b. vrsnak and s. lulic- formation of coronal mhd shock waves ii: the pressure pulse mechanism

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

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FORMATION OF CORONAL MHD SHOCK WAVES

II. The Pressure Pulse Mechanism

B. VRŠNAK and S. LULIC Hvar Observatory, Faculty of Geodesy, Kaˇ ci´ ceva 26, HR-10000 Zagreb, Croatia

(Received 12 February 2000; accepted 11 April 2000)

Abstract. The ignition of coronal shock waves by flares is investigated. It is assumed that an explo-sive expansion of the source region caused by impulsive heating generates a fast-mode MHD blastwave which subsequently transforms into a shock wave. The solutions of 1-D MHD equations for theflaring region and for the external region are matched at their boundary. The obtained results showunder what conditions flares can ignite shock waves that excite the metric type II bursts. The heatinput rate per unit mass has to be sufficiently high and the preflare value of the plasma parameter β

in the flaring region has to be larger than βcrit

0. The critical values depend on the flare dimensions

and impulsiveness. Larger and more impulsive flares are more effective in generating type II bursts.Shock waves of a higher Mach number require a higher preflare value of β and a more powerfulheating per unit mass. The results demonstrate why only a small fraction of flares is associated withtype II bursts and why the association rate increases with the flare importance.

1. Introduction

Metric type II bursts (Nelson and Melrose, 1985) reveal the propagation of fast-mode MHD shock waves in the solar corona (Uchida, 1974). The question whetherthese shocks are caused by flares or fast material ejecta is the subject of numer-

ous studies (for a review, see Cliver, Webb, and Howard, 1999). Let us brieflysummarize some of the observational results regarding this problem.The majority of metric type II bursts starts at frequencies close to or below

100 MHz, several minutes after the peak of the associated microwave burst (Har-vey, 1965; Švestka and Fritzová-Švestková, 1974). The radio emission usuallyceases at frequencies higher than 20 MHz (Nelson and Melrose, 1985). Occasion-ally, type II bursts start in the decimetric wavelength range, and their beginning canprecede the peak of the associated microwave burst (Vršnak et al., 1995).

It is well known that there is an association between metric type II bursts andflares and that the association rate increases with the flare importance (Cliver,Webb, and Howard, 1999). Drago and Tagliaferri (1967) found a correlation be-tween the rise times of the Hα flare emission and the time delays of the associatedtype II bursts. More recently, Pearson et al. (1989) have shown that a weak cor-relation exists between the impulsiveness of hard X-ray bursts and the onset fre-quencies and time delays of the type II bursts. Vršnak et al. (1995) have shown thatmicrowave bursts associated with type II bursts occurring in the dm–m wavelengthrange have in average about two times higher peak fluxes than flares of the same

Solar Physics 196: 181–197, 2000.© 2000 Kluwer Academic Publishers. Printed in the Netherlands.



182 B. VRŠNAK AND S. LULIC

importance not associated with type II bursts. Vršnak (2000) has found a relationbetween the time delay of type II bursts and the impulsiveness of the associatedmicrowave bursts.

Recently, Klassen et al. (1999) showed that the occurrence of a type II burstis often preceded by a type II burst precursor , consisting of numerous impulsivefast-drifting bursts in the decimetric wavelength range. This feature is usuallycharacterized by a slowly drifting high frequency edge and is associated with animpulsive microwave and hard X-ray burst (Klassen et al., 1999).

Observations indicate that type II burst shock waves have velocities in the orderof 1000 km s−1 and low Mach numbers, usually between 1.2 and 1.7 (Nelsonand Melrose, 1985). Similar velocities are observed in the case of chromosphericMoreton waves spreading out from the flare site (Moreton, 1960; Smith and Har-vey, 1971). Uchida (1974) has shown that both phenomena are probably causedby a common shock wave ignited by a flare. Vršnak and Luli c (2000, hereafterPaper I) have shown that such shock waves can be generated by an abrupt, but not

necessarily superalfvénic, expansion of the source region.In Paper I the process driving the expansion of the source region was not spec-

ified. However, considering the time/distance scales it was inferred that the ‘high-frequency’ type II bursts starting in the dm–m wavelength range are most prob-ably caused by solar flares, since even the most abrupt ejections – flare sprays(Tandberg-Hanssen, Martin, and Hansen, 1980) – are not impulsive enough to cre-ate such a type II burst. In this paper an explosive expansion of the flaring volumewill be considered as a source of the shock wave. Signatures of such a flare-ignitedprocess will be discussed and confronted with observations. Furthermore, an orderof magnitude analysis will be presented, to show under what conditions flares canignite shock waves that excite metric type II bursts. The results can explain why

only a small fraction of flares is associated with type II bursts.

2. Expansion of the Flaring Volume

2.1. THE MODEL

In the following, a 1-D model will be considered, where all quantities are uniformin the y- and z-directions and the magnetic field is oriented in the y-direction (Fig-ure 1). The magnetic field, plasma density, temperature and pressure are taken to beuniform initially, having the values B0, ρ0, T 0, and p0, respectively. Furthermore,it will be assumed that the plasma is perfectly conducting (magnetic diffusivity

η = 0) implying that the ‘frozen-in’ condition is satisfied (Priest, 1982). Theinitial value of the ratio of the plasma and the magnetic pressure is taken to beβ0 = p0/pB0 1, and it is assumed that the plasma is at rest.

Let us assume that an impulsive heating of the source region begins at t = 0 andthat the heat input rate per unit volume is described by the function q(x,t). The



FORMATION OF CORONAL MHD SHOCK WAVES, II 183

Figure 1. Definition of the flaring region (shaded) and the external region: (a) before the onset of heating; (b) after the heating had started. The magnetic field Bi and the density ρi in the internalregion decrease due to the expansion, whereas in the external region Be and ρe increase.

behaviour of the system is governed by the MHD equations (see Paper I) whichcan be written in a 1-D situation as:

ρ

∂u∂t + u ∂u

∂x

= −∂p

tot

∂x, (1)

ρ

∂e

∂t + u

∂e

∂x

+ p

∂u

∂x= q , (2)

∂B

∂t +

∂(Bu)

∂x= 0 , (3)

∂ρ

∂t +

∂(ρu)

∂x= 0 , (4)

p =ρ

mpkT = (γ − 1)ρe . (5)

Here ptot = p+B2/2µ0 = p+pB is the total pressure, mp is the proton mass andγ = (s+ 2)/s is the ratio of specific heats determined by the number of degrees of freedom s. Furthermore, according to Figure 1, the magnetic field and the plasmaflow velocity are denoted as B = By, u = ux , respectively.

It will be assumed that the heat is deposited only into the region between x = 0and the marginal magnetic field line initially located at x = x0, i.e. to the sameplasma element (Figure 1). The boundary at x = 0 is considered fixed, whereasthe other boundary (further on simply the boundary) can move freely and its co-ordinate and velocity will be denoted as xt and ut :

ut =∂xt

∂t , xt = x0 +xt = x0 +

t 0

ut dt . (6)

The velocity ut will be called the expansion velocity.




At any moment, the heat is released only within 0 < x < xt (further on flaring

volume or i-region), and all the quantities describing a physical state of this volumewill be denoted by the subscript ‘i’. The expansion of the flaring volume causes acompression of the plasma in the ambient region (further on external region ore-region – the related quantities will be denoted by the subscript ‘e’). A large am-plitude MHD perturbation is created, spreading through the e-region. The leadingedge of the perturbation forms during the time interval 0 < t < t m in which theexpansion velocity increases from u = 0 to the maximum value um. The spatialprofile of the leading edge subsequently steepens and a shock wave forms after atime/distance determined by the expansion velocity time profile (see Paper I).

2.2. THE BEHAVIOUR OF THE INTERNAL REGION

For a provisional function q(x,t) the evolution of the system can be determinedonly numerically. In this paper a particular family of q(x,t) functions will be

considered, allowing for an analytical description of the heated volume expansion.In order to obtain the explicit expressions relating the heat input rate and the ex-pansion velocity, the problem will be inverted. Instead of specifying the functionq(x,t) and searching for the response of the system, it will be demanded that theflaring volume expands ‘uniformly’, meaning that the plasma density within theflaring volume remains uniform during the expansion. Then, the solution of thesystem of Equations (1)–(5) will be found for a prescribed function ut , i.e. theevolution of the system will be completely determined imposing the kinematics of the flaring volume boundary.

Since the frozen-in condition is satisfied (η = 0), in the 1-D situation theconservation of magnetic flux and mass (Equations (3) and (4)), together with theimposed ‘uniform’ expansion constraint (∂ρi/∂x = 0), imply that the magnetic

field Bi also remains uniform during the expansion (∂Bi /∂x = 0) and one canwrite:

Bi

B0=

ρi

ρ0=

x0

xt

. (7)

So, within the flaring volume the magnetic pressure pBi= B2

i /2µ0, normalizedwith respect to the initial value pB0 = B2

0 /2µ0 = ρ0v2A0

/2, can be expressed as

P Bi≡

pBi

pB0

=

Bi

B0

2

=

x0

xt

2

. (8)

Going back to Equation (4) and taking into account ∂ρi/∂x = 0 one finds:

1ρi

∂ρi

∂t = −∂u

∂x. (9)

Since ρi does not depend on x one can write ∂u/∂x = C, where C does not dependon x. This implies that the plasma flow velocity within the flaring volume (x < xt )can be expressed as




u =x

xt

ut , (10)

since it is assumed that u = 0 at x = 0.

Using Equations (6) and (10), Equation (1) can be transformed to obtain

ρi

∂ut

∂t

x

xt

= −∂pi

∂x, (11)

where it was taken into account that ∂Bi/∂x = 0 so that ∂pBi/∂x = 0. Since

ρi ∂ut /∂t does not depend on x, the integration of Equation (11) with respect to x

implies that the gas pressure in the i-region is of the form pi = C1 − C2(x/xi )2,where C1 and C2 do not depend on x. Denoting the gas pressure at x = 0, at themoment t , as p(x = 0, t) ≡ p0

t and the difference of the gas pressure at x = 0 andat x = xt as pt one finds:

pi= p0

t −p

t x

xt 2

.(12)

The superscript ‘0’ will be used further on to denote quantities at x = 0.The plasma pressure pi at the flaring volume boundary at the moment t is

pi(x = xt , t) ≡ pbt = p0

t − pt . The superscript ‘b’ will be used further on todenote quantities at the flaring volume boundary. Substituting Equation (12) intoEquation (11) one finds

ρi

2

∂ut

∂t =

pt

xt

, (13)

which determines the acceleration of the boundary:

∂ut

∂t = 2pt

x0ρ0. (14)

Here, Equation (7) was used to eliminate ρi.Using Equations (6), (7), and (10), as well as Equation (5) with the ratio of

specific heat capacities γ = 53 , i.e., using e = 3p/2ρ, Equation (2) can be written

as

q =3

2

∂pi

∂t +

3

2

pi

xt

ut +3

2

∂pi

∂x

x

xt

ut +pi

xt

ut . (15)

Taking into account Equation (12) one finds for 0 < x < xt

q(x,t) = q0t −qt x

xt

2

, (16)

where

q0t =

3

2

∂p0t

∂t +

5

2

p0t

xt

ut , (17)




and

qt =3

2

∂pt

∂t +

5

2

pt

xt

ut (18)

are the rate at which the heat is deposited at x = 0 and the difference betweenthe heat released at x = 0 and x = xt , respectively. The expansion velocity ut isrelated to pt by Equation (14), and xt is defined by Equation (6). Equation (16)shows that the heat deposited at the boundary q(x = xt , t) ≡ qb

t = q0t −qt can

be expressed as

qbt =

3

2

∂pbt

∂t +

5

2

pbt

xt

ut . (19)

3. Results

3.1. ‘MATCHING’ OF THE INTERNAL AND EXTERNAL REGION

Fast expansion of the flaring region causes a compression of the magneto-plasmaahead the boundary (Figure 1(b)). In Paper I it was shown that starting with aβ0 0, the gas pressure in the e-region remains much smaller than the magneticpressure (βe 0). Furthermore, it was shown that the factor of compression ≡

ρe/ρ0 = Be/B0 depends on the velocity of the boundary:

t =

1+

U t

2

2

, (20)

where U t = u

t /v

A0

. The highest value of compression m= (U

m) is related to

the final Mach number of the shock wave as

M =

5+ m

8− 2m

m (21)

(see Paper I).Since the total pressure ptot

= p+pB must be a continuous function, ptoti (xt ) =

ptote (xt ) must hold at the flaring volume boundary, and one can write

B2i

2µ0+ pb

t = ptote (xt ) ≈

B2e

2µ0, (22)

since βe 1 is assumed. Normalizing Equation (22) with respect to the initialmagnetic pressure pB0 = B20 /2µ0 and using the relations Bi/B0 = x0/xt and

Be/B0 = one finds

P bτ = 2−X−2

τ =

1+

U τ

2

4

−X−2τ , (23)




where the normalized time τ = t /t m and the normalized co-ordinate of the bound-ary Xτ = xt /x0 were introduced.

Equation (6) can be written in the normalized form as

U τ = ∂Xτ

∂τ , Xτ = 1+ 1

τ

0U τ dτ , (24)

where = t A/t m, and t A = x0/vA0 represents the Alfvén travel time across theflaring region. Similarly, Equation (14) can be normalized such that

P τ = ∂U τ

∂τ , (25)

where P τ = pt /pB0 . One can also write

T 0τ

T 0= Xτ

P 0τ

β0, (26)

where T 0τ is the plasma temperature at x = 0 (further on the central temperature)and P 0τ = p0

t /pB0 .Prescribing the function U τ Equations (23), (24), and (25) determine P bτ , Xτ ,

and P τ , respectively. Furthermore, P bτ and P τ determine P 0τ = P bτ + P τ

(Equation (12)). Normalizing Equations (17) and (19) with respect to the initialmagnetic field energy density (Q0

τ = q0t /pB0 and Qb

τ = qbt /pB0) one can write

Q0τ =

ϕ0τ

t A=

φ0τ

t m;Qb

τ =ϕb

τ

t A=

φbτ

t m, (27)

where the dimensionless functions φ0τ and φb

τ read

φ0τ =

3

2

∂P 0τ

∂τ +

5

2

P 0τ U τ

Xτ , (28)

and

φbτ =

3

2

∂P bτ

∂τ +

5

2

P bτ U τ

Xτ

. (29)

The functions ϕ can be expressed as ϕ = φ. The functions ϕ0τ , ϕb

τ , φ0τ and φb

τ

depend on the parameter = t A/t m and on the highest value of the expansionvelocity U m. Large flares (large t A = x0/vA0 ) and short-duration flares (short t m)have a larger . Bearing in mind that the energy release process caused by themagnetic field reconnection develops on a time scale larger than the Alfvén timescale (Priest and Forbes, 1986), i.e., t A < t m, it will be assumed that < 1. Takingas an example a very large and fast developing event characterized by x0 105 kmand t m ≈ 100 s, and using for the Alfvén velocity vA0 ≈ 1000 km s−1, one finds 1.

Knowing Q0τ and Qb

τ one can evaluate Qτ = Q0τ − Qb

τ , determining theheating in the flaring volume:




Q(x,τ) = Q0τ −Qτ

x

xτ

2

(30)

(see Equation (16)). In this way the solution of the system of Equations (1)–(5) is

completed. Integrating Equation (28) over the interval 0 < x < xτ one finds therate at which the energy is released in the entire flaring region:

Qtot(τ ) = ( 23 Q0

τ +13 Qb

τ )xt . (31)

Using the relations xt = Xτ x0 and vA0 = x0/t A one finally finds

Qtot(τ ) = vA0 ( 23 ϕ0

τ +13 ϕb

τ )Xτ ≡ vA0 (τ) , (32)

where the introduced dimensionless function (τ) depends on the parameter , aswell as on U m.

3.2. AN EXAMPLE

Let us consider as an example the flare whose evolution is governed by the gener-ating function denoted as F1 in Paper I. It is defined as

U = U m sin2π

2τ

(33)

and its behaviour in the time interval 0 < τ < 1 determines the spatial profile of the leading edge of the associated blast wave.

In Figure 2 the evolution of the heating of the flaring region is shown for severalvalues of the parameters and U m, illustrating properties of flares developing ondifferent spatial/time scales and characterized by different expansion velocities. A

faster expansion (larger value of U m) implies a higher Mach number of the shockwave generated by the flare (see Paper I). Spatially large flares (large x0, i.e., larget A = x0/vA0) and fast developing flares (short t m) are characterized by a larger .

The dimensionless function (τ) defined by Equation (32) is shown in Fig-ure 2(a). The rate at which the heat is released in the flaring volume can be evalu-ated as q tot(τ ) = pB0 vA0 (τ). Figure 2(a) indicates that the maximum value max

and the time when it is achieved (τ max) primarily depend on the applied value of U m. The values of max do not depend significantly on the value of the parameter. So, it can concluded that a higher value of U m (and thus the higher Mach num-ber of the associated shock wave) requires more powerful heating. Furthermore,a stronger preflare magnetic field and a higher Alfvén velocity require a morepowerful heating qtot(τ ) for a given U m.

The behaviour of the central temperature is shown in Figure 2(b), where β0 =

0.1 was used to evaluate Equation (26). A lower value of β0 would give highertemperatures. One finds that the central temperature is higher in the case of a small and a large U m. This implies that small flares have to be hotter than large flares inorder to generate an expansion of the same U m. Such behaviour is consistent with




Figure 2. Heating of the flaring region shown for different values of U m and = t A/t m. (a) Thedimensionless function (τ) shown for U m = 0.8 (thin) and U m = 0.4 (thick ) obtained using = 0.2 (dotted ), = 0.1 ( full) and = 0.01(dashed ). (b) The ‘central’ temperature ratio T 0τ /T 0obtained using U m = 0.8 (thick ) and U m = 0.4 (thin) for = 0.05 (dotted ), = 0.1 ( full)and = 0.2 (dashed ). (c) The dimensionless function φ0(τ ) shown for U m = 0.8 and = 0.1(thick-full); U m = 0.4 and = 0.1 (dotted ); U m = 0.4 and = 0.05 (dashed ). The function φb(τ )

is shown for U m = 0.8 and = 0.1 by the thin-full line. (d) The difference φ = φ0− φb for the

same values of parameters as in (c).

the results shown in Figure 2(a). For a given U m, the total heating rate Qtot doesnot depend significantly on the dimensions of the flare. So, the rate at which theheat is released per unit volume is higher in small flares, resulting in higher plasmatemperatures. Figure 2(b) shows that the temperature maximum is achieved afterτ = 1.

The dimensionless function φ0(τ ) defined by Equation (28) is shown in Fig-ure 2(c). It describes the heat input rate at X = 0. The heating rate per unit volumecan be evaluated according to Equation (27) as q(τ) = (pB0 /t m)φ(τ). For thematter of illustration, the function φb(τ ) (heating rate at the boundary) is alsoshown for U m = 0.8 and = 0.1 (thin full line). In Figure 2(d) the differencebetween the heating rate at X = 0 and at the boundary φ(τ) = φ0(τ ) − φb(τ )

is presented for the same values of parameters as used in Figure 2(c). One finds

that in the case of the uniform expansion considered here, the heat is releasedalmost uniformly across the flaring volume (see the thick and thin full lines inFigure 2(c)); comparing Figures 2(c) and 2(d) one finds that the differences φ

are much smaller than the peak value of φ. Furthermore, it is evident that the heatrelease spreads from the central region towards the boundary during the process.




Figure 3. (a) The highest value of the heating rate of the flaring region represented by max as afunction of U m for = 1 (shaded ), = 0.5 (dashed ), = 0.2 (dotted ) and = 0.1 ( full). (b)max as a function of for U m = 0.4 (dotted ), U m = 0.8 ( full) and U m = 1.2 (dashed ). (c) τ max

as a function of for the same values of U m as used in (b). (d) The impulsiveness of the heatingimp = max/t max shown as a function of U m/t m for t m = 100 s and for the same values of as in(a).

In the beginning the heating is more powerful at X = 0, whereas later it is morepowerful at the boundary.

Figure 2(c) shows that a higher value of U m requires a more powerful heatingper unit volume q(τ) for a given B0 and t m. Furthermore, q(τ) depends on theparameter . The maximum value of q needed to achieve a given U m is higher inspatially small flares (small t A, i.e., small ).

In Figure 3 the dependence of the highest value of the total heating rate of theflaring volume and the impulsiveness of the heating process on the parameters U mand is depicted. max is presented as a function of U m and in Figures 3(a) and3(b), respectively. One finds that the value of the highest heat input rate Qmax

=

vA0 max depends primarily on U m and vA0 .

Figure 3(c) shows the dependence of the time of the maximum heat input rateτ max on the parameter for different values of U m. Figure 3(d) depicts the impul-siveness of heating imp = max/τ max as a function of U m/t m. One finds that imp isa monotonic function of U m/t m that does not significantly depend on in the rangeof values < 0.2. This indicates that the starting frequency and the time delay of




the type II burst excited by a flare are related primarily to the impulsiveness of theheat input rate in the flaring volume (see Figure 7 in Paper I).

3.3. CONDITIONS FOR TYPE II BURST FORMATION

3.3.1. The Plasma Parameter β

Using Equations (5) and (7), Equation (23) can be rewritten in the formx0

xt

2

+ β0x0

xt

T i

T 0= 2 , (34)

which gives

T i

T 0=

1

β0

xt

x02−

x0

xt

, (35)

In the following, the quantity xt m = umt m/2 will be used as an order of magnitude estimate of the increase of the flaring volume due to its expansion inthe interval 0 < t < t m. In the normalized form it can be expressed as Xτ =1 ≡

Xm = U m/2 where Xm = xt m /x0.Let us now consider in more detail two extreme types of events. The first type of

events are those satisfying the condition xt m /x0 1 (i.e., Xm ≡ xt m /x0 1).We will call such an event the small flare since the stated condition is easilyachieved for a small value of x0, i.e., small value of . The other extreme is anevent satisfying xt m /x0 1 (i.e., xt m /x0 ≈ 1). Since Xm = U m/2 onefinds that such events have to be large and fast developing (small t m and large x0,implying large ). We will call such an event simply the large flare.

In the small-flare case, Equation (35) can be written in an order of magnitude

form as

β0 ≈ 2m

xt m

x0

T 0

T i. (36)

In the large-flare case, Equation (35) reduces to

β0 ≈ (2m − 1)

T 0

T i. (37)

The compression m is related to U m by Equation (20) and to the Mach number M

of the associated shock wave by Equation (21).Equations (36) and (37) can be combined with one constraint imposed by ob-

servations. In the hottest parts of a flare the temperature usually ranges between107 and 108 K, and is rarely exceeding 108 K (Dennis and Schwartz, 1989; Kosugi,1994). So, taking T i < 2×108 K, one can use the limit T i /T 0 < 100. In Figure 4(a)β0 = βcrit

0 is shown as a function of the Mach number of the associated shock waveusing T i /T 0 = 100. A flare can generate a type II burst of a given Mach numberonly if it occurs in a coronal region having β0 above βcrit

0 (M). The large-flare case,




depicted by the thick line, is obtained straightforwardly using Equations (20), (21),and (37). In the case of a small flare one can approximately take xt m ≈ xt m sincext m x0. Estimating xt m approximately as xt m ≈ umt m/2, Equation (36) canbe transformed into an order of magnitude form which reads

β0 ≈ 2 U m

2

T 0

T i. (38)

The results obtained using Equations (38) and taking T i /T 0 = 100 are shown inFigure 4(a) by thin lines for different values of . Figure 4(a) shows that for a givenvalue of a higher Mach number requires a higher value of β0. Furthermore, itcan be concluded that for a given β0 and M , small flares (small t A) have to developfaster (shorter t m). Finally, for the same M , a less impulsive flare must occur in ahigher β0 region.

Let us compare an extremely small flare characterized by x0 = 1000 km, witha larger flare characterized by x0 = 10 000 km. Assuming that the impulsive heat

release in both flares is lasting for t m=

100 s, and taking v0=

1000 km s

−1

one finds values of as 0.01 and 0.1, respectively. Observations of type II burstsindicate that usually M ≥ 1.2 holds (Nelson and Melrose, 1985) which corre-sponds to U m ≥ 0.25 and ≥ 1.26. Using these values and taking for the ratioof temperatures T i/T 0 = 100, Equation (38) gives the values βcrit

0 = 0.2 for thesmaller flare and βcrit

0 = 0.02 for the larger one. In the low corona, where flaresoccur, β 1 is usually valid (Dulk and McLean, 1978). This implies that verysmall flares can produce a type II burst only under exceptional circumstances whencoronal conditions deviate from β 1.

Equations (37) and (38) show that the flares producing type II bursts have to bevery hot and impulsive. Figure 4(a) demonstrates that large flares can produce typeII bursts more easily, i.e., the critical values are more convenient for the coronal

conditions. Furthermore, one finds that larger and more impulsive flares can gen-erate shock waves of higher Mach numbers, and thus the type II bursts of higherstarting frequencies and shorter onset time delays.

3.3.2. Shear of the Preflare Magnetic Field

Let us consider another aspect of Equation (23). The heat release rate per unitmass can be related to the heat release rate per unit volume q as q = ρi.Applying Equation (7) this relation can be written as q = ρ0x0/xt . Substitutingthis expression into Equation (19) one finds

bρ0 =3

2

∂pbt

∂t

xt

x0+

5

2

pbt

x0ut . (39)

In the small-flare case (xt m ≈ xt m ) Equation (39) can be written in an order-of-magnitude form expressing the average heat release rate per unit mass in theinterval 0 < t < t m as

b≈

pbmum

ρ0x0, (40)




Figure 4. The lower limit of (a) the preflare value of the plasma parameter β0; (b) the total heatreleased per unit mass; (c) the critical value of the shear ϑ . The large-flare case is depicted by the

thick line and the small-flare cases are shown by thin lines denoted by the applied values of . Thetwo dotted thick vertical lines represent the Mach number range inferred from observations and thethick vertical line shows the Mach number corresponding to U m = 1, i.e., um = vA0 .

where pbm is the highest pressure attained at the flaring volume boundary. Equa-

tion (40) was obtained replacing xt , pbt , and ut in Equation (39) by the average

values xt , pbt , and ut , respectively. Then, we used the approximations ∂pb

t /∂t ≈

pbm/t m, xt ≈ xt m /2, xt m ≈ ut t m, ut ≈ um/2, and pb

t ≈ pbm/2.

When the condition xt m ≈ xt m is satisfied, Equation (23) can be writtenapproximately as

P bm ≈ 2 , (41)

where P bm = pbm/pB0 . Multiplying Equation (40) by t mρ0/pB0 and substituting P bm

from Equation (41), the total heat (Eb= bt m) liberated per unit mass at the flaring

volume boundary during t m can be expressed in the normalized form as




Eb

EB0

≈ 2 U m

, (42)

where EB0 = pB0 /ρ0 is the magnetic energy associated with B0 that is contained

in an unit plasma mass element. The dependence of Eb/EB0 on U m is shown inFigure 4(b) for different values of .Let us presume that the heat is liberated in situ as a part of the energy release

process causing the flare. This means that it is provided by the preflare magneticfield component associated with electric currents, i.e., with the stored free energy.Denoting this component of the magnetic field as Bz, the ratio Bz/B0 can be relatedto the total heat released per unit mass during the time t m as (Bz/B0)2 = Eb/EB0 .On the other hand, one can express the ratio Bz/B0 introducing the shear angle ϑ

as Bz/B0 = tan ϑ . Equation (42) then implies

ϑ > arctan

U m

, (43)

since Equation (16) implies that E0 > Eb.The dependence of the lower limit of the shear angle ϑ on the Mach number of

the shock wave as determined by Equation (43) for the small-flare case is presentedin Figure 4(c) by the thin lines. Assuming that the shear angle ϑ is related to theshear of the photospheric magnetic field, Figure 4c indicates that for M ≥ 1.2small flares must occur in regions of extremely large shear. Bearing in mind thatthere is also a non-thermal component of the energy release, the critical values areeven higher.

When xt m /x0 1 (large flare) Equation (19) can be written in an order of magnitude form as

b ≈ 32p

b

m

t m. (44)

For xt m /x0 1 Equation (23) can be approximately written as

P bm ≈ 2− 1 . (45)

Following the same procedure as in deriving Equation (43) one finds:

ϑ > arctan

1.5(2 − 1) . (46)

Equation (46) gives a lower limit of the shear ranging from ϑ > 45◦ for M = 1.2to ϑ > 70◦ for M = 2. The result given by Equation (46) is shown in Figure 4(c)by the thick line. Figure 4(c) indicates that larger flares are more effective ingenerating perturbations that cause type II radio bursts than the smaller ones.




4. Discussion and Conclusion

The energy released in solar flares is provided by free magnetic energy accumu-lated in the non-potential preflare magnetic field structure. It can be presumed thatthe energy is liberated by the fast reconnection of the magnetic field, proceedingat some 10% of the Alfvén velocity (Priest and Forbes, 1986). The heat neededfor the flaring volume expansion that generates the blast can be released by twomechanisms. It can be liberated in situ by the reconnection process itself (Vrš-nak, 1989), simultaneously with the non-thermal component. This is sometimesobserved as a very hot, hard X-ray emitting plasma, at the summits of flaringloops (Kosugi, 1994). The other possible mechanism is the thermalization of theaccelerated particles. In this case the blast wave ignition should take place in lowerlayers of the solar atmosphere (Karlický and Odstrcil, 1994). In both cases one canexpect that the heat release (qtot(t)) is similar to the hard X-ray or microwave burst.

The back-extrapolations of the type II emission lanes indicate that the blast is

ignited close to the peak of an impulsive microwave and hard X-ray burst (Vrš-nak et al., 1995). Simultaneously, numerous fast-drifting bursts occur in the dm-m wavelength range, forming a type II burst precursor (Klassen et al., 1999).The analysis presented in this paper shows that the highest heat input rate oc-curs somewhat earlier than the maximum expansion velocity is achieved (Fig-ure 3(c)). Inspecting the examples shown in Figure 6 in Paper I, one finds thatback-extrapolations of the synthesized type II burst harmonic lanes point to thesegment of the ‘precursor’ corresponding to the time of the most powerful heating.Assuming that the hard X-ray and microwave bursts reveal processes that also pro-vide the plasma heating, one finds a good correspondence between the observationsand the model.

Taking as an example the generating function defined by Equation (33) it wasshown that the average acceleration U m/t m is higher for a higher impulsivenessof the heat input rate Qmax/τ max (Figure 3(d)). This means that the starting fre-quencies and time delays of type II bursts should depend on the energy releaseimpulsiveness (see Figure 7 in Paper I). Such a dependence was found by Pearsonet al. (1989) for hard X-ray bursts and by Vršnak (2000) for microwave bursts.

Two conditions for the type II burst onset were inferred in Section 3.3. Firstly,the preflare value of the parameter β0 has to be above some critical value, as shownby Equations (37) and (38). The critical value is higher for smaller and less im-pulsive flares. For example a flare characterized by = 0.1 can generate a shockwave of the Mach number M = 1.2 only if it occurs in a β0 > 0.02 region. Such avalue of the parameter is a characteristic of e.g., a flare in a vA0 = 1000 km s−1

environment, developing on the time scale of t m = 100 s and having the lengthscale of x0 = 104 km. A lower value of (e.g., a longer time scale or a smallerlength scale) implies a higher necessary value of β0. Similarly, a higher value of M

requires a higher β0 or a larger .




Secondly, the heat released per unit mass (E) must be sufficiently high (Fig-ure 4(b)). It has to be at least several times higher than the magnetic field energycontained in an unit mass plasma element. The ratio is higher for smaller and lessimpulsive flares (Figure 4(b)). This also implies that such flares have to be hotterconsidering the same value of B0 (Figure 2(b)). Furthermore, if the heat is producedin situ, a higher value of E requires that a larger amount of the free magnetic fieldenergy is stored in an unit mass. So, the shear of the magnetic field has to be largerin smaller and less impulsive flares.

The inferred conditions for the type II burst occurrence can be confronted withobservations analysing well observed events like in Aurass et al. (1999). The com-parison of the two events presented there shows that the smaller 1N/C4.7 flare(named E2) was associated with a type II burst, whereas the larger 2B/M4.4 flare(named E1) was not. The event E1 took place in a region characterized by a modestshear of only 40◦–50◦, whereas the event E2 occurred in a region of a strong shearof 70◦–80◦. This is consistent with the conditions given by Equations (43) and

(46) and the results presented in Figure 4(c). A much stronger shear in the case of the event E2 provided a larger amount of the stored energy density, i.e., a highervalue of E was provided, although the magnetic field was somewhat weaker thanin the event E1. Furthermore, it was estimated that although the flare E1 releasedabout ten times more energy during the non-thermal energy release phase, the rateat which it was released per unit mass was more than four times higher in the flareE2 since it was released in a smaller volume and in a shorter time.

Regarding the condition on β0 let us note that in some cases homologous flaringcan be essential for type II burst formation. The first flare in the sequence causesan increase of the coronal density due to the ‘evaporation’ process, so that the nextflare occurs in a higher β environment (Strong et al., 1984). So, it is possible that

the first of the two successive flares in the same active region is not associated witha type II burst, whereas the second one is. Such an effect can presumably explainthe apparent contradictions found by Cliver, Webb, and Howard (1999) in a criticalre-examination of the ‘Alfvén velocity condition’. It was demonstrated there thatsome of flares occurring successively in the same active region produced type IIbursts and some did not. They concluded that since the Alfvén velocity cannotchange significantly within few hours, the type II burst excitation by a flare can notdepend critically on the value of the Alfvén velocity. However, the value of β0 canchange easily due to the ‘evaporation’ process, changing the conditions for type IIburst occurrence.

Finally, let us summarize the conditions favourable for the flare-ignited shockwave formation:

– a relatively high preflare plasma β environment, characterized by low Alfvénvelocity;

– a highly sheared preflare magnetic field structure;




– a preflare state that provides a fast developing instability which can drive theenergy release process on a time scale comparable with the Alfvén travel time andwhich is efficient in releasing the heat;

– a large spatial extent of the unstable magnetic structure.The variety of necessary conditions for the shock wave formation may explain

why only a small fraction of flares is associated with type II bursts. Even largeflares are not always associated with type II bursts since some of the necessaryconditions may not be satisfied. The estimates presented in Section 3.3. explainthe tendency that the association rate of type II bursts increases with the flareimportance: the criteria are less restrictive for larger flares. Finally the presentedanalysis demonstrates why type II bursts are characterized by low Mach numbers.A large Mach number would require an extremely powerful heating, causing anunreasonably large temperature increase.

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b. vrsnak and s. lulic- formation of coronal mhd shock waves ii: the pressure pulse mechanism

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