24 research outputs found
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
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
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
A Search for Chaotic Behavior in Northern Hemisphere Stratospheric Variability
Northern Hemisphere stratospheric variability is investigated with respect to chaotic behavior using time series from three different variables extracted from four different reanalysis products and two numerical model runs with different forcing. The time series show red spectra at all frequencies and the probability distribution functions show persistent deviations from a Gaussian distribution. An exception is given by the numerical model forced with perpetual winter conditions—a case that shows more variability and follows a Gaussian distribution, suggesting that the deviation from Gaussianity found in the observations is due to the transition between summer and winter variability. To search for the presence of a chaotic attractor the correlation dimension and entropy, the Lyapunov spectrum, and the associated Kaplan–Yorke dimension are estimated. A finite value of the dimensions can be computed for each variable and data product, with the correlation dimension ranging between 3.0 and 4.0 and the Kaplan–Yorke dimension between 3.3 and 5.5. The correlation entropy varies between 0.6 and 1.1. The model runs show similar values for the correlation and Lyapunov dimensions for both the seasonally forced run and the perpetual-winter run, suggesting that the structure of a possible chaotic attractor is not determined by the seasonality in the forcing, but must be given by other mechanisms
Collapse of generalized Euler and surface quasi-geostrophic point-vortices
Point vortex models are presented for the generalized Euler equations, which
are characterized by a fractional Laplacian relation between the active scalar
and the streamfunction. Special focus is given to the case of the surface
quasi-geostrophic (SQG) equations, for which the existence of finite-time
singularities is still a matter of debate. Point vortex trajectories are
expressed using Nambu dynamics. The formulation is based on a noncanonical
bracket and allows for a geometrical interpretation of trajectories as
intersections of level sets of the Hamiltonian and Casimir. Within this
setting, we focus on the collapse of solutions for the three point vortex
model. In particular, we show that for SQG the collapse can be either
self-similar or non-self-similar. Self-similarity occurs only when the
Hamiltonian is zero, while non-self-similarity appears for non-zero values of
the same. For both cases, collapse is allowed for any choice of circulations
within a permitted interval. These results differ strikingly from the classical
point vortex model, where collapse is self-similar for any value of the
Hamiltonian, but the vortex circulations must satisfy a strict relationship.
Results may also shed a light on the formation of singularities in the SQG
partial differential equations, where the singularity is thought to be reached
only in a self-similar way
Water Mass Transformations in the Southern Ocean Diagnosed from Observations: Contrasting Effects of Air-Sea Fluxes and Diapycnal Mixing
Abstract
Transformation and formation rates of water masses in the Southern Ocean are estimated in a neutral-surface framework using air–sea fluxes of heat and freshwater together with in situ estimates of diapycnal mixing. The air–sea fluxes are taken from two different climatologies and a reanalysis dataset, while the diapycnal mixing is estimated from a mixing parameterization applied to five years of Argo float data. Air–sea fluxes lead to a large transformation directed toward lighter waters, typically from −45 to −63 Sv (1 Sv ≡ 106 m3 s−1) centered at γ = 27.2, while interior diapycnal mixing leads to two weaker peaks in transformation, directed toward denser waters, 8 Sv centered at γ = 27.8, and directed toward lighter waters, −16 Sv centered at γ = 28.3. Hence, air–sea fluxes and interior diapycnal mixing are important in transforming different water masses within the Southern Ocean. The transformation of dense to lighter waters by diapycnal mixing within the Southern Ocean is slightly larger, though comparable in magnitude, to the transformation of lighter to dense waters by air–sea fluxes in the North Atlantic. However, there are significant uncertainties in the authors' estimates with errors of at least ±5 W m−2 in air–sea fluxes, a factor 4 uncertainty in diapycnal mixing and limited coverage of air–sea fluxes in the high latitudes and Argo data in the Pacific. These water mass transformations partly relate to the circulation in density space: air–sea fluxes provide a general lightening along the core of the Antarctic Circumpolar Current and diapycnal diffusivity is enhanced at middepths along the current.</jats:p