TY - JOUR
T1 - Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics
JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science
M3 - 10.1098/rspa.2017.0388
VL - 473
IS - 2205
AU - Cotter, C. J.
AU - Gottwald, G. A.
AU - Holm, D. D.
Y1 - 2017/09/01
UR - http://rspa.royalsocietypublishing.org/content/473/2205/20170388.abstract
N2 - In Holm (Holm 2015 Proc. R. Soc. A 471, 20140963. (doi:10.1098/rspa.2014.0963)), stochastic fluid equations were derived by employing a variational principle with an assumed stochastic Lagrangian particle dynamics. Here we show that the same stochastic Lagrangian dynamics naturally arises in a multi-scale decomposition of the deterministic Lagrangian flow map into a slow large-scale mean and a rapidly fluctuating small-scale map. We employ homogenization theory to derive effective slow stochastic particle dynamics for the resolved mean part, thereby obtaining stochastic fluid partial equations in the Eulerian formulation. To justify the application of rigorous homogenization theory, we assume mildly chaotic fast small-scale dynamics, as well as a centring condition. The latter requires that the mean of the fluctuating deviations is small, when pulled back to the mean flow.
ER -