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Ring-shaped velocity distribution functions in energy-dispersed structures formed at the boundaries of a proton stream injected into a transverse magnetic field: Test-kinetic results
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10.1063/1.3686134
/content/aip/journal/pop/19/2/10.1063/1.3686134
http://aip.metastore.ingenta.com/content/aip/journal/pop/19/2/10.1063/1.3686134

Figures

Image of FIG. 1.
FIG. 1.

(Color online) Example of ring-shaped and non-gyrotropic (crescent-like) ion velocity distribution functions observed by Cluster (C1) Cluster Ion Spectrometry (CIS) instruments (Ref. 45) at 09:25:40 UT (left panel), 09:36:32 UT (middle panel), and 09:46:11 UT (right panel) on October 1, 2001 (Ref. 7). All panels illustrate sections in the velocity plane perpendicular to the local magnetic field; the units are km/s on both axes (from Lee et al., Ref. 7).

Image of FIG. 2.
FIG. 2.

(Color online) Magnetic field distribution in the simulation domain for case I. The field is unidirectional and changes orientation at x=0.

Image of FIG. 3.
FIG. 3.

(Color online) Electric field distribution in the simulation domain for case I: (left panel) Ex component, (middle panel) Ey component, and (right panel) magnitude of the E-field. The integration domain is defined by: [−40 000, +40 000] km × [−30 000, +30 000] km.

Image of FIG. 4.
FIG. 4.

(Color online) Schematic diagram of the simulation geometry.

Image of FIG. 5.
FIG. 5.

(Color online) Distribution of protons in the xOy plane after 120 s (∼55TL ) from injection in the electromagnetic field illustrated in Figures 2 and 3. The local value of the number density is color coded. The cloud moved in a region of virtually uniform electric and magnetic field. The spatial mesh on which the VDF is reconstructed is also shown; each bin is identified by a combination of letters and numbers as shown on the left side and bottom side of the figure.

Image of FIG. 6.
FIG. 6.

(Color online) Projection in the space of perpendicular velocities, for vz =0, of the Liouville mapped velocity distribution functions of protons in the cloud at Δt = 120 s. The spatial bins are defined in Fig. 5. One notes the drifting Maxwellian VDF obtained in the central core of the cloud and non-gyrotropic VDFs at the edges of the cloud; the latter result from large Larmor radius particle whose gyrocenters are inside the cloud. Note the different regions of the velocity space populated in bins C1−I1 compared to C5−I5.

Image of FIG. 7.
FIG. 7.

(Color online) Distribution of protons in the xOy plane after 225 s (∼100TL ) from injection in the electromagnetic field illustrated in Figures 2 and 3. The local value of the number density is color coded. The cloud has spent some time in the region of non-uniform fields and its shape is elongated into the +y-direction under the action of the gradient-B drift. The spatial mesh on which the VDF is reconstructed is also shown.

Image of FIG. 8.
FIG. 8.

(Color online) Projection in the space of perpendicular velocities, for vz =0, of the Liouville mapped velocity distribution functions of protons in the cloud at Δt = 225 s. The spatial bins are defined in Fig. 7. A drifting Maxwellian VDF is obtained in the core of the cloud, in bins D1−F1, D2−F2. Ring-shaped VDFs are obtained close to the edges of the cloud, bins D4−F4, E5−F5. Crescent-like VDF are obtained close to the trailing and leading edges, A2−A4, B4−B5, C4−C5, G4−G5, H4−H5, and I4−I5.

Image of FIG. 9.
FIG. 9.

(Color online) Distribution of protons in the xOy plane after 275 s (∼125TL ) from injection in the electromagnetic field illustrated in Figures 2 and 3. This snapshot illustrates the initial stage of the interaction between the proton cloud and the region of the most rapid variation of B ; some parts of the cloud intersected the plane x=0 where B =0. The local value of the number density is color coded. The spatial mesh on which the VDF is reconstructed is also shown.

Image of FIG. 10.
FIG. 10.

(Color online) Projection in the space of perpendicular velocities, for vz =0, of the Liouville mapped velocity distribution functions of protons in the cloud at Δt = 275 s. The spatial bins are defined in Fig. 9. During this initial stage of the interaction of the cloud with the discontinuity, one identifies the Maxwellian core of the cloud (bins B1−B2) and non-gyrotropic VDFs at the leading edge (e.g., bins D5−I5).

Image of FIG. 11.
FIG. 11.

(Color online) Distribution of protons in the xOy plane after 300 s (∼135TL ) from injection in the electromagnetic field illustrated in Figures 2 and 3. The local value of the number density is color coded. The spatial mesh on which the VDF is reconstructed is also shown. The figure illustrates a later stage of the interaction between the cloud and the central region of the discontinuity where the magnetic field vanishes. A significant number of protons moved in the region of positive Bz .

Image of FIG. 12.
FIG. 12.

(Color online) Projection in the space of perpendicular velocities, for vz =0, of the Liouville mapped velocity distribution functions of protons at Δt = 300 s. The spatial bins are defined in Fig. 11. We note that in the region of positive Bz , the VDFs of protons are ring-shaped (bins G4−I4 and H5−I5) or crescent-like (bins G3−I3 and E5−G5).

Image of FIG. 13.
FIG. 13.

(Color online) Distribution of protons in the xOy plane after 350 s (∼160TL ) from injection in the electromagnetic field illustrated in Figures 2 and 3. The local value of the number density is color coded. The spatial mesh on which the VDF is reconstructed is also shown. One notes the splitting of the cloud into two populations: population P1 that does not cross the surface where B  = 0 and remains trapped in some region on the left side of the discontinuity (x < 0) and, respectively, population P2 that penetrates into the right side of the magnetic discontinuity. At later stages, the two populations disconnect. In the reminder of the paper, we follow only P2.

Image of FIG. 14.
FIG. 14.

(Color online) Projection in the space of perpendicular velocities, for vz =0, of the Liouville mapped velocity distribution functions of protons in the cloud at Δt = 350 s. The spatial bins are defined in Fig. 13. Only VDFs of the P2 population are shown. Ring-shaped and crescent-like VDFs are observed in the large majority of spatial bins.

Image of FIG. 15.
FIG. 15.

(Color online) Distribution of protons of the population P2 in the xOy plane after 600 s (∼270TL ) from injection in the electromagnetic field illustrated in Figures 2 and 3. The local value of the number density is color coded. The spatial mesh on which the VDF is reconstructed is also shown. The protons move in a region of uniform magnetic and electric field, on the right side of the discontinuity.

Image of FIG. 16.
FIG. 16.

(Color online) Projection in the space of perpendicular velocities, for vz =0, of the Liouville mapped velocity distribution functions of protons in the cloud at Δt = 600 s. The spatial bins are defined in Fig. 15. All the VDFs obtained for this stage of propagation are either ring-shaped or crescent-like, a signature of the interaction of the cloud with the region of magnetic field gradient.

Image of FIG. 17.
FIG. 17.

(Color online) Distribution of protons in the xOy plane after 360 s (∼160TL ) from injection on the left side in the case of a unidirectional, non-uniform, parallel magnetic field, and a uniform electric field. The local value of the number density is color coded. The spatial mesh on which the VDF is reconstructed is also shown. The deformation of the shape of the cloud is due to the gradient-B drift.

Image of FIG. 18.
FIG. 18.

(Color online) Projection in the space of perpendicular velocities, for vz =0, of the Liouville mapped velocity distribution functions of protons in the cloud at Δt = 360 s for a parallel magnetic field and a uniform electric field. Spatial bins are defined in Fig. 17. Note, the formation of the central cavity due to the gradient-B drift in bins of the upper three rows; non-gyrotropic VDFs are obtained in bins from the column A, B, C, G, H, and I.

Tables

Generic image for table
Table I.

Input parameters of the test-kinetic simulations: N 0, kT 0, and V 0 are the density, thermal energy, and average velocity of the drifting Maxwellian given by Eq. (6); B 1 z and B 2 z are the asymptotic values of the magnetic field; L is the length scale of the discontinuity; RL is the Larmor radius of the thermal protons in the frame of reference moving with V 0; TL is the proton cyclotron period; Nx  × Ny is the number of injection sources; n is the number of test-particles injected from each source; x 0 and y 0 are the coordinates of the first source; and dx 0 and dy 0 are the separation distances between sources along Ox and Oy.

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/content/aip/journal/pop/19/2/10.1063/1.3686134
2012-02-23
2014-04-18
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Ring-shaped velocity distribution functions in energy-dispersed structures formed at the boundaries of a proton stream injected into a transverse magnetic field: Test-kinetic results
http://aip.metastore.ingenta.com/content/aip/journal/pop/19/2/10.1063/1.3686134
10.1063/1.3686134
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