Bounds on solutions of the rotating, stratified, incompressible, non-hydrostatic, three-dimensional Boussinesq equations

We study the three-dimensional, incompressible, non-hydrostatic Boussinesq fluid equations, which are applicable to the dynamics of the oceans and atmosphere. These equations describe the interplay between velocity and buoyancy in a rotating frame. A hierarchy of dynamical variables is introduced whose members $\Omega_{m}(t)$ ($1 \leq m < \infty$) are made up from the respective sum of the $L^{2m}$-norms of vorticity and the density gradient. Each $\Omega_{m}(t)$ has a lower bound in terms of the inverse Rossby number, $Ro^{-1}$, that turns out to be crucial to the argument. For convenience, the $\Omega_{m}$ are also scaled into a new set of variables $D_{m}(t)$. By assuming the existence and uniqueness of solutions, conditional upper bounds are found on the $D_{m}(t)$ in terms of $Ro^{-1}$ and the Reynolds number $Re$. These upper bounds vary across bands in the $\{D_{1},\,D_{m}\}$ phase plane. The boundaries of these bands depend subtly upon $Ro^{-1}$, $Re$, and the inverse Froude number $Fr^{-1}$. For example, solutions in the lower band conditionally live in an absorbing ball in which the maximum value of $\Omega_{1}$ deviates from $Re^{3/4}$ as a function of $Ro^{-1},\,Re$ and $Fr^{-1}$.Comment: 24 pages, 3 figures and 1 tabl

The 3D incompressible Euler equations with a passive scalar: a road to blow-up?

The 3D incompressible Euler equations with a passive scalar $\theta$ are considered in a smooth domain $\Omega\subset \mathbb{R}^{3}$ with no-normal-flow boundary conditions \bu\cdot\bhn|_{\partial\Omega} = 0. It is shown that smooth solutions blow up in a finite time if a null (zero) point develops in the vector \bB = \nabla q\times\nabla\theta, provided \bB has no null points initially\,: \bom = \mbox{curl}\,\bu is the vorticity and q = \bom\cdot\nabla\theta is a potential vorticity. The presence of the passive scalar concentration $\theta$ is an essential component of this criterion in detecting the formation of a singularity. The problem is discussed in the light of a kinematic result by Graham and Henyey (2000) on the non-existence of Clebsch potentials in the neighbourhood of null points.Comment: 5 pages, no figure

A Bound on Mixing Efficiency for the Advection-Diffusion Equation

An upper bound on the mixing efficiency is derived for a passive scalar under the influence of advection and diffusion with a body source. For a given stirring velocity field, the mixing efficiency is measured in terms of an equivalent diffusivity, which is the molecular diffusivity that would be required to achieve the same level of fluctuations in the scalar concentration in the absence of stirring, for the same source distribution. The bound on the equivalent diffusivity depends only on the functional "shape" of both the source and the advecting field. Direct numerical simulations performed for a simple advecting flow to test the bounds are reported.Comment: 10 pages, 2 figures, JFM format (included

A regularity criterion for solutions of the three-dimensional Cahn-Hilliard-Navier-Stokes equations and associated computations

We consider the 3D Cahn-Hilliard equations coupled to, and driven by, the forced, incompressible 3D Navier-Stokes equations. The combination, known as the Cahn-Hilliard-Navier-Stokes (CHNS) equations, is used in statistical mechanics to model the motion of a binary fluid. The potential development of singularities (blow-up) in the contours of the order parameter $\phi$ is an open problem. To address this we have proved a theorem that closely mimics the Beale-Kato-Majda theorem for the $3D$ incompressible Euler equations [Beale et al. Commun. Math. Phys., Commun. Math. Phys., ${\rm 94}$, $61-66 ({\rm 1984})$]. By taking an $L^{\infty}$ norm of the energy of the full binary system, designated as $E_{\infty}$, we have shown that $\int_{0}^{t}E_{\infty}(\tau)\,d\tau$ governs the regularity of solutions of the full 3D system. Our direct numerical simulations (DNSs), of the 3D CHNS equations, for (a) a gravity-driven Rayleigh Taylor instability and (b) a constant-energy-injection forcing, with $128^3$ to $512^3$ collocation points and over the duration of our DNSs, confirm that $E_{\infty}$ remains bounded as far as our computations allow.Comment: 11 pages, 3 figure

The role of BKM-type theorems in 3D3D Euler, Navier-Stokes and Cahn-Hilliard-Navier-Stokes analysis

The Beale-Kato-Majda theorem contains a single criterion that controls the behaviour of solutions of the $3D$ incompressible Euler equations. Versions of this theorem are discussed in terms of the regularity issues surrounding the $3D$ incompressible Euler and Navier-Stokes equations together with a phase-field model for the statistical mechanics of binary mixtures called the $3D$ Cahn-Hilliard-Navier-Stokes (CHNS) equations. A theorem of BKM-type is established for the CHNS equations for the full parameter range. Moreover, for this latter set, it is shown that there exists a Reynolds number and a bound on the energy-dissipation rate that, remarkably, reproduces the $Re^{3/4}$ upper bound on the inverse Kolmogorov length normally associated with the Navier-Stokes equations alone. An alternative length-scale is introduced and discussed, together with a set of pseudo-spectral computations on a $128^{3}$ grid.Comment: 3 figures and 3 table

Phase transitions in the fractional three-dimensional Navier-Stokes equations

The fractional Navier-Stokes equations on a periodic domain $[0,\,L]^{3}$ differ from their conventional counterpart by the replacement of the $-\nu\Delta\mathbf{u}$ Laplacian term by $\nu_{s}A^{s}\mathbf{u}$, where $A= - \Delta$ is the Stokes operator and $\nu_{s} = \nu L^{2(s-1)}$ is the viscosity parameter. Four critical values of the exponent $s$ have been identified where functional properties of solutions of the fractional Navier-Stokes equations change. These values are: $s=\frac{1}{3}$; $s=\frac{3}{4}$; $s=\frac{5}{6}$ and $s=\frac{5}{4}$. In particular, in the fractional setting we prove an analogue of one of the Prodi-Serrin regularity criteria ($s > \frac{1}{3}$), an equation of local energy balance ($s \geq \frac{3}{4}$) and an infinite hierarchy of weak solution time averages ($s > \frac{5}{6}$). The existence of our analogue of the Prodi-Serrin criterion for $s > \frac{1}{3}$ suggests that the convex integration schemes that construct H\"older-continuous solutions with epochs of regularity for $s < \frac{1}{3}$ are sharp with respect to the value of $s$

An analytical and computational study of the incompressible Toner-Tu Equations

The incompressible Toner-Tu (ITT) partial differential equations (PDEs) are an important example of a set of active-fluid PDEs. While they share certain properties with the Navier-Stokes equations (NSEs), such as the same scaling invariance, there are also important differences. The NSEs are usually considered in either the decaying or the additively forced cases, whereas the ITT equations have no additive forcing. Instead, they include a linear, activity term \alpha \bu (\bu is the velocity field) which pumps energy into the system, but also a negative \bu|\bu|^{2}-term which provides a platform for either frozen or statistically steady states. Taken together, these differences make the ITT equations an intriguing candidate for study using a combination of PDE analysis and pseudo-spectral direct numerical simulations (DNSs). In the $d=2$ case, we have established global regularity of solutions, but we have also shown the existence of bounded hierarchies of weighted, time-averaged norms of both higher derivatives and higher moments of the velocity field. Similar bounded hierarchies for Leray-type weak solutions have also been established in the $d=3$ case. We present results for these norms from our DNSs in both $d=2$ and $d=3$, and contrast them with their Navier-Stokes counterparts

Vorticity moments in four numerical simulations of the 3D Navier–Stokes equations

The issue of intermittency in numerical solutions of the 3D Navier–Stokes equations on a periodic box [0,L]3 is addressed through four sets of numerical simulations that calculate a new set of variables defined by Dm(t)=(ϖ−10Ωm)αm for 1≤m≤∞ where αm=2m/(4m−3) and [Ωm(t)]2m=L−3∫V|ω|2mdV with ϖ0=νL−2. All four simulations unexpectedly show that the Dm are ordered for m=1,…,9 such that Dm+1<Dm. Moreover, the Dm squeeze together such that Dm+1/Dm↗1 as m increases. The values of D1 lie far above the values of the rest of the Dm, giving rise to a suggestion that a depletion of nonlinearity is occurring which could be the cause of Navier–Stokes regularity. The first simulation is of very anisotropic decaying turbulence; the second and third are of decaying isotropic turbulence from random initial conditions and forced isotropic turbulence at fixed Grashof number respectively; the fourth is of very-high-Reynolds-number forced, stationary, isotropic turbulence at up to resolutions of 40963

Efficient transport of femtosecond laser-generated fast electrons in a millimeter thick graphite

We demonstrate efficient transport of fast electrons generated by ∼1018 W/cm2, 30 fs, 800 nm laser pulses through a millimeter thick polycrystalline graphite. Measurements of hot electron spectra at the front side of the graphite target show enhancement in terms of the electron flux and temperature, while the spectra at the rear confirm the ability of the graphite to transport large electron currents over a macroscopic distance of a millimeter. In addition, protons of keV energies are observed at the rear side of such a macroscopically thick target and attributed to the target-normal-sheath-acceleration mechanism

Probing ultrafast dynamics in a solid-density plasma created by an intense femtosecond laser

We report a study on the dynamics of a near-solid density plasma using an ultraviolet (266 nm) femtosecond probe laser pulse, which can penetrate to densities of ∼ 1022 cm-3, nearly an order of magnitude higher than the critical density of the 800 nm, femtosecond pump laser. Time-resolved probe-reflectivity from the plasma shows a rapid decay (picosecond- timescale) while the time-resolved reflected probe spectra show red shifts at early temporal delays and blue shifts at longer delays. This spectral behaviour of the reflected probe can be explained by a laser-driven shock moving inward and a subsequent hydrodynamic free expansion in the outward direction