Raymond Pierrehumbert FRS

Abstract:

We examine the decay of passive scalars with small, but nonzero, diffusivity in bounded two-dimensional (2D) domains. The velocity fields responsible for advection are smooth (i.e., they have bounded gradients) and of a single large scale. Moreover, the scale of the velocity field is taken to be similar to the size of the entire domain. The importance of the initial scale of variation of the scalar field with respect to that of the velocity field is strongly emphasized. If these scales are comparable and the velocity field is time periodic, we see the formation of a periodic scalar eigenmode. The eigenmode is numerically realized by means of a deterministic 2D map on a lattice. Analytical justification for the eigenmode is available from theorems in the dynamo literature. Weakening the notion of an eigenmode to mean statistical stationarity, we provide numerical evidence that the eigenmode solution also holds for aperiodic flows (represented by random maps). Turning to the evolution of an initially small scale scalar field, we demonstrate the transition from an evolving (i.e., non-self-similar) probability distribution function (pdf) to a stationary (self-similar) pdf as the scale of variation of the scalar field progresses from being small to being comparable to that of the velocity field (and of the domain). Furthermore, the non-self-similar regime itself consists of two stages. Both stages are examined and the coupling between diffusion and the distribution of the finite time Lyapunov exponents is shown to be responsible for the pdf evolution.

Abstract:

Hydrology refers to the whole panoply of effects the water molecule has on climate and on the land surface during its journey there and back again between ocean and atmosphere. On its way, it is cycled through vapour, cloud water, snow, sea ice and glacier ice, as well as acting as a catalyst for silicate-carbonate weathering reactions governing atmospheric carbon dioxide. Because carbon dioxide affects the hydrologic cycle through temperature, climate is a pas des deux between carbon dioxide and water, with important guest appearances by surface ice cover.

Abstract:

We study the statistical and geometrical properties of the potential temperature (PT) field in the surface quasigeostrophic (SQG) system of equations. In addition to extracting information in a global sense via tools such as the power spectrum, the g-beta spectrum, and the structure functions we explore the local nature of the PT field by means of the wavelet transform method. The primary indication is that an initially smooth PT field becomes rough (within specified scales), though in a qualitatively sparse fashion. Similarly, initially one-dimensional iso-PT contours (i.e., PT level sets) are seen to acquire a fractal nature. Moreover, the dimensions of the iso-PT contours satisfy existing analytical bounds. The expectation that the roughness will manifest itself in the singular nature of the gradient fields is confirmed via the multifractal nature of the dissipation field. Following earlier work on the subject, the singular and oscillatory nature of the gradient field is investigated by examining the scaling of a probability measure and a sign singular measure, respectively. A physically motivated derivation of the relations between the variety of scaling exponents is presented, the aim being to bring out some of the underlying assumptions which seem to have gone unnoticed in previous presentations. Apart from concentrating on specific properties of the SQG system, a broader theme of the paper is a comparison of the diagnostic inertial range properties of the SQG system with both the two- and three-dimensional Euler equations. (c) 2002 American Institute of Physics.