͑ a ͒ H. L. Dryden, the distinguished American aero- dynamicist. His wind-tunnel experiments were a critical influence on Onsager’s theories of 3D turbulence. ͑ b ͒ A smoke visualization of turbulent flow in a wind tunnel, where the grid is at the top and the mean flow is downward. ͑ a ͒ courtesy of the National Archives and Records Administration, and ͑ b ͒ with permission of Thomas Corke and Hassan Nagib. 

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... Z ͑ r ͒ , ͑ ␴ ͒ , are Lagrange multipliers to enforce constraints ͑ 15 ͒ , ͑ 17 ͒ , and ͑ 19 ͒ , respectively. The stream function satisfies the generalized mean-field equation − ⌬␺ ̄ ͑ r ͒ = 1 d ␴␴ exp ͕ − ␤ ̄ ͓ ␴␺ ̄ ͑ r ͒ − ␮ ͑ ␴ ͔͖͒ ; ͑ 21 ͒ see Miller ͑ 1990 ͒ and Robert ͑ 1990 ͒ . This theory is an application to 2D Euler of the method worked out by Lynden-Bell ͑ 1967 ͒ to describe gravitational equilibrium after “violent relaxation” in stellar systems. The Robert-Miller theory solves the problems discussed by Onsager, in the passage quoted above, with respect to the point-vortex assumption. The new theory incorporates infinitely many conservation laws of 2D Euler, although in our opinion that is not the critical difference. In fact, the point-vortex model, in the gener- ality considered by Onsager, also has infinitely many conserved quantities, i.e., the total number of vortices of a given circulation. 8 More importantly, the Robert- Miller theory includes information about the area of the vorticity level sets, which is lacking in the point-vortex model. As remarked by Miller et al. ͑ 1992 ͒ , the Joyce- Montgomery mean-field equation is formally recovered in a “dilute-vorticity limit” in which the area of the level sets shrinks to zero keeping the net circulation fixed. This corresponds well with the conditions suggested by Onsager for the validity of the point-vortex model that “vorticity is mostly concentrated in small regions.” The second main assumption invoked in Onsager’s theory is the ergodicity of the point vortex dynamics. This is a standard assumption invoked in justifying Gibbsian statistical theory. It has, however, proved to be false! Khanin ͑ 1982 ͒ showed that a part of the phase space of the system of N point vortices in the infinite plane consists of integrable tori. His proof used the fact that the three-vortex system is exactly integrable ͑ Novikov, 1975 ͒ . By adding additional vortices successively at further and further distances and using the fact that these additional vortices only weakly perturb the previous system, one can apply Kolmogorov-Arnold-Moser theory iteratively to establish integrability of the N -vortex system. Of course, statistical mechanics does not require strict ergodicity because macroscopic ob- servables are nearly constant over the energy surface. Thus any reasonable mixing over the energy surface will suffice to justify the use of a microcanonical ensemble. Of more serious concern are the possible slow time scales of this mixing. Onsager also worried about this point when he wrote to Lin that: “I still have to find out whether the processes anticipated by these considerations are rapid enough to play a dominant role in the evolution of vortex sheets, and just how the conservation of momentum will modify the conclusions.” Lundgren and Pointin ͑ 1977 ͒ performed numerical simulations of the point-vortex model with initial conditions corresponding to several local clusters of vortices at some distance from each other. The equilibrium theory predicts their final coalescence into a single large super- vortex. Instead, it was found that the clusters individu- ally reach a “local equilibrium,” not coalescing over the time scale of the simulation. Lundgren and Pointin argued theoretically that the vortices will eventually reach the equilibrium, single-vortex state. Similar metastable states of several large vortices have been seen in experiments with magnetically confined, pure electron col- umns and dubbed “vortex crystals” by Fine et al. ͑ 1995 ͒ and Jin and Dubin ͑ 2000 ͒ . These states have been explained by a regional maximum-entropy theory in which entropy is maximized assuming a fixed number of the strong vortices ͑ Jin and Dubin, 1998, 2000 ͒ . Clearly, Onsager’s ergodicity hypothesis is nontrivial and open to question. Despite these caveats, equilibrium theories of large- scale vortices have had some notable successes. Onsager himself considered decaying wake turbulence in an “in- finite vortex trail,” as he wrote to Lin. Indeed pp. 28–31 of Folder 11:129 contain detailed calculations, 9 similar to those in Lamb ͑ 1932 ͒ , Chap. 156, pp. 224 and 225. Analytical solutions of the mean-field Poisson-Boltzmann equation for vortex street geometries were later discovered ͑ Chow et al. , 1998; Kuvshinov and Shep, 2000 ͒ . Final states of freely decaying 2D Navier-Stokes simulations at high Reynolds number, started from fully turbulent initial conditions, have also been found to be in remarkable agreement with the predictions of the Joyce-Montgomery or sinh-Poisson mean-field equation ͑ Montgomery et al. , 1992, 1993 ͒ . Similar simulations started from a single band of vorticity, periodically modulated to induce Kelvin-Helmholtz instability, show good agreement with the generalized Robert-Miller theory ͑ Sommeria et al. , 1991 ͒ . In the limit of a thin initial band, the original Joyce-Montgomery mean-field theory is found to give identical results and agrees well with the simulations. Furthermore, the process is much as Onsager anticipated when he wrote to Lin: “ 1⁄4 the sheet will roll up and possibly contract into concentrated vortices in some places, and at the same time the remaining sections of the sheet will be stretched into feeble, more or less haphazard distributed discontinuities of velocity.” For further comparisons of mean-field equations with results of numerical simulations, see Yin et al. ͑ 2003 ͒ . A number of natural phenomena have been tentatively described by equilibrium vortex models of the sort proposed by Onsager. A fascinating example that was mentioned earlier is the Great Red Spot of Jupiter. For some recent work on this topic, see Turkington et al. ͑ 2001 ͒ and Bouchet and Sommeria ͑ 2002 ͒ . The second half of Onsager ͑ 1949d ͒ , titled “Turbu- lence,” deals with three-dimensional and fully developed turbulence. The second halves of the Pauling and Lin notes also discuss 3D turbulence. The Gibbsian statistical theory discussed in the first half of these documents does not describe a turbulent cascade process. As Onsager wrote at the end of the first section of Onsager ͑ 1949d ͒ on two dimensions, “How soon will the vortices discover that there are three dimensions rather than two? The latter question is important because in three dimensions a mechanism for complete dissipation of all kinetic energy, even without the aid of viscosity, is available.” Of course, it is no surprise that equilibrium statistical mechanics is inapplicable to a dissipative, irreversible process such as turbulence. More startling is Onsager’s conclusion that turbulent motion remains dissipative even in the limit as molecular viscosity tends to zero. In the Pauling note of March 1945, he had already made a similar assertion: “The energy is gradually divided up among ρ degrees of freedom, only for sufficiently large k គ the viscosity disposes of it for good; but it does not seem to matter much just how large this k គ is.” This remark was repeated at greater length in the Lin note of June 1945 as well: “We anticipate a mechanism of dissipation in which the role of the viscosity is altogether second- ary, as suggested by G. I. Taylor: a smaller viscosity is automatically compensated by a reduced micro- scale of the motion, in such a way that most of the vorticity will belong to the micro-motion, but only a small fraction of the energy.” Again, in the abstract of his APS talk in November, he wrote: “In actual liquids this subdivision of energy is in- tercepted by the action of viscosity, which destroys the energy more rapidly the greater the wave number. However, various experiments indicate that the viscosity has a negligible effect on the primary process; hence one may inquire about the laws of turbulent dissipation in an ideal fluid.” For good measure, similar remarks were made no less than four times in the published paper ͑ Onsager, 1949d ͒ . Considering the economy Onsager routinely prized in stating his results, it would appear that explaining the inviscid mechanism of energy dissipation in 3D turbulence was a chief preoccupation of Onsager’s work on statistical hydrodynamics. We can ask what evidence may have pushed Onsager in that direction. One reference in the 1949 paper was Dryden’s review article ͑ Dryden, 1943 ͒ on the statistical theory of turbulence. At the time, Dryden ͑ see Fig. 6 ͒ was a researcher in aerodynamics at the National Bu- reau of Standards in Washington, D.C. Starting in 1929, he published a series of papers on the measurement of turbulence in wind tunnels. A problem he had studied was the decay of nearly homogeneous and isotropic turbulence behind a wire-mesh screen. Dryden used hot- wire anemometry techniques to take accurate measurements of turbulence levels v in the tunnel, where v denotes the velocity fluctuation away from the mean. This permitted him to determine the rate of decay of the turbulent kinetic energy 10 Q as Q = − 1 ͑ d / dt ͒ v 2 , where d / dt denotes the convective derivative. Let V = ͑ v ͒ be the root-mean-square velocity fluctuation and L the spatial correlation length of the velocity, usually called the integral length scale. By simple dimensional ...