1. Introduction
The generation of baroclinic (internal) tides via barotropic tidal flow over topography is a principal source of mechanical energy that ultimately drives ocean interior mixing (Munk and Wunsch 1998). As such, several different analytical and numerical models have been developed to investigate this process (e.g., Bell 1975; Baines 1982; Holloway and Merrifield 1999; Balmforth et al. 2002; Zarroug et al. 2010). These models have contributed significantly to the current understanding of internal tide generation, and practically they provide reasonable quantitative estimates of the rate of barotropic-to-baroclinic tidal conversion for many important ocean settings.
A popular analytical model of internal tide generation is a Green function–based approach that was initially developed to study internal tide scattering by two-dimensional, knife-edge topography in a uniformly stratified fluid (Robinson 1969). Llewellyn-Smith and Young (2002) employed Green functions to estimate the barotropic-to-baroclinic conversion rate at weak, three-dimensional topography in a nonuniformly stratified, finite-depth ocean in the limit of hydrostatic waves and subcritical topography, invoking the WKB approximation, which assumes a vertically varying, local vertical wavenumber for the internal wave field and requires variations in the stratification to occur on significantly larger scales than the local vertical wavelength. It was Llewellyn-Smith and Young (2003), however, who directly extended the approach of Robinson (1969) to analytically calculate conversion rates for barotropic tidal flow past a finite-height, two-dimensional, knife-edge barrier in a finite-depth ocean, again in the limit of the WKB approximation. Thereafter, Pétrélis et al. (2006) adapted the approach for finite-height, two-dimensional topographies of any symmetric shape in a uniformly stratified, finite-depth ocean. Balmforth and Peacock (2009) modified the approach for studies of periodic supercritical topography in a uniformly stratified ocean of infinite depth, and Echeverri and Peacock (2010) advanced the Green function approach to handle two-dimensional topographies of arbitrary shape in a finite-depth ocean of arbitrary stratification, albeit assuming the WKB approximation. Most recently, Falahat et al. (2014) performed global linear computations of tidal energy conversion by small-amplitude, subcritical topography without the WKB approximation, comparing how the accuracy depends on the topographic length scale.
In this paper, we further advance the Green function method to enable the study of internal tide generation by arbitrary, two-dimensional topography in arbitrary nonuniform stratifications without the restriction of the WKB approximation. The analytical method and corresponding WKB approximations are presented in section 2, followed by a brief description of our supporting numerical model in section 3. In section 4, we consider isolated Gaussian topographies in a nonuniform stratification and investigate the shortcomings of WKB-based predictions. We then apply the method to two realistic scenarios in section 5, considering transects from two very different ocean ridge systems with very different background stratifications, again comparing the results to those of WKB-based predictions. In all cases, our results are validated by comparisons with the results of the numerical model described in section 3. Finally, in section 6, we present our conclusions.
2. Theory
We define a streamfunction ψ(x, z, t) so that the associated velocity field is (u, w) = (−ψz, ψx), with x and z as the horizontal and vertical coordinates, respectively. The streamfunction of the total wave field is ψ(x, z, t) = ψb(x, z, t) + ψ′(x, z, t), where ψb = Re(−Uze−iωt) represents the background barotropic forcing of velocity amplitude U, and ψ′ is the internal wave field generated by the barotropic forcing. It is assumed that the response is at the same frequency ω as the barotropic tide, that is, ψb = Re[ϕb(x, z)e−iωt] and ψ′(x, z, t) = Re[ϕ′(x, z)e−iωt], where t is time, ϕb and ϕ′ are complex amplitudes, and Re denotes the real part.
3. Numerics
To validate the solutions obtained using the complete Green function approach and further confirm the errors associated with the WKB approach, we ran several corresponding numerical simulations based on the nonlinear, hydrostatic, terrain-following (σ coordinate) Princeton Ocean Model (Blumberg and Mellor 1987). Simulations were conducted in a narrow channel with the topography of interest in the center of the domain and periodic boundary conditions in the across-channel direction, which is equivalent to a 2D x–z domain (results are constant with y). The Princeton Ocean Model uses a Mellor–Yamada turbulence closure scheme (Mellor and Yamada 1982; Blumberg and Mellor 1987) in the vertical. Although it is possible to add a background diffusivity level, this was not applied here. In the horizontal, the model uses a Smagorinsky diffusivity (Smagorinsky 1963; Blumberg and Mellor 1987), which is set to zero in our simulations (and hence no explicit horizontal diffusivity is applied). However, as with all numerical models, there is numerical diffusion.
Modal decomposition was performed over the flat bottom sections upstream and downstream of the topographic feature. The boundary forcing at the ends of the channel was through the Flather condition (Flather 1976; Carter and Merrifield 2007) with the surface elevation chosen to correspond to an along-channel velocity of U = 2.5 × 10−4 m s−1. At the ends of the channel the relaxation boundary condition of Carter and Merrifield (2007) prevented the reflection of baroclinic energy in all modes. The low forcing velocities ensured the velocity remained sinusoidal and linear. The simulations had a horizontal resolution of 200 m and had 300 layers in the vertical. The resulting wave field was then linearly scaled to obtain the modal distribution of the energy fluxes at other forcing amplitudes. Each numerical simulation took about 9 h on a machine with 2×Intel Xeon E5-2670, 2.6 GHz (32 CPUs effectively), and 128 GB RAM. As a comparison, the computation of 400 modes for a given stratification, forcing, and Coriolis frequencies took around 12 h on a laptop with an Intel(R) Core(TM) i7, 2.8-GHz CPU, and 4 GB RAM; once the mode shapes are computed, the Green function code takes around 20 min (on the same laptop) for each of the topographic transects.
4. Gaussian topography
(a) The stratification profile N(z) considered in section 4. The streamfunction Φ(z) for (b) mode 1 and (c) mode 4 for the stratification in (a), as determined by solving (6) numerically (solid line) and by using the WKB approximation (dashed line). The streamfunctions are normalized so that 
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0145.1
The total energy flux E (kW m−1) associated with the baroclinic internal tide generated (in the right or left direction) by barotropic forcing of amplitude U = 0.02 m s−1 as a function of the criticality ε and the height ratio 
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0145.1
For comparison, the WKB-predicted energy fluxes are presented in Fig. 2b, with the overall qualitative form of the results being similar to those of the complete Green function approach, presented in Fig. 2a. The WKB-predicted energy fluxes are notably different, however, and the differences between the two sets of results are highlighted in Fig. 2c, which presents the absolute value of the difference in energy flux between the two methods, and Fig. 2d, which presents the relative difference. The standout features are that the WKB method predicts energy fluxes that are O(10) kW m−1 higher for tall topography, and for subcritical topography and topography with 
Insight into why the WKB approach is inaccurate for certain topographies is obtained by plotting the horizontal velocity fields for two different cases, namely, (i) the supercritical topography with ε = 2 and 
The instantaneous horizontal velocity field predicted by the (a) CGF and (b) WKB approaches for the supercritical topography (ε = 2 and 
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0145.1
As in Fig. 3, but for the subcritical topography (ε = 0.45 and 
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0145.1
To validate our findings, Fig. 5a presents comparisons between the two analytical approaches and the results of numerical simulations for E1 and E4 for 
The terms E1 and E4 as a function of ε for (a) 
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0145.1
Having established the validity of the complete Green function approach in Fig. 5, we proceed to identify regions in the ε–
An overall summary of the important fundamental scenario of Gaussian topography in an oceanographically reasonable, nonuniform stratification is that while the complete Green function approach consistently agrees with the results of the full numerical simulations, thus validating the method, WKB predictions for the overall energy flux are really only accurate for strongly supercritical topographies of modest-to-small height ratio. Significantly, the WKB approach is inaccurate for very subcritical topography with even small height ratios. An alternate but equivalent summary is that WKB predictions for the overall energy flux are accurate only if the topographic length scale is less than half of the horizontal scale of the first mode, and the height ratio is modest to small.
5. Realistic topography
The capability of the Green function method to determine internal tide generation for realistic oceanic scenarios was investigated by Echeverri and Peacock (2010) in the limit of the WKB approximation. In this section, we similarly use the complete Green function method to investigate internal tide generation along 2D transects across the Kaena Ridge near the Hawaiian Islands and the Wyville Thomson Ridge near the Faroe–Shetland Channel; these two scenarios were chosen because they present quite different configurations in terms of the background stratification profile and the topographic shapes. The predictions are compared with those of WKB theory and the results of our numerical model.
a. Kaena Ridge
The topography and stratification considered in this section are those used in section 4.1 in Echeverri and Peacock (2010). Specifically, as shown in Fig. 6a, we consider 10 evenly spaced 140-km-long transects across the Kaena Ridge. The stratification profile N(z), as shown in Fig. 6b, comprises a pycnocline, where N attains a maximum value of 0.0191 rad s−1 at 55 m below the sea surface in an ocean of total depth of around 4800 m (the specific value depending weakly on the transect number) and a deep-ocean stratification of 7.63 × 10−4 rad s−1. We consider semidiurnal forcing (ω = 1.4053 × 10−4 rad s−1), and the local Coriolis parameter is f = 5 × 10−5 rad s−1. Figure 6c shows a discernable difference between the mode-1 shapes computed with and without the WKB approximation.
(a) A contour plot of the depth of the topography (measured from the sea surface) at the Kaena Ridge. Investigations are performed for the 10 transects shown using double-headed arrows; the size of the arrows is directly proportional to the total, generated, internal tide energy flux calculated using the WKB approximation (Echeverri and Peacock 2010). (b) A typical stratification for the region shown in (a). (c) Mode-1 streamfunction Φ1(z), as obtained by solving (6) numerically (thick line) and the WKB approach (thin line), for the stratification presented in (b), with ω = 1.4053 × 10−4 rad s−1 (M2) and f = 5 × 10−5 rad s−1. The term Φ1 is normalized so that 
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0145.1
Figure 7a presents the complete Green function model prediction for a snapshot of the total horizontal velocity field for 1 of the 10 transects’ sections we consider at the Kaena Ridge. The corresponding wave field determined by numerical simulations is presented in Fig. 7b. There is a remarkable similarity between the results of theory and numerics, with wave beams emanating from the same locations on the topography in both the plots. In the numerical model, the wavebeams become diffuse as they radiate away from the topography because of the numerical dissipation, whereas the inviscid wavebeams in the analytic solution remain indefinitely sharp. A similar level of agreement was obtained for the other nine cross sections as well.
A snapshot of the total horizontal velocity field u at 1 of the 10 topographic sections of the Kaena Ridge predicted by (a) the complete Green function method and (b) numerical simulations. Both the wave fields correspond to a (linear) barotropic forcing amplitude of 2.5 × 10−4 m s−1 for the horizontal velocity; x is the distance measured along the transect under consideration. The northeast and southwest directions correspond to x > 0 and x < 0, respectively.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0145.1
As a more quantitative comparison between the results of the theory without and with the WKB approximation, and with the predictions of the numerical model, Figs. 8a–c present the variation of the internal tide energy flux in the first three modes in the southwest direction across the 10 topographic transects. We observe large differences between the theoretical model without and with the WKB approximation, and there is generally excellent quantitative agreement between the results of the complete Green function method and the numerical simulations. The mean, standard deviation, and maximum of the relative error (with respect to the complete Green function method) of the WKB predictions across all 10 transects for 
Energy fluxes in the southwestward-propagating internal tides generated by barotropic forcing of amplitude U = 0.02 m s−1 at the Kaena Ridge. Plotted in the various figures are the energy fluxes in the first three modes: (a) 
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0145.1
b. Wyville Thomson Ridge
The Wyville Thomson Ridge is an ~140-km-long topographic feature located at the southwestern end of the Faroe–Shetland Channel. A contour plot of a roughly 2° latitude by 2.5° longitude section of this ridge system, centered near 60°N and 7.5°W, is presented in Fig. 9a. Field observations reveal strong internal tides and subsequent mixing in the region, with corresponding numerical modeling attributing a part of the source of these internal tides to generation at the Wyville Thomson Ridge (Hall et al. 2011). Figure 9b presents a typical stratification, comprising two pycnoclines: a relatively strong one at 100-m depth and a relatively weak one at 600-m depth, beneath a nearly constant stratification of N = 4.59 × 10−4 rad s−1. For this stratification, and using ω = 1.4053 × 10−4 rad s−1 (M2) and f = 1.26 × 10−4 rad s−1 (corresponding to a latitude of 60°N), plots of the vertical structure of mode 1 (Φ1) based on (6) and the WKB approach are presented in Fig. 9c. The WKB mode 1, which contains small-scale noise similar to those in N(z), is notably different from Φ1. We choose to work with the noisy N(z) profile (instead of smoothening it to retain only the large-scale features) so as to demonstrate the robustness and accuracy of the complete Green function approach even in the presence of small-scale features in N(z). We note that stratification profiles with a double pycnocline structure are found in other parts of the ocean too, the Arctic Circle (Pinkel 2005) and the Bay of Bengal (Sengupta et al. 2006), for example.
(a) A contour plot of the topography at the Wyville Thomson Ridge (D is the depth of the topography measured from the sea surface). Investigations are performed for the 25 equispaced transects shown in black. (b) A typical stratification for the region shown in (a). (c) Mode-1 streamfunction Φ1(z), as obtained by solving (6) numerically (thick line) and the WKB approach (thin line), for the stratification presented in (b), with ω = 1.4053 × 10−4 rad s−1 (M2) and f = 1.26 × 10−4 rad s−1 (60°N). The term Φ1 is normalized so that 
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0145.1
We consider 25 different two-dimensional topographic cross sections, indicated in Fig. 9a, and the barotropic tidal forcing is assumed to be in the northeast/southwest direction with a magnitude of 0.104 m s−1, corresponding to an amplitude of 104 m2 s−1 for the volume flow per unit width of the barotropic tide in a 1000-m deep ocean (Sherwin 1991). Topography reaches within 300–500 m of the ocean surface in some locations, and outside this domain topography is assumed to smoothly descend to a constant ocean depth of 900–1200 m, the specific value depending on the cross section. The start and the end points for every cross section are chosen such that z > 0 throughout, as the analysis requires z = 0 on either side of the topography and z ≥ 0 everywhere in the topographic domain.
Figure 10a presents a snapshot of the theoretical prediction (based on the complete Green function approach) of the horizontal velocity field for one of the 25 topographic sections; the corresponding wave field determined by the numerical model is presented in Fig. 10b. The resulting wave field is highly complex, resulting from multiple reflections that occur where the stratification changes significantly, but there is nevertheless striking qualitative agreement between the two wave fields. As a more quantitative comparison, the variations of the northeastward-propagating energy flux for the first three modes are plotted for the 25 transects in Figs. 11a–c. There are very large differences between the complete Green function and WKB approaches, especially for modes 2 and 3, and again there is excellent quantitative agreement in 
As in Fig. 7, but at one of the 25 topographic transects of the Wyville Thomson Ridge.
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0145.1
Energy fluxes in the northeastward-propagating internal tides generated at the Wyville Thomson Ridge. Plotted in the various figures are the energy fluxes in the first three modes: (a) 
Citation: Journal of Physical Oceanography 46, 7; 10.1175/JPO-D-15-0145.1
6. Conclusions
The Green function method, which to date has been used to study internal tide generation by barotropic flow over topography in constant stratifications and nonuniform stratifications in the limit of the WKB approximation, has been advanced to account for fully nonuniform stratifications. To validate the method and to test the inaccuracies of WKB approximate solutions, tidal generation was first studied for idealized isolated Gaussian topography in a realistic ocean stratification. Consistently excellent agreement was obtained between the complete Green function solution and the results of numerical simulations. The WKB approach, however, was found to give substantial errors in the subcritical regime even for small height ratios, and as the height ratio becomes larger and the topography starts to impinge on the pycnocline, there are furthermore large errors for both subcritical and supercritical scenarios.
Studies were then performed for realistic ocean scenarios using cross-sectional profiles of the Kaena and Wyville Thomson Ridges. Again, in all cases, the solutions of the complete Green function approach and numerical simulations were in excellent agreement. The relative errors of the WKB approach for the Kaena Ridge, however, were on average around 25% for the energy fluxes in the first two modes, although with much larger errors for some specific cross sections. For the Wyville Thomson Ridge, which has a more complex double pycnocline stratification, the WKB approach was highly inaccurate; the mean total generated energy flux based on the WKB approach is 10.77 kW m−1, whereas the value based on the complete Green function approach is 5.37 kW m−1, thus resulting in a 100% error for the WKB approach. As with the Kaena Ridge scenario, the WKB errors in the energy fluxes in the individual modes are much larger.
We conclude that in order to reasonably estimate the energy flux in individual modes of the internal tide generated by barotropic tidal flow over ocean ridges it is necessary to use the complete Green function approach developed in this paper and not to resort to the WKB approximate solution. We have thoroughly investigated whether there are conditions under which one can reliably use the WKB approximation for idealized Gaussian topography and for realistic topography. Our conclusion is that WKB predictions can be quite sensitive to errors in the mode shapes, and given that the computational demands of the complete Green function method are significantly less than those of the numerical simulations (see section 3 for a quantitative comparison), the former should be used at all times, safe in the knowledge that this approach has now been thoroughly validated by direct comparison with numerical simulations. As a follow-up study, it would be intriguing to perform global estimates of internal tide generation along the lines of Falahat et al. (2014) using the complete Green function approach, albeit in the two-dimensional limit but with the capability to model supercritical topographies also.
Finally, we note that a variation of the approach that we present can be used to study topographic scattering of low-mode internal tides, a mechanism believed to be important to identify and quantify the processes by which internal waves dissipate in the ocean. The results of such a study appear in Mathur et al. (2014). With that study and the present study in hand, linear, two-dimensional, internal tide generation and scattering by deep-ocean topography are now quite well considered, although scenarios such as the impact of steady flow and nonlinearities (Lamb 2004) still need to be fully addressed. Moving forward, attention also needs to be directed toward developing reliable analytical methods to study internal tide generation and scattering by three-dimensional topography (e.g., Munroe and Lamb 2005; King et al. 2010).
Acknowledgments
MM thanks the Ministry of Earth Sciences, Government of India for financial support under the Monsoon Mission Grant MM/2014/IND-002. TP acknowledges funding from National Science Foundation Grants OCE 0645529 and OCE1129757. GSC acknowledges funding from the National Science Foundation Grant OCE 0825266. The comments from the reviewers, which have helped improve the manuscript, are also acknowledged.
REFERENCES
Baines, P. G., 1982: On internal tide generation models. Deep-Sea Res., 29A, 307–338, doi:10.1016/0198-0149(82)90098-X.
Balmforth, N. J., and T. Peacock, 2009: Tidal conversion by supercritical topography. J. Phys. Oceanogr., 39, 1965–1974, doi:10.1175/2009JPO4057.1.
Balmforth, N. J., G. R. Ierley, and W. R. Young, 2002: Tidal conversion by subcritical topography. J. Phys. Oceanogr., 32, 2900–2914, doi:10.1175/1520-0485(2002)032<2900:TCBST>2.0.CO;2.
Bell, T. H., 1975: Topographically generated internal waves in the open ocean. J. Geophys. Res., 80, 320–327, doi:10.1029/JC080i003p00320.
Blumberg, A. F., and G. L. Mellor, 1987: A description of a three-dimensional coastal ocean circulation model. Three-Dimensional Coastal Ocean Models, Coastal Estuarine Sci., Vol. 4, Amer. Geophys. Union, 1–16.
Carter, G. S., and M. A. Merrifield, 2007: Open boundary conditions for regional tidal simulations. Ocean Modell., 18, 194–209, doi:10.1016/j.ocemod.2007.04.003.
Echeverri, P., and T. Peacock, 2010: Internal tide generation by arbitrary two-dimensional topography. J. Fluid Mech., 659, 247–266, doi:10.1017/S0022112010002417.
Falahat, S., J. Nycander, F. Roquet, and M. Zarroug, 2014: Global calculation of tidal energy conversion into vertical normal modes. J. Phys. Oceanogr., 44, 3225–3244, doi:10.1175/JPO-D-14-0002.1.
Flather, R. A., 1976: A tidal model of the north-west European continental shelf. Mem. Soc. Roy. Sci. Liege, 6, 141–164.
Hall, R. A., J. M. Huthnance, and R. G. Williams, 2011: Internal tides, nonlinear internal wave trains, and mixing in the Faroe-Shetland Channel. J. Geophys. Res., 116, C03008, doi:10.1029/2010JC006213.
Holloway, P. E., and M. A. Merrifield, 1999: Internal tide generation by seamounts, ridges, and islands. J. Geophys. Res., 104, 25 937–25 951, doi:10.1029/1999JC900207.
King, B., H. P. Zhang, and H. L. Swinney, 2010: Tidal flow over three-dimensional topography generates out-of-forcing-plane harmonics. Geophys. Res. Lett., 37, L14606, doi:10.1029/2010GL043221.
Lamb, K. G., 2004: Nonlinear interaction among internal wave beams generated by tidal flow over supercritical topography. Geophys. Res. Lett., 31, L09313, doi:10.1029/2003GL019393.
Llewellyn-Smith, S. G., and W. R. Young, 2002: Conversion of the barotropic tide. J. Phys. Oceanogr., 32, 1554–1566, doi:10.1175/1520-0485(2002)032<1554:COTBT>2.0.CO;2.
Llewellyn-Smith, S. G., and W. R. Young, 2003: Tidal conversion at a very steep ridge. J. Fluid Mech., 495, 175–191, doi:10.1017/S0022112003006098.
Mathur, M., G. S. Carter, and T. Peacock, 2014: Topographic scattering of the low mode internal tide in the deep ocean. J. Geophys. Res. Oceans, 119, 2165–2182, doi:10.1002/2013JC009152.
Mellor, G. L., and T. Yamada, 1982: Development of a turbulence closure model for geophysical fluid problems. Rev. Geophys., 20, 851–875, doi:10.1029/RG020i004p00851.
Munk, W., and C. Wunsch, 1998: Abyssal recipes II: Energetics of tidal and wind mixing. Deep-Sea Res., 45, 1977–2010, doi:10.1016/S0967-0637(98)00070-3.
Munroe, J. R., and K. G. Lamb, 2005: Topographic amplitude dependence of internal wave generation by tidal forcing over idealized three-dimensional topography. J. Geophys. Res., 110, C02001, doi:10.1029/2004JC002537.
Pétrélis, F., S. G. Llewellyn-Smith, and W. R. Young, 2006: Tidal conversion at a submarine ridge. J. Phys. Oceanogr., 36, 1053–1071, doi:10.1175/JPO2879.1.
Pinkel, R., 2005: Near-inertial wave propagation in the western Arctic. J. Phys. Oceanogr., 35, 645–665, doi:10.1175/JPO2715.1.
Robinson, R. M., 1969: The effects of a vertical barrier on internal waves. Deep-Sea Res. Oceanogr. Abstr., 16, 421–429, doi:10.1016/0011-7471(69)90030-8.
Sengupta, D., G. N. B. Raj, and S. S. C. Shenoi, 2006: Surface freshwater from Bay of Bengal runoff and Indonesian Throughflow in the tropical Indian Ocean. Geophys. Res. Lett., 33, L22609, doi:10.1029/2006GL027573.
Sherwin, T. J., 1991: Evidence of a deep internal tide in the Faeroe–Shetland Channel. Tidal Hydrodynamics, B. B. Parker, Ed., 469–488.
Smagorinsky, J., 1963: General circulation experiments with the primitive equations. I. The basic experiment. Mon. Wea. Rev., 91, 99–164, doi:10.1175/1520-0493(1963)091<0099:GCEWTP>2.3.CO;2.
Zarroug, M., J. Nycander, and K. Döös, 2010: Energetics of tidally generated internal waves for nonuniform stratification. Tellus, 62A, 71–79, doi:10.1111/j.1600-0870.2009.00415.x.
