Generalized flows, intrinsic stochasticity, and turbulent transport

The study of passive scalar transport in a turbulent velocity field leads naturally to the notion of generalized flows, which are families of probability distributions on the space of solutions to the associated ordinary differential equations which no longer satisfy the uniqueness theorem for ordinary differential equations. Two most natural regularizations of this problem, namely the regularization via adding small molecular diffusion and the regularization via smoothing out the velocity field, are considered. White-in-time random velocity fields are used as an example to examine the variety of phenomena that take place when the velocity field is not spatially regular. Three different regimes, characterized by their degrees of compressibility, are isolated in the parameter space. In the regime of intermediate compressibility, the two different regularizations give rise to two different scaling behaviors for the structure functions of the passive scalar. Physically, this means that the scaling depends on Prandtl number. In the other two regimes, the two different regularizations give rise to the same generalized flows even though the sense of convergence can be very different. The “one force, one solution” principle is established for the scalar field in the weakly compressible regime, and for the difference of the scalar in the strongly compressible regime, which is the regime of inverse cascade. Existence and uniqueness of an invariant measure are also proved in these regimes when the transport equation is suitably forced. Finally incomplete self similarity in the sense of Barenblatt and Chorin is established.

Small viscosity asymptotics for the inertial range of local structure and for the wall region of wall-bounded turbulent shear flow.

The small viscosity asymptotics of the inertial range of local structure and of the wall region in wallbounded turbulent shear flow are compared. The comparison leads to a sharpening of the dichotomy between Reynolds number dependent scaling (power-type) laws and the universal Reynolds number independent logarithmic law in wall turbulence. It further leads to a quantitative prediction of an essential difference between them, which is confirmed by the results of a recent experimental investigation. These results lend support to recent work on the zero viscosity limit of the inertial range in turbulence.

Successful external fertilization in turbulent environments.

Mathematical and experimental simulations predict that external fertilization is unsuccessful in habitats characterized by high water motion. A key assumption of such predictions is that gametes are released in hydrodynamic regimes that quickly dilute gametes. We used fucoid seaweeds to examine whether marine organisms in intertidal and subtidal habitats might achieve high levels of fertilization by restricting their release of gametes to calm intervals. Fucus vesiculosus L. (Baltic Sea) released high numbers of gametes only when maximal water velocities were below ca. 0.2 m/s immediately prior to natural periods of release, which occur in early evening in association with lunar cues. Natural fertilization success measured at two sites was always close to 100%. Laboratory experiments confirmed that (i) high water motion inhibits gamete release by F. vesiculosus and by the intertidal fucoids Fucus distichus L. (Maine) and Pelvetia fastigiata (J. Ag.) DeToni (California), and (ii) showed that photosynthesis is required for high gamete release. These data suggest that chemical changes in the boundary layer surrounding adults during photosynthesis and/or mechanosensitive channels may modulate gamete release in response to changing hydrodynamic conditions. Therefore, sensitivity to environmental factors can lead to successful external fertilization, even for species living in turbulent habitats.

Turbulent diffusion phase transition is due to singular energy spectrum.

The phase transition for turbulent diffusion, reported by Avellaneda and Majda [Avellaneda, M. & Majda, A. J. (1994) Philos. Trans. R. Soc. London A 346, 205-233, and several earlier papers], is traced to a modeling assumption in which the energy spectrum of the turbulent fluid is singularly dependent on the viscosity in the inertial range. Phenomenological models of turbulence and intermittency, by contrast, require that the energy spectrum be independent of the viscosity in the inertial range. When the energy spectrum is assumed to be consistent with the phenomenological models, there is no phase transition for turbulent diffusion.