The development of orographic wakes and vortices is revisited from the dynamical perspective of a three-dimensional (3D) vorticity-vector potential formulation. Particular emphasis is given to the role of upstream blocking in the formation of the wake.
Scaling arguments are first presented to explore the limiting form of the 3D vorticity inversion for the case of flow at small dynamical aspect ratio δ. It is shown that in the limit of small δ the inversion is determined completely by the two horizontal vorticity components—that is, the part of the velocity induced by the vertical component of vorticity vanishes in the small-δ limit. This result leads to an approximate formulation of small-δ fluid mechanics in which the three governing prognostic variables are the two horizontal vorticity components and the potential temperature. The remainder of the study then revisits the problem of orographic wake formation from the perspective of this small-δ vorticity dynamics framework.
Previous studies have suggested that one of the potential routes to stratified wake formation is through the blocking of flow on the upstream side of the barrier. This apparent link between blocking and wake formation is shown to be relatively straightforward in the small-δ vorticity context. In particular, it is shown that blocking of the flow inevitably leads to a horizontal vorticity distribution that favors deceleration of the leeside flow at the ground. This process of leeside flow deceleration, as well as the subsequent time evolution of the wake, is illustrated through a series of numerical initial-value problems involving flows past 2D and 3D barriers. It is proposed that the initiation of the wake flow in these stratified problems resembles the flow produced by a retracting piston in shallow-water theory.
Researchers have long suspected that a causal link exists between the blocking of flow upstream of a mountain barrier and the formation of vortices in the lee. Early indirect evidence for such a link was provided by the original satellite images that first documented atmospheric vortex streets downwind of mountainous islands (see review in Chopra 1973). A survey of sounding data showed that the occurrence of these vortices was almost always tied to a sharp temperature inversion below the obstacle crest. The high stability of this inversion causes the low-level flow to be blocked and thus pass laterally around the barrier rather than ascend to the peak. A perhaps more direct indication of the importance of upstream blocking was provided later by towing tank experiments involving stratified flows past obstacles (see review in Baines 1995, section 6.6). These experiments showed that for flows with relatively weak stratification, the effect of increasing static stability is to suppress wake formation relative to the homogenous case. Once the stratification is increased to the point of upstream blocking, however, the trend is reversed and the eddies rapidly intensify.
Early attempts to explain this apparent dependence on upstream blocking were focused primarily on the role of surface friction and the effects of blocking on boundary layer separation. However, thinking on this topic was significantly revised with the numerical experiments of Smolarkiewicz and Rotunno (1989a, hereafter SR89). SR89 computed a series of flows past obstacles at increasing stratification with the boundary condition applied at the surface of the obstacle being free-slip. As expected with the absence of surface friction, the simulations with relatively weak stratification produced no evidence of leeside flow separation or associated wakes. With the onset of upstream blocking, however, the situation changed: the simulations produced realistic wakes and vortices that showed a surprising qualitative resemblance to those observed previously in the laboratory. This unexpected result provided a rather strong indication that the role of upstream blocking in wake formation is essentially independent of boundary layer processes. A series of quantitative comparisons between experiments and free-slip calculations has recently provided further evidence to support this conclusion (Vosper 2000).
Subsequent theoretical studies have recognized that blocking plays a critical role in determining the overall structure of stratified wakes. For instance, Smith (1989b) has pointed out that blocking causes the low-level isentropic surfaces in the flow to intersect the terrain, thus allowing the mature eddies in the wake to have a positive surface potential temperature anomaly at their core. Similarly, Schär and Durran (1997) have suggested that blocking alters the fluxes of potential vorticity (PV) in the wake by allowing a flux of PV to occur across the obstacle surface. Even so, these previous studies have been directed almost exclusively toward understanding the implications of upstream blocking for the eventual mature structure of the wake. The role of blocking in the actual process of wake formation has by comparison received relatively little attention. With this in mind, a fundamental question that remains poorly understood is: why does upstream blocking cause the wake eddies to form in the first place?
The present study attempts to address this basic question by revisiting the problem of wake formation from the perspective of a three-dimensional (3D) vorticity inversion. Our approach is fundamentally similar to the vorticity-vector potential framework used in some numerical modeling studies (e.g., Weinan and Liu 1997; Fletcher 2000, section 17.4), except that here we apply the inversion diagnostically to numerical results computed in the pressure-velocity framework. Section 2 describes the physical context for the study and provides an overview of the numerical model formulation and experimental setup. Section 3 then reviews the vorticity-vector potential framework and explores the limiting form of the 3D vorticity inversion for the case of flow past small-aspect-ratio terrain. Interestingly, we show that for small-aspect-ratio flow the inversion is determined completely by the two horizontal vorticity components.
This emphasis on horizontal vorticity motivates an analysis of the two-dimensional (2D) problem in section 4. Wakes produced by both upstream blocking and breaking waves are considered and some basic structural differences between the two types of wakes are discussed. The upstream blocking case is then examined in further detail through consideration of a simplified initial-value problem involving the flow of a low-level stratified layer. The evolution of the leeside wake in this problem is shown to be qualitatively very similar to the flow produced by a retracting piston in shallow-water theory. This analysis is then extended to the corresponding 3D case through consideration of the 3D vortex-line evolution and the associated induced velocity fields in section 5. We show that the basic insights gained from analysis of the 2D case extend (with some elaboration) to the 3D problem as well. A summary and concluding remarks are then given in section 6.
2. Experimental setup and scale analysis
a. Physical framework
The physical framework for our study is similar to that considered by Epifanio and Durran (2002a) but is repeated here for ease of reference below. We consider nonrotating, compressible Boussinesq flow as described by
with lower boundary condition
Here h(x, y) is the terrain height; P is the Boussinesq disturbance pressure; b is the buoyancy; N is the basic-state buoyancy frequency; and cs is the constant Boussinesq sound speed. For fluid velocities much smaller than cs, topographic disturbances described by (1)–(4) can be considered essentially incompressible (as discussed below in section 2b). A brief justification of (1)–(3) is given in appendix C for readers unfamiliar with the compressible Boussinesq system.
The Boussinesq potential temperature variable is defined to be θ = θ0(z) + b, where
is the basic-state profile. The stress tensor 𝗧 and diffusive heat flux B are then given by
where we have assumed the kinematic viscosity and thermal diffusivity to be equal and uniform. (A variable eddy viscosity is introduced in section 2c for use in the numerical experiments.) The surface of the terrain is taken to be free-slip and thermally insulating apart from a small heat flux needed to maintain a reference potential temperature profile in the absence of a disturbance.1 The domain is assumed to be unbounded both in the horizontal and aloft.
The disturbance is initiated by impulsively accelerating the fluid from rest to a constant free-stream velocity U = (U, 0, 0) at the initial time t = 0. This impulsive start-up closely resembles the rapid acceleration of an obstacle from rest at the beginning of a towing tank experiment. Under the assumption of incompressibility, the impulsively started fluid would adjust immediately to potential flow during its instantaneous acceleration [see discussion in Crook et al. (1990) or Rotunno and Smolarkiewicz (1991)], and we therefore adopt the potential flow solution as the initial state for our numerical calculations.
For the 3D computations we consider flow past a simple topographic obstacle described by
Here h0 is the maximum terrain height and β ≥ 1 determines the width of the barrier in the direction across the stream. For β = 1 the terrain is similar to an axisymmetric bell-shaped obstacle with length scale L but decays to zero height over the finite distance 4L. For the 2D computations we use the cross-sectional profile of (7) and (8) in the centerline plane y = 0.
b. Scale analysis
Here we develop a scale analysis aimed specifically at the range of obstacle heights of interest in the present study—namely nondimensional heights ε = Nh0/U in the range 2 ≤ ε ≤ 6. The underlying assumptions behind our scaling results will be as follows:
(i) Vertical displacements in the fluid are limited to roughly U/N. This assumption is motivated by previous laboratory (Hunt and Snyder 1980) and numerical (Ólafsson and Bougeault 1996) studies showing that the fluid originating below a height of roughly zs = h0 − U/N far upstream of the obstacle tends to pass primarily around the sides of the barrier rather than ascending to the obstacle crest.
(ii) The length scale is set by the horizontal distance moved by a fluid element along the lower boundary during a vertical displacement of U/N. If the terrain slope is measured by h0/L, then the associated length scale must be (U/N) × (L/h0) = L/ε.
(iii) The time scale is set by the time needed for a fluid element moving at the free-stream speed U to move the horizontal length scale L/ε; that is, L/εU.
These scaling assumptions imply that for fixed basic-state parameters U and N and fixed obstacle half-width L, the characteristic length and time scales of the disturbance contract with increasing obstacle height. The physical basis for this prediction is that as h0 is increased while keeping the remaining parameters fixed, a greater fraction of the upstream flow is diverted laterally around the barrier (i.e., zs/h0 increases). The portion of the obstacle remaining above this layer of deflected flow then becomes progressively narrower, thus decreasing the associated length and time scales for flow past the mountain.
Figure 1 shows a test of the above scaling assumptions for a series of flows with uniform N and U past axisymmetric obstacles (i.e., β = 1) of varying height. In all calculations the parameters U, N, and L are held fixed with U/NL = 0.033. The computations are initialized by impulsive start-up as described previously in section 2a. Further details of the calculations are as given in the following subsection.
The characteristic time scale of the flows shown in Fig. 1 (and of the process of orographic wake formation in particular) is measured by the time of first flow reversal along the lee slope, denoted by ts. As predicted, the leeside flow reversal occurs more quickly as the obstacle height is increased; indeed, the log–log plot of ts versus ε reveals the −1 slope predicted by the scale analysis (solid line). The circles in Fig. 1 show the position xs of the closest upwind stagnation point from the crest (i.e., the node of attachment) at a time when the wake has reached a mature and locally quasi-steady state (specifically εUt/L = 12). The distance from this node to the crest is taken as a rough measure of the characteristic length scales involved in the flow. As with the time scale, the length scale is seen to decrease steadily with increasing ε. The log–log plot for xs reveals a slope slightly greater than the assumed −1 value.
where hats indicate dimensionless quantities. We then define
with lower boundary condition
where we have defined x = Lx̃ and y = Lỹ to indicate that the terrain length scale remains L even though the disturbance length scale contracts to L/ε. The control parameters for this scaling are thus the nondimensional obstacle height ε = Nh0/U, the vertical aspect ratio δ = (U/N) (L/ε)−1 = h0/L, the Reynolds number Re = (UL/K0ε) δ2, the Mach number Ma = U/cs, and any further parameters needed to specify the terrain shape and/or complexities of the upstream sounding. For small Ma the disturbance is essentially incompressible, while for small terrain slopes the flow is predominantly hydrostatic.
c. Numerical model
Numerical solutions to (1)–(3) are obtained using an updated version of the nonhydrostatic, compressible Boussinesq model described by Epifanio and Durran (2001) and Epifanio and Durran (2002a). The principal model update from those studies is the replacement of approximate free-slip and thermal insulation conditions valid for small terrain slopes only (e.g., Epifanio and Durran 2002a) by exact free-slip and thermal insulation conditions valid for arbitrary topography. The implementation of these new conditions is easily extended to include surface drags and surface heat fluxes and will be described in greater detail in a forthcoming publication (C. C. Epifanio 2005, unpublished manuscript).
The model employs a two-time-step technique in which terms associated with acoustic propagation are integrated with a small time step Δts, while the remaining terms are advanced on a larger time step Δt (e.g., Durran 1999, section 7.3.2). Terrain is incorporated through use of the terrain-following vertical coordinate
where zT is the model domain depth (Gal-Chen and Somerville 1975). Turbulent fluxes are represented by augmenting the background viscosity/diffusivity K0 in (5) and (6) by the addition of a turbulent eddy viscosity Ke; that is, K = K0 + Ke where Ke is defined by the first-order closure of Lilly (1962). We also include a weak fourth-order horizontal smoother to help stabilize sharp gradients that form along the lee slope of the terrain. Both lateral and upper boundaries are represented by sponge layers.
All numerical results are for Ma = 0.03 and Re = 150, while ε, δ, and β are varied. We consider flows both with uniform basic-state wind U and stability N and with uniform wind and two layers of different stability. In all cases, the horizontal extent of the model domain is at least −(8 + β) L ≤ x ≤ (10 + β) L, |y| ≤ (9 + β) L, with Δx = Δy = 0.09 L except where otherwise noted. The vertical domain depth is 3.2 L for the scaling verification tests in sections 2b and 3b, and 2L for all remaining computations. (The dependence of the domain depth on L is necessary to minimize corruption of the 3D vorticity inversion calculations described in section 3.) The vertical grid spacing takes its minimum value at the lower boundary and increases geometrically with height with stretching factor 1.005. For the simulations with constant N and U we set Δz = 0.09 U/N at the boundary (unless otherwise noted); for simulations with two-layer stability we set Δz = zint/20, where zint is the height of the interface between the layers. The outer 4L of the domain in the horizontal and the upper half of the domain in the vertical are dedicated to the damping layers.
3. Vorticity dynamics at small aspect ratio
Our vorticity-based description of orographic wake formation requires that we unambiguously associate certain aspects of the observed velocity field with corresponding structures in the distribution of vorticity—that is, we need to invert the vorticity distribution to obtain the flow field. Here we briefly review the familiar concepts of 3D vorticity dynamics and kinematic inversions. Some simplifications of these ideas for flows with small vertical aspect ratio are then introduced and discussed in sections 3b and 3c.
a. Vorticity dynamics and inversions
For simplicity we consider the flow to be essentially incompressible (consistent with the smallness of Ma in our numerical simulations). The curl of (1) then gives the vorticity equation
where ζ = (ξ, η, ζ) is the vorticity, k is the vertical unit vector, and F = −∇ · 𝗧 is the viscous/turbulent force per unit mass vector. Vorticity is produced and/or diffused in the interior of the fluid through baroclinicity and through the action of viscous or turbulent stresses. Vorticity is also present along the boundary of the fluid due to the specified surface stress or no-slip conditions imposed there, but for free-slip flows at small δ this boundary production of vorticity is expected to be negligible (e.g., Rotunno et al. 1999).
Together, Eqs. (11) and (2) govern the time evolution of both ζ and b. To close the system we need only a kinematic inversion specifying u in terms of ζ. A simple inversion algorithm valid for arbitrary flow fields can be found in Hirasaki and Hellums (1970, hereafter HH). Here we apply their method to the specific case of a bounded numerical model domain V with a rigid lid at the top and uniform wind u = U at the lateral boundaries (consistent with the use of sponge layers in our numerical calculations). Extensions to include sheared basic-state winds at the boundary are relatively straightforward. For clarity we let the lateral boundary surfaces be represented by S while the rigid top and bottom are represented by R.
Following HH, we first decompose the total velocity u into an irrotational part ui = ∇ϕ associated with the boundary conditions and a vortical part ur = ∇ × ψ induced by the interior distribution of vorticity; i.e.,
where ϕ and ψ refer to the scalar and vector potentials defining the irrotational and vortical parts of the flow, respectively (see, e.g., Batchelor 1967, chapter 2). This separation into vortical and irrotational parts is not unique, and it proves convenient to let ϕ be defined by
where n is the unit normal to the boundary. The irrotational flow then clearly accounts for the externally imposed normal velocity at the lateral boundaries of the model domain. The corresponding vortical flow must therefore have no normal component at the boundaries, and a vector potential satisfying this constraint can be uniquely defined by
where condition (14b) enforces a choice of gauge requiring ∇ · ψ = 0 everywhere, while (14c) guarantees (∇ × ψ) · n = 0 on S and R. Proof that (14) yields the desired vortical flow ur is straightforward and can be found in HH. Given ζ on V, we can then solve (14) and combine with the solution to (13) to recover the velocity field (12).
With appropriate surface stress and heat flux conditions, Eqs. (2) and (11) with (12), (13), and (14) form a complete set that can in principle be integrated in time to determine the time evolution of the disturbance. However, here we employ this system only as a diagnostic tool, providing an alternative conceptual framework to the equations of motion in pressure-velocity form. Our approach is thus to first compute ζ from nonlinear numerical simulations using the numerical model described in section 2c (or else to first specify an idealized ζ distribution as in Figs. 5 and B2) and then solve the above elliptic systems numerically to obtain the vector and scalar potentials. Some details of the numerical implementation are provided in appendix A.
The use of (13) and (14) as a diagnostic tool has several attractive features at the level of conceptual interpretation. The first is that by solving (14) for a given vorticity distribution [or part of a vorticity distribution since (14) is linear] we can unambiguously associate the observed features of the velocity field with structures in the interior vorticity. As shown in the following subsection, this sense of attribution takes on a somewhat simplified form in the limit of small vertical aspect ratio. A second advantage is that the solution to (13) yields the familiar time-independent potential flow that serves as the initial condition for our numerical model calculations. We can therefore interpret the early evolution of the disturbance in terms of the vertical displacement of the isentropes by the initial potential flow and the consequent production of gradients in buoyancy and hence vorticity.
b. The small-δ limit
Orographic wakes in the atmosphere are typically the result of flow past mesoscale topographic features with relatively gently sloping sides (see, e.g., Smith and Grubišić 1993, for the case of Hawaii). As such, here we consider the limiting behavior of the vorticity inversion (14) in the limit of small vertical aspect ratio δ. We focus on the particular case in which the vorticity distribution is concentrated primarily near the lower boundary, as is appropriate for orographic wakes at the surface. However, in other respects the analysis is quite general. Note in particular that the main results of this section are not critically dependent on the particular scaling factors derived in section 2b. All that is required is that the dynamical aspect ratio of the flow—however this is determined—and the terrain slope be small. Indeed, to simplify the notation we will here refer to the depth scale as ℋ and the length scale as ℒ so that δ = ℋ/ℒ. The scaling of section 2b is then recovered by setting ℋ = U/N and ℒ = L/ε.
We first note that the three components of the vorticity scale as
thus demonstrating the familiar results that in the limit of small δ: a) the horizontal components of vorticity are determined by the vertical shear of the horizontal winds; and b) the vertical component of vorticity is an order of magnitude smaller than the horizontal components.
To proceed further we need appropriate scaling factors for the vector potential ψ. Suppose we let the three components of ψ be written ψ = (F, G, H) so that the inversion relation (14a) becomes
The remaining boundary conditions are then simple Neumann or Dirichlet conditions applied at the lateral boundaries and lid.
As mentioned above, we are primarily interested in the case where the maxima of vorticity are localized in the vicinity of the lower boundary. Scaling the three components of ψ is then somewhat subtle, since the boundary conditions (17) then play an important role in setting the appropriate scaling factors. One approach to help guide our intuition is to first consider solutions to a simple analytic inversion problem in which the specified vorticity distribution mimics the pattern of vorticity observed in orographic wakes. In particular, we consider a vorticity distribution that is maximized at the lower boundary and has a horizontal length scale of ℒ and a vertical decay scale of ℋ. Details of the analysis can be found in appendix B, but the essential inferences with regard to our scaling results are as follows:
(i) The two horizontal components of ψ both have a depth scale of ℋ near the lower boundary, while the vertical component H has a depth scale of ℒ (cf. Figs. B1a,b). This difference in vertical scales results from the lower boundary conditions (17), which in the limit of small terrain slopes reduce to Dirichlet conditions on F and G, and a Neumann condition on H. The two horizontal components thus feel an image source of opposite sign below the lower boundary, and this distribution of sources and images naturally gives F and G the depth scale of the near-boundary source distribution. By contrast, the vertical component H feels an image source of like sign, and the depth scale is then set by the isotropy property of the Laplacian operator in (16).
(ii) Even though the vertical scale of H is ℒ, the first derivative ∂H/∂z recovers the source depth scale of ℋ near the boundary (cf. Figs. B1b,d). This stems from the fact that if H satisfies the Poisson equation (16) with a Neumann condition at the boundary, then ∂H/∂z must satisfy the vertical derivative of this equation with a Dirichlet boundary condition.
Comparison of Figs. B1 and B3 shows that the depth scales inferred from our simple analytic argument do in fact provide an accurate description of the observed depth scales for the vector potential in our simulated topographic wakes. Using this result along with (15) in (16), we then find that all three components of the vector potential must scale similarly; that is,
where we have defined z = ℒz̃ to reflect the fact that H has depth scale of ℒ rather than ℋ. The scaled lower boundary conditions are then
where x̃ and ỹ are as defined previously and δT = h0/L. (Recall that for the scaling of section 2b we have δT = δ.) Finally, the curl of the vector potential takes the form
Equations (19)–(21) reveal several interesting scaling predictions for flows with small dynamical aspect ratio δ and small terrain slope δT. First, (21) predicts that in the limit of small aspect ratio the potential component H associated with the vertical component of vorticity ζ vanishes from the expressions for the velocity—that is, the flow field is completely determined by F and G. Similarly, the first two of (20) reduce to homogenous Dirichlet conditions on F and G as δT → 0; the horizontal components of ψ are then fully decoupled both from each other and from H and are thus determined independently by their respective components of vorticity. Taken together, these two scaling results imply that in the limit of small aspect ratio δ and small terrain slope δT, the flow field is determined completely by the two horizontal vorticity components. That is, the vertical vorticity ζ does not induce any velocity in the small-aspect-ratio and terrain-slope limit. Further, the x component of velocity is determined solely by η in this limit, while the y component depends solely on ξ.
Suppose we let urδ be the result of solving (19)–(21) with the terms multiplied by δ or δT in (20) and (21) set explicitly equal to zero.2 We will refer to such a calculation as a small-δ inversion of the vorticity. We can then test the scaling predictions given above by measuring the error in the small-δ inversion relative to the full vorticity inversion for flows past topographic obstacles with gradually decreasing δ = δT. (Here and for the remainder of this study we explicitly adopt the moderate-ε scaling results given in section 2b.) To be specific, we define a normalized measure of the small-δ inversion error to be
where the angle brackets indicate a root-mean-square (rms) average over all grid points in the region −2L ≤ x ≤ 3L, −2L ≤ y ≤ 2L, h ≤ z ≤ 1.5h0. Here the normalization factor 〈ur(0)〉 represents the rms value of ur extrapolated to δ = 0.
Figure 2 shows the normalized inversion error Einv as a function of δ for a series of uniform basic-state flows past axisymmetric obstacles with fixed ε and gradually increasing L. As anticipated, the small-δ inversion reproduces the full vorticity inversion in the limit of decreasing δ; moreover, the asymptotic convergence rate is clearly linear as predicted by (19)–(21). Note that for δ = 0.05 the normalized error in the small-δ inversion is less than 0.07. A qualitative comparison between urδ and ur for this case is shown in Fig. 3.
Finally, we note that while the scaling results derived above were for the specific case of a flow localized near the boundary, the end result (i.e., the small-δ limit) can in fact be shown to hold more generally as long as the appropriate length scales in the flow satisfy ℋ/ℒ ≪ 1.
c. Discussion: Vertical circulation at small δ
Note that the small-δ vorticity inversion clearly captures the circulation about a vertical axis that is inherent in the mature stage of the orographic wake shown in Fig. 3d. This ability to reproduce the vertical circulation may seem somewhat surprising at first glance given that the small-δ inversion does not explicitly make use of any information about the vertical vorticity. A heuristic argument that explains this result is shown schematically in Fig. 4. Figure 4a shows a portion of a vortex line running parallel to the y axis with the associated induced flow field encircling the line as determined by right-hand rule. Gradients in vertical motion may subsequently tilt this vortex line to produce vertical vorticity as shown in Fig. 4b. However, within the constraints of small-δ dynamics, the angle of inclination of the line is limited to be O(δ) so that ζ ≪ η [cf. (15)]. The induced flow field at every point along the line then remains dominated by the contribution from the horizontal component of vorticity; but because the line remains tilted over a large distance in the horizontal, the flow on a level surface features evident circulation about a vertical axis.
This basic heuristic argument is illustrated more concretely in Fig. 5. Figure 5a shows the surface velocity field induced by a specified vorticity distribution in which the bounding vortex tube forms an arc oriented tangent to the yz plane. The vortex line at the core of the distribution is indicated by the purple line in the figure; the vorticity magnitude takes its maximum value along this line and then decays outward with radial distance from the core. The geometry of the distribution is specified such that the vortex lines comprising the arc intersect the lower boundary at an angle of δ ≈ 0.05, thus effectively determining the aspect ratio of the distribution to be small. (The geometric aspect ratio of the figure has been stretched by a factor of 10 in the vertical for ease of visual interpretation.) The vertical vorticity at the intersection of the arc with the boundary is contoured in color. As seen in the figure, the velocity field induced by this vorticity distribution passes directly under the core of the vortex arc in the direction anticipated by the right-hand rule applied to the horizontal component of vorticity. Consistent with our previous scaling arguments, the flow is almost entirely in the x direction with very little curling of the flow around the vertical vorticity centers at the lateral edges. Figure 5b shows a small-δ inversion of this same vorticity distribution. The color contours in Fig. 5b show the vertical vorticity as diagnosed from the small-δ inverted flow field. As anticipated by Fig. 4, the small-δ inversion accurately reproduces the true flow field including its circulation about a vertical axis [to within an O(δ) correction].
To conclude this section, we briefly summarize the full set of approximate equations governing the dynamics of small-aspect-ratio flows in the vorticity-vector potential framework. The vertical vorticity becomes a diagnostic variable in this limit so that we no longer need a predictive equation for ζ. The full set of prognostic equations then reduces to
where we have neglected terms of order δ in the viscous and thermal diffusion terms. The kinematic vorticity inversion is specified by
with lower boundary conditions
and the associated diagnostic variables are then
4. Two-dimensional wake dynamics
The scaling analysis of the previous section shows that for flow at small aspect ratio δ the streamwise component of velocity u is determined completely by the cross-stream component of vorticity η. This result suggests that the process of leeside flow deceleration in orographic wakes should likely take place in the context of flow past a 2D ridge as well. Here we briefly consider wake formation associated with breaking mountain waves and upstream blocking in both 2D and 3D flows past a ridge of infinite cross-stream extent. Particular attention is given to the upstream blocking case: a more thorough analysis of wakes associated with breaking waves will be presented in future work. We note for reference that for 2D flow the approximations (23) and (24) reduce to the standard vorticity-streamfunction formulation for a stratified incompressible fluid.
Figure 6 shows a pair of high-resolution numerical calculations illustrating the basic types of orographic wake structures produced in flows past a 2D ridge. The simulations include 30 grid points in the along-ridge direction and periodic along-ridge boundary conditions so as to allow fully 3D turbulence and mixing to develop in any regions unstable to spanwise perturbations. [A similar approach has been adopted previously by Afanasyev and Peltier (1998).] Three-dimensionality is introduced by means of a small random potential temperature perturbation added shortly following the impulsive start-up (specifically at Ut/L = 2). The grid spacing for both simulations is given by Δx = Δy = L/50 with NΔz/U = 0.1 (or NΔz/2πU ≈ 1/62) at the lower boundary. (Here N is taken to be the upper-layer stability for the two-layer flow shown in Figs. 6b,c.) All results are averaged both in the along-ridge direction and over a time interval UT/L = 2 in an effort to smooth out the vigorous small-scale turbulent fluctuations present in the instantaneous fields.
Figure 6a shows results for flow of uniform basic-state wind and stability past a ridge of nondimensional height ε = 2. At this value of ε, the flow features vigorous breaking of the mountain wave aloft but only weak blocking of the surface flow upstream of the barrier (see also Pierrehumbert and Wyman 1985). Note that the averaged isentropes in the breaking wave are steepened to the point of overturning. Inspection of the instantaneous fields confirms that this region of overturned isentropes is associated with intense small-scale turbulent motions (not shown). Downstream of the turbulent region is an elevated layer of well-mixed fluid evidently produced by turbulent mixing in the breaking wave. The flow in this well-mixed layer is nearly (but not quite) stagnant and thus constitutes an elevated topographic wake structure extending downstream at approximately the height of the obstacle crest.
The color contours show that the elevated wake in Fig. 6a is associated with a dipole pattern of horizontal vorticity, with negative η below the well-mixed region and positive η above. The close correlation of the vorticity anomalies with the isentrope displacements further suggests that this vorticity pattern is primarily the result of baroclinic vorticity generation. In particular, the isentropes passing below the well-mixed region are displaced strongly downward relative to their upstream heights; fluid elements descending the lee slope must therefore experience a net positive ∂b/∂x along their trajectories and thus develop negative η. By contrast, the isentropes above the well-mixed region are displaced upward and the fluid elements above the wake thus develop positive η. This dipole pattern of horizontal vorticity then implies a net deceleration of the intervening fluid, as evidenced by the nearly stagnant flow in the wake.
The strong negative η anomaly below the well-mixed region in Fig. 6a suggests that wave breaking should in general be associated with accelerated flow at the surface and decelerated flow aloft (see the extensive literature on downslope windstorms as reviewed in Baines 1995, chapter 5). However, observations (Schär et al. 2003; Jiang et al. 2003), laboratory studies (Hunt and Snyder 1980; Baines 1995, chapter 6), and numerical simulations (Ólafsson and Bougeault 1996; Schär and Durran 1997, among many others) all show that orographic wake flows are often (if not usually) most strongly reversed near the ground. Comparison of Figs. 6a and 6b suggests that this development of leeside flow reversal at the ground most likely stems from the blocking of flow upstream of the mountain barrier rather than from gravity wave breaking above the lee slope. The simulation shown in Fig. 6b features a basic state identical to that considered in Fig. 6a, except that the static stability N has been increased by a factor of 4 throughout a layer of depth 0.6 h0. This increase in N prevents most of the fluid in the stable layer from traversing the obstacle crest, and the result is then a warm anomaly over the lee slope as air from aloft descends to replace the colder surface air blocked upstream. On the lee side, the temperature gradient between the cold stable layer and the warmer lee-slope air has collapsed to a front and the surface flow behind the front is reversed.
Steep leeside temperature gradients like the one shown in Fig. 6b are often interpreted as low-level wave breaking or hydraulic jumps because of the nearly vertical isentropes seen emerging from the boundary. However, consideration of this feature in a true 1:1 geometric aspect ratio (Fig. 6c) reveals that the steep gradient in θ more closely resembles the front of a density current that propagates into the warmer leeside air produced by the blocking. This result then suggests a close analogy between the dynamics of the surface wake in Fig. 6b and that of a density current that propagates continuously upstream against the flow. In terms of vorticity, the steep gradient in buoyancy across the front produces a strongly positive η anomaly at the top of the stable layer (Fig. 6c), which is then slowly advected away downstream. This positive η then implies deceleration of the flow at the surface, as manifested by the wake of reversed fluid above the lower boundary.
The formation of the surface wake in Fig. 6b is clearly the result of processes occurring primarily in the stable layer below the obstacle crest. We might therefore try to simplify the problem still further by replacing the upper weak stability layer by a fluid of uniform potential temperature. Such a model has been referred to by Baines (1995) as flow of a stratified fluid with a pliant upper boundary; here we refer to it simply as a stratified-layer flow. Furthermore, while the wave-breaking case in Fig. 6a exhibits vigorous 3D turbulence as soon as the isentropes overturn, the corresponding surface wake problem remains essentially 2D until long after the wake is well established. We can therefore address the dynamics of the surface wake problem in the context of a strictly 2D model.
Figure 7 shows a pair of steady-state numerical calculations illustrating the flow of a 2D stratified fluid layer with initial depth 0.6 h0 in cases where the upstream flow is either mostly blocked (ε = 6) or unblocked (ε = 1). The wind vectors shown in Fig. 7 are relative to a reference frame in which the background wind speed is zero and the obstacle is towed through the fluid at speed U. For ε = 1 (Fig. 7a), the stratified layer is simply deflected upward and then back downward as the obstacle passes. This upward deflection implies negative ∂b/∂x and positive Dη/Dt over the windward slope, with the opposite being true over the lee side. The net result is then a maximum in η over the obstacle crest that serves both to decelerate the low-level fluid near the peak of the obstacle as well as to deepen the layer by enhancing the (otherwise irrotational) lifting and lowering at the layer top. The overall structure of the flow is seen to be qualitatively very similar to supercritical shallow-water flow past an obstacle (Houghton and Kasahara 1968).
The corresponding flow for the case ε = 6 is shown in Fig. 7b. For this value of ε, the layer is mostly blocked upstream of the ridge and is therefore simply pushed along by the obstacle as it moves through the fluid. On the lee side, the temperature gradient has collapsed to a front and the fluid downstream of the front streams in the direction of obstacle motion at a speed slightly exceeding U. As with Fig. 6b, this leeside wake structure supports a qualitative interpretation in terms of the colder surface wake air propagating into the lee-slope warm anomaly and thus effectively following the obstacle as it moves through the fluid. On both sides of the obstacle the vorticity pattern is localized near the top of the stratified layer and is of the appropriate sense to induce a flow in the layer in the direction of obstacle motion.
The early time evolution leading to the steady-state shown in Fig. 7b is illustrated in Fig. 8. As the ridge is set in motion the isentropes initially ascend over the windward slope due to the instantaneous adjustment to potential flow associated with impulsively starting the obstacle (Fig. 8b). This lifting of the layer is then quickly followed by the ground-relative stagnation of the surface flow (Fig. 8c) and the associated formation of a borelike disturbance that propagates upstream (Fig. 8d). On the lee side the isentropes initially slide downward along the terrain slope as the obstacle is pulled away (Fig. 8b); this depression is then communicated leeward by means of a rarefaction wave (or wave of depression; Figs. 8c–e). At the leading edge of the layer the isentropes collapse into a front (Figs. 8c,d) and the denser fluid begins to propagate into the space evacuated by the obstacle. Note that in many respects this leeside evolution shows a close resemblance to the dam-break and retracting-piston problems from classical shallow-water theory (e.g., Stoker 1957, chapter 10).
5. The dynamics of wake formation in 3D
The results of the previous section show that surface wake formation in 2D results from the blocking of low-level flow upstream of the ridge and the consequent production of a thermal gradient and associated vorticity anomaly above the lee slope. Based on the small-δ scaling results of section 3, we might expect that a similar description applies to 3D flow at small aspect ratio as well. Here we consider the dynamics of wake formation in 3D from the general perspective of the small-δ vorticity inversion developed in sections 3b and 3c. For the sake of brevity, we restrict our attention to a single representative case: the flow of a stratified layer with initial depth 0.6h0 past a slightly elongated obstacle with ε = 4, δ = 0.1, and β = 3.
Figure 9 shows the early time evolution of the 3D flow as seen in vertical cross section through the centerline plane of the ridge at y = 0. Comparison with the corresponding figure for the 2D case (Fig. 8) suggests that the basic processes leading to leeside deceleration and flow reversal in 3D are essentially the same as those described previously for 2D. As in 2D, the initial adjustment to potential flow leads to lifting of the isentropes over the windward slope of the ridge and depression of the isentropes on the leeward side (Fig. 9a). This raising/lowering of the isentropes results in negative buoyancy gradients over both slopes of the ridge and thus leads to the baroclinic generation of positive η anomalies both upstream and downstream of the obstacle (Fig. 9b). Associated with this η distribution is an induced surface flow in the direction of obstacle propagation, as required by the small-δ vorticity inversion (24).
On the windward side of the obstacle, the induced flow leads to horizontal convergence in the stable layer and the formation of a borelike disturbance that propagates upstream (Fig. 9c). As the bore passes it leaves behind positive y vorticity in the layer and an associated surface current in the direction of obstacle motion (Fig. 9d). On the downstream side, the induced flow has the opposite effect in that it leads to horizontal divergence in the stable layer and the associated formation of a rarefaction wave that propagates to the lee (Figs. 9b,c). As with the upstream-propagating bore, passage of the rarefaction wave leaves behind positive η in the stable layer and a corresponding surface flow that effectively follows the obstacle. At the leading edge of the layer the isentropes quickly collapse to form a front (Figs. 9c,d) and the denser air begins to propagate into the leeside warm anomaly much as seen previously in the 2D case (cf. Figs. 8d,e). Indeed, for the early times shown in Fig. 9, the centerline flow is essentially indistinguishable from the corresponding 2D flow with the same value of ε (Fig. 9e).
The time evolution of the wake is illustrated from a more fully 3D perspective in Figs. 10 and 11. The color field in Fig. 10 shows a 3D rendering of the isentropic surface in the flow located initially at a height of 0.8 times the depth of the stratified layer. The purple and red tubes show selected 3D vortex lines running approximately tangent to the θ surface, while the vectors show the vortical component of the velocity field at the ground. The associated evolution of dye lines [computed as in Epifanio and Durran (2001)] at the terrain surface is illustrated in Fig. 11. The purple lines in Fig. 11 show dye lines that are initially parallel to the x axis, while the black contour shows the dye line with initial position x = 0 (see Fig. 11a). For convenience we refer to this latter contour as the dividing dye line, since it separates the fluid originating upstream of the obstacle from that originating on the downstream side.
Early in the evolution of the wake, the structure of the 3D vortex lines in Fig. 10 is qualitatively very similar to that deduced previously by Rotunno and Smolarkiewicz (1991). Following the impulsive start-up, the lifting and lowering of isentropes by the potential flow leads to negative buoyancy over the windward slope of the obstacle and positive buoyancy in the lee (Fig. 10a). The resulting buoyancy gradients then in turn lead to baroclinic vorticity generation. Initially the baroclinically produced vorticity is horizontal and oriented parallel to the buoyancy contours on surfaces of constant height. Upstream of the ridge the vortex lines circle clockwise around the cold anomaly, while downstream the vorticity encircles the warm anomaly in the counterclockwise direction. The flow field induced by this vorticity distribution is initially weak but is decelerative over both slopes of the ridge (as expected from Fig. 9b) and implies lateral spreading on the upstream face of the obstacle and lateral convergence in the lee. The surface dye lines have correspondingly begun to bend outward upstream of the dividing line and inward on the downstream side (Fig. 11b).
A short time later the distribution of potential temperature at the terrain surface has been shifted downstream slightly due to advection (Fig. 11c). On the downwind side the temperature gradient has contracted to a front (cf. Fig. 9c) and the flow behind the front is reversed (in the ground-relative frame). Downstream of the front the isentropic surfaces show a marked depression as the wave of rarefaction associated with flow reversal has begun propagating leeward (Fig. 10b). As with the θ distribution, the pattern of clockwise and counterclockwise vortex loops has also shifted downstream. The vortex lines have been tilted downward over the lee slope to produce an antisymmetric pattern of vertical vorticity at the wake edges, as deduced previously by Rotunno and Smolarkiewicz (1991). This vorticity pattern has then been amplified by vertical stretching in the frontal zone as illustrated recently by Epifanio and Durran (2002b). Even so, from an inversion and attribution standpoint the flow remains well approximated by the small-δ inversion of section 3. (And this remains true for later times as well.) Note that as the leeside flow is drawn back toward the obstacle and inward, the region of positive surface-θ anomaly downstream of the dividing line in Fig. 11c has begun to collapse.
The shape of the potential temperature surface in Fig. 10b shows that the leeside flow deceleration and associated rarefaction of the stratified layer has led to a noticeable lateral gradient in buoyancy at the edges of the wake. As time evolves, this lateral buoyancy gradient is gradually adjusted through a pair of inward propagating waves of elevation that start at the sides of the wake and progress to the center (Figs. 10c –e). As the adjustment waves pass they leave a baroclinically generated x component of vorticity in the wake and an associated velocity component directed toward the centerline plane (Figs. 10c, d). By the time the waves meet at the center of the wake in Fig. 10e, the vortex lines bend markedly in the downstream direction and the associated surface flow features evident lateral convergence. Note that this inward adjustment of the buoyancy gradient has lead to a significant lateral contraction of the surface dye lines downstream of the dividing line in Figs. 11d–f. At the same time, the flow upstream of the dividing line has continued to be deflected outward as it passes around the ridge. The end result is then that the stratified wake air is drawn into a localized tongue over the lee slope while warmer surface fluid from upstream is deflected laterally around the front.
As time evolves further the waves of adjustment continue to progress to the opposite sides of the wake (Fig. 10f) and the associated baroclinic generation straightens the vortex lines somewhat from the bent structure seen in Fig. 10e. By Fig. 10g, the vortex lines in the front part of the wake have begun to bend in the upstream direction and the relatively cold wake air thus begins to spread laterally into the warmer air at the sides. Ultimately the spreading wake air rejoins the fluid passing around the barrier from upstream and a pair of isolated surface potential temperature anomalies are shed permanently downstream into the wake (Fig. 10h). At this stage of development the wake has reached its mature state in which the surface streamlines form closed paths (in the ground-relative frame) with the pinched-off warm anomalies at the center. The structure of the vortex lines in this mature state is shown in Fig. 10h.
6. Summary and concluding remarks
This study has explored the dynamics of orographic wake formation in free-slip stratified flows as viewed from the general dynamical perspective of a 3D vorticity-vector potential formulation. Particular emphasis has been given to the role of upstream blocking and flow splitting in the development of the wake.
Scaling arguments were developed based on simple analytic calculations to explore the limiting form of the 3D vorticity inversion for flow at small dynamical aspect ratio δ. The scalings show that for flow at sufficiently small values of δ, the inversion is determined completely by the two horizontal vorticity components—that is, the velocity field induced by the vertical component of vorticity vanishes in the limit of small δ. The across-ridge component of velocity u is determined solely by the y component of vorticity η in this limit, while the along-ridge component υ is determined solely by the x component of vorticity ξ. This approximate small-δ inversion combined with the prognostic equations for horizontal vorticity and potential temperature yields a closed formulation of small-δ fluid mechanics with three governing prognostic variables. The applicability of this framework to the problem of orographic wake formation has been established by showing that the error in the small-δ approximation decreases to zero as the width of the obstacle increases.
In small-δ flow, the onset of leeside flow reversal at the ground depends on the production of sufficient y vorticity in the vicinity of the boundary. From this perspective the role of upstream blocking in the formation of the wake is readily apparent. A simple conceptual model that highlights the importance of upstream blocking is shown in Fig. 12. Blocking of the flow leads to the formation of a warm anomaly over the lee slope as potentially warmer air descends from aloft to replace the colder air deflected laterally around the barrier. The resulting negative temperature gradient between the warm lee-slope air and the colder fluid downstream then leads to the baroclinic generation of positive η above the boundary. This production of positive η implies a deceleration of the underlying surface flow, and the associated horizontal convergence ultimately contracts the temperature gradient to form a front. If the baroclinic generation across the front remains sufficiently strong, then the flow behind the front eventually reverses and the colder air begins to move back toward the obstacle.
The importance of upstream blocking was illustrated in greater detail through a series of simplified numerical initial-value problems involving the flow of a low-level stratified layer. In the case of 2D flow the dynamics of the stratified-layer wake was found to be relatively straightforward: the colder wake air effectively propagates into the leeside warm anomaly as a gravity current and is thus continually drawn upstream behind the obstacle as the barrier passes through the fluid. Indeed, the early time evolution of the 2D flow resembles that of the 2D retracting-piston and dam-break problems from classical shallow-water theory. The dynamics of the corresponding 3D case was then analyzed in terms of the production and evolution of the vortex lines near the top of the stratified layer. While the details of the 3D flow are somewhat more involved, the fundamental conclusion still applies: the essential role of upstream blocking is to produce a warm anomaly over the lee slope and to thus cause the colder surface air downstream to be drawn back toward the obstacle.
Finally, the high-resolution calculations presented in section 4 show that for the case of 3D flow past an infinitely long ridge, the wake structure produced by upstream blocking is distinct from that produced by breaking waves. In particular, blocking of the flow leads to wake formation at the surface, while breaking mountain waves lead to acceleration of the surface flow and the formation of a wake aloft. This difference between the two wake structures is perhaps most easily interpreted in terms of the difference in the sense of the horizontal vorticity near the ground. As described above, blocking inevitably leads to low-level baroclinic generation with the appropriate sense to decelerate the surface flow in the lee. By contrast, the baroclinic generation below a breaking mountain wave is exactly the opposite and the surface flow thus tends to be accelerated. It would appear that these arguments extend naturally to the case of flow past a 3D ridge as well, although this has not been explicitly tested.
a. Concluding remarks
The present analysis supports the general proposition that the formation of stratified wakes and vortices in blocked flow is essentially an adjustment under gravity (in the sense of Gill 1982, chapter 5) caused by the presence of a topographic obstacle moving through the fluid. The motion of such an obstacle causes the vertical displacement of density surfaces in the vicinity of the barrier and thus creates gradients in the density between the fluid near the obstacle and the fluid further downstream. The adjustment of the density surfaces causes air to first be drawn back toward the obstacle and inward and then ultimately to spread over the lee slope to form recirculating vortices (as shown in Fig. 10). It would thus appear that while the formation of such vortices perhaps inevitably leads to the onset of dissipation, the basic tendency to produce the vortices is nonetheless a property of the inviscid and adiabatic equations governing stably stratified flow.
There has been and continues to be a discussion in the literature on the role of PV generation in orographic wakes (e.g., Smith 1989b, a; Smolarkiewicz and Rotunno 1989b; Schär and Smith 1993; Schär and Durran 1997; Rotunno et al. 1999; Epifanio and Durran 2002b; Schneider et al. 2003). The production of PV—and perhaps more importantly, surface potential temperature anomalies—in the wake is clearly of great importance in that it allows the structure of the wake to be maintained indefinitely downstream.3 Nonetheless, it is well known that to be able to infer the structure of the flow field from the distribution of PV requires an assumption of balance (e.g., Hoskins et al. 1985). While it appears that far downstream of the obstacle the known constraints of balance models (all of which involve the assumption |∂u/∂x + ∂υ/∂y| ≪ |ζ| at some level) are well satisfied, in the near field of the wake these balance constraints are at all times strongly violated due to the presence of the surface front over the lee slope—and this is especially true during the early stages of wake formation. We must therefore conclude that while the generation of PV may have important consequences for the downstream evolution of orographic wakes, an understanding of the process of wake formation requires more fundamental considerations, such as those developed here.
This research was supported by NSF Grant ATM-0242228. Part of this work was begun while C. Epifanio was the recipient of a postdoctoral fellowship in the Advanced Study Program at the National Center for Atmospheric Research (NCAR).
The Vorticity Inversion: Implementation Details and Verification
The vorticity inversion (14) yields the three component equations (16) along with the coupled boundary conditions (17) applied at the lower boundary of the model domain. The Laplacians are discretized using second-order finite differences in the terrain-following coordinate (10), and the resulting difference equations are solved by means of simultaneous iteration using the conjugate-residual algorithm outlined by Smolarkiewicz and Margolin (1994). The boundary conditions are applied by first discretizing the last of (17) as
where the subscripts 0 and 1 refer to the lower boundary and first interior grid points, respectively, and the height of the first interior grid level is Δq/2. Substituting for Fi,j,0 and Gi,j,0 in terms of Hi,j,0 using (17) then yields a linear system for H at the boundary, and we solve this system iteratively for each outer iteration of the coupled elliptic solvers for (16). Similar methods are used to solve (13) for the scalar potential ϕ.
Figure A1 shows a sample verification of the vorticity inversion technique for flow past an axisymmtric obstacle with ε = 3, Re = 150, and δ = 0.6. The characteristic errors in the inverted flow fields are in the vicinity of 1%.
Scale Analysis for the Vector Potential Components
Here we consider the appropriate amplitude and spatial scales for the three components of the vector potential ψ. To help guide our intuition, we first consider a prototype inversion problem for a specified vorticity distribution resembling the pattern of vorticity seen in our numerical wake calculations. We then show that the results of our prototype calculation describe the vector potential scales for the full wake problem with a high degree of accuracy.
Suppose as our simplified inversion problem we first consider
where G may be either of the two horizontal components of the vector potential ψ. We consider the case of a semi-infinite domain with a flat horizontal boundary at z = 0, so that (17) implies
and substituting into (B1) then yields
where 𝒦 = k2 + l2 and where the approximation is valid in the small-aspect-ratio limit δ = 𝒦ℋ ≪ 1 with z/ℋ ∼ 1. The solution to our prototype problem (B1) with the Dirichlet boundary condition (B2) thus has a characteristic amplitude scale given by Uℋ. To find the vertical fluctuation scale, we take
showing that for z/ℋ ∼ 1 the appropriate depth scale for G is clearly ℋ. Similarly, ∂G/∂z also has a depth scale given by ℋ. The associated spatial structures of G and ∂G/∂z are shown in Figs. B1a,c.
Now consider an analogous problem for the vertical component of ψ; namely
where the length scale ℒ is given by ℒ = 1/𝒦. The boundary condition (17) in this case implies
as before and then solving for 𝗛 gives
showing that the amplitude scale for our prototype problem is again Uℋ. However, the vertical gradient of the solution in the present case is
implying that in contrast to the result for G, the appropriate depth scale for H is actually ℒ rather than ℋ. The corresponding difference in the vertical fluctuation scales between the two components is clearly seen in Figs. B1a,b.
The difference in the depth scales for G and H in our prototype inversion problem clearly stems from the difference in the lower boundary conditions applied to the potentials. Specifically, the Dirichlet condition (B2) imposed on G implies an image vorticity distribution of opposite sign below the boundary, and the vertical scale of the solution is then set by the depth scale of the source distribution and its image. By contrast, the Neumann condition (B5) implies an image source of like sign, and the vertical scale for H is then determined by the spatial isotropy of the Laplacian. Nonetheless, note that while (B6) and (B7) suggest a vertical scale of ℒ for H, the second derivative,
shows that ∂H/∂z recovers the source distribution depth scale of ℋ for z/ℋ ∼ 1. This change in depth scale between H and ∂H/∂z is apparent in Figs. B1b,d.
Finally, to conclude our idealized scale analysis, we must consider the effect of introducing small but nonzero terrain slopes at the boundary. For sufficiently small boundary deformations we expect the solution of the modified inversion problem to be well approximated by the corresponding solution for the half-space (i.e., we expect the dependence on the boundary conditions to be continuous). Indeed, the scalings derived above suggest that the first two of (17) reduce to homogenous Dirichlet conditions on F and G whenever h0/L ≪ 1 [cf. (20)]. Similarly, the last of (20) reduces to a homogenous Neumann condition on H whenever h0/L is much smaller than the dynamical aspect ratio δ. However, if h0/L is of the same order of magnitude as δ (as for the wake scaling of section 2b), then all three terms of the last of (17) must be retained and the boundary condition on H is no longer a simple Neumann condition [cf. (20)]. Even so, nothing in the modified condition demands a rescaling of H [i.e., the scalings derived above are completely consistent with the full boundary conditions in (17)] and we therefore expect that the prototype potential scales derived for the flat boundary problem extend to the case of small terrain slopes as well. The numerical calculation described in Fig. B2 confirms that this is in fact the case.
Figure B3 shows the y and z components of the vector potential ψ for a numerical topographic wake simulation with ε = 3, β = 1, and δ = h0/L = 0.1. Comparison with Fig. B1 shows that the vector potential spatial and amplitude scales for the wake flow are in fact accurately predicted by the results of the simple analytic inversion problems described above. In particular, note that both G and H are well described by the amplitude scale Uℋ, where ℋ is taken to be U/N as predicted in section 2b. Similarly, the depth scale for G is well approximated by ℋ, while the vertical fluctuation scale for H is clearly much larger. And as in Fig. B1d, the first derivative ∂H/∂z recovers the source depth scale of ℋ near the lower boundary. Indeed, the most substantial discrepancy between Figs. B1 and B3 is that ∂G/∂z reveals the vertically propagating mountain wave above the lee slope as well as the more concentrated wake flow near the lower boundary in Fig. B3c. However, as the depth scale of the mountain wave is also U/N, the vorticity distribution in the wave does not modify the general scaling results for the vector potential components given above. Note in particular that ∂G/∂z in Fig. B3c is well described by the amplitude scale U, as anticipated.
The Compressible Boussinesq Framework
Let the standard potential temperature and Exner function variables be denoted by Θ = Θ0(z) + Θ′ and Π = Π0(z) + Π′, respectively. Here primes indicate deviations from the hydrostatically balanced background state. We assume that viscous and diabatic effects are of secondary importance and thus temporarily restrict attention to the case of inviscid and adiabatic flow. The full compressible equations for nonrotating stratified fluid dynamics can then be written
where cs is the speed of sound. Consistent with the conventional Boussinesq approximation, we first assume that the total and basic-state potential temperature variables appearing as coefficients and denominators in (C1)–(C3) can be replaced by a constant reference value Θs. This requires that the total depth of the disturbance be significantly smaller than the characteristic height scale of the Θ0 distribution. Having made this assumption, (C1) and (C2) then immediately reduce to the inviscid and adiabatic forms of (1) and (2) with P = cpΘsΠ′, b = gΘ′/Θs, and N 2 = (g/Θs) dΘ0/dz.
We next characterize our disturbance of interest in terms of a fluctuation length scale ℒ, a fluctuation depth scale ℋ, a horizontal velocity scale U, a vertical velocity scale Uℋ/ℒ, and a time scale ℒ/U. The nondimensional form of the pressure-divergence equation (C3) can then be written as
where Ma = U/cs is the Mach number, ℋ0 = c2s/g, and where dimensionless quantities are denoted by hats. The compressible Boussinesq version of (C3) then follows directly from the assumption that ℋ ≪ ℋ0, in which case the dimensional form of the pressure-divergence equation becomes
The compressible Boussinesq system thus relies on exactly the same assumptions about depth scales as the conventional Boussinesq system but does not make the further assumption of small Ma that filters the acoustic modes. The corresponding retention of a prognostic pressure equation then enables the use of split-explicit time-integration methods for the numerical solution of the system, as described briefly in section 2c.
For the case in which Ma ≪ 1, the solutions to (1) and (2) with (C4) are effectively incompressible and the compressible Boussinesq and conventional Boussinesq systems thus yield essentially identical results. Two further approximations often applied in this limit are (i) the speed of sound cs is taken to be constant, since variations in cs are of little consequence at small Ma; and (ii) the nonlinear pressure-divergence equation (C4) is replaced by its linearized form (3) (see, e.g., Durran and Klemp 1987; Skamarock and Klemp 1993; Klemp et al. 1994; Epifanio and Durran 2001). To appreciate this latter approximation, note that the slow-mode solutions remain exactly incompressible through two orders in Ma regardless of whether (C4) or (3) is used as the pressure-divergence equation. Differences between the two formulations must therefore be limited to O(Ma2) weak compressibility effects.4 Practical experience shows that omitting the pressure advection terms in (C4) in fact has little discernible impact on most small-Ma atmospheric motions—see Klemp and Wilhelmson (1978) and Durran and Klemp (1983) for further discussion of this point—and we have therefore adopted the linearized form (3) in the present study out of computational convenience.
* The National Center for Atmospheric Research is sponsored by the National Science Foundation
Corresponding author address: Craig C. Epifanio, Dept. of Atmospheric Sciences, Texas A&M University, College Station, TX 77843. Email: email@example.com
Specifically, we assume a reference potential temperature profile θr(z) = N 2rz + θs where Nr and θs are constants. We then allow a heat flux −K0N 2r/(∂h/∂x)2 + (∂h/∂y)2 + 1 across the lower boundary so as to maintain this reference profile in the presence of the diffusive term (6). In our experiments with uniform N, we simply set Nr = N. In the cases with two-layer stability, we let Nr equal N in the upper layer, thus implying that the lower layer is to be considered a departure from our assumed reference state.
An analysis similar to that found in appendix B shows that neglecting the small horizontal derivatives in (19a) and (19b) leads to O(1) errors in ŵ (but not û and υ̂) for heights exceeding z/ℒ ≃ 1. (This error can be prevented if the domain depth is relatively shallow so that zT/ℒ ≪ 1.) Here we avoid this problem by simply including these terms in our definition of the approximate small-δ vorticity inversion. Note that retaining these terms does not compromise the small-δ inversion as a test of our main scaling predictions, since those predictions are based entirely upon neglecting terms in (20) and (21).
The far-field balanced structure of the wakes described here and in Fig. 16 of Schär and Durran (1997) seems likely to be determined as much by the anomalies of surface θ shed downstream as by the PV in the fluid interior. This point appears to have been underemphasized in previous work.