### Abstract

Multiple scale perturbation methods are used to study the transport and acceleration of energetic charged particles in quasi-periodic, fluid velocity structures in one, two, or three space dimensions, with spatial period l _{u}, where l_{u} is much less than the diffusion scale length l_{d} = κ_{0}/u_{0} and κ_{0} and u_{0} are characteristic values of the energetic particle diffusion coefficients and fluid speed, respectively. The particle diffusion tensor K is also allowed to vary periodically on the scale l_{u}. In the case in which the perturbation parameter ε = l_{u}/l_{d} = u _{0}l_{u} is small (0 < ≪ 1), the long space and time behavior of the energetic particle distribution function 〈 f 〉 at lowest order is shown to satisfy a modified Fokker-Planck equation. This equation arises from compatibility conditions imposed on the perturbation equations in order to obtain a consistent perturbation expansion that is free of secular terms. The analysis shows that the particles are accelerated stochastically on the large scale as a result of the divergence ∇ δu of the background fluid velocity perturbation δu. The net acceleration of the particles due to the velocity variations can be described in part by a second-order Fermi-like momentum space diffusion term in the long-scale transport equation obtained by averaging over the short-scale variations. The momentum space diffusion coefficient DT describing the effect depends on the two-point correlation of the fluid velocity divergence ∇ δu at different points in the flow. There is also a further energization term in the long-scale transport equation, corresponding to the work done by the scattering center fluid against the differential cosmic-ray pressure gradient that is modified as a result of the short-scale variations. The convective particle streaming is also modified as a result of the short-scale variations. The analysis shows that the effective spatial diffusion tensor for low-energy particles can be significantly modified as a result of turbulent diffusion, whereas higher energy particles with much larger diffusion tensor elements are not significantly affected by turbulent diffusion. Averaging over a random ensemble of short-scale, quasi-periodic velocity structures generalizes the turbulent transport coefficients obtained by previous authors.

Original language | English (US) |
---|---|

Pages (from-to) | 195-226 |

Number of pages | 32 |

Journal | Astrophysical Journal |

Volume | 595 |

Issue number | 1 I |

DOIs | |

State | Published - Sep 20 2003 |

### Fingerprint

### Keywords

- Acceleration of particles
- Cosmic rays
- Turbulence

### ASJC Scopus subject areas

- Space and Planetary Science

### Cite this

*Astrophysical Journal*,

*595*(1 I), 195-226. https://doi.org/10.1086/377355

**Diffusive-Compression Acceleration and Turbulent Diffusion of Cosmic Rays in Quasi-Periodic and Turbulent Flows.** / Webb, G. M.; Ko, C. M.; Zank, G. P.; Jokipii, J. Randy.

Research output: Contribution to journal › Article

*Astrophysical Journal*, vol. 595, no. 1 I, pp. 195-226. https://doi.org/10.1086/377355

}

TY - JOUR

T1 - Diffusive-Compression Acceleration and Turbulent Diffusion of Cosmic Rays in Quasi-Periodic and Turbulent Flows

AU - Webb, G. M.

AU - Ko, C. M.

AU - Zank, G. P.

AU - Jokipii, J. Randy

PY - 2003/9/20

Y1 - 2003/9/20

N2 - Multiple scale perturbation methods are used to study the transport and acceleration of energetic charged particles in quasi-periodic, fluid velocity structures in one, two, or three space dimensions, with spatial period l u, where lu is much less than the diffusion scale length ld = κ0/u0 and κ0 and u0 are characteristic values of the energetic particle diffusion coefficients and fluid speed, respectively. The particle diffusion tensor K is also allowed to vary periodically on the scale lu. In the case in which the perturbation parameter ε = lu/ld = u 0lu is small (0 < ≪ 1), the long space and time behavior of the energetic particle distribution function 〈 f 〉 at lowest order is shown to satisfy a modified Fokker-Planck equation. This equation arises from compatibility conditions imposed on the perturbation equations in order to obtain a consistent perturbation expansion that is free of secular terms. The analysis shows that the particles are accelerated stochastically on the large scale as a result of the divergence ∇ δu of the background fluid velocity perturbation δu. The net acceleration of the particles due to the velocity variations can be described in part by a second-order Fermi-like momentum space diffusion term in the long-scale transport equation obtained by averaging over the short-scale variations. The momentum space diffusion coefficient DT describing the effect depends on the two-point correlation of the fluid velocity divergence ∇ δu at different points in the flow. There is also a further energization term in the long-scale transport equation, corresponding to the work done by the scattering center fluid against the differential cosmic-ray pressure gradient that is modified as a result of the short-scale variations. The convective particle streaming is also modified as a result of the short-scale variations. The analysis shows that the effective spatial diffusion tensor for low-energy particles can be significantly modified as a result of turbulent diffusion, whereas higher energy particles with much larger diffusion tensor elements are not significantly affected by turbulent diffusion. Averaging over a random ensemble of short-scale, quasi-periodic velocity structures generalizes the turbulent transport coefficients obtained by previous authors.

AB - Multiple scale perturbation methods are used to study the transport and acceleration of energetic charged particles in quasi-periodic, fluid velocity structures in one, two, or three space dimensions, with spatial period l u, where lu is much less than the diffusion scale length ld = κ0/u0 and κ0 and u0 are characteristic values of the energetic particle diffusion coefficients and fluid speed, respectively. The particle diffusion tensor K is also allowed to vary periodically on the scale lu. In the case in which the perturbation parameter ε = lu/ld = u 0lu is small (0 < ≪ 1), the long space and time behavior of the energetic particle distribution function 〈 f 〉 at lowest order is shown to satisfy a modified Fokker-Planck equation. This equation arises from compatibility conditions imposed on the perturbation equations in order to obtain a consistent perturbation expansion that is free of secular terms. The analysis shows that the particles are accelerated stochastically on the large scale as a result of the divergence ∇ δu of the background fluid velocity perturbation δu. The net acceleration of the particles due to the velocity variations can be described in part by a second-order Fermi-like momentum space diffusion term in the long-scale transport equation obtained by averaging over the short-scale variations. The momentum space diffusion coefficient DT describing the effect depends on the two-point correlation of the fluid velocity divergence ∇ δu at different points in the flow. There is also a further energization term in the long-scale transport equation, corresponding to the work done by the scattering center fluid against the differential cosmic-ray pressure gradient that is modified as a result of the short-scale variations. The convective particle streaming is also modified as a result of the short-scale variations. The analysis shows that the effective spatial diffusion tensor for low-energy particles can be significantly modified as a result of turbulent diffusion, whereas higher energy particles with much larger diffusion tensor elements are not significantly affected by turbulent diffusion. Averaging over a random ensemble of short-scale, quasi-periodic velocity structures generalizes the turbulent transport coefficients obtained by previous authors.

KW - Acceleration of particles

KW - Cosmic rays

KW - Turbulence

UR - http://www.scopus.com/inward/record.url?scp=0141484416&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0141484416&partnerID=8YFLogxK

U2 - 10.1086/377355

DO - 10.1086/377355

M3 - Article

AN - SCOPUS:0141484416

VL - 595

SP - 195

EP - 226

JO - Astrophysical Journal

JF - Astrophysical Journal

SN - 0004-637X

IS - 1 I

ER -