DOCSLIB.ORG

SCHISM Numerical Formulation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
SCHISM Numerical Formulation 1 SCHISM numerical formulation Joseph Zhang 2 Horizontal grid: hybrid 3 4 Side 2 3 Side 2 Side 1 Side 3 Side 1 1 Side 4 2 1 Side 3 2 3 2 1 1 Counter-clockwise convention 3 2 3 ys 2 2 x i 2 1 s N 1 i i 1 2 Ni N i 1 Ni 1 i • Internal ‘ball’: starting elem is arbitrary • Boundary ball: starting elem is unique Vertical grid (1): SZ 3 Terrain-following zone s zone S zone SZ zone z s=0 Nz hc h hs s S-levels kz s=-1 kz-1 Z-levels 2 1 z(1 s ) h s ( h - h ) C ( s ) ( - 1 s 0) cc tanh ( s- 1/ 2) tanh( / 2) sinh(sf ) ff C(s ) (1 - b ) b (0 b 1; 0 f 20) sinhff 2tanh( / 2) h min( h , hs ) S-coordinates 4 -3 f=10 , any b f=10, b=1 hc f=10, b=0 f=10, b=0.7 Vertical grid: SZ 5 z=-hs The bottom cells are shaved – no staircases! Vertical grid (2): LSC2 via VQS Dukhovskoy et al. (2009) Procedure for VQS • Different # of levels are used at different depths • At a given depth, the # of levels is interpolated from the 2 adjacent master grids • Thin layers near the bottom are masked • Staircases appear near the bottom (like Z), but otherwise terrain following What to do with the staircases? A u u P 1 B 2 Q R 1 2 LSC2= Localized Sigma Coordinates with Shaved Cells (ivcor =1 in vgrid.in) *No masking of thin layers (c/o implicit scheme) Vertical grid: LSC2 8 z (m) Degenerate prism Vertical grid (3) 9 • 3D computational unit: uneven prisms • Equations are not transformed into S- or s-coordinates, but solved in the original Z-space • Pressure gradient d z •Z-method with cubic spline (extrapolation on surface/bottom) nk nk k k (whole level) Nz N -1 ……. z uj,k+1 vj,k+1 uj,k+1 vj,k+1 k Ci,k ……. k-1 Ci,k k-1 k-1 k +1 wi,k-1 wi,k-1 b kb Polymorphism Zhang et al. (2016) Bathymetry (m) Polymorphism flood z (m) x (m) S (PSU) *Make sure there is no jump in Cd between 2D/3D zones Semi-implicit scheme 12 nn1 - Continuity uunn1dz (1 - ) dz 0 t --hh 풖풏+ퟏ − 풖∗ = 풇 − 휃훻휂푛+1 − (1 − 휃)훻휂푛 + 풎풏+ퟏ − 훼 풖 풖풏+ퟏ퐿 푥, 푦, 푧 Momentum Δ푡 풛 n1 nu n1 n τw , at z ; b.c. (3D) z un1 n nu n1, at zh - , nn C ||u z b Db Du uun1 - Advection: ≈ * ; u (x , y , z , t t , t ) Dt t * Implicit treatment of divergence, pressure gradient and SAV form drag terms by-passes most severe stability constraints: 0.5 ≤≤1 Explicit treatment: Coriolis, baroclinicity, horizontal viscosity, radiation stress… Eulerian-Lagrangian Method (ELM) 13 ELM: takes advantage of both Lagrangian and Eulerian methods Grid is fixed in time, and time step is not limited by Courant number condition Advections are evaluated by following a particle that starts at certain point at time t and ends right at a pre-given point at time t+t. The process of finding the starting point of the path (foot of characteristic line) is called backtracking, which is done by integrating dx/dt=u3 backward in 3D space and time. To better capture the particle movement, the backward integration is often carried out in small sub-time steps Simple backward Euler method 2nd-order R-K method Interpolation-ELM does not conserve; integration ELM does Interpolation: numerical diffusion vs. dispersion ut 1 x Characteristic line t+t t advection Flow dispersion diffusion Operating range of SCHISM 14 …. for a fixed grid ( 풖 + ℎ)∆푡 퐶퐹퐿 = ∆푥 • ELM prefers larger t (truncation error ~1/t) • As a result, SCHISM requires CFL>0.4 • Convergence: CFL=const, t0 • For barotropic field applications: [100,450] sec • For baroclinic field applications: [100,200] sec (e.g., start with 150 sec) ELM with high-order Kriging 15 • Best linear unbiased estimator for a random function f(x) • “Exact” interpolator • Works well on unstructured grid (efficient) • Needs filter (ELAD) to reduce dispersion N f( x , y )1 2 x 3 y ii K (|xx - |) i1 Cubic spline Drift function Fluctuation K is called generalized covariance function K( h )- h , h23ln h , h , Minimizing the variance of the fluctuation we get N i 0 i1 N 2-tier Kriging cloud iix 0 i1 N iiy 0 i1 x f(,) xi y i f i • The numerical dispersion generated by kriging is reduced by a diffusion ELAD filter (Zhang et al. 2016) Semi-implicit scheme 16 nn1 - Continuity uunn1dz (1 - ) dz 0 t --hh 풖풏+ퟏ − 풖∗ = 풇 − 휃훻휂푛+1 − (1 − 휃)훻휂푛 + 풎풏+ퟏ − 훼 풖 풖풏+ퟏ퐿 푥, 푦, 푧 Momentum Δ푡 풛 n1 nu n1 n τw , at z ; b.c. z un1 n nu n1, at zh - , nn C ||u z b Db Du uun1 - ≈ * ; u (x , y , z , t t , t ) Dt t * Implicit treatment of divergence and pressure gradient terms by-passes most severe CFL condition : 0.5 ≤≤1 Explicit treatment: Coriolis, baroclinicity, horizontal viscosity, radiation stress… Galerkin finite-element formulation 17 A Galerkin weighted residual statement for the continuity equation: 휂푛+1 − 휂푛 න 휙 푑Ω + 휃 − න 훻휙 ∙ 푼푛+1푑Ω + න 휙 푈푛+1푑Γ + 1 − 휃 − න 훻휙 ∙ 푼푛푑Ω + න 휙 푈푛푑Γ = 0, 푖 Δ푡 푖 푖 푛 푖 푖 푛 Ω Ω Γ Ω Γ ( = 1, ⋯ , 푁푝) 휙푖: shape/weighting function; U: depth-integrated velocity; :implicitness factor Need to eliminate Un+1 to get an equation for alone. We’ll do this with the aid from momentum equation: 풖풏+ퟏ − 풖∗ = 풇 − 휃훻휂푛+1 − (1 − 휃)훻휂푛 + 풎풏+ퟏ − 훼 풖 풖풏+ퟏ퐿 푥, 푦, 푧 Δ푡 풛 Shape function (global) 2D case 18 The unknown velocity can be easily found as: 퐻2 푼푛+1 = 푮ෙ − 휃Δ푡 훻휂푛+1 퐻෩ 퐻 푮ෙ = 푼∗ + ∆푡 푭 + 흉 − (1 − 휃)퐻훻휂푛 (explicit terms) 퐻෩ 푤 퐻෩ = 퐻 + (휒 + 훼|풖|퐻)Δ푡 3D case 19 Vertical integration of the momentum equation 푼풏+ퟏ − 푼∗ 푛+1 푛 푛+1 휶 ub = 푭 − 휃훻휂 − 1 − 휃 훻휂 + 흉푤 − 휒풖푏 − 훼 풖 푼 Δ푡 db 푧푣 풛풗 Assuming: න |풖|풖푑푧 ≅ 풖 න 풖푑푧 ≡ 풖 푼휶 z=-h −ℎ −풉 Momentum equation applied to the bottom boundary layer 풖풏+ퟏ − 풖∗ 휕 휕풖 풃 풃 = 풇 − 휃훻휂푛+1 − 1 − 휃 훻휂푛 − 훼 풖 풖풏+ퟏ + 휈 Δ푡 풃 풃 풃 휕푧 휕푧 Therefore 푛+1 1 ∗ 푛 휃∆푡 푛+1 풖푏 = 풖푏 + 풇풃∆푡 − (1 − 휃)∆푡훻휂 − 훻휂 1 + 훼|풖푏|∆푡 1 + 훼|풖푏|∆푡 Now still need to find U 3D case: locally emergent vegetation 20 We have 푼훼 = 푼n+1 So: ෡ 푛+1 휃퐻Δ푡 푛+1 푼 = 푮ퟏ − 훻휂 1 + 훼|풖|∆푡 ∗ ෡ 푛 ∗ 푼 + 푭 + 흉풘 ∆푡 − 1 − 휃 퐻∆푡훻휂 − 휒෤∆푡(풖푏 + 풇푏∆푡) 푮1 = 1 + 훼|풖|∆푡 퐻෡ = 퐻 − 휒෤∆푡 휒 휒෤ = 1 + 훼|풖풃|∆푡 3D case: locally submerged vegetation 21 Integrating from bottom to top of canopy: 푼훼 휕풖 푧푣 = 푼∗훼 + 푭휶∆푡 − 휃퐻훼∆푡훻휂푛+1 − (1 − 휃)퐻훼∆푡훻휂푛 − 훼∆푡 풖 푼훼 + ∆푡휈 ቤ 푧 푧 휕푧 푣 푣 −ℎ 푼∗훼 = න 풖∗푑푧, 푭휶 = න 풇푑푧 −ℎ −ℎ Inside vegetation layer, the stress obeys exponential law (Shimizu and Tsujimoto 1994) 휕풖 휈 ≡ 푢′푤′ = 푹 푒훽2(푧−푧푣), (푧 ≤ 푧 ) 휕푧 0 푣 훽2 is a constant found from empirical formula: 푁 퐻 − 퐻훼 휒|풖푏| 훽 = 푣 −0.32 − 0.85 log 퐼 퐼 = 2 퐻훼 10 퐻훼 퐻 휕풖 푧푣 Therefore 휈 ቤ = 훽휒풖푛+1 (훽 = 푒훽2 푧푣−푧푏 − 1) 휕푧 푏 휃−ℎ 퐻෡훼∆푡 훼 푛+1 푼∗훼 + 푭휶∆푡 + 훽휒෤∆푡 풖∗ + 풇 ∆푡 − (1 − 휃)퐻෡훼∆푡훻휂푛 푼 = 푮3 − 훻휂 푏 푏 훼 훼 1 + 훼|풖|∆푡 푮3 = 퐻෡ = 퐻 + 훽휒෤∆푡 1 + 훼|풖|∆푡 Finally: 훼|풖|∆푡 푛+1 푛+1 훼 푼 = 푮ퟐ − 휃퐻നΔ푡훻휂 퐻ന = 퐻 − 휒෤Δ푡 − 푐퐻෡ 푐 = 1 + 훼|풖|∆푡 General case 22 In compact form: 푼푛+1 = 푬 − 휃퐻෱Δ푡훻휂푛+1 퐻2 , locally 2D 퐻෩ 퐻෱ = 퐻෡ , locally 3D emergent 1 + 훼|풖|∆푡 (Note the difference in 퐻ෙ between 2D/3D) 퐻ന, locally 3D submerged 푮෱, 2D 푬 = ൞ 푮ퟏ, 3D emergent 푮ퟐ, 3D submerged Finite-element formulation (cont’d) 23 Finally, substituting this eq. back to continuity eq. we get one equation for elevations alone: I1 푛+1 2 2 푛+1 න 휙푖휂 + 휃 Δ푡 퐻ෙ훻휙푖 ∙ 훻휂 푑Ω I3 Ω 푛 푛 푛+1 푛 푛+1 = න 휙푖휂 + 휃Δ푡훻휙푖 ∙ 푬 + (1 − 휃)Δ푡훻휙푖 ∙ 퐔 푑Ω − 휃Δ푡 න 휙푖푈෡푛 푑Γ푣 − (1 − 휃) Δ푡 න 휙푖푈푛 푑Γ − 휃Δ푡 න 휙푖푈푛 푑Γത푣 + 퐼7 Ω Γ푣 Γ Γഥ푣 I5 I6 I4 sources (i=1,…,Np) Notes: • When node i is on boundaries where essential b.c. is imposed, is known and so no equations are needed, and so I2 and I6 need not be evaluated there • When node i is on boundaries where natural b.c. is imposed, the velocity is known and so the last term on LHS is known only the first term is truly unknown! • b.c. is free lunch in FE model (cf. staggering in FV) • The inherent numerical dispersion in FE nicely balances out the numerical diffusion inherent in the implicit time stepping scheme! Shape function 24 Used to approximate the unknown function Although usually used within each individual elements, shape function is global not local! N u ui i, i( x j ) d ij i1 Must be sufficiently smooth to allow integration by part in weak formulation Mapping between local and global coordinates Assembly of global matrix NMN L u d L u d i i j i i j m 0, jN1,..., i1 m 1 i 1 m x 1 j-1 j j+1 N x 1 j-1 j j+1 N Conformal Non-conformal (Elevation, velocity) (optional for velocity: indvel=0 -> less dissipation and needs further stabilization (MB* schemes) Matrices: I1 25 Ni I()ˆˆ nn1 g 2 t 2 H퐻ෙˆ 1 d 1 i ' i ' j j1 j i34(j) Ni 3 2 n1 1 dil', gt A Hi퐻ෙˆ ' l j j, l12 j 4A jl11 j i i’ is local index of node i in j.
Load more

Recommended publications
  • Acoustic Nonlinearity in Dispersive Solids
    ACOUSTIC NONLINEARITY IN DISPERSIVE SOLIDS John H. Cantrell and William T. Yost NASA Langley Research Center Mail Stop 231 Hampton, VA 23665-5225 INTRODUCTION It is well known that the interatomic anharmonicity of the propagation medium gives rise to the generation of harmonics of an initially sinusoidal acoustic waveform. It is, perhaps, less well appreciated that the anharmonicity also gives rise to the generation of a static or "dc" component of the propagating waveform. We have previously shown [1,2] that the static component is intrinsically linked to the acoustic (Boussinesq) radiation stress in the material. For nondispersive solids theory predicts that a propagating gated continuous waveform (acoustic toneburst) generates a static displacement pulse having the shape of a right-angled triangle, the slope of which is linearly proportional to the magnitude and sign of the acoustic nonlinearity parameter of the propagation medium. Such static displacement pulses have been experimentally verified in single crystal silicon [3] and germanium [4]. The purpose of the present investigation is to consider the effects of dispersion on the generation of the static acoustic wave component. It is well known that an initial disturbance in media which have both sufficiently large dispersion and nonlinearity can evolve into a series of solitary waves or solitons [5]. We consider here that an acoustic tone burst may be modeled as a modulated continuous waveform and that the generated initial static displacement pulse may be viewed as a modulation-confined disturbance. In media with sufficiently large dispersion and nonlinearity the static displacement pulse may be expected to evolve into a series of modulation solitons.
    [Show full text]
  • Nearshore Current Pattern and Rip Current Occurrence at Jungmun Beach, Jeju by Numerical Computation 1. Introduction
    한국항해항만학회지 제41권 제2호 J. Navig. Port Res. Vol. 41, No. 2 : 55-62, April 2017 (ISSN:1598-5725(Print)/ISSN:2093-8470(Online)) DOI : http://dx.doi.org/10.5394/KINPR.2017.41.2.55 Nearshore Current Pattern and Rip Current Occurrence at Jungmun Beach, Jeju by Numerical Computation Seung-Hyun An*․†Nam-Hyeong Kim *Coastal, Harbor and Disaster Prevention Research Institute, Sekwang Engineering Consultants Co., LTD. †Department of Civil Engineering, J eju National University, Jeju 63243, Korea Abstract : A nearshore current or a wave-induced current is an important phenomenon in a nearshore zone, which is composed of longshore, cross-shore, and rip currents. The nearshore current is closely related to the occurrence of coastal accidents by beachgoers. A considerable number of coastal accidents by beachgoers involving the rip current have been reported at Jungmun Beach. However, in studies and observations of the nearshore current of Jungmun Beach, understanding of the rip current pattern remains unclear. In this study, a scientific approach is taken to understand the nearshore current and the rip current patterns at Jungmun Beach by numerical computation for year of 2015. From results of numerical computation, the occurrence and spatial characteristics of the rip current, and the similarities between the rip current and incident wave conditions are analyzed. The primary results of this study reveal that the rip currents are frequently generated at Jungmun Beach, especially in the western parts of the beach, and that the rip currents often occur with a wave breaking height of around 0.5 ~ 0.7 m, a wave period of around 6 ~ 8 seconds, and a breaking angle of around 0 ~ 15 degrees.
    [Show full text]
  • CHAPTER 100 Another Approach to Longshore Current Evaluation M.A
    CHAPTER 100 Another approach to longshore current evaluation M.A. Losada, A. Sanchez-Arcilla and C. Vidal A simple model to predict the longshore current velocity at the breaker line on a beach with oblique wave incidence, is presented. The model balances driving and resistance terms (gradients of radiation and turbulent Reynolds stresses and bottom friction) to get a general expression for the velocity. This equation shows explicitely the influence of Iribarren's parameter on longshore current generation. It has been tested with field and laboratory data, obtaining a reasonable fit to measured values. The resulting (predictive) model is expected to be valid for any type of breakers though the calibration has been mainly done for spilling and plunging types, due to the scarcity of results for other breakers. 1.- INTRODUCTION Longshore currents in the surf zone have been acceptably modelled using the radiation stress concept (Longuet-Higgins,1970). The longshore-trust (Nw/m ) due to an oblique wave approach, given by the gradient of the radiation stress, is balanced (in stationary and longshore-uniform conditions) by bottom friction and horizontal mixing (Bowen, 1969), (Longuet-Higgins, 1970). The gradient of the radiation stress is evaluated using sinusoidal theory (as a first approximation for slowly varying depths) and turns out to be proportional to the local rate of energy dissipation, D (joules/(m x sec)), regardless of its origin. Inside the surf zone a significant fraction of D comes from wave breaking because the turbulence associated to the breaking process is responsible for most of the dissipated energy. Bottom friction plays a minor role in this context, being important only in special cases (e.g.
    [Show full text]
  • Chapter 187 Wave Breaking and Induced Nearshore
    CHAPTER 187 WAVE BREAKING AND INDUCED NEARSHORE CIRCULATIONS Ole R. S0rensen f, Hemming A. Schafferf, Per A. Madsenf, and Rolf DeigaardJ ABSTRACT Wave breaking and wave induced currents are studied in two horizontal dimen- sions. The wave-current motion is modelled using a set of Boussinesq type equations including the effects of wave breaking and runup. This is done with- out the traditional splitting of the phenomenon into a wave problem and a current problem. Wave breaking is included using the surface roller concept and runup is simulated by use of a modified slot-technique. In the environment of a plane sloping beach two different situations are studied. One is the case of a rip channel and the other concerns a detached breakwater parallel to the shoreline. In both situations wave driven currents are generated and circulation cells appear. This and many other nearshore physical phenomena are described qualitatively correct in the simulations. 1. INTRODUCTION State of the art models for wave driven currents are based on a splitting of the phenomenon into a linear wave problem, described e.g. by the mild-slope equation, and a current problem, described by depth-integrated flow equations including radiation stress. Wave-current interaction effects such as current re- fraction and wave blocking may be included in these systems by successive and iterative model executions. A more direct approach to the problem would be the use of a time do- main Boussinesq model which automatically includes the combined effects of wave-wave and wave-current interaction in shallow water without the need for explicit formulations of the radiation stress.
    [Show full text]
  • A Nonlinear Formulation of Radiation Stress and Applications to Cnoidal
    A Nonlinear Formulation of Radiation Stress and Applications to Cnoidal Shoaling Martin O. Paulsen Henrik Kalisch · g Abstract In this article we provide formulations of energy flux and radiation stress consistent with the scaling regime of the Korteweg-de Vries (KdV) equation. These quantities can be used to describe the shoaling of cnoidal waves approaching a gently sloping beach. The transformation of these waves along the slope can be described using the shoaling equations, a set of three nonlinear equations in three unknowns: the wave height H, the set-downη ¯ and the elliptic parameter m. We define a numerical algorithm for the efficient solution of the shoaling equations, and we verify our shoaling formulation by comparing with experimental data from two sets of experiments as well as shoaling curves obtained in previous works. Keywords: Surface waves KdV equation Conservation laws Radiation stress Set-down · · · · 1 Introduction In the present work, the development of surface waves across a gently sloping bottom is in view. Our main goal is the prediction of the wave height and the set-down of a periodic wave as it enters an area of shallower depth. This problem is known as the shoaling problem and has a long history, with contributions from a large number of authors going back to Green and Boussinesq. In order to reduce the problem to the essential factors, we make a number of simplifying as- sumptions. First, the bottom slope is small enough to allow the waves to continuously adjust to the changing depth without altering their basic shape or breaking up.
    [Show full text]
  • The Three-Dimensional Current and Surface Wave Equations
    1978 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 The Three-Dimensional Current and Surface Wave Equations GEORGE MELLOR Princeton University, Princeton, New Jersey (Manuscript received 26 April 2002, in ®nal form 4 March 2003) ABSTRACT Surface wave equations appropriate to three-dimensional ocean models apparently have not been presented in the literature. It is the intent of this paper to correct that de®ciency. Thus, expressions for vertically dependent radiation stresses and a de®nition of the Doppler velocity for a vertically dependent current ®eld are obtained. Other quantities such as vertically dependent surface pressure forcing are derived for inclusion in the momentum and wave energy equations. The equations include terms that represent the production of turbulence energy by currents and waves. These results are a necessary precursor for three-dimensional ocean models that handle surface waves together with wind- and buoyancy-driven currents. Although the third dimension has been added here, the analysis is based on the assumption that the depth dependence of wave motions is provided by linear theory, an assumption that is the basis of much of the wave literature. 1. Introduction pressure so that their results differ from mine and, after vertical integration, differ from the corresponding terms In the last several decades there has been signi®cant in Phillips (1977). They did not address three-dimen- progress in the understanding and numerical modeling sional effects on the wave energy equation. of ocean surface waves (e.g., Longue-Higgins 1953; To contrast the developments in this paper with con- Phillips 1977; WAMDI Group 1988; Komen et al.
    [Show full text]
  • 7 X 11 Long.P65
    Cambridge University Press 978-0-521-86028-4 - Waves in Oceanic and Coastal Waters Leo H. Holthuijsen Index More information Index absorption 294, 296 barrier 221 accelerometer 13 basin (wave-, tank) 181, 183 acquisition (spectrum) 51 Battjes method 101 action 221 Beaufort wind scale 190 balance (equation) 257, 261, 288, 335 Berkhoff mild-slope equation 217 density (spectrum) 261 Bernoulli equation 117 advective 111, 343 linearised 117 air 177 nonlinear 323 bubble 108 biphase 271, 273, 274 flow 178, 182, 183 blocking, current-induced 264 motion 182 Boltzmann integral 185 pressure 18, 178, 179 bore 281 airplane 19, 20, 21 bottom Airy (wave theory) 53, 107, 138 friction 109, 276, 295 aliasing 333 roughness 279, 296 alphabet, Greek 31 slope 74, 224, 243, 282 altimeter boundary condition 115, 116, 320, 323 acoustic 22 boundary condition laser 21 kinematic, bottom 115, 116, 320, 323 radar 22 kinematic, surface 114, 116, 319, 323 altimeter (radar) dynamic, surface, bottom 115, 117, 130, 142, 170, footprint (satellite) 22 320 wave height 22 boundary-element method 215, 240 amplitude boundary layer random 33, 45 atmospheric 147, 183 response 53, 54, 55 atmospheric (internal) 148 spectrum 34, 45 bottom 109, 276, 278 analysis Boussinesq (equations, model) 198, 241 cross-spectral 324 Bragg scatter 20, 270 spectral 324 breaking 188, 242, 276 Anderson–Darling test 89 criterion 189 anemometer, wind 146, 151 index 282 annual-maximum approach (long-term statistics) 85, wave types 242 98 breakwater 210, 213, 266, 288, 296 array breeze 151, 181 phase 17
    [Show full text]
  • Wave–Current Interaction: a Comparison of Radiation-Stress and Vortex-Force Representations
    1122 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 37 Wave–Current Interaction: A Comparison of Radiation-Stress and Vortex-Force Representations E. M. LANE Institute of Geophysics and Planetary Physics, University of California, Los Angeles, Los Angeles, California J. M. RESTREPO Department of Mathematics, and Department of Physics, The University of Arizona, Tucson, Arizona J. C. MCWILLIAMS Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, California (Manuscript received 10 May 2006, in final form 8 August 2006) ABSTRACT The vortex-force representation of the wave-averaged effects on currents is compared to the radiation- stress representation in a scaling regime appropriate to coastal and shelf waters. Three-dimensional and vertically integrated expressions for the conservative current equations are obtained in both representa- tions. The vortex-force representation decomposes the main wave-averaged effects into two physically understandable concepts—a vortex force and a Bernoulli head. The vortex force is shown to be the dominant wave-averaged effect on currents. This effect can occur at higher order than the apparent leading order for the radiation-stress representation. Excluding nonconservative effects such as wave breaking, the lowest-order radiation or interaction stress can be completely characterized in terms of wave setup, forcing of long (infragravity) waves, and an Eulerian current whose divergence cancels that of the primary wave Stokes drift. The leading-order, wave-averaged dynamical effects incorporate the vortex force together with material advection by Stokes drift, modified pressure-continuity and kinematic surface boundary condi- tions, and parameterized representations of wave generation by the wind and breaking near the shoreline.
    [Show full text]
  • Rogue Waves and Their Generating Mechanisms in Different Physical Contexts
    Physics Reports 528 (2013) 47–89 Contents lists available at SciVerse ScienceDirect Physics Reports journal homepage: www.elsevier.com/locate/physrep Rogue waves and their generating mechanisms in different physical contexts M. Onorato a,b, S. Residori c,∗, U. Bortolozzo c, A. Montina d, F.T. Arecchi e,f a Dipartimento di Fisica Generale, Università degli Studi di Torino, Via Pietro Giuria 1, 10125 Torino, Italy b INFN, Sezione di Torino, Via Pietro Giuria 1, 10125 Torino, Italy c INLN, Université de Nice Sophia-Antipolis, CNRS, 1361 route des Lucioles, 06560 Valbonne, France d Perimeter Institute for Theoretical Physics, Waterloo, Ontario, N2L 2Y5 Canada e Dipartimento di Fisica, Università di Firenze, Italy f CNR-INO, largo E. Fermi 6, 50125 Firenze, Italy article info a b s t r a c t Article history: Rogue waves is the name given by oceanographers to isolated large amplitude waves, Accepted 19 December 2012 that occur more frequently than expected for normal, Gaussian distributed, statistical Available online 14 March 2013 events. Rogue waves are ubiquitous in nature and appear in a variety of different contexts. editor: G.I. Stegeman Besides water waves, they have been recently reported in liquid Helium, in nonlinear optics, microwave cavities, etc. The first part of the review is dedicated to rogue waves in the oceans and to their laboratory counterpart with experiments performed in water basins. Most of the work and interpretation of the experimental results will be based on the nonlinear Schrödinger equation, an universal model, that rules the dynamics of weakly nonlinear, narrow band surface gravity waves.
    [Show full text]
  • Numerical Simulation of Linear Wave Propagation, Wave-Induced Circulation, Sediment Transport and Beach Evolution Theofanis V. K
    Transactions on the Built Environment vol 40 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509 Numerical simulation of linear wave propagation, wave-induced circulation, sediment transport and beach evolution Theofanis V. Karambas Aristotle University of Thessaloniki, Department of Civil Engineering, Division of Hydraulics and Environmental Eng. Thessaloniki, 54006, Greece Email: [email protected] Abstract In the present work two numerical models for linear wave propagation, wave induced circulation and sediment transport are presented. The wave model WAVE-L is based on the hyperbolic type mild slope equation and is valid for a compound wave field near coastal structures where the waves are subjected to the combined effects of shoaling, refraction, diffraction, reflection (total and partial) and breaking. Radiation stress components estimated from the hyperbolic wave model drive the depth averaged circulation model COAST for the description of the nearshore currents and sediment transport in the surf and swash zone. The model COAST is coupled with a 3D bed evolution model or with an one-line model to provide bathymetry or shoreline changes. 1. Introduction Many models exist for the evaluation of wave deformation in the coastal region but the most of them are based on the progressive wave assumption (period averaged refraction wave models) and employ elliptic or parabolic type differential equations which are in general difficult to numerically solve. Besides they are not valid for a compound wave field near coastal structures where the waves subject to the combined effects of shoaling, refraction, diffraction, reflection and breaking. Transactions on the Built Environment vol 40 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509 264 Coastal Engineering and Marina Developments The evaluation of the wave field only is not sufficient for the design of coastal structures.
    [Show full text]
  • Formulation of the Undertow Using Linear Wave Theory
    Formulation of the undertow using linear wave theory Guannel, G., & Özkan-Haller, H. T. (2014). Formulation of the undertow using linear wave theory. Physics of Fluids, 26(5), 056604. doi:10.1063/1.4872160 10.1063/1.4872160 American Institute of Physics Publishing Version of Record http://cdss.library.oregonstate.edu/sa-termsofuse PHYSICS OF FLUIDS 26, 056604 (2014) Formulation of the undertow using linear wave theory G. Guannel1,a) and H. T. Ozkan-Haller¨ 2 1School of Civil and Construction Engineering, Oregon State University, Corvallis, Oregon 97331, USA 2College of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Corvallis, Oregon 97331, USA (Received 6 September 2013; accepted 7 February 2014; published online 13 May 2014) The undertow is one of the most important mechanisms for sediment transport in nearshore regions. As such, its formulation has been an active subject of research for at least the past 40 years. Still, much debate persists on the exact nature of the forcing and theoretical expression of this current. Here, assuming linear wave theory and keeping most terms in the wave momentum equations, a solution to the undertow in the surf zone is derived, and it is shown that it is unique. It is also shown that, unless they are erroneous, most solutions presented in the literature are identical, albeit simplified versions of the solution presented herein. Finally, it is demonstrated that errors in past derivations of the undertow profile stem from inconsistencies between (1) the treatment of advective terms in the momentum equations and the wave action equation, (2) the expression of the mean current equation and the surface shear stress, and (3) the omission of bottom shear stress in the momentum equation.
    [Show full text]
  • Effect of Wave Radiation Stress in Storm Surge-Induced Inundation: a Case Study for the East Coast of India
    Pure Appl. Geophys. Ó 2019 Springer Nature Switzerland AG https://doi.org/10.1007/s00024-019-02379-x Pure and Applied Geophysics Effect of Wave Radiation Stress in Storm Surge-Induced Inundation: A Case Study for the East Coast of India 1 2 1 1 3 P. L. N. MURTY, A. D. RAO, K. SIVA SRINIVAS, E. PATTABHI RAMA RAO, and PRASAD K. BHASKARAN Abstract—Tropical cyclone-induced coastal inundation is a ecosystem and thickly populated coastal cities. Low- potential hazard for the east coast of India. In the present study, two lying coastal areas along the ECI are highly suscep- case studies are presented to examine the significance and impor- tance of wave radiation stress in storm surge modeling during two tible and vulnerable to the impact of storm surges due extreme weather events associated with the Phailin and Hudhud to landfalling tropical cyclones that originate in the cyclones. Model computations were performed using the advanced Bay of Bengal basin. The national weather agency circulation (ADCIRC) model and the coupled ADCIRC ? SWAN (Simulating Waves Nearshore) model for these two events. Mete- has categorized both Phailin and Hudhud cyclones as orological and astronomical forcing were used to simulate the very severe cyclonic storms (VSCS) that made hydrodynamic fields using the ADCIRC model run in a stand-alone landfall along the ECI near Gopalpur, Odisha and mode, whereas the coupled ADCIRC ? SWAN model also incorporated the wave radiation stress attributed from wave Visakhapatnam, Andhra Pradesh, respectively. These breaking effects. Cyclonic wind fields were generated using the two VSCS resulted in a trail of destruction during revised Holland model.
    [Show full text]