BibTex format
@article{Crisan:2022,
author = {Crisan, D and Holm, DD and Leahy, J-M and Nilssen, T},
journal = {arXiv},
title = {Variational principles for fluid dynamics on rough paths},
url = {http://arxiv.org/abs/2004.07829v2},
year = {2022}
}
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Citation
RIS format (EndNote, RefMan)
TY - JOUR
AB - In this paper, we introduce a new framework for parametrization schemes (PS)in GFD. Using the theory of controlled rough paths, we derive a class of roughgeophysical fluid dynamics (RGFD) models as critical points of rough actionfunctionals. These RGFD models characterize Lagrangian trajectories in fluiddynamics as geometric rough paths (GRP) on the manifold of diffeomorphic maps.Three constrained variational approaches are formulated for the derivation ofthese models. The first is the Clebsch formulation, in which the constraintsare imposed as rough advection laws. The second is the Hamilton-Pontryaginformulation, in which the constraints are imposed as right-invariant roughvector fields. The third is the Euler--Poincar\'e formulation in which thevariations are constrained. These variational principles lead directly to theLie--Poisson Hamiltonian formulation of fluid dynamics on geometric roughpaths. The GRP framework preserves the geometric structure of fluid dynamicsobtained by using Lie group reduction to pass from Lagrangian to Eulerianvariational principles, thereby yielding a rough formulation of the Kelvincirculation theorem. The rough-path variational approach includes non-Markovianperturbations of the Lagrangian fluid trajectories. In particular, memoryeffects can be introduced through this formulation through a judicious choiceof the rough path (e.g. a realization of a fractional Brownian motion). In thespecial case when the rough path is a realization of a semimartingale, werecover the SGFD models in Holm (2015). However, by eliminating the need forstochastic variational tools, we retain a pathwise interpretation of theLagrangian trajectories. In contrast, the Lagrangian trajectories in thestochastic framework are described by stochastic integrals which do not have apathwise interpretation. Thus, the rough path formulation restores thisproperty.
AU - Crisan,D
AU - Holm,DD
AU - Leahy,J-M
AU - Nilssen,T
PY - 2022///
TI - Variational principles for fluid dynamics on rough paths
T2 - arXiv
UR - http://arxiv.org/abs/2004.07829v2
UR - http://hdl.handle.net/10044/1/95336
ER -