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.

Grass genomes

For the most part, studies of grass genome structure have been limited to the generation of whole-genome genetic maps or the fine structure and sequence analysis of single genes or gene clusters. We have investigated large contiguous segments of the genomes of maize, sorghum, and rice, primarily focusing on intergenic spaces. Our data indicate that much (>50%) of the maize genome is composed of interspersed repetitive DNAs, primarily nested retrotransposons that insert between genes. These retroelements are less abundant in smaller genome plants, including rice and sorghum. Although 5- to 200-kb blocks of methylated, presumably heterochromatic, retrotransposons flank most maize genes, rice and sorghum genes are often adjacent. Similar genes are commonly found in the same relative chromosomal locations and orientations in each of these three species, although there are numerous exceptions to this collinearity (i.e., rearrangements) that can be detected at the levels of both the recombinational map and cloned DNA. Evolutionarily conserved sequences are largely confined to genes and their regulatory elements. Our results indicate that a knowledge of grass genome structure will be a useful tool for gene discovery and isolation, but the general rules and biological significance of grass genome organization remain to be determined. Moreover, the nature and frequency of exceptions to the general patterns of grass genome structure and collinearity are still largely unknown and will require extensive further investigation.

Slip complexity in earthquake fault models.

We summarize studies of earthquake fault models that give rise to slip complexities like those in natural earthquakes. For models of smooth faults between elastically deformable continua, it is critical that the friction laws involve a characteristic distance for slip weakening or evolution of surface state. That results in a finite nucleation size, or coherent slip patch size, h*. Models of smooth faults, using numerical cell size properly small compared to h*, show periodic response or complex and apparently chaotic histories of large events but have not been found to show small event complexity like the self-similar (power law) Gutenberg-Richter frequency-size statistics. This conclusion is supported in the present paper by fully inertial elastodynamic modeling of earthquake sequences. In contrast, some models of locally heterogeneous faults with quasi-independent fault segments, represented approximately by simulations with cell size larger than h* so that the model becomes "inherently discrete," do show small event complexity of the Gutenberg-Richter type. Models based on classical friction laws without a weakening length scale or for which the numerical procedure imposes an abrupt strength drop at the onset of slip have h* = 0 and hence always fall into the inherently discrete class. We suggest that the small-event complexity that some such models show will not survive regularization of the constitutive description, by inclusion of an appropriate length scale leading to a finite h*, and a corresponding reduction of numerical grid size.