Vorticity Gradient Research Articles

Persistent presence of small vertical scale velocity features during three-dimensional equilibration of equatorial inertial instability

Small vertical scale velocity features (SVSs) are ubiquitous in the upper equatorial ocean but their lifecycle and role in the large scale dynamics are only beginning to be understood. In this article, we study the development of SVSs generated through inertial instability of a prototypical equatorial zonal flow with a uniform meridional shear, U(y) = Λy. While previous studies employ a zonally symmetric setting, in which the flow is constrained to remain invariant in the streamwise direction throughout its evolution, here we use a fully three-dimensional framework. We choose a setting, in which the fastest growing linear modes are zonally symmetric and the initial perturbation is almost zonally symmetric, so that the flow remains nearly two-dimensional until the symmetric instability is fully neutralized. The secondary instabilities of the modified zonal mean flow favor zonally nonsymmetric modes, which leads to three-dimensionalization of the flow. It has been previously conjectured in the literature that the dominant secondary instability is an inflection point-type barotropic instability that favors disturbances with a large vertical scale and the necessary condition for which is the reversal of the meridional gradient of the background potential vorticity. We show that although the conditions necessary for barotropic instability indeed arise after the neutralization of the symmetric instability, the dominant secondary instabilities form a sequence of high vertical mode disturbances excited through zonally nonsymmetric inertial instability of the evolving zonal mean flow. The barotropic eddies, therefore, remain underdeveloped and do not play an important role in the dynamics. We cannot rule out the possible influence of the applied lateral hyperdiffusion (used to control near-grid point noise in the integrations) on the relative roles of barotropic and inertial instability. The competition between these two possible secondary instabilities needs further investigation with higher resolution experiments. We note, however, that when zonally asymmetric inertial instability is not possible (achieved here by shortening the zonal extent of the integration region while keeping the hyperdiffusion coefficient the same), barotropic instability fully develops and the flow is dominated by the barotropic mode.

An experimental study of multiple zonal jet formation in rotating, thermally driven convective flows on a topographic beta-plane

A series of rotating, thermal convection experiments were carried out on the Coriolis platform in Grenoble, France, to investigate the formation and energetics of systems of zonal jets through nonlinear eddy/wave-zonal flow interactions on a topographic β-plane. The latter was produced by a combination of a rigid, conically sloping bottom and the rotational deformation of the free upper surface. Convection was driven by a system of electrical heaters laid under the (thermally conducting) sloping bottom and led to the production of intense, convective vortices. These were observed to grow in size as each experiment proceeded and led to the development of weak but clear azimuthal jet-like flows, with a radial scale that varied according to the rotation speed of the platform. Detailed analyses reveal that the kinetic energy-weighted radial wavenumber of the zonal jets, kJy, scales quite closely either with the Rhines wavenumber as kJy ≃ 2(βT/2urms)1/2, where urms is the rms total or eddy velocity and βT is the vorticity gradient produced by the sloping topography, or the anisotropy wavenumber as kJy≃1.25(βT3/ϵ)1/5, where ϵ is the upscale turbulent energy transfer rate. Jets are primarily produced by the direct quasi-linear action of horizontal Reynolds stresses produced by trains of topographic Rossby waves. The nonlinear production rate of zonal kinetic energy is found to be strongly unsteady, however, with fluctuations of order 10-100 times the amplitude of the mean production rate for all cases considered. The time scale of such fluctuations is found to scale consistently with either an inertial time scale, τp∼1./urmsβT, or the Ekman spin-down time scale. Kinetic energy spectra show some evidence for a k−5/3 inertial subrange in the isotropic component, suggestive of a classical Kolmogorov-Batchelor-Kraichnan upscale energy cascade and a steeper spectrum in the zonal mean flow, though not as steep as k−5, as anticipated for fully zonostrophic flow. This is consistent with a classification of all of these flows as marginally zonostrophic, as expected for values of the zonostrophy parameter Rβ ≃ 1.6–1.7, though a number of properties related to flow anisotropy were found to vary significantly and systematically within this range.

Atmospheric responses to oceanic eddies in the Kuroshio Extension region

AbstractWe examined atmospheric responses to 35,000+ oceanic eddies in the Kuroshio Extension region during the period of 2006–2009. Using satellite data, we showed that cold (warm) eddies cause surface winds to decelerate (accelerate) and reduce (increase) latent and sensible heat fluxes, cloud liquid water, water vapor content, and rain rate; all of these changes are quantified. Both the linear correlation between wind divergence and downwind sea surface temperature (SST) gradient and the correspondence between vorticity and crosswind SST gradient support the vertical momentum mixing mechanism, which indicates that SST perturbations modify surface winds by changing the vertical turbulent mixing in the marine atmospheric boundary layer (MABL). High‐resolution National Centers for Environmental Prediction Climate Forecast System Reanalysis (CFSR) data can reproduce the atmospheric responses to the oceanic eddies in the MABL albeit with some differences in intensity. In addition, the CFSR data reveal that the atmospheric responses to these oceanic eddies are not confined in the MABL. MABL deepens (shoals) over the warm (cold) eddies; enhanced (reduced) vertical transport of transient zonal momentum occurs over the warm (cold) eddies from the sea surface to about 850 hPa level; vertical velocity anomalies over oceanic eddies penetrate beyond the MABL into free atmosphere; there exists a positive correlated relationship between SST and convective rain rate anomalies, indicative of ocean eddies' impact on the free troposphere. However, the composites of cloud liquid water and rain rate are different from the results based on the satellite data.

Why compensated cold‐core rings look stable

AbstractIn contrast to many real‐ocean rings and eddies, circular vortices in idealized two‐layer models tend to be highly unstable, unless their radius is made small or their baroclinicity is made artificially weak. The addition of a middle layer with uniform potential vorticity weakens vertical coupling between the upper and lower layers that enhances vortex stability and makes the vortex lifespan more realistic. It is demonstrated here that such a three‐layer vortex model in so‐called compensated state (without rotation in the deep layer) possesses smaller lower interface slope than the two‐layer model that reduces the potential vorticity gradient in the lower layer and provides less unstable configurations. Numerical simulations of an unstable cyclone with parameters typical for cold‐core ring Bob reveal strong deformations and pulsations of the vortex core in the two‐layer setup, while the ring becomes only slightly deformed and looks stable when the middle layer with uniform potential vorticity is added to be consistent with observations. Enhanced vortex stability in such three‐layer setup has important implications for further studies of real‐ocean eddies.

Parameterization of eddy fluxes based on a mesoscale energy budget

It has recently been proposed to formulate eddy diffusivities in ocean models based on a mesoscale eddy kinetic energy (EKE) budget. Given an appropriate length scale, the mesoscale EKE can be used to estimate an eddy diffusivity based on mixing length theory. This paper discusses some of the open questions associated with the formulation of an EKE budget and mixing length, and proposes an improved energy budget-based parameterization for the mesoscale eddy diffusivity. A series of numerical simulations is performed, using an idealized flat-bottomed β-plane channel configuration with quadratic bottom drag. The results stress the importance of the mixing length formulation, as well as the formulation for the bottom signature of the mesoscale EKE, which is important in determining the rate of EKE dissipation. In the limit of vanishing planetary vorticity gradient, the mixing length is ultimately controlled by bottom drag, though the frictional arrest scale predicted by barotropic turbulence theory needs to be modified to account for the effects of baroclinicity. Any significant planetary vorticity gradient, β, is shown to suppress mixing, and limit the effective mixing length to the Rhines scale. While the EKE remains moderated by bottom friction, the bottom signature of EKE is shown to decrease as the appropriately non-dimensionalized friction increases, which considerably weakens the impact of changes in the bottom friction compared to barotropic turbulence. For moderate changes in the bottom-friction, eddy fluxes are thus reasonably well approximated by the scaling relation proposed by Held and Larichev (1996), which ignores the effect of bottom friction.

Evaluation of a Heuristic Model for Tropical Cyclone Resilience

Abstract This work examines the applicability of a previously postulated heuristic model for the temporal evolution of the small-amplitude tilt of a tropical cyclone–like vortex under vertical shear forcing for both a dry and cloudy atmosphere. The heuristic model hinges on the existence of a quasi-discrete vortex Rossby wave and its ability to represent the coherent precession and tilt decay of a stable vortex in the free-alignment problem. Linearized numerical solutions for a dry and cloudy vortex confirm the model predictions that an increase in the magnitude of the radial potential vorticity (PV) gradient within the vortex skirt surrounding the core yields a more rapid evolution of a sheared vortex toward the equilibrium, left-of-shear tilt configuration. However, in the moist-neutral limit, in which the effective static stability vanishes in rising and sinking regions, the heuristic model yields a poor approximation to the simulated vortex core evolution, but a left-of-shear tilt of the near-core vortex, radially beyond the heating region, remains the preferred long-time solution. Within the near-core skirt, the PV perturbation generated by vertical shearing exhibits continuous-spectrum-type vortex Rossby waves, features that are not captured by the heuristic model. Nevertheless, the heuristic model continues to predict the rapid vertical alignment and equilibrium, left-of-shear tilt configuration of the simulated near-core vortex in the moist-neutral limit.

S3T Stability of the Homogeneous State of Barotropic Beta-Plane Turbulence

Abstract Zonal jets and nonzonal large-scale flows are often present in forced–dissipative barotropic turbulence on a beta plane. The dynamics underlying the formation of both zonal and nonzonal coherent structures is investigated in this work within the statistical framework of stochastic structural stability theory (S3T). Previous S3T studies have shown that the homogeneous turbulent state undergoes a bifurcation at a critical parameter and becomes inhomogeneous with the emergence of zonal and/or large-scale nonzonal flows and that these statistical predictions of S3T are reflected in direct numerical simulations. In this paper, the dynamics underlying the S3T statistical instability of the homogeneous state as a function of parameters is studied. It is shown that, for weak planetary vorticity gradient β, both zonal jets and nonzonal large-scale structures form from upgradient momentum fluxes due to shearing of the eddies by the emerging infinitesimal flow. For large β, the dynamics of the S3T instability differs for zonal and nonzonal flows but in both the destabilizing vorticity fluxes decrease with increasing β. Shearing of the eddies by the mean flow continues to be the mechanism for the emergence of zonal jets while nonzonal large-scale flows emerge from resonant and near-resonant triad interactions between the large-scale flow and the stochastically forced eddies. The relation between the formation of large-scale structure through modulational instability and the S3T instability of the homogeneous state is also investigated and it is shown that the modulational instability results are subsumed by the S3T results.

Baroclinic instability and the mesoscale eddy field around the Lofoten Basin

AbstractThe vigorous mesoscale eddy field around the Lofoten Basin west of northern Norway is thought to be related to eddy shedding from the Norwegian Atlantic Current flowing along the Norwegian coast. Here we study baroclinic instability in the current with a particular focus on the influence of topography. The flow over the steepest part of the continental slope is found to be the most unstable. The growth characteristics cannot be understood from Eady theory alone but require the consideration of interior potential vorticity gradients. A study of the fully developed macroturbulent field shows that eddy kinetic energy is advected away from the generation regions and that nonlinear effects likely draw the eddy statistics away from the linear growth regime.

Interaction between Rossby and Gravity Waves in a Simple Analytical Model

The interaction between Rossby and gravity waves is examined in a vertical-zonal two-dimensional model, in which the basic state has an upward gradient of buoyancy at a lower level and a downward gradient of horizontal vorticity at a higher level. Because of the gradients, there exist westward-propagating upper Rossby and westward- and eastward-propagating lower gravity waves, where the propagation is relative to the fluid. The initial value problem for the disturbance is analytically solved. The temporal evolution of the analytical solution from an initial value shows the following characteristics. Resonant interaction between the westward-propagating upper Rossby wave and the eastward-propagating lower gravity wave is possible, in the same way as between two counter-propagating Rossby waves in barotropic and baroclinic problems. On the boundary in the parameter space between the unstable region, where resonant exponential growth occurs, and the stable region, where the solution oscillates, the marginal solution grows as a linear function of time. As in other instability problems, the marginal linear growth in the present model is not trivial. For small horizontal wave numbers, the westward-propagating gravity wave makes a non-negligible contribution to the resonant interaction between the westward-propagating Rossby wave and the eastward-propagating gravity waves.

Satellite Observations of Mesoscale Eddy-Induced Ekman Pumping

AbstractThree mechanisms for self-induced Ekman pumping in the interiors of mesoscale ocean eddies are investigated. The first arises from the surface stress that occurs because of differences between surface wind and ocean velocities, resulting in Ekman upwelling and downwelling in the cores of anticyclones and cyclones, respectively. The second mechanism arises from the interaction of the surface stress with the surface current vorticity gradient, resulting in dipoles of Ekman upwelling and downwelling. The third mechanism arises from eddy-induced spatial variability of sea surface temperature (SST), which generates a curl of the stress and therefore Ekman pumping in regions of crosswind SST gradients. The spatial structures and relative magnitudes of the three contributions to eddy-induced Ekman pumping are investigated by collocating satellite-based measurements of SST, geostrophic velocity, and surface winds to the interiors of eddies identified from their sea surface height signatures. On average, eddy-induced Ekman pumping velocities approach O(10) cm day−1. SST-induced Ekman pumping is usually secondary to the two current-induced mechanisms for Ekman pumping. Notable exceptions are the midlatitude extensions of western boundary currents and the Antarctic Circumpolar Current, where SST gradients are strong and all three mechanisms for eddy-induced Ekman pumping are comparable in magnitude. Because the polarity of current-induced curl of the surface stress opposes that of the eddy, the associated Ekman pumping attenuates the eddies. The decay time scale of this attenuation is proportional to the vertical scale of the eddy and inversely proportional to the wind speed. For typical values of these parameters, the decay time scale is about 1.3 yr.

Comments on “How Does the Boundary Layer Contribute to Eyewall Replacement Cycles in Axisymmetric Tropical Cyclones?”

Abstract : In a recent paper, Kepert (2013, hereafter K13) investigated the theoretical role of the boundary layer in eyewall replacement cycles. Specifically, he used a family of steady-state, axisymmetric hurricane boundary layer models to examine the boundary layer response to an imposed radial profile of tangential winds with two wind maxima. Based on these solutions, he proposed a new feedback mechanism for secondary eyewall formation (SEF) and pointed to the role of the underlying boundary layer dynamics in this process. Specifically, he proposed (abstract) that the boundary layer contributes to the formation of outer eyewalls through a positive feedback among the local enhancement of the radial vorticity gradient, the frictional updraft, and convection, and concluded (section 6) that supergradient flow . . . is not essential to SEF.

Small scale creation for solutions of the incompressible two-dimensional Euler equation

We construct an initial data for the two-dimensional Euler equation in a disk for which the gradient of vorticity exhibits double exponential growth in time for all times. This estimate is known to be sharp | the double exponential growth is the fastest possible growth rate.

Temporal evolution of vorticity staircases in randomly strained two-dimensional vortices

The evolution of a Gaussian vortex subject to a weak-external-random n-fold multipolar strain field is examined using fully nonlinear simulations. The simulations show that at large Reynolds numbers, fine scale steps form at the periphery of the vortex, before merging, generally leaving one large step, which acts as a barrier between the vorticity within the coherent core and the surrounding, well mixed, “surf zone.” It is shown for n = 2 that the width and the number of fine scale steps which initially form at the periphery of the vortex is dependent on the strain parameters, but that the range of radial values for which steps initially occur is only dependent on n and the amplitude of the strain field. A criteria is developed which can predict this range of radial values using the linear stability results of Le Dizès [“Non-axisymmetric vortices in two-dimensional flows,” J. Fluid Mech. 406, 175 (2000)]. This criteria is based upon the perturbation vorticity needing to be larger than some fraction of the vorticity gradient to flatten the vortex profile. For n = 3 and 4, the radial step range is again predicted, and it is observed that for these higher wavenumbers the long lasting steps are narrower than the n = 2 case. For n = 4 the steps which form are so narrow that they do not persist very long before they are destroyed by the strain field and viscosity.

The stratospheric wintertime response to applied extratropical torques and its relationship with the annular mode

The response of the wintertime Northern Hemisphere (NH) stratosphere to applied extratropical zonally symmetric zonal torques, simulated by a primitive equation model of the middle atmosphere, is presented. This is relevant to understanding the effect of gravity wave drag (GWD) in models and the influence of natural forcings such as the quasi-biennial oscillation (QBO), El Nino-Southern Oscillation (ENSO), solar cycle and volcanic eruptions on the polar vortex. There is a strong feedback due to planetary waves, which approximately cancels the direct effect of the torque on the zonal acceleration in the steady state and leads to an EP flux convergence response above the torque’s location. The residual circulation response is very different to that predicted assuming wave feedbacks are negligible. The results are consistent with the predictions of ray theory, with applied westerly torques increasing the meridional potential vorticity gradient, thus encouraging greater upward planetary wave propagation into the stratosphere. The steady state circulation response to torques applied at high latitudes closely resembles the Northern annular mode (NAM) in perpetual January simulations. This behaviour is analogous to that shown by the Lorenz system and tropospheric models. Imposed westerly high-latitude torques lead counter-intuitively to an easterly zonal mean zonal wind $$(\overline{u})$$ response at high latitudes, due to the wave feedbacks. However, in simulations with a seasonal cycle, the feedbacks are qualitatively similar but weaker, and the long-term response is less NAM-like and no longer easterly at high latitudes. This is consistent with ray theory and differences in climatological $$\overline{u}$$ between the two types of simulations. The response to a tropospheric wave forcing perturbation is also NAM-like. These results suggest that dynamical feedbacks tend to make the long-term NH extratropical stratospheric response to arbitrary external forcings NAM-like, but only if the feedbacks are sufficiently strong. This may explain why the observed polar vortex responses to natural forcings such as the QBO and ENSO are NAM-like. The results imply that wave feedbacks must be understood and accurately modelled in order to understand and predict the influence of GWD and other external forcings on the polar vortex, and that biases in a model’s climatology will cause biases in these feedbacks.

Exponential growth of the vorticity gradient for the Euler equation on the torus

We prove that there are solutions to the Euler equation on the torus with C1,α vorticity and smooth except at one point such that the vorticity gradient grows in L∞ at least exponentially as t→∞. The same result is shown to hold for the vorticity Hessian and smooth solutions. Our proofs use a version of a recent result by Kiselev and Šverák [6].

Double exponential growth of the vorticity gradient for the two-dimensional Euler equation

For the two-dimensional Euler equation on the torus, we prove that the $L^\infty$–norm of the vorticity gradient can grow as double exponential over arbitrary long but finite time provided that at time zero it is already sufficiently large. The method is based on the perturbative analysis around the singular stationary solution studied by Bahouri and Chemin in 1994. Our result on the growth of the vorticity gradient is equivalent to the statement that the operator of Euler evolution is not bounded in the linear sense in Lipschitz norm for any time $t>0$.

Absolute/Convective Instability Transition in a Backward Facing Step Combustor: Fundamental Mechanism and Influence of Density Gradient

Hydrodynamic instabilities of the flow field in lean premixed gas turbine combustors can generate velocity perturbations that wrinkle and distort the flame sheet over length scales that are smaller than the flame length. The resultant heat release oscillations can then potentially result in combustion instability. Thus, it is essential to understand the hydrodynamic instability characteristics of the combustor flow field in order to understand its overall influence on combustion instability characteristics. To this end, this paper elucidates the role of fluctuating vorticity production from a linear hydrodynamic stability analysis as the key mechanism promoting absolute/convective instability transitions in shear layers occurring in the flow behind a backward facing step. These results are obtained within the framework of an inviscid, incompressible, local temporal and spatio-temporal stability analysis. Vorticity fluctuations in this limit result from interaction between two competing mechanisms—(1) production from interaction between velocity perturbations and the base flow vorticity gradient and (2) baroclinic torque in the presence of base flow density gradients. This interaction has a significant effect on hydrodynamic instability characteristics when the base flow density and velocity gradients are colocated. Regions in the space of parameters characterizing the base flow velocity profile, i.e., shear layer thickness and ratio of forward to reverse flow velocity, corresponding to convective and absolute instability are identified. The implications of the present results on understanding prior experimental studies of combustion instability in backward facing step combustors and hydrodynamic instability in other flows such as heated jets and bluff body stabilized flames is discussed.

Second order potential vorticity and its potential applications

A new invariant, the second order potential vorticity (SPV), is derived in this paper. SPV is the dot product of vorticity and the potential vorticity (PV) gradient, and is proven conservative for a compressible, adiabatic and frictionless atmosphere. Research shows that the new invariant may be used to indicate the evolution of PV, because SPV includes all the information that determines PV evolution: the wind field, and the PV gradient. Furthermore, SPV is capable of diagnosing heavy precipitation because of the strong signals it presents in areas of heavy rainfall. SPV also shows great potential as a comprehensive conserved quantity for indicating the dynamical tropopause and baroclinic instability.

Influence of the Barotropic Mean Flow on the Width and the Structure of the Atlantic Equatorial Deep Jets

AbstractA representation of an equatorial basin mode excited in a shallow-water model for a single high-order baroclinic vertical normal mode is used as a simple model for the equatorial deep jets. The model is linearized about both a state of rest and a barotropic mean flow corresponding to the observed Atlantic Equatorial Intermediate Current System. It was found that the eastward mean flow associated with the North and South Intermediate Counter Currents (NICC and SICC, respectively) effectively shields the equator from off-equatorial Rossby waves. The westward propagation of these waves is blocked, and focusing on the equator due to beta dispersion is prevented. This leads to less energetic jets along the equator. On the other hand, the westward barotropic mean flow along the equator reduces the gradient of absolute vorticity and hence widens the cross-equatorial structure of the basin mode. Increasing lateral viscosity predominantly affects the width of the basin modes’ Kelvin wave component in the presence of the mean flow, while the Rossby wave is confined by the flanking NICC and SICC. Independent of the presence of the mean flow, the application of sufficient lateral mixing also hinders the focusing of off-equatorial Rossby waves, which is hence an unlikely feature of a low-frequency basin mode in the real ocean.

On the Observability of Oceanic Gyres

AbstractThis study examines the observability of a stratified ocean in a square flat basin on a midlatitude beta plane. Here, “observability” means the ability to establish, in a finite interval of time, the time-dependent ocean state given density observations over the same interval and with no regard for errors. The dynamics is linearized and hydrostatic, so that the motion can be decomposed into normal modes and the observability analysis is simplified. An observability Gramian (a symmetric matrix) is determined for the flows in an inviscid interior, in frictional boundary layers, and in a closed basin. Its properties are used to establish the condition for complete observability and to identify optimal data locations for each of these flows.It is found that complete observability of an oceanic interior in time-dependent Sverdrup balance requires that the observations originate from the westernmost location at each considered latitude. The degree of observability increases westward due to westward propagation of long baroclinic Rossby waves: data collected in the west are more informative than data collected in the east. Likewise, the best locations for observing variability in the western (eastern) boundary layer are near (far from) the boundary. The observability of a closed basin is influenced by the westward propagation and the boundaries. Optimal data locations that are identified for different resolutions (0.01 to 1 yr) and lengths of data records (0.2 to 20 yr) show a variable influence of the planetary vorticity gradient. Data collected near the meridional boundaries appear always less informative, from the viewpoint of basin observability, than data collected away from these boundaries.