Coronal heating by Nanoflares and Alfvén
waves, predicting observational features
by
Patrick Antolin Tobos
Master Thesis
Supervisor: Kazunari Shibata
Kyoto University
Graduate School of Science
Department of Astronomy
Japan
July, 2006
i
À ma famille, particulièrement les petits, Maya et Manu.
§0.0
1
Contents
Introduction
3
1 Steady shock heating by fast MHD waves and
1.1 MHD Waves in the solar atmosphere . . . . . .
1.1.1 Slow MHD mode and acoustic waves . .
1.1.2 Fast MHD mode and Alfvén waves . . .
1.1.3 Structure and purpose . . . . . . . . . .
1.2 Shock heating theory . . . . . . . . . . . . . . .
1.2.1 Shock formation distance . . . . . . . .
1.2.2 Variation of the shock amplitude . . . .
1.3 Model for the corona . . . . . . . . . . . . . . .
1.3.1 Loop geometry and equations . . . . . .
1.3.2 Boundary values and numerical model .
1.4 Results . . . . . . . . . . . . . . . . . . . . . . .
acoustic waves
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2 Nanoflare heating simulations
2.1 Magnetic reconnection in the solar atmosphere . . . . . . . . . . . . . .
2.2 Nanoflare heating model . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Loop geometry and basic equations . . . . . . . . . . . . . . . . .
2.2.2 Nanoflare heating function . . . . . . . . . . . . . . . . . . . . . .
2.3 Initial conditions and numerical model . . . . . . . . . . . . . . . . . . .
2.4 Nanoflare heating results . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Basic common features . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Different heating models sharing a loop oscillation pattern . . . .
2.4.3 The effect of longer heating events, a quasi-steady evolution . . .
2.4.4 Energetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Differences between Alfvén wave heating and Nanoflare heating mechanisms
2.5.1 Completely different coronas . . . . . . . . . . . . . . . . . . . .
2.5.2 Predicting observables . . . . . . . . . . . . . . . . . . . . . . . .
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3 Nanoflare and Alfvén
3.1 Model . . . . . . .
3.2 A quasi-steady and
3.3 Energetics . . . . .
3.4 Intensity fluxes . .
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wave heating combined
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dynamic corona . . . . . . .
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Conclusions
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Bibliography
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§0.0
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Aknowledgements
In the roots of my childhood lies the first thought about Japan. Distant, different,
interesting. A bubble in space and time. Studying Astronomy in Japan was the product
of a dream, a lot of effort and the help of many dear people. For helping me make this
dream come true, my most sincere thanks. To my family and friends, especially, who
even if on the other side of the Earth, are in my heart. In Japan, very special thanks to
my supervisor, Kazunari Shibata, without whom none of this could have been possible.
His motivation and insight in solar physics have been most valuable in my learning. I am
very grateful for all the fruitful discussions with Shibata sensei, T. Suzuki, T. Kudoh,
T. Magara and L. Heggland about the fascinating coronal heating problem.
I wish to thank the people in the Department of Astronomy, maybe the most hardworking people I have ever met. I wish to encourage the students to go abroad, hopefully to
Colombia, and experience other ways of thinking. Special thanks to Takahashi san for
being patient with me and my poor japanese understanding, and to Uemura san and
Sampa san, who always cheer up the Department. Ato san nenkan yoroshiku onegaishimasu.
Studying abroad is a challenge, especially in a country as Japan I believe, so different
from every place I have been before. Fortunately, here in Japan, I have met wonderful
people with whom I share the same challenge. Fabricio, Joseph, Takeshi, thank you so
much for always being there.
Me faltan palabras para poder expresar la felicidad que es poder compartir este reto con
David. Tenerte aquí en Japón es una de las más increíbles sorpresas y dichas que me ha
dado la vida. Gracias por tu fuerza hermano.
A Noriko chan, mi pulguita querida, mi segundo sol, aette hontou ni ureshii. Noriko
chan to iru aida ni motto utsukushii Nihon ga mieru. Todo se vuelve más hermoso
cuando estoy a tu lado. Estoy feliz de haberte encontrado.
Arigatou.
The numerical calculations were carried out on Altix3700 BX2 at YITP in Kyoto University.
§0.0
3
Introduction
Since its discovery fifty years ago, the few million degree corona still keeps many puzzling
phenomena unsolved. The most famous one, the coronal heating problem consists in
identifying and understanding the physical mechanism responsible for the high coronal
temperatures. Indeed, observational evidence (see for example Alfvén, 1941) correctly
interpreted by Edlén (1943) pointed out the existence of a 106 K, tenuous plasma. This
fact, contradicting our common sense based on the second law of thermodynamics has
remained a big mystery for decades. It is important however to note that the corona
being so tenuous, its heating to observable values is accomplished with a relatively small
energy flux of ∼ 105.5 erg cm−2 s−1 (Withbroe & Noyes, 1977) compared to the large
radiative energy flux from the Sun (∼ 5.4 × 1010 erg cm−2 s−1 ). This implies that the
energy reservoir in the deeper layers in the Sun is sufficient for coronal heating. The
true problem consists then in finding a mechanism by which a sufficient portion of that
energy reservoir can be transported through the different layers in the Sun, namely,
the photosphere, chromosphere and transition region, and deposit it in the corona over
reasonable length and time scales.
The coronal heating problem consists then of a series of different steps that need to
be solved. First, an energy source and a mechanism that converts that energy into
heat must be specified. Then, the plasma response to the heating must be determined.
Finally, the spectrum of emitted radiation and its manifestation as observable quantities
must be predicted. For a review of the coronal heating problem please refer to Klimchuk
(2006), Narain & Ulmschneider (1996) or Aschwanden (2004).
Coronal heating mechanisms proposed so far belong to two categories: magnetic waves
(i.e. Alfvén wave heating mechanism, Alfvén, 1947; Narain & Ulmschneider, 1996;
Moriyasu et al., 2004) and magnetic reconnection (i.e. Nanoflare heating mechanism,
Parker, 1988; Priest, Heyvaerts & Title, 2002). In spite of their different nature, it is
hard to differentiate between these two classes of mechanisms during observations as they
don’t exclude each other and they can occur at the same time, or even more, one could
induce the other mechanism to happen in the corona. It is important however to see
which one is the principal heating scenario in the corona. In the first case, Alfvén waves,
generated somehow in the lower layers of the solar atmosphere, transport the magnetic
energy into the corona and can dissipate it through many different ways, like steepening
into shocks (Hollweg, 1982; Stein & Schwartz, 1972) in which plasma is heated through
collisions at the shock front, or through a process of phase mixing (Heyvaerts & Priest
1983; Sakurai & Granik, 1984), in which resistivity comes into play, or by resonant
absorption (Poedts, Kerner & Goossens, 1989). In the case of magnetic reconnection,
the idea of a magnetic flux being composed by many magnetic threads is taken into
§0.0
4
account (Priest, Heyvaerts & Title, 2002). Due to sub-photospheric motions (footpoint
braiding; Parker, 1994) or other interactions among magnetic fluxes at different layers of
the solar atmosphere, current sheets are formed and many reconnections of these small
magnetic threads occur, producing impulsive and sporadic releases of magnetic energy
from the stressed fields. Each reconnection event drives plasma inflow into the current
sheet heating it through resistivity, thus producing a small flare (Tajima & Shibata,
2002). More over, these perturbations of the magnetic field through reconnection events
generate slow and fast MHD shocks that propagate and heat the surrounding plasma.
Discerning between these two heating models is not an easy task and in order to achieve
this it is necessary to analyze them first separately and answer how feasible heating mechanisms they are. For instance, any standing heating theory has to provide a mechanism
which is able to balance three types of energy losses: downward conduction, radiative
losses and the solar wind. As previously stated, the required energy is estimated as
& 105.5 erg cm−2 s−1 .
Observations from Yohkoh, SoHO and TRACE have provided high resolution data that
show a very dynamic Sun. In the fall of year 2006 Solar-B will be launched equipped
with unprecedented state-of-the-art instruments as SOT (Solar Optical Telescope), XRT
(X-ray Telescope) and EIS (EUV Imaging Spectrometer) that will provide a new insight
into the different layers of the Sun. In this context it is important to discuss at present
which are the observational signatures that can distinguish among the Alfvén wave and
Nanoflare heating mechanisms.
Whichever the principal coronal heating mechanism, additionally to the necessary released energy, it should happen in relatively small time and space scales. It should
remain dormant long enough to let magnetic stresses build-up to the necessary energy
level (in order to heat the corona), and when its onset occurs it should happen rapidly
and efficiently. It should happen ubiquitously in most regions of the Sun and in a small
space scale as it has so far avoided high resolution and cadence observations (1’'700 km
and in the order of minutes). In this picture, at present, the heating mechanism that
fits best is considered to be the nanoflare heating scenario (Klimchuk, 2005; Dahlburg,
Klimchuk & Antiochos, 2005) for its sporadic and impulsive character. However, some
works have shown that the Alfvén wave heating mechanism can also possess the same
sporadic, impulsive and ubiquitous character (Moriyasu et al, 2004, hereafter M04).
The present work constitutes a first step in the attempt to elucidate these observational
signatures unique of each heating mechanism. The main aim in this work is to compare
these two different heating mechanisms in the case of a coronal loop. In order to do this,
we set up two different coronas (§2 and §3), one created from the heating of nanoflares,
in which case we perform 1-D hydrodynamical (HD) numerical simulations, and another
corona created from the heating of Alfvén waves, already presented in M04, in which
1.5-D MHD numerical simulations were performed. The numerical scheme used is the
§0.0
5
CIP scheme (with MOCCT scheme for the 1.5-D MHD simulations, Yabe & Aoki 1991,
Evans & Hawley 1988). The quasi-steady stage of the simulations are compared to
the analytic model of a steady corona heated by fast and slow MHD shocks (switch-on
shock trains and N-waves respectively). The observational consequences of each model
are investigated predicting the XRT intensity fluxes for Solar-B. The different observable
signatures of these two heating models are discussed in order to distinguish which heating
mechanism is operating in the corona when it is observed.
The structure of the work is organized as follows. We start in chapter §1 by considering
a steady corona which is continuously heated by fast and slow MHD shocks (switch-on
shock trains and N-waves respectively). A brief review of useful results in MHD shock
theory is given and numerical solutions of the steady loop heating are presented. The
energy balance and behavior of physical quantities along the loop are analyzed.
In chapter §2 time-dependent models are considered. We construct a possible heating
scenario for nanoflares and carry out 1-D HD simulations of the heating process. Different heating functions are considered and we discuss the basic common features and main
differences among them. As a general result, a quasi-steady state is achieved rapidly in
which all dynamical effects (shocks) are damped. The obtained uniform corona is compared with the steady corona of chapter §1. We then obtain Alfvén wave heating model
results (following M04) and conclude the chapter with a comparison among models. The
Solar-B/XRT intensity fluxes are constructed for each case which allows to determine
possible unique observational features of each mechanism.
In chapter §3 we consider a third model, including both Alfvén wave and nanoflare
heating mechanisms. Adopting the same idea as in M04, 1.5-D MHD simulations are
carried out in which torsional Alfvén waves are generated at the footpoints of the loop.
These waves, by non-linear effects, convert into linearly polarized Alfvén waves that
steepen into shocks and heat the corona. Additionally nanoflares are input randomly in
time and space along the loop. As radial components are not considered, nanoflares do
not perturb the magnetic field and thus do not produce Alfvén waves. There is however
an interaction between these two heating mechanisms that is worth analyzing. The
result is a loop sharing features of both heating mechanisms and which agrees well with
observations. Finally, we end the work providing a brief summary and conclusions.
§1.1
6
1 Steady shock heating by fast MHD waves
and acoustic waves
Any heating mechanism, after its onset, has to be able to reach a quasi-steady state
that corresponds to the averagely observed corona (which is however very dynamic in a
small spatial scale range). We will start with the analysis of a steady corona heated by
shocks resulting from the steepening of slow (acoustic) and fast MHD waves (N-waves
and switch-on shock trains respectively).
1.1 MHD Waves in the solar atmosphere
1.1.1
Slow MHD mode and acoustic waves
In the solar corona the main acting forces are the gas pressure gradient and the Lorentz
force (magnetic pressure gradient and magnetic tension), as well as the gravitational force
near the Sun, in the photosphere or chromosphere. The interaction between these three
kinds of forces dictate the characteristics of waves (dispersive or non-dispersive, isotropic
or anisotropic, compressible or incompressible, etc.) which are known as MHD waves.
When we concentrate in a closed magnetic structure as a coronal loop, the MHD waves
can be either oscillating (standing) waves (kink mode, sausage mode) or propagating
waves. The generation place of these waves is very important, as, if generated in the
photosphere, they have to be able to propagate up to the corona and dissipate their
energy there in order to be accounted for coronal heating. Acoustic waves are known
to dissipate their energy very rapidly, being strongly damped in the chromosphere and
transition region due to the stratification of the atmosphere (Stein & Schwartz, 1972).
This is also the case for the slow MHD mode, which becomes purely longitudinal and
propagates as an acoustic wave at the sound speed when considering displacement along
a magnetic field line in a low-β plasma. In most of this work we will consider only this
kind of displacement, i.e. the one dimensional case of a magnetic flux in which quantities
vary only along its length. Hence, we will use the term acoustic wave indifferently for
both purely acoustic waves and slow mode MHD waves.
As acoustic waves propagate they inevitably steepen to form shocks thus dissipating
their energy. If these acoustic waves are formed at the photosphere they reach the
corona with only 0.01% of their original energy (Stein & Schwartz, 1972). This is the
reason why acoustic waves have not been regarded as a major coronal heating source,
unless they can be generated directly in the corona. This possibility has been considered
in the nanoflare scenario (cf. §2). Another possible scenario for coronal generation of
§1.1
7
acoustic waves has been proposed by Kudoh & Shibata (1999), in which torsional Alfvén
waves effectively transport the energy of random motions at the photosphere into the
corona, creating spicules in the open magnetic field region and, through nonlinear effects,
producing acoustic waves there.
The solar atmosphere is continuously regenerated with new magnetic flux that emerges
from the sub-photospheric regions after becoming magnetically buoyant due to many
magnetic instabilities, as for example Parker instability (Parker, 1969; Shibata et al.,
1989a,b). These new magnetic flux tubes have to pass through many different layers,
photosphere, chromosphere and transition region until reaching the corona. The continuous flux emergence implies a continuous reorganization of the magnetic field at different
levels of the atmosphere. The interaction between ’new’ and ’old’ magnetic flux stresses
the fields, and induces changes in topology towards more stable, lower energy configurations (i.e. potential field, Sakurai, 1979). These changes of topology are achieved
through the process of magnetic reconnection. A consequence of magnetic reconnection
between new and old flux tubes (even small events as nanoflares), is the oscillation of
these tubes around their equilibrium state (Sturrock, 1999; Roald, Sturrock & Wolfson,
2000; Sakai et al. 2000). This oscillation can excite at coronal heights various modes
of waves traveling through the surrounding plasma, which could dissipate and heat the
corona. The period of these oscillations would be τ = 2L/vA , where L is the length of
the newly emerged magnetic flux and vA , the Alfvén speed, is the speed at which the perturbations would propagate along the structure. Quoting the same values, L∼1.5×109
cm, a magnetic field B ∼ 100 G, and a mean plasma density ρ ∼ 10−12 g cm−3 , the
period of oscillation would be ∼ 100 s. As discussed in Sturrock (1999), longitudinal
MHD waves would be excited by these oscillations (being the tubes perpendicular to the
overlying magnetic field).
Acoustic waves in coronal loops have been observed, for example, in EUV images by
SOHO/EIT or TRACE, with periods around 100-200 s, and speeds in the range 75-150
km s−1 (DeForest & Gurman, 1998; De Moortel et al, 2002), corresponding to temperatures in the range 1−1.5×106 K. However, the energy associated with these propagating
waves was estimated to 3.5±1.2 × 102 erg cm−2 s−1 , far below the requirement for coronal heating.
1.1.2
Fast MHD mode and Alfvén waves
In the linear regime, when transverse (with respect to the magnetic field line) velocity
amplitude δv⊥ is much smaller than the Alfvén speed, vA =
√Bq ,
4πρ
the fast MHD mode
and the Alfvén mode degenerate and propagate along the field line as transverse waves
at same speed vA . When non-linear effects and higher dimensions are considered the
degeneracy is removed. Hence, in this work we will use the term fast MHD mode (or
§1.1
8
transverse waves) indifferently for both the fast MHD mode and the Alfvén mode unless
stated otherwise. Before treating these two waves as the same wave let’s remember a
couple of properties of each mode that will be useful later when analyzing the results.
Both the slow MHD mode and the fast MHD mode perturb the plasma velocity, in
contrast to the Alfvén mode which does not. These perturbations cause positive and
negative Doppler shifts that can be detected as line broadening. Nonthermal broadening
of UV and EUV coronal lines in the open-field corona have been measured with Skylab
and with SoHO/SUMER where nonthermal velocities of δv ≈ 20-30 km s−1 were reported
in coronal holes and quiet Sun regions (Doschek & Feldman, 1977; Spadaro, 1999).
These spectroscopic measurements seem to indicate a strong presence of fast mode or
Alfvén mode MHD waves in the open-field structures of the solar corona. However,
there have been very few observations so far of the fast MHD mode in closed magnetic
structures as coronal loops, and none seem to have been reported from SoHO/EIT or
TRACE, probably because of the necessary high cadence of seconds. Williams et al.
(2001, 2002) and Katsiyannis et al. (2003) reported the observational discovery of fast
MHD modes propagating in coronal loops with SECIS. The waves were observed to
have a quasi-periodic pattern with a mean period of about 6 s. Interpretation of these
observations in terms of fast wave trains was suggested. Fast MHD modes have been
theoretically predicted to form quasi-periodic wave trains that steepen into shocks (called
switch-on shock trains) as a consequence of Alfvén waves continuously produced from
the photosphere (Roberts, Edwin & Benz, 1983). A model of coronal heating by these
periodic trains of switch-on shocks was proposed by Hollweg (1982) for an open field
region, concluding that the model could become a plausible source of coronal heating.
For the coronal heating problem we need upward propagating waves that have been
generated by a chromospheric or sub-photospheric energy source (unless the waves are
generated and dissipate somehow locally in the corona). However, one problem is that
MHD waves (especially the fast MHD mode) tend to be totally internally reflected at
some height in the chromosphere and transition region, due to the high density gradients,
and thus are evanescent there (Hollweg, 1978; Heggland, 2003). The (shear) Alfvén
waves (Alfvén mode) however, as they do not modulate the plasma, do not suffer these
difficulties and a substantial fraction can easily propagate into the corona. Given the
observational evidence for MHD waves in coronal holes, we can estimate the energy flux
(Poynting flux) that is carried by the Alfvén waves,
2
6
FA = ρ < δv > vA = 10 erg
cm−2
s
−1
ρ
10−15 cm3
δv
30km/s
2 vA
1000km/s
,
which is sufficient to heat coronal holes, quiet Sun regions and a substantial fraction of
active region loops, if it can dissipate over an appropriate height range. However, (shear)
Alfvén waves are notoriously difficult to dissipate in a homogeneous media as the corona
because of the low coronal shear viscosity (Hollweg 1991). A possible scenario in which
§1.1
9
this viscosity is enhanced is considered by the resonant absorption mechanism (Poedts,
Kerner & Goossens, 1989). Another possible way by which Alfvén waves could dissipate
their energy is by mode conversion, that is, by converting non-linearly to another MHD
mode as the slow mode (Kudoh & Shibata, 1999) or the fast MHD mode, which can
dissipate. For further details about the Alfvén wave dissipation problem we refer the
interested reader to Nakariakov, Roberts & Murawski (1997) or Heggland (2003), and
references therein. In the present work we will concentrate only on shocks as an energy
dissipation mechanism.
1.1.3
Structure and purpose
Most of the analytical work concerning waves has been focused mainly on open magnetic
field regions (see for example Stein & Schwartz, 1972; Foukal & Smart, 1981; Hollweg,
1982; Suzuki, 2004). In this work we will concentrate on the heating of closed magnetic
field structures. In this first chapter we will follow an analytical procedure in order to
estimate the heating of a magnetic flux tube (coronal plasma loop) by fast MHD waves
(including Alfvén waves) and acoustic waves (including slow MHD waves). However,
considering that in our work the fast MHD mode and the Alfvén mode are degenerate,
we won’t be able to differentiate the energy transported by each of these waves. We will
first briefly summarize analytic results for the heating from the dissipation of acoustic
and fast mode MHD waves that steepen into shocks (N waves type of shock and switchon shocks respectively) when traveling along magnetic field lines. With these results we
will then (§1.2) construct a self-consistent model of a loop heated by these type of shocks.
The structure of the work in this first part follows roughly the steps taken in Suzuki
(2004) in which a similar analysis is carried out for the case of an open magnetic field
region, where not only the coronal heating problem but also the acceleration of the solar
wind is studied in a self-consistent manner. Suzuki (2002a, 2002b, 2004, 2005, 2006)
found positive results in which the coronal heating and acceleration of the solar wind
agrees well with observations, showing that these processes may be due to an interplay
between heating by acoustic and fast MHD waves. The heating by acoustic waves is
important in the inner corona, while the fast MHD waves heat and accelerate efficiently
the plasma in the outer corona.
§1.2
10
1.2 Shock heating theory
1.2.1
Shock formation distance
a) N-waves
After being generated somewhere in the solar atmosphere probably due to convective
motions, compressional MHD waves propagate upward steepening their wave fronts into
shocks (Suzuki, 2002b). Along a magnetic field line, acoustic waves of wavelength λ
and amplitude δvq,0 at the footpoints, starting with a sinusoidal velocity profile δvq =
δvq,0 sin(2πz/λ) (where z is the distance traveled upward), have a phase speed:
1
vp = cs + (γ + 1)δvq ,
2
(1.1)
where cs is the sound speed and γ = 5/3 is the ratio of specific heats. The shock is
formed when the peak catches up with the shock front (and with the trough). This
distance is estimated to be (Stein & Schwartz 1972)
1
λ cs
z − z0 = 2H ln 1 +
,
4(γ + 1) H δvq,0
(1.2)
where H is the density scale height (∼ 107 cm in the photosphere, to ∼ 109 cm in
the corona) and z0 is the height at which the waves are created. Taking values for the
boundary (at footpoint) amplitude δvq,0 /cs = 0.1 ∼ 1, we can see that the distance of
shock formation is of the order of a wavelength, which takes the values 108 ∼ 109 cm.
b) Switch-on shock trains
In the case of the fast MHD mode (linearly polarized Alfvén waves) propagating upwards,
the perturbation it produces in density is both longitudinal and transverse. Its phase
speed varies as a function of the fluctuating field, δB⊥ , normal to the background field,
Bq (Kulikovsky & Lyubimov, 1965),
q
vp =
2
Bq2 + δB⊥
√
4πρ
(1.3)
The speed by which the wave crest overtakes the wave front is then just the difference
√
between the phase speed and the Alfvén speed, vA = Bq / 4πρ0 (where ρ0 is the ambient
density at the wave fronts). The distance of shock formation is calculated by integrating
over the lapse of time that the wave takes in traveling a distance of λ⊥ /4, and remembering the requirement of the plasma compressibility, satisfied by the above equation
(Montgomery, 1959):
q
2
Bq2 + δB⊥
ρ
' const.
(1.4)
§1.2
11
Nonlinear effects can be quite large for waves propagating from the photosphere up to the
corona. Numerical simulations of torsional Alfvén waves generated in the photosphere
and propagating into the corona nonlinearly convert to the slow and fast MHD modes
which subsequently steep into shocks (Moriyasu, 2004). The fluctuating field δB⊥ at the
footpoints is shown to be sometimes as large as the longitudinal field ∼ Bq . Therefore,
supposing the waves are generated in the photosphere or chromosphere, taking appropriate values, Bq ∼ 100 − 1000 G, δB⊥ ∼ 0.7Bq G, the distance results ∼ 5 × 109 cm, for
transversal waves with period of 100 s. This is illustrated in Moriyasu (2004), Kudoh &
Shibata (1999) and Heggland (2002), where transverse waves created at the photosphere
propagate into the corona as fast shocks.
1.2.2
Variation of the shock amplitude
a) N-waves
We can see that in a distance of the order of a few ten thousand kilometers (much less
however for the acoustic waves) both types of waves have already steepen into shocks. As
shocks, acoustic (and slow mode) waves travel as N-waves, and fast mode waves travel as
switch-on shock trains (see Fig. 1 in Suzuki, 2002a or 2004). The later are called switchon shock trains due to the fact that it ’switches’ on to a non-zero value the transverse
component δB⊥ (with respect to the propagation direction) of a magnetic field line as
it propagates. The main difference in shock geometry between N-waves and switch-on
shock trains is that while N-waves form one shock per wavelength (characteristic sawtooth shape) switch-on shock trains can form two, due to the fast propagation speed
of the crest and the trough. The amplitude of the shock, αq ≡ δvq /cs for the N-waves
and α⊥ ≡ δv⊥ /vA for the switch-on shock trains, is directly related to the dissipation of
the wave, and hence to the heating. Analytical calculations relating this quantity to the
entropy change through the shock can be found in Stein & Schwartz (1972) in the case
of the N-waves, and in Priest (1982) in the case of the switch-on shock trains. Assuming
that the shock amplitude is small in both cases (weak-shock approximation, i.e., αq 1
and α⊥ 1), the entropy generation at each shock front is given by the jump condition
across the shock as
∆sq
γ+1
=
R
12γ 2
∆p
p
3
=
2γ(γ + 1) 3
αq ,
3
(1.5)
where R is the gas constant, ∆p denotes the pressure difference at the front and p is
the pressure in the upstream region (unshocked region). Using the perfect gas law,
T = p/(ρR), in (1.5) the energy loss rate Qq per volume (erg cm−3 s−1 ) at the shocks
for waves with period τq results in
Qq =
ρ0 T ∆sq
2γ(γ + 1)p0 3
=
αq ,
τq
3τq
(1.6)
§1.2
12
where p0 is the ambient pressure (average of upstream and downstream pressures). For
the case of a sawtooth-shape shock, the wave energy per wavelength is written as Eλ,q =
1
2
3 ρ(δvq ) λq
= 13 γpαq2 λq , and its variation can be estimated as
∇ · Eλ,q =
dEλ,q Eλ,q dA
+
= −Qq τq ,
ds
A ds
(1.7)
where s denotes distance along the magnetic field and A(s) denotes the cross-section area
of the magnetic flux tube. Taking the logarithmic derivative of Eλ,q and substituting
in (1.7), we get the spatial variation of the shock amplitude in static media along a
magnetic flux tube:
αq
dαq
=
ds
2
1 dρ 2(γ + 1)
1 dA
3 dT
−
−
αq −
−
ρ ds
cs τq
A ds
2T ds
.
(1.8)
The first term on the right hand side of (1.8) is positive so that the more decreasing
the density in the solar atmosphere and the stronger the shock is. The second term
is the entropy generation (heating) term and is negative. These first two terms always
dominate the other two terms, namely, geometrical expansion and temperature variation
in the (1-2)R region (Suzuki, 2004). Before discussing the importance of (1.8), let’s
follow the same procedure and obtain the shock amplitude variation equation for the
switch-on shock trains.
b) Switch-on shock trains
In the case of the switch-on shock trains, the entropy generation across the shock front
in the case of a weak shock (σ − 1 1), can be expressed as (Boyd & Sanderson, 1969)
B2
∆s⊥
' q (σ − 1)2
R
8πp
(1.9)
(Hollweg, 1982), where p is the pressure in the upstream region and σ is the ratio of the
downstream to the upstream densities. The transversal field component δB⊥ generated
by the passage of the shock can be calculated as (Priest, 1982)
c2s
2
2
δB⊥ ' 2(σ − 1)Bq 1 − 2 ,
vA
(1.10)
where cs and vA denote respectively the sound speed and the Aflvén speed in the upstream region. Before continuing it is important to note that eq.(1.10) implies that
vA > cs , so that a switch-on shock can only exist when the Alfvén speed exceeds the
sound speed in the upstream region. That is, switch-on shocks can only exist for low-β
plasmas. This imposes a limit on the model, namely, that it is only valid for regions
from the chromosphere up into the corona, but not valid for the photosphere where we
§1.2
13
have high-β plasma1 . Eliminating (σ − 1) from equations (1.9) and (1.10), we arrive to
the following expression for the generation of entropy,
∆s⊥
'
R
Bq2
32p 1 −
4
c2s
2
vA
2 α⊥ ,
(1.11)
√
where the relation δv⊥ = δB⊥ / 4πρ is used (for the downstream region). Now, recalling
that two shock fronts are formed for each wavelength, the energy loss rate Q⊥ per volume
(erg cm−3 s−1 ) for waves with period τ⊥ becomes
Q⊥ =
Bq2
2ρ0 T ∆s⊥
=
τ⊥
16πτ⊥
4
α⊥
1−
c2s
2
vA
2 .
(1.12)
It is important to note the difference in heating from the N-waves (1.6) and the switch-on
shock trains (1.12). The heating from the N-waves presents a cubic dependence on the
shock amplitude (∝ αq3 ), while in the case of the switch-on shock trains, the dependence
4 ). As we are considering weak shocks only, this means that
is on the fourth power (∝ α⊥
the damping of the waves due to the switch-on shocks is weaker than the damping due
to the N-waves so that they can carry more energy into the corona. If dissipation of the
waves occur, then they can contribute more to the heating of the corona. In the case
of the Alfvén waves, the wave energy per wavelength Eλ,⊥ is not conserved in moving
media. Instead, the wave action constant turns out to be conserved (Jacques, 1977)
Sw,⊥ ≡
2 v (v + v )(v + v )
ρα⊥
A
A
A
,
ς⊥
(1.13)
where ς⊥ depends on the shape constant of the waves. In the case of the switch-on shock
trains considered (saw-tooth shape, see Fig. 1 in Suzuki, 2002a or 2004), ς⊥ = 3. The
variation of the wave action at the shock fronts (which also describes the variation of
the wave energy flux) results
vA
∇ · Sw,⊥ = −Q⊥
vA + v
(1.14)
Using eq.(1.12), eq.(1.14) results
1 d
vA + v Bq2
(ASw,⊥ ) = −
A ds
vA 16πτ⊥
4
α⊥
1−
c2s
2
vA
2 .
(1.15)
Taking the logarithmic derivative of (1.13), using eq.(1.15) and the conservation of magnetic flux Bq A =const., we get the equation describing the variation of the shock ampli1
However, inside a flux tube, it is possible to have low-β plasma further down into the chromosphere.
The reason is due to the fact that inside a flux tube magnetic and gas pressures balance (approximately)
the external gas pressure, so that the inside gas pressure is lower than the external high gas pressure.
§1.2
14
tude in steady flow for the case of the switch-on shock trains:

dα⊥
α⊥  vA + 3v 1 dρ
=
−

ds
2
2(vA + v) ρ ds
2
3α⊥
4τ⊥ (vA + v) 1 −
2 +
2
cs
2
vA

2 dA 
.
A ds
(1.16)
c) Discussion
In the model we consider, flow is not important as it is everywhere subsonic (even in the
corona). The coefficient of the first term on the right hand side of (1.16) can then be
approximated to 1/2. Comparing with the case of the N-waves, eq.(1.8), we see that the
amplitude of the shocks differ very much from the case of the N-waves to the case of the
switch-on shock trains. In this case, the dependence on the density change is opposite
than the case for the N-waves. The decrease in density decreases the amplitude of the
switch-on shock trains. This is due to the fact that as density decreases the Alfvén speed
increases which would increase also the wave action constant. However, this quantity is
always decreasing as eq.(1.14) reflects. Hence the strength of the shock has to decrease
too (eq. 1.13). As noted before, the energy loss of the switch-on shock trains is less
than the energy loss for the N-waves. This is reflected also in the second term of (1.16)
3,
which shows that the variation of the amplitude of the shock is proportional to α⊥
one order higher than in the case of the N-waves (∝ αq2 in eq. 1.8). Consequently, the
dissipation at the switch-on shocks is weaker than that at the hydrodynamical shocks
of the N-waves. As stated previously, the switch-on shocks can potentially heat more
efficiently the corona than the N-waves because they caarry more energy. Conversely,
the N-waves seem more efficient for chromospheric heating than the switch-on shock
trains. This result is also important for solar wind acceleration (Suzuki, 2004).
In order to see the importance of (1.8) and (1.16), let’s suppose we have a 1010 cm long
coronal loop being heated by N-waves and switch-on shock trains continuously and let’s
take the typical wavelength of acoustic waves and Alfvén waves in the corona to be 109
cm and 3 × 109 cm respectively. This implies that we have in average 10 acoustic shocks
and 6 fast MHD shocks heating the loop. The average of these shock amplitudes will
be described by αq and α⊥ respectively for acoustic and fast shocks, and the variation
across the loop of this average amplitudes will be described by eq.(1.8) and (1.16). As
these equations are static in time, they apply physically to the case of a loop heated by
N-waves and switch-on shock trains that has reached a steady state. It is interesting to
compare this steady state with the quasi-steady state of a (time-dependent) simulation
in which acoustic and fast shocks are produced in a coronal loop (cf. ch.§2 and §3).
c) Limitations of the model
This steady coronal heating model is based on several assumptions that we now discuss.
§1.2
15
In the case of the N-waves, the propagation is only valid for a static media. This
assumption can be justified by the fact that N-waves with τq ∼ 100 s dissipate most of
their energy very rapidly (in a distance on the order of ∼ 103 km), where flow speeds
are very low. We will see also (cf. §1.4) that in all the calculations the flow velocities
are very low everywhere (subsonic) so that the static limit is also valid for the switch-on
shock trains (v vA , this is not valid however when modeling open field regions).
The second limitation is that we do not take into account the effect of gravity, that is,
we suppose that the acoustic cut-off frequency is well below the frequency of the waves
considered. As discussed in Stein & Schwartz (1972), weak-shock theory gives reasonable
estimates for waves having a period τ < π/ωac where ωac = γg/(2cs ) is the acoustic cutoff frequency in a gravitationally stratified atmosphere. Replacing by typical values in
the high chromosphere and corona we see that this assumption is valid for waves with
periods τ . 2000 s.
The third assumption is the weak-shock approximation, that is, αq 1 and α⊥ 1.
This assumption can be justified as well considering that in all our calculations αq and
α⊥ are smaller than 0.5 (the boundary value of αq is taken as 0.6, but this quantity shows
always a steep decrease, cf. §1.4). We have to remember however that in the calculation
for shock formation we considered shocks with strength α⊥ = δB⊥ /Bq > 0.5 in order
for shocks to form in the loop. These two arguments can be reconciled supposing that
strong shocks are formed mostly near the footpoints (in the transition region) where the
Alfvén speed is high, and quickly become weak shocks in the corona. This implies that
our model is only valid starting from the high chromosphere2 .
The WKB approximation needs also to be satisfied (this is needed for deriving the wave
action conservation). This means that the wavelengths of the waves considered need to
be smaller than the variation length-scale of the amplitudes of the waves. The Alfvén
speed scale height (HA ∼ 2 × 105 km in the low corona) then needs to be larger than the
wavelength of the Alfvén wave (with vA ∼ 103 km s−1 , the threshold would be τ⊥ ' 200
s). In our calculations the Alfvén speed has a maximum velocity of ∼ 1300 km s−1 , so
that the WKB approximation holds for waves with τ⊥ . 150 s.
A severe assumption is that we assume the transverse waves (fast mode) only to dissipate
through the switch-on shocks. However there are many ways in which waves could
dissipate, as we discussed at the beginning of this chapter. Taking into account transverse
variation of the field strength, other dissipation processes have to be considered, as
phase mixing (Heyvaerts & Priest,1983), or mode conversion (Nakariakov, Roberts &
Murawski, 1997). In the presence of turbulence, turbulent cascade might occur (Hollweg,
1986; Hu, Habbal & Li, 1999). However, in this work only dissipation through shocks is
2
If the waves are supposed to have periods much less than 100 s, on the order of 10 s, then the
weak-shock approximation gains validity as shocks are formed in a distance around 108 − 109 km.
16
Height above Photosphere @´103 kmD
§1.3
35
30
25
20
15
10
5
-30
-20
-10
10
20
3
Projected length @´10 kmD
30
Figure 1.1: Geometry of the coronal loop of total length 105 km and flux expansion of
∼ 12 (base-to-apex ratio of area cross-section). The radius of the loop at the footpoint
is 1000 km.
considered3 .
1.3 Model for the corona
1.3.1
Loop geometry and equations
In order to build a self-consistent model, we consider the basic MHD equations which,
together with (1.8) and (1.16), form a closed system of equations. Here, we consider
only variation along the magnetic field of a flux tube, neglecting toroidal components
and changes in the radial direction of the flux tube. The constructed model is then
purely one dimensional but we consider the flux tube expansion A(s), the cross-section
area of the flux tube, s denoting length along the loop of total length L (Fig. 1.1). Here
we model only the high portion of the loop, starting from the high chromosphere up
to the corona. The expansion of the loop is taken accordingly to this situation, with
a base-to-apex area ratio of ∼ 12, as seen in figure 1.1. More precisely, the equations
of cross-section area variation are given in the appendix (eq. 2.1-2.6). The basic MHD
equations for this model (one dimensional, time-independent) are the following:
3
Viscosity is considered implicitly through shocks. Compressive viscosity is not considered and could
be important in the corona (Ofman et al., 2000) contributing to the dissipation of the acoustic waves.
However, the dissipated energy is lost mostly through downward conduction
§1.3
17
The equation of mass conservation:
ρvA = const.,
(1.17)
where mass density ρ = mn, n being the total particle density (cm−3 ) and m the average
particle mass (g). For a fully ionized gas such as the corona, n/2 ' ne ' ni , where ne
and ni are the electron density and the ion density respectively. The average mass is
approximately 2.12 × 10−24 g. The electron and ion components having equal outflow
velocity v, the equation of momentum conservation becomes:
v
1 dp 1 dpw,q 1 dpw,⊥
dv
= −gef f −
−
−
,
ds
ρ ds ρ ds
ρ ds
(1.18)
where gef f is the effective gravity term along the loop,
π
GM cos( L
s)
gef f =
R+
L
π
2 ,
π
sin( L
s)
(1.19)
(R = R +h, where h is the height above the photosphere where the boundary conditions
are set) pw,q is the wave pressure of the N-waves, from which the wave pressure gradient
can be written
dpw,q
1
1 d
=
(AFq ),
γ+1
ds
cs (1 + 2 αq ) A ds
(1.20)
with Fq = 13 ραq2 c3s (1 + γ+1
2 αq ), the wave energy flux (magnitude) along the loop from the
N-waves. pw,⊥ is the wave pressure of the switch-on shock trains, which is essentially
identical to that of the usual Alfvén waves. The wave pressure gradient results (Jacques,
1977)
dpw,⊥
d
=
ds
ds
2
δB⊥
8πς⊥
=
d
ds
2 B2
α⊥
q
8πς⊥
.
(1.21)
Expanding the momentum equation (1.18), it can be written as
gef f
dρ
ρ
c2s 1 αq2 1 dT
c2s 4 + 3(γ + 1)αq dαq
=
+
+
+
α
+
q 2
2
2
α2
ds
v2 γ
2 T ds
v
6 + 3(γ + 1)αq ds
1 − vcs2 ( γ1 + 3q ) v
2
2
vA
α⊥ vA
dα⊥
c2s
1 dA
2
2
+
− 1 − 2 αq − 2 α⊥
. (1.22)
3 v 2 ds
3v
cs
A ds
The energy equation is
nγ d
γ − 1 ds
kB T
nγ−1
= Qq + Q⊥ − ∇ · Fc − R,
(1.23)
where Qq and Q⊥ are the heating terms due to the N-waves (1.6) and switch-on shock
trains (1.12) respectively. Fc = −κ0 T 5/2 dT /ds is the thermal conduction flux, with
κ0 = 9 × 10−7 erg s−1 K−1 cm−1 the Spitzer conductivity, and
R=
n2
φ(T )
4
(1.24)
§1.3
18
T (K)
4×
104
1×
105
3×
105
γ
χ
Reference
8.00 ×
10−37
Sterling et al., 1993
8.00 ×
10−22
Sterling et al., 1993
10−8
Hildner, 1974
Hildner, 1974
<T <1×
105
<T <3×
105
<T <8×
105
-2.5
3.94 ×
8 × 105 < T < 2 × 107
-1.0
5.51 × 10−17
0.4
10−27
T >2×
107
3.0
0.0
3.31 ×
Nagai, 1980
Table 1.1: Parameter values of the radiative loss function φ(T ) = χT γ (erg cm3 s−1 ) for
different temperature domains.
is the volumetric energy loss due to radiation. φ(T ) is the radiative loss function for
optically thin plasmas (Landini & Monsignori-Fossi, 1990) (see table 1.1). For temperatures below 4 × 104 K the plasma becomes optically thick. In this case, the radiative
loss R can be approximated by (Sterling et al., 1993; Anderson & Athay, 1989),
R = 4.9 × 109 ρ
(1.25)
Expanding the energy equation, we get
d2 T
ds2
=−
51
2T
dT
ds
2

2 ρα4
vA
1 dA dT
1  ρ2
⊥
+
φ(T
)
−
−
2
A ds ds
κ0 T 5/2 4m2
4τ⊥ (1 − vc2s )2
A
2
3
2
2c ρα (γ + 1)
dρ
2cs ρv dT
− 2c2s v
.
− s q
+
3τq
(γ − 1)T ds
ds
−
(1.26)
Equations (1.8), (1.16), (1.17), (1.22) and (1.26) form a closed system of equations
together with the ideal gas law,
kB
T,
m
(where kB is the Boltzmann constant), and the conservation of magnetic flux
p=ρ
∇ · B = 0 ⇔ Bq A = const.,
(1.27)
(1.28)
The energy flux in static media is used for the slow magneto-acoustic waves (these waves
dissipate quickly, as discussed previously),
1
γ+1
2 2
αq .
Fq ' ρ(αq ) cs cs 1 +
3
2
(1.29)
For the Alfvén waves (and fast magneto-acoustic waves), the wave energy flux can be
written
1
3
2 2
F⊥ ' ρ(α⊥ ) vA vA + v ,
3
2
which can also be well approximated to the case of static media (v vA ).
(1.30)
19
ΦHTL@erg cm3 s-1D
§1.3
10-22
10-23
105
107
106
Temperature @KD
108
Figure 1.2: Radiative loss function φ(T ) = χT γ , given by table (1.1).
1.3.2
Boundary values and numerical model
The system of equations (1.8), (1.16), (1.17), (1.22), (1.26), (1.27) and (1.28) have
as unknowns αq , α⊥ , ρ, p, T , v and B (together with A, given by eq. 2.1 to 2.6
which represent two differential equations). The model of the loop starts in the high
chromosphere, 2000 km above the photosphere (h = 2000 in eq. 1.19), and ends in the
second footpoint. The loop at first glance might seem symmetrical but is not totally so
due to the asymmetrical assumption that waves are supposed to be generated at only
one footpoint (left footpoint). This creates a slight asymmetry in the heating reflected
in the equations with the form of the heating terms of the waves4 . We take accordingly
the following boundary conditions: a chromospheric magnetic field of Bq,0 = 28 G,
which decreases to ∼ 3 G in the corona; a boundary flow speed of v0 = 1 km s−1
(consistent with observations of Teriaca et al., 1999); we take at the boundary a radius
of the loop of 1000 km (which is close to 5000 km in the corona); the gradient of the
temperature is set to 0 at the boundary, corresponding to the characteristic temperature
plateau in the chromosphere. As a typical chromospherical temperature we take 15000
K. The boundary amplitude of the N-waves is set to αq,0 = 0.6. We do not change
this parameter, as it is found that its variation affects very little the heating of the
corona. However, there is a big dependence of the state of the system on the boundary
4
In order to create a perfectly symmetric loop we would have to consider wave propagation and flow
also from the right footpoint. However, in such a case we would have to assume that the transversal
waves do not interact at the apex of the loop for the sake of consistency with the model. A weak
interaction would imply α⊥ ∼ 0.1 at the apex, which according to our results is too marginal, cf.
Fig.1.3
§1.4
20
amplitude of the switch-on shock trains, α⊥,0 . We will then show the results of a few
cases varying this parameter in the range 0.09 − 0.17 and its influence on the coronal
loop physical conditions. The boundary density is set subject to the condition that at
the second footpoint (right) the physical conditions are (approximately) the same as at
the left footpoint. In this model, for the sake of simplicity and stability (of the numerical
calculations), we suppose that at the chromospheric height at which we set the footpoint
of the loop (2000 km), waves have already steepened into shocks and are dissipating their
energy. As discussed in §1.2.1, this assumption is marginally correct.
The system of equations was solved using Mathematica 5.0r and solutions were checked
with the Rosenbrock method (Press et al., 1992) which adopts implicit spatial grids to
keep numerical stability and allows a correct treatment of stiff systems.
1.4 Results
In this section we show the results of the heating of the loop from shock dissipation of
the acoustic waves (N-waves) and fast MHD waves (switch-on shock trains).
Each value of α⊥,0 defines a case. In table 1.2 several values of interest showing properties
of the resulting corona are presented for each case. The value denoted by hT R in the
table corresponds to the approximate height of the transition region and is calculated
supposing an exponential decrease in density from the photosphere. Namely, ρ0 =
ρph exp(−hT R /H), where ρP h is a typical value of density at the photosphere (we take
2.5 × 10−7 g cm−3 ), H is the density scale height at the photosphere (H ∼ 200 km) and
ρ0 is the value of the density where our calculation starts.
Figure 1.3 (a) shows the variation of the shock amplitudes of the N-waves and switch-on
shock trains for different boundary values of α⊥,0 . The first aspect to be noticed is the
quite different behavior between the amplitudes of the two kinds. From one part, αq is
an always decreasing quantity which after the first few thousand kilometers is just ∼ 1%
of its original value (as expected from Schwartz & Stein, 1972), reflecting pretty much
the trend in the density profile. Also, we can see that it varies very little with respect
to α⊥,0 , due to the relatively low energy flux carried by the acoustic waves compared to
the transversal waves (this shows at some extent that the interaction between N-waves
and switch-on shock trains is small). On the other hand, the switch-on shock trains
amplitude exhibits for all cases a gradual increase along the loop (except in the first few
hundred kilometers where the decrease in density is the most weighting term in eq. 1.16)
due to the less strong dissipation at the shocks and the positive influence from the flux
expansion. Actually, the flux expansion term turns out to be very important, since, as
seen in eq.(1.16), it is the only positive term accounting for the growth of the amplitude.
The behavior of the switch-on shock amplitude has great influence in the physical quantities considered. If we take a look to (d) and (f) in Fig. 1.3 and at the values tabulated
§1.4
21
input
ouput
α⊥,0
0.09
0.12
0.15
0.17
F⊥,0
(×107 )
1.00
1.22
1.43
1.59
Fq,0
(×104 )
0.75
1.60
2.84
3.82
Tmax
(×106 )
1.27
1.68
2.05
2.23
ρ0 (×10−14 )
1.77
3.78
6.71
9.0
hT R
3295
3143
3028
2970
vA,0
593
406
305
263
vA,L/2
358
260
203
177
cs,L/2
115
132
146
153
vL/2
4.39
4.94
5.38
5.45
Table 1.2: The main input parameter is the switch-on shock amplitude α⊥,0 , from which
values of the other parameters at the footpoint (left footpoint, denoted by sub-index
0) and loop apex (denoted by the half-length L/2) are obtained. Wave energy fluxes
Fq,0 and F⊥,0 are in erg cm−2 s−1 . Temperature and density ρ0 are in K and g cm−3
respectively. As the heating is not symmetric the maximum temperature Tmax does not
occur at the apex. The quantity hT R (in km) denotes the height of the transition region
and is calculated assuming an exponentially decreasing density at the footpoint (refer
to §1.4 for details). The Alfvén speed of the switch-on shocks vA , the sound speed cs of
the N-waves type of shock and the flow speed v are all in km s−1 .
§1.4
22
Α!# ∆v#vA
1
vA , cs , v !km s"1"
Α$# ∆v#cs
1000
500
0.5
0.2
100
50
0.1
0.05
10
5
0.02
20
40
60
80
Length along the loop !!103 km"
20
40
60
80
Length along the loop !!103 km"
(a)
(b)
F!!erg cm"2 s"1"
107
T !K"
2!106
106
1.5!106
105
106
104
5!105
103
105
20
40
60
80
Length along the loop !!103 km"
(c)
Qt !erg cm
10
"3
(d)
Ρ !g cm"3"
s "
"1
10"13
"4
10"14
10"5
10"6
10
20
40
60
80
Length along the loop !!103 km"
10"15
"7
10"16
20
40
60
80
Length along the loop !!103 km"
(e)
20
40
60
80
Length along the loop !!103 km"
(f )
Figure 1.3: Variation of physical quantities along a magnetic flux tube heated by slow
(acoustic) and fast MHD shocks (time-independent model). Blue and red color correspond to the N-waves (αq ) and switch-on shock trains (α⊥ ) respectively. The panels are:
(a) variation of N-waves and switch-on shock trains amplitudes, (b) Alfvén (orange),
sound (pink) and flow (purple) speeds, (c) wave energy fluxes (eq. 1.29 and 1.30), (d)
temperature, (e) heating from the dissipation of the N-waves (eq. 1.6) and switch-on
shock trains (eq. 1.12) together with radiation losses (green), and (f) density. Different
cases are plotted in different line styles. The case with α⊥,0 = (0.09, 0.12, 0.15, 0.17)
corresponds to the (short-long-short dashed, dotted-dashed-dotted, dashed, solid) line
respectively.
§1.4
23
in table 1.2, we see that the maximum temperature (apex temperature) almost doubles,
reaching 2.2 × 106 K, and the coronal density increases more than 5 times (passing from
∼ 4 × 10−16 to ∼ 2 × 10−15 ) when changing α⊥,0 from 0.09 to 0.17. This strong influence can be understood by considering the values of wave energy flux input from the
switch-on shock trains (cf. table 1.2). As it is shown, the energy input increases ∼ 60%
from its first value of ∼ 1 × 107 erg cm−2 s−1 . As discussed in the Introduction, in order
to balance energy losses due to downward thermal conduction and radiation, the energy
input should be superior to ∼ 105.5 erg cm−2 s−1 . We see that the wave energy flux
from the fast MHS waves easily satisfies the energy requirement in this model. However, the wave energy flux from the acoustic waves is below this threshold (∼ 104 erg
cm−2 s−1 , cf. table 1.2). This can be confirmed by Fig. 1.3e, which shows the energy
balance (volumetric heating) along the loop. We can see that in the first few thousand
kilometers there is an efficient release of energy through radiation. The heating from
the acoustic waves decreases along the loop in a similar way as radiative cooling, being
always approximately 10-20 times smaller than the radiation losses. On the other hand,
the heating from the Alfvén waves behaves very differently, decreasing in the first few
hundred kilometers (not shown in the figure) but then increasing considerably in the rest
of the loop. At a height of approximately 3000 km and a value of ∼ 2 × 10−5 erg cm−3
s−1 the heating caused by the switch-on shock trains gets superior to the radiative losses
and continues increasing reaching a maximum in the corona after propagating ∼ 30000
km. The maximum in the most energetic case (α⊥ = 0.17) is ∼ 5 × 10−4 erg cm−3
s−1 . The behavior of the heating term Q⊥ is actually very similar to α⊥ , exhibiting
also an increase in the same portion of the loop and a maximum (bump) at the same
location (Fig. 1.3a). The location of the heating maximum corresponds well with the
sharp decrease in wave energy flux from the Alfvén waves as seen in Fig. 1.3c. This
model exhibits a corona heated efficiently by switch-on shocks which dissipate most of
their energy in the corona. The resulting corona cannot be maintained with only heating from acoustic waves. The energy carried by these waves is relatively small due to
the low boundary density. In a model treating adequately the chromosphere (here the
temperature plateau is almost absent) the density would be higher implying a higher
energy for acoustic waves. We can however guess that acoustic wave heating plays an
important role in the chromosphere and transition region, where 99% of the energy is
deposited.
The dependence on α⊥,0 is clearly shown in Fig. 1.4, where the coronal values of temperature (a) (left) footpoint densities (b), the boundary values of the wave energy fluxes (c),
the apex Alfvén, sound and flow velocities (d) and the height of the transition region are
plotted with respect to the footpoint switch-on shock amplitude value. An important
fact to notice in (d) (that can also be noticed in Fig. 1.3b), is the decrease in Alfvén speed
the higher the magnitude of α⊥ gets, which contrasts with the corresponding increase in
§1.4
24
(a)
(b)
(c)
(d)
(e)
Figure 1.4: Dependence of (a) maximum temperature, (b) footpoint density, (c) footpoint wave energy flux, (d) apex velocities and (e) height of transition region on the
footpoint amplitude of the switch-on shocks α⊥,0 . In (c), red and blue colors denote
wave energy flux from the acoustic and fast MHD waves respectively (which are normalized by 107 and 104 respectively). In (d) orange (solid line), pink (dashed line) and
purple (dotted-dashed line) colors correspond respectively to Alfvén speed, sound speed
and flow speed.
§1.4
25
sound speed. This is explained by the fact that as the amplitude of the switch-on shocks
increase, so does the momentum carried by these waves. This momentum input in the
coronal plasma enhances the density there (Fig. 1.4b) thus decreasing the Alfvén speed.
The amplitude increase enhances the dissipation of the waves, mainly the dissipation
of the fast MHD waves in the corona, as seen in Fig. 1.3e, thus increasing the heating
in the corona. This produces an enhancement of downward thermal conduction which
carries the excess of heat down to the chromosphere, where it can be more efficiently
radiated away. The density at the footpoint consequently increases (this can be seen in
the increase of radiation losses at the footpoint in Fig. 1.3e) producing the phenomenon
of chromospheric evaporation, thus increasing the coronal density and decreasing the
Alfvén speed. A slow flow of plasma of ∼ 5 km s−1 into the corona and a decrease in
height of the transition region (Fig. 1.4e) can be seen in Fig. 1.3b, as a consequence
of a rather mild chromospheric evaporation (not explosive as in solar flares or as in the
spicule model of Kudoh & Shibata, 1999).
From Fig. 1.3b and 1.4d we see then that as the wave energy input increases the Alfvén
speed decreases and the sound speed increases in the coronal part of the loop, so that
the speeds become very similar at the apex. Hence, the value of β(= 8πp/Bq2 , the
ratio between gas and magnetic pressures) reaches values close to unity at the apex.
As discussed in §1.2.2b, this model is valid only in low-β plasmas, as switch-on shocks
can only exist in such regions. Hence, the value of α⊥,0 cannot be increased any further
without changing other boundary values such as the magnetic field at the footpoints, the
magnetic flux expansion and the boundary value of the temperature. A change in these
parameters should be accompanied with a lower height of the footpoints, starting from
the chromosphere, in order for the weak-shock approximation to remain valid. Such a
model is not considered in the present work.
Before continuing to the following chapter we would like to add a comment stressing the
sensibility of the coronal conditions to the boundary amplitude of the switch-on shock
trains. Such a big sensibility reflects the fact that chromosphere and corona are linked
very strongly and should always be modeled together. In this case the model does not
reproduce well the chromospheric temperature plateau which makes the system even
more sensible to input values, and which could lead to non-physical solutions. This
might be the main difficulty of time independent calculations, namely, that the modeler
has to know very well before hand the possible range of solutions of the system (trial and
error strategies are only advised for patient people). Let’s now pass to time-dependent
calculations, i.e., simulations, of a corona heated by nanoflares and Alfvén waves.
§2.1
26
2 Nanoflare heating simulations
2.1 Magnetic reconnection in the solar atmosphere
The solar atmosphere is continuously regenerated with new magnetic flux that emerges
from the sub-photospheric regions after becoming magnetically buoyant due to many
magnetic instabilities, as for example Parker instability (Parker, 1969; Shibata et al.,
1989a,b). These new magnetic flux tubes have to pass through many different layers,
photosphere, chromosphere and transition region until reaching the corona. The continuous flux emergence implies a continuous reorganization of the magnetic field at different
levels of the atmosphere. The interaction between ’new’ and ’old’ magnetic flux stresses
the fields, and induces changes in topology towards more stable, lower energy configurations (i.e. potential field, Sakurai, 1979). These changes of topology are achieved
through the process of magnetic reconnection. In layers where the magnetic field gradient is large, resistivity can become important and be anomalously large. Such layers are
called current sheets, and it’s in these regions where cutting and pasting of field lines
can take place, thus changing locally and globally the magnetic field topology (Parker,
1963; Sweet, 1958; Petschek, 1964). It is now well established that solar flares, the most
energetic phenomena in the solar atmosphere, has their roots in magnetic reconnection.
Parker (1988) proposed the idea of ubiquitous magnetic reconnection events leading to
much smaller energetic phenomena and heating the solar corona. The energy released
in these events estimated to be around nine orders of magnitude less than the solar flare
energy range (1030 -1033 erg), defines them as nanoflares (1024 -1027 erg).
Because of the relatively faint energy release associated with nanoflares, they are difficult
to be observed. Measurements in EUV (with SoHO/EIT and TRACE) have revealed
nonetheless the existence of such events (Krucker & Benz, 1998; Parnell & Jupp, 2000;
Aschwanden & Parnell, 2002; Shimizu, 2005). The observed nanoflares share many
properties of the energetic solar flares, for example, they exhibit an impulsive rise and
decay (on time scales of a few minutes) consistent with plasma heating and subsequent
cooling. A fundamental difference between them, however, is that solar flares only occur
in active regions, where magnetic fields are strong, while nanoflares can occur everywhere
in the quiet Sun or even in coronal holes. The observed occurrence frequency as a
function of energy of nanoflares has shown a power-law distribution with an index that
varies between -1.5 (Aschwanden et al. 2000b) and -2.5 (Parnell & Jupp, 2000). This
index is an important parameter for coronal heating, as an energy budget calculation
shows that in order for the nanoflares to be efficient heating mechanisms, the powerlaw index of their distribution has to be steeper than -2 (Hudson, 1991). Observations,
§2.1
27
then, have not yet provided conclusive arguments about the importance of nanoflares in
coronal heating.
It is estimated that 95% of the photospheric magnetic flux closes within the transition
region, while only 5% forms large-scale coronal loops (fact known as the ’magnetic
carpet’, Priest et al., 2002). Hence, the biggest concentration of magnetic flux lies
in the canopy geometry of the transition region. This flux concentration is ideal for
magnetic dissipation events which lead to the observed EUV nanoflares. The location
of nanoflares would then mainly be confined to the lowest layer of the corona, just
above the transition region. There have been however many theoretical models which
propose nanoflares throughout the corona (Taroyan, Bradshaw & Doyle, 2005, hereafter
T05; Parker, 1988; Levine, 1974). In this chapter we will concentrate on models where
nanoflares are produced randomly throughout the corona (the comparison with a model
in which nanoflares are concentrated in the canopy region is left as future work).
In the first chapter we studied a time-independent model of a magnetic flux tube heated
by shock dissipation of fast (switch-on shock trains) and slow (N-waves) MHD waves. As
discussed, such a case could be applied in the solar corona to a magnetic flux tube that is
constantly heated by acoustic and Alfvén waves generated at the footpoints, propagating
across the loop and dissipating their energy when steepening into shocks. The relaxation
of such a loop to a quasi-steady state would then correspond to the case illustrated in
chapter §1. Such quasi-steady states are more likely to happen in quiet Sun regions,
where magnetic field strengths are low. However, near active regions, where magnetic
field strengths are strong, the Sun exhibits highly dynamic and energetic phenomena,
and magnetohydrodynamic processes could be maintained in a non-equilibrium state.
As described in the Introduction, non-uniformity seems to happen in the heating process
of the corona, and nanoflares, being ubiquitous, sporadic, impulsive releases of energy,
seem to reflect this characteristic of the Sun. However, M04 proposed the model of a
loop heated by Alfvén waves generated at the footpoints (by sub-photospheric random
motions) producing shocks all along the loop and exhibiting an X-ray intensity profile
that is usually attributed to nanoflares. Hence it was suggested that nanoflares may
not be a result of magnetic reconnection but in fact may be due to (non-linear) Alfvén
waves, contrary to widespread opinion.
In M04 1.5-D MHD simulations were carried out in which additionally to the poloidal
component along the loop, the toroidal component (around the loop) was considered
(without however considering any change in that component, i.e. 1.5 dimensional). Each
footpoint of the loop is perturbed (in a non-correlated way) by transverse (toroidal) motions imitating random sub-photospheric motions that generate torsional Alfvén waves.
Through the process of non-linear mode-coupling these waves produce fast and slow
MHD waves (i.e. linearly polarized Alfvén waves) that steepen into fast and slow shocks
respectively. As a result, these shocks propagate along the loop reflecting in the tran-
§2.2
28
sition region and colliding in the corona, dissipating energy and producing ubiquitous,
sporadic and very localized heating events suggesting nanoflares.
In this chapter we propose a model of a loop with the same characteristics as in M04 but
being heated by nanoflares caused by magnetic reconnection. In this case however we
don’t take into account the effect of the magnetic field and we concentrate in the hydrodynamic response of the loop to the nanoflares. Hence we perform 1-D hydrodynamic
simulations in which the nanoflares are input artificially along the loop. This type of
modelling has already been done in the past (Walsh et al., 1997; Mendoza-Briceño et
al., 2002; Taroyan, Bradshaw & Doyle, 2005). In this chapter we investigate the characteristic features caused by the nanoflares so as to compare with the Alfvén wave model
proposed by M04.
2.2 Nanoflare heating model
2.2.1
Loop geometry and basic equations
The geometry of the magnetic flux loop considered in this case is similar to the geometry
of the loop in Fig.1.1 considered for the steady case, however, as the performed simulations start from the photosphere, we consider a larger magnetic flux expansion of 1000
(with the expansion occurring mostly in the transition region), and an initial loop radius
of 200 km (see Fig.2.1). The equations describing the variation of the cross-section area
along the loop are the same as in M04. Taking s the length along an external magnetic
field line of the magnetic flux loop, r the radius of the loop, and z the length along the
central axis of the loop we have,
Z
r=
cos θds
(2.1)
sin θds
(2.2)
Z
z=
where θ is the angle that the external magnetic field line makes with the radius of the
loop (as a vector),
θ = θt + (θr − θt )fn
−4H0
θr = − arctan
r
z
θt = arctan k cosh
11zd
1
z − zd
fn = −
tanh
−1 .
2
H0
(2.3)
(2.4)
(2.5)
(2.6)
In these set of equations k and zd are parameters describing the shape of the loop,
namely, the rate of expansion at the footpoints (we take k = 3.48 and zd = 6H0 ). H0
§2.2
29
Figure 2.1: Loop Geometry. The length of the loop is 1010 cm and the base-to-apex
area-cross section ratio is 1000. We can see the canopy structure of the expansion in the
chromosphere and transition region.
corresponds to the scale height at the photosphere, H0 = 200 km. θr in eq.(2.4) is a
value that inhibits the area from expanding near the photosphere. θt in eq.(2.5) controls
the expansion of the loop at the apex. In order to link smoothly these two parameters
that describe different parts of the loop, eq.(2.6) is defined. The parameters chosen for
the loop geometry assure a base-to-apex area ratio of 1000. Conservation of magnetic
flux, eq.(1.28), can be written here,
Bs = B0
r 2
0
r
(2.7)
where B0 is the value of the magnetic field at the photosphere and r0 is the initial
radius of the loop. We set r0 = H0 . In the photosphere the value of β = 8πp/Bs2 is
approximately unity.
In this model we write gravity in a slightly different way as in eq.(1.19), which nonetheless
is a good approximation.
z π ,
(2.8)
L
where g is the gravity at the photosphere, and L is the total length of the loop. Namely,
gef f (z) = g cos
g = 2.74 × 104 cm s−2 , and L = 1010 cm, same length as the loop considered in §1.3.1.
Taking into account the field line whose length is measured by s, gravity along it can be
written
gs = gef f
dz
.
ds
(2.9)
§2.2
30
For this model we consider 1-D hydrodynamical equations (now time-dependent).
The equation of mass conservation:
∂ρ
∂ρ
∂
+v
= −ρBs
∂t
∂s
∂s
v
Bs
(2.10)
The momentum equation:
∂v
∂v
1 ∂p
+v
=−
− gs
∂t
∂s
ρ ∂s
(2.11)
The energy equation, taking into account thermal conduction and radiative cooling:
∂e
v
R−S −H
∂e
∂
1 ∂
2 ∂T
−
r κ
,
(2.12)
+v
= −(γ − 1)eBs
+ 2
∂t
∂s
∂s Bs
ρ
ρr ∂s
∂s
where e denotes the internal energy of the ideal gas,
e=
1 p
.
γ−1ρ
(2.13)
S accounts for background heating, maintaining the loop initially at a certain temperature distribution balancing radiative losses, R, whose function is defined in §1.3.1
(eq.1.24 and 1.25).
H denotes the heating from the nanoflares that we will now describe.
2.2.2
Nanoflare heating function
Nanoflares are input artificially along the loop and are represented in the form of episodic
heating events which occur randomly in time and space and have a random duration.
The heating rate due to the nanoflares is represented as
H=
n
X
Hi (t, s)
(2.14)
i=1
where Hi (t, s), i=1,...,n are the discrete episodic heating events, and n is the total number
of events. Despite the fact that here we consider a background source of heating, it is
important to consider a case in which the entire heating is due to the nanoflares, and
address various important aspects such as the evolution of the loop from chromospheric
to coronal temperatures, the formation of the transition region and its dynamic behavior
in time. This is done for example in Taroyan et al. (2005, hereafter T05). As the aim
in this work is to compare with the Alfvén wave heating done by M04, in which a
background source of the same type is considered, we choose also to include such a
heating in our model so as to reduce differences for the sake of comparison.
The space distribution of nanoflares has been a controversial subject in recent years.
Uniform distribution of the heating, is supported in some studies (Priest et al., 1998),
while TRACE observations of isothermal loops favor the idea that the heating is concentrated near the footpoints (Aschwanden et al., 2001). The later however seems to have
§2.2
31
gained more credibility in later studies (Gudiksen et al., 2005; Ugarte-Urra et al., 2005),
where best fit cases are obtained with the total heating input directed preferentially at
the footpoints. It is possible however that different loops are heated in different ways.
In this work we assume for simplicity that the heating events are distributed randomly
along the loop above the chromosphere. We adopt the same form of the heating function
of each event as in T05,
(
Hi (t, s) =
E0 sin
π(t−ti )
τi
i|
, ti <t<ti + τi ;
exp − |s−s
sh
0,
(2.15)
otherwise,
where E0 is the maximum heat input and sh is the heating scale length. The offset time
ti , the duration τi and the location si of each event are written in the following way
ti = αi ttotal , τi = βi τmax , si = smin + γi (L − 2smin ),
(2.16)
where αi , βi , γi are random numbers between 0 and 1, ttotal is the maximum duration
of a single heating event and smin is the minimum height below which heating events
are switched off. We thus assume that the minimum height of each event is at the top
of the chromosphere.
• Parameters for the heating function
In order to set the values to the parameters of the heating function, let’s try to estimate
the nanoflare duration time. One of the hardest parameters to estimate in magnetic
reconnection theory is the dimension of the current sheet. The thickness is believed to
be extremely thin (even of the order of 10 m) and the width, ∆ (in the case of the
Sweet-Parker model; Sweet, 1958; Parker, 1963) is of the order of ∼ 1000 km. The
length across the reconnection region is then estimated to be of the order of ∼ 1000
km. As the Alfvén speed in the corona is ∼ 1000 km s−1 , the time scale of a (small)
reconnection event leading to a nanoflare should oscillate around 1s. This value however
is not established. In the present work we propose two models essentially differing in the
maximum duration time of a heating event. Following are the values of the parameters
involved in the heating function (eq.2.14 - 2.16) for these two cases.
1. E0 = 0.5 erg cm−3 s−1 , τmax = 10 s, n=681.
2. E0 = 1.0 erg cm−3 s−1 , τmax = 2 s, n=340.
For both cases we set the minimum height of the heating events to smin = 2 × 108 cm
and a short heating scale height of sh = 2 × 107 cm to account for the localization of
the events. The total time of the simulation is 568 min for both cases. In Fig. 2.2 we
can see an interval of time of the nanoflare distribution. In the first case the average
§2.3
32
of occurrence of events is 1 event each 50 s, and in the second case it’s 1 event each
100s. In order to easily differentiate between the two heating models we will refer to the
longer and shorter nanoflare duration models shortly as τ10 and τ2 respectively. Also,
subindexes with ’10’ or ’2’ will correspond to τ10 or τ2 respectively.
Setting these parameters we can calculate the total energy of each event, as well as
the total energy flux from the nanoflares. Integrating in time and space eq.(2.15) and
considering the cross-area where the heating occurs we get for the total energy Ei of each
event,
Z
L Z ti +τi
Ei = A
Hi (t, s) dtds
0
(2.17)
ti
with A = π(∆/2)2 ' ∆2 and ∆ ∼ 108 cm, it results,
Ei ' 2∆2 E0 τmax sh (2 − e−si /sh − e−(L−si )/sh )
' 4 × 1016 E0 τmax sh erg.
(2.18)
The total energy flux F can be calculated in the following way,
F=
1
ttotal
L Z ttotal
Z
H(t, s) dtds '
0
0
4n
ttotal
E0 τmax sh ,
(2.19)
Replacing with the values of the parameters for each case, we find the total energy of
each event and the total energy flux to be:
1. Hi,10 = 4 × 1024 erg,
2. Hi,2 = 1.6 × 1024 erg,
F10 = 8 × 106 erg cm−2 s−1 .
F2 = 1.6 × 106 erg cm−2 s−1 .
In both cases the energy of each event is in the nanoflare range (cf. §2.1) and the total
energy flux should be enough to maintain the corona. The second case is a little less
energetic than the first one, but as the energy maximum, E0 , of each event is higher, it
can exhibit sharper observational features (cf. §2.4.2).
2.3 Initial conditions and numerical model
The initial temperature of the loop is set at 104 K, corresponding to the temperature
resulting from background heating. As for the density, it is assumed to follow hydrostatic
pressure balance from the photosphere to a height of 4H0 = 800 km. From that height
up to the loop apex the density decreases with the fourth power of height ∝ h−4 . This
is based on the work by Shibata et al. (1989a, 1989b), in which the results of 2-D MHD
simulations of emerging flux by Parker instability exhibit such pressure distribution.
The density at the photosphere is set at ρ0 = 2.53 × 10−7 g cm−3 , and, correspondingly,
the photospheric pressure is p0 = 2.09 × 105 dyn cm−2 . As the plasma β parameter is
§2.4
33
Figure 2.2: A sample of the time distribution of nanoflares in the case τmax = 10 s.
unity in the photosphere, the value adopted for the magnetic field at the photosphere
is B0 = 2.29 × 103 G. The expansion of the cross-section area from base to apex being
1000, the value of the magnetic field at the apex is Btop = 2.29 G.
The grid in the numerical scheme accounts for a spatial resolution of ds = 0.1H0 = 20
km, with a total grid number of 5070. The boundary conditions at both footpoints are
set to be symmetric so as reflect the (acoustic) waves. The numerical scheme adopted
is called CIP (Yabe & Aoki, 1991) which treats the advective terms (left hand side)
of equations (2.10), (2.11) and (2.12) after solving the non-advective terms (right hand
side) of these equations by finite-difference method. Numerical viscosity is added to the
pressure term for a proper treatment of the shocks. The total time of the simulation is
568 min.
2.4 Nanoflare heating results
2.4.1
Basic common features
In Fig. 2.5 and Fig. 2.6 we show results of the nanoflare heating simulation for τ10
and τ2 respectively. The evolution of (a) temperature, (b) density, (c) pressure and (d)
flow velocity in the loop is shown from t = 0 min to t = 283 min. In Fig. 2.3 and
Fig. 2.4 snapshots of the evolution of the same quantities are shown for cases τ10 and τ2
respectively. Temperature (a), density (b), pressure (c) and flow velocity (d) are shown
§2.4
34
(a)
(b)
(c)
(d)
Figure 2.3: Snapshots of the time evolution of different quantities in the case of Nanoflare
heating, with τmax = 10 s. Temperature (a), density (b), pressure (c) and flow velocity
(c) along the loop are plotted for the initial distribution (blue), at 28 min (green) and
142 min (red).
§2.4
35
(a)
(b)
(c)
(d)
Figure 2.4: Same as Fig. 2.3 but for the case in which the heating events last a maximum
of τmax = 2 s.
§2.4
36
(a)
(b)
§2.4
37
(c)
(d)
Figure 2.5: Evolution of temperature (a), density (b), pressure (c) and velocity (d) inside
the loop from t=0 min to t=283 min for the case of Nanoflare heating with τmax = 10
s. Bright colors indicate high values (positive velocities for (d)). In cases (b) and (c)
logarithmic values are plotted and the color bar only shows the gradient corresponding
to values in the region above the chromosphere (the color bar is saturated for the first
and the last ∼ 2000 km). The color bar ranges are the same as in Fig. 2.6. Crosses
indicate the locations of the randomly input heating events.
§2.4
38
(a)
(b)
§2.4
39
(c)
(d)
Figure 2.6: Same case as in Fig. 2.5 but for heating events lasting a maximum of 2 s.
§2.4
40
at three different times: initial distribution (blue line), at 28 min (green line) and at
148 min (red line). We can see from the time evolution figures and from the snapshots
that the system evolves very rapidly in both heating cases. In less that 30 min the
temperature not only in the apex of the loop but also near the footpoints is already a
typical coronal temperature (∼ 1 − 2 × 106 K). This is due mainly to two reasons: first,
the energy from the nanoflares is input artificially so that it is used directly to heat the
plasma (not through some dissipation process first), and second, localized energy input
is efficiently distributed along the loop due to thermal conduction (the high thermal
conduction coefficient in the corona implies a conduction cooling time in this case of a
little more than an hour). Density and pressure also exhibit a rapid change from their
initial distribution in both heating cases. In the early stage of the simulation density
is around 10−16 g cm−3 , and rapidly increases to values above 10−15 g cm−3 . Pressure,
initially close to 10−3 dyn cm−2 in the apex, increases about 2 orders of magnitude
throughout the simulation. From Fig. 2.5 and Fig. 2.6 we can see that both density and
pressure have a profile which is rather constant in the corona, with the characteristic
steep increase in the canopy region of the transition region and chromosphere. The flow
velocity along the loop exhibits dynamic behavior at the beginning of the simulation,
reaching speeds of ∼ 100 km s−1 but the (acoustic) shocks due to the nanoflares rapidly
decrease their magnitudes to values around ∼ 10 km s−1 . Shock formation at an early
stage of the simulation is easy due to the fact that temperature is still low, so that sound
speed is low. Hence we have numerous supersonic flows steepening into (acoustic) shocks
and heating the surrounding plasma. Nanoflares increase locally temperature and gas
pressure, thus creating shocks constantly. The gradual rise in temperature in the corona
reduces the amplitudes of the shocks, which however are still produced, reducing flow
speeds even more.
Fig. 2.7, and Fig. 2.8 show, respectively, the detailed temperature and density profiles
for the first 140 minutes. The time difference between each plotted line is 34 s. We
can see from these figures that the initial strong shocks are strongly damped and after a
couple of reflections the initial amplitude is reduced (this can also be appreciated looking
carefully at panels (d) of Fig. 2.5 and Fig. 2.6). Shocks going from one footpoint to
the other in the lapse of 20 minutes can be observed, which sets a speed of ∼ 90 km
s−1 , agreeing well with the sound speed value in a ∼ 1.2 × 106 K plasma. The shocks
propagate and reflect near the footpoints, since the steep gradients in the transition
region act as a wall (also, the boundary conditions at the footpoints are symmetric).
From the figures we can see that these shocks quickly dissipate their energy and their
amplitude decreases by at least a factor of 10 in a lapse of 20 minutes.
As it is appreciated, in both cases, τ10 and τ2 , the loop, after approximately 50 min,
reaches a quasi-steady state in which all flows are subsonic, shock amplitudes are low,
and hence also pressure and density variations. Only sporadic significant changes in
§2.4
41
temperature can be clearly appreciated. The obtained loops are hence quite uniform.
We will discuss this result in §2.4.3.
2.4.2
Different heating models sharing a loop oscillation pattern
Differences between the two nanoflare heating cases are easily seen. Panel to panel
comparison between Fig. 2.3 and Fig. 2.4 shows sharper density and pressure variations
in the short nanoflare duration case, corresponding to stronger shocks. Shock speeds are
also higher in this case, as seen in the (d) panels. At same times temperature, density
and pressure appear with lower values in τ2 (near to one order of magnitude difference
in the case of pressure and density). The average pressure in the corona is pave,10 = 0.20
dyn cm−2 and pave,2 = 0.04 dyn cm−2 respectively for τ10 and τ2 . Fig. 2.6a shows
darker colors with respect to Fig. 2.5a, indicating a lower average temperature for
the shorter duration nanoflare case. Indeed, the average coronal temperature for τ10 is
Tave = 1.67 × 106 K, while for τ2 is Tave = 1.10 × 106 K. We can also see sharper heating
events in the case of the shorter duration nanoflare case. The temperature panels indeed
show more localized bright events, which reflect the higher energy input per nanoflare
and the shorter time duration. The maximum temperature throughout the simulation
is 2.56 × 106 K and 2.35 × 106 K for τ10 and τ2 respectively.
These results agree well with the energy flux calculations (eq. 2.19), where we found the
estimated energy flux in τ2 to be 7 times less than the energy flux in τ10 . The energy
balance in time along the loop is intimately related with the dynamic features of the
loop. We postpone the energy balance analysis until §2.4.4.
The nanoflare heating leads to the formation of a very thin transition region separating
the dark cooler chromospheric region from the bright hot corona, as seen in Fig. 2.5a
and Fig. 2.6a. These figures also show a rather static transition region (considering
the total simulation time until 568 min) which however exhibits an oscillation pattern
throughout the simulation but mostly present at the beginning and gradually damping.
This oscillation is present in all panels, and indicates an oscillation from side to side
of the whole loop due to the nanoflares. In both heating cases the transition region
oscillates with an average amplitude of ∼ 1000 km (reaching a maximum of ∼ 5000 km
at the initial stage of the simulation) and an average of 12 times each 50 min (setting
an oscillation period of ∼ 250 seconds). If this oscillation was generated by a single
propagating wave (or shock), it would imply a propagation speed of ∼ 700 km s−1 (the
length between the two transition regions of the loop is ∼ 90000 km), much too high
for a shock speed. As a single nanoflare has too little energy in order to be seen sharply
(as a narrow gaussian distribution) in the temperature evolution, and to cause a major
perturbation, the biggest perturbations in the loop are caused by groups of nanoflares, or
clusters, which happen at times which are relatively shortly separated. In one oscillation
§2.4
42
period we have in average 5 (2 or 3) heating events in the case of τ10 (τ2 ). Groups of 2
or 3 nanoflares in the case of τ10 and 1 or 2 nanoflares in the case of τ2 seem to generate
sufficiently strong shock trains (for some time during the simulation) in order to produce
a displacement of the transition region. At the beginning of the simulation we can see
various of these groups of nanoflares, and because each one of these large heating events
propagates (as acoustic shocks, or N-waves) in two directions (towards both footpoints),
we could say that one oscillation of the loop is caused, in average, by two of these large
heating events (2 groups of nanoflares per period), thus 4 propagating waves. Hence,
the propagation speed of each of the shocks produced by the heating events, inferred
from the loop oscillation period, is ∼ 90 km s−1 , which correctly corresponds to the
sound speed of the medium. The acoustic shocks reflect at the transition region and
are rapidly damped. In the second half of the simulation the oscillation has almost
completely disappeared. At this stage the shocks caused by the heating events have
lower amplitudes (due to the higher temperatures) and are damped quickly, explaining
the gradual damping and eventual disappearance of the oscillations. The periodicity
of these and propagation of the shocks can clearly be seen in the intensity flux images
shown in §2.5.2, Fig. 2.18. We will comment on these figures in §2.5.2.
2.4.3
The effect of longer heating events, a quasi-steady evolution
The simulation carried out in T05 has same maximum energy input per event and
same frequency of events as in τ10 (0.5 erg cm−3 s−1 , 1 event each 50 s). However, the
maximum nanoflare duration adopted in that work is 40 s. Another important difference
is the absence in their simulation of any background heating, so that the heating of the
loop is entirely due to the nanoflares. Using a background heating source accounts for
thermal stability, avoiding possible dramatic cooling of the loop, however, in the present
work background heating only plays a role at the beginning of the simulation, when
temperatures are low. Still, as previously analyzed, the temperature along the loop
quickly rises to coronal values. Hence, any effect due to background heating in this case
is minimal. In T05, the transition region is found to be dynamic as well. Indeed, a
shift in location of ∼ 1000 km can be appreciated often during the simulation. However,
the shift of location of the transition region is not periodic. This reinforces the idea
that an oscillation of the loop is possible when nanoflares form groups (happen close
together in time) and the generated acoustic shocks add up. That is, when the duration
of the nanoflares is sufficiently small and the frequency is sufficiently high so that when a
nanoflare occurs in some part of the loop the traveling acoustic shock can meet a second
nanoflare along its way and add up with the second shock. Another difference with T05
are the supersonic flows that can be seen throughout their simulation, not only at the
early stage as in the present work (it is important to note however that in T05 the total
§2.4
43
(a)
(b)
Figure 2.7: Temperature profiles for, (a), ≤ 10 s and, (b), ≤ 2 s duration nanoflare
heating models respectively. The time interval from t = 0 min to t = 140 min is shown
with the profile lines being plotted each 34 s.
§2.4
44
(a)
(b)
Figure 2.8: Density profiles for, (a), ≤ 10 s and, (b), ≤ 2 s duration nanoflare heating
models respectively. The time interval from t = 0 min to t = 140 min is shown with the
profile lines being plotted each 34 s.
§2.4
45
simulation time is ∼ 5200 s, less than one sixth the total simulation time in the present
work). Despite the difference in spatial resolution (100 m in their case, 20 km here)
which allows in T05 to correctly observe the dynamics of the transition region, it is clear
that the heating duration of the nanoflares has a drastic influence on the dynamic and
energetic features of the loop. As previously stated, the loops obtained in the present
case have a quite uniform nature, characteristic of a quasi-steady state. From τ10 and
τ2 we can say that the effect of having shorter and multiple heating duration events of
the same magnitude throughout the corona has a smoothing effect. The multiplicity of
these randomly distributed and uniformly shaped events in the corona produces non-stop
(acoustic) shock collisions of the same magnitude and hence rapid energy dissipation and
plasma heating.
From the previous results it can be said that the nanoflare heating, when uniformly
distributed along the loop, succeeds in creating and maintaining a corona. The obtained
corona in the case of short duration and randomly distributed nanoflares appears rather
uniform and the loop quickly reaches a quasi-steady state. Uniformity in the corona
is only achieved when energy losses, here due to radiative cooling and thermal conduction, are balanced efficiently by energy input, here from the nanoflares. Famous tests
which can quantify how well is the energy balance achieved are scaling laws which relate
thermodynamic properties of the loop with its geometry (namely, its length). The most
famous of these is the RTV scaling law (Rosner, Tucker & Vaiana, 1978), which relates
the maximum temperature occurring at the apex of the loop, Tm , to the product of its
pressure p and length L,
1
Tm ' 1.4 × 103 (pL) 3 .
(2.20)
This scaling law is known to be quite insensitive to changes in the assumed geometry of
the magnetic field, and has proved to agree well with observations (Rosner et al, 1978).
Inserting the mean coronal pressures 0.203 dyn cm−2 and 0.04 dyn cm−2 for the heating
cases τ10 and τ2 respectively (calculated in the time interval (100-568) min, assuring the
quasi-steady phase of the loop), we obtain
Tm,10 ' 1.77 × 106 K,
Tm,2 ' 1.04 × 106 K.
(2.21)
Now, the average coronal temperatures at the apex for τ10 and τ2 are 1.82 × 106 K and
1.20 × 106 K, respectively, setting a difference of 2.9% in the first case, and 15.6% in the
second case.
We can see a good agreement with the RTV law for both cases but mainly with τ10 .
This agreements would imply that our quasi-steady assumption is indeed correct and
that the energy balance is maintained throughout the simulation. Let’s take a closer
look at the energetics.
§2.4
2.4.4
46
Energetics
The energy equation (2.12) dictates the energetics in the loop at all times. Energy balance in the corona is determined by the interplay between energy losses due to downward
thermal conduction and radiation, and the energy input determined by the nanoflares
(and background heating, which only has a small effect in the initial stage of the simulation, as discussed previously). The energy input of each heating event, described by
eq.(2.15), is random in time and space and is transferred to the plasma in several forms,
thermally, in a potential or kinetic form. The total energy flux rate resulting from this
transfer is named here flow energy flux rate and can be written as
γ
1 2
ρv +
p + ρgef f y .
F =v
2
γ+1
(2.22)
The terms in eq.(2.22) are from left to right kinetic energy, enthalpy and potential
energy. The contribution from each one of these terms to the total flow energy flux
rate is plotted in Fig. 2.9 for cases (a) τ10 and (b) τ2 . The fluxes are averaged in time
for the time interval (100, 568) min. The first thing to note is that most of the flux
is directed upwards, for coronal heating. For τ2 actually none downward flux can be
seen. In both cases most of the energy flux is concentrated into heating the upper part
of the loop. Also, more than 90% of the total energy is in thermal form all along the
loop with few exceptions near the footpoints, where potential energy attains a maximum
and is of equal order to thermal energy. Energy flux in the form of kinetic energy is
negligible all throughout the loop for all times, except for τ10 , near the footpoints where
some downflow can be seen. It’s important to remember that the energy flux rates that
are showed are calculated only for the quasi-steady stage of the simulation. That is,
all the initial supersonic flows and shocks have been damped and only weak dynamic
phenomena can be seen. This explains the total absence of kinetic energy fluxes.
We note a completely different shape between cases τ10 and τ2 . While the former exhibits
a rather constant (upward) in magnitude and oscillatory energy flux rate with values
close to 105 erg cm−2 s−1 , the latter shows a smoothly decreasing energy flux rate
directed upwards, with a maximum near the footpoints of ∼ 105 erg cm−2 s−1 , and a
negligible minimum value at the apex. This implies that in the apex of the loop most of
the energy of the loop has been dissipated for coronal heating, hence, the energy input
for τ2 is barely enough to maintain the corona. We could say that we have a loop which
is in a slowly cooling state (in the order of a few conduction cooling times).
A(z)
To account for the effects of loop expansion we plot the total energy flux rate F (z) A(z=0)
(where z accounts for length along the loop’s central axis) in Fig. 2.10 for both heating
cases. We can see that for τ10 the energy flux rate is rather constant all along the
loop with an average value around 107 erg cm−2 s−1 . For τ2 the energy flux rate has a
minimum at the apex of the loop, as previously noticed. However the value in the rest
§2.4
47
(a)
(b)
Figure 2.9: Flow energy flux rates (eq. 2.22) for cases (a) τ10 and (b) τ2 . Different colors
denote different terms accounting for the flow energy flux rate. Black, red, pink and
green correspond to total flow energy flux rate, enthalpy, potential energy and kinetic
energy respectively. Thick lines correspond to positive flux (upflows), while dashed lines
correspond to negative flux (downflows). The energy fluxes are averaged in the time
interval (100, 568) min.
§2.4
48
(a)
(b)
Figure 2.10: Total flow energy flux rates (eq. 2.22) for cases (a) τ10 and (b) τ2 taking
into account the loop cross-section area expansion A(z).
§2.4
49
of the corona is in the range between 106 and 107 erg cm−2 s−1 . These results agree
well with the energy calculations made previously (eq. 2.19) and show that the energy
released from the nanoflares is enough to heat and sustain the corona.
In order to correctly estimate the energy balance in the loop it is necessary to see how
much of the energy flux is actually converted into heat and how much is lost as downward
thermal conduction and radiation. Hence we plot the variation of the flow energy flux
rate ∇ · F together with conduction flux ∇ · Fc (last term of eq. 2.12, see also eq. 1.23)
and radiative cooling (eq. 1.24 and 1.25) in order to see the volumetric heating rate (erg
cm−3 s−1 ). This is plotted in Fig. 2.11 for (a) τ10 and (b) τ2 . The quantities shown
in this figure have also been averaged omitting the initial time up to t = 100 min. We
can appreciate in the upper plot of each panel that in the coronal part of the loop the
energy input from the nanoflares dominates and heats the plasma. On the other hand
radiative cooling dominates near the footpoints up to a height that varies between the 2
heating cases. In the long nanoflare duration case the point where the total heating rate
equals radiative losses is above the 10000 km (∼ 14000 km), while in the short nanoflare
duration case the two quantities meet at a height below 10000 km (∼ 7000 km, half
the other height). The volumetric heating rate has nonetheless a maximum near the
footpoints and this has to do with the fact that density is 3 to 4 orders of magnitude
higher there, hence, more energy is needed to heat the large amount of plasma. In the
lower panels we see conduction flux dominating in the upper part of the loop, roughly
constant, transporting all the excess heat into the lower parts. The flow heating rate,
which is mostly the thermal heating rate has a maximum of order unity at the footpoints
and quickly decreases and looses 90% of its initial heating rate efficiency in the first 2000
km. As the heating comes mostly from acoustic heating, we can see that the acoustic
waves created by the nanoflares loose more than 90% of their energy in the first few
thousand kilometers when traveling from one footpoint to the other. This agrees with
the rapid damping of the initially observed acoustic shocks.
In Fig. 2.12 we show the heating rate per unit mass,
1
ρvA ∇
· (AF), for cases (a) τ10
and (b) τ2 . This quantity shows where along the loop is the plasma heated in the most
efficient manner. We can see that in both cases the corona is the region where the energy
per unit mass is a maximum (hence, where the energy is mostly dissipated).
At this point it is interesting to make a link with the analysis made in chapter §1.
Although in that case energy balance along the loop is achieved mostly by the fast
mode MHD waves (that steepen and form the switch-on shock trains), we saw that
the acoustic waves played a role in heating the plasma near the footpoints (and it is
of course meaningful to make a comparison between heating mechanisms, which is the
most important matter in the present work). As the nanoflares occur with a frequency
of 1 each 50 s and 1 each 100 s respectively for τ10 and τ2 , we could say that the acoustic
waves generated by these propagate, respectively, as trains with a period of the order of
§2.4
50
(a)
(b)
Figure 2.11: Volumetric heating and losses along the loop for cases (a) τ10 , and (b),
τ2 . In the upper plot in each panel the total volumetric heating rate is shown in black,
together with radiative losses in blue. In the lower plot thermal conduction flux is plotted
in red and the variation of flow energy flux rate is plotted in green. The quantities are
averaged in the time interval (100, 568) min.
§2.4
51
(a)
(b)
Figure 2.12: Heating rate per unit mass along the loop for cases (a) τ10 , and (b), τ2
averaged in the time interval (100, 568) min.
§2.5
52
50 s or 100 s, similar to the case of the N-waves. Two important differences between these
models however is, first, the fact that in the steady model only the region starting from
the high chromosphere is considered, and the acoustic waves generated at the footpoints
have lost most of their energy when reaching the corona. The second important difference
is that in the nanoflare simulations acoustic shocks are created directly in the corona
and hence can dissipate and heat the corona in-situ. Comparing Fig. 2.11 with Fig.
1.3e, we see that the volumetric heating rates behave in approximately the same way.
Weak shock amplitudes for the switch-on shock trains produce lower coronal densities
and temperatures and accordingly the point where radiative losses equals the heating
rate is pushed farther up into the corona at approximately a height of 7000 km for
the weakest considered case. We can see that the weakest switch-on shock case shares
common features with τ2 . Coronal values of temperature, density and wave energy flux
have a very similar shape. The wave energy flux in both cases decreases steeply along
the loop, losing 90% of the initial value in the first 5000 km. The similarities are due
to the fact that the amount of energy input in both cases is very similar. The stronger
amplitude cases for the switch-on shock trains show higher temperatures and densities
corresponding to higher energy fluxes and hence show more similarities with τ10 .
The steady solutions found in §1.4 for the case of N-waves and switch-on shock trains
heating show similarities with the uniform nanoflare heating reinforcing the idea that
loops heated by ubiquitous, equally energetic and short duration nanoflares produce
quasi-steady and uniform coronal loops. Let’s now compare these results with the simulation results of a loop heated by (nonlinear) Alfvén waves.
2.5 Differences between Alfvén wave heating and Nanoflare
heating mechanisms
2.5.1
Completely different coronas
In the case of nonlinear Alfvén wave heating in M04, the energy input is located in the
footpoints of the loops. A mechanical motion transversal to the loop field lines applies a
torque in the frozen magnetic field, one at each footpoint (in toroidal
q direction) with a
randomly determined direction and a magnitude corresponding to < vφ2 > = 2.06 km
s−1 in average. The torques at each footpoints are uncorrelated and generate nonlinearly
Alfvén waves that propagate as linearly polarized Alfvén waves that steepen into shocks
along the loop. These shocks collide at various locations producing nanoflare-like heating
events. In this section some of the results obtained by M04 are reproduced and extended
in order to compare with the Nanoflare heating scenario. In the next chapter we will
consider a model that includes both heating mechanisms, nanoflares and Alfvén waves.
§2.5
53
(a)
(b)
(c)
(d)
Figure 2.13: Snapshots of the time evolution of different quantities in the case of Alfvén
wave heating. (a) Temperature, (b) density, (c) pressure and (d) flow velocity along the
loop are plotted for the initial distribution (blue), at 28 min (green) and 142 min (red).
§2.5
54
(a)
(b)
§2.5
55
(c)
(d)
Figure 2.14: Evolution of (a) temperature, (b) density, (c) pressure and (d) velocity
inside the loop from t=0 min to t=283 min for the case of Alfvén wave heating. Bright
colors indicate high values (positive velocities for (d)). In cases (b) and (c) logarithmic
values are plotted and the color bar only shows the gradient corresponding to values in
the region above the chromosphere (the color bar is saturated for the first and the last
∼ 2000 km). The color bar ranges are the same as in Fig.2.5 and 2.6
§2.5
56
The simulations carried out in the Alfvén wave cases (here and in §3) are 1.5-D MHD,
that is, poloidal and toroidal directions in the magnetic flux loop are taken into account
but no change in the toroidal direction is allowed. In this case the same numerical scheme
(CIP) is used to solve the equations, except the induction equation which is treated with
the MOC-CT method (Evans & Hawley, 1988; Stone & Norman, 1992). For more detail
about the equations and model please refer to M04.
Let’s begin our analysis with the observed thermodynamic properties along the loop. In
Fig. 2.13 we show snapshots in time of the evolution of (a) temperature, (b) density, (c)
pressure and (d) flow velocity at the same times as in Fig. 2.3. We can appreciate that
the evolution of the quantities is totally different as for τ10 and τ2 . After half an hour the
temperature along the loop is still very oscillatory. Also pressure and density exhibit ∼2
orders of magnitude variations, i.e. strong shocks. The velocity at that time confirms the
presence of shocks showing flows along the loop with supersonic velocities above 100 km
s−1 . After 140 min the system seems to have reached a more stable configuration in which
temperature, density and pressure in the corona have typical coronal values. However,
comparing with the quasi-steady cases of τ10 and τ2 it is clear that the corona produced
by this model is much more dynamic. This fact is confirmed by Fig. 2.14, which shows
the evolution in the loop of the same physical quantities for the same interval of time
as in Fig. 2.5 and 2.6, that is, from 0 to 280 min. We can see from this figure that very
dynamic events are maintained throughout the simulation. The transition region is in
constant movement and drastic shifts of more than 10000 km can be appreciated. Fig.
2.15a, which shows the temperature profile for the first 140 min, shows clearly the large
shifts in position of the transition region. While in the Nanoflare case coronal values
are reached after only 30 minutes in the Alfvén wave case it takes much more time,
approximately 100 minutes are needed to have a corona. This constitutes an important
difference between the two heating mechanisms. The highly unsteady behavior of the
loop in the Alfvén wave case is due to the constant torque exerted at both footpoints
which produces torsional Alfvén waves that non-linearly convert to linearly polarized
Alfvén waves. These waves can compress the plasma and through shocks a large amount
of energy is input at the footpoints. This momentum input produces upward flows of
cool plasma, or spicules, as discussed in M04 and Kudoh & Shibata (1999). The large
energy input at the footpoints can be appreciated by plotting the total energy flux. We
refer the interested reader to M04 and Moriyasu (2004) for further details. The linearlypolarized Alfvén waves travel as fast and slow magneto-acoustic waves in the corona.
The shocks produced by these waves can be appreciated in Fig. 2.15b which shows the
density profile for the first 140 min. The panels in this figure are plotted with the same
scale as for Fig. 2.8 and 2.7. Comparing with these figures we can effectively see that
the amplitude of the shocks created by the Alfvén waves are much more stronger than
the shocks produced by the nanoflares that are input throughout the corona. Also, we
§2.5
57
can see that the amplitudes decrease when traveling from one footpoint to the other,
but not as much as in the case of the acoustic waves generated by the nanoflares. This
is due to the fact that Alfvén waves are difficult to dissipate their energy, hence a strong
shock remains a strong shock for many Alfvén traveling times from one footpoint to
the other. This agrees with the analytic model of chapter §1, where the amplitudes of
the switch-on shock trains remained large throughout the corona. This lack of shock
dissipation together with the non-stop torsion of the magnetic field lines (and thus nonstop generation of Alfvén waves) are responsible for the ubiquitous presence of shocks
in the corona. Another fact reflecting the lack of dissipation of the fast MHD shocks
is the relatively low average coronal temperature throughout the simulation, namely,
1.19 × 106 K, quite low compared to 1.82 × 106 K in the case of τ10 (for τ2 the average
coronal temperature is 1.10 × 106 K, similar to the case of the Alfvén waves).
Hence, from one side, ubiquitous, equally energetic, short duration Nanoflares that are
input artificially (simulating small reconnection events) and randomly along the loop
succeed in rapidly creating and maintaining a corona which is in a uniform and quasisteady state in which the transition region exhibits a periodic motion. On the other side
the continuous torques produced by sub-photospheric motions generate Alfvén waves
that nonlinearly steepen into (strong) shocks that propagate continuously throughout
the corona heating it impulsively and setting it into continuous motion by shifting considerably the transition region in a somewhat chaotic form. A corona is obtained but
after considerably more time. It is hard to judge whether the resulting corona is in a
quasi-steady state or not. Judging from the coronal dynamics point of view we have
a very unsteady corona. However judging from the energetics point of view we have a
corona in which the energy balance is well established. Indeed, the average temperature
(at the apex of the loop) and pressure (in the corona) for the Alfvén wave case are
respectively, 1.5 × 106 K and p = 0.116 dyn cm−2 , and the temperature corresponding
to thermal equilibrium (obtained with the RTV scaling law, eq. 2.20) is Tm = 1.47 × 106
K. That is, we have only a 2.48 % difference with the theoretical energy balance value.
Hence, even if we have a very dynamical corona it is a quasi-steady corona from the
energetics point of view.
We could say that the considered Nanoflare model agrees more with quiet-Sun region
loops, where dynamics are low, while the Alfvén wave model agrees better with active
region loops, where highly dynamical events occur, for instance, continuous mechanical
sub-photospheric motions.
Let’s now study the observational characteristic features of such different coronas.
§2.5
58
(a)
(b)
Figure 2.15: (a) Temperature and (b) density profiles for the Alfvén wave heating mechanism. The time interval from t = 0 min to t = 140 min is shown with the profile lines
being plotted each 34 s, the same as in Fig. 2.7 and Fig. 2.8. The scale is also the same
§2.5
2.5.2
59
Predicting observables
In this section we simulate what would be observed by Solar-B/XRT (X-Ray Telescope)
when looking at the previously modelled loops. These calculations were done with an
IDL based program that uses the Solar-B/XRT response function data base for different
filters which was kindly given by D. Shiota, N. Narukage and D. Brooks.
In Fig. 2.17 we show the calculated XRT intensity fluxes at the loop apex from t = 160
minutes to t = 280 minutes for the three studied models: (upper row) τ2 , (middle row)
τ10 and (lower row) the Alfvén wave model. As each filter has a defined sensibility
towards temperature (i.e., a different response function), the results obtained with each
filter can be very different. Here we choose filters Ti poly (first column to the left) and
Thin Be (second column), which are sensible to temperatures a little bit higher than
1 MK, as seen in Fig. 2.16, which shows the Solar-B/XRT response function. Fig.
2.17 shows that the temporal variation of the intensity is very similar to those actually
observed with the X-Ray intensities with Yohkoh. The best fit with observations is
maybe provided by the Alfvén wave model, mainly for observations of active regions.
In the nanoflare cases we can appreciate the characteristic impulsive rise and decay (on
time scales of a few minutes), consistent with plasma heating and subsequent cooling
(Aschwanden et al. 2000).
In Fig. 2.18 we show the evolution of the XRT intensity fluxes in the loop from t = 85
min to t = 142 min for the case of τ10 (panels a and b) and τ2 (panels c and d), and
t = 85 min to t = 142 min for the Alfvén wave case (panels e and f). In the case of
τ2 , as the achieved coronal temperatures are lower than in the other cases (a coronal
temperature average of 1.2 × 106 K.), the intensity flux calculated with the Thin Be
filter appears completely black. We then show instead the intensity flux calculated with
Thin Al mesh (panel a). The gradient scale in each panel is different and is chosen so
as to enhance the contrast in order to see more clearly the features of each plot. The
maximum intensity fluxes are quite different, in order of increasing intensity we have τ2 ,
the Alfvén wave case and τ10 . In all cases, however, wave traces can be seen. In the
Nanoflare cases (panels a,b,c,d) these traces form a uniformly threaded pattern. The
oscillation of the transition region can clearly be seen in these panels. Looking carefully
we can see that the maximums in the oscillation coincide with the reflection points of
the acoustic waves, and that each one of these waves traces back to a nanoflare, after a
few reflections at both transition regions or collisions with other shocks. This threaded
pattern allows to understand how is the uniformity achieved. The (weak) acoustic shocks
caused by the nanoflares propagate, colliding one another, reflecting at the transition
regions and rapidly damping and dissipating their energy in a uniform way throughout
the corona. In panel (d) (Thin Be) we can see clearly the nanoflares and the acoustic
shock propagation. In the Alfvén wave model (panels e and f) the features produced
§2.5
60
Figure 2.16: Temperature response function for the different filters of Solar-B/XRT.
Courtesy: D. Shiota, N. Narukage, D. Brooks
in the intensity flux are stronger and the wave patterns are clearly appreciated. We
can also easily distinguish roughly two different patterns with different inclinations,
accounting for the fast and slow MHD modes (the fast MHD mode making a lower angle
with the length axis). Another characteristic feature is the location of the brightest
points. The nanoflare heating model exhibits bright points that are located more or
less randomly in the corona with however brighter points at the footpoints (transition
region), where the waves reflect. The Alfvén wave heating model has its brightest points
(very localized brighter points than in the nanoflare cases, indicating stronger shocks)
near the footpoints with strong wave-like features across the loop (panel e) both with
actually no strong localized brightenings throughout the corona. This feature agrees
with the low dissipation feature from the fast shocks in the corona, showing a maximum
amplitude near the footpoints and slowly decreasing through reflections, as seen in Fig.
2.15. In the Nanoflare case, however, the acoustic shocks generated in the corona rapidly
dissipate, becoming weak shocks, as seen also in Fig. 2.8.
Let’s now construct a model of a loop heated by both nanoflares and Alfven waves. As
expected, the resulting loop shares common features with both heating mechanisms.
§2.5
61
(a)
(b)
(c)
(d)
(e)
(f )
Figure 2.17: XRT intensity fluxes calculated at the loop apex from t = 160 minutes to
t = 280 minutes for different models: (upper row) τ2 , (middle row) τ10 and (lower row)
the Alfvén wave model. The intensity is calculated using different filters. From left to
right, the columns correspond to filters Ti poly and Thin Be respectively.
§2.5
62
(a)
(b)
§2.5
63
(c)
(d)
§2.5
64
(e)
(f )
Figure 2.18: Evolution of the XRT intensity fluxes in the loop apex for different models:
from t = 85 min. to t = 142 min.,(upper row) τ2 and (middle row) τ10 ; from t = 227
min. to t = 284 min. (lower row) the Alfvén wave model. The intensity is calculated
using different filters. From left to right, the columns correspond to filters Ti poly and
Thin Be respectively, except for case (a) where filter Thin Al mesh was used.
§3.2
65
3 Nanoflare and Alfvén wave heating combined
3.1 Model
In this chapter we consider a third heating model, one that includes both, nanoflares
and Alfvén waves. Physically it would correspond to a loop in which sub-photospheric
motions generate torsional Alfvén waves that non-linearly convert into fast and slow
MHD waves which steepen into shocks. Additionally nanoflares occur throughout the
loop, generated by external factors that are not considered1 . When a nanoflare occurs,
outflow from the reconnection site produces a gas pressure gradient that is directed
radially outwards. As the magnetic field lines are magnetically frozen to the plasma,
the magnetic field lines are locally distorted, expanding radially. Hence, nanoflares
normally should perturb the transversal component of the magnetic field, Br (i.e., the
radial component) generating also fast and slow MHD waves (by means of the gradient
of the radial component) that can steepen into shocks and contribute even further to the
heating of the corona. However, in this study the radial component is not considered.
The present model is 1.5 dimensional, namely, only the longitudinal (poloidal, along the
magnetic field line) and the toroidal components (the later however with no variation,
∂
∂φ
= 0) are considered. Hence, in the present model there can be no generation of
transversal waves (i.e., Alfvén waves) that can non-linearly steepen into shocks. The
only effect of the nanoflares is then in the gas pressure (compressing the plasma and
producing acoustic waves), and only through this quantity there can be an interaction
between the effects of the nanoflares and the effects of the Alfvén waves.
The 1.5-D MHD simulations that are carried out in this model are just as in the Alfvén
wave case. The only difference here is then the addition of nanoflares all along the loop.
The nanoflare heating function considered is exactly the same as in τ10 . The case of
Alfvén waves together with τ2 was also analyzed but results are not presented, as they
do not apport significantly to the discussion. We will simply comment briefly on them.
1
they could be generated, for example, where the shear of the magnetic field is high, that is, where
the Alfvén wavelength is very small, mostly at the transition region. Current sheets could be then
produced, leading to magnetic reconnection events.
§3.2
66
(a)
(b)
(c)
(d)
Figure 3.1: Snapshots of the time evolution of different quantities in the case of
Nanoflares and Alfvén wave heating combined. (a) Temperature, (b) density, (c) pressure and (d) flow velocity along the loop are plotted for the initial distribution (blue),
at 28 min (green) and 142 min (red).
§3.2
67
(a)
(b)
§3.2
68
(c)
(d)
Figure 3.2: Evolution of (a) temperature, (b) density, (c) pressure and (d) velocity
inside the loop from t=0 min to t=283 min for the case of Nanoflare and Alfvén wave
heating combined. Bright colors indicate high values (positive velocities for (d)). In
cases (b) and (c) logarithmic values are plotted and the color bar only shows the gradient
corresponding to values in the region above the chromosphere (the color bar is saturated
for the first and the last ∼ 2000 km). The color bar ranges are the same as in figures
2.5, 2.6 and 2.14.
§3.2
69
3.2 A quasi-steady and dynamic corona
In Fig. 3.2 the time evolution along the loop of (a) temperature, (b) density, (c) pressure
and (d) velocity is shown from t = 0 min. to t = 283 min. (same as in figures 2.5, 2.6
and 2.14). Snapshots of the time evolution of these quantities can be appreciated in
Fig. 3.1. As expected, we can see that this model shares features of the nanoflare model
and the Alfvén wave model. Indeed, coronal values for all quantities are reached after
only half an hour from the start of the simulation (which did not happen in the Alfvén
wave case), accounting for the fast dissipation of the energy due to the nanoflares. Also,
shocks propagating in the corona can be seen at all times, although weaker than in the
pure Alfvén wave case, these are stronger than in τ10 . Temperature, pressure and density
show in average higher values than in the previous cases due to the higher energy input
in the present model. For instance, the average coronal temperature and pressure are
respectively 1.76 × 106 K and 0.297 dyn cm−2 , much higher than the average values
found for τ10 and for the pure Alfvén wave case (cf. §2.5.1). Velocity (d panels in both
previous figures) shows lower values. The supersonic speeds in the Alfvén wave case are
replaced here by flows with speeds lower than 40 km s−1 , i.e. subsonic in the present
model. This reflects the presence of a high number of (weaker) shocks in the corona
which reduce flow speeds. As in τ10 and τ2 , nanoflares increase locally temperature and
gas pressure, thus creating shocks constantly. As the temperature rises in the corona,
the shocks have lower amplitudes but are still produced, reducing flow speeds even more.
In figures 3.2 and 3.3 we can appreciate a dynamic transition region with height shifts
as big as 3000 km. These are however not as big as in the pure Alfvén wave case but
are still more important than in τ10 . The cool upflows of plasma due to the torsional
Alfvén waves are rapidly heated by the nanoflares and the dissipation of the generated
acoustic waves. The periodic oscillation of the transition region which was present in τ10
seems to have disappeared in the present case due to the constant generation of Alfvén
waves at the footpoints (the cool upflows move the transition region upward in a rather
random way, accordingly to the way in which Alfvén waves are produced).
The resulting corona has many fast, slow and acoustic weak shocks (the two later behaving however in almost the same way in this 1.5 dimensional model)2 that together
with the dissipation from the nanoflares contribute to coronal heating. The amplitudes
of all these shocks being small during most of the simulation reflect a quasi-steady and
2
In this 1.5-D model the way to differentiate between these three types of shock is to consider
the variation of the magnetic field, mainly the transverse (toroidal) component. In the case of a fast
shock, the magnetic field lines are compressed in the transversal direction in the upstream region, i.e.,
magnetic pressure is enhanced after the shock has passed. In the case of a slow shock the contrary
effect is produced. That is, the magnetic pressure is reduced after the advent of the shock. An acoustic
shock however does not affect the toroidal component of the magnetic field. In this 1.5-D model only
gas pressure is affected by the acoustic shocks.
§3.3
70
uniform state. Hence, it seems that from the dynamics point of view a quasi-steady and
uniform corona is achieved quite rapidly as in the case of τ10 . To quantify the energy
balance in the loop we again invoke the RTV scaling law, eq. (2.20). The average temperature at the apex of the loop is found to be 2.08 × 106 K, and the corresponding
value with the RTV law results 2.01 × 106 K, setting a difference of only 3.5%. The idea
of a quasi-steady and uniform corona is then supported also from the energetics point
of view.
In the case of τ2 together with Alfvén waves the same overall effects are appreciated.
Initially stronger shocks and higher flow speeds are observed due to the more energetic
nanoflares. The shock amplitudes are also rapidly damped and the loop reaches a quasisteady and uniform state in which coronal quantities have lower values than in the
case with τ10 . The theoretical temperature resulting from the RTV scaling law and the
average apex temperature differ by less than 10%.
Let’s take a closer look at the energetics for τ10 and the non-linear Alfvén wave heating
combined.
3.3 Energetics
In this 1.5-D model magnetic energy comes into play. The flow energy flux (eq. 2.22)
due to nanoflares and Alfvén waves in this case can be written in the following way,
1
1
γ
2
2
F = vs
ρ(vs + vφ ) +
p + ρgs y − (v × B) × B .
(3.1)
2
γ+1
4π
The new element in F , the term involving B, denotes Poynting flux and can be divided
1
(v ×B)×B| =
into two parts: | 4π
1
2
4π (Bφ vs −Bφ Bs vφ ).
The first term corresponds to the
advection of magnetic energy, that is, energy flux transported by the compressive fast
and slow MHD waves. The second term indicates shear of magnetic field, corresponding
to the (incompressive) Alfvén waves.
In Fig. 3.4 we plot the flow energy flux (3.1). Comparing with the case of the nanoflares,
Fig. 2.9 several differences stand clear. First, the common quantities, thermal energy
(in red), potential energy (in pink) and kinetic energy (in green) show higher values in
the present case. For instance, kinetic energy that can hardly be seen in Fig. 2.9 is now
present throughout the corona with values close to 104 erg cm−2 s−1 and a maximum
at the footpoints of ∼ 108 erg cm−2 s−1 . This trend and values are also seen in the
potential energy. These quantities coming from the compression of the plasma show
higher values due to the additional compressive waves, namely, the slow and fast MHD
waves. The apport from the magnetic waves, namely the Poynting flux (blue colors),
constitutes the biggest difference with τ10 . Dark and light blue colors correspond to
the energy flux from the compressive (fast and slow modes) and incompressive (Alfvén
§3.3
71
(a)
(b)
Figure 3.3: (a) Temperature and (b) density profiles for the Nanoflare and Alfvén wave
heating mechanisms combined. The time interval from t = 0 min to t = 140 min is
shown with the profile lines being plotted each 34 s, the same as in figures 2.7, 2.8 and
§3.3
72
Figure 3.4: Flow energy flux rates (eq. 3.1) for the case of Alfvén waves and τ10 combined.
Different colors denote different terms accounting for the flow energy flux rate. Black,
red, pink, green, correspond to total flow energy flux rate, enthalpy, potential energy
and kinetic energy respectively. Dark blue and light blue correspond, respectively, to
advection of magnetic energy and to shear of the magnetic field (corresponding to Alfvén
waves). Both terms constitute the Poynting flux. Thick lines correspond to positive flux
(upflows), while dashed lines correspond to negative flux (downflows). The energy fluxes
are averaged in the time interval (100, 568) min.
§3.4
73
mode) magnetic waves respectively. The advected magnetic energy flux has its origin in
the energy carried by the (shear) Alfvén waves which cannot modulate the plasma and
consequently cannot heat it directly. This component (light blue) is clearly the biggest
source of energy among the flow energy flux components, reaching a value of a few 109
erg cm−2 s−1 at the footpoints. Alfvén waves can however convert through non-linear
effects into the fast and slow modes, which compress the plasma, steepen into shocks
and dissipate the energy, finally resulting in thermal energy. The energy carried by these
modes (dark blue color) shows values close to thermal energy (enthalpy), indicating that
a significant part of the heating is of magnetic wave origin. Thermal energy, when it does
not come from the compressive MHD modes, comes from the compressive acoustic waves
and dissipated energy due to the nanoflares. In Moriyasu (2002) it is shown that fast
and slow MHD shocks (acoustic kind of shock) contribute approximately in an equal way
to the heating of the corona. It is interesting to determine in the present model which
contribute more to the heating of the corona, fast and slow MHD shocks or nanoflares.
This contribution is measured by the variation of the energy flux, namely ∇ · F. In Fig.
3.5 we plot ∇ · F − ∇ · Fc − R where the term with Fc denotes heating rate due to
thermal conduction, and R denotes radiative cooling (cf. §1.3.1). Now, F in the case
of the nanoflares is taken only as enthalpy. For the magnetic waves, F represents the
Poynting flux. We note that the difference in heating from the two mechanisms is not so
important. However, the heating rate coming from enthalpy seems to have a little more
weight. Enthalpy represents mostly nanoflares, but it also includes energy from Alfvén
waves that has been converted into thermal energy. It is thus difficult to estimate which
mechanism contributes the most to the heating in the present case. It can only be said
that the difference is not so important.
Let’s analyze the observed XRT intensity flux for the present model.
3.4 Intensity fluxes
Following the study made in §2.5.2, we plot now the XRT intensity fluxes corresponding
to the Alfvén wave and nanoflare models combined. In Fig. 3.6 we show in (a) and (b)
the intensity fluxes for filters Ti poly and Thin Be respectively, at the apex of the loop
for times t = 160 min to t = 280 min. Comparing with Fig. 2.17, we note that the
values of intensity in this case are approximately one order in magnitude higher than
in the independent cases. This is due to the presence now of two sources of energy,
and consequently, two sources of shocks, Alfvén waves and nanoflares. Features of both
heating mechanisms can be appreciated. Indeed, the numerous steep rises in the case of
the Alfvén waves are also appreciated here. The following decay (cooling) after the rise
is however more gradual, as in τ10 .
In Fig. 3.7 we show the time evolution of the XRT intensity flux. Again, common
§3.4
74
Figure 3.5: Volumetric heating rate (∇ · F − ∇ · Fc − R) along the loop for different
quantities. The volumetric heating rate from the nanoflares (F = enthalpy) and the
MHD waves (F = Poynting flux) are plotted in pink and blue respectively. The total
volumetric heating rate is shown in black. The quantities are averaged in the time
interval (100, 568) min.
features with the previous models can be seen. For instance, the uniformity created by
the acoustic shocks of the nanoflares is present, as in Fig. 2.18, panels (c) and (d). The
shock propagation traces of the fast and slow modes are clearer in the present plots than
in 2.18, panels (e) and (f), since here the temperature is much higher throughout the
simulation. The nanoflares are appreciable here as very localized brightenings, and as
the generated acoustic shocks are weak and dissipate rapidly, their traces are hard to
see. On the other hand the brightenings due to the MHD modes are less localized and
brighter. The traces of the MHD shocks appear bright with maxima at collision points
with other shocks. The lack of dissipation of these shocks (fast shocks) is also clearly
appreciated from the numerous reflections at the transition region.
In the present model we can see that most of the dynamical phenomena are due to
the generation of Alfvén waves at the footpoints. The resulting dynamical feature of
the corona is mainly caused by the lack of dissipation of the MHD shocks. It could
be compared with big ocean waves that hardly dissipate their energy and happen to be
reflected between two cliffs, here, the transition region at both footpoints. The nanoflares
although been less observable than the dynamic effects of the torsional Alfvén waves, play
a very important role in the present model. The numerous weak acoustic shocks created
by them reduce flow speeds and dissipate energy in a more effective way, setting rapidly
a quasi-steady and uniform state in the corona. Nanoflares have thus a smoothing effect.
§3.4
75
(a)
(b)
Figure 3.6: XRT intensity fluxes calculated at the loop apex from t = 160 minutes to
t = 280 minutes for the Nanoflare and Alfvén wave models combined. The intensity is
calculated using different filters. From left to right, the columns correspond to filters Ti
poly and Thin Be respectively.
§3.4
76
(a)
(b)
Figure 3.7: Evolution of the XRT intensity fluxes in the loop apex from t = 85 min.
to t = 142 min. for the Nanoflare and Alfvén wave models combined. The intensity is
calculated using different filters. From left to right, the columns correspond to filters Ti
poly and Thin Be respectively.
§3.4
77
Conclusions
In this work two popular heating scenarios were investigated: Nanoflare heating and
Alfvén wave heating. The former is based on the idea that multiple magnetic reconnection events take place ubiquitously in the Sun, and that heating is released sporadically
an impulsively from the reconnection region, liberating the energy from the stressed
magnetic field. On the other hand Alfvén wave heating focus on the idea that energy
is transported by MHD waves and is released in the corona through some process of
dissipation, for example, through shocks.
The coronal heating mechanism whichever it might be has to remain dormant long
enough so that the magnetic stresses built to the required energy level. The onset must
be rapid and efficient, liberating large amounts of energy in the proper time and length
scales in the corona. This picture has been matched in recent years mostly by the
Nanoflare heating scenario. However some works, as M04 have pointed out that the
observed nanoflares can actually be caused by Alfvén waves steepening in some way and
generating shocks which collide and release nanoflare-like energy amounts.
In order to correctly assess the validity of each one of these heating events their manifestation in terms of observable quantities must be predicted.
In the present work we examined first the analytic case of a steady-corona heated by
trains of acoustic shocks (N-waves) and trains of fast MHD shocks (switch-on shock
trains). It was found that, supposing the fast MHD mode to be able to propagate into
the corona, most of the heating comes from the fast MHD shocks, which can conserve a
sizeable amount of energy during their propagation. The basic features of the obtained
coronal loop were dictated by the initial shock amplitude of the switch-on shock trains
(i.e. initial energy input), and we observed that coronal values were actually quite
sensible to this parameter. However, this picture is far from complete. One of the main
problems that cannot be considered in a one dimensional model is the refraction and
reflection of the fast MHD mode in the chromosphere and transition region, due to the
high physical gradients there. Acoustic waves generated at the footpoints, even if less
effected by reflection or refraction, played a negligible role in coronal heating, dissipating
most of their energy near the footpoints, hence contributing only slightly to the overall
energy balance.
In chapter 2 we concentrated on time-dependent simulations. 1-D HD and 1.5-D MHD
simulations of a loop heated by nanoflares in one model and by nanoflares and Alfv’en
waves in a second model were realized. For the nanoflares we considered different heating
functions, differing mainly by the maximum duration of heating events. One model in
which this time is set to 10 s and another with a lower duration of 2s, but with higher
energy per event. Although some differences in the observed features of the loop, we
§3.4
78
found that both cases produce a quasi-steady corona that is highly uniform, sharing
common features with the analytic model of chapter 1. In the uniformly distributed
nanoflares case, the generated acoustic waves can dissipate directly in the corona, thus
bringing a high heating rate per unit mass. An oscillation of the loop was found (which
was not found later for the models with Alfvén wave heating) that revealed the presence
of many of these weak acoustic shocks generated by the nanoflares. The high number
of shocks propagate in the corona damping quickly through collisions with other shocks
or through reflection with the footpoints, and consequently heating the plasma in a
uniform manner. On the other hand, in the Alfvén wave and nanoflare heating model
the torsional Alfvén waves generated near the footpoints convert into linearly polarized
MHD waves that can steepen into shocks. A large energy input from the footpoints
caused a compression of the plasma there causing upward flows of cool material, hence
shifting the transition region considerably. The highly dynamic features in this case
are maintained throughout the simulation. We noticed that the fast MHD shocks do
not dissipate as easily as the acoustic waves, conserving their amplitude through a long
distance, thus agreeing with the observed behavior of the switch-on shock amplitude in
the analytic model of chapter 1. The obtained corona in the Alfvén case is far from
being uniform and the quasi-steady state is also dubious. Even if very different, both
heating scenarios can create and maintain a corona. We could say then that the uniform
nanoflare heating model applies better to quiet Sun regions, while the Alfvén wave
heating would be more suitable to describe active region loops.
The observed XRT intensity fluxes for each model were calculated in order to compare
with (future) observations and be able to distinguish between both heating scenarios.
All time variations of the intensity display characteristic features that are commonly
observed with TRACE or Yohkoh. For instance, the slower decrease of the intensity
after its rapid onset in the case of nanoflares, or the highly variable intensity flux in the
case of the Alfvén wave heating. In the evolution plot of the intensity flux nanoflares can
be seen to create weak acoustic shocks which rapidly damp. Also the brightest points
seem to be distributed in the coronal part of the loop in the case of uniform nanoflare
input, these seem to concentrate near the footpoints in the case of the Alfvén wave
heating.
It would be interesting to consider a third model in which both heating scenarios can
happen. It could be a loop subject to sub-photospheric motions that create fast and
slow magneto-acoustic waves that steepen into shocks, together with artificially input
nanoflares that would perturb the magnetic field producing Alfvén waves as well. As the
region of greatest motion along the loop is the transition region this could be the region
where magnetic reconnection events are more likely to happen (in the canopy region).
Hence a heating function that concentrates near the footpoints could be more realistic
and provide further interesting observable signatures. This is left as future work.
§3.4
79
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