Edward Goldsmith

We employ a multiple-scales asymptotic analysis to derive a simple model for the mesoscale tropical atmosphere interacting with a field of cloud-scale convective circulations. Most importantly, we take account of the fact that cloud-scale convection in the tropics experiences a westward tilt under the influence of the non-traditional Coriolis force. The systematic approach uncovers a two-way coupling between the mesoscale and cloud-scales and provides a physically consistent closure via the specification of averaged flux terms. This closure takes the form of a non-local vertical diffusion of absolute mesoscale horizontal momentum, with a diffusion kernel containing all details of the underlying convective circulations. Ultimately, it is shown that the westward tilt in convection creates up-scale fluxes of momentum and drives a self-regulating vertical shear of zonal wind at the mesoscale. In the tropics where vertical shear is typically weak, the effect of the non-traditional Coriolis terms (NCT) is maximized, and therefore has a significant effect on mesoscale dynamics. The mechanism for self-regulation uncovers an underlying tendency for the tropical atmosphere to adjust towards a state where the vertical shear of the mesoscale zonal wind balances the non-traditional component of Earth’s rotation. Ultimately, this study indicates that the NCT plays a significant role in tropical dynamics and suggests that we reassess the validity of omitting its effect in meteorological models.

In the study of subgrid-scale tropical convection, the importance of retaining the frequently omitted "nontraditional" component of the Coriolis force is increasingly being recognized. A number of recent papers have developed linear theories examining the behavior of a diabatic heat-source-driven convective circulation in the presence of the full Coriolis force, and it was shown that the nontraditional Coriolis terms drive vertical shears on the large scales through upscale fluxes of momentum. In the present work, we generalize these results to the nonlinear regime, using a formal asymptotic theory based upon the fact that rotation is a second-order effect compared with advection by the vertical component of velocity at subgrid scales. Ultimately, we demonstrate that the same basic flow structures persist, with a particular emphasis on the counterrotating vortex pair induced by the nontraditional Coriolis terms which drive a westward tilt in convection. We compute the form of the upscale momentum flux convergence in the nonlinear regime, greatly extending the regimes of validity provided by the simple analytical expressions previously given in the linear case. This study constitutes an important step toward being able to accurately and consistently parameterize the large-scale vertical shear driven by nonlinear, subgrid convective processes under the influence of the nontraditional Coriolis force terms.

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 & beta;$$ \beta $$-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.image

The question of how finite-amplitude, small-scale topography affects small-amplitude motions in the ocean is addressed in the framework of the rotating shallow water equations. The extent to which the dispersion relations of Poincaré, Kelvin and Rossby waves are modified in the presence of topography is illuminated, using a range of numerical and analytical techniques based on the method of homogenisation. Both random and regular periodic arrays of topography are considered, with the special case of regular cylinders studied in detail, because this case allows for highly accurate analytical results. The results show that, for waves in a $\beta$-channel bounded by sidewalls, and for steep topographies outside of the quasi-geostrophic regime, topography acts to slow Poincaré waves slightly, Rossby waves are slowed significantly and Kelvin waves are accelerated for long waves and slowed for short waves, with the two regimes being separated by a narrow band of resonant wavelengths. The resonant band, which is due to the excitation of trapped topographic Rossby waves on each seamount, may affect any of the three wave types under the right conditions, and for physically reasonable results requires regularisation by Ekman friction. At larger topographic amplitudes, for cylindrical topography, a simple and accurate formula is given for the correction to the Rossby wave dispersion relation, which extends previous results for the quasi-geostrophic regime.

The unsteady ascent of a buoyant, turbulent line plume through a quiescent, uniform environment is modelled in terms of the width-averaged vertical velocity and density deficit. It is demonstrated that for a well-posed, linearly stable model, account must be made for the horizontal variation of the velocity and the density deficit; in particular the variance of the velocity field and the covariance of the density deficit and velocity fields, represented through shape factors, must exceed threshold values, and that models based upon ‘top-hat’ distributions in which the dependent fields are piecewise constant are ill-posed. Numerical solutions of the nonlinear governing equations are computed to reveal that the transient response of the system to an instantaneous change in buoyancy flux at the source may be captured through new similarity solutions, the form of which depend upon both the ratio of the old to new buoyancy fluxes and the shape factors.