5,600 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
Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
We give an elementary proof of the global well-posedness for the critical 2D
dissipative quasi-geostrophic equation. The argument is based on a non-local
maximum principle involving appropriate moduli of continuity.Comment: 7 page
Particle trajectories in linearized irrotational shallow water flows
We investigate the particle trajectories in an irrotational shallow water
flow over a flat bed as periodic waves propagate on the water's free surface.
Within the linear water wave theory, we show that there are no closed orbits
for the water particles beneath the irrotational shallow water waves. Depending
on the strength of underlying uniform current, we obtain that some particle
trajectories are undulating path to the right or to the left, some are looping
curves with a drift to the right and others are parabolic curves or curves
which have only one loop
On periodic water waves with Coriolis effects and isobaric streamlines
In this paper we prove that solutions of the f-plane approximation for
equatorial geophysical deep water waves, which have the property that the
pressure is constant along the streamlines and do not possess stagnation
points,are Gerstner-type waves. Furthermore, for waves traveling over a flat
bed, we prove that there are only laminar flow solutions with these properties.Comment: To appear in Journal of Nonlinear Mathematical Physics; 15 page
On the critical dissipative quasi-geostrophic equation
The 2D quasi-geostrophic (QG) equation is a two dimensional model of the 3D
incompressible Euler equations. When dissipation is included in the model then
solutions always exist if the dissipation's wave number dependence is
super-linear. Below this critical power the dissipation appears to be
insufficient. For instance, it is not known if the critical dissipative QG
equation has global smooth solutions for arbitrary large initial data. In this
paper we prove existence and uniqueness of global classical solutions of the
critical dissipative QG equation for initial data that have small
norm. The importance of an smallness condition is due to the fact
that is a conserved norm for the non-dissipative QG equation and
is non-increasing on all solutions of the dissipative QG., irrespective of
size.Comment: 12 page
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
The Degasperis-Procesi equation as a non-metric Euler equation
In this paper we present a geometric interpretation of the periodic
Degasperis-Procesi equation as the geodesic flow of a right invariant symmetric
linear connection on the diffeomorphism group of the circle. We also show that
for any evolution in the family of -equations there is neither gain nor loss
of the spatial regularity of solutions. This in turn allows us to view the
Degasperis-Procesi and the Camassa-Holm equation as an ODE on the Fr\'echet
space of all smooth functions on the circle.Comment: 17 page
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
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