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Register a new userMinimum Quadratic Helicity States
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Building on previous results on the quadratic helicity in magnetohydrodynamics (MHD) we investigate particular minimum helicity states. Those are eigenfunctions of the curl operator and are shown to constitute solutions of the quasi-stationary incompressible ideal MHD equations. We then show that these states have indeed minimum quadratic helicity.
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The relative importance of the helicity and cross-helicity electromotive dynamo effects for self-sustained magnetic field generation by chaotic thermal convection in rotating spherical shells is investigated as a function of shell thickness. Two distinct branches of dynamo solutions are found to coexist in direct numerical simulations for shell aspect ratios between 0.25 and 0.6 - a mean-field dipolar regime and a fluctuating dipolar regime. The properties characterising the coexisting dynamo attractors are compared and contrasted, including differences in temporal behavior and spatial structures of both the magnetic field and rotating thermal convection. The helicity $alpha$-effect and the cross-helicity $gamma$-effect are found to be comparable in intensity within the fluctuating dipolar dynamo regime, where their ratio does not vary significantly with the shell thickness. In contrast, within the mean-field dipolar dynamo regime the helicity $alpha$-effect dominates by approximately two orders of magnitude and becomes stronger with decreasing shell thickness.
On the basis of solutions of the Bragg-Hawthorne equations we discuss the helicity of thin toroidal vortices with the swirl - the orbital motion along the torus diretrix. It is shown that relationship of the helicity with circulations along the small and large linked circles - directrix and generatrix of the torus - depends on distribution of the azimuthal velocity in the core of the swirling vortex ring. In the case of non-homogeneous swirl this relationship differs from the well-known Moffat relationship - the doubled product of such circulations multiplied by the number of links. The results can be applied to vortices in planetary atmospheres and to vortex movements in the vicinity of active galactic nuclei.
Vortex breakdown phenomena in the axial vortices is an important feature which occurs frequently in geophysical flows (tornadoes and hurricanes) and in engineering flows (flow past delta wings, Von-Kerman vortex dynamo). We analyze helicity for axisymmetric vortex breakdown and propose a simplified formulation. For such cases, negative helicity is shown to conform to the vortex breakdown. A model problem has been analyzed to verify the results. The topology of the vortex breakdown is governed entirely by helicity density in the vertical plane. Our proposed methodology may be regarded as the prototype for identifying and characterize the breakdowns/eye in more complicated large-scale flows such as tornadoes/hurricanes.
We solve the Navier-Stokes equations with two simultaneous forcings. One forcing is applied at a given large-scale and it injects energy. The other forcing is applied at all scales belonging to the inertial range and it injects helicity. In this way we can vary the degree of turbulence helicity from non helical to maximally helical. We find that increasing the rate of helicity injection does not change the energy flux. On the other hand the level of total energy is strongly increased and the energy spectrum gets steeper. The energy spectrum spans from a Kolmogorov scaling law $k^{-5/3}$ for a non-helical turbulence, to a non-Kolmogorov scaling law $k^{-7/3}$ for a maximally helical turbulence. In the later case we find that the characteristic time of the turbulence is not the turnover time but a time based on the helicity injection rate. We also analyse the results in terms of helical modes decomposition. For a maximally helical turbulence one type of helical mode is found to be much more energetic than the other one, by several orders of magnitude. The energy cascade of the most energetic type of helical mode results from the sum of two fluxes. One flux is negative and can be understood in terms of a decimated model. This negative flux is however not sufficient to lead an inverse energy cascade. Indeed the other flux involving the least energetic type of helical mode is positive and the largest. The least energetic type of helical mode is then essential and cannot be neglected.
Helicity, as one of only two inviscid invariants in three-dimensional turbulence, plays an important role in the generation and evolution of turbulence. From the traditional viewpoint, there exists only one channel of helicity cascade similar to that of kinetic energy cascade. Through theoretical analysis, we find that there are two channels in helicity cascade process. The first channel mainly originates from vortex twisting process, and the second channel mainly originates from vortex stretching process. By analysing the data of direct numerical simulations of typical turbulent flows, we find that these two channels behave differently. The ensemble averages of helicity flux in different channels are equal in homogeneous and isotropic turbulence, while they are different in other type of turbulent flows. The second channel is more intermittent and acts more like a scalar, especially on small scales. Besides, we find a novel mechanism of hindered even inverse energy cascade, which could be attributed to the second-channel helicity flux with large amplitude.
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