323,366 research outputs found
Frictionally decaying frontal warm-core eddies
No full textPurpose. The dynamics of nonstationary, nonlinear, axisymmetric, warm-core geophysical surface frontal vortices affected by Rayleigh friction is investigated semi-analytically using the nonlinear, nonstationary reduced-gravity shallow-water equations. The scope is to enlarge the number of known (semi)analytical solutions of nonstationary, nonlinear problems referring to geophysical problems and even to pave the way to their extension to broader geometries and/or velocity fields. Methods and Results. The used method to obtain the solutions is based on the decomposition of the original equations in a part expressing their prescribed spatial structure, so that they can be trans-formed into ordinary differential equations depending on time only. Based on that analytical proce-dure, the solutions are then found numerically. In this frame, it is found that vortices characterized by linear distributions of their radial velocity and arbitrary structures of their section and azimuthal velocity can be described exactly by a set of nonstationary, nonlinear coupled ordinary differential equa-tions. The first-order problem (i. e., that describing vortices characterized by a linear azimuthal velocity field and a quadratic section) consists of a system of 4 differential equations, and each further order introduces in the system three additional ordinary differential equations and two algebraic equa-tions. In order to illustrate the behavior of the nonstationary decaying vortices and to put them in the context of observed dynamics in the World Ocean, the system's solution for the first-order and for the second-order problem is then obtained numerically using a Runge-Kutta method. The solutions demonstrate that inertial oscillations and an exponential attenuation dominate the vortex dynamics: expansions and shallowings, contractions and deepenings alternate during an exact inertial period while the vortex decays. The dependence of the vortex dissipation rate on its initial radius is found to be non-monotonic: it is higher for small and large radii. The possibility of solving (semi)analytically complex systems of differential equations representing observed physical phenomena is rare and very valuable. Conclusions. Our analysis adds realism to previous theoretical investigations on mesoscale vortices, represents an ideal tool for testing the accuracy of numerical models in simulating nonlinear, nonsta-tionary frictional frontal phenomena in a rotating ocean, and paves the way to further extensions of (semi-) analytical solutions of hydrodynamical geophysical problems to more arbitrary forms and more complex density stratifications
Analytical Solutions for Circular Stratified Eddies of the Reduced–Gravity Shallow-Water Equations
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REPLICA-SYMMETRY BREAKING IN NEURAL NETWORKS
No full textReplica-symmetry breaking is studied in fully connected neural networks with modified pseudo-inverse interactions. The interaction matrix has an intermediate form between the Hebb learning rule and the pseudo-inverse one. At low temperature there is a region of parameters where the replica-symmetric solution is stable while its entropy is negative. It indicates the existence of the alternative solution in which the replica symmetry is broken. A one-step replica-symmetry breaking solution is found and its properties are analyzed
Exact analytical solutions of the nonlinear shallow-water equations for a case of axisymmetric oscillations of a fluid in a rotating paraboloidal basin
No full textNonlinear radial oscillations of a fluid in a parabolic basin with regard for the external action
No full textWithin the framework of the theory of long waves, we find a class of exact analytic solutions of the problem of description of nonlinear axially symmetric oscillations of a fluid in a parabolic basin with regard for the action of stationary radial bulk forces. The radial projection of the velocity of these oscillations (seiches) is a linear function of the radial coordinate, whereas the azimuthal velocity and displacements of the free surface of the fluid are polynomials in the radial coordinate with time-dependent coefficients. The method of finding solutions is based on the exact replacement of the original problem by a system of ordinary differential and algebraic equations. The action of the bulk forces may result either in the increase in the frequency of oscillations of the fluid and in the decrease in this frequency and affect the motion of the water edge, the characteristics of waves, and the velocity field
Nonstationary westward translation of nonlinear frontal warm-core eddies
No full textFor the first time, an analytical theory and a very high-resolution, frontal numerical model, both based on the unsteady, nonlinear, reduced-gravity shallow water equations on a beta plane, have been used to investigate aspects of the migration of homogeneous surface, frontal warm-core eddies on a beta plane. Under the assumption that, initially, such vortices are surface circular anticyclones of paraboloidal shape and having both radial and azimuthal velocities that are linearly dependent on the radial coordinate (i.e., circular pulsons of the first order), approximate analytical expressions are found that describe the nonstationary trajectories of their centers of mass for an initial stage as well as for a mature stage of their westward migration. In particular, near-inertial oscillations are evident in the initial migration stage, whose amplitude linearly increases with time, as a result of the unbalanced vortex initial state on a beta plane. Such an initial amplification of the vortex oscillations is actually found in the first stage of the evolution of warm-core frontal eddies simulated numerically by means of a frontal numerical model initialized using the shape and velocity fields of circular pulsons of the first order. In the numerical simulations, this stage is followed by an adjusted, complex nonstationary state characterized by a noticeable asymmetry in the meridional component of the vortex's horizontal pressure gradient, which develops to compensate for the variations of the Coriolis parameter with latitude. Accordingly, the location of the simulated vortex's maximum depth is always found poleward of the location of the simulated vortex's center of mass. Moreover, during the adjusted stage, near-inertial oscillations emerge that largely deviate from the exactly inertial ones characterizing analytical circular pulsons: a superinertial and a subinertial oscillation in fact appear, and their frequency difference is found to be an increasing function of latitude. A comparison between vortex westward drifts simulated numerically at different latitudes for different vortex radii and pulsation strengths and the corresponding drifts obtained using existing formulas shows that, initially, the simulated vortex drifts correspond to the fastest predicted ones in many realistic cases. As time elapses, however, the development of a beta-adjusted vortex structure, together with the effects of numerical dissipation, tend to slow down the simulated vortex drift
MODIFIED PSEUDO-INVERSE NEURAL NETWORKS STORING CORRELATED PATTERNS
No full textNeural networks with symmetric couplings which have an intermediate form between the Hebb learning rule and the pseudo-inverse one, storing strongly correlated patterns, are studied. Signal-to-noise analysis is made and replica-symmetric thermodynamic Calculations are performed. Both approaches show that both in the Hopfield model limit and in the Pseudo-inverse model limit the maximal capacity of the order of (2p/In(1/p)-1 (where p << 1 is the average neural activity) can be achieved by appropriate adjustment of the threshold term of the Hamiltonian
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