Wave-induced mean flows in rotating shallow water with uniform potential vorticity

2018 ◽  
Vol 839 ◽  
pp. 408-429 ◽  
Author(s):  
Jim Thomas ◽  
Oliver Bühler ◽  
K. Shafer Smith
Keyword(s):  
Shallow Water ◽  
Potential Vorticity ◽  
Rotation Rate ◽  
Standing Waves ◽  
Leading Order ◽  
Mean Flow ◽  
Wave Fields ◽  
The Mean ◽  
Rotating Shallow Water ◽  
Wave Induced

Theoretical and numerical computations of the wave-induced mean flow in rotating shallow water with uniform potential vorticity are presented, with an eye towards applications in small-scale oceanography where potential-vorticity anomalies are often weak compared to the waves. The asymptotic computations are based on small-amplitude expansions and time averaging over the fast wave scale to define the mean flow. Importantly, we do not assume that the mean flow is balanced, i.e. we compute the full mean-flow response at leading order. Particular attention is paid to the concept of modified diagnostic relations, which link the leading-order Lagrangian-mean velocity field to certain wave properties known from the linear solution. Both steady and unsteady wave fields are considered, with specific examples that include propagating wavepackets and monochromatic standing waves. Very good agreement between the theoretical predictions and direct numerical simulations of the nonlinear system is demonstrated. In particular, we extend previous studies by considering the impact of unsteady wave fields on the mean flow, and by considering the total kinetic energy of the mean flow as a function of the rotation rate. Notably, monochromatic standing waves provide an explicit counterexample to the often observed tendency of the mean flow to decrease monotonically with the background rotation rate.

Instabilities of buoyancy-driven coastal currents and their nonlinear evolution in the two-layer rotating shallow water model. Part 2. Active lower layer

2010 ◽  
Vol 665 ◽  
pp. 209-237 ◽  
Author(s):  
J. GULA ◽  
V. ZEITLIN ◽  
F. BOUCHUT
Keyword(s):  
Shallow Water ◽  
Linear Stability ◽  
Nonlinear Evolution ◽  
Lower Layer ◽  
Short Wave ◽  
Finite Volume Scheme ◽  
Coastal Currents ◽  
Mean Flow ◽  
The Mean ◽  
Rotating Shallow Water

This paper is the second part of the work on linear and nonlinear stability of buoyancy-driven coastal currents. Part 1, concerning a passive lower layer, was presented in the companion paper Gula & Zeitlin (J. Fluid Mech., vol. 659, 2010, p. 69). In this part, we use a fully baroclinic two-layer model, with active lower layer. We revisit the linear stability problem for coastal currents and study the nonlinear evolution of the instabilities with the help of high-resolution direct numerical simulations. We show how nonlinear saturation of the ageostrophic instabilities leads to reorganization of the mean flow and emergence of coherent vortices. We follow the same lines as in Part 1 and, first, perform a complete linear stability analysis of the baroclinic coastal currents for various depths and density ratios. We then study the nonlinear evolution of the unstable modes with the help of the recent efficient two-layer generalization of the one-layer well-balanced finite-volume scheme for rotating shallow water equations, which allows the treatment of outcropping and loss of hyperbolicity associated with shear, Kelvin–Helmholtz type, instabilities. The previous single-layer results are recovered in the limit of large depth ratios. For depth ratios of order one, new baroclinic long-wave instabilities come into play due to the resonances among Rossby and frontal- or coastal-trapped waves. These instabilities saturate by forming coherent baroclinic vortices, and lead to a complete reorganization of the initial current. As in Part 1, Kelvin fronts play an important role in this process. For even smaller depth ratios, short-wave shear instabilities with large growth rates rapidly develop. We show that at the nonlinear stage they produce short-wave meanders with enhanced dissipation. However, they do not change, globally, the structure of the mean flow which undergoes secondary large-scale instabilities leading to coherent vortex formation and cutoff.

A Coupled-Mode Technique for the Prediction of Wave-Induced Set-Up and Mean Flow in Variable Bathymetry Domains

2007 ◽  
Author(s):  
K. A. Belibassakis ◽  
Th. P. Gerostathis ◽  
G. A. Athanassoulis
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Water Environment ◽  
Wave Model ◽  
Mean Flow ◽  
Flow Equations ◽  
Coupled Mode ◽  
The Mean ◽  
Set Up ◽  
Wave Induced

In the present work, a complete, phase-resolving wave model is coupled with an iterative solver of the mean-flow equations in intermediate and shallow water depth, permitting an accurate calculation of wave set-up and wave-induced current in intermediate and shallow water environment with possibly steep bathymetric variations. The wave model is based on the consistent coupled-mode system of equations, developed by Athanassoulis & Belibassakis (1999) for the propagation of water waves in variable bathymetry regions. This model improves the predictions of the mild-slope equation, permitting the treatment of wave propagation in regions with steep bottom slope and/or large curvature. In addition, it supports the consistent calculation of wave velocity up to and including the bottom boundary. The above wave model has been further extended to include the effects of bottom friction and wave breaking, which are important factors for the calculation of radiation stresses on decreasing depth. The latter have been used as forcing terms to the mean flow equations in order to predict wave-induced set up and mean flow in open and closed domains. Numerical results obtained by the present model are presented and compared with predictions obtained by the mild-slope approximation (Massel & Gourlay 2000), and experimental data (Gourlay 1996).

Stable and unstable shear modes of rotating parallel flows in shallow water

1987 ◽  
Vol 184 ◽  
pp. 477-504 ◽  
Author(s):  
Y.-Y. Hayashi ◽  
W. R. Young
Keyword(s):  
Energy Density ◽  
Shallow Water ◽  
Wave Energy ◽  
Potential Vorticity ◽  
The Other ◽  
Shear Mode ◽  
Mean Flow ◽  
Shear Modes ◽  
The Mean ◽  
Energy And Momentum

This article considers the instabilities of rotating, shallow-water, shear flows on an equatorial β-plane. Because of the free surface, the motion is horizontally divergent and the energy density is cubic in the field variables (i.e. in standard notation the kinetic energy density is ½ h(u2 + v2)). Marinone & Ripa (1984) observed that as a consequence of this the wave energy is no longer positive definite (there is a cross-term Uh′u′). A wave with negative wave energy can grow by transferring energy to the mean flow. Of course total (mean plus wave) energy is conserved in this process. Further, when the basic state has constant potential vorticity, we show that there are no exchanges of energy and momentum between a growing wave and the mean flow. Consequently when the basic state has no potential vorticity gradients an unstable wave has zero wave energy and the mean flow is modified so that its energy is unchanged. This result strikingly shows that energy and momentum exchanges between a growing wave and the mean flow are not generally characteristic of, or essential to, instability.A useful conceptual tool in understanding these counterintuitive results is that of disturbance energy (or pseudoenergy) of a shear mode. This is the amount of energy in the fluid when the mode is excited minus the amount in the unperturbed medium. Equivalently, the disturbance energy is the sum of the wave energy and that in the modified mean flow. The disturbance momentum (or pseudomomentum) is defined analogously.For an unstable mode, which grows without external sources, the disturbance energy must be zero. On the other hand the wave energy may increase to plus infinity, remain zero, or decrease to minus infinity. Thus there is a tripartite classification of instabilities. We suggest that one common feature in all three cases is that the unstable shear mode is roughly a linear combination of resonating shear modes each of which would be stable if the other were somehow suppressed. The two resonating constituents must have opposite-signed disturbance energies in order that the unstable alliance has zero disturbance energy. The instability is a transfer of disturbance energy from the member with negative disturbance energy to the one with positive disturbance energy.

scholarly journals Influence of Bottom Topography on Integral Constraints in Zonal Flows with Parameterized Potential Vorticity Fluxes

2013 ◽  
Vol 43 (2) ◽  
pp. 311-323 ◽  
Author(s):  
V. O. Ivchenko ◽  
B. Sinha ◽  
V. B. Zalesny ◽  
R. Marsh ◽  
A. T. Blaker
Keyword(s):  
Potential Vorticity ◽  
Bottom Topography ◽  
Momentum Conservation ◽  
Integral Constraint ◽  
Mean Flow ◽  
Transfer Coefficients ◽  
Flat Bottom ◽  
Zonal Flows ◽  
Eddy Fluxes ◽  
The Mean

Abstract An integral constraint for eddy fluxes of potential vorticity (PV), corresponding to global momentum conservation, is applied to two-layer zonal quasigeostrophic channel flow. This constraint must be satisfied for any type of parameterization of eddy PV fluxes. Bottom topography strongly influences the integral constraint compared to a flat bottom channel. An analytical solution for the mean flow solution has been found by using asymptotic expansion in a small parameter, which is the ratio of the Rossby radius to the meridional extent of the channel. Applying the integral constraint to this solution, one can find restrictions for eddy PV transfer coefficients that relate the eddy fluxes of PV to the mean flow. These restrictions strongly deviate from restrictions for the channel with flat bottom topography.

scholarly journals Wave field and zonal flow of a librating disk

2015 ◽  
Vol 782 ◽  
pp. 178-208 ◽  
Author(s):  
Stéphane Le Dizès
Keyword(s):  
Wave Field ◽  
Rotation Rate ◽  
Zonal Flow ◽  
Shear Layers ◽  
Buoyancy Frequency ◽  
Frequency Interval ◽  
Mean Flow ◽  
Disk Radius ◽  
Harmonic Response ◽  
The Mean

In this work, we provide a viscous solution of the wave field generated by librating a disk (harmonic oscillation of the rotation rate) in a stably stratified rotating fluid. The zonal flow (mean flow correction) generated by the nonlinear interaction of the wave field is also calculated in the weakly nonlinear framework. We focus on the low dissipative limit relevant for geophysical applications and for which the wave field and the zonal flow exhibit generic features (Ekman scaling, universal structures, etc.). General expressions are obtained which depend on the disk radius $a^{\ast }$, the libration frequency ${\it\omega}^{\ast }$, the rotation rate ${\it\Omega}^{\ast }$ of the frame, the buoyancy frequency $N^{\ast }$ of the fluid, its kinematic diffusion ${\it\nu}^{\ast }$ and its thermal diffusivity ${\it\kappa}^{\ast }$. When the libration frequency is in the inertia-gravity frequency interval ($\min ({\it\Omega}^{\ast },N^{\ast })<{\it\omega}^{\ast }<\max ({\it\Omega}^{\ast },N^{\ast })$), the presence of conical internal shear layers is observed in which the spatial structures of the harmonic response and of the mean flow correction are provided. At the point of focus of these internal shear layers on the rotation axis, the largest amplitudes are obtained: the angular velocity of the harmonic response and the mean flow correction are found to be $O({\it\varepsilon}E^{-1/3})$ and $({\it\varepsilon}^{2}E^{-2/3})$ respectively, where ${\it\varepsilon}$ is the libration amplitude and $E={\it\nu}^{\ast }/({\it\Omega}^{\ast }a^{\ast 2})$ is the Ekman number. We show that the solution in the internal shear layers and in the focus region is at leading order the same as that generated by an oscillating source of axial flow localized at the edge of the disk (oscillating Dirac ring source).

scholarly journals Flow Kinematics in Variable-Height Rotating Cylinder Arrays

2016 ◽  
Vol 138 (11) ◽  
Author(s):  
Anna E. Craig ◽  
John O. Dabiri ◽  
Jeffrey R. Koseff
Keyword(s):  
Power Output ◽  
Rotation Rate ◽  
Three Dimensional ◽  
Vertical Axis ◽  
Streamwise Velocity ◽  
Vertical Flow ◽  
Mean Flow ◽  
Freestream Velocity ◽  
Cylinder Arrays ◽  
The Mean

Experimental data are presented for large arrays of rotating, variable-height cylinders in order to study the dependence of the three-dimensional mean flows on the height heterogeneity of the array. Elements in the examined arrays were spatially arranged in the same staggered paired configuration, and the heights of each element pair varied up to ±37.5% from the mean height (kept constant across all arrays), such that the arrays were vertically structured. Four vertical structuring configurations were examined at a nominal Reynolds number (based on freestream velocity and cylinder diameter) of 600 and nominal tip-speed ratios of 0, 2, and 4. It was found that the vertical structuring of the array could significantly alter the mean flow patterns. Most notably, a net vertical flow into the array from above was observed, which was augmented by the arrays' vertical structuring, showing a 75% increase from the lowest to highest vertical flows (as evaluated at the maximum element height, at a single rotation rate). This vertical flow into the arrays is of particular interest as it represents an additional mechanism by which high streamwise momentum can be transported from above the array down into the array. An evaluation of the streamwise momentum resource within the array indicates up to a 56% increase in the incoming streamwise velocity to the elements (from the lowest to highest ranking arrays, at a single rotation rate). These arrays of rotating cylinders may provide insight into the flow kinematics of arrays of vertical axis wind turbines (VAWTs). In a physical VAWT array, an increase in incoming streamwise flow velocity to a turbine corresponds to a (cubic) increase in the power output of the turbine. Thus, these results suggest a promising approach to increasing the power output of a VAWT array.

scholarly journals A Framework for Parameterizing Eddy Potential Vorticity Fluxes

2012 ◽  
Vol 42 (4) ◽  
pp. 539-557 ◽  
Author(s):  
David P. Marshall ◽  
James R. Maddison ◽  
Pavel S. Berloff
Keyword(s):  
Stress Tensor ◽  
Potential Vorticity ◽  
Large Scale ◽  
Eddy Diffusivity ◽  
Linear Instability ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Conservation Of Energy ◽  
The Mean ◽  
Energy And Momentum

Abstract A framework for parameterizing eddy potential vorticity fluxes is developed that is consistent with conservation of energy and momentum while retaining the symmetries of the original eddy flux. The framework involves rewriting the residual-mean eddy force, or equivalently the eddy potential vorticity flux, as the divergence of an eddy stress tensor. A norm of this tensor is bounded by the eddy energy, allowing the components of the stress tensor to be rewritten in terms of the eddy energy and nondimensional parameters describing the mean shape and orientation of the eddies. If a prognostic equation is solved for the eddy energy, the remaining unknowns are nondimensional and bounded in magnitude by unity. Moreover, these nondimensional geometric parameters have strong connections with classical stability theory. When applied to the Eady problem, it is shown that the new framework preserves the functional form of the Eady growth rate for linear instability. Moreover, in the limit in which Reynolds stresses are neglected, the framework reduces to a Gent and McWilliams type of eddy closure where the eddy diffusivity can be interpreted as the form proposed by Visbeck et al. Simulations of three-layer wind-driven gyres are used to diagnose the eddy shape and orientations in fully developed geostrophic turbulence. These fields are found to have large-scale structure that appears related to the structure of the mean flow. The eddy energy sets the magnitude of the eddy stress tensor and hence the eddy potential vorticity fluxes. Possible extensions of the framework to ensure potential vorticity is mixed on average are discussed.

scholarly journals Jets and Orography: Idealized Experiments with Tip Jets and Lighthill Blocking

2007 ◽  
Vol 64 (10) ◽  
pp. 3627-3639 ◽  
Author(s):  
P. B. Rhines
Keyword(s):  
Numerical Simulations ◽  
Shallow Water ◽  
Rossby Wave ◽  
Potential Vorticity ◽  
Rossby Waves ◽  
Zonal Flow ◽  
Flow Acceleration ◽  
Westerly Flow ◽  
The Mean ◽  
Strong Control

Abstract This paper describes qualitative features of the generation of jetlike concentrated circulations, wakes, and blocks by simple mountainlike orography, both from idealized laboratory experiments and shallow-water numerical simulations on a sphere. The experiments are unstratified with barotropic lee Rossby waves, and jets induced by mountain orography. A persistent pattern of lee jet formation and lee cyclogenesis owes its origins to arrested topographic Rossby waves above the mountain and potential vorticity (PV) advection through them. The wake jet occurs on the equatorward, eastern flank of the topography. A strong upstream blocking of the westerly flow occurs in a Lighthill mode of long Rossby wave propagation, which depends on βa2/U, the ratio of Rossby wave speed based on the scale of the mountain, to zonal advection speed, U (β is the meridional potential vorticity gradient, f is the Coriolis frequency, and a is the diameter of the mountain). Mountains wider (north–south) than the east–west length scale of stationary Rossby waves will tend to block the oncoming westerly flow. These blocks are essentially β plumes, which are illustrated by their linear Green function. For large βa2/U, upwind blocking is strong; the mountain wake can be unstable, filling the fluid with transient Rossby waves as in the numerical simulations of Polvani et al. For small values, βa2/U ≪ 1 classic lee Rossby waves with large wavelength compared to the mountain diameter are the dominant process. The mountain height, δh, relative to the mean fluid depth, H, affects these transitions as well. Simple lee Rossby waves occur only for such small heights, δh/h ≪ aβ/f, that the f/h contours are not greatly distorted by the mountain. Nongeostrophic dynamics are seen in inertial waves generated by geostrophic shear, and ducted by it, and also in a texture of finescale, inadvertent convection. Weakly damped circulations induced in a shallow-water numerical model on a sphere by a lone mountain in an initially simple westerly wind are also described. Here, with βa2/U ∼1, potential vorticity stirring and transient Rossby waves dominate, and drive zonal flow acceleration. Low-latitude critical layers, when present, exert strong control on the high-latitude waves, and with no restorative damping of the mean zonal flow, they migrate poleward toward the source of waves. While these experiments with homogeneous fluid are very simplified, the baroclinic atmosphere and ocean have many tall or equivalent barotropic eddy structures owing to the barotropization process of geostrophic turbulence.

scholarly journals Incidence and reflection of internal waves and wave-induced currents at a jump in buoyancy frequency

2014 ◽  
Vol 1 (1) ◽  
pp. 269-315
Author(s):  
J. P. McHugh
Keyword(s):  
Wave Packet ◽  
Internal Gravity Waves ◽  
Buoyancy Frequency ◽  
Mean Flow ◽  
Weakly Nonlinear ◽  
Transmitted Waves ◽  
The Mean ◽  
Induced Currents ◽  
Wave Induced ◽  
The Waves

Abstract. Weakly nonlinear internal gravity waves are treated in a two-layer fluid with a set of nonlinear Schrodinger equations. The layers have a sharp interface with a jump in buoyance frequency approximately modelling the tropopause. The waves are periodic in the horizontal but modulated in the vertical and Boussinesq flow is assumed. The equation governing the incident wave packet is directly coupled to the equation for the reflected packet, while the equation governing transmitted waves is only coupled at the interface. Solutions are obtained numerically. The results indicate that the waves create a mean flow that is strong near and underneath the interface, and discontinuous at the interface. Furthermore, the mean flow has an oscillatory component with a vertical wavelength that decreases as the wave packet interacts with the interface.

scholarly journals CURRENT CHARACTERISTICS IN THE PRESENCE OF NEAR-ORTHOGONAL WAVES

2012 ◽  
Vol 1 (33) ◽  
pp. 41
Author(s):  
Kian Yew Lim ◽  
Ole Secher Madsen ◽  
Hin Fatt Cheong
Keyword(s):  
Profile Analysis ◽  
Numerical Study ◽  
Flow Profile ◽  
Wave Direction ◽  
Mean Flow ◽  
Wave Basin ◽  
Current Interaction ◽  
The Mean ◽  
Current Flows ◽  
Wave Induced

An experimental study involving near-orthogonal wave-current interaction in a wave basin is reported in this paper. Due to previous shortcomings associated with 2D bottom configurations, i.e. occurrence of ripple-induced turning of flows close to the bed, the present experiments were conducted with the bottom covered by closely packed ceramic marbles (mean diameter of 1.25cm). Three types of flows were generated over this bottom: current-alone, wave-alone and combined wave-current flow. For current-alone and wave-current cases, the log-profile analysis was used to resolve the equivalent Nikuradse sand grain roughness, kn, while the energy dissipation method was used to estimate kn for wave-alone case. The results show that kn obtained for current- and wave-alone tests is roughly 2.2 times the diameter of the marbles. For orthogonal wave-current flows, the kn value, when used in combination with the Grant-Madsen (GM) model to reproduce the experimental apparent roughness, is found to be smaller than the measured current-alone and wave-alone kn. Similar under-prediction of bottom roughness is also observed when the GM model is compared with a numerical study, thus supporting the conjecture that when the current is weak compared to the waves, simple theoretical models like GM are not sufficiently sensitive to the angle of wave-current interaction. Experiments with currents at angles of 60° and 120° to the wave direction yield apparent roughness smaller than the 90° case, which is counter-intuitive since one would expect the mean flow to experience a stronger wave-induced turbulence when it is more aligned with the wave direction. This result indicates a possible contamination from wave-induced mass transport to the mean flow profile for non-orthogonal combined flow cases, and therefore highlights the need for other alternatives to the log-profile analysis when attempting to resolve kn from current velocity profiles from combined wave-current flows.

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