Plasticity as a Link Between Spatially Explicit, Distance-Independent, and Whole-Stand Forest Growth Models

Models at various levels of resolution are commonly used for both forest management and ecological research. They all have comparative advantages and disadvantages, making desirable a better understanding of the relationships between various approaches. Accounting for crown and root morphological plasticity in the limit where equilibrium among neighbors is reached (perfect plasticity) transforms spatial models into nonspatial, distance-independent versions. The links between spatial and nonspatial models obtained through a perfect plasticity assumption are more realistic than ignoring spatial structure by a mean field approximation. This article also reviews the connection between distance-independent models and size distributions and how distributions evolve over time and relate to whole-stand descriptions. In addition, some ways in which stand-level knowledge feeds back into detailed individual-tree formulations are demonstrated. This presentation is intended to be accessible to nonspecialists. Study Implications Introducing plasticity improves the representation of physio-ecological processes in spatial modelling. Plasticity explains in part the practical success of distance-independent models. The nature of size distributions and their relationship to individual-tree and whole-stand models are discussed. A size distribution is a one-variable distribution; joint distributions for two or more trees depend on the distances between them unless spatial structure is negligible. Limitations of current individual-tree models and questions for future research are discussed.

Rigid-perfectly-plastic materials. Two extremal principles

This chapter introduces the notion of rigid-perfect plasticity, and provides proofs of the extremum principles of applied loads and velocities. The variational principles are applied to generate the upper- and lower-bound theorems for collapse loads in rigid-perfect plasticity models. The method of sliding rigid blocks is presented for the solution of practical problems in forming operations and structural collapse. To facilitate such modelling the hodograph graphical device is also discussed.

Ice-mass survival for approximately 800 Myr in the tropical Kasei Valles region, Mars

<p>High obliquity excursions on Mars are hypothesised to have redistributed water from the poles to nourish mid-latitude glaciers. Evidence of this process is provided by a variety of viscous flow features—ice-rich deposits buried beneath sediment mantle—located there today, including ‘lobate debris aprons’, or LDAs. During high obliquity extremes, ice may have persisted even nearer the equator, as indicated by numerous enigmatic moat-like depressions in the tropical Kasei Valles region. Numerous depressions surround isolated mesas and demarcate the past interaction between flowing lava and what were presumably ice-rich radial flows resembling today’s LDAs, but which have long since disappeared. Little is known about ‘ghost lobate debris aprons’ (ghost LDAs), besides their spatial extent as recorded by these depressions. This collection of ghost LDAs implies tropical ice loss over an area ~100,000 km<sup>2</sup>. To constrain their history in Kasei Valles we derive model ages of different terrain types from crater counts. To constrain the volume of ice loss, we use a 2D perfect-plasticity model of ice flow to reconstruct the ghost LDA surfaces. Parametrised by the present surface topography and the range of yield stresses derived from radar interrogation of mid-latitude ice masses, the model reconstructs former ice surfaces along multiple flowlines orientated normal to ghost LDA boundaries. This reconstruction indicates between 1,300–3,300 km<sup>3</sup> of ice—similar to that present in Iceland on Earth—was lost since lava emplacement ~1.4 Ga. Dating of these depressions shows that the ghost LDAs survived for ~800 million years following lava emplacement in the Kasei Valles region before their final demise.</p>