Nonlocal Velocity Research Articles

Subtidal Response of Scotian Shelf Circulation to Local and Remote Forcing. Part II: Barotropic Model

A linear, barotropic numerical model that features realistic bathymetry of the Scotian Shelf provides solutions forced by steady and periodic wind stress that are generally incoherent on spatial scales of the shelf width. Closed circulation cells occur in association with, and on the scales of, major bathymetric features. Bathymetrically steered flow is prevalent, and more evident at lower frequencies. Model transport forced by a coastally trapped wave propagating across the backward boundary is not sensitive to the model structure of the incident wave; bathymetry rapidly modifies the wave form as it propagates in the free direction. Higher modes and reflected waves are probably dissipated over short distances by mattering and strong bottom stress. Patterns seen in the model solutions are consistent with results from more fundamental models, and are readily explained by relatively simple vorticity balances. Model results generally reproduce the diverse local and nonlocal velocity responses observed on the Scotian Shelf during winter. The model is more effective in accounting for the remote contribution to circulation. Wind-forced results generally reflect the observed directional variability, and are similar in magnitude to observations inside the 100-m isobath but underestimate the actual current response at deeper stations. It is speculated that spatial variability in wind stress, an effect not addressed in this work, may be responsible for some of this difference. The significant of baroclinic effects in deeper water, which cannot be discounted, is also beyond the scope of this model. Nevertheless, the fairly good agreement between the model and observed current responses to local and nonlocal forcing confirms the importance of bathymetric modifications to Scotian Shelf circulation at subtidal frequencies.

On the torsional waves in an elastic bar with long-range cohesive forces

With reference to some of the results obtained in [1], the equation of motion and the equations of nonlocal stress components in the form of Kroener-Eringen are transformed into the Fourier space. A lengthy calculation using a separable form of the nonlocal elastic moduli leads to the governing equation of the problem with two types of solutions. In each case the nonlocal phase velocity and, respectively, the nonlocal group velocity of the very short waves turn out to be by about 36% less than the velocities of their classical counterparts.

A dilemma in the physics of gravitational fields

It is shown that improper use of local quantities for nonlocal situations in fields leads to traditional errors. Nonlocal theoretical quantities referred to standards in a fixed field are defined in order to obtain reliable results. Nonlocal properties of gravitational fields and matter located in it are deduced with the help of physical principles and an electromagnetic model for matter. In spite of the fact that the local velocity of light should be constant, the field is a space of variable nonlocal velocity of light, which accounts for its properties. Matter and light virtually propagate themselves without exchanging energy with the external field, in disagreement with traditional assumptions. Matter becomes contracted by the field. The results are self-consistent and consistent with the observed facts. Bodies withr⩽2GM would be different from black holes and they may account for the peak of highest energy of cosmic radiation and other astronomical facts.

Singularity formation for a fluid mechanics model with nonlocal velocity

We study a 1D fluid mechanics model with nonlocal velocity. The equation can be viewed as a fractional porous medium flow, a 1D model of the quasi-geostrophic equation, and also a special case of Euler-Alignment system. For strictly positive smooth initial data, global regularity has been proved by Do, Kiselev, Ryzhik and Tan. We construct a family of non-negative smooth initial data so that solution loses $C^1$ regularity. Our result indicates that strict positivity is a critical condition to ensure global regularity of the system. We also extend our construction to the corresponding models in multi-dimensions.

Infinite energy solutions for a 1D transport equation with nonlocal velocity

We study a one dimensional dissipative transport equation with nonlocal velocity and critical dissipation. We consider the Cauchy problem for initial values with infinite energy. The control we shall use involves some weighted Lebesgue or Sobolev spaces. More precisely, we consider the familly of weights given by w β (x) = (1+|x| 2) −β/2 where β is a real parameter in (0, 1) and we treat the Cauchy problem for the cases θ0 ∈ H 1/2 (w β) and θ0 ∈ H 1 (w β) for which we prove global existence results (under smallness assumptions on the L ∞ norm of θ0). The key step in the proof of our theorems is based on the use of two new commutator estimates involving fractional differential operators and the family of Muckenhoupt weights.

Decay estimate to a compressible Euler system with non-local velocity alignment

Decay estimate to a compressible Euler system with non-local velocity alignment