R Soler and J Terradas ldquoOnset of theKelvin-Helmholtz instability in partially ionized magnetic fluxtubesrdquo Astronomy and Astrophysics vol 578 article A104 2015

[55] N F Loureiro A A Schekochihin and D A UzdenskyldquoPlasmoid and Kelvin-Helmholtz instabilities in Sweet-Parkercurrent sheetsrdquoPhysical ReviewE Statistical Nonlinear and SoftMatter Physics vol 87 no 1 article 013102 2013

[56] L Feng B Inhester and W Q Gan ldquoKelvinndashHelmholtzinstability of a coronal streamerrdquoAstrophysical Journal vol 774no 2 article 141 2013

[57] C Foullon E Verwichte V M Nakariakov K Nykyri and CJ Farrugia ldquoMagnetic KelvinndashHelmholtz instability at the SunrdquoAstrophysical Journal Letters vol 729 no 1 article L8 2011

[58] L Ofman and B J Thompson ldquoSDOAIA observation ofKelvinndashHelmholtz instability in the solar coronardquo The Astro-physical Journal Letters vol 734 no 1 article L11 2011

[59] C Foullon E Verwichte KNykyriM J Aschwanden and I GHannah ldquoKelvinndashHelmholtz instability of the CME reconnec-tion outflow layer in the low coronardquoThe Astrophysical Journalvol 767 no 2 article 170 2013

[60] U V Mostl M Temmer and A M Veronig ldquoThe KelvinndashHelmholtz instability at coronal mass ejection boundaries in

the solar corona Observations and 25D MHD simulationsrdquoAstrophysical Journal Letters vol 766 no 1 article 12 2013

[61] K Nykyri and C Foullon ldquoFirst magnetic seismology ofthe CME reconnection outflow layer in the low corona with25-D MHD simulations of the Kelvin-Helmholtz instabilityrdquoGeophysical Research Letters vol 40 no 16 pp 4154ndash4159 2013

[62] T V Zaqarashvili A J Dıaz R Oliver and J L BallesterldquoInstability of twisted magnetic tubes with axial mass flowsrdquoAstronomy amp Astrophysics vol 516 article A84 2010

[63] R Soler A J Dıaz J L Ballester and M Goossens ldquoKelvinndashHelmholtz instability in partially ionized compressible plas-masrdquoThe Astrophysical Journal vol 749 no 2 article 163 2012

[64] T V Zaqarashvili ldquoSolar spicules recent challenges in observa-tions and theoryrdquo in Proceedings of the 3rd School andWorkshopon Space Plasma Physics I Zhelyazkov and T Mishonov Edsvol 1356 of AIP Conference Proceedings pp 106ndash116 AIPPublishing LLC Melville NY USA 2011

[65] I Zhelyazkov ldquoMagnetohydrodynamic waves and their stabilitystatus in solar spiculesrdquo Astronomy amp Astrophysics vol 537article A124 2012

[66] A Ajabshirizadeh H Ebadi R E Vekalati and K Molaverdi-khani ldquoThe possibility of Kelvin-Helmholtz instability in solarspiculesrdquo Astrophysics and Space Science vol 357 no 1 article33 2015

[67] I Zhelyazkov and T V Zaqarashvili ldquoKelvin-Helmholtz insta-bility of kink waves in photospheric twisted flux tubesrdquo Astron-omy amp Astrophysics vol 547 article A14 2012

[68] I Zhelyazkov R Chandra A K Srivastava and T MishonovldquoKelvinndashHelmholtz instability of magnetohydrodynamic wavespropagating on solar surgesrdquoAstrophysics and Space Science vol356 no 2 pp 231ndash240 2015

[69] I Zhelyazkov T V Zaqarashvili R Chandra A K Srivastavaand T Mishonov ldquoKelvinndashHelmholtz instability in solar coolsurgesrdquo Advances in Space Research vol 56 no 12 pp 2727ndash2737 2015

[70] I Zhelyazkov R Chandra and A K Srivastava ldquoCan magne-tohydrodynamic waves traveling on solar dark mottles becomeunstablerdquo Bulgarian Journal of Physics vol 42 no 1 pp 68ndash872015

[71] I Zhelyazkov T V Zaqarashvili and R Chandra ldquoKelvin-Helmholtz instability in coronal mass ejecta in the lowercoronardquo Astronomy and Astrophysics vol 574 2015

[72] I Zhelyazkov R Chandra and A K Srivastava ldquoKelvinndashHelmholtz instability in coronal mass ejections and solarsurgesrdquo in Proceedings of the 5th School and Workshop on SpacePlasma Physics I Zhelyazkov and TMishonov Eds vol 1714 ofAIP Conference Proceedings AIP Publishing LLC Melville NYUSA article 030005 2016

[73] D Kuridze V Henriques M Mathioudakis et al ldquoThe dynam-ics of rapid redshifted andblueshifted excursions in the solarH120572linerdquoThe Astrophysical Journal vol 802 no 1 article 26 2015

[74] I Zhelyazkov ldquoOn modeling the KelvinndashHelmholtz instabilityin solar atmosphererdquo Journal of Astrophysics and Astronomyvol 36 no 1 pp 233ndash254 2015

[75] S Vasheghani Farahani T Van Doorsselaere E Verwichte andV M Nakariakov ldquoPropagating transverse waves in soft X-raycoronal jetsrdquoAstronomyampAstrophysics vol 498 no 2 pp L29ndashL32 2009

[76] I Zhelyazkov ldquoReview of themagnetohydrodynamic waves andtheir stability in solar spicules and X-ray jetsrdquo in Topics inMagnethydrodynamics L Zheng Ed chapter 6 InTech RijekaCroatia 2012


[77] I Zhelyazkov ldquoKelvinndashHelmholtz instability of kink waves inphotospheric chromospheric and X-ray solar jetsrdquo in Proceed-ings of the 4th School and Workshop on Space Plasma Physics IZhelyazkov and T Mishonov Eds vol 1551 pp 150ndash164 AIPPublishing LLC Melville NY USA 2013

[78] M Goossens J V Hollweg and T Sakurai ldquoResonant behav-iour of MHD waves on magnetic flux tubes III Effect ofequilibrium flowrdquo Solar Physics vol 138 no 2 pp 233ndash2551992

[79] K Hain and R Lust ldquoZur Stabilitat zylindersymmetrischerPlasmakonfigurationen mit Volumenstromenrdquo Zeitschrift furNaturforschung vol 13 pp 936ndash940 1958

[80] J P Goedbloed ldquoStabilization of magnetohydrodynamic insta-bilities by force-free magnetic fields I Plane plasma layerrdquoPhysica vol 53 no 3 pp 412ndash444 1971

[81] T Sakurai M Goossens and J V Hollweg ldquoResonant behav-iour of MHD waves on magnetic flux tubes I Connectionformulae at the resonant surfacesrdquo Solar Physics vol 133 no 2pp 227ndash245 1991

[82] P M Edwin and B Roberts ldquoWave propagation in a magneticcylinderrdquo Solar Physics vol 88 no 1-2 pp 179ndash191 1983

[83] M Terra-Homem R Erdelyi and I Ballai ldquoLinear and non-linear MHD wave propagation in steady-state magnetic cylin-dersrdquo Solar Physics vol 217 no 2 pp 199ndash223 2003

[84] V M Nakariakov ldquoMHD oscillations in solar and stellarcoronae current results and perspectivesrdquo Advances in SpaceResearch vol 39 no 12 pp 1804ndash1813 2007

[85] K Bennett B Roberts and U Narain ldquoWaves in twistedmagnetic flux tubesrdquo Solar Physics vol 185 no 1 pp 41ndash59 1999

[86] P S Cally ldquoLeaky and non-leaky oscillations in magnetic fluxtubesrdquo Solar Physics vol 103 no 2 pp 277ndash298 1986

[87] I Zhelyazkov R Chandra and A K Srivastava ldquoKelvin-Helmholtz instability in an active region jet observed withHinoderdquo Astrophysics and Space Science vol 361 no 2 article51 2016

[88] T V Zaqarashvili Z Voros and I Zhelyazkov ldquoKelvin-Helmholtz instability of twisted magnetic flux tubes in the solarwindrdquo Astronomy amp Astrophysics vol 561 article A62 2014

[89] J W Dungey and R E Loughhead ldquoTwisted magnetic fields inconducting fluidsrdquoAustralian Journal of Physics vol 7 no 1 pp5ndash13 1954

[90] A R Paraschiv A Bemporad and A C Sterling ldquoPhysicalproperties of solar polar jets A statistical study with HinodeXRTdatardquoAstronomyampAstrophysics vol 579 article A96 2015

[91] T Miyagoshi and T Yokoyama ldquoMagnetohydrodynamic nu-merical simulations of solar X-ray jets based on the magneticreconnection model that includes chromospheric evaporationrdquoThe Astrophysical Journal vol 593 no 2 pp L133ndashL136 2003

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


the morphological and kinematic characteristics of threesmall-scale X-rayEUV jets simultaneously observed by theHinode XRT and the Transition Region and Coronal Explorer(TRACE) While observing the coronal jets for two differentwavelength bands they obtain matching characteristics fortheir projected speed (90ndash310 km sminus1) lifetime (100ndash2000 s)and size (11ndash5 times 105 km) Chifor at al [12] have reported2007 January 1516 observations of a recurring jet situatedon the west side of NOAA active region 10938 A strongblue-shifted component and an indication of a weak red-shifted component at the base of the jet were observed around119879e = 16MK in these jets The upflow velocities wereobserved exceeding up to 150 km sminus1 These jets were seenover a range of temperatures between 025 and 25MK whiletheir estimated electron densities lie above 1011 cmminus3 for thehigh-velocity upflow components Yang et al [16] presentedsimultaneous observations of three recurring jets in EUV andsoft X-ray (SXR) which occurred in an active region on 2007June 5 On comparing their morphological and kinematicproperties the authors have found that EUVand SXR jets hadsimilar onset locations directions size and terminal veloci-ties The three observed jets were having maximum Dopplervelocities ranging from 25 to 121 km sminus1 in the Fe xii 120582195 lineand from 115 to 232 km sminus1 in the He ii 120582256 line Extensivemulti-instrument observations obtained simultaneously withthe SUMER spectrometer on board the Solar andHeliosphericObservatory (SoHO) with EIS and XRT on board Hinodeand with the Extreme-ultraviolet imagers (EUVI) of the SunndashEarth Connection Coronal and Heliospheric Investigation(SECCHI) instrument suite on board the Ahead and BehindSTEREO spacecrafts were performed by Madjarska [17] Thedynamic process of X-ray jet formation and evolution hasbeen derived in great detail In particular for the first timethere was found spectroscopically a temperature of 12MK(Fe xxiii 26376 A) and density of 4 times 1010 cmminus3 in the quietSun The author has clearly identified two types of upflowsin which the first one was the collimated upflow along theopen magnetic fields and the second was the formation ofa plasma cloud from the expelled bright point small-scaleloops Chandrashekhar et al [18] studied the dynamics oftwo jets seen in a polar coronal hole with a combination ofEIS and XRTHinode data They found no evidence of helicalmotions in these events but detected a significant shift of thejet position in a direction normal to the jet axis with a driftvelocity of about 27 and 7 km sminus1 respectively

The launch of the Solar Dynamics Observatory (SDO)[19] with the Atmospheric Imaging Assembly (AIA) [20 21]opens a new page in observing the solar jets Moschou etal [22] have reported high cadence observations of solarcoronal jets observed in the extreme-ultraviolet (EUV) 304 Ausing Atmospheric Imaging Assembly (AIA) instrument onboard SDO They registered in fact coronal hole jets withspeeds of 94 to 760 km sminus1 and lifetimes of the order of severaltens of minutes A detailed description of the dynamicalbehavior of a jet in an on-disk coronal hole observed withAIASDOwas presented by Chandrashekhar et al [23]Theirstudy reveals new evidence of plasma flows prior to thejetrsquos initiation along the small-scale loops at the base of the

jet The authors have also found further evidence that flowsalong the jet consisting of multiple quasi-periodic small-scale plasma ejections In addition spectroscopic analysisestimates temperature as Log 589 plusmn 008K and electron den-sities as Log 875 plusmn 005 cmminus3 in the observed jet Measuredproperties of the registered transverse wave have providedevidence that strong damping of the wave occurred as itpropagates along the jet with speeds of sim110 km sminus1 Usingthe magnetoseismological inversion observed plasma andwave parameters the jetrsquos magnetic field is estimated as 119861= 121 plusmn 02G Recently Sterling et al [24] have reportedhigh-resolution X-ray and extreme-ultraviolet observationsof 20 randomly selected X-ray jets that form in coronal holesat the solar polar caps In each jet converse to the widelyaccepted emerging magnetic flux model a miniature versionof the filament eruptions that initiated coronal mass ejectionsdrove the jet producing reconnection process Formation ofa rotating jet during the filament eruption on 2013 April10ndash11 on the base of multiwavelength and multiviewpointobservations with STEREOSECCHIEUVI and SDOAIAwas reported by Filippov et al [25] The confined eruptionof the filament within a null-point topology which is alsoknown as an Eiffel tower magnetic field configuration formsa twisted jet after magnetic reconnection near the null pointThe sign of the helicity in the jet is observed the same as thatof the sign of the helicity in the filament It is noteworthy thatthe untwistingmotion of the reconnectedmagnetic field linesgives rise to the accelerating plasma along the jet

It is well established that themagnetic reconnection at dif-ferent heights in the solar atmosphere plays a key role in trig-gering jet-like events The magnetic reconnection betweenopen and closed fields (standard reconnection scenario) isone of the well-known processes of the jetrsquos occurrence [1526] The jets emerging by this kind of mechanism are knownas standard jets [3 7 27 28] Some other observationaland simulation studies showed that the reconnection at themagnetic null in a fan-spine magnetic topology can alsotrigger jet-like events [29ndash32] Eruptions of small archesfilaments and flux ropes from within this type of magneticfield configurations can be responsible for reconnection andjetsrsquo occurring These types of jets are known as blowoutjets [28 33 34] Moore et al [35] used the full-disk He ii304 A movies from the Atmospheric Imaging Assembly onSDO to study the cool (119879 sim 105 K) component of X-rayjets observed in polar coronal holes by XRT The AIA 304 Amovies revealed that most polar X-ray jets spin as they eruptThe authors examined 54 X-ray jets that were found in polarcoronal holes in XRT movies sporadically taken during thefirst year of continuous operation of AIA (2010 May through2011 April) These 54 jets were big and bright enough in theXRT images to be categorized as a standard jet or as a blowoutjet From the X-ray movies 19 of the 54 jets appeared likestandard jets 32 appeared as blowout jets and three wereambiguous and were not falling in any category Moore et al[36] have studied 14 large-scale solar coronal jets observedin Sunrsquos pole In EUV movies from the SDOAIA each jetwas very similar to most X-ray and EUV jets erupting incoronal holes However each was exceptional in that it went












Bi

v0

Be

(a)

Bi

v0

Be

(b)




119901i + 1198612i2120583 = 119901e + 1198612e2120583 (1)




119863 dd119903 (119903120585119903) = 1198621119903120585119903 minus 1198622119903119901tot119863d119901tot

d119903 = 1198623120585119903 minus 1198621119901tot(2)


119863 = 1205880 (Ω2 minus 1205962A) 11986241198621 = 21198610120601120583119903 (Ω41198610120601 minus 119898119903 119891B1198624)


1198622 = Ω4 minus (1198962119911 + 11989821199032 )11986241198623 = 1205880119863[Ω2 minus 1205962A + 211986101206011205831205880

dd119903 (1198610120601119903 )]

+ 4Ω4(11986120120601120583119903 )2

minus 12058801198624 4119861201206011205831199032 1205962A

(3)where


(4)



d2119901totd1199032 + [1198623119903119863 d

d119903 (1199031198631198623 )] d119901totd119903

+ [1198623119903119863 dd119903 (11990311986211198623 ) + 11198632 (11986221198623 minus 11986221)] 119901tot = 0

(5)



d2119901totd1199032 + 1119903 d119901tot

d119903 minus (11989820 + 11989821199032 )119901tot = 0 (6)

where



119888T = 119888sVAradic1198882s + V2A

(8)




1015840119898 (1198980e119886)119870119898 (1198980e119886) = 0

(9)





1198980i = 1198961199111198980e = (1198962119911V2Ae minus 1205962)12

VAe (11)

respectively


119891B = 119898119860 + 119896119911119861i119911120596Ai = 119898119860 + 119896119911119861i119911radic120583120588i (12)



d119903 (119861120601119903 ) minus 411986121206011205831199032 1205881205962A

(13)



120585119903 = 1198631198623d119901totd119903 + 11986211198623119901tot (14)



[ d2

d1199032 + 1119903 dd119903 minus (11989820i + 11989821199032 )]119901tot = 0 (15)

where



119901tot (119903 ⩽ 119886) = 120572i119868119898 (1198980i119903) (17)




(18)




(19)


11989820e = 1198962119911 (1 minus 12059621205962Ae) (20)


119901tot (119903 gt 119886) = 120572e119870119898 (1198980e119903) (21)



120585e119903 = 120572e1199031198980e1199031198701015840119898 (1198980e119903)120588e (1205962 minus 1205962Ae) (22)


120596Ae = 119896119911119861e119911radic120583120588e = 119896119911VAe (23)



120585i1199031003816100381610038161003816119903=119886 = 120585e1199031003816100381610038161003816119903=119886 119901tot i minus 1198612i120601120583119886 120585i119903

1003816100381610038161003816100381610038161003816100381610038161003816119903=119886 = 119901tot e1003816100381610038161003816119903=119886 (24)



= 119875119898 (1198980e119886)(120588e120588i) (1205962 minus 1205962Ae) + 1198602119875119898 (1198980e119886) 120583120588i (25)


119865119898 (1198980i119886) = 1198980i1198861198681015840119898 (1198980i119886)119868119898 (1198980i119886) 119875119898 (1198980e119886) = 1198980e1198861198701015840119898 (1198980e119886)119870119898 (1198980e119886)

(26)








= 19166(27)




119860 = 06687119862 = 16566

119860e = 17229119862T = 062119862Te = 08649

(29)



|119898|1198722A gt (1 + 1120578) (|119898| 1198872 + 1) (30)



3943865

3775

3785 38 3825 385

MA = 375

MA = 375

m = 1 120576 = 016

18

2

22

24

26Re

(ph

A

i)

05 1 15 2 25 3 35 40kza

(a)

3893915

MA = 3865

MA = 394

m = 1 120576 = 0

0

01

02

03

04

05

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)


406

3985

3785

38 3845 3885


MA = 375

MA = 375

m = 1 120576 = 0

14

16

18

2

22

24

26

Re(

ph

Ai)

05 1 15 2 25 3 35 40kza

(a)


MA = 406

m = 1 120576 = 0

MA = 3985

0

01

02

03

04

05

06

07

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)









Re(

ph

Ai)

MA = 6625 MA = 6668

MA = 67

MA = 66704

m = 1 120576 = 0

25

3

35

4

45

05 1 15 2 25 3 35 40kza

(a)

MA = 67

MA = 66704

m = 1 120576 = 0

0

01

02

03

04

05

06

07

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)







23

225

Re(

ph

Ai)

MA = 2325

MA = 2275

MA = 239

MA = 2325

MA = 2359305

MA = 225

m = 1 120576 = 0

08

09

1

11

12

13

14

15

16

17

05 1 15 2 25 3 35 40kza

(a)

MA = 2359305

MA = 238

MA = 239

MA = 237

m = 1 120576 = 0

0

005

01

015

02

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)




119886120596AiVAi

= 119898119860119886119861i119911+ 119896119911119886 = 119898120576 + 119896119911119886

119886120596AeVAi








Re(

ph

Ai)

MA = 4022

MA = 3989

m = 1 120576 = 0025

21

215

22

225

23

235

24

05 1 15 2 25 3 35 40kza

(a)

MA = 4022

MA = 3989

m = 1 120576 = 0025

05 1 15 2 25 3 35 40kza

0

01

02

03

04

05

06

Im(

ph

Ai)

(b)


Re(

ph

Ai)

MA = 395MA = 3926

m = 4 120576 = 0025

206

207

208

209

21

211

212

05 1 15 2 25 3 35 40kza

(a)

MA = 395

MA = 3926

m = 4 120576 = 0025

0

005

01

015

02

025

03

035

04

045Im

(ph

A

i)

05 1 15 2 25 3 35 40kza

(b)





Re(

ph

Ai)

MA = 404

MA = 413

m = 1 120576 = 04

21

215

22

225

23

235

24

245

05 1 15 2 25 3 35 40kza

(a)

MA = 404 MA = 413

m = 1 120576 = 04

0

01

02

03

04

05

06

07

08

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)


Re(

ph

Ai)

MA = 402

MA = 39565

m = minus1 120576 = 04

205

21

215

22

225

23

235

24

05 1 15 2 25 3 35 40kza

(a)

MA = 402

MA = 39565

m = minus1 120576 = 04

0

01

02

03

04

05

06

07Im

(ph

A

i)

05 1 15 2 25 3 35 40kza

(b)






Re(

ph

Ai)

MA = 42

MA = 4062

m = 2 120576 = 04

215

22

225

23

235

24

245

25

1 15 2 25 3 35 405kza

(a)

MA = 42

MA = 415

MA = 41

MA = 4075

MA = 4062

m = 2 120576 = 04

0

01

02

03

04

05

Im(

ph

Ai)

1 15 2 25 3 35 405kza

(b)


Re(

ph

Ai)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

2175

218

2185

219

2195

22

2205

26 28 3 32 34 36 38 424kza

(a)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

0

005

01

015

02

025Im

(ph

A

i)

26 28 3 32 34 36 38 424kza

(b)









(ph

A

i)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

216

2165

217

2175

218

2185

219

36 38 4 42 44 46 48 534kza

(a)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

0

005

01

015

02

025

Im(

ph

Ai)

36 38 4 42 44 46 48 534kza

(b)










119901jet = 119901cor + 11986122120583 (32)






Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics














Bi

v0

Be

(a)

Bi

v0

Be

(b)




119901i + 1198612i2120583 = 119901e + 1198612e2120583 (1)




119863 dd119903 (119903120585119903) = 1198621119903120585119903 minus 1198622119903119901tot119863d119901tot

d119903 = 1198623120585119903 minus 1198621119901tot(2)


119863 = 1205880 (Ω2 minus 1205962A) 11986241198621 = 21198610120601120583119903 (Ω41198610120601 minus 119898119903 119891B1198624)


1198622 = Ω4 minus (1198962119911 + 11989821199032 )11986241198623 = 1205880119863[Ω2 minus 1205962A + 211986101206011205831205880

dd119903 (1198610120601119903 )]

+ 4Ω4(11986120120601120583119903 )2

minus 12058801198624 4119861201206011205831199032 1205962A

(3)where


(4)



d2119901totd1199032 + [1198623119903119863 d

d119903 (1199031198631198623 )] d119901totd119903

+ [1198623119903119863 dd119903 (11990311986211198623 ) + 11198632 (11986221198623 minus 11986221)] 119901tot = 0

(5)



d2119901totd1199032 + 1119903 d119901tot

d119903 minus (11989820 + 11989821199032 )119901tot = 0 (6)

where



119888T = 119888sVAradic1198882s + V2A

(8)




1015840119898 (1198980e119886)119870119898 (1198980e119886) = 0

(9)





1198980i = 1198961199111198980e = (1198962119911V2Ae minus 1205962)12

VAe (11)

respectively


119891B = 119898119860 + 119896119911119861i119911120596Ai = 119898119860 + 119896119911119861i119911radic120583120588i (12)



d119903 (119861120601119903 ) minus 411986121206011205831199032 1205881205962A

(13)



120585119903 = 1198631198623d119901totd119903 + 11986211198623119901tot (14)



[ d2

d1199032 + 1119903 dd119903 minus (11989820i + 11989821199032 )]119901tot = 0 (15)

where



119901tot (119903 ⩽ 119886) = 120572i119868119898 (1198980i119903) (17)




(18)




(19)


11989820e = 1198962119911 (1 minus 12059621205962Ae) (20)


119901tot (119903 gt 119886) = 120572e119870119898 (1198980e119903) (21)



120585e119903 = 120572e1199031198980e1199031198701015840119898 (1198980e119903)120588e (1205962 minus 1205962Ae) (22)


120596Ae = 119896119911119861e119911radic120583120588e = 119896119911VAe (23)



120585i1199031003816100381610038161003816119903=119886 = 120585e1199031003816100381610038161003816119903=119886 119901tot i minus 1198612i120601120583119886 120585i119903

1003816100381610038161003816100381610038161003816100381610038161003816119903=119886 = 119901tot e1003816100381610038161003816119903=119886 (24)



= 119875119898 (1198980e119886)(120588e120588i) (1205962 minus 1205962Ae) + 1198602119875119898 (1198980e119886) 120583120588i (25)


119865119898 (1198980i119886) = 1198980i1198861198681015840119898 (1198980i119886)119868119898 (1198980i119886) 119875119898 (1198980e119886) = 1198980e1198861198701015840119898 (1198980e119886)119870119898 (1198980e119886)

(26)








= 19166(27)




119860 = 06687119862 = 16566

119860e = 17229119862T = 062119862Te = 08649

(29)



|119898|1198722A gt (1 + 1120578) (|119898| 1198872 + 1) (30)



3943865

3775

3785 38 3825 385

MA = 375

MA = 375

m = 1 120576 = 016

18

2

22

24

26Re

(ph

A

i)

05 1 15 2 25 3 35 40kza

(a)

3893915

MA = 3865

MA = 394

m = 1 120576 = 0

0

01

02

03

04

05

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)


406

3985

3785

38 3845 3885


MA = 375

MA = 375

m = 1 120576 = 0

14

16

18

2

22

24

26

Re(

ph

Ai)

05 1 15 2 25 3 35 40kza

(a)


MA = 406

m = 1 120576 = 0

MA = 3985

0

01

02

03

04

05

06

07

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)









Re(

ph

Ai)

MA = 6625 MA = 6668

MA = 67

MA = 66704

m = 1 120576 = 0

25

3

35

4

45

05 1 15 2 25 3 35 40kza

(a)

MA = 67

MA = 66704

m = 1 120576 = 0

0

01

02

03

04

05

06

07

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)







23

225

Re(

ph

Ai)

MA = 2325

MA = 2275

MA = 239

MA = 2325

MA = 2359305

MA = 225

m = 1 120576 = 0

08

09

1

11

12

13

14

15

16

17

05 1 15 2 25 3 35 40kza

(a)

MA = 2359305

MA = 238

MA = 239

MA = 237

m = 1 120576 = 0

0

005

01

015

02

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)




119886120596AiVAi

= 119898119860119886119861i119911+ 119896119911119886 = 119898120576 + 119896119911119886

119886120596AeVAi








Re(

ph

Ai)

MA = 4022

MA = 3989

m = 1 120576 = 0025

21

215

22

225

23

235

24

05 1 15 2 25 3 35 40kza

(a)

MA = 4022

MA = 3989

m = 1 120576 = 0025

05 1 15 2 25 3 35 40kza

0

01

02

03

04

05

06

Im(

ph

Ai)

(b)


Re(

ph

Ai)

MA = 395MA = 3926

m = 4 120576 = 0025

206

207

208

209

21

211

212

05 1 15 2 25 3 35 40kza

(a)

MA = 395

MA = 3926

m = 4 120576 = 0025

0

005

01

015

02

025

03

035

04

045Im

(ph

A

i)

05 1 15 2 25 3 35 40kza

(b)





Re(

ph

Ai)

MA = 404

MA = 413

m = 1 120576 = 04

21

215

22

225

23

235

24

245

05 1 15 2 25 3 35 40kza

(a)

MA = 404 MA = 413

m = 1 120576 = 04

0

01

02

03

04

05

06

07

08

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)


Re(

ph

Ai)

MA = 402

MA = 39565

m = minus1 120576 = 04

205

21

215

22

225

23

235

24

05 1 15 2 25 3 35 40kza

(a)

MA = 402

MA = 39565

m = minus1 120576 = 04

0

01

02

03

04

05

06

07Im

(ph

A

i)

05 1 15 2 25 3 35 40kza

(b)






Re(

ph

Ai)

MA = 42

MA = 4062

m = 2 120576 = 04

215

22

225

23

235

24

245

25

1 15 2 25 3 35 405kza

(a)

MA = 42

MA = 415

MA = 41

MA = 4075

MA = 4062

m = 2 120576 = 04

0

01

02

03

04

05

Im(

ph

Ai)

1 15 2 25 3 35 405kza

(b)


Re(

ph

Ai)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

2175

218

2185

219

2195

22

2205

26 28 3 32 34 36 38 424kza

(a)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

0

005

01

015

02

025Im

(ph

A

i)

26 28 3 32 34 36 38 424kza

(b)









(ph

A

i)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

216

2165

217

2175

218

2185

219

36 38 4 42 44 46 48 534kza

(a)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

0

005

01

015

02

025

Im(

ph

Ai)

36 38 4 42 44 46 48 534kza

(b)










119901jet = 119901cor + 11986122120583 (32)






Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




1198622 = Ω4 minus (1198962119911 + 11989821199032 )11986241198623 = 1205880119863[Ω2 minus 1205962A + 211986101206011205831205880

dd119903 (1198610120601119903 )]

+ 4Ω4(11986120120601120583119903 )2

minus 12058801198624 4119861201206011205831199032 1205962A

(3)where


(4)



d2119901totd1199032 + [1198623119903119863 d

d119903 (1199031198631198623 )] d119901totd119903

+ [1198623119903119863 dd119903 (11990311986211198623 ) + 11198632 (11986221198623 minus 11986221)] 119901tot = 0

(5)



d2119901totd1199032 + 1119903 d119901tot

d119903 minus (11989820 + 11989821199032 )119901tot = 0 (6)

where



119888T = 119888sVAradic1198882s + V2A

(8)




1015840119898 (1198980e119886)119870119898 (1198980e119886) = 0

(9)





1198980i = 1198961199111198980e = (1198962119911V2Ae minus 1205962)12

VAe (11)

respectively


119891B = 119898119860 + 119896119911119861i119911120596Ai = 119898119860 + 119896119911119861i119911radic120583120588i (12)



d119903 (119861120601119903 ) minus 411986121206011205831199032 1205881205962A

(13)



120585119903 = 1198631198623d119901totd119903 + 11986211198623119901tot (14)



[ d2

d1199032 + 1119903 dd119903 minus (11989820i + 11989821199032 )]119901tot = 0 (15)

where



119901tot (119903 ⩽ 119886) = 120572i119868119898 (1198980i119903) (17)




(18)




(19)


11989820e = 1198962119911 (1 minus 12059621205962Ae) (20)


119901tot (119903 gt 119886) = 120572e119870119898 (1198980e119903) (21)



120585e119903 = 120572e1199031198980e1199031198701015840119898 (1198980e119903)120588e (1205962 minus 1205962Ae) (22)


120596Ae = 119896119911119861e119911radic120583120588e = 119896119911VAe (23)



120585i1199031003816100381610038161003816119903=119886 = 120585e1199031003816100381610038161003816119903=119886 119901tot i minus 1198612i120601120583119886 120585i119903

1003816100381610038161003816100381610038161003816100381610038161003816119903=119886 = 119901tot e1003816100381610038161003816119903=119886 (24)



= 119875119898 (1198980e119886)(120588e120588i) (1205962 minus 1205962Ae) + 1198602119875119898 (1198980e119886) 120583120588i (25)


119865119898 (1198980i119886) = 1198980i1198861198681015840119898 (1198980i119886)119868119898 (1198980i119886) 119875119898 (1198980e119886) = 1198980e1198861198701015840119898 (1198980e119886)119870119898 (1198980e119886)

(26)








= 19166(27)




119860 = 06687119862 = 16566

119860e = 17229119862T = 062119862Te = 08649

(29)



|119898|1198722A gt (1 + 1120578) (|119898| 1198872 + 1) (30)



3943865

3775

3785 38 3825 385

MA = 375

MA = 375

m = 1 120576 = 016

18

2

22

24

26Re

(ph

A

i)

05 1 15 2 25 3 35 40kza

(a)

3893915

MA = 3865

MA = 394

m = 1 120576 = 0

0

01

02

03

04

05

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)


406

3985

3785

38 3845 3885


MA = 375

MA = 375

m = 1 120576 = 0

14

16

18

2

22

24

26

Re(

ph

Ai)

05 1 15 2 25 3 35 40kza

(a)


MA = 406

m = 1 120576 = 0

MA = 3985

0

01

02

03

04

05

06

07

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)









Re(

ph

Ai)

MA = 6625 MA = 6668

MA = 67

MA = 66704

m = 1 120576 = 0

25

3

35

4

45

05 1 15 2 25 3 35 40kza

(a)

MA = 67

MA = 66704

m = 1 120576 = 0

0

01

02

03

04

05

06

07

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)







23

225

Re(

ph

Ai)

MA = 2325

MA = 2275

MA = 239

MA = 2325

MA = 2359305

MA = 225

m = 1 120576 = 0

08

09

1

11

12

13

14

15

16

17

05 1 15 2 25 3 35 40kza

(a)

MA = 2359305

MA = 238

MA = 239

MA = 237

m = 1 120576 = 0

0

005

01

015

02

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)




119886120596AiVAi

= 119898119860119886119861i119911+ 119896119911119886 = 119898120576 + 119896119911119886

119886120596AeVAi








Re(

ph

Ai)

MA = 4022

MA = 3989

m = 1 120576 = 0025

21

215

22

225

23

235

24

05 1 15 2 25 3 35 40kza

(a)

MA = 4022

MA = 3989

m = 1 120576 = 0025

05 1 15 2 25 3 35 40kza

0

01

02

03

04

05

06

Im(

ph

Ai)

(b)


Re(

ph

Ai)

MA = 395MA = 3926

m = 4 120576 = 0025

206

207

208

209

21

211

212

05 1 15 2 25 3 35 40kza

(a)

MA = 395

MA = 3926

m = 4 120576 = 0025

0

005

01

015

02

025

03

035

04

045Im

(ph

A

i)

05 1 15 2 25 3 35 40kza

(b)





Re(

ph

Ai)

MA = 404

MA = 413

m = 1 120576 = 04

21

215

22

225

23

235

24

245

05 1 15 2 25 3 35 40kza

(a)

MA = 404 MA = 413

m = 1 120576 = 04

0

01

02

03

04

05

06

07

08

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)


Re(

ph

Ai)

MA = 402

MA = 39565

m = minus1 120576 = 04

205

21

215

22

225

23

235

24

05 1 15 2 25 3 35 40kza

(a)

MA = 402

MA = 39565

m = minus1 120576 = 04

0

01

02

03

04

05

06

07Im

(ph

A

i)

05 1 15 2 25 3 35 40kza

(b)






Re(

ph

Ai)

MA = 42

MA = 4062

m = 2 120576 = 04

215

22

225

23

235

24

245

25

1 15 2 25 3 35 405kza

(a)

MA = 42

MA = 415

MA = 41

MA = 4075

MA = 4062

m = 2 120576 = 04

0

01

02

03

04

05

Im(

ph

Ai)

1 15 2 25 3 35 405kza

(b)


Re(

ph

Ai)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

2175

218

2185

219

2195

22

2205

26 28 3 32 34 36 38 424kza

(a)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

0

005

01

015

02

025Im

(ph

A

i)

26 28 3 32 34 36 38 424kza

(b)









(ph

A

i)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

216

2165

217

2175

218

2185

219

36 38 4 42 44 46 48 534kza

(a)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

0

005

01

015

02

025

Im(

ph

Ai)

36 38 4 42 44 46 48 534kza

(b)










119901jet = 119901cor + 11986122120583 (32)






Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120585119903 = 1198631198623d119901totd119903 + 11986211198623119901tot (14)



[ d2

d1199032 + 1119903 dd119903 minus (11989820i + 11989821199032 )]119901tot = 0 (15)

where



119901tot (119903 ⩽ 119886) = 120572i119868119898 (1198980i119903) (17)




(18)




(19)


11989820e = 1198962119911 (1 minus 12059621205962Ae) (20)


119901tot (119903 gt 119886) = 120572e119870119898 (1198980e119903) (21)



120585e119903 = 120572e1199031198980e1199031198701015840119898 (1198980e119903)120588e (1205962 minus 1205962Ae) (22)


120596Ae = 119896119911119861e119911radic120583120588e = 119896119911VAe (23)



120585i1199031003816100381610038161003816119903=119886 = 120585e1199031003816100381610038161003816119903=119886 119901tot i minus 1198612i120601120583119886 120585i119903

1003816100381610038161003816100381610038161003816100381610038161003816119903=119886 = 119901tot e1003816100381610038161003816119903=119886 (24)



= 119875119898 (1198980e119886)(120588e120588i) (1205962 minus 1205962Ae) + 1198602119875119898 (1198980e119886) 120583120588i (25)


119865119898 (1198980i119886) = 1198980i1198861198681015840119898 (1198980i119886)119868119898 (1198980i119886) 119875119898 (1198980e119886) = 1198980e1198861198701015840119898 (1198980e119886)119870119898 (1198980e119886)

(26)








= 19166(27)




119860 = 06687119862 = 16566

119860e = 17229119862T = 062119862Te = 08649

(29)



|119898|1198722A gt (1 + 1120578) (|119898| 1198872 + 1) (30)



3943865

3775

3785 38 3825 385

MA = 375

MA = 375

m = 1 120576 = 016

18

2

22

24

26Re

(ph

A

i)

05 1 15 2 25 3 35 40kza

(a)

3893915

MA = 3865

MA = 394

m = 1 120576 = 0

0

01

02

03

04

05

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)


406

3985

3785

38 3845 3885


MA = 375

MA = 375

m = 1 120576 = 0

14

16

18

2

22

24

26

Re(

ph

Ai)

05 1 15 2 25 3 35 40kza

(a)


MA = 406

m = 1 120576 = 0

MA = 3985

0

01

02

03

04

05

06

07

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)









Re(

ph

Ai)

MA = 6625 MA = 6668

MA = 67

MA = 66704

m = 1 120576 = 0

25

3

35

4

45

05 1 15 2 25 3 35 40kza

(a)

MA = 67

MA = 66704

m = 1 120576 = 0

0

01

02

03

04

05

06

07

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)







23

225

Re(

ph

Ai)

MA = 2325

MA = 2275

MA = 239

MA = 2325

MA = 2359305

MA = 225

m = 1 120576 = 0

08

09

1

11

12

13

14

15

16

17

05 1 15 2 25 3 35 40kza

(a)

MA = 2359305

MA = 238

MA = 239

MA = 237

m = 1 120576 = 0

0

005

01

015

02

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)




119886120596AiVAi

= 119898119860119886119861i119911+ 119896119911119886 = 119898120576 + 119896119911119886

119886120596AeVAi








Re(

ph

Ai)

MA = 4022

MA = 3989

m = 1 120576 = 0025

21

215

22

225

23

235

24

05 1 15 2 25 3 35 40kza

(a)

MA = 4022

MA = 3989

m = 1 120576 = 0025

05 1 15 2 25 3 35 40kza

0

01

02

03

04

05

06

Im(

ph

Ai)

(b)


Re(

ph

Ai)

MA = 395MA = 3926

m = 4 120576 = 0025

206

207

208

209

21

211

212

05 1 15 2 25 3 35 40kza

(a)

MA = 395

MA = 3926

m = 4 120576 = 0025

0

005

01

015

02

025

03

035

04

045Im

(ph

A

i)

05 1 15 2 25 3 35 40kza

(b)





Re(

ph

Ai)

MA = 404

MA = 413

m = 1 120576 = 04

21

215

22

225

23

235

24

245

05 1 15 2 25 3 35 40kza

(a)

MA = 404 MA = 413

m = 1 120576 = 04

0

01

02

03

04

05

06

07

08

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)


Re(

ph

Ai)

MA = 402

MA = 39565

m = minus1 120576 = 04

205

21

215

22

225

23

235

24

05 1 15 2 25 3 35 40kza

(a)

MA = 402

MA = 39565

m = minus1 120576 = 04

0

01

02

03

04

05

06

07Im

(ph

A

i)

05 1 15 2 25 3 35 40kza

(b)






Re(

ph

Ai)

MA = 42

MA = 4062

m = 2 120576 = 04

215

22

225

23

235

24

245

25

1 15 2 25 3 35 405kza

(a)

MA = 42

MA = 415

MA = 41

MA = 4075

MA = 4062

m = 2 120576 = 04

0

01

02

03

04

05

Im(

ph

Ai)

1 15 2 25 3 35 405kza

(b)


Re(

ph

Ai)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

2175

218

2185

219

2195

22

2205

26 28 3 32 34 36 38 424kza

(a)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

0

005

01

015

02

025Im

(ph

A

i)

26 28 3 32 34 36 38 424kza

(b)









(ph

A

i)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

216

2165

217

2175

218

2185

219

36 38 4 42 44 46 48 534kza

(a)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

0

005

01

015

02

025

Im(

ph

Ai)

36 38 4 42 44 46 48 534kza

(b)










119901jet = 119901cor + 11986122120583 (32)






Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








= 19166(27)




119860 = 06687119862 = 16566

119860e = 17229119862T = 062119862Te = 08649

(29)



|119898|1198722A gt (1 + 1120578) (|119898| 1198872 + 1) (30)



3943865

3775

3785 38 3825 385

MA = 375

MA = 375

m = 1 120576 = 016

18

2

22

24

26Re

(ph

A

i)

05 1 15 2 25 3 35 40kza

(a)

3893915

MA = 3865

MA = 394

m = 1 120576 = 0

0

01

02

03

04

05

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)


406

3985

3785

38 3845 3885


MA = 375

MA = 375

m = 1 120576 = 0

14

16

18

2

22

24

26

Re(

ph

Ai)

05 1 15 2 25 3 35 40kza

(a)


MA = 406

m = 1 120576 = 0

MA = 3985

0

01

02

03

04

05

06

07

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)









Re(

ph

Ai)

MA = 6625 MA = 6668

MA = 67

MA = 66704

m = 1 120576 = 0

25

3

35

4

45

05 1 15 2 25 3 35 40kza

(a)

MA = 67

MA = 66704

m = 1 120576 = 0

0

01

02

03

04

05

06

07

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)







23

225

Re(

ph

Ai)

MA = 2325

MA = 2275

MA = 239

MA = 2325

MA = 2359305

MA = 225

m = 1 120576 = 0

08

09

1

11

12

13

14

15

16

17

05 1 15 2 25 3 35 40kza

(a)

MA = 2359305

MA = 238

MA = 239

MA = 237

m = 1 120576 = 0

0

005

01

015

02

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)




119886120596AiVAi

= 119898119860119886119861i119911+ 119896119911119886 = 119898120576 + 119896119911119886

119886120596AeVAi








Re(

ph

Ai)

MA = 4022

MA = 3989

m = 1 120576 = 0025

21

215

22

225

23

235

24

05 1 15 2 25 3 35 40kza

(a)

MA = 4022

MA = 3989

m = 1 120576 = 0025

05 1 15 2 25 3 35 40kza

0

01

02

03

04

05

06

Im(

ph

Ai)

(b)


Re(

ph

Ai)

MA = 395MA = 3926

m = 4 120576 = 0025

206

207

208

209

21

211

212

05 1 15 2 25 3 35 40kza

(a)

MA = 395

MA = 3926

m = 4 120576 = 0025

0

005

01

015

02

025

03

035

04

045Im

(ph

A

i)

05 1 15 2 25 3 35 40kza

(b)





Re(

ph

Ai)

MA = 404

MA = 413

m = 1 120576 = 04

21

215

22

225

23

235

24

245

05 1 15 2 25 3 35 40kza

(a)

MA = 404 MA = 413

m = 1 120576 = 04

0

01

02

03

04

05

06

07

08

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)


Re(

ph

Ai)

MA = 402

MA = 39565

m = minus1 120576 = 04

205

21

215

22

225

23

235

24

05 1 15 2 25 3 35 40kza

(a)

MA = 402

MA = 39565

m = minus1 120576 = 04

0

01

02

03

04

05

06

07Im

(ph

A

i)

05 1 15 2 25 3 35 40kza

(b)






Re(

ph

Ai)

MA = 42

MA = 4062

m = 2 120576 = 04

215

22

225

23

235

24

245

25

1 15 2 25 3 35 405kza

(a)

MA = 42

MA = 415

MA = 41

MA = 4075

MA = 4062

m = 2 120576 = 04

0

01

02

03

04

05

Im(

ph

Ai)

1 15 2 25 3 35 405kza

(b)


Re(

ph

Ai)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

2175

218

2185

219

2195

22

2205

26 28 3 32 34 36 38 424kza

(a)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

0

005

01

015

02

025Im

(ph

A

i)

26 28 3 32 34 36 38 424kza

(b)









(ph

A

i)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

216

2165

217

2175

218

2185

219

36 38 4 42 44 46 48 534kza

(a)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

0

005

01

015

02

025

Im(

ph

Ai)

36 38 4 42 44 46 48 534kza

(b)










119901jet = 119901cor + 11986122120583 (32)






Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




3943865

3775

3785 38 3825 385

MA = 375

MA = 375

m = 1 120576 = 016

18

2

22

24

26Re

(ph

A

i)

05 1 15 2 25 3 35 40kza

(a)

3893915

MA = 3865

MA = 394

m = 1 120576 = 0

0

01

02

03

04

05

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)


406

3985

3785

38 3845 3885


MA = 375

MA = 375

m = 1 120576 = 0

14

16

18

2

22

24

26

Re(

ph

Ai)

05 1 15 2 25 3 35 40kza

(a)


MA = 406

m = 1 120576 = 0

MA = 3985

0

01

02

03

04

05

06

07

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)









Re(

ph

Ai)

MA = 6625 MA = 6668

MA = 67

MA = 66704

m = 1 120576 = 0

25

3

35

4

45

05 1 15 2 25 3 35 40kza

(a)

MA = 67

MA = 66704

m = 1 120576 = 0

0

01

02

03

04

05

06

07

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)







23

225

Re(

ph

Ai)

MA = 2325

MA = 2275

MA = 239

MA = 2325

MA = 2359305

MA = 225

m = 1 120576 = 0

08

09

1

11

12

13

14

15

16

17

05 1 15 2 25 3 35 40kza

(a)

MA = 2359305

MA = 238

MA = 239

MA = 237

m = 1 120576 = 0

0

005

01

015

02

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)




119886120596AiVAi

= 119898119860119886119861i119911+ 119896119911119886 = 119898120576 + 119896119911119886

119886120596AeVAi








Re(

ph

Ai)

MA = 4022

MA = 3989

m = 1 120576 = 0025

21

215

22

225

23

235

24

05 1 15 2 25 3 35 40kza

(a)

MA = 4022

MA = 3989

m = 1 120576 = 0025

05 1 15 2 25 3 35 40kza

0

01

02

03

04

05

06

Im(

ph

Ai)

(b)


Re(

ph

Ai)

MA = 395MA = 3926

m = 4 120576 = 0025

206

207

208

209

21

211

212

05 1 15 2 25 3 35 40kza

(a)

MA = 395

MA = 3926

m = 4 120576 = 0025

0

005

01

015

02

025

03

035

04

045Im

(ph

A

i)

05 1 15 2 25 3 35 40kza

(b)





Re(

ph

Ai)

MA = 404

MA = 413

m = 1 120576 = 04

21

215

22

225

23

235

24

245

05 1 15 2 25 3 35 40kza

(a)

MA = 404 MA = 413

m = 1 120576 = 04

0

01

02

03

04

05

06

07

08

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)


Re(

ph

Ai)

MA = 402

MA = 39565

m = minus1 120576 = 04

205

21

215

22

225

23

235

24

05 1 15 2 25 3 35 40kza

(a)

MA = 402

MA = 39565

m = minus1 120576 = 04

0

01

02

03

04

05

06

07Im

(ph

A

i)

05 1 15 2 25 3 35 40kza

(b)






Re(

ph

Ai)

MA = 42

MA = 4062

m = 2 120576 = 04

215

22

225

23

235

24

245

25

1 15 2 25 3 35 405kza

(a)

MA = 42

MA = 415

MA = 41

MA = 4075

MA = 4062

m = 2 120576 = 04

0

01

02

03

04

05

Im(

ph

Ai)

1 15 2 25 3 35 405kza

(b)


Re(

ph

Ai)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

2175

218

2185

219

2195

22

2205

26 28 3 32 34 36 38 424kza

(a)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

0

005

01

015

02

025Im

(ph

A

i)

26 28 3 32 34 36 38 424kza

(b)









(ph

A

i)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

216

2165

217

2175

218

2185

219

36 38 4 42 44 46 48 534kza

(a)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

0

005

01

015

02

025

Im(

ph

Ai)

36 38 4 42 44 46 48 534kza

(b)










119901jet = 119901cor + 11986122120583 (32)






Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Re(

ph

Ai)

MA = 6625 MA = 6668

MA = 67

MA = 66704

m = 1 120576 = 0

25

3

35

4

45

05 1 15 2 25 3 35 40kza

(a)

MA = 67

MA = 66704

m = 1 120576 = 0

0

01

02

03

04

05

06

07

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)







23

225

Re(

ph

Ai)

MA = 2325

MA = 2275

MA = 239

MA = 2325

MA = 2359305

MA = 225

m = 1 120576 = 0

08

09

1

11

12

13

14

15

16

17

05 1 15 2 25 3 35 40kza

(a)

MA = 2359305

MA = 238

MA = 239

MA = 237

m = 1 120576 = 0

0

005

01

015

02

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)




119886120596AiVAi

= 119898119860119886119861i119911+ 119896119911119886 = 119898120576 + 119896119911119886

119886120596AeVAi








Re(

ph

Ai)

MA = 4022

MA = 3989

m = 1 120576 = 0025

21

215

22

225

23

235

24

05 1 15 2 25 3 35 40kza

(a)

MA = 4022

MA = 3989

m = 1 120576 = 0025

05 1 15 2 25 3 35 40kza

0

01

02

03

04

05

06

Im(

ph

Ai)

(b)


Re(

ph

Ai)

MA = 395MA = 3926

m = 4 120576 = 0025

206

207

208

209

21

211

212

05 1 15 2 25 3 35 40kza

(a)

MA = 395

MA = 3926

m = 4 120576 = 0025

0

005

01

015

02

025

03

035

04

045Im

(ph

A

i)

05 1 15 2 25 3 35 40kza

(b)





Re(

ph

Ai)

MA = 404

MA = 413

m = 1 120576 = 04

21

215

22

225

23

235

24

245

05 1 15 2 25 3 35 40kza

(a)

MA = 404 MA = 413

m = 1 120576 = 04

0

01

02

03

04

05

06

07

08

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)


Re(

ph

Ai)

MA = 402

MA = 39565

m = minus1 120576 = 04

205

21

215

22

225

23

235

24

05 1 15 2 25 3 35 40kza

(a)

MA = 402

MA = 39565

m = minus1 120576 = 04

0

01

02

03

04

05

06

07Im

(ph

A

i)

05 1 15 2 25 3 35 40kza

(b)






Re(

ph

Ai)

MA = 42

MA = 4062

m = 2 120576 = 04

215

22

225

23

235

24

245

25

1 15 2 25 3 35 405kza

(a)

MA = 42

MA = 415

MA = 41

MA = 4075

MA = 4062

m = 2 120576 = 04

0

01

02

03

04

05

Im(

ph

Ai)

1 15 2 25 3 35 405kza

(b)


Re(

ph

Ai)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

2175

218

2185

219

2195

22

2205

26 28 3 32 34 36 38 424kza

(a)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

0

005

01

015

02

025Im

(ph

A

i)

26 28 3 32 34 36 38 424kza

(b)









(ph

A

i)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

216

2165

217

2175

218

2185

219

36 38 4 42 44 46 48 534kza

(a)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

0

005

01

015

02

025

Im(

ph

Ai)

36 38 4 42 44 46 48 534kza

(b)










119901jet = 119901cor + 11986122120583 (32)






Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




23

225

Re(

ph

Ai)

MA = 2325

MA = 2275

MA = 239

MA = 2325

MA = 2359305

MA = 225

m = 1 120576 = 0

08

09

1

11

12

13

14

15

16

17

05 1 15 2 25 3 35 40kza

(a)

MA = 2359305

MA = 238

MA = 239

MA = 237

m = 1 120576 = 0

0

005

01

015

02

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)




119886120596AiVAi

= 119898119860119886119861i119911+ 119896119911119886 = 119898120576 + 119896119911119886

119886120596AeVAi








Re(

ph

Ai)

MA = 4022

MA = 3989

m = 1 120576 = 0025

21

215

22

225

23

235

24

05 1 15 2 25 3 35 40kza

(a)

MA = 4022

MA = 3989

m = 1 120576 = 0025

05 1 15 2 25 3 35 40kza

0

01

02

03

04

05

06

Im(

ph

Ai)

(b)


Re(

ph

Ai)

MA = 395MA = 3926

m = 4 120576 = 0025

206

207

208

209

21

211

212

05 1 15 2 25 3 35 40kza

(a)

MA = 395

MA = 3926

m = 4 120576 = 0025

0

005

01

015

02

025

03

035

04

045Im

(ph

A

i)

05 1 15 2 25 3 35 40kza

(b)





Re(

ph

Ai)

MA = 404

MA = 413

m = 1 120576 = 04

21

215

22

225

23

235

24

245

05 1 15 2 25 3 35 40kza

(a)

MA = 404 MA = 413

m = 1 120576 = 04

0

01

02

03

04

05

06

07

08

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)


Re(

ph

Ai)

MA = 402

MA = 39565

m = minus1 120576 = 04

205

21

215

22

225

23

235

24

05 1 15 2 25 3 35 40kza

(a)

MA = 402

MA = 39565

m = minus1 120576 = 04

0

01

02

03

04

05

06

07Im

(ph

A

i)

05 1 15 2 25 3 35 40kza

(b)






Re(

ph

Ai)

MA = 42

MA = 4062

m = 2 120576 = 04

215

22

225

23

235

24

245

25

1 15 2 25 3 35 405kza

(a)

MA = 42

MA = 415

MA = 41

MA = 4075

MA = 4062

m = 2 120576 = 04

0

01

02

03

04

05

Im(

ph

Ai)

1 15 2 25 3 35 405kza

(b)


Re(

ph

Ai)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

2175

218

2185

219

2195

22

2205

26 28 3 32 34 36 38 424kza

(a)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

0

005

01

015

02

025Im

(ph

A

i)

26 28 3 32 34 36 38 424kza

(b)









(ph

A

i)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

216

2165

217

2175

218

2185

219

36 38 4 42 44 46 48 534kza

(a)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

0

005

01

015

02

025

Im(

ph

Ai)

36 38 4 42 44 46 48 534kza

(b)










119901jet = 119901cor + 11986122120583 (32)






Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Re(

ph

Ai)

MA = 4022

MA = 3989

m = 1 120576 = 0025

21

215

22

225

23

235

24

05 1 15 2 25 3 35 40kza

(a)

MA = 4022

MA = 3989

m = 1 120576 = 0025

05 1 15 2 25 3 35 40kza

0

01

02

03

04

05

06

Im(

ph

Ai)

(b)


Re(

ph

Ai)

MA = 395MA = 3926

m = 4 120576 = 0025

206

207

208

209

21

211

212

05 1 15 2 25 3 35 40kza

(a)

MA = 395

MA = 3926

m = 4 120576 = 0025

0

005

01

015

02

025

03

035

04

045Im

(ph

A

i)

05 1 15 2 25 3 35 40kza

(b)





Re(

ph

Ai)

MA = 404

MA = 413

m = 1 120576 = 04

21

215

22

225

23

235

24

245

05 1 15 2 25 3 35 40kza

(a)

MA = 404 MA = 413

m = 1 120576 = 04

0

01

02

03

04

05

06

07

08

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)


Re(

ph

Ai)

MA = 402

MA = 39565

m = minus1 120576 = 04

205

21

215

22

225

23

235

24

05 1 15 2 25 3 35 40kza

(a)

MA = 402

MA = 39565

m = minus1 120576 = 04

0

01

02

03

04

05

06

07Im

(ph

A

i)

05 1 15 2 25 3 35 40kza

(b)






Re(

ph

Ai)

MA = 42

MA = 4062

m = 2 120576 = 04

215

22

225

23

235

24

245

25

1 15 2 25 3 35 405kza

(a)

MA = 42

MA = 415

MA = 41

MA = 4075

MA = 4062

m = 2 120576 = 04

0

01

02

03

04

05

Im(

ph

Ai)

1 15 2 25 3 35 405kza

(b)


Re(

ph

Ai)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

2175

218

2185

219

2195

22

2205

26 28 3 32 34 36 38 424kza

(a)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

0

005

01

015

02

025Im

(ph

A

i)

26 28 3 32 34 36 38 424kza

(b)









(ph

A

i)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

216

2165

217

2175

218

2185

219

36 38 4 42 44 46 48 534kza

(a)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

0

005

01

015

02

025

Im(

ph

Ai)

36 38 4 42 44 46 48 534kza

(b)










119901jet = 119901cor + 11986122120583 (32)






Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Re(

ph

Ai)

MA = 404

MA = 413

m = 1 120576 = 04

21

215

22

225

23

235

24

245

05 1 15 2 25 3 35 40kza

(a)

MA = 404 MA = 413

m = 1 120576 = 04

0

01

02

03

04

05

06

07

08

Im(

ph

Ai)

05 1 15 2 25 3 35 40kza

(b)


Re(

ph

Ai)

MA = 402

MA = 39565

m = minus1 120576 = 04

205

21

215

22

225

23

235

24

05 1 15 2 25 3 35 40kza

(a)

MA = 402

MA = 39565

m = minus1 120576 = 04

0

01

02

03

04

05

06

07Im

(ph

A

i)

05 1 15 2 25 3 35 40kza

(b)






Re(

ph

Ai)

MA = 42

MA = 4062

m = 2 120576 = 04

215

22

225

23

235

24

245

25

1 15 2 25 3 35 405kza

(a)

MA = 42

MA = 415

MA = 41

MA = 4075

MA = 4062

m = 2 120576 = 04

0

01

02

03

04

05

Im(

ph

Ai)

1 15 2 25 3 35 405kza

(b)


Re(

ph

Ai)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

2175

218

2185

219

2195

22

2205

26 28 3 32 34 36 38 424kza

(a)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

0

005

01

015

02

025Im

(ph

A

i)

26 28 3 32 34 36 38 424kza

(b)









(ph

A

i)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

216

2165

217

2175

218

2185

219

36 38 4 42 44 46 48 534kza

(a)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

0

005

01

015

02

025

Im(

ph

Ai)

36 38 4 42 44 46 48 534kza

(b)










119901jet = 119901cor + 11986122120583 (32)






Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




Re(

ph

Ai)

MA = 42

MA = 4062

m = 2 120576 = 04

215

22

225

23

235

24

245

25

1 15 2 25 3 35 405kza

(a)

MA = 42

MA = 415

MA = 41

MA = 4075

MA = 4062

m = 2 120576 = 04

0

01

02

03

04

05

Im(

ph

Ai)

1 15 2 25 3 35 405kza

(b)


Re(

ph

Ai)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

2175

218

2185

219

2195

22

2205

26 28 3 32 34 36 38 424kza

(a)

MA = 4125

MA = 4109

MA = 41

MA = 4093

m = 3 120576 = 04

0

005

01

015

02

025Im

(ph

A

i)

26 28 3 32 34 36 38 424kza

(b)









(ph

A

i)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

216

2165

217

2175

218

2185

219

36 38 4 42 44 46 48 534kza

(a)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

0

005

01

015

02

025

Im(

ph

Ai)

36 38 4 42 44 46 48 534kza

(b)










119901jet = 119901cor + 11986122120583 (32)






Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




(ph

A

i)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

216

2165

217

2175

218

2185

219

36 38 4 42 44 46 48 534kza

(a)

MA = 415

MA = 414

MA = 413

MA = 412

m = 4 120576 = 04

0

005

01

015

02

025

Im(

ph

Ai)

36 38 4 42 44 46 48 534kza

(b)










119901jet = 119901cor + 11986122120583 (32)






Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119901jet = 119901cor + 11986122120583 (32)






Acknowledgments


References







































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics
































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



reviewarticle modeling kelvin–helmholtz instability in soft x-ray...

Documents