This book introduces a mathematical theory for the interaction of oceanic surface gravity waves and oceanic currents. This theory is formulated using the quasi-linear approximation for a uniform density fluid with a free surface and it provides wave-averaged expressions for the wave amplitudes and for the dynamical evolution of the currents. The surface gravity wave–current interaction theory is a more complete theory than previous with respect to an asymptotic expansion in the small parameter V/C, where V is a current speed and C is a wave speed. This book also illustrates the formal theory with several examples, and the path for its implementation in more realistic wave and circulation models is envisioned.
This book is appealing to oceanic research scientists and mathematicians interested in geophysical fluid dynamics.
Author(s): James C. McWilliams
Series: SpringerBriefs in Mathematics of Planet Earth
Publisher: Springer
Year: 2022
Language: English
Pages: 124
City: Singapore
Preface
Acknowledgments
Contents
1 Introduction
2 General Balances
3 Quasi-Linear Wave and Current Balances
3.1 Wave Quasi-Linear Equations: WQL
3.2 Current Quasi-Linear Equations: CQL
3.3 Quasi-Linear Energy Balances
4 Scaling Analysis
5 Quasi-Linear Wave Theory: CEW
5.1 Waves at Leading Order
5.2 Waves with CEW
6 Quasi-Linear Current Theory: WEC
6.1 WEC at Leading Order
6.2 Bernoulli Head and Surface Elevation
6.3 Radiation Stress
6.4 Energy Conversion at Leading Order
6.5 CEW Stokes Drift
6.6 CEW–WEC Force and Energy Conversion
7 Wave–Current Resonance
8 Examples
9 A Program for Broad-Band Wave and Current Spectra
10 Material Concentration and Buoyancy
11 Summary and Prospects
A Navier-Stokes and Quasi-Linear Energy Balances
A.1 Navier-Stokes Energetics
A.2 Quasi-Linear Wave–Current Energetics
B Quasi-Linear Surface Boundary Conditions
C Non-divergence of uSt in Quasi-Linear Theory
D Lagrangian Trajectories and Stokes Drift
E Solvability Condition for Φ"0365Φ
F CEW Stokes Drift
G Irrotational Companion to the Stokes Vortex Force
H CEW–WEC Force A
I CEW–WEC Force B
J Illustrative Examples
J.1 Horizontally-Uniform Currents, uh(z)
J.2 2D Jet Currents v(x,z)
J.2.1 Non-resonant Solutions
J.2.2 Resonant Solutions
J.2.3 Broad Jets
J.3 2D Roll-Cell Currents
J.3.1 Non-resonant Solutions
J.3.2 Resonant Solutions
J.3.3 Broad Roll Cells
J.4 Depth-Uniform Eddy Currents, uh(x,y)
J.4.1 Non-resonant Solutions
J.4.2 Resonant Solutions
J.4.3 Broad Eddy Currents
K Eddy Wave–Current Resonances
K.1 3D Rotational Eddies
K.2 3D Overturning Eddy Cells
L CEW and WEC for Material Concentration and Buoyancy
References
