The omega equation and potential vorticity

Two fundamental perspectives on the dynamics of midlatitude weather systems are provided by potential vorticity (PV) and the omega equation. The aim of this paper is to investigate the link between the two perspectives, which has so far received very little attention in the meteorological literature. It also aims to give a quantitative basis for discussion of quasi-geostrophic vertical motion in terms of components associated with system movement, maintaining a constant thermal structure, and with the development of that structure. The former links with the isentropic relative-flow analysis technique. Viewed in a moving frame of reference, the measured development of a system depends on the velocity of that frame of reference. The requirement that the development should be a minimum provides a quantitative method for determining the optimum system velocity. The component of vertical velocity associated with development is shown to satisfy an omega equation with forcing determined from the relative advection of interior PV and boundary temperature. The analysis carries through in the presence of diabatic heating provided the omega equation forcing is based on the interior PV and boundary thermal tendencies, including the heating effect. The analysis is shown to be possible also at the level of the semi-geostrophic approximation. The analysis technique is applied to a number of idealized problems that can be considered to be building blocks for midlatitude synoptic-scale dynamics. They focus on the influences of interior PV, boundary temperature, an interior boundary, baroclinic instability associated with two boundaries, and also diabatic heating. In each case, insights yielded by the new perspective are sought into the dynamical behaviour, especially that related to vertical motion. Copyright © 2003 Royal Meteorological Society

Empirical evidence for a nonlinear effect of galactic cosmic rays on clouds

Galactic cosmic ray (GCR) changes have been suggested to affect weather and climate, and new evidence is presented here directly linking GCRs with clouds. Clouds increase the diffuse solar radiation, measured continuously at UK surface meteorological sites since 1947. The ratio of diffuse to total solar radiation-the diffuse fraction, (DF)-is used to infer cloud, and is compared with the daily mean neutron count rate measured at Climax; Colorado from 1951-2000, which provides a globally representative indicator of cosmic rays. Across the UK, oil days of high cosmic ray flux (above 3600 X 10(2) neutron counts h(-1), which occur 87% of the time on average) compared with low cosmic ray flux, (i) the chance of an overcast day increases by (19 +/- 4)%; and (ii) the diffuse fraction increases by (2 +/- 0.3)%. During sudden transient reductions in cosmic rays (e.g. Forbush events), simultaneous decreases occur in the diffuse fraction. The diffuse radiation changes are; therefore; unambiguously due to cosmic rays. Although the statistically significant nonlinear cosmic ray effect is small, it will have a considerably larger aggregate effect on longer timescale (e.g. centennial) climate variations when day-to-day variability averages out.

The influence of hilly terrain on canopy-atmosphere carbon dioxide exchange

Topography influences many aspects of forest-atmosphere carbon exchange; yet only a small number of studies have considered the role of topography on the structure of turbulence within and above vegetation and its effect on canopy photosynthesis and the measurement of net ecosystem exchange of CO2 (N-ee) using flux towers. Here, we focus on the interplay between radiative transfer, flow dynamics for neutral stratification, and ecophysiological controls on CO2 sources and sinks within a canopy on a gentle cosine hill. We examine how topography alters the forest-atmosphere CO2 exchange rate when compared to uniform flat terrain using a newly developed first-order closure model that explicitly accounts for the flow dynamics, radiative transfer, and nonlinear eco physiological processes within a plant canopy. We show that variation in radiation and airflow due to topography causes only a minor departure in horizontally averaged and vertically integrated photosynthesis from their flat terrain values. However, topography perturbs the airflow and concentration fields in and above plant canopies, leading to significant horizontal and vertical advection of CO2. Advection terms in the conservation equation may be neglected in flow over homogeneous, flat terrain, and then N-ee = F-c, the vertical turbulent flux of CO2. Model results suggest that vertical and horizontal advection terms are generally of opposite sign and of the same order as the biological sources and sinks. We show that, close to the hilltop, F-c departs by a factor of three compared to its flat terrain counterpart and that the horizontally averaged F-c-at canopy top differs by more than 20% compared to the flat-terrain case.

Linear and nonlinear microrheology of dense colloidal suspensions

The length and time scales accessible to optical tweezers make them an ideal tool for the examination of colloidal systems. Embedded high-refractive-index tracer particles in an index-matched hard sphere suspension provide 'handles' within the system to investigate the mechanical behaviour. Passive observations of the motion of a single probe particle give information about the linear response behaviour of the system, which can be linked to the macroscopic frequency-dependent viscous and elastic moduli of the suspension. Separate 'dragging' experiments allow observation of a sample's nonlinear response to an applied stress on a particle-by particle basis. Optical force measurements have given new data about the dynamics of phase transitions and particle interactions; an example in this study is the transition from liquid-like to solid-like behaviour, and the emergence of a yield stress and other effects attributable to nearest-neighbour caging effects. The forces needed to break such cages and the frequency of these cage breaking events are investigated in detail for systems close to the glass transition.

The Klein-Gordon equation on the half line: a Riemann-Hilbert approach

We solve an initial-boundary problem for the Klein-Gordon equation on the half line using the Riemann-Hilbert approach to solving linear boundary value problems advocated by Fokas. The approach we present can be also used to solve more complicated boundary value problems for this equation, such as problems posed on time-dependent domains. Furthermore, it can be extended to treat integrable nonlinearisations of the Klein-Gordon equation. In this respect, we briefly discuss how our results could motivate a novel treatment of the sine-Gordon equation.