Vorticity Model Research Articles

Numerical study of a regularized barotropic vorticity model of geophysical flow

We study a Crank–Nicolson in time, finite element in space, numerical scheme for a Bardina regularization of the barotropic vorticity (BV) model. We derive the regularized model from the simplified Bardina model in primitive variables, present a numerical algorithm for it, and prove the algorithm is unconditionally stable with respect to the timestep size and optimally convergent in both space and time. Numerical experiments are provided that verify the theoretical convergence rates, and also that test the model/scheme on a benchmark double-gyre wind forcing experiment. For the latter test, we find the proposed model/scheme gives a good coarse mesh approximation to the highly resolved direct numerical simulation of the BV model, and compares favorably to related regularization model results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1492–1514, 2015

Water Velocity, Vorticity and Bed Deviation Modeling for a Reach from Damietta Branch Using K-ε Turbulence Model Solved by Cubic Interpolated Pseudo (CIP) Method

A multi-meander reach with length of 20 km located between km 130.0 and km 150.0 downstream of Delta barrages - Damietta branch was selected and numerically studied. This study aimed to simulate and study numerically the water velocity, vorticity and bed deviation of the curved zones for the reach under study and illustrating the relationship between them. Consequently, the vulnerable zones subjected to maximum velocities were accurately determined. Field data were collected and analyzed for the modeling process. A 3-D model called iRIC (International River Interface Corporative) based on an explicit finite difference method (Abbott-Ionescu scheme) was applied. Therefore, in order to fulfill such objective, standard K-ɛ turbulence model was employed using Cubic Interpolated Pseudo (CIP) method for solving the advection terms. For illustrating obviously the variation of water velocity with vorticity and deviation of bed elevation, two important zones consisted of several meanders were selected, the first zone located from km: 132.00 to km: 137.33 while the second located between km: 137.33 and km: 142.67. Through the modeling process, it was assumed that the sediment particles move in the bed layer zone only. From this study, it was found that for both selected zones, the velocity value was ranged between 0.13 m/sec and 0.24 m/sec, and it could be considered as a small range to make scouring process. It was observed also, that there was a noticeable relationship between water velocity, vorticity and deviation of bed elevation.

Improvement of a near wake model for trailing vorticity

A near wake model, originally proposed by Beddoes, is further developed. The purpose of the model is to account for the radially dependent time constants of the fast aerodynamic response and to provide a tip loss correction. It is based on lifting line theory and models the downwash due to roughly the first 90 degrees of rotation. This restriction of the model to the near wake allows for using a computationally efficient indicial function algorithm. The aim of this study is to improve the accuracy of the downwash close to the root and tip of the blade and to decrease the sensitivity of the model to temporal discretization, both regarding numerical stability and quality of the results. The modified near wake model is coupled to an aerodynamics model, which consists of a blade element momentum model with dynamic inflow for the far wake and a 2D shed vorticity model that simulates the unsteady buildup of both lift and circulation in the attached flow region. The near wake model is validated against the test case of a finite wing with constant elliptical bound circulation. An unsteady simulation of the NREL 5 MW rotor shows the functionality of the coupled model.

Pressure induced breather overturning on deep water: Exact solution

Abstract A vortical model of breather overturning on deep water is proposed. The action of wind is simulated by nonuniform pressure on the free surface. The fluid motion is described by an exact solution of 2D hydrodynamic equations for an inviscid fluid in Lagrangian variables. Fluid particles rotate in circles of different radii. Formation of contraflexure points on the breather profile is studied. The mechanism of wave breaking and the role of flow vorticity are discussed.

Long-term variability of the Kuroshio path south of Japan

This study investigates the long-term variability of the Kuroshio path south of Japan. Sensitivity experiments using a data-assimilative model suggest that the duration of the large meander (LM) strongly depends on the Kuroshio transport; specifically, low transport leads to a long duration of the LM. Actually, we find a good correlation between the duration of the past LMs and the Sverdrup transport estimated by a wind-driven linear baroclinic vorticity model. Then we explore favorable conditions for the LM and find a close relationship between the Kuroshio Extension (KE) state and the LM. That is, a precondition for the LM that the Kuroshio path on the Izu Ridge is fixed at a deep channel located around 34°N is achieved during a stable KE state. In addition, westward propagating signals with negative anomalies in the Kuroshio region and high sea-surface height (SSH) state east of Taiwan are key for generation of a small meander southeast of Kyushu that triggers a subsequent LM. The signals related to the above conditions change the upstream Kuroshio transport and velocity, which are consistent with features indicated by the former observational studies. Using reanalysis data, we construct long-time series of indices for the three conditions, which explain well the past LMs. The indices suggest that long-term non-LM states around 1970 and in the 1990s were attributed to a low-SSH state east of Taiwan and an unstable KE state, respectively.

Unforced Oscillation of Rip-Current Vortex Cells

Abstract A numerical simulation of a monochromatic surface gravity wave–driven flow over an alongshore quasi-periodic rip-channeled beach using the wave-resolving model Funwave is used to investigate coherent, very low-frequency (VLF) motions with characteristic frequencies f < 4.0 mHz inside of the surf zone generated by wave breaking. These oscillations of the nearshore cellular vorticity pattern occur for shore-normal waves over a wide range of amplitudes of the incident wave field and occur despite the wave forcing being essentially constant. The oscillations occur at the lower end of the VLF spectrum or around fp = 0.55 mHz. For small incident wave amplitudes, an equilibrium state consisting of a staggered counterrotating vortex array generates a net weak alongshore current that is also seen in drifter trajectories observed in the field. Using a simpler pseudospectral vorticity model of a single dipole generated by a smooth, stationary in time forcing function sb, this study shows show that the Strouhal number of the vortex shedding process responsible for the oscillation is dependent on the circulation strength of the vortices in the dipole, as well as the bottom friction parameter. This process includes the pinching off, advection, and eventual regeneration of the vortex in the dipole. A simple scaling argument shows good agreement with the frequencies observed in the simulations.

Vortical Freak Waves in Water Under External Pressure Action

A vortical model for freak wave formation in water is presented. The wind action is simulated by nonuniform pressure on the free surface. The motion of the fluid is described by an exact solution of 2D hydrodynamic equations for ideal inviscid fluid in Lagrangian variables. Fluid particles rotate in circles of different radius. The model describes the appearance of a freak wave in the field of the Gerstner wave. The physical parameters of the wave and feasibility of the proposed scenario are discussed.

RESTRICTIONS ON THE GEOMETRY OF THE PERIODIC VORTICITY EQUATION

We prove that several evolution equations arising as mathematical models for fluid motion cannot be realized as metric Euler equations on the Lie group DIFF∞(𝕊1) of all smooth and orientation-preserving diffeomorphisms on the circle. These include the quasi-geostrophic model equation, cf. [A. Córdoba, D. Córdoba and M. A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. 162 (2005) 1377–1389], the axisymmetric Euler flow in ℝd(see [H. Okamoto and J. Zhu, Some similarity solutions of the Navier–Stokes equations and related topics, Taiwanese J. Math. 4 (2000) 65–103]), and De Gregorio's vorticity model equation as introduced in [S. De Gregorio, On a one-dimensional model for the three-dimensional vorticity equation, J. Stat. Phys. 59 (1990) 1251–1263].

Long‐term change of the Pacific North Equatorial Current bifurcation in SODA

The long‐term change of the North Equatorial Current (NEC) bifurcation in the Pacific Ocean is assessed based on the recently developed Simple Ocean Data Assimilation (SODA, version 2.2.4). It is found that the NEC bifurcation latitude (NBL) has shifted southward over the past 60 years, although it displayed a slight northward migration from 1970 to 1992. This southward shift of the bifurcation latitude is associated with changes in the wind stress curl over the tropical Pacific Ocean between 10°N and 20°N, leading to the strengthening of the Kuroshio at its origin. The conclusion is further supported by simulations of Intergovernmental Panel on Climate Change models. It is demonstrated that the long‐term change of the seasonal south–north migration of the bifurcation is modulated by the southward shift of the mean position. Over the past 6 decades, the phase speed of first‐mode baroclinic Rossby waves (CR) at the latitude of the bifurcation increases from 13 cm s−1 in 1950 to 18 cm s−1 in 2005, and the corresponding seasonal amplitude increases (decreases) before (after) the mid‐1980s. Using a linear vorticity model, it is found that the long‐term modulation of the NBL seasonal migration amplitude is associated with the increase of CR in responses to the southward shift of the mean NBL. It is expected that the seasonal amplitude will decrease moderately in the following decades if the ocean continues warming.

Vortex generation and flow pattern development after a solitary wave passing over a bottom cavity

Several vortices are generated immediately inside the cavity for a short duration when a solitary wave passes over a bottom cavity in shallow water. Afterward, the primary vortex grows gradually in size and then is transported outward to breed another counter-rotating vortex to form a jet in the tranquil environment. In this study, a two-dimensional streamfunction–vorticity model was treated in a transient grid to conform the free surface and formulated by the Finite Analytic (FA) discretization to seek a time-accurate numerical solution. We used an overset grid to refine the small region around the cavity to improve the solution accuracy and obtain computational efficiency in the analysis. The model was first validated for a quasi-steady boundary-layer flow, induced by a solitary wave passing over a flat bottom, through the comparison of velocity profiles with Liu et al. [J Fluid Mech 2007;574:449–63]. Then, the evolved vortical flow for the cavity problem was investigated under the effects of various sizes of a cavity and various amplitudes of the incident solitary wave. These simulated results showed consistent agreement in the evolved vortex patterns with the visualization experiment in laboratory.

On the Linear Relationship between Loop Current Retreat Latitude and Eddy Separation Period

Abstract A linear correlation exists between the retreat latitude of the Loop Current following eddy separation and the subsequent eddy separation period. This empirical relationship was first identified in satellite altimeter-derived Loop Current metrics. In this paper, a simple vorticity model of the Loop Current is used to provide a semitheoretical basis for this relationship. After suitable scaling approximations, the theory predicts that the LC separation period is a linear function of retreat latitude, which agrees well with altimeter-derived empirical results. Specifically, the predicted slope and y intercept agree to within 9% and 2%, respectively, with the altimetry-derived values.

The geometry of a vorticity model equation

We provide rigorous evidence of the fact that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics describes the geodesic flow on the subgroup of orientation-preserving diffeomorphisms fixing one point, with respect to right-invariant metric induced by the homogeneous Sobolev norm $H^{1/2}$ and show the local existence of the geodesics in the extended group of diffeomorphisms of Sobolev class $H^{k}$ with $k\ge 2$.

Two-dimensionality of gravity water flows of constant nonzero vorticity beneath a surface wave train

We show that a free surface water flow of constant nonzero vorticity beneath a wave train and above a flat bed has to be two-dimensional and the vorticity must have only one nonzero component which points in the horizontal direction orthogonal to the direction of wave propagation. The obtained results are of relevance to studies of resonant wave train interactions in flows of constant nonzero vorticity: in contrast to irrotational flows, all wave trains have to propagate in the same direction. An important practical aspect of these considerations lies in that wave trains of constant vorticity model the interaction of swell with tidal currents, negative vorticity being appropriate for the flood current and positive vorticity for the ebb current.

The Role of Moist Processes in the Formation and Evolution of Mesoscale Snowbands within the Comma Head of Northeast U.S. Cyclones

Abstract The role of moist processes in regulating mesoscale snowband life cycle within the comma head portion of three northeast U.S. cyclones is investigated using piecewise potential vorticity (PV) inversion, modeling experiments, and potential temperature tendency budgets. Snowband formation in each case occurred along a mesoscale trough that extended poleward of a 700-hPa low. This 700-hPa trough was associated with intense frontogenetical forcing for ascent. A variety of PV evolutions among the cases contributed to midlevel trough formation and associated frontogenesis. However, in each case the induced flow from diabatic PV anomalies accounted for a majority of the midlevel frontogenesis during the band’s life cycle, highlighting the important role that latent heat release plays in band evolution. Simulations with varying degrees of latent heating show that diabatic processes associated with the band itself were critical to the development and maintenance of the band. However, changes in the meso-α-scale flow associated with the development of diabatic PV anomalies east of the band contributed to frontolysis and band dissipation. Conditional stability was reduced near 500 hPa in each case several hours prior to band formation. This stability remained small until band formation, when the stratification generally increased in association with the release of conditional instability. Previous studies have suggested that the dry slot is important for the initial stability reduction at midlevels, but this was not evident for the three banding cases examined. Rather, differential horizontal temperature advection in moist southwest flow ahead of the upper trough was the dominant process that reduced the midlevel conditional stability.

A depth-integrated model for weakly dispersive, turbulent, and rotational fluid flows

A set of weakly dispersive Boussinesq-type equations, derived to include viscosity and vorticity terms in a physically consistent manner, is presented in conservative form. The model includes the approximate effects of bottom-induced turbulence, in a depth-integrated sense, as a second-order correction. Associated with this turbulence, vertical and horizontal rotational effects are captured. While the turbulence and horizontal vorticity models are simplified, a model with known physical limitations has been derived that includes the quadratic bottom friction term commonly added in an ad hoc manner to the inviscid equations. An interesting result of this derivation is that one should take care when adding such ad hoc models; it is clear from this exercise that (1) it is not necessary to do so – the terms can be included through a consistent derivation from the viscous primitive equations – and (2) one cannot properly add the quadratic bottom friction term without also adding a number of additional terms in the integrated governing equations. To solve these equations numerically, a highly accurate and stable model is developed. The numerical method uses a fourth-order MUSCL-TVD scheme to solve the leading order (shallow water) terms. For the dispersive terms, a cell averaged finite volume method is implemented. To verify the derived equations and the numerical model, four cases of verifications are given. First, solitary wave propagation is examined as a basic, yet fundamental, test of the models ability to predict dispersive and nonlinear wave propagation with minimal numerical error. Vertical velocity distributions of spatially uniform flows are compared with existing theory to investigate the effects of the newly included horizontal vorticity terms. Other test cases include comparisons with experiments that generate strong vorticity by the change of bottom bathymetry as well as by tidal jets through inlet structures. Very reasonable agreements are observed for the four cases, and the results provide some information as to the importance of dispersion and horizontal vorticity.

Radial velocity dependence in the spatial orientations of galaxies in and around the local supercluster

We study the radial velocity dependence in the spatial orientations of 10 562 spiral galaxies that have radial velocity <5000 km s −1 . The inclination angle and intrinsic flatness of galaxy are used to covert two-dimensional given parameters into three-dimensional spin vectors of the galaxy. We have performed Kolmogorov–Smirnov, Kuiper and Fourier tests in order to examine non-random effects in the expected isotropic distributions. The galaxies that have radial velocities 1500 to 2000 and 3000 to 3500 km s −1 show preferred alignments in both the two- and three-dimensional analysis. The possible explanations of the results will be discussed. The origin of the angular momentum and the Hubble recessional velocity of galaxies is closely related to the physics of the early universe. The dynamical ‘fossil’ nature of the alignment of rotational axes of galaxies in superclusters provides us with the information on the formation process of galaxy at the early epoch (Djorgovski 1987). There have been contradictory theoretical models predicting various galaxy cluster formation scenarios. The ‘pancake model’ (Doroshkevich 1973; Shandarin 1974) predicts that the angular momentum of galaxies tends to lie within the cluster plane whereas the ‘primordial vorticity model’ (Ozernoy 1978) says that the angular momentum of galaxies tends to be oriented perpendicular the cluster plane. The ‘hierarchy model’ (Peebles 1969) predicts that the direction of angular momentum is entirely random. According to this scenario, galaxies were first formed and then obtained their angular momentum by tidal forces while they were gathering gravitationally to form a cluster with no dissipation. Lee & Pen (2002) found the observational evidence for the alignment of galaxy spin axes with the local tidal share field. Their result is consistent with the linear tidal-torque model based on gravitational instability. In an analytical study, Lee (2004) concluded a large-scale coherence in the orientation of galaxies in the two-dimensional sheet-like structures in the frame of the tidal-torque theory. Godlowski (1994) studied the dependence of galaxy alignment on radial velocity (RV) of 2227 galaxies in the local supercluster (LSC) and found anisotropy for the galaxies that have RVs 1000 to 1500 km s −1 . He concluded the dominance of Virgo cluster

Negative Specific Heat in a Quasi-2D Generalized Vorticity Model

Negative specific heat is a dramatic phenomenon where processes decrease in temperature when adding energy. It has been observed in gravo-thermal collapse of globular clusters. We now report finding this phenomenon in bundles of nearly parallel, periodic, single-sign generalized vortex filaments in the electron magnetohydrodynamic model for the unbounded plane under strong magnetic confinement. We derive the specific heat using a steepest-descent method and a mean-field property. Our derivations show that as temperature increases, the overall size of the system increases exponentially and the energy drops. The implication of negative specific heat is a runaway reaction, resulting in a collapsing inner core surrounded by an expanding halo of filaments.

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.

Two-layer quasigeostrophic potential vorticity model

We formulate the two-layer quasigeostrophic potential vortex (QGPV) model as an infinite-dimensional Hamiltonian system and apply Noether’s theorem to derive the conservation laws for these equations. The point (singular) vortex QGPV model is also analyzed as a finite-dimensional Hamiltonian system. By virtue of its point symmetries, the invariants for the finite-dimensional system are obtained. The existence of these point symmetries is used to prove the integrability of the three-vortex QGPV problem. We analyze the relative equilibrium configurations for the three-vortex, two-layer problem and study collapsing and merging (collapse in different layers) configurations. We also study two special classes of symmetrical four-vortex QGPV problems and establish their integrability and present some simulations for one of these classes of integrable QGPV four-vortex problems.

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.