RT Journal Article
SR Electronic
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
FD The Royal Society
DO 10.1098/rspa.2017.0388
VO 473
IS 2205
A1 Cotter, C. J.
A1 Gottwald, G. A.
A1 Holm, D. D.
YR 2017
UL http://rspa.royalsocietypublishing.org/content/473/2205/20170388.abstract
AB 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.