1. Introduction

The rain curtain associated with the hook-shaped appendage to a supercell’s radar echo is usually regarded as a passive indicator of a possible tornado. Close-range airborne and mobile radar observations made during the Verification of the Origins of Rotation in Tornadoes Experiment (VORTEX; Rasmussen et al. 1994) in 1994–95 and in subsequent follow-up experiments have revealed the presence of a hook echo prior to tornadogenesis in every case. Development of the hook is generally thought to result from horizontal advection of precipitation in the mesocyclone (Glickman 2000). In at least some cases, however, hook-echo formation seems to be due to a rain curtain descending in the rear-flank downdraft (Markowski 2002). Can this rain curtain, as it descends in the mesocyclone, trigger a tornado? If so, the hook echo would be more than just an indicator of tornadogenesis; its associated rain curtain would be an instigator of it (Davies-Jones 1998; Davies-Jones et al. 2001) through a baroclinic or a barotropic mechanism or a combination of both effects.

In the baroclinic mechanism, the tornado derives its spin from vorticity generated baroclinically near the ground as suggested by diagnoses of supercell simulations (Rotunno and Klemp 1985; Davies-Jones and Brooks 1993; Trapp and Fiedler 1995; Adlerman et al. 1999). Davies-Jones (2000a) devised an analytical Lagrangian model to demonstrate how an air current passing though a rain-cooled downdraft acquires significant baroclinic vorticity during descent (see also Davies-Jones et al. 2001). Air on the left (right) side of the current exits the downdraft with streamwise (antistreamwise) horizontal and cyclonic (anticyclonic) vertical vorticity. In a supercell’s rear-flank downdraft (RFD), air on the left passes out of the downdraft near the ground and is entrained into the storm’s main updraft. Its cyclonic spin is greatly amplified by vertical stretching of cyclonic vorticity and tilting up of streamwise vorticity. In contrast, air exiting the RFD on its right side usually encounters lesser or no updraft, and so is only occasionally spun up into a generally weaker anticyclonic tornado. According to this theory, there should be a thin region of rain-cooled air roughly coincident with the hook-associated rain curtain (or simply “hook” from now on). Although mobile mesonet observations have detected the presence of cold hooks in the vicinity of a few tornadoes, surprisingly, rain-cooled low-θ_E air was not detected at the surface within a few hundred meters to several kilometers of several strong and violent tornadoes intercepted during the VORTEX and follow-on field programs (Markowski et al. 2002). Incidentally, the bias of numerical simulations with simple microphysical schemes to cold hooks can be due to the autoconversion scheme producing rainwater too soon and too low in the cloud (Straka and Rasmussen 1997). The production and also evaporation of rain are excessive, and can cause cold pools to form too quickly (Gilmore et al. 2004).

One explanation for the lack of a surface baroclinic zone near some major tornadoes is that the air entering the tornado passed through a baroclinic zone that is either aloft or at a surface location that is several kilometers away from the tornado (in a storm-relative reference frame). Another possibility, the one investigated here, is that the mechanism in these cases is barotropic rather than baroclinic. An axisymmetric numerical model (Davies-Jones 2000b, 2006; hereafter called the D-J model) is devised that demonstrates how a tornado can form from barotropic vorticity alone. It largely simulates Fujita’s (1973, 1975) recycling hypothesis of tornadogenesis (Fig. 1). Fujita observed that the overshooting top collapses prior to a major tornado and that the rotating rain curtain curves toward the tornado, suggesting that a downdraft and its outflow may be transporting angular momentum (AM) to the tornado. In his conceptual model, the collapsing top triggers the tornado by intensifying the water-loaded cyclonically twisting downdraft. Air with high AM flows out of the downdraft near the ground. Some of it converges toward the axis and enters the tornado, thus maintaining it. Fujita originally thought that the high-AM air at the surfaces originates from near storm top (Fig. 1), but he later conceded that this was unrealistic from a thermodynamic viewpoint (Forbes and Bluestein 2001) as pointed out by Davies-Jones (1986; hereafter DJ86). Markowski et al. (2003, hereafter MSR03, p. 196) hypothesize that the air that enters the tornado may have risen from low levels and then come back down to the ground.

The most realistic way of simulating a tornado-producing supercell is with a complex 3D cloud model. However, in a very intricate flow it may not be possible to identify the tornadogenesis mechanism (Rotunno 1986, p. 427). A “bare-bones” approach is adopted here instead. The model is made as simple as possible to isolate the essential physics and make the cause of the simulated vortex formation clear-cut. Emanuel (1991, p. 189) has used this approach to construct a minimal hurricane model. In helical environments (i.e., ones in which the storm-relative environmental winds are strong and veer markedly with height), updrafts rotate at midlevels owing to upward tilting of environmental streamwise vorticity, and a midlevel mesocyclone forms (Lilly 1982, 1983; Davies-Jones 1984; Rotunno and Klemp 1985). After the updraft becomes water loaded, downdrafts are induced by the drag exerted on the air by falling hydrometeors (hereafter called particles) and by evaporative chilling of dry air. The aim of the model is to demonstrate whether precipitation drag alone can lower rotation to near the ground, and thus provide an explanation for the puzzling lack of surface baroclinicity around some observed strong and violent tornadoes. In the axisymmetric model, torques associated with variation in the drag forces in the radial direction generate azimuthal vorticity. However, the constraint of axisymmetry prevents tilting of this vorticity toward the vertical. Diffusion of angular momentum is a relatively slow process at the Reynolds number of 2000 used in this paper. Hence, angular momentum is nearly conserved following a parcel. Any tornado-like vortex that forms in the model can originate only from inward and downward transport of angular momentum from the initial mesocyclone aloft, not from the baroclinic mechanism. In a less constrained three-dimensional model with a rain curtain that is arc-shaped (like a radar hook) instead of completely circular, a mixture of the barotropic and baroclinic mechanisms can occur.

2. Review of axisymmetric tornado models and design of this model

Lewellen (1976, 1993) has reviewed axisymmetric laminar tornado models of tornadoes prior to 1993. These models have been designed for different purposes, such as reproductions of the flow in laboratory tornado chambers (e.g., Rotunno 1977), demonstrations of tornado intensity (e.g., Fiedler 1993, 1994) and tornado structure (e.g., Walko 1988; Nolan and Farrell 1999), and simulations of tornadogenesis in a central updraft (e.g., Proctor 1979; Leslie and Smith 1978; Smith and Leslie 1978, 1979; Eskridge and Das 1976; Das 1983; MSR03).

The tornadogenesis models most like the present one are those built by Das (1983) and MSR03. In the Das model, a gush of precipitation is released on the axis at the low top of an open cylindrical domain, and centrifuging of particles forms a curtain of small radius (∼700 m). (Incidentally, the drag exerted on the air in the radial direction was omitted.) This simulation produces a vortex by the recycling process. However, the mechanism works only for a very large maximum mixing ratio (16 g kg⁻¹) and with centrifuging. Das admitted that such intense gushes of precipitation are not observed along the axes of developing tornadoes.

Since the MSR03 model is an extension of the D-J one, we will describe the D-J model design first. The intent of the D-J model is not to replicate the detailed structure of a tornado, which is best done using models with sophisticated turbulence parameterizations (e.g., Lewellen et al. 2000), but rather to demonstrate a tornadogenesis mechanism that may occur in supercells. Therefore, the domain is both large enough to contain a complete storm-scale circulation and closed. This avoids uncoupling the tornado from its parent storm and simplifies global energy and other budgets. Fiedler (1995) has demonstrated the dangers in modeling a tornado in isolation from its parent storm. He found profound changes in vortex intensity when the domain is reduced from a closed domain encompassing the entire storm to a much smaller open domain with restricted characteristics. The smaller open domain includes only the lower regions of the tornado and its immediate surroundings and artificially isolates the lower portion of the updraft from the updraft aloft and the compensating downdraft. The flow in a small open domain is controlled largely by conditions at the open boundaries, which are often selected quite arbitrarily. The interior flow does not interact freely with the implied storm outside the domain. MSR03 provide additional reasons for making the domain large.

The domain is nonrotating because the earth’s vorticity is insignificant compared to horizontal vorticity in supercell environments (Davies-Jones 1984). The swirl is present in the initial conditions and is not maintained externally by rotating the domain or by imposing a tangential wind as an inflow side-boundary condition. In fact, there is no tangential wind at the side and hence no circulation around any circuit contained in the sidewall. This vanishing of circulation around a closed convective cell is consistent with the generation of vertical vorticity within the cell through tilting of initially horizontal vorticity by the cell’s drafts.

The initial state is a Beltrami flow (BF) that describes a rotating updraft surrounded by downdraft. This state is plausible because it includes the convection that has generated the storm-scale rotation. The BF is balanced (apart from a slow viscous decay of amplitude). In other words, if it is left unperturbed, its pattern does not change. A vortex will not form simply as an unintended consequence of a particular choice of unbalanced initial and boundary conditions. Baroclinic Beltrami flows are dynamically impossible (Yih 1969, p. 80), so the flow in the model is statically neutral and there is no latent heat release. These are not large drawbacks for our demonstration because we want to isolate a barotropic mechanism for tornadogenesis and to preclude thermal baroclinic zones, buoyancy oscillations, gravity waves, and recirculation of thermal anomalies in the closed domain.

The D-J model is also set up to produce a realistic rain curtain (albeit axisymmetric). Particles are released into the divergent flow at the top of a mature updraft so they fall near the updraft/downdraft interface in a curtain on the scale of a hook echo (roughly 5 km in radius). The liquid water mixing ratio is moderate.

MSR03 extended the D-J model by including stratification and latent heat, effects that are also represented in the models of Proctor (1979), Smith and Leslie (1979), and Walko (1988), none of which have precipitation. MSR03 (797–800) compared their model design with those of Leslie and Smith (1978), Smith and Leslie (1978, 1979), Das (1983), and Walko (1988). As in the D-J model, a precipitation-induced downdraft transports air rich in angular momentum to the ground, and a tornado develops as a result of convergence of this air beneath the central updraft. MSR03 demonstrated successfully that, in agreement with the VORTEX observations, simulated tornadoes are stronger and longer lived when the annular downdraft is relatively warmer and the circulation-rich air is more easily lifted by the updraft.

Compared to the D-J model, the MSR03 model is more realistic regarding inclusion of stratification and latent heat. However, it also has some added unrealistic features. For example, the annular downdraft is predominantly cyclonic, whereas downdrafts are primarily anticyclonic in helical environments. This is due to the initial tangential wind field, a height-varying Rankine combined vortex, having a circulation that increases outward to an asymptotic value instead of returning toward zero in the compensating downdraft. MSR03 state that this “makes vortex intensity less sensitive to the trajectories of the parcels that enter it.” Moreover, the updraft in their model persists only if the initial core radius of the vortex is less than about 800 m. Thus, the initial vortex is much narrower than a mesocyclone. Similarly, the updraft that evolves after passage of the initial thermal bubble is much narrower than observed supercell updrafts. The location of the precipitation-induced downdraft is controlled to a much greater extent in the MSR03 model. In the D-J model, precipitation particles are released through the top above the updraft and are advected around the updraft by the winds as they fall though a great depth, forming a rain curtain that falls from near the storm summit (Forbes 1981). Rain could not be introduced high in the storm in the MSR03 model because of excessive recycling of rainwater in the updraft. Instead, rain was introduced in a toroidal region with maximum intensity at a height of 3000 m and a radial distance of 1750 m from the model’s axis.

3. The initial Beltrami flow

The model uses a Beltrami-flow representation of a midlevel mesocyclone (Davies-Jones 1985, 2002) as its initial condition (IC). This initial flow is chosen because it mimics nature with regards to establishment of a rotating updraft prior to tornadogenesis and it is balanced (apart from a slow viscous decay of amplitude). Henceforth, all variables are nondimensionalized by scaling lengths by the height of the domain H, velocities by the initial maximum vertical velocity W₀, times by the advective time scale H/W₀, accelerations and forces per unit mass by W₀²/H, and specific energies by W₀². In a Beltrami flow the vorticity vector ω is parallel everywhere to the velocity vector v; that is,

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where λ is the abnormality (Aris 1962, p. 72). The Boussinesq continuity equation,

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is used to simplify the equations and to avoid use of modified diffusion terms. Since ∇ · v = 0 and ∇ · ω ≡ 0, λ is made a constant. In this case the flow is called Trkalian (Aris 1962, p. 73). The equation of motion governing Boussinesq flow with constant eddy viscosity ν in a nonrotating adiabatic atmosphere is

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where t is time, V ≡ |v| is the wind speed, σ is the dry static energy minus its constant value in the resting adiabatic atmosphere, −∇σ is the nonhydrostatic pressure-gradient force (NHPGF) per unit mass, and Re ≡ H W₀/ν is the Reynolds number. Here σ is given by

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where p is pressure, p₀ is the surface hydrostatic pressure, g is the gravitational acceleration, c_p is the specific heat at constant pressure, R is the gas constant for dry air, κ ≡ R/c_p = 2/7, θ₀ and H₀ = c_pθ₀/g are the potential temperature and height of the adiabatic atmosphere, G ≡ gH/W₀² is the nondimensional gravitational acceleration, and the domain is bounded by rigid nonporous bottom and top surfaces at z = 0 and z = 1. Provided that it is small compared to the hydrostatic pressure, the dimensional nonhydrostatic pressure p*_nh is related to σ by p*_nh = ρ₀W²₀(1 − Hz/H₀)^5/2σ, where ρ₀ is the surface density.

Inserting (3.1) into (3.3) reduces the equation of motion to

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The associated vorticity equation is

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owing to cancellation between the vorticity–advection and the vortex-stretching and tilting terms in the full vector vorticity equation ∂ω/∂t + v · ∇ω = ω · ∇v − Re⁻¹∇ × ∇ × ω. Together (3.6) and (3.1) imply that

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(Shapiro 1993), where x is the position vector. The inclusion of constant viscosity causes the amplitudes to decay exponentially, but does not affect the flow pattern. Subtracting (3.7) from (3.5) shows that the NHPGF balances the inertia force and yields the Bernoulli relationship,

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The sum of static and kinetic energies is constant in space and time in a Beltrami flow.

Trkalian solutions are found as follows. Taking the curl of (3.1) gives

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After use of (3.2) the vertical component of (3.10) becomes the Helmholtz boundary-value problem

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Given a solution of (3.11) for w, the other velocity components and the pressure are found from (3.1) and (3.9). Since the velocity components satisfy linear equations, the superposition of two Trkalian flows (one denoted by a subscript \\, the other by subscript o) with the same abnormality is also a Trkalian flow (unsubscripted) with the same λ. The static energy, given by

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has a term that involves interaction between the two flows.

Davies-Jones (1985, 2002) presented a steady inviscid Trkalian solution in cylindrical coordinates (r, ϕ, z) that describes a central axisymmetric nonbuoyant updraft in an unstratified environment superposed on winds characterized by steady veering and constant speed V_\\ with height. Here we use Shapiro’s generalization (3.8) to obtain the corresponding viscously decaying solution. In strongly sheared environments the earth’s vorticity is negligible (Davies-Jones 1984). Thus the vorticity of the environmental wind v_\\(ϕ, z) is ω_\\(ϕ, z) = (−∂υ_\\/∂z, ∂u_\\/∂z, υ_\\/r − ∂u_\\/r∂ϕ). The Trkalian constraint ω_\\(ϕ, z) = λv_\\(ϕ, z) is satisfied by

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where λ is the constant rate at which the wind is veering with height and V_\\(t) = V_\\(0) exp(−Re⁻¹λ²t). In Cartesian coordinates v_\\ = V_\\(t)(−sinλz, cosλz, 0) so the hodograph is a clockwise-turning circular arc. Let the deviations of wind and vorticity from the environmental state be given in cylindrical coordinates by v_o ≡ v − v_\\ ≡ (u_o, υ_o, w_o), ω_o ≡ ω − ω_\\ ≡ (ξ_o, η_o, ζ_o), and impose the boundary conditions w_o = 0 at z = 0 and 1. An axisymmetric solution of (3.11) is

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where J_m is the Bessel function of order m, μ = π, k = (λ² − μ²)^1/2, and λ² > μ². From integration of ζ_o = λw_o where w_o = ∂(rυ_o)/r∂r for an axisymmetric wind,

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and from integration of the axisymmetric continuity equation,

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The axisymmetric solution consists of a central updraft that is rotating cyclonically at midlevels surrounded by a lower-amplitude concentric anticyclonic downdraft, which in turn is surrounded by a lesser cyclonic updraft, and so on out to radial infinity. A convective cell developing in the helical environment would naturally approximate the central part of this Beltrami flow prior to becoming water loaded (Davies-Jones 2002).

From (3.9) and (3.13) the static energy,

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has an asymmetric part σ_o\\ owing to the interaction between the environmental and deviation winds and an axisymmetric part σ_oo (Davies-Jones 2002).

Incidentally, the angular momentum about the axis relates only to the axisymmetric flow. The environmental flow has no circulation around a horizontal circuit, so we can define angular momentum M as υ_or. It is easily shown that the axisymmetric flow has a Stokes streamfunction ψ_o = M/λ and an azimuthal vorticity η_o = λM/r.

The axisymmetric model takes the inner axisymmetric part (v = v_o, σ = σ_oo) of the above Beltrami solution as its IC (Fig. 2). It is assumed implicitly that the rotating updraft and surrounding concentric anticyclonic downdraft are established through interaction with the veering environmental wind field. In other words, the environmental flow imposes its abnormality on the axisymmetric flow. The model subsequently discards the (asymmetric) advection by the environmental wind and the asymmetric part of the pressure field arising from the interaction of the convective cell and the veering wind field, omissions that are frequently made by tornado modelers (e.g., Lewellen 1993; Fiedler 1993). These effects will be included in the future in more realistic, but still idealized, three-dimensional simulations. The o subscript is now unnecessary and is dropped henceforth.

The nonrotating domain is closed by insertion of a lateral boundary at the first zero (apart from the axis) in the radial profiles of radial velocity and tangential velocity. The model’s IC thus consists of a closed radial–vertical circulation with a central updraft that rotates cyclonically at midaltitudes and so resembles a mesocyclone aloft, surrounded by a compensating downdraft. Even though it contains air that is revolving about the axis in a cyclonic direction, the downdraft is a region of anticyclonic vorticity because the shear vorticity there is negative and larger in magnitude than the positive curvature vorticity.

We let the environmental wind that establishes the initial axisymmetric flow have a full-circle hodograph (λ = 2π) centered at the origin. Lilly (1982, 1983) used such a hodograph in his 3D numerical simulations and found that the storm motion is almost stationary. Some properties of the initial axisymmetric solution are presented in Table 1. The aspect ratio (radius/height) of the domain R = 0.7042 is determined by the constraint that λ = 2π and the boundary condition u = 0 at r = R [J₁(kR) = 0 ⇒ R = 3.8317/k where k = (λ² − μ²)^1/2 = √3π]. For illustrative purposes, the values in Table 1 are scaled to an updraft 12 km high with an initial maximum updraft speed of 34 m s⁻¹ even though the Boussinesq approximation is strictly valid only for shallow motions (≤0.1H₀ = 3 km deep). Values for the corresponding anelastic solution (Davies-Jones and Richardson 2002) are comparable, but the maxima of υ and w are lower down in the storm (roughly 0.4H instead of 0.5H). Moncrieff (1978) and Seitter and Kuo (1983) found that anelastic and Boussinesq solutions for deep convection are qualitatively similar. The Boussinesq equations are sufficient here because the aim of this paper is simply to demonstrate a mechanism. The above values for W₀ and H produce a typical mesocyclone with an updraft radius of 5.3 km, maximum tangential wind of 23 m s⁻¹ at r = 4.1 km, and a maximum pressure deficit of 4 mb (Table 1). With this scaling, one unit of time is 5.8 min and the side is at 8.4 km.

4. The model’s speed limit

A measure of a model’s success in producing a strong tornado is whether the maximum winds in the vortex significantly break the thermodynamic speed limit (Fiedler 1993, 1994). Fiedler concluded that the thermodynamic speed limit is only exceeded in supercritical end-wall vortices or in two-celled vortices. The speed limit is equal to the maximum updraft speed according to parcel theory. The analogous quantity in the D-J model is the maximum updraft speed (=1) in the initial Beltrami representation of the midlevel mesocyclone.

5. The numerical model

The bare-bones model is constructed as follows: The nondimensional equations are cast in the vorticity-streamfunction formulation. The streamfunction ψ satisfies u = −∂ψ/r∂z, w = ∂ψ/r∂r, and the azimuthal vorticity η ≡ ∂u/∂z − ∂w/∂r. The boundary conditions on velocity in the principal simulation are chosen to fit in with the initial Beltrami flow; namely,

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In terms of ψ and η, these boundary conditions are

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The surface of the closed domain is a vorticity surface (on which ω · n = 0). Thus helicity would be conserved but for the presence of nonconservative forces (Moffatt and Tsinober 1992). The cost of accommodating the Beltrami flow is the unrealistic boundary conditions on velocity (Shapiro 1993), namely, no slip for the azimuthal (tangential) motion and no stress for the radial–vertical motion. Note, however, that Lewellen (1993, p. 24) did advocate using these conditions at the ground in tornado models if a constant eddy viscosity is used because of the strong, almost radial flow into the vortex in the lower part of the turbulent boundary layer of a tornado. Compared to a laminar boundary layer, a turbulent boundary layer has much sharper velocity gradients near the surface. A constant-viscosity model with a no-slip condition applied to u as well as υ produces flow into the vortex that is weaker, deeper, more elevated and less radial than would occur in nature. At the other extreme, free-slip conditions on both u and υ would eliminate the boundary layer and an end-wall vortex would not form.

The model was tested by running it undisturbed (i.e., without rain) and comparing results with the exact Beltrami solution for Re = 2000. This Beltrami-flow simulation provided a valuable check of the model itself (Shapiro 1993) and of most of the terms in the global budgets. The agreement was very good. The flow pattern did not change discernibly. The maximum rotation remained at the midlevel, and the velocity components decayed almost precisely as exp(−λ²t/Re) without changing form. Pressure and global quadratic quantities such as kinetic energy and helicity decayed as exp(−2λ²t/Re) as in the exact solution. By time t = 6 the velocity and pressure amplitudes decay by 11% and 21%, respectively. Even though the downdraft transports angular momentum downward, there is no intensification of rotation at low elevations because there is no advection of angular momentum M (the streamlines and M contours coincide in Fig. 2). Without rain, the twisting downdraft does not lead to vortex formation.

The initial balance of the BF is upset by introducing a realistic amount of precipitation particles at the closed top boundary, a procedure used at an open top by Eskridge and Das (1976) and Das (1983) in tornadogenesis models and by Proctor (1988) in microburst simulations. The liquid water mixing ratio q at the top is given as a specified function of time and radial distance; namely,

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where τ = 0.5, s = 0 or 2, and the maximum mixing ratio Q = 5 g kg⁻¹. According to (5.4), particles are released at the top of the initial updraft with the mixing ratio having the same radial variation as the updraft and no particles are let loose above the initial downdraft. The mixing ratio at the top of the axis approaches 5 g kg⁻¹ by t = 1. When s = 0 the mixing ratio amplitude is maintained through the rest of the simulation. When s = 2 the amplitude declines to zero between t ≈ 4 and t = 7. The remaining boundary conditions on q are

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The latter is chosen because it is a weak constraint on q at the ground, which is an outflow boundary for water. Applying ∂q/∂z = 0 instead, as used by Eskridge and Das (1976), leads to an inaccurate water budget.

Inside the domain q is governed by a continuity equation (Kessler 1969). The fall velocity of the particles V_F (<0) is assumed to be a specified constant. Provision (C = 1) is made for the centrifuging of particles with a radial terminal velocity of (υ²/rG)|V_F|, where G ≡ gH/W₀² is the nondimensional gravitational acceleration. This effect is switched off by setting C = 0. The drag force (Cqυ²/r, 0, −Gq) exerted by the particles on the air is added to the equation of motion (3.3). The full set of nondimensional equations for the model is

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where Pr is the Prandtl number (eddy viscosity divided by the diffusivity of q). Static energy is an auxiliary variable that is governed by the diagnostic equation

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which is obtained by adding the drag term to (3.3), taking its divergence, and using (3.2). The boundary conditions for pressure are (from symmetry and the equations of motion),

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We make σ = 0 at r = R, z = 0 in order to obtain a unique solution.

In supplementary simulations, the lower boundary condition (LBC) was changed from ∂u/∂z = υ = 0 (hereafter called the mixed LBC) to free slip (∂u/∂z = ∂υ/∂z = 0) or no slip (u = υ = 0) to investigate its effect on vortex formation and intensity. The IC was altered near the ground to fit the LBC. The modified initial flow is given by

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(b = 0 for the original IC). This IC satisfies u = ∂u/∂z = υ = ∂υ/∂z = 0 at z = 0 and so can be used as a common beginning for all of the above LBCs. It satisfies the Beltrami criterion in two directions [ξ = λu, ζ = λw, but η = (k²υ − ∂²υ/∂z²)/λ ≠ λυ if b ≠ 0].

The numerical methods used to solve the model equations are described in the appendix.

6. The tornadogenesis simulation

In the simulations, G = 100, Re = 2000, Pr = 1, τ = 0.5, Q = 5 g kg⁻¹, s = 2 (precipitation tapers off), V_F = −0.25, and C = 0 (no centrifuging), unless stated otherwise. The unstaggered grid consisted of a square mesh of 201 × 285 points with Δr = Δz = 42.25 m when scaled to the mesocyclone. Doubling or halving this resolution affects the results minimally.

In the principal simulation, particles are released only above the updraft, the LBC is mixed, and the initial flow is Beltrami (Fig. 2). How the descending rain curtain upsets the Beltrami flow is evident at t = 3.5 (Fig. 3). The drag torque has generated positive (negative) azimuthal vorticity on the inward (outward) side of the curtain, causing the downdraft axis to move inward from the side and an inflow maximum to form on the updraft–downdraft interface just inward of the rain curtain at a height of H/4. This elevated inflow conveys air with higher angular momentum to the mesocyclone aloft and causes it to contract. The drag has hardly altered the surface inflow.

Figure 4 shows the fields at t = 4.5, a time when there is still only a mesocyclone. [A mesocyclone is defined here as a cyclonic vortex with core radius greater than 2 km, a tornado cyclone (TC) as a vortex with core radius less than 2 km that does not break the speed limit, and a tornado as a vortex that does break the speed limit.] The inflow maximum aloft is now more pronounced (Fig. 4). Particles have now reached the ground in an annular curtain about 5 km in radius with a little of the rain being recirculated by the updraft. The associated drag forces induce a secondary downdraft that intrudes into the updraft at low levels. The winds flowing out of this downdraft are weaker than the inflow to the updraft and so all the subsiding air near the ground flows toward the axis. This causes the precipitation streamer to curve inward near the ground and the maximum surface-inflow velocity to increase and move inward. In the atmosphere, this downdraft encroachment is visible as a clear slot and as cloud elements cascading down the side of the storm tower. The updraft narrows at mid- to low levels as it is squeezed by the downdraft. This squeezing and the downward transport of some air with fairly high angular momentum causes the mesocyclone to contract, descend, and spin up a bit. The mesocyclone is now within about 10% of being in cyclostrophic balance between the heights of H/8 and H/2 (1.5 and 6 km). The updraft maximum and pressure minima are now at a slightly lower height but their amplitudes have barely changed.

By t = 5.25 (Fig. 5), the mesocyclone has lowered, strengthened, and contracted into a tornado cyclone. A secondary maximum in tangential velocity has formed at low levels and is poised to contract and intensify into a tornado. Note that this maximum tangential velocity does not descend steadily from aloft as it would in the dynamic pipe effect (Smith and Leslie 1978; Trapp and Davies-Jones 1997). The downdraft has penetrated further into the updraft at low levels and brought air with moderately high angular momentum even closer to the foot of the axis. The maximum inward surface velocity continues to move inward and intensify. The vertical velocity is still upward all along the axis with a maximum that has descended slightly and is near the speed limit. The minimum pressure deficit on the axis remains aloft but is now larger. Pressures have fallen at low levels in the updraft.

Next, the air with relatively high angular momentum penetrates inward into the base of the central updraft where it is stretched upward in the increasingly convergent flow and spins up into a tornado, which is capped by a vortex breakdown and a transition to the broader tornado cyclone aloft. The axial¹ flow immediately above the breakdown is reduced but still upward. Downdraft is present very near (2 km from) the tornado as observed in Doppler radar and photogrammetric analyses (DJ86). Maximum tangential winds in the still-contracting tornado cyclone are almost equal to the speed limit, but are now located at quite low levels. At t = 6.07 (Fig. 6), the tornado is at its peak. It has a corner region with an intense axial jet. At low levels, practically all the rising air is now contained in the tornado. At the surface, radial flow into the corner region is 1.53 (∼52 m s⁻¹). Vertical and tangential winds in the axial jet far exceed the speed limit of 1 (the initial maximum updraft speed), reaching speeds of 3.49 (∼119 m s⁻¹) and 2.27 (∼77 m s⁻¹), respectively, while the tangential wind in the TC aloft stays roughly at the limit (Table 2). The nondimensional extreme tangential speed υ_m and vertical speed w_m are comparable to the theoretical values for the end-wall vortex deduced by Fiedler and Rotunno (1986) and values for some numerically simulated vortices (Lewellen et al. 2000). The ratios w_max/υ_max and −u_min/υ_max, where u_min is the minimum radial velocity, are 1.54 and 0.67, respectively. These values are slightly higher than the ones in Lewellen et al. 2000. The Bernoulli effect (w²_m/2 = 6.09) accounts for 93% of the extreme static energy deficit Δσ of 6.54 (Δp = 83 mb) in the axial jet. The maximum and minimum nonhydrostatic vertical pressure-gradient force (NHVPGF) per unit mass are 195 (or 1.95g) in the axial jet and −73 just above the axial jet. In response to large downward axial pressure-gradient force aloft, the upper half of the updraft has turned to downdraft. This is the model counterpart of collapse of the overshooting top in supercells. Outside the rain curtain, there is now outward flow along the surface.

The ratios υ_m/υ_c = 2.08 and (Δσ_max/Δσ_c)^1/2 = 2.34, where υ_c is the maximum tangential velocity above 2 km, Δσ_max is the maximum static energy deficit, and Δσ_c is the deficit on the axis at the height of υ_c. These measures of the near-surface intensification of the vortex lie in the ranges found by Lewellen et al. (2000). In the TC aloft (υ_c/Δσ_c)^1/2 = 1. This is close to the corresponding value of 0.95 in the initial Beltrami mesocyclone.

From a vorticity perspective, baroclinic vorticity is generated as azimuthal vorticity by drag-associated torques, but remains in this component because the axisymmetry prohibits tilting of azimuthal vorticity. Since angular momentum serves as a streamfunction for the other components of vorticity, the radial and vertical vorticity can be deduced from the contour plots for M. At a fixed height the areal average of vertical vorticity is zero because M = 0 at the side. Thus increasing cyclonic vorticity at a low elevation must be accompanied by more anticyclonic vorticity at the same level. This constraint is satisfied because the intensifying cyclonically revolving downdraft is wider and still anticyclonic. The penetration of this downdraft toward the axis surrounds the vortex with anticyclonic vorticity in the “clear slot.” This vorticity is tilted radially inward in the inward-directed outflow from the downdraft and then turned abruptly upward in the axial updraft. Low-level spinup results from stretching of this cyclonic vertical barotropic vorticity.

At first, the inertia of the updraft inhibits filling of the tornado from above and acts like a buoyant cork (i.e., a buoyancy force that prevents backflow; Fiedler 1995) for a while. The tornado decays at t = 6.9 as the pressure-driven axial downdraft penetrates almost to the surface and eliminates the updraft (and cork).

7. Time–height and time–radius diagrams

Time–height and time–radius diagrams show how quickly the tornado forms and how difficult it can be to provide tornado warning with much lead time. Contours of the maximum mixing ratio as a function of time and height are shown in Fig. 7. The specified mixing ratio at the top approaches maximum amplitude (0.005) in roughly 0.8 of a time unit. Significant surface rainfall (at rates ∼60 mm h⁻¹) begins at around 4 time units because the fall speed in still air is 0.25 and the domain height is 1. The maximum precipitation mixing ratio at low levels is less than 2 g kg⁻¹ prior to and during tornadogenesis with the heavier concentrations of water staying aloft.

The history of the axial velocity w_axis (t, z) is shown in Fig. 8. Between t = 4 and 5, the axial updraft weakens at upper levels and the maximum axial velocity w_max(t) descends from mid- to low altitudes without intensification as the mesocyclone enters its collapse stage (DJ86, p. 223). The maximum then lowers further as an end-wall vortex and associated axial jet form at around t = 5.4. The speed limit is exceeded significantly only in the axial jet. An axial downdraft begins at mid–upper levels just before t = 6 and reaches the surface at t = 7.

The maximum pressure deficit on the axis begins increasing and descending from midlevels at t = 4.5 (Fig. 9). It lowers to near the ground as tornadogenesis proceeds. The pressure falls very rapidly as the axial jet intensifies because it is roughly proportional to w²_max.

The associated axial NHVPGF evolution is shown in Fig. 10. The NHVPGF in the mesocyclone/TC becomes increasingly negative at progressively lower levels (above 1 km) as the cyclone contracts and descends (DJ86, p. 224). In the end-wall vortex itself, intense upward (downward) NHVPGFs below (above) the vertical velocity maximum accelerate (decelerate) air entering (exiting) the axial jet.

The air with the highest angular momentum (>0.200) stays between 3 and 9 km AGL throughout the simulation (Fig. 11). The precipitation-induced downdraft transports air with angular momentum of around 0.125 to near the surface between t = 3 and 6. Near-conservation of parcel angular momentum tells us that this air must descend at least 2 km.

The contours of maximum tangential velocity υ_max(t, z) (Fig. 12) provide indications of the two vortices: the mesocyclone/TC with maximum strength increasing and lowering from 6 to 3 km between t = 4 and 6; and a tornadic vortex at low levels that forms at t = 5.4, peaks at t = 6.1, and decays at t = 6.9. The locations of the maximum tangential velocity and minimum pressure drop suddenly to near the ground around t = 5.3 as the lower vortex becomes the stronger one (Figs. 12 and 9). The time–radius diagrams for the maximum tangential velocity at each radius in the lowest 2 km (Fig. 13a) and above 2 km (Fig. 13b) reveal that the mesocyclone contracts into a TC at around t = 5. The radii of maximum tangential wind in the lowest 2 km (Fig. 13a) and maximum surface inflow (Fig. 14) decrease steadily between t = 4 and t = 6 even though these parameters pertain to the mesocyclone and TC at first and then to the tornado. In this experiment, the evolution of the near-surface flow to mesocyclone to tornado cyclone to tornado is due to a continuous process of downward and inward AM advection and forced convergence. Low-level mesocyclogenesis and tornadogenesis are consequences of a single mechanism (as in Lewellen and Lewellen 2007a, p. 2188).

There is considerable inflow along the surface owing to the free-slip lower boundary condition on radial velocity. All through the simulation, because of the tendency for the pressure on a streamline to decrease as the velocity increases and vice versa, the maximum inflow is nearly collocated with the lowest pressure at the surface (an inflow low; see Davies-Jones 2002) and the pressure deficit at the foot of the axis (a stagnation point) is considerably smaller. When a tornado is present, this unrealistically small surface pressure deficit is a consequence of the free-slip condition on u.

8. Energy and other budgets

Radar is unable to detect tornadoes in supercells at medium to long ranges because beam spreading limits resolution and because low altitudes in the storm are below the radar horizon. Should a radar meteorologist decide not to issue a tornado warning because the supercell’s updraft seems to be declining? The answer is no based on the simulated evolution of global quantities, which provide bulk measures of how the storm is affected by tornadogenesis. Global kinetic energy (GKE) is partitioned into the kinetic energy (KE) of the u, υ, and w wind components (UKE, ZKE, WKE, respectively). An energy diagram is shown in Fig. 15. Frictional dissipation is a sink for all KE parts. The vertical KE (WKE) gains energy from the work done by the precipitation drag force. This is the only source of GKE. Also included in Fig. 15 are the values at t = 6 for the no-rain case (BF), in which there are no energy transformations apart from frictional dissipation. The Beltrami flow has minimal gradients of velocity. Therefore, in the flow with rain there is significantly more dissipation. This increase in dissipation is less than the work input so the GKE is slightly larger in the simulation with rain than in the no-rain case. Because the domain is closed, the pressure-gradient force (PGF) does no net work, but converts WKE into radial KE (UKE) when rain is present. The centrifugal force acts at right angles to the motion and so never does any work, but does change UKE into azimuthal KE (ZKE). Figures 16 –18 show how the KE parts evolve from their initial values to their values at maximum vortex intensity (t = 6) and beyond. During tornadogenesis, the NHVPGF becomes downward in most of the updraft. This causes the vertical KE to lose a lot more energy to UKE than it gains from work done by the drag force. Consequently, WKE is drastically reduced by the time the tornadic vortex is strongest, despite the extremely high WKE density in the axial jet. The jet contributes little to the overall WKE because its volume is small compared to the volume of the updraft. Tornado formation as the updraft declines has been observed often in Doppler radar analyses (DJ86, p. 223). There is considerably more UKE at t = 6 than present initially owing to the gain from WKE more than compensating for losses to ZKE and dissipation. The azimuthal KE gains from UKE through the centrifugal term but loses more energy to dissipation, so there is a slight loss of ZKE from t = 0 to t = 6. However, there is more ZKE in the simulation with rain than in the BF. These KE transformations only partly confirm the vortex-valve hypothesis (Lemon et al. 1975), which predicts that WKE is lost ultimately to ZKE, not to UKE.

After t = 6, the vortex weakens and the WKE recovers somewhat as the central storm-scale updraft is replaced by intensifying axial downdraft at progressively lower levels. Helicity declines precipitously during tornado genesis and decay owing to losses associated with the diffusion term in the helicity equation. The term containing the drag force is positive but quite insignificant. At the same time, enstrophy increases dramatically because dynamical production associated with stretching of vortex filaments overcomes large diffusive losses. Mean angular momentum in the domain declines slowly owing to the frictional torque exerted at the boundaries.

9. Supplementary experiments

In a preliminary experiment to see if a vortex would form no matter how the Beltrami flow was perturbed, the precipitation particles were replaced by light bubbles with a constant ascent rate that were released at the ground below the updraft to mimic the effect of buoyancy. The bubbles caused the maxima of rotation and updraft to ascend rather than lower. Winds near the ground remained moderate.

Turning centrifuging on (C = 1) or changing the Prandtl number to 1/3 has virtually no effect on the principal simulation. Maintaining the precipitation at the top (s = 0) instead of allowing it to taper off (s = 2) makes little difference up to the time of maximum tornado intensity (Table 2; also compare Fig. 6 herein with Fig. 3 in Davies-Jones 2006). However, it leads to a stronger central downdraft afterward as more rain falls along the axis.

The initial condition was changed to the slightly non-Beltrami one [(5.14); Fig. 19] to provide a common start that is compatible with a variety of LBCs. The initial flow is no longer balanced everywhere, but with a mixed or no-slip LBC a vortex does not form if rain is not introduced. With a free-slip LBC a weak tornado cyclone (υ_max = 0.5) develops at the surface even without rain. [This also happens when the Beltrami IC is used instead of (5.14).]

Table 2 summarizes vortex intensities in a series of experiments with rain and different LBCs or ICs. With the new IC and the mixed LBC, an intense tornado develops, but it is weaker than in the principal simulation because there is no inflow along the ground initially. When the LBC is altered to free slip, the centrifugal force is unreduced near the ground and, as the fluid spins up, is able in time to balance most of the inward PGF. Consequently, the radial inflow and the maximum AM advection are much weaker in this case. A short-lived tornado forms as its central downdraft descends to the ground and produces a two-celled structure with a widening inner cell instead of an end-wall vortex with an axial jet (Fig. 20). However, its maximum tangential wind (1.15) just exceeds the speed limit. The tornado does not have the frictional interaction with the ground that it needs to become strong. It is significant, however, that collapse to the tornado cyclone scale occurs even without interaction with the ground. In this model at least, the near-ground mesocyclone is just a passing phase during the collapse.

With the no-slip LBC, an end-wall vortex forms but it is considerably weaker than with a mixed LBC because the inflow into the vortex is slower, deeper, more elevated, and less radial, and the contours of angular momentum do not descend as far (Fig. 21). Consequently, the tornado develops slowly. It is stronger than the tornado in the free-slip case because the radial-velocity minimum, which is now just above the ground, moves inward further and drives air with relatively high AM closer to the axis (Figs. 20 and 21). In the simulations, the viscosity is low enough that this effect outweighs the loss of AM owing to the frictional torque (Rotunno 1979; Howells et al. 1988).

In a variation of the principal simulation, particles were released above the initial tangential wind maximum with a J₁(kr) radial distribution (as in the initial υ) instead of above the initial updraft [with the positive part of a J₀(kr) profile]. The maximum mixing ratio is the same as before so more rain is released in this case. It falls mainly in the original downdraft near the side boundary and drags down to near the ground air with less angular momentum (0.075) than in the principal simulation (0.125). Although a tornado cyclone forms near the ground (Table 2), it fails to spin up into a tornado.

10. Conditions for corner flow collapse

The surface-inflow maximum (u_min < 0) in the Beltrami flow is stationary and slowly fills owing to viscous decay. The following analysis describes the force imbalance needed for this minimum of u to move inward and deepen as it does during the corner flow collapse (CFC; Lewellen and Lewellen 2007b) that follows the introduction of rain. Since there is no surface inflow with the no-slip LBC, the method only applies to the mixed and free-slip LBCs.

We start with the formula for the slope of an isopleth of a scalar a in (t, r) space. This is

View Expanded

Choosing an isopleth of u (say, U < 0) and the zero isopleth of ∂u/∂r, we obtain Petterssen’s formulas (Petterssen 1956, p. 48) for the motion of a u isopleth,

View Expanded

and for the speed of the u minimum

View Expanded

where the subscript min denotes evaluation at the minimum. At the ground, ∂u/∂t = I + F, where I ≡ −u∂u/∂r is the radial inertial force (excluding the centrifugal force) and F is the rest of the radial force there [= −∂σ/∂r + υ²/r + Re⁻¹(∇²u − u/r ²)]. After substituting for the partial derivatives of u, the motion formulas become

View Expanded

View Expanded

Since u_min < 0, ∂I/∂r > 0 at the minimum. The minimum (where I = 0) will deepen if F is negative at its center (Petterssen 1956, p. 51). The radial friction force is relatively small and has a maximum near the u minimum. Thus it has almost no influence on the motion of the minimum. If we neglect radial friction, then I + F in (10.4) and (10.5) can be replaced by −∂B/∂r, where B = σ + u²/2 is the surface Bernoulli function, when the LBC is mixed.

For the Beltrami flow, c_min = 0 because the radial PGF (RPGF) and inertial force balance, the centrifugal force vanishes at the ground, and radial friction = −Re⁻¹λ²u. The individual isopleths move slowly into the minimum and disappear as it fills.

During CFC, the u minimum deepens, which needs F < 0 at the minimum, and moves inward slowly compared to the steering speed |u_min|, which requires that −∂I/∂r < ∂F/∂r < 0 at the minimum. Just inside the deepening minimum, the negative isopleths must also move inward so the net radial force I + F < 0 there. These are all necessary conditions for CFC to occur with mixed or free LBCs.

In the Beltrami flow, the Bernoulli function is zero with the KE cancelled by negative static energy. In the principal simulation, the rain-induced downdraft initially adds energy to the surface flow (B > 0) with u²/2 > |σ| inside the rain curtain except near the axis (Fig. 22). The inertial force is now slightly larger in magnitude than the RPGF everywhere except near the axis and the side boundary. The u minimum is deepening because the RPGF is inward there. The (negative) isopleths of u on the axis side of the minimum (where I < 0) are moving inward because ∂B/∂r < 0 there, those just on the outside are moving outward (consistent with a deepening minimum) because ∂B/∂r < 0, and those on the far outside are moving inward because ∂B/∂r > 0. The minimum moves inward because it is located where ∂²B/∂r ² < 0. Near the axis the RPGF is outward, so the minimum is not moving inward in response to a drop in axial pressure at the surface.

When the LBC is free, the centrifugal force enters into the equation for surface flow. It is an outward force and so acts to fill the u minimum. It opposes the inward motion of isopleths on the axis side of the minimum. The surface centrifugal force is small at the onset of CFC, but eventually becomes large, blocks further collapse, and prevents a strong tornado.

11. Conclusions

A simple axisymmetric model without buoyancy can simulate several observed features of tornadogenesis. The model reproduces on the mesocyclone scale an anticyclonic clear-slot intrusion of descending air and a collapsing top. On the small scale, it generates a tornado with winds that easily exceed the speed limit by concentrating barotropic vorticity. The tornado has an axial jet capped by a vortex breakdown that is an abrupt transition to a tornado cyclone aloft. Maximum winds in the TC are at roughly the speed limit.

The model thus demonstrates that a rain curtain descending along the updraft–downdraft interface can indeed instigate tornadogenesis barotropically. The tornadogenesis culprit in the model is undeniably the rain-induced downdraft/outflow, which transports some air with fairly high angular momentum downward and inward toward the axis and causes the corner flow of the tornado cyclone to collapse. A strong tornado does not form when the lower boundary condition is free slip. Some necessary conditions for corner flow collapse are established.

Global budgets indicate that a small conversion of work to kinetic energy (KE) associated with the descent of precipitation alters the flow in 30 min from the initial mesocyclone aloft with moderate surface winds to a tornado cyclone and tornado. The “altered state” has less vertical KE and helicity and more horizontal KE, enstrophy, dissipation, and diffusion. The most noteworthy global precursor of tornado formation is a dramatic decline in vertical kinetic energy despite the conversion of work by the drag force initially into WKE and the imminent formation of a high-speed axial jet. This decline is accompanied by rapid intensification of the global maxima in vertical, inflow, and tangential velocity and pressure deficit, as the penetration of air with higher angular momentum to near the foot of the axis initiates a tornado with an axial jet. During this process, the updraft maximum and pressure minimum move down the axis from mid- to low levels, and the tangential-velocity maximum moves steadily inward and jumps down to near the ground.