1,762 research outputs found
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
-norms of the vorticity, denoted by , and particularly
on , where for . The first result, more appropriate for the unforced case, can be stated
simply : if there exists an for which the integral condition
is satisfied () then no singularity can occur on . The
constant for large . Secondly, for the forced case, by
imposing a critical \textit{lower} bound on , no
singularity can occur in 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
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
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- moments, , of , where
and are, respectively, the vorticity and current density in
three-dimensional magnetohydrodynamics (MHD). We show by mathematical analysis,
for unit magnetic Prandtl number , how these moments can be used to
identify three possible regimes for solutions of the MHD equations; these
regimes are specified by inequalities for and . 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 that characterize the power-law dependences of the energy
spectra on the wave number , in the inertial range of
scales. We also comment on (a) the generalization of our results to the case
and (b) the relation between and the order- moments
of gradients of hydrodynamic fields, which are used in characterizing
intermittency in turbulent flows.Comment: 14 pages, 3 figure
Depletion of nonlinearity in magnetohydrodynamic turbulence: insights from analysis and simulations
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 , 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 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 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
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
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