Dynamics of scaled norms of vorticity for the three-dimensional Navier-Stokes and Euler equations

A series of numerical experiments is suggested for the three-dimensional Navier-Stokes and Euler equations on a periodic domain based on a set of $L^{2m}$-norms of vorticity $\Omega_{m}$ for $m\geq 1$. These are scaled to form the dimensionless sequence $D_{m}= (\varpi_{0}^{-1}\Omega_{m})^{\alpha_{m}}$ where $\varpi_{0}$ is a constant frequency and $\alpha_{m} = 2m/(4m-3)$. A numerically testable Navier-Stokes regularity criterion comes from comparing the relative magnitudes of $D_{m}$ and $D_{m+1}$ while another is furnished by imposing a critical lower bound on $\int_{0}^{t}D_{m}\,d\tau$. The behaviour of the $D_{m}$ is also important in the Euler case in suggesting a method by which possible singular behaviour might also be tested.Comment: To appear in the Procedia IUTAM volume of papers Topological Fluid Dynamic

Quasi-conservation laws for compressible 3D Navier-Stokes flow

We formulate the quasi-Lagrangian fluid transport dynamics of mass density $\rho$ and the projection q=\bom\cdot\nabla\rho of the vorticity \bom onto the density gradient, as determined by the 3D compressible Navier-Stokes equations for an ideal gas, although the results apply for an arbitrary equation of state. It turns out that the quasi-Lagrangian transport of $q$ cannot cross a level set of $\rho$. That is, in this formulation, level sets of $\rho$ (isopychnals) are impermeable to the transport of the projection $q$.Comment: 2 page note, to appear in Phys Rev

Stretching and folding processes in the 3D Euler and Navier-Stokes equations

Stretching and folding dynamics in the incompressible, stratified 3D Euler and Navier-Stokes equations are reviewed in the context of the vector \bdB = \nabla q\times\nabla\theta where q=\bom\cdot\nabla\theta. The variable $\theta$ is the temperature and \bdB satisfies \partial_{t}\bdB = \mbox{curl}\,(\bu\times\bdB). These ideas are then discussed in the context of the full compressible Navier-Stokes equations where $q$ takes the two forms q = \bom\cdot\nabla\rho and q = \bom\cdot\nabla(\ln\rho).Comment: UTAM Symposium on Understanding Common Aspects of Extreme Events in Fluid

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

Knowledge development for organic systems: An example of weed management

Despite the large amount information on weed biology and specific weed control measures produced by researchers, organic farmers still prioritise weeds as an important area for further research. A recent project investigating weed management in organic farming systems has established that knowledge and learning are key requirements for this to be effective. Development of relevant, practically useful knowledge depends on access to information generated ‘scientifically’ by researchers and also to knowledge generated as a result of farmer experience with weeds. This requires that farmers, advisors and researchers take a participatory approach to collecting and processing information on weed management, using it to develop new and relevant knowledge. The appropriate framework for knowledge development is thus a collegiate one in which all stakeholders’ value and learn from the observations and experience of others. These findings have implications for the way in which research is conducted and funded

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