29th Symposium on Naval Hydrodynamics Gothenburg, Sweden, 26-31 August 2012

Numerical Simulation of Internal Generation at a Break Laura K. Brandt, James W. Rottman, Kyle A. Brucker and Douglas G.Dommermuth Naval Hydrodynamics Division, Science Applications International Corporation 10260 Campus Point Drive, San Diego, CA 92121, USA

ABSTRACT cline, as described by Gerkema (2001) and Mauge´ and Gerkema (2008). A fully nonlinear, three-dimensional numerical model In this paper, we develop a numerical scheme is developed for the simulation of tidal flow over arbi- capable of accurately simulating the generation of these trary bottom topography in an with realistic strat- beams with realistic ocean stratification and bottom to- ification. The model is capable of simulating accurately pography. The numerical method is fully nonlinear, the generation of fine-scale internal tidal beams, and uses a technique similar to the Cartesian-grid free- their interaction with an ocean and the sub- surface capturing code called Numerical Flow Analysis sequent generation of solitary internal that propa- (NFA) of Dommermuth, OShea, Wyatt, Ratcliffe, Wey- gate on this thermocline. Several preliminary simulation mouth, Hendrickson, Yue, Sussman, Adams and Va- results are shown for uniform and non-uniform flow over lenciano (2007) and Rottman, Brucker, Dommermuth an idealized two-dimensional ridge, which are compared and Broutman (2010a), but modified to handle nonlinear with linear theory, and for flow over an idealized two- background stratification and ocean bottom topography. dimensional continental shelf. The scheme is sufficiently robust to simulate the genera- tion of the beams, the interaction of these internal wave beams with an ocean thermocline, and the INTRODUCTION subsequent propagation of the generated internal solitary waves shoreward over realistic bottom topography. It is important to have a good understanding of the ocean This paper also describes some preliminary environment in which surface and subsurface op- simulations for stratified flow over idealized two- erate. In particular, operating in the littoral dimensional ridges, which are compared with linear the- ocean environment can be significantly affected by the ory, and a two-dimensional shelf break, including the presence of large-amplitude internal waves. A gener- generation of internal wave beams and the interaction of ation mechanism for these waves is the motion of the these beams with idealized ocean . barotropic tide over continental shelf breaks, as dis- cussed, for example, by Pingree and New (1989), Hol- lowy, Chatwin and Craig (2001), Lien and Gregg (2001), THE NUMERICAL MODEL and Garrett and Kunze (2007). The ultimate objective is to produce a forecast model for the generation and prop- We consider nonlinear, three-dimensional, stratified fluid agation of large amplitude internal waves in a realistic flow over bottom topography. A Cartesian coordinate ocean in the regions about a continental shelf. system (x, y, z) is used with z as the vertical coordinate arXiv:1410.1896v1 [.flu-dyn] 7 Oct 2014 The numerical modeling of this generation pro- and (x, y) as the horizontal coordinates. The background cess is difficult because of the complexity of the topogra- stratification is assumed variable in the vertical but ho- phy, the complicated structure of the ocean stratification mogeneous in the horizontal. Typically, we will impose and currents, and the wide range of spatial and tempo- a background stratification that represents an ocean with ral scales. When the continental slope is near a criti- a seasonal thermocline. The background is as- cal value, which occurs often, a fine-scale internal wave sumed to be forced by a barotropic tide. beam is generated that must be accurately resolved in a The computer code Numerical Flow Analy- domain that has the very large length scale of the ocean sis (NFA), Dommermuth et al. (2007), originally de- shelf region. Also, it is possible for this internal wave signed to provide turnkey capabilities to simulate the beam to produce internal solitary waves by the interac- free-surface flow around ships, has been extended to have tion of the beam with a moderately strong ocean thermo- the ability to perform high-fidelity stratified sub-surface calculations. The governing equations are formulated on respectively : a Cartesian grid thereby eliminating complications asso- L0 ∂ρ ciated with body-fitted grids. The sole geometric input γC = (5) into NFA is a surface panelization of the and/or bot- ρ0 ∂x3 0 tom. No additional gridding beyond what is used already ∆ρ γJ = . (6) in potential-flow methods and calculations ρ0 is required. The ease of input in combination with a Here, (∂ρ/∂x ) is the dimensional characteristic mean- flow solver that is implemented using parallel-computing 3 0 gradient and ∆ρ is the dimensional density jump. methods permit the rapid turnaround of numerical simu- The density fluctuations are split into two parts because lations of high-Re stratified fluid interactions with a com- they require different theoretical and numerical treat- plex bottom. ments. The grid is stretched along the Cartesian axes The splitting requires an additional equation using one-dimensional elliptic equations to improve res- that we choose as follows. olution near the bottom and the mixing layer. Away from ∂ρ˜ ∂u ρ˜ the bottom and the mixing layer, where the flow is less J + j J = 0 . (7) complicated, the mesh is coarser. Details of the grid- ∂t ∂xj stretching algorithm, which uses functions that Substituting (4) and (7) into (1) gives are specified in physical , are provided in Knupp and Steinberg (1993). ∂ρ˜ ∂u (ρ +ρ ˜ ) C + j C C = 0 . (8) ∂t ∂xj

GOVERNING EQUATIONS For an infinite Reynolds number, viscous stresses are negligible, and the conservation of momen- Consider a turbulent flow in a stratified fluid. tum is Physical quantities are normalized by characteristic ve- ∂ρui ∂ ∂p δi3 locity (U0), length (L0), (L0/U0), density (ρ0), and + (ρujui) = − − 2 − τi , (9) 2 ∂t ∂xj ∂xi Fr (ρ0U0 ) scales. Let ρ and ui, respectively de- note the normalized density and three-dimensional ve- where p is the normalized pressure and τi is a normalized locity field as a function of normalized space (xi) and that will act tangential to the surface of the bottom. normalized time (t). The is δij is the Kronecker delta function. Fr is a Froude num- ber: ∂ρ ∂ujρ gL0 + = 0 . (1) Fr ≡ 2 , (10) ∂t ∂xj U0 where g is the acceleration of . The Froude num- For incompressible flow with no , ber is the ratio of inertial to gravitational . As O’Shea, Brucker, Dommermuth and Wyatt (2008) dis- ∂ρ ∂ρ cuss, the sub-grid scale stresses are modeled implicitly + uj = 0 . (2) ∂t ∂xj in9. The pressure, p is then decomposed into the dy- Subtracting (2) from (1) gives a solenoidal condition for namic, p , and hydrostatic, p , components as the velocity: d h ∂u p = pd + ph . (11) i = 0 . (3) ∂xi The hydrostatic pressure is defined in terms of the refer- ence density and the density stratification as follows. The normalized density is decomposed in terms of the constant reference density plus small departures ∂ph δi3 = −(1 + ρC ) 2 . (12) that are further split into a known mean perturbation ∂xi Fr (ρC ), a continuous fluctuation (ρ˜C ) due to the mean den- sity gradient, and a fluctuation with a discontinuous jump The substitution of (4) and (12) into (9) and us- ing (7) and (8) to simplify terms gives a new expression in the density (ρ˜J ) corresponding to the bottom of the mixing layer: for the conservation of :

∂ui ∂ 1 + (ujui) = − ρ = 1 + γC ρC (x3) + γC ρ˜C (xi, t) + γJ ρ˜J (xi, t) . (4) ∂t ∂xj 1 + γC (ρC +ρ ˜C ) + γJ ρ˜J   ∂pd δi3 γC and γJ quantify the magnitudes of the density fluc- + (γC ρ˜C + γJ ρ˜J ) 2 + τi . (13) tuations for the continuous and discontinuous portions, ∂xi Fr

2 ∗ If γC << 1 and γJ << 1, a Boussinesq approximation The first prediction for the velocity field (ui ) is may be employed in the preceding equation to yield  u∗ = uk + ∆t Rk ∂u ∂ i i i i + (u u ) = ∂t ∂x j i   k  j 1 ∂pd   − k k . (18) ∂pd 1 + γC (ρC +ρ ˜C ) + γJ ρ˜J ∂xi − + (RiBC ρ˜C + RiBJ ρ˜J ) δi3 + τi . (14) ∂xi The are advanced using the mass conservation equations (7) and (8): where RiBC and RiBJ are bulk Richardson numbers de- fined as ∗ k ∂ k k  ρ˜C =ρ ˜C − ∆t uj (ρC +ρ ˜C ) (19) ∂xj γC L0 ∂ρ gL0 Ri ≡ ρ˜ = (15) ∗ k k k  BC 2 C 2 ρ˜ =ρ ˜ − VOF u , ρ˜ , ∆t . (20) Fr ρ0 ∂x3 0 U0 J J j J γJ ∆ρ gL0 The advective terms for ρ˜C are calculated using a third- RiBJ ≡ 2 ρ˜J = 2 , (16) Fr ρ0 U0 order finite-volume approximation, whereas the advec- tion of ρ˜J is calculated using the Volume of (VOF) The bulk Richardson numbers are the ratios of buoyant to method. A Poisson equation for the pressure is solved inertial forces for continuous and discontinuous density again during the second stage of the Runge-Kutta algo- fluctuations. rithm: The momentum equations using either (13) or   ∗ (14) and the mass conservation equations (7) and (8) are ∂ 1 ∂pd ∗ ∗ = integrated with respect to time. The divergence of the ∂xi 1 + γC (ρC +ρ ˜C ) + γJ ρ˜J ∂xi momentum equations in combination with the solenoidal  ∗ k  ∂ ui + ui ∗ condition (3) provides a Poisson equation for the dy- + Ri , (21) ∂xi ∆t namic pressure. The dynamic pressure is used to project the velocity onto a solenoidal field and to impose a no- ui is advanced to the next step to complete one cycle of flux condition on the surface of the body. The details of the Runge-Kutta algorithm: the time integration, the pressure projection, the formu- lations of the body boundary conditions, and the formu- 1 ∆t uk+1 = u∗ + uk + R∗ lations of the inflow and outflow boundary conditions are i 2 i i 2 i described in the next three sections.   ∗  1 ∂pd − ∗ ∗ , (22) 1 + γC (ρC +ρ ˜C ) + γJ ρ˜J ∂xi TIME INTEGRATION and the densities are advanced to complete the algorithm:

A second-order Runge-Kutta scheme is used to ρ˜∗ +ρ ˜k ∆t ∂ ρ˜k+1 = C C − u∗(ρ +ρ ˜∗ ) integrate with respect to time the field equations for the C 2 2 ∂x j C C velocity and density. During the first stage of the Runge- j ∗ k ! Kutta algorithm, a Poisson equation for the pressure is uj + uj ρ˜k+1 =ρ ˜k − VOF , ρ˜∗ , ∆t . (23) solved: J J 2 J

∂  1  ∂pk d ENFORCEMENTOF NO-FLUX BOUNDARY k k = ∂xi 1 + γC (ρC +ρ ˜C ) + γJ ρ˜J ∂xi CONDITIONS ∂  uk  i + Rk , (17) A no-flux condition is satisfied on the surface of ∂x ∆t i i the bottom using a finite-volume technique.

k where Ri denotes the nonlinear convective, , uini = vini , (24) and stress terms in the momentum equation, (13), at time k k k k step k. ui , ρ˜C , ρ˜J , and pd are, respectively, the veloc- where ni denotes the unit normal to the body that points ity components, continuous fluctuating density, discon- into the fluid and vi is the velocity of the bottom. If the tinuous fluctuating density, and dynamic pressure at time bottom is not moving, vi = 0. Cells near the bottom may step k. ∆t is the time step. For the next step, this pres- have an irregular shape, depending on how the surface of sure is used to project the velocity onto a solenoidal field. the bottom cuts the cell. Let Sb denote the portion of the

3 cell whose surface is on the bottom, and let S0 denote body- formulation as follows. the other bounding surfaces of the cell that are not on the + bottom. τi = β(ui − vi)δ(x − xb ) , (29)

Gauss’s theorem is applied to the volume inte- where β is a body-force coefficient, vi is the velocity + gral of (17): of the body, x is a point in the fluid, and xb is a point slightly outside the body. Equation (29) forces the fluid Z  n  ∂pk velocity to match the velocity of the body. Note that free- ds i d = k k slip boundary conditions are recovered with β = 0 and 1 + γC (ρC +ρ ˜C ) + γJ ρ˜J ∂xi S0+Sb no-slip boundary conditions are imposed as β → ∞. Z  k  ui ni Dommermuth, Innis, Luth, Novikov, Schlageter ds + Rini . (25) ∆t and Talcott (1998) discuss modeling using body-force S0+Sb formulations. Recently, there have been several stud- Here, ni denotes the components of the unit normal on ies which use similar body boundary conditions in both the surfaces that bound the cell. Based on (18), a Neu- finite volume (Meyer, Devesa, Hickel, Hu and Adams mann condition is derived for the pressure on Sb as fol- (2010a), Meyer, Hickel and Adams (2010b) and finite el- lows ement (Hoffman (2006a), Hoffman (2006b), John (2002) simulations. The finite element simulations of Hoffman   k ni ∂pd (2006b) at very high Reynolds numbers are able to pre- ∗ k = 1 + γC (ρC +ρ ˜C ) + γJ ρ˜J ∂xi dict the lift and drag to within a few percent of the con- ∗ k sensus values from experiments, using a very economi- u ni u ni − i + i + R n . (26) cal number of grid points. The finite volume implemen- ∆t ∆t i i tations have also shown promise albeit at more modest The Neumann condition for the velocity (24) is substi- Reynolds numbers (Re = 3, 900). Rottman, Brucker, tuted into the preceding equation to complete the Neu- Dommermuth and Broutman (2010b) show that finite mann condition for the pressure on Sb: volume simulations with Re → ∞ are able to accu- rately predict flow about bluff-bodies using a partial-slip   k ni ∂pd type boundary condition to model the effects of the un- ∗ k = 1 + γC (ρC +ρ ˜C ) + γJ ρ˜J ∂xi resolved turbulent boundary layer. v∗n ukn − i i + i i + R n . (27) ∆t ∆t i i FORMULATION OF INFLOW AND OUTFLOW This Neumann condition for the pressure is substituted BOUNDARY CONDITIONS into the integral formulation in 25: The effects of oscillating tidal currents can be Z  n  ∂pk modeled by including an oscillatory current in velocity: ds i d = ∗ k ∂x c 1 + γC (ρC +ρ ˜C ) + γJ ρ˜J i u1 ← u1 + U cos(ωt). (30) S0 Z  k  Z ∗ c ui ni vi ni U is the amplitude of the current at inflow normalized ds + Rini + ds . (28) ∆t ∆t by U0 and ω is the frequency of the current normalized S0 Sb by L0/U0. This equation is solved using the method of fractional ar- The velocities near the inflow and outflow eas. Details associated with the calculation of the area boundaries are nudged every N time steps toward target fractions are provided in Sussman and Dommermuth velocities that conserve flux: (2001) along with additional references. Cells with a cut u ← u − βN∆tf I (x)(u − U I ) for x ∈ ∨ volume of less than 2% of the full volume of the cell 1 1 1 I O O are merged with neighbors. The merging occurs along u1 ← u1 − βN∆tf (x)(u1 − U ) for x ∈ ∨0. the direction of the normal to the body. This improves (31) the conditioning of the Poisson equation for the pressure. I O As a result, the stability of the projection operator for the β is a relaxation parameter, and f (x) and f (x) are velocity is also improved (see Equations (18) and (22)). tapering functions at the inflow and outflow regions, re- spectively. U I and U O are the target velocities at the inflow and outflow boundaries, respectively. ∨I and ∨0 ENFORCEMENTOF NO-SLIP BOUNDARY denote regions near the inflow and outflow boundaries, CONDITIONS respectively. I The stress τi is used to impose partial-slip and Away from the inflow boundary, f (x) = 0. no-slip conditions on the surface of the bottom using a At the inflow boundary, f I (x) = 1. f I (x) smoothly

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−2 −2 1 1.5 2 2.5 3 1 1.5 2 2.5 3 3.5 4 4.5 ρ /ρo ρ /ρo (a) (b)

Figure 1: Normalized density ρ/ρ0 as a function of normalized depth z/L0, where L0 = H − h0: (a) uniform stratification; and (b) nonuniform stratification representing a typical seasonal thermocline.

transitions between the two regions. Similarly, f O(x) = the ratio of the slope of the topography to the slope of the O 0 away from the outflow boundary, and f (x) = 1 at the tidally generated internal waves, S = h0/(Ls), where s outflow boundary. = pω2/(N 2 − ω2). At inflow, the target velocity is set to zero, We performed two-dimensional numerical sim- U I = 0, then by conservation of flux, ulations for tidal flow over an asymmetrical ridge with uniform stratification, as shown in figure1a, or nonuni- S  U O = I − 1 U c cos(ωt) (32) form stratification, as shown in figure1b. The nonuni- S0 form stratification represents an idealized seasonal ther- mocline. The simulations have ω/N = 0.67 and where S0 is the surface area at the outflow boundary, and h0/H = 0.5. For linear theory to be accurate, the flow SI is the surface area at the inflow boundary. must have a subcritical value for the steepness parameter (S < 1) and a small value for the excursion parameter VALIDATION (ex << 1). In figures2 and3, S = 0.42, 0.75 and ex = 0.029, 0.062, for the shallow slope side of the to- pography and steep slope side of the topography, respec- We desribe several preliminary simulations using NFA to tively. The steepness parameters are subcritical, but close assess the accuracy of the numerical scheme for simulat- enough to critical for some nonlinear effects to exist. ing realistic internal wave tidal beams and their interac- tions with an ocean thermocline. Figure2a shows results from an NFA simula- tion of uniformly stratified flow over a ridge after 10 tidal periods of adjustment. Visually, the comparison between FLOWOVERATWO-DIMENSIONALRIDGE: NFA and linear theory (figure2b) is very good. The tidal COMPARISON WITH LINEAR THEORY beam is generated at the steepest part of the slope and is seen as a straight line in the flow, since the background A first step to validate the numerical technique stratification is homogeneous. The dashed line in figure is to compare its results for an idealized two-dimensional 2a shows the ray path predicted by linear theory, which problem with linear theory. Echeverri and Peacock agrees well with the primary beam shown in the nonlin- (2010) developed a linear model using a Greens func- ear simulation. Figure2c compares the nonlinear and tion approach to simulate internal tidal beams generated linear results at z/L = −0.64, where L = H − h . by tidal flow over two-dimensional topography with non- 0 0 0 While the nonlinear results show some nonlinearly gen- uniform stratification. The relevant parameters in this erated harmonics, which of course do not exist in the lin- case are ω, the tidal frequency; N, the buoyancy fre- ear theory, the magnitudes of the beams and the large- quency; h , the topographic height; H, the depth; 0 scale behavior compare very well. L, a horizontal length scale representative of the topogra- phy; and U0, the barotropic tidal current amplitude. The Figure3 shows a similar computation for non- governing nondimensional parameters are: the frequency uniform stratification. This stratification is more realis- ratio ω/N; the relative height of the topography h0/H; tic, with an idealized thermocline profile near the surface. the tidal excursion parameter, defined as the tidal excur- Figure3a shows results from NFA after 18 tidal peri- sion distance normalized by the topographic length scale, ods of adjustment. When comparing it with linear theory ex = U0/(ωL); and the steepness parameter, defined as (figure3b), the NFA results have a slightly stronger mag-

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Figure 2: Normalized horizontal velocity u/U0 for tidal flow Figure 3: Normalized horizontal velocity u/U0 for tidal flow over isolated topography with uniform stratification: (a) ver- over isolated topography with non-uniform stratification: (a) tical cross-section of nonlinear simulation results using NFA. vertical cross-section of nonlinear simulation results using The dashed line shows the beam path predicted by linear the- NFA. The dashed line shows the beam path predicted by lin- ory; (b) vertical cross-section of linear theory results; and (c) ear theory; (b) vertical cross-section of linear theory results; comparison of NFA with linear theory for a horizontal line at and (c) comparison of NFA with linear theory for a horizontal z/L0 = −0.64, where L0 = H − h0. line at z/L0 = −0.64, where L0 = H − h0. nitude. We believe this stronger amplitude is due to non- tion. Figure3c compares the nonlinear and linear re- linearities in the flow caused by the strong stratification sults at z/L0 = −0.64. Linear theory captures the main near the surface. The dashed line in figure3a shows the flow behavior, but the nonlinear simulation shows some ray path predicted by linear theory, which agrees well higher frequency perturbations about the main flow. with the primary beam shown in the nonlinear simula- The process of transfer from semi-

6 diurnal internal (M2) to subharmonic internal tides in the z-direction and x-direction, respec- (M1) and higher harmonics (M4, M6, etc.) has been a tively. topic of recent interest. Gerkema, Staquet and Bouruet- The tidal beams associated with M1, M2 and Aubertot (2006) discusses how energy can transfer to M4 harmonics, according to linear theory, are plotted as subharmonic internal tides which produce features that white dashed lines. Close to the generation site, para- appear as “slices” in the generated internal tidal beam. metric subharmonic instability causes the observed fan As time goes on, more and more “slices” appear. The of weaker beams about the main beam, while further slices are the troughs and crests of the M1 beams. Higher away where the beam reflects from the thermocline, or harmonics are generated when the beam reflects off the the lower boundary, higher harmonics are generated due bottom due to the nonlinear interaction of the incoming to the nonlinear interaction of incoming and reflected and outgoing beams. We will show that this is also true beams. These instabilities grow over time and can even- for beams that reflect off the surface and thermocline. tually cause the internal wave tidal beam to break. 0 The interaction of the internal wave tidal beam

−0.2 with the thermocline also generates internal solitary waves (ISW) that propagate horizontally along the ther- −0.4 mocline. Figure5b shows the vertical displacement of −0.6 the thermocline as a function of x at two successive . Comparing the red line with the the blue line in −0.8 z/Lo figure5b at x/L0 ≈ −13, a wave and successive −1 smaller waves are seen propagating up the shelf. A rough C/C = 1.123 −1.2 estimate of the wave velocity is 0 , where C0 is the linear long-wave speed for internal waves in a −1.4 two-layer fluid, which is consistent for a two-layer soli-

−1.6 tary wave with the shown amplitude. 1 1.5 2 2.5 (a) ρ /ρo CONCLUSIONS Figure 4: Normalized density ρ/ρ0 as a function of normal- ized depth z/L0, where L0 = H − h0: nonuniform stratifica- We have described the formulation and implementation tion for a thermocline (approximated as a density jump) over a of a fully nonlinear, three-dimensional numerical model continental shelf. that is capable of simulating with great accuracy tidal flow of a stratified ocean over arbitrary bottom topogra- phy, even though the range of scales from the long scales FLOWOVERATWO-DIMENSIONALCONTINENTAL of the shelf topography to the fine scales of the resonantly SHELF generated internal wave tidal beams is very large. This new numerical model is an extension of the existing NFA NFA results for the internal wave tidal beam code, which originally was designed for the computation generated by the interaction of the barotropic tide with an of the motion of ships in a free-surface flow. Further- idealized, two-dimensional continental shelf break, with more, we have used the newly modified numerical model the vertical density profile shown in figure4, is shown in to compute tidal flow over an idealized two-dimensional figure5a. In this example, ω/N = 0.13; h0/H = 0.375; ridge and found the simulations to agree very well with the tidal excursion parameter is ex = 0.0046; and the linear theory for both uniform and nonlinear stratifica- peak steepness parameter is S = 0.81. Note that the tion. Finally, the model was used to generate a tidal beam tidal frequency in this example is two orders of magni- by the interaction of the barotropic tide with an idealized tude higher than it is in the real ocean. This was done two-dimensional continental shelf break. The simulation as a computationaly inexpensive way to obtain a regime results show the generated beam and its harmonics, gen- in which internal solitary waves are formed by internal erated by nonlinear interactions, as well as the solitary wave tidal beams (for testing purposes). In future stud- waves generated by the beam’s interactions with the ther- ies, other flow parameters will be adjusted to reach this mocline. regime with the tidal frequency set to the semi-diurnal tidal frequency. The NFA numerical domain was 8192x512 Acknowledgements grid points with ∆x/L0 = 0.007 and the minimum ∆z/L0 = 0.0007 and maximum ∆z/L0 = 0.007. This was sponsored by Dr. Ron Joslin at the There were approximately 188 grid points across the in- Office of Naval Research (contract number N00014-08- terface. There were 170 and 703 grid points across the C-0508), Dr. Tom C. Fu at the Naval Surface War-

7 −0.055

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−25 −20 −15 −10 −5 0 5 10 15 x/Lo (a) (b)

Figure 5: (a) Contours of u/U0 after 8.25 tidal periods, depicting the generation of an internal wave tidal beam and its interaction with the thermocline. The beam paths associated with the M1, M2 and M4 tidal modes are shown as white dashed lines. (b) Elevation of thermocline: (red) at 8.25 tidal periods (a); (blue) at 8.5 tidal periods fare Center, Carderock Division, and SAIC’s Research Garrett, C. and Kunze, E., “ generation in the deep and Development program. The numerical simulations ocean,” Ann. Rev. Fluid Mech., Vol. 39, 2007, pp. 57–87. were supported in part by a grant from the Department of Defense High Performance Computing Moderniza- Gerkema, T., “Internal and interfacial tides: beam scattering tion Program (http://www.hpcmo.hpc.mil/). The numer- and local generation of solitary waves,” J. Mar. Res., Vol. 59, ical simulations were performed on the SGI R Altix R 2001, pp. 227–255. ICE 8200LX (Silicon Graphics International Corpora- Gerkema, T., Staquet, C., and Bouruet-Aubertot, P., “Decay of tion) at the U.S. Army Research and De- semi-diurnal internal-tide beams due to subharmonic velopment Center. Animated versions of the numeri- cal simulations described here as well as many others resonance,” Geophys. Res. Let., Vol. 33, 2006, p. L08,604. are available online at http://www.youtube.com/ Hoffman, J., “Adaptive simulation of the subcritical flow past waveanimations a sphere.” J. Fluid Mech., Vol. 568, 2006a, pp. 77–88. Hoffman, J., “Simulation of turbulent flow past bluff bodies on References coarse meshes using general galerkin methods: drag crisis and turbulent euler .” Comput. Mech., Vol. 38, 2006b, pp. 390–402. Dommermuth, D., Innis, G., Luth, T., Novikov, E., Schlageter, E., and Talcott, J., “Numerical simulation of bow waves,” Hollowy, P., Chatwin, P., and Craig, P., “Internal tide Proceedings of the 22nd Symposium on Naval observations from the australian northwest shelf in summer Hydrodynamics, Washington, D.C., 1998, pp. 508–521. 1995,” J. Phys. Ocean., Vol. 31, 2001, pp. 1182–1199. Dommermuth, D. G., OShea, T. T., Wyatt, D. C., Ratcliffe, T., John, V., “Slip with friction and penetration with resistance Weymouth, G. D., Hendrickson, K. L., Yue, D. K. P., boundary conditions for the navier-stokes equations - Sussman, M., Adams, P., and Valenciano, M., “An application numerical tests and aspects of the implementation,” J. Comp. of cartesian-grid and volume-of-fluid methods to numerical Appl. Math., Vol. 147, 2002, pp. 287–300. ship hydrodynamics,” Proceedings of the 9th International Knupp, P. M. and Steinberg, S., Fundamentals of grid Symposium on Numerical Ship Hydrodynamics, held 5 – 8 generation, CRC Press, 1993. August 2007 in Ann Arbor, Michigan, 2007. Lien, R.-C. and Gregg, M. C., “Observations of in a Echeverri, P. and Peacock, T., “Internal tide generation by tidal beam and across a coastal ridge,” J. Geophys. Res., Vol. arbitrary two-dimensional topography,” J. Fluid Mech., Vol. 106, 2001, pp. 4575–4591. 659, 2010, pp. 247–266. URL: http://dx.doi.org/10.1017/S0022112010002417 Mauge´, R. and Gerkema, T., “Generation of weakly nonlinear

8 nonhydrostatic internal tides over large topography: a multi-modal approach,” Nonlin. Processes Geopgys., Vol. 15, 2008, pp. 233–244. Meyer, M., Devesa, A., Hickel, S., Hu, X. Y., and Adams, N. A., “A conservative immersed interface method for large- simulation of incompressible flows,” J. Comput. Phys., Vol. in press. Meyer, M., Hickel, S., and Adams, N. A., “Assessment of implicit large-eddy simulation with a conservative immersed interface method for turbulent cylinder flow,” Int. J. Heat Fluid Flow, Vol. in press. O’Shea, T. T., Brucker, K. A., Dommermuth, D. G., and Wyatt, D. C., “A numerical formulation for simulating free-surface hydrodynamics,” Proceedings of the 27th Symposium on Naval Hydrodynamics, Seoul, , 2008. Pingree, R. D. and New, A. L., “Downward propagation of internal tidal energy into the ,” Deep- Res. I, Vol. 36, 1989, pp. 735–758. Rottman, J. W., Brucker, K. A., Dommermuth, D., and Broutman, D., “Parameterization of the near-field internal wave field generated by a ,” Proceedings of the 28th International Symposium on Naval Hydrodynamics, held 12 – 17 September 2010 in Pasadena, CA, 2010a. Rottman, J. W., Brucker, K. A., Dommermuth, D. G., and Broutman, D., “Parameterization of the internal-wave field generated by a submarine and its turbulent wake in a uniformly stratified fluid,” Proceedings of the 28th Symposium on Naval Hydrodynamics, Pasadena, , USA, 2010b. Sussman, M. and Dommermuth, D., “The numerical simulation of ship waves using cartesian-grid methods,” Proceedings of the 23rd Symposium on Naval Ship Hydrodynamics, Nantes, , 2001, pp. 762–779.

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