Academic literature on the topic 'Linear Vorticity Balance'

Abstract This study develops a linear numerical model to address the source mechanism of the gravity waves generated within a vortex dipole simulated in a fully nonlinear nonhydrostatic mesoscale model. The background flow for this linear model is obtained from potential vorticity inversion constrained by the nonlinear balance equation. The forcing imposed in the linear model is derived from an imbalance in the large-scale flow—that is, the forcing or imbalance in the vorticity, divergence, and thermodynamic equations, respectively. The response from the sum of these imbalanced forcings obtained from the linear dynamics shows well-defined gravity wave signals, which compare reasonably well in terms of location, phase, and amplitude with the gravity waves simulated in a fully nonlinear nonhydrostatic mesoscale model. It is found that the vorticity forcing, largely due to the advection of balanced relative vorticity, is the leading contributor to the gravity waves in the exit region of the vortex-dipole jet.

It is now well established that two distinct types of motion occur in geophysical turbulence: slow motions associated with potential vorticity advection and fast oscillations due to inertia–gravity waves (or acoustic waves). Many studies have theorized the existence of a flow for which the entire motion is controlled by the potential vorticity (or one ‘master variable’) – this is known as balance. In real geophysical flows, deviations from balance in the form of inertia–gravity waves or ‘imbalance’ have often been found to be small. Here we examine the extent to which balance holds in rotating stratified turbulence which is nearly balanced initially.Using the non-hydrostatic fluid dynamical equations under the Boussinesq approximation, we analyse properties of rotating stratified turbulence spanning a range of Rossby numbers (Ro≡|ζ|max/f) and the frequency ratios (c≡N/f) where ζ is the relative vertical vorticity, f is the Coriolis frequency and N is the buoyancy frequency. Using a recently introduced diagnostic procedure, called ‘optimal potential vorticity balance’, we extract the balanced part of the flow in the simulations and assess how the degree of imbalance varies with the above parameters.We also introduce a new and more efficient procedure, building upon a quasi-geostrophic scaling analysis of the complete non-hydrostatic equations. This ‘nonlinear quasi-geostrophic balance’ procedure expands the equations of motion to second order in Rossby number but retains the exact (unexpanded) definition of potential vorticity. This proves crucial for obtaining an accurate estimate of balanced motions. In the analysis of rotating stratified turbulence at Ro≲1 and N/f≫1, this procedure captures a significantly greater fraction of the underlying balance than standard (linear) quasi-geostrophic balance (which is based on the linearized equations about a state of rest). Nonlinear quasi-geostrophic balance also compares well with optimal potential vorticity balance, which captures the greatest fraction of the underlying balance overall.More fundamentally, the results of these analyses indicate that balance dominates in carefully initialized simulations of freely decaying rotating stratified turbulence up to O(1) Rossby numbers when N/f≫1. The fluid motion exhibits important quasi-geostrophic features with, in particular, typical height-to-width scale ratios remaining comparable to f/N.

Abstract Balance dynamics are proposed in a probabilistic framework, assuming that the state variables and the master, or control, variables are random variables described by continuous probability density functions. Balance inversion, defined as recovering the state variables from the control variables, is achieved through Bayes’ theorem. Balance dynamics are defined by the propagation of the joint probability of the state and control variables through the Liouville equation. Assuming Gaussian statistics, balance inversion reduces to linear regression of the state variables onto the control variables, and assuming linear dynamics, balance dynamics reduces to a Kalman filter subject to perfect observations given by the control variables. Example solutions are given for an elliptical vortex in shallow water having unity Rossby and Froude numbers, which produce an outward-propagating pulse of inertia–gravity wave activity. Applying balance inversion to the potential vorticity reveals that, because potential vorticity and divergence share well-defined patterns of covariability, the inertia–gravity wave field is recovered in addition to the vortical field. Solutions for a probabilistic balance dynamics model applied to the elliptical vortex reveal smaller errors (“imbalance”) for height control compared to potential vorticity control. Important attributes of the probabilistic balance theory include quantification of the concept of balance manifold “fuzziness,” and clear state-independent definitions of balance and imbalance in terms of the range of the probabilistic inversion operators. Moreover, the theory provides a generalization of the notion of balance that may prove useful for problems involving moist physics, chemistry, and tropical circulations.

Abstract Beta, the meridional gradient of planetary vorticity, causes tropical cyclones to propagate poleward and westward at approximately 2 m s−1. In a previous shallow-water linear model, the simulated vortex accelerated without limit, ostensibly because beta forced a free linear mode. In the analogous nonlinear model, wave–wave interaction limited the propagation speed. Subsequent work based upon the asymmetric balance (AB) approximation was unable to replicate the linear result. The present barotropic nondivergent model replicates the linear beta gyres as a streamfunction dipole with a uniform southeasterly ventilation flow across the vortex. The simulated storm accelerates to unphysical, but finite, speeds that are limited by vorticity filamentation. In the analogous nonlinear model, nonlinearly forced wavenumber-1 gyres have opposite phase to the linear gyres so that their ventilation flow counteracts advection by the linear gyres to limit the overall vortex speed to approximately 3 m s−1. A bounded mean vortex with zero circulation at large radius must contain an outer annulus of anticyclonic vorticity to satisfy the circulation theorem. The resulting positive mean vorticity gradient constitutes an outer waveguide that supports downstream-propagating, very-low-frequency vortex Rossby waves. It is confined between an inner critical radius where the waves are absorbed and an outer turning point where they are reflected. Vorticity filamentation at the critical radius limits the beta-drift acceleration. The original unlimited linear acceleration stemmed from too-weak dissipation caused by second-order diffusion applied to velocity components instead of vorticity. Fourth-order diffusion and no outer waveguide in the Rankine-like vortex of the AB simulations plausibly explain the different results.

Abstract An exact solution of the primitive equations (PEs) and the corresponding exact solutions of the alternative balance (AB), geostrophic momentum (GM), and quasigeostrophic (QG) equations are presented. The PE solution illustrates how the temperature and horizontal vorticity fields evolve in a linear horizontal flow with constant deformation and vertical vorticity when the initial temperature field is also linear, as well as how ageostrophic circulations are produced. The other exact solutions show the errors produced by the various approximations and confirm that the AB solution is more accurate than the GM one and that the QG solution is almost always the most inexact. The utility of the Q vector and similar vectors is examined for each solution. The PE solution verifies that in a hyperbolic wind field (i) the isotherms eventually parallel the outflow axis, (ii) the ageostrophic circulation ultimately becomes normal to the outflow axis, (iii) thermal-wind balance becomes established in the direction normal to the isotherms, and (iv) the rotational component of the vector frontogenetical function decays.

The trajectory of a non-isolated monopole on the beta-plane is calculated as an asymptotic expansion in the ratio of the strength of the vortex to the beta-effect. The method of matched asymptotic expansions is used to solve the equations of motion in two regions of the flow: a near field where the beta-effect enters as a first-order forcing in relative vorticity, and a wave field in which the dominant balance is a linear one between the beta-effect and the rate of change of relative vorticity. The resulting trajectory is computed for Gaussian and Rankine vortices.

Abstract Sverdrup balance underlies much of the theory of ocean circulation and provides a potential tool for describing the interior ocean transport from only the wind stress. Using both a model state estimate and an eddy-permitting coupled climate model, this study assesses to what extent and over what spatial and temporal scales Sverdrup balance describes the meridional transport. The authors find that Sverdrup balance holds to first order in the interior subtropical ocean when considered at spatial scales greater than approximately 5°. Outside the subtropics, in western boundary currents and at short spatial scales, significant departures occur due to failures in both the assumptions that there is a level of no motion at some depth and that the vorticity equation is linear. Despite the ocean transport adjustment occurring on time scales consistent with the basin-crossing times for Rossby waves, as predicted by theory, Sverdrup balance gives a useful measure of the subtropical circulation after only a few years. This is because the interannual transport variability is small compared to the mean transports. The vorticity input to the deep ocean by the interaction between deep currents and topography is found to be very large in both models. These deep transports, however, are separated from upper-layer transports that are in Sverdrup balance when considered over large scales.

AbstractA time-dependent, inviscid, linear theory for the generation of poleward undercurrent flow under upwelling conditions along midlatitude ocean eastern boundaries is proposed. The theory relies on a conceptual separation of time scales between the rapid, coastal-trapped wave response to upwelling winds and the subsequent slow, interior, quasigeostrophic evolution. Solutions are obtained under idealized conditions in which the coastal boundary and the continental-slope topography are uniform alongshore, and the time-dependent wind-stress forcing is applied over a limited meridional range, uniform cross-shore, and directed alongshore. A time-dependent coastal boundary condition on the slow-time-scale interior flow, consisting of the low-frequency, geostrophically balanced sea surface height disturbance over the outer shelf, is obtained from consideration of the fast-time-scale, coastal-trapped response. A quasigeostrophic potential vorticity equation is then solved to determine the interior response to this time-dependent boundary condition. Under upwelling conditions, the results show the formation of a localized region of subsurface poleward flow over the upper continental slope that is qualitatively consistent in amplitude, location, and timing with observations of poleward undercurrents on eastern boundaries. Despite its origin as a sea surface height anomaly, the coastal-boundary condition drives a baroclinic planetary wave response, in which the poleward subsurface flow evolves in planetary vorticity balance with induced subsurface upwelling.

Vorticity distributions in axisymmetric vortex rings produced by a piston–pipe apparatus are numerically studied over a range of Reynolds numbers, $Re$, and stroke-to-diameter ratios, $L/D$. It is found that a state of advective balance, such that $\unicode[STIX]{x1D701}\equiv \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}/r\approx F(\unicode[STIX]{x1D713},t)$, is achieved within the region (called the vortex ring bubble) enclosed by the dividing streamline. Here $\unicode[STIX]{x1D701}\equiv \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D719}}/r$ is the ratio of azimuthal vorticity to cylindrical radius, and $\unicode[STIX]{x1D713}$ is the Stokes streamfunction in the frame of the ring. Some, but not all, of the $Re$ dependence in the time evolution of $F(\unicode[STIX]{x1D713},t)$ can be captured by introducing a scaled time $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D708}t$, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity. When $\unicode[STIX]{x1D708}t/D^{2}\gtrsim 0.02$, the shape of $F(\unicode[STIX]{x1D713})$ is dominated by the linear-in-$\unicode[STIX]{x1D713}$ component, the coefficient of the quadratic term being an order of magnitude smaller. An important feature is that, as the dividing streamline ($\unicode[STIX]{x1D713}=0$) is approached, $F(\unicode[STIX]{x1D713})$ tends to a non-zero intercept which exhibits an extra $Re$ dependence. This and other features are explained by a simple toy model consisting of the one-dimensional cylindrical diffusion equation. The key ingredient in the model responsible for the extra $Re$ dependence is a Robin-type boundary condition, similar to Newton’s law of cooling, that accounts for the edge layer at the dividing streamline.

We consider a disturbance that evolves from a strictly linear finite-growth-rate instability wave, with nonlinear effects first becoming important in the critical layer. The local Reynolds number is assumed to be just small enough so that the spatial-evolution, nonlinear-convection, and viscous-diffusion terms are of the same order of magnitude in the interactive critical-layer vorticity equation. The numerical results show that viscous effects eventually become important even when the viscosity is very small due to continually decreasing scales generated by the nonlinear effects. The vorticity distribution diffuses into a more regular pattern vis-a-vis the inviscid case, and the instability-wave growth ultimately becomes algebraic. This leads to a new dominant balance between linear- and nonlinear-convection terms and an equilibrium critical layer of the Benney & Bergeron (1969) type begins to emerge, but the detailed flow field, which has variable vorticity within the cat's-eye boundary, turns out to be somewhat different from theirs. The solution to this rescaled problem is compared with the numerical results and is then used to infer the scaling for the next stage of evolution of the flow. The instability-wave growth is simultaneously affected by mean-flow divergence and nonlinear critical-layer effects in this latter stage of development and is eventually converted to decay. The neutral stability point is the same as in the corresponding linear case, however.

À l'échelle des bassins océaniques, les vitesses verticales présentent des valeurs nettement inférieures à celles des vitesses horizontales, imposant ainsi un défi considérable en ce qui concerne leur mesure directe dans l'océan. Par conséquent, leur évaluation nécessite une combinaison d'ensembles de données observationnelles et de considérations théoriques. Diverses méthodes ont été tentées, allant de celles qui se fondent sur la divergence du courant horizontal in situ à celles qui reposent sur des équations complexes de type oméga. Cependant, l'équilibre de Sverdrup a attiré l'attention des chercheurs, y compris la nôtre, en raison de sa description robuste et simple de la dynamique des océans. L'une de ses composantes fondamentales est l'équilibre de vorticité linéaire (LVB: βv = f ∂z w). Celle-ci introduit une dimension verticale dans l'équilibre de Sverdrup conventionnel, en établissant un lien entre le mouvement vertical et le transport méridien au-dessus de lui. Afin de progresser dans la perspective théorique de l'estimation des vitesses verticales, on analyse la validité de cet équilibre linéaire dans une simulation de modèle de circulation générale océanique (OGCM) eddy-permitting. Au départ, cette analyse est effectuée dans la région de l'océan Atlantique Nord, puis étendue à l'ensemble de l'océan mondial, en mettant l'accent sur des échelles supérieures à des échelles plus grandes que 5 degrés. L'analyse a révélé la faisabilité du calcul d'un champ de vitesse verticale robuste sous la couche de mélange en utilisant l'approche LVB pour de grandes fractions de la colonne d'eau dans les régions intérieures des gyres tropicaux et subtropicaux, ainsi que dans certaines couches de la circulation subpolaire et australe à des échelles de temps annuelles et interannuelles. Des déviations par rapport à la LVB se produisent dans les courants de la frontière occidentale, les flux tropicaux zonaux forts, les gyres subpolaires et les échelles plus petites en raison des non-linéarités, des mélanges et des contributions au bilan de vorticité induites par la bathymétrie. L'étude de la validité de la LVB dans l'océan global fournit une base relativement simple pour l'estimation des vitesses verticales à travers de la LVB intégrée indéfinie en profondeur. Grâce à l'utilisation d'un OGCM, il a été démontré que ces estimations ont la capacité de reproduire avec précision l'amplitude temporelle moyenne et de la variabilité interannuelle des vitesses verticales dans des portions substantielles de l'océan global, en comparaison avec le modèle de référence. Nous construisons ici le produit DIOLIVE (Depth-Indefinitive integrated Observation-based LInear Vorticity Estimates) dérivé des vitesses géostrophiques ARMOR3D basées sur des observations et appliquées à la LVB intégrée indéfinie en profondeur, avec les données de forçage du vent ERA5 comme conditions limites à la surface. Ce produit contient des vitesses verticales couvrant l'ensemble de la thermocline globale à une résolution horizontale de 5 degrés et 40 niveaux isopycnaux pendant la période 1993-2018.Une analyse comparative entre le produit DIOLIVE et quatre autres produits de vitesse verticale, comprenant une simulation OGCM, deux réanalyses et une reconstruction basée sur l'observation de l'équation oméga, est proposée. Diverses métriques sont utilisées pour évaluer les caractéristiques multidimensionnelles de la circulation verticale de l'océan. Le produit basé sur l'équation oméga révèle d'importantes divergences par rapport à la synchronisation et à la baroclinicité reproduites par l'ensemble de validation. Mais, dans les régions où la LVB est une hypothèse valide, le produit DIOLIVE démontre une capacité remarquable à reproduire la structure barocline de l'océan, présentant une cohérence spatiale satisfaisante et un accord notable en termes de variabilité temporelle lorsqu'il est comparé aux deux réanalyses et à la simulation OGCM
At oceanic basin scales, vertical velocities are several orders of magnitude smaller than their horizontal counterparts, rendering a formidable challenge for their direct measurement in the real ocean. Therefore, their estimations need a combination of observation-based datasets and theoretical considerations.Historically, scientists have employed various techniques to estimate vertical velocities across different scales constrained by the available observations of their time. Various approaches have been attempted, ranging from methods utilizing in situ horizontal current divergence to those based on intricate omega-type equations. However, the Sverdrup balance has captured the attention of researchers and ours due to its robust and straightforward description of ocean dynamics. One of the fundamental components of the Sverdrup balance is the linear vorticity balance (LVB: βv = f ∂z w). It introduces a novel vertical dimension to the conventional Sverdrup balance, establishing a connection between vertical movement and the meridional transport above it.In order to advance on the theoretical prospect of estimating the vertical velocities, it is primarily identified the annual and interannual timescales patterns governing the linear vorticity balance within an eddy-permitting OGCM simulation. Initially, this analysis is conducted over the North Atlantic Ocean, and subsequently expanded to encompass the entire global ocean, focusing on larger scales than 5 degrees. The analysis revealed the feasibility of computing a robust vertical velocity field beneath the mixed layer using the LVB approach across large fractions of the water column in the interior regions of tropical and subtropical gyres and within some layers of the subpolar and austral circulation. Departures from the LVB occur in the western boundary currents, strong zonal tropical flows, subpolar gyres and smaller scales due to the nonlinearities, mixing and bathymetry-driven contributions to the vorticity budget.The extensive validity of the LVB description of the global ocean provides a relatively simple foundation for estimating the vertical velocities through the indefinite depth-integrated LVB. Using an OGCM, it has demonstrated that the estimates possess the capability to accurately reproduce the time-mean amplitude and interannual variability of the vertical velocity field within substantial portions of the global ocean when compared to the reference model. Here, we build the DIOLIVE (indefinite Depth-Integrated Observation-based LInear Vorticity Estimates) product by applying the observation-based geostrophic velocities from ARMOR3D into the indefinite depth-integrated LVB formalism, with wind stress data from ERA5 serving as boundary condition at the surface. This product contains vertical velocities spanning the global ocean's thermocline at 5 degrees horizontal resolution and 40 isopycnal levels during the 1993-2018 period.A comparative analysis between the DIOLIVE product and four alternative products, including one OGCM simulation, two reanalyses and an observation-based reconstruction based on the omega equation, is conducted using various metrics assessing the vertical circulation's multidimensional features of the ocean vertical flow. The omega equation-based product displays large departures from the synchronicity and baroclinicity reproduced by the validation ensemble. However, in regions where the LVB holds as a valid assumption, the DIOLIVE product demonstrates a remarkable ability to replicate the baroclinic structure of the ocean, exhibiting satisfactory spatial consistency and notable agreement in terms of temporal variability when compared to the two reanalyses and the OGCM simulation