An approximate analytic theory of acoustic shock propagation in nonuniform media is used to determine the motion of a flare-generated shock wave in the corona. The shock is followed from the time it strikes the chromosphere-corona transition region (density interface) out to 5 solar radii under the assumption that the corona in this region is approximately in hydrostatic equilibrium. The strength of the shock incident on the transition region from below determines the ejection velocity of eruptive prominence material, as well as the initial velocity of the coronal shock. The calculation is applied to one well-documented case of a related flare spray, moving type 4 isolated source, and type 2 burst. It is shown that a chromospheric shock of the appropriate strength to produce the observed prominence and type 4 velocities strengthens as it moves out in the corona by an amount sufficient to account for the observed high velocity of the type 2 burst

Kopp, R. A.

English

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A GAS-DYNAMIC CALCULATION OF TYPE I1 SHOCK PROPAGATION 
THROUGH THE CORONA Roger A. Kopp 
ABSTRACT An approximate analytic theory of acoustic shock propagation in nonuniform media is 
used to determine the motion of a flare-generated shock wave in the corona. The shock is 
followed from the time it strikes the chromosphere-corona transition region (density 
interface) out to 5R,  under the assumption that the corona in this region is 
approximately in hydrostatic equilibrium. In the actual corona, the situation would apply 
to the case of slow-mode shock propagation along a radially diverging magnetic field. 
The strength of the shock incident on the transition region from below determines the 
ejection velocity of eruptive prominence material, as well as the initial velocity of the 
coronal shock. The calculation is applied to one well-documented case of a related flare 
spray, moving type IV isolated source, and type I1 burst. It is shown that a chromospheric 
shock of the appropriate strength to produce the observed prominence and type IV 
velocities strengthens as it moves out in the corona by an amount sufficient to account 
for the observed high velocity of the type I1 burst. 
INTRODUCTION 
It is now regarded as a well-established fact that type I1 
solar radio bursts are the direct result of flare-generated 
shock waves traveling outward through the corona 
[Wild, 1969, 19701. Whether such waves are primarily 
acoustic (slow mode) or magnetohydrodynamic (fast 
mode) in character, however, is still an open question. 
Many workers have relied on magnetic effects to explain 
certain features of the bursts, such as band-splitting 
[Sturrock, 19611 , although nonmagnetic explanations 
are frequently available as well [McLean, 19671 . 
If the waves are acoustic in nature, then rather large 
shock Mach numbers are required to produce the 
observed burst velocities of the order of 1000 kmlsec 
[Weiss, 19651. Nevertheless, at least low in the corona 
the material motions following a flare, as evidenced by 
the underlying eruptive prominence (or so-called “flare 
The author is at the High Altitude Observatory, National Center 
for Atmospheric Research, Boulder, Colorado. The National 
Center for Atmospheric Research is sponsored by the National 
Science Foundation. 
spray”), appear to be largely channeled along magnetic 
lines of force. Observational evidence for magnetic fields 
influencing the motion of a type I1 source at higher 
levels has been presented by Kui [1969]. However, at 
feast in some cases shock propagation must occur across 
field lines if one is to explain the Moreton-wave 
phenomenon as the intersection of a type I1 wave front 
with the chromosphere [Wild, 1969; Uchida, 19701. 
Occasionally associated with a type I1 burst and 
prominence eruption, but occurring later in time and 
having a much longer duration, is a moving burst of type 
IV. Smerd and Dulk [ 1970) distinguish three subclasses 
of type IV: the isolated source, the expanding arch, and 
the advancing front. Here we are concerned only with 
the first of these, for which it appears that the source 
moves outward along predominantly open magnetic field 
lines [Smerd and Dulk, 1970; Dulk and Altschuler, 
19711. 
Type IV bursts of the “isolated source” variety are 
relatively infrequent; Smerd and Dulk [ 19701 have 
identified possibly eight examples observed with the 
80-MHz radioheliograph during the period February 
252 
https://ntrs.nasa.gov/search.jsp?R=19730002056 2020-03-17T07:21:59+00:00Z
1968 to April 1970. Particularly well observed was the 
event of March 21, 1970, reported by Sheridan [ 19701 : 
Following a flare of importance 2N near the east limb at 
OOh 23m, a large eruptive prominence was seen in Ha! to 
leave the flare location with a velocity of 260 krnlsec. It 
reached a height of 0.3 R ,  above the photosphere before 
fading from view. Simultaneously, radiospectrograph 
records indicated a type I1 burst with a velocity of order 
1500 km/sec [Smerd and Dulk, 19701. Type I1 bursts 
are frequently observed to occur in pairs after large 
flares, presumably the result of multiple shocks being 
emitted by the flaring region. This was the case with the 
event being described. We refer here to the second, and 
stronger, burst.) A short time after the prominence 
faded, a type IV source appeared at r =  1.9 R ,  and 
subsequently moved outward to a distance of more than 
5 R,  with a constant 29O-km/sec velocity, whereupon it 
fragmented and slowly faded away. The source velocity 
was comparable to that measured for the prominence 
eruption; extrapolation backwards in time showed that 
both left the sun during the flash phase of the flare. 
Sheridan [ 19701 suggested that the outward-moving 
prominence material excited the type IV source near the 
r = 1.9 R, levei However, such an interpretation does 
not account for the nearly equal velocities of the source 
and the prominence ejecta. Alternatively, we may 
conjecture that the flare spray and the type IV source 
are simply different manifestations of the same material, 
the ejected “blob” becoming visible at 80 MHz only 
after the prominence fades in Ha! and rises significantly 
above the 80-MHz plasma level in the corona. 
Here we show, by means of a simple model, that a 
relationship exists between the velocities of the eruptive 
prominence and moving type IV isolated source (here 
assumed to be the same) and that of the type I1 burst. 
We postulate that these three phenomena are the natural 
consequence of a single flare-initiated shock traversing 
the solar atmosphere. In the model, the velocity of the 
flare spray is taken as a measure of the strength of the 
shock at the top of the chromosphere. With this 
information and knowledge of the coronal density 
structure, the shock strength and velocity may then be 
calculated as functions of height in the corona. AS an 
example, we apply the calculation to the event observed 
by Sheridan [ 19701. 
SHOC K-W AV E CALCULATIONS 
The shock wave emitted by the flare during its flash 
phase is assumed to propagate vertically upward, or 
nearly so, through the chromosphere. Here we will be 
concerned with its motion only after the shock reaches 
the “top” of the chromosphere. By that time we assume 
it to be a slow mode shock traveling along the magnetic 
field. 
The atmosphere ahead of the shock is characterized by 
large gradients of temperature and density. It is conven- 
ient to divide the calculation of the subsequent shock 
motion into two parts, corresponding roughly to regions 
with large or small temperature gradients. 
The shock immediately encounters the chromosphere- 
corona transition region, where the preshock atmos- 
pheric density drops abruptly by more than 2 orders of 
magnitude over a radial distance of less than a scale 
height. We idealize this region to be a sharp density 
interface (infinite temperature gradient), with equal 
pressures above and below. 
The interaction of a slow-mode shock with a density 
interface is shown schematically by an (x, t )  diagram in 
figure 1. Regions 0 and 3 represent the undisturbed 
chromosphere and corona, respectively, separated by the 
contact discontinuity CC‘. When the chromospheric 
Shock Interaction with the Transition Region 
t 
I d 
Figure 1. A distance-time (x,t)  diugram of a plane 
shock wave,& impinging normally upan a contact 
discontinuity Cc‘. The case shown here is that of the 
shock entering a region of lower density, for which 
(assuming equal specific heat ratios) the reflected wave, 
d, is a rarefaction. 
253 
shock$ strikes the interface, it sets the latter in motion. 
At the same time a transmitted shock3and a reflected 
rarefaction wave 61 propagate away from the contact 
surface in the directions shown. Region 1 indicates the 
shocked chromospheric gas, region 4 the shocked coro- 
nal gas, and region 5 the shocked chromospheric gas, 
which has been somewhat recooled by the downward- 
propagating rarefaction wave. 
In this picture the H a  prominence eruption is identi- 
fied with chromospheric material immediately below the 
ascending interface (region 5); the velocity of the 
interface v s  corresponds to that of the prominence. 
This shock wave-density interface model of a flare 
spray is similar to that proposed by Osterbrock [1961] 
for spicules, except that a much stronger disturbance is 
involved here. There are six nontrivial, independent 
algebraic relations between the state variables in the five 
regions of the ( x ,  t) diagram (the trivial ones being 
po = p3, vo = v 3  = 0, p4 = ps , and v4 = v5 ). Apart from 
differences in notation, they have been given by Oster- 
brock. Together with the known properties of the 
undisturbed chromosphere and corona, these relations 
completely determine the shock-interface interaction in 
terms of the strength of the incident shock. Solution by 
iteration is straightforward. 
As a numerical example, we assume a chromospheric 
temperature To = 7X103 O K ,  a coronal temperature 
T3 = 1.5X106 OK, and a 1O:l hydrogen-helium ratio 
(mean molecular weight p = 0.609). The specific heat 
obtain an interface velocity of about 290 kmlsec corre- 
sponding to the eruptive prominence-moving type IV 
source velocity of the event of March 21, 1970 
[Sheridan, 19701 - 
As the transmitted coronal shock moves away from the 
sun it traverses a medium with continuously varying 
properties, which cause additional changes in its 
strength. To describe the motion of the shock in the 
corona we use the method of Whitham [1958], which 
was applied to the case of a spherical shock in a 
gravitational atmosphere by Kopp [ 19681. For a corona 
in hydrostatic equilibrium this theory yields the follow- 
ing total differential equation' for shock Mach number 
M: 
Shock Propagation in the Corona 
(1) 
In this equation, r denotes distance from the cen- 
ter of the sun, g(r) = GM,/r2 is the gravitational accel- 
eration, a = (yRT/p)In is the velocity of sound, and 
(l/A)/(dA/dr) is the rate of geometric spreading of the 
wave. In addition, 
ostshock gas velocit 
Y + l  pEshock sound veloci; 
postshock sound velocity 
preshock sound velocity 
ratio y is taken to be 5/3 on both sides of the interface. 112 
Table 1 shows the results of a calculation in which the 
strength of the chromospheric shock was adjusted to 
5 = 
Table 1.  Shock-density interface calculation 
Chromosphere 
Corona 
Preshock 
To = 7 X l d  OK 
T3 = 1.5X1$ OK 
Flare spray V S  = 288.1 km/sec 
T, = l.10X105 "K 
Chromospheric shock Mach number 
Coronal shock Mach number 
M = 14.26 
M = 2.48 
(V = 179.8 km/sec) 
(V = 457.8 km/sec) 
254 
and 
where 
27 * 2 - Y  - 1 = postshock pressure 
4 7 + 1 preshock pressure 
Since E > O  and t> 1, we have that 5’ + E {  1 > 0 ,  so 
that the first term in equation (1) corresponds to  a 
strengthening of the shock as it moves into regions of 
lower density. Similarly, the second term yields a 
weakening of the wave due to geometric attenuation, 
and the third term indicates the effect of a coronal 
temperature gradient on the shock. 
Equation (1) predicts changes in shock strength 
resulting only from nonuniform properties of the pre- 
shock gas. In addition, changes occur in response to 
disturbances overtaking the shock from the rear. These 
include (1) re-reflected waves generated by the changing 
shock strength itself, and (2) waves produced by 
variations in the motions of the shock “driver.” The 
basis of Whitham’s [I9581 method is that the first of 
thep disturbances can be neglected in an approximate 
description of the shock motion. For the coronal shock 
under consideration here, we suggest that the “driver” 
waves can be neglected as well, since the observed 
constapcy of the type IV velocity implies a continuously 
driven type 11 shock. 
Equation (1) can be numerically integrated for a 
corona of known temperature and for a shock of known 
geometrical spreading. Again, as an example, we adopt a 
constant coronal temperature of 1.5X 106’K (a = 184.6 
kmlsec). Using the transmitted shock strength, from the 
interface calculation, Mo = 2.48, as an initial condition, 
and assuming a slow-mode shock channeled along a 
radially diverging magnetic field [(l /A)(dA/dr)= 2 / r ] ,  we 
obtain the radial dependence of shock speed shown in 
figure 2. Then the path of the shock in the (r, t )  diagram 
can be found from 
6oo t/ 
400 
I 2 3 4 5 
q r/Re 
Figure 2. Shock velocity versus height in a 
I . 5 X I 0 6  “K corona, assuming a radially diverging 
geometry and a driven shock. 
0 500 1000 1500 2000 2500 3000 
t(sec) 
Figure 3. Shock path r(t) corresponding to figure 2. 
(2)  
percent of it8 maximum value of 1 1  65 kmlsec, whi& 
occurs near 4.8 R,. Evidently, the gravitational and area 
terms in equation (1)  nearly balance each other over a 
rather broad region. 
Considering the uncertainties inherent in deducing 
type I1 source velocities from clynamic spectxa [Zaitsev, 
1969; Van Bueren and Kupems, 19701, ?he maximum 
where t = 0 corresponds to the time when the shock 
leaves the transition region (r = R,). The shock path for 
the above example is shown in figure 3. 
The most striking aspect of this calculation is the rapid 
strengthening of the shock below about 2 R,. Beyond 
r =  2 R ,  the shock speed varies by only about 20 
255 
shock speed obtained by this calculation is in reasonable 
agreement with the observed type I1 velocity for the 
event of March 21, 1970 [Smerd and Dklk, 1970; cf. 
previous section]. 
DISCUSSION 
We have seen that the eruptive prominence, moving type 
IV isolated source, and type I1 burst associated with a 
particular flare may all be the result of a flare-initiated 
shock wave traveling outward along the magnetic field. 
When the shock strikes the transition region, the 
chromospheric material set in motion by the shock 
continues to rise into the corona and initially becomes 
visible as the ascending H a  prominence, in the manner 
proposed by Usterbrock [1961]. Later, as the Ha 
emission fades and the material passes the 80-MHz 
plasma level in the corona, it appears as a moving source 
of type IV emission. 
The major predictions of the shock-interface calcula- 
tion are that: 
1 .  A strong slow-mode shock is required in the upper 
chromosphere to produce a flare spray of the 
observed velocity. Such a shock could result either 
from intense heating in the flaring region or from 
rapid growth of a weak flare-initiated shock as it runs 
outward through chromospheric layers of succes- 
sively lower density. 
2. The kinetic temperature of the rising prominence 
(-lo5 OK) greatly exceeds that of the chromosphere, 
whereas the density is somewhat less. This may 
account for the eventual disappearance of the spray 
in Ha as a new statistical equilibrium becomes 
established at the higher temperature. 
3. The transmitted shock is of intermediate strength. In 
particular, its velocity is too low to match that of a 
type I1 burst. 
The flare-excited shock continues into the corona, 
where the outward-decreasing density leads to its re- 
strengthening and acceleration to typical type I1 veloci- 
ties. The observed constancy of the type IV source 
velocity out to distances of several solar radii lends 
credence to the postulate that the type I1 Shock is 
steadily driven from below. 
The derived shock speeds depend, of course, on the 
assumed values of the coronal temperature and the 
geometrical spreading factor. For example, a higher 
coronal temperature would decrease the initial rate of 
strengthening of the shock through the “scale-height” 
term in equation (1). Moreover, the shock Mach number 
would then reach its maximum value closer to the sun. 
In spite of the increased speed of sound in the corona, 
the result would be a lower shock velocity. 
It would be extremely interesting to follow the motion 
of such a shock to large distances from the sun 
(r> 5 Re),  as it is possible that type I1 shocks may be 
detected as collisionless interplanetary shock waves at 
1 AU. Such a calculation is tractable at the present time, 
although it requires explicit consideration of the expan- 
sion state of the preshock atmosphere. 
ACKNOWLEDGMENTS 
The author wishes to thank Drs. G. Dulk and G. New- 
kirk, Jr., for several illuminating discussions. Mrs. Nancy 
Werner kindly assisted with the numerical calculations, 
and Mr. V. Pizzo offered valuable technical assistance. 
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Kopp, R.A.: The Dynamical Structure of a Shock- 
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257 


A gas-dynamic calculation of type 2 shock propagation through the corona

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