29th Symposium on Naval Hydrodynamics Gothenburg, Sweden, 26-31 August 2012 Numerical Simulation of Internal Tide Generation at a Continental Shelf 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 ocean 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 wave tidal beams, and uses a technique similar to the Cartesian-grid free- their interaction with an ocean thermocline and the sub- surface capturing code called Numerical Flow Analysis sequent generation of solitary internal waves 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 internal wave 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 ships op- simulations for stratified flow over idealized two- erate. In particular, submarines 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 thermoclines. 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 [physics.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 current 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 ship and/or bot- ρ0 @x3 0 tom. No additional gridding beyond what is used already ∆ρ γJ = : (6) in potential-flow methods and hydrostatics 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 density 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 weight functions that Substituting (4) and (7) into (1) gives are specified in physical space, 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), time (L0=U0), density (ρ0), and + (ρujui) = − − 2 − τi ; (9) 2 @t @xj @xi Fr pressure (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 stress that will act tangential to the surface of the bottom. normalized time (t). The conservation of mass 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 gravity. The Froude num- For incompressible flow with no diffusion, ber is the ratio of inertial to gravitational forces. 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 momentum: @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 densities 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 Fluid (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.