Real-World Turbulence as Dissipative Euler Solutions: A Physics Perspective

Gregory Eyink
Johns Hopkins University

Since Onsager’s proposal in 1949 that turbulent velocity fields at high Reynolds number
may be considered as dissipative weak solutions of the Euler equations, there has been extensive
work in the mathematics community, but almost no exploitation of the theory by physicists. This seems
to be due to the fact that its physical meaning remains obscure to most fluids scientists. However, as
we show here, Onsager’s theory can be understood most intuitively as an application of the concept
of “renormalization-group invariance” to the problem of explaining the available experimental data on
anomalous energy dissipation. Such anomalies imply diverging velocity gradients in the inviscid limit,
or a ``violet catastrophe’’ in Onsager’s own words. Regularizing this ultraviolet divergence, e.g. by a
spatial coarse-graining at length-scale ??, leads to a description of turbulent velocities as “coarse-grained
Euler solutions” at scales ?? in the inertial range. As we show, this notion of “coarse-grained solution’'
is equivalent to the mathematical concept of “weak solution” when the length-scale ?? can be taken
arbitrarily small; it also underlies the practical turbulence modeling method of Large-Eddy Simulation.
Since spatial coarse-graining is a purely passive operation (“removing one’s spectacles”), no physics
can depend upon the arbitrary scale ??. A consequence of this arbitrariness is Onsager’s experimentally
confirmed prediction of (near) singularities with Hölder exponent h<1/3. The fecundity of Onager’s theory
is then demonstrated by extending it to special-relativistic turbulence described by dissipative fluid models,
such as the Israel-Stewart class. It is shown that a dissipative anomaly can appear in the internal energy
balance, which is due both to forward energy cascade and a new compressive anomaly called “pressure-work
defect”. Furthermore, there is a forward cascade of negentropy which is fed by the pressure-work defect,
and which leads to an even more fundamental anomaly in the entropy balance. A renormalization-group
invariance argument like Onsager’s establishes the fluid singularities required to sustain such cascades.
An interesting feature of the special-relativistic case is that there is no possible space-time coarse-graining
which preserves Lorentz symmetry, but, similar to lattice quantum field-theory, the symmetry is restored in
the limit ???0. Thus cascade rates can be to some extent frame-dependent at finite Reynolds numbers.

This talk is based on joint work with Theodore D. Drivas.

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