Exact Resummations in the Theory of Hydrodynamic Turbulence: II A Ladder to Anomalous Scaling

In paper I of this series on fluid turbulence we showed that exact resummations of the perturbative theory of the structure functions of velocity differences result in a finite (order by order) theory. These findings exclude any known perturbative mechanism for anomalous scaling of the velocity structure functions. In this paper we continue to build the theory of turbulence and commence the analysis of nonperturbative effects that form the analytic basis of anomalous scaling. Starting from the Navier-Stokes equations (at high Reynolds number Re) we discuss the simplest examples of the appearance of anomalous exponents in fluid mechanics. These examples are the nonlinear (four-point) Green's function and related quantities. We show that the renormalized perturbation theory for these functions contains ``ladder`` diagrams with (convergent!) logarithmic terms that sum up to anomalous exponents. Using a new sum rule which is derived here we calculate the leading anomalous exponent and show that it is critical in a sense made precise below. This result opens up the possibility of multiscaling of the structure functions with the outer scale of turbulence as the renormalization length. This possibility will be discussed in detail in the concluding paper III of this series.Comment: PRE in press, 15 pages + 21 figures, REVTeX, The Eps files of figures will be FTPed by request to [email protected]

Saturation of Turbulent Drag Reduction in Dilute Polymer Solutions

Drag reduction by polymers in turbulent wall-bounded flows exhibits universal and non-universal aspects. The universal maximal mean velocity profile was explained in a recent theory. The saturation of this profile and the crossover back to the Newtonian plug are non-universal, depending on Reynolds number Re, concentration of polymer $c_p$ and the degree of polymerization $N_p$. We explain the mechanism of saturation stemming from the finiteness of extensibility of the polymers, predict its dependence on $c_p$ and $N$ in the limit of small $c_p$ and large Re, and present the excellent comparison of our predictions to experiments on drag reduction by DNA.Comment: 4 pages, 4 figs., included, PRL, submitte

Fusion Rules in Turbulent Systems with Flux Equilibrium

Fusion rules in turbulence specify the analytic structure of many-point correlation functions of the turbulent field when a group of coordinates coalesce. We show that the existence of flux equilibrium in fully developed turbulent systems combined with a direct cascade induces universal fusion rules. In certain examples these fusion rules suffice to compute the multiscaling exponents exactly, and in other examples they give rise to an infinite number of scaling relations that constrain enormously the structure of the allowed theory.Comment: Submitted to PRL on July 95, 4 pages, REVTe

Parametric Generation of Second Sound by First Sound in Superfluid Helium

We report the first experimental observation of parametric generation of second sound (SS) by first sound (FS) in superfluid helium in a narrow temperature range in the vicinity of $T_\lambda$. The temperature dependence of the threshold FS amplitude is found to be in a good quantitative agreement with the theory suggested long time ago and corrected for a finite geometry. Strong amplitude fluctuations and two types of the SS spectra are observed above the bifurcation. The latter effect is quantitatively explained by the discreteness of the wave vector space and the strong temperature dependence of the SS dissipation length.Comment: 4 pages, 4 postscript figures, REVTE

Cooper pair turbulence in atomic Fermi gases

We investigate the stability of spatially uniform solutions for the collisionless dynamics of a fermionic superfluid. We demonstrate that, if the system size is larger than the superfluid coherence length, the solution characterized by a periodic in time order parameter is unstable with respect to spatial fluctuations. The instability is due to the parametric excitations of pairing modes with opposite momenta. The growth of spatial modulations is suppressed by nonlinear effects resulting in a state characterized by a random superposition of wave packets of the superfluid order parameter. We suggest that this state can be probed by spectroscopic noise measurements.Comment: 4 pages, 2 figure

Inertial- and Dissipation-Range Asymptotics in Fluid Turbulence

We propose and verify a wave-vector-space version of generalized extended self similarity and broaden its applicability to uncover intriguing, universal scaling in the far dissipation range by computing high-order (\leq 20\/) structure functions numerically for: (1) the three-dimensional, incompressible Navier Stokes equation (with and without hyperviscosity); and (2) the GOY shell model for turbulence. Also, in case (2), with Taylor-microscale Reynolds numbers 4 \times 10^{4} \leq Re_{\lambda} \leq 3 \times 10^{6}\/, we find that the inertial-range exponents (\zeta_{p}\/) of the order - p\/ structure functions do not approach their Kolmogorov value p/3\/ as Re_{\lambda}\/ increases.Comment: RevTeX file, with six postscript figures. epsf.tex macro is used for figure insertion. Packaged using the 'uufiles' utilit

Normal and Anomalous Scaling of the Fourth-Order Correlation Function of a Randomly Advected Passive Scalar

For a delta-correlated velocity field, simultaneous correlation functions of a passive scalar satisfy closed equations. We analyze the equation for the four-point function. To describe a solution completely, one has to solve the matching problems at the scale of the source and at the diffusion scale. We solve both the matching problems and thus find the dependence of the four-point correlation function on the diffusion and pumping scale for large space dimensionality $d$. It is shown that anomalous scaling appears in the first order of $1/d$ perturbation theory. Anomalous dimensions are found analytically both for the scalar field and for it's derivatives, in particular, for the dissipation field.Comment: 19 pages, RevTex 3.0, Submitted to Phys.Rev. E, revised versio

Measurement of the Electric and Magnetic Polarizabilities of the Proton

The Compton scattering cross section on the proton has been measured at laboratory angles of 90$^\circ$ and 135$^\circ$ using tagged photons in the energy range 70--100 MeV and simultaneously using untagged photons in the range 100--148~MeV. With the aid of dispersion relations, these cross sections were used to extract the electric and magnetic polarizabilities, $\bar{\alpha}$ and $\bar{\beta}$ respectively, of the proton. We find $$\bar{\alpha}+\bar{\beta} = ( 15.0 \pm 2.9 \pm 1.1 \pm 0.4 ) \times 10^{-4} \: {\rm fm}^3,$$ in agreement with a model-independent dispersion sum rule, and $$\bar{\alpha}-\bar{\beta} = ( 10.8 \pm 1.1 \pm 1.4 \pm 1.0 ) \times 10^{-4} \: {\rm fm}^3,$$ where the errors shown are statistical, systematic, and model-dependent, respectively. A comparison with previous experiments is given and global values for the polarizabilities are extracted.Comment: 35 pages, 11 PostScript figures, uses RevTex 3.