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 (Δx = Δy = Δz = 20 m). For this study, the obstacle height H(x, y) is specified following (7) and (8) of Epifanio and Rotunno (2005) with parameters h₀ = 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.

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Domain, topography, and initial conditions (at y = +3.84 km) for the present numerical solutions. Isolines of potential temperature θ in intervals of 1 K starting with 289 K (green contour lines, with the 293 K contour in red) and vectors representing the initial velocity (u, υ, w) = (5, 0, 0) m s⁻¹ are shown. The domain extends 15.36 km in the along-flow (x), 7.68 km in the spanwise (y), and 3.84 km in the vertical (z) direction.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0142.1

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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 m² s⁻¹), one can estimate the Reynolds number (Uh₀/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.

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Isolines of the potential temperature θ along the flow through the obstacle centerline (y = 0) in intervals of 1 K starting with 289 K (green contour lines, with the 293 K contour in red) and vectors representing (u, w) for t = (a) 1, (b) 8, (c) 11, (d) 50, and (e) 60 min.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0142.1

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Vectors representing (u, υ) at z = 0.4 km for t = (a) 1 and (b) 11 min along with vertical vorticity ζ (color shading) and isolines of potential temperature θ as in Fig. 3.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0142.1

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Figures 5 and 6 show a windowed-in view of the centerline x–z cross sections of θ, flow vectors, and spanwise vorticity η = ∂_zu − ∂_xw (top panels) along with x–y sections of θ, flow vectors, and vertical vorticity ζ = ∂_xυ − ∂_yu at z = 0.4 km.

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(top) Windowed-in view of centerline θ and flow vectors as in Fig. 3 with the addition of the spanwise vorticity η in color shading and (bottom) θ (green contours), flow vectors, and vertical vorticity ζ at z = 0.4 km for t = (a) 11 and (b) 17 min. The horizontal line in the top panels shows the z = 0.4 km level displayed in the bottom panels.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0142.1

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As in Fig. 5, but for t = (a) 23 and (b) 29 min.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0142.1

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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.

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(a) As in Fig. 5, but for the 60–120-min average and (b) the tke averaged over the same period; note that the horizontal section is shown at z = 0.2 km. The square in bottom panel of (a) shows the analysis region for Fig. 8.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0142.1

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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).

a. Vorticity

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.

The flow in the square shown in the bottom panel of Fig. 7a is chosen for detailed study since $\overline{\zeta }=0$ on its upstream side at 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

${\displaystyle {\int }_{y=-1.5}^{y=0}{Z_{zx}|}_{x=0.2}\hspace{0.17em}dy}=-{\displaystyle {\int }_{x=-1.3}^{x=0.2}{Z_{zy}|}_{y=0}\hspace{0.17em}dx}$(3)

since the fluxes are zero on the other two sides and the limits are given in kilometers. Numerical evaluation of the rhs yields a value of 28.65 m² s⁻² while the integral on the lhs yields 32.86 m² s⁻². Recognizing the noisy fields near the right side of the square, we consider a square with a variable right-side position $\widehat{x}$ so that integration of (2) over the area of the square gives

${\displaystyle {\int }_{y=-1.5}^{y=0}{Z_{zx}|}_{x=\widehat{x}}\hspace{0.17em}dy}=-{\displaystyle {\int }_{x=-1.3}^{x=\widehat{x}}{Z_{zy}|}_{y=0}\hspace{0.17em}dx};$(4)

the average of (4) from $\widehat{x}=-0.2$ to 0.2 gives

${\displaystyle {\int }_{y=-1.5}^{y=0}\langle Z_{zx}\rangle \hspace{0.17em}dy}=-\langle {\displaystyle {\int }_{x=-1.3}^{x=\widehat{x}}{Z_{zy}|}_{y=0}\hspace{0.17em}dx}\rangle$(5)

where the angle brackets indicate the average. With this adjustment, the integral on the lhs of (5) then yields a value of 28.35 m² s⁻² while the integral on the rhs is 29.21 m² s⁻² which indicates that the equality (5) holds to an accuracy of ~3%.

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Vertical vorticity $\overline{\zeta }$ (shading) and vertical vorticity flux vectors (Z_zx, Z_zy) at z = 0.4 km in the analysis square shown in Fig. 7a based on (a) the exact form (2) and (b) the approximate form $\left(\overline{u}\hspace{0.17em}\overline{\zeta },-\overline{w}\hspace{0.17em}\overline{\eta }\right)$.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0142.1

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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 m² s⁻² while the lhs yields 30.25 m² s⁻² which indicates that (7) holds to an accuracy of ~10%. From the foregoing analysis we conclude the $\overline{\zeta }$ exiting on the downstream side of the square is almost entirely due to the cross-stream flux at y = 0 approximated by $-\overline{w}\overline{\eta }$.

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(a) Cross-stream flux of vertical vorticity Z_zy|_y=0 and (b) averaged downstream flux of vertical vorticity ⟨Z_zx⟩ using the exact form (black lines) and the approximate form (green lines) corresponding to the integrands in (7).

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0142.1

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From Part I (section 2c) the equation for $\overline{\eta }$ in statistically steady flow can by be written as

${\partial }_x{\overline{Z}}_{yx}+{\partial }_z{\overline{Z}}_{yz}=-{\partial }_x\overline{b}-\left({\partial }_x{Z}_{yx}^{\prime }+{\partial }_z{Z}_{yz}^{\prime }\right),$(8)

where b = gθ/θ₀₀, $\left(Z_{yx},\hspace{0.17em}{Z}_{yz}\right)=\left(\overline{u\hspace{0.17em}\eta }-\overline{\upsilon \hspace{0.17em}\xi }+\nu {\nabla }^2\overline{w},\overline{w\hspace{0.17em}\eta }-\overline{\upsilon \hspace{0.17em}\zeta }-\nu {\nabla }^2\overline{u}\right)$ and viscous effects are small as shown below.³ Consistent with (6), the eddy-correlation term is equivalent to ∂_zF_x − ∂_xF_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 $\overline{\eta }$ reaches a minimum (marked by η_m) in Fig. 10, the flow is essentially laminar, hence (8) may be written along y = 0 as

$u{\partial }_x\eta +w{\partial }_z\eta =-{\partial }_{x}b.$(9)

Equation (9) can be expressed as u_sdη/ds = −∂_xb where s is the distance and u_s is the speed along a streamline; this equation may be formally integrated to give $\eta ={\displaystyle {\int }_s-{\partial }_{x}bds/u_s}$, along a streamline from an upstream point where η = 0. Choosing the streamline that goes down through the zone where −∂_xb < −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 ≃ −∂_xbds/u_s ≃ −0.05 s⁻¹, which is close to the values shown in Fig. 7a.

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Budget for $\overline{\eta }$ at y = 0 with the time-mean transport term, $-{\partial }_x{\overline{Z}}_{yx}-{\partial }_z{\overline{Z}}_{yz}$, in blue-to-red color shading; the eddy-flux term, $-{\partial }_x{Z}_{yx}^{\prime }-{\partial }_z{Z}_{yz}^{\prime }$, in the color contours (−0.002, −0.001, and −0.005 in blue and 0.0005, 0.001, and 0.002 in red); and the baroclinic term, $-{\partial }_x\overline{b}$, indicated by the single dashed black contour line of −0.5 × 10⁻³ s⁻². The η_m on the leeside of the obstacle indicates the location of the negative minimum of $\overline{\eta }$. The red line corresponds to the location of the square shown in the bottom panel of Fig. 7a. The rectangle indicates the window for the x-averaged terms shown in Fig. 11.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0142.1

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As one can see in the top panel of Fig. 7a, the field of $\overline{\eta }$ becomes more diffuse with distance downstream; thus $\overline{w}\overline{\eta }$ is nearly zero at z = 0.4 km and therefore cannot cancel out the positive contributions from the downward transport of $\overline{\eta }<0$ in the integral on the rhs of (5). In the turbulent layer below z ≃ 0.4 km, Fig. 10 shows that the main balance in (8) is between mean and turbulence transport of $\overline{\eta }$. For clarity, Fig. 11 shows the x-averaged values of the terms in (8) which indicates that the time-mean downstream transport of negative $\overline{\eta }$ (<0) is balanced by turbulent diffusion (>0). A similar analysis of (6) downstream of the square shown in Fig. 7a (not shown) indicates a balance of time-mean downstream advection and turbulent diffusion of $\overline{\zeta }$.

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Budget for $\overline{\eta }$ at y = 0 averaged from x = −0.5 to x = 0.5 km as a function of z showing the transport due to the time-mean fields, $-{\partial }_x\left(\overline{u}\hspace{0.17em}\overline{\eta }-\overline{\upsilon }\overline{\xi }\right)-{\partial }_z\left(\overline{w}\hspace{0.17em}\overline{\eta }-\overline{\upsilon }\overline{\zeta }\right)$; the time-mean eddy term, $-{\partial }_x\left(\overline{u^{\prime }\hspace{0.17em}{\eta }^{\prime }}-\overline{{\upsilon }^{\prime }\hspace{0.17em}{\xi }^{\prime }}\right)-{\partial }_z\left(\overline{w^{\prime }\hspace{0.17em}{\eta }^{\prime }}-\overline{{\upsilon }^{\prime }\hspace{0.17em}{\zeta }^{\prime }}\right)$; the baroclinic term, $-{\partial }_x\overline{b}$; and the viscous term, $\nu \left({\partial }_{xx}+{\partial }_{zz}\right)\overline{\eta }$, where ν = 1 m² s⁻¹.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0142.1

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In summary, the analysis leading to (7) shows that the vertical vorticity $\overline{\zeta }$ of the lee vortices is directly related to the time-mean downward transport of negative spanwise vorticity $\overline{w}\overline{\eta }>0$ while the analysis of the flow using (9) and Fig. 10 show that the downward-transported negative $\overline{\eta }$ is baroclinically generated. Therefore the vertical vorticity of the lee vortices $\overline{\zeta }$ must originate through the inviscid, adiabatic process of baroclinic generation of spanwise vorticity $\overline{\eta }$ in the leeside descending stratified flow. Turbulent diffusion of $\overline{\eta }$ downstream is important as it reduces contributions from negative values of $\overline{w}\overline{\eta }$ that would, in the absence of turbulent diffusion, bring the integral on the rhs of (5) to zero, and therefore, the leeside vertical vorticity $\overline{\zeta }$ would not extend downstream. This conclusion is consistent with that from Rotunno et al. (1999, section 5) based on a simulation of lee vortices in a continuously stratified fluid with parameterized turbulent diffusion.

Before leaving this section, the conclusion from the analysis of (6) that the Reynolds stress terms are small allows a simple computation of the terms in the advective form of the vertical vorticity equation in terms of the time-mean fields, namely,

$\overline{\mathbf{u}}\cdot \nabla \zeta =\overline{\xi }{\partial }_x\overline{w}+\overline{\eta }{\partial }_y\overline{w}+\overline{\zeta }{\partial }_z\overline{w}$(10)

where the first two terms on the rhs are called the “tilting” terms and the last term on the rhs is called the “stretching” term. Figure 12 shows the tilting and stretching terms at 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.

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Tilting and stretching terms in the advective form of the vertical vorticity Eq. (10) at (a),(b) z = 0.4 km and (c),(d) z = 0.2 km. Contours are plotted in increments of 1 × 10⁻⁴ s⁻² (zero line not shown, negative values denoted by dashed lines) while the color shading shows the time-averaged vertical vorticity. The thick blue line in (c) and (d) shows the outline of the obstacle at z = 0.2 km.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0142.1

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b. Potential vorticity

With the present direct numerical solutions of the NSE, we are in position to make a connection between the foregoing analysis of the lee-vortex ω and PV. Following Epifanio and Qian (2008, section 5b), we look at the potential vorticity (1) defined as

$\overline{\text{PV}}=\overline{\mathit{\omega }}\cdot \nabla \overline{\theta }.$(11)

We note that the potential vorticity as defined in (11) is the quantity whose conservation equation results from the curl of the time-averaged momentum equation and the gradient of the time-averaged potential-temperature equation. Figure 13 shows $\overline{\text{PV}}$ at 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 $\overline{\text{PV}}$ associated with the lee vortices.

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$\overline{\text{PV}}$ as defined in (11) at z = 0.4 km along with the vectors $\left(\overline{u},\overline{\upsilon }\right)$ and the isoline of $\overline{\theta }=289\hspace{0.17em}\text{K}$ (green contour).

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0142.1

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The evolution equation for $\overline{\text{PV}}$ is

${\partial }_t\overline{\text{PV}}+\nabla \cdot \mathbf{J}=0$(12)

where the $\overline{\text{PV}}$-flux vector is defined as

$\mathbf{J}=\overline{\mathrm{u}}\overline{\text{PV}}+\nabla \overline{\theta }\times \mathbf{F}-\overline{\mathit{\omega }}\overline{Q}.$(13)

[Epifanio and Qian 2008, their Eq. (24)]; in this definition 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 $\overline{\zeta }$, the y component of J is a central element in understanding the origin of the $\overline{\text{PV}}$ dipole. For steady-state flow, (12) reduces to ∇ ⋅ J = 0 and therefore a cross stream flux J_y < 0 at y = 0 is required to maintain a $\overline{\text{PV}}$ dipole downstream with $\overline{\text{PV}}>0$ for y < 0. At y = 0, we expect by symmetry that $\overline{\upsilon }=0$ and the equation for $\overline{\theta }$ is

$\overline{u}{\partial }_x\overline{\theta }+\overline{w}{\partial }_z\overline{\theta }=\overline{Q}.$(14)

Under these conditions we have at y = 0

$J_y=\left(F_x-\overline{w}\overline{\eta }\right){\partial }_z\overline{\theta }-\left(F_z+\overline{u}\overline{\eta }\right){\partial }_x\overline{\theta }.$(15)

Figure 14a shows that 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, $J_y^F$, and $J_y^Q$. Although these two components both contribute to the magnitude of J_y, the structure $J_y^Q$ seems to best explain that of J_y. Figure 14 taken together with (15) suggests a close connection with the analysis of the $\overline{\zeta }$ from Fig. 9, which is the importance of time-mean transport of $\overline{\eta }$. In the following we explore this connection from a geometrical point of view.

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Cross-stream flux of $\overline{\text{PV}}$ at y = 0 for (a) J_y, (b) $J_y^F$ (the frictional part of J_y), and (c) $J_y^Q$ (the heat-transfer part of J_y). Vectors and isentropes are as in the top panel of Fig. 7a.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0142.1

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Figure 15 shows a series of vortex lines that go through the locations of minimum $\overline{\eta }$ at y = 0 (see the top panel of Fig. 7a); the color of the vortex line indicates the local value of $\overline{\theta }$. Vertical (x–z) cross sections of $\overline{\theta }$ (shaded) for z < 1.25 km in the planes at y = 0 and y = 1.9 km allow one to see the location and $\overline{\theta }$ of three of the vortex lines as they enter the display domain at 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 $\overline{\zeta }<0$) and then ascend for y < 0 (implying $\overline{\zeta }>0$). All four of these vortex lines display little variation of $\overline{\theta }$ along their lengths and thus, according to (16), $\overline{\text{PV}}\approx 0$.

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Vortex lines that go through the strip of minimum $\overline{\eta }$ at y = 0 shown in the top panel of Fig. 7a. Vertical (x–z) cross sections of $\overline{\theta }$ (shaded) for z < 1.25 km in the planes y = 0 and y = 1.9 km are shown with the color scale in the upper left. Color of the vortex line varies along their lengths according to the same color scale. These vortex lines have only small variations of $\overline{\theta }$ which, according to (16), indicates small values of $\overline{\text{PV}}$ along its length. The horizontal display domain is as in Fig. 13 (−3 ≤ x ≤ +1 km, −2 ≤ y ≤ +2 km).

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0142.1

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In contrast, Fig. 16 shows a vortex line that passes through the $\overline{\text{PV}}$ extrema of Fig. 13. In this case, $\overline{\theta }$ increases along the vortex line $\left(\overline{\text{PV}}>0\right)$ for y < 0 and decreases along the vortex line $\left(\overline{\text{PV}}<0\right)$ for y > 0. Note that, as with the vortex lines of Fig. 15, this vortex line passes close to location of minimum $\overline{\eta }$ at y = 0, which indicates the basic baroclinic source of $\overline{\eta }$ is common to vortex lines with and without significant $\overline{\text{PV}}$. Note also that the vortex line in this case forms a loop that connects the lower distribution of $\overline{\eta }<0$ to the upper distribution where $\overline{\eta }>0$ shown in the upper panel of Fig. 7a. Also shown for comparison is a vortex line from Fig. 15.

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As in Fig. 15, but for a vortex line that has a large variation of $\overline{\theta }$ along its length indicating large $\overline{\text{PV}}$ where the variations are largest according to (16). Also included for comparison is a vortex line from Fig. 15.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0142.1

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Figure 17 shows the vortex lines from Fig. 16 together with the isosurface $\overline{\theta }=289\hspace{0.17em}\text{K}$. Figure 17a suggests the following interpretation. The baroclinically produced $\overline{\eta }<0$, which forms the lower portion of these vortex lines, induces the spanwise rotary motion that continually transports the isosurface of $\overline{\theta }$ back toward the obstacle, thus wrapping the isosurface around the lower portion of the vortex line and thereby producing a $\overline{\theta }$ gradient along it. The vortex line thus descends through the $\overline{\theta }$ surface for y > 0 and ascends through it for y < 0 (Fig. 17b) which, according to (16), implies a dipole in $\overline{\text{PV}}$.

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Vortex lines from Fig. 16 together with the isosurface $\overline{\theta }=289\hspace{0.17em}\text{K}$ from a downstream viewpoint looking (a) down at an angle to the flow and (b) from a lower level, more directly into the flow.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0142.1

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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.

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Vortex lines that go through the incipient wave-breaking zone at y = 0 at t = 8 min (Fig. 3b). Vertical (x–z) cross sections of θ (shaded) for z < 1.25 km in the planes y = 0 and y = 1.9 km are shown with the color scale in the lower left. Color of the vortex line varies along their lengths according to the same color scale. These vortex lines have only small variations of θ, which, according to (16), indicates small values of PV along its length. The horizontal display domain is as in Fig. 15.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0142.1

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Vortex lines from Fig. 18 together with the isosurface θ = 289 K from a viewpoint looking down and across flow.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0142.1

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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.

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Evolution of an isentropic surface with vortex lines lying in the surface for a flow with the stratified layer below the obstacle top. Colors on the surface indicate the perturbation temperatures (blues: negative; oranges: positive). In this case the vortex lines of the evolving lee vortices loop over the reversed flow and terminate at the lower surface. Figure 6 in Epifanio (2014) modified from Epifanio and Rotunno (2005). Arrows added to indicate the sense of the vortex lines.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0142.1

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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)].