Energetics and Dynamics of Internal Tides in Monterey Bay Using Numerical Simulations

ENERGETICS AND DYNAMICS OF INTERNAL TIDES IN MONTEREY BAY USING NUMERICAL SIMULATIONS

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Dujuan Kang November 2010

This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/sv691gk5449

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Oliver Fringer, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Stephen Monismith

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Robert Street

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

iii Abstract

Mixing processes in the ocean play a key role in controlling the large-scale circulation and energy distribution of the ocean. Internal tide-driven mixing is most important among the processes to mix the ocean interior. In the past decade, significant efforts have been made to understand tidal mixing processes. However, more details and better understanding are still required for some fundamental problems, such as the mechanisms that govern internal tide generation, radiation, and dissipation processes and the associated energy partitioning. This research aims to understand the energetics and dynamics of tidal mixing processes through both theoretical analysis and numerical simulations. The complete form of barotropic and baroclinic energy equations are derived and employed as the theoretical framework for analyzing the tidal energy budget. These equations provide a more accurate and detailed energy analysis because they include the full nonlinear and nonhydrostatic energy flux contributions as well as an improved evaluation of the available potential energy. This approach has been implemented in the hydrodynamic SUNTANS model, which is being employed to study the energetics of barotropic-to- baroclinic tidal conversion over complex bathymetry in the real ocean. Three-dimensional, high-resolution simulations of the barotropic and baroclinic tides in the Monterey Bay area are conducted using the SUNTANS model. A detailed analysis of the energy budget is performed to address the question of how the barotropic tidal energy is partitioned between local barotropic dissipation and local generation of baroclinic energy. After that, we then assess how much of this generated baroclinic energy is lost locally versus how much is radiated away and made available for open-ocean mixing. The mechanism of internal tide generation is investigated

iv by examining the dependence of barotropic-to-baroclinic energy conversion on three nondimensional parameters, namely the steepness parameter, the tidal excursion parameter, and the Froude number. Finally, a simple parametric model is presented to estimate the barotropic-to-baroclinic energy conversion.

v Acknowledgements

It is a pleasure to thank everyone who provided me support and made this dissertation possible. My sincerest gratitude goes to my advisor, Dr. Oliver Fringer, for his excellent guidance, encouragement, and support throughout my PhD study. He taught me numerous modelling skills beyond textbooks and inspired me with his in- novative thinking. Oliver was always available to discuss research and give advice. His patience and encouragement helped me overcome many diﬃculties and achieve my best. I am truly grateful to have such a wonderful mentor. I am especially grateful to Dr. Robert Street for leading me into the area of ocean sciences and giving me invaluable advice on my studies, research, and future career path. His careful dissertation reading and detailed comments are sincerely appreci- ated. I am very thankful to Dr. Stephen Monismith for helping me develop theoretical background. Many thanks to his insightful comments and useful suggestions to improve my dissertation. I would also like to thank Dr. Leif Thomas for serving as one of my oral exam committee members, and for his great ideas to broaden my research horizons. Finally, I thank my defense chair, Dr. Gianluca Iaccarino for his time and kind help. A special thank you goes to Dr. Rocky Geyer for his encouragement and insightful advice on my research and papers. I would also like to thank Dr. Karan Venayag- amoorthy and Dr. Alan Blumberg for providing useful comments and suggestions on my papers. I am very thankful to Dr. Steven Jachec, who has helped me with simulation setup and data analysis. His pioneer work in modelling internal tides in Monterey Bay has been an inspiration to me. I also thank Dr. Jody Klymak, Dr. James Girton,

vi Dr. Eric Kunze and Samantha Brody for kindly providing field data in support of this dissertation. Many thanks to Dr. Edward Gross and the UCLA ROMS group for their great help with my research work on the ROMS-SUNTANS coupling. I would like to thank Dr. Jeffrey Koseff, who helped me adapt to the new study environment when I first came to Stanford and gave me great support when I was searching for future career opportunities. I greatly appreciate Dr. Dale Haidvogel and Dr. Enrique Curchitser for their support and understanding, which allow me to focus on completing this dissertation. My sincere thanks also go to Dr. Andreas Thurnherr and Dr. Sonya Legg for useful discussions and suggestions on my research. The Environmental Fluid Mechanics Laboratory (EFML) is a comfortable place to work in. I feel very fortunate to have been part of it and would like to thank all the faculty, researchers, post-docs, students, and staffs for creating such a big warm family. A special thank you goes to our research group, including the current members: Bing Wang, Yi-Ju Chou, Subbayya Sankaranarayanan, Vivien Chua, Phillip Wolfram, Sean Vitousek, Goncalo Gil, Sergey Koltakov, and the former: Zhonghua Zhang, Steven Jachec, Karan Venayagamoorthy, Jan Wang, Gang Zhao, Sheng Chen, Mike Barad, and Jun Lee. I also thank Jill Nomura and Sandra Wetzel for the kind help. I have been very lucky to have many good friends during my study at Stanford. I really enjoyed knowing all of them. I am particularly thankful to Yifang Chen for her support and care when I was writing this dissertation. My deepest gratitude is to my parents, whose endless love and constant encouragement have always been the strongest power supporting me. Finally, I give my unique thanks to my husband, Wei Li. This PhD dissertation is dedicated to him for his love, care, patience, and support. Thank you. My research and dissertation were funded by ONR Grant N00014-05-1-0294 (Sci- entific officers: Dr. C. Linwood Vincent, Dr. Terri Paluszkiewicz and Dr. Scott Harper). This support is gratefully acknowledged.

vii Dedication

To my husband, Wei Li, for his love, care, patience, and support.

viii Contents

Abstract iv

Acknowledgements vi

Dedication viii

1 Introduction 1 1.1 Motivation and Background ...... 1 1.2 Project Objectives ...... 4 1.3 Dissertation Layout ...... 5

2 Literature Review 6 2.1 Linear Internal Wave Theory ...... 6 2.2 Internal Tide Generation Mechanisms ...... 10 2.3 Numerical Modelling of Internal Tides ...... 15 2.4 Internal Tides in Monterey Bay ...... 16 2.4.1 Field Observations ...... 17 2.4.2 Numerical Simulations ...... 17 2.5 Summary ...... 20

3 Numerical Methodology 22 3.1 SUNTANS ...... 22 3.2 Nonhydrostatic Pressure Solver: Time Accuracy ...... 24 3.2.1 Governing Equations ...... 25

ix 3.2.2 Numerical Discretization ...... 26 3.2.3 Time Accuracy of Pressure Methods ...... 27 3.2.4 Temporal Convergence Test ...... 29 3.2.5 Conclusions ...... 34 3.3 Nonhydrostatic Pressure Solver: Computational Eﬃciency ...... 34 3.3.1 Pressure-Poisson Equation ...... 35 3.3.2 Preconditioned Conjugate Gradient Method ...... 37 3.3.3 Convergence Test of CG and PCG ...... 38 3.3.4 Conclusions ...... 41 3.4 One-way ROMS-SUNTANS Coupling ...... 41 3.4.1 ROMS vs. SUNTANS ...... 41 3.4.2 One- and Two-way Coupling ...... 42 3.4.3 Interpolation Algorithm ...... 45 3.4.4 Implementation of One-way Nesting ...... 49 3.5 Summary ...... 50

4 Calculation of Available Potential Energy in Internal Wave Fields 51 4.1 Introduction ...... 52 4.2 Interpretation of APE ...... 53 4.3 Energy Conservation Laws ...... 55 4.4 Energetics of Progressive Internal Waves ...... 57 4.4.1 Numerical Setup ...... 57 4.4.2 Evolution of First-mode Internal Waves ...... 58 4.4.3 Energetics ...... 60 4.5 Conclusions ...... 63

5 Theoretical Framework: Barotropic and Baroclinic Energy Equa- tions 64 5.1 Governing Equations ...... 64 5.1.1 Boundary Conditions ...... 66 5.1.2 Deﬁnitions and Assumptions ...... 68 5.2 Energy Equations ...... 72

x 5.2.1 Kinetic Energy Equation ...... 73 5.2.2 Perturbation Potential Energy Equation ...... 74 5.2.3 Available Potential Energy Equation ...... 74 5.2.4 Total Energy Equation ...... 75 5.3 Barotropic Energy Equations ...... 77 5.4 Baroclinic Energy Equations ...... 79

6 Numerical Simulations: Energetics and Dynamics of Internal Tides in the Monterey Bay Area 84 6.1 Introduction ...... 85 6.2 Theoretical Framework ...... 88 6.3 Simulation Setup ...... 92 6.3.1 The SUNTANS Model ...... 92 6.3.2 Domain and Grid ...... 94 6.3.3 Initial Conditions ...... 95 6.3.4 Boundary Conditions ...... 96 6.3.5 Simulation Parameters ...... 98 6.4 Dynamics ...... 98 6.4.1 Free Surface and Barotropic Velocities ...... 99 6.4.2 Depth-averaged Velocities ...... 102 6.4.3 Baroclinic Velocities ...... 107 6.5 Energetics ...... 110 6.5.1 Horizontal Structure ...... 110 6.5.2 Energy Flux Budget ...... 115 6.5.3 Energy Flux Contributions ...... 121 6.5.4 Sensitivity Test of Simulation Parameters ...... 124 6.6 Generation Mechanism ...... 125 6.6.1 Parameter Space ...... 125 6.6.2 Energy Distribution vs. Parameters ...... 128 6.6.3 A Parametric Model to Estimate Energy Conversion . . . . . 131 6.7 Conclusions ...... 134

xi 7 Summary and Future Directions 136 7.1 Summary ...... 136 7.2 Future Directions ...... 139

References 140

xii List of Tables

4.1 Comparison of the imbalance of equation (4.12) at x=λ for the three APE formulations under diﬀerent conditions...... 60

6.1 Four types of simulation performed in this study...... 98

6.2 M2 tidal energy budget for the five subdomains indicated in Figure 6.17. The energy in estimated in MW (and percentage)...... 119 6.3 Contributions of different energy fluxes to the total baroclinic energy flux divergence for subdomain (a). The contributions are estimated in percentages...... 121 6.4 Sensitivity test of various simulation parameters...... 125 6.5 Key physical parameters governing internal tide generation and propagation...... 126

xiii List of Figures

1.1 The global energy ﬂux budget proposed by Munk & Wunsch (1998). The winds and tides are the two major sources of energy to mix the ocean. The tides contribute 3.5 TW of energy with 2.6 TW dissipated in shallow marginal seas and 0.9 TW lost in the deep ocean. The winds provide 1.2 TW of additional mixing power to maintain the global abyssal density distribution...... 2

2.1 Reflection of internal waves on varying slope in uniform stratification. 9 2.2 Cartoon showing the internal tide response to barotropic tides flowing

over topography characterized by diﬀerent ϵ1 and ϵ2 (St. Laurent & Garrett, 2002)...... 12

2.3 The parameter space deﬁned by the steepness parameter ϵ1 and the

tidal excursion parameter ϵ2 (Garrett & Kunze, 2007)...... 14 2.4 Bathymetry of Monterey Bay and the surrounding area...... 16 2.5 Observed depth-integrated station-averaged horizontal energy flux (red arrows) in the Monterey Sumarine Cayon (Kunze et al., 2002). Light yellow patches indicate areas of critical bottom slope for semidiurnal frequencies...... 18 2.6 Simulated depth-integrated, period-averaged baroclinic energy flux (blue arrows) in Monterey Bay and the surrounding area (Jachec et al., 2006). Red arrows indicate the observed energy flux in Figure 2.5, while green arrows represent field energy fluxes that are less reliable due to the influence of California Undercurrent...... 19

xiv 3.1 Depiction of the three dimensional SUNTANS grid structure as described by Fringer et al. (2006a). Figure from Zhang (2010)...... 23 3.2 Staggered grid for the test problem. ◦ indicates the locations of η and p, × indicates the locations of u, and indicates the locations of w. There are ghost points for η and p around the real domain...... 29 3.3 Contours of the nonhydrostatic pressure field (a) and the velocity field (b) after 100 time steps with ∆t = 0.05T using the projection method along with the iteration for P ...... 30 3.4 Temporal convergence results for the free surface (a), u-velocity (b), and nonhydrostatic pressure (c). Legend: Pressure projection without the P term ∗, pressure correction without the P term ⋄, pressure projection with iteration for P +, pressure correction with iteration for P ◦...... 31 3.5 Comparison of the free-surface evolution at x = 0 for the methods. Legend: Pressure projection without the P term (−−), pressure correction without the P term (−·), pressure projection with iteration for P (−), pressure correction with iteration for P (··)...... 32 3.6 Comparison of efficiency for the methods. Legend: Pressure projection without the P term ∗, pressure correction without the P term ⋄, pressure projection with iteration for P +, pressure correction with iteration for P ◦...... 33 √ 3.7 Effects of the aspect ratio on κ (a) and the number of iterations N (b) required to solve the discrete nonhydrostatic pressure-Poisson

equation. Legend: CG ∗, PCG with preconditioner M1 ◦, PCG with

preconditioner M2 ⋄...... 39 3.8 The convergence behavior with diﬀerent initial conditions: (a) Linear initial free surface, (b) Gaussian initial free surface. Legend: CG ∗,

PCG with preconditioner M1 ◦, PCG with preconditioner M2 ⋄. . . . 40 3.9 Depiction of a one-dimensional simulation involving one-way nesting (a) and two-way nesting (b). (Fringer et al. 2006b)...... 43

xv 3.10 Example of the domain setting for a nested ROMS-SUNTANS simulation. ROMS is run at the United States West Coast domain (a) and SUNTANS is applied to Monterey Bay subdomain (b). The red line represent the SUNTANS open boundaries...... 44 3.11 Planview (a) and vertical slice (b) of a ROMS-SUNTANS intergrid boundary. The red lines represent the unstructured, z-level SUNTANS grid, while the blue lines represent the curvilinear-coordinate, bottom- following ROMS grid. The four black points indicate the locations where ROMS data are needed for the bilinear interpolation in the horizontal; the green points indicate the locations where data are needed for the linear interpolation in the vertical; the black arrows indicate the locations on SUNTANS grid where the above interpolated data are transferred into SUNTANS (Fringer et al. (2006b), with modiﬁca- tions)...... 45 3.12 Depiction of the bilinear interpolation for a rectangular grid (a) and an orthogonal curvilinear grid with small curvature (b)...... 47 3.13 Depiction of how to identify the nearest ROMS cell for a given SUN-

TANS point S1. First Interpolate from the four corner points of the

ROMS domain to obtain R0, a ROMS grid point very near to S1 (a),

then approach R1, the ROMS grid point nearest to S1 (b), and ﬁnally

identify the ROMS cell with four ROMS grid points nearest to S1. . . 47 3.14 Procedure of the one-way nesting between ROMS and SUNTANS . . 49

4.1 Schematic diagram depicting the diﬀerent formulations of the APE

density. The rectangular area AEF D stands for AP E1, the light

shaded area ACD represents AP E2, while the triangle area ACD

stands for AP E3. The dark shaded area is err3, the diﬀerence be-

tween AP E2 and AP E3...... 54 4.2 Diagram of the energy budget for adiabatic, Boussinesq flow. Solid arrow lines: traditional energy flux budget; dashed arrow lines: active energy flux budget...... 56

xvi 4.3 Schematic of the two-dimensional (x-z) domain with periodic boundary conditions imposed at the left and right boundaries. Two initial background density profiles for the simulations are shown: (a) linear stratification (blue solid line); (b) nonlinear stratification (red dash line). 58 4.4 Evolution of a first-mode internal wave under different conditions: (a)

linear ρr, λ/D = 100; (b) nonlinear ρr, λ/D = 100; (c) nonlinear −1 ρr, λ/D = 4. The linear phase speed c1 = 1 m s and F r = 0.2 for all three cases. In each upper panel the four color plots show the

distributions of ρ − ρ0 at t/T =0, 1, 2, and 3. The lower panels (d)-(f) show the corresponding energy balance (4.12) as a function of time t/T at x = λ. In each of the lower panels, there are three indistinguishable ′ ′ curves ρ gw (solid), −sumKE (dashed), and sumAP E2 (dots), all normalized by the maximum value of ρ′gw over the ﬁrst three wave periods...... 59 4.5 Comparison of the energy balance (4.12) for diﬀerent APE formulations as a function of t/T at x = λ. The embedded plot is a zoomed-in view

to provide more detail. This case is with nonlinear ρr, λ/D = 100, and F r = 0.2. All terms are normalized by the maximum value of ρ′gw over the ﬁrst three wave periods...... 61 4.6 Comparison of the tendency of the kinetic energy and the three APE (a) and comparison of the depth-integrated terms in the APE equation

(4.9) for AP E = AP E1 (b). The simulation parameters and the value for normalization are the same as those in Figure 4.4(e)...... 62

5.1 Energy exchange diagram. Orange solid boxes represent the energy components within a fixed water column. Solid arrows indicate the energy conversion among different energy components within the control volume, while dashed arrows represent the divergence of energy fluxes transferring energy into or out of a fixed volume...... 76

xvii 5.2 Energy exchange diagram illustrating the barotropic-to-baroclinc energy conversion. Orange round boxes represent the energy components within a fixed water column. Solid arrows indicate the energy conversion among different energy components within the control volume, while dashed arrows represent the divergence of energy fluxes transferring energy into or out of a fixed volume...... 83

6.1 Mixing processes that transfer barotropic tidal energy into heat in the ocean. The associated energy distributions are indicated by the energy terms in the barotropic and baroclinic energy equations...... 85 6.2 Bathymetry map of Monterey Bay and the surrounding open ocean. The location of Station K is indicated by a black ∗. The domain outside the white box indicates the area aﬀected by the sponge layers in the simulations...... 87 6.3 The unstructured grid of the computational domain (upper). The lower two zoomed-in plots are for the subdomain (a) indicated in Figure 6.17 (lower-left), and Monterey Bay (lower-right). In the upper and lower- left plots, only cell centers are shown, while in the lower-right plot, cell edges are shown...... 93 6.4 Horizontal grid spacing histogram and the cumulative fraction for the grid depicted in Figure 6.3...... 94 6.5 Initial temperature and salinity proﬁles for the simulations (Courtesy of Dr. J. B. Girton, University of Washington and Dr. J. M. Klymak, University of Victoria). The black dots represent the vertical grid spacing for the simulations...... 95

6.6 M2 barotropic tidal ellipses along open boundaries. Bathymetry contours are spaced at -200, -500, -1000, -1500, -2000, -2500, -3000, -3500 m...... 97 6.7 Comparison of the free-surface elevation η, the East-West U and North- South V barotropic velocities between SUNTANS predictions (red dashed) and OTIS solutions (black solid) at Station K...... 99

xviii 6.8 Free-surface elevation at t = 18TM2 from an M2 forced barotropic run (Type III in Table 6.1). Bathymetry contours are spaced at -200, -500, -1000, -1500, -2000, -2500, -3000, -3500 m...... 100

6.9 Free-surface elevation at t = 18TM2 from an M2 forced baroclinic run (Type IV in Table 6.1). Bathymetry contours are spaced at -200, -500, -1000, -1500, -2000, -2500, -3000, -3500 m...... 101 6.10 Comparison of the depth-averaged velocities between SUNTANS predictions (red dashed) and field observations (black solid) at Station K: (a) original field data; (b) high-pass filtered field data...... 102 6.11 Power spectra of the observed East-West velocity at Station K. Upper: depth-averaged velocity; Lower: velocity. The vertical dashed lines

indicate the Coriolis (f) and M2 frequencies...... 103 6.12 Power spectra of the observed North-South velocity at Station K. Up- per: depth-averaged velocity; Lower: velocity. The vertical dashed

lines indicate the Coriolis (f) and M2 frequencies...... 104 6.13 Time series of the East-West u velocity at Station K: (a) ﬁeld obser-

vations; (b) M2 filtered field observations; (c) SUNTANS predictions. The white areas indicate regions without observations data...... 105 6.14 Time series of the North-South v velocity at Station K: (a) field obser-

vations; (b) M2 filtered field observations; (c) SUNTANS predictions. The white areas indicate regions without observation data...... 106 6.15 Vertical transect along which baroclinic velocities are shown. . . . . 108 6.16 Vertical structure of the East-West (a) and North-South (b) baroclinic velicities along the transect indicated in Figure 6.15. . . . .⟨ . .⟩ . . . . 109 6.17 Depth-integrated, period-averaged baroclinic energy flux, F′ . The five boxes (a)-(e) indicate the subdomains in which energetics are investigated in detail. Bathymetry contours are spaced at -200, -500, -1000, -1500, -2000, -2500, -3000, -3500 m...... 111 6.18 Depth-integrated, period-averaged barotropic-to-baroclinic conversion ⟨ ⟩ rate, C . Bathymetry contours are spaced at -200, -500, -1000, -1500, -2000, -2500, -3000, -3500 m...... 112

xix 6.19 Depth-integrated,⟨ ⟩ period-averaged baroclinic energy ﬂux divergence, ′ ∇H · F . Bathymetry contours are spaced at -200, -500, -1000, - 1500, -2000, -2500, -3000, -3500 m...... 113 6.20 Depth-integrated, period-averaged baroclinic dissipation rate, ∇ · ⟨ ⟩ ⟨ ⟩ H F′ − C . Bathymetry contours are spaced at -200, -500, -1000, -1500, -2000, -2500, -3000, -3500 m...... 114 6.21 Energy distribution as a function of depth. Lower: energy terms (6.34)- (6.36) in 200-m isobath bounded bins; Upper: cumulative sum of the lower...... 116 6.22 Energy distribution as a function of depth. Lower: energy terms (6.37)- (6.38) in 200-m isobath bounded bins; Upper: cumulative sum of the lower...... 117

6.23 Schematic of the M2 tidal energy budget for the two subdomains bounded by the 0-m, 200-m and 3000-m isobaths...... 118

6.24 Schematic of the M2 tidal energy budget in percentages for subdomain (a) in Figure 6.17...... 119

6.25 Schematic of the M2 tidal energy budget in percentages for subdomain (e) in Figure 6.17...... 120 6.26 Baroclinic energy ﬂux contribution from hydrostatic pressure work. Bathymetry contours are spaced at -100, -500, -1000, -1500, -2000 m. 122 6.27 Baroclinic energy ﬂux contribution from nonhydrostatic pressure work. Bathymetry contours are spaced at -100, -500, -1000, -1500, -2000 m. 123 6.28 Distribution of the conversion and the barotropic and baroclinic energy

as a function of (a) ϵ1 = γ/s, (b) ϵ2 = U0kb/ω, and (c) F r = U0/c for subdomain (a) in Figure 6.17...... 129 6.29 Distribution of the barotropic and baroclinic dissipation terms as a

function of (a) ϵ1 = γ/s, (b) ϵ2 = U0kb/ω, and (c) F r = U0/c for subdomain (a) in Figure 6.17...... 130

6.30 Distribution of the energy terms (in MW) as a function of (a) ϵ1 = γ/s

and ϵ2 = U0kb/ω, (b) ϵ1 = γ/s and F r = U0/c for subdomain (a) in Figure 6.17...... 131

xx 6.31 Distribution of the tidal energy conversion (in MW) as a function of

ϵ2 = U0kb/ω with diﬀerent values of ϵ1 = γ/s for subdomain (a) in Figure 6.17...... 132 6.32 Regression of the tidal energy conversion using (a) function (6.52) based on the linear relation (6.56), and (b) function (6.53) based on the quadratic relation (6.57). The upper panels are the SUNTANS results, and the lower panels are the best-ﬁt functions. On each plot, the bold

line indicates the relation between ϵ1 and ϵ2, while the two thin lines indicate one standard deviation from the bold line...... 134

xxi Chapter 1

Introduction

1.1 Motivation and Background

The ocean is a key component of the Earth’s climate system, and mixing processes in the ocean are critical in determining its thermohaline circulation and distribution of properties. The tides are one of the major sources of energy to mix the interior ocean. The problem of how and where the ocean tides dissipate their energy has received significant attention in the past decade. Munk & Wunsch (1998) proposed a global tidal energy flux budget as shown in Figure 1.1. Of the 3.5 TW (1 TW = 1012 W) total tidal energy lost in the ocean, approximately 2.6 TW is dissipated in shallow marginal seas through bottom friction, while the remaining portion is lost in the deep ocean. Egbert & Ray (2000, 2001) have confirmed that approximately 1 TW, or 25–30% of the global total tidal energy is lost in the deep ocean by inferring dissipation from a global tidal model. They found the tides lose much more energy in the open ocean, generally in regions with rough topographic features. Field observations also show that turbulent mixing is several order of magnitude larger over rough topography than over smooth abyssal plains (Polzin et al., 1997; Ledwell et al., 2000). This evidence has led to the renewed interest in internal tides as a major source of energy for deep-ocean mixing (St. Laurent & Garrett, 2002; Garrett, 2003; St. Laurent et al., 2003; Khatiwala, 2003; Legg, 2004a, b; Garrett & Kunze, 2007).

1 CHAPTER 1. INTRODUCTION 2

Figure 1.1: The global energy ﬂux budget proposed by Munk & Wunsch (1998). The winds and tides are the two major sources of energy to mix the ocean. The tides contribute 3.5 TW of energy with 2.6 TW dissipated in shallow marginal seas and 0.9 TW lost in the deep ocean. The winds provide 1.2 TW of additional mixing power to maintain the global abyssal density distribution. CHAPTER 1. INTRODUCTION 3

The barotropic tidal energy is converted into heat through a series of important mixing processes. When the barotropic tides flow over rough topographic features, a portion of the barotropic energy is lost directly through local mixing, while the other portion is converted into baroclinic energy through the generation of internal (baroclinic) tides. This generated baroclinic energy either dissipates locally or radiates into the open ocean, and then cascades into smaller scales along the internal wave spectrum and finally turns into deep ocean turbulence (Figure 1.1). In the past few years significant efforts have been made to understand these tidal mixing processes and the associated energy distributions. The generation of internal tide over idealized topography have been studied both analytically (Llewellyn Smith & Young, 2002; St. Laurent & Garrett, 2002; Balmforth et al., 2002; St. Laurent et al., 2003; Balmforth & Peacock, 2009) and numerically (Holloway & Merrifield, 1999; Khatiwala, 2003; Legg, 2004a, b; Vlasenko et al., 2005; Legg & Huijts, 2006). A great number of numerical investigations have also been performed to estimate the tidal energy flux budget for the real topography (Holloway, 2001; Niwa & Hibiya, 2001,2004; Merrifield & Holloway, 2002; Simmons et al., 2004; Jachec et al., 2006; Carter et al., 2008; Zilberman et al., 2009). Moreover, a series of field experiments have been conducted to provide more details of the mixing processes such as the Brazil Basin Tracer Release Experiment (http://www.whoi.edu/science/AOPE/cofdl/bbasin/), the Hawaii Ocean Mixing Ex- periment (HOME; http://www.nsf.gov/od/lpa/news/03/pr0374.htm) and the experiment for Assessing the Effects of Submesoscale Ocean Parameterizations (AESOP; http://www.apl.washington.edu/projects/AESOP/agenda.html). Tidal mixing processes are of key importance in the large-scale ocean circulation. However, they cannot be resolved in coarse-resolution ocean/climate models. Therefore, developing realistic tidal mixing parameterizations has become one of the greatest challenges in ocean and climate studies. Current tidal mixing parameterizations in ocean/climate models can capture the gross picture of the interior-ocean mixing (St. Laurent et al., 2002; Polzin, 2004; Jayne, 2009). However, many important details still require improvement. For example, the ratio of dissipated energy to total generated baroclinic energy, which is often assumed to be constant, actually varies in space and time. Therefore, more analytical studies and high-resolution CHAPTER 1. INTRODUCTION 4

simulations are needed to better understand the details and improve tidal mixing parameterizations. Monterey Bay lies along the Central U. S. West Coast. It is featured by the prominent Monterey Submarine Canyon (MSC), numerous ridges, smaller canyons, and a continental slope and break region. This area is exposed to the large- and meso-scale variations of the California Current System as well as the tidal currents. Energetic internal wave activity has been observed in MSC and the surrounding region (Petruncio et al., 1998; Lien & Gregg, 2001; Kunze et al., 2002; Carter et al., 2005). Due to the complex bathymetry, tidal mixing processes in this area are of great interest. In this dissertation, the nonhydrostatic SUNTANS model (Fringer et al. , 2006a) is employed to investigate the internal tide generation and energetics in the Monterey Bay area.

1.2 Project Objectives

The ultimate goal of this research is to contribute to the development of a physically based, energetically consistent parameterization of tidal mixing for large-scale climate simulations. To achieve this goal, this study focuses on understanding the dynamics and energetics of tidal mixing processes through both theoretical analyses and numerical simulations. The core objectives of this research are:

1. To derive a theoretical framework for the accurate evaluation of the tidal energy budget.

2. To investigate internal tide generation mechanisms in the Monterey Bay area.

3. To evaluate the nonlinear and nonhydrostatic contributions to the total energy budget.

4. To examine the parameters that determine the tidal energy partitioning.

5. To provide a parametric model to estimate the barotropic-to-baroclinic tidal energy conversion. CHAPTER 1. INTRODUCTION 5

1.3 Dissertation Layout

This dissertation is organized as follows: Chapter 2 first introduces linear internal wave theory. It then presents a literature review on theoretical and numerical studies of internal tide generation and energetics. Observations and simulations of internal tides in Monterey Bay are also reviewed. In Chapter 3, my work on the numerical method and model development is described, which includes a study of the time accuracy and computational efficiency of the nonhydrostatic pressure solver as well as the preliminary development of a multi- scale simulation tool based on the one-way coupling between ROMS and SUNTANS. Chapter 4 presents a comparison of three common formulations for calculating the available potential energy (APE) in internal wave fields. Both theoretical analysis and numerical simulations are employed to examine their differences. Advantages and limitations of each formulation under different nonlinear and nonhydrostatic conditions are discussed. Chapter 5 provides a detailed derivation of the barotropic and baroclinic energy equations. These equations with the full nonlinear and nonhydrostatic contributions are employed as the theoretical framework for analyzing the tidal energy flux budget. Chapter 6 focuses on numerical simulations of the internal tide in the Monterey Bay area. The energy budget is estimated based on the theoretical framework provided in Chapter 5. The internal tide generation mechanism is also investigated. Finally, a dissertation summary and future work recommendations are given in Chapter 7. The contents of Chapters 3-6 have been written as journal papers or conference proceedings, and therefore there may be some repeated information. Chapter 2

Literature Review

2.1 Linear Internal Wave Theory

Internal waves are gravity waves that exist either at the interface between two fluids of different densities or in a continuously stratified fluid. In the former case, they propagate horizontally along the interface, while in the latter, they are free to propagate in any direction. To understand the internal wave in a continuously stratified fluid, we analyze the linearized Euler equations with the Boussinesq approximation following Kundu & Cohen (2002)

∂u 1 ∂p − fv = − , (2.1) ∂t ρ0 ∂x ∂v 1 ∂p + fu = − , (2.2) ∂t ρ0 ∂y ∂w 1 ∂p g = − − ρ , (2.3) ∂t ρ0 ∂z ρ0 ∂ρ ρ N 2 = 0 w , (2.4) ∂t g ∂u ∂v ∂w + + = 0 , (2.5) ∂x ∂y ∂z where ρ0 is the constant reference density, and all variables represent perturbations

6 CHAPTER 2. LITERATURE REVIEW 7

due to internal waves, f is the Coriolis frequency, and N is the buoyancy frequency defined by g dρ N 2(z) = − r , (2.6) ρ0 dz where ρr(z) is the reference density field in the absence of wave perturbations. For 2 stable stratification (dρr/dz ≤ 0), N is non-negative and thus N is real. Solving equations (2.1)-(2.5) for w gives

∂2 ∂2w ∇2w + N 2∇2 w + f 2 = 0 , (2.7) ∂t2 H ∂z2

∇2 ∇2 where the Laplace operator and its horizontal component H are deﬁned by

∂2 ∂2 ∂2 ∇2 = + + , (2.8) ∂x2 ∂y2 ∂z2 ∂2 ∂2 ∇2 = + . (2.9) H ∂x2 ∂y2

We assume a solution to (2.7) of the form

w =we ˆ −i(kx+ly+mz−ωt) , (2.10) where ω is the internal wave frequency. k and l are the horizontal wavenumbers in the x- and y-directions, respectively, and m is the vertical wavenumber. The wavenumber vector is defined as K = (k, l, m) and its magnitude is given by K2 = k2 + l2 + m2. Substituting (2.10) into (2.7), we obtain the dispersion relation for an internal wave in a continuously stratified fluid as

k2 + l2 m2 ω2 = N 2 + f 2 . (2.11) K2 K2

There are several limiting cases for this relation:

• Nonrotating case (f = 0): the dispersion relation (2.11) reduces to

k2 + l2 ω2 = N 2 , (2.12) K2 CHAPTER 2. LITERATURE REVIEW 8

which is also suitable to describe high-frequency internal waves (ω ∼ N ≫ f).

• Hydrostatic case (k ≪ m): the dispersion relation becomes (2.11)

k2 + l2 ω2 = N 2 + f 2 , (2.13) m2

which is the same as for the low-frequency case (ω ∼ f ≪ N).

• Nonrotating and hydrostatic case (f = 0 & k ≪ m): the dispersion relation (2.11) reduces to k2 + l2 ω2 = N 2 , (2.14) m2 which is applicable to the mid-frequency case (f ≫ ω ≫ N).

Following Kundu & Cohen (2002), the phase and group velocities in vector form are

ω ω c = = (ke + le + me ) , (2.15) K K2 x y z ∂ω (N 2 − f 2)m c = = [kme + lme − (k2 + l2)e ] , (2.16) g ∂K ωK4 x y z from which we have

c · cg = 0 . (2.17)

Equations (2.15)-(2.17) indicate that in a continuously stratified fluid, the group velocity cg and phase velocity c are perpendicular to each other, and their horizontal components are in the same direction while vertical components are opposite (Figure 2.1). This property is quite different from that of surface waves or internal waves at a density discontinuity, for which cg and c are in the same direction. Furthermore, substituting (2.10) into the continuity equation (2.5) yields

K · u = 0 , (2.18) showing that the velocity vectors are aligned with the group velocity. Let θ be the angle of group velocity with respect to the horizontal, the dispersion relation (2.11) CHAPTER 2. LITERATURE REVIEW 9

can also be expressed as √ √ k2 + l2 ω2 − f 2 tan θ = = , (2.19) m N 2 − ω2 which indicates that a propagating internal wave exists only when f ≤ ω < N.

0 c

α c g

Figure 2.1: Reﬂection of internal waves on varying slope in uniform stratiﬁcation.

The reflection property of internal waves is very special in that the angles of incidence and reflection are preserved with respect to the direction of gravity (Phillips, 1977; Dauxios & Young, 1999). Figure 2.1 depicts idealized reflection situations when internal waves are incident upon a slope, with θ the angle of group velocity and α the angle of the slope, with respect to the horizontal. Waves hitting a slope with supercritical (α > θ) or subcritical (α < θ) angle will be focused into a more narrow ray toward deeper or shallower water with an increase in energy density (Eriksen, 1982). In the case of critical reflection (α = θ), significant concentration of the wave energy leads to enhanced boundary layer mixing (Cacchione & Wunsch, 1974; Ivey & Nokes, 1989; Slinn & Riley, 1996, 1998a,b; White, 1994). CHAPTER 2. LITERATURE REVIEW 10

2.2 Internal Tide Generation Mechanisms

Internal, or baroclinic, tides are internal waves of tidal frequency. They are generated by the interaction of barotropic tidal currents with rough bottom topography in strat- iﬁed waters (Baines, 1982). The ideal bathymetry for internal tides generation include the continental shelf-slope, the ridges, seamounts, and other rough topographic features. Barotropic tides typically cause the vertical movement of the ocean surface of O (1) m, while baroclinic tides can cause the vertical displacements of density surfaces of O (10) m or even O (100) m. Early studies of internal tides were carried out several decades ago both in theory (Baines, 1973, 1982; Bell, 1975) and by observations (Wunsch, 1975; Hendry, 1977; Torgrimson & Hickey, 1979). However, the important role of internal tides in deep- ocean mixing was not recognized until roughly one decade ago (Munk & Wunsch, 1998; Egbert & Ray, 2000, 2001; St. Laurent & Garrett, 2002; Garrett, 2003). In this section, we investigate the mechanisms that govern internal tide generation. Four key nondimensional parameters are generally employed to discuss the character of internal tide generation (Baines, 1982; St. Laurent & Garrett, 2002; Legg, 2004a,b; Legg & Huijts, 2006; Garrett & Kunze, 2007):

• The ﬁrst parameter is the steepness parameter deﬁned by

γ ϵ = , (2.20) 1 s

where the topographic slope is given by √( ) ( ) ∂d 2 ∂d 2 γ = tan α = + , (2.21) ∂x ∂y

and the internal wave characteristic slope is given by √ ω2 − f 2 s = tan θ = , (2.22) N 2 − ω2

where α and θ are the angles discussed in previous section (see Figure 2.1), and CHAPTER 2. LITERATURE REVIEW 11

z = −d(x, y) is the vertical location of the bottom topography. This steepness parameter is used to distinguish subcritical topography when its slope is ﬂat

compared to the wave characteristic slope (ϵ1 < 1) from supercritical topogra-

phy when its slope is steeper than the wave characteristic slope (ϵ1 > 1). The topography is referred to as critical when its slope matches the wave character-

istic slope (ϵ1 = 1). Recently, more studies have been carried out to understand the dependence of the barotropic to baroclinic energy conversion on this parameter (Balmforth et al., 2002; Khatiwala, 2003; Balforth & Peacock, 2009).

• The second nondimensional parameter is the tidal excursion parameter deﬁned by U k ϵ = 0 b , (2.23) 2 ω

where U0 is the amplitude of the barotropic tide and kb is the horizontal wavenumber of the topography. This parameter measures the ratio of the tidal −1 excursion amplitude U0/ω to the horizontal scale of the topography kb . When

the tidal excursion is less than the topographic scale (ϵ2 < 1), internal tides are generated mainly at the forcing frequency ω. When the tidal excursion is

larger than the topographic scale (ϵ2 > 1), lee waves can form (Figure 2.2). This parameter has been examined widely in studying internal tide generation over idealized topography (St. Laurent & Garrett, 2002; Legg & Huijts, 2006; Garrett & Kunze, 2007).

• The third nondimensional parameter is the Froude number deﬁned by

U0 U0k k F r = = = ϵ2 , (2.24) c ω kb

where k and c = ω/k are the horizontal wavenumber and phase speed of the internal tide, respectively. The Froude number measures the ratio of barotropic

tidal speed U0 to the baroclinic wave speed c. If the horizontal scales of the

topography and the internal tide match (k = kb), this parameter is the same as the tidal excursion parameter, which is the case in some analytical studies (St. Laurent & Garrett, 2002; Garrett & Kunze, 2007). The Froude number can be CHAPTER 2. LITERATURE REVIEW 12 1 ε

ε2

Figure 2.2: Cartoon showing the internal tide response to barotropic tides ﬂowing over topography characterized by diﬀerent ϵ1 and ϵ2 (St. Laurent & Garrett, 2002).

used to estimate the nonlinearity of the wave (Vlasenko et al., 2005). When F r ≪ 1, linear theory is valid for analyzing internal tide generation. First-mode

internal tides are generated over subcritical topography (ϵ1 < 1), while internal

tidal beams are generated over critical or supercritical topography (ϵ1 ≥ 1). At intermediate Froude numbers (F r ∼ 1), nonlinearity becomes important. Nonlinear internal wave bores, weak unsteady lee waves, and solitary internal waves may be generated depending on the topographic features. At high Froude numbers (F r > 1), in addition to bores and solitary internal waves, strong unsteady lee waves may form. Furthermore, under nonrotating (f = 0) and CHAPTER 2. LITERATURE REVIEW 13

hydrostatic (k ≪ m) conditions, the Froude number (2.24) reduces to

U m F r = 0 . (2.25) N

• The fourth nondimensional parameter is the ratio of the topographic amplitude

h0 to the total water depth H, which is given by

h δ = 0 , (2.26) H

where H = η + d with η(x, y) the vertical elevation of the water surface. This parameter has been used to investigate the role of ﬁnite ocean depth in the

barotropic to baroclinic tidal conversion. For topography with small ϵ1, the energy conversion rate decreases when δ increases. While for very steep topographic features, the energy conversion rate for δ → 1 is much larger than that for δ ≪ 1 (Llewellyn Smith & Young, 2002; Khatiwala, 2003; St. Laurent et al., 2003; Vlasenko et al., 2005).

From definitions (2.20)-(2.24) we can see that the first three parameters are related to each other via ϵ k 2 = F r b . (2.27) ϵ1 mγ For nonrotating and hydrostatic flows, this relation can be written as

ϵ U k h 2 = 0 b 0 , (2.28) ϵ1 Nh0 γ where the parameter U0/Nh0 has been used to examine the wave nonlinearity in lien of the Froude number (2.24) in some previous studies (Legg, 2004a, b; Legg & Huijts, 2006; Garrett & Kunze, 2007). Garrett & Kunze (2007) summarized the conditions for linearization for the ﬁve regions in the ϵ1 − ϵ2 parameter space (Figure 2.3). They deﬁned the steepness parameter as ϵ1 = kbh0/s, which is the same as (2.20) if γ = kbh0. Linear theory is valid for region 1 of Figure 2.3, in which both ϵ1 and ϵ2 are small. First mode internal tides are generated in this region. As ϵ1 increases but ϵ2 remains small (region 5 of Figure CHAPTER 2. LITERATURE REVIEW 14

2.3), higher spatial harmonics are generated although all with the same frequency ω. In this region the generated baroclinic tides are in the form of internal tidal beams due to the superposition of numerous vertical modes. As ϵ2 increases but ϵ1 remains small (region 2 of Figure 2.3), higher temporal harmonics are generated although all with the same wavenumber k, and resonantly reinforced lee waves are generated in this region. As both ϵ1 and ϵ2 become larger (regions 4 and 5 of Figure 2.3), simple analytical models breaks down and numerical models are necessary. Nonlinear hydraulic eﬀects were seen in numerical simulations for both regions and signiﬁcant local mixing occurs for narrow topographic features in region 3 (Legg & Huijts, 2006). 2 ε

1 ε1

Figure 2.3: The parameter space deﬁned by the steepness parameter ϵ1 and the tidal excursion parameter ϵ2 (Garrett & Kunze, 2007). CHAPTER 2. LITERATURE REVIEW 15

2.3 Numerical Modelling of Internal Tides

Numerical modelling is a useful tool to study internal wave dynamics and energetics. In this section we will review numerical studies of internal tide generation and the associated energy distribution over idealized and real bathymetry. Idealized simulations are necessary and instructive for understanding internal tide generation mechanisms, especially for nonlinear cases. Such investigations have focused on the generation efficiency of different topographic features (Holloway & Mer- rifield, 1999), the role of finite ocean depth (Khatiwala, 2003), the nonlinear response under various tidal excursion parameters and the Froude number (Legg, 2004a, b; Legg & Huijts, 2006), and an exploration of the full parameter space (Vlasenko et al. , 2005). Significant efforts have also been made to numerically estimate the internal tide generation and energetics for real bathymetry. Hotspots of internal wave activity include the Northern British Columbia Coast (Cummins & Oey, 1997), the Hawaiian Ridge (Merrifield et al., 2001; Merrifield & Holloway, 2002; Carter et al., 2008), the East China Sea (Niwa & Hibiya, 2004), the Monterey Bay region (Jachec et al., 2006; Carter, 2010; Hall & Carter, 2010), and the Mid-Atlantic Ridge (Zilberman et al. , 2009). All these studies employed the hydrostatic Princeton Ocean Model (POM) (Blumberg & Mellor, 1987) except for the work by Jachec et al. (2006) which employed the nonhydrostatic SUNTANS model (Fringer et al. , 2006a). All these studies contribute to refining the global tidal energy flux budget. In most of the previous numerical studies, the barotropic to baroclinic energy conversion was estimated by the baroclinic energy flux divergence ∇ · (u′p′) (e.g., Cummins & Oey, 1997; Merrifield et al., 2001; Merrifield & Holloway, 2002; Jachec et al., 2006). This term actually represents the radiated portion of the total energy conversion due to hydrostatic pressure work. It represents a good estimate of the total generated baroclinic energy for linear and hydrostatic cases. However in the presence of strong nonlinear and nonhydrostatic effects, a more accurate estimation is needed. CHAPTER 2. LITERATURE REVIEW 16

2.4 Internal Tides in Monterey Bay

Monterey Bay lies along the Central U. S. West Coast and is bisected by the Monterey Submarine Canyon (MSC) in the cross-shore direction (Figure 2.4). This area is exposed to the large- and mesoscale variations of the California Current System as well as the tidal currents. Due to the strong density stratiﬁcation and sharp topography, the barotropic tidal currents generate substantial energetic internal tides, which lead to local mixing as well as baroclinic energy radiation into the deep ocean. In this section, we will review the observations and numerical modelling of internal tides in the Monterey Bay area.

Elevation (m)

Moss Smooth Landing Ridge MSC Latitude Sur Platform Sur Ridge Sur

Longitude

Figure 2.4: Bathymetry of Monterey Bay and the surrounding area. CHAPTER 2. LITERATURE REVIEW 17

2.4.1 Field Observations

Monterey Bay is exposed to a mixed, predominantly semidiurnal tide, with the M2, K1, and S2 constituents contributing most of the total amplitude. The current variance in Monterey Bay is mainly associated with tidal motions. Internal, or baroclinic tidal currents can reach 15-20 cm s−1, which is an order of magnitude larger than barotropic tidal currents in the Bay (Paduan & Cook, 1997; Petruncio et al., 1998). The internal tidal currents are comparable in magnitude to those associated with the mesoscale eddies and meanders of the CCS (Ramp et al., 1997; Strub & James, 2000) and local coastal upwelling jets (Rosenfeld et al. , 1994). Fluctuations of internal tides in the MSC are found to be much larger than those in the surrounding area (Petruncio et al. , 1998). Nonlinear tidally-driven processes were also observed such as internal tidal bores within the MSC (Key, 1999), internal hydraulic jumps over the Smooth Ridge (Lien & Gregg, 2001), the offshore propagating solitary internal waves (Carter et al. , 2005). Identification of the source of the internal tides observed within the canyon is of great interest. Petruncio et al. (1998) hypothesized that the observed semidiurnal internal tide may be generated at the Smooth Ridge. However, Kunze et al. (2002) suggested that the generation occurs both within the canyon and from outside deeper region because the energy fluxes do not decrease monotonically (Figure 2.5), which indicates local generation along the canyon axis. Carter & Gregg (2002) and Carter et al. (2005) also present observations indicating both local and remote internal tide generation.

2.4.2 Numerical Simulations

Several numerical investigations have been conducted to understand internal tide generation and energetics in the Monterey Bay area. The hydrostatic POM has been employed to simulate the generation of internal tides in Monterey Submarine Canyon (Rosenfeld et al., 1999; Petruncio et al., 2002). These hydrostatic simulations replicate many features of the observed internal tides, such as the bottom-intensiﬁed current within the MSC and shoreward energy propagation along the canyon ﬂoor. CHAPTER 2. LITERATURE REVIEW 18 Latitude(36N)

Longitude

Figure 2.5: Observed depth-integrated station-averaged horizontal energy ﬂux (red arrows) in the Monterey Sumarine Cayon (Kunze et al., 2002). Light yellow patches indicate areas of critical bottom slope for semidiurnal frequencies.

However, there is limitation in modelling the energy cascade of internal waves with higher frequency. Rosenfeld et al. (1999) and Petruncio et al. (2002) suggested using nonhydrostatic models to study the energy cascade and including the alongshore component of the tidal velocity. Rosenfeld et al. (2006) used a nonhydrostatic version of POM to perform tidal simulations for a region oﬀ central California, including Mon- terey Bay. Their results suggested that long time series of currents should be used to examine the vertical structure of tidal velocities. Jachec et al. (2006) carried out three-dimensional simulations of the internal tides in Monterey Bay using the nonhydrostatic SUNTANS model (Fringer et al. , 2006a). They used a high-resolution grid, speciﬁed the alongshore component of the barotropic tidal currents, and employed the CHAPTER 2. LITERATURE REVIEW 19

field observed stratification as initial condition. Their results of the depth-integrated baroclinic energy flux indicate that the Sur Platform region is the main generation site for the baroclinic energy flux observed in the MSC (Figure 2.6).

Figure 2.6: Simulated depth-integrated, period-averaged baroclinic energy flux (blue arrows) in Monterey Bay and the surrounding area (Jachec et al., 2006). Red arrows indicate the observed energy flux in Figure 2.5, while green arrows represent field energy fluxes that are less reliable due to the influence of California Undercurrent. CHAPTER 2. LITERATURE REVIEW 20

2.5 Summary

Although much has been learned about the internal wave ﬁeld in Monterey Bay, many questions remain unanswered, speciﬁcally:

• How is barotropic energy partitioned between local barotropic dissipation and local generation of baroclinic energy?

• How is the generated baroclinic energy further partitioned between local baroclinic dissipation and baroclinic radiation?

• What are the nonlinear and nonhydrostatic contributions to the total energy budget?

• What parameters determine the tidal energy partitioning?

• How do you estimate the barotropic-to-barocloinic energy conversion based on the governing parameters?

In this dissertation I will answer these questions using the SUNTANS model. The simulations are similar to those of Jachec et al. (2006), although the following changes are made:

• Domain: I use a much larger domain to allow the evolution of oﬀshore propagating waves.

• Grid: I use a grid with each Voronoi center lying inside the corresponding grid cell to reduce the errors caused by grid correction.

• Tidal forcing: I use the original tidal currents computed by OTIS to force the run, while Jachec et al. adjusted them to give a better match in shallow water.

• Boundary conditions: I employ sponge layers at open boundaries to reduce the reﬂection of baroclinic waves and energy, while they used the radiation boundary condition. CHAPTER 2. LITERATURE REVIEW 21

• Stratiﬁcations: I specify both initial temperature and salinity and use nonlinear state function to compute density during simulations, while they speciﬁed only initial rho and use linear state function.

• Simulation time: my simulations are run for a much longer time (18 M2 tidal cycles).

In addition to these changes in simulation setup, the major improvement is that I derive the complete form of barotropic and baroclinic energy equations and implemented them into the SUNTNANS model. These equations provide a more accurate and detailed energy analysis because they include the full nonlinear and nonhydrostatic energy ﬂux contributions as well as an improved evaluation of the available potential energy. Chapter 3

Numerical Methodology 1 2

In this chapter, the SUNTANS model is ﬁrst introduced, and then my work on the numerical method and model development is presented, including the study on time accuracy and computational eﬃciency of the nonhydrostatic pressure solver as well as some preliminary work on the development of a multi-scale simulation tool based on one-way coupling between the ROMS and SUNTANS models.

3.1 SUNTANS

To investigate the internal tide dynamics and energetics, we employ the Stanford Un- structured Nonhydrostatic Terrain-following Adaptive Navier-Stokes Simulator (SUN- TANS) developed by Fringer et al. (2006a). SUNTANS solves the three-dimensional, nonhydrostatic equations (see Section 5) on a ﬁnite-volume grid which is unstructured in the horizontal and structured (z-level) in the vertical (Figure 3.1). The grid is staggered such that the velocity components are located at the cell faces, while all other variables are stored at the Voronoi points (the analogue of cell centers).

1Section 3.2 has been peer-reviewed and published in substantial part as “Time accuracy of pressure methods for non-hydrostatic free-surface flows” by D. Kang & O. B. Fringer in Proceedings of the 9th International Estuarine and Coastal Modeling Conference. 2Section 3.3 has been published in substantial part as “Efficient computation of the nonhydrostatic pressure” by D. Kang & O. B. Fringer in Proceedings of the 16th International Offshore and Polar Engineering Conference.

22 CHAPTER 3. NUMERICAL METHODOLOGY 23

Figure 3.1: Depiction of the three dimensional SUNTANS grid structure as described by Fringer et al. (2006a). Figure from Zhang (2010).

The nonhydrostatic formulation in SUNTANS is based on that of Casulli (1999), in which the pressure is split into its hydrostatic and nonhydrostatic components. The free-surface is advanced semi-implicitly to avoid the stability limitations associated with fast surface gravity waves, and the pressure correction method of Armfield & Street (2002) is implemented to solve the nonhydrostatic pressure on a staggered grid. It has been shown that this method maintains second-order time accuracy when extended to free-surface flows (Kang & Fringer, 2006a). Solution of the pressure- Poisson equation is the most computationally demanding part of the procedure. In SUNTANS it is solved by employing the preconditioned conjugate gradient algorithm with the block-Jacobi preconditioner, which is most efficient for problems with small aspect ratios (Kang & Fringer, 2006b). These studies on the nonhydrostatic pressure solver will be discussed in detail in Section 3.2 and 3.3. The other terms in the momentum equations are discretized with the second-order Adams-Bashforth method as in Zang et al. (1994). For scalar advection, a second- order accurate total variation diminishing (TVD) scheme is employed to reduce the numerical diffusion. Finally, SUNTANS is a parallel code with the message-passing interface (MPI), which makes the high-resolution and large-scale simulations possible. CHAPTER 3. NUMERICAL METHODOLOGY 24

3.2 Nonhydrostatic Pressure Solver: Time Accu- racy

Primitive hydrostatic models of ocean and coastal regions are widely and appropriately used for many situations (Blumberg & Mellor, 1987; Dietrich et al., 1987; Shchepetkin & McWilliams, 2003). They are capable of predicting the vertical structure of mainly horizontal flow. However, for some problems the hydrostatic approximation is no longer valid and a nonhydrostatic model is needed to represent important physical processes, such as flow over rapidly varying slopes, upwelling and downwelling motion, internal wave evolution, and short waves where the ratio of the vertical to horizontal scales of motion is not sufficiently small (Marshall et al., 1997; Casulli, 1999). When developing the nonhydrostatic models, the prominent challenges are the accurate and efficient computations of the nonhydrostatic pressure. In this and the following sections we will address these two challenges both analytically and numerically. The simulations in these two sections are conducted using a simplified code that has the same numerical schemes as SUNTANS but a different (structured rectangular) grid. The focus of this section is to examine the time accuracy of the free-surface nonhydrostatic solver. Two fractional-step pressure methods (pressure projection and pressure correction) are extended to nonhydrostatic free-surface flows. We apply the semi-implicit time discretization to the free-surface elevation (hydrostatic part), and employ the two pressure methods to discretize the nonhydrostatic pressure on a staggered grid. The time accuracy and efficiency for each pressure method are investigated in detail. Using a sloshing free-surface wave as a test case, we show analytically and numerically that the correction method is always second-order accurate in time, while the projection method is second-order accurate in time only if the nonhydrostatic pressure is included in the depth-averaged continuity equation through the use of an iterative procedure. We demonstrate that this convergence behavior holds for the linearized equations. CHAPTER 3. NUMERICAL METHODOLOGY 25

3.2.1 Governing Equations

The two-dimensional (x − z) linearized, incompressible, and inviscid Navier-Stokes equations with a free surface are studied. These simplifications are made because the viscous and nonlinear terms do not affect the analysis of temporal accuracy and efficiency. Therefore, the inviscid momentum equations in the absence of nonlinear advection are given by

∂u ∂η ∂p = −g − , (3.1) ∂t ∂x ∂x ∂w ∂p = − . (3.2) ∂t ∂z

The total reduced pressure pt/ρ0 has been decomposed into a sum of its hydrostatic and nonhydrostatic components pt/ρ0 = gη + p, where η(x, t) is the free-surface elevation and p(x, z, t) is the reduced nonhydrostatic pressure. Continuity is expressed by the incompressibility condition

∂u ∂w + = 0 . (3.3) ∂x ∂z

Integrating equation (3.3) over the depth leads to the free-surface equation ∫ ∂η ∂ η = − udz , (3.4) ∂t ∂x −d where d(x, y) is the depth of the domain. We have employed the kinematic free-surface and bottom boundary conditions,

∂η ∂η + u = w at z = η , ∂t ∂x ∂d −u = w at z = −d . (3.5) ∂x

In its linearized form, the depth-averaged continuity equation (3.4) is given by ∫ ∂η ∂ 0 = − udz . (3.6) ∂t ∂x −d CHAPTER 3. NUMERICAL METHODOLOGY 26

3.2.2 Numerical Discretization

Most popular methods of discretizing the nonhydrostatic equations are based on a fractional-step theme. At each time step, there are two substeps: in the first (or predictor) step, a predicted velocity field is obtained by including an approximate pressure field in the momentum equations. Then in the second (or corrector) step, a Poisson equation is solved to provide a pressure correction field, which is applied to the predicted velocity field to enforce continuity at the new time step (Armfield & Street, 2000, 2002). Here we extend two commonly used pressure methods (pressure projection and pressure correction) to nonhydrostatic free-surface flows. Applying the semi-implicit time discretization of Casulli (1999) to the free-surface elevation, we obtain the discrete form of the momentum and depth-averaged continuity equations as

un+1 − un [ ] = −g θG ηn+1 + (1 − θ)G ηn − G pn+1/2 , (3.7) ∆t x x x wn+1 − wn = −G pn+1/2 , (3.8) ∆t z ηn+1 − ηn ∑ ∑ = −θ∆zD un+1 − (1 − θ)∆zD un , (3.9) ∆t x x where ∆x and ∆z are the horizontal and vertical grid spacing, and ∆t is the time step size. Gx and Gz are the discrete gradient operators in the x- and z-directions, Dx is the discrete horizontal divergence operator, and summation is over k = 1,...,Nk. ()n, ()n+1, and ()n+1/2 represent the variable at corresponding time steps. This discretization is second-order accurate when θ = 1/2 and ﬁrst-order accurate otherwise. Employing the pressure methods to discretize the nonhydrostatic pressure, we obtain the predictor

u∗ − un [ ] = −g θG ηn+1 + (1 − θ)G ηn − αG pn−1/2 , (3.10) ∆t x x x w∗ − wn = −αG pn−1/2 , (3.11) ∆t z ηn+1 − ηn ∑ ∑ = −θ∆zD u∗ − (1 − θ)∆zD un ∆t x x CHAPTER 3. NUMERICAL METHODOLOGY 27

∑ + θ∆t∆zDx GxP, (3.12) where α and P for the two pressure methods are summarized as:

Method α P Projection 0 pn+1/2 n+1/2 n−1/2 Correction 1 pc (= p − p )

In the corrector step, a Poisson equation is ﬁrst solved with the divergence of the predicted velocity as a source term

1 LP = D · u∗ . (3.13) ∆t

The predicted velocity ﬁeld is then corrected with the gradient of the pressure correction obtained from (3.13) to produce a divergence-free velocity with

un+1 − u∗ = −GP, (3.14) ∆t where G and L are the discrete gradient (∇) and Laplacian (∇2) operators.

3.2.3 Time Accuracy of Pressure Methods

Substituting the corrector (3.14) into the predictor (3.10) - (3.12) yields

un+1 − un [ ] = −g θG ηn+1 + (1 − θ)G ηn − G (P + αpn−1/2) , (3.15) ∆t x x x wn+1 − wn = −G (P + αpn−1/2) , (3.16) ∆t z ηn+1 − ηn ∑ ∑ = −θ∆zD un+1 − (1 − θ)∆zD un . (3.17) ∆t x x

We can see that equation (3.17) is exactly the same as equation (3.9) exactly, and thus the free-surface term in the momentum equation (3.15) will not introduce additional error compared to the original discrete equation (3.7) when the diﬀerent pressure algorithms are employed. By subtracting the original discrete equations (3.7) - (3.8), CHAPTER 3. NUMERICAL METHODOLOGY 28

which are second-order accurate in time when θ = 1/2, from above equations (3.15) - (3.16), the error in the momentum equations is given by ( ) Error = −G(P + αpn−1/2 − pn+1/2) + O ∆t2 , (3.18) which is actually a second-order error for either pressure method, since the ﬁrst term vanishes when the values of α and P are substituted. Special care should be given to the third term on the right hand side of equation (3.12), which will aﬀect the time accuracy of the pressure methods. P is used as part of the source term when solving ηn+1 in the predictor step. However, P is the new pressure or pressure correction, and cannot be obtained until ηn+1 has been solved. There are two ways to handle this term:

1) Dropping P Term

By dropping the third term on the right hand side of equation (3.12) and substituting the corrector, we obtain

n+1 − n ∑ ∑ η η n+1 n = −θ∆zDx u − (1 − θ)∆zDx u ∆t ∑ − θ∆t∆zDx GxP, (3.19) where the error term, compared with (3.9), is { ∑ ∑ n+1/2 −θ∆t∆zDx Gxp Projection , −θ∆t∆zDx GxP = ∑ − 2 ∂p n+1/2 θ∆t ∆zDx Gx ∂t Correction .

We can see that the projection method introduces a ﬁrst-order error term in the free- surface equation, and thus reduces the time accuracy to O (∆t). For the correction method the error term is second order, so the correction method remains O (∆t2). CHAPTER 3. NUMERICAL METHODOLOGY 29

2) Iteration

In the second method, at each time step the pressure or pressure correction at the previous time step is ﬁrst used as an approximation for P , and the predictor and the Poisson equation are solved iteratively until P has converged to its new time step value, so that an accurate solution for ηn+1 can be obtained. The corrector is then carried out and integration proceeds to the next time step. Using this iterative procedure, because P has converged to its value at the new time step, the time accuracy of the pressure methods is not aﬀected and remains second-order accurate.

3.2.4 Temporal Convergence Test

Figure 3.2: Staggered grid for the test problem. ◦ indicates the locations of η and p, × indicates the locations of u, and indicates the locations of w. There are ghost points for η and p around the real domain. CHAPTER 3. NUMERICAL METHODOLOGY 30

Simulations of a 2-D sloshing free-surface wave are performed on a staggered grid, where η and p are cell-centered and u and w are face-centered (Figure 3.2). The rectangular domain of length L = 1 m and depth D = 1 m is subdivided into a uniform Cartesian grid with Ni ×Nk cells, where Ni = Nk = 32. The ﬂow is initialized with a cosine free-surface h(x, t = 0) = a cos(πx/L), where a = ∆z/10. We compute the evolution of the free-surface seiche for a total of tmax = T/10 seconds, where T = 2π/ω is the period of the wave, and ω is given by the linearized dispersion relation for irrotational water waves ω2 = πg/L tanh(πD/L). The temporal convergence rate 5 n of each scheme is analyzed by integrating the ﬂow over Nmax = 10×2 /2 time steps, with n = 1,..., 5, and comparing each solution to the reference solution which is 5 computed with Nmax = 10 × 2 time steps. The value of theta is set to θ = 0.5 to ensure second-order accuracy of the theta method, and the tolerance for the conjugate gradient pressure solver is 10−10. We employ the pressure methods to integrate the equations along with the two methods of calculating the P term in the predicted free-surface equation (3.12), which is discussed in Section 3.2.3.

Figure 3.3: Contours of the nonhydrostatic pressure ﬁeld (a) and the velocity ﬁeld (b) after 100 time steps with ∆t = 0.05T using the projection method along with the iteration for P . CHAPTER 3. NUMERICAL METHODOLOGY 31

At the free-surface, we apply a Dirichlet boundary condition, which is linearized by applying it at z = 0 rather than at z = h. The nonhydrostatic pressure is set to zero at the free-surface. At the other three boundaries, a Neumann boundary condition is applied, such that there is no normal ﬂux through the solid impenetrable boundaries. Since v · n = 0, where n is a vector of unit length normal to the boundaries, then n · ∇p = 0 on the solid boundaries. The boundary condition for the vertical velocity at the free surface can be obtained from the incompressibility condition. Figures 3.3 depicts the simulated nonhydrostatic pressure contours and velocity ﬁeld using the projection method with iteration for P after 100 time steps with ∆t = 0.05T .

(a) (b) (c) 10 0

O( ∆ t) O( ∆ t) O( ∆ t) 10 −2

t) −4 ∆ 10 E(

2 10 −6 O( ∆ t ) O( ∆ t2) O( ∆ t2)

10 −8 10 −4 10 −2 10 −4 10 −2 10 −4 10 −2 ∆ t ∆ t ∆ t

Figure 3.4: Temporal convergence results for the free surface (a), u-velocity (b), and nonhydrostatic pressure (c). Legend: Pressure projection without the P term ∗, pressure correction without the P term ⋄, pressure projection with iteration for P +, pressure correction with iteration for P ◦. CHAPTER 3. NUMERICAL METHODOLOGY 32

Figure 3.4 shows the temporal convergence results for the free surface, u-velocity, and nonhydrostatic pressure for the four schemes, i.e. the two pressure methods along with two methods to compute the P term. We can see that all errors for the projection method without the P term converge with ﬁrst-order accuracy, while the other three methods are second-order accurate in time for all variables. The errors for the two iterative schemes are almost identical and are much smaller than that of the correction method without the P term. Here only three iterations are required to achieve a converged solution at each time step. (m) η

t /

Figure 3.5: Comparison of the free-surface evolution at x = 0 for the methods. Legend: Pressure projection without the P term (−−), pressure correction without the P term (−·), pressure projection with iteration for P (−), pressure correction with iteration for P (··).

We also compare the evolution of the free-surface elevation at x = 0 over the ﬁrst two periods for the four schemes (Figure 3.5). This comparison is made on a 100×100 grid with ∆t = T/100. There is an obvious amplitude damping and a phase shift for the projection method without the P term. The accuracy of the correction method CHAPTER 3. NUMERICAL METHODOLOGY 33

without the P term is much better but there is still a phase error. The projection and correction methods with iteration for P give the best results. We finally compare the computational efficiency for the four schemes (Figure 3.6). The error is the free-surface error, and the runtime is in CPU seconds and represents the total wall-clock time required to integrate the equations for a total of tmax = T/10 seconds using Matlab on 1.4GHz Intel Pentium processor. The projection method without the P term is the least efficient scheme since the resulting error is two orders of magnitude larger than the other schemes for the same CPU time. When including the P term in the continuity equation, despite an apparent increase in the computational expense resulting from the iterative procedure, less CPU time is required to achieve the same error as the non-iterative procedure.

10 −1

10 −3 Error

10 −5

10 −7 10 0 10 2 CPU time (s)

Figure 3.6: Comparison of eﬃciency for the methods. Legend: Pressure projection without the P term ∗, pressure correction without the P term ⋄, pressure projection with iteration for P +, pressure correction with iteration for P ◦. CHAPTER 3. NUMERICAL METHODOLOGY 34

3.2.5 Conclusions

For inviscid free-surface flows, both the projection and correction methods demonstrate second-order time accuracy for the linearized equations. However, second-order accuracy is attained for the projection method only if the pressure is included in the depth-averaged continuity equation. Special care should be taken when solving the new free-surface ηn+1 by including the pressure in the source term. P is the new pressure or pressure correction, and cannot be obtained until ηn+1 has been solved. Therefore, a special procedure is required to update P to approximate its value at the new time step so as to accurately solve for ηn+1. Iteration can be used to update P without affecting the time accuracy for the two pressure methods. However, if the P term is dropped from the free-surface equation (3.12), the projection method becomes first-order accurate, while the correction method remains second-order accurate in time. The two pressure methods with iteration for the P term are the most efficient schemes, and the projection method without the P term is the least efficient scheme.

3.3 Nonhydrostatic Pressure Solver: Computational Eﬃciency

Solving the Poisson equation (3.13) is the most computationally demanding portion of fractional-step pressure methods. Zang et al. (1994) employed a multigrid solver to solve the pressure-Poisson equations in their laboratory-scale code. For ﬁeld-scale applications, preconditioning is required because the linear system resulting from the discretization of the pressure-Poisson equation is poorly conditioned. Casulli (1999) employed the preconditioned conjugate gradient algorithm both for the nonhydrostatic pressure as well as the free-surface, and Marshall et al. (1997) used a block-Jacobi preconditioner for the preconditioned conjugate gradient algorithm for the solution of the nonhydrostatic pressure. In this section, we derive two simple block diagonal preconditioners for the preconditioned conjugate gradient algorithm based on the structure of a matrix resulting CHAPTER 3. NUMERICAL METHODOLOGY 35

from the discretizaton of the linearized, nonhydrostatic Navier-Stokes equations. Us- ing a sloshing free-surface wave as a test case, we demonstrate how the effectiveness of the preconditioners is strongly dependent on the grid aspect ratio. The effect of different initial conditions on the convergence behavior of the conjugate gradient and preconditioned conjugate gradient algorithms is also presented.

3.3.1 Pressure-Poisson Equation

In this study, we use the same two-dimensional test case as in previous section, and thus the governing equations and discretization are the same as in Section 3.2.1 and 3.2.2. The pressure Poisson equation (3.13) can be written in matrix-vector form as

AP = S, (3.20) where A is a symmetric, positive-deﬁnite matrix which is made up of the two- or three-dimensional operators for the representative Poisson equation, P is a column vector of the pressure correction, and S is a column vector representing the source term of equation (3.13). If the cells in the water column do not have equal vertical grid spacing ∆z, then A is not symmetric, but it can be symmetrized by multiplication of a symmetrization matrix (Marshall et al. , 1997). Matrix A has ﬁve (2-D) or seven (3-D) diagonals which can be arranged in a block structure. For the 2-D test case depicted in Section 3.2.4, A can be wirtten as a NiNk × NiNk block tridiagonal matrix    C1 B   . .   B .. ..     . .  A =  .. C ..  , (3.21)  i   . .   .. .. B 

BCNi CHAPTER 3. NUMERICAL METHODOLOGY 36

where B and Ci are square matrices of size Nk × Nk. B is a diagonal matrix representing the horizontal second derivatives   l  0   .   ..      B =  l0  , (3.22)  .   .. 

l0 and Ci is a tridiagonal matrix    l2 l1   . .   l .. ..   1   . .  Ci =  .. l ..  , (i = 1, ··· ,Ni) , (3.23)  2   .. ..   . . l1 

l1 l2

1 1 − 1 1 where l0 = ∆x2 , l1 = ∆z2 , and l2 = 2( ∆x2 + ∆z2 ). Applying the boundary conditions gives    l1 + l2 l1   . .   l .. ..   1   . .  Ci =  .. l ..  , (i = 2, ··· ,Ni − 1) , (3.24)  2   .. ..   . . l1  l l − l  1 2 1   l0 + l1 + l2 l1   . .   l .. ..   1   . .  =  .. l + l ..  , (i = 1,Ni) . (3.25)  0 2   .. ..   . . l1 

l1 l0 + l2 − l1 CHAPTER 3. NUMERICAL METHODOLOGY 37

3.3.2 Preconditioned Conjugate Gradient Method

The most popular iterative method for solving the elliptic system (3.20) is the conjugate gradient method (CG). The convergence rate of CG is a strong function of the 2-norm condition number κ of matrix A. It is shown theoretically that the CG √ algorithm converges in approximately κ iterations (Demmel, 1997). Therefore, the larger the value of κ (i.e. the more ill-conditioned the matrix), the more slowly CG converges. For poorly conditioned problems, the preconditioned conjugate gradient algorithm (PCG) is employed to improve the convergence performence. The idea be- hind preconditioning is to devise a preconditioner, M, which is a symmetric, positive deﬁnite matrix that approximates A, but is substantially easier to invert. Using a preconditioner, equation (3.20) can be solved indirectly with a solution of

M −1AP = M −1S. (3.26)

A careful, problem-dependent choice of M can often make κ(M −1A) << κ(A) and thus accelerate convergence dramatically. We can choose a simple and efficient preconditioner for the present problem by exploiting the sparseness and diagonal dominance of A. For most environmental flows 1 1 of interest the aspect ratio ϵ = D/L is very small, thus ∆x2 << ∆z2 , which suggests 1 the ∆x2 term be dropped from matrix A. Based on this feature, we can devise two simple and efficient preconditioners for the test case in Section 3.2.4. By dropping the blocks B from matrix A, we can obtain a block diagonal preconditioner as   M  11   .   ..      M1 =  M1i  , (3.27)  .   .. 

M1Ni CHAPTER 3. NUMERICAL METHODOLOGY 38

where the diagonal blocks are identical to those of A, i.e.

M1i = Ci , (i = 1, ··· ,Ni) . (3.28)

1 If we not only drop B, but also neglect the eﬀect of ∆x2 in matrix C, we can reach an even simpler block diagonal preconditioner as   M  21   .   ..      M2 =  M2i  , (3.29)  .   .. 

M2Ni where the diagonal blocks are given by   −  1 1   .   1 −2 ..    1  . . .  M2i =  ......  , (i = 1, ··· ,Ni) . (3.30) ∆z2    .   .. −2 1  1 −3

The upper-left and bottom-right elements of matrix M2i are not −2 because of the application of boundary conditions. Returning to the original Poisson equation (3.13), we can see that as the aspect ratio decreases, the ratio of ∂2P/∂x2 to ∂2P/∂z2 becomes small, such that the ∂2P/∂x2 term in equation (3.13) can be neglected. The second preconditioner (3.29) reﬂects this feature and is good for the problems with small aspect ratios.

3.3.3 Convergence Test of CG and PCG

We study the convergence performance of CG and PCG using the two-dimensional test case depicted in Section 3.2.4. Here the ﬂow is initialized with a cosine free CHAPTER 3. NUMERICAL METHODOLOGY 39

surface

η(x, t = 0) = η0 cos(πx/L) , (3.31) where η0 = ∆z/10. We employ Ni = Nk = 20. In order to analyze how the convergence behavior depends on the problem aspect ratio, we ﬁx the length of the domain 1 1 at L = 10 m, and let the depth be D = 4 , 2 , 1, 2, 4, 8, 16, 32 m.

N N (a) (b)

κ N

ε ε √ Figure 3.7: Eﬀects of the aspect ratio on κ (a) and the number of iterations N (b) required to solve the discrete nonhydrostatic pressure-Poisson equation. Legend: CG ∗, PCG with preconditioner M1 ◦, PCG with preconditioner M2 ⋄. √ As shown in Figure 3.7, both κ and N decrease with a decrease in the aspect √ ratio for either the original or preconditioned problem. As ϵ decreases, κ and N for CG converge to a constant value while those for PCG decrease rapidly. The curves of √ κ for both preconditioners are identical and they are always below the curve for CG

(Figure 3.7 (a)). The curve of N for preconditioner M1 is always below that for CG.

However, the behavior of N with preconditioner M2 depends greatly on the aspect ratio; behavior is better for small ϵ but even worse than that for CG when ϵ is large (Figure 3.7 (b)). This result shows clearly that the eﬀectiveness of the preconditioners strongly depends on the grid aspect ratio. By employing the preconditioners, a bulk of the computational expense is alleviated for low-aspect ratio problems when the CHAPTER 3. NUMERICAL METHODOLOGY 40

flow is predominantly hydrostatic. Furthermore, the form of the initial condition for the free surface has a pronounced effect on the convergence behavior because of its effect on the starting point for the CG and PCG algorithms. Figure 3.8 demonstrates the convergence behavior of CG and PCG with two different initial conditions, namely the linear free surface

x − x η(x, t = 0) = −2η ( c ) , (3.32) 0 L and the Gaussian free surface

x − x η(x, t = 0) = −2η exp[−( c )2] , (3.33) 0 σ where η0 = ∆z/10, xc = L/2, and σ = ∆x.

(a) (b)

Figure 3.8: The convergence behavior with diﬀerent initial conditions: (a) Linear initial free surface, (b) Gaussian initial free surface. Legend: CG ∗, PCG with preconditioner M1 ◦, PCG with preconditioner M2 ⋄. CHAPTER 3. NUMERICAL METHODOLOGY 41

3.3.4 Conclusions

Solving the Poisson equation is the most computationally demanding portion of the fractional-step pressure methods. The most popular iterative method for solving this equation is the conjugate gradient (CG) method. For poorly conditioned problems, the preconditioned conjugate gradient (PCG) method is employed to improve the convergence performance. We present two simple block diagonal preconditioners for the PCG algorithm and demonstrate their convergence performance in simulations of a linearized surface wave. For a cosine initial free surface, the iteration number using CG decreases and converges to a constant value with a decrease in the problem aspect ratio. Using PCG, the iteration number decreases rapidly with a decrease in the aspect ratio, thereby alleviating a bulk of the computational expense associated with the nonhydrostatic pressure for low aspect ratio problems when the flow is predominantly hydrostatic. Furthermore, the form of the initial condition for the free surface has a pronounced effect on the convergence behavior because of its effect on the starting point for the CG and PCG algorithms.

3.4 One-way ROMS-SUNTANS Coupling

In this section, we describe the development of a multi-scale and multi-physics simulation tool based on the one-way coupling between ROMS and SUNTANS.

3.4.1 ROMS vs. SUNTANS

The Regional Oceanic Modeling System (ROMS) is a regional-scale hydrostatic parallel ocean model (Song & Haidvogel, 1994; Haidvogel & Beckmann, 1999; Shchep- etkin & McWilliams, 2003, 2005). It solves the incompressible, primitive equations with a free surface. ROMS employs orthogonal curvilinear coordinates in the horizontal and a generalized topography-following coordinate in the vertical. The open boundary conditions combine the outward radiation with a ﬂow-adaptive nudging toward external data (Marchesiello et al. , 2001). The time-stepping algorithm is split- explicit, where the free-surface elevation and barotropic momentum are advanced CHAPTER 3. NUMERICAL METHODOLOGY 42

using small time steps while a much larger time step is used for temperature, salinity, and baroclinic momentum equations. This algorithm allows a substantial increase in the time-step size without aﬀecting its accuracy (Shchepetkin & McWilliams, 2005). To avoid the σ-coordinate pressure-gradient error, ROMS reconstructs both the density ﬁeld and the geopotential height gz as continuous function of the transformed horizontal and σ coordinate, and then uses analytical integration to compute the pressure-gradient term. (Shchepetkin & McWilliams, 2003). SUNTANS is a local-scale nonhydrostatic parallel ocean model. As introduced in Section 3.1, it solves the three-dimensional, nonhydrostatic Navier-Stokes equations under the Boussinesq approximation with a free surface. SUNTANS employs an unstructured, staggered, z-level grid. ROMS is written in Fortran 90, while SUNTANS is written in the C programming language. Both ROMS and SUNTANS are designed to use parallel computer architectures. ROMS employs both the Open-MP directives for shared memory architectures and the message-passing interface (MPI) for distributed memory environment. SUN- TANS, on the other hand, employs MPI exclusively.

3.4.2 One- and Two-way Coupling

Physical processes in the oceans span a large range of scales, which cannot be simulated using one model due to the computing power limit. Therefore, different ocean models are developed and employed to study specific physical processes from large scale ocean circulation down to fine-scale mixing and dissipation. When two or more models of distinct spatio-temporal scales are coupled, a hybrid simulation tool is generated that can capture a broad range of scales and physics (Fringer et al. , 2006b). There are two common nesting strategies for the transfer of information between grids in a coupled simulation. In one-way nesting, information from the coarse grid is interpolated to supply boundary data for the fine grid, but the coarse-grid solution is not in turn affected by information from the fine grid. In two-way nesting, however, there is feedback information from the fine grid that will affect the coarse-grid solution directly. Figure 3.9 depicts the procedure to advance the nested simulation CHAPTER 3. NUMERICAL METHODOLOGY 43

(a)

(b)

Figure 3.9: Depiction of a one-dimensional simulation involving one-way nesting (a) and two-way nesting (b). (Fringer et al. 2006b). CHAPTER 3. NUMERICAL METHODOLOGY 44

(one-dimensional). For one-way nesting, as shown in Figure 3.9 (a), the solution at time t proceeds to time 2t as follows: 1a) information from the simulation at time t on the coarse grid is transferred to the boundary of the fine grid. 2a) The solution then progresses on the fine grid over two time steps with the fine-grid time step size t/2. 3a) The solution on the coarse grid is advanced forward in time one coarse-grid time step. 4a) This solution is then transferred again to the boundary of the fine grid. The procedure then repeats itself in steps 5a and 6a. For two-way nesting, as shown in Figure 3.9 (b), the procedure is altered because information will be transferred from the fine grid onto the coarse grid in steps 4b and 8b.

−125 −124 −123 −122 −121

(a) (b)

Figure 3.10: Example of the domain setting for a nested ROMS-SUNTANS simulation. ROMS is run at the United States West Coast domain (a) and SUNTANS is applied to Monterey Bay subdomain (b). The red line represent the SUNTANS open boundaries.

Because of the ease of implementation, one-way nesting has been employed suc- cessfully in many coupling simulations (Tseng et al., 2005; Penven et al., 2006). Coupling of ROMS and SUNTANS forms a hybrid simulation tool that can capture a broad range of spatial-temporal scales as well as diﬀerent ﬂow physics (Fringer et al. 2006b; Kang et al. 2006). In a nested simulation, the larger-scale model computes CHAPTER 3. NUMERICAL METHODOLOGY 45

the regional-scale, hydrostatic circulation, while the smaller-scale model computes the coastal-scale, nonhydrostatic processes. A typical nested domain setting is shown in Figure 3.10. ROMS runs at the West Coast domain to simulate the California Current System. SUNTANS, on the other hand, is applied to Monterey Bay subdomain to study the smaller-scale nonhydrostatic coastal processes such as the tide-topography interaction and the subsequent generation of internal tides.

3.4.3 Interpolation Algorithm

(a) (b)

Figure 3.11: Planview (a) and vertical slice (b) of a ROMS-SUNTANS intergrid boundary. The red lines represent the unstructured, z-level SUNTANS grid, while the blue lines represent the curvilinear-coordinate, bottom-following ROMS grid. The four black points indicate the locations where ROMS data are needed for the bilinear interpolation in the horizontal; the green points indicate the locations where data are needed for the linear interpolation in the vertical; the black arrows indicate the locations on SUNTANS grid where the above interpolated data are transferred into SUNTANS (Fringer et al. (2006b), with modiﬁcations).

ROMS and SUNTANS use diﬀerent grids and coordinate systems. ROMS employs orthogonal curvilinear-coordinate, topography-following grids, while SUNTANS employs unstructured, z-level grids. This diﬀerence complicates the information transfer between the grids and thus requires complex interpolation procedures both in the CHAPTER 3. NUMERICAL METHODOLOGY 46

horizontal as well as the vertical. In our present interpolation scheme, bilinear interpolation is ﬁrst carried out in the horizontal (Figure 3.11 (a)) for all ROMS vertical layers, and the interpolated data are then used to obtain the variables on the SUN- TANS grid through linear interpolation in the vertical (Figure 3.11 (b)). For a rectangular grid, as depicted in Figure 3.12 (a), the value of point C is determined basing on a weighted average of the values of the four vertices by

C = (1 − α)(1 − β)C1 + (1 − α)βC2 + α(1 − β)C3 + αβC4 , (3.34) where α and β are weights for the bilinear interpolation. This formulation can be extended to an orthogonal curvilinear grid with a very small curvature. For example, the coordinates of the green point in Figure 3.12 (b) is obtained approximately by { x = (1 − α)(1 − β)x + (1 − α)βx + α(1 − β)x + αβx , 1 2 3 4 (3.35) y = (1 − α)(1 − β)y1 + (1 − α)βy2 + α(1 − β)y3 + αβy4.

If the coordinates of the green point and the four vertices are all known, then by solving the pair of equations (3.35) we can obtain the two weights as { √ α = 1 − (A − B2 − C)/D, D ≠ 0 √ (3.36) β = 1 − (B + B2 − C)/E, E ≠ 0 where   A = x (y − y) + x(y − y − y + y ) − x (y − 2y + y) + x (y − y ) + x (y − 2y + y),  1 4 1 2 3 4 4 1 3 2 3 3 2 4   B = x1(y4 − y) + x(y1 − y2 − y3 + y4) − x4(y1 − 2y2 + y) + x3(y − y2) + x2(y3 − 2y4 + y), C = 4[(x − x )(y − y ) − (x − x )(y − y )][x(y − y ) + x (y − y) + x (y − y )],  3 4 1 2 1 2 3 4 4 2 4 2 2 4  − − − − −  D = 2[(x2 x4)(y1 y3) (x1 x3)(y2 y4)],  E = 2[(x3 − x4)(y1 − y2) − (x1 − x2)(y3 − y4)].

The information transferred between ROMS and SUNTANS is only a small portion of the ROMS output, therefore we need to identify ROMS cells where information must be saved and perform interpolations from them to obtain the variables on the CHAPTER 3. NUMERICAL METHODOLOGY 47

SUNTANS grid. Basing on the above discussion of bilinear interpolation, we design an algorithm that can quickly identify the nearest ROMS cell for a given SUNTANS point (Figure 3.13).

(a) (b)

Figure 3.12: Depiction of the bilinear interpolation for a rectangular grid (a) and an orthogonal curvilinear grid with small curvature (b).

R0 R0

(a) (b) (c)

Figure 3.13: Depiction of how to identify the nearest ROMS cell for a given SUNTANS point S1. First Interpolate from the four corner points of the ROMS domain to obtain R0, a ROMS grid point very near to S1 (a), then approach R1, the ROMS grid point nearest to S1 (b), and ﬁnally identify the ROMS cell with four ROMS grid points nearest to S1. CHAPTER 3. NUMERICAL METHODOLOGY 48

Because the whole ROMS domain consists of actually an orthogonal curvilinear grid, then for its four vertices and the given SUNTANS point there should be a relation in the form of equation (3.35). By solving this pair of equations we obtain the interpolation weights (α, β) from equation (3.36). Applying the weights to the

ROMS grid size (Nx,Ny) will give a pair of indices { I = round(α(N − 1) + 1), x (3.37) J = round(β(Ny − 1) + 1), where round represents a function that rounds the element to the nearest integer. As shown in Figure 3.13 (a), (I,J) indicates a point R0 on the ROMS grid, which is very near to S1 but not necessarily the nearest neighbor because of the curvilinear feature of the ROMS grid. R0 then approaches the next grid point that is nearer to C and

ﬁnally reaches R1, the grid point nearest to S1 (Figure 3.13 (b)). The next step is −→ to identify the cell containing S1 from the four cells around R1. Let v represent the −→ vector pointing from R1 to S1, and vi (i = 1, ··· , 4) represent the vectors pointing −→ from R1 to its four nearest neighbor points. The vector v can then be expressed as −→ −→ a linear combination of a pair of vectors (vi , vj ) as ( ) ( ) a −→ −→ −→ ··· ̸ v = vi vj , (i, j = 1, , 4, i = j) (3.38) b where (a, b) are the corresponding coeﬃcients, whose values can be obtained by solving equation (3.38). The desired cell is that with positive values of (a, b) (Figure 3.13 (c)). We then move on to identify the nearest ROMS cell for the next SUNTANS point.

If this point is very near to S1, we can skip the first step, i.e. step (a) in Figure 3.13, and repeat step (b) and (c) starting from R1. This algorithm is very efficient since it is of order O (N), where N is the total number of the SUNTANS points to be identified, and does not require a search algorithm with a computational complexity of, at best, O (N log N). CHAPTER 3. NUMERICAL METHODOLOGY 49

3.4.4 Implementation of One-way Nesting

SUNTANS

Figure 3.14: Procedure of the one-way nesting between ROMS and SUNTANS

In one-way nesting, the coarse-grid solution is not affected by the solution on the fine grid. Therefore, we can first obtain the coarse-grid solution for all time steps inde- pendently, and then compute the fine-grid solution. Our one-way nesting procedure is designed based on this feature (Figure 3.14). The first step is to identify ROMS cells where data are needed for the bilinear interpolation in the horizontal. A Matlab code based on the algorithm presented in Section 3.4.3 is employed to return the indices of the identified ROMS cells and the corresponding bilinear interpolation weights. Two Fortran codes are added to ROMS to read in the output from Matlab and output the information for SUNTANS. The horizontal interpolation is performed within ROMS using the weights computed previously. Furthermore, in order to simplify the vertical interpolation, the data from above horizontal interpolation are fixed from the ROMS σ levels to corresponding z levels. Thus at each ROMS time step, the output from CHAPTER 3. NUMERICAL METHODOLOGY 50

ROMS contains data for all required SUNTANS grid points on each fixed ROMS z level. The output is then read into SUNTANS by transfer.c, a new file added to SUNTANS for the nesting case. This file performs temporal and vertical interpolations as well as sets boundary and initial conditions for SUNTANS. This framework forms the basis for on-going ROMS-SUNTANS simulations of internal waves on the Australian North West Shelf (Fringer et al. , 2006c) and of exchange flow between San Francisco Bay and the coastal Pacific Ocean.

3.5 Summary

In this chapter, we first introduced the SUNTANS model briefly, and then presented my study on the time accuracy and computational efficiency of the nonhydrostatic pressure solver. Using a sloshing free-surface wave as a test case, we demonstrated both analytically and numerically that the correction method is always second-order accurate in time, while the projection method maintains second-order time accuracy only if the nonhydrostatic pressure is included in the depth-averaged continuity equation through the use of an iterative procedure. We also found that the effectiveness of preconditioners for the PCG algorithm strongly depends on the grid aspect ratio. By employing the preconditioners, a bulk of the computational expense is alleviated for low-aspect ratio problems when the flow is predominantly hydrostatic. In the last section of this chapter, we described the development and implementation of the one-way coupling between ROMS and SUNTANS. Chapter 4

Calculation of Available Potential Energy in Internal Wave Fields 3

The appropriate and accurate evaluation of available potential energy (APE) has important ramiﬁcations for analyses of internal wave energetics. In this chapter, three common formulations for calculating the available potential energy (APE) in internal wave ﬁelds are compared, namely the perturbation APE, AP E1, the exact local APE,

AP E2, and its approximation for linear stratification, AP E3. The relationship among these formulations is illustrated through a graphical interpretation and a derivation of the energy conservation laws. Numerical simulations are carried out to quantitatively assess the performance of each APE under the influence of different nonlinear and nonhydrostatic effects. The results show that AP E2 is the most attractive in evaluating the local APE, especially for nonlinear internal waves, since use of AP E2 introduces the smallest errors when computing the energy conservation laws. Larger errors arise when using AP E1 because of the large disparity in magnitude between the kinetic energy and AP E1. We show that the disparity in the tendency of AP E1 is compensated by a large flux arising from the reference pressure and density fields.

Because the tendency of the kinetic energy is close to that of AP E3, computational

3This has been published in substantial part as “On the calculation of available potential energy in internal wave ﬁelds” by D. Kang & O. B. Fringer in Journal of Physical Oceanography, 40(11), 2539–2545.

51 CHAPTER 4. AVAILABLE POTENTIAL ENERGY 52

errors arise when using AP E3 only in the presence of nonlinear stratiﬁcation, and these errors increase for stronger ﬂow nonlinearity.

4.1 Introduction

Not all potential energy can be converted into kinetic energy, which ultimately con- tributes to mixing. The small active portion of the potential energy that is available for this conversion is referred to as the available potential energy (APE). The concept of APE has been widely used to study the energetics of internal waves (Klymak & Moum, 2003; Venayagamoorthy & Fringer, 2005; Klymak et al., 2006; Scotti et al., 2006; Lamb, 2007; Moum et al., 2007; Carter et al., 2008; Lamb & Nguyen, 2009) and other mixing processes in stratified fluids (Winters et al., 1995; Huang, 1998; Molemaker & McWilliams, 2010). The domain-integrated APE for an incompressible fluid is defined as the difference in the potential energy between the perturbed state and the reference state, which is the minimum potential energy state obtained through adiabatic processes (Lorenz, 1955). Following this definition, different formulations of the APE density have been employed. A classic definition is the perturbation potential energy den- ′ ′ sity AP E1 = ρ gz, where ρ is the perturbation density, which has been widely used to calculate the depth-integrated (Venayagamoorthy & Fringer, 2005; Moum et al., 2007) or domain-integrated APE (Klymak & Moum, 2003; Klymak et al., 2006) in analyzing internal wave energetics. Another well-known expression for the APE den- 2 2 2 ′2 2 sity is AP E3 = ρ0N ζ /2=g ρ /2ρ0N , where ρ0 is the constant reference density associated with the Boussinesq approximation, ζ is the vertical displacement of a fluid particle and N is the buoyancy frequency (Gill, 1982; Kundu, 1990). Although

AP E3 is derived from linear theory, it is commonly used for internal wave calculations in which the stratification is slowly-varying (Carter et al., 2008). A positive-definite expression for arbitrary stratifications was proposed by Holliday & McIntyre (1981) ∫ z − ′ ′ as AP E2 = z−ζ g[ρ(z) ρr(z )]dz , where ρr is the reference density. More recently, this formulation was employed in analyzing the energetics of nonlinear internal waves (Scotti et al., 2006; Lamb, 2007; Lamb & Nguyen, 2009). Lamb (2008) compared CHAPTER 4. AVAILABLE POTENTIAL ENERGY 53

the calculation of AP E1 and AP E2 for an isolated perturbation and pointed out that their integrals over a finite domain are identical. Typically, when assessing the energy flux budget for a linear, hydrostatic wave, only the dominant kinetic energy flux term up′ is calculated, where p′ is the perturbation hydrostatic pressure (Kunze et al., 2002; Merrifield & Holloway, 2002; Nash et al., 2005). However, in the presence of strong nonlinear and nonhydrostatic effects, it is important to include the nonlinear and nonhydrostatic terms in the kinetic energy flux as well as the APE flux term (Venayagamoorthy & Fringer, 2005; Lamb, 2007; Moum et al., 2007). Therefore, an appropriate evaluation of the APE has important ramifications for analyses of internal wave energetics. In this paper, we provide a comparison of these three formulations for calculating the APE in internal wave fields. Both theoretical analysis and numerical simulations are employed to highlight their differences. In particular, we compare their performance in the numerical simulations under different nonlinear and nonhydrostatic conditions. Advantages and limitations of each formulation in analyzing the energetics of internal waves are discussed.

4.2 Interpretation of APE

We consider a stratified incompressible fluid with a stable reference stratification

2 g dρr ρr(z). The buoyancy frequency is defined by N (z) = − . Figure 4.1 shows ρ0 dz the vertical distribution of the reference density ρr(z) and the perturbed density ′ ρ(x, y, z, t) = ρr(z) + ρ (x, y, z, t) for a horizontal location (x, y) and time t. Due to a perturbation, the fluid particle experiences a vertical displacement ζ = z −z∗, moving from z∗(x, y, z, t) in the reference state to z in the perturbed state. As a result, the density of the fluid parcel must satisfy ρ(x, y, z, t) = ρr(z∗). The APE densities at point D(x, y, z, t) in Figure 4.1 can be interpreted graphically in terms of areas as

′ × AP E1 = ∫ρ gz = g Area(AEF D) , (4.1) z ′ ′ AP E2 = g [ρ(z) − ρr(z )] dz = g × Area(ACD) , (4.2) z∗ CHAPTER 4. AVAILABLE POTENTIAL ENERGY 54

where Area(ACD) is the lightly-shaded region in Figure 4.1. Because the portion of the potential energy between the perturbed and the reference density proﬁles is the true active potential energy, or the energy that is available for conversion to kinetic energy, AP E2 is an exact expression to evaluate the local APE (Holliday & McIntyre,

1981; Shepherd, 1993; Lamb, 2007, 2008). AP E1 includes some area associated with the inactive portion of the potential energy and thus is larger in magnitude than the kinetic energy. Furthermore, AP E1 is coordinate dependent since its value depends on the height at which z = 0. Therefore, AP E1 is not a good choice to evaluate

APE on a local basis. A coordinate-independent formulation of AP E1, given by ρgζ = g × Area(GHCD), was used by Winters et al. (1995) to obtain the volume- integrated APE.

G AD

H C

EF 0

Figure 4.1: Schematic diagram depicting the diﬀerent formulations of the APE density. The rectangular area AEF D stands for AP E1, the light shaded area ACD represents AP E2, while the triangle area ACD stands for AP E3. The dark shaded area is err3, the diﬀerence between AP E2 and AP E3. CHAPTER 4. AVAILABLE POTENTIAL ENERGY 55

′ Using the Taylor series expansion in ρ , the formulation of AP E2 can be expressed as

2 ′2 3 2 ′3 g ρ g (N )zρ ′4 AP E2 = 2 + 2 6 + O(ρ ) . (4.3) 2ρ0N 6ρ0N

If the ﬂuid is linearly stratiﬁed (with constant N), only the leading term on the right hand side of equation (4.3) remains. In this limit we obtain

g2ρ′2 AP E3 = 2 = AP E2 + err3 . (4.4) 2ρ0N

This expression is the well-known linear APE density (Gill, 1982; Kundu, 1990). For nonlinear stratifications, AP E3 is not an exact expression of the APE density and differs to leading order by an amount that can be estimated by the second term on the right hand side of equation (4.3). Graphically, AP E3 can be interpreted as g times the area of triangle (ACD) in Figure 4.1. The dark shaded region is err3, the difference between AP E2 and AP E3, which vanishes for linear stratification.

4.3 Energy Conservation Laws

For an inviscid, non-diﬀusive, Boussinesq ﬂuid, the evolution equations for the kinetic energy density KE = ρ0u · u/2 and the potential energy density PE = ρgz are given by

∂KE + ∇ · [u (KE + p + p′ + p + p )] = −ρgw , (4.5) ∂t | nh {z s r } Fk ∂P E + ∇ · (uPE) = ρgw , (4.6) ∂t | {z } Fp where the pressure is split into its hydrostatic and nonhydrostatic components with p = ph + pnh. The hydrostatic pressure is further decomposed as ph = pr(z) + p′(x, y, z, t) + p (x, y, t), with p the free-surface pressure. Using the rigid-lid approx- s s ∫ 0 ′ imation, the reference pressure is pr = g z ρr dz and the perturbation hydrostatic CHAPTER 4. AVAILABLE POTENTIAL ENERGY 56

∫ ′ 0 ′ ′ pressure is p = g z ρ dz . Summing the above two energy equations gives the total energy conservation law as

∂(KE + PE) + ∇ · [u (KE + PE + p + p′ + p + p )] = 0 . (4.7) ∂t nh s r

These energy equations show that energy is transferred into and out of a particular control volume via the kinetic energy flux Fk and the potential energy flux Fp, while within the control volume energy is converted between Ep and Ek via the buoyancy flux −ρgw. This energy budget is illustrated by the solid arrows in Figure 4.2.

External energy

KE APE PE

Figure 4.2: Diagram of the energy budget for adiabatic, Boussinesq flow. Solid arrow lines: traditional energy flux budget; dashed arrow lines: active energy flux budget.

We now consider the energy budget between the active energy components. Based on the deﬁnition of each APE density, and along with the expression of pr, we obtain the KE and PE equations with active energy ﬂuxes as

∂KE + ∇ · [u (KE + p + p′ + p )] = −ρ′gw , (4.8) ∂t | {znh s } ′ Fk ∂AP E + ∇ · [u (AP E + f)] = ρ′gw , (4.9) ∂t | {z } ′ Fp

′ − where Fk = Fk upr is the active kinetic energy ﬂux because it excludes work done CHAPTER 4. AVAILABLE POTENTIAL ENERGY 57

by the reference pressure from the total kinetic energy ﬂux. The function f is deﬁned as    pr + ρrgz , for AP E1 f =  0 , for AP E2 (4.10)  0 , for AP E3 with linear ρr .

′ For AP E2, the available potential energy flux Fp is just uAP E2 which is the truly active potential energy flux, while for AP E1, a reference energy flux term uf is in- ′ cluded, which can be much larger than the active energy flux terms given ρr ≫ ρ

(Venayagamoorthy & Fringer, 2005). For AP E3 in nonlinear-stratiﬁed ﬂuids, equation (4.9) does not hold due to the error discussed in Section 4.2, which requires inclusion of the term ∂(err3)/∂t on the right-hand side of equation (4.9) to ensure a balance. The energy conservation between the active components is obtained by taking the sum of (4.8) and (4.9) to give

∂(KE + AP E ) n + ∇ · [u (KE + AP E + p + p′ + p + f)] = 0 . (4.11) ∂t n nh s

The dashed arrows in Figure 4.2 illustrate the energy transfer and conversion through active energy ﬂuxes. Equations (4.8)-(4.11) show that the conservation laws for AP E2 are independent of the reference state and thus appropriately describe the energy transfer between the kinetic energy and the available potential energy.

4.4 Energetics of Progressive Internal Waves

4.4.1 Numerical Setup

We study the evolution of a first-mode internal wave over flat, frictionless topography in a two-dimensional (x − z) domain of length L = 2λ and depth D, where λ is the wavelength (Figure 4.3). Periodic boundary conditions are imposed in the direction of wave propagation, which extends the domain to an infinite length in the x-direction. Numerical simulations are performed using the SUNTANS code of Fringer et al. (2006a) to assess the influence of the stratification, nonlinearity, and CHAPTER 4. AVAILABLE POTENTIAL ENERGY 58

nonhydrostatic eﬀects on diﬀerent APE formulations.

(b) D (a)

Figure 4.3: Schematic of the two-dimensional (x-z) domain with periodic boundary conditions imposed at the left and right boundaries. Two initial background density profiles for the simulations are shown: (a) linear stratification (blue solid line); (b) nonlinear stratification (red dash line).

Two different initial reference stratifications are considered, namely linear stratification given by ρr(z) = ρ0 − ∆ρ1(z/D), and nonlinear stratification given by −3 ρr(z) = ρ0 − ∆ρ2[tanh(5z/D + 1) − tanh(1)], where ρ0 = 1000 kg m . We as- −1 sume a first-mode internal wave phase speed c1 = 1 m s , a depth D = 500 m and a −3 −3 wavelength λ = 50 km, which requires ∆ρ1 = 2.01 kg m and ∆ρ2 = 1.01 kg m . To assess the influence of internal wave nonlinearity, for each stratification we consider two different Froude numbers, namely F r = u0/c1 = 0.05 and F r = 0.2, where u0 is the horizontal velocity amplitude. Nonhydrostatic effects are assessed by shortening the wavelength to λ = 2 km so that λ/D = 4, which is more nonhydrostatic than the longer wave case which has λ/D = 100. With λ/D = 4 we set F r = 0.2 and perform two additional simulations using the linear and nonlinear stratifications. In total, there are six simulations, represented by the results in each column of Table 4.1.

4.4.2 Evolution of First-mode Internal Waves

The upper three panels (a)-(c) in Figure 4.4 depict a time sequence of the evolution of waves with a fixed Froude number F r = 0.2 but with different stratifications and aspect ratios λ/D. Despite having the same Froude number, the wave in the linear stratification (Figure 4.4(a)) does not steepen into a train of rank-ordered solitary-like CHAPTER 4. AVAILABLE POTENTIAL ENERGY 59

waves as it does in the nonlinear stratification as shown in Figure 4.4(b). Although both waves travel at the same linear phase speed, isopycnal displacements for the nonlinear stratification have a stronger effect on the amplitude dispersion of the wave, thereby causing nonlinear steepening. This effect can be reduced by increasing the relative importance of the nonhydrostatic pressure (Figure 4.4(c)). Decreasing the value of λ/D increases the relative effect of the nonhydrostatic pressure, thereby re- ducing the rate at which the waves steepen.

(a) (b) (c) t/T=0 0

-1 t/T=1 0

-1 t/T=2 1 0

-1 t/T=3 0

z/D z/D z/D z/D z/D z/D z/D 0 -1 0 1 2 0 1 2 0 1 2 (d) (e) (f) 1

-1 0 1 2 3 0 1 2 3 0 1 2 3

NormalizedTerms t/T t/T t/T

Figure 4.4: Evolution of a first-mode internal wave under different conditions: (a) linear ρr, λ/D = 100; (b) nonlinear ρr, λ/D = 100; (c) nonlinear ρr, λ/D = 4. The −1 linear phase speed c1 = 1 m s and F r = 0.2 for all three cases. In each upper panel the four color plots show the distributions of ρ − ρ0 at t/T =0, 1, 2, and 3. The lower panels (d)-(f) show the corresponding energy balance (4.12) as a function of time t/T at x = λ. In each of the lower panels, there are three indistinguishable curves ρ′gw ′ (solid), −sumKE (dashed), and sumAP E2 (dots), all normalized by the maximum value of ρ′gw over the first three wave periods. CHAPTER 4. AVAILABLE POTENTIAL ENERGY 60

Table 4.1: Comparison of the imbalance of equation (4.12) at x=λ for the three APE formulations under diﬀerent conditions.

linear ρb, λ/D=100 nonlinear ρb, λ/D=100 nonlinear ρb, λ/D=4 Fr=0.05 Fr=0.2 Fr=0.05 Fr=0.2 Fr=0.05 Fr=0.2 Imb1 0.0019 0.0025 0.0092 0.0529 0.0038 0.0065 Imb2 0.0016 0.0015 0.0052 0.0241 0.0036 0.0064 Imb3 0.0014 0.0014 0.0209 0.1220 0.0171 0.0377

4.4.3 Energetics

In practice, the depth- and volume-integrated energy budgets are of primary interest, particularly for internal waves. In what follows we focus on the depth-integrated budget, and we note that a similar analysis was performed for the volume-integrated budget that yielded identical results. The depth-integration of the left-hand side of equations (4.8) and (4.9) are represented by sumKE′ and sumAP E, respectively. Here () represents the depth-integration of a quantity, while sum stands for the summation of all terms on the left-hand side of an equation. Using this notation, we must have −sumKE′(x, t) = sumAP E(x, t) = ρ′gw(x, t), (4.12) which holds for all APE formulations except for AP E3 in the presence of nonlinear stratiﬁcation. The lower three panels (d)-(f) in Figure 4.4 illustrate this balance relation as a function of time t/T at x = λ for AP E2. On each panel, three curves ′ ′ representing normalized ρ gw, −sumKE , and sumAP E2 are indistinguishable, im- plying a precise balance given by (4.12).

A quantitative measure of the imbalance in computing equation (4.12) is given by [ ] std sumAP E (x, t) − ρ′gw(x, t) Imb (x) = [ n ] , (4.13) n std ρ′gw(x, t) where n = 1, 2, 3, and std( ) represents the standard deviation of a quantity over the ﬁrst six wave periods. The results at x = λ are presented in Table 4.1. Errors are CHAPTER 4. AVAILABLE POTENTIAL ENERGY 61

incurred both due to the theoretical imbalance when using AP E3 and from computational errors when solving the equations on discrete grid. In general, AP E2 performs the best, although it yields a larger imbalance than AP E3 in linear stratification due to numerical errors in computing z∗ on a discrete grid. As expected, in the presence of nonlinear stratification AP E3 does not satisfy equation (4.9) and thus shows significant imbalance. Although in theory AP E1 should satisfy the balance relation (4.12) well, it demonstrates relatively large imbalances, which cannot be improved (in a relative sense) with more numerical accuracy. The numerical imbalance for AP E1 ′ is larger than that for AP E2 because small errors in computing ρ are magnified for ′ ′ AP E1 = ρ gz relative to AP E2 ∼ ρ gζ since, in general, z >> ζ. A comparison of the energy balance (4.12) for different APE formulations is shown in Figure 4.5 for the case with nonlinear ρr, λ/D = 100, and F r = 0.2.

1 1

0.5

2.49 t/T 2.5 0

ρ’gw

NormalizedTerms sumAPE 1

sumAPE 2

-1 sumAPE 3

2.5 2.6 2.7 2.8 2.9 t/T

Figure 4.5: Comparison of the energy balance (4.12) for diﬀerent APE formulations as a function of t/T at x = λ. The embedded plot is a zoomed-in view to provide more detail. This case is with nonlinear ρr, λ/D = 100, and F r = 0.2. All terms are normalized by the maximum value of ρ′gw over the ﬁrst three wave periods. CHAPTER 4. AVAILABLE POTENTIAL ENERGY 62

For the same case as in Figure 4.5, Figure 4.6 (a) compares the tendency terms of KE and APE. ∂AP E1/∂t is roughly one order of magnitude larger than ∂KE/∂t.

While for AP E2 and AP E3, the tendency terms of APE and KE are of the same order of magnitude. Figure 4.6(b) presents the contributions from all terms in equation

(4.9) for AP E1. The reference energy flux term ∇ · (uf), although having little physical significance, is roughly one order of magnitude larger than the active energy flux term ∇ · (uAP E1). This large reference energy flux compensates the large tendency of AP E1 in the conservation law. These results show how AP E1 does not represent the exact local APE and highlight the role of the ∇ · (uf) term in evaluating the true energy flux budget for AP E1.

6 (a) KE / t APE 1 / t 0 APE 2 / t

APE 3 / t -6

6 (b) ρ’gw

APE 1 / t

Normalized Terms Normalized 0 (uAPE 1) (uf) -6 2.45 2.5 2.55 t/T

Figure 4.6: Comparison of the tendency of the kinetic energy and the three APE (a) and comparison of the depth-integrated terms in the APE equation (4.9) for AP E = AP E1 (b). The simulation parameters and the value for normalization are the same as those in Figure 4.4(e). CHAPTER 4. AVAILABLE POTENTIAL ENERGY 63

4.5 Conclusions

We have compared three different APE formulations and assessed their performance in numerical simulations of a progressive internal wave under different nonlinear and nonhydrostatic conditions. A theoretical analysis and numerical simulations clearly show that AP E2 (and AP E3 in the presence of linear stratification) is more attractive in evaluating the local APE because the size and tendency of AP E2 are of the same order of magnitude as those of KE, while the size and tendency of AP E1 are much larger. The disparity in the tendency is compensated by the presence of a large reference energy flux term uf in the conservation law for AP E1. In computing the conservation laws, the imbalance for AP E1 is related to its large tendency term and the large reference energy flux term which accentuate numerical errors. While the imbalance for AP E3 is related to the nonlinearity of the stratification, and the errors increase for stronger internal wave nonlinearity. Overall, AP E2 shows the best numerical performance in computing the conservation laws, particularly for nonlinear and nonhydrostatic cases. Chapter 5

Theoretical Framework: Barotropic and Baroclinic Energy Equations

In this chapter, energy evolution equations for the barotropic and baroclinc flows are derived, respectively. By evaluating different energy flux terms in these equations, we can assess how the barotropic tidal energy is partitioned between local barotropic dissipation due to bottom drag and local generation of baroclinic energy. Further- more, we can evaluate how much of this generated baroclinic energy is lost locally versus how much is radiated away and possibly made available for deep-ocean mixing. These energy equations will be employed as the theoretical framework for tidal energy analysis in the following chapter.

5.1 Governing Equations

We begin the derivation of the energy equations with the Reynolds-averaged form of the Navier-Stokes equations with the Boussinesq approximation in a rotating frame of reference, which are given by ( ) ∂u 1 ∂p ∂ ∂u + u · ∇u − fv + bw = − + ∇H · (νH ∇H u) + νV , (5.1) ∂t ρ0 ∂x ∂z ( ∂z) ∂v 1 ∂p ∂ ∂v + u · ∇v + fu = − + ∇H · (νH ∇H v) + νV , (5.2) ∂t ρ0 ∂y ∂z ∂z

64 CHAPTER 5. BAROTROPIC & BAROCLINIC ENERGY EQUATIONS 65

( ) ∂w 1 ∂p ∂ ∂w g + u · ∇w − bu = − + ∇H · (νH ∇H w) + νV − ρ , ∂t ρ0 ∂z ∂z ∂z ρ0 (5.3) subject to the incompressibility constraint

∇ · u = 0 , (5.4)

2 −1 where νH and νV , in units of m s , are the horizontal and vertical eddy viscosities, respectively, and the sine and cosine of latitude Coriolis terms are given by f = 2Ω sin ϕ and b = 2Ω cos ϕ, respectively, where ϕ is the latitude, and Ω is the Earth’s angular velocity. ρ0 is the constant reference density and the total density is given by

′ ρ(x, y, z, t) = ρ0 + ρb(z) + ρ (x, y, z, t) , (5.5)

′ where ρb is the background density, and ρ is the perturbation density associated with motion.

The pressure term is split into its hydrostatic, ph, and nonhydrostatic, q, components with p = ph + q, where the hydrostatic pressure is deﬁned by

∂p h = − (ρ + ρ + ρ′) g . (5.6) ∂z 0 b

Integrating this equation from z to the free surface, η, gives

′ ph = ps + p0 + pb + p ∫ ∫ η η ′ = ps + ρ0g(η − z) + g ρb dz + g ρ dz , (5.7) z z where ps is the atmosphere pressure at the free surface. Substitution of (5.5) and (5.7) into the momentum equations (5.1)-(5.3) yields

∂u 1 ∂q 1 ∂p ∂η 1 ∂p′ + u · ∇u − fv + bw = − − s − g − ∂t ρ0 ∂x ρ0 ∂x (∂x ρ)0 ∂x ∂ ∂u + ∇ · (ν ∇ u) + ν , (5.8) H H H ∂z V ∂z CHAPTER 5. BAROTROPIC & BAROCLINIC ENERGY EQUATIONS 66

∂v 1 ∂q 1 ∂p ∂η 1 ∂p′ + u · ∇v + fu = − − s − g − ∂t ρ0 ∂y ρ0 ∂y (∂y ρ)0 ∂y ∂ ∂v + ∇H · (νH ∇H v) + νV , (5.9) ∂z ∂z ( ) ∂w 1 ∂q ∂ ∂w + u · ∇w − bu = − + ∇H · (νH ∇H w) + νV . (5.10) ∂t ρ0 ∂z ∂z ∂z

Integrating the continuity equation (5.4) from the bottom, z = −d(x, y), to the free surface, z = η(x, y, t), yields the depth-averaged continuity equation (∫ ) (∫ ) ∂η ∂ η ∂ η + u dz + v dz = 0 . (5.11) ∂t ∂x −d ∂y −d

The transport equation for density is given by ( ) ∂ρ ∂ ∂ρ + u · ∇ρ = ∇ · (κ ∇ ρ) + κ , (5.12) ∂t H H H ∂z V ∂z

2 −1 where κH and κV , in units of m s , are the horizontal and vertical mass diﬀusivities, respectively. Substitution of (5.5) into this equation yields [ ] ∂ρ′ ∂ ∂ ρ N 2 + u · ∇ρ′ = ∇ · (κ ∇ ρ′) + κ (ρ′ + ρ ) + 0 w , (5.13) ∂t H H H ∂z V ∂z b g where the buoyancy frequency is deﬁned by

g dρ N 2 ≡ − b . (5.14) ρ0 dz

5.1.1 Boundary Conditions

1. The kinematic free-surface and bottom-boundary conditions are given by

∂h + u · ∇ η = w at z = η , (5.15) ∂t H H −uH · ∇H d = w at z = −d . (5.16)

2. The dynamic condition at the surface is that the pressure just below the free CHAPTER 5. BAROTROPIC & BAROCLINIC ENERGY EQUATIONS 67

surface is always equal to the ambient pressure. Taking the atmosphere pressure

ps to be zero, the condition is

′ p = p0 = pb = p = q = 0 at z = η . (5.17)

3. The free-surface drag (stop = 0) or stress (stop = 1) at z = η is given by

∂u ν = −(1 − s )C |u | u + s τ s , (5.18) V ∂z top d,T H top x ∂v ν = −(1 − s )C |u | v + s τ s , (5.19) V ∂z top d,T H top y

s s − − where τx and τy are the imposed wind shear stress in the x and y directions,

respectively, and Cd,T is the drag coeﬃcient at the surface.

4. The bottom drag at z = −d is given by

∂u ν = C |u | u , (5.20) V ∂z d,B H ∂v ν = C |u | v , (5.21) V ∂z d,B H

where Cd,B is the drag coeﬃcient at bottom.

5. The vertical velocity ﬂux at the free surface and bottom can be obtained from the kinetic and drag conditions

= Cd,B |uH | w at z = −d . (5.23) CHAPTER 5. BAROTROPIC & BAROCLINIC ENERGY EQUATIONS 68

6. The vertical mass ﬂux at the free surface and bottom are given by

∂ρ κ = 0 at z = η , (5.24) V ∂z ∂ρ κ = 0 at z = −d . (5.25) V ∂z

5.1.2 Deﬁnitions and Assumptions

In order to derive the barotropic and baroclinic energy equations, we will use the following deﬁnitions and assumptions:

1. The depth-integration of some quantity ϕ(x, y, z, t) is given by ∫ η ϕ = ϕ(x, y, z, t) dz . (5.26) −d

2. The time-averaged quantity is given by ∫ 1 t+T ⟨ϕ⟩ = ϕ(x, y, z, τ) dτ . (5.27) T t

3. Leibniz’ rule, where ϕ is either u, v, p or E, and s is either x, y, or t, then gives

∂ϕ ∂ϕ ∂η ∂d = − ϕ(η) − ϕ(−d) . (5.28) ∂s ∂s ∂s ∂s

4. The depth-integration and time-averaging operations commute, which implies that ∫ ∫ ⟨ ⟩ 1 t+T η ϕ = ϕ(x, y, z, τ) dz dτ T ∫t ∫−d 1 t+T 0 ≈ ϕ(x, y, z, τ) dz dτ T ∫t ∫ −d 1 0 t+T = ϕ(x, y, z, τ) dz dτ T −d t = ⟨ϕ⟩ , (5.29) CHAPTER 5. BAROTROPIC & BAROCLINIC ENERGY EQUATIONS 69

and this holds when η << d or H = η + d ≈ d.

5. The horizontal velocity is decomposed into its barotropic and baroclinic components

′ uH = UH + uH , (5.30)

where the depth-averaged (or barotropic) horizontal velocities are deﬁned as ∫ 1 η 1 U = u dz = u , (5.31) H ∫−d H 1 η 1 V = v dz = v , (5.32) H −d H

where the total water depth is given by H = η(x, y, t) + d(x, y). The above deﬁnitions imply that u′ = 0 and v′ = 0. Using these deﬁnitions we can write the depth-integrated continuity equation (5.11) as

∂H + ∇ · (HU ) = 0 . (5.33) ∂t H H

6. The vertical velocity is also decomposed as

w = W + w′ , (5.34)

where W and w′ both satisfy the continuity equation

∂U ∂V ∂W + + = 0 , (5.35) ∂x ∂y ∂z ∂u′ ∂v′ ∂w′ + + = 0 , (5.36) ∂x ∂y ∂z

and the boundary conditions

∂η + U · ∇ η = W at z = η , (5.37) ∂t H H −UH · ∇H d = W at z = −d , (5.38) CHAPTER 5. BAROTROPIC & BAROCLINIC ENERGY EQUATIONS 70

and

′ · ∇ ′ uH H η = w at z = η , (5.39) − ′ · ∇ ′ − uH H d = w at z = d . (5.40)

From (5.35) we have

∂W = −∇ · U , (5.41) ∂z H H

integrating of which from the bottom z = −d to z and imposing the boundary conditions (5.38) yields ∫ z ∂W W = dz + W |−d −d ∂z = −∇H · [(z + d)UH ]

= −∇H · (dUH ) − z∇H · UH . (5.42)

The physical meaning of W is the convergence of the barotropic ﬂow in the water column from z = −d to z.

7. In the hydrostatic case, we only consider the horizontal kinetic energy (HKE). The HKE per unit volume, in units of Jm−3, is deﬁned as

1 E = ρ u · u hk 2 0 H H ′ ′ = Ehk0 + Ehk + Ehk0 , (5.43)

where

1 1 ( ) E = ρ U · U = ρ U 2 + V 2 , (5.44) hk0 2 0 H H 2 0 1 1 ( ) E′ = ρ u′ · u′ = ρ u′2 + v′2 , (5.45) hk 2 0 H H 2 0 ′ · ′ ′ ′ Ehk0 = ρ0UH uH = ρ0 (Uu + V v ) , (5.46)

′ where Ehk0 and Ehk represent the barotropic and baroclinic HKE, respectively, CHAPTER 5. BAROTROPIC & BAROCLINIC ENERGY EQUATIONS 71

′ while Ehk0 is the cross term. The depth-integrated HKE is given by

′ ′ Ehk = Ehk0 + Ehk + Ehk0 ′ = HEhk0 + Ehk + 0 . (5.47)

8. In the full nonhydrostatic case, The kinetic energy per unit volume, in units of Jm−3, is deﬁned as

1 E = ρ u · u k 2 0 ′ ′ = Ehk0 + Ek + Ehk0 , (5.48)

where

1 1 ( ) E = ρ U · U = ρ U 2 + V 2 , (5.49) hk0 2 0 H H 2 0 1 ( ) 1 ( ) E′ = ρ u′ · u′ + w2 = ρ u′2 + v′2 + w2 , (5.50) k 2 0 H H 2 0 ′ · ′ ′ ′ Ehk0 = ρ0U u = ρ0 (Uu + V v ) , (5.51)

′ where Ehk0 and Ek represent the barotropic HKE and baroclinic KE, respec- ′ tively, while Ehk0 is the cross term. The depth-integrated kinetic energy is given by

′ ′ Ek = Ehk0 + Ek + Ehk0 ′ = HEhk0 + Ek + 0 . (5.52)

9. Following Gill (1982), the perturbation potential energy per unit area, in units of Jm−2, is deﬁned as the diﬀerence in the potential energies of the system between the disturbed and undisturbed states, namely ∫ ∫ η 0 Ep0 = ρ0gz dz − ρ0gz dz ∫−d −d η = ρ0gz dz 0 CHAPTER 5. BAROTROPIC & BAROCLINIC ENERGY EQUATIONS 72

1 = ρ gη2 . (5.53) 2 0

10. The available potential energy (APE) per unit volume, in units of Jm−3, is deﬁned as ∫ z ′ − ′ ′ Ep = g [ρ(z) ρr(z )] dz , (5.54) z−ζ

where ρr = ρ0 + ρb is the reference density, ζ is the vertical displacement of a

ﬂuid particle. This deﬁnition is AP E2 as discussed in Chapter 4.

5.2 Energy Equations

We now derive the energy evolution equations. We do not consider the eﬀect of atmosphere pressure (ps = 0) and neglect the bw component of the Coriolis force since w cos ϕ ≪ v sin ϕ. Therefore, we have the following momentum equations ( ) ∂u 1 ∂q ∂η 1 ∂p′ ∂ ∂u + u · ∇u − fv = − − g − + ∇H · (νH ∇H u) + νV , ∂t ρ0 ∂x ∂x ρ0 ∂x ∂z ∂z ( (5.55)) ∂v 1 ∂q ∂η 1 ∂p′ ∂ ∂v + u · ∇v + fu = − − g − + ∇H · (νH ∇H v) + νV , ∂t ρ0 ∂y ∂y ρ0 ∂y ∂z ∂z ( ) (5.56) ∂w 1 ∂q ∂ ∂w + u · ∇w = − + ∇H · (νH ∇H w) + νV . (5.57) ∂t ρ0 ∂z ∂z ∂z

These momentum equations along with the continuity equation (5.4), the free-surface equation (5.33), and the transport equation for density (5.13) comprise a set of equations for u, v, w, q, h, and ρ′. CHAPTER 5. BAROTROPIC & BAROCLINIC ENERGY EQUATIONS 73

5.2.1 Kinetic Energy Equation

The kinetic energy equation is obtained by multiplying momentum equations (5.55)-

(5.57) by ρ0u, ρ0v, and ρ0w respectively, and then adding them together to give

∂Ek ′ + ∇ · (uEk) + uH · ∇H (ρ0gη) + ∇ · (up ) + ∇ · (uq) ∂t ( ) ∂ ∂E = −ρ′gw + ∇ · (ν ∇ E ) + ν k − ϵ . (5.58) H H H k ∂z V ∂z where the dissipation rate ϵ, in units of W m−3, is given by

∂u ∂u ϵ = ρ ν ∇ u · ∇ u + ρ ν · . (5.59) 0 H H H 0 V ∂z ∂z

Depth integration of equation (5.58) is given by

∂Ek ′ + ∇ · (uEk) + uH · ∇H (ρ0gη) + ∇ · (up ) + ∇ · (uq) ∂t ( ) ∂ ∂E = −ρ′gw + ∇ · (ν ∇ E ) + ν k − ϵ , (5.60) H H H k ∂z V ∂z and after applying Leibniz’s rule and the boundary conditions it becomes

∂E ( ) ( ) k + ∇ · u E + ∇ · (U Hρ gη) + ∇ · u p′ + ∇ · (u q) ∂t H H k H H 0 H H H H ′ = ρ0gη∇H · (HUH ) − ρ gw + ∇H · (νH ∇H Ek) − D − ϵ , (5.61) where the drag term D is given by [ ] 2 2 2 D = ρ0Cd,T |uH (η)| u (η) + v (η) + w (η) − w(η)ηt [ ] 2 2 2 + ρ0Cd,B |uH (−d)| u (−d) + v (−d) + w (−d) [ ] 2 = ρ0Cd,T |uH (η)| |u(η)| − w(η)ηt 2 + ρ0Cd,B |uH (−d)| |u(−d)| . (5.62)

Here we use free-surface drag (stop = 0) in boundary conditions (5.18) and (5.19). CHAPTER 5. BAROTROPIC & BAROCLINIC ENERGY EQUATIONS 74

5.2.2 Perturbation Potential Energy Equation

The equation for perturbation potential energy is derived by multiplying equation

(5.33) by ρ0gη to give

∂E p0 = −ρ gη∇ · (HU ) ∂t 0 H H = ρ0gηηt . (5.63)

5.2.3 Available Potential Energy Equation

Taking the total derivative of APE (5.54) gives ( ) ( ) ( ) ∂ E′ ( ) ∂ρ ∂ζ p + ∇ · uE′ = gζ + u · ∇ρ + ρg + u · ∇ζ − ρ gw . (5.64) ∂t p ∂t ∂t r

Since the vertical displacement ζ satisﬁes

∂ζ + u · ∇ζ = w , (5.65) ∂t substitution of (5.12) and (5.65) into (5.64) yields ( ) [ ] ∂ E′ ( ) ∂ ∂ p + ∇ · uE′ = ρ′gw + ∇ · (κ gζ∇ ρ′) + κ gζ (ρ′ + ρ ) − ϵ , ∂t p H H H ∂z V ∂z b p (5.66)

−3 where the dissipation rate of APE, ϵp, in units of W m , is given by

∂(ρ + ρ′) ∂ζ ϵ = gκ ∇ ρ′ · ∇ ζ + gκ b . (5.67) p H H H V ∂z ∂z

The depth-integration of (5.66) is given by ( ) [ ] ∂ E′ ( ) ∂ ∂ p + ∇ · uE′ = ρ′gw + ∇ · (κ gζ∇ ρ′) + κ gζ (ρ′ + ρ ) − ϵ , ∂t p H H H ∂z V ∂z b p (5.68) CHAPTER 5. BAROTROPIC & BAROCLINIC ENERGY EQUATIONS 75

and, after applying Leibniz’s rule and the boundary conditions, it becomes ( ) ∂ E′ ( ) p + ∇ · u E′ = ρ′gw + ∇ · (κ gζ∇ ρ′) − ϵ . (5.69) ∂t H H p H H H p

5.2.4 Total Energy Equation

The depth-integrated total energy equation is obtained by the sum of equations (5.61), (5.63), and (5.69), vis.

( ) ∂ ′ Unsteadiness Ek + E + Ep0 ∂t ( p ) ′ ∇ · Flux (Advection) + ∇H · uH Ek + uH E ( p ) ′ ∇ · Flux (Pressure) + ∇H · UH Hρ0gη + uH p + uH q ′ ∇ · Flux (Diﬀusion) − ∇H · (νH ∇H Ek) − ∇H · (κH gζ∇H ρ )

Dissipation = −ϵ − ϵp Drag − D. (5.70)

The time-averaging of equation (5.70) along with the assumption (5.29) gives the time-averaged and depth-integrated barotropic total energy equation as

1 Unsteadiness ∆E T ⟨ ⟩ ′ ∇ · Flux (Advection) + ∇H · uH Ek + uH E ⟨ p ⟩ ′ ∇ · Flux (Pressure) + ∇H · UH Hρ0gη + uH p + uH q ′ ∇ · Flux (Diﬀusion) − ∇H ·⟨νH ∇H Ek⟩ − ∇H ·⟨κH gζ∇H ρ ⟩

Dissipation = − ⟨ϵ⟩ − ⟨ϵp⟩ Drag − ⟨D⟩ , (5.71) where the net change in the depth-integrated total energy is given by

∆E = E(t + T ) − E(t) ( ) ( ) = E + E′ + E − E + E′ + E . (5.72) k p p0 t+T k p p0 t CHAPTER 5. BAROTROPIC & BAROCLINIC ENERGY EQUATIONS 76

External Energy

Internal Energy

Figure 5.1: Energy exchange diagram. Orange solid boxes represent the energy components within a fixed water column. Solid arrows indicate the energy conversion among different energy components within the control volume, while dashed arrows represent the divergence of energy fluxes transferring energy into or out of a fixed volume.

The above evolution equations of kinetic energy (5.61), perturbation potential energy (5.63), and available potential energy (5.69) show the energy transfer and conversion among diﬀerent energy components, while equation (5.70) gives the energy conservation law. These relations are illustrated in Figure 5.1. The energy within a

fixed water column can be stored as kinetic energy Ek, perturbation potential energy ′ Ep0, available potential energy Ep, and internal energy (heat), and these are indicated by the orange solid boxes as in the figure. These energy components exchange energy with the external environment through the divergence of energy fluxes that result from pressure work, advection and viscous diffusion of energy. The kinetic and available potential energy fluxes are given by ( ) ′ ∇H · Fk = ∇H · uH Ek + UH Hρ0gη + uH p + uH q − ∇H · (νH ∇H Ek) , (5.73) ( ) ∇ · ′ ∇ · ′ − ∇ · ∇ ′ H Fp = H uH Ep H (κH gζ H ρ ) , (5.74) and they are indicated by the dashed arrows in Figure 5.1. The solid arrows in the CHAPTER 5. BAROTROPIC & BAROCLINIC ENERGY EQUATIONS 77

figure illustrate the exchanges of energy among different components within a water column. The kinetic and potential energy are converted into heat through viscous dissipation, while the conversion between kinetic and potential energy occurs through buoyancy flux.

5.3 Barotropic Energy Equations

Employing Leibniz’s rule (5.28) and the boundary conditions (5.15)-(5.21) with stop = 0, the depth-averaged horizontal momentum equations are given by

∂U ∂η 1 ∂p′ 1 ∂q + UH · ∇H U + Ax − fV = −g − − ∂t ∂x ρ0H ∂x ρ0H ∂x + ∇H · (νH ∇H U) − Fx − Dx , (5.75) ∂V ∂η 1 ∂p′ 1 ∂q + UH · ∇H V + Ay + fU = −g − − ∂t ∂y ρ0H ∂y ρ0H ∂y + ∇H · (νH ∇H V ) − Fy − Dy , (5.76) where the surface and bottom drag terms are

C C D = d,T |u (η)| u(η) + d,B |u (−d)| u(−d) , (5.77) x H H H H C C D = d,T |u (η)| v(η) + d,B |u (−d)| v(−d) , (5.78) y H H H H and the unclosed advection and horizontal diﬀusion terms are

1 ( ) A = ∇ · u′ u′ , (5.79) x H H H 1 ( ) A = ∇ · u′ v′ , (5.80) y H H H ν F = H [u′(η)η + 2u′ (η)η + 2u′ (−d)d + u′(−d)d ] x H xx x x x x xx ν [ ] + H u′(η)η + 2u′ (η)η + 2u′ (−d)d + u′(−d)d , (5.81) H yy y y y y yy ν F = H [v′(η)η + 2v′ (η)η + 2v′ (−d)d + v′(−d)d ] y H xx x x x x xx ν [ ] + H v′(η)η + 2v′ (η)η + 2v′ (−d)d + v′(−d)d . (5.82) H yy y y y y yy CHAPTER 5. BAROTROPIC & BAROCLINIC ENERGY EQUATIONS 78

The barotropic HKE equation is obtained by multiplying equations (5.75) and (5.76) by ρ0U and ρ0V , respectively, and then adding them together to give

∂E 1 ( ) 1 hk0 + U · ∇ E + U · ∇ (ρ gη) + ∇ · U p′ + ∇ · (U q) ∂t H H hk0 H H 0 H H H H H H 1 1 = − ρ′gW + q W − ρ (UA + VA ) H H z 0 x y + ∇H · (νH ∇H Ehk0) − ρ0 (UFx + VFy) − ϵh0 − ρ0 (UDx + VDy) , (5.83)

where the barotropic dissipation rate ϵh0 is given by

ϵh0 = ρ0νH (∇H UH · ∇H UH ) . (5.84)

Multiplying equation (5.83) by H and applying Leibniz’s rule and boundary conditions, we then obtain the depth-integrated barotropic HKE equation as

∂HE ( ) hk0 + ∇ · (U HE ) + ∇ · (U Hρ gη) + ∇ · U p′ + ∇ · (U q) ∂t H H hk0 H H 0 H H H H ′ = ρ0gη∇H · (HUH ) − ρ gW + qzW − Ah0

+ H∇H · (νH ∇H Ehk0) − Fh0 − ϵh0 − Dh0 , (5.85) where the unclosed advection and diﬀusion terms and the barotropic drag term are given by

Ah0 = ρ0H (UAx + VAy) , (5.86)

Fh0 = ρ0H (UFx + VFy) , (5.87)

Dh0 = ρ0H (UDx + VDy)

= ρ0Cd,T |uH (η)| [Uu(η) + V v(η)]

+ ρ0Cd,B |uH (−d)| [Uu(−d) + V v(−d)] . (5.88)

The depth-integrated barotropic total energy equation is then obtained by adding (5.85) and (5.63), namely,

∂ ( ) Unsteadiness E + E ∂t hk0 p0 CHAPTER 5. BAROTROPIC & BAROCLINIC ENERGY EQUATIONS 79

( ) ∇ · Flux - Advection + ∇H · UH Ehk0 ( ) ′ ∇ · Flux - Pressure + ∇H · UH Hρ0gη + UH p + UH q

∇ · Flux - Diﬀusion − H∇H · (νH ∇H Ehk0) + Fh0 ′ Bt-Bc Conversion = −ρ gW + qzW − Ah0

Dissipation − ϵh0

Drag − Dh0 . (5.89)

Time-averaging equation (5.89) along with assumption (5.29) gives the time-averaged and depth-integrated total energy equation as

1 Unsteadiness ∆Eh0 T ⟨ ⟩ ∇ · Flux - Advection + ∇H · UH Ehk0 ⟨ ⟩ ′ ∇ · Flux - Pressure + ∇H · UH Hρ0gη + UH p + UH q ⟨ ⟩ ∇ · Flux - Diﬀusion − H∇H · νH ∇H Ehk0 + ⟨Fh0⟩ ⟨ ⟩ ⟨ ⟩ ′ Bt-Bc Conversion = − ρ gW + qzW − ⟨Ah0⟩

Dissipation − ⟨ϵh0⟩

Drag − ⟨Dh0⟩ , (5.90) where the net change in the depth-integrated barotropic total energy is given by

∆Eh0 = Eh0(t + T ) − Eh0(t) ( ) ( ) − = Ehk0 + Ep0 t+T Ehk0 + Ep0 t . (5.91)

5.4 Baroclinic Energy Equations

Subtracting the depth-averaged momentum equations (5.75) and (5.76) from the momentum equations (5.55) and (5.56) gives

′ ′ ∂u · ∇ ′ ′ · ∇ − − ′ 1 ∂p 1 ∂q + u u + uH H U Ax fv + + ∂t ρ0 ∂x ρ0 ∂x CHAPTER 5. BAROTROPIC & BAROCLINIC ENERGY EQUATIONS 80

( ) ′ ′ 1 ∂p 1 ∂q ′ ∂ ∂u = + + ∇H · (νH ∇H u ) + νV + Fx + Dx , (5.92) ρ0H ∂x ρ0H ∂x ∂z ∂z ′ ′ ∂v · ∇ ′ ′ · ∇ − ′ 1 ∂p 1 ∂q + u v + uH H V Ay + fu + + ∂t ρ0(∂y ρ)0 ∂y ′ ′ 1 ∂p 1 ∂q ′ ∂ ∂v = + + ∇H · (νH ∇H v ) + νV + Fy + Dy . (5.93) ρ0H ∂y ρ0H ∂y ∂z ∂z

′ The baroclinic HKE equation is obtained by multiplying the above equations by ρ0u ′ and ρ0v , respectively, and then adding them together to give

∂E′ hk + ∇ · (uE′ ) + ∇ · (u′ E′ ) − ρ U∇ · (u′ u′) − ρ V ∇ · (u′ v′) ∂t hk H H hk0 0 H H 0 H H 1 1 −∇ · ′ ′ − ∇ · ′ − ′ ′ ′ ′ · ∇ ′ ′ · ∇ = (u p ) (u q) ρ gw + qzw + uH H p + uH H q ( ) H H ∂ ∂E′ + ∇ · (ν ∇ E′ ) + ν hk − ϵ′ H H H hk ∂z V ∂z h ′ ′ ′ ′ ′ ′ + ρ0 (u Ax + v Ay) + ρ0 (u Fx + v Fy) + ρ0 (u Dx + v Dy) , (5.94)

′ where the horizontal baroclinic dissipation rate ϵh is given by ( ) ∂u′ ∂u′ ϵ′ = ρ ν (∇ u′ · ∇ u′ ) + ρ ν H · H . (5.95) h 0 H H H H H 0 V ∂z ∂z

Multiplying the vertical momentum equation (5.57) by ρ0w gives ( ) [ ( )] ∂ 1 2 1 2 ρ0w + ∇ · u ρ0w ∂t 2 [ ( 2 )] [ ( )] 1 ∂ ∂ 1 = −q w + ∇ · ν ∇ ρ w2 + ν ρ w2 − ϵ′ , (5.96) z H H H 2 0 ∂z V ∂z 2 0 v

′ where the vertical baroclinic dissipation rate ϵv is given by

∂w ∂w ϵ′ = ρ ν (∇ w · ∇ w) + ρ ν · . (5.97) v 0 H H H 0 V ∂z ∂z CHAPTER 5. BAROTROPIC & BAROCLINIC ENERGY EQUATIONS 81

The depth-integrated baroclinic kinetic energy equation is obtained by adding (5.94) and (5.96) and then depth integrating to give

∂E′ k + ∇ · (uE′ ) + ∇ · (u′ E′ ) − ρ U∇ · (u′ u′) − ρ V ∇ · (u′ v′) ∂t k H H hk0 0 H H 0 H H = −∇ · (u′p′) − ∇ · (u′q) − ρ′gw′ − q W ( ) z ∂ ∂E′ + ∇ · (ν ∇ E′ ) + ν k − ϵ′ , (5.98) H H H k ∂z V ∂z and after applying Leibniz’s rule and the boundary conditions it becomes

∂E′ ( ) ( ) ( ) ( ) k + ∇ · u E′ + ∇ · u′ E′ + ∇ · u′ p′ + ∇ · u′ q ∂t H H k H H hk0 H H H H − ′ ′ − ∇ · ∇ ′ − ′ − ′ = ρ gw qzW + Ah0 + H (νH H Ek) D ϵ , (5.99) where the baroclinic dissipation rate and drag terms are given by

ϵ′ = ϵ′ + ϵ′ , (5.100) h v [ ] ′ ′ ′ 2 D = ρ0Cd,T |uH (η)| u (η)u(η) + v (η)v(η) + w (η) − w(η)ηt [ ] ′ ′ 2 + ρ0Cd,B |uH (−d)| u (−d)u(−d) + v (−d)v(−d) + w (−d) . (5.101)

The depth-integrated baroclinic total energy equation is then given by the sum of (5.99) and (5.69), namely,

( ) ∂ ′ ′ Unsteadiness Ek + Ep ∂t ( ) ′ ′ ′ ∇ · Flux - Advection + ∇H · uH E + uH E + uH E ( k ) hk0 p ∇ · ∇ · ′ ′ ′ Flux - Pressure + H uH p + uH q ∇ · − ∇ · ∇ ′ ∇ · ∇ ′ Flux - Diﬀusion H (νH H Ek) + H (κH gζ H ρ ) ′ Bt-Bc Conversion = ρ gW − qzW + Ah0 − ′ − ′ Dissipation ϵ ϵp Drag − D′ . (5.102) CHAPTER 5. BAROTROPIC & BAROCLINIC ENERGY EQUATIONS 82

The time-average of equation (5.102) along with the assumption (5.29) gives the time-averaged and depth-integrated baroclinic total energy equation as

1 Unsteadiness ∆E′ T ⟨ ⟩ ′ ′ ′ ∇ · Flux - Advection + ∇H · uH E + uH E + uH E ⟨ k ⟩ hk0 p ∇ · ∇ · ′ ′ ′ Flux - Pressure + H uH p + uH q ′ ′ ∇ · Flux - Diﬀusion − ∇H ·⟨νH ∇H E ⟩ + ∇H ·⟨κH gζ∇H ρ ⟩ ⟨ ⟩ ⟨ k ⟩ ′ Bt-Bc Conversion = ρ gW − qzW + ⟨Ah0⟩ ⟨ ⟩ ⟨ ⟩ − ′ − ′ Dissipation ϵ ϵp Drag − ⟨D′⟩ , (5.103) where the net change in the depth-integrated barotropic total energy is given by

∆E′ = E′(t + T ) − E′(t) ( ) ( ) = E′ + E′ − E′ + E′ . (5.104) k p t+T k p t

We have obtained two sets of energy evolution equations for barotropic (5.89) and baroclinic (5.102) ﬂows, respectively. Similar to the diagram in Figure 5.1, a new diagram in Figure 5.2 is employed to illustrate the barotropic-to-baroclinc energy conversion. In the new diagram, the kinetic energy Ek is split into barotropic Ehk0 ′ and baroclinic Ek components. The energy conversion between them occurs through ′ the buoyancy ﬂux ρ gW − qzW . Correspondingly, the dissipation term −ϵ − D is also ′ ′ split into two parts −ϵh0 − Dh0 and −ϵ − D . The divergence of the kinetic energy

ﬂux ∇H · Fk is decomposed as ( ) ′ ∇H · Fhk0 = ∇H · UH HEhk0 + UH Hρ0gη + UH p + UH q

− H∇H · (νH ∇H Ehk0) + Fh0 , (5.105) ( ) ∇ · ′ ∇ · ′ ′ ′ H Fk = H uH Ek + uH Ehk0 + uH p + uH q − ∇ · ∇ ′ H (νH H Ek) . (5.106) CHAPTER 5. BAROTROPIC & BAROCLINIC ENERGY EQUATIONS 83

The relationship among these new components is illustrated by the arrows in Figure 5.2.

External Energy

Internal Energy

Figure 5.2: Energy exchange diagram illustrating the barotropic-to-baroclinc energy conversion. Orange round boxes represent the energy components within a fixed water column. Solid arrows indicate the energy conversion among different energy components within the control volume, while dashed arrows represent the divergence of energy fluxes transferring energy into or out of a fixed volume. Chapter 6

Numerical Simulations: Energetics and Dynamics of Internal Tides in the Monterey Bay Area 4

This chapter focuses on the numerical investigation of internal tide energetics and dynamics in the Monterey Bay area. Three-dimensional, high-resolution simulations of the barotropic and baroclinic tides in this area are conducted using the SUNTANS model. Model results are presented and compared to the ﬁeld observations. Based on the theoretical framework in Chapter 5, a detailed tidal energy analysis is performed to address the question of how the barotropic tidal energy is partitioned between local barotropic dissipation and local generation of baroclinic energy, and then how much of this generated baroclinic energy is lost locally versus how much is radiated away for open-ocean mixing. We also investigate the internal tide generation mechanism by examining the dependence of barotropic-to-baroclinic energy conversion on three nondimensional parameters, namely the steepness parameter, the tidal excursion parameter, and the Froude number. Finally, a simple parametric model is presented to estimate the barotropic-to-baroclinic energy conversion.

4This chapter has been reproduced in substantial part as a submitted paper entitled “Energetics of barotropic and baroclinic tides in the Monterey Bay area”.

84 CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 85

6.1 Introduction

The ocean is a key component of the Earth’s climate system, and mixing processes in the ocean are critical in determining its thermohaline circulation and distribution of properties. The tides are one of the major sources of energy to mix the ocean. Recent studies have shown that 25-30% of the global barotropic tidal energy is lost in the deep ocean (Munk & Wunsch, 1998; Egbert & Ray, 2000, 2001). Internal tides are believed to play an important role in transferring this energy into deep ocean turbulence (Figure 6.1). When the barotropic tide ﬂows over rough topographic features, a portion of the barotropic energy is lost directly through local dissipation and mixing, while the other portion is lost to the generation of internal (baroclinic) tides. This generated baroclinic energy either dissipates locally or radiates out into the open ocean.

Barotropic Tides

BT input

Topography

conversion BT dissipation Baroclinic Tides BC radiation

Internal Waves BC dissipation cascade

Deep Turbulence Local Mixing

Figure 6.1: Mixing processes that transfer barotropic tidal energy into heat in the ocean. The associated energy distributions are indicated by the energy terms in the barotropic and baroclinic energy equations.

In the past few years signiﬁcant eﬀort has been made to estimate regional internal tide energetics using numerical simulations. Regions studied include the Northern CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 86

British Columbia Coast (Cummins & Oey, 1997), the Hawaiian Ridge (Merrifield et al., 2001; Merrifield & Holloway, 2002; Carter et al., 2008), the East China Sea (Niwa & Hibiya, 2004), the Monterey Bay region (Jachec et al., 2006; Carter, 2010; Hall & Carter, 2010), and the Mid-Atlantic Ridge (Zilberman et al. , 2009). All of these studies employed the hydrostatic Princeton Ocean Model (POM) (Blumberg & Mellor, 1987) except for the work by Jachec et al. (2006), which employed the nonhydrostatic SUNTANS model (Fringer et al. , 2006a). In some previous numerical studies (e.g., Cummins & Oey, 1997; Merrifield et al., 2001; Merrifield & Holloway, 2002; Jachec et al., 2006), the barotropic-to-baroclinic energy conversion was estimated by the baroclinic energy flux divergence, which only represents the radiated portion of the total conversion. Monterey Bay lies along the Central U. S. West Coast. It consists of the prominent Monterey Submarine Canyon (MSC), numerous ridges and smaller canyons to the north and south, and a continental slope and break region (Figure 6.2). The complex bathymetry is favorable for internal tide generation. Energetic internal wave activity has been observed in the submarine canyon (Petruncio et al., 1998; Kunze et al., 2002; Carter et al., 2005). Jachec et al. (2006) performed high-resolution simulations (∼290 m within the Bay) using the nonhydrostatic SUNTANS model (Fringer et al. , 2006a) to simulate internal tides in the Monterey Bay area. They determined that the Sur Platform region is the primary source for the M2 internal energy flux observed within MSC. The total baroclinic energy generation rate within their domain (∼200 km alongshore × 90 km offshore) is approximately 52 MW. The purpose of this work is twofold: first, to provide a theoretical framework for the accurate evaluation of tidal energy flux budget with full nonlinear and nonhydrostatic contributions and, second, to conduct numerical simulations of internal tides in the Monterey Bay region and estimate the tidal energy budget based on the theoretical framework. We first present a brief derivation of the barotropic and baroclinic energy equations in Section 6.2. This is followed by the simulation setup in Section 6.3. Subsequent sections demonstrate the simulation results of dynamics

(Section 6.4) and energetics (Section 6.5) of the M2 barotropic and baroclinic tides in the Monterey Bay area. In Section 6.6 we investigate the internal tide generation CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 87

Elevation (m)

38 o N

37 o Moss Landing K *

36 o

35 o

34 o

126 o 125 o 124 o 123 o 122 o 121 o W

Figure 6.2: Bathymetry map of Monterey Bay and the surrounding open ocean. The location of Station K is indicated by a black ∗. The domain outside the white box indicates the area aﬀected by the sponge layers in the simulations. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 88

mechanism and present a parametric model to estimate the barotropic-to-baroclinic conversion. Conclusions are summarized in Section 6.7.

6.2 Theoretical Framework

In order to study the energetics of barotropic and baroclinic tides, we derive the barotropic and baroclinic energy equations with full energy ﬂuxes. These equations provide a theoretical framework for the numerical evaluation of the tidal energy budget in subsequent sections. Here we provide a brief derivation of the equations. More detail is provided in Chapter 5. The derivation is based on the governing equations of SUNTANS, which include the three-dimensional Reynolds-averaged Navier-Stokes equations under the Boussi- nesq approximation, along with the density transport equation and the continuity equation, ( ) ∂u 1 g ∂ ∂u + u · ∇u = −2Ω × u − ∇p − ρk + ∇H · (νH ∇H u) + νV , ∂t ρ0 ρ0 ∂z ∂z ( ) (6.1) ∂ρ ∂ ∂ρ + u · ∇ρ = ∇ · (κ ∇ ρ) + κ , (6.2) ∂t H H H ∂z V ∂z ∇ · u = 0 , (6.3) where u = (u, v, w) is the velocity vector and Ω is the Earth’s angular velocity vector. ν and κ, in units of m2s−1, are the eddy viscosity and eddy diﬀusivity, respectively.

()H and ()V stand for the horizontal and vertical components of a variable or operator. The total density is given by

′ ρ(x, y, z, t) = ρ0 + ρb(z) + ρ (x, y, z, t) , (6.4)

′ where ρ0 is the constant reference density, ρb is the background density, and ρ is the perturbation density due to wave motions. The pressure term is split into its hydrostatic ph and nonhydrostatic q parts as p = ph + q, where the hydrostatic CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 89

pressure can be further decomposed as ∫ ∫ η η − ′ ph = ρ|0g({zη z}) + g ρb dz + g ρ dz , (6.5) | z {z } | z{z } p0 ′ pb p where η is the surface elevation. To obtain the barotropic and baroclinic equations, we ﬁrst split the velocity into its barotropic and baroclinic parts with u = U + u′. The barotropic velocities are deﬁned as ∫ 1 η 1 UH = uH dz = uH , (6.6) H −d H W = −∇H · [(d + z)UH ] , (6.7) ∫ · η · where ( ) = −d( ) dz stands for the depth-integration of a quantity from the bottom z = −d(x, y) to the surface z = η(x, y, t), and the total water depth is H = η + d. Based on the velocity decomposition, the kinetic energy density, in units of J m−3, is ′ ′ decomposed with Ek = Ek0 + Ek + Ek0, where

1 ( ) E = ρ U 2 + V 2 , (6.8) k0 2 0 1 ( ) E′ = ρ u′2 + v′2 + w2 , (6.9) k 2 0 ′ ′ ′ Ek0 = ρ0 (Uu + V v ) . (6.10)

′ Here Ek0 is the barotropic horizontal kinetic energy density, Ek is the baroclinic kinetic ′ energy density, and Ek0 is the cross term which vanishes upon depth-integration. Following Gill (1982), the perturbation potential energy due to surface elevation, in units of J m−2, is given by 1 E = ρ gη2 . (6.11) p0 2 0 The available potential energy density, in units of J m−3, is deﬁned as ∫ z ′ ′ − − ′ ′ Ep = ρ gζ g [ρb(z) ρb(z )] dz , (6.12) z−ζ CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 90

where ζ is the vertical displacement of a ﬂuid particle due to wave motions. This deﬁnition is an exact expression of the local APE and has been employed in analyzing internal wave energetics by numerous authors (Scotti et al., 2006; Lamb, 2007; Lamb & Nguyen, 2009; Kang & Fringer, 2010). Applying these variable decompositions and the boundary conditions, we obtain the depth-integrated barotropic and baroclinic energy equations as

∂ ( ) E + E + ∇ · F = −C − ϵ − D , (6.13) ∂t k0 p0 H 0 0 0 ∂ ( ) E′ + E′ + ∇ · F′ = C − ϵ′ − D′ , (6.14) ∂t k p H where the depth-integrated barotropic and baroclinic energy ﬂux terms, with the small unclosed terms neglected (see Chapter 5), are given by

′ − ∇ F0 = |UH{zEk}0 |+UH ρ0gη +{zUH p + UH q} | νH {zH Ek}0 , (6.15) Advection Pressure work Diﬀusion

F′ = u E′ + u E′ + u E′ +u′ p′ + u′ q −ν ∇ E′ − κ ∇ E′ , (6.16) | H k H{z k0 H }p | H {z H } | H H k{z H H }p Advection Pressure work Diﬀusion where the contributions from energy advection, pressure work, and diﬀusion have been labeled. The barotropic-to-baroclinic conversion rate, the dissipation rates and the bottom drag terms are given by

∂q C = ρ′gW − W, (6.17) ∂z ϵ0 = ρ0νH ∇H UH · ∇H UH , (6.18) ′ ′ ′ ϵ = ϵk + ϵp , (6.19) ∂u′ ∂u′ ϵ′ = ρ ν ∇ u′ · ∇ u′ + ρ ν H · H k 0 H H H H H 0 V ∂z ∂z ∂w ∂w + ρ ν ∇ w · ∇ w + ρ ν , (6.20) 0 H H H 0 V ∂z ∂z ∂(ρ + ρ′) ∂ζ ϵ′ = gκ ∇ ρ′ · ∇ ζ + gκ b , (6.21) p H H H V ∂z ∂z CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 91

D0 = ρ0Cd |uH | (uU + vV ) , at z = −d (6.22) ( ) ′ ′ ′ 2 D = ρ0Cd |uH | uu + vv + w , at z = −d (6.23)

where Cd is the bottom drag coeﬃcient. The time-averaged forms of equations (6.13) and (6.14) under the assumption (5.29) are given by

1 ⟨ ⟩ ⟨ ⟩ ∆E0 + ∇H · F0 = − C − ⟨ϵ0 + D0⟩ , (6.24) T ⟨ ⟩ 1 ⟨ ⟩ ⟨ ⟩ ∆E′ + ∇ · F′ = C − ϵ′ + D′ , (6.25) T H ∫ ⟨⟩ 1 t+T where = T t () dτ stands for the time-average of a quantity over a time interval T . Therefore net changes of the depth-integrated barotropic and baroclinic total energy are given by ( ) ( )

∆E0 = Ek0 + Ep0 − Ek0 + Ep0 , (6.26) ( ) t+T ( ) t ∆E′ = E′ + E′ − E′ + E′ . (6.27) k p t+T k p t

′ For a periodic system with period T , ∆E0 and ∆E tend to zero and thus the ﬁrst terms in equation (6.24) and (6.25) vanish. The remaining balanced terms describe ⟨ ⟩ the energy distribution associated with tidal mixing processes. The ∇H · F0 term represents the total barotropic energy that is available for conversion to baroclinic ⟨ ⟩ energy, C represents the actual portion⟨ ⟩ of the barotropic energy that is converted ′ into baroclinic energy, and the ∇H · F term represents the portion of the converted baroclinic energy that radiates from the conversion site. Local mixing occurs along with the conversion and radiation processes, and they are measured by the barotopic ⟨ ⟩ ′ ′ (− ⟨ϵ0 + D0⟩) and baroclinic (− ϵ + D ) dissipation terms, respectively. Figure 6.1 illustrates the tidal energy budget using these terms. This approach presents an exact measure of the barotopic-to-baroclinic tidal energy conversion and highlights its relation to the total convertible barotropic energy, as well as the radiated baroclinic energy. The conversion term includes two parts CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 92

′ − ∂q representing the hydrostatic ρ gW and nonhydrostatic ∂z W contributions, respectively. Furthermore, we consider the contributions to the energy flux budget from the available potential energy flux and the kinetic energy fluxes due to nonlinear and nonhydrostatic effects. In most of the previous studies, only the dominant hydrostatic ′ ′ baroclinic energy term, ∇H · (u p ), was calculated to represent the conversion (e.g. Cummins & Oey, 1997; Merrifield et al., 2001; Merrifield & Holloway, 2002; Jachec et al., 2006). Niwa & Hibiya (2004) evaluated the conversion using a similar term as ρ′gW , which is the hydrostatic part of our conversion term (6.17). Carter et al. (2008) derived barotropic and baroclinic equations from POM’s hydrostatic governing equations and thus did not include the nonhydrostatic contribution. Their approach distinguishes between the conversion from the barotropic input and the baroclinic radiation. Moreover, they used a linear APE definition, which may cause error in the presence of strongly nonlinear stratification (Kang & Fringer, 2010).

6.3 Simulation Setup

6.3.1 The SUNTANS Model

In this study we employ the SUNTANS model of (Fringer et al. , 2006a) to simulate internal tides in the Monterey Bay region, and the above theoretical framework for tidal energy analysis has been implemented in the numerical model. SUNTANS solves the three-dimensional, nonhydrostatic equations (6.1)-(6.3) on a finite-volume grid which is unstructured in the horizontal and structured (z-level) in the vertical. The nonhydrostatic formulation in SUNTANS is based on that of (Casulli, 1999), in which the pressure is split into its hydrostatic and nonhydrostatic components. The free- surface is advanced semi-implicitly to avoid the stability limitations associated with fast surface gravity waves, and the pressure correction method of Armfield & Street (2002) is implemented to solve the nonhydrostatic pressure on a staggered grid. It has been shown that this method maintains second-order time accuracy when extended to free-surface flows (Kang & Fringer, 2006a). To reduce the numerical diffusion, we employed a second-order accurate total variation diminishing (TVD) scheme for scalar CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 93

Figure 6.3: The unstructured grid of the computational domain (upper). The lower two zoomed-in plots are for the subdomain (a) indicated in Figure 6.17 (lower-left), and Monterey Bay (lower-right). In the upper and lower-left plots, only cell centers are shown, while in the lower-right plot, cell edges are shown. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 94

advection. SUNTANS is a parallel code and employs the message-passing interface (MPI). All simulations are performed on the MJM Linux Networx Intel Xeon EM64T Cluster at the Army Research Laboratory (ARL).

6.3.2 Domain and Grid

10 100 8 80 6 60 4 40 Cumulative%

Number1000 / 2 20 0 0 0 1 2 3 4 5 6 Grid Spacing (km)

Figure 6.4: Horizontal grid spacing histogram and the cumulative fraction for the grid depicted in Figure 6.3.

The simulation domain extends approximately 200 km north and south of Moss Landing, and 400 km offshore (Figure 6.2). Compared to the domain of Jachec et al. (2006), our domain covers a broader area of internal tides generation and allows the evolution of offshore-propagating waves. The horizontal unstructured grid for our simulations is depicted in Figure 6.3. The grid resolution smoothly transitions from roughly 80 m within the Bay to 11 km along the offshore boundary. Approximately 60% of the grid cells have a resolution smaller than 1000 m, and 80% of the grid cells have a resolution smaller than 1600 m (Figure 6.4). In the vertical, there are 120 z-levels with thickness stretching from roughly 6.6 m at the surface to 124 m in the deepest location, which provides better resolution in the shallow regions. The vertical locations of grid centers are indicated by the black dots in Figure 6.5. In total, the mesh consists of approximately 6 million grid cells. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 95

6.3.3 Initial Conditions

The initial free-surface and velocity field are set to be quiescent throughout the domain. The initial stratification is specified with horizontally-homogeneous temperature and salinity profiles obtained from the 2006 AESOP field experiment (Figure 6.5). We use linear extrapolation to extend the observation profiles from 2000 m to 4800 m. Stratification can be turned off for the barotropic run when simulating barotropic tides.

1000

2000

Depth(m) 3000

4000

0 5 10 15 33.5 34 34.5 Temperature ( o C) Salinity (psu)

Figure 6.5: Initial temperature and salinity proﬁles for the simulations (Courtesy of Dr. J. B. Girton, University of Washington and Dr. J. M. Klymak, University of Victoria). The black dots represent the vertical grid spacing for the simulations. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 96

6.3.4 Boundary Conditions

At the coastline, we apply the no-ﬂow condition, while at the three open boundaries the barotropic velocities are speciﬁed as

Ub = U0 cos (ωt + ϕ) , (6.28)

where ω is the tidal frequency, and the corresponding amplitude U0 and phase ϕ are obtained with the OTIS global tidal model (Egbert & Erofeeva, 2002). In this research, the tidal forcing is either M2 (Figure 6.6) or a sum of the ﬁrst eight tidal constituents (M2, K1, S2, O1, K2, P1, N2, Q1). In order to prevent transient oscil- lations associated with impulsively starting the tidal forcing, the boundary velocities are spun up over a time scale of τramp to approach the imposed forcing with

− Ubactual = Ub [1 exp (t/τramp)] , (6.29)

where τramp is set to one day in the simulations. Furthermore, at all three open boundaries a sponge layer is imposed to absorb the internal waves and minimize the reﬂection of baroclinic energy into the domain. Following Zhang (2010), the sponge layer is applied as a damping term on the right-hand side of the horizontal momentum equation of the form ( ) ′ uH 4r SH = − exp − , (6.30) τsponge Ls where r is the distance to the closest open boundaries, and Ls is the width of the sponge layer. The damping time scale τsponge is determined by

L τ = − s , (6.31) sponge 4c log(1 − α) where c is the internal wave speed and α is the decay rate. The ﬁrst mode M2 −1 internal wave speed in a depth of 3000 m is c1 = 1.98 ms and the corresponding wave length is approximately 88 km. In our simulations this value is used as the width of the sponge layer and α = 99.99% of a baroclinic wave is absorbed by the sponge layer over a time scale of τsponge = 2750 s. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 97

38 o N 3.72 cm s -1

37 o

36 o

35 o

34 o

126 o 125 o 124 o 123 o 122 o 121 o W

Figure 6.6: M2 barotropic tidal ellipses along open boundaries. Bathymetry contours are spaced at -200, -500, -1000, -1500, -2000, -2500, -3000, -3500 m. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 98

Table 6.1: Four types of simulation performed in this study.

Type Forcing Stratiﬁcation Run Time For Section I sum of 8 tides no 30 days 6.4.1 II sum of 8 tides yes 30 days 6.4.2

III M2 only no 18 TM2 6.4.1

IV M2 only yes 18 TM2 6.4.1, 6.4.3, 6.5, 6.6

6.3.5 Simulation Parameters

We perform four sets of simulations for this study (Table 6.1). All of them begin on August 18th, 2006 (year day 229). A time step size of ∆t = 18 s is used to ensure stability. In the simulations no turbulence model is employed. Diﬀusion of scalars is 2 −1 ignored by setting κH = κV = 0. A horizontal eddy-viscosity of νH = 1 m s and a −3 2 −1 vertical viscosity of νV = 5×10 m s are applied uniformly throughout the domain. In a sensitivity test of the magnitude of viscosities, we found minor diﬀerence in the energy analysis results when these values increase or decrease one order of magnitude

(see Appendix). A constant bottom drag coeﬃcient of CD = 0.0025 and a constant Coriolis frequency of f = 8.7 × 10−5 rad s−1 are speciﬁed for the whole domain. For an 18-M2-cycle simulation, the model runs 1.25 wall-clock days, or 3830 CPU hours using 128 processors on the MJM Linux EM64T cluster at the ARL.

6.4 Dynamics

In this section, we present the simulation results of the barotropic and baroclinic tides in the Monterey Bay area. In previous numerical studies of internal tides in the Monterey Bay region, SUNTANS has shown a high level of skill in predicting the water surface and velocities in the canyon (Jachec et al., 2006; Jachec, 2007). Here model predictions are further compared with the field observations at Station K during the 2006 AESOP (Assessing the Effectiveness of Submesoscale Ocean Pa- rameterizations) field experiment. The field data was provided by Dr. J. M. Klymak CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 99

(http://hornby.seos.uvic.ca/ jklymak/). Station K is located north of the Sur Plat- form as indicated in Figure 6.2.

6.4.1 Free Surface and Barotropic Velocities

1 ) 0 (m η

−1

0.02 ) −1 0 U (m s −0.02

0.05 ) −1 0 V (m s −0.05 230 235 240 245 250 255 Days in 2006

Figure 6.7: Comparison of the free-surface elevation η, the East-West U and North- South V barotropic velocities between SUNTANS predictions (red dashed) and OTIS solutions (black solid) at Station K.

The free-surface elevation and barotropic tidal currents are ﬁrst validated with the OTIS solutions. This barotropic simulation is forced by a sum of the ﬁrst eight tidal constituents from OTIS and run for 30 days (Type I in Table 6.1). Figure 6.7 compares CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 100

(m) 0.5 38 o N

37 o

0.45

36 o

35 o 0.4

34 o

0.35 126 o 125 o 124 o 123 o 122 o 121 o W

Figure 6.8: Free-surface elevation at t = 18TM2 from an M2 forced barotropic run (Type III in Table 6.1). Bathymetry contours are spaced at -200, -500, -1000, -1500, -2000, -2500, -3000, -3500 m. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 101

(m) 0.5 38 o N

37 o

0.45

36 o

35 o 0.4

34 o

0.35 126 o 125 o 124 o 123 o 122 o 121 o W

Figure 6.9: Free-surface elevation at t = 18TM2 from an M2 forced baroclinic run (Type IV in Table 6.1). Bathymetry contours are spaced at -200, -500, -1000, -1500, -2000, -2500, -3000, -3500 m. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 102

the SUNTANS output to the OTIS solutions at Station K. The surface elevation and the north-south barotropic current predictions by SUNTANS and OTIS are in good agreement. However, the east-west barotropic velocities are quite diﬀerent in both the amplitude and the phase. This may be caused by the coarse grid resolution (∼ 1o ) of the OTIS model which is unable to resolve the complex bathymetry. Comparisons in next section validate that SUNTANS result is much closer to the real observations. Overall, the model shows a good performance in simulating the barotropic tides.

Sea-surface elevations from an M2 forced barotropic run (Type III in Table 6.1) and an M2 forced baroclinic run (Type IV in Table 6.1) are illustrated in Figure 6.8 and Figure 6.9, respectively.

6.4.2 Depth-averaged Velocities

(a) (b) 0.05 ) −1 0 U (m s U(m

−0.05 0.1

) 0.05 −1 0

V (m s (m V −0.05

−0.1 235 240 245 235 240 245 Days in 2006

Figure 6.10: Comparison of the depth-averaged velocities between SUNTANS predictions (red dashed) and field observations (black solid) at Station K: (a) original field data; (b) high-pass filtered field data. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 103 2 -2 -1 -2 2 m s cpd s m

Figure 6.11: Power spectra of the observed East-West velocity at Station K. Up- per: depth-averaged velocity; Lower: velocity. The vertical dashed lines indicate the Coriolis (f) and M2 frequencies.

To further validate the model, the depth-averaged velocities are compared between SUNTANS predictions and the AESOP field observations at Station K (Figure 6.10). For this comparison we employ a baroclinic simulation of type II in Table 6.1. There are missing field observations at some depths, which are indicated by the white areas in Figure 6.13. Therefore we only average the velocities over depths with field observations. A strong slowly-varying trend can be seen from the observed depth- averaged velocities (Figure 6.10(a)), which may be due to low-frequency currents not included in the model. Velocity spectra (Figure 6.11 and 6.12) clearly show those low- frequency bands. In order to compare the tidal currents, a high-pass filter is applied CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 104 2 -2 -1 -2 2 m s cpd s m

Figure 6.12: Power spectra of the observed North-South velocity at Station K. Up- per: depth-averaged velocity; Lower: velocity. The vertical dashed lines indicate the Coriolis (f) and M2 frequencies. to the field data to remove the low-frequency signal. We implement the high-pass filtering in this way: an observed time series is first mapped to the frequency space via Fourier transform, then the amplitude of low frequencies are reduced to zero, and finally an inverse Fourier transform is employed to obtain a new time series with the low-frequency components removed. Figure 6.10(b) compares the depth-averaged velocities between SUNTANS predictions and the filtered observations. They are in good agreement in both amplitude and phase. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 105

(a)

(m s -1 ) 0.15

0.1

(b) 0.05

-0.05

-0.15

Days in 2006

Figure 6.13: Time series of the East-West u velocity at Station K: (a) field observations; (b) M2 filtered field observations; (c) SUNTANS predictions. The white areas indicate regions without observations data. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 106

(a)

(m s -1 ) 0.15

0.1

(b) 0.05

-0.05

-0.15

Days in 2006

Figure 6.14: Time series of the North-South v velocity at Station K: (a) field observations; (b) M2 filtered field observations; (c) SUNTANS predictions. The white areas indicate regions without observation data. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 107

6.4.3 Baroclinic Velocities

We examine the model performance in predicting the baroclinic features. An M2- forced baroclinic simulation (Type IV in Table 6.1) is employed for this comparison. Figure 6.13(a) depicts the depth-time profiles of the observed East-West velocity. The profile shows multi-frequency features that are not easy to compare. We therefore apply a band-pass filter to the observations and obtain the M2-fit velocity profile (Figure 6.13(b)) to be compared with the SUNTANS prediction (Figure 6.13(c)). The implementation of the M2 band-pass filter is similar to that of the high-pass filter. In the second step, amplitude of all frequencies are reduced to zero except for the band within 5% of M2. Figure 6.14 show the similar comparison for the North-South velocity. The model predictions capture the vertical structure of the velocities. The multi-mode feature can be seen in both field observations and model predictions, which indicates the existence of higher-mode baroclinic tides at this location, as discussed in Section 6.6 which shows that baroclinic tides in this region are mainly in the form of tidal beams. Figure 6.16 demonstrates the vertical distribution of baroclinic velocities along a transect indicated in Figure 6.15. It can be seen clearly that baroclinic tides are generated at Sur Platform and then radiate to the north and south in the form of tidal beams. Jachec (2007) has shown that the generated baroclinic tides radiate in all directions and thus may be better described as internal tidal “surface” rather than internal tidal “beams”. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 108

Elevation (m)

MSC

K ∗ Latitude Sur Platform Sur Ridge Sur

Longitude

Figure 6.15: Vertical transect along which baroclinic velocities are shown. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 109

(a) (b)

y - y K (km) y - y K (km)

-15 -10 -5 0 5 10 15 (cm s -1 )

Figure 6.16: Vertical structure of the East-West (a) and North-South (b) baroclinic velicities along the transect indicated in Figure 6.15. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 110

6.5 Energetics

We evaluate the depth-integrated, time-averaged barotropic and baroclinic energy equations (6.24) and (6.25) for the energy analysis in this section. They are averaged over the last six M2 tidal cycles of an 18-tidal-cycle simulation (Type IV in Table 6.1). Because the system is periodic, the ﬁrst term, or the tendency term, in each equation tends to zero upon period-averaging. We therefore obtain the balance relations between the three dominant parts as ⟨ ⟩ ⟨ ⟩ ∇H · F0 = − C − ⟨ϵ0 + D0⟩ , (6.32) ⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ′ ′ ′ ∇H · F = C − ϵ + D , (6.33) ⟨ ⟩ In the following analysis, we use the model-computed conversion rate ( C ) and the ⟨ ⟩ ⟨ ⟩ ′ energy ﬂux divergence terms (∇H · F0 and ∇H · F ) directly, while the barotropic and baroclinic dissipation rates are inferred from the above balance relations.

6.5.1 Horizontal Structure

Figure 6.17 illustrates⟨ the⟩ horizontal distribution of the depth-integrated baroclinic energy flux vectors, F′ . Large fluxes are seen in the vicinity of four typical topo- graphical features, which include a northern shelf-slope region (b), the MSC (c), the Sur Ridge-Platform region (d), and the Davidson Seamount (e). The energy budget within each subdomain is discussed in the next section. Figure 6.18 shows the horizontal distribution of the depth-integrated barotropic-to-baroclinic conversion rate, ⟨ ⟩ C , for region (a) in Figure 6.17. Red color represents positive energy conversion rate, which implies generation of internal tides, and the figure shows that most of the generation is contained within the 200-m and 3000-m isobaths. Negative energy conversion rate (blue color) represents energy transfer from the baroclinic tide to the barotropic tide. It is due to the phase difference between locally and remotely generated internal tides (Zilberman et al. , 2009). Significant negative conversion occurs within the MSC because large baroclinic energy generated at the North Sur Plat- form region radiates into the MSC and is steered by the canyon bathymetry (Figure CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 111

2 kW m -1

30’ (a)

(b) 37 o N

(c)

30’

36 o N (d)

30’ (e)

35 o N

124 o W 30’ 123 o W 30’ 122 o W 30’ ⟨ ⟩ Figure 6.17: Depth-integrated, period-averaged baroclinic energy ﬂux, F′ . The ﬁve boxes (a)-(e) indicate the subdomains in which energetics are investigated in detail. Bathymetry contours are spaced at -200, -500, -1000, -1500, -2000, -2500, -3000, -3500 m. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 112

3000 m 200 m (W m -2 ) 0.2

0.15 30’

0.1

37 o N

0.05

30’ 0

36 o N −0.05

−0.1 30’

−0.15

35 o N −0.2 124 o W 30’ 123 o W 30’ 122 o W 30’

Figure⟨ 6.18:⟩ Depth-integrated, period-averaged barotropic-to-baroclinic conversion rate, C . Bathymetry contours are spaced at -200, -500, -1000, -1500, -2000, -2500, -3000, -3500 m. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 113

(W m -2 ) 0.2

0.15 30’

0.1

37 o N

0.05

30’ 0

36 o N −0.05

−0.1 30’

−0.15

35 o N −0.2 124 o W 30’ 123 o W 30’ 122 o W 30’

Figure⟨ 6.19:⟩ Depth-integrated, period-averaged baroclinic energy ﬂux divergence, ′ ∇H · F . Bathymetry contours are spaced at -200, -500, -1000, -1500, -2000, -2500, -3000, -3500 m. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 114

ε -2 log10 ' (W m ) −0.5

30’

−1

37 o N

30’ −1.5

36 o N

−2 30’

35 o N −2.5 124 o W 30’ 123 o W 30’ 122 o W 30’

∇ · ⟨Figure⟩ 6.20:⟨ ⟩ Depth-integrated, period-averaged baroclinic dissipation rate, H F′ − C . Bathymetry contours are spaced at -200, -500, -1000, -1500, -2000, -2500, -3000, -3500 m. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 115

6.17). This effect was also demonstrated by Jachec et al. (2006) and Carter (2010). Figure⟨ 6.19⟩ illustrates the divergence of the depth-integrated baroclinic energy flux, ′ ∇H · F . The difference between Figure 6.19 and Figure 6.18, which represents the ⟨ ⟩ ⟨ ⟩ ′ baroclinic dissipation rate, ∇H · F − C , is shown in Figure 6.20. Large baroclinic energy dissipation occurs near the locations of strong internal tide generation.

6.5.2 Energy Flux Budget

The total power within a region is obtained by area-integrating the period-averaged and depth-integrated energy terms to give ∑ ( ⟨ ⟩) BT Input = − ∇H · F0 ∆A, (6.34) ∑ l(⟨ ⟩) Conversion = C ∆A, (6.35) ∑l ( ⟨ ⟩) ′ BC Radiation = ∇H · F ∆A, (6.36) ∑l ( ⟨ ⟩ ⟨ ⟩) BT Dissipation = ∇H · F0 + C ∆A, (6.37) ∑l ( ⟨ ⟩ ⟨ ⟩) ′ BC Dissipation = ∇H · F − C ∆A, (6.38) ∑l ( ⟨ ⟩ ⟨ ⟩) ′ Total Dissipation = ∇H · F0 + ∇H · F ∆A, (6.39) l ∑ where l implies summation of the grid cells within an area that is bounded by condition l and ∆A is the area of each grid cell. We ﬁrst study the energy distribution as a function of depth. The summation areas for (6.34)-(6.39) are bounded by the condition

ld : −200(n − 1) ≤ z ≤ −200n , n = 1, ··· , 17 . (6.40)

The lower panel of Figure 6.21 compares barotropic energy input, baroclinic energy radiation and barotropic-to-barolinic energy conversion in 200-m isobath bounded bins, while the upper panel compares their cumulative sum. In the region shallower CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 116

200

100

BT Input −100 Conversion BC Radiation

CumulativePower (MW) −200 20

BT Input Power(MW) −10 Conversion BC Radiation −20 0 0.5 1 1.5 2 2.5 3 Depth (km)

Figure 6.21: Energy distribution as a function of depth. Lower: energy terms (6.34)- (6.36) in 200-m isobath bounded bins; Upper: cumulative sum of the lower. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 117

200 BT Dissipation 100 BC Dissipation

−100

CumulativePower (MW) −200 20 BT Dissipation 10 BC Dissipation

Power(MW) −10

−20 0 0.5 1 1.5 2 2.5 3 Depth (km)

Figure 6.22: Energy distribution as a function of depth. Lower: energy terms (6.37)- (6.38) in 200-m isobath bounded bins; Upper: cumulative sum of the lower. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 118

142 MW 9.4 MW BT Input

BT Dissipation - 19 MW - 7 MW

Conversion 123 MW 2.4 MW BC Radiation 41.4 MW 5.2 MW

BC Dissipation - 76.4 MW - 7.6 MW

Deep Slope Shelf -3000m -200m

Figure 6.23: Schematic of the M2 tidal energy budget for the two subdomains bounded by the 0-m, 200-m and 3000-m isobaths. than 200 m, only a small portion of the input barotropic energy is converted into baroclinic energy, and the negative baroclinic energy radiation implies that baroclinic energy generated in deeper regions ﬂows into this shallow region and is then dissipated. In the region deeper than 2200 m, almost all input barotropic energy is converted into baroclinic energy. The radiated baroclinic energy is large in the region between the 1400-m and 2600-m isobaths, and is much smaller in other regions. Fig- ure 6.22 compares the barotropic and baroclinic dissipations. Most of the barotropic energy dissipation occurs in the region shallower than 2000 m, while the baroclinic energy dissipation occurs at all depths with two peaks near 1200 m and within 200 m. Figure 6.23 illustrates a schematic of the energy budget for the shelf and slope regions bounded by the 200-m and 3000-m isobaths. 142 MW barotropic energy is added to the slope region and approximately 87% of this energy is converted into baroclinic energy. Most of this generated baroclinic energy is dissipated locally, while the remaining portion (38%) radiates into deeper and shallower regions. The shelf region acts as a baroclinic energy sink because it dissipates both the energy generated locally and the portion ﬂowing into it from the slope region. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 119

Table 6.2: M2 tidal energy budget for the ﬁve subdomains indicated in Figure 6.17. The energy in estimated in MW (and percentage).

Domains (a) (b) (c) (d) (e) BT Input 151.95 31.04 8.30 44.80 29.82 (100%) (100%) (100%) (100%) (100%) BT-BC Conversion 133.39 29.17 5.85 39.20 28.24 (88%) (94%) (70.5%) (87.5%) (94.7%) BC Radiation 56.20 22.05 -3.64 24.02 22.91 (37%) (71%) (-44%) (53.6%) (76.8%) BT Dissipation -18.56 -1.87 -2.45 -5.60 -1.58 (12%) (6%) (29.5%) (12.5%) (5.3%) BC Dissipation -77.19 -7.12 -9.49 -18.42 -5.33 (51%) (23%) (114.5%) (33.9%) (17.9%) Total Dissipation -95.75 -8.99 -11.94 -20.78 -6.91 (63%) (29%) (144%) (46.4%) (23.2%)

Barotropic Tides 100%

Topography 88% (100%) 12%

Baroclinic Tides 37% (42%)

Internal Waves (58%) 51%

Deep Turbulence Local Mixing

Figure 6.24: Schematic of the M2 tidal energy budget in percentages for subdomain (a) in Figure 6.17. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 120

Barotropic Tides 100%

Topography 95% (100%) 5%

Baroclinic Tides 77% (81%)

Internal Waves (19%) 18%

Deep Turbulence Local Mixing

Figure 6.25: Schematic of the M2 tidal energy budget in percentages for subdomain (e) in Figure 6.17.

We list the detailed energy budget (Table 6.2) for the five subdomains (a)-(e) indicated in Figure 6.17 as these represent regions with topographic features of the coastal shelf break. Subdomain (a), a 200 km × 230 km domain, is used to represent the Monterey Bay area as it includes all typical topographic features in this area. For this area, approximately 133 MW (88%) of the 152 MW barotropic energy is converted into baroclinic energy, and 56 MW (42%) of this generated baroclinic energy radiates away. In the previous study by Jachec et al. (2006), the positive and negative ′ ′ portions of the hydrostatic baroclinic energy flux divergence (∇H ·(u p )) were used to estimate the baroclinic energy generation and dissipation, respectively. For a 200 km (alongshore) × 90km (offshore) domain, they saw 52 MW baroclinic energy available for deep ocean mixing. The tidal energy budget highly depends on topographic features as shown in Ta- ble 6.2. The Davidson Seamount and the Northern shlef-break region are the most efficient topographic features to convert (∼ 94%) barotropic energy into baroclinic energy and then let it radiate out into the open ocean (>70%). The Sur Platform region also converts a large portion (87.5%) and radiates about half of the barotropic CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 121

Table 6.3: Contributions of different energy fluxes to the total baroclinic energy flux divergence for subdomain (a). The contributions are estimated in percentages.

′ ∇H · F 100% KE Advection -0.54% APE Advection 0.18% Hydrostatic Work 100.87% Nonhydrostatic Work -0.51% Diﬀusion 0.002%

energy as baroclinic energy. The MSC acts as an energy sink because it does not radiate energy but instead absorbs the baroclinic energy from the Sur Platform region (Figure 6.17). In particular, the energy budget for the Davidson Seamount (subdomain (e)) is quite similar to that for the Hawaiian Islands by Carter et al. (2008), which shows that 85% of the barotropic energy that is lost is converted into baroclinic energy, and 74% of this baroclinic energy radiates into the open ocean. 74% of this baroclinic energy radiates into the open ocean.

6.5.3 Energy Flux Contributions

As discussed in Section 6.2, our method computes the full energy fluxes and thus allows us to compare the contributions of different components which include the energy advection, diffusion, nonhydrostatic pressure work, as well as the traditional hydrostatic pressure work. Table 6.3 lists the energy flux contributions to the total baroclinic energy flux divergence for the subdomain (a). The traditional energy flux ′ ′ due to hydrostatic pressure work uH p is the dominant term, whose divergence ac- counts for ∼ 101% of the total divergence, while all the other terms account for ∼ -1% together. The nonhydrstatic contribution, although small in this region, acts to reduce the hydrostatic energy flux. This can be seen clearly in Figure 6.26-6.27, where the hydrostatic and nonhydrostatic energy flux flow in opposite directions within the Monterey Submarine Canyon. The small advection and nonhydrostatic contributions imply that the flow in the Monterey Bay area is mainly linear and hydrostatic. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 122

37 o N 2 kW m -1

50’50'

40’

30’30'

20’20' 30’30' 20’20' 10’10' 122-122 W o W 50’50'

Figure 6.26: Baroclinic energy ﬂux contribution from hydrostatic pressure work. Bathymetry contours are spaced at -100, -500, -1000, -1500, -2000 m. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 123

37 o N 10 W m -1

50’50'

40’40'

30’30'

20’20' 30’30' 20’20' 10’10' 122-122 W o W 50’

Figure 6.27: Baroclinic energy ﬂux contribution from nonhydrostatic pressure work. Bathymetry contours are spaced at -100, -500, -1000, -1500, -2000 m. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 124

6.5.4 Sensitivity Test of Simulation Parameters

A set of M2-forced numerical simulations are carried out to examine the sensitivity of model results to the dissipation parameters and the resolution of grid and bathymetry. For each simulation, we calculate the barotropic energy input (6.34), the barotropic- to-baroclinic conversion (6.35), and the baroclinic energy radiation (6.36) for subdomain (a) in Figure 6.17. The reference simulation is that discussed in Section 2 −1 −3 6.5.2, which employed constant eddy viscosities (νH = 1 m s and νV = 5 × 10 m2s−1) and a fine-resolution grid and bathymetry described in Section 6.3. Detailed comparisons are listed in Table 6.4. Simulations 1-4 are used to test the sensitivity to the dissipation parameters. The value of horizontal or vertical eddy viscosity is increased or decreased by one order of magnitude each at a time. These changes have minor effect on the integrated power for the domain. Simulations 5-6 are employed to examine the sensitivity to the resolution of bottom topography and grid. The fine grid in Table 6.4 represents the grid described in Section 6.3, which consists of approximately 50 thousand horizontal grid cells. The horizontal resolution within subdomain (a) smoothly transitions from 80 m in the bay to 1.5 km along the offshore boundary of subdomain (a). The coarse grid in Table 6.4 represents a lower-resolution grid consisting of 7500 horizontal grid cells but the same vertical levels as the fine grid. The horizontal grid resolution within subdomain (a) varies from 480 m in the Bay to 4 km along the offshore boundary of subdomain (a). The fine and coarse bathymetry have the same resolutions as the fine and coarse grids, respectively. With the same grid but coarser bathymetry (Simulation 5), the barotropic energy input and barotropic-to-baroclinic energy conversion are underestimated by roughly 10%, and the baroclinic energy radiation is underestimated by roughly 20%. When the grid and bathymetry are both coarse (Simulation 6), there is a further ∼2% underestimation of each energy term compared to Simulation 5. These results show that a high-resolution bathymetry plays an important role in the accurate calculation of the tidal energy budget. Jachec (2007) examined the effect of grid resolution from the point of view of dynamics. He demonstrated that the velocity and isopycnal displacement are underpredicted using a coarse grid and bathymetry. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 125

Table 6.4: Sensitivity test of various simulation parameters.

Sim. νH νV Grid Bathy. BT Input Conversion BC Radiation [m2s−1] [MW] (relative change to Sim. 0) 0 1 5×10−3 Fine Fine 151.95 133.39 56.20 1 0.1 5×10−3 Fine Fine 150.29 133.90 56.40 (-1.09%) (+0.38%) (+0.36%) 2 10 5×10−3 Fine Fine 155.02 132.39 54.71 (+2.02%) (-0.75%) (-2.65%) 3 1 5×10−4 Fine Fine 152.10 133.19 56.48 (+0.10%) (-0.15%) (+0.50%) 4 1 5×10−2 Fine Fine 151.78 133.19 54.34 (-0.11%) (-0.15%) (-3.31%) 5 1 5×10−3 Fine Coarse 137.88 120.20 45.59 (-9.26%) (-9.89%) (-18.88%) 6 1 5×10−3 Coarse Coarse 134.73 116.29 44.76 (-11.33%) (-12.82%) (-20.36%)

Besides the above tests of dissipation parameters and grid resolution, we also ﬁnd simulations with or without nonhydrostatic pressure have minor eﬀect on the energetics results in the Monterey Bay area. Detailed comparisons can be seen in the study of Jachec (2007).

6.6 Generation Mechanism

6.6.1 Parameter Space

Three important nondimensional parameters are generally employed to discuss the character of internal tide generation (Baines, 1982; St. Laurent & Garrett, 2002; Legg & Huijts, 2006; Garrett & Kunze, 2007), which is governed by the dimensional parameters listed in Table 6.5. The ﬁrst parameter is the steepness parameter deﬁned by

γ ϵ = , (6.41) 1 s CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 126

Table 6.5: Key physical parameters governing internal tide generation and propagation.

Parameters Descriptions ω tidal frequency f Coriolis frequency N buoyancy frequency U0 amplitude of the barotropic tidal current k horizontal wavenumber of the internal tide m vertical wavenumber of the internal tide c phase speed of the internal tide (c = ω/k) kb horizontal wavenumber of the topography d depth of the topography

where the topographic slope is given by √( ) ( ) ∂d 2 ∂d 2 γ = + , (6.42) ∂x ∂y and the internal wave characteristic slope is given by ( ) k ω2 − f 2 1/2 s = = . (6.43) m N 2 − ω2

This nondimensional parameter is used to distinguish between subcritical topography when the topographic slope is ﬂat compared to the wave characteristic slope (ϵ1 < 1) and supercritical topography when the topographic slope is steeper than the wave characteristic slope (ϵ1 > 1). The topography is referred to as critical when its slope matches the wave characteristic slope (ϵ1 = 1). Recently, more studies have been carried out to understand the dependence of the barotropic to baroclinic energy conversion on this parameter (Balmforth et al., 2002; Khatiwala, 2003; Balmforth & Peacock, 2009). The second nondimensional parameter is the tidal excursion parameter deﬁned by

U k ϵ = 0 b , (6.44) 2 ω CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 127

which measures the ratio of the tidal excursion U0/ω to the horizontal scale of the −1 topography kb . When the tidal excursion is less than the topographic scale (ϵ2 < 1), linear internal tides are generated mainly at the forcing frequency ω. When the tidal excursion is larger than the topographic scale (ϵ2 > 1), lee waves can form. This parameter has been examined widely in studying internal tide generation over idealized topography (St. Laurent & Garrett, 2002; Legg & Huijts, 2006; Garrett & Kunze, 2007). The third nondimensional parameter is the Froude number deﬁned by

U0 U0k k F r = = = ϵ2 , (6.45) c ω kb which measures the ratio of barotropic current speed U0 to the baroclinic wave speed c = ω/k. If the horizontal scales of the topography and the internal tide match (k = kb), this parameter is the same as the tidal excursion parameter, which is the case for some analytical studies (St. Laurent & Garrett, 2002; Garrett & Kunze, 2007). The Froude number can be used to estimate the nonlinearity of the wave (Vlasenko et al. , 2005). When F r ≪ 1, linear theory is valid for analyzing internal tide generation.

First-mode internal tides are generated over subcritical topography (ϵ1 < 1), while internal tidal beams are generated over critical or supercritical topography (ϵ1 ≥ 1). At intermediate Froude numbers (F r ∼ 1), nonlinearity becomes important, and nonlinear internal wave bores, weak unsteady lee waves, and solitary internal waves may be generated depending on the topographic features. At high Froude numbers (F r > 1), in addition to the bores and solitary internal waves, strong unsteady lee waves may form. Furthermore, under nonrotating (f = 0) and hydrostatic (k ≪ m) conditions, the deﬁnition (6.45) reduces to F r = U0m/N. From deﬁnitions (6.41)-(6.45) we can see that the three parameters are related to one another via ϵ k 2 = F r b . (6.46) ϵ1 mγ

If we estimate the topographic length scale as kb = πγ/d, then this relation becomes

ϵ π 2 = F r . (6.47) ϵ1 md CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 128

For nonrotating and hydrostatic ﬂows, it can be written as

ϵ U 2 = π 0 , (6.48) ϵ1 Nd where the parameter U0/Nd has been used to examine the wave nonlinearity in lien of the Froude number (6.45) in some previous studies (Legg, 2004a, b; Legg & Huijts, 2006; Garrett & Kunze, 2007).

6.6.2 Energy Distribution vs. Parameters

We now examine the internal tide generation and radiation in the Monterey Bay region. Subdomain (a) is chosen as our study domain since it includes all of the typical topographic features in the region. Figure 6.28 and Figure 6.29 show the energy distribution as a function of the above three parameters. Energy terms (6.34)- (6.38) are computed with the following conditions to bound the summation areas

− ≤ ≤ ··· lϵ1 : 0.4(n 1) ϵ1 0.4n , n = 1, , 20 , (6.49) − ≤ ≤ ··· lϵ2 : 0.015(n 1) ϵ1 0.015n , n = 1, , 20 , (6.50)

lF r : 0.0025(n − 1) ≤ ϵ1 ≤ 0.0025n , n = 1, ··· , 20 . (6.51)

As shown in Figure 6.28, barotropic-to-baroclinic conversion (green bins) occurs predominantly in regions deﬁned by ϵ1 = γ/s < 5, ϵ2 = U0kb/ω < 0.2, and 0.006 ≤ F r ≤

0.03, with peaks at the critical topography (ϵ1 = 1) and small values of the excursion parameter (ϵ2 = 0.038) and Froude number (F r = 0.012), respectively. Under these conditions, baroclinic tides generated in this region are mainly linear and in the form of internal tidal beams (St. Laurent & Garrett, 2002; Vlasenko et al., 2005). The input barotropic energy (blue bins) shows a similar distribution as the conversion for all three parameters. However, the radiated baroclinic energy (red bins) behaves diﬀerently. Supercritical topography (ϵ1 > 1) is more eﬃcient in radiating the generated baroclinic energy, while subcritical topography (ϵ1 < 1) acts as an energy sink because it absorbs more baroclinic energy. Similarly, the region with smaller excursion (ϵ2 < 0.02) absorbs more baroclinic energy and then dissipates the energy within CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 129

(a) 20

0 BT Input Conversion Power(MW) −20 BC Radiation

0 1 2 3 4 5 6 7 8 1

(b) 20

Power(MW) −20 0 0.05 0.1 0.15 0.2 0.25 0.3 2

Power(MW) −20 0 0.01 0.02 0.03 0.04 0.05 Fr

Figure 6.28: Distribution of the conversion and the barotropic and baroclinic energy as a function of (a) ϵ1 = γ/s, (b) ϵ2 = U0kb/ω, and (c) F r = U0/c for subdomain (a) in Figure 6.17. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 130

(a) 20 BT Dissipation BC Dissipation

Power(MW) −20

0 1 2 3 4 5 6 7 8 1

(b) 20

Power(MW) −20 0 0.05 0.1 0.15 0.2 0.25 0.3 2

Power(MW) −20 0 0.01 0.02 0.03 0.04 0.05 Fr

Figure 6.29: Distribution of the barotropic and baroclinic dissipation terms as a function of (a) ϵ1 = γ/s, (b) ϵ2 = U0kb/ω, and (c) F r = U0/c for subdomain (a) in Figure 6.17. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 131

the region. In the region with 0.01 ≤ F r ≤ 0.02 both baroclinic energy generation and radiation are much larger. Figure 6.29 shows the distribution of barotropic and baroclinic dissipations with respect to the three parameters. Baroclinic dissipation (orange bins) mainly occurs in regions with subcritical topography, smaller excursion, and 0.01 ≤ F r ≤ 0.02. From Figure 6.28(c) and Figure 6.29(c), we can see the baroclinic energy generation, radiation, and dissipation are all focused within the same region (0.01 ≤ F r ≤ 0.02) and peak at roufhly F r = 0.012.

6.6.3 A Parametric Model to Estimate Energy Conversion

BT Input BT-BC Conversion BC Radiation

0.15

2 0.1

0.05

0 0.04

0.03

Fr 0.02

0.01

0 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 1 1 1

-4 -3 -2 -1 0 1 2 3 4 (MW)

Figure 6.30: Distribution of the energy terms (in MW) as a function of (a) ϵ1 = γ/s and ϵ2 = U0kb/ω, (b) ϵ1 = γ/s and F r = U0/c for subdomain (a) in Figure 6.17.

Figure 6.30 illustrates the energy distribution as a function of two parameters.

Energy conversion occurs mainly in regions where ϵ1 and ϵ2 satisfy a particular relation. When ϵ1 and ϵ2 are small, this relation is linear. As the values of ϵ1 and ϵ2 increase, the departure is weakly quadratic. We also compare the energy conversion CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 132

4 1 = 0.5

1 = 1 3 1 = 1.5

1 = 2 2

Conversion(MW) 1

0 0 0.05 0.1 0.15 2

Figure 6.31: Distribution of the tidal energy conversion (in MW) as a function of ϵ2 = U0kb/ω with diﬀerent values of ϵ1 = γ/s for subdomain (a) in Figure 6.17.

as a function of ϵ2 with different values of ϵ1 (Figure 6.31). For a fixed ϵ1, the distribution of energy conversion with respect to ϵ2 resembles normal distribution. The mean and variance of these curves increase as the value of ϵ1 increase, while the peak of these curves has a maximum value when ϵ1 = 1. Based on these features, we estimate the energy conversion using two functions, one reflects the linear relation between ϵ1 and ϵ2 [ ] [ ] 2 − 2 − ϵ1 −(ϵ2 aLϵ1) CL(ϵ1, ϵ2) = C0L exp 2 exp 2 , (6.52) 2σ2L 2(σ1Lϵ1) and the other reflects the quadratic relation between ϵ1 and ϵ2 [ ] [ ] 2 − − 2 − 2 − ϵ1 −(ϵ2 aϵ1 bϵ1 c) CQ(ϵ1, ϵ2) = C0 exp 2 exp r 2 . (6.53) 2σ2 2(σ1ϵ1)

The coeﬃcients C0L, aL, σ1L, and σ2L for function (6.52), and C0, a, b, c, r, σ1, and

σ2 for function (6.53) are obtained by minimizing the least square error between the function and the BT-to-BC conversion in Figure 6.30. For the linear function (6.52), CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 133

the regression coeﬃcients are given by

C0L = 4.914 a = 3.855 × 10−2 L (6.54) −3 σ1L = 9.665 × 10

σ2L = 1.223 , with an R-squared value of 0.8. For the quadratic function (6.53), the regression coeﬃcients are given by

C0 = 4.914 a = 3.037 × 10−2 b = 1.953 × 10−3 c = 4.793 × 10−3 (6.55) r = 1.237 −3 σ1 = 8.658 × 10

σ2 = 1.240 , with an R-squared value of 0.9. Figure 6.32(a) illustrates the estimate of barotropic-to-baroclinic conversion using function (6.52) based on the linear relation between ϵ1 and ϵ2, vis.

ϵ2 = aLϵ1 , (6.56) while Figure 6.32(b) shows the estimate using function (6.53) based on the quadratic relation between ϵ1 and ϵ2, vis.

2 ϵ2 = aϵ1 + bϵ1 + c . (6.57)

The upper panels display results from numerical simulations, while the lower panels display functions (6.52) and (6.53). Both functions capture the features of barotropic- to-baroclinic conversion with small values of ϵ1 and ϵ2. For larger values, the quadratic function (6.53) is better. CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 134

(a) (b)

0.15 (MW) 4

2 0.1

0.05 3

0 2 0.15 1

2 0.1

0 0.05

0 0 1 2 3 4 5 0 1 2 3 4 5 1 1

Figure 6.32: Regression of the tidal energy conversion using (a) function (6.52) based on the linear relation (6.56), and (b) function (6.53) based on the quadratic relation (6.57). The upper panels are the SUNTANS results, and the lower panels are the best-ﬁt functions. On each plot, the bold line indicates the relation between ϵ1 and ϵ2, while the two thin lines indicate one standard deviation from the bold line. 6.7 Conclusions

Three-dimensional, high-resolution simulations of the barotropic and baroclinic tides in the Monterey Bay area have been conducted using the SUNTANS model. The model predicted sea surface, barotropic and baroclinic velocities agree with observations. Based on the theoretical framework in Chapter 5, a detailed tidal energy analysis is performed to address the question of how the barotropic tidal energy is partitioned between local barotropic dissipation and local generation of baroclinic energy, and then how much of this generated baroclinic energy is lost locally versus how much is radiated away for open-ocean mixing. For subdomain (a), a 200 km CHAPTER 6. INTERNAL TIDE IN MONTEREY BAY 135

× 230 km domain including all of the typical topographic features in the Monterey Bay area, approximately 133 MW (88%) of the 152 MW barotropic energy that is lost is converted to baroclinic energy, and 56 MW (42%) of this baroclinic energy radiates away. The tidal energy partitioning depends greatly on the topographic features. For example, the Davidson Seamount is most efficient in baroclinic energy generation (95%) and radiation (81%), while the MSC acts as an energy sink because it does not radiate energy but instead absorbs the baroclinic energy from the Sur Platform region as shown by Jachec et al. (2006). We also examine the energy flux contributions from nonlinear and nonhydrostatic effects. The small advection and nonhydrostatic contributions imply that the flow in the Monterey Bay area is mainly linear and hydrostatic. The internal tide generation mechanism is investigated by examining the dependence of barotropic-to-baroclinic energy conversion on three nondimensional parameters, namely the steepness parameter, the tidal excursion parameter, and the Froude number. It is found that barotropic-to-baroclinic conversion is large when the steepness parameter and tidal excursion parameter satisfy a certain relation. When both parameters are small, this relation is linear. As the values increase, the departure is weakly quadratic. Based on this feature, we obtain a simple parametric model to estimate the barotropic-to-baroclinic energy conversion, which only needs a few dimensionless parameters without running high-resolution simulations. This may be useful for tidal mixing parameterization in large-scale ocean and climate simulations. Chapter 7

Summary and Future Directions

7.1 Summary

Mixing in the ocean is critical in determining the large-scale circulation and energy distribution of the ocean. Internal tides, which can propagate vertically in the continuously stratified ocean, play a key role in deep-ocean mixing. This research focus on understanding the dynamics and energetics of tidal mixing processes through theoretical analysis and numerical simulations. The theoretical framework for analyzing internal tide energetics is based on the complete form of barotropic and baroclinic energy equations. This approach is implemented in the SUNTANS model and employed to study the internal tide driven mixing in the real ocean. In this research, the Monterey Bay area is chosen as the simulation region. The energy budget in this area is analyzed in detail and the mechanisms of internal tide generation are investigated. A simple parametric model is also provided to estimate the barotropic-to-baroclinic tidal energy conversion. In this research we consider the full nonhydrostatic equations and employ a nonhydrostatic model. The prominent challenges for developing nonhydrostatic models are the accurate and efficient computations of the nonhydrostatic pressure. My first study focuses on addressing these two challenges for the free-surface nonhydrostatic solver (Chapter 3). This accuracy study demonstrates that the correction method is always second-order accurate in time, while the projection method maintains the

136 CHAPTER 7. SUMMARY AND FUTURE DIRECTIONS 137

second-order time accuracy only if the nonhydrostatic pressure is included in the depth-averaged continuity equation through the use of an iterative procedure. The efficiency study shows that the effectiveness of the block diagonal preconditioners strongly depends on the grid aspect ratio and the initial conditions. These features have been reflected in the nonhydrostatic SUNTANS model. An appropriate and accurate evaluation of the available potential energy (APE) is of great importance for analyzing the internal wave energetics. In Chapter 4, three commonly used APE formulations are compared. The relationship among them is illustrated through a graphical interpretation and a derivation of the energy conservation laws. Numerical simulations are also carried out to quantitatively assess the performance of each APE under the influence of different nonlinear and nonhydrostatic effects. The results show that the exact APE density AP E2 is the most attractive in evaluating the local APE, especially for nonlinear internal waves. Larger errors arise when using the perturbation potential energy density AP E1 because of the large disparity in tendency between the kinetic energy and AP E1. This disparity is compensated by a large flux arising from the reference pressure and density fields.

AP E3 is the linear form of AP E2 and thus shows signiﬁcant errors in the presence of nonlinear stratiﬁcation. The complete form of barotropic and baroclinic energy equations are employed as the theoretical framework for analyzing internal tide energetics. Chapter 5 provides a detailed derivation of these equations. The full nonlinear and nonhydrostatic energy

ﬂux contributions as well as an accurate evaluation of the APE (AP E2) are included in this approach. Therefore, we can obtain a more accurate and detailed energy analysis using this approach. This chapter also presents two energy exchange diagrams, which clearly illustrate how to evaluate the energy exchange and conversion using diﬀerent terms in these equations. The theoretical approach has been implemented in the SUNTANS model and is being employed to study the internal tide driven mixing in the real ocean. Chapter 6 focuses on investigating the dynamics and energetics of the tidal mixing processes using this approach. Three-dimensional, high-resolution simulations of the barotropic and baroclinic tides in the Monterey Bay area are conducted using the SUNTANS CHAPTER 7. SUMMARY AND FUTURE DIRECTIONS 138

model. The model predicted sea surface, barotropic and baroclinic velocities agree with observations. Based on the theoretical framework in Chapter 5, a detailed tidal energy analysis is performed to address the question of how the barotropic tidal energy is partitioned between local barotropic dissipation and local generation of baroclinic energy, and then how much of this generated baroclinic energy is lost locally versus how much is radiated away for open-ocean mixing. For subdomain (a), a 200 km × 230 km domain including all of the typical topographic features in the Monterey Bay area, approximately 133 MW (88%) of the 152 MW barotropic energy is converted into baroclinic enerngy, and 56 MW (42%) of this generated baroclinic enerngy radiates away for deep-ocean mixing. The tidal energy partitioning depends greatly on the topographic features. For example, the Davidson Seamount is most efficient in baroclinic energy generation (95%) and radiation (81%), while the MSC acts as an energy sink because it does not radiate energy but instead absorbs the baroclinic energy from the Sur Platform region. We also examine the energy flux contributions from nonlinear and nonhydrostatic effects. The small advection and nonhydrostatic contributions imply that the flow in the Monterey Bay area is mainly linear and hydrostatic. Internal tide generation mechanism is investigated by examining the dependence of barotropic-to-baroclinic energy conversion on three nondimensional parameters, namely the steepness parameter, the tidal excursion parameter, and the Froude number. It is found that barotropic-to-baroclinic conversion is large when the steepness parameter and tidal excursion parameter satisfy a certain relation. When both parameters are small, this relation is linear. As the values of them increase, the departure is weakly quadratic. Based on this feature, we obtain a simple parametric model to estimate the barotropic-to-baroclinic energy conversion, which only needs a few dimensionless parameters without running high-resolution simulations. This is useful to the improvement of tidal mixing parameterization for large-scale ocean and climate simulations. CHAPTER 7. SUMMARY AND FUTURE DIRECTIONS 139

7.2 Future Directions

Tidal mixing processes are of key importance in the large-scale ocean circulation. However, they cannot be resolved in coarse-resolution ocean/climate models. There- fore, developing realistic tidal mixing parameterizations has become one of the greatest challenges in ocean and climate studies. Current tidal mixing parameterizations in ocean/climate models can capture the gross picture of the interior-ocean mixing (St. Laurent et al., 2002; Polzin, 2004; Jayne, 2009). However, many important details still require improvement. For example, the ratio of dissipated energy to total generated baroclinic energy, which is often assumed to be constant, actually varies in space and time. Therefore, more analytical studies and high-resolution simulations are needed to better understand the details and improve tidal mixing parameterizations. The main goal of future study is to develop a physically based, energetically consistent parameterization of tidal mixing for large-scale climate simulations. In this research we have proposed a simple parametric model to estimate the the barotropic- to-baroclinic energy conversion based on an accurate energetics analysis. However a better understanding of its physical basis and more tests in other locations are still needed. High-resolution simulations will be focused on investigating important controlling parameters. Coarse-resolution and large-scale GCM (general circulation Model) simulations are to test the parameterization. More ﬁeld experiments and data will be important to test and improve the parameterization. A combination of theoretical analysis, numerical simulations and ﬁeld-data analysis will be required to achieve the goal. References

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