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A new mathematical formulation for the constitutive laws governing elastic perfectly plastic materials is proposed here. In particular, it is shown that the elastic strain rate and the plastic strain rate form an orthogonal decomposition with respect to the tangent cone and the normal cone of the yield domain. It is also shown that the stress rate can be seen as the projection on the tangent cone of the elastic stress tensor. This approach leads to a coherent mathematical formulation of the elasto-plastic laws and simplifies the resulting system for the associated flow evolution equations. The cases of one or two yields functions are treated in detail. The practical examples of the von Mises and Tresca yield criteria are worked out in detail to demonstrate the usefulness of the new formalism in applications.

Quasi-geostrophic (QG) theory describes the dynamics of synoptic scale flows in the trophosphere that are balanced with respect to both acoustic and internal gravity waves. Within this framework, effects of (turbulent) friction near the ground are usually represented by Ekman Layer theory. The troposphere covers roughly the lowest ten kilometers of the atmosphere while Ekman layer heights are typically just a few hundred meters. However, this two-layer asymptotic theory does not explicitly account for substantial changes of the potential temperature stratification due to diabatic heating associated with cloud formation or with radiative and turbulent heat fluxes, which, in the middle latitudes, can be particularly important in about the lowest three kilometers. To address this deficiency, this paper extends the classical QG-Ekman layer model by introducing an intermediate, dynamically and thermodynamically active layer, called the "diabatic layer" (DL) from here on. The flow in this layer is also in acoustic, hydrostatic, and geostrophic balance but, in contrast to QG flow, variations of potential temperature are not restricted to small deviations from a stable and time independent background stratification. Instead, within the diabatic layer, diabatic processes are allowed to affect the leading-order stratification. As a consequence, the diabatic layer modifies the pressure field at the top of the Ekman layer, and with it the intensity of Ekman pumping seen by the quasi-geostrophic bulk flow. The result is the proposed extended quasi-geostrophic three-layer QG-DL-Ekman model for mid-latitude (dry and moist) dynamics., Comment: 12 pages, 5 figures

Recent research has suggested that the overall dependence of convection near coasts on large-scale atmospheric conditions is weaker than over the open ocean or inland areas. This is due to the fact that in coastal regions convection is often supported by meso-scale land-sea interactions and the topography of coastal areas. As these effects are not resolved and not included in standard cumulus parametrization schemes, coastal convection is among the most poorly simulated phenomena in global models. To outline a possible parametrization framework for coastal convection we develop an idealized modeling approach and test its ability to capture the main characteristics of coastal convection. The new approach first develops a decision algorithm, or trigger function, for the existence of coastal convection. The function is then applied in a stochastic cloud model to increase the occurrence probability of deep convection when land-sea interactions are diagnosed to be important. The results suggest that the combination of the trigger function with a stochastic model is able to capture the occurrence of deep convection in atmospheric conditions often found for coastal convection. When coastal effects are deemed to be present the spatial and temporal organization of clouds that has been documented form observations is well captured by the model. The presented modeling approach has therefore potential to improve the representation of clouds and convection in global numerical weather forecasting and climate models., Comment: Manuscript submitted for publication in Journal of Advances in Modeling Earth Systems

The representation of clouds and associated processes of rain and snow formation remains one of the major uncertainties in climate and weather prediction models. In a companion paper (Part I), we systematically derived a two moment bulk cloud microphysics model for collision and coalescence in warm rain based on the kinetic coalescence equation (KCE) and used stochastic approximations to close the higher order moment terms, and do so independently of the collision kernel. Conservation of mass and consistency of droplet number concentration of the evolving cloud properties were combined with numerical simulations to reduce the parametrization problem to three key parameters. Here, we constrain these three parameters based on the physics of collision and coalescence resulting in a "region of validity." Furthermore, we theoretically validate the new bulk model by deriving a subset of the "region of validity" that contains stochastic parameters that skillfully reproduces an existing model based on an a priori droplet size distribution by Seifert and Beheng (2001). The stochastic bulk model is empirically validated against this model, and parameter values that faithfully reproduce detailed KCE results are identified. Furthermore, sensitivity tests indicate that the stochastically derived model can be used with a time step as large as 30 seconds without significantly compromising accuracy, which makes it very attractive to use in medium to long range weather prediction models. These results can be explored in the future to select parameters in the "region of validity" that are conditional on environmental conditions and the age of the cloud., Comment: Part II of two parts

We propose a mathematical methodology to derive a stochastic parameterization of bulk warm cloud micro-physics properties. Unlike previous bulk parameterizations, the stochastic parameterization does not assume any particular droplet size distribution, all parameters have physical meanings which are recoverable from data, and the resultant parameterization has the flexibility to utilize a variety of collision kernels. Our strategy is a new two-fold approach to modelling the kinetic collection equation. Partitioning the droplet spectrum into cloud and rain aggregates, we represent droplet densities as the sum of a mean and a random fluctuation. Moreover, we use a Taylor approximation for the collision kernel which allows the resulting parameterization to be independent of the collision kernel. To address the two-moment closure for bulk microphysical equations, we represent the higher (third) order terms as points in an Ornstein-Uhlenbeck-like stochastic process. These higher order terms are aggregate number concentration, and aggregate mixing ratio, fluctuations and product fluctuations on regions of a 2D space of pre-collision droplets. This 2D space is naturally partitioned into four domains, each associated with a collision process: cloud and rain self-collection, auto-conversion, and accretion. The stochastic processes in the solution to the kinetic collection equation are defined as the sum of a mean and the product of a standard deviation and a random fluctuation. An order of magnitude argument on the temporal fluctuations of the evolving cloud properties eliminates the terms containing a standard deviation and a random fluctuation from the mean evolution equations. The remaining terms form a coupled set of ODE's without any ad-hoc parameters or any assumed droplet size distributions and flexible enough to accept any collision kernel without further derivations., Comment: Part I of two parts

We present a numerical method to solve the optimal transport problem with a quadratic cost when the source and target measures are periodic probability densities. This method is based on a numerical resolution of the corresponding Monge-Amp\`ere equation. We extend the damped Newton algorithm of Loeper and Rapetti \cite{LR} to the more general case of a non uniform density which is relevant to the optimal transport problem, and we show that our algorithm converges for sufficiently large damping coefficients. The main idea consists of designing an iterative scheme where the fully nonlinear equation is approximated by a non-constant coefficient linear elliptic PDE that we solve numerically. We introduce several improvements and some new techniques for the numerical resolution of the corresponding linear system. Namely, we use a Fast Fourier Transform (FFT) method by Strain \cite{St}, which allows to increase the efficiency of our algorithm against the standard finite difference method. Moreover, we use a fourth order finite difference scheme to approximate the partial derivatives involved in the nonlinear terms of the Newton algorithm, which are evaluated once at each iteration; this leads to a significant improvement of the accuracy of the method, but does not sacrifice its efficiency. Finally, we present some numerical experiments which demonstrate the robustness and efficiency of our method on several examples of image processing, including an application to multiple sclerosis disease detection., Comment: 30 pages, 18 figures

Inspired by recent developments in Berdina-like models for turbulence, we propose an inviscid regularization for the surface quasi-geostrophic (SQG) equations. We are particularly interested in the celebrated question of blowup in finite time of the solution gradient of the SQG equations. The new regularization yields a necessary and sufficient condition, satisfied by the regularized solution, when a regularization parameter $\alpha$ tends to zero, for the solution of the original SQG equations to develop a singularity in finite time. As opposed to the commonly used viscous regularization, the inviscid equations derived here conserve a modified energy. Therefore, the new regularization provides an attractive numerical procedure for finite time blow up testing. In particular, we prove that, if the initial condition is smooth, then the regularized solution remains as smooth as the initial data for all times. Moreover, much like the original problem, the inviscid regularization admits a maximum principle.