Spectral Energy Balance Equation Research Articles

Instability of energy spectrum disturbance for ship turbulent wakes: SAR imaging simulation and analysis

We present a ship turbulent wake synthetic aperture radar (SAR) simulator (STW-SARSim) producing two-dimensional true-scale turbulent wakes in SAR images that extend for kilometers. In this paper, we innovatively overcome the sea surface time-varying problem in solving the spectral energy balance equation by transforming the unsteady problem into a steady problem and considering the dark trailing centerline region of the turbulent wake in SAR images as a texture modulation effect of the Bragg wave for sea surface scattering. We also present SAR images simulated in three different bands: (i) Sentinel-1A C-band, (ii) TerraSAR X-band, and (iii) ALOS PALSAR L-band to validate the reliability of our proposed method and analyze the effects of wind speed, wind direction, and dissipation rates on the turbulent wake simulation. This paper provides new insights into the simulation of turbulent wakes in SAR images of large scenes inspired by recent research on detection algorithms. It sets the stage for future dataset generation for detection.

Accuracy aspects of conventional discretization methods for scalar transport with nonzero divergence velocity field arising from the energy balance equation

AbstractWe are concerned with the numerical solution of a linear transport problem with nonzero divergence velocity field that originates from the spectral energy balance equation describing the evolution of wind waves and swells in coastal seas. The discretization error of the commonly used first‐order upwind finite difference and first‐order vertex‐centered upwind finite volume schemes in one space dimension is analyzed. Smoothness of nondivergent velocity field plays a crucial role in this. No such analysis has been attempted to date for such problems. The two schemes studied differ in the manner in which they treat the scalar flux numerically. The finite difference variant is shock captured, whereas the vertex‐centered finite volume approximation employs an arithmetic mean of the velocity and appears not to be flux conservative. The methods are subsequently extended to two dimensions on triangular meshes. Numerical experiments are provided to verify the convergence analysis. The main finding is that the finite difference scheme displays optimal rates of convergence and offers higher accuracy over the finite volume scheme, regardless the regularity of the velocity field. The latter scheme notably yields convergence rates of 0.5 and 0 in L2‐norm and L∞‐norm, respectively, when the velocity field is not smooth. A test case illustrating wave shoaling and refraction over submerged shoals is also presented and demonstrates the practical importance of flux conservation.

A modified Lin equation for the energy balance in isotropic turbulence

• Resolution of the scale-invariance paradox is used to modify the Lin equation. • Exact symmetries justify the division of wavenumber space into different regions. • The roles of these different regions can be established by energy conservation. • Equivalent decompositions can be applied to the Navier–Stokes equations. • This could be the basis of new theoretical approaches to the turbulence problem. At sufficiently large Reynolds numbers, turbulence is expected to exhibit scale-invariance in an intermediate (“inertial”) range of wavenumbers, as shown by power law behavior of the energy spectrum and also by a constant rate of energy transfer through wavenumber. However, there is an apparent contradiction between the definition of the energy flux (i.e., the integral of the transfer spectrum) and the observed behavior of the transfer spectrum itself. This is because the transfer spectrum T ( k ) is invariably found to have a zero-crossing at a single point (at k = k * ), implying that the corresponding energy flux cannot have an extended plateau but must instead have a maximum value at k = k * . This behavior was formulated as a paradox and resolved by the introduction of filtered/partitioned transfer spectra, which exploited the symmetries of the triadic interactions (J. Phys. A: Math. Theor., 2008). In this paper we consider the more general implications of that procedure for the spectral energy balance equation, also known as the Lin equation. It is argued that the resulting modified Lin equations (and their corresponding Navier–Stokes equations) offer a new starting point for both numerical and theoretical methods, which may lead to a better understanding of the underlying energy transfer processes in turbulence. In particular the filtered partitioned transfer spectra could provide a basis for a hybrid approach to the statistical closure problem, with the different spectra being tackled using different methods.

A Two-Dimensional Wave Prediction Model Along the Best Track of Typhoon Linda 1997

Problem statement: A two-dimensional wave prediction model along the best track of Typhoon Linda 1997 was interested to study the impact of typhoon wind-wave characteristics. The dynamical wave model with deep water condition was used to predict the wave height (Hs) of Typhoon Linda before and after entering into the Gulf of Thailand (GoT). Approach: The standard one-way nested grid for a regional scale of the third generation WAve Model Cycle 4 (WAMC4) is scrutinized in the present study. This model is enabled to solve the spectral energy balance equation on a coarse resolution grid in order to produce boundary conditions for a small area by the nested grid technique along the best track of typhoon. The model takes full advantage of the fine resolution wind fields in space and time produced by the available US Navy Operational Global Atmospheric Prediction System (NOGAPS) model with 1° resolution. The nested grid application was developed in order to gradually increase the resolution from the open ocean towards the South China Sea (SCS) and the Gulf of Thailand (GoT) respectively. Results: The model results were predicted at five stations which were before and during the typhoon entering into the GoT. The wind speeds of the stations 1-5 were in ranges of 5.14-29.81, 4.11-28.27, 0.51-24.67, 0.51-31.35 and 0.51-33.41 m sec-1, respectively. While the Hs of these stations were found in ranges of 0.54-2.99, 0.68-2.85, 0.11-1.57, 0.12-2.92 and 0.09-2.76 m, respectively. The model results were compared with buoy observations at Ko-Chang and Rayong locations in the GoT which were obtained from the Seawatch project. The comparison of those results at Ko-Chang and Rayong showed the percentage errors of 11.20 and 15.12% respectively. Conclusion: The model results presented the relationship of typhoon wind-induced ocean wave at five stations along the best track. The tendency of the Hs from the model in the spherical coordinate propagation with deep water condition in the fine grid domain was in good agreement with the Hs from the observations.

Topographical Scattering of Waves: Spectral Approach

The topographical scattering of gravity waves is investigated using a spectral energy balance equation that accounts for first-order wave-bottom Bragg scattering. This model represents the bottom topography and surface waves with spectra, and evaluates a Bragg scattering source term that is theoretically valid for small bottom and surface slopes and slowly varying spectral properties. The robustness of the model is tested for a variety of topographies uniform along one horizontal dimension including nearly sinusoidal, linear ramp, and step profiles. Results are compared with reflections computed using an accurate method that applies integral matching along vertical boundaries of a series of steps. For small bottom amplitudes, the source term representation yields accurate reflection estimates even for a localized scatterer. This result is proved for small bottom amplitudes h relative to the mean water depth H. Wave reflection by small amplitude bottom topography thus depends primarily on the bottom elevation variance at the Bragg resonance scales, and is insensitive to the detailed shape of the bottom profile. Relative errors in the energy reflection coefficient are found to be typically 2h∕H.

Numerical and Physical Diffusion: Can Wave Prediction Models Resolve Directional Spread?

Abstract A new semi-Lagrangian advection scheme called multistep ray advection is proposed for solving the spectral energy balance equation of ocean surface gravity waves. Existing so-called piecewise ray methods advect wave energy over a single time step using “pieces” of ray trajectories, after which the spectrum is updated with source terms representing various physical processes. The generalized scheme presented here allows for an arbitrary number N of advection time steps along the same rays, thus reducing numerical diffusion, and still including source-term variations every time step. Tests are performed for alongshore uniform bottom topography, and the effects of two types of discretizations of the wave spectrum are investigated, a finite-bandwidth representation and a single frequency and direction per spectral band. In the limit of large N, both the accuracy and computation cost of the method increase, approaching a nondiffusive fully Lagrangian scheme. Even for N = 1 the semi-Lagrangian scheme test results show less numerical diffusion than predictions of the commonly used first-order upwind finite-difference scheme. Application to the refraction and shoaling of narrow swell spectra across a continental shelf illustrates the importance of controlling numerical diffusion. Numerical errors in a single-step (Δt = 600 s) scheme implemented on the North Carolina continental shelf (typical swell propagation time across the shelf is about 3 h) are shown to be comparable to the angular diffusion predicted by the wave–bottom Bragg scattering theory, in particular for narrow directional spectra, suggesting that the true directional spread of swell may not always be resolved in existing wave prediction models, because of excessive numerical diffusion. This diffusion is effectively suppressed in cases presented here with a four-step semi-Lagrangian scheme, using the same value of Δt.

Swell Transformation across the Continental Shelf. Part II: Validation of a Spectral Energy Balance Equation

Abstract State-of-the-art parameterizations of the interactions of waves with a sandy bottom are evaluated using extensive field observations of swell evolution across the North Carolina continental shelf and hindcasts performed with the spectral wave prediction model CREST. The spectral energy balance equation, including bottom friction and wave–bottom scattering source terms, was integrated numerically for selected time periods with swell-dominated conditions. Incident wave spectra at the model boundary were estimated from buoy measurements near the shelf break, assuming weak spatial variations in the offshore wave field. The observed strong and variable decay of the significant wave height across the shelf is predicted accurately with an overall scatter index of 0.15. Predicted wave directional properties at the peak frequency also agree well with observations, with a 5° root-mean-square error on the mean direction at the peak frequency and a 0.22 scatter index for the directional spread. Slight modifi...

Shoaling transformation of wave frequency‐directional spectra

A Boussinesq model for the nonlinear transformation of the frequency‐directional spectrum and bispectrum of surface gravity waves propagating over a gently sloping, alongshore uniform beach is compared with field and laboratory observations. Outside the surf zone the model predicts the observed spectral evolution, including energy transfers to harmonic components traveling in the direction of the dominant waves, and the cross‐interactions of waves traveling in different directions that transfer energy to components with the vector sum wavenumber. The sea surface elevation skewness and asymmetry, third‐order moments believed to be important for sediment transport, also are predicted well. Effects of surf zone wave breaking are incorporated with a heuristic frequency‐dependent dissipation term in the spectral energy balance equation and an empirical relaxation of the bispectrum to Gaussian statistics. The associated coefficients are calibrated with observations that span a wide range of surf zone conditions. With calibrated coefficients, the model predicts observed surf zone frequency spectra well and surf zone skewness and asymmetry fairly well. The observed directional spectra inside the surf zone are broader than the predicted spectra, suggesting that neglected scattering effects associated with the random onset of wave breaking or with higher‐order nonlinearity may be important.

Bragg scattering of random surface gravity waves by irregular seabed topography

The Bragg scattering of random, non-stationary surface gravity waves by random topography on a gently sloping bottom is investigated. A correction is given of previously published expressions for the triad wave–wave–bottom interaction source term in the spectral energy balance equation, and the result is reconciled with deterministic theories for the reflection of waves from sinusoidal seabed undulations. For both normal and oblique incidence, the stochastic and deterministic theories are equivalent in the limit of long propagation distances. Even for relatively short distances (for example two bottom undulations), the reflected energy predicted by the stochastic source term formulation is generally within 15% of values predicted by deterministic theories. The detuning of Bragg resonance by refraction and shoaling is discussed, suggesting practical validity conditions for the stochastic theory. The effect of bottom scattering on swell propagation is illustrated with numerical model computations for the North Carolina continental shelf using high-resolution bathymetry and an efficient semi-implicit scheme to evaluate the bottom scattering source term and integrate the energy balance equation. Model results demonstrate the importance of forward scattering of waves that propagate at large oblique angles over bottom features with typical scales of one to several surface wavelengths. This process contributes significantly to the directional spread of swell on the continental shelf by diffusing energy, in the spectrum, around the mean wave direction. Back-scattering, caused by bottom features with crests parallel to those of the surface waves and wavelengths close to half the surface wavelength, is weak, owing to the sharp roll-off of the bottom elevation spectrum at high wavenumbers. Model predictions are consistent with field measurements.

A Hybrid Eulerian–Lagrangian Model for Spectral Wave Evolution with Application to Bottom Friction on the Continental Shelf

Abstract A hybrid Eulerian–Lagrangian wave model is presented that solves the spectral energy balance equation for surface gravity waves in varying depth. The energy of each spectral component is advected along (Lagrangian) ray trajectories. The source terms in the energy balance equation (e.g., interactions between wave components and nonconservative processes) are computed on a fixed Eulerian grid and interpolated onto the ray trajectories. The source terms are integrated in time along the rays. This integration is performed in parallel over the entire model domain. The main advantage of this new model, named CREST (Coupled Rays with Eulerian Source Terms), is that refraction of waves by subgrid-scale depth variations is evaluated accurately using precomputed rays, and thus the model can be applied with relatively coarse source term grids to large coastal areas. Hindcasts of swell evolution across the North Carolina continental shelf are presented for a source term restricted to energy dissipation in th...

Modeling short sea wave energy distributions in the far wakes of ships

A mathematical model based on the spectral energy balance equation for suppression of shortwave energy in centerline ship wakes is developed. At distances behind a ship of more than a few ship lengths, the dominant influences on short wave energy are the growing effects of wind and nonlinear energy transfer from longer waves, and the attenuating effects of turbulence and viscous attenuation which can be greatly enhanced by surface film distributions which are modified by passage of the ship. These influences are modeled as “source terms” in the energy balance equation. The model is exercised using turbulence data from scale model experiments and surface film distributions measured in an actual ship wake. Agreement between the resulting distribution of Bragg‐scattering wave energy and radar image intensity in the far wake is excellent. The influences of variations in the source term model parameters are explored numerically.

Application of a hindcast model to waves in Bass Strait, Australia

This paper describes the application of a spectral hindcast model in computing the wave climate in Bass Strait, Australia, from historical wind fields. The model is based on the numerical solution of the spectral energy balance equation. The conclusion reached is that the physical data available are inadequate to calibrate the model, and that less elaborate models may suffice.

A Theory of Gaps in the Turbulence Spectra of Stably Stratified Shear Flows

A gap and adjacent bump is predicted to occur in the velocity spectrum of stratified shear flows when the flux Richardson number Rf, ≲1/2. This prediction is based on a straightforward solution of the spectral energy balance equation with the assumption that the turbulence is locally inertial and nearly stationary. The limitations of the locally inertial assumption are examined critically. The gap occurs at Spectral wavelengths varying from 30 m to more than 1 km, depending on Rf, the turbulence intensity, and the Brunt-Väisälä frequency. In general, the gap occurs near a wavenumber where the spectral transfer function ε(k) vanishes. Both the gap and the zero of ε(k) are caused by shear and/or buoyancy production of energy. In addition, an alternative derivation of ε(k) is made from a more general consideration that avoids the locally inertial assumption. Both expressions predict that ε(k) will vanish at a value of k when Rf ≲1/2. A comparison is made with the gaps and adjacent peaks frequently observed in the wind spectra of clear air turbulence (CAT).

Le calcul numérique de l'état de la mer

This type of calculation is still basically empirical and none of the attempts to bring it within the scope of hydrodynamics theory had so far been successful. Forms of spectral representation of the state of sea in terms of proposed by various specialists had been showing discrepancies for a long time, mainly on the following points a) Total spectrum energy. b) The tendency of the spectrum to develop into a more or less hypothetical limited spectrum. These discrepancies have now been cleared up, as follows : a) Wind conditions should be referred to a standard height, as with the currently applied meteorological relationships between surface wind and pressure gradient. b) Recent theory has confirmed the more or less simultaneous growth of the various spectrum components. These theories, which are mainly based on linearisation of the hydrodynamics equations, involve average conditions, the turbulence spectrum above the waves and the average profile in the lower layers, all of which are unfortunately still difficult to establish. With the equations, however, five simultaneous change processes can be considered, which are combined into a spectral energy balance equation. They are the following : 1. Propagation (or advection). 2. The Phillips growth term. 3. The Miles growth term. 4. The third-order interaction term. 5. Passive resistance damping. Hence, a digital model is characterised by its spectral energy equation. Five digital models are described and a certain number of their responses are classified into groups in terms of schematic special situations (Figs. 1 to 3 and 10 to 32 inclusive.) The results of a computation with model DSA representing the North Atlantic are compared with various observed and estimated data (Figs. 4 to 9 inclusive).