Structure functions and cancellation exponent in MHD: DNS and Lagrangian averaged modeling

Pablo D. Mininni1,*

Jonathan Pietarila Graham1, Annick Pouquet1, David Montgomery2, and Darryl Holm3

1. NCAR, Boulder, CO

2. Dartmouth College, Hanover, NH

3. LANL, and Imperial College, UK

* mininni@ucar.edu

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]

RV RM

Earth 1,9 1 109 102

Jupiter 5,3 0,41 1012 106

Sun 104 27 1015 108

Disks 10-2 0,1 1011 1010

Galaxy 10-6 7·1010 109 106 V

MM

M

V

RR

P

LUR

LUR

=

=

=

η

ν

The low PM dynamo problem

• PM « 1 and RV »1, the flow is highly complex and turbulent• If stretching and folding can overcome dissipation, dynamo action takes

place above a critical RM

PM = 510-2

2 j2

Lagrangian averaged MHD and LESLeads to a drastic reduction in the degrees of freedom by the introduction of

smoothing lengths M and V.

From Zhao & Mohseni arXiv:physics/0408113

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, 035112 (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, 035112; PRE 71, 046304 (2005)

MHD and LAMHD simulations

5122 DNS, R = 280 20482 LAMHD, R = 5200 ν = η = 10-3 ν = η = 210-5

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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

ν = η = 1.610-4

• The Kolmogorov wavenumber in the DNS is kK = 332

• In the LAMHD simulations we filter at -1 = 80 (5122 run) and 40 (2562 run)

• A good agreement is observed up to -1

DNS 5122 LAMHD 2562 LAMHD D

jz 0.51 0.07 0.60 0.07 0.60 0.10 0.98

z 0.81 0.15 0.89 0.11 0.85 0.10 0.38

Graham, Mininni, & Pouquet, arXiv:physics/0506088

Cancellation exponent in free decaying runs• Free decaying runs, energy

initially loaded in the ring in Fourier space between k = 1 and 3

ν = η = 1.610-4

• The alpha-model is able to capture the time evolution of the cancellation exponent

• 20482 LAMHD free decaying simulation with R = 5200, ν = η = 210-5

• 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 10242 MHD simulations,

and LAMHD simulations with -1 = 80 (5122 run), 40 (2562 run), and 20 (2562 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, 5122 LAMHD, and two sets of 2562 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.