A. Maassen van den Brink

Quasinormal-mode expansion for waves in open systems

An open system is not conservative because energy can escape to the outside. An open system by itself is thus not conservative. As a result, the time-evolution operator is not hermitian in the usual sense and the eigenfunctions (factorized solutions in space and time) are no longer normal modes but quasinormal modes (QNMs) whose frequencies

Nonlocal stochastic mixing-length theory and the velocity profile in the turbulent boundary layer

\omega

Stochastic Theory of Turbulence Mixing by Finite Eddies in the Turbulent Boundary Layer

are complex. QNM analysis has been a powerful tool for investigating open systems. Previous studies have been mostly system specific, and use a few QNMs to provide approximate descriptions. Here we review recent developments which aim at a unifying treatment. The formulation leads to a mathematical structure in close analogy to that in conservative, hermitian systems. Many of the mathematical tools for the latter can hence be transcribed. Emphasis is placed on those cases in which the QNMs form a complete set for outgoing wavefunctions, so that in principle all the QNMs together give an exact description of the dynamics. Applications to optics in microspheres and to gravitational waves from black holes are reviewed, and directions for further development are outlined.

Periodic oscillations of the critical temperature versus the applied magnetic flux in a four-terminal SQUID

Turbulence mixing by finite size eddies will be treated by means of a novel formulation of nonlocal K-theory, involving sample paths and a stochastic closure hypothesis, which implies a well defined recipe for the calculation of sampling and transition rates. The connection with the general theory of stochastic processes will be established. The relation with other nonlocal turbulence models (e.g. transilience and spectral diffusivity theory) is also discussed. Using an analytical sampling rate model (satisfying exchange) the theory is applied to the boundary layer (using a scaling hypothesis), which maps boundary layer turbulence mixing of scalar densities onto a nondiffusive (Kubo-Anderson or kangaroo) type stochastic process. The resulting transpport equation for longitudinal momentum Px ≡ ϱU is solved for a unified description of both the inertial and the viscous sublayer including the crossover. With a scaling exponent e ≈ 0.58 (while local turbulence would amount to e → ∞) the velocity profile U+ = ƒ(y+) is found to be in excellent agreement with the experimental data. Inter alia (i) the significance of e as a turbulence Cantor set dimension, (ii) the value of the integration constant in the logarithmic region (i.e. if y+ → ∞), (iii) linear timescaling, and (iv) finite Reynolds number effects will be investigated. The (analytical) predictions of the theory for near-wall behaviour (i.e. if y+ → 0) of fluctuating quantities also perfectly agree with recent direct numerical simulations.

KUBO-ANDERSON MIXING IN THE TURBULENT BOUNDARY LAYER

Turbulence mixing is treated by means of a novel formulation of nonlocal K-theory, involving sample paths and a stochastic hypothesis. The theory simplifies for mixing by exchange (strong-eddies) and is then applied to the boundary layer (involving scaling). This maps boundary layer turbulence onto a nondiffusive (Kubo-Anderson or kangaroo) type stochastic process. The theory involves an exponent epsilon (with the significance of a Cantor set dimension if epsilon is less than 1). With expsilon approximately equal to 0.58 (epsilon approaches infinity in the diffusion limit) the ensuing mean velocity profile U-bar+ = f(y+) is in perfect agreement with experimental data. The near-wall (y approaches 0) velocity fluctuations agree with recent direct numerical simulations

Nonisothermal activation : nonlinear transport theory

Small changes in the critical temperature, which are periodic in the applied magnetic flux, are observed in a four-terminal SQUID, i.e. a SQUID interrupted by a four-terminal Josephson junction. This novel effect is discussed for a conventional two-terminal SQUID, consisting of a weak constriction, by applying thermodynamics in the Ginzburg-Landau regime. Special attention is paid to the case of a SQUID ring with very small self-inductance in an applied magnetic flux of an odd amount of half magnetic flux quanta so that no circulating current is present. An extension is made to the case of the four-terminal SQUID to demonstrate that the theory for the two-terminal SQUID also applies to the experimentally used four-terminal case.

Boundary-layer turbulence as a kangaroo process

A novel ab initio analysis of the Reynolds stress is presented in order to model non-local turbulence transport. The theory involves a sample path space and a stochastic hypothesis. A scaling relation maps the path space onto the boundary layer. Analytical sampling rates are shown to model mixing by exchange. Nonlocal mixing involves a scaling exponent e≈0.58 (e→∞ in the diffusion limit). The resulting transport equation represents a nondiffusive (Kubo-Anderson or kangaroo) type stochastic process.