8.2 NONAXISYMMETRIC, UNSTEADY TORNADIC CORNER FLOWS

D. C. Lewellen,  W. S. Lewellen, and J. Xia
West Virginia University, Morgantown, WV.

INTRODUCTION

The corner flow is that region of a swirling flow where the surface layer inflow turns up into the vortex core. For tornadoes this region is of critical interest for at least two reasons, one simple, the other more profound: first, since we live just above the surface the damage potential in this region is of greatest interest, and second, due to the interaction of the vortex with the surface, the corner flow is generally the site of the highest velocities, shears, and pressure drops in the entire flow. In previous work, (referred to as LLX hereafter), we used large-eddy simulations to study the corner flow for quasi-steady conditions, generally with axisymmetric boundary conditions. Here we present some preliminary results of ongoing work aimed in part to extend that study to nonaxisymmetric, time dependent boundary conditions, as well as larger scale (mesocyclone) corner flows, in order to address tornadogenesis and variability. After a brief review of basic corner flow dynamics in section 2, we present examples of time transient behavior in section 3 (leading at times to dramatic increases in tornado intensification near the surface) and some effects of translation in section 4 (leading to more modest increases in intensification). The simulations are fully 3D and unsteady, and employ a stretched grid to simultaneously allow fine grid resolution in the corner flow region while imposing boundary conditions far from the corner flow. Details of the numerical model used and the simulation procedures employed can be found in LLX.
 

BASIC CORNER FLOW BEHAVIOR

Outside of the core in the surface layer underneath a swirling flow, the magnitude of the pressure is to a good approximation inherited from that in the flow above since normal pressure gradients tend to be small near a surface. This implies a strong radial pressure gradient since this is required to balance the strongly swirling flow. On the other hand, the swirl velocity near the surface necessarily drops to zero at least in a thin layer due to surface friction and perhaps in a deeper layer depending on the time development of the vortex. The resulting deviation from cyclostrophic balance in the surface layer drives a strong radial inflow which can overshoot its equilibrium point in the corner flow before turning up into the core flow. This radial overshoot brings angular momentum levels to smaller radii then elsewhere in the flow leading to larger swirl velocities and lower pressures.
 
 

Figure 1: Summary of the near surface intensification of the vortex for a set of 50 simulations as a function of corner flow swirl ratio Sc.

The structure of the corner flow, in particular the degree of overshoot, is critically dependent on the magnitude of the flux of low swirl fluid in the surface layer, what we called the ``depleted angular momentum flux'' in LLX, as well as on the upper core size, rc, and the angular momentum level in the outer flow, $\Gamma _{\infty }$. In LLX we formed a corner flow swirl ratio from these quantities (Sc, which can be interpreted as the ratio of a characteristic swirl velocity to a characteristic flow-through velocity for the surface-layer/corner/core flow) which successfully parameterizes the most basic corner flow variations induced by changing a variety of physical variables. Figure 1, from LLX, shows two measures of near surface intensification: the ratio of peak time averaged swirl velocity to maximum upper core velocity (well off the surface), and the square root of the analogous ratio for pressure drop. The largest intensification is for a critical value of corner flow swirl ratio, $S_c^* \approx 1.2$, for which the radial overshoot reaches the axis, turning into an intense vertical velocity jet off the surface capped by a vortex breakdown. Below Sc* the unfavorable pressure gradient induced by the stagnating inflow forces the surface flow to turn upwards even outside of the upper core radius; above Sc* the radial overshoot is only a fraction of the core radius, the centrifugal barrier preventing further radial penetration.
 
 

TEMPORAL OVERSHOOTS

One of the most important features of fig. 1 is the narrowness of the low swirl intensification peak. It would seem that conditions must be rather finely tuned to sit on this peak in a quasi-steady state. On the other hand, if the boundary conditions on the vortex change in time, then it is easy to sweep through a range of corner flow behaviors including the low swirl peak. The possibility of temporal overshoots (non-steady state behavior) also arises. Figure 2 shows relative intensification results as a function of time for two such cases. For simplicity and to increase the possibilities for temporal overshoots we begin in each case with a quasi-steady vortex, abruptly make a change in the boundary conditions, and then follow the simulations until they reach their new quasi-steady states. In case 1 we start with a very low swirl corner flow (Sc=.8) achieved by including lower swirl inflow through the side boundaries in a layer above the surface. This layer of inflow was abruptly shut off in case 1, producing conditions which eventually lead to a high swirl quasi-steady state (Sc=12). This end point was used as the starting point for case 2, where the low swirl inflow layer was now abruptly reinstated. The two cases are thus in some sense time reversals of each other, but the differences between them are dramatic. Case 1 includes a period of near surface intensification far greater than anything found for the quasi-steady state cases of fig. 1 (even accounting for the time averaging in fig. 1), while case 2 does not. It should be noted that the square root of the pressure drop near the domain top varies less than 20% from the initial value ($\sqrt{p^*}$ used for normalization in the figure) for case 1 for t<1.4t*, before rising by a factor of 1.7 for the high swirl end condition.
 
 

Figure 2: Relative intensification as a function of time for two cases described in the text. pmin is the minimum pressure drop within the domain; p* the average minimum pressure drop near the top of the domain for the low swirl quasi-steady state; and $t^*=r_d^2/\Gamma _{\infty }$ represents a characteristic time scale for the low swirl surface layer flow to be exhausted after the lateral boundary condition at radius rd is changed.

 

Figure 3: Vertical cross section of azimuthally averaged fields in the corner flow region for case 1 at time of peak pressure drop. Solid line: angular momentum fraction (relative to $\Gamma _{\infty }$); dashed: pressure normalized by minimum average pressure drop near domain top at the given time.

We have performed related simulations restoring swirl to the low level inflow rather than shutting off the low swirl inflow, or varying the low-level inflow profile, top boundary conditions, or grid resolution, and have even switched the boundary conditions sequentially (beginning at one corner of the domain, working its way around in time). The depth of the intensification spike in fig. 2 and details of its shape vary with these changes, but the qualitative behavior encountered appears to be robust. At the very low swirl starting point the lowest pressures and highest velocities are well above the surface. This remains the case until the low swirl fluid near the surface has been mostly removed by the strong convergence, whereupon the lowest pressure region (as well as the vortex breakdown lying above it) descends. The peak intensification (fig. 3) occurs when the (now conical) breakdown lies just above the surface capped by a narrow central downdraft. This point is followed rapidly by this downdraft reaching the surface, opening the core somewhat (to produce a ``medium swirl'' corner flow configuration). The intensification level has dropped sharply at this point but still greatly exceeds the levels of fig. 1. In the final stages a much wider downdraft descends to the surface, generally pushing the surface level vortex to the side rather than opening up its core. What was the primary vortex on the surface now appears as a secondary vortex rotating about a much larger core. It weakens in the process, leaving finally a high swirl corner flow with the updraft in a large annulus above the surface and with very weak multiple secondary vortices in evidence. The final stages of the once intense low level vortex are at least qualitatively suggestive of the ``roping out'' stage of tornado evolution.

A full range of corner flow configurations are also identifiable in the ``reverse'' simulation, case 2, again with the peak intensification point occurring for a low swirl configuration with breakdown just above the surface (fig. 4). There is a crucial difference however. In case 1 we have low swirl fluid preceding the higher swirl fluid through the corner flow, pulling the latter in to smaller radii in its wake to increase intensification (a temporal overshoot). In case 2 high swirl fluid precedes lower swirl fluid into the corner, impeding the radial overshoot of the latter.
 
 

Figure 4: As in fig. 3 but for case 2. Note the order of magnitude change in pressure contour interval.

 

TRANSLATION EFFECTS

Figure 5: Cross section (400m$\times$400m) at 26m height of high swirl corner flow showing perturbation pressure contours (500 Pa intervals) and horizontal velocity arrows (interpolated to a uniform 10m grid for clarity, with maximum length representing 74ms-1).

 

Figure 6: As in fig. 5 but with the addition of a 30ms-1 surface translation (max arrow length = 82ms-1).

Figures 5 and 6 illustrate some of the changes induced by adding a large (30ms-1) surface translation to a high swirl vortex (Sc initially 6.9). The case is the same as that described in LLX except a more modest 15ms-1 translation was applied there. The translation has three principle effects: shifting the center of the vortex at the surface relative to that above, reducing the corner flow swirl ratio by effectively increasing the surface depleted angular momentum flux into the core, and breaking the axisymmetry seen by the secondary vortices revolving about the main vortex. The latter two effects lead to fewer secondary vortices, with some decidedly stronger than others; for the strong translation illustrated there is effectively only one on the surface. Note how far from axisymmetric the near surface vortex becomes; this possibility should be kept in mind when trying to interpret near surface pressure traces or Doppler wind measurements in terms of idealized axisymmetric profiles. In related simulations we have added sufficient translation to a ``medium swirl'' (Sc= 2.6) vortex to effectively drive it to Sc*, giving a large surface intensification and a vortex with a sharp bend between the vortex on the surface and that above.

Perhaps more surprising is that the addition of surface translation to a very low swirl vortex (Sc below Sc*) can also lead to intensification (both in the steady state and through temporal overshoots). In contrast to the high swirl case, the translation now effectively adds angular momentum in the near surface layer (about the now shifted vortex center on the surface), raising Sc, and increasing the near surface vortex intensification. That the introduction of surface translation tends to drive Sc towards Sc* from either the high or low swirl side, suggests that maybe the conditions necessary to be near Sc* in a quasi-steady state do not require as much fine tuning as it would at first seem.
 
 

CONCLUDING REMARKS

In an ongoing study of nonaxisymmetric and/or time varying corner flows we have found different mechanisms for enhancing the near surface tornado intensification relative to the vortex strength far from the surface. The introduction of a large surface translation speed can lead to an increase by shifting the corner flow swirl ratio towards Sc*, that value which gives the greatest intensification for quasi-steady state conditions. More significantly, a change in the near surface inflow conditions which allows the flow to evolve from a very low swirl corner flow (Sc < Sc*) to a high swirl corner flow gives rise to a temporal overshoot producing dramatically larger near surface intensification factors. Both results reinforce one of our main conclusions in LLX: the critical role that the near surface inflow plays in tornado vortex intensification.

These results on corner flow structure should be relevant for both the tornado scale and mesocyclone scale corner flows, impacting tornado structure evolution in the former case and tornadogenesis in the latter. The changes in near surface flow conditions could arise from changes in surface topography, roughness, translation speed, or, probably most importantly, from the impact of other flow features such as a rear flank downdraft wrapping around and cutting off the low swirl inflow near the surface. This latter scenario presents an attractive possibility for one type of tornadogenesis. A high swirl (referring to the vortex as a whole) mesocyclone which does not extend to the surface early in its evolution, overlies a layer of low swirl fluid, thus giving rise to a very low swirl mesocyclone corner flow. This gives little or no recognizable surface vortex but, crucially, provides strong low level central convergence. If a downdraft wrapping around this system either cuts off much of the low swirl inflow, or introduces angular momentum to the surface layer itself, then an intense, though transient, vortex on the surface can naturally arise. For example dimensioning the simulation of case 1 for a reasonable mesocyclone scale ($\Gamma_{\infty}=4\cdot 10^4m^2s^{-1}, r_d=5$km), produces near surface swirl velocities exceeding 60ms-1 (9min after switching the boundary conditions), which last for 6min and reach a peak of 100ms-1.
 
 

ACKNOWLEDGEMENTS

This research was supported by the National Science Foundation with S.P. Nelson as technical monitor.
 
 

Bibliography

Lewellen, D. C., W. S. Lewellen, and J. Xia, 2000:

The influence of a local swirl ratio on tornado intensification near the surface.

J. Atmos. Sci., 57, 527-544.