Numerical Simulations of Torsional Alfv\'en Waves in Axisymmetric Solar Magnetic Flux Tubes

0.51 MB·

by  D. Wójcik

SolarPhysics
DOI:10.1007/•••••-•••-•••-••••-•
Numerical Simulations of Torsional Alfv´en Waves in
Axisymmetric Solar Magnetic Flux Tubes
D.W´ojcik1 · K.Murawski1 ·
Z.E.Musielak2,3 · P.Konkol1 · A.Mignone4
7
(cid:13)c Springer••••
1
0
2
Abstract

n We investigate numerically Alfv´en waves propagating along an axisymmetric
a
andnon-isothermalsolarflux tube embeddedinthe solaratmosphere.The tube
J
magneticfieldiscurrent-freeanddivergeswithheight,andthewavesareexcited
7
by a periodic driver along the tube magnetic field lines. The main results are
1
that the two wave variables, the velocity and magnetic field perturbations in

] the azimuthal direction, behave differently as a result of gradients of physical
R
parametersalongthetube.Toexplainthesedifferencesinthewavebehavior,the
S timeevolutionofthewavevariablesandtheresultingcutoffperiodforeachwave
.
h variablearecalculated,andusedtodetermineregionsinthesolarchromosphere
p where strong wave reflection may occur.
-
o
Keywords: Waves: Alfv´en
r
t
s
a
[ 1. Introduction

1
Observations by severalrecently launched spacecrafts have revealed the ubiqui-
v
4 touspresenceofoscillationsinthesolaratmosphere,whichcanbeinterpretedas
9 magnetohydrodynamics(MHD) waves(e.g. NakariakovandVerwichte,2005)or
5 specifically as Alfv´en waves whose signatures were observed in prominences,
4
spicules and X-ray jets by Okamoto (2007), De Pontieu (2007) and Cirtain
0
.
1
0
B
7 D.Wo´jcik
1 e-mail:[email protected]
:
v
1 GroupofAstrophysics,FacultyofMathematics, PhysicsandInformatics,UMCS,ul.
i
X Radziszewskiego20-031Lublin,Poland
ar 2 DepartmentofPhysics,UniversityofTexasatArlington,Arlington,TX76019, USA
3 Kiepenheuer-Institut fu¨rSonnenphysikSch¨oneckstr. 6,79104Freiburg,Germany
4 DipartimentodiFisicaGeneraleFacolt´adiScienzeM.F.N.,Universit´adegliStudidi
Torino,10125Torino,Italy
SOLA: driven_Alfven_final.tex; 18 January 2017; 1:23; p. 1
Wo´jciketal.
(2007), respectively. Moreover, observational evidence for the existence of tor-
sionalAlfv´enwavesinthesolaratmospherewasgivenbyJess(2009),seehowever
Dwivedi and Srivastava (2010) and Okamoto and De Pontieu (2011) who re-
ported the presence of propagatingAlfv´en waves in solar spicules. Alfv´en waves
havebeenofaparticularinterestbecausetheycancarryenergyandmomentum
along magnetic field lines to the solar corona, where the wave energy can heat
the corona and the wave momentum may accelerate the solar wind.
ThereisalargebodyofliteraturedevotedtoAlfv´enwavesandtheirpropaga-
tioninthesolaratmosphere(ZhugzhdaandLocans,1982;Hollweg,Jacksonand
Galloway, 1982; An et al., 1989; Hollweg, 1990; Musielak, 1992; Musielak and
Moore, 1995; Kudoh and Shibata, 1999; Hollweg and Isenberg, 2007; Musielak,
Routh and Hammer, 2007; Murawskiand Musielak, 2010; Routh, Musielak and
Hammer, 2010; Webb et al., 2012; Chmielewski et al., 2013; Murawski, 2014;
Perera,MusielakandMurawski,2015).Therearetwomainproblemsconsidered
in these papers, namely, the propagation conditions for Alfv´en waves and the
dissipation of energy and momentum carried by these waves. The first problem
involves the concept of cutoff frequency whose existence is caused by the pres-
ence of strong gradients of physical parameters in the solar atmosphere, and it
has been explored for linear Alfv´en waves by Murawski and Musielak (2010)
and Perera, Musielak and Murawski (2015), and for torsional Alfv´en waves in
thin magnetic flux tubes by Musielak, Routh and Hammer (2007) and Routh,
MusielakandHammer (2010).The secondproblemdealswith the coronalheat-
ing (Suzuki and Inutsuka, 2005) and it involves different mechanisms of Alfv´en
wavedissipation,likephase-mixing(OfmanandDavila,1995)ornonlinearmode
coupling (Ulmschneider, Za¨hringer and Musielak, 1991), and wave momentum
deposition(DwivediandSrivastava,2006;Chmielewskietal., 2013, 2014); how-
ever, a realistic modeling of both Alfv´en wave propagation and dissipation is
difficult to perform(Banerjee,HasanandChristensen-Dalsgaard,1998;O’Shea,
BanerjeeandDoyle,2005;BemporadandAbbo, 2012;Chmielewskietal., 2013;
Jel´ınek et al., 2015).
MurawskiandMusielak(2010)consideredimpulsivelygeneratedAlfv´enwaves
intheone-dimensionalsolaratmospherewithasmoothedstep-wisetemperature
profile and a vertical magnetic field. Perera, Musielak and Murawski (2015)
studiedanalyticallyandnumericallythecaseofperiodicallydrivenAlfv´enwaves
and their propagation in an isothermal solar atmosphere. Musielak, Routh and
Hammer(2007)andRouth,MusielakandHammer(2010)investigatedtorsional
Alfv´en waves propagating in thin magnetic flux tubes embedded in the isother-
mal and non-isothermal solar atmosphere, respectively. The main aim of this
paper is to extend the work of Murawski and Musielak (2010) and Perera,
Musielak and Murawski (2015) to an axi-symmetric solar magnetic flux tube
embedded in the solaratmosphere with a realistictemperature profile of Avrett
andLoeser(2008)andcurvedmagneticfieldlines (Low,1985),performnumeri-
calsimulationsofthepropagationoftorsionalAlfv´enwaves,andtocomparethe
obtained numerical results to the analytical ones obtained by Routh, Musielak
and Hammer (2010).
This paper is organized as follows. The numerical model of the solar at-
mosphere is described in Section 2. The numerical and analytical results are
SOLA: driven_Alfven_final.tex; 18 January 2017; 1:23; p. 2
DrivenAlfv´enwaves
presented and discussed in Sections 3 and 4, respectively. Our discussion of the
analytical and numerical result is given in Section 5, and a short summary of
the results and some concluding remarks are presented in Section 6.
2. Numerical Model
We considera magneticallystructuredandgravitationallystratifiedsolaratmo-
sphere, which is described by the following set of ideal MHD equations:
∂̺
+ (̺V)=0, (1)
∂t ∇·
∂V
̺ +̺(V )V = p+ 1( B) B+̺g, (2)
∂t ·∇ −∇ µ ∇× ×
∂B
= (V B), B =0, (3)
∂t ∇× × ∇·
∂p
+V p = γp V , p= kB̺T , (4)
∂t ·∇ − ∇· m
where ̺ is mass density, p is gas pressure, V is the plasma velocity, B the
magnetic field, and g =(0, g,0) is the gravitationalacceleration of magnitude
274 m s−2. The symbol T d−enotes temperature, m is particle mass, specified by
a meanmolecular weight of 1.24, k is the Boltzmann’s constant, γ =1.4 is the
B
adiabatic index, and µ the magnetic permeability of plasma.
Weconsidertheaxisymmetriccaseinwhichallplasmavariablesareinvariant
in the azimuthal direction θ, but the θ-components of the perturbed magnetic
fieldB andvelocityV arenotidenticallyzero.Weassumethattheequilibrium
θ θ
magnetic field is current-free, B = 0, and its components along the radial
∇×
(r), azimuthal (θ), and vertical (y), directions are specified by the following
expressions:
3Sr(y a)
B = − , (5)
er 5
(r2+(y a)2)2
−
B = 0, (6)
eθ
S r2 2(y a)2
B = − − +B , (7)
ey 5 e0
(r(cid:0)2+(y a)2)2(cid:1)
−
where a and S are free parameters which denote the vertical location of the
magnetic moment and the vertical magnetic field strength, respectively. Equa-
tions (5)–(7) comprise a special (potential and axisymmetric) case of the three-
dimensional model which was developed by Low (1985). We set a= 0.75 Mm
−
choose S such that at r = 0 Mm and y = 6.0 Mm the magnetic field is
r
B = B = 9.5 Gs. Here, y = y denotes the reference level. The magnetic
e ey r
fieldvectorsresultingfromtheseequationsaredrawninFigure1.Themagnetic
field lines are curved and the curvature decays with height y, it grows with the
radial distance r, and the magnitude of B decays with y and r.
e
SOLA: driven_Alfven_final.tex; 18 January 2017; 1:23; p. 3
Wo´jciketal.
Figure 1 Magneticfieldvectors.
As the equilibrium magnetic field, specified by Equations (5)–(7) is current-
free,the equilibriumgaspressureandmassdensityarehydrostaticandthey are
given as (e.g. Murawski et al., 2015):
p (y) =p exp y dy′ , ̺ (y)= 1∂ph , (8)
h 0 − yr Λ(y′) h −g ∂y
h R i
where p =0.01 Pa is the gas pressure evaluated at y =y ,
0 r
k T (y)
B h
Λ(y)= (9)
mg
is the pressure scale-height, and the symbol T (y) denotes the hydrostatic tem-
h
perature profile specified in the model of Avrett and Loeser (2008).
In the model, the solar photosphere occupies the region 0 < y < 0.5 Mm,
the chromosphere resides in 0.5 Mm < y < 2.1 Mm, and the solar corona is
representedby higheratmosphericlayers,whichstartfromthe transitionregion
that is located at y 2.1 Mm. Note that in equilibrium the gas pressure and
≈
mass density do not vary with the radial distance r, while they fall off with the
height y (not shown).
Figure 2 shows the vertical profile of the Alfv´en speed cA = Be/√µ̺h along
the central axis, r = 0 Mm, with c = 12 km s−1 at y = 0 Mm. Higher up c
A A
growswithy,reachingmaximalvalueof2715kms−1 aty =2.44Mm,andthen
SOLA: driven_Alfven_final.tex; 18 January 2017; 1:23; p. 4
DrivenAlfv´enwaves
✸✿✵
✷✿✺
✷✿✵
☎✆s✝
✁✂✄ ✶✿✺
❝❆
✶✿✵
✵✿✺
✵✿✵
✵ ✺ ✶✵ ✶✺ ✷✵ ✷✺ ✸✵
②❬(cid:0)♠❪
Figure 2 Vertical profileofAlfv´enspeed cA forr=0 Mmatthe equilibriumconditions of
Equations.(5)-(8).
c falls off with height. This characteristic shape of the Alfv´en velocity profile
A
will have a prominent effect on the wave behavior in our model as shown and
discussed in Section 3.
3. Numerical Results
Numericalsimulationsareperformedusing the PLUTO code,whichis basedon
a finite-volume/finite-difference method (Mignone et al., 2007, 2011), designed
to solve a system of conservation laws on structured meshes. For all considered
caseswesetthesimulationboxas0to5.12Mminther directionand 0.1Mm
−
to 30 Mm in the y direction. We impose the boundary conditions by fixing
at the top and bottom of the simulation region all plasma quantities to their
equilibrium values, while at the right boundary, outflow boundary conditions
areimplemented.Theseoutflowboundaryconditionsdonoleadtoanyincoming
signal reflection as we use the background magnetic field splitting (Mignone et
al., 2007). At r = 0 Mm axisymmetrical boundaries are implemented. Along
the r-direction this simulation box is divided into 1024 cells and along the y-
direction into 1536 cells for 0.1<y <7.58 Mm and into 512cells in the range
−
of 7.58<y <30 Mm.
For our problem the Courant-Friedrichs-Lewy number is set to 0.3 and the
Harten-Lax-van Leer Discontinuities (HLLD) approximate Riemann solver
(Toro, 2009) is adopted. At the bottom boundary the periodic driver is addi-
tionally set as
r r2 2π
V (r,t)= A exp sin t , (10)
θ w V −w2 P
(cid:18) (cid:19) (cid:18) d (cid:19)
where P is the period of the driver, A is the amplitude of the driver and w is
d V
its spatial width. We set and hold fixed w =100 km and A =5 km s−1, while
V
allowing P to vary.
d
Figure 4 illustrates the r-y spatial profiles of V (left) and B (right) for
θ θ
P = 35 s which corresponds to an effective amplitude of the driver of about
d
SOLA: driven_Alfven_final.tex; 18 January 2017; 1:23; p. 5
Wo´jciketal.
✹✿✵ ✹✿✵
✸✿✺ ✸✿✺
✸✿✵ ✸✿✵
✷✿✺ ✷✿✺
✄ ✄
✁✂♠ ✷✿✵ ✁✂♠ ✷✿✵
② ✶✿✺ ② ✶✿✺
✶✿✵ ✶✿✵
✵✿✺ ✵✿✺
✵✿✵ ✵✿✵
✵ ✺✵ ✶✵✵ ✶✺✵ ✷✵✵ ✷✺✵ ✵ ✺✵ ✶✵✵ ✶✺✵ ✷✵✵ ✷✺✵
t❬(cid:0)❪ t❬(cid:0)❪
☎✵✿✵✽ ☎✵✿✵✻ ☎✵✿✵✹ ☎✵✿✵✷ ✵✿✵✵ ✵✿✵✷ ✵✿✵✹ ✵✿✵✻ ✵✿✵✽ ☎✹✵ ☎✸✵ ☎✷✵ ☎✶✵ ✵ ✶✵ ✷✵ ✸✵ ✹✵
❱✒❬✆✝(cid:0)✞✟❪ ❇✒❬✆(cid:0)❪
Figure 3 VerticalprofilesofVθ (left)andBθ (right)forr=0.1Mmvs timeforPd=35s.
Figure 4 Spatial profileofVθ (left)andBθ (right)forPd=35satt=340s.
0.2 km s−1. It is seen in V (r,y) that this driver excites Alfv´en waves which
θ
penetrate the chromosphere,and att 70 s reachthe transitionregionandthe
≈
solar corona. As a result of the Alfv´en speed profile, displayed in Figure 2, the
Alfv´en waves accelerate up to the altitude where c attains its maximal value,
A
and higher up the Alfv´en waves decelerate.
Notethatasaconsequenceofthedivergenceofthemagneticfieldwithheight
(Figure 1), the wave front spreads with height and magnetic shells develop in
time (Murawski et al., 2015) and a standing wave pattern is present in the low
photosphere, below y 0.5 Mm. A qualitatively similar scenario can be seen in
≈
the low atmospheric layersin profiles of B (r,y) (Figure 4, right). However,the
θ
perturbations in B decay with altitude, remaining largest in low atmospheric
θ
regions.This conclusionconfirmsthe formerfindingsofMurawskiandMusielak
(2010).Thepatternsofstandingwavesandmagneticshellsareeasilyseenbelow
the transition region.
SOLA: driven_Alfven_final.tex; 18 January 2017; 1:23; p. 6
DrivenAlfv´enwaves
Figure3displaystheverticalprofilesofV (left)andB (right)atr=0.1Mm.
θ θ
The penetration of Alfv´en waves into the solar corona is seen in the V profiles
θ
(left). The presented results clearly show that the wave variables behave differ-
ently, namely the gradients in physical parameters have stronger effects on B
θ
thanonV becausetheformerdecaysrapidlywithheightwhilethelatterreaches
θ
the solar corona; the fact that the Alfv´en wave variables behave differently is a
well-known phenomenon (e.g. Hollweg, Jackson and Galloway, 1982; Musielak,
1992;MusielakandMoore,1995;Musielak,RouthandHammer,2007;Murawski
and Musielak, 2010; Routh, Musielak and Hammer, 2010; Perera,Musielak and
Murawski,2015). The presence ofgradientsis the mainphysicalreasonfor such
a different behavior of V and B , and it also leads to wave reflection, which
θ θ
occurs below y 1.75 Mm and decays with time; the waves that undergo
≈
reflection become trapped and this trapping dominates in the upper parts of
the solarchromosphere.Similarto V (r =0.1Mm,y,t),the profilesofB reveal
θ θ
the transient phase, which occurs for t . 150 s but later on the oscillations in
B reach a quasi-stationary stage (bottom).
θ
The above numerical results have important physical implications, namely,
they show that Alfv´en waves do lose their identity because their wave velocity
and magnetic field variables behave differently. Since the wave velocity variable
reachesthecoronaandthewavemagneticfieldvariablecannot,theAlfv´enwave
in the solar corona does not obey any more its equipartition of energy between
the two wave variables.This means that the physicalnature of the Alfv´en wave
in the solar corona is different than in the solar chromosphere. The problem is
out of the scope of this paper but it does require future studies.
4. Cutoff Periods for Torsional Waves in Thin Flux Tubes
To compare the results of our numerical simulations to the analytically de-
termined conditions for the propagation of torsional Alfv´en waves in a solar
magnetic flux tube, we follow Musielak, Routh and Hammer (2007) and Routh,
Musielak and Hammer (2010). With the wave variables ~v = v (r,y,t)θˆ and~b
θ
= b (r,y,t)θˆ, we follow Musielak, Routh and Hammer (2007) and write the
θ
θ-components of the linearized induction equation of motion and induction as
∂(v /r) 1 ∂ ∂
θ
B (r,y) +B (r,y) (rb )=0 , (11)
∂t − ρ r2 er ∂r ey ∂y θ
h (cid:20) (cid:21)
and
∂(rb ) ∂ ∂
θ r2 B (r,y) +B (r,y) (v /r)=0 . (12)
er ey θ
∂t − ∂r ∂y
(cid:20) (cid:21)
According to Musielak, Routh and Hammer (2007), the thin flux tube ap-
proximation gives the following relationship:
rdB
ey
B (r,y)= , (13)
er
−2 dy
SOLA: driven_Alfven_final.tex; 18 January 2017; 1:23; p. 7
Wo´jciketal.
which allows writing Equations (11) and (12) as
∂v 1 ∂b dB B ∂b
θ θ ey ey θ
+ r +b =0 , (14)
θ
∂t 2ρ ∂r dy − ρ ∂y
h (cid:18) (cid:19) h
and
∂b 1 ∂v dB ∂v
θ θ ey θ
+ r v B =0 . (15)
θ ey
∂t 2 ∂r − dy − ∂y
(cid:18) (cid:19)
Assuming the tube being in temperature equilibrium with its surroundings,
Routh, Musielak and Hammer (2010) used horizontal pressure balance and
obtained the following wave equations for torsional Alfv´en waves:
∂2v ∂2v c2 ∂v c2 1 dH
θ c2 θ + A θ A + v =0 , (16)
∂t2 − A ∂y2 2H ∂y − 4H2 4 dy θ
(cid:18) (cid:19)
and
∂2b ∂2b c2 4H dc ∂b c2 1 2H dc dH
θ c2 θ A 1+ A θ A + A b =0,
∂t2 − A ∂y2 − 2H c dy ∂y − 4H2 4 c dy − dy θ
(cid:18) A (cid:19) (cid:18) A (cid:19)
(17)
where H(y) = c2(y)/γg is the pressure scale height with c (y) being the sound
s s
speed, and c (y)=B (y)/ ρ (y) is the Alfv´en velocity.
A ey h
As demonstrated by Routh, Musielak and Hammer (2010), the critical fre-
p
quencies are given by
1 1 dc 2 d2c
Ω2 (y)= A c A , (18)
cr,v 2"2(cid:18) dy (cid:19) − A(cid:18) dy2 (cid:19)#
and
1 1 dc 2 d2c
Ω2 (y)= A +c A , (19)
cr,b 2 2 dy A dy2
" (cid:18) (cid:19) (cid:18) (cid:19)#
and the resulting turning-point frequencies become
1
Ω2 (y)=Ω2 (y)+ , (20)
tp,v cr,v 4t2 (y)
ac
and
1
Ω2 (y)=Ω2 (y)+ , (21)
tp,b cr,b 4t2 (y)
ac
where t is actual wave travel time expressed by
ac
y dy
t = , (22)
ac
c (y)
Zyb A
e
with y being an atmospheric height at which the wave is initially generated.
b e
Finally, the cutoff frequency for torsional Alfv´en waves propagating in a thin
SOLA: driven_Alfven_final.tex; 18 January 2017; 1:23; p. 8
DrivenAlfv´enwaves
and non-isothermal magnetic flux tube embedded in the solar atmosphere is
Ω (y)=max[Ω (y),Ω (y)] . (23)
cut,τ tp,τ,v tp,τ,b
Havingobtainedthe turning-pointfrequenciesandthe cutofffrequency,we may
define the turning-point periods as
2π 2π
P = , P = , (24)
tp,v tp,b
Ω Ω
tp,v tp,b
and the cutoff period is
2π
P = . (25)
cut
Ω
cut
✶✵☎
✶✵✁
❞✆s✝ ✶✵✂
✉t✲♦❢❢♣❡r✐♦ ✶✵✄
❈
✶✵(cid:0)✂
✶✵(cid:0)✁
✵✿✵ ✵✿✺ ✶✿✵ ✶✿✺ ✷✿✵ ✷✿✺ ✸✿✵ ✸✿✺ ✹✿✵
②❬▼♠❪
Figure 5 Cutoff period for Alfv´en waves vs height y calculated from Equations (23) and
(25).
5. Discussion of Analytical and Numerical Results
Figure5displaysthecutoffperiod,P ,vs heighty.Aty =0,whichisthebot-
cut
tom of the photosphere, P =10 s. Higher up it grows,reaching its maximum
cut
value of about 300 s at y 0.2 Mm, and subsequently it falls off with height to
≈
its global minimum of about 0.03 s, which occurs at the solar transition region.
In the solar corona P increases again with y but its values are lower than 1
cut
s. This general trend of P (y) is consistent with that found by Murawski and
cut
Musielak (2010), however, the exact values of P (y) are different in the two
cut
cases. In particular, Murawski and Musielak (2010) found that for the straight
vertical equilibrium magnetic field P was higher than 40 s, while Figure 5
cut
reveals the corresponding values being a fraction of 1 s. This clearly shows how
strongly the cutoff depends on the geometry and strength of the background
magnetic field.
OurnumericalresultsshowthatAlfv´enwaves-oratleasttheirwavevelocity
perturbation - penetrate the transition region, whereas the magnetic field per-
turbation does not, as shown in Figures 3 and 4. The analytical results imply
SOLA: driven_Alfven_final.tex; 18 January 2017; 1:23; p. 9
Wo´jciketal.
✵✿✵✵✸✺ ✶✿✷
✵✿✵✵✸✵ ✶✿✵
✵✿✵✵✷✺
✵✿✽
❡r✵✿✵✵✷✵ ❡r
♦✇ ♦✇✵✿✻
P✵✿✵✵✶✺ P
✵✿✹
✵✿✵✵✶✵
✵✿✵✵✵✺ ✵✿✷
✵✿✵✵✵✵ ✵✿✵
✵✿✵ ✵✿✺ ✶✿✵ ✶✿✺ ✷✿✵ ✷✿✺ ✸✿✵ ✸✿✺ ✹✿✵ ✵✿✵ ✵✿✺ ✶✿✵ ✶✿✺ ✷✿✵ ✷✿✺ ✸✿✵ ✸✿✺ ✹✿✵
②❬▼♠❪ ②❬▼♠❪
Figure 6 Spectral power (in arbitrary units) of Alfv´en waves for wave period P = 35 s
obtainedfromthetimesignaturesofFigure3forr=0.1Mmandt=250s;forVθ (left)and
Bθ (right).
the same asthey alsoshowthatthe two wavevariablesbehave differently.How-
ever, there are some discrepancies between the analytical and numerical results
because the thin flux tube approximation, which is the basis for obtaining the
cutoff frequency, does not apply to our model of the solar corona because the
tubebecomesthickandtheconditiongivenbyEquation(13)isnotsatisfiedany
longer. The agreement between the analytical theory and our numerical model
is better in the solar chromosphere where the tube remains thin.
Because of the different behavior of the two wave variables, one may look
directly at the values of the turning-point frequencies given by Equations (20)
and (21) for each wave variable. Since in the photosphere for 0.2 Mm < y <
2.5Mm,P =0,asignalinmagneticfieldisnotabletopenetratethisregion,
tp,b
which explains why B is absent in the solar corona (Figure 3, right); this is
θ
an important result, which clearly shows the connections between the turning-
pointfrequencies andthe different behaviorofthe twowavevariables.However,
asignalcanbeseeninthisregioninV (Figure3,left).Indeed,forP =35sthe
θ d
condition of P <P is satisfiedonly up to y 0.3 Mm, while for y 0.1 Mm
tp,v
≈ ≈
P < P . For larger values of y, we have P > P and the Alfv´en waves
tp,b tp,bv
become evanescent.
Figure6illustratesspectralpowercorrespondingtoP =35sforV (left)and
d θ
B (right).Asuddenfall-offofpoweraty 0.2Mmcorrespondstoenergybeing
θ
≈
spentonexcitationofanon-zerosignalinB .ThusenergygoesfromV ,inwhich
θ θ
it was originally generated, into B . We point out that for linear Alfv´en waves
θ
propagating in homogeneous media, there is a perfect equipartition of the wave
kineticenergyassociatedwithV andthe wavemagneticenergyassociatedwith
θ
B ; our results show that this perfect equipartition of wave energy is strongly
θ
violated in inhomogeneous media (see our comments at the end of Section 3).
InthesolarcoronathevalueoftheP islargerthenalocalvalueofP and
d tp,v
P so the waves are evanescent in this region. However,our numerical results
tp,b
show a tunneling of V (left) to the solar corona with the signal in B (right)
θ θ
falling off with height more strongly, with and B being practically zero above
θ
y =1.5 Mm.
SOLA: driven_Alfven_final.tex; 18 January 2017; 1:23; p. 10