1. 7-13 September 2009 Coronal Shock Formation in Various Ambient Media IHY-ISWI Regional Meeting Heliophysical phenomena and Earth's environment 7-13 September 2009, Šibenik, Croatia 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. 7-13 September 20092T. Žic et al. 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 t max, and the maximum expansion velocity v max ) ○environment dependent on radial distance! ○speed of low-amplitude perturbation w 0 (r) : constant 1/r 1/r 2 ○two cases: high   sound & low   MHD

3. 7-13 September 20093T. Žic et al. 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)] Piston expansion and wave-front propagation

4. 7-13 September 20094T. Žic et al. Model Source-surface speed, v(t), at certain time t is defined by: ○initial velocity v 0, ○final velocity v max ○acceleration time t max Kinetic energy conservation has been taken into account; e.g. for  >> 1:  u 2 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

5. 7-13 September 20095T. Žic et al. Non-linear wavefront evolution velocity and position of a given wavefront segment (“signal”) are defined by: w(t) = dr w (t)/dt w(t) = w 0 (r) + k u(t) discontinuity = shock rw*rw*

6. 7-13 September 20096T. Žic et al. Solving differential equations Taking into account the energy conservation and w(u) we find: ○with the flow velocity boundary condition: u 0 ≡ u(t 0 ) = v(t 0 ); [the source velocity at the moment t 0 is equal to the speed of the source- surface, v(t 0 )] ○where: u 0, 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

7. 7-13 September 20097T. Žic et al. Example of the wave-front propagation and determination of the time/distance shock formation for w 0 = 500 km/s

8. 7-13 September 20098T. Žic et al. Shock-formation time (t * ) and distance (r w ) for w 00 (r)

9. 7-13 September 20099T. Žic et al. Shock-formation time (t * ) and distance (r w ) for w 01 (r)

10. 7-13 September 200910T. Žic et al. Shock-formation time (t * ) and distance (r w ) for w 02 (r)

11. 7-13 September 200911T. Žic et al. Results and conclusion The results show that the shock-formation time t ∗ and the shock-formation distance r w ∗ are: ○approximately proportional to the acceleration phase duration t max, ○shorter for a higher source speed v max, ○only weakly dependent on the initial source size r p0, ○shorter for a higher source acceleration a, and ○lower in an environment characterized by steeper decrease of w 0

12. Questions?