Pitch-angle dependent transport of particles through discontinuities

Pitch-angle dependent Pitch-angle dependent transport of particles through transport of particles through

discontinuitiesdiscontinuities

Y.Y. KartavykhKartavykh1,21,2

ISSI, Berne, 10-14 February, 2014

(1) Ioffe Physical-Technical Institute(2) University of Würzburg

ISSI, Berne, 10-14 September 2014ISSI, Berne, 10-14 September 2014


From Kartavykh , Dröge, Klecker, JGR, 2013

This event was considered by L.Wang et al., 2011


From Baumjohann&Treumann „Basic Plasma Physics“

Structure of the magnetosphere of the Earth

Bi-modal electron pitch angle distributions observed in SEP event on June, 4, 2000 by s/c Wind


Pitch-angle distributions measured by ACE and

Wind

Energy spectra of outward and inward streaming electrons


The observed by Wind and ACE time profiles



As an example to fit – event on June, 4, 2000One-dimensional propagation model First – finite

differences approach

s(t): coordinate along the magnetic field spiral(t): pitch angleW(t): Wiener process

Solving equation of focused transport via Monte-Carlo approach

System of SDE


There is no delay between solar and backstreaming elections within the accuracy of measurements (12 s)


Reflecting Boundary Model

Somewhat arbitrary assumptions about the reflectivity‘ shape

Distance Wind – bow shock = 450 000 km (0.003 AU)


Adiabatic Transmission Model

Mirrow condition:particles are reflected if:

Magnetic moment of a gyrating particle


Adiabatic transmission and magnetosheath diffusion model

Magnetosheath size

Adiabatic changes of pitch angles when go through the bow shock



Corrected omnidirectional and sectored data

Reflection from beyond the Earth


Similar model, but reflection boundary is located beyond the Earth , at a heliocentric distance , e.g. 1.5 AU(0.8 AU from the Earth along the mfl)


Fokker-Planck equation for the particle’s propagation which can be applied to consider also shock-like structures

Stochastic differential equations for changes in s and

The pitch angle diffusion coefficient

Convection is included implicitly


At certain conditions the solution of the kinetic equation reproduces the results for the diffusive shock acceleration (DSA).

Cyan – prediction from the DAS approach

Continuous and homogenious in PA cosine injection of particles on the SF

t1=10 -20 hours, t2=30-40 , t3=50-60, t4=70-80, t5=90-100, t6=110-120, t7=130-140, t8=150-160 , t9 =170-180, t12=230-240 , t18=350-360, t21=410-420 hours.


Some examples of simulated energy spectra


Pitch angle distributions at different locations in upstream (left) and downstream (right) regions. Compression ratio R = 3.0, 1= 0.01 AU, 2 = 0.001 AU, zup = 0.7 AU, zdown = 0.1 AU. Continuous (in time ) injection of protons at the shock front for 700 hours. Time of counting: 200 - 700 hours


Spatial dependence of accelerated protons

From DSA theory density in upstream is proportional

where U is the speed of fluid


In certain range of parametrs, depending on the boundary and initial conditions, this approach gives results similar to the DSA theory

On the other hand, the results which are different from DSA approach look more realistic


C O N C L U S I O N S

-- Model of pitch angle diffusion (PAD) transport is flexible and allows relatively easily track particles through discontinuities

-- Equivalence of the results with the DSA theory depends on the initial and boundary conditions

-- Model o f PAD transport gives more realistic results for shock acceleration.

Thank you for your attention!

Pitch-angle dependent transport of particles through discontinuities

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