Journal articles on the topic 'Linear Vorticity Balance'
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1
Wang, Shuguang, and Fuqing Zhang. "Source of Gravity Waves within a Vortex-Dipole Jet Revealed by a Linear Model." Journal of the Atmospheric Sciences 67, no. 5 (May 1, 2010): 1438–55. http://dx.doi.org/10.1175/2010jas3327.1.
Abstract:
Abstract This study develops a linear numerical model to address the source mechanism of the gravity waves generated within a vortex dipole simulated in a fully nonlinear nonhydrostatic mesoscale model. The background flow for this linear model is obtained from potential vorticity inversion constrained by the nonlinear balance equation. The forcing imposed in the linear model is derived from an imbalance in the large-scale flow—that is, the forcing or imbalance in the vorticity, divergence, and thermodynamic equations, respectively. The response from the sum of these imbalanced forcings obtained from the linear dynamics shows well-defined gravity wave signals, which compare reasonably well in terms of location, phase, and amplitude with the gravity waves simulated in a fully nonlinear nonhydrostatic mesoscale model. It is found that the vorticity forcing, largely due to the advection of balanced relative vorticity, is the leading contributor to the gravity waves in the exit region of the vortex-dipole jet.
2
McKIVER, WILLIAM J., and DAVID G. DRITSCHEL. "Balance in non-hydrostatic rotating stratified turbulence." Journal of Fluid Mechanics 596 (January 17, 2008): 201–19. http://dx.doi.org/10.1017/s0022112007009421.
Abstract:
It is now well established that two distinct types of motion occur in geophysical turbulence: slow motions associated with potential vorticity advection and fast oscillations due to inertia–gravity waves (or acoustic waves). Many studies have theorized the existence of a flow for which the entire motion is controlled by the potential vorticity (or one ‘master variable’) – this is known as balance. In real geophysical flows, deviations from balance in the form of inertia–gravity waves or ‘imbalance’ have often been found to be small. Here we examine the extent to which balance holds in rotating stratified turbulence which is nearly balanced initially.Using the non-hydrostatic fluid dynamical equations under the Boussinesq approximation, we analyse properties of rotating stratified turbulence spanning a range of Rossby numbers (Ro≡|ζ|max/f) and the frequency ratios (c≡N/f) where ζ is the relative vertical vorticity, f is the Coriolis frequency and N is the buoyancy frequency. Using a recently introduced diagnostic procedure, called ‘optimal potential vorticity balance’, we extract the balanced part of the flow in the simulations and assess how the degree of imbalance varies with the above parameters.We also introduce a new and more efficient procedure, building upon a quasi-geostrophic scaling analysis of the complete non-hydrostatic equations. This ‘nonlinear quasi-geostrophic balance’ procedure expands the equations of motion to second order in Rossby number but retains the exact (unexpanded) definition of potential vorticity. This proves crucial for obtaining an accurate estimate of balanced motions. In the analysis of rotating stratified turbulence at Ro≲1 and N/f≫1, this procedure captures a significantly greater fraction of the underlying balance than standard (linear) quasi-geostrophic balance (which is based on the linearized equations about a state of rest). Nonlinear quasi-geostrophic balance also compares well with optimal potential vorticity balance, which captures the greatest fraction of the underlying balance overall.More fundamentally, the results of these analyses indicate that balance dominates in carefully initialized simulations of freely decaying rotating stratified turbulence up to O(1) Rossby numbers when N/f≫1. The fluid motion exhibits important quasi-geostrophic features with, in particular, typical height-to-width scale ratios remaining comparable to f/N.
3
Hakim, Gregory J. "A Probabilistic Theory for Balance Dynamics." Journal of the Atmospheric Sciences 65, no. 9 (September 1, 2008): 2949–60. http://dx.doi.org/10.1175/2007jas2499.1.
Abstract:
Abstract Balance dynamics are proposed in a probabilistic framework, assuming that the state variables and the master, or control, variables are random variables described by continuous probability density functions. Balance inversion, defined as recovering the state variables from the control variables, is achieved through Bayes’ theorem. Balance dynamics are defined by the propagation of the joint probability of the state and control variables through the Liouville equation. Assuming Gaussian statistics, balance inversion reduces to linear regression of the state variables onto the control variables, and assuming linear dynamics, balance dynamics reduces to a Kalman filter subject to perfect observations given by the control variables. Example solutions are given for an elliptical vortex in shallow water having unity Rossby and Froude numbers, which produce an outward-propagating pulse of inertia–gravity wave activity. Applying balance inversion to the potential vorticity reveals that, because potential vorticity and divergence share well-defined patterns of covariability, the inertia–gravity wave field is recovered in addition to the vortical field. Solutions for a probabilistic balance dynamics model applied to the elliptical vortex reveal smaller errors (“imbalance”) for height control compared to potential vorticity control. Important attributes of the probabilistic balance theory include quantification of the concept of balance manifold “fuzziness,” and clear state-independent definitions of balance and imbalance in terms of the range of the probabilistic inversion operators. Moreover, the theory provides a generalization of the notion of balance that may prove useful for problems involving moist physics, chemistry, and tropical circulations.
4
Gonzalez, Israel, Amaryllis Cotto, and Hugh E. Willoughby. "Synthesis of Vortex Rossby Waves. Part II: Vortex Motion and Waves in the Outer Waveguide." Journal of the Atmospheric Sciences 72, no. 10 (October 1, 2015): 3958–74. http://dx.doi.org/10.1175/jas-d-15-0005.1.
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Abstract Beta, the meridional gradient of planetary vorticity, causes tropical cyclones to propagate poleward and westward at approximately 2 m s−1. In a previous shallow-water linear model, the simulated vortex accelerated without limit, ostensibly because beta forced a free linear mode. In the analogous nonlinear model, wave–wave interaction limited the propagation speed. Subsequent work based upon the asymmetric balance (AB) approximation was unable to replicate the linear result. The present barotropic nondivergent model replicates the linear beta gyres as a streamfunction dipole with a uniform southeasterly ventilation flow across the vortex. The simulated storm accelerates to unphysical, but finite, speeds that are limited by vorticity filamentation. In the analogous nonlinear model, nonlinearly forced wavenumber-1 gyres have opposite phase to the linear gyres so that their ventilation flow counteracts advection by the linear gyres to limit the overall vortex speed to approximately 3 m s−1. A bounded mean vortex with zero circulation at large radius must contain an outer annulus of anticyclonic vorticity to satisfy the circulation theorem. The resulting positive mean vorticity gradient constitutes an outer waveguide that supports downstream-propagating, very-low-frequency vortex Rossby waves. It is confined between an inner critical radius where the waves are absorbed and an outer turning point where they are reflected. Vorticity filamentation at the critical radius limits the beta-drift acceleration. The original unlimited linear acceleration stemmed from too-weak dissipation caused by second-order diffusion applied to velocity components instead of vorticity. Fourth-order diffusion and no outer waveguide in the Rankine-like vortex of the AB simulations plausibly explain the different results.
5
Davies-Jones, Robert. "The Frontogenetical Forcing of Secondary Circulations. Part II: Properties of Q Vectors in Exact Linear Solutions." Journal of the Atmospheric Sciences 66, no. 2 (February 1, 2009): 244–60. http://dx.doi.org/10.1175/2008jas2803.1.
Abstract:
Abstract An exact solution of the primitive equations (PEs) and the corresponding exact solutions of the alternative balance (AB), geostrophic momentum (GM), and quasigeostrophic (QG) equations are presented. The PE solution illustrates how the temperature and horizontal vorticity fields evolve in a linear horizontal flow with constant deformation and vertical vorticity when the initial temperature field is also linear, as well as how ageostrophic circulations are produced. The other exact solutions show the errors produced by the various approximations and confirm that the AB solution is more accurate than the GM one and that the QG solution is almost always the most inexact. The utility of the Q vector and similar vectors is examined for each solution. The PE solution verifies that in a hyperbolic wind field (i) the isotherms eventually parallel the outflow axis, (ii) the ageostrophic circulation ultimately becomes normal to the outflow axis, (iii) thermal-wind balance becomes established in the direction normal to the isotherms, and (iv) the rotational component of the vector frontogenetical function decays.
6
LLEWELLYN SMITH, STEFAN G. "The motion of a non-isolated vortex on the beta-plane." Journal of Fluid Mechanics 346 (September 10, 1997): 149–79. http://dx.doi.org/10.1017/s0022112097006290.
Abstract:
The trajectory of a non-isolated monopole on the beta-plane is calculated as an asymptotic expansion in the ratio of the strength of the vortex to the beta-effect. The method of matched asymptotic expansions is used to solve the equations of motion in two regions of the flow: a near field where the beta-effect enters as a first-order forcing in relative vorticity, and a wave field in which the dominant balance is a linear one between the beta-effect and the rate of change of relative vorticity. The resulting trajectory is computed for Gaussian and Rankine vortices.
7
Thomas, Matthew D., Agatha M. De Boer, Helen L. Johnson, and David P. Stevens. "Spatial and Temporal Scales of Sverdrup Balance*." Journal of Physical Oceanography 44, no. 10 (October 1, 2014): 2644–60. http://dx.doi.org/10.1175/jpo-d-13-0192.1.
Abstract:
Abstract Sverdrup balance underlies much of the theory of ocean circulation and provides a potential tool for describing the interior ocean transport from only the wind stress. Using both a model state estimate and an eddy-permitting coupled climate model, this study assesses to what extent and over what spatial and temporal scales Sverdrup balance describes the meridional transport. The authors find that Sverdrup balance holds to first order in the interior subtropical ocean when considered at spatial scales greater than approximately 5°. Outside the subtropics, in western boundary currents and at short spatial scales, significant departures occur due to failures in both the assumptions that there is a level of no motion at some depth and that the vorticity equation is linear. Despite the ocean transport adjustment occurring on time scales consistent with the basin-crossing times for Rossby waves, as predicted by theory, Sverdrup balance gives a useful measure of the subtropical circulation after only a few years. This is because the interannual transport variability is small compared to the mean transports. The vorticity input to the deep ocean by the interaction between deep currents and topography is found to be very large in both models. These deep transports, however, are separated from upper-layer transports that are in Sverdrup balance when considered over large scales.
8
Samelson, R. M. "Time-Dependent Linear Theory for the Generation of Poleward Undercurrents on Eastern Boundaries." Journal of Physical Oceanography 47, no. 12 (December 2017): 3037–59. http://dx.doi.org/10.1175/jpo-d-17-0077.1.
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AbstractA time-dependent, inviscid, linear theory for the generation of poleward undercurrent flow under upwelling conditions along midlatitude ocean eastern boundaries is proposed. The theory relies on a conceptual separation of time scales between the rapid, coastal-trapped wave response to upwelling winds and the subsequent slow, interior, quasigeostrophic evolution. Solutions are obtained under idealized conditions in which the coastal boundary and the continental-slope topography are uniform alongshore, and the time-dependent wind-stress forcing is applied over a limited meridional range, uniform cross-shore, and directed alongshore. A time-dependent coastal boundary condition on the slow-time-scale interior flow, consisting of the low-frequency, geostrophically balanced sea surface height disturbance over the outer shelf, is obtained from consideration of the fast-time-scale, coastal-trapped response. A quasigeostrophic potential vorticity equation is then solved to determine the interior response to this time-dependent boundary condition. Under upwelling conditions, the results show the formation of a localized region of subsurface poleward flow over the upper continental slope that is qualitatively consistent in amplitude, location, and timing with observations of poleward undercurrents on eastern boundaries. Despite its origin as a sea surface height anomaly, the coastal-boundary condition drives a baroclinic planetary wave response, in which the poleward subsurface flow evolves in planetary vorticity balance with induced subsurface upwelling.
9
Shariff, Karim, and Paul S. Krueger. "Advective balance in pipe-formed vortex rings." Journal of Fluid Mechanics 836 (December 12, 2017): 773–96. http://dx.doi.org/10.1017/jfm.2017.814.
Abstract:
Vorticity distributions in axisymmetric vortex rings produced by a piston–pipe apparatus are numerically studied over a range of Reynolds numbers, $Re$, and stroke-to-diameter ratios, $L/D$. It is found that a state of advective balance, such that $\unicode[STIX]{x1D701}\equiv \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}/r\approx F(\unicode[STIX]{x1D713},t)$, is achieved within the region (called the vortex ring bubble) enclosed by the dividing streamline. Here $\unicode[STIX]{x1D701}\equiv \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}/r$ is the ratio of azimuthal vorticity to cylindrical radius, and $\unicode[STIX]{x1D713}$ is the Stokes streamfunction in the frame of the ring. Some, but not all, of the $Re$ dependence in the time evolution of $F(\unicode[STIX]{x1D713},t)$ can be captured by introducing a scaled time $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D708}t$, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity. When $\unicode[STIX]{x1D708}t/D^{2}\gtrsim 0.02$, the shape of $F(\unicode[STIX]{x1D713})$ is dominated by the linear-in-$\unicode[STIX]{x1D713}$ component, the coefficient of the quadratic term being an order of magnitude smaller. An important feature is that, as the dividing streamline ($\unicode[STIX]{x1D713}=0$) is approached, $F(\unicode[STIX]{x1D713})$ tends to a non-zero intercept which exhibits an extra $Re$ dependence. This and other features are explained by a simple toy model consisting of the one-dimensional cylindrical diffusion equation. The key ingredient in the model responsible for the extra $Re$ dependence is a Robin-type boundary condition, similar to Newton’s law of cooling, that accounts for the edge layer at the dividing streamline.
10
Goldstein, M. E., and Lennart S. Hultgren. "Nonlinear spatial evolution of an externally excited instability wave in a free shear layer." Journal of Fluid Mechanics 197 (December 1988): 295–330. http://dx.doi.org/10.1017/s002211208800326x.
Abstract:
We consider a disturbance that evolves from a strictly linear finite-growth-rate instability wave, with nonlinear effects first becoming important in the critical layer. The local Reynolds number is assumed to be just small enough so that the spatial-evolution, nonlinear-convection, and viscous-diffusion terms are of the same order of magnitude in the interactive critical-layer vorticity equation. The numerical results show that viscous effects eventually become important even when the viscosity is very small due to continually decreasing scales generated by the nonlinear effects. The vorticity distribution diffuses into a more regular pattern vis-a-vis the inviscid case, and the instability-wave growth ultimately becomes algebraic. This leads to a new dominant balance between linear- and nonlinear-convection terms and an equilibrium critical layer of the Benney & Bergeron (1969) type begins to emerge, but the detailed flow field, which has variable vorticity within the cat's-eye boundary, turns out to be somewhat different from theirs. The solution to this rescaled problem is compared with the numerical results and is then used to infer the scaling for the next stage of evolution of the flow. The instability-wave growth is simultaneously affected by mean-flow divergence and nonlinear critical-layer effects in this latter stage of development and is eventually converted to decay. The neutral stability point is the same as in the corresponding linear case, however.
11
Lien, Ren-Chieh, and Thomas B. Sanford. "Small-Scale Potential Vorticity in the Upper-Ocean Thermocline." Journal of Physical Oceanography 49, no. 7 (July 2019): 1845–72. http://dx.doi.org/10.1175/jpo-d-18-0052.1.
Abstract:
AbstractTwenty Electromagnetic Autonomous Profiling Explorer (EM-APEX) floats in the upper-ocean thermocline of the summer Sargasso Sea observed the temporal and vertical variations of Ertel potential vorticity (PV) at 7–70-m vertical scale, averaged over O(4–8)-km horizontal scale. PV is dominated by its linear components—vertical vorticity and vortex stretching, each with an rms value of ~0.15f. In the internal wave frequency band, they are coherent and in phase, as expected for linear internal waves. Packets of strong, >0.2f, vertical vorticity and vortex stretching balance closely with a small net rms PV. The PV spectrum peaks at the highest resolvable vertical wavenumber, ~0.1 cpm. The PV frequency spectrum has a red spectral shape, a −1 spectral slope in the internal wave frequency band, and a small peak at the inertial frequency. PV measured at near-inertial frequencies is partially attributed to the non-Lagrangian nature of float measurements. Measurement errors and the vortical mode also contribute to PV in the internal wave frequency band. The vortical mode Burger number, computed using time rates of change of vertical vorticity and vortex stretching, is 0.2–0.4, implying a horizontal kinetic energy to available potential energy ratio of ~0.1. The vortical mode energy frequency spectrum is 1–2 decades less than the observed energy spectrum. Vortical mode energy is likely underestimated because its energy at vertical scales > 70 m was not measured. The vortical mode to total energy ratio increases with vertical wavenumber, implying its importance at small vertical scales.
12
McWilliams, James C., Jonathan Gula, M. Jeroen Molemaker, Lionel Renault, and Alexander F. Shchepetkin. "Filament Frontogenesis by Boundary Layer Turbulence." Journal of Physical Oceanography 45, no. 8 (August 2015): 1988–2005. http://dx.doi.org/10.1175/jpo-d-14-0211.1.
Abstract:
AbstractA submesoscale filament of dense water in the oceanic surface layer can undergo frontogenesis with a secondary circulation that has a surface horizontal convergence and downwelling in its center. This occurs either because of the mesoscale straining deformation or because of the surface boundary layer turbulence that causes vertical eddy momentum flux divergence or, more briefly, vertical momentum mixing. In the latter case the circulation approximately has a linear horizontal momentum balance among the baroclinic pressure gradient, Coriolis force, and vertical momentum mixing, that is, a turbulent thermal wind. The frontogenetic evolution induced by the turbulent mixing sharpens the transverse gradient of the longitudinal velocity (i.e., it increases the vertical vorticity) through convergent advection by the secondary circulation. In an approximate model based on the turbulent thermal wind, the central vorticity approaches a finite-time singularity, and in a more general hydrostatic model, the central vorticity and horizontal convergence are amplified by shrinking the transverse scale to near the model’s resolution limit within a short advective period on the order of a day.
13
Ménesguen, Claire, J. C. McWilliams, and M. J. Molemaker. "Ageostrophic instability in a rotating stratified interior jet." Journal of Fluid Mechanics 711 (September 28, 2012): 599–619. http://dx.doi.org/10.1017/jfm.2012.412.
Abstract:
AbstractOceanic large- and meso-scale flows are nearly balanced in forces between Earth’s rotation and density stratification effects (i.e. geostrophic, hydrostatic balance associated with small Rossby and Froude numbers). In this regime advective cross-scale interactions mostly drive energy toward larger scales (i.e. inverse cascade). However, viscous energy dissipation occurs at small scales. So how does the energy reservoir at larger scales leak toward small-scale dissipation to arrive at climate equilibrium? Here we solve the linear instability problem of a balanced flow in a rotating and continuously stratified fluid far away from any boundaries (i.e. an interior jet). The basic flow is unstable not only to geostrophic baroclinic and barotropic instabilities, but also to ageostrophic instabilities, leading to the growth of small-scale motions that we hypothesize are less constrained by geostrophic cascade behaviours in a nonlinear regime and thus could escape from the inverse energy cascade. This instability is investigated in the parameter regime of moderate Rossby and Froude numbers, below the well-known regimes of gravitational, centrifugal, and Kelvin–Helmholtz instability. The ageostrophic instability modes arise with increasing Rossby number through a near-degeneracy of two unstable modes with coincident phase speeds. The near-degeneracy occurs in the neighbourhood of an identified criterion for the non-integrability of the ‘isentropic balance equations’ (namely $A\ensuremath{-} S= 0$ with $A$ the absolute vertical vorticity and $S$ the horizontal strain rate associated with the basic flow), beyond which development of an unbalanced component of the flow is expected. These modes extract energy from the basic state with large vertical Reynolds stress work (unlike geostrophic instabilities) and act locally to modify the basic flow by reducing the isopycnal Ertel potential vorticity gradient near both its zero surface and its critical surface (phase speed equal to basic flow speed).
14
Durland, Theodore S., Joseph Pedlosky, and Michael A. Spall. "Response to a Steady Poleward Outflow. Part I: The Linear, Quasigeostrophic Problem." Journal of Physical Oceanography 39, no. 7 (July 1, 2009): 1541–50. http://dx.doi.org/10.1175/2008jpo3999.1.
Abstract:
Abstract The response of a zonal channel to a uniform, switched-on but subsequently steady poleward outflow is presented. An eastward coastal current with a Kelvin wave’s cross-shore structure is found to be generated instantly upon initiation of the outflow. The current is essentially in geostrophic balance everywhere except for the vicinity of the outflow channel mouth, where the streamlines must cross planetary vorticity contours to feed the current. The adjustment of this region generates a plume that propagates westward at Rossby wave speeds. The cross-shore structure of the plume varies with longitude, and at any given longitude it evolves with time. The authors show that the plume evolution can be understood both conceptually and quantitatively as the westward propagation of the Kelvin current’s meridional spectrum, with each spectral element propagating at its own Rossby wave group velocity.
15
OLBERS, DIRK, DANIEL BOROWSKI, CHRISTOPH VÖLKER, and JORG-OLAF WÖLFF. "The dynamical balance, transport and circulation of the Antarctic Circumpolar Current." Antarctic Science 16, no. 4 (November 30, 2004): 439–70. http://dx.doi.org/10.1017/s0954102004002251.
Abstract:
The physical elements of the circulation of the Antarctic Circumpolar Current (ACC) are reviewed. A picture of the circulation is sketched by means of recent observations from the WOCE decade. We present and discuss the role of forcing functions (wind stress, surface buoyancy flux) in the dynamical balance of the flow and in the meridional circulation and study their relation to the ACC transport. The physics of form stress at tilted isopycnals and at the ocean bottom are elucidated as central mechanisms in the momentum balance. We explain the failure of the Sverdrup balance in the ACC circulation and highlight the role of geostrophic contours in the balance of vorticity. Emphasis is on the interrelation of the zonal momentum balance and the meridional circulation, the importance of diapycnal mixing and eddy processes. Finally, new model concepts are described: a model of the ACC transport dependence on wind stress and buoyancy flux, based on linear wave theory; and a model of the meridional overturning and the mean density structure of the Southern Ocean, based on zonally averaged dynamics and thermodynamics with eddy parametrization.
16
Lu, Jian, Lantao Sun, Yutian Wu, and Gang Chen. "The Role of Subtropical Irreversible PV Mixing in the Zonal Mean Circulation Response to Global Warming–Like Thermal Forcing." Journal of Climate 27, no. 6 (March 13, 2014): 2297–316. http://dx.doi.org/10.1175/jcli-d-13-00372.1.
Abstract:
Abstract The atmospheric circulation response to the global warming–like tropical upper tropospheric heating is revisited using a dry atmospheric general circulation model (AGCM) in light of new diagnostics based on the concept of finite-amplitude wave activity (FAWA) on equivalent latitude. For a given tropical heating profile, the linear Wentzel–Kramers–Brillouin (WKB) wave refraction analysis sometimes gives a very different and even opposite prediction of the eddy momentum flux response to that of the actual full model simulation, exposing the limitation of the traditional linear approach in understanding the full dynamics of the atmospheric response under global warming. The implementation of the FAWA diagnostics reveals that in response to the upper tropospheric heating, effective diffusivity—a measure of the mixing efficiency—increases and advances upward and poleward in the subtropics and the resultant enhancement and the poleward encroachment of eddy potential vorticity mixing leads to a poleward displaced potential vorticity (PV) gradient peak in the upper troposphere. The anomalous eddy PV flux, in balance with the PV dissipation, gives rise to a poleward shift in the eddy-driven jet and eddy-driven mean meridional circulation. Sensitivity experiments show that these irreversible dissipation processes in the upper troposphere are robust, regardless of the width of the tropical heating.
17
Woollings, Tim, Camille Li, Marie Drouard, Etienne Dunn-Sigouin, Karim A. Elmestekawy, Momme Hell, Brian Hoskins, Cheikh Mbengue, Matthew Patterson, and Thomas Spengler. "The role of Rossby waves in polar weather and climate." Weather and Climate Dynamics 4, no. 1 (January 13, 2023): 61–80. http://dx.doi.org/10.5194/wcd-4-61-2023.
Abstract:
Abstract. Recent Arctic warming has fuelled interest in the weather and climate of the polar regions and how this interacts with lower latitudes. Several interesting theories of polar-midlatitude linkages involve Rossby wave propagation as a key process even though the meridional gradient in planetary vorticity, crucial for these waves, is weak at high latitudes. Here we review some basic theory and suggest that Rossby waves can indeed explain some features of polar variability, especially when relative vorticity gradients are present. We suggest that large-scale polar flow can be conceptualised as a mix of geostrophic turbulence and Rossby wave propagation, as in the midlatitudes, but with the balance tipped further in favour of turbulent flow. Hence, isolated vortices often dominate but some wavelike features remain. As an example, quasi-stationary or weakly westward-propagating subpolar anomalies emerge from statistical analysis of observed data, and these are consistent with some role for wave propagation. The noted persistence of polar cyclones and anticyclones is attributed in part to the weakened effects of wave dispersion, the mechanism responsible for the decay of midlatitude anomalies in downstream development. We also suggest that the vortex-dominated nature of polar dynamics encourages the emergence of annular mode structures in principal component analyses of extratropical circulation. Finally, we consider how Rossby waves may be triggered from high latitudes. The linear mechanisms known to balance localised heating at lower latitudes are shown to be less efficient in the polar regions. Instead, we suggest the direct response to sea ice loss often manifests as a heat low, with radiative cooling balancing the heating. If the relative vorticity gradient is favourable this does have the potential to trigger a Rossby wave response, although this will often be weak compared to waves forced from lower latitudes.
18
Mestel, A. J. "On the stability of high-Reynolds-number flows with closed streamlines." Journal of Fluid Mechanics 200 (March 1989): 19–38. http://dx.doi.org/10.1017/s0022112089000546.
Abstract:
In steady, two-dimensional, inertia-dominated flows it is well known that the vorticity is constant along the streamlines, which, in a bounded domain, are necessarily closed. For inviscid flows, the variation of vorticity across the streamlines is arbitrary, while for forced, weakly dissipitative flows, it is determined by the balance between viscous diffusion and the forcing. This paper discusses the linear stability of flows of this type to two-dimensional disturbances. Arnol'd's stability theorems are discussed. An alternative functional to Arnol'd's is found, which gives the same stability criteria and which permits a representation of the problem in terms of a Schrödinger equation. Conditions for stability are derived from this functional. In particular it is shown that total flow reversals are potentially unstable. The results are illustrated with respect to the geometrically simple case when the streamlines are circular and the forcing is due to a rotating magnetic field, for which case the stability regions are calculated as a function of two parameters. It is shown that the entire theory, including Arnol'd's theorems, applies also to poloidal axisymmetric flows.
19
Swaters, Gordon E. "On the baroclinic instability of cold-core coupled density fronts on a sloping continental shelf." Journal of Fluid Mechanics 224 (March 1991): 361–82. http://dx.doi.org/10.1017/s0022112091001799.
Abstract:
A theory is presented to describe the linear baroclinic instability of coupled density fronts on a sloping continental shelf. The new baroclinic model equations used to study the instability process correspond to an ‘intermediate lengthscale’ dynamical balance. Specifically, the frontal dynamics, while geostrophic, is not quasigeostrophic because frontal height deflections are not small in comparison with the frontal scale height. The evolution of the frontal height is strongly coupled to the geostrophic pressure in the surrounding slope water through the hydrostatic balance which expresses the continuity of the dynamic pressures across the frontal interface. The deeper surrounding slope water evolves quasi-geostrophically and is coupled to the front by baroclinic vortex-tube stretching/compression associated with the perturbed density front (allowing the release of mean frontal potential energy) and the topographic vorticity gradient associated with the sloping bottom. It is shown that the baroclinic stability characteristics are principally determined by a so-called non-dimensional interaction parameter (denoted μ) which physically measures the ratio of the destabilizing baroclinic vortex-tube stretching/compression to the stabilizing topographic vorticity gradient. For a given along-front mode wavenumber it is shown that a minimum μ is required for instability. Several other general stability results are presented: necessary conditions for instability, growth rate and phase speed bounds, the existence of a high wavenumber cutoff, and a semicircle theorem for the unstable modes. The linear stability equations are solved exactly for a parabolic coupled density front and a detailed description of the spatial and temporal characteristics of the instabilities is given. For physically realistic parameter values the instabilities are manifested as amplifying topographic Rossby waves in the slope water, and on the density front the unstable perturbations take the form of amplifying anticyclones which have maximum amplitude on the offshore side.
20
Lo, Kin-Hing, Rengarajan Sriram, and Konstantinos Kontis. "Wake Flow Characteristics over an Articulated Lorry Model with/without AC-DBD Plasma Actuation." Applied Sciences 9, no. 12 (June 14, 2019): 2426. http://dx.doi.org/10.3390/app9122426.
Abstract:
The wake flow characteristics of a 1:20 scale articulated lorry model with a linear Alternate Current Dielectric Barrier Discharge (AC-DBD) plasma actuation implemented was experimentally investigated. Time-averaged velocity, turbulence, and vorticity information along the centreline of the model were constructed using a two-component particle image velocimetry technique. In addition, force balance was used to measure the time-average drag force acting on the model with and without the use of AC-DBD plasma actuation. In general, the AC-DBD plasma actuation showed negligible effect in changing the drag coefficient of the test model. Moreover, implementing the AC-DBD plasma actuation around the rear end of the trailer model could neither alter the size nor the reverse flow velocity in the wake region. In contrast, the AC-DBD plasma actuation increased the levels of fluctuation in the flow turbulence kinetic energy and vorticity but showed no observable effect to alter the frequency response of the flow in the wake region. It is deduced that the use of AC-DBD plasma actuation indeed generated no flow control effect at the rear end of an articulated lorry trailer.
21
Crowe, Matthew N., and John R. Taylor. "Baroclinic Instability with a Simple Model for Vertical Mixing." Journal of Physical Oceanography 49, no. 12 (December 2019): 3273–300. http://dx.doi.org/10.1175/jpo-d-18-0270.1.
Abstract:
AbstractHere, we examine baroclinic instability in the presence of vertical mixing in an idealized setting. Specifically, we use a simple model for vertical mixing of momentum and buoyancy and expand the buoyancy and vorticity in a series for small Rossby numbers. A flow in subinertial mixed layer (SML) balance (see the study by Young in 1994) exhibits a normal mode linear instability, which is studied here using linear stability analysis and numerical simulations. The most unstable modes grow by converting potential energy associated with the basic state into kinetic energy of the growing perturbations. However, unlike the inviscid Eady problem, the dominant energy balance is between the buoyancy flux and the energy dissipated by vertical mixing. Vertical mixing reduces the growth rate and changes the orientation of the most unstable modes with respect to the front. By comparing with numerical simulations, we find that the predicted scale of the most unstable mode matches the simulations for small Rossby numbers while the growth rate and orientation agree for a broader range of parameters. A stability analysis of a basic state in SML balance using the inviscid QG equations shows that the angle of the unstable modes is controlled by the orientation of the SML flow, while stratification associated with an advection/diffusion balance controls the size of growing perturbations for small Ekman numbers and/or large Rossby numbers. These results imply that baroclinic instability can be inhibited by small-scale turbulence when the Ekman number is sufficiently large and might explain the lack of submesoscale eddies in observations and numerical models of the ocean surface mixed layer during summer.
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TAYLOR, JOHN R., and RAFFAELE FERRARI. "On the equilibration of a symmetrically unstable front via a secondary shear instability." Journal of Fluid Mechanics 622 (March 10, 2009): 103–13. http://dx.doi.org/10.1017/s0022112008005272.
Abstract:
The equilibration of a symmetrically unstable density front is examined using linear stability theory and nonlinear numerical simulations. The initial state, chosen to approximate conditions in the surface ocean, consists of a weakly stratified mixed layer above a strongly stratified thermocline. Each layer has a uniform horizontal density gradient and a velocity field in thermal wind balance. The potential vorticity (PV) in the mixed layer is negative, indicating conditions favourable for symmetric instability. Once the instability reaches finite amplitude, a secondary Kelvin–Helmholtz (K-H) instability forms. Linear theory accurately predicts the time and the wavenumber at which the secondary instability occurs. Following the secondary instability, small-scale turbulence injects positive PV into the mixed layer from the thermocline and from the upper boundary, resulting in a rapid equilibration of the flow as the PV is brought back to zero. While the physical parameters used in this study correspond to typical conditions near a surface ocean front, many of the conclusions apply to symmetric instabilities in the atmosphere.
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Chen, Chih-Chieh, Gregory J. Hakim, and Dale R. Durran. "Transient Mountain Waves and Their Interaction with Large Scales." Journal of the Atmospheric Sciences 64, no. 7 (July 1, 2007): 2378–400. http://dx.doi.org/10.1175/jas3972.1.
Abstract:
Abstract The impact of transient mountain waves on a large-scale flow is examined through idealized numerical simulations of the passage of a time-evolving synoptic-scale jet over an isolated 3D mountain. Both the global momentum budget and the spatial flow response are examined to illustrate the impact of transient mountain waves on the large-scale flow. Additionally, aspects of the spatial response are quantified by potential vorticity inversion. Nearly linear cases exhibit a weak loss of domain-averaged absolute momentum despite the absence of wave breaking. This transient effect occurs because, over the time period of the large-scale flow, the momentum flux through the top boundary does not balance the surface pressure drag. Moreover, an adiabatic spatial redistribution of momentum is observed in these cases, which results in an increase (decrease) of zonally averaged zonal momentum south (north) of the mountain. For highly nonlinear cases, the zonally averaged momentum field shows a region of flow deceleration downstream of the mountain, flanked by broader regions of weak flow acceleration. Cancellation between the accelerating and decelerating regions results in weak fluctuations in the volume-averaged zonal momentum, suggesting that the mountain-induced circulations are primarily redistributing momentum. Potential vorticity anomalies develop in a region of wave breaking near the mountain, and induce local regions of flow acceleration and deceleration that alter the large-scale flow. A “perfect” conventional gravity wave–drag parameterization is implemented on a coarser domain not having a mountain, forced by the momentum flux distribution from the fully nonlinear simulation. This parameterization scheme produces a much weaker spatial response in the momentum field and it fails to produce enough flow deceleration near the 20 m s−1 jet. These results suggest that the potential vorticity sources attributable to the gravity wave–drag parameterization have a controlling effect on the longtime downstream influence of the mountain.
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Jones, M. S., L. Le Baron, and T. J. Pedley. "Biflagellate gyrotaxis in a shear flow." Journal of Fluid Mechanics 281 (December 25, 1994): 137–58. http://dx.doi.org/10.1017/s002211209400306x.
Abstract:
A flagellated, bottom-heavy micro-organism's swimming direction in a shear flow is determined from a balance between the gravitational and viscous torques (gyrotaxis). Hitherto, the cell has been assumed to be a spheroid and the flagella have been neglected. Here we use resistive-force theory to calculate both the magnitude and the direction of a biflagellate cell's swimming velocity and angular velocity relative to the fluid when there is an arbitrary linear flow far from the cell. We present an idealized model for the flagellar beat but, in calculating the velocity of the fluid relative to an element of a flagellum, the presence of the cell body is not neglected. Results are given for the case of a spherical cell body whose flagella beat in a vertical plane, when the ambient linear flow is in the same vertical plane. Results show that resistive-force theory can be used for organisms where the cell body has significant effect on the flow past the flagella and that the viscous torque on the flagella is a significant term in the torque balance equations. A model is presented for the calculation of a cell's velocity and angular velocity in a shear flow which is valid up to high magnitudes of rate of strain or vorticity. The main application of the results will be to modify a recent continuum model for suspensions of gyrotactic micro-organisms (Pedley & Kessler 1990).
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Ripa, P. "On improving a one-layer ocean model with thermodynamics." Journal of Fluid Mechanics 303 (November 25, 1995): 169–201. http://dx.doi.org/10.1017/s0022112095004228.
Abstract:
A popular method used to incorporate thermodynamic processes in a shallow water model (e.g. one used to study the upper layer of the ocean) is to allow for density variations in time and horizontal position, but keep all dynamical fields as depth independent. This is achieved by replacing the horizontal pressure gradient by its vertical average. These models have limitations, for instance they cannot represent the ‘thermal wind’ balance (between the horizontal density gradient and the vertical shear of the velocity) which dominates at low frequencies. A new model is now proposed which uses velocity and density fields varying linearly with depth, with coefficients that are functions of horizontal position and time. This model can explicitly represent the thermal wind balance, but its use is not restricted to low- frequency dynamics.Volume, mass, buoyancy variance, energy and momentum are conserved in the new model. Furthermore, these integrals of motion have the same dependence on the dynamical fields as the exact (continuously stratified) case. The evolution of the three components of the absolute vorticity field are correctly represented. Conservation of density–potential vorticity is not fulfilled, though, owing to artificial removal of the vertical curvature of the velocity field.The integrals of motion are used to construct a ‘free energy’ [Escr ]f, which is quadratic to the lowest order in the deviation from a steady state with (at most) a uniform velocity field. [Escr ]f is positive definite, and therefore the free evolution of the system cannot lead to an ‘explosion’ of the dynamical fields. (This is not the case if the velocity shear and/or the density vertical gradient is excluded in the model, which results in a non-negative definite free energy.)In a model with one active layer, linear waves on top of a steady state with no currents are, to a very good approximation, those of the first two vertical modes of the continuously stratified model. These are the familiar geophysical gravity and vortical waves (e.g. Poincaré, Rossby, and coastal Kelvin waves at mid-latitudes, equatorial waves, etc.).Finally, baroclinic instability is well represented in the new model. For long perturbations (wavelengths of the order of the deformation radius of the first mode) the agreement with more precise calculations is excellent. On the other hand, the comparison with the eigenvalues of Eady's problem (which corresponds to wavelengths of the order of the deformation radius of the second mode) shows differences of the order of 40%. Nevertheless, the new model does have a high-wavenumber cutoff, even though it is constrained to linear profiles in depth and therefore cannot reproduce the exponential trapping of Eady's problem eigensolutions.In sum, the integrals of motion, vorticity dynamics, free waves and baroclinic instability results all give confidence in the new model. Its main novelty, however, lies in the ability to incorporate thermodynamic processes.
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Roulstone, I., B. Banos, J. D. Gibbon, and V. N. Roubtsov. "A geometric interpretation of coherent structures in Navier–Stokes flows." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2107 (April 2009): 2015–21. http://dx.doi.org/10.1098/rspa.2008.0483.
Abstract:
The pressure in the incompressible three-dimensional Navier–Stokes and Euler equations is governed by Poisson's equation: this equation is studied using the geometry of three-forms in six dimensions. By studying the linear algebra of the vector space of three-forms Λ 3 W * where W is a six-dimensional real vector space, we relate the characterization of non-degenerate elements of Λ 3 W * to the sign of the Laplacian of the pressure—and hence to the balance between the vorticity and the rate of strain. When the Laplacian of the pressure, Δ p , satisfies Δ p >0, the three-form associated with Poisson's equation is the real part of a decomposable complex form and an almost-complex structure can be identified. When Δ p <0, a real decomposable structure is identified. These results are discussed in the context of coherent structures in turbulence.
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Zurita-Gotor, Pablo. "The Impact of Divergence Tilt and Meridional Flow for Cross-Equatorial Eddy Momentum Transport in Gill-Like Settings." Journal of the Atmospheric Sciences 77, no. 6 (May 15, 2020): 1933–53. http://dx.doi.org/10.1175/jas-d-19-0158.1.
Abstract:
Abstract This work investigates the sensitivity of the cross-equatorial eddy momentum flux and its rotational and divergent components to Hadley cell strength in simple variants of the Gill problem. An expression is derived linking the divergent momentum flux to the mean meridional wavenumber weighted by the spectrum of divergent eddy kinetic energy, supporting the relation between divergence phase tilt and momentum flux suggested by a previous study. Newtonian cooling makes the divergence tilt eastward moving away from the equator as observed, but this tilt is also sensitive to the Hadley cell. As the divergence tilt is enhanced in the downstream direction of the flow, wave propagation increases along that direction when the Hadley cell strengthens. The meridional flow also plays a second, important role for cross-equatorial propagation. With no Hadley cell, inviscid Sverdrup balance requires perfect compensation between the divergent and rotational momentum fluxes at the equator. The model can only produce cross-equatorial propagation when Sverdrup balance is violated, which in the linear, nearly inviscid limit requires vorticity advection by the mean flow. As the Hadley cell attenuates the geopotential tilt imparted by the divergent forcing, the compensation by the rotational momentum flux is reduced. The linear model can reproduce reasonably well previous nonlinear results by Kraucunas and Hartmann when linearized about their zonal-mean climatologies. The sensitivity of the cross-equatorial momentum fluxes to Hadley cell strength in these solutions is dominated by changes in the divergent flux and consistent with diagnosed changes in the divergence tilt.
28
Yuan, Li, and Kevin Hamilton. "Equilibrium dynamics in a forced-dissipative f-plane shallow-water system." Journal of Fluid Mechanics 280 (December 10, 1994): 369–94. http://dx.doi.org/10.1017/s0022112094002971.
Abstract:
The equilibrium dynamics in a homogeneous forced-dissipative f-plane shallow-water system is investigated through numerical simulations. In addition to classical two-dimensional turbulence, inertio-gravity waves also exist in this system. The dynamics is examined by decomposing the full flow field into a dynamically balanced potential-vortical component and a residual ‘free’ component. Here the potential-vortical component is defined as part of the flow that satisfies the gradient-wind balance equation and that contains all the linear potential vorticity of the system. The residual component is found to behave very nearly as linear inertio-gravity waves. The forcing employed is a mass and momentum source balanced so that only the large-scale potential-vortical component modes are directly excited. The dissipation is provided by a linear relaxation applied to the large scales and by an eighth-order linear hyperdiffusion. The statistical properties of the potential-vortical component in the fully developed flow were found to be very similar to those of classical two-dimensional turbulence. In particular, the energy spectrum of the potential-vortical component at scales smaller than the forcing is close to the ∼ k−3 expected for a purely two-dimensional system. Detailed analysis shows that the downscale enstrophy cascade into any wavenumber is dominated by very elongated triads involving interactions with large scales. Although not directly forced, a substantial amount of energy is found in the inertio-gravity modes and interactions among inertio-gravity modes are principally responsible for transferring energy to the small scales. The contribution of the inertio-gravity modes to the flow leads to a shallow tail at the high-wavenumber end of the total energy spectrum. For parameters roughly appropriate for the midlatitude atmosphere (notably Rossby number ∼ 0.5), the break between the roughly ∼ k−3 regime and this shallower regime occurs at scales of a few hundred km. This is similar to the observed mesoscale regime in the atmosphere. The nonlinear interactions among the inertio-gravity modes are extremely broadband in spectral space. The implications of this result for the subgrid-scale closure in the shallow-water model are discussed.
29
Kushner, Paul J., and Theodore G. Shepherd. "Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics. Part 1. Pseudomomentum-based theory." Journal of Fluid Mechanics 290 (May 10, 1995): 67–104. http://dx.doi.org/10.1017/s0022112095002424.
Abstract:
There exists a well-developed body of theory based on quasi-geostrophic (QG) dynamics that is central to our present understanding of large-scale atmospheric and oceanic dynamics. An important question is the extent to which this body of theory may generalize to more accurate dynamical models. As a first step in this process, we here generalize a set of theoretical results, concerning the evolution of disturbances to prescribed basic states, to semi-geostrophic (SG) dynamics. SG dynamics, like QG dynamics, is a Hamiltonian balanced model whose evolution is described by the material conservation of potential vorticity, together with an invertibility principle relating the potential vorticity to the advecting fields. SG dynamics has features that make it a good prototype for balanced models that are more accurate than QG dynamics.In the first part of this two-part study, we derive a pseudomomentum invariant for the SG equations, and use it to obtain: (i) linear and nonlinear generalized Charney–Stern theorems for disturbances to parallel flows; (ii) a finite-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit; and (iii) a wave-mean-flow interaction theorem consisting of generalized Eliassen–Palm flux diagnostics, an elliptic equation for the stream-function tendency, and a non-acceleration theorem. All these results are analogous to their QG forms.The pseudomomentum invariant – a conserved second-order disturbance quantity that is associated with zonal symmetry – is constructed using a variational principle in a similar manner to the QG calculations. Such an approach is possible when the equations of motion under the geostrophic momentum approximation are transformed to isentropic and geostrophic coordinates, in which the ageostrophic advection terms are no longer explicit. Symmetry-related wave-activity invariants such as the pseudomomentum then arise naturally from the Hamiltonian structure of the SG equations. We avoid use of the so-called ‘massless layer’ approach to the modelling of isentropic gradients at the lower boundary, preferring instead to incorporate explicitly those boundary contributions into the wave-activity and stability results. This makes the analogy with QG dynamics most transparent.This paper treats the f-plane Boussinesq form of SG dynamics, and its recent extension to β-plane, compressible flow by Magnusdottir & Schubert. In the limit of small Rossby number, the results reduce to their respective QG forms. Novel features particular to SG dynamics include apparently unnoticed lateral boundary stability criteria in (i), and the necessity of including additional zonal-mean eddy correlation terms besides the zonal-mean potential vorticity fluxes in the wave-mean-flow balance in (iii).In the companion paper, wave-activity conservation laws and stability theorems based on the SG form of the pseudoenergy are presented.
30
Itano, Toshihisa, and Akira Kasahara. "Effect of Top and Bottom Boundary Conditions on Symmetric Instability under Full-Component Coriolis Force." Journal of the Atmospheric Sciences 68, no. 11 (November 1, 2011): 2771–82. http://dx.doi.org/10.1175/jas-d-11-09.1.
Abstract:
Abstract The linear stability of a zonal flow confined in a domain within horizontal top and bottom boundaries is examined under full consideration of the Coriolis force. The basic zonal flow is assumed to be in thermal wind balance with the density field and to be sheared in both vertical and horizontal directions under statically and inertially stable conditions. By imposing top and bottom boundary conditions in this framework, the number of wave modes increases to four, instead of two in an unbounded domain, as already reported in studies on internal gravity waves. The four modes are classified into two pairs of high- and low-frequency modes: the high modes are superinertial and the low modes are subinertial. The discriminant of symmetric instability is nevertheless determined by the sign of the potential vorticity of the basic zonal flow, as in the case of an unbounded domain. The solutions satisfying the top and bottom boundary conditions are interpreted as the superposition of incident and reflected waves, revealing that the neutral solutions consist of two neutral plane waves with oppositely directed vertical group velocities. This may explain why the properties of wave behavior, such as the instability criteria, remain the same in both the bounded and unbounded domains, although the manifestation of wave activity, such as the order of dispersion relation, is quite different in the two cases. Furthermore, the slope of the constant momentum surface, the slope of the isopycnic surface including the nontraditional effect of the Coriolis force, and the ratio between the frequencies of gravity and inertial waves form an essential set of parameters for symmetric motion. The combination of these dimensionless quantities determines the fundamental nature of symmetric motions, such as stability, regardless of boundary conditions with and without the horizontal component of the planetary vorticity.
31
Shyu, Jinn-Hwa, and O. M. Phillips. "The blockage of gravity and capillary waves by longer waves and currents." Journal of Fluid Mechanics 217 (August 1990): 115–41. http://dx.doi.org/10.1017/s0022112090000659.
Abstract:
Surface waves superimposed upon a larger-scale flow are blocked at the points where the group velocities balance the convection by the larger-scale flow. Two types of blockage, capillary and gravity, are investigated by using a new multiple-scale technique, in which the short waves are treated linearly and the underlying larger-scale flows are assumed steady but can have a considerably curved surface and uniform vorticity. The technique first provides a uniformly valid second-order ordinary differential equation, from which a consistent uniform asymptotic solution can readily be obtained by using a treatment suggested by the result of Smith (1975) who described the phenomenon of gravity blockage in an unsteady current with finite depth.The corresponding WKBJ solution is also derived as a consistent asymptotic expansion of the uniform solution, which is valid at points away from the blockage point. This solution is obviously represented by a linear combination of the incident and reflected waves, and their amplitudes take explicit forms so that it can be shown that even with a significantly varied effective gravity g’ and constant vorticity, wave action will remain conserved for each wave. Furthermore, from the relative amplitudes of the incident and reflected waves, we clearly demonstrate that the action fluxes carried by the two waves towards and away from the blockage point are equal within the present approximation.The blockage of gravity–capillary waves can occur at the forward slopes of a finite-amplitude dominant wave as suggested by Phillips (1981). The results show that the blocked waves will be reflected as extremely short capillaries and then dissipated rapidly by viscosity. Therefore, for a fixed dominant wave, all wavelets shorter than a limiting wavelength will be suppressed by this process. The minimum wavelengths coexisting with the long waves of various wavelengths and slopes are estimated.
32
Ren, Shuzhan, and Dehai Luo. "Coupling of Wind and Potential Temperature in an Ekman Model in the Stratified Atmospheric Boundary Layer." Journal of the Atmospheric Sciences 79, no. 3 (March 2022): 649–62. http://dx.doi.org/10.1175/jas-d-21-0049.1.
Abstract:
Abstract A linear Ekman model in the stratified atmospheric boundary layer (ABL) is proposed based on the steady-state version of the linearized three-dimensional primitive equations with the inclusion of the vertical diffusivity. Due to the inclusion of the potential temperature equation and hydrostatic equation, pressure and potential temperature couple with wind in the proposed model, and thus are not arbitrarily specified variables as in previous studies on the baroclinicity in the Ekman model. The extended thermal wind balance equation and the Ekman potential vorticity equation are derived to describe the coupling. The two equations, along with the equation describing the constraint on potential temperature, are employed to derive the analytical solutions of the proposed Ekman model. Because potential temperature is not a specified variable but part of the solution, the derived analytical solutions have very different forms from those derived in previous studies. The differences illustrate the impact of the inclusion of the potential temperature equation and hydrostatic equation on wind, pressure, and potential temperature in the proposed Ekman model. It is found that the computed wind profiles based on the proposed model can capture some important features of the observed wind profiles.
33
Smyth, W. D., J. N. Moum, and J. D. Nash. "Narrowband Oscillations in the Upper Equatorial Ocean. Part II: Properties of Shear Instabilities." Journal of Physical Oceanography 41, no. 3 (March 1, 2011): 412–28. http://dx.doi.org/10.1175/2010jpo4451.1.
Abstract:
Abstract Narrowband oscillations observed in the upper equatorial Pacific are interpreted in terms of a random ensemble of shear instability events. Linear perturbation analysis is applied to hourly averaged profiles of velocity and density over a 54-day interval, yielding a total of 337 unstable modes. Composite profiles of mean states and eigenfunctions surrounding the critical levels suggest that the standard hyperbolic tangent model of Kelvin–Helmholtz (KH) instability is a reasonable approximation, but the symmetry of the composite perturbation is broken by the stratification and vorticity gradient of the underlying equatorial undercurrent. Unstable modes are found to occupy a range of frequencies with a peak near 1.4 mHz, consistent with the frequency content of the observed oscillations. A probabilistic theory of random instabilities predicts this peak frequency closely. An order of magnitude estimate suggests that the peak frequency is of order N, in accord with the observations. This results not from gravity wave physics but from the balance of shear and stratification that governs shear instability in geophysical flows. More generally, it is concluded that oscillatory signals with frequency bounded by N can result from a process that has nothing to do with gravity waves.
34
Nakamura, Noboru, and Da Zhu. "Formation of Jets through Mixing and Forcing of Potential Vorticity: Analysis and Parameterization of Beta-Plane Turbulence." Journal of the Atmospheric Sciences 67, no. 9 (September 1, 2010): 2717–33. http://dx.doi.org/10.1175/2009jas3159.1.
Abstract:
Abstract Formation of multiple jets in forced beta-plane turbulence is studied from the perspective of nonuniform nonconservative arrangement of potential vorticity (PV). Numerical simulations are analyzed to show that mixing and forcing reinforce jets by concentrating PV gradients at the axes of prograde jets. Based on the formalism developed in the companion paper, the nonconservative driving of jets is diagnosed and parameterized through the diffusive flux of PV and the source of wave activity. It is found that the two terms nearly balance on a long time scale, and they are both strongly anticorrelated with the PV gradient, which suggests that PV controls the nonconservative processes and that these processes could be parameterized as functions of the PV gradient. The flux is modeled using the effective diffusivity formula recently obtained by Ferrari and Nikurashin. Consistent with the PV barrier concept, the nonlinear diffusivity is a decreasing function of the squared PV gradient and agrees well with the diffusivity diagnosed from the numerical simulation. The source term is assumed to be inversely proportional to the PV gradient. The parameterization gives rise to a nonlinear partial differential equation (PDE) for the mean flow. A finite-difference model of the PDE predicts formation of a piecewise linear PV (staircase) and piecewise parabolic jets from a near-uniform initial condition when anisotropy and mixing of the flow are sufficiently strong. The origin of the discontinuities is antidiffusive instability of PV gradients, and although nonlinearity allows the discrete model to integrate stably, the solution is sensitive to the initial condition and resolution. The emerging jets in the 1D model have similar characteristics to those in the numerical simulation, but the details of the transient behavior are distinct. Similar discrete models of ill-posed PDEs in which discontinuities form also appear in image processing and granular matter dynamics.
35
Spreen, Gunnar, Ron Kwok, Dimitris Menemenlis, and An T. Nguyen. "Sea-ice deformation in a coupled ocean–sea-ice model and in satellite remote sensing data." Cryosphere 11, no. 4 (July 4, 2017): 1553–73. http://dx.doi.org/10.5194/tc-11-1553-2017.
Abstract:
Abstract. A realistic representation of sea-ice deformation in models is important for accurate simulation of the sea-ice mass balance. Simulated sea-ice deformation from numerical simulations with 4.5, 9, and 18 km horizontal grid spacing and a viscous–plastic (VP) sea-ice rheology are compared with synthetic aperture radar (SAR) satellite observations (RGPS, RADARSAT Geophysical Processor System) for the time period 1996–2008. All three simulations can reproduce the large-scale ice deformation patterns, but small-scale sea-ice deformations and linear kinematic features (LKFs) are not adequately reproduced. The mean sea-ice total deformation rate is about 40 % lower in all model solutions than in the satellite observations, especially in the seasonal sea-ice zone. A decrease in model grid spacing, however, produces a higher density and more localized ice deformation features. The 4.5 km simulation produces some linear kinematic features, but not with the right frequency. The dependence on length scale and probability density functions (PDFs) of absolute divergence and shear for all three model solutions show a power-law scaling behavior similar to RGPS observations, contrary to what was found in some previous studies. Overall, the 4.5 km simulation produces the most realistic divergence, vorticity, and shear when compared with RGPS data. This study provides an evaluation of high and coarse-resolution viscous–plastic sea-ice simulations based on spatial distribution, time series, and power-law scaling metrics.
36
KARSTEN, RICHARD H., and GORDON E. SWATERS. "Nonlinear effects in two-layer large-amplitude geostrophic dynamics. Part 1. The strong-beta case." Journal of Fluid Mechanics 412 (June 10, 2000): 125–60. http://dx.doi.org/10.1017/s0022112000008260.
Abstract:
Baroclinic large-amplitude geostrophic (LAG) models, which assume a leading-order geostrophic balance but allow for large-amplitude isopycnal deflections, provide a suitable framework to model the large-amplitude motions exhibited in frontal regions. The qualitative dynamical characterization of LAG models depends critically on the underlying length scale. If the length scale is sufficiently large, the effect of differential rotation, i.e. the β-effect, enters the dynamics at leading order. For smaller length scales, the β-effect, while non-negligible, does not enter the dynamics at leading order. These two dynamical limits are referred to as strong-β and weak-β models, respectively.A comprehensive description of the nonlinear dynamics associated with the strong- β models is given. In addition to establishing two new nonlinear stability theorems, we extend previous linear stability analyses to account for the finite-amplitude development of perturbed fronts. We determine whether the linear solutions are subject to nonlinear secondary instabilities and, in particular, a new long-wave–short-wave (LWSW) resonance, which is a possible source of rapid unstable growth at long length scales, is identified. The theoretical analyses are tested against numerical simulations. The simulations confirm the importance of the LWSW resonance in the development of the flow. Simulations show that instabilities associated with vanishing potential- vorticity gradients can develop into stable meanders, eddies or breaking waves. By examining models with different layer depths, we reveal how the dynamics associated with strong-β models qualitatively changes as the strength of the dynamic coupling between the barotropic and baroclinic motions varies.
37
BECKERS, M., R. VERZICCO, H. J. H. CLERCX, and G. J. F. VAN HEIJST. "Dynamics of pancake-like vortices in a stratified fluid: experiments, model and numerical simulations." Journal of Fluid Mechanics 433 (April 25, 2001): 1–27. http://dx.doi.org/10.1017/s0022112001003482.
Abstract:
The dynamics and the three-dimensional structure of vortices in a linearly stratified, non-rotating fluid are investigated by means of laboratory experiments, an analytical model and through numerical simulations. The laboratory experiments show that such vortices have a thin pancake-like appearance. Due to vertical diffusion of momentum the strength of these vortices decreases rapidly and their thickness increases in time. Also it is found that inside a vortex the linear ambient density profile becomes perturbed, resulting in a local steepening of the density gradient. Based on the assumption of a quasi-two-dimensional axisymmetric flow (i.e. with zero vertical velocity) a model is derived from the Boussinesq equations that illustrates that the velocity field of the vortex decays due to diffusion and that the vortex is in so-called cyclostrophic balance. This means that the centrifugal force inside the vortex is balanced by a pressure gradient force that is provided by a perturbation of the density profile in a way that is observed in the experiments. Numerical simulations are performed, using a finite difference method in a cylindrical coordinate system. As an initial condition the three-dimensional vorticity and density structure of the vortex, found with the diffusion model, are used. The influence of the Froude number, Schmidt number and Reynolds number, as well as the initial thickness of the vortex, on the evolution of the flow are investigated. For a specific combination of flow parameters it is found that during the decay of the vortex the relaxation of the isopycnals back to their undisturbed positions can result in a stretching of the vortex. Potential energy of the perturbed isopycnals is then converted into kinetic energy of the vortex. However, when the stratification is strong enough (i.e. for small Froude numbers), the evolution of the vortex can be described almost perfectly by the diffusion model alone.
38
Kessler, William S. "Mean Three-Dimensional Circulation in the Northeast Tropical Pacific*." Journal of Physical Oceanography 32, no. 9 (September 1, 2002): 2457–71. http://dx.doi.org/10.1175/1520-0485-32.9.2457.
Abstract:
Abstract Historical XBT data are used to construct a mean climatology of the three-dimensional geostrophic circulation in the northeast tropical Pacific (southwest of Mexico and Central America) and are diagnosed based on linear dynamics forced with satellite scatterometer winds. Unlike the familiar central tropical Pacific, where the zonal scales are very large and the wind forcing nearly a function of latitude alone, the North Pacific east of about 120°W is strongly influenced by wind jets blowing through gaps in the Central American cordillera. The curl imposed by these wind jets imprints on the ocean, producing a distinctive pattern of thermocline topography and geostrophic currents that are consistent with the Sverdrup balance. Notably, the weakening of the North Equatorial Countercurrent near 110°W is due to the wind forcing. Given the observed stratification and wind stress curl, planetary vorticity conservation also determines the distribution of vertical velocity in the region, with about 3.5 Sv (Sv ≡ 106 m3 s−1) of upwelling through the base of the thermocline under the Costa Rica Dome. This upwelling is associated with stretching of the water column under the dome, which thereby causes the northern “Subsurface Counter Current” (SSCC or Tsuchiya Jet) to turn away from the equator; about half the transport of the SSCC upwells through the thermocline via this mechanism. This may be part of the process by which intermediate-depth water, flowing into the Pacific from the south, is brought to the surface and into the Northern Hemisphere.
39
ZAKI, TAMER A., and SANDEEP SAHA. "On shear sheltering and the structure of vortical modes in single- and two-fluid boundary layers." Journal of Fluid Mechanics 626 (May 10, 2009): 111–47. http://dx.doi.org/10.1017/s0022112008005648.
Abstract:
Studies of vortical interactions in boundary layers have often invoked the continuous spectrum of the Orr–Sommerfeld (O-S) equation. These vortical eigenmodes provide a link between free-stream disturbances and the boundary-layer shear – a link which is absent in the inviscid limit due to shear sheltering. In the presence of viscosity, however, a shift in the dominant balance in the operator determines the structure of these eigenfunctions inside the mean shear. In order to explain the mechanics of shear sheltering and the structure of the continuous modes, both numerical and asymptotic solutions of the linear perturbation equation are presented in single- and two-fluid boundary layers. The asymptotic analysis identifies three limits: a convective shear-sheltering regime, a convective–diffusive regime and a diffusive regime. In the shear-dominated limit, the vorticity eigenfunction possesses a three-layer structure, the topmost being a region of exponential decay. The role of viscosity is most pronounced in the diffusive regime, where the boundary layer becomes ‘transparent’ to the oscillatory eigenfunctions. Finally, the convective–diffusive regime demonstrates the interplay between the the accumulative effect of the shear and the role of viscosity. The analyses are complemented by a physical interpretation of shear-sheltering mechanism. The influence of a wallfilm, in particular viscosity and density stratification, and surface tension are also evaluated. It is shown that a modified wavenumber emerges across the interface and influences the penetration of vortical disturbances into the two-fluid shear flow.
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MANELA, A., and M. S. HOWE. "The forced motion of a flag." Journal of Fluid Mechanics 635 (August 25, 2009): 439–54. http://dx.doi.org/10.1017/s0022112009007770.
Abstract:
The prevailing view of the dynamics of flapping flags is that the onset of motion is caused by temporal instability of the initial planar state. This view is re-examined by considering the linearized two-dimensional motion of a flag immersed in a high-Reynolds-number flow and taking account of forcing by a ‘street’ of vortices shed periodically from its cylindrical pole. The zone of nominal instability is determined by analysis of the self-induced motion in the absence of shed vorticity, including the balance between flag inertia, bending rigidity, varying tension and fluid loading. Forced motion is then investigated by separating the flag deflection into ‘vortex-induced’ and ‘self’ components. The former is related directly to the motion that would be generated by the shed vortices if the flag were absent. This component serves as an inhomogeneous forcing term in the equation satisfied by the ‘self’ motion. It is found that forced flapping is possible whenever the Reynolds number based on the pole diameter ReD ≳ 100, such that a wake of distinct vortex structures is established behind the pole. Such conditions typically prevail at mean flow velocities significantly lower than the critical threshold values predicted by the linear theory. It is therefore argued that analyses of the onset of flag motion that are based on ideal, homogeneous flag theory are incomplete and that consideration of the pole-induced fluid flow is essential at all relevant wind speeds.
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Wang, Peng, James C. McWilliams, and Claire Ménesguen. "Ageostrophic instability in rotating, stratified interior vertical shear flows." Journal of Fluid Mechanics 755 (August 19, 2014): 397–428. http://dx.doi.org/10.1017/jfm.2014.426.
Abstract:
AbstractThe linear instability of several rotating, stably stratified, interior vertical shear flows $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\overline{U}(z)$ is calculated in Boussinesq equations. Two types of baroclinic, ageostrophic instability, AI1 and AI2, are found in odd-symmetric $\overline{U}(z)$ for intermediate Rossby number ($\mathit{Ro}$). AI1 has zero frequency; it appears in a continuous transformation of the unstable mode properties between classic baroclinic instability (BCI) and centrifugal instability (CI). It begins to occur at intermediate $\mathit{Ro}$ values and horizontal wavenumbers ($k,l$) that are far from $l= 0$ or $k = 0$, where the growth rate of BCI or CI is the strongest. AI1 grows by drawing kinetic energy from the mean flow, and the perturbation converts kinetic energy to potential energy. The instability AI2 has inertia critical layers (ICL); hence it is associated with inertia-gravity waves. For an unstable AI2 mode, the coupling is either between an interior balanced shear wave and an inertia-gravity wave (BG), or between two inertia-gravity waves (GG). The main energy source for an unstable BG mode is the mean kinetic energy, while the main energy source for an unstable GG mode is the mean available potential energy. AI1 and BG type AI2 occur in the neighbourhood of $A-S= 0$ (a sign change in the difference between absolute vertical vorticity and horizontal strain rate in isentropic coordinates; see McWilliams et al., Phys. Fluids, vol. 10, 1998, pp. 3178–3184), while GG type AI2 arises beyond this condition. Both AI1 and AI2 are unbalanced instabilities; they serve as an initiation of a possible local route for the loss of balance in 3D interior flows, leading to an efficient energy transfer to small scales.
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Adames, Ángel F., and John M. Wallace. "On the Tropical Atmospheric Signature of El Niño." Journal of the Atmospheric Sciences 74, no. 6 (May 24, 2017): 1923–39. http://dx.doi.org/10.1175/jas-d-16-0309.1.
Abstract:
Abstract The linear atmospheric signature of ENSO, obtained by regressing fields of geopotential height Z, wind, vertical velocity, and rainfall upon the Niño-3.4 sea surface temperature (SST) index, is partitioned into zonally symmetric and eddy components. The zonally symmetric component is thermally forced by the narrowing and intensification of the zonally averaged equatorial rain belt during El Niño and mechanically forced by the weakening of the upper-tropospheric equatorial stationary waves and their associated flux of wave activity. The eddy component of the ENSO signature is decomposed into barotropic (BT) and baroclinic (BC) contributions, the latter into first and second modal structures BC1 and BC2, separable functions of space (x, y), and pressure p, using eigenvector analysis. BC1 exhibits a nearly equatorially symmetric planetary wave structure comprising three dumbbell-shaped features suggestive of equatorial Rossby waves, with out-of-phase wind and geopotential height perturbations in the upper and lower troposphere. BC1 and BT exhibit coincident centers of action. In regions of the tropics where the flow in the climatological-mean stationary waves is cyclonic, BT reinforces BC1, and vice versa, in accordance with vorticity balance considerations. BC1 and BT dominate the eddy ENSO signature in the free atmosphere. Most of the residual is captured by BC2, which exhibits a shallow, convergent boundary layer signature forced by the weakening of the equatorial cold tongue in SST. The anomalous boundary layer convergence drives a deep convection signature whose upper-tropospheric outflow is an integral part of the BC1 contribution to the ENSO signature.
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Hassanzadeh, Pedram, and Zhiming Kuang. "Quantifying the Annular Mode Dynamics in an Idealized Atmosphere." Journal of the Atmospheric Sciences 76, no. 4 (April 1, 2019): 1107–24. http://dx.doi.org/10.1175/jas-d-18-0268.1.
Abstract:
Abstract The linear response function (LRF) of an idealized GCM, the dry dynamical core with Held–Suarez physics, is used to accurately compute how eddy momentum and heat fluxes change in response to the zonal wind and temperature anomalies of the annular mode at the quasi-steady limit. Using these results and knowing the parameterizations of surface friction and thermal radiation in Held–Suarez physics, the contribution of each physical process (meridional and vertical eddy fluxes, surface friction, thermal radiation, and meridional advection) to the annular mode dynamics is quantified. Examining the quasigeostrophic potential vorticity balance, it is shown that the eddy feedback is positive and increases the persistence of the annular mode by a factor of more than 2. Furthermore, how eddy fluxes change in response to only the barotropic component of the annular mode, that is, vertically averaged zonal wind (and no temperature) anomaly, is also calculated similarly. The response of eddy fluxes to the barotropic-only component of the annular mode is found to be drastically different from the response to the full (i.e., barotropic + baroclinic) annular mode anomaly. In the former, the eddy generation is significantly suppressed, leading to a negative eddy feedback that decreases the persistence of the annular mode by nearly a factor of 3. These results suggest that the baroclinic component of the annular mode anomaly, that is, the increased low-level baroclinicity, is essential for the persistence of the annular mode, consistent with the baroclinic mechanism but not the barotropic mechanism proposed in the previous studies.
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Negretti, M. Eletta, and Paul Billant. "Stability of a Gaussian pancake vortex in a stratified fluid." Journal of Fluid Mechanics 718 (February 8, 2013): 457–80. http://dx.doi.org/10.1017/jfm.2012.624.
Abstract:
AbstractVortices in stably stratified fluids generally have a pancake shape with a small vertical thickness compared with their horizontal size. In order to understand what mechanism determines their minimum thickness, the linear stability of an axisymmetric pancake vortex is investigated as a function of its aspect ratio $\alpha $, the horizontal Froude number ${F}_{h} $, the Reynolds number $\mathit{Re}$ and the Schmidt number $\mathit{Sc}$. The vertical vorticity profile of the base state is chosen to be Gaussian in both radial and vertical directions. The vortex is unstable when the aspect ratio is below a critical value, which scales with the Froude number: ${\alpha }_{c} \sim 1. 1{F}_{h} $ for sufficiently large Reynolds numbers. The most unstable perturbation has an azimuthal wavenumber either $m= 0$, $\vert m\vert = 1$ or $\vert m\vert = 2$ depending on the control parameters. We show that the threshold corresponds to the appearance of gravitationally unstable regions in the vortex core due to the thermal wind balance. The Richardson criterion for shear instability based on the vertical shear is never satisfied alone. The dominance of the gravitational instability over the shear instability is shown to hold for a general class of pancake vortices with angular velocity of the form $\tilde {\Omega } (r, z)= \Omega (r)f(z)$ provided that $r\partial \Omega / \partial r\lt 3\Omega $ everywhere. Finally, the growth rate and azimuthal wavenumber selection of the gravitational instability are accounted well by considering an unstably stratified viscous and diffusive layer in solid body rotation with a parabolic density gradient.
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Shaw, Tiffany A., and William R. Boos. "The Tropospheric Response to Tropical and Subtropical Zonally Asymmetric Torques: Analytical and Idealized Numerical Model Results." Journal of the Atmospheric Sciences 69, no. 1 (January 1, 2012): 214–35. http://dx.doi.org/10.1175/jas-d-11-0139.1.
Abstract:
Abstract The tropospheric response to prescribed tropical and subtropical zonally asymmetric torques, which can be considered as idealizations of vertical momentum transfers by orographic gravity waves or convection, is investigated. The linear analytical Gill model response to westward upper-tropospheric torques is compared to the response to a midtropospheric heating, which is a familiar point of reference. The response to an equatorial torque projects onto a Kelvin wave response to the east that is of opposite sign to the response to the east of the heating at upper levels. In contrast, the torque and heating both produce Rossby gyres of the same sign to the west of the forcing and the zonal-mean streamfunction responses are identical. When the forcings are shifted into the Northern Hemisphere, the streamfunction responses have opposite signs: there is upwelling in the Southern (Northern) Hemisphere in response to the torque (heating). The nonlinear response to westward torques was explored in idealized general circulation model experiments. In the absence of a large-scale meridional temperature gradient, the response to an equatorial torque was confined to the tropics and was qualitatively similar to the linear solutions. When the torque was moved into the subtropics, the vorticity budget response was similar to a downward control–type balance in the zonal mean. In the presence of a meridional temperature gradient, the response to an equatorial torque involved a poleward shift of the midlatitude tropospheric jet and Ferrel cell. The response in midlatitudes was associated with a poleward shift of the regions of horizontal eddy momentum flux convergence, which coincided with a shift in the upper-tropospheric critical line for baroclinic waves. The shift in the critical line was caused (in part) by the zonal wind response to the prescribed torque, suggesting a possible cause of the response in midlatitudes. Overall, this hierarchy of analytical and numerical results highlights robust aspects of the response to tropical and subtropical zonally asymmetric torques and represents the first step toward understanding the response in fully comprehensive general circulation models.
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Sekma, H., Y. H. Park, and F. Vivier. "Time-Mean Flow as the Prevailing Contribution to the Poleward Heat Flux across the Southern Flank of the Antarctic Circumpolar Current: A Case Study in the Fawn Trough, Kerguelen Plateau." Journal of Physical Oceanography 43, no. 3 (March 1, 2013): 583–601. http://dx.doi.org/10.1175/jpo-d-12-0125.1.
Abstract:
Abstract The major mechanisms of the oceanic poleward heat flux in the Southern Ocean are still in debate. The long-standing belief stipulates that the poleward heat flux across the Antarctic Circumpolar Current (ACC) is mainly due to mesoscale transient eddies and the cross-stream heat flux by time-mean flow is insignificant. This belief has recently been challenged by several numerical modeling studies, which stress the importance of mean flow for the meridional heat flux in the Southern Ocean. Here, this study analyzes moored current meter data obtained recently in the Fawn Trough, Kerguelen Plateau, to estimate the cross-stream heat flux caused by the time-mean flow and transient eddies. It is shown that the poleward eddy heat flux in this southern part of the ACC is negligible, while that from the mean flow is overwhelming by two orders of magnitude. This is due to the unusual anticlockwise turning of currents with decreasing depth, which is associated with significant bottom upwelling engendered by strong bottom currents flowing over the sloping topography of the trough. The circumpolar implications of these local observations are discussed in terms of the depth-integrated linear vorticity budget, which suggests that the six topographic features along the southern flank of the ACC equivalent to the Fawn Trough case would yield sufficient poleward heat flux to balance the oceanic heat loss in the subpolar region. As eddy activity on the southern flank of the ACC is too weak to transport sufficient heat poleward, the nonequivalent barotropic structure of the mean flow in several topographically constricted passages should accomplish the required task.
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REZNIK, G. M., and R. GRIMSHAW. "Nonlinear geostrophic adjustment in the presence of a boundary." Journal of Fluid Mechanics 471 (November 5, 2002): 257–83. http://dx.doi.org/10.1017/s0022112002002148.
Abstract:
The process of nonlinear geostrophic adjustment in the presence of a boundary (i.e. in a half-plane bounded by a rigid wall) is examined in the framework of a rotating shallow water model, using an asymptotic multiple-time-scale theory based on the assumed smallness of the Rossby number ε. The spatial scale is of the order of the Rossby scale. Different initial states are considered: periodic, ‘step’-like, and localized. In all cases the initial perturbation is split in a unique way into slow and fast components evolving with characteristic time scales f−1 and (εf)−1, respectively. The slow component is not influenced by the fast one, at least for times t [les ] (fε)−1, and remains close to geostrophic balance. The fast component consists mainly of linear inertia–gravity waves rapidly propagating outward from the initial disturbance and Kelvin waves confined near the boundary.The theory provides simple formulae allowing us to construct the initial profile of the Kelvin wave, given arbitrary initial conditions. With increasing time, the Kelvin wave profile gradually distorts due to nonlinear-wave self-interaction, the distortion being described by the equation of a simple wave. The presence of Kelvin waves does not prevent the fast–slow splitting, in spite of the fact that the frequency gap between the Kelvin waves and slow motion is absent. The possibility of such splitting is explained by the special structure of the Kelvin waves in each case considered.The slow motion on time scales t [les ] (εf)−1 is governed by the well-known quasigeostrophic potential vorticity equation for the elevation. The theory provides an algorithm to determine initial slow and fast fields, and the boundary conditions to any order in ε. For the periodic and step-like initial conditions, the slow component behaves in the usual way, conserving mass, energy and enstrophy. In the case of a localized initial disturbance the total mass of the lowest-order slow component is not conserved, and conservation of the total mass is provided by the first-order slow correction and the Kelvin wave.On longer time scales t [les ] (ε2f)−1 the slow motion obeys the so-called modified quasi-geostrophic potential vorticity (QGPV) equation. The theory provides initial and boundary conditions for this equation. This modified equation coincides exactly with the ‘improved’ QGPV equation, derived by Reznik, Zeitlin & Ben Jelloul (2001), in the step-like and localized cases. In the periodic case this equation contains an additional term due to the Kelvin-wave self-interaction, this term depending on the initial Kelvin wave profile.
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Tsang, Yue-Kin, and David G. Dritschel. "Ellipsoidal vortices in rotating stratified fluids: beyond the quasi-geostrophic approximation." Journal of Fluid Mechanics 762 (December 2, 2014): 196–231. http://dx.doi.org/10.1017/jfm.2014.630.
Abstract:
AbstractWe examine the basic properties and stability of isolated vortices having uniform potential vorticity (PV) in a non-hydrostatic rotating stratified fluid, under the Boussinesq approximation. For simplicity, we consider a uniform background rotation and a linear basic-state stratification for which both the Coriolis and buoyancy frequencies, $f$ and $N$, are constant. Moreover, we take $f/N\ll 1$, as typically observed in the Earth’s atmosphere and oceans. In the small Rossby number ‘quasi-geostrophic’ (QG) limit, when the flow is weak compared to the background rotation, there exist exact solutions for steadily rotating ellipsoidal volumes of uniform PV in an unbounded flow (Zhmur & Shchepetkin, Izv. Akad. Nauk SSSR Atmos. Ocean. Phys., vol. 27, 1991, pp. 492–503; Meacham, Dyn. Atmos. Oceans, vol. 16, 1992, pp. 189–223). Furthermore, a wide range of these solutions are stable as long as the horizontal and vertical aspect ratios ${\it\lambda}$ and ${\it\mu}$ do not depart greatly from unity (Dritschel et al.,J. Fluid Mech., vol. 536, 2005, pp. 401–421). In the present study, we examine the behaviour of ellipsoidal vortices at Rossby numbers up to near unity in magnitude. We find that there is a monotonic increase in stability as one varies the Rossby number from nearly $-1$ (anticyclone) to nearly $+1$ (cyclone). That is, QG vortices are more stable than anticyclones at finite negative Rossby number, and generally less stable than cyclones at finite positive Rossby number. Ageostrophic effects strengthen both the rotation and the stratification within a cyclone, enhancing its stability. The converse is true for an anticyclone. For all Rossby numbers, stability is reinforced by increasing ${\it\lambda}$ towards unity or decreasing ${\it\mu}$. An unstable vortex often restabilises by developing a near-circular cross-section, typically resulting in a roughly ellipsoidal vortex, but occasionally a binary system is formed. Throughout the nonlinear evolution of a vortex, the emission of inertia–gravity waves (IGWs) is negligible across the entire parameter space investigated. Thus, vortices at small to moderate Rossby numbers, and any associated instabilities, are (ageostrophically) balanced. A manifestation of this balance is that, at finite Rossby number, an anticyclone rotates faster than a cyclone.
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REZNIK, G. M., V. ZEITLIN, and M. BEN JELLOUL. "Nonlinear theory of geostrophic adjustment. Part 1. Rotating shallow-water model." Journal of Fluid Mechanics 445 (October 16, 2001): 93–120. http://dx.doi.org/10.1017/s002211200100550x.
Abstract:
We develop a theory of nonlinear geostrophic adjustment of arbitrary localized (i.e. finite-energy) disturbances in the framework of the non-dissipative rotating shallow-water dynamics. The only assumptions made are the well-defined scale of disturbance and the smallness of the Rossby number Ro. By systematically using the multi-time-scale perturbation expansions in Rossby number it is shown that the resulting field is split in a unique way into slow and fast components evolving with characteristic time scales f−10 and (f0Ro)−1 respectively, where f0 is the Coriolis parameter. The slow component is not influenced by the fast one and remains close to the geostrophic balance. The algorithm of its initialization readily follows by construction.The scenario of adjustment depends on the characteristic scale and/or initial relative elevation of the free surface ΔH/H0, where ΔH and H0 are typical values of the initial elevation and the mean depth, respectively. For small relative elevations (ΔH/H0 = O(Ro)) the evolution of the slow motion is governed by the well-known quasi-geostrophic potential vorticity equation for times t [les ] (f0Ro)−1. We find modifications to this equation for longer times t [les ] (f0Ro2)−1. The fast component consists mainly of linear inertia–gravity waves rapidly propagating outward from the initial disturbance.For large relative elevations (ΔH/H0 [Gt ] Ro) the slow field is governed by the frontal geostrophic dynamics equation. The fast component in this case is a spatially localized packet of inertial oscillations coupled to the slow component of the flow. Its envelope experiences slow modulation and obeys a Schrödinger-type modulation equation describing advection and dispersion of the packet. A case of intermediate elevation is also considered.
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STANISLAS, MICHEL, LAURENT PERRET, and JEAN-MARC FOUCAUT. "Vortical structures in the turbulent boundary layer: a possible route to a universal representation." Journal of Fluid Mechanics 602 (April 25, 2008): 327–82. http://dx.doi.org/10.1017/s0022112008000803.
Abstract:
A study of streamwise oriented vortical structures embedded in turbulent boundary layers is performed by investigating an experimental database acquired by stereoscopic particle image velocimetry (SPIV) in a plane normal both to the mean flow and the wall. The characteristics of the experimental data allow us to focus on the spatial organization within the logarithmic region for Reynolds numbers Reθ up to 15000. On the basis of the now accepted hairpin model, relationships and interaction between streamwise vortices are first investigated via computation of two-point spatial correlations and the use of linear stochastic estimation (LSE). These analyses confirm that the shape of the most probable coherent structures corresponds to an asymmetric one-legged hairpin vortex. Moreover, two regions of different dynamics can be distinguished: the near-wall region below y+=150, densely populated with strongly interacting vortices; and the region above y+=150 where interactions between eddies happen less frequently. Characteristics of the detected eddies, such as probability density functions of their radius and intensity, are then studied. It appears that Reynolds number as well as wall-normal independences of these quantities are achieved when scaling with the local Kolmogorov scales. The most probable size of the detected vortices is found to be about 10 times the Kolmogorov length scale. These results lead us to revisit the equation for the mean square vorticity fluctuations, and to propose a new balance of this equation in the near-wall region. This analysis and the above results allow us to propose a new description of the near-wall region, leading to a new scaling which seems to have a good universality in the Reynolds-number range investigated. The possibility of reaching a universal scaling at high enough Reynolds number, based on the external velocity and the Kolmogorov length scale is suggested.