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Axisymmetric Tornado Simulations at High Reynolds Number

Richard Rotunno National Center for Atmospheric Research, Boulder, Colorado

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George H. Bryan National Center for Atmospheric Research, Boulder, Colorado

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David S. Nolan Rosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, Florida

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Nathan A. Dahl Rosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, Florida

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Print Publication:
01 Oct 2016
DOI:
https://doi.org/10.1175/JAS-D-16-0038.1
Page(s):
3843–3854
Full access

Abstract

This study is the first in a series that investigates the effects of turbulence in the boundary layer of a tornado vortex. In this part, axisymmetric simulations with constant viscosity are used to explore the relationships between vortex structure, intensity, and unsteadiness as functions of diffusion (measured by a Reynolds number Rer) and rotation (measured by a swirl ratio Sr). A deep upper-level damping zone is used to prevent upper-level disturbances from affecting the low-level vortex. The damping zone is most effective when it overlaps with the specified convective forcing, causing a reduction to the effective convective velocity scale We. With this damping in place, the tornado-vortex boundary layer shows no sign of unsteadiness for a wide range of parameters, suggesting that turbulence in the tornado boundary layer is inherently a three-dimensional phenomenon. For high Rer, the most intense vortices have maximum mean tangential winds well in excess of We, and maximum mean vertical velocity exceeds 3 times We. In parameter space, the most intense vortices fall along a line that follows , in agreement with previous analytical predictions by Fiedler and Rotunno. These results are used to inform the design of three-dimensional large-eddy simulations in subsequent papers.

Denotes Open Access content.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: Richard Rotunno, National Center for Atmospheric Research, P. O. Box 3000, Boulder, CO 80307. E-mail: rotunno@ucar.edu

Abstract

This study is the first in a series that investigates the effects of turbulence in the boundary layer of a tornado vortex. In this part, axisymmetric simulations with constant viscosity are used to explore the relationships between vortex structure, intensity, and unsteadiness as functions of diffusion (measured by a Reynolds number Rer) and rotation (measured by a swirl ratio Sr). A deep upper-level damping zone is used to prevent upper-level disturbances from affecting the low-level vortex. The damping zone is most effective when it overlaps with the specified convective forcing, causing a reduction to the effective convective velocity scale We. With this damping in place, the tornado-vortex boundary layer shows no sign of unsteadiness for a wide range of parameters, suggesting that turbulence in the tornado boundary layer is inherently a three-dimensional phenomenon. For high Rer, the most intense vortices have maximum mean tangential winds well in excess of We, and maximum mean vertical velocity exceeds 3 times We. In parameter space, the most intense vortices fall along a line that follows , in agreement with previous analytical predictions by Fiedler and Rotunno. These results are used to inform the design of three-dimensional large-eddy simulations in subsequent papers.

Denotes Open Access content.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: Richard Rotunno, National Center for Atmospheric Research, P. O. Box 3000, Boulder, CO 80307. E-mail: rotunno@ucar.edu
Keywords: Circulation/ Dynamics; Dynamics; Tornadogenesis; Turbulence

1. Introduction

The recent review of tornado dynamics by Rotunno (2013, hereafter R13) put heavy emphasis on the low-Reynolds-number, mostly laminar flow seen in laboratory experiments. Figure 1 illustrates the basic model: The flow at some distance from the ground (the outer flow) is in rotation about a vertical axis; at the lower end of the vertical axis is the “end-wall boundary layer,” over which the outer flow comes to satisfy the no-slip condition on the lower bounding surface; the reduction of centrifugal force in the boundary layer allows the radial pressure gradient force to accelerate boundary layer fluid toward the center, whereupon it turns to the vertical and achieves the largest vertical and tangential wind speeds in the “end-wall vortex”; the latter transitions through a “vortex breakdown” to a more slowly rotating, “two-celled vortex” (downdraft at the center). Turbulent flow occurs downstream (upward) of the vortex breakdown but not in the end-wall boundary layer. As the end-wall boundary layer directly influences end-wall-vortex intensity, it is important to know the conditions under which the end-wall boundary layer may become turbulent. This paper is the first in a series aimed at understanding the nature of turbulence in the end-wall boundary layer and how that turbulence affects vortex intensity.

Fig. 1.
Fig. 1.

A schematic diagram of the streamsurfaces in the radial–vertical plane of a laboratory vortex. The various parts of the vortex are labeled and described in the text. A photograph of a laboratory vortex (Pauley and Snow 1988) is shown in the background.

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-16-0038.1

According to the review in R13, the Reynolds number for laboratory experiments and numerical simulations thereof is O(104), which is much lower than that in natural flows, which may be O(109). Fiedler and Garfield (2010) carried out axisymmetric tornado simulations for atmospherically relevant Reynolds numbers with several different turbulence parameterizations, and, in each case, the parameterizations indicated small turbulence intensities in the end-wall boundary layer (see their Fig. 8). Lewellen et al. (2000), using large-eddy simulations (LESs) (which, in principle, attempt to simulate flow at infinite Reynolds number), found structures similar to that schematized in Fig. 1; their Fig. 5 and the analyses in their Figs. 6a, 12a, and 15a show little evidence of resolved turbulent flow in the end-wall boundary layer. Although there is parameterized subgrid-scale turbulence in LES, one must rely on its ability to represent faithfully the effects of turbulence. However, in the absence of direct turbulence measurements from real tornadoes, there is no way to determine the efficacy of such parameterizations. In the sequel to this work, we report on LES of tornado-like vortices with special attention to the requirements of resolving turbulence in the end-wall boundary layer. In this first part, we describe the numerical setup for constant-viscosity, axisymmetric simulations, which were used to help design our LES experiments. In the course of setting up the axisymmetric simulations, we took advantage of the opportunity to explore much higher Reynolds numbers than previously achieved in such numerical simulations to investigate the possibility of axisymmetric instability of the end-wall boundary layer.

As in the numerical experiments described in R13, the present experiments are also carried out in a closed domain. Numerical simulations of tornado-like vortices in a closed domain have the advantage that boundary conditions are unambiguous and put definite constraints on the solution. On the other hand, one desires the domain size to not significantly influence the simulated vortex dynamics. Thus, one must use a domain large enough for artificially enhanced viscous effects to damp disturbances originating near the vortex top (which is of little physical interest) to prevent them from propagating downward and/or recirculating to the region of interest. In the course of the present investigation, it became clear that simulations at higher Reynolds numbers than used previously would require even more damping for a reasonable domain size. We find the required damping to be a significant drain on the prescribed forcing that should be accounted for when estimating the thermodynamic speed limit (TSL; Fiedler and Rotunno 1986) on vortex intensity. When this is taken into account, the effective TSL is much lower and easily exceeded by the present simulated vortices.

For ease of comparison with atmospheric observations, spatial scales will be given in dimensional terms. However, the present experiments are guided by previous studies, pointing to the importance of the nondimensional input parameters characterizing the imposed rotation, updraft forcing, and viscous effects: namely, a swirl ratio Sr and the Reynolds number Rer. The present series of numerical experiments allow the construction of a vortex-type regime diagram in (Sr, Rer) extending over a large range of Rer. (The subscript r refers to use of the radial length scale of the updraft forcing in the definitions.) These experiments cover a range of Rer that is nearly two orders of magnitude greater than in previous studies. This extended range in Rer, together with a large number of simulations with fine increments in Sr, adds further support for the theoretical relation for the optimal state, [Eq. (10) of Fiedler (2009)].

The plan of this paper is to first describe in section 2 the physical problem, put it in its meteorological context, and consider the necessary trade-offs involved in its numerical solution. The governing equations and simulation design are described in section 3; sensitivity tests demonstrating the need for and effects of the damping layer are described in the appendix. Examples of the numerical solutions are described in section 4 and summarized in a vortex-type regime diagram for a wide range of the control parameters (Sr, Rer). A summary is given in section 5.

2. Physical problem

Figure 2 shows a schematic diagram of the physical problem following the basic design of Fiedler (1995). The entire domain rotates at the rate Ω, with nonslip, impermeable walls at the bottom and top boundaries and an impermeable free-slip wall at r = R. The solution for the three velocity components in the rotating reference frame and in the cylindrical coordinates (r, θ, z) is (u, υ, w) = (0, 0, 0) in the absence of forcing. The prescribed forcing F(r, z) is placed in the vertical momentum equation as a surrogate for the buoyancy and/or dynamic pressure gradient forcing in a supercell thunderstorm (Klemp 1987), while the domain rotation is intended to represent the rotation of the supercell. With F(r, z) > 0, an in–up–out circulation is created, which, in turn, transports angular momentum inwards below the forcing maximum and locally intensifies the tangential velocity υ. A boundary layer forms at the bottom and top boundaries to bring the fluid into zero motion relative to the rotating domain.

Fig. 2.
Fig. 2.

Definition of the physical problem for the numerical simulations. The small red square is the 1 km × 1 km display domain used in Figs. 6 and 9.

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-16-0038.1

The conceptual model embodied in Fig. 2 is that the in–up meridional flow brings angular momentum inwards in the lower portion of the domain in analogy to the low-level flow (below cloud base) in a rotating thunderstorm. The flow in the upper and outer portions of the domain is, however, a much poorer analog for the complex processes occurring in a real thunderstorm, as the actual up–out flow is in cloud, subsequently exits to a stratified atmosphere, and does not return to the low-level inflow during the lifetime of the thunderstorm. Hence, a modeling device must be used to make sure that disturbances near the domain top Z do not make their way back to the simulated vortex (near the origin). In Fiedler (1995), the fluid viscosity was enhanced near the domain top, which required resolution of a topside boundary layer. In the present study, we choose to use a linear relaxation in time (with time constant τ) of the flow back to its unforced solution above the height zd (Fig. 2).

3. Governing equations and numerical setup

a. Governing equations

The governing equations for a constant-density, effectively incompressible fluid in the rotating domain reference frame are
e1a
e1b
e1c
e1d
where ϕ ≡ p/ρ, p is the pressure, ρ is the (constant) density, ν is the kinematic viscosity, and cs = 300 m s−1 is the speed of sound in air. Although maximum simulated wind speeds Vmax ≃ 100 m s−1, the flow is effectively solenoidal (i.e., ∇ ⋅ u = 0) since (Vmax/cs)2 ≪ 1; the assumption of solenoidal u is used in the formulation of the diffusion terms (e.g., Batchelor 1967, p. 604).

The equations above describe the motions of a fluid that is compressible and for which density is assumed to be constant. This equation set was chosen for two main reasons. First, we are interested primarily in flow in the lowest ~1 km AGL for which the constant-density assumption is valid. Second, this set of equations allows us to use existing numerical techniques in the modeling framework used for this study—Cloud Model 1 (CM1) and, in particular, the split-explicit time integration technique for compressible flows (e.g., Wicker and Skamarock 2002)—as well as existing parallelization methods for distributed-memory supercomputers for three-dimensional simulations that will be reported in future papers. In addition, there are several ancillary benefits, such as a simpler equation set for analysis purposes and a weaker upper-level response to the updraft forcing that does not need to be damped as aggressively.

The last terms on the right-hand sides of Eqs. (1a)–(1c) are the linear damping terms in which the coefficient α(z) regulates the distance over which the full damping with time constant τ is achieved. The damping function
e2
where 0 ≤ zd ≤ Z defines the damping layer.
Finally, the updraft forcing is defined following Nolan (2005) as
e3
where
e4
The forcing function F(r, z) is prescribed such that the maximum Fmax occurs at (r, z) = (0, zb), which defines the center of an elliptically shaped region (vertical and horizontal axes lz and lr, respectively), over which the forcing goes to zero. The basic velocity scale W is given by the vertical integral
e5
With Eq. (3) substituted into Eq. (5), the velocity scale
e6

The boundary conditions are u = 0 on the upper and lower bounding surfaces, while the normal velocity and stress components are zero at r = R.

Altogether there are 10 input parameters, Ω, W, lr, lz, zb, R, Z, ν, τ, and zd, and, by Buckingham’s Π theorem, 8 nondimensional parameters that determine the solution. With some hindsight, we choose the following:
e7
The first parameter is a swirl ratio Sr, and the second is the Reynolds number Rer, which respectively represent system rotation and diffusive effects (the subscript r signifies that we use lr for the length scale in the swirl ratio and Reynolds number instead of Z, as used in previous studies); these are the two principle solution control parameters to be varied in the present work. The third and fourth parameters characterize the geometry of the forcing and will be fixed in rough analogy to the forcing of vertical acceleration in a supercell thunderstorm. The fifth parameter measures the forcing horizontal scale against domain width, and small values will be used to ensure there are no significant domain-size effects. The sixth parameter measures forcing location against domain depth. Ideally, one would like this parameter to be small; however, computational expense militates against it. Thus, the seventh and eighth parameters are chosen to damp disturbances before they can reflect from the domain top and/or recirculate to the lower inflow layer.

b. Numerical-solution method

The prognostic Eqs. (1a)–(1d) are integrated in time using a third-order Runge–Kutta scheme, using split-explicit integration for the acoustic modes following Wicker and Skamarock (2002). To improve the stability of the split-explicit time integration method, a weak three-dimensional divergence damper on the acoustic time steps is included following Skamarock and Klemp (1992).

The radial grid spacing is 5 m for r < 1 km and increases gradually to 495 m between r = 1 and r = 20 km. For most simulations, the vertical grid spacing is 5 m for z < 1 km and increases gradually to 495 m between z = 1 km and z = 15 km. An exception is that most simulations with Rer ≥ 320 000 were run with vertical grid spacing of 2.5 m for z < 0.5 km, which better resolves the shallow boundary layers for these cases. The time step varies throughout each simulation to maintain numerical stability, taking into account both advective and diffusive processes.

c. Parameter settings

The dimensional parameters settings are given in Table 1; the fixed values are chosen to conform to the physical considerations in section 2 (details on the damping layer are given in the appendix); these values thus determine six of the eight nondimensional parameters given in Table 2. The variable dimensional parameters Ω and ν are chosen to explore the range of solutions in the nondimensional parameter space (Sr, Rer). With the fixed dimensional values of W = 80 m s−1 and lr = 3000 m, we have therefore Ω = Sr × 0.026 s−1 and ν = × 2.4 × 105 m2 s−1.

Table 1.

Parameter settings for the domain shown in Fig. 2.

Table 1.
Table 2.

Nondimensional parameters based on the dimensional parameters in Table 1.

Table 2.

4. Results

Figures 3 and 4 contain matrices in (Sr, Rer) showing the respective maxima of the tangential and vertical velocities averaged from 5 × 104 to 6 × 104 s in the lowest 1 km; unless otherwise mentioned, the velocities reported herein are nondimensionalized by the effective forcing value We = 66 m s−1 (see the appendix). We note that the present experimental range of Rer is much greater than in previous studies. Specifically, the highest Reynolds number, Reh = Wh/ν, where h is the height of the domain, used in Fiedler (2009) is 40 000. Estimating from lr/h = from Eq. (1) of Fiedler (1998), we find that the highest Rer = Reh × lr/h ≃ 12 800 in Fiedler (2009). Comparison with the highest value of Rer = 640 000 used here indicates a factor of 50 increase in the present experiments.

Fig. 3.
Fig. 3.

Solution matrix in (Sr, Rer) for the maximum tangential velocity divided by We; contour lines are overlaid in intervals of 0.2. The black solid line in this figure and the following two figures shows dependence.

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-16-0038.1

Fig. 4.
Fig. 4.

Solution matrix in (Sr, Rer) for the maximum vertical velocity divided by We; contour lines are overlaid in intervals of 0.5.

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-16-0038.1

Figure 5 shows the pressure minimum (nondimensionalized by ) averaged over the same time interval. Focusing first on the latter, there is a clearly an optimal combination of Sr and Rer that produces the greatest pressure drop; these solutions are the optimal solutions that correspond to the vortex shown in Fig. 1, in which the pressure minimum occurs above the lower surface in the end-wall vortex (Church and Snow 1985). These optimal solutions tend to occur along a diagonal line in the (Sr, Rer) matrix; solutions below this line in the matrix are single-cell solutions, while those above the line are predominantly two-celled solutions. Figure 3 indicates that the optimal solutions can exceed the TSL, while Fig. 4 shows that the vertical velocity maxima are about twice the corresponding tangential velocity maxima, consistent with the theory of Fiedler and Rotunno (1986).

Fig. 5.
Fig. 5.

Solution matrix in (Sr, Rer) for the minimum pressure divided by ; contour lines are overlaid in intervals of 1.0.

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-16-0038.1

Figure 6 depicts the flow (display domain indicated in Fig. 2) for several solutions that span across the optimal solutions indicated in Figs. 3–5. These solutions generally conform to the behavior expected from previous work. As reviewed in R13, the boundary layer thickness δ is proportional to , which can be expressed as δ/lr ∝ in the present notation; scanning Fig. 6 across (constant Sr, varying Rer) or vertically (constant Rer, varying Sr) generally shows this expected behavior of the vortex boundary layer. Also consistent with the theory reviewed in R13, conservation of angular momentum applied to the two-celled vortex gives , where rc is the radius and υc the tangential velocity of the two-celled vortex; with υc ≃ W based on energetics, one expects therefore that rc/lr ∝ Sr; this too is generally consistent with the behavior seen by scanning Fig. 6 vertically (constant Rer, varying Sr). The optimal solution at the middle of Fig. 6 is the result of the solution finding the appropriate relation between the radius of the end-wall vortex (∝ δ/lr ∝ ) and that of the two-celled vortex (∝Sr): that is, by finding the combination in Sr–Rer space where
e8
[Fiedler (2009), his Eq. (10)]. As the (Sr, Rer) matrices are constructed on a log–log scale, a power law is represented by a straight line; the line drawn in Figs. 3–5 corresponds to a −⅓ dependence in basic agreement with Eq. (8). Note that the constant of proportionality implied in Eq. (8) is not universal and is expected to change for parameters settings different from those given in Table 2. For example, changes in domain size or upper-level damping could change the wind speeds and pressures shown in Figs. 3–5, although we expect the relation given by Eq. (8) to hold true.
Fig. 6.
Fig. 6.

Selected solutions showing the time-averaged tangential velocity divided by We (red shades) and radial–vertical velocity vectors. For clarity, the radial velocity component has been magnified by a factor of 2.

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-16-0038.1

To obtain a more refined estimate for the Sr = Sr(Rer) that produces the optimal vortex, additional simulations were conducted holding Rer fixed but with finer intervals of Sr than were used in Figs. 3–5. An example is shown for Rer = 640 000 in Fig. 7. From a series of such figures (not shown), the minimum value of pressure was used to define the optimal vortex. Overall results are shown in shown in Fig. 8; the agreement of the data with Eq. (8) adds further confidence in this theoretical estimate.

Fig. 7.
Fig. 7.

Results for Rer = 640 000: (a) maximum tangential velocity divided by We; (b) maximum vertical velocity divided by We; (c) minimum pressure divided by .

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-16-0038.1

Fig. 8.
Fig. 8.

A refined estimate of the validity of optimal vortex criterion [Eq. (8)]. The red crosses indicate ϕmin(Sr, Rer) from the numerical solutions, and the blue line is a reference line based on Eq. (8).

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-16-0038.1

In earlier studies, Nolan and Farrell (1999) and Nolan (2005) claimed that the optimal configuration should follow along lines of , which would appear as a one-to-one diagonal line on Figs. 3–5. They argued that vortex structure was largely controlled by the boundary layer, which the scaling analysis shown in Nolan (2005) is controlled by . While their numerical results seemed to support this claim, their simulations were confined mostly to the range 0.02 < Sr < 0.1 and 400 < Rer < 1600. In fact, some of the contours on the left (low Rer) sides of Figs. 3–5 appear to be bending upward, suggesting some agreement in this range. The vastly higher Reynolds numbers used in the present simulations find much better agreement with the analytical predictions of Fiedler (2009) and also produce sustained wind speeds well above the convective velocity scale.

A feature of primary importance to the present work and its sequel is the effects of turbulence. The present axisymmetric model is, of course, incapable of simulating turbulent flow; however, axisymmetric-solution unsteadiness is an indication of an axisymmetric instability that would likely lead to three-dimensional turbulence in an LES context. Figure 9 shows the standard deviation away from the time-averaged tangential velocity for three of the cases shown in Fig. 6, corresponding to the two-celled and optimal solutions (the two single-cell solutions are steady). It is clear that the vortex column is unsteady; however, there is no indication of unsteadiness in the end-wall boundary layer. Further tests with a fourfold reduction of vertical grid size (not shown) confirm the latter conclusion. The axisymmetric and three-dimensional instabilities associated with vortex breakdown and the two-celled vortex have been documented in the literature [most recently by Nolan (2012)]; however, Fig. 9 suggests the absence of an axisymmetric instability of the end-wall vortex.

Fig. 9.
Fig. 9.

Standard deviation of the tangential velocity divided by We, corresponding to three of the cases shown in Fig. 6. The cases with Rer = 160 000, Sr = 0.0025 and Rer = 20 000, Sr = 0.005 were essentially steady with zero standard deviation.

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-16-0038.1

The present results suggest that turbulence in the end-wall boundary layer of actual tornadoes must originate through some combination of three-dimensional instabilities and flow separation from surface roughness elements. We expect the effects of the consequent turbulent diffusion of momentum on the end-wall boundary layer to conform qualitatively to the present case of laminar diffusion over a smooth surface. However, for quantitative estimates, some other approach is required. In the following companion papers, the focus will be on investigating the effects on mean vortex intensity of three-dimensional turbulence over rough surfaces in the end-wall boundary layer using LES.

5. Conclusions

The present study of axisymmetric tornado simulations has established the basic model rationale and numerical setup for our companion studies using the technique of large-eddy simulation (LES), in which the effects of three-dimensional turbulence can be explicitly calculated. Working within the closed-domain design of Fiedler (1995), we find for simulations with much reduced physical diffusion that an enhanced upper-level damping is generally required to prevent spurious reflections and/or recycling of disturbances from affecting the solutions in the region of physical interest. This damping, when taken into account, lowers the estimate for the thermodynamic speed limit (TSL; Fiedler and Rotunno 1986) in the simulations, making the degree to which the maximum wind exceeds the TSL (Fig. 3) under the “optimal” condition [Eq. (8)] all the more impressive. The optimal condition [Eq. (10) of Fiedler (2009)] is validated here over a range of Reynolds numbers that is almost two orders of magnitude greater than previously demonstrated.

With respect to our companion studies, the most important result is that, even with Reynolds numbers O(106), there is no indication of axisymmetric instability in the vortex boundary layer in the present solutions. The implication is that instability and turbulence in the high-Reynolds-number vortex boundary layer must arise through three-dimensional effects. Currently, these effects are totally or largely parameterized even in LES-type studies (Lewellen et al. 2000). The authors are unaware of any practical way to evaluate the efficacy of such parameterizations other than with an LES model capable of resolving the large eddies in the vortex boundary layer. The latter is the subject of our following companion papers.

Acknowledgments

D. Nolan and N. Dahl were supported by the NSF through Grant AGS-1265899. The authors acknowledge high-performance computing support from Yellowstone (ark:/85065/d7wd3xhc), project UMIA0009, provided by NCAR’s Computational and Information Systems Laboratory.

APPENDIX

The Damping Layer

As discussed in Fiedler (1995), given the necessarily finite numerical-model domain, effects of wave reflection from the upper and/or outer boundaries should be controlled through enhanced dissipation. Within the current model setup, in which grid spacing is relatively small and Reynolds number is relatively high compared to recent studies, the most convenient method to achieve this outcome was to use the linear damping terms in Eqs. (1a)–(1c). After experimenting with several configurations, we decided to overlap the updraft forcing and damping layer, as illustrated in Fig. 2, which acts to draw eddies up into the damping layer. A consequence of this configuration on the effective forcing velocity is discussed below.

To demonstrate the problem with insufficient upper-level dissipation, a simulation without the upper-level damper is shown in Figs. A1a–c. In this case, Rer = 10 000 and Sr = 0.01. A low-angular-momentum “eddy” is triggered along the upper boundary by the initial updraft forcing, which then propagates along the outer boundary, and later the lower boundary. Although not shown here, there are also eddies that can propagate up the main updraft, reflect off the upper boundary, and propagate downward into the area of interest near the surface. A Hovmöller plot at z = 500 m (Fig. A1d) shows highly unsteady behavior in this case. In contrast, when the upper-level damper is used, the aforementioned eddies do not propagate into the lower-left corner of the domain, and the resulting flow is nearly steady (Fig. A1e).

Fig. A1.
Fig. A1.

(top) Tangential velocity from a simulation without an upper-level damper at indicated times. The contour interval is 2 m s−1, the zero contour is excluded, and negative values are dashed. (bottom) Hovmöller diagrams of tangential velocity, normalized by We, at 500 m AGL from (d) a simulation without an upper-level damper and (e) a simulation with an upper-level damper.

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-16-0038.1

With the present damping layer [or with enhanced viscosity near the upper boundary used by Fiedler (1995)], energy is removed from the flow. To get a quantitative estimate of this effect, Fig. A2 shows the dimensional vertical velocity in the Ω = 0 case, both with and without the upper damping layer. In the case without the damping layer, in which the upper boundary was placed at 25 km AGL to minimize its impact on the flow, the vertical velocity reaches a peak value of 80 m s−1 (black), precisely the value calculated from Eqs. (5) or (6) (Table 1). However with the upper-layer damper, and our nominal domain depth of 15 km, the peak vertical velocity ≃ 66 m s−1 (red). This latter velocity is the effective driving velocity for the tornado-like vortex solutions found here. Hence, solution velocities are reported herein nondimensionalized by the effective forcing velocity We = 66 m s−1.

Fig. A2.
Fig. A2.

Nonrotating solution for vertical velocity at r = 0 with (red) and without (black) upper-layer damping.

Citation: Journal of the Atmospheric Sciences 73, 10; 10.1175/JAS-D-16-0038.1

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  • Rotunno, R., 2013: The fluid dynamics of tornadoes. Annu. Rev. Fluid Mech., 45, 59–84, doi:10.1146/annurev-fluid-011212-140639.

  • Skamarock, W. C., and J. B. Klemp, 1992: The stability of time-split numerical methods for the hydrostatic and the nonhydrostatic elastic equations. Mon. Wea. Rev., 120, 2109–2127, doi:10.1175/1520-0493(1992)120<2109:TSOTSN>2.0.CO;2.

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  • Wicker, L. J., and W. C. Skamarock, 2002: Time-splitting methods for elastic models using forward time schemes. Mon. Wea. Rev., 130, 2088–2097, doi:10.1175/1520-0493(2002)130<2088:TSMFEM>2.0.CO;2.

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  • Rotunno, R., 2013: The fluid dynamics of tornadoes. Annu. Rev. Fluid Mech., 45, 59–84, doi:10.1146/annurev-fluid-011212-140639.

  • Skamarock, W. C., and J. B. Klemp, 1992: The stability of time-split numerical methods for the hydrostatic and the nonhydrostatic elastic equations. Mon. Wea. Rev., 120, 2109–2127, doi:10.1175/1520-0493(1992)120<2109:TSOTSN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Wicker, L. J., and W. C. Skamarock, 2002: Time-splitting methods for elastic models using forward time schemes. Mon. Wea. Rev., 130, 2088–2097, doi:10.1175/1520-0493(2002)130<2088:TSMFEM>2.0.CO;2.

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  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    A schematic diagram of the streamsurfaces in the radial–vertical plane of a laboratory vortex. The various parts of the vortex are labeled and described in the text. A photograph of a laboratory vortex (Pauley and Snow 1988) is shown in the background.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Definition of the physical problem for the numerical simulations. The small red square is the 1 km × 1 km display domain used in Figs. 6 and 9.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Solution matrix in (Sr, Rer) for the maximum tangential velocity divided by We; contour lines are overlaid in intervals of 0.2. The black solid line in this figure and the following two figures shows dependence.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Solution matrix in (Sr, Rer) for the maximum vertical velocity divided by We; contour lines are overlaid in intervals of 0.5.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Solution matrix in (Sr, Rer) for the minimum pressure divided by ; contour lines are overlaid in intervals of 1.0.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Selected solutions showing the time-averaged tangential velocity divided by We (red shades) and radial–vertical velocity vectors. For clarity, the radial velocity component has been magnified by a factor of 2.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    Results for Rer = 640 000: (a) maximum tangential velocity divided by We; (b) maximum vertical velocity divided by We; (c) minimum pressure divided by .

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    A refined estimate of the validity of optimal vortex criterion [Eq. (8)]. The red crosses indicate ϕmin(Sr, Rer) from the numerical solutions, and the blue line is a reference line based on Eq. (8).

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    Standard deviation of the tangential velocity divided by We, corresponding to three of the cases shown in Fig. 6. The cases with Rer = 160 000, Sr = 0.0025 and Rer = 20 000, Sr = 0.005 were essentially steady with zero standard deviation.

  • <sc>Fig</sc>. A1.
    View in gallery
    Fig. A1.

    (top) Tangential velocity from a simulation without an upper-level damper at indicated times. The contour interval is 2 m s−1, the zero contour is excluded, and negative values are dashed. (bottom) Hovmöller diagrams of tangential velocity, normalized by We, at 500 m AGL from (d) a simulation without an upper-level damper and (e) a simulation with an upper-level damper.

  • <sc>Fig</sc>. A2.
    View in gallery
    Fig. A2.

    Nonrotating solution for vertical velocity at r = 0 with (red) and without (black) upper-layer damping.

Fig. 1.

A schematic diagram of the streamsurfaces in the radial–vertical plane of a laboratory vortex. The various parts of the vortex are labeled and described in the text. A photograph of a laboratory vortex (Pauley and Snow 1988) is shown in the background.
Fig. 2.

Definition of the physical problem for the numerical simulations. The small red square is the 1 km × 1 km display domain used in Figs. 6 and 9.
Fig. 3.

Solution matrix in (Sr, Rer) for the maximum tangential velocity divided by We; contour lines are overlaid in intervals of 0.2. The black solid line in this figure and the following two figures shows dependence.
Fig. 4.

Solution matrix in (Sr, Rer) for the maximum vertical velocity divided by We; contour lines are overlaid in intervals of 0.5.
Fig. 5.

Solution matrix in (Sr, Rer) for the minimum pressure divided by ; contour lines are overlaid in intervals of 1.0.
Fig. 6.

Selected solutions showing the time-averaged tangential velocity divided by We (red shades) and radial–vertical velocity vectors. For clarity, the radial velocity component has been magnified by a factor of 2.
Fig. 7.

Results for Rer = 640 000: (a) maximum tangential velocity divided by We; (b) maximum vertical velocity divided by We; (c) minimum pressure divided by .
Fig. 8.

A refined estimate of the validity of optimal vortex criterion [Eq. (8)]. The red crosses indicate ϕmin(Sr, Rer) from the numerical solutions, and the blue line is a reference line based on Eq. (8).
Fig. 9.

Standard deviation of the tangential velocity divided by We, corresponding to three of the cases shown in Fig. 6. The cases with Rer = 160 000, Sr = 0.0025 and Rer = 20 000, Sr = 0.005 were essentially steady with zero standard deviation.
Fig. A1.

(top) Tangential velocity from a simulation without an upper-level damper at indicated times. The contour interval is 2 m s−1, the zero contour is excluded, and negative values are dashed. (bottom) Hovmöller diagrams of tangential velocity, normalized by We, at 500 m AGL from (d) a simulation without an upper-level damper and (e) a simulation with an upper-level damper.
Fig. A2.

Nonrotating solution for vertical velocity at r = 0 with (red) and without (black) upper-layer damping.
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