1. Introduction
A number of reviews of the dynamics of the tornado vortex are available in the literature (Bengtsson and Lighthill 1982; Davies-Jones 1986; Rotunno 1986; Snow 1987). The basis of our modeling studies was laid in the review by Lewellen (1993), which updated the earlier review by Lewellen (1976). Both reviews attempted to provide a critical assessment of the existing theoretical models to see how well they described the wind and pressure field in a tornado, to see what understanding they provided as to the parameters that govern the flow, and to clarify some of the essential questions that remain to be answered by future research.
Some essential elements of the tornado vortex are fairly well established. The flow spirals radially inward into a core flow that is basically a swirling, rising plume or jet, but often includes an inner downward jet along the axis. The radial flow toward the center of the vortex is greatly intensified in the surface layer. The vortex flow is driven by the energy generated from latent heat release in a thunderstorm with appreciable rotation. Davies-Jones (1995) has given a good review of what is known about the characteristics of a tornadic thunderstorm and how a tornado fits within such a storm. The tornado vortex allows a significant fraction of the available potential energy of the parent thunderstorm to be converted into wind kinetic energy very close to the surface where it can cause great damage. Local concentration of the vertical vorticity can lead to wind speeds exceeding 100 m s−1 within an annular region of approximately 100 m radius. There is even the opportunity for the unique interaction between the vortex and the surface in the central “corner flow region” (Lewellen 1976) to permit maximum low-level wind speeds to exceed that achievable by a simple exchange of storm potential energy for kinetic energy, that is, to exceed the so-called thermodynamic speed limit (Fiedler and Rotunno 1986).
One of the critical questions related to tornado vortex dynamics is the role of turbulence in determining the interaction with the surface. In particular, how is the location and magnitude of the maximum swirl velocity set? A high-resolution, fully three-dimensional, unsteady simulation of the interaction of a tornado vortex with the surface is one way of alleviating the problems associated with modeling turbulent transport within such a flow. This may be accomplished when it is possible to provide sufficient resolution to simulate the largest eddies responsible for the principle transport within the flow. It is still not possible to do a direct simulation of turbulence at full-scale Reynolds numbers, so some subgrid closure model is still required. However, the more resolution that can be incorporated into the simulation, the less relative importance the subgrid model assumes. In this paper we present a large-eddy simulation (LES) of turbulent transport in the corner flow region of the tornado vortex for one particular set of physical conditions and show that it is relatively independent of resolution.
The fundamental numerical difficulty is that a grid spacing of a few meters is required to capture some of the important turbulent eddies in the core of the tornado near the surface, while the boundary conditions of the vortex are set by the parent thunderstorm on a scale of tens of kilometers. Obviously, the total thunderstorm cannot be resolved on the fine scale required for the tornado eddies. Since we are concentrating on the interaction of the tornado vortex with the surface, we limit our domain to 1 km × 1 km × 2 km, with boundary conditions intended to represent conditions that might be obtained from an inner nest of a thunderstorm simulation such as that of Wicker and Wilhelmson (1995) or Grasso and Cotton (1995). This is large enough to encompass the strongest velocities in the tornado, and small enough that we can utilize a reasonably fine grid resolution to resolve the dominant turbulence in the corner flow region of the tornado. We use grids independently stretched in the horizontal and vertical directions to obtain smallest grid spacing of 1.5 m in the vertical and 2.5 m in the horizontal. The resolution is varied in section 6 to allow us to answer questions about how sensitive our results are to resolution.
The penalty we must pay for this small domain is that the primary energy and vorticity sources supplied by the thunderstorm must be given through the boundary conditions for the simulation. Thus our results do not address the question of tornadogenesis but deal with the question of how the details of the low-level, tornado flow depend upon larger-scale features in the thunderstorm represented by the boundary conditions on our domain. Different thunderstorms undoubtedly exhibit a wide variety of detailed velocity distributions on an inner domain boundary corresponding to the edges of our computational domain. In this paper, we restrict our attention to one possible set of physical conditions, and address how well the low-level turbulence within such a tornado can be simulated and how it controls the low-level central dynamics. We compare our results with recent radar observations (Wurman et al. 1996) to confirm that the model exhibits several features that are present, at least at some times, in actual tornadoes. Results from ongoing investigations of the sensitivity of the tornado vortex flow to variations in the boundary conditions, questions related to the sensitivity of the tornado vortex flow to unsteadiness in the boundary conditions, surface roughness, etc., will be deferred to a future paper. We first analyze one set of boundary conditions in some detail before moving on to the more general problem. The only physical variation considered in the present paper is the translation of the tornado.
2. Model review
The basic fluid dynamic code has been developed over a number of years under Office of Naval Research (ONR) and Electric Power Research Institute (EPRI) sponsorship by W. S. Lewellen, R. I. Sykes, and their colleagues. The model is based on the spatially filtered, incompressible Navier–Stokes equations (Sykes and Henn 1992). Second-order accurate, finite-difference equations with leapfrog time differencing are used to solve for the velocity components. Continuity is imposed by the Poisson equation for the pressure using a direct solver. The subgrid fluxes are computed using an effective eddy viscosity, which is a function of the subgrid turbulent kinetic energy, and a turbulent length scale, which acts as an effective spatial filter on the simulation. The details of the subgrid model as applied for this study are given in the appendix. Results of the ONR and EPRI studies have been published in a number of papers, for example, Sykes et al. (1990, 1994), Lewellen and Yoh (1993), and Lewellen et al. (1996).
Extending the code to permit a variable grid in both horizontal directions, allowed us to achieve relatively high resolution in the neighborhood of the center of the tornado. This was accomplished using the technique of Farnell (1980). This technique is not as computationally efficient as the fast Fourier transform method possible for uniform grids, but the fact that it allows a fine grid mesh in the central region of most interest more than compensates. The high-resolution results to be discussed below utilized meshes that varied from 2.5 m to 50 m in the horizontal and from 1.5 m to 100 m in the vertical. The maximum degree of stretching allowed for any neighboring grids was 16%. The behavior of the computation is very much like a fine grid nested within a coarse grid, with the gradual change allowing for a fully two-way interaction between the fine and coarse grids.
3. Boundary conditions
Our intent here is to provide relatively simple boundary conditions that are representative of the conditions that can occur on approximately a 1 km3 domain within tornadic thunderstorms. We apply axisymmetric boundary conditions to our horizontal boundaries, since we expect the flow on this small domain to be nearly axisymmetric on the average, and because these are the simplest boundary conditions to apply. The circulation 2πΓ = 2πυr is held constant in the inflow at the horizontal boundaries (Γ = 104 m2 s−1), except in a surface layer, and the radial inflow 2πQ = 2πur is also held constant [Q = (8/3) × 103 m2 s−1] below 1 km, except in the surface layer. Above 1 km, u = 0 on the horizontal boundaries. Our inflow boundary conditions yield ω = 0.08 s−1 and ac = 0.021 s−1 within a cylinder with a 1 km diameter and height, where ω is the volume average vertical vorticity and ac is the average horizontal convergence. These values are within the range of values obtained on low-level domains of order 1 km3 in both the thunderstorm simulations by Wicker and Wilhelmson (1995) and Grasso and Cotton (1995).
Our simulation is carried out in a frame of reference moving with the tornado. The surface boundary conditions assume a surface moving with a constant translation velocity = −15 m s−1. (Some comparison results are presented in section 5 with this translation velocity set to zero.) The horizontal boundary conditions impose a turbulent surface layer below 200 m. The tangential velocity on the horizontal boundaries was taken to have a vertical variation proportional to ln(z/zo) below 200 m, as expected in a fully turbulent surface layer; zo is set equal to 0.2 m. The radial velocity within this surface layer was given a modified logarithmic distribution following Lewellen (1977), with the maximum inward radial velocity at approximately 30-m height. This surface layer specification roughly approximates what would be obtained if the domain were doubled with the same values of circulation and convergence, and the surface layer started with zero thickness at this larger radius.
The boundary conditions are completed by applying a uniform updraft velocity of 21.9 m s−1 within a 1-km diameter disk at a 2-km height. We wish to simulate the dynamics of the tornado vortex within the lowest few hundred meters, but place the top boundary condition much higher to allow for adjustment between the region of interest and the top where the boundary condition is applied. As will be seen, this allows a relatively strong vertical downdraft to develop naturally within the lower part of the domain, without specifying any in our top boundary condition. This condition is somewhat arbitrary, since we could obtain a stronger or weaker central downdraft by specifying different conditions at the top of our domain; nonetheless, it satisfies our goal of establishing conditions on the lower 1 km3 that are representative of what may be found in a real tornado. Extending our domain height to 2 km essentially allows us to obtain a more realistic boundary condition at 1-km height than we could easily apply there directly.
We consider our boundary conditions on a 1-km cube to be representative of the conditions that can occur on this size domain within tornadic thunderstorms. These are unlikely to exactly agree with any particular radar observation or storm simulation, which are likely to be highly variable and have somewhat more complicated variations on the domain boundary surface, including some nonaxially symmetric features. For our purpose of examining the turbulence within the corner flow region, it is sufficient to provide relatively simple boundary conditions that contain the most essential features found in the tornado vortex and that fall within the range of what is physically possible.
We look for quasi-steady state conditions for the above boundary conditions, so the initial conditions are not critical. For computational efficiency, we allow the flow to spin up on a relatively coarse grid first before refining the inner mesh. The quasi-steady assumption is an important constraint on the flow. In nature and in the thunderstorm simulations the unsteadiness in the boundary conditions on this tornado vortex domain may be one of the reasons that a strong tornado is a relatively rare event. We will explore this further in our future work.
4. Simulation results for one swirl ratio
In this section we present results of our nominal high-resolution simulation. In section 5, we remove the asymmetry in the mean motion imposed by the translation to make it easier to analyze the role of the secondary eddies, and in section 6 we consider the sensitivity to resolution and subgrid modeling questions.
a. Time-averaged distributions
Since we are holding the boundary conditions steady, we can start up the simulation for an appropriate spinup time (∼ 5 minutes), and then collect “numerical statistics” on the stationary turbulence in our simulation. Figure 1 shows the time average (over 45 s) of the pressure perturbation due to the dynamics on a vertical slice of the total computational domain aligned with the direction of translation of the vortex. It shows why it is reasonable to concentrate our grid points within a few hundred meters of the center of the ground surface within our domain. Above this region, variations with respect to z are relatively weak, and radial gradients are relatively weak outside of this inner core flow. Above approximately 300 m, the region of strongest radial pressure gradients occur in the neighborhood of 100-m radius. This radial core size is determined internally, rather than being directly specified by the top boundary condition. The tilt and twist of the vortex at low levels is a direct result of the translation of the tornado. The minimum pressure in Fig. 1 occurs approximately 30 m above the surface. However, Fig. 1 slightly underestimates the magnitude of the low level pressure deficits due to the fact that the center of the vortex at the surface is approximately 15 m to the right (facing in the direction of the tornado motion) of the slice shown. The minimum average pressure in the center of the vortex is about 75% lower than it is in the quasi-cylindrical region over the upper part of the domain.
Figure 2 shows the average vertical velocity on the same slice as that shown for the dynamic pressure in Fig. 1. Our upper boundary condition has imposed a uniform velocity across the top of the domain, but at lower heights the strongest updraft is in an annulus. A primary feature of this flow is the local minimum in the radius of this annulus that occurs a short distance above the surface. This is caused primarily by the strong inflow layer induced by the surface friction. Above a few hundred meters, the annulus is centered about a nearly constant radius that is close to the radius of maximum radial pressure gradients identified in Fig. 1. Inside this core, the vertical velocity decreases with decreasing height passing through zero to define a downdraft below approximately 1 km. The vertical velocity reaches a minimum of less than −30 m s−1 a little above the surface. The approximate linear variation of w with z inside the core above a height of approximately 300 m is consistent with the swirl velocity being nearly independent of z in this region. The approximate cylindrical symmetry here is forced by the domination of the cyclostrophic pressure balance. The quasi-cylindrical region in this simulation occurs in spite of our removing the net inflow through the horizontal boundaries above 1 km height and in spite of the relatively large variation in the subgrid effective viscosity as our grid size expands with increasing height.
The maximum tangential velocity in this quasi-cylindrical region is 50–55 m s−1, and occurs at a core radius, rc, of approximately 150 m. Thus this simulation implies that the F[] of Eq. (2) is ∼ 0.2 for the present conditions. If we compare the turbulent flow in the quasi-cylindrical region with simple laminar converging flow, which has a core radius of (ν/ac)1/2 (Rott 1958), then this value of F[] would imply an effective eddy viscosity, ν, of 4% of Γ. Of course, the mean flow in the present simulation is far from a simple converging flow as seen in Fig. 2 and, as will be seen, the effective value of ν varies greatly across the domain.
The arrows in the plot represent the direction and magnitude of the “radial” and vertical velocity in this plane, interpolated to a uniform 10-m spacing for clarity. The converging flow penetrates closer to the center of the vortex near the surface. Although this strong inward flow is a direct result of the reduced swirl velocity immediately adjacent to the surface, it has the paradoxical effect of producing the highest swirl velocities in the upper part of the surface layer. The flow is not quite symmetric, with the region of maximum velocities being somewhat larger and occurring slightly higher above the surface on the upwind side of the vortex than on the downwind side. This is consistent with the higher inflow on this side of the moving vortex. Because of the translation-induced tilt and twist of the bottom of the tornado vortex, the plane shown in Fig. 3 does not cut precisely though the center of the vortex everywhere. Thus a small part of the tangential velocity is included as radial velocity in this representation and the arrows in the figure tend to slant toward the left near the bottom (where the slice cuts behind the precise center of the mean vortex) and toward the right near the top (where the slice cuts slightly in front of the center of the vortex).
The in-the-plane velocity arrows show that the strongest updraft is in an annulus centered slightly inside of the radius of maximum tangential velocity. The local minimum in the radius of this annulus that occurs a short distance above the surface implies that the strong favorable pressure gradient tends to keep the more intense radial flow in the boundary layer attached to the surface until it flows to a much smaller radius than the equilibrium core radius above the boundary layer. The similarity of this feature in our LES results to the ensemble mean, axisymmetric simulations of Lewellen and Sheng (1980) also suggests that the increase in turbulent length scale with height may be a contributing factor. In the earlier results, this turbulent feature was assumed, in analogy with that found in simpler turbulent boundary layers, but in the current simulation it is a result of the large-eddy dynamics. The quantitative analysis of the transport of angular momentum by the resolved eddies is deferred to section 5 where the translation of the tornado is removed to simplify the analysis of this turbulent transport.
Figure 4 presents a horizontal slice of the same time-averaged data as shown in Fig. 3, at a height of 27 m above the surface. Here the vertical velocity is presented in the form of contours and the horizontal velocity by the arrows. The strongest updrafts are concentrated in an annulus with a radius of ∼40 m, with a general downdraft over the central region. The strongest upward velocity is 50 m s−1, and the downdraft in the center of the vortex reaches 30 m s−1. Again the asymmetry in the flow caused by the 15 m s−1 translation of the tornado is readily apparent with the center of the low-level tornado displaced backward about 20 m and about 15 m to the right of the center of the domain. The maximum vertical velocity occurs on the left, downwind side of the vortex since the low-level tilt of the vortex allows part of the tangential velocity to add to the vertical downwind of the vortex and to subtract from it upwind. An inspection of the horizontal plane 1.5 m above the surface (not shown) shows an upward jet at the center of the vortex of ∼ 15 m s−1.
The distribution of the time-averaged velocity variance, q2 (q2 = twice the turbulent kinetic energy, TKE) is shown in Figs. 5 and 6 for the same vertical and horizontal slices shown in Figs. 3 and 4. Contours of the total are shown solid and contours of the subgrid component are shown dashed. It is clear that the resolved turbulence dominates the subgrid component everywhere in the corner flow except in a thin layer next to the surface. The resolved turbulence exceeds the unresolved, modeled part, by an order of magnitude over most of the region. Comparisons between the TKE plots and the mean velocity plots show that the highest values of TKE coincide with the largest gradients in vertical velocity, which lie slightly inside the largest gradients in tangential velocity, and well inside the largest mean tangential velocities.
b. Instantaneous snapshots
The relatively large values of TKE in Figs. 5 and 6 make it clear that instantaneous velocities must depart significantly from their time-averaged values. This strong variability also makes it difficult to present a “typical” sample of the data. We attempt this by showing instantaneous velocities on the same vertical and horizontal slices with the horizontal slice shown at two times, 2 s apart in Figs. 7–9. We choose to add a gray scale to the out-of-the plane velocity contours in these figures to help delineate the contours. The position of the arrows here coincides with the vertices of the stretched grid used for this simulation. The horizontal slices indicate that the updraft annulus seen in Fig. 4 is made up of a few intense updrafts rotating about a weaker downdraft in the central region of the tornado. These updrafts are associated with strong secondary vortices that rotate about the primary vortex with a speed that is significantly lower than that associated with the maximum mean swirl velocities. The vertical slice also shows the intersection of a few of these eddies as they spiral upwards. This basic turbulent eddy structure is a distinctive feature of many tornado observations. They were identified in field observations by Fujita (1970), who named them “suction vortices,” and first observed in a laboratory tornado model by Ward (1972).
Another way to view this primary eddy structure is by looking at 3D perspectives of instantaneous constant pressure surfaces. Figure 10 provides several illustrations of such a perspective at different pressure levels, with velocity vectors superimposed on the grid points that the pressure contours intersect. The simulation time coincides with that used in Fig. 9. The −5000-Pa pressure surface is shown within a 300-m cubic domain in Fig. 10a. For purposes of this presentation, the hydrostatic component of the pressure distribution has been added to the dynamic perturbation component shown in Fig. 1. This combination yields the familiar funnel shape of the tornado cloud. The maximum velocities on the isosurface are of the order of 100 m s−1. A strong eddy spiralling upward counter to the tornado tangential velocity is evidenced. The exhibited structure is somewhat reminiscent of that seen in the dust cloud of some strong tornadoes, for example, the 1974 Great Bend tornado (Golden and Purcell 1977).
The eddy structure itself is somewhat clearer when more intense pressure contours are used for the field at one instant of time, as shown in Figs. 10b–d. As the pressure level is reduced in units of 1000 from −5000 in Fig. 9a to −8000 in Fig. 10d, the viewer is allowed to look deeper within the eddy structure. The tilt and twist of the vortex introduced by the 15 m s−1 translation of the tornado shows up clearly in these views. Although the arrows tend to get rather sparse as only the center structure of the eddies shown at lower values of pressure, they do confirm that these eddies are individual secondary vortices. The velocity vectors are absent from Fig. 10d for clarity.
We wish to emphasis that this eddy structure is highly variable. A time series of the eddy structure, as represented by the −7000 Pa pressure isosurface, is shown in Fig. 11 at half-second intervals for approximately two-thirds of a rotation of the mean vortex. Figure 10c follows one half-second after Fig. 11d. The time of Fig. 11a coincides with that for Fig. 8. These eddies undergo fairly rapid evolution as they rotate about the center of the vortex, with a typical lifetime of order one rotation period, and rotation velocity well below the maximum mean swirl velocity. As Fig. 11 shows, these eddies consistently spiral upward in the direction that is counter to the tornado rotation. The primary location of the secondary vortices are well inside of the cone of maximum average swirl velocity; they are centered between the annular updraft and the central downdraft within the tornado. Consequently, the observed direction of winding permits the flow around the secondary vortex to be aligned with both the main tornado circulation and the up- and downdrafts. This is consistent with Rotunno’s (1984) explanation of alignment with the combined tangential and vertical vorticity distributions.
c. Sample pressure traces
If a fast pressure response instrument were to successfully intercept our simulated, translating tornado, it would exhibit a variety of pressure traces as illustrated in Fig. 12. The traces shown simulate the results of a number of surface instruments placed 20 m apart in the direct path of the tornado. Although the mean center of the tornado passes over each of the instruments in turn, the minimum pressure seen by the instruments varies from −5200 Pa to −10 300 Pa. These values compare with an average minimum value of −7200 Pa somewhere on the surface over the 45-s period of quasi-steady simulation. The pressure drops much lower when one of the primary eddies passes directly over the instruments. Since these unsteady eddies are rotating about the center as the tornado passes over the instrument, each instrument sees a different flow structure. This figure also demonstrates that in order for an instrument to capture these minimums it will be necessary for it to sample at least at the rate of a few Hertz.
The pressure traces would show even stronger variations if collected at 27-m height as illustrated in Fig. 13. The minimum varies from −7000 Pa to almost −16 000 Pa in this set. The average minimum at this height is 13 000 Pa, 80% lower than the average minimum on the surface. These simulated pressure traces demonstrate that it will be difficult to unambiguously interpret individual pressure traces that may be obtained in a real tornado.
d. Qualitative comparisons with Doppler radar observations
Finescale Doppler radar observations of tornadoes, with 50–130-m resolution volume for ranges from 2 to 6 km, have recently become available (Wurman et al. 1996). These data appear to confirm that the tornado flow within approximately 0.5 km of the tornado center is nearly axisymmetric. Further, they report that, in the annular region between 0.5 km and a couple of radar resolution volumes outside the maximum tangential velocity, the circulation correlates well with a constant value of circulation, particularly at levels above 200-m height. The circulation assumed for the horizontal boundary condition on our 1-km2 domain is sufficiently close, approximately three-fourths of their observed circulation at the corresponding radius, that it is interesting to compare some of the features exhibited in our model results with some of their observations. The overall flow patterns are similar. Their observation of a velocity difference of 140 m s−1 across a diameter of 120–200 m at a height of 110 m compares with our mean difference of 130 m s−1 across a diameter of 200 m at the same height. Our results show a higher difference of approximately 160 m s−1 across a smaller diameter of 100 m at height of 30 m, where they report the radar results may have been contaminated by ground clutter. For their tornado moving at 5–15 m s−1, they report a backward shift of the tornado below 200 m similar to that exhibited in Fig. 1. They did not observe the strong radial inflow near the surface indicated in our Fig. 3, but they did report that “photogrammetric analyses suggest that there was strong inflow in a layer 10–20 m deep near the surface.”
More important differences may be associated with the debris-free eye, which they observe to have a 400 m diameter at a height of 330 m. The downdraft with which they suggest this may be associated is considerably larger than the ∼ 150 m diameter of the recirculating inner cell obtained at this height in our simulation. As they state, the weak reflectivity eye may be increased in size by centrifuging effects arising from the strong tangential flow. Their estimate of the presence of a downdraft of >25 m s−1 in the eye region at heights between 400 m and 1 km is considerably stronger than our central downdraft over the same region (as shown in Fig. 1), although our results do show a minimum vertical velocity of −30 m s−1 at a lower altitude.
The resemblance of the radar observations to the present simulation may be little more than coincidental since the single Doppler data does not supply sufficient information about the through flow to determine how well our assumed through flow boundary conditions compare to those occurring at the time of the radar observations. Possible variations in the boundary conditions on our limited computational domain, that is, changes in the swirl ratio and details of the inflow and outflow boundary conditions, are expected to yield significant variations in the range of possible tornado flows. Nevertheless, this comparison does confirm that the model exhibits several features that are present, at least at some times, in actual tornadoes.
5. Axisymmetric case with no translation
The slight tilt and twist of the tornado due to its translation has only minor influence on the local dynamics, but makes it difficult to analyze the angular momentum balance of this flow. In order to make this easier, we performed a similar calculation with the translation removed so that the average flow is axisymmetric. This also permits us to make more definitive statements regarding the influence of translation on the dynamics. For ease of computation, the finest horizontal resolution was relaxed to 4 m, since as discussed in section 6, the results are quite similar for 2.5-m and 4-m resolution and the resolved turbulence still dominates over the region of most interest.
Figure 14 shows the mean flow obtained for this nontranslating case. The slight asymmetries in this figure give some indication of how well the 1-min average represents an ensemble average. For the present steady boundary conditions, deleting the 15 m s−1 translation velocity used in Fig. 3 results in essentially no change in the core radius above ∼ 300 m; however, it does decrease the low-level, maximum mean velocity by approximately 5 m s−1 and shows a reduction in the fluctuating extreme velocities. The maximum total velocity variance decreases by approximately one-third. The peak mean pressure deficit of 5600 Pa for this nontranslating case is about 30% less than the peak mean deficit of 7800 Pa in the translating case. The slight velocity increase resulting from the translating case appears to be brought about by the tilting of the horizontal vorticity generated from the translation producing additional vertical vorticity, while the asymmetry is responsible for the stronger fluctuations.
Figure 15 shows a plot of the pressure variance for this nontranslating case, which clearly marks the location of the strongest secondary vortices. The contour interval is 2 × 105 Pa2, so the standard deviation of the pressure fluctuation reaches up to 23% of the peak mean pressure deficit. The contours of average vertical velocity are superimposed on the left (Fig. 15a) and the contours of average tangential velocity are superimposed on the right (Fig. 15b) to show the relation of the location of the secondary vortices to the average velocity. This figure shows that the location of the secondary vortices coincides with the region of maximum vertical velocity gradients, which is centered a little inside of the cone of maximum vertical velocity, which in turn is slightly inside the maximum average swirl velocity. The radius of the local maximum in the pressure variance also coincides with the boundary between the inner recirculating flow and the outer through flow.
The dynamics of these secondary vortices are strongly influenced by the vertical velocity components, which are of the same order of magnitude as the tangential component. The strength and location of the secondary vortices are strong indications that these features are driven even more by the vertical velocity gradients than by the tangential velocity gradients. Thus they are more related to the instabilities on a cylindrical vorticity sheet with the vorticity sheet containing both vertical and tangential components of vorticity analyzed by Rotunno (1978) than to the simpler instabilities on a pure rotating cylindrical shear layer as analyzed by Snow (1978). However, as shown by Rotunno (1984), the three-dimensionality of these low-wavenumber modes is essential for the growth of these particular vortex instabilities. They are further complicated by the conical nature of the flow as the streamlines erupt from the radial inflow layer next to the surface.
The turbulent transport of angular momentum is represented in terms of the axisymmetric velocity covariances in Fig. 16. The vertical–tangential velocity covariance
The turbulent transport of angular momentum arising from the secondary vortices is inward so that the angular momentum penetrates closer to the center where it acts to spin up the recirculating flow in the center of the vortex. In the upper–central part of the corner flow, there is a reversal of the role of the eddies in the core region where the turbulent flux of angular momentum is directed outward, balancing the inward mean radial flow. In the regions where there are the strongest crossings between the streamlines and constant angular momentum lines, it may be seen that the existing gradients of angular momentum are flatter than those that would exist if the angular momentum followed the streamlines, as it would in the absence of turbulent transport.
The actual magnitudes of the turbulent covariances in Fig. 16 are not large in comparison with the corresponding product of the mean velocities, but the cumulative effect is to transport sufficient angular momentum into the central recirculating flow to have the peak values of Γ in the recirculating flow reach more than 30% of the outer boundary inflow value. This allows the mean flow in this recirculating cell to be significantly different than the stagnant core, which might be expected to occur if there were no transfer of momentum across streamlines in the central core. This spinup of the central recirculating flow enhances the radial pressure gradient and thus also has the nonlinear effect of enhancing the mean flow radial penetration near the surface by delaying boundary layer separation to a smaller radius. Thus, although the turbulent contribution to the angular momentum budget is negative in the immediate vicinity of the maximum swirl velocity, as seen in Fig. 17a, the role of the turbulent transport in setting up the strong recirculation contributes to the intensification of the vortex by the surface interaction.
6. Sensitivity to resolution and subgrid model
In order to have confidence in these results, it is necessary to investigate their sensitivity to modeling choices. Here we wish to separate the sensitivity to physical variables from that of modeling variables. Here we look at the influence of changing the numerical resolution and changing the subgrid model. We believe that independence of the time-averaged results to details of the small-scale resolution and of the subgrid model is an effective test of whether the turbulent eddies are sufficiently resolved in our LES.
a. Relaxed resolution
Figures 18 and 19 are repeats of Figs. 2 and 3, with the fine-scale resolution relaxed from 2.5 m to 4 m. The averaging time period used for Figs. 18 and 19 is 1 min. There is no significant difference between the velocity fields represented by these two sets of plots. There are a few discernable differences in some of the wiggles in the contours, which are more likely due to insufficient averaging time rather than to the difference in resolution. The contours obtained for a 30-s time average were the same except for a similar level of difference in the small-scale wiggles. The distribution of the time-averaged velocity variance shown in Fig. 20 does show that the resolved q2 still dominates the subgrid in this coarser resolution run, but not quite to the same extent that it did in Fig. 5 for the higher resolution run. The total velocity variance is quite similar but the maximum unresolved q2 exceeds 400 m2 s−2, while in Fig. 5 it remained below 300 m2 s−2. As should be expected, there are differences in the instantaneous distributions, not shown for the coarser resolution. The qualitative features of the secondary vortices are the same as observed at the higher resolution.
Figures 21–23 show the results of repeating the resolution test, by relaxing the finescale resolution to 8 m. Again the averaging time period is 1 min. The results remain remarkably similar to those shown in Figs. 3 and 4, but we do begin to see a few differences. The region of negative vertical velocity is smaller and does not reach as small a value. The maximum tangential velocity occurs at a slightly higher position on the downwind side of the vortex. The total velocity variance shown in Fig. 23 also remains close to that shown in Fig. 5, but the subgrid portion is getting large enough that it is no longer clear that the resolved turbulent transport completely dominates the unresolved modeled component in the corner region of the vortex.
b. Modified subgrid-scale parameter
Another way of testing the effectiveness of the LES results is to modify the subgrid turbulence model and see how much this influences the results. Figure 24 in comparison with Fig. 18 shows the result of increasing the subgrid turbulent length scale coefficient from 0.21 to 0.3 (at the 4-m finest horizontal grid spacing). This increase in Λ with its resulting increase in subgrid TKE would increase the effective subgrid diffusivity by approximately a factor of 2 if the resolved flow remained unchanged. There is now a slight decrease in the average maximum velocity. The maximum tangential velocity on the downwind side has dropped below the 80 m s−1 contour value, and the region exceeding this contour value is significantly smaller on the upwind side. Closer examination of the velocity values in this region show that they are reduced by approximately 5 m s−1. The peak velocity in the plane represented by the arrows changes slightly more, going from 62 to 55 m s−1. This is a more sensitive test of the impact of the subgrid model than making the same increase in the turbulent length scale in the finer resolution run. (It also takes far less computing resources.) This run further increases the ratio of the subgrid TKE to the resolved TKE as seen by comparing Figs. 25 and 20. The resolved still dominates over most of the flow, but the total velocity variance has been reduced.
In this test, the subgrid Λ was changed throughout the domain, not just in the well-resolved corner flow region (as for our resolution test). This had the effect of slightly increasing the core radius observed in the upper parts of the domain. We believe that this is responsible for the somewhat larger differences found upon varying the subgrid model versus those found for doubling the fine-grid resolution to 8 m. Nevertheless, the relative insensitivity of the results to this subgrid diffusion model provides further confidence in the high-resolution results.
7. Concluding remarks
We have simulated flow in the lower level of a tornado for a particular set of physical conditions that lead to several features that resemble features found in recent radar observations (Wurman et al. 1996) and have demonstrated that sufficient resolution has been achieved for our example that the average velocity and pressure distributions show little sensitivity in the region of maximum velocities to either finer resolution or modified subgrid turbulent model. Thus, this model provides a good framework for investigating the sensitivity of this unique flow to physical variations in the boundary conditions. The only physical variation considered in the present paper is the translation of the tornado. The sensitivity to variables such as details of the inflow and outflow velocity distributions, surface effective roughness, and time-varying boundary conditions will be the subject of a future paper. Some conclusions that follow for this particular set of physical boundary conditions representative of a strong tornado are the following.
The time-averaged velocity distributions show the maximum swirl velocity magnitude occurring within 50 m of the surface to be approximately 60% greater than the maximum swirl velocity in the quasi-cylindrical region above a few hundred meters.
The instantaneous velocity distributions show the turbulence dominated by a small number of strong secondary vortices spiralling around the main vortex, leading to maximum instantaneous velocities typically one-third larger than the maximum time-averaged velocity. These eddies are located in the regions of the flow with greatest vertical velocity gradients, well inside of the cone of maximum mean swirl velocity, and spiral around the mean vortex at velocities less than the average maximum velocity.
The turbulent transport of angular momentum induced by the secondary vortices is predominantly inward at low levels, allowing the inner recirculating flow to acquire values of angular momentum of up to 30% of that provided by the inflow boundary conditions, thus enhancing the surface intensification of the velocities.
To record the minimum pressure on a surface trace for the conditions of this simulation it would be necessary to have a time resolution of at least a few Hertz. If a successful high-frequency recording of this simulated tornado were made while it passed directly over the surface instrument, it would still underestimate the minimum instantaneous pressures occurring at a height of ∼ 30 m by an average of 80%.
Comparisons between simulations with a translation velocity of 15 m s−1 and those with no translation show that the interaction of the translating tornado with the surface slightly increases the maximum mean velocity ∼ 5 m s−1. The translation has a larger effect on the velocity fluctuations, yielding approximately a one-third increase in maximum total velocity variance. It also forces the bottom of the mean vortex to lag and be slightly to the right of the direction of translation of the upper vortex.
Acknowledgments
This research was supported by National Science Foundation Grant ATM-9317599 and Pittsburgh Supercomputer Center Grant ATM940025P. We would also like to acknowledge Aytekin Gel’s help with generating Figs. 10 and 11, and acknowledge the role of two anonymous reviewers in forcing us to tighten up our discussion of several points.
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APPENDIX
Subgrid Model
Time-averaged, dynamic pressure on the vertical plane through the center of the full 1 km × 2 km domain aligned with the direction of translation. Tick marks represent 100-m intervals.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Time-averaged vertical velocity on the vertical plane through the center of the full 1 km × 2 km domain aligned with the direction of translation.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Time-averaged velocities on a 300 m × 300 m vertical plane through the center of the domain in the direction of translation. Solid (dashed) curves represent contours of tangential velocity out of (into) the plane. The arrows represent the direction and magnitude of the radial and vertical velocity in the plane with the maximum length corresponding to 64 m s−1. Tick marks represent 10-m intervals.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Time-averaged velocities on a 200 m × 200 m horizontal plane at a height of 27 m above the surface. Solid (dashed) curves represent contours of positive (negative) vertical velocities. The arrows represent the direction and magnitude of the horizontal velocity in the plane with the maximum length corresponding to 88 m s−1.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Time-averaged total velocity variance (=2 × turbulent kinetic energy) on a 200 m × 200 m vertical plane through the center of the domain in the direction of translation. Solid curves represent contours of the total, resolved plus unresolved, while the dashed curves represent the unresolved component.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Time-averaged total velocity variance on a 200 m × 200 m horizontal plane at a height of 27 m above the surface. Solid curves represent contours of the total, resolved plus unresolved, while the dashed curves represent the unresolved component.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Instantaneous velocities on a 150 m × 150 m vertical plane through the center of the domain in the direction of translation. The gray scale represents contours of tangential velocity out of (into) the plane. The arrows represent the direction and magnitude of the radial and vertical velocity in the plane with the maximum length corresponding to 104 m s−1.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Instantaneous velocities on a 150 m × 150 m horizontal plane at a height of 27 m above the surface. The gray scale represents contours of positive (negative) vertical velocities. The arrows represent the direction and magnitude of the horizontal velocity in the plane with the maximum length corresponding to 110 m s−1.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Instantaneous velocities on a 150 m × 150 m horizontal plane at a height of 27 m above the surface 2 s after that in Fig. 8. The maximum length of the arrow corresponds to 110 m s−1.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Three-dimensional perspectives of the pressure contours within a 300 m × 300 m × 300 m domain, with velocity vectors super imposed for the instant of time corresponding to that for Fig. 9. The pressure levels in Pascals are (a) −5000, (b) −6000, (c) −7000, and (d) −8000. [Vectors are absent from (d) for clarity.]
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Time evolution of the three-dimensional perspectives of the −7000 Pa pressure contour at half-second intervals for the time between Figs. 7 and 9.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Simulated surface pressure traces for samplers located 20 m apart along the path of the mean center of the tornado on the surface.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Simulated pressure traces for samplers at a 27-m height located 20 m apart along the path of the mean center of the tornado at that height.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Time-averaged velocity distributions similar to Fig. 3, but with the ground translation removed to yield an axisymmetric vortex and fine resolution relaxed to 4 m. The maximum arrow length corresponds to 60 m s−1. See Fig. 3 for notation.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Time-averaged pressure variance on a 200 m × 200 m domain for the nontranslating simulation with contour intervals of 2 × 105 Pa2, superposed on average vertical velocity contours (a) and average tangential velocity contours (b).
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Axisymmetric covariances on a 200 m × 200 m domain of (a) the vertical velocity and tangential velocity w′υθ′ and (b) the radial velocity and the tangential velocity. Contour intervals are in units of 25 m2 s−2.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
(a) Turbulent transport contribution to the angular momentum balance on a 200 m × 200 m domain with contour intervals in units of 100 m2 s−2. (b) Superposition of axisymmetric streamlines and contours of constant specific angular momentum Γ. The streamlines are denoted by arrows and the specific angular momentum contours are labeled in m2 s−1.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Time-averaged velocities as in Fig. 3 but with the fine resolution relaxed to 4 m. The maximum length of the arrow corresponds to 62 m s−1.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Time-averaged velocities as in Fig. 4 but with the fine resolution relaxed to 4 m. The maximum length of the arrow corresponds to 87 m s−1.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Total velocity variance as in Fig. 5 but with the fine resolution relaxed to 4 m.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Time-averaged velocities as in Fig. 18 but with the fine resolution relaxed to 8 m. The maximum length of the arrow corresponds to 63 m s−1.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Time-averaged velocities as in Fig. 19 but with the fine resolution relaxed to 8 m. The maximum length of the arrow corresponds to 88 m s−1.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Total velocity variance as in Fig. 20 but with the fine resolution relaxed to 8 m.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Time-averaged velocities as in Fig. 18 but with the subgrid turbulent diffusivity increased by approximately a factor of 2. The maximum length of the arrow corresponds to 55 m s−1.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
Total velocity variance as in Fig. 20 but with the subgrid turbulent diffusivity increased by approximately a factor of 2.
Citation: Journal of the Atmospheric Sciences 54, 5; 10.1175/1520-0469(1997)054<0581:LESOAT>2.0.CO;2
