1. Introduction
The continuous propagation of an isolated thunderstorm updraft is determined largely by the linear and nonlinear interactions between the updraft and the environmental wind field. Conceptual models of these interactions have been used to deduce the dynamic pressure field around the flanks of an axisymmetric updraft in a sheared environment. Anomalous storm motion (relative to the mean tropospheric wind) has been attributed to asymmetrically distributed pressure or vertical pressure-gradient forces. The best-known models are those of Newton and Newton (1959, hereafter NN), Fujita and Grandoso (1968, hereafter FG), Rotunno and Klemp (1982, hereafter RK82), and Rotunno and Klemp (1985, hereafter RK85).
Newton and Newton (1959) postulated that the updraft acts like a solid obstacle, diverting the environmental flow around it. The updraft propagates toward the quadrant where there is an upward pressure-gradient force (PGF) and new updraft growth, and away from the opposite quadrant of downward PGF and updraft suppression. The obstacle analogy has been criticized by RK82 and Davies-Jones et al. (1994). The pressure distribution in NN is wrong for several reasons. At low levels, the updraft is not an obstacle because air accelerates toward it and enters it rather than slowing and going around it. At midlevels, the updraft is a quite porous obstacle, and numerical models reveal that the pressure gradient across the updraft is aligned with the environmental shear, not the storm-relative wind (RK82). The effect of possible midlevel updraft rotation on the pressure field around the obstacle was not considered by NN. The obstacle analogy predicts the direction of the horizontal pressure gradient correctly only near the equilibrium level, where the flow resembles a source in a uniform stream (Davies-Jones 1985, hereafter DJ85; Davies-Jones et al. 1994).
Fujita and Grandoso (1968) attributed the rightward propagation, relative to the mean wind, of a cyclonically rotating updraft to the effect of the Kutta–Joukowski lift force (here sideways), or the horizontal pressure force, acting on a hypothetical rotating rigid updraft column. This model is invalid because forces in fact act on individual air parcels flowing three-dimensionally through the storm.
RK82 presented a more realistic model, based on the linearized diagnostic pressure equation (a Poisson equation for perturbation pressure with a forcing function with two terms, one equal to minus twice the scalar product of environmental shear and horizontal gradient of updraft and the other equal to minus the vertical buoyancy gradient). Since the pressure equation is linear, its solution may be decomposed into a shear-induced pressure and a buoyancy-induced pressure. Their conceptual model is based on a heuristic rather than analytic solution of the Poisson equation. In their words, “the qualitative behavior of the solution [of the Poisson equation] may be found by noting that for a function consisting of a narrow band of Fourier components, the Laplacian of a function is negatively proportional to the function itself … we expect [the heuristic solution] to be approximately correct away from boundaries.” In spite of their disclaimer, several authors have used their model to deduce pressure close to the ground (e.g., LeMone et al. 1988). RK82 (p. 139) themselves used the predicted pressure patterns to identify the flank of the updraft where latently unstable subcloud air will be lifted to its level of free convection by upward pressure-gradient forces, and hence to determine the preferred flank for updraft growth. By definition, the heuristic solution satisfies Helmholtz's equation.
RK82 found that there is a high–low couplet of shear-induced pressure that straddles an axisymmetric updraft and is aligned with the shear vector, and that contours of buoyancy-induced pressure are nearly centered on the updraft. When the vector shear is constant, the highs and lows are stacked vertically and are most intense at the level of maximum updraft. Below this level, the vertical pressure-gradient force (VPGF) is upward (downward) on the downshear (upshear) side of the updraft. The converse is true at upper levels. The updraft–shear interaction cannot cause the updraft to move off a straight hodograph. When the shear has constant magnitude but turns clockwise with height, the pressure pattern turns with the shear, resulting in upward VPGF and a bias in updraft growth on the updraft's right side (relative to the shear vector).
The RK82 model has some validity because it passed some qualitative testing by numerical simulations. However, the apparent success of their theory is somewhat fortuitous because their solution for pressure is invalid at low levels as it vanishes at the ground instead of satisfying the Neumann boundary condition (BC) that is dictated by the vertical equation of motion (and that RK82 used in their numerical model). Other shortcomings are that it is a particular solution of the Poisson equation only when the forcing function has a purely sinusoidal dependence on height, and that it is inaccurate near levels where the shear changes rapidly with height (section 5).
RK82, RK85, and Klemp (1987) introduced the concept of (nonlinear) rotationally induced propagation in addition to the linear shear-induced bias discussed above. In strong unidirectional shear, the initial storm motion is on the hodograph owing to the lack of any bias. Because the ambient vorticity is perpendicular to the storm-relative winds, midlevel vortices develop on the right and left flanks (looking downshear) of the initial updraft. The combined effects of water loading in the center of the initial updraft and of low-level air being lifted by the flanking vortices causes the updraft to split into two mirror-image counter-rotating halves that propagate away from each other and evolve into severe right- and left-moving supercells. Propagation off a straight hodograph is induced rotationally by the vortex-associated upward nonhydrostatic pressure-gradient forces on the right and left flanks of the right- and left-moving updrafts, respectively. For hodographs that turn clockwise with height at low levels and are otherwise nearly straight, the right-moving cyclonic updraft propagates further off the hodograph owing to linear updraft–shear interaction and becomes the stronger supercell. For the entire spectrum of supercells (not just ones in nearly unidirectional shear), RK85, Klemp (1987), Rotunno (1993), and Weisman and Rotunno (2000, hereafter WR) regarded rotationally induced propagation as the larger effect with linear updraft–shear interaction being of secondary importance, merely providing the bias that allows right-moving storms to become the dominant ones in clockwise-turning shear.
Updrafts propagate discretely as well as continuously. Discrete propagation is generally associated with multicell storms (Chappell 1986) and is not amenable to mathematical analysis. Although fine-resolution observations and modeling reveal that discrete new updrafts form in supercells during cyclic mesocyclogenesis (Adlerman et al. 1999), the updraft propagation still appears continuous at coarser scales. Therefore, discrete propagation is discussed in this paper only in the context of splitting or bifurcation of the initial nonsupercellular updraft in strong unidirectional (or “straight”) shear.
Lilly (1982, 1986b), DJ85, and Brooks and Wilhelmson (1993) considered circular hodographs (or “circular shear”) instead of nearly straight shear. DJ85 presented an exact steady-state solution of the Euler equations of motion for a special case of a nonbuoyant axisymmetric updraft in an environmental wind with a circular hodograph. The updraft rotates at midlevels because the vorticity is purely streamwise (Davies-Jones 1984, hereafter DJ84). Such a flow is called a Beltrami flow and obeys a universal Bernoulli relationship; thus, pressure is low where total wind speeds or the vorticity magnitude is high, and vice versa. Although the lack of buoyancy is problematic, the Beltrami model does reproduce some features of simulated supercells in unstable environments with circular shear and is an excellent case for testing theories.
Weisman and Rotunno (2000) used results from a set of simulations with shear varying from straight to circular to argue that (i) “the physical processes that promote storm maintenance, rotation, and propagation are similar for all hodographs shapes employed, and are due primarily to nonlinear interactions between the updraft and ambient shear, associated with the localized development of rotation on the storm's flank” (WR, abstract); (ii) “Beltrami solutions do not capture the essence of supercell dynamics” (WR, p. 1454), although they did “find that the relevance of the Beltrami solutions improves as hodograph curvature increases” (WR, p. 1470); (iii) propagation is a result of the development of storm rotation; and hence (iv) their paradigm, based on rotationally induced propagation, is more complete than a supposedly competing one based on storm-relative streamwise vorticity and storm-relative environmental helicity (SREH) for understanding the full spectrum of supercell storms. According to WR (p. 1452), the difference between the two viewpoints is “Whether a storm generates [overall updraft] rotation by virtue of its propagation [off the hodograph; the streamwise-vorticity perspective] or whether the propagation is, in fact, a result of the development of storm rotation [on the flanks; the vertical-wind-shear perspective].” Streamwise-vorticity theory explains the origins of overall updraft rotation, given a known well-defined environment and observed or estimated storm motion [see Lilly (1982) for the circular shear case; DJ84 for general shear]. It represented an improvement on previous theories of updraft rotation because it did not depend (i) on the RK82 heuristic solution or (ii) on use of a constant instead of the actual height-dependent wind to advect vertical vorticity horizontally (Rotunno 1981). DJ84 used linear theory and an integral theorem to corroborate RK82's conclusion that an axisymmetric updraft should move on the concave side of a curved hodograph. In practice, it is advantageous to use actual storm motion, readily observed by radar, for evaluating SREH because storm motion sometimes is affected by external influences such as outflow boundaries, fronts, and topography (DJ84; Bunkers et al. 2000). In circular shear, the streamwise-vorticity paradigm is the better one because the cyclonic vortex nearly coincides with the updraft, the anticyclonic vortex migrates to the downdraft, and rotationally induced propagation is secondary to shear-induced propagation. On the other hand, the shear paradigm is better for strong fairly straight shear because the vortex pair on the flanks straddles the initial updraft and the initial storm splits as a result of rotationally induced propagation. Weisman and Rotunno (2000) (p. 1454) criticize the streamwise-vorticity theory because it assumes for calculation of updraft rotation that either storm motion is a given or it is determined by its linear component. For nearly straight shear where nonlinear propagation dominates, it has to be given the updraft's propagation in order to predict its rotation. But the WR theory is no more complete since it fails to account for the decline in nonlinear propagation and increase in rightward linear propagation with increasing clockwise turning of the hodograph as the cyclonic vortex moves inward toward the center of the updraft. According to WR (p. 1454), “…nonlinear forcing is important to updraft … propagation for straight and curved hodographs alike, with the linear hodograph-curvature effects merely biasing a particular storm flank.”
A supercell tends to retain its large-scale shape over long time intervals, and thus it may be idealized as a form-preserving disturbance (FPD). A form-preserving disturbance is a useful concept in the context of supercell propagation because it has an unambiguous motion vector. DJ84 and Kanehisa (2002, hereafter K02) used this approximation, in linear theory and for finite-amplitude motions, respectively, to obtain formulas for the covariance of vertical velocity and vertical vorticity.
In this paper, formulas for the motion of updraft, vorticity, and other maxima are derived from Petterssen's (1956) formula and the relevant prognostic equations in section 2, and checked in section 4 by showing that they correctly predict the propagation of all form-preserving updrafts including the axisymmetric Beltrami updraft. Since updraft motion depends to a large extent on the distribution of the nonhydrostatic vertical pressure-gradient force (NHVPGF) around the updraft, the formal solution of the nonlinear diagnostic equation for nonhydrostatic pressure subject to homogeneous Neumann BCs at the ground (and top, if present) is presented in section 3. Hydrostatic propagation (owing to large horizontal differences in hydrostatic pressure across the updraft) is found to be potentially significant at low levels. An explanation is given in section 4 for the presence of low pressure at the ground in storm inflows (“inflow lows”). Solutions for the shear-induced pressure and propagation are found in section 5 for an assumed axisymmetric updraft in straight and circular shear. Section 6 addresses the rotation associated with shear-induced propagation. In section 7, nonlinear propagation is explored, an explanation is given for the continued deviant motion of supercells long after the storm split that gave birth to them, and WR's claim (p. 1471) that their theory is the “best paradigm for understanding the full spectrum of supercell storms, independent of hodograph shape” is challenged.
2. Nonlinear formulas for motion of updrafts and vorticity maxima
The motions given by Petterssen's formula are instantaneous and apply equally as well to narrow towering cumulus clouds that cannot resist shear as to supercells. Hence they are naturally functions of height. Previous papers have deduced only the direction of propagation of the maximum of one field (w) at one level. For a 3D storm or cell, a mean value
3. Determination of pressure and components of motion
The general effect of the forcing terms in (3.3b) on nonhydrostatic pressure is as follows. Downward (upward) acceleration caused by the NHVPGF [the only force in (2.10)], generally occurs at points with heavy (light) overlying columns of air and associated positive (negative) values of FM. Hence, the nonhydrostatic pressure is generally low (high) beneath heavy (light) columns. Spin and splat forcing are associated with parcel spin and changing parcel shape, respectively. Since dzz lnρs ≤ 0 in atmospheres with constant nonnegative lapse rates, splat forcing is positive practically everywhere. In high Rossby number flows such as supercells, spin forcing is negative almost everywhere. Therefore, the splat and spin terms generally contribute to high and low pressure, respectively, consistent with the general association of vortices with low pressure and regions of high strain rates with high pressure (e.g., stagnation highs).
4. Propagation of form-preserving disturbances
The author is unaware of any nonlinear form-preserving analytical solutions in an unstably stratified fluid. However, a simple example of an exact nonlinear form-preserving updraft without buoyancy is a Beltrami flow (DJ85), albeit one with A constant. A flow is Beltrami if the vorticity ω and relative velocity vectors v − C0 are parallel everywhere in a reference frame moving with a constant velocity C0, that is, ω = λ(v − C0), where the scalar λ is the abnormality. Hence, w and ζ are perfectly correlated. In the limit of vanishing buoyancy, the inviscid Boussinesq equations have exact steady-state Beltrami solutions (DJ85; Lilly 1986b), which are quasi-linear because pressure has a nonlinear component, but each Cartesian velocity component is the solution of a single linear partial differential equation (a Helmholtz equation if λ is constant).
5. Linear propagation
In supercell dynamics, the linear terms in the equations include the first-order effects of the environmental winds. Even though the nonlinear terms are comparable to or larger than the linear ones, the linear terms may provide most of the bias that causes the updraft to propagate in a certain direction and to rotate as a whole. As demonstrated in section 7, this is the case for circular hodographs. Hence, we now examine the linear components of motion.
a. General shear
b. Shear-induced solutions for special shear flows
Five examples are considered:
(a) constant vector shear S from 0 to 12 km, R = 5.3 km;
(b) same but S = 0 above 6 km;
(c) full circle hodograph, R = 5.3 km;
(d) full circle hodograph, R = 4.0 km; and
(e) half-circle hodograph, R = 4.0 km.
1) Constant-shear cases
For an initial axisymmetric updraft, the deviation spin and splat, and hence pnhNL, are axisymmetric, and pnhM is almost axisymmetric because buoyancy is nearly coincident with updraft owing to the release of latent heat in the updraft. In constant vector shear, say S0 = Si, tilting of vortex tubes produces a vortex pair that straddles the updraft axis in the cross-shear direction and makes the deviation wind asymmetric with a strong component in azimuthal wavenumber two. This nonlinear effect, which plays an important role in storm splitting, is considered in section 6. For now we consider the wavenumber-one asymmetry arising in pnhL from the initial interaction of an axisymmetric updraft with constant environmental shear.
The first hodograph considered is the straight one in WR, but with the layer of constant shear (S = 5.83 × 10−3 s−1) extending from 0 to 12 km instead of from 0 to 6 km. The fact that mesoanticyclones are 50 times more common than mesoanticyclones (Davies-Jones 1986) indicates that virtually straight hodographs (without appreciable clockwise turning near the ground) are uncommon in nature. Example (a) is the only one where the heuristic (RK) solution, FL/(k2 ;pl μ2), is a particular solution of the (inhomogeneous) diagnostic pressure equation. An additive complementary solution of the homogeneous solution is needed, however, for satisfaction of the BCs (DJ96). The Green's function solution satisfies the BCs automatically. The solutions (Fig. 6) are shown for an updraft radius R of 5.3 km (k = 0.45 km−1). Since the storm-relative environmental wind is from the east (west) in the lower (upper) half of the domain (Fig. 6b), the Green's function solution features an inflow low at the ground and an outflow low at the top, both on the east side of the updraft (Fig. 6c). The RK solution is qualitatively correct near the midlevel because the contributions to pressure at z = H/2 from z = H/2 ± Δz cancel each other for all Δz ∈ (0, H/2). However, it grossly underestimates the magnitude of pressure and overestimates |NHVPGF| near the boundaries. In both solutions the NHVPGF (Fig. 6d) is upward (downward) on the storm-relative upwind (downwind) side of the updraft. The shear-induced NHVPGF changes sign at the maximum ρsw in agreement with the RK82 conceptual model (RK82, pp. 139, 143). As pointed out by RK82 (see their Fig. 3a), this distribution of NHVPGF mitigates the tilting and shearing apart of the updraft. Figure 8 shows that the maximum forcing for surface pressure is located at 2 km. The propagation velocity cL(z, t0) − u0(z) at each level is about half that required to keep the initial updraft vertical (Fig. 6b). Thus, the initial updraft is tilted rapidly downshear as in a 2D simulation (where the complication of splitting is absent) of dry convective overturning in constant shear (SK). A form-preserving updraft was not obtained in SK as the updraft continued to tilt slowly after 15 min. As shown by SK, moist convection may attain a steady configuration in which the updraft leans upshear. The upshear tilting of the initial erect updraft is due to the hydrostatic forcing associated with a cold pool and precipitation loading.
In example (b), the hodograph is the straight one in Fig. 3a of WR (with shear confined to the lowest 6 km). In this case I(r, ϕ, z, ẑ) is the same as in Fig. 8, except that I(r, ϕ, z, ẑ) = 0 for ẑ > 6 km. The nonhydrostatic pressure at mid- and (especially) at upper levels is smaller (Fig. 7c) than in the first example owing to lack of forcing from levels above 6 km. Consequently, the top of the updraft initially drifts downstream with the upper winds (Fig. 7b). In the absence of upper-level forcing, the RK solution fails at mid- and upper levels as well as near the boundaries. At 3 km, the shear-induced NHVPGF (Fig. 7d) is downward (upward) on the upshear (downshear) flank in agreement with the simulation in WR (see their Fig. 10g). But the NHVPGF contradicts the RK82 conceptual model by changing sign at 4 km, well below the maximum of ρsw at 6 km.
2) Circular-hodograph cases
Figure 11 compares the Green's function and RK distributions around the updraft of pressure and NHVPGF at the radius of maximum |∂rw|. The azimuth angles of the extrema of pnhL and −∂zpnhL in the Green's function (or Beltrami flow) solutions are compared also to the prediction of the NN conceptual model (see Fig. 8 of DJ96). The asymmetric component of dynamic pressure agrees with RK82 at the midlevel and with NN at the top. Even though the shear and wind vectors veer through 360°, the Green's function high/low couplet turns with height only through 180° because of the weighted sum in (5.15). Both RK82 and NN predict 360° twisting of the highs and lows. RK82 predicts the orientation of the couplet correctly at the midlevel because of symmetry (cancellation of the tendencies of the weighted-sum contributions, from below and above, respectively, to turn the couplet to the left and right of the midlevel shear vector). There is an inflow low at the ground and outflow low at the top (see section 4). The outflow low is predicted correctly by NN. There is a net southward pressure force on the updraft column that does not move the updraft southward of the mean wind (contra FG). The maximum (minimum) VPGF turns through 360° and is located on the right (left) side (looking downshear at each level) of the updraft. At 3 km, the maximum of shear-induced NHVPGF is located on the southern side of the updraft, in good agreement with the corresponding simulation at Ri = 1.8 in WR (see their Fig. 11c). The VPGF is incorrect at every level in the NN theory, and correct only at the midlevel in the RK82 conceptual model. At the ground (top), the RK propagation is northward (southward) and infinite, in contrast to the actual eastward propagation there. In example (d) where the updraft is made narrower than the form-preserving one, the NHVPGF changes little. The curve cL (z, t0) (instead of a single point C0) is due to the error in assuming that (5.20) with R = 4.0 km is a form-preserving updraft. Even with this deliberate error, it is still apparent that the actual motion should lie close to C0. With a smaller (larger) R than the Beltrami one, the shear-induced propagation off the hodograph is too little (much) to produce a height-independent cL (z, t0) for the assumed updraft (Fig. 9).
Since the relevance to supercells of nonbuoyant Beltrami flows has been questioned by WR, we now compare the Beltrami solution with another non-Beltrami one in example (e), which is for the half-circle hodograph with an updraft radius of 4.0 km (Fig. 12). Even if the flow were steady and barotropic, the stagnation pressure pt would still vary significantly across streamlines and vortex lines according to Crocco's theorem, ∇pt = ρsv × ω. Comparison of Figs. 12 and 11 shows little change in the azimuthal distribution of shear-induced NHVPGF at low levels, indicating that the Beltrami updraft model yields a good qualitative prediction of the low-level NHVPGF for circular hodographs even when the shear is terminated at 6 km to make a more realistic hodograph. At 3 km, the RK solution has the maximum NHVPGF on the east-southeast (ESE) side of the updraft, but the Green's function solution has it on the SSE side in accord with the corresponding simulation in WR (see their Figs. 10i and 12i). The surface pnhL field still features an inflow low. The RK solution is again inaccurate near the ground, and in and near the no-shear layer. The motions of the updraft maxima no longer lie at the hodograph's center of curvature (Fig. 10) owing to the reduced diameter of the updraft (as in Fig. 9) and the lack of forcing above 6 km. However, they still lie on the concave side of the hodograph, far from the low-elevations part of the hodograph curve. At low levels, cL (z, t0) lies close to the motion of the corresponding simulated storm (WR, Fig. 3c), suggesting that much of the modeled low-level propagation is shear-induced. This motion is associated with mainly streamwise vorticity and large SREH. At upper levels, cL (z, t0) lies far from the simulated storm motion and seems to be influenced too much by the constant east wind above 6 km (cf. Fig. 10 with WR, Fig. 3c). This may be caused by the assumed erect updraft not being a good representation above 6 km of a form-preserving updraft, which would probably lean to the east with height. For a shape-changing updraft, the nonlinear −∇H(w∂zw) term in (3.9d) could mitigate the tendency of the upper part of the updraft to drift downstream.
6. Rotation associated with shear-induced propagation
Once the shear-induced updraft motion has been determined, the associated updraft rotation can be deduced. If the shear turns clockwise with height, then (5.17) and (5.18) show that the storm-relative wind v0(z) − CL is directed to the left of S0(z), that is in the same general direction as the environmental vorticity vector ω0(z) = k × S0(z). Hence the vorticity in the inflow is at least partly streamwise in a reference frame moving with the velocity CL (or CLN if CM ≈ 0).
How is the covariance of ζ and w established? Multiplying the vertical-motion equation Lw̃ = −α0∂zp̃nhLN by
Although form preservation is a useful concept and is perhaps needed for strictly height-independent motions, it is an overconstraint on supercell dynamics. This is least so for supercells in circular shear where storm splitting is insignificant and the updrafts are described to some extent by form-preserving Beltrami models. We now show that supercell dynamics in circular shear are indeed different from those in straight shear.
7. Nonlinear versus linear propagation
We now explore the effects on updraft motion of nonlinear terms (wind deviation at the updraft center, and horizontal variations of nonlinear NHVPGF and of vertical advection of vertical velocity across the updraft). The dominance of nonlinear rotationally induced propagation in strong, nearly unidirectional shear after initial storms split into right- and left-moving supercells is discussed in section 7a. Also, the cause of the persistent deviate motions of split storms is explained. In contrast, the linear terms provide the updraft with most of the bias for propagation off a circular hodograph and for the concomitant overall rotation (see section 7b).
a. Storm motion in straight shear
The straight-shear case is more nonlinear and hence more complicated than the circular-shear case, but strangely it is the one discussed (heuristically) in texts (e.g., Holton 1992, 298–303). It represents one extreme, where the motion of storms off the hodograph is due to nonlinear (instead of shear induced) propagation and where updrafts derive their vertical helicity from covariance of ζ and nonlinear NHVPGF. In say, westerly shear with f = 0, the initial updraft lifts up loops of the northward-oriented vortex tubes, forming a midlevel vortex pair that straddles the initial updraft with the cyclonic vortex on the right side of the plane of symmetry (when viewed from upshear) and the anticyclonic vortex on the left side. Thus the southern (northern) half of the updraft rotates cyclonically (anticyclonically). As shown in section 5b, linear propagation is east or west. Because of the north–south symmetry, an axisymmetric updraft cannot propagate off the hodograph either linearly or nonlinearly. If the two vortices were directly north and south of the updraft, they would induce a westward motion on each other (Prandtl and Tietjens 1957, p. 209), and a larger westward motion on the updraft maximum through the easterly flow perturbation between the vortices and the
Provided that the shear is sufficiently strong, the initial updraft in a simulation bifurcates or splits at about 40 min into a severe right-moving (SR) supercell with a cyclonic updraft and a mirror-image severe left-moving (SL) supercell with an anticyclonic updraft. The bifurcation occurs if Ri is below 2 to 2.5, which is roughly the threshold at which the nonlinear term FNL becomes significant according to the scale analysis in (3.3d). The updraft splits as a result of two effects. First, precipitation accumulates near the plane of symmetry, and the associated downward drag forces decelerate the center of the initial updraft, causing the original updraft maximum to disappear (Klemp and Wilhelmson 1978). Even though splitting occurs in simulations without precipitation (RK82), this process accelerates the splitting process. Second, the midlevel vortices are centers of low pressure because the centripetal acceleration acting on air parcels in their circulation can be provided only by a pressure-gradient force directed inward toward the axes of the vortices (Schlesinger 1980; RK82). The upward NHVPGF below the vortices lift up low-level air by the vortex-suction mechanism, thus growing the updraft on the cyclonic and anticyclonic flanks of the initial storm. The bifurcation may be regarded as discrete propagation. The original maximum is replaced by two new updraft maxima off the symmetry plane (but closer to it than the southern cyclonic and northern anticyclonic vortices, which migrate, respectively, toward the extrema of tilting on the southern flank of the southern updraft and the northern flank of the northern updraft). The new southern (northern) updraft already has overall cyclonic (anticyclonic) rotation because it has developed in rising air with preexisting cyclonic (anticyclonic) vorticity. The forcing field has minima above and on the far side (relative to the symmetry plane) of the updraft centers owing to the presence of the outermost midlevel vortices and their associated large negative
Why do the rightward and leftward motions persist after the storms become separated by tens of kilometers? Since the supercells are mirror images, we may concentrate on the SR one. As a result of vortex-tube tilting there is still an anticyclonic vortex on the north flank and a cyclonic vortex on the south flank of the right mover (Fig. 7 of Weisman and Klemp 1982). If the two midlevel vortices were equally strong and equidistant from the maximum updraft, the horizontal gradient of the nonlinear “spin-related” NHVPGF would vanish at the center, the southward propagation would cease, and then splitting would occur a second time. This does not happen because the anticyclonic vortex is less intense than the cyclonic one just after the split and is further away from the center. The cyclonic vortex is quite close to the (cyclonic) updraft on its southern side, and its suction [associated with
b. Storm motion in circular shear
The above quasi-linear, steady, Beltrami model of a rotating nonbuoyant updraft in circular shear does not have to be radically modified in order to incorporate flow evolution and buoyancy forces. The linear theory of DJ84 shows that buoyancy and initial updraft growth reduce the magnitude of the correlation coefficient r between vertical velocity and vertical vorticity, but the mechanisms responsible for updraft rotation and propagation off the hodograph in the steady Beltrami flow are still present in the evolving linear solutions. For clockwise-turning circular shear, r decreases from 1 to ∼(1 + Ri)−1/2, according to the scale analysis in DJ84 (p. 2998) if we estimate the growth rate as (CAPE)1/2/H, the storm-relative wind speed as Us/π, and the horizontal length scale as H/π. For Ri = 1.8, r ∼ 0.6, in rough agreement with WR's results (their Figs. 9c–d). Therefore, the cyclonic vortex should be near the updraft maximum at each level, while the anticyclonic vortex must be located in lesser updraft or in downdraft. The cyclonic vortex may be close enough to the updraft core to make rotationally induced propagation a secondary effect. This is indeed the case in WR's simulation with a circle hodograph (WR's Figs. 8d, 11b). The supercell in this case decays after an hour owing to a combination of (a) storm-relative winds of only 10 m s−1 (Droegemeier et al. 1993), (b) precipitation falling in the inflow, and possibly (c) the microphysics package because it has (i) no frozen hydrometeors (Gilmore et al. 2002, manuscript submitted to Mon. Wea. Rev., hereafter GSR), (ii) a fast evaporation rate owing to an error in the ventilation term (GSR), and (iii) conversion of excess cloud water at a very small threshold to rain water (Straka and Rasmussen 1997). At 3 km, the updraft is propagating off the circle hodograph southward or to the right of the shear vector (Fig. 3 of WR). This is more in the direction at the updraft center of the horizontal gradient of the shear-induced NHVPGF than in the direction predicted by the nonlinear NHVPGF field (Fig. 11 of WR). Furthermore, the magnitude of the gradient of nonlinear NHVPGF is smaller at the center. Only the shear-induced NHVPGF suppresses updraft on its northern side to counteract the northward advection of vertical velocity there. It is also evident from WR (Figs. 10–12) that the covariance of ∂yw and ∂zpnhL in (4.6c) is larger than that of ∂yw and ∂zpnhNL in (4.6d) in both the full- and half-circle cases. Thus the rightward propagation is largely linear. Rotunno (1993, p. 69) claimed that the mechanism for propagation is cancelled when the updraft catches up to the vorticity center. This is true only of nonlinear propagation because the updraft still propagates linearly off the hodograph to the right of the environmental shear at each level. As shown in section 5, the distribution at low levels of linear NHVPGF around a non-Beltrami updraft in circular shear is basically the same as around a Beltrami updraft. The motion vector in the WR simulation is close to the center of the circle as in Beltrami flow. Consequently, the environmental vorticity is highly streamwise in the updraft-relative reference frame, and the cyclonic vorticity acquired by air parcels entering the updraft originates essentially from a linear term, tilting of environmental streamwise vorticity, and is amplified by nonlinear stretching and reorientation of vortex tubes.
The relevance of Beltrami flows to supercells has been questioned by WR, who used in their Eq. (18) the vector identity v · ∇v = ∇(v · v/2) − v × ω to decompose advection of vertical velocity −v · ∇w in the storm's reference frame into Bernoulli β ≡ −∂z(|v|2/2) and Lamb Λ ≡ k · v × ω contributions. Their criticisms are based on their numerical simulations in which they observe that (i) “the components of the Lamb vector are … at least as large as the components of the Bernoulli term” and (ii) the “dynamic pressure forcing on the right flank of the storm … is associated almost entirely with the vertical component of the Lamb vector, with the vertical component of the Bernoulli term contributing positively mostly on the left flank of the updraft, working counter to the observed updraft propagational tendencies.” The first observation, while true, is predictable from DJ84, where the Lamb vector is included. For the circle hodograph, r decreases from 1 to ∼0.6 as Ri increases from 0 to 1.8 (see above), and naturally the Lamb vector becomes significant. As shown above, the shear-induced motion vector still lies near the center of the circle and the correlation between updraft and cyclonic rotation is still highly significant. Moreover, the pressure field is still qualitatively similar to that of a Beltrami flow [section 5b(2)]. Elementary considerations also show that the Lamb vector grows rapidly with minor departure from Beltrami flow. Consider, for example, an angle of 30° between v and ω. The local flow is highly helical as indicated by a normalized helicity density v · ω/(|v‖ω|) of 0.87 (Droegemeier et al. 1993), yet the normalized Lamb vector |v × ω|/(|v‖ω|) is 0.5, which implies significant variations in stagnation pressure. With regard to (ii), WR apparently are concerned that, in the reference frame of the storm (the one for which c = 0) in the full-circle simulation, −∇H∂zpnh|W is balanced by −∇HΛ|W rather than by −∇Hβ|W, as would be expected based on a Beltrami flow (Λ ≡ 0) solution. Equivalently, the steering current arises from Λ instead of from β. Seemingly, this renders Beltrami flow irrelevant to supercell propagation. Weisman and Rotunno (2000) conclude that “it is the specific lack of a perfect correlation between the vertical velocity and vertical velocity fields that creates the nonlinear dynamic forcing demonstrated to be critical for … updraft propagation.” Paradoxically, the Beltrami model of a rotating updraft provides a counterexample that refutes this conclusion. Obviously, the steering current must be provided by the β term in the Beltrami limit Λ → 0. The paradox is resolved in the appendix where it is shown that the horizontal advection of w appears explicitly in the Lamb term (which explains WR's second observation) but transfers to the β term if Λ ≡ 0. The switch is caused by terms that appear in Λ and β with opposite signs. The motion formula (2.11) is derived using (2.10). Hence, updraft motion depends on the full advection of vertical wind Λ + β, not on Λ or β separately. The steering current is naturally vH|W(z, t) and whether it originates from the Λ or the β part of the advection term is immaterial. Despite the significant Lamb vector, a buoyant updraft in circular shear has rotational and propagational characteristics that are qualitatively similar to those of a Beltrami updraft.
c. Intermediate shears
With increasing clockwise turning of the shear vector, storm splitting becomes less significant. In simulations with semicircular shear from 0 to 5 km, the nonsupercellular storms that develop on the left flank of the initial storm move on the concave side of the hodograph and have little overall updraft rotation (Weisman and Klemp 1984). In the case with the most shear, the initial storm evolves continuously into the right-flank SR supercell while the left-flank updrafts develop as discrete new cells rather than as the products of storm splitting, and move slightly to the right of the mean wind. With even more turning of the shear vector, “growth on the left flank may be suppressed to the extent that there would be no apparent splitting at all; the initial storm just begins moving to the right of the mean winds” (Klemp 1987, p. 385). Unfortunately, there is not a simulation in WR with a three-quarter circle hodograph. Lilly's (1982, 1983) simulation with a circle hodograph of radius 20 m s−1 generated an intense quasi-steady storm with no splitting. These changes in storm behavior as the shear becomes more circular are consistent with the present finding that the horizontal gradient of the nonlinear (linear) NHVPGF at the updraft center largely determines updraft propagation in environments with nearly straight (highly curved) hodographs. For example, in the WR simulations (their Figs. 8 and 10–12), the observed propagation at 3 km (WR, Fig. 3) is attributable principally to the nonlinear rotationally induced mechanism in the cases with straight and quarter-circle hodographs, and to the linear shear–updraft interaction in the semicircle and circle cases where nonlinear propagation is smaller, owing to near coincidence of updraft and cyclonic vortex. Once the storm motion is determined, the overall rotation of the updraft can be deduced from the streamwise-vorticity theory, which is valid for finite as well as infinitesimal vertical displacements (DJ84; RK85; K02). In confirmation of this theory, the correlation at low levels between vertical velocity and vertical vorticity in the updraft of the right-moving storm increases with increasing low-level streamwise vorticity (WR, Figs. 3, 9).
8. Conclusions
A nonlinear formula for updraft motion has been derived from Petterssen's formula and the vertical equation of motion, and tested on form-preserving updrafts. Continuous propagation of an updraft maximum is determined largely by the horizontal gradient of NHVPGF at the updraft center. The formula for NHVPGF is derived, not from an inaccurate heuristic solution as in past studies, but from formal solution of the Poisson equation for nonhydrostatic pressure subject to homogeneous Neumann BCs at horizontal boundaries. As previously found, propagation of supercell updrafts is due mainly to the NHVPGF distribution arising from the linear interaction between the shear and updraft and from nonlinear dynamical effects.
A combination of the nonlinear updraft–shear interaction and streamwise-vorticity/helicity concepts is required to understand the propagation of supercell updrafts in all wind shears, because the dynamics in straight and circular shear are different. The nonlinear rotationally induced propagation discussed in RK85 and WR is important in straight shear, where the vortex pair formed by lifting of environmental vortex tubes straddles the initial updraft. Even in this case the anticyclonic vortex on the left side of the right-moving (SR) supercell and the cyclonic vortex on the right side of the left-moving (SL) supercell migrate to the downdrafts and, owing to their remoteness from the updraft centers, play little part in the anomalous propagations. The deviate motions of the supercells are maintained by the distribution of NHVPGF [upward ahead of each updraft owing to
Acknowledgments
This work was supported in part by NSF Grant ATM-0003869. The reviewers' thoughtful comments led to improvements in the paper. I am indebted to Dr. H. Kanehisa for kindly allowing me to use his results prior to their publication. Joan O'Bannon drafted Figs. 1 and 2.
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Appendix
Resolution of the Paradox
A schematic of sources and their images below the ground. The abscissa represents amplitude along a given vertical of the forcing function F in −∇2pnh = F. The images are sources (even extension of the forcing function below the ground; solid) and sinks (odd extension; dashed) for homogeneous Neumann and Dirichlet BCs, respectively
Citation: Journal of the Atmospheric Sciences 59, 22; 10.1175/1520-0469(2003)059<3178:LANPOS>2.0.CO;2
(left) The nonhydrostatic pressure field (solid contours) associated with a point source and its image source below the ground. The dashed lines with arrows are lines of pressure-gradient force. The NHVPGF is zero at the ground. (right) Same for the Dirichlet condition satisfied by the heuristic solution of −∇2pnhL = FL. The image is now a sink and the NHVPGF at the ground is nonzero
Citation: Journal of the Atmospheric Sciences 59, 22; 10.1175/1520-0469(2003)059<3178:LANPOS>2.0.CO;2
Three-dimensional schematic indicating a distribution of the forcing function F (in − ∇2pnh = F) at neighboring source points (x̂, ŷ, ẑ) that is associated with an eastward gradient of NHVPGF (see west–east vertical section) and a southward gradient of NHVPGF (see north–south vertical section) at the field point (x, y, z) located at the origin. Hs (Ls) denote higher (lower) values of F below and to the east (west) of the field point, and above and to the west (east) of the field point, respectively. Similarly, the h's (l's) denote higher (lower) values of F below and to the south (north) of the field point and above and to the north (south) of the field point, respectively. Hidden lines are dashed and hidden letters are in outline format
Citation: Journal of the Atmospheric Sciences 59, 22; 10.1175/1520-0469(2003)059<3178:LANPOS>2.0.CO;2
G(z, ẑ) as a function of ẑ for z = 0, 3, 6, 9, 12 km, H = 12 km, and k =
Citation: Journal of the Atmospheric Sciences 59, 22; 10.1175/1520-0469(2003)059<3178:LANPOS>2.0.CO;2
Spatial relationships between different maxima of an FPD on a level surface in (a) westerly straight shear, and (b) shear that turns clockwise with height. Maxima and minima are denoted generally by ⊕ and ⊝, respectively. For 
Citation: Journal of the Atmospheric Sciences 59, 22; 10.1175/1520-0469(2003)059<3178:LANPOS>2.0.CO;2
Shear-induced solutions for the case with constant westerly shear of 5.83 × 10−4 s−1 from 0 to 12 km. Values of parameters listed in bottom label (here and in subsequent figures). (a) The updraft profile W(z). The base-state density ρs(z) is shown in Fig. 7a. (b) Variation with height of the environmental wind u0(z) and of the instantaneous linear eastward motion cL(z, t0) of an initially erect updraft according to Petterssen's formula. (c) Maximum values of pnhL at each level according to the GF and RK solutions. The maximum values are located at the radius r = 1.84/k of largest updraft gradient on the western side of the updraft where ϕ = π. Pressures at other ϕ are the same except multiplied by −cosϕ; for example, pressures on the eastern side (ϕ = 0) of the updraft are the same but negative. (d) The linear NHVPGF solutions along the same vertical as in (c). The NHVPGF at r = 1.84/k, ϕ = 0 (eastern side) has the same magnitude but opposite sign.
Citation: Journal of the Atmospheric Sciences 59, 22; 10.1175/1520-0469(2003)059<3178:LANPOS>2.0.CO;2
Same as Fig. 6, except there is no shear from 6 to 12 km, and (a) the base-state density ρs(z) is graphed. Also, W(z) is the same as in Fig. 6a. The RK solutions are zero above 6 km
Citation: Journal of the Atmospheric Sciences 59, 22; 10.1175/1520-0469(2003)059<3178:LANPOS>2.0.CO;2
Same as for Fig. 4, but for the integrand I(r, ϕ, z, ẑ) in (5.15) evaluated at r = 1.84/k, ϕ = π on the western side of the updraft in the same case as in Fig. 6. Physically, I(r, ϕ, z, ẑ) represents the contribution (in mb km−1) to pnhL at the field point (1.84/k, π, z) from the source point at (1.84/k, π, ẑ). The area between the ẑ axis and each curve is pnhL(z). For the case of Fig. 7, I(r, ϕ, z, ẑ) is the same except that I(r, ϕ, z, ẑ) = 0 for ẑ > 6 km
Citation: Journal of the Atmospheric Sciences 59, 22; 10.1175/1520-0469(2003)059<3178:LANPOS>2.0.CO;2
The full-circle hodograph, centered at C0. Also shown is the instantaneous linear motion cL(z, t0) of an initially erect updraft of radius R = 4.0 km. Numbers along the curves denote heights (in km). For R = 5.3 km, cL(z, t) = CL = C0
Citation: Journal of the Atmospheric Sciences 59, 22; 10.1175/1520-0469(2003)059<3178:LANPOS>2.0.CO;2
As in Fig. 9, but for the half-circle hodograph
Citation: Journal of the Atmospheric Sciences 59, 22; 10.1175/1520-0469(2003)059<3178:LANPOS>2.0.CO;2
Contours of pnhL and NHVPGF in the cylindrical surface located at the radius r = 1.84/k of strongest updraft gradient for the case of the full-circle hodograph and R = 5.3 km. The numbers in the parentheses at top right of each plot list, in order, are the lowest and highest contour levels, and the contour interval. The ordinate is height z with tick marks at 1-km intervals and the abscissa is azimuth ϕ with tick marks at 30° intervals and with E, N, S, W denoting the cardinal directions. Left panels contain pnhL (in mb) for the GF solution (bottom) and for the RK solution (top). Right panels show NHVPGF (in mm s−2) for the GF solution (bottom) and for the RK solution (top)
Citation: Journal of the Atmospheric Sciences 59, 22; 10.1175/1520-0469(2003)059<3178:LANPOS>2.0.CO;2
Same as in Fig. 11, but for the case of the half-circle hodograph and R = 4.0 km. The RK solutions are zero above 6 km
Citation: Journal of the Atmospheric Sciences 59, 22; 10.1175/1520-0469(2003)059<3178:LANPOS>2.0.CO;2
