Comment on ``Piecewise potential vorticity inversion: Elementary tests''

Egger (2008) constructs some idealised experiments to test the usefulness of piecewise potential vorticity inversion (PPVI) in the diagnosis of Rossby wave dynamics and baroclinic development. He concludes that, ``PPVI does not help us to understand the dynamics of linear Rossby waves. It provides local tendencies of the streamfunction which are unrelated to the true ones. The same way, the motion of baroclinic waves in shear flow cannot be understood by using PPVI. Moreover, the effect of boundary temperatures as determined by PPVI is unrelated to the flow evolution.'' He goes further in arguing that we should not consider velocities as ``induced'' by PV anomalies defined by carving up the global domain. However, these conclusions partly reflect the limitations of his idealised experiments and the manner in which the PV components were partitioned from one another.

An integral equation method for a boundary value problem arising in unsteady water wave problems

In this paper we consider the 2D Dirichlet boundary value problem for Laplace’s equation in a non-locally perturbed half-plane, with data in the space of bounded and continuous functions. We show uniqueness of solution, using standard Phragmen-Lindelof arguments. The main result is to propose a boundary integral equation formulation, to prove equivalence with the boundary value problem, and to show that the integral equation is well posed by applying a recent partial generalisation of the Fredholm alternative in Arens et al [J. Int. Equ. Appl. 15 (2003) pp. 1-35]. This then leads to an existence proof for the boundary value problem. Keywords. Boundary integral equation method, Water waves, Laplace’s

Consistent Dirichlet Boundary Conditions for Numerical Solution of Moving Boundary Problems

We consider the imposition of Dirichlet boundary conditions in the finite element modelling of moving boundary problems in one and two dimensions for which the total mass is prescribed. A modification of the standard linear finite element test space allows the boundary conditions to be imposed strongly whilst simultaneously conserving a discrete mass. The validity of the technique is assessed for a specific moving mesh finite element method, although the approach is more general. Numerical comparisons are carried out for mass-conserving solutions of the porous medium equation with Dirichlet boundary conditions and for a moving boundary problem with a source term and time-varying mass.

Boundary-layer ventilation by baroclinic lifecycles

Ventilation of the boundary layer has an important effect on local and regional air quality and is a prerequisite for long-range pollution transport. Once in the free troposphere, pollutants can alter the chemical composition of the troposphere and impact on the Earth's radiative forcing. Idealised baroclinic life cycles, LC1 and LC2, have been simulated in a three-dimensional dry hemispheric model in the presence of boundary-layer turbulent fluxes. A passive tracer is added to the simulations to represent pollution emitted at, or near, the surface. A simple conveyor-belt diagnostic is developed to objectively identify regions of the boundary layer that can be ventilated by either warm or cold conveyor belts. Transport of pollutants within and above the boundary layer is examined on synoptic scales. Three different physical mechanisms are found to interact with each other to ventilate pollutants out of the boundary layer. These mechanisms are turbulent mixing within the boundary layer, horizontal advection due to Ekman convergence and divergence within the boundary layer, and advection by the warm conveyor belt. The mass of tracer ventilated by the two life cycles is remarkably similar given the differences in frontal structure, suggesting that the large-scale baroclinicity is an effective constraint on ventilation.

Boundary value problems for third-order linear PDEs in time-dependent domains

We present the extension of a methodology to solve moving boundary value problems from the second-order case to the case of the third-order linear evolution PDE qt + qxxx = 0. This extension is the crucial step needed to generalize this methodology to PDEs of arbitrary order. The methodology is based on the derivation of inversion formulae for a class of integral transforms that generalize the Fourier transform and on the analysis of the global relation associated with the PDE. The study of this relation and its inversion using the appropriate generalized transform are the main elements of the proof of our results.

Cloud-radar observations of insects in the UK convective boundary layer

Radar has been applied to the study of insect migration for almost 40 years, but most entomological radars operate at X-band (9.4 GHz, 3.2 cm wavelength), and can only detect individuals of relatively large species, such as migratory grasshoppers and noctuid moths, over all of their flight altitudes. Many insects (including economically important species) are much smaller than this, but development of the requisite higher power and/or higher frequency radar systems to detect these species is often prohibitively expensive. In this paper, attention is focussed upon the uses of some recently-deployed meteorological sensing devices to investigate insect migratory flight behaviour, and especially its interactions with boundary layer processes. Records were examined from the vertically-pointing 35 GHz ‘Copernicus’ and 94 GHz ‘Galileo’ cloud radars at Chilbolton (Hampshire, England) for 12 cloudless and convective occasions in summer 2003, and one of these occasions (13 July) is presented in detail. Insects were frequently found at heights above aerosol particles, which represent passive tracers, indicating active insect movement. It was found that insect flight above the convective boundary layer occurs most often during the morning. The maximum radar reflectivity (an indicator of aerial insect biomass) was found to be positively correlated with maximum screen temperature.

Boundary integral equations on unbounded rough surfaces: Fredholmness and the finite section method

We consider a class of boundary integral equations that arise in the study of strongly elliptic BVPs in unbounded domains of the form $D = \{(x, z)\in \mathbb{R}^{n+1} : x\in \mathbb{R}^n, z > f(x)\}$ where $f : \mathbb{R}^n \to\mathbb{R}$ is a sufficiently smooth bounded and continuous function. A number of specific problems of this type, for example acoustic scattering problems, problems involving elastic waves, and problems in potential theory, have been reformulated as second kind integral equations $u+Ku = v$ in the space $BC$ of bounded, continuous functions. Having recourse to the so-called limit operator method, we address two questions for the operator $A = I + K$ under consideration, with an emphasis on the function space setting $BC$. Firstly, under which conditions is $A$ a Fredholm operator, and, secondly, when is the finite section method applicable to $A$?

Condition number estimates for combined potential boundary integral operators in acoustic scattering

We study the classical combined field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to Brakhage-Werner/Leis/Panic, and the direct formulation associated with the names of Burton and Miller. We obtain lower and upper bounds on the condition numbers for these formulations, emphasising dependence on the frequency, the geometry of the scatterer, and the coupling parameter. Of independent interest we also obtain upper and lower bounds on the norms of two oscillatory integral operators, namely the classical acoustic single- and double-layer potential operators.

The influence of the atmospheric boundary layer on nocturnal layers of noctuids and other moths migrating over southern Britain

Insects migrating at high altitude over southern Britain have been continuously monitored by automatically-operating, vertical-looking radars over a period of several years. During some occasions in the summer months, the migrants were observed to form well-defined layer concentrations, typically at heights of 200-400 m, in the stable night-time atmosphere. Under these conditions, insects are likely to have control over their vertical movements and are selecting flight heights which are favourable for long-range migration. We therefore investigated the factors influencing the formation of these insect layers by comparing radar measurements of the vertical distribution of insect density with meteorological profiles generated by the UK Met. Office’s Unified Model (UM). Radar-derived measurements of mass and displacement speed, along with data from Rothamsted Insect Survey light traps provided information on the identity of the migrants. We present here three case studies where noctuid and pyralid moths contributed substantially to the observed layers. The major meteorological factors influencing the layer concentrations appeared to be: (a) the altitude of the warmest air, (b) heights corresponding to temperature preferences or thresholds for sustained migration and (c), on nights when air temperatures are relatively high, wind-speed maxima associated with the nocturnal jet. Back-trajectories indicated that layer duration may have been determined by the distance to the coast. Overall, the unique combination of meteorological data from the UM and insect data from entomological radar described here show considerable promise for systematic studies of high-altitude insect layering.

The moist boundary layer under a mid-latitude weather system

Mid-latitude weather systems are key contributors to the transport of atmospheric water vapour, but less is known about the role of the boundary layer in this transport. We expand a conceptual model of dry boundary-layer structure under synoptic systems to include moist processes, using idealised simulations of cyclone waves to investigate the three-way interaction between the boundary layer, atmospheric moisture and large-scale dynamics. Forced by large-scale thermal advection, boundary-layer structures develop over large areas, analogous to the daytime convective boundary layer, the nocturnal stable boundary layer and transitional regimes between these extremes.

On the dependence of boundary-layer ventilation on frontal type

Case study simulations with idealized tracers have been used to determine the relationship between the dynamics and conceptual representations of different midlatitude frontal systems and the amount, distribution, and time scale of boundary layer ventilation by these systems. The key features of ventilation by a kata– and ana–cold frontal system are found to be quantitatively and also often qualitatively similar to the main ventilation pathways, which are the conveyor belts, cloud head, and other convection. The conveyor belts and cloud head occur within cloud, implying that they can be identified using satellite imagery. Differences in the transport by the two systems can be related to their conceptual representations and include a sensitive dependence on the diurnal cycle for the kata- but not the ana-cold frontal case.

Storm track sensitivity to sea surface temperature resolution in a regional atmosphere model

A high resolution regional atmosphere model is used to investigate the sensitivity of the North Atlantic storm track to the spatial and temporal resolution of the sea surface temperature (SST) data used as a lower boundary condition. The model is run over an unusually large domain covering all of the North Atlantic and Europe, and is shown to produce a very good simulation of the observed storm track structure. The model is forced at the lateral boundaries with 15–20 years of data from the ERA-40 reanalysis, and at the lower boundary by SST data of differing resolution. The impacts of increasing spatial and temporal resolution are assessed separately, and in both cases increasing the resolution leads to subtle, but significant changes in the storm track. In some, but not all cases these changes act to reduce the small storm track biases seen in the model when it is forced with low-resolution SSTs. In addition there are several clear mesoscale responses to increased spatial SST resolution, with surface heat fluxes and convective precipitation increasing by 10–20% along the Gulf Stream SST gradient.