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The interaction of a barotropic flow with topography generates baroclinic motion that exerts a stress on the barotropic flow. Here, explicit solutions are calculated for the spatial-mean flow (i.e., the barotropic tide) resulting from a spatially uniform but time-varying body force (i.e., astronomical forcing) acting over rough topography. This approach of prescribing the force contrasts with that of previous authors who have prescribed the barotropic flow. It is found that the topographic stress, and thus the impact on the spatial-mean flow, depend on the nature of the baroclinic motion that is generated. Two types of stress are identified: (i) a “wave drag” force associated with propagating wave motion, which extracts energy from the spatial-mean flow, and (ii) a topographic “spring” force associated with standing motion at the seafloor, including bottom-trapped internal tides and propagating low-mode internal tides, which significantly damps the time-mean kinetic energy of the spatial-mean flow but extracts no energy in the time-mean. The topographic spring force is shown to be analogous to the force exerted by a mechanical spring in a forced-dissipative harmonic oscillator. Expressions for the topographic stresses appropriate for implementation as baroclinic drag parameterizations in global models are presented.

We consider the power-law spectra of internal gravity waves in a rotating and stratified ocean. Field measurements have shown considerable variability of spectral slopes compared to the high-wavenumber, high-frequency portion of the Garrett–Munk (GM) spectrum. Theoretical explanations have been developed through wave turbulence theory (WTT), where different power-law solutions of the kinetic equation can be found depending on the mechanisms underlying the nonlinear interactions. Mathematically, these are reflected by the convergence properties of the so-called collision integral (CL) at low- and high-frequency limits. In this work, we study the mechanisms in the formation of the power-law spectra of internal gravity waves, utilizing numerical data from the high-resolution modeling of internal waves (HRMIW) in a region northwest of Hawaii. The model captures the power-law spectra in broad ranges of space and time scales, with scalings ω−2.05±0.2 in frequency and m−2.58±0.4 in vertical wavenumber. The latter clearly deviates from the GM76 spectrum but is closer to a family of induced-diffusion-dominated solutions predicted by WTT. Our analysis of nonlinear interactions is performed directly on these model outputs, which is fundamentally different from previous work assuming a GM76 spectrum. By applying a bicoherence analysis and evaluations of modal energy transfer, we show that the CL is dominated by nonlocal interactions between modes in the power-law range and low-frequency inertial motions. We further identify induced diffusion and the near-resonances at its spectral vicinity as dominating the formation of power-law spectrum.

Internal waves generated at the seafloor propagate through the interior of the ocean, driving mixing where they break and dissipate. However, existing theories only describe these waves in two limiting cases. In one limit, the presence of an upper boundary permits bottom-generated waves to reflect from the ocean surface back to the seafloor, and all the energy flux is at discrete wavenumbers corresponding to resonant modes. In the other limit, waves are strongly dissipated such that they do not interact with the upper boundary and the energy flux is continuous over wavenumber. Here, a novel linear theory is developed for internal tides and lee waves that spans the parameter space in between these two limits. The linear theory is compared with a set of numerical simulations of internal tide and lee wave generation at realistic abyssal hill topography. The linear theory is able to replicate the spatially-averaged kinetic energy and dissipation of even highly non-linear wave fields in the numerical simulations via an appropriate choice of the linear dissipation operator, which represents turbulent wave breaking processes.

The evidence for, baroclinic energetics of, and geographic distribution of parametric subharmonic instability (PSI) arising from both diurnal and semidiurnal tides in a global ocean general circulation model is investigated using 1/12.5° and 1/25° simulations that are forced by both atmospheric analysis fields and the astronomical tidal potential. The paper examines whether PSI occurs in the model, and whether it accounts for a significant fraction of the tidal baroclinic energy loss. Using energy transfer calculations and bispectral analyses, evidence is found for PSI around the critical latitudes of the tides. The intensity of both diurnal and semidiurnal PSI in the simulations is greatest in the upper ocean, consistent with previous results from idealized simulations, and quickly drops off about 5° from the critical latitudes. The sign of energy transfer depends on location; the transfer is positive (from the tides to subharmonic waves) in some locations and negative in others. The net globally integrated energy transfer is positive in all simulations and is 0.5%–10% of the amount of energy required to close the baroclinic energy budget in the model. The net amount of energy transfer is about an order of magnitude larger in the 1/25° semidiurnal simulation than the 1/12.5° one, implying the dependence of the rate of energy transfer on model resolution.

Motivated by the substantial sensitivity of eddies in two-layer quasigeostrophic (QG) turbulence models to the strength of bottom drag, this study explores the sensitivity of eddies in more realistic ocean general circulation model (OGCM) simulations to bottom drag strength. The OGCM results are interpreted using previous results from horizontally homogeneous, two-layer, flat-bottom, f-plane, doubly periodic QG turbulence simulations and new results from two-layer, β-plane QG turbulence simulations run in a basin geometry with both flat and rough bottoms. Baroclinicity in all of the simulations varies greatly with drag strength, with weak drag corresponding to more barotropic flow and strong drag corresponding to more baroclinic flow. The sensitivity of the baroclinicity in the QG basin simulations to bottom drag is considerably reduced, however, when rough topography is used in lieu of a flat bottom. Rough topography reduces the sensitivity of the eddy kinetic energy amplitude and horizontal length scales in the QG basin simulations to bottom drag to an even greater degree. The OGCM simulation behavior is qualitatively similar to that in the QG rough-bottom basin simulations, in that baroclinicity is more sensitive to bottom drag strength than are eddy amplitudes or horizontal length scales. Rough topography therefore appears to mediate the sensitivity of eddies in models to the strength of bottom drag. The sensitivity of eddies to parameterized topographic internal lee wave drag, which has recently been introduced into some OGCMs, is also briefly discussed. Wave drag acts like a strong bottom drag in that it increases the baroclinicity of the flow, without strongly affecting eddy horizontal length scales.

The effects of a parameterized linear internal wave drag on the semidiurnal barotropic and baroclinic energetics of a realistically forced, three-dimensional global ocean model are analyzed. Although the main purpose of the parameterization is to improve the surface tides, it also influences the internal tides. The relatively coarse resolution of the model of ~8 km only permits the generation and propagation of the first three vertical modes. Hence, this wave drag parameterization represents the energy conversion to and the subsequent breaking of the unresolved high modes. The total tidal energy input and the spatial distribution of the barotropic energy loss agree with the Ocean Topography Experiment (TOPEX)/Poseidon (TPXO) tidal inversion model. The wave drag overestimates the high-mode conversion at ocean ridges as measured against regional high-resolution models. The wave drag also damps the low-mode internal tides as they propagate away from their generation sites. Hence, it can be considered a scattering parameterization, causing more than 50% of the deep-water dissipation of the internal tides. In the near field, most of the baroclinic dissipation is attributed to viscous and numerical dissipation. The far-field decay of the simulated internal tides is in agreement with satellite altimetry and falls within the broad range of Argo-inferred dissipation rates. In the simulation, about 12% of the semidiurnal internal tide energy generated in deep water reaches the continental margins.

The resonance of diurnal tidal elevations is investigated with a forward ocean tide model run in a realistic near-global domain and a synthesis of free oscillations (normal modes) computed for realistic global ocean geometries and ocean physics. As a prelude to performing the forward ocean tide simulations, the topographic wave drag, which is now commonly employed in forward ocean tide models, is tuned specifically for diurnal tides. The synthesis of global free oscillations predicts reasonably well the forward ocean diurnal tide model sensitivity to changes in the frequency, zonal structure, and meridional structure of the astronomical diurnal tidal forcing. Three global free oscillations that are important for understanding diurnal tides as a superposition of forced-damped, resonant, free oscillations are identified. An admittance analysis of the frequency sweep experiments demonstrates that some coastal locations such as the Sea of Okhotsk are resonant to diurnal tidal forcing. As in earlier work done with semidiurnal tides, a series of simulations are performed in which regions possessing significant coastal diurnal tides are blocked out. The largest perturbations to the open-ocean diurnal tides take place in Blocked Sea of Okhotsk experiments. Lesser but still significant perturbations also arise from the blocking out of other regions of large diurnal tidal elevations or dissipation. Interpretation of the results is made more complex, however, by the fact that substantial perturbations also arise from blocking out regions where neither tidal elevations nor dissipation are large. The “blocking” experiments are relevant to understanding tides of the ice age, during which lower sea levels entail a reduced area of continental shelves.

Many investigators have idealized the oceanic mesoscale eddy field with numerical simulations of geostrophic turbulence forced by a horizontally homogeneous, baroclinically unstable mean flow. To date such studies have employed linear bottom Ekman friction (hereinafter, linear drag). This paper presents simulations of two-layer baroclinically unstable geostrophic turbulence damped by quadratic bottom drag, which is generally thought to be more realistic. The goals of the paper are 1) to describe the behavior of quadratically damped turbulence as drag strength changes, using previously reported behaviors of linearly damped turbulence as a point of comparison, and 2) to compare the eddy energies, baroclinicities, and horizontal scales in both quadratic and linear drag simulations with observations and to discuss the constraints these comparisons place on the form and strength of bottom drag in the ocean. In both quadratic and linear drag simulations, large barotropic eddies develop with weak damping, large equivalent barotropic eddies develop with strong damping, and the comparison in goal 2 above is closest when the nondimensional friction strength parameter is of order 1. Typical values of the quadratic drag coefficient (cd ∼ 0.0025) and of boundary layer depths (Hb ∼ 50 m) imply that the quadratic friction strength parameter cdLd/Hb, where Ld is the deformation radius, may indeed be of order 1 in the ocean. Model eddies are realistic over a wider range of friction strengths when drag is quadratic, because of a reduced sensitivity to friction strength in that case. The quadratic parameter is independent of the mean shear, in contrast to the linear parameter. Plots of eddy length scales, computed from satellite altimeter data, versus mean shear and versus rough estimates of the friction strength parameters suggest that both linear and quadratic bottom drag may be active in the ocean. Topographic wave drag contains terms that are linear in the bottom flow, thus providing some justification for the use of linear bottom drag in models.

Analysis of spectral kinetic energy fluxes in satellite altimetry data has demonstrated that an inverse cascade of kinetic energy is ubiquitous in the ocean. In geostrophic turbulence models, a fully developed inverse cascade results in barotropic eddies with large horizontal scales. However, midocean eddies contain substantial energy in the baroclinic mode and in compact horizontal scales (scales comparable to the deformation radius Ld). This paper examines the possibility that relatively strong bottom friction prevents the oceanic cascade from becoming fully developed. The importance of the vertical structure of friction is demonstrated by contrasting numerical simulations of two-layer quasigeostrophic turbulence forced by a baroclinically unstable mean flow and damped by bottom Ekman friction with turbulence damped by vertically symmetric Ekman friction (equal decay rates in the two layers). “Cascade inequalities” derived from the energy and enstrophy equations are used to interpret the numerical results. In the symmetric system, the inequality formally requires a cascade to large-scale barotropic flow, independent of the stratification. The inequality is less strict when friction is in the bottom layer only, especially when stratification is surface intensified. Accordingly, model runs with surface-intensified stratification and relatively strong bottom friction retain substantial small-scale baroclinic energy. Altimetric data show that the symmetric inequality is violated in the low- and midlatitude ocean, again suggesting the potential impact of the “bottomness” of friction on eddies. Inequalities developed for multilayer turbulence suggest that high baroclinic modes in the mean shear also enhance small-scale baroclinic eddy energy. The inequalities motivate a new interpretation of barotropization in weakly damped turbulence. In that limit the barotropic mode dominates the spatial average of kinetic energy density because large values of barotropic density are found throughout the model domain, consistent with the barotropic cascade to large horizontal scales, while baroclinic density is spatially localized.

The energy pathways in geostrophic turbulence are explored using a two-layer, flat-bottom, f-plane, quasigeostrophic model forced by an imposed, horizontally homogenous, baroclinically unstable mean flow and damped by bottom Ekman friction. A systematic presentation of the spectral energy fluxes, the mean flow forcing, and dissipation terms allows for a comprehensive understanding of the sources and sinks for baroclinic and barotropic energy as a function of length scale. The key new result is a robust inverse cascade of kinetic energy for both the baroclinic mode and the upper layer. This is consistent with recent observations of satellite altimeter data over the South Pacific Ocean. The well-known forward cascade of baroclinic potential and total energy was found to be very robust. Decomposing the spectral fluxes into contributions from different terms provided further insight. The inverse baroclinic kinetic energy cascade is driven mostly by an efficient interaction between the baroclinic velocity and the barotropic vorticity, the latter playing a crucial catalytic role. This cascade can be further enhanced by the baroclinic mode self-interaction, which is only present with nonuniform stratification (unequal layer depths). When model parameters are set such that modeled eddies compare favorably with observations, the inverse baroclinic kinetic energy cascade is actually much stronger than the well-known inverse cascade in the barotropic mode. The upper-layer kinetic energy cascade was found to dominate the lower-layer cascade over a wide range of parameters, suggesting that the surface cascade and time mean density stratification may be sufficient for estimating the depth-integrated cascade from ocean observations. This may find useful application in inferring the kinetic to gravitational potential energy conversion rate from satellite measurements.

The effects of mean flow direction on statistically steady, baroclinically unstable, beta-plane quasigeostrophic (QG) turbulence are examined in a two-layer numerical model. The turbulence is forced by an imposed, horizontally homogeneous, vertically sheared mean flow and dissipated by bottom Ekman friction. The model is meant to be an idealization of the midocean eddy field, which generally has kinetic energies larger than the mean and is isotropic. Energetic eddies can be generated even when planetary beta (b) dominates gradients of mean potential vorticity (PV; also, q), as long as the mean shear has a substantial meridional component. However, eddies are isotropic only when the angle between layer mean PV gradients exceeds approximately 908. This occurs when planetary and shear-induced gradients are comparable. Maps of PV indicate that these gradients may indeed be comparable over much of the midocean. Coherent jets form when the mean flow has a substantial meridional component and b is large. When b is nonzero, but small enough to permit isotropy, and the zonal component of the mean flow is not strongly eastward, lattices of like-signed coherent vortices develop. Likesigned vortex formation from initial and forcing conditions without a vorticity preference has not been observed before in QG systems. The vortex arrays are sensitive to the details of small-scale dissipation. Both cyclonic and anticyclonic fields arise in the simulations, depending on initial conditions, but they have different energies, consistent with broken symmetries in the governing equations.