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Recently, the Nash-style convex integration has been becoming the main scheme for the mathematical study of turbulence, and the main building block of it has been either Beltrami flow (finite mode) or Mikado flow (compactly supported in the physical side). On the other hand, in physics, it is observed that turbulence is composed of a hierarchy of scale-by-scale vortex stretching. Thus our mathematical motivation in this study is to find another type of building blocks accompanied by vortex stretching and scale locality (possibly finitely many Fourier modes). In this paper, we give a complete list of solutions to the 3D Euler equations with finitely many Fourier modes, which is an extension of the corresponding 2D result by Elgindi et al. (Comm Math Phys 355(1): 145–159, 2017). In particular, we show that there are no 3D Euler flows with finitely many Fourier modes, except for stationary 2D-like flows and Beltrami flows. We also discuss the case when viscosity and Coriolis effect are present.

In this paper, a new framework for iterative speckle noise reduction in polarimetric synthetic aperture radar (SAR) data is introduced. Speckle is inherent to all coherent imaging systems and affects SAR imagery in the form of strong intensity variations in pixels with similar backscattering coefficients. This makes the interpretation of SAR data in several applications a difficult task. The proposed framework includes a preprocessing step capable of dealing with noise correlation usually found in single-look data. The general filtering approach is based on the Beltrami flow for denoising manifolds or images painted on manifolds. The principal contribution of this work is to adapt this approach to deal with covariance or coherency matrices instead of optical imagery. The evaluation presented suggests that this approach allows for good spatial and radiometric preservation compared to other state-of-the-art methods. Experiments are performed on the basis of synthetic and real-world experimental data. The validation of the proposed framework is accomplished using two refined error performance measures and the well-known effective number of looks measured. The source code of a parallel implementation of the proposed framework is released under the MPL 2.0 ( https://www.mozilla.org/en-US/MPL/2.0/ ) alongside this paper.

In this work we show that when an inviscid axisymmetric Rankine flow experiences a soft expansion, rotating Kelvin waves can be excited. Downstream of the region where the expansion occurs (the transition region) the resulting flow can be expressed as the addition of a Rankine and a Beltrami flow. The Beltrami constant is determined from the Rankine upstream flow, and the helix pitch of the n=1 mode results from the boundary conditions downstream. Finally, a discussion of the process leading to oscillatory flow and a conjecture about the topological background that sustains the Beltrami flow structure are offered. © 2008 American Institute of Physics. Fil:González, R. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.