Eddy fluxes and topography in stratified quasi-geostrophic models

1999 ◽  
Vol 380 ◽  
pp. 59-80 ◽  
Author(s):  
WILLIAM J. MERRYFIELD ◽  
GREG HOLLOWAY
Keyword(s):  
Layer Thickness ◽  
Stream Function ◽  
Potential Vorticity ◽  
Lower Layer ◽  
Stratified Flow ◽  
Mean Flow ◽  
Random Forcing ◽  
Flow Over Topography ◽  
Eddy Fluxes ◽  
The Mean

Turbulent stratified flow over topography is studied using layered quasi-geostrophic models. Mean flows develop under random forcing, with lower-layer mean stream-function positively correlated with topography. When friction is sufficiently small, upper-layer mean flow is weaker than, but otherwise resembles, lower-layer mean flow. When lower-layer friction is larger, upper-layer mean flow reverses and can exceed lower-layer mean flow in strength. The mean interface between layers is domed over topographic elevations. Eddy fluxes of potential vorticity and layer thickness act in the sense of driving the flow toward higher entropy. Such behaviour contradicts usual eddy parameterizations, to which modifications are suggested.

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 On strongly nonlinear gravity waves in a vertically sheared atmosphere

2020 ◽  
Vol 6 (1) ◽  
pp. 63-74
Author(s):  
Mark Schlutow ◽  
Georg S. Voelker
Keyword(s):  
Gravity Waves ◽  
Lower Layer ◽  
Vertical Shear ◽  
Horizontal Wind ◽  
Necessary Condition ◽  
Mean Flow ◽  
Individual Layer ◽  
Strongly Nonlinear ◽  
The Mean ◽  
The Stability

Abstract We investigate strongly nonlinear stationary gravity waves which experience refraction due to a thin vertical shear layer of horizontal background wind. The velocity amplitude of the waves is of the same order of magnitude as the background flow and hence the self-induced mean flow alters the modulation properties to leading order. In this theoretical study, we show that the stability of such a refracted wave depends on the classical modulation stability criterion for each individual layer, above and below the shearing. Additionally, the stability is conditioned by novel instability criteria providing bounds on the mean-flow horizontal wind and the amplitude of the wave. A necessary condition for instability is that the mean-flow horizontal wind in the upper layer is stronger than the wind in the lower layer.

The structure of a highly decelerated axisymmetric turbulent boundary layer

2021 ◽  
Vol 929 ◽  
Author(s):  
N. Agastya Balantrapu ◽  
Christopher Hickling ◽  
W. Nathan Alexander ◽  
William Devenport
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Shear Layer ◽  
Layer Thickness ◽  
Boundary Layer Thickness ◽  
Large Scale ◽  
Convection Velocity ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean

Experiments were performed over a body of revolution at a length-based Reynolds number of 1.9 million. While the lateral curvature parameters are moderate ( $\delta /r_s < 2, r_s^+>500$ , where $\delta$ is the boundary layer thickness and r s is the radius of curvature), the pressure gradient is increasingly adverse ( $\beta _{C} \in [5 \text {--} 18]$ where $\beta_{C}$ is Clauser’s pressure gradient parameter), representative of vehicle-relevant conditions. The mean flow in the outer regions of this fully attached boundary layer displays some properties of a free-shear layer, with the mean-velocity and turbulence intensity profiles attaining self-similarity with the ‘embedded shear layer’ scaling (Schatzman & Thomas, J. Fluid Mech., vol. 815, 2017, pp. 592–642). Spectral analysis of the streamwise turbulence revealed that, as the mean flow decelerates, the large-scale motions energize across the boundary layer, growing proportionally with the boundary layer thickness. When scaled with the shear layer parameters, the distribution of the energy in the low-frequency region is approximately self-similar, emphasizing the role of the embedded shear layer in the large-scale motions. The correlation structure of the boundary layer is discussed at length to supply information towards the development of turbulence and aeroacoustic models. One major finding is that the estimation of integral turbulence length scales from single-point measurements, via Taylor's hypothesis, requires significant corrections to the convection velocity in the inner 50 % of the boundary layer. The apparent convection velocity (estimated from the ratio of integral length scale to the time scale), is approximately 40 % greater than the local mean velocity, suggesting the turbulence is convected much faster than previously thought. Closer to the wall even higher corrections are required.

scholarly journals The Emergence of Shallow Easterly Jets within QBO Westerlies

2018 ◽  
Vol 75 (1) ◽  
pp. 21-40 ◽  
Author(s):  
Peter Hitchcock ◽  
Peter H. Haynes ◽  
William J. Randel ◽  
Thomas Birner
Keyword(s):  
Potential Vorticity ◽  
General Circulation ◽  
Circulation Model ◽  
Shear Zones ◽  
Wave Activity ◽  
Mean Flow ◽  
Quasi Biennial Oscillation ◽  
Westerly Jet ◽  
Eddy Fluxes ◽  
Biennial Oscillation

A configuration of an idealized general circulation model has been obtained in which a deep, stratospheric, equatorial, westerly jet is established that is spontaneously and quasi-periodically disrupted by shallow easterly jets. Similar to the disruption of the quasi-biennial oscillation (QBO) observed in early 2016, meridional fluxes of wave activity are found to play a central role. The possible relevance of two feedback mechanisms to these disruptions is considered. The first involves the secondary circulation produced in the shear zones on the upper and lower flanks of the easterly jet. This is found to play a role in maintaining the aspect ratio of the emerging easterly jet. The second involves the organization of the eddy fluxes by the mean flow: the presence of a weak easterly anomaly within a tall, tropical, westerly jet is demonstrated to produce enhanced and highly focused wave activity fluxes that reinforce and strengthen the easterly anomalies. The eddies appear to be organized by the formation of strong potential vorticity gradients on the subtropical flanks of the easterly anomaly. Similar wave activity and potential vorticity structures are found in the ERA-Interim for the observed QBO disruption, indicating this second feedback was active then.

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.

Tidal Currents and Estuarine-Type Circulation in Johnstone Strait, British Columbia

1976 ◽  
Vol 33 (10) ◽  
pp. 2242-2264 ◽  
Author(s):  
Richard E. Thomson
Keyword(s):  
Lower Layer ◽  
Small Variation ◽  
Tidal Mixing ◽  
Mean Flow ◽  
Western Basin ◽  
Nonlinear Advection ◽  
Tide Height ◽  
The Mean ◽  
Stratified Estuary ◽  
Tidal Signal

Four months of current meter observations across the western basin of Johnstone Strait have been examined, with particular attention given to the mean flow and to variations at tidal frequencies. We show that the time-averaged motions are typical of a moderately stratified estuary driven by tidal mixing and nonlinear advection. Steady currents are nearly unidirectional at all depths with the net outflow in the upper layer essentially balanced by a net inflow in the lower layer to order 103 m3∙s−1. In addition, the relatively small variation in residual current speed is found to decrease with depth and to be associated mostly with the quasi-fortnightly tidal cycle. Near the surface the variance in the residual flow appears to be related to along-channel winds whose speeds and duration exceed 6 m∙s−1 and 24 h, respectively. Time-dependent motions are dominated by the tidal signal which is mixed, predominately semidiurnal. Maximum speeds of order 1 m∙s−1 are found at depth and are generally 1.5–1.7 times larger than in the upper layer. There is also a strong correlation between the tidal current speeds below 150-m depth and the local tide height lagged by 6 h. It is suggested that these large lower layer currents are associated with baroclinic motions being generated by the barotropic tide propagating over the rapidly shoaling bathymetry to the east of the observation region.

Ageostrophic instability of ocean currents

1982 ◽  
Vol 117 ◽  
pp. 343-377 ◽  
Author(s):  
R. W. Griffiths ◽  
Peter D. Killworth ◽  
Melvin E. Stern
Keyword(s):  
Potential Vorticity ◽  
Lower Layer ◽  
Rotation Period ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Inertial Effects ◽  
Mean Flow ◽  
Special Cases ◽  
Free Streamlines ◽  
A Current

We investigate the stability of gravity currents, in a rotating system, that are infinitely long and uniform in the direction of flow and for which the current depth vanishes on both sides of the flow. Thus, owing to the role of the Earth's rotation in restraining horizontal motions, the currents are bounded on both sides by free streamlines, or sharp density fronts. A model is used in which only one layer of fluid is dynamically important, with a second layer being infinitely deep and passive. The analysis includes the influence of vanishing layer depth and large inertial effects near the edges of the current, and shows that such currents are always unstable to linearized perturbations (except possibly in very special cases), even when there is no extremum (or gradient) in the potential vorticity profile. Hence the established Rayleigh condition for instability in quasi-geostrophic models, where inertial effects are assumed to be vanishingly small relative to Coriolis effects, does not apply. The instability does not depend upon the vorticity profile but instead relies upon a coupling of the two free streamlines. The waves permit the release of both kinetic and potential energy from the mean flow. They can have rapid growth rates, the e-folding time for waves on a current with zero potential vorticity, for example, being close to one-half of a rotation period. Though they are not discussed here, there are other unstable solutions to this same model when the potential vorticity varies monotonically across the stream, verifying that flows involving a sharp density front are much more likely to be unstable than flows with a small ratio of inertial to Coriolis forces.Experiments with a current of buoyant fluid at the free surface of a lower layer are described, and the observations are compared with the computed mode of maximum growth rate for a flow with a uniform potential vorticity. The current is observed to be always unstable, but, contrary to the predicted behaviour of the one-layer coupled mode, the dominant length scale of growing disturbances is independent of current width. On the other hand, the structure of the observed disturbances does vary: when the current is sufficiently narrow compared with the Rossby deformation radius (and the lower layer is deep) disturbances have the structure predicted by our one-layer model. The flow then breaks up into a chain of anticyclonic eddies. When the current is wide, unstable waves appear to grow independently on each edge of the current and, at large amplitude, form both anticyclonic and cyclonic eddies in the two-layer fluid. This behaviour is attributed to another unstable mode.

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.

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.

Overtransmission of Rossby Waves at a Lower-Layer Critical Latitude in the Two-Layer Model

2020 ◽  
Vol 77 (3) ◽  
pp. 859-870 ◽  
Author(s):  
Matthew T. Gliatto ◽  
Isaac M. Held
Keyword(s):  
Rossby Waves ◽  
Lower Layer ◽  
Scattering Problem ◽  
Lower Troposphere ◽  
Vertical Shear ◽  
Mean Flow ◽  
External Mode ◽  
The Mean ◽  
Critical Latitude ◽  
Pv Gradient

Abstract Rossby waves, propagating from the midlatitudes toward the tropics, are typically absorbed by critical latitudes (CLs) in the upper troposphere. However, these waves typically encounter CLs in the lower troposphere first. We study a two-layer linear scattering problem to examine the effects of lower CLs on these waves. We begin with a review of the simpler barotropic case to orient the reader. We then progress to the baroclinic case using a two-layer quasigeostrophic model in which there is vertical shear in the mean flow on which the waves propagate, and in which the incident wave is assumed to be an external-mode Rossby wave. We use linearized equations and add small damping to remove the critical-latitude singularities. We consider cases in which either there is only one CL, in the lower layer, or there are CLs in both layers, with the lower-layer CL encountered first. If there is only a CL in the lower layer, the wave’s response depends on the sign of the mean potential vorticity gradient at this lower-layer CL: if the PV gradient is positive, then the CL partially absorbs the wave, as in the barotropic case, while for a negative PV gradient, the CL is a wave emitter, and can potentially produce overreflection and/or overtransmission. Our numerical results indicate that overtransmission is by far the dominant response in these cases. When an upper-layer absorbing CL is encountered, following the lower-layer encounter, one can still see the signature of overtransmission at the lower-layer CL.

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