The effect of time-dependent coronal heating on the solar
wind from coronal holes
Øystein Lie-Svendsen∗ , Viggo H. Hansteen† and Egil Leer†
∗
Norwegian Defence Research Establishment, div. for electronics, P.O. Box 25, NO–2027 Kjeller, Norway
†
Institute of Theoretical Astrophysics, Univ. of Oslo, P.O. Box 1029, Blindern, NO–0315 Oslo, Norway
Abstract. We have modelled the solar wind response to a time-dependent energy input in the corona. The model, which
extends from the upper chromosphere to 1 AU, solves the time-dependent transport equations based on the gyrotropic
approximation to the 16-moment set of transport equations, which allow for temperature anisotropies. Protons are heated
perpendicularly to the magnetic field, assuming a coronal heating function that varies sinusoidally in time. We find that heating
with periods less than about 3 hours does not leave visible manifestations in the solar wind (the oscillations are efficiently
damped near the Sun); heating with periods of order 10 hours leads to perturbations comparable to Ulysses observations;
while heating with periods of order 100 hours results in a series of steady-state solutions. Mass flux perturbations tend to
be larger than perturbations in wind speed. Heating in coronal holes with periods of order 30 hours leads to large mass flux
perturbations near Earth, even when the amplitude of the change in heating rate in the corona is small.
INTRODUCTION
THE MODEL
Does the fairly steady-state high speed wind that, e.g.,
Ulysses observes from polar coronal holes [1] indicate a
steady-state coronal heating process, or merely that the
solar wind has filtered out the high-frequency variations
before they reached Ulysses’ orbit? Several of the proposed coronal heating mechanisms, e.g., by “nanoflares”
[e.g. 2] or by “jets” [e.g. 3, 4], are inherently time dependent. The level of, or lack of, variability in the solar wind
may put severe constraints on such mechanisms, or even
rule out some of them.
To study the solar wind response to a time-dependent
energy input in the corona, we have employed a fluid solar wind model based on the 16-moment approximation,
a model that extends from the chromosphere to 1 AU [5].
In addition to yielding constraints on the coronal heating
mechanism, modelling a time-dependent solar wind can
also provide information about the evolution of shocks in
the solar wind.
Only a few previous studies [6, 7, 8] have considered
the effect of time-dependent energy input in the corona.
The main improvement of the model used here is that
the coupling between the chromosphere and corona is
included, and that it provides a better description of
energy transport between the chromosphere, transition
region and corona.
The model includes neutral hydrogen, protons (which
are produced dynamically through ionization of HI), and
electrons. For each species s the coupled equations for
density ns , drift speed us , temperature parallel and perpendicular to the magnetic field (Tsk and Ts⊥ ), and radial
heat flux densities of parallel and perpendicular thermal
motion (qsk and qs⊥ ), are solved. These equations are described elsewhere [5, 9]. We choose a rapidly expanding
flow geometry, in which the flux tube area A(r) increases
by a factor 5 relative to radial (r2 ) expansion near the
Sun, and increases radially beyond a few solar radii.
Except for a small energy input in the transition region
in order to maintain a minimum pressure in the corona,
the main heating is contained in the proton perpendicular
heating term, which is specified as the divergence of a
“mechanical” energy flux,
Q pm⊥ (r,t) = −
1 ∂ Fm (r,t)
.
A ∂r
(1)
We choose a simple analytical form for the applied energy flux,
r − RS
Fm (r,t) = Fm0 (t) exp −
,
(2)
Hm
where t denotes time and we choose Hm = 0.5 RS where
RS is the solar radius. With the chosen parameters most
of the energy will be deposited within a solar radius
CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference,
edited by M. Velli, R. Bruno, and F. Malara
© 2003 American Institute of Physics 0-7354-0148-9/03/$20.00
299
above the solar surface. For the time-dependence we
choose a purely sinusoidal variation in time,
2πt
Fm0 (t) = F0 1 + a p sin
.
(3)
tp
We choose F0 = 400 W m−2 , which in the case of a
constant heating rate (a p = 0) leads to a reasonable highspeed wind with a speed of about 700 km/s and a mass
flux scaled to Earth (nu)E ' 2.4 × 1012 m−2 s−1 . For the
amplitude of the change in heating rate we choose a p =
0.75, so that Fm0 (t) varies between 100 and 700 W m−2
over a period t p .
RESULTS
Figure 1 shows a snapshot of the solution after the model
has been run for so long that the perturbations have had
time to reach 1 AU. As can be seen, periods of order
103 s are filtered out close to the Sun and are therefore
not visible in the solar wind. This is also the case with
shorter periods; choosing, e.g., t p = 300 s to simulate
the so-called 5-minute oscillations, the solution is essentially steady state for r > 2 RS . Heating with a period
of 104 s leads to an essentially steady-state wind beyond
about 1/2 AU. For longer periods the solar wind is no
longer able to smear out the perturbations by 1 AU, and
for t p = 105 s the figure shows that the mass flux perturbations even increase with distance. We attribute this
to a “snowplow” effect in which faster parcels of plasma
overtake slower parcels so that matter piles up behind
the forward shock while behind the reversed shock the
mass flux becomes very low. This process only works
as long as there is sufficient difference in speed between
the different regions. When this is no longer the case, in
the outer solar wind, the enhanced pressure in the highdensity regions will cause this plasma to expand. Eventually this expansion will damp the oscillations. Note also
that, in agreement with e.g., Ulysses observations, the
perturbations in mass flux are larger than the perturbations in wind speed. The temperature panels show that
the perturbations in electron temperature are smaller than
the proton temperature perturbation, indicating that electron heat conduction can smear out the small-scale electron perturbations. For even longer heating periods than
105 s the solution essentially turns into a series of steadystate solutions corresponding to the different values for
the coronal heating rate.
Figure 2 shows how the mass flux evolves as a function
of time and distance for heating periods t p = 104 s and
105 s. We note again that the 104 s perturbations are
gradually damped and disappear around 0.5 AU, while
105 s perturbations even grow in amplitude and remain
clearly visible at 1 AU. Note also that the perturbations
300
behave like “linear” waves, despite that the magnitude
of the perturbations can be very large: If we launch
perturbations in the corona with a period of 105 s, say,
we shall find exactly the same period in the solar wind (as
long as the perturbations are detectable). In that sense the
waves maintain their identity, and we find no tendency
for the solar wind to generate “higher harmonics.”
A time-dependent coronal heating will have a similar
effect in the slower solar wind generated in a radially
expanding flow geometry. The main difference is that
shocks tend to be somewhat less pronounced in a radially
expanding flow, and that the lower wind speed allows the
high-pressure interaction regions more time to expand
before they reach Earth orbit.
What mechanism damps the oscillations in the solar
wind? As alluded to above, the perturbations are damped
mainly by expansion of the high-pressure regions (and
compression of the low pressure regions). Such pressure differences are generated directly by changing the
amount of coronal heating; for the shortest periods the
gas does not have time to expand within one period of the
heating. Pressure perturbations are also created indirectly
in the solar wind when faster parcels of plasma overtake
slower parcels. As the fluid moves outwards these highpressure regions expand and cool, hence damping the oscillations.
In addition, electron heat conduction can in principle
smear out temperature differences in the electron gas. To
study the role of electron heat conduction, we have rerun the model with t p = 104 s, but effectively switching
off electron heat conduction beyond r = 5 RS . As expected, the electron temperature perturbations then become much larger (than they do in the t p = 104 s panel
of Figure 1), with the relative perturbations in Te being
comparable to the relative perturbations in Tp seen in the
figure. The perturbations in Te are still damped however,
even in the absence of heat conduction, and disappear
beyond about 0.5 AU, as do the perturbations in Tp . But
most importantly, switching off electron heat conduction
has essentially no effect whatsoever on the perturbations
in wind speed u p and mass flux (nu)E . Hence electron
heat conduction is of little importance in damping the
solar wind oscillations. This is to be expected since the
electron temperatures in the models are much lower than
the proton temperatures, so that the dynamics will be
dominated by the proton pressure gradients.
In order to damp the oscillations significantly the highpressure regions must have time to expand a distance of
order the distance between the high- and low-pressure
regions. Hence for rapid changes in the coronal heating
rate, and hence small distances between the high- and
low-pressure regions, the required time will be shorter
than for the long periods of the heating. Since this damping takes place as the solar wind propagates outwards, a
more rapid damping for small values of t p means that the
up
600
400
4.0•1012
(nu)E
50
100
150
Distance [RS]
200
Flux density [m-2s-1]
1.0•1013
up
(nu)E
0
0
50
100
150
Distance [RS]
tp=3 104 s
1.2•1013
1.0•1013
up
8.0•1012
200
(nu)E
2.0•1012
0
0
50
100
150
Distance [RS]
tp=105 s
1.2•1013
1.0•1013
up
8.0•1012
(nu)E
0
0
50
100
150
Distance [RS]
200
0
250
250
tp=3 104 s
104
Te
103
0
107
200
2.0•1012
200
Tp
800
400
100
150
Distance [RS]
105
108
600
50
106
1000
6.0•1012
4.0•1012
Te
107
0
250
200
250
tp=104 s
104
103
0
200
200
Tp
800
400
100
150
Distance [RS]
105
108
600
50
106
1000
6.0•1012
4.0•1012
Te
107
0
250
200
104
800
400
2.0•1012
Tp
108
600
4.0•1012
105
1000
6.0•1012
tp=103 s
106
103
0
0
250
tp=104 s
8.0•1012
200
Speed [km/s]
Temperature [K]
2.0•1012
1.2•1013
Flux density [m-2s-1]
107
6.0•1012
0
0
Flux density [m-2s-1]
800
Speed [km/s]
Temperature [K]
8.0•1012
108
Speed [km/s]
Temperature [K]
Flux density [m-2s-1]
1.0•1013
1000
Speed [km/s]
Temperature [K]
tp=103 s
1.2•1013
50
100
150
Distance [RS]
200
250
tp=105 s
106
Tp
105
Te
104
103
0
50
100
150
Distance [RS]
200
250
FIGURE 1. Snapshot of the solution for t p = 103 s, 104 s, 3 × 104 s, and 105 s. Here (nu)E (solid curves) denote the proton
particle flux scaled to 1 AU; u p (dashed curves) is the proton speed, Te (solid curves) is the mean electron temperature, and Tp
(dashed curves) is the mean proton speed (Tp ≡ (Tpk + 2Tp⊥ )/3). The dotted curves denote the corresponding steady-state values
(a p = 0).
301
Time [s]
5.0•104
4.0•104
3.0•104
2.0•104
1.0•104
0
0
50
100
Distance [RS]
150
200
100
Distance [RS]
150
200
2.0•105
Time [s]
1.5•105
1.0•105
5.0•104
0
0
50
FIGURE 2. Proton particle flux scaled to 1 AU for t p = 104 s (top panel) and t p = 105 s (bottom panel) as a function of time and
distance.
damping occurs closer to the Sun. For the longest period
shown in Figure 1, t p = 105 s, the plasma has not come
to approximate pressure equilibrium even at 1 AU. For
very short periods, t p ≤ 103 s, the coronal plasma hardly
has time to respond to the changes and the initial perturbations in the corona will therefore be smaller, too.
A more detailed presentation of this study is given by
Lie-Svendsen et al. [9].
ACKNOWLEDGMENTS
This work was supported in part by the Research Council
of Norway under grants 121076/420 and 136030/431.
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