b. vrsnak and s. lulic- formation of coronal mhd shock waves i: the basic mechanism

8/3/2019 B. Vrsnak and S. Lulic- Formation of Coronal MHD Shock Waves I: The Basic Mechanism

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FORMATION OF CORONAL MHD SHOCK WAVES

I. The Basic Mechanism

B. VRNAK and S. LULICHvar Observatory, Faculty of Geodesy, Kaciceva 26, HR-10000 Zagreb, Croatia

(Received 14 February 2000; accepted 26 May 2000)

Abstract. The formation and evolution of a large amplitude MHD perturbation propagating per-pendicular to the magnetic field in a perfectly conducting low plasma is studied. The perturbationis generated by an abrupt expansion of the source region. Explicit expressions for the time and thedistance needed for the transformation of the perturbations leading edge into a shock wave arederived. The results are applied to coronal conditions and the dynamic spectra of the radio emissionexcited by the shock are synthesized, reproducing metric and kilometric type II bursts. The featurescorresponding to the metric type II burst precursor and the moving type IV burst in the case ofkilometric type II bursts are identified. A specific radio signature that is sometimes observed at theonset of a metric type II burst is found to appear immediately before the shock wave formationdue to the associated growth of the magnetic field gradient. Time delays and starting frequencies ofbursts onsets are calculated and presented as a function of the impulsiveness of the source-regionexpansion, using different values of the ambient Alfvn velocity and various time profiles of theexpansion velocity. The results are confronted with the observations of metric and kilometric type IIsolar radio bursts.

1. Introduction

Various phenomena reveal the propagation of MHD shock waves through the solarcorona. Type II radio bursts observed in the metric wavelength range (Nelson andMelrose, 1985) are caused by MHD shock waves spreading out over distancesof several solar radii (Bougeret, 1985). Analogous radio events are observed atkilometric wavelengths, tracing shock waves in the interplanetary space from 10solar radii to 1 AU (Gopalswamy et al., 1999). On the other hand, chains of type Isolar radio bursts observed in the metric wavelength range during noise stormsindicate the coronal shock waves propagating at much smaller time/distance scales(Karlick and Odstrcil, 1997).

The interplanetary shock waves traced by the type II bursts in the frequencyrange of 30 kHz 2 MHz, are also observed by interplanetary scintillation measure-ments (Manoharan, 1997) and by in situ measurements onboard several spacecraft

(Kallenrode et al., 1993). These events are associated with fast coronal mass ejec-tions and long-lived solar energetic (E > 1 Mev) particle events (Kahler, 1994;Cane, 1997).

The coronal shock waves exciting metric type II bursts sometimes also causechromospheric Moreton waves (Moreton, 1960; Smith and Harvey, 1971). Uchida

Solar Physics 196: 157180, 2000. 2000 Kluwer Academic Publishers. Printed in the Netherlands.


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158 B. VRNAK AND S. LULIC

(1974) has shown that both phenomena can be ascribed to a fast-mode MHD shockwave propagating away from active regions along valleys of a low Alfvn ve-locity. The EIT instrument onboard SOHO disclosed similar disturbances in thecorona (Thompson et al., 1998; Klassen et al., 2000).

Observations indicate that type II burst shock waves have velocities in the orderof 1000 km s1 and have low Mach numbers, usually between 1.2 and 1.7 (Nelsonand Melrose, 1985). Mann, Classen, and Aurass (1995) found that the frequencydrift of type II bursts depends on the starting frequency and that there is a correla-tion between the bandwidth of the band splitting and the drift rate of the burst. Theyinferred that type II bursts are excited either by subcritical quasi-perpendicular, orweak supercritical quasi-parallel fast magnetosonic shocks.

Detailed examinations of high-sensitivity digital radio spectra revealed two phe-nomena associated with metric type II bursts, shedding a new light on the formationof coronal shock waves. Klassen et al. (1999a) showed that the occurence of atype II burst is often preceded by a cluster of numerous impulsive fast-drifting

bursts in the decimetric wavelength range. The pattern is usually characterizedby a slowly drifting high-frequency edge and is associated with an impulsive mi-crowave and hard X-ray burst (Klassen et al., 1999a). This radio signature, namedby Klassen et al. (1999a) type II burst precursor, was in fact noted also by Karlick(1984). On the other hand, immediately before the onset of a type II burst, closeto its starting frequency, a feature having a triangular or inverted-U spectral shapeis sometimes observed (Aurass, Magun, and Mann, 1994). Due to its shape thisfeature was called arc by Klassen et al. (1999a).

Whereas coronal mass ejections (CMEs) were unambigously identified as asource of interplanetary shock waves, there are still doubts whether metric type IIbursts are generated by flare-ignited blasts or by fast CMEs (for a review see Cliver,

Webb, and Howard, 1999). Consequently, there are two actual scenaria for theformation of coronal shock waves. One is the piston mechanism (Maxwell andDryer, 1982; Maxwell, Dryer, and McIntosh, 1985) and the other is the pressurepulse mechanism (Cane and White, 1989). In the first case the shock should beproduced by a supersonic mass ejection acting as a moving piston (e.g., an eruptiveprominence, spray or CME). In the second case a sudden pressure pulse produces aMHD blast wave which subsequently transforms into a shock wave. Mann (1995)has studied such a transformation, motivated by the discovery of large-amplitudelow-frequency magnetic field fluctuations in the foreshock of the Earth. Karlickand Odstricil (1994) have studied such a process numerically, simulating coronalconditions and demonstrated that a type II burst and the associated phenomenaobserved in the microwave wavelength range could be the consequences of a MHD

perturbation caused by a flare.In the following it will be shown that an abrupt, fast expansion of the source

region can generate a fast-mode MHD shock wave, regardless of the cause of theexpansion. It may be a pressure pulse caused by an impulsive heating (flare), as wellas an ejection of matter driven by some ideal MHD instability (CME). A simple


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FORMATION OF CORONAL MHD SHOCK WAVES, I 159

analytical 1-D model will be used to investigate the formation and evolution of theleading edge of a large amplitude magnetosonic perturbation in general. The goal isto estimate the time/distance at which the shock wave is created, and to relate thesequantities with the time profile of the source-region expansion velocity. Then, themodel will be applied to the propagation and evolution of the perturbation throughthe corona and interplanetary space and the results will be confronted with theobservations.

In general, the magnetic field in active regions has a bipolar structure. So ini-tially, the perturbation travelling upwards from the core of an active region hasto propagate perpendicular to the field lines. A similar situation holds for theazimuthal spread away from the active region. Since the intention of this studyis to analyse the evolution of the perturbation from the moment of its formation,only MHD perturbations propagating perpendicular to the magnetic field will beconsidered. Parallel-propagating waves were treated numerically by Cohen andKulsrud (1975). A more general treatment of large amplitude MHD perturbations

can be found in Mann (1995).

2. Formation and Evolution of the Perturbation

2.1. THE MODEL

Let us consider a general 1-D situation in which the magnetic field is alignedin the y-direction, and all quantities are uniform in the y- and z-directions. Thesource region (further on also the internal region, or i-region) is initially confinedto x < x0. At t = 0 it starts to expand, pushing the ambient region (further onthe external region, or e-region). At a moment t > 0, the boundary between thei- and e-region (further on the boundary) has the coordinate xt and the velocityut, representing the Lagrangian coordinate and velocity of a plasma element at theboundary, and one can write:

ut =xt

t, xt = x0 + xt = x0 +

t0

ut dt . (1)

The ambient magnetoplasma is initially (t < 0) at rest and homogeneous, charac-terized by the magnetic field strength B0, the plasma density 0, the Alfvn velocityvA0 , the plasma temperature T0, the gas pressure p0 and the magnetic pressurepB0 = B

20 /20 (Figure 1).

The plasma in front of the boundary is pushed and compressed by the fastexpansion of the i-region. Denoting the quantities that describe the external region(x > xt) by the subscript e (Figure 1), one can write Be > B0, Te > T0 ande > 0, implying also pBe > pB0 and pe > p0. So, the Alfvn and the soundvelocity are increased with respect to the initial values (vAe > vA0 , ce > c0).


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Figure 1. A general definition of the internal region (shaded) and the external region: (a) before the

onset of expansion; (b) after it.

In the following it will be assumed that the plasma is perfectly conducting(i.e., magnetic diffusivity = 0) which implies that the frozen-in condition issatisfied (Priest, 1982). Furthermore, it will be assumed that the ratio of the gas andmagnetic pressure is 0 = p0/pB0 1 (Dulk and McLean, 1978). This impliesthat c0 vA0 , meaning that the fast magnetosonic velocity v0 in the unperturbedplasma is approximately equal to the Alfvn velocity, v0 vA0 . In the Appendix(see Equation (A.12)) it is shown that under these approximations the conditione 1 holds also after t = 0. Then, the behaviour of the e-region is governed bythe equation (see Appendix)

u

t+ (v(u) + u)

u

x= 0 . (2)

Here u = u(x,t) is the plasma flow velocity, and x and t are the Eulerian vari-ables. The velocity v(u) is related to v0 and u as v = v0 + u/2 (see Appendix,Equation (A.21)).

In the case of a large amplitude perturbation Equation (2) shows that the ve-locity of an element of the perturbation depends on the associated plasma flowvelocity, which means that each point of the wave is propagating at its own restframe speed w = v + u (see Landau and Lifshitz, 1987). The compression ofthe magnetoplasma in the e-region causes an increase of the local Alfvn velocity(i.e., v0 v). As the magnetoplasma is pushed outwards faster, the associatedperturbation travels outwards at an accordingly higher velocity. A newcoming el-ement of the perturbation (further on signal) travels faster, and after some time itwill reach the previous signal. A discontinuity forms in the function u(x), meaningthat a shock develops at the considered segment of the perturbation.


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Figure 2. (a) Schematic presentation of the evolution of the perturbation profile. (b) Motion of theboundary (xt xt ) and formation of the perturbation (ut (x)) the relation between the Lagrangianco-ordinates and velocities (xt, xt , ut, ut ) and the Eulerian ones (xt (u),ut (x)). (c)The shock waveformation. The quantities in front of the discontinuity are denoted by the subscript 1 and behind itby 2. The motion of the discontinuity relative to the upstream plasma (flowing at the velocity u

1)

is characterized by the local Mach number M12.

Knowing the function u(x) (further on the perturbation profile) at the momentt1 it is possible to determine further evolution of the profile. (Figure 2a). Theperturbation profile at the moment t2 > t1 is defined by its inverse function x(u):

xt2 (u) = xt1 (u) + (v + u)(t2 t1) (3)

(Landau and Lifshitz, 1987; Mann, 1995). Here xt1 (u) and xt2 (u) are Eulerian co-ordinates defining the location of the signal characterized by the flow velocity u, atthe moments t1 and t2, respectively (Figure 2(a)). The second term on the right hand

side of Equation (3) represents the distance travelled by the signal during the timeinterval t2 t1, propagating at the rest frame velocity w(u) = v + u = v0 + 3u/2.

2.2. SOURCE-REGION EXPANSION AND THE PERTURBATION PROFILE

Let us relate the motion of the source-region boundary with the perturbation profile.It will be assumed that the motion of the boundary is prescribed by the functionuL(t) ut representing the Lagrangian velocity of a plasma element located at theboundary. It will be assumed that ut is a monotonically increasing function in thetime interval 0 < t < tm. The Lagrangian coordinate xL(t) xt of the boundary isdefined by Equation (1). Let us suppose that during the infinitesimal time intervalt = t t the Lagrangian co-ordinate and velocity of the boundary change from

xt and ut to xt and ut , respectively. This change (xt xt , ut ut ) is shown inthe u(x) graph in Figure 2(b) by the dotted arrow. At the moment t the Lagrangiancoordinate of a plasma element corresponds to the Eulerian coordinate xt(u) ofan element of the perturbation profile characterized by the plasma flow velocityu = ut that is located at x = xt, which means xt = xt(ut). During the time


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interval t this element of the perturbation profile advances to xt (ut) accordingto Equation (3). This displacement is drawn by the dashed arrow in Figure 2(b).Taking into account xt(ut) = xt one can write:

xt (ut) = xt + (vt + ut)(t t) , (4)

where vt = v0 + ut/2 (Equation (A.21)). Equation (4) determines parametricallythe profile of the perturbations leading edge at the moment t, using t as the para-meter in the interval 0 < t < t m. The parameter t defines the value of the plasmaflow velocity u = ut and the location xt from where the signal characterized by theflow velocity ut was emitted.

The relation between the slope u/x of the spatial perturbation profile definedby the function u(x) and the acceleration of the boundary (ut/t) defined by thetime profile ut can be found applying

xu

=

x

tu

t

, (5)

since u and x are defined by the parameter t. Taking the derivative of Equation (4)with respect to t at t = t and taking into account xt/t = ut, one finds x/t =v. Bearing in mind that u(x) is a monotonic function, one can apply u/x =(x/u)1. Using Equation (5) and x/t = v one gets:

u

x=

1

v

u

t. (6)

This result can be obtained less formally, but more transparently, by the geometric

consideration shown in Figure 2(b). In the infinitesimal time interval t the velocityof the boundary increases for u = (ut/t)t and it advances in the x-directionfor xt = ut t. At the same time the element of the perturbation characterized bythe flow velocity ut advances for xw = wtt = (ut + vt)t. From Figure 2(b)one finds that the perturbation elements characterized by the flow velocities ut andut + u are separated by x = vtt. So, one finds u/x = (1/v)(u/t).

3. Shock Wave Formation

3.1. FORMATION OF THE DISCONTINUITY IN THE PERTURBATION PROFILE

Let us consider the propagation of two adjacent elements of the perturbation profile(signals) in the case of a monotonically decreasing function u(x) generated by amonotonically increasing function ut. The signals are initially separated by x . Thesignal located at a larger x (the earlier signal) is characterized by the flow velocityu, whereas the signal located at a smaller x (the later signal) by the flow velocity


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u + u. The corresponding rest frame signal velocities differ by w = 3u/2, sincew = v + u = v0 + 3u/2. The later (faster) element will reach the former one anda discontinuity will be created in the perturbation profile, after the time

t = xw

= xw

xx

= 13

2

u

x

, (7)

where it was taken into account that the function u(x), and thus also w(x), has anegative slope (Figure 2(a)). The time interval t is shortest for the segment of theperturbation having the steepest gradient (u/x = min.), i.e., the discontinuitywill begin to develop at the segment where 2u/x2 = 0.

Substituting Equation (6) into Equation (7) the time interval needed for anelement of the perturbation to create a discontinuity can be written as

tt =v

3

2u

t

t=t

, (8)

where t = t is the moment at which the signal is emitted. Taking into accountv = v0 + u/2 one gets

tt = v

3v

t=t

=

2v0 + u

3u

t=t

, (9)

where u = u/t and v = v/t = u/2. For a given function ut and for aparticular value ofv0, Equation (9) determines the time needed for an element ofthe perturbation generated at t = t to form a discontinuity in the perturbationprofile.

After the normalization with respect to tm, Equation (9) becomes

=

V

3V

=

=

2 + U

3U

=

, (10)

where = t/tm, = t/tm, V = v/v0, V = V/,U = u/v0 and U =U/. Using Equation (10) one can determine the moment s at which the signalemitted at the moment will form a discontinuity:

s = + = +2 + U

3 U. (11)

The shock formation starts at the segment of the perturbation for which the function

s has a minimum, i.e., s / = 0. Let us denote the moment at which this per-turbation element is created as 0 = t0/tm. Taking the derivative of Equation (11)and demanding s / = 0, one gets the condition

(2 + U )U = 4U2 , (12)


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which determinates 0. Here we denoted U = 2U/2. Once 0 is known, themoment b at which the shock formation begins can be evaluated substituting =0 into Equation (11):

b = 0 + V0

3(V )0= 0 + 2 + U

0

3(U )0. (13)

Let us denote by db the distance between the location of the boundary at themoment 0 and the location at which the shock formation starts. Since db =wtt0 , where tt0 is defined by Equation (9) for t

= t0, one can write:

Db = W0 0 = (V0 + U0 )0 =

1 +

3

2U0

(2 + U0 )

3U0, (14)

where Db is the distance db normalized with respect to the distance d0 = v0tm.We denoted 0 = tt0 /tm, whereas the normalized rest frame velocity of thesignal launched at

0can be written as W

0= w

0/v

0= V

0+ U

0. The distance

from the initial position of the source-region boundary can be expressed in thenormalized form as

Db = X0 + Db =

00

U d +

2 + 4U + 32 U

2

3U

0

, (15)

where X0 is the distance (normalized with respect to d0 = v0tm) that the bound-ary travelled till the moment 0 (see Equation (1)).

Equations (13) and (15) show that the discontinuity appears sooner and that thestarting distance db = Dbv0tm is shorter for the events characterized by a highervalue ofU (more impulsive events). The same holds for a lower ambient Alfvn

velocity. The time delay and starting distance depend also on the time profile ofthe boundary expansion velocity, since 0 depends on the form of the function U(Equation (12)).

3.2. COMPLETION OF THE SHOCK

When the discontinuity in the perturbation profile is created, the jump conditions atthe shocked segment of the perturbation must be satisfied (see, e.g., Benz, 1993).The situation is presented schematically in Figure 2(c). Let us denote the normal-ized plasma flow velocities in front and behind the discontinuity as U1 = u1()/v0and U2 = u2()/v0, respectively, and the corresponding normalized local wavepropagation velocities as V1 = v1()/v0 and V2 = v2()/v0. For the perpendicular

shock in a low plasma, the RankineHugoniot relations reduce to a rather simpleform. Neglecting the plasma pressure ( 1) and using = /0 = V2 andpB =

2V20 /20 (see Appendix, Equations (A.13) and (A.14), respectively), theratio of the plasma density behind and in front of the discontinuity, 12 = 2/1,can be expressed as


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12 =

( 52 + M

212)

2 + 8M212 (52 + M

212) . (16)

Here, M12 = vs /v1 is the local Mach number at the moment , and vs is thevelocity of the shocked segment of the perturbation in the frame moving at thevelocity u1. The shock moves in the rest frame at the normalized velocity M() =Vs ( )+U1( ), where Vs = vs /v0. Equation (16) shows that the compression 12 =2/1 = (V2/V1)

2 cannot be larger than 4, i.e., V2/V1 cannot be larger than 2,implying Vm < 2 and thus, Um < 2 (see Appendix, Equation (A.22)).

Equation (16) provides the expression for the local Mach number M12:

M12 =

(5 + 12)12

8 212, (17)

which determines the velocity of the shocked segment of the perturbation relativeto the upstream plasma flowing at the velocity U1. The discontinuity propagates inthe rest frame at the Mach number M:

M = Vs ( ) + U1( ) = M12()V1( ) + U1() . (18)

The position of the discontinuity can be expressed as

Xs = X0 +

00

Ud +

2 + 4U + 32 U

2

3U

0

+

0

M d , (19)

where Xs = xs /d0 is the normalized co-ordinate of the discontinuity at the moment. The first term on the right-hand side of Equation (19), X0 = x0/d0, is the normal-ized coordinate of the initial ( = 0) location of the boundary, whereas the secondone represents the distance that the boundary travelled until = 0 (Equation (1)).

Thus, the first two terms give the coordinate of the boundary at = 0. The distancefrom this point to the point at which the discontinuity forms (Equation (14)) isdescribed by the third term on the right hand side of Equation (19). The last termrepresents the distance travelled by the shock after its formation had started.

Equation (17) shows that during the phase of the shock formation, the localMach number of the discontinuity and the associated local compression increasefrom M12 = 1 and 12 = 1, as the ratio V2/V1 increases from V2/V1 = 1 afterthe discontinuity occurs at b. The initial rest frame velocity of the discontinuityis Mb = V0 + U0 . The total compression at the moment is defined by =2/0 = V

22 , and the rest frame velocity of the discontinuity by Equation (18).

The shock is completed at the moment f, achieving the final compression:

f = V2m =

1 + Um2

2, (20)

where Um is the highest expansion velocity of the boundary, i.e., the highest plasmaflow-velocity associated with the perturbation. The final velocity of the shock wavehas the Mach number


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Mf = Vm

5 + V2m

8 2V2m. (21)

So, there is a phase during which the rest frame Mach number of the discontinuitychanges from Mb to Mf, and during which the compression of the plasma behindthe discontinuity increases from b = V20 to f. The time = f b needed forthe shock completion depends on the form of the function U that also determineswhether the shock accelerates (Mf > Mb) or decelerates (Mf < Mb) duringthis period.

4. Results

4.1. FORMATION OF THE SHOCK

The formation and evolution of the leading edge of the perturbation, and its steep-ening into a shock wave will be illustrated using three different forms of the gen-erating function defined in the time interval 0 < < 1 (i.e., 0 < t < t m)as

F1 U = Um sin2

2

, (22)

F2 U = Um2 , (23)

and

F3 U = 21 f

1/3

2 . (24)

In the case of F1 the acceleration has a symmetric time profile. In F2 it is a linearfunction of time in the 0 < < 1 interval after which it sharply drops to zero.If the expansion of the source region is generated by a pressure pulse caused byheating, F1 reproduces a symmetric pulse, whereas F2 describes a sharply peakedpulse. In the case of F2 the time of the appearance of the discontinuity can beexpressed in a simple form: b = (14/9Um)1/2. It is created at the segment of theperturbation characterized by U = 27 Um. The third function (F3) generates a linearU(X) perturbation profile. This implies (Equation (7)) that the shock is completedinstantaneously at the moment f = b which is related to Um as

f =

1

1 + Um231

. (25)

Figure 3 exhibits the evolution of the leading edge of the perturbation profileU(X) for F1, F2 and F3, in the time steps of = 0.1. The normalized plasmaflow velocity U = u/v0 (y-axis) is presented as a function of the normalized


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Figure 3. Evolution of the leading edge of perturbations generated by different motions of the i-regionboundary: (a) F1, Um = 0.4; (b) F1, Um = 0.8; (c) F1, Um = 1.2; (d) F2, Um = 0.8; (e) F3,Um = 0.8. The starting point of the shock formation is indicated by a cross (in the case of F3, allsignals pile-up at the same moment).

distance from the initial position of the source-region boundary (x-axis). Beforethe appearance of the discontinuity the positions of the perturbation elements aredetermined using Equation (4). The motion of the discontinuity is followed ap-plying Equations (18) and (19). The first three panels exhibit the evolution of theleading edge of the perturbation profile generated by F1 using Um = 0.4, Um = 0.8and Um = 1.2. The corresponding final Mach numbers are M = 1.35, M = 1.83and M = 2.59, respectively. The remaining two panels show the evolution of theperturbation generated by F2 and F3 using Um = 0.8. The wave profiles steepen

in time and at the moment b a discontinuity is created at the distance defined byEquation (15). The amplitude of the discontinuity increases in time. The shockformation is completed when all segments of the perturbations leading edge pile-up into the discontinity extending from U = 0 to U = Um. The vertical lines at


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Figure 4. Kinematics of the i-region expansion (B position, A acceleration) and the motion of theshock (S) generated by F1 using: (a) Um = 0.4 and (b) Um = 1.2.

the left hand side of the graphs exhibit the location of the i-region boundary during0 < < 1.

Figure 4 shows the propagation of the discontinuity (the thick line denoted as S)generated by F1, using Um = 0.4 and Um = 1.2. The motion of the i-regionboundary is drawn by the thin line denoted as B, whereas the curve denoted as Aexhibits its acceleration. It was assumed that after = 1 the boundary propagatesat the constant velocity Um. The i-region expansion is much slower than the shockpropagation (M/ Um = 3.4 and 2.2, respectively). In the case of a high value of Umthe discontinuity appears before = 1.

Figure 5(a) shows the behaviour of the Mach number M (Equation (18)) afterthe appearance of the discontinuity for F1, F2 and F3 using Um = 0.8, and for F2using Um = 0.4. One finds that during the shock growth, M can be increasing ordecreasing, depending on the form of the function U() used. Applying the func-tion F3, the acceleration-phase does not show up since the shock is completedinstantenously. In Figure 5(b) the ratio of the plasma density behind the shock andin the unperturbed medium, () = 2/0 = B2/B0 = V22 , is shown for the casesconsidered in Figure 5(a).

4.2. SYNTHESIZED DYNAMIC RADIO SPECTRA

Let us now apply the model to the solar corona, i.e., the x-axis of Figure 1 willnow be set upwards from the core of an active region. Figure 6 exhibits severalexamples of dynamic spectra synthesized using the generating function F1. Theradio emission at the plasma frequency and its harmonic are drawn by the thick


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Figure 5. Evolution of the shock during its formation: (a) Mach number M(); (b) the compression = 2/0.

lines. It was assumed that the emission is excited in the regions ahead and behindthe shock, causing the band-splitting (Mann, Classen, and Aurass, 1995). Apartfrom the fundamental and harmonic emission lanes corresponding to a type II burst,the upward motion of the source-region boundary is depicted conceiving the radioemission to be at the harmonic of the local plasma frequency.

The value ofv0 = 1000 km s1 was used for the coronal Alfvn velocity in allthe cases. The spectrum in Figure 6(a) reproduces a hectometric/kilometric type II

burst caused by a CME accelerating for tm = 2 hr. The RAE interplanetary densitymodel in which the electron density decreases with the radial distance R as ne R2.63 (Feinberg and Stone, 1971) was applied and it was assumed that the source-region boundary was initially located at the height of 0.1 solar radii. Figure 6(b)reproduces a type II burst in the dekametric wavelength range, whereas Figures 6(ce) illustrate metric type II bursts of various starting frequencies and time delaysobtained using subalfvenic expansion velocities. In Figure 6(f) a metric type IIburst generated by an extremely abrupt expansion (tm = 1 min; Um = 1.2) issynthesized. In the examples shown in Figures 6(bf) the source-region boundarywas set initially at 0.02 solar radii. A twofold Newkirk coronal density model,behaving as ne 104.32/R (Newkirk, 1961) was assumed to reproduce the coronaldensity above an active region which is higher than in the quiet corona, as inferred

from the radio observations (Gergely, 1982). Let us note that the 2Newkirk modelcorresponds roughly to the 5Saito model which shows at low heights a densitydecrease of the form ne R6(Saito, 1970).

The onset of the burst emission is shown enlarged in the boxes attached to thespectra shown in Figures 6(b), 6(d), and 6(f). It was taken into account that the


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Figure 6. Dynamic spectra f ( t) synthesized using F1 and the following values:(a) Um = 0.4, tm = 2h (hectometric type II burst); (b) Um = 0.2, tm = 5 min (dekametrictype II burst); (c) Um = 0.4, tm = 5 min; (d) Um = 0.4, tm = 2 min; (e) Um = 0.8, tm = 2min;(f) Um = 1.2, tm = 1 min (decimetric type II burst). The crosses trace the motion of the i-regionboundary in the interval 0 < t < tm.

emission appears already before the shock appearance, from the segments of theperturbation characterized by a sufficiently high magnetic field gradient B/x. As

the leading edge of the perturbation steepens the associated magnetic field gradientincreases, causing an intensification of the related electric current density. After thethreshold value is achieved (Spicer and Brown, 1981) some of the high-frequencyturbulences are excited, causing the electromagnetic emission (Benz, 1993). Theresult of such a scenario is the appearance of a specific spectral feature at the onsetof the synthesized type II bursts. The feature has a shape which reproduces wellthe analogous arc spectral features observed by Aurass, Magun, and Mann (1994)and Klassen et al. (1999a).

The synthesized spectra shown in Figure 6 demonstrate that a typical type IIburst can be generated by a source-region expansion characterized by subalfvnicvelocities Um 0.40.8. Such an expansion corresponds to Mach numbers M 1.351.83, which is consistent with the values 1.21.7 inferred from the obser-

vations of metric type II bursts (Nelson and Melrose, 1985). A more impulsivesource-region expansion generates a type II burst of a higher starting frequency,a shorter time delay and a higher Mach number (compare Figures 6(b) and 6(c),or Figures 6(d) and 6(e)). A typical burst occurs few minutes after t = tm, buthigh-frequency bursts can appear even before t = tm (Figures 6(e) and 6(f)). The


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Figure 7. Time delays t (after the beginning of the i-region expansion) and starting frequenciesf (fundamental band) of type II bursts, presented as a function of the impulsiveness of the i-regionexpansion (Um/tm). (a) F1 (plus-signs), F2 (squares) and F3 (crosses); v0 = 1000 km s

1 andtm = 2 min. (b) F1; v0 = 500 km s

1 and tm = 1 min (squares), v0 = 500 km s1 and tm = 3 min

(plus-signs), v0 = 2000 km s1 and tm = 1 min (crosses), v0 = 2000 km s

1 and tm = 3 min

(circles). (c) The superposition of the results obtained for F1, F2 and F3, combining v0 = 500 km s1and v0 = 2000 km s

1 with tm = 1 min and tm = 5 min. (d) The same as in (c) but for tm = 1 hand tm = 5 h. The twofold Newkirk coronal density model is used in (a) (b), and (c), and the RAEinterplanetary density model in (d).


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duration of the arc feature at the onset of the burst ranges from 10 s to 20 s,representing some 10% of the time delay.

Figure 7 shows the time delays (left-hand side panels) and starting frequencies(right-hand side panels) as a function of the parameter U

m/t

mrepresenting the

impulsiveness of the i-region expansion. One finds that the time delays depend notonly on the impulsiveness but also on the form of the generating function andon the ambient Alfvn velocity. The scatter of the points in Figures 7(c) and 7(d)increases even more if the coronal density models are varied.

5. Discussion

5.1. TYPE II BURST MORPHOLOGY

Metric and kilometric type II bursts can be generated by a subalfvnic expansion of

the source-region as demonstrated by Figure 6. In the case of metric type II bursts,for a tm in the range 25 min, the onset time delays (tb) are in the range of 2 -15 minfor the expansion velocities between Um = 0.4 and Um = 0.8, corresponding tothe shock wave Mach numbers 1.351.83. Taking 1000 km s1 for the ambientAlfvn velocity, one finds the starting frequencies between 60 MHz and 300 MHzin the harmonic band, consistent with observations. In the case of a very impul-sive expansion the shock can occur even before the end of the acceleration phase(tb < tm), very soon after the moment of the highest acceleration of the boundary(Figure 6(f)). The starting frequency is then considerably higher the harmonicband of type II burst occurs in the decimetric wavelength range, between 500 MHzand 600 MHz (compare, e.g., with Figure 3 in Gopalswamy et al., 1998, or Figure 3in Klassen et al., 1999b).

The synthesized spectra shown in Figure 6 depict also the motion of the source-region boundary, assuming the radio emission is at the harmonic of the local plasmafrequency. Comparing the synthesized metric type II burst spectra with the ob-served ones (see, e.g., Klassen et al., 1999a) one finds that this trace correspondsto the high frequency cut-off of the type II burst precursor (Klassen et al., 1999a).Analogously, in the synthesized hectometric/kilometric type II burst spectra, thetrace of the boundary motion is similar to the moving type IV emission (Benz,1993) associated with the CME onset (Aurass et al., 1999).

The synthesized spectra reproduce qualitatively and quantitatively the other twophenomena observed in metric type II bursts: the arc structure at the onset of thetype II burst, and the band splitting. The relative bandwidths f/f and durations of

the synthesized arc features are 1%10% and 10 s20 s, respectively, reproducingwell the examples shown by Klassen et al. (1999a). The arc feature is immediatelyfollowed by the band split lanes of the type II burst emission, as in the observedspectra. Here it was assumed that the plasma ahead and behind the shock radiates atthe plasma frequency (Mann, Classen, and Aurass, 1995; for other mechanisms see


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Krger, 1979). The obtained relative bandwidths are between 10% and 30%. Thesplitting is larger in the bursts of a higher starting frequency and of a higher Machnumber, consistent with the observations shown by Mann, Classen, and Aurass(1995).

During the phase of the shock completion (b < < f) the compression ofthe plasma behind the discontinuity increases from b to f and the shock velocitychanges (Figure 5). An increase of implies an intensification of the associatedradio emission. Furthermore, the emission pattern in the dynamic spectrum departsthe constant velocity lane. The duration of this phase is shorter or comparablewith the duration of the acceleration phase of the source-region boundary (Fig-ure 5). In the case of the generating function F3 defined by Equation (24) theacceleration phase is absent, since in that case the shock forms instantaneously.

Let us consider a source rising at a constant speed through a Newkirk-modelcorona. In the standard dynamic spectrum, where at the y-axis the frequency isshown logarithmically, the radio emission associated with the source motion is

depicted by a curved lane of the form y t1. Thus, a straight lane represents anacceleration. The effect can be caused by a real acceleration of the source, butthe same can happen even in the case of a constant velocity of the source if thecompression in the source increases. The dynamic spectrum of the type II burstobserved on 27 December 1993 (09:15 UT), presented by Klassen et al. (1999a)in their Figure 5, shows such a morphology. The burst starts by the appearance ofthe arc feature, lasting for some 20 s, after which a weak, band split, type II burstemission starts. In the following 100 s the emission intensifies, forming straightemission lanes. Thus, this event presumably reveals (under assumption of the mo-tion along the density gradient through a Newkirk-type of corona) the process ofthe shock wave growth the phase of increasing compression and increasing Mach

number (Figure 5).5.2. TYPE II BURST SOURCES

The distance at which the shock forms (and thus also the starting frequency ofthe type II burst) depends on the duration of the acceleration phase tm and on theambient Alfvn velocity. Assuming vA0 1000 km s

1 one finds that the shockappearing at about 100 MHz has to be generated by the expansion of the source-region developing on a time scale of minutes (Figure 6(c)). Shocks appearingaround 1 MHz should be related to the events developing on the time scale ofhours (Figure 6(a)).

A high-frequency metric type II burst, starting in the dmm wavelength range,

requires a more impulsive expansion of the source-region (e.g., Um 0.8, andtm 2 min see Figure 6(e)). Type II bursts of still higher starting frequencies andshorter time delays can be generated only by an extremely impulsive expansion: asuperalfvnic velocity Um 1 has to be achieved in about 1 min (Figure 6(f)).


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The presented consideration of time scales indicates that metric type II burstsare most likely generated by solar flares, since CMEs develop on a considerablylonger time scale. From the velocity/distance characteristics of the CME motion(see, e.g., Gosling et al., 1976) one finds that the growth rate of the instabilitydriving the eruption (Vrnak, 1998) is in the order of = 103 s1 correspondingto a time scale of hours. Thus, it is more appropriate to identify CMEs as thesource of hectometric/kilometric type II bursts. Since the growth rate in the case oferuptive prominences is of the same order (Vrnak, 1998) they can also be excludedas a possible source of metric type II bursts. Even the ejections having the fastestgrowth rate the flare sprays can hardly develop fast enough to produce high-frequency metric type II bursts. The growth rate of a flare spray eruption is in theorder of = 102 s1, corresponding to the time scale of ten minutes (Tandberg-Hanssen, Martin, and Hansen, 1980). So, one can estimate that flare sprays arecapable to generate only the metric type II bursts starting at frequencies below100 MHz.

Yet another aspect of the shock formation should be mentioned. The analysispresented in this paper is based on a 1-D model. In a 3-D situation the energyflux density of the wave must decrease as the wave spreads out radially from thesource, due to the energy flux conservation. This effect becomes important at dis-tances larger than the source dimensions. Taking into account also the dissipativeprocesses associated with the shock, it is obvious that the shock has to cease aftersome time/distance. On the other hand, this effect also prevents a too small andnot sufficiently impulsive source to produce a shock since the perturbation ceasesbefore transforming into a shock.

Thus, the smallest source expansions can account only for events like type Iburst chains in noise storms. For the same reason, it is not likely that the flare-

associated shocks, producing metric type II bursts, could reach the interplanetaryspace and cause kilometric bursts. This is consistent with the fact that metric type IIbursts usually cease at frequencies higher than 20 MHz (Nelson and Melrose,1985). On the other hand, in the case of a large source such as a CME, the effect isless pronounced and the associated shock can propagate across the interplanetaryspace and excite a kilometric type II burst.

6. Conclusion

The formation of MHD shock waves at different time/distance scales can be ex-plained by the same mechanism: steepening of a large amplitude MHD perturba-

tion generated by an abrupt expansion of the source-region. The duration of theacceleration process and the maximum expansion velocity of the source-regiondetermine the time/distance characteristics of the shock formation process. Al-though only perpendicular shocks were considered in this paper, a similar behav-iour should be expected for the oblique shocks (Mann, 1995).


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Various solar phenomena, occurring on different scales, can be explained ap-plying the model to coronal conditions: interplanetary shock waves generatingkilometric type II bursts, coronal shock waves causing metric type II bursts, aswell as smaller scale coronal shock waves causing chains of type I bursts duringnoise storms.

The presented results indicate that the possible sources of the shocks generatingmetric type II bursts are the flares and the extremely abrupt ejections. However, thehigh-frequency type II bursts starting in the dmm wavelength range are charac-terized by such time/distance scales that can not be related to material ejections. Incontrast, the kilometric type II bursts are caused by fast CMEs of large dimensionsand long acceleration times. On the other hand, the shocks exciting chains of type Ibursts are most probably caused by nano-flares (Karlick and Odstrcil, 1997)associated with small scale restructuring of the coronal magnetic field caused bythe emerging flux process usually associated with noise storms (Benz and Zlobec,1982).

A comparison of the model with the observations indicates that a subalfvnicsource-region expansion (Um < 1) can reproduce most of the observed phenomena.The value ofUm < 1 corresponds to M < 2.16 (Equation (21)). An average type IIburst spectrum constructed by Ledenev and Urbarz (1992) exhibits a harmonicband starting close to 300 MHz, some 3 min after the impulsive phase of a flare.Such a spectrum is well reproduced by the one shown in Figure 6(d), representingthe shock created by an expansion with Um = 0.4 and tm = 2 min. The range ofMach numbers 1.2 < M < 1.7 inferred from the observations of metric type IIbursts (Nelson and Melrose, 1985) implies 0.25 < Um < 0.71 in the case ofa perpendicular shock. In the case of oblique shocks these numbers should besomewhat higher (Mann, Classen, and Aurass (1995)).

Appendix

The processes that are considered in this paper develop on large spatial and timescales and can be described by MHD equations (Priest, 1982). Neglecting gravityand viscosity terms, the equation of motion can be written as

Du

Dt= p +

1

0( B) B , (A.1)

where denotes the plasma mass density, u the plasma flow velocity, p the plasmapressure, and B the magnetic field, whereas D/Dt = /t+ u is the convective

derivative. Neglecting energy losses, the heat equation can be represented in theform

De

Dt+ p u = q , (A.2)


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where e denotes the internal energy per unit mass, and q is the rate at which theheat is released into the unit volume. In a perfectly conducting medium (magneticdiffusivity = 0) the induction equation becomes

Bt

= (u B) . (A.3)

In combination with B = 0, Equation (A.3) implies that the magnetic field isfrozen-in the plasma (Priest, 1982).

Equations (A.1)(A.3) have to be supplemented by the equation of continuity

t+ (u) = 0 , (A.4)

and the equation of state, which for the ideal gas reads

p =

mpkT = ( 1)e , (A.5)

where mp is the proton mass and = (s + 2)/s is the ratio of specific heatsdetermined by the number of degrees of freedom s.

In the considered 1-D situation, where all quantities are assumed to be trans-lation invariant in the y- and z-directions, and assuming that the processes takingplace in the e-region are adiabatic, Equations (A.1)(A.4) reduce to:

u

t+ u

u

x

=

x

p +

B2

2o

=

ptot

x, (A.6)

pe

e

= constant , (A.7)

B

t+

(Bu)

x= 0 , (A.8)

t+

(u)

x= 0 , (A.9)

where B = By, u = ux and ptot = p + (B2/20) = p + pB is the total pres-sure. Equations (A.8) and (A.9) show that in the 1-D situation, when the frozen-incondition is satisfied, one can write:

B

=

B0

0= constant . (A.10)

In the direction perpendicular to the magnetic field, perturbations of the totalpressure ptot travel at the fast magnetosonic velocity v defined by

v2 = v2Ae + c2e =

1 +

2e

v2Ae , (A.11)


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where v2Ae = B2e /0e = 2pBe /e is the Alfvn velocity, c

2e = pe/e is the sound

velocity, and e = pe/pBe is the ratio of gas and magnetic pressures. Presumingthat the plasma in the e-region is compressed adiabatically and using = 53 , onefinds:

e = 0

0

e

1/3, (A.12)

where Equations (A.5) and (A.10) were used. Assuming 0 1, Equation (A.12)shows that after the adiabatic compression the plasma pressure is still much smallerthan the magnetic field pressure: e 1 since e > 0. This also implies ce vAe ,meaning that v vAe , and one can write:

V2

v

v0

2

vAe

vA0

2=

e

0= . (A.13)

Here, v0 is the initial magnetosonic velocity, and v0 vA0 is valid due to 0 1. represents the factor of the compression. Furthermore, the total pressure ptote =pBe + pe can be approximated as

ptote pBe = pB0

e

0

2

v20

202e , (A.14)

where the expressions Be/B0 = e/0 and pB0 0v20 /2 were used.

The equation of motion for the external region can be written using Equa-tion (A.14) as

u

t+ u

u

t=

v20

0

e

x. (A.15)

Equation (A.15) has to be supplemented by the equation of continuity,

e

t+ u

e

x+ e

u

x= 0 , (A.16)

and by pe/e = const. Eliminating e/x and substituting e = v0/v0 (Equa-

tion (A.13)) one can write Equations (A.15) and (A.16) as

u

x=

uu

t 2v

v

t

v2 u2(A.17)

and

2v

t+ v

u

x+ 2u

v

x= 0 , (A.18)

respectively. Taking into account the fact that the compression in the e-region,and so the perturbation velocity v, is a function of the plasma flow velocity u, i.e.,


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v

t=

v

u

u

tand

v

x=

v

u

u

x,

Equation (A.18) becomes:

2 vu

u

t+ v

u

x+ 2u v

u

u

x= 0 . (A.19)

Substituting the expression for u/x from Equation (A.17) into Equation (A.19),and using v/t = (v/u)(u/t), one finds a very simple equation relating uand v:

dv

du=

1

2. (A.20)

Taking into account that v = v0 for u = 0, one gets the solution of Equation (A.20):

v = v0 +u

2. (A.21)

Equation (A.21) can be written in the normalized form:

V = 1 +U

2, (A.22)

where V = v/v0 and U = u/v0. Since Equation (A.17) demands u = v0, Equa-tion (A.22) implies U < 2 (i.e. u < 2v0), because U = 2 means also u = v. Thecondition U < 2 is consistent with Equation (17) in Section 3.2 which exposes anintrinsic property of the perpendicular MHD shock wave that can be expressed as = /0 < 4 (see, e.g., Benz, 1993). Since U < 2 implies V < 2, one finds/0 = V

2 < 4. Equations (A.22) and (A.13) imply

=

1 +U

22

. (A.23)

Substituting Equation (A.21) into Equation (A.18) one gets

u

t+

v0 +

3

2u

u

x= 0 , (A.24)

i.e.,

u

t+ (v(u) + u)

u

x= 0 , (A.25)

which governs the behaviour of the external region. Linearization of Equation(A.24) for u v0 leads to the wave equation for a small amplitude perturba-

tion propagating at the velocity v0. Considering a plasma initially moving at u =constant, one finds that Equation (A.25) governs the propagation of the smallamplitude wave excited by the source that travels together with the medium: thewave travels at the velocity v relative to the medium, and its rest frame velocity isw = v + u.


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b. vrsnak and s. lulic- formation of coronal mhd shock waves i: the basic mechanism

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