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Two different types of perturbations of the Lorenz 63 dynamical system forRayleigh-Benard convection by multiplicative noise -- called stochasticadvection by Lie transport (SALT) noise and fluctuation-dissipation (FD) noise-- are found to produce qualitatively different effects, possibly because thetotal phase-space volume contraction rates are different. In the process ofmaking this comparison between effects of SALT and FD noise on the Lorenz 63system, a stochastic version of a robust deterministic numerical algorithm forobtaining the individual numerical Lyapunov exponents was developed. With thisstochastic version of the algorithm, the value of the sum of the Lyapunovexponents for the FD noise was found to differ significantly from the value ofthe deterministic Lorenz 63 system, whereas the SALT noise preserves the Lorenz63 value with high accuracy. The Lagrangian averaged version of the SALTequations (LA SALT) is found to yield a closed deterministic subsystem for theexpected solutions which is found to be isomorphic to the original Lorenz 63dynamical system. The solutions of the closed chaotic subsystem, in turn, drivea linear stochastic system for the fluctuations of the LA SALT solutions aroundtheir expected values.

Lyapunov Exponents of Two Stochastic Lorenz 63 Systems