Introduction to Nearshore Hydrodynamics

The following sections are included:

• Prologue

• Preface

• Contents

The following sections are included:

• A brief historical overview

• Summary of content

• References - Chapter 1

The following sections are included:

• Introduction

• Kinematics of fluid flow
  • Eulerian versus Lagrangian description
  • Streamlines, pathlines, streaklines
  • Vorticity ω_i and deformation tensor e_ij
  • Gauss' theorem, Green's theorems
  • The kinematic transport theorem, Leibnia rule

• Dynamics of fluid flow
  • Conservation of mass
  • Conservation of momentum
  • Stokes' viscosity law, the Navier-Stokes equations
  • The boundary layer approximation
  • Energy dissipation in viscous flow
  • The Euler equations, irrotational flow

• Conditions at fixed and moving boundaries
  • Kinematic conditions
  • Dynamic conditions

• Basic ideas for turbulent flow
  • Reynolds' decomposition of physical quantities
  • Determination of the turbulent mean flow
  • The Reynolds equations
  • Modelling of turbulent stresses

• Energy flux in a flow

• Appendix: Tensor notation

• References - Chapter 2

The following sections are included:

• Assumptions and the simplified equations

• Basic solution for linear waves
  • Solution for ø and η
  • Evaluation of linear waves
  • Particle motion
  • The pressure variation
  • Deep water and shallow water approximations

• Time averaged properties of linear waves in one horizontal dimension (1DH)
  • Introduction
  • Mass and volume flux
  • Momentum flux—radiation stress
  • Energy density
  • Energy flux
  • Dimensionless functions for wave averaged quantities

• Superposition of linear waves
  • Standing waves
  • Wave groups
  • Wave spectra

• Linear wave propagation over uneven bottom
  • Introduction
  • Shoaling and refraction
    • Simple shoaling
    • Determination of the refraction pattern
  • Refraction by ray tracing
  • The geometrical optics approximation
  • Kinematic wave theory

• Wave modification by currents
  • Introduction
  • Waves on a steady, locally uniform current
  • Vertically varying currents
  • The kinematics and dynamics of wave propagation on current fields

• Combined refraction-diffraction
  • Introduction
  • The wave equation for linear long waves
  • The mild slope equation
  • Further developments of the MSE
  • The parabolic approximation

• References - Chapter 3

The following sections are included:

• Introduction

• The energy equation

• The energy balance for periodic waves
  • Introduction of dimensionless parameters for 1-D wave motion
  • A closed form solution of the energy equation
  • The energy equation for steady irregular waves

• The general energy equation: Unsteady wave-current motion

• The wave action equation

• References - Chapter 4

The following sections are included:

• Introduction

• The highest possible wave on constant depth

• Qualitative description of wave breaking
  • Analysis of the momentum variation in a transition (Or: Why do the waves break?)

• Wave characteristics at the breakpoint

• Experimental results for surfzone waves
  • Qualitative surfzone characteristics
  • The phase velocity c
  • Surface profiles η(t)
  • The surface shape parameter B_o
  • The crest elevation η_c/H
  • The roller area
  • Measurements of particle velocities
  • Turbulence intensities
  • The values of P_xx, B_x and D
  • The wave generated shear stress

• Surfzone wave modelling
  • Surfzone assumptions
  • Energy flux E_f for surfzone waves
  • Radiation stress in surfzone waves
  • Volume flux in surfzone waves
  • The phase velocities for quasi-steady breaking waves
  • The energy dissipation in quasi steady surfzone waves

• Further analysis of the energy dissipation
  • Energy dissipation for random waves
  • Energy dissipation with a threshold
  • A model for roller energy decay
  • Advanced computational methods for surfzone waves

• Swash

• References - Chapter 5

The following sections are included:

• Introduction

• 1DH shoaling-breaking model

• 2DH refraction models
  • The wave propagation pattern
  • Determination of the wave amplitude variation

• Wave action models

• Models based on the mild slope equation and the parabolic approximation

• References - Chapter 6

The following sections are included:

• Introduction

• The equations for the classical nonlinear wave theories

• The system of dimensionless variables used

• Stokes waves

• Long waves
  • The Stokes or Ursell parameter
  • Long waves of moderate amplitude
  • Long waves of small amplitude
  • Long waves of large amplitude

• Conclusion

• References - Chapter 7

The following sections are included:

• Introduction

• Second order Stokes waves
  • Development of the perturbation expansion
  • First order approximation
  • Second order approximation
  • The solution for ø₂
  • The surface elevation η
  • The pressure p
  • The volume flux and determination of K
  • Stokes' two definitions of the phase velocity
  • The particle motion
  • Convergence and accuracy

• Higher order Stokes waves
  • Introduction
  • Stokes third order theory
  • Waves with currents
  • Stokes fifth order theory
  • Very high order Stokes waves

• The stream function method
  • Introduction
  • Description of the stream function method
  • Comparison of stream function results with a Stokes 5th order solution

• References - Chapter 8

The following sections are included:

• Introduction

• Solution for the Laplace equation

• The Boussinesq equations

• Boussinesq equations in one variable
  • The fourth order Boussinesq equation
  • The third order Korteweg-de Vries (KdV) equation

• Cnoidal waves — Solitary waves
  • The periodic case: Cnoidal waves
  • Final cnoidal wave expressions
  • Infinitely long waves: Solitary waves

• Analysis of cnoidal waves for practical applications
  • Specification of the wave motion
  • Velocities and pressures
  • Wave averaged properties of cnoidal waves
  • Limitations for cnoidal waves

• Alternative forms of the Boussinesq equations - The linear dispersion relation
  • Equations in terms of the velocity u_o at the bottom
  • Equations in terms of the velocity u_s at the MWS
  • The equations in terms of the depth averaged velocity ū
  • The equations in terms of Q
  • The linear dispersion relation

• Equations for 2DH and varying depth

• Equations with enhanced deep water properties
  • Introduction
  • Improvement of the linear dispersion properties
  • Improvement of other properties

• Further developments of Boussinesq modelling
  • Fully nonlinear models
  • Extension of equations to O(μ⁴) accuracy
  • Waves with currents
  • Models of high order
  • Robust numerical methods
  • Frequency domain methods for solving the equations

• Boussinesq models for breaking waves
  • Eddy viscosity models
  • Models with roller enhancement
  • Vorticity models
  • Wave breaking modelled by the nonlinear shallow water equations

• Large amplitude long waves U ≫ 1: The nonlinear shallow water equations (NSW)

• References - Chapter 9

• Appendix

The following sections are included:

• Introduction
  • The boundary layer equations. Formulation of the problem
  • Perturbation expansion for u
  • The 1st order solution
  • The 2nd order solution
  • The steady streaming u_s in wave boundary layers
  • Results for u_s

• Energy dissipation in a linear wave boundary layer

• Turbulent wave boundary layers
  • Rough turbulent flow
  • Energy dissipation in turbulent wave boundary layers

• Bottom shear stress in 3D wave-current boundary layers
  • Introduction
  • Formulation of the problem
  • The mean shear stress
  • Special cases

• References - Chapter 10

The following sections are included:

• Introduction

• Depth integrated conservation of mass
  • Separation of waves and currents

• Conditions at fixed and moving boundaries, II
  • Kinematic conditions
  • Dynamic conditions

• Depth integrated momentum equation
  • Integration of horizontal equations
  • Integration of the vertical momentum equation

• The nearshore circulation equations
  • The time averaged momentum equation
  • The equations for depth uniform currents

• Analysis of the radiation stress in two horizontal dimensions, 2DH
  • S_αβ expressed in terms of S_m and S_p
  • Radiation stress for linear waves in two horizontal dimensions (2DH)

• Examples on a long straight beach
  • The momentum balance
  • The cross-shore momentum balance: Setdown and setup
  • Longshore currents
  • Longshore current solution for a plane beach
  • Discussion of the examples

• Wave drivers

• Conditions along open boundaries
  • Introduction about open boundaries
  • Absorbing-generating boundary conditions
  • Boundary conditions along cross-shore boundaries

• References - Chapter 11

The following sections are included:

• The vertical variation of currents
  • Introduction
  • The governing equations for the variation over depth of the 3D currents

• The cross-shore circulation, undertow
  • Formulation of the 2-D problem and general solution
  • Boundary conditions
  • Solution for the undertow profiles with depth uniform ν_t and α₁
  • Discussion of results and comparison with measurements
  • Solutions including the effect of the boundary layer
  • Undertow outside the surfzone
  • Conclusions

• References - Chapter 12

The following sections are included:

• Introduction

• Governing equations
  • Time-averaged depth-integrated equations
  • Choices for splitting the current

• Solution for the vertical velocity profiles

• Calculation of the integral terms: The dispersive mixing coefficients
  • Final form of the basic equations

• Example: Longshore currents on a long straight coast

• Applications and further developments of quasi-3D modelling
  • The start-up of a longshore current
  • Rip currents
  • Curvilinear version of the SHORECIRC model
  • The nearshore community model, NearCoM

• References - Chapter 13

• Appendix

The following sections are included:

• Infragravity waves
  • Introduction
  • Basic equations for infragravity waves
  • Homogeneous solutions – Free edge waves
  • IG wave generation

• Shear instabilities of longshore currents
  • Introduction
  • The discovery of shear waves
  • Derivation of the basic equations
  • Stability analysis of the equations
  • Further analyses of the initial instability
  • Numerical analysis of fully developed shear waves

• References - Chapter 14

The following sections are included:

• Author Index

• Subject Index