Björn Birnir

Excitonic Dynamical Franz-Keldysh Effect

The dynamical Franz-Keldysh effect is exposed by exploring near-band-gap absorption in the presence of intense THz electric fields. It bridges the gap between the dc Franz-Keldysh effect and multiphoton absorption and competes with the THz ac Stark effect in shifting the energy of the excitonic resonance. A theoretical model which includes the strong THz field nonperturbatively via a nonequilibrium Green functions technique is able to describe the dynamical Franz-Keldysh effect in the presence of excitonic absorption. [S0031-9007(98)06611-3]

Towards an elementary theory of drainage basin evolution: I. the theoretical basis

Abstract In this and an accompanying paper, we show that a relatively simple family of continuous models provides a satisfactory, albeit elementary, theory of the evolution of fluvial landscapes in terms of (1) the emergence of channelized flows; (2) the development of stable surfaces with ridges and valleys; (3) the decline of the surfaces; (4) relationships between surface forms and surface flows; and (5) environmentally caused landform variability. In this paper, we provide a theoretical basis for modeling the evolution of drainage basins in terms of a set of PDEs that generalize the model of Smith and Bretherton, in part with conditions specifying a freewater surface. We show that under boundary conditions involving a fixed base level, there exist solutions to the model that converge towards the satisfaction of an optimality principle by which “mature” surface forms evolve to minimize a function of the sediment flux over the surface, subject to two constraints. We show that self-organizing drainage surfaces determined by this principle are “separable” in terms of their spatial and temporal components and possess a characteristic set of observable geomorphic properties. These properties include self-similar decline and a related “law of height-proportional erosion” by which the erosion in any surface patch is proportional to the average elevation of the patch above the base level. The equations governing this optimality principle are equivalent to those of Smith and Bretherton but, in contrast to the severe instability that solutions to these equations exhibit when used to model the evolution of channelized flows from initially planar surfaces, we show that certain classes of non-planar surfaces provide stable solutions to the equations. In particular, solutions are stable if the direction of the surface gradient at the boundary of a drainage basin is not too closely aligned with the direction of the boundary. This condition implies that stable solutions possess “valleys”. We also present the results of a linearized stability analysis of solutions to a sub-family of our equations in order to prepare the basis for an evaluation of such approaches in modeling the emergence of channelized flows. In the second paper, we examine numerical solutions to our equations, showing that they provide realistic characterizations of drainage basin evolution in accordance with our theory, and show the inadequacy of linearized analyses as models of initial channel growth.

Discrete and continuous models of the dynamics of pelagic fish: Application to the capelin

In this paper, we study simulations of the schooling and swarming behavior of a mathematical model for the motion of pelagic fish. We use a derivative of a discrete model of interacting particles originated by Vicsek, Czir´ok et al. [6] [5] [23] [24]. Recently, a system of ODEs was derived from this model [2], and using these ODEs, we find transitory and long-term behavior of the discrete system. In particular, we numerically find stationary, migratory, and circling behavior in both the discrete and the ODE model and two types of swarming behavior in the discrete model. The migratory solutions are numerically stable and the circling solutions are metastable. We find a stable circulating ring solution of the discrete system where the fish travel in opposite directions within an annulus. We also find the origin of noise-driven swarming when repulsion and attraction are absent and the fish interact solely via orientation.

Transient attractors: towards a theory of the graded stream for alluvial and bedrock channels

Abstract We construct a theory in which we interpret stable, self-similar, intermediate asymptotic solutions to PDEs representing conservation of mass over stream long profiles as graded streams . The theory applies to alluvial channels for which the transport of sediment is modeled by q s ∝ q γ w S δ , where q w is the discharge of water and S the slope, and to bedrock channels for which the transport of sediment is modeled by ( ∂q s / ∂x )∝ q γ w S δ , where x is the distance downstream. The parameters α and β of these similarity solutions, z ( x, t )= τ α F ( x / τ β ), where τ=τ 0 + ωt and t is time, may be derived using dimensional analysis and are representable in terms of conditions of sediment removal at the lower boundary of the profile and ( γ, δ ). Conditions on the physical realizability of profiles lead to constraints on admissible values of ( γ, δ, α, β, ω ). All alluvial self-similar profiles and an important subset of bedrock self-similar profiles are stable and act as transient attractors when boundary conditions are unchanging. Their forms are independent of the details of their initial conditions. The remaining bedrock channels are unstable because of the spontaneous emergence of shocks that migrate upstream as breaks in slope. Two regimes of profile behavior exist: for ω ω >0, corresponding to low energy environments, profiles have decelerating relative loss rates and infinite life. Changes in profile elevations over time may be decomposed into upper boundary and lower boundary effects controlled, respectively, by ωα and ωβ and depending ultimately on γ and δ . We examine explicit profiles for geomorphically important sets of boundary conditions: desert mountain/pediment ( α + β =0); hanging valley ( α =0); fixed Davisian base-level ( β =0); and steady state ( α =1), for alluvial and bedrock channels and various tectonic conditions. We investigate numerically the effects of changes from hanging valley to fixed base level boundary conditions, showing that profiles associated with the hanging valley attractor retreat upstream as profiles associated with the fixed based level attractor replace them from below. We demonstrate numerically the extension of the theory to streams with tributaries. The theory provides the natural definition of grade for our models, and indicates that it is a real and deep property of the associated profiles. The theory also resolves many contentious issues concerning the concept of grade. We conjecture that defining grade in terms of stable, self-similar, intermediate asymptotic solutions to conservation equations generalizes over reasonable extensions to the transport conditions of our models.

Towards an elementary theory of drainage basin evolution: II. a computational evaluation

Abstract Numerical solutions to a recently-introduced family of continuous models provide realistic representations of the evolution of fluvial landscapes. The simplest subfamily of the models offers a characterization of the evolution of “badlands” as a process involving (1) a first, transient stage in which branching valleys emerge from unchanneled surfaces; (2) a second, “equilibrium” stage in which a fully-developed surface with branching valleys and ridges declines in a stable, self-similar mode; and (3) a final, dissipative stage in which regularities in the landscape break down. In the transient stage of development, small perturbations to the surface induce local variations in water flow, differential erosion, and the rapid emergence of a coherent, fine-scale structure of channelized flow patterns. The small-scale features evolve into larger scale features by a process in which small flows intersect and grow. Standard linearized analyses of the equations are inadequate for characterizing this process, which appears to be initially dominated by random effects and non-linear saturation. In the second stage, the numerical solutions converge towards satisfaction of an optimality principle by which the patterns of ridges, valleys, and surface concavities minimize a function of the sediment flux over the surface, subject to two constraints. This stage is in accordance with a theoretical analysis of the model presented in a previous paper, and the numerical solutions are stable in accordance with this analysis. The optimality principle is associated with both the emergence of separable solutions to the conservation equations and a variety of regularities in landscape form and evolution, including self-similar decline of forms and a “law of height-proportional erosion”. The numerical solutions provide detailed insight into the co-evolution of landforms and flows of water and sediment. The family of models provides an elementary theory characterizing the evolution of drainage basin phenomena, and in particular (1) possesses interpretations in terms of various geomorphological concepts and observations; (2) appears capable of explaining variations in geomorphic forms over a wide variety of environments; and (3) unifies certain aspects of the continuous, discrete, and variational approaches to landscape modeling.

The scaling of fluvial landscapes

Abstract The analysis of a family of physically based landscape models leads to the analysis of two stochastic processes that seem to determine the shape and structure of river basins. The partial differential equation determine the scaling invariances of the landscape through these processes. The models bridge the gap between the stochastic and deterministic approach to landscape evolution because they produce noise by sediment divergences seeded by instabilities in the water flow. The first process is a channelization process corresponding to Brownian motion of the initial slopes. It is driven by white noise and characterized by the spatial roughness coefficient of 0.5. The second process, driven by colored noise, is a maturation process where the landscape moves closer to a mature landscape determined by separable solutions. This process is characterized by the spatial roughness coefficient of 0.75 and is analogous to an interface driven through random media with quenched noise. The values of the two scaling exponents, which are interpreted as reflecting universal, but distinct, physical mechanisms involving diffusion driven by noise, correspond well with field measurements from areas for which the advective sediment transport processes of our models are applicable. Various other scaling laws, such as Hacks law and the law of exceedence probabilities, are shown to result from the two scalings, and Hortons laws for a river network are derived from the first one.

An example of blow-up, for the complex KdV equation and existence beyond the blow-up

The KdV equation with smooth complex initial data that blows up in a finite amount of time is studied. First, examples are given using the elliptic solutions of Airault, McKean and Moser. Second, the KdV flow of spectral coordinates is described showing that blow-up occurs when eigenvalues run to infinity in finite time, but eigenvalue collisions are harmless and do not cause a singularity. Eigenvalue collisions and blow-up are studied numerically. Finally, the solutions are continued through the singularity by continuing the coordinate flows on the appropriate Riemann surface.

An explicit description of the global attractor of the damped and driven sine-Gordon equation

We prove that the size of the finite-dimensional attractor of the damped and driven sine-Gordon equation stays small as the damping and driving amplitude become small. A decomposition of finite-dimensional attractors in Banach space is found, into a partℬ that attracts all of phase space, except sets whose finitedimensional projections have Lebesgue measure zero, and a partC that only attracts sets whose finite-dimensional projections have Lebesgue measure zero. We describe the components of the ℬ-attractor andC, which is called the “hyperbolic” structure, for the damped and driven sine-Gordon equation. ℬ is low-dimensional but the dimension ofC, which is associated with transients, is much larger. We verify numerically that this is a complete description of the attractor for small enough damping and driving parameters and describe the bifurcations of the ℬ-attractor in this small parameter region.

The Kolmogorov-Obukhov Statistical Theory of Turbulence

In 1941 Kolmogorov and Obukhov postulated the existence of a statistical theory of turbulence, which allows the computation of statistical quantities that can be simulated and measured in a turbulent system. These are quantities such as the moments, the structure functions and the probability density functions (PDFs) of the turbulent velocity field. In this paper we will outline how to construct this statistical theory from the stochastic Navier–Stokes equation. The additive noise in the stochastic Navier–Stokes equation is generic noise given by the central limit theorem and the large deviation principle. The multiplicative noise consists of jumps multiplying the velocity, modeling jumps in the velocity gradient. We first estimate the structure functions of turbulence and establish the Kolmogorov–Obukhov 1962 scaling hypothesis with the She–Leveque intermittency corrections. Then we compute the invariant measure of turbulence, writing the stochastic Navier–Stokes equation as an infinite-dimensional Ito process, and solving the linear Kolmogorov–Hopf functional differential equation for the invariant measure. Finally we project the invariant measure onto the PDF. The PDFs turn out to be the normalized inverse Gaussian (NIG) distributions of Barndorff-Nilsen, and compare well with PDFs from simulations and experiments.

The local ill-posedness of the modified KdV equation

Abstract We find a new method for proving the local ill-posedness of the Cauchy problem for non-linear partial differential equations. The method is used to prove that the Cauchy problem for the Modified KdV equation is ill-posed in Sobolev spaces H s ( R ), s