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 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

Conditional regularity of solutions of the three dimensional Navier-Stokes equations and implications for intermittency

Two unusual time-integral conditional regularity results are presented for the three-dimensional Navier-Stokes equations. The ideas are based on $L^{2m}$-norms of the vorticity, denoted by $\Omega_{m}(t)$, and particularly on $D_{m} = \Omega_{m}^{\alpha_{m}}$, where $\alpha_{m} = 2m/(4m-3)$ for $m\geq 1$. The first result, more appropriate for the unforced case, can be stated simply : if there exists an $1\leq m < \infty$ for which the integral condition is satisfied ($Z_{m}=D_{m+1}/D_{m}$) $$\int_{0}^{t}\ln (\frac{1 + Z_{m}}{c_{4,m}}) d\tau \geq 0$$ then no singularity can occur on $[0, t]$. The constant $c_{4,m} \searrow 2$ for large $m$. Secondly, for the forced case, by imposing a critical \textit{lower} bound on $\int_{0}^{t}D_{m} d\tau$, no singularity can occur in $D_{m}(t)$ for \textit{large} initial data. Movement across this critical lower bound shows how solutions can behave intermittently, in analogy with a relaxation oscillator. Potential singularities that drive $\int_{0}^{t}D_{m} d\tau$ over this critical value can be ruled out whereas other types cannot.Comment: A frequency was missing in the definition of D_{m} in (I5) v3. 11 pages, 1 figur

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