A Nonhydrostatic Finite-Element Model for Three-Dimensional Stratified Oceanic Flows. Part II: Model Validation

Transcription

1 2832 MONTHLY WEATHER REVIEW A Nonhydrostatic Finite-Element Model for Three-Dimensional Stratified Oceanic Flows. Part II: Model Validation R. FORD Department of Mathematics, Imperial College, London, United Kingdom C. C. PAIN, M.D.PIGGOTT, A.J.H.GODDARD, C.R.E.DE OLIVEIRA, AND A. P. UMPLEBY Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College, London, United Kingdom (Manuscript received 14 July 2003, in final form 22 January 2004) ABSTRACT A three-dimensional nonhydrostatic ocean model that utilizes finite-element discretizations on structured or unstructured meshes has been formulated. In the current part of this work a first step in the validation of this model is performed. Due to the inherent strengths of the modeling approach in being able to fully simulate three-dimensional flows in arbitrary domains, it has been decided here to focus attention on a single well-studied collection of problems. As such the flow of stratified fluid past an isolated Gaussian seamount is considered. It is shown that at moderate resolutions the model is relatively insensitive to the use of an increasingly fine mesh. In addition, qualitative as well as rough quantitative comparisons are made between results from the current model and previous studies carried out using alternate numerical models. Close agreement is demonstrated, both in the eddy-shedding and flow-trapping structure of the flow, as well as in the generation of internal trapped lee waves. Finally, time-periodic forced flow is examined, where resonantly generated trapped waves on the seamount are shown to be produced. The current model predicts resonant interactions occurring at parameter values consistent with previous numerical as well as linearized analytical studies. 1. Introduction In Part I of this work (Ford et al. 2004, hereafter Part I), a three-dimensional finite-element model was introduced, which has been specifically designed to simulate nonhydrostatic stratified oceanic flows in arbitrary domains, especially those involving strong topographic features. The model has at its foundations an efficient and robust finite-element approach for solving the Navier Stokes equations, with suites of discretization options available to handle a wide variety of fluid flow properties. However, in addition to this, further developments were required in order to enable the simulation of flows with characteristics intrinsic to oceanic situations. In Part I, the generation of spurious solution behavior on distorted meshes (e.g., those required to conform to complex bathymetry) was highlighted, and addressed, through the simulation of flow around a Gaussian seamount. One of the great strengths of this new modeling approach hinges on its ability to allow computations in Corresponding author address: Dr. M. D. Piggott, Applied Modelling and Computation Group, Department of Earth Science and Engineering, Imperial College, Prince Consort Road, London, SW7 2BP, United Kingdom. m.d.piggott@imperial.ac.uk very general geometries. The use of unstructured meshes allows an extremely accurate discrete representation of complex coastlines and bathymetry, which is of paramount importance for the correct representation of many physical process (e.g., those considered here). Therefore it has been decided to further investigate the ability of the model to accurately simulate a variety of problems based upon the isolated seamount geometry. An additional reason for focusing on this collection of problems, as well as allowing a uniform and straightforward language to be used throughout this paper, is the excellent range of extensive numerical and theoretical work available upon which comparisons with the results from the current model may be made. A large part of the background material used to perform these comparisons is due to Haidvogel and coworkers (Beckmann and Haidvogel 1997, 1993; Chapman and Haidvogel 1992, 1993; Haidvogel et al. 1993), the majority of which are based upon the use of the semispectral primitive equation model (SPEM; Haidvogel et al. 1991). This model is hydrostatic, uses second-order finite differences in the horizontal, and Chebyshev polynomials based on sigma coordinates in the vertical. It uses fourth-order hyperdiffusion in the horizontal for both mass and momentum. In contrast, the model employed here uses hexahedral finite elements, 2004 American Meteorological Society

2 DECEMBER 2004 FORD ET AL is nonhydrostatic, with second-order diffusion for mass and momentum. It is impossible therefore to compare resolution, or solutions, exactly between the two models. However, in both cases the view is taken that the resolution employed should be sufficient to ensure that dissipative effects are minimized as far as possible. In Chapman and Haidvogel (1992) the conditions required for the formation of Taylor caps on an isolated seamount were investigated with attention paid to the role of stratification and seamount height. The process of eddy formation, eddy shedding, and subsequent particle trapping were highlighted in detail, yielding an appropriate set of base experiments with which to proceed here. Over tall seamounts the phenomenon of internal lee wave generation was briefly examined, primarily in relation to their impact on the fluid-trapping property of the Taylor caps. The generation of internal lee waves was considered in much further detail in Chapman and Haidvogel (1993) where again the importance of stratification, seamount height, and inflow conditions were examined. In particular, internal waves trapped to the seamount are considered where local acceleration due to the seamount geometry s affect on the flow structure are sufficient to support them. These waves subsequently die away as they move from the seamount since the background flow is itself not sufficiently strong to support them. Additional results and simulations may be found in Xing and Davies (1996) where the enhancement of turbulent mixing by internal lee waves is considered. It is also shown that the magnitude and spatial distribution of the lee waves are strongly dependent on the form of horizontal diffusion employed, with significant differences observed between the use of Laplacian and biharmonic parameterizations. Again, this highlights that care must be taken in drawing anything other than qualitative comparisons. Additional details of the numerical model SPEM used in the majority of the above studies, as well as the seamount problem, and specifics of the sigma-coordinate vertical structure of the discretization may be found in Beckmann and Haidvogel (1993). The second part of that work (Haidvogel et al. 1993) looks at a slightly alternate problem where a weak oscillatory background flow is imposed on the seamount. Resonant interaction of this flow with seamount-trapped waves is studied, with attention on the matching between the frequency of the imposed flow and that of the trapped wave. Again this is an ideal problem with which to compare the current numerical model, because a different part of parameter space is examined in the course of the comparison, but also because there exist linearized analytical results for use in additional model validation. The finite-element model described in Part I is shown here to display close agreement with results available in the literature, as summarized above. The range of phenomena and nondimensional parameters considered represents a stringent first validation of the model. However, it is important to note that comparisons are being made primarily with hydrostatic models, and hence small discrepancies may be ascribed to the importance of nonhydrostatic effects. This may be investigated in further detail through direct comparison with alternate hydrostatic and nonhydrostatic models, and the adjustment of problem aspect ratios, for example. These issues, along with further analysis of the hydrostatic pressure solver presented in Part I, shall not be considered here and are reserved for future presentation. The remainder of this paper is set out as follows. In section 2 the geometry for the problems considered here is described, a mesh for this is constructed, and useful nondimensional numbers for describing various problem regimes are given. In section 3 various simulations of flow past a seamount are carried out with the new finite-element model. As well as comparing the eddyshedding and fluid-trapping properties of the flow with previous studies, a qualitative examination of the sensitivity of the model with respect to mesh resolution is made. This is in order to test an obviously desirable, or rather, necessary property of any model, as well as demonstrating that extremely high resolution is not required here in order to obtain satisfactory simulations. Following this the internal wave generation property of the flow is compared with previous studies. Finally the resonant generation of trapped waves is considered, results presented show agreement with other numerical as well as analytical results. In section 4 some comments and conclusions from this work are given. 2. Problem setup a. Physical parameters and initial configuration Following Chapman and Haidvogel (1992, 1993) and Haidvogel et al. (1993), the computational domain is a channel of length L and width D, with solid vertical boundaries at y D /2. Here generally values of L 560 km and D 514 km are taken. A rigid lid is imposed at z 0, and flow past an isolated seamount with Gaussian geometry is considered. The bottom of the domain is given by z h(x, y), where h(x, y) H{1 exp[ (x y )/L ]}, (1) with the nondimensional parameter representing the fractional seamount height, and L representing a horizontal length scale for the seamount. Away from the seamount the ocean is of depth H, and the maximum height of the seamount is H (so 1 corresponds to a seamount that just reaches the surface). There is an inflow boundary at x x 0 and an outflow boundary at x x 1. A uniform upstream velocity U is imposed that is independent of the vertical coordinate z. The problem is nondimensionalized using the horizontal scale L for length, and the advection time scale L/U for time. Following the previously cited works, a seamount with a horizontal scale of 25 km is considered, in an ocean of depth 4.5 km, so the value of the aspect ratio H/Lis fixed at 0.18.

3 2834 MONTHLY WEATHER REVIEW Three further nondimensional numbers now specify the problem, the fractional seamount height, the Rossby number U Ro, fl and the Burger number NH Bu, fl where f is the Coriolis parameter and the buoyancy (or Brunt Väisälä) frequency N is given via g z 2 N, H 0 here g is the acceleration due to gravity, z is the density difference in the vertical between the sea surface and sea floor, and 0 is a reference value for density. Related to these nondimensional parameters, and also of importance here, is the Froude number Ro Fr. Bu Following nondimensionalization is controlled simply by altering the height of the seamount, Ro is controlled by varying f, and then B may be controlled by varying N, or equivalently here, as z / 0 is taken as fixed, by varying g. The density is initially linear in z, so the buoyancy frequency N is constant. This stratification is prescribed at the inflow boundary. It is assumed that L is sufficiently small so that the f-plane approximation may be made, and so the Coriolis parameter f is a constant. A horizontal Reynolds number (Re UL/, with the viscosity as defined in Part I) of 1000 is employed, based on the seamount s horizontal length scale L, with the vertical viscosity being an order of magnitude smaller. The horizontal thermal diffusivity is set equal to the horizontal viscosity, and the vertical thermal diffusivity is zero. The choice of parameters here matches those that are used as standard in the cited literature. However as mentioned previously, due to the different forms for diffusive processes employed by the models, exact comparison is impossible and the choice of viscosities and diffusivities has been made to allow at least sensible comparions to be made. b. Computational domain and mesh Since the model as presented in Part I uses meshes based upon hexahedral or tetrahedral elements, an important aspect of any simulation is the design of the finite-element mesh that conforms to the desired topography. Although mesh generation algorithms and fluid solution techniques for more versatile tetrahedral elements are available, in this work, as a first step, a simple hexahedral mesh is used. A hexahedral mesh is most readily constructed by first generating a mesh of quadrilateral elements on the planar surface z 0. In fact, for simplicity, it has been chosen to use rectangular surface elements, with the edges of the surface elements corresponding to straight lines at constant values of x or y. This surface corresponds to the upper surface of the ocean, which is take to be a rigid lid in these simulations. Three-dimensional elements are then generated by dropping vertical lines from each node down to the surface z h(x, y). On each such line nodes are placed at the same sigma-coordinate level, that is, at i z i /h(x, y), where the set of values of i is the same for all vertical lines (with 0 0 corresponding to the surface and V 1 corresponding to the bottom topography, there being V levels in the vertical). In this way, the top and bottom faces of each element are the projection of the surface element down to two adjacent sigma levels. Nodes at the same value of i for i 0 are then connected with the same connectivity as the surface mesh. This defines a mesh of hexahedral elements, in which the edges of the elements resemble the coordinate lines of a sigma-coordinate mesh. Note that the faces of this mesh are not planar the coordinates are known at the nodes, and then interpolated using trilinear interpolation across each element (as in the basis {N i } defined in Part I). The geometry of the problem, together with one of the meshes used in this study, is shown in Fig. 1. There is a region of uniform resolution ([ 1.5 x, y 1.5]) centered on the seamount. The resolution in this region is x y (corresponding to a dimensional resolution of km), where x and y represent the distance between mesh points in the x and y directions, respectively. Then, for 1.5 x 6.0, there is a transition region across which the mesh spacing x in the x direction increases exponentially, such that ( x) i 1 ( x) i, for some constant 1. The same applies to the mesh spacing in the y direction over 1.5 y 6.0 (only y 0 is shown in Fig. 1). There are 24 mesh intervals in the x direction between x 1.5 and x 6.0, and the mesh spacing is continuous at x 1.5 so that, from a mesh point at x 1.5, the next mesh point (at the same value of y) isatx ( here is approximately 1.034, and in a later case where horizontal resolution is increased by a factor of 1.5, takes the value 1.022). Again, the same applies in the y direction. Given the number of mesh points in the transition region and the resolution across the seamount, the resolution in all other regions are then determined on the condition that the mesh spacing at the outer limit of the transition regions is equal to the uniform mesh spacing in the outer regions. In the case shown, the mesh spacing in the outer regions is (corresponding to km in dimensional units). The limits of the computational domain are x and y , corresponding to approximately 560 km by 514 km. There are 12 intervals in the vertical direction, resulting in a total of elements and nodes. In the

4 DECEMBER 2004 FORD ET AL FIG. 1. (left) The geometrical configuration and mesh for flow past a seamount. The perspective is from an elevation above the surface of the fluid, and one-half of the domain is excluded in order to illustrate the seamount topography as well as the conforming mesh. The inflow boundary is to the left, and the outflow boundary to the right. (right) The decomposition of the domain into 32 subdomains for parallel processing. mesh shown in Fig. 1, the mesh points on each of the vertical lines are uniformly spaced from the top at z 0 to the bottom at z h(x, y). c. Domain decomposition and parallel simulations The simulations presented in this work have been carried out both on serial machines as well as a Beowulf cluster. The finite-element code is parallelized using MPI (Message Passing Interface) and standard domain decomposition methods, where individual processors handle overlapping subdomains. Calculating the subdomains involves the solution of a graph-partitioning problem. Here this is achieved using a k-way multilevel approach where an initial partitioning is calculated on the coarsest graph. As the initial coarse partition is projected up through the hierarchy, a constrained optimization problem is solved using recurrent neural network techniques to refine the partitioning. The constrained optimization problem is constructed to minimize interprocessor communication cost while balancing the computational load on each processor (for further details see Pain et al. 1999). An important aspect of the finite-element model will be its use with adaptive mesh algorithms, indeed this will generally be required before the model will be competitive with standard models. However, with an adapting mesh the issues of element conformity and dynamic load balancing must be addressed to obtain a robust and efficient method. The required algorithm is extremely important and complex and shall be described in detail when adaptive simulations mimicking those computed here are presented. An example of a decomposed domain for the seamount geometry is shown in Fig. 1 where the mesh on the sea floor and lateral boundaries are displayed, the number of subdomains is 32 here. d. Sponge regions For the simulations conducted here an imposed uniform velocity field is applied at both the inflow and outflow boundaries of the domain. The impusively started nature of the flow incident upon the seamount causes internal waves to be generated early in the simulations. Sponge regions are employed over the first and last 53.4 km of the domain as a means to damp out and avoid reflection of these waves at boundaries. This also allows for the use of simple outflow conditions rather than more complex open-boundary conditions. The distance of these regions from the seamount is sufficient to ensure that the solution behavior of interest is not affected. In the outflow region viscosity is as in the interior of the domain but a Rayleigh damping term is included here. In the inflow region again a Rayleigh damping term is included, but now viscosity is set at two orders of magnitude larger than that used in the interior of the domain. These choices have been found sufficient for the simulations conducted here. In the case of time-periodic velocity boundary conditions used is section 3e to model the generation of resonant waves over the seamount, Rayleigh damping is used in sponge regions 50 km long at each end of the channel with no enhanced viscous damping. Again this was found to be sufficient for these simulations. 3. Simulation of flow past an isolated seamount a. Impulsively started steady inflow In this section the performance of the model in simulating the problem of impulsively started flow past an isolated seamount is considered. At small Rossby numbers a cyclonic eddy is initially shed from the mount, and as time proceeds a strongly bottom-intensified anticyclonic circulation is trapped to the mount. The aim in this paper is not to explore the space of

5 2836 MONTHLY WEATHER REVIEW FIG. 2. The density at a level (top) 10% from the sea surface and (bottom) 10% off the seafloor, at (left) times t or 4 days, and (right) t or 24 days. Domain shown is 3.3 x 9, 4 y 4 in nondimensional units. physical parameters (,, B, Ro), instead attention is focused on just several choices of these parameters. Particular parameters have been chosen to provide a stern test of the numerical model; seamount heights of up to 0.9 are used since they lead to the greatest distortion of the sigma-coordinate-type mesh, which is one potential source of error in the current simulations; a relatively large Burger number of Bu 1.5 is used to ensure that this phenomenon may be significant (there is no such error in an unstratified fluid; see Part I). Initially the density field is linear in z, varying from 1 at the surface to 0 at the domain s maximum depth. Finally, a relatively small Rossby number of Ro 0.1 is used, which means that the Coriolis terms must be treated accurately. The configuration and mesh are as described in section 2b, a uniform inflow boundary condition is imposed at x 8.136, an outflow boundary condition of equal magnitude is imposed at x , and 0.9. Throughout these steadily forced problems, as well as the internal lee-wave simulations to be presented presently, a time step of t or 4320 s is used, an order of magnitude larger than those used in, for example, Adcroft et al. (1997), Chapman and Haidvogel (1992), and Xing and Davies (1996). This corresponds to a Courant number of approximately 0.45 in both the horizontal and vertical. The grid Reynolds number is of the order O (10 2 ), and hence the need for a Petrov Galerkin treatment of advection as described in Part I. Initially a cyclonic eddy is shed from the seamount; this can be seen in the density field at a depth level of z (i.e., 10% of the fluid depth), at a time of t or 4 days, shown in Fig. 2. The same field at t or 24 days is also shown, demonstrating that the eddy is swept out of the domain, a quasi-steady solution developing. Recall that the height of the seamount is 90% of the ocean depth, so the top of the seamount just touches this level. The formation of a cyclonic eddy is clearly visible in the density field. The corresponding density field at 90% of the model depth is also shown in Fig. 2. The flow trapped to the seamount can be seen in Fig. 3, where the density perturbation (following the removal of the initial linear profile), the velocity normal to the plane, and the vertical velocity in the plane y 0, which passes through the center of the seamount, are shown at time t or 24 days. The velocity normal to the plane y 0 can be taken as a measure of the strength of the circulation induced by the presence of the seamount. b. Sensitivity to resolution To investigate the sensitivity of numerical results to resolution in the current model, two additional simulations have been conducted, in which (from the mesh constructed in section 2, denoted by resolution A), (i) the vertical resolution has been independently increased, from 12 to 20 levels, denote this resolution B, and (ii) the horizontal resolution has been independently increased uniformly, by a factor of 1.5 in both the x and y directions, denote this resolution C. The total number of nodes and elements for the three resolutions are A: elements, nodes; B: elements, nodes; and C: elements, nodes. In Fig. 4 the density perturbation, the velocity normal to the plane and the vertical velocity in the plane y 0 at time t are given for the simulation with resolution B, and in Fig. 5 the same fields for the simulation with resolution C are given. In general, there appears to be little difference between these simulations, indicating that the initial res-

6 DECEMBER 2004 FORD ET AL FIG. 3. The density perturbation, normal velocity, and vertical velocity through the midplane y 0att , or 24 days; resolution A. FIG. 4. The density perturbation, normal velocity, and vertical velocity through the midplane y 0att , or 24 days; resolution B: 20 vertical levels.

7 2838 MONTHLY WEATHER REVIEW FIG. 5. The density perturbation, normal velocity, and vertical velocity through the midplane y 0att , or 24 days; resolution C: increased horizontal resolution. FIG. 6. (top left), (top right), (bottom left) Flux through the planes y ( 1, 0, 1), respectively, as a function of the along-channel coordinate, for resolution A, B, and C respectively. (bottom right) Total circulation through alongchannel planes from y 3toy 3, for resolutions A, B, and C. All results taken at time t or 24 days.

8 DECEMBER 2004 FORD ET AL FIG. 7. Vertical component of velocity at a depth of 5% at time t or 24 days, for the three resolutions A, B, and C, respectively. Domain shown is 2 x, y 2 in nondimensional units. olution is adequate to resolve the problem at hand. Of course a quantitative sensitivity analysis of this matter would be useful, with comparisons made against results from a very high-resolution simulation. This shall be performed in the future where adaptive meshes will be used to obtain high-resolution results. To help quantify this statement here, however, in the first three panels of Fig. 6 (top left to bottom right) the depth-integrated normal velocity through the planes y ( 1, 0, 1) is displayed as a function of x for the three resolutions, from which it can be seen that there is little difference between the three. A second quantitative measure of the flow is the total integral of the modulus of the normal velocity through planes at various values of the crosschannel coordinate y. This gives a measure of the total circulation through each plane and is shown for the three resolutions at t in the fourth panel (bottom right) in Fig. 6, again indicating that the resolution of the coarsest of these appears to be adequate here. However, this statement is a qualitative one, in the sense that the degree to which one may regard a solution as adequately resolved depends to an extent upon the field of interest, and in these simulations the vertical velocity near the seamount top appears to depend sensitively on the resolution. In Fig. 7 the vertical velocity at a depth of 5% of the model domain at time t is given for the three resolutions. Qualitatively, this field appears to be more sensitive to horizontal resolution than the cross-channel fluxes. All three fields have qualitatively the same form, but the field for resolution in C appears better resolved than the other two. This suggests that a further increase in horizontal resolution would be more likely to further improve this aspect of the simulation than an increase in vertical resolution. Note however that the vertical velocity here is of very small magnitude compared with the vertical velocity at depth (this was the reason that it was chosen as a good indicator of sensitivity to resolution), and so depending upon one s interests, this component of the flow may, or may not, be relatively unimportant. c. Model comparisons In this section a comparison is made between results from the current model using resolution A, and results

9 2840 MONTHLY WEATHER REVIEW FIG. 8. Density perturbation at a depth of 400 m after (left) 1 and (right) 2 days of model integration. Domain shown is 3 x, y 3 in nondimensional units. Domain shown is 3 x, y 3 in nondimensional units. from Chapman and Haidvogel (1992) as well as Adcroft et al. (1997). The Burger number has been reduced to Bu 1.0 here to match prior studies, and attention is paid first to the eddy-shedding structure of the flow. To start, the initial transient response of the flow is compared with that obtained by Chapman and Haidvogel (1992). Figure 2 of their paper shows the density field at a depth of 400 m, at times of up to two days into the simulation. In our Fig. 8 the density perturbation at the same depth is presented for this model, at times of 1 and 2 days into the simulation. It can be seen at once that there is excellent qualitative agreement between the two models. Columns of fluid are compressed on the upstream side of the seamount, and stretched on the downstream side, and this initially induces a vortex pair on the seamount. A positive density anomaly is associated with anticyclonic vorticity, since the compressed columns advect dense fluid up from deep water, and conversely a negative density anomaly is associated with cyclonic vorticity. Both models show that a positive density anomaly (anticyclonic vorticity) is trapped on top of the seamount, while a negative density anomaly (cyclonic vorticity) is advected downstream. One qualitative difference between the two calculations is the small region of the negative density anomaly that appears to be trapped on the upwind side of the seamount after 2 days in the current simulation, which is not apparent in the simulation of Chapman and Haidvogel (1992). A similar phenomenon is observed in a calculation of flow past a seamount in the nonhydrostatic Massachusetts Institute of Technology (MIT) ocean model in Adcroft et al. (1997); it seems appropriate to conjecture that this phenomenon might be related to nonhydrostatic effects. Quantitatively, it is found that the range of values of density agrees with Chapman and Haidvogel (1992) reasonably well. For example, comparing the values at a time of 1 day, the current simulation shows a distribution in the density perturbation of of to ; that is, a range of values of This corresponds to a range, in the units of Chapman and Haidvogel (1992), of kg m 3, which agrees well with the contour intervals employed in Chapman and Haidvogel (1992), which at t 1 were to kg m 3. Although without detailed reference to their solution fields, it is still possible to conclude that the current model simulates this feature of the flow with good quantitative agreement with the model of Chapman and Haidvogel (1992), bearing in mind that differences exist not only in the discretization, but also in the form and magnitude of the frictional terms. In order to follow the cyclonic eddy as it breaks away from the seamount the density field at a depth of 4000 m was presented in Fig. 3 of Chapman and Haidvogel (1992). In our Fig. 9 the same field obtained with the current finite-element model at times of 2 12 days is given. Both simulations show that the eddy starts to break away from the seamount at about t 4, and is fully developed by about t 6. Again, there is good qualitative agreement, and quantitative agreement is as good as can be determined by comparison of contour levels alone. A difference between the current simulation and that of Chapman and Haidvogel (1992) might be that here the flow in the neighborhood of the seamount appears to be more intense at time t 12. d. Internal lee waves It is well known that, for at least some parameter values, a significant field of internal waves can be established downstream of the seamount. In Chapman and Haidvogel (1993) it is demonstrated that, for sufficiently large Froude numbers, and sufficiently small seamount, a significant field of stationary internal waves could be found in steady-state numerical solutions of flow over a seamount. In Fig. 10 density fields and vertical velocities are shown for steadily forced flow past seamounts of vary-

10 DECEMBER 2004 FORD ET AL FIG. 9. Density at a depth of 4000 m after 2, 4, 6, 8, 10, and 12 days. Note that a well-defined eddy-shedding event occurs between t 4 and t 6. Domain shown is 3 x 12, 6 y 4 in nondimensional units. ing heights. The results are shown at t or 24 days, and the flow is essentially in steady state at this time. In these calculations the Rossby number Ro 0.2 and the Burger number Bu 1.0. The fractional seamount height takes the values (0.3, 0.5, 0.75), and the aspect ratio is 0.18, as in previous cases. Here, for a small fractional seamount height of 0.3, a marked field of trapped internal wave forms can be seen in the region downstream of the seamount. As the seamount height increases, however, the amplitude of the internal waves diminishes, and the regions of significant vertical velocity become trapped to the seamount itself. In Fig. 11 the along-stream and vertical components of velocity are plotted at a depth of 1125 m for the seamount with a fractional height of 0.5. In agreement with Chapman and Haidvogel (1993) (see also Xing and Davies 1996), since the inflow employed here is fairly weak, local acceleration of the flow due to the presence of the seamount, and an accompanying increase in the local Froude number, leads to an enhanced wave structure trapped to the seamount in the region of higher along-stream velocity. Also in agreement with Chapman and Haidvogel (1993), the internal wave structure tends to follow the path of the current and curl around the back of the seamount.

11 2842 MONTHLY WEATHER REVIEW FIG. 10. (left) Density and (right) vertical velocity in an along-stream section through the seamount for (top to bottom) fractional seamount heights 0.3, 0.5, Domain shown is 2.87 x 2.87 in nondimensional units. e. Time-periodic inflow The case of resonantly forced flow past a seamount has also been simulated here, this has previously been investigated in Haidvogel et al. (1993). The essential point here is that there exist trapped neutral waves on the seamount. If the inflow is not steady, as in the previous examples, but time periodic (e.g., a tidal flow), then there exists the possibility that the time-periodic forcing may resonate with the trapped waves. It is desirable that the current numerical model should be able to simulate this phenomenon. For this problem, the fractional seamount height takes the value is 0.9. As in all previous problems, linear stratification in z (i.e., constant buoyancy frequency) is employed. For this flow configuration, the dispersion calculations of Brink (1989) have been repeated by Haidvogel et al. (1993) to find that the maximum resonant interaction should occur at a Burger number of approximately Bu Clearly the linearized theory presented in Brink (1989; see also Chapman 1989; Brink 1990) assumes that the amplitude of the inflow is infinitesimal. In current calculations the Rossby number of the maximum inlet velocity is taken to be Ro , this corresponds to a maximum inlet velocity of 2 mm s 1. Such a small inlet velocity represents a stern test of the model numerics, since presumably if any artificial velocity field were present in the model due to the type of discretization used (cf. the pressure gradient type error discussed in Part I), the small inlet velocity could be swamped by model error. The forcing used here is sinusoidal with a period of 1 day in dimensional units and all results are presented 25 days into the simulation. A time step of 500 s is used and the horizontal viscosity is such that the Reynolds number at the inflow is Re 1, increasing over the seamount to the value of the maximum fluid velocity, the vertical viscosity being an order of magnitude smaller. The horizontal thermal diffusivity is two orders of magnitude smaller than the horizontal viscosity, and the vertical thermal diffusivity is zero. Another feature of this problem that distinguishes it

12 DECEMBER 2004 FORD ET AL FIG. 11. (left) Along-stream velocity and (right) vertical velocity at a depth of 25% or 1125 m for a fractional seamount height of 0.5 in the lee-wave simulation. Domain shown is 3 x, y 3 in nondimensional units. from the steadily forced case is that the region of dynamical activity is concentrated around the seamount. Consequently, a mesh is employed in which the resolution is centered on top of the mount, and there is no need for a large region of high-resolution downstream of the mount, as was the case in the mesh shown in Fig. 1. In Fig. 12 the mesh used for the resonantcase simulations is given. The domain is 300 km wide and 400 km long with the seamount at the very center. The resolution varies from approximately 2.15 km over the seamount, stretching to 12.5 km away from the seamount, there are 8 levels in the vertical, leading to a total of elements and nodes. Simulations at several Burger numbers have been undertaken to establish whether a resonant response can be obtained and, if so, whether it occurs at the correct Burger number. In Fig. 13 the maximum velocity amplification obtained during the simulation is shown as a function of Burger number. It is evident that the model correctly captures a resonant peak, and places it between B 1.5 and B 1.55, with an amplification of velocity in the trapped wave of almost 75 times that of the inflow. This is comparable to the results obtained in Haidvogel et al. (1993) where typically a maximum amplification by a factor of between 70 and 90 was observed at a Burger number somewhere between Bu 1.4 and Bu 1.6 for the many simulations presented. The discrepancy of approximately 5% in the location of the maximum resonant response as compared to the linear analytical results can be partially ascribed to the nonlinear nature of the equations modeled here. In addition, the results of Haidvogel et al. (1993) showed some sensitivity to the magnitude and choice of viscous parameters, which has not been considered too closely in the current work. 4. Concluding remarks In Part I of this work (Ford et al. 2004) a new threedimensional nonhydrostatic finite-element ocean model was presented. In Part II a series of calculations have been performed with the model of flow past an isolated Gaussian seamount. This has been undertaken as a first step in a thorough validation of both the new modeling approach as well as the model itself. In simulations of steadily forced flow past a seamount the model has been shown to be reasonably insensitive FIG. 12. The mesh used for the resonant-case simulations. FIG. 13. Maximum velocity vs Burger number, demonstrating resonant response.

13 2844 MONTHLY WEATHER REVIEW to increases in resolution from a fairly moderate grid spacing. This is of course a crucial property of any numerical model and has important implications for larger-scale simulations. Results obtained with the model have been seen to be qualitatively very similar to those obtained by others. This is also the case for the generation of trapped internal lee waves, and the resonant interaction of trapped waves with a time-periodic inflow where the model additionally agrees well with linearized analytical results. Through these successful simulations of reasonably large-scale stratified flow past an isolated seamount, it can be claimed that the model appears to be more than capable of simulating oceanic-scale flows over topography. The smooth manner in which topography may be represented additionally removes another possible source of model error. Further experimentation is being carried out with the model. For example the use of meshes fully unstructured in both the horizontal and vertical, as well as meshes that adapt dynamically to the computed solution field, are being investigated. As a continuation of the work presented here, for example, resolution and computational resources may be focused around the seamount in regions of trapped fluid motion, eddy shedding, and internal wave structures. These, along with further consideration of nonhydrostatic issues, both computational and in relation to results from flow past tall seamounts, will be the subject of a further investigation. Acknowledgments. We acknowledge the contributions of the lead author, Rupert Ford, who died on 30 March 2001 when he and his research were flourishing. His enthusiasm and excitement at each achievement made in this project is greatly missed. This paper was completed in his absence, though not without his vision. We hope that it does justice to one aspect of his work. We would like to thank both Dr. Marco Luchini and Mr. Nick James for initiatives that gave us access to a distributed parallel machine that greatly assisted the development of the model. This research was carried out under funding from the National Environmental Research Council of the United Kingdom under Grants NER/A/S/2000/00375 and GR9/02518 for which we are grateful. We are very grateful to three anonymous referees whose comments helped improve the quality of this paper. REFERENCES Adcroft, A., C. Hill, and J. Marshall, 1997: Representation of topography by shaved cells in a height coordinate ocean model. Mon. Wea. Rev., 125, Beckmann, A., and D. B. Haidvogel, 1993: Numerical simulation of flow around a tall isolated seamount. Part I: Problem formulation and model accuracy. J. Phys. Oceanogr., 23, , and, 1997: A numerical simulation of flow at Fieberling Guyot. J. Geophys. Res., 102 (C3), Brink, K. H., 1989: The effect of stratification on seamount-trapped waves. Deep-Sea Res., 36, , 1990: On the generation of seamount-trapped waves. Deep-Sea Res., 37, Chapman, D. C., 1989: Enhanced subinertial diurnal tides over isolated topographic features. Deep-Sea Res., 36, , and D. B. Haidvogel, 1992: Formation of Taylor caps over a tall isolated seamount in a stratified ocean. Geophys. Astrophys. Fluid Dyn., 64, , and, 1993: Generation of internal lee waves trapped over a tall isolated seamount. Geophys. Astrophys. Fluid Dyn., 69, Ford, R., C. C. Pain, M. D. Piggott, A. J. H. Goddard, C. R. E. de Oliveira, and A. P. Umbleby, 2004: A nonhydrostatic finite-element model for three-dimensional stratified oceanic flows. Part I: Model formulation. Mon. Wea. Rev., 132, Haidvogel, D. B., J. L. Wilkin, and R. Young, 1991: A semi-spectral primitive equation ocean circulation model using vertical sigma coordinates and orthogonal curvilinear horizontal coordinates. J. Comput. Phys., 94, , A. Beckmann, D. C. Chapman, and R.-Q. Lin, 1993: Numerical simulation of flow around a tall isolated seamount. Part II: Resonant generation of trapped waves. J. Phys. Oceanogr., 23, Pain, C. C., C. R. E. de Oliveira, and A. J. H. Goddard, 1999: A neural network graph partitioning procedure for grid-based domain decomposition. Int. J. Numer. Methods Eng., 44, Xing, J., and A. M. Davies, 1996: Internal lee waves and turbulence mixing over an isolated seamount: Results from turbulence energy models. Int. J. Numer. Methods Fluids, 23,