Alfv\'enic Wave Heating of the Upper Chromosphere in Flares

0.32 MB·

by  Jeffrey W. Reep

Draftversion January11,2016
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
ALFVE´NIC WAVE HEATING OF THE UPPER CHROMOSPHERE IN FLARES
J. W. Reep
NationalResearchCouncilPost-DocProgram,NavalResearchLaboratory,Washington, DC20375USA
and
A.J.B. Russell
DivisionofMathematics,UniversityofDundee,Nethergate, Dundee,DD14HN,Scotland,UK
Draft version January 11, 2016
6
1 ABSTRACT
0
We have developed a numerical model of flare heating due to the dissipation of Alfv´enic waves
2
propagating from the corona to the chromosphere. With this model, we present an investigation of

n the key parameters of these waves on the energy transport, heating, and subsequent dynamics. For
a sufficiently high frequencies and perpendicular wave numbers, the waves dissipate significantly in the
J upper chromosphere, strongly heating it to flare temperatures. This heating can then drive strong

8 chromospheric evaporation, bringing hot and dense plasma to the corona. We therefore find three
important conclusions: (1) Alfv´enic waves, propagating from the corona to the chromosphere, are

] capable of heating the upper chromosphere and the corona, (2) the atmospheric response to heating
R
due to the dissipationofAlfv´enic wavescanbe strikinglysimilar to heatingby anelectronbeam, and
S (3) this heating can produce explosive evaporation.
.
h Subject headings: Sun: chromosphere, Sun: corona, Sun: flares, Sun: atmosphere, Sun: transition
p region, waves
-
o
r 1. INTRODUCTION to the classic CTTM, discussed by Fletcher & Hudson
t
s (2008), Brown et al. (2009), Krucker et al. (2011), and
The standard model of solar flares, referred to as
a
Melrose & Wheatland (2014). It has been suggested
[ the CSHKP model (Carmichael 1964; Sturrock 1966;
Hirayama1974;Kopp & Pneuman1976),explainsmany that these issues could potentially be resolvedif some of
1 observational features of flares, assuming that they are the flare energy were transportedthroughthe coronaby
v waves, and used to either accelerate electrons in higher
driven by magnetic reconnection. After the reconnec-
9 tioneventtriggers,betweenapproximately1030–1033erg density regions (Fletcher & Hudson 2008) or reacceler-
6 ate energetic particles (Brown et al. 2009; Varady et al.
is released into the plasma, driving intense heating
9 2014).
and brightening across the electromagnetic spectrum
1 In this paper, we examine Alfv´enic waves as a heat-
(Fletcher et al. 2011). It is not clear, however, how
0 ing mechanism that may act separately or in addi-
that energy is partitioned between in situ heating of the
1. corona, particle acceleration, and wave generation, nor tion to the thick-target model. Magnetohydrodynamic
0 to what extent the observable features of a flare depend waves (Alfv´en 1942) are observed ubiquitously in the
6 on the balance between different types of coronalenergy corona(Tomczyk et al.2007; McIntosh et al.2011), and
1 transport. often considered a leading candidate to explain coronal
: heating (Klimchuk 2006) and the FIP effect (Laming
v The collisional thick-target model (CTTM) (Brown
2004, 2015). For flares, Alfv´en and guided fast waves
i 1971) assumes that the released energy goes into ac-
X produced during reconnection can deliver concentrated
celeration of coronal particles, primarily electrons, to
Poynting flux to the chromosphere (Birn et al. 2009;
r extremely high energies. Down-going particles stream
a Russell & Stackhouse 2013), where they damp in the
through the corona, eventually colliding with the much
cool, partially ionized plasma (De Pontieu et al. 2001;
denser chromosphere where they lose energy through
Khodachenko et al. 2004; Soler et al. 2015). Simula-
Coulomb collisions. This energy loss in turn heats the
tions of magnetic reconnection show that Alfv´en waves
chromosphere,drivingevaporationintothecorona,heat-
carry a large fraction of the released energy (> 30%) in
ing the loop and producing the sharp rises in intensi-
low β plasma (Kigure et al. 2010). Previous studies by
ties in the soft X-rays and extreme ultraviolet. Ob-
Emslie & Sturrock(1982)andRussell & Fletcher(2013)
servations of hard X-ray (HXR) bursts in flares show
haveshownthatenergytransportby Alfv´enicwavescan
without question the braking of high energy electrons
in the chromosphere, with as many as 1036s−1 in- explain temperature minimum heating observed in so-
lar flares, where temperature rises approximately 100K
ferred for large flares (e.g. Holman et al. 2003), how-
(Machado et al. 1978; Emslie & Machado 1979). Here,
ever, these observations do not exclude the possibility
we emphasize the ability of Alfv´enic waves to heat the
ofadditionalenergy transportby means other than run-
upper chromosphere.
away particles. Indeed, the flux of high energy electrons
Following the formalism of Russell & Fletcher (2013)
braking in the chromosphere presents several challenges
and references therein, we combine a hydrodynamic
model with energy transport through Alfv´enic waves,
jeff[email protected] whereby the waves propagate from the reconnection site
[email protected]
2 Reep & Russell
in the coronatowards the chromosphere. We present re- whichreducestothestandardexpressioninthehighfre-
sults from simulations that vary the wave parameters in quency limit and in the fully ionized case. The wave
order to show directly that not only can waves heat the dampingisduetocollisionsbetweenthedifferentspecies,
temperatureminimumregion,buttheycanalsoheatthe which decreases the wave amplitude and heats the local
upper chromosphere. Further, the heating can appear plasma as the waves propagate along the flux tube. As
extremelysimilartoanelectronbeam,andcandriveex- in Emslie & Sturrock (1982), the heating term Q(z) is
plosive evaporation. We comment on the implications calculated from the decreasing Poynting flux as
for the interpretation of observations of solar flares.
dS
Q(z)= (5)
2. HEATINGBYALFVE´NICWAVES −dz
wherethefrictionduetotheCowlingtermheatstheions
Emslie & Sturrock (1982) developed a model of
and the rest heats the electrons.
Alfv´en wave heating to explain the observed temper-
The WKB model is reasonably easy to implement
ature minimum region heating observed in solar flares
within existing codes, accommodates a wide range of
(Machado et al. 1978). Using a WKB approximation,
wave properties, has a small computational overhead,
theyderiveanexpressionfortheperiod-averagedPoynt-
and is an accurate approximation when used appropri-
ing flux as a function of distance along a magnetic flux
ately. The main restriction is that the derivation as-
tube with a finite resistivity, where decreases in the
sumes that the parallel wavelength, 2πv (z)/ω, is less
Poyntingfluxareassumedtoheattheplasma. Weadopt A
than or similar to the gradient length scale of v (z),
theirmodel,althoughwederivetheWKBresultusingan A
making reflection negligible. This is a good approxima-
ambipolarresistivityinsteadoftheclassicalresistivityto
tion for Alfv´en waves with frequencies of 1Hz or higher
better account for ion-neutral collisions in the chromo-
once they are in the chromosphere. Since the model
sphere, which are vitally important for wave damping
does not accountfor wavereflection atthe transitionre-
(Piddington 1956; Leake et al. 2014).
gion,whichcanbe substantial(Emslie & Sturrock1982;
TheAlfv´enicwavesareinjectedatthetopofthemodel
Russell & Fletcher2013),wesetS toproduceasuitable
with period-averagedPoyntingflux S , whichdamps ac- 0
0
Poyntingfluximmediatelybelowthetransitionregionso
cording to Equation 2.18 of Emslie & Sturrock (1982):
thatappropriateheatingratesareobtainedforthe chro-
z dz′ mosphere and acceptthat the model underestimates the
S(z)=S exp (1) coronal heating associated with a given level of chromo-
0 −Z0 LD(z′)! spheric heating.
where z is the curvilinear coordinate along the loop and 3. NUMERICALMODEL
L (z) is an effective damping length, given by
D We have implemented the Alfv´en heating model out-
lined in Section 2 in the state-of-the-art Hydrodynam-
−1
1 1 icsandRadiationCode(HYDRAD; Bradshaw & Mason
L (z)= +
D L (z) L (z) 2003), which solves the one-dimensional equations de-
⊥ k !
scribing conservation of mass, momentum, and energy
−1
η k2c2 η ω2c2 for a two-fluid plasma confined to an isolated magnetic
= k x + ⊥ flux tube (the currentversion’sequations are detailed in
4πv 4πv3
A A ! Bradshaw & Cargill2013). Thecodedoesnotevolvethe
4πv3 magneticfield,sothatratherthanevaluatingthepropa-
= A (2) gationanddampingofAlfv´enicwaves,thecodeemulates
c2(η k2v2 +η ω2)
k x A ⊥ wave heating with the WKB approximation.
The model chromosphere is based on the VAL C
wherev isthelocalAlfv´enspeed,cthespeedoflight,k
A x
model (Vernazza et al. 1981), along with the approxi-
theperpendicularwavenumber,ωtheangularfrequency,
mation to optically thick radiative losses prescribed by
andη andη theperpendicularandparallelresistivities.
⊥ k
Carlsson& Leenaarts (2012), and the effects of neutrals
The perpendicular wavedamping termincludes Cowling
resistivity (e.g. Soler et al. 2013), such that as detailed in Reep et al. (2013). The electron beam
heating model used in Section 4 is based on Emslie
η =η +η (3) (1978). For all of the simulations here, we employ a full
⊥ k C
loop with length 2L = 60Mm, initially tenuous and in
m (ν +ν ) B2ρ
= e ei en + n hydrostatic equilibrium, semi-circular and oriented ver-
nee2 c2νniρ2t(1+ξ2θ2) tically, and assume that the heating is symmetric about
the apex.
We use the subscripts e, i, n, and t to refer to elec-
An important feature of HYDRAD is its abil-
trons, ions, neutrals, and total, respectively. Each ν
ity to solve for non-equilibrium ionization states
referstoacollisionfrequency(the equationsarelistedin
(Bradshaw & Mason 2003). Since the resistivities de-
Russell & Fletcher 2013, and note a typo in that work:
pend on collisions between ions and electrons as well
ν should scale as T1/2), while n is the number density,
ni i as between ions and neutrals, it is critically important
ρ the mass density, ξ the ionization fractionof hydrogen
to properly treat the ionization state of the plasma.
(ξ = ρ /ρ ), and θ = ω/ν . The local Alfv´en speed is
i t ni In particular, if the plasma is rapidly heated, the
modified by the presence of neutrals:
ionization state may lag behind the actual tempera-
B 1+ξθ2 1/2 ture. HYDRAD usesionizationandrecombinationrates
v (z)= (4) takenfromtheCHIANTIv.8database(Dere et al.1997;
A √4πρ 1+ξ2θ2
t
(cid:16) (cid:17)
Alfv´enic Wave Heating 3
Del Zanna et al.2015)tosolvefortheionizationfraction chromosphere is strongly heated, as the Poynting flux
of a given element. We treat radiative losses using a full sharply decreases across this layer so that only minus-
calculation with CHIANTI. cule amounts of energy are carried to the temperature
Equation 5 is taken as the heating term in the energy minimum region. Deeper inthe chromosphere,the heat-
equation. The ionization balance is solved with the fol- ing falls off as the Poynting flux dissipates, and is not
lowing equation (Bradshaw & Mason 2003): a smooth function primarily due to changes in the ion-
ization state of the plasma. The pressure increase in
∂Yi ∂ the upper chromosphere is sharp enough that material
+ (Y v)=n (I Y +R Y I Y R Yi)
∂t ∂s i e i−1 i−1 i i+1− i i− i−1 explosively evaporates, reaching over 200km s−1 in 10
(6) seconds of heating. If the heating were sustained, the
where Yi is the fractional population of an ionization density in the corona would increase significantly, caus-
state i of element Y, and Ii and Ri are the ionization ing brightening across the extreme ultraviolet and soft
and recombination rates from and to i (respectively). X-rays characteristic of flares.
Solving this equation for hydrogen gives the fractional For comparison,consider heating by an electron beam
population of ion and neutral densities in the chromo- in the CTTM (using the model of Emslie 1978 with a
sphere that determine the damping length. sharp cut-off). We adopt a low-energy cut-off E =
c
To study the effects of chromospheric heating by 20keV, spectral index δ = 5, and energy flux F =
0
Alfv´enic waves, we have run 24 numerical experiments. 1010erg s−1 cm−2 (equal to the Poynting flux consid-
These simulations cover a wide range of possible val- ered), shown in the bottom row of Figure 1. Compared
ues, and allow for systematic investigationof wave heat- to the previous simulation, slightly more energy is de-
ing. We vary the wave frequency f = ω/2π between [1, posited in the corona as the electrons collide with ambi-
10, 100]Hz and the perpendicular wave number at the entparticlesthere,andacomparableamountofenergyin
loop apex kx,a between [0, 10−5, 10−4, 4 10−4]cm−1. the chromosphere. The temperature in the upper chro-
×
For evaluation of LD from Equation 2, we assume that mosphere and corona rises slightly higher than in the
the magnetic field has a photospheric value B0 = 1000 previoussimulation,while the evaporationagainreaches
G, decreasing along the flux tube with the pressure as about 200km s−1. The atmospheric response is nearly
B(z) = B (P(z))0.139 (as in Russell & Fletcher 2013), identical, and without a direct measure of the energy
0 P0 input, would be difficult to distinguish observationally.
which is constant in time. Since the density and mag-
The explanation for the different wave-driven behav-
netic field vary with position, the Alfv´en speed v also
A
iors – namely gentle versus explosive evaporation – is
varies. We also adopt two different dependencies for k
x
straight-forward. As seen in Equation 2, the damp-
asafunctionofposition: withk (z)=k (B(z))(linear
x x,a Ba ing length is shorter for higher frequencies (which in-
in B) for magnetic expansion in one dimension as in an crease the damping by perpendicular resistivity) or
arcade geometry, and k (z)=k B(z) (as the square higher perpendicular wave numbers (which increase the
x x,a Ba damping by parallel resistivity). In this regard, our
root) for magnetic expansion in twqo dimensions as in a simulation results are consistent with the findings of
flux tube that expands radially with height. We do not
Emslie & Sturrock (1982), who showed that waves with
scale the flux density with the changing cross-sectional
higher frequency or perpendicular wave number do not
area implied from the expansion of B(z).
penetrate the deep chromosphere because they dissipate
higher in the atmosphere.
4. RESULTS The top row of Figure 2 explicitly shows the change
We consider first a simulation with wave heating for with wave frequency, with plots of heating from three
k = 10−5cm−1, f = 10Hz, and k scaling linearly simulations with frequencies of 1, 10, 100Hz, all for the
x,a x
with the magnetic field. The top row of Figure 1 shows same wave number, k =0. Note that without damping
x
the atmospheric response to 10 seconds of heating, with from k , waves barely heat the corona. Similarly, the
x
an initial Poynting flux of 1010erg s−1 cm−2 (note the effect of increasing k for a fixed frequency is seen by
x
x-axisislogarithmicandextendsfromfoot-pointtofoot- comparing the top center plot in Figure 2 (k = 0), the
x
point). With increasing depth into the chromosphere, top row of Figure 1 (k =10−5 cm−1), the bottom left
x,a
thedensityrisesandtheionizationfractionfalls,increas- panel of Figure 2 (k = 10−4 cm−1), and the middle
x,a
ing the effectiveness of ion-neutral friction. The temper- row of Figure 1 (k =4 10−4 cm−1).
ature minimum region is strongly heated, with a non- x,a ×
Finally, supposing that k (z) does not vary linearly,
negligibleamountofheatinginthe upperchromosphere. x
but as the square root of the magnetic field (k (z) =
As the temperature rises slowly, so does the pressure, x
k B(z)/B ), the last two plots of Figure 2 repeat
causingagentleevaporationtoforminthetransitionre- x,a a
gion,reachingabout50kms−1in10seconds(theplotde- the simulations in Figure 1, with otherwise identical pa-
rampeters. Themaindifferenceisthatbecausethedepen-
fines right-movingflowsas positive, left-moving asnega-
denceonthe magnetic fieldis reduced,the wavenumber
tive). Thecoronaisessentiallyunaffectedbythesewaves,
is higher at larger heights, so that the waves dissipate
asthey propagatethroughwithonly minimaldamping.
more of their energy there.
Contrast this now with a simulation that has a much
higherperpendicularwavenumber,k =4 10−4cm−1,
x,a × 5. CONCLUSIONS
but otherwise equalproperties, shownin the middle row
of Figure 1. Due to the increase in the wave number, The main conclusions are as follows:
the waves are strongly damped in the upper atmosphere
by collisions between ions and electrons. The upper (1) Flare-generated Alfv´enic waves can heat not only
4 Reep & Russell
Figure 1. A comparison between three scenarios: heating by Alfv´enic waves with f = 10Hz, kx,a = 10−5cm−1 (top row), kx,a =
4×10−4cm−1 (middlerow),andanelectronbeamwithEc=20keV(bottomrow). Atafewselectedtimeperiods,thefirstcolumnshows
thetotalenergydepositedintheloop,thesecondcolumntheelectrontemperature,andthethirdcolumnthebulkflowvelocity. Notethat
thex-axisislogarithmicintheseplots,withtheloopapexat30Mm. Thevelocityplotsdefineright-movingflowsaspositive,left-moving
negative. Thedashedverticallinesmarktheinitialtransitionregionboundary.
Figure 2. Energy deposition plots for wave heating with various parameters. The top row shows three simulations with kx = 0, i.e.
without perpendicular damping, and f = 1,10,100Hz, respectively. The bottom left plot shows kx,a = 10−4cm−1 and f =10Hz. The
lasttwoplotsarethesameasthewaveheatingsimulationsinFigure1,exceptthat theirwavenumberscalesaskx(z)=kx,apB(z)/Ba.
Notethatthex-axisislogarithmic,asbefore.
Alfv´enic Wave Heating 5
the temperature minimum region,but also the up- burst alone does not rule out Alfv´enic wave heating.
per chromosphere and corona. The similarity of the heating signatures is particularly
problematic for studies of nanoflares, where HXR emis-
(2) Heating by dissipation of Alfv´enic waves can be sion,if present, is too faint to be detected. For example,
very similar to heating by electron beams (e.g. Testa et al. (2014) recently investigated chromospheric
Brown 1971; Emslie 1978). heating during nanoflares and found that IRIS observa-
tions are consistent with heat input by nonthermal par-
(3) Since the upper chromosphere can be strongly
ticles; our results suggest that similar signatures could
heated, Alfv´enic waves can cause explosive evapo-
also be produced by wave heating. New HXR instru-
ration. Aswithbeamheatingmodels,the pressure
ments such as FOXSI (Krucker et al. 2014) and NuStar
in the chromosphere rises sharply, causing a rapid
(Harrison et al.2013),withimprovedsensitivityandspa-
expansion of material.
tial resolution, may help resolve this.
It seems possible that Alfv´enic waves can play an im-
Ion-neutral friction damps down-going waves at
portant role in flare heating. It is undeniable, however,
the temperature minimum region (as found by
thattherearemanyelectronsbeingacceleratedinflares,
Emslie & Sturrock 1982 and Russell & Fletcher 2013)
producing strong HXR bursts, which are well correlated
and it becomes important in the upper atmosphere for
with the rise in SXR emission (Dennis & Zarro 1993).
high frequencies. The frequencies required to heat the
Therefore, future work needs to further establish the vi-
upper chromosphere this way are sensitive to the field
ability of this heating mechanism, but also to what ex-
strength, but in our simulations 10Hz waves produced
tentitoperatesin tandemwithelectronbeams,andhow
a pronounced heating and 100Hz waves were almost en-
energy might be partitioned between them.
tirely absorbed there. Millisecond spikes in radio and
HXRs(Kiplinger et al.1983;Benz1986)showthatflares
produce such frequencies, and in situ observations of This research was performed while JWR held an NRC
ResearchAssociateshipawardat the US NavalResearch
magnetosphericreconnectionshowgenerationofAlfv´enic
Laboratory with the support of NASA. AJBR was sup-
waves with high frequencies (Keiling 2009). It is there-
ported by the U.K. Science and Technology Facilities
fore credible that Alfv´enic waves excited during flares
Council under grant ST/K000993/1 to the University
would also include a component that heats the upper
of Dundee. The authors benefited from participation in
chromosphere by ion-neutral friction.
theInternationalSpaceScienceInstitute teamon“Mag-
Parallel resistivity can also have a significant role if
neticWavesinSolarFlares: Beyondthe‘Standard’Flare
the incident waves have fine structure perpendicular to
Model,” led by Alex Russell and Lyndsay Fletcher. We
the magnetic field(notconsideredby Russell & Fletcher
acknowledge particularly helpful comments about resis-
2013). In our simulations, incident waves with k
x,a
≥ tive damping of waves with large k from James Leake
10−4cm−1 produced considerable heating in the upper ⊥
and Peter Damiano, which influenced this work. We
chromosphere. This wavenumber correspondsto a scale
thank Martin Laming and Graham Kerr for comments
of 600m, which is two orders of magnitude larger than
that helped to improve this paper.
the coronal proton inertial length (assuming n 109
≈
cm−3) and at least two orders of magnitude larger than
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