Lagrangian analysis of alignment dynamics for isentropic compressible magnetohydrodynamics

After a review of the isentropic compressible magnetohydrodynamics (ICMHD) equations, a quaternionic framework for studying the alignment dynamics of a general fluid flow is explained and applied to the ICMHD equations.Comment: 12 pages, 2 figures, submitted to a Focus Issue of New Journal of Physics on "Magnetohydrodynamics and the Dynamo Problem" J-F Pinton, A Pouquet, E Dormy and S Cowley, editor

Depletion of Nonlinearity in Magnetohydrodynamic Turbulence: Insights from Analysis and Simulations

We build on recent developments in the study of fluid turbulence [Gibbon \textit{et al.} Nonlinearity 27, 2605 (2014)] to define suitably scaled, order-$m$ moments, $D_m^{\pm}$, of $\omega^\pm= \omega \pm j$, where $\omega$ and $j$ are, respectively, the vorticity and current density in three-dimensional magnetohydrodynamics (MHD). We show by mathematical analysis, for unit magnetic Prandtl number $P_M$, how these moments can be used to identify three possible regimes for solutions of the MHD equations; these regimes are specified by inequalities for $D_m^{\pm}$ and $D_1^{\pm}$. We then compare our mathematical results with those from our direct numerical simulations (DNSs) and thus demonstrate that 3D MHD turbulence is like its fluid-turbulence counterpart insofar as all solutions, which we have investigated, remain in \textit{only one of these regimes}; this regime has depleted nonlinearity. We examine the implications of our results for the exponents $q^{\pm}$ that characterize the power-law dependences of the energy spectra $\mathcal{E}^{\pm}(k)$ on the wave number $k$, in the inertial range of scales. We also comment on (a) the generalization of our results to the case $P_M \neq 1$ and (b) the relation between $D_m^{\pm}$ and the order-$m$ moments of gradients of hydrodynamic fields, which are used in characterizing intermittency in turbulent flows.Comment: 14 pages, 3 figure

Lagrangian particle paths and ortho-normal quaternion frames

Experimentalists now measure intense rotations of Lagrangian particles in turbulent flows by tracking their trajectories and Lagrangian-average velocity gradients at high Reynolds numbers. This paper formulates the dynamics of an orthonormal frame attached to each Lagrangian fluid particle undergoing three-axis rotations, by using quaternions in combination with Ertel's theorem for frozen-in vorticity. The method is applicable to a wide range of Lagrangian flows including the three-dimensional Euler equations and its variants such as ideal MHD. The applicability of the quaterionic frame description to Lagrangian averaged velocity gradient dynamics is also demonstrated.Comment: 9 pages, one figure, revise

Estimates for the two-dimensional Navier-Stokes equations in terms of the Reynolds number

The tradition in Navier-Stokes analysis of finding estimates in terms of the Grashof number \bG, whose character depends on the ratio of the forcing to the viscosity $\nu$, means that it is difficult to make comparisons with other results expressed in terms of Reynolds number \Rey, whose character depends on the fluid response to the forcing. The first task of this paper is to apply the approach of Doering and Foias \cite{DF} to the two-dimensional Navier-Stokes equations on a periodic domain $[0,L]^{2}$ by estimating quantities of physical relevance, particularly long-time averages \left, in terms of the Reynolds number \Rey = U\ell/\nu, where U^{2}= L^{-2}\left and $\ell$ is the forcing scale. In particular, the Constantin-Foias-Temam upper bound \cite{CFT} on the attractor dimension converts to a_{\ell}^{2}\Rey(1 + \ln\Rey)^{1/3}, while the estimate for the inverse Kraichnan length is (a_{\ell}^{2}\Rey)^{1/2}, where $a_{\ell}$ is the aspect ratio of the forcing. Other inverse length scales, based on time averages, and associated with higher derivatives, are estimated in a similar manner. The second task is to address the issue of intermittency : it is shown how the time axis is broken up into very short intervals on which various quantities have lower bounds, larger than long time-averages, which are themselves interspersed by longer, more quiescent, intervals of time.Comment: 21 pages, 1 figure, accepted for publication from J. Math. Phys. for the special issue on mathematical fluid mechanic

Exponentially growing solutions in homogeneous Rayleigh-Benard convection

It is shown that homogeneous Rayleigh-Benard flow, i.e., Rayleigh-Benard turbulence with periodic boundary conditions in all directions and a volume forcing of the temperature field by a mean gradient, has a family of exact, exponentially growing, separable solutions of the full non-linear system of equations. These solutions are clearly manifest in numerical simulations above a computable critical value of the Rayleigh number. In our numerical simulations they are subject to secondary numerical noise and resolution dependent instabilities that limit their growth to produce statistically steady turbulent transport.Comment: 4 pages, 3 figures, to be published in Phys. Rev. E - rapid communication

Two center multipole expansion method: application to macromolecular systems

We propose a new theoretical method for the calculation of the interaction energy between macromolecular systems at large distances. The method provides a linear scaling of the computing time with the system size and is considered as an alternative to the well known fast multipole method. Its efficiency, accuracy and applicability to macromolecular systems is analyzed and discussed in detail.Comment: 23 pages, 7 figures, 1 tabl

Blocking by fixed and variable stimuli: effects of stimulus distribution on blocking

An experiment with rats compared the ability of fixed and variable duration cues to produce blocking. Rats in Group B (Blocking) were trained that both fixed- (F) and variable- (V) duration cues would be followed by food delivery. In a subsequent training stage F and V continued to be reinforced, but F was accompanied by X, and V by Y. In the test phase responding to X and Y was examined. Control Group O (Overshadowing) received identical treatment, except that F and V were nonreinforced in the first training stage. In Group B there was evidence for blocking, but only of X which had been conditioned in compound with the fixed-duration F; there was no evidence for blocking of Y, which had been conditioned in compound with the variable duration V. It is suggested that this result may occur because fixed cues reach a higher, more stable asymptote of associative strength than their variable equivalents