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Register a new userThe Lagrangian-averaged model for magnetohydrodynamics turbulence and the absence of bottleneck
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Added by
Jonathan Pietarila Graham
Publication date
2009
fields
Physics
and research's language is
English
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We demonstrate that, for the case of quasi-equipartition between the velocity and the magnetic field, the Lagrangian-averaged magnetohydrodynamics alpha-model (LAMHD) reproduces well both the large-scale and small-scale properties of turbulent flows; in particular, it displays no increased (super-filter) bottleneck effect with its ensuing enhanced energy spectrum at the onset of the sub-filter-scales. This is in contrast to the case of the neutral fluid in which the Lagrangian-averaged Navier-Stokes alpha-model is somewhat limited in its applications because of the formation of spatial regions with no internal degrees of freedom and subsequent contamination of super-filter-scale spectral properties. No such regions are found in LAMHD, making this method capable of large reductions in required numerical degrees of freedom; specifically, we find a reduction factor of 200 when compared to a direct numerical simulation on a large grid of 1536^3 points at the same Reynolds number.
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The theory of turbulent transport of parallel ion momentum and heat by the interaction of stochastic magnetic fields and turbulence is presented. Attention is focused on determining the kinetic stress and the compressive energy flux. A critical parameter is identified as the ratio of the turbulent scattering rate to the rate of parallel acoustic dispersion. For the parameter large, the kinetic stress takes the form of a viscous stress. For the parameter small, the quasilinear residual stress is recovered. In practice, the viscous stress is the relevant form, and the quasilinear limit is not observable. This is the principal prediction of this paper. A simple physical picture is developed and shown to recover the results of the detailed analysis.
A linearly unstable, sinusoidal $E times B$ shear flow is examined in the gyrokinetic framework in both the linear and nonlinear regimes. In the linear regime, it is shown that the eigenmode spectrum is nearly identical to hydrodynamic shear flows, with a conjugate stable mode found at every unstable wavenumber. In the nonlinear regime, turbulent saturation of the instability is examined with and without the inclusion of a driving term that prevents nonlinear flattening of the mean flow, and a scale-independent radiative damping term that suppresses the excitation of conjugate stable modes. A simple fluid model for how momentum transport and partial flattening of the mean flow scale with the driving term is constructed, from which it is shown that, except at high radiative damping, stable modes play an important role in the turbulent state and yield significantly improved quantitative predictions when compared with corresponding models neglecting stable modes.
A dynamic procedure for the Lagrangian Averaged Navier-Stokes-$alpha$ (LANS-$alpha$) equations is developed where the variation in the parameter $alpha$ in the direction of anisotropy is determined in a self-consistent way from data contained in the simulation itself. The dynamic model is initially tested in forced and decaying isotropic turbulent flows where $alpha$ is constant in space but it is allowed to vary in time. It is observed that by using the dynamic LANS-$alpha$ procedure a more accurate simulation of the isotropic homogeneous turbulence is achieved. The energy spectra and the total kinetic energy decay are captured more accurately as compared with the LANS-$alpha$ simulations using a fixed $alpha$. In order to evaluate the applicability of the dynamic LANS-$alpha$ model in anisotropic turbulence, a priori test of a turbulent channel flow is performed. It is found that the parameter $alpha$ changes in the wall normal direction. Near a solid wall, the length scale $alpha$ is seen to depend on the distance from the wall with a vanishing value at the wall. On the other hand, away from the wall, where the turbulence is more isotropic, $alpha$ approaches an almost constant value. Furthermore, the behavior of the subgrid scale stresses in the near wall region is captured accurately by the dynamic LANS-$alpha$ model. The dynamic LANS-$alpha$ model has the potential to extend the applicability of the LANS-$alpha$ equations to more complicated anisotropic flows.
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