## 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

*t*of updraft maxima and other extrema on an

*f*plane. On a constant-height chart (at height

*z*) of an arbitrary field variable, say

*q,*consider an extremum at a horizontally moving point Q(

*z,*

*t*), the “center.” Every extremum is approximately elliptical in a sufficiently small neighborhood of Q(

*z,*

*t*) in the horizontal (

*x*–

*y*) plane. Since a moving extremum of

*q*satisfies ∂

_{x}

*q*|

_{Q(z,t)}= ∂

_{y}

*q*|

_{Q(z,t)}= 0,

_{t}

**c**

*z,*

*t*

_{H}

_{H}

*q*

_{Q(z,t)}

**c**(

*z,*

*t*) ≡

*c*

_{1}

**i**+

*c*

_{2}

**j**is the motion vector of the extremum;

**i**,

**j**,

**k**are the unit basis vectors of the Cartesian coordinate system; and ∂

_{x}

*q*≡ ∂

*q*/∂

*x,*etc. The general solution of (2.1) is

*x*and

*y*axes are chosen to lie parallel to the principal axes of the ellipse,

_{xy}

*q*

_{Q(z,t)}

**c**

*z,*

*t*

_{xt}

*q*

_{xx}

*q,*

_{yt}

*q*

_{yy}

*q*

_{Q(z,t)}

*q*is controlled by a prognostic equation of the form

*dq*

*dt*

_{xx}

*q*|

_{Q(z,t)}, and ∂

_{yy}

*q*|

_{Q(z,t)}, are both negative (positive) for a maximum (minimum). The last term on the right of (2.8) represents the continuous-propagation velocity of the extremum of

*q*at each level, that is, the deviation of the extremum's motion from the steering wind at the center. The propagation velocity depends (i) on the gradients in the

*x*and

*y*directions of the forcing

*x*and

*y*directions of the vertical advection of

*q,*(iii) inversely on the intensity

^{2}

_{H}

*q*|

_{Q(z,t)}of the extremum, and (iv) on the ellipticity of the extremum, which causes the propagation velocity to turn toward the long axis of the ellipse (Petterssen 1956, 49–52; Stewart 1945, 421–422). For a circular extremum, ∂

_{xx}

*q*and ∂

_{yy}

*q*at the center Q(

*z,*

*t*) are equal to ∂

_{rr}

*q*at Q where

*r*is radial distance, and the orientation of the coordinate axes is arbitrary because every axis of a circle is a principal axis. Then (2.8) can be expressed in vector form as

*f*plane. For brevity the listed expressions are given for circular maxima. The generalizations to elliptical maxima follow simply from application of (2.8) instead of (2.9). In hydrostatic flow, there is no equation of the form (2.5) for

*w,*which is obtained instead from a diagnostic omega equation. In frictionless nonhydrostatic flow, the equation for vertical motion is

*w*

*t*

**v**

_{H}

*w*

*α*

_{z}

*p*

_{nh}

*α*≡ 1/

*ρ*is the specific volume,

*ρ*is density and

*p*

_{nh}is the nonhydrostatic (or hydrodynamic) pressure. [Note that

*p*

_{nh}is equal to the pressure minus the

*local*hydrostatic pressure

*p*

_{h}(

*x,*

*y,*

*z,*

*t*) ≡

*g*

^{∞}

_{z}

*ρ*(

*x,*

*y,*

*ẑ,*

*t*)

*dẑ*; Das (1979).] Applying (2.9) to a circular vertical velocity maximum at a point W(

*z,*

*t*) with reference to (2.5) and (2.10) yields its motion vector:

*w.*For a stationary updraft, ∇

_{H}(NHVPGF −

**v**· ∇

*w*)|

_{W(z,t)}= 0.

*s*≡

*c*

_{p}ln

*θ*is

*ds*

*dt*

*Q̇*

*T,*

*Q̇*is the net rate of (diabatic) heating per unit mass,

*T*is temperature, and

*θ*is potential temperature. The motion of a circular entropy maximum at a point S(

*z,*

*t*) is

_{z}

*s*≡ (

*c*

_{p}/

*g*)

*N*

^{2}in terms of

*N*

^{2}, the square of the local Brunt–Väisälä frequency. The maximum propagates in the direction of the greatest sum of diabatic heating and adiabatic warming.

*ζ*

_{a}≡

*f*+

*ζ*on an

*f*or

*β*plane is

**ω**_{a}≡ (

*ξ,*

*η,*

*ζ*

_{a}) is the absolute vorticity vector, and −

*J*

_{xy}(

*α,*

*p*) ≡ −∂

_{x}

*α*∂

_{y}

*p*+ ∂

_{y}

*α*∂

_{x}

*p*is the vertical component of the solenoid torque. The first term on the right represents production and amplification of vertical vorticity through tilting and vertical stretching of vortex tubes. The motion of a circular maximum of

*αζ*

_{a}at a point Z(

*z,*

*t*) is given by

*αζ*

_{a}maximum propagates in the direction of the greatest sum of vertical stretching, tilting of horizontal vorticity into the vertical, vertical solenoidal torque, and vertical advection of

*αζ*

_{a}.

*δ*on an

*f*plane may be written in the form

*dδ*/

*dt*is omitted in order to filter out gravity waves (Thompson 1961, chapter 11). In unbalanced models a circular divergence extremum centered at D(

*z,*

*t*) will move with the velocity

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 **c***t*) may be regarded as the instantaneous motion of a core (e.g., updraft core, warm core, vortex core) with **c**(*z,* *t*) − **c***t*) expressing the rate at which the axis is tilting. In contrast to narrow forms of convection in shear, a supercell changes shape slowly and its maxima move at roughly the same velocity at all heights, thus giving rise to a meaningful storm motion. For a form-preserving disturbance, the motions of the extrema of all variables at all levels have to be identical by definition. This is verified in section 4. The form of a supercell is not preserved perfectly, however, as evidenced for example by significant cyclic variations (Adlerman et al. 1999).

## 3. Determination of pressure and components of motion

_{H}∂

_{z}

*p*

_{nh}|

_{W(z,t)}where the nonhydrostatic pressure is determined by a diagnostic equation in anelastic flow. In this section, we present solutions

*p*

_{nh}of this equation in terms of the other variables

*u,*

*υ,*

*w,*

*p*

_{h}. In the anelastic approximation, the density

*ρ*can be replaced by

*ρ*

_{s}(

*z*), the density of an appropriate static base-state atmosphere, except when multiplied by gravity (Bannon 1996). This means that horizontal density variations cannot be neglected in the computation of hydrostatic pressure. Anelastic flow on an

*f*plane is governed by the equations

^{∞}

_{z}

*ρq*

_{l}

*dz*of liquid water and ice in the overlying column of mass

*q*

_{l}is the mixing ratio for water plus ice). Taking the three-dimensional divergence of (3.1) and using (3.2) results in a diagnostic equation for

*p*

_{nh}on an

*f*plane:

*d*

_{z}≡

*d*/

*dz,*

*e*

_{ij}≡ (∂

*υ*

_{i}/∂

*x*

_{j}+ ∂

*υ*

_{j}/∂

*x*

_{i})/2 is the rate-of-deformation tensor,

*≡ ∇ ×*

**ω****v**is the relative vorticity vector, and the Einstein summation convention is used. The minus sign has been placed on the left of (3.3a) because ∇

^{2}is an essentially negative operator (Duff and Naylor 1966, p. 270); that is, ∇

^{2}

*p*

_{nh}and

*p*

_{nh}are negatively correlated, according to Green's theorem, in a domain with ∂

*p*

_{nh}/∂

*n*= 0 on the boundary. The forcing function

*F*consists of (i) hydrostatic forcing

*F*

_{M}arising from horizontal differences in the weights of overlying atmospheric columns and (ii) dynamical forcing due to “splat” and “spin” (

*F*

_{SPLAT}and

*F*

_{SPIN}), following Rowland (1880) and Bradshaw and Koh (1981). The splat/spin decomposition is invariant to Galilean transformations and to three-dimensional rotations of the coordinate axes, and hence is physically meaningful. RK85 decomposed the dynamical forcing (with

*f*= 0) into a shear term, 2

*ρ*

_{s}(∂

_{x}

*υ*∂

_{y}

*u*+ ∂

_{y}

*w*∂

_{z}

*υ*+ ∂

_{z}

*u*∂

_{x}

*w*) ≡ 2

*ρ*

_{s}(

*e*

^{2}

_{12}

*e*

^{2}

_{23}

*e*

^{2}

_{31}

*F*

_{SPIN}, and an extension term,

*ρ*

_{s}[(∂

_{x}

*u*)

^{2}+ (∂

_{y}

*υ*)

^{2}+ (∂

_{z}

*w*)

^{2}−

*w*

^{2}

*d*

_{zz}ln

*ρ*

_{s}] ≡

*ρ*

_{s}[

*e*

^{2}

_{11}

*e*

^{2}

_{22}

*e*

^{2}

_{33}

*w*

^{2}

*d*

_{zz}ln

*ρ*

_{s}]. Their decomposition has less partial cancellation between terms, but is unphysical because it is not invariant to rotation of the

*x*and

*y*axes about the

*z*axis. The shear term is not negative definite, and so their claim that the shear term contributes to negative pressure perturbations (e.g., WR, p. 1462; Klemp 1987, p. 377) is questionable. Negative values of the shear term arise from the negative-definite

*F*

_{SPIN}.

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 *F*_{M}. 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 *d*_{zz} ln*ρ*_{s} ≤ 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).

*F*

_{L}, as in RK82. Then

**S**

_{0}≡

*d*

_{z}

**v**

_{0}is the environmental shear,

*e*

^{′}

_{ij}

*υ*

^{′}

_{i}

*x*

_{j}+ ∂

*υ*

^{′}

_{j}

*x*

_{i})/2 is the deviation rate-of-deformation tensor,

*′ ≡ ∇ ×*

**ω****v**′ is the deviation relative vorticity, the linear forcing is

*F*

_{LN}≡

*F*

_{M}+

*F*

_{L}, and the nonlinear forcing

*F*

_{NL}is comprised of nonlinear splat and spin forcing

*F*

^{″}

_{SPLAT}

*F*

^{″}

_{SPIN}

*z*→ ∞. In a horizontally infinite domain, the solution

*p*

_{nh}of (3.3c) can be decomposed into hydrostatic, shear-induced, and nonlinear parts corresponding to the different forcings; that is,

*p*

_{nh}=

*p*

_{nhLN}+

*p*

_{nhNL}=

*p*

_{nhM}+

*p*

_{nhL}+

*p*

_{nhNL}, where −∇

^{2}

*p*

_{nhM}=

*F*

_{M}and ∂

_{z}

*p*

_{nhM}= 0 at horizontal boundaries, etc.

**S**| be the scale of the low- and midlevel shear,

*D*and

*L*be length scales for vertical and horizontal gradients, respectively,

*H*be the height of the equilibrium level,

*ρ*

_{s}≡

*H*

^{−1}

^{H}

_{0}

*ρ*

_{s}(

*z*)

*dz,*and

*U*

_{S}≡ |

**S**|

*D.*Note that

*U*

_{s}can be interpreted as the length of the lower portion of a smooth hodograph and that Ri ≡ CAPE/

*U*

^{2}

_{S}

*g*

^{H}

_{0}

*ρ*′/

*ρ*

_{s})

*dz,*

*w*∼ (CAPE)

^{1/2}, and

*p*

_{h}=

*g*

^{H}

_{z}

*ρ*

_{s}(

*ẑ*)

*dẑ*+

*g*

^{H}

_{z}

*ρ*′

*dẑ*+

*p*

_{h}|

_{z=H}, we estimate that

*F*

_{M}∼

*ρ*

_{s}

*L*

^{2}and

*F*

_{L}∼ 2

*ρ*

_{s}

**S**|(CAPE)

^{1/2}/

*L.*From the conservation of (zero) potential vorticity, the vertical vorticity of a parcel is

*ζ*=

*· ∇*

**ω**_{H}

*h,*where

*h*is the finite vertical displacement of the parcel (DJ84; RK85). Therefore,

*ζ*≡ |

**S**|

*D*/

*L*and

*F*

_{NL}∼ 2

*ρ*

_{s}

*ζ*

^{2}∼

*ρ*

_{s}

*U*

^{2}

_{S}

*L*

^{2}. If we let

*L*∼

*D*/2 for a supercell updraft, the relative magnitudes of the forcing terms in (3.3c) are

*and*their distribution relative to the updraft.

*p*′ ≡

*p*−

*p*

_{s}(

*z*) from a base-state hydrostatic pressure

*p*

_{s}(

*z*) instead of for the local nonhydrostatic pressure

*p*

_{nh}. Their equation is obtained by adding ∇

^{2}(

*p*

_{s}−

*p*

_{h}) to both sides of (3.3a), resulting in

^{2}

*p*

*g*

_{z}

*ρ*

*F*

_{L}

*F*

_{NL}

_{z}(

*p*

_{s}−

*p*

_{h}) =

*gρ*′. This form is not used here because the solution of (3.3e) contains a surface-integral term owing to the BCs being inhomogeneous (i.e., ∂

_{z}

*p*′ = −

*gρ*′ at

*z*= 0,

*H*).

*ρ,*

*q*

_{l}, and wind fields at a particular time, the Poisson equation (3.3), subject to Neumann BCs

*z*≥ 0 with the homogeneous Neumann boundary condition

_{z}

*p*

_{nh}

*z*

*w*= 0 at

*z*= 0 and (3.4). The diagnostic pressure equation is a derivative of the equations of motion. Since differentiation entails a loss of information and uniqueness, the BC on pressure must be consistent with the equations of motion in order to provide enough of the “missing information” to recover the pressure field to within the usual arbitrary additive constant. Failure to consider the BC results in an erroneous pressure solution. In (3.3a)

*p*

_{nh}may be regarded as the potential at a field point associated with a distribution of point sources (present where

*F*> 0) and sinks (present where

*F*< 0). Recall that a point source has a far-reaching influence on the potential field (“action at a distance”). According to the method of images (Duff and Naylor 1966, p. 272), the ground

*z*= 0 (where ∂

_{z}

*p*

_{nh}= 0) has the same effect as an even extension of the forcing function below the ground, that is, as giving each source or sink a fictitious identical image or reflection of the same sign and magnitude below the ground (Fig. 1). On the other hand, the homogeneous Dirichlet BC

*p*

_{nh}= 0 is consistent with odd extension of

*F*below the ground and mirror images of the opposite sign (Fig. 1). The pressure field induced by a source of high pressure and its image source below the ground is shown in Fig. 2, along with the Dirichlet case where the image is a pressure sink (i.e., a source of low pressure). The correct BC at the ground radically alters the pressure gradient near the ground, and allows for surface pressure features in spite of the zero forcing there. These features are forced from above (Dudhia and Moncrieff 1989; Davies-Jones 1996, hereafter DJ96). The Dirichlet BC is discussed here because RK's heuristic solution

*p*

_{nhL}∼

*F*

_{L}of −∇

^{2}

*p*

_{nhL}=

*F*

_{L}satisfies it instead of the correct one.

*z*≥ 0 is equivalent to the one over the full space −∞ <

*x,*

*y,*

*z*< ∞, where an even extension of

*F*is made below the ground; that is,

*F*(

*x,*

*y,*−

*z*) =

*F*(

*x,*

*y,*

*z*). The formal solution is thus,

*π*[(

*x̂*−

*x*)

^{2}+ (

*ŷ*−

*y*)

^{2}+ (

*ẑ*−

*z*)

^{2}]

^{1/2}is a Green's function (GF; Duff and Naylor 1966, pp. 272, 282). This is the correct way to invert the Laplacian operator in (3.3a) subject to the BC (3.5). The pressure field is not just a facsimile of

*F.*From differentiation of (3.6),

*x̂,*

*ŷ,*

*ẑ*) on the pressure at a field point P ≡ (

*x,*

*y,*

*z*) varies inversely with distance

*d*

_{PQ}= [(

*x*−

*x̂*)

^{2}+ (

*y*−

*ŷ*)

^{2}+ (

*z*−

*ẑ*)

^{2}]

^{1/2}. In contrast, the influence of unit forcing from Q on −∇

_{H}∂

_{z}

*p*

_{nh}at P decreases as (

*d*

_{PQ})

^{−3}for large

*d*

_{PQ}. Action at a distance is less far-reaching for −∇

_{H}∂

_{z}

*p*

_{nh}than for

*p*

_{nh}itself. From (3.8) we may deduce a distribution of

*F*that forces, say, an southeast (SE) gradient of NHVPGF at the field point (Fig. 3).

*p*

_{nh}=

*p*

_{nhL}+

*p*

_{nhM}+

*p*

_{nhNL}into (2.11), we see that updraft motion can be decomposed into linear shear-induced and weight-induced parts, and a nonlinear dynamical part:

**c**

_{L}, from an assumed initial updraft (section 5). For an updraft that changes form, all the motions are functions of time and height. Equation (3.9c) shows that cold pools can affect updraft propagation at low levels (Droegemeier et al. 1993), because shear-induced propagation owing to a gradient in VPGF of 3 × 10

^{−2}m s

^{−2}km

^{−1}at the updraft center (a typical value in the simulations of WR) is negated entirely by an opposing temperature gradient there of 1 K km

^{−1}. We assume henceforth that |

**c**

_{M}| is small compared to |

**c**

_{L}| and |

**c**

_{NL}| (as is the case in RK82, p. 143; RK85, p. 284; Klemp 1987, pp. 383, 389; WR, p. 1462). This is true prior to the development of intense asymmetric water loading and cold pool.

## 4. Propagation of form-preserving disturbances

*A*(

*t*) is the amplification (initial value 1),

*X*and

*Y*are storm-relative coordinates defined by

**C**(

*t*) ≡ (

*C*

_{1},

*C*

_{2}) is the propagation velocity (K02). It follows from the chain rule, the anelastic continuity equation and the definitions of relative vorticity and divergence that

*q*′ =

*w*′,

*ζ*′, or

*δ*′ [and

*q*

_{0}(

*z*) ≡ 0 for these variables]. In the linear theory of dry inviscid convection in a sheared nonrotating environment (DJ84), potential vorticity is conserved and is zero, pressure satisfies a Poisson equation with linear forcing, and hence the shapes of the buoyancy field (K02) and the pressure field are preserved also. For nonlinear motions, it is apparent from (3.3c) that

*p*

_{nhL}=

*A*(

*t*)

*p̃*

_{nhL}and

*p*

_{nhNL}=

*A*

^{2}(

*t*)

*p̃*

_{nhNL}. In the context of this paper, K02 showed that (i) the growth rate of an FPD,

*d*ln

*A*/

*dt*≡

*σ*(

*t*) =

*σ*

_{LN}+

*σ*

_{NL}(

*t*), where

*σ*

_{LN}= (

*σ*−

*d*ln

*σ*/

*dt*) is constant (>0) and the nonlinear part

*σ*

_{NL}(

*t*) =

*A*(

*t*)

*d*ln

*σ*/

*dt*|

_{t=0}< 0; (ii)

*A*(

*t*) =

*σ*

_{LN}[

*σ*(0) exp(−

*σ*

_{LN}

*t*) −

*σ*(0) +

*σ*

_{LN}]

^{−1}; (iii)

**C**(

*t*) =

**C**

_{LN}+

**C**

_{NL}(

*t*), where

**C**

_{LN}is constant and

**C**

_{NL}=

*A*(

*t*)

**C**

_{NL}(0). As

*t*→ ∞,

*A*(

*t*) →

*σ*

_{LN}/[−

*σ*

_{NL}(0)].

_{X}

*q̃*= ∂

_{Y}

*q̃*= 0 at Q, yields the desired result that

**c**(

*z,*

*t*) =

**C**(

*t*) for the motions of all the extrema. Inserting (4.3) into the (2.8) version of Petterssen's formula and equating terms in

*A*and

*A*

^{2}, respectively, yields the following linear and nonlinear motions for the extrema of any variable

*q*of an FPD in the principal axes coordinate system.

*Q*=

*A*(

*t*)

*Q̃*

_{LN}+

*A*

^{2}(

*t*)

*Q̃*

_{NL}. For example, in the case where (2.5) is taken to be the anelastic version of (2.10),

*q̃*→

*w̃,*

*Q̃*

_{LN}→ −

*α*

_{s}∂

_{z}

*p̃*

_{nhLN}, and

*Q̃*

_{NL}→ −

*α*

_{s}∂

_{z}

*p̃*

_{nhNL}, where

*α*

_{s}≡

*ρ*

^{−1}

_{s}

**C**

_{LN}and

**C**

_{NL}can be obtained as follows. Inserting (4.3) into (2.5) and equating powers of

*A*yields the linear and nonlinear equations

*L*is now the linear material-derivative operator for

*q*′. Multiplying (4.5a) by ∂

_{X}

*q̃*and ∂

_{Y}

*q̃,*integrating the resulting equations over an area of the

*X*–

*Y*plane with

*q̃*= 0 on its lateral boundary, and solving for

*C*

_{LN1}and

*C*

_{LN2}yields (for all

*z*)

_{X}

*q̃*∂

_{Y}

*q̃*〉 terms vanish. From (4.6a,b) for

*w̃,*the shear-induced motion of an updraft of circular cross section is

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**−

**C**

_{0}are parallel everywhere in a reference frame moving with a constant velocity

**C**

_{0}, that is,

*=*

**ω***λ*(

**v**−

**C**

_{0}), 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).

*θ*

_{0}with a clockwise-turning circular hodograph centered on

**C**

_{0}. Nondimensional pressure, Π ≡ (

*p*/

*p*

_{0})

^{κ}, where

*p*

_{0}= 1000 mb, is introduced here to write the pressure-gradient force per unit mass as a potential field [−

*α*∇

*p*= −∇(

*c*

_{p}

*θ*

_{0}Π)]. The circular hodograph has to be greater than a semicircle to satisfy the BCs

*w*= 0 at

*z*= 0,

*H.*In cylindrical coordinates (

*r,*

*ϕ,*

*z*) in the moving reference frame, let the relative environmental wind,

**v**

_{0}−

**C**

_{0}≡ (

*U*

_{0},

*V*

_{0}), be given by

*M*and the rate of wind veering

*λ,*are constant. The Beltrami flow solutions for the cylindrical components of the wind deviations and the pressure are

*W*

_{max}is the maximum vertical velocity, the abnormality

*λ*= (

*k*

^{2}+

*μ*

^{2})

^{1/2},

*J*

_{0}and

*J*

_{1}are the zero- and first-order Bessel functions, and we assume for simplicity that

*p*=

*p*

_{0}at

*z*= 0. In (4.8d), Π is the sum of the hydrostatic pressure Π

_{H}≡ 1 −

*gz*/

*c*

_{p}

*θ*

_{0}and the nonhydrostatic pressure Π

_{nh}≡ Π

_{0}+ Π

_{nhL}+ Π

_{nhNL}, where Π

_{0}≡ −

*M*

^{2}/2

*c*

_{p}

*θ*

_{0}is a constant, Π

_{nhL}≡ −(

**v**

_{0}−

**C**

_{0}) ·

**v**′/

*c*

_{p}

*θ*

_{0}is a linear asymmetric term, and Π

_{nhNL}≡ −

**v**′ ·

**v**′/2

*c*

_{p}

*θ*

_{0}is a nonlinear axisymmetric term. Since the Boussinesq and anelastic solutions for deep convection are qualitatively similar (Moncrieff 1978; Seitter and Kuo 1983, hereafter SK), DJ85 applied (4.8) to an updraft 12 km high even though the Boussinesq equations are strictly valid only for shallow motions (≤3 km deep). The inviscid anelastic equations actually have an exact solution of similar form to (4.8) except that the vertical dependence is not purely sinusoidal (Davies-Jones and Richardson 2002). For both Boussinesq and anelastic flows, (2.11) reduces to

**C**

_{NL}= 0. Thus, at each level, the updraft motion deviates from the environmental flow

**v**

_{0}(

*z*) in the direction of ∇

_{H}(−∂

_{z}Π

_{nhL})|

_{r=0}(RK82). From the steady vertical equation of motion, the linear part of −

*c*

_{p}

*θ*

_{0}∂

_{z}Π

_{nhL}equals

*U*

_{0}∂

_{r}

*w*′. It follows from this and the relationship ∂

_{ϕ}

*U*

_{0}=

*V*

_{0}implied by (4.7) that −

*c*

_{p}

*θ*

_{0}∇

_{H}∂

_{z}Π

_{nhL}evaluated at

*r*= 0 using L'Hôpital's rule is equal to [

**v**

_{0}−

**C**

_{0}]∂

_{rr}

*w*

^{′}

_{r=0}. Thus, (2.11) and its specialization (4.9) successfully predict that the updraft is moving with the velocity

**C**

_{0}. The alternative formula (4.6) also yields

**C**

_{L}=

**C**

_{0}. When the hodograph is a full circle, the mean environmental wind is also at the center, which demonstrates that an updraft does not have to move relative to the mean wind in order to rotate as a whole. If the hodograph is less than a full circle, the updraft moves to the right (relative to the midlevel environmental shear vector) of the mean wind. Depending on the choice of the free parameters

*M*and

*W*

_{max}in the Beltrami solution, the nonlinear VPGF,

**v**′ · ∂

_{z}

**v**′, can dominate the linear VPGF ∂

_{z}[(

**v**

_{0}−

**C**

_{0}) ·

**v**′]; hence, one might draw the conclusion that the processes that contribute to updraft maintenance and propagation are significantly nonlinear, regardless of hodograph curvature (WR, p. 1470, based on discussion on p. 1462). In Beltrami flow, the nonlinear VPGF does maintain the updraft, but it is concentric with the updraft and does not cause the updraft to propagate because ∇

_{H}∂

_{z}Π

_{nhNL}|

_{r=0}= 0. Even at moderate instability (CAPE of 2200 m

^{2}s

^{−2}), instead of the Beltrami flow's neutral stability, the tendencies in circular shear for the cyclonic vortex to lie close to the updraft center and for linear propagation to dominate nonlinear propagation may be deduced from Figs. 9–12 in WR (see section 7).

*r*= 0) to

_{LN}≡ −∂

_{z}

*V*

_{0}∂

_{r}

*w*′ is the linear (asymmetric) tilting term in the vertical vorticity equation and −∂

_{z}

*V*

_{0}is the radial environmental vorticity. Thus, at each level, the motion of the cyclonic vortex deviates from the environmental flow

**v**

_{0}(

*z*) in the direction of ∇

_{H}

_{LN}|

_{r=0}and the nonlinear (axisymmetric) part of the tilting term plays no role in vortex propagation in Beltrami flow. Since −∂

_{z}

*V*

_{0}=

*λU*

_{0}from (4.7) and

*ζ*′ =

*λw*′,

_{LN}=

*λU*

_{0}∂

_{r}

*w*′ = −

*λc*

_{p}

*θ*

_{0}∂

_{z}Π

_{nhL}. Hence (2.15) yields the same motion (

**C**

_{0}) for the vortex as (2.11) does for the updraft. The updraft and cyclonic vortex stay together (on the axis) because the linear NHVPGF and linear tilting term are collocated.

_{z}

**v**

_{0}· ∇

_{H}

*w*′ = −

*k*

^{2}

*V*

_{0}

*V*′,

^{2}

_{H}

_{nhL}= −

*k*

^{2}

^{2}

_{H}

_{nhL}=

*k*

^{2}(

*U*

_{0}

*U*′ +

*V*

_{0}

*V*′)/

*c*

_{p}

*θ*

_{0}, and ∂

_{rr}

*δ*′|

_{r=0}= −

*k*

^{2}

*δ*′|

_{r=0}/2. Thus, (4.11) becomes

_{H}(

*U*

_{0}

*U*′)|

_{r=0}= (

**v**

_{0}−

**C**

_{0})

*δ*′|

_{r=0}/2,

**c**(

*z*) =

**C**

_{0}again.

*ζ*= 0 at

*z*= 0) is approximately valid for the first 40 min of numerical simulations with

*f*= 0 (RK85; Davies-Jones and Brooks 1993). Rotation at the ground is virtually absent until a significant downdraft develops because air entering the updraft acquires vertical vorticity through tilting of horizontal vorticity only as it rises away from the ground (Davies-Jones 1982). In a reference frame moving with the propagation velocity

**C**, the horizontal equations of motion for irrotational flow on the ground have the integral (∂

_{t}

*ϕ*′ +

*p*/

*ρ*

_{0}+ |

**v**

_{H}−

**C**|

^{2}/2)|

_{z=0}=

*f*(

*t*) [Bernoulli's law for unsteady irrotational flow; see Prandtl and Tietjens (1957, p. 127)], where

**v**

^{′}

_{H}

_{H}

*φ*′ = (∂

*φ*′/∂

*r*)

**e**

_{r}defines the surface velocity potential

*φ*′,

**e**

_{r}is the unit vector in the radial direction, and

*f*(

*t*) is an arbitrary function of time. The asymmetric part of the surface pressure field is given by

*p*

_{L}

*ρ*

_{0}

**v**

_{0}

**C**

**v**

**v**′ is radially inward, the lowest surface pressure occurs on the side of the updraft with the greatest inflow. Thus we have an explanation for “inflow lows,” which have been detected at the surface in mesonet observations by Charba and Sasaki (1971), Lemon (1976), and Barnes (1978a,b), and at low levels in aircraft data by LeMone et al. (1988), in pressure fields calculated from Doppler radar–observed winds by Bonesteele and Lin (1978), and in numerical simulations (e.g., Wilhelmson and Klemp 1981; Klemp and Rotunno 1983; Brooks et al. 1993). At the top boundary where

*w*again vanishes,

**v**′ is radially outward above the axisymmetric updraft if the flow there is also irrotational, and there is an “outflow low.”

## 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

**c**

_{L}(

*z,*

*t*) and

**c**

_{M}(

*z,*

*t*). From (2.15), the linear motion of a circular vorticity extremum is

*ω*_{0}=

**k**×

**S**

_{0}= (−

*d*

_{z}

*υ*

_{0},

*d*

_{z}

*u*

_{0}, 0) is the environmental vorticity, and

**S**

_{0}≡

*d*

_{z}

**v**

_{0}≡

*S*

_{1}

**i**+

*S*

_{2}

**j**is the environmental shear. At a given level a vorticity maximum deviates from the environmental wind in the direction of the greatest linear tilting.

_{z}

*p*

_{nhLN}, where

_{z}

*p*

_{nhLN}, on the axis (0, 0,

*z*) in cylindrical coordinates (

*r,*

*ϕ,*

*z*), where

*r*≡ (

*x*

^{2}+

*y*

^{2})

^{1/2},

*ϕ*≡ tan

^{−1}(

*y*/

*x*), is

*F*

_{M}should be nearly collocated with the updraft (RK82; RK85; WR), and

*F*

_{L}should provide the main contribution to the left side of (5.3) as in RK82. In cylindrical coordinates,

**c**

_{L}(

*z,*

*t*

_{0}) of a deforming updraft at an initial time

*t*

_{0}or to estimate the height-independent shear-induced motion

**C**

_{L}of a form-preserving updraft. The axisymmetric form

*w*

*r,*

*ϕ,*

*z*

*W*

*z*

*J*

_{0}

*kr*

*W*(

*z*) = 0 at

*z*= 0,

*H.*In spite of its rather spurious oscillations, the Bessel function radial dependence is preferred over other possible functions (such as a Gaussian) for the following reasons: (i) it is compatible with the RK heuristic solution because Bessel functions arise in solutions of Helmholtz's equation in cylindrical coordinates; (ii) it permits separation of variables and tractable mathematics; (iii) the Beltrami flow exact solution presented in section 4 has this radial profile; (iv) as in real life, the compensating descent is close to the updraft, not at infinity; and (v) any axisymmetric updraft

*w*(

*r,*

*ϕ,*

*z*) =

*f*(

*r*)

*W*(

*z*) can be expressed in terms of the Fourier–Bessel transform

*g*(

*k̂*) of

*f*(

*r*) and the Fourier sine series of

*ρ*

_{s}(

*z*)

*W*(

*z*) as

*J*

_{0}(

*kr*) sin(

*πz*/

*H*) represents one wave component in this expansion.

_{rr}

*w*|

_{r=0}= (

*k*

^{2}/2)

*W*(

*z*) here. From (3.9b) and (5.9), the shear-induced motion of the updraft at the level

*z*is

*w*(

*r,*

*z*) =

*W*(

*z*)

*J*

_{0}(

*kr*) from the start, and solving the equation −∇

^{2}

*p*

_{nhL}=

*F*

_{L}by separation of variables. The linear forcing is given by

*p*

_{nhL}

*J*

_{1}

*kr*

*P*

_{1}

*z*

*ϕ*

*P*

_{2}

*z*

*ϕ*

*G*(

*z,*

*ẑ*) is the Green's function, and

*a*and

*b*are limits of integration specified below. The Green's function is symmetric in

*z*and

*ẑ,*satisfies the ODE

*d*

^{2}

*G*/

*dẑ*

^{2}−

*k*

^{2}

*G*= 0 everywhere except at

*ẑ*=

*z,*the BCs

*dG*/

*dẑ*= 0 at

*z*=

*a,*

*b,*and has a first-order jump discontinuity of 1 at

*ẑ*=

*z*described by

*dG*/

*dẑ*(

*z,*

*z*−) −

*dG*/

*dẑ*(

*z,*

*z*+) = 1. The nonhydrostatic pressure induced by the updraft–shear interaction is therefore

*r,*

*ϕ,*

*z*) is the sum, weighted by

*G*(

*z,*

*ẑ*), of the sources and sinks at the points (

*r,*

*ϕ,*

*ẑ*),

*a*≤

*ẑ*≤

*b,*along the vertical line through (

*r,*

*ϕ,*

*z*). Note that the closer to the ground strong shear and updraft are located, the more influence they have on surface pressure. The shear-induced NHVPG is given by

_{H}of (5.16), the shear-induced propagation velocity in anelastic flow is

*ρ*

_{s}

*W*

**S**

_{0}) (

*ẑ*) is extended evenly below the ground,

*a*= −∞,

*b*= ∞, and

*G*

*z,*

*ẑ*

*k*

*ẑ*

*z*

*k.*

*z*=

*H*(i.e.,

*w*= 0 at

*z*= 0,

*H*), a source at

*ẑ*has an infinite number of images at −

*ẑ,*±(2

*nH*+

*ẑ*), ±(2

*nH*−

*ẑ*),

*n*= 1, 2, … , owing to multiple reflections. It is now simpler to restrict the integration to the physical domain by setting

*a*= 0,

*b*=

*H,*and by including the combined influences of a source and all its images in the Green's function; that is,

*G*(

*z,*

*ẑ*) =

^{∞}

_{i=1}

*k*|

*ẑ*

_{i}−

*z*|)/2

*k,*where the

*z*

_{i}are the heights of the source and its images (listed above). Summation of the series yields the expression

*ẑ*for selected values of

*z.*In both (5.18a) and (5.18b)

*G*(

*z,*

*ẑ*) is positive and has a maximum at

*F*

_{L}from a level

*ẑ*≠

*z*causes the updraft motion at level

*z,*

**c**

_{L}(

*z,*

*t*

_{0}), to deviate from the environmental wind at that level

**v**

_{0}(

*z*) in the direction of

**S**

_{0}(

*ẑ*) if

*ẑ*>

*z,*and in the direction of −

**S**

_{0}(

*ẑ*) if

*ẑ*<

*z*(Fig. 5). For a simple hodograph that turns only in one direction (either clockwise or counterclockwise), the integrated forcing (over

*ẑ*) clearly moves the motion off the hodograph to the concave side (Fig. 5b). From (5.10) it is clear that the influence of a given remote level

*ẑ*on

**c**

_{L}(

*z,*

*t*

_{0}) varies as 1/

*k.*Since the radius

*R*of the central updraft [the distance from the axis to the first zero of

*J*

_{0}(

*kr*)] is 2.40/

*k,*the propagation velocities in highly sheared environments increase with updraft diameter, in agreement with Newton and Fankhauser (1975), who found that large-diameter storms deviate far more from the mean winds than small- and medium-sized storms. RK82 solved the Poisson equation heuristically by supposing that in the interior of the flow, that is, far away from boundaries, −∇

^{2}

*p*′ ∼

*k*

^{2}

*p*′. LeMone et al. (1988) inserted the

*k*

^{2}factor for dimensional consistency and magnitude estimates after assuming that the vertical variation of

*p*′ was small compared to the horizontal variation. The heuristic solution of RK82 can be obtained in fact from (5.15) and (5.18) by taking the limit

*k*→ ∞ since

*G*(

*z,*

*ẑ*) →

*δ*(

*ẑ*−

*z*)/

*k*

^{2}in this limit where

*δ*( ) is the delta function. Therefore the heuristic solution is valid only for infinitessimally narrow erect updrafts and is least applicable to the wide updrafts found in supercells. In the limit

*k*→ ∞ (5.17) becomes

*G*(

*z,*

*ẑ*)/∂

*z*= −∂

*G*(

*z,*

*ẑ*)/∂

*ẑ,*integrating by parts, and using (3.2). Thus, infinitely narrow axisymmetric updrafts fail to propagate at any level relative to the environmental wind and hence are unable to stand erect in wind shear. In an updraft composed initially of multiple horizontal Fourier components, the short waves will shear apart as they tend to move with the environmental wind at each level, and very long waves also will deform because they propagate too much at each level to yield a height-independent

**c**[as in example (d) below]. Only a narrow band of large-wavelength components that move duly off the hodograph will persist.

### b. Shear-induced solutions for special shear flows

*W*(

*z*) =

*ρ*

_{0}

*W*

_{0}sin

*μz*/

*ρ*

_{s}(

*z*),

*μ*=

*π*/

*H,*

*H*= 12 km, and

*W*

_{0}= 30 m s

^{−1}(Fig. 6a). The base-state density is approximated as

*ρ*

_{s}(

*z*) =

*ρ*

_{0}exp(−

*z*/

*H*

_{ρ}), where

*ρ*

_{0}= 1.2 kg m

^{−3}and

*H*

_{ρ}= 9.7 km (Fig. 7a). The maximum vertical velocity of 60 m s

^{−1}occurs at a height of 7.4 km. Although ∇

^{2}(

*ρ*

_{s}

*w*) = −(

*k*

^{2}+

*μ*

^{2})

*ρ*

_{s}

*w*in all cases, the solution of the linear boundary-value problem for pressure is never

*p*

^{}

_{nh}

^{2}

*p*

^{}

_{nh}

*k*

^{2}+

*μ*

^{2}).

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 *p*_{nhNL}, are axisymmetric, and *p*_{nhM} 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 **S**_{0} = *S***i**, 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 *p*_{nhL} 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, *F*_{L}/(*k*^{2} ;pl *μ*^{2}), 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 ± Δ*z* cancel each other for all Δ*z* ∈ (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 *ρ*_{s}*w* 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 *c*_{L}(*z,* *t*_{0}) − *u*_{0}(*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 *ρ*_{s}*w* at 6 km.

#### 2) Circular-hodograph cases

**C**

_{0}= (10, 0) m s

^{−1}is the wind at the center of the hodograph,

*M*= 10 m s

^{−1}is the constant wind speed relative to the center, and

*λ*= 30° km

^{−1}is the rate at which the relative winds veer with height. The half-circle hodograph is obtained from the circle one by holding the winds constant above 6 km (Fig. 10). In (c), the initial updraft with

*R*= 5.3 km matches the nonbuoyant Beltrami flow solution (4.8c) in the Boussinesq limit (where

*ρ*

_{s}≈

*ρ*

_{0}). This is the only case where the updraft is a form-preserving one. Even though steadiness and updraft rotation have not been assumed, Bernoulli's law has not been invoked, and baroclinity has not been excluded, the Green's function and the Beltrami flow solutions for

*p*

_{nhL}are the same. Owing to balance between the shear-induced NHVPGF and horizontal advection of vertical velocity in the exact steady solution,

**c**

_{L}(

*z,*

*t*) =

**C**

_{0}; that is, the updraft maximum propagates off the hodograph at each level to the center of the circle. The influence of −∇

_{H}∂

_{z}

*p*

_{nhM}|

_{r=0}on propagation has not been included, but this effect is small since the extrema of

*p*

_{nhM}are near the updraft axis and the motion is nearly

**C**

_{0}in WR's simulation of this case. New updraft generation by the shear-induced VPGF is collocated with cyclonic-vorticity generation by linear tilting of the purely streamwise environmental vorticity: thus the updraft and cyclonic vortex are collocated.

Figure 11 compares the Green's function and RK distributions around the updraft of pressure and NHVPGF at the radius of maximum |∂_{r}*w*|. The azimuth angles of the extrema of *p*_{nhL} and −∂_{z}*p*_{nhL} 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 **c**_{L} (*z,* *t*_{0}) (instead of a single point **C**_{0}) 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 **C**_{0}. 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 **c**_{L} (*z,* *t*_{0}) 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 *p*_{t} would still vary significantly across streamlines and vortex lines according to Crocco's theorem, ∇*p*_{t} = *ρ*_{s}**v** × * 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

*p*

_{nhL}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,

**c**

_{L}(

*z,*

*t*

_{0}) 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,

**c**

_{L}(

*z,*

*t*

_{0}) 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*∂

_{z}

*w*) 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 **v**_{0}(*z*) − **C**_{L} is directed to the left of **S**_{0}(*z*), that is in the same general direction as the environmental vorticity vector **ω**_{0}(*z*) = **k** × **S**_{0}(*z*). Hence the vorticity in the inflow is at least partly streamwise in a reference frame moving with the velocity **C**_{L} (or **C**_{LN} if **C**_{M} ≈ 0).

*L*

*ζ̃*

**ω**_{0}· ∇

*w̃*from (2.14) and (4.5a). K02 showed that a vertical-displacement variable

*h̃*exists such that

*Lh̃*=

*w̃.*Therefore,

*ζ̃*

**ω**_{0}· ∇

*h̃*(conservation of zero potential vorticity), if

*w̃*and

*ζ̃*

*w*and

*ζ*is thus

**s**and

**n**are unit vectors in the direction of and 90° to the left of

**v**

_{0}(

*z*) −

**C**

_{LN}, respectively,

*ω*

_{s}≡

**s**·

**ω**_{0}and

*ω*

_{c}≡

**n**·

**ω**_{0}are the streamwise and crosswise vorticity, and ∂

*h̃*/∂

*s*≡

**s**· ∇

*h̃,*etc. Assuming either that

*h̃*is circular in cross section (as in DJ84; K02) or that

*w̃*≡

*dh̃*/

*dt*is circular (as in this paper) eliminates the crosswise term by virtue of symmetry of

*h̃*about the streamwise axis through the updraft center. Note that oblique orientation of an highly elliptical updraft maximum relative to the storm-relative wind affects rotation and so is a complicating factor (DJ84). For a circular updraft, (6.1) states that the covariance of

*ζ*and

*w*is proportional to the helicity density |

**v**

_{0}−

**C**

_{LN}|

*ω*

_{s}. The covariance of the linear tilting and NHVPGF terms (or equivalently, the covariance of

*L*

*ζ̃*

*Lw̃*) also depends on the helicity density [not on the superhelicity density

**ω**_{0}· ∇ ×

**ω**_{0}(Hide 1989) as indicated by RK82 Eq. (13), which is derived using the heuristic pressure]. The relative locations of the extremes of variables on a level surface in the cases of straight shear and circular clockwise-turning shear are summarized in Fig. 5.

How is the covariance of *ζ* and *w* established? Multiplying the vertical-motion equation *Lw̃* = −*α*_{0}∂_{z}*p̃*_{nhLN} by *ζ̃**L**ζ̃***ω**_{0} · ∇*w̃* by *w̃,* adding the resultant equations together, and integrating over the domain yields ∂_{t}〈〈*ζw*〉〉 = [*σ*(*t*)/*σ*_{LN}]〈〈*ζ*(−*α*_{0}∂_{z}*p*_{nhLN})〉〉, where 〈〈 〉〉 denotes domain integral, and the relationship ∂_{t}〈〈*ζw*〉〉 = 2*σ*(*t*)〈〈*ζw*〉〉 has been used. The equivalent result, using the full equations, is ∂_{t}〈〈*ζw*〉〉 = 〈〈*ζ*(−*α*_{0}∂_{z}*p*_{nh})〉〉, where the right side represents the conversion of horizontal helicity into vertical helicity (Lilly 1986b). It follows that ∂_{t}〈〈*ζw*〉〉 = −{*σ*(*t*)/[−*σ*_{NL}(*t*)]}〈〈*ζ*(−*α*_{0}∂_{z}*p*_{nhNL})〉〉. Thus, conversion of horizontal to vertical helicity in an FPD requires a positive covariance between *ζ* and the linear NHVPGF to establish convection with positive vertical helicity and a negative one between *ζ* and nonlinear NHVPGF to decrease the conversion to zero as the asymptotic steady state is approached. This result appears to be contradicted by the right-moving supercell in straight shear, where the correlation between *ζ* and nonlinear NHVPGF is in fact positive. However, the covariances are zero in this case because they are taken over the whole domain, which includes both supercells. Furthermore, FPD theory does not apply to storm splitting and the two resulting supercells with divergent paths.

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 **v**^{′}_{H}_{W(z,t)} steering term in (3.9d). For a steady configuration where all the above induced velocities must be equal, the vortices must be on the east-northeast and east-southeast sides of the updraft as in Brown (1992). Note that the presence of the vortex pair shifts the updraft motion toward the lower part of the hodograph. The −∇_{H}(*w*∂_{z}*w*) nonlinear term in (3.9d) also contributes an upshear motion if the updraft leans downshear.

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 *F*_{NL} 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 *F*^{″}_{SPIN}*F* induces a horizontal gradient of the NHVPGF at each updraft center that has a component away from the plane of symmetry. The split storms thus propagate continuously away from one another. Low-level convergence along the leading edges of the storms' cool-air outflow may help to maintain the anomalous propagation (Brown 1992). The rotation of the updrafts is sustained due to tilting of the storm-relative streamwise (antistreamwise) vorticity in the inflow of the right (left) mover. Thus the rotation is predicted qualitatively by linear theory if the storm motion is given (DJ84).

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 *F*^{″}_{SPIN}*F*_{M} + *F*^{″}_{SPLAT}

### b. Storm motion in circular shear

*ζ*and

*linear*NHVPGF. The updraft is wide, erect, and circular in cross section even in the presence of strong shear. These are all attributes of supercell updrafts in circular shear. The streamlines and vortex lines coincide, the cyclonic vortex is collocated with the central updraft and is a midlevel mesocyclone, and there is an annular region of anticyclonic vorticity surrounding the updraft instead of an anticyclonic vortex. Nonhydrostatic pressure in steady Beltrami flow is given by the universal Bernoulli relationship,

*c*

_{p}

*θ*

_{0}Π

_{nh}= −|

**v**−

**c**|

^{2}/2, which may be written as

*c*

_{p}

*θ*

_{0}

_{nh}

**ω**^{2}

*λ*

^{2}

*=*

**ω***λ*(

**v**−

**c**). (Note that this is the solution, not of −

*c*

_{p}

*θ*

_{0}∇

^{2}Π

_{nh}= −|

*|*

**ω**^{2}/2, as would be expected heuristically, but of −

*c*

_{p}

*θ*

_{0}∇

^{2}Π

_{nh}=

*e*

_{ij}

*e*

_{ij}− |

*|*

**ω**^{2}/2.) Vertical differentiation of (7.1) yields

*c*

_{p}

*θ*

_{0}

_{z}

_{nh}

*λ*

^{−2}

_{z}

**ω**_{0}

**ω**

**ω**_{z}

**ω**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 *U*_{s}/*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 ∂_{y}*w* and ∂_{z}*p*_{nhL} in (4.6c) is larger than that of ∂_{y}*w* and ∂_{z}*p*_{nhNL} 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**×

*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,*

**ω***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

*The local flow is highly helical as indicated by a normalized helicity density*

**ω**.**v**·

*/(|*

**ω****v**‖

*|) of 0.87 (Droegemeier et al. 1993), yet the normalized Lamb vector |*

**ω****v**×

*|/(|*

**ω****v**‖

*|) 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*

**ω****c**=

**0**) in the full-circle simulation, −∇

_{H}∂

_{z}

*p*

_{nh}|

_{W}is balanced by −∇

_{H}Λ|

_{W}rather than by −∇

_{H}

*β*|

_{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

**v**

_{H}|

_{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 *F*^{″}_{SPIN}*F*_{M} + *F*^{″}_{SPLAT}

## 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.

## REFERENCES

Abramowitz, M., and I. A. Stegun, Eds. . 1964:

*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*. Applied Mathematics Series, No. 55, National Bureau of Standards, 1046 pp.Adlerman, E. J., K. K. Droegemeier, and R. Davies-Jones, 1999: A numerical simulation of cyclic mesocyclogenesis.

,*J. Atmos. Sci.***56****,**2045–2069.Bannon, P. R., 1996: On the anelastic approximation for a compressible atmosphere.

,*J. Atmos. Sci.***53****,**3618–3628.Barnes, S. L., 1978a: Oklahoma thunderstorms on 29–30 April 1970. Part I: Morphology of a tornadic thunderstorm.

,*Mon. Wea. Rev.***106****,**673–684.Barnes, S. L., 1978b: Oklahoma thunderstorms on 29–30 April 1970. Part II: Radar-observed merger of twin hook echoes.

,*Mon. Wea. Rev.***106****,**685–696.Bonesteele, R. G., and Y. J. Lin, 1978: A study of updraft–downdraft interaction based on perturbation pressure and single-Doppler radar data.

,*Mon. Wea. Rev.***106****,**62–68.Bradshaw, P., and Y. M. Koh, 1981: A note on Poisson's equation in a turbulent flow.

,*Phys. Fluids***24****,**777.Brooks, H. E., and R. B. Wilhelmson, 1993: Hodograph curvature and updraft intensity in numerically modeled supercells.

,*J. Atmos. Sci.***50****,**1824–1833.Brooks, H. E., C. A. Doswell III,, and R. Davies-Jones, 1993: Environmental helicity and the maintenance and evolution of low-level mesocyclones.

*The Tornado: Its Structure, Dynamics, Prediction, and Hazards, Geophys. Monogr.,*No. 79, Amer. Geophys. Union, 97–104.Brown, R. A., 1992: Initiation and evolution of updraft rotation within an incipient supercell thunderstorm.

,*J. Atmos. Sci.***49****,**1997–2014.Bunkers, M. J., B. A. Klimowski, J. W. Zeitler, R. L. Thompson, and M. L. Weisman, 2000: Predicting supercell motion using a new hodograph technique.

,*Wea. Forecasting***15****,**61–79.Chappell, C. F., 1986: Quasi-stationary convective events.

*Mesoscale Meteorology and Forecasting,*P. S. Ray, Ed., Amer. Meteor. Soc., 289–310.Charba, J., and Y. Sasaki, 1971: Structure and movement of the severe thunderstorms of 3 April 1964 as revealed from radar and surface mesonetwork data analysis.

,*J. Meteor. Soc. Japan***49****,**191–214.Das, P., 1979: A non-Archimedean approach to the equations of convection dynamics.

,*J. Atmos. Sci.***36****,**2183–2190.Davies-Jones, R., 1982: Observational and theoretical aspects of tornadogenesis.

*Intense Atmospheric Vortices,*L. Bengtsson and J. Lighthill, Eds., Springer-Verlag, 175–189.Davies-Jones, R., 1984: Streamwise vorticity: The origin of updraft rotation in supercell storms.

,*J. Atmos. Sci.***41****,**2991–3006.Davies-Jones, R., 1985: Dynamical interaction between an isolated convective cell and a veering environmental wind. Preprints,

*14th Conf. on Severe Local Storms,*Indianapolis, IN, Amer. Meteor. Soc., 216–219.Davies-Jones, R., 1986: Tornado dynamics.

*Thunderstorm Morphology and Dynamics,*2d ed., E. Kessler, Ed., University of Oklahoma Press, 197–236.Davies-Jones, R., 1996: Inclusion of boundary conditions on pressure in conceptual models of updraft-environment interaction. Preprints,

*18th Conf. on Severe Local Storms,*San Francisco, CA, Amer. Meteor. Soc., 713–717.Davies-Jones, R., and H. E. Brooks, 1993: Mesocyclogenesis from a theoretical perspective.

*The Tornado: Its Structure, Dynamics, Prediction, and Hazards, Geophys. Monogr.,*No. 79, Amer. Geophys. Union, 105–114.Davies-Jones, R., and Y. P. Richardson, 2002: An exact anelastic Beltrami-flow solution for use in model validation. Preprints,

*19th Conf. on Weather Analysis and Forecasting/15th Conf. on Numerical Weather Prediction,*San Antonio, TX, Amer. Meteor. Soc., 43–46.Davies-Jones, R., C. A. Doswell III,, and H. E. Brooks, 1994: Comments on “Initiation and evolution of updraft rotation within an incipient supercell thunderstorm.”.

,*J. Atmos. Sci.***51****,**326–331.Droegemeier, K. K., S. M. Lazarus, and R. Davies-Jones, 1993: The influence of helicity on numerically simulated convective storms.

,*Mon. Wea. Rev.***121****,**2005–2029.Dudhia, J., and M. W. Moncrieff, 1989: A three-dimensional numerical study of an Oklahoma squall line containing right-flank supercells.

,*J. Atmos. Sci.***46****,**3363–3391.Duff, G. F. D., and D. Naylor, 1966:

*Differential Equations of Applied Mathematics*. Wiley, 423 pp.Fujita, T. T., and H. Grandoso, 1968: Split of a thunderstorm into anticyclonic and cyclonic storms and their motion as determined from numerical model experiments.

,*J. Atmos. Sci.***25****,**416–439.Hide, R., 1989: Superhelicity, helicity and potential vorticity.

,*Geophys. Astrophys. Fluid Dyn.***48****,**68–79.Holton, J. R., 1992:

*An Introduction to Dynamic Meteorology.*3d ed. Academic Press, 511 pp.Kanehisa, H., 2002: A nonlinear extension of the helicity formula for convective storms.

*J. Meteor. Soc. Japan,*in press.Klemp, J. B., 1987: Dynamics of tornadic thunderstorms.

,*Annu. Rev. Fluid Mech.***19****,**369–402.Klemp, J. B., and R. B. Wilhelmson, 1978: The simulation of three-dimensional convective storm dynamics.

,*J. Atmos. Sci.***35****,**1070–1096.Klemp, J. B., and R. Rotunno, 1983: A study of the tornadic region within a supercell thunderstorm.

,*J. Atmos. Sci.***40****,**359–377.Lemon, L. R., 1976: The flanking line, a severe thunderstorm intensification source.

,*J. Atmos. Sci.***33****,**686–694.LeMone, M. A., G. M. Barnes, J. C. Fankhauser, and L. F. Tarleton, 1988: Perturbation pressure fields measured by aircraft around the cloud-base updraft of deep convective clouds.

,*Mon. Wea. Rev.***116****,**313–327.Lilly, D. K., 1982: The development and maintenance of rotation in convective storms.

*Intense Atmospheric Vortices,*L. Bengtsson and J. Lighthill, Eds., Springer-Verlag, 149–160.Lilly, D. K., 1983: Dynamics of rotating thunderstorms.

*Mesoscale Meteorology—Theories, Observations and Models.*D. K. Lilly and T. Gal-Chen, Eds., Reidel, 531–543.Lilly, D. K., 1986a: The structure, energetics and propagation of rotating convective storms. Part I: Energy exchange with the mean flow.

,*J. Atmos. Sci.***43****,**113–125.Lilly, D. K., 1986b: The structure, energetics and propagation of rotating convective storms. Part II: Helicity and storm stabilization.

,*J. Atmos. Sci.***43****,**126–140.Margenau, H., and G. M. Murphy, 1956:

*The Mathematics of Physics and Chemistry*2d ed. Van Nostrand, 604 pp.Mathews, J., and R. L. Walker, 1965:

*Mathematical Methods of Physics*. Benjamin, 475 pp.Moncrieff, M. W., 1978: The dynamical structure of two-dimensional steady convection in constant vertical shear.

,*Quart. J. Roy. Meteor. Soc.***104****,**543–567.Newton, C. W., and H. R. Newton, 1959: Dynamical interactions between large convective clouds and environment with vertical shear.

,*J. Meteor.***16****,**483–496.Newton, C. W., and J. C. Fankhauser, 1975: Movement and propagation of multicellular convective storms.

,*Pure Appl. Geophys.***113****,**747–764.Petterssen, S., 1956: .

*Weather Analysis and Forecasting.*Vol. 1,*Motion and Motion Systems,*2d ed., McGraw-Hill, 428 pp.Prandtl, L., and O. G. Tietjens, 1957:

*Fundamentals of Hydro- and Aeromechanics*. Dover, 270 pp.Rotunno, R., 1981: On the evolution of thunderstorm rotation.

,*Mon. Wea. Rev.***109****,**577–586.Rotunno, R., 1993: Supercell thunderstorm modeling and theory.

*The Tornado: Its Structure, Dynamics, Prediction, and Hazards, Geophys. Monogr.,*No. 79, Amer. Geophys. Union, 57–73.Rotunno, R., and J. B. Klemp, 1982: The influence of the shear-induced pressure gradient on thunderstorm motion.

,*Mon. Wea. Rev.***110****,**136–151.Rotunno, R., and J. B. Klemp, 1985: On the rotation and propagation of simulated supercell thunderstorms.

,*J. Atmos. Sci.***42****,**271–292.Rowland, H. A., 1880: On the motion of a perfectly incompressible fluid when no solid bodies are present.

,*Amer. J. Math.***3****,**226–268.Schlesinger, R. E., 1980: A three-dimensional numerical model of an isolated thunderstorm. Part II: Dynamics of updraft splitting and mesovortex couplet evolution.

,*J. Atmos. Sci.***37****,**395–420.Seitter, K. L., and H-L. Kuo, 1983: The dynamical structure of squall-line type thunderstorms.

,*J. Atmos. Sci.***40****,**2831–2854.Stewart, H. J., 1945: Kinematics and dynamics of fluid flow.

*Handbook of Meteorology,*F. A. Berry Jr., E. Bollay, and N. R. Beers, Eds., McGraw-Hill, 411–500.Straka, J. M., and E. N. Rasmussen, 1997: Toward improving microphysical parameterizations of conversion parameterizations.

,*J. Appl. Meteor.***36****,**896–902.Thompson, P. D., 1961:

*Numerical Weather Analysis and Prediction*. Macmillan, 170 pp.Weisman, M. L., and J. B. Klemp, 1982: The dependence of numerically simulated storms on vertical wind shear and buoyancy.

,*Mon. Wea. Rev.***110****,**504–520.Weisman, M. L., and J. B. Klemp, 1984: The structure and classification of numerically simulated convective storms in directionally varying wind shears.

,*Mon. Wea. Rev.***112****,**2479–2498.Weisman, M. L., and R. Rotunno, 2000: The use of vertical wind shear versus helicity in interpreting supercell dynamics.

,*J. Atmos. Sci.***57****,**1452–1472.Wilhelmson, R. W., and J. B. Klemp, 1981: A three-dimensional numerical simulation of splitting severe storms on 3 April 1964.

,*J. Atmos. Sci.***38****,**1581–1600.

## Appendix

### Resolution of the Paradox

*β*and Λ, (2.10) is

*V*

^{2}

_{H}

**v**

_{H}·

**v**

_{H}. Labels

*β̂*

_{z}

*V*

^{2}

_{H}

*β.*

_{z}

*V*

^{2}

_{H}

**v**

_{H}· ∇

_{H}

*w*and

*β*= −

**v**

_{H}· ∇

_{H}

*w*−

*w*∂

_{z}

*w.*Substituting

**v**

_{H}· ∇

_{H}

*w*for ∂

_{z}

*V*

^{2}

_{H}

*β*cause the label on the steering current to switch from

*β̂*

(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 −∇^{2}*p*_{nhL} = *F*_{L}. 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

(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 −∇^{2}*p*_{nhL} = *F*_{L}. 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

(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 −∇^{2}*p*_{nhL} = *F*_{L}. 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 − ∇^{2}*p*_{nh} = *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

Three-dimensional schematic indicating a distribution of the forcing function *F* (in − ∇^{2}*p*_{nh} = *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

Three-dimensional schematic indicating a distribution of the forcing function *F* (in − ∇^{2}*p*_{nh} = *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

*G*(*z,* *ẑ*) as a function of *ẑ* for *z* = 0, 3, 6, 9, 12 km, *H* = 12 km, and *k* = *π*/*H*

*G*(*z,* *ẑ*) as a function of *ẑ* for *z* = 0, 3, 6, 9, 12 km, *H* = 12 km, and *k* = *π*/*H*

*G*(*z,* *ẑ*) as a function of *ẑ* for *z* = 0, 3, 6, 9, 12 km, *H* = 12 km, and *k* = *π*/*H*

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 *ζ̃*_{LN},the maxima and minima are denoted by **S**_{0}, **ω**_{0}, and **v**_{0} − **C**_{LN} indicate the environmental shear, vorticity and storm-relative wind at *z*_{0}, the height of the surface. Solid arrows **S**_{1} and **S**_{2} are the shear vectors at heights *z*_{1} = *z*_{0} − Δ*z* and *z*_{2} = *z*_{0} + Δ*z* where Δ*z* > 0. The dashed arrows −**S**_{1} and **S**_{2} at *z*_{0} indicate the directions in which the linear forcings from *z*_{1} and *z*_{2}, respectively, are tending to move the updraft. In (a), the maxima on a low (field) level *z*_{0} propagate to the east because the net forcing from all source levels is in this direction. In (b), the forcing from all levels is causing the updraft to propagate to the concave side of the hodograph. At bottom of each figure are shown the relative locations of the centers with symbols as in text. The minimum of ∇_{H}(−*α*_{s}∂_{z}*p̃*_{nhLN}) can be on either the downstream or upstream side (with respect to the relative wind **v**_{0} − **C**_{LN}) of the maximum of *h̃.* The vorticity is cyclonic (anticyclonic) on the right (left) side of the vertical displacement maximum owing to the drawing up of loops of vortex tubes (as in Figs. 7–8 of DJ84). At each level, the displacement peak, updraft and cyclonic (anticyclonic) vortices deviate from the environmental wind in the directions at their centers of ∇_{H}* w̃,* ∇

_{H}(−∂

_{z}

*p̃*

_{nhLN}), ∇

_{H}

_{LN}, and −∇

_{H}

_{LN}, respectively, as indicated by the dashed arrows. In (a), the environmental vorticity is crosswise and there is no correlation between

*w̃*and

*ζ̃*

*w̃*and

*ζ̃*

*α*

_{s}∂

_{z}

*p̃*

_{nhLN}and

_{LN}

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 *ζ̃*_{LN},the maxima and minima are denoted by **S**_{0}, **ω**_{0}, and **v**_{0} − **C**_{LN} indicate the environmental shear, vorticity and storm-relative wind at *z*_{0}, the height of the surface. Solid arrows **S**_{1} and **S**_{2} are the shear vectors at heights *z*_{1} = *z*_{0} − Δ*z* and *z*_{2} = *z*_{0} + Δ*z* where Δ*z* > 0. The dashed arrows −**S**_{1} and **S**_{2} at *z*_{0} indicate the directions in which the linear forcings from *z*_{1} and *z*_{2}, respectively, are tending to move the updraft. In (a), the maxima on a low (field) level *z*_{0} propagate to the east because the net forcing from all source levels is in this direction. In (b), the forcing from all levels is causing the updraft to propagate to the concave side of the hodograph. At bottom of each figure are shown the relative locations of the centers with symbols as in text. The minimum of ∇_{H}(−*α*_{s}∂_{z}*p̃*_{nhLN}) can be on either the downstream or upstream side (with respect to the relative wind **v**_{0} − **C**_{LN}) of the maximum of *h̃.* The vorticity is cyclonic (anticyclonic) on the right (left) side of the vertical displacement maximum owing to the drawing up of loops of vortex tubes (as in Figs. 7–8 of DJ84). At each level, the displacement peak, updraft and cyclonic (anticyclonic) vortices deviate from the environmental wind in the directions at their centers of ∇_{H}* w̃,* ∇

_{H}(−∂

_{z}

*p̃*

_{nhLN}), ∇

_{H}

_{LN}, and −∇

_{H}

_{LN}, respectively, as indicated by the dashed arrows. In (a), the environmental vorticity is crosswise and there is no correlation between

*w̃*and

*ζ̃*

*w̃*and

*ζ̃*

*α*

_{s}∂

_{z}

*p̃*

_{nhLN}and

_{LN}

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 *ζ̃*_{LN},the maxima and minima are denoted by **S**_{0}, **ω**_{0}, and **v**_{0} − **C**_{LN} indicate the environmental shear, vorticity and storm-relative wind at *z*_{0}, the height of the surface. Solid arrows **S**_{1} and **S**_{2} are the shear vectors at heights *z*_{1} = *z*_{0} − Δ*z* and *z*_{2} = *z*_{0} + Δ*z* where Δ*z* > 0. The dashed arrows −**S**_{1} and **S**_{2} at *z*_{0} indicate the directions in which the linear forcings from *z*_{1} and *z*_{2}, respectively, are tending to move the updraft. In (a), the maxima on a low (field) level *z*_{0} propagate to the east because the net forcing from all source levels is in this direction. In (b), the forcing from all levels is causing the updraft to propagate to the concave side of the hodograph. At bottom of each figure are shown the relative locations of the centers with symbols as in text. The minimum of ∇_{H}(−*α*_{s}∂_{z}*p̃*_{nhLN}) can be on either the downstream or upstream side (with respect to the relative wind **v**_{0} − **C**_{LN}) of the maximum of *h̃.* The vorticity is cyclonic (anticyclonic) on the right (left) side of the vertical displacement maximum owing to the drawing up of loops of vortex tubes (as in Figs. 7–8 of DJ84). At each level, the displacement peak, updraft and cyclonic (anticyclonic) vortices deviate from the environmental wind in the directions at their centers of ∇_{H}* w̃,* ∇

_{H}(−∂

_{z}

*p̃*

_{nhLN}), ∇

_{H}

_{LN}, and −∇

_{H}

_{LN}, respectively, as indicated by the dashed arrows. In (a), the environmental vorticity is crosswise and there is no correlation between

*w̃*and

*ζ̃*

*w̃*and

*ζ̃*

*α*

_{s}∂

_{z}

*p̃*

_{nhLN}and

_{LN}

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 *ρ*_{s}(*z*) is shown in Fig. 7a. (b) Variation with height of the environmental wind *u*_{0}(*z*) and of the instantaneous linear eastward motion **c**_{L}(*z,* *t*_{0}) of an initially erect updraft according to Petterssen's formula. (c) Maximum values of *p*_{nhL} 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 −cos*ϕ*; 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.

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 *ρ*_{s}(*z*) is shown in Fig. 7a. (b) Variation with height of the environmental wind *u*_{0}(*z*) and of the instantaneous linear eastward motion **c**_{L}(*z,* *t*_{0}) of an initially erect updraft according to Petterssen's formula. (c) Maximum values of *p*_{nhL} 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 −cos*ϕ*; 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.

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 *ρ*_{s}(*z*) is shown in Fig. 7a. (b) Variation with height of the environmental wind *u*_{0}(*z*) and of the instantaneous linear eastward motion **c**_{L}(*z,* *t*_{0}) of an initially erect updraft according to Petterssen's formula. (c) Maximum values of *p*_{nhL} 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 −cos*ϕ*; 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.

Same as Fig. 6, except there is no shear from 6 to 12 km, and (a) the base-state density *ρ*_{s}(*z*) is graphed. Also, *W*(*z*) is the same as in Fig. 6a. The RK solutions are zero above 6 km

Same as Fig. 6, except there is no shear from 6 to 12 km, and (a) the base-state density *ρ*_{s}(*z*) is graphed. Also, *W*(*z*) is the same as in Fig. 6a. The RK solutions are zero above 6 km

Same as Fig. 6, except there is no shear from 6 to 12 km, and (a) the base-state density *ρ*_{s}(*z*) is graphed. Also, *W*(*z*) is the same as in Fig. 6a. The RK solutions are zero above 6 km

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 *p*_{nhL} 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 *p*_{nhL}(*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

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 *p*_{nhL} 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 *p*_{nhL}(*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

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 *p*_{nhL} 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 *p*_{nhL}(*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

The full-circle hodograph, centered at **C**_{0}. Also shown is the instantaneous linear motion **c**_{L}(*z,* *t*_{0}) of an initially erect updraft of radius *R* = 4.0 km. Numbers along the curves denote heights (in km). For *R* = 5.3 km, **c**_{L}(*z,* *t*) = **C**_{L} = **C**_{0}

The full-circle hodograph, centered at **C**_{0}. Also shown is the instantaneous linear motion **c**_{L}(*z,* *t*_{0}) of an initially erect updraft of radius *R* = 4.0 km. Numbers along the curves denote heights (in km). For *R* = 5.3 km, **c**_{L}(*z,* *t*) = **C**_{L} = **C**_{0}

The full-circle hodograph, centered at **C**_{0}. Also shown is the instantaneous linear motion **c**_{L}(*z,* *t*_{0}) of an initially erect updraft of radius *R* = 4.0 km. Numbers along the curves denote heights (in km). For *R* = 5.3 km, **c**_{L}(*z,* *t*) = **C**_{L} = **C**_{0}

As in Fig. 9, but for the half-circle hodograph

As in Fig. 9, but for the half-circle hodograph

As in Fig. 9, but for the half-circle hodograph

Contours of *p*_{nhL} 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 *p*_{nhL} (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)

Contours of *p*_{nhL} 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 *p*_{nhL} (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)

Contours of *p*_{nhL} 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 *p*_{nhL} (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)

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

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

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