Wind forcing in the equilibrium range of wind-wave spectra

2002 ◽  
Vol 470 ◽  
pp. 223-245 ◽  
Author(s):  
TETSU HARA ◽  
STEPHEN E. BELCHER
Keyword(s):  
Energy Levels ◽  
Momentum Conservation ◽  
Wind Forcing ◽  
Degree Of Saturation ◽  
Spectral Energy ◽  
Turbulent Stress ◽  
Linear Waves ◽  
Conservation Of Momentum ◽  
Wind Sea ◽  
Equilibrium Range

A new analytical model is developed for the equilibrium range of the spectrum of wind-forced ocean surface gravity waves. We first show that the existing model of Phillips (1985) does not satisfy overall momentum conservation at high winds. This constraint is satisfied by applying recent understanding of the wind forcing of waves. Waves exert a drag on the air flow so that they support a fraction of the applied wind stress, which thus leaves a smaller turbulent stress near the surface to force growth of shorter wavelength waves. Formulation of the momentum budget accounting for this sheltering constrains the overall conservation of momentum and leads to a local turbulent stress that reduces as the wavenumber increases. This local turbulent stress then forces wind-induced wave growth. Following Phillips (1985), the wind sea is taken to be a superposition of linear waves, and equilibrium is maintained by a balance between the three sources and sinks of wave action.These assumptions lead to analytical formulae for the local turbulent stress and the degree of saturation, B(k), of waves in the equilibrium range. We identify a sheltering wavenumber, ks, over which the local turbulent stress is significantly reduced by longer waves. At low wavenumbers or at low winds, when k [Lt ] ks, the sheltering is weak and B(k) has a similar form to the model of Phillips (1985). At higher wavenumbers or at higher winds, ks, B(k) makes a transition to being proportional to k0. The additional constraint of conservation of momentum also yields a formula for the coefficient that appears in the solution for B(k). The spectra for mature seas are calculated from the model and are shown to agree with field observations. In particular, our model predicts more realistic spectral levels toward the high wavenumber limit compared to the previous model of Phillips (1985).We suggest that the model may explain the overshoot phenomena observed in the spectral energy levels as the fetch increases.

Wave Breaking Dissipation in a Young Wind Sea

2014 ◽  
Vol 44 (1) ◽  
pp. 104-127 ◽  
Author(s):  
Michael Schwendeman ◽  
Jim Thomson ◽  
Johannes R. Gemmrich
Keyword(s):  
Wind Waves ◽  
Phase Speed ◽  
Field Studies ◽  
Turbulent Energy ◽  
Strength Parameter ◽  
Video Recordings ◽  
Sonar System ◽  
Wind Sea ◽  
Equilibrium Range

Abstract Coupled in situ and remote sensing measurements of young, strongly forced wind waves are applied to assess the role of breaking in an evolving wave field. In situ measurements of turbulent energy dissipation from wave-following Surface Wave Instrument Float with Tracking (SWIFT) drifters and a tethered acoustic Doppler sonar system are consistent with wave evolution and wind input (as estimated using the radiative transfer equation). The Phillips breaking crest distribution Λ(c) is calculated using stabilized shipboard video recordings and the Fourier-based method of Thomson and Jessup, with minor modifications. The resulting Λ(c) are unimodal distributions centered around half of the phase speed of the dominant waves, consistent with several recent studies. Breaking rates from Λ(c) increase with slope, similar to in situ dissipation. However, comparison of the breaking rate estimates from the shipboard video recordings with the SWIFT video recordings show that the breaking rate is likely underestimated in the shipboard video when wave conditions are calmer and breaking crests are small. The breaking strength parameter b is calculated by comparison of the fifth moment of Λ(c) with the measured dissipation rates. Neglecting recordings with inconsistent breaking rates, the resulting b data do not display any clear trends and are in the range of other reported values. The Λ(c) distributions are compared with the Phillips equilibrium range prediction and previous laboratory and field studies, leading to the identification of several inconsistencies.

scholarly journals Time-dependent radiation hydrodynamics on a moving mesh

2020 ◽  
Vol 493 (4) ◽  
pp. 5397-5407 ◽  
Author(s):  
Philip Chang ◽  
Shane W Davis ◽  
Yan-Fei Jiang(姜燕飞)
Keyword(s):  
Large Scale ◽  
Momentum Conservation ◽  
Conservation Equation ◽  
Time Dependent ◽  
Moving Mesh ◽  
Test Cases ◽  
Multiple Sources ◽  
Linear Waves ◽  
Implicit Solver ◽  
Coupling Time

ABSTRACT We describe the structure and implementation of a radiation hydrodynamic solver for manga, the moving-mesh hydrodynamics module of the large-scale parallel code, Charm N-body GrAvity solver (changa). We solve the equations of time-dependent radiative transfer (RT) using a reduced speed of light approximation following the algorithm of Jiang et al. By writing the RT equations as a generalized conservation equation, we solve the transport part of these equations on an unstructured Voronoi mesh. We then solve the source part of the RT equations following Jiang et al. using an implicit solver, and couple this to the hydrodynamic equations. The use of an implicit solver ensures reliable convergence and preserves the conservation properties of these equations even in situations where the source terms are stiff due to the small coupling time-scales between radiation and matter. We present the results of a limited number of test cases (energy conservation, momentum conservation, dynamic diffusion, linear waves, crossing beams, and multiple shadows) to show convergence with analytic results and numerical stability. We also show that it produces qualitatively the correct results in the presence of multiple sources in the optically thin case.

scholarly journals Nonlinear Horizontal Diffusion for GCMs

2007 ◽  
Vol 135 (4) ◽  
pp. 1439-1454 ◽  
Author(s):  
Erich Becker ◽  
Ulrike Burkhardt
Keyword(s):  
General Circulation ◽  
Frictional Heating ◽  
Circulation Model ◽  
Spherical Geometry ◽  
Momentum Conservation ◽  
Horizontal Resolution ◽  
Spectral Energy ◽  
Vertical Diffusion ◽  
Linear Diffusion ◽  
Horizontal Diffusion

Abstract The mixing-length-based parameterization of horizontal diffusion, which was originally proposed by Smagorinsky, is revisited. The complete tendencies of horizontal momentum diffusion, the associated frictional heating, and horizontal diffusion of sensible heat in spherical geometry are derived. The formulations are modified for the terrain-following vertical-hybrid-coordinate system in a way that ensures energy and angular momentum conservation at each layer. Test simulations with a simple general circulation model, run at T42 horizontal resolution and for permanent January conditions, confirm the conservation properties and highlight the enhancement of nonlinear horizontal diffusion in areas of high baroclinic activity. The simulated internal variability is dependent on the nature of the horizontal diffusion, with high-frequency variability being enhanced over the northern continents and low-frequency variability being increased (decreased) over the Pacific (Atlantic) Ocean when using nonlinear rather than linear diffusion. Locally reduced horizontal dissipation over Europe is compensated by increased dissipation owing to vertical diffusion, indicating the potential importance of nonlinear horizontal diffusion for gravity wave–resolving simulations. Inspection of the spectral energy reveals that the scheme needs to be modified in order to damp unbalanced ageostrophic motions at the smallest resolved scales more efficiently. A corresponding empirical modification is proposed and proves to work properly.

scholarly journals A Model of the Air–Sea Momentum Flux and Breaking-Wave Distribution for Strongly Forced Wind Waves

2007 ◽  
Vol 37 (7) ◽  
pp. 1811-1828 ◽  
Author(s):  
Tobias Kukulka ◽  
Tetsu Hara ◽  
Stephen E. Belcher
Keyword(s):  
Momentum Flux ◽  
Breaking Waves ◽  
Wind Forcing ◽  
Small Scale ◽  
Breaking Wave ◽  
Unit Surface Area ◽  
Wave Age ◽  
Coupled Equations ◽  
Wave Distribution ◽  
Equilibrium Range

Abstract Under high-wind conditions, breaking surface waves likely play an important role in the air–sea momentum flux. A coupled wind–wave model is developed based on the assumption that in the equilibrium range of surface wave spectra the wind stress is dominated by the form drag of breaking waves. By conserving both momentum and energy in the air and also imposing the wave energy balance, coupled equations are derived governing the turbulent stress, wind speed, and the breaking-wave distribution (total breaking crest length per unit surface area as a function of wavenumber). It is assumed that smaller-scale breaking waves are sheltered from wind forcing if they are in airflow separation regions of longer breaking waves (spatial sheltering effect). Without this spatial sheltering, exact analytic solutions are obtained; with spatial sheltering asymptotic solutions for small- and large-scale breakers are derived. In both cases, the breaking-wave distribution approaches a constant value for large wavenumbers (small-scale breakers). For low wavenumbers, the breaking-wave distribution strongly depends on wind forcing. If the equilibrium range model is extended to the spectral peak, the model yields the normalized roughness length (Charnock coefficient) of growing seas, which increases with wave age and is roughly consistent with earlier laboratory observations. However, the model does not yield physical solutions beyond a critical wave age, implying that the wind input to the wave field cannot be dominated by breaking waves at all wavenumbers for developed seas (including field conditions).

A Generic Method to Model Frequency-Direction Wave Spectra for FPSO Motions

2008 ◽  
Author(s):  
Hermione J. van Zutphen ◽  
Philip Jonathan ◽  
Kevin C. Ewans
Keyword(s):  
Frequency Spectrum ◽  
Two Dimensions ◽  
Wind Forcing ◽  
Wave Spectra ◽  
Spectral Form ◽  
New Approach ◽  
Spreading Function ◽  
Wind Sea ◽  
Model Frequency ◽  
Sea Wave

We report a new approach to model the frequency-direction spectrum, in which the frequency-direction spectra from measurements or hindcast studies are fitted simultaneously in two dimensions, frequency and direction. Depending on the amount of wind forcing on the partition, either a unimodal (swell) or bimodal (wind-sea) wave spreading function is adopted together with the spectral form which best fits the frequency spectrum. This paper describes the new method and presents the results on a measured dataset.

Spectral energy density for quarry explosions

1963 ◽  
Vol 53 (5) ◽  
pp. 989-996 ◽  
Author(s):  
G. E. Frantti
Keyword(s):  
Energy Density ◽  
Seismic Waves ◽  
Energy Levels ◽  
Total Duration ◽  
Seismic Energy ◽  
Body Waves ◽  
Propagation Path ◽  
Spectral Energy ◽  
Time Duration ◽  
Duration Time

Abstract Several explosions of varying time duration have been recorded at 156 km along a constant propagation path from a central Michigan limestone quarry. Energy density for body waves and surface waves is examined as a function of frequency and observed to peak between 1 and 10 cps. A correlation between spectral amplitudes and source duration time is revealed and is emphasized at shot durations which approximate the dominant period of seismic waves. A study of the data suggests that seismic energy levels may be controlled, in part, by regulating the time duration of delayed quarry blasts. This parameter (total duration time) has been generally neglected in published studies involving commercial blasts.

scholarly journals Head-on collision of internal waves with trapped cores

2017 ◽  
Vol 24 (4) ◽  
pp. 751-762 ◽  
Author(s):  
Vladimir Maderich ◽  
Kyung Tae Jung ◽  
Kateryna Terletska ◽  
Kyeong Ok Kim
Keyword(s):  
Wave Amplitude ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Class Iii ◽  
Navier Stokes Equations ◽  
Linear Waves ◽  
Non Linear ◽  
Conservation Of Momentum ◽  
Run Up ◽  
The Waves

Abstract. The dynamics and energetics of a head-on collision of internal solitary waves (ISWs) with trapped cores propagating in a thin pycnocline were studied numerically within the framework of the Navier–Stokes equations for a stratified fluid. The peculiarity of this collision is that it involves trapped masses of a fluid. The interaction of ISWs differs for three classes of ISWs: (i) weakly non-linear waves without trapped cores, (ii) stable strongly non-linear waves with trapped cores, and (iii) shear unstable strongly non-linear waves. The wave phase shift of the colliding waves with equal amplitude grows as the amplitudes increase for colliding waves of classes (i) and (ii) and remains almost constant for those of class (iii). The excess of the maximum run-up amplitude, normalized by the amplitude of the waves, over the sum of the amplitudes of the equal colliding waves increases almost linearly with increasing amplitude of the interacting waves belonging to classes (i) and (ii); however, it decreases somewhat for those of class (iii). The colliding waves of class (ii) lose fluid trapped by the wave cores when amplitudes normalized by the thickness of the pycnocline are in the range of approximately between 1 and 1.75. The interacting stable waves of higher amplitude capture cores and carry trapped fluid in opposite directions with little mass loss. The collision of locally shear unstable waves of class (iii) is accompanied by the development of instability. The dependence of loss of energy on the wave amplitude is not monotonic. Initially, the energy loss due to the interaction increases as the wave amplitude increases. Then, the energy losses reach a maximum due to the loss of potential energy of the cores upon collision and then start to decrease. With further amplitude growth, collision is accompanied by the development of instability and an increase in the loss of energy. The collision process is modified for waves of different amplitudes because of the exchange of trapped fluid between colliding waves due to the conservation of momentum.

On the scaling of stress-driven entrainment experiments

1979 ◽  
Vol 90 (3) ◽  
pp. 509-529 ◽  
Author(s):  
James F. Price
Keyword(s):  
Mixed Layer ◽  
Mixed Layer Depth ◽  
Side Wall ◽  
Momentum Conservation ◽  
Conservation Equation ◽  
Entrainment Rate ◽  
Surface Mixed Layer ◽  
Mean Velocity ◽  
Wide Range ◽  
Conservation Of Momentum

The entrainment experiments of Kato & Phillips (1969) and Kantha, Phillips & Azad (1977) (hereafter KP and KPA) are analysed to demonstrate a more general and effective scaling of the entrainment observations. The preferred scaling is \[ V^{-1} dh/dt = E(R_v), \] where h is the mixed-layer depth, V is the mean velocity of the mixed layer, Rv = B/V2 and B is the total mixed-layer buoyancy. This scaling effectively collapses entrainment data taken at various h/L, where L is the tank width, and in cases in which the interior is density stratified (KP) or homogeneous (KPA). The entrainment law E(Rv) is computed from the KP and KPA observations using the conservation equations for mean momentum and buoyancy. A side-wall drag term is included in the momentum conservation equation. In the range 0·5 < Rv < 1·0, which includes nearly all of the KP, KPA data, E ≃ 5 × 10−4R−4v. This is very similar to the entrainment law followed by a surface half-jet (Ellison & Turner 1959) and by the wind-driven ocean surface mixed layer (Price, Mooers & Van Leer 1978).The analysis shows that, when forcing is steady, Rv is quasi-steady and, provided that side-wall drag is not large, Rv ≃ 0·6 over a wide range of RT = B/U2*, where U* is the friction velocity of the imposed stress. In the absence of side-wall drag (vanishing h/L) the conservation of momentum then leads to U−1*dh/dt = n(0·6)½R−½T, where n = ½ or 1 if the interior is linearly stratified or homogeneous. The KP, KPA data show this dependence throughout the range 17 < RT < 160 where the effect of side-wall drag is negligible or can be removed by a linear extrapolation. This result, together with the form and magnitude of the observed side-wall effect, suggests that mean momentum conservation is a key constraint upon the entrainment rate in the KP, KPA experiments.

scholarly journals Sensitivity of Simulated Climate to Conservation of Momentum in Gravity Wave Drag Parameterization

2009 ◽  
Vol 22 (10) ◽  
pp. 2726-2742 ◽  
Author(s):  
Tiffany A. Shaw ◽  
Michael Sigmond ◽  
Theodore G. Shepherd ◽  
John F. Scinocca
Keyword(s):  
Gravity Wave ◽  
Southern Hemisphere ◽  
Momentum Flux ◽  
Middle Atmosphere ◽  
Momentum Conservation ◽  
Wave Drag ◽  
Conservation Of Momentum ◽  
Simulated Climate ◽  
The Impact ◽  
Wave Momentum

Abstract The Canadian Middle Atmosphere Model is used to examine the sensitivity of simulated climate to conservation of momentum in gravity wave drag parameterization. Momentum conservation requires that the parameterized gravity wave momentum flux at the top of the model be zero and corresponds to the physical boundary condition of no momentum flux at the top of the atmosphere. Allowing momentum flux to escape the model domain violates momentum conservation. Here the impact of momentum conservation in two sets of model simulations is investigated. In the first set, the simulation of present-day climate for two model-lid height configurations, 0.001 and 10 hPa, which are identical below 10 hPa, is considered. The impact of momentum conservation on the climate with the model lid at 0.001 hPa is minimal, which is expected because of the small amount of gravity wave momentum flux reaching 0.001 hPa. When the lid is lowered to 10 hPa and momentum is conserved, there is only a modest impact on the climate in the Northern Hemisphere; however, the Southern Hemisphere climate is more adversely affected by the deflection of resolved waves near the model lid. When momentum is not conserved in the 10-hPa model the climate is further degraded in both hemispheres, particularly in winter at high latitudes, and the impact of momentum conservation extends all the way to the surface. In the second set of simulations, the impact of momentum conservation and model-lid height on the modeled response to ozone depletion in the Southern Hemisphere is considered, and it is found that the response can display significant sensitivity to both factors. In particular, both the lower-stratospheric polar temperature and surface responses are significantly altered when the lid is lowered, with the effect being most severe when momentum is not conserved. The implications with regard to the current round of Intergovernmental Panel on Climate Change model projections are discussed.

scholarly journals Inter-annual variations in wave spectral characteristics at a location off the central west coast of India

2015 ◽  
Vol 33 (2) ◽  
pp. 159-167 ◽  
Author(s):  
V. Sanil Kumar ◽  
M. Anjali Nair
Keyword(s):  
West Coast ◽  
Spectral Characteristics ◽  
Wave Spectrum ◽  
Spectral Energy ◽  
Monsoon Period ◽  
West Coast Of India ◽  
Annual Variations ◽  
The Indian Ocean ◽  
Wind Sea ◽  
Two Peaks

Abstract. The inter-annual variations in wave spectrum are examined based on the wave data measured at 9 m water depth off the central west coast of India from 2009 to 2012 using a wave rider buoy. The temporal variation of the spectral energy density over a calendar year indicates similar variation in all the four years studied. The inter-annual variations in wave spectrum are observed in all months with larger variations during January to February, May and October to November due to the changes in wind-sea. The seasonal average wave spectrum during the monsoon (June–September) is single-peaked and the swell component is high in 2011 compared to other years. The annual averaged wave spectrum had higher peak energy during 2011 due to the higher spectral energy present during the monsoon period. During the non-monsoon period, two peaks are predominantly observed in the wave spectra; with the average peak at 0.07 Hz corresponding to the swells from the Indian Ocean and another at 0.17 Hz due to the local wind field.

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