399 research outputs found
Conservation Properties of the Hamiltonian Particle-Mesh method for the Quasi-Geostrophic Equations on a sphere
The Hamiltonian particle-mesh (HPM) method is used to
solve the Quasi-Geostrophic model generalized to a sphere, using
the Spherepack modeling package to solve the Helmholtz
equation on a colatitude-longitude grid with spherical harmonics.
The predicted energy conservation of a Poisson system is
shown to be approximately retained and statistical mean-eld
theory is veried
Equilibrium statistical mechanics and energy partition for the shallow water model
The aim of this paper is to use large deviation theory in order to compute
the entropy of macrostates for the microcanonical measure of the shallow water
system. The main prediction of this full statistical mechanics computation is
the energy partition between a large scale vortical flow and small scale
fluctuations related to inertia-gravity waves. We introduce for that purpose a
discretized model of the continuous shallow water system, and compute the
corresponding statistical equilibria. We argue that microcanonical equilibrium
states of the discretized model in the continuous limit are equilibrium states
of the actual shallow water system. We show that the presence of small scale
fluctuations selects a subclass of equilibria among the states that were
previously computed by phenomenological approaches that were neglecting such
fluctuations. In the limit of weak height fluctuations, the equilibrium state
can be interpreted as two subsystems in thermal contact: one subsystem
corresponds to the large scale vortical flow, the other subsystem corresponds
to small scale height and velocity fluctuations. It is shown that either a
non-zero circulation or rotation and bottom topography are required to sustain
a non-zero large scale flow at equilibrium. Explicit computation of the
equilibria and their energy partition is presented in the quasi-geostrophic
limit for the energy-enstrophy ensemble. The possible role of small scale
dissipation and shocks is discussed. A geophysical application to the Zapiola
anticyclone is presented.Comment: Journal of Statistical Physics, Springer Verlag, 201
Least-biased correction of extended dynamical systems using observational data
We consider dynamical systems evolving near an equilibrium statistical state
where the interest is in modelling long term behavior that is consistent with
thermodynamic constraints. We adjust the distribution using an
entropy-optimizing formulation that can be computed on-the- fly, making
possible partial corrections using incomplete information, for example measured
data or data computed from a different model (or the same model at a different
scale). We employ a thermostatting technique to sample the target distribution
with the aim of capturing relavant statistical features while introducing mild
dynamical perturbation (thermostats). The method is tested for a point vortex
fluid model on the sphere, and we demonstrate both convergence of equilibrium
quantities and the ability of the formulation to balance stationary and
transient- regime errors.Comment: 27 page
Oceanic rings and jets as statistical equilibrium states
Equilibrium statistical mechanics of two-dimensional flows provides an
explanation and a prediction for the self-organization of large scale coherent
structures. This theory is applied in this paper to the description of oceanic
rings and jets, in the framework of a 1.5 layer quasi-geostrophic model. The
theory predicts the spontaneous formation of regions where the potential
vorticity is homogenized, with strong and localized jets at their interface.
Mesoscale rings are shown to be close to a statistical equilibrium: the theory
accounts for their shape, their drift, and their ubiquity in the ocean,
independently of the underlying generation mechanism. At basin scale, inertial
states presenting mid basin eastward jets (and then different from the
classical Fofonoff solution) are described as marginally unstable states. These
states are shown to be marginally unstable for the equilibrium statistical
theory. In that case, considering a purely inertial limit is a first step
toward more comprehensive out of equilibrium studies that would take into
account other essential aspects, such as wind forcing.Comment: 15 pages, submitted to Journal of Physical Oceanograph
Compatible finite element methods for geophysical fluid dynamics
This article surveys research on the application of compatible finite element
methods to large scale atmosphere and ocean simulation. Compatible finite
element methods extend Arakawa's C-grid finite difference scheme to the finite
element world. They are constructed from a discrete de Rham complex, which is a
sequence of finite element spaces which are linked by the operators of
differential calculus. The use of discrete de Rham complexes to solve partial
differential equations is well established, but in this article we focus on the
specifics of dynamical cores for simulating weather, oceans and climate. The
most important consequence of the discrete de Rham complex is the
Hodge-Helmholtz decomposition, which has been used to exclude the possibility
of several types of spurious oscillations from linear equations of geophysical
flow. This means that compatible finite element spaces provide a useful
framework for building dynamical cores. In this article we introduce the main
concepts of compatible finite element spaces, and discuss their wave
propagation properties. We survey some methods for discretising the transport
terms that arise in dynamical core equation systems, and provide some example
discretisations, briefly discussing their iterative solution. Then we focus on
the recent use of compatible finite element spaces in designing structure
preserving methods, surveying variational discretisations, Poisson bracket
discretisations, and consistent vorticity transport.Comment: correction of some typo
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