The small-scale solar surface dynamo

The existence of a turbulent small-scale solar surface dynamo is likely, considering existing numerical and laboratory experiments, as well as comparisons of a small-scale dynamo in MURaM simulations with Hinode observations. We find the observed peaked probability distribution function (PDF) from Stokes-V magnetograms is consistent with a monotonic PDF of the actual vertical field strength. The cancellation function of the vertical flux density from a Hinode SP observation is found to follow a self-similar power law over two decades in length scales down to the ~200 km resolution limit. This provides observational evidence that the scales of magnetic structuring in the photosphere extend at least down to 20 km. From the power law, we determine a lower bound for the true quiet-Sun mean vertical unsigned flux density of ~43 G, consistent with our numerically-based estimates that 80% or more of the vertical unsigned flux should be invisible to Stokes-V observations at a resolution of 200 km owing to cancellation. Our estimates significantly reduce the order-of-magnitude discrepancy between Zeeman- and Hanle-based estimates.Comment: Proceedings of the Second Hinode Science Meeting, ASP Series 2009. 8 pages, 4 figure

Cancellation exponent and multifractal structure in two-dimensional magnetohydrodynamics: direct numerical simulations and Lagrangian averaged modeling

We present direct numerical simulations and Lagrangian averaged (also known as alpha-model) simulations of forced and free decaying magnetohydrodynamic turbulence in two dimensions. The statistics of sign cancellations of the current at small scales is studied using both the cancellation exponent and the fractal dimension of the structures. The alpha-model is found to have the same scaling behavior between positive and negative contributions as the direct numerical simulations. The alpha-model is also able to reproduce the time evolution of these quantities in free decaying turbulence. At large Reynolds numbers, an independence of the cancellation exponent with the Reynolds numbers is observed.Comment: Finite size box effects have been taken into account in the definition of the partition function. This has resulted in a more clear scaling in all figures. Several points are clarified in the tex

A framework for the evaluation of turbulence closures used in mesoscale ocean large-eddy simulations

We present a methodology to determine the best turbulence closure for an eddy-permitting ocean model through measurement of the error-landscape of the closure's subgrid spectral transfers and flux. We apply this method to 6 different closures for forced-dissipative simulations of the barotropic vorticity equation on a f-plane (2D Navier-Stokes equation). Using a high-resolution benchmark, we compare each closure's model of energy and enstrophy transfer to the actual transfer observed in the benchmark run. The error-landscape norms enable us to both make objective comparisons between the closures and to optimize each closure's free parameter for a fair comparison. The hyper-viscous closure most closely reproduces the enstrophy cascade, especially at larger scales due to the concentration of its dissipative effects to the very smallest scales. The viscous and Leith closures perform nearly as well, especially at smaller scales where all three models were dissipative. The Smagorinsky closure dissipates enstrophy at the wrong scales. The anticipated potential vorticity closure was the only model to reproduce the upscale transfer of kinetic energy from the unresolved scales, but would require high-order Laplacian corrections in order to concentrate dissipation at the smallest scales. The Lagrangian-averaged alpha-model closure did not perform successfully for forced 2D isotropic Navier-Stokes: small-scale filamentation is only slightly reduced by the model while small-scale roll-up is prevented. Together, this reduces the effects of diffusion.Comment: 44 pages, 21 figures, 1 Appendix, submitted to Ocean Modelin

Not Much Helicity is Needed to Drive Large Scale Dynamos

Understanding the in situ amplification of large scale magnetic fields in turbulent astrophysical rotators has been a core subject of dynamo theory. When turbulent velocities are helical, large scale dynamos that substantially amplify fields on scales that exceed the turbulent forcing scale arise, but the minimum sufficient fractional kinetic helicity f_h,C has not been previously well quantified. Using direct numerical simulations for a simple helical dynamo, we show that f_h,C decreases as the ratio of forcing to large scale wave numbers k_F/k_min increases. From the condition that a large scale helical dynamo must overcome the backreaction from any non-helical field on the large scales, we develop a theory that can explain the simulations. For k_F/k_min>8 we find f_h,C< 3%, implying that very small helicity fractions strongly influence magnetic spectra for even moderate scale separation.Comment: 5 pages, 4 figure

Turbulent small-scale dynamo action in solar surface simulations

We demonstrate that a magneto-convection simulation incorporating essential physical processes governing solar surface convection exhibits turbulent small-scale dynamo action. By presenting a derivation of the energy balance equation and transfer functions for compressible magnetohydrodynamics (MHD), we quantify the source of magnetic energy on a scale-by-scale basis. We rule out the two alternative mechanisms for the generation of small-scale magnetic field in the simulations: the tangling of magnetic field lines associated with the turbulent cascade and Alfvenization of small-scale velocity fluctuations ("turbulent induction"). Instead, we find the dominant source of small-scale magnetic energy is stretching by inertial-range fluid motions of small-scale magnetic field lines against the magnetic tension force to produce (against Ohmic dissipation) more small-scale magnetic field. The scales involved become smaller with increasing Reynolds number, which identifies the dynamo as a small-scale turbulent dynamo.Comment: accepted by Ap

The Lagrangian-averaged model for magnetohydrodynamics turbulence and the absence of bottleneck

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.Comment: 22 pages, 9 figures; accepted Phys.Rev.

Three regularization models of the Navier-Stokes equations

We determine how the differences in the treatment of the subfilter-scale physics affect the properties of the flow for three closely related regularizations of Navier-Stokes. The consequences on the applicability of the regularizations as SGS models are also shown by examining their effects on superfilter-scale properties. Numerical solutions of the Clark-alpha model are compared to two previously employed regularizations, LANS-alpha and Leray-alpha (at Re ~ 3300, Taylor Re ~ 790) and to a DNS. We derive the Karman-Howarth equation for both the Clark-alpha and Leray-alpha models. We confirm one of two possible scalings resulting from this equation for Clark as well as its associated k^(-1) energy spectrum. At sub-filter scales, Clark-alpha possesses similar total dissipation and characteristic time to reach a statistical turbulent steady-state as Navier-Stokes, but exhibits greater intermittency. As a SGS model, Clark reproduces the energy spectrum and intermittency properties of the DNS. For the Leray model, increasing the filter width decreases the nonlinearity and the effective Re is substantially decreased. Even for the smallest value of alpha studied, Leray-alpha was inadequate as a SGS model. The LANS energy spectrum k^1, consistent with its so-called "rigid bodies," precludes a reproduction of the large-scale energy spectrum of the DNS at high Re while achieving a large reduction in resolution. However, that this same feature reduces its intermittency compared to Clark-alpha (which shares a similar Karman-Howarth equation). Clark is found to be the best approximation for reproducing the total dissipation rate and the energy spectrum at scales larger than alpha, whereas high-order intermittency properties for larger values of alpha are best reproduced by LANS-alpha.Comment: 21 pages, 8 figure

Highly turbulent solutions of LANS-alpha and their LES potential

We compute solutions of the Lagrangian-Averaged Navier-Stokes alpha-model (LANS) for significantly higher Reynolds numbers (up to Re 8300) than have previously been accomplished. This allows sufficient separation of scales to observe a Navier-Stokes (NS) inertial range followed by a 2nd LANS inertial range. The analysis of the third-order structure function scaling supports the predicted l^3 scaling; it corresponds to a k^(-1) scaling of the energy spectrum. The energy spectrum itself shows a different scaling which goes as k^1. This latter spectrum is consistent with the absence of stretching in the sub-filter scales due to the Taylor frozen-in hypothesis employed as a closure in the derivation of LANS. These two scalings are conjectured to coexist in different spatial portions of the flow. The l^3 (E(k) k^(-1)) scaling is subdominant to k^1 in the energy spectrum, but the l^3 scaling is responsible for the direct energy cascade, as no cascade can result from motions with no internal degrees of freedom. We verify the prediction for the size of the LANS attractor resulting from this scaling. From this, we give a methodology either for arriving at grid-independent solutions for LANS, or for obtaining a formulation of a LES optimal in the context of the alpha models. The fully converged grid-independent LANS may not be the best approximation to a direct numerical simulation of the NS equations since the minimum error is a balance between truncation errors and the approximation error due to using LANS instead of the primitive equations. Furthermore, the small-scale behavior of LANS contributes to a reduction of flux at constant energy, leading to a shallower energy spectrum for large alpha. These small-scale features, do not preclude LANS to reproduce correctly the intermittency properties of high Re flow.Comment: 37 pages, 17 figure

Spectral flux and error-landscape of 2D LES

Non UBCUnreviewedAuthor affiliation: Los Alamos National LaboratoryPostdoctora