Sign-singular measure and its association with turbulent scalings

Dynamic Phase Alignment in Navier-Stokes Turbulence

In Navier-Stokes turbulence, energy and helicity injected at large scales are subject to a joint direct cascade, with both quantities exhibiting a spectral scaling ∝k^{-5/3}. We demonstrate via direct numerical simulations that the two cascades are compatible due to the existence of a strong scale-dependent phase alignment between velocity and vorticity fluctuations, with the phase alignment angle scaling as cosα_{k}∝k^{-1}.

Cancellation exponents in isotropic turbulence and magnetohydrodynamic turbulence

Small-scale characteristics of turbulence such as velocity gradients and vorticity fluctuate rapidly in magnitude and oscillate in sign. Much work exists on the characterization of magnitude variations, but far less on sign oscillations. While in homogeneous turbulence averages performed on large scales tend to zero because of the oscillatory character, those performed on increasingly smaller scales will vary with the averaging scale in some characteristic way. This characteristic variation at high Reynolds numbers is captured by the so-called cancellation exponent, which measures how local averages tend to cancel out as the averaging scale increases, in space or time. Past experimental work suggests that the exponents in turbulence depend on whether one considers quantities in full three-dimensional (3D) space or uses their one- or two-dimensional cuts. We compute cancellation exponents of vorticity and longitudinal as well as transverse velocity gradients in isotropic turbulence at Taylor-scale Reynolds numbers up to 1300 on 8192^{3} grids. The 2D cuts yield the same exponents as those for full 3D, while the 1D cuts yield smaller numbers, suggesting that the results in higher dimensions are more reliable. We make the case that the presence of vortical filaments in isotropic turbulence leads to this conclusion. This effect is particularly conspicuous in magnetohydrodynamic turbulence, where an increased degree of spatial coherence develops along the direction of an imposed magnetic field.

Cancellation properties in Hall magnetohydrodynamics with a strong guide magnetic field

We present a signed measure analysis of compressible Hall magnetohydrodynamic turbulence with an external guide field. Signed measure analysis allows us to characterize the scaling behavior of the sign-oscillating flow structures and their geometrical properties (fractal dimensions of structures). A reduced numerical model, valid when a strong guide magnetic field is present, is used here. In order to discuss the effect of the Hall term, different values for the ion skin depth are considered in the simulations. Results show that as the Hall term is increased, the fractal dimension of the current and vorticity sheets decreases. This observation, together with previous analysis of the same fields, provides a comprehensive description of the effect of the Hall force on the formation of structures. Two main processes are identified, namely, the widening and unraveling of the sheets.

Sign-singularity of the reduced magnetic helicity in the solar wind plasma

We investigate the scaling laws of a signed measure derived from the reduced magnetic helicity which has been determined from Cluster data in the solar wind. This quantifies the handedness of the magnetic field; namely, it can be related to the polarization of the magnetic field fluctuations (right or left hand). The measure results to be sign-singular; that is, we do not observe any scale-dependent effect at the ion- and at electron-cyclotron frequencies. Cancellations between right- and left-hand polarizations go on in the dispersive or dissipative range, beyond the electron-cyclotron frequency. This means that the mechanism responsible for the generation of the dispersive or dissipative range is rather insensitive to the polarization of the magnetic field fluctuations.