Progress on Coronal, Interplanetary, Foreshock, and Outer Heliospheric Radio Emissions
Iver H. Cairns , P. A. Robinson , and G. P. Zank, PASA, 17 (1), 22.
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STOCHASTIC GROWTH THEORY

Stochastic growth theory describes situations in which an unstable distribution of particles interacts self-consistently with its driven waves in an inhomogeneous plasma environment and evolves to a state in which (1) the particle distribution fluctuates stochastically about a state close to time- and volume-averaged marginal stability, and (2) the fluctuations in the distribution drive waves so that the wave gain $G = 2 \ln (E / E_{0})$ is a stochastic variable (Robinson 1992, 1995, Robinson et al. 1993, Cairns & Robinson 1997, 1999). (Put another way, the wave gain is the time integral of the wave growth rate, being related to the time-varying wave electric field E(t) by $E^{2}(t) = E_{0}^{2} \exp[ G(t) ]$ where E₀ is a constant field.) At a given location the postulated stochastic nature of the wave gain means that the wave fields undergo a random walk in $\log E$, whence SGT predicts that the waves occur in bursts with irregular, widely variable fields. Moreover, the closeness to marginal stability means that SGT predicts that the unstable particle distribution and driven waves will persist far from the region where the unstable distribution was first created. SGT is therefore a natural theory to explain the bursty and irregular plasma waves and associated persistence of (marginally) unstable particle distributions that are characteristic of observations in space.

One focus of our current research program is to determine how widely applicable SGT is, with a view to ascertaining whether the combination of SGT and nonlinear wave processes is a more broadly applicable paradigm for wave growth in space plasmas. Note that current interpretations of some astrophysical emissions (e.g., radio emissions from pulsars and AGNs) implicitly require preservation of the driving electron distributions for distances much greater than predicted by standard theory to relax the distribution function, thereby perhaps pointing to a role for SGT there.

As required for a theory involving a stochastic variable, the primary observational tests of SGT involve the statistics of the observed wave fields. In particular, for simple SGT systems (in which thermal and nonlinear effects can be neglected and many fluctuations in the distribution occur during a characteristic time for wave growth), SGT predicts via the Central Limit Theorem that the probability distributions of G and $\log E$ should be Gaussian in G and $\log E$, respectively, (Robinson 1992, 1995, Robinson et al. 1993, Cairns & Robinson 1997, 1999); that is,

$$P(\log E) = \frac{1}{\sqrt{2\pi} \sigma} \exp{\left[ -\frac{ (\log E - \mu)^{2} }{2 \sigma^{2}} \right] } \ .$$ (1)

Here $\mu = \langle \log E \rangle$ is the average of $\log E$ while $\sigma$ is the standard deviation of $\log E$. Theoretical predictions for the distribution $P(\log E)$ are also known for situations when thermal effects, net linear growth, and nonlinear processes are important (Robinson et al. 1993, Robinson 1995) and have been tested successfully (Robinson et al. 1993, Cairns, Robinson & Anderson 2000). It is appropriate to contrast the predictions of SGT with the standard model for wave growth in plasmas (e.g., Stix 1962, Krall & Trivelpiece 1973, Melrose 1986): in the standard model the plasma and unstable particle distribution are homogeneous and the waves undergo exponential growth with a constant growth rate given by homogeneous ``linear'' instability theory until the waves reach a level (the threshold) at which one or more nonlinear processes can proceed to saturate the instability and limit the wave fields. The standard model therefore predicts that the $P(\log E)$ distribution should be uniform (or flat) from thermal fields up to the threshold field for the nonlinear processes. [The predictions for $P(\log E)$ above the nonlinear threshold depend on the nature of the nonlinear processes (e.g., Cairns & Robinson 1997, 1999), being either flat, peaked above the nonlinear threshold, or a decreasing power-law tail.] The thresholds for known nonlinear processes can be calculated from analytic plasma theory. It is therefore easy to compare the SGT predictions for the $P(\log E)$ distribution with those of the standard model so as to determine which model, if either, is consistent with observations. A final comment is that the burstiness of the observed waves has no obvious explanation in the standard model, requiring the development of detailed models on a case by case basis, while SGT explains this basic characteristic directly. Conversely, SGT requires an explanation for why the growth is stochastic, typically involving the growth of waves and associated modifications of the unstable particle distribution in an inhomogeneous plasma. Semi-quantitative models for this exist for the Langmuir waves in type III bursts and Earth's foreshock.

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