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

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 3. Ocean Data 4. Summary and Future Work 1 Outline

Introduction

Plunger frequency: 3.33 Hz Length: 43 ft 13 wave gauges 2 Thanks to Dr. Diane Henderson for conducting the wave tank experiments. Waves Gauge 1 Gauge 2 Gauge 3 Gauge 13

3 Wave Data

4 Superposition

5 Wave Spectrum

6 Wave Spectrum

7 Wave Spectrum

8 Wave Spectrum

energy to lower sidebands, causing either the wave’s spectral mean A phenomenon that occurs when the carrier wave loses and transfers or spectral peak to decrease monotonically. 9 Frequency Downshift

10 Frequency Downshift →

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

system, as they are successful at modeling waves. Notation: • u : Modulating envelope We use sixth-order operator splitting in time to perform simulations. 12 Mathematical Models We use nonlinear partial differential equations to model the • u χ : ”Spatial” derivative • u ξ : ”Temporal” derivative • | u | 2 : Nonlinear term

• Dissipative Nonlinear Schrödinger Equation (dNLS) 4 u 2 u • Dysthe Equation (Dysthe) 4 u 2 u 32 i u 2 u u 2 • Viscous Dysthe Equation (vDysthe) 4 u 2 u 32 i u 2 u u 2 i u 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. 8 iu 2 u 8 iu 5 u 0 13 u 8 iu iu 0 8 iu 2 u u iu 0 i u u iu Mathematical Models • Nonlinear Schrödinger Equation (NLS) iu χ + u ξξ + 4 | u | 2 u = 0

• Dysthe Equation (Dysthe) 4 u 2 u 32 i u 2 u u 2 • Viscous Dysthe Equation (vDysthe) 4 u 2 u 32 i u 2 u u 2 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. 13 0 5 u 8 iu 8 iu 2 u i u 0 u iu 8 iu 8 iu 2 u u iu Mathematical Models • Nonlinear Schrödinger Equation (NLS) iu χ + u ξξ + 4 | u | 2 u = 0 • Dissipative Nonlinear Schrödinger Equation (dNLS) iu χ + u ξξ + 4 | u | 2 u + i δ u = 0

• Viscous Dysthe Equation (vDysthe) 4 u 2 u 32 i u 2 u u 2 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. 13 0 5 u 8 iu 8 iu 2 u i u u iu Mathematical Models • Nonlinear Schrödinger Equation (NLS) iu χ + u ξξ + 4 | u | 2 u = 0 • Dissipative Nonlinear Schrödinger Equation (dNLS) iu χ + u ξξ + 4 | u | 2 u + i δ u = 0 • Dysthe Equation (Dysthe) iu χ + u ξξ + 4 | u | 2 u + ϵ ( − 8 iu 2 u ∗ ξ − 32 i | u | 2 u ξ − 8 iu ( H ( | u | 2 )) ξ = 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. Mathematical Models • Nonlinear Schrödinger Equation (NLS) iu χ + u ξξ + 4 | u | 2 u = 0 • Dissipative Nonlinear Schrödinger Equation (dNLS) iu χ + u ξξ + 4 | u | 2 u + i δ u = 0 • Dysthe Equation (Dysthe) iu χ + u ξξ + 4 | u | 2 u + ϵ ( − 8 iu 2 u ∗ ξ − 32 i | u | 2 u ξ − 8 iu ( H ( | u | 2 )) ξ = 0 • Viscous Dysthe Equation (vDysthe) iu χ + u ξξ + 4 | u | 2 u + i δ u + ϵ ( − 8 iu 2 u ∗ ξ − 32 i | u | 2 u ξ − 8 iu ( H ( | u | 2 )) ξ + 5 δ u ξ ) = 0

efficacy of our models. L M largest amplitude 14 Model Assessment We use conserved quantities and Fourier amplitudes to assess the • Mass: M = 1 ∫ L 0 | u | 2 d ξ • Spectral Mean: ω m = P • Spectral Peak: ω p , the frequency of the Fourier mode with the

Wave Tank Experiments

J. D. Carter, A. Govan. Frequency downshift in a viscous fluid. European Journal of Mechanics - B/Fluids , 59:177 – 185, 2016. 15 Experimental Setup Gauge 1 Gauge 2 Gauge 3 Gauge 13

16 Collected Data

17 Simulation Results: Fourier Amplitudes

18 Simulation Results: Conserved Quantities

Ocean Data

Google Maps, The Pacific Ocean, Google Maps , 2018. 19 Snodgrass et al. Ocean Measurement Stations

D. M. Henderson, H. Segur, The role of dissipation in the evolution of ocean swell, Journal of Geophysical Research: Oceans 118:5074 - 5091, 2013. 20 Collected Data

• 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 21 Collected Data: Challenges

Results for which viscous Dysthe is successful: • Wave phase: Random 22 Simulation Results: Fourier Amplitudes • ∆ f = 6 . 02 mHz • Wave period: L = 5 h

23 Results for which viscous Dysthe is successful Simulation Results: Conserved Quantities

Summary and Future Work

The choice of phase, period, and discretization affect which model 24 Summary The viscous Dysthe equation is the most successful at modeling the wave tank data. most successfully models the ocean data .

• 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) 25 Future Work

25 Questions?