Earthquake Complexity

Encyclopedia of earth sciences series/Encyclopedia of earth sciences · 2023-01-01

Earthquake Complexity

Encyclopedia of earth sciences series/Encyclopedia of earth sciences · 2022-01-01

Charged particle isotropization in collisionless environments: turbulent magnetic fields as a conduit for random walks

Journal of Plasma Physics · 2020 · 1 citations

• Physics
• Classical mechanics
• Mechanics

We develop a simple model for the kinematics of charged particles in regions of magnetic turbulence. We approximate the local magnetic field as smoothly varying in strength and direction, where adiabatic invariance prevails, or as presenting rapid changes in direction or ‘kinks’. Particles execute guiding centre gyromotion around a field line. However, in analogy to kinetic theory for collisional environments, when the particle undergoes a rapid change in direction by some angle $\unicode[STIX]{x1D703}$ , it would instantaneously transition to Larmor motion around the new field line. This mimics Brownian motion wherein we replace collisions with other particles by rapid transitions or ‘collisions’ with other field lines. Using standard methods drawn from Brownian motion, we follow the evolution of the parallel and perpendicular components of the velocity, namely $v_{\Vert }$ and $v_{\bot }$ , and rigorously show that kinetic energy isotropization necessarily emerges.

SIXTEEN. Extracting Our Venus

Princeton University Press eBooks · 2019-05-12

Emergence of patterns in random processes. III. Clustering in higher dimensions

Physical review. E · 2019-07-08

Newman et al. [Phys. Rev. E 86, 026103 (2012)10.1103/PhysRevE.86.026103] showed that points uniformly distributed as independent and identically distributed random variables with nearest-neighbor interactions form clusters with a mean number of three points in each. Here, we extend our analysis to higher dimensions, ultimately going to infinite dimensions, and we show that the mean number of points per cluster rises monotonically with a limiting value of four.

Mathematical Methods for Geophysics and Space Physics

Princeton University Press eBooks · 2016-09-29 · 5 citations

Graduate students in the natural sciences—including not only geophysics and space physics but also atmospheric and planetary physics, ocean sciences, and astronomy—need a broad-based mathematical toolbox to facilitate their research. In addition, they need to survey a wider array of mathematical methods that, while outside their particular areas of expertise, are important in related ones. While it is unrealistic to expect them to develop an encyclopedic knowledge of all the methods that are out there, they need to know how and where to obtain reliable and effective insights into these broader areas. Here at last is a graduate textbook that provides these students with the mathematical skills they need to succeed in today’s highly interdisciplinary research environment. This authoritative and accessible book covers everything from the elements of vector and tensor analysis to ordinary differential equations, special functions, and chaos and fractals. Other topics include integral transforms, complex analysis, and inverse theory; partial differential equations of mathematical geophysics; probability, statistics, and computational methods; and much more. Proven in the classroom, Mathematical Methods for Geophysics and Space Physics features numerous exercises throughout as well as suggestions for further reading. Provides an authoritative and accessible introduction to the subject Covers vector and tensor analysis, ordinary differential equations, integrals and approximations, Fourier transforms, diffusion and dispersion, sound waves and perturbation theory, randomness in data, and a host of other topics Features numerous exercises throughout Ideal for students and researchers alike An online illustration package is available to professors

A MULTIRATE VARIABLE-TIMESTEP ALGORITHM FOR N-BODY SOLAR SYSTEM SIMULATIONS WITH COLLISIONS

The Astronomical Journal · 2016-02-18 · 1 citations

ABSTRACT We present and analyze the performance of a new algorithm for performing accurate simulations of the solar system when collisions between massive bodies and test particles are permitted. The orbital motion of all bodies at all times is integrated using a high-order variable-timestep explicit Runge–Kutta Nyström (ERKN) method. The variation in the timestep ensures that the orbital motion of test particles on eccentric orbits or close to the Sun is calculated accurately. The test particles are divided into groups and each group is integrated using a different sequence of timesteps, giving a multirate algorithm. The ERKN method uses a high-order continuous approximation to the position and velocity when checking for collisions across a step. We give a summary of the extensive testing of our algorithm. In our largest simulation—that of the Sun, the planets Earth to Neptune and 100,000 test particles over 100 million years—the relative error in the energy after 100 million years was of the order of 10 −11 .

Chapter 1. Mathematical Preliminaries

Princeton University Press eBooks · 2016-09-29

Mathematical Methods for Geophysics and Space Physics

Princeton University Press eBooks · 2016-05-03

Chapter 2. Ordinary Differential Equations

Princeton University Press eBooks · 2016-09-29