Non-hydrostatic Terms Research Articles

Vertically averaged and moment equations: New derivation, efficient numerical solution and comparison with other physical approximations for modeling non-hydrostatic free surface flows

Efficient modeling of flow physics is a prerequisite for a reliable computation of free-surface environmental flows. Non-hydrostatic flows are often present in shallow water environments, making the task challenging. In this work, we use the method of weighted residuals for modeling non-hydrostatic free surface flows in a depth-averaged framework. In particular, we focus on the Vertically Averaged and Moment (VAM) equations model. First, a new derivation of the model is presented using expansions of the field variables in sigma-coordinates with Legendre polynomials basis. Second, an efficient two-step numerical scheme is proposed: the first step corresponds to solving the hyperbolic part with a second-order path-conservative PVM scheme. Then, in a second step, non-hydrostatic terms are corrected by solving a linear Poisson-like system using an iterative method, thereby resulting in an accurate and efficient algorithm. The computational effort is similar to the one required for the well-known Serre-Green-Naghdi (SGN) system, while the results are largely improved. Finally, the physical aspects of the model are compared to the SGN system and a multilayer model, demonstrating that VAM is comparable in physical accuracy to a two-layer model.

Geodetic investigations of the mission concept MAGIC to reveal Callisto's internal structure

Geodetic and geophysical investigations of the Galilean moon Callisto address fundamental questions regarding the formation and evolution of the Jovian system. Callisto's evolution and internal structure appear to significantly differ from the other Jovian satellites. Similarly-sized Ganymede is a highly evolved ice-rock moon with a differentiated interior, intrinsic magnetic field, and abundant surface evidence of internal activity. In contrast, Callisto's surface is ancient, and Galileo spacecraft data suggest its interior is only incompletely differentiated, despite the presumed presence of a sub-surface ocean. These properties make Callisto uniquely able to constrain the timing and nature of the Jovian system formation. The Magnetics, Altimetry, Gravity, and Imaging of Callisto (MAGIC) mission concept is conceived to fully characterize the properties of this enigmatic moon from its deep interior to the icy shell. Three main instruments are included as a scientific payload. Highly accurate measurements of Callisto's topography, magnetic field, and morphology are obtained by the onboard laser altimeter, magnetometer, and camera, respectively. The telecommunication system supports an additional gravity and radio science investigation. Long- and short-wavelength gravity anomalies afford powerful constraints on internal differentiation and the properties of the hydrosphere (water and ice). Comprehensive numerical simulations and covariance analyses of MAGIC mission scenarios presented in this paper show that the gravitational degree-2 normalized coefficients and the pole obliquity enable the determination of the moment of inertia with an accuracy better than 0.015%. The combination of gravity and altimetry measurements acquired by MAGIC are essential to the characterization of Callisto's interior if – as is likely – the degree-2 gravity includes non-hydrostatic terms. MAGIC's radio science data yield the estimation of Callisto's gravity field with spatial resolutions of <100 km. The combination of gravitational and deformation tides that are retrieved by the radio science and altimetry investigations, respectively, leads to the recovery of the rigid ice shell thickness to within ∼3 km. Together these datasets would resolve ambiguities inherent in Galileo flyby data, revealing Callisto's interior structure as well as the existence and properties of its postulated internal ocean.

A Weakly Non-hydrostatic Shallow Model for Dry Granular Flows

A non-hydrostatic depth-averaged model for dry granular flows is proposed, taking into account vertical acceleration. A variable friction coefficient based on the $\mu(I)$ rheology is considered. The model is obtained from an asymptotic analysis in a local reference system, where the non-hydrostatic contribution is supposed to be small compared to the hydrostatic one. The non-hydrostatic counterpart of the pressure may be written as the sum of two terms: one corresponding to the stress tensor and the other to the vertical acceleration. The model introduced here is weakly non-hydrostatic, in the sense that the non-hydrostatic contribution related to the stress tensor is not taken into account due to its complex implementation. A simple and efficient numerical scheme is proposed. It consists of a three-step splitting procedure, and it is based on a hydrostatic reconstruction. Two key points are: (i) the friction force has to be taken into account before solving the non-hydrostatic pressure. Otherwise, the incompressibility condition is not ensured; (ii) both the hydrostatic and the non-hydrostatic pressure are taken into account when dealing with the friction force. The model and numerical scheme are then validated based on several numerical tests, including laboratory experiments of granular collapse. The influence of non-hydrostatic terms and of the choice of the coordinate system (Cartesian or local) is analyzed. We show that non-hydrostatic models are less sensitive to the choice of the coordinate system. In general, the non-hydrostatic model introduced here much better reproduces granular collapse experiments compared to hydrostatic models. An important result is that the simulated mass profiles up to the deposit and the front velocity are greatly improved. As expected, the influence of the non-hydrostatic pressure is shown to be larger for small values of the slope.

Extension of artificial viscosity technique for solving 2D non-hydrostatic shallow water equations

Abstract This study highlights the solution of the 2D non-hydrostatic shallow water equations using an artificial viscosity (AV) technique. The model recently proposed by the first author in Ginting (2017) is extended in this paper by reconstructing the flow variables using the Monotonic Upwind Scheme for Conservation Laws (MUSCL) technique to achieve second-order accuracy and by adding the non-hydrostatic terms, where the Runge–Kutta second-order scheme is used for temporal discretization. To reduce computational time, a novel way is proposed, where the AV technique as well as the non-hydrostatic terms are computed in a hybrid manner, only once per single time step. The solution of the non-hydrostatic terms forms a system of linear equations and the conjugate gradient method is therefore employed to solve it. The results show this technique is robust and accurate for simulating a set of problems of coastal engineering, thus could become a promising method for non-hydrostatic shallow flow applications.

Numerical dispersion in non-hydrostatic modeling of long-wave propagation

Numerical discretization with a finite-difference scheme is known to introduce frequency dispersion in depth-integrated long-wave models commonly used in tsunami research and hazard mapping. While prior studies on numerical dispersion focused on the linear shallow-water equations, we include the non-hydrostatic pressure and vertical velocity through a Keller box scheme and investigate the properties of the resulting system in relation to a hydrostatic model. Fourier analysis of the discretized governing equations gives rise to a dispersion relation in terms of the water-depth parameter as well as the time step, grid size, and wave direction. The dispersion relation is illustrated by its lead-order approximation derived from Taylor series expansions as well as numerical experiments involving standing and progressive waves with uniform water depth. Interaction between the spatial discretization and non-hydrostatic terms leads to significant reduction of numerical dispersion outside the shallow-water range. Numerical dispersion also decreases for wave propagation oblique to the computational grid due to effective increase in spatial resolution. The time step, which counteracts numerical dispersion from spatial discretization, only has secondary effects within the applicable range of Courant numbers. Since the governing equations of the non-hydrostatic model derived from the Keller box scheme tend to underestimate dispersion in shoaling water, the numerical effects are complementary in producing a solution closer to Airy wave theory. A case study of the 2011 Tohoku tsunami illustrates the grid sensitivity and convergence properties in real-world applications. A properly selected grid size can achieve an accurate description of wave propagation over a wide range of water-depth parameters across the ocean.

Multi-layer non-hydrostatic free surface modelling using the discontinuous Galerkin method

A multi-layer non-hydrostatic version of the unstructured mesh, discontinuous Galerkin finite element based coastal ocean model, Thetis, is developed. This is accomplished using the PDE solver framework, Firedrake, which is used to automatically produce the code for the discretised model equations in a rapid and efficient manner. The motivation for this work is a need to accurately simulate dispersive nearshore free surface processes.In order to resolve both frequency dispersion and non-linear effects accurately, additional non-hydrostatic terms are included in the layer-integrated hydrostatic equations, producing a form similar to the layered non-linear shallow water equations, but with extra vertical velocities at the layer interfaces. An implementation process is adopted to easily handle the inter-layer connection, i.e. the governing equations are transformed into a depth-integrated system through the introduction of depth-averaged variables.The model is verified and validated through comparisons against several idealised and experimentally-based test cases. All the comparisons demonstrate good agreement, showing that the developed non-hydrostatic model has excellent capabilities in representing coastal wave phenomena including shoaling, refraction and diffraction of dispersive short waves, as well as propagation, run-up and inundation of non-linear tsunami waves.

On plane internal waves: their amplification, and potential to break in a rotating stratified quiescent fluid

ABSTRACTIn this article, we assess the potential of plane internal waves to break in the interior of a rotating but otherwise quiescent stratified fluid. We show that wave energy density of a monochromatic wave is uniform in space and time, and calculate two heuristic stability measures: the isopycnal steepness (for overturning instability) and the Froude number (for shear instability) as functions of the wave energy density and phase to determine the regions where fast perturbation growth and subsequent wave breaking are likely to occur. For purely inertial waves, the Froude number field is uniform, and wave breaking is equally probable at any point. For non-inertial waves the locations of the minimal convective and shear stability always coincide, but the onset of shear instability almost always occurs at lower energy densities than the onset of overturning for all wave frequencies. In a rotating fluid, the separation between the two critical energy densities is most pronounced at low frequencies. At high frequencies, the critical energies for the instabilities nearly coalesce. In a non-rotating fluid, in contrast, the two critical energies coalesce at all frequencies. When a wave travels through a jump from weaker to stronger stratification, it amplifies in both steepness and Froude number. For the transmitted wave, the amplification factors for both stability measures are a monotonic function of the ratio of the two buoyancy frequencies. The Froude number amplification factor is a nonlinear function of the wave energy density: waves entering a region with a stronger stratification with larger initial energy amplify to a larger Froude number. The amplification upon refraction is stronger in both steepness and Froude number when nonhydrostatic terms are retained in the governing equations. In contrast to the transmitted wave, the superposition of the incident and the reflected waves on the other side of the stratification jump can become destabilised for both scenarios (the incident wave entering either stronger or weaker stratification). When the sufficiently energetic incident wave enters stronger stratification, the flow is destabilised simultaneously on both sides of the interface. When the incident wave enters weaker stratification, only the superposition of the incident and reflected waves can become destabilised, but not the transmitted wave.

Extension of the Dupuit–Forchheimer Model for Non-Hydrostatic Flows in Unconfined Aquifers

The classical Dupuit–Forchheimer approach, commonly used in analysing unconfined groundwater-flow systems, relies on the assumption of a negligible vertical component of the flow. This approximation is valid only when the convergence of streamlines is very limited and the drawdown of the phreatic surface is small, or the thickness of the horizontal layer of the heterogeneous aquifers is sufficiently small. In this study, a higher-order one-dimensional model is proposed for groundwater-flow problems with significant inclination and curvature of the phreatic surface. The model incorporates non-hydrostatic terms that take into account the effects of the vertical velocity of the flow, and was solved with an implicit finite-difference scheme. The accuracy of the proposed model was demonstrated by simulating various unconfined seepage- and groundwater-flow problems with moderate curvilinear effects. The computational results for steady-state flows were compared with the results of the full two-dimensional potential-flow methods and experimental data, resulting in a reasonably good agreement. In general, the comparison results exhibited the efficiency and validity of the model in simulating complex unconfined flows over curved bedrock and curvilinear flows over planar bedrock with a steep slope.

Development of BFMCOUPLER (v1.0), the coupling scheme that links the MITgcm and BFM models for ocean biogeochemistry simulations

Abstract. In this paper, we present a coupling scheme between the Massachusetts Institute of Technology general circulation model (MITgcm) and the Biogeochemical Flux Model (BFM). The MITgcm and BFM are widely used models for geophysical fluid dynamics and for ocean biogeochemistry, respectively, and they benefit from the support of active developers and user communities. The MITgcm is a state-of-the-art general circulation model for simulating the ocean and the atmosphere. This model is fully 3-D (including the non-hydrostatic term of momentum equations) and is characterized by a finite-volume discretization and a number of additional features enabling simulations from global (O(107) m) to local scales (O(100) m). The BFM is a biogeochemical model based on plankton functional type formulations, and it simulates the cycling of a number of constituents and nutrients within marine ecosystems. The online coupling presented in this paper is based on an open-source code, and it is characterized by a modular structure. Modularity preserves the potentials of the two models, allowing for a sustainable programming effort to handle future evolutions in the two codes. We also tested specific model options and integration schemes to balance the numerical accuracy against the computational performance. The coupling scheme allows us to solve several processes that are not considered by each of the models alone, including light attenuation parameterizations along the water column, phytoplankton and detritus sinking, external inputs, and surface and bottom fluxes. Moreover, this new coupled hydrodynamic–biogeochemical model has been configured and tested against an idealized problem (a cyclonic gyre in a mid-latitude closed basin) and a realistic case study (central part of the Mediterranean Sea in 2006–2012). The numerical results consistently reproduce the interplay of hydrodynamics and biogeochemistry in both the idealized case and Mediterranean Sea experiments. The former reproduces correctly the alternation of surface bloom and deep chlorophyll maximum dynamics driven by the seasonal cycle of winter vertical mixing and summer stratification; the latter simulates the main basin-wide and mesoscale spatial features of the physical and biochemical variables in the Mediterranean, thus demonstrating the applicability of the new coupled model to a wide range of ocean biogeochemistry problems.

Nonhydrostatic Dam Break Flows. II: One-Dimensional Depth-Averaged Modeling for Movable Bed Flows

AbstractThe dam break flow over a movable bed is an important problem in fluvial flow processes. These flows are usually predicted by a one-dimensional (1D) approach based on a hydrostatic pressure distribution. Recent three-dimensional (3D) nonhydrostatic simulations of dam break waves over movable beds based on the Reynolds-averaged Navier-Stokes (RANS) equations revealed that the Saint Venant theory is not accurate in predicting the flow dynamics within the scour hole developed. In this work, a generalized 1D nonhydrostatic model for flow over movable beds is proposed assuming a linear, nonhydrostatic, pressure distribution. The new set of 1D equations accounts for the vertical acceleration, which is important in dam break waves over movable beds given the instantaneous curved beds generated over the erodible terrain. These equations account for both the bed and suspended-load transport modes. A high-resolution finite volume numerical scheme with a semi-implicit treatment of nonhydrostatic terms is dev...

Dispersion and kinematics of multi-layer non-hydrostatic models

Multi-layer non-hydrostatic models are gaining popularity in studies of coastal wave processes owing to the resolution of the flow kinematics, but the linear dispersion relation remains the primary criterion for assessment of model convergence. In this paper, we reformulate the linear governing equations of an N-layer model into Boussinesq form by writing the non-hydrostatic terms as high-order derivatives of the horizontal flow velocity. The equation structure allows implementation of Fourier analysis to provide a [2N−2, 2N] expansion of the velocity at each layer. A variable transformation converts the governing equations into separate flux- and dispersion-dominated systems, which explicitly give an equivalent Pade´ expansion of the wave celerity for examination of the convergence and asymptotic properties. Flow continuity equates the depth-integrated horizontal velocity to the celerity and verifies the analytical solution. The surface-layer velocity, which is driven by the kinematic free surface boundary condition, shows a positive error and converges monotonically to the solution of Airy wave theory. When the depth parameter kd > 2N, flow reversal occurs in the sub-surface layers to offset overestimation of the surface velocity and to better approximate the flux. This model internal mechanism facilitates convergence of the celerity at large kd and benefits applications on wave transformation. Such non-physical flow reversal, however, might complicate studies that require detailed wave kinematics.

Evaluation of nearshore wave models in steep reef environments

To provide coastal engineers and scientists with a quantitative evaluation of nearshore numerical wave models in reef environments, we review and compare three common- ly used models with detailed laboratory observations. These models are the following: (1) SWASH (Simulating WAves till SHore) (Zijlema et al. 2011), a phase-resolving nonlinear shallow-water wave model with added nonhydrostatic terms; (2) SWAN (Simulating WAve Nearshore) (Booij et al. 1999), a phase-averaged spectral wave model; and (3) XBeach (Roelvink et al. 2009), a coupled phase-averaged spectral wave model (applied to modeling sea-swell waves) and a nonlinear shallow-water model (applied to modeling infragravity waves). A quantitative assessment was made of each model's ability to predict sea-swell (SS) wave height, infragravity (IG) wave height, wave spectra, and wave setup (η) at fivelocationsacross the laboratory fringingreefprofile of Demirbilek et al. (2007). Simulations were performed with the "recommended"empirical coefficients as documented for each model, and then the key wave-breaking parameter for each model (α in SWASH and γ in both SWAN and XBeach) was optimized to most accurately reproduce the observations. SWASH, SWAN, and XBeach were found to be capable of predicting SS wave height variations across the steep fringing reef profile with reasonable accuracy using the default coeffi- cients. Nevertheless, tuning of the key wave-breaking parameter improved the accuracy of each model's predictions. SWASH and XBeach were also able to predict IG wave height and spectral transformation. Although SWAN was capable of modeling the SS wave height, in its current form, it was not capable of modeling the spectral transformation into lower frequencies, as evident in the underprediction of the low- frequency waves.

Implementation of a direct nonhydrostatic pressure gradient discretisation into a layered ocean model

The increasing demand for the investigation of nonhydrostatic effects in ocean modelling requires crucial modifications to models applying the hydrostatic pressure assumption. Within many studies the capability of the pressure-correcting projection method for the inclusion of the missing nonhydrostatic pressure contribution into an existing hydrostatic model kernel was verified. It provides accurate results but also requires computational (and implementational) effort for solving a Poisson equation for the pressure correction.In contrast, some studies were based on an alternative approach that does not require the inversion of a Poisson equation. Within this alternative approach the nonhydrostatic pressure contribution is calculated by an explicit vertical integration of the additional nonhydrostatic terms in the balance of vertical momentum. Since ocean models are not intended to replace classical engineering tools and are usually applied to nearly hydrostatic flows, this straight-forward extension of the hydrostatic procedure is an interesting option. However, the feasibility of the alternative approach was not tested within a full 3D explicit mode-splitting model yet. Furthermore, the nonhydrostatic capability of the alternative approach has so far not been validated against well known nonhydrostatic benchmark test cases. In order to assess the potential of the alternative approach for future ocean modelling applications, in the present study these required but still missing investigations are carried out.To demonstrate the necessary modifications to an explicit mode-splitting hydrostatic model kernel, the extension of the General Estuarine Transport Model (GETM) is outlined. The simulation results of laboratory and idealised oceanic test cases are presented and compared to analytical theory, laboratory experiments and other numerical simulations. This validation indicates the nonhydrostatic capability of the extended GETM. However, stability issues limited the efficiency and the applicability of the straight-forward inclusion of nonhydrostatic dynamics.

Importance of capturing heliospheric variability for studies of thermospheric vertical winds

Using the Global Ionosphere Thermosphere Model with observed real‐time heliospheric input data, the magnitude and variability of thermospheric neutral vertical winds are investigated. In order to determine the role of variability in the Interplanetary Magnetic Field (IMF) and solar wind density on the neutral wind variability, the heliospheric input data are smoothed. The effects of smoothing the IMF and solar wind and density on the vertical winds are simulated for the cases of no smoothing, 5‐minute, and 12‐minute smoothing. Various vertical wind acceleration terms, such as the nonhydrostatic acceleration, are quantified. Polar stereographic projections of the variabilities of vertical wind and ion flows are compared to highlight existing correlations. Overall, the smoother, that is, the less variable the IMF and solar wind parameters are, the weaker are the magnitude and the variability of the thermospheric vertical winds. Weaker IMF variability leads to smaller variability in ion flows, which in turn negatively impacts the variability and the magnitude of Joule heating. Small‐scale temporal variation of the vertical wind acceleration, and thus the variability of the vertical wind, is dominated by the nonhydrostatic term that is controlled primarily by the temporal variation of the Joule heating, which in turn is related to ion flow variations that are shaped by the IMF in the high‐latitude thermosphere. Wavelet analysis of the vertical wind data shows that gravity waves of ∼5 and ∼10‐minute periods are more prominent when the model is run with high‐resolution real‐time IMF and solar wind data. Better capturing of the temporal variation of the IMF and solar wind parameters is crucial for modeling the variability and magnitude of thermospheric vertical winds.

Transient wave behaviour over an underwater sliding hump from experiments and analytical and numerical modelling

Flume measurements of a one-dimensional sliding hump starting from rest in quiescence fresh water indicate that when the hump travels at speed less than the shallow-water wave celerity, three waves emerge, travelling in two directions. One wave travels in the opposite direction to the sliding hump at approximately the shallow-water wave celerity (backward free wave). Another wave travels approximately in step with the hump (forced wave), and the remaining wave travels in the direction of the hump at approximately the shallow-water wave celerity (forward free wave). These experiments were completed for a range of sliding hump speed relative to the shallow-water wave celerity, up to unity of this ratio, to investigate possible derivation from solutions of the Euler equation with non-linear and non-hydrostatic terms being included or excluded. For the experimental arrangements tested, the forced waves were negative (depression or reduced water surface elevation) waves while the free waves were positive (bulges or increased water surface elevation). For experiments where the sliding hump travelled at less than 80% of the shallow-water wave celerity did not include transient behaviour measurements (i.e. when the three waves still overlapped). The three wave framework was partially supported by these measurements in that the separated forward and forced waves were compared to measurements. For the laboratory scale experiments, the forward free wave height was predicted reasonably by the long-wave equation (ignoring non-linear and non-hydrostatic terms) when the sliding hump speed was less than 80% of the shallow-water wave celerity. The forced wave depression magnitude required the Euler equations for all hump speed tested. The long-wave solution, while being valid in a limited parameter range, does predict the existence of the three waves as found in these experiments (forward travelling waves measured quantitatively while the backward travelling waves visually by video). Nevertheless, the forced wave transient development required non-linear and non-hydrostatic terms for higher sliding hump speeds. The forward free wave, controversially, does not need non-linear and non-hydrostatic terms until much higher hump speeds, suggesting that the forward free wave falls in the parameter space where long-wave equations apply whereas the forced wave more often falls into the parameter space requiring non-linear and non-hydrostatic terms. It does raise the question of why the forced wave transient dynamics does not impact on the initial transient dynamics where the forward free wave is in the long-wave theory, escaping the forced wave (at least for speeds less than 80% of the shallow-water wave celerity).

Depth‐integrated free‐surface flow with a two‐layer non‐hydrostatic formulation

SUMMARYThis paper presents the derivation of a depth‐integrated wave propagation and runup model from a system of governing equations for two‐layer non‐hydrostatic flows. The governing equations are transformed into an equivalent, depth‐integrated system, which separately describes the flux‐dominated and dispersion‐dominated processes. The depth‐integrated system reproduces the linear dispersion relation within a 5 error for water depth parameter up to kd = 11, while allowing direct implementation of a momentum conservation scheme to model wave breaking and a moving‐waterline technique for runup calculation. A staggered finite‐difference scheme discretizes the governing equations in the horizontal dimension and the Keller box scheme reconstructs the non‐hydrostatic terms in the vertical direction. An semi‐implicit scheme integrates the depth‐integrated flow in time with the non‐hydrostatic pressure determined from a Poisson‐type equation. The model is verified with solitary wave propagation in a channel of uniform depth and validated with previous laboratory experiments for wave transformation over a submerged bar, a plane beach, and fringing reefs. Copyright © 2011 John Wiley &amp; Sons, Ltd.

Numerical investigation of internal solitary waves from the Luzon Strait: Generation process, mechanism and three-dimensional effects

A fully nonlinear, non-hydrostatic model, MITgcm, is used to investigate internal solitary waves (ISWs) from the Luzon Strait (LS). As the ISWs in the South China Sea (SCS) have drawn more and more attention in recent years, they are studied in various ways, i.e., via remote sensing images, in situ measurements, and numerical simulations. The inspiration of this paper derived from the potential flaws of different numerical models that were employed to examine ISWs. In this study, we performed three-dimensional (3D) experiments with realistic topography and stratification, as well as with fully non-hydrostatic terms in the model, which was rather important for investigating the ISWs. Modeling results showed that baroclinic tides in the LS were essentially three-dimensional (3D), and that wave structures around two ridges in the strait were complicated with interesting internal oceanic phenomena. Several zonal cross-sections were chosen to illustrate vertical structures of zonal velocity field, and to show their meridional variances together with surface horizontal velocity gradients in order to highlight the advantages of 3D modeling with fully nonlinear, non-hydrostatic terms. Following Vlasenko et al. (2005), analysis of two parameters (Froude number and slope parameter that is defined as the ratio of inclination of topography to slope of radiated rays) that govern generation regime indicated that internal waves produced in the LS were subject to a mixed lee wave regime rather than baroclinic tide regime or unsteady lee wave regime. The propagation of ISWs beyond the generation area showed that manifestation of 3D effects was not very obvious, which, through further analysis, was mainly attributed to homogeneity of topography, inaccuracy of barotropic forcing, and Kuroshio intrusion in the LS. To better understand the necessity of 3D modeling, we chose several zonal cross-sections and performed various sensitivity experiments to show discrepancies between 2D and 3D cases.

The Effect of Tilted Rotation on Shear Instabilities at Low Stratifications

Abstract A linear stability analysis of the inviscid stratified Boussinesq equations is presented given a steady zonal flow with constant vertical shear in a tilted f plane. Full nonhydrostatic terms are included: 1) acceleration of vertical velocity and 2) Coriolis force terms arising from the meridional component of earth’s rotation vector. Calculations of growth rates, critical wavenumbers, and dominance regimes for baroclinic and symmetric instabilities are compared with results from the traditional nonhydrostatic equations, which include a strictly vertical rotation vector, as well as results from the hydrostatic equations. The authors find that for positive zonal z shear, tilted rotation enhances the dominance regime of symmetric instabilities at the expense of baroclinic instabilities and maintains symmetric instabilities at larger scales than previously indicated. Furthermore, in contrast to former studies, it is determined that hydrostatic growth rates for both instabilities are not maximal. Rather, growth rates peak in the fully nonhydrostatic equations for parameter regimes physically relevant and consistent with abyssal ocean stratifications and weak zonal z shears and oceanic measurements of the Labrador Sea and Southern Ocean. In addition, the authors find that zonal shear modifies the frequency range of subinertial inertio–gravity waves. Tilted rotation effects break the base flow shear reflection symmetry present in the traditional and hydrostatic models. Thus, only in the fully nonhydrostatic model does weak negative zonal z shear stabilize the flow and decrease the subinertial frequency range.

Non-hydrostatic and non-linear contributions to the internal wave energy flux in sill regions

A three-dimensional non-linear, non-hydrostatic model in cross-sectional form is used to determine the factors influencing the relative importance of the linear, non-hydrostatic and non-linear contributions to the internal wave energy flux in sill regions due to tidal forcing. The importance of the free surface elevation term is also considered. Idealised topography representing the sill at the entrance to Loch Etive, the site of a recent measurement programme, is used. Calculations show that the non-linear terms in the energy flux become increasingly important as the sill Froude Number (Fs) increases and the sill aspect ratio is increased. The vertical profile of the stratification, in particular its value close to the sill crest where internal waves are generated, has a significant influence on unsteady lee wave and mixed tidal–lee wave generation and the non-linear contribution to the energy flux. Calculations show that as Fs increases, the energy flux due to the non-linear and non-hydrostatic terms increases more rapidly than the linear term. The importance of the non-linear terms in the energy flux also increases as the sill aspect ratio is increased. Increasing the buoyancy frequency reduces the contribution of the non-hydrostatic and non-linear terms to the total energy flux. Also, as the buoyancy frequency is increased, this reduces unsteady lee wave and mixed tidal–lee wave generation. In essence, these calculations show that the energy flux due to the non-hydrostatic and non-linear terms is appreciable in sill regions.

Godunov-Based Model for Nonhydrostatic Wave Dynamics

A new numerical model based on the incompressible, nonhydrostatic Navier-Stokes equations for free surface flow is developed. The equations are transformed vertically to the σ coordinate system and laterally to an orthogonal curvilinear system and solved in a fractional step manner in which the pressure is split into hydrostatic and nonhydrostatic components. The model treats the nonhydrostatic term implicitly and uses a collocated grid and pressure interpolation to prevent checkerboard solutions that occur when the velocity and pressure become decoupled. Advection and hydrostatic pressure terms are integrated explicitly with a second-order accurate predictor-corrector scheme. The corrector utilizes fluxes that are computed in a Godunov-based manner by solving a Riemann problem at each cell face. Flow variables are reconstructed at each cell face to obtain second-order spatial accuracy. Numerical simulations of Stokes, cnoidal, and solitary waves with the proposed method and a reference method in which th...