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The Helmholtz scattering problem with high wave number is truncated by the perfectly matched layer (PML) technique and then discretized by the linear continuous interior penalty-finite element method (CIP-FEM). It is proved that the truncated PML problem satisfies the 1 inf-sup condition with inf-sup constant of order O(k⁻¹). Stability and convergence of the truncated PML problem are discussed. In particular, the convergence rate is twice the previous result. The preasymptotic error estimates in the energy norm of the linear CIP-FEM as well as FEM are proved to be C₁kh + C₂k³h² under the mesh condition that k³ h² is sufficiently small. Numerical tests are provided to illustrate the preasymptotic error estimates and show that the penalty parameter in the CIP-FEM may be tuned to reduce greatly the pollution error.

Several blending methods have been developed in dynamic downscaling and rapid cycled data assimilation. Blending the large‐scale part of the global model (GM) analysis or forecast has led to improvement in regional model (RM) simulations. However, in previous studies the blended waveband of the GM has generally been determined using a fixed, arbitrarily chosen cutoff wave number. Here we introduce a new dynamic blending (DB) scheme with a dynamic cutoff wave number computed according to the spectral characteristics of GM forecast quality and the spectral distribution of errors in the RM. The DB scheme is described and applied to eight‐day summertime and seven‐day wintertime cycled Weather Research and Forecasting Model forecasts over a regional domain in the continental United States. The scheme can determine a cutoff wave number with significant temporal variations. The temporal variation results from the error growth property of the RM and has a clear diurnal oscillation, suggesting that fewer (more) GM waves should be introduced into the RM at noon (night). The cutoff wave number difference between the two periods indicates seasonal variation of the cutoff wave number with larger day‐to‐day change in winter. Comparison among no blending experiment, two fixed wave number blending experiments, and two DB experiments with and without vertically varying cutoff wave number suggests that the DB scheme with vertically averaged but temporally varying cutoff wave number results in less model bias and less disturbance to the RM dynamic balance. By reducing the forecast background error, the DB scheme can potentially provide improved first guess for a rapid‐update‐cycle weather forecast system. Plain Language Summary Blending is a process to introduce large‐scale information from a global model to mitigate bias of weather forecasting in a regional model. However, in previous studies, the large‐scale waveband that should be introduced into regional model is difficult to determine. This study introduces a new scheme with a dynamic‐scale selection process, named dynamic blending. The dynamic blending method can potentially provide improved first guess for a rapid‐update‐cycle weather forecast system. Key Points Blending the large‐scale part of the global model leads to improvement in the performance of regional model simulations We introduce a new scheme to dynamically determine the cutoff scale in blending process of global and regional model Dynamic blending scheme results in less model bias and less disturbance to the RM dynamic balance

In this paper, Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation. First of all, the interpolation basis function is applied to treat the spatial variables and their partial derivatives, and the collocation method for solving the second order differential equations is established. Secondly, the differential matrix is used to simplify the given differential equations on a given test node. Finally, based on three kinds of test nodes, numerical experiments show that the present scheme can not only calculate the high wave numbers problems, but also calculate the variable wave numbers problems. In addition, the algorithm has the advantages of high calculation accuracy, good numerical stability and less time consuming.

PurposeThis meshless collocation method is applicable not only to the Helmholtz equation with Dirichlet boundary condition but also mixed boundary conditions. It can calculate not only the high wavenumber problems, but also the variable wave number problems.Design/methodology/approachIn this paper, the authors developed a meshless collocation method by using barycentric Lagrange interpolation basis function based on the Chebyshev nodes to deduce the scheme for solving the three-dimensional Helmholtz equation. First, the spatial variables and their partial derivatives are treated by interpolation basis functions, and the collocation method is established for solving second order differential equations. Then the differential matrix is employed to simplify the differential equations which is on a given test node. Finally, numerical experiments show the accuracy and effectiveness of the proposed method.FindingsThe numerical experiments show the advantages of the present method, such as less number of collocation nodes needed, shorter calculation time, higher precision, smaller error and higher efficiency. What is more, the numerical solutions agree well with the exact solutions.Research limitations/implicationsCompared with finite element method, finite difference method and other traditional numerical methods based on grid solution, meshless method can reduce or eliminate the dependence on grid and make the numerical implementation more flexible.Practical implicationsThe Helmholtz equation has a wide application background in many fields, such as physics, mechanics, engineering and so on.Originality/valueThis meshless method is first time applied for solving the 3D Helmholtz equation. What is more the present work not only gives the relationship of interpolation nodes but also the test nodes.

On 24 May 2014, a Mw 6.9 earthquake occurred in the west of Gokceada Island, northern Aegean Sea. The earthquake was close to Canakkale, Enez, Tekirdag cities, and damaged 300 buildings in the Marmara Region, NW Turkey. We simulated its broadband (0.1–10 Hz) ground motions including 1D deep and shallow structures soil amplification effects at the 12 strong ground motion stations in the western Marmara Region. The 1D deep velocity structures from the focal layer to the engineering bedrock with an S-wave velocity of 0.78 km/s in different azimuthal directions were tuned by comparing the observed group-velocity dispersion curves of Rayleigh and Love waves from the mainshock with theoretical ones. We also added the shallow parts from previous surveys into the 1D models. Synthetic seismograms on the engineering bedrock were generated using the discrete wave number method with a source model and the 1D deep velocity structures. Then the surface motion was generated considering shallow soil amplification. The synthetic seismograms are generally in good agreement with the observed low and high-frequency parts at most of the stations indicating an appropriateness of the source model and the 1D structural model.

We report a unique type of ULF waves observed by low‐altitude Space Technology 5 (ST‐5) constellation mission. ST‐5 is a three‐microsatellite constellation deployed into a 300 × 4500 km dawn‐dusk and Sun‐synchronous polar orbit with 105.6° inclination angle. Because of the Earth's rotation and the dipole tilt effect, the spacecraft's dawn‐dusk orbit track can reach as low as subauroral latitudes during the course of a day. Whenever the spacecraft traverse the dayside closed field line region at subauroral latitudes, they frequently observe strong transverse oscillations at 30–200 mHz, or in the Pc2–3 frequency range. These Pc2–3 waves appear as wave packets with durations in the order of 5–10 min. As the maximum separations of the ST‐5 spacecraft are in the order of 10 min, the three ST‐5 satellites often observe very similar wave packets, implying these wave oscillations occur in a localized region. The coordinated ground‐based magnetic observations at the spacecraft footprints, however, do not see waves in the Pc2–3 band; instead, the waves appear to be the common Pc4–5 waves associated with field line resonances. We suggest that these unique Pc2–3 waves seen by ST‐5 are in fact the Doppler‐shifted Pc4–5 waves as a result of rapid traverse of the spacecraft across the resonant field lines azimuthally at low altitudes. The observations with the unique spacecraft dawn‐dusk orbits at proper altitudes and magnetic latitudes reveal the azimuthal characteristics of field line resonances. Key Points A unique type of ULF waves is observed by ST‐5 at dayside subauroral latitudes The waves are found to be associated with high‐m number field line resonances

The trajectory of surface gravity waves in the potential flow regime is affected by the gravitational acceleration, water density and sea bed depth. Although the gravitational acceleration and water density are approximately constant, the effect of water depth on surface gravity waves exponentially decreases as the water depth increases. In shallow water, cloaking an object from surface waves by varying the sea bed topography is possible, however, as the water depth increases, cloaking becomes a challenge because there is no physical parameter to be engineered and subsequently affects the wave propagation. In order to create an omnidirectional cylindrical cloaking device for finite-depth/deep-water waves, we propose an elastic composite plate that floats on the surface around a to-be-cloaked cylinder. The composite plate is made of axisymmetric, homogeneous and isotropic annular thin rings which provide adjustable degrees of freedom to engineer and affect the wave propagation. We first develop a pseudo-spectral method to efficiently determine the wave solution for a floating composite plate. Next, we optimise the physical parameters of the plate (i.e. flexural rigidity and mass of every ring) using an evolutionary algorithm to minimise the energy of scattered waves from the object and therefore cloak the inner cylinder from incident waves. We show that the optimised cloak reduces the energy of scattered waves as high as 99 % for the target wave number. We quantify the effectiveness of our cloak with different parameters of the plate and show that varying the flexural rigidity is essential to control wave propagation and the cloaking structure needs to be at least made of four rings with a radius of at least three times of the cloaked region. We quantify the wave drift force exerted on the structures and show that the optimised plate reduces the exerted force by 99.9 %. The proposed cloak, due to its structural simplicity and effectiveness in reducing the wave drift force, may have potential applications in cloaking offshore structures from water waves.

Despite their adverse impacts, definitions and measurements of heat waves are ambiguous and inconsistent, generally being endemic to only the group affected, or the respective study reporting the analysis. The present study addresses this issue by employing a set of three heat wave definitions, derived from surveying heat-related indices in the climate science literature. The definitions include three or more consecutive days above one of the following: the 90th percentile for maximum temperature, the 90th percentile for minimum temperature, and positive extreme heat factor (EHF) conditions. Additionally, each index is studied using a multiaspect framework measuring heat wave number, duration, participating days, and the peak and mean magnitudes. Observed climatologies and trends computed by Sen’s Kendall slope estimator are presented for the Australian continent for two time periods (1951–2008 and 1971–2008). Trends in all aspects and definitions are smaller in magnitude but more significant for 1951–2008 than for 1971–2008. Considerable similarities exist in trends of the yearly number of days participating in a heat wave and yearly heat wave frequency, suggesting that the number of available heat wave days drives the number of events. Larger trends in the hottest part of a heat wave suggest that heat wave intensity is increasing faster than the mean magnitude. Although the direct results of this study cannot be inferred for other regions, the methodology has been designed as such that it is widely applicable. Furthermore, it includes a range of definitions that may be useful for a wide range of systems impacted by heat waves.

A simplified model of the Earth's atmosphere is considered, which on the one hand exactly reproduces the real characteristics of the Earth, and on the other hand extremely simplifies the configuration of continents and oceans and completely ignores the topology of the real planet. The aim of this formulation is to analyze the generation of large-scale mid-latitude eddies and waves on a zonally homogeneous flat planet with energy balance and atmospheric properties similar to those of the Earth. It is shown that the preservation of a relatively small equatorial ocean on the background of a flat desert planet is sufficient to reproduce the general atmospheric circulation and the dynamics of mid-latitude wave and vortex structures that are realistic for the Earth. The considered idealized system correctly reproduces not only the spatial structure of the emerging baroclinic waves, but also their seasonal dynamics, including the variability of the spectral composition of the observed waves. The modes with wave numbers 3-8 contain most of the wave energy. The spectra for higher modes are characterized by a power law [Formula omitted] with a slope [Formula omitted] for the winter season and [Formula omitted] for the summer season. During the baroclinic season, there is a change in the dominant wave number from 7 to 4. Another particular feature of baroclinic waves in a considered configuration is a significant temporal variation of the phase velocity with two distinct stages of monotonic increase and decrease. There is a time lag between beginning of the gradual increase of the phase velocity for different modes. The higher modes begin to accelerate earlier and as a result lower modes move slower than higher modes. The temporal evolution of the baroclinic wave intensity is well described by the Eady parameter.

An experimental investigation of wave propagation coefficients determination of the granite was presented in the present study. Firstly, a series of pendulum impact tests were performed to investigate the stress wave properties of the granite. High-speed digital image correlation (HDIC) was utilized to capture the displacement and velocity at the free end of the impacted granite bar. Subsequently, the HDIC-based non-contact method was introduced for the determination of wave propagation coefficients of the granite. Finally, experimental studies based on the traditional contact method using strain gauges were performed to validate the present HDIC-based non-contact method. The results show that both the attenuation coefficient and wave number increase as frequency increases. Moreover, the propagation coefficients (attenuation coefficient and wave number) determined by the present HDIC-based non-contact method agree well with that determined by the traditional contact method using strain gauges. The present HDIC-based non-contact method can be used to predict the stress wave propagation through the granite effectively.HighlightsA non-contact method by high-speed digital image correlation (HDIC) was proposed.The HDIC-based non-contact method was validated by traditional strain gauges measurement.The propagation coefficients can be determined by the present method efficiently.The HDIC-based non-contact method can predict the stress wave propagation.