arXiv:1011.5834v1 [astro-ph.SR] 26 Nov 2010 uhmr nry n hsrltdpoesis process related this and requires energy, the more of ad- much In heating reviews). the or recent dition, for 2006, 2003, Klimchuk, Ireland, & e.g. Walsh of (see, number interpretation data challenge the observational A that and descriptions remain understood. theoretical questions well fundamental yet of not is heating Introduction 1. 2 1 h ealdpyia ehns fcoronal of mechanism physical detailed The loUiest fGagw KG28QQ G12 UK Glasgow, of 94720 University USA also CA, Berkeley, UC SSL, also hspoescnpa ao oei h etn ftechromospher the of heating the in role major 0.1- as a proce large play as heating can altitude f This process with this becomes increases ions. temperature plasma and typical the in electrons result when heat even to dynamics continue continues the dissipation motions alter heatin turbulent not electron quit gas does resistive neutral of the impact to generation the electron-neutral leads the with by currents in t Ionization moving these results frozen-in of ions ions is dissipation of and The it electrons frame act of if reference the motion as the relative under in acts su drift induced electrons field Under magnetized fields magnetic However, collisions. the gas. ion-neutral and th neutral by than neutrals, dominated larger by becomes be dragged gyrofrequency to their continue is, magnetiz motions that become electrons G); ambient (30 that show fields We low. typically is level ecnie h eeaino lcrccret nteslrchromo solar the in currents electric of generation the consider We eeaino lcrccret nthe in currents electric of Generation hoopeevanurlindrag neutral-ion via chromosphere .S Hudson S. H. lnTrn ulig acetrM39LUK 9PL M13 Manchester Building, Turing Alan AAeu el ehrh Scientifique Recherche la de Avenue 3A P2,CR-nvriyo Orl´eans of CNRS-University LPC2E, 57 r´asCDX2FRANCE 2 Orl´eans CEDEX 45071 S,U ekly A S 94720 USA CA, Berkeley, UC SSL, colo hsc n Astronomy and Physics of School h nvriyo Manchester of University The .Krasnoselskikh V. 2 .D ae n .P Abbett P. W. and Bale, D. S. , ABSTRACT .Vekstein G. 1 akrivne pca aefrteeelemen- events, these release for energy name tary special particles. a non-thermal field invented heat- of Parker acceleration magnetic plasma the in the and result ing of hypothetically changes – topology sudden – These twisted events motions. and footpoint braided photospheric random becomes by it coronal as the field, in magnetic dis- dissipa- tangential spontaneously small-scale episodic arising many continuities the at by energy of heated tion be could corona understand. to difficult equally akr(98 rpsdteie httesolar the that idea the proposed (1988) Parker 1 o fteeeti n magnetic and electric the of ion olso rqec hl ion while frequency collision e n ffiin a ionization. gas efficient and g . Vk.W ocuethat conclude We eV/km. 0.3 ftehayprils thus, particles; heavy the of n corona and e si oecetta tcan it that efficient so is ss peeweeteionization the where sphere nes lcrccurrents. electric intense e lyinzd n resistive and ionized, ully dee o ekmagnetic weak for even ed a.W n htthis that find We gas. hcniin,in are ions conditions, ch h yaiso the of dynamics the o nanoflares h inspi- The . ration for this was the discovery of the hard X- (which replace the tangential discontinuities) are ray microflares by Lin et al. (1984); the energy of continuously formed and dissipated. In this mod- one nanoflare was to be roughly 10−9 times the ification of the initial scenario the current sheets energy of a major flare, and thus orders of mag- are the result of the turbulent cascade. The ini- nitude weaker than even the microflares. Parker’s tial energy reservoir in such a view is contained in idea stimulated intensive searches for any obser- large scale magnetic-field structures. vational signatures of nanoflares (Cargill 1994; Here, we discuss another possibility: the direct Sakamoto et al. 2008; Vekstein 2009) and their generation of relatively small-scale electric cur- possible contribution to the overall energy budget rents by neutral gas motions. Clearly, a huge of the solar corona (Klimchuk 2006). energy reservoir exists in the form of turbulent Microflares were first detected in hard X-rays motion of neutral gas at and beneath the pho- in a balloon experiment by Lin et al. (1984). tosphere, supported by the underlying convection The subsequent development of new instrumenta- zone. It is widely accepted that this energy can tion allowed the multi-wave satellite and ground be partially transformed into the excess magnetic based high-resolution observations of smaller-scale energy in the chromosphere and corona. How- (about a thousand of kms or even smaller) and ever, there is no quantitative model that describes lower energy phenomena. Soft X-ray imaging the physical mechanism of such energy transfer. revealed abundant microflares in active regions The development of a model that describes the en- (Shimizu et al. 1994), and RHESSI observations ergy transfer from quasi-neutral gas motions to the found that virtually all of a sample of some 25,000 magnetic field in a step-by-step manner inevitably hard X-ray microflares occurred in active regions. raises questions about the spatial and temporal Krucker et al. (1997) found flare-like brightenings scales at which such a transfer can occur and about in areas of the quiet , and observations at its location. Recent observations provide strong EUV wavelengths (e.g., the “blinkers” of Har- indications that the energy reservoir is indeed the rison, 1997) reveal bursting activity above the dynamic turbulent , and that the en- magnetic network borders. Similar phenomena ergy transfer from the turbulent gas motions to that form small X-ray jets at the limb were re- various kinds of trapped and transient magnetic ported by Koutchmy et al. (1997). From these and field oscillations takes place at the chromospheric other EUV observations (Berghmans et al. 1998; level. Benz & Krucker 1998), if not the hard X-rays, we Analysis of the high resolution spatial (∼150 km conclude that Parker’s idea of episodic heating of at the Sun) and temporal (few sec) data obtained the apparently steady quiet corona should not be by the SOT (Solar Optical Telescope) aboard discarded, even though no convincing evidence for Hinode (De Pontieu et al., 2007) revealed that the required steepening (Hudson 1991) of the en- the chromosphere is dominated by a multitude of ergy distribution function has yet been presented. thin (∼200 km wide) dynamic, jetlike “type II” However, the idea of coronal heating via tan- spicules. They are ejected upwards with charac- gential discontinuities that arise spontaneously teristic velocities of 20-150 km/s and reach heights (nanoflares) does not address two important ques- of 2000-10000 km before disappearing from the tions. Namely, where does the excess magnetic en- chromospheric passband (in this case, the H line ergy come from, and what was its original source? of Ca ii). The type II spicules have quite short life- Parker’s analysis considered ideal MHD statisti- times of 10-300 s (according to the authors, most cal equilibria that contain multiple discontinuities. of them last less than 100 s) and many of them Later Rappazzo et al. (2008) showed that a large- undergo substantial transverse displacements of scale MHD energy source perturbed by slow mo- the order of 500-1000 km. Moreover, the large- tions on its boundary, supposed to be induced scale long-living spicules display oscillatory mo- from the photosphere, results in the generation of tions in the direction perpendicular to their own a Poynting flux. This drives an anisotropic turbu- axes. Since the structure can be taken to lent cascade dominated by magnetic energy. The outline the direction of the magnetic field, this led result looks similar to Parker’s tangling of mag- the authors to the conclution that the observed netic field lines but the small-scale current sheets motions indicate Alfv´enic oscillations.

2 Furthermore, the spatio-temporal variations of of wave turbulence introduced into the . the chromosphere have always revealed the great- We propose a mechanism to generate electric est complexity, and prominences also consist of nu- currents that can be viewed as Alfv´en or magne- merous threadlike features with strong and mixed tosonic waves that result from strong ion-neutral flows along these threads (Lin et al. 2005). Obser- drag. We assume that photospheric motions vations with SOT have provided an exceedingly are turbulent, and consist of both compressional variable and dynamic picture of these flows and and rotational flows with energies exceeding that field structures (Okamoto et al. 2009). The SOT which is necessary to drive the observed oscilla- chromospheric data are in the H-line (Ca ii), show- tions of suspended threads or jets in the chro- 4 ing plasma at roughly 2 × 10 K. The movie pre- mosphere. The question then becomes how these sented by Okamoto et al (available on the Hinode oscillations can be transported from the photo- web site http://solarb.msfc.nasa.gov/) shows sphere to the chromosphere, and how the flow of ubiquitous continuous motions along the promi- neutrals is transferred into the motion of charged nence thread lines. The oscillatory motions ob- particles and, ultimately, into the generation of served might be interpreted in terms of propagat- electric current. ing or standing Alfv´en waves on the magnetic field There are two physical processes that are nec- lines that presumably compose the prominence. essary to convert the energy of the neutral gas The typical transverse spatial scale of threads was motion into the magnetic field oscillations. Pho- found to be of the order of 600 km, with a charac- tospheric motions of charged particles are domi- teristic length of the order of several Mm. The nated by frequent electron-neutral and ion-neutral characteristic temporal scale was found to vary collisions, so that ions and electrons tend to follow from 100 to several hundred seconds. the neutral gas motion with the zero net electric SOT observations near the limb led to the current. discovery of another small scale dynamic phe- Since density falls off rapidly with height in nomenon in the chromosphere: tiny chromo- the solar chromosphere, the electron collision fre- spheric “anemone jets,” named for the similar quency with neutrals and ions decreases to the X-ray features (Shibata et al. 2007). They also point where it becomes smaller than the elec- resemble larger-scale features well known in Hα tron gyrofrequency. However, the ion-neutral col- data and called surges or sprays, which often oc- lision frequency at this height still remains sub- cur near and in association with flares stantially larger than the ion gyrofrequency, since or other transient activity (e.g. Rust, 1968). All ion-neutral and electron-neutral collision rates are of these observations can be considered as indica- not substantially different (see, e.g., De Pontieu tions of the generation of small-scale perturbations et al., 2001). Therefore, the motions of ions and and oscillations of the magnetic field in the chro- electrons differ. Electrons tend to move along the mosphere, where the degree of ionization is still magnetic field lines and drift due to E × B in relatively small. the transverse direction, while the ions continue to move together with the neutrals. This differ- 2. The coupling problem ence results in the generation of electric currents. In spite of greatly improved observational data, To view this situation from a different perspec- the physics of chromospheric and coronal heating tive, if ions move together with neutrals (due to is not yet well understood. While the underlying the strong drag) with some angle to the back- mechanism must be associated with the magnetic ground magnetic field, an inductive electric field field, the details of how efficiently the energy of v × B appears in the reference frame of the convective motion is transformed into magnetic plasma. This induced electric field inevitably gen- energy in the solar atmosphere is an open ques- erates electric currents that can be calculated us- tion. The importance of this process should not ing the plasma conductivity tensor. When elec- be understated. It is relevant not only to coro- trons and ions are both demagnetized, this tensor nal heating and the acceleration of the solar wind, reduces to the simple scalar conductivity, and the but also to the formation of the initial spectrum resulting current is very small. When electrons become magnetized, the motions of electrons and

3 ions become decoupled, and the efficiency of the an MHD approach to the physics (e.g., Steiner current generation is substantially higher. 2007). Figure 1 compares collision frequencies Thus, we conclude that there is an efficient from the Fontenla et al. (2009) quiet-Sun model chromospheric dynamo operating in the layer be- (Model 1001) with the gyrofrequencies of electrons tween the level of electron magnetization and the and ions. For this model we assume a constant height where either the degree of ionization be- magnetic field of 30 G (e.g. Harvey et al. 2007; comes high (comparable with unity) or where the Lites et al. 2008). ions become magnetized. Forced oscillating cur- rents are generated most effectively when the in- 10 10 ductive electric field has characteristic frequencies and wavelengths close to the eigenmodes of the system, i.e. the MHD waves in the magnetized 108 weakly ionized plasma. At higher altitudes, where -1 ions are also magnetized, the motion of ions and 106 electrons can still differ. However, under these conditions, both components will mainly move Rate, s along the magnetic field lines and perform drift 104 motions across the field. Thus, the current is di- rected mainly along the magnetic field lines, while 102 in the intermediate region it can be generated in -0.5 0.0 0.5 1.0 1.5 2.0 2.5 an arbitrary direction. Height, Mm In summary, understanding how currents are generated by the turbulent motions of neutral gas Fig. 1.— Collision frequencies in the quiet-Sun in the photosphere can be broken down into two model of Fontenla et al. (2009). Blue (red) curves steps. First, we must understand how the energy represent proton (electron) collision frequencies, and vorticity of photospheric motions is trans- with the lower lines showing neutral rates only ported upward to the level where electron and ion (e.g. De Pontieu et al. 2001); the dotted horizon- motions become decoupled. Second, we must de- tal lines show the gyrofrequencies (ions and elec- rive a self-consistent system of equations that de- trons) for an assumed field strength of 30 G. scribe the inductive electric field and the result- ing deformation of the background magnetic field. The Fontenla models cover a range of physical Once we obtain the electric field and estimates of features in the solar atmosphere, and were origi- current density, we can then determine the effi- nally intended for irradiance modeling. Here, we ciency of electron heating due to collisions and can use these models as a guide to set a range of pa- investigate the efficacy of other possible heating rameters that are reasonably consistent – in this mechanisms. limited theoretical framework – with solar struc- ture. Figure 2 shows the location of the chromo- 3. Atmospheric models spheric dynamo layer for the full range of models. Electrons become magnetized, for all models, es- We appeal to standard semi-empirical model at- sentially throughout the solar atmosphere. Thus mospheres for approximate values of the physical the chromospheric dynamo layer begins close to parameters of the plasma in the photosphere and the photosphere and extends high into the chro- chromosphere. Fontenla et al. (2009) provide a re- mosphere. This is true for all of the Fontenla cent series of eight such models representing fea- models, ranging from umbra to bright fac- tures such as the quiet Sun, faculae, and sunspots. ula, for which the transition-region pressures range Such models are based on macroscopic radiative 2 from 0.2 to about 2.0 dyne cm− . Note that in transport theory and are adjusted to recreate so- such models, substantial hydrogen ionization does lar spectroscopic observations. They do not repre- not take place below the transition region between sent the dynamics, nor the plasma physics, since chromospheric and coronal temperatures. computer technology thus far allows at best only We emphasize that these models do not have

4 any plasma physics in them as such, and only serve where the density is presented as the sum of hy- as references at the order-of-magnitude scale. In- drostatic equilibrium density and a perturbation deed, the mechanism discussed in this paper would (note that in hydrostatic equilibrium there are no point to areas in which models of the solar atmo- flows). sphere can be greatly improved. By taking into account the equilibrium density dependence upon height this can be re-written as Dynamo layer

Bright facula ∂ ρ ∂vx ∂vy ∂ 1 8 + + + − vz ∂t ρ0 ∂x ∂y ∂z H Sunspot penumbra    

Sunspot umbra 1 ρv ρ 6 = z − ∇ −→v , (2) H ρ0 ρ0 Facula  

4 (not facula) where we have separated the linear and nonlinear

Fontenla model parts and put the latter on the right-hand side. Enhanced network The equation of motion 2 Quiet-Sun network lane ∂−→v Quiet-Sun inter-network (ρ + ρ0) ( + (−→v ∇)−→v )= −∇P − (ρ + ρ0) Gz ∂t 0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 Height, Mm takes the form b ∂−→v P 1 P ρ Fig. 2.— Chromospheric dynamo layers for the + ∇( ) − z ( ) + ( )Gz ∂t ρ0 H ρ0 ρ0 various solar features modeled by Fontenla et al. (2009). The field strengths assumed are 30 G for bρ ∂v b = −(v∇)v − ( + (v∇)v). (3) the quiet Sun (the lower two models), and 1500 G ρ0 ∂t for the others. We shall also use the equation of state taken in the simplified form of the equation for adiabatic 4. Energy transfer from the photosphere motion, dP γPph d to the chromosphere = (ρ + ρ0) . dt ρph dt We first consider the physics of mass, momen- This is tum, and energy transport in the lower chro- dP 2 d = C (ρ + ρ0) . mosphere. In order to formulate a mathemati- dt S dt cal description of the problem we need to take It also can be re-written for perturbations as into account physical conditions between the pho- tosphere and lower chromosphere, as described 2 ∂ P 2 ∂ ρ γ − 1 CS above. ( ) − CS ( )+ vz = ∂t ρ0 ∂t ρ0 γ H We begin with an isothermal atmosphere, i.e., a gas having a constant sound speed and strati- 2 −→ ρ vz ρ −→ P vz P CS[( v ∇) − ( )]−( v ∇)( )+ ( ). (4) fied by a uniform gravitational force acting in the ρ0 H ρ0 ρ0 H ρ0 negative z-direction, for which Our objective here is the analysis of propaga- tion of the photospheric perturbations upwards ρ0 (z)= ρph exp(−z/H) . (1) into the chromosphere, and for this purpose we

Here ρph is the density at the level of the photo- analyze the characteristics of linear perturbations. sphere corresponding to z = 0, and H is the den- To obtain the linear wave mode dependencies, we sity scale height. The gas motions are described determine the eigenmodes of the linearized system: by the equation of continuity ∂ ρ −→ 1 ∂ ρ −→ 1 +∇· v − vz =0 ⇒ = −∇· v + vz, ∂ρ ∂t ρ0 H ∂t ρ0 H + ∇·{(ρ0 + ρ)−→v } =0, ∂t (5)

5 density is well known in the terrestrial atmo- sphere. It plays an important role in the prop- ∂−→v P 1 P ρ + ∇( ) − z ( ) + ( )Gz =0, (6) agation of infrasonic perturbations induced by ∂t ρ0 H ρ0 ρ0 explosions in the atmosphere (e.g. Klostermeyer, and b b 1969a, 1969b) . The same physical phenomena play important roles in the solar photosphere 2 ∂ P 2 −→ 1 CS and chromosphere and have been the subject ( )+ CS∇ · v − vz =0. (7) ∂t ρ0 γ H of many observational, theoretical, and simula- tion studies. At low frequencies acoustic-gravity We choose standard solutions of the form: waves (purely hydrodynamic) play a role (e.g. Straus et al. 2008, 2009), and at high frequencies P,ρ,v ∼ exp i(ωt − kxx − kyy − kzz), MHD modes (e.g., Hollweg 1986). The simula- tions (e.g., Stein & Nordlund 1998) can include and analyze the dispersion relations that deter- essential physics such as optically-thick radiative mine the dependencies of the frequency ω upon −→ transfer. components of the wave vector k . But our prob- The results of the more realistic simulations lem is formulated as a boundary-value problem highlight important features of neutral gas motion. where we assume the perturbations and their time For example, vorticity is primarily generated due dependences to be prescribed at the boundary to baroclinic forces, and tends to concentrate in z = 0. We want to describe the evolution of per- tube-like structures whose widths are comparable turbations with height z. In this case the very to the numerical resolution. In addition, plasma same equations should be used to find out the heating depends on the formation of supersonic dependence of the component k of the k-vector z flows and shock-like structures. If shocks form upon the frequency ω and other components k , x systematically, they could well be an important k . Substituting dependencies defined above into y factor in the physics of coronal heating. However the linearized equations one can obtain the disper- according to Stein & Nordlund (1998), shocks are sion equation in the form indeed observed at the edges of integranular lanes,

2 2 2 2 2 iK γ − 1 but are a rare occurrence. At any one time, su- [ω − iKg]{ω − C [(k + K )+ )]}− S H γ personic flow occurs in only 3-4% of the surface area. i γ − 1 iGC2 (k2 + K2)(K + )=0, The role of acoustic gravity waves has also been S H γ intensively studied experimentally. Recently new 2 2 2 2 i experimental studies combining the SOT/NFI and where C = γGH, k = k + k ,K = kz − . S x y H SOT/SP instruments on Hinode and the Michel- 4 2 2 2 1 2 2 son Doppler Imager (MDI) on SOHO, with the ω − ω C (k + 2 ) + (γ − 1)k g i S 4H aid of 3D computer simulations, were performed kz = ± 2 2 . 2H s ω CS by Straus et al. (2009). The authors came to the conclusion that the gravity waves are the This analysis demonstrates that a wide class of dominant phenomenon in the quiet middle/upper perturbations satisfying the condition photosphere and that they transport sufficiently more mechanical energy than the high frequency 4 2 2 2 1 2 2 ω − ω C (k + ) + (γ − 1)k G > 0 (>5 mHz) acoustic waves. In addition they con- S 4H2 clude that the acoustic-gravity wave flux is 3- increase with height, and the characteristic growth 5 times larger than the upper-limit estimate of rate is Fossum & Carlsson (2005). These observations 1 Γ = Im(kz)= . together with the numerical models allow us to 2H consider acoustic gravity waves as one of the most This phenomenon, the “effective growth” of important sources from which energy may be the perturbations with altitude in a hydrostatic transformed to the magnetic field, and then to gas equilibrium with exponentially decreasing heat via the action of electric currents.

6 We note that that the absolute magnitude of trons the density perturbation actually decreases with −→ dVe −→ −→ −→ −→ −→ height, but it drops more slowly than the back- me = −e{E +[Ve×B ]}−(νen+νei)me(Ve−Vn), ground density. This results in the growth of the dt (8) relative density perturbation, written as ρ/ρ0 in 8 10 1 where ν ≈ 5 × 10 − 10 s− is the frequency our notation. On the other hand the velocity per- en of electron-neutral collisions, the major source of turbations really grow exponentially. An impor- the electron drag. Here ν is electron-ion collision tant issue here is the characteristic vertical scale ei frequency. of this growth. Taking the gas temperature to be 5000 K we find Ions are supposed to be dragged by neutrals, thus the velocity of electrons can be expressed as k T H = B = 140 km. −→ −→ −→ j MG V = V − . e n en This linear analysis includes only the effect of the growth of the velocity and relative density pertur- The physical processes we describe occur in a bations with the altitude, hence, the perturbation parameter range corresponding to the transition scales do not vary with altitude. Of course, there in electron motions from unmagnetized, collision- are a wide range of phenomena, such as vortical dominated, to magnetized, when initially |ωBe| < flows, that are not accounted for in this formalism. (νen+ νei), and then with the growth of altitude For example, the vortex radius can decrease |ωBe| becomes larger than (νen+ νei). We consider with altitude. Indeed, assuming conservation of hereafter slow motions ω << (νen+νei), |ωBe| sim- angular momentum the increase of the velocity ilar to those we described in the previous para- should result in the shrinking of the transverse di- graph. In this case one can neglect the electron ameter of the vortex. This effect is relevant for inertia in the left-hand side of the equation 8. It Rossby vortices in a multi-layer atmosphere and can then be simplified to obtain: it is in a good agreement with the observation of Stein and Nordlund (1998) that the vorticity con- −→ −→ −→ −→ 1 −→ −→ j centrates in small scale tubes. −e{E +[V ×B ]− [ j ×B ]}+(ν +ν )m =0 n ne en ei e ne (9) 5. Magnetic perturbations produced by Taking the curl and replacing ∇× B via the turbulent motions of the neutral −→ gas ∂ B −→ = −∇× E Turbulent convective fluid motions at the pho- ∂t tospheric level extend upwards into the lower chro- one can obtain the final equation mosphere. The latter is a weakly ionized plasma −→ with a typical temperature of 6000-7000 K, neu- ∂ B −→ −→ 1 1 −→ −→ 12 14 −3 = ∇× [Vn × B ] − ∇× [∇× B × B ] tral hydrogen density nH ∼ 10 −10 cm , and ∂t eµ0 n 10 11 −3 electron density ne ∼ 10 − 10 cm . Under −→ (ν + ν )m ∇× B these conditions the frequency of the ion-neutral − en ei e ∇× , (10) 7 9 −1 2 collisions is as high as νin ≈ 10 −10 s which, as e µ0 n −→ will be confirmed below, produces quite a strong where j is replaced by drag effect resulting in the bulk velocity of ions, −→ 1 −→ Vi closely matching that of the neutral gas, Vn. j = ∇× B. (11) Then the magnitude of the generated electric cur- µ0 −→ −→ −→ −→ −→ rent j = nee(Vi − Ve) ≈ nee(Vn − Ve) is deter- Equation 10 together with the system of equa- mined by the electron bulk velocity Ve. The latter is governed by the equation of motion for the elec- tions 2, 3, 4 make up a closed system describing an interaction of neutral gas perturbations with the ionized electron component that can result in gen- eration of electric currents. We neglect hereafter

7 −→ −→ −→ the influence of electron motions on neutral gas j = nee(Vn − Ve). It can be easily found from motions by assuming that the degree of ionization the expression 13 that is low. −→ 1 −→ In what follows it is assumed that the chromo- j = ∇× b = µ0 spheric magnetic field can be represented as −→ −→ −→ B0 −→ −→ −→ −→ B = B0 + b , i ∇×{( h · ∇)Vn − h (∇ · Vn)} µ0ω −→ where B0 is a background field, which for the sake B0 −→ −→ ≈ i ( h · ∇)∇× Vn. of simplicity is assumed here to be just a uniform µ0ω b field, while is a relatively small field deformation To evaluate the difference of the ion and neutral caused by a prescribed flow of neutral gas with −→ velocities let us consider the equation of motion for velocity Vn and frequency ω. ions: b Then the linearized (with respect to ) version −→ of Eq. 10 takes the form dVi −→ −→ −→ −→ −→ M = e{E +[V × B ]}−M ν (V −V ). (14) i dt i i in i n −→ 2 | ωBe | −→ −→ 2 νen 2−→ b +ide ( h ·∇)∇× b −de(1+i )∇ b = As can be seen from Eq. 14, the sought-after ve- ω ω −−→ −→ −→ (12) locity deviation ∆Vi = (Vi − Vn) is such that B0 −→ −→ −→ −→ the ion-neutral drag balances the other forces ex- i {( h · ∇)V − h (∇ · V )}. ω n n erted upon the ions. A simple estimate shows that where h is the unit vector along the background the net electromagnetic force is the dominant one there, and Eq. 14 yields magnetic field, and de = c/ωpe is the skin depth. Let us now estimate the relative roles of the dif- −−→ −→ −→ −→ Miνin∆Vi ≈ e{E + [Vn × B ]} ferent terms in the left-hand side of Equation 12. 11 −3 For ne = 10 cm the electron plasma frequency 1 −→ −→ me −→ 10 −1 ∼ [ j × B ]+ νen j ωpe = 10 s , which yields the de ≈ 3 cm. The n en typical frequency of the neutral gas flows under It follows then from Eq. 14 that consideration is ω ≈ (10−2 − 10−3) s−1 and their 2 3 −→ −→ −→ −→ length scale is L ∼ 10 − 10 km. For B0 ≈ 100G −−→ e |{E + [Vn × B ]} | Ωi | j | 9 −1 ∆Vi ≈ ∼ (15) the electron gyrofrequency ωBe = 2 × 10 s , Miνin νin ne while the characteristic electron-neutral collision 8 9 −1 and frequency νen ∼ (10 − 10 )sec . Therefore the −−→ ωBi ∆Vi ∼ δve < δve, ratio of the second and third terms compared to νin the first can be considered small. Thus Equa- where δv is the velocity of current carrying elec- tion 12 can be reduced to e trons, and ions are assumed to be demagnetized. −→ B0 −→ −→ −→ −→ b ≈ i {( h · ∇)V − h (∇ · V )}, (13) On the other hand, by using Eq. 13, the electric ω n n current can be estimated as which means that in the lower chromosphere the b B0V j ∼ ∼ n . magnetic field is effectively frozen into the neutral- µ0L µ0ωHL gas flow. The velocity of the current -carrying electrons is This describes the magnetic field generation due to the motions of conductive fluid neglect- 2 j eB0Vn m ωBe de δve ∼ ∼ 2 ∼ Vn < Vn. ing small scale effects scaling as the electron iner- ne µ0mωHL ne ω HL tial length, and assuming the motions to be slow: (16) ω << Ωe. Let us now verify that the assump- It is seen now from Eqs. 15 and 16 that, indeed, tion made above, namely that the bulk velocity −→ δVi << Vn, and δVi < δve. of the ions V is so close to that of the neutrals −→ i A simple estimate of the currents generated at V that the electric current can be written as n different altitudes and for different values of the

8 magnetic field can be obtained by assuming the is much smaller than the electron neutral collision neutral gas motions to be rotational. In this case requency. This leads to the resistivity along the the characteristic velocity in the vortex Vn of char- magnetic field to be classically collisional, where acteristic spatial scale L rotating with the charac- the dominant effect is due to electron-neutral col- teristic frequency ω can be evaluated as follows: lisions

νenme νen Vn ∼ ωL. Ek = − 2 jk = − 2 jk = ne ε0ωp Thus, the characteristic magnetic field perturba- tion is iνen B0 −→ −→ −→ −→ −→ − 2 ( h ·∇×{( h · ∇)Vn − h (∇ · Vn)}) L µ0ε0ω ω δB ∼ B0 , p H d2 and consequently the current density is ∼ ν δB. en L δB B0 j ∼ ∼ . To evaluate the parallel electric field one should µ0L µ0H take into account that the electron-neutral col- lision frequency varies with height from ν ∼ One can see that the magnetic field perturba- en 5×108 to 1010 s−1 and the plasma frequency varies tions can become quite large; the current densi- in turn from 1010 to 1011. Taking the characteris- ties for magnetic fields of the order of 100G can tic current densities obtained above we find elec- become as great as tric fields in the range B0 2 1 A − − −2 −1 j ∼ ∼ 10 − 10 2 . E ≈ (10 − 10 ) V/m, µ0H m k

It is worth noting that this mechanism is de- with the electron heating written (Braginskii, pendent upon the velocity shear. An interesting 1965) as feature of this process is the very high efficiency 3 deTe −→ −→ of the field generation around the boundaries of Ne + pe∇ · Ve = −∇ · qe + Qe. (17) 2 dt the neighbouring vortices, where the characteris- tic shear of the velocity of the gas motions is large. Here pe is the electron pressure, and the electron In this case the characteristic velocity shear is heating Qe is determined by two “friction forces,” one due to the relative velocity between electrons Vn −→ −−→ >> ω; and ions/neutrals (Rei, Ren), and another due to L −→ the electron temperature gradient RT : thus smaller-scale magnetic-field structures can be −→ −→ −→ −−→ −→ −→ −→ −→ −→ generated quite efficiently and magnetic fields gen- Qe = (Rei, Ve −Vi )+(Ren, Ve −Vn)+(RT , Ve −Vn), erated and current densities can become tens or with hundreds of times higher than the estimate above. −→ −→ −−→ −→ Rei = −νeimeNe{0.5(Vek − Vik) h + (Ve⊥ − Vi⊥)}, 6. Electron and ion resistive heating. −−→ −→ −−→ −−→ R = −ν m N {0.5(V −V ) h +(V −V )}, To evaluate the electron heating efficiency one en en e e ek nk e⊥ n⊥ should find the induced electric fields parallel and and perpendicular to the magnetic field. This can be −→ −→ 3 ν −→ done in a straightforward way by making use of the RT = −0.7Ne( h , ∇)Te − Ne [ h , ∇Te]. 2 Ωe component of the equation of motion for electrons along the magnetic field (Eq. 8). As a result

(ν + ν )m N 2 2 dVek νenme Q = − ei en e e {0.5j + j + (18) me = −eEk − jk; e 2 2 k ⊥ dt ne Ne e 2 in the lowest-order approximation one should take 0.7Ne e −→ −→ −→ ( h ·∇× B )( h , ∇)Te}; into account that the frequency ω of wave motions µ0

9 here we neglect the diference between the ion and where κz is the electron thermal conductivity in neutral velocities. The last term in the equation of the parallel direction. For our conditions when energy balance to be kept describes the heat flux electron-neutral conditions are dominant in the that is written as parallel direction, −→ −→ −→ qe = que + qT e, e neTe κz ≈ γ , meνen −→ 3 ν −→ −−→ −→ que =0.7NeTe(Vek−Vnk)+ NeTe [ h , Ve⊥−Vi⊥], 2 Ωe where γ is a coefficient of order unity (according to and Braginskii, 1965) for electron-ion collisions; when ions are singly ionized it is equal to ∼3.8. This −→ NeTe qT e = −3.16 ∇kTe− leads to the following estimate of the characteristic me(νen + νei) heat loss rate NeTe(νen + νei) 5 NeTe −→ ∂ k n T ∂k T k2 n T 2 4.66 2 ∇⊥Te − [ h , ∇⊥Te]. ∇ · −→q ≈ γ B e e B e ∼ B e e , meΩe 2 meΩe e 2 ∂z meνen ∂z meνenLT We shall begin by evaluating the energy supply provided by resistive dissipation that is deter- where (Te/LT ) is the characteristic variation of mined by first two terms in the right hand side temperature with altitude. Under the physical of the equation of energy balance. The volumetric conditions of the chromosphere this will lead to heating power of the electrons can be estimated as Te jkEkmeνen 1/2 2 ( ) ∼ ( 2 ) ∼ 0.1 − 0.3 eV/km. 3 dTe νen d 2 LT kBne nkB ∼ jkEk ∼ 2 δB 2 dt µ0 L Taking into account an exponential decrease −4 −3 W of the plasma density one can find that it can ∼ (10 ∼ 10 ) 3 . m heat plasma to the hydrogen ionization energy In a thermal equilibrium when the macroscopic (13.6 eV) on a characteristic distance of the order flow evacuates the power supplied by collisional of a few hundred km. It should be noted however resistive dissipation, one can estimate the temper- that the thermal conductivity increases with the ature variation with altitude as temperature and this can result in some flattening of the temperature profile. dTe ∂Te ∂Te jkEk = Vez , ∼ 3 , Another important physical effect is related to dt ∂z ∂z nk V 2 B ez the ion heating due to the perpendicular electric

where Vez is the characteristic macroscopic verti- fields. It is known that the major component of cal velocity of electron (plasma) motion that can the current along the electric field perpendicular be evaluated to be of the order of the sound veloc- to the magnetic (the Pedersen current) can be car- ity. Then ried by ions. This can lead to the more efficient heating of ions than of electrons. To evaluate the ∂Te jkEk Pedersen current we must estimate the ion veloc- ∼ 3 ∼ (0.01 − 0.1) V/km. ∂z 2 nkB VS ity deviation from the velocity of neutrals in the direction of the electric field, as was already done It is easy to see that the collisional heating of in the previous paragraph the electrons becomes rather efficient. Another −→ −→ −→ possible evaluation of the temperature variation −−→ e |{E + [Vn × B ]} | ∆Vi ≈ ∼ with altitude can be obtained by assuming a sta- Miνin tionary static thermal equilibrium. Under such 1 −→ −→ jB0 conditions the power supply will be balanced by [ j × B ] ∼ . the divergence of the heat flux in Equation 17, Miνinn nMiνin which can be estimated as It follows then that ∂ ∂T ∇ · −→q ≈ κe e , −−→ −→ −→ −→ e ∂z z ∂z (ne∆Vi ·{E + [Vn × B ]}) ∼

10 2 2 2 2 j B0 Mi Ωi d 2 values. This gives rise to an estimate of the charac- ∼ 2 δB . nMiνin me νin L teristic magnetic field generated by such motions Ion heating due to the perpendicular electric field readily becoming comparable to or even larger component can be evaluated as than the background field. Their relative mag- nitude is determined by the ratio of the charac- 2 dTi −→ −→ −→ −→ Mi Ωi dTe teristic velocity of neutral motions at the altitude ∼ (j⊥i ·{E + [Vn × B ]}) ∼ dt me νenνin dt of electron magnetization to the characteristic pa- rameter ωH. One can see that for characteris- This estimate shows that the ion collisional tic frequencies of the order of 10−2 − 10−3 s−1, heating can be several times smaller than, or com- and characteristic heights of the order of 140 km, parable to, the electron heating. the field generation becomes quite efficient already for the velocities of the order of ten-several tens 7. Discussion of km/s. The helical magnetic field component can become larger than the background field. A sim- The current generation mechanism that we pro- ilar estimate shows that amplification/weakening pose here is strongly dependent upon four basic of the background magnetic field component can parameters: the velocity of the turbulent gas mo- be evaluated to be of the order of tions, the background magnetic field, the char- b δn acteristic frequency of the spectrum of turbulent ∼ . motions of the gas, and the characteristic spa- B0 n tial scale of velocity shear. The physical process ””Under such conditions, we show that the elec- of the generation of electric currents and mag- tron heating can become rather efficient due to netic fields is actually nothing other than the well- the collisional resistive dissipation of the electric known turbulent-dynamo mechanism operating as current. These processes take place in the regions a result of the the freezing of the magnetic field where the magnetic field vorticity increases. into the flow of the neutral gas. It is worth noting that the efficiency of the The altitudes where this mechanism will oper- electron and ion heating can become substantially ate are strongly dependent upon the local mag- more important than that due to the collisional netic field strength: the stronger the field, the resistive dissipation described above. The electric lower the altitude of electron magnetization. The current and magnetic field generation we have dis- efficiency of the electric current generation is pro- cussed takes place at the chromospheric footpoints portional to the magnitude of the velocity shear. of flux tubes that may extend into the corona. The The characteristic spatial scales are proportional direction of the generated field is determined by to the characteristic scales of this velocity shear, the characteristics of the gas velocity shear and i.e. the flow vorticity. It is interesting to note that can lead to the formation of the magnetic field con- in this formalism, magnetic fields and currents figurations that can become unstable, or can form can be generated by both compressible and incom- new structures by interaction with neighbouring pressible motions (see the first and second terms fields with different orientations. The increase of in equation 13). Compressive motion tends to the perpendicular component of the magnetic field amplify or weaken the background magnetic field, can result in kink instability if this component sat- while the rotational component effectively gener- isfies the Kruskal-Shafranov condtion ates the perpendicular or helical components of the field. This means that compressional motions δB l > . tend to generate transverse currents, while non- B R compressional motions can generate field aligned currents as well. Here l is the characteristic scale of the cross sec- tion of the tube and R is its characteristic length. Since density tends to decrease exponentially Under such conditions the tube cannot keep its with height in the chromosphere, the character- configuration intact; it will become unstable and istic velocity of chromospheric flows can increase then different kinds of perturbations will result in by a factor of ten or more over their photospheric its reconfiguration. Other types of emerging con-

11 figurations with magnetic field inversions will re- physical process is proportional to the character- sult in local reconnection. The detailed study of istic velocity shear of the gas turbulent motions these effects lies beyond the scope of our paper and inversely proportional to their characteristic and will be carried out elsewhere. frequency. We also found that the magnetic fields The result of the heating would be a rapid thus generated can be comparable to or even larger increase of the degree of ionization of the gas. than the background magnetic field for motions It is worth remembering that our whole descrip- having characteristic scales of the order of several tion is valid only when the degree of ionization is hundred km and characteristic time scales of the low. This rapid ionization process can lead, ac- order of several minutes. This can produce a sub- cording to most radiative-convective models (e.g., stantial restructuring of magnetic field configura- Stein & Nordlund 1998) to an efficient decrease of tions and an opportunity to create multiple sites the radiative cooling rate of the gas. Thus the gen- of reconnection. It also results in an efficient in- eration of electric currents results not only in the crease of collisional resistive heating of electrons plasma heating itself, but also causes a decrease and ions, and hence the rapid ionization of the gas, of the cooling process that will then strengthen thus unstably altering its thermal equilibrium. the effect of radiative heating. We cannot make any quantitative estimate of these effects, but they The authors are grateful to CNES for financial may be adressed in future simulations. support in the frame of CNES “Solar Orbiter Re- The coupling of acoustic-type waves and MHD search grant” and to SSL for the financial support waves can also be treated in the MHD approxima- of the visit of V. K. to the University of Califor- tion (e.g., Hollweg et al. 1982). The major differ- nia at Berkeley. H. H. thanks NASA for support ence in our approach here is that the neutral colli- under NAG5-12878. V. K. acknowledges very use- sions reduces the feedback effect on the motion of ful discussions with M. Ruderman, B. De Pontieu, the gas resulting from the newly generated mag- O. Podladchikova, and X. Valliers. The authors netic field. This produces a more efficient transfer also thank the referee, whose comments helped in of energy from gas motions to current generation improving the article. and magnetic field amplification. REFERENCES It seems possible to incorporate certain aspects of this formalism into a system of conservation Abbett, W. P. 2007, ApJ, 665, 1469 equations similar to those used in Abbett (2007). In that work, the resistive MHD equations were Benz, A. O. & Krucker, S. 1998, Sol. Phys., 182, solved numerically within a computational do- 349 main that includes both a turbulent model convec- Berghmans, D., Clette, F., & Moses, D. 1998, tion zone and corona. We are currently updating A&A, 336, 1039 this system of equations to incorporate additional physics, and are planning to perform a compar- Braginskii, S. I. 1965, Reviews of Plasma Physics, ison between a standard resistive radiative-MHD 1, 205 model and our new results. We hope to report on the results of this study in the near future. Cargill, P. J. 1994, ApJ, 422, 381 De Pontieu, B., Martens, P. C. H., & Hudson, 8. Conclusion H. S. 2001, ApJ, 558, 859 We have presented an analysis of the effect of De Pontieu, B., McIntosh, S., Hansteen, V. H., electric current and associated magnetic fields gen- Carlsson, M., Schrijver, C. J., Tarbell, T. D., eration in a “chromospheric dynamo layer” where Title, A. M., Shine, R. A., Suematsu, Y., electrons become magnetized while ions remain Tsuneta, S., Katsukawa, Y., Ichimoto, K., collisionally coupled to the neutrals. We have Shimizu, T., & Nagata, S. 2007, PASJ, 59, 655 shown that electric currents and magnetic fields can be generated very efficiently due to turbulent Fontenla, J. M., Curdt, W., Haberreiter, M., motions of the neutral gas. The efficiency of this Harder, J., & Tian, H. 2009, ApJ, 707, 482

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