Perturbation Of Boundary Conditions Research Articles

Sensitivity of regional ensemble data assimilation spread to perturbations of lateral boundary conditions

The implementation of a regional ensemble data assimilation and forecasting system requires the specification of appropriate perturbations of lateral boundary conditions (LBCs), in order to simulate associated errors. The sensitivity of analysis and 6-h forecast ensemble spread to these perturbations is studied here formally and experimentally by comparing three different LBC configurations for the ensemble data assimilation system of the ALADIN-France limited-area model (LAM). While perturbed initial LBCs are provided by the perturbed LAM analyses in each ensemble, the three ensemble configurations differ with respect to LBCs used at 3- and 6-h forecast ranges, which respectively correspond to: (1) perturbed LBCs provided by the operational global ensemble data assimilation system (GLBC), which is considered as a reference configuration; (2) unperturbed LBCs (ULBC) obtained from the global deterministic model; (3) perturbed LBCs obtained by adding random draws of an error covariance model (PLBC) to the global deterministic system. A formal analysis of error and perturbation equations is first carried out, in order to provide an insight of the relative effects of observation perturbations and of LBC perturbations at different ranges, in the various ensemble configurations. Horizontal variations of time-averaged ensemble spread are then examined for 6-h forecasts. Despite the use of perturbed initial LBCs, the regional ensemble ULBC is underdispersive not only near the lateral boundaries, but also in approximately one-third of the inner area, due to advection during the data assimilation cycle. This artefact is avoided in PLBC through the additional use of non-zero LBC perturbations at 3- and 6-h ranges, and the sensitivity to the amplitude scaling of the covariance model is illustrated for this configuration. Some aspects of the temporal variation of ensemble spread and associated sensitivities to LBC perturbations are also studied. These results confirm the importance of LBC perturbations for regional ensemble data assimilation. They also indicate that perturbing initial LBC is not sufficient to obtain realistic ensemble spread, whereas this can be achieved approximately by using random covariance draws for simulating LBC errors during the forecast and associated data assimilation cycling.

Perturbative solution for terahertz two-wire metallic waveguides with different radii.

We introduce the technique of perturbation of boundary condition to the problem of terahertz two-wire metallic waveguides with different radii. Based on the quasi-TEM analytical mode fields derived by use of Möbius transformation, a concise expression for the complex effective index is obtained analytically. The expression is in good agreement with the simulation result. Further, the dispersion and attenuation are obtained from the expression. In addition, we find a zero value point of the group velocity dispersion around 1.268 THz. The results show that the technique of perturbation of boundary condition is helpful in the analysis and design of terahertz metal waveguide.

A kinematic wave theory of capacity drop

Capacity drop at active bottlenecks is one of the most puzzling traffic phenomena, but a thorough understanding of its mechanism is critical for designing variable speed limit and ramp metering strategies. In this study, within the framework of the kinematic wave theory, we propose a simple model of capacity drop based on the observation that capacity drop occurs when an upstream queue forms at an active bottleneck. Different from existing models, the new model still uses continuous fundamental diagrams but employs an entropy condition defined by a discontinuous boundary flux function, which introduces a traffic state-dependent capacity constraint. For a lane-drop area, we demonstrate that the model is well-defined, and its Riemann problem can be uniquely solved. After deriving the flow-density relations upstream and downstream to a bottleneck location, we find that the model can replicate the following three characteristics of capacity drop: the maximum discharge flow-rate can be reached only when both upstream and downstream traffic conditions are uncongested, capacity drop occurs when the bottleneck is activated, and some steady traffic states cannot be observed at both locations. We show that the new model is bistable subject to perturbations in initial and boundary conditions. With empirical observations at a merging bottleneck we also verify the three characteristics of capacity drop predicted by the new model. Through this study, we establish that the new model is physically meaningful, conceptually simple, computationally efficient, and mathematically tractable. We finally discuss future extensions and potential applications of the new model.

Multiscale EnKF Assimilation of Radar and Conventional Observations and Ensemble Forecasting for a Tornadic Mesoscale Convective System

Abstract In recent studies, the authors have successfully demonstrated the ability of an ensemble Kalman filter (EnKF), assimilating real radar observations, to produce skillful analyses and subsequent ensemble-based probabilistic forecasts for a tornadic mesoscale convective system (MCS) that occurred over Oklahoma and Texas on 9 May 2007. The current study expands upon this prior work, performing experiments for this case on a larger domain using a nested-grid EnKF, which accounts for mesoscale uncertainties through the initial ensemble and lateral boundary condition perturbations. In these new experiments, conventional observations (including surface, wind profiler, and upper-air observations) are assimilated in addition to the WSR-88D and the Center for Collaborative Adaptive Sensing of the Atmosphere (CASA) radar data used in the previous studies, better representing meso- and convective-scale features. The relative impacts of conventional and radar data on analyses and forecasts are examined, and biases within the ensemble are investigated. The new experiments produce a substantially improved forecast, including better representation of the convective lines of the MCS. Assimilation of radar data substantially improves the ensemble precipitation forecast. Assimilation of conventional data together with radar observations substantially improves the forecast of near-surface mesovortices within the MCS, improves forecasts of surface temperature and dewpoint, and imparts a slight but noticeable improvement to short-term precipitation forecasts. Furthermore, ensemble analyses and forecasts are found to be sensitive to the localization radius applied to conventional data within the EnKF.

Changes in U.S. East Coast Cyclone Dynamics with Climate Change

Abstract Previous studies investigating the impacts of climate change on extratropical cyclones have primarily focused on changes in the frequency, intensity, and distribution of these events. Fewer studies have directly investigated changes in the storm-scale dynamics of individual cyclones. Precipitation associated with these events is projected to increase with warming owing to increased atmospheric water vapor content. This presents the potential for enhancement of cyclone intensity through increased lower-tropospheric diabatic potential vorticity generation. This hypothesis is tested using the Weather Research and Forecasting Model to simulate individual wintertime extratropical cyclone events along the United States East Coast in present-day and future thermodynamic environments. Thermodynamic changes derived from an ensemble of GCMs for the IPCC Fourth Assessment Report (AR4) A2 emissions scenario are applied to analyzed initial and lateral boundary conditions of observed strongly developing cyclone events, holding relative humidity constant. The perturbed boundary conditions are then used to drive future simulations of these strongly developing events. Present-to-future changes in the storm-scale dynamics are assessed using Earth-relative and storm-relative compositing. Precipitation increases at a rate slightly less than that dictated by the Clausius–Clapeyron relation with warming. Increases in cyclone intensity are seen in the form of minimum sea level pressure decreases and a strengthened 10-m wind field. Amplification of the low-level jet occurs because of the enhancement of latent heating. Storm-relative potential vorticity diagnostics indicate a strengthening of diabatic potential vorticity near the cyclone center, thus supporting the hypothesis that enhanced latent heat release is responsible for this regional increase in future cyclone intensity.

On Stability of Basis Property of Root Vectors System of the Sturm-Liouville Operator with an Integral Perturbation of Conditions in Nonstrongly Regular Samarskii-Ionkin Type Problems

We study a question on stability and instability of basis property of system of eigenfunctions and associated functions of the double differentiation operator with an integral perturbation of Samarskii-Ionkin type boundary conditions.

Climate change impacts on wave and surge processes in a Pacific Northwest (USA) estuary

AbstractTotal water levels (TWLs) within estuaries are influenced by tides, wind, offshore waves, and streamflow, all of which are uniquely affected by climate change. The magnitude of TWL associated with various return periods is relevant to understanding how the hydrodynamics of a bay or estuary may evolve under distinct climate scenarios. A methodology for assessing the hydrodynamic response of a small estuary under major boundary condition perturbations is presented in this study. The coupled Advanced Circulation (ADCIRC) and Simulating Waves Nearshore (SWAN) model was used to simulate wave and water elevation conditions within Tillamook Bay, OR, USA for two long‐term scenarios; 1979–1998 and 2041–2060. The model output provided multidecadal time series of TWLs for statistical analysis. Regional climate data from the North American Regional Climate Change Assessment Program (NARCCAP) were used to drive streamflow modeling (MicroMet/SnowModel/HydroFlow) and meteorological forcing within ADCIRC‐SWAN. WAVEWATCH III, which was forced with global climate data from the Community Climate Science Model (CCSM, a contributing model to NARCCAP), was used to produce open boundary wave forcing. Latitudinal and seasonal gradients were found in TWLs associated with varying return periods for both the hindcast and forecast. Changes in TWLs from hindcast to forecast included the sea level rise component and were also modulated by changes in boundary conditions.

Sensitivity analysis of transonic flow past a NASA airfoil/wing with spoiler deployments

Transonic flow past a NASA SC(2)-0710 airfoil with deployments of a spoiler up to 6° was studied numerically. We consider angles of attack from -0.6° to 0.6° and free-stream Mach numbers from 0.81 to 0.86. Solutions of the unsteady Reynoldsaveraged Navier-Stokes equations were obtained with a finite-volume solver using several turbulence models. Both stationary and time-dependent deployments of the spoiler were examined. The study revealed the existence of narrow bands of the Mach number, angle of attack, and spoiler deflection angle, in which the flow was extremely sensitive to small perturbations. Simulations of 3D flow past a swept wing confirmed the flow sensitivity to small perturbations of boundary conditions.

Rigorous evaluation of propagation losses in arbitrarily shaped waveguide structures using boundary integral – resonant mode expansion and perturbation of boundary conditions

The accurate consideration of propagation losses in arbitrarily shaped waveguide-based structures is studied in this paper. For such a purpose, a software tool based on the perturbation of the boundary conditions on the waveguide metallic walls and on the boundary integral – resonant mode expansion method has been developed. To show the advantages of the proposed technique with respect to the classic power-loss method, the complex propagation wavenumbers of a double ridge and an elliptical waveguide have been first computed and compared with results of a commercial software based on the finite element technique. Next a circular, a sectorial shaped and a triangular shaped waveguide have been considered. Then, a computer-aided design software package based on this modal analysis tool has been applied to predict the propagation loss effects in complex waveguide structures, such as an evanescent mode ridge waveguide filter, a traditional dual mode filter with circular cavities and a twist component for K-band applications.

A New Method for Generating Initial Condition Perturbations in a Regional Ensemble Prediction System: Blending

Abstract A blending method for generating initial condition (IC) perturbations in a regional ensemble prediction system is proposed. The blending is to combine the large-scale IC perturbations from a global ensemble prediction system (EPS) with the small-scale IC perturbations from a regional EPS by using a digital filter and the spectral analysis technique. The IC perturbations generated by blending can well represent both large-scale and small-scale uncertainties in the analysis, and are more consistent with the lateral boundary condition (LBC) perturbations provided by global EPS. The blending method is implemented in the regional ensemble system Aire Limitée Adaptation Dynamique Développement International-Limited Area Ensemble Forecasting (ALADIN-LAEF), in which the large-scale IC perturbations are provided by the European Centre for Medium-Range Weather Forecasts (ECMWF-EPS), and the small-scale IC perturbations are generated by breeding in ALADIN-LAEF. Blending is compared with dynamical downscaling and breeding over a 2-month period in summer 2007. The comparison clearly shows impact on the growth of forecast spread if the regional IC perturbations are not consistent with the perturbations coming through LBC provided by the global EPS. Blending can cure the problem largely, and it performs better than dynamical downscaling and breeding.

The impact of downscaled initial condition perturbations on convective‐scale ensemble forecasts of precipitation

AbstractA major issue in convective‐scale ensemble prediction systems (EPSs) is the specification of effective initial condition perturbations (ICPs). The present work considers the suitability of downscaled ICPs from a multi‐model global EPS for short‐range regional ensemble forecasts of convective precipitation at convection‐permitting resolution. Previous studies indicated the importance of convective‐scale initial condition uncertainties, with the most pronounced impact in weather conditions characterised by weak versus strong synoptic‐scale forcing of convection. However, the downscaled ICPs do not explicitly represent small‐scale uncertainty, which questions their effectiveness in convective‐scale EPSs. To investigate the issue, the high‐resolution ensemble system of the Deutscher Wetterdienst, COSMO‐DE‐EPS, which includes physics perturbations and lateral boundary condition perturbations in addition to ICPs, is employed. Forecasts are compared with a second EPS, identical but without ICPs, for a period of 3.5 months in the central European warm season. Weakly forced conditions are considered separately from strongly forced conditions, using an objective classification based on the area‐averaged convective adjustment time‐scale.Generally for all EPSs, forecast quality measures show a distinct behaviour in strong versus weak forcing conditions. However, the impact of the ICPs is found to be similar in the two regimes. The impact of the ICPs is clearly largest and positive (consistently in terms of ensemble variance and probabilistic forecast quality, but negative for the equitable threat score) in the first six forecast hours when the ICPs dominate the physics perturbations and lateral boundary condition perturbations. The ICPs then decay relatively quickly with lead time as the physics perturbations and lateral boundary condition perturbations start to become important and later dominant. Probabilistic precipitation forecasts by the EPSs outperform the deterministic COSMO‐DE at the same convection‐permitting resolution, and this more strongly in the first nine forecast hours with the EPS applying the ICPs.

Discrete self similarity in filled type I strong explosions

We present new solutions to the strong explosion problem in a non power law density profile. The unperturbed self similar solutions developed by Sedov, Taylor, and Von Neumann describe strong Newtonian shocks propagating into a cold gas with a density profile falling off as r−ω, where \documentclass[12pt]{minimal}\begin{document}$\omega \le \frac{7-\gamma }{\gamma +1}$\end{document}ω≤7−γγ+1 (filled type I solutions), and γ is the adiabatic index of the gas. The perturbations we consider are spherically symmetric and log periodic with respect to the radius. While the unperturbed solutions are continuously self similar, the log periodicity of the density perturbations leads to a discrete self similarity of the perturbations, i.e., the solution repeats itself up to a scaling at discrete time intervals. We discuss these solutions and verify them against numerical integrations of the time dependent hydrodynamic equations. This is an extension of a previous investigation on type II solutions and helps clarifying boundary conditions for perturbations to type I self similar solutions.

Boundary conditions for Kerr-AdS perturbations

Abstract The Teukolsky master equation and its associated spin-weighted spheroidal harmonic decomposition simplify considerably the study of linear gravitational perturbations of the Kerr(-AdS) black hole. However, the formulation of the problem is not complete before we assign the physically relevant boundary conditions. We find a set of two Robin boundary conditions (BCs) that must be imposed on the Teukolsky master variables to get perturbations that are asymptotically global AdS, i.e. that asymptotes to the Einstein Static Universe. In the context of the AdS/CFT correspondence, these BCs allow a non-zero expectation value for the CFT stress-energy tensor while keeping fixed the boundary metric. When the rotation vanishes, we also find the gauge invariant differential map between the Teukolsky and the Kodama-Ishisbashi (Regge-Wheeler-Zerilli) formalisms. One of our Robin BCs maps to the scalar sector and the other to the vector sector of the Kodama-Ishisbashi decomposition. The Robin BCs on the Teukolsky variables will allow for a quantitative study of instability timescales and quasinormal mode spectrum of the Kerr-AdS black hole. As a warm-up for this programme, we use the Teukolsky formalism to recover the quasinormal mode spectrum of global AdS-Schwarzschild, complementing previous analysis in the literature.

Erratum to: Perturbation and interpolation theorems for the H ∞-calculus with applications to differential operators

We prove comparison theorems for the H ∞-calculus that allow to transfer the property of having a bounded H ∞-calculus from one sectorial operator to another. The basic technical ingredient are suitable square function estimates. These comparison results provide a new approach to perturbation theorems for the H ∞-calculus in a variety of situations suitable for applications. Our square function estimates also give rise to a new interpolation method, the Rademacher interpolation. We show that a bounded H ∞-calculus is characterized by interpolation of the domains of fractional powers with respect to Rademacher interpolation. This leads to comparison and perturbation results for operators defined in interpolation scales such as the L p -scale. We apply the results to give new proofs on the H ∞-calculus for elliptic differential operators, including Schrodinger operators and perturbed boundary conditions. As new results we prove that elliptic boundary value problems with bounded uniformly coefficients have a bounded H ∞-calculus in certain Sobolev spaces and that the Stokes operator on bounded domains Ω with ∂Ω ∈ C 1,1 has a bounded H ∞-calculus in the Helmholtz scale L p,σ (Ω), p ∈ (1,∞).

General Theory for Multiple Input-Output Perturbations in Complex Molecular Systems. 1. Linear QSPR Electronegativity Models in Physical, Organic, and Medicinal Chemistry

In general perturbation methods starts with a known exact solution of a problem and add "small" variation terms in order to approach to a solution for a related problem without known exact solution. Perturbation theory has been widely used in almost all areas of science. Bhor's quantum model, Heisenberg's matrix mechanincs, Feyman diagrams, and Poincare's chaos model or "butterfly effect" in complex systems are examples of perturbation theories. On the other hand, the study of Quantitative Structure-Property Relationships (QSPR) in molecular complex systems is an ideal area for the application of perturbation theory. There are several problems with exact experimental solutions (new chemical reactions, physicochemical properties, drug activity and distribution, metabolic networks, etc.) in public databases like CHEMBL. However, in all these cases, we have an even larger list of related problems without known solutions. We need to know the change in all these properties after a perturbation of initial boundary conditions. It means, when we test large sets of similar, but different, compounds and/or chemical reactions under the slightly different conditions (temperature, time, solvents, enzymes, assays, protein targets, tissues, partition systems, organisms, etc.). However, to the best of our knowledge, there is no QSPR general-purpose perturbation theory to solve this problem. In this work, firstly we review general aspects and applications of both perturbation theory and QSPR models. Secondly, we formulate a general-purpose perturbation theory for multiple-boundary QSPR problems. Last, we develop three new QSPR-Perturbation theory models. The first model classify correctly >100,000 pairs of intra-molecular carbolithiations with 75-95% of Accuracy (Ac), Sensitivity (Sn), and Specificity (Sp). The model predicts probabilities of variations in the yield and enantiomeric excess of reactions due to at least one perturbation in boundary conditions (solvent, temperature, temperature of addition, or time of reaction). The model also account for changes in chemical structure (connectivity structure and/or chirality paterns in substrate, product, electrophile agent, organolithium, and ligand of the asymmetric catalyst). The second model classifies more than 150,000 cases with 85-100% of Ac, Sn, and Sp. The data contains experimental shifts in up to 18 different pharmacological parameters determined in >3000 assays of ADMET (Absorption, Distribution, Metabolism, Elimination, and Toxicity) properties and/or interactions between 31723 drugs and 100 targets (metabolizing enzymes, drug transporters, or organisms). The third model classifies more than 260,000 cases of perturbations in the self-aggregation of drugs and surfactants to form micelles with Ac, Sn, and Sp of 94-95%. The model predicts changes in 8 physicochemical and/or thermodynamics output parameters (critic micelle concentration, aggregation number, degree of ionization, surface area, enthalpy, free energy, entropy, heat capacity) of self-aggregation due to perturbations. The perturbations refers to changes in initial temperature, solvent, salt, salt concentration, solvent, and/or structure of the anion or cation of more than 150 different drugs and surfactants. QSPR-Perturbation Theory models may be useful for multi-objective optimization of organic synthesis, physicochemical properties, biological activity, metabolism, and distribution profiles towards the design of new drugs, surfactants, asymmetric ligands for catalysts, and other materials.

Direct radiative forcing and climate effects of anthropogenic aerosols with different mixing states over China

An online coupled regional climate and chemistry model was used to investigate the direct effects of anthropogenic aerosols (sulfate, nitrate, black carbon BC and organic carbon OC) with different mixing states over China. Three mixing assumptions were considered, including external (EM), internal (IM, BC-core surrounded by well mixed scattering-shells) and partially internal (PIM, 32.2% of sulfate and nitrate, 35.5% of BC and 48.5% of OC were internally mixed) mixtures. Results indicated that high levels of anthropogenic aerosols were found in Southwest and Central to East China. Regional mean surface loadings of sulfate, nitrate, BC, primary OC over China were 9.56, 3.64, 2.30, and 2.99 μg m−3, respectively. PIM-aerosol optical depth and single scattering albedo, which were consistent with AERONET and satellite observations, were 0.51 ± 0.37 and 0.95 ± 0.02 in Central to East China, implying that proportions of internally mixed aerosols in PIM were reasonable to some degrees. Both aerosol direct radiative forcing (DRF) and corresponding climate responses were sensitive to aerosol mixing states and BC/OC hygroscopicities. The more BC was internally mixed or hydrophilic, the more solar radiation was absorbed, thus leading to more decreases in cloud amount (CA) and subsequently less surface cooling. Combining with the uncertainties of BC/OC hygroscopicities, regional mean PIM-aerosol DRF at the top of atmosphere ranged from −0.78 to −0.61 W m−2 in all-sky and from −5.24 to −4.95 W m−2 in clear-sky. Additionally, responses of cloud amount and water path, total column absorbed solar radiation (TCASR), surface air temperature and precipitation (TP) to PIM-aerosol DRFs over China were about −0.45∼ −0.37%, −0.44 ∼ −0.32 g m−2, +0.69 ∼ +0.72 W m−2, −0.13 ∼ −0.11 K and −4.56 ∼ −4.29%, respectively. These responses were also sensitive to the lateral boundary condition perturbations especially for CA, TCASR and TP, while DRFs themselves were not.

Uniform analyticity and exponential decay of the semigroup associated with a thermoelastic plate equation with perturbed boundary conditions

In a bounded domain, we consider an Euler–Bernoulli-type thermoelastic plate equation with perturbed boundary conditions. The boundary conditions are such that when the perturbation parameter goes to infinity, we recover the hinged boundary conditions, while one recovers the clamped boundary conditions when the perturbation parameter goes to zero. Relying on resolvent estimates, we show that the underlying semigroup is uniformly, with respect to the perturbation parameter, analytic and exponentially stable. The main features of our proof are appropriate decompositions of the components of the system and the use of Lionsʼ interpolation inequalities.

Recent changes in the flow of the Ross Ice Shelf, West Antarctica

Comparison of surface velocities measured during the Ross Ice Shelf Geophysical and Glaciological Survey (RIGGS, 1973 to 1978) and velocities measured via feature tracking between two Moderate-resolution Imaging Spectroradiometer (MODIS) mosaics (compiled from 2003/4 and 2008/9 images) reveals widespread slowing and minor areas of acceleration in the Ross Ice Shelf (RIS) over the approximately 30 year interval. The largest changes (−13 ma−2) occur near the Whillans and Mercer Ice Streams grounding line in the southernmost part of the ice shelf. Speed has increased over the interval (up to 5 ma−2) between the MacAyeal Ice Stream grounding line and the shelf front, and along the eastern shelf front. Changes in ice thickness computed using ICESat laser altimetry are used together with a well-tested model of the ice shelf to investigate underlying causes of change in the flow of the ice shelf over time. The observed transients represent a combination of recent forcings and ongoing response to ice stream discharge variations over the past millennium. While evidence of older events may be present, the modern signal is dominated by shorter time scale events, including the stagnation of Kamb Ice Stream about 160 years ago, recent changes in basal drag on the Whillans Ice Stream ice plain and, perhaps, iceberg calving. Details in embayment geometry, for example the shallow sea floor below Crary Ice Rise, modulate the spatial pattern of ice shelf response to boundary condition perturbations.

Perturbation of initial and boundary conditions for a limited‐area ensemble: multi‐model versus single‐model approach

AbstractWith the aim of developing of an ensemble forecasting system for the short‐range, the validity of a multi‐analysis multi‐model approach for initial‐ and boundary‐condition perturbations has been explored by the Consortium for Small‐scale Modelling (COSMO). This approach has been adopted in the COSMO Short‐Range Ensemble Prediction System (COSMO–SREPS), in which members are driven by a few operational deterministic runs. In this work, we show the extent to which this approach is beneficial with respect to the downscaling of a single‐model global ensemble, adopted by the operational COSMO Limited‐Area Ensemble Prediction System (COSMO–LEPS). Both COSMO–SREPS and COSMO–LEPS are made up of 16 integrations of the COSMO model with 7 km horizontal mesh size. Results based on the 2010–2011 winter season show that the 16 member COSMO–LEPS outperforms the 16 member COSMO–SREPS, which has boundaries perturbed by three global model runs only, but the reverse is true when the three‐member versions of the two systems are compared. Hence, if only a small‐size Limited‐Area Model (LAM) ensemble is feasible, then the multimodel approach for boundaries gives better results. The dependence of the scores on ensemble size is also studied, showing that all the quality measures tend to saturate before the full size is reached. Finally, the impact of combining the two ensembles is assessed, showing that a positive impact results from adding multi‐model‐driven members to COSMO–LEPS.

Calculation of the spectrum of whispering gallery modes in cylindrical resonators with perturbed boundary conditions

The spectrum of whispering gallery modes for resonators with a small deformation of the boundary is calculated analytically. Cylindrical resonators with two different cross sections (segment close to a circle and segment close to a semicircle) are considered. The calculation is performed for resonators with metal boundaries, but the obtained result is a good approximation for dielectric resonators as well. The applicability limits of the found expressions for the spectra are analysed. It is shown that the spectra calculated using the obtained expressions coincide well with computer-calculated spectra. The perturbation-induces changes in the field distribution are qualitatively studied using numerical simulation.