A model was develped for the mass, energy, and magnetic field transport into the corona. The focus is on the flow below the photosphere which allows the energy to pass into, and be dissipated within, the solar atmosphere. The high flow velocities observed in spicules are explained. A treatment following the work of Bailyn et al. (1985) is examined. It was concluded that within the framework of the model, energy may dissipate at a temperature comparable to the temperature where the waves originated, allowing for an equipartition solution of atmospheric flow, departing the sun at velocities approaching the maximum Alfven speed

Mayr, Hans G.

Schatten, Kenneth H.

English

NASA Technical Reports Server

PHLEGETHON FLOW - A PROPOSED ORIGIN FOR SPICULES AND CORONAL HEATING
Kenneth H. Schatten
Hans G. Mayr
Laboratory for Atmospheres
NASA/Goddard Space Flight Center
Greenbelt, Maryland
INTRODUCTION
We draw upon existing work to put together a model of the mass, energy and
magnetic field transport into the corona. This transport originates, in our
model, from a deep-seated flow issuing out of concentrated field structures.
The fluid is ejected into the solar atmosphere with the help of magnetic
forces on fibril field structures. We refer to the subsurface flow as a
"phlegethon", named after the river of fire that flowed in Hades in Greek
mythology. A new term to describe the gas is suggested because the flow
appears to carry significantly greater amounts of energy in its non-potential
magnetic fields than in its kinetic energy. As fire is a process which
unleashes the stored (chemical) energy to create heat, the term is used here
generically to describe heating through the unleashing of a magnetic form of
stored energy. As with other models, the wave and current energy flux leads
to the dramatic heating of the solar atmosphere. We shall concentrate on the
flow below the photosphere (the phlegethon flow), which allows the energy to
pass into, and be dissipated within, the solar atmosphere. We do not treat
the details of the physical processes (wave breaking, heat conduction, etc.)
above the photosphere, nor on the coronal field geometry, which usually forms
loops in the solar atmosphere.
Network field structures are already considered good candidates for supplying
coronal mass as they carry plasma Jets, known as spicules (e.g._ Beckers,
1973), having densities from 3 x 1010 to 3 x 1011 particles cm -3 (compared to
~10 _ particles cm -_ in the surrounding corona), with diameters of 500 to 1200
km_ extending in height 10,000 to 20,000 km, with temperatures from I to 2 x
107 K. They contain material with upward supersonic flow velocities of 20 to
30 km s-" with neither sign of material falling nor the upward velocity abated
by gravity, make them ideal for supplying coronal material (Withbroe and
Noyes, 1977). Thus we consider these structures as sources of material and
energy for the solar atmosphere. A number of reviews have been written on the
nature of spicules, with Beckers (1972) providing a comprehensive review of
their observational and theoretical understanding.
THE ORIGIN OF SPICULES
We consider spicules to represent highly evacuated flux tubes from which gases
pour supersonically into the solar atmosphere. Observationally, the high
degree of evacuation of fluxtubes may manifest itself in the large nearly
uniform field strength of 1400-2000 Gauss (Beckers and Schroter, 1968),
comparable to the photospheric gas pressure, observed in nearly all
photospheric magnetic features. This leaves little room for significant gas
pressure on these field structures. Parker (1984) discusses theoretical
https://ntrs.nasa.gov/search.jsp?R=19870011500 2020-03-20T12:03:21+00:00Z
reasons how in a superadiabatic environment, free energy is utilized to
concentrate the magnetic field into localized bundles of large field strength
(fibrils), consistent with evacuated fluxtubes.
The differing behavior of magnetic buoyancy from gaseous buoyancy may play a
major role in the evacuation of fluxtubes, since buoyancy is a critical force
for the fluid dynamics within the convection zone. Buoyancy is required for
this material to overturn since there is insufficient thermal energy (4 x
10-10 ergs per hydrogen ion) without buoyancy to allow the bulk of the
material to rise from the convection zone base to the photosphere against the
gravitational potential (-2 x 10-9 ergs per hydrogen ion). The magnetic
buoyancy force differs from the thermal buoyancy force, because the latter is
proportional to the weight of the displaced material and is therefore
comparable to the actual weight of the material within a specific volume. The
magnetic buoyancy force, however, originates from the non-potential field
configuration, which the surrounding fluid pressures have contorted the
magnetic field into. This bears little relation to the weight of material on
the magnetic field, but rather is related to the external fluid pressures,
modified by the Maxwell stress tensor. Weenvision a fibril fluxtube
extending downto the base of the convection zone, decreasing in radius as it
is examined at greater depths. At the lowest altitudes the magnetic field
will no longer contract indefinitely, so that at somealtitude the magnetic
field lines will be nearly parallel. At these altitudes the magnetic buoyancy
force will be muchless than the gravitational or thermal buoyancy force. In
the unmagnetized convection zone fluid, the buoyancy force balances the
gravity force since the equations of stellar structure relate the
gravitational force to the pressure gradient. Thus the gas density within a
deep seated fluxtube decreases with altitude faster than the surrounding gas,
where the thermally buoyant forces operate, and the fluxtube can achieve a
relatively low density during its passage to the solar surface. Examining
conditions higher in the convection zone, neaT the photosphere, for example,
the magnetic stresses are near 10-2 dynes cm-_ for 300 km wide fibrils of
field strength 2000 Gauss, with a 300 km _cale height. This is comparable to
th_gravitational _orce of 10-2 dynes cm-J for photospheric densities of 2 x
10 "1 particles cm -_.
To summarize our view, for typical deep convection zone densities and fields,
gravitational or buoyancy forces exceed magnetic forces for field aligned
fluid flow. For shallow conditions or highly evacuated fluxtubes, the Maxwell
stresses can exceed the gravitational force. This also can occur at any
height, given a sufficiently low density on the fluxtube, because the Lorentz
force is governed by the field geometry, not by the amount of material on the
fluxtube. Thus, near the solar surface the vertical Lorentz force per unit
mass is comparable to, or can exceed the gravitational or buoyant force. This
can allow the high flow velocities observed in spicules to occur as a "solar
wind" type solution to the Bernoulli equations including the Lorentz force.
ATMOSPHERIC HEATING AND ACCELERATION
We shall examine a treatment following the work of Bailyn et a1.(1985). They
have considered the solution topologies for polytropic winds associated with
momentum deposition and find the general solution can have multiple critical
points. Nevertheless, an integral of the motion can be obtained from the
402
magnetic Bernoulli equation as:
2 in(_)=E+_ D(r')dr'- @G (I)KEp _ ½v2+c° _ o
where E is the constant of integration (which without the magnetic stress term
would be the total energy per particle), D (r)_[z/p = Lz/Nm is the rate of
non-thermal momentum addition per unit mass; c e yP/p, the square of the
sound speed; KE_, the kinetic energy per parti_le; ¢_, the gravitational
• r , ,potentlal; and _here E_=_ D (r) dr may be calculated for a known
m n " " _. r .ag etlc fleld and denslty structure, including the wave field.
_ow_examining the terms of equation (1), for the spicule flow we have
v_ ~ 2 x-1012 c_ 2 sZ 2, and cL_ _ 1012 cm2 s -2, both small compared
¢_ _ 1015 cmL s-L, and _B _ 1014 cm2 s-2 for 10 Gauss twisted fieldsith
containing 1011 particles cm -3 integrated over a I000 km height. Thus for
spicule material a small distance above the photosphere, the terms on the left
14
of equation (I) may be small, and the terms on the right, between 10 and
1015 cm 2 s -2. Figure I shows these terms in the solar atmosphere for a simple
field geometry.
The Maxwell stress, Lz includes not only the static component which cannot
provide heating, but also contributions from waves (in particular, their
decay). If we consider a small volume of space containing an arbitrary
collection of Alfven waves, one solution to the dynamical equations, as Parker
showed, is for the plasma to travel along the field lines at the Alfven
speed. In this case the waves (in this reference frame) can remain
stationary. Thus we can consider a damped set of waves travelling into a
region of space, to add energy and momentum to the plasma within. A solution
in which the plasma flow velocity equals the Alfven velocity, we refer to as
the "equipartition solution". For sinusoidal waves, the energy transport is
divided equally between mechanical and electromagnetic terms. Above the
photosphere, the magnetic contribution (per unit mass) to the vertical
momentum equation, EB, monotonically rises, until it exceeds all other
physical contributors (eg. e_). The flow gradually accedes to the demands of
this relentless magnetic for_e, until the flow rate approaches the local
Alfven velocity. This occurs at the "equipartition solution", when the wave
energy and particle energy are comparable. At this point, the coronal plasma
can no longer acquire energy or momentum from the waves (the curve KE
approaches an asymptotic value) in the same manner that a sailboat "r_nning
with the wind" cannot surpass the wind speed.
Coronal heating problems are now considered. Kuperus, Ionson, and Spicer
(1981) _ugg_st that the quiet corona requires a heat source of a_out 3 x _05
erg cm -_ s-" and the quiet _romosph_re, a heat source of 4 x 10_ erg cm -_ s-I
for global values of 2 x 10_ erg s-" and 2.4 x 1029 erg s-I, respectively.
The Maxwell stress may be transported at a flow rate of S = VA (BoBI/4_).
Taking spicule number densities at 1011 cm -3, field strengths of
B° _ B _ 10 G, e h ve V of order 100 km s -I, yielding an energy flux of
_ lOBerg cm -_ s-_, providing for a global value of 2 x 102_ er_ s-I,
when integrated over phlegethon structures with a total area of 2 x 10_
cm-. This value is sufficient for coronal heating, but 10 times too low for
chromospheric heating. These flow energies appear also sufficient, when
4O3
IOO0
1015 107
300
1012 104
10
1011
CONVECTION
ZONEBASE
.Ncz Tq
• ,..",,.
"_ NCz
%.
%
%'" ........... .N.pf
PHOTOSPHERE
-2 -3 -4 -5 -6 -7
LOG (DEPTH/ME))
DEPTH
PHLEGETHON FLOW,
KEp
SOLAR
ATMOSPHERE
SOLAR RADII
1.53 13 1AU
IIIII II I
26 140
-E
EB
VA
B2/8_ -
\
\
k
_-,-°,.O,o
103 105 107
-3-2-1 0 1 2 KILOMETERS
SCALE HEIGHTS
I ALTITUDE*-------
Figure I. Shown are pressures, energies and densities in the convection zone
and solar atmosphere. The subscript cz refers to normal convection zone
fluid, and pf to "phlegethon flow" - material flow embedded on field
structures. The magnetic pressure exceeds gas pressure above the
photosphere. The kinetic energy per particle KE D related to the temperature
and flow speed squared, the magnetic contributioh to the flow energy, EB, the
constant, E, of the motion, and gravitational potential, $, on fluxtubes are
also provided in the convection zone and solar atmosphere. The Alfven
velocity, VA, allows the base energy quality to reach the solar atmosphere.
To allow for the rapidly changing conditions near the photosphere, doubly
logarithmic abscissa scales are chosen which provide magnification within the
region.
4O4
thermalized, to elevate spicule material to coronal temperatures. As the
waves "break", they contain an energy flux of S and a mass flux of NV_.
Dividing the former by the latter provides for 5 x 10-10 ergs particl_ -I, or
~2xiO K thermal energy per particle.
DISCUSSION
We have examined the following scenario for solar atmospheric heating. Fluid
on rising fluxtubes deep within the convection zone obtains insufficient
buoyancy to rise en masse to the photosphere. In this fashion, rising
fluxtubes within the Sun's convection zone can become highly evacuated. Near
the photosphere, the magnetic buoyancy force becomes large, ejecting material
upwards in accordance with the magnetic Bernoulli equation. This leads to an
outpouring of supersonic gases into the solar atmosphere. These gases contain
insufficient kinetic energy to heat the corona or escape the gravitational
pull of the Sun. In our view they are "wicked" into the solar atmosphere,
where an energization of the material by concurrent wave field stresses
occurs.
Examining these gases as they ascend on fluxtubes to greater altitudes, their
density decreases, owing to gravity and the widening field channel that these
gases flow along. _n typical field geometries, when the fluid velocity
exceeds a few km s-', hydromagnetic waves dominate the vertical momentum
equation, relative to gravity (Lz>_g). This allows for the velocity to
increase rather than decrease with altitude, consistent with the observed
behavior of spicules rising supersonically into the solar atmosphere, with
their motion unabated by gravity. Although the gases contain insufficient
kinetic energy to supply coronal heating, they carry concurrent Maxwell stress
and wave energy in their non-potential field configuration. With the
explosive release of this energy, the gases erupt into the solar atmosphere to
form a hot corona and a dynamically expanding wind. We find that these
structures can provide the magnetic field, mass flow, and energy budget
necessary to maintain the conventional solar atmosphere. Within the framework
of this model, energy may dissipate at a temperature comparable to the
temperature where the waves originated, allowing for an "equipartition
solution" of atmospheric flow, departing the Sun at velocities approaching the
maximum Alfven speed.
REFERENCES
Bailyn, C., Rosner, R. and Tsinganos, K., 1985, Ap. J., 296, 696.
Beckers, J. M., 1972, Ann. Rev. Astron. Astrophys., 73.
Beckers, J. M. and Schroter E. H., 1968, Solar Phys., 4, 142.
Kuperus, M., Ionson, J. A., and Spicer, D. S. 1981, Ann. Rev. Astron.
Astrophys., 19, 7.
Landau, L. D., 1951, Classical Theory of Fields, (Addison-
Wesley Reading, Mass.).
Parker, E. N. 1984, Ap. J., 283, 343.
Withbroe, G. L. and Noyes, R. W. 1977, Ann. Rev. Astron. Astrophys., 15, 363.
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Phlegethon flow: A proposed origin for spicules and coronal heating

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