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Brief Summary of Stochastic Growth Theory and its Applications in Space Physics

Iver H. Cairns and P. A. Robinson,
School of Physics, University of Sydney, NSW 2006, Australia
Tel: +61-2-9351-3961, Fax: +61-2-9351-7726, e-mail: cairnsphysics.usyd.edu.au

(16 January 2001).

Most waves observed in space plasmas are bursty, with widely and irregularly varying fields that are typically weak compared with the thresholds for relevant nonlinear processes. In addition, it is common for the waves and driving distributions to persist together unexpectedly far from the source of the unstable particle distribution. Well-known examples include the Langmuir waves and electron beams in interplanetary type III solar radio bursts and planetary foreshocks. The burstiness, wide variability, typical weakness, and persistence of the waves pose fundamental, longstanding theoretical problems for the standard ``secular growth'' model [1] for wave growth in plasmas, in which the background plasma is homogeneous and the waves grow exponentially with constant growth rate until the lowest threshold for a nonlinear process is exceeded and the wave growth is then saturated by the nonlinear process.

Stochastic growth theory (SGT), a relatively new theory, addresses the development of fluctuations in the waves and unstable distribution due to their self-consistent interaction in the pre-existing inhomogeneous background plasma [2-9]. In SGT the particle distribution is close to time and volume-averaged marginal stability but with fluctuations about this state that cause the wave gain G (the time integral of the growth rate) to be a stochastic variable. Here $G = \int dt \Gamma$ and E²(t) = E₀² e^G(t), where E(t) is the wave field at time t and E₀ is a reference field. SGT then naturally explains bursty, irregular waves (since the gain is proportional to the logarithm of the wave field and is a stochastic variable) that persist with their driving distribution (since the system is close to marginal stability).

Importantly, SGT quantitatively predicts the statistics of the wave fields [3-9]. For pure SGT the probability distribution P(E) of wave fields E is lognormal. Put another way, binning the wave fields logarithmically and constructing the distribution $P(\log E)$ , the SGT prediction is that

$\begin{displaymath}P(\log E) = (\sqrt{2\pi\ \sigma^{2}})^{-1/2} e^{- ( \log E - \mu )^{2} / 2 \sigma^{2}}\ , \end{displaymath}$

(1)

Spatial or temporal variations in the quantities $\mu$ and $\sigma$ can modify the probability distribution and prevent a straightforward test of the theory - for instance, in Earth's foreshock both $\mu$ and $\sigma$ vary with the foreshock depth parameter D_f [e.g., 6,7]. In this case, it is necessary to either (1) calculate the $P(\log E)$ distribution and test SGT for short intervals of time or ranges of position - so that appropriate values of $\mu$ and $\sigma$ can be used to test the theory [e.g., 3,6,8,9], or (2) extract trends in $\mu$ and $\sigma$ with time or position and then test SGT using a suitably renormalized field variable [e.g. 7]. Both procedures have been used successfully.

How should the field variable be renormalized? The most obvious choice is the new variable (ordering the data using time t rather than position merely for convenience)

$\begin{displaymath}X(t) = [\ \log E(t) - \mu(t)\ ] / \sigma(t)\ , \end{displaymath}$

(2)

$\begin{displaymath}P(X) = {2 \pi}^{-1/2} e^{-X^{2} / 2}\ . \end{displaymath}$

(3)

Two ways to extract the trends in $\mu$ and $\sigma$ are clear. The first is to fit an analytic forms, such as a power-law in position for example [7]. Otherwise, where the experimental or theoretical situation is less well understood, one can construct sliding averages of $\mu$ or $\sigma$ for the dataset and extract those. This second path is followed below.

Models for why the growth is stochastic and for the parameters $\mu$ and $\sigma$ can be constructed in some cases [3,5-8], so as to provide a complete SGT description of the phenomena. In other circumstances, the observational data may be shown to be consistent with SGT, but without an extant model for the observed values of $\mu$ and $\sigma$ [e.g., 9].

At the time of writing, SGT appears to be widely applicable to space plasmas, being consistent with the available data in all seven cases yet analyzed: Langmuir waves in type III sources [3], thermal Langmuir waves in the solar wind [5], Langmuir waves in the edge [5] and major volume of Earth's foreshock [6,7], PF waves over Earth's polar cap [8], and mirror mode and electromagnetic ion cyclotron waves in Earth's magnetosheath [9]. These results imply that SGT is widely applicable in space and astrophysical plasmas. Of course, it is not relevant in every case.

We are interested in giving people tools to determine whether SGT is relevant. Related web pages to this one give a simple IDL code, instructions for running it and interpreting the results, and both a sample data file and the associated output (for Langmuir waves in Earth's foreshock). Please contact us if you have any questions or if you would like us to participate or otherwise advise you in your analysis. An acknowledgement of use of this material is required in any publications to which it contributes. This is the only formal requirement for use, but we would also like to be told what your results are and of any improvements that should be made to the IDL code.

Those desiring a greater understanding of SGT than this brief summary can provide should read the references below or contact either Iver Cairns (cairns@physics.usyd.edu.au) or Peter Robinson (p.robinson@physics.usyd.edu.au). Their fax number is +61-2-9351-7726 and their telephone numbers are +61-2-9351-3961 and +61-2-9351-3779, respectively.