Barotropic Vorticity Research Articles

On the transition process of a swirling vortex generated in a rotating tank

In this study, we investigate the transition of a swirling vortex from a one-celled to a two-celled vortex structure in a rotating tank. The main idea is to initiate the flow by siphoning fluid out of the tank and then to lift the siphoning mechanism out of the water within a short period of time. Before it reaches a state of quasi-two-dimensionality, the core region of the vortex can be roughly divided into three stages. (1) A siphoning stage induces the formation of the one-celled vortex. (2) A downward jet impingement stage triggers the transition of the vortex into the two-celled one. (3) A detachment stage of the inner cell leads to a cup-like recirculation zone, which is pushed upward by an axial flow from the boundary layer. This eventually develops into a stable quasi-two-dimensional barotropic vortex. The core region is enclosed by an outer region, which is in cyclostrophic balance. In the siphoning stage, the flow pattern can be well fitted by Burgers’ vortex model. However, in the post-siphoning stage, the present data show a flow pattern different from the existing two-celled models of Sullivan and Bellamy-Knights. Flow details, including flow patterns, velocity profiles, and surface depressions were measured and visualized by particle tracking velocimetry and the dye-injection method with various colors. The one-celled and two-celled flow structures are also similar to the conceptual images of the one- and two-celled tornadoes proposed in the literature.

A Note on Similarity Reductions of Barotropic and Quasi-geostrophic Potential Vorticity Equation

Applying the classical Lie symmetry approach to the barotropic and quasi-geostrophic potential vorticity equation without forcing and dissipation on a β-plane channel, we find a new symmetry, which is not presented in a previous work [F. Huang, Commun. Theor. Phys. (Beijing, China) 42 (2004) 903]. A general finite transformation group is obtained based on the full Lie point symmetry. Furthermore, two new types of similarity reduction solutions are obtained.

The ENIAC Forecasts: A Re-creation

The numerical forecasts made in 1950 using the Electronic Numerical Integrator and Computer (ENIAC) paved the way for the remarkable advances that have been made over the past half-century in weather prediction and climate modeling. We review the circumstances in which the forecasts were made, the nature of the ENIAC machine, and the roles of the people involved. The basis for the forecasts was the barotropic vorticity equation, and the initial data were prepared manually from standard weather charts. Now that the NCEP-NCAR reanalysis extends back to 1948, the initial height fields for the forecasts are readily available in digital form. We describe the reconstruction of the forecasts using reanalyzed data. Were the ENIAC forecasts any good? To date, no objective verification of the four integrations has been available. A comparison of the original and reconstructed forecasts shows them to be in good agreement. Quantitative verification of the forecasts yields surprising results. On the basis of root-mean-square errors, persistence beats the forecast in three of the four cases. The mean error, or bias, is smaller for persistence in all four cases. However, when SI scores are compared, all four forecasts show skill and three are substantially better than persistence. A small modification of the prediction equation, in which the streamfunction replaces the geopotential height as the prognostic variable, can be implemented without any computational penalty. The modified equation yields slightly improved forecasts in all four cases.

Comparison of the Life Cycles of the NAO Using Different Definitions

Abstract This study investigates the North Atlantic Oscillation (NAO) on an intraseasonal time scale. The authors investigate the question of how the characteristics of NAO events are influenced by the choice of its definitions using daily NCEP–NCAR reanalysis data spanning 51 boreal winters. Four different NAO indexes are used in this study, including one station/gridpoint–based index and three pattern-based indexes. It is found that the NAO events obtained using pattern–based indexes are quite similar to each other, while some notable differences are observed when the NAO is defined using the station/gridpoint–based index (NAO1). The characteristics of the pattern-based NAO are found to be more antisymmetric for its two phases, including its time-averaged spatial structures, its lifetime distributions, and time-evolving spatial structures. The NAO1, on the other hand, reveals some asymmetric characteristics between the two phases. Emphasis is placed on comparing the characteristics of the NAO events obtained using the NAO1 index and one of the pattern-based indices, that is, NAO2. The time-averaged spatial structures for the NAO2 expand across more of the polar region than the NAO1. The positive NAO1 shows a wave train signal over the Pacific–North American region during the setup phase, while the negative NAO1 is found to develop more locally over northern Europe and the North Atlantic. The wave activity flux for the NAO2 is primarily in the zonal direction while for the NAO1, on the other hand, it is mostly concentrated over the North Atlantic with a pronounced southward component. The barotropic vorticity equation is used to examine the physical mechanisms that drive the life cycle of the NAO.

Nonlinear and linear instability of the Rossby‐Haurwitz wave

AbstractAs is known, the large‐scale dynamics of barotropic atmosphere can approximately be described by the nonlinear barotropic vorticity equation. It is also well known that the Rossby‐Haurwitz (RH) waves, being exact solutions to this equation, represent one of the main features of meteorological fields. Therefore the stability properties of the RH wave are of considerable interest for deeper understanding of the low‐frequency variability of the atmosphere. Many works has been devoted to the barotropic instability of flows on a beta‐plane and a sphere. However, mathematically, the nonlinear stability problem of the RH wave is still far from its complete solution. Indeed, some of the stability results have been obtained numerically, and hence, contain calculation errors. Severe truncation of perturbations used in the spectral stability analysis, though leads to interesting and useful conclusions, does not allow obtaining comprehensive results. The weak point of some analytical nonlinear instability studies consists in using inappropriate norms for perturbations. It should also be noted that a necessary condition for the linear instability of the RH wave was obtained only recently (Skiba, 2000).In the present work, the nonlinear stability of the RH wave in an ideal incompressible fluid on a rotating sphere is analytically studied. Let H(n) be a subspace of homogeneous spherical polynomials of degree n. Mathematically, a RH wave of degree n is the sum of a super‐rotating flow of subspace H(1) and a homogeneous spherical polynomial of subspace H(n). First, we derive a conservation law for arbitrary RH‐wave perturbations which asserts that any perturbation evolves in such a way that its kinetic energy E(t) and enstrophy q(t) decrease, remain constant or increase simultaneously. The law is used to divide all the perturbations into three invariant sets depending on the value of their mean spectral number k(t)=q(t)/E(t) introduced by Fjortoft (1953). These sets are denoted as M where k(t)¡n(n+1) (large‐scale perturbations), N where k(t)¿n(n+1) (small‐scale perturbations), and Z where k(t)=n(n+1) (boundary surface between the sets M and N). Note that Z includes one more invariant set, namely, the subspace H(n). The existence of invariant sets of perturbations allows us to study the RH wave instability in each set separately. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerical Simulation of Baroclinic Waves with a Parameterized Boundary Layer

Abstract A dry three-dimensional baroclinic life cycle model is used to investigate the role of turbulent fluxes of heat and momentum within the boundary layer on midlatitude cyclones. Simulations are performed of life cycles for two basic states: with and without turbulent fluxes. The different basic states produce cyclones with contrasting frontal and mesoscale flow structures. The analysis focuses on the generation of potential vorticity (PV) in the boundary layer and its subsequent transport into the free troposphere. The dynamic mechanism through which friction mitigates a barotropic vortex is that of Ekman pumping. This has often been assumed to also be the dominant mechanism for baroclinic developments. The PV framework highlights an additional, baroclinic mechanism. Positive PV is generated baroclinically due to friction to the northeast of a surface low and is transported out of the boundary layer by a cyclonic conveyor belt flow. The result is an anomaly of increased static stability in the lower troposphere, which restricts the growth of the baroclinic wave. The reduced coupling between lower and upper levels can be sufficient to change the character of the upper-level evolution of the mature wave. The basic features of the baroclinic damping mechanism are robust for different frontal structures, with and without turbulent heat fluxes, and for the range of surface roughness found over the oceans.

Preemptive Forecasts Using an Ensemble Kalman Filter

Abstract An ensemble Kalman filter (EnKF) estimates the error statistics of a model forecast using an ensemble of model forecasts. One use of an EnKF is data assimilation, resulting in the creation of an increment to the first-guess field at the observation time. Another use of an EnKF is to propagate error statistics of a model forecast forward in time, such as is done for optimizing the location of adaptive observations. Combining these two uses of an ensemble Kalman filter, a “preemptive forecast” can be generated. In a preemptive forecast, the increment to the first-guess field is, using ensembles, propagated to some future time and added to the future control forecast, resulting in a new forecast. This new forecast requires no more time to produce than the time needed to run a data assimilation scheme, as no model integration is necessary. In an observing system simulation experiment (OSSE), a barotropic vorticity model was run to produce a 300-day “nature run.” The same model, run with a different vorticity forcing scheme, served as the forecast model. The model produced 24- and 48-h forecasts for each of the 300 days. The model was initialized every 24 h by assimilating observations of the nature run using a hybrid ensemble Kalman filter–three-dimensional variational data assimilation (3DVAR) scheme. In addition to the control forecast, a 64-member forecast ensemble was generated for each of the 300 days. Every 24 h, given a set of observations, the 64-member ensemble, and the control run, an EnKF was used to create 24-h preemptive forecasts. The preemptive forecasts were more accurate than the unmodified, original 48-h forecasts, though not quite as accurate as the 24-h forecast obtained from a new model integration initialized by assimilating the same observations as were used in the preemptive forecasts. The accuracy of the preemptive forecasts improved significantly when 1) the ensemble-based error statistics used by the EnKF were localized using a Schur product and 2) a model error term was included in the background error covariance matrices.

Inertial instability in rotating and stratified fluids: barotropic vortices

The unfolding of inertial instability in intially barotropic vortices in a uniformly rotating and stratified fluid is studied through numerical simulations. The vortex dynamics during the instability is examined in detail. We demonstrate that the instability is stabilized via redistribution of angular momentum in a way that produces a new equilibrated barotropic vortex with a stable velocity profile. Based on extrapolations from the results of a series of simulations in which the Reynolds number and strength of stratification are varied, we arrive at a construction based on angular momentum mixing that predicts the infinite-Reynolds-number form of the equilibrated vortex toward which inertial instability drives an unstable vortex. The essential constraint is conservation of total absolute angular momentum. The construction can be used to predict the total energy loss during the equilibration process. It also shows that the equilibration process can result in anticyclones that are more susceptible to horizontal shear instabilities than they were initially, a phenomenon previously observed in laboratory and numerical studies.

Energy extremals and nonlinear stability in a variational theory of a coupled barotropic flow - Rotating solid sphere system

A new variational principle—extremizing the fixed frame kinetic energy under constant relative enstrophy—for a coupled barotropic flow—rotating solid sphere system is introduced with the following consequences. In particular, angular momentum is transferred between the fluid and the solid sphere through a modeled torque mechanism. The fluid’s angular momentum is therefore not fixed but only bounded by the relative enstrophy, as is required of any model that supports super-rotation. The main results are: At any rate of spin Ω and relative enstrophy, the unique global energy maximizer for fixed relative enstrophy corresponds to solid-body super-rotation; the counter-rotating solid-body flow state is a constrained energy minimum provided the relative enstrophy is small enough, otherwise, it is a saddle point. For all energy below a threshold value which depends on the relative enstrophy and solid spin Ω, the constrained energy extremals consist of only minimizers and saddles in the form of counter-rotating states. Only when the energy exceeds this threshold value can pro-rotating states arise as global maximizers. Unlike the standard barotropic vorticity model which conserves angular momentum of the fluid, the counter-rotating state is rigorously shown to be nonlinearly stable only when it is a local constrained minima. The global constrained maximizer corresponding to super-rotation is always nonlinearly stable.

Summertime ENSO–North African–Asian Jet teleconnection and implications for the Indian monsoons

Recent findings indicate that ENSO may modulate the strength of the Indian monsoons by altering the meridional tropospheric temperature gradient between Asia and the equatorial Indian Ocean. Here we show that during northern hemisphere summertime El Niño events both colder upper tropospheric temperatures and increased vorticity anomalies are observed within the North African‐Asian (NAA) jet. The temperature response within the NAA jet is the dominant ENSO‐related change to the meridional tropospheric temperature gradient between Asia and the equatorial Indian Ocean. Forced solutions of the linearized barotropic vorticity equation indicate that these anomalies within the NAA jet are a response to northern hemisphere summertime El Niño‐related convective activity over the equatorial Pacific Ocean. Additional experiments indicate that westward propagating anomalies in both the tropics and middle latitudes produce the anomaly response within the NAA jet. These findings suggest that the teleconnection linking ENSO and the monsoons is mediated by the response of the NAA jet to westward propagating Rossby waves.

Phase transitions of barotropic flow coupled to a massive rotating sphere—Derivation of a fixed point equation by the Bragg method

The kinetic energy of barotropic flow coupled to an infinitely massive rotating sphere by an unresolved complex torque mechanism is approximated by a discrete spin–lattice model of fluid vorticity on a rotating sphere, analogous to a one-step renormalized Ising model on a sphere with global interactions. The constrained energy functional is a function of spin–spin coupling and spin coupling with the rotation of the sphere. A mean field approximation similar to the Curie–Weiss theory, modeled after that used by Bragg and Williams to treat a 2D Ising model of ferromagnetism, is used to find the barotropic vorticity states at thermal equilibrium for a given temperature and rotational frequency of the sphere. A fixed point equation for the most probable barotropic flow state is one of the main results. This provides a crude model of super- and sub-rotating planetary atmospheres in which the barotropic flow can be considered to be the vertically averaged rotating stratified atmosphere and where a key order parameter is the changeable amount of angular momentum in the barotropic fluid. Using the crudest two domains partition of the resulting fixed point equation, we find that in positive temperatures associated with low-energy flows, for fixed planetary spin larger than Ω c > 0 there is a continuous transition from a disordered state in higher temperatures to a counter-rotating solid-body flow state in lower positive temperatures. The most probable state is a weakly counter-rotating mixed state for all positive temperatures when planetary spin is smaller than Ω c . For sufficiently large spins Ω > 2 Ω c , there is a single smooth change from slightly pro-rotating mixed states to a strongly pro-rotating ordered state as the negative value of T increases (or decreases in absolute value). But for smaller spins Ω < 2 Ω c there is a transition from a predominantly mixed state (for T ⪡ - 1 ) to a pro-rotating state at β - ( Ω ) < 0 plus a second β c Ω , for which the fixed point equation has three fixed points when β < β c Ω instead of just the pro-rotating one when β c Ω < β . An argument based on comparing free energy shows to that the pro-rotating state is preferred when there are three fixed points because it has the highest free energy—at negative temperatures the thermodynamically stable state is the one with the maximum free energy. In the non-rotating case ( Ω = 0 ) the most probable state changes from a mixed state for all positive and large absolute-valued negative temperatures to an ordered state of solid-body flow at small absolute-valued negative temperatures through a standard symmetry-breaking second-order phase transition. The predictions of this model for the non-rotating problem and the rotating problem agree with the predictions of the simple mean field model and the spherical model. This model differs from previous mean field theories for quasi-2D turbulence in not fixing angular momentum nor relative enstrophy—a property which increases its applicability to coupled fluid–sphere systems and by extension to 2D turbulent flows in complex domains such as no-slip square boundaries where only the total circulation is fixed—as opposed to classical statistical equilibrium models such as the vortex gas model and Miller–Robert theories that fix all the vorticity moments. Furthermore, this Bragg mean field theory is well-defined for all positive and negative temperatures unlike the classical energy–enstrophy theories.

Spectral Energy Fluxes in Geostrophic Turbulence: Implications for Ocean Energetics

AbstractThe energy pathways in geostrophic turbulence are explored using a two-layer, flat-bottom, f-plane, quasigeostrophic model forced by an imposed, horizontally homogenous, baroclinically unstable mean flow and damped by bottom Ekman friction. A systematic presentation of the spectral energy fluxes, the mean flow forcing, and dissipation terms allows for a comprehensive understanding of the sources and sinks for baroclinic and barotropic energy as a function of length scale. The key new result is a robust inverse cascade of kinetic energy for both the baroclinic mode and the upper layer. This is consistent with recent observations of satellite altimeter data over the South Pacific Ocean. The well-known forward cascade of baroclinic potential and total energy was found to be very robust. Decomposing the spectral fluxes into contributions from different terms provided further insight. The inverse baroclinic kinetic energy cascade is driven mostly by an efficient interaction between the baroclinic velocity and the barotropic vorticity, the latter playing a crucial catalytic role. This cascade can be further enhanced by the baroclinic mode self-interaction, which is only present with nonuniform stratification (unequal layer depths). When model parameters are set such that modeled eddies compare favorably with observations, the inverse baroclinic kinetic energy cascade is actually much stronger than the well-known inverse cascade in the barotropic mode. The upper-layer kinetic energy cascade was found to dominate the lower-layer cascade over a wide range of parameters, suggesting that the surface cascade and time mean density stratification may be sufficient for estimating the depth-integrated cascade from ocean observations. This may find useful application in inferring the kinetic to gravitational potential energy conversion rate from satellite measurements.

Dynamics of the Extratropical Response to a Tropical Atlantic SST Anomaly

Abstract Previous atmospheric general circulation model (AGCM) experiments revealed that atmospheric responses to a tropical Atlantic sea surface temperature anomaly (SSTA) were asymmetric with respect to the sign of the SSTA. A positive SSTA produced a south–north dipole in geopotential heights, much like the North Atlantic Oscillation (NAO), while a negative SSTA yielded an eastward-propagating wave train, with the northern lobe of the NAO absent. Here these height responses are decomposed into components that are symmetric or antisymmetric with respect to the sign of the SSTA. The symmetric, or notionally linear, component is a nearly south–north dipole projecting on the NAO, while the antisymmetric, or notionally nonlinear, component is a different dipole. Experiments with a diagnostic linear baroclinic model (LBM) suggest that both components are maintained primarily by transient-eddy forcing. Dynamical mechanisms for the formation of the two components are explored using the LBM and a nonlinear barotropic vorticity equation model (BVM). Transient-eddy feedback is sufficient to explain the linear response. The NAO-like linear response occurs when the initial heating induces transient-eddy forcing in the exit of the Atlantic jet. The structure of the background absolute vorticity in this region is such that this transient-eddy forcing induces a nearly north–south dipole in anomalous geopotential heights. When the nonlinear self-interaction of this transient-induced low-frequency perturbation is included in the BVM, the dipole axis tilts to the east or west, resulting in a response that is nonlinear about the sign of the forcing.

Numerical stability of one method of asymptotic stabilization

A problem of numerical projection onto a stable manifold in a neighborhood of a fixed point of hyperbolic type is considered. It is shown that a nonlinear iterative method of projection construction is stable with respect to errors occurring in solutions of intermediate problems. A formulation of the corresponding assertion and results of numerical simulations for solution of the problem on asymptotic stabilization over initial data for an equation of barotropic vortex type on a semisphere are presented.

The generalized energy equation and instability in the two-layer barotropic vortex

The linear two-layer barotropic primitive equations in cylindrical coordinates are used to derive a generalized energy equation, which is subsequently applied to explain the instability of the spiral wave in the model. In the two-layer model, there are not only the generalized barotropic instability and the super high-speed instability, but also some other new instabilities, which fall into the range of the Kelvin-Helmholtz instability and the generalized baroclinic instability, when the upper and lower basic flows are different. They are perhaps the mechanisms of the generation of spiral cloud bands in tropical cyclones as well.

Periodic Structures of Rossby Wave under Influenceof Dissipation

A simple barotropic potential vorticity equation with theinfluence of dissipation is applied to investigate the nonlinearRossby wave in a shear flow in the tropical atmophere. By thereductive perturbation method, we derive the rotational KdV (rKdVfor short) equation. And then, with the help of Jacobi ellipticfunctions, we obtain various periodic structures for these Rossby waves. It is shown that dissipationis very important for these periodic structures of rational form.

Evolution of Dipole-Type Blocking Life Cycles: Analytical Diagnoses and Observations

Abstract A variable coefficient Korteweg–de Vries (VCKdV) system is derived by considering the time-dependent background flow and boundary conditions from a nonlinear, inviscid, nondissipative, and equivalent barotropic vorticity equation in a beta plane. One analytical solution obtained from the VCKdV equation can be successfully used to explain the evolution of atmospheric dipole-type blocking (DB) life cycles. Analytical diagnoses show that the background mean westerlies have great influence on evolution of DB during its life cycle. A weak westerly is necessary for blocking development and the blocking life period shortens, accompanied with the enhanced westerlies. The shear of the background westerlies also plays an important role in the evolution of blocking. The cyclonic shear is preferable for the development of blocking but when the cyclonic shear increases, the intensity of blocking decreases and the life period of DB becomes shorter. Weak anticyclonic shear below a critical threshold is also favorable for DB formation. Time-dependent background westerly (TDW) in the life cycle of DB has some modulations on the blocking life period and intensity due to the behavior of the mean westerlies. Statistical analysis regarding the climatological features of observed DB is also investigated. Observational results show that the Pacific is a preferred region for DB, especially at high latitudes. These features may associate with the weakest westerlies and the particular westerly shear structure over the northwestern Pacific.

Laboratory experiments on vortices colliding with multiple islands

AbstractLaboratory experiments have been performed to investigate the physical processes that govern the interaction of a self‐propagating barotropic cyclonic vortex with aligned circular cylinders. The motivation was to understand the dynamics which controls the interaction of North Brazil Current (NBC) rings with the Lesser Antilles (LA) in the Eastern Caribbean Sea. The geometrical parameter regulating the flow in the experiments was the ratio of the gap size to the diameter of the incident vortex, G /d. The range 0.03 ≤ G /d ≤ 0.4 was studied. Since the passages of the LA have values of 0.07 ≤ G /d ≤ 0.3 [2], the experimental results might explain recent oceanic observations [3]. (© 2006 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Six circumpolar currents—on the forcing of the Antarctic Circumpolar Current by wind and mixing

The transport of the Antarctic Circumpolar Current (ACC) is influenced by a variety of processes and parameters. A proper implementation of basin geometry, ocean topography and baroclinicity is known to be a fundamental requisite for a realistic simulation of the circulation and transport. Other, more subtle parameters are those of eddy-induced transports and diapycnal mixing of thermohaline tracers or buoyancy, either treated by eddy resolution or by a proper parameterization. Quite a number of realistic numerical simulations of the circulation in the Southern Ocean have recently been published. Many concepts on relations of the ACC transport to model parameters and forcing function are in discussion, however, without much generality and little success. We present a series of numerical simulations of circumpolar flow with a simplified numerical model, ranging from flat-bottom wind-driven flow to baroclinic flow with realistic topography and wind and buoyancy forcing. Analysis of the balances of momentum, vorticity, and baroclinic potential energy enables us to develop a new transport theory, which combines the most important mechanisms driving the circulation of the ACC and determining its zonal transport. The theory is based on the importance of the bottom vertical velocity in generating vorticity and shaping the baroclinic potential energy of the ACC. It explains the breaking of the \(f/h\)-constraint by baroclinicity and brings together in one equation the wind and buoyancy forcing of the current. The theory emphasizes the role of Ekman pumping and eddy diffusion of buoyancy to determine the transport. It also demonstrates that eddy viscosity effects are irrelevant in the barotropic vorticity balance and that friction arises via eddy diffusion of density. In this regime, the classical Stommel model of vorticity balance is revived where the bottom friction coefficient is replaced by \(K/\lambda^2\) (with the Gent–McWilliams coefficient \(K\) and the baroclinic Rossby radius \(\lambda\)) and a modified wind curl forcing appears.

Dynamical Subgrid-Scale Parameterizations from Direct Numerical Simulations

AbstractDynamical subgrid-scale parameterizations of stochastic backscatter, eddy drain viscosity, and net eddy viscosity have been formulated and calculated for two-dimensional turbulent flows on the sphere based on the statistics of direct numerical simulations (DNSs) with the barotropic vorticity equation. A relatively simple methodology based on a stochastic model representation of the subgrid-scale eddies, but which takes into account the memory effects of turbulent eddies, has been employed. The parameterizations have a cusp behavior at the cutoff wavenumber of the retained scales and have closely similar forms to those based on eddy damped quasi-normal Markovian (EDQNM) and direct interaction approximation (DIA) closure models. Large-eddy simulations (LESs) incorporating DNS-based subgrid-scale parameterizations are found to have kinetic energy spectra that compare closely with the results of higher-resolution DNS at the scales of LES for both isotropic turbulence and Rossby wave turbulence. The methodology presented is general and should be equally applicable to parameterizations of baroclinic processes and convective processes. Applications of the parameterizations to climate models and prediction models are discussed.