Journal articles on the topic 'Nonlinear barotropic flow over topography'
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Zavala Sansón, Luis. "Nonlinear and time-dependent equivalent-barotropic flows." Journal of Fluid Mechanics 871 (May 30, 2019): 925–51. http://dx.doi.org/10.1017/jfm.2019.354.
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Some oceanic and atmospheric flows may be modelled as equivalent-barotropic systems, in which the horizontal fluid velocity varies in magnitude at different vertical levels while keeping the same direction. The governing equations at a specific level are identical to those of a homogeneous flow over an equivalent depth, determined by a pre-defined vertical structure. The idea was proposed by Charney (J. Met., vol. 6 (6), 1949, pp. 371–385) for modelling a barotropic atmosphere. More recently, steady, linear formulations have been used to study oceanic flows. In this paper, the nonlinear, time-dependent model with variable topography is examined. To include nonlinear terms, we assume suitable approximations and evaluate the associated error in the dynamical vorticity equation. The model is solved numerically to investigate the equivalent-barotropic dynamics in comparison with a purely barotropic flow. We consider three problems in which the behaviour of homogeneous flows has been well established either experimentally, analytically or observationally in past studies. First, the nonlinear evolution of cyclonic vortices around a topographic seamount is examined. It is found that the vortex drift induced by the mountain is modified according to the vertical structure of the flow. When the vertical structure is abrupt, the model effectively isolates the surface flow from both inviscid and viscous topographic effects (due to the shape of the bottom and Ekman friction, respectively). Second, the wind-driven flow in a closed basin with variable topography is studied (for a flat bottom this is the well-known Stommel problem). For a zonally uniform, negative wind-stress curl in the homogeneous case, a large-scale, anticyclonic gyre is formed and displaced southward due to topographic effects at the western slope of the basin. The flow reaches a steady state due to the balance between topographic,$\unicode[STIX]{x1D6FD}$, wind-stress and bottom friction effects. However, in the equivalent-barotropic simulations with abrupt vertical structure, such an equilibrium cannot be reached because the forcing effects at the surface are enhanced, while bottom friction effects are reduced. As a result, the unsteady flow is decomposed as a set of planetary waves. A third problem consists of performing simulations of the wind-driven flow over realistic bottom topography in the Gulf of Mexico. The formation of the so-called Campeche gyre is explored. It is found that such circulation may be consistent with the equivalent-barotropic dynamics.
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Pfeffer, Richard L., Robin Kung, Wen Ding, and Guo-Qing Li. "Barotropic flow over bottom topography— experiments and nonlinear theory." Dynamics of Atmospheres and Oceans 19, no. 1-4 (October 1993): 101–14. http://dx.doi.org/10.1016/0377-0265(93)90033-4.
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Ibanez, Ruy, Joseph Kuehl, Kalyan Shrestha, and William Anderson. "Brief communication: A nonlinear self-similar solution to barotropic flow over varying topography." Nonlinear Processes in Geophysics 25, no. 1 (March 6, 2018): 201–5. http://dx.doi.org/10.5194/npg-25-201-2018.
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Abstract. Beginning from the shallow water equations (SWEs), a nonlinear self-similar analytic solution is derived for barotropic flow over varying topography. We study conditions relevant to the ocean slope where the flow is dominated by Earth's rotation and topography. The solution is found to extend the topographic β-plume solution of Kuehl (2014) in two ways. (1) The solution is valid for intensifying jets. (2) The influence of nonlinear advection is included. The SWEs are scaled to the case of a topographically controlled jet, and then solved by introducing a similarity variable, η = cxnxyny. The nonlinear solution, valid for topographies h = h0 − αxy3, takes the form of the Lambert W-function for pseudo velocity. The linear solution, valid for topographies h = h0 − αxy−γ, takes the form of the error function for transport. Kuehl's results considered the case −1 ≤ γ < 1 which admits expanding jets, while the new result considers the case γ < −1 which admits intensifying jets and a nonlinear case with γ = −3.
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Musgrave, R. C., R. Pinkel, J. A. MacKinnon, Matthew R. Mazloff, and W. R. Young. "Stratified tidal flow over a tall ridge above and below the turning latitude." Journal of Fluid Mechanics 793 (March 29, 2016): 933–57. http://dx.doi.org/10.1017/jfm.2016.150.
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The interaction of the barotropic tide with a tall, two-dimensional ridge is examined analytically and numerically at latitudes where the tide is subinertial, and contrasted to when the tide is superinertial. When the tide is subinertial, the energy density associated with the response grows with latitude as both the oscillatory along-ridge flow and near-ridge isopycnal displacement become large. Where $f\neq 0$, nonlinear processes lead to the formation of along-ridge jets, which become faster at high latitudes. Dissipation and mixing is larger, and peaks later in the tidal cycle when the tide is subinertial compared with when the tide is superinertial. Mixing occurs mainly on the flanks of the topography in both cases, though a superinertial tide may additionally generate mixing above topography arising from convective breaking of radiating waves.
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Le Corre, Mathieu, Jonathan Gula, and Anne-Marie Tréguier. "Barotropic vorticity balance of the North Atlantic subpolar gyre in an eddy-resolving model." Ocean Science 16, no. 2 (April 20, 2020): 451–68. http://dx.doi.org/10.5194/os-16-451-2020.
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Abstract. The circulation in the North Atlantic subpolar gyre is complex and strongly influenced by the topography. The gyre dynamics are traditionally understood as the result of a topographic Sverdrup balance, which corresponds to a first-order balance between the planetary vorticity advection, the bottom pressure torque, and the wind stress curl. However, these dynamics have been studied mostly with non-eddy-resolving models and a crude representation of the bottom topography. Here we revisit the barotropic vorticity balance of the North Atlantic subpolar gyre using a new eddy-resolving simulation (with a grid space of ≈2 km) with topography-following vertical coordinates to better represent the mesoscale turbulence and flow–topography interactions. Our findings highlight that, locally, there is a first-order balance between the bottom pressure torque and the nonlinear terms, albeit with a high degree of cancellation between them. However, balances integrated over different regions of the gyre – shelf, slope, and interior – still highlight the important role played by nonlinearities and bottom drag curls. In particular, the Sverdrup balance cannot describe the dynamics in the interior of the gyre. The main sources of cyclonic vorticity are nonlinear terms due to eddies generated along eastern boundary currents and time-mean nonlinear terms in the northwest corner. Our results suggest that a good representation of the mesoscale activity and a good positioning of mean currents are two important conditions for a better representation of the circulation in the North Atlantic subpolar gyre.
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Aguiar-González, Borja, and Theo Gerkema. "Limiting amplitudes of fully nonlinear interfacial tides and solitons." Nonlinear Processes in Geophysics 23, no. 4 (August 18, 2016): 285–305. http://dx.doi.org/10.5194/npg-23-285-2016.
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Abstract. A new two-fluid layer model consisting of forced rotation-modified Boussinesq equations is derived for studying tidally generated fully nonlinear, weakly nonhydrostatic dispersive interfacial waves. This set is a generalization of the Choi–Camassa equations, extended here with forcing terms and Coriolis effects. The forcing is represented by a horizontally oscillating sill, mimicking a barotropic tidal flow over topography. Solitons are generated by a disintegration of the interfacial tide. Because of strong nonlinearity, solitons may attain a limiting table-shaped form, in accordance with soliton theory. In addition, we use a quasi-linear version of the model (i.e. including barotropic advection but linear in the baroclinic fields) to investigate the role of the initial stages of the internal tide prior to its nonlinear disintegration. Numerical solutions reveal that the internal tide then reaches a limiting amplitude under increasing barotropic forcing. In the fully nonlinear regime, numerical experiments suggest that this limiting amplitude in the underlying internal tide extends to the nonlinear case in that internal solitons formed by a disintegration of the internal tide may not reach their table-shaped form with increased forcing, but appear limited well below that state.
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Nikurashin, Maxim, and Raffaele Ferrari. "Radiation and Dissipation of Internal Waves Generated by Geostrophic Motions Impinging on Small-Scale Topography: Theory." Journal of Physical Oceanography 40, no. 5 (May 1, 2010): 1055–74. http://dx.doi.org/10.1175/2009jpo4199.1.
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Abstract Observations and inverse models suggest that small-scale turbulent mixing is enhanced in the Southern Ocean in regions above rough topography. The enhancement extends O(1) km above the topography, suggesting that mixing is supported by the breaking of gravity waves radiated from the ocean bottom. In this study, it is shown that the observed mixing rates can be sustained by internal waves generated by geostrophic motions flowing over bottom topography. Weakly nonlinear theory is used to describe the internal wave generation and the feedback of the waves on the zonally averaged flow. Vigorous inertial oscillations are driven at the ocean bottom by waves generated at steep topography. The wave radiation and dissipation at equilibrium is therefore the result of both geostrophic flow and inertial oscillations differing substantially from the classical lee-wave problem. The theoretical predictions are tested versus two-dimensional high-resolution numerical simulations with parameters representative of Drake Passage. This work suggests that mixing in Drake Passage can be supported by geostrophic motions impinging on rough topography rather than by barotropic tidal motions, as is commonly assumed.
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Vlasenko, Vasiliy, and Nataliya Stashchuk. "Amplification and Suppression of Internal Waves by Tides over Variable Bottom Topography." Journal of Physical Oceanography 36, no. 10 (October 1, 2006): 1959–73. http://dx.doi.org/10.1175/jpo2958.1.
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Abstract The energy exchange between internal waves and barotropic currents over inclined bottom topography is studied theoretically and in the framework of the numerical model. The energy balance equation derived for a continuously stratified fluid predicts that energy can either be transferred toward or away from the internal wave depending on the direction of propagation of both the wave and current. Four scenarios of wave–flow interaction over the inclined bottom were identified. An internal wave extracts energy from the background tidal flow during its propagation upslope–upstream or downslope–downstream and its amplitude grows. The wave loses energy propagating downslope–upstream or upslope–downstream and reduces in amplitude. This mechanism of suppression or amplification of internal waves by a current over an inclined bottom is verified numerically. When applied to the area of the Knight Inlet sill, a high-resolution fully nonlinear, nonhydrostatic model reproduces the packets of internal waves generated by supercritical tidal flow over the sill. Careful inspection of the wave fields revealed the presence of an irregular wave structure within wave packets—namely, internal waves are not arranged by amplitude. This phenomenon, obtained numerically and observed in situ, is treated in terms of the mechanism of wave–flow interaction: the energy exchange between the tidal current and generated internal waves over the inclined bottom topography is the reason for the absence of traditional rank-ordered waves in the packet.
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Durski, S. M., R. M. Samelson, J. S. Allen, and G. D. Egbert. "Normal-Mode Instabilities of a Time-Dependent Coastal Upwelling Jet." Journal of Physical Oceanography 38, no. 9 (September 1, 2008): 2056–71. http://dx.doi.org/10.1175/2008jpo3803.1.
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Abstract The linear stability of a nearly time-periodic, nonlinear, coastal upwelling–downwelling circulation, over alongshore-uniform topography, driven by a time-periodic wind stress is investigated using numerical methods. The near-periodic alongshore-uniform basic flow is obtained by forcing a primitive equation numerical model of coastal ocean circulation with periodic wind stress. Disturbance growth on this near-periodic flow is explored in linear and nonlinear model simulations. Numerous growing normal modes are found in the linear analyses at alongshore scales between 4 and 24 km. These modes vary in cross-shore structure and timing of maximum disturbance growth rate. One group of modes, in the 6.5–8.5-km alongshore-scale range, bears strong resemblance to the ensemble average disturbance structures observed in perturbed nonlinear model simulations. These modes are of a mixed type, exhibiting both strong baroclinic and barotropic energy exchange mechanisms, with maximum disturbance growth occurring during the transition from upwelling favorable to downwelling favorable winds. Nonlinear disturbance growth is characterized by similar structures at these same scales, but with significant exchange of energy between disturbances at different alongshore scales, such that overall disturbance energy accumulates at the longest (domain) scales, and gradually propagates offshore mainly in the pycnocline over numerous forcing cycles.
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VOISIN, B., E. V. ERMANYUK, and J. B. FLÓR. "Internal wave generation by oscillation of a sphere, with application to internal tides." Journal of Fluid Mechanics 666 (November 25, 2010): 308–57. http://dx.doi.org/10.1017/s0022112010004209.
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A joint theoretical and experimental study is performed on the generation of internal gravity waves by an oscillating sphere, as a paradigm of the generation of internal tides by barotropic tidal flow over three-dimensional supercritical topography. The theory is linear and three-dimensional, applies both near and far from the sphere, and takes into account viscosity and the unsteadiness arising from the interference with transients generated at the start-up. The waves propagate in conical beams, evolving with distance from a bimodal to unimodal wave profile. In the near field, the profile is asymmetric with its major peak towards the axis of the cones. The experiments involve horizontal oscillations and develop a cross-correlation technique for the measurement of the deformation of fluorescent dye planes to sub-pixel accuracy. At an oscillation amplitude of one fifth of the radius of the sphere, the waves are linear and the agreement between experiment and theory is excellent. As the amplitude increases to half the radius, nonlinear effects cause the wave amplitude to saturate at a value that is 20% lower than its linear estimate. Application of the theory to the conversion rate of barotropic tidal energy into internal tides confirms the expected scaling for flat topography, and shows its transformation for hemispherical topography. In the ocean, viscous and unsteady effects have an essentially local role, in keeping the wave amplitude finite at the edges of the beams, and otherwise dissipate energy on such large distances that they hardly induce any decay.
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Rapaka, Narsimha R., Bishakhdatta Gayen, and Sutanu Sarkar. "Tidal conversion and turbulence at a model ridge: direct and large eddy simulations." Journal of Fluid Mechanics 715 (January 9, 2013): 181–209. http://dx.doi.org/10.1017/jfm.2012.513.
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AbstractDirect and large eddy simulations are performed to study the internal waves generated by the oscillation of a barotropic tide over a model ridge of triangular shape. The objective is to go beyond linear theory and assess the role of nonlinear interactions including turbulence in situations with low tidal excursion number. The criticality parameter, defined as the ratio of the topographic slope to the characteristic slope of the tidal rays, is varied from subcritical to supercritical values. The barotropic tidal forcing is also systematically increased. Numerical results of the energy conversion are compared with linear theory and, in laminar flow at low forcing, they agree well in subcritical and supercritical cases but not at critical slope angle. In critical and supercritical cases with higher forcing, there are convective overturns, turbulence and significant reduction (as much as 25 %) of the radiated wave flux with respect to laminar flow results. Analysis of the baroclinic energy budget and spatial modal analysis are performed to understand the reduction. The near-bottom velocity is intensified at critical angle slope leading to a radiated internal wave beam as well as an upslope bore of cold water with a thermal front. In the critical case, the entire slope has turbulence while, in the supercritical case, turbulence originates near the top of the topography where the slope angle transitions through the critical value. The phase dependence of turbulence within a tidal cycle is examined and found to differ substantially between the ridge slope and the ridge top where the beams from the two sides cross.
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REZNIK, GREGORY M., and GEORGI G. SUTYRIN. "Baroclinic topographic modons." Journal of Fluid Mechanics 437 (June 22, 2001): 121–42. http://dx.doi.org/10.1017/s0022112001004062.
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The theory of solitary topographic Rossby waves (modons) in a uniformly rotating two-layer ocean over a constant slope is developed. The modon is described by an exact, form-preserving, uniformly translating, horizontally localized, nonlinear solution to the inviscid quasi-geostrophic equations. Baroclinic topographic modons are found to translate steadily along contours of constant depth in both directions: either with negative speed (within the range of the phase velocities of linear topographic waves) or with positive speed (outside the range of the phase velocities of linear topographic waves). The lack of resonant wave radiation in the first case is due to the orthogonality of the flow field in the modon exterior to the linear topographic wave field propagating with the modon translation speed, that is impossible for barotropic modons. Another important property of a baroclinic topographic modon is that its integral angular momentum must be zero only in the bottom layer; the total angular momentum can be non-zero unlike for the beta-plane modons over flat bottom. This feature allows modon solutions superimposed by intense monopolar vortices in the surface layer to exist. Explicit analytical solutions for the baroclinic topographic modons with piecewise linear dependence of the potential vorticity on the streamfunction are presented and analysed.
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AKYLAS, T. R., R. H. J. GRIMSHAW, S. R. CLARKE, and ALI TABAEI. "Reflecting tidal wave beams and local generation of solitary waves in the ocean thermocline." Journal of Fluid Mechanics 593 (November 23, 2007): 297–313. http://dx.doi.org/10.1017/s0022112007008786.
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It is generally accepted that ocean internal solitary waves can arise from the interaction of the barotropic tide with the continental shelf, which generates an internal tide that in turn steepens and forms solitary waves as it propagates shorewards. Some field observations, however, reveal large-amplitude internal solitary waves in deep water, hundreds of kilometres away from the continental shelf, suggesting an alternative generation mechanism: tidal flow over steep topography forces a propagating beam of internal tidal wave energy which impacts the thermocline at a considerable distance from the forcing site and gives rise to internal solitary waves there. Motivated by this possibility, a simple nonlinear long-wave model is proposed for the interaction of a tidal wave beam with the thermocline and the ensuing local generation of solitary waves. The thermocline is modelled as a density jump across the interface of a shallow homogeneous fluid layer on top of a deep uniformly stratified fluid, and a finite-amplitude propagating internal wave beam of tidal frequency in the lower fluid is assumed to be incident and reflected at the interface. The induced weakly nonlinear long-wave disturbance on the interface is governed in the far field by an integral-differential equation which accounts for nonlinear and dispersive effects as well as energy loss owing to radiation into the lower fluid. Depending on the strength of the thermocline and the intensity of the incident beam, nonlinear wave steepening can overcome radiation damping so a series of solitary waves may arise in the thermocline. Sample numerical solutions of the governing evolution equation suggest that this mechanism is quite robust for typical oceanic conditions.
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GAYEN, BISHAKHDATTA, and SUTANU SARKAR. "Direct and large-eddy simulations of internal tide generation at a near-critical slope." Journal of Fluid Mechanics 681 (May 25, 2011): 48–79. http://dx.doi.org/10.1017/jfm.2011.170.
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A numerical study is performed to investigate nonlinear processes during internal wave generation by the oscillation of a background barotropic tide over a sloping bottom. The focus is on the near-critical case where the slope angle is equal to the natural internal wave propagation angle and, consequently, there is a resonant wave response that leads to an intense boundary flow. The resonant wave undergoes both convective and shear instabilities that lead to turbulence with a broad range of scales over the entire slope. A thermal bore is found during upslope flow. Spectra of the baroclinic velocity, both inside the boundary layer and in the external region with free wave propagation, exhibit discrete peaks at the fundamental tidal frequency, higher harmonics of the fundamental, subharmonics and inter-harmonics in addition to a significant continuous part. The internal wave flux and its distribution between the fundamental and harmonics is obtained. Turbulence statistics in the boundary layer including turbulent kinetic energy and dissipation rate are quantified. The slope length is varied with the smaller lengths examined by direct numerical simulation (DNS) and the larger with large-eddy simulation (LES). The peak value of the near-bottom velocity increases with the length of the critical region of the topography. The scaling law that is observed to link the near-bottom peak velocity to slope length is explained by an analytical boundary-layer solution that incorporates an empirically obtained turbulent viscosity. The slope length is also found to have a strong impact on quantities such as the wave energy flux, wave energy spectra, turbulent kinetic energy, turbulent production and turbulent dissipation.
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Vlasenko, V., N. Stashchuk, C. Guo, and X. Chen. "Multimodal structure of baroclinic tides in the South China Sea." Nonlinear Processes in Geophysics 17, no. 5 (October 7, 2010): 529–43. http://dx.doi.org/10.5194/npg-17-529-2010.
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Abstract. The modelling of baroclinic tides generated in the northern South China Sea is studied using a fully-nonlinear non-hydrostatic numerical model. The focus of the modelling efforts was on the vertical structure of internal waves in the vicinity of the Luzon Strait. The barotropic tidal flow interacting with a two-ridge bottom topography in the area of the Luzon Strait produces a complex baroclinic tidal signal. A multimodal baroclinic bore with counter-phase displacement of isopycnals generated over the ridges and propagating westward disintegrates into a series of large-amplitude solitary internal waves. The leading first-mode solitary wave of depression is followed by a second mode solitary wave coupled with a packet of short-scale internal waves riding it. Scrutiny of the characteristics of the both wave forms, i.e. the carrier second-mode solitary wave and the packet of short waves, revealed that the short-scale waves are basically concentrated in the upper 500 m layer and attenuate exponentially below it. The short waves exist only thanks to a specific structure of horizontal velocity produced by the second-mode solitary wave. Having equal phase speeds and propagating together for a long distance, this coupled system produces quite a remarkable signal at the free surface, which can be detected by means of remote sensing technique. It was found in a series of sensitivity experiments that the eastern ridge is responsible for the generation of progressive first-mode tidal waves disintegrated into packets of first-mode ISWs. The western ridge produces quite a strong higher-mode signal. The waves generated over the eastern and western ridges interfere in the near-field, and their nonlinear superposition enhances the multimodal signal in the whole domain.
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Gray, Alison R., and Stephen C. Riser. "A Global Analysis of Sverdrup Balance Using Absolute Geostrophic Velocities from Argo." Journal of Physical Oceanography 44, no. 4 (April 1, 2014): 1213–29. http://dx.doi.org/10.1175/jpo-d-12-0206.1.
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Abstract Using observations from the Argo array of profiling floats, the large-scale circulation of the upper 2000 decibars (db) of the global ocean is computed for the period from December 2004 to November 2010. The geostrophic velocity relative to a reference level of 900 db is estimated from temperature and salinity profiles, and the absolute geostrophic velocity at the reference level is estimated from the trajectory data provided by the floats. Combining the two gives the absolute geostrophic velocity on 29 pressure surfaces spanning the upper 2000 db of the global ocean. These velocities, together with satellite observations of wind stress, are then used to evaluate Sverdrup balance, the simple canonical theory relating meridional geostrophic transport to wind forcing. Observed transports agree well with predictions based on the wind field over large areas, primarily in the tropics and subtropics. Elsewhere, especially at higher latitudes and in boundary regions, Sverdrup balance does not accurately describe meridional geostrophic transports, possibly due to the increased importance of the barotropic flow, nonlinear dynamics, and topographic influence. Thus, while it provides an effective framework for understanding the zero-order wind-driven circulation in much of the global ocean, Sverdrup balance should not be regarded as axiomatic.
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Merryfield, William J., Patrick F. Cummins, and Greg Holloway. "Equilibrium Statistical Mechanics of Barotropic Flow over Finite Topography." Journal of Physical Oceanography 31, no. 7 (July 2001): 1880–90. http://dx.doi.org/10.1175/1520-0485(2001)031<1880:esmobf>2.0.co;2.
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Carnevale, G. F., R. Purini, P. Orlandi, and P. Cavazza. "Barotropic quasi-geostrophic f -plane flow over anisotropic topography." Journal of Fluid Mechanics 285, no. -1 (February 1995): 329. http://dx.doi.org/10.1017/s0022112095000565.
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Egger, Joseph. "A modified quasigeostrophic equation for barotropic mean flow over topography." Meteorologische Zeitschrift 12, no. 1 (March 17, 2003): 43–46. http://dx.doi.org/10.1127/0941-2948/2003/0012-0043.
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Boland, Emma J. D., Andrew F. Thompson, Emily Shuckburgh, and Peter H. Haynes. "The Formation of Nonzonal Jets over Sloped Topography." Journal of Physical Oceanography 42, no. 10 (July 9, 2012): 1635–51. http://dx.doi.org/10.1175/jpo-d-11-0152.1.
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Abstract Coherent jets are ubiquitous features of the ocean’s circulation, and their characteristics, such as orientation and energetics, may be influenced by topography. In this study, the authors introduce a large-scale, topographic slope with an arbitrary orientation into quasigeostrophic, doubly periodic, barotropic and baroclinic systems. In both systems, the flow organizes itself into coherent tilted nonzonal jets that are aligned perpendicular to the barotropic potential vorticity (PV) gradient. In the two-layer system, the upper layer, the lower layer, and the barotropic PV gradients all have different orientations and therefore the jets cross the layer-wise PV gradients. The fact that the jets cross layer-wise PV gradients and the requirement of conservation of PV for fluid parcels together results in the drift of the tilted jets across the domain. Like their zonal counterparts, the tilted jets exhibit strong transport anisotropy. The dynamical response to jet deflection is very strong in the two-layer baroclinic case, with eddy energy production increasing by orders of magnitude as the topographic slope becomes more zonal. This increase in eddy energy is also reflected in an increase in jet spacing and a reduction in strength of the across-jet transport barriers, shown using an effective diffusivity diagnostic. The dynamics identified here, while formally valid within the constraints of quasigeostrophic scalings, provide important insight into the sensitive relationship between flow orientation and flow stability in regions with broad topographic slopes.
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SANSÓN, L. ZAVALA, and G. J. F. VAN HEIJST. "Ekman effects in a rotating flow over bottom topography." Journal of Fluid Mechanics 471 (November 5, 2002): 239–55. http://dx.doi.org/10.1017/s0022112002002239.
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This paper presents a general two-dimensional model for rotating barotropic flows over topography. The model incorporates in a vorticity–stream function formulation both inviscid topography effects, associated with stretching and squeezing of fluid columns enforced by their motion over variable topography, and viscous effects, due to the Ekman boundary layer at the solid bottom. From the present formulation, conventional two-dimensional models can be recovered. The model is tested by means of laboratory experiments on homogeneous vortices encountering irregular topographies. The experimental observations are then compared with the corresponding numerical simulations based on the general model. The results suggest that such a formulation incorporates both inviscid and viscous topography effects correctly.
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Yoden, Shigeo. "Multiple Stable States of Quasi-geostrophic Barotropic Flow over Sinusoidal Topography." Journal of the Meteorological Society of Japan. Ser. II 63, no. 6 (1985): 1031–45. http://dx.doi.org/10.2151/jmsj1965.63.6_1031.
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Carnevale, G. F., R. C. Kloosterziel, and G. J. F. Van Heijst. "Propagation of barotropic vortices over topography in a rotating tank." Journal of Fluid Mechanics 233 (December 1991): 119–39. http://dx.doi.org/10.1017/s0022112091000411.
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A small-scale cyclonic vortex in a relatively broad valley tends to climb up and out of the valley in a cyclonic spiral about the centre, and when over a relatively broad hill it tends to climb toward the top in an anticyclonic spiral around the peak. This phenomenon is examined here through two-dimensional numerical simulations and rotating-tank experiments. The basic mechanism involved is shown to be the same as that which accounts for the northwest propagation of cyclones on a β-plane. This inviscid nonlinear effect is also shown to be responsible for the observed translationary motion of barotropic vortices in a free-surface rotating tank. The behaviour of isolated vortices is contrasted with that of vortices with non-vanishing circulation.
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ALPAR, B., E. DOGAN, H. YUCE, and H. ALTIOK. "Sea level changes along the Turkish coasts of the Black Sea, the Aegean Sea and the Eastern Mediterranean." Mediterranean Marine Science 1, no. 1 (June 1, 2000): 141. http://dx.doi.org/10.12681/mms.285.
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Short, tidal, subtidal, seasonal, secular sea-level variations, sea-level differences and interactions between the basins have been studied, based on the data collected at some permanent and temporary tide gauges located along the Turkish coasts, mostly along the Straits connecting the Marmara Sea to outer seas. Even though the deficiency of sufficient information prevented us to reach the desired results, many pre-existed studies have been improved. Short-period oscillations were clearly identified along the Turkish Strait System and related to their natu-ral periods. The tidal amplitudes are low along the Turkish coasts, except northern Aegean and eastern Mediterranean. The stability of harmonic constants of Samsun and Antalya were examined and most of the long period constituents were found to be unstable. Even the Marmara Sea is not affected from the tidal oscillations of Black and Aegean Seas, some interactions in low frequency band have been detected. Subtidal sea level fluctuations (3-14 day) have relations with the large-scale cyclic atmospheric patterns passing over the Turkish Straits System. Short-term effects of wind on sea level are evident.Seasonal sea-level fluctuations along the Turkish Straits System are in accord with Black Sea's hydrological cycle. The differential range of the monthly mean sea levels between the Black Sea and the Marmara Sea is highly variable; high during spring and early summer and low during fall and winter.On the average, there is a pronounced sea-level difference (55 cm) along the Turkish Straits System. However, the slope is nonlinear, being much steeper in the Strait of Istanbul. This barotrophic pressure difference is one of the most important factors causing the two-layer flow through the system. The topography and hydrodynamics of the straits, the dominant wind systems and their seasonal variations make this flow more complicated. For secular sea level changes, a rise of 3.2 mm/a was computed for Karsiyaka (1935-71) and a steady trend (-0.4 mm /a) has been observed for annual sea levels at Antalya (1935-77). The decreasing trend (-6.9 mm/a) at Samsun is contrary to the secular rising trend of the Black Sea probably because of its rather short monitoring period (1963-77).
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Constantinou, Navid C. "A Barotropic Model of Eddy Saturation." Journal of Physical Oceanography 48, no. 2 (February 2018): 397–411. http://dx.doi.org/10.1175/jpo-d-17-0182.1.
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AbstractEddy saturation refers to a regime in which the total volume transport of an oceanic current is insensitive to the wind stress strength. Baroclinicity is currently believed to be the key to the development of an eddy-saturated state. In this paper, it is shown that eddy saturation can also occur in a purely barotropic flow over topography, without baroclinicity. Thus, eddy saturation is a fundamental property of barotropic dynamics above topography. It is demonstrated that the main factor controlling the appearance or not of eddy-saturated states in the barotropic setting is the structure of geostrophic contours, that is, the contours of f/H (the ratio of the Coriolis parameter to the ocean’s depth). Eddy-saturated states occur when the geostrophic contours are open, that is, when the geostrophic contours span the whole zonal extent of the domain. This minimal requirement for eddy-saturated states is demonstrated using numerical integrations of a single-layer quasigeostrophic flow over two different topographies characterized by either open or closed geostrophic contours with parameter values loosely inspired by the Southern Ocean. In this setting, transient eddies are produced through a barotropic–topographic instability that occurs because of the interaction of the large-scale zonal flow with the topography. By studying this barotropic–topographic instability insight is gained on how eddy-saturated states are established.
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Yao, Wenqi, and Weiqing Ren. "Noise-induced transition in barotropic flow over topography and application to Kuroshio." Journal of Computational Physics 300 (November 2015): 352–64. http://dx.doi.org/10.1016/j.jcp.2015.07.059.
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TIAN, YUDONG, ERIC R. WEEKS, KAYO IDE, J. S. URBACH, CHARLES N. BAROUD, MICHAEL GHIL, and HARRY L. SWINNEY. "Experimental and numerical studies of an eastward jet over topography." Journal of Fluid Mechanics 438 (July 5, 2001): 129–57. http://dx.doi.org/10.1017/s0022112001004372.
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Motivated by the phenomena of blocked and zonal flows in Earth's atmosphere, we conducted laboratory experiments and numerical simulations to study the dynamics of an eastward jet flowing over wavenumber-two topography. The laboratory experiments studied the dynamical behaviour of the flow in a barotropic rotating annulus as a function of the experimental Rossby and Ekman numbers. Two distinct flow patterns, resembling blocked and zonal flows in the atmosphere, were observed to persist for long time intervals.Earlier model studies had suggested that the atmosphere's normally upstream- propagating Rossby waves can resonantly lock to the underlying topography, and that this topographic resonance separates zonal from blocked flows. In the annulus, the zonal flows did indeed have super-resonant mean zonal velocities, while the blocked flows appear subresonant. Low-frequency variability, periodic or irregular, was present in the measured time series of azimuthal velocity in the blocked regime, with dominant periodicities in the range of 6–25 annulus rotations. Oscillations have also been detected in zonal states, with smaller amplitude and similar frequency. In addition, over a large region of parameter space the two flow states exhibited spontaneous, intermittent transitions from the one to the other.We numerically simulated the laboratory flow geometry in a quasi-geostrophic barotropic model over a similar range of parameters. Both flow regimes, blocked and zonal, were reproduced in the simulations, with similar spatial and temporal characteristics, including the low-frequency oscillations associated with the blocked flow. The blocked and zonal flow patterns are present over wide ranges of forcing, topographic height, and bottom friction. For a significant portion of parameter space, both model flows are stable. Depending on the initial state, either the blocked or the zonal flow is obtained and persists indefinitely, showing the existence of multiple equilibria.
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Thompson, Luanne, and Glenn R. Flierl. "Barotropic flow over finite isolated topography: steady solutions on the beta-plane and the initial value problem." Journal of Fluid Mechanics 250 (May 1993): 553–86. http://dx.doi.org/10.1017/s0022112093001569.
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Solutions for inviscid rotating flow over a right circular cylinder of finite height are studied, and comparisons are made to quasi-geostrophic solutions. To study the combined effects of finite topography and the variation of the Coriolis parameter with latitude a steady inviscid model is used. The analytical solution consists of one part which is similar to the quasi-geostrophic solution that is driven by the potential vorticity anomaly over the topography, and another, similar to the solution of potential flow around a cylinder, that is driven by the matching conditions on the edge of the topography. When the characteristic Rossby wave speed is much larger than the background flow velocity, the transport over the topography is enhanced as the streamlines follow lines of constant background potential vorticity. For eastward flow, the Rossby wave drag can be very much larger than that predicted by quasi-geostrophic theory. The combined effects of finite height topography and time-dependence are studied in the inviscid initial value problem on the f-plane using the method of contour dynamics. The method is modified to handle finite topography. When the topography takes up most of the layer depth, a stable oscillation exists with all of the fluid which originates over the topography rotating around the topography. When the Rossby number is order one, a steady trapped vortex solution similar to the one described by Johnson (1978) may be reached.
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Broutman, Dave, and Roger Grimshaw. "Spectral multigrid and collocation methods for barotropic nondivergent flow over irregular coastal topography." Geophysical & Astrophysical Fluid Dynamics 52, no. 1-3 (April 1990): 1–23. http://dx.doi.org/10.1080/03091929008219837.
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Shakespeare, Callum J., Brian K. Arbic, and Andrew McC. Hogg. "The Drag on the Barotropic Tide due to the Generation of Baroclinic Motion." Journal of Physical Oceanography 50, no. 12 (December 2020): 3467–81. http://dx.doi.org/10.1175/jpo-d-19-0167.1.
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AbstractThe 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.
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Grotto, Francesco, and Umberto Pappalettera. "Equilibrium statistical mechanics of barotropic quasi-geostrophic equations." Infinite Dimensional Analysis, Quantum Probability and Related Topics 24, no. 01 (March 2021): 2150007. http://dx.doi.org/10.1142/s0219025721500077.
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We consider equations describing a barotropic inviscid flow in a channel with topography effects and beta-plane approximation of Coriolis force, in which a large-scale mean flow interacts with smaller scales. Gibbsian measures associated to the first integrals energy and enstrophy are Gaussian measures supported by distributional spaces. We define a suitable weak formulation for barotropic equations, and prove existence of a solution preserving Gibbsian measures, thus providing a rigorous infinite-dimensional framework for the equilibrium statistical mechanics of the model.
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Solodoch, Aviv, Andrew L. Stewart, and James C. McWilliams. "Baroclinic instability of axially symmetric flow over sloping bathymetry." Journal of Fluid Mechanics 799 (June 22, 2016): 265–96. http://dx.doi.org/10.1017/jfm.2016.376.
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Observations and models of deep ocean boundary currents show that they exhibit complex variability, instabilities and eddy shedding, particularly over continental slopes that curve horizontally, for example around coastal peninsulas. In this article the authors investigate the source of this variability by characterizing the properties of baroclinic instability in mean flows over horizontally curved bottom slopes. The classical two-layer quasi-geostrophic solution for linear baroclinic instability over sloping bottom topography is extended to the case of azimuthal mean flow in an annular channel. To facilitate comparison with the classical straight channel instability problem of uniform mean flow, the authors focus on comparatively simple flows in an annulus, namely uniform azimuthal velocity and solid-body rotation. Baroclinic instability in solid-body rotation flow is analytically analogous to the instability in uniform straight channel flow due to several identical properties of the mean flow, including vanishing strain rate and vorticity gradient. The instability of uniform azimuthal flow is numerically similar to straight channel flow instability as long as the mean barotropic azimuthal velocity is zero. Non-zero barotropic flow generally suppresses the instability via horizontal curvature-induced strain and Reynolds stress work. An exception occurs when the ratio of the bathymetric to isopycnal slopes is close to (positive) one, as is often observed in the ocean, in which case the instability is enhanced. A non-vanishing mean barotropic flow component also results in a larger number of growing eigenmodes and in increased non-normal growth. The implications of these findings for variability in deep western boundary currents are discussed.
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Verron, J., and C. Le Provost. "A numerical study of quasi-geostrophic flow over isolated topography." Journal of Fluid Mechanics 154 (May 1985): 231–52. http://dx.doi.org/10.1017/s0022112085001501.
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An extensive set of numerical simulations is performed to synthesize the behaviour of a barotropic flow over isolated topography on an f-plane and on a β-plane. The model is based on the quasi-geostrophic vorticity equation, where the dissipation terms have been retained. The use of open boundary conditions. following the method described by Orlanski (1976), allows detailed simulation of time-dependent flows over long periods.On the f-plane, the ultimate solution is always characterized by a typical vorticity field with an anticyclonic vortex trapped over the topography, but different transient regimes occur, related to the importance of advection versus topography effect: direct advection of the positive vortex for strong flows; eddy interactions and double-vortex-structure appearance for weaker flows; oscillatory regimes with topographic trapped-waves generation for very strong vorticity-interaction cases.On the β-plane, and for prograde flows, the situation is complicated by a Rossby wave pattern extending mainly downstream but also having an upstream component corresponding to zonal waves. For retrograde flows the obstacle does not excite Rossby waves but a transient response with zonal waves whose lifetime depends on the nonlinearity.
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LIM, K., G. N. IVEY, and N. L. JONES. "Experiments on the generation of internal waves over continental shelf topography." Journal of Fluid Mechanics 663 (September 8, 2010): 385–400. http://dx.doi.org/10.1017/s002211201000354x.
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Experiments were performed to examine the generation of internal waves by a barotropic tide forcing a continuously stratified fluid over idealized continental shelf/slope topography. A range of responses was observed, including the generation of both internal wave beams and boundary layer boluses, primarily dependent on the values of both the Reynolds number and the topographic steepness parameter. The formation of beams required a critical bottom slope, whilst for bolus formation a large vertical fluid excursion was necessary. A bolus formed when the non-dimensional vertical excursion parameter ΔhN/W0 > 3.2. Here Δh is the vertical excursion, N is the buoyancy frequency and W0 is the near-bottom vertical velocity associated with the local depth-averaged velocity. We simplified the classification of the observed flow regimes using a generation parameter G, defined as the ratio of a Reynolds number to the topographic steepness parameter. The estimated flow regime boundaries were: for G < 3 only a beam was observed, for 3 < G < 50 there was a transitional regime with both a beam and a bolus observed, for 50 < G < 400 there was another transitional regime with no beam but a bolus observed, and finally for the regime with G > 400 there was no bolus observed. We estimated that approximately 4% of the barotropic energy was converted to baroclinic energy when beams were generated.
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Carnevale, George F., and Jorgen S. Frederiksen. "Nonlinear stability and statistical mechanics of flow over topography." Journal of Fluid Mechanics 175, no. -1 (February 1987): 157. http://dx.doi.org/10.1017/s002211208700034x.
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Zilberman, N. V., J. M. Becker, M. A. Merrifield, and G. S. Carter. "Model Estimates of M2 Internal Tide Generation over Mid-Atlantic Ridge Topography." Journal of Physical Oceanography 39, no. 10 (October 1, 2009): 2635–51. http://dx.doi.org/10.1175/2008jpo4136.1.
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Abstract The conversion of barotropic to baroclinic M2 tidal energy is examined for a section of the Mid-Atlantic Ridge in the Brazil Basin using a primitive equation model. Model runs are made with different horizontal smoothing (1.5, 6, and 15 km) applied to a 192 km × 183 km section of multibeam bathymetry to characterize the influence of topographic resolution on the model conversion rates. In all model simulations, barotropic to baroclinic conversion is highest over near- and supercritical slopes on the flanks of abyssal hills and discordant zones. From these generation sites, internal tides propagate upward and downward as tidal beams. The most energetic internal tide mode generated is mode 2, consistent with the dominant length scales of the topographic slope spectrum (50 km). The topographic smoothing significantly affects the model conversion amplitudes, with the domain-averaged conversion rate from the 1.5-km run (15.1 mW m−2) 4% and 19% higher than for the 6-km (14.5 mW m−2) and 15-km runs (12.2 mW m−2), respectively. Analytical models for internal tide generation by subcritical topography predict conversion rates with modal dependence and spatial patterns qualitatively similar to the Princeton Ocean Model (POM) and also show a decrease in conversion with smoother topography. The POM conversion rates are approximately 20% higher than the analytical estimates for all model grids, which is attributed to spatial variations in the barotropic flow and near-bottom stratification over generation sites, which are incorporated in the model but not in the analytical estimates.
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Grimshaw, R., and K. R. Helfrich. "Internal solitary wave generation by tidal flow over topography." Journal of Fluid Mechanics 839 (January 29, 2018): 387–407. http://dx.doi.org/10.1017/jfm.2018.21.
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Oceanic internal solitary waves are typically generated by barotropic tidal flow over localised topography. Wave generation can be characterised by the Froude number $F=U/c_{0}$, where $U$ is the tidal flow amplitude and $c_{0}$ is the intrinsic linear long wave phase speed, that is the speed in the absence of the tidal current. For steady tidal flow in the resonant regime, $\unicode[STIX]{x1D6E5}_{m}<F-1<\unicode[STIX]{x1D6E5}_{M}$, a theory based on the forced Korteweg–de Vries equation shows that upstream and downstream propagating undular bores are produced. The bandwidth limits $\unicode[STIX]{x1D6E5}_{m,M}$ depend on the height (or depth) of the topographic forcing term, which can be either positive or negative depending on whether the topography is equivalent to a hole or a sill. Here the wave generation process is studied numerically using a forced Korteweg–de Vries equation model with time-dependent Froude number, $F(t)$, representative of realistic tidal flow. The response depends on $\unicode[STIX]{x1D6E5}_{max}=F_{max}-1$, where $F_{max}$ is the maximum of $F(t)$ over half of a tidal cycle. When $\unicode[STIX]{x1D6E5}_{max}<\unicode[STIX]{x1D6E5}_{m}$ the flow is always subcritical and internal solitary waves appear after release of the downstream disturbance. When $\unicode[STIX]{x1D6E5}_{m}<\unicode[STIX]{x1D6E5}_{max}<\unicode[STIX]{x1D6E5}_{M}$ the flow reaches criticality at its peak, producing upstream and downstream undular bores that are released as the tide slackens. When $\unicode[STIX]{x1D6E5}_{max}>\unicode[STIX]{x1D6E5}_{M}$ the tidal flow goes through the resonant regime twice, producing undular bores with each passage. The numerical simulations are for both symmetrical topography, and for asymmetric topography representative of Stellwagen Bank and Knight Inlet.
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ESLER, J. G., O. J. RUMP, and E. R. JOHNSON. "Transcritical rotating flow over topography." Journal of Fluid Mechanics 590 (October 15, 2007): 81–106. http://dx.doi.org/10.1017/s0022112007007719.
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The flow of a one-and-a-half layer fluid over a three-dimensional obstacle of non-dimensional height M, relative to the lower layer depth, is investigated in the presence of rotation, the magnitude of which is measured by a non-dimensional parameter B (inverse Burger number). The transcritical regime in which the Froude number F, the ratio of the flow speed to the interfacial gravity wave speed, is close to unity is considered in the shallow-water (small-aspect-ratio) limit. For weakly rotating flow over a small isolated obstacle (M → 0) a similarity theory is developed in which the behaviour is shown to depend on the parameters Γ = (F−1)M−2/3 and ν = B1/2M−1/3. The flow pattern in this regime is determined by a nonlinear equation in which Γ and ν appear explicitly, termed here the ‘rotating transcritical small-disturbance equation’ (rTSD equation, following the analogy with compressible gasdynamics). The rTSD equation is forced by ‘equivalent aerofoil’ boundary conditions specific to each obstacle. Several qualitatively new flow behaviours are exhibited, and the parameter reduction afforded by the theory allows a (Γ, ν) regime diagram describing these behaviours to be constructed numerically. One important result is that, in a supercritical oncoming flow in the presence of sufficient rotation (ν ≳ 2), hydraulic jumps can appear downstream of the obstacle even in the absence of an upstream jump. Rotation is found to have the general effect of increasing the amplitude of any existing downstream hydraulic jumps and reducing the lateral extent and amplitude of upstream jumps. Numerical results are compared with results from a shock-capturing shallow-water model, and the (Γ, ν) regime diagram is found to give good qualitative and quantitative predictions of flow patterns at finite obstacle height (at least for M ≲ 0.4). Results are compared and contrasted with those for a two-dimensional obstacle or ridge, for which rotation also causes hydraulic jumps to form downstream of the obstacle and acts to attenuate upstream jumps.
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Gemmrich, Johannes, and Jody M. Klymak. "Dissipation of Internal Wave Energy Generated on a Critical Slope." Journal of Physical Oceanography 45, no. 9 (September 2015): 2221–38. http://dx.doi.org/10.1175/jpo-d-14-0236.1.
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AbstractTwo-dimensional simulations of stratified flow over an isolated ridge are used to evaluate energy dissipation associated with barotropic tidal flow over topography with critical or near-critical slope. In the midslope region, a shallow borelike flow forms along the bottom in a layer where dissipation rates are increased by several orders of magnitude, and the flow speed is about twice the barotropic background velocity. The height and turbulence in this layer depend on predictable functions of stratification, rotation, and the characteristic forcing speed. A physically sound power-law parameterization of the total energy dissipation associated with this turbulent layer is presented. This simple parameterization is also applicable to coarser-resolution models, where it may be included to compute energy dissipation above continental slopes, even for cases where the slope angle differs somewhat from criticality.
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Zavala Sansón, L., A. C. Barbosa Aguiar, and G. J. F. van Heijst. "Horizontal and vertical motions of barotropic vortices over a submarine mountain." Journal of Fluid Mechanics 695 (February 8, 2012): 173–98. http://dx.doi.org/10.1017/jfm.2012.9.
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AbstractThe evolution of barotropic vortices over a topographic, axisymmetric mountain in a homogeneous rotating fluid is studied experimentally. The aim is to identify the main physical processes observed in (i) a horizontal plane of motion, perpendicular to the rotation axis of the system, and (ii) a vertical plane across the diameter of the mountain. The vortices are monopolar cyclones initially generated near or over the topography. Initially, the vortices drift towards the mountain due to the $\ensuremath{\beta} $-effect associated with the topographic slope. On arriving, they turn around the obstacle in an anticyclonic direction, whilst anticyclonic vorticity is generated over the summit. The long-term vorticity distribution is dominated by the original cyclone elongated around the topographic contours and the generated anticyclone over the tip of the topography. In the vertical plane an oscillatory uphill–downhill flow is generated, which is directly related to the drift of the cyclone around the mountain. Depending on the vortex characteristics, the period of the oscillation ranges from 4 to 10 times the rotation period of the system. The horizontal and vertical flow fields are reproduced numerically by using a shallow-water formulation, which allows a detailed view of the vertical motions, hence facilitating the interpretation of the experimental results. In addition, the cyclone–anticyclone pair over the mountain is compared with analytical solutions of topographically trapped waves. A general conclusion is that vertical motions persist for several days (or rotation periods), which implies that this mechanism might be potentially important for the vertical transport over seamounts.
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Melville, W. K., and Karl R. Helfrich. "Transcritical two-layer flow over topography." Journal of Fluid Mechanics 178 (May 1987): 31–52. http://dx.doi.org/10.1017/s0022112087001101.
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The evolution of weakly-nonlinear two-layer flow over topography is considered. The governing equations are formulated to consider the effects of quadratic and cubic nonlinearity in the transcritical regime of the internal mode. In the absence of cubic nonlinearity an inhomogeneous Korteweg-de Vries equation describes the interfacial displacement. Numerical solutions of this equation exhibit undular bores or sequences of Boussinesq solitary waves upstream in a transcritical regime. For sufficiently large supercritical Froude numbers, a locally steady flow is attained over the topography. In that regime in which both quadratic and cubic nonlinearity are comparable, the evolution of the interface is described by an inhomogeneous extended Kortewegde Vries (EKdV) equation. This equation displays undular bores upstream in a subcritical regime, but monotonic bores in a transcritical regime. The monotonic bores are solitary wave solutions of the corresponding homogeneous EKdV equation. Again, locally steady flow is attained for sufficiently large supercritical Froude numbers. The predictions of the numerical solutions are compared with laboratory experiments which show good agreement with the solutions of the forced EKdV equation for some range of parameters. It is shown that a recent result of Miles (1986), which predicts an unsteady transcritical regime for single-layer flows, may readily be extended to two-layer flows (described by the forced KdV equation) and is in agreement with the results presented here.Numerical experiments exploiting the symmetry of the homogeneous EKdV equation show that solitary waves of fixed amplitude but arbitrary length may be generated in systems described by the inhomogeneous EKdV equation.
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Keeler, J. S., B. J. Binder, and M. G. Blyth. "On the critical free-surface flow over localised topography." Journal of Fluid Mechanics 832 (October 26, 2017): 73–96. http://dx.doi.org/10.1017/jfm.2017.639.
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Flow over bottom topography at critical Froude number is examined with a focus on steady, forced solitary wave solutions with algebraic decay in the far field, and their stability. Using the forced Korteweg–de Vries (fKdV) equation the weakly nonlinear steady solution space is examined in detail for the particular case of a Gaussian dip using a combination of asymptotic analysis and numerical computations. Non-uniqueness is established and a seemingly infinite set of steady solutions is uncovered. Non-uniqueness is also demonstrated for the fully nonlinear problem via boundary-integral calculations. It is shown analytically that critical flow solutions have algebraic decay in the far field both for the fKdV equation and for the fully nonlinear problem and, moreover, that the leading-order form of the decay is the same in both cases. The linear stability of the steady fKdV solutions is examined via eigenvalue computations and by a numerical study of the initial value fKdV problem. It is shown that there exists a linearly stable steady solution in which the deflection from the otherwise uniform surface level is everywhere negative.
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Buttle, Nicholas R., Ravindra Pethiyagoda, Timothy J. Moroney, and Scott W. McCue. "Three-dimensional free-surface flow over arbitrary bottom topography." Journal of Fluid Mechanics 846 (May 3, 2018): 166–89. http://dx.doi.org/10.1017/jfm.2018.254.
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We consider steady nonlinear free surface flow past an arbitrary bottom topography in three dimensions, concentrating on the shape of the wave pattern that forms on the surface of the fluid. Assuming ideal fluid flow, the problem is formulated using a boundary integral method and discretised to produce a nonlinear system of algebraic equations. The Jacobian of this system is dense due to integrals being evaluated over the entire free surface. To overcome the computational difficulty and large memory requirements, a Jacobian-free Newton–Krylov (JFNK) method is utilised. Using a block-banded approximation of the Jacobian from the linearised system as a preconditioner for the JFNK scheme, we find significant reductions in computational time and memory required for generating numerical solutions. These improvements also allow for a larger number of mesh points over the free surface and the bottom topography. We present a range of numerical solutions for both subcritical and supercritical regimes, and for a variety of bottom configurations. We discuss nonlinear features of the wave patterns as well as their relationship to ship wakes.
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Griffiths, Stephen D., and R. H. J. Grimshaw. "Internal Tide Generation at the Continental Shelf Modeled Using a Modal Decomposition: Two-Dimensional Results." Journal of Physical Oceanography 37, no. 3 (March 1, 2007): 428–51. http://dx.doi.org/10.1175/jpo3068.1.
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Abstract Stratified flow over topography is studied, with oceanic applications in mind. A model is developed for a fluid with arbitrary vertical stratification and a free surface, flowing over three-dimensional topography of arbitrary size and steepness, with background rotation, in the linear hydrostatic regime. The model uses an expansion of the flow fields in terms of a set of basis functions, which efficiently capture the vertical dependence of the flow. The horizontal structure may then be found by solving a set of coupled partial differential equations in two horizontal directions and time, subject to simple boundary conditions. In some cases, these equations may be solved analytically, but, in general, simple numerical procedures are required. Using this formulation, the internal tide generated by a time-periodic barotropic tidal flow over a continental shelf and slope is calculated in various idealized configurations. The topography and fluid motion are taken to be independent of one coordinate direction and the fluid to be either two-layer or uniformly stratified. For the two-layer case, expressions for the shoreward and oceanward energy fluxes associated with the internal tide are derived. For the uniformly stratified case, it is studied numerically how the accuracy of the solutions depends upon the number of basis functions used, and it is shown that good solutions and energy flux estimates can often be obtained with only a few basis functions. In both cases, the results show that the position of the coastline, through its effect on the form of the barotropic tide, significantly influences the strength of the internal tide generation.
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LaCasce, J. H., J. Escartin, Eric P. Chassignet, and Xiaobiao Xu. "Jet Instability over Smooth, Corrugated, and Realistic Bathymetry." Journal of Physical Oceanography 49, no. 2 (February 2019): 585–605. http://dx.doi.org/10.1175/jpo-d-18-0129.1.
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AbstractThe stability of a horizontally and vertically sheared surface jet is examined, with a focus on the vertical structure of the resultant eddies. Over a flat bottom, the instability is mixed baroclinic/barotropic, producing strong eddies at depth that are characteristically shifted downstream relative to the surface eddies. Baroclinic instability is suppressed over a large slope for retrograde jets (with a flow antiparallel to topographic wave propagation) and to a lesser extent for prograde jets (with flow parallel to topographic wave propagation), as seen previously. In such cases, barotropic (lateral) instability dominates if the jet is sufficiently narrow. This yields surface eddies whose size is independent of the slope but proportional to the jet width. Deep eddies still form, forced by interfacial motion associated with the surface eddies, but they are weaker than under baroclinic instability and are vertically aligned with the surface eddies. A sinusoidal ridge acts similarly, suppressing baroclinic instability and favoring lateral instability in the upper layer. A ridge with a 1-km wavelength and an amplitude of roughly 10 m is sufficient to suppress baroclinic instability. Surveys of bottom roughness from bathymetry acquired with shipboard multibeam echo sounding reveal that such heights are common beneath the Kuroshio, the Antarctic Circumpolar Current, and, to a lesser extent, the Gulf Stream. Consistent with this, vorticity and velocity cross sections from a 1/50° HYCOM simulation suggest that Gulf Stream eddies are vertically aligned, as in the linear stability calculations with strong topography. Thus, lateral instability may be more common than previously thought, owing to topography hindering vertical energy transfer.
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Song, Jian, and ShaoXia Liu. "The barotropic Rossby waves with topography on the earth’s δ-surface." International Journal of Nonlinear Sciences and Numerical Simulation 21, no. 7-8 (November 18, 2020): 781–88. http://dx.doi.org/10.1515/ijnsns-2019-0178.
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AbstractThe Rossby solitary waves in the barotropic vorticity model which contains the topography on the earth’s δ-surface is investigated. First, applying scale analysis method, obtained the generalized quasi-geostrophic potential vorticity equation (QGPVE). Using The Wentzel–Kramers–Brillouin (WKB) theory, the evolution equation of Rossby waves is the variable-coefficient Korteweg–de Vries (KdV) equation for the barotropic atmospheric model. In order to study the Rossby waves structural change to exist in some basic flow and topography on the δ-surface approximation, the variable coefficient of KdV equation must be explicitly, Chebyshev polynomials is used to solve a Sturm-Liouville-type eigenvalue problem and the eigenvalue Rossby waves, these solutions show that the parameter δ usually plays the stable part in Rossby waves and slow down the growing or decaying of Rossby waves with the parameter β.
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VANNESTE, JACQUES. "Nonlinear dynamics over rough topography: homogeneous and stratified quasi-geostrophic theory." Journal of Fluid Mechanics 474 (January 10, 2003): 299–318. http://dx.doi.org/10.1017/s0022112002002707.
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The weakly nonlinear dynamics of quasi-geostrophic flows over a one-dimensional, periodic or random, small-scale topography is investigated using an asymptotic approach. Averaged (or homogenized) evolution equations which account for the flow–topography interaction are derived for both homogeneous and continuously stratified quasi-geostrophic fluids. The scaling assumptions are detailed in each case; for stratified fluids, they imply that the direct influence of the topography is confined within a thin bottom boundary layer, so that it is through a new bottom boundary condition that the topography affects the large-scale flow. For both homogeneous and stratified fluids, a single scalar function entirely encapsulates the properties of the topography that are relevant to the large-scale flow: it is the correlation function of the topographic height in the homogeneous case, and a linear transform thereof in the continuously stratified case.Some properties of the averaged equations are discussed. Explicit nonlinear solutions in the form of one-dimensional travelling waves can be found. In the homogeneous case, previously studied by Volosov, they obey a second-order differential equation; in the stratified case on which we focus they obey a nonlinear pseudodifferential equation, which reduces to the Peierls–Nabarro equation for sinusoidal topography. The known solutions to this equation provide examples of nonlinear periodic and solitary waves in continuously stratified fluid over topography.The influence of bottom topography on large-scale baroclinic instability is also examined using the averaged equations: they allow a straightforward extension of Eady's model which demonstrates the stabilizing effect of topography on baroclinic instability.
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Rottman, James W., Dave Broutman, and Roger Grimshaw. "Numerical simulations of uniformly stratified fluid flow over topography." Journal of Fluid Mechanics 306 (January 10, 1996): 1–30. http://dx.doi.org/10.1017/s0022112096001206.
Full textAbstract:
We use a high-resolution spectral numerical scheme to solve the two-dimensional equations of motion for the flow of a uniformly stratified Boussinesq fluid over isolated bottom topography in a channel of finite depth. The focus is on topography of small to moderate amplitude and slope and for conditions such that the flow is near linear resonance of either of the first two internal wave modes. The results are compared with existing inviscid theories: the steady hydrostatic analysis of Long (1955), time-dependent linear long-wave theory, and the fully nonlinear, weakly dispersive resonant theory of Grimshaw & Yi (1991). For the latter, we use a spectral numerical technique, with improved accuracy over previously used methods, to solve the approximate evolution equation for the amplitude of the resonant mode. Also, we present some new results on the modal similarity of the solutions of Long and of Grimshaw & Yi. For flow conditions close to linear resonance, solutions of Grimshaw & Yi's evolution equation compare very well with our fully nonlinear numerical solutions, except for very steep topography. For flow conditions between the first two resonances, Long's steady solution is approached asymptotically in time when the slope of the topography is sufficiently small. For steeper topography, the flow remains unsteady. This unsteadiness is manifested very clearly as periodic oscillations in the drag, which have been observed in previous numerical simulations and tow-tank experiments. We explain these oscillations as mainly due to the internal waves that according to linear theory persist longest in the neighbourhood of the topography.
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ZOU, QINGPING. "A viscoelastic model for turbulent flow over undulating topography." Journal of Fluid Mechanics 355 (January 25, 1998): 81–112. http://dx.doi.org/10.1017/s0022112097007386.
Full textAbstract:
Aviscoelastic model (a mixing-length model with relaxation) is developed to investigate the effect of turbulent advection on the mean flow perturbation and the drag force induced by turbulent shear flow over an undulating surface. The relaxation term is proportional to the ratio of eddy turnover time to travelling time; accordingly, near the surface, the relaxation model reduces to an eddy-viscosity or mixing-length model, while far from the surface it reduces to a rapid-distortion model.The linear governing equations are transformed into streamline coordinates and solved through matched asymptotic expansions. According to order-of-magnitude estimates in Belcher, Newley & Hunt (1993), the drag force contributed by nonlinear shear stress is of the same order as that contributed by asymmetric pressure arising from the leeward thickening of the perturbed boundary layer. The nonlinear analysis in the present model confirms this estimate in most cases. Our analytical results show a dip in shear stress at the interface between the inner and outer layers and provide evidence that this dynamical feature is related to eddy advection. Numerical calculation using a shooting method gives results that compare well with the analysis.
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TSELUIKO, D., M. G. BLYTH, D. T. PAPAGEORGIOU, and J. M. VANDEN-BROECK. "Electrified viscous thin film flow over topography." Journal of Fluid Mechanics 597 (February 1, 2008): 449–75. http://dx.doi.org/10.1017/s002211200700986x.
Full textAbstract:
The gravity-driven flow of a liquid film down an inclined wall with periodic indentations in the presence of a normal electric field is investigated. The film is assumed to be a perfect conductor, and the bounding region of air above the film is taken to be a perfect dielectric. In particular, the interaction between the electric field and the topography is examined by predicting the shape of the film surface under steady conditions. A nonlinear, non-local evolution equation for the thickness of the liquid film is derived using a long-wave asymptotic analysis. Steady solutions are computed for flow into a rectangular trench and over a rectangular mound, whose shapes are approximated with smooth functions. The limiting behaviour of the film profile as the steepness of the wall geometry is increased is discussed. Using substantial numerical evidence, it is established that as the topography steepness increases towards rectangular steps, trenches, or mounds, the interfacial slope remains bounded, and the film does not touch the wall. In the absence of an electric field, the film develops a capillary ridge above a downward step and a slight depression in front of an upward step. It is demonstrated how an electric field may be used to completely eliminate the capillary ridge at a downward step. In contrast, imposing an electric field leads to the creation of a free-surface ridge at an upward step. The effect of the electric field on film flow into relatively narrow trenches, over relatively narrow mounds, and down slightly inclined substrates is also considered.