Structure functions and cancellation exponent in MHD: DNS and Lagrangian averaged modeling Pablo D. Mininni 1,* Jonathan Pietarila Graham 1, Annick Pouquet 1, David Montgomery 2, and Darryl Holm 3 1.NCAR, Boulder, CO 2.Dartmouth College, Hanover, NH 3.LANL, and Imperial College, UK *

Magnetic fields in astrophysics Generation of magnetic fields occurs in media for which the viscosity and the magnetic diffusivity are vastly different. B [Gauss] T [days] RVRV RMRM Earth1, Jupiter5,30, Sun Disks , Galaxy ·

The low P M dynamo problem P M « 1 and R V »1, the flow is highly complex and turbulent If stretching and folding can overcome dissipation, dynamo action takes place above a critical R M P M = 5  22 j2j2

Lagrangian averaged MHD and LES Leads to a drastic reduction in the degrees of freedom by the introduction of smoothing lengths  M and  V. From Zhao & Mohseni arXiv:physics/

Lagrangian averaged MHD (  -model) The fields are written as the sum of filtered and fluctuating components The velocity and magnetic field are smoothed, but not the fields’ sources The 3D invariants are Holm, Chaos 12, 518 (2002); Mininni, Montgomery, & Pouquet, Phys. Fluids 1, (2005)

Lagrangian averaged MHD (  -model) Tested against four MHD problems: selective decay, dynamic alignment, inverse cascade, and dynamo action. The growth rate of the inverse cascade is wrong in 2D, but works in 3D.

Lagrangian averaged MHD (  -model) Structures in LAMHD are thicker due to the introduction of the filtering length . The model correctly captures the spectral behavior up to the wavenumner  -1. The tails in the PDFs are captured by the model. Mininni, Montgomery, & Pouquet, Phys. Fluids 1, ; PRE 71, (2005)

MHD and LAMHD simulations DNS, R = LAMHD, R = 5200 =  = =  = 2  10 -5

Cancellation exponent Measures fast oscillations in sign on arbitrary small scales Is also a measure of sign singularity Given a component of the field (e.g. the z component of the current density), we introduce a signed measure on a set Q(L) of size L as Then we define the partition function measuring cancellations at a given lengthscale l We study the scaling behavior of the cancellation: A positive exponent indicates fast changes in sign on small scales. A smooth field has  = 0. We can also define a fractal dimension of the turbulent structures as

Cancellation exponent From Sorriso-Valvo, Carbone, Noullez, Politano, Pouquet, & Veltri, Phys. Plasmas 9, 89 (2002).

Cancellation exponent in forced runs Random forcing at k=1 and 2 =  = 1.6  The Kolmogorov wavenumber in the DNS is k K = 332 In the LAMHD simulations we filter at  -1 = 80 (512 2 run) and 40 (256 2 run) A good agreement is observed up to  -1 DNS512 2 LAMHD256 2 LAMHDD jzjz 0.51    zz 0.81    Graham, Mininni, & Pouquet, arXiv:physics/

Cancellation exponent in free decaying runs Free decaying runs, energy initially loaded in the ring in Fourier space between k = 1 and 3 =  = 1.6  The alpha-model is able to capture the time evolution of the cancellation exponent LAMHD free decaying simulation with R = 5200, =  = 2  Cancellations at small scales are persistent, even several turnover times after the peak in the dissipation

Structure functions For a component of a field f we define the structure functions of order p as where the increment is given by If the flow is self similar we expect a behavior IK assumes and K41 In MHD the data can be fitted by the generalized She- Leveque formula

Results from simulations Forced MHD simulations, and LAMHD simulations with  -1 = 80 (512 2 run), 40 (256 2 run), and 20 (256 2 run) In MHD we have an exact result in the inertial range where z  = u  B are the Elsasser variables In LAMHD we replace by

Results from simulations Unfiltered results Filtered results Ensamble average using four simulations (DNS, LAMHD, and two sets of LAMHD), 50 turnover times in each simulation

Results from simulations

Conclusions The model works well down to the cut-off length. The alpha-model is able to reproduce the statistical properties of the large-scale energy spectra. Although structures in LAMHD are thicker due to the introduction of the filtering length , the LAMHD simulations correctly capture the behavior of the cancellation exponent in forced and free decaying turbulence. An equivalent of the Karman-Howarth theorem exists for LAMHD. Intermittency in the inertial range (as reflected by the anomalous scaling of the structure functions) is present in the alpha-model. The values of the  p exponents are within the DNS error bars up to p = 8 for z + and B, and up to p = 6 for u.