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.

Ectopic mineralization of articular cartilage in the bullfrog Rana catesbeiana and its possible involvement in bone closure

Mineralization of the articular cartilage is a pathological condition associated with age and certain joint diseases in humans and other mammals. In this work, we describe a physiological process of articular cartilage mineralization in bullfrogs. Articular cartilage of the proximal and distal ends of the femur and of the proximal end of the tibia-fibula was studied in animals of different ages. Mineralization of the articular cartilage was detected in animals at 1 month post-transformation. This mineralization, which appeared before the hypertrophic cartilage showed any calcium deposition, began at a restricted site in the lateral expansion of the cartilage and then progressed to other areas of the epiphyseal cartilage. Mineralized structures were identified by von Kossa's staining and by in vivo incorporation of calcein green. Element analysis showed that calcium crystals consisted of poorly crystalline hydroxyapatite. Mineralized matrix was initially spherical structures that generally coalesced after a certain size to occupy larger areas of the cartilage. Alkaline phosphatase activity was detected at the plasma membrane of nearby chondrocytes and in extracellular matrix. Apoptosis was detected by the TUNEL (TDT-mediated dUTP-biotin nick end-labeling) reaction in some articular chondrocytes from mineralized areas. The area occupied by calcium crystals increased significantly in older animals, especially in areas under compression. Ultrastructural analyses showed clusters of needle-like crystals in the extracellular matrix around the chondrocytes and large blocks of mineralized matrix. In 4-year-old animals, some lamellar bone (containing bone marrow) occurred in the same area as articular cartilage mineralization. These results show that the articular cartilage of R. catesbeiana undergoes precocious and progressive mineralization that is apparently stimulated by compressive forces. We suggest that this mineralization is involved in the closure of bone extremities, since mineralization appears to precede the formation of a rudimentary secondary center of ossification in older animals.

Integrable aspects of the scaling q-state Potts models I: bound states and bootstrap closure

We discuss the q-state Potts models for q less than or equal to 4, in the scaling regimes close to their critical or tricritical points. Starting from the kink S-matrix elements proposed by Chim and Zamolodchikov, the bootstrap is closed for the scaling regions of all critical points, and for the tricritical points when 4 > q greater than or equal to 2. We also note a curious appearance of the extended last line of Freudenthal's magic square in connection with the Potts models. (C) 2003 Elsevier B.V. B.V. All rights reserved.