Torsional Alfven Waves in Solar Magnetic Flux Tubes of Axial Symmetry

0.96 MB·English

by  K. Murawski

Astronomy & Astrophysics manuscript no. MSMSK˙rev˙final˙4a (cid:13)c ESO 2015
January 5, 2015
Torsional Alfv´en Waves in Solar Magnetic Flux Tubes of Axial
Symmetry
K. Murawski1, A. Solov’ev2, Z.E. Musielak3,4, A.K. Srivastava5, and J. Kra´skiewicz1
1 Group of Astrophysics, University of Maria Curie-Sk lodowska, ul. Radziszewskiego 10, 20-031 Lublin, Poland
e-mail: [email protected]
2 Central (Pulkovo) Astronomical Observatory,Russian Academy of Sciences, St.Petersburg, Russia
e-mail: [email protected]
3 Department of Physics, Universityof Texas at Arlington, Arlington, TX 76019, USA
5
e-mail: [email protected]
1
4 Kiepenheuer-Institutfu¨r Sonnenphysik,Sch¨oneckstr.6, 79104 Freiburg, Germany
0
e-mail: [email protected]
2
5 Department of Physics, Indian Instituteof Technology (Banaras Hindu University),Varanasi-221005, India

n e-mail: [email protected]
a
J Received ¡date¿ / Accepted ¡date¿

1
ABSTRACT

]
R Aims. Propagation and energy transfer of torsional Alfv´en waves in solar magnetic flux tubes of axial symmetry is
studied.
S
Methods. An analytical model of a solar magnetic flux tube of axial symmetry is developed by specifying a magnetic
.
h fluxandderivinggeneralanalyticalformulae fortheequilibriummassdensityandagaspressure.Themainadvantage
p of this model is that it can be easily adopted to any axisymmetric magnetic structure. The model is used to simulate
- numerically thepropagation of nonlinearAlfv´en wavesin such 2D fluxtubesof axial symmetryembeddedin thesolar
o
atmosphere. The waves are excited by a localized pulse in the azimuthal component of velocity and launched at the
r
topof thesolar photosphere,and theypropagate throughthesolar chromosphere,transition region, andintothesolar
t
s corona.
a Results. The results of our numerical simulations reveal a complex scenario of twisted magnetic field lines and flows
[
associated with torsional Alfv´en waves as well as energy transfer to the magnetoacoustic waves that are triggered by
1 the Alfv´en waves and are akin to the vertical jet flows. Alfv´en waves experience about 5% amplitude reflection at the
v transitionregion.Magnetic(velocity)fieldperturbationsexperienceattenuation(growth)withheightisagreementwith
2 analytical findings. Kinetic energy of magnetoacoustic waves consists of 25% of the total energy of Alfv´en waves. The
5 energy transfermay lead tolocalized mass transport in theform ofvertical jets, aswell as tolocalized heating asslow
2 magnetoacoustic waves are proneto dissipation in theinnercorona.
0 Key words.Sun:atmosphere – (magnetohydrodynamics) MHD – waves
0
.
1
0 1. Introduction ready observedoutwardly-propagatingAlfv´enwavesin the
5
quiescent solar atmosphere in the localized solar magnetic
1 The role of magnetohydrodynamic (MHD) waves in trans-
fluxtubes (e.g.,Jessetal.2009;Fujimura&Tsuneta2009;
v: portingenergyandheatingvariouslayersofthesolaratmo- Okamoto&DePontieu2011;De Pontieuetal.2012;Sekse
i sphere,andaccelerationofthesolarwind,hasbeenstudied etal.2013),whileVanDoorsselaereetal.(2008)questioned
X
by many authors (e.g., Hollweg 1981; Hollweg et al. 1982;
the interpretation of Alfv´en wave observations in the solar
r Priest 1982; Hollweg 1992; Musielak 1998; Ulmschneider
a atmosphere. Specifically, Jess et al. (2009) interpreted the
& Musielak 1998, 2003; Dwivedi & Srivastava 2006, 2010;
HαdataintermsoftorsionalAlfv´enwavesinthesolarchro-
Zaqarashvili & Murawski 2007; Gruszecki et al. 2008;
mosphere,withperiodsrangingfrom2 mintonear12 min,
Murawski & Musielak 2010; Ofman 2009; Chmielewski et
and with the maximum power near 6−7 min. Torsional
al.2013).AmongthethreebasicMHDmodes,Alfv´enwaves
and swirl-like motions were also reported by respectively
are of a particular interest as they can carry their energy
Bonet et al. (2008) and Wedemeyer-Bo¨hm & Rouppe van
along magnetic field lines to higher atmospheric altitudes
derVoort(2009).AccordingtoBonetetal.(2008),thereare
on fast temporal scales (e.g., Hollweg 1978, 1981, 1992;
vortex motions of G band bright points around downflow
Musielak & Moore 1995; Cargill et al. 1997; Vasheghani
zones in the photosphere, and lifetimes of these motions
Farahani et al. 2011, 2012).
areofthe orderof5min.Wedemeyer-Bo¨hm&Rouppevan
The interest in Alfv´en waves has recently significantly
der Voort (2009)observed disorganizedrelative motions of
increasedbecause oftheir observationalevidence in the so-
photospheric brightpoints and concludedthat they induce
lar atmosphere. Several authors claim that they have al-
swirl-like motions in the solar chromosphere.
Send offprint requests to: K. Murawski
2 K.Murawski et al.: Non-linear Alfv´en Waves in Magnetic Flux Tubes
Different theoretical aspects of the generation, propa-
gation and dissipation of torsional Alfv´en waves in the so-
lar atmosphere were investigated by Hollweg (1978, 1981),
Heinemann & Olbert (1980), and Ferriz-Mas et al. (1989),
who solved the Alfv´en wave equations using different sets
ofwavevariables.Musielaketal.(2007)demonstratedthat
the propagation of these waves along isothermal and thin
magnetic flux tubes is cutoff-free. However, Routh et al.
(2007,2010)showedthat gradientsofphysicalparameters,
suchastemperature,areresponsibleforthe originofcutoff
frequencies, which restrict the wave propagationto certain
frequency intervals. In most of the above work only linear
(small-amplitude) Alfv´en waves were considered.
Numerical studies of linear and nonlinear torsional
Alfv´en waves in solar magnetic flux tubes were performed
byHollwegetal.(1982),whoconsideredaone-dimensional Fig.1. Hydrostaticequilibrium profile of solar temperature.
(1D) model. The original work of Holweg et al. was ex-
tended to higher dimensions by Kudoh & Shibata (1999)
and Saito et al. (2001),who investigatedthe effects caused
by nonlinearities. Fedun et al. (2011) showed by means of
numerical simulations that chromospheric magnetic flux-
tubes can act as a frequency filter for torsional Alfv´en
waves. Wedemeyer-Bo¨hm et al. (2012) used the observa-
tional results originally reported by Wedemeyer-Bo¨hm &
RouppevanderVoort(2009)todemonstratethattheswirl-
like motions observed in the lower parts of the solar at-
mosphere produce magnetic tornado-like motions in the
solar transition region and corona. However, more recent
studies performed by Shelyag et al. (2013) seem to imply
that the tornado-like motions actually do not exist once
time dependence of the local velocity field is taken into
account. Instead, the tornado-like motions were identified
as torsional Alfv´en waves propagating along solar mag-
netic flux tubes. Recently, Wedemeyer-Bo¨hm & Rouppe
vander Voort(2009)reportedonsmall,rotatingswirlsob-
served in the solar chromosphere, and interpreted them as
plasma spiraling upwards in a funnel-like magnetic struc-
ture. Chmielewski et al. (2014)and Murawskiet al. (2014)
modeled such short-livedswirls and associatedplasma and
torsional motions in coronal magnetic structures.
As a result of the expanding, curved and strong mag-
neticfieldlinesofsolarmagneticfluxtubesembeddedinthe
solaratmosphere,formulatingamodelofsuchfluxtubes is Fig.2. Equilibrium magnetic field lines. Red, green, and blue
arrowscorrespondtothex-,y-,andz-axis,respectively.Thesize
always a formidable task. Nevertheless, many efforts were
of the box shown is (−0.75,0.75)×(0,4)×(−0.75,0.75) Mm.
undertakeninthepasttodeveloprealisticfluxtubemodels.
Thecolormapcorrespondstothemagnitudeofamagneticfield.
Among many others, let us mention Low (1980) and Gent
etal.(2013,andreferencestherein),who constructedmag-
netic flux tube models in the solar atmosphere. In partic- gion, which are significant in constituting vertical jet flows
ular, Low (1980) formulated an analytical sunspot model, (mass) and localized energy transport processes in the in-
whileGentetal.(2013)solvedanalyticallythemagnetohy- nercorona.Thepaperisorganizedasfollows.Ourmodelof
drostatic equilibrium problem. In this paper, we develop a a solar magnetic flux tube of axial symmetry embedded in
modelofasolarmagneticfluxtube ofaxialsymmetry.The the solaratmosphere is developedin Section 2.The results
model requires specifying a magnetic flux and gives ana- ofournumericalsimulationsarepresentedanddiscussedin
lyticalformulae for the equilibrium mass density anda gas Section 3. Conclusions are given in Section 4.
pressure; these formulae are general enough to be applied
to any axisymmetric magnetic structure.
2. A model of a solar magnetic flux tube of axial
The main goal of this paper is to perform numerical
simulations of nonlinear torsional Alfv´en waves in our de- symmetry
veloped model of magnetic flux of axial symmetry tube
We considera model solaratmosphere thatis describedby
embedded in the solar atmosphere with the temperature
the following ideal, adiabatic, 3D magnetohydrodynamic
profile givenby Avrett & Loeser(2008).We alsostudy the
(MHD) equations:
energy transfer from Alfv´en waves to slow magnetoacous-
tic waves in the non-linear regime, as well as their reflec- ∂̺
tion and transmission through realistic solar transition re- +∇·(̺V)=0, (1)
∂t
K.Murawski et al.: Non-linear Alfv´en Waves in Magnetic Flux Tubes 3
Fig.3. EquilibriumprofilesoftheAlfv´en(solid)andsound(dashed)speeds(theleft panel)andplasmaβ (therightpanel)along
theflux tubeaxis.
We assume that the solar atmosphere is in static equi-
librium (V = 0) with the Lorentz force balanced by the
e
gravity force and the gas pressure gradient, which means
that
1
(∇×B )×B +̺ g−∇p =0 , (5)
e e e e
µ
where the subscript′e′ correspondsto the equilibriumcon-
figuration. We consider an axisymmetric flux tube whose
equilibrium is described by Eq. (5) and its magnetic field
satisfies the solenoidal condition. We now introduce the
magnetic flux (Ψ) normalized by 2π, and obtain
r
Ψ(r,y)= B r′dr′, (6)
ey
Z0
where B is a vertical magnetic field component and r is
ey
the radial distance from the flux tube axis given by
r = x2+z2. (7)
ApsaresultofEq.(6),themagneticfieldisautomatically
divergence-free and its radial (B ), azimuthal (B ) and
er eθ
vertical (B ) components are expressed as
Fig.4.Numericalblockswiththeirboundaries(solidlines)and ey
the pulse in velocity Vz of Eq. (30) (color maps) in the x−y 1∂Ψ 1∂Ψ
plane for z =0 Mm at t=0 seconds. A part of the simulation Ber =− , Beθ =0, Bey = . (8)
r ∂y r ∂r
region is only displayed.
Weconsideramagneticfluxtube ofaxialsymmetry,which
is initially non-twisted, and represent the tube magnetic
∂V 1
̺ +̺(V·∇)V=−∇p+ (∇×B)×B+̺g, (2) field in terms of its azimuthal component of the vector-
∂t µ potential (Aθˆ) as
∂p k
+∇·(pV)=(1−γ)p∇·V, p= B̺T , (3) B =∇×Aθˆ, (9)
∂t m e
∂B =∇×(V×B), ∇·B=0 , (4) where θˆis a unit vector along the azimuthal direction. In
∂t this case, we have
where ̺ is mass density, V is the flow velocity, and B ∂A 1∂(rA)
B =− , B =0, B = . (10)
is the magnetic field. The standard notation is used for er eθ ey
∂y r ∂r
the other physical parameters, namely, p, T, γ = 5/3, µ,
g = (0,−g,0), m, and k are the gas pressure, tempera- Comparing Eqs. (8) and (10), we find
B
ture,adiabaticindex,magneticpermeability,meanparticle
1
mass, and Boltzmann’s constant, respectively. The magni- A(r,y)= Ψ(r,y). (11)
tude of the gravitational acceleration is g = 274 m s−2. r
The value of m is specified by the mean molecular weight, In Cartesian coordinates the magnetic vector potential
which is 1.24 in the solar photosphere (Oskar Steiner, pri- components are
vate communication) and assumed to be constant in the z x
entire flux tube model. [Ax,Ay,Az]=[A ,0,−A ]. (12)
r r
4 K.Murawski et al.: Non-linear Alfv´en Waves in Magnetic Flux Tubes
Theseformulaeareusefulfornumericalsimulationsbecause However,itfollowsfromEq.(16)thatweneedtospecify
they identically satisfy the selenoidal condition, which is S(r,y) as
important in controlling numerical errors.
Multiplying now Eq. (5) by B , we obtain 1 ∂Ψ 2 ∂2Ψ∂Ψdr
e S(r,y)= + . (21)
2r2 ∂r ∂y2 ∂r r2
B ·(∇p −̺ g)=0. (13) (cid:18) (cid:19) Z
e e e
We calculate first ∂S(r,y)/∂r, and then use Eq. (19) to
Sincep =p (Ψ,y),therefore,fromtheaboveequationwith
e e determine ∂S(y,Ψ)/∂Ψ. The result is
the use of Eq. (8), we have
∂p (y,Ψ) 1 1 ∂ 1∂Ψ 1 ∂2Ψ
e
∂p (Ψ,y) =− + . (22)
̺ g =− e , (14) ∂Ψ µ r∂r r ∂r r2 ∂y2
e ∂y (cid:20) (cid:18) (cid:19) (cid:21)
Using Eq. (20), we get
which means that the hydrostatic condition is satisfied
along magnetic field lines, and that Ψ = const. ∂p (y,Ψ) ∂p (r,y) ∂p (y,Ψ)∂Ψ
e e e
= − . (23)
Rewriting Eq. (5) in its components andtaking Eq.(8)
∂y ∂y ∂Ψ ∂y
intoaccount,weobtaintheso-calledGrad-Shafranovequa-
tion (e.g., Low 1975; Priest 1982) given by Finally, with help of Eq. (22), we write
∂2Ψ − 1∂Ψ + ∂2Ψ =−µr2∂pe(Ψ,y). (15) ̺e(r,y)=̺h(y)+
∂r2 r ∂r ∂y2 ∂Ψ 1 ∂ 1 ∂Ψ 2 ∂Ψ 2 ∂2Ψ∂Ψdr
{ − +
If we consider the equilibrium gas pressure as a given µg ∂y "2r2 (cid:18)∂r(cid:19) (cid:18)∂y(cid:19) ! Z ∂y2 ∂r r2#
function,thenEq.(15)becomesanonlinearDirichletprob-
1∂Ψ ∂ 1∂Ψ
lem for Ψ. In contrast, Low (1980) proposed the inverse − },(24)
r ∂y ∂r r ∂r
problemby specifying magneticfieldthrougha choiceofΨ (cid:18) (cid:19)
by integrating Eq. (15) and deriving the general formulas with
forp and̺ givenintermsofΨanditsderivatives.Inour
e e
approach,weregardyasaparameter,andafterintegrating ph(y)
̺ (y)= (25)
h
Eq. (15) over Ψ, we find (Solov’ev 2010) gΛ(y)
1 1 ∂Ψ 2 ∂2Ψ∂Ψdr being the hydrostatic mass density.
p (r,y)=p (y)− + , (16) Note that in the above formulas the magnetic flux (Ψ)
e h µ 2r2 ∂r ∂y2 ∂r r2
" (cid:18) (cid:19) Z # andconsequentlythemagneticfluxfunction(A)arefreeto
choose. For a flux tube, we can specify them as
where
r 1
y dy′ A(r,y)=B0exp(−ky2y2)1+k4r4 + 2By0r, (26)
p (y)=p exp − , (17) r
h 0 Λ(y′)
" Zyr # Ψ(r,y)=rA(r,y), (27)
is the hydrostatic gas pressure, and where B is the external magnetic field along the vertical
y0
direction, k and k are inverse length scales along the ra-
r y
kBTh(y) dialandverticaldirections,respectively.Themagneticfield
Λ(y)= (18)
mg lines,whichfollowfromEqs.(10)and(26)aredisplayedin
Fig. 2 for k =k =4 Mm−1. We choose the magnitude of
r y
is the pressurescale-heightand T (y) is a hydrostatictem-
h the reference magnetic field B in such way that the mag-
0
perature profile.
netic field within the flux tube, at (x = 0,y = 0.5,z = 0)
We adopt T (y) for the solar atmosphere that is speci-
h Mm, is about 285 Gauss, and B ≈11.4 Gauss. For these
y0
fied by the model developed by Avrett & Loeser (2008).
values,the resultingmagneticfieldlines arepredominantly
This temperature profile is smoothly extended into the
vertical around the line, x = z = 0 Mm, while further out
corona (Fig. 1). It should be noted that in our model the they are bent and B decays with distance from this line.
e
solarphotosphereoccupies the region0<y <0.5Mm, the
It must be also noted that the magnetic field at the top of
chromosphere is sandwiched between y = 0.5 Mm and the
the simulation region is essentially uniform, with its value
transitionregion,whichislocatedaty ≃2.1Mm,andthat
of B .
y0
abovethisheighttheatmosphericlayersrepresentthesolar
Figure3illustratesthesoundspeed,c ,(thedashedline
s
corona.
in the left panel), Alfv´en speed, c , (the solid line in the
A
According to Eq. (14), the equilibrium mass density same panel), and plasma β (in the right panel) given by
canbecalculatedif∂p (y,Ψ)/∂y isknown.Moreover,from
e
Eq. (16), we get p = p (r,y). Thus, in order to find
e e γp (r,y) B2(r,y)
∂p (y,Ψ)/∂y, we use the following relations that are valid c (r,y)= e , c (r,y)= e , (28)
e s A
for any differentiable function (S): s ̺e(r,y) sµ̺e(r,y)
2µp (r,y)
e
∂S(r,y) ∂S(Ψ,y)∂Ψ β(r,y)= . (29)
= , (19) B2(r,y)
∂r ∂Ψ ∂r e
∂S(r,y) ∂S(Ψ,y) ∂S(Ψ,y)∂Ψ Thesoundspeedbelowthetransitionregionissmallerthan
= + . (20) 10kms−1,however,itabruptlyraisesinthesolartransition
∂y ∂y ∂Ψ ∂y
K.Murawski et al.: Non-linear Alfv´en Waves in Magnetic Flux Tubes 5
region,and then reachesits coronalvalue of 100km s−1 at netic field, and s is the coordinate alongthe magnetic field
theheighty =10Mm(notshowninthefigure).TheAlfv´en lines.
speed revealssimilar trends, reaching a value of more than Moran (2001) showed that V ∝ ̺−1/4/AB , where A
700 km s−1 at y = 10 Mm (also not shown). The coronal θ e e
is the cross-sectional area of the flux tube. Near the flux
plasma β is ≈ 3×10−2 at y = 4 Mm (the right panel). It
tube axis we can take AB approximately constant, hence
e
increases with the depth but remains lower than 1 above in that region V ∝ ̺−1/4. On the other hand, B ∝ ̺1/4
the height y ≈0.8 Mm. It should be noted that as a result θ e θ e
(Moran 2001). Hence, near the flux tube axis, these quan-
of strong magnetic field, β is lower within the flux tube
tities are only dependent on the variation of ̺ , with V
than in the surrounding medium (not shown). e θ
and B respectively growing and decreasing with height.
θ
As aresultofthatthe perturbationsofmagneticfieldlines
3. Results of numerical simulations remain weak in the solar corona (see also Murawski &
Musielak 2010 for the corresponding analysis in the case
Equations (1)-(4) are solved numerically using the code
of the straight vertical magnetic field lines).
FLASH (Lee & Deane 2009; Lee 2013). This code imple-
Ratio of magnetic to kinetic energies (evaluated in the
ments a third-order unsplit Godunov solver (Murawski &
entire computational domain) of Alfv´en waves is shown in
Lee 2011)with variousslope limiters andRiemannsolvers,
Fig. 7 (top). Initially, at t = 0 s, as we launch the initial
aswellasadaptivemeshrefinement.WechoosethevanLeer
pulse in the azimuthal velocity alone the magnetic energy
slopelimiterandtheRoeRiemannsolver(Murawski&Lee
is zero at t = 0 s but, according to our numerical simula-
2012). For all the considered cases, we set the simulation
tions,contributionsofthemagneticenergygrowintime.It
box as (−1.25,1.25)Mm×(0,20)×(−1.25,1.25)Mm, and
is well known that for torsional Alfv´en waves, there is the
impose boundary conditions by fixing in time all plasma
equipartitionbetweenpotentialandkineticenergies.Hence
quantities at all six boundary surfaces to their equilibrium
accordingtothetoppanelofFig.7,thisequipartitionofen-
values. In the present numerical studies, we use a non-
ergyis notreachedinthe numericalsimulationuntil about
uniform grid with a minimum (maximum) level of refine-
t=200s (where the energyratio equalsunity). This is the
ment set to 2 (5). The grid system at t = 0 s is shown in
naturalexplanationfor the limiting value of the energy ra-
Fig.4withtheblocksbeingdisplayedonlyuptoy =6Mm.
tiobeingapproximatelyunityatt=200s.Att=225sthe
Abovethisaltitudetheblocksystemremainshomogeneous.
magnetic energy is higher than the kinetic energy with the
Aseachblockconsistsof8×8×8identicalnumericalcells,
ratio equalto ≈1.1.Ratios of slow magnetoacousticwaves
we reach the effective finest spatial resolution of 19.53 km,
energy to Alfv´en waves energy (asterisks) and fast magne-
below the altitude y =4.25 Mm.
toacousticwavesenergytoAlfv´enwavesenergy(diamonds)
We initially perturb the azimuthal componentof veloc-
is illustrated in Fig.7 (bottom). At t=0 s kinetic energies
ity, V , by using
θ
of both the slow and fast magnetoacoustic waves are zero
z x r2+(y−y )2 but as these waves are driven by the ponderomotive force
ˆ 0
V θ =[V ,V ]=[ ,− ] A exp − , (30)
θ x z w w v w2 which results from Alfv´en waves the energy ratio grows in
(cid:20) (cid:21) time reachinga maximum level of about 0.25 which means
where A denotes the amplitude of the pulse, θˆis the unit that about 25% of the Alfv´en wave energy was converted
v
vectoralong the azimuthal direction, y is its verticalposi- intokineticenergiesofmagnetoacousticwaves.Thesewaves
0
tion,andw isthewidth.WesetA =150kms−1,w =150 propagatealongmagneticfieldlinesandpowerthe vertical
v
km, and y = 500 km, and hold them fixed. This value of plasma flows in form of the jets.
0
Av results in the effective maximum velocity of about 9.6 Figure 8 illustrates maximum of V (left-panel) and B
km s−1 (Fig. 4). (right-panel) as evaluated from the θnumerical data. Weθ
The initial velocity pulse of Eq. (30) triggers torsional clearly see that V grows with height as predicted on the
θ
Alfv´en waves. These waves are illustrated in Fig. 5, which basis of the liner theory approximationreported by Moran
shows magnetic field lines at t = 50 s (the top-left panel), (2001).Ontheotherhand,B exhibitsafall-offwithywith
θ
t = 150 s (the top-right panel), t = 200 s (the bottom-left the numerical data points being close to the linear theory
panel), and t = 250 s (the bottom-right panel). At t = data (solid-line) as reportedby Moran(2001).It must also
50 s (the top-left panel) the upwardly propagating Alfv´en be notedthatthe relationshipsshownbyMoran(2001)are
waves,while observed in Bθ, reachthe altitude y ≈1 Mm, onlyvalidwhenthereisthe equipartitionofenergyfortor-
and at a later time Alfv´en waves spread in space while sional Alfv´en waves. According to the top panel of Fig. 7,
propagating along diverged magnetic field lines (the top- this occurs only the time of our numerical simulations is
right and bottom panels). t = 200 s or longer. As the results of Fig. 8 are given for
Note that small-amplitude magnetic field and velocity the times shorter than t=200 s, then comparison of these
perturbations that include the effect of magnetic field ex- results with the analytical findings of Moran (2001) is not
pansion,aregovernedbythefollowingwaveequations(Eqs. strictly valid.This fall-offis alsoclearlyevidentinthe spa-
(3) & (4) in Hollweg 1981): tial profiles of B (x,y,z =0) as displayed at t=150 s and
z
t=200 s (Fig. 6, bottom panels). At t=50 s and t=100
∂2 r2v B ∂ ∂v
∂(cid:0)t2 (cid:1) = µ̺ee∂s(cid:20)r2Be∂s(cid:21) , (31) csa(ltizoepdpbaenleolws),ththeetrpaenrstiutriobnatiroengsioonfbBuzt(xa,tyl,azte=r t0i)maersetlhoe-
∂2b ∂ B ∂b perturbations that penetrate the inner solar corona are of
∂t2 =r2Be∂s µ̺er2∂s , (32) significantly lower magnitude than those remaining below
(cid:18) e (cid:19) the transition region. Consequently, these profiles demon-
where v = V /r, b = rB , V and B are the azimuthal strate clearly that Alfv´en waves,while traced in perturbed
θ θ θ θ
components of respectively perturbed velocity and mag- magneticfieldlines,areessentiallylocatedbelowthetransi-
6 K.Murawski et al.: Non-linear Alfv´en Waves in Magnetic Flux Tubes
Fig.5. Magnetic field lines at t = 50 s (top-left), t = 150 s (top-right), t = 200 s (bottom-left), and t = 250 s (bottom-
right). Red, green and blue arrows design the directions of the x-, y-, and z-axis, respectively. The size of the box shown is
(−0.75,0.75)×(0,4)×(−0.75,0.75) Mm. The color map corresponds to the magnitudeof a magnetic field.
tionregion,whichagreeswith the conclusionsofMurawski perturbationspenetratethetransitionregionandenterthe
& Musielak (2010) who used the analytical and numerical solar corona, leading to a generation of the main centrally
techniques for the 1D waveequations and found that mag- located vortex, which starts developing at t=100 s.
neticfieldperturbations,whileexcitedbelowthetransition
region, remain week in the solar corona in comparison to
the perturbations in lower atmospheric layers. Our results clearly show that the velocity perturba-
Figure 9 shows temporal snapshots of velocity stream- tions are significantly enhanced in the solar transition re-
lines at t=100 s (the left-top panel), t=150 s (the right- gion as a result of very steep temperature gradient. The
toppanel),t=200s(theleft-bottompanel),andt=225s main physical consequence of this decrease is that in the
(the right-bottompanel). These streamlines are defined by inner corona, just above the transition region, we see the
the following equations: more velocity streamline perturbations (Figs. 9), while we
donotseemuchperturbationsoftheazimuthalcomponent
dx dy dz ofthemagneticfields(cf.,Figs.5-6).Notethatinthelinear
= = . (33)
Vx Vy Vz limit,Vθ isgovernedbythewaveequationofEq.(31)which
was derived by Hollweg (1981). This equation differs from
Notethatasignalintheazimuthalvelocityisclearlyseenin Eq. (32), and this difference affects the evolution scenario
thesolarcorona.Indeed,thestreamlinesshowthatvelocity of velocity perturbations.
K.Murawski et al.: Non-linear Alfv´en Waves in Magnetic Flux Tubes 7
Fig.6. Spatial profiles of Bz(x,y,z = 0) associated with torsional Alfv´en waves, at t = 50 s (top-left), t = 100 s (top-right),
t=150 s (bottom-left), and t=200 s (bottom-right). A part of thesimulation region is displayed only.
Torsional Alfv´en waves can be traced in the vertical As a resultof nonlinearity,torsionalAlfv´enwavesdrive
profiles of V (x,y,z = 0) = V (x,y,z = 0) (see Fig. 10). a vertical flow through a ponderomotive force (Hollweg et
z θ
At t=200 s (the bottom-right panel), the upwardly prop- al. 1982;Nakariakovet al. 1997,1998).We refer to Eq.(6)
agating Alfv´en waves resulting from the initial pulse are inHollwegetal.(1982)whichillustratesallthetermswhich
well seen, with their amplitudes of |V | ≈ 16 km s−1. The contribute to the component of velocity that is parallel to
z
downwardly propagating waves subside as it is discernible a magnetic field line, V . Denoting by s distance measured
k
at t = 100 s (the top-right panel) and at t = 150 s (the along this line and by r(s) a distance of any point on this
bottom-leftpanel).Att=100s,theupwardlypropagating line from the y-axis we restate Eq. (6) in Hollweg et al.
wavesreachthealtitudeofmorethany =1.6Mmasshown
in the top-right panel of the figure.
8 K.Murawski et al.: Non-linear Alfv´en Waves in Magnetic Flux Tubes
Fig.8. Maximum of Vθ (left-panel) and Bθ (right-panel) vs height. The numerical (analytical) data points are represented by
asterisks (solid line).
1 1 ∂ lnr ∂ B2
̺V2− B2 − θ , (34)
B θ µ θ ∂s ∂s 2µ
k (cid:20)(cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
where g and B are the components of, respectively,
k k
thegravitationalaccelerationandperturbedmagneticfield
along the magnetic field line, and V andB are azimuthal
θ θ
components of velocity and magnetic field, respectively.
Following Hollweg et al. (1982), we state that the terms
in the square bracket result from the s-component of the
Lorentz force: the first term denotes the s-component of
the centrifugal force that corresponds to twisting motions;
the second term represents the magnetic tension; the third
term is the magnetic pressure.
The vertical flow can be seen in the spatial profiles of
V (x,y,z =0)(Fig.11,middle). At t=400s,slowmagne-
y
toacoustic waves in a low plasma β region of the magnetic
flux tube of axialsymmetry as displayedin the right-panel
ofFig.3areweaklycoupledtofastmagnetoacousticwaves,
which are described by essentially only V component of
y
thevelocity.Oursimulationsdemonstratethatatt=175s
slow magnetoacoustic waves already penetrated the solar
corona, reaching the level higher than y =4.5 Mm.
It is noteworthy that non-linear torsional Alfv´en waves
driveperturbationsin amass density,whichareassociated
with fast (Fig. 11, top) and predominantly slow (Fig. 11,
middle)magnetoacousticwaves.Theverticalspatialprofile
oflog(̺(x,y,z =0)/̺ )isshownatt=225sinthebottom
0
panelofFig.11.Here̺ =10−12 kgm−3 isthehydrostatic
0
massdensityatthelevely =10Mm.Atthistimethedense
plasmajetsreachthealtitudeofy ≈3.5Mm.Theplasmais
lifted up due to the magnetic pressure gradientoriginating
fromthe Alfv´enwavesandactsagainstthe gravitytopush
Fig.7. The top panel: ratio of magnetic to kinetic energies of the chromospheric material up to the inner solar corona.
Alfv´en waves; the bottom panel: ratio of slow magnetoacoustic
Therefore, such predominant vertical pressure can create
waveskineticenergytoAlfv´enwavesenergy(asterisks)andratio
the cool plasma jet flows typically observed in the solar
of fast magnetoacoustic waves kinetic energy to total energy
corona.
of Alfv´en waves (diamonds). These energies are evaluated by
integration in space over theentire computational domain. Figure 12 displays the time-signatures of Vz (the solid
line) and V (the dashed line) collected at the detection
y
point, (x = 0.2,y = 3.5,z = 0.2) Mm. We infer that slow
magnetoacousticwaves,representedbyV ,reachthedetec-
y
tion point at t ≈ 130 s, while Alfv´en waves arrive at this
(1982) as
point about 30 s earlier.
Noting that up to y ≈ 0.7 Mm Alfv´en speed is lower
∂ ̺V ∂ ̺V2 1 ∂p ̺g
k k k than the sound speed (Fig. 3, left), the travel times of the
+ =− + +
∂t(cid:18)Bk (cid:19) ∂s Bk ! Bk ∂s Bk torsional Alfv´en, tAt(y), and slow magnetoacoustic waves,
K.Murawski et al.: Non-linear Alfv´en Waves in Magnetic Flux Tubes 9
Fig.9.Streamlinesatt=100s(top-left),t=150s(top-right),t=200s(bottom-left),andt=225s(bottom-right).Red,green
and blue arrows design the directions of the x-, y-, and z-axis, respectively. The size of the box shown is (−0.75,0.75)×(0,6)×
(−0.75,0.75) Mm. The color map corresponds to the magnitudeof a total velocity.
t (y), are given by than the arrival times of Alfv´en and slow magnetoacoustic
st
waves. This dispatch results from nonlinearity.
y dy˜
tAt(y)≈ , (35) The amplitude of the Alfv´en waves, seen in Vθ, pene-
c (r,y˜)
Z0.5 A tratingintothe upperregionofthe atmospheredropsfrom
y dy˜ about V =9 kms−1 at t=100 s just under the transition
tst(y)≈Z0.5 cT(r,y˜), (36) rreeggiioonn,. WtθherocuagnheVvθal=ua8te.4thkemtrsa−n1smjuisstsiaobnocvoeetffihceietnrat,nsition
wheretheintegrationisperformedalongthecorresponding
magnetic field line, and c = c c / c2+c2 is the tube
T s A s A
speed. Using Eqs. (35) and (36), we find tAt(y = 1.9) ≈ At
p C = , (37)
157.7 s and t (y = 1.9)≈ 173.3 s. These values are larger t
st A
i
10 K.Murawski et al.: Non-linear Alfv´en Waves in Magnetic Flux Tubes
Fig.10. Spatial profiles of Vz(x,y,z =0) associated with torsional Alfv´en waves at t =50 s (the top-left panel), t= 100 s (the
top-right panel), t=150 s (thebottom-left panel) and t=200 s (thebottom-right panel).
whereA (A )istheamplitudeoftheincident(transmitted) 4. Discussion and Conclusions
i t
waves. Substituting A = 9 km s−1 and A = 8.4 km s−1
i t
In this paper, we present an analytical model of a mag-
intotheaboveformulawegetC ≈0.95,whichmeansthat
t
netic flux tube of axial symmetry embedded in the solar
about than 5% of the waves amplitude became reflected in
atmosphere whose photosphere, chromosphere and transi-
the transition region and about 95% was transmitted into
tion region are described by the model of Avrett & Loeser
upper atmospheric layers. These are rough estimates due
(2008). Our flux tube model can easily be adopted to any
to difficulties with precise estimation of wave amplitudes
axisymmetric magnetic structure by specifying a different
which highly vary in time at the transition region.
magnetic flux. The model was used to perform numerical
simulations of the propagation of torsional Alfv´en waves
and their coupling to magnetoacoustic waves by using the
FLASH code. Torsional Alfv´en waves are launched at the