Spin-wave interactions in quantum antiferromagnets

We study spin-wave interactions in quantum antiferromagnets by expressing the usual magnon annihilation and creation operators in terms of Hermitian field operators representing transverse staggered and ferromagnetic spin fluctuations. In this parameterization, which was anticipated by Anderson in 1952, the two-body interaction vertex between staggered spin fluctuations vanishes at long wavelengths. We derive a new effective action for the staggered fluctuations only by tracing out the ferromagnetic fluctuations. To one loop order, the renormalization group flow agrees with the nonlinear-$\sigma$-model approach.Comment: 7 pages, no figures; new references added; extended discussion on vertex structure. To appear in Europhysics Letter

On the nonlinear mapping of an ocean wave spectrum into a synthetic aperture radar image spectrum and its inversion

A new, closed nonlinear integral transformation relation is derived describing the mapping of a two-dimensional ocean wave spectrum into a synthetic aperture radar (SAR) image spectrum. The general integral relation is expanded in a power series with respect to orders of nonlinearity and velocity bunching. The individual terms of the series can be readily computed using fast Fourier transforms. The convergence of the series is rapid. The series expansion is also useful in identifying the different contributions to the net imaging process, consisting of the real aperture radar (RAR) cross-section modulation, the nonlinear motion (velocity bunching) effects, and their various interaction products. The lowest term of the expansion with respect to nonlinearity order yields a simple quasi-linear approximate mapping relation consisting of the standard linear SAR modulation expression multiplied by an additional nonlinear Gaussian azimuthal cutoff factor. The cutoff scale is given by the rms azimuthal (velocity bunching) displacement. The same cutoff factor applies to all terms of the power series expansion. The nonlinear mapping relation is inverted using a standard first-guess wave spectrum as regularization term. This is needed to overcome the basic 180° mapping ambiguity and the loss of information beyond the azimuthal cutoff. The inversion is solved numerically using an iteration technique based on the successive application of the explicit solution for the quasi-linear mapping approximation, with interposed corrections invoking the full nonlinear mapping expression. A straightforward application of this technique, however, generally yields unrealistic discontinuities of the best fit wave spectrum in the transition region separating the low azimuthal wave number domain, in which useful SAR information is available and the wave spectrum is modified, from the high azimuthal wave number region beyond the azimuthal cutoff, where the first-guess wave spectrum is retained. This difficulty is overcome by applying a two-step inversion procedure. In the first step the energy level of the wave spectrum is adjusted, and the wave number plane rotated and rescaled, without altering the shape of the spectrum. Using the resulting globally fitted spectrum as the new first-guess input spectrum, the original inversion method is then applied without further constraints in a second step to obtain a final fine-scale optimized spectrum. The forward mapping relation and inversion algorithms are illustrated for three Seasat cases representing different wave conditions corresponding to weakly, moderately, and strongly nonlinear imaging conditions

Computations and parameterizations of the nonlinear energy transfer in a gravity-wave spectrum. Part I: A new method for efficient computations of the exact nonlinear transfer integral

A more efficient method of computing the nonlinear transfer in a surface wave spectrum is developed which is symmetrical with respect to all wavenumbers of the resonant interaction quadruplets. This enables a large number of computations to be carried out, as required for investigations of the spectral energy balance or the development of parameterizations. New results are presented for finite-depth surface waves. By filtering out regions in interaction phase space, the assumptions involved in the narrow-peak and local-interaction approx-imations are investigated. Both approximations are found to be useful but are generally not sufficiently accurate to replace exact computations or provide adequate parameterizations for wave models

On the existence of a fully developed wind-sea spectrum

We consider the energy transfer equation for well-developed ocean waves under the influence of wind, and study the conditions for the existence of an equilibrium solution in which wind input, wave-wave interaction and dissipation balance each other. For the wind input we take the parameterization proposed by Snyder and others, which was based on their measurements in the Bight of Abaco and which agrees with Miles's theory. The wave-wave interaction is computed with an algorithm given recently by S. Hasselmann and others. The dissipation is less well-known, but we will make the general assumption that it is quasi-linear in the wave spectrum with a factor coefficient depending only on frequency and integral spectral parameters. In the first part of this paper we investigate whether the assumption that the equilibrium spectrum exits and is given by the Pierson-Moskowitz spectrum with a standard type of angular distribution leads to a reasonable dissipation function. We find that this is not the case. Even if one balances the total rate of change for each frequency (which is possible), a strong angular imbalance remains. Thus the assumed source terms are not consistent with this type of asymptotic spectrum. In the second part of the paper we choose a different approach. We assume that the dissipation is given and perform numerical experiments simulating fetch-limited growth, to see under which conditions a stationary solution can be reached. For the dissipation we take K. Haseelmann's form with two unknown parameters. From our analysis it follows that for a certain range of values of these parameters, a quasi-equilibrium solution results. We estimate the relation between dissipation parameters and asymptotic growth rates. For equilibrium spectra, the input, dissipation and nonlinear-transfer source functions are all significant in the energy-containing range of the spectrum. The energy balance proposed by Zakharov and Filonenko in 1966 and Kitaigorodskii in 1983, in which dissipation is assumed to be significant only at high frequencies, yields a spectrum that grows too rapidly and does not approach equilibrium. One of our equilibrium solutions has a one-dimensional spectrum that lies close to the Pierson-Moskowitz spectrum. However, the angular distribution differs in some important features from standard spreading functions. The energy balance of this equilibrium spectrum is analysed in detail

Computations of the response of a wave spectrum to a sudden change in wind direction

The response of a wind-sea spectrum to a step function change in wind direction is investigated theoretically for a sequence of direction changes ranging from 30° to 180°, in increments of 30°. Two spectral energy balance models are used: the model EXACT-NL, in which the nonlinear transfer is represented exactly, and the model 3G-WAM, in which the nonlinear transfer is approximated by the discrete interaction parameterization. In both modes the input and dissipation source functions are taken from the energy balance proposed by Komen et al. The operational model 3G-WAM reproduces fairly closely the EXACT-NL results. For wind direction changes less than 60°, the wind-sea direction adjusts smoothly. The high-frequency components relax more rapidly to the new wind direction than the low-frequency components. The computed relaxation rates are generally consistent with the analysis of measured directional spectra by D.E. Hasselman et al. and Allender et al. However, the relaxation rate is found to be a function of wind speed as well as frequency. For wind direction changes greater than 60°, a second, independent wind-sea spectrum is generated in the new wind direction, while the old wind-sea gradually decays as swell

Three year global intercomparison of ERS-1 SAR wave mode spectral retrievals with WAM model data

A global statistical intercomparison was carried out for the period January 1993 to December 1995 between wave spectra retrieved from ERS-1 SAR Wave Mode (SWM) data using an inversion algorithm of the closed nonlinear wave-to-SAR spectral mapping relation and wave spectra computed with the wave model WAM. A combined quality analysis of the satellite data and a performance analysis of the retrieval algorithm was carried out. The assessment yielded about 75 percent successful retrievals. Time series of significant wave heights in different parts of the world oceans showed good overall agreement. However, a more detailed investigation exploring the distinct spectral properties of the windsea and swell content of the wave spectra revealed a small but systematic model overprediction of windsea and an underprediction of swell systems while the overpredicted windsea can be attributed to incorrect wind fields, the underpredicted swell could be caused by deficiencies in the model

Statistical analysis and intercomparison of WAM model data with global ERS-1 SAR wave mode spectral retrievals over 3 years

Ocean wave spectra were retrieved from a set of ERS-1 synthetic aperture radar (SAR) wave mode (SWM) spectra between January 1993 and December 1995. An assessment is given of the SWM data quality and the retrieval performance as well as the operational feasibility of the retrieval algorithm. Sensitivity studies are performed to demonstrate the weak residual dependence of the retrieval on the first-guess input spectrum. The mean spectral parameters of the SWM retrievals are compared with spectral parameters from collocated wave model (WAM) spectra. The time series of SWM-retrieved and WAM-derived monthly mean significant wave heights H-s in various ocean basins show good overall agreement but with a small systematic underestimation of H-s by the WAM. A decomposition of the wave spectra into wind sea and swell reveals an average 10% overprediction of the wind sea by the WAM while swell is underpredicted by 20-30%. The positive wind-sea bias exhibits no clear wave height dependence, while the negative swell bias decreases with swell wave height. This could be due to a too strong damping in the WAM at low frequencies. Detailed regional investigations point to the existence of smaller-scale phenomena, which may not be adequately reproduced by the WAM at the present resolution of the wind forcing. Finally, an intercomparison is made of the observed and modeled azimuthal cutoff length scales, and global distributions are investigated. Ratios of the observed azimuthal cutoff wavenumber to the mean azimuthal wavenumber component indicate that about 75% of the swell can be directly resolved by the SAR, while about 70% of the wind sea lies at least partially beyond the cutoff

Computations and parameterizations of the nonlinear energy transfer in a gravity-wave spectrum. Part II: Parameterizations of the nonlinear energy transfer for application in wave models

Four different parameterizations of the nonlinear energy transfer Snl in a surface wave spectrum are in investigated. Two parameterizations are based on a relatively small number of parameters and are useful primarily for application in parametrical or hybrid wave models. In the first parameterization, shape-distortion parameters are introduced to relate the distribution Snl for different values of the peak-enhancement parameter γ. The second parameterization is based on an EOF expansion of a set of Snl computed for a number of different spectral distributions. The remaining two parameterizations represent operator forms that contain the same number of free parameters as used to describe he wave spectrum. Such parameterizations with a matched number of input and output parameters are required for numerical stability in high-resolution discrete spectral models. A cubic, fourth-order diffusion-operator expression derived by a local-interaction expansion is found to be useful for understanding many of the properties of Snl, but is regarded as too inaccurate in detail for application in most wave models. The best results are achieved with a discrete-interaction operator parameterization, in which a single interaction configuration, together with its mirror image (representing a two-dimensional continuum of interactions with respect to a variable reference wavenumber scale and direction) is used to simulate the net effect of the full five-dimensional interaction continuum

Assimilation of wave data into the wave model WAM using an impulse response function method

A new method for the assimilation of wave data into a third-generation wave model is presented, Deviations between observed and modeled wave spectra are used to derive corrections of the wind field which drives the wave model, The wave field can then be subsequently corrected by a new integration of the wave model with the improved wind field, A basic difficulty of such dynamically consistent wave data assimilations schemes which correct both wind and wave data is the nonsynchronous and nonlocal nature of the wind field corrections: errors observed in the wave spectrum at a given measurement time and location can be produced by errors in the wind field at much earlier times and far distant locations, Formally, these problems can be rigorously resolved by the adjoint modeling method, However, in practice, the adjoint technique requires an order of magnitude more computer time than the integration of the wave model itself, Here an alternative method is developed, The linearized wave model equation which relates small wind to wave spectrum changes is inverted, The central assumption of the inversion is that the wind impact functions representing the impulse response (Green's) function of the wave evolution can be approximated by a S-function, Physically, this implies that the wind field perturbations responsible for observed perturbations in the wave spectrum can be regarded as strongly localized in space and time for any given component of the spectrum, To obtain stable estimates, the corrections for different wave components are averaged over wavenumber clusters representing different wave systems, For cases in which the linear approximation is inadequate, the method can be applied iteratively, Tests of the concept and application of the method for a number of synthetic wind field cases are encouraging