Lee Vortices. Craig C. Epifanio. Introduction. Stratified Flow Past Topography: Basic Phenomenology

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1 Epifanio, C. C.: Lee Vortices in Encyclopedia of the Atmospheric Sciences, Cambridge University Press, Lee Vortices Craig C. Epifanio National Center for Atmospheric Research, Boulder, Colorado Introduction Flow of the atmosphere past a high mountain barrier often results in a low-level wake of reversed flow immediately downstream of the obstacle. Viewed from above such a wake is manifest as a pair of counter-rotating vortices circulating about vertical axes. Observations suggest that this pattern of counter-rotating lee vortices can sometimes persist over a time scale of days. Flow past the island of Hawaii often provides examples of such quasi-steady recirculating wakes (Fig. 1a). In other cases the counter-rotating wake pattern is unstable and a transition occurs to a state in which vortices of alternating sign are periodically shed downstream to form a vortex street. The imprint of vortex streets in cloud layers downstream of mountainous islands is occasionally captured in satellite images. A particularly striking example is shown in Fig. 1b. Observational studies suggest that wakes and vortices are a common feature of atmospheric flow in mountainous regions. Lee vortices that form in the vicinity of cities tend to recirculate pollutants and thus have important consequences for local air quality. Examples include the Melbourne Eddy (near Melbourne, Australia), the Santa Barbara Eddy (near Santa Barbara, California), and the Denver Cyclone (near Denver, Colorado). In some cases orographic vortices are linked with the initiation and intensification of severe weather. A wellstudied example is the Denver Cyclone, which is often associated with the development of severe storms producing hail, flooding and tornados. Studies of flow past the Alps suggest that low-level mountain wakes may also interact with upper-level troughs to produce larger, synoptic-scale lee cyclones. Lee vortices generally develop on time scales that are short compared to a day and have length scales on the order of km. As a result, the rotation of the Earth has only a secondary effect on the motion. Most theoretical studies of mountain wakes have neglected the Coriolis force (i.e., have considered non- Corresponding author address: Craig C. Epifanio; NCAR, MMM-FL3; 3450 Mitchell Lane; Boulder, CO cepi@mmm.mmm.ucar.edu rotating flow) and we focus on such studies in the following sections. There is some evidence to suggest that on longer time scales and larger spatial scales the influence of the Earth s rotation tends to suppress the formation lee vortices; but in general the effect of planetary rotation on lee vortex formation is a topic in need of further study. Stratified Flow Past Topography: Basic Phenomenology Overview Here we briefly review the basic phenomenology of non-rotating stratified flow past an isolated ridge in three-dimensions (3D) as revealed by theoretical and numerical investigations and laboratory experiments. Most idealized studies of 3D flow over orography have assumed a uniform upstream flow speed U and constant upstream buoyancy frequency (or Brunt-Väisälä frequency) N and we make the same restrictions here. Such a model gives a rough first approximation to many atmospheric flows but excludes phenomena such as trapped lee waves (or so-called ship waves) which depend on vertical variations in N and U. We suppose that the ridge shape may be characterized by a streamwise length scale a, a cross-stream length scale b, and a maximum height h. The non-dimensional control parameters governing the behavior of the flow for constant N and U are then: a) the non-dimensional mountain height ɛ = N h/u, which measures the amplitude of the disturbance; b) the vertical aspect ratio δ = U/Na, which measures the importance of non-hydrostatic effects; and c) the horizontal aspect ratio β = b/a. For most atmospheric flows of interest the vertical aspect ratio δ is small (less than 0.1) so that the flow is essentially hydrostatic; the set of control parameters then reduces to ɛ and β. Laboratory, numerical, and theoretical studies suggest roughly four classes of phenomena of importance in hydrostatic flow over topography with uniform upstream N and U: (i) small-amplitude waves; (ii) wavebreaking; (iii) upstream stagnation and flow-splitting; and (iv) lee vortices. The schematic regime diagram in 1

2 ENCYCLOPEDIA OF THE ATMOSPHERIC SCIENCES: LEE VORTICES 2 (a) (b) Figure 1. (a) Aerial photo of cloud layer in flow past the island of Hawaii. The cores of lee vortices feature warmer air than the surrounding flow and are typically manifest in aerial and satellite photos as holes in the cloud layer. The two lobes of clear air extending downstream of the island in (a) are the signature of a pair of counter-rotating vortices. Arrows suggest the flow field as inferred from the cloud pattern. (Photo courtesy of Vanda Grubišić) (b) Satellite image of a vortex street downstream of Alejandro Selkirk Island off the coast of Chile as seen by the Landsat7 satellite. The island is in the bottom left part of the figure. g replacements Fig. 2 summarizes the occurrence of these phenomena as a function of ɛ and β. Detailed descriptions of the flow classes are as follows: Small-amplitude waves. When ɛ 1 the mountaininduced disturbance for all β takes the form of a smallamplitude mountain wave (see Lee Waves and Mountain Waves for examples of small-amplitude waves). The ɛ Flow Splitting, Lee Vortices LV Wave Breaking, Flow Splitting, Lee Vortices Small-Amplitude Waves Wave Breaking Figure 2. Schematic regime diagram for steady stratified flow past an isolated ridge as a function of ɛ and β. Note that the actual shapes and positions of the curves will depend on obstacle shape. β flow in this regime is well described by theoretical approaches valid in the limit of small ɛ. As ɛ increases the streamlines in the wave pattern above the lee slope steepen somewhat in the vertical due to nonlinear (or finite-amplitude) effects. Wave breaking. For β > 1 (i.e., for elongated ridges) streamlines above the lee slope overturn when ɛ exceeds a critical value ɛ b (β) usually in the range depending on obstacle shape. Overturning of the streamlines places more dense fluid over less dense fluid causing the wave to break and become locally turbulent. An example of overturning streamlines and wave breaking in a laboratory flow with ɛ = 2.5 and β = 3 is shown in Fig. 3a. Wave breaking produces a well-mixed region above the lee slope which tends to decouple the low-level lee-slope flow from the flow farther aloft. The resulting low-level flow is strongly accelerated and is similar to supercritical shallow-water (or single-layer hydraulic) flow (see Downslope Winds for a discussion of shallow-water flow over a ridge; see also Hydraulic Flow for a basic discussion of shallow-water dynamics). Downstream of the obstacle the depressed lee-slope streamlines abruptly return to their upstream heights in a structure resembling a shallow-water hydraulic jump (Fig. 3a). Flow splitting. For all β there exists a critical mountain height ɛ s (β) at which the flow stagnates on the up-

3 ENCYCLOPEDIA OF THE ATMOSPHERIC SCIENCES: LEE VORTICES 3 (b) (a) (c) stream face of the ridge. This flow stagnation is due to the positive upstream pressure anomaly associated with the disturbance and occurs at a finite height z s on the upstream face. Below z s the flow splits and passes primarily around the obstacle rather than over while above z s the flow ascends the crest as for ɛ < ɛ s. Figure 3b illustrates upstream stagnation and flow splitting in a laboratory flow with ɛ = 2.5 and β = 2; Fig. 3c shows surface streamlines from a numerical computation with the same ɛ and β. Upstream splitting reduces vertical displacements in the flow and thus tends to supress wave breaking. For β < 1 upstream stagnation occurs at smaller ɛ than wave breaking and the latter is suppressed for all ɛ. By contrast, for elongated ridges ɛ b < ɛ s and the phenomena are observed to coexist over some range in ɛ (cf. Fig. 2). Lee vortices. A counter-rotating pair of eddies appears in the lee of the obstacle when ɛ exceeds a value ɛ e (β) (Figs. 3b,c). In laboratory studies extensive lee vortices are typically observed only when upstream stagnation is also present, as in Fig. 3b. However, numerical simulations indicate that for β < 1 vortices form at mountain heights smaller than ɛ s, suggesting that vortex formation is essentially independent of flow splitting. In any case, it seems to be true that for most obstacle shapes ɛ e (β) ɛ s (β) for β > 1 and the two curves are taken to be coincident over this range in β in Fig. 2. Counter-rotating vortex pairs with weak flow reversal are likely to be stable and quasi-steady. Vortex pairs with extensive regions of vigorous reversed flow are expected to be unstable and transition to a vortexshedding state. Boundary Layers and Free-slip Models Figure 3. (a) Dye lines in the centerline plane for flow over a 3D ridge with ɛ = 2.5 and β = 3. The streamline above the lee slope has steepened to the point of overturning, causing the wave to break and the flow to become turbulent. (From Castro and Snyder, 1993) (b) Dye lines in the centerline plane and on the obstacle surface for flow over a ridge with ɛ = 2.5 and β = 2. The basic flow is from left to right. Flow-splitting is apparent on the windward slope below the second height contour. Also apparent is a wake in the lee with a hint of possible reversed flow along the centerline. (From Baines, 1995) (c) Surface streamlines from a numerical computation using an obstacle identical to that shown in (b). Note that the dye lines in (b) are suggestive of the streamlines in (c). Fluid flowing past a stationary solid topographic obstacle is constrained to be motionless at the obstacle surface due to frictional coupling between the fluid and the solid boundary. The character of the flow immediately above the boundary depends on the influence of viscous effects, as measured by the Reynold s number Re = Ua/ν, where ν is the kinematic viscosity. In laboratory models of atmospheric flows, Re is usually quite large (> 10 3 ). The flow immediately above the obstacle surface then takes the form of a thin boundary layer over which the flow speed increases from zero at the surface to roughly the free-stream speed U away from the surface (Fig. 4a). The boundary layer in the atmosphere is generally more complex than in laboratory experiments (see Boundary Layers) but the basic premise of a thin shear zone still applies. For our purposes we might define the thickness of the boundary

4 ENCYCLOPEDIA OF THE ATMOSPHERIC SCIENCES: LEE VORTICES 4 layer to be the depth over which shear effects dominate the effects of stratification. In the stably stratified atmosphere this depth is typically several tens of meters. Above the boundary layer the flow speed varies relatively little with height and viscous effects are negligible. In principle, the flow in this region could be determined by displacing the effective obstacle surface upwards to a flow surface (or material surface) at the top of the boundary layer and assuming that the fluid passes freely over this surface with no frictional coupling. If the boundary layer is thin and remains everywhere attached to the obstacle, a reasonable approximation is to neglect the boundary layer altogether and consider the obstacle surface to be frictionless. This approximation is referred to as a free-slip condition. When atmospheric vortex streets were first observed in satellite imagery in the early 1960s, it was commonly assumed that the vorticity of the wake derives from the shear in the boundary layer. The process by which the vorticity of the boundary layer enters the interior of the flow is well-established (if not completely understood) for high Re homogenous flows. If the boundary layer fluid traversing the obstacle surface at some point encounters an adverse pressure gradient, the boundary layer tends to separate from the obstacle and shed its vorticity into the fluid interior (Fig. 4b). Boundarylayer separation is responsible for the production of wakes in most flows of interest in engineering applications (e.g., flow past airplanes, cars, etc.). Laboratory studies refined this view by demonstrating that in stratified flow over topography, boundarylayer separation is strongly influenced by the stratified wave dynamics above the boundary layer. In particular, it was found that the mountain-wave pattern of depressed streamlines and accelerated flow above the lee slope tends to suppress lee-slope boundary-layer separation. For obstacles of gentle slope (h/a = ɛδ < 0.2), as is typical in the atmospheric context, boundary-layer separation does not occur at all for small-amplitude hydrostatic waves (unless lee waves are present due to re- Figure 4. (a) Schematic close-up of the boundary layer at the surface of a solid obstacle. (b) Separation of the boundary layer. flection from the tank top). However, separation may occur in connection with hydraulic jumps (as in Fig. 3a) or similar structures when wave steepening and breaking are present. Flow-splitting is also believed to promote boundary-layer separation. In this case the fluid below the stagnation height z s passes nearly horizontally around the obstacle and separates from the boundary in essentially two-dimensional fashion (i.e., as in flow past a vertically oriented cylinder). This separation of the boundary layer in nearly horizontal planes has often been invoked to explain the observed eddies circulating about vertical axes in stratified wakes. By the late 1980s computing power had increased to the extent that numerical computations of 3D flow over topography were relatively common. Researchers soon discovered that numerical models with free-slip boundary conditions reproduce the observed features of topographic wakes in stratified laboratory experiments with surprising accuracy, at least for small δ and moderate ɛ (say, ɛ < 10). As discussed above, free-slip models completely neglect the boundary layer so that the numerically simulated lee vortices are clearly not due to boundary-layer separation. Similarly, researchers modelling observed atmospheric wakes have found that reducing or neglecting surface friction often results in intensification of the vortices. On the basis of these results it is reasonable to suppose that for hydrostatic flow at moderate ɛ the formation of lee wakes and vortices is not intrinsically dependent on boundary-layer separation. The vorticity of the wake is instead generated in the interior of the fluid either through buoyancy gradients (i.e., baroclinicity) or turbulent stresses. Most of the theoretical work on topographic wakes in the past decade has focused on wakes in free-slip flows. The following sections review some of the basic results of this effort. However, it should be kept in mind that for δ > 1 or ɛ > 10 boundary-layer separation may in fact dominate the behavior of the wake. Shallow-water theory As discussed in the overview above, analogs to supercritical shallow-water flow and hydraulic jumps may occur in stratified flows with uniform N and U when the obstacle is sufficiently high to force overturning waves. Similarities to hydraulic flow also occur without breaking waves when the upstream conditions include a strong low-level inversion (or layer of large N), as is often the case in trade-wind flow. To a first approximation the inversion acts as a free surface when large-amplitude waves are present. Indeed, observa-

5 ENCYCLOPEDIA OF THE ATMOSPHERIC SCIENCES: LEE VORTICES 5 tions of lee wakes in trade-wind flow past the island of Hawaii are suggestive of hydraulic-type behavior including weak hydraulic jumps. Considerations such as these have led to attempts to model lee vortex formation in terms of shallow-water theory. Note that the PSfrag shallow- replacements water results described here have close analogues for stratified flow as discussed in the following section. We consider flow of a thin layer of homogenous fluid past a submerged obstacle of gentle slope. Such a flow is well described by the shallow water equations (see Hydraulic Flow for details) u + (u )u = g η t (1) η + (u(d + η)) = 0. t (2) Here d(x, y) is the resting depth of the fluid, η(x, y) is the displacement of the free surface from the rest position, u = (u(x, y), v(x, y)) is the horizontal flow field, and = ( / x, / y). The geometry of the fluid layer is indicated in Fig. 5. From (1) and (2) are derived the useful relations ( ) t + u B = g η (3) t and ( ) ζ t + u d + η = 0 (4) where B = u u/2 + gη is the Bernoulli function and ζ is the vertical vorticity. At steady state both B and ζ/(d + η) are unchanged following the flow. The vorticity equation (4) is often usefully considered in the equivalent conservation form ζ t + J = 0 (5) with J = uζ denoting the flux of vorticity. From (1) it is straightforward to show ( J = uζ = ˆk B + u ) (6) t where ˆk is the vertical unit vector. The Bernoulli function thus serves as the effective streamfunction for the vorticity flux at steady state. Let the upstream flow be given by constant speed U and depth D and let the maximum obstacle height be h. The non-dimensional control parameters for the flow are then the upstream Froude number F 0 = U/(gD) 1/2 and d Figure 5. Side view of shallow-water flow through a hydraulic jump. In the limit of ideal flow the width of the jump shrinks to zero resulting in a discontinuity. This is indicated by the dotted vertical line in the figure. the non-dimensional mountain height M = h/d. Numerical calculations suggest that vigorous wakes with recirculating vortices are possible only when the upstream flow is subcritical (i.e., F 0 < 1) and we restrict attention to this case. For simplicity we assume that the flow has adjusted to be steady everywhere except possibly far downstream of the obstacle. Both theory and numerical computations show that for obstacle heights M less than a critical value M c (depending on F 0 ) the flow remains subcritical and continuous everywhere. In this case (4) shows that the vorticity is everywhere zero (since it is zero upstream) and no wake forms. However, when M exceeds M c the flow becomes supercritical at the crest and over the lee slope of the obstacle (cf. Downslope Winds) and continuous solutions no longer exist. A hydraulic jump then forms in the lee as shown in Fig. 5. As discussed below, the formation of a hydraulic jump in general implies vertical vorticity production and consequent wake development. A shallow-water hydraulic jump is essentially a thin transition zone between two regions of fluid where (1) and (2) apply. In the jump zone the dynamics differs from that described by (1) and (2) and some sort of dissipation occurs. Note that details of the flow in the jump need not be specified. All that is required to determine the flow outside the jump is that mass and momentum be conserved and energy dissipated in the jump. Since (1) and (2) do not hold in the jump the material relations (3) and (4) are violated in the jump as well. In particular, the presence of dissipation produces a decrease in the Bernoulli function for particles crossing the jump. This results in a wake of fluid with decreased Bernoulli function extending downstream of the jump as shown in Fig. 6. For a steady jump of finite length, such a wake entails gradients of the Bernoulli function which, according to (6), are necessarily associated with nonzero advective fluxes of ζ. The sense of the vorticity advection implies ζ > 0 on the right side η

6 ENCYCLOPEDIA OF THE ATMOSPHERIC SCIENCES: LEE VORTICES 6 of the flow (facing downstream) and ζ < 0 on the left. Note that the conservation equation (5) can in fact be generalized to hold for arbitrary (i.e., dissipative and nonhydrostatic) flow conditions. It is then consistent to suppose that in addition to conserving mass and momentum the flow in the hydraulic jump also conserves ζ (in the sense given by (5)). It can be shown that for steady flow the conservation of ζ implies a flux of vorticity in the jump directed tangent to the jump with the net flux given by the local Bernoulli difference B across the jump. This flux is denoted by the open arrow in Fig. 6. Since B varies along the jump there is in general nonzero divergence (or convergence) of the vorticity flux in the jump. This divergence/convergence is exactly balanced by the local advective flux of ζ into/away from the jump as given by (6). We thus find that in a hydraulic jump of finite extent, vertical vorticity is produced and then advected away downstream. If the vorticity production is sufficiently strong the resulting wake will feature reversed flow along the centerline and an associated pair of recirculating lee vortices. An example of such a flow is shown in Figs. 7a,b. The structure of the wake is modified somewhat when the height of the obstacle is sufficient to penetrate the fluid surface. Instead of a single hydraulic jump in the lee of the mountain a pair of jumps forms, B=B B=B hydraulic jump B<B B=B Figure 6. Schematic illustration of vorticity production in steady flow past a hydraulic jump. The heavy line over the lee slope indicates the position of the jump. Solid lines are streamlines and shading represents regions of reduced Bernoulli function with darker shading indicating greater reduction. (Here B is the value of the Bernoulli function upstream.) Heavy dark arrows show the advective vorticity flux uζ as inferred from the Bernoulli gradient using (6). The open arrow indicates the flux of ζ in the jump. (From Schär and Smith, 1993a) one on each side of the obstacle. Figures 7c,d give an example of this case. The detailed time evolution leading to the steady state shown in Figs. 7c,d is somewhat complex and beyond the scope of the present discussion. However, at steady state, at least, the basic principles of Bernoulli reduction and vorticity generation in the jumps still apply. Shallow-water theory allows some basic insight into the nonlinear dynamics of orographic flows when hydrauliclike conditions are present. Such flows are commonly observed to form lee vortices. The drawback of this approach is that the processes responsible for wake formation occur in hydraulic jump regions where the details of the flow are not specified. In particular, shallow-water theory does not explicitly address the dynamics of vorticity generation in the wake. A deeper understanding of wake formation then necessitates the consideration of more complete models. Stratified Theory The dynamics of stratified fluids is significantly more complex than that of a shallow layer of homogenous fluid. Nonetheless, an analysis similar in spirit to that of the previous section can be based on analogs of (5) and (6) for stratified flow. The stratified generalization of the conservation equation (5) takes the form Q t + J = 0 (7) where Q = ζ θ is the potential vorticity 1 (see Potential Vorticity). Here ζ is the three-dimensional vorticity vector and θ is the potential temperature. The potential vorticity flux J may be divided into an advective part uq and a dissipative part J D, the latter resulting from viscous and diabatic effects. It can then be shown that the stratified version of (6) is ( J = uq + J D = θ B + u ) ζ θ (8) t t which is clearly analogous to (6) at steady state. Here B = u u/2 + c p T + gz (with T the sensible temperature, c p the specific heat at constant pressure, and g the gravitational constant) is the Bernoulli function which at steady state is constant following the flow (cf. (3)) 1 The potential vorticity is usually defined as Π = ζ θ/ρ where ρ is the density. Then Q = ρπ is the potential vorticity per unit volume in the same sense that ρu is the x momentum per unit volume. As this is cumbersome terminology we refer to Q simply as the potential vorticity.

7 ENCYCLOPEDIA OF THE ATMOSPHERIC SCIENCES: LEE VORTICES 7 (a) (b) (c) (d) Figure 7. Lee vortices in steady shallow-water flow past an isolated obstacle. (a) Streamlines and (b) nondimensional vorticity aζ/(gd) 1/2 (contour interval 0.3 with negative contours dashed and zero contour suppressed) for M = 0.8 and F 0 = 0.5. (c) and (d) as in (a) and (b) except for the case M = 2 and F 0 = 0.5 in which the obstacle protrudes through the fluid surface. Heavy lines in all panels indicate the positions of hydraulic jumps. (Modified from Schär and Smith, 1993a) except where modified through dissipation. The relation (8) shows that for steady flow gradients of B on an isentropic surface (or surface of constant θ) imply fluxes of potential vorticity. Figure 8 depicts the flow fields on an isentropic surface passing over an obstacle with a thin dissipative region over the lee slope. The dissipation may be due to wave breaking, a hydraulic jump, or a similar disturbance. As in the shallow-water case, the localized dissipative region produces a wake of fluid with reduced Bernoulli function extending downstream. At steady state the Bernoulli gradients implicit in such a wake are necessarily associated with nonzero fluxes of potential vorticity as given by (8). Since J D = 0 away from the dissipative region, the potential vorticity fluxes downstream are advective and imply nonzero Q. We thus find that potential vorticity is generated in the dissipative region and advected downstream at the lateral edges of the wake. Downstream of the dissipative region the isentropic surface is essentially horizontal so that Figure 8. Schematic depiction of potential vorticity production in steady stratified flow past a topographic obstacle. Flow fields are shown on a low-level isentropic surface passing over the obstacle. Thin lines are streamlines. Dark shading over the lee slope indicates a localized region of dissipation due to wave breaking or a hydraulic jump. The shaded area extending downstream represents reduced values of the Bernoulli function with darker shading indicating greater reduction. Open arrows show the potential vorticity flux J associated with the Bernoulli gradient on the isentropic surface. (From Schär and Durran, 1997)

8 ENCYCLOPEDIA OF THE ATMOSPHERIC SCIENCES: LEE VORTICES 8 the presence of nonzero Q also implies nonzero vertical vorticity. If the vertical vorticity is sufficiently strong the associated wake will possess reversed flow and a pair of lee vortices. The above analysis nicely demonstrates the close relationship between dissipative processes and wake structure at steady state. However, it should be noted that this steady-state relationship does not imply a sense of causality. At present it is not clear whether viscous and diabatic effects play a primary role in vortex development or are instead the by-product of an essentially inviscid and adiabatic tendency to form vortices (or whether such a distinction can even be made). Note also that while the steady-state Bernoulli analysis predicts the production of potential vorticity in the wake due to dissipative processes, it does not address in detail the dynamics of vorticity and potential vorticity generation. As such, the approach has the same limitation as the shallow-water analysis, namely, that it does not explain how the individual air parcels in the wake acquire vorticity or potential vorticity. The most fundamental attempt to explicitly account for vorticity generation in orographic wakes is based essentially on the vorticity dynamics of inviscid threedimensional mountain waves. In incompressible (as is approximately the case for most atmospheric flows of interest) and inviscid stratified flow, the vorticity is governed by (see Vorticity) ζ ρ p + (u )ζ = (ζ )u + t ρ 2 (9) where ρ is the density and p the pressure. The second term on the right gives the generation of vorticity by baroclinicity while the first term describes stretching and tilting of the vorticity by the flow field. In most orographic disturbances the baroclinic term is well represented by ρ p ρ 2 k b (10) where k is again the vertical unit vector and b is the buoyancy (see Buoyancy and Buoyancy Waves). In this approximation vorticity is generated through horizontal gradients in the buoyancy and the associated tendency toward vertical motion. Note that the vorticity generation is entirely horizontal. To produce vertical vorticity the baroclinically generated horizontal vorticity must be tilted by the flow field so as to have a vertical component. For inviscid and adiabatic flow the potential vorticity Q remains everywhere zero (if it is zero upstream; see Potential Vorticity) and the vorticity is thus tangent to isentropic surfaces. Vertical vorticity can then be diagnosed by tracing vortex lines 2 along a constant-θ surface while accounting for the vertical deflections of the surface. If the vortex lines cross contours of height on the surface then vertical vorticity is necessarily present. Note that for adiabatic flow the potential temperature θ is unchanged following the motion of fluid particles; in this case isentropic surfaces are also stream-surfaces in the flow. Figure 9 summarizes the basic dynamics leading to vertical vorticity production in inviscid and adiabatic mountain waves. The heavy solid line in Fig. 9a represents a streamline in the wave field above the mountain at a height Nz/U = π/2. Above the windward slope fluid particles in the wave ascend and become negatively buoyant, leading to a negative x-gradient of buoyancy upstream of the obstacle. According to (10) this generates vorticity of the sense pointing into the plane of the figure (i.e., in the positive y direction). As the flow descends above the lee slope the opposite occurs and a negative y-component of vorticity is generated. In three dimensions these vortex lines running into and out of the figure loop around the lateral ends of the obstacle and connect. Figure 9b shows a vortex line on the isentropic surface corresponding to the streamline in Fig. 9a. Downstream of the obstacle both the surface and the vortex line are displaced downward by the descending flow while upstream the displacement is upward. As a result, there are regions at the ends of the obstacle where the vortex line ascends or descends the sloping isentropic surface and the vorticity has a nonzero vertical component. Note that the sign of the vorticity is of the appropriate sense to describe the flow in lee vortices. We thus find that baroclinically generated horizontal vorticity inherent in the mountain wave is tilted by the ascending and descending flow in the wave to produce vertical vorticity at the lateral ends of the obstacle. This vorticity may then contribute to the formation of a counter-rotating vortex pair. Further analysis suggests that dissipation may act to extend the vorticity pattern downstream in a pair of potential vorticity streamers as depicted in Fig. 8. Recent work suggests a possible link between the mountain-wave analysis described above and the shallowwater approach of the previous section. Figure 10 shows 2 A vortex line is a curve in space along which the vorticity is everywhere tangent. Vortex lines are thus the analogs for the vorticity field of streamlines for the velocity field.

9 ENCYCLOPEDIA OF THE ATMOSPHERIC SCIENCES: LEE VORTICES 9 rag replacements replacements Nz/2πU x/a Nz/2πU x/a Figure 9. Vertical vorticity production in a 3D mountain wave. (a) Streamline with upstream height Nz/U = π/2 in the centerline plane for flow over an obstacle with ɛ = 0.5 and β = 2.4. Circular arrows indicate horizontal vorticity generated through buoyancy gradients. (b) Schematic vortex line on the isentropic surface corresponding to the streamline in (a). Circular arrows indicate vertical vorticity in regions where the vortex line ascends or descends the sloping isentropic surface. (After Smolarkiewicz and Rotunno, 1989) (a) (b) the pronounced vorticity anomalies at the edges of the wake. In the flow of Fig. 10 the viscosity is sufficiently high to suppress the onset of small-scale turbulence. Real flows are likely to be turbulent in both the steepened mountain wave and the hydraulic jump, raising the possibility of vorticity generation by turbulent stresses. It remains to be seen how well the mechanisms of vorticity generation in the viscous laminar model extend to more realistic flows with turbulence. The details of vorticity generation in some large-ɛ (ɛ > 3, say) cases with prominent flow-splitting are also uncertain at present. Numerical simulations of flow at large ɛ often show jump-like features on the lateral slopes of the obstacle as in the shallow-water calculations of Figs. 7c,d. The tilting and stretching mechanism described above likely plays an important role in creating the vertical vorticity of the wake in such cases. However, in other large-ɛ flows the jumps on the lateral slopes are weak or apparently absent (at least at steady state). One possibility is that the Bernoulli deficit of the wake in such cases is determined not by jump-like features but rather by weak dissipation distributed throughout the length of the wake. A Bernoulli gradient and associated potential vorticity flux are then produced where the recirculating (low-bernoulli) wake flow joins the in- y/a results of a numerical simulation of stratified flow past -4 an isolated ridge with ɛ = 1.8 and β = 5. A hydraulic PSfrag replacements jump similar to that in Fig. 3a forms downstream of the -6 obstacle; the shading in Fig. 10 shows the position of the jump. Behind the jump the flow is weakly reversed x/a indicating the early stages of vortex formation. From a macroscopic perspective the flow in Fig. 10 is similar to the shallow-water calculation of Figs. 7a,b with ified flow past a long ridge with ɛ = 1.8, β = 5 and Figure 10. Incipient vortex formation in viscous strat- streams of vertical vorticity extending downstream of Re = Fields are shown on a low-level terrainfollowing surface. Vectors show the horizontal velocity the lateral ends of the jump. However, in the case of Fig. 10 it can be shown that the vertical vorticity of the and contours give the non-dimensional vertical vorticity wake originates in the mountain wave upstream of the ζ/ɛ 2 δn (contour interval 0.21 with negative contours jump through a mechanism similar to that described in dashed and zero contour suppressed). Shaded area indicates the position of a hydraulic jump similar to that Fig. 9. Upon reaching the jump the vorticity is amplified several-fold through vertical streching to produce shown in Fig. 3a. (From Epifanio and Durran, 2000) -2

10 ENCYCLOPEDIA OF THE ATMOSPHERIC SCIENCES: LEE VORTICES 10 cident flow on the flanks of the obstacle. Further work is needed to understand the nature of vorticity generation in this latter type of flow. Vortex Shedding Vortex shedding and vortex streets are thought to result in most cases from the instability of an attached (i.e., non-shedding) counter-rotating wake flow. Any noise or impulsive disturbance in the flow tends to trigger the instability and leads to growing modes which disrupt the attached wake pattern (see, e.g., Barotropic Flow and Barotropic Instability). The instability is a property of the shear flow in the wake and has little dependence on either the mechanisms of wake formation or the details of the flow near the obstacle. As such, the transition to vortex shedding in geophysical flows is expected to be somewhat similar to that found in homogenous flows past submerged obstacles, for which there is an extensive literature. Some of the more illustrative results on vortex shedding in 2d homogenous flows have been extended in detail to shallow-water flow over an isolated obstacle. Subsequent studies of wake stability in stratified flow have revealed instability structures similar to those found in the shallow-water case. However, it should be noted that the literature on wake stability and vortex shedding in stratified flows is rather limited. Figure 11 describes the instability and transition to vortex shedding of an elongated wake in shallow-water flow. The wake in Figs. 11a,b supports essentially two distinct types of instability. The shear lines at the lateral edges of the wake each in isolation support unstable modes associated with the local vorticity extremum. However, the energy of these modes propagates rapidly downstream away from the obstacle before the disturbance has an opportunity to grow significantly. As a result, these modes have little impact on the overall (or global) stability of the wake. A second type of unstable mode is anti-symmetric about the wake centerline and depends on a coupling of the two shear lines with the reversed flow in between. Modes of this type grow in place and disrupt the counter-rotating wake pattern. The disturbance is typically first manifest as a wavelike oscillation near the downstream stagnation point (Fig. 11c). At later times (Fig. 11d) isolated patches of vorticity are shed from the downstream end of the wake. Eventually the disturbance becomes evident in the immediate lee of the mountain as well and vortices of alternating sign are shed from either side of the obstacle with nearly perfect periodicity. In this final (or saturation) state the flow is similar to that shown in Fig. 1b. Detailed stability analyses of wake flows show that the growing disturbance tends to emanate from the part of the wake featuring reversed flow. Wakes without reversed flow are expected to be stable. Similarly, if the wake features only a limited region of weak reversed flow then the dispersion of disturbance energy away from this region may be sufficient to suppress the growth of the mode. The transition (or bifurcation) from a stable wake to an unstable wake with increasing obstacle height has yet to be explored for shallow-water flow. It seems likely that in the free-slip inviscid case virtually any wake with flow reversal will eventually become unstable as the wake elongates in the downstream direction. However, the addition of viscosity and/or bottom friction tends to stabilize wakes with weak reversed flow. This stabilization is effected primarily through a decrease in the downstream length of the wake (thus reducing the extent of the flow reversal) and to a lesser degree through direct reduction in the growth rates of the disturbance modes. Numerical computations with realistic parameters suggest that bottom friction may indeed exert an important stabilizing influence on atmospheric wakes, such as that behind the island of Hawaii (Fig. 1a). See Also: Lee Waves and Mountain Waves, Downslope Winds, Vorticity, Potential Vorticity, Buoyancy and Buoyancy Waves, Hydraulic Flow, Boundary Layers, Mountain Meteorology, Orographic Effects Keywords: lee vortices, orographic wakes, wake stability, vortex shedding, baroclinicity, mountain waves, gravity waves, wave breaking, flow splitting, shallowwater flow, hydraulic jumps, boundary-layer separation References Baines, P. G., 1995: Topographic Effects in Stratified Flows. Cambridge University Press, 482 pp. Chopra, K. P., 1973: Atmospheric and oceanic flow problems introduced by islands. Advances in Geophysics, 16, Epifanio, C. C. and D. R. Durran, 2002: Lee vortex formation in free-slip stratified flow over ridges. Part I: Comparison of weakly nonlinear inviscid theory and fully nonlinear viscous simulations. J. Atmos. Sci., 59, Huerre, P. and A. M. Monkewitz, 1990: Local and

11 ENCYCLOPEDIA OF THE ATMOSPHERIC SCIENCES: LEE VORTICES 11 global instabilities in spatially developing flows. Annu. Rev. Fluid Mech., 22, Time = 0.0 Time = 0.0 (a) (b) Schär, C. and R. B. Smith, 1993: Shallow-water flow past isolated topography. Part I: Vorticity production and wake formation. J. Atmos. Sci., 50, Smith, R. B., 1989: Hydrostatic airflow over mountains. Advances in Geophysics, 31, Smith, R. B. and V. Grubišić, 1993: Aerial observations of Hawaii s wake. J. Atmos. Sci., 50, Smolarkiewicz, P. K. and R. Rotunno, 1989: Low Froude number flow past three-dimensional obstacles. Part I: Baroclinically generated lee vortices. J. Atmos. Sci., 46, Time = 36.0 (c) Time = 72.0 (d) Figure 11. Wake instability and vortex shedding in shallow-water flow past an isolated obstacle. (a) Streamlines and (b) non-dimensional vorticity aζ/(gd) 1/2 (contour interval 0.25 with negative contours dashed and zero contour suppressed) of initial state. The initial state was obtained through a numerical computation in which symmetry about the wake centerline was explicitly enforced, thus inhibiting the growth of anti-symmetric disturbance modes. The symmetry condition was then removed allowing the wake to become unstable. (c) and (d) show the subsequent evolution of the wake in terms of the vorticity distribution at times (gd) 1/2 t/a = 36 and (gd) 1/2 t/a = 72. (Adapted from Schär and Smith, 1993b)