Experimental investigation of precession driven flows in a triaxial ellipsoid

<p>Precession driven flows are relevant for geo- and astrophysical fluid dynamics as well as industrial applications. In the context of planetary core dynamics, they are attributed to the generation of magnetic fields and/or anomalous dissipation. While precession driven flows have been frequently studied in a cylindrical, spherical or spheroidal container shape, the geometry of a triaxial ellipsoid, representing the geophysical case of core mantle boundary deformation in a tidally locked planet, has received less attention.</p><p>Here, we present results from an experimental study in a triaxial ellipsoid. The main focus of our study is on the base flow of uniform vorticity and we report a good agreement between experimental data and existing semi-analytical models. The amplitude of the time averaged uniform vorticity displays a hysteresis loop as a function of the precession forcing and we demonstrate that this observation depends on the ellipticity of the container. Our study also comprises experiments where the boundary layer is expected to be in a turbulent state. Therefore, we discuss the applicability of an effective damping coefficient in the semi-analytical models to account for the dissipation in a turbulent boundary layer. </p>

Inviscid limit for the damped generalized incompressible Navier-Stokes equations on $ \mathbb{T}^2 $

<p style='text-indent:20px;'>In this paper, for the damped generalized incompressible Navier-Stokes equations on <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{T}^{2} $\end{document}</tex-math></inline-formula> as the index <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> of the general dissipative operator <inline-formula><tex-math id="M4">\begin{document}$ (-\Delta)^{\alpha} $\end{document}</tex-math></inline-formula> belongs to <inline-formula><tex-math id="M5">\begin{document}$ (0,\frac{1}{2}] $\end{document}</tex-math></inline-formula>, we prove the absence of anomalous dissipation of the long time averages of entropy. We also give a note to show that, by using the <inline-formula><tex-math id="M6">\begin{document}$ L^{\infty} $\end{document}</tex-math></inline-formula> bounds given in Caffarelli et al. [<xref ref-type="bibr" rid="b4">4</xref>], the absence of anomalous dissipation of the long time averages of energy for the forced SQG equations established in Constantin et al. [<xref ref-type="bibr" rid="b12">12</xref>] still holds under a slightly weaker conditions <inline-formula><tex-math id="M7">\begin{document}$ \theta_{0}\in L^{1}(\mathbb{R}^{2})\cap L^{2}(\mathbb{R}^{2}) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ f \in L^{1}(\mathbb{R}^{2})\cap L^{p}(\mathbb{R}^{2}) $\end{document}</tex-math></inline-formula> with some <inline-formula><tex-math id="M9">\begin{document}$ p&gt;2 $\end{document}</tex-math></inline-formula>.</p>

Brownian motion of classical spins: Anomalous dissipation and generalized Langevin equation

In this work, we derive the Langevin equation (LE) of a classical spin interacting with a heat bath through momentum variables, starting from the fully dynamical Hamiltonian description. The derived LE with anomalous dissipation is analyzed in detail. The obtained LE is non-Markovian with multiplicative noise terms. The concomitant dissipative terms obey the fluctuation–dissipation theorem. The Markovian limit correctly produces the Kubo and Hashitsume equation. The perturbative treatment of our equations produces the Landau–Lifshitz equation and the Seshadri–Lindenberg equation. Then we derive the Fokker–Planck equation corresponding to LE and the concept of equilibrium probability distribution is analyzed.

A Lagrangian fluctuation–dissipation relation for scalar turbulence. Part II. Wall-bounded flows

We derive here Lagrangian fluctuation–dissipation relations for advected scalars in wall-bounded flows. The relations equate the dissipation rate for either passive or active scalars to the variance of scalar inputs from the initial values, boundary values and internal sources, as those are sampled backward in time by stochastic Lagrangian trajectories. New probabilistic concepts are required to represent scalar boundary conditions at the walls: the boundary local-time density at points on the wall where scalar fluxes are imposed and the boundary first hitting time at points where scalar values are imposed. These concepts are illustrated both by analytical results for the problem of pure heat conduction and by numerical results from a database of channel-flow turbulence, which also demonstrate the scalar mixing properties of near-wall turbulence. As an application of the fluctuation–dissipation relation, we examine for wall-bounded flows the relation between anomalous scalar dissipation and Lagrangian spontaneous stochasticity, i.e. the persistent non-determinism of Lagrangian particle trajectories in the limit of vanishing viscosity and diffusivity. In Part I of this series, we showed that spontaneous stochasticity is the only possible mechanism for anomalous dissipation of passive or active scalars, away from walls. Here it is shown that this remains true when there are no scalar fluxes through walls. Simple examples show, on the other hand, that a distinct mechanism of non-vanishing scalar dissipation can be thin scalar boundary layers near the walls. Nevertheless, we prove for general wall-bounded flows that spontaneous stochasticity is another possible mechanism of anomalous scalar dissipation.