1,475 research outputs found
Hydrodynamic Nambu Brackets derived by Geometric Constraints
A geometric approach to derive the Nambu brackets for ideal two-dimensional
(2D) hydrodynamics is suggested. The derivation is based on two-forms with
vanishing integrals in a periodic domain, and with resulting dynamics
constrained by an orthogonality condition. As a result, 2D hydrodynamics with
vorticity as dynamic variable emerges as a generic model, with conservation
laws which can be interpreted as enstrophy and energy functionals. Generalized
forms like surface quasi-geostrophy and fractional Poisson equations for the
stream-function are also included as results from the derivation. The formalism
is extended to a hydrodynamic system coupled to a second degree of freedom,
with the Rayleigh-B\'{e}nard convection as an example. This system is
reformulated in terms of constitutive conservation laws with two additive
brackets which represent individual processes: a first representing inviscid 2D
hydrodynamics, and a second representing the coupling between hydrodynamics and
thermodynamics. The results can be used for the formulation of conservative
numerical algorithms that can be employed, for example, for the study of fronts
and singularities.Comment: 12 page
Construction of Hamiltonian and Nambu forms for the shallow water equations
A systematic method to derive the Hamiltonian and Nambu form for the shallow
water equations, using the conservation for energy and potential enstrophy, is
presented. Different mechanisms, such as vortical flows and emission of gravity
waves, emerge from different conservation laws (CLs) for total energy and
potential enstrophy. The equations are constructed using exterior differential
forms and self-adjoint operators and result in the sum of two Nambu brackets,
one for the vortical flow and one for the wave-mean flow interaction, and a
Poisson bracket representing the interaction between divergence and geostrophic
imbalance. The advantage of this approach is that the Hamiltonian and Nambu
forms can be here written in a coordinate independent form
Hyperbolic Covariant Coherent Structures in two dimensional flows
A new method to describe hyperbolic patterns in two dimensional flows is
proposed. The method is based on the Covariant Lyapunov Vectors (CLVs), which
have the properties to be covariant with the dynamics, and thus being mapped by
the tangent linear operator into another CLVs basis, they are norm independent,
invariant under time reversal and can be not orthonormal. CLVs can thus give a
more detailed information on the expansion and contraction directions of the
flow than the Lyapunov Vector bases, that are instead always orthogonal. We
suggest a definition of Hyperbolic Covariant Coherent Structures (HCCSs), that
can be defined on the scalar field representing the angle between the CLVs.
HCCSs can be defined for every time instant and could be useful to understand
the long term behaviour of particle tracers.
We consider three examples: a simple autonomous Hamiltonian system, as well
as the non-autonomous "double gyre" and Bickley jet, to see how well the angle
is able to describe particular patterns and barriers. We compare the results
from the HCCSs with other coherent patterns defined on finite time by the
Finite Time Lyapunov Exponents (FTLEs), to see how the behaviour of these
structures change asymptotically
Nonlinear stratospheric variability: multifractal detrended fluctuation analysis and singularity spectra
Characterising the stratosphere as a turbulent system, temporal fluctuations
often show different correlations for different time scales as well as
intermittent behaviour that cannot be captured by a single scaling exponent. In
this study, the different scaling laws in the long term stratospheric
variability are studied using Multifractal de-trended Fluctuation Analysis. The
analysis is performed comparing four re-analysis products and different
realisations of an idealised numerical model, isolating the role of topographic
forcing and seasonal variability, as well as the absence of climate
teleconnections and small-scale forcing. The Northern Hemisphere (NH) shows a
transition of scaling exponents for time scales shorter than about one year,
for which the variability is multifractal and scales in time with a power law
corresponding to a red spectrum, to longer time scales, for which the
variability is monofractal and scales in time with a power law corresponding to
white noise. Southern Hemisphere (SH) variability also shows a transition at
annual scales. The SH also shows a narrower dynamical range in multifractality
than the NH, as seen in the generalised Hurst exponent and in the singularity
spectra. The numerical integrations show that the models are able to reproduce
the low-frequency variability but are not able to fully capture the shorter
term variability of the stratosphere
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