Post on 10-Jan-2016
description
transcript
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
7-13 September 2009
2T. 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 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
7-13 September 2009
3T. 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)]
( )x t( )er t
( )r t
( )wr t
Piston expansion and wave-front propagation
7-13 September 2009
4T. Žic et al.
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
7-13 September 2009
5T. Žic et al.
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
*
7-13 September 2009
6T. Žic et al.
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
7-13 September 2009
7T. Žic et al.
Example of the wave-front propagation and determination of the time/distance shock formation for w0 = 500 km/s
7-13 September 2009
8T. Žic et al.
Shock-formation time (t*) and distance (rw) for w00(r)
10 500 kmsw
7-13 September 2009
9T. Žic et al.
Shock-formation time (t*) and distance (rw) for w01(r)
10 p0( ) 500 kmsw r r r
7-13 September 2009
10T. Žic et al.
Shock-formation time (t*) and distance (rw) for w02(r)
2 10 p0( ) 500 kmsw r r r
7-13 September 2009
11T. Žic et al.
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?