36 research outputs found
Asymptotics for moist deep convection I: Refined scalings and self-sustaining updrafts
Moist processes are among the most important drivers of atmospheric dynamics, and scale analysis and asymptotics are cornerstones of theoretical meteorology. Accounting for moist processes in systematic scale analyses therefore seems of considerable importance for the field. Klein & Majda (TCFD, vol. 20, 525--552, (2006)) proposed a scaling regime for the incorporation of moist bulk microphysics closures in multi-scale asymptotic analyses of tropical deep convection. This regime is refined here to allow for mixtures of ideal gases and to establish consistency with a more general multiple scales modelling framework for atmospheric flows.
Deep narrow updrafts, so-called "hot towers", constitute principal building blocks of larger scale storm systems. They are analysed here in a sample application of the new scaling regime. A single quasi-onedimensional columnar cloud is considered on the vertical advective (or tower life cycle) time scale. The refined asymptotic scaling regime is essential for this example as it reveals a new mechanism for the self-sustainance of such updrafts. Even for strongly positive convectively available potential energy (CAPE), a vertical balance of buoyancy forces is found in the presence of precipitation. This balance induces a diagnostic equation for the vertical velocity and it is responsible for the generation of self-sustained balanced updrafts. The time dependent updraft
structure is encoded in a Hamilton-Jacobi equation for the precipitation mixing ratio. Numerical solutions of this equation suggest that the self-sustained updrafts may strongly enhance hot tower life cycles
On nonlinear conservation laws with a nonlocal diffusion term
AbstractScalar one-dimensional conservation laws with a nonlocal diffusion term corresponding to a Riesz–Feller differential operator are considered. Solvability results for the Cauchy problem in L∞ are adapted from the case of a fractional derivative with homogeneous symbol. The main interest of this work is the investigation of smooth shock profiles. In the case of a genuinely nonlinear smooth flux function we prove the existence of such travelling waves, which are monotone and satisfy the standard entropy condition. Moreover, the dynamic nonlinear stability of the travelling waves under small perturbations is proven, similarly to the case of the standard diffusive regularisation, by constructing a Lyapunov functional
Nonlinear Aggregation-Diffusion Equations: Radial Symmetry and Long Time Asymptotics
We analyze under which conditions equilibration between two competing
effects, repulsion modeled by nonlinear diffusion and attraction modeled by
nonlocal interaction, occurs. This balance leads to continuous compactly
supported radially decreasing equilibrium configurations for all masses. All
stationary states with suitable regularity are shown to be radially symmetric
by means of continuous Steiner symmetrization techniques. Calculus of
variations tools allow us to show the existence of global minimizers among
these equilibria. Finally, in the particular case of Newtonian interaction in
two dimensions they lead to uniqueness of equilibria for any given mass up to
translation and to the convergence of solutions of the associated nonlinear
aggregation-diffusion equations towards this unique equilibrium profile up to
translations as
Global well-posedness for passively transported nonlinear moisture dynamics with phase changes
We study a moisture model for warm clouds that has been used by Klein and
Majda as a basis for multiscale asymptotic expansions for deep convective
phenomena. These moisture balance equations correspond to a bulk microphysics
closure in the spirit of Kessler and of Grabowski and Smolarkiewicz, in which
water is present in the gaseous state as water vapor and in the liquid phase as
cloud water and rain water. It thereby contains closures for the phase changes
condensation and evaporation, as well as the processes of autoconversion of
cloud water into rainwater and the collection of cloud water by the falling
rain droplets. Phase changes are associated with enormous amounts of latent
heat and therefore provide a strong coupling to the thermodynamic equation.
In this work we assume the velocity field to be given and prove rigorously
the global existence and uniqueness of uniformly bounded solutions of the
moisture model with viscosity, diffusion and heat conduction. To guarantee
local well-posedness we first need to establish local existence results for
linear parabolic equations, subject to the Robin boundary conditions on the
cylindric type of domains under consideration. We then derive a priori
estimates, for proving the maximum principle, using the Stampacchia method, as
well as the iterative method by Alikakos to obtain uniform boundedness. The
evaporation term is of power law type, with an exponent in general less or
equal to one and therefore making the proof of uniqueness more challenging.
However, these difficulties can be circumvented by introducing new unknowns,
which satisfy the required cancellation and monotonicity properties in the
source terms
A conservative reconstruction scheme for the interpolation of extensive quantities in the Lagrangian particle dispersion model FLEXPART
Lagrangian particle dispersion models require interpolation of all
meteorological input variables to the position in space and time of
computational particles. The widely used model FLEXPART uses linear
interpolation for this purpose, implying that the discrete input fields
contain point values. As this is not the case for precipitation (and other
fluxes) which represent cell averages or integrals, a preprocessing scheme is
applied which ensures the conservation of the integral quantity with the
linear interpolation in FLEXPART, at least for the temporal dimension.
However, this mass conservation is not ensured per grid cell, and the scheme
thus has undesirable properties such as temporal smoothing of the
precipitation rates. Therefore, a new reconstruction algorithm was developed,
in two variants. It introduces additional supporting grid points in each time
interval and is to be used with a piecewise linear interpolation to
reconstruct the precipitation time series in FLEXPART. It fulfils the desired
requirements by preserving the integral precipitation in each time interval,
guaranteeing continuity at interval boundaries, and maintaining
non-negativity. The function values of the reconstruction algorithm at the
sub-grid and boundary grid points constitute the degrees of freedom, which
can be prescribed in various ways. With the requirements mentioned it was
possible to derive a suitable piecewise linear reconstruction. To improve the
monotonicity behaviour, two versions of a filter were also developed that
form a part of the final algorithm. Currently, the algorithm is meant
primarily for the temporal dimension. It was shown to significantly improve
the reconstruction of hourly precipitation time series from 3-hourly input
data. Preliminary considerations for the extension to additional dimensions
are also included as well as suggestions for a range of possible applications
beyond the case of precipitation in a Lagrangian particle model
Recommended from our members
Global well-posedness for the primitive equations coupled to nonlinear moisture dynamics with phase changes
In this work we study the global solvability of the primitive equations for
the atmosphere coupled to moisture dynamics with phase changes for warm clouds, where water is present in the form of water vapor and in the liquid state as cloud water and rain water. This moisture model contains closures for the phase changes condensation and evaporation, as well as the processes of autoconversion of cloud water into rainwater and the collection of cloud water by the falling rain droplets. It has been used by Klein & Majda and corresponds to a basic form of the bulk microphysics closure in the spirit of Kessler, and Grabowski & Smolarkiewicz. The moisture balances are strongly coupled to the thermodynamic equation via the latent heat associated to the phase changes. In our previous paper we assumed the velocity field to be given and proved rigorously the global existence and uniqueness of uniformly bounded solutions of the moisture balances coupled to the thermodynamic equation. In this paper we present the solvability of a full moist atmospheric flow model, where the moisture model is coupled to the primitive equations of atmospherical dynamics governing the velocity field. For the derivation of a priori estimates for the velocity field we thereby use the ideas of Cao & Titi, who succeeded in proving the global solvability of the primitive equations.Austrian Science Fund via the Hertha-Firnberg project T-764.
Deutsche Forschungsgemeinschaft through Grant CRC 1114 ``Scaling Cascades in Complex Systems'', projects A02 and C06.
National Natural Science Foundation of China grants 11971009, 11871005, and 11771156.
Natural Science Foundation of Guangdong Province grant 2019A1515011621
South China Normal University start-up grant 550-8S0315.
Hong Kong RGC grant CUHK 14302917.
Einstein Stiftung/Foundation - Berlin, through the Einstein Visiting Fellow Program
John Simon Guggenheim Memorial Foundation
The Dynamic State Index with Moisture and Phase Changes
The dynamic state index (DSI) is a scalar field that combines variational information on the total energy and enstrophy of a flow field with the second law of thermodynamics. Its magnitude is a combined local measure for non-stationarity, diabaticity, and dissipation in the flow, and it has been shown to provide good qualitative indications for the onset and presence of precipitation and the organization of storms.
The index has been derived thus far for ideal fluid models only, however, so that one may expect improved and quantitative insights from a revised definition of the quantity that includes more complex aerothermodynamics. The present paper suggests definitions of the DSI for flows of moist air with phase changes and precipitation
Kinetic modelling of colonies of myxobacteria
A new kinetic model for the dynamics of myxobacteria colonies on flat surfaces is derived formally, and first analytical and numerical results are presented. The model is based on the assumption of hard binary collisions of two different types: alignment and reversal. We investigate two different versions: a) realistic rod-shaped bacteria and b) artificial circular shaped bacteria called Maxwellian myxos in reference to the similar simplification of the gas dynamics Boltzmann equation for Maxwellian molecules. The sum of the corresponding collision operators produces relaxation towards nematically aligned equilibria, i.e. two groups of bacteria polarized in opposite directions.
For the spatially homogeneous model a global existence and uniqueness result is proved as well as exponential decay to equilibrium for special initial conditions and for Maxwellian myxos. Only partial results are available for the rod-shaped case. These results are illustrated by numerical simulations, and a formal discussion of the macroscopic limit is presented
Global well-posedness for the primitive equations coupled to nonlinear moisture dynamics with phase changes
Funder: John Simon Guggenheim Memorial Foundation; doi: https://doi.org/10.13039/100005851Abstract: In this work we study the global solvability of the primitive equations for the atmosphere coupled to moisture dynamics with phase changes for warm clouds, where water is present in the form of water vapor and in the liquid state as cloud water and rain water. This moisture model contains closures for the phase changes condensation and evaporation, as well as the processes of auto conversion of cloud water into rainwater and the collection of cloud water by the falling rain droplets. It has been used by Klein and Majda in [17] and corresponds to a basic form of the bulk microphysics closure in the spirit of Kessler [16] and Grabowski and Smolarkiewicz [12]. The moisture balances are strongly coupled to the thermodynamic equation via the latent heat associated to the phase changes. In [14] we assumed the velocity field to be given and proved rigorously the global existence and uniqueness of uniformly bounded solutions of the moisture balances coupled to the thermodynamic equation. In this paper we present the solvability of a full moist atmospheric flow model, where the moisture model is coupled to the primitive equations of atmospherical dynamics governing the velocity field. For the derivation of a priori estimates for the velocity field we thereby use the ideas of Cao and Titi [6], who succeeded in proving the global solvability of the primitive equations