670 research outputs found
Turbulent diffusion in rapidly rotating flows with and without stable stratification
In this work, three different approaches are used for evaluating some Lagrangian properties of homogeneous turbulence containing anisotropy due to the application of a stable stratification and a solid-body rotation. The two external frequencies are the magnitude of the system vorticity, chosen vertical here, and the Brunt–Väisälä frequency, which gives the strength of the vertical stratification. Analytical results are derived using linear theory for the Eulerian velocity correlations (single-point, two-time) in the vertical and the horizontal directions, and Lagrangian ones are assumed to be equivalent, in agreement with an additional Corrsin assumption used by Kaneda (2000). They are compared with results from the kinematic simulation model (KS) by Nicolleau & Vassilicos (2000), which also incorporates the wave–vortex dynamics inherited from linear theory, and directly yields Lagrangian correlations as well as Eulerian ones.
Finally, results from direct numerical simulations (DNS) are obtained and compared We address the question of the validity of Corrsin's simplified hypothesis, which states the equivalence between Eulerian and Lagrangian correlations. Vertical correlations are found to follow this postulate, but not the horizontal ones. Consequences for the vertical and horizontal one-particle dispersion are examined. In the analytical model, the squared excursion lengths are calculated by time integrating the Lagrangian (equal to the Eulerian) two-time correlations, according to Taylor's procedure. These quantities are directly computed from fluctuating trajectories by both KS and DNS. In the case of pure rotation, the analytical procedure allows us to relate Brownian asymptotic laws of dispersion in both the horizontal and vertical directions to the angular phase-mixing properties of the inertial waves. If stratification is present, the inertia–gravity wave dynamics, which affects the vertical motion, yields a suppressed vertical diffusivity, but not a suppressed horizontal diffusivity, since part of the horizontal velocity field escapes wavy motion
Solutions and Singularities of the Semigeostrophic Equations via the Geometry of Lagrangian Submanifolds
Using Monge-Amp\`ere geometry, we study the singular structure of a class of
nonlinear Monge-Amp\`ere equations in three dimensions, arising in geophysical
fluid dynamics. We extend seminal earlier work on Monge-Amp\`ere geometry by
examining the role of an induced metric on Lagrangian submanifolds of the
cotangent bundle. In particular, we show that the signature of the metric
serves as a classification of the Monge-Amp\`ere equation, while singularities
and elliptic-hyperbolic transitions are revealed by the degeneracies of the
metric. The theory is illustrated by application to an example solution of the
semigeostrophic equations.Comment: 22 pages, 5 figure
Uniqueness of the solution to the Vlasov-Poisson system with bounded density
In this note, we show uniqueness of weak solutions to the Vlasov-Poisson
system on the only condition that the macroscopic density defined by
\rho(t,x) = \int_{\Rd} f(t,x,\xi)d\xi is bounded in \Linf. Our proof is
based on optimal transportation
Asymptotic limit analysis for numerical models of atmospheric frontogenesis
Accurate prediction of the future state of the atmosphere is important
throughout society, ranging from the weather forecast in a few days time
to modelling the effects of a changing climate over decades and generations.
The equations which govern how the atmosphere evolves have long been
known; these are the Navier-Stokes equations, the laws of thermodynamics
and the equation of state. Unfortunately the nonlinearity of the equations
prohibits analytic solutions, so simplified models of particular
flow phenomena have historically been, and continue to be, used alongside numerical
models of the full equations.
In this thesis, the two-dimensional Eady model of shear-driven frontogenesis
(the creation of atmospheric fronts) was used to investigate how errors
made in a localised region can affect the global solution. Atmospheric fronts
are the boundary of two different air masses, typically characterised by a
sharp change in air temperature and wind direction. This occurs across
a small length of O(10 km), whereas the extent of the front itself can be
O(1000 km). Fronts are a prominent feature of mid-latitude weather systems
and, despite their narrow width, are part of the large-scale, global
solution. Any errors made locally in the treatment of fronts will therefore
affect the global solution.
This thesis uses the convergence of the Euler equations to the semigeostrophic
equations, a simplified model which is representative of the
large-scale flow, including fronts. The Euler equations were solved numerically
using current operational techniques. It was shown that highly predictable
solutions could be obtained, and the theoretical convergence rate
maintained, even with the presence of near-discontinuous solutions given by
intense fronts.
Numerical solutions with successively increased resolution showed that
the potential vorticity, which is a fundamental quantity in determining the
large-scale, balanced flow, approached the semigeostrophic limit solution.
Regions of negative potential vorticity, indicative of local areas of instability,
were reduced at high resolution. In all cases, the width of the front reduced
to the grid-scale.
While qualitative features of the limit solution were reproduced, a stark
contrast in amplitude was found. The results of this thesis were approximately
half in amplitude of the limit solution. Some attempts were made at
increasing the intensity of the front through spatial- and temporal-averaging.
A scheme was proposed that conserves the potential vorticity within the
Eady model.Open Acces
The Lagrangian description of aperiodic flows: a case study of the Kuroshio Current
This article reviews several recently developed Lagrangian tools and shows
how their combined use succeeds in obtaining a detailed description of purely
advective transport events in general aperiodic flows. In particular, because
of the climate impact of ocean transport processes, we illustrate a 2D
application on altimeter data sets over the area of the Kuroshio Current,
although the proposed techniques are general and applicable to arbitrary time
dependent aperiodic flows. The first challenge for describing transport in
aperiodical time dependent flows is obtaining a representation of the phase
portrait where the most relevant dynamical features may be identified. This
representation is accomplished by using global Lagrangian descriptors that when
applied for instance to the altimeter data sets retrieve over the ocean surface
a phase portrait where the geometry of interconnected dynamical systems is
visible. The phase portrait picture is essential because it evinces which
transport routes are acting on the whole flow. Once these routes are roughly
recognised it is possible to complete a detailed description by the direct
computation of the finite time stable and unstable manifolds of special
hyperbolic trajectories that act as organising centres of the flow.Comment: 40 pages, 24 figure
The Euler-Poincaré Equations in Geophysical Fluid Dynamics
Recent theoretical work has developed the Hamilton's-principle analog of Lie-Poisson Hamiltonian systems defined on semidirect products. The main theoretical results are twofold:
1. Euler-Poincaré equations (the Lagrangian analog of Lie-Poisson Hamiltonian equations) are derived for a parameter dependent Lagrangian from a general variational principle of Lagrange d'Alembert type in which variations are constrained;
2. an abstract Kelvin-Noether theorem is derived for such systems.
By imposing suitable constraints on the variations and by using invariance properties of the Lagrangian, as one does for the Euler equations for the rigid body and ideal fluids, we cast several standard Eulerian models of geophysical fluid dynamics (GFD) at various levels of approximation into Euler-Poincaré form and discuss their corresponding Kelvin-Noether theorems and potential
vorticity conservation laws. The various levels of GFD approximation are related by substituting a sequence of velocity decompositions and asymptotic expansions into Hamilton's principle for the Euler equations of a rotating
stratified ideal incompressible fluid. We emphasize that the shared properties of this sequence of approximate ideal GFD models follow directly from their Euler-Poincaré formulations. New modifications of the Euler-Boussinesq
equations and primitive equations are also proposed in which nonlinear dispersion adaptively filters high wavenumbers and thereby enhances stability and regularity without compromising either low wavenumber behavior or geophysical balances
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