There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

A comprehensive review of zonal flow phenomena in plasmas is presented. While the emphasis is on zonal flows in laboratory plasmas, planetary zonal flows are discussed as well. The review presents the status of theory, numerical simulation and experiments relevant to zonal flows. The emphasis is on developing an integrated understanding of the dynamics of drift wave–zonal flow turbulence by combining detailed studies of the generation of zonal flows by drift waves, the back-interaction of zonal flows on the drift waves, and the various feedback loops by which the system regulates and organizes itself. The implications of zonal flow phenomena for confinement in, and the phenomena of fusion devices are discussed. Special attention is given to the comparison of experiment with theory and to identifying directions for progress in future research.

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Zonal flows in plasma—a review

The progress in our understanding of several aspects of turbulent Rayleigh-Benard convection is reviewed. The focus is on the question of how the Nusselt number and the Reynolds number depend on the Rayleigh number Ra and the Prandtl number Pr, and on how the thicknesses of the thermal and the kinetic boundary layers scale with Ra and Pr. Non-Oberbeck-Boussinesq effects and the dynamics of the large scale convection roll are addressed as well. The review ends with a list of challenges for future research on the turbulent Rayleigh-Benard system.

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Heat transfer and large scale dynamics in turbulent Rayleigh-Bénard convection

Advanced Mathematical Methods for Scientists and Engineers

An Introduction to Magnetohydrodynamics

We describe a noncontact technique for velocity measurement in electrically conducting fluids. The technique, which we term Lorentz force velocimetry (LFV), is based on exposing the fluid to a magnetic field and measuring the drag force acting upon the magnetic field lines. Two series of measurements are reported, one in which the force is determined through the angular velocity of a rotary magnet system and one in which the force on a fixed magnet system is measured directly. Both experiments confirm that the measured signal is a linear function of the flow velocity. We then derive the scaling law that relates the force on a localized distribution of magnetized material to the velocity of an electrically conducting fluid. This law shows that LFV, if properly designed, has a wide range of potential applications in metallurgy, semiconductor crystal growth, and glass manufacturing.

Lorentz Force Velocimetry

A Lorentz force flowmeter is a device for the contactless measurement of flow rates in electrically conducting fluids. It is based on the measurement of a force on a magnet system that acts upon the flow. We formulate the theory of the Lorentz force flowmeter which connects the measured force to the unknown flow rate. We first apply the theory to three specific cases, namely (i) pipe flow exposed to a longitudinal magnetic field, (ii) pipe flow under the influence of a transverse magnetic field and (iii) interaction of a localized distribution of magnetic material with a uniformly moving sheet of metal. These examples provide the key scaling laws of the method and illustrate how the force depends on the shape of the velocity profile and the presence of turbulent fluctuations in the flow. Moreover, we formulate the general kinematic theory which holds for arbitrary distributions of magnetic material or electric currents and for any velocity distribution and which provides a rational framework for the prediction of the sensitivity of Lorentz force flowmeters in laboratory experiments and in industrial practice.

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Theory of the Lorentz force flowmeter

(Received 7 March 2003 and in revised form 1 July 2003) Direct numerical simulation is applied to investigate instability and transition to turbulence in the flow of an electrically conducting incompressible fluid between two parallel unbounded insulating walls affected by a wall-normal magnetic field (the Hartmann flow). The linear stability analysis of this flow provided unrealistically high critical Reynolds numbers, about two orders of magnitude higher than those observed in experiments. We propose an explanation based on the streak growth and breakdown mechanism described earlier for other shear flows. The mechanism is investigated using a two-step procedure that includes transient growth of two-dimensional optimal perturbations and the subsequent three-dimensional instability of the modulated streaky flow. In agreement with recent experimental investigations the calculations produce a critical range between 350 and 400 for the Hartmann thickness based Reynolds number, where the transition occurs at realistic amplitudes of two- and three-dimensional perturbations.

Numerical study of the instability of the Hartmann layer

The linear stability of the parallel flow ψ0=sin(y) (Kolmogorov flow) is considered, taking into account viscosity, linear friction, and confinement (lateral walls). The computations provide neutral stability curves in the parameter space, wave numbers, and wave speeds, as well as the spatial structure of first unstable modes. Evidence is presented that stability parameters depend nonuniformly on the confinement. It is shown that already weak transverse confinement significantly decreases the longitudinal wavelength of perturbations at instability onset. Strong confinement changes the character of the instability into an oscillatory one instead of a purely exponential growing mode, which is obtained for weakly confined systems. Theoretical predictions of critical parameters are in reasonable agreement with experimental results in electromagnetically driven flows of conducting fluids.

Instabilities in two‐dimensional spatially periodic flows. Part I: Kolmogorov flow

Pumped heat electricity storage (PHES) has been recently suggested as a potential solution to the large-scale energy storage problem. PHES requires neither underground caverns as compressed air energy storage (CAES) nor kilometer-sized water reservoirs like pumped hydrostorage and can therefore be constructed anywhere in the world. However, since no large PHES system exists yet, and theoretical predictions are scarce, the efficiency of such systems is unknown. Here we formulate a simple thermodynamic model that predicts the efficiency of PHES as a function of the temperature of the thermal energy storage at maximum output power. The resulting equation is free of adjustable parameters and nearly as simple as the well-known Carnot formula. Our theory predicts that for storage temperatures above $400\text{ }\ifmmode^\circ\else\textdegree\fi{}\mathrm{C}$ PHES has a higher efficiency than existing CAES and that PHES can even compete with the efficiencies predicted for advanced-adiabatic CAES.

André Thess

Papers

Lorentz Force Velocimetry

Theory of the Lorentz force flowmeter

Numerical study of the instability of the Hartmann layer

Instabilities in two‐dimensional spatially periodic flows. Part I: Kolmogorov flow

Thermodynamic efficiency of pumped heat electricity storage.