Internal wave modeling in oceanic numerical models: …coaps.fsu.edu/~gouillon/PAPER/prospectus_flav.pdf · representation of the key phenomena of these dynamics, numerical models

Internal wave modeling in oceanic numericalmodels: impact of the model resolution on thewave dynamic, energetic and associated mixing

Flavien Gouillon

COAPS, Florida Sate [email protected]

March 13, 2009

A Dissertation Proposal submitted to theDissertation Committee:

Dr. E. ChassignetDr. J.J. O’Brien

Dr. S. MoreyDr. L.L. St-LaurentDr. C.A ClaysonDr. M. Huettel

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Contents

1 Introduction 3

2 Literature Review 52.1 General Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Internal wave modeling in OGCMs . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Internal wave induced mixing representation in numerical models . . . . . . . . 7

3 Open issues and Objectives 10

4 Proposed work 12

5 Preliminary results 16

6 Time line 20

A Analytical Solution of a Baroclinic Tidal Flow over a Small Bottom Topog-raphy 21

B Numerical Models 22B.1 The Hybrid Coordinate Ocean Model (HYCOM) . . . . . . . . . . . . . . . . . 23B.2 The Regional Ocean Modeling System (ROMS) . . . . . . . . . . . . . . . . . . 24B.3 The MIT general circulation model . . . . . . . . . . . . . . . . . . . . . . . . . 24

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Numerical Simulation of Internal Waves March 13, 2009

1 Introduction

Numerical models have become an indispensable tool for describing and understanding thedynamics of the ocean thermohaline circulation. In order to give an accurate and realisticrepresentation of the key phenomena of these dynamics, numerical models need to 1) producewater masses with realistic properties, 2) transport them correctly and 3) maintain these watermass characteristics [Lee et al, 2001]. The latter requirement is mainly achieved by operatingthe model with realistic mixing rates.

The tides, along with the winds, are now accepted as one of the most significant sources ofenergy for that mixing. Recently, there have been many studies linking internal wave generationto tidal flow over topography in the deep ocean [Egbert and Ray, 2001; Merrifield and Holloway,2002]. In these regions, where the bottom is considered to be rough, measurements show hugeamounts of mixing [Polzin et al., 1997; Toole et al., 1997]. This interaction between tidal flowand bathymetry results in a net transfer of energy from the barotropic (depth averaged) tide tothe baroclinic tide (internal mode). This baroclinic tidal energy must dissipate via turbulentprocesses. Munk and Wunsch [1998] suggest that this dissipation accounts for roughly half ofthe energy necessary to vertically mix the abyssal ocean and to maintain the strength of theglobal thermohaline circulation. It also plays a crucial role in transferring the momentum fromthe surface to the deep ocean. Internal wave observations are sparse and are a major constraintin obtaining a detailed understanding of the baroclinic tides and their associated mixing. Asa result, it is necessary to rely on numerical modeling. However, while the theory of internalwaves generated by interaction of a barotropic flow with topography is well established [e.g.Baines, 1982; Bell, 1975; Hibiya, 1986], their representation and the mixing induced by theirpropagation in Oceanic General Circulation Models (OGCMs) is still not fully understood [DiLorenzo et al., 2006].

In the oceanic numerical models, the approximations commonly made (i.e. hydrostatic,Boussinesq), the choice of horizontal and vertical resolution used, the vertical discretizationof the model, and the numerical method employed to solve the governing equations make itimpossible to perfectly resolve the scales and the dynamic of the internal waves. It has beendemonstrated that the choice of the horizontal and vertical grid spacing of the model influencesthe representation of these waves [Robertson, 2006; Jachec, 2007]. However, only a narrowrange of values for ∆x, ∆y (1 km to 5 km) and vertical level grid spacing have been considered(25 to 40 levels). None of these studies investigate the simulation of internal waves using avery high spatial resolution to a very coarse model resolution (from few meters to hundred ofkilometers). The model vertical discretization and the model numerical algorithms (advection,time-stepping, etc.) choice play a crucial role in the ability of the model to represent internalwaves as well as their induced mixing. In numerical models, the adiabatic property of advectionis a challenge to maintain unless it is explicitly built into the model’s algorithmic framework[Griffies, 1999], such as in isopycnic (constant potential density surfaces) models. In an isopy-cnal model, the diapycnal component is calculated by a vertical mixing parameterization (ex:K-Profile, Mellor-Yamada, Kraus-Turner). This controlled diapycnal mixing represents an im-portant advantage for isopycnal models over fixed coordinate models such as z -level and σ-level

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models. In the latter models, conserving the adiabatic properties of the advection scheme is achallenge as the advection numerical scheme creates a numerical diapycnal mixing component,regardless of how the mixing term in the governing equations is formulated. This numericalinduced diapycnal mixing is likely to overshadow the common vertical mixing that occurs innature [Chassignet et al., 1996] and can potentially lead to serious errors in global ocean cir-culation simulations [Bryan, 1987] or long term climate simulations [Toggweiler, 1994]. Only afew studies have been conducted to document, investigate, and quantify this spurious diapycnalmixing [Griffies, 1999, Lee et al., 2002, Marchesiello et al., 2008].

Detailed in this proposal is a numerical study aimed at understanding the different mech-anisms that could affect the representation of internal waves in numerical models, as well asdocumenting and quantifying the numerically induced mixing in fixed coordinate ocean models.The study focuses particularly on these main issues:

1. The ability of OGCMs to represent the internal wave as a function of model grid spacing,

2. Quantifying and documenting the spurious diapycnal mixing in the fixed coordinate oceanmodel,

3. How the hydrostatic approximation made in OGCMs affects the simulation of these wavesfor these different choices of model resolution.

The primary tools for this investigation will be two hydrostatic models, the HYbrid Co-ordinate Ocean Model (HYCOM) and the Regional Ocean Modeling System (ROMS) and anon-hydrostatic model, the MIT general circulation model (gcm). The study consists of the re-alization of the same numerical idealized simulations with all three models and the comparisonof the simulated internal wave dynamical properties, energy fields, and induced mixing withanalytical solutions. The results of this study are expected to be innovative and of broad inter-est for the ocean modeling and the tidal mixing community as they are linked to recent effortsto find a physically derived parameterization of mixing near rough topographic features. Theseresults are also expected to add to our understanding of internal wave properties (dynamicand energetic), behaviors, and associated mixing in numerical ocean models, an understandingparticularly important for the large scale modeling community. The proposal is organized asfollows: Section 2 gives a detailed background on the past literature dealing with the numericalmodeling of internal waves and their induced mixing. In section 3, specific open issues and theoriginality of the works are underlined. The proposed work is described in section 4. Finally,in section 5 some preliminary results are shown. The remainder of this prospectus provides anestimated time line to accomplish the project.

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2 Literature Review

2.1 General Background

As already stated, a better knowledge of the representation of internal waves and theirassociated mixing in oceanic numerical models is crucial to understand the global ocean cir-culation. In the deep ocean a complex series of steps are required to transfer the energy fromthe barotropic flow into the turbulent response (or mixing). Legg (2006) summarized thesesteps as: 1) conversion of the barotropic energy into baroclinic energy by pushing the stratifiedflow over topographic obstacles; 2) a local (at the tip of the rough topography) mixing due tohigh shear in the baroclinic flow; 3) radiation of energy away from the topography in the formof internal waves; 4) non-linear wave-wave interactions causing a cascade of energy to smallerscales; 5) wave-topography interactions leading to further cascade of energy to small scales;and finally 6) mixing when shear is sufficiently high (when the vertical length scale are smallenough) away from the ridge. Each of these steps has recently received significant attention.In particular, many studies have focused on the theory of step 1) which shows the dependenceof the conversion of barotropic to baroclinic energy on the topographic steepness [Blamsforthet al., 2002; Llewellyn Smith and Young, 2002; St Laurent et al., 2003]. The theory of the waveturbulence (step 2), that leads to the cascade to smaller scale, has been made by Winters andD’Asaro (1997), Lvov et al. (2004), and Polzin (2004). The theory of internal wave generationand propagation (step 3) has been well established throughout the 20th century, [Baines, 1973,1982; Bell, 1975; and more recently Miropol’sky, 2001; Khatiwala, 2003; Vlasenko et al., 2005].The study of steps 4), 5) and 6) are ongoing, especially the wave breaking through reflectionand scattering on topography [Legg and Adcroft, 2003; Zikanov and Slinn, 2001; Nash et al.,2004]. These latter steps are still not theoretically well understood. A useful alternative toexplore the internal wave behavior and properties which are not readily measurable via fieldtechniques (due to time, space, equipment, and environmental constraints), or impossible toderive analytically, is the use of numerical models. However, as seen above, the processesin step 1-6 involve a wide and large spectrum of time and space scales. This makes it verychallenging for numerical models to detail internal wave generation, propagation, breaking andmixing accurately. In the following section, we provide a brief but more specific overview of theliterature that deal with the representation of the internal wave dynamic and their associatedmixing processes in numerical models, separating it for simplicity according to: a) internal wavemodeling in OGCMs and b) Internal wave induced mixing representation in numerical models.

2.2 Internal wave modeling in OGCMs

Internal waves have horizontal scales that range from meters to a few kilometers and verticalscales that range from a few meters to one hundred meters. Then, a simple question arises:how can numerical model with horizontal grid spacing commonly greater than 5 km and a finitenumber of vertical levels (usually between 20 and 35) accurately represent these waves ?

Only recently studies have been conducted to investigate and map the flux of energy intothe tidal internal wave field. As pioneering studies, two three-dimensional primitive equations

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models have been used in a global tidal simulation: the Princeton Ocean Model (POM) andthe Hallberg isopycnal model (HIM) [Merrifield et al., 2001; Simmons, 2004; respectively]. Thefirst estimates of tidal conversion at a number of locations were derived (a specific interestwas in the Pacific with the Hawaiian ridge). They found that the numerical models resultswere qualitatively in good agreement with observational evidence. For example, the simulatedsurface expression of the internal tides was in good agreement with the first mode internal tidemapped with altimeter data by Ray and Mitchum (1996). However, quantitative comparison ofmodel outputs with observations of internal tides and localized ocean mixing demonstrate thatthere are still many unresolved issues. Recent studies have also tried to seek the impact of themodel grid resolution on the representation of the internal waves using regional models ratherthan OGCMs [Robertson, 2005; Di Lorenzo et al., 2006; Jachec et al., 2007; Bernsten, 2008].It has been shown that in order to properly simulate the internal wave one needs an accuraterepresentation of the bottom topography as well as a sufficient (not determined) model gridresolution [Jachec et al., 2007]. Di Lorenzo et al. (2006) and Zaron and Egbert (2006) showthat errors due to topography misrepresentation or inadequate model vertical discretizationcould lead to an underestimation of the tidal conversion by 50%, which could potentially bea source of important errors on the representation of the deep ocean dynamics. In a paperby Robertson (2006), it has been demonstrated (by comparison with observations) that whenusing a 1km model grid resolution, the baroclinic flow field was accurately reproduced in termsof velocities magnitude and direction and a 4km model resolution was sufficient to have aqualitative estimate of the tidal conversion mechanism. In the same paper, the author alsostates that most of the errors are attributed to the topographic errors. Recently, Bernsten etal. (2008) state that even when using coarse model grid, internal waves are present. However,their wavelength, periods and amplitudes are strongly affected and the waves are inaccuratelysimulated. In their conclusions, the authors also state that the grid spacing choice becomesincreasingly important as the Froude number (baroclinic response) increases. So the modelperformance is obviously dependent on the chosen grid resolution. The low internal wave modesare generally comprised between 20 km and 50 km (but depend on stratification, tidal frequency,latitude and water depth). Thus, it is expected that model with high enough resolution (<10km) will capture these low modes, and thus are able to represent most of the energetic field,but will not resolve the higher wave mode responsible for the turbulent response.

In OGCMs, based on practical computer limits and justified a priori scaling analyses, thehydrostatic approximation is applied. This approximation is acceptable for large-scale phe-nomena and predominantly horizontal phenomena [Kantha and Clayson, 2000]. However, thisapproximation is expected to fail in localized region where vertical and horizontal velocitiesare similar. Moreover, the real world non-linear evolution of the internal wave is strongly non-hydrostatic and thus cannot be accurately represented in hydrostatic model. Indeed, the neglectof the non-hydrostatic pressure in a hydrostatic model is problematic since the non-hydrostaticpressure plays a significant role in the internal wave evolution balancing the nonlinear wavesteepening. In other words, the governing equations solved by the model are missing a piece ofthe physics that control the internal wave evolution, thus greatly affect the prediction of thewave dissipation. Many authors have pointed out the need to include nonlinear contributions

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to the wave energy fluxes in order to be more accurate [Venayagamoorthy and Fringer, 2005;Moum, 2006]. Daily and Imberger (2003) showed that by neglecting the dispersive processesgoverning the internal wave evolution the model favor the wave steepening. This could poten-tially lead to errors in the internal wave energetic field as the wave propagates and thus leadto inaccurate representation of these waves in hydrostatic models. Hodges et al. (2006) showedthat for fine model resolution a temporal horizon (time for which the internal wave solutiondiverge from the ’true’ solution) for model skill to represent the wave can be derived, and that,for coarser grids, model results depend mostly on the diffusion and dissipation prior to thesteepening of the wave.

2.3 Internal wave induced mixing representation in numerical models

The dissipation of energy that occurs in the internal tidal beams is controlled by the ver-tical mixing rate, and thus it will control the wave evolution. These dissipation processes,as previously stated, are not resolvable in models without an excessive (i.e. greater than thecommonly used number of vertical levels, usually between 25 and 40 vertical levels) number oflevels [Robertson, 2006] although they are crucial to keep accurate water masses characteristics.In order to address this problem, the vertical mixing is parameterized in the model to representthis unresolved sub-grid scale mixing. The unresolved processes are represented by formulasinvolving parameters that govern the resolved flow. These formulas are derived using resultsfrom measurement, theory, and modeling. Many different parameterizations exist and give awide range of vertical mixing rate upon the choice made initially. The performance of 9 ofthese vertical mixing schemes (that are commonly implemented in most of the OGCMs such asMellor-Yamada, KPP, GISS, and other generic length scales parameterizations) on simulatingaccurate estimation of vertical diffusivity has been discussed in Robertson (2006) (the readeris referred to that paper for the vertical mixing parameterizations description). The authorconcludes that there is significant divergence between the temperature and salinity verticaldiffusivity rates of each model and thus in their resulting water masses distribution. Most ofthese parameterizations use the gradient Richardson (Ri) number to determine spatially andtemporally varying vertical mixing coefficients for momentum and tracers. Although they showsome agreement with observations in previous studies, most of these schemes are not appro-priate to account for the vertical mixing generated by internal wave propagation [Robertson,2006]. The need to derive and implement tidal mixing parameterization in numerical models inorder to have better agreement with observations has recently been demonstrated by the GISSmixing model that has been upgraded to include this enhanced mixing due to the break-uptide against rough bottom topography as well as bottom tidal drag and enhanced bottom shear[Canuto et al., to be submitted]. The new mixing model has been assessed using a stand-alone3D OGCM and preliminary results show improvement on the water masses characteristic ina global configuration. In Jayne and St Laurent (2001), a first diffusivity parameterizationbased on estimate of the global distribution of tidal energy available for enhanced turbulentmixing was presented and is regarded as a preliminary formulation. More specifically, in thisformulation, the vertical diffusivity resulting from the internal tide breaking is expressed as a

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function of the energy transfer from the barotropic tide to the baroclinic tide which is a functionof space and stratification. Although this parameterization accounts for stratification and bot-tom roughness (2 of the main contributor for internal wave generation), it neglects a frequencydependant factor as well as the critical latitude effect. This mixing scheme has been developedand intended for application in OGCMs, particularly those used at coarse resolution to studythe long time scales associated with the thermohaline circulation. A first implementation of theparameterization in such a model is discussed by Simmons et al. (2002), whose initial resultssuggest that the equilibrium behavior of OGCM simulation is significantly improved by thisnew mixing scheme. More recently, Koch-Larrouy et al. (2007) simulated baroclinic tides inthe Indonesian region in a regional OGCM using this specific parameterization. The authorsshow that the parameterization improves the water mass characteristic and distribution bycomparing to the available observations in that region. Thus it suggests that horizontal andvertical mixing rates were adequately prescribed.

As previously stated, accurately simulating baroclinic velocity field and vertical mixing rateis crucial to understanding the ocean dynamic. When using fixed coordinate ocean numericalmodels (that is: σ- or z -level models) a traditional error associated with the steepness ofthe vertical coordinate arises: a spurious diapycnal mixing that is likely to give unrealisticvertical mixing rate and thus is likely to produce unrealistic distribution of the water masses.This problem was quickly identified [Barnier et al., 1998] and some preliminary solutions wereproposed [Mellor et al., 1998]. Recently, Marchesiello et al. (2008) state that this type of errorin sigma coordinate model was indeed leading to unacceptable spurious diapycnal mixing andthat increasing the model resolution was not part of the solution. Example of such spuriousdiapycnal mixing problem in a σ-level model (ROMS) is shown in Figure 1. It is clear that after2 years, the model, with no forcing, should stay very close to the climatology. Instead there isapproximately a 0.7 psu difference. The author found that the diffusive term in the third-orderupstream-biased advection scheme, the most commonly used advection scheme for ROMS, wasthe source of this numerically induced vertical mixing. In the same study, they developeda new numerical scheme that is based on the splitting of the advection and diffusion termsin the advection numerical algorithm. This splitting allows for a direct access to the lateralmixing terms and therefore, an opportunity to correct for the diapycnal part of the mixing byusing rotated diffusive operators. They show that, when applied to a realistic configuration,their new advection numerical scheme reduces the spurious mixing (to within 20% of the initialerror) and simulates more accurately the water masses distribution as well as preserves thedynamical properties of the region (such as the energy spectrum showing that the energy at allscales is well simulated). The authors, using scaling analysis (allowed because of the splittingbetween the advection and diffusion terms in the numerical advection algorithm), derived tracerdiapycnal harmonic diffusivity as a function of the model resolution, the slope parameter, thedepth of the basin and the velocity of the flow. For specific choices of these parameters (choicebased on common values when running realistic configurations), they found that the spuriousmixing was increasing as the resolution becomes coarser but that there is a change of regimeat medium resolution (∼ 20 km) and that it appears to decrease when ∆x > 20 km. Thusit shows that the common solution to increase the model resolution to alleviate the spurious

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mixing present in the model is generally false. However, the use of rotated diffusion operatorsto correct for the spurious mixing introduce new errors (larger truncation errors in the sumof the diffusive term) that were not present in the original scheme and a new constraint: thevertical discretization needs to be finer (increase of the number of vertical levels). The abilityof this new advection scheme to reduce the spurious mixing still needs to be better addressedand quantified with other schemes used in ROMS.

Figure 1: Two-year evolution of salinity at 1000 m in the standard ROMS simulation (1/6◦),using the standard advection scheme. This phase corresponds to the model spinup, startingfrom ORCA05 mean state which is close to the Levitus climatology. Salinity at this depth ischaracterized by a minimum corresponding to the equatorward spreading of AAIW. a) initialstate, b) solution at 6 months and c) solution at 2 years. From Marchesiello et al. (2008).

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3 Open issues and Objectives

As reported in the above section, the literature about internal wave generation, propagationand dissipation (evolution) in OGCMs is extensive. However, many issues appear to be stillopen. In particular, the tidal mixing community’s ultimate goal is to derive a physically basedparameterization of the diapycnal mixing in the ocean interior, a key process in the globalthermohaline circulation. In order to reach this Grail, one needs to understand that there isinsufficient knowledge about the fundamentals of internal wave dynamics and energetic repre-sentation in numerical models. These latter points are mostly controlled by the choice of thegrid resolution and the choice of the turbulent closure schemes. Thus the general goal of thisproposal is designed to improve our understanding of internal wave generation and propagationin OGCMs so that, ultimately, we can better represent their associated small-scale turbulentprocesses. The general work hypothesis is that, in OGCMs, the choice of the model grid reso-lution and the hydrostatic approximation strongly impacts the dynamics and energetics of theinternal wave field, and that a new specific internal wave breaking mixing parameterizationbased on the model resolution needs to be derived. The following points will be original:

A. The investigation of internal wave generation, propagation and evolution in OGCMs as afunction of a wide range of model grid spacing (from meters to hundreds of kilometers)in the horizontal and vertical directions. Recent work by Roberston, (2005), Di Lorenzo,(2006), Zaron and Egbert (2006), and Berntsen (2008) considered a very narrow rangeof model grid resolution (from 1 m to 5 km) and regional models. In this study, themodel grid resolution will be of the order of meters to hundreds of kilometers. It will givean important and necessary insight to the ocean modeling community on how commonlyused model resolutions for more global simulations represent the dynamic and energetic ofthese waves and what is a sufficient enough resolution to capture the linear and nonlinearinternal wave dynamic. The analytical solution derived by Khatiwala (2003) will be usedfor comparison and quantitative analysis of the results.

B. The documentation of how the model partitions the energy between the baroclinic en-ergy and the induced local diapycnal mixing for these different choices of resolution. Afirst attempt by Simmons et al. (2003) to implement such partitioning in a tidal mixingmodel parameterization has been done. However, due to the absence of sufficient knowl-edge about the process, they simply assume that approximately 70% of the energy wasradiating away from the topography while nearly 30% was dissipated locally by turbu-lence. The present study will document this tidal conversion process and provide betterinsights into the internal wave energetic field as a function of the model grid spacing.This could potentially lead to the development of a new internal wave breaking mixingparameterization that will also be a function of the model grid spacing.

C. The quantification and documentation of the spurious diapycnal mixing in fixed coordi-nate ocean models. The problem of numerically induced diapycnal mixing due to theadvection scheme (transport of the density) in fixed coordinate ocean models is not newand has been widely commented (see scientific background section). However, there is

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not a single attempt in the literature to quantify this mixing and document how it evolveswith respect to the model grid spacing.

D. The quantification of the difference between hydrostatic and non-hydrostatic simulationof baroclinic tides in the linear and nonlinear regime for the wide range (meters to hundredkilometers) of model resolution.

E. Investigate the possibility of deriving from the above findings a better internal wavebreaking mixing parameterization to implement in numerical models which will dependon the model grid spacing.

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4 Proposed work

In order to address the above objectives, idealized configurations are analytically and numer-ically set up: the process of internal tide generation and dissipation by an oscillating backgroundflow interacting with a polynomial ridge. The advantage of using this type of process studyconfiguration is that, as seen in section 2, extensive work has been previously made, whereanalytical and numerical solutions are provided, allowing further comparisons with our experi-ments. In this section, we review our method to derive the analytical expression from Khatiwala(2003), as well as a thorough description of the three numerical models used, and conclude bydescribing specifically the proposed work. In this section we give a detailed description of thework we propose in order to answer the objectives points described in section 3.

Objectives A and B: qualitatively validate numerical simulation of internal wavesagainst the analytical solution of Khatiwala (2003) for an idealized configurationand determine the impact of the model grid resolution on simulating internal wavedynamical properties and energetic budget

The main goal of these two objectives is to validate the numerical simulations using HYCOMand ROMS against the analytical solution found by Khatiwala (2003) as well as investigate theimpact of model grid resolution on the representation of the dynamic and energetic of theinternal wave. The numerical setup reproduces the situation being described by Figure 2. Adomain example would be a channel of 1200 km long and 45 km wide, discretized by a uniformlyspaced grid of 801 x 31 (x, y) in the case of the 1.5 km model resolution. The obstacle is apolynomial ridge of height h (depending on the configuration) defined by Di Lorenzo et al.(2006):

h = hmax

(

1− x2

a2

)2

if |x| < a

0 if |x| > a

(1)

With a being half of the width of the ridge, hmax the maximum height of the topography, andx the distance from the ridge. The open boundaries are located at the west and east of thedomain while there is a free-slip condition at the bottom and side of the north and south walls.No bottom drag is specified to isolate the energy loss by the tidal conversion. The verticalresolution is 25 layers (uniformly spaced, 80 m) for HYCOM and 45 sigma layers for ROMSwith surface and bottom stretching coefficients being 1. The open boundaries are periodic(and thus avoid reflection and a pilled up of energy, although time integration computationsare done before the baroclinic response reaches the model boundaries) and a barotropic tidalflow of 0.02 m.s−1 is prescribed for both models with a semi-diurnal M2 frequency (12.421hr). A uniform stratification is chosen: N = 2x10−3s−1 and only depends on the temperaturechange (the salinity is set up at 35 psu in all the domain). There is no Coriolis accelerationconsidered in these idealized experiments. The choice of these parameters satisfies the lineartheory conditions (see Appendix A). The simulations are performed for 8 days (16 tidal cycles)and analyzed only over the last 4 tidal cycles. The spin-up is achieved after approximately

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one day and a half (3 tidal cycles). The model outputs are saved every 15 minutes to avoidtidal aliasing. This study focuses on the sensitivity of internal waves to the topographic height,the width of the ridge and the model grid resolution (vertical and horizontal). Thus theseparameters are the only ones to change. We also focus on a simple linear wave regime forthe comparison to the analytical solution and we deliberately ignore complications that arisewith critical latitudes and parametric subharmonic instabilities. The flow is initially at rest.There is no diapycnal mixing scheme prescribed in HYCOM while an analytical constant verticaldiffusivity of 10−8m2s−1 is used for ROMS. In order to be certain that the configuration satisfiesthe linear generation regime, the Froude number should be much smaller than 1, and the internalwave slope characteristic should be much greater than the gradient of the slope (AppendixA). To investigate the sensitivity of the stability of the water masses in the area around thesill to the various parameters used and described above (such as length and width of theridge, stratification, barotropic forcing, etc.), the time averaged value of the area of the watermasses with Richardson numbers (Ri) less than a critical value will be computed at eachtime step to indicate the potential for mixing in the different configurations (method based onBerntsen et al., 2008). In order to compare the simulation results to the analytical solutionwe will first focus on the qualitative features such as comparing snapshots of the baroclinicvelocity field and structure of the internal wave beams. By varying the length and width ofthe ridge, we will investigate how the models simulate these waves and their velocities for thesubcritical regime (linear regime). A more quantitative comparison can be obtained computingthe frequency spectra of the simulated vertical velocity [Legg and Huijts, 2006] as well as thespectra computed from Khatiwala (2003). This computation investigates the ability of themodel on decomposing the wave into vertical modes. This is crucial for understanding thetidal partitioning process and how the energy is generated at the ridge for each numericalmodels. Recently, significant attention has been paid to the energy conversion and dissipationprocesses, particularly the change in this conversion rate as the topography changes [Balmforthet al., 2002; Llewellyn Smith and Young, 2002, 2003]. Thus, we will use previous numericalexperiment and analytical formulation available to assess the quality of HYCOM and ROMSon representing the tidal partitioning process for different model resolution. In this study, wecompute the tidal conversion rate following Khatiwala (2003) using:

P =

∫L

p′((x, z = h(x), t)Ubt(x, t)

dh

dxdx (2)

where L = 2a (total length of the ridge), p ′ is the perturbation pressure ate the height ofthe ridge and Ubt the barotropic velocity. This is an accurate formulation since the verticalstructure of the water column is uniform [Lamb, 2007]. We intend to conduct the same analysisfor numerous horizontal and vertical different grid spacing. First, we concentrate on modelresolution that is typically used within the ocean modeling community such as ∆x = ∆y = 0.5km, 1.5 km, 10 km, 30 km (1/3◦) and approximately 100 km (1◦) as well as various number ofvertical layers (5 to 45). These model resolutions are widely used and range from high resolu-tion coastal modeling to global ocean circulation simulations. Thus, the results of our studyare expected to add knowledge on the numerical simulation of internal waves for the all ocean

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modeling community.

Objective C: quantify the spurious mixing in fixed coordinate models with respectto the model grid resolution

The primary goal of this objective is to quantify and document the errors associated withthe steepness of the model vertical coordinates and the transport of density. As seen in sec-tion 2, this spurious diapycnal mixing can dramatically change the water masses characteristicsand distribution within the considered domain. In the context of no diapycnal mixing schemeprescribed and no dissipation, the volume of the different water masses initially prescribedshould be conserved. This is not true for fixed coordinate models (i.e. terrain following andgeopotential models) due to the diffusive part of the higher order advection schemes commonlyused. The numerical configuration set up for ROMS (see above) reproduces this context of noprescribed diapycnal mixing and low diffusion. Thus, in estimating the evolution of these watermasses volumes from the initial state, we could determine, in ROMS, the volume loss for eachspecific density (water mass) at a specific time. This loss would only be due to the spuriousmixing, assuming that we remove or neglect the diffusion component (set at 10−8m2s−1 in theROMS configuration and diagnose every time step) as well as quantify the pressure gradienterror associated with the vertical coordinate. In our configuration, to quantify the associatedpressure gradient errors, some sensitivity tests are conducted by running the model withoutany tidal forcing and at rest in order to chose the best pre-processing options (choice of themodel numerical algorithms that gives the less numerically induced circulation considering thepressure gradient errors as well as the spurious mixing associated errors) to run our modelconfiguration. We also intend to investigate the impact of the model grid spacing on thisspurious diapycnal mixing and determine whether or not increasing the resolution can allevi-ate this problem which has recently been contested by Marchesiello et al., (2008). Recently,these authors proposed and validated a new advection scheme (implemented only in ROMS sofar, but designed for all sigma level models) where the diffusive component is split from theadvection terms in the advection numerical algorithm and thus does not seem to generate asmuch spurious diapycnal mixing. In this study, we will also use this new scheme to investigateits properties in simulating the internal waves and document if it better preserves the watermasses characteristics and distribution by comparing simulation results to the previous advec-tion scheme simulations.

Objective D: document the impact of the hydrostatic approximation on the evolu-tion of the internal wave for different model grid resolution and wave regime

To address this objective, the same idealized experiments seen in section 4 are conductedwith the MIT-gcm configured in both hydrostatic and non-hydrostatic mode. By comparingthe simulation results to the previous numerical experiment and the analytical solution, wedocument the impact of the hydrostatic approximation on the dynamic and energetic of thesimulated internal waves. On doing so, we expect to find some convergence criteria for the modelgrid spacing where, for linear wave regime, the hydrostatic approximation does not impact thenumerical solution and accurately reproduces internal wave properties and behavior. We also

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intend to set up the MIT-gcm with and without the hydrostatic approximation for configu-rations where the internal wave regime is strongly non-linear (higher and steeper topography,stronger barotropic forcing) and for the model resolution range seen in section 4. The primarygoal of conducting these simulations is to investigate how the model resolution can impact theinternal wave generation and propagation processes for highly non-linear (supercritical) regime.

Objective E: investigate the possibility of deriving from the above points a betterinternal wave diapycnal mixing parameterization to implement in numerical models

Based on the previous findings we will investigate the possibility to develop a better param-eterization in regards to the model resolution chosen. We expect that coarser model resolutionwill change the simulated baroclinic velocity field as well as the density of the water column,thus will change the internal wave energy field and the tidal partitioning process. This willdramatically influence the preliminary formulation from Jayne and St Laurent (2001) that ac-counts for tidal dissipation in numerical models and depends on the tidal conversion process(energy available), local density and space. Thus, we expect this formulation to also vary infunction of the model grid spacing and thus to be a function of the model resolution.

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5 Preliminary results

In this section we present preliminary results obtained so far for the objectives described insection 3.

A. Validation of the HYCOM and ROMS numerical simulations against theanalytical solution and previous numerical simulations

In this paragraph, we present some qualitative features of the results obtained with theanalytical solution, HYCOM and ROMS. Figure 2 shows snapshots of cross-vertical sections ofthe baroclinic velocity field taken at the same time for a) the analytical solution, b) HYCOMand c) ROMS, set up as the idealized configuration described in section 4. Since the numericalresults are compared against the analytical solution, the linear theory must be satisfied andthus, the steepness parameter must be less than 1 (see Appendix A). In this particular case, theheight of the bump is 200 m and thus satisfy ε � 1. From visual inspection of the Figure 2 it isclear that both models are similar to the analytical solution. The magnitude of the baroclinicresponse is very similar in all cases and reaches a maximum at the surface and the bottom(∼ 0.01 m.s−1). It seems that, both models slighlty underestimate these maxima especiallywhen the signal has propagated far away from the ridge. This could potentially come fromsome numerical diffusion/dissipation since no bottom friction has been prescribed. The onlysignificant difference is in the group speed of the wave (internal beam), that appears to belagging (delayed). Indeed, the analytical solution gives a maximum surface baroclinic velocityat 84 km. In HYCOM, the wave has travelled 78 km to reach the same maximum and thus isvery close to what the analytical solution predicted (3 grid points) and in ROMS, the wave hastravelled approximately 66 km (12 grid points) (Figure 2). Since the group speed depends onlyon the stratification, coriolis force and the frequency of the tidal forcing, the hypothesis is that,in ROMS, the spurious mixing changes the stratification and thus changes the dynamic of theinternal wave by changing one of the controlling parameters of the internal wave characteristic(beam).

Figure 2: Cross vertical sections of baroclinic velocities for a) the analytical solution, b)HYCOM and (c) ROMS. The black solid lines are the analytical wave characteristic (waveslope/beam).

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B. Impact of the model resolution on the simulation of internal waves

This section describes how the grid resolution choice affects the representation of the in-ternal wave in the numerical model. Figure 3 shows snapshots of cross vertical sections of thebaroclinic velocities simulated with ROMS where the only parameters to vary are the bumpheight and the model grid spacing (from 1.5 km to 30 km). The ridge in Figure 3a and 3b hasa width of 300 km and a height of 200 m. The wave regime is linear (given the configurationparameter the steepness parameter is satisfied). It appears that the 2 smulations are very simi-lar. When ∆x = 1.5 km the baroclinic response appears to be slightly stronger except near theridge. The mode 1 internal wave that propagate away from the ridg is clearly visible and is atthe same location for both simulations. Thus, we surmize that the resolution in a subcriticalwave regime does not seem to affect the internal wave group speed but only the magnitude ofthe baroclinic response. This is in agreement with previous studies from Robertson (2006) andDi Lorenzo et al. (2006). The ridge in Figure 3c and 3d has a width of 300 km and a height of1600 m. Thus the wave regime is considered to be supercritical and the linear theroy does notapply anymore. The 2 simualtions present significant differences at the ridge. When ∆x = 1.5km the baroclinic response at the top of the ridge is strong and baroclinic horizontal velocitiesgrdaient are high switching from negative to positive magnitude. This is due to the higher wavemodes (high frequency) that interact between each other. At coarser resolution, these highermodes are not present. The mode 1 internal wave that propagate away from the ridg is alsoclearly visible and is at the same location for both simulations. Thus, we surmize that, in asupercritical wave regime, the model resolution does not seem to affect the internal wave groupspeed for low wave modes but does strongly affect the higher modes simulation typically foundnear the ridge. This is mostly due to the fact that, since lower mode have wavelength greaterthan the grid spacing the model is able to capture and represent them. However, the highermodes with high frequency and wavelength that the model is unable to capture when ∆x = 30km is unable to represent them. If these higher modes, that are responsible for the turbulentresponse (and thus the enhanced local mixing), are not resolved in the coarse resolution simu-lation, it could strongly affect existing mixing parameterization that should probably take intoaccount the model resolution.

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Figure 3: Impact of the resolution on the internal wave generation and propagation using ROMSfor different wave regime. In the left panels a) and c) are the high resolution simulations ∆x= 1.5 km and b) and d) are the coarse resolution simulations with ∆x = 30 km. The modelgrid spacing and the bump height are the only parameters to vary in these configurations.

C. Quantification of the spurious mixing in the fixed coordinate ocean model(ROMS)

This section quantifies the spurious mixing associated with the propagation of the simulatedinternal wave in ROMS (the fixed coordinate ocean model), based on the method described insection 4. In Figure 4a, the left panel shows the density field after 5 days of simulation, andthe right panel shows the time evolution of a volume for a specific density interval (representedby the black solid dot in the density field graphic and corrsponding to the density class 25.285-25.295). The volume of this water mass is clearly decreasing with time by approximatelyhalf of its initial volume. In the context of no diapycnal mixing (as we prescribed in the modelconfiguration) this should not be the case and thus we associate this volume loss to the spuriousdiapycnal mixing only. It appears that the water mass characterized by the density class [25.985-25.295] loses half of its initial volume to other density interval. This is shown in Figure 4b.The left panel shows the volume as a function of the density at the initial time (blue solid line)and after 5 days (black solid line). After 5 days, the initial prescribed water masses volumehave leaked into other density intervals. This leakage corresponds to 0.01 kg.m−3 (equivalent

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to 10 different intervals). In Figure 4b right panel, histograms of the number of intervals with aparticular volume are shown. At the initial time, 45 different density intervals are representedthat corresponds to the 45 initial water masses (number of vertical levels) prescribe (as theinitial stratification). After 5 days, a considerable number of new density intervals (that arisesfrom the leakage due to the spurious diapycnal mixing) with small volume appears. This changein the stratification is only due to the numerics of the model and through time, it will affectconsiderably the dynamic of the internal wave by changing the wave group speed (see part Aof this section). As an example, if the density changes locally by 0.01 kg.m−3 it corresponds toa change of 3 meters in the internal wave beam slope (a loss or gain of 3 meters in the verticalfor every 1 km the wave travelled horizontally).

Figure 4: a) the left panel is the density field after 5 days (left panel) and the right panel isthe evolution of the water mass volume within density class [25.285-25.295] represented by theblack solid dot. b) The left panel quantifies the volume of some specific density with the bluesolid line being at the initial time and the black line after 5 days of the simulation. The densityinterval is 0.001 kg.m−3. The right panel shows the histograms of the number of specific densityinterval at the initial time and after 5 days.

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6 Time line

This project will be completed in approximately one year and four months according tothe timeline summarized in Table 1. In the first 3 months, the idealized experiments thatcorrespond to objective points A, B and C (internal wave dynamic, energetic and associatedmixing in OGCMs as a function of model grid spacing) will be finished and will result in thepreparation of a written paper. The remaining year will be devoted to the points describedby objectives D (non-hydrostatic numerical experiment) and E (investigate the possibility ofa new internal wave breaking parameterization). The remaining time will be devoted to thepreparation of a written article as well as the preparation of the PhD dissertation.

Table 1: Time schedule to accomplish this project

Timeline Task to accomplishEnd of March Defend prospectus

April 2009 Finish idealized experiment with HYCOM and ROMSEnd of April Talk at the EGU meeting in Vienna

May 2009 to December 2009 Non-hydrostatic simulation with the MIT-gcmMay 2009 Start writing paper about idealized configuration

End of December 2009 Finish all numerical experimentsJanuary 2010 - April 2010 Write the PhD dissertation, review and defense

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Appendix

A Analytical Solution of a Baroclinic Tidal Flow over a

Small Bottom Topography

In this section, we describe the analytical solution derived by Khatiwala (2003) for abarotropic tide interacting with an oceanic ridge. This derivation is based on earlier workfrom Bell (1975) and the linear theory of internal waves. However, this solution has a reflectingfree surface which fundamentally changes the nature of Bell’s solution and is proven to be amore valid hypothesis than a radiation condition. For a complete description of the solutionderivation, the reader is referred to Khatiwala (2003). In this study, we consider the followingidealized configuration for both the analytical and numerical models: an oscillating backgroundbarotropic flow (representing the tidal flow) interacting with an oceanic ridge (periodic topog-raphy) as shown in Figure 5.

Figure 5: Schematic used to illustrate the situation being modeled and establishing the notation(From Kathiwala, 2003)

In a frame that moves with the background flow, the governing equation for the vertical velocityis given by:

∂

∂t2∇2W ′ + f 2∂2W ′

∂z2+ N2∂2W ′

∂ξ2+ r

∂

∂t∇2W ′ = 0, (3)

where W ′ is the vertical velocity, f the Coriolis force, r is an inverse time scale for damping thewave, N is the buoyancy frequency (constant), t the time, z the depth, and ξ is the horizontalcoordinate in the moving frame defined by:

ξ(x, t) = x−∫ t

0

U0 cos ω0t′dt′ = x− U0

ω0

sin ω0t, (4)

Where ω0 is the forcing frequency and U0 the velocity of the flow. Then applying a Fouriertransform to the topography and the vertical velocity expressions, in doing so, going to thespectral space instead of the physical space, we can write:

d2Wn

dz2+ k2

[N2 − n2ω20 + imω0

n2ω20 − f 2 − imω0

]Wn = 0, (5)

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where Wn is the vertical velocity in the spectral space of order n, and k is the topographywavelength.The solution of this equation can be written as:

w′n(x, z, t) ≈

n0∑n=−n0

∫ ∞

−∞w(k)dk, (6)

wherew(k) =

1

2πh(k)inω0

sin µn(H − z)

sin µnH× Jn(

U0k

ω0

)ei(nω0t+kξ), (7)

where ξ is defined in equation 4. Using boundary conditions and the Cauchy’s theorem it leadsto the final expression of the vertical velocity in the moving frame ξ: if ξ>0 then, from Eq. 4we have:

w′(x, z, t) ≈∞∑

j=1

− 1

jπ

n0∑n=1

(−1)nkjnnω0Jn(U0kjn

ω0

) sinjπz

H×

[h(−kjn)ei(nω0t−kjnξ)+h(kjn)e−i(nω0t−kjnξ)

](8)

if ξ<0 then, from Eq. 4 we have:

w′(x, z, t) ≈∞∑

j=1

− 1

jπ

n0∑n=1

kjnnω0Jn(U0kjn

ω0

) sinjπz

H×

[h(kjn)ei(nω0t+kjnξ)+h(−kjn)e−i(nω0t+kjnξ)

],

(9)where

kjn =jπ

H

[ n2ω20 − f 2

N2 − n2ω20

]1/2, (10)

where kjn represent the poles in the k -plane, H is the maximum depth, j are the modes number,is the Fourier transform of the topography and Jn is the Bessel function of order n. It is clearthat Eq. (6) and Eq. (7) consist of a wave with a vertical structure that propagates horizontallyaway from the ridge. In the above derivation, a number of approximations have been made. Inparticular, the solution is based on the linear theory of the internal wave thus requiring thatthe baroclinic response is much smaller than the magnitude of the barotropic forcing. Anothernecessary condition for the above theory to be valid concerns a well-known parameter, thesteepness parameter ε and is expressed as:

ε =h0k√

(ω20 − f 2)/(N2 − ω2

0)� 1, (11)

When the above analytical formulation is used to validate the model, these conditions areapplied in order to satisfy the linear theory.

B Numerical Models

The choice of vertical coordinates is a key feature in ocean modeling. Currently, one has3 options: geopotential, terrain-following, and isopycnal vertical discretization. However, none

22

provides a universal utility [Griffies, 2004]. One of the strong advantages of isopycnic modelsover fixed vertical coordinate models is that no spurious diapycnal mixing exists, thus conservingwater masses properties through time. The main advantage of terrain-following models is theretaining of high resolution in shallow waters. In this study, we employed two hydrostaticmodels: the HYbrid Coordinate Ocean Model (HYCOM) used in a fully isopycnal mode andthe Regional Ocean Modeling System (ROMS) used in sigma coordinates. A third model isused and provides the additional non-hydrostatic capability: the MIT-gcm in order to accuratelyresolve the evolution of the wave. This section is an overview of the three models.

B.1 The Hybrid Coordinate Ocean Model (HYCOM)

The awareness that there is no optimal vertical coordinate has led to the development ofa generalized-coordinate ocean model, the HYbrid Coordinate Ocean Model (HYCOM, Bleck,2002). The implementation of the generalized coordinate in HYCOM follows the theoreticalfoundation set forth in Bleck and Boudra, (1981) and Bleck and Benjamin (1993), that is, eachcoordinate surface is assigned a reference isopycnal. The model continually checks whethergrid points lie on their reference isopycnals and, if not, attempts to move them verticallytoward the reference position. However, the grid points are not allowed to migrate when thiswould lead to excessive crowding of coordinate surfaces. Thus, vertical grid points can begeometrically constrained to remain at a fixed depth while being allowed to join and followtheir reference isopycnals in adjacent areas [Bleck, 2002]. The default configuration in HYCOMis isopycnal in the open stratified ocean but smoothly reverts to a terrain-following coordinatein shallow coastal regions and to fixed pressure-level coordinates in the surface mixed layerand/or unstratified seas [Chassignet et al., 2003]. HYCOM is in this way able to combine theadvantages of the different types of coordinates in optimally simulating coastal and open-oceancirculation features. Even if it is a hybrid model, HYCOM can be still configured with onlyone coordinate, that is, isopycnic, pressure-level, or terrain-following. Moreover several verticalmixing parameterizations have been implemented in HYCOM. The model can be integratedwith a Kraus-Turner (KT, Kraus and Turner, 1967) slab mixed layer, a KPP [Large et al., 1994,1997] non-slab vertical mixing formulation, a Price-Weller-Pinkel (PWP, Price et al., 1986) slabmixed layer, a MellorYamada level-2.5 turbulence closure model [Mellor and Yamada, 1974,1982], or a Goddard Institute for Space Studies (GISS) level-2.0 turbulence closure model[Canuto et al., 2001, 2002]. A detailed discussion of the relative performance of these fivemixed layers can be found in Halliwell, (2004). Moreover, in the continuous effort to improverepresentation of the real word, other parameterizations are under development and testing[Chang et al., 2005; Xu et al., 2006]. The main advantage in using HYCOM in assessing thegeneration and propagation of internal waves is that, when run in fully isopycnic coordinate,the only mixing present in the domain is the one prescribed by the user ( i.e. a ’controlled’mixing as opposed to a spurious one).

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B.2 The Regional Ocean Modeling System (ROMS)

The other hydrostatic model employed in this study is the Regional Oceanic Modeling Sys-tem (ROMS). For a complete description, the reader is referred to Shchepetkin and McWilliams(2003, 2005). The model is widely used in the ocean modeling community (www.myroms.org),especially for coastal ocean modeling. Here we briefly describe the model. ROMS is a split-explicit, free-surface and terrain-following vertical coordinate oceanic model, where short timesteps are used to advance the surface elevation and barotropic momentum equation and wherea much larger time step is used for temperature, salinity, and baroclinic momentum. ROMSemploys a two-way time-averaging procedure for the barotropic mode which satisfies the 3Dcontinuity equation. A third-order, upstream-biased, dissipative advection scheme for momen-tum, implemented with a specially designed predictor-corrector time-step algorithm (Leapfrog-Adams-Moulton III, LF-AM3; Shchepetkin and McWilliams, 2005), allows the generation ofsteep gradients, enhancing the effective resolution of the solution for a given grid size [Shchep-etkin and McWilliams, 1998]. A K-profile parameterization (KPP) boundary layer scheme[Large et al., 1994] usually parameterizes the subgrid-scale vertical mixing processes but otheroption are available that are similar to the one that HYCOM contains.

B.3 The MIT general circulation model

The MIT general circulation model (gcm) solves the non-hydrostatic, non-linear primitiveequations using a finite volume formulation [Marshall et al., 1997]. This feature is essentialto adequately represent the dynamic and evolution of the internal waves. In the model, thetopography is represented by lopped cell (equivalent of partial steps) [Adcroft et al., 1997]which is also necessary for accurately representing its interaction with the tide as it allows fora more precise diagnosis of the topographic drag and power input. The model has an implicitfree-surface formulation. The vertical mixing schemes implemented in the model are similar tothe ones present in HYCOM or ROMS such as KPP, Mellor-Yamada, etc. The unique abilityof MIT-gcm to treat non-hydrostatic dynamics in the presence of complex geometry makes itan ideal tool to study internal wave dynamics and mixing in oceanic canyons and ridges drivenby large amplitude barotropic tidal currents imposed through open boundary conditions.

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Internal wave modeling in oceanic numerical models: …coaps.fsu.edu/~gouillon/PAPER/prospectus_flav.pdf · representation of the key phenomena of these dynamics, numerical models