Wave propagation through a stationary field of clouds: A homogenisation approach

The effect of a subgrid‐scale cloud field on the propagation of long atmospheric waves is investigated using a new scale‐consistent formulation based on the asymptotic theory of homogenisation. A key aim is to quantify potential model errors in wave propagation speeds, introduced by using averaged fields in place of the fully resolved circulation, in the setting of a simple stratified Boussinesq midlatitude β$$ \beta $$‐channel model. The effect of the cloud field, represented here by a random array of strongly nonlinear axisymmetric circulations, is found to appear in the large‐scale governing equations through new terms which redistribute the large‐scale buoyancy and horizontal momentum fields in the vertical. These new terms, which have the form of nonlocal integral operators, are linear in the cloud number density and are fully determined by the solution of a linear elliptic equation known as a cell problem. The cell problem in turn depends on the details of the nonlinear cloud circulations. The integral operators are calculated explicitly for example cloud fields and then dispersion relations are compared for different waves in the presence of clouds at realistic densities. The main finding is that baroclinic Rossby waves are significantly slowed and damped by the clouds, whilst inertia–gravity waves are affected almost exclusively by damping, most strongly at the lowest frequencies. In contrast, all waves with a barotropic structure are found to be almost unaffected by the presence of clouds, even at the highest realistic cloud densities. An important consequence of this study is a new approach to the closure of subgrid‐scale cloud fields in the parameterisation of convection in large‐scale atmospheric models. A new scale‐consistent asymptotic formulation for the effect of subgrid‐scale convection on the large‐scale atmosphere is derived. The effect of many randomly distributed nonlinear cloud circulations, such as the one shown in the image, is shown to enter the large‐scale equations through nonlocal operators involving “transilient kernels”. The influence of a cloud field on the dispersive characteristics of midlatitude inertia–gravity and Rossby waves is investigated.