A new frequency-uniform coercive boundary integral equation for acoustic scattering

A new boundary integral operator is introduced for the solution of the soundsoft acoustic scattering problem, i.e., for the exterior problem for the Helmholtz equation with Dirichlet boundary conditions. We prove that this integral operator is coercive in L2(Γ) (where Γ is the surface of the scatterer) for all Lipschitz star-shaped domains. Moreover, the coercivity is uniform in the wavenumber k = ω/c, where ω is the frequency and c is the speed of sound. The new boundary integral operator, which we call the “star-combined” potential operator, is a slight modification of the standard combined potential operator, and is shown to be as easy to implement as the standard one. Additionally, to the authors' knowledge, it is the only second-kind integral operator for which convergence of the Galerkin method in L2(Γ) is proved without smoothness assumptions on Γ except that it is Lipschitz. The coercivity of the star-combined operator implies frequency-explicit error bounds for the Galerkin method for any approximation space. In particular, these error estimates apply to several hybrid asymptoticnumerical methods developed recently that provide robust approximations in the high-frequency case. The proof of coercivity of the star-combined operator critically relies on an identity first introduced by Morawetz and Ludwig in 1968, supplemented further by more recent harmonic analysis techniques for Lipschitz domains.

Boundary layer dynamics over London, UK, as observed using Doppler lidar during REPARTEE-II

Urban boundary layers (UBLs) can be highly complex due to the heterogeneous roughness and heating of the surface, particularly at night. Due to a general lack of observations, it is not clear whether canonical models of boundary layer mixing are appropriate in modelling air quality in urban areas. This paper reports Doppler lidar observations of turbulence profiles in the centre of London, UK, as part of the second REPARTEE campaign in autumn 2007. Lidar-measured standard deviation of vertical velocity averaged over 30 min intervals generally compared well with in situ sonic anemometer measurements at 190 m on the BT telecommunications Tower. During calm, nocturnal periods, the lidar underestimated turbulent mixing due mainly to limited sampling rate. Mixing height derived from the turbulence, and aerosol layer height from the backscatter profiles, showed similar diurnal cycles ranging from c. 300 to 800 m, increasing to c. 200 to 850 m under clear skies. The aerosol layer height was sometimes significantly different to the mixing height, particularly at night under clear skies. For convective and neutral cases, the scaled turbulence profiles resembled canonical results; this was less clear for the stable case. Lidar observations clearly showed enhanced mixing beneath stratocumulus clouds reaching down on occasion to approximately half daytime boundary layer depth. On one occasion the nocturnal turbulent structure was consistent with a nocturnal jet, suggesting a stable layer. Given the general agreement between observations and canonical turbulence profiles, mixing timescales were calculated for passive scalars released at street level to reach the BT Tower using existing models of turbulent mixing. It was estimated to take c. 10 min to diffuse up to 190 m, rising to between 20 and 50 min at night, depending on stability. Determination of mixing timescales is important when comparing to physico-chemical processes acting on pollutant species measured simultaneously at both the ground and at the BT Tower during the campaign. From the 3 week autumnal data-set there is evidence for occasional stable layers in central London, effectively decoupling surface emissions from air aloft.

Advances in continuously profiling the thermodynamic state of the boundary layer: integration of measurements and methods

This paper describes advances in ground-based thermodynamic profiling of the lower troposphere through sensor synergy. The well-documented integrated profiling technique (IPT), which uses a microwave profiler, a cloud radar, and a ceilometer to simultaneously retrieve vertical profiles of temperature, humidity, and liquid water content (LWC) of nonprecipitating clouds, is further developed toward an enhanced performance in the boundary layer and lower troposphere. For a more accurate temperature profile, this is accomplished by including an elevation scanning measurement modus of the microwave profiler. Height-dependent RMS accuracies of temperature (humidity) ranging from 0.3 to 0.9 K (0.5–0.8 g m−3) in the boundary layer are derived from retrieval simulations and confirmed experimentally with measurements at distinct heights taken during the 2005 International Lindenberg Campaign for Assessment of Humidity and Cloud Profiling Systems and its Impact on High-Resolution Modeling (LAUNCH) of the German Weather Service. Temperature inversions, especially of the lower boundary layer, are captured in a very satisfactory way by using the elevation scanning mode. To improve the quality of liquid water content measurements in clouds the authors incorporate a sophisticated target classification scheme developed within the European cloud observing network CloudNet. It allows the detailed discrimination between different types of backscatterers detected by cloud radar and ceilometer. Finally, to allow IPT application also to drizzling cases, an LWC profiling method is integrated. This technique classifies the detected hydrometeors into three different size classes using certain thresholds determined by radar reflectivity and/or ceilometer extinction profiles. By inclusion into IPT, the retrieved profiles are made consistent with the measurements of the microwave profiler and an LWC a priori profile. Results of IPT application to 13 days of the LAUNCH campaign are analyzed, and the importance of integrated profiling for model evaluation is underlined.

Determining the contribution of volcanic ash and boundary layer aerosol in backscatter lidar returns: a three‐component atmosphere approach

A solution of the lidar equation is discussed, that permits combining backscatter and depolarization measurements to quantitatively distinguish two different aerosol types with different depolarization properties. The method has been successfully applied to simultaneous observations of volcanic ash and boundary layer aerosol obtained in Exeter, United Kingdom, on 16 and 18 April 2010, permitting the contribution of the two aerosols to be quantified separately. First a subset of the atmospheric profiles is used where the two aerosol types belong to clearly distinguished layers, for the purpose of characterizing the ash in terms of lidar ratio and depolarization. These quantities are then used in a three‐component atmosphere solution scheme of the lidar equation applied to the full data set, in order to compute the optical properties of both aerosol types separately. On 16 April a thin ash layer, 100–400 m deep, is observed (average and maximum estimated ash optical depth: 0.11 and 0.2); it descends from ∼2800 to ∼1400 m altitude over a 6‐hour period. On 18 April a double ash layer, ∼400 m deep, is observed just above the morning boundary layer (average and maximum estimated ash optical depth: 0.19 and 0.27). In the afternoon the ash is entrained into the boundary layer, and the latter reaches a depth of ∼1800 m (average and maximum estimated ash optical depth: 0.1 and 0.15). An additional ash layer, with a very small optical depth, was observed on 18 April at an altitude of 3500–4000 m. By converting the lidar optical measurements using estimates of volcanic ash specific extinction, derived from other works, the observations seem to suggest approximate peak ash concentrations of ∼1500 and ∼1000 mg/m3,respectively, on the two observations dates.

Boundary integral methods for singularly perturbed boundary value problems

In this paper we consider boundary integral methods applied to boundary value problems for the positive definite Helmholtz-type problem -DeltaU + alpha U-2 = 0 in a bounded or unbounded domain, with the parameter alpha real and possibly large. Applications arise in the implementation of space-time boundary integral methods for the heat equation, where alpha is proportional to 1/root deltat, and deltat is the time step. The corresponding layer potentials arising from this problem depend nonlinearly on the parameter alpha and have kernels which become highly peaked as alpha --> infinity, causing standard discretization schemes to fail. We propose a new collocation method with a robust convergence rate as alpha --> infinity. Numerical experiments on a model problem verify the theoretical results.

The key role of the western boundary in linking the AMOC strength to the north-south pressure gradient

A key idea in the study of the Atlantic meridional overturning circulation (AMOC) is that its strength is proportional to the meridional density gradient, or more precisely, to the strength of the meridional pressure gradient. A physical basis that would tell us how to estimate the relevant meridional pressure gradient locally from the density distribution in numerical ocean models to test such an idea, has been lacking however. Recently, studies of ocean energetics have suggested that the AMOC is driven by the release of available potential energy (APE) into kinetic energy (KE), and that such a conversion takes place primarily in the deep western boundary currents. In this paper, we develop an analytical description linking the western boundary current circulation below the interface separating the North Atlantic Deep Water (NADW) and Antarctic Intermediate Water (AAIW) to the shape of this interface. The simple analytical model also shows how available potential energy is converted into kinetic energy at each location, and that the strength of the transport within the western boundary current is proportional to the local meridional pressure gradient at low latitudes. The present results suggest, therefore, that the conversion rate of potential energy may provide the necessary physical basis for linking the strength of the AMOC to the meridional pressure gradient, and that this could be achieved by a detailed study of the APE to KE conversion in the western boundary current.

Well-posed two-point initial-boundary value problems with arbitrary boundary conditions

We study initial-boundary value problems for linear evolution equations of arbitrary spatial order, subject to arbitrary linear boundary conditions and posed on a rectangular 1-space, 1-time domain. We give a new characterisation of the boundary conditions that specify well-posed problems using Fokas' transform method. We also give a sufficient condition guaranteeing that the solution can be represented using a series. The relevant condition, the analyticity at infinity of certain meromorphic functions within particular sectors, is significantly more concrete and easier to test than the previous criterion, based on the existence of admissible functions.