Barotropic Vorticity Research Articles

The role of higher vorticity moments in a variational formulation of Barotropic flows on a rotating sphere

The effects of a higher vorticity moment on a variational problemfor barotropic vorticity on a rotating sphere is examined rigorouslyin the framework of the Direct Method. This variational modeldiffers from previous work on the Barotropic Vorticity Equation(BVE) in relaxing the angular momentum constraint, which then allowsus to state and prove theorems that give necessary and sufficientconditions for the existence and stability of constrained energyextremals in the form of super and sub-rotating solid-body steadyflows. Relaxation of angular momentum is a necessary step in themodeling of the important tilt instability where the rotational axisof the barotropic atmosphere tilts away from the fixed north-southaxis of planetary spin. These conditions on a minimal set ofparameters consisting of the planetary spin, relative enstrophy andthe fourth vorticity moment, extend the results of previous work and clarify the role of the higher vorticity moments in models of geophysical flows.

On resonant Rossby—Haurwitz triads

The dynamics of non-divergent flow on a rotating sphere are described by the conservation of absolute vorticity. The analytical study of the non-linear barotropic vorticity equation is greatly facilitated by the expansion of the solution in spherical harmonics and truncation at low order. The normal modes are the well-known Rossby—Haurwitz (RH) waves, which represent the natural oscillations of the system. Triads of RH waves, which satisfy conditions for resonance, are of critical importance for the distribution of energy in the atmosphere.We show how non-linear interactions of resonant RH triads may result in dynamic instability of large-scale components. We also demonstrate a mathematical equivalence between the equations for an orographically forced triad and a simple mechanical system, the forced-damped swinging spring. This equivalence yields insight concerning the bounded response to a constant forcing in the absence of damping. An examination of triad interactions in atmospheric reanalysis data would be of great interest.

Laboratory Modeling of Geophysical Vortices

Investigators have modeled oceanic and atmospheric vortices in the laboratory in a number of different ways, employing background rotation, density effects, and geometrical confinement. In this article, we address barotropic vortices in a rotating fluid, emphasizing generation techniques, instability issues, and topography effects in particular. We then review work on vortices in shallow fluid layers, including topography effects on vortices in coastal areas and the role of vortices in tidal exchange between two connected basins.

Diurnal-period internal waves near point conception, California

Diurnal-period internal waves were observed near Point Conception California, using an array of moorings extending 120 km along the inner shelf. The waves have an along-shelf coherence scale of at least 50 km, and appear to propagate nearly straight onshore. Wave amplitudes vary over time, depending on thermal stratification and the amplitude of the diurnal sea breeze oscillation. Barotropic tides and vorticity over the mid-shelf are not correlated with internal wave amplitude. Large amplitude internal waves, with supercritical Froude numbers, are observed in mid-summer. Although such waves may drive vertical mixing and cross-shelf transport of passive particles, there is no significant correlation between wave amplitude and invertebrate settlement in the Santa Barbara Channel.

Northern Hemisphere Stationary Waves in Future Climate Projections

Abstract The response of the atmospheric large-scale circulation to an enhanced greenhouse gas (GHG) forcing varies among coupled global climate model (CGCM) simulations. In this study, 16 CGCM simulations of the response of the climate system to a 1% yr−1 increase in the atmospheric CO2 concentration to quadrupling are analyzed with focus on Northern Hemisphere winter. A common signal in 14 out of the 16 simulations is an increased or unchanged stationary wave amplitude. A majority of the simulations may be categorized into one of three groups based on the GHG-induced changes in the atmospheric stationary waves. The response of the zonal mean barotropic wind is similar within each group. Fifty percent of the simulations belong to the first group, which is categorized by a stationary wave with five waves encompassing the entire NH and a strengthening of the zonal mean barotropic wind. The second and third groups, respectively consisting of three and two simulations, are characterized by a broadening and a northward shift of the zonal mean barotropic wind, respectively. A linear model of barotropic vorticity is employed to study the importance of these mean flow changes to the stationary wave response. The linear calculations indicate that the GHG-induced mean wind changes explain 50%, 4%, and 37% of the stationary wave changes in each group, respectively. Thus, for the majority of simulations the zonal mean wind changes do significantly explain the stationary wave response.

Passage of a Barotropic Vortex through a Gap

Abstract The flow of a vortex advected by a uniform current toward a gap in a straight barrier is studied. This is an idealization of flows observed in the world’s oceans, for example, in the Yucatan Channel and in the various passages of the Lesser Antilles. The vortex evolution and the transport properties are studied as a function of three nondimensional parameters related to the vortex intensity, the vortex initial position, and the gap’s span. The flow evolution is computed numerically with a two-dimensional inviscid model. The vortex is observed to behave in one of the following ways: it passes completely through the gap, it splits and only a fraction passes, or it stays entirely in the upstream side of the wall. In each of these regimes, transport and mixing are analyzed using the Lagrangian flow geometry, finite-size Lyapunov exponents, and residence times of fluid particles. Laboratory experiments are performed in a homogeneous, rotating fluid. In the region of parameter space that leads to total passage of the vortex, there is good agreement between the evolution observed in the laboratory and that obtained numerically. In the regions of parameter space, which lead to partial passage and full blockage, the agreement is good only until the generation of vorticity on the walls becomes important.

Forecasts by PHONIAC

The first computer weather forecasts were made in 1950, using the ENIAC (Electronic Numerical Integrator and Computer). The ENIAC forecasts led to operational numerical weather prediction within five years, and paved the way for the remarkable advances in weather prediction and climate modelling that have been made over the past half century. The basis for the forecasts was the barotropic vorticity equation (BVE). In the present study, we describe the solution of the BVE on a mobile phone (cell-phone), and repeat one of the ENIAC forecasts. We speculate on the possible applications of mobile phones for micro-scale numerical weather prediction.

Algorithm 888

A collection of MATLAB classes for computing and using spherical harmonic transforms is presented. Methods of these classes compute differential operators on the sphere and are used to solve simple partial differential equations in a spherical geometry. The spectral synthesis and analysis algorithms using fast Fourier transforms and Legendre transforms with the associated Legendre functions are presented in detail. A set of methods associated with a spectral_field class provides spectral approximation to the differential operators ∇ ⋯, ∇ ×, ∇, and ∇ 2 in spherical geometry. Laplace inversion and Helmholtz equation solvers are also methods for this class. The use of the class and methods in MATLAB is demonstrated by the solution of the barotropic vorticity equation on the sphere. A survey of alternative algorithms is given and implementations for parallel high performance computers are discussed in the context of global climate and weather models.

Vortex Lines within Low-Level Mesocyclones Obtained from Pseudo-Dual-Doppler Radar Observations

Abstract Vortex lines passing through the low-level mesocyclone regions of six supercell thunderstorms (three nontornadic, three tornadic) are computed from pseudo-dual-Doppler airborne radar observations obtained during the Verification of the Origins of Rotation in Tornadoes Experiment (VORTEX). In every case, at least some of the vortex lines emanating from the low-level mesocyclones form arches, that is, they extend vertically from the cyclonic vorticity maximum, then turn horizontally (usually toward the south or southwest) and descend into a broad region of anticyclonic vertical vorticity. This region of anticyclonic vorticity is the same one that has been observed almost invariably to accompany the cyclonic vorticity maximum associated with the low-level mesocyclone; the vorticity couplet straddles the hook echo of the supercell thunderstorm. The arching of the vortex lines and the orientation of the vorticity vector along the vortex line arches, compared to the orientation of the ambient (barotropic) vorticity, are strongly suggestive of baroclinic vorticity generation within the hook echo and associated rear-flank downdraft region of the supercells, and subsequent lifting of the baroclinically altered/generated vortex lines by an updraft. Discussion on the generality of these findings, possible implications for tornadogenesis, and the similarity of the observed vortex lines to vortex lines in larger-scale convective systems are included as well.

Analytical solution of nonlinear barotropic vorticity equation

The C κ (κ ≥ 3) stability of nonlinear barotropic vorticity equation was proved. The necessary and sufficient conditions for the initial value problem to be well-posed were presented. Under the conditions of well-posedness, the corresponding analytical solution was also gained.

A high‐order element‐based Galerkin method for the barotropic vorticity equation

AbstractA high‐order element‐based Galerkin method is developed to solve the non‐divergent barotropic vorticity equation (BVE). The solution process involves solving a conservative transport equation for the vorticity fields and a Poisson equation for the stream function fields. The discontinuous Galerkin method is employed for solving the transport equation and a spectral element method (continuous Galerkin) is used for the Poisson equation. A third‐order strong stability preserving explicit Runge–Kutta scheme is used for time integration.A series of tests have been performed to validate the model, which include the evolution of an idealized tropical cyclone and interaction of dual vortices in close proximity. The numerical convergence study is performed by solving the BVE on the sphere where the analytic solution is known. The test results are consistent with physical observations, and the model exhibits exponential convergence. Copyright © 2008 John Wiley & Sons, Ltd.

Dependence of vortex axisymmetrization on the characteristics of the asymmetry

AbstractThis study investigates how different characteristics of initial asymmetries, including their positions and profiles, can impact on the vortex axisymmetrization process for barotropic vortices. When an initial disturbance is placed near the core of a vortex, a new asymmetry is generated inside the original asymmetry and grows due to its upshear tilt. Differential basic‐state rotation then shifts the phase to a downshear tilt and the asymmetry weakens. As the initial radius of the imposed asymmetry is increased, the initial upshear tilt of the asymmetry decreases. There is also a decrease in the efficiency with which the differential rotation shifts the phase tilt from upshear to downshear. The latter is related to differential radial propagation of the asymmetry in the form of vortex Rossby waves. These two mechanisms that are position‐dependent act against each other. There is an optimal radius at which the energy exchange between the symmetric and asymmetric flows is maximized. For a range of very different basic‐state profiles examined here, the optimal radius is around 1.5 to 2 times the radius of the maximum wind. The initial growth of asymmetries with higher azimuthal wavenumbers is weaker than their lower‐wavenumber counterparts due to a smaller upshear phase tilt with their smaller azimuthal length‐scales.Nonlinearity reduces the magnitude and multiple perturbations of the newly induced inner asymmetry, and also limits the radial propagation of the asymmetry. The further the asymmetry is away from the core, the slower the axisymmetrization is. Depending on the position of the initial asymmetry, the basic state can have an increase of the maximum wind, a double‐peak profile, or an increase of its outer wind profile through axisymmetrization. Copyright © 2008 Royal Meteorological Society

The Recent Decline of the Arctic Summer Sea-Ice Cover in the Context of Internal Climate Variability

By means of a 21-year simulation of a coupled regional pan-Arctic atmosphere-ocean-ice model for the 1980's and 1990's and comparison of the model results with SSM/I satellite-derived sea-ice concentrations, the patterns of maximum amplitude of interannual variability of the Arctic summer sea-ice cover are revealed. They are shown to concentrate beyond an area enclosed by an isopleth of barotropic planetary potential vorticity that marks the edge of the cyclonic rim current around the deep inner Arctic basin. It is argued that the propagation of the interannual variability signal farther into the inner Arctic basin is hindered by the dynamic isolation of upper Arctic Ocean and the high summer cloudiness usually appearing in the central Arctic. The thinning of the Arctic sea-ice cover in recent years is likely to be jointly responsible for its exceptionally strong decrease in summer 2007 when sea-ice decline was favored by anomalously high atmospheric pressure over the western Arctic Ocean, which can be regarded as a typical feature for years with low sea-ice extent. In addition, unusually low cloud cover appeared in summer 2007, which led to substantial warming of the upper ocean. It is hypothesized that the coincidence of several favorable factors for low sea-ice extent is responsible for this extreme event. Owing to the important role of internal climate variability in the recent decline of sea ice, a temporal return to previous conditions or stabilization at the current level can not be excluded just as further decline.

A Form of Potential Vorticity Equation for Depth-Integrated Flow with a Free Surface

Abstract A form of linear, barotropic potential vorticity equation is derived for an ocean with a free surface, in which only one scalar variable appears (ocean bottom pressure, or subsurface pressure). Unlike quasigeostrophic or rigid-lid derivations, the only approximation made (apart from linearization) is that changes in the circulation must be slow compared with the inertial frequency. Effects of stratification are included, but only parametrically in the sense that density is treated as a given quantity or forcing term rather than a variable.

Northward propagation of the subseasonal variability over the eastern Pacific warm pool

Previous studies suggest that easterly vertical shear of summer mean flow could be a key factor in the northward propagation of the subseasonal variability (SSV) over the Asian monsoon region (e.g., Jiang et al., 2004). Analysis indicates that this same mechanism may also be dictating propagation features over the eastern Pacific (EPAC). Consistent with the presence of local easterly vertical wind shear, northward propagation of the SSV is also evident over the EPAC with a phase speed of about 0.6 deg day−1. Moreover, similar to its counterpart in the Indian Ocean, the northward propagating SSV over this region is also characterized by positive anomalies of equivalent barotropic vorticity and lower‐tropospheric specific humidity, and planetary boundary layer (PBL) convergence, to the north of the convection center. Thus, the present study provides another independent example of the essential role that easterly vertical wind shear may be playing in regulating the meridional migration of the SSV.

A Barotropic Model of the Angular Momentum–Conserving Potential Vorticity Staircase in Spherical Geometry

Abstract An idealized analytical model of the barotropic potential vorticity (PV) staircase is constructed, constrained by global conservation of absolute angular momentum, perfect homogenization of PV in mixing zones between (prograde) westerly jets, and the requirement of barotropic stability. An imposed functional relationship is also assumed between jet speed and latitudinal separation using a multiple of the “dynamical Rossby wave” Rhines scale inferred from the strength of westerly jets. The relative simplicity of the barotropic system provides a simple relation between absolute angular momentum and PV (or absolute vorticity). A family of solutions comprising an arbitrary number of jets is constructed and is used to illustrate the restriction of jet spacing and strength imposed by the constraints of global conservation of angular momentum and barotropic stability. Asymptotic analysis of the theoretical solution indicates a limiting ratio of jet spacing to the dynamical Rhines scale equal to the square root of 6, meaning that westerly jets are spaced farther apart than predicted by the dynamical Rhines scale. It is inferred that an alternative “geometrical” Rhines scale for jet spacing can be obtained from conservation of absolute angular momentum on the sphere if the strength of zonal jets is known from other considerations. Numerical simulations of the full (nonaxisymmetric) equations reveal a pattern of zonal jet evolution that is consistent with our construction of ideal PV staircases in spherical geometry (which can be considered as limiting cases), as well as with the asymptotic analysis of a geometrical Rhines scale. The evolution of the PV staircase originating from an upscale cascade of energy in the barotropic model is therefore seen to depend on conservation of energy (for the strength of jets) and conservation of absolute angular momentum (for the spacing and number of jets). Further analysis of the numerical results confirms a “Taylor identity” relating the flux of eddy potential vorticity to mean-flow acceleration. Eddy fluxes are responsible for the occasional transitions between mode number as well as for maintaining the sharp westerly jets against small-scale dissipation. Suggestions are made for extending the theoretical model to PV staircases that are asymmetric between hemispheres or with latitudinal variation of amplitude, as modeled in the shallow-water system.

Laboratory experiments on mesoscale vortices colliding with an island chain

The present laboratory study investigates the behavior of a self‐propagating barotropic cyclonic vortex colliding perpendicularly with aligned circular cylinders and determines the condition for a vortex to bifurcate and split into multiple vortices and/or to generate dipoles downstream of the cylinders. During the experiments, four parameters were varied: G, the gap width between the cylinders; d, the diameter of the incident vortex; Ydis, a parameter expressing the initial vortex positions; and Disl, the total length of the “middle” island. It has been observed that as long as 0.1 < G/d ≤ 0.4, the vortex fluid was funneled between two cylinders at one of the gaps and a dipole generally formed, much like water ejected from a circular nozzle generates a dipole ring. After the dipole formed, the cyclonic part of the dipole became dominant and self propagated away from the cylinders. Furthermore, in some experiments having 0.2 < Disl/d ≤ 0.5, after a weak dipole formed, the remnant of the original vortex moved zonally “south.” When the remnant of the vortex came in contact with a new cylinder, fluid peeled off the outer edge of the vortex and a so‐called “streamer” went around the cylinder in a counterclockwise direction. Under the right conditions, this fluid formed a new cyclonic vortex in the wake of the cylinder, causing bifurcation of the original vortex into two vortices, as observed in previous studies. In general, independently of the configurations and Ydis, the number of cyclonic vortices downstream of the cylinders was one, either originating from the dipole or generated by the bifurcation of the original vortex. The vortex center position, radius, and circulation, before and after the interaction, were computed from its velocity field. It was found that for 0.1 < G/d ≤ 0.4, intense vortices experienced greater amplitude loss than weak vortices. The formation of both a dominant cyclone and an anticyclone (i.e., a dipole) downstream of the aligned cylinders, representing an island chain, is in agreement with recent oceanic observations of North Brazil Current (NBC) rings interacting with the Lesser Antilles in the Eastern Caribbean Sea. Since the passages of the Lesser Antilles have values of 0.07 ≤ G/d ≤ 0.3, the oceanic observations might be explained by the experimental results reported in this paper.

Stability of some exact solutions of the shallow‐water equations for testing numerical models in spherical geometry

AbstractFive families of exact axisymmetric solutions of the nonlinear shallow‐water equations in spherical geometry have recently been proposed as an aid to the development and testing of global numerical models. Sufficient conditions for the stability of these solutions are here derived to guide the choice of values for the family parameters. Thus it can be ensured that any significant time evolution occurring in a numerical model initialised with one of these exact solutions is of numerical origin, and does not reflect an inherent physical instability. With the caveat that only sufficient conditions for stability are examined, it appears that planetary rotation stabilises the solutions (as would be so if the flow were governed by barotropic vorticity dynamics), and that low Rossby and Froude numbers favour their stability. © Crown Copyright 2008. Reproduced with the permission of the Controller of HMSO. Published by John Wiley & Sons, Ltd

Non-linear dynamics of barotropic Rossby waves in a meridional shear flow

The barotropic vorticity equation is solved numerically and analytically in order to study the non-linear evolution of a barotropic forced Rossby wave in a meridional shear flow. The flow configuration is an idealised model for a western boundary current in an oceanic basin. There is a critical layer at the longitude where the shear flow speed is equal to the wave phase speed, and the wave is incident on the critical layer from the east or west. The numerical solutions show that a non-linear eastward-propagating Rossby wave is absorbed by the mean flow at the critical layer at early times and reflected at later times, but a westward-propagating wave is transmitted through the critical layer. The asymptotic analyses show that the critical layer thickness is O(ϵ1/2), where ϵ is the relative magnitude of the perturbation to the basic flow, and that if the solution includes an eastward-propagating component the wave-induced mean flow increases with time in the critical layer. The westward-propagating part of the solution is non-singular and does not on its own affect the mean flow in the critical layer.

An Analysis of West African Dynamics Using a Linearized GCM*

Abstract This study utilizes a linear, primitive equation spherical model to study the development and propagation of easterly wave disturbances over West Africa. Perturbations are started from an initial disturbance consisting of a barotropic vortex and the governing equations are integrated forward in time. The perturbations are introduced into basic states corresponding to the observed dynamical and thermodynamical characteristics of two wet years in the Sahel and two dry years. The model simulations show consistent contrasts in wave activity between the wet and dry years. The waves are markedly stronger in the wet years and show a barotropic structure throughout the troposphere. The waves tend to extend throughout the troposphere to the level of the tropical easterly jet (TEJ) in the wet years, but not in the dry years. The upper-tropospheric shear, which is stronger in wet years, appears to be a key factor in wave development. This shear is dependent on the intensity of the TEJ, suggesting that the TEJ is an important factor in interannual variability in the Sahel. When the overall shear is weak, vertical development is suppressed. Another contrast is that in the dry years the growth rates show a single maximum around 3000–4000 km, but in the wet years there is a second, around 6000–7000 km. This suggests that both synoptic-scale and planetary-scale waves are active in the rainy season of some wet years. Imposing considerations of potential vorticity, the generation of planetary-scale waves implies a strong link between the surface and the TEJ in wet years. Such a link is absent in the dry years. This is likely a major factor in the interannual variability of rainfall in the Sahel.