Pair Of Eddies Research Articles

Ship and satellite observations of chlorophyll stocks in interacting cyclone‐anticyclone eddy pairs in the western Gulf of Mexico

When anticyclonic eddies shed by the Loop Current of the Gulf of Mexico reach the western margin of the gulf, they influence the surface circulation over the continental slope and rise. Of particular interest is the generation of cyclone (cold‐core)‐anticyclone (warm‐core) pairs when aging Loop Current eddies interact with the continental margin. In this paper we describe the physical and biological characteristics of these cyclone‐anticyclone pairs. Our objective was to determine how eddy pairs affect the distribution of phytoplankton in the region and how satellite ocean color measurements are applicable to tracing of the eddies. We present shipboard data collected between 1980 and 1982 on the hydrography, chlorophyll stocks, and nutrient concentrations of eddy pairs in the western Gulf of Mexico and compare these data with coastal zone color scanner (CZCS) images collected during the time frame of the cruises. Surface pigment concentrations followed a seasonal cycle, with low concentrations (0.05–0.1 mg m−3) found within cyclones and anticyclones from April through early November and higher concentrations (>0.1 mg m−3) found in the winter. CZCS pigment concentrations were locally high in the flow confluence of cyclone‐anticyclone pairs. The CZCS imagery shows that some cyclone‐anticyclone geometries transport high‐chlorophyll shelf water seaward at least 100–200 km off‐shelf.

Flows generated by the periodic horizontal oscillations of a sphere in a linearly stratified fluid

The flow field induced by a sphere oscillating horizontally in a linearly stratified fluid is studied using a series of laboratory experiments. The resulting flows are shown to depend on the Stokes number β, the Keulegan–Carpenter number KC and the internal Froude number Fr. For Fr [clubs ] 0.2, it is shown that the nature of the resulting flow field is approximately independent of Fr and, based on this observation, a flow regime diagram is developed in the (β, KC)-plane. The flow regimes include: (i) fully-attached flow; (ii) attached vortices; (iii) local vortex shedding; and (iv) standing eddy pair. An internal-wave flow regime is also identified but, for such flows, the motion field is a function of Fr as well. Some quantitative measures are given to allow for future comparisons of the present results with analytical and/or numerical models. Wherever possible, the results are compared with the experiments of Tatsuno & Bearman (1990) on right circular cylinders oscillating in homogeneous fluids.

Unsteady flow and heat transfer for cylinder pairs in a channel

The incompressible, two-dimensional, unsteady flow past a pair of cylinders of square cross-section, placed in tandem normal to the flow in a channel, has been investigated by dye visualization and direct numerical simulation. The objective has been to evaluate the effect of cylinder separation distance, λ, on the flow behavior and heat transfer, for cylinder diameter ratios, D d , of 1 and 2 over a range of Reynolds numbers 200 ⩽ Re ⩽ 1600, based on the larger (downstream) cylinder diameter. A comparison between the experimental and numerical results for cylinders of equal cross-section dimensions shows very good agreement. The results for D d = 1 reveal three distinct flow patterns as a function of λ and Re which, apparently, have not been previously reported: (1) For 0.25 ⩽ λ ⩽ 4 with Re ⩽ 200, the inter-cylinder flow consists of a pair of steady counter-rotating eddies which do not exchange fluid with the surrounding flow and eddy shedding is observed only for the downstream cylinder. (2) For 0.25 ⩽ λ ⩽ 1.0 with 400 ⩽ Re ⩽ 1600, vertical flow oscillations arise in the inter-cylinder space, and the periodic ingestion of backward-jetting fluid from the top and bottom walls of the downstream cylinder into the inter-cylinder space is observed. For fixed λ the unsteadiness increases with Re but only the downstream cylinder sheds large eddies. (3) At a critical inter-cylinder spacing related to Re according to λ c ∼ Re − 2 3 , the shedding of large eddies also occurs at the upstream cylinder and this results in a highly mixed inter-cylinder flow. For a cylinder diameter ratio of D d = 2 , with the smaller cylinder located upstream of the larger heated cylinder, it is shown that for Re fixed an optimal location exists for the upstream cylinder such that the heat transfer from the downstream cylinder is maximized. The optimal location corresponds to a spacing smaller than the critical inter-cylinder spacing.

Wind-induced eddy motion in the lower St. Lawrence Estuary

The lower St. Lawrence Estuary is a wide estuary, in the sense of being several internal Rossby radii in breadth. Thus, the Coriolis effects are significant and complex circulation patterns are to be expected. We analyse the response of this body to a strong up-channel pulse of wind, using data from an array of current meters and sea level gauges. Interesting features of the flow are found, such as coastal jets accelerating into the wind as the wind speed relaxes (apparently due to the establishment of an along-channel pressure gradient), and a pair of synoptic-scale eddies developing in the current pattern.

Hurricane‐induced surge and currents on the Texas‐Louisiana shelf

This study consists of numerical model simulations of hurricane‐induced surge and currents on the Texas‐Louisiana shelf. The numerical experiment includes the simulation of multiple hurricane tracks with landfalling points along the Texas‐Louisiana coast. A parallel storm that traverses the entire Texas‐Louisiana coastline is also modeled to assess the difference in shelf response between landfalling and parallel storms. The grid extends from the Texas‐Mexico border to the Gulf Coast of Florida, with the ocean open boundary seaward of the shelf break. Along‐shelf and cross‐shelf surge and current variability are assessed as a function of shoreline geometry and bottom topography. A complementary one‐dimensional mixed layer model is used to evaluate the vertical structure of the currents and the maximum depth of hurricane influence. The choice of landfalling storm tracks is based principally on historical storm tracks for the Texas‐Louisiana area. Storm intensity, i.e., central pressure depression, for all storms is set equal to the 10‐year recurrence interval central pressure. Strong cyclonic flow is induced by the hurricane wind field within a distance from the storm's center equal to 3 to 5 times the storm's radius of maximum winds. Far‐field hurricane winds generate a northward moving coastal jet along the west coast of Florida, particularly for the landfalling storms. The far‐field continental shelf currents are topographically steered by the shoreline and bathymetry. There is strong evidence of a cyclonic‐anticyclonic eddy pair being generated in the wake of the parallel storm traveling eastward over the continental shelf. An intercomparison of the one‐ and two‐dimensional models shows that for depths less than 50 m, the combined use of the surge and mixed layer models permits a description of the vertical and horizontal hurricane‐induced current patterns. The surge height computations for a given storm strength point to the importance of several factors that determine surge levels as a function of offshore distance. These factors are bathymetry, topography, angle of approach of the storm, distance to landfall location, and shelf width. The most dramatic increase in surge levels occurs in depths less than 10 m, where the surge profile mirrors the changes in the bottom slope. In contrast, the sensitivity of surge dynamics to shelf width is neither strong nor consistent. A narrow spread of maximum surge values is observed. Each transect is examined individually to explain local behavior and the reason for the modest variation in surge values among dissimilar geographical areas. No critical dependence on the exact storm angle exists for storm angles within the range of angles associated with maximum surge generation (within 20° counterclockwise from the shore‐normal direction). Since the landfalling storms used in this study fall into this category, no significant dependence of surge height on the landfalling angle is observed.

A numerical investigation of jets and eddies near an eastern ocean boundary

The dynamics of the circulation near an eastern‐ocean boundary are investigated using a 2½‐layer numerical model that includes entrainment of cool water into the upper layer. Solutions are found in a regional ocean basin and are forced by an upwelling‐favorable, alongshore wind field without curl. Dynamically simpler versions of the model are also utilized in order to isolate the effects of various physical processes; in particular, a linearized instability model is used to study the unstable waves associated with various background coastal circulations. The model is quite successful in simulating many features of the observed flow and SST fields. Initially, the “main run” solution spins up as in a linear model, generating a surface jet, an undercurrent, and an upwelling front. Subsequently, small‐scale, “fingerlike” disturbances appear along the coast and grow rapidly in amplitude and scale. The growth in scale has two causes: the slower development of larger‐scale disturbances, and eddy coalescence via nonlinear interactions. Eventually, the solution adjusts to a realistic equilibrium state that contains upwelling filaments, squirts, eddies, dipole eddy pairs, and a realistic mean circulation. Entrainment is a crucial process in the dynamics of the mean flow. In particular, entrainment damps the westward propagation of Rossby waves, thereby ensuring that the currents remain coastally trapped. (In this aspect entrainment plays a dynamical role similar to vertical mixing in continuously stratified models.) Eddies influence the mean flow primarily by providing a strong heat source near the coast, but they have very little effect on the mean momentum field. The small‐scale, fingerlike disturbances are caused by a frontal instability that requires the existence of an upper‐layer temperature gradient. Some dynamically simpler solutions indicate that the larger‐scale disturbances are caused by baroclinic instability. On the other hand, unstable‐wave solutions to the linearized instability model suggest that their cause is frontal instability, becoming pure baroclinic instability only when there is no temperature gradient.

Calculation of turbulent convection between corotating disks in axisymmetric enclosures

A numerical investigation is conducted for the case of turbulent flow in the unobstructed space between a pair of centrally clamped coaxial disks corotating in a fixed axisymmetric enclosure. The finite difference procedure of Chang et al. ( J. Heat Transfer 111, 625–632 (1989)) is extended to include a standard two-equation ( κ- ε) model of turbulence in the core of the flow. A van Driest relation, that accounts for the effects of streamline curvature and wall shear on the energy-containing length scales, is used in conjunction with Prandtl's mixing length hypothesis to model the near wall flow. The set of equations is solved assuming a constant property, circumferentially symmetric, statistically stationary flow. This approach predicts mean velocity and heat transfer results that are in good agreement with timeaveraged experimental data. The predictions reveal a flow that, in non-dimensional variables, tends to a limiting asymptotic state at high Reynolds numbers. In the absence of blowing a symmetrical pair of crossstream eddies appear near the enclosure wall the rotation of which increases with increasing disk rotational speed. High rotational speeds, in excess of 2400 rpm in the configuration studied, and small disk separations induce large values of shear and temperature (due to viscous dissipation) in the vicinity of the enclosure wall. The flow and its heat transfer characteristics can be drastically altered by the combined effects of radial and axial blowing. Specifically, it is shown that axial blowing significantly reduces the shear and heat transfer at the curved enclosure wall.

Incorporation of a large‐scale wind‐driven turbulence model into a range‐dependent parabolic equation

Atmospheric turbulence contains numerous scales ranging from a few centimeters to a few kilometers. As shown by Daigle et al. [J. Acoust. Soc. Am. 64, 622–630 (1978)], the effects of small scale turbulence (L<10 m) can be handled fairly well from a statistical standpoint. This approach works because for most of the time, small‐scale turbulence can be treated as isotropic over short ranges. However, large‐scale turbulence is anisotropic in nature. This results in variations in temperature and wind speed with range. In order to account for the effects of wind‐driven large‐scale turbulence on acoustic propagation, a large‐scale turbulence model was developed. This large‐scale turbulence model assumes elongated longitudinal vortex pairs roughly aligned with the mean wind speed with height and range due to the passage of an eddy pair. The wind speed variations are used to calculate range‐dependent sound‐speed profiles. The sound‐speed profiles are used by a range‐dependent parabolic equation to calculate variations in the received amplitude and phase of an acoustic wave as the eddy pair passes through the field of propagation.

A mesoscale eddy dipole in the offshore California Current

There is a transition zone in the California Current system between inshore coastal and offshore oceanic flow. This transition zone is associated with the main core of the California Current and is populated with offshore mesoscale eddies. The structure of isolated eddies and dipole eddy pairs found within the transition zone is examined using water mass analysis (spiciness and dissolved oxygen). The analyses show that these offshore mesoscale eddies have cores which contain California Undercurrent water. These observations and independent observations of the Sitka eddy near Sitka, Alaska, support the conclusion that a bathymetrically induced instability of the California Undercurrent is a likely generation mechanism for the offshore eddies. This conclusion also is consistent with mathematical models of eddy generation in a subsurface, poleward baroclinic current. The similarity between long‐term mean conditions (CaICOFI), conditions observed during January 1981 and July 1985, and those of the Sitka eddy, coupled with the bathymetric effects, provides a plausible explanation for the recurrence of offshore mesoscale eddies at preferred locations within the transition zone.

Flow past a circular cylinder in a rotating stratified fluid

Experimental results are presented to show the development of the flow past a cylinder in a rotating stratified fluid. Both transient and fully developed flow states are investigated, for a range of values of Rossby number Ro and Burger number Bu and for full depth and truncated obstacles. The fully developed flows are shown to be of three main types, delineated respectively as attached eddy pairs, transitional, and vortex wakes, with the boundaries between each flow type being determined by the values of the Reynolds number, the Burger number, the aspect ratio and the ratio of the height of the obstacle to the depth of the fluid. For cases of homogeneous fluid flow, the critical Rossby and Reynolds numbers for vortex shedding are significantly lower than the values found by Boyer. The discrepancy is ascribed to aspect ratio effects and differences in the cylinder shear layer structures in the two studies. The weakening effects of density stratification upon characteristic features of rapidly rotating homogeneous flows are seen to be two-fold: firstly, the wake asymmetries associated with the unstratified flows are seen to be destroyed by the stratification, and, secondly, the retention times of starting eddies, the eddy growth times and the isolated eddy formation times are shown to no longer tune with the background rotation timescale but to scale with the advective timescale of the motion and the aspect ratio of the cylinder. Cross-stream velocity profiles taken at several downstream stations in the wake of the cylinder are used to show a linear increase in the momentum flux coefficient with increasing Re , in the range 55 &lt; Re &lt; 135. Velocity profile measurements taken above and downstream of the truncated mcylinder are presented. These show that the vertical distance over which the upper level fluid is influenced by the presence of the obstacle depends strongly upon the value of Bu , and is typically one-tenth of the obstacle height at the highest values of Bu . Finally, flow visualization studies of the motion in the vertical shear layers for Bu = 0 are compared and contrasted with other similar observations, and with theoretical understandings of the flow structure. The importance of the upper boundary condition is clarified.

Spatio-temporal mosaic of the phytoplankton spring bloom in the open Baltic Sea in 1986

The spatio-temporal variability of the phytoplankton spring bloom in the open Baltic Sea during an international joint study (PEX'86) is described in relation to the physical and chemical environment. Simultaneous participation of 14 research vessels enabled highly synoptic sampling of the study area of 20 X 40 n mile. Onset of the bloom was controlled by mesoscale hydrodynamic features a pair of cyclonic and anticyclonic eddies; water masses with different salinities. The eddies supported an exceptionally high amount of chlorophyll (> 700 mg m-' in the cyclonic eddy). The temporal development of the bloom was different in each of the mesoscale features. Chlorophyll levels in the eddies started to decrease due to se&mentation before formation of the thermocline and the depletion of inorganic nutrients. Variability of the chlorophyll distribution was highest just before the peak in the spatially averaged concentrahon was attained.

An experimental study of flow past a triangular cape in a linearly stratified fluid

Data are presented from a series of laboratory investigations into the steady uniform channel flow of a stratified fluid past a triangular cape. The spatial and temporal developments of the flow in the wake of the cape are described for a range of the principal controlling dimensionless parameters Re and S (the Reynolds number and stratification parameter respectively). It is shown that the characteristics of the primary eddies generated at the cape tip are determined essentially by the scales of the secondary eddies formed between the lee side of the cape and the lateral wall. In turn, the scales of these secondary eddies are shown to depend upon the Re and S through the influence of these parameters upon the vigour of the vertical motions within the disturbance flow. When vertical motions are weak (for example, with high values of S and/or low intermediate value of Re), the primary and secondary eddies have relatively large horizontal scales: in consequence, strong mutual interactions occur between them, resulting in the formation of eddy pairs in the far wake. As the vigour of the vertical motions increases (for example, with low values of S and/or high values of Re), the size of the secondary eddies is reduced, and the mutual interactions between primary and secondary eddies are weakened. In such circumstances (and for cases in which the length scale of the cape is reduced), the frictional effects of the lateral side wall play a dominant role in the flow development. The mode of eddy shedding from the cape is also shown to depend upon both Re and S. In particular, the transition from the shedding of relatively large eddies to the appearance of small intense tip eddies, and the dependence of this transition upon both Re and S, are demonstrated. It is shown that this transition is accompanied by increases in the value of the Strouhal number from about 0.09 for the large eddies to values for the tip eddies which depen upon both Re and S. The spin-down process for the far-field eddies is shown to be dominated by the frictional influence of the upper and lower solid bounding surfaces of the fluid. Specifically, measurements of the temporal decay of the maximum vorticity in the primary eddies indicate that the spin-down periods of such features are much shorter than the viscous timescales D2v. Energy dissipation by internal wave generation is negligible in the system considered.

An experimental study of flow past a triangular cape in a linearly stratified fluid

Data are presented from a series of laboratory investigations into the steady uniform channel flow of a stratified fluid past a triangular cape. The spatial and temporal developments of the flow in the wake of the cape are described for a range of the principal controlling dimensionless parameters Re and S (the Reynolds number and stratification parameter respectively). It is shown that the characteristics of the primary eddies generated at the cape tip are determined essentially by the scales of the secondary eddies formed between the lee side of the cape and the lateral wall. In turn, the scales of these secondary eddies are shown to depend upon the Re and S through the influence of these parameters upon the vigour of the vertical motions within the disturbance flow. When vertical motions are weak (for example, with high values of S and/or low intermediate value of Re), the primary and secondary eddies have relatively large horizontal scales: in consequence, strong mutual interactions occur between them, resulting in the formation of eddy pairs in the far wake. As the vigour of the vertical motions increases (for example, with low values of S and/or high values of Re), the size of the secondary eddies is reduced, and the mutual interactions between primary and secondary eddies are weakened. In such circumstances (and for cases in which the length scale of the cape is reduced), the frictional effects of the lateral side wall play a dominant role in the flow development. The mode of eddy shedding from the cape is also shown to depend upon both Re and S. In particular, the transition from the shedding of relatively large eddies to the appearance of small intense tip eddies, and the dependence of this transition upon both Re and S, are demonstrated. It is shown that this transition is accompanied by increases in the value of the Strouhal number from about 0.09 for the large eddies to values for the tip eddies which depen upon both Re and S. The spin-down process for the far-field eddies is shown to be dominated by the frictional influence of the upper and lower solid bounding surfaces of the fluid. Specifically, measurements of the temporal decay of the maximum vorticity in the primary eddies indicate that the spin-down periods of such features are much shorter than the viscous timescales D 2 v . Energy dissipation by internal wave generation is negligible in the system considered.

Two contra-rotating eddies of the Mozambique Ridge Current

The combined existence of a cyclonic and anticyclonic eddy pair on the Mozambique Ridge tends to confirm the hypothesis that they are created by a westward-flowing Mozambique Ridge Current. The eddies are mutually comparable in size but smaller than cyclonic eddies previously observed in the region.

A preliminary study of the basic hydrographic features and the current structures off the Changjiang River mouth in the dry season

The structure of the Changjiang River plume in the dry season is discussed by using data from CTD and anchor stations obtained since 1980 as well as historical hydrographic data. The main results are as follows. (1) In the dry season the Changjiang Diluted Water is confined into a narrow band west of 122°30′E, extends to the south or southeast off the South Passage, and at the surface layer behaves like a quasi-geostrophic current instead of a jet-like current, which is different from what it does in the flood season. (2) The southward Huanghai Coastal Current in winter does not seem to keep its path through the Changjiang Estuary and does not become the southward Donghai Coastal Current as conventionally believed. In fact the Huanghai Coastal Current Water could mix up with the Changjiang Diluted Water and the offshore water going northward there. (3) A small warmer tongue in the south of the study area was found in Dec. 1985 and Jan. 1986, which is generated by a northward upwind current east of the southward Donghai Coastal Current. (4) The surface current in the study area is much affected by winds varying with short time scales but not so much by the monthly mean wind field. (5) A pair of small surface eddies was found in Jan. 1986, which were caused by the Taiwan Warm Current and northerly wind.

Instability and eddy pairs on the Leeuwin current south of Australia

Infra-red images taken by the Advanced Very High Resolution Radiometer onboard NOAA7 reveal much of the surface structure of the Leeuwin Current along the south coast of Australia during spring of 1982. Between Cape Leeuwin and Cape Arid, large eruptions from the outer edge of the current temporarily blocked the flow of warm water along the coast and diverted it into unstable baroclinic waves. The waves had zero phase speed relative to the coast. They also led to formation of cyclone-anticyclone eddy pairs, a phenomenon previously discovered in laboratory experiments with density fronts and buoyancy driven boundary currents. The dimensions, separation distances, growth-rates and life-times of the waves and eddies indicate that these structures were the result of baroclinic instability. Although some eddy pairs carried a large volume of warm water away from the current, tentative estimates of heat transport resulting from baroclinic instability show only a small net heat loss from the current.

Large Eddy Structure of the Turbulent Wake behind a Circular Cylinder

Application of Lumley's definition of ``large eddy'' to Grant's measurements in a cylinder wake yields, for the dominant mode, a counter-rotating eddy pair having cores approximately normal to the wake plane, in a plane parallel to the cylinder, and planes of circulation roughly normal to the maximum positive strain rate.

Viscous Sublayer and Adjacent Wall Region in Turbulent Pipe Flow

The boundary-layer research facility utilizing the highly viscous fluid, glycerine, was constructed to permit detailed experimental investigation of the viscous sublayer. At a pipe Reynolds number of 8700 the sublayer thickness corresponding to a nondimensional distance from the wall of yuτ/ν = 5.00 was 0.110 in. Detailed measurements of the streamwise fluctuating velocities were obtained with hot-film anemometers within the viscous sublayer as well as in the transition region between the linear and logarithmic mean velocity profiles. These data were used to form the space-time correlation function of the streamwise fluctuating velocities. An eigenfunction decomposition of the streamwise fluctuating velocity into a sum of products of eigenfunctions in the inhomogeneous coordinate direction, with random coefficients dependent on the homogeneous and stationary variables, was obtained from the correlation data. One dominant eigenfunction with a structure nearly identical to the mean velocity in the wall region was found. The dominant large scale structure of the flow in the wall region, obtained with the aid of a mixing length approximation, consisted of randomly distributed counterrotating eddy pairs of elongated streamwise extent.

Mesoscale Eddies in Wake of Islands

The properties of mesoscale eddies in the wake of islands are investigated in the light of their apparent resemblance to the Kármán vortex-street pattern. By use of the theoretical results from drag theory and the observed quantities such as cross-stream diameter of the obstacle, wind speed and the dimensions of the pattern, estimates are made for the rate of shedding of eddy pairs, the lateral and longitudinal spacing between the vortices, and the life-time of the eddies and drag of the island. These estimates for the various characteristic parameters of the eddy pattern are in good agreement with their corresponding values in the laboratory experiments where stable vortex streets become discernible. Effects of the tip, ground, and neighboring obstacles are neglected and the problem is treated essentially as two-dimensional flow. Use of this theory makes possible a computation of the lateral coefficient of eddy viscosity. The results provide support for the usual assumption made in meteorological studies that viscosity which is used in the equations of motion and which enters into various dimensionless parameters such as Reynolds number, is the eddy viscosity rather than the kinematic viscosity used in hydrodynamic theory. General meteorological considerations applied to these results indicate that the vertical flux of momentum must be included to account for the lifetime of the eddies.

Karman vortex streets in wakes of islands.

W satellites equipped with narrow-angle cameras have revealed the existence of mesoscale eddies, which are on a scale too small to be delineated by the standard weather observation network and are too large to be seen by an observer on the ground or even pictured from a high-flying aircraft. Hubert and Krueger have described the characteristic properties of mesoscale eddies formed in the wakes of islands that are steep-sided and extend far above the lowlying inversion. These eddies are made visible by patterns in stratocumulus clouds lying beneath a strong inversion about 0.5 to 2.0 km above the ocean surface and have been observed to persist several hundred kilometers downstream. The bandwidth of the eddies is of the order of the cross-stream diameter of the island. Under favorable conditions, the pattern of the eddies bears a strong resemblance to the classical Karman vortex street pattern observed in laboratory studies of wakes behind obstacles. Figure 1 is a typical example of these eddies in the vicinity of the Madeira Islands. Recently, Chopra and Hubert examined the properties of the eddies downstream of the Tenerife and the Gran Canaria Islands in the light of this apparent resemblance between the meteorological mesoscale eddies and the classical vortex street pattern of Fig. 2. This analogy is analyzed further in this note. Viscosity plays two roles in the flow pattern of a Karman vortex street. First, it leads to the formation of the boundary layer in which vorticity is generated and the vortex pairs are formed, the vortices being shed alternately near each edge of the obstacle. Second, the strength of the eddies is dissipated by the mechanism of diffusion; the size of an eddy increases, and its strength decreases as it propagates downstream, until its region of influence overlaps that of a neighboring eddy of opposite circulation. In all of the other respects, the fluid may be regarded as perfect (inviscid). The rate N of shedding of the eddy pair is given by .V ue/a = (l/a)foo (k/2a) taDh(irh/a)] (1) where ue is the speed of propagation downstream of the eddies in a coordinate system fixed to the obstacle, UQ is the speed of the undisturbed flow, a is the longitudinal spacing (pitch) between two eddies of similar circulation of strength k, and h is the lateral spacing (bandwidth) between the two vortex rows. Drag and Spacing Ratio