First-order turbulence closure for modelling complex canopy flows

Simple first-order closure remains an attractive way of formulating equations for complex canopy flows when the aim is to find analytic or simple numerical solutions to illustrate fundamental physical processes. Nevertheless, the limitations of such closures must be understood if the resulting models are to illuminate rather than mislead. We propose five conditions that first-order closures must satisfy then test two widely used closures against them. The first is the eddy diffusivity based on a mixing length. We discuss the origins of this approach, its use in simple canopy flows and extensions to more complex flows. We find that it satisfies most of the conditions and, because the reasons for its failures are well understood, it is a reliable methodology. The second is the velocity-squared closure that relates shear stress to the square of mean velocity. Again we discuss the origins of this closure and show that it is based on incorrect physical principles and fails to satisfy any of the five conditions in complex canopy flows; consequently its use can lead to actively misleading conclusions.

Shadow lines in the arithmetic of elliptic curves

Let E/Q be an elliptic curve and p a rational prime of good ordinary reduction. For every imaginary quadratic field K/Q satisfying the Heegner hypothesis for E we have a corresponding line in E(K)\otimes Q_p, known as a shadow line. When E/Q has analytic rank 2 and E/K has analytic rank 3, shadow lines are expected to lie in E(Q)\otimes Qp. If, in addition, p splits in K/Q, then shadow lines can be determined using the anticyclotomic p-adic height pairing. We develop an algorithm to compute anticyclotomic p-adic heights which we then use to provide an algorithm to compute shadow lines. We conclude by illustrating these algorithms in a collection of examples.

Exploring smart grid possibilities: a complex systems modelling approach

Smart grid research has tended to be compartmentalised, with notable contributions from economics, electrical engineering and science and technology studies. However, there is an acknowledged and growing need for an integrated systems approach to the evaluation of smart grid initiatives. The capacity to simulate and explore smart grid possibilities on various scales is key to such an integrated approach but existing models – even if multidisciplinary – tend to have a limited focus. This paper describes an innovative and flexible framework that has been developed to facilitate the simulation of various smart grid scenarios and the interconnected social, technical and economic networks from a complex systems perspective. The architecture is described and related to realised examples of its use, both to model the electricity system as it is today and to model futures that have been envisioned in the literature. Potential future applications of the framework are explored, along with its utility as an analytic and decision support tool for smart grid stakeholders.