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S. Hasselmann and K. Hasselmann

Abstract

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.

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K. Hasselmann

Abstract

An optimal linear filter (fingerprint) is derived for the detection of a given time-dependent, multivariate climate change signal in the presence of natural climate variability noise. Application of the fingerprint to the observed (or model simulated) climate data yields a climate change detection variable (detector) with maximal signal-to-noise ratio. The optimal fingerprint is given by the product of the assumed signal pattern and the inverse of the climate variability covariance matrix. The data can consist of any, not necessarily dynamically complete, climate dataset for which estimates of the natural variability covariance matrix exist. The single-pattern analysis readily generalizes to the multipattern case of a climate change signal lying in a prescribed (in practice relatively low dimensional) signal pattern space: the single-pattern result is simply applied separately to each individual base pattern spanning the signal pattern space. Multipattern detection methods can be applied either to test the statistical significance of individual components of a predicted multicomponent climate change response, using separate single-pattern detection tests, or to determine the statistical significance of the complete signal, using a multivariate test. Both detection modes make use of the same set of detectors. The difference in direction of the assumed signal pattern and computed optimal fingerprint vector allows alternative interpretations of the estimated signal associated with the set of optimal detectors. The present analysis yields an estimated signal lying in the assumed signal space, whereas an earlier analysis of the time-independent detection problem by Hasselmann yielded an estimated signal in the computed fingerprint space. The different interpretations can be explained by different choices of the metric used to relate the signal space to the fingerprint space (inverse covariance matrix versus standard Euclidean metric, respectively). Two simple natural variability models are considered: a space-time separability model, and an expansion in terms of P0Ps (principal oscillation patterns). For each model the application of the optimal fingerprint method is illustrated by an example.

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K. Herterich and K. Hasselmann

Abstract

The statistical properties of observed North Pacific sea surface temperature (SST) anomalies are simulated by a simple mixed layer advection and diffusion model with stabilizing feedback and local stochastic forcing by the atmosphere. An optimal fit of the model to the SST auto- and cross-spectra yields the effective temperature advection velocities and diffusion coefficients in the mixed layer, the local feedback factors and the strength and scales of the atmospheric forcing. The results obtained by model fitting are in general agreement with independent direct estimates, where such data are available. The analysis supports previous models in which the origin of midlatitude SST anomalies on time scales of months to a few years is attributed to stochastic forcing by the atmosphere.

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K. Herterich and K. Hasselmann

Abstract

The horizontal dispersion of tracers in the presence of a random field of ocean surface waves is examined. Random fluctuations in the local Stokes-drift current cause a water particle to follow a random-walk path. The associated diffusion coefficients for individual particles, particle pairs and a continuous tracer patch can be calculated rigorously within the framework of perturbation analysis. For a fully developed Pierson-Moskowitz wave spectrum all diffusion coefficients scale as the third power of the wind speed and are typically of the order 10−2 m2 s−1 for a wind speed of 10 m s−1. The diffusion coefficients are strongly anisotropic and decrease approximately exponentially with depth below the sea surface.

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I. R. Young, S. Hasselmann, and K. Hasselmann

Abstract

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.

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G. J. Komen, S. Hasselmann, and K. Hasselmann

Abstract

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.

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K. Hasselmann and T. P. Barnett

Abstract

Many parameters that measure climatic variability have nonstationary statistics, that is, they depend strongly on the phase of the annual cycle. In this case normal statistical analysis techniques based on time-invariant models are inappropriate. Generalized methods accounting for seasonal nonstationarity (phase averaged or cyclostationary models) have been developed to treat such data.

The methods are applied to the problem of predicting El Niño off South America. It is shown that El Niños may be predicted up to a year in advance with considerably more confidence and accuracy using phase-averaged models than with time-invariant models.

In a second application surface air temperature anomalies are predicted over North America from Pacific Ocean sea surface temperatures. Again, the phase-averaged models consistently outperform models based on standard statistical procedures.

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S. Hasselmann, K. Hasselmann, J. H. Allender, and T. P. Barnett

Abstract

Four different parameterizations of the nonlinear energy transfer S nl 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 S nl for different values of the peak-enhancement parameter γ. The second parameterization is based on an EOF expansion of a set of S nl 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 S nl, 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.

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P. Lemke, E. W. Trinkl, and K. Hasselmann

Abstract

The analysis of Arctic (1966–76) and Antarctic (1973–79) sea ice data is presented, and a dynamical model based on white noise atmospheric forcing, local stabilizing relaxation and lateral diffusion and advection is constructed to explain the observations. Longitudinal dependent forcing. feedback, lateral diffusion and advection parameters are derived by fitting the model to the observed cross-spectral matrix of the sea ice anomaly fields. It is inferred that diffusion and advection of sea ice anomalies play an important role in sea ice dynamics. The model advection patterns agree reasonably well with the observed ocean surface circulation in the Arctic Ocean and around Antarctica.

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K. Hasselmann, D. B. Ross, P. Müller, and W. Sell

Abstract

No abstract available.

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