Two-dimensional Moist Stratified Turbulence and the Emergence of Vertically Sheared Horizontal Flows

Moist stratified turbulence is studied in a two-dimensional Boussinesq system influenced by condensation and evaporation. The problem is set in a periodic domain and employs simple evaporation and condensation schemes, wherein both the processes push parcels towards saturation. Numerical simulations demonstrate the emergence of a moist turbulent state consisting of ordered structures with a clear power-law type spectral scaling from initially spatially uncorrelated conditions. An asymptotic analysis in the limit of rapid condensation and strong stratification shows that, for initial conditions with enough water substance to saturate the domain, the equations support a straightforward state of moist balance characterized by a hydrostatic, saturated, vertically sheared horizontal flow (VSHF). For such initial conditions, by means of long time numerical simulations, the emergence of moist balance is verified. Specifically, starting from uncorrelated data, subsequent to the development of a moist turbulent state, the system experiences a rather abrupt transition to a regime which is close to saturation and dominated by a strong VSHF. On the other hand, initial conditions which do not have enough water substance to saturate the domain, do not attain moist balance. Rather, the system remains in a turbulent state and oscillates about moist balance. Even though balance is not achieved with these general initial conditions, the time scale of oscillation about moist balance is much larger than the imposed time scale of condensation and evaporation, thus indicating a distinct dominant slow component in the moist stratified two-dimensional turbulent system.Comment: 23 pages. 9 figure

Nonlinear stability of the ensemble Kalman filter with adaptive covariance inflation

The Ensemble Kalman filter and Ensemble square root filters are data assimilation methods used to combine high dimensional nonlinear models with observed data. These methods have proved to be indispensable tools in science and engineering as they allow computationally cheap, low dimensional ensemble state approximation for extremely high dimensional turbulent forecast models. From a theoretical perspective, these methods are poorly understood, with the exception of a recently established but still incomplete nonlinear stability theory. Moreover, recent numerical and theoretical studies of catastrophic filter divergence have indicated that stability is a genuine mathematical concern and can not be taken for granted in implementation. In this article we propose a simple modification of ensemble based methods which resolves these stability issues entirely. The method involves a new type of adaptive covariance inflation, which comes with minimal additional cost. We develop a complete nonlinear stability theory for the adaptive method, yielding Lyapunov functions and geometric ergodicity under weak assumptions. We present numerical evidence which suggests the adaptive methods have improved accuracy over standard methods and completely eliminate catastrophic filter divergence. This enhanced stability allows for the use of extremely cheap, unstable forecast integrators, which would otherwise lead to widespread filter malfunction.Comment: 34 pages. 4 figure