Anomalous exponents at the onset of an instability

TL;DR

This paper calculates critical exponents at the onset of instability, revealing anomalous scaling behavior due to multiplicative noise, supported by numerical simulations and explaining recent experimental findings.

Contribution

It introduces a method to compute critical exponents exactly at instability onset and demonstrates how multiplicative noise causes anomalous scaling behavior.

Findings

Critical exponents can be anomalous under multiplicative noise.

Numerical simulations confirm the anomalous scaling behavior.

Results explain recent experimental observations of dynamo instability.

Abstract

Critical exponents are calculated exactly at the onset of an instability, using asymptotic expansiontechniques. When the unstable mode is subject to multiplicative noise whose spectrum at zero frequency vanishes, we show that the critical behavior can be anomalous, i.e. the mode amplitude X scales with departure from onset \mu as $<X> ~ \mu^\beta$ with an exponent $\beta$ different from its deterministic value. This behavior is observed in a direct numerical simulation of the dynamo instability and our results provide a possible explanation to recent experimental observations.