Mem. S.A.It. Vol. 74, 675 c
° SAIt 2003 Memorie
dellaModels for magnetic heating of the solar atmosphere
L. Del Zanna
Dipartimento di Astronomia e Scienza dello Spazio, Universit`a degli Studi di Firenze, Largo E. Fermi 2, 50125 Firenze
Abstract. Sustaining the hot solar corona requires an energy flux of up to about 10
7erg cm
−2s
−1whose source must be the photospheric motions below. However, the precise way this energy is transferred and damped still remains an open ques- tion. Here a review of recent advances in theoretical models for coronal heating, in the light of the fundamental observational results of the last decade, will be presented.
1. Introduction
The longstanding coronal heating prob- lem may be summarized as follows: why is the coronal plasma at temperatures which are hundreds of times higher than those in the photosphere below? Nowadays there is little doubt that the old pic- ture of heating by shocked sound waves produced by turbulent convective motions below the photosphere may be good for the low chromosphere, but that the heat- ing at larger distances of the low-β (i.e.
magnetically dominated), almost collision- less coronal plasma must be of intrinsic magnetic nature. Well known reviews of the several proposed mechanisms are those of Narain & Ulmschneider (1990; 1996);
see also Chiuderi (1997), Malara & Velli (2001), Del Zanna & Velli (2002).
A quantitative analysis of enery losses and requirements in the various layers of the solar atmosphere, and in differ-
Send offprint requests to: L. Del Zanna Correspondence to: ldz@arcetri.astro.it
ent regions, was pursued for the first time by Withbroe & Noyes (1977) (see also Withbroe, 1988, and Parker, 1991).
According to these authors, the energy
losses in conduction and UV radiation in
the chromosphere and in the transition re-
gion require an energy flux input of about
F ∼ 5 × 10
5erg cm
−2s
−1for quiet regions
and coronal holes (where extra mechanical
losses to drive the solar wind must be taken
into account), and F ∼ 2×10
7erg cm
−2s
−1for active regions, where energy losses are
the highest, also at coronal levels. Since we
know that the temperature profile is an in-
creasing function of height, moving from
the photosphere to the corona, reaching
maxima of T ∼ 1 − 1.5 × 10
6K in quiet re-
gions and coronal holes and of T > 2×10
6K
in active regions, the fundamental ques-
tions are how this energy is transported
up in the corona (if it is not generated
there directly) and how it can be damped
there. Concerning this last point, since col-
lisions are very rare in the coronal plasma
due to the low density, the question could
be translated in how small scales can be reached, in order for dissipation to be effi- cient.
Any model for (magnetic) heating of the solar corona should therefore be able to account for the three basic aspects: gen- eration (where?, how much?), transport (how?), and dissipation (how?, how effi- ciently?). Once the fundamental role of magnetic fields is estabilished, the first as- pect is usually explained by the shuffling of magnetic fieldlines due to granular and supergranular photospheric velocity fields.
The generated energy flux is then given by the vertical component of the Poynting vec- tor that, assuming flux-freezing in the pho- tosphere, is
F ∼ S
k= c
4π E
⊥B
k= 1
4π B
kB
⊥v
⊥(1) that is about 10
7erg cm
−2s
−1for B
k∼ 100 G, B
⊥∼ 10 G, and v
⊥∼ 1 km s
−1, (here the perpendicular components actu- ally refer to root mean square values), so more than enough at least for quiet regions and coronal holes, to which this review is particularly devoted especially in its last part, where the connections to solar wind properties will be discussed.
To be able to disentangle ourselves in the jungle of different theories and physical mechanisms proposed for energy transport and damping in the corona, here the usual distinction between DC and AC mecha- nisms will be maintained. In active regions, where large magnetic loops are present, the Alfv´enic time scale τ
A= L/v
A(L is the typical length scale) of response to foot- point motions is usually very short, lead- ing to a quasi-static series of magnetic equi- libria that dissipate energy through Ohmic diffusion in thin current-sheets (flare and nanoflare models). Coronal holes are in- stead characterized by open magnetic field- lines, along which the solar wind is be- lieved to escape, so that L is very large and footpoint perturbations must propa- gate upwards in the form of MHD waves (AC mechanisms).
2. DC flaring activity, the Parker scenario and MHD turbulence In the photosphere magnetic flux is contin- uously replaced by the emerging, fragment- ing and cancelling of ephemeral regions bipoles, which usually are born at super- granule cell centers and then migrate to the cell boundaries, where cancellation with neighbouring fields occur. The SoHO/MDI instrument have recently shown that this process (magnetic carpet, Schrijver et al.
1998) is much more rapid and frequent than previously believed, so that 90% of the total quiet Sun magnetic flux is replaced in ap- proximately one day! It is clear that in this way miriads of reconnection events could occur in the overlying corona (see the tec- tonics model by Priest et al., 2002), pro- viding a continuous release of energy in the corona.
Similar models were previously applied to explain X-ray bright points (Priest et al.
1994), where a 2-D force-free field evolves quasi-statically due to the converging mo- tions of magnetic fragments of opposite po- larity: during the interaction phase, recon- nection occurs at the null point, the plasma is heated (giving rise to the observed X- ray emission), and later the fragments an- nihilate and the flux is cancelled. Going in 3-D, a wild variety of different geome- tries suitable for quasi-static reconnection of force-free fields naturally arise (Priest
& Schrijver 1999). Related problems are those of the energy release by large flares (Priest & Forbes, 2002, for a review) and also the heating and temperature profiles along coronal loops (see Reale’s contribu- tion in these proceedings). DC dissipation in current sheets formed due to the mo- tion of magnetic structures is certainly a viable method for energy release in corona.
However, due to the huge value of mag- netic Reynolds numbers there (∼ 10
12) ex- tremely small scales must form in the pro- cess, unless the dissipation is not Ohmic but rather of some non collisional type.
Another basic aspect to consider is the
frequency of the flaring events as a function
of the energy released. Parker (1988) was the first to suggest that the corona might be heated through a moltitude of small flar- ing events, rather than thanks to rare large flares of the kind described in the previ- ous section. Let us consider the distribu- tion of energy release events f (E) ∝ E
α, supposed to be a power law, integrated be- tween E
min= 10
24erg, corresponding to the lowest energy events that can be mea- sured in EUV (nanoflares), and E
max= 10
32erg, corresponding to a large flare:
E
tot= Z
EmaxEmin