Coronal Shock Formation in Various Ambient Media

1. IHY-ISWI Regional MeetingHeliophysical phenomena and Earth's environment7-13 September 2009, Šibenik, Croatia Coronal Shock Formation in Various Ambient Media Tomislav Žic, Bojan Vršnak Hvar Observatory, Faculty of Geodesy, Kačićeva 26, HR-10000 Zagreb Manuela Temmer, Astrid Veronig Institute of Physics, University of Graz, Universitätsplatz 5/II, 8010 Graz, Austria

2. 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 T. Žic et al.

3. Intention Piston expansion and wave-front propagation • 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)] T. Žic et al.

4. 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: r u2w Ra = const.g(u) Ra = 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 T. Žic et al.

5. discontinuity = shock rw* 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) T. Žic et al.

6. 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, r0 and g0 stand for values at initial moment t0; when a given wave segment is created • a= 1 in the cylindrical coordinate system • a= 2 in the spherical coordinate system T. Žic et al.

7. Example of the wave-front propagation and determination of the time/distanceshock formation for w0 = 500 km/s T. Žic et al.

8. Shock-formation time (t*) and distance (rw) for w00(r) T. Žic et al.

9. Shock-formation time (t*) and distance (rw) for w01(r) T. Žic et al.

10. Shock-formation time (t*) and distance (rw) for w02(r) T. Žic et al.

11. 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 T. Žic et al.

12. Questions? Thank you for your attention