Normal Mode Model Research Articles

A study of vibrational anharmonicity, fermi resonance interactions, and local mode behavior in CH 3Cl

The infrared spectrum of CH 3Cl has been studied up to 16 500 cm −1 and vibrational assignments made for up to six quanta of excitation in CH stretching. Analysis of a total of 66 vibrational levels has been made in terms of a joint local mode and normal mode model, local with respect to CH stretching vibrations and normal with respect to other vibrations. Fermi resonance interactions with CH 3 deformation overtones must be included in the analysis. Reproduction of the observed vibration levels with a rms error of 2.61 cm −1 is achieved in terms of a 37-parameter model, with the four anharmonicity constants and two Darling-Dennison constants associated with CH 3 group stretchings constrained to be interrelated through local mode relations. The parameters obtained are in excellent accord with those available for the other methyl halides, and with established structural differences between the molecules.

A Normal-Mode Approach to Jovian Atmospheric Dynamics

We propose a nonlinear, quasi-geostrophic, baroclinic model of Jovian atmospheric dynamics, in which vertical variations of velocity are represented by a truncated sum over a complete set of orthogonal functions obtained by a separation of variables of the linearized quasi-geostrophic potential vorticity equation. A set of equations for the time variation of the mode amplitudes in the nonlinear case is then derived. We show that for a planet with a neutrally stable, fluid interior instead of a solid lower boundary, the baroclinic mode represents motions in the interior, and is not affected by the baroclinic modes. One consequence of this is that a normal-mode model with one baroclinic mode is dynamically equivalent to a one layer model with solid lower topography. We also show that for motions in Jupiter's cloudy lower troposphere, the stratosphere behaves nearly as a rigid lid, so that the normal-mode model is applicable to Jupiter. We test the accuracy of the normal-mode model for Jupiter using two simple problem forced, vertically propagating Rossby waves, using two and three baroclinic modes and baroclinic instability, using two baroclinic modes. We find that the normal-road model provide qualitatively correct results, even with only a very limited number of vertical degrees of freedom.

Matched field processing in shallow water for range, depth, and bearing determination: Results of experiment and simulation

Recent theoretical studies have investigated high-resolution methods for estimating source position in a shallow-water waveguide. In this paper, an eigenvector method and a geophysical model based on several types of seismic data are used to estimate the bearing, depth, and range of a moving submerged source in the presence of an interfering surface source at the same range and bearing. The receiver was a sparse horizontal array of four sensors on the bottom (depth 82 m). A variant of the eigenvector method known as the MUSIC method was employed for measured and test field matching and a normal mode model was employed to calculate the test field. The data analyzed were obtained by towing a submerged source directly below the stern of the CFAV Endeavour, so that the surface and submerged sources were essentially at the same range and bearing. Source bearing estimates for the data showed that source bearings were usually well defined although somewhat ambiguous. The presence of the deep source was indicated at 15 and 19.5 Hz but the ranges were completely ambiguous. Simulations produced similar results for the same small number of sensors and a small number of propagating modes. However, simulations also showed that, with a horizontal array of 15 hydrophones and a signal-to-noise ratio of 10 dB, unambiguous resolution of depth, bearing, and range would be obtained.

The equatorial Pacific Ocean prior to and during El Niño of 1982/83-a normal mode model view

The equatorial Pacific Ocean prior to and during El Niño of 1982/83-a normal mode model view

The equatorial Pacific Ocean prior to and during El Niño of 1982/83—a normal mode model view

AbstractAn equatorial ocean model with multiple vertical and horizontal modes is developed for the Pacific Ocean, and run for the five years 1979 to 1983. In the year prior to the warm event, or El Niño, of 1982/83 the model shows an unusually strong and delayed seasonal Kelvin wave generated in the western Pacific. By analogy with the ocean‐atmosphere coupling mechanism postulated by several recent authors for explaining the El Niño/Southern Oscillation cycle, it is suggested that this wave could ultimately have been instrumental in causing the subsequent El Niño to be so strong. The course of events during El Niño is described in terms of equatorial normal modes and related to the evolving wind field. The strength and importance of the second vertical mode during the event is emphasized. Evidence for a coupled system decay mechanism is also found. The model is compared with several types of observations, but principally the expendable bathythermograph (XBT) data of the Pacific Ship of Opportunity Programme. These data are analysed from a normal mode perspective which is related to that of the model, supporting the main conclusions derived from it. Such data have not been analysed previously in this manner, converting observed variables directly to theoretical equatorial wave amplitudes. The adequacy of the data for such analysis is discussed.

Sensitivity of matched field processing to sound-speed profile mismatch for vertical arrays in a deep water Pacific environment

In this paper, the sensitivity of matched field processing to sound-speed profile mismatch will be examined (based upon archival profiles resulting in various degrees of mismatch). A 10-Hz source is considered whose field is generated by a normal mode model and only the water-borne energy is used, thereby eliminating issues relating to the estimation of bottom parameters. This paper will examine how array parameters, i.e., number of phones and array depth, affect range and depth localization for various degrees of mismatch. In particular, it will be seen where an array is most and least sensitive to sound-speed mismatch as a function of depth from the surface and range from the source, and the degree to which range and depth resolution are possible under ideal, as well as under likely, mismatch conditions.

The effective depth of a Pekeris ocean waveguide, including shear wave effects

Weston’s [J. Acoust. Soc. Am. 32, 647–654 (1960)] concept of the effective depth of a Pekeris-type shallow water waveguide has been extended to admit seabeds that support shear waves. The amended effective depth formula leads to estimates of normal mode phase speeds that are surprisingly accurate, without numerical iteration. Results for six hypothetical, though realistic, bottom types are compared with an ‘‘exact’’ normal mode model that includes shear wave effects and another normal mode model that does not.

Maximum-likelihood passive localization using mode filtering

The maximum-likelihood estimator for passive range and depth estimation of an acoustic point source in a shallow-water waveguide is presented. The data from a vertical array of hydrophones are passed through a modal filter, the output of which is the set of complex modal amplitudes associated with the normal-mode model of acoustic propagation. The range and depth estimates are then found by a maximum-likelihood estimation procedure that uses these modal amplitudes as inputs. This technique is compared to the matched-field procedure and is shown to have better signal-to-noise and sidelobe behavior for a given scenario. Results are given for both synthetic and real data. The results with the real data demonstrate the importance of the mode-filtering property of the maximum-likelihood estimator presented in this work.

Sensitivity of matched‐field processing to sound‐speed profile mismatch for vertical arrays in a deep water Pacific environment

In this paper, the sensitivity of matched field processing to sound‐speed profile mismatch will be examined (based upon archival profiles resulting in various degrees of mismatch). A 10‐Hz source is considered whose field is generated by a normal mode model and only the water‐borne energy is used, thereby eliminating issues relating to the estimation of bottom parameters. This paper will examine how array parameters, i.e., number of phones and array depth, affect range and depth localization for various degrees of mismatch. In particular, it will be seen where an array is most and least sensitive to sound‐speed mismatch as a function of depth from the surface and range from the source, and the degree to which range and depth resolution are possible under ideal, as well as under likely, mismatch conditions.

An analysis of the response of the thermospheric normal modes to temporally varying convection at high latitudes

A normal modes model that includes realistic temperature and electron density profiles, and the effects of viscosity, heat conduction, and Lorentz forcing has been used to study the response of the high‐latitude thermosphere to temporal variations in the forcing due to the plasma convection pattern. In particular, the rotation rate of the two‐cell convection pattern was varied from the usual 2π in 24 hours to produce a crude simulation of the effects of temporal variations in the forcing. The model calculation proceeds from an initial steady state at 48 hours through four 6 hour integrations, where the rotation rate is alternately decreased and increased, corresponding to decelerated and accelerated flow. The results show that the rotational wind component is only slightly modified when the plasma rotation has been varied. The divergent wind component responds more strongly to variation of the rotation rate, an effect which is greatest in the E region. Variation of the rotation rate then results in greater magnitude and phase variation with height of the divergent wind component in the E region, indicating that vertically propagating gravity waves are of importance in that height range. To the extent to which varying the rotation rate can simulate variations in the configuration of the plasma convection, as are due to substorms or variations in the interplanetary magnetic field, the model indicates that detailed consideration of such geophysical effects is necessary if the transient response of the neutral wind is to be better understood.

Pulse response measurements over three 45‐km propagation paths in the Florida Straits

Three point reciprocal transmission experiments, undertaken in the Florida Straits, have yielded a 35‐day time history of shallow water pulse responses for 45‐km ranges in a depth of approximately 550 m. The transmitted pulses consist of 4 cycles of a 460‐Hz carrier. Several distinct and separable arrivals persist throughout the 35‐day period for all three ranges. The sound‐speed profile has a 90‐m surface duct above a negative gradient of approximately 0.15 (1/s). Both the sound speed and bathymetry have complicated range dependence and the medium is highly variable in time owing to meanders of the core of the Florida Current. Yet, the arrival patterns are remarkably stable. The observed time spread of all arrivals is nearly a full second. Travel times of the earliest arrivals are consistent with surface duct transmission, whereas later arrivals are consistent with ray model predictions for refracted bottom reflected (RBR) rays. The RBR arrivals are spread over approximately 0.3 s. Ray models, normal mode, and PE models are used to predict and identify intermediate arrivals. [Work supported by ONR.]

Normal modes and local modes in H2 X: beyond the x, K relations

The so-called ‘x, K relations’, which interrelate the normal mode and local mode descriptions of the stretching vibrations of polyatomic molecules, demand simple numerical relationships between the normal mode anharmonicity constants xij and the Darling-Dennison resonance constants Kijkl . In this paper, we examine the consequences of relaxing these demands for the case of H2 X molecules (or groups). Extra terms appear in the equivalent local mode representation which is shown to be similar in form to the recently-described approximate anharmonically coupled anharmonic oscillator model. The normal mode model is also extended to include the effects of Fermi resonance between symmetric stretch and bend, and the equivalent local mode model is derived. The effects of relaxing the x, K relations are examined by reference to the available experimental data for H2O and D2O. It is shown that the extra terms introduced into the model allow a more reliable interpretation of the local mode spectroscopic constants for the stretching vibrations.

Performance modeling for hydrophone arrays

A capability to model the performance of volumentric hydrophone arrays in different acoustic environments has been developed. A normal mode model that uses a WKB approximation is used to calculate acoustic propagation. Ambient noise is not included in the model. The model uses an algorithm that can efficiently calculate signal gain for a large-scale array for a large number of source positions. Performance measures derived from the signal gain calculations have been used to compare the multipath losses for different array geometries in a convergence zone environment.

Gaussian beam tracing for computing ocean acoustic fields

The method of Gaussian beam tracing has recently received a great deal of attention in the seismological community. In comparison to standard ray tracing, the method has the advantage of being free of certain ray-tracing artifacts such as perfect shadows and infinitely high energy at caustics. It also obviates the need for eigenray computations. The technique is especially attractive for high-frequency, range-dependent problems where normal mode, FFP, or parabolic models are not practical alternatives. The Gaussian beam method associates with each ray a beam with a Gaussian intensity profile normal to the ray. The beamwidth and curvature are governed by an additional pair of differential equations, which are integrated along with the usual ray equations to compute the beam field in the vicinity of the central ray of the beam. We have adapted the beam-tracing method to the typical ocean acoustic problem of a point source in a cylindrically symmetric waveguide with depth-dependent sound speed. We present an overview of the method and a comparison of results obtained by conventional ray-tracing, beam-tracing, and full-wave theories. These results suggest that beam tracing is markedly superior to conventional ray tracing.

Experimental determination of modal depth functions from covariance matrix eigenfunction analysis

A low-frequency sound field in a shallow-water duct is usually represented by a set of discrete normal modes, each of which is characterized by a unique vertical pressure amplitude depth function. In many applications, such as matched-field signal processing, it is important to have accurate determinations of these depth functions, which are usually calculated from normal-mode models. By using a densely populated vertical array spanning the water column and a large time-bandwidth test signal, in some circumstances the normal-mode functions of the duct may easily be obtained by diagonalizing the cross-spectral density function of the received signal. The condition required for using this technique is that, over the bandwidth used, the fields of individual modes must be mutually incoherent so that the cross-spectral density matrix is the sum of individual modal diads, making the modal depth functions the same as the matrix eigenfunctions. Experimentally determined mode functions of orders 1 through 4 are compared with theoretical results.

Low-frequency acoustic/seismic propagation in a sloping ocean environment: Comparison between measured results and numerical predictions

The propagation of low-frequency signals and noise in shallow water, and more generally in a bottom-limited environment, is of considerable interest to the Navy. A particular aspect of this problem, namely the relative merits of waterborne versus seismic propagation paths, has been investigated using the results of at-sea measurements and predictions based on geoacoustic and numerical models of the test environment. The experiment was conducted by the Naval Ocean Research and Development Activity in a shallow water region off the southeastern coast of the U. S. A vertical array of hydrophones in the water column and a distribution of tri-axial geophones on the ocean bottom were used to monitor the response to both cw and broadband (explosives) sound sources. Predictions of the propagation were made using the SAFARI version of FFP, the wide-angle version of the Parabolic Equation Method, and the SNAP normal mode model. Preliminary results of the measurements are discussed and comparisons are made with the predictions based on the numerical models.

Modifying normal mode models to permit special range dependence

Benchmarks for testing range‐dependent computer models are very scarce. We suggest that well validated normal mode models having no range dependence can be modified to permit their modeling a range dependence of a special form having parallel boundaries. This would permit modeling a three‐dimensional index of refraction of the special form n2 = a(z) + c + b /(x2 + y2), where a(z) is any depth‐dependent function that an existing normal model will accept. Existing normal mode models are found to need only minor coding changes to permit generating special three‐dimensional solutions of the Helmholtz equation or solutions of the standard parabolic equation.

A range‐dependent normal mode model with full mode coupling

A preliminary version of a range‐dependent transmission loss model is presented. It assumes that the ocean is piecewise constant in range. We use full mode matching at the interfaces to compute the acoustic field. Our approach is similar to that used by Evans and Gilbert [“Acoustic propagation in a refracting ocean waveguide with an irregular interface,” Comp. Math. Appls. 11, 795–805 (1985)] in that we also expand the desired normal modes as a weighted sum of depth dependent basis functions. As is well known, this “Galerkin method” leads to a linear algebraic problem for the unknown weights. However, our approach is different because we insist on using other basis functions that lead to structured algebraic problems, where fast techniques are available. The future development of a fast version of this model will exploit this special algebraic structure. We implement within the generic sonar model because this allows for considerable flexibility in updating developments and the choice of supporting submodels.

Acoustic scattering in an inhomogeneous waveguide: Theory

A transition matrix formalism is developed for the acoustic scattering from a target in a layered, inhomogeneous waveguide. The connection is made with the normal mode model of propagation and the total acoustic wave is expressed as a sum over waveguide modes. For a target in a deep ocean environment, the scattering solution may be expressed in terms of the free-field T matrix and the normal mode wavefunctions for the empty waveguide. Both homogeneous and inhomogeneous layers are considered.

A range‐dependent normal mode model with full mode coupling

A preliminary version of a range‐dependent transmission loss model is presented. It assumes that the ocean is piecewise constant in range. Full mode matching at the interfaces is used to compute the acoustic field. Our approach is similar to that used by Evans and Gilbert [Comput. Math. Appl. 11, 795–805 (1985)] in that we also expand the desired normal modes as a weighted sum of depth‐dependent basis functions. As is well known, this “Galerkin method” leads to a linear algebraic problem for the unknown weights. However, our approach is different because we insist on using other basis functions that lead to structured algebraic problems, where fast techniques are available. The future development of a fast version of this model will exploit this special algebraic structure. Implementation within the generic sonar model is used because this allows for considerable flexibility in updating developments and the choice of supporting submodels.