Power-law Wavenumber Spectrum Research Articles

Three dimensional finite element modeling of scattering by objects at rough interfaces

Fully three-dimensional finite element models for buried objects and objects at interfaces have been extended to incorporate interface roughness. Near field results are solved using a fully scattered field formulation, in which all fields in the absence of the target are applied and the target scattering is solved. Results are extended to the far field through the Helmholtz-Kirchhoff integral, using a numerically determined Green’s function approach. Interface roughness is applied using a modified power law wavenumber spectrum. Previous model demonstrations evaluated the scattering in the flat interface limit, in which background fields were prescribed analytically. Now, the presence of the surface roughness forces background fields to be evaluated and prescribed numerically. Such models have strong applicability for predicting scattering by objects buried within or resting on a rough seafloor interface. Model results are compared with various experimental and other modeled results, demonstrating model validity over a wide frequency range. [Work supported by Applied Research Laboratories IR&D and ONR, Ocean Acoustics.]

A NUMERICAL STUDY OF ROUGH-SURFACE SCATTER

A computationally efficient formulation of the method of moments is developed and applied to the problem of scattering from one-dimensional random pressure-release surfaces. Comparisons are made between the predictions of small-slope and operator expansions and a reference solution obtained by matrix inversion for surfaces characterized by a power-law wavenumber spectrum. For Fresnel parameters greater than unity, a significant increase in backscatter over a wide range of grazing angles is found that is inconsistent with composite-roughness theory, and which is only partially predicted by the various approximations. The small-slope approximation is then applied to the three-dimensional scattering problem, and it is shown that the failure of such approximations carries over from the two-dimensional case to three dimensions.

A Numerical Study of Rough-Surface Scatter

A computationally efficient formulation of the method of moments is developed and applied to the problem of scattering from one-dimensional random pressure-release surfaces. Comparisons are made between the predictions of small-slope and operator expansions and a reference solution obtained by matrix inversion for surfaces characterized by a power-law wavenumber spectrum. For Fresnel parameters greater than unity, a significant increase in backscatter over a wide range of grazing angles is found that is inconsistent with composite-roughness theory, and which is only partially predicted by the various approximations. The small-slope approximation is then applied to the three-dimensional scattering problem, and it is shown that the failure of such approximations carries over from the two-dimensional case to three dimensions.

Eulerian and Lagrangian scaling properties of randomly advected vortex tubes

We investigate the Eulerian and Lagrangian spectral scaling properties of vortex tubes, and the consistency of these properties with Tennekes’ (1975) statistical advection analysis and universal equilibrium arguments. We consider three different vortex tubes with power-law wavenumber spectra: a Burgers vortex tube, an inviscid Lundgren single spiral vortex sheet, and a vortex tube solution of the Euler equation. While the Burgers vortex is a steady solution of the Navier–Stokes equation, the other two are unsteady solutions of, respectively, the Navier–Stokes and the Euler equations. In our numerical experiments we study the vortex tubes by subjecting each of them to external ‘large-scale’ sinusoidal advection of characteristic frequency f and length scale ρ.Not only do we find that the Eulerian frequency spectrum ϕE(ω) can be derived from the wavenumber spectrum E(k) using the simple Tennekes advection relation ω ∼ k for all finite advection frequencies f when the vortex is steady, but also when the vortex is unsteady, and in the Lundgren case even when f = 0 owing to the self-advection of the Lundgren vortex by its own differential rotation.An analytical calculation using the method of stationary phases for f = 0 shows that for large enough Reynolds numbers the combination of strain with differential rotation implies that ϕL(ω) ∼ ω−2+Const for large values of ω. We verify numerically that ϕL(ω) does not change when f ≠ 0. With the Burgers vortex tube we are in a position to investigate the spectral broadening of the Eulerian frequency spectrum with respect to the Lagrangian frequency spectrum. A spectral broadening does exist but is different from the spectral broadening predicted by Tennekes (1975).

Power law wave number spectra of fractal particle distributions advected by flowing fluids

Recently, it has been shown that an initial cloud of particles advected by a fluid may, under common circumstances (e.g., when the particles float on the fluid surface), eventually becomes distributed on a fractal set in space. This paper considers the characterization of such fractal spatial patterns by wave number spectra. If a scaling range exists in which the spectrum has an observable power law dependence, k−ρ, then the exponent ρ is given by ρ=D2+1−M, where D2 is the correlation dimension of the fractal attractor and M is the dimension of the relevant space. We find that observability of the power law may be obscured by fluctuations in the k-spectrum, but that averaging can be employed to compensate for this. Theoretical results and supporting numerical computations utilizing a random map are presented. In the companion paper by Sommerer [Phys. Fluids 8, 2441 (1996)], an experimental example utilizing particles floating on the surface of a flowing fluid is given. (More generally we note that our result for the k-spectrum power law exponent ρ should apply to fractal patterns in other physical contexts.)

Power law wave-number spectra of scum on the surface of a flowing fluid.

An initial cloud of particles floating on the surface of a flowing fluid will often tend to a fractal pattern. If the wave-number spectrum of the pattern has an observable power law dependence ${k}^{\ensuremath{-}\ensuremath{\rho}}$, then the exponent $\ensuremath{\rho}$ is predicted to be $\ensuremath{\rho}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}D}_{2}\ensuremath{-}1$, where ${D}_{2}$ is the correlation dimension of the fractal attractor. Numerical, experimental, and theoretical results are shown to support this prediction, but it is also found that, when the observable range in $k$ is limited, the predicted power law scaling can be obscured by fluctuations in the $k$ spectrum. The expected behavior can, however, be restored by use of averaging.

Bounded cascade clouds: albedo and effective thickness

Abstract. If climate models produced clouds having liquid water amounts close to those observed, they would compute a mean albedo that is often much too large, due to the treatment of clouds as plane-parallel. An approximate lower-bound for this "plane-parallel albedo bias" may be obtained from a fractal model having a range of optical thicknesses similar to those observed in marine stratocumulus, since they are more nearly plane-parallel than most other cloud types. We review and extend results from a model which produces a distribution of liquid water path having a lognormal-like probability density and a power-law wavenumber spectrum, with parameters determined by stratocumulus observations. As the spectral exponent approaches -1, the simulated cloud approaches a well-known multifractal, referred to as the "singular model", but when the exponent is -5/3, similar to what is observed, the cloud exhibits qualitatively different scaling properties, the socalled "bounded model". The mean albedo for bounded cascade clouds is a function of a fractal parameter, 0 << 1, as well as the usual plane-parallel parameters such as single scattering albedo, asymmetry, solar zenith angle, and mean vertical optical thickness. A simple expression is derived to determine from the variance of the logarithm of the vertically-integrated liquid water. The albedo is shown to be approximated well by the plane-parallel albedo of a cloud having an "effective" vertical optical thickness, smaller than the mean thickness by a factor χ(f), which is given as an analytic function of f. California stratocumulus have a mean fractal parameter (f) ≈ 0.5, relative albedo bias of 15%, and an effective thickness 30% smaller than the mean thickness (χ ≈ 0.7). For typical observed values of mean liquid water and (f), the effective thickness approximation gives a plane-parallel albedo within 3% of the mean albedo.

Scattering from a rough sedimental seafloor containing shear and layering

A simple first-order perturbation approach is presented, which directly determines the acoustic field scattered by a rough seafloor. The theory is based on the Born approximation. The seafloor is allowed to be shear-supporting or to be layered. Scattering strengths are derived for both monostatic and bistatic scattering, and discussed with respect to their dependence on seafloor parameters and roughness spectrum. Power-law wave-number spectra are found to be appropriate for explaining important features of backscatter, which are the magnitude, the frequency, and grazing-angle dependence. For grazing angles below critical, shear-wave velocities exceeding some 350 m s−1 significantly modify the scattering strength. With respect to narrow-band acoustic signals layering of the seafloor may be of importance. Numerical results are presented for the frequency range 10–100 kHz.

Near-normal incidence scattering from rough, finite surfaces: Kirchhoff theory and data comparison for arctic sea ice

The Kirchhoff theory is applied for the target strength of a rough, circular surface whose roughness is characterized by a two-dimensional, isotropic power-law wave number spectrum, W2(κ) = η2κ−p2. The reflection coefficient for ice and three nondimensional parameters are found to govern the target strength. These parameters are ζ=κ0a, η≡η2ap2−4, and p1=p2−1, where κ0 is the acoustic wave number, a is the radius of the surface, and p1 is the spectral exponent of the one-dimensional power-law wave-number spectrum from which W2(κ) is derived. The general influence of ζ, p1, and η on the target strength is discussed. Calculations of average target strength of the ice/water interface of a submerged cylindrical block of ice are shown, which are then compared with individual realizations of measured target strengths of ice blocks for ζ between 25 and 100, corresponding to frequencies between 20 and 80 kHz for a=0.29 m. Data and theory show that the (smooth surface) form function for a finite surface does not describe the observed diffraction pattern. Instead, the lobes of the pattern diminish and the nulls fill in—i.e., the total backscatter becomes more incoherent—as frequency increases or as the large wave-number components of the roughness spectrum contribute more to the total acoustic return. These comparisons also allowed us to infer the rough surface statistics of the ice surface and the compressional sound-speed structure within the skeletal zone of the ice.

Solar Radiative Fluxes for Stochastic, Scale-invariant Broken Cloud Fields

Solar radiative fluxes for broken, cumuloform cloud fields are examined from the point of view of subgrid parameterization for general circulation models (GCMs). A simple stochastic scaling model is used to simulate extensive broken cloud fields having horizontal variation of optical depth τ characterized by power-law wave-number spectra and power-law area/perimeter and cloud-size distribution properties. Fluxes are computed with the Monte Carlo method of photon transport. Accurate flux estimates for extensive cloud fields are attainable with as few as 50 000 photons/simulation. Radiative fluxes for individual realizations of the cloud model represent the population well. Also, the effect of anisotropic scaling on azimuthally averaged fluxes may often be minimal. The latter two points are beneficial for flux parameterization purposes. Solar fluxes for various scaling, regular, and plane-parallel broken cloud fields are compared. Scaling cloud fields the size of GCM grid boxes often produce significantly different reflectances from those produced by the extreme cases of plane-parallel and white noise arrays. When calculating fluxes at low sun periods, abundant small clouds should not be neglected. Reflectances for model cloud fields with horizontally variable τ are about 10%–15% smaller than those produced by the same cloud field but with all clouds having τ equal to the mean cloudy value of τ for the variable field. In some conditions, fluxes for extensive cloud fields are approximated well by both regular arrays and when horizontal transfer of photons is neglected.

Fractal characterization of the near-surface oceanic bubble layer from deep-ocean reverberation measurements.

At high sea states and low grazing angles, it has been hypothesized that surface reverberation is dominated by backscatter from the bubble layer beneath the sea surface. Assuming the hypothesis is true, a simple stochastic bubble-layer backscatter model has been used to analyze surface reverberation data from four different deep-ocean areas. In every area, the inferred wave-number spectrum for the horizontal structure of the bubble layer is an inverse power law of the form P(K)=AK−β, where A is a constant, K is the horizontal wave number of the structure, and 3≲β≲4. The existence of power-law wave-number spectra indicates that the horizontal structure of the bubble layer is fractal. Using D=4−β/2 for the fractal dimension, 2≲D≲2.5 is obtained for measurements analyzed so far. Fractal structure in the bubble layer implies that horizontal inhomogeneities exist on a wide range of scales and possess scale invariance. Hence, on the average, the medium looks the same at all scales between some ‘‘inner scale’’ and ‘‘outer scale.’’ (Presently, the inner and outer scales are not known.) In this paper some physical processes are considered that could generate fractal structure in the bubble layer and some experimental means are suggested for identifying the processes. [Work supported by ONR.]

Equilibrium Geostrophic Turbulence I: A Reference Solution in a β-Plane Channel

Abstract A numerical solution is calculated for quasi-geostrophic, adiabatic, baroclinic, wind-driven flow in a β channel. The rates of driving and dissipation are such that the solution is turbulent in equilibrium. The equilibrium state is characterized by vigorous turbulent transports of momentum (by isopycnal form drag and horizontal Reynolds stress divergence), mass and heat. Eddy generation is by baroclinic conversion of mean potential energy. A well-defined inertial range with power-law wavenumber spectra develops in a numerical approximation of the limit of vanishing rates of interior diffusion. In that same limit, mean potential vorticity gradients and eddy enstrophy also vanish. The diagnostic eddy diffusivity for heat (i.e., the ratio −v′T/Ty) is found to be nearly uniform in space. Eddy energies are in approximate equipartition between kinetic and potential over a wide range of horizontal scales. The mew jet is only slightly supercritical, by a global linear instability criterion, but it is hi...

Nonlinear theory of type II irregularities in the equatorial electrojet

The two fluid equations of motion describing type II irregularities in the equatorial electrojet are reformulated in terms of a scalar and vector potential for the fluid momentum densities. Elimination of the vector potential yields a single equation, first order in time, for the time evolution of the ionization density. Following Sudan and Keskinen, this equation is solved in the direct interaction approximation for the nonlinear spectral width of the two-dimensional, steady-state turbulence. Numerical results are given for the power law wavenumber spectrum which results from theoretical considerations, computer simulations, and observations of type II irregularities. These results extend the work of Sudan and Keskinen to smaller wavenumbers and higher levels of turbulence.

Power-law wavenumber spectrum deduced from ionospheric scintillation observations

Ionospheric scintillation observations near Boulder, Colorado, from the radio source Cygnus A were spectral analyzed at 26 MHz. The scintillation spectral features are used to calculate the diffraction effects and hence deduce the F-region irregularity wavenumber spectrum for sizes from about 4 to 0.6 km. The wavenumber spectrum on many occasions follows a power-law variation PN(κx) ∝ κx−βH+1, where βH typically varies from 3 to 4. An example based on the Gaussian spectrum shows that the extrapolation to radio frequencies much higher than 26 MHz will not explain the observed equatorial and auroral scintillations. Abrupt changes in electron density are required to support the higher-wavenumber spectral densities that cause these higher-frequency scintillations. These abrupt density changes can be represented by a power-law wavenumber spectrum. The scintillation-deduced power-law variation agrees with in-situ density spectra measurements from the Ogo 6 satellite. Several production mechanisms that follow a power-law variation are discussed. The measurement of the wavenumber spectrum will help explain the mechanism that produces small-scale irregularities.