Properties of conditionally filtered equations: Conservation, normal modes, and variational formulation

John Thuburn,Geoffrey K Vallis

doi:10.1002/qj.3307

John Thuburn, Geoffrey K Vallis

Open Access

https://doi.org/10.1002/qj.3307

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Abstract

Conditionally filtered equations have recently been proposed as a basis for modeling the atmospheric boundary layer and convection. Conditional filtering decomposes the fluid into a number of categories or components, such as convective updraughts and the background environment, and derives governing equations for the dynamics of each component. Because of the novelty and unfamiliarity of these equations, it is important to establish some of their physical and mathematical properties and to examine whether their solutions might behave in counterintuitive or even unphysical ways. It is also important to understand the properties of the equations in order to develop suitable numerical solution methods. The conditionally filtered equations are shown to have conservation laws for mass, entropy, momentum or axial angular momentum, energy, and potential vorticity. The normal modes of the conditionally filtered equations include the usual acoustic, inertio‐gravity, and Rossby modes of the standard compressible Euler equations. In addition, the equations support modes with different perturbations in the different fluid components that resemble gravity modes and inertial modes but with zero pressure perturbation. These modes make no contribution to the total filter‐scale fluid motion, and their amplitude diminishes as the filter scale diminishes. Finally, it is shown that the conditionally filtered equations have a natural variational formulation, which can be used as a basis for systematically deriving consistent approximations.

Highlights

Filtered equations have recently been proposed as a basis for mathematical and numerical modelling of the atmospheric boundary layer and convection (Thuburn et al 2018).Conditional filtering itself is an extension of coarse-graining ideas that are commonly used in large-eddy turbulence modelling, and that enable one to write down equations of motion valid c 2017 Royal Meteorological SocietyPrepared using qjrms4.cls [Version: 2013/10/14 v1.1]for a particular scale of motion, with the subgrid-scale terms appearing on the right-hand side and in need of parameterization – see Leonard (1975), Frisch (1995) and Aluie et al (2018)for a range of examples
We have identified inertia and gravity modes with zero pressure perturbation in which the fluid components move separately, and in general in opposite vertical and horizontal directions. This is precisely a property one might wish for when modelling subgrid-scale convection, in which some of the subgrid-scale fluid ascends while some of it descends
The amplitude of these modes goes to zero as the filter scale diminishes, which is an attractive property when considering how the fluid system might behave as the model resolution increases

Summary

Introduction

Filtered equations have recently been proposed as a basis for mathematical and numerical modelling of the atmospheric boundary layer and convection (Thuburn et al 2018).Conditional filtering itself is an extension of coarse-graining ideas that are commonly used in large-eddy turbulence modelling, and that enable one to write down equations of motion valid c 2017 Royal Meteorological SocietyPrepared using qjrms4.cls [Version: 2013/10/14 v1.1]for a particular scale of motion, with the subgrid-scale terms appearing on the right-hand side and in need of parameterization – see Leonard (1975), Frisch (1995) and Aluie et al (2018)for a range of examples. That ζi × ui Equations (112), (116), (114), and (133) derived from the variational method agree with the conditionally filtered equations (8), (9), (18), and (34). Summary and discussion of the conditionally filtered equations.

Results

Conclusion