An efficient numerical method for compressible flows of a real gas using arithmetic averaging

An efficient numerical method is presented for the solution of the Euler equations governing the compressible flow of a real gas. The scheme is based on the approximate solution of a specially constructed set of linearised Riemann problems. An average of the flow variables across the interface between cells is required, and this is chosen to be the arithmetic mean for computational efficiency, which is in contrast to the usual square root averaging. The scheme is applied to a test problem for five different equations of state.

An efficient numerical scheme for the Euler equations

A finite difference scheme based on flux difference splitting is presented for the solution of the Euler equations for the compressible flow of an ideal gas. A linearised Riemann problem is defined, and a scheme based on numerical characteristic decomposition is presented for obtaining approximate solutions to the linearised problem. An average of the flow variables across the interface between cells is required, and this average is chosen to be the arithmetic mean for computational efficiency, leading to arithmetic averaging. This is in contrast to the usual ‘square root’ averages found in this type of Riemann solver, where the computational expense can be prohibitive. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second order scheme which avoids nonphysical, spurious oscillations. The scheme is applied to a shock tube problem and a blast wave problem. Each approximate solution compares well with those given by other schemes, and for the shock tube problem is in agreement with the exact solution.

An efficient numerical scheme for the two-dimensional shallow water equations using arithmetic averaging

A finite difference scheme based on flux difference splitting is presented for the solution of the two-dimensional shallow water equations of ideal fluid flow. A linearised problem, analogous to that of Riemann for gas dynamics is defined, and a scheme, based on numerical characteristic decomposition is presented for obtaining approximate solutions to the linearised problem, and incorporates the technique of operator splitting. An average of the flow variables across the interface between cells is required, and this average is chosen to be the arithmetic mean for computational efficiency leading to arithmetic averaging. This is in contrast to usual ‘square root’ averages found in this type of Riemann solver, where the computational expense can be prohibitive. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second order scheme which avoids nonphysical, spurious oscillations. An extension to the two-dimensional equations with source terms is included. The scheme is applied to the one-dimensional problems of a breaking dam and reflection of a bore, and in each case the approximate solution is compared to the exact solution of ideal fluid flow. The scheme is also applied to a problem of stationary bore generation in a channel of variable cross-section. Finally, the scheme is applied to two other dam-break problems, this time in two dimensions with one having cylindrical symmetry. Each approximate solution compares well with those given by other authors.

A numerical scheme for two-dimensional, open channel flows with non-rectangular geometries

An algorithm based on flux difference splitting is presented for the solution of two-dimensional, open channel flows. A transformation maps a non-rectangular, physical domain into a rectangular one. The governing equations are then the shallow water equations, including terms of slope and friction, in a generalized coordinate system. A regular mesh on a rectangular computational domain can then be employed. The resulting scheme has good jump capturing properties and the advantage of using boundary/body-fitted meshes. The scheme is applied to a problem of flow in a river whose geometry induces a region of supercritical flow.

Numerical solution of some nonlocal, nonlinear dispersive wave equations

We use a spectral method to solve numerically two nonlocal, nonlinear, dispersive, integrable wave equations, the Benjamin-Ono and the Intermediate Long Wave equations. The proposed numerical method is able to capture well the dynamics of the solutions; we use it to investigate the behaviour of solitary wave solutions of the equations with special attention to those, among the properties usually connected with integrability, for which there is at present no analytic proof. Thus we study in particular the resolution property of arbitrary initial profiles into sequences of solitary waves for both equations and clean interaction of Benjamin-Ono solitary waves. We also verify numerically that the behaviour of the solution of the Intermediate Long Wave equation as the model parameter tends to the infinite depth limit is the one predicted by the theory.

Numerical modelling of two-way propagation of non-linear dispersive waves

We describe and implement a fully discrete spectral method for the numerical solution of a class of non-linear, dispersive systems of Boussinesq type, modelling two-way propagation of long water waves of small amplitude in a channel. For three particular systems, we investigate properties of the numerically computed solutions; in particular we study the generation and interaction of approximate solitary waves.

Multiple-scattering in radar systems: a review

Although extensively studied within the lidar community, the multiple scattering phenomenon has always been considered a rare curiosity by radar meteorologists. Up to few years ago its appearance has only been associated with two- or three-body-scattering features (e.g. hail flares and mirror images) involving highly reflective surfaces. Recent atmospheric research aimed at better understanding of the water cycle and the role played by clouds and precipitation in affecting the Earth's climate has driven the deployment of high frequency radars in space. Examples are the TRMM 13.5 GHz, the CloudSat 94 GHz, the upcoming EarthCARE 94 GHz, and the GPM dual 13-35 GHz radars. These systems are able to detect the vertical distribution of hydrometeors and thus provide crucial feedbacks for radiation and climate studies. The shift towards higher frequencies increases the sensitivity to hydrometeors, improves the spatial resolution and reduces the size and weight of the radar systems. On the other hand, higher frequency radars are affected by stronger extinction, especially in the presence of large precipitating particles (e.g. raindrops or hail particles), which may eventually drive the signal below the minimum detection threshold. In such circumstances the interpretation of the radar equation via the single scattering approximation may be problematic. Errors will be large when the radiation emitted from the radar after interacting more than once with the medium still contributes substantially to the received power. This is the case if the transport mean-free-path becomes comparable with the instrument footprint (determined by the antenna beam-width and the platform altitude). This situation resembles to what has already been experienced in lidar observations, but with a predominance of wide- versus small-angle scattering events. At millimeter wavelengths, hydrometeors diffuse radiation rather isotropically compared to the visible or near infrared region where scattering is predominantly in the forward direction. A complete understanding of radiation transport modeling and data analysis methods under wide-angle multiple scattering conditions is mandatory for a correct interpretation of echoes observed by space-borne millimeter radars. This paper reviews the status of research in this field. Different numerical techniques currently implemented to account for higher order scattering are reviewed and their weaknesses and strengths highlighted. Examples of simulated radar backscattering profiles are provided with particular emphasis given to situations in which the multiple scattering contributions become comparable or overwhelm the single scattering signal. We show evidences of multiple scattering effects from air-borne and from CloudSat observations, i.e. unique signatures which cannot be explained by single scattering theory. Ideas how to identify and tackle the multiple scattering effects are discussed. Finally perspectives and suggestions for future work are outlined. This work represents a reference-guide for studies focused at modeling the radiation transport and at interpreting data from high frequency space-borne radar systems that probe highly opaque scattering media such as thick ice clouds or precipitating clouds.

The price of risk in construction projects: contingency approximation model (CAM)

Little attention has been focussed on a precise definition and evaluation mechanism for project management risk specifically related to contractors. When bidding, contractors traditionally price risks using unsystematic approaches. The high business failure rate our industry records may indicate that the current unsystematic mechanisms contractors use for building up contingencies may be inadequate. The reluctance of some contractors to include a price for risk in their tenders when bidding for work competitively may also not be a useful approach. Here, instead, we first define the meaning of contractor contingency, and then we develop a facile quantitative technique that contractors can use to estimate a price for project risk. This model will help contractors analyse their exposure to project risks; and help them express the risk in monetary terms for management action. When bidding for work, they can decide how to allocate contingencies strategically in a way that balances risk and reward.

Evaluating the ability of a numerical weather prediction model to forecast tracer concentrations during ETEX 2

In this paper the meteorological processes responsible for transporting tracer during the second ETEX (European Tracer EXperiment) release are determined using the UK Met Office Unified Model (UM). The UM predicted distribution of tracer is also compared with observations from the ETEX campaign. The dominant meteorological process is a warm conveyor belt which transports large amounts of tracer away from the surface up to a height of 4 km over a 36 h period. Convection is also an important process, transporting tracer to heights of up to 8 km. Potential sources of error when using an operational numerical weather prediction model to forecast air quality are also investigated. These potential sources of error include model dynamics, model resolution and model physics. In the UM a semi-Lagrangian monotonic advection scheme is used with cubic polynomial interpolation. This can predict unrealistic negative values of tracer which are subsequently set to zero, and hence results in an overprediction of tracer concentrations. In order to conserve mass in the UM tracer simulations it was necessary to include a flux corrected transport method. Model resolution can also affect the accuracy of predicted tracer distributions. Low resolution simulations (50 km grid length) were unable to resolve a change in wind direction observed during ETEX 2, this led to an error in the transport direction and hence an error in tracer distribution. High resolution simulations (12 km grid length) captured the change in wind direction and hence produced a tracer distribution that compared better with the observations. The representation of convective mixing was found to have a large effect on the vertical transport of tracer. Turning off the convective mixing parameterisation in the UM significantly reduced the vertical transport of tracer. Finally, air quality forecasts were found to be sensitive to the timing of synoptic scale features. Errors in the position of the cold front relative to the tracer release location of only 1 h resulted in changes in the predicted tracer concentrations that were of the same order of magnitude as the absolute tracer concentrations.

Numerical modeling of the propagation environment in the atmospheric boundary layer over the Persian Gulf

Strong vertical gradients at the top of the atmospheric boundary layer affect the propagation of electromagnetic waves and can produce radar ducts. A three-dimensional, time-dependent, nonhydrostatic numerical model was used to simulate the propagation environment in the atmosphere over the Persian Gulf when aircraft observations of ducting had been made. A division of the observations into high- and low-wind cases was used as a framework for the simulations. Three sets of simulations were conducted with initial conditions of varying degrees of idealization and were compared with the observations taken in the Ship Antisubmarine Warfare Readiness/Effectiveness Measuring (SHAREM-115) program. The best results occurred with the initialization based on a sounding taken over the coast modified by the inclusion of data on low-level atmospheric conditions over the Gulf waters. The development of moist, cool, stable marine internal boundary layers (MIBL) in air flowing from land over the waters of the Gulf was simulated. The MIBLs were capped by temperature inversions and associated lapses of humidity and refractivity. The low-wind MIBL was shallower and the gradients at its top were sharper than in the high-wind case, in agreement with the observations. Because it is also forced by land–sea contrasts, a sea-breeze circulation frequently occurs in association with the MIBL. The size, location, and internal structure of the sea-breeze circulation were realistically simulated. The gradients of temperature and humidity that bound the MIBL cause perturbations in the refractivity distribution that, in turn, lead to trapping layers and ducts. The existence, location, and surface character of the ducts were well captured. Horizontal variations in duct characteristics due to the sea-breeze circulation were also evident. The simulations successfully distinguished between high- and low-wind occasions, a notable feature of the SHAREM-115 observations. The modeled magnitudes of duct depth and strength, although leaving scope for improvement, were most encouraging.

Identifying and quantifying nonconservative energy production/destruction terms in hydrostatic Boussinesq primitive equation models

This paper seeks to illustrate the point that physical inconsistencies between thermodynamics and dynamics usually introduce nonconservative production/destruction terms in the local total energy balance equation in numerical ocean general circulation models (OGCMs). Such terms potentially give rise to undesirable forces and/or diabatic terms in the momentum and thermodynamic equations, respectively, which could explain some of the observed errors in simulated ocean currents and water masses. In this paper, a theoretical framework is developed to provide a practical method to determine such nonconservative terms, which is illustrated in the context of a relatively simple form of the hydrostatic Boussinesq primitive equation used in early versions of OGCMs, for which at least four main potential sources of energy nonconservation are identified; they arise from: (1) the “hanging” kinetic energy dissipation term; (2) assuming potential or conservative temperature to be a conservative quantity; (3) the interaction of the Boussinesq approximation with the parameterizations of turbulent mixing of temperature and salinity; (4) some adiabatic compressibility effects due to the Boussinesq approximation. In practice, OGCMs also possess spurious numerical energy sources and sinks, but they are not explicitly addressed here. Apart from (1), the identified nonconservative energy sources/sinks are not sign definite, allowing for possible widespread cancellation when integrated globally. Locally, however, these terms may be of the same order of magnitude as actual energy conversion terms thought to occur in the oceans. Although the actual impact of these nonconservative energy terms on the overall accuracy and physical realism of the oceans is difficult to ascertain, an important issue is whether they could impact on transient simulations, and on the transition toward different circulation regimes associated with a significant reorganization of the different energy reservoirs. Some possible solutions for improvement are examined. It is thus found that the term (2) can be substantially reduced by at least one order of magnitude by using conservative temperature instead of potential temperature. Using the anelastic approximation, however, which was initially thought as a possible way to greatly improve the accuracy of the energy budget, would only marginally reduce the term (4) with no impact on the terms (1), (2) and (3).