Ralf W. Wittenberg

Low-dimensional models of coherent structures in turbulence

Abstract For fluid flow one has a well-accepted mathematical model: the Navier-Stokes equations. Why, then, is the problem of turbulence so intractable? One major difficulty is that the equations appear insoluble in any reasonable sense. (A direct numerical simulation certainly yields a “solution”, but it provides little understanding of the process per se .) However, three developments are beginning to bear fruit: (1) The discovery, by experimental fluid mechanicians, of coherent structures in certain fully developed turbulent flows; (2) the suggestion, by Ruelle, Takens and others, that strange attractors and other ideas from dynamical systems theory might play a role in the analysis of the governing equations, and (3) the introduction of the statistical technique of Karhunen-Loeve or proper orthogonal decomposition, by Lumley in the case of turbulence. Drawing on work on modeling the dynamics of coherent structures in turbulent flows done over the past ten years, and concentrating on the near-wall region of the fully developed boundary layer, we describe how these three threads can be drawn together to weave low-dimensional models which yield new qualitative understanding. We focus on low wave number phenomena of turbulence generation, appealing to simple, conventional modeling of inertial range transport and energy dissipation.

Dissipativity, analyticity and viscous shocks in the (de)stabilized Kuramoto-Sivashinsky equation

Abstract We investigate an extension of the Kuramoto–Sivashinsky (KS) model for complex spatiotemporal dynamics, containing an additional linear stabilizing or destabilizing term. Generalizing previous results, we prove dissipativity and analyticity. In the destabilized case, a stable, attracting shock-like transition layer solution is observed numerically, and we obtain its asymptotic scaling. This “viscous shock” solution does not exhibit extensive scaling, which sheds light on the difficulties in obtaining optimal bounds and proving extensivity for the KS equation.

A rigorous bound on the vertical transport of heat in Rayleigh-Bénard convection at infinite Prandtl number with mixed thermal boundary conditions

A rigorous upper bound on the Nusselt number is derived for infinite Prandtl number Rayleigh-Benard convection for a fluid constrained between no-slip, mixed thermal vertical boundaries. The result suggests that the thermal boundary condition does not affect the qualitative nature of the heat transport. The bound is obtained with the use of a nonlinear, stably stratified background temperature profile in the bulk, notwithstanding the lack of boundary control of the temperature due to the Robin boundary conditions.

Changing Risk Behaviours and the HIV Epidemic: A Mathematical Analysis in the Context of Treatment as Prevention

Background Expanding access to highly active antiretroviral therapy (HAART) has become an important approach to HIV prevention in recent years. Previous studies suggest that concomitant changes in risk behaviours may either help or hinder programs that use a Treatment as Prevention strategy. Analysis We consider HIV-related risk behaviour as a social contagion in a deterministic compartmental model, which treats risk behaviour and HIV infection as linked processes, where acquiring risk behaviour is a prerequisite for contracting HIV. The equilibrium behaviour of the model is analysed to determine epidemic outcomes under conditions of expanding HAART coverage along with risk behaviours that change with HAART coverage. We determined the potential impact of changes in risk behaviour on the outcomes of Treatment as Prevention strategies. Model results show that HIV incidence and prevalence decline only above threshold levels of HAART coverage, which depends strongly on risk behaviour parameter values. Expanding HAART coverage with simultaneous reduction in risk behaviour act synergistically to accelerate the drop in HIV incidence and prevalence. Above the thresholds, additional HAART coverage is always sufficient to reverse the impact of HAART optimism on incidence and prevalence. Applying the model to an HIV epidemic in Vancouver, Canada, showed no evidence of HAART optimism in that setting. Conclusions Our results suggest that Treatment as Prevention has significant potential for controlling the HIV epidemic once HAART coverage reaches a threshold. Furthermore, expanding HAART coverage combined with interventions targeting risk behaviours amplify the preventive impact, potentially driving the HIV epidemic to elimination.

Spatially Localized Models of Extended Systems

We investigate the construction of low-dimensional spatially localized models of extended systems. Specifically, the Kuramoto—Sivashinsky (KS) equation on large one-dimensional domains displays spatiotemporally complex dynamics that are remarkably well-localized in both real and Fourier space, as demonstrated by a (spline) wavelet representation. We show how wavelet projections may be used to construct various localized, relatively low-dimensional models of KS spatiotemporal chaos. There is persuasive evidence that short, periodized systems, internally forced at their largest scales, form minimal models for chaotic dynamics in arbitrarily large domains. Such models assist in the understanding of extended systems.

Matching models of HIV-1 viral dynamics to clinical data

Creating individualized within-host multiple phase disease models of HIV-1 infection has long been a goal of mathematicians and biologists. The challenge is in trying to build models that are representative of the disease, include realistic parameter estimates, and are able to incorporate a changing model structure. In this paper, we propose a fitting procedure, motivated by the biology of the disease, for matching parameters of differential equation and stochastic models of HIV-1 infection to data, which leverages high performance computing resources. The search uses knowledge of the biological set points to restrict the search domain, and parallel simulated annealing to match the model to acute and early chronic phase patient data. We highlight this method by finding parameters for two interconnected models of HIV-1 infection which we have developed. The high quality of our data allows us to model not only viral data, but also CD4 count data through the acute and chronic phases of the disease. The time span of our model exceeds that of previous models. The algorithm is able to find parameter values for four patients consistent with literature ranges and display individual set point equilibration and disease progression for both clinical markers.