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1 Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2010 Internal Wave Propagation and Numerically Induced Diapycnal Mixing in Oceanic General Circulation Models Flavien Gouillon Follow this and additional works at the FSU Digital Library. For more information, please contact

2 THE FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES INTERNAL WAVE PROPAGATION AND NUMERICALLY INDUCED DIAPYCNAL MIXING IN OCEANIC GENERAL CIRCULATION MODELS By FLAVIEN GOUILLON A Dissertation submitted to the Department of Oceanography in partial fulfillment of the requirements for the degree of Doctor of Philosophy Degree Awarded: Fall Semester, 2010

3 The members of the committee approve the dissertation of Flavien Gouillon defended on September 29, Eric Chassignet Professor Directing Dissertation Carol Anne Clayson University Representative Louis St Laurent Committee Member Steven Morey Committee Member Markus Huettel Committee Member Approved: James O Brien Committee Member Jeff Chanton, Area Chair, Department of Oceanography Joseph Travis, Dean, College of Arts and Sciences The Graduate School has verified and approved the above-named committee members. ii

4 A ma famille... iii

5 ACKNOWLEDGEMENTS Many people contributed and provided me support in producing this dissertation. My first thanks goes to Eric Chassignet, my main advisor. One of the best thing I shall remember about Eric is his unconditional support, belief in me and his constant help. Merci encore Eric. My next thanks goes to all my committee members: Lou St Laurent, Carol Ann Clayson, James J. O Brien, Markus Huettel and Steve Morey, who, constantly helped me throughout the progress of this work. Steve, be certain that I will miss you and all the Tsunami soccer. I am also especially indebted to Dr. O Brien for bringing me at the Florida State University after my short internship in I also want to thank the people from the Center for Ocean-Atmospheric Prediction Studies for providing a warm and friendly environment and a special thanks to Austin (a.k.a. Ouaich!), Henry (mass removal expert), my favorite Russian of all time: Dmitry and my HYCOM geeky big sister: Alexandra. I also feel very fortunate to have been a part of the Florida State University Department of Oceanography (you shall be regretted), thus, I extend my gratitude to all the department faculty (and the big cheese), staff and students. Finally, on the course of these five years, I have benefited from the support of many people that, in their own special way, greatly contributed to the achievement of this dissertation: Nicolas Wienders, Thierry Penduff, Nicolas Choplain, the HPC facility staff, Mohamed Jardak, Florian Lemarié, Alistair Adcroft, Catherine Edwards, Brian Arbic, my interns: Nico, Manu, Benoit, and Gaelle. I would like to thank le groupe de Luminy, a big part of where I am and who I am is because of you. Huge thanks to the Tally soccer circle with special thanks for the Turbulence team, Jimmy, Jess, Kel, Jeff and Cath. Merci to the Buchholz family and Michelle who adopted ze French guy. Un immense merci to gros Matt and to Mumu. I want to thank my family for their continuous support, love and so much more. Finally, I could have never done anything without Marie... I love y all! iv

6 TABLE OF CONTENTS List of Tables List of Figures vi vii Abstract viii 1. INTRODUCTION BACKGROUND Internal tides and Ocean Mixing Internal wave representation in OGCMs Spurious diapycnal mixing in fixed coordinate ocean models Research Overview A MULTI-MODEL COMPARISON OF INTERNAL WAVE GENERATION AND PROPAGATION Introduction Numerical Models Results Summary and discussion DIAGNOSING THE SPURIOUS MIXING IN FIXED COORDINATE OCEAN MODELS Introduction Numerical models Spurious diapycnal mixing associated with a lock exchange problem Spurious diapycnal mixing associated with internal wave propagation Summary CONCLUSION AND PERSPECTIVES Summary of work Perspective of the work Future work v

7 A. IMPACT OF MODEL GRID SPACING IN A REALISTIC APPLICATION: BAROCLINIC TIDES IN THE GULF OF MEXICO A.1 Model configurations A.2 Validation of the tides in HYCOM A.3 Impact of varying the model grid spacing on the internal wave field B. THE TRACER FLUX METHOD REFERENCES BIOGRAPHICAL SKETCH vi

8 LIST OF TABLES 3.1 Description of the experiments conducted. x (km) is the model horizontal resolution, Npt is the number of points defining the ridge, N is the number of levels, and z (m) the model vertical resolution. These experiments are repeated with varying the ridge height to change the wave regime (i.e., subcritical, critical or supercritical) Description of experiments conducted for the lock exchange scenario. x is the horizontal model grid spacing resolution in the x-direction, N is the number of σ-levels, and z is the horizontal model grid spacing resolution in the z-direction Description of Experiments conducted for the internal wave scenario. x is the horizontal model grid length resolution in the x-direction, N x is the number of points that defines the ridge width, N is the number of z- or σ- levels, and z is the horizontal model grid length resolution in the z-direction 49 vii

9 LIST OF FIGURES 2.1 Dissipation rates derived from satellite altimetry sea-surface elevation from Egbert and Ray (2000). Enhanced dissipation is colored in red and attenuated dissipation is colored in blue Turbulent diapycnal eddy diffusivity in the Brazil basin from Polzin et al. (1997) Energy budget proposed by Munk and Wunsch (1998) Baroclinic velocities simulated with the Stanford Unstructured Nonhydrostatic Terrain-following Adaptative Navier-Stokes Simulator (SUNTANS) model for a transect across Monterey Bay with a coarse model resolution ( 3 km) and a fine model resolution ( 0.3 km) from Jachec (2007) West-East Pacific vertical cross-section showing isopycnals (red contours) and the sigma levels (dash lines). Regions of interest are areas where the sigma levels intersect with isopycnals. From Florian Lemarié (personal communication, 2010) Two-year evolution of the salinity field at 1000 m in a ROMS simulation (1/6 ) using the third-order upstream advection scheme (U3H) forced only by seasonal forcing. The minimum salinity observed at 1000 m should be almost time invariant in the absence of spurious mixing. The top panel shows the topography, Panel (a) is the initial state, Panel (b) is the solution after 6 months and Panel (c) is the solution after 2 years where strong anomalies can be seen, due to spurious mixing. From Marchesiello et al. (2009) Schematic of the configuration used for all models Wavelength in km (ordinate) of the different internal wave modes (abscissa) for our particular scenario. The solid gray lines represent the different model resolution used in our experiments Snapshot (t = 6 days) of the cross-vertical section the zonal U baroclinic velocity for a subcritical wave regime for all three models: HYCOM (left panels),roms (right panels) and the MITgcm (right panels) and for all model resolutions experiments viii

10 3.4 Spatial wavelets of the surface zonal U baroclinic velocity for a subcritical wave regime for all three models: HYCOM (left panels), ROMS (middle panels) and the MITgcm and (right panels)for all model resolution experiments Snapshot (t = 6 days) of the cross-vertical section the zonal U baroclinic velocity for a critical wave regime for all three models: HYCOM (left panels), ROMS (right panels) and the MITgcm (right panels) and for all model resolution experiments Spatial wavelets of the surface zonal U baroclinic velocity for a critical wave regime for all three models: HYCOM (left panels), ROMS (middle panels) and the MITgcm (right panels) and for all model resolution experiments Snapshot (t = 6 days) of the cross-vertical section the zonal U baroclinic velocity for a supercritical wave regime for all three models: HYCOM (left panels), ROMS (right panels) and the MITgcm (right panels) and for all model resolution experiments Spatial wavelets of the surface zonal U baroclinic velocity for a supercritical wave regime for all three Models: HYCOM (left panels), ROMS (middle panels) and the MITgcm (right panels) and for all model resolution experiments The vertical velocity power spectrum, integrated over the whole depth, scaled by U 2 for all experiments: left columns are for x = 1.5 km, middle columns are for x = 5 km and right columns are for x = 10 km. The top raws is for a subcritical wave regime (ridge height is 200 m), the middle raws are for a critical wave regime (ridge height is 600 m) and the bottom raws are for a supercritical wave regime (ridge height is 1600 m). The red solid line is HYCOM run in a fully isopycnal fashion, the blue solid line is HYCOM run in a fully σ fashion, the black solid line is ROMS, and the dashed black solid line is the analytical solution The nondimensional energy fluxes (normalized by the energy flux computed from the linear theory) above the tip of the ridge for a finite number of modes for the critical (top panels) and supercritical (bottom panels) wave regimes and the two finest model resolutions ( x = 1.5 km on the left panels and x = 5 km on the right panels. The dash line is the analytical solution computed from the knife-edge theory and the solid color lines represent the energy fluxes simluated for each models: black is using HYCOM, red is using ROMS, and blue is using the MITgcm Initial temperature field for the lock exchange problem ix

11 4.2 Cross-vertical section of the temperature field for the lock exchange problem at t = 20 hr using ROMS for the U3H (left panels) and the MPDATA (right panels) advection schemes. The top panels are the COARSE experiments, the middle panels are the MEDIUM experiments, and the bottom panels are the FINE experiments Cross-vertical section of the temperature field for the lock exchange problem at t = 20 hr using the MITgcm for the third-order upwind DST (left panels) and the Superbee (right panels) advection schemes. The top panels are the COARSE experiments, the middle panels are the MEDIUM experiments, and the bottom panels are the FINE experiments Total volume (normalized) of the density classes volume distribution at t = 15 hr for ROMS (top panels) and the MITgcm (bottom panels) and for all the different tracer advection schemes. Solid gray lines are the two third-order schemes and solid black lines are schemes using the flux corrector methods. The abscissa represent the density classes (binning is 0.7 kg m 3 for the COARSE experiment, 0.08 kg m 3 for the MEDIUM experiment, 0.08 kg m 3 for the FINE experiment) and the ordinate is the normalized volume. The vertical solid black lines represent the location of the two initial prescribe densities Time series (hr) of volume-integrated density variance (kg 2 m 3 ) for the lock exchange scenario calculated for ROMS (top panels) and the MITgcm (bottom panels) and all advection schemes (solid gray lines are the high-order advection schemes; black solid lines are the schemes using flux corrector methods Time series (hr) of volume-integrated density variance for the lock exchange scenario calculated for the GETM with four different resolutions and a firstorder upstream advection scheme (left panel) and the Superbee advection scheme (right panel). From Burchard and Rennau (2008) Hovmöller-type diagram (time-density) of diapycnal diffusivity (κ eff ) using ROMS. The left panels use the U3H advection scheme and the right panels use the MPDATA advection scheme. The top panels are the COARSE experiment, the middle panels are the MEDIUM experiment, and the bottom panels are the FINE experiment. The solid black horizontal lines represent the two initial densities Hovmöller-type diagrams (time-density) of diapycnal diffusivity (κ eff ) using the MITgcm. The left panels use the third-order upwind DST advection scheme and the right panels use the Superbee advection scheme. The top panels are the COARSE experiment, the middle panels are the MEDIUM experiment, and the bottom panels are the FINE experiment. The solid black horizontal lines represent the two initial densities x

12 4.9 Time series of the volume-integrated numerical tracer variance decay for ROMS (top panels) and the MITgcm (bottom panels). The left panels are the COARSE experiment, the middle panels are the MEDIUM experiment, and the right panels are the FINE experiment Time series of the volume-integrated numerical tracer variance decay from the Burchard and Rennaud paper (2008) for four different resolutions and two different tracer advection schemes (left panel uses the U1H and the right panel uses the Superbee Normalized density class volume distribution (%) at t = 18 hr, for x = 1.5 km and zoom over a specific interval for: subcritical wave regime (top panels), critical wave regime (middle panels), supercritical wave regime (bottom panels) Normalized density class volume distribution (%)at t = 18 hr, for x = 5 km and zoom over a specific interval for: subcritical wave regime (top panels), critical wave regime (middle panels), supercritical wave regime (bottom panels) Normalized density class volume distribution (%) at t = 18 hr, for x = 10 km and zoom over a specific interval for: subcritical wave regime (top panels), critical wave regime (middle panels), supercritical wave regime (bottom panels) Tidally averaged (over tidal cycle 16) buoyancy frequency change (s 2 ) from initial distribution for all model resolutions and all wave regimes Tidally averaged (over tidal cycle 16) diapycnal diffusivity (m 2 /s, logarithmic scale) for all model resolutions and all wave regimes A.1 SSH (m) of the Gulf of Mexico for: the initial time (left panels), after 20 days with only nesting (no tidal forcing and no atmospheric forcing) (right panels). The top panels are for the 1/12 experiments and bottom panels are for the 1/24 experiments A.2 SSH (m) of the Gulf of Mexico for: nesting with atmospheric forcing (left panels), after 20 days with all forcing including tides (right panels). The top panels are for the 1/12 experiments and bottom panels are for the 1/24 experiments A.3 Tidal amplitude (colored) and phases (contoured) spatial maps for: M 2 using the 1/12 configuration (top left panel), M 2 using the 1/24 configuration (bottom left panel), O 1 using the 1/12 configuration (top left panel), and O 1 using the 1/24 configuration (bottom left panel) A.4 Tidal amplitudes (colored) and phases (contoured) spatial maps for: the GOT99 M 2, M 2 simulated with the Egbert tidal model, the GOT99 O 1, and O 1 simulated with the Egbert tidal model xi

13 A.5 Vertical velocity snapshots (t = 17 days) of two cross-vertical sections for: the coarser resolution (1/12 ) (top panels) and the finer resolution (1/24 ) A.6 Zoom of the vertical velocity snapshots (t = 17 days) at 22.7 N for both configurations B.1 The tracer flux method, some parts are from Griffies et al. (2000) xii

14 ABSTRACT Numerical ocean models have become powerful tools for providing a realistic view of the ocean state and for describing ocean processes that are difficult to observe. Recent improvements in model performance focus on simulating realistic ocean interior mixing rates, as ocean mixing is the main physical process that creates water masses and maintains their properties. Below the mixed layer, diapycnal mixing primarily arises from the breaking of internal waves, whose energy is largely supplied by winds and tides. This is particularly true in abyssal regions, where the barotropic tide interacts with rough topography and where high levels of diapycnal mixing have been recorded (e.g., the Hawaiian Archipelago). Many studies have discussed the representation of internal wave generation, propagation, and evolution in ocean numerical models. Expanding on these studies, this work seeks to better understand the representation of internal wave dynamics, energetics, and their associated mixing in several different classes of widely used ocean models (e.g., the HYbrid Coordinate Ocean Model, HYCOM; the Regional Ocean Modeling System, ROMS; and the MIT general circulation model, MITgcm). First, a multi-model study investigates the representation of internal waves for a wide spectrum of numerical choices, such as the horizontal and vertical resolution, the vertical coordinate, and the choice of the numerical advection scheme. Idealized configurations are compared to their corresponding analytical solutions. Some preliminary results of realistic baroclinic tidal simulations are shown for the Gulf of Mexico. Second, the spurious diapycnal mixing that exists in models with fixed vertical coordinates (i.e., geopotential and terrain following) is documented and quantified. This purely numerical error arises because, in fixed-coordinate models, the numerical framework cannot properly maintain the adiabatic properties of an advected water parcel. This unrealistic mixing of water masses can be a source of major error in both regional and global ocean models. We use the tracer flux method to compute the spurious diapycnal diffusivities for both a lock xiii

15 exchange scenario and a propagating internal wave field using all three models. Results for the lock exchange experiments are compared to the results of a recent study. Our results, obtained by using three different model classes, provide a comprehensive analysis of the impact of model resolution choice and numerical framework on the magnitude of the spurious diapycnal mixing and the representation of internal waves. xiv

16 CHAPTER 1 INTRODUCTION This research aims at providing knowledge in two related areas. First, it helps document the impact of numerical choices (e.g., model resolution, advection scheme choice) on the representation of the dynamics and energetics of internal waves in Oceanic General Circulation Models (hereafter, OGCMs). Second, it gives new estimates of the spurious (i.e., numerical) diapycnal mixing for different idealized simulations conducted with three widely used numerical ocean models. From a dynamical perspective, it is necessary to address these points to better understand the processes of conversion of tidal barotropic to baroclinic energy and internal wave generation and evolution. These processes are sensitive to many environmental factors (e.g., stratification, topography) that can have an important feedback on the internal wave field and local mixing. Better understanding of these processes lays the foundation for potentially deriving a physically based parameterization of internal wave breaking. Such has been the quest of tidal mixing research efforts for the past decade. Numerically, it is critical that OGCMs represent the mixing induced by the generation and propagation of internal waves. Indeed, the propagation of internal waves creates high vertical velocities and thus can locally generate large diapycnal diffusivity. In coarse resolution numerical simulations, models miss the dynamics and energetics of the internal waves because of their inability to resolve the higher vertical/horizontal wave modes. In addition, the mixing associated with internal propagation is generally crudely parameterized. A better knowledge of the impact of model numerics will provide additional benefit to tidal mixing research by fostering developments for the improvement of internal wave breaking parameterizations for ocean models. In this study, we focus on documenting and quantifying the implicit mixing generated 1

17 during two idealized scenarios: (1) a lock exchange scenario and (2) the propagation of internal waves, using fixed-coordinate ocean models (in this context, fixed-coordinates means vertical coordinate depths that are time-independent, except for possible small perturbations due to changes in the ocean surface elevation). The inability of fixed-coordinate models to preserve the adiabatic properties of a water parcel results in a spurious mixing that can overshadow the naturally occurring mixing. This causes the affected water masses to be irreversibly and unrealistically modified, which is a major drawback of fixed-coordinate ocean models. From a general and multidisciplinary perspective, internal waves strongly affect the dispersion and mixing of different tracers/substances as well as living organisms. The large vertical displacement of internal waves lead to the trapping of water masses, dissolved pollutants, planktonic organisms, and suspended sediments that can be transported (and thus dispersed) by the flow. The challenge of understanding internal wave dynamics, energetics, and their associated mixing is largely motivated by the difficulty and constraints of obtaining measurements. Data are sparse and expensive. Numerical simulations offer a useful alternative to explore these processes. Although several recent studies focus on tidal mixing and internal waves representation in numerical models, many questions remain unanswered. In this study, steps are taken to address these questions by conducting idealized experiments. The following section provides a background of previous studies, a statement of the remaining open questions, and the specific objectives addressed in this dissertation. 2

18 CHAPTER 2 BACKGROUND 2.1 Internal tides and Ocean Mixing Numerical models have become an indispensable tool for describing and understanding the dynamics of ocean thermohaline circulation. To give an accurate and realistic representation of the key phenomena of these dynamics, numerical models need to (1) produce water masses with realistic properties, (2) transport them correctly, and (3) maintain these water mass properties (Lee et al., 2002). The latter requirement is mainly met by operating the model with realistic mixing rates. Without this deep mixing, within a few thousand years the ocean would turn into a stagnant pool of cold and salty water with an equilibrium maintained by near-surface mixing (Munk and Wunsch, 1998). The tides and the winds are now accepted as the most (but not the only) significant sources of energy for such mixing. Recently, many studies have linked internal wave generation to tidal flow over topography in the deep ocean (Egbert and Ray, 2001, Figure 2.1; Merrifield and Holloway, 2002). In these regions, where the bottom is considered to be rough, measurements show large amounts of mixing (Polzin et al., 1997, Figure 2.2; Toole et al., 1997). This interaction between tidal flow and bathymetry results in a net transfer of energy from the barotropic (depth-averaged) tide to baroclinic (internal modes) tides. This baroclinic tidal energy must dissipate via turbulent processes. Munk and Wunsch (1998) suggested that this dissipation accounts for roughly half of the energy necessary to vertically mix the abyssal ocean and to maintain the strength of the global thermohaline circulation. It also plays a crucial role in transferring momentum from the surface to the deep ocean (Figure 2.3). However, internal wave observations are sparse and the lack of observations is a major constraint in obtaining a detailed understanding of baroclinic tides 3

19 Figure 2.1: Dissipation rates derived from satellite altimetry sea-surface elevation from Egbert and Ray (2000). Enhanced dissipation is colored in red and attenuated dissipation is colored in blue. Figure 2.2: Turbulent diapycnal eddy diffusivity in the Brazil basin from Polzin et al. (1997) 4

20 Figure 2.3: Energy budget proposed by Munk and Wunsch (1998) and their associated mixing. As a result, it is necessary to rely on numerical modeling. While the theory of internal waves generated by interaction of a barotropic flow with topography is well established (e.g. Baines, 1982; Bell, 1975; Hibiya, 1986), both the representation and the mixing induced by their propagation in OGCMs is still not fully understood (Di Lorenzo et al., 2006). In this chapter, we first provide a detailed background on the extensive literature documenting the representation of internal wave dynamics and energetics in OGCMs. Second, we present a literature review of the spurious diapycnal mixing in fixed coordinate ocean models. Finally, we describe an overview of the research, clearly state the remaining open issues that have not been answered by the previous studies, and describe the scientific questions that this dissertation intends to address. 5

21 2.2 Internal wave representation in OGCMs In oceanic numerical models, the commonly made approximations (e.g., hydrostatic and Boussinesq), the choice of horizontal and vertical resolution, vertical discretization approach, and the numerical methods employed to solve the governing equations make it impossible to perfectly resolve the scales (few meters to a kilometer) and the dynamics of internal waves. It has been demonstrated that the choice of the horizontal and vertical grid spacing of the model influences the representation of these waves (Robertson, 2006; Jachec, 2007, Figure 2.4). However, only a narrow range of values for horizontal grid spacing ( x = y = 1 km to 5 km) and vertical level grid spacing has been considered (25 to 40 levels) generally using a unique model for each study. None of these studies has investigated the simulation of internal waves using varying spatial resolutions (from a few meters to hundreds of kilometers) for different classes of models. As previously mentioned, direct observation of the baroclinic tidal field is prohibitively expensive, so numerical ocean models provide a useful alternative. The model results need to be accurate to rigorously assess the connection between tidal mixing and internal wave dynamics. Achieving this accuracy partly relies on the choice of both the grid resolution and the advection scheme. A poor choice of grid resolution can lead to unrealistically low velocities and other scalar quantities. This can lead to inaccurate estimates of the energy wave field and dissipation rate due to internal waves. Many studies have simulated baroclinic tides for idealized and realistic applications, using various model grid spacing (always using the hydrostatic approximation) (e.g., Petruncio (1996, Monterey Bay, x = y = 1 km), Holloway (1996, 2001, Northern Australian slope, x = y = 4 km), Cummins and Oey (1997, British Columbia, x = y = 4 km), Robertson (2005, 2006, Ross Sea and Fieberling Guyot, x = y = 1 km to 5 km). All of these studies showed that their respective model resolutions were unable to accurately reproduce tidal velocities and that additional evaluation of the model performances was needed. It is evident that the model performances heavily relied on the model grid spacing and other considerations, such as the advection scheme or vertical mixing parameterizations. So far, the required resolution for accurate performance has been determined by the available computational and storage resources. Internal tide wavelengths vary with latitude, depth, stratification, and the tidal frequency 6

22 Figure 2.4: Baroclinic velocities simulated with the Stanford Unstructured Nonhydrostatic Terrain-following Adaptative Navier-Stokes Simulator (SUNTANS) model for a transect across Monterey Bay with a coarse model resolution ( 3 km) and a fine model resolution ( 0.3 km) from Jachec (2007). considered. For example, a typical mode one internal wave length for water depth shallower than 2000 m is generally between 20 and 50 km. For diurnal frequencies, wavelengths of this scale would be found at water depth shallower than 1000 m. The studies cited above all used horizontal grid spacing of less than 7 km and thus were able to capture only the semidiurnal tidal constituents in shallow water. Robertson (2006) concluded that typical resolutions employed for regional applications (e.g., 4 km) were able to give only qualitative estimates of where internal waves are generated. Robertson (2006) also concluded that the choice of vertical mixing parameterizations had little impact on the tidal velocity and internal tide generation compared to the model grid spacing. It is worthwhile to mention that only one study, Ezer et al. (2002) compared two models (Princeton Ocean Model, POM; and the Regional Ocean Modeling System, ROMS) with the same vertical coordinate system for several realistic applications. No study has intended to document the impact of this vertical coordinate system by carrying out similar simulations with different model classes. This study addresses exactly this point. 7

23 2.3 Spurious diapycnal mixing in fixed coordinate ocean models The numerical discretizations of the advection terms are either dispersive (i.e., they lead to numerically induced oscillations) or diffusive (i.e., they introduce numerical mixing to suppress the numerical dispersion). Most models are now equipped with and use highorder advection schemes (e.g., ROMS; the HYbrid Coordinate Ocean Model, HYCOM; the MITgcm). Indeed, the transport of biogeochemical tracers relies on non-dispersive advection schemes on all scales, mostly because strong gradients of these tracers would lead to (unphysical) negative concentrations. In numerical models, the adiabatic property of advection is challenging to maintain unless it is explicitly built into the model s algorithmic framework (Griffies, 2000), such as in isopycnic (vertical coordinates that follow constant potential density surfaces) models. In an isopycnal model, the diapycnal component is calculated by a vertical mixing parameterization (e.g., K-Profile, Mellor-Yamada, Kraus- Turner). This controlled diapycnal mixing represents an important advantage for isopycnal models over fixed coordinate models, such as z-level (geopotential-following) and sigma-level (terrain-following) models. Strong horizontal mixing along iso-sigma surfaces can lead to strong spurious diapycnal fluxes of heat and salt (and hence density) when it is applied within regions with sloping isopycnal or sloping sigma levels (Figure 2.5). This numerically induced diapycnal mixing component arises regardless of how the mixing term in the governing equations is formulated because of the non-conservation of the adiabatic properties of these water masses. This numerically induced diapycnal mixing is likely to overshadow the common vertical mixing that occurs naturally (Chassignet et al., 1996) and can potentially lead to serious errors when considering global ocean circulation simulations (Bryan, 1987) or long-term climate simulations (Toggweiler, 1994; Marchesiello et al., 2009, Figure 2.6). Numerous studies have assessed both performance and the amount of numerically induced mixing in various advection schemes by conducting idealized experiments (Haidvogel and Beckmann, 1999) or by comparing identical model simulations, changing only the advection scheme (Gerdes et al., 1991; Matear, 2001). Only a few studies have been conducted to document, investigate, and quantify this spurious diapycnal mixing (Griffies, 2000, Lee et al., 2002, Marchesiello et al., 2009). The results of these studies showed that, to some extent, 8

24 Figure 2.5: West-East Pacific vertical cross-section showing isopycnals (red contours) and the sigma levels (dash lines). Regions of interest are areas where the sigma levels intersect with isopycnals. From Florian Lemarié (personal communication, 2010). the spurious mixing can be minimized for a specific configuration by carefully choosing the right advection scheme and model resolution. Griffies et al. (2000) developed a method for diagnosing the effective diapycnal mixing (i.e., the sum of the physical and numerical mixing) based on the earlier work of Winters and D Asaro (1996). This method is named the tracer flux method. The result of their diagnosis is a vertical profile of horizontally averaged diapycnal diffusivity for each model time step. Marshall et al. (2006) used a numerical analysis of the lengthening of idealized tracer contours, strained by the geostrophic flow, to derive effective diffusivities at the surface due to the mesoscale activity. In a realistic simulation of the Faroe Bank Channel overflow, Riemenscheider and Legg (2007) did not prescribe any physical mixing and used a Total Variation Diminishing (TVD) advection scheme with a low level of numerical diffusion to study the effective mixing. Using a horizontal grid spacing of 2.5 km and a vertical grid spacing of 25 m, they found that the magnitude of the numerically induced mixing 9

25 Figure 2.6: Two-year evolution of the salinity field at 1000 m in a ROMS simulation (1/6 ) using the third-order upstream advection scheme (U3H) forced only by seasonal forcing. The minimum salinity observed at 1000 m should be almost time invariant in the absence of spurious mixing. The top panel shows the topography, Panel (a) is the initial state, Panel (b) is the solution after 6 months and Panel (c) is the solution after 2 years where strong anomalies can be seen, due to spurious mixing. From Marchesiello et al. (2009) 10

26 is equivalent to the naturally occurring mixing. If coarser resolutions are used, then the magnitude of the spurious diapycnal mixing is actually greater than that occurring in these highly adiabatic (weak mixing) regions. Morales Maqueda and Holloway (2006) computed the tracer variance decay due to the numerical mixing. The numerical diffusivities were then estimated by multiplying the diffusivity of the considered advection scheme (first-order upstream for their study) with the quotient of the variance decay induced by a secondorder advection scheme and the variance that would have been removed by the considered advection scheme. In a more recent study, Marschesiello et al. (2009) showed a dimensional analysis of the numerical mixing depending on the model resolution, as well as a clear demonstration that the diffusion terms for the third-order upstream scheme in ROMS cause the numerically induced mixing. Burchard and Rennau (2009) relied on a straightforward technique that consists of calculating the rate of change between the advected square of the tracer and the square of the advected tracer (so the diffusion part is isolated and can be explicitly quantified). All of these techniques have constraints. For example, many of the recent techniques are limited to a single advection scheme. Also, the tracer flux method computes only horizontally averaged diapycnal diffusivities. In this dissertation, we will focus on applying the robust tracer flux method from Griffies et al. (2000). 2.4 Research Overview This study aims to provide a better understanding of the different mechanisms that can affect the representation of internal waves in numerical models, as well as to document and quantify the numerically induced mixing by the propagation of these waves in fixed coordinate ocean models. The study focuses particularly on the following outstanding questions: 1. How well do OGCMs represent internal wave dynamics and energetics as a function of model grid spacing and vertical coordinate system (i.e., model class)? 2. What is the magnitude of the spurious diapycnal mixing for a simple gravitational adjustment scenario in the model configurations applied for this study? 3. Is the impact of the model resolution and/or the choice of the advection scheme on the magnitude of the spurious mixing dynamically important? 11

27 4. What is the magnitude of the spurious diapycnal mixing induced by the propagation of these waves in fixed coordinate ocean models? 5. Does increasing the model horizontal and/or vertical resolution alleviate this numerically induced diapycnal mixing? The purpose of this study is to document the performance of several OGCM configurations with different numerical frameworks for representing the internal wave field as well as to document when the level of diapycnal spurious mixing becomes too large (i.e., comparable to the ocean interior diapycnal mixing). The questions cited above are answered by comparing the results of three numerical ocean models for idealized simulations to the results of their reciprocal analytical solutions. The results of this study are expected to be innovative and of broad interest for the ocean modeling and the tidal mixing research communities as they are linked to recent efforts to find (i.e., derive) a physically based parameterization of mixing near rough topographic features. These results are also expected to add to our understanding of internal wave properties, dynamic, energetic, and associated mixing in numerical ocean models. This understanding is particularly important for the large-scale modeling community. In Chapter 3, the impact of the model horizontal and vertical resolution choice on representing internal waves is described. In Chapter 4, the numerically induced mixing in fixed coordinate models is quantified for different numerical configurations. Finally, conclusions and future of this work are addressed in Chapter 5. 12

28 CHAPTER 3 A MULTI-MODEL COMPARISON OF INTERNAL WAVE GENERATION AND PROPAGATION The aim of this chapter is to investigate the internal wave field representation in three Oceanic General Numerical Models (hereafter, OGCMs) as a function of vertical and horizontal resolution. A common idealized numerical study of a barotropic tide interacting with an oceanic ridge is configured for the HYbrid Coordinate Ocean Models (HYCOM), the Regional Ocean Modeling System (ROMS) and the MIT general circulation model (MITgcm). The impact of the model grid spacing on the representation of the internal wave dynamics and energetics is investigated by running the same experiments with varying horizontal and vertical resolutions. The results obtained using the three numerical models are compared to analytical solutions. 3.1 Introduction The generation of internal waves has recently gained considerable attention in the oceanography community for several reasons. First, observations have shown that the amplitude of these waves might be larger than expected in certain locations (e.g., the Hawaiian ridge, Ray and Mitchum, 1997; Rudnick et al., 2003). Second, in conjunction with rough topography, internal waves are considered a key component in providing the necessary energy to vertically mix the deep ocean. The interaction of the barotropic tide in the stratified ocean with the bottom topography generates internal tides and results in a net transfer of energy from the barotropic (i.e., depth-averaged) mode to the baroclinic mode. This tidal conversion process has been the focus of numerous analytical and numerical studies (Bell, 1975; Balmforth et al., 2002; Llewelyn Smith and Young, 2002; Holloway and Merrifield, 1999; Munroe and Lamb, 2005; St Laurent, 2003; Pétrélis et al., 2003). The baroclinic tidal energy eventually dissipates 13

29 via turbulent mixing. This mixing is believed to account for half of the energy necessary to maintain the strength of the thermohaline circulation. Using numerical simulations, Simmons et al. (2004) showed that the global ocean circulation and thermohaline circulation are sensitive to this tidally driven mixing. The wavelength of internal waves is short (a few kilometers) compared to the basin scale and the barotropic tide wavelength. Because of this short wavelength, in situ data must be sampled at a high frequency. This is an expensive method for gaining insight into the physics of internal waves. As a complementary approach, ocean numerical models are used to fill in the gaps in data. However, recent studies comparing available in situ data to model output demonstrate that many issues remain unresolved. Documenting the performance of all model classes (i.e., isopycnic, geopotential, and terrain-following) on simulating internal waves will provide much-needed information for resolving these issues. In this chapter, the ability of several numerical models to accurately reproduce internal wave field dynamics and energetics is investigated and further documented, expanding on recent efforts. The focus is on assessing the impact of both the model resolution (horizontal and vertical) and vertical coordinate system choice on the representation of internal waves. All models are subject to specific numerical errors. For example, the numerical timestepping algorithm (decoupling the fast barotropic mode to the slow baroclinic mode) used in most OGCMs introduces an error in the energy conservation properties for both the isopycnal and terrain-following models. This error can lead to a 25% underestimate of the nominal baroclinic conversion. Errors also arise because of the discretization of the pressure gradient for nongeopotential model classes although these errors are thought to be minimal (Hallberg and Rhines, 1996). Because the wavelength of internal waves is short compared to the barotropic tidal wavelength (a few meters compared to a hundred meters), the model grid size is a key factor in determining the models performance in representing internal waves. The low internal wave mode wavelengths are generally between 20 km and 50 km (depending on stratification, tidal frequency, latitude, and water depth). Therefore, it is expected that models with high enough resolution (< 10 km) can capture these low modes and thus are able to represent most of the energetic field. However, at coarser resolution, the models are not resolving the higher wave modes responsible for the turbulent response. The choice of a finite number of vertical levels is also problematic in resolving all the higher vertical wave modes of the baroclinic 14

30 wave. Recent studies have investigated the impact of the model grid resolution on the representation of the internal waves using regional models (Robertson, 2005; Di Lorenzo et al., 2006; Jachec et al., 2007; Bernsten et al., 2008). To properly simulate internal waves, one needs an accurate representation of the bottom topography as well as a sufficient model grid resolution (Jachec et al., 2007). This sufficient model grid spacing has not yet been determined. These studies investigated only a small spectrum of model resolution choices. Di Lorenzo et al. (2006) and Zaron and Egbert (2006) showed that errors due to topography misrepresentation (e.g., topography smoothing methods for sigma-level models to ensure model stability or stepwise type topography for geopotential models) or inadequate model vertical discretization can lead to a 50% reduction in the transfer of barotropic energy to baroclinic energy. Underestimating the tidal conversion process could be a potential source of large errors in the representation of the deep ocean dynamics and processes. By directly comparing numerical results to observations, Robertson (2006) demonstrated that the magnitude and direction of the baroclinic velocity are accurately reproduced when using a 1-km model grid resolution. Conversely, a model resolution of 4 km provides only qualitative estimates of the tidal conversion mechanism. Robertson also noted that most of the errors can be attributed to topographic errors. Recently, Bernsten et al. (2008) stated that internal waves are present even in a coarse model grid. However, internal wave wavelengths, periods, and amplitudes are strongly affected when the grid spacing is reduced. As a result, the waves are inaccurately simulated. The authors also observed that the choice of grid spacing becomes increasingly important as the Froude number (baroclinic response) increases. It is evident that the model s performance is dependent on the grid resolution. The research presented in this chapter is motivated by (1) the lack of documentation and knowledge of the strengths and weaknesses of the three model classes on resolving the internal wave field and (2) the unresolved issue of internal wave representation in OGCMs when the model grid spacing is varied. 15

31 3.2 Numerical Models Common configuration of the models To generate and propagate an internal wave field, a semidiurnal (i.e., tidal period is ω = hr) barotropic tidal flow is prescribed at the western boundary of a 1200-km-long, flat channel with a Gaussian ridge centered at 600 km (Figure 3.1). The model is set up with a nonrotating (i.e., f = 0 s 1 ), stratified ocean (N = s 1 ), following the numerical experiments described in Di Lorenzo et al. (2006). The interaction between the barotropic tide and the topography generates perturbations to isopycnals via the creation of vertical velocities and thus generates internal waves. These internal waves are freely propagating as the condition f < ω < N is satisfied. A radiation condition for the barotropic components is applied to avoid boundary reflection and to alleviate energy accumulation in the basin. The simulations are run for eight days. In this time frame of eight days, the baroclinic component has not yet reached the boundaries and thus there is no need to prescribe a sponge layer or a baroclinic open boundary radiation condition. To isolate the tidal conversion process and remove other physical sources of dissipation, the models are run with both lateral and bottom free-slip (i.e., no bottom drag) conditions. Temperature is the only active tracer in these configurations (salinity is passive). All tracer diffusion parameters are set to zero; the viscosity parameter is set to the minimum value for which the model remains stable. The model uses a linear equation of state to remove complex processes such as cabbeling and thermobaricity. No explicit mixing is prescribed. The horizontal and vertical resolutions and vertical coordinates used in the numerical experiments are given in Table 3.1. Each experiment is run for the three types of wave regimes: subcritical (linear theory of internal wave can be applied), critical, and supercritical (highly nonlinear wave regime, linear theory of internal wave cannot be applied). These wave regimes (γ) are defined by the ratio of the topographic slope and the internal wave beam characteristics (s): subcritical if γ 1 γ = (dh/dx)/s = critical if γ = 1 supercritical if γ 1 (3.1) where s is given by: s = [(ω 2 f 2 )/(N 2 ω 2 )] 1/2 (3.2) 16

32 Figure 3.1: Schematic of the configuration used for all models Table 3.1: Description of the experiments conducted. x (km) is the model horizontal resolution, Npt is the number of points defining the ridge, N is the number of levels, and z (m) the model vertical resolution. These experiments are repeated with varying the ridge height to change the wave regime (i.e., subcritical, critical or supercritical) Experiment Model, vertical coordinate x (km) Npt N z (m) 1 HYCOM, ρ HYCOM, ρ HYCOM, ρ ROMS, σ ROMS, σ ROMS, σ MITgcm, z MITgcm, z MITgcm, z In our experiments, no Coriolis force (f) is prescribed and the stratification and the forcing frequency (ω) are both invariant. Thus, the wave regime is a factor only of the ridge height (h). The wave regime is subcritical when the ridge height is 200 m, critical when the ridge height is 600 m, and supercritical when the ridge height is 1600 m. 17

33 3.2.2 Characteristics of the models The three numerical models introduced in this section represent some of the most commonly used models in the oceanographic community and offer a wide variety of numerical choices presently available. Each model solves the hydrostatic primitive equations and is classified with respect to its vertical coordinate system. The following subsection describes the salient aspects (e.g., numerical advection schemes) of the models and their horizontal and vertical discretizations. The HYCOM model (Bleck, 2002; Chassignet et al., 2003; Halliwell, 2004) is a hybrid (i.e., linear combination of two or more conventional vertical coordinates) ocean numerical model. The advantage of using a hybrid model resides in its ability to simplify the numerical implementation of several physical processes (e.g., mixed layer detrainment, convective adjustment, sea ice modeling) without harming the model basics. It also allows for an efficient vertical resolution throughout the water column. In this study, HYCOM is run in a fully isopycnic fashion. The isopycnic coordinate is used to avoid spurious mixing arising from the Eulerian vertical coordinates (i.e., the diapycnal mixing is explicitly prescribed). HYCOM uses a leap-frog, second-order centered scheme for the momentum advection scheme and a flux corrector transport advection scheme for the tracers. The model is run with no explicit viscosity. ROMS is a terrain-following model commonly used for coastal applications. In this study, ROMS uses the third-order upstream-biased advection scheme (U3H hereafter) for the momentum equations (Shchepetkin and McWilliams, 2005). The same advection scheme is used for the tracer advection scheme. The U3H is known to potentially generate large levels of spurious mixing (Marchesiello et al., 2009). The vertical advection scheme is the fourthorder Akima scheme. These advection schemes are the most commonly used by the ROMS community. All vertical advection schemes in ROMS are implemented to be nondiffusive. The model is run with no explicit viscosity although a viscosity is inherently implemented in the advection schemes. Smoothing the bathymetry is a common method for avoiding spurious jets created by pressure gradient errors in a terrain-following model. As noted in the introduction to this chapter, this technique can reduce the tidal conversion by half, so in this study there is no smoothing method applied to the topography. This results in topographic parameters (i.e., Haney number and Beckmann and Haidvogel number) that are 18

34 considered too large for the critical and supercritical wave regimes (e.g., 20 for the Haney number). Therefore, some of the experiments using ROMS will be subject to spurious jets. The MITgcm (Marshall et al., 1997) is a geopotential (z-) model using the shaved cell capability to resolve complicated geometries and topography. The geopotential models have been widely used because they are the simplest numerical discretization. These models also have an accurate representation of the equation of state and the pressure gradient. In this study, the MITgcm is run with a quasi-second order, Adams-Bashforth scheme and a thirdorder, direct spatial time advection scheme for the tracers. The MITgcm is run with common viscosity values to provide smooth solutions. 3.3 Results In this section, the results obtained by the three models are described. An internal wave wavelength of mode n is given by the relation NH/(2n(ω/N)), where N is the buoyancy frequency, H the maximum depth of the basin, ω the forcing frequency, and n the mode number. Figure 3.2 shows the modes resolved by the different model resolutions employed in this study (i.e., using the specific configuration described in Figure 3.1). It is clear that when x = 10 km, only modes 1 and 2 are captured; at x = 5 km, the first four modes are resolved by the model. However, the finer resolution ( x = 1.5 km) is able to capture more than the ten internal wave horizontal modes which account for more than 95% of the total energy in the internal wave field (St Laurent et al., 2003). A configuration conducted with x = 0.75 km (not shown here) shows only a slight improvement for both the dynamics and energetics of the wave. Thus, for computational purposes the finest experiments are conducted with x = 1.5 km and are assumed to be the best solution. The baroclinic response for all experiments is investigated by looking at the magnitude of the generated zonal baroclinic velocities in the basin and the maximum amplitude of the internal waves generated. A spatial wavelet method is used to quantitatively assess the wave dynamics. The surface baroclinic velocity for a particular time (t = 6 days) is transformed using the Morlet wavelet transformation, defined as: g(x, τ,a) = Ce (x τ)2 2 cos(5( x τ )) (3.3) a 19

35 Figure 3.2: Wavelength in km (ordinate) of the different internal wave modes (abscissa) for our particular scenario. The solid gray lines represent the different model resolution used in our experiments. where C is the normalization constant, a the wavelet coefficient (dilatation parameter), τ the translation parameter and x the distance from the western boundary. The resulting wavelet map has a distance from the ridge in the x-direction (in km) and a wavelength (in km) in the y-direction. Here, the baroclinic response for a subcritical wave regime (i.e., ridge height of 200 m) is described. As a qualitative assessment, the model internal wave representation is compared to an analytical solution developed by Khatiwala (2003). As shown in Figure 3.3, the three model simulations are similar and present the same baroclinic response. Indeed, the baroclinic velocity magnitudes are comparable and represent approximately 50% of the initial forcing. These magnitudes are also well conserved when x = 5 km, but decrease when x = 10 km away from the ridge in both ROMS and the MITgcm. This can be explained by the viscosity present in these two models that dissipates energy as the internal waves propagate. Most of the baroclinic velocity magnitude differences are observed at the tip of the ridge when the model grid spacing is varied. This is mainly due to the higher wave modes that are not resolved when using coarser horizontal and vertical resolutions. When x = 10 km, only the first two modes are resolved. The two resolved modes are not enough to generate an internal wave beam structure, which arises when higher wave modes 20

36 are present. The velocity extremes at the surface and bottom occur at different distances for all three models, demonstrating that the simulated wavelengths and wave phases are different. The maximum amplitude of the internal waves (i.e., maximum isopycnal vertical displacement) for this scenario, observed at the tip of the ridge, is approximately 10 m for all three models. The internal wave amplitudes are not sensitive to the grid resolution; the coarser resolution experiments also have a 10-m amplitude. To clearly show the simulated wavelengths and their locations, a spatial wavelet analysis (given by equation 3.3 and describe in above paragraph) of the surface baroclinic velocities is shown in Figure 3.4. For the fine resolution, HYCOM, ROMS, and the MITgcm mode-1 wavelength is close to the analytical solution (black solid line). ROMS has the shortest wavelength ( 54 km), and HYCOM and the MITgcm both have a 56 km wavelength; the analytical solution gives a 56.8 km wavelength. The difference between the simulated and analytical wavelengths (i.e., at most 2.8 km) represents less than two model grid points. The spatial wavelet analysis also shows that, as the resolution gets coarser and coarser, the loss of energy is large and can reach a 50% underestimation error in the simulated baroclinic velocity, results that are consistent with the findings of Di Lorenzo et al. (2006). An interesting finding is the difference of the simulated wavelength at the tip of the ridge. This is attributed to the different vertical coordinate system; the z- and isopycnic models have fewer levels above the ridge and thus resolve fewer vertical modes. For a 600-m-high ridge, the wave regime is considered critical. The magnitude of the baroclinic response for all three models is equivalent to the magnitude of the initial forcing. Figure 3.5 shows that all three models have a similar internal wave vertical structure for x = 1.5 km, but not for coarser resolutions. For the two coarser resolutions ( x = 5 km and x = 10 km), the topographic parameters for ROMS are violated. Thus, strong spurious jets are likely to appear due to the pressure gradient error near the ridge. The slope of the sigma levels near the ridge is too steep and a topography smoothing would be required to avoid such error. All three models simulate the same internal wave field horizontal and vertical structure away from the ridge. HYCOM conserves the qualitative features of the wave field at coarser resolution; however, spurious noises near the ridge in ROMS overshadow the internal wave field. The MITgcm simulates an internal wave structure similar to that in HYCOM. The maximum internal wave amplitude for this scenario is observed at the tip of the ridge and 21

37 22 Figure 3.3: Snapshot (t = 6 days) of the cross-vertical section the zonal U baroclinic velocity for a subcritical wave regime for all three models: HYCOM (left panels),roms (right panels) and the MITgcm (right panels) and for all model resolutions experiments

38 23 Figure 3.4: Spatial wavelets of the surface zonal U baroclinic velocity for a subcritical wave regime for all three models: HYCOM (left panels), ROMS (middle panels) and the MITgcm and (right panels)for all model resolution experiments

39 24 Figure 3.5: Snapshot (t = 6 days) of the cross-vertical section the zonal U baroclinic velocity for a critical wave regime for all three models: HYCOM (left panels), ROMS (right panels) and the MITgcm (right panels) and for all model resolution experiments

40 is approximately 40 m for all three models. Figure 3.6 shows the surface baroclinic velocity spatial wavelet analysis for this critical wave regime, confirming that both the wave and phase speed values are close to the analytical values for all three models and that the simulated wavelengths are quite similar to the subcritical wave regime wavelengths. At the tip of the ridge, the simulated wavelengths between HYCOM and the MITgcm are similar; ROMS has a shorter simulated mode-1 wavelength because of the vertical coordinate, which is able to capture the higher wave modes. HYCOM and the MITgcm have the same number of levels above the tip of the ridge (which is less than what is initially prescribed, e.g., 35 levels for a ridge height of 600 m and for the finer resolution). ROMS conserves all sigma levels above the tip of the ride (e.g., 50 levels for the finer resolution). For the supercritical wave regime (i.e., ridge height of 1600 km), the baroclinic response for all three models is about three times the initial forcing. For the fine resolution ( x = 1.5 km) all three models have similar simulated horizontal and vertical internal wave structure. The magnitude of the baroclinic response is weaker in ROMS and the MITgcm because of the viscosity present in the model. HYCOM conserves the structure of the internal wave field at coarser resolutions. When x = 5 km, the ROMS simulation has a strong pressure gradient error and the internal wave field near the ridge is corrupted. The MITgcm simulation has a noisy solution at the tip of the ridge, but away from the ridge, the model conserves the internal wave structure. When coarser resolutions are used, energy is redistributed in the lower modes because higher modes are unresolved by the model horizontal and vertical grid, and the magnitude of the velocities in the internal wave beam is largely decreased. When x = 10 km, HYCOM still simulates accurately the locations of the velocity extremes (at the surface and at the bottom). Approximately half of the magnitude of the higher resolution extremes is captured at coarser resolution, and the energy in the basin is greatly reduced. The ROMS simulation also has a strong pressure gradient error and the spurious jets created overwhelm the internal wave response. The MITgcm exhibits strong velocities at the tip of the ridge. The viscosity parameters might be too weak to get a smooth solution. At coarser resolution, the MITgcm still simulates accurately the locations of the extremes baroclinic velocities. The maximum amplitude of the internal waves for the supercritical scenario is approximately 100 m. This magnitude is not affected by the coarser resolution choice since for x = 10 km, the inetrnal wave maximum amplitude is 25

41 26 Figure 3.6: Spatial wavelets of the surface zonal U baroclinic velocity for a critical wave regime for all three models: HYCOM (left panels), ROMS (middle panels) and the MITgcm (right panels) and for all model resolution experiments

42 approximately 90 m for all three models. Figure 3.8 shows the spatial wavelet analysis for the supercritical wave regime. All three models accurately simulate the mode-1 wave away from the ridge. At the tip of the ridge, all three models resolve the same wavelength, which is a different result from that found with the previous wave regimes. At coarser resolution, surface velocities are much weaker because of the lesser modes resolved by the model. For the coarser resolution, HYCOM and the MITgcm still simulate well the mode-1 wave; however, the ROMS simulation has pressure gradient errors. The vertical velocity frequency spectra taken at 10 km away from the ridge for all simulations are shown in Figure 3.9 scaled by U 2 following Legg and Huijts (2006). Our time sampling is sufficient to resolve up to a frequency of 8ω although even higher frequency can be generated (N/ω 15, so that propagating waves of frequency up to 15ω is possible). The power is shown on a logarithmic scale so that both large and small orders of magnitude are visible; the frequency is on a linear scale to highlight the different harmonics. The diagnosed spectra from the subcritical wave regime show a linear response (the dominant energy is in the frequency of the primary forcing) that compares well with the linear theory for all three models. For the critical and supercritical wave regimes the diagnosed spectra have more energy onto the higher harmonics for all three models. The model resolution is clearly affecting the magnitude of the energy contained in each harmonic. The finer resolution ( x = 1.5 km) compares well with the analytical solution for the subcritical wave regime. As the model resolution becomes coarser, the energy is well conserved for the first harmonic but quickly decreases for the higher harmonics. For all harmonics, the drop in the energy magnitude is more visible within ROMS and the MITgcm than within the HYCOM run in a fully isopycnic fashion. The energy fluxes above the tip of the ridge are computed for all simulations using the knife-edge theory derived in St. Laurent et al. (2003) and compare to their analogous analytical solution. The model results provide output of the pressure levels, thus the horizontal and vertical modal structures of the internal wave velocity field can be easily determined. These baroclinic velocities may be written as: u l = N n=1 a nφ n (l), where n is the mode number, l denote the levels, u the baroclinic velocity, and Φ the modal structure of the velocity. Each a n coefficient, which directly relate to the modal energy flux, are computed using a least square fit method. Normalizing the depth-integrated energy fluxes normalized 27

43 28 Figure 3.7: Snapshot (t = 6 days) of the cross-vertical section the zonal U baroclinic velocity for a supercritical wave regime for all three models: HYCOM (left panels), ROMS (right panels) and the MITgcm (right panels) and for all model resolution experiments

44 29 Figure 3.8: Spatial wavelets of the surface zonal U baroclinic velocity for a supercritical wave regime for all three Models: HYCOM (left panels), ROMS (middle panels) and the MITgcm (right panels) and for all model resolution experiments

45 Figure 3.9: The vertical velocity power spectrum, integrated over the whole depth, scaled by U 2 for all experiments: left columns are for x = 1.5 km, middle columns are for x = 5 km and right columns are for x = 10 km. The top raws is for a subcritical wave regime (ridge height is 200 m), the middle raws are for a critical wave regime (ridge height is 600 m) and the bottom raws are for a supercritical wave regime (ridge height is 1600 m). The red solid line is HYCOM run in a fully isopycnal fashion, the blue solid line is HYCOM run in a fully σ fashion, the black solid line is ROMS, and the dashed black solid line is the analytical solution. by the energy fluxes from linear theory gives: F knife = F 0 N n=1 n 1 a 2 n, where F 0 is defined as: F 0 = 1 2π ρ(n2 ω 2 )(ω 2 f 2 ) ω U 2 0H 2. The nondimensional energy fluxes at the tip of the ridge are computed for all models, for critical and supercritical wave regimes, for the two finest model resolutions, and also for the analytical solution (Figure 3.10). The results obtained with the 10 km model resolutions are not included because only the two first modes are present, results obtained at higher modes are not significant because these modes are not resolved. At the highest resolution (i.e., x = 1.5 km), all models significantly capture the energy in the lowest mode and agree well with the analytical solution. The differences can be attributed to non-linear effects that are not included in the linear theory, and to the fact that models only resolve a finite number of horizontal and vertical modes. ROMS sufficiently captures the energy even at the highest modes, since its vertical coordinate allows for more modes to be resolved above the ridge. Results from both HYCOM and the MITgcm run at coarser resolution reveal that higher modes diverge strongly from the analytical solution, due to the limited number 30

46 of levels above the tip of the ridge. In that respect, energy fluxes obtained with both the HYCOM model and the MITgcm are very similar. For all models, the energy contained in the unresolved higher modes, due to the horizontal and vertical grid spacing, needs to be redistributed in the lowest resolved mode. This redistribution of energy in the resolved modes remains an open issue and a challenge to understand. For example, the HYCOM model and the MITgcm have more energy in modes four to six than it should. This energy drops near mode ten, which corresponds to the maximum number of modes that these models can resolve. The redistribution of energy of these non-resolved higher modes into the lower modes could explain why these models have too much energy. This can be confirmed, because ROMS does not show the same behavior due to its better vertical discretization. When x = 5 km, all three models show a drop of energy at mode three, since it approaches what the models are able to resolve in the horizontal direction. 3.4 Summary and discussion In this chapter, the importance of the model resolution choice on the representation of internal waves in an idealized scenario has been demonstrated and quantified. The dynamics of the internal waves are generally well simulated in all three models. The baroclinic velocities generated at the ridge for the subcritical simulations compare well to the analytical solution derived from the linear theory for all model resolutions. The mode one wavelength is well resolved in all models and for all grid spacing choices. A spatial wavelet analysis shows that varying the model resolution does not significantly affect the simulation of the wavelength for HYCOM and the MITgcm. When using coarser resolution, the internal wave propagation is modified in ROMS due to pressure gradient errors and the spurious diapycnal mixing induced by the internal wave propagation (topic described in the next chapter). The internal wave field is contaminated by spurious jets and the baroclinic response is greater than what is shown with higher resolutions. The power spectrums show that the response at the fundamental frequency (i.e., frequency of the forcing) is well captured for all model simulations and that the supercritical wave regimes also have energy in other harmonics due to non-linear processes. Most of the differences between models are seen when investigating the energy above 31

47 Figure 3.10: The nondimensional energy fluxes (normalized by the energy flux computed from the linear theory) above the tip of the ridge for a finite number of modes for the critical (top panels) and supercritical (bottom panels) wave regimes and the two finest model resolutions ( x = 1.5 km on the left panels and x = 5 km on the right panels. The dash line is the analytical solution computed from the knife-edge theory and the solid color lines represent the energy fluxes simluated for each models: black is using HYCOM, red is using ROMS, and blue is using the MITgcm. the ridge and due to the vertical discretization. The energy fluxes above the ridge that are contained in the lower modes are well resolved in all three models, since they compare well to the analytical solution derived from the knife-edge theory. The energy contained in the nonresolved higher modes is redistributed in the lower modes by the model. To understand how the model redistributes this energy is challenging and remains an open question. Because ROMS keeps the same number of vertical levels at the tip of the ridge, the simulated energy fluxes better correlate with the energy fluxes from the analytical solution (knife-edge theory). HYCOM and the MITgcm have only a finite number of levels, which does not permit a sufficient resolution to resolve the higher vertical wave modes. In appendix A we provide some preliminary results on the impact of varying the model 32

48 grid spacing for a realistic application (with tidal forcing) within the Gulf of Mexico (GoM). Specifically, the HYCOM model is used to describe the impact of the model resolution on the generation, propagation, and horizontal and vertical structure of internal waves in the GoM. 33

49 CHAPTER 4 DIAGNOSING THE SPURIOUS MIXING IN FIXED COORDINATE OCEAN MODELS The aim of this chapter is to quantify the spurious diapycnal mixing for idealized model simulations using ROMS and the MITgcm. Both models have fixed vertical coordinates, sigma- and z-levels, respectively. The problem of spurious mixing is first investigated using a lock exchange simulation that allows a comparison of the results using the tracerflux methodology discussed by Griffies et al. (2000) with the results of a recent study by Burchard and Rennau (2008). After evaluating and validating the tracer flux method for the lock exchange scenario, we quantify the spurious diapycnal mixing for an idealized model simulation containing a purely propagating internal wave field in both ROMS and the MITgcm. 4.1 Introduction Numerical ocean models provide an approximate solution to the oceans state. Thus it is necessary to evaluate the performance of these models. A criterion commonly used to evaluate the performance of a numerical ocean model is the ability of the respective model to accurately reproduce the properties and evolution of water masses. This ability is achieved by obtaining a realistic rate of diapycnal mixing in the ocean interior (Lee et al., 2002). Numerical ocean models using terrain-following (sigma-level) and geopotential (z-level) vertical coordinates suffer from spurious diapycnal mixing resulting from the inability of their advection schemes (particularly higher-order schemes) to preserve the adiabatic properties of the water parcel. Over long-term simulations, the result of spurious diapycnal mixing is unrealistic mixing of the water masses that may overshadow the naturally occurring mixing. An example of such numerically induced mixing using ROMS is illustrated in Figure

50 In this example, significant salinity anomalies (± 0.3 psu) are generated near steep slopes in the models bathymetry (i.e., near the continental shelf slope). These steep slopes lead to the sigma-level vertical coordinate reaching a maximum and thus favors the creation of spurious mixing. Using a finer model resolution, a classic approach, helps to alleviate the problem of numerical diapycnal mixing. Marchesiello et al. (2009) performed a scaling of the diapycnal mixing, and their results suggest that increasing the models horizontal resolution leads to a reduction in the numerical diffusion. However, they also found that for medium resolution ( 20 km horizontal grid spacings), there is a decrease in the magnitude of the spurious mixing (the cause is unknown). As a result, the full impact of the model horizontal grid resolution upon the numerically induced mixing remains an open issue. In recent years, various attempts have been made to quantify this spurious diapycnal mixing. Griffies et al. (2000) sought to compute the fluxes across a particular isopycnal surface (see Appendix B). This method computed the temporal change of the tracer concentration above the isopycnal. However, this approach has limitations. While rearranging and sorting the densities of each model grid cell, one loses information regarding the location of the diapycnal diffusion; only horizontally averaged estimates of the diapycnal diffusivity are retained. Morales Maqueda (2007), Holloway (2007) and Riemenscheider and Legg (2007) used different approaches to diagnose the spurious mixing. Each of these approaches largely relied on the computation of the variance decay for a tracer within a constant volume. However, these methods also have limitations because they are restricted to a single advection scheme and provide only diffusivity estimates (approximations) for other possible advection schemes such as the Flux Corrected Tracer (FCT) method. Burchard and Rennau (2008) circumvented the aforementioned limitations by developing a new method, also based on the tracer variance decay, which can be applied to any advection scheme and within any numerical framework. This methodology was applied to several idealized cases, including the lock exchange problem (Haidvogel and Beckmann, 1999). In this study, the tracer flux method is first applied to the lock exchange problem for both ROMS and the MITgcm. The results of these simulations are then compared with the results of Burchard and Rennau (2008). Next is an investigation of the impact of the model resolution on the magnitude of this spurious mixing by conducting three different experiments with two different advection schemes. After demonstrating in section 4.3 that the results obtain by using the tracer flux method in this study compare well with the results 35

51 obtained by Burchard and Rennau (2008), the method is applied to an idealized simulation of propagating internal waves (section 4.4). 4.2 Numerical models The idealized numerical ocean model simulations are conducted using two Oceanic General Circulation Models (OGCMs hereafter): ROMS and the MITgcm. For a more complete description of ROMS, the reader is referred to Shchepetkin and McWilliams (2005). For a description of the MITgcm, the reader is referred to Marshall et al. (1997). ROMS is a split-explicit, free-surface and terrain-following vertical coordinate oceanic model. The barotropic and baroclinic momentum equations are solved using different time steps (i.e., a shorter time step for the barotropic component) while assuming the hydrostatic approximation. ROMS employs a two-way, time-averaging procedure for the barotropic mode that satisfies the 3D continuity equation. The momentum equations are solved in the applications presented here using a third-order upstream biased scheme (Shchepetkin and McWilliams, 2005). ROMS has a plethora of tracer advection scheme options; however, this study focuses only on the third-order upstream biased (U3H) and the MPDATA tracer advection schemes. These two schemes are the most commonly used advection schemes within the ROMS community ( The U3H scheme is known to be accurate (Shchepetkin and McWilliams, 2005) but still generates high levels of spurious mixing and may produce over- and undershooting of tracer values. The MPDATA scheme prevents such over- and undershooting by preserving the positive-definite characteristics of the tracers at the cost of being more computationally expensive than U3H (John Warner, personal communication). Indeed, for a 20-hour ROMS simulation run on a single 1.6 GHz processor with 2Gb of cache memory (Linux-Ubuntu machine, no job was run but the model), the U3H experiments ran in 5 minutes while the MPDATA experiments ran in 12 minutes. The vertical advection scheme used in the ROMS experiments is the fourth-order Akima scheme. The vertical advection schemes in ROMS have been developed to be non-diffusive. No explicit mixing is prescribed and free-slip boundary conditions are applied. All diffusion coefficients are turned off or set to small ( 10 8 ) values. The MITgcm integrates the hydrostatic Boussinesq equations (but also has a nonhydrostatic capability, Marshall et al., 1997). It is a geopotential-following (z-level) verti- 36

52 cal grid model that represents the topography through a piece-wise linear representation (shaved cells, also known as the intersecting boundary method) and includes free-surface displacements. The two advection schemes used for the MITgcm are the third-order upwind Direct Spatial Temporal (DST hereafter) and second-order flux corrector (Superbee). The computational cost for the MITgcm is five minutes using the third-order upwind DST and approximately seven minutes using the Superbee on the one-processor machine described above. The vertical velocity is explicitly computed from the continuity equation. The choice of these advection schemes of both models was motivated by the need (1) to reproduce the Burchard and Rennau experiments as closely as possible and (2) to use the preferred advection schemes in each of these models. The aim is to diagnose the effective diapycnal mixing, which is the sum of the physical (explicit, not prescribed in these experiments) and numerical (implicit) mixing. To accomplish this, we use the tracer flux method, which is based on the amount of tracer flux across a particular isopycnal surface (Griffies et al., 2000). This value is found by analyzing the temporal change of the tracer above the isopycnal. The same flux is analyzed using the cumulative integral of the advection-diffusion equation. A more thorough description of the tracer flux method is provided in Appendix B. 4.3 Spurious diapycnal mixing associated with a lock exchange problem Experimental set-up Model experiments are designed to reproduce the lock exchange problem described in Haidvogel and Beckmann (1999) as well as Burchard and Rennau (2008) using both ROMS and the MITgcm. The scenario is a closed, two-dimensional (x, z) domain with a constant depth of H = 20 m and a length of L = 64 km. Initially the right half of the domain (L > 0 km) has a density of ρ = 1025 kg m 3 while the left side of the domain (L < 0 km) has a density of ρ = 1020 kg m 3. The two initial densities represent two water masses separated by a vertical barrier (Figure 4.1). Initializing in this manner simulates the removal of the barrier at time t = 0. The experiments use a linear equation of state (to avoid cabbeling and thermobaricity processes) with salinity as a passive tracer, and the initial surface elevation 37

53 Figure 4.1: Initial temperature field for the lock exchange problem Table 4.1: Description of experiments conducted for the lock exchange scenario. x is the horizontal model grid spacing resolution in the x-direction, N is the number of σ-levels, and z is the horizontal model grid spacing resolution in the z-direction Experiment Model x(m) N, z(m) COARSE-U3H ROMS , 2 COARSE-MPDATA ROMS , 2 COARSE-3 rd upwind DST MITgcm , 2 COARSE-Superbee MITgcm , 2 MEDIUM-U3H ROMS , 0.5 MEDIUM-MPDATA ROMS , 0.5 MEDIUM-3 rd upwind DST MITgcm , 0.5 MEDIUM-Superbee MITgcm , 0.5 FINE-U3H ROMS , FINE-MPDATA ROMS , FINE-3 rd upwind DST MITgcm , FINE-Superbee MITgcm , and velocity are equal to zero. The Coriolis force, bottom friction, and mixing are neglected (i.e., set to zero or infinitesimally small), such that the only effective density mixing is due to the advection of the density (i.e., numerically induced mixing). To investigate the impact of the model resolution and the choice of advection scheme on the spurious mixing, the experiments employ three different horizontal and vertical model grid spacings, each with two advection schemes (description in section 4.2). The experiments are summarized in Table

54 4.3.2 Spurious mixing identification Figure 4.2 shows the density/temperature distribution at t = 20 hr (salinity is held constant in the domain at 35 psu) using ROMS for the COARSE experiment (top panels), the MEDIUM experiment (middle panels), and the FINE experiment (bottom panels). The left panels display results using the U3H advection scheme and the right panels display results using the MPDATA advection scheme. Figure 4.3 follows the same display convention using the MITgcm, where the left panels display results using the third-order upwind DST advection scheme and the right panels display results using the Superbee scheme. In the absence of numerical diffusion, mixing, or other tracer sources/sinks, only the two initial prescribed densities are expected to be present at all times (e.g., an example would be any isopycnic models run with no explicit mixing prescribed). Figures 4.2 and 4.3 show that this is not the case for ROMS and the MITgcm. In each of the experiments, new temperatures appear in the domain. The horizontal grid resolution plays an important role in shaping the sharpness of the density front although the wave speed is preserved for all experiments. The results also indicate that, for the U3H and the third-order upwind DST advection schemes, temperature under/overshooting exists near the front. The flux-limiter method used in both the MPDATA and Superbee advection schemes avoids such occurrences of extreme values by conserving positive-definite characteristic for all tracers. ROMS exhibits less diffusion than the MITgcm because the spreading of new temperatures along the interface is smaller. This can be explained by the improved numerics applied in estimating the momentum equations and the difference in the vertical coordinate system of ROMS (Martin Losch, personal communication). The representation of the front is different in both models. In ROMS, the sharpness of the front is conserved for both advection schemes. However, in the MITgcm, the vertical extension of the front is more noticeable. As the model resolution increases, the results for all advection schemes converge to the same solution. The exception is the FINE experiment for the MITgcm, which uses the third-order upwind DST scheme and illustrates some oscillations along the interface that can be attributed to the dispersion effects of the scheme. To increase stability, the MITgcm requires viscosity parameters to be set to higher values than in ROMS or the General Estuarine Turbulence Model (GETM, employed by Burchard and Rennau, 2008 and Rennau and Burchard, 2009), especially at finer resolution. Similar results were 39

55 Figure 4.2: Cross-vertical section of the temperature field for the lock exchange problem at t = 20 hr using ROMS for the U3H (left panels) and the MPDATA (right panels) advection schemes. The top panels are the COARSE experiments, the middle panels are the MEDIUM experiments, and the bottom panels are the FINE experiments. found in the earlier study by Haidvogel and Beckmann (1999), where, using the Miami Isopycninc Coordinate Ocean Model (MICOM) and the S-Coordinates Rutgers University Model (SCRUM), sufficient explicit viscosity was required for this problem to have a smooth solution. Figure 4.4 shows the normalized total volume distribution of density classes in the domain after t = 15 hr for both models and all aforementioned advection schemes. At t = 0, only two initial density classes are present (1020 kg m 3 and 1025 kg m 3, denoted by the vertical solid black line), each with a normalized volume of 0.5 (i.e., 50%). At t = 15 hr, new density classes have appeared due to the leakage of volume from the initial water masses to new density classes. The COARSE experiments contain a larger volume of water masses in these new density classes when compared to the MEDIUM and FINE experiments suggesting that more mixing occurs when using coarser model grids. The advection schemes that use flux corrector methods (i.e., MPDATA and Superbee) always have less volume within the new water masses when compared to the high-order schemes and indicate that such advection schemes are less diffusive. The over/undershoots created by these higher-order schemes are defined by the densities created outside the initial range. The flux limiter schemes create 40

56 Figure 4.3: Cross-vertical section of the temperature field for the lock exchange problem at t = 20 hr using the MITgcm for the third-order upwind DST (left panels) and the Superbee (right panels) advection schemes. The top panels are the COARSE experiments, the middle panels are the MEDIUM experiments, and the bottom panels are the FINE experiments. new densities that are confined between the initial densities. For the MEDIUM and FINE experiments, the MITgcm exhibits more leakage than ROMS shown by the larger volume of newly created density classes. This indicates that the MITgcm is more diffusive (i.e., the spurious mixing magnitude is greater) than ROMS. For the COARSE experiments, both models are very similar. Time series of the volume-integrated density variance for ROMS and the MITgcm (Figure 4.6) are compared to those of the Burchard and Rennau study (Figure 4.5). Results from both models and those of the Burchard and Rennau study compare well since, for all experiments, the time series for the volume-integrated density variance are similar. The model resolution has a stronger impact on the spurious mixing in ROMS than in the MITgcm. The initial volume-integrated tracer variance is kg 2 /m 3. For the MEDIUM experiment with the ROMS-U3H, the volume-integrated tracer variance decay at t = 12 hr equals kg 2 /m 3, while for the MITgcm third-upwind DST it is kg 2 /m 3. All advection schemes behave similarly, i.e., the higher-order schemes are more diffusive than the flux corrector schemes. 41

57 Figure 4.4: Total volume (normalized) of the density classes volume distribution at t = 15 hr for ROMS (top panels) and the MITgcm (bottom panels) and for all the different tracer advection schemes. Solid gray lines are the two third-order schemes and solid black lines are schemes using the flux corrector methods. The abscissa represent the density classes (binning is 0.7 kg m 3 for the COARSE experiment, 0.08 kg m 3 for the MEDIUM experiment, 0.08 kg m 3 for the FINE experiment) and the ordinate is the normalized volume. The vertical solid black lines represent the location of the two initial prescribe densities Effective diapycnal diffusivity (κ eff ) This section describes the application of the tracer flux method from Griffies et al. (2000) (described in section 4.2 and detailed in Appendix B) to compute the effective diapycnal diffusivity (κ eff ). In the experiments, κ eff should only be interpreted as the implicit diapycnal diffusion component since no explicit mixing is prescribed. The resulting diapycnal diffusivities are vertically averaged to retrieve the initial vertical resolution (see Appendix B), and thus the binning of density classes for each of the experiments is not the same (COARSE = 0.7 kg/m 3, MEDIUM = 0.08 kg/m 3, FINE = 0.02 kg/m 3 ). Hovmöller diagrams (time-density) of κ eff for ROMS and for the MITgcm are presented in Figure 4.7 and 4.8, respectively. These figures show, for all experiments, one value of κ eff at all times and for each density class. The observed periodicity (2.5 hr) in the values of κ eff for both models is due to the seiche of the basin (natural period of the basin). The magnitude of κ eff using the MITgcm is always greater than κ eff when using ROMS. For example, at t = 42

58 Figure 4.5: Time series (hr) of volume-integrated density variance (kg 2 m 3 ) for the lock exchange scenario calculated for ROMS (top panels) and the MITgcm (bottom panels) and all advection schemes (solid gray lines are the high-order advection schemes; black solid lines are the schemes using flux corrector methods Figure 4.6: Time series (hr) of volume-integrated density variance for the lock exchange scenario calculated for the GETM with four different resolutions and a first-order upstream advection scheme (left panel) and the Superbee advection scheme (right panel). From Burchard and Rennau (2008). 43

59 Figure 4.7: Hovmöller-type diagram (time-density) of diapycnal diffusivity (κ eff ) using ROMS. The left panels use the U3H advection scheme and the right panels use the MPDATA advection scheme. The top panels are the COARSE experiment, the middle panels are the MEDIUM experiment, and the bottom panels are the FINE experiment. The solid black horizontal lines represent the two initial densities. 10 hr, for the MITgcm κ eff = m 2 /s and for ROMS κ eff = m 2 /s. The model resolution choice has a stronger impact when ROMS is used than when the MITgcm is used. It is clear that when the MITgcm is used, the MEDIUM and FINE resolution diapycnal diffusivity are very similar. When ROMS is used, the refinement of the grid induces a net decrease in the magnitude of κ eff between the two experiments. The MITgcm reaches a convergent solution where spurious diffusion cannot be reduced even as the grid is refined. Advection schemes using the flux corrector method (MPDATA and Superbee) have larger κ eff at density classes near the mean of the initial two prescribed densities. This could be explained by the anti-diffusive fluxes near the initial densities, which redistribute the density fluxes inside the initial range of densities. 44

60 The two higher-order schemes (U3H and third-order upwind DST) clearly exhibit some diapycnal diffusivity at extreme density classes. For the COARSE experiment, negative diapycnal diffusivities can be locally generated, especially near the initial densities and at their mean. These negative values of κ eff are associated with the creation or destruction of water masses and up-gradient fluxes. ROMS generates more negative diffusivities when using the MPDATA scheme at a coarse model resolution. There is a strong diapycnal diffusivity signal for the first three hours of the simulation due to the shock after releasing the vertical separation. The high magnitude of κ eff using the MITgcm is essentially due to the lack of development and complexity of the momentum advection schemes (Martin Losch, personal communication). It is worthwhile to compare these magnitudes of diapycnal diffusivity to the magnitude of the ocean interior diapycnal diffusivity. In general, a background diapycnal diffusivity value commonly used in ocean modeling is κ eff = m 2 /s although recent observations show that a value of m 2 /s would be more appropriate. At rough topography diapycnal diffusivity can reach m 2 /s, which is of the order of diapycnal diffusivity found for this lock exchange problem. It clearly shows that the spurious diapycnal mixing is a serious problem in fixed coordinate ocean models as spurious diapycnal diffusivity magnitude are larger than the observed diapycnal diffusivity Numerical tracer variance decay The tracer variance decay, defined as: D num = 2κ eff ( ρ z ) 2, is computed using the nonvertically averaged diapycnal diffusivity (computed in above section) This metric has proven to be useful for quantifying the spurious mixing (Morales Maqueda, 2006; Holloway, 2007; Burchard and Rennau, 2008). Time series for the volume-integrated numerical tracer variance decay are computed for both models (Figure 4.9) and compared to those of Burchard and Rennau study (Figure 4.10). For the coarse resolution simulations, the volumeintegrated numerical tracer variances do not match those of Burchard and Rennau (2008) particularly well, although the order of magnitude is comparable. This discrepancy may be explained by the different momentum advection schemes used in the different models. The magnitude of the volume-integrated numerical tracer variance decay for the MEDIUM and FINE experiments compares well with the results of the similar experiments in the Burchard 45

61 Figure 4.8: Hovmöller-type diagrams (time-density) of diapycnal diffusivity (κ eff ) using the MITgcm. The left panels use the third-order upwind DST advection scheme and the right panels use the Superbee advection scheme. The top panels are the COARSE experiment, the middle panels are the MEDIUM experiment, and the bottom panels are the FINE experiment. The solid black horizontal lines represent the two initial densities. and Rennau study. For both models, the overall spurious mixing decreases when the model resolution becomes finer. ROMS behaves similarly to the GETM, while the MITgcm has a decreasing trend that does not appear in ROMS or the GETM. This trend does not appear to be sensitive to either the model resolution or the advection scheme choice and might be due to an external diffusive source that is time dependant (i.e., spurious diffusion arising from the time stepping scheme). At fine resolution, the behavior of the different advection schemes is similar. Models that use a flux-corrector advection scheme present less spurious mixing as the time series of the volume-integrated numerical tracer variances have a weaker decreasing trend. Results in this section suggest that the tracer flux and the numerical tracer variance decay methods are robust and can be applied in more complex scenarios, such as a 46

62 Figure 4.9: Time series of the volume-integrated numerical tracer variance decay for ROMS (top panels) and the MITgcm (bottom panels). The left panels are the COARSE experiment, the middle panels are the MEDIUM experiment, and the right panels are the FINE experiment. pure-propagating internal wave field. 4.4 Spurious diapycnal mixing associated with internal wave propagation As illustrated in Chapter 3, the interaction of the barotropic tide with rough topography has been extensively studied because of the enhanced generated mixing at these locations. The induced generation of internal waves can potentially induce strong vertical velocity and thus large displacement of isopycnals. Although many studies have focused on such scenarios in order to quantify the effective mixing (i.e., local dissipation of the internal wave), no studies have attempted to quantify the implicit (i.e., numerical) mixing which could potentially be of the same magnitude as the explicit one. The main goal of this section is to document and to quantify (using the tracer flux method used in section 4.3 and detailed in Appendix B) the spurious mixing induced by the propagation of a pure internal wave field using the ROMS 47

63 Figure 4.10: Time series of the volume-integrated numerical tracer variance decay from the Burchard and Rennaud paper (2008) for four different resolutions and two different tracer advection schemes (left panel uses the U1H and the right panel uses the Superbee. configurations described in Chapter Experimental set-up The numerical configurations used in this section are all the ROMS experiments described in Table 3.1 and Figure 3.1. The Table 4.2 summarizes the list of all experiments used in this section. To investigate the impact of the model resolution on the magnitude of the internal wave propagation induced spurious mixing, we use the three different model resolutions seen previously (i.e., x = 1.5 km, x = 5 km, x = 10 km). By varying the ridge height, we also conduct the same experiments for three different wave regimes: subcritical, critical and supercritical. For the critical and supercritical wave regimes, the baroclinic response is similar or greater than the initial forcing. The internal wave dynamics is then considered non-linear, which could be an important factor on the spurious diapycnal mixing magnitudes Results For subcritical flows, the baroclinic response (e.g., creation of vertical velocity) is weak and thus we expect less spurious mixing signal at this wave regime. Based on the configurations 48

64 Table 4.2: Description of Experiments conducted for the internal wave scenario. x is the horizontal model grid length resolution in the x-direction, N x is the number of points that defines the ridge width, N is the number of z- or σ-levels, and z is the horizontal model grid length resolution in the z-direction Experiment Model x(km), Nx N, z(m) Wave regime 1 ROMS 1.5, 14 50, 40 sub-critical 2 ROMS 1.5, 14 50, 40 critical 3 ROMS 1.5, 14 50, 40 super-critical 5 ROMS 5, 4 15, sub-critical 6 ROMS 5, 4 15, critical 7 ROMS 5, 4 15, super-critical 8 ROMS 10, 1 7, sub-critical 9 ROMS 10, 1 7, critical 10 ROMS 10, 1 7, super-critical described in Table 4.2, we compute the normalized density class volume distribution for all experiment. Figure 4.11 shows the ROMS experiments with x = 1.5 km for all wave regimes (top is subcritical and bottom is supercritical). Since 50 levels are used, initially there are 50 water masses that occupied 100% of the total volume of the basin and each represent 2% of that total. This is represented by the red dots in Figure After six days, histograms of the volume distribution of these water masses are plotted. If the model had no spurious mixing, the volume in each initial density class would still represent 2%. Figure 4.11 shows that the normalized volume contained on the initial density has decreased (left panels) and that volume in new density classes appears in between the initial density classes. A zoom on specific density classes intervals (right panels) illustrate this leakage of initial density classes to neighborhood density classes. The leakage to other density classes increases as the bump height increases as the volume in newly created density classes is greater in the supercritical regime than the subcritical one. This is mainly due to both the generation of stronger vertical velocities and the presence of steeper slope in the vertical coordinate system. For the subcritical wave regime, the maximum loss from the initial volume occupied by a water mass (2%) is seen at kg/m 3 and is characterized by a 0.6% decrease. For the critical wave regime, it is clear that the leakage is greater for all model resolutions and can reach up to 1% for intermediate density such as kg/m 3. When the bump is 49

65 Figure 4.11: Normalized density class volume distribution (%) at t = 18 hr, for x = 1.5 km and zoom over a specific interval for: subcritical wave regime (top panels), critical wave regime (middle panels), supercritical wave regime (bottom panels) m (supercritical wave regime) the volume change in the initial density classes reaches a maximum also around the intermediate density (middle depth) and is clearly greater than the other wave regimes. The zoom on specific density intervals sows that the spread of this leakage is more important at coarser resolution and that strong numerical diffusion exists as volume of the newly created density classes is greater (Figure 4.12 and 4.13). For example, the experiments using x = 5 km, show that the volume of the initial density classes is reduced by approximately 45%. When x = 10 km, the volume of the initial density classes is reduced by 60%. It means that the domain contains more newly created density classes than density classes that were initially present. The tracer flux method computes horizontally averaged estimate of the effective diapycnal diffusivity. In this study the effective diffusivity is define as only the numerical 50

66 Figure 4.12: Normalized density class volume distribution (%)at t = 18 hr, for x = 5 km and zoom over a specific interval for: subcritical wave regime (top panels), critical wave regime (middle panels), supercritical wave regime (bottom panels). diffusivity since there is no explicit diffusion prescribed. The information about the location of where this numerical mixing occurs thus cannot be retrieved. A method to address this problem is to compute the tidally averaged change in the buoyancy frequency from a reference state (the initial vertical profile in this study) following Xing and Davies (2009). The results of these computations are shown in Figure The colors show where the water being transformed with the red contours being the formation of denser water and the blue contours being where lighter water is formed. The white contours are where no significant (i.e., too small to distinguish from computer precision) change has been made with respect to the initial vertical profile. Figure 4.14 shows that when the model has a coarser resolution and taller (rougher) bottom topography, the anomaly in the buoyancy frequency field is larger (i.e., more spurious mixing) than at finer model resolution. This confirms the above 51

67 Figure 4.13: Normalized density class volume distribution (%) at t = 18 hr, for x = 10 km and zoom over a specific interval for: subcritical wave regime (top panels), critical wave regime (middle panels), supercritical wave regime (bottom panels). results that show that coarser model resolutions and taller ridge have more leakage to newly formed density classes. For the subcritical wave regime, results show some persistent change near the ridge with a change in the buoyancy frequency of approximately s 2. Water masses modifications also occurrs away from the ridge but are not persistent features. This could be due to the scattering of the internal creating higher harmonics and different wave frequencies that are not properly filtered in our tidal averaging method. At the finest model resolution, the magnitude of the buoyancy frequency anomalies is approximately the same for all the considered wave regimes. However, if the water masses are modified with approximately the same rate, Figure 4.14 also shows that the water masses affected by the spurious mixing 52

68 spread further away from the ridge. At coarser model resolution, the misrepresentation of internal wave can lead to large level of spurious mixing as anomalies of the buoyancy frequency field are larger as the grid spacing gets larger. For all numerical experiments, the maximum change in the buoyancy frequency field from the initial one is located near the ridge and more specifically at the tip of the ridge for the supercritical wave regime. This result is consistent with the numerical study conducted by Xing and Davies (2009) that shows that the maximum explicit mixing is located in areas where the energy of the internal wave field is concentrated. This is also explained by the fact that the angle between the vertical coordinate (sigma-grid) and the displaced isopycnal is maximal. The maximum change ( s 2 ) in the vertical profile from the initial uniform stratification is used to analytically compute the change in the mode-1 wavelength. The result show that, after eight days, the persistent buoyancy anomalies generated by the spurious mixing are responsible for shortening the mode-1 wavelength by approximately five kilometers. The change in the internal waves propagation represent about three grid points in the fine model resolution and one grid point in the coarser ones. This is consistent with the results found in Chapter 3 using the spatial wavelet analysis where ROMS showed shorter wavelengths. To quantify the diapycnal diffusivities, which are responsible for the buoyancy frequency field anomalies, we use the tracer flux method (validated in section 4.3 and detailed in Appendix B). Figure 4.15 shows the result of the computation with the tidally averaged diapycnal diffusivity for all ROMS experiments. The diapycnal diffusivities are tidally averaged in order to eliminate the buoyancy flux coming from the western open boundary (where the tidal forcing is prescribed) and artificial vertical diffusion due to the isopycnals displacement. The diapycnal diffusivity increases as the model resolution gets coarser and the ridge higher. At fine resolution ( x = 1.5 km), the vertical structure of the diapycnal diffusivity is similar and only the magnitude slightly varies as the ridge gets taller. The largest level of spurious diapycnal mixing is seen at denser density which is where the sigma-levels are slopping the most. A peak is seen at an intermediate density for all finer resolution and can be interpreted as the energetic mode-2 wave that contains about 30% of the total energy within the field. At coarser resolution this peak does not appear and the vertical structure 53

69 Figure 4.14: Tidally averaged (over tidal cycle 16) buoyancy frequency change (s 2 ) from initial distribution for all model resolutions and all wave regimes. looks smoother because of the vertical resolution employed but also because higher wave modes are not resolved. For the subcritical wave regime, the magnitude of the diapycnal diffusivity is quite similar for all model resolutions (less than m 2 /s). In contrast, for the supercritical wave regime, the diapycnal diffusivity magnitude rapidly increases as the model resolution gets coarser and reach a maximum of m 2 /s when x = 10 km and in denser water (which relates to the deep part of the domain). The order of these diapycnal diffusivity magnitudes is consistent with a quantitative scaling of spurious diapycnal diffusivity done by Marchesiello et al. (2009). 4.5 Summary In this chapter, we first use a gravitational adjustment problem where no explicit mixing was prescribed to document and quantify the spurious mixing for two model classes: a terrainfollowing model with ROMS and a geopotential model with the MITgcm. To compute the resulting spurious diapycnal diffusivity, the tracer flux method, described in Winters and D Asaro (1996) and Griffies et al. (2000), was applied. Results show that the spurious diapycnal is one order greater than what is observed in the abyssal ocean for the coarser resolution. Using finer model resolutions alleviates the problem but does not solve it. The different advection schemes used in the study also show that advection schemes using 54

70 Figure 4.15: Tidally averaged (over tidal cycle 16) diapycnal diffusivity (m 2 /s, logarithmic scale) for all model resolutions and all wave regimes. flux corrector methods have less spurious diffusion than high odd-order advection scheme. However, the additional computational cost might not be worth the small reduction in the spurious diapycnal diffusivity. These results are validated by comparison to findings from a previous numerical experiment done by Burchard and Rennau (2008). Second, the tracer flux method was applied to a more realistic scenario of internal tide freely propagating simulate with ROMS. Results show that, after eight days and at coarse model resolution, the volume of newly formed density classes due to the spurious diapycnal mixing is very large and corresponds to 50% of the total domain volume. At fine model resolution, the spurious mixing induced weaker leakage to adjacent density classes. The buoyancy frequency anomaly field show that most of the water mass modifications occur near the ridge, where the vertical coordinate is strongly sloping. At fine resolution, the wave regime does not impact the magnitude of these water mass properties modifications but their spread within the domain as they extend further away from the ridge. At coarse model resolution, the initial vertical profile is strongly modified. The change in stratification due to the spurious mixing affects the internal wave propagation. Specifically, for the extreme case of supercritical wave regime and coarse resolution (i.e., x = 10 km), the mode-1 wavelength can be shortened by three grid points. For long term model integrations, this could be a dramatic source of errors in the representation of internal wave dynamics. The diapycnal diffusivity estimates show that at coarse model resolution, the diapycnal diffusivity can be 55

71 one or two order of magnitudes higher than the observed diapycnal diffusivity in the calm ocean interior. These diapycnal diffusivities have similar order of magnitude to what is observed in regions where intense internal wave breaking occurs and thus where enhanced mixing is recorded. These results suggest that at coarse model resolution, physical (explicit) mixing through the choice of a mixing parameterization is not necessarily needed and that a better quantification of the spurious mixing for specific numerical applications is needed. 56

72 CHAPTER 5 CONCLUSION AND PERSPECTIVES 5.1 Summary of work First, this research work, documents the internal gravity wave representation for idealized model configurations as a function of the model grid spacing and numerical choices. Second, it quantifies, for two fixed vertical-coordinate models (terrain-following and geopotential), the spurious (i.e., numerical) diapycnal mixing arising from internal gravity wave propagation. The multimodel study of internal wave propagation in OGCMs shows that coarse model resolutions are able to reproduce qualitatively the internal wave generation and propagation but fail to reproduce an accurate baroclinic response in the ocean interior. This is mainly because coarse model resolutions cannot resolve the higher wave modes that greatly contribute to the energy field propagating away from the generation site. However, all model experiments well simulate the mode-1 wavelength and phase of the internal gravity wave when compared to the wavelength and phase of the analytical solution. Both HYCOM and the MITgcm simulate an internal wave wavelength similar to the analytical solution and thus realistically reproduce the generation and propagtion of the internal wave. ROMS, the terrain-following model, simulates a shorter wavelength. This is due to (1) the inherent pressure gradient error arising from the unsmooth topography and (2) the stronger level of spurious diapycnal mixing that modifies the initial stratification near the ridge where steep slopes in the vertical grid are present. However, ROMS energy fluxes above the tip of the ridge agree better with the predicted energy fluxes (from the knife-edge theory) than with the HYCOM and the MITgcm energy fluxes. This is due to the vertical coordinate that conserves all levels above the ridge. Indeed, both HYCOM and the MITgcm vertical coordinate outcrop the topography, losing the vertical resolution when the topography is tall. ROMS conserves all initial vertical levels and thus is able to resolve 57

73 more vertical wave modes. For all experiments, the aliasing between the model grid spacing and the internal wave propgation (i.e., internal wave wavelength) is not significant. At large grid spacing and tall topography (supercritical wave regime), the terrain-following model creates strong spurious jets due to pressure gradient errors that overshadow the baroclinic tidal response in the domain. To accurately represent an internal wave in a terrain-following model with model parameters similar to aforementioned configuration (e.g., no Coriolis force and a typical uniform stratification), the model resolution needs to be less than 5 km. Using geopotential or isopycnic models is more suitable for coarser resolution simulation and for long-term climate scale simulations. The main advantage of the sigma-level models resides in the simulation of the higher frequency processes above the rough topography, and thus the higher vertical wave modes, as the model vertical resolution does not change. To investigate the spurious diapycnal mixing generated by internal wave propagation, numerical experiments with ROMS and the MITgcm are conducted for a gravitational adjustment problem. This scenario represents an extreme case of internal gravity wave since it involves a strong discontinuity and a large amplitude internal wave. The wave s amplitude is about half of the total depth of the basin. The model configurations of the so-called lock-exchange problem are set up following the study of Burchard and Rennau (2008). The use of similar model configurations allows for a validation of the model results by comparing the results from the present study and the Burchard study. The key point is to prescribe no explicit mixing; thus, only the two prescribed initial densities should be present at any time during the simulations. Section 4 of this study clearly shows that this is not the case as new density classes are formed. The only way to create mixing and modify the water masses in the model is by numerical mixing if no physical mixing is present. The results show how these newly formed densities are distributed, and using the tracer flux method, the study quantifies the corresponding spurious diapycnal diffusivities for all numerical experiments. Considering all experiments, the spurious diapycnal diffusivity magnitudes range from 10 4 to m 2 /s for the gravitational adjustment problem, which is greater than the diapycnal diffusivities found in the abyssal ocean (10 5 m 2 /s). Results also show that, even at fine model resolutions, the induced spurious diapycnal mixing is one order of magnitude higher than the background mixing observed in the ocean interior. Using coarse model resolution can increase the magnitude of this spurious diapycnal mixing by five times. The study also shows that the choice betwee high odd-order advection 58

74 scheme orand an advection scheme that uses flux corrector method does not impact the magnitude of the spurious mixing compared to varying the grid spacing. Additionally, the computational cost of flux corrector methods may not be justified. When using advection schemes with flux corrector methods, unrealistic temperatures (and salinity) are not created and thus it might be of importance when using coarser model resolutions to avoid drift in water mass properties. Thehigher odd-order schemes (e.g., The Prather advection scheme) could reduce significantly the spurious diapycnal mixing but they have not been yet tested. The diapycnal diffusivities computed from the tracer flux method are used to compute the numerical tracer variance decay (metric for the spurious mixing) following the study of Burchard and Rennau (2008). The estimates of this numerical tracer variance decay computed in this study compare well to the results found in the previous study by Burchard and Rennau (2008). These results validates the application of the tracer flux method for this scenario and gives confidence on these new estimates of spurious diapycnal mixing magnitudes in ROMS and the MITgcm for this lock-exchange scenario. The tracer flux method is then applied to the idealized internal ROMS experiments (tide interacting with a ridge) described in section 3. The results show that for all experiments the volume of the newly formed density classes is large and can represent half of the total volume of the basin. The spurious diapycnal mixing leads to a strong leakage of initial density to neighboring density classes. The numerical experiments using coarse resolution and tall ridge (supercritical regime) have a larger leakage (high vertical diffusivity and thus strong spurious mixing) than the subcritical and low topography which have a weak leakage. The leakage to other density and thus the creation of other density smooths out the initial vertical profile and thus strongly changes the initial stratification which, in term, is changing the internal gravity wave dynamics. Computing tidally averaged buoyancy field anomalies also shows the stratification change. The spurious mixing is mostly located near the ridge where high vertical velocity exists and where the slope between the isopycnal and vertical coordinate is maximal. The maximum stratification change after 8 days (at coarse resolution and tall topography) induces the wavelength of the internal wave to be shorter by about 5 km, which represent less than a grid point. The tidally averaged diapycnal diffusivities are also computed. Results show that the internal wave propagation induces a level of spurious diffusion that is comparable to or larger 59

75 than the naturally occurring diapycnal diffusion. At fine model grid spacing, the spurious diapycnal mixing levels are small at the surface (where the slope between vertical coordinate and isopycnal is weak) even for supercritical wave regimes. At depth, the diapycnal diffusivities are comparable to what is observed in the abyssal ocean. When the model resolution is coarse, the wave regime has a strong impact on the spurious mixing levels, as their magnitudes strongly increase as the ridge gets taller. The coarser numerical experiments exhibit light levels of spurious mixing that are comparable to diapycnal diffusivity in regions of intense internal wave breaking (and thus enhanced turbulent processes). The estimates found in this study are in agreement with preliminary scaling done by Marchesiello et al. (2009). In this dissertation, it has been shown that the choice of the vertical coordinates for representing the internal gravity wave propagation matters most when the model resolution is coarse. New estimates of spurious diapycnal mixing are given for a wide variety of numerical choices using ROMS and the MITgcm. Qualitative features of the spurious mixing associated with the misrepresentation of internal wave and internal wave propagation have been documented. The results demonstrate and confirm that the numerically induced mixing can potentially be a source of large drift in water mass properties for long-term climate simulations. Although the community has recently focused on finding accurate tidal mixing parameterizations to reproduce realistic mixing rates in the ocean interior, this study clearly shows that, alleviating the spurious diapycnal mixing problem and ensuring adiabatic properties of water parcel advection is also vital. 5.2 Perspective of the work This dissertation adds to the understanding and knowledge of the representation of internal gravity waves in OGCMs. The study, though, needs to be put in the context of tidal mixing. Indeed, the various results presented in this work show that for most of the OGCMs, typical resolutions used for regional or global applications exhibit spurious diapycnal diffusivities larger than the observed value. The misrepresentation of internal wave dynamics and the aliasing between the wavelength of the internal wave and the model grid spacing induces a large numerical (i.e., unrealistic) diapycnal component. These unrealistic vertical diffusion fluxes cannot be controlled by the user, and thus the vertical mixing in numerical simulations 60

76 cannot be diagnosed and prescribed to the desired values (i.e., local diapycnal mixing). This has a dramatic impact for long-term simulation such as climate runs. The vertical spurious diffusion is large where rough topography exists, and in these locations the water mass properties are changed in an unrealistic and irreversible way. The water mass properties drift can, in turn, lead to large inaccuracies and deficiencies in the representation of water mass pathways and the thermohaline circulation. The recent effort in the numerical ocean modeling community to find better parameterization for the tidal mixing always implies that the numerical mixing in the model is smaller than the physical vertical mixing. The abyssal ocean diapycnal diffusivities (i.e., m 2 /s) are one order of magnitude higher than the machine precision only when using single precision for a 32 bit machine (i.e., ). This study is a first step toward a quantitative assessment and documentation of the spurious mixing. So far, the regime of the flow and the resolution have been characterized as the main parameters to control the spurious diapycnal mixing magnitude. In the case of internal gravity wave propagation, the stratification should also act strongly on the location of the generated spurious mixing. The effects of these three parameters (grid spacing, regime of the flow, and stratification) could eventually be incorporated into existing tidal mixing parameterizations to compute a reference value for diapycnal fluxes to be added to or subtracted from what the physical parameterization has computed for vertical diffusivities. 5.3 Future work This section briefly describes future work to be done as a follow-up to this study and issues to be addressed that could be important for the ocean modeling community. Although the results in the dissertation are not presented, many sensitivity experiments have been conducted for all three models. As an example, when using a terrain-following vertical coordinate, an evaluation of the pressure gradient errors has initially been made by looking at the magnitude of the spurious jets when no forcing is prescribed in the model configuration. The impact of several parameters, such as viscosity and diffusivity coefficients, for all three models has been investigated, as has the impact of layer thicker diffusion for HYCOM. Although a wide spectrum of values for these parameters has been considered, the study investigate only the commonly used coefficients for standard applications. However, 61

77 more numerical experiments with varying the viscosity parameters are needed, as the baroclinic response is very sensitive, depending on the choice of the model. A scaling of spurious diapycnal diffusion with respect to the baroclinic response would be an interesting computation. Numerous numerical experiments are performed withvarying either the horizontal or the vertical resolution. Because of the limits on dissertation length, only those results obtained while keeping a constant ratio of horizontal to vertical grid spacing are retained. However, conducting more numerical experiments in which one model grid spacing direction remains constant (horizontal or vertical) could potentially tackle the problem of redistribution of the non-resolved higher modes into lower modes. Although the numerical approaches, methods, and parameters in this work were selected primarily because of the need to have similar configurations between the models, they do not necessarily represent the best options available in these models and thus many experiments remain to be conducted. This work would also benefit from additional numerical experiments being conducted using the MITgcm and its non-hydrostatic capability. Direct Numerical Simulations (DNS) and Large Eddy Simulation (LES) would be useful as a direct comparison to the presentstudy results could lead to important findings of a physicallybased tidal mixing parameterization. The method recently developed and employed by Burchard and Rennau (2008) to compute the spurious diapycnal mixing in fixed vertical-coordinate ocean models needs to be implemented in the three numerical models used in this study. The results using the Burchard and Rennau (2008) method could be compared to the result when using the tracer flux method. The implementation in ROMS is in progress (collaboration with Florian Lemarié) and some preliminary results have already been obtained for the lock-exchange scenario. Achieving a better understanding of the compatibility of the momentum and tracer advection schemes with the potential vorticity equation is also of interest. Indeed, spurious diffusion sources could potentially appear in the stretching terms (Yves Morel, personal communication). This dissertation has treated only idealized scenarios with specific configurations. The quantification of the spurious diapycnal mixing in realistic numerical applications (with open boundaries) has recently received considerable attention. A work in progress that could advance the understanding of the impact of model resolution on internal gravity wave 62

78 generation and propagation and further document the spurious mixing magnitude is made using HYCOM for realistic numerical experiments conducted in the Gulf of Mexico with tidal forcing (Appendix A). The three experiments being conducted are similar except in the choice of the vertical coordinate. The HYCOM model solutions obtained in a fully isopycnic mode (no spurious mixing by definition, water parcel advection property is totally adiabatic) are compared to solutions obtained when HYCOM is run in a fully geopotential or terrainfollowing fashion. The preliminary results confirm that when higher model resolution is used, the terrain-following vertical levels is better suited to simulate internal gravity wave dynami and energy but at low model resolution, an isopycnic framework is required for an accurate representation of these waves. 63

79 APPENDIX A IMPACT OF MODEL GRID SPACING IN A REALISTIC APPLICATION: BAROCLINIC TIDES IN THE GULF OF MEXICO The main purpose of this appendix is to show preliminary results of the impact of varying the grid spacing on the internal tide propagation in a more realistic application. This is accomplished by using two similar HYCOM Gulf of Mexico (GoM) configurations that differ only by the horizontal model resolution. A.1 Model configurations Two model configurations are used: a 1/12 ( 9 km) and a 1/24 ( 4.5 km). They are based on the Navy Research Laboratory (NRL) GoM configuration (Figure A.3) and differ only by their horizontal model grid spacing. HYCOM is run in an hybrid fashion: terrain-following in shallow waters, geopotential in the homogeneous layers and isopycnal in the ocean interior. The initial conditions of both configurations are taken from a restart of an assimilated simulation of the NRL 1/24 GoM configuration which is currently available on the HYCOM website ( 01pt1). These initial temperatures, salinities and velocities correspond to those of August 1 st The model is forced with the Navy Operational Global Atmospheric Prediction System (NOGAPS, a 0.5 product) and includes wind stress, wind speed, and heat flux (using bulk formula). The boundaries of the domain are nested to bi-monthly fields from an HYCOM Atlantic 1/12 free-run, averaged over as the NRL GoM simulations. The tidal forcing is prescribed by a local tidal potential (based on Newton s tidal equilibrium theory) and remotely by prescribing the tidal barotropic transport and elevation at the open boundaries (Eastern boundaries on Figure A.1). The tidal transport and elevation at these 64

80 Figure A.1: SSH (m) of the Gulf of Mexico for: the initial time (left panels), after 20 days with only nesting (no tidal forcing and no atmospheric forcing) (right panels). The top panels are for the 1/12 experiments and bottom panels are for the 1/24 experiments. locations are extracted from the Egbert tidal model. The two forcing are crucial to simulate accurately the tides in this semi-enclosed basin due to their non-linear interactions (Gouillon et al., 2010). The main eight tidal constituents are considered (M 2, S 2, K 2, N 2 which are semidiurnal and O 1, K 1, P 1, Q 1 which are diurnal). A quadratic bottom friction is used for bottom friction and no explicit mixing is prescribed to isolate the two dissipation mechanism (bottom friction and tidal conversion). Figures A.1 and A.2 show SSH snapshots at similar time of different configurations that differ only by which forcing is included. A.2 Validation of the tides in HYCOM This brief section validates the implementation of the tides in the GoM by comparison to available observations. The t-tide program (Pawlowicz, 2002) is used to extract tidal 65

81 Figure A.2: SSH (m) of the Gulf of Mexico for: nesting with atmospheric forcing (left panels), after 20 days with all forcing including tides (right panels). The top panels are for the 1/12 experiments and bottom panels are for the 1/24 experiments. harmonics from the simulated sea surface elevation. Figure A.3 shows the spatial tidal amplitude (colored) and phase (contoured) maps for the main semidiurnal tidal constituent, M 2, and the main diurnal tidal constituent, O 1 for both model resolutions. The same maps (with same colorbar scale) are plotted using the GOT99 altimetry data and the Egbert inverse tidal model and shown in Figure A.4 for comparison. For the semidiurnal tidal constituent, the observed amphidromic point (zero tidal elevation at all time) location is well simulated for both configurations. The tidal amplitudes well compare to observations and the Egbert tidal inverse model (e.g., the amplification process of the M 2 tide at the West Florida Shelf (Clarke and Battisti, 1981) is well reproduced. The semidiurnal co-tidal lines also agree with osberved data. There are no major differences 66

82 Figure A.3: Tidal amplitude (colored) and phases (contoured) spatial maps for: M 2 using the 1/12 configuration (top left panel), M 2 using the 1/24 configuration (bottom left panel), O 1 using the 1/12 configuration (top left panel), and O 1 using the 1/24 configuration (bottom left panel). between the two model configurations for the semidiurnal tide. For the diurnal constituent, the co-oscillating resonance phenomena (or port phaselocking process, Reid and Whitaker, 1981) is well simulated as the tidal amplitudes and phases are spatially homogeneous within the basin for both model configurations. The diurnal tide amplitudes for the 1/12 configuration are higher than the 1/24 in the West part of the basin, indicating that there might not be the same dissipation near the entrance of the GoM thus reinforcing the resonance mechanism. These results validate the implementation of the tides in HYCOM and thus give confidence in the analysis of the baroclinic tides in the basin. 67

83 Figure A.4: Tidal amplitudes (colored) and phases (contoured) spatial maps for: the GOT99 M 2, M 2 simulated with the Egbert tidal model, the GOT99 O 1, and O 1 simulated with the Egbert tidal model. A.3 Impact of varying the model grid spacing on the internal wave field The West-East cross-vertical sections (left panels in Figure A.5) show that, West of the Yucatan Peninsula (hereafter, YP), the vertical velocities are about m/s and are stronger to the East of the YP as vertical velocities reach m/s). The periodical pattern suggests that internal waves are propagating away from their generation sites (YP shelf). For both model configurations the strongest velocities are observed near 84 W because of the abrupt change in the topography (presence of the shelf). In both configurations, a transition zone is clearly seen around 800 m deep. It corresponds to an abrupt change of density in the vertical profile, thus changing the internal wave dynamics. Although the 68

84 Figure A.5: Vertical velocity snapshots (t = 17 days) of two cross-vertical sections for: the coarser resolution (1/12 ) (top panels) and the finer resolution (1/24 ). vertical velocity magnitudes are very similar in both configurations, their horizontal and vertical structure differs. In the 1/12 configuration, the internal wave beams are larger than in the 1/24, where narrower internal wave beams and finer structure are simulated. This is also confirmed by the vertical velocity structure seen at the West Florida Shelf (Figure A.6), where the main internal wave beam characteristics are not change but that higher frequency internal wave exists in the 1/24 configuration. In the coarse model resolution, where the number of wave modes resolved is less than in the fine model resolution, there is a clear shift of positive and negative vertical velocities while the finer model resolution has only positive or very weak negative vertical velocities. The internal wave propagation is clearly impacted by the model resolution as, for the same longitude (e.g., 91 W), negative vertical velocities are simulated in the coarse simulation while positive are simulated using the finer grid. East of the YP, the magnitude of the vertical velocities simulated with the coarser grid are greater than the ones simulated with the fine grid. This could be explained by the higher modes that faster locally dissipate while the coarse configuration needs to redistribute the energy of the non-resolved wave modes to the lower ones. Similar results are found for the North-South cross-vertical sections (right panels in 69

85 Figure A.6: Zoom of the vertical velocity snapshots (t = 17 days) at 22.7 N for both configurations. Figure A.5). A striking difference between the two configurations is the number of internal wave beams simulated between 24 N and 27 N. The coarse resolution has two distinct beams of positive vertical velocities while the fine model resolution has four internal wave beams with positive vertical velocities. There is clearly a strong aliasing between the model grid spacing and the wavelength of the simulated internal waves. 70