Reconnection-Driven Magnetohydrodynamic Turbulence in a Simulated Coronal-Hole Jet

27 Pages·2017·0.79 MB·English

https://ntrs.nasa.gov/search.jsp?R=20170000252 2019-04-11T11:44:43+00:00Z
Reconnection-Driven Magnetohydrodynamic Turbulence
in a Simulated Coronal-Hole Jet
Vadim M. Uritsky1,2, Merrill A. Roberts1,2, C. Richard DeVore2, and Judith T. Karpen2
1Catholic University of America, 620 Michigan Avenue NE, Washington, DC 20064 USA
2Heliophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771 USA
[email protected]
ABSTRACT
Extreme-ultravioletandX-rayjetsoccurfrequentlyinmagneticallyopencoronalholesonthe
Sun,especiallyathighsolarlatitudes. Someofthesejetsareobservedbywhite-lightcoronagraphs
astheypropagatethroughtheoutercoronatowardtheinnerheliosphere,andithasbeenproposed
thattheygiverisetomicrostreamsandtorsionalAlfv´enwavesdetectedin situinthesolarwind.
To predict and understand the signatures of coronal-hole jets, we have performed a detailed
statistical analysis of such a jet simulated with an adaptively refined magnetohydrodynamics
model. The results confirm the generation and persistence of three-dimensional, reconnection-
driven magnetic turbulence in the simulation. We calculate the spatial correlations of magnetic
fluctuations within the jet and find that they agree best with the Mu¨ller-Biskamp scaling model
includingintermittentcurrentsheetsofvarioussizescoupledviahydrodynamicturbulentcascade.
The anisotropy of the magnetic fluctuations and the spatial orientation of the current sheets
are consistent with an ensemble of nonlinear Alfv´en waves. These properties also reflect the
overall collimated jet structure imposed by the geometry of the reconnecting magnetic field. A
comparison with Ulysses observations shows that turbulence in the jet wake is in quantitative
agreement with that in the fast solar wind.
Subject headings: Magnetic reconnection – Sun: corona – Sun: magnetic fields – Turbulence – Waves
1. Introduction Magneticreconnectionorresistivediffusionacross
these sheets converts magnetic free energy to ki-
Magnetohydrodynamic(MHD)turbulenceplays
neticandthermalenergiesofthebulkplasmaand
a fundamental role in numerous systems of crit-
to kinetic energy of highly accelerated particles.
ical interest to heliophysics: the solar convective
Inlow-β plasmas,wherethemagneticfieldisdom-
zoneandatmosphere; theinterplanetarymedium;
inant, the current sheets form and the associated
the magnetotails of the Earth and other magne-
reconnection/diffusion processes occur preferen-
tized planets; and the heliopause at the inter-
tially near null points of the field. Null regions
face between the heliosphere and the interstellar
therefore act as generators of reconnection-driven
medium. In plasmas characterized by moderate
turbulence in highly conducting, low-β plasmas
to high values of plasma β ≡ nkBT/(B2/2µ0), such as the solar corona.
where n and T are plasma number density and
In general, turbulence occurs naturally in mov-
temperature, B is magnetic field strength, k is
B ing fluids characterized by a high Reynolds num-
Boltzmann’s constant, and µ0 is the permeability
ber, Re = v l/ν, where v is a typical ambient
of free space, thermal pressure reaches or exceeds l l
flow speed at the scale l and ν is the kinematic
the pressure exerted by the magnetic field, and
viscosity. (In resistive MHD plasmas, the mag-
the turbulent flows readily bend and fold the field
netic Reynolds or Lunquist number is defined us-
to produce current sheets throughout the volume.
ing the magnetic diffusivity η in place of the vis-
1
cosity.) The condition Re ≫ 1, satisfied in many timesareobservedbywhite-light coronagraphsto
astrophysical systems, ensures that a broad range propagate several solar radii from the Sun into
of inertial spatial scales l, bounded by the large the inner heliosphere. Neugebauer (2012) sug-
driving scale l and the small dissipative scale l , gested that these jets may be the origin of small-
D d
l ≪ l ≪ l , is supported. While the physical scale microstreams detected in situ in the solar
d D
mechanisms underlying energy dissipation at the wind(Neugebaueretal.1995). Becausemanyjets
dissipativescalel playacriticalrolebyproviding sensed remotely in the corona exhibit a distinc-
d
a sink for the energy injected into the system at tively helical structure that traverses the corona
thedrivingscalel ,theinertial-rangebehaviorin at highly supersonic speeds, leading to its iden-
D
between is, to a large extent, independent of the tification as a nonlinear Alfv´en wave, it is plau-
details of the dissipation process. This leads to a sible that coronal hole jets also are the source of
statistical self-similarity of fluctuations in the ve- suchwavesdetectedintheinterplanetarymedium
locityandmagneticfieldsacrossthosescales,even (Gosling et al. 2010; Marubashi et al. 2010).
inweaklycollisionaltocollisionlessplasmaswhere In this paper, we analyze a first-principles, 3D
the dissipation is dominated by wave-particle in- numerical simulation of the initiation and propa-
teractions or other kinetic effects (Schekochihin gation of a coronal hole jet (Karpen et al. 2016)
et al. 2009; Daughton et al. 2011; Leonardis et al. to establish and characterize its turbulent na-
2013). ture. The physical model underlying the simu-
The turbulent outflows from reconnection- lation is null-point magnetic reconnection occur-
driven systems can provide important clues about ringattheinterfacebetweentheambientcoronal-
the geometry of the reconnection region and en- hole flux of one polarity and a concentrated patch
able remote sensing of the driving mechanism of opposite-polarity flux provided by an embed-
through its characteristic signatures in the flow. ded bipole (Lau & Finn 1990; Antiochos 1990).
Such features include unstable velocity shear The topology of this configuration, in which an
and ensuing multiscale vorticity in the recon- inner system of flux that closes to the solar sur-
nection exhaust of terrestrial substorms (Keiling face is embedded within an outer system of flux
et al. 2009), fragmented current sheets and fila- that opens to the heliosphere, supports strong
mentsembeddedinlarger-scaleoutflowfromfield- electric currents associated with steep gradients
reversed configurations (Uritsky et al. 2001; Kli- in the magnetic-field direction at the interface be-
mas et al. 2004), and topological markers of the tweenthetwofluxsystems(Antiochos1996). Pre-
underlying magnetic-field configuration through viousCartesian, gravity-free, uniform-background
the three-dimensional (3D) geometry of the tur- 3D numerical simulations of such configurations
bulent flows (e.g. Biskamp 2003). These and (Pariat et al. 2009, 2010, 2015, 2016) demon-
other observational hallmarks of the reconnection strated that this model produces explosive jets
process have been identified in extreme ultravio- with helical structure, density-enhanced outflows,
let (EUV) images of the corona from Solar and and Alfv´enic wave fronts, in accord with observa-
Heliospheric Observatory and Solar Terrestrial tions. Theenergysourceforthejetsisthetwisted
Relations Observatory spacecraft (Uritsky et al. magnetic flux under the separatrix. Recent work
2007, 2013), flyby time-series data from MErcury (Karpen et al. 2016) extends those investigations
Surface, Space ENvironment, GEochemistry, and by including the effects of spherical geometry, so-
Ranging probe (Uritsky et al. 2011), and numer- lar gravity, density and magnetic-field stratifica-
ical data from high-resolution 3D simulations of tion, and an isothermal solar wind. The ensuing
prescribed, vorticity-laden MHD flows (Uritsky reconnection-driven jet wave front propagates un-
et al. 2010b). hindered into the outer corona, reaching 5 solar
Solar EUV and X-ray jets (Raouafi et al. 2016, radii in 1250 s, and its duration, length, diam-
and references therein) are transient, highly dy- eter, plasma-outflow speed, leading-front speed,
namic brightenings of the low solar corona that plane-of-the-sky transverse speed, kinetic energy,
producefastcollimatedoutflowsofplasma. When and helical morphology are all typical of observed
these events occur in the open magnetic fields of coronal hole jets.
coronal holes, the quasi-radial jet outflows some- We review the relevant statistical hierarchical
2
models of turbulent hydrodynamic and magneto- K41theoryassumesthatǫp isindependentofl for
hydrodynamic cascades, with and without inter- all orders p, with the result that
mittency, in §2. A concise summary of the nu-
mericalsimulation, whichisdescribedindetailby δvlp ∝lζp (4)
Karpen et al. (2016), is given in §3. The grid reg-
and
ularization used to facilitate our analysis is de-
ζ =p/3. (5)
scribed in §4. The main results of our analysis of p
the jet are presented in the succeeding three sec- The ζ are known as the structure function ex-
p
tions. Namely, we discuss noteworthy large-scale ponents. The first-order exponent replicates the
features of its radial structure in §5, analyze the velocity scaling in Equation (2), ζ1 = 1/3, while
statisticsofitssmall-scalevelocity-andmagnetic- the second-order exponent relates to the energy
field fluctuations in §6, and illustrate its filamen- spectral index in Equation (3), α=ζ2+1=5/3.
tary electric-current structures in §7. The paper
In practice, however, it is often found that the
concludes with a brief discussion of the implica-
linear approximation in Equation (5) is invalid.
tions of our results in §8.
Turbulent intermittency causes the energy dissi-
pation rate ǫ to vary with spatial scale l.
l
2. Hierarchical MHD Turbulence Models
Intermittency in turbulent flows reflects the
presence of intense small-scale dissipative struc-
A statistical framework for describing inertial-
turesbreakingglobalscale-invariance(She&Lev-
range turbulence was introduced by Kolmogorov
eque 1994; Boratav 1997; Uritsky et al. 2007;
(1941, hereafter referred to as the K41 model) for
Abramenko & Yurchyshyn 2010). Such structures
hydrodynamicfluids. Heassumedthattheenergy
are commonly observed both in laboratory ex-
dissipation is spatially uniform and isotropic, and
periments and in nature, and are successfully re-
that cross-scale interactions take place through a
producedinhigh-resolutionnumericalsimulations
local cascade process constantly breaking turbu-
(for a review, see She & Zhang 2009). To account
lent eddies into smaller pieces. Consequently the
for these effects, the energy dissipation rate is as-
(uniform) energy dissipation rate ǫ per unit mass
sumedtofollowtheKolmogorovrefinedsimilarity
scales as follows:
(KRS) hypothesis (Kolmogorov 1962; Stolovitzky
2 3
ǫ∝(δv ) /τ ∝(δv ) /l. (1) & Sreenivasan 1994),
l l l
Here δv is the characteristic velocity magnitude ǫp ∝lτp. (6)
l l
and τ is the typical eddy turnover time at scale l.
l
The new exponents τ are related to ζ via
This relation leads directly to the primary result p p
of the K41 theory:
ζp =p/3+τp/3. (7)
δv ∝ǫ1/3l1/3. (2)
l The particular values
An important consequence of the l1/3 scaling of
ζ0 =0 τ0 =0,
the turbulent velocity field is that the kinetic en-
ergy content E , obtained by integrating the ki- ζ3 =1 τ1 =0, (8)
k
neticenergyoverallwavenumbersbeyondk,obeys
are exact results of the conservation of energy in
the familiar law
steady incompressible fluids under the assump-
E ∝ǫ2/3k−α (3) tions of isotropy and homogeneity (Kolmogorov
k
1962;Politano&Pouquet1995). Thesevaluesalso
with spectral index α=5/3. are approximately correct for a much wider class
The K41 scaling relations have proved highly offlowsinwhichthelocaldissipationratesandthe
successful in analyses of turbulent energy spec- velocityfluctuationsexhibitthestrongcorrelation
tra (e.g. Bruno & Carbone 2013, and references underlying the KRS scaling.
therein). However, typically they fail to explain The key step in deriving the appropriate sta-
the higher-order statistics of turbulent flows. The tisticaldescriptionofintermittentturbulenceisto
3
determine the p dependence of τ (i.e., the depar- &Zhang2009). Thisleadstotheenergyspectrum
p
tureofζ fromthep/3law)forp≥2. Forinertial-
p E ∝(ǫv )1/2k−3/2, (13)
range hydrodynamic turbulence that is fully de- k A
scribed in terms of moment ratios of the energy
with spectral index α=3/2.
dissipationrateǫ ,She&Leveque(1994,hereafter
l
To describe the scale-dependent dissipation
the SL model) obtained the closed-form solution
regimes that typically occur in magnetized plas-
τp/3 =−2p/9+2[1−(2/3)p/3]. (9) mas, thestatisticalformalismdevelopedbyShe&
Leveque (1994) has been extended to include the
This immediately yields, from Equation (7),
IK phenomenology (Grauer et al. 1994; Politano
ζ =p/9+2[1−(2/3)p/3]. (10) & Pouquet 1995). The KRS then takes the form
p
TheSLmodelsatisfiesEquations(8)andpredicts ζp =p/4+τp/4 (14)
the energy spectrum exponent α ≈ 1.696, within
2% of the K41 prediction in Equation (5). How- with
ever, SL deviates strongly from K41 for higher
ζ0 =0 τ0 =0,
statistical moments affected by intermittent en-
ergy dissipation in quasi-one-dimensional vortex ζ4 =1 τ1 =0, (15)
filaments. The scaling corrections introduced by
in analogy with Equations (8) from hydrodynam-
SLtheoryhavebeenverifiedinmanyexperimental
ics. The dissipation rate exponents describing the
and theoretical studies (She & Zhang 2009). The
intermittent turbulence follow the rule
degree of intermittency is evaluated using higher
order (p>3) structure functions requiring exten- τp/4 =−p/8+[1−(1/2)p/4]. (16)
sive spatial averaging (Abramenko et al. 2008).
By comparison with hydrodynamic turbu- This results in (Politano & Pouquet 1995, here-
lence, magnetohydrodynamic turbulence exhibits after PP)
a rather different form of scaling, influenced by ζ =p/8+1−(1/2)p/4. (17)
the coupling of magnetic and velocity pertur- p
bations by Alfv´en waves. These effects in non- The energy spectrum associated with the second-
intermittent (uniform ǫl) magnetofluid turbulence order exponent ζ2 in the intermittent PP model
were first investigated independently by Irosh- has α ≈ 1.543. This correction is slightly larger
nikov (1963) and Kraichnan (1965); hereafter we (about 3%) than that of the SL model for non-
refer to them jointly as the IK model. This cou- magnetized fluids, while the higher moments dis-
pling is usually described in terms of Els¨asser playstillmoresignificantdeparturesfromthenon-
variables z∓ =v±B. In the IK framework, z+ intermittent scaling.
andz−eddiesinteractwhentheymeetwhiletrav-
Subsequent 3D magnetohydrodynamic turbu-
eling in opposite directions along the large-scale
lence simulations (Mu¨ller & Biskamp 2000, here-
magneticfieldB . Relativetokinematiceddiesin
0 after the MB model) have exhibited a hybrid be-
hydrodynamic turbulence, Alfv´enic eddies in the
havior between the SL and PP models, in which
Els¨asser field have a cross-scale energy transfer
K41-like scaling supported by a vortex cascade
ratereducedbythefactorτ /τ ,whereτ =l/v
A l A A combinedwithIK-likedissipativestructuresinthe
is the Alfv´en time at scale l and v is the Alfv´en
A form of current sheets. In this case, the ζ expo-
p
speed. Due to this weak interaction,
nents take the form in Equation (7) with
2 4
ǫ∝(τ /τ )(δz ) /τ ∝τ (δz ) /l (11)
A l l l A l τp/3 =−2p/9+[1−(1/3)p/3]. (18)
so that the Els¨asser perturbations scale as
Then
δzl ∝(ǫvA)1/4l1/4. (12) ζp =p/9+1−(1/3)p/3, (19)
Inweaklycompressibleregimes(Zank&Matthaeus and the energy spectrum in the intermittent MB
1993), the scaling behavior of velocity and mag- model has α ≈ 1.741. This model also satisfies
netic field perturbations repeats that of δz (She Equations (8).
l
4
The three hierarchical intermittency models in its top. Two still images and a corresponding on-
Equations (10), (17), and (19), as well as a num- line animation from the simulation comprise Fig-
ber of models proposed more recently, are conve- ure2, which shows the spherical domain toheight
niently represented by a unifying scaling ansatz r ≈ 5.4R⊙, selected magnetic field lines (white
containing three adjustable parameters q, s, and lines), and plasma velocity magnitude against the
d (Mu¨ller & Biskamp 2000): plane of the sky (color shading) at times t = 0 s
(top) and 3650 s (bottom). The jet onset was at
ζ =(p/q)(1−s)+d[1−(1−s/d)p/q]. (20)
p t≈ 2750 s, and the simulation was stopped at t =
4000 s.
Here q is a spatial exponent describing the basic
scaling δv ∝ l1/q, s is a temporal exponent re- A spherical domain of extent [1R⊙,9R⊙] ×
l [−9◦,+9◦]×[−9◦,+9◦]inradius,latitude,andlon-
flecting energy transfer time at the smallest iner-
tial scales, t ∝ls, andthe codimension d=3−D gitude was assumed, with open inner and outer
l
radial boundaries and closed side boundaries in
isdefinedbythedimensionD ofdissipativestruc-
tures embedded in three-dimensional space (Ta- both transverse directions. R⊙ = 7 × 108 m is
the radius of the Sun. The grid was exponentially
ble 1). The SL hydrodynamic model with vor-
stretched radially and linearly spaced in angle to
tex filaments (D =1) is recovered by substituting
divide the domain into nearly cubic cells through-
q = 3, s = 2/3, and d = 2 into Equation (20).
out. Adaptive mesh refinement, managed by the
The PP magnetohydrodynamic model is obtained
PARAMESH toolkit optimized for parallel com-
for q = 4, s = 1/2, and d = 1, with the dis-
puter architectures (MacNeice et al. 2000), was
sipative structures interpreted as current sheets.
usedtotargetlocalregionswheretheelectriccur-
The combination q = 3, s = 2/3, and d = 1
rent density was relatively strong. The volume of
yieldstheMBmodel. Morecomplexintermittency
maximallyrefinedgridblocksexpandedwithtime
regimes can be represented using Equation (20),
toresolvethesurfacecurrentsthatbuiltupatthe
including anisotropic scaling of velocity, magnetic
interfacebetweentheopenandclosedfluxsystems
and Els¨asserfield perturbations (see She& Zhang
and, after jet initiation, the filamentary volume
2009, and references therein).
currents associated with the propagating nonlin-
Bywayofcontrast,theclassicalBrowniannoise
earAlfv´en wavesgeneratedbytheonsetofimpul-
model(hereaftertheBNmodel)withuncorrelated
sivereconnection. Thecombinationofrefinements
fluctuations corresponds to q = 2 and s = 0 (d
increasedthetotalnumberofgridcellsbyanorder
arbitrary) in Equation (20), i.e.,
ofmagnitudeduringthesimulation,whichenabled
ζ =p/2. (21) us to resolve small intermittent structures.
p
The time-dependent, ideal, magnetohydrody-
Aswillbedemonstratedin§6,ourcoronal-jetsim-
namicequationsforconservationofmass,momen-
ulations deviate strongly from the uncorrelated
tum, and magnetic flux were solved in spherical
BN model, as well as from non-intermittent mag-
coordinates using the Adaptively Refined MHD
netic turbulence. They adhere most closely to the
Solver (ARMS; DeVore & Antiochos 2008). The
MB model expressed in Equation (19).
MHD energy equation was eliminated by assum-
ingthattheplasmaevolutionisisothermal,which
3. Coronal Jet Simulation
is the simplest and most computationally efficient
way to generate a supersonic solar wind (Parker
The coronal-jet numerical simulation that we
1958). ARMS employs Flux-Corrected Transport
analyze is described in detail by Karpen et al.
methods (DeVore 1991) to advance the resulting
(2016). Here, we summarize its main features to
setoffinite-volumeequationsontheadaptedgrid.
provide context for the turbulence analysis pre-
Theoretical developments in implicit large-eddy
sented in the following sections. The magnetic
simulation techniques (Grinstein et al. 2007) have
structure of the jet-producing region is shown at
demonstrated that certain numerical algorithms,
timet=0inFigure1,illustratingtheinnerclosed
when applied to ideal hydrodynamic systems, in-
(gray curves) and outer open (white curves) flux
corporate an implicit sub-grid-scale model that
systems and the dome-shaped separatrix between
emulates the viscous diffusion explicitly included
them with a magnetic null point (red spheroid) at
5
Table 1: Classification of turbulence models based on Equation (20)
Model q s d Cascading structures Dissipative structures
Non-intermittent
Kolmogorov 1941 (K41) 3 0 - Fluid vortices -
Iroshnikov & Kraichnan (IK) 4 0 - Alfv´en wave packets -
Brownian noise (BN) 2 0 - - -
Intermittent
She & Leveque (SL) 3 2/3 2 Fluid vortices Vortex filaments
Politano & Pouquet (PP) 4 1/2 1 Alfv´en wave packets Current sheets
Mu¨ller & Biskamp (MB) 3 2/3 1 Fluid vortices Current sheets
Fig. 1.— Close view of jet-producing region at t = 0 s, showing magnetic field lines in flux systems that
are open (white curves) and closed (gray curves), the sign and magnitude of the radial component of the
magneticfieldonthesolarsurface(10−4T,colorshading),andthehigh-β regionencompassingthemagnetic
null point (suspended spheroid)
.
6
in the Navier-Stokes equations. Such schemes the jet material trailing the Alfv´enic wave front
are particularly well-suited for simulating large- above the local, ambient solar-wind value. The
Returbulentflows. Flux-CorrectedTransportisa front progresses at the coronal Alfv´en speed (up
member of this class of algorithms (Drikakis et al. to 3000 km s−1) in the frame moving with the so-
2007). lar wind, so it can traverse one solar radius in less
Parameter values chosen for the simulation are than 250 s, as indicated by Figure 2.
representativeofcoronalholejetsobservedonthe
Sun. The base mass density ρ⊙ = 2.0 × 10−13 4. Grid Regularization
kg m−3 and uniform temperature T⊙ = 1.0×106
AlthoughtheadaptivegridutilizedbyARMSis
K are typical of the tenuous, relatively cool at-
essential toresolvingimportant jetfeaturesinthe
mosphere of coronal holes. Combined with solar
model while maintaining manageably sized data
gravity, the resulting thermal pressure causes the
files, it presents a formidable challenge when it
outflowing wind to become transonic at r ≈ 6R⊙
where it reaches 130 km s−1. A locally uniform, comes to analyzing those same features. In order
toexaminethepropertiesofthejets,especiallythe
ambient coronal-hole field was provided by a Sun-
centered monopole of strength B =−2.5×10−4 statistical self-similarity of the velocity and mag-
m netic fields, a regular grid is necessary. We there-
T at the surface. The embedded polarity of the
fore developed a method of regularizing the grid
jet-generating region was provided by a radially
while maintaining the integrity of the addaptively
oriented dipole of surface strength B = +3.5×
10−3 T, placed at (0◦,0◦) in latitudedand longi- gridded data. Due to its simplicity and bias to-
tude and at depth 107 m below the surface. As wards nearest neighbor points in determining the
valuesofthenewgridpoints,alinearinterpolation
shown in Figure 1, the superposition of these two
fieldshasamagneticnullpointatheight1.5×107 based on a Delaunay triangulation of the original
data points was chosen.
m above the surface. The footprint of the dome-
The regularization was implemented in the In-
shaped separatrix on the surface is a circle of ra-
dius2.2×107 m,whichistypicalofjet-generating teractive Data Language (IDL) using the Trian-
gulate and Trigrid functions as follows. After se-
regions on the Sun.
lecting a given two-dimensional slice of the model
The magnetic field within the dome is slowly
space with φ = 0, the adaptively gridded r and θ
energized by subsonic rotational motions imposed
coordinates were read into IDL. From this set of
at the base to mimic the photospheric motions.
The peak speed v = 25 km s−1, about 20% co-planar points, the Triangulate procedure con-
max structs a Delaunay triangulation, using the “di-
of the sound speed and less than 1% of the lo-
vide and conquer method” described in Lee &
cal Alfv´en speed. These flows were ramped up
Schachter (1980). The Delaunay triangulation
smoothly from zero at t = 0 s to full speed at t
produces a set of triangles from a set of arbitrary
= 1000 s, then held fixed thereafter. By time t≈
points,suchthatacircleformedbyconnectingthe
2750 s, when the reconnection started and began
verticesofanytriangledoesnotcontainanyother
to generate the jet, the motions had introduced
point. This maximizes the minimum angle of the
about 1.5 turns of maximum twist into the mag-
triangles,andthusensuresthatonlynearbypoints
netic field beneath the dome. The reconnection
are used to construct them, making it an optimal
onset was driven by an ideal kink-like instability
choice for the correct interpolation and display of
of the twisted magnetic field beneath the dome,
our adaptively-gridded data. After the triangles
as established by preceding Cartesian simulations
have been created, a spatially regular grid is over-
(Pariat et al. 2009; Rachmeler et al. 2010). The
laidonthembytheTrigrid function,andthevalue
resulting buckling of the dome drives together in-
at each point of the regular grid is given by a lin-
ternal twisted field and external untwisted field
earinterpolationofthevaluesoftheverticesofthe
across the separatrix electric current sheet, trans-
Delaunay triangle into which it falls. This again
ferring twist onto the external field lines where it
ensures that, since the circumcircle of any Delau-
can propagate away freely into the heliosphere in
nay triangle contains no other points, the values
theformofnonlinearAlfv´enwaves. Wavepressure
used in the interpolation are those closest to the
compresses the plasma, enhancing the density of
7
Fig. 2.— Far view of jet cross-section (at θ = 0◦) showing magnetic field lines (white lines) and velocity
magnitude (km s−1, color shading) against the plane of the sky at t = 0 s (top) and t = 3650 s (bottom).
An animation of this figure for t∈[0,4000] s at 25 s cadence is provided online.
8
desiredregulargridpoint, therebyminimizingthe tuations at low altitudes r < 1.4R⊙. Beyond
distortion of the original dataset. r ≈ 2R⊙, on the other hand, the maximum β
value converges rapidly to the average value, in-
5. Radial Structure of the Jet dicating that the local perturbations become neg-
ligible, and both values are very small. The large
The structure of the jet is substantially non-
spike in βmax near the photospheric boundary
uniform at small and intermediate spatial scales,
(r ≈ 1.03R⊙) coincides with the location of the
and exhibits a large-scale variability in the radial
null region near the separatrix, as is evident from
direction. In this section, we investigate both of
Figure 5.
theseeffectsatt=3650s,providingacharacteris-
Figure 4b shows the normalized mass-density
ticexampleofafullydevelopedturbulentcascade
fluctuation amplitude δρ, which we define as the
inthemodel. Temporalevolutionofthejetfluctu-
ratioofthestandarddeviationofthemassdensity
ationswillbeaddressedinthesubsequentsection.
σ to its mean value hρi,
ρ
Figure 3 shows constant-latitude cross-sections
ofthetransverseandradialcomponentsoftheve- 2 1/2
locity and magnetic fields at t = 3650 s as an ρ−hρi
σ θ,φ
exampleoffullydevelopedturbulence. Thetrans- δρ(r)≡ ρ = (cid:28)(cid:16) (cid:17) (cid:29) . (23)
hρi hρi
verse components are defined as θ,φ θ,φ
v ≡ v2+v2 1/2, The averaging denoted by h...iθ,φ and the calcula-
θφ θ φ tion of σ are done over latitude and longitude at
ρ
B ≡(cid:0)B2+B(cid:1)2 1/2. (22) eachradialdistancer. Atr <2R⊙,themassden-
θφ θ φ
sity has a relatively large fluctuation amplitude
The range of r c(cid:0)oordinates(cid:1) plotted here ex- that becomes particularly strong at r < 1.4R⊙,
cludes the strongest magnetic field near the sur- showing that the mass density is quite variable in
face, for better visibility of less intense structures this inner region and that the flows are compres-
forming at higher altitudes. The velocity field sional. At r > 2R⊙, in contrast, the fluctuation
inside the jet volume varies from less than 200 amplitude approacheszero, indicating that inthis
to more than 1700 km s−1, exhibiting compact outer region the flows are incompressible.
regions of high-speed perturbations embedded in The Wal´en (1944) number R is defined as the
w
a slower background flow. The magnetic field ratio of the transverse plasma flow speed to the
perturbation also varies by a significant amount. Alfv´en speed associated with the transverse mag-
The transverse components of the velocity and netic field,
the magnetic field have a similar spatial struc-
ture, spanning across multiple scales as expected V
R (r)≡ θφ . (24)
fpolrotassthuorbwusletnhtatfltohwe.reAsecmlobslearnicnevbesettiwgeaetnionthoefstphae- w *Bθφ/(µ0ρ)1/2+θ,φ
tial patterns of vθφ and Bθφ increases with r. As Foranideal,linearornonlinearAlfv´enwaveinan
demonstratedbelow,thisreflectsthedominanceof isotropic plasma, R = ±1. As Figure 4c shows,
w
Alfv´eniccouplingathigheraltitudes. Thecorrela- in our simulation R > 1 over the lower com-
w
tionbetweentheradialfieldcomponentsplottedin pressional region below 1.4R⊙, where the plasma
the bottom panels of Figure 3, on the other hand, ejected by the reconnection events generates sub-
ismuchlesspronounced. Itisnotshownhere,but stantial nonuniformities in the mass density. Be-
this lack of strong correlation persists even when yond r ≈ 2R⊙, on the other hand, Rw converges
the large-scale background v and B fields are re- to 1 and remains there at higher altitudes. This
moved. indicates that the upper region is strongly domi-
Figure 4a shows that the average value of nated by Alfv´en waves. The intermediate region
plasma β is small at all altitudes, signaling that between 1.4 and 2.0R⊙ in which the Wal´en num-
magnetic pressure prevails over thermal pressure. ber is greater than one but density fluctuations
Its maximum value βmax, however, is significantly are relatively small, is likely controlled by both
larger than the average and has substantial fluc- compressible and shear Alfv´enic modes.
9
V (r,ϕ), km/s B (r,ϕ), gauss
θϕ θϕ
4 1723 1.33
2 1292 0.99
s
e
e
gr 0 861 0.66
e
d
ϕ, −2
430 0.33
−4
0 0.00
1 .5 2 .0 2 .5 3 .0 1 .5 2 .0 2 .5 3 .0
V(r,ϕ), km/s B(r,ϕ), gauss
r r
4 738 -0.19
2 528 -0.69
s
e
e
gr 0 318 -1.18
e
d
ϕ, -2
109 -1.68
-4
-100 -2.17
1 .5 2 .0 2 .5 3 .0 1 .5 2 .0 2 .5 3 .0
r, R r, R
O • O •
Fig. 3.— Constant-latitude cross-sections (at θ = 0◦) of transverse (top) and radial (bottom) components
of velocity (left) and magnetic (right) fields at t=3650 s.
10