Energy transfer in two-dimensional magnetohydrodynamic turbulence: formalism and numerical results

The basic entity of nonlinear interaction in Navier-Stokes and the Magnetohydrodynamic (MHD) equations is a wavenumber triad ({\bf k,p,q}) satisfying ${\bf k+p+q=0}$. The expression for the combined energy transfer from two of these wavenumbers to the third wavenumber is known. In this paper we introduce the idea of an effective energy transfer between a pair of modes by the mediation of the third mode, and find an expression for it. Then we apply this formalism to compute the energy transfer in the quasi-steady-state of two-dimensional MHD turbulence with large-scale kinetic forcing. The computation of energy fluxes and the energy transfer between different wavenumber shells is done using the data generated by the pseudo-spectral direct numerical simulation. The picture of energy flux that emerges is quite complex---there is a forward cascade of magnetic energy, an inverse cascade of kinetic energy, a flux of energy from the kinetic to the magnetic field, and a reverse flux which transfers the energy back to the kinetic from the magnetic. The energy transfer between different wavenumber shells is also complex---local and nonlocal transfers often possess opposing features, i.e., energy transfer between some wavenumber shells occurs from kinetic to magnetic, and between other wavenumber shells this transfer is reversed. The net transfer of energy is from kinetic to magnetic. The results obtained from the studies of flux and shell-to-shell energy transfer are consistent with each other.Comment: 27 pages REVTEX; 14 ps figure

Incompressible Turbulence as Nonlocal Field Theory

It is well known that incompressible turbulence is nonlocal in real space because sound speed is infinite in incompressible fluids. The equation in Fourier space indicates that it is nonlocal in Fourier space as well. Contrast this with Burgers equation which is local in real space. Note that the sound speed in Burgers equation is zero. In our presentation we will contrast these two equations using nonlocal field theory. Energy spectrum and renormalized parameters will be discussed.Comment: 7 pages; Talk presented in Conference on "Perspectives in Nonlinear Dynamics (PNLD 2004)" held in Chennai, 200

Variational Principles for Lagrangian Averaged Fluid Dynamics

The Lagrangian average (LA) of the ideal fluid equations preserves their transport structure. This transport structure is responsible for the Kelvin circulation theorem of the LA flow and, hence, for its convection of potential vorticity and its conservation of helicity. Lagrangian averaging also preserves the Euler-Poincar\'e (EP) variational framework that implies the LA fluid equations. This is expressed in the Lagrangian-averaged Euler-Poincar\'e (LAEP) theorem proven here and illustrated for the Lagrangian average Euler (LAE) equations.Comment: 23 pages, 3 figure

Local shell-to-shell energy transfer via nonlocal Interactions in fluid turbulence

In this paper we analytically compute the strength of nonlinear interactions in a triad, and the energy exchanges between wavenumber shells in incompressible fluid turbulence. The computation has been done using first-order perturbative field theory. In three dimension, magnitude of triad interactions is large for nonlocal triads, and small for local triads. However, the shell-to-shell energy transfer rate is found to be local and forward. This result is due to the fact that the nonlocal triads occupy much less Fourier space volume than the local ones. The analytical results on three-dimensional shell-to-shell energy transfer match with their numerical counterparts. In two-dimensional turbulence, the energy transfer rates to the near-by shells are forward, but to the distant shells are backward; the cumulative effect is an inverse cascade of energy.Comment: 10 pages, Revtex

Large eddy simulation of two-dimensional isotropic turbulence

Large eddy simulation (LES) of forced, homogeneous, isotropic, two-dimensional (2D) turbulence in the energy transfer subrange is the subject of this paper. A difficulty specific to this LES and its subgrid scale (SGS) representation is in that the energy source resides in high wave number modes excluded in simulations. Therefore, the SGS scheme in this case should assume the function of the energy source. In addition, the controversial requirements to ensure direct enstrophy transfer and inverse energy transfer make the conventional scheme of positive and dissipative eddy viscosity inapplicable to 2D turbulence. It is shown that these requirements can be reconciled by utilizing a two-parametric viscosity introduced by Kraichnan (1976) that accounts for the energy and enstrophy exchange between the resolved and subgrid scale modes in a way consistent with the dynamics of 2D turbulence; it is negative on large scales, positive on small scales and complies with the basic conservation laws for energy and enstrophy. Different implementations of the two-parametric viscosity for LES of 2D turbulence were considered. It was found that if kept constant, this viscosity results in unstable numerical scheme. Therefore, another scheme was advanced in which the two-parametric viscosity depends on the flow field. In addition, to extend simulations beyond the limits imposed by the finiteness of computational domain, a large scale drag was introduced. The resulting LES exhibited remarkable and fast convergence to the solution obtained in the preceding direct numerical simulations (DNS) by Chekhlov et al. (1994) while the flow parameters were in good agreement with their DNS counterparts. Also, good agreement with the Kolmogorov theory was found. This LES could be continued virtually indefinitely. Then, a simplifiedComment: 34 pages plain tex + 18 postscript figures separately, uses auxilary djnlx.tex fil

Leray and LANS-α\alpha modeling of turbulent mixing

Mathematical regularisation of the nonlinear terms in the Navier-Stokes equations provides a systematic approach to deriving subgrid closures for numerical simulations of turbulent flow. By construction, these subgrid closures imply existence and uniqueness of strong solutions to the corresponding modelled system of equations. We will consider the large eddy interpretation of two such mathematical regularisation principles, i.e., Leray and LANS$-\alpha$ regularisation. The Leray principle introduces a {\bfi smoothed transport velocity} as part of the regularised convective nonlinearity. The LANS$-\alpha$ principle extends the Leray formulation in a natural way in which a {\bfi filtered Kelvin circulation theorem}, incorporating the smoothed transport velocity, is explicitly satisfied. These regularisation principles give rise to implied subgrid closures which will be applied in large eddy simulation of turbulent mixing. Comparison with filtered direct numerical simulation data, and with predictions obtained from popular dynamic eddy-viscosity modelling, shows that these mathematical regularisation models are considerably more accurate, at a lower computational cost.Comment: 42 pages, 12 figure

Biophysical forcing of particle production and distribution during a spring bloom in the North Atlantic

Abstract: The beam attenuation serves as a proxy for particulate matter and is a key parameter in visibility algorithms for the aquatic environment. It is well known, however, that the beam attenuation is a function of the acceptance angle of the transmissometer used to measure it. Here we compare eight different transmissometers with four different acceptance angles using four different deployment strategies and sites, and find that their mean attenuation values differ markedly and in a consistent way with instrument acceptance angle: smaller acceptance angles provide higher beam attenuation values. This difference is due to variations in scattered light collected with different acceptance angles and is neither constant nor easy to parameterize. Variability (in space or time) in the ratios of beam attenuations measured by two different instruments correlates, in most cases, with the particle size parameter (as expected from Mie theory), but this correlation is often weak and can be the opposite of expectations based on particle size changes. We recommended careful consideration of acceptance angle in applications of beam transmission data especially when comparing data from different instruments

Biophysical forcing of particle production and distribution during a spring bloom in the North Atlantic

Abstract: The beam attenuation serves as a proxy for particulate matter and is a key parameter in visibility algorithms for the aquatic environment. It is well known, however, that the beam attenuation is a function of the acceptance angle of the transmissometer used to measure it. Here we compare eight different transmissometers with four different acceptance angles using four different deployment strategies and sites, and find that their mean attenuation values differ markedly and in a consistent way with instrument acceptance angle: smaller acceptance angles provide higher beam attenuation values. This difference is due to variations in scattered light collected with different acceptance angles and is neither constant nor easy to parameterize. Variability (in space or time) in the ratios of beam attenuations measured by two different instruments correlates, in most cases, with the particle size parameter (as expected from Mie theory), but this correlation is often weak and can be the opposite of expectations based on particle size changes. We recommended careful consideration of acceptance angle in applications of beam transmission data especially when comparing data from different instruments