Connecting the Sun and the Solar Wind: The First 2.5 Dimensional Self-consistent MHD Simulation under the Alfvén Wave Scenario

0.95 MB·English

by  Takuma Matsumoto

Connecting the Sun and the Solar Wind: The First 2.5 Dimensional
Self-consistent MHD Simulation under the Alfv´en Wave Scenario
Takuma Matsumoto and Takeru Ken Suzuki
Department of Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8602, Japan
[email protected]
2
1
0 ABSTRACT
2
The solar wind emanates from the hot and tenuous solar corona. Earlier studies using 1.5 dimensional

n simulations show that Alfv´en waves generated in the photosphere play an important role in coronal heating
a
through the process of non-linear mode conversion. In order to understand the physics of coronal heating and
J
solarwindaccelerationtogether,itisimportantto considerthe regionsfromphotosphereto interplanetaryspace
5
as a single system. We performed 2.5 dimensional, self-consistent magnetohydrodynamic simulations, covering
2
from the photosphere to the interplanetary space for the first time. We carefully set up the grid points with

spherical coordinate to treat the Alfv´en waves in the atmosphere with huge density contrast, and successfully
]
R simulatethesolarwindstreamingoutfromthe hotsolarcoronaasaresultofthe surfaceconvectivemotion. The
footpoint motion excites Alfv´en waves along an open magnetic flux tube, and these waves traveling upwards in
S
the non-uniform medium undergo wave reflection, nonlinear mode conversion from Alfv´en mode to slow mode,
.
h and turbulent cascade. These processes leads to the dissipation of Alfv´en waves and acceleration of the solar
p wind. It is found that the shock heating by the dissipation of the slow mode wave plays a fundamental role in
-
the coronal heating process whereas the turbulent cascade and shock heating drive the solar wind.
o
r
Subject headings: Sun: photosphere— Sun: chromosphere — Sun: corona — stars: mass-loss
t
s
a
1. Introduction tialscalesfinallyheatsthecoronaanddrivesthesolarwind
[
(Matthaeus et al. 1999). Various phenomenological ap-

2 The coronal heating and solar wind acceleration are proacheshavebeenconsideredtoavoidthecomplexitiesof
vfundamental problems in solar physics. Although various theturbulence(Cranmer et al.2007;Verdini & Velli2007;
7physicalmechanismshavebeenproposedforcoronalheat- Bigot et al. 2008). In numerical simulations an incom-
0ing,itremainsunclearwhythehotcoronaexistsabovethe pressibleapproximationis usuallyadopted(Einaudi et al.
7cool photosphere and the high-speed solar wind emanates 1996;Dmitruk & Matthaeus2003;Van Ballegooijen et al.
6from there. The main difficulty arises due to the rapid 2011), although the mode conversion from Alfv´en to slow
.
9decreaseofthe density,amountingto morethan15orders mode seems to play an important role (Kudoh & Shibata
0of magnitude in the interplanetaryspace comparedto the 1999; Suzuki & Inutsuka 2005, 2006; Antolin & Shibata
1photospheric value. The huge density contrast between 2010; Matsumoto & Shibata 2010). One of the major
1the photosphere and interplanetary space has made the challenges in numerical simulation is to consider the huge
v:problem difficult to understand the energy transfer from density contrastbetween the solar photosphere and inter-
ithe Sun to the solar wind as a single system even within planetary space. In this paper, we show results of two-
Xthe magnetohydrodynamic (MHD) framework. dimensional MHD simulations, considering region from
r Alfv´en waveis believed to be a primary candidate that the solar photosphere and solar wind as a single system
a
drives the solar wind (e.g. McIntosh et al. 2011). Direct and include the details of wave reflection from the tran-
observations of propagating Alfv´en waves have been re- sition region, nonlinear mode conversion as well as the
portedafterthelaunchofHinodesatellite(De Pontieu et al. turbulent cascade for the first time.
2007;Nishizuka et al.2008;Okamoto & De Pontieu2011).
Since Alfv´en waves are notoriously difficult to dissipate, 2. Method
various physical processes have been proposed. The dissi-
pation of Alfv´en waves is the key behind the acceleration We have performed two-dimensional MHD simulation
of solar wind and the essential problem is to dissipate the with radiative cooling, thermal conduction, and grav-
Alfv´en wave which eventually will transfer the energy to ity. Our numerical model includes a single flux tube ex-
accelerate the solar wind. Recently, the mechanism of tended from a kilo Gauss (kG) patch in the polar region
turbulent cascade has been proposed in which the down- (Tsuneta et al. 2008) to the interplanetary space ( 20
∼
wardwavethatisgeneratedduetothereflectionofAlfv´en Rs). Our basic equations are
waveinthe gravitationallystratifiedatmosphere interacts
with the upward propagating Alfv´en wave and develops ∂ρ
+ (ρV)=0, (1)
Alfv´enic turbulence. Once the Alfv´enic turbulence is gen- ∂t ∇·
erated,theenergycascadeoftheturbulencetosmallerspa-
1
(a) 1.0
s
0.8 R
5 0.6 1
.
x 0.4 0
m ∆ρ/ρ 0.2
0.0
o
o 0 5 10 15 20
Z
(b) 1.0
s
0.8 R
0

0 0.6 2
0
1 0.4 .
0

x ∆ρ/ρ 0.2
0.0
m
1 2 3 4
o
o (c) log T [K] 6 (d) log ρ [g cm-3]
Z 10 10 -10 s
R
5
-12 3
-0
1
4
-14 x
6
3 -16
1.00 1.01 1.02 1.03 1.04 1.00 1.01 1.02 1.03 1.04
Fig. 1.— Results of MHD simulation of Alfv´en wave propagation from the solar photosphere to the interplanetary space.
(a) Normalized density fluctuation, (b) region that is magnified 5 times. The red squared region in (a) is equivalent to (b).
(c)Temperaturedistribution, (d) densitydistribution, the regionsshownaremagnified100times from(b). The redsquared
region in (b) is equivalent to (c) and (d). The white solid lines in each panel represent the magnetic field lines. Lengths are
shown in units of Rs = 6.96 105 km.
×
∂ρV BB (r,θ = π/2,φ) and ∂ = 0. Initial magnetic field is ex-
+ ρVV+P =ρg, (2) θ
∂t ∇·(cid:18) T − 8π (cid:19) trapolatedusing potential field approximationso that the
open field lines are extended from a kilo Gauss strength
∂B
+ (VB BV)=0, (3) to 10 Gauss at the height of 2,000 km. Initially, we
∂t ∇· − set an isothermal (104 K) atmosphere. We input ve-
∂ 1 locity disturbance in θ direction at the footpoint of the
E + ( +PT)V (V B)B+κ T flux tube in order to generate Alfv´en waves. We assume
∂t ∇·(cid:20) E − 4π · ∇ (cid:21) (4) white noise power spectrum with (1 km s−1)2 in total
=ρg V R, power, which is rather a good approximation to the ob-
· − served spectrum (Matsumoto & Kitai 2010). Since the
where, ρ,V,B, and T are mass density, velocity, mag- velocitydisturbancesarepre-determined,anyfeedbackef-
netic field, and temperature, respectively. PT indicates fects (Grappin et al. 2008) of the coronal disturbances on
total pressure, Pg +B2/8π, where Pg is gas pressure. To- the photospheric motions are ignored. Periodic bound-
tal energy per unit volume is described as = ρV2/2+ ary condition is posed in the φ direction. Our numerical
Pg/(γ 1)+B2/8π with specific heat ratioEγ = 5/3. g simulation is based on HLLD scheme (Miyoshi & Kusano
is grav−itational acceleration, GMˆr/r2, where G and M 2005) that is robust and efficient among the various kind
−
arethegravitationalconstantandsolarmass,respectively. of approximate Riemann solvers. The solenoidal condi-
We adopt anisotropic thermal conduction tensor κ along tion , B = 0, is satisfied within a round-off error by
∇·
magnetic field lines (Yokoyama & Shibata 2001). Radia- using flux-CT method (T´oth 2000). The TVD-MUSCL
tive cooling term R is assumed to be a function of local scheme enables us to archive the second order accuracy
density and temperature (Suzuki & Inutsuka 2005). Note in space, while the Runge-Kutta method gives the second
thatnootheradhocsourcetermsforheatingareincluded order accuracy in time. We use min-mod limiter in order
in the energy equation. tosuppressthenumericaloscillationaroundshocks,which
The vector form of our basic equations are appropri- is one of the standardtechnique in TVD-scheme. Our nu-
ately transformed into spherical coordinate system with merical domain extends from the photosphere to R = 20
Rs radially. The spacialresolutionis 25 km at the surface
2
and increasing with radius. The horizonal length is 3,000
1000
km at the photosphere with spacial resolution of 100 (a)
∼
km. Total grid points in our simulation are 8198 in radial
]
direction and 32 in horizontal direction. -1s
The turbulentheatingrate isestimatedby dimensional m 100
analysissinceviscosityandresistivityarenotincludedex- k
plicitlyinourbasicequations. First,wederivetheFourier [
r
component,vˆ ,oftheAlfv´enicdisturbance(v )asfollows. V
θ θ
10
vˆ (r,k )= v (r,φ)e−ikφrφrdφ (5) (b)
θ φ θ
Z ]
-1s 102
Then, energy spectral density, E(r,kφ), becomes m
k 101
1 [
E(r,kφ)= 2πr∆φ |vˆθ(r,kφ)|2+|vˆθ(r,−kφ)|2 , (6) V θ
(cid:2) (cid:3) 1
where∆φindicatestheangularsystemsizeinφdirection.
10-1
By using k and E(r,k ), we can estimate the energy ex- ]
changingraφte,ǫ(r,kφ),fφoracertainwavenumber,kφ,with [K 106 (c)
neighboring Fourier modes. Then ǫ(r,k ) becomes e
φ r
u
ǫ(r,k ) ρ¯E(r,k )3/2k5/2, (7) at 105
φ ∼ φ φ er
p
where ρ¯denotes the mean density averagedovertime and m
φ direction. As a turbulent heating rate, we use ǫ(r,kφ) e 104
whose wave number is larger than a critical wave number T
that is determined by numerical resolution. We choose
1015
k r∆φ/2π = 4 as the critical wave number that corre-
φ
sponds to the spatial resolution covering one wave length
]
3
by 8 grid points in our simulation. -m 1010
c
3. Results & Discussions [
Ne 105
The coupling between the magnetic field and the sur-
(d)
face convection excites upward propagating Alfv´en wave
(Steiner et al. 1998), and it could be an efficient energy 10-4 10-3 10-2 10-1 1 101
carrierinthesolaratmosphere. Consideringsuchscenario,
(R-Rs)/Rs
weperformed2.5D MHDsimulationscoveringregionsdi-
rectly from the photosphere to the interplanetary space.
Once the Alfv´en wave is forced to excite, the numerical
system attains quasi-steady state within 1,800 minutes. Fig. 2.— Comparison of the simulation and the ob-
Due to the dissipation of the Alfv´en wave, the initially servation is summarized below. Red solid lines in each
static and isothermal (104 K) atmospheres eventually de- panel represent results of the simulations; these values
velops a hot corona (106 K) and a high-speed (& 500 km are averaged over distance and over 30 minutes. (a)
s−1) solar wind (Figs. 1 and 2). The radial profiles of ve- The green crosses (Teriaca et al. 2003) and the blue tri-
locity, temperature, and density are quite consistent with angles (Zangrilli et al. 2002) represent the proton out-
the spectroscopic and interplanetary scintillationobserva- flow speeds in the polar region observed by SOHO. The
tions (Fig. 2). Even though previous one dimensional black circles with crossed error bars (Grall et al. 1996)
simulations (Suzuki & Inutsuka 2005, 2006) show similar are obtained by VLBA. The black circles with verticaler-
radial variations, the coronal heating and solar wind ac- ror bars (Habbal et al. 1995) indicate measurements by
celerationmechanisminourtwodimensionalsimulationis SPARTAN 201-01. (b) The blue circles (Banerjee et al.
essentially different from the previous ones. 1998) show the nonthermal broadening inferred from
SUMER/SOHO. The black crosses (Esser et al. 1999) are
The energy losses such as radiative cooling, thermal
derived empirically from nonthermal broadening based
conduction, and adiabatic cooling due to the solar wind
on the UVCS/SOHO measurements. (c) The black cir-
are the main cooling processes in the solar atmosphere.
cles (Fludra et al. 1999) show electron temperature by
In order to maintain the solar corona, heating processes
CDS/SOHO. (d) The black circles (Wilhelm et al. 1998)
are necessary to balance the cooling processes. As shown
and the blue stars (Teriaca et al. 2003) are data based
in the panel (a) of figure 3, the energy flux of the Alfv´en
on observations by SUMER/SOHO and by CDS/SOHO,
wave decreases monotonically, a part of which is trans-
respectively. The green triangles (Teriaca et al. 2003)
ferred to the solar wind. Although a sizable fraction of
and orange squares (Lamy et al. 1997) are observed by
fluxisdecreasedinthechromospherebythereflectionand
LASCO/SOHO.
dissipation processes, the energy flux that is supplied to
3
] 1.00
1
-s 108
2 (a)
-m 107
c
g a|
r 106 V
[e +/ 0.10
a
ux 105 v
d
l
f |

gy 104
r
e
n 103 0.01
E
0.001 0.010 0.100 1.000 10.000
102
(R-Rs)/Rs
e (b)
t
a Fig. 4.— Alfv´enwavenonlinearityasafunctionofradius.
r
g 10-5 Alfv’en wave nonlinearity is determined as (dva+)=(Vθ
olin-1s] aBrθe/a√r4epπrρe)s/e2n,tsantdheVtArainssAitlifovn´enresgpioened.. The black-hatche−d

o3
C-m 10-10
/ c
g g Shock heating the turbulent heating rate from the Fourier component of
n r Alfv´enic disturbances.
ati [e 10-15 Turbulent heating In the chromosphere, both shock and turbulence con-
e Radiative cooling tributetotheheating. Thewavenonlinearity,whichisde-
H
10-20 Adiabatic cooling fined as wave amplitude divided by phase speed, quickly
increases in the chromosphere with the rapid expansion
0.001 0.010 0.100 1.000 10.000 of the magnetic flux tube (Fig. 4). As a result, com-
(R-Rs)/Rs pressive waves are generated effectively by the nonlinear
modeconversionofAlfv´enwaves. Theturbulentheatingis
also important in the chromosphere because the Alfv´enic
Fig. 3.— Alfv´en wave energy flux and its dissipation
turbulence is developed efficiently due to both phase mix-
processes. (a) Alfv´enwaveenergyflux asa function ofra-
ing (Heyvaerts & Priest 1983) as well as wave reflection
dius. (b) Heating and coolingrate as a function ofradius.
(Matthaeus et al.1999). Sincethefluxtuberapidlyopens
The green solid line shows shock-heating rate estimated
near the photosphere, the Alfv´en speeds of the neighbor-
by counting sudden entropy jumps. The red-hatched area
ing field lines are different with each other. Due to the
indicates turbulent heating rate estimated from Fourier
difference in Alfv´en speeds across the magnetic field, the
and dimensional analysis. The black solid line and the
Alfv´en waves along the neighboring field lines gradually
blue solid line show radiative and adiabatic cooling rates,
become out of phase, even though the waves are in phase
respectively. The black-hatched area represents the tran-
at the photosphere, which creates strong shear to dissi-
sition region.
pate their wave energy. In addition to the phase mixing,
the rapiddecrease ofthe density in the chromosphereand
the transition region causes increase in the Alfv´en speed
the corona is wellabove 105 erg cm−2 s−1, a typical num-
thatfinallyleadstothe reflectionoftheAlfv´enwave. The
ber that is required to maintain the corona and the solar
nonlinear wave-wave interaction between the pre-existing
wind(Hansteen & Leer1995). Thedissipationmechanism
outwardcomponentandthereflectedcomponentdevelops
of the Alfv´en wave should be different in various regions
turbulent cascade.
depending on the background medium.
In the corona, the shock dissipation is the main con-
Thepanel(b)offigure3showscomparisonofeachcom-
tributor to the heating although the turbulence is also
ponent of the heating and cooling rates. The heating is
effective in the lower corona. Passing through the tran-
separated into the compressive (dilatational) and incom-
sition region, the wave nonlinearity decreases rapidly be-
pressible (shearing) parts. The green solid line shows the
cause of the large Alfv´en speed in the corona, so the local
compressive component; compressive waves are generated
shock formation in the corona is not significant. Instead,
duetothenonlinearmodeconversionfromtheAlfv´enwave
compressive waves are generated by the vertically fluctu-
to compressive (or slow mode) wave (Kudoh & Shibata
ating motion of the transition region; the nonlinearly ex-
1999)andthesecompressivewaveseventuallysteepeninto
cites longitudinalwavesin the chromospherecontinuously
shocks. We estimate the heating rate by counting the
tap the transition region (Kudoh & Shibata 1999), which
entropy jumps at shock fronts in the simulations. The
further excites upward propagating compressive waves in
red-hatchedarea shows the incompressible heating, which
the corona. The reflected wave component drops off sig-
is done by the dissipation of Alfv´enic turbulence owing
nificantly above the transition region (Fig. 5) because
to strong shearing motion at small scales. We estimate
the Alfv´en speed does not change so much. As a result,
4
2> er 1
+) w (a)
a o
v p 10-1
d
<( 0.10 zed 10-2
> / ali 0.01 Rs
2) m 10-3 0.1 Rs
-
a r 1 Rs
v o
d N 10-4 10 Rs
(
< 0.01 1 10
k R∆φ/2π
0.001 0.010 0.100 1.000 10.000 φ
(R-Rs)/Rs
0
Fig. 5.— Energy ratio of downward propagating Alfv´en
wave to upward propagating Alfv´en wave. The energy of -1
downward(+)/upward(-)propagating Alfv´en wave is pro- x
hpaotrcthioendalarteoa(rdevpar±es)e2nt=s t(hVeθt∓ranBsθit/i√on4πreρg)i/o4n.. The black- nde -2
i

r
e
w -3
the turbulent cascadeis suppressedin the subsonic region o
(1.02 to 4 Rs). The power index of the energy spectral P
-4
density of Alfv´enic disturbance is significantly softer than
(b)
-5/3,whichindicates that the energycascadingto smaller
-5
scales is not effective in this region (Fig. 6). The shock
heating compensates for the absence of turbulent heating 0.001 0.010 0.100 1.000 10.000
in order to balance the cooling there. Generally, heating
(R-Rs)/Rs
below the sonic point controls mass loading to the solar
wind, so we suggest that the shock heating mechanism
works efficiently to determine the mass loss rate from the Fig. 6.— (a) Normalized power spectral density with
sun. In our simulation, mass loss rate is of the order of respect to normalized the horizontal wave number. The
10−14 M⊙ yr−1 which agreesreasonablywell with the ob- symbols of plus, asterisk, diamond, and triangle show the
served value. power spectral density at 0.01, 0.1, 1, and 10 Rs, respec-
tively. (b) Power index of the turbulent power spectral
Inthesolarwindaccelerationregion(R&4R ),again,
s
densitywithrespecttoheight. Theverticalaxisrepresents
both turbulent heating and shock heating are compara-
thepowerindexofAlfv´enicturbulentspectrum. Thehori-
ble. The wave nonlinearity, once dropped above the tran-
zontaldottedlineindicatesthevaluesofKolmogorovtype
sition region, increases gradually with radius due to the
turbulence, 5/3. The black-hatched area represents the
decrease in Alfv´en speed. Even though the nonlinearity
−
transition region.
is still small, Alfv´en waves suffer nonlinear effects due to
theirlongtravelingdistance. Then,compressivewavesare
locally excited by the mode conversion, and these waves
(e.g., Perez & Boldyrev 2008).
finally get dissipated by the shocks. The turbulent heat-
ing works effectively in the solar wind. The signature of Our two-dimensionalMHD simulationshowsthat both
turbulent cascade can be seen clearly in the power spec- the shock and the turbulence are important for the coro-
tra of Alfv´en waves (Fig. 7). Moreover, the turbulent nal heating and the solar wind acceleration. We showed
cascade is not isotropic but anisotropic (Shebalin et al. that the energy exchange between the Alfv´en mode and
1983; Goldreich & Sridhar 1995); the direction of turbu- slowmodeiseffective,althoughpreviousMHDsimulations
lent cascading is perpendicular to the background mag- of turbulence in homogeneous media show the decoupling
netic field. The energy cascading is triggered by the non- of Alfv´en and compressivemodes (Cho & Lazarian2003).
linearwave-waveinteractionbetweentheoutgoingandre- The inhomogeneity of the backgroundmedium due to the
flectionwaves(Matthaeus et al.1999). Theincreaseofthe density stratification and the rapidly expanding flux tube
reflectioncomponentofAlfv´enwavescanbe seeninfigure is essential to understand the energy transfer processes
7. The work done by the gas and the magnetic pressure inthe solaratmosphere. Inthe previous1DMHD simula-
from the turbulence is also of the same order and found tions(Suzuki & Inutsuka2005,2006),theshockheatingis
tobe sufficienttoacceleratethe solarwind. Figures5and over-estimated because the geometrical expansion dilutes
7 indicate that the critical balance state, krvA kφvθ, the shocksin a multidimensionalsystem. We showedthat
of Alfv´enic turbulence (Goldreich & Sridhar 1995∼) is at- the shock dilution is not so significant in a 2D system.
tained as R increases from the wave-like state, or weak The turbulent cascading process in our simulation results
turbulencestate,(krvA >kφvθ,atmost)inthelowcorona from 2D nonlinear terms while the previous studies (e.g.
5
Banerjee, D., Teriaca, L., Doyle, J. G., & Wilhelm, K.
1998, A&A, 339, 208
R= 1.1 Rs R= 6.0 Rs
Bigot,B., Galtier, S., & Politano,P.2008,A&A, 490,325
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