Mixing with applications to inertial-confinement-fusion implosions

Approximate one-dimensional (1D) as well as 2D and 3D simulations are playing an important supporting role in the design and analysis of future experiments at National Ignition Facility. This paper is mainly concerned with 1D simulations, used extensively in design and optimization. We couple a 1D buoyancy-drag mix model for the mixing zone edges with a 1D inertial confinement fusion simulation code. This analysis predicts that National Ignition Campaign (NIC) designs are located close to a performance cliff, so modeling errors, design features (fill tube and tent) and additional, unmodeled instabilities could lead to significant levels of mix. The performance cliff we identify is associated with multimode plastic ablator (CH) mix into the hot-spot deuterium and tritium (DT). The buoyancy-drag mix model is mode number independent and selects implicitly a range of maximum growth modes. Our main conclusion is that single effect instabilities are predicted not to lead to hot-spot mix, while combined mode mixing effects are predicted to affect hot-spot thermodynamics and possibly hot-spot mix. Combined with the stagnation Rayleigh-Taylor instability, we find the potential for mix effects in combination with the ice-to-gas DT boundary, numerical effects of Eulerian species CH concentration diffusion, and ablation-driven instabilities. With the help of a convenient package of plasma transport parameters developed here, we give an approximate determination of these quantities in the regime relevant to the NIC experiments, while ruling out a variety of mix possibilities. Plasma transport parameters affect the 1D buoyancy-drag mix model primarily through its phenomenological drag coefficient as well as the 1D hydro model to which the buoyancy-drag equation is coupled.

Dynamical evolution of Rayleigh-Taylor and Richtmyer-Meshkov mixing fronts

Dynamic behavior of mixing fronts plays a crucial role in multifluid turbulent mixing. In this paper, we derive an analytic solution for the dynamic evolution of mixing fronts driven by constant acceleration Rayleigh-Taylor (RT) and impulsive acceleration Richtmyer-Meshkov instabilities, from a simple physics model expressed as a pair of ordinary differential equations. An approximate closed form asymptotic evaluation of the RT solution is obtained, through terms of order O(1), as t--> infinity. This three term expansion, including lower order terms, is used to interpret experimental and simulation data. Our solutions improve on previous analyses in their agreement with experimental data, in that we can fit both the slope and the intercept of the Z(b) vs Agt(2) experimental plots by adjusting parameters in our model. Since the experimental data are close to self-similar, the improvement due to the lower order contributions in the asymptotic expansion is modest. We also apply this analysis to simulation data, for which preasymptotic data exist. We reexamine previous simulation data and determine an improved growth rate alpha(b)=0.0625. The present paper provides concepts and tools to explore the preasymptotic aspects of these data.

A three-dimensional renormalization group bubble merger model for Rayleigh-Taylor mixing

In this paper we formulate a model for the merger of bubbles at the edge of an unstable acceleration driven (Rayleigh-Taylor) mixing layer. Steady acceleration defines a self-similar mixing process, with a time-dependent inverse cascade of structures of increasing size. The time evolution is itself a renormalization group (RNG) evolution, and so the large time asymptotics define a RNG fixed point. We solve the model introduced here at this fixed point. The model predicts the growth rate of a Rayleigh-Taylor chaotic fluid mixing layer. The model has three main components: the velocity of a single bubble in this unstable flow regime, an envelope velocity, which describes collective excitations in the mixing region, and a merger process, which drives an inverse cascade, with a steady increase of bubble size. The present model differs from an earlier two-dimensional (2-D) merger model in several important ways. Beyond the extension of the model to three dimensions, the present model contains one phenomenological parameter, the variance of the bubble radii at fixed time. The model also predicts several experimental numbers: the bubble mixing rate, alpha(b)=h(b)/Agt(2) approximately 0.05-0.06, the mean bubble radius, and the bubble height separation at the time of merger. From these we also obtain the bubble height to the radius aspect ratio. Using the experimental results of Smeeton and Youngs (AWE Report No. O 35/87, 1987) to fix a value for the radius variance, we determine alpha(b) within the range of experimental uncertainty. We also obtain the experimental values for the bubble height to width aspect ratio in agreement with experimental values. (c) 2002 American Institute of Physics.

Stripe forming architecture of the gap gene system

In this report, we show that gap genes encode exactly one set of pair-rule stripes, which occur in the native even-skipped position. The core of this work is a detailed analysis that shows how this conclusion follows from the arrangement of gap domains in the embryo. This analysis shows that: (1) pattern forming information is transmitted from gap to pair-rule genes by means of a nonredundant set of morphogenetic gradients, and (2) the stripe forming capability of the gap genes is constrained by the arrangement of these gradients and by the fact that each gap domain consists of a pair of correlated gradients. We also show that in the blastoderm, the regulatory sign of a transcriptional regulator is unlikely to change in a concentration dependent manner. The principal analytic tool used to establish these results is the gene circuit method. Here, this method is applied to examine hybrid data sets consisting of real gene expression data for four gap genes and hypothetical pair-rule expression data generated by translating native even-skipped data along the anterior-posterior axis. In this way, we are able to investigate the stripe forming capabilities of the gap gene system in the complete absence of pair-rule cross regulation. We close with an inference about evolutionary development. It is argued that the constraints on gap gene architecture identified here are a consequence of selective pressures that minimize the number of gap genes required to determine segments in long-germ band insects.

Automated assay of gene expression at cellular resolution

We have recently developed an automated image processing method for obtaining quantitative values for average levels of gene expression at the resolution of a single cell. This method is described in the present paper. We place this method within a larger framework for the study of gene regulation in Drosophila, stressing that gene circuit models and improved data processing methods are mutually reinforcing approaches to this problem.

Prediction of mutant expression patterns using gene circuits

Networks of interacting transcription factors, or gene circuits, form an essential part of the metabolic pathways controlling macromolecular synthesis. This paper conveys two new results about gene circuits. We first show how a gene circuit for mutant phenotypes can be constructed from the wild type gene circuit for the same organism. We then present results of computational studies that show that mutant expression patterns can be correctly predicted using gene circuits whose parameters have been determined from wild type data only. Further computational studies demonstrate that this property is insensitive to errors as large as a factor of two in the input data. Together, these results show that gene circuits can be used to identify the regulatory mechanisms governing an entire family of genotypes from a knowledge of the wild type genotype alone. It is argued that this fact forms the basis for a new paradigm in genetics.

Mechanism of eve stripe formation

In this paper we analyze the formation of stripes of expression of the pair-rule gene eve. We identify detailed mechanisms which control the formation of stripes 2-5. Each stripe is formed as a result of generalized activation by bcd and ubiquitous transcription factors combined with localized repression by gap genes. Each of the eight stripe borders of these four stripes is shown to be under the control of a particular gap gene expression domain. Protein synthesis from eve and its controlling gap genes begins at the same time, but localized eve expression is substantially delayed relative to localized expression of gap domains. We show that this delay results from a change in the spatial balance between activation and repression due to the intensification and refinement of gap domains during cleavage cycle 14. eve stripe formation is ordered in time; stripe 2 appears earlier than stripes 3-5. We show that this happens because the formation of stripe 2 is less dependent on gap domain refinement than is the case for stripes 3-5: Each of stripes 3-5 is controlled by a pair of overlapping gap domains, whereas stripe 2 is controlled by a disjoint pair of gap domains. Finally, we observe that eve stripes do not form unless Eve protein has an extremely small diffusivity, and argue that this low diffusivity is a result of the apical localization of pair-rule message. This implies that localization of pair-rule message is required for stripe formation. The essential tool used to obtain these results is the method of gene circuits, which is a new approach to the analysis of gene expression data. Its purpose is to provide a way to use this data to infer how concentrations of products of a given gene change with time and how these changes are influenced by the activating or repressing effects of the products of other genes. The gene circuit method is based on three main ideas, explained in the paper. First is the choice of protein concentrations as state variables for the description of gene regulation. Second is the summary of chemical reaction kinetics by coarse-grained rate equations for protein concentrations. Third is the use of least squares fits to gene expression data to measure phenomenological parameters occurring in the gene circuit.

Model for cooperative control of positional information in Drosophila by bicoid and maternal hunchback

The blastoderm of the fruit fly Drosophila melanogaster is unusually well suited for analysis of fundamental questions in animal development. One such question is how genes specify the positional information which determines the developmental pathways (fate) of cells at appropriate spatial locations. In this paper we propose a dynamical model of gene regulation which explicitly describes how positional information is used in the blastoderm. The model is applied to analyze important experimental findings on the dependence of cell fate on the concentration of the Bicoid morphogen. The model shows that positional information in the presumptive middle body is cooperatively determined by maternal products of the bicoid and hunchback genes.

A connectionist model of development

We present a phenomenological modeling framework for development. Our purpose is to provide a systematic method for discovering and expressing correlations in experimental data on gene expression and other developmental processes. The modeling framework is based on a connectionist or "neural net" dynamics for biochemical regulators, coupled to "grammatical rules" which describe certain features of the birth, growth, and death of cells, synapses and other biological entities. We outline how spatial geometry can be included, although this part of the model is not complete. As an example of the application of our results to a specific biological system, we show in detail how to derive a rigorously testable model of the network of segmentation genes operating in the blastoderm of Drosophila. To further illustrate our methods, we sketch how they could be applied to two other important developmental processes: cell cycle control and cell-cell induction. We also present a simple biochemical model leading to our assumed connectionist dynamics which shows that the dynamics used is at least compatible with known chemical mechanisms.

Neural nets

The brain is one of the most highly organized structures in the known universe. It is a biological computer which has evolved over a billion years to program, monitor and control all bodily functions. It is also the organ of knowing, feeling and thinking. To understand how the brain works is perhaps the most difficult of all scientific problems. A scientific theory of the brain would provide a comprehensive understanding of a substantial body of facts on the basis of a few fundamental principles. In this sense, there is no theory of the brain.

Perspectives on Supercomputing

This article provides a brief look at the current status of supercomputers and supercomputing in the United States. It addresses a variety of applications of supercomputers and the characteristics of a large modern supercomputing facility, the radical changes in the design of supercomputers that are impending, and the conditions that are necessary for a conducive climate for the further development and application of supercomputers.