Conditionally filtered equations have recently been proposed as a basis for modelling
the atmospheric boundary layer and convection. Conditional filtering decomposes the
fluid into a number of categories or components, such as convective updrafts and the
background environment, and derives governing equations for the dynamics of ...
Conditionally filtered equations have recently been proposed as a basis for modelling
the atmospheric boundary layer and convection. Conditional filtering decomposes the
fluid into a number of categories or components, such as convective updrafts 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 counter-intuitive 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, they posses 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.