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Advances in Space Research 43 (2009) 612–617

A MHD-turbulence model for solar corona

Z. Romeou a,*,1, M. Velli b, G. Einaudi c

a Department of Physics, University of Patras, 26500 Patras, Greeceb Department of Astronomy and Space Sciences, University of Florence, Largo E. Fermi 2, 50125 Florence, Italy

c Department of Physics, University of Pisa, 56100 Pisa, Italy

Received 29 December 2007; received in revised form 7 October 2008; accepted 19 October 2008

Abstract

The disposition of energy in the solar corona has always been a problem of great interest. It remains an open question how the lowtemperature photosphere supports the occurence of solar extreme phenomena. In this work, a turbulent heating mechanism for the solarcorona through the framework of reduced magnetohydrodynamics (RMHD) is proposed. Two-dimensional incompressible long timesimulations of the average energy disposition have been carried out with the aim to reveal the characteristics of the long time statisticalbehavior of a two-dimensional cross-section of a coronal loop and the importance of the photospheric time scales in the understanding ofthe underlying mechanisms. It was found that for a slow, shear type photospheric driving the magnetic field in the loop self-organizes atlarge scales via an inverse MHD cascade. The system undergoes three distinct evolutionary phases. The initial forcing conditions arequickly ‘‘forgotten” giving way to an inverse cascade accompanied with and ending up to electric current dissipation. Scaling lawsare being proposed in order to quantify the nonlinearity of the system response which seems to become more impulsive for decreasingresistivity. It is also shown that few, if any, qualitative changes in the above results occur by increasing spatial resolution.� 2008 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Magnetohydrodynamics (MHD); Turbulence; Sun: activity; Sun: corona; Sun: magnetic fields

1. Introduction

Mechanisms of magnetohydrodynamic (MHD) turbu-lence have played over the latest years a fundamendal rolein the understanding of a number of extreme events relatedwith the solar magnetic field and also in providing someanswers to the so far unresolved problem of coronal heat-ing. Different MHD turbulent scenarios effective in thedevelopment of the fine scales in the solar corona have beendiscussed by many authors (Cargill, 1994; Priest et al.,2000; Vekstein and Katsukawa, 2000). Numerical model-ling in two and three dimensions (Dahlburg and Einaudi,2002; Hendrix and van Hooven, 1996) has been developed

0273-1177/$34.00 � 2008 COSPAR. Published by Elsevier Ltd. All rights rese

doi:10.1016/j.asr.2008.10.014

* Corresponding author.E-mail addresses: zrom@gge.gr (Z. Romeou), velli@arcetri.astro.it

(M. Velli), einaudi@pdf.unifi.it (G. Einaudi).1 On move to the Directory of Metrology, Ministry of Development,

Athens, Greece.

to validate or disprove the ideas of topological dissipationby Parker (1988) further developed into the so-callednanoflare heating theory. Einaudi et al. (1996) and Geor-goulis et al. (1998) were the first to show how the dissipa-tion time-series of the two dimensional magneticallyforced simulations of MHD turbulence displayed intermit-tent behaviour with events following a power-low behav-iour. Gomez et al. (2000) also showed how this model ofturbulence of a coronal loop could be relevant to the coro-nal heating problem. See also related work in Einaudi andVelli (1999) and Milano and Gomez (1997). Here we aremaking a step forward by studying how the dissipationend evolution of the 2D system depends on the time-scalesassociated with the photospheric forcing. To begin, we willstudy the case of a constant forcing, resembling a photo-spheric shearing motion. We then carry out simulationswith various random generated but spatially large scale-forcing profiles and variable in time over a range of time-scales compared with the internal dynamical time-scale of

rved.

Z. Romeou et al. / Advances in Space Research 43 (2009) 612–617 613

the turbulence. The paper is constructed as follows. In Sec-tion 2 we present the assumptions and the details of themodel we have used, and in Section 3 we present and dis-cuss the numerical results. We summarise the results andconclude in Section 4.

2. The numerical model – initial conditions and setup

We consider a cross-section of a coronal loop threadedby a strong axial magnetic field BO with footpoints rootedin the photosphere. Boundary disturbances propagatealong the axial direction z with the associated Alfvenicvelocity giving rise to perpendicular magnetic and velocityfields b? and u?. In the limit of a large loop aspect ratio,one may follow the evolution by using the reduced MHDequations (Strauss, 1976):

qo~u?otþ ~u? � ~r~u?

� �¼ � ~r?

�p þ 1

2b2?

�þ ~b? � ~r~b?

þ B0

o~b?ozþ mr2 ~u?; ð1Þ

o~b?ot¼ ~b? � ~r~u? � ~u? � ~r~b? þ B0

o~u?ozþ gr2 ~b?; ð2Þ

where q is the mass density, p is the plasma pressure andm and g are the collisional dissipation coefficients, the kine-matic viscosity and the resistivity respectively. These equa-tions are valid for a plasma with small ratio of kinetic tomagnetic pressures, in the limit of a large loop-aspect ratio(� � l=L� 1, L being the length of the loop and l being theminor radius of the loop) and of a small ratio of poloidal toaxial magnetic field ðb?=B0 6 �Þ. Incompressibility of plas-ma motion on the transverse plane follows from the lattercondition. Consequently, the density, initially consideredto be uniform remains so, allowing use of the same unitsfor velocities and magnetic fields via the normalizationb! b=q1=2. We can alternatively write the above set ofequations as follows:

ota ¼ B0ozwþ ½w; a� þ gr2a; ð3Þotx ¼ B0ozjþ ½w;x� � ½a; j� þ mr2w; ð4Þ

where the brackets ½c; d� ¼~ez � ðrc�rdÞ are the standardPoisson brackets. The scalar potentials w and a aredefined by ~u ¼ r� ðw~ezÞ and ~b ¼ r� ða~ezÞ respectively,and they relate to vorticity x and current density j asx ¼ �r2w and j ¼ �r2a respectively. Of special interestare the terms B0oz which represent the coupling betweenneighbouring z ¼ const. planes transferring energy fromthe footpoints into the coronal part of the loop thus en-abling communication across planes at different z. If as inour case we are interested in only following the evolutionof the transverse fields on the planes instead of the fullythree dimensional propagation along z, we can considerthem as forcing terms that can be modeled consistentlywith our understanding of the photospheric driver.

Thus we write

B0

o~u?oz¼ ~F mðx; y; tÞ ð5Þ

and

B0

o~b?oz¼ ~F uðx; y; tÞ; ð6Þ

where ~F m and ~F u are unknown external forcing functions.Following Einaudi et al. (1996) and Georgoulis et al.

(1998), we proceed in a further assumption by imposing~F u ¼ 0 and by rewriting ~F m in the more convenient form

~F m ¼ r� fm~ez: ð7Þ

These assumptions constitute of course a further reductionof the RMHD system. The first assumption however can bejustified by the argument that in the more realistic threedimensional case the plasma pressure is much smaller thanthe magnetic pressure. Consequently the rms velocity fieldsremain consistently much smaller than the magnetic fields.The assumption about ~F m facilitates our numerical setupand will be discussed further in the following section. Notethat in this case we need to impose a functional profile forfm in contrast with the three dimensional case where forc-ing is imposed by the boundary conditions for the velocityfield. For a more detailed discussion on the assumptionsunder which these equations are valid and the final formin which they are used in this model see Romeou et al.(2004), Romeou et al. (2006), Einaudi et al. (1996) andGeorgoulis et al. (1998).

3. Numerical results

First we investigate the effect of a time constant forcing.Therefore we substitute fm with a non-time dependent func-tional profile. Thus we assume

fm ¼X

ij

aij sinðkixþ kjy þ fijÞ: ð8Þ

The coefficients aij satisfy the condition hf 2mi ¼ 1 which

fixes our physical units in terms of the large scale field B0

(in velocity units), the typical photospheric velocity uph,the loop length L and the aspect ratio 1=�. Denoting byb0, l? and s our magnetic field, length and time unitsrespectively we have s ¼ l?

B0and

1 � hfmi �B0

b0

uphsL

ð9Þ

which leads to

b0 � B0

l?uph

B0L

� �1=2

: ð10Þ

On the assumption of an aspect ratio of 10 and a ratio ofphotospheric velocity to coronal Alfven speed of about1/1000, it follows that b0=B0 ’ 0:01 and s � 10L=B0s. Anestimation of B0 � 1000 km s�1 and L ¼ 104 km leads tos � 100 s. The unit for the energy dissipation follows

Fig. 1. Dependence of dissipation hgj2i on scb, scv, namely the amplitudesof the initial profiles given for a and w, for a range of g’s from 0.005 to0.08. In each graph, timeseries of averaged current dissipation of a fixed gare plotted for successively smaller values of scb, scv (scb of a dashedline > scb of a dashed-dot line > scb of a dashed-triple dot line). In theupper graph, it is shown that for g ¼ 0:005, in the range of g < 0:0625 andfor successively smaller values of scb, scv, the peak is reached later in timeand the dissipation peak gets bigger, while for g ¼ 0:0625 (middle graph)and g > 0:0625 (bottom graph) it reaches an asymptotic value for allvalues of scb, scv. For comparison, the linear approximation solution isoverplotted, dotted line in all graphs. It can be easily shown that the lateris given by hgj2i � ð1=gÞð1� expð�gk2tÞÞ2.

614 Z. Romeou et al. / Advances in Space Research 43 (2009) 612–617

directly from the parameters given above and an averagevalue of 109 cm�3 for the coronal density:

½gj2� ¼ 4pqb20=s ð11Þ

which for a density of 109cm�3 yields a dissipation unit of2.1 � 10�4 erg/cm3/s. Photospheric boundary motion is

confined within the ring 3 6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2

i þ k2j

q6 4 being zero else-

where, to simulate a large scale forcing, compared to thescales at which dissipation occurs. The explicit profile we as-sumed for fm in our calculations was fm ¼

ffiffiffi2p

sin ð4xþ 1Þ. Inthis time-constant case the field we chose corresponds to ashearing motion with a scale equal to 4. The initial profileswe used for the fields w and a were of the well studied Ors-zag–Tang type Orszag and Tang, 1979 as follows:

wðx; yÞ ¼ 2scvðcos xþ cos yÞ ð12Þ

and

aðx; yÞ ¼ scbð2 cos xþ cos 2yÞ; ð13Þ

where scv and scb denote the initial amplitudes of the fields.To facilitate our calculations, we have further assumedscv ¼ scb and g ¼ m. We have calculated time-series ofthe spatial average of the current dissipation gj2 with vary-ing resolutions of 64� 64, 128� 128 and 256� 256. Wehave shown (see Romeou et al., 2006) that passing from128� 128 to 256� 256 does not cause a considerablechange. In the following runs we kept a resolution of128� 128 grid points with uniform spacing in both direc-tions. We next investigated how the evolution at long timesdepends on the initial triggering. For that reason we plot-ted timeseries of the mean dissipation of each g keepingthe amplitudes of the initial profiles of the fields, namelyscb, scv as parameters, and successively reducing them.Fig. 1 shows clearly a strong dependence on these twoparameters which can be regarded as the seeds of the trig-gering necessary for reconnection to start up.

Decreasing g seems to suppress the contribution of thelinear terms and strengthen nonlinearities. These conclu-sions only regard the initial phase of the evolution. The longtime evolution is not very much affected as long as scb andscv are not too small to allow instabilities to develop. Whatchanges significantly though with resistivity is the value ofthe peak and the time it is reached. The time at which dissi-pation leaves off the linear approximation solution is indica-tive of the time it takes for the nonlinearities to evolve andfor the energy to redistribute and be released via resistivemodes. The appearance of a strong sharp peak thereforereveals the tendency of the system to respond rapidly andintensively to what in the absence of resistivity would havebeen an ideal MHD energy release. We then plotted thisasymptotic value of the maximum of the mean dissipationagainst g. We used these values to plot the dissipation peakagainst resistivity, the difference of this peak from the linearapproximation asymptotic value again as a function of g,and finally the time it takes to reach this peak after it leavesoff the linear approximation solution. We used five points,

Fig. 2. Graphs 1–3: a-fieldlines contours for g ¼ 0:01. Three distinct phases are demonstrated clearly: An initial linear phase followed by the formation of acompact islands structure which at later times disentagles giving way to a more diffused magnetic fieldlines structure. Graph 4: spectra of the square of themagnetic potential at a time before the peak (solid line), at a time right after the peak (dotted line), and for times long after the peak (dashed and dashed-dottedlines). It is shown the redistribution of the magnetic energy before, during, after, and long after the peak in modes smaller than the k ¼ 4 initially imposed bythe forcing, demonstrating the expected appearance of a secondary inverse cascade at longer times where the system seems to relax to larger scales.

Z. Romeou et al. / Advances in Space Research 43 (2009) 612–617 615

see Romeou et al. (2006), to draw the best fit for the lines inthe corresponding log–log graphs, from which we obtainedthe following scaling laws:

for the dissipation peak, dismax ¼ 0:23g�1:56, for the difference of this peak from the linear approxi-

mation asymptotic value as a function of g,dðdismaxÞ ¼ 0:04g�2:4,

and finally for the time it takes to reach this peak after itleaves off the linear approximation solution,dtpeak ¼ 0:64g�0:1.

It is evident from these results that with lower resistivity thedissipation peak increases rapidly from the point of view ofthe order of the magnitude. Nonlinear effects become moreeffective while dtpeak also increases with decreasing resistiv-

Fig. 3. Typical long time timeseries of hgj2i with random forcing, forg ¼ 0:004, 128 � 128 resolution, and tkink ¼ 64. The full run is shown onthe top, two intervals of which are zoomed in the middle and at thebottom. The relationship between tkink and the intervals of appearance ofdissipation events would be important to understand the role of the timescaling and it is a work under study.

616 Z. Romeou et al. / Advances in Space Research 43 (2009) 612–617

ity. We further investigated the structure of the fields andthe currents. In all cases there were three distinct phasesdeveloping.

This response is typical for the whole range of g’s we tried,see Fig. 2 (graphs 1–3). Repeating the runs for increased

resolution 256 � 256 and g ¼ 0:01, showed similar results,making us draw the conclusion that any discontinuitiesappearing in the last phase of the evolution are not a numer-ical effect, but are probably inherited to the system and forthis reason may need closer attention.

Another interesting point comes from the study of thespectra. Initially, before the peak is reached the dominantmode for the emission of magnetic energy is the oneimposed by the forcing profile, namely k ¼ 4. Right after,energy is allowed to redistribute in other modes as well.In the long run, the system seems to develop a secondaryinverse cascade activity as it is expected for 2D MHD mag-netic invariants, see Fig. 2 (graph 4).

To compare these results with a random time dependentforcing profile, we followed Georgoulis et al. (1998), Einau-di et al. (1996) and Einaudi and Velli (1999) and assumedfor the forcing profile a form where the eddy turnover timetkink is introduced. We first investigated the dependencewith resolution and concluded that it seems not to beimportant when resolution is high enough, for a discussionsee Romeou et al. (2006). We confirmed that initial condi-tions are soon forgotten and the profiles of the timeseriesresemble the later (after the dissipation peak) stages ofthe constant forcing case. Finally, we calculated long timetimeseries for different tkink, a statistical study of whichwould show how the system responds to the photosphericdriver. A systematical study for quantifying this depen-dence is ongoing work (see Fig. 3).

4. Conclusions

Under the assumptions of the Reduced MHD approxi-mation, we have investigated the spatial and temporalresponse of a coronal loop system in two different cases.Firstly by assuming constant time forcing and secondly byapplying random time dependent forcing. Using a numericalpseudospectral method for the calculation of timeseries ofhgj2i we have showed: the dependence with g, the effects ofresolution, the effects of the initial conditions, and the roleof the time scaling in the mechanisms involved. We haveshown that there are three phases appearing, indicated bythe development of a strong and fast dissipation peak whichfalls off rapidly. The system quickly ‘‘forgets” its initial mag-netic and velocity conditions and the fields are organised intoa compact magnetic island structure, which gradually disen-tagle before a secondary inverse cascade takes over. Study ofthe spectral response corroborates this behaviour. Thestrength and the duration of the dissipation peak have beenscaled with g in order to quantify the non-linear response ofthe system which becomes more evident and impulsive fordecreasing g. Few, if any, qualitative changes in the aboveare seen when the spatial resolution is increased. Simulationsof long time timeseries for the second case where we set thesystem subject to random forcing for a range of differenttkink and g’s were produced in which the relationshipbetween the coronal response to the photospheric driver

Z. Romeou et al. / Advances in Space Research 43 (2009) 612–617 617

was demonstrated. Looking for scaling laws also in this case,is work under study.

References

Cargill, P.J. Some implications of the nanoflare concept. Astrophys. J.422, 381–393, 1994.

Dahlburg, R.B., Einaudi, G. MHD unstable modes in the 3D evolution of2D MHD Structures and the diminished role of coalescence instabil-ities. Phys. Lett. A 294, 101–107, 2002.

Einaudi, G., Velli, M. The distribution of flares, statistics of magnetohy-drodynamic turbulence and coronal heating. Phys. Plasmas 6 (11),4146–4153, 1999.

Einaudi, G., Velli, M., Politano, H., Pouquet, A. Energy release in aturbulent corona. Astrophys. J. 457, L113–L116, 1996.

Georgoulis, M., Velli, M., Einaudi, G. Statistical properties of magneticactivity in the solar corona. Astrophys. J. 497, 957–966, 1998.

Gomez, D.O., Dmitruk, P.A., Milano, L.J. Recent theoretical results oncoronal heating. Solar Phys. 195, 299–318, 2000.

Hendrix, D.L., van Hooven, G. Magnetohydrodynamic turbulence andimplications for solar coronal heating. Astrophys. J. 467, 887–893, 1996.

Milano, L.J., Gomez, D.O. Solar coronal heating: AC versus DC.Astrophys. J. 490, 442–451, 1997.

Orszag, S.A., Tang, C.M. Small-scale structure of two-dimensionalmagnetohydrodynamic turbulence. J. Fluid Mech. 90, 129–143, 1979.

Parker, E.N. Nanoflares and the solar X-ray corona. Astrophys. J. 330,474–479, 1988.

Priest, E.R., Foley, C.R., Heyvaerts, J., et al. A method to determine the heatingmechanisms of the solar corona. Astrophys. J. 539, 1002–1022, 2000.

Romeou, Z., Velli, M., Einaudi, G. Long time incompressible 2D MHDsimulations of coronal loop heating: the role of photospheric time-scales. In: Proceedings of the SOHO 15 Workshop-Coronal Heating,ESA SP-575, vol. 539, pp. 1002–1022, 2004.

Romeou, Z., Velli, M., Einaudi, G. Forced MHD turbulence simulations forcoronal loop heating. Recent Adv. Astron. Astrophys. 848, 105–114, 2006.

Strauss, H. Non-linear, three-dimensional magnetohydrodynamics ofnon-circular tokamacs. Phys. Fluids 19, 134–140, 1976.

Vekstein, G., Katsukawa, Y. Scaling laws for a nanoflare-heated solarcorona. Astrophys. J. 541, 1096–1103, 2000.