Finite-amplitude Topography Research Articles

The Impact of Representations of Realistic Topography on Parameterized Oceanic Lee Wave Energy Flux

AbstractOceanic lee waves are generated when quasi‐steady flows interact with rough topography at the bottom of the ocean, providing an important sink of energy and momentum from the mean flow and a source of turbulent kinetic energy. Linear theory with a spectral representation of topography is typically used to inform parameterizations of lee wave generation. Here, we use a realistic wave resolving simulation of the Drake Passage, a hot‐spot of lee wave generation, to investigate the utility of such parameterizations for areas of complex large scale topography. The flow is often blocked and split by large amplitude topographic features, creating an “effective topography,” and calling into question the spectral representation of small scale topography for lee wave generation. By comparing the resolved modeled wavefield to parameterizations employing various representations of topography, we show that spectral methods may not be appropriate in areas of rough topography. We develop a simple topographic representation consisting of an ensemble of topographic peaks, which allows physical treatment of flow blocking at finite amplitude topography. This method effectively predicts bottom vertical velocities and lee wave energy flux, whereas spectral methods overestimate the energy flux by approximately 4 times. Our results also imply that the nature of lee waves in such regions can be misrepresented by a spectral approach to topographic representation. This leads to both an overestimate of wave energy flux and an underestimate of wave nonlinearity, with implications for the mechanisms by which lee waves break and mix in the abyssal ocean.

Forward modelling of global gravity fields with 3D density structures and an application to the high-resolution (~ 2 km) gravity fields of the Moon

Rigorous modelling of the spherical gravitational potential spectra from the volumetric density and geometry of an attracting body is discussed. Firstly, we derive mathematical formulas for the spatial analysis of spherical harmonic coefficients. Secondly, we present a numerically efficient algorithm for rigorous forward modelling. We consider the finite-amplitude topographic modelling methods as special cases, with additional postulates on the volumetric density and geometry. Thirdly, we implement our algorithm in the form of computer programs and test their correctness with respect to the finite-amplitude topography routines. For this purpose, synthetic and realistic numerical experiments, applied to the gravitational field and geometry of the Moon, are performed. We also investigate the optimal choice of input parameters for the finite-amplitude modelling methods. Fourth, we exploit the rigorous forward modelling for the determination of the spherical gravitational potential spectra inferred by lunar crustal models with uniform, laterally variable, radially variable, and spatially (3D) variable bulk density. Also, we analyse these four different crustal models in terms of their spectral characteristics and band-limited radial gravitation. We demonstrate applicability of the rigorous forward modelling using currently available computational resources up to degree and order 2519 of the spherical harmonic expansion, which corresponds to a resolution of ~ 2.2 km on the surface of the Moon. Computer codes, a user manual and scripts developed for the purposes of this study are publicly available to potential users.

The Impact of Finite-Amplitude Bottom Topography on Internal Wave Generation in the Southern Ocean

Abstract Direct observations in the Southern Ocean report enhanced internal wave activity and turbulence in a kilometer-thick layer above rough bottom topography collocated with the deep-reaching fronts of the Antarctic Circumpolar Current. Linear theory, corrected for finite-amplitude topography based on idealized, two-dimensional numerical simulations, has been recently used to estimate the global distribution of internal wave generation by oceanic currents and eddies. The global estimate shows that the topographic wave generation is a significant sink of energy for geostrophic flows and a source of energy for turbulent mixing in the deep ocean. However, comparison with recent observations from the Diapycnal and Isopycnal Mixing Experiment in the Southern Ocean shows that the linear theory predictions and idealized two-dimensional simulations grossly overestimate the observed levels of turbulent energy dissipation. This study presents two- and three-dimensional, realistic topography simulations of internal lee-wave generation from a steady flow interacting with topography with parameters typical of Drake Passage. The results demonstrate that internal wave generation at three-dimensional, finite bottom topography is reduced compared to the two-dimensional case. The reduction is primarily associated with finite-amplitude bottom topography effects that suppress vertical motions and thus reduce the amplitude of the internal waves radiated from topography. The implication of these results for the global lee-wave generation is discussed.

The effects of the spatial distribution of bottom topography and bottom drag on seiche-induced wave train formation

We consider the nonlinear, non-hydrostatic dynamics of seiches in small to medium-sized lakes. Using numerical simulations of shallow water equations modified to include weakly non-hydrostatic effects, we illustrate how spatially varying bottom drag and finite amplitude topography lead to the bending of wave trains that develop from the initial standing wave. For the case of variable topography, we discuss how the seiche and the wave trains that develop can resuspend material (e.g. nutrients) from the bottom of the lake and redistribute it in space. The numerical methods employed are spectrally accurate in space and second-order in time, yielding excellent accuracy and little numerical dissipation. We find that while the resuspension itself is largely due to the long standing waves at early times, the redistribution of nutrient distribution that is seen at later times is profoundly influenced by the development of the wave trains; a fundamentally non-hydrostatic effect.

Global rate and spectral characteristics of internal gravity wave generation by geostrophic flow over topography

The rate of generation of internal gravity waves in the lee of small length scale topography by geostrophic flow in the World Ocean was estimated using linear theory with corrections for finite amplitude topography. Several global data sets were combined for the calculation including an ocean circulation model for the near-bottom geostrophic flow statistics, over 500 abyssal current meter records, historical climatological data for the buoyancy frequency, and two independent estimates of the small scale topographic statistical properties. The first topography estimate was based on an empirically-derived relationship between paleo-spreading rates and abyssal hill roughness, with corrections for sedimentation. The second estimate was based on small-scale (<100 km) roughness of satellite altimetry-derived gravity field, using upward continuation relationships to derive estimates of abyssal hill roughness at the seafloor at scales less than approximately 20 km. The lee wave generation rate was found to be between 0.34 to 0.49 TW. The Southern Hemisphere produced 92% of the lee wave energy, with the Southern Ocean dominating. Strength of the bottom flow was the most important factor in producing the global pattern of generation rate, except in the Indian Ocean where extremely rough topography produced strong lee wave generation despite only moderate bottom flows. The results imply about one half of the mechanical power input to the ocean general circulation from the extra-equatorial wind stress of the World Ocean results from abyssal lee wave generation. Topographic length scales between 176 m and 2.5 km (horizontal wavelengths between 1 and 16 km) accounted for 90% of the globally integrated generation.

Decay of an internal tide due to random topography in the ocean

We present a theoretical and numerical study of the decay of an internal wave caused by scattering at undulating sea-floor topography, with an eye towards building a simple model in which the decay of internal tides in the ocean can be estimated. As is well known, the interactions of internal waves with irregular boundary shapes lead to a mathematically ill-posed problem, so care needs to be taken to extract meaningful information from this problem. Here, we restrict the problem to two spatial dimensions and build a numerical tool that combines a real-space computation based on the characteristics of the underlying partial differential equation with a spectral computation that satisfies the relevant radiation conditions. Our tool works for finite-amplitude topography but is restricted to subcritical topography slopes. Detailed results are presented for the decay of the gravest vertical internal wave mode as it encounters finite stretches of either sinusoidal topography or random topography defined as a Gaussian random process with a simple power spectrum. A number of scaling laws are identified and a simple expression for the decay rate in terms of the power spectrum is given. Finally, the resulting formulae are applied to an idealized model of sea-floor topography in the ocean, which seems to indicate that this scattering process can provide a rapid decay mechanism for internal tides. However, the present results are restricted to linear fluid dynamics in two spatial dimensions and to uniform stratification, which restricts their direct application to the real ocean.

Decay of an internal tide due to random topography in the ocean

We present a theoretical and numerical study of the decay of an internal wave caused by scattering at undulating sea-floor topography, with an eye towards building a simple model in which the decay of internal tides in the ocean can be estimated. As is well known, the interactions of internal waves with irregular boundary shapes lead to a mathematically ill-posed problem, so care needs to be taken to extract meaningful information from this problem. Here, we restrict the problem to two spatial dimensions and build a numerical tool that combines a real-space computation based on the characteristics of the underlying partial differential equation with a spectral computation that satisfies the relevant radiation conditions. Our tool works for finite-amplitude topography but is restricted to subcritical topography slopes. Detailed results are presented for the decay of the gravest vertical internal wave mode as it encounters finite stretches of either sinusoidal topography or random topography defined as a Gaussian random process with a simple power spectrum. A number of scaling laws are identified and a simple expression for the decay rate in terms of the power spectrum is given. Finally, the resulting formulae are applied to an idealized model of sea-floor topography in the ocean, which seems to indicate that this scattering process can provide a rapid decay mechanism for internal tides. However, the present results are restricted to linear fluid dynamics in two spatial dimensions and to uniform stratification, which restricts their direct application to the real ocean.

A laboratory study of low-mode internal tide scattering by finite-amplitude topography

We present the first laboratory experimental results concerning the scattering of a low-mode internal tide by finite-amplitude Gaussian topography. Experiments performed at the Coriolis Platform in Grenoble used a recently conceived internal wave generator as a means of producing a high-quality mode-1 wave field. The evolution of the wave field in the absence and presence of a Gaussian was studied by performing spatiotemporal modal decompositions of velocity field data obtained using particle image velocimetry. The results support the belief that finite-amplitude topography produces significant reflection of the internal tide and transfer of energy from low to high modes.

Constraints on the current rate of deformation and surface uplift of the Australian continent from a new seismic database and low-T thermochronological data

Estimates of the current rate of deformation and surface uplift for the Australian continent are derived by integration of a new seismic database and show that parts of the continent are currently experiencing deformation at a rate of 1– 5 × 10−16/s and uplifting at a rate of 10–50 m/Ma. In the east, these regions coincide with the regions of maximum topography, suggesting that, if this uplift rate is long-term, up to 50% of the present-day topographic relief in the southeastern Highlands and Flinders Ranges has formed in the last 10 Ma, i.e. the time we estimate for the onset of the present-day stress field experienced by the Indo-Australian Plate. These estimates are supported by fission-track data from the Snowy Mountains, which indicate that a non-negligible proportion of the present-day relief is the remnant of a much older topography formed during the various accretion or breakup events along the eastern margin of the continent in Late Paleozoic to Early Mesozoic time and that younger relief growth (i.e. younger than 100 Ma) must be limited to less than a kilometer in amplitude. By contrast, in the western part of the continent no such correlation exists between present-day topography and uplift predicted by integrating seismic strain rate over 10 Ma. This suggests that the apparently high level of seismic activity observed in the southwestern part of the Yilgarn Craton and along Proterozoic mobile belts, such as the Albany–Fraser Province of southeastern Western Australia and the Fitzroy Trough of northern Western Australia, is transient or that, contrary to what is happening in the east, erosional processes are able to remove surface relief created at the relatively slow rate of 10 m/Ma, potentially because there existed no finite amplitude topography prior to the onset of the present-day compressional stress field.

On nonlinear baroclinic trapped waves over abrupt topography

Nonlinear properties of baroclinic trapped topographic waves are examined analytically and numerically. Confined topography of finite amplitude that is intersected by an isopycnal surface is considered. A weakly nonlinear theory originally developed by Dewar and Leonov [2004. Variability on steep, confined topography. Deep-Sea Research II 51, 2973–2993] is extended here to include a term neglected in the original study. The analysis shows that the waves should break backward (pseudo-eastward) and that the steepening is stronger for off-shelf initial disturbances. A modified quasigeostrophic (QG) numerical model that allows finite-amplitude topography is developed. The model is first applied to initial disturbances of moderate steepness; the results are compared with the frontal theory predictions and reasonable agreement is demonstrated, which simultaneously justifies the extended theory and verifies the new numerical procedure. The numerical model is also applied to disturbances of stronger steepness, a case where the weakly nonlinear analysis is not applicable. Further, on-shelf/off-shelf asymmetry in steepening results in different structures produced by breaking of initial disturbances of different signs: an on-shelf initial disturbance results in a filamentation of the on-shelf fluid into the off-shelf regime, whereas an off-shelf disturbance produces anticyclones, which detach and transport off-shelf fluid onto the shelf.

Verification studies for a z-coordinate primitive-equation model: Tidal conversion at a mid-ocean ridge

We document the accuracy and convergence of solutions for a z-coordinate primitive-equation model of internal tide generation and propagation. The model, which is based on MOM3 numerics, is linearized around a state of rest to facilitate comparison with analytic estimates of baroclinic generation at finite-amplitude topography in a channel forced by barotropic tidal flow at its boundaries. Unlike the analytical model, the numerical model includes mixing of both buoyancy and momentum, and several definitions of “baroclinic conversion” are possible. These are clarified by writing out the energetics of the linearized equations in terms of barotropic kinetic energy, baroclinic kinetic energy, and available potential energy. The tidal conversion computed from the model, defined as the rate of conversion of barotropic kinetic energy into available potential energy, agrees well with analytical predictions. A comparison of different treatments of bottom topography (full-cells, partial-cells, and ghost-cells) indicates that the partial-cell treatment is the most accurate in this application. Convergence studies of flow over a smooth supercritical ridge show that the dissipation along tidal characteristics is, apparently, an integrable singularity. When the ocean bottom is not smooth, the accuracy and convergence of the model depend on the power spectrum of the topography. A numerical experiment suggests that the power spectrum of the resolved topography must roll off faster than k −2 to obtain convergent results from a linear numerical model of this type.

Equilibrium Statistical Mechanics of Barotropic Flow over Finite Topography

Inviscid equilibria of barotropic flows over finite-amplitude topography are determined by means of statistical mechanics, extending previous quasigeostrophic theory. Imposing constraints of energy and enstrophy conservation leads to a linear relation between equilibrium mean potential vorticity and mean transport streamfunction. This relation is tested numerically and is found to hold over a wide range of topographic amplitudes. Implications for improving parameterizations of entropy generation by eddies are discussed.

Three-dimensional aspects of nonlinear stratified flow over topography near the hydrostatic limit

Steady, finite-amplitude internal-wave disturbances, induced by nearly hydrostatic stratified flow over locally confined topography that is more elongated in the spanwise than the streamwise direction, are discussed. The nonlinear three-dimensional equations of motion are handled via a matched-asymptotics procedure: in an ‘inner’ region close to the topography, the flow is nonlinear but weakly three-dimensional, while far upstream and downstream the ‘outer’ flow is governed, to leading order, by the fully three-dimensional linear hydrostatic equations, subject to matching conditions from the inner flow. Based on this approach, non-resonant flow of general (stable) stratification over finite-amplitude topography in a channel of finite depth is analysed first. Three-dimensional effects are found to inhibit wave breaking in the nonlinear flow over the topography, and the downstream disturbance comprises multiple small-amplitude oblique wavetrains, forming supercritical wakes, akin to the supercritical free-surface wake induced by linear hydrostatic flow of a homogeneous fluid. Downstream wakes of a similar nature are also present when the flow is uniformly stratified and resonant (i.e. the flow speed is close to the long-wave speed of one of the modes in the channel), but, in this instance, they are induced by nonlinear interactions precipitated by three-dimensional effects in the inner flow and are significantly stronger than their linear counterparts. Finally, owing to this nonlinear-interaction mechanism, vertically unbounded uniformly stratified hydrostatic flow over finite-amplitude topography also features downstream wakes, in contrast to the corresponding linear disturbance that is entirely locally confined.

Mean flow generation along a sloping region in a rotating homogeneous fluid

The mean flow generated by an oscillatory (tidal) current normal to a long sloping region of constant cross section and finite height in a rotating homogeneous fluid is investigated analytically and by laboratory experiments. The geometry is characterized by constant fluid depths hs and hd in the shallow (shelf) and deep regions, respectively, with the sloping region between being smooth and having a characteristic width L. This physical system is characterized by the temporal Rossby number Rot = ω/f, the Ekman number E, the geometric parameters hs/hd and hs/L, and the normalized tidal excursion as/L (or, alternatively, the Rossby number); here ω is the forcing frequency of the oscillatory current, f is the Coriolis parameter, and as is the characteristic tidal excursion. The analysis assumes Rot ∼ O(1), E « 1, hs/L « O(1), (hd − hs)/hd ∼ O(1) (i.e., finite amplitude topography), and as/L ∼ O(1) (i.e., finite amplitude tidal excursions). The analysis is based upon a spectral method [Zimmerman, 1978] and predicts mean flows along the depth contours of the topography with the shallow region on the right, facing downstream, for vertically upward, or northern hemisphere, rotation. The theoretical approaches used by other investigators using different frictional parameterizations for infinitesimal tidal excursions are also presented. The laboratory experiments were conducted in circular rotating test cells of 1.8 and 13.0 m diameters in which continuous annular topographies of constant cross section were installed along the tank peripheries. The various theories were in qualitative support of the laboratory observations with respect to (1) mean flow direction, (2) location of most intense currents, and (3) dependence on the principal system parameters. Owing to the very small velocities associated with the mean flows generated and the inherent errors of the experiments, quantitative comparison could not be conclusive. Predictions from the various theories give reasonable estimates of such quantities as mean Lagrangian displacements and Eulerian surface transports.

Stability of stratified flow of large depth over finite-amplitude topography

The flow of a Boussinesq density-stratified fluid of large depth past the algebraic mountain (Witch of Agnesi) is studied in the hydrostatic limit using the asymptotic theory of Kantzios & Akylas (1993). The upstream conditions are those of constant velocity and Brunt-Vaisala frequency. On the further assumptions that the flow is steady and there is no permanent alteration of the upstream flow conditions (no upstream influence), Long's model (Long 1953) predicts a critical amplitude of the mountain (e = 0.85) above which local density inversions occur, leading to convective overturning. Linear stability analysis demonstrates that Long's steady flow is in fact unstable to infinitesimal modulations at topography amplitudes below this critical value, 0.65 ≤ e < 0.85. This instability grows at the expense of the mean flow and may be attributed to a discrete spectrum of modes that become trapped over the mountain in the streamwise direction. The transient problem is also solved numerically, mimicking impulsive startup conditions. In the absence of instability, Long's steady flow is reached. For topography amplitudes in the unstable range 0.65 ≤ e < 0.85, however, the flow fluctuates about Long's steady state over a long timescale ; there is no significant upstream influence and no evidence of transient wave breaking is found for e ≤ 0.75.

Generation of Nonlinear Internal Tides and Solitary Waves

Abstract A set of generalized Boussinesq equations is derived for internal waves in a two-layer system; the effects of earth's rotation are included by using the f-plane approximation; a forcing mechanism due to the interaction between a tidal flow and a small finite-amplitude topography is included as well. These equations thus enable the study of the generation of nonlinear internal tides and the effects of nonhydrostatic and Coriolis dispersion on the evolution of the internal tide. Numerical solutions of the equations are presented. The three main appearances of the internal tide are considered: a linear periodic wave, a cnoidal-like wave, and a train of solitary wave sequences. In the latter case the internal tide appears in a disintegrated form. Special attention is paid to the conditions in which such a disintegration occurs, in particular to the influence of earth's rotation (Coriolis dispersion). Critical parameters of dispersion are derived in terms of the strength of the forcing; these paramete...

Numerical Simulations of Stratified Rotating Flow over Finite Amplitude Topography

Abstract A primitive equation numerical ocean model is used to investigate the response of an basic flow, initially horizontally and vertically uniform, to the presence of an isolated seamount at the center of a 4000-m-deep periodic channel. The model is configured with 5-km horizontal resolution and 15 constant-density layer in the vertical. The Coriolis parameter, fluid stratification, coefficient of lateral friction, and length scale or the topography are held constant while the men downstream flow speed and height of the obstacle vary up to an extreme case of a 20 cm s−1 current impinging on a 3500 seamount. The results include the familiar anticyclonic vortex straddling the topograhy while a cyconic vortex propagates downgstream. Both advective and vorticity interaction effects are noted in the experiment with the extent of perturbations to the density field being influenced by variations in the mean flow speed and obstacle height. Possible modifications to the model are discussed in relation to obse...

Quasi-geostrophic stratified flow over isolated finite amplitude topography

The quasi-geostrophic response of a stratified stream incident upon isolated finite amplitude topography on a f-plane is examined in the limit of a Boussinesq, incompressible, inviscid fluid. Compact solutions are derived subject to the following stipulations: uniform upstream velocity and stratification, a circular obstacle and an entirely isentropic/isopycnic lower surface. It is shown that for a semi-infinite flow domain the criterion for Taylor cap formation (i.e., a region of closed streamlines) is F−1 > 332. However, for the isentropic lower boundary condition the solutions exist (i.e., have physical validity) only if R0F−1 < 0.5. (Here R0 and F refer to the Rossby and Froude numbers defined respectively in terms of the mountain half-width and height.) Also considered are the modifications both to the flow response and to the foregoing existence criterion that are induced by the introduction of an upstream profile comprising two layers of uniform but different stratification. In addition, the relationship of the derived solutions to the results obtained in previous studies is explored, and in particular an outline is given of the impact of adopting the ‘traditional’ simplified lower boundary condition.

Effects of Finite Height Topography on Nongeostrophic Baroclinic Instability: Implications to Theories of Lee Cyclogenesis

Baroclinic instability in the presence of steep finite amplitude topography is studied in the primitive equation model. The quasi-geostrophic theory of Alpine cyclogenesis of Speranza et al. is reanalyzed and discussed in this context. The present model is a generalization of the one used by Stone to include topographic effects, lateral shear of the basic wind, and/or lateral walls. We focus in particular on the differences between this formulation and the quasi-geostrophic one when the meridional scale of the topography is very small (of the order of 100 km). We find that only in the primitive equation model does a small-volume mountain, of height and width comparable with those of the Alps, introduce significant large-scale modifications to the baroclinic modes. The most unstable mode attains its maximum amplitude to the southern side of the mountain. We show that these results do not depend upon the specification of the lateral boundary conditions provided the basic state baroclinicity is meridionally confined.

A Theory of Deep Cyclogenesis in the Lee of the Alps. Part II: Effects of Finite Topographic Slope and Height

The theory of cyclogenesis in the lee of the Alps presented by Speranza et al. in Part I is reexamined here (Part II) in the context of models dealing with finite amplitude topography. This generalization leads to the specification of the limits of validity of the approximations made in Part I and, at the same time, allows us to calculate the stability properties and normal mode structure of the model atmosphere with realistic topography. The effects of finite slope and finite geometrical height of the mountain are considered separately and their relative importance is evaluated. In the case of a three-dimensional obstacle with an idealized shape simulating the Alps, the modifications induced by the orography on the free, baroclinically unstable modes show essentially the same features observed in numerical experiments, including the orientation of the dipolar structure in the pressure perturbation across the mountain.