Continuity equations in the Generalised Lagrangian Mean theory

Abstract

Generalised Lagrangian Mean (GLM) theory describes the joint evolution of the mean flow and its perturbations. This paper analyses the exact GLM continuity equations (GLM-CEs), presents them in new forms, and adds new equations to complete them. Our three-dimensional results include: (i) A modified approach to GLM theory that uses only the basic notions of classical fluid dynamics. The only tools used are Lagrangian X, Eulerian x, averaged (mean) Eulerian x' coordinates of fluid particles, and an averaging parameter a. The targeted forms of GLM-CEs are expressed with functions x(x',t, a) and r(x',t, a), where r is the fluid density. (ii) Actual velocity divergence div u expressed through div'u', where u and u' are the actual and mean fluid velocities. (iii) Three new forms of GLM-CEs. Each version represents a sum of a mean and an a-dependent (tilde) part. The latter addition makes each form mathematically complete. (iv) The original Andrews-McIntyre transformation (AMT) is also complemented by its tilde part. Then it also gives the full version of the exact GLM-CEs. However, it works only for a special class of fluid flows. Its structure explains why the tilde equations have been overlooked. (v) Our generalisation of AMT, which repeats the results (i)-(iii) but requires a more complex derivation. (vi) Examples of flows with small but finite Lagrangian perturbations, linking our exposition to the classical GLM theory. Keywords: continuity equations, general Lagrangian mean theory, average (mean) coordinates, Eulerian description, tilde equations, Andrews-McIntyre transformation.