Abstract

Suppose the observations of Lagrangian trajectories for fluid flow in some physical situation can be modelled sufficiently accurately by a spatially correlated Itô stochastic process (with zero mean) obtained from data which is taken in fixed Eulerian space. Suppose we also want to apply Hamilton’s principle to derive the stochastic fluid equations for this situation. Now, the variational calculus for applying Hamilton’s principle requires the Stratonovich process, so we must transform from Itô noise in the data frame to the equivalent Stratonovich noise. However, the transformation from the Itô process in the data frame to the corresponding Stratonovich process shifts the drift velocity of the transformed Lagrangian fluid trajectory out of the data frame into a non-inertial frame obtained from the Itô correction. The issue is, ‘Will non-inertial forces arising from this transformation of reference frames make a difference in the interpretation of the solution behaviour of the resulting stochastic equations?’ This issue will be resolved by elementary considerations.

Highlights

  • Suppose the Lagrangian trajectories for fluid flow in some physical situation are modelled sufficiently accurately by a spatially correlated Itô stochastic process obtained from data which is taken to be statistically stationary with zero mean in the inertial frame of fixed Eulerian space

  • The present note investigates how to deal with non-inertial forces in stochastic dynamics which arise from Itô corrections as changes of frame when applying mixed Itô and Stratonovich stochastic modelling in 3D stochastic Euler–Boussinesq equations (SEB) fluid dynamics

  • The integrand is in a fixed inertial frame and the circulation loop is in the moving frame of the Lagrangian fluid parcels

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Summary

Introduction

The present note investigates how to deal with non-inertial forces in stochastic dynamics which arise from Itô corrections as changes of frame when applying mixed Itô and Stratonovich stochastic modelling in 3D SEB fluid dynamics The resolution of this issue has already been given above in the comparison between equations (1.3) and (1.4). Langmuir circulations would be viewed in the relative drift frame of the Lagrangian fluid parcels with velocity uLt (x), as being caused by the non-inertial force felt in the moving frame of the Itô correction uS(x) The presence of this sort of fictitious force is why Newton’s law of motion F = ma only applies in an inertial frame. Ut is the Eulerian momentum per unit mass in Newton’s second law and uLt is the transport drift velocity for the corresponding equivalent Stratonovich representation of the Lagrangian trajectory. With the constraints in (2.1) Hamilton’s principle will result in a motion equation in the Euler–Poincaré form [36]

D δ δb db
Conclusion
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