An idealized analytical model and numerical large-eddy simulations are used to explore fluid-dynamic mechanisms by which tornadoes may be intensified near the surface relative to conditions aloft. The analytical model generalizes a simple model of Barcilon and Fiedler and Rotunno for a steady supercritical end-wall vortex to more general vortex corner flows, angular momentum distributions, and time dependence. The model illustrates the role played by the corner flow swirl ratio in determining corner flow structure and intensification; predicts an intensification of near-surface swirl velocities relative to conditions aloft of Iυ ∼ 2 for supercritical end-wall vortices in agreement with earlier analytical, numerical, and laboratory results; and suggests how larger intensification factors might be achieved in some more general corner flows. Examples of the latter are presented using large-eddy simulations. By tuning the lateral inflow boundary conditions near the surface, quasi-steady vortices exhibiting nested inner and outer corner flows and Iυ ∼ 4 are produced. More significantly, these features can be produced without fine tuning, along with an additional doubling (or more) of the intensification, in a broad class of unsteady evolutions producing a dynamic corner flow collapse. These scenarios, triggered purely by changes in the far-field near-surface flow, provide an attractive mechanism for naturally achieving an intense near-surface vortex from a much larger-scale less-intense swirling flow. It is argued that, applied on different scales, this may sometimes play a role in tornadogenesis and/or tornado variability. This phenomenon of corner flow collapse is considered further in a companion paper.
In a tornadic supercell velocities are intensified within a small region of the storm, within a few tens of meters of the surface where they can do the most damage. Determining the maximum possible tornado intensity near the surface and what storm conditions produce them are longstanding goals of severe storm research. Important components interact on a large range of spatial scales. To deal with the resulting complexity, the process has generally been studied in somewhat idealized steps progressing from large scales to small: the origin of rotation at midlevels within the storm, low-level mesocyclogenesis, tornadogenesis, and tornado structure [see, e.g., Davies-Jones et al. (2001) for a recent review]. The subject of the present work—the near-surface intensification of a vortex due to fluid-dynamic effects—is potentially important in the last three of these. In the basic mechanism the combination of reduced swirl velocity near the surface and a strong radial pressure gradient inherited from the faster swirling flow above drives an inward radial flow; the inertia of this radial flow can bring larger angular momentum levels into smaller radii, with a corresponding increase in swirl velocity and central pressure drop relative to conditions aloft.
This mechanism has been studied most extensively for the tornado corner flow: that region where the vortex core meets the surface and near-surface inflow turns upward into core flow. Results from laboratory, theoretical, and numerical models have shown that peak mean swirl velocities in the corner flow can exceed those aloft by a factor ∼2 (e.g., Baker 1981; Baker and Church 1979; Fiedler and Rotunno 1986; Lewellen et al. 2000a). Typically these studies assume approximately steady-state axisymmetric conditions, a surface layer produced purely by friction, and approximate cyclostrophic balance in the flow above. These conditions are not required for near-surface intensification to be important, however. One of the main goals of the present work is to study corner flow structure and near-surface intensification for more general conditions, particularly of the reduced angular momentum layer above the surface. Both low-level mesocyclogenesis and tornadogenesis (regardless of the mechanisms that produce them) will involve a nontrivial evolution of their respective near-surface layers, which will affect the evolving mesocyclone and tornado corner flows in potentially critical ways and likely obscure the separation between the two processes. Even in a “mature” tornado, terrain changes, encroaching gust fronts, etc. will likely lead to a rich variety of low-level inflow conditions. The present work continues to explore one of the themes from our previous studies: that corner flow structure and near-surface intensification are highly sensitive to properties of the near-surface inflow layer (Lewellen et al. 2000a, b; Lewellen and Lewellen 2002).
The intensification represented by a mesocyclone spawning a tornado critically involves interaction with the surface, but the dynamics are still a matter of debate (e.g., Davies-Jones et al. 2001) and are probably not unique. Some evidence of near-surface intensification of the mesocyclone is provided by airborne Doppler observations (e.g., Wakimoto et al. 1998; Ziegler et al. 2001) and of the tornado by models of tornadic thunderstorms (e.g., Wilhelmsen and Wicker 2001) and finescale Doppler radar observations (e.g., Wurman and Gill 2000; Wurman and Alexander 2005; Bluestein et al. 2003, 2004). Study of the phenomena in these cases is complicated by the limited grid resolution that can be utilized near the surface in the full storm simulations; the limited beam resolution, ground clutter problems, and difference between air and scatterer velocities involved in the radar measurements; and the inability, given the complexity of the entire thunder storm and uncertainty of tornado occurrence, to systematically explore the parameters directly responsible for the phenomena with either tool.
In this work we consider idealized analytical models and numerical experiments designed with a limited goal of understanding some of the basic ingredients involved in near-surface vortex intensification. This limited focus allows dynamical relationships and the parameters governing them to be more clearly identified; these results can then be used to help interpret field observations and storm model simulations. Only fluid-dynamics effects are considered. Buoyancy is not included nor is the possibility for added intensification due to subsidence warming in the tornado core. Intensification is considered only relative to conditions in the core flow aloft; that is, no attempt is made to connect that to a thermodynamic speed limit (or, more accurately, a thermodynamic velocity scale). We also concentrate on the intensification of the primary vortex; we do not consider here the added intensification that can occur in secondary vortices as demonstrated in observations, laboratory, and 3D numerical modeling studies.
In section 2 of this paper an idealized analytical model is developed for the corner flow based on conservation of mass, vertical momentum, and angular momentum. This generalizes a simple model of an end-wall vortex employed by Barcilon (1967) and Fiedler and Rotunno (1986). This model is used in different levels of approximation to better understand what levels of near-surface intensification can be straightforwardly realized and how even greater levels might be achieved. It also illustrates in a simpler context the corner flow’s sensitivity to the near-surface inflow and the significance of the corner flow swirl ratio, Sc, introduced in Lewellen et al. (2000a). Some of the derivations involved are set aside in appendix A.
Section 3 presents results from large-eddy simulations (LES) with enhanced near-surface intensification. Considered first are quasi-steady examples where the near-surface inflow conditions have been adjusted to achieve higher intensification than has been reported previously for quasi-steady states. Second are examples of a type of unsteady vortex evolution, dubbed “corner flow collapse” in previous studies (Lewellen et al. 2000b; Lewellen and Lewellen 2002), that without any fine tuning can naturally achieve near-surface velocities an order of magnitude greater than those aloft. One motivation for studying the quasi-steady cases is that it helps in understanding the origins of the greater intensification found for the unsteady evolution. A discussion of the role that these types of corner flows may play in tornadogenesis and tornado structure is given in section 4. Summaries of the LES model, simulation issues, and a table of simulations are given in appendix B. In a companion paper (Lewellen and Lewellen 2007), corner flow collapse and its numerical simulation are considered in greater depth.
Other examples of large intensification in unsteady vortices have been reported in the literature (e.g., Walko 1988; Fiedler 1994; Davies-Jones 2000; Markowski et al. 2003) in generally axisymmetric simulations. Some of these share features of corner flow collapse evolution, though identifying the latter mechanism as the origin of the observed intensification is uncertain given other effects involved: for example, buoyancy, perfect axisymmetry, and the particular starts from idealized initial conditions that are employed.
2. An idealized corner flow model
We begin with a vortex corner flow (Fig. 1) that is assumed to be axisymmetric in the mean and make a series of assumptions with an eye toward representing the key features present in simulation results while providing some level of analytic tractability. A constant value is assumed for angular momentum (Γ∞, defined about the central axis) above the surface boundary layer and outside of an outer core radius ro. Below this region a layer of fluid with Γ < Γ∞ is drawn radially inward before turning upward and becoming the main core flow. This flows between ro and an inner core radius, ri (which may be zero for some range of heights). Inside of ri we assume Γ ≈ 0; the flow here may be recirculating, turbulent, or nearly stagnant. For simplicity buoyancy is ignored, the hydrostatic part of the equations of motion separated out, and density is assumed constant (and its value drops out of the equation set).1
We wish to estimate the peak pressure and velocity at a lower level in the corner flow (zl) relative to their values well above at an upper level zu. In between these levels the flow is expected to be highly turbulent, strongly dissipative, and possibly including a vortex breakdown. To proceed we follow previous authors (e.g., Barcilon 1967; Fiedler and Rotunno 1986) in utilizing the analogy between a vortex breakdown and a hydraulic jump, and adapting the textbook treatment of the latter (matching momentum and mass fluxes across the jump). In this way some basic relations between conditions at zl and zu can be deduced without need of solving for (or making further assumptions about) conditions in between such as the shape of ri(z) or the pressure distribution within the heart of the turbulent corner flow.
Consider the control volume shown in Fig. 1 bounded by zl and zu in the vertical, and by r = 0 and ro in the radial. At any instant in time, the balance of vertical momentum within the control volume can be expressed as
where the first three terms represent, respectively, fluxes of vertical momentum into the volume from below, out from above, and in through the sides. The last terms represent the vertical component of the hydrodynamic pressure force acting on the boundary of the control volume, and the rate of change of the total vertical momentum within the volume.
Assuming quasi-steady conditions for now and performing a time average, the last term drops. Further, it is assumed that the turbulent momentum flux across the boundaries (and the smaller viscous transport) can be neglected relative to the mean flux, in part because zl, zu, and ro are assumed sufficiently far upstream, downstream, and outside of the regions of largest turbulence in the corner flow, respectively. With this assumption, the third term in (1) drops because the r = ro boundary will then be a streamline of the mean flow.
where βj ≡ r 2ij/(r 2oj − r 2ij) for j = l, u, and a fractional area coordinate A has been employed to parameterize how Γ̂ ≡ Γ/Γ∞ varies from 0 to 1 in the annulus between ri and ro. The chief assumptions made to arrive at (2) are (appendix A): similar profile shapes at zl and zu for Γ and for w within the updraft annulus (a way to satisfy mass and angular momentum conservation); approximately inviscid flow outside of ro and below zl so that Bernoulli’s law holds; and approximate cyclostrophic balance (to relate p to Γ) on the planes z = zl, zu, and outside of ro between these planes (but not in the boundary layer or the interior of the control volume). Some particular limits of (2) are now considered in turn.
a. Γ(A) = ril = riu = 0
Then (2) reduces to
with a solution of rou/rol ≈ 2.22 (along with the trivial solution rou = rol). This represents probably the simplest model of vortex breakdown, first considered by Barcilon (1967) and applied to the tornado corner flow by Fiedler and Rotunno (1986). The core flow upstream and downstream of the breakdown is assumed radially uniform with a jump in angular momentum from 0 to Γ∞ at ro. The near-surface intensification relative to the core conditions above, defined either in terms of the swirl velocity (Iυ ≡ υmax/υc) or pressure drop (Ip ≡ pmin/pc), is given simply by the radius ratio, Iυ = Ip = rou/rol. The value is in quite reasonable agreement with that estimated from a more realistic treatment by Fiedler and Rotunno (1986) (∼1.7), that inferred by them from the laboratory measurements of Baker (1981) and Baker and Church (1979) (∼2.3), and from 3D simulation results (∼2.5: Lewellen et al. 2000a).
b. ril = riu = 0
The Γ(A) dependence drops out of (2), since βu = βl = 0, leading again to (3). The profiles of the pressure and swirl velocity in the core will depend on Γ(A) and the location of the peak swirl velocity may lie inside of rol; however, because the profiles have the same functional form at zu and zl, the relative intensification remains as above, Iυ = Ip = rou/rol ≈ 2.22.
c. Γ(A) = 0
Sc is defined so that it represents a ratio of a characteristic swirl velocity to a characteristic flow-through velocity specifically for the surface/corner/core portion of the flow. This is achieved by defining a depleted angular momentum flux, ϒ, as the total flux of the local flow variable Γ∞ − Γ. The depleted angular momentum is a useful construct because, to good approximation, it is a conserved variable [which follows in an incompressible flow from mass and angular momentum conservation (Lewellen et al. 2000a)] and because it has a significant magnitude only within the surface/corner/core flow (where Γ < Γ∞). For quasi-steady conditions the integrated flux, ϒ, is constant as the flow enters the corner, turns, and heads up the core. In the Γ(A) = 0 limit it is easily computed, for example, across the z = zl plane,
This relates the quantity of most interest, the intensification I = rou/rol, to Sc and a dimensionless measure of the upper core structure, βu. In addition there are constraints that ri < ro, that rou ≥ rol, and that the total head cannot increase in passing downstream from zl to zu. Combined, these require
Figure 2 reproduces Fig. 8 from Lewellen et al. (2000a), plotting near-surface intensification as a function of Sc for a large set of simulated tornadoes, together with the predicted curve from (5) and (6). The solution branch shown is the one that maximizes I, realized by taking ril = 0 for Sc ≤ S*c and riu = 0 for Sc ≥ S*c, where S*c is the value of the swirl ratio at peak intensification. There is a solid central updraft (ril = 0) above the surface for Sc < S*c, becoming an annular updraft (ril > 0) for Sc > S*c, just as observed in the simulated corner flows in Lewellen et al. (2000a).
The structure of the simple model solution matches the behavior (e.g., the narrow low-swirl intensification peak) fairly well, with the biggest differences being the value of S*c, and that Iυ and Ip differ for larger Sc in the simulation data. These differences are results of the simple step-function Γ distribution with radius assumed at zl and zu. While we saw above that choosing a different form for Γ(A) does not increase the peak I, it does affect the computation of ϒ and, hence, Sc. A more realistic profile (i.e., not discontinuous) increases S*c. For example considering the family of profiles Γ(A) = Γ∞Aa, the step approximation above corresponds to the limit a → ∞ and produces the lowest value S*c ≈ 0.70; the limit a → ½ (p is unbounded for a ≤ ½) produces S*c ≈ 1.78; and a = 1 (uniform radial distribution of vertical vorticity in the core updraft annulus, representing a Rankine combined vortex when ri = 0) gives S*c ≈ 1.25. The latter is shown by the thin lines in Fig. 2, which also reproduce the Iυ > Ip behavior seen for larger Sc in the simulations. The same solution branch is taken as in the simple model of (5), given in the figure, but now for (2) solved numerically.
d. ril = 0
In this case (2) reduces to
The leading “1” (originating from the momentum flux into the control volume from below) represents the only positive term; rou/rol is clearly bounded and maximized for βu = 0 (where upon it takes the same ∼2.2 value as above). For riu > 0, however, it is possible for Iυ and/or Ip to exceed rou/rol for some choices of Γ(A). This represents a possibility for enhanced near surface intensification, which will be discussed further below.
e. Time variation
Time variation represents another possibility for enhanced near-surface intensification. A modest time variation in the mean flow (i.e., representing velocities small compared to the peak swirl velocities present at any time) affects the treatment of momentum conservation given above in three ways: the last term in (1) must be included; the rate of change of the control volume itself must be included when mass conservation is used to relate w(r, zu) to w(r, zl); and Bernoulli’s equation [used to eliminate w(r, zu)] picks up an additional term owing to the rate of change of pressure at a fixed point in space (a nondissipative head loss). Returning to the case above with Γ(A) = 0, the time varying extension of (5) can be written (appendix A) as
where 𝒥 ≤ 1 arises from the unsteady Bernoulli term, and and are normalized rates of change of the control volume and vertical momentum within the control volume, respectively. For the most interesting case of a shrinking control volume, (8) has an additional positive term (−) that can permit enhanced intensification. The terms and 𝒥 factor work toward reducing the intensification, however. As discussed in more depth in the companion paper, a dynamic corner flow collapse can lead to much greater intensification levels than are achieved in quasi–steady state, but that potential falls off if the rate of attempted collapse is either too slow (so − contributes little) or too fast (so the term dominates).
An estimate of the potential unsteady intensification (appendix A) can be obtained by approximating (8) for conditions where we expect (based on the quasi-steady analysis) the largest intensification—that is, ril, βu = 0, and 𝒥 = 1; postulating a family of shapes for the collapsing corner flow; and integrating the ODE that results. This procedure effectively fine tunes the evolution of ϒ flowing into the corner region so as to maximize the intensification—indeed producing a singular result as ϒ → 0 (appendix A, Fig. A1). In actuality, the evolution of ϒ into the corner will be governed by physics in the inflow layer and will not be expected to follow this optimal history. Nonetheless, the results demonstrate that the requirement set by the vertical momentum equation, mass and angular momentum conservation, which in the Γ(A) = 0 approximation limited quasi-steady intensification factors to ∼2.2, does not preclude even very large near-surface intensification for some unsteady evolutions.
3. Simulated corner flows with enhanced intensification
The solutions to the corner flow model considered above support the prevailing view that near-surface intensification is generally greatest, with a value ∼2, given a supercritical end-wall vortex capped by a vortex breakdown just above the surface. The model also suggests ways in which even greater intensification levels can be achieved, for example, by involving more complex Γ distributions or time evolution. We now consider progressively more complex simulation examples, demonstrating that these possibilities can be realized. The LES model and basic simulation setup are as in Lewellen et al. (2000a). These are summarized in appendix B along with comments on some critical modeling issues and a more detailed specification of the simulations discussed below. In the figures below quantities are nondimensionalized using the angular momentum level in the far field (Γ∞) and the domain “radius” (rd ≡ half the lateral extent of the square domain) to form length (rd), time (ts ≡ r 2d/Γ∞), and velocity (Vs ≡ Γ∞/rd) scales.
a. Nested corner flows
Figures 3 and 4 summarize results from quasi-steady simulation S1, which exhibits distinct “nested” inner and outer corner flows on different spatial scales. Identifying the corner flow region by the strong radial-to-vertical-turning Γ gradients that bound it, we see in Fig. 3 a large-scale corner flow bounded by Γ/Γ∞ contours between ∼0.1 and 1 and, inside of this, a much smaller-scale corner flow (best seen in the zoomed in view of Fig. 4) with Γ/Γ∞ in the range ∼0–0.1. To realize this configuration, the near-surface inflow at the domain boundary was constructed with varying angular momentum in layers: a thin layer with no swirl just above the surface, a thin layer above it with Γ = Γ∞/2, a thick no-swirl layer above them, and finally Γ = Γ∞ above. Note that in the azimuthally averaged results of the figure the Γ distribution is smooth even at rd because the discontinuous inflow conditions are imposed on the square domain boundary circumscribing the radius rd circle (so there is some mixing at work at large radii smoothing the distribution). The layer thicknesses were adjusted to produce near-critical low-swirl corner flows on both inner (Fig. 4) and outer (Fig. 3) corner scales, the former with an abrupt quasi-axisymmetric vortex breakdown, the latter with a spiral-mode vortex breakdown best seen in the instantaneous pressure field (Fig. 5). Each is accompanied by a significant intensification factor, with the combination producing Iυ ≈ 3.2 and Ip ≈ 4.2 as measured in the time and azimuthally averaged flow. This measure underestimates the true intensification because the lateral wander of the inner vortex below the first breakdown is nonnegligible in comparison to its small core size. A time average of peak values at each time gives instead, Iυ ≈ 5.3 and Ip ≈ 5.2.
The two vortex breakdowns were diagnosed from the observed behavior—sharp transitions from states with strong upward axial flows to ones with significantly larger core radii, reduced axial velocities, and increased turbulence levels. The presence of a second vortex breakdown following a first might seem inconsistent with the interpretation of a vortex breakdown as the transition from a supercritical flow to a subcritical one (Benjamin 1962). In the present example, however, two different fluid populations are involved: the small-scale central jet flow can transition from super to subcritical in the inner vortex breakdown, while the much larger annular updraft accelerates to supercritical above before undergoing its breakdown.
This simulation illustrates two other points stressed in Lewellen et al. (2000a): the inadequacy in some cases of any single parameter (e.g., a swirl ratio) to characterize the interaction of a vortex with the surface and the extreme sensitivity of the corner flow structure and intensity to the properties of the near-surface inflow. At least three different swirl ratios are relevant in the present case: the domain-scale swirl ratio and corner flow swirl ratios on inner and outer scales. The sensitivity to the inflow structure has been confirmed by related simulations: the elimination of all low swirl inflow at the lateral boundaries produces a quasi-steady high-swirl corner flow with multiple secondary vortices, as one would expect from the high domain-scale swirl ratio; eliminating only the lowest thin layer of no-swirl inflow maintains nested corner flows, but now a high-swirl corner inside of a large-scale low-swirl corner; most dramatically, eliminating just the thin layer of Γ = Γ∞/2 inflow produces a very low swirl corner on the large scale (similar to that in Fig. 8a below), replacing a strong near-surface intensification with a strong deintensification.
b. A continuously nested corner flow
In the previous simulation the radial overshoot of the lowest layers of fluid prevent the narrow central downdraft above from reaching the surface, just as the radial overshoot into the core of the deeper low-swirl layer above prevents a much larger-scale downdraft from opening up the core and relieving the low pressures below. Figures 6 and 7 summarize results from simulation S2 taking this result further, producing a “continuously nested” extended conical corner flow. In the now extended near-surface inflow layer at the side boundary, both the angular momentum and horizontal convergence are increased linearly from the surface. Each successive inflow layer down to the surface exhibits a “low swirl” character on a progressively decreasing scale—initially forced radially inward by a cyclostrophic imbalance but later turned upward by a central stagnation so that the radial overshoot relative to the core above occurs well off the surface. The radial overshoot of each successive layer limits the ability of the core downdraft above to open up the smaller core below. Aloft, the result is reminiscent of the conical similarity solution of Long (1958), but now smoothly connected to a nonsingular boundary layer flow below. Larger Γ/Γ∞ is more effectively drawn into the corner than in the previous example, leading to higher velocities. The levels of near-surface intensification are modestly increased (Iυ ≈ 3.5, Ip ≈ 3.6 from azimuthal time averages; Iυ ≈ 6.8, Ip ≈ 5.2 from time averages of peak values).
As in the simpler nested corner flow above, this example evades the intensification limit suggested by the simplest approximations to the corner flow model of section 2 via a thick boundary layer. At the heights at which the inner flow undergoes a vortex breakdown, the outer flow has not yet turned the corner to achieve its maximum vertical velocity. Accordingly the peak swirl velocities occur in fluid with angular momentum that is only a small fraction of Γ∞.
c. Dynamic corner flow collapse
Figures 8 –11 are from a simulation, A1, in which the mean flow is unsteady. It is an example of what we have previously dubbed “corner flow collapse” (Lewellen et al. 2000b; Lewellen and Lewellen 2002). The evolution of Γ and w are summarized in Fig. 8, the peak velocities and pressure drop in Fig. 9, and different vertical profiles at the time of peak intensification in Figs. 10 and 11. The initial state (Fig. 8a) is from a quasi-steady simulation in which the domain swirl ratio is large but the corner flow swirl ratio is low, a consequence of a zero-swirl inflow layer above the surface. There is initially a large-scale central updraft, a vortex breakdown at about two-thirds of the domain height, and no large velocities near the surface. The evolution in Fig. 8 is triggered by shutting off the near-surface zero-swirl inflow at the domain boundary. Subsequently the low-swirl fluid in the surface layer is steadily exhausted up the core; its flux through the corner, ϒ, drops in time until Sc approaches S*c. At this point the corner flow collapses rapidly to smaller radii, driven both from above (by the inertia in the upper core flow removing low-swirl fluid from the corner) and from below (by the radial overshoot of near-surface flow for Sc near S*c). As a consequence, the peak velocities and pressure drops increase dramatically and drop in height to just above the surface (Fig. 9). At the time of peak intensification the structure, both on larger (Fig. 10) and smaller (Fig. 11) scales, exhibits many of the features of the continuously nested corner flow considered above, including the extended conical core and the inner-scale low-swirl corner with surface jet and vortex breakdown. An important difference is that the outer Γ contours are drawn into significantly smaller radii leading to enhanced near-surface intensification; they do not stagnate at larger radii, as in Fig. 6, because ϒ into the corner is now reduced relative to that in the core above. The intensification levels are increased correspondingly: at the time of peak intensification Iυ ≈ 8.7 and Ip ≈ 11.9 as measured from the azimuthal averages and Iυ ≈ 15.4 and Ip ≈ 18.0 taken from the local instantaneous peaks.
The highest intensification does not persist. The conical core produces a strong vertical pressure gradient that first decelerates the axial flow, then drives a narrow central downdraft to the surface, opening the core somewhat to produce a medium swirl configuration with a smaller but still significant near-surface intensification (Fig. 8d). In the final stages a much wider downdraft descends to the surface opening the core further and pushing the low-level vortex aside. This weakens and contorts the low-level vortex; animations of the simulation in these stages are qualitatively suggestive of the “roping out” stage of tornado evolution.
We note in passing the appearance of multiple local maxima in the swirl velocity located at different radii in Fig. 10b, even while the Γ distribution (Fig. 10a) indicates a single coherent circulation. Modest changes in Γ gradients can easily lead to multiple swirl velocity maxima, suggesting that the appearance of the latter is not conclusive evidence for identifying distinct circulation scales (e.g., mesocyclone, tornado cyclone, and tornado). Note also that, while Sc changes dramatically during the evolution (from below S*c to well above), the swirl ratio of the vortex as a whole (i.e., the domain swirl ratio in the simulation) changes very little because the inflow conditions over the bulk of the boundary remained unchanged.
The basic features of this simulation do not require fine tuning of either the initial near-surface inflow layer or of how it is shut off. Figure 12 shows the evolution of the peak pressure drop during two variant simulations, B1 and C3. In the first, the depth of the low-swirl inflow layer at the outer boundary is doubled, but with Γ dropping linearly from Γ∞ to 0 at the surface (changing the distribution of the depleted angular momentum flux but leaving its total, ϒ, virtually unchanged). The subsequent evolution from this quasi-steady initial state was again forced by shutting off all the inflow in this near-surface layer at the outer boundary. The second case was started from this same initial state, but the near-surface inflow was shut off only along one quadrant (centered at a corner) of the outer boundary. Both simulations produced large transient near-surface intensification (Fig. 12), with the same basic stages of evolution identifiable in the full velocity fields as in simulation A1. In the asymmetric case, the high Γ fluid that is drawn down toward the surface in the quadrant where the low-level inflow was shut off, spirals around the vortex as it approaches smaller radii—effectively blocking the low-swirl inflow from the other three-fourths of the boundary from reaching the vortex core. The inside of the spiral represents a strong shear layer that is subject to breaking into secondary vortices. The onset of the corner flow collapse is delayed and the inner corner flow occurs off center and translates, but the intensification remains strong. It should be noted that this simulation was performed at a lower central resolution than A1 and B1; because of the location and movement of the intensifying vortex at the surface, the fine grid had to cover a much greater area in the domain and therefore could not be as refined without increasing the computation significantly.
There are three identifiable ingredients in the near-surface intensification produced transiently during corner flow collapse, each contributing (for favorable conditions) approximately a factor of 2 in intensification over conditions aloft: 1) sweeping Sc over S*c; 2) nested corner flows on different scales; and 3) a true unsteady contribution with the total momentum flux through the corner decreasing in time. Ingredient 1 could be achieved by beginning with Sc < S*c and reducing ϒ (as done here) or beginning with Sc > S*c and increasing ϒ. Ingredient 3, however, requires ϒ decreasing. In this case, 1 and 3 work together to increase the intensification (the inertia of the low swirl fluid ahead helps to pull the high-swirl fluid following it into smaller radii) while for ϒ increasing they conflict (high swirl fluid precedes low-swirl fluid, impeding its radial overshoot) and, as illustrated in Lewellen et al. (2000b), little or no intensification results. If ∂ϒ/∂t is such that ingredients 1 and 3 are well correlated for large intensification then ingredient 2 tends to be produced automatically to some degree, though its extent and the resulting total intensification will depend somewhat on the details of the angular momentum gradients present. In a companion paper (Lewellen and Lewellen 2007) we survey results from scores of corner flow collapse simulations, exploring the dependence of its onset, magnitude, duration, and structure on several variables, as well as addressing important numerical/modeling issues. The basic scenario proves quite robust; however, in accordance with expectations from the unsteady analysis of section 2 and quasi-steady simulations above, there is a rich dependence on both the rate at which the low-swirl inflow is reduced and its distribution near the surface.
In Fig. 2 the low-swirl peak representing the largest mean intensification occurs for only a narrow range of Sc. The frequency of occurrence of such configurations might naturally be expected to be reduced accordingly. The quasi-steady nested corner flows presented above require an even higher degree of tuning of the conditions to achieve near critical low-swirl corner flows on different scales. This alone does not preclude their relevance to real tornadoes: only a small fraction of mesocyclones seem to produce severe tornadoes, some fine tuning of the circumstances is likely required. Nonetheless the potential importance of nested corner flows as mechanisms for intensification in real tornadoes likely rests on the natural occurrence of the inflow structure required for them during the scenario of corner flow collapse.
Corner flow collapse can occur over a wide range of conditions and naturally lead to large levels of near-surface intensification relative to conditions aloft (e.g., an order of magnitude in velocity scale). It seems to us likely that it will sometimes play a role on both tornado and mesocyclone scales, with the developing tornado/tornado cyclone corner flow naturally nested inside of the large-scale mesocyclone corner flow. On the tornado scale, corner flow collapse demonstrates how fairly modest changes in the low-level flow environment at larger radii can lead to rapid and significant changes in near-surface structure and intensity. While the mechanisms have not been identified, evidence of such changes is clearly seen in the damage tracks of many destructive tornadoes.
On larger scales corner flow collapse provides a possible mechanism for tornadogenesis and suggests the role that the rear-flank downdraft may play in it. The RFD has long been thought to sometimes play a critical role in tornadogenesis, but the mechanism is still a matter of debate [see Markowski (2002) for a recent review]. In the present scenario, an RFD wrapping around the mesocyclone and reaching the surface could impede the low-swirl inflow beneath the elevated mesocyclone, triggering the corner flow collapse process—first dropping the mesocyclone circulation to low levels and then rapidly producing an intense tornado. The critical role of the RFD in this picture is not in providing angular momentum and/or convergence that directly drives the tornado (though it might contribute), but rather to trigger the tornado indirectly by blocking access to low-swirl fluid at large radii thereby setting off corner flow collapse. In this scenario, low-level mesocyclogenesis and tornadogenesis are consequences of a single mechanism. The exhausting of the low-swirl fluid from the surface layer leading to the first is related to the dynamic pipe effect (Leslie 1971; Smith and Leslie 1979; Trapp and Davies-Jones 1997), though unlike the usual description of DPE it does not require cyclostrophic balance in the swirling flow, but requires the presence of the surface and an impediment to the low-swirl inflow at some outer radius. As discussed in Lewellen and Lewellen (2007), the strength of the tornado produced by the final corner flow collapse (or even its occurrence) is still dependent on additional factors (e.g., the rate at which the low-swirl inflow is impeded).
This scenario requires a particular configuration for the initial mesocyclone: a large swirl ratio for the vortex as a whole, but a low corner flow swirl ratio. The cyclostrophic imbalance in the excess low-swirl inflow producing low Sc ultimately provides the strong central convergence required for a tornado, while the high swirl above ultimately provides the necessary circulation. That the interaction with the surface provides the convergence for the final amplification of a tornado cyclone into a tornado was suggested earlier by Rotunno (1986). This mesocyclone configuration would generally include a vortex breakdown aloft, contrary to the suggestion of Trapp (2000). It is true, as Trapp argues, that a high-swirl vortex with a surface layer set up purely by surface friction (or a mature mesocyclone that has a thin surface layer relative to its core size) will not have the corner flow dynamics required to produce a supercritical flow and hence vortex breakdown aloft. This is circumvented by the possibility that the high-swirl flow has yet to reach all the way to the surface, permitting a mesocyclone state with a low Sc (and hence a vortex breakdown) even though the mesocyclone as a whole has a high swirl ratio.
Bluestein et al. (2003) suggest that some of their Doppler radar measurements (particularly the appearance of a small-scale bow-shaped echo preceding a rapid tornadogenesis by only several minutes) may indicate corner flow collapse at work. The appearance of an arc of vortex signatures with the tornado at its tip in their observations is also consistent with the shear layer development found in the asymmetric type corner flow collapse.
The appearance of nested corner flow structure on different scales in the simulations shows that, in general, any single parameter (e.g., Sc) is insufficient to completely characterize a corner flow. Given a Γ distribution that roughly approximates a step function, Sc provides a reasonable categorization (cf. section 2c), but for more complex Γ distributions generalizing Sc to a profile (e.g., using corner flows defined within successive Γ contours) may be better. Generalizing the angular momentum accounting to nonaxisymmetric conditions (e.g., tilted, twisting, or translating vortices) would be of even greater utility.
Determining whether, or how frequently, the mechanisms considered for enhanced intensification play a role in tornadic storms will require more complete observations of evolving wind fields (particularly at low levels) and/or analysis of severe storm simulations with high resolution in the lower layers. It should be interesting and feasible to study some of these scenarios in laboratory tornado models as well. In any case, the relatively strong dependence of tornado intensity on variations in the near-surface flow likely make its predictability in terms of larger-scale mesocyclone observables more challenging than previously hoped.
There is a physical feedback that tends to limit the magnitude of the near-surface intensification of a vortex: higher swirl velocities and pressure drops near the surface relative to conditions aloft imply a vertical pressure gradient that tends to drive a core downdraft, driving the vortex toward cylindrical symmetry and reducing the intensification. One means to circumvent this is to balance the vertical pressure gradient, partially or entirely, with buoyant forces. Our focus here has been another well-known, but purely fluid dynamic, mechanism that can work along with buoyant forcing: balancing the vertical pressure gradient with core updrafts (either central or annular) that decelerate with height, with the enhanced updraft at low levels supported by a radial overshoot in the vortex corner flow produced by cyclostrophic imbalance in the surface layer. A simple analytical model (Barcilon 1967; Fiedler and Rotunno 1986) suggests that the level of intensification that can be supported by this mechanism is limited by the conservation of mass, angular momentum, and vertical momentum to a factor of ∼2, a result supported by many laboratory and numerical studies. In the first part of the present work we generalized this model to consider corner flows besides the supercritical end-wall vortex, more general angular momentum distributions, and time dependence. The model illuminates the importance of the corner flow swirl ratio in determining quasi-steady corner flow structure and intensification. It also suggests how much larger intensification levels can be made consistent with mass conservation, angular momentum conservation, and the vertical momentum equation.
In the second part of the paper some of these corner flows with enhanced intensification were realized using large-eddy simulations. Quasi-steady nested inner and outer corner flows with intensification factors approximately twice any previously reported in quasi–steady state were realized by tuning the near-surface inflow layer at the outer boundary of the simulations. Many of the features observed in these simulations, and additional intensification, were realized without fine tuning in a class of unsteady evolutions producing a dynamic corner flow collapse.
Beginning with a vortex with a high domain swirl ratio but a very low Sc due to a large low-swirl flux in the surface layer, very large (∼10) transient intensification relative to conditions aloft were produced by reducing that low-swirl near-surface flux. These scenarios are considered in more detail in a companion paper (Lewellen and Lewellen 2007). A wide range of perturbations proves capable of triggering this phenomena including both complete and partial shutoff of the near-surface inflow at some outer radius or an increase in the swirl component of that flux. Given its robustness and the magnitude of the near-surface intensification achieved, corner flow collapse is an attractive possible mechanism that may sometimes contribute on the tornado scale to tornado variability and on the mesocyclone scale to tornadogenesis.
This work was supported by National Science Foundation Grant ATM236667.
Derivation of Analytic Model Equations
To obtain (2) from (1) we proceed as sketched in the main text. Assume quasi-steady conditions, take time and azimuthal averages, neglect turbulent fluxes into the control volume in comparison with mean fluxes, and split the control surface pressure integral, then (1) becomes
On the lower surface we use a fractional area coordinate, A, to write
We have neglected w(r, zl) inside of ri and outside of ro in comparison to its value within the main updraft annulus for simplicity. Mass and angular momentum conservation are satisfied by choosing (again for simplicity) profiles of the same form at zu,
Assuming that the swirl velocity dominates the radial outside of ro between zl and zu and on the zl and zu planes, we use approximate cyclostrophic balance, ∂p/∂r = −Γ2/r3, to rewrite the pressure terms in (A1). On the outer boundary p(ro, z(ro)) = −Γ2∞/(2r 2o), so the last term becomes −Γ2∞ ln (rou/rol). The other pressure terms can be integrated by parts to give
where j = l or u and βj ≡ r 2ij/(r 2oj − r 2ij).
Outside of ro and upstream of zl the flow is assumed approximately inviscid so Bernoulli’s equation applies with Bernoulli constant ≈0. The first term in (A1) can then be written
where an integration by parts has again been used and the dependence on the Γ profile found to cancel. The inside pressure is found by integrating the cyclostrophic condition,
Using (A5) it is easy to show that
Collecting the pieces together and canceling out a common factor of Γ2∞ then results in (2).
The ril dependence is eliminated using πSc = rolrou/ (r 2ol − r 2il) and (5) follows given the definitions of I and βu.
Mass conservation is again used to relate the vertical velocity at zu to that at zl,
In the unsteady case Bernoulli’s “constant” is no longer constant. It changes, following a fluid parcel, by the local rate of change of the pressure. Representing the accumulated difference up to zl by −δH, the unsteady version of Bernoulli can still be used to substitute for wl,
where 𝒥 ≡ 1 − 2δHr 2ol/Γ2∞ gives the change from the steady case. Similarly we now have, πSc𝒥 = rolrou/(r 2ol − r 2il). Using these relations in (A9) with the additional definitions,
To obtain an idea of the possible intensification we assume some functional form for r0(z, t) in order to compute and . For simplicity, we assume that rou, zl, and zu remain unchanged in time and define dimensionless variables ζ ≡ (z − zl)/(zu − zl), R ≡ rol(t)/rou, and λ ≡ Γ∞t/[rou(zu − zl)]. Then we can write a general form ro(z, t) = rouG(ℛ, ζ) and compute,
We can use mass conservation [the generalization of (A10) with control volume defined up to height z] to reexpress r 20(z)w(z) in order to evaluate , giving
Since I = 1/R, (A15) together with (A16) and (A17) defines a second-order ODE for R that may be integrated numerically once a form for G is provided. Figure A1 gives results from a representative example (c = 6, b = 1) of the form G (R, ζ) = R(1 + bζc) + [1 − (1 + b)R]ζ (a form, motivated by corner flow collapse simulations like Fig. 8, that as R drops to zero interpolates smoothly between a near cylindrical form with a flared top to a cone). The solution does not include a central downdraft; instead, an acceleration of w at low levels balances the increasing vertical pressure gradient. Maintaining this structure throughout the evolution (leading to a singular intensification) requires a particular ϒ evolution into the corner that would not actually arise without fine tuning. Note that dR/dλ = (zu − zl)/(π Γ3∞)(∂ϒl/∂t) represents a nondimensional rate of change of ϒ flowing through the corner, which will be shown in Lewellen and Lewellen (2007) to play an important role in the corner flow collapse phenomena.
LES Model and Simulation Parameters
Equations and numerics
The LES model used for the present study is as described in Lewellen et al. (1997) and Lewellen et al. (2000a). It is a fully 3D unsteady implementation of the incompressible Navier–Stokes equations on staggered grids, with the grid spacing stretched independently in the three directions. A leapfrog scheme in time and second-order central differences in space (Piacsek and Williams 1970) are employed with a variable time step. The Poisson equation for pressure is solved directly on the nonuniform grid using the method of Farnell (1980) in the horizontal directions and a tridiagonal matrix inversion in the vertical. The current code has been extended to allow temperature and humidity variables, moist thermodynamics, and single or multiple debris species, though none of these features has been employed for the present simulations. There is as well a compressible version of the code.
The subgrid model includes a prognostic equation for subgrid turbulence kinetic energy (q2/2) and a subgrid turbulence length scale given diagnostically by
The terms included depend on (in order) the local grid spacing, distance from the surface, Brunt–Väisälä frequency N, and an incorporation of local rotational damping effects where
V is the velocity vector, p the pressure, and repeated indices are summed.2
The surface boundary condition is no-slip with a specified aerodynamic roughness length z0. The lateral and top boundary conditions are imposed on the velocities: fixed values except for tangential components on outflow, which are taken zero slope. Generally, at a given height the lateral inflow on the square boundary is specified so that it has constant values of angular momentum and horizontal convergence. Simulations are run until they have reached a quasi-steady state (so that the initial condition is immaterial) or, for simulations studying time evolution, begun from a quasi-steady state of some prior simulation.
Comments on the use of open boundary conditions
It is important in performing limited “open” domain simulations to insure that the boundary conditions are compatible with a realistic flow field outside of the domain. The influence of the flows inside and outside of the simulated domain on each other can, in principle, be rigorously included through the imposed boundary conditions, if the correct flow field can be imposed there, because the underlying physics is local in space and time. Approximating this can, nonetheless, be problematic if the flow development inside the domain is such that it would force a strong (and a priori unknown) variation on the boundary and hence in the flow outside the domain (e.g., Fiedler 1995).
Here, as in prior work, we have concentrated on two general classes of simulations in which this difficulty is, to good approximation, avoided: quasi-steady simulations and evolving simulations in which strong time variation occurs only within an interior subdomain. In the first case, the feedback between the interior and exterior of the domain is included implicitly: the two are assumed to be in equilibrium with each other, resulting in the given steady boundary condition. The self-consistency of this approach (which does not account for turbulent fluctuations across the boundary) has been checked by 1) first-time averaging 3D results from a quasi-steady simulation, 2) using this average to define boundary conditions on a much smaller subdomain, 3) simulating the flow within the subdomain, and 4) finally comparing the time average results in the large and small domains for the flow within the subdomain. For quasi-steady corner flow simulations the results are found to agree admirably as long as the boundaries are not within the corner flow region itself (with its very strong gradients and turbulent fluctuations). In the second case, the flow near the boundaries changes little during the development studied (provided the domains chosen are large enough) so the potential influence on the exterior flow is minimal. That corner flow collapse simulations fall into this category is demonstrated in Lewellen and Lewellen (2007). In either the quasi-steady or time-varying cases the physical results considered are restricted to correlations of flow variables within the simulated domain (e.g., surface intensification relative to core conditions aloft), but not outside of it (e.g., maximum wind speeds possible for a given CAPE).
The conditions for the principal simulations considered in the text are summarized in Table B1. The conditions are nondimensionalized using the far-field angular momentum level (Γ∞) and the domain “radius” (rd ≡ half the lateral domain size) to form length (rd), time (ts ≡ r 2d/Γ∞), and velocity (Vs ≡ Γ∞/rd) scales. As examples, a dimensionalization on the tornado scale could be Γ∞ = 104 m2 s−1 and rd = 1 km giving ts = 100 s and Vs = 10 m s−1; or, on the mesocyclone scale, Γ∞ = 4 × 104 m2 s−1 and rd = 5 km produce ts = 625 s and Vs = 8 m s−1.
Corresponding author address: D. C. Lewellen, MAE Dept., P.O. Box 6106, West Virginia University, Morgantown, WV 26506-6106. Email: email@example.com
Accordingly, when we refer to, for example, momentum within a given volume, it may more properly be thought of as momentum per unit mass.