7-13 September 2009

Coronal Shock Coronal Shock Formation in Various Formation in Various

Ambient MediaAmbient Media

IHY-ISWI Regional MeetingHeliophysical phenomena and Earth's environment

7-13 September 2009, Šibenik, Croatia

Tomislav Žic, Bojan VršnakHvar Observatory, Faculty of Geodesy, Kačićeva 26, HR-10000 Zagreb

Manuela Temmer, Astrid VeronigInstitute of Physics, University of Graz, Universitätsplatz 5/II, 8010 Graz, Austria

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Introduction

Coronal MHD shock waves are closely associated with flares or CMEs

Necessary requirement: a motion perpendicular to the magnetic field lines (the source volume-expansion) large amplitude perturbation in the ambient plasma

the source region expansion is investigated in the cylindrical and spherical coordinate system

2D & 3D piston driver of an MHD shock wave○ constant piston acceleration (duration of an acceleration phase is tmax, and the

maximum expansion velocity vmax)○ environment dependent on radial distance!○ speed of low-amplitude perturbation w0(r) :

• constant• 1/r • 1/r2

○ two cases: high sound & low MHD

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Intention

Our interest: the shock-formation time/distance due to the non-linear wavefront evolution larger-amplitude elements propagate faster;[Landau, L.D. and Lifshitz, E.M.: Fluid Mechanics, (Pergamon Press, 1987)]

Energy conservation signal amplitude is decreasing with distance difference from 1D model (!)[Vršnak, B. and Lulić, S., Solar Phys., 196 (2000) 157-180(24)]

( )x t( )er t

( )r t

( )wr t

Piston expansion and wave-front propagation

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Model

Source-surface speed, v(t), at certain time t is defined by:○ initial velocity v0,

○ final velocity vmax

○ acceleration time tmax

Kinetic energy conservation has been taken into account; e.g. for >> 1:

 u2 w R = const. g(u) R = const.○ ( = 1 cylindrical; = 2 spherical)

generally, g(u) depends on characteristics of the ambient plasma, primarily on the value of ; we consider << 1 and >> 1

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Non-linear wavefront evolution

velocity and position of a given wavefront segment (“signal”) are defined by:

w(t) = drw(t)/dt

w(t) = w0(r) + k u(t)w

x

discontinuity = shock

0Av

u

u

w

w

rw

*

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Solving differential equations

Taking into account the energy conservation and w(u) we find:

○ with the flow velocity boundary condition: u0 ≡ u(t0) = v(t0);[the source velocity at the moment t0 is equal to the speed of the source-surface, v(t0)]

○ where:

• u0, r

0 and g

0 stand for values at initial moment t

0; when a given wave

segment is created◦ = 1 in the cylindrical coordinate system

◦ = 2 in the spherical coordinate system

1/( 1)0 0 d d

0d dw w w

w w

r g g gr g r r

r r

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Example of the wave-front propagation and determination of the time/distance shock formation for w0 = 500 km/s

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Shock-formation time (t*) and distance (rw) for w00(r)

10 500 kmsw

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Shock-formation time (t*) and distance (rw) for w01(r)

10 p0( ) 500 kmsw r r r

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Shock-formation time (t*) and distance (rw) for w02(r)

2 10 p0( ) 500 kmsw r r r

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Results and conclusion

The results show that the shock-formation time t∗ and the shock-formation distance rw

∗ are:○ approximately proportional to the acceleration phase

duration tmax,

○ shorter for a higher source speed vmax,

○ only weakly dependent on the initial source size rp0,

○ shorter for a higher source acceleration a, and

○ lower in an environment characterized by steeper decrease of w0

Questions?