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Quasi-linear Theory for Surface Wave-Current Interactions

Part of the Mathematics of Planet Earth Book Series

DOI: 10.1007/978-981-19-2876-5

ISBN: 978-981-19-2875-8

Publisher: Springer Nature Singapore - Springer Journals

Jun 21, 2022

1 - 10 of 11 Chapters

LitStream

Chapter 1Introduction

McWilliams, James C.

Pages: 1 –5

[The physics problem of the mutual influences of rapidly oscillating surface gravity waves and more slowly evolving upper-ocean currents is well-studied.]
LitStream

Chapter 2General Balances

McWilliams, James C.

Pages: 7 –8

[Consider a uniform-density fluid with an upper free surface at z = η and a flat bottom at z = − H.]
LitStream

Chapter 3Quasi-Linear Wave and Current Balances

McWilliams, James C.

Pages: 9 –15

[In this chapter the QL theory for wave–current interaction is developed with only the assumptions that η is small enough to allow a Taylor series expansion about the mean surface elevation of z = 0]
LitStream

Chapter 4Scaling Analysis

McWilliams, James C.

Pages: 17 –20

[As discussed in Chap. 1, a QL theory, on the one hand, is an ansatz, where wave-wave nonlinear advection is neglected because its effects, while important for small- but finite-𝜖 wave evolution, are not necessarily of primary importance for the WEC and CEW interactions, even when wave...
LitStream

Chapter 5Quasi-Linear Wave Theory: CEW

McWilliams, James C.

Pages: 21 –26

[To solve the WQL equations (3.3)–(3.4), a perturbative approach is taken assuming small μ.]
LitStream

Chapter 6Quasi-Linear Current Theory: WEC

McWilliams, James C.

Pages: 27 –38

[The WQL and CQL equations can be integrated on the slow time of current evolution (n.b., Eq. (4.2)) after solving the elliptical problem (5.18) for P. This latter step is what distinguishes this wave–current interaction theory from previous ones (e.g., CL76 et seq.) that only include the...
LitStream

Chapter 7Wave–Current Resonance

McWilliams, James C.

Pages: 39 –41

[For a given current field u and primary wave specified by (a, k), consider an expanded representation of the leading-order wave field]
LitStream

Chapter 8Examples

McWilliams, James C.

Pages: 43 –45

[Four examples of different current shapes are considered here and, at greater length, in Appendix J. In each case a current is specified and presumed to remain “frozen” in time, and the corresponding CEW and WEC effects are evaluated.]
LitStream

Chapter 9A Program for Broad-Band Wave and Current Spectra

McWilliams, James C.

Pages: 47 –48

[The examples in Chap. 8 are all for narrow-banded wave and current fields, i.e., with only a few horizontal wavenumber k components]
LitStream

Chapter 10Material Concentration and Buoyancy

McWilliams, James C.

Pages: 49 –52

[Surface gravity waves are mostly uninfluenced by interior density and material concentration variations. A scale estimate (Chap. 4) of the buoyancy force is smaller than the leading-order wave pressure-gradient force by the ratio Δρ∕ρ near the surface, i.e., ∼ 10−3.]

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