**By:** Oliver Allanson

**Figure 1: **A particle undergoes Brownian motion.

**The short answer: probably not, at least not all of the time. **

In our state-of-the-art and physics-based numerical experiments, we analyse the motion of 100 million individual high-energy electrons that evolve within conditions like that found within the Earth’s hazardous ‘radiation belt’ environment. We observe that electrons do not always behave according to the manner that is most typically used by scientists to describe their evolution. The standard mathematical description that is most commonly used is based upon diffusion proceeding in a manner that is analogous to ‘Brownian motion’, e.g. the familiar high-school experiment showing the random motion of particles suspended within a fluid. The random motion of an individual particle undergoing Brownian motion is illustrated in Figure 1 [1]. In contrast, we observe that the electrons sometimes spread apart at rates that either ‘accelerate’ or ‘decelerate’ in time. This could have implications for the modelling of high-energy electrons in our magnetosphere, and hence for satellite safety.

**Figure 2: **The Earth’s Radiation Belts.

**Figure 3: **Not all diffusion is Brownian! The ‘mean-squared-displacement’ can evolve at rates that either increase (‘super-diffusion’) or decrease (‘sub-diffusion’) with time.

**The Earth’s outer radiation belt **

The Earth’s outer radiation belt is a dynamic and spatially extended radiation environment within the Earth’s inner magnetosphere, composed of energetic plasma that is trapped by the geomagnetic field (see Figure 2 [2]). The size and location of the outer radiation belt varies dramatically in response to solar wind variability. The lifetime of some individual energetic particles can be long (~years). However, orders of magnitude changes in the particle flux can occur on much shorter timescales (~hours). Whilst we know that the radiation belt environment is ultimately driven by the solar wind and the pre-existing state of the magnetosphere, it is very challenging to accurately predict, or model, fluxes within the radiation belt. This difficulty arises from the fact that the magnetosphere can store and transport energy in many different ways, and over a range of different time and length scales. This difficulty in prediction is a pressing concern given the hundreds of satellites that orbit within this hazardous environment. The highly variable and energetic electron environment poses critical space weather hazards for Low, Medium, and Geosynchronous Earth Orbiting (LEO, MEO, and GEO) spacecraft; thus, the ability to predict its variability is a key goal of the magnetospheric space weather community. Most physics-based computer models of particle dynamics in the radiation belts rely upon a specific version of the ‘quasilinear theory’. This approach is founded upon a number of physical assumptions that are now known not to always hold in the radiation belt. Furthermore, the mathematics that is used to describe this quasilinear theory is based upon ‘normal diffusion’ equations, i.e. equations that (in a given space) describe ‘stochastic’ Brownian motion. This stochastic assumption is also considered to be uncertain in given circumstances. Our work tries to test these assumptions, by processing data from state-of-the-art and fully self-consistent numerical experiments. Electron diffusion characteristics are directly extracted from particle data. The ‘nature’ of the diffusive response is not always constant in time, i.e. we observe a time dependent ‘rate of diffusion’, that is inconsistent with Brownian motion (see Figure 3 [3]). However, after an initial transient phase, the rate of diffusion does tend to a constant, in a manner that is consistent with the assumptions of quasilinear diffusion theory. This work establishes a framework for future investigations on the nature of diffusion due to in the Earth’s outer radiation belts, using physics-based numerical experiments.

**How much, and when, does this matter?**

All of the work described here pertains to a ‘benchmarking’ scenario in which we prove the concept of our experimental technique, and under which conditions one is least likely to observe particularly exotic behaviour [4]. In future experiments we will: (i) make more quantitative assessments; (ii) subject the plasma to more extreme conditions (we therefore expect to find a more sustained ‘non-Brownian’ response); (iii) assess the implications on current models.

[1] A particle undergoes Brownian motion.

Reproduced from https://commons.wikimedia.org/wiki/File:Csm_Brownian-Motion_f99de6516a.png.

[2] The Earth’s Radiation Belts.

Reproduced from https://www.nasa.gov/mission_pages/sunearth/news/gallery/20130228-radiationbelts.html.

[3] Not all diffusion is Brownian! The ‘mean-squared-displacement’ can evolve at rates that either increase (‘super-diffusion’) or decrease (‘sub-diffusion’) with time.

Reproduced from https://commons.wikimedia.org/wiki/File:Msd_anomalous_diffusion.svg.

[4] O. Allanson, C. E. J. Watt, H. Ratcliffe, N. P. Meredith, H. J. Allison, S. N. Bentley, T.

Bloch and S. A. Glauert, Particle-in-cell experiments examine electron diffusion by whistler-mode waves: 1. Benchmarking with a cold plasma, Journal of Geophysical Research: Space Physics (in press).