Local Rossby Number Research Articles

Understanding coherent vortices through computational fluid dynamics

A new definition of coherent vortices in turbulence is proposed, where the vorticity equation reduces to a cyclostrophic balance. Afterward, we describe five fundamental vortex interactions, the sheet, the spiral, the pairing, the even longitudinal, and the odd longitudinal modes. Numerous examples of these interactions are provided from direct numerical or large-eddy simulations. The resulting vortices are responsible for the internal intermittent character of turbulence, with highly nongaussian tails for the probability density functions of vorticity, passive scalar, and low pressure. In a mixing layer, the combination of the odd longitudinal and the pairing modes (helical pairing) is inhibited by compressibility, above a convective Mach number of ≈0.7. When turbulence is submitted to a solid-body rotation, anticyclonic vortices of local Rossby number of the order of 1 transform into intense perpendicular Gortler-type alternate longitudinal vortices.

A Three-Dimensional Balance Theory for Rapidly Rotating Vortices

Abstract A three-dimensional balance formulation for rapidly rotating vortices, such as hurricanes, is presented. The asymmetric balance (AB) theory represents a new mathematical framework for studying the slow evolution of rapidly rotating fluid systems. The AB theory is valid for large Rossby number; it makes no formal restriction on the magnitude of the divergence or vertical advection, which need not be small. The AB is an ordered expansion in the square of the ratio of orbital to inertial frequencies, the square of a local Rossby number. The approximation filters gravity and inertial waves from the system. Advantage is taken of the weak asymmetries near the vortex care as well as the tendency for low azimuthal wavenumber asymmetries to dominate. Linearization about a symmetric balanced vortex allows the three-dimensional asymmetric dynamics to be deduced properly. The AB formulation has a geopotential tendency equation with a three-dimensional elliptic operator. The AB system has a uniformly valid co...

The structure of a turbulent free shear layer in a rotating fluid

An experimental investigation has been carried out on the effects of rotation on the development and structure of turbulence in a free shear layer, oriented so that its mean vorticity is parallel or antiparallel to the system vorticity. The effective local Rossby number extended down to about 1/3. The experimental methods were hydrogen-bubble flow visualization and hot-film anemometry.In summarizing the results we refer to stabilized flow when the system vorticity has the same sign as the shear vorticity and destabilized and subsequently restabilized when it has the opposite sign (Tritton 1992). The roller eddy pattern, familiar in non-rotating flow, was observed in all stabilized flows, but was almost completely disrupted by even weak destabilization. Notable features of the quantitative results were: reorientation by Coriolis effects of the Reynolds stress tensor (inferred from the ratio of the cross-stream to longitudinal turbulence intensity and the normalized shear stress); changes in the ratio of spanwise to longitudinal intensity similar to but weaker than changes in the ratio of cross-stream to longitudinal; a gradual decrease, with increasing stabilization, of the Reynolds shear stress leading ultimately to its changing sign; an increase of the Reynolds shear stress in the destabilized range followed by rapid collapse to almost zero with restabilization. Absolute intensities did not change in line with the turbulence energy production, implying enhancement of dissipation in destabilized flow and inhibition in stabilized and restabilized. Correlation measurements indicated changes of lengthscale in the spanwise direction, and spectra indicated changes in the longitudinal direction that suggest that this enhancement and inhibition are associated with variations between fully three-dimensional and partially two-dimensional turbulence. Data for a wake in a rotating fluid (Witt & Joubert 1985) show similarities to some of the above observations and can be incorporated into the interpretation.

On the interaction of an isolated obstacle with a rotating stratified shear flow

Abstract Consideration is given to the effects of upstream shear upon the flow of a rotating stratified fluid past an isolated obstacle. The vorticity equation governing the interaction of the flow with the obstacle is derived for the inertially-dominated regime, and it is demonstrated that the effect of increasing vertical shear in the undisturbed flow is to reduce the vertical penetration length of the topographic disturbance. Following Huppert (1975) and Johnson (1977), limiting cases of high and low stratification are investigated. Laboratory data are presented in qualitative support of the analytical results, and it is demonstrated that the vertical attenuation of the topographic disturbance is increased by an increase in the shear parameter. The effect is explained in terms of the increasing attenuating role of the local Rossby number at each successive elevation above the obstacle.

Stabilizing and destabilizing effects of a solid-body rotation on quasi-two-dimensional shear layers

The effect of a solid-body rotation, characterized by an angular velocity Ω, on a two-dimensional mixing layer (in a plane perpendicular to Ω) of relative vorticity ω2D, upon which is superposed a small three-dimensional turbulent perturbation, is considered. Using the Kelvin theorem in the frame rotating with Ω, and with the aid of arguments based on the straining of absolute vortex filaments by the basic velocity, it is shown that the rotation is always stabilizing (with respect to the nonrotating case) in the cyclonic case. In the anticyclonic case, a slight rotation is destabilizing. At a local Rossby number R0=‖ω2D‖/2‖Ω‖ of the order of 1, the anticyclonic rotation disrupts catastrophically the coherent structures of the mixing layer. Anticyclonic rotation becomes stabilizing again for R0<0.5. Also presented are three-dimensional numerical simulations which support the theory, and agree qualitatively with experimental results. The consequences for oceanic and atmospheric vortices are briefly discussed.

On Intermediate Models for Barotropic Continental Shelf and Slope Flow Fields. Part III: Comparison of Numerical Model Solutions in Periodic Channels

The study of intermediate models for barotropic continental shelf and slope flow fields initiated in Parts I and II is continued. The objective is to investigate the possible use of intermediate models for process and data assimilation studies of nonlinear mesoscale eddy and jet current fields over the continental shelf and slope. Intermediate models contain physics between that in the primitive equations and that in the quasi-geostrophic equations and are capable of representing subinertial frequency motion over the O(1) topographic variations typical of the continental slope while filtering out high-frequency gravity–inertial waves. We concentrate on the application of intermediate models to the f-plane shallow-water equations. The accuracy of several intermediate models is evaluated here by a comparison of numerical finite-difference solutions with those of the primitive shallow-water equations (SWE) and with those of the quasi-geostrophic equations (QG) for flow in a periodic channel. The intermediate models that we consider are based on the balance equations (BE), the balance equations derived from momentum equations (BEM), the potential vorticity conserving linear balance equations (LQBE), the hybrid balance equations (HBE), the near balance equation (NBE), a geostrophic vorticity (GV) approximation, the geostrophic momentum (GM) approximation, and the quasi-geostrophic momentum and full continuity equations (IM). The periodic channel provides a basic geometry for the study of physical flow processes over the continental shelf and slope. Wall boundary conditions are formulated for the intermediate models and implemented in the numerical finite-difference approximations. The ability of intermediate models to represent linear ageostrophic coastally trapped waves, i.e., Kelvin and continental shelf waves, is verified by numerical experiments. The results of numerical solution intercomparisons for initial-value problems involving O(1) topographic variations are as follows. For flow at small local Rossby number |εL| < 0.2, where εL is given by the magnitude of the vorticity divided by f, all of the intermediate models do well, while the QG model does poorly. For flows with larger values of |εL|, e.g., |εL| ≈ 0.5, the performance of the different intermediate models varies. BEM and BE consistently give extremely accurate solutions while the solutions from LQBE are almost as good. The other models are substantially less accurate with errors generally increasing in the order NBE, HBE, GV, GM, IM. The QG solution always has the largest errors. Consistent with the results from the studies in Part II in a doubly periodic domain, the balance equations BE and BEM, followed closely by LQBE, appear to be the most accurate intermediate models.

On Intermediate Models for Barotropic Continental Shelf and Slope Flow Fields. Part II: Comparison of Numerical Model Solutions in Doubly Periodic Domains

As part of a program to improve understanding of the dynamics of the complicated, vigorous eddy and jet flow fields recently observed over the continental shelf and slope, we investigate the potential of intermediate models for use in both process and data assimilation studies of these flows. Intermediate models incorporate physics simpler than that contained in the full primitive equations yet more complete than in the quasi-geostrophic equations, and are capable of representing subinertial flows over O(1) bottom topographic variations and/or with O(1) isopycnal slopes. In addition, intermediate models dynamically filter out high-frequency gravity- inertial motions leading, potentially, to higher computational efficiency and well-posed limited area forecast/hindcast models. Initial studies focus on single layer flows on an f-plane with a free surface, governed by the shallow-water equations. In Part I, various intermediate models are formulated and their accuracy assessed by comparing some exact nonlinear analytical solutions that exist for the shallow-water equations with corresponding analytical solutions of the intermediate models. Here in Part II, an extensive set of numerical finite-difference solutions to initial-value problems in doubly periodic domains (to isolate model differences from the influence of boundary condition implementation on solid walls) is used to determine the accuracy of various intermediate models by comparing their predictions with those of a shallow-water equation model that uses a potential enstrophy and energy conserving numerical scheme (SWE). Intermediate model results are also contrasted with those from a quasi-geostrophic (QG) model. The intermediate models considered are based on the geostrophic momentum (GM) approximation, the derivation of Salmon utilizing Hamilton's principle (HP), a geostrophic vorticity (GV) approximation, a combination of the quasi-geostrophic momentum and full continuity equations (IM), the linear balance equations (LBE), the balance equations (BE), the related balance-type (HBE, BEM, NBE) and modified linear balance equations (LQBE), and on Lynch's slow equations in their original form (SE) and in a modified form (MSE). In addition, a semi-implicit version (SEMI) of the shallow-water equations, which numerically filters high-frequency motions, is included in the study. The basic initial-value problem used to test the various intermediate models involves sinusoidal flow over a symmetric Gaussian-shaped bottom topographic feature. Comparisons are made for a range of the relevant dimensionless model parameters including the strength of the flow (as measured by the Rossby number), the square of the ratio of a characteristic horizontal length scale to the Rossby radius of deformation, and the height of the topographic feature. A second initial-value problem involves the evolution of a rotating elliptical vortex over both flat and variable bottom topography. Results show that in cases with low local Rossby number flow, but with large topographic height, most of the intermediate models are substantially more accurate than QG. Even for flows with O(1) local Rossby numbers, some of the intermediate models continue to give excellent results. Specifically, BE, BEM and SEMI consistently do the best in the comparisons. Although the relative ordering of the remaining models is somewhat parameter-dependent, in general the next most accurate models are MSE and LQBE followed by HP and NBE. These are followed in quality by GV and GM while the remaining models, HBE, IM and LBE, perform least accurately for the range of parameters studied. Generally, intermediate models that have an integral invariant corresponding to conservation of potential enstrophy do better than those without, with the best results coming from models which have this property and a nonlinear balance equation.

A two-dimensional model of mesoscale frontogenesis in the ocean

A two-dimensional primitive equation model conserving potential vorticity is used to simulate the effect of a horizontal barotropic deformation field on a pre-existing isopycnic potential vorticity gradient. The novel feature of the prognostic model is that it accurately describes the development of the density and velocity distributions up to local Rossby number 0.5 and beyond. The model coordinates are based in the vertical on quasi-isopycnic vertical coordinates and in the horizontal on geostrophic coordinates. The model is integrated forward in time for three days creating a density front and a jet of 0.4 m s−1 in the surface layers. Cross-jet velocities are one order of magnitude smaller. Conservation of potential vorticity leads to a modulation of spacing between isopycnals as both the cyclonic and anticyclonic vorticity associated with the horizontal shear of the accelerating jet increase. Vortex stretching and compression sustain a strong vertical circulation. The paper shows how the jet kinetic energy is limited by the amount of available potential energy in a catchment area defined by the extent of the region of confluence. The jet becomes shallower as frontogenesis approaches this limit. The model is also applicable to the development of permanent streams in the large-scale circulation of the upper ocean, by slow (order one month) frontogenesis due to gyre-scale deformation acting on an isopycnic potential vorticity gradient in the seasonal thermocline. The vigorous upwelling on the anticyclonic side of the jet raises nutrients through the euphotic zone, increasing primary productivity. This explains observed meoscale modulations of biomass.

A two‐dimensional model of mesoscale frontogenesis in the ocean

AbstractA two‐dimensional primitive equation model conserving potential vorticity is used to simulate the effect of a horizontal barotropic deformation field on a pre‐existing isopycnic potential vorticity gradient. The novel feature of the prognostic model is that it accurately describes the development of the density and velocity distributions up to local Rossby number 0.5 and beyond. The model coordinates are based in the vertical on quasi‐isopycnic vertical coordinates and in the horizontal on geostrophic coordinates. The model is integrated forward in time for three days creating a density front and a jet of 0.4 m s−1 in the surface layers. Cross‐jet velocities are one order of magnitude smaller. Conservation of potential vorticity leads to a modulation of spacing between isopycnals as both the cyclonic and anticyclonic vorticity associated with the horizontal shear of the accelerating jet increase. Vortex stretching and compression sustain a strong vertical circulation. The paper shows how the jet kinetic energy is limited by the amount of available potential energy in a catchment area defined by the extent of the region of confluence. The jet becomes shallower as frontogenesis approaches this limit.The model is also applicable to the development of permanent streams in the large‐scale circulation of the upper ocean, by slow (order one month) frontogenesis due to gyre‐scale deformation acting on an isopycnic potential vorticity gradient in the seasonal thermocline.The vigorous upwelling on the anticyclonic side of the jet raises nutrients through the euphotic zone, increasing primary productivity. This explains observed meoscale modulations of biomass.

A criterion for vortex breakdown

A criterion for the onset of vortex breakdown over a wide range of the Reynolds number is proposed. Based upon previous experimental, theoretical, and numerical studies, as well as a new numerical study, an appropriately defined local Rossby number is used to delineate the region where breakdown occurs. Comparisons are made with previously suggested criticality parameters and the unique features of the proposed Rossby number parameter are shown. A number of previous theoretical studies concentrating on inviscid standing-wave analyses for trailing wing-tip vortices are reviewed and reinterpreted, along with the previous numerical and experimental studies, in terms of the Rossby number parameter. For the case of trailing wing-tip-type vortices, it is shown that previous numerical studies were performed at lower Reynolds numbers than the corresponding laboratory experiments. Utilizing a consistently defined Reynolds number, Rossby number–Reynolds number plots of these previous vortex breakdown studies, for both trailing wing-tip and leading-edge vortices, are obtained. A criticality condition is identified for both types of vortices.

On the Flow Through Broad Gaps with Application to the Windward Passage

The flow through broad passage connecting oceans and marginal seas is examined by a simplified two-layer analytical model. Attention is focused on the flow resulting from the difference between the upper layer depth in the two basis which imposes a pressure gradient along the passage. The land masses separating the ocean from the adjacent marginal sea are represented by two portions of an infinitely long wall extending from the free surface to the bottom of the ocean. The passage, whose width is larger than the deformation radius, is represented by a gap separating the two portions of the wall. The model is frictionless, hydrostatic and nondiffusive but the movements within the gap are not constrained to be quasi-geostrophic in the sense that the local Rossby number is not necessarily small and the interface displacements are of the same order as the upper layer depth. Steady solutions for the upstream and downstream fields are obtained analytically using the momentum equation in an integrated form, the Bernoulli integral and conservation of potential vorticity. It is found that, surprisingly, the transport through the gap is independent of the gap's width. Upstream the oceanic water approaches the gap only from one direction; upon reaching the gap, the approaching current splits into two branches. One continues to flow in the oceanic basin and never enters the gap whereas the other passes through the gap and penetrates into the marginal sea. Downstream, the pentrating flow forms a boundary current which is confined between the wall on the right (looking downstream) and a free bounding streamline on the left. A possible application of this theory to the flow from the Atlantic Ocean into the Caribbean Sea via the Windward Passage is discussed. The observed locations, positions and directions of both upstream and downstream flows agree with the model predictions. In addition, the predicted transport through the passage (∼12 × 106 m3 s−1) is approximately equal to the observed transport.

Turbulence and waves in a rotating tank

A turbulent field is produced with an oscillating grid in a deep, rotating tank. Near the grid, the Rossby number is kept large, 0(3-33), and the turbulence is locally unaffected by rotation. Away from the grid, the scale of the turbulence increases, the r.m.s. turbulent velocity decreases, and rotation becomes increasingly important. The flow field changes dramatically at a local Rossby number of about 0.20, and thereafter remains independent of depth. The flow consists of concentrated vortices having axes approximately parallel to the rotation axis, and extending throughout the depth of the fluid above the turbulent Ekman layer. The number of vortices per unit area is a function of the grid Rossby number. The local vorticity within cores can be a factor of 50 larger than the tank vorticity 2Ω. The total relative circulation contained in the vortices remains, however, a small fraction of the tank circulation.The concentrated vortex cores support waves consisting of helical distortions, which travel along the axes of individual vortices. Isolated, travelling waves seem well-described by the vortex-soliton theory of Hasimoto (1972). The nonlinear waves transport mass, momentum and energy from the vigorously turbulent region near the grid to the rotation-dominated flow above. Interactions between waves, which are frequent occurrences, almost always result in a local breakdown of the vortex core, and small-scale turbulence production. Usually the portions of broken core reform within ½−1 rotation periods, but occasionally cores are destroyed and reformed on a much longer timescale.

Inertial effects in an oceanic circulation model

Abstract The flow in a mechanically driven thin barotropic rotating fluid system is analysed. The linear theory of Baker and Robinson (1969) is modified and extended into the non-linear regime. An internal parameter, the “local Rossby number”, is indicative of the onset of nonlinear effects. If this parameter is 0(1) then inertial effects are as important as Coriolis accelerations in the interior of the transport-turning western boundary layer and both of its Ekman layers. The inertial effects in the Ekman layers, ignored in previous explorations of non-linear wind driven oceanic circulation, are retained here and calculated using an approximation of the Oseen type. The circulation problem is reduced to a system of scalar equations in only two independent variables; the system is valid for non-small local Rossby number provided only that the approximate total vorticity is positive. To complete the solution for small Rossby number a boundary condition for the inertially induced transport is needed. It is f...