Nonlinear stability and the saturation of instabilities to axisymmetric vortices

Nonlinear stability theorems are presented for axisymmetric vortices under the restriction that the disturbance is independent of either the azimuthal or the axial coordinate. These stability theorems are then used, in both cases, to derive rigorous upper bounds on the saturation amplitudes of instabilities. Explicit examples of such bounds are worked out for some canonical profiles. The results establish a minimum order for the dependence of saturation amplitude on supercriticality, and are thereby suggestive as to the nature of the bifurcation at the stability threshold.

Spatial variability of liquid cloud and rain: observations and microphysical effects

Spatial variability of liquid cloud water content and rainwater content is analysed from three different observational platforms: in situ measurements from research aircraft, land-based remote sensing techniques using radar and lidar, and spaceborne remote sensing from CloudSat. The variance is found to increase with spatial scale, but also depends strongly on the cloud or rain fraction regime, with overcast regions containing less variability than broken cloud fields. This variability is shown to lead to large biases, up to a factor of 4, in both the autoconversion and accretion rates estimated at a model grid scale of ≈40 km by a typical microphysical parametrization using in-cloud mean values. A parametrization for the subgrid variability of liquid cloud and rainwater content is developed, based on the observations, which varies with both the grid scale and cloud or rain fraction, and is applicable for all model grid scales. It is then shown that if this parametrization of the variability is analytically incorporated into the autoconversion and accretion rate calculations, the bias is significantly reduced.