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

Spectral dissipation of finite-depth gravity waves due to turbulent bottom friction

The spectral dissipation of finite-depth random gravity waves is evaluated for a given empirical relationship between the local bottom stress and the wave field. The dissipation is quasilinear. Computations for a quadratic friction law yield satisfactory agreement with wave measurements at two stations in the Gulf of Mexico

Functional renormalization group in the broken symmetry phase: momentum dependence and two-parameter scaling of the self-energy

We include spontaneous symmetry breaking into the functional renormalization group (RG) equations for the irreducible vertices of Ginzburg-Landau theories by augmenting these equations by a flow equation for the order parameter, which is determined from the requirement that at each RG step the vertex with one external leg vanishes identically. Using this strategy, we propose a simple truncation of the coupled RG flow equations for the vertices in the broken symmetry phase of the Ising universality class in D dimensions. Our truncation yields the full momentum dependence of the self-energy Sigma (k) and interpolates between lowest order perturbation theory at large momenta k and the critical scaling regime for small k. Close to the critical point, our method yields the self-energy in the scaling form Sigma (k) = k_c^2 sigma^{-} (k | xi, k / k_c), where xi is the order parameter correlation length, k_c is the Ginzburg scale, and sigma^{-} (x, y) is a dimensionless two-parameter scaling function for the broken symmetry phase which we explicitly calculate within our truncation.Comment: 9 pages, 4 figures, puplished versio

Directional wave spectra observed during JONSWAP 1973

Estimates of the directional wave spectrum obtained from the meteorological buoy of the University of Hamburg and a pitch-and-roll buoy of the Institute of Oceanographic Sciences are reported from a series of measurements made within the framework of the Joint North Sea Wave Project during September 1973. Three main aspects were considered. First, the properties and parameterization of the directional spectrum were studied when the waves were generated by steady winds without any significant swell contribution. The results do not support the parameterization proposed by Mitsuyasu et al. (1975) and are in agreement with a parameterization in which the peak frequency is the relevant scale parameter. Second, comparisons are made between two independent methods of fitting the data exactly by means of a maximum likelihood technique (Long and Hasselmann, 1979) and a least-squares technique. The two methods give very similar fits to the observed data. Finally, the response of the directional wave spectrum to veering winds is considered and a simple model is constructed as a first attempt to describe some of the observations

Critical behavior of weakly interacting bosons: A functional renormalization group approach

We present a detailed investigation of the momentum-dependent self-energy Sigma(k) at zero frequency of weakly interacting bosons at the critical temperature T_c of Bose-Einstein condensation in dimensions 3<=D<4. Applying the functional renormalization group, we calculate the universal scaling function for the self-energy at zero frequency but at all wave vectors within an approximation which truncates the flow equations of the irreducible vertices at the four-point level. The self-energy interpolates between the critical regime k > k_c, where k_c is the crossover scale. In the critical regime, the self-energy correctly approaches the asymptotic behavior Sigma(k) \propto k^{2 - eta}, and in the short-wavelength regime the behavior is Sigma(k) \propto k^{2(D-3)} in D>3. In D=3, we recover the logarithmic divergence Sigma(k) \propto ln(k/k_c) encountered in perturbation theory. Our approach yields the crossover scale k_c as well as a reasonable estimate for the critical exponent eta in D=3. From our scaling function we find for the interaction-induced shift in T_c in three dimensions, Delta T_c / T_c = 1.23 a n^{1/3}, where a is the s-wave scattering length and n is the density, in excellent agreement with other approaches. We also discuss the flow of marginal parameters in D=3 and extend our truncation scheme of the renormalization group equations by including the six- and eight-point vertex, which yields an improved estimate for the anomalous dimension eta \approx 0.0513. We further calculate the constant lim_{k->0} Sigma(k)/k^{2-eta} and find good agreement with recent Monte-Carlo data.Comment: 23 pages, 7 figure

Hamiltonian formalism and the Garrett-Munk spectrum of internal waves in the ocean

Wave turbulence formalism for long internal waves in a stratified fluid is developed, based on a natural Hamiltonian description. A kinetic equation appropriate for the description of spectral energy transfer is derived, and its self-similar stationary solution corresponding to a direct cascade of energy toward the short scales is found. This solution is very close to the high wavenumber limit of the Garrett-Munk spectrum of long internal waves in the ocean. In fact, a small modification of the Garrett-Munk formalism includes a spectrum consistent with the one predicted by wave turbulence.Comment: 4 pages latex fil

Self-energy and critical temperature of weakly interacting bosons

Using the exact renormalization group we calculate the momentum-dependent self-energy Sigma (k) at zero frequency of weakly interacting bosons at the critical temperature T_c of Bose-Einstein condensation in dimensions 3 <= D < 4. We obtain the complete crossover function interpolating between the critical regime k << k_c, where Sigma (k) propto k^{2 - eta}, and the short-wavelength regime k >> k_c, where Sigma (k) propto k^{2 (D-3)} in D> 3 and Sigma (k) \propto ln (k/k_c) in D=3. Our approach yields the crossover scale k_c on the same footing with a reasonable estimate for the critical exponent eta in D=3. From our Sigma (k) we find for the interaction-induced shift of T_c in three dimensions Delta T_c / T_c approx 1.23 a n^{1/3}, where a is the s-wave scattering length and n is the density.Comment: 4 pages,1 figur

Topological Defects and the Spin Glass Phase of Cuprates

We propose that the spin glass phase of cuprates is due to the proliferation of topological defects of a spiral distortion of the antiferromagnet order. Our theory explains straightforwardly the simultaneous existence of short range incommensurate magnetic correlations and complete a-b symmetry breaking in this phase. We show via a renormalization group calculation that the collinear O(3)/O(2) symmetry is unstable towards the formation of local non-collinear correlations. A critical disorder strength is identified beyond which topological defects proliferate already at zero temperature.Comment: 7 pages, 2 figures. Final version with some changes and one replaced figur