A nanoflare distribution generated by repeated relaxations triggered by kink instability

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0Astronomy & Astrophysicsmanuscript no. 14067 c© ESO 2010September 20, 2010

A Nanoflare Distribution Generated by Repeated RelaxationsTriggered by Kink Instability

M. R. Bareford1, P. K. Browning1, and R. A. M. Van der Linden2

1 Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, The University of Manchester,Oxford Road, Manchester M13 9PL, U.K. e-mail:[email protected]

2 SIDC, Royal Observatory of Belgium, Ringlaan 3, B-1180 Brussels, Belgium

Received ; accepted

ABSTRACT

Context. It is thought likely that vast numbers of nanoflares are responsible for the corona having a temperature of millions ofdegrees. Current observational technologies lack the resolving power to confirm the nanoflare hypothesis. An alternative approach isto construct a magnetohydrodynamic coronal loop model thathas the ability to predict nanoflare energy distributions.Aims. This paper presents the initial results generated by a coronal loop model that flares whenever it becomes unstable to an idealMHD kink mode. A feature of the model is that it predicts heating events with a range of sizes, depending on where the instabilitythreshold for linear kink modes is encountered. The aims areto calculate the distribution of event energies and to investigate whetherkink instability can be predicted from a single parameter.Methods. The loop is represented as a straight line-tied cylinder. The twisting caused by random photospheric motions is captured bytwo parameters, representing the ratio of current density to field strength for specific regions of the loop. Instabilityonset is mappedas a closed boundary in the 2D parameter space. Dissipation of the loop’s magnetic energy begins during the nonlinear stage of theinstability, which develops as a consequence of current sheet reconnection. After flaring, the loop evolves to the stateof lowest energywhere, in accordance with relaxation theory, the ratio of current to field is constant throughout the loop and helicity isconserved.Results. There exists substantial variation in the radial magnetic twist profiles for the loop states along the instability threshold. Theseresults suggest that instability cannot be predicted by anysimple twist-derived property reaching a critical value. The model is appliedsuch that the loop undergoes repeated episodes of instability followed by energy-releasing relaxation. Hence, an energy distributionof the nanoflares produced is collated. This paper also presents the calculated relaxation states and energy releases for all instabilitythreshold points.Conclusions. The final energy distribution features two nanoflare populations that follow different power laws. The power law indexfor the higher energy population is more than sufficient for coronal heating.

Key words. Instabilities — Magnetic fields — Magnetic reconnection — Magnetohydrodynamics (MHD) — Plasmas — Sun: corona

1. Introduction

The idea that nanoflares are sufficiently numerous to maintaincoronal temperatures was first proposed by Parker (1988). Thistheory implies that the coronal background emission is the resultof nanoflares occurring continually throughout the solar atmo-sphere. Hence, it is probably unfeasible to observe these flaresindividually. This perhaps explains why observational studieshave failed to agree on the importance of nanoflares with re-gard to coronal heating (Krucker & Benz 1998; Parnell & Jupp2000; Aschwanden & Parnell 2002; Parnell 2004). The imprac-ticality of nanoflare detection has motivated the development ofmodels that attempt to show how coronal loops might dissipatetheir magnetic energy in the form of discrete heating eventsandthereby replenish coronal heating losses. The primary dissipa-tion mechanism is thought to be magnetic reconnection, sincestrong evidence of this process has been found in observationsof large-scale flares (Fletcher 2009; Qiu 2009). Coronal heatingrequires fast dissipation of magnetic energy, a requirement thatis compatible with reconnection timescales.

A loop’s excess magnetic energy is introduced via the ran-dom convective motions that occur at or below the loop’s pho-tospheric boundaries (i.e., footpoints). There are two possibil-

Send offprint requests to: P. K. Browning

ities for how the kinetic energy of these motions is dissipatedwithin the loop (Klimchuk 2006). Direct current (DC) heat-ing occurs when the Alfven time is small compared to thetimescale of photospheric turbulence. Thus, the loop movesquasi-statically through a series of force-free equilibria until theaccumulating magnetic stresses are dissipated as heat. If theAlfven time is slow compared to photospheric motions, alternat-ing current (AC) heating takes place: footpoint motions generatewaves (acoustic and MHD), some of which may deposit energywithin the coronal loop. The waves that can penetrate the coronaare thought to be restricted by type and frequency (Narain &Ulmschieder 1996; Hollweg 1984). Whether or not these wavescarry sufficient energy to heat the corona is not certain (Porteret al. 1994). There is considerably less doubt, however, over thesufficiency of DC heating, which is the basis for the model pre-sented here. (AC heating may still be partly reponsible for coro-nal heating: the reconnection mechanism mentioned above maygenerate waves within the corona.)

The coronal part of the loop is defined by its magneticfield (plasma beta,β≈ 0.01) and, since the loop’s magnetic fluxand plasma are frozen together, its magnetic field is continu-ally twisted by random swirling motions at the photosphere.Ifthe driving is slow compared to Alfven timescales, the loop’sforce-free state can be represented by∇ × B = α(r)B, where

http://arxiv.org/abs/1005.5249v2

2 M. R. Bareford et al.: The Energy Distribution of Nanoflares

α = (µ0 j · B)/(|B|2) is the ratio of current density to magneticfield andr is a position vector (Woltjer 1958).

Many 3D MHD models have shown how such coronal loopsexhibit current sheet formation during the nonlinear phaseofan ideal kink instability (Baty & Heyvaerts 1996; Velli et al.1997; Arber et al. 1999; Baty 2000). Essentially, helical currentsheets become the site of Ohmic dissipation, resulting in a heat-ing event. Further simulations have revealed the appropriate cor-relation between magnetic energy and Ohmic heating (Browning& Van der Linden 2003; Browning et al. 2008; Hood et al. 2009).The energy released by an instability depends on the complexdynamics of the magnetic reconnection events that occur insidethe loop. Obviously, this is difficult to model - although this hasbeen achieved by 3D MHD simulations. The computational ex-pense of these simulations rules them out as a means of explor-ing fully the relationship between theα-profile and the amountof energy released. Fortunately, there is another way to calculatethe energy release.

Relaxation theory states that when a magnetic field reachesinstability (or is otherwise disrupted) it will evolve towards aminimum energy state such that the total magnetic axial flux andtheglobal magnetic helicity are conserved (Taylor 1974, 1986).The relaxed state is the well known constant-α or linear force-free field:

∇ × B = αB. (1)

The original intention of this theory was to explain laboratoryplasma phenomena; but latterly, it has been frequently applied tothe solar corona (Heyvaerts & Priest 1984; Browning et al. 1986;Vekstein et al. 1993; Zhang & Low 2003; Priest et al. 2005).The helicity measures the self-linkage of the magnetic field, seeBerger (1999). A modified expression for this quantity needstobe used since the field lines cross the photospheric boundaries,which creates non-zero normal flux and therefore removes gaugeinvariance.

K =∫

V(A + A′) · (B − B ′) dV, (2)

whereA is the magnetic potential,B ′ is the potential field withthe same boundary conditions andA′ is the corresponding vec-tor potential. This relative helicity (Berger & Field 1984;Finn& Antonsen 1985) is the difference between the helicity of theactual field and that of a potential field (represented by the dashterms) with the same normal flux distribution. In ideal MHD,the helicity of every flux region is conserved. Taylor proposedthat in the presence of reconnection all such local invariantsare destroyed, whilst the global helicity is conserved. Helicityis still affected by global resistive diffusion, however, it can stillbe treated as a conserved quantity, if the change in helicityissubstantially smaller than the change in magnetic energy. Theresults of one aforementioned MHD simulation (Browning et al.2008) have shown thatδK/K ∼ 10−4 andδW/W ∼ 10−2 (it shouldbe noted that these figures are limited by the coarseness of thenumerical grid used in the simulation). During relaxation,helic-ity is redistributed more evenly within the loop; as a consequenceα becomes invariant with radius. These properties enable onetolocate a loop’s relaxed state and hence calculate the energyre-leased during relaxation, i.e., the upper limit of the flare energy.

In applying relaxation theory to coronal heating, it becameclear that the effectiveness of the heating depends on how muchfree energy is stored before a relaxation event occurs (Heyvaerts& Priest 1984). Browning & Van der Linden (2003) proposedthat relaxation is triggered by the onset of ideal MHD instabil-ity. Thus, the coronal field evolves quasi-statically in response

to slow photospheric driving, building up free magnetic energy- until the field becomes unstable (i.e., the threshold for linearinstability is reached). At this point, a dynamic heating eventensues. It should be noted that ideal instabilities are relevant (asopposed to resistive instabilities) because the time-scales are suf-ficiently fast. Dissipation and energy release can occur duringthe nonlinear phase of the instability, as has been demonstratedextensively by numerical simulations of the nonlinear kinkinsta-bility (Galsgaard & Nordlund 1997; Velli et al. 1997; Lionello etal. 1998; Baty 2000; Gerrard et al. 2001). The helical deforma-tion of the kink instability generates current sheets in thenon-linear regime, in which fast magnetic reconnection rapidlydissi-pates magnetic energy. Recently, it has been demonstrated using3D MHD simulations that this process causes the field to relaxtowards a state which is closely-approximated as a constant-αstate, and the energy release is in good agreement with relaxationtheory (Browning et al. 2008; Hood et al. 2009). These papershave investigated field profiles, based on the model of Browning& Van der Linden (2003), in the unstable region of parameterspace and have shown that fast reconnection develops in a cur-rent sheet, with dissipation of magnetic energy and relaxation toa new, lower-energy state. Efficient ”mixing” of theα profile isfacilitated in the later phases of the evolution, during which thecurrent sheet fragments (Hood et al. 2009).

Supported by this evidence, the model developed here isbased on the idea that coronal magnetic fields respond to pho-tospheric driving by evolving through force-free equilibria, witha heating event triggered whenever the ideal instability thresh-old is reached. Following Browning & Van der Linden (2003),we use a simple cylindrical field model, and represent the nonlin-ear force-free field using a two-parameter family of currentpro-files in whichα is a piecewise constant. Browning and Van derLinden considered only individual heating events. The primaryaim of this paper is to predict adistribution of heating events,generated by an ongoing stress-relax cycle. The photosphericdriving stresses the coronal magnetic field until it becomesun-stable and relaxes; then the driving resumes, and the process re-peats, leading to a series of heating events as expected in thenanoflare coronal heating scenario. This is modelled through amonte-carlo approach, with the photospheric footpoint motionstreated as random. Recent observational evidence (Abramenkoet al. 2006) lends support to the idea that random, turbulentpho-tospheric motions can provide an energy source for coronal heat-ing in Active Regions.

In calculating the relaxation and distribution of heatingevents, we need to know the linear instability threshold forline-tied coronal loops over a family of field profiles. As a conse-quence, we obtain some interesting new results concerning sta-bility properties of cylindrical loops, for a much more extensiverange of current profiles than previously studied (including pro-files for which the sense of the twist of the field varies acrosstheloop). A secondary aim is thus to explore the properties of thelinear kink instability on this family of fields, and in particular,to determine the extent to which stability can be defined by anysingle quantity such as ”critical twist”.

The paper is structured in the following manner. Section 2describes the composition of the loop model, along with theequations used to express the loop’s magnetic field. The calcula-tion of the loop’s instability threshold is also explained,as wellas the procedure for allowing the loop to repeatedly undergoin-stability and the equations used to determine the energy releaseassociated with each relaxation. Section 3 outlines the analysisof possible critical parameter values for the onset of loop insta-bility. The results of the simulations of energy release arepre-

M. R. Bareford et al.: The Energy Distribution of Nanoflares 3

sented in Sect. 4 as nanoflare energy distributions. Finally, inthe last section, the results are discussed and our conclusions aregiven.

2. Model

The model discussed here was used by Browning & Van derLinden (2003) and then extended by Browning et al. (2008) toinclude a potential envelope (Fig. 1). First, a loop is consid-

Fig. 1. Schematic of a straightened coronal loop in ther-θ plane (left)and in ther-z plane (right). The loop, comprises a core (dark grey)and an outer layer (light grey); it is embedded in a potentialenvelope(white). The core radius is half the loop radius and 1/6 the enveloperadius (R1:R2:R3 = 0.5:1:3). The loop’s aspect ratio (L/R2) is 20.

ered to evolve through equilibria as it is driven by photosphericfootpoint motions. An idealised model of a straight cylindricalloop is used with the photosphere represented by two planesat z=0,L; however, the essential physics should apply to morecomplex geometries. The stressed field is line-tied with (ingen-eral) a non-uniformα(r) (whereα = µ0 j‖/B). This is repre-sented by a two-parameter family of piecewise-constant-α pro-files. Secondly, it is proposed that a relaxation event is triggeredwhen the loop’s field becomes linearly unstable. The energy re-leased, due to fast magnetic reconnection during the nonlineardevelopment of the instability, can then be calculated using re-laxation theory.

The MHD kink instability needs to be ideal in order tobe consistent with the observed rapidity of flare occurrences.However, ideal conditions mean there is no resitivity to dissi-pate magnetic energy; the 3D MHD simulations discussed in theIntroduction provide a solution to this impasse. Once a linearinstability has achieved a positive growth rate, it will soon be-come nonlinear: at this point, current sheets will form whereinfast reconnection of the magnetic field can take place. Theseexpectations are justified by the results of the cited numeri-cal simulations. The linear perturbation can be represented asf = f (r, z)eiθeγt, where the azimuthal mode number is set to 1;this mode has been found to be the least stable (Van der Linden& Hood 1999). The effect of such perturbations on the coronalloop are represented by the standard set of linearised idealMHDequations.

2.1. Equilibrium fields

The loop’s radialα-profile is approximated by a piecewise-constant function featuring two parameters. This design, firstproposed by Melrose et al. (1994), is readily extensible: extralayers of constantα can be inserted to obtain more realistic pro-files. The ratio of current to magnetic field isα1 in the core,α2in the outer layer and zero in the potential envelope. Note thatthe magnetic field is continuous everywhere (though the currenthas discontinuities). Recent work indicates that theseα discon-tinuities have little discernable effect when compared to similarbut continuousα-profiles (Hood et al. 2009).

Without an envelope, the loop’s outer surface (located atR2)acts as a conducting wall. This is unrealistic in the contextofthe solar corona; the loop would be more stable than it mightbe otherwise. Browning et al. (2008) plotted the relationship be-tween the growth rate of the instability and the distance to theouter surface of the potential envelope (i.e., a more distant con-ducting wall). They found that for six unstable loop states thegrowth rate was invariant once the outer surface of the envelope(R3) exceeded3

2 R2. TheR3 boundary was placed at twice thisvalue.

The fields are expressed in terms of the well-known Besselfunction model, generalised to the concentric layer geome-try (Melrose et al. 1994; Browning & Van der Linden 2003;Browning et al. 2008). These expressions will change slightlywheneverα1 or α2 become negative; these alterations are cap-tured byσ symbols:σ1 =

α1|α1|

, σ2 =α2|α2|

andσ1,2 = σ1σ2.(The sign of anα term merely denotes the orientation of theazimuthal field). Thus, the field equations for the three regions(core, outer layer and potential envelope) are as follows:

B1z = B1J0(|α1|r), (3)

B1θ = σ1B1J1(|α1|r), 0 ≤ r ≤ R1, (4)

B2z = B2J0(|α2|r) +C2Y0(|α2|r), (5)

B2θ = σ2(B2J1(|α2|r) +C2Y1(|α2|r)), R1 ≤ r ≤ R2, (6)

B3z = B3, (7)

B3θ = σ2C3

rR2, R2 ≤ r ≤ R3. (8)

The fields must be continuous at the inner radial boundaries,R1andR2. Therefore, the constantsB2, B3, C2 andC3 can be ex-pressed like so:

B2 = B1σ1,2J1(|α1|R1)Y0(|α2|R1) − J0(|α1|R1)Y1(|α2|R1)

∆, (9)

C2 = B1J0(|α1|R1)J1(|α2|R1) − σ1,2J1(|α1|R1)J0(|α2|R1)

∆, (10)

B3 = B2F0(|α2|R2), (11)

C3 = B2F1(|α2|R2), (12)

where

∆ =2

π|α2|R1, (13)

F0,1(x) = J0,1(x) +C2

B2Y0,1(x). (14)


At all times, the magnetic flux through the loop and envelope isconserved:

ψ∗ =

∫ R3

02πr∗B∗z dr∗ =

2πB2

|α2|R2F1(|α2|R2)

+ 2πR1B1J1(|α1|R1)

(

1|α1|−σ1,2

|α2|

)

+ πB2F0(|α2|R2)(

R 23 − R 2

2

)

, (15)

where the asterisks denote dimensionless quantities. Hence, inthe model,ψ∗ is normalised to 1 andB1 can be determined (not-ing that, in equation 15,B2 is a function ofB1). We also nor-malise with respect to the coronal loop radius,R2, see Fig. 1.

As the random motions of the photosphere proceed, the loopevolves through a series of force-free equilibrium states until itbecomes linearly unstable. We now discuss the calculation of theinstability onset.

2.2. Linear kink instability threshold

A coronal loop’s instability is constrained by the line-tying ofthe photospheric footpoints (Hood 1992). Hence, all perturba-tions are required to vanish at the loop ends (z= 0,L). Whenthe growth rate of a perturbation transitions from a negativevalue to a positive one, the loop has reached the threshold ofan ideal linear instability. The instability threshold is acurvein 2-dimensionalα-space (α1, α2). The properties of the loop(e.g.,α1 andα2) at these threshold points can be found by sub-stituting the perturbation function into the linearised MHD equa-tions, leading to an eigenvalue equation for the growth rates(Chap. 7, Priest 1987). The growth rates and eigenfunctionsofthe most unstable modes are found numerically, for line-tiedfields, with the CILTS code, described in Browning & Van derLinden (2003) and Browning et al. (2008). CILTS can be config-ured such that one of the loop’sα parameters is fixed whilst theother is incremented. The code terminates as soon as the realpartof the eigenfunction falls below zero, i.e., the loop is no longerunstable to kink perturbations. The left panel of Fig. 2 shows theclosed instability threshold curve mapped by the CILTS code(see also Fig. 5 of Browning et al. 2008). The threshold curvehas symmetry: it is invariant when rotated byπ radians. Thus,it is sufficient to show how various properties (e.g., magnetictwist and energy release) vary along the top half of the thresh-old curve. For ease of plotting we can convert this half of thethreshold curve to a one dimensional form: the filled circlesandbold numbers shown in the right panel of Fig. 2 represent the ticmarks and labels for the 1D threshold point axis, see Figs. 8 -11.

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Fig. 2. The left panel shows the closed instability threshold (solid) withtheBz reversal lines (dashed). The top half of the threshold (whereα2 >

0), annotated with threshold point numbers, is shown in the right panel.

It is important to remember that the threshold only appliesto the specific loop geometry outlined above. A new thresholdwould need to be calculated should the loop’s proportions, com-position or envelope change as a consequence of some activ-

ity. Another caveat is that there are some points inα-space thatyield singularities when calculating quantities such as helicityor magnetic energy. These arise when the axial field (Bz) has asignificant region of reversal; sinceψ∗ is normalised to 1 andconserved, the helicity and energy terms will diverge when un-normalisedψ∗ → 0 (see Browning & Van der Linden 2003).Fortunately, the instability threshold does not enter the regionwhereBz begins to pass through zero (see dashed lines in Fig.2), so these singularities are not encountered.

2.3. Random walk

When a loop is twisted by turbulent photospheric motions, itperforms a random walk through theα-space enclosed by theinstability threshold (Fig. 3). This traversal ofα-space is ran-dom in direction but constant in length (we set the dimension-less step-length to beδα=0.1 for most of the results presentedhere). Clearly, the nature of this random walk depends on thestatistical properties of the driving photospheric motions; in fu-ture, different forms of this random driving will be investigated,but for now, we take the simplest assumptions.

The time unitτ is the step time, the time taken forα to changeby 0.1/R2 (in dimensional units). We may estimate, roughly,a timescale for this process as follows. Based on axial val-ues, a changeδα corresponds to a change in magnetic twistδφ≈ (L/2)(δα/R2); takingL/R2= 20 givesδφ≈1. If this is causedby photospheric twisting motions of magnitudevθ for a time in-tervalτ, we findτ≈ (δφ)R f /vθ, whereR f is the footpoint radius.With typical values ofR f =200 km andvθ = 1 km s−1, we obtainτ≈ 200 s; note that this is consistent with quasi-static evolution,justifying a posteriori our choice of random-walk step size. Thestep time (τ) may be identified with the correlation time of pho-tospheric motions, which is likely to be rather longer than thevalue given above; for example, a granule lifetime of 1000 s maybe appropriate (Zirker & Cleveland 1993). The effect of increas-ing the step length (δα) is considered in Sect. 4.2.3

Eventually, the field will reach the instability threshold:itwill become linearly unstable. At this point, the field releasesenergy and transitions to a lower-energy state defined by Taylorrelaxation: helicity is conserved and theα-profile relaxes to asingle value.

2.4. Energy release calculation

We have extended the model (Browning & Van der Linden 2003;Browning et al. 2008) to allow a loop torepeatedly undergo re-laxation as it evolves withinα-space. Initially, a loop starts froma randomly-selected stable state. The field profile then undergoesa random walk until it crosses the instability threshold; where-upon, the loop relaxes and the profile transitions to the relax-ation line (α1=α2). The constantα-value (αrx) will, of course,vary depending on where the threshold was crossed;αrx is foundby helicity conservation (Browning & Van der Linden 2003). Inmathematical terms, we find the roots of the following equation:

K(αrx) − K(αit1, αit2) = 0, (16)

whereαit1 andαit2 are the coordinates of the instability thresh-old crossing (conservation of axial flux is assured through thenormalisationψ∗ = 1). The helicity can be expressed as follows:

K = 2L∫ R3

0

Iψ(r)r

dr, (17)

whereI is the current andL is the loop length (Finn & Antonsen1985). The equation for the magnetic energy contained within


the loop and envelope is straightfoward:

W =Lπµ0

∫ R3

0rB 2 dr, (18)

whereL is normalised to 20 (sinceR2=1) andµ0 is set to 1.See Appendix A for the full expressions. The energy differencebetween the unstable and relaxed states can be calculated thus:

δW = W(αit1, αit2) −W(αrx). (19)

This is the relaxation energy: the energy released as heat duringthe event.

After relaxation, the loop resumes its random walk until itreaches the threshold and the process repeats. Fig. 3 gives an il-lustration of this process. We thus generate a sequence of energy

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Fig. 3. The instability threshold encloses the relaxation line, which isa subsection of theα1=α2 line centred on the origin. The annotationsillustrate the initial stages of a simulation that begins atthe positionmarked by the cross. The first random walk is shown in light grey; itends at the threshold position markedN and the associated relaxationpoint is indicated by△, which is the starting point for a second ran-dom walk (dark grey). This walk attains instability onset atthe positionmarked� and relaxes to the point labelled� - the starting point for athird walk.

release events, which are collated to produce a nanoflare energydistribution, see Sect. 4.2.

When the loop relaxes, theα-profile throughout the loop andenvelope becomes constant, i.e.,α1=α2=α3,0. The envelopeis no longer potential; it has acquired a residual current. In prin-ciple, as the main portion of the loop is twisted again by on-going motions, a new equilibrium would develop with varyingcurrent (α1,α2) in the loop and a non-zero current (α3) in theenvelope. For some threshold sections the consequent residualcurrent is so small that the threshold shape would remain un-changed. However, the validity of the simulation process canonly be assured by including an extra stage, wherein the enve-lope dissipates its helicity so that it becomes potential again. Theloop can now resume its random walk with respect to the samethreshold.

Although the primary purpose of this model is to calculatethe distribution of energy releases, in achieving this we also ob-tain some interesting new results on linear stability, which aresummarised in the next section. This is because, in order to ex-plore the full parameter space of equilibrium current profiles, wehave calculated the linear stability properties of a much widerfamily of fields than previously investigated: in particular, fieldswith reversed twists.

3. Instability threshold and critical twist

The evolution of the field profile throughα-space is determinedby photospheric perturbations which map to changes in the mag-netic field and hence to changes inα1 andα2. A loop’s magnetictwist is directly related to rotational photospheric motions:

ϕ =LBθ(r)rBz(r)

, 0 ≤ r ≤ R3. (20)

Thus, the photospheric motions directly determine a loop’sϕ-profile, which in turn determinesα(r) (and hence, in our model,α1 andα2). The magnetic twist of coronal loops is an observablefeature (Kwon & Chae 2008). Portier-Fozzani et al. (2001) haveeven observed a loop’s twist decreasing over time - evidenceper-haps of a loop evolving towards a state of minimum energy.

3.1. Criteria for instability

Many workers have looked at the idea that the magnetic twist ofa loop can be used as a proxy for the onset of kink instability withsomecritical twist parameter determining instability. Hood &Priest (1979) performed a MHD stability analysis on a variety ofstraightened line-tied coronal loops. They showed that thecriti-cal twist,ϕcrit, varies according to the aspect ratio (L/R2), plasmabeta (β) and the transverse magnetic structure (i.e., the nature ofthe variable twist profile). In general, short fat loops (lowaspectratio) have low critical twists, whereas long thin loops (high as-pect ratio) have high critical twists. Subsequent work (Hood &Priest 1981) revealed that a loop ofuniform twist, aspect ratio 10and zeroβ possessed a critical twist of 2.49π. This figure is oftenquoted as a general result.

The question arises as to whether there is any single param-eter (such as peak or average twist) which determines instabilityonset forall twist profiles. Indeed, more generally, it would bedesirable to have a single quantity determining instability onseteven for more complex (non-cylinderical) fields (Malanushenkoet al. 2009). Incidental to our task, we have calculated the sta-bility properties of an extensive family of equilibria, includingfields with reversed twist (as well as simple monotonic-twist pro-files as used by other authors). This provides a very useful testfor any proposed criteria for instability onset.

Loops with variable-twist profiles have been previously stud-ied; however, all such profiles generally have a similar form(Velli et al. 1990; Mikic et al. 1990; Baty 2001): the axial twistis the maximum, then the twist declines to a negligible valueat the loop boundary. Velli et al. (1990) calculated that insta-bility occurred whenϕ0= ϕ(r=0)=2.5π; this agrees with Hoodand Priest’s result for a uniform twist profile. Mikic et al.(1990)calculated a critical axial twist of 4.8π, the loop’s average twisthowever, was∼2.5π.

The idea of using magnetic twist as a marker for instabil-ity relies on the existence of some twist-derived parameterhav-ing a constant value for all the points on the instability thresh-old. Baty (2001) used a MHD stability code to show that fora small set of equilibria the average twist at instability isthesame for several different magnetic configurations. However,there are several differences between Baty’s work and the modelpresented here. Firstly, the twist profiles defined for each equi-librium are all positive and none contain multiple peaks (Fig.1,Baty 2001). Furthermore, critical twist convergence arises whenthe normalised distance,d, is greater than 5 (Fig.5, Baty 2001).

d =ϕ0R2

L. (21)


In our equilibria,d = ϕ0/20, the threshold values of the absoluteaxial magnetic twist vary from zero to 9π, which meansd variesfrom 0 to 1.42, thus the average twists will not be in the regimewhere the critical axial twist converges to 2.5π. In fact, the larged regime cannot be attained.

The idea of a single critical average twist seems unlikelywhen one examines the threshold presented here. Clearly, atsome threshold pointsα1 andα2 are of opposite sign; thus, oneor two places on the threshold will have an average twist equal tozero, and yet they are unstable. Perhaps critical twist is achievedwithin a subsection of the loop; this idea is explored further inSects. 3.2 and 3.3.

3.2. Radial twist profiles and linear eigenfunctions

In order to understand the nature of the instability, we investi-gate the twist profiles and eigenfunctions of the unstable mode,for different parts of the instability threshold. The magnetic twistprofiles for a selection of points just outside the thresholdcurveexhibit considerable variation, as Fig. 4 illustrates. Forthe un-

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D

0

1

2

3

4

5

6

7

8

9

0 0.5 1 1.5 2 2.5 3

ϕ

r

E

F

Fig. 4. The loop’s radial twist profile at specific points along the thresh-old (labelled A-F).

stable equilibrium labelled as point B (α1< 0, α2> 0), the cor-responding twist profile (also labelled B) shows that the corefield has a strong negative twist, whilst in most of the outer layerand in all of the envelope the field has positive twist. As onemoves from A to B, the twist in the core becomes more neg-ative whilst the twist in the outer layer moves in the oppositedirection. It appears that the increase inα2 stabilises the neg-ative core twist by providing additional reversed (i.e., positive)twist in the outer layer. The sharp corner at the top left of thethreshold marks the point where instabilities driven within thecore intersect those that originate from within the outer layer.Profiles C to D therefore, suggest instabilities driven in the outerlayer, since|α2| > |α1|. The peak twist in the outer layer reducesas the core twist moves from negative to positive. Further alongthe threshold, whereα1 > α2, the instabilities are likely to bedriven in the core. Note that profiles D, E and F are always posi-tive in sign; D has a twist peak near the loop edge while E and Fare roughly monotonically decreasing. The corresponding mag-netic field profiles for the six points A-F are given in AppendixB.

It seems that instabilities are driven mainly on or near thepeak of largest absolute twist. A twisted field region may be sta-bilised by an enclosing region of opposite twist. Furthermore,a twisted outer layer may need less twist to achieve instability

if the core has the same twist orientation. To investigate theseideas further, we plot the unstable eigenfunctions obtained fromCILTS for the sameα-space points, A to F.

Fig. 5. The linear eigenfunction,V x(x, y=0,z), for theα-space pointsA (left) and B (right) profiled in Fig. 4. Cartesian coordinates are used,hence, the x-axis is equivalent to the radial axis.

The eigenfunctions for profiles A and B (Fig. 5) show thatthe amplitude is strongest in the core. Interestingly, the ampli-tude for profile A has dropped to zero long before the envelopeboundary atR3, which suggests that in the subsequent relaxationonly inner regions will be affected, with little change in the po-tential envelope. There is a strong similarity between the eigen-

Fig. 6. The linear eigenfunction,V x(x, y=0,z), for theα-space points C(left) and D (right) profiled in Fig. 4.

functions for profiles C to D (Fig. 6) and the amplitude is high-est near theR2 boundary, indicating an outer layer instability.Finally, the two plots in Fig. 7 (profiles E and F), clearly showa progression towards the eigenfunction calculated for A (albeitwith a V x of opposite sign). Notice also that the form of the

Fig. 7. The linear eigenfunction,V x(x, y=0,z), for theα-space points E(left) and F (right) profiled in Fig. 4.

eigenfunction changes significantly between B and C, indicatingthat different modes are going unstable. This is to be expected,since theα-space positions labelled B and C (Fig. 4, top left)are either side of the intersection point formed by the two curvesthat describe the instability threshold.


3.3. Critical twist parameters

Next, we look for a twist-related parameter that takes on a criti-cal value whenever the loop reaches the threshold. As expected,the variation in axial twist,ϕ0, is similar to the variation inϕ(R1),see Fig. 8. None of the quantities suggest any single (constant)

-15

-10

-5

0

5

10

0 20 40 60 80 100 120

Mag

netic

Tw

ist

Threshold Point

Fig. 8. The variation in magnetic twist around the instability threshold(the threshold points are defined in Fig. 2) for three radial positions.The solid line represents the variation in axial twist,ϕ0; the dashed lineis the variation in twist at the boundary between the core andthe outerlayer,ϕ(R1); and the long-short dashed line is the variation in twist atthe boundary between the outer layer and the potential envelope,ϕ(R2).The twist values are plotted in units ofπ.

critical value. Perhaps, the average twist is less variablearoundthe threshold? There are several ways to calculate this expres-sion;

〈ϕ〉Ri0 =

∫ R

0LBθ(r) dr

∫ R

0rBz(r) dr

, (22)

〈ϕ〉Ri0 =

1R

∫ R

0

LBθ(r)rBz(r)

dr , (23)

〈ϕ〉Ri0 =

1πR2

∫ R

02πr

LBθ(r)rBz(r)

dr , (24)

whereRi is a radial upper bound (e.g.,R2 or R3). Equation 24 isthe average twist weighted by area. The other two equations (22and 23) have been used by Baty (2001) and Velli et al. (1990).Note, Equation 22 can be calculated analytically, see AppendixA. 〈ϕ〉R2

0 denotes the average twist, weighted by area, over thecore and outer layer. Similarly,〈ϕ〉R3

0 denotes the same quantitybut over the loop and potential envelope. The tilde (∼) and hat(∧) symbols are used to indicate the other equations.

Whenα1 andα2 are equal there is no distinction between theloop regions; this occurs on the threshold whenα1=α2≈ 1.4 - atthis point〈ϕ〉R2

0 ≈ 5π. Whenα2= 0 andα1≈ 2.2 the loop is identi-cal to the core with a bigger potential envelope and〈ϕ〉R1

0 ≈7.7π.Thus, as expected, fatter loops like the first case, have lower in-stability twists than thinner ones. When the loop’s core is poten-tial (i.e., whenα1=0 andα2≈2.2), 〈ϕ〉R2

R1≈4.5π is the average

outer layer twist. In this configuration, the loop is less stable thanthe case when only the core is non-potential.

None of the twist averages (Fig. 9) is invariant along the en-tire threshold. Although,〈ϕ〉R3

0 and〈ϕ〉R30 have approximately the

same value (≈ 2.2π) between threshold points 40 and 90. Allof these points show a positive peak twist atR2. Hence, the en-velope’s contribution dominates the overall average twist. This

-10

-5

0

5

0 20 40 60 80 100 120

Ave

rage

Mag

netic

Tw

ist [

0-R

1]

Threshold Point

-10

-5

0

5

0 20 40 60 80 100 120

Ave

rage

Mag

netic

Tw

ist [

R1-

R2]

Threshold Point

-10

-5

0

5

0 20 40 60 80 100 120

Ave

rage

Mag

netic

Tw

ist [

0-R

2]

Threshold Point

-10

-5

0

5

0 20 40 60 80 100 120

Ave

rage

Mag

netic

Tw

ist [

0-R

3]

Threshold Point

Fig. 9. The variation in the average twist over the core, 0 -R1 (top left),the outer layer,R1 - R2 (top right), the loop, 0 -R2 (bottom left) and theloop and envelope, 0 -R3 (bottom right). The solid lines were calculatedaccording to Eqn. 24; the dashed according to Eqn. 23 and the long-short dashed according to Eqn. 22. Again, the twist values are plottedin units ofπ.

is especially true for the〈ϕ〉R3

0 case: the higher the radial coor-dinate the greater the weight of the calculated twist. Within theenvelope, the twist declines as 1/r and so, the inclusion of theenvelope twist averages out the final result.

Notice also, that all the twist averages go through zero whenthe core and outer layer have opposite twists. It seems that thesequantities do not reveal the detail necessary to understandwhya particular loop configuration is on the point of instability, norwhere in the loop that instability originates.

Finally, we consider the proposal of Malanushenko et al.(2009), that a critical value of normalised helicity (equivalent, inour terms, to the normalised loop helicity,K/ψ2, over the range0 -R2) indicates instability onset. In fact, the normalised helicity

-2-1.5

-1-0.5

0 0.5

1 1.5

2 2.5

3

0 20 40 60 80 100 120

K/ψ

2 [0-R

2]

Threshold Point

Fig. 10. The variation inK/ψ2 (over the range 0 -R2) along the insta-bility threshold. The threshold states between the vertical dashed linesfeature reverse twist; outside the lines the twist is single-signed.

is certainly not the same for every threshold point; this quantitypasses through zero becauseα1 andα2 take on values of oppo-site sign along some sections of the threshold. For fields withsingle-signed twist, the normalised helicity gives an approxi-mate threshold, but even here, the (absolute) critical value rangesfrom about 1.5 - 2.5. Figure 10 shows that the idea of such a crit-ical value breaks down for loops that feature regions of reversedtwist.


4. Distribution of energies and coronal heatingconsiderations

We now proceed to the main task of our work, which is to cal-culate the distribution of magnitudes of the sequence of heatingevents generated by random photospheric driving.

4.1. Helicity and energy

The top left panel of Fig. 11 plots total helicities of the thresh-old states. A total helicity (or flux) is one calculated over therange 0 -R3, i.e., the loop and envelope. None of the threshold

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120

K

Threshold Point

0.34 0.36 0.38

0.4 0.42 0.44 0.46 0.48

0.5 0.52 0.54 0.56

0 20 40 60 80 100 120

W*

Threshold Point

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 20 40 60 80 100 120

α rx

Threshold Point

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 20 40 60 80 100 120

δW*

Threshold Point

Fig. 11. Helicity (top left), magnetic energy (top right), relaxed alpha(bottom left) and energy release (bottom right) along the 1Drepresenta-tion of the instability threshold. The energies,W* and δW∗, are dimen-sionless quantities.

states have sufficient helicity for the relaxed state to feature he-lical modes (Taylor 1986) and so all relaxed states are cylin-derically symmetric. The bottom left panel confirms that eachthreshold state corresponds to a relaxed state. The maximumαrxis about 0.48. Hence, there is a good chance that the envelopecurrent after relaxation will not be insignificant. This result con-firms the need to investigate how this current could be dissipated(although the maximum value is still significantly less thantyp-ical α1 andα2 threshold values).

The energies shown in Fig. 11 (top right and bottom right)are given as dimensionless quantities, to calculate the dimen-sional energy we first give the dimensional flux though the loopand envelope,

ψ =

∫ 3Rc

02πrBz dr = 9πR2

c Bc , (25)

whereRc is the coronal loop radius (the dimensionalisedR2) andBc is themean axial coronal field. Since we have normalised themagnetic field, such thatψ∗ = 1, the field strength can be dimen-sionalised thus,

B =ψ

R2c

B∗ = 9πBcB∗, (26)

where the asterisk is again used to denote dimensionless values.Hence the dimensional energy release becomes

δW =1µ0

(

ψ

R2c

)2

R3c δW ∗ =

81π2

µ0R3

c B2c δW ∗ . (27)

This expression differs slightly from the one used by Browning& Van der Linden (2003): their dimensionless energy releaseiscalculated per unit length andR3=R2. Assuming typical values(Rc = 1 Mm andBc = 0.01 T), we obtain dimensional energy val-ues of 6× 1022δW∗ J ≡ 6 × 1029δW∗ erg. Thus, the top end oftheδW∗ scale (≈0.073) is equivalent to 4× 1028 erg. This is inthe microflare range, but nanoflare energies will be obtainedforweaker fields or for smaller loops.

4.2. Flare energy distribution

Every time a loop’s random walk reaches the instability thresh-old, the loop’s relaxed state (i.e., its position on the relaxationline) and energy release are calculated. The relaxed state is thestart of a new random walk which will lead to another relaxation(and energy release). If this process is repeated often enough itcan be shown that certain energy release sizes are more commonthan others. The flare energy distributions converge as the num-ber of relaxation events simulated (see Sect. 2.4) is increased.

0

0.02

0.04

0.06

0.08

0 0.02 0.04 0.06 0.08

Fla

re C

ount

(no

rmal

ised

)

Flare Energy (dimensionless)

0

0.02

0.04

0.06

0.08

0 0.02 0.04 0.06 0.08

Fla

re C

ount

(no

rmal

ised

)


0

0.02

0.04

0.06

0.08

0 0.02 0.04 0.06 0.08

Fla

re C

ount

(no

rmal

ised

)


0

0.02

0.04

0.06

0.08

0 0.02 0.04 0.06 0.08F

lare

Cou

nt (

norm

alis

ed)


Fig. 12. Flare energy distributions for 100 (top left), 1000 (top right),104 (bottom left) and 105 (bottom right) relaxation events.

The gross features of the converged energy distribution canbe explained by presenting the energy distribution for the high-est number of flare events (Fig. 12, bottom right) alongside theinstability threshold in Fig. 13. Both the distribution andthethreshold are colour-coded according to event energy. The colourof the threshold point encountered by a coronal loop indicatesthe energy of the resulting flare and also the part of the distri-bution where the heating event falls. Dimensionalised values forthe flare energies have been recovered by using typical loop pa-rameters (see above).

The energy release changes as one moves along the thresh-old, i.e., there is an energy release gradient. This gradient issmall for low energies, therefore, only a few bins cover the lowenergy sections of the threshold, see Fig. 13. The low energyre-lease events are divided amongst a small number of bins, hence,these bins contain many more events. The threshold sectionscor-responding to the profile minima are slightly longer than thoseassociated with the first peak (≈ 1.5 times), however, the minimasections have a much higher energy release gradient. The energyrelease events are divided amongst a higher number of bins andso each of these bins contain fewer events. As one moves intothe threshold sections that correspond with the second profilepeak (denoted by blue-green shades) the energy release gradientdecreases. Hence, one would expect the associated bins to havehigher event counts. This increase is accentuated however by the


proximity of the corresponding relaxation points. Once a loopachieves a blue-green instability, it will relax to a point (alsocoloured blue-green), which happens to be closest to the greensection of the threshold. Subsequent walks will have a higherchance of crossing that section, thus, the corresponding distri-bution bins will contain even more events. The highest energyreleases available on the threshold are farthest away from therelaxation line. In this part of the distribution, the chance of anenergy release event is inversely proportional to the size of theenergy release.

δW*

0 0.02 0.04 0.06 0.08


0

0.02

0.04

0.06

0.08

Fla

re C

ount

(no

rmal

ised

)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

δW*

-3 -2 -1 0 1 2 3α1

-4

-3

-2

-1

0

1

2

3

4

α 2

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Fig. 13. The flare energy distribution for 105 relaxation events and theinstability threshold (with relaxation line). All lines are colour-codedaccording to energy release.

0

0.02

0.04

0.06

0.08

0 0.02 0.04 0.06 0.08

Fla

re C

ount

(no

rmal

ised

)


0

0.02

0.04

0.06

0.08

0 0.02 0.04 0.06 0.08

Fla

re C

ount

(no

rmal

ised

)


0

0.02

0.04

0.06

0.08

0 0.02 0.04 0.06 0.08

Fla

re C

ount

(no

rmal

ised

)


0

0.02

0.04

0.06

0.08

0 0.02 0.04 0.06 0.08

Fla

re C

ount

(no

rmal

ised

)


Fig. 14. Flare energy distributions over 105 relaxation events for a vari-ety of loop lifetimes. The lifetimes are 1000 relaxation events (top left),100 relaxation events (top right), 10 relaxation events (bottom left) and1 relaxation event (bottom right).

These results are not strongly tied to the loop lifetime, whichis the number of relaxations a loop undergoes during the simu-lation. We can simulate loop replacement by randomly selectinga new position within the stability region after a certain number

of relaxations. The top left distribution of Fig. 14 is generatedfrom a loop that is replaced after every 1000 relaxation events.As the loop lifetime is reduced so is the height of the secondpeak. However, this peak reduction is only noticeable for smalllifetimes, e.g.,< 10 relaxations. A minimum lifetime of 1 relax-ation (Fig. 14, bottom right) still yields a two-peaked distribu-tion, albeit with a reduced second peak. This result correspondsto finding the energy release from an ensemble of identical loops.It is also the most conservative in terms of total energy released.

The concept of a loop ensemble (a collection of 105 loopsflaring simultaneously) is particularly useful, since it allows usto sidestep the complications that come with allowing loopstosurvive many relaxations. Otherwise, we would need to under-stand how a loop is affected by its energy release. A loop mayshrink or implode after flaring (Janse & Low 2007). The instabil-ity threshold would be invalidated should the loop’s aspectratiobe altered as a consequence of post-flare shrinkage. Therefore,the next section will examine in more detail the ensemble distri-bution.

4.2.1. Distribution of ”nanoflares”

The ensemble distribution (Fig. 14, bottom right) does not yielda simple inverse power-law when converted to a log scale: thereare two peaks in the profile. The trailing edge of the first peakequates to a power-law slope of≈ -1.5. Its internal structure can-not currently be resolved since one would need to increase thethreshold point density. The trailing edge of the second peakgives a slope of≈ -8.3; this is much greater than the critical gra-dient for nanoflare heating,m ≤ −2 (Hudson 1991).

These power-law figures are provisional and are likely tochange as the model is enhanced. For example, a more realisticα-space traversal function - one whereδα2 is correlated withδα1- will prefer walks parallel to the relaxation line; this will alterthe distribution of heating events. Furthermore, we consider hereonly a single loop of fixed dimensions (or an ensemble of iden-tical loops). In reality, in any given large-scale loop structure,sub-loops of varying radii will be generated, depending on thehorizontal scale-length of the driving photospheric motions. Thedistribution must then be averaged over a distribution of radii, aswell as considering variations in length and field strength.

4.2.2. Heating Flux

The primary aim of our model is to calculate the distributionof nanoflares. In general terms, the heating rate will be similarto other calculations in the literature based on random photo-spheric twisting (Abramenko et al. 2006; Zirker & Cleveland1993; Berger 1991; Sturrock & Uchida 1981). Within our ap-proach, all the energy input from the photosphere must be dissi-pated, in a long-term time-average over many events, since thebuild up of coronal magnetic field is limited by the instabilitythreshold.

The energy flux,F, can be expressed using Eqn. 27;

F =81π2

µ0

1Nτ

12πR2

cR3

c B2c 〈δW ∗

〉 =81π2µ0

RcB2c

τ

〈δW ∗〉

N, (28)

whereN is the average number of steps taken to reach the thresh-old,τ is the time taken to complete each step in the random walkand〈δW ∗〉 is the average dimensionless energy release, given by

〈δW ∗〉 =1

105

105∑

i=1

δW ∗ . (29)


For the ensemble case,〈δW ∗〉 ≈ 0.0293 and N ≈ 264.Applying previously used values (Bc = 0.01 T, τ= 200 s andRc = 1 Mm), yields a total flux of 6×106 erg cm−2 s−1. Thisresult is applicable to Active Regions (a value for the QuietSun can be obtained by settingBc =0.001 T; this simply low-ersF to 6×104 erg cm−2 s−1). These results are slightly lowerthan Withbroe & Noyes (1977) measurements of coronal heatinglosses, which are 107 erg cm−2 s−1 (Active Regions) and 3× 105

erg cm−2 s−1 (Quiet Sun). This shortfall vanishes however if therandom walk is conducted using larger step sizes, see followingsection.

4.2.3. Random walk step size

So far, we have assumed photospheric motions to be somewhattemporally incoherent, with a short random walk step,δα= 0.1,corresponding toτ=200 s. Here, we consider the effect of vary-ing this step length; in other words, varying the coherence timeof the photospheric motions. Fig. 15, which should be comparedwith the bottom right panel of Fig. 14, shows the distributionsthat result whenδα is increased by factors of 10 and 40. Forthe latter case, the step is sufficiently long that, on average, justone step is required to reach the instability threshold: this corre-sponds to a temporally coherent twisting of the loop (althoughthe spatial profile of the twisting varies randomly between heat-ing events).

0

0.02

0.04

0.06

0.08

0 0.02 0.04 0.06 0.08

Fla

re C

ount

(no

rmal

ised

)


0

0.02

0.04

0.06

0.08

0 0.02 0.04 0.06 0.08

Fla

re C

ount

(no

rmal

ised

)


Fig. 15. Flare energy distributions over 105 relaxation events for a stepsize ofδα= 1 (left) andδα= 4 (right). Loop lifetime is one relaxationevent.

It can be seen that the energy distribution is virtually inde-pendent of the random walk step size. However, the heating fluxdoes depend on this quantity. It is expected that the heating fluxshould be proportional toτ (Berger 1991; Zirker & Cleveland1993). Eqn. 28 shows this to be the case, since the average num-ber of steps,N, for the random walk to reach the threshold scalesasN ∝ τ−2. In particular, if we consider the limiting case of co-herent twisting (τ= 8000 s,N ≈ 1.13 and〈δW ∗〉 ≈0.0305) wefind that the flux increases significantly,F ≈ 3× 107 erg cm−2

s−1 for Active Regions.

4.2.4. Random walks in twist space

The instability threshold can be expressed in terms ofϕ(R1) andϕ(R2). This is more representative of the twist profile (see Sect.3), which is directly generated by photospheric motions. Weuseϕ(R1) andϕ(R2) to obtain a 2-parameter representation of thefamily ϕ(r), since the twist profiles of Fig. 4 usually show peaktwists at the radial boundariesR1 (0.5) andR2 (1.0). The instabil-ity threshold can thus be plotted intwist space rather thanalphaspace, as in Fig. 3. Given that the process is driven by turbu-lent photospheric motions, it is more realistic to considerthatthe twist randomly evolves, rather than theα profile. We thusrepeat our calculations with a random walk inϕ-space.

-10

-5

0

5

10

-15 -10 -5 0 5 10 15

ϕ(R

2) /

π

ϕ(R1) / π 0

0.02

0.04

0.06

0.08

0 0.02 0.04 0.06 0.08

Fla

re C

ount

(no

rmal

ised

)


Fig. 16. The instability threshold and relaxation line inϕ-space (left),alongside the flare energy distribution for a 105 loop ensemble per-formed withinϕ-space (right). The random walk step size,δϕ, is ap-proximately 0.32π, this corresponds to a step time of 200 s.

When a simulation is run, for a sequence of heating events,in ϕ-space, the resulting energy profile is more or less identicalto that generated by anα-space simulation, see Fig. 16 (right).The translation toϕ-space results in a slight reorientation of therelaxation line with respect to the threshold. Consequently, therelaxation line is closer to the part of the threshold associatedwith the highest energies; this explains why the distribution hasa thicker tail. The energy flux isF ≈ 4× 106 erg cm−2 s−1, whichincreases with step size in the same manner as shown for theα-space simulations.

4.3. Temporal properties

We can gain some insight into the temporal distribution of heat-ing events by assuming that each step of a random walk inα-space takes the same arbitrary unit of time, corresponding tophotospheric driving. The time axes of Fig. 17 are shown interms of random walk step number (i.e., step count or runningstep count). The time unitτ is the time taken for one step, whichwas estimated to be 200 s, see Sect. 2.3.

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 2000 4000 6000 8000 10000

Ste

p C

ount

Event Number

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 0.02 0.04 0.06 0.08

Ste

p C

ount


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

500000 1.5e+06 2.5e+06

Fla

re E

nerg

y (d

imen

sion

less

)

Time (running step count)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 50 100 150 200 250

Fla

re E

nerg

y (1

024 e

rg)

Time (hours)

Fig. 17. Top Left: the number of steps between relaxation and insta-bility (i.e., time interval between heating events) for each event of asimulation comprising 104 events.Top Right: the number of steps takento reach the threshold against the energy released.Bottom Left: the en-ergy release as a function of time (running step count).Bottom Right:the 105 event simulation produced 105 flares of varying energies. Thesize of the flares are shown in a way that is reminiscent of actualflare/microflare/nanoflare observations: the bigger the event, the widerthe base of the triangle used to represent that flare. The figure covers atime sequence equal to 5000 steps, taken from a random position withinthe simulation data.

The probability that a flare event will have occurred aftera particular number of steps inα-space is invariant with time,see Fig. 17 (top left). This is also true for the probability that a


flare event will have a particular energy: the flare energy is notdependent on where in the simulation it occurs. The two hori-zontal bands shown in the bottom left panel are consistent withthe peaks shown in Figs. 14-15.

There does appear to be a slight relationship between theflare interval time (i.e., the number of steps taken to reach thethreshold) and the flare energy, see Fig. 17 (top right); but,nopositive correlation is evident. Observations strongly suggestthat these two properties are uncorrelated (Wheatland 2000).The top right panel echoes the shape of the energy release distri-butions. The instability threshold has an energy release gradient,which means that sections of threshold that have a more or lessconstant energy release (low gradient) will be visited by moreloops. The higher the number of visiting loops, the wider therange of step counts associated with that section of the thresh-old. Thus, Fig. 17 (top right) is in agreement with observations.

Finally, the simulated energy releases can also be representedas a time series of flare energies, again see Fig. 17 (bottom left).

4.4. Critical magnetic shear and coronal heatingrequirements

Parker (1988) explains that a magnetic flux bundle rooted inthe photosphere (i.e., a coronal loop) will be stressed as pho-tospheric motions move the footpoints in a direction transverseto the original magnetic field. The coronal heating requirementsfor Active Regions and for the Quiet Sun can be converted toequivalent magnetic stresses, which can in turn be converted tomagnetic shears (B⊥/Bz);

B⊥Bz≈

4πFB2

z vθ, (30)

whereB⊥ is the transverse field,Bz is the original axial field,F isthe coronal heating requirement andvθ is the photospheric flowvelocity. We setvθ = 1 km s−1 for consistency and find that thenecessary magnetic shear is approximately 0.12 (∼7◦) for ActiveRegions and 0.38 (∼21◦) for the Quiet Sun.

Thus, for coronal heating towork, there must be some mech-anism which restricts the shear to around these levels. If dis-sipation occurs at lower levels of shear the energy flux cannotmaintain coronal temperatures. Conversely, if the shear can buildup to much larger levels, the energy input would be higher thanrequired. Our model provides an explanation for these criticalshear values. We calculate two types of mean magnetic shearalong the threshold, the mean absolute shear and the root meansquare shear:

S MABS =2

R22

∫ R2

0r

∣

∣

∣

∣

∣

∣

Bθ

Bz

∣

∣

∣

∣

∣

∣

dr , (31)

S RMS =

√

√

2

R22

∫ R2

0r

(

Bθ

Bz

)2

dr . (32)

Both quantities are area-averaged. The plot (Fig. 18) showsthatthe values ofSMABS and SRMS are comparable with those de-rived above. In general, we find that the average shear at thethreshold is slightly higher than the limit suggested by Parker.Perhaps more shear is required, since not all of the magneticen-ergy released during relaxation will be converted to heat. Fig. 18suggests that the apparent limiting value for magnetic shear isdetermined by the linear instability.

We should mention the work of Dahlburg et al. (2009), whoprovide an alternative explanation of this shear dependency in-volving an explosive ”secondary instability”. They have used a

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100 120

Mag

netic

She

ar [0

-R2]

Threshold Point

Fig. 18. The mean absolute (solid) and root mean square (dashed) ofthe magnetic shear along the instability threshold. The shears are calcu-lated over the loop volume, 0-R2.

3D viscoresistive MHD code to simulate the shearing of a line-tied flux tube. The results of this code have revealed that thesecondary instability requires a critical shear for onset.

5. Summary and conclusions

We have used a model based on relaxation theory to determinethe energy release of a sequence of heating events generatedby random photospheric footpoint motions. This is a ”stress-relax” scenario, in which free energy is continually built-up inthe corona due to photospheric driving - repeated relaxations oc-cur whenever a critical condition is reached. We postulate thatheating events are triggered by the onset of an ideal kink insta-bility. Energy release then occurs by fast magnetic reconnectionin current sheets generated during the nonlinear phase of the in-stability, and this is calculated by assuming a helicity-conservingrelaxation to a minimum energy state. The energy release of anyindividual heating event depends on the current profile at thepoint where the instability threshold is crossed - thus heatingevents of a range of sizes are found.

The initial stressed fields are modelled using piecewise-constant profiles ofα, yielding a two-parameter family of cur-rent profiles. Photospheric driving thus yields a random walk ofthe current profile. In order to pursue the calculation, the idealinstability threshold of a line-tied loop has been determined.The modelled current profiles incorporate regions of positive andnegative twist as well as monotonic twist profiles, as have beenmore commonly studied. We have shown that it is not possibleto determine the onset of instability by any simple criterion suchas achieving a critical twist or critical helicity. Nevertheless, Fig.4 suggests that the location of the instability is coincident withthe peak twist (the largest absolute twist).

A distribution of heating events has been calculated, bothfor the case of a single loop which undergoes repeated stressingand relaxation, and for an ensemble of identical loops whicharerandomly stressed (and for intermediate cases). For a sufficientlylarge number of events, a statistically-stable distribution of eventsizes is obtained. As expected, the smallest events are the mostcommon, and there is (as noted by Browning & Van der Linden2003) a minimum event size for given loop parameters, perhapscorresponding to an ”elemental nanoflare”. More surprisingly,there is a second peak of event frequency at intermediate magni-tudes, although this is somewhat reduced in size if an ensembleof loops is considered. This can be explained as follows. Thefirst peak (at minimum energy) occurs simply because the rangeof energies near the minimum naturally encompasses the largestpart of the instability threshold curve (because of the flatness ofthe minimum). The second peak is found because the instabil-ity threshold is most likely to be crossed in the part near to theregion of constant-α.


An ensemble of identical loops - in which each individualloop starts from a randomly chosen initial state and is stresseduntil it undergoes a single relaxation event - is investigated inmore detail. The distribution of heating events is qualitativelysimilar, although, the secondary peak of event frequency ismuchlower, and the decay of frequency with increasing energy ismuch flatter. A significant result is that, although the distributionis not a simple power law, the high-energy part of the distributionis well approximated by a power law with an index of around -8.3, which is considerably steeper than the minimum requiredfor nanoflare heating to be effective (-2).

The model requires that the field is sometimes unstable - al-though most of the time, the field profile will be well within thestable region. Typically, the dimensionalα value is of magni-tude 1 - 2, leading to dimensional values ofα≈ 1 - 2 Mm−1 (fora loop radius of 1 Mm). Note that this is the maximum valuewhich could be found, and usually we would expect lower val-ues. This is consistent with observations; for example Regnier &Priest (2007) findα magnitudes around 1 Mm−1. Furthermore,a consequence of the fact that the fields are predicted to fluc-tuate between stable and unstable states is that we predict avalue for the average horizontal field component (on average,this will be somewhat less than the value at marginal stability).At threshold, we findB⊥/Bz is around 0.5, which means that itsaverage value should be around 0.25. This agrees very well withthe limit on this quantity required by Parker (1988), in order forthe Poynting flux from the photosphere to match coronal heat-ing requirements. Hence, we have an explanation for the criticalshear value predicted by Parker. This also implies that the av-erage heating flux from our model will be sufficient for coronalheating. Indeed, the fluxes derived from our results agree (espe-cially if we raise the correlation time for photospheric motions)with the required values.

Relaxation theory makes no prediction about the spatial dis-tribution of energy dissipation. However, this can be determinedfrom numerical simulations. Recently, Hood et al. (2009) haveshown that heating is well-distributed across the loop volume, asthe current sheet, associated with the nonlinear kink instability,stretches and fragments, thereby filling the loop cross section.This has implications for the observed emission.

A further important consideration is that we study a singleloop of fixed dimensions, with repeated stressing and relaxation.In practice, coronal heating events will occur in regions offieldof varying sizes. A large flare will inevitably involve a large mag-netic field volume, whereas nanoflares may involve small setsoffieldlines. This could be accounted for in our model by allowingthe loop radius to be randomly distributed also; the effect wouldbe to convolve a set of our energy distributions. A second limita-tion of the calculation is that we have assumed a uniform randomwalk inα space. This is not realistic, since photospheric twistingmotions are likely to be correlated across the loop cross sectionand therefore changes in which the twist in the outer later issim-ilar to that in the core are much more likely: in other words, thereshould be a positive correlation between changes inα1 (the corecurrent) and changes inα2 (the current in the outer layer), ratherthan these being independent random variables as assumed here.This will be considered further in future work.

As well as the improvements mentioned above, the modelwill be enhanced such that there is, more realistically, zero net-current carried by the loop. At present, in most cases the po-tential layer outside the loop contains azimuthal field. A currentneutralisation layer will be inserted between the loop and the en-velope - external magnetic fields will have an axial direction only

(Hood et al. 2009), representing a response to localised photo-spheric twist motions.

The model has included a number of simplifications, andat this stage is more of a ”proof of principle” showing thata distribution of heating events can be produced from an(almost)ab initio coronal heating model. These initial resultsdemonstrate the viability of the model for further research.

Acknowledgements. We are grateful to Jim Klimchuk for helpfulcomments and M. R. B. thanks STFC for studentship support.

Appendix A: Expressions for loop properties

Expressions for some key quantities (〈ϕ〉, K and W) are given here.For compactness, these are given only forα1 , 0 andα2 , 0, while spe-cial cases (e.g.,α1 = 0) must be dealt with separately. Expressions forconstant-α fields can be recovered by settingα1 =α2, which gives morefamiliar formulae. The superscripts and subscripts that accompany eachquantity term denote the upper and lower radial bounds over which thequantity is calculated.

A.1. Average magnetic twist

〈ϕ〉R10 =

σ1L[

1− J0(|α1|R1)]

R1J1(|α1|R1)(A.1)

〈ϕ〉R2R1=

σ2L[

F0(|α2|R1) − F0(|α2|R2)]

[

R2F1(|α2|R2) − R1F1(|α2|R1)] (A.2)

〈ϕ〉R3R2=

2σ2LR2

[

B2J1(|α2|R2) +C2Y1(|α2|R2)]

log(

R3R2

)

B2F0(|α2|R2)[

R 23 − R 2

2

] (A.3)

A.2. Magnetic helicity

KR10 = σ1

2πLB 21

|α1|

[

R 21 J 2

0 (|α1|R1) + R 21 J 2

1 (|α1|R1)

− 2R1

|α1|J0(|α1|R1)J1(|α1|R1)

]

(A.4)

KR2R1=

2σ2πLB 22

|α2|

[

R 22 F 2

0 (|α2|R2) + R 22 F 2

1 (|α2|R2)

− 2R2

|α2|F0(|α2|R2)F1(|α2|R2)

−R 21 F 2

0 (|α2|R1) − R 21 F 2

1 (|α2|R1)

+ 2R1

|α2|F0(|α2|R1)F1(|α2|R1)

+2B1R1J1(|α1|R1)

B2

[

F0(|α2|R1) − F0(|α2|R2)]

×

(

1|α1|−σ1,2

|α2|

)]

(A.5)

KR3R2= 2σ2LC3R2

[

(

ψR2 − πB3R 22

)

log

(

R3

R2

)

+πB3

2

(

R 23 − R 2

2

)

]

(A.6)


A.3. Magnetic energy

W R10 =

LπB 21

µ0

[

R 21 J 2

0 (|α1|R1) + R 21 J 2

1 (|α1|R1)

−R1

|α1|J0(|α1|R1)J1(|α1|R1)

]

(A.7)

W R2R1=

LπB 22

µ0

[

R 22 F 2

0 (|α2|R2) + R 22 F 2

1 (|α2|R2)

−R2

|α2|F0(|α2|R2)F1(|α2|R2)

−R 21 F 2

0 (|α2|R1) − R 21 F 2

1 (|α2|R1)

+R1

|α2|F0(|α2|R1)F1(|α2|R1)

]

(A.8)

W R3R2=

Lπµ0

[

B 23

2

(

R 23 − R 2

2

)

+C 23 R 2

2 log

(

R3

R2

)]

(A.9)

Appendix B: Magnetic field profiles for a selectionof α-space points

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0 0.5 1 1.5 2 2.5 3

B

r

A: α1 = -2.36, α2 = 0.4

-0.04-0.03-0.02-0.01

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0 0.5 1 1.5 2 2.5 3

B

r

B: α1 = -2.77, α2 = 2.8

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 0.5 1 1.5 2 2.5 3

B

r

C: α1 = -2.4, α2 = 3.29

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.5 1 1.5 2 2.5 3

B

r

D: α1 = 0.78, α2 = 1.8

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.5 1 1.5 2 2.5 3

B

r

E: α1 = 1.78, α2 = 1.0

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.5 1 1.5 2 2.5 3

B

r

F: α1 = 2.15, α2 = 0.2

Fig. B.1. The magnetic field profiles,Bz (solid) andBθ (dashed), forthe sixα-space points identified in the top left panel of Fig. 4.

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A nanoflare distribution generated by repeated relaxations triggered by kink instability

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