ASTR/ATOC-5410: Fluid Instabilities, Waves, and Turbulence November 2, 2016, Axel Brandenburg

Handout 15b: Intermittency in turbulence & shell models

Turbulence is intermittent, which means it is bursty. Mathematically, this means different moments scale differently. In turbulence, this applies to velocity differences, δu_{r} = u(x) − u(x + r), not the actual velocities u(x). We skip for now the fact that u is really a vector and demonstrate what it means in practice by studying a simple model of turbulence—a shell model!

1 Shell models

Shell models assume nearest (or next-nearest) neighbor coupling. Let us start with nearest neighbor coupling. So, a toy model of turbulence is the following:

\frac{du_{n}}{dt} = −k_{n} ∑_{i=−1}^{1} ∑_{j=−1}^{1} C_{ij}u_{n−i}u_{n+1} − νk_{n}^{2}u_{n},  (1)

where, for further simplicity, we have assumed that u_{n} is real. The last term on the right corresponds to the dissipative term, and the the first one on the right is a quadratic nonlinearity, representing u · ∇u.

The most important aspect in constraining the coefficients C_{ij} is the assumption that some particular conservation law should be obeyed. Energy conservation is obvious, and this means that

\frac{d}{dt} (\frac{1}{2}u_{n}^{2}) = −u_{n}k_{n} ∑_{i=−1}^{1} ∑_{j=−1}^{1} C_{ij}u_{n−i}u_{n+1} − νk_{n}^{2}u_{n}^{2}.  (2)

By writing out all 9 terms, we see that there are only 2 pairs that can balance, so

\frac{du_{n}}{dt} = A (−u_{n−1}^{2} + 2u_{n}u_{n+1}) + B (−u_{n−1}u_{n} + 2u_{n+1}^{2}) − νk_{n}^{2}u_{n}.  (3)

Remarkably, such a model reproduces the expected Kolmogorv scaling. For logarithmically spaced shells, the energy spectrum is E(k_{n}) = u_{n}^{2}/k_{n}. Furthermore, the energy in each shell fluctuates and exhibits intermittency!

Intermittency is linked to the property that the ζ_{p} deviate from a linear dependence on p. A completely non-intermittent behavior would mean ζ_{p} = p/3. A phenomenological relation that describes the behavior observed in experiments and simulations is given by the She–Leveque relation She & Leveque (1994)

ζ_{p} = \frac{p}{9} + C [1 − (1 − \frac{2/3}{C})^{p/3}] ,  (4)

where C is interpreted as the co-dimension of the dissipative structures. For weakly compressible or incompressible turbulence the dissipative structures are one-dimensional tube-like structures, so the codimension is C = 2. Under compressible conditions the dissipative structures tend to become twodimensional sheet-like structures, so C = 1, which is also borne out by simulations of highly supersonic turbulence. Sheet-like dissipative structures are also expected in hydromagnetic turbulence, where these structures correspond to current sheets. In that case one expects the same scaling as for supersonic turbulence.

References

She, Z.-S., & Leveque, E., “Universal scaling laws in fully developed turbulence,” Phys. Rev. Lett. 72, 336-339 (94).

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