Frequency Downshift in the Ocean Camille Zaug (in collaboration with - - PowerPoint PPT Presentation

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Frequency Downshift in the Ocean Camille Zaug (in collaboration with - - PowerPoint PPT Presentation

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This work was supported in part by the National Science Foundation grant DMS-1716120 Frequency Downshift in the Ocean Camille Zaug (in collaboration with Dr. John Carter) January 2019 Seattle University 1. Introduction 2. Wave Tank Experiments

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Frequency Downshift in the Ocean

Camille Zaug (in collaboration with Dr. John Carter) January 2019

Seattle University This work was supported in part by the National Science Foundation grant DMS-1716120

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SLIDE 2

Outline

  • 1. Introduction
  • 2. Wave Tank Experiments
  • 3. Ocean Data
  • 4. Summary and Future Work

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Introduction

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Waves

Gauge 1 Gauge 2 Gauge 3 Gauge 13

Length: 43 ft Plunger frequency: 3.33 Hz 13 wave gauges

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Thanks to Dr. Diane Henderson for conducting the wave tank experiments.

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SLIDE 5

Wave Data

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Superposition

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Wave Spectrum

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Wave Spectrum

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Wave Spectrum

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Wave Spectrum

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Frequency Downshift

A phenomenon that occurs when the carrier wave loses and transfers energy to lower sidebands, causing either the wave’s spectral mean

  • r spectral peak to decrease monotonically.

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Frequency Downshift

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→

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SLIDE 13

Goal

We wish to model frequency downshift in data collected from a wave tank and from the Pacific Ocean (collected by Snodgrass et al.).

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  • F. E. Snodgrass, K. F. Hasselmann, G. R. Miller, W. H. Munk, W. H. Powers, Propagation of ocean swell across the Pacific,

Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 259:431 - 497, 1966.

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Mathematical Models

We use nonlinear partial differential equations to model the system, as they are successful at modeling waves. Notation:

  • u: Modulating envelope
  • uχ: ”Spatial” derivative
  • uξ: ”Temporal” derivative
  • |u|2: Nonlinear term

We use sixth-order operator splitting in time to perform simulations.

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Mathematical Models

  • Nonlinear Schrödinger Equation (NLS)

iuχ + uξξ + 4|u|2u = 0

  • Dissipative Nonlinear Schrödinger Equation (dNLS)

iu u 4 u 2u i u

  • Dysthe Equation (Dysthe)

iu u 4 u 2u 8iu2u 32i u 2u 8iu u 2

  • Viscous Dysthe Equation (vDysthe)

iu u 4 u 2u i u 8iu2u 32i u 2u 8iu u 2 5 u

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  • J. D. Carter, D. Henderson, and I. Butterfield. A comparison of frequency downshift models of wave trains on deep water. Physics of Fluids, 31: 013103, 2019.
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Mathematical Models

  • Nonlinear Schrödinger Equation (NLS)

iuχ + uξξ + 4|u|2u = 0

  • Dissipative Nonlinear Schrödinger Equation (dNLS)

iuχ + uξξ + 4|u|2u + iδu = 0

  • Dysthe Equation (Dysthe)

iu u 4 u 2u 8iu2u 32i u 2u 8iu u 2

  • Viscous Dysthe Equation (vDysthe)

iu u 4 u 2u i u 8iu2u 32i u 2u 8iu u 2 5 u

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  • J. D. Carter, D. Henderson, and I. Butterfield. A comparison of frequency downshift models of wave trains on deep water. Physics of Fluids, 31: 013103, 2019.
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Mathematical Models

  • Nonlinear Schrödinger Equation (NLS)

iuχ + uξξ + 4|u|2u = 0

  • Dissipative Nonlinear Schrödinger Equation (dNLS)

iuχ + uξξ + 4|u|2u + iδu = 0

  • Dysthe Equation (Dysthe)

iuχ + uξξ + 4|u|2u + ϵ(−8iu2u∗

ξ − 32i|u|2uξ − 8iu(H(|u|2))ξ = 0

  • Viscous Dysthe Equation (vDysthe)

iu u 4 u 2u i u 8iu2u 32i u 2u 8iu u 2 5 u

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  • J. D. Carter, D. Henderson, and I. Butterfield. A comparison of frequency downshift models of wave trains on deep water. Physics of Fluids, 31: 013103, 2019.
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SLIDE 18

Mathematical Models

  • Nonlinear Schrödinger Equation (NLS)

iuχ + uξξ + 4|u|2u = 0

  • Dissipative Nonlinear Schrödinger Equation (dNLS)

iuχ + uξξ + 4|u|2u + iδu = 0

  • Dysthe Equation (Dysthe)

iuχ + uξξ + 4|u|2u + ϵ(−8iu2u∗

ξ − 32i|u|2uξ − 8iu(H(|u|2))ξ = 0

  • Viscous Dysthe Equation (vDysthe)

iuχ + uξξ + 4|u|2u + iδu + ϵ(−8iu2u∗

ξ − 32i|u|2uξ − 8iu(H(|u|2))ξ + 5δuξ) = 0 13

  • J. D. Carter, D. Henderson, and I. Butterfield. A comparison of frequency downshift models of wave trains on deep water. Physics of Fluids, 31: 013103, 2019.
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Model Assessment

We use conserved quantities and Fourier amplitudes to assess the efficacy of our models.

  • Mass: M = 1

L

∫ L

0 |u|2dξ

  • Spectral Mean: ωm = P

M

  • Spectral Peak: ωp, the frequency of the Fourier mode with the

largest amplitude

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Wave Tank Experiments

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Experimental Setup

Gauge 1 Gauge 2 Gauge 3 Gauge 13

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  • J. D. Carter, A. Govan. Frequency downshift in a viscous fluid. European Journal of Mechanics - B/Fluids, 59:177 – 185, 2016.
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Collected Data

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Simulation Results: Fourier Amplitudes

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Simulation Results: Conserved Quantities

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Ocean Data

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Snodgrass et al. Ocean Measurement Stations

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Google Maps, The Pacific Ocean, Google Maps, 2018.

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Collected Data

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  • D. M. Henderson, H. Segur, The role of dissipation in the evolution of ocean swell, Journal of Geophysical Research: Oceans 118:5074 - 5091, 2013.
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Collected Data: Challenges

  • Scale differences
  • Only four wave gauges
  • No phase data presented in spectra
  • We must discretize the data to achieve desired units
  • Period of waves is ambiguous

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Simulation Results: Fourier Amplitudes

Results for which viscous Dysthe is successful:

  • ∆f = 6.02 mHz
  • Wave period: L = 5 h
  • Wave phase: Random

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Simulation Results: Conserved Quantities

Results for which viscous Dysthe is successful

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Summary and Future Work

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Summary

The viscous Dysthe equation is the most successful at modeling the wave tank data. The choice of phase, period, and discretization affect which model most successfully models the ocean data.

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Future Work

  • Determine the most reasonable parameters for the ocean data
  • Run ocean simulations with the dissipative Gramstad-Trulsen

equation (which will eliminate sideband growth not observed in the ocean data)

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Questions?

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