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Collisionless shock formation, spontaneous electromagnetic fluctuations, and streaming instabilities
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View: Figures


Image of FIG. 1.
FIG. 1.

Phases of the shock formation. Two identical pair plasmas interpenetrate. The overlapping region turns unstable, and two shocks form near the border of each shell. The simulation box contains half of the system.

Image of FIG. 2.
FIG. 2.

Integrated density in the direction normal to the flow for 3 instants of a typical shock formation simulation. The last plot shows the growth of the magnetic energy integrated over the transverse direction, and . The dashed line is the theoretical growth-rate. The initial Lorentz factor was . All the field growth plots look qualitatively the same until . The saturation time is . The field at saturation is .

Image of FIG. 3.
FIG. 3.

Same as Fig. 2 , but using 800 particles per cell.

Image of FIG. 4.
FIG. 4.

Growth-rate in units, in terms of for (left) and (right).

Image of FIG. 5.
FIG. 5.

Field at saturation from the simulations (circles), compared with Eqs. (9)–(11) .

Image of FIG. 6.
FIG. 6.

Magnetic field energy evolution for different simulation setups and . Black: piston-wall method, Red: full shock picture, Blue: periodic system of counter-streaming beams, Black dashed: theoretical growth rate. A detailed description of the models is given in the text.

Image of FIG. 7.
FIG. 7.

Saturation time from the PIC simulations, circles, from the fluctuations near Eq. (31) , bold line, from the -integrated fluctuations Eq. (26) , thin line, and from the 2D formula (32) , thin dashed line. The 3D -integrated and the 2D theories give almost the same result.


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Scitation: Collisionless shock formation, spontaneous electromagnetic fluctuations, and streaming instabilities