Academic literature on the topic 'Nonlinear barotropic flow over topography'

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.

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.

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.

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.

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.

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.

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.

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.

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.

This thesis presents a numerical exploration of the dynamics governing rotating flow driven by a surface stress in the " sliced cylinder " model of Pedlosky & Greenspan (1967) and Beardsley (1969), and its close relative, the " sliced cone " model introduced by Griffiths & Veronis (1997). The sliced cylinder model simulates the barotropic wind-driven circulation in a circular basin with vertical sidewalls, using a depth gradient to mimic the effects of a gradient in Coriolis parameter. In the sliced cone the vertical sidewalls are replaced by an azimuthally uniform slope around the perimeter of the basin to simulate a continental slope. Since these models can be implemented in the laboratory, their dynamics can be explored by a complementary interplay of analysis and numerical and laboratory experiments. ¶ In this thesis a derivation is presented of a generalised quasigeostrophic formulation which is valid for linear and moderately nonlinear barotropic flows over large-amplitude topography on an f-plane, yet retains the simplicity and conservation properties of the standard quasigeostrophic vorticity equation (which is valid only for small depth variations). This formulation is implemented in a numerical model based on a code developed by Page (1982) and Becker & Page (1990). ¶ The accuracy of the formulation and its implementation are confirmed by detailed comparisons with the laboratory sliced cylinder and sliced cone results of Griffiths (Griffiths & Kiss, 1999) and Griffiths & Veronis (1997), respectively. The numerical model is then used to provide insight into the dynamics responsible for the observed laboratory flows. In the linear limit the numerical model reveals shortcomings in the sliced cone analysis by Griffiths & Veronis (1998) in the region where the slope and interior join, and shows that the potential vorticity is dissipated in an extended region at the bottom of the slope rather than a localised region at the east as suggested by Griffiths & Veronis (1997, 1998). Welander's thermal analogy (Welander, 1968) is used to explain the linear circulation pattern, and demonstrates that the broadly distributed potential vorticity dissipation is due to the closure of geostrophic contours in this geometry. ¶ The numerical results also provide insight into features of the flow at finite Rossby number. It is demonstrated that separation of the western boundary current in the sliced cylinder is closely associated with a " crisis " due to excessive potential vorticity dissipation in the viscous sublayer, rather than insufficient dissipation in the outer western boundary current as suggested by Holland & Lin (1975) and Pedlosky (1987). The stability boundaries in both models are refined using the numerical results, clarifying in particular the way in which the western boundary current instability in the sliced cone disappears at large Rossby and/or Ekman number. A flow regime is also revealed in the sliced cylinder in which the boundary current separates without reversed flow, consistent with the potential vorticity " crisis " mechanism. In addition the location of the stability boundary is determined as a function of the aspect ratio of the sliced cylinder, which demonstrates that the flow is stabilised in narrow basins such as those used by Beardsley (1969, 1972, 1973) and Becker & Page (1990) relative to the much wider basin used by Griffiths & Kiss (1999). ¶ Laboratory studies of the sliced cone by Griffiths & Veronis (1997) showed that the flow became unstable only under anticyclonic forcing. It is shown in this thesis that the contrast between flow under cyclonic and anticyclonic forcing is due to the combined effects of the relative vorticity and topography in determining the shape of the potential vorticity contours. The vorticity at the bottom of the sidewall smooths out the potential vorticity contours under cyclonic forcing, but distorts them into highly contorted shapes under anticyclonic forcing. In addition, the flow is dominated by inertial boundary layers under cyclonic forcing and by standing Rossby waves under anticyclonic forcing due to the differing flow direction relative to the direction of Rossby wave phase propagation. The changes to the potential vorticity structure under strong cyclonic forcing reduce the potential vorticity changes experienced by fluid columns, and the flow approaches a steady free inertial circulation. In contrast, the complexity of the flow structure under anticyclonic forcing results in strong potential vorticity changes and also leads to barotropic instability under strong forcing. ¶ The numerical results indicate that the instabilities in both models arise through supercritical Hopf bifurcations. The two types of instability observed by Griffiths & Veronis (1997) in the sliced cone are shown to be related to the western boundary current instability and " interior instability " identified by Meacham & Berloff (1997). The western boundary current instability is trapped at the western side of the interior because its northward phase speed exceeds that of the fastest interior Rossby wave with the same meridional wavenumber, as discussed by Ierley & Young (1991). ¶ Numerical experiments with different lateral boundary conditions are also undertaken. These show that the flow in the sliced cylinder is dramatically altered when the free-slip boundary condition is used instead of the no-slip condition, as expected from the work of Blandford (1971). There is no separated jet, because the flow cannot experience a potential vorticity " crisis " with this boundary condition, so the western boundary current overshoots and enters the interior from the east. In contrast, the flow in the sliced cone is identical whether no-slip, free-slip or super-slip boundary conditions are applied to the horizontal flow at the top of the sloping sidewall, except in the immediate vicinity of this region. This insensitivity results from the extremely strong topographic steering near the edge of the basin due to the vanishing depth, which demands a balance between wind forcing and Ekman pumping on the upper slope, regardless of the lateral boundary condition. The sensitivity to the lateral boundary condition is related to the importance of lateral friction in the global vorticity balance. The integrated vorticity must vanish under the no-slip condition, so in the sliced cylinder the overall vorticity budget is dominated by lateral viscosity and Ekman friction is negligible. Under the free-slip condition the Ekman friction assumes a dominant role in the dissipation, leading to a dramatic change in the flow structure. In contrast, the much larger depth variation in the sliced cone leads to a global vorticity balance in which Ekman friction is always dominant, regardless of the boundary condition.

A finite element shallow-water model is tested with two types of surface topography. The model uses rectangular subdivisions in a vorticity-divergence formulation, and a semi-implicit time discretization. In the first experiment an east-west ridge or valley is placed in a channel with east-west periodic conditions. Linear quasi-geostrophic solutions are derived with the rigid lid assumption. The Rossby waves are successfully simulated in the model with linear solutions as the initial conditions. The model phase speeds are very close to the analytic values when the latter are properly corrected. In the second experiment a ridge is placed across the channel and the Coriolis parameter is set to zero. The initial conditions consist of a uniform flow through the channel and constant free-surface height. The numerical simulations agree with hydraulic jump theory. In the jump cases the model predicts increasing wind speeds and decreasing free surface heights. Higher spatial resolution would be required to properly simulate the details of the hydraulic jump formation. (fr)

This thesis investigates several aspects of stratified flow over isolated topography in ocean, lake, and atmospheric settings. Three major sub-topics are addressed: steady, inviscid internal waves trapped over topography in a pycnocline stratification, topographically generated internal waves and their interaction with the viscous bottom boundary layer, and flow over large-scale crater topography in the atmosphere. The first topic examines the conditions that lead to very large internal waves trapped over topography in a fluid with a pycnocline stratification. This type of stratification is connected to ocean or lake settings. The steady-state Euler equations of motion are used to derive a single partial differential equation for the isopycnal displacement in supercritical flows under two conditions: a vertically varying background current under the Boussinesq approximation and a constant background current under non-Boussinesq conditions. A numerical method is developed to solve these equations for an efficient exploration of parameter space. Very large waves are found over depression topography when the background flow speed is close to a limiting value. Variations in the background current are examined, as well as comparisons between Boussinesq and non-Boussinesq results. The second topic aims to extend the above subject by considering unsteady, viscous flows. Once again, supercritical flow over topography in a pycnocline stratification generates internal waves. These internal waves interact with the viscous bottom boundary layer to produce bottom boundary instabilities. The three-dimensional aspects of these instabilities are studied under changes in viscosity. The boundary layer instabilities have important implications for sediment transport in the coastal oceans or lakes. Lastly, the final topic is motivated by the connection between dust streaks on the Martian surface and crater topography. Flow over a large 100-km diameter crater is examined with numerical simulations conducted using the Weather Research and Forecasting model. Modifications to the stratification and topography are applied. It is found that a large hydraulic structure of amplitude comparable to the crater depth forms in many cases. This structure may have important implications for dust transport in the atmosphere. In addition, Martian atmospheric parameters are used to study the flow properties under Mars-like conditions.

After analysis of general properties of horizontal motion in primitive equations and introduction of principal parameters, the key notion of geostrophic equilibrium is introduced. Quasi-geostrophic reductions of one- and two-layer rotating shallow-water models are obtained by a direct filtering of fast inertia–gravity waves through a choice of the time scale of motions of interest, and by asymptotic expansions in Rossby number. Properties of quasi-geostrophic models are established. It is shown that in the beta-plane approximations the models describe Rossby waves. The first idea of the classical baroclinic instability is given, and its relation to Rossby waves is explained. Modifications of quasi-geostrophic dynamics in the presence of coastal, topographic, and equatorial wave-guides are analysed. Emission of mountain Rossby waves by a flow over topography is demonstrated. The phenomena of Kelvin wave breaking, and of soliton formation by long equatorial and topographic Rossby waves due to nonlinear effects are explained.

Strongly nonlinear, rapidly rotating, stably stratified flow is called geostrophic turbulence. This subject, which blends ideas from chapters 2,4, and 5, is relevant to the large-scale flow in the Earth’s oceans and atmosphere. The quasigeostrophic equations form the basis of the study of geostrophic turbulence. We view the quasigeostrophic equations as a generalization of the vorticity equation for two-dimensional turbulence to include the important effects of stratification, bottom topography, and varying Coriolis parameter. Thus the theory of geostrophic turbulence represents an extension of the theory of two dimensional turbulence. However, its richer physics and greater applicability to real geophysical flows make geostrophic turbulence a much more interesting and important subject. This chapter offers a very brief introduction to the theory of geostrophic turbulence. We illustrate the principal ideas by separately considering the effects of bottom topography, varying Coriolis parameter, and density stratification on highly nonlinear, quasigeostrophic flow. We make no attempt at a comprehensive review. In every case, the theory of geostrophic turbulence relies almost solely on two now-familiar components: a conservation principle that energy and potential vorticity are (nearly) conserved and an irreversibility principle in the form of an appealing assumption that breaks the time-reversal symmetry of the exact (inviscid) dynamics. This irreversibility assumption takes a great many superficially dissimilar forms, fostering the misleading impression of a great many competing explanations for the same phenomena. However, broadminded analysis inevitably reveals that these competing explanations are virtually equivalent. We begin by considering the quasigeostrophic flow of a single layer of homogeneous fluid over a bumpy bottom. No case better illustrates how diverse forms of the irreversibility principle lead to the same conclusions.

A numerical model is presented for the simulation of the two-dimensional, inviscid, free-surface flow developing by the propagation and breaking of water waves over a flat bottom of steep slope. The simulation is based on the numerical solution of the unsteady, two-dimensional, Euler equations subject to the fully-nonlinear free-surface boundary conditions, the non-penetration condition at the bottom and appropriate inflow and outflow conditions. A boundary-fitted transformation, which includes both the time-dependent free surface and the arbitrary bottom shape, is applied. For the numerical solution of the Euler equations, a two-stage fractional time-step method is employed for the temporal discretization, while a hybrid scheme is used for the spatial discretization. Finite differences are used in the streamwise direction and a pseudo-spectral method in the vertical direction. An absorption zone is placed at the outflow region in order to minimize wave reflection by the outflow boundary. Wave breaking is modeled by a surface roller breaking model, which modifies the dynamic free-surface condition. The simulation results are in very good agreement with available experimental results for the wave propagation and breaking over bottom with slope 1:35. Results, from the simulations over bottom with steeper slopes of 1:15 and 1:10, which generate strong spilling and mild plunging breakers, respectively, are also in very good agreement with available predictions for the breaking depth and wave height. In all cases, a vortex is formed under the breaking wave front and convected in the surf zone.

A non-linear coupled-mode system of horizontal equations is presented, as derived from Luke’s (1967) variational principle, which models the evolution of nonlinear water waves in intermediate depth over a general bottom topography. The vertical structure of the wave field is represented by means of a complete local-mode series expansion of the wave potential. This series contains the usual propagating and evanescent modes, plus two additional terms, the free-surface mode and the sloping-bottom mode, enabling to consistently treat the non-vertical end-conditions at the free-surface and the bottom boundaries. The present coupled-mode system fully accounts for the effects of non-linearity and dispersion, and has the following main features: (i) various standard models of water-wave propagation are recovered by appropriate simplifications, and (ii) it exhibits fast convergenge, and thus, a small number of modes (up to 5) are usually enough for the precise numerical solution, provided that the two new modes (the free-surface and the sloping-bottom ones) are included in the local-mode series. In the present work, the coupled-mode system is applied to the numerical investigation of families of steady traveling wave solutions in constant depth, corresponding to a wide range of water depths, ranging from intermediate to shallow-water wave conditions and its results are compared vs. Stokes and cnoidal wave theories, respectively. Also, numerical results are presented for waves propagating over variable bathymetry regions and compared with second-order Stokes theory and experimental data.

Correctly predicting the track of a ship when impacted by incident waves is one of the most important tasks in accurately simulating seakeeping performance. This is especially difficult during a storm surge in coastal regions, where the bottom topography rapidly changes. In fact accurately predicting a ship track in calm water has not been completely solved. Most recently, Lin et al. integrated the total forces and moments over an actual ship hull and rudders in a fully nonlinear ship motion model named the Digital, Self-consistent Ship Experimental Laboratory (DiSSEL) by using finite element/finite difference in calm water (Lin et al. 2005 [1], Lin and Kuang, 2006 [2], 2008 [3], 2009 [4]; Lin et al., 2009 [5]). The new method is not only accurate, but also computationally efficient. As part of a continuing effort, in this study, the incident wave impact is now included. The incident wave impact on the ship hull and rudders is integrated over the actual wetted surfaces at each time step, based on the ship position. First, the numerical simulations are benchmarked using existing experimental data for an experimental hull form. The DiSSEL simulations agree well with the experimental data. Based on the simulations, we explore the physics to gain an understanding into the possible factors that cause changes to the ship tracks. For example, why will the turning circle of a ship with constant rudder angle and ship speed, in the presence of waves, sometimes be a spiral while other times remain a circle? Finally DiSSEL will be coupled with the coastal wave model by Lin and Perrie (1997, 1999) [6], [7], to predict the ship track in a coastal region impacted by a storm surge. Future improvements to DiSSEL will include the incorporation of a propeller model. Since a model for the propellers has not yet been implemented, the influence of the propeller slipstream on the rudder is currently handled by empirically modifying the effective inflow velocity into the rudder to account for the acceleration of the flow by the propeller.