1076 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18

Internal Generation over Topography: Experiments with a Free-Surface z-Level Model

YOUYU LU Department of , Dalhousie University, Halifax, Nova Scotia, Canada

DANIEL G. WRIGHT Bedford Institute of Oceanography, Fisheries and Canada, Dartmouth, Nova Scotia, Canada

DAVID BRICKMAN Bedford Institute of Oceanography, Fisheries and Oceans Canada, Dartmouth, Nova Scotia, Canada

(Manuscript received 16 November 1999, in ®nal form 7 July 2000)

ABSTRACT A three-dimensional, z-level, primitive-equation ocean circulation model (DieCAST) is modi®ed to include a free-surface and partial cells. The updating of free-surface elevation is implicit in time so that the extra computational cost is minimal compared with the original DieCAST code, which uses the rigid-lid approximation. The addition of partial cells allows the bottom cell of the model to have variable thickness, hence improving the ability to accurately represent topographic variations. The modi®ed model is tested by solving a two- dimensional, linearized problem of internal tide generation over topography. Baines' method is modi®ed to more cleanly separate the internal tide from the full solution. The model results compare favorably with the semianalytic solution of Craig. In particular, the model reproduces the predicted variation of internal tide energy ¯ux as a function of the ratio of bottom slope to characteristic slope.

1. Introduction larly limits the application of the model in coastal waters if one wants to simulate motions driven by and DieCAST is a ®nite-difference numerical model that synoptic atmospheric forcings. To overcome this weak- solves the three-dimensional primitive equations gov- ness, the surface elevation must be free to change in erning the ocean circulation. It is a z-level model, that response to divergence and convergence of the depth- is, the vertical space is discretized into cells that do not vary in thickness over the horizontal. The original code integrated ¯ow. Various schemes have been proposed includes the rigid-lid approximation with the surface for incorporating a free-surface into general circulation pressure updated by solving a two-dimensional (2D) models. In this study, we choose to implement the time- elliptic equation. The DieCAST model has been suc- implicit scheme of Dukowicz and Smith (1994) rather cessfully applied to simulate the circulation in oceans than an explicit scheme such as that considered by Kill- and lakes driven by wind, buoyancy ¯ux, and boundary worth et al. (1991). This choice is motivated by the mixing (e.g., Dietrich and Lin 1994; Davidson et al. desire to avoid major changes to the code while retaining 1998; Sheng et al. 1998; and Sheng 2001, hereafter the computational ef®ciency of the rigid-lid model. Both SHE). goals are achieved satisfactorily using Dukowicz and Two limiting features of the model are 1) the exclu- Smith's (1994) method. sion of surface gravity waves by the rigid-lid approxi- With ®xed-height z levels, topographic changes such mation and 2) the use of ®xed z levels in the vertical as a continental slope are represented by a staircase discretization, potentially causing a poor representation approximation. Obviously, the accuracy of this approx- of topographic variations. The ®rst weakness particu- imation increases with an increase in the number of levels and a decrease in the thickness of the levels, hence, an increase in computational cost. By contrast, Corresponding author address: Dr. Youyu Lu, Physical Ocean- terrain-following coordinates (e.g., the ␴ coordinate) ography Research Division, Scripps Institution of Oceanography, can accurately resolve topographic changes even with University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0230. a single layer. To minimize this weakness, we use the E-mail: [email protected] method of partial cells (Adcroft et al. 1997; Pacanowski

᭧ 2001 American Meteorological Society

Unauthenticated | Downloaded 09/29/21 02:09 PM UTC JUNE 2001 LU ET AL. 1077 and Gnanadesikan 1998) in which the bottom cell of ments previous modeling efforts focused on the internal each vertical column has variable thickness (uniform tide generation problem (e.g., Holloway 1996; Cummins over each cell) in order to more accurately represent the and Oey 1997; Xing and Davies 1998; Holloway and true water depth. We will show that the use of partial Merri®eld 1999). The results of these previous studies cells signi®cantly improves the model solution above reveal that the generation of waves strongly depends on the sloping bottom. Terrain-following coordinates have topographic and strati®cation variations. The wave ®eld another advantage, that is, the boundary layer near the undergoes large variations over relatively small spatial bed can be better resolved by increasing the number scales, consistent with observations. It has long been of layers near the bottom. The use of partial cells does speculated that the energy transfered from surface to not help to resolve the bottom boundary layer over var- internal tides plays an important role in both coastal and iable topography; however, it does help in situations deep ocean mixing (e.g., Sandstrom and Oakey 1995; where topographic variations play an important role, but Munk 1997). On the other hand, there is observational the topography is not well represented without using an evidence, for example, from satellite altimetry data (Ray undesirably large number of ®xed-height z levels. Ad- and Mitchum 1996), that shows the propagation of in- vantages of using a z-level model include the conve- ternal tides over thousands of kilometers from their nience of easier analysis of results and the elimination source regions. The development and application of of the pressure gradient error associated with ␴-coor- models that can deal with this problem and the analysis dinate models when both density strati®cation and to- of the results remain challenging issues in oceanogra- pographic gradients are present (Haney 1991; Mellor et phy. al. 1994). As part of this study, we must separate the internal It should be noted that neither ®xed z levels nor ␴ tide ®eld from the full model solution in order to es- coordinates are well suited to focusing vertical resolu- tion in the vicinity of the where shear pro- timate the energy ¯ux carried by the internal tide. We duction of turbulence may be particularly important will discuss the necessary modi®cations to previously (e.g., Xing and Davies 1996, 1998). Fixed z levels may proposed methods (e.g., Baines 1982; Holloway 1986). have some advantage over ␴ coordinates for this prob- The remainder of the paper is arranged as follows. In lem, but isopycnal models are clearly better suited to the next section, we introduce the idealized study of thermocline problems than either of the other formu- internal tide generation and present the method used to lations. separate the internal tide from the full model solution. The model modi®cations are tested using an idealized In section 3, we introduce the DieCAST ocean model study of internal tide generation over sloping topogra- and its setup to solve this test case. Appendix A dis- phy. For this problem, terrain-following coordinates cusses details of the free surface formulation, appendix have an additional advantage over ®xed z-level models B gives some model details that are unique to the A- that was not mentioned above. In order to accurately grid formulation, and appendix C discusses details of represent the internal waves that propagate onto the the partial cell implementation. In section 4, we present shelf, at least several layers are required on the shallow the model results in terms of the ¯ow ®eld and energy side of the slope. Obviously terrain-following coordi- ¯uxes and make comparisons with theoretical expec- nates are well suited to handling this problem. In our tations. A summary and conclusions are presented in z-level model, we will choose our vertical grid resolu- section 5. tion to be ®ner near the surface in order to reasonably resolve the vertical structure of the ®eld on the shelf. It is noteworthy that most previous studies of internal tide generation (e.g., Holloway 1996; Xing 2. Internal tide generation: A test problem and Davies 1998) have been conducted with models using terrain-following coordinates; it is not obvious a. The 2D linear problem how ®ne the horizontal and vertical resolutions of a z- level model will have to be in order to successfully solve this problem. The linear, 2D problem of internal tide generation, The internal tide generation problem is particularly solved using a semianalytic method by Craig (1987), is relevant to the validation of the model extensions con- formulated as follows. The 2D model geometry is il- sidered here because the surface tide serves as the en- lustrated in Fig. 1. It consists of deep and shallow re- ergy source, and the bottom topography plays an es- gions of uniform depth, separated by a linearly sloping sential role in the barotropic-to-baroclinic energy trans- region. The model coordinates are chosen such that the fer. Further, the speci®c problem considered here is ap- topography varies in the x direction; all gradients in the propriate for model validation because results can be y direction vanish. The depth of the ocean, measured readily compared with those from a semianalytic so- from the undisturbed surface, is H(x). Under the hy- lution documented by Craig (1987). drostatic and Boussinesq approximations, the linearized This modeling study, although idealized, comple- equations for this problem are

Unauthenticated | Downloaded 09/29/21 02:09 PM UTC 1078 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18

supercritical (␣ Ͼ 1) cases. In the following discussion, we consider the case f ϭ 0. However, we note that the solution of (6) for nonzero f can be obtained simply by including the appropriate value of f in the de®nition of c. The results presented here as a function of ␣ would not be changed for nonzero f, except for the interpre- tation of ␣.

b. Separating surface and internal tide components

An internal tide is generated when a barotropic sur- face tide propagates in strati®ed water into a region with spatially varying topography. To examine the generation of an internal tide using results from a primitive equation numerical model, one must be able to separate the in- ternal tide ®eld from the full solution. Following Baines (1982), we ®rst de®ne the ``surface tide'' (denoted by a superscript ``s'') as the solution of FIG. 1. The model domain, consisting of shallow and deep regions with a sloping bottom in between. the following equations:

ss utxϩ p /␳* ϭ 0, (7)

utxϪ f␷ ϩ p /␳* ϭ 0, (1) s pzoϩ ␳ g ϭ 0, (8)

␷ t ϩ fu ϭ 0, (2) ss uxzϩ w ϭ 0. (9)

pzoϩ (␳ ϩ ␳Ä )g ϭ 0, (3) Note that the surface tide solution is calculated in the absence of horizontal density gradients. Baines uses this u ϩ w ϭ 0, (4) xz solution as an estimate of the external mode and sub-

␳Ätozϩ w␳ ϭ 0, (5) tracts it from the full solution to estimate the internal tide. Following this procedure, we ®nd that the resulting where the subscripts x and t denote the partial deriva- estimate of the baroclinic energy ¯ux exhibits large am- tives with respect to the x coordinate and time, respec- plitude periodic variations in x over the ¯at regions (see tively; u, ␷, and w are the x, y, and z velocity compo- Fig. 6a). These anomalous periodic variations indicate nents; p is pressure; ␳ is a reference density (a constant * that the baroclinic mode is not cleanly separated from value); ␳o(z) is the background time-mean density ®eld (chosen to have no horizontal variation);␳Ä (x, z, t) is the the external mode using this method. density perturbation due to advection by the tidal ¯ow; We observe that in the presence of background den- and f is the parameter. sity strati®cation, ␳o(z), the isopycnals are heaved by Because the 2D velocity ®eld is nondivergent, one the vertical velocity associated with the surface tide can de®ne a stream-function ␺ such that u ϭ even in the absence of internal wave motion. This results Ϫi␻t Ϫi␻t in changes of density and pressure, denoted asà and pÃ, ϪRe{␺ze }, w ϭ Re{␺xe }, where ␻ is the tidal ␳ frequency and i ϭ ͙Ϫ1. From (1), (2), (3), and (5), one respectively, which satisfy can derive an equation for ␺: s ␳à tozϩ w ␳ ϭ 0, (10) 2 ␺xxϪ c ␺ zz ϭ 0, (6) pà ϩ (␳ ϩ ␳à )g ϭ 0; pà ϭ p s at z ϭ 0. (11) where N ϭ (Ϫg␳ /␳ )1/2 is the buoyancy frequency and zo oz * c2 ϭ (␻ 2 Ϫ f 2)/N 2 is the characteristic slope of the Note that advection by ws alone determines␳à and pÃ; internal tide. Craig (1987) seeks the solution for (6) advection by baroclinic ¯ow is excluded. Clearly,␳à and under the following conditions: ␺ ϭ 0atz ϭϪH and pà are part of the external mode of response and should

␺xx ϭ 0atz ϭ 0 (the assumption of an x-independent be accounted for in the estimation of the surface tide. vertical velocity through z ϭ 0 is justi®ed when the x- The internal tide ®elds (denoted by a superscript ``i'') extent of the model domain is small compared with the are now estimated by wavelength of the surface tide). Craig shows that the solution structure is determined by a nondimensional (uii, w ) ϭ (u Ϫ u s, w Ϫ w s), (12) parameter, ␣ ϭ H /c, the ratio of bottom slope to the x (p ii, ␳ ) ϭ (p Ϫ pÃ, ␳Ä Ϫ ␳Ã ), (13) characteristic slope of the internal tide. The value of ␣ ϭ 1 divides the problem into subcritical (␣ Ͻ 1) and which satisfy the following equations:

Unauthenticated | Downloaded 09/29/21 02:09 PM UTC JUNE 2001 LU ET AL. 1079

ii s utxϩ p /␳* ϭ (p Ϫ pÃ)/ x␳*, (14) according to (17). Hence, the time-averaged energy equation is ii i s pz ϩ ␳ g ϭ 0; p ϭ p Ϫ p at z ϭ 0, (15) ii ii i s ͗up͘ϩ͗xzwp͘ϭ͗u (p Ϫ pÃ) x͘, (21) uiiϩ w ϭ 0, (16) xz which represents a balance between the divergence of ii ␳tozϩ w ␳ ϭ 0. (17) internal energy ¯ux and the rate of energy conversion from surface to internal tide averaged over a tidal cycle. The above approach to separating out the internal tide The right side represents the energy source for the in- is introduced in order to account for the pressure effect ternal tide, and the left side represents the energy re- associated with the heaving of the density ®eld by the distribution by the internal tide. In section 4, we will barotropic ¯ow, an effect that was not accounted for in show that the energy source is concentrated over the the early work of Baines (1982). An alternative ap- slope as expected and that the baroclinic energy ¯ux proach to dealing with this issue has been taken, for over the ¯at regions is well de®ned. example, by Holloway (1996) and Holloway and Mer- The key to the above procedure for separating the ri®eld (1999). In these studies, pi is calculated using internal tide from the full solution lies in determining 0 the pressure ®eld with baroclinic motions eliminated. In p iiϭϪ␳* wN2 dzЈ, (18) the above discussion, this has been achieved by con- t ͵ z sidering the tidal ¯ow that would occur in the absence where ␻ 2 has been neglected compared to N 2, consistent of density strati®cation. While this approach gives a with the hydrostatic approximation. With this method, reasonable result, one must bear in mind that the energy pressure perturbations due to isopycnal heaving by the loss from surface to internal tide is neglected. The po- barotropic tide are excluded from the estimates of pi. tentially signi®cant in¯uence of nonlinear bottom stress However, the assumption that pi vanishes at z ϭ 0 gives is also not considered. The latter effect could be ac- rise to a different error in the estimation of the internal counted for by using an appropriately linearized bottom energy ¯ux (see Fig. 5 and the discussion in section stress formulation (e.g., see Wright and Thompson 2c). 1983). The term on the right side of (14), (p Ϫ pÃ) /␳ , may One approach that can account for the energy loss to s x * be thought of as a ``body force'' that generates the in- the internal tide is as follows. Consider a supplementary ternal tide. It is very small over the ¯at regions (ideally, run, which differs from the basic model run by the ad- it should vanish), but its magnitude is ampli®ed over dition of very strong vertical mixing in the horizontal variable topography. momentum equations (using an implicit numerical for- Multiplying (14) by ui and (15) by wi and adding, we mulation), but with the surface and bottom stress set to obtain the kinetic energy budget for the internal tide: zero. By mixing momentum so strongly that baroclinic motions are eliminated over just one or two model time i 2 ii ii [␳*(u )]txϩ (up) ϩ (wp) z steps, baroclinic motions are effectively eliminated, and the results of this run can be used in place of the results ii i s ϭϪg␳ w ϩ u (p Ϫ pÃ)x . (19) obtained from Eqs. (7)±(11) as an estimate of the ``sur- The terms on the left side of (19) are the change of face tide.'' The advantage of this approach is that the kinetic energy with time and the divergences of hori- energy transfer from the surface tide to the internal tide zontal and vertical energy ¯uxes, respectively. The is accounted for, and the pressure variations due to the terms on the right side are the rate of exchange between heaving of the background density ®eld by the surface internal potential energy and internal kinetic energy and tide (without any heaving by the baroclinic tide) are the rate of energy transfer from the external to the in- automatically determined. ternal mode. Of course, a disadvantage of both of the approaches i discussed above is that a supplementary model run is Similarly, multiplying (17) by g␳ /␳oz, we obtain the potential energy equation for the internal tide: required to estimate the surface tide so that the baro- clinic tide can be obtained by differencing. However, 2 2 this is a relatively minor price to pay for an improved ␳*N ␳i ϭ g␳ iiw . (20) separation of the surface and internal tidal motions. In 2 ␳ []΂΃oz t the next section, we consider what information about The left side represents the rate of change of internal the baroclinic energy ¯ux can be reliably obtained with- potential energy, while the term on the right side rep- out making any additional runs. resents the conversion between internal potential and internal kinetic energy. c. Decomposition using only the full model solution Averaging the energy equations over a tidal cycle (denoted by an angle bracket ͗͘), the time-derivative In the previous section, we presented methods for terms drop out, since the tidal forcing is constant, and separating the internal tide from the total solution that ͗g␳iwi͘ϭ0 because ␳i and wi are 90Њ out of phase involved an auxiliary model run in which baroclinic

Unauthenticated | Downloaded 09/29/21 02:09 PM UTC 1080 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18 motion was suppressed. Here we consider whether or clinic energy ¯ux can be obtained from the results of a not it is possible to separate the energy ¯uxes associated single run of the full model using either of these two with the surface and internal tides using only the so- approaches. lution of the full model, which includes the combination of the two tides, without doing an additional model run. The horizontal velocity associated with the surface 3. Model description tide is usually taken as the depth-average of the full a. Model modi®cation velocity (e.g., Cummins and Oey 1997), that is, The DieCAST model solves the three-dimensional 0 (3D) nonlinear equations governing temperature, salin- s u ϭ ͵ udz/H. (22) ity, density, pressure, and velocity. The Boussinesq and ϪH hydrostatic approximations are made. The numerical Once us is determined, the vertical velocity associated methods used in earlier versions of the DieCAST model with the surface tide can be estimated using are documented by, for example, Dietrich et al. (1987), Dietrich (1992), and Sheng et al. (1998). These papers ss w ϭ (1 ϩ z/H)␩txϩ z/HH u , (23) use the Arakawa C grid for the spatial discretization s (e.g., Sheng et al. 1998). In the present study, we work where Hxu is the vertical velocity just above the bottom associated with us. The relative differences between the with the A-grid version of DieCAST (see appendix B (us, ws) calculated using (22) and (23) and those ob- for some numerical details on the A-grid). The major tained by solving (7)±(9) are small. We note that the modi®cations made to the model code for the purpose approximation of us by (22) assumes that ui has a neg- of the present work are the additions of a free-surface 0 i formulation and allowance for partial thicknesses of the ligible depth-mean (#ϪH u dz ϭ 0), and the determination of ws by (23) assumes that the contribution ␩i ϭ ␩ Ϫ bottom cells of the model. The method used to allow a s s s s free surface is the time-implicit scheme suggested by ␩ [where ␩ ϭ p /(␳*g)atz ϭ 0 and p is determined as in the previous section] to the surface elevation is Dukowicz and Smith (1994). Appendix A provides a also negligible. Equations (22) and (23) effectively cor- brief account of how to implement this method in the respond to making the rigid-lid approximation for the DieCAST model. The primary advantage of including internal tide. free-surface effects is that gravity waves associated with In reality, the surface elevation ␩i associated with the the free surface are explicitly represented. The incor- internal tide makes a ®nite contribution to pi. By sep- poration of partial cells into the DieCAST code is dis- arating the surface elevation as ␩ ϭ ␩s ϩ ␩i, the vertical cussed in appendix C. By using this method, a z-level integration of (15) gives model can accurately resolve the changes in the topog- raphy (as with terrain-following coordinates, the accu- 0 racy remains limited by horizontal resolution). p iiϭ ␳*g␩ ϩ g ͵ ␳ idzЈ. (24) z b. Model setup The second term on the right is well approximated by the approach taken by Holloway (1996). The ®rst term To solve the 2D internal tide generation problem, as on the right is neglected in Holloway's approach. Ne- outlined in the previous section, we choose the model glecting the surface pressure associated with the internal domain as a straight channel with a single grid in the wave ®eld introduces an inaccuracy into the estimation cross-channel (y) direction and set the velocity, ¯uxes, of the vertical structure of the baroclinic energy ¯ux, and gradients of all quantities to be zero in this direction. particularly near the surface. Of course, as the depth In the along-channel (x) direction, the model con®gu- increases, the second term becomes increasingly im- ration is identical to that of Craig (1987). The shallow portant and can dominate the surface pressure contri- water region is 200 m deep and 50 km wide, the sloping bution, especially for large-amplitude internal tides. region has a constant slope of 0.01 over a distance of The depth-integrated energy ¯ux consists of two 80 km, and the deep water is 1000 m deep and 500 km parts: the part associated with ␩i is 0 ␳ g ͗ui␩i͘ dz, wide. The x axis is discretized using uniform grid spac- #ϪH * and that associated with density changes in the interior ing (⌬x). We shall show that the model solutions are 00i i is ##ϪHzg ͗u ␳ dzЈ͘ dz. Because the depth-average of sensitive to the choice of ⌬x. We discretize the vertical ui is much smaller than the typical magnitude of ui, the space using 25 nonuniformly spaced layers. The thick- ®rst part is negligible compared with the second part. ness of these layers ranges from 20 m near the surface Consequently, the depth-integrated horizontal energy to 69 m near the bottom, with 8 layers in the upper 200 0 ¯ux can be calculated from the vertical integral #ϪH g m. As discussed later, such a choice of vertical levels i 0 i i i ͗u #z ␳ dzЈ͘ dz, using the estimates of u and ␳ obtained ensures that there is suf®cient vertical resolution to re- from (22), (23), and (10)±(13). This is equivalent to solve the baroclinic mode in both the deep and shallow calculating pi using (18), following Holloway (1996). regions. Thus, a reliable estimate of the depth-integrated baro- At the open boundary in the deep water (at x ϭ 0in

Unauthenticated | Downloaded 09/29/21 02:09 PM UTC JUNE 2001 LU ET AL. 1081

FIG. 2. Snapshots of the vertical velocity in a model run (a) with and (b) without partial cells. Negative values are plotted using broken lines; zero and positive values are plotted using solid lines. The contour interval is 1 ϫ 10Ϫ4 m sϪ1.

Fig. 1) we introduce a surface tide, which propagates face, bottom, and side boundaries. A radiation condition into the model domain and sets up a surface elevation is applied to the temperature at x ϭ 0 and x ϭ L. of 1 m in magnitude at the coast (at x ϭ L in Fig. 1). To be consistent with Craig (1987), momentum ad- The surface tide is re¯ected at x ϭ L by enforcing zero vection is neglected, heat advection is linearized about depth-integrated ¯ow there, but it is allowed to prop- the initial state, and minimal subgrid scale mixing is agate out of the model domain through x ϭ 0 using a used. No explicit mixing of heat is included in either radiation boundary condition [the radiation phase speed the horizontal or vertical directions. A slip condition is for the surface tide is (gH)1/2]. The internal tide is al- applied to the velocity at the side boundaries, there is lowed to propagate out through both x ϭ 0 and x ϭ L no explicit mixing of momentum in the vertical, and the using radiation boundary conditions for the baroclinic bottom stress is set to zero. A Laplacian formulation is mode (the radiation phase speed is taken equal to that used for the horizontal mixing of momentum, with the of the ®rst baroclinic mode). The tidal frequency is set diffusion coef®cient set to 10 m2 sϪ1 (momentum dif- at ␻ ϭ 10Ϫ4 sϪ1. fusion over a horizontal distance of 3 km requires ap- The DieCAST model solves the advection±diffusion proximately 9 days, comparable with the maximum time equation for temperature and salinity. In this idealized required for the internal waves generated at the sloping test, we initialize the salinity to be uniform and set zero topography to propagate out of the model domain). salt ¯uxes at all the model boundaries. The salinity ®elds are thus uniform in both space and time. The density variation is purely due to changes in temperature. The 4. Model results: Validation initial temperature ®eld varies linearly in the vertical a. Internal tide ¯ow ®elds and is uniform in the horizontal. A linear equation of state is applied, hence the buoyancy frequency (N)is We ®rst illustrate the improvement in the model so- uniform. We run the model with different values of the lution achieved by using partial cells. Figure 2 shows buoyancy frequency, hence different values of ␣ ϭ HxN/ two snapshots of the vertical velocity ®eld obtained ␻. Insulation conditions for heat are applied at the sur- from simulations with and without partial cells. When

Unauthenticated | Downloaded 09/29/21 02:09 PM UTC 1082 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18

FIG. 3. A time sequence of internal tide streamfunction (␺i) plots. Starting from the second panel from the top, each panel lags the one above it by 10.9 (the tidal period is 17.45 h). Negative ␺i (broken lines) correspond to counterclockwise baroclinic circulation. Positive ␺i (solid lines) correspond to clockwise circulation. The contour interval is 0.25 m 2 sϪ1. The results are for ␣ ϭ 0.6 and were simulated using ⌬x ϭ 1 km. only standard vertical cells are used, the poor represen- a bottom-intensi®ed in a strati®ed re-en- tation of the sloping bottom leads to substantial grid- trant channel. This test of partial cells has been ex- scale noise in the velocity ®eld. The noise is not limited amined previously for an implementation in the GFDL over the slope, since the vertical motion generates in- Modular Ocean Model (Pacanowski and Gnanadesikan ternal waves that propagate into the ¯at regions. When 1998). Although this test case serves as an additional partial cells are used, the grid-scale noise is almost com- validation of the model, the results obtained with our pletely eliminated, and a smoothly varying velocity ®eld implementation of partial cells in the DieCAST model is obtained. The improvements are also evident in the are similar to those presented by Pacanowski and Gnan- results for other variables and the subsequent calculation adesikan and will not be reproduced here. Partial cells of energy ¯uxes. are used throughout the rest of the model experiments The implementation of partial cells has also been val- presented here. idated against the analytic solution of Rhines (1975) for Figure 3 shows a time sequence of streamfunction ␺i

Unauthenticated | Downloaded 09/29/21 02:09 PM UTC JUNE 2001 LU ET AL. 1083

FIG. 4. Same as in Fig. 3 but with ⌬x ϭ 4 km. Note the weaker baroclinic velocity in shallow water. plots for the internal tide. We de®ne ␺i such that ui ϭ It is of interest to consider whether we can achieve iii i Ϫ␺␺zx, w ϭ with ␺ ϭ 0atz ϭϪH. The results shown similar results to those presented in Fig. 3 with coarser in Fig. 3 are for the subcritical case ␣ ϭ 0.6, obtained resolution. For ␣ ϭ 0.6, we have considered ⌬x ϭ 2 using a horizontal grid spacing of ⌬x ϭ 1 km. The time km and ⌬x ϭ 4 km. Results show that the solution with sequence of ␺i clearly shows that the internal tide is ⌬x ϭ 2 km is very close to that with ⌬x ϭ 1 km. For predominantly generated at the bottom of the slope, after ⌬x ϭ 4 km (Fig. 4), the solution does not change sig- the surface tide arrives there, and propagates toward ni®cantly in the deep water, but the internal tide in the both the deep ocean and the shelf. The propagation shallow region is greatly reduced. These conclusions speed is approximately equal to the local phase speed based on idealized experiments are generally consistent of the ®rst baroclinic mode. After the solution reaches with results obtained in much more realistic 3D con®g- a periodic state, the evolution of ␺i over a tidal cycle urations (e.g., Cummins and Oey 1997; Xing and Davies (not shown here) agrees, both qualitatively and quan- 1996, 1998). titatively, with the semianalytic solution of Craig (1987; The spatial resolution required to accurately repro- see his Fig. 5). duce the nature of the internal wave ®eld is related to

Unauthenticated | Downloaded 09/29/21 02:09 PM UTC 1084 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18

FIG. 5. (a) The horizontal energy ¯ux of the internal tide, ͗uipi͘, for ␣ ϭ 0.6. Zero and positive values are plotted as solid lines. Negative values are plotted as broken lines. The contour interval is 0.005 W m Ϫ2. (b) Same as (a) i 0 i except for contour of ͗u #z ␳ dzЈ͘. the wavelength of the internal tide. In the vertical di- sociated with the internal tide includes a contribution rection, the wavelength of the ®rst mode equals twice from ␩i, the change associated with the internal the water depth (for this constant N case), hence 25 tide [Eq. (24)]. While this contribution is tiny compared levels in deep water and 8 levels in shallow water are with ␩s, the in¯uence on the estimation of the baroclinic adequate to resolve the vertical variations associated energy ¯ux is signi®cant. This point is emphasized by i 0 i with the ®rst internal mode and marginally adequate to Fig. 5b, which shows the quantity ͗u g #z ␳ dzЈ͘, ob- resolve the vertical variations in the second mode. For tained by neglecting ␩i in the calculation of the energy a given characteristic slope, c ϭ ␻/N, the horizontal ¯ux shown in Fig. 5a. This is effectively equivalent to wavelength of the ®rst baroclinic mode is 2H/c ϭ 2␣H/ the result that would be obtained using the approach of

Hx. For ␣ ϭ 0.6 and Hx ϭ 0.01, the wavelength is 120 Holloway (1996). As noted earlier, the fact that the depth km in deep water (H ϭ 1000 m) and 24 km in shallow integral of ui is small results in negligible contribution water (H ϭ 200 m). Hence ⌬x ϭ 4 km is easily suf®cient from ␩i to the depth-integrated energy ¯ux (Fig. 6a). to resolve the internal tide in the deep water but marginal However, if one wishes to examine the depth distribu- at best in the shallow water. The situation is exacerbated tion of the internal energy ¯ux, the surface pressure if higher modes are involved in the energy transfers over variation associated with the internal tide should be ac- the slope. For larger ␣ values, the spatial resolution is counted for unless one is con®dent that the internal tide less of a constraint in the ¯at regions but more of a does not extend to the surface. constraint over the sloping region. Note that if the density variations associated with heaving of the density ®eld by the surface tide are not removed from ␳i (and hence also from pi), then the b. The energy budget depth-integrated energy ¯ux would include large am- In Fig. 5a, we show the contour lines of the horizontal plitude variations over the length scale of the internal energy ¯ux, ͗uipi͘, for ␣ ϭ 0.6. In the deep water, the tide. This erroneous result is shown as the thinner bro- energy ¯ux is directed to the left (negative values). In ken line in Fig. 6a. Such variations are a clear indication the shallow water, the ¯ux is directed to the right (pos- of poor separation between the surface and internal itive values). We noted earlier that the pressure pi as- wave modes.

Unauthenticated | Downloaded 09/29/21 02:09 PM UTC JUNE 2001 LU ET AL. 1085

FIG. 6. (a) The depth-integrated horizontal energy ¯uxes calculated using ͗uipi͘ (solid line) and i 0 i ͗u #z ␳ dzЈ͘ (thicker broken line, essentially overlying the solid line). The thinner broken line shows the erroneous result that is obtained when heaving of the density ®eld is not accounted for in the estimation of the pressure variations associated with the surface tide. (b) The depth- i s integrated values of ͗u (p Ϫ pÃ)x͘, the rate of energy conversion from surface to internal tides. Both (a) and (b) are for the case ␣ ϭ 0.6.

Figure 6a shows that the values of the depth-inte- ground density ®eld ␳o hence the buoyancy frequency grated internal tide energy ¯ux jumps from negative to N). The energy ¯uxes are calculated for each individual positive at the bottom of the slope. This ¯ux divergence run. In Fig. 8, we plot the values of the internal tide i s is balanced by the source term ͗u (p Ϫ pÃ)x͘, the con- energy ¯ux toward the deep ocean [denoted by F1 as in version rate from surface to internal tidal energy. In Fig. Craig (1987)] as a function of ␣. Superimposed on this i s 6b, we plot the depth-integrated values of ͗u (p Ϫ pÃ)x͘. plot are the values of F1 calculated by Craig. The agree- This plot shows that the energy conversion takes place ment between the two sets of values, both in the sub- mainly at the bottom of the slope. critical (␣ Ͻ 1) and supercritical (␣ Ͼ 1) regimes, are i s The other region with nonzero ͗u (p Ϫ pÃ)x͘ is at the quite good. An interesting feature of the model calculated top of the slope where the depth-integrated conversion values is the presence of two regions of reduced energy rate is negative for this geometry and choice of param- ¯ux near ␣ ϭ 0.4 and ␣ ϭ 0.7. These two anomalies eters. are not indicated by Craig's values (although he did not Except near the bottom and the top of the slope, the calculate energy ¯uxes for as many values of ␣ as pre- i s magnitude of ͗u (p Ϫ pÃ)x͘ is small, and the energy sented here). Interestingly, the occurrence of these two ¯uxes (͗uipi͘, ͗wipi͘) have almost no divergence. Con- reductions of the energy ¯ux approximately coincide with sequently, one can de®ne a scaler ⌿ such that ͗uipi͘ϭ two special cases in terms of internal tide characteristics, i i Ϫ⌿z, ͗w p ͘ϭ⌿x, and ⌿ can be used to illustrate the as shown in Fig. 9. For the ®rst case, which occurs at ␣ pathway of the baroclinic energy ¯ux, as shown in Fig. ϭ 2/3 for the present model geometry, the wave char- i i i i 7. The directions of the ¯ux vectors (͗u p ͘, ͗w p ͘) are acteristics link the bottom and top corners of the slope parallel to the contour lines. Starting from the bottom with one re¯ection at the surface. For the second case, of the slope, internal tide energy propagates toward the which occurs at ␣ ഠ 0.38, the wave characteristics link deep water along ``rays'' that bounce back and forth the same two regions with two re¯ections from the sur- between sea surface and sea bed. Similar rays also prop- face. These results are at least qualitatively consistent agate up the slope toward the shallow water. with the prediction of MuÈller and Liu (2000) that a linear slope will be transparent to internal waves with this prop- c. Energy ¯ux as a function of ␣ϭHxN/␻ erty. [Note that there is a misprint in the sign of the last We now consider the results from a sequence of model term in Eq. (39) of MuÈller and Liu (2000).] In this case, runs using different values of ␣ (by changing the back- forward-propagating internal waves generated at the base

Unauthenticated | Downloaded 09/29/21 02:09 PM UTC 1086 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18

i i i i FIG. 7. Contours of ⌿ de®ned as ͗u p ͘ϭϪ⌿z, ͗w p ͘ϭ⌿x, for ␣ ϭ 0.6. Negative values are plotted as broken lines with a contour interval of 0.5 W mϪ1. Zero and positive values are plotted as solid lines with a contour interval of1WmϪ1. The vectorial direction of the internal tide energy ¯ux (͗uipi͘, ͗wipi͘) is parallel to the contour lines. The two arrows indicate the directions of the depth-integrated energy ¯uxes in the deep and shallow waters, respectively. of the slope should traverse the slope without re¯ection, amount of work required to modify the rigid-lid code resulting in reduced total re¯ected internal wave energy. and the extra computational cost of including a free- The spreading of the anomalies in the ␣ axis is probably surface are minimal. We note that the same free-surface in¯uenced by a combination of nonzero diffusion in our method has been implemented in the Los Alamos Par- model and the approximate representation of the bottom allel Ocean Program (Smith et al. 1992). The method topography by the ®nite model resolution. Regarding the of including partial cells is that of Adcroft et al. (1997), in¯uence of ®nite model resolution, it is noteworthy that and an implementation in the GFDL Modular Ocean analytic solutions have singularities at sharp corners Model has been discussed previously by Pacanowski (Sandstrom 1976). and Gnanadesikan (1998). With a free-surface included, the forcing of the surface tide can be explicitly included. The use of partial cells signi®cantly improves the model 5. Conclusions solution, compared with using standard, ®xed-thickness The addition of a free-surface and partial cells makes z levels. two important modi®cations to the DieCAST ocean cir- To test these new developments, the modi®ed culation model. The time-implicit, free-surface scheme DieCAST model is applied to simulate the 2D, linear of Dukowicz and Smith (1994) is similar to the implicit problem of internal tide generation over topography. method of updating the surface pressure ®eld in the rigid-lid DieCAST model (appendix A). Hence, the

FIG. 8. Magnitude of the internal tide energy ¯ux into the deep FIG. 9. Internal tide characteristics (broken lines) that link the bot- ocean as a function of ␣ ϭ HxN/␻. Squares denote the semianalytical tom and top corners of the shelf break. The two cases shown are (a) solution of Craig (1987), and solid circles are calculated using the ␣ ϭ 2/3 and (b) ␣ ϭ 0.38 for the model geometry speci®ed in section free-surface DieCAST model. 4a.

Unauthenticated | Downloaded 09/29/21 02:09 PM UTC JUNE 2001 LU ET AL. 1087

The method of separating the internal tide from the full Canadian Institute for Climate Studies. We are grateful model solution, which includes the motions and density to David Dietrich for providing us with the original C- variations associated with the surface tide, is discussed. grid version of DieCAST and for helpful discussions It is shown that a consistent determination of the energy about the conversion to an A-grid formulation. We are budget for the internal tide requires accounting for the also grateful to Richard Greatbatch for suggesting the fact that the density variations due to heaving by the internal tide generation problem as a test case and to surface tide and the associated pressure variations are Chris Mooers, Alan Davies, and two anonymous re- part of the external mode and not part of the internal viewers for numerous helpful comments on the manu- mode. Attention to this point is essential to achieve zero script. divergence of the depth-integrated internal tide energy ¯ux away from its generation regions. It is also shown APPENDIX A that the surface pressure variations associated with the internal tide must be accounted for to accurately esti- Implicit Free-Surface Method mate the vertical structure of the baroclinic energy ¯ux. To relax the rigid lid approximation, we linearize the This latter effect is not critical to the determination of kinematic boundary condition at the sea surface to give the depth-integrated energy ¯ux associated with the in- ternal tide. w ϭ ␩t,atz ϭ 0. (A1) The model is used to determine numerical solutions Vertical integration of the nondivergent continuity equa- corresponding to the problem considered by Craig tion then gives (1987), and results are compared with his semianalytical solution. Close agreement between the numerical model 0 (u dz ϭ 0, (A2 ´ ١ and analytical results are obtained when the spatial res- ␩ ϩ t ͵ olution of the model is adequate to resolve the wave- ΂΃ϪH length of the lowest mode. By running the model with and integration of the hydrostatic equation gives different background density gradients, the model pre- 0 dicts a variation of the internal tide energy ¯ux (F1)as a function of the ratio of bottom slope to characteristic p ϭ ␳*g␩ ϩ g ͵ ␳ dzЈ. (A3) z slope (␣). The model results reveal two dips in F1 within the subcritical range of ␣, but otherwise shows a mono- The horizontal momentum equation is tonic increase of F1 with increasing ␣, in agreement with the results presented by Craig (1987). The two dips ut ϩ L (u) ϩ f ϫ u in the energy ¯ux occur for the parameter values at 0 (␳/␳* dzЈϪg١␩ ϩ D (u which the results of MuÈller and Liu (2000) predict that ϭϪg١ ͵ m a linear slope will be transparent to the internal waves z generated at the base of the slope. It is likely that this ϩ (A u ) . (A4) effect accounts for the reduction in internal wave energy ␷ zz re¯ected backward from the slope. This question war- In the above equations, we de®ne u as the horizontal rants further investigation but is beyond the scope of velocity vector; ␩ is the surface elevation measured the present work. from the position of the undisturbed sea surface (z ϭ In summary, the results from this study demonstrate 0); z ϭϪH(x, y) is the position of the bottom; f is an the successful addition of a free-surface and partial cells upward directed vector with magnitude equal to the Cor-

,L, Dm) are the horizontal gradient ,١) ;to a z-level ocean model (DieCAST). The modi®cations iolis parameter extend the applicability of the model to include prob- advection, and horizontal diffusion operators, respec- lems that require the relaxation of the rigid-lid approx- tively; and A␷ is the vertical viscosity. imation and more accurate representation of topographic The implicit time-differencing scheme of Dukowicz changes. The idealized test case of internal tide gen- and Smith (1994) is applied to discretize (A2) and (A4). eration can easily be extended to more general situations This leads to that include nonlinear effects, turbulent mixing, and 0 (un dz ϭ 0, (A5 ´ ١ three-dimensionality. Potential problems with previous (␩ nϩ1 Ϫ ␩ n)/⌬t ϩ approaches to separating the surface and internal tides ͵ ϪH have been identi®ed, and simple corrective measures have been suggested. (unϩ1 Ϫ unϪ1)/(2⌬t) ϩ L (unn) ϩ f ϫ u 0 (␳ nn/␳* dzЈϪg١␩ n ϩ D (u Ϫ1 Acknowledgments. Youyu Lu thanks Keith Thompson ϭϪg١ ͵ m for his support of this work through the GLOBEC± z Canada project. Daniel G. Wright's work on the ϩ (A unϩ1) , (A6) DieCAST ocean model has been supported by the At- ␷ zz mospheric Environment Service of Canada through the where

Unauthenticated | Downloaded 09/29/21 02:09 PM UTC 1088 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18

un ϭ ␪unϩ1 ϩ (1 Ϫ ␪)un, (A7)

n nnϪ1 ␩ ϭ ␣␦␩112ϩ (␣ ϩ ␣ )␩ ϩ (1 Ϫ ␣ 12Ϫ ␣ )␩ , (A8)

nϩ1 n and ␦␩ ϭ ␩ Ϫ ␩ . The values of ␪, ␣1, and ␣ 2 determine whether an explicit or implicit scheme is em- ployed to deal with the free surface elevation. ␪ ϭ 1,

␣1 ϭ 1, and ␣ 2 ϭ 0 give a fully implicit scheme while ␪ ϭ 0.5, ␣1 ϭ ␣ 2 ϭ 1/3 give a time-centered explicit scheme. Dukowicz and Smith (1994) give a detailed discussion of the in¯uences of different values of ␪ and ␣ on the physical and computational modes. They also show that the linear approximation (A1) can be replaced by its nonlinear counterpart, and the vertical integral in (A2) can be performed from z ϭϪH to z ϭ ␩. Also, a variable thickness top-layer could be implemented into the DieCAST model following Dukowicz and Smith's FIG. B1. A control volume center on A , where (p, T, S, ␳,u, formulation. However, the linear formulation is justi®ed i,j,k ␷)i,j,k are calculated. For the Arakawa A-grid, (p, T, S, ␳,u,␷)i,j,k when the change of surface elevation is much smaller actually represent averages over the cell volume. The positions of than the total water depth. the staggered C-grid velocities are denoted by (Ui,j,k, Vi,j,k, wi,j,k). The surface elevation and velocity ®elds are updated according to the following procedure. ®elds follows the same procedure as in the rigid-lid 1) The velocity u is partially updated based on explicit model (e.g., Sheng et al. 1998). treatment of advection, , the baroclinic pressure gradient, and ␩ at previous time steps as APPENDIX B well as horizontal mixing; the trial ®eld obtained is denoted as uà nϩ1; A-Grid Details nϩ1 2) the in¯uence of vertical mixing on uà is accounted The version of DieCAST used in the present study for using an implicit scheme; the new trial ®eld is uses the Arakawa A-grid formulation. It was developed nϩ1 denoted as uÄ ; as a modi®cation of the C-grid version of the DieCAST 3) ␩ is implicitly updated by solving the following el- model discussed by Sheng et al. (1998) and has bene- liptic equation for P ϭ 2⌬t␣1g␦␩: ®ted from numerous consultations with Dr. David Die- .H١P) Ϫ ␥P ϭ F ϩ F*, (A9) trich, the developer of the original version of DieCAST)´١ Details of the C-grid version of DieCAST are presented 00nϩ1 n ϪH u by Sheng et al. and will not be repeated here. We discuss ´١(ϪH uà dz, F* ϭ (1/␪ Ϫ 1##´١ where F ϭ 2 dz, and ␥ ϭ 1/[2(⌬t) g␣1␪]; and only the differences between the C-grid and the A-grid 4) the velocity ®eld is ®nally updated using formulations. The most important differences between the A-grid (١P. (A10 unϩ1 ϭ uà nϩ1 ϩ and C-grid formulations are associated with the spatial When the rigid-lid approximation is made, the po- discretization of the horizontal velocity components u sition of the surface is ®xed at z ϭ 0; ␩t in (A1) and and ␷. On the A-grid, u and ␷ are de®ned at the center (A2) is zero, and ␩ in (A3) and (A4) is replaced by po/ of each cell (i.e., colocated with T, S, ␳, and p), whereas (␳ g), where p is the pressure under the rigid-lid. In * o in the C-grid formulation, u and ␷ are de®ned on the the rigid-lid code, the updating procedure is similar to bounding faces of the control volume. Figure B1 shows that described above except for the following changes an isolated control volume centered at an A-grid point. (cf., Sheng et al. 1998): 1) in step 1), the updating of The A-grid point and the locations at which the C-grid u is based on the surface pressure at the previous time velocity components are de®ned are each indicated. The n nϪ1 n step,po , instead of ␩ and ␩ ; and 2) the elliptic equa- colocation of u and ␷ on the A-grid overcomes the dif- tion is constructed for P ϭ ␦po, and (A9) is replaced by ®culty with the C-grid in estimating the Coriolis terms. However, the use of a control volume approach requires (ϭ F. (A11 ( H١)´١ P knowledge of the velocity components normal to the Except for the change of forcing terms, the most sig- faces of the model grid cells, that is, at the C-grid lo- ni®cant difference between (A9) and (A11) is the ap- cations. Thus interpolations of the horizontal velocity pearance of the ␥ P term in (A9). Because the value of components from the A-grid to the C-grid locations are

␥ is a constant as long as ⌬t, ␪, and ␣1 are ®xed, the required. extra computations required in updating ␩ are negligi- As discussed in appendix A, a trial velocity uà nϩ1 is ble. ®rst determined at the A-grid points, which includes all The updating of the temperature (T) and salinity (S) effects except for a corrective step to ensure continuity.

Unauthenticated | Downloaded 09/29/21 02:09 PM UTC JUNE 2001 LU ET AL. 1089

The correction for continuity is done on the C-grid ve- locity components, and the required velocity adjust- ments are then interpolated to the A-grid. The ®rst step in this procedure is to interpolate the trial velocity uà nϩ1 from A-grid points to the C-grid cell surfaces. To do this, we use a straightforward, level by level, linear interpolation in the horizontal (the adjustment required for partial cells is discussed in appendix C). The depth- integrated velocity, or transport, at the C-grid locations is then calculated, and F, the term on the right side of (A9), is obtained. After solving (A9), the C-grid veloc- ities are adjusted to be consistent with the modi®cation of the surface pressure gradient. After this adjustment, the C-grid transport satis®es the depth-integrated con- tinuity equation. The vertical velocity (w) is determined at the z levels between the levels where T, S, u, and ␷ are de®ned by integrating the divergence equation ex- pressing incompressibility. This yields a nondivergent velocity ®eld which, in general, is subsequently used in a control volume treatment of the advective terms for heat, salt, and momentum. For the linearized problem considered here, only heat advection is considered. Once the velocity has been corrected to satisfy the continuity equation on the C-grid, the changes required FIG. C1. A partial cell Ai,k in the x±z plane with a regular cell Aiϩ1,k by this correction are interpolated back to the A-grid. to its right. The thickness of the partial cell is reduced by an amount Since these changes are associated entirely with a equal to the height of the shaded area. A solid circle denotes the change in the surface pressure gradient, the velocity center of each cell. The open circle denotes the position at which the * changes on the A-grid are taken to be depth-independent values of the A-grid variablesAiϩ1,k are calculated by vertical inter- polation. The variables at A andA* are either averaged or dif- and determined by interpolating the depth-integrated i,k iϩ1,k ferenced to provide estimates of velocity, ¯ux, or pressure gradient transports from the C-grid to the A-grid. This completes at the center of the intervening cell face (indicated by a ϫ). the updating of the velocity on both the A- and C-grids. One other interpolation is required by the A-grid for- mulation. Since the velocity is updated on the A-grid Figure C1 shows the con®guration of a partial cell, but the pressure gradient is most naturally determined centered at Ai,k, at the level k. Its thickness is reduced at the C-grid locations, the pressure gradient must be by an amount equal to the height of the shaded area in interpolated from the C-grid to the A-grid. To do this, order to more accurately represent the real topography. we use simple linear interpolation with the normal com- The cells at upper levels (the level k Ϫ 1 and above) ponent of the pressure gradient set to zero at all solid are full cells with no side boundaries adjoining partial boundaries. The justi®cation for the latter choice is sim- cells. Hence, all ¯ux and conservation formulae are un- ple. It is equivalent to determining the changes in the changed for cells at the level k Ϫ 1 and above. The cell C-grid velocity associated with the pressure gradient and at the right of the partial cell, centered at Ai ϩ 1,k, has then linearly interpolating from the C-grid to the A- full thickness but has an adjoining side in common with grid. the partial cell. Both Ai,k and Ai ϩ 1,k need special treat- ment. The following factors must be properly accounted APPENDIX C for in order to correctly implement partial cells. Partial Cells: Improving the Representation 1) The ¯uxes across cell boundaries affected by partial of Topography cells must account for the reduced surface area of the face. This is a simple matter of multiplying by A detailed discussion of the implementation of partial the appropriate cell height when calculating the cells in the B-grid GFDL MOM2 model is given by boundary ¯ux. Pacanowski and Gnanadesikan (1998). Although we use 2) The horizontal velocity component normal to a cell an A-grid version of DieCAST, making some algebraic face with reduced height must be interpolated to the and numerical details different, the basic considerations midlevel of the cell face (e.g., the position indicated of the two implementations are the same. Here we brief- by ϫ in Fig. C1) in order to maintain second order ly review the changes in the model code that are required accuracy in the advective ¯uxes. For our A-grid when adding partial cells and refer readers to Paca- model, the vertical interpolations are done at the A- nowski and Gnanadesikan (1998) for further details. grid locations (to the position of the open circle in

Unauthenticated | Downloaded 09/29/21 02:09 PM UTC 1090 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 18

Fig. C1), and then horizontal interpolation is used cell, we had a cell that was slightly thicker than the to determine the component of velocity normal to normal cell depth, the stability problem would not the intervening cell face. occur. Unfortunately, this approach would introduce 3) The calculation of vertical velocity from the conti- substantial additional complications, and hence we nuity equation must be based on the same interpo- take a much simpler approach. We accomplish es- lated horizontal velocity components as used in the sentially the same effect by using an implicit scheme calculation of advective ¯uxes. Note that the discre- for the vertical momentum mixing and increasing tized continuity equation ensures the balance of vol- the vertical viscosity between partial cells and the ume ¯uxes across all surfaces of a control volume. overlying cell when the partial cell is so thin that The volume ¯ux at the intervening cell face is the stability considerations might be a concern. In the interpolated velocity at ϫ multiplied by the area of present study, we have increased the vertical mixing that surface. at the top of partial cells when their thickness is less 4) Horizontal diffusive ¯uxes must be based on differ- than 10 m. When this occurs, the vertical viscosity ences between quantities that have been interpolated is increased to (⌬z)2/⌬t, where ⌬z is the distance to the appropriate middepth level of the face. Once between the center of the partial cell and the center again, vertical interpolations are done on the A-grid, of the overlying cell. No increase in the vertical dif- and then ®rst differences are used to estimate the fusivity for heat and salinity was needed or used in diffusive ¯uxes. If the quantity being interpolated is the present study. a simple linear function of depth, its horizontal dif- fusive ¯ux is zero to within roundoff error in our model. REFERENCES 5) An estimate of the average pressure gradients over Adcroft, A., C. Hill, and J. Marshall, 1997: Representation of to- each cell volume is required. In the A-grid DieCAST pography by shaved cells in a height coordinate ocean model. model, the components of the pressure gradient nor- Mon. Wea. Rev., 125, 2293±2315. mal to the cell faces are ®rst determined at the mid- Baines, P. G., 1982: On internal tide generation models. Deep-Sea depth level of each cell face, and then these values Res., 29, 307±338. Craig, P. D., 1987: Solutions for internal tidal generation over coastal are used to estimate the average over the cell volume topography. J. Mar. Res., 45, 83±105. by taking appropriate weighted averages. To deter- Cummins, P. F., and L.-Y. Oey, 1997: Simulation of barotropic and mine the pressure gradient normal to a cell face, we baroclinic tides off northern British Columbia. J. Phys. Ocean- ®rst estimate the pressure at the appropriate level on ogr., 27, 762±781. each side of the face and then take the difference. Davidson, F. J. M., R. J. Greatbatch, and A. D. Goulding, 1998: On the net cyclonic circulation in large strati®ed lakes. J. Phys. To determine the pressure at the midlevel of the cell Oceanogr., 28, 527±534. face, we ®rst interpolate temperature and salinity to Dietrich, D. E., 1992: The Sadie ocean modeling system programmers this level and use these to determine the in situ den- guide and users' manual. Sandia National Laboratories sity. The density is then assumed to vary linearly SAND92-7386, 64 pp. [Available from Sandia National Labo- between this level and the overlying level, and the ratories, P.O. Box 5800, Albuquerque, NM 87185.] ÐÐ, and C. Lin, 1994: Numerical studies of shedding in the hydrostatic equation is used to determine the change Gulf of Mexico. J. Geophys. Res., 99, 7599±7615. in pressure between these two levels. By using this ÐÐ, M. G. Marietta, and P. J. Roache, 1987: An ocean modeling approach, the horizontal pressure gradients are zero system with turbulent boundary layers and topography: Numer- to within roundoff error, when T and S are linear ical description. Int. J. Numer. Methods Fluids, 7, 833±855. functions of depth, even when using a nonlinear Dukowicz, J. K., and R. D. Smith, 1994: Implicit free-surface method for the Bryan±Cox±Semtner ocean model. J. Geophys. Res., 99, equation of state. Once the components of the pres- 7991±8014. sure gradient normal to the cell faces are determined, Haney, R. L., 1991: On the pressure gradient force over steep to- a depth-weighted average [i.e., Ai ϭ (hiϪ1/2AiϪ1/2 ϩ pography in sigma coordinate ocean models. J. Phys. Oceanogr., 21, 610±619. hiϩ1/2Aiϩ1/2)/(2hi)] is used to estimate the gradient at the center of the cell. Note that the depth weighting Holloway, P. E., 1996: A numerical model of internal tides with application to the Australia North West Shelf. J. Phys. Ocean- makes the pressure gradient at a face have dimin- ogr., 26, 21±37. ishing in¯uence as the area of the face decreases. In ÐÐ, and M. A. Merri®eld, 1999: Internal tide generation by sea- the limit as the vertical extent of a cell face ap- mounts, ridges, and islands. J. Geophys. Res., 104, 25 937±25 proaches either zero or the full cell thickness, we 951. recover the standard formulae used to estimate the Killworth, P. D., D. Stainforth, D. J. Webb, and S. M. Paterson, 1991: The development of a free-surface of Bryan±Cox±Semtner ocean cell-centered pressure gradient on an A-grid. model. J. Phys. Oceanogr., 21, 1333±1348. 6) Finally, it is possible for partial cells to be very thin, Mellor, G. L., T. Ezer, and L.-Y. Oey, 1994: On the pressure gradient which can result in local numerical instabilities. One conundrum of sigma coordinate ocean models. J. Atmos. Oceanic way to avoid this possibility would be to limit the Technol., 11, 1126±1134. minimum thickness of a partial cell such that the MuÈller, P., and X. Liu, 2000: Scattering of internal waves at ®nite topography in two dimensions. Part I: Theory and case studies. stability criterion is not violated. However, we prefer J. Phys. Oceanogr., 30, 532±563. to avoid the resulting modi®cation of the bottom Munk, W. H., 1997: Once again: Once again-tidal friction. Progress topography. If, instead of having a very thin partial in Oceanography, Vol. 40, Pergamon, 7±35.

Unauthenticated | Downloaded 09/29/21 02:09 PM UTC JUNE 2001 LU ET AL. 1091

Pacanowski, R. C., and A. Gnanadesikan, 1998: Transient response ÐÐ, D. G. Wright, R. J. Greatbatch, and D. E. Dietrich, 1998: in a z-level ocean model that resolves topography with partial CANDIE: A new version of the DieCAST ocean circulation cells. Mon. Wea. Rev., 126, 3248±3270. model. J. Atmos. Oceanic Technol., 15, 1414±1432. Ray, R. D., and G. T. Mitchum, 1996: Surface manifestation of in- Smith, R. D., J. K. Dukowicz, and R. C. Malone, 1992: Parallel ocean ternal tides generated near Hawaii. Geophys. Res. Lett., 23, circulation modeling. Physics, 60D, 38±61. 2101±2104. Wright, D. G., and K. R. Thompson, 1983: Time-averaged forms of Rhines, P. B., 1975: Edge-, bottom- and Rossby Waves in a rotating the nonlinear stress law. J. Phys. Oceanogr., 13, 341±345. strati®ed ¯uid. Geophys. Fluid Dyn., 1, 273±302. Xing, J., and A. M. Davies, 1996: Processes in¯uencing the internal Sandstrom, H., 1976: On topographic generation and coupling of tide, its higher harmonics, and tidally induced mixing on the internal waves. Geophys. Fluid Dyn., 7, 231±270. Malin±Hebrides Shelf. Progress in Oceanography, Vol. 38, Per- ÐÐ, and N. S. Oakey, 1995: Dissipation in internal tides and solitary gamon, 155±204. waves. J. Phys. Oceanogr., 25, 604±614. ÐÐ, and ÐÐ, 1998: Formulation of a three-dimensional shelf edge Sheng, J., 2001: Dynamics of a buoyancy-driven coastal jet: The model and its application to internal tide generation. Contin. Gaspe Current. J. Phys. Oceanogr., in press. Shelf Res., 18, 405±440.

Unauthenticated | Downloaded 09/29/21 02:09 PM UTC