4,146 research outputs found
Energy Spectrum of Quasi-Geostrophic Turbulence
We consider the energy spectrum of a quasi-geostrophic model of forced,
rotating turbulent flow. We provide a rigorous a priori bound E(k) <= Ck^{-2}
valid for wave numbers that are smaller than a wave number associated to the
forcing injection scale. This upper bound separates this spectrum from the
Kolmogorov-Kraichnan k^{-{5/3}} energy spectrum that is expected in a
two-dimensional Navier-Stokes inverse cascade. Our bound provides theoretical
support for the k^{-2} spectrum observed in recent experiments
Steady water waves with multiple critical layers: interior dynamics
We study small-amplitude steady water waves with multiple critical layers.
Those are rotational two-dimensional gravity-waves propagating over a perfect
fluid of finite depth. It is found that arbitrarily many critical layers with
cat's-eye vortices are possible, with different structure at different levels
within the fluid. The corresponding vorticity depends linearly on the stream
function.Comment: 14 pages, 3 figures. As accepted for publication in J. Math. Fluid
Mec
Position-dependent exact-exchange energy for slabs and semi-infinite jellium
The position-dependent exact-exchange energy per particle
(defined as the interaction between a given electron at and its
exact-exchange hole) at metal surfaces is investigated, by using either jellium
slabs or the semi-infinite (SI) jellium model. For jellium slabs, we prove
analytically and numerically that in the vacuum region far away from the
surface , {\it
independent} of the bulk electron density, which is exactly half the
corresponding exact-exchange potential [Phys.
Rev. Lett. {\bf 97}, 026802 (2006)] of density-functional theory, as occurs in
the case of finite systems. The fitting of
to a physically motivated image-like expression is feasible, but the resulting
location of the image plane shows strong finite-size oscillations every time a
slab discrete energy level becomes occupied. For a semi-infinite jellium, the
asymptotic behavior of is somehow different.
As in the case of jellium slabs has
an image-like behavior of the form , but now with a
density-dependent coefficient that in general differs from the slab universal
coefficient 1/2. Our numerical estimates for this coefficient agree with two
previous analytical estimates for the same. For an arbitrary finite thickness
of a jellium slab, we find that the asymptotic limits of
and only
coincide in the low-density limit (), where the
density-dependent coefficient of the semi-infinite jellium approaches the slab
{\it universal} coefficient 1/2.Comment: 26 pages, 7 figures, to appear in Phys. Rev.
Note on Global Regularity for 2D Oldroyd-B Fluids with Diffusive Stress
We prove global regularity of solutions of Oldroyd-B equations in 2 spatial
dimensions with spatial diffusion of the polymeric stresses
Semilocal density functional theory with correct surface asymptotics
Semilocal density functional theory is the most used computational method for
electronic structure calculations in theoretical solid-state physics and
quantum chemistry of large systems, providing good accuracy with a very
attractive computational cost. Nevertheless, because of the non-locality of the
exchange-correlation hole outside a metal surface, it was always considered
inappropriate to describe the correct surface asymptotics. Here, we derive,
within the semilocal density functional theory formalism, an exact condition
for the image-like surface asymptotics of both the exchange-correlation energy
per particle and potential. We show that this condition can be easily
incorporated into a practical computational tool, at the simple
meta-generalized-gradient approximation level of theory. Using this tool, we
also show that the Airy-gas model exhibits asymptotic properties that are
closely related to the ones at metal surfaces. This result highlights the
relevance of the linear effective potential model to the metal surface
asymptotics.Comment: 6 pages, 4 figure
On the particle paths and the stagnation points in small-amplitude deep-water waves
In order to obtain quite precise information about the shape of the particle
paths below small-amplitude gravity waves travelling on irrotational deep
water, analytic solutions of the nonlinear differential equation system
describing the particle motion are provided. All these solutions are not closed
curves. Some particle trajectories are peakon-like, others can be expressed
with the aid of the Jacobi elliptic functions or with the aid of the
hyperelliptic functions. Remarks on the stagnation points of the
small-amplitude irrotational deep-water waves are also made.Comment: to appear in J. Math. Fluid Mech. arXiv admin note: text overlap with
arXiv:1106.382
On a novel integrable generalization of the nonlinear Schr\"odinger equation
We consider an integrable generalization of the nonlinear Schr\"odinger (NLS)
equation that was recently derived by one of the authors using bi-Hamiltonian
methods. This equation is related to the NLS equation in the same way that the
Camassa Holm equation is related to the KdV equation. In this paper we: (a) Use
the bi-Hamiltonian structure to write down the first few conservation laws. (b)
Derive a Lax pair. (c) Use the Lax pair to solve the initial value problem. (d)
Analyze solitons.Comment: 20 pages, 1 figur
Equations of the Camassa-Holm Hierarchy
The squared eigenfunctions of the spectral problem associated with the
Camassa-Holm (CH) equation represent a complete basis of functions, which helps
to describe the inverse scattering transform for the CH hierarchy as a
generalized Fourier transform (GFT). All the fundamental properties of the CH
equation, such as the integrals of motion, the description of the equations of
the whole hierarchy, and their Hamiltonian structures, can be naturally
expressed using the completeness relation and the recursion operator, whose
eigenfunctions are the squared solutions. Using the GFT, we explicitly describe
some members of the CH hierarchy, including integrable deformations for the CH
equation. We also show that solutions of some - dimensional members of
the CH hierarchy can be constructed using results for the inverse scattering
transform for the CH equation. We give an example of the peakon solution of one
such equation.Comment: 10 page
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