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07submitted to The Astrophysical JournalPreprint typeset using LATEX style emulateapj v. 10/09/06

NONLINEAR DYNAMICS OF THE PARKER SCENARIO FOR CORONAL HEATING

A.F. Rappazzo1, M. Velli2

Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA

G. Einaudi3

Dipartimento di Fisica “E. Fermi”, Universita di Pisa, 56127 Pisa, Italy

and

R.B. Dahlburg4

Laboratory for Computational Physics and Fluid Dynamics,Naval Research Laboratory, Washington, DC 20375, USA

submitted to The Astrophysical Journal

ABSTRACT

The Parker or field line tangling model of coronal heating is studied comprehensively via long-time high-resolution simulations of the dynamics of a coronal loop in cartesian geometry within theframework of reduced magnetohydrodynamics (RMHD). Slow photospheric motions induce a Poyntingflux which saturates by driving an anisotropic turbulent cascade dominated by magnetic energy. Inphysical space this corresponds to a magnetic topology where magnetic field lines are barely entangled,nevertheless current sheets (corresponding to the original tangential discontinuities hypothesized byParker) are continuously formed and dissipated.Current sheets are the result of the nonlinear cascade that transfers energy from the scale of con-

vective motions (∼ 1, 000 km) down to the dissipative scales, where it is finally converted to heatand/or particle acceleration. Current sheets constitute the dissipative structure of the system, andthe associated magnetic reconnection gives rise to impulsive “bursty” heating events at the smallscales. This picture is consistent with the slender loops observed by state-of-the-art (E)UV and X-rayimagers which, although apparently quiescent, shine bright in these wavelengths with little evidenceof entangled features.The different regimes of weak and strong MHD turbulence that develop, and their influence on

coronal heating scalings, are shown to depend on the loop parameters, and this dependence is quan-titatively characterized.

Subject headings: MHD — Sun: corona — Sun: magnetic fields — turbulence

1. INTRODUCTION

In a previous letter (Rappazzo et al. 2007) we de-scribed simulations, within the framework of RMHD incartesian geometry, aimed at solving the Parker field-linetangling (coronal heating) problem (Parker 1972, 1988).We also developed a phenomenological model for nonlin-ear interactions, taking into account the inertial photo-spheric line-tying effect, which explained how the aver-age coronal heating rate would depend on the only freeparameter present in the simulations, namely the ratioof the coronal loop Alfven crossing time and the pho-tospheric eddy turnover time. This paper is devoted toa more detailed discussion of the numerical simulationsand of the relationship between this work, the originalParker conjecture, and the nanoflare scenario of coronalheating.Parker’s book (Parker 1994) is devoted to an exami-

nation of the basic theorem of magnetostatics, namelythat the lowest available energy state of a magnetic fieldin an infinitely conducting fluid contains surfaces of tan-gential discontinuity, or current sheets. It is Parker’s

1 NASA Postdoctoral Fellow; rappazzo@jpl.nasa.gov2 Also at Dipartimento di Astronomia e Scienza dello Spazio,

Universita di Firenze, 50125 Florence, Italy; mvelli@jpl.nasa.gov3 einaudi@df.unipi.it4 rdahlbur@lcp.nrl.navy.mil

conjecture that the continuous footpoint displacement ofcoronal magnetic field lines must lead to the developmentof such discontinuities as the field continuously tries torelax to its equilibrium state, and it is the dynamicalinterplay of energy accumulation via footpoint motionand the bursty dissipation in the forming current sheetswhich gives rise to the phenomenon of the high temper-ature solar corona, heated by the individual bursts ofreconnection, or nanoflares.What then does turbulence have to do with the

nanoflare heating scenario? Parker himself strongly criti-cizes the use of the “t” word, the formation of the currentsheets being due in his opinion to the “requirement forultimate static balance of the Maxwell stresses”. Butwhat better way is there to describe the nonlinear globaldynamics of a magnetically dominated plasma in whichthe formation of an equilibrium state containing currentsheets is an inevitable asymptotic state if the photo-spheric driver were turned off?The striving of the global magnetic field toward a state

containing current sheets must occur through local vio-lations of the force-free condition, the induction of localflows, the collapse of the currents into ever thinner layers:a nonlinear process generating ever smaller scales. Fromthe spectral point of view, a power law distribution ofenergy as a function of scale is expected, even though

2 Rappazzo et al.

the kinetic energy is much smaller than the magnetic en-ergy. The last two statements are clear indications thatthe word turbulence provides a correct description of thedynamical process.A final important issue is whether the overall dissi-

pated power tends to a finite value as the resistivity andviscosity of the coronal plasma become arbitrarily small.That this must be the case is easy to understand (see§ 3.3). For suppose that for an arbitrary, continuous,foot-point displacement the coronal field were only tomap the foot-point motion, and that there were no non-linear interactions, i.e. the Lorenz force and convectivederivatives were negligible everywhere. In this case, themagnetic field and the currents in the corona would thengrow linearly in time, until the coronal dissipation atthe scale of photospheric motions balanced the forcing.The amplitudes of the coronal fields and currents wouldthen be inversely proportional to resistivity (eqs. (30)-(31)), and the dissipated power, product of resistivityand square of the current, would also scale as the in-verse power of the resistivity (eq. (33)). In other words,the smaller the resistivity in the corona, the higher thepower dissipated would be. But the amplitudes can notbecome arbitrarily large, because non-linear effects inter-vene to stop the increase in field amplitudes, increasingthe effective dissipation at a given resistivity. Since thepower can not continue to increase monotonically as theresistivity is decreased, it is clear that at some point non-linear interactions must limit the dissipated power to afinite value, regardless of the value of the resistivity. Fi-nite dissipation at arbitrarily small values of dissipativecoefficients is another definition of a turbulent system.All this assuming that a statistically stationary state

may be reached in a finite time, a question closely re-lated to the presence of finite time singularities in 3Dmagnetohydrodynamics. It now appears that magneticfield relaxation in an unforced situation does not lead tothe development of infinitely thin current sheets in a fi-nite time, but rather the current development appears tobe only exponential in time (Grauer and Marliani 2000).In forced numerical simulations, as the ones we will de-scribe in detail here, this is a mute point: for all intentsand purposes a statistically stationary state is achievedat a finite time independent of resistivity for sufficientlyhigh resolution. In fact, even if the growth is exponen-tial, we can estimate that the width of the current sheetsreaches the meter-scale in a few tenths Alfven crossingtimes τA. A typical value is τA = 40 s, so that this ini-tial time is not only finite, but also short compared withan active region timescale. Once the steady state hasbeen reached this phenomenon is no longer important.The nonlinear regime is in fact characterized by the pres-ence of numerous current sheets, so that while some ofthem are being dissipated others are being formed, anda steady state is maintained.It therefore seems that the Parker field-line tan-

gling scenario of coronal heating may be describedas a particular instance of magnetically dominatedMHD turbulence. Numerous analytical and numericalmodels of this process have been presented in thepast, each discussing in some detail aspects of thegeneral problem as presented above (Parker 1972,1988; Heyvaerts &Priest 1992; van Ballegooijen1986; Berger 1991; Sturrock and Uchida 1981;

Gomez & Ferro-Fontan 1992; Mikic et al. 1989;Hendrix & Van Hoven 1996; Longcope & Sudan1994; Dmitruk & Gomez 1999; Einaudi et al.1996; Georgoulis et al. 1998; Dmitruk et al. 1998;Einaudi & Velli 1999).The numerical simulations presented here bring closure

to the question as posed in cartesian geometry, startingfrom a uniform axial magnetic field straddling from oneboundary plane to another, rather than the more realisticcase of a single photosphere with curved coronal loops.Simulations of such full 3D sections of the solar coronahave been presented recently by Gudiksen & Nordlund(2005). While this approach has advantages when in-vestigating the coronal loop dynamics within its coronalneighborhood, modeling a larger part of the solar coronanumerically drastically reduces the number of points oc-cupied by the coronal loops. At the moment the very lowresolution attainable with this kind of simulations doesnot allow the development of turbulence. The transferof energy from the scale of convection cells ∼ 1000 kmtoward smaller scales is in fact inhibited, because thesmaller scales are not resolved (their linear resolution isin fact ∼ 500 km). Thus, these simulations have not beenable to shed light on the detailed coronal statistical re-sponse nor on the different regimes which may developand how they depend on the coronal magnetic field cross-ing time and the photospheric eddy turnover time.In § 2 we introduce the coronal loop model, whose

properties are qualitatively analyzed in § 3. The resultsof our simulations are described in § 4, and their turbu-lence properties are analyze in more detail in § 5. Finallyin § 6 we summarize and discuss our results.

2. PHYSICAL MODEL

A coronal loop is a closed magnetic structure threadedby a strong axial field, with the footpoints rooted inthe photosphere. This makes it a strongly anisotropicsystem, as measured by the relative magnitude of theAlfven velocity associated with the axial magnetic fieldvA ∼ 2000 km s−1 compared to the typical photosphericvelocity uph ∼ 1 km s−1. This means that the relativeamplitude of magnetic field perturbation generated inthe corona by the photospheric dragging process remainsvery small, as an efficient energy cascade in this inducedmagnetic field occurs.We study the loop dynamics in a simplified Cartesian

geometry, neglecting field line curvature i.e. the toroidal-ity of loops, as a “straightened out” box, with an or-thogonal square cross section of size ℓ (along which thex-y directions lie), and an axial length L (along the zdirection) embedded in an axial homogeneous uniformmagnetic field B0 = B0 ez. This simplified geometry al-lows us to perform simulations with both high numericalresolution and long-time duration.In § 2.1 we introduce the equations used to model the

dynamics, while in § 2.2 we give the boundary and initialconditions used in our numerical simulations.

2.1. Governing Equations

The dynamics of a plasma embedded in a strong ax-ial magnetic field are well described by the equationsof reduced MHD (RMHD) (Kadomtsev & Pogutse 1974;Strauss 1976; Montgomery 1982). This simplified set ofequations is obtained performing a series expansion of

Coronal Heating 3

the fields in the small parameter ǫ, inserting them in thefull set of MHD equations, and then retaining only thefirst order leading contributions (the zeroth order is givenby the big axial magnetic field B0). It is found that atthe first order the velocity and magnetic fields have onlyperpendicular components, and that their temporal evo-lution is given by the RMHD equations. In dimensionlessform they can be written as:

∂u⊥

∂t+ (u⊥ ·∇⊥)u⊥ = −∇⊥

(

p+b2⊥2

)

+ (b⊥ ·∇⊥) b⊥ + cA∂b⊥∂z

+(−1)n+1

Ren∇

2n⊥ u⊥, (1)

∂b⊥∂t

= (b⊥ ·∇⊥)u⊥ − (u⊥ ·∇⊥) b⊥

+ cA∂u⊥

∂z+

(−1)n+1

Ren∇

2n⊥ b⊥, (2)

∇⊥ · u⊥ = 0, ∇⊥ · b⊥ = 0, (3)

where u⊥ and b⊥ are the orthogonal components of thevelocity and magnetic fields, and p is the kinetic pressure.The gradient operator has only components in the x-yplane perpendicular to the axial direction z, i.e.

∇⊥ = ex∂

∂x+ ey

∂

∂y, (4)

and the dynamics in the orthogonal planes is coupled tothe axial direction through the linear terms ∝ ∂z .To render the RMHD equations in the dimensionless

form (1)-(3), we have first expressed the magnetic fieldsin velocity dimensions dividing by

√4πρ0 (where ρ0 is a

density supposed homogeneous and constant), i.e. con-sidering the associated Alfven velocities (b→ b/

√4πρ0).

We have then used the typical photospheric velocityuph to render the magnetic and velocity fields in non-dimensional form, while lengths and times have been ex-pressed in units of the perpendicular length of the com-putational box ℓ and its related crossing time t⊥ = ℓ/uph.We use a computational box with an aspect ratio of 10,which spans

0 ≤ x, y ≤ 1, 0 ≤ z ≤ 10. (5)

As a result, in equations (1)-(2), the linear terms ∝∂z are multiplied by the dimensionless Alfven velocitycA = vA/uph, i.e the ratio between the Alfven velocityassociated with the axial magnetic field vA = B0/

√4πρ0,

and the photospheric velocity uph.The index n is called dissipativity, and depending on

its value the diffusive terms adopted in equations (1)-(2)correspond to ordinary diffusion for n = 1 and to so-called hyperdiffusion for n > 1. When n = 1 the ∇2

⊥/Rediffusive operator is recovered, so that Re1 = Re = Remcorresponds to the kinetic and magnetic Reynolds num-ber (considered of equal and uniform value):

Re =ρ0 ℓuphν

, Rem =4πρ0 ℓuph

ηc2, (6)

where viscosity ν and resistivity η are supposed to beconstant and uniform (c is the speed of light).We have performed numerical simulations with both

n = 1 and n = 4. Hyperdiffusion is used because with a

limited resolution the diffusive timescales associated withordinary diffusion are small enough to affect the largescale dynamics and render very difficult the resolution ofan inertial range, even with a grid with 512x512 pointsin the x-y plane (which is the highest resolution grid weused for the plane). The diffusive time τn at the scaleλ associated with the dissipative terms used in (1)-(2) isgiven by:

τn ∼ Ren λ2n (7)

While for n = 1 the diffusive time decreases relativelyslowly towards smaller scales, for n = 4 it decreasesfar more rapidly. This allows to have longer diffu-sive timescales at large spatial scales and similar dif-fusive timescales at the resolution scale. Numericallywe require that the diffusion time at the resolution scaleλmin = 1/N , where N is the number of grid points, tobe of the same order of magnitude for both normal andhyperdiffusion, i.e.

Re1N2

∼ RenN2n

−→ Ren ∼ Re1N2(n−1) (8)

For instance a numerical grid with N = 512 points whichrequires a Reynolds number Re1 = 800 with ordinarydiffusion, can implement Re4 ∼ 1019, removing diffusiveeffects at the large scales, and allowing (if present) theresolution of an inertial range.The numerical integration of the RMHD equations (1)-

(3) is substantially simplified by using the potentials ofthe velocity (ϕ) and magnetic field (ψ),

u⊥ = ∇× (ϕez) , b⊥ = ∇× (ψ ez) , (9)

linked to vorticity and current by ω = −∇2⊥ϕ and j =

−∇2⊥ψ.

We solve numerically equations (1)-(3) written in termsof the potentials (see Rappazzo et al. (2007)) in Fourierspace, i.e. we advance the Fourier components in the x-ydirections of the scalar potentials ϕ and ψ. Along thez direction no Fourier transform is performed so thatwe can impose non-periodic boundary conditions (speci-fied in § 2.2), and a central second-order finite differencescheme is used. In the x-y plane a Fourier pseudospec-tral method is implemented. Time is discretized with athird-order Runge-Kutta method.

2.2. Boundary and Initial Conditions

As boundary conditions at the photospheric surfaces(z = 0, L) we impose two independent velocity pat-terns, intended to mimic photospheric motions, made upof large spatial scale projected convection cell flow pat-terns constant in time. The velocity potential at eachboundary is given by:

ϕ(x, y) =1

√∑

mn α2mn

∑

k,l

ℓ αkl

2π√k2 + l2

sin

[

2π

ℓ(kx+ ly) + 2π ξkl

]

(10)

We excite all the wave number values (k, l) ∈ Z2 in-

cluded in the range 3 ≤(

k2 + l2)1/2 ≤ 4, so that the

resulting average injection wavenumber is kc ∼ 3.4, andthe average injection scale ℓc, the convection cell scale,

4 Rappazzo et al.

is given by ℓc = ℓ/kc. αkl and ξkl are two sets of ran-dom numbers whose values range between 0 and 1, andare independently chosen for the two boundary surfaces.The normalization adopted in eq. (10) sets the value of

the corresponding velocity rms (see eq. (9)) to 1/√2, i.e.

∫ ℓ

0

∫ ℓ

0

dxdy(

u2x + u2y)

=1

2(11)

At time t = 0 no perturbation is imposed inside thecomputational box, i.e. b⊥ = u⊥ = 0, and only the axialmagnetic field B0 is present: the subsequent dynamicsare then the effect of the photospheric forcing (10) onthe system, as described in the following sections.

3. ANALYSIS

In order to clarify aspects of the linear and nonlinearproperties of the RMHD system, we provide an equiv-alent form of the equations (1)-(3). In terms of theElsasser variables z± = u⊥ ± b⊥, which bring out thebasic symmetry of the equations in terms of parallel andanti-parallel propagating Alfven waves, they can be writ-ten as

∂z+

∂t= −

(

z− ·∇⊥

)

z+ + cA∂z+

∂z

+(−1)n+1

Ren∇

2n⊥ z+ −∇⊥P, (12)

∂z−

∂t= −

(

z+ ·∇⊥

)

z− − cA∂z−

∂z

+(−1)n+1

Ren∇

2n⊥ z− −∇⊥P, (13)

∇⊥ · z± = 0, (14)

where P = p + b2⊥/2 is the total pressure, and is linkedto the nonlinear terms by incompressibility (14):

∇2⊥P = −

2∑

i,j=1

(

∂iz−j

)(

∂jz+i

)

. (15)

In terms of the Elsasser variables z± = u⊥ ± b⊥, a

velocity pattern u0,L⊥ at upper or lower boundary surface

becomes the constraint z+ + z− = 2u0,L⊥ at that bound-

ary. Since, in terms of characteristics (which in this caseare simply z± themselves), we can specify only the in-coming wave (while the outgoing wave is determined bythe dynamics inside the computational box), this velocitypattern implies a reflecting condition at the top (z = L)and bottom (z = 0) planes:

z− = −z+ + 2u0⊥ at z = 0, (16)

z+ = −z− + 2uL⊥ at z = L. (17)

The linear terms (∝ ∂z) in equations (12)-(13) give riseto two distinct wave equations for the z± fields, whichdescribe Alfven waves propagating along the axial direc-tion z. This wave propagation, which is present dur-ing both the linear and nonlinear regimes, is responsiblefor the continuous energy influx on large perpendicularscales (see eq. (10)) from the boundaries into the loop.The nonlinear terms (z∓ ·∇⊥)z

± are then responsible

for the transport of this energy from the large scales to-ward the small scales, where energy is finally dissipated,i.e. converted to heat and/or particle acceleration.A well-known important feature of the nonlinear terms

in equations (12)-(14) is the absence of self-coupling, i.e.only counterpropagating waves interact non-linearly, andif one of the two fields z± is zero, there are no non-linear interactions at all. This fact, i.e. that counter-propagating wave-packets may interact only while theyare crossing each other, lies at the basis of the so-calledAlfven effect (Iroshnikov 1964; Kraichnan 1965), whichultimately renders the nonlinear timescales longer andslows down the dynamics.From this description three different timescales arise

naturally: τA, τph and τnl. τA = L/vA is the crossingtime of the Alfven waves along the axial direction z, i.e.the time it takes for an Alfven wave to cover the looplength L. τph ∼ 5 m is the characteristic time associ-ated with photospheric motions, while τnl is the nonlin-ear timescale.For a typical coronal loop τA ≪ τph. For instance for

a coronal loop long L = 40, 000 km and with an Alfvenvelocity vA = 2, 000 kms−1 we obtain τA = 20 s, whichis small compared to τph ∼ 5 m = 300 s. For this reasonwe consider in this paper a forcing which is constant intime (see eq. (10)), i.e. for which formally τph = ∞.In the RMHD ordering the nonlinear timescale τnl is

bigger than the Alfven crossing time τA. As we shall seethis ordering is maintained during our simulations andwe will give analytical estimates of the value of τnl as afunction of the characteristic parameters of the system.An important feature of equations (12)-(14) that we

will use to generalize our results is that, apart from theReynolds numbers, there is only one fundamental non-dimensional parameter:

f =ℓc vALuph

. (18)

Hence all the physical quantities which result from thedynamical evolution, e.g. energy, Poynting flux, heatingrate, timescales, etc., must depend on this single param-eter f .

3.1. Energy Equation

From equations (1)-(3), with n = 1, and consideringthe Reynolds numbers equal, the following energy equa-tion can be derived:

∂

∂t

(

1

2u2⊥ +

1

2b2⊥

)

= −∇ · S − 1

Re

(

j2 + ω2)

, (19)

where S = B × (u × B) is the Poynting vector. Asexpected the energy balance of the system described byeq. (19) is due to the competition between the energy(Poynting) flux flowing into the computational box andthe ohmic and viscous dissipation. Integrating eq (19)over the whole box the only relevant component of thePoynting vector is the component along the axial direc-tion z, because in the x-y plane periodic boundary con-ditions are used and their contribution to the Poyntingflux is null. As B = cA ez+b⊥ and u = u⊥, this is givenby

Sz = S · ez = −cA (u⊥ · b⊥) . (20)

Considering that the velocity fields at the photosphericboundaries are given by u0

⊥ and uL⊥, for the integrated

Coronal Heating 5

energy flux we obtain

S = cA

∫

z=L

da(

uL⊥ · b⊥

)

− cA

∫

z=0

da(

u0⊥ · b⊥

)

. (21)

The injected energy flux therefore depends not only onthe photospheric forcing and the axial Alfven velocity(which have fixed values), but also on the value of themagnetic fields at the boundaries, which is determinedby the dynamics of the system inside the computationalbox: the injection of energy depends on the nonlineardynamics which develops, and viceversa.The simplified topology investigated in this paper,

i.e. a strong axial magnetic field whose footpoints aredragged by 2D orthogonal motions, more properly ap-plies to regions where emerging flux is not dominant.In particular it applies to the stage of an active region,when the initial emergence has been completed, and theactive region is completely formed. Nevertheless also inthis mature stage, the axial component of the velocity uzfield may bring new magnetic field (bem⊥ ) in the corona.The associated Poynting flux is

Sefz =

(

bef⊥

)2

uz. (22)

This additional flux component is negligible respect to(19), which represent the flux associated to the self-consistent field-line dragging, when Sef

z < Sz , i.e. asin convective motions the axial and orthogonal velocityfields are of the same order, uz ∼ uph:

(

bef⊥

)2

< B0 bturb⊥ . (23)

In § 5.2 we give an estimate of the value of the fieldbturb⊥ generated by the field-line dragging, and will be

able to quantify for which value of bef⊥ the emerging fluxcan be neglected.

3.2. Linear Stage

For t < τnl nonlinear terms, which are quadratic inthe physical fields, can be neglected. Neglecting also thediffusion terms, which at the large scale play no role,equations (1)-(3) reduce to two simple wave equations.Coupled with the boundary conditions (10) they lead, ontimescales longer than the crossing time τA (at smallertimescales the wave propagation must be explicitly ac-counted) to the following solution:

b⊥(x, y, z, t) =[

uL(x, y)− u0(x, y)] t

τA, (24)

u⊥(x, y, z, t) = uL(x, y)z

L+ u0(x, y)

(

1− z

L

)

, (25)

as can be verified by substitution in equations (1)-(3)(retaining only the linear terms). While the velocityfield is of the order of the imposed photospheric veloc-ity, the magnetic field grows linearly in time and duringthis phase is just a mapping of the photospheric veloc-ity fields. However, for a generic choice of the forcingvelocities uL and u0, the resulting fields (24)-(25) giverise to non-vanishing forces in the perpendicular planes,and these must grow quadratically in time, so that whileinitially these terms can be neglected, they become dy-namically important after a certain time. This occurs

naturally because there is no reason the current struc-ture set up by the velocity field forcing should give riseto vanishing forces in the perpendicular planes.For a particular, singular set of velocity forcing pat-

terns, the generated coronal field, while still growing lin-early in time, may give rise to vanishing forces: we deferthe discussion of the numerical simulations of these par-ticular cases to another work.For instance consider the simple case uL = 0, in terms

of potentials we have that ψ = −ϕ0 t/τA and ϕ = ϕ0 (1−z/L), i.e. they are both proportional to ϕ0 (where u0

⊥ =∇×

(

ϕ0 ez)

), with ψ growing linearly in time.In this case both b⊥ and u⊥ are proportional to ∝

∇⊥ × (ϕ0 ez), so that taking the curl of the nonlinearterms the condition for the vanishing of nonlinear forcesis given by

∇(

∇2 ϕ0

)

×∇ϕ0 = 0, with ϕ0 = ϕ0 (x, y) . (26)

This condition is then satisfied by those fields for whichthe gradient of the laplacian of ϕ0 is parallel to its gradi-ent, i.e. the functions for which the laplacian is constantalong the streamlines of the field. As ω = −∇

2 ϕ this isalso equivalent to require that the vorticity is constantalong the streamlines. This condition is in general notverified, unless very symmetric functions are chosen. Forinstance, the vanishing force condition is verified in carte-sian geometry by any 1D function like ϕ0 = f(x), and inpolar coordinates by any radial function ϕ0 = g(r).In particular our forcing function (10) does give rise

to non-vanishing forces . In fact the gradient and thelaplacian of the gradient are given by the sum of the samevectors but with different amplitudes. The laplacian hasthe amplitudes multiplied by 2πk2, and we are summingover all the wavenumbers with 3 ≤ |k| ≤ 4.Inserting equation (24) in the expression for the inte-

grated energy flux (21), during the linear stage the energyentering the computational box for unitary time is givenby

S = cA

∫

da |uL − u0|2 · t

τA, (27)

i.e. S grows linearly in time until when the physical fieldsbecome big enough and the nonlinear stage begins.A similar linear analysis was already performed by

Parker (1988), who noted that if this is the mechanismresponsible for coronal heating, then the energy fluxSz ∼ S/ℓ2 must approach the value Sz ∼ 107 erg cm2 s−1

necessary to sustain an active region. On the other handthe value reached by Sz depends on the nonlinear dynam-ics, and it is crucial to estimate its value self-consistently,investigating and solving the nonlinear problem. An Sz

too small compared with the required observational con-straint would then rule out the Parker model.

3.3. Effects of Diffusion

The linear solution (24)-(25) has been obtained with-out taking into account the diffusive terms. This is jus-tified, given the large value of the Reynolds numbers forthe solar corona. But numerically it can be important.At very low resolution diffusion is so important that littleor no nonlinear dynamics develop and the system reachesa balance between the photospheric forcing and diffusionof the large scale fields.

6 Rappazzo et al.

Fig. 1.—: Streamlines of the velocity field u0⊥, the

boundary forcing at the bottom plane z = 0 for run A.In lighter vortices the velocity field is directed anti-clockwise while in darker vortices it is directed clock-wise. The cross-section shown in the figure is roughly4000× 4000 km2, where the typical scale of a convectivecell is 1000 km.

This situation is clearly an artifact and does not repre-sent any realistic physical phenomenon, but it will be in-teresting to compare with the results of our simulations.For an “ad hoc” and sufficiently low value of the Reynoldsnumbers the average dissipation can be the same as inthe high Reynolds number limit. But in one case thesystem has not developed any nonlinear dynamics andthe currents are present only at the large scales, whilein the other case the nature of the “steady state” thatis reached is completely different and not due to the dif-fusion at the large scales of the fields. The currents inthe last case are only present at the small scales, and thesystem reaches a “statistically steady state” because theenergy which flows for unitary time in the computationalbox at the large orthogonal scales (due to the Poyntingflux associated to the photospheric motions) balances theflow of energy that from the large scales cascades to thesmall scales (see discussion in § 5.2).While diffusion will only slightly change the shape in

z of the velocity field in (25), it has a stronger effect onthe magnetic field which would otherwise grow linearlyin time. Here we describe briefly such effect. Linearizingequation (2) retaining the diffusive term (with n = 1 andRe1 = Re), we obtain for the magnetic field

∂b⊥∂t

= cA∂u⊥

∂z+

1

Re∇

2⊥b⊥. (28)

We use the general boundary conditions u⊥ = u0,u⊥ = uL respectively in z = 0 and z = L, but we nowtake into account that the forcing velocities have onlycomponents at the injection scale ℓc (see eq. (10)). For

this pattern the relation ∇2⊥ϕ = − (2π/ℓc)

2 ϕ, whereℓc = ℓ/kc with the average wavenumber kc ∼ 3.4, is

Fig. 2.—: Streamlines of the velocity field uL⊥, the

boundary forcing at the top plane z = L for run A. Thenumerical grid has 512x512 points in the x-y planes, witha linear resolution of ∼ 8 km.

approximately valid. Hence for the integration of equa-tion (28) we use the ansatz ∇

2⊥b⊥ = − (2π/ℓc)

2b⊥. This

is justified because the magnetic field is always the re-sult of the mapping of the boundary velocity forcing, asfound in the calculations without diffusion (24). Inte-grating then eq. (28) over z and dividing by the lengthL, we obtain for b⊥ averaged along z:

∂b⊥∂t

=cAL

[

uL (x, y)− u0 (x, y)]

− (2π)2

ℓ2cReb⊥. (29)

Indicating with uph = uL − u0, with τR = ℓ2cRe/ (2π)2

the diffusive time-scale and with τA = L/cA the Alfvencrossing time, the solution is given by:

b⊥ (x, y, t) = uph (x, y)τRτA

[

1− exp

(

− t

τR

)]

, (30)

|j (x, y, t)| = |uph (x, y)|(

2π

ℓc

)

τRτA

×

×[

1− exp

(

− t

τR

)]

. (31)

So that the magnetic energy EM and the ohmic dissipa-tion rate J are given by

EM =1

2

∫

V

d3x b2⊥ =

=1

2ℓ2 Lu2ph

(

τRτA

)2 [

1− exp

(

− t

τR

)]2

, (32)

J =1

Re

∫

V

d3x j2 =

= ℓ2Lu2phτRτ2A

[

1− exp

(

− t

τR

)]2

, (33)

Coronal Heating 7

Run cA nx × ny × nz n Re, Re4 tmax/τAA 200 512 × 512 × 200 1 8 · 102 548B 200 256 × 256 × 100 1 4 · 102 1061C 200 128 × 128 × 100 1 2 · 102 2172D 200 128 × 128 × 100 1 1 · 102 658E 200 128 × 128 × 100 1 1 · 101 1272F 50 512 × 512 × 200 4 3 · 1020 196G 200 512 × 512 × 200 4 1019 453H 400 512 × 512 × 200 4 1020 77I 1000 512 × 512 × 200 4 1019 502

TABLE 1: Summary of the simulations. cA is the ax-ial Alfven velocity and nx × ny × nz is number of pointsfor the numerical grid. n is the dissipativity, n = 1 indi-cates normal diffusion, n = 4 hyperdiffusion. Re (= Re1)or Re4 indicates respectively the value of the Reynoldsnumber or of the hyperdiffusion coefficient (see eq.(12)-(13)). The duration of the simulation tmax/τA is givenin Alfven crossing time unit τA = L/vA.

where uph is the rms of uph, and with the rms of theboundary velocities u0 and uL fixed to 1/2 (11) we haveuph ∼ 1. Both total magnetic energy (32) and ohmic dis-sipation (33) grow quadratically in time for time smallerthan the resistive time τR, while on the diffusive timescale they saturate to the values

EsatM =

ℓ6 c2A u2phRe

2

L (2πkc)4 , Jsat =

ℓ4 c2A u2phRe

L (2πkc)2 , (34)

written explicitly in terms of the loop parameters andReynolds number.Magnetic energy saturates to a value proportional to

the square of both the Reynolds number and the Alfvenvelocity, while the heating rate saturates to a value thatis proportional to the Reynolds number and the squareof the axial Alfven velocity. We have also used equa-tions (32)-(33) as a check in our numerical simulations,and during the linear stage, before nonlinearity sets inthey are well satisfied.From equation (32)-(33) we can estimate the satura-

tion time as the time at which the functions (32)-(33)reach 2/3 of the saturation values. It is approximatelygiven by

τsat ∼ 2 τR =2 ℓ2Re

(2πkc)2 (35)

In the next section we describe the results of our sim-ulations, which investigate the linear and nonlinear dy-namics.

4. NUMERICAL SIMULATIONS

In this section we present a series of numerical simula-tions, summarized in Table 1, modeling a coronal layerdriven by a forcing velocity pattern constant in time. Onthe bottom and top planes we impose two independentvelocity forcings as described in § 2.2, which result fromthe linear combination of large-scale eddies with randomamplitudes, normalized so that the rms of the photo-spheric velocity is uph ∼ 1 kms−1. For each simulation adifferent set of random amplitudes is chosen, correspond-ing to different patterns of the forcing velocities. A re-alization of this forcing with a specific choice (run A) ofthe random amplitudes is shown in Figures 1-2.The length of a coronal section is taken as the unitary

length. As we excite all the wavenumbers between 3 and

4, and the typical convection cell scale is∼ 1, 000 km, thisimplies that each side of our section is roughly 4, 000 kmlong. Our typical grid for the cross-sections has 512x512grid points, corresponding to ∼ 1282 points per convec-tive cell, and hence a linear resolution of ∼ 8 km.Between the top and bottom plates a uniform magnetic

field B = B0 ez is present. The subsequent evolution isdue to the shuffling of the footpoints of the magnetic fieldlines by the photospheric forcing.In the different numerical simulations, keeping fixed

the cross-section length (∼ 4, 000 km) and axial length(∼ 40, 000 km), we explore the behavior of the system fordifferent values of cA, i.e. the ratio between the Alfvenvelocity associated with the axial magnetic field and therms of the photospheric motions (density is supposed uni-form and constant).Nevertheless, as shown in (18) the fundamental param-

eter is f = ℓcvA/Luph, so that changing cA = vA/uphis equivalent to explore the behaviour of the system fordifferent values of f , where the same value of f can be re-alized with a different choice of the quantities, providedthat the RMHD approximation is valid, i.e. we are de-scribing a slender loop threaded by a strong magneticfield.We also perform simulations with different numerical

resolutions, i.e. different Reynolds numbers, and bothnormal (n = 1) and hyper-diffusion (n = 4).The qualitative behaviour of the system is the same

for all the simulations performed. In the next sectionwe describe these qualitative features in detail for run A,and then describe the quantitative differences found inthe other simulations.

4.1. Run A

In this section we present the results of a simula-tion performed with a numerical grid with 512x512x200points, normal (n = 1) diffusion with a Reynolds numberRe = 800, and the Alfven velocity vA = 200 kms−1 cor-responding to a ratio cA = vA/uph = 200. The stream-lines of the forcing velocities applied in the top (z = L)and bottom (z = 0) planes are shown in Figures 1-2. Thetotal duration is roughly 550 axial Alfven crossing times(τA = L/vA).Plots of the total magnetic and kinetic energies

EM =1

2

∫

dV b2⊥, EK =1

2

∫

dV u2⊥, (36)

and of the total ohmic and viscous dissipation rates

J =1

Re

∫

dV j2, Ω =1

Re

∫

dV ω2, (37)

along with the incoming energy rate (integrated Poyntingflux) S (see eq. (21)), are shown in Figures 3-4. At thebeginning the system has a linear behavior (see eqs. (24)-(25), and (27)), characterized by a linear growth in timefor the magnetic energy, the Poynting flux and the elec-tric current, which implies a quadratic growth for theohmic dissipation ∝ (t/τA)

2, until time t ∼ 6 τA, whennonlinearity sets in. We can identify this time as thenonlinear timescale, i.e. τnl ∼ 6 τA. The timescales ofthe system will be analyzed in more details in §5.5.After this time, in the fully nonlinear stage, a statisti-

cally steady state is reached, in which the Poynting flux,

8 Rappazzo et al.

Fig. 3.—: Run A: High-resolution simulation withvA/uph = 200, 512x512x200 grid points and Re = 800.Magnetic (EM ) and kinetic (EK) energies as a functionof time (τA = L/vA is the axial Alfven crossing time).

i.e. the energy that is entering the system for unitarytime, balances on time average the total dissipation rate(J +Ω). As a result there is no average accumulation ofenergy in the box, beyond what has been accumulatedduring the linear stage, while a detailed examination ofthe dissipation time series (see inset in Figure 4) showsthat the Poynting flux and total dissipations are decor-related around dissipation peaks.In the diffusive case from eqs. (32)-(35), with the values

of this simulation we would obtain τsat ∼ 50 τA, EsatM ∼

6100 and Jsat ∼ 7100; all values well beyond those of thesimulation. A value of Re = 85 would fit the simulatedaverage dissipation, while Re = 140 would approximatelyfit the average magnetic energy. In any case this wouldonly fit the curves, but the physical phenomena wouldbe completely different, as we describe in the followingsections.An important characteristic of the system is the mag-

netic predominance for both energy and dissipation (Fig-ures 3 and 4). In the linear stage (§ 3.2) while the mag-netic field grows linearly in time, the velocity field doesnot, and its value is roughly the sum of the boundaryforcing fields. The physical interpretation is that be-cause we are bending the axial magnetic field with a con-stant forcing, as a result the perpendicular magnetic fieldgrows linearly in time, while the velocity remains limited.More formally this is a consequence of the fact that, whileon the perpendicular magnetic field no boundary condi-tion is imposed, the velocity field must approach the im-posed boundary values at the photosphere both duringthe linear and nonlinear stages.In Figure 5 the 2D averages in the x-y planes of the

magnetic and velocity fields and of the ohmic dissipationj2/Re, are plotted as a function of z at different times.These macroscopic quantities are smooth and present al-most no structure along the axial direction. The reason isthat every disturbance or gradient along the axial direc-tion, at least considering the large perpendicular scales(for the small scales behavior see § 5 ), is smoothed outby the fast propagation of Alfven waves along the axial

Fig. 4.—: Run A: The integrated Poynting flux S dy-namically balances the Ohmic (J) and viscous (Ω) dissi-pation. Inset shows a magnification of total dissipationand S for 150 ≤ t/τA ≤ 250.

direction, their propagation time τA is in fact the fastesttimescale present (in particular τA < τnl), and then thesystem tends to be homogeneous along this direction.The predominance of the ohmic over the viscous dissi-

pation is due to the fact that, as we show in the next sec-tions, the dissipative structures are current sheets, wheremagnetic reconnection takes place.The phenomenology described in this section is general

and we have found it in all the simulations that we haveperformed, in particular we have always found that inthe nonlinear stage a statistical steady state is reachedwhere energies fluctuate around a mean value and to-tal dissipation and Poynting flux on the average balancewhile on small timescales decorrelate. In particular, tocheck the temporal stability of these features, which arefully confirmed, we have performed a numerical simula-tion (run C) with the same parameters as run A, butwith a lower resolution (128x128x100), a Reynolds num-ber Re = 200 and a longer duration (t ∼ 2, 000 τA). Onthe opposite the average levels of the energies and of totaldissipation depend on the parameters used as we describein the next sections.Before describing these features, in the next section

we describe the current sheets formation, their temporalevolution and other properties.

4.1.1. Current Sheets, Magnetic Reconnection, GlobalMagnetic Field Topology and Self-Organization

The nonlinear stage is characterized by the presence ofcurrent sheets elongated along the axial direction (Fig-ures (18a)-(18b)), which exhibit temporal dynamics andare the dissipative structures of the system. We nowshow that they are the result of a nonlinear cascade.Figure 6 shows the time evolution of the first 11 modesof magnetic energy for the first 20 crossing times τA forrun A. During the linear stage the magnetic field is givenby eq. (24) and is the mapping of the difference betweenthe top (z = 10) and bottom (z = 0) photospheric ve-locities uL(x, y)−u0(x, y), whose streamlines are shownin Figure 17a. The field lines of the orthogonal magnetic

Coronal Heating 9

Fig. 5.—: Run A: 2D averages in the x-y planes of theohmic dissipation j2/Re, the magnetic and velocity fields

b2⊥, and u2⊥, as a function of z. The different colours

represent 10 different times separated by ∆t = 50 τA inthe interval 30 τA ≤ t ≤ 480 τA.

field in the midplane (z = 5) at time t = 0.63 τA areshown in Figure 17b, and as expected they map the ve-locity field. The same figure shows in colour the axialcurrent j. As shown by eq. (24) (taking the curl) thelarge scale motions that we have imposed at the photo-sphere induce large scale currents in all the volume and,as described in the previous section, if there was nota nonlinear dynamics a balance between diffusion andforcing would be reached, where no small scale wouldbe formed and the magnetic field would always map thephotospheric velocities.As time proceeds the magnetic field grows and a cas-

cade transfers energy from the large scales, where thephotospheric forcing (10) injects energy at the wavenum-bers n = 3 and 4, to the small scales (Figure 6). Inphysical space this cascade corresponds to the collapseof the large scale currents which lead to the formationof current sheets, as shown in Figures 17c and 17d. InFigures 17e and 17f we show the magnetic field lines attime t = 18.47 τA, in the fully nonlinear stage, withrespectively the axial component of the current j andof the vorticity ω. The resulting magnetic topology isquiet complex, X and Y-points are not in fact easily dis-tinguished. They are distorted and very often a com-ponent of the magnetic field orthogonal to the currentsheet length is present, so that the sites of reconnectionare more easily identified by the corresponding vorticityquadrupoles. As shown in Figures 17e and 17f, the moreor less distorted current sheets are always embedded inquadrupolar structures for the vorticity, a characteris-tic maintained throughout the whole simulation, and aclear indication that nonlinear magnetic reconnection istaking place.Figures 18a and 18b show a view from the side and

the top of the 3D current sheets at time t = 18.47 τA.When looked from the side the current sheets, which areelongated along the axial direction, look space filling, butthe view from the top shows that the filling factor is

Fig. 6.—: Run A: First 11 magnetic energy modes asa function of time covering the first 20 Alfven crossingtimes τA. Photospheric motions inject energy at n = 3and 4.

actually small (see also Figure 17).Another aspect of the dynamics is self-organization:

while until time t = 4.79 τA the magnetic field lines arestill approximately a mapping of the photospheric veloc-ities, in the fully nonlinear stage they depart from it andhave an independent topology that evolves dynamicallyin time (see the associated movie for the time evolutioncovering 40 crossing times from ∼ 508 τA up to ∼ 548 τA;notations and simulation are the same used in Figure 17).The reason for which the photospheric forcing does notdetermine the spatial shape of the magnetic field linesis due to the bigger value of the rms of the magneticfield b⊥ =< b2⊥ >1/2 in the volume respect to the rms ofthe photospheric forcings uph =< (u0

⊥ − uL⊥)

2 >1/2∼ 1(eqs. (16)-(17)).This means that the contribution to the dynamics of

the Alfvenic perturbations propagating from the bound-ary are much smaller, over short periods of time, than theself-consistent non-linear evolution due to the magneticfields inside the domain, and therefore can not deter-mine the topology of the magnetic field. For run A andG, both with cA = 200, the ratio is b⊥/uph ∼ 6 and itincreases up to b⊥/uph ∼ 27 in run I with cA = 1000. Onthe other hand these waves continuously transport fromthe boundaries the energy that sustain the system in amagnetically dominated statistically steady state.All the facts presented in this section, and the proper-

ties of the cascade and of the resulting current sheets inpresence of a magnetic guide field outlined in § 5, lead tothe conclusion that the current sheets do not generallyresult directly from a geometrical misalignment of neigh-boring magnetic field lines stirred by their footpoints mo-tions, but that they are the result of a nonlinear cascadein a self-organized system.Although the magnetic energy dominates over the ki-

netic energy, the ratio of the rms of the orthogonal mag-netic field over the axial dominant field B0 is quite small.For cA = 200, 400 and 1000 it is ∼ 3%, so that the aver-age inclination of the magnetic fieldlines respect to theaxial direction is just ∼ 2, it is only for the lower value

10 Rappazzo et al.

cA = 50 that b⊥/B0 ∼ 4% and the angle is ∼ 4. Thefield lines of the total magnetic field at time t = 18.47τAare shown in Figures 18c and 18d. The computationalbox has been rescaled for an improved viewing, and to at-tain the original aspect ratio the box should be stretched10 times along the axial direction. The magnetic topol-ogy for the total field is quiete simple, as the line ap-pear slightly bended. It is only in correspondence of thesmall scale current-sheets that field lines on the oppositeside may show a relative inclination. But as the currentsheets are very tiny (and their width decreases at higherReynolds numbers), they occupy only a very small frac-tion of the volume, so that the bulk of the magnetic fieldlines appears only slightly bended.It is often suggested, or implicitly assumed, that cur-

rent sheets are formed because the magnetic field linefootpoints are subject to a random walk. The complexityof the footpoint trajectory would then be a necessary in-gredient. In fact it would give rise to a complex topologyfor the coronal magnetic field, leading either to tangledfield lines which would then release energy via fast mag-netic reconnection, or to turbulence. So that the “com-plexity” of the footpoint motions would be responsiblefor the “complex” dynamics in the corona.On the opposite our simulations show that this system

in inherently turbulent, and that “simple” footpoint mo-tions give rise to turbulent dynamics characterize by thepresence of an inertial range (§ 5) and dynamical currentsheets. In fact our photospheric forcing velocities (Fig-ures 1-2) are constant in time and have only large-scalecomponents (eq. (10)), so that the footpoint motions are“ordered” and do not follow any random walk. Duringthe linear stage this gives rise to a magnetic field thatgrows linearly in time (eq. (24)) and that is a mappingof the velocity fields (see eq. (24) and Figures 17a and17b), i.e. both the magnetic field and the current haveonly large-scale components. The footpoint motions ofour photospheric velocities never bring two magnetic fieldlines close to one another, i.e. they never geometricallyproduce a current sheet. Current sheets are produced onan ideal timescale, the nonlinear timescale, by the cas-cade. Furthermore, as we show in the next section, thestatistically steady state that characterizes the nonlinearstage results from the balance at the large-scales betweenthe injection of energy and the flow of this energy fromthe large scales toward the small scales, where it is finallydissipated.As the system is self-organized and the magnetic en-

ergy increases at higher values of the axial magneticfields, very likely different static or time-dependent (withthe characteristic photospheric time ∼ 300 s) forcingfunctions, will not be able to determine the spatial shapeof the orthogonal magnetic field. In our more realisticsimulation with cA = 1000 the ratio b⊥/uph is in fact∼ 27. Other forcing functions are currently being inves-tigated, and time-dependent forcing functions are likelyto modulate with their associated timescale the rms ofthe system, like total energy and dissipation.

5. TURBULENCE

Before analyzing in detail the turbulent properties ofthe system, in this brief introduction we show why thetime-dependent Parker problem, i.e. the dynamics of amagnetofluid threaded by a strong axial field whose foot-

points are stirred by a velocity field, is an MHD turbu-lence problem.The fact that at the large orthogonal scales the Alfven

crossing time τA is the fastest timescale, and in particu-lar it is smaller than the nonlinear timescale τnl (whichcan be identified with the energy transfer time at thedriving scale), implies that the Alfven waves that con-tinuously propagate and reflect from the boundaries to-ward the interior, and that during the linear stage giverise to the fields (24)-(25), are basically equivalent to ananisotropic magnetic forcing function that stirs the fluid,whose orthogonal length is that of the convective cells(∼ 1000 km) and whose axial length is given by the looplength L.The typical forced MHD turbulence simulation (e.g.

see Biskamp (2003) and references therein) is performedusing a 3-periodic numerical cube, and introducing in theMHD equations a forcing function. The forcing functionis supposed to mimic some physical process that injectsenergy at the large scales.Solutions (24)-(25) can be approximately obtained in-

troducing the magnetic forcing function Fm in the mag-netic field equation (2)

Fm =uL(x, y)− u0(x, y)

τA, (38)

and implementing 3-periodic boundary conditions in ourelongated (0 ≤ x, y ≤ 1, 0 ≤ z ≤ L) computational box.During the linear stage this forcing would give rise, apartfrom the small velocity field (25), to the same magneticfield. During the nonlinear stage, as τA < τnl, it wouldstill give rise to a similar injection of energy. This prop-erty justifies also, to a certain extent, the previous 2Dcalculations (Einaudi et al. 1996).We want to underline the strong similarity between

the 3-periodic simulations of forced MHD turbulence de-scribed in the previous paragraph and the time depen-dent Parker problem. In particular the photospheric mo-tions imposed at the boundaries for the Parker problemtake the place of, and represent, one of the many physi-cal realizations of the forcing function generally used forthe 3-periodic MHD turbulence box. The time-dependentParker problem is then, for the presence of the dominantaxial magnetic field, and the independence of the forcingfunction (38) by the axial coordinate z, the anisotropiclimit of the 3-periodic magnetically forced MHD turbu-lence problem. The main difference between the two sys-tems is the presence of lying-tying for the Parker prob-lem, which is lost with 3-periodic boundary conditions.The effect of lying-tying is mainly that to inhibit an in-verse cascade for the magnetic field, as described later inthis section (§ 5.4).The presence of dynamical current sheets elongated

along the direction of the strong axial magnetic field,that are continuously formed and dissipated, is mostproperly accounted for in the framework of MHD tur-bulence. In fact, two important characteristic of MHDturbulence are that the cascade takes place mainly inthe plane orthogonal to the local mean magnetic field(Shebalin et al. (1983)), where small scales form, andthat the small scales are not uniformly distributed inthis plane. Rather they are organized in current-vortexsheets aligned along the direction of the local main field,and constitute the dissipative structures of MHD tur-

Coronal Heating 11

Fig. 7.—: Run G: Ratio between cross-helicity HC andtotal energy E as a function of time. HC ≪ E showsthat the system is in a regime of balanced turbulence.

bulence (e.g. Biskamp & Muller (2000), Biskamp (2003)and references therein). When the system is threadedby a strong axial magnetic field, as in our case, the cas-cade takes place mainly in the orthogonal planes, andthe current sheets are elongated along the axial direction(Figure 18).

5.1. Spectral Properties

In order to resolve the inertial range and investigate thepower law spectra, we have carried out four simulations(runs F, G, H and I in Table 1) with a resolution of512x512x200 grid points using a mild power (n = 4) forhyperdiffusion (12)-(13).In turbulence the fundamental physical fields are the

Elsasser variables z± = u⊥ ± b⊥. Their associated ener-gies

E± =1

2

∫

dV(

z±)2, (39)

are linked to kinetic and magnetic energies EK , EM andto the cross helicity HC

HC =1

2

∫

dV u⊥ · b⊥ (40)

byE± = EK + EM ±HC (41)

Nonlinear terms in equations (12)-(15) are symmetric un-der the exchange z+ ↔ z−, so as substantially are alsoboundary conditions (16)-(17), given that the two forc-ing velocities are different but have the same rms values(= 1/

√2). It is than expected that HC ≪ E so that

none of the two energies prevails E+ ∼ E− ∼ E, whereE = EK + EM is total energy. In Figure 7 the ratioHC/E is shown as a function of time for run G. Crosshelicity has a maximum value of 5% of total energy, andits time average is ∼ 1%, and similar values are found forall the simulations. Furthermore perpendicular spectraof E and E± in simulations F, G, H and I, overlap eachother, so that as expected we can also assume that

δz+λ ∼ δz−λ ∼ δzλ, (42)

Fig. 8.—: Total energy spectra as a function of thewavenumber n for simulations F, G, H and I. To highervalues of cA = vA/uph, the ratio between the Alfven andphotospheric velocities, correspond steeper spectra, withspectral index respectively 1.8, 2, 2.3 and 2.7.

where δzλ is the rms value of the Elsasser fields z± atthe perpendicular scale λ.In the following we always consider the spectra in the

orthogonal plane x-y integrated along the axial direc-tion z, unless otherwise noted. Furthermore as they areisotropic in the Fourier kx-ky plane, we will consider theintegrated 1D spectra, so that for total energy

E =1

2

∫ L

0

dz

∫

ℓ∫

0

dxdy(

u2 + b2)

=

=1

2

∫ L

0

dz ℓ2∑

k

(

|u|2 + |b|2)

(k, z) =∑

n

En,

n = 1, 2, . . . (43)

Figure 8 shows the total energy spectra En averaged intime, obtained from the hyperdiffusive simulations F, G,H and I with dissipativity n = 4 (eqs. (12)-(13)) andrespectively cA = 50, 200, 400 and 1000. An inertialrange displaying a power law behaviour is clearly re-solved. The spectra visibly steepens increasing the valueof cA, with spectral index ranging from 1.8 for cA = 50up to ∼ 2.7 for cA = 1000. The spectra are clearly al-

ways steeper than the strong turbulence k−5/3⊥ or k

−3/2⊥

spectra. This steepening is certainly not a numerical ar-tifact: the use of hyperdiffusion gives rise to a hump athigh wave-number values, known as the bottleneck effect(Falkovich 1994), which when present flattens the spec-tra. Furthermore we use the same value of dissipativity(n = 4) used by Maron & Goldreich (2001), who find thesame IK spectral slope (−3/2) as in the recent higher-resolution simulations performed by Muller & Grappin(2005) with standard n = 1 diffusion. In our simula-tions, a hump at high wavenumbers is more visible inrun H with cA = 400, which may be due to the bottle-neck effect, but might also be due to a transition fromweak to strong turbulence as discussed below.

12 Rappazzo et al.

Fig. 9.—: Run I : Snapshot of the 2D spectrumE(n⊥, nz) in bilogarithmic scale at time t ∼ 145 τA.n⊥ and nz are respectively the orthogonal and axialwavenumbers. The 2D spectrum is shown as a functionof n⊥ and nz + 1, to allow the display of the nz = 0component.

Recently a lot of progress has been made in theunderstanding of MHD turbulence for a system em-bedded in a strong magnetic field, both in the con-dition of so-called strong (Goldreich & Sridhar 1995,1997; Cho & Vishniac 2000; Biskamp & Muller 2000;Muller et al. 2003; Muller & Grappin 2005; Boldyrev2005, 2006; Mason et al. 2006) and weak turbulence(Ng & Bhattacharjee 1997; Goldreich & Sridhar 1997;Galtier et al. 2000; Galtier & Chandran 2006).In the presence of a strong field, nonlinear dynam-

ics take place in the planes orthogonal to the fields(Shebalin et al. 1983). The nonlinear terms in equa-tions (12)-(13), or equivalently (1)-(2), are associatedwith a timescale Tλ, the energy-transfer time at the cor-responding scale λ, characterizing the nonlinear dynam-ics at that scale. Tλ does not necessarily coincide withthe eddy turnover time τλ = λ/δzλ (see later in thissection). Spatial structures along the axial direction re-sults from wave convection (at the Alfven speed cA) ofthe orthogonal fluctuations. Hence their axial length ℓ‖is linked to the fluctuations at the orthogonal scale λby the so-called “critical balance” (Goldreich & Sridhar1995; Cho et al. 2002; Oughton et al. 1994)

ℓ‖(λ) ∼ cATλ. (44)

Tλ will be smaller at smaller scales, so that smaller per-pendicular scales create smaller axial scales. For a systemembedded in a strong enough field the length of the axialstructures can be longer than the characteristic length ofthe system, in our case the length of the coronal loop L.So that in the range of perpendicular wavenumber forwhich

ℓ‖(λ) > L, (45)

it is expected that the cascade along the axial directionis strongly inhibited. Figure 9 shows a snapshot at timet ∼ 145 τA of the 2D spectrum E(n⊥, nz) for run I inbilogarithmic scale, where n⊥ and nz are respectively

Fig. 10.—: Total energy at the injection scale (modes3 and 4), time-averaged for the four simulations F, G, Hand I with different Alfven velocities. The dashed lineshows the curve Ein ∝ c2A, while the continuous lineshows Ein as a function of cA as obtained from equa-tion (67) for α = 0 corresponding to a Kolmogorov spec-trum.The actual growth of Ein, both simulated and derivedfrom (67), show that the growth is less than quadraticalbut higher than in the simple Kolmogorov case. .

the axial and orthogonal wavenumbers. Following alongnz the cartesian lines n⊥ = const it is clearly visible thatfrom n⊥ = 1 up to n⊥ ∼ 20 the wavenumbers with nz >1 (the parallel spectrum has also the nz = 0 component,in Figure 9 the vertical coordinate is nz +1) are scarcelypopulated compared to the respective wavenumbers withnz ≤ 1.On the other hand, beyond n⊥ ∼ 20 the spectrum is

roughly constant along n⊥ = const up to a critical valuewhere it drops. Hence for n⊥ . 20 the system is in aweak turbulent regime, while for n⊥ & 20 a transition tostrong turbulence is observed. Interestingly enough, theslope of the 1D spectrum for run I (Figure 8) diminishesits value around n⊥ ∼ 20 (though we can not exclude acontribution from the bottleneck effect).In the strong turbulence regime ℓ‖(λ) < L so that,

while the cascade proceeds, each scale λ adds the cor-responding axial length ℓ‖(λ). On the other hand theenergy cascades from the large to the small scales, andthe larger eddies convect with their characteristic (larger)timescales the smallest eddies, so that the small scales“vibrate” with their characteristic timescales but modu-lated by the timescales of all the previous larger scales.For this reason at high perpendicular wavenumbers thevalue of the spectrum is roughly constant before drop-ping down in correspondence to the frequency associatedwith that particular scale.The nonlinear terms in equations (12)-(15) are dimen-

sionally associated with the timescale

τλ =λ

δzλ, (46)

the eddy turnover time at the orthogonal scale λ, whilethe linear terms are associated with the Alfven crossing

Coronal Heating 13

time τA = L/vA. The ratio between the two timescalesat the injection scale λ ∼ ℓc

χ =τAτℓc

=L δzinℓc cA

(47)

gives a measure of their relative strength.Increasing cA, i.e. the strength of the axial magnetic

field, the magnetic energy and total energy increase,while kinetic energy is always smaller than magnetic en-ergy and increases much less (increasing its value by afactor of 6 from cA = 50 to cA = 1000). In particularFigure 10 shows total energy at the injection scales (see§ 2.2), i.e. the sum of the modes n = 3 and n = 4 (seeeq. 43) of total energy,

Ein = E3 + E4 (48)

as a function of the non-dimensional Alfven velocity cA.Their growth is less than quadratical in cA, which im-plies that the rms of the velocity and magnetic fieldsat the injection scale (or equivalently the Elsasser fieldsδzin) grow less than linearly. Hence increasing cA theratio χ, which can be considered a measure of the rela-tive strength of the nonlinear interactions at the injectionscale, decreases. We therefore expect that at differentvalues of cA different regimes of weak turbulence are re-alized.

5.2. Phenomenology of the Inertial Range and CoronalHeating Scalings

We now introduce a phenomenological model, basedon a dimensional analysis, in order to determine thetimescales Tλ and the properties of the cascade. Thespectra that we have found can in fact be easily derivedby order of magnitude considerations, with some physicalinsight.In this section we use only dimensional quantities for

the scalings, so that we can compare the coronal heatingrates with the observational constraint. The magneticfield is measured in velocity unities, i.e. the correspond-ing Alfven velocity is used. The same unit is used for theElsasser variables z± = u⊥ ± b⊥.For weak turbulence ℓ‖ > L, so that the new scale

L must be introduced in the scalings, along with thetimescale τA = L/vA. Iroshnikov (1964) and Kraichnan(1965) proposed that the energy transfer time Tλ, be-cause of the Alfven effect, is longer than the eddyturnover time, and is given by

Tλ ∼ τλτλτA, (49)

But they were considering isotropic turbulence, so thatthe scale L is not present in τA. We now generalizethe IK scaling (49) to the anisotropic case. The eddyturnover time is the fundamental nonlinear timescale, hasthe dimension of a time, and hence Tλ should be propor-tional to it. This quantity can of course be multipliedby a non-dimensional quantity, where vA and L must bepresent. As the only remaining quantities are λ and δzλ,the only meaningful dimensionless quantity is the ratioτλ/τA, which is non-dimensional and can hence be raisedto the power α. The IK scaling is then generalized to

Tλ ∼ τλ

(

τλτA

)α

, with α > 0, (50)

where now τλ = λ/δzλ but τA = L/vA, where L is theaxial finite length of the system, for us the coronal loop.Simply with dimensional considerations α is only con-strained to be a positive number, so that the propertyTλ > τλ > τA is preserved. Note that for α = 0 theeddy turnover time and the energy transfer time coin-cide Tλ = τλ. This circumstance corresponds to strongturbulence for which ℓ‖ < L and for which our scalingarguments are no longer valid.Dimensionally, and integrating over the whole volume

(ℓ× ℓ×L), the energy cascade rate, supposed to be con-stant along the inertial range, may be written as

ǫ ∼ ℓ2 Lρδzλ

2

Tλ, (51)

Using (50) the energy transfer rate is given by

ǫ ∼ ℓ2L · ρ δz2λ

Tλ∼ ℓ2L · ρ

(

L

vA

)αδzα+3

λ

λα+1. (52)

Identifying, as usual, the eddy energy with the band-integrated Fourier spectrum δz2λ ∼ k⊥Ek⊥

, where k⊥ ∼ℓ/λ, from eq. (52) we obtain the spectrum

Ek⊥∝ k

− 3α+5α+3

⊥ , (53)

where for α = 0 the −5/3 slope for the “anisotropicKolmogorov” spectrum is recovered, and to α = 1 corre-sponds the −2 slope. To higher values of α correspondsteeper spectral slopes up to the asymptotic value of −3.Correspondingly, from eqs. (51)-(52), we have the fol-

lowing scaling relations for δzλ and Tλ:

δzλ ∼(

ǫ

ℓ2Lρ

)1

α+3 (vAL

)α

α+3

λα+1α+3 (54)

Tλ ∼(

ℓ2Lρ

ǫ

)

α+1α+3 (vA

L

)2α

α+3

λ2α+1α+3 (55)

As expected both δzλ and the energy transfer time de-crease at smaller scales. In particular when Tλ becomessmaller than the crossing time τA, small scales along theaxial direction starts to form and a transition to strongturbulence (where our scalings are not valid) realizes.Recently Boldyrev (2005) has proposed a promising

model for the cascade of strong turbulence, which seemsto overcome some discrepancies between previous mod-els and numerical simulations, and that self-consistentlyaccounts for the formation of current sheets. His energytransfer time is given by

Tλ =λ

δzλ

(

vAδzλ

)α

where 0 ≤ α ≤ 1 (56)

Very interestingly our scaling (50), which is valid forweak turbulence, when ℓ‖ > L and the external length Lmust be kept into account, reduces to the Boldyrev scal-ing for strong turbulence. In this case in fact ℓ‖ < L andthe scale along the axial direction are autonomously cre-ated by the system without interference from the exter-nal scale L. L cannot now be introduced in the scalings,and the only available dimensional quantities are δzλ, vAand λ. The scale λ cannot now be used, and the onlynon-dimensional factor that can be formed is vA/δzλ, sothat

(

λ vAL δzλ

)α

−→(

vAδzλ

)α

. (57)

14 Rappazzo et al.

As pointed out in the last paragraph of § 3, forthe problem that we are considering in this paper, thesolutions of equations (12)-(14) depend on the non-dimensional parameter f = ℓc vA/L uph (eq. (18)). Inparticular also the parameter α (50) must be a functionof f

α = α

(

ℓc vALuph

)

, (58)

i.e. it will depend on the characteristic parameters ofthe coronal loop: ℓc, the length of a convection cell,which represents the driving scale of the forcing, vA, theAlfven velocity associated to the DC magnetic field, L,the length of the loop and uph, the rms of the photo-spheric forcing velocity.With our simulations we estimate the value of α from

the slope of the total energy spectra (53), as described inRappazzo et al. (2007). As shown in Figure 8 to differentvalues of cA = vA/uph, (i.e. changing f) ranging from50 up to 1000 correspond spectral slopes ranging from∼ −1.8 up to ∼ −2.7. To this values of the spectralindex correspond (through eq. (53)) values of α rangingfrom ∼ 0.33 up to ∼ 10.33.This continuum of spectra might also take place in the

strong case, although the range would be very limited,from −3/2 up to −5/3. Boldyrev (2006) has given aphenomenological argument (based on dynamical align-ment) for which in presence of a strong field, but stillin the strong turbulence case (to which correspond thescaling (56)), α = 1, and correspondingly the spectralslope is −3/2. Decreasing the value of the mean mag-netic field, the Alfven velocity vA becomes of the orderof magnitude of the fluctuations, so that it cannot beused in the dimensional analysis, so that α = 0 in (56),corresponding to a Kolmogorov spectrum k−5/3, withthe consistent hypothesis that at intermediate intensitiesof the magnetic field intermediate spectral slopes mightdevelop, although very difficult to identify numericallygiven the small difference among them.We now discuss the bearing of the cascade properties

on the coronal heating scalings. The energy that is in-jected at the large scales by photospheric motions, andwhose energy rate (ǫin) is given quantitatively by thePoynting flux (21), is then transported (without beingdissipated) along the inertial range at the rate ǫ (52),and finally dissipated at the dissipative scales at the rateǫd. For balance, in a stationary cascade all these fluxesmust be equal

ǫin = ǫ = ǫd (59)

In dimensional unities the injection energy rate (21) isgiven by S, the Poynting flux integrated over the photo-spheric surfaces:

ǫin = S =

= ρvA

[∫

z=L

da(

uL⊥ · b⊥

)

−∫

z=0

da(

u0⊥ · b⊥

)

]

. (60)

2D spatial periodicity in the orthogonal planes allowsus to expand the velocity and magnetic fields in Fourierseries, e.g.

u⊥ (x, y) =∑

r,s

ur,s eikr,s·x, (61)

where

kr,s =2π

ℓ(r, s, 0) r, s ∈ Z (62)

Using this expansion the surface integrated scalar prod-uct of u⊥ and b⊥ (that are real fields) at the boundaryis given by

∫

dau⊥ · b⊥ =∑

r,s

ur,s ·∫ ℓ

0

∫ ℓ

0

dxdy b⊥eikr,s·x =

= ℓ2∑

r,s

ur,s · b−r,−s, r, s ∈ Z (63)

This integral is clearly dominated by the large scale com-ponents of the fields, as b⊥ is subject to a cascade, andfrom observations also photospheric motions exhibit aninertial range with an approximately Kolmogorov powerlaw. In particular in our case (eq. (10)) boundary ve-locities have only components for those wave numbers(r, s) ∈ Z

2 whose module ranges between 3 and 4,

3 ≤(

r2 + s2)1/2 ≤ 4. Then in (63) only the correspond-

ing components of b⊥ are selected.At the injection scale, which is the scale of convective

motions ℓc ∼ 1, 000 km, a weak turbulence regime de-velops, so that the cascade along the axial direction z islimited and in particular the magnetic field b⊥ can beconsidered approximately uniform along z at the largeorthogonal scales. Then from eq. (60) we obtain

ǫin = S ∼ ρvA

∫

da(

uL⊥ − u0

⊥

)

· b⊥ (64)

Introducing uph = uL⊥ − u0

⊥, using relation (63), andintegrating over the surface, we can now write

ǫin = S ∼ ℓ2ρvAuphδzℓc , (65)

where we have approximated the value of δbℓc , the rmsof the magnetic field at the injection scale ℓc, with therms of the Elsasser variable because the system is mag-

netically dominated, i.e. δzℓc =(

δu2ℓc + δb2ℓc)1/2 ∼ δbℓc .

We now have an expression for ǫin, where the onlyunknown variable is δzℓc , as ℓc, ρ, vA and uph are theparameters characterizing our model of a coronal loop.The transfer energy rate ǫ is supposed to be constant

along the inertial range, so that its value does not dependon λ. Considering then λ = ℓc in equation (52), we have

ǫ ∼ ρ ℓ2Lα+1

ℓα+1c vαA

δzα+3ℓc

. (66)

Equations (65) and (66) show another aspect of self-organization. Both ǫin and ǫ, respectively the rate of theenergy flowing in the system at the large scales, and therate of the energy flowing from the large scales towardthe small scales depend on δzℓc , the rms of the fields atthe large scale. This shows that the energetic balanceof the system is determined by the balance of the en-ergy fluxes ǫ and ǫin at the large scales. The small scaleswill then dissipate the energy that is transported alongthe inertial range (see eq. (59)). This implies that be-yond a numerical threshold total dissipation (dissipationintegrated over the whole volume) is independent of theReynolds number. In fact beyond a value of the Reynoldsnumber for which the diffusive time at the large scale is

Coronal Heating 15

Fig. 11.—: Analytical (69) and numerically computeddissipated flux as a function of the axial Alfven velocityvA. The continuous line shows the Poynting flux (69)as a function of vA in the case α = 0, correspondingto a Kolmogorov-like cascade. To higher values of vAcorrespond a higher dissipation rate because a weak tur-bulence regime develops.

negligible, i.e. when the resolution is high enough to re-solve an inertial range, the large-scale balance betweenǫ and ǫin is no longer influenced by diffusive processes.Of course this threshold is quite low respect to the highvalues of the Reynolds numbers for the solar corona, butit is still computationally very demanding.In order to have an analytical expression for the coronal

heating scalings we determine from (65) and (66) thevalue of δz∗ℓc for which the balance ǫin = ǫ is realized. Itis given by

δz∗ℓcuph

∼(

ℓcvALuph

)

α+1α+2

(67)

Substituting this value in (66) or equivalently in (65) weobtain the energy flux

S∗ ∼ ℓ2ρ vAu2ph

(

ℓcvALuph

)

α+1α+2

. (68)

As stated in (59) in a stationary cascade all energy fluxesare equal on the average. S∗ is then the energy that forunitary time flows through the boundaries in the coronalloop at the convection cell scale, and that from thesescales flows towards the small scales. This is also thedissipation rate, and hence the coronal heating scaling,i.e. the energy which is dissipated in the whole volumefor unitary time. As shown in equation (58) the power αdepends on the parameters of the coronal loop, and itsvalue is determined numerically with the aforementionedtechnique.The observational constraint with which to compare

our results is the energy flux sustaining an active region.The energy flux at the boundary is the axial componentof the Poynting vector Sz (see § 3.1). This is obtaineddividing S∗ (68), the Poynting flux integrated over the

surface, by the surface ℓ2:

Sz =S∗

ℓ2∼ ρ vAu

2ph

(

ℓcvALuph

)α+1α+2

, (69)

where α is not a constant, but a function of the loopparameters (58). The exponent in (69) goes from 0.5 forα = 0 up to the asymptotic value 1 for larger α. Wedetermine α numerically, measuring the slope of the in-ertial range (Figure 8), and inverting the spectral powerindex (53). We have used simulations F, G, H and I tocompute the values of α, because they implement hy-perdiffusion, resolve the inertial range, and then are be-yond the numerical threshold below which total dissipa-tion does not depend on the Reynolds number. Thesesimulations implement vA = 50, 200, 400 and 1, 000, andthe corresponding α are ∼ 0.33, 1, 3, 10.33. The corre-sponding values for the power (α + 1)/(α + 2) (69) are∼ 0.58, 0.67, 0.8 and 0.91, close to the asymptotic value1. Sz is shown in Figure 11 (diamond points) as a func-tion of the axial Alfven velocity vA. To compute thevalue of Sz for vA = 2, 000 kms−1 we have estimatedα ∼ .95, although for values close to 1 Sz does not havea critical dependence on the value of the exponent.In Figure 11 we compare the analytical function Sz (69)

with the respective value determined from our numer-ical simulations (star points), i.e. with the total dis-sipation rate by the surface and converted to dimen-sional units ((J + Ω)/ℓ2, see (37)). For the numericalsimulation values, the error-bar is defined as 1 stan-dard deviation of the temporal signal. The analyticaland computational values are in good agreement for allthe 4 simulations considered, and for the more realis-tical value vA = 2, 000 kms−1 the dissipated flux is∼ 1.6×106 erg cm−2 s−1. This value is in the lower rangeof the observed constraint 107 erg cm−2 s−1.The continuous line in Figure 11 corresponds to the

function Sz for α = 0 (which is approximately realizedfor vA . 50 kms−1), in correspondence of which a Kol-

mogorov spectrum would be present, and Sz ∝ v3/2A . The

computed and analytical values of Sz for higher vA arealways beyond this curve, because α increases its values,and a more efficient dissipation takes place. This is dueto the fact that to higher values of α correspond highervalues of the energy transfer time, and consequently alonger linear stage, higher values of the fields at the largescales (67), and hence a higher value of the energy rates(see (65), (66) and (68)). So that it is realized the onlyapparently paradox that to a weaker turbulent regime, towhich corresponds less efficiency in the nonlinear terms,corresponds a higher total dissipation.In the last paragraph of § 3.1 we have shown that when

the condition (23) is satisfied the emerging flux can beneglected. But in eq. (23) we have to specify the value ofthe magnetic field bturb⊥ self-consistently generated by thenon-linear dynamics. This value is given by (67) as themagnetic field dominates (δz∗ℓc ∼ bturb⊥ ). By substitutionwe can now estimate that the emerging flux is negligi-ble when the emerging component of the magnetic fieldsatisfies

bef⊥ < B0

√

(

ℓcL

)α+1α+2

(

uphvA

)1

α+2

(70)

16 Rappazzo et al.

Fig. 12.—: Transition to turbulence: Total ohmicand viscous dissipation as a function of time for simula-tions A, B, C, G (displayed on the same scales). All thesimulations implement cA = 200, but different Reynoldsnumbers, from Re = 200 up to 800. Run G implementshyperdiffusion. For Reynolds numbers lower than 100the signal is completely flat and displays no dynamics, athigher Reynolds smaller temporal structures are present.

In the asymptotic state α ≫ 1 the condition reduces

to bef⊥ /B0 <√

ℓc/L. For a coronal loop with L ∼40, 000 km, as ℓc ∼ 1, 000 km this implies that emerg-

ing flux does not play a role if bef⊥ /B0 < 1/6.

5.3. Transition to Turbulence and Dissipation vs.Reynolds Number

Turbulence is a characteristic of high Reynolds numbersystems (e.g. Frisch (1995)). For a sufficiently high vis-cosity nonlinear dynamics is strongly suppressed, and oursystem relaxes to a diffusive equilibrium (§3.3), and nosignificant small scale is formed. Increasing the Reynoldsnumber, the diffusive time at the injection scale (7)τd ∼ Re ℓ2c increases. At a certain point it will be bigenough not to influence the dynamics as the large scales,an inertial range will then be resolved and total dissipa-tion will not depend any longer from the Reynolds num-bers. In fact for higher values of Re the intertial rangewill extend to higher wave-numbers, but the energy fluxwill remain the same.At higher Reynolds numbers smaller scales are re-

solved, and each scale will contribute with its charac-teristic time Tλ to the temporal structure of the rmsof the system. Figure 12 shows total dissipation as afunction of time for simulations A, B, C and G, on thesame time interval, and on the same scale. At increas-ingly higher values of the Reynolds numbers smaller andsmaller temporal structures are added to the signal. Ide-ally the temporal structure of total dissipation at higherReynolds numbers is well described by shell-model simu-lations. For smaller values of Re the signal is completelyflat (see Figure 13). This behaviour identifies a transitionto turbulence.Figure 13 shows total dissipation as a function of time

for the same 4 simulations shown in Figure 12, plus other

Fig. 13.—: Total ohmic and viscous dissipation as afunction of time for simulations A, B, C, D, E and G, allof them implement cA = 200 but different Reynolds num-bers. The threshold beyond which dissipation is indepen-dent of the Reynolds number can be identified aroundRe = 800, corresponding to a numerical resolution of512x512 points in the orthogonal planes.

2 simulations at lower Reynolds number, respectivelyRe = 100 and Re = 10 for the complete time interval.For the lowest value of Re no dynamics is present, sothat the threshold value for the transition to turbulencecan be set to Re ∼ 100. To higher values of Re dissi-pation grows. An inertial range is barely solved with aresolution of 512x512 grid points in the x-y planes, sothat simulation with Re = 800 can be considered at thethreshold. On the other hand simulation G implementshyperdiffusion, so that an inertial range is solved, andthe dynamics is not affected by diffusion. The presenceof a sufficiently extended inertial range implies in factthat we are beyond the numerical threshold where dissi-pation does not depend on the Reynolds number (§ 5.2).The threshold value can be identified to a sufficient ex-tent at Re = 800, i.e. for a numerical grid of 512x512points. The number of points to use along the axial di-rection should be enough to allow the formation of all thesmall scales due to the “critical balance” (Figure 9), buta larger number of points would only result in a waste ofcomputational time.

5.4. Inverse Cascade and Line-tying

Two dimensional simulations (Einaudi et al. 1996;Georgoulis et al. 1998) have shown an inverse cascade forthe magnetic energy, corresponding in physical space tothe coalescence of magnetic islands. In the 3D case theDC magnetic field along the axial direction is present,giving rise to a field line tension that tends to inhibitan inverse cascade, as motions linked to the coalescencewould bend the field lines of the total magnetic field,which are mostly elongated along the axial direction (Fig-ure 18). On the other hand field line tension depends onthe strength of the axial field, becoming stronger for astronger field.In Figures 14 and 15 the first 4 modes of magnetic en-

ergy for simulations F and I, respectively with cA = 50

Coronal Heating 17

Fig. 14.—: Run F: In this simulation with cA = 50an inverse cascade at the wavenumbers n = 1 and 2 isrealized. Energy is injected at wavenumbers n = 3 and4.

and 1000, are plotted as a function of time. Energy isinjected at wave-numbers n = 3 and 4. Modes associatedto wave-numbers 1 and 2 grow to higher values than atthe injection scale in run F, while in run I they are alwayslimited to lower values. In runs G and H, with respec-tively cA = 200 and 400 an intermediate behaviour isfound, but none of the modes n = 1 or 2 never becomesbigger than the injection energy modes.

5.5. Timescales

In the previous sections we have always affirmed thatthe Alfven crossing time τA = L/vA is the fastesttimescale in the system, and that in particular it issmaller than the nonlinear timescale τnl, which we canidentify with the energy transfer time (55) at the injec-tion scale τnl = Tℓc .In Figure 3 it is already clear that the nonlinear

timescale is longer that τA, in fact it shows that thetimescale over which energy has substantial variations isbigger than the Alfven crossing time.The same behaviour is identified in Figures 14-15,

which show the time evolution of the magnetic energymodes for runs F and I. These are more relevant quanti-ties, because to realize a weak MHD turbulence regimeit is required that the energy transfer time Tλ is biggerthan the crossing time τA at the injection scale λ = ℓcand for a limited range of smaller scales down to somelower bound λ∗: λ∗ ≤ λ ≤ ℓc. The magnetic energymodes at the injection scale (n = 3 and 4) change theirvalues on scales bigger than τA, and for a larger valueof the Alfven velocity the nonlinear timescale is longerrespect to the crossing time (Figures 14-15). We canroughly estimate τnl ∼ 5τA for run F with cA = 50 andτnl ∼ 20τA for run I with cA = 1, 000.Using our scaling relations we can derive an analytical

estimate for the energy transfer time Tλ. Substituting

Fig. 15.—: Run I: Simulation performed with cA =1000. The increased magnetic field line tension inhibitsan inverse cascade for the orthogonal magnetic field.

the energy rate (68) in equation (55) we obtain:

Tλ ∼(

τAτα+1c

)1

α+2

(

λ

ℓc

)2α+1α+3

, (71)

where τc = ℓc/uph. In particular the ratio over the Alfvencrossing time is:

TλτA

∼(

τcτA

)α+1α+2

(

λ

ℓc

)2α+1α+3

, (72)

and as τc > τA then self-consistently Tλ > τA. Forour loop ℓc ∼ 1, 000 km and uph ∼ 1 kms−1, so thatτc ∼ 1, 000 s. For runs F and I shown in Figures 14 and15, the loop length is always L = 40, 000 km, while theAlfven velocity is respectively vA = 50 and 1, 000 kms−1,and the corresponding crossing times τA = 800 and 40 s.Using the values of α computed in §5.2 (respectivelyα = 0.33 and 10.33) we can then roughly estimate from(72), the nonlinear timescale τnl = Tλ=ℓc and its ratiowith the Alfven crossing time:

τnlτA

=TℓcτA

∼(

τcτA

)

α+1α+2

. (73)

For runs F and I we find τnl/τA = 1.2 and 22.3 in agree-ment with the simulations.Equation (72) can also be used to estimate the exten-

sion of the weak turbulence inertial range. The region forwhich the weak turbulence condition Tλ > τA is satisfiedis:

λ > λ∗ = ℓc

(

τAτc

)

α+32(α+2)

(74)

Figure 16 shows the temporal spectrum of magneticenergy for run G with cA = 200, i.e. we perform theFourier transform of the magnetic energy as a functionof time, and then plot its squared modulus. We userun G because it is the one for which we have savedmore frequently the rms quantity and then the plot cov-ers a wider range at high frequencies. The power spec-trum is roughly constant up to ν/νA ∼ 0.2, which cor-responds to t/τA ∼ 5 in agreement with our scaling (73)

18 Rappazzo et al.

Fig. 16.—: Temporal spectrum of magnetic energy forrun G. νA = 1/τA is the frequency corresponding to theAlfven crossing time. The intermediate part of the spec-trum exhibits a ν−2 power law.

which for this case gives τnl/τA ∼ 3.3. Beyond thiscritical point the power spectrum exhibits a power lawwhich fits ν−2, in agreement with shell-model simulations(Buchlin & Velli 2007).

6. DISCUSSION AND CONCLUSIONS

We would like first to clarify a few concepts that mightotherwise result in misunderstandings of the work thatwe have presented. The concept of turbulence is usedto describe different processes in different research fields,so that its use, without specifications, can result vagueand misleading. It is in fact very often used to describechaotic behaviors at the small scales, often linked to theintermittent dissipation of energy. Although this aspectis present in our simulations, when we say that the Parkerproblem is an MHD turbulence problem, we refer mainlyto the property of turbulence to transfer energy fromlarge to small scales. Namely to its ability to trans-port the energy from the scale of photospheric motions(∼ 1000 km), where it is injected, down to the small dis-sipative scales (meters?), without dissipating it at theintermediate scales. This property is clearly identifiedby the presence of an inertial range with a power lawspectrum, which extends from the injection scale to thedissipative scale.Furthermore turbulence, magnetic reconnection and

ohmic heating associated to currents are sometime pre-sented as alternative and/or mutually exclusive coronalheating models. This contraposition is artificial. Cur-rent sheets are in fact the dissipative structures ofMHD turbulence, and magnetic reconnection at the lociof current sheets is observed in virtually every MHD tur-bulence simulation in both 2D and 3D (see e.g., Biskamp(2003) and references therein ). Nanoflares are then nat-urally associated with the time and space intermittencyof the small scale deposition of energy (as shown in the2D case by Georgoulis et al. (1998)), which is due to thecascade which leads to the formation and dissipation ofcurrent sheets, and to which we refer collectively with

the term MHD turbulence.In summary, the main results presented in this paper

are the following:

• The time-dependent Parker problem may be seenas a weak and strong MHD turbulence problem,where the large scale forcing function is realized bythe photospheric motions.

• This system is genuinely turbulent, in the sensethat small scale formation is not driven passivelyby the random walk of the footpoints, rather itis a property of the maxwell stresses developingin the coronal volume. Current sheets thereforedo not generally result directly from a “geomet-rical” misalignment of neighboring magnetic fieldlines stirred by their footpoint (random) motions,they are the result of a nonlinear cascade in a self-organized system.

• Nanoflares are naturally associated with the inter-mittent dissipation of the energy that, injected atthe large scales by photospheric motions, is trans-ported to the dissipative scales through a cascade,and is finally dissipated through nonlinear mag-netic reconnection.

• Beyond a threshold, that is low compared to thecoronal Reynolds numbers, but still computation-ally very demanding, total dissipation is indepen-dent of the Reynolds numbers. This threshold cor-responds to a numerical resolution of ∼ 512× 512grid points in the planes orthogonal to the domi-nant DC magnetic field.

• Scalings for total dissipation (coronal heating scal-ings) are derived using the properties of the turbu-lent cascade. It is shown that there is no single uni-versal scaling law (see (69)), rather the small-scaleenergy deposition rate depends on the parametersof the coronal loop. For a loop long 40, 000 km,with an Alfven velocity vA = 2, 000 kms−1 and anumerical density of 1010 cm−3, whose footpointsare subject to photospheric motions of uph ∼1 kms−1 on a scale of ℓc ∼ 1, 000 km, the en-ergy flux entering the system and being dissipatedis Sz ∼ 1.6 × 106 erg cm−2 s−1. On the otherhand, for a coronal loop typical of a quiet-Sunregion, that has the same parameter of the pre-vious case but with a length of 100, 000 km andvA = 500 kms−1, the resulting Poynting flux isSz ∼ 7× 104 erg cm−2 s−1.

The most advanced EUV and X-RAY imagers (e.g.those onboard SOHO, TRACE, STEREO and HINODE)have space resolutions (∼ 800 km) of the order of thegranulation cells. Hence they do not resolve the small-scales where current sheets, magnetic reconnection andall the dynamical features of the system take place. Theirresolution is roughly 1/5 the length of the perpendicularcross-section of our numerical box (∼ 4000 km). Hence,even if the system is highly dynamical on small-scales(see Figure 17 and the associated movie), integratingover these scales has the effect to “averaging” the smallscale dynamics. In particular small scale reconnection

Coronal Heating 19

cannot be detected, magnetic fieldlines will appear onlyslightly bended (Figure 18), and their dynamics will ap-pear slower (a modulation of the nonlinear timescale withthe thermodinamical timescales).The topological and dynamical effects associated with

magnetic reconnection should be taken into accountwhen modeling the thermodynamical and observationalproperties of coronal loops (Schrijver 2007), recallingthat most of the dynamics take place at sub-resolutionscales while we observe the integrated emission.Two density current fields that have the same “steady”

integrated ohmic dissipation, balanced by a correspond-ing Poynting Flux (see § 3.3, equation (37) and Figure 4),but with different spatial distributions will have differentemissions. Consider the first with only large scale com-ponents, as the one that would result from a diffusiveprocess (§ 3.3), while in the second the current has onlysmall scale components, as in the simulations that wehave presented. In the second case the filling factor issmall (Figures 17 and 18) so that the density of currenthas a far larger value, and this would correspond to twovery different thermodynamical and observational out-comes. But the highly dynamical effects associated withthe second case will be averaged and result less dynamicalwhen integrated. Still the integrated observables shouldbe very distinct between the two cases.Finally, while our simulations give an accurate descrip-

tion of the time-dependent Parker problem, the use ofthe reduced MHD equation is justified only for slenderloops threaded by a strong axial magnetic field. For shortloops, or loops that have orthogonal component of themagnetic field comparable to the axial component, thefull set of MHD equation should be implemented. Forthe slender loops that we have simulated we observe amodest accumulation of energy, which subsequently is re-leased via nanoflares. On the other hand shorter loops,or loops in a more complicated geometry, or subject toloop-loop interactions, and more generally loops affectedby the neighboring coronal environment, might exhibitthe ability to accumulate more energy (e.g. Low (2006))and then release it in larger flares, possibly via a “sec-ondary instability” (Dahlburg et al. 2005) or fast mag-

netic reconnection (Cassak et al. 2006).Magnetohydrodynamics (MHD) has proved to be a

useful tool to investigate the properties of the turbulentcascade (Biskamp 2003). MHD is very well known togive an approximate description of the plasma dynamicsat large scales and low frequencies. In MHD turbulenceit is generally supposed that at the small scales a “dis-sipative mechanism” is present. Most of the propertiesof the turbulent cascade do not depend on the details ofthe dissipative mechanism, whether it is described by thediffusive operator present in equations (1)-(2), or moreproperly by a kinetic mechanism.In particular in our case, the timescales associated at

the scale λ ((71) for weak turbulence and (56) for thestrong case) decrease for smaller scales. In this way thesmall-scale dynamics is characterized by high-frequencyphenomena, and then it is not well described by MHD,but rather a kinetic model would be more appropriate. Itis then possible that (self-consistently) at the small scalesparticle acceleration plays an important role in the dissi-pation of energy, a physical process that should be inves-tigated through kinetic models. Nevertheless the coronalheating rates (69), like the cascade properties over an ex-tended range of scales, are independent of the details ofthe dissipation mechanism. They are determined by thebalance, at the large scales (see § 5.2), between the rateof the energy flowing into the loop from the boundariesdue to the work done by photospheric motions on themagnetic field line footpoints at the scale of the convec-tive cells, and the rate at which the energy flows alongthe inertial range from the large scales towards the smallscales.

The authors thank Bill Matthaeus for useful discus-sions. A.F.R. is supported by the NASA PostdoctoralProgram, M.V. is supported by NASA LWS-TR&Tand SR&T, and R.B.D. is supported by NASA SPTP.A.F.R. and M.V. thank the IPAM program “GrandChallenge Problems in Computational Astrophysics” atUCLA.

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Coronal Heating 21

(a) (b)

(c) (d)

(e) (f)

Fig. 17.—: Run A: (a) Streamlines of the boundary velocity fields u0⊥−uL

⊥ constant in time. (b)-(e) Axial componentof the current j (in color) and field-lines of the orthogonal magnetic field in the mid-plane (z = 5), at selected timescovering the linear and nonlinear regimes up to t = 18.47 τA. (f) Axial component of the vorticity ω (in color) andfield-lines of the orthogonal magnetic field in the mid-plane at time t = 18.47 τA.During the linear stage the orthogonal magnetic field is a mapping of the boundary forcing (cfr. a and b). After thecollaps of the large-scale currents (b, c, d), which in Fourier space correspond to a cascade of energy (see Figure 6),the topology of the magnetic field departs from the boundary velocity mapping and evolves dynamically in time(see movie). (e)-(f) Current sheets are embedded in quadrupolar vorticity structure, a clear indication of nonlinearmagnetic reconnection.

22 Rappazzo et al.

(a) (b)

(c) (d)

Fig. 18.—: Run A: Side (a,c) and top (b,d) views of current sheets (a,b) and field lines of the total magnetic field(c,d) at time t = 18.47 τA (same time as in Figures 17e-17f). For an improved visualization the box size has beenrescaled, but the axial length of the computational box is 10 times longer that the perpendicular cross-section length.The rescaling of the box artificially enhances the structures inclination. To restore the original aspect ratio the boxshould be stretched 10 time along z.(a)-(b) Two isosurfaces of the squared current j2. The isosurface at the value j2 = 2.8 · 105 is represented in partiallytransparent yellow, while red displays the isosurface with j2 = 8 · 105, well below the maximum value of the currentat this time j2max = 8.4 · 106. As typical of current sheets, isosurfaces corresponding to higher values of j2 are nestedinside those corresponding to lower values. For this reason the red isosurface appears pink. Although from the sideview the sheets appear space-filling, the top view shows that the filling factor is small.(c)-(d) Field-lines of the total magnetic field (orthogonal plus axial), and in the mid-plane (z = 5) field-lines of theorthogonal component of the magnetic field.