Environ Fluid MechDOI 10.1007/s10652-014-9340-9

ORIGINAL ARTICLE

On the periodicity of atmospheric von Kármán vortexstreets

Christopher G. Nunalee · Sukanta Basu

Received: 22 July 2013 / Accepted: 15 January 2014© Springer Science+Business Media Dordrecht 2014

Abstract For over 100 years, laboratory-scale von Kármán vortex streets (VKVSs) havebeen one of the most studied phenomena within the field of fluid dynamics. During thisperiod, countless publications have highlighted a number of interesting underpinnings ofVKVSs; nevertheless, a universal equation for the vortex shedding frequency (N ) has yetto be identified. In this study, we have investigated N for mesoscale atmospheric VKVSsand some of its dependencies through the use of realistic numerical simulations. We findthat vortex shedding frequency associated with mountainous islands, generally demonstratesan inverse relationship to cross-stream obstacle length (L) at the thermal inversion heightof the atmospheric boundary layer. As a secondary motive, we attempt to quantify the rela-tionship between N and L for atmospheric VKVSs in the context of the popular Strouhalnumber (Sr )–Reynolds number (Re) similarity theory developed through laboratory exper-imentation. By employing numerical simulation to document the Sr−Re relationship ofmesoscale atmospheric VKVSs (i.e., in the extremely high Re regime) we present insightinto an extended regime of the similarity theory which has been neglected in the past. Inessence, we observe mesoscale VKVSs demonstrating a consistent Sr range of 0.15–0.22while varying L (i.e, effectively varying Re).

Keywords Island wakes · Marine boundary layer · Stably stratified flows ·Strouhal number · Von Kármán vortex street

1 Introduction

Atmospheric von Kármán vortex streets (VKVSs), such as the one shown in Fig. 1, areone of the most intriguing phenomena within the field of fluid dynamics. These spectacularpatterns are commonly referred to as the atmospheric analog of classical VKVSs which

C. G. Nunalee (B) · S. BasuDepartment of Marine, Earth, and Atmospheric Sciences, North Carolina State University,Raleigh, NC 27695, USAe-mail: cgnunale@ncsu.edu

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Fig. 1 MODIS satellite imagery of gravity waves and a VKVS in the wake of the Crozet Islands in thesouthern Indian Ocean on November 2nd, 2004. The shorter island upstream produces continuous gravitywaves while the taller island downstream produces a VKVS. Image courtesy of NASA’s MODIS website [32]

have been observed in the laboratory for over 100 years [3,28,46]. While VKVSs have beenheavily studied in the laboratory [34,35,37], our scientific understanding of the behavior andcharacteristics of their atmospheric cousins is generally confined to what has been gleanedfrom visible satellite imageries [9,20,44,49], idealized numerical simulations [16,36,38,42],and a handful of realistic numerical modeling studies [7,10,26]. Based on the facts thatobservational satellite images simply use marine stratocumulus clouds as tracers for theunderlying wind pattern and only provide 2-dimensional plan views with coarse resolution,we have learned little about the quantitative differences between atmospheric and classicalvortex street properties. The chronic challenges associated with sparse observational data, andthe inherent shortcomings of numerical modeling, have translated into a vague understandingof the properties of mesoscale atmospheric VKVSs. This in turn provokes one to questionwhether the sound knowledge-base concerning classical VKVSs can be justly applied toatmospheric-scale vortex streets.

While it was once thought that atmospheric VKVSs form in the same way as classicalVKVSs (i.e., through boundary layer separation), it has been shown that atmospheric VKVSsare inherently different as they may form through the tilting of baroclinically generatedhorizontal vorticity into the vertical axis [36,42] in the absence of a frictional boundarylayer. On the other hand, others have advocated the importance of turbulent diffusion andpotential vorticity (PV) non-conservation [38,39] for atmospheric VKVS formation. Thepopularity of multiple lines of thought such as these gives credence to the elusive nature ofmesoscale wake vortices resembling von Kármán vortices. Irrespective of their generationmechanisms, it is well justified to speculate that structures as large as mesoscale VKVSs (i.e.,hundreds of km) may have significant implications on other meso/micro scale atmosphericprocesses (i.e., vertical and horizontal wind shear, thermal stability, turbulent kinetic energydissipation, etc.). Such interactions, in addition to their implications for societal operations(i.e., astronomy, dispersion, aviation, etc.), drives the need to understand mesoscale VKVScharacteristics and behaviors.

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In this study, we present new information pertaining to the temporal behavior ofatmospheric VKVSs (i.e., vortex shedding period) and compare our findings to classicalVKVSs. Through a series of numerical simulations, we highlight some of the factors whichgovern the vortex shedding frequency (N ) of mesoscale VKVSs and lay the foundation forempirical relationships between N and these factors. This study is organized as follows:Sect. 2 presents background information on the atmospheric VKVS problem; in Sect. 3 weoutline the scientific questions addressed in this work and the experiments conducted. Resultsfrom all modeling experiments are presented in Sects. 4 and 5; and finally in Sects. 6 and 7we provide a discussion of the results and directions for future work, respectively.

2 Background

In 1912, von Kármán and Rubach made the remarkable discovery that the local circulationswithin VKVS-type wakes are theoretically stable when a certain vortex spacing is observed[47]. This condition of stability means that VKVSs have the potential to concentrate vorticityin narrow wake zones for extremely long distances. Later in the early 1960s, when satelliteobservations first became readily accessible, Hubert and Krueger [20] reinforced this stabilityconcept by observing atmospheric VKVSs existing hundreds of km downstream of the bluffbody that invoked them. In their report, they compared individual island wakes to laboratoryVKVSs and emphasized the fact that these mesoscale eddies seem to maintain energy forvery long periods of time, despite being separated from their mechanical sources by verylarge distances (i.e. over 100 km). More recently, advanced satellite observing systems haveidentified that the large size of these structures is typical and that the presence of atmosphericVKVSs is not necessarily rare [32].

2.1 Formation mechanisms

Although constrained by limited observational data, several recent studies have been devotedto understanding the conditions which support the formation of VKVSs. Most notably, Etling[14] emphasized the importance of an elevated layer of stable stratification below the obstacleheight for vortex formation. This requirement is based on the popular dividing streamlineconcept introduced by Synder et al. [43]. The dividing streamline (Hc) is defined as thecritical height separating flow over the crest of a 3-dimensional barrier from flow around thesides of the barrier. Mathematically, Hc can be estimated by Eq. 1 where H is the height ofthe obstacle and Fr is the Froude Number defined in Eq. 2. Here, U is the upstream windvelocity and fBV is the Brunt–Väisälä frequency (Eq. 3) where g is the gravitational constant,�o is a reference potential temperature, and θ is mean potential temperature as a function ofheight (z) above ground level.

Hc = H(1 − Fr) (1)

Fr = U

H fBV(2)

fBV =√

g

�o

dθ

dz(3)

Below the elevated capping inversion layer, environments conducive to VKVS formationare typically characterized by a well-mixed atmospheric boundary layer (ABL) [26]. Appli-cation of the dividing streamline concept to VKVSs implies lateral flow around the obstacle

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within the stratified layer between the ABL inversion and Hc [21]. If, for example, Hc of theisland is beneath the elevated inversion then a fully over-obstacle flow may develop alongwith potential downstream gravity waves [7,31]. This effect is clearly visible in Fig. 1 as theshorter island more upstream produces a gravity wave and the higher obstacle downstreamproduces a VKVS.

Mesoscale VKVSs can also be conceptualized using a PV framework in which each vortexrepresents a positive or negative PV anomaly. In this case, PV is not conserved within thesystem and is instead generated through two primary and interconnected mechanisms. First,enhanced horizontal vorticity is originally produced by dissipation of the mountain wakethrough the manifestation of a hydraulic jump [39]. Furthermore, Epifanio and Durran [12,13]explored the properties of this hydraulic jump in stratified flows past obstacles and foundthat PV generation was possible in nonlinear viscous wakes even in the absence of surfacefriction. In conjunction with energy dissipation, baroclinic tilting of horizontal vorticity intothe vertical axis is the second dominant process in lee PV production [36].

From a broader perspective, positive and negative PV banners develop downstream of eachside of the obstacle and then are amplified through interaction with a breaking gravity wavealoft, above the dividing streamline [38,42]. In this sense, the gravity wave acts to amplifyvertical vorticity by tilting and stretching horizontal vorticity into the vertical axis on thelee slope of the island [2]. Although research dedicated to understanding the formation oforganized vorticity anomalies in the wake of mountainous terrain has progressed significantlyin recent decades, significantly less is known about the characteristics of these vortices incomparison to vortices produced in the laboratory. Therefore, the focus of this study lies inthe fundamental behavior of VKVSs in the atmosphere, irrespective of forcing subtleties.

2.2 Vortex pattern analysis

Over the years, quantitative metrics have been used to characterize VKVSs, some of whichwere developed by von Kármán himself. Two such metrics are the aspect ratio,

h

a(4)

and the dimensionless width,h

d(5)

where h is the distance between the two counter-rotating vortex trains, a is the distancebetween two vortices in the same vortex train and d is the diameter of the obstacle. Followingvon Kármán’s theoretical framework for inviscid flow around a cylinder, the aspect ratio yieldsa theoretically stable value of 0.28 while the dimensionless width ratio has been observedin the laboratory to be 1.2 [45]. The importance of these dimensionless parameters lies inthe fact that they are generally considered to be the universal properties of VKVSs with nodependence on exterior factors. Knowledge of these typical values for laboratory flows hasprovided some sense of predictability for the turbulent wake of bluff bodies for engineeringapplications such as aerodynamically-induced vibration [29].

With regards to atmospheric VKVSs, Young and Zawislak [49] analyzed 30 individualVKVS events and determined typical values of ≈0.42 for the aspect ratio and ≈1.61 forthe dimensionless width, albeit with a certain margin of error. These differences from lab-oratory VKVS parameters motivates the speculation that perhaps the universality of thesemetrics is not applicable to mesoscale atmospheric flows. After all, classical VKVSs havegenerally been studied in a purely two-dimensional sense; however, in the ABL an additional

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non-homogeneous dimension is introduced. This extra dimensionality adds gravity to thesystem and also introduces variability through momentum, heat, and moisture. In addition,the bluff body responsible for the VKVSs varies vertically and can not be idealized as acylinder. To the best of our knowledge, studies devoted to understanding the influence ofthese changes on the characteristics of atmospheric VKVSs have not been reported in thescientific literature. This means that dependencies of universal metrics (e.g., aspect ratio) onenvironmental conditions (i.e., free-stream wind speed, length scale, viscosity, sea surfacetemperature, etc.) may be present given the significant differences between laboratory andatmospheric VKVSs.

2.3 Vortex shedding frequency

One key to understanding the aspect ratio, and other qualities of the vortex street geometry,is grasping the behavior of the vortex shedding frequency (N ) or period (P = 1/N ). Thisis because, intuitively, N ≈ Ue/a where Ue is defined as the downstream eddy propagationspeed and a is the distance between two vortices of the same sign of rotation used to determinethe aspect ratio. Soon after the discovery of mesoscale VKVSs, Chopra and Hubert [9] studieda particular vortex street induced by the island of Madeira and estimated a period of a littleover 7 h. This estimate was based on the assumption that Ue was ∼85 % of the freestreamvelocity (Uo) which was well justified based on laboratory observations [6]. As satellitetechnology advanced, the ability to directly measure N also became more feasible. In 1977,Thomson et al. [44] studied wakes generated by the Aleutian Islands and determined, usingsatellite pictures and limited meteorological data, vortex shedding periods ranging from 3.8to 6.2 h. Even more recently Li et al. [26] used a numerical weather prediction (NWP) modelknown as MM5 to simulate a specific VKVS in the wake of the Aleutian Islands and foundP to be about 3 h and 40 min for that particular event.

While there is not a deterministic universal model for N , much work has been dedicatedto characterizing its relationship to the Reynolds number (Re). Since the pioneering work ofRoshko [35], many researchers have observed a similarity relationship between the dimen-sionless Strouhal number (Sr ) and Re, where Sr = N L/V and Re = V L/ν. In this case,ν, V , and L represent kinematic viscosity, a relevant velocity scale, and a relevant lengthscale (i.e., obstacle diameter), respectively. Essentially, Sr can be thought of as a normal-ized shedding frequency value. This similarity relationship has illustrated the fact that Srbehaves uniquely for different ranges of Re, the asymptotic approach of Sr to 0.21 withincreasing Re is one such characteristic of the 102 < Re < 104 regime [35]. Nevertheless,radically different behavior has been observed at higher Re values such as a Sr ‘jump’ around105 < Re < 106 [1]. The formulation of empirical relationships between Sr and Re hasbeen pursued by many scholars over the past six decades and a few empirical classificationsfor Sr and N do exist [34]. While beneficial for certain applications, these relationships areonly valid for certain Re ranges and only up to Re ≈ O

(108

). However, Re can commonly

range from O(108

)< Re < O

(1011

)in the ABL1 and yet coherent VKVSs are still clearly

visible from satellite imageries. In other words, the presence of naturally occurring VKVSsin the ABL provides the opportunity to document the Sr−Re relationship in the extremelyhigh Re regime.

In this paper, we have employed a rigorous technique to document N in an effort to gaininsight into the temporal behavior of mesoscale atmospheric VKVSs. In addition, we also

1 We arrive at these values for Re based on magnitudinal analysis assuming ν ≈ 10−5 m2 s−1, O(1)

< V <

O(10

)m s−1, and O

(1)

< L < O(100

)km.

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aim to put our findings into the context of historical laboratory observations by commentingon the natural Sr−Re relationship in the atmosphere. Therefore, the value of this researchis twofold: first and primarily, we determine the relationship of N with L and secondly,we compare atmospheric VKVSs to classical VKVSs with regards to their inherent Sr−Rerelationship.

3 Methodology and simulation details

In the present study, we have used one of the most advanced mesoscale NWP models, theWeather Research and Forecasting (WRF) model [41], to simulate and study the intrin-sic properties of atmospheric VKVSs. Recently, several studies have demonstrated thatmesoscale numerical models can accurately simulate atmospheric VKVSs [7,10,26]. Par-ticularly, Couvelard et al. [10] found the WRF model to realistically capture atmosphericVKVSs with a spatial resolution of only 6 km. However, they noted the need for increasedresolution in future studies; thus we have selected a resolution of 2 km in this study (Table 1).

After validating the model’s numerical representation of a specific VKVS event, we mea-sure the vortex shedding frequency based on the timing of the characteristic transverse jetswhich develop between the discharge of two counter rotating vortices [37]. Combining thisestimate for the vortex shedding frequency with approximate length and velocity scales, weput forth deterministic values for Sr in the atmosphere. Next, we adjust the length scale of thesystem by varying the diameter and height of a quasi-idealized island surrogate; this effec-tively modifies Re. While keeping all other conditions constant, we document the Sr−Rerelationship for atmospheric VKVS events. In parallel, we also explore the dominance ofthe horizontal and vertical length scales (i.e., island diameter and height) of the system withrespect to the observed Sr−Re relationship. Finally, we test the universality of the observedrelationship by simulating additional VKVS events invoked under diverse settings in twoother locations across the globe.

The Portuguese island of Madeira was selected as the primary island studied in thiswork due to its rugged terrain perpendicular to the regional Northeastern trade winds andpopularity among previous VKVS researchers [4,7,9,10,14]. The island of Madeira is fairlymountainous and covers an area of about 823 km2 (Fig. 2) with a maximum elevation of1,862 m above mean sea level (msl). It is a very rugged landmass with a central chain ofmountains that essentially runs west/east, maintaining a high elevation throughout. Madeirais one of the most suitable islands for numerical studies considering it is among the largestislands known to produce a clearly discernible VKVS pattern in the marine stratocumulusclouds. This is important for this study because even the geometry of Madeira is near the lowerresolvable limit of modern mesoscale models such as the WRF model [40]. From NASA’smoderate resolution imaging spectroradiometer (MODIS; [24,25,33]) image gallery a controlVKVS event was identified in the wake of the Portuguese island of Madeira on July 5th, 2002(left panel of Fig. 3).

Table 1 Domain configurations

Domain �x, �y (km) �t (s) Nx Ny Dimensions (km2)

d01 10 45 125 125 1,240×1,240

d02 2 9 451 451 900×900

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Fig. 2 Spatial dimensions of the island of Madeira

Fig. 3 MODIS satellite imagery of multiple VKVSs in the wake of Madeira Island and the Canary Islands(left) and WRF model simulated cloud cover (right) near Madeira for July 5, 2002. Left image is a courtesyof NASA’s MODIS website [32]

In this study, we have used a similar modeling configuration to that used in the successfulMadeira simulations of Couvelard et al. [10]. However, in the present simulations we haveused higher horizontal resolution of 2 km by 2 km (see Table 1). Two domains were used, anouter and an inner nested domain; and in both domains there were 61 pressure-based, terrainfollowing vertical coordinates between the surface and 100 mb with ∼28 levels below 1 kmabove ground level (AGL). This vertical resolution is similar to typical operational NWPmodels; however, the horizontal resolution is significantly higher than most NWP models[15]. The simulations conducted for this study were run for a total of 48 h with the first 24 hbeing used as spin-up time for the model. The final 24 h of the simulations were used foranalysis and enabled us to study the temporal evolution of the vortex street, specifically theshedding rate. The physics parameterizations used in these simulations included: Mellor-Yamada-Janjic PBL scheme [22]; WRF Single-Moment 5-class Microphysics [18], RRTMLongwave Radiation [30], Dudhia Shortwave Radiation [11], Kain-Fritsch Cumulus in theouter domain [23], and Noah Land Surface Model [8]. For initialization and lateral boundary

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conditions, the ERA-Interim reanalysis dataset produced by the European Centre for MediumRange Weather Forecasts [5] was used. It contains environmental data records from 1979through the present at 6-h intervals, and includes a horizontal resolution of ∼79 km with 60vertical coordinate levels. In this particular study, we have not explored the sensitivity of ournumerical VKVS simulations to physics parameterization schemes or model configuration,but this would be an interesting subject for future research.

4 Real-terrain case study

For this particular event (Fig. 3), the wind in the lower troposphere was impinging on Madeirafrom the north-northeast as shown by the orientation of the lee vortices in Fig. 3; these vor-tices are traced by the marine stratocumulus cloud deck present at the top of the boundarylayer. Most importantly, the wind direction was nearly perfectly perpendicular to Madeira’smountain range. This caused maximal horizontal flow blockage sufficient to invoke theobserved VKVS. Also in Fig. 3, the VKVS is observed to maintain a coherent pattern for avery long distance downstream of Madeira (i.e., hundreds of km). Additionally, smaller vonKármán vortices can be seen shedding from the nearby Canary Islands to the southeast ofMadeira.

The WRF model qualitatively simulated the atmospheric VKVS associated with the realMadeira terrain fairly well (right panel of Fig. 3). The diameter of each simulated vortexclosely resembled the actual vortex size (i.e., roughly the cross-stream diameter of Madeira)immediately downstream of Madeira. Further downstream, the two vortex trains steadilydiverged from one another and each vortex became larger and more diffused. From thesimulated cloud pattern (Fig. 3) the distance between the vortices of the same rotation wassimilar to the satellite observations; however, the overall length of the vortex street waschronically underestimated by the WRF model. This could perhaps be a result of numericaldiffusion or the PBL parameterization used.

As with most numerical modeling efforts, quantitative inaccuracies were present in theWRF simulation. Radiosonde profiles of wind speed and potential temperature at Madeira’scapital of Fuchal are shown in Fig. 4 in comparison with simulated profiles. On this day,Fuchal was on the leeward side of the island which contributed to the strong decreasein wind speed beneath about 1,000 m msl. From the observations, a thermal inversion

0 5 10 150

500

1000

1500

2000

2500

3000

3500

Wind Speed (m/s)

Hei

ght A

GL

(m)

Simulated 12ZSimulated 16ZObserved 12Z

15 20 25 30 35 40 45 500

500

1000

1500

2000

2500

3000

3500

Potential Temperature (C)

Hei

ght A

GL

(m)

Observed 12ZSimulated 12Z FuchalSimulated 12Z Upstream

Fig. 4 (Left) Observed and simulated wind speed vertical profiles at Fuchal, Madeira. (Right) Observed andsimulated potential temperature vertical profiles at Fuchal, Madeira and at a point just upstream of Madeira.Both plots are valid for July 5th, 2002

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Fig. 5 Streamwise vertical cross-section of simulated potential temperature (K) through the island of Madeira(flow is from the left). Gravity wave and hydraulic jump signatures are identifiable along with a distinct ABLcapping inversion above 600 m msl upstream of the island

was apparent around 800 m msl which was an important environmental component forthe development of the vortex street; below the inversion the radiosonde recorded a well-mixed boundary layer. The WRF model struggled to capture these features and instead pro-duced a fairly stratified ABL at the location of Fuchal. Also, the simulated and observedfree-atmosphere thermal profile was noticeably different above 2,000 m with the WRFproducing a warm bias of about 5 K. However, the thermal profile depicted by the WRFmodel just upstream of Madeira was much more similar to that recorded by the radiosondewith a well-mixed boundary layer beneath a strong inversion. Although the simulatedinversion was a few hundred meters lower than what was observed in the radiosonde,the radiosonde temperature measurements may have been altered by the thermal effectsinduced by the island itself while the upstream modeled profile corresponded to an entirelymarine ABL.

Nonetheless, it is important to note that the highly-dynamic nature of island wakes makesit difficult to compare model solutions to instantaneous observations. Figure 5 displays avertical cross-section through the island of Madeira in the streamwise direction; here theunsettled recirculation zone just downstream of the island is clearly distorting the thermalprofile from the incoming upstream profile. In this figure, the WRF-simulated low-levelthermal profile upstream of Madeira generally matches what was observed by the radiosondewith a capping inversion above 600 m msl along with a well-mixed ABL beneath. Also,Fig. 5 illustrates the presence of a gravity wave signature embedded within the downstreamVKVS in addition to a hydraulic jump above the crest of the island. These features areexpected according to the mesoscale dynamics known to be associated with atmosphericVKVSs [38,39,42].

The stratified layers above the inversion acted to divert the flow around the flanks ofthe island. Since most of the flow traversed the island laterally, we assumed that the flowspeed-up over the crest of the island was minimal. Therefore, we took the wind speed ataround 1,500 m msl to be roughly the free-stream, pressure-driven (i.e., geostrophic) windspeed and characteristic velocity scale of the flow (Uo). In this case, we will assume Uo tobe on the order of 10 m s−1 which is consistent between both the numerical simulation andthe observation. Using this velocity scale, and the thermal profile previously discussed, we

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arrived at a Brunt-Väisälä frequency of ∼0.017 Hz and a Froude number of ∼0.23 for thesimulated 0–1,800 m msl layer. From here, we estimated the dividing streamline height forthis real case to be ∼1,212 m msl.

The WRF model simulated the vortex street as having a lateral spacing (h) of about 54 kmand a longitudinal spacing (a) of about 187 km. This produced an aspect ratio of 0.29.This value was considerably smaller than the average aspect ratio observed by Young andZawaislak [49] for atmospheric VKVSs of 0.42, but closer to the theoretically derived stableaspect ratio of 0.28 [47]. Using the assumed values of the upstream wind speed (Uo) andthe vortex spacing (a), we can estimate 1/N ≈ 6 h using the assumption made by Chopraand Hubert [9] that the eddy propagation speed (Ue) is ∼85 % of (Uo) [6]. Furthermore,this yields Sr approximately equal to 0.20 assuming a length scale equal to the cross-streamlength of the island (≈45 km) at its base. However, it should be noted that the validity ofthe eddy propagation speed assumption is questionable in the atmosphere as it was derivedbased on laboratory VKVSs; in our simulations we observed Ue to be as low as 60 % of Uo.

The wind perturbations associated with the VKVS propagated downstream in a regularpattern. Figure 6 shows that the presence of Madeira’s steep topography primarily redirectsupstream momentum around the island laterally. This increases wind speeds on the flanksof the island up to nearly 20 m s−1, a 100 % increase from the upstream wind flow whichwas about 10 m s−1. These two high wind speed zones were separated by somewhat of arecirculation zone where wind speeds were generally <1.5 m s−1. Although the recirculationzone meanders with the shedding of each counter-rotating vortex, its edges may easily have awind difference of 15 m s−1 over 6 km or less. Additionally, the enhancement of wind shearin this region directly leads to enhanced turbulence and narrow ribbons of anomalously highturbulent kinetic energy (not shown).

High momentum transverse jets are also visible in the lee of Madeira and can be observedpenetrating across the wake from alternating sides. Generally, these jets changed in theirorientation by about 90◦ from one another but in some cases up to 180◦ from one another.Essentially, the momentum withheld in these jets evolve to form the outer rings of eachpropagating vortex. This is what initially fuels the high tangential wind speeds in the rimof each vortex. These wind speed maxima can be seen in Fig. 6 maintaining themselves forhundreds of km downstream, nearly all the way south to the Canary Island archipelago. Thealternating jets occur at regular intervals and correspond to the vortex shedding frequency.The two snapshots in Fig. 6 illustrate nearly identical vortex streets separated in time by 5 h.During these 5 h, the vortex pattern has completed one entire period and has returned back toits original phase. The passage of two transverse jets across one particular location representsthe shedding frequency of the vortex street, and will thus be used in Sect. 5 as a measurablepredictor of Sr .

5 Quasi-idealized-terrain case study

Next, additional simulations were ran for the same VKVS event described in Sect. 4. But,in each new simulation the island of Madeira was substituted with an idealized bell-shapedisland of predefined diameter and height (see Eq. 6). Efforts to systematically vary these twoaspects of the island exposed the natural sensitivity of the VKVS shedding frequency to thefundamental length scale of the system. The experimentation presented in this section hastwo primary components: (1) the height of the island is kept constant (i.e., similar height tothe real island of Madeira; 1,800 m msl) and the diameter is varied and (2) the height ofthe island is varied while the diameter is held constant for a few cases and varied for others.

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Fig. 6 Wind speed contours (colored) and horizontal wind velocity vectors (black) at ≈110 m msl at 4 UTC(top) and at 9 UTC (bottom). Notice that the vortex street appears nearly identical at these two times despitebe separated in time by 5 h; this highlights the roughly 5 h period of the street

Next, in Sect. 5.2 two unrelated VKVS events were simulated in other parts of the worldunder different conditions but with the control, idealized Madeira island (i.e., H = 1,800 mand base diameter = 40 km) replacing the actual islands.

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Fig. 7 Terrain height profiles through the center of each idealized island. Note that at the height of the ABLinversion the diameters of all islands are very similar regardless of differing H

The geometry of the idealized islands were defined by

Zxy = H

4

[1 + cos

(2π(x − d)

d+ π

)][1 + cos

(2π(y − d)

d+ π

)](6)

where Zxy is the local terrain height of the island as a function of zonal (x) and meridional(y) position, H is the maximum terrain height at the center of the island, and d is thediameter of the base of the island. These new islands used the modal land-use classification ofcentral Madeira and were symmetrical in both horizontal axes but had varying diameters andmaximum heights. The control quasi-idealized simulation essentially duplicated the majorgeometry of Madeira with H = 1800 m and base diameter of 40 km. The new idealized islandshad base diameters of 20, 60, 80, 100, 140, and 200 km and at the height of the upstreamsimulated thermal inversion (≈800 m msl; see Sect. 4) the islands were roughly 60 % of theirbase diameter for the control height for the Madeira event. As will be subsequently shown inSect. 5.1, the two cases with island diameters of 140 and 200 km did not produce any formof VKVS signature. This was important because it provides some of the first evidence of theupper Re range for atmospheric VKVSs. On the other hand, we tested the smallest lengthscales possible given our mesoscale modeling approach. Since this study is concentrated onmesoscale VKVSs we essentially captured the entire range of possible mesoscale VKVSs,albeit that covers only about one order of magnitude of Re.

In addition, the maximum height of the islands were also adjusted to new heights of1,400, 2,200, and 2,800 m in order to test the sensitivity of the Sr−Re relationship to this(vertical) length scale. It is critically important to note that the formulation for idealizedterrain height used here (Eq. 6) only weakly adjusted island diameter at the inversion heightwhen H is varied and base diameter is held constant. That is, the difference in island diameterat inversion height for the H = 1,400 and 2,800 m cases is only about 7 km (Fig. 7). Thisbehavior is desirable because it demonstrates if and how Sr responds to varying H when thelargest horizontal length scale above the inversion is held nearly constant.

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Additionally, two other VKVS events were identified from the MODIS satellite imageries.These events were associated with the rugged Pacific Ocean islands of Guadalupe off of thewest coast of Baja Mexico, and the westernmost island in the Juan Fernández archipelago,Alejandro Selkirk Island. For both cases, the actual islands were replaced with the controlidealized island used in the Madeira case and the shedding frequency of the simulated VKVSswere documented. Idealized islands were used instead of real terrain for these cases in order toavoid introducing terrain-induced bias into our periodicity sensitivity tests and also becauseGuadalupe and Alejandro Selkirk Islands are considerably smaller than Madeira which wouldhave required adjusting the resolution of our numerical grid in order to simulate them withhigh fidelity. By examining the characteristics of these two additional events we were ableto asses whether background meteorological and/or geographical conditions (e.g., globalpositioning, boundary layer height, upstream wind speed, etc.) affect Sr for atmosphericVKVSs.

5.1 Madeira event

In order to quantitatively document the shedding frequency in each simulation, U and V windspeed time series were taken from locations downstream of the islands which experienced thepassage of each transverse jet. The downstream locations where the time series were takencorrespond to roughly one island diameter downstream and at heights from the surface up to2000 m msl. From these time series, Fourier spectral analysis and autocorrelation functionalanalysis were used to compute the VKVS period, and eventually Sr , at each grid level withinthe lower troposphere. This approach produced ∼50 P (i.e., Sr ) samples per simulation orcase; the mean and standard deviation of these values were computed for analysis and areeventually presented in Table 2.

Using the convention described in Sect. 2.2, and assuming that the islands’ diameters at theheight of the inversion represented an effective length scale, the respective Reynolds numbersfor each Madeira simulation (in increasing order) were 1.2 × 1010, 2.4 × 1010, 3.6 × 1010,4.8 × 1010, 6.0 × 1010, 8.4 × 1010, and 1.2 × 1011. Please note that the WRF model neglectsmolecular viscosity in the governing momentum equations; however, the effects of moleculardissipation are parameterized within the TKE equation. By assuming the standard molecularviscosity of air, we were able to estimate Re based on the simulated wind speed. On theother hand, the WRF model computes eddy viscosity (KM ) which provides the opportunityto estimate modified Reynolds numbers (ReK ) using an eddy viscosity approach instead ofthe traditionally molecular viscosity approach. The computed ReK values are presented inTable 2 for comparison to Re and the values of KM are presented later in this section alongwith a brief discussion.

Figure 8 displays how N varied with island diameter for the quasi-idealized Madeiraevents. The passage of the transverse jets act as proxies for the periods of the VKVSs andare demarcated by the times with the strong wind speeds shown by the hot colors and longervectors. Additionally, the jets alternate in their orientation by 90◦–180◦ (refer to Sect. 4) withthe larger vortex streets favoring the 90◦ end of the spectrum and vice versa for the smallervortex streets. The time span between the passage of two jets signifies one full period of thevortex street.

In Fig. 9, the autocorrelation analyses for the control case are shown. It is clear to see thatfor this case the periods estimated by the two techniques yield very similar values of around4 h and also for all heights plotted. In addition, the periods for the U and V winds agreefairly well. To be concise, similar analysis for the other Madeira cases are not shown here,but their results are summarized by cases M1−M13 in Table 2.

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Table 2 Simulation diagnostics for quasi-idealized cases assuming a length scale defined by the islanddiameter at the ABL inversion (L) and constant wind velocity scale of 10 m s−1. L∗, H∗, and P∗ representthe control simulation island diameter, height, and observed period, respectively (i.e., L∗ = 24 km, H∗ =1,800 m, and P∗ = 3.93 h). ReK is the modified Reynolds number computed by substituting eddy viscosity formolecular viscosity. Finally, M , G, and J F represent simulations of Madeira, Guadalupe, and Juan Fernándezevents, respectively

Case Hc (m) L/L∗ H/H∗ Re ReK P (h) P/P∗ Sr

M1 1212 0.5 1.0 1.2 × 1010 48900 2.20 ± 0.00 0.56 ± 0.00 0.150 ± 0.000

M2 1212 1.0 1.0 2.4 × 1010 97800 3.93 ± 0.10 1.01 ± 0.03 0.185 ± 0.005

M3 1212 1.5 1.0 3.6 × 1010 146700 5.85 ± 0.34 1.49 ± 0.09 0.170 ± 0.010

M4 1212 2.0 1.0 4.8 × 1010 195600 7.45 ± 0.62 1.89 ± 0.16 0.185 ± 0.015

M5 1212 2.5 1.0 6.0 × 1010 244500 8.25 ± 0.75 2.10 ± 0.19 0.205 ± 0.015

M6 1212 3.5 1.0 8.4 × 1010 342300 − − −M7 1212 5.0 1.0 1.2 × 1011 489000 − − −M8 812 0.9 0.8 2.2 × 1010 88020 3.65 ± .32 0.93 ± 0.08 0.167 ± 0.015

M9 1612 1.1 1.2 2.6 × 1010 107580 4.17 ± 0.17 1.06 ± 0.04 0.176 ± 0.010

M10 2012 1.1 1.4 2.6 × 1010 107580 4.15 ± 0.18 1.06 ± 0.05 0.193 ± 0.025

M11 812 1.8 0.8 4.3 × 1010 2152080 7.37 ± 0.70 1.88 ± 0.18 0.167 ± 0.015

M12 1612 2.2 1.2 5.3 × 1010 2630320 8.08 ± 0.25 2.06 ± 0.07 0.182 ± 0.005

M13 2012 2.3 1.4 5.5 × 1010 2749880 8.34 ± 0.84 2.12 ± 0.21 0.190 ± 0.015

G1 1274 1.0 1.0 4.0 × 1010 98700 4.50 ± 0.33 1.15 ± 0.09 0.160 ± 0.010

J F1 1244 0.5 1.0 1.2 × 1010 417391 1.92 ± 0.25 0.50 ± 0.05 0.175 ± 0.025

From Table 2, it is evident that increasing the island diameter resulted in essentially amonotonic increase in VKVS period, up to a particular threshold that is. Here L representsthe effective diameter of the islands (assumed to be the cross-stream diameter at the ABLinversion), P is the measured vortex period, and Re and Sr are the dimensionless Reynoldsand Strouhal numbers. For cases M1−M5, VKVS signatures were very obvious as the winddirection made regular oscillations, and 2-dimensional plan view images of the systems (notshown here) clearly showed repeating pattens of vortices downwind of the islands. On theother hand, for cases M6 and M7 no von Kármán type vortex streets were ever observed. Inthese cases, the vortices shed behind each flank of the island (i.e., vortex train) never appearedto grow large enough to interact with the opposite train. This leads one to believe that perhapsthis is near the upper Re limit for VKVS formation in the atmosphere. Nevertheless, for casesM1−M5 VKVS periods of 2–9 h were observed which led to Strouhal numbers in the rangeof 0.15 to 0.22 for all cases. Interestingly, this is very similar to what has been documented forlower Reynolds number flows containing VKVSs. For case M5, N was difficult to identify asthis Reynolds number appeared to fall within a sort of transition zone; in this case the islandwake appeared to alternate between a definitive VKVS and a non-coherent wake structureleading one to question the validity of the Sr estimate for this case.

Additionally, the heights of the idealized islands were varied both greater than and less thanthe original Madeira maximum height of 1,800 m msl while the base diameter was kept 40 km(cases M8−M10 in Table 2). These new heights reached 1,400, 2,200, and 2,600 m msl;however, interestingly we observed little differences in periodicity between these simulationsand the corresponding simulation of 40 km diameter, H = 1,800 m idealized Madeira case.

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Fig. 8 Time series of wind speed and direction corresponding to a location approximately one diameterdownstream from the island and at 100 m msl. From top to bottom, the time series represent cases M1–M7shown in Table 2. The vertical lines represent 2 h time intervals and the magnitude of the wind speed isdenoted by the relative length of the vectors and their colors (i.e., cool colors represent weak wind and hotcolors represent strong wind)

For a more in-depth perspective of this observation, we also conducted additional simulationsof all three new island heights but with 80 km diameters (cases M11−M13, Table 2). In thesecases, we also observed the shedding frequency to demonstrate little variability betweeneach simulation of varying height and to be very similar to case M4. Essentially, theseresults indicate that the island diameter represents the dominant, intrinsic length scale ofthe system (more so than the height) and thus primarily drives the Sr−Re relationship.Fundamentally, this is expected as the mesoscale VKVS phenomenon is generally assumedto be the atmospheric analog of classical 2-D VKVSs. In essence, this assumption of two-dimensionality is supported by our findings here. The two-dimensional nature of VKVSs is

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10−5

10−4

10−3

100

105

1010

frequency (Hz)

Pow

er S

pect

ral D

ensi

ty (

m2 s

−1 )

USlope=−3Slope=−5/3

10−5

10−4

10−3

100

105

1010

frequency (Hz)

Pow

er S

pect

ral D

ensi

ty (

m2 s

−1 )

VSlope=−3Slope=−5/3

Fig. 9 Fourier spectral analysis of the U (upper left) and V (upper right) wind components for the controlcase and autocorrelation function analysis (bottom). Multiple lines of the same color and symbol representanalysis for different heights above the surface. The green lines represent peaks in spectral energy and theanalyzed energy frequency

also evident in the velocity spectra (Fig. 9, top panels). The −3 spectral scaling is a hallmarkfor 2-D turbulence [27,40]. A detailed spectral characterization of VKVSs is beyond thescope of this paper. However, we would like to point out that a −5/3 regime (i.e., inertialrange) was also evident in some of the simulations in addition to (sometimes in lieu of) the−3 spectral regime. These results will be reported in a future publication.

Based on the concept of two-dimensionality, the thermally stable layer between the ABLinversion and the island dividing streamline acts as the critical zone harboring lateral flowaround the island, thus fueling the VKVSs. From this point, one may consider that themaximum horizontal length scale of the island, within this critical zone, is related to the largestand most dominant scales of motion in the VKVS. Additionally, islands which typicallyinstigate VKVSs are, roughly speaking, cone-shaped which leaves the largest diameter withinthe critical zone to be at the ABL inversion height. This island diameter can be thought of asthe effective length scale of the VKVS system which dictates the Sr−Re relationship. Thishypothesis is supported by the results found here as the formulation used here for creatingidealized island terrain height lead to little change in island diameter at the ABL height whenchanging maximum island height.

Table 2 also introduces a modified Reynolds number (ReK ) which is formulated the sameas the traditional Re except that eddy viscosity (KM ) is used in lieu of molecular viscosity.For atmospheric applications, the use of ReK is perhaps more appropriate than the use ofRe considering the fact that atmospheric flows are dominated by turbulent diffusion overmolecular diffusion by many orders of magnitude. With regards to mesoscale VKVSs, theuse of ReK has been occasionally used in previous studies [16,26,44]. In the present work,eddy viscosity within the ABL was volume averaged throughout the upstream portion of thedomain. Then, these KM values were averaged in time throughout the simulation period andincorporated into Table 2. Figure 10 illustrates the mean profiles of KM for each event. Using

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Fig. 10 Upstream planar averaged vertical profiles of eddy viscosity for the Madeira event (left), theGuadalupe event (middle), and the Juan Fernández event (right)

mean KM values we find Reynolds numbers ranging from ≈5×104 to 2.7 × 106; comparingour Sr measurements with laboratory measurements of VKVSs in this Re range we findclose similarities. This implies that the periodicity of atmospheric VKVSs behave similar toclassical VKVSs when the concept of eddy viscosity is employed.

5.2 Guadalupe and Juan Fernández events

The conclusions from cases M1−M13 in Table 2 directly support the hypothesis that thedimensionless period of mesoscale VKVSs is essentially not a function of a vertical lengthscale and instead primarily a function of a horizontal length scale and a velocity scale. Whilethe concept of a universal Sr number for atmospheric VKVSs, virtually irrespective of Re,is an intriguing conclusion; unfortunately, the data presented here are too imprecise to makesuch a bold conclusion. Nevertheless, the information displayed in Table 2 does seem toindicate that the dimensionless periods of mesoscale VKVSs do fall in a relatively compactrange (i.e., 0.15–0.22). In order to support this finding, we considered the Guadalupe and JuanFernández events previously described and compared their simulated shedding frequency tothat of the quasi-idealized Madeira cases (G1 and J F1).

Although some free-stream conditions were similar between all three events, as typical forVKVS formation (i.e., a thermal inversion below the crest of the island, steady free-streamwind flow from a constant direction, etc.), these simulations had entirely different initial andboundary conditions and thus acted as acceptable trial candidates for testing the universalityof the atmospheric Sr−Re relationship. In Fig. 11, time-height plots of potential temperature

Fig. 11 Upstream potential temperature profiles for the Guadalupe (left panel) and Juan Fernández (rightpanel) events corresponding to 24–36 h of the simulations

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profiles can be seen for cases G1 and J F1. Comparing these figures to what was illustratedin Sec. 4, it can be noted that the Guadalupe event had a slightly lower thermal inversionfrom the Madeira event. This increased the effective length scale to about 65 % of the basediameter of the idealized island, compared to 60 % for the Madeira cases. Alternatively, theJuan Fernández event had a much higher thermal inversion height than the control Madeiraevent. For this case the effective length scale was about 30 % of the island base diameterwhich was about half that of the control Madeira case. With these length scales in mind, wewould expect case G1 to have a slightly longer period than the control case (M2) and caseJ F1 to have a period of about half that of the control case in order to keep Sr generallywithin the same range as found for the Madeira case. This is indeed what was found. It shouldbe pointed out that these additional simulations (cases G1 and J F1) had free-stream windspeeds similar to the Madeira case of near 10 m s−1 at the boundary layer height. This meansthat we did not test scenarios where the velocity scale was significantly higher or lower than10 m s−1; but on the other hand, we were not aware of any VKVS event which occurred inextremely higher or lower wind circumstances.

6 Discussion and conclusion

In this paper, we have used the WRF model as a tool to understand the behavior of quasi-idealized VKVSs induced by mountainous islands within the marine ABL. While the WRFmodel underestimated the length of these particular streets, other characteristics of the VKVSswere qualitatively, and somewhat quantitatively, similar to observations; namely the vortexspacing and consequently shedding frequency appeared realistic. The period was estimatedfor the control case to be about 4 h yielding a Strouhal number of ∼0.185 when assuming avelocity scale of 10 m s−1 and a length scale defined by the island diameter at the height of thethermal inversion (i.e., about 24 km). Furthermore, subsequent experimentation conductedby systematically adjusting the length scale (i.e., Re) of the islands produced little identifiablechange in this value of Sr while within a Re regime conducive to VKVS formation. Thisindicates that the period of mesoscale atmospheric VKVSs is a direct and linear functionof L . Also, it was found that the island diameter, specifically at the height of the ABLinversion, was the dominant length scale (L) associated with VKVS frequency modification.This conclusion was supported by the fact that (based on the island shape equation usedhere; Eq. 6) changing the maximum height of the island resulted in little change to the islanddiameter at the ABL inversion and therefore the frequency of the VKVS remained generallyconstant. In addition, numerical analysis of two other, very unique cases conformed to theSr−Re relationship found for the control event. This supports the quasi-universality of therange of Sr put forth in Table 2 for mesoscale atmospheric flows.

Although somewhat broad, the range of Strouhal numbers put forth here for mesoscaleVKVSs generally resembles what has been observed in laboratory flows of much lowerReynolds number. The weakly fluctuating values of Sr with respect to Re leads the authors tobelieve that atmospheric VKVSs behave similarly to VKVSs in the range of 102 < Re < 104

studied by Roshko [35]. If this is indeed the case, one might conclude that the use of eddyviscosity, as opposed to molecular viscosity, could be used as an instrument to concep-tualize similarity theories gleaned from laboratory VKVSs in the context of atmosphericVKVSs.

It should be noted that while the Sr measurement method used here provided a reasonableassessment of the general Sr−Re relationship for atmospheric VKVSs, it was limited bythe fact that the selection of the reference point was a somewhat subjective process. Iden-

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tifying a downstream point that is perfectly in the center of the wake is a difficult task andadditionally the wake itself may not be symmetrical nor entirely steady throughout the day.These slight asymmetries and time dependencies may arise from changes in wind direc-tion or ABL height and eventually contribute to the uncertainty of the values presented inTable 2.

On a final note, understanding the behavior of vortex shedding frequency in atmosphericflows is another step forward in identifying and understanding the universal Sr−Re similarityrelationship. From this new knowledge, we can better understand the placement of embeddedwake vortices as a function of time and the spacing between atmospheric vortices. Froma practicality perspective, the ability to estimate the vortex shedding frequency, and vortexspacing, from commonly known environmental parameters may prove advantageous througha number of applications varying from optical wave propagation [48] to long-range pollutantdispersion [17].

7 Future directions

Moving forward, future studies should be carried out to explicitly assess mesoscale modelsensitivity to physical parameterization schemes (surface layer, planetary boundary layer,etc.) and validity when simulating atmospheric VKVSs. This will help to identify any modelbias associated with defining the Sr−Re relationship in the turbulent atmosphere. Suchassessments will be challenged by limited observational data; however, evolving remotesensing technology such as stereo satellite-derived winds at cloud height may offer uniqueopportunities to address this issue [19]. Additionally, more advanced numerical modelingtechniques, such as large-eddy simulation (LES), can be used to study atmospheric VKVSson the meteorological microscale. Perhaps studies of this nature could provide better insightinto the bridge between atmospheric and classical vortex streets. To the knowledge of theauthors, only one such study ([16]) has been published which was devoted to investigatingthe structure of von Kármán vortices produced by somewhat smaller obstacles than those inthis study. In that work, the authors calculated slightly lower Sr values than we have here,near 0.12–0.13; this provokes the question that perhaps the dynamics of atmospheric VKVSsvaries throughout the meso-micro scale transition regime.

Also, LES should produce a more realistic turbulence regime in the lee of the islandand possibly provide insight into interaction of turbulence and mesoscale dynamics in thisregion. In addition, projects should be undertaken to more accurately measure the vortexshedding frequency, possibly by using meteorological measurements as benchmarks [19].The current study is limited by the fact that the only validation of simulated Sr against realisticSr was possible by comparing the spatial distances between eddies in the satellite imageryto the simulated eddy patterns. This weakness, combined with the unavoidable limitationsof numerical modeling, introduces an important source of uncertainty which will need tobe addressed in future studies before an atmospheric Sr−Re similarity relationship can besolidified.

Acknowledgments The authors acknowledge financial support received from the Department of DefenseAFOSR grant under award number (FA9550-12-1-0449). Any opinions, findings and conclusions or recom-mendations expressed in this material are those of the authors and do not necessarily reflect the views of theDepartment of Defense. Additionally, computational resources were generously provided by the RenaissanceComputing Institute of Chapel Hill, NC. Finally, the authors would like to thank Branko Kosovic, Gary Lack-mann, Ákos Horváth, Kevin Mueller, and Mikhail Vorontsov for important dialogue and also Bert Holtslagfor pointing out a relevant internal report by Annick Terpstra.

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