@ Pergamon Press plc Chin.Astron.Astrophys.(l990)14/2,119-128 Printed in Great Britain A translation of Chjn.J.Space Sci. (1990110/1,8-16 02751062/90$10.00+.00
SOLAR METER NOISESTORM : A WHISTLER MODEL LIU Xu-ehao Beijing
Observatory,
Academia Sinica
Received 1987 June 22
ABSTRACT I propose a whistler model of solar meter-wave noise storm driven by evolution of magnetic structures in coronal active regions where magnetic inhomogeneitiesin the form of strong magnetic filaments are present. The evolution generates weak shocks. Their passage through the filaments heats up some of the electrons which acquire a loss cone distribution. In the storm region the loss cone distribution of the accelerated electrons then generates bright whistler and Langmuir waves. Through induced scattering the Langmuir wave slips to lower wave number region, there to interact strongly with the whistler wave, generating narrow beam electromagneticradiation (Type I bursts). The field and configuration of the strong magnetic filaments determines the bandwidth and duration of the Type I bursts. The noise continuum is taken to be the product of coupling between the Langmuir wave from isotropic electrons and low frequency waves.
Key Words: Solar meter wave. Weak shock. Loss cone instability. Langmuir wave. 1. INTRODUCTION Although the first solar radio observations already discovered noise storms that are mainly in the meter wave band Cl], there is yet no widely recognized radiation mechanism, because of their complexity. In the recent solar activity peak year, Benz and Wentzel [21, Benz and Huba ISI and Wentzel [3] successively put forward noise storm models with similar structures. In these models, the driving force is provided by the emergence of new magnetic flux. Evolution of the magnetic field generates weak shocks or local current instability;nonthermal anisotropic electrons or currents then produce high frequency waves (Langmuir waves [31, high mixed waves [2,41) and low frequency waves (ion-acoustic 131, low mixed [2,41); coupling between high and low frequency waves then produces the observed electromagneticwaves (Type I bursts). Unfortunately none of these models can explain the directivity of the Type I bursts.
120
LIU Xu-ehao
At the 1977 Huangshan Astrophysics Colloquium, I proposed [5] the coupling of whistlers and Langmuir waves as the emission mechanism for Type I bursts. Elgaroy [6] also considered this process as a candidate mechanism. Chiu 171, Kuijpers IS] and thernov [S] have applied similar mechanisms for modulating other types of bursts. Anv theory of Type I bursts
must explain
the following
features
rsj:
1. Duration between 0.1 and 5 seconds. 2. Relative bandwidth Af/f% l/50. 3. Strong normal mode circular polarieation (degree of polarization almost 100%). 4. Directivity: half-power width is only 25’. 5. Brightness temperature Ibr~10’~-10~~K, and there is no observable harmonics ( &r/If t 10m3). 6. Relation with the noise continuum. 2. MODELOF CORONA Our working model refers to a baseline photosphere and is as follows: 1. The noise
source
is located
height
at 0.1-0.5&
of 0.3&
above the
above the photosphere.
2. The electron density in the active region is 6-10 times the average value of the quiet R-corona at the same height [6]; at 0.3& above photosphere, h= 2.9~10scm-~. 3. The magnetic field B - O.S[R/R.
-
of the active lP’G%
(l.O2SR/Ra
region
is,
[lo],
G 10)
(1)
where R is the radial distance from the center of the sun in units of I@. At R= 1.3, E%3G and the electron gyro frequency is oH/2n=6.4MBz. 4. When the plasma density vA = B/(41cp)~‘~ IJ 380 km/s.
is p, the Alfven velocity
is
5. In the region l.l-2.5@, the temperature of the solar atmosphere is relatively stable [ll], the electron temperature T, .., 2 x 10’ K.
is
In reality, coronal active regions are not homogeneous. No clearer indication of the coronal nonhomogeneity is provided than by the shortness of the duration (0.29) and the narrowness of the fractional bandwidth (0.02). In the present model I wish to emphasiee the magnetic nonhomogeneity of coronal active regions. Under the assumption of plasma emission, the fractional bandwidth of Type I bursts, Ar&1~0.02, gives a radial scale length of about 5000km for the sources. Interactions among the three waves (see next Section) require us to assume the existence of a magnetic field of about 10G in the magnetic filaments. The
Whistler
Model
121
corresponding Alfven velocity is 350-12OOkm/s. The size of the transverse section of the filaments is determined by the Alfven velocity and the duration of the Type I bursts of a few tenths of a second. It is roughly equal to the product of the velocity of weak shocks with the burst duration, or about 10Dkn. These strong-field, thousands-km-long filaments twist and entwine almost into plaits. The intervals in a Type I burst chain reflect the distances between the filaments. The intervals in a burst chain are about a few tenthe of a second, hence the filaments are separated by distances comparable to their diameters. Benz et al. 12-41 have studied the weak shocks through magnetic field evolution in the active region and the relevant unstable dynamical processes in the plasma. Emergence of new magnetic flux causes disturbances in the active region. When such disturbances reach a region where the Alfven Mach number satisfies yhkl, weak shocks will be formed. The shock gives rise to electric current linked to the free magnetic energy and random acceleration of the electrons 1121. For a shock wake thickness of ALrSkm, we can get a temperature as high as -1O’Tp while the number density of the statistically accelerated particles will be [3] nr/nes10’3. When the accelerated particles are captured by strong-field filaments with a suitable configuration, a loss cone distribution may be formed. The superheated electrons in localized magnetic filaments will eventually all become superheated electrons with an almost isotropic distribution in the coronal active region (or coronal loop) *
3. COUPLING OF LANGMUIRAND WHISTLER WAVES TO PRODUCE ELECTROMAGNETIC WAVE The relevant resonance conditions among the three waves place stringent requirements on both the Langmuir and the whistler waves. Perhaps it is because of this circumstance that Type I bursts have the characteristics of short duration, narrow frequency band, high directivity etc.. 1. Dispersion
Relation
For the whistler [131,
wave in cold
plassa
the dispersion
relation
is,
where c is the velocity of light, I+ is the plasma frequency, and uw and KU are the frequency and wave number of the whistler wave. This dispersion relation under the conditions given in Section 2 is shown in Fig. 1. Fast electrons
with a loss
cone distribution
are superposed
on
122
LIU Xu-zhao
the plasma of finite distribution
temperature
and the combined electron IO-‘I-
Fig.
Fig.
1
The dispersion relation for whistlers condition of Section 2
under the
Fig. 2 The whistler growth curves (solid) and damping curves (dotted). cosa=1/4, nk=O.OOOl, I%= 7 x 10scm/s, q&,=1/20, l/15, l/10, l/5 (from left to right).
is f -1. + F/i 1‘ - [n,/(ZnY)‘“lurP[-l/2(V/Y,Yl: F, -qtl/(2nV,)‘“1U(iai -d U(lal
-00
-
l; if
u(lal
-0~)
~1.
Ial -
if. Ia1 -
* LIP{_
4 [w,Iv,)‘+
itan-‘(VA/V,)1
>n.
Itan-VSIVI)I
< *
w”/v,Yl}:
(4)
~(o is the half angle of the loss cone , nr is the density of fast electrons when ao=O, V and vt are the electron velocity of the plasma and the maxwellian velocity, and V,, V,, and Vf are the perpendicular, parallel and maxwellian velocities of the fast electron. The dispersion relation for the whistlers propagating in the parallel direction in a plasma with such a distribution is, 1141,
z, =
Z(A) [4, -
((oy-wH)l(~yv,J1coaa)l
2
Whistler
Model
123
are the plasma dispersion functions [151. Under the approximation weah growth rate (weak damping rate), 171~~ (or=q+iy) and l7la Ioh-orl, we expand (7) and neglecting second order small quantities, we obtain
of
The calculated whistler growth and damping curves for the active region parameters of Section 2 are shown in Fig. 2. Fig. 3 gives the growth curves for different loss cone distributions. These two figures show the effect of the magnetic field on the stability character of the whistler-variation in the loss cone distribution being an effect of magnetic configuration. We can clearly see from these diagrams that, as the magnetic field increases, the frequency region producing whistlers moves towards higher frequencies and becomes broader. This is in agreement with Kuijpers’ results [8]. The effect of a variation in the loss cone distribution on the instability frequency region is not so strong.
10-
10
-I
7. 310
10 Fig.
Fig. 3
The whistler growth rate for loss-cone apertures
Fig. 4 The whistler growth rate fractional densities of fast
4
different
for different electrons
Figure 4 gives the whistler growth rate for different fractional densities of the fast electrons. We see that the growth rate is almost proportional to the fractional density, while the width of the region of instability is almost independent of it. Both theoretical
studies
and space observations
show that
large
124
LIU Xu-ehao
amplitude whistlers occur easily [16-M]. The magnetic energy associated with the whistlers may account for a fraction of the total energy of the anisotropic system. The saturation energy of whistlers can be estimated by the following formulae: KA(#)+ K,(&c)-'I(&?O:/C~K~)+ 2lBkJ:)- Cl, 1 K&) --",u?s(Inl, . y:P,(V,. t)d’Y. 2 5
(7)
where &w'(t) represents the turbulent magnetic energy of the whistler with wave number &. Assume the spectrum of the whistler is flat and that oP2/csKs%;2 (o~=~R. 1.69 .lOsHe; &,nrO.OPcm-'1.Then at saturation the whistler reaches a maximum energy of IV;-t [KL(O)-KKl(r)]
= IO-'ergcm-'.
(81
where z is the characteristictime of isotropisation.The diffusion coefficient of the angle of throw is D. * um@./c~a.
(9)
pw is the ratio of wave energy over total magnetic energy. Taking
the magnetic field in the filaments to be 1OG and cosa=1/3, Aa~l, we then have r = (&)‘/D.
- (aO)z/(~,,,!7,cosa)
rrr 7 x lo-%,
(10)
From Fig. 2 we see that yv is approximately equal to 3x10-30u, and rxl/2yw, so large amplitude whistlers can form. The time scale of the whistlers remaining in the instability region is z' = L/Q. Over the frequency range O.l-O,5ou, vg is approximately 0.5-0.65 times the Alfven velocity and L*5OOOkm, so z’@O.2s*z. t’ can be comparable to the growth time of Type I bursts. 2. Langmuir and ElectromagneticWaves The dispersion relations for Langmuir, whistler and electromagneticwaves under the conditions stated in Section 2 are shown in Fig. 5. Instability of the loss cone can generate Langmuir waves with a brightness temperature nZIO'°K 1191. We can make the following estimation: The growth rate of the Langmuir wave can be taken to be lo-so,. From Eq. (10) we see that its instability time scale is about 10 times its growth time. Also, if we estimate the time of isotropieationz by using the possible saturation energy of the whistler wave, then z will be underestimated.Hence we can take it that the instability time scale of the Langmuir wave is equal to several tens of the growth time. And this is is sufficient to generate the large amplitude Langmuir waves necessary for Type I bursts.
Whistler Model
125
10-a
w.-0.5wulr.
-I
o,+o.bon
0
Fig. 5 The three-wave interaction.Shown are the dispersion curves of Langmuir (L), 0- and X-mode waves
Even though the maximum growth rate of the loss cone instability is almost perpendicular to the magnetic field [ZOI, differential scattering will cause the Langmuir wave to slip to lower wave numbers and to be confined to the following cone 1211: (11) In the present case 0 should be less than 14". Of course, the Langauir wave may be directly scattered into electromagnetic wave, but the efficiency of this process is far less than that of it being scattered into the lower wave number region [8]. The time scale of the latter process is, [a],
cue -7 1089keSiF: * 0.03s. x
*?
This is far less than the sojourn time of the whistlers in the filaments stated above. 3. The 1 + W + t Process We follow the ideas of Htiijerand Wilhelnsson [221 and of Kuijpers [8] and calculate the locus of the electromagneticwave resulting from the three-wave interaction in the Kw diagram (Fig.5). Fig. 2 shows that the whistler wave generated by the loss cone instability in strog magnetic field regions is in the frequency range 0.2-0.40s. Here, we take ow=O.30~ for the calculation. For &=0.04, the electromagneticwave produceable by the coupling of the whistler and Langmuir waves is in the shaded area of Fig. 5; for k&=0.03, it is in the larger
126
LIU Xu-zhao
triangle marked by dashed lines. The probability of three-wave interaction involving a whistler is, 1231, u(K, L, IL) - ~%.
(12)
This shows that the waves couple only if the whistler wave packets are not strictly along the magnetic field. As pointed out above, the wave packets are basically along the field, being inside a 19.5' cone. Hence, the whistlers that can participate in the three-wave interaction are along a hollow cone with the magnetic field as axis. Note that in Fig. 5 the whistler is taken to be along the magnetic field. When the whistler wave packets are inclined at a finite angle to the field, the final transverse wave number will be increased, that is, the whole area will be compressed upward and the escaping transverse waves will tend to move more along the field.
4. COMPARISON WITH OBSERVATIONS In a magnetized plasma, plasma emission is generally polarised. It is easily seen from Fig. 5 that it is difficult for the X-mode waves to satisfy the resonance condition. Hence, in a whistler model, Type I bursts are strongly O-mode circularly polarized waves. In this work we regard high directivity of noise storms to be characteristicof Type I bursts. It is a criterion that distinguishes the different low- and high-frequencydisturbances that coupled to generate the observed radiation. The diagram of three-wave interaction affords a qualitative analysis of the directivity of the Type I bursts generated. First, the diagram shows that electromagneticwaves along the magnetic field are the easiest to generate. Next, the Langmuir and whistler waves both have high brightness temperatures, as the Langmuir wave through differential scattering slips into low wave number region, as soon as it enters the region of possible interaction,strong coupling will take place and there will be effectively no further degradation into lower wave number region. In this model, Type I burst sources originate in coronal filaments, the axes of different bursts will differ while their directional diagrams will roughly agree [241. The discussion in Section 2 on the structure of coronal filaments already gave a good explanation of the characteristic features of Type I bursts, their narrow bandwidth and short duration. The upper limit to the brightness temperature of the electromagnetic wave generated by the three-wave interaction here is determined by the brightness temperature of the Langmuir wave. Section 3 shows that the fast electrons with the loss cone distribution are capable of generating Langmuir waves of the
Whistler
required
127
Model
temperature.
In the present study, the brightness temperature of the lowfrequency wave will ensure the following two circumstances: one is that it will permit the Langmuir wave to be relatively weak so that the second harwnic will not appear, one is that as soon as the Langmuir wave slips into a region of interaction, a strong coupling will take place, preventing it from slipping into still lower wave number region. In this case the radiation will be highly directional. The co-existence (association in space and time) of the noise continuum and Type I bursts and certain similar features (such as polarieationf show that they have the same origin. On the other hand, some of the main features of Type I bursts are quite radically different from the noise continuum and this shows that their emission mechanisms are different. We cannot regard Type I bursts as acre enhancements in the fluctuations in the noise storm. The fact electrons generated by weak shocks passing through coronal loops will eventually acquire an isotropic distribution. Such fast electrons can generate Langmuir waves with nslO’°K. A different low-frequency wave (such an ion-acoustic wave) combines with the Langmuir wave to generate the continuum emission. The directivity of this radiation (about 60*) can be explained by propagation effects.
5. INCLUSION This paper qualitatively describes the background and plasma processes in the generation of aeter wave noise storms in the corona. iI Filaments with strong magnetic fields are present in the coronal region that produces the meter wave noise storms, The size and configuration of the filaments determine the duration and bandwidth of Type I bursts. The strong magnetic field in the filaments, about 3-4 times the average field in an active region, causes the loss cone distributed fast electrons to generate whistlers with a rather high frequency (-0.30~). ii) Magnetic evolution of the active region generates weak shocks. The latter pass through a coronal active region (particularly strong-field filaments) and generate fast electrons with a loss cone distribution. iii) Instability of the loss cone distribution generates high brightness temperature Langauir waves and whistlers. These coalesce into escaping radiation (Type I bursts). iv) Finally, the fast electrons throughout the active region acquire an almost isotropic distribution. These electrons then generate Langmuir waves with brightness temperatures less than 10°K. These then couple with low-frequency waves such as ion-acoustic waves to produce the observed noise continuum. The model proposed here can hopefully of solar meter band noise storms.
explain
the main features
128
LIU xd-shaa
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