## 1. Introduction

This two-part study is motivated by the desire to develop a better understanding of the physical relation between the shallow-water equations (SWE) and the Navier–Stokes equations (NSE) in their representations of two-layer stratified flow past topography. In a fluid composed of two layers of different, but constant, densities, all the vorticity is concentrated at the interface between the two layers; this view of vorticity is complicated in cases where the generally horizontally oriented density interface forms a discontinuity in its vertical displacement (i.e., a “hydraulic jump”), such as can occur in the SWE (Rotunno and Smolarkiewicz 1995). In this two-part study, we examine numerical solutions of the three-dimensional NSE for a two-layer fluid, with a thin connecting layer of continuous density variation, flowing past topography for situations in which solutions to the SWE indicate hydraulic jumps.

Hydraulic jumps play a critical role in solutions to the SWE exhibiting lee vortices (counterrotating eddies immediately downstream of a mountain barrier), as shown in Fig. 1 (from Epifanio 2014). These solutions are classified into two distinct types: one where the interfacial height remains above the obstacle (Figs. 1a,b), and the other, where the density interface is pierced by the obstacle (Figs. 1c,d). In the former case, two oppositely signed bands of vertical vorticity extend downstream from the spanwise termination points of a leeside hydraulic jump (Fig. 1b); in the latter case, these bands extend downstream from hydraulic jumps on the flanks of the obstacle where it pierces the interfacial layer (Fig. 1d). Owing to the added complexity of the numerical solution to the SWE for the case where the obstacle pierces the interface (e.g., Schär and Smith 1993, their appendix B), the present two-part study is restricted to the case where the interface remains above the obstacle as in Figs. 1a and 1b. The pierced-layer case is briefly discussed in section 3c.

In the representation of two-layer flow in the SWE, the vertical vorticity *ζ* is that of the lower layer (of thickness *h*) and is only nonzero if the potential vorticity *ζ*/*h* is nonzero, which, in the flows over frictionless topography under consideration, can only occur when there is internal energy dissipation. In the case of the interfacial layer remaining above the obstacle, the solution to SWE has *ζ*/*h* = 0 until the flow passes through the horizontal discontinuity in the interfacial-layer height (the hydraulic jump); energy dissipation in the flow through the hydraulic jump produces a dipole of *ζ*/*h* and thus (because *h* > 0) oppositely signed bands of *ζ* extending downstream from the spanwise extremities of the hydraulic jump (Fig. 1b). Hence the conclusion from the SWE is that lee vortices are produced by dissipative processes associated with the hydraulic jump (Schär and Smith 1993, section 2d).

The critical role of hydraulic jumps in the SWE for lee-vortex formation motivates the present study. Rotunno and Bryan (2018, hereafter Part I) examined the flow past a two-dimensional (no spanwise variation) obstacle for cases in which the SWE produce a leeside hydraulic jump. The solutions to the NSE for this flow indicate the hydraulic jump is a leeside wave in the thin density-transition layer that overturns backward (against the ambient flow) toward the obstacle leading to a time- and spanwise-mean rotary motion and turbulence; the spanwise vorticity for the mean rotary motion is supplied by baroclinic vorticity production in the thin interfacial layer as it descends on the leeside of the obstacle and flows into the wave (Fig. 11 of Part I). Here in Part II we examine direct numerical solutions of the NSE to study the same two-layer fluid as in Part I, but for flow past a three-dimensional obstacle (limited spanwise extent). These solutions allow an analysis of the origin of lee-vortex vorticity and its relation to the continuous-fluid version of the hydraulic jump in a fully resolved turbulent flow.

In section 2a, we briefly review the experimental setup for the numerical solutions of the NSE which, except for the three-dimensional obstacle, is identical to that in Part I (where the differences between hydraulic jumps in the traditional water–air system and the atmospheric case are discussed). In section 2b we present the results. Our analysis of the lee vortices found in the NSE in terms of vorticity and Ertel potential vorticity is given in section 3 and concluding remarks in section 4.

## 2. Two-layer flow past a submerged obstacle in the NSE

### a. Numerical setup

The numerical setup for the present numerical experiment is described in Part I; the only changes here are the spanwise-varying topography and a spanwise doubling of the integration domain keeping the same grid resolution (Δ*x* = Δ*y* = Δ*z* = 20 m). For this study, the obstacle height *H*(*x*, *y*) is specified following (7) and (8) of Epifanio and Rotunno (2005) with parameters *h*_{0} = 1000 m (maximum obstacle height), *L* = 500 m (a horizontal scale) and *β* = 3 (which determines the spanwise extent of the obstacle; *β* = 1 produces a circular obstacle). Figure 2 shows the obstacle and integration domain in a three-dimensional perspective. As in Part I, the domain in the along-flow (*x*) direction extends 15.36 km with “open” boundary conditions applied at both ends, while the domain in the vertical (*z*) direction extends from the height of the zero-stress topography *z* = *H*(*x*, *y*) to *z* = 3.84 km where there is a zero-stress, rigid lid. The domain in the spanwise (*y*) direction extends 7.68 km with periodicity assumed at the lateral boundaries.

The initial potential-temperature profile is given by (9) of Part I and is shown in the plane *y* = 3.84 km in Fig. 2; the profile is similar to the two-layer idealization of the SWE except for the finite-thickness transition layer connecting the lower-fluid to the upper-fluid constant potential temperatures. The initial velocity field (*u*, *υ*, *w*) = (*U*, 0, 0), in the Cartesian coordinates (*x*, *y*, *z*), quickly adjusts to potential flow past the obstacle. As discussed in Part I, the initial-condition parameters for the potential-temperature profile, the mountain height and the upstream velocity, *U* = 5 m s^{−1}, are chosen based on shallow-water-theory predictions for a stationary leeside hydraulic jump.

As described in Part I, the numerical simulations performed for this study are in the category of direct numerical simulations (DNSs) in which no subgrid turbulence parameterization is used. Using a constant kinematic viscosity and thermometric conductivity (*ν* = *κ* = 1 m^{2} s^{−1}), one can estimate the Reynolds number (*Uh*_{0}/*ν*) to be approximately 5000—small compared to the real atmosphere, but large enough to capture turbulence motions without approximation. A spectrum analysis (not shown) of the vertical velocity in the zone of high turbulent kinetic energy (Fig. 7b of Part I) indicates that the energy in the largest turbulent scale is at a wavelength of approximately 1000 m and that the spectrum roughly follows a −5/3 decrease with scale down to the smallest resolved turbulent scale which is approximately 100 m. The ratio of the grid length (20 m) to the largest turbulent scale is therefore 0.02, which is within the recommended upper limit of 15 × (Reynolds number)^{−3/4} ≈ 0.025 for accurate DNS of free-shear layers (Moin and Mahesh 1998, p. 554).

### b. Results

Figure 3 shows a time sequence of the flow (*u*, *w*) and potential temperature *θ* in a vertical (*x*–*z*) cross section taken along the flow on the obstacle centerline (*y* = 0). Figure 4 shows the same fields except in a horizontal (*x*–*y*) cross section at *z* = 0.4 km. Figures 3a and 4a show the flow at time *t* = 1 min after initialization so that the rapid adjustment to potential flow around the obstacle is in evidence. Consistent with shallow-water theory and with the flow past a two-dimensional obstacle (Fig. 3 of Part I), the flow in this regime exhibits a leeside steepening of the isentropes (Fig. 3b) that continues to isentrope overturning and local flow reversal (Figs. 3c, 4b). As in the 2D case in Fig. 3 of Part I, warm air is extruded downward and mixes with the cooler lower-level air; however, in the present case, the initial warm-air bubble is much smaller and mixes more rapidly. The flow adjusts to a sequence of such episodes of warm-air extrusion into, and mixing with, the lower-layer cool air as illustrated in Figs. 3d and 3e.

Figures 5 and 6 show a windowed-in view of the centerline *x*–*z* cross sections of *θ*, flow vectors, and spanwise vorticity *η* = ∂_{z}*u* − ∂_{x}*w* (top panels) along with *x*–*y* sections of *θ*, flow vectors, and vertical vorticity *ζ* = ∂_{x}*υ* − ∂_{y}*u* at *z* = 0.4 km.

Figure 5a shows a snapshot of the flow at *t* = 11 min; the flow shown in the upper panel is qualitatively similar to that expected from the simulations past a 2D obstacle shown in Fig. 5 of Part I in which *η* < 0, baroclinically generated along the descending isentropes, induces a rotary motion with reversed flow. The reversed flow splits the layer vertically and elevates the upper isentropes which then baroclinically produce *η* > 0 above the reversed flow. The bottom panel of Fig. 5a shows the isentropes that extend below *z* = 0.4 km and the flow that advects *θ* into two local maxima collocated with the two members of the dipole in *ζ*. We note that at this time there is no turbulence in evidence and that, therefore, the flow that results from this initial-value problem produces lee vortices in the absence of turbulent dissipation. Figure 5b shows that turbulent motions follow subsequently as the extruded warm air becomes cut off from its leeside source (top panel) and the initial *ζ*/*θ* anomalies drift downstream (bottom panel).

Figures 6a and 6b show the next episode of extruded warm air with baroclinic production of *η* < 0 and subsequent mixing (top panels) and the associated formation and downstream drift of a *ζ* dipole (bottom panels). At this stage the wake turbulence from the previous episode leads to a much less orderly flow. Episodes like those illustrated in the previous two figures continue throughout the simulation. After approximately 60 min, the flow settles into a fully developed statistically steady turbulent flow shown in Fig. 7a.

Figure 7b shows the turbulent kinetic energy (tke) based on the 60–120-min average of 1-min output data.^{1} The top panels in Fig. 7 showing the centerline flow and tke are qualitatively consistent with Fig. 7 of Part I in the 2D-obstacle case. The bottom panels of Fig. 7 show that the most intense turbulence is well correlated with the time-averaged vertical vorticity downstream of the location where tke and *ζ* are generated (near *x* ≈ −1.0 km).

## 3. Analysis

### a. Vorticity

*L*/

*U*≃ 6.67 min in the present case). This first period is called the “inviscid phase” (analogous to the results shown in Figs. 3a–c and 5a) while the second and much longer period analyzed by Schär and Durran (1997) is called the “dissipative phase” (analogous to the present results shown in Figs. 3d and 3e and Figs. 6 and 7). In this latter phase, Schär and Durran (1997) analyze the lee vortices in terms of the Ertel potential vorticity, which in its Boussinesq form is

**= (**

*ω**ξ*,

*η*,

*ζ*) is the vorticity vector. As argued in Rotunno et al. (1999), analysis of the lee vortices solely in terms of PV leaves unanswered the question of how

**is created in the steady-state flow as fluid parcels traverse the obstacle from upstream where**

*ω***= 0. This section provides an answer to this question for the case at hand based on the present numerical simulations.**

*ω*To isolate the essential source(s) of the vertical vorticity of the lee eddies, it is convenient to work with the flux form of the vector vorticity equation (section 5 of Haynes and McIntyre 1987) which has the property that the flux of vorticity in its own direction is zero. Consequently, the vertical flux of vertical vorticity is identically zero and the present analysis reduces to determining the horizontal fluxes of vertical vorticity into and out of an area surrounding a lee vortex.

*y*< 0.

*x*= −1.3 km and flank side at

*y*= −1.5 km, and therefore, all the vertical vorticity in the square must flow through its borders at

*y*= 0 and

*x*= 0.2 km. The vertical vorticity flux vector (

*Z*

_{zx},

*Z*

_{zy}) in this square is shown in Fig. 8a. This figure shows a vertical vorticity flux into the square along

*y*= 0 and out of the square along

*x*= 0.2 km. Integrating (2) over the area of the square at

*z*= 0.4 km gives

^{2}s

^{−2}while the integral on the lhs yields 32.86 m

^{2}s

^{−2}. Recognizing the noisy fields near the right side of the square, we consider a square with a variable right-side position

^{2}s

^{−2}while the integral on the rhs is 29.21 m

^{2}s

^{−2}which indicates that the equality (5) holds to an accuracy of ~3%.

_{x}

*F*

_{y}− ∂

_{y}

*F*

_{x}where

^{2}

*z*= 0.4 km, Fig. 7a) as well as at the level of maximum tke (

*z*= 0.2 km, Fig. 7b) and the main balance is between the terms on the lhs. This conclusion is consistent with the finding of Epifanio and Qian (2008, their section 5a) in a large-eddy simulation of a large-amplitude mountain wave that the direct effects of the Reynolds stress on the mean-flow modification are negligible. Furthermore, our analysis indicates that

Figure 9 shows the integrands in (5) and (7); the cross-stream flux of vertical vorticity (Fig. 9a) is closely followed by its approximation while the downstream flux is more variable (Fig. 9b). Numerical evaluation of (7) yields a rhs value of 33.66 m^{2} s^{−2} while the lhs yields 30.25 m^{2} s^{−2} which indicates that (7) holds to an accuracy of ~10%. From the foregoing analysis we conclude the *y* = 0 approximated by

*b*=

*gθ*/

*θ*

_{00},

^{3}Consistent with (6), the eddy-correlation term is equivalent to ∂

_{z}

*F*

_{x}− ∂

_{x}

*F*

_{z}. Figure 10 shows the terms in (8) along

*y*= 0 and the flow vectors; the only place where there is significant baroclinic generation is within the leeside descent region. In the region upstream of the location where

*η*

_{m}) in Fig. 10, the flow is essentially laminar, hence (8) may be written along

*y*= 0 as

*u*

_{s}

*dη*/

*ds*= −∂

_{x}

*b*where

*s*is the distance and

*u*

_{s}is the speed along a streamline; this equation may be formally integrated to give

*η*= 0. Choosing the streamline that goes down through the zone where −∂

_{x}

*b*< −0.5 × 10

^{−3}s

^{−2}in Fig. 10, one can estimate that

*ds*≃ 500 m and

*u*

_{s}≃ 5 m s

^{−1}and therefore

*η*≃ −∂

_{x}

*bds*/

*u*

_{s}≃ −0.05 s

^{−1}, which is close to the values shown in Fig. 7a.

As one can see in the top panel of Fig. 7a, the field of *z* = 0.4 km and therefore cannot cancel out the positive contributions from the downward transport of *z* ≃ 0.4 km, Fig. 10 shows that the main balance in (8) is between mean and turbulence transport of *x*-averaged values of the terms in (8) which indicates that the time-mean downstream transport of negative

In summary, the analysis leading to (7) shows that the vertical vorticity

*z*= 0.2 and

*z*= 0.4 km for the statistically steady flow; the stretching term is dominant at

*z*= 0.4 km while the tilting term is negative; at

*z*= 0.2 km, the tilting term is positive and dominant while the stretching term is small. These outcomes are expected based on the analysis of Epifanio and Durran (2002) of a leeside breaking mountain wave (identified as a hydraulic jump) in a continuously stratified flow. Their analysis of fluid-parcel trajectories passing through the leeside vertical vorticity maxima indicate that during the parcel’s leeside descent, it acquires horizontal vorticity through baroclinicity which is subsequently tilted to become vertical vorticity which is, in turn, stretched at the across-stream end points of the hydraulic jump (Epifanio and Durran 2002, p. 1174). The next section will provide further information on the structure of the vorticity distribution in the present simulated hydraulic jump.

### b. Potential vorticity

**and PV. Following Epifanio and Qian (2008, section 5b), we look at the potential vorticity (1) defined as**

*ω**z*= 0.4 km in the same window as the bottom panel of Fig. 7a. As expected from the SWE solutions (Fig. 1b), there is a dipole of

**F**is (minus) the divergence of eddy-momentum flux and

*Q*is (minus) the divergence of the eddy-heat flux. As with the equation for

*y*component of

**J**is a central element in understanding the origin of the

**J**= 0 and therefore a cross stream flux

*J*

_{y}< 0 at

*y*= 0 is required to maintain a

*y*< 0. At

*y*= 0, we expect by symmetry that

*y*= 0

*J*

_{y}(

*y*= 0) reaches a negative minimum along the axis of the maximum downward excursion of the leeside isentropes. Figures 14b and 14c show the frictional and heating contributions to

*J*

_{y},

*J*

_{y}, the structure

*J*

_{y}. Figure 14 taken together with (15) suggests a close connection with the analysis of the

**intersects surfaces of constant**

*ω**θ*, or in other words, the degree to which

*θ*varies along a vortex line, namely,

*δl*is an infinitesimal distance along a vortex line. The following three-dimensional analysis makes extensive use of (16), but using

Figure 15 shows a series of vortex lines that go through the locations of minimum *y* = 0 (see the top panel of Fig. 7a); the color of the vortex line indicates the local value of *x*–*z*) cross sections of *z* < 1.25 km in the planes at *y* = 0 and *y* = 1.9 km allow one to see the location and *y* = 1.9 km and *z* ≈ 1 km (the height of the undisturbed layer of strong potential-temperature increase shown in Fig. 2) and then descend for *y* > 0 (implying *y* < 0 (implying

In contrast, Fig. 16 shows a vortex line that passes through the *y* < 0 and decreases along the vortex line *y* > 0. Note that, as with the vortex lines of Fig. 15, this vortex line passes close to location of minimum *y* = 0, which indicates the basic baroclinic source of

Figure 17 shows the vortex lines from Fig. 16 together with the isosurface *y* > 0 and ascends through it for *y* < 0 (Fig. 17b) which, according to (16), implies a dipole in

The vortex lines shown in Fig. 15 are consistent with those found in the inviscid, adiabatic linear theory of Smolarkiewicz and Rotunno (1989, their Fig. 2b); however, the one in Fig. 16 is not. As demonstrated above, this vortex line loops more or less vertically through an isentropic surface and thus has significant gradients of potential temperature along its length and therefore significant potential vorticity. To understand the origin of this looping vortex line, we examine the vortex lines and isosurfaces of *θ* at the time when the leeside reversed flow first forms (Fig. 3b).

Figure 18 shows several vortex lines passing through *y* = 0 at *t* = 8 min together with *θ*(*x*, 0, *z*) and *θ*(*x*, *y* = 1.9 km, *z*). Each vortex line has little variation of *θ* along its length and so by (16), PV = 0. Figure 19 shows the same vortex lines together with the isosurface *θ* = 289 K. These figures together suggest that the looping, potential-vorticity-bearing vortex line seen in the steady-state configuration (Figs. 16, 17) has its origin in the inviscid, adiabatic, but nonlinear, fluid dynamics of wave overturning, which in turn, lead to local static instability and turbulence. PV generation occurs subsequently as the crest of the wave moves through the loop and reconnects through turbulent mixing with fluid in the wave trough. As discussed in relation to Figs. 3c and 3d, this process of wave formation and overturning occurs repeatedly through the simulation and thus the vortex-loop structure emerges in the time average of the statistically stationary solution.

### c. The case where an obstacle pierces the two-layer-fluid interface in the SWE

Epifanio and Rotunno (2005) consider the case where the fluid is stably stratified below the height of the obstacle crest with much weaker stratification above; this case is the continuous-fluid analog to the pierced-layer case in the SWE (Figs. 1c,d). Figure 20 (Epifanio 2014) depicts the evolution of lee vortices in this flow started from rest; the flow regime is one in which the upstream flow below the obstacle crest is blocked and must flow around, rather than over, the obstacle. Figure 20 shows the vortex lines on an evolving isentropic surface with centers of leeside vertical vorticity forming as the isentropes and vortex lines bend downward on the leeside flanks of the obstacle. We note that in this case the vortex lines associated with the return flow loop over it and begin and terminate at the terrain. These experiments show that lee vortices can also form for cases in which the obstacle extends above the stratified layer with PV = 0. PV is subsequently created by diffusive effects as the evolving isentropes of each vortex wrap in a spiraling motion. While the evolution of this flow differs from that of the present case of flow of the stratified layer over the obstacle, the conclusion is the same: lee vortices can form under conditions of PV = 0; PV is created subsequently as a result of the motions engendered by the vorticity distributions created by the inviscid, adiabatic processes in a density-stratified fluid.

We note that the “looping-over” structure of the vortex lines has been commented on in Schär and Durran (1997, their Fig. 12) and in Rotunno et al. (1999, their Fig. 4) in their respective analyses of continuously stratified flow. The structure of these solutions in the *y*–*z* plane suggests that the lee vortices in these cases are associated with blocking of the upstream flow [Fig. 4c in Schär and Durran (1997) and Fig. 3a in Rotunno et al. (1999)].

## 4. Discussion and conclusions

As a model for atmospheric motions, the shallow-water equations (SWE) have two limitations that complicate interpretation in terms of concepts based on the full Navier–Stokes equations (NSE). First, the SWE treat fluids with layers of constant density which implies a discontinuity in the vertical derivative of density at the interface between the layers. Second, these equations are only valid for motions with horizontal length scales much greater than vertical length scales.

The steepening of the height of the fluid layer *h*(*x*, *t*) toward a discontinuity (interpreted as the hydraulic jump; see Fig. 2 of Part I) in the SWE violates the second limitation. The formation of a discontinuity indicates a breakdown of some approximation to the physics; to “save” the solution one typically allows the discontinuity and then matches the solutions on either side based on physical principles to obtain a “weak” solution (Whitham 1974, p. 26). In the case of a hydraulic jump, the conservation of mass and momentum in the layer provide the matching conditions (Whitham 1974, 458–460). Alternatively, Schär and Smith (1993, their section 2e) formulate a diffusion term that prevents a discontinuity while conserving mass and momentum. With either method, energy is dissipated through the zone of large height and velocity gradients, which leads to the creation of shallow-water-equation potential vorticity, *ζ*/*h*, and therefore, vorticity *ζ*, since *h* > 0. Hence their conclusion that dissipation creates lee-vortex vorticity (Schär and Smith 1993, their section 2d).

As with the two-dimensional case of Part I, the present study of flow past a three-dimensional obstacle produces a leeside spanwise rotary motion in the along-stream-vertical plane but of limited spanwise extent and therefore, we have shown, lee vortices. Ertel potential vorticity (PV) is produced as a consequence of the dissipative processes that result from the frictionless, adiabatic dynamics giving rise to leeside rotary motion/lee vortices. Hence the view from the NSE is that the three-dimensional flow associated with lee vortices leads to PV production, not vice versa.

The reconciliation of these contrasting conclusions from the SWE and the NSE concerning lee vortices and potential vorticity production can be summarized as follows. The internal dynamics of the hydraulic jump is not accessible to the SWE since irreversible processes must be parameterized at the leeside discontinuity (identified with the physical hydraulic jump) (Schär and Smith 1993, their section 2e). The potential vorticity in the SWE, and therefore the vorticity, is created by the parameterized dissipation at the discontinuity. Hence, the view of lee-vortex formation from the SWE focuses on the leeside vorticity downstream of the discontinuity/hydraulic jump (Fig. 1b). Solutions to the NSE, on the other hand, reveal the hydraulic jump’s internal dynamics is driven by frictionless, adiabatic processes that result in the overturning of the thin transition layer which leads to local static instability and turbulence (Fig. 11 of Part I). With the present simulation using a three-dimensional obstacle, the rotary motion/reversed flow has a limited spanwise extent, which we have shown is consistent with a leeside dipole in the vertical vorticity. The latter turbulent motions give rise to energy dissipation which, consistent with the SWE, generate the potential vorticity that flows downstream from the hydraulic jump.

In summary, the principal conclusions are as follows:

Analysis of the NSE indicates the original source of the vertical vorticity of the lee vortices is the descent of negative spanwise vorticity, expressed analytically in (7). The role of turbulent dissipation, emphasized in the SWE, prevents the re-ascent of negative spanwise vorticity that would limit the downwind transport of vertical vorticity.

Analysis of the NSE shows that leeside negative spanwise vorticity is created baroclinically in the descending leeside flow [Fig. 10 and (9)]. Figure 15 shows that tilting of the vortex lines with negative spanwise vorticity produces the dipole of vertical vorticity associated with the lee vortices. In the SWE, and more generally, in typical formulations of water-wave equations (Whitham 1974, chapter 10), baroclinic vorticity generation does not appear explicitly, but rather, is implicit at the two-layer interface (Rotunno and Smolarkiewicz 1995, their section 2c).

The leeside hydraulic jump is critical to lee-vortex formation in SWE as it implies energy dissipation and the creation of a potential-vorticity dipole. The internal dynamics is not, however, accessible to the SWE. According to the NSE, the hydraulic jump is a backward overturning isentrope with the reversed flow induced by baroclinically generated negative spanwise vorticity. The static instability and mixing induced by the overturning isentrope leads to energy dissipation and potential-vorticity creation.

Analysis of the NSE leads to a geometrical interpretation of the potential-vorticity dipole associated with the lee vortices in which the overturning isentropes envelope the baroclinically produced spanwise vorticity (Fig. 17); this process originates in the inviscid, adiabatic, nonlinear dynamics leading to isentrope overturning (Fig. 19).

## Acknowledgments

The authors thank Dr. Vanda Grubišić (NCAR) for her comments on the first draft of this manuscript. We gratefully acknowledge the high-performance computing support from Cheyenne (doi:10.5065/D6RX99HX) and the three-dimensional images (Figs. 15–19) created with VAPOR (www.vapor.ucar.edu) provided by NCAR’s Computational and Information Systems Laboratory, sponsored by the National Science Foundation. This material is based upon work supported by the National Center for Atmospheric Research, which is a major facility sponsored by the National Science Foundation under Cooperative Agreement 1852977.

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. John Wiley and Sons, 636 pp.*Linear and Nonlinear Waves*

^{1}

Clearly a longer averaging period and/or more frequent outputs would be required to produce more spatially homogeneous distributions of these fields.

^{2}

The indices 1, 2, and 3 for *i* or *j* correspond to the Cartesian coordinates (*x*, *y*, *z*) and velocity components (*u*, *υ*, *w*).

^{3}

Note that *Z*_{yz} = −*Z*_{zy} as a consequence of the antisymmetry of the vorticity flux tensor *Z*_{ij} (Haynes and McIntyre 1987).