A simple model of dry convective helical vortices (with applications to the atmospheric dust devil)

Dynamics of Atmospheres and Oceans 40 (2005) 151–162

A simple model of dry convective helical vortices (with applications to the atmospheric dust devil) Michael V. Kurgansky a,b,∗ a

Department of Geophysics, Faculty of Physical and Mathematical Sciences, University of Concepci´on, Avda. Esteban Iturra s/n, Barrio Universitario, Casilla 160-C, Concepci´on, Chile b A.M. Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences, Moscow, Russia Received 12 March 2004; accepted 8 March 2005 Available online 12 May 2005

Abstract An asymptotic solution of inviscid Boussinesq equations for a ‘dry convective Rankine vortex’ with prescribed buoyant forcing is given. The obtained vortex solution demonstrates monotonic growth with height of the vortex core radius, which becomes infinite at a certain critical altitude, and the corresponding attenuation of the vertical vorticity. This idealized vortex is then embedded in a convectively unstable boundary layer; the resulting approximate vortex solution has been applied to determine the maximum rotational wind speed and diameter of dry convective dust-devil-like vortices. © 2005 Elsevier B.V. All rights reserved. Keywords: Boussinesq fluid; Intense atmospheric vortices; Dust devil; Rankine vortex

1. Introduction This paper addresses fluid dynamics of the intense atmospheric vortices, which for decades has been in the focus of interest of many fluid dynamists and meteorologists (cf. Bengtsson and Lighthill, 1982). In addition to the vortex models earlier proposed in the literature (e.g. Lugt, 1983), an asymptotic solution of inviscid Boussinesq equations for a ‘dry convective Rankine vortex’ with prescribed buoyant forcing is given. The ∗

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obtained vortex solution demonstrates monotonic growth with height of the vortex core radius, which reaches infinite values at a certain critical level, and the corresponding attenuation of the vertical vorticity. This vortex model has been initially designed to mimic, in a crude sense, the structure of dust devils, at least for an idealized axisymmetric steady case. It might also be relevant to a larger thermal and/or a concentrated updraft (associated with Rayleigh–B´enard convective cells), at the bottom of which dust devils are formed. The assumptions made to construct such a vortex model will be divided into two groups: those which seem obvious, if we deal with a simple picture of a dust-devil-like vortex or, alternatively, a larger thermal in which this vortex is embedded (group A), and those which determine the particular model to be considered (group B). The latter are consistent with the ideas by Carroll and Ryan (1970) and Kanak et al. (2000), who proposed that the maintenance of a dust devil originates in the convective activity, and are suggested by their simplicity. A1 : The vortex is assumed to be in quasi-steady equilibrium, i.e. all partial derivatives with respect to time are put equal to zero. A2 : The maximum of velocity in the vortex is a short distance—a few meters often—above the surface and the velocity vector vanishes at large distances above it. B1 : We consider a circularly shaped vortex. Rotational symmetry around the vertical axis coinciding with the vortex center is assumed. Consistent with the hypothesis on pure convective origin of dust devils, when no external wind shears are imposed, the tilt of the vortex axis with height is disregarded. B2 : The dust devil vortex is embedded in a strongly convective boundary layer. In this layer, all superadiabatic temperature lapse rates and vertical wind shears are concentrated within a thin surface-adjacent layer, having the thickness of order of the Monin–Obukhov length-scale, that is a few meters for desert conditions (Hess and Spillane, 1990; Kanak et al., 2000). Consistent with this observational picture, we treat the vortex flow beyond this surface-adjacent layer as inviscid and assume that the ambient atmosphere is neutrally stratified. B3 : We assume that the relative distribution of the velocity components is the same across the vortex at all altitudes (the similarity assumption). In particular, we assume that the rotational velocity at each horizontal level has a profile, which is characteristic for a Rankine vortex with irrotational flow periphery. The latter assumption results in a spatially uniform angular momentum distribution outside the vortex core, which is fairly consistent with the conceptual picture in Hess and Spillane (1990) and Kanak et al. (2000), showing no vertical wind shears in strongly convective environments beyond a thin surface-adjacent layer. In this first attempt, the vertical velocity in the vortex core is assumed to have everywhere the same sign, corresponding to an updraft flow, while in reality there is evidence that a weaker downdraft is often present in the dust devil center (Sinclair, 1966, 1973; Kaimal and Businger, 1970). Because of the similarity assumption, which attracts by its simplicity and leads to a manageable solution of governing equations, our vortex model is incapable of reproducing stagnant or even downward motion in the vortex center, starting from a certain critical height, namely the effect, which is observed for laboratory vortices (cf. Lugt, 1983). More precisely, our model accounts for this dynamic effect only

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in a well-defined, restricted sense, by placing the stagnation point at the uppermost vortex level, where the radius of the vortex core is infinite. Nevertheless, despite a number of assumptions have been made, the proposed approach proves to be useful, at least conceptually, because an exact asymptotic solution of Boussinesq equations can be constructed and compared to observations. The paper is organized as follows. In Section 2, an idealized inviscid vortex model is constructed by the similarity method, which eventually goes back to Schl¨uter and Temesv´ary (1958), who solved a mathematically similar magnetohydrostatic problem. In Section 3, the realistic case is considered when the vortex is embedded in a convective boundary layer; comparisons of some model predictions with observational data on dust devils are also given. At last, our results are summarized and discussed in Section 4. 2. Steady helical vortex In a Boussinesq fluid, the complete set of governing equations, which include the momentum balance, mass continuity, and heat transfer equations, reads as Dv = −∇π∗ + b + νT ∇ 2 v, Dt

(1)

∇ · v = 0,

(2)

Db g + wN 2 = kT ∇ 2 b + q. Dt cp T0

(3)

Here, v is the velocity and νT the small-scale turbulent viscosity; D/Dt = ∂/∂t + v · ∇. Buoyant force b = −g(θ * /θ 0 ) is determined by the potential temperature deviation θ * from its reference values θ 0 = θ 0 (z) and by the gravity acceleration g = −
φ, with φ = gz as the geopotential and z the height; π* = cp θ 0 (p/p00 )κ − cp θ 0 (p0 /p00 )κ , where p0 = p0 (z) is the hydrostatic pressure p, p00 = 1000 hPa and κ = R/cp , R being the gas constant and cp the specific heat at constant pressure. In Eq. (3), b = g(θ * /θ 0 ) is the buoyancy, w the vertical velocity, N2 = g(d log θ 0 (z))/dz the Brunt–V¨ais¨al¨a frequency squared, kT the turbulent thermal diffusivity, q is equal to the rate of diabatic heating, per unit mass, and T0 = T0 (z) the reference temperature profile. Consistent with the assumptions A1 and B1 of Section 1, consider a steady axisymmetric vortex with the vector g being parallel to the vortex axis. The vortex flow is inviscid, and the surrounding atmosphere is neutrally stratified, N2 ≡ 0 (the assumption B2 ). Cylindrical polar (r, ϕ, z) coordinates are applied, where r = 0 coincides with the axis of symmetry. The velocity v and vorticity ω vector components are (u, v, w) and (ωr , ωϕ , ωz ), respectively. These components and the buoyancy field b are the functions of (r, z). We take the r- and z-component of (1) u

∂u ∂u v2 ∂π∗ +w − =− , ∂r ∂z r ∂r

(4)

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u

∂w ∂w ∂π∗ +w =− + b, ∂r ∂z ∂z

(5)

respectively. The azimuthal ϕ-component of (1) is identically satisfied. Following the assumptions A2 and B3 , we consider a Rankine-like vortex with irrotational peripheral flow. The radius of the vortex core rm (z) is assumed to be a monotonic increasing function of altitude z. The swirl velocity reaches its maximum vm (z) at r = rm (z) and is taken in the form
−1 (z)r, r ≤ rm (z) vm (z)rm v= . (6) vm (z)rm (z)r −1 , r > rm (z) The peripheral flow, outside the vortex core, has spatially uniform specific angular momentum distribution Γ = vr = vm (z)rm (z) = constant. Without any loss, only cyclonic vortices with Γ > 0 are considered. Consistent with initial vigorous horizontal mixing in the vortex core, which eventually leads to the establishment of the swirl velocity profile (6), it is anticipated that at each horizontal level the vertical velocity w is uniformly distributed inside the vortex core and all radial gradients of w are concentrated at its edge; in the peripheral flow w ≡ 0. It is reminiscent of the velocity profile in a circular pipe for the turbulent state of flow (cf. Sommerfeld, 1964, p. 272, Fig. 65). So, we apply
Qπ−1 rm −2 (z), r ≤ rm (z) , (7) w= 0, r > rm (z) where Q is the constant vertical mass flux. Mechanical equilibrium within the vortex core is −2 (z) + maintained by a poloidal circulation having the streamfunction ψ = −(Q/2π)r 2 rm C, C is a constant, such that ru = ∂ψ/∂z and rw = −∂ψ/∂r; see Eq. (2) for the considered axisymmetric case. The radial velocity u vanishes at the vortex axis and linearly increases along with the radius r within the vortex core; on the vortex core sidewalls, u falls down to zero values and remains nil everywhere in the peripheral flow domain. Therefore, the poloidal circulation is absent outside the vortex core; the streamfunction ψ is constant and can always be taken as −(Q/2π) + C, without any loss. Consequently, ψ is continuous on the vortex core edge, at r = rm (z). First, we substitute (6), (7) and the corresponding u-component into (4) and integrate the resulting equation over r between 0 and ∞. It yields Q2 −1 2 Q2

y (y ) − y − Γ 2 y = π∗ (0, z), 4π2 8π2

(8)

−2 (z) was introduced and a prime denotes the derivative with where a variable y(z) = rm respect to the height z. It was used that π* (∞, z) ≡ 0, i.e. the non-hydrostatic pressure component vanishes at r = ∞. Second, we (a) take (5) at r = 0, (b) assume that w(0, z) → 0 and π* (0, z) → 0 at z → ∞, (c) apply (7), and (d) integrate (5) over the height between z and ∞ to get  ∞ Q2 π∗ (0, z) = − b(0, z) dz − 2 y2 . (9) 2π z

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Eliminating of π* (0, z) between (8) and (9) yields Q2 Q2 Q2 −1 2 y (y ) − 2 y

+ 2 y2 − Γ 2 y = − 2 8π 4π 2π

 z

∞

b(0, z) dz.

(10)

It is immediate to verify Eq. (10) by using the similarity assumption and seeking the solution of (4) and (5) in a more general form consistent with the mass continuity equation (2), namely as u = −F(x)ry , v = G(x)ry and w = 2F (x)y, where x ≡ r2 y, and F and G are arbitrary differentiable functions, regular at x → ∞ (cf. Schl¨uter and Temesv´ary, 1958). By literally repeating the above-described two steps, we arrive at a non-linear equation   ∞  ∞ 1 1 −1 2 ∞ 2

2 2 2 F (x) dx − y F (x) dx + 2F (0)y − y G2 (x) dx y (y ) 2 2 0 0 0  ∞ b(0, z) dz =− z

that generalizes (10). Finally, we make use of    Q, x≤1  Γ, F (x) = 2π and G(x) = Γ   , 0, x>1 x

x≤1 x>1

,

which corresponds to (7) and (6), respectively, and arrive at (10). Substituting y = ζ 2 into (10) yields a simpler equation  ∞ Q2 4 Q2

2 2 b(0, z) dz, (11) − 2 ζζ + 2 ζ − Γ ζ = − 2π 2π z but the original y-variable is advantageous in intermediate calculations. When Γ = 0, then (11) is reminiscent of a non-linear equation, derived by Schl¨uter and Temesv´ary (1958) to describe the sunspot structure. If b(0, z) were known and provided the boundary conditions for the ζ-variable were specified, one could integrate (11) in order to find ζ = ζ(z) and fully determine the velocity field in the vortex, given Γ and Q values. In a thermodynamically consistent model, however, b(0, z) must be determined by the vortex motion itself and also by the right-hand side forcing term in (3). It is the latter part of the model, which is difficult to formulate. In this attempt, we are focusing on a vortex model having very simple kinematic structure—although it may need a complex diabatic forcing for its support—and are using for this purpose an exact solution ζ

= 0 of Eq. (11). It implies ζ = a−2 (h − z),

a = constant,

(12)

2Q2 2 2 Γ (h − z) − (h − z)3 . a4 π 2 a8

(13)

where b(0, z) =

This solution (12) describes a vortex with its core unlimitedly expanding with height and with accompanying attenuation of vertical vorticity ωz = 2Γ ζ 2 = (2Γ /a4 )(h − z)2 . To arrive

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at such a solution, it has been assumed that the vortex radius becomes infinite at some height z = h. Schl¨uter and Temesv´ary (1958) have already noticed this particular solution (12) and attributed it to an essential singularity at the point z = h. The solution (12) is valid for the entire lower half-space −∞ < z < h and also reads as rm (z) =

a2 . h−z

(14)

So far, it was sufficient to consider the buoyancy field b(0, z) only at the vortex axis. To specify the b-function everywhere within the vortex core, we make use of the ‘cyclostrophic thermal wind’ equation that follows from (4) and (5)  ∂ψ ∂  ωϕ  ∂ψ ∂  ωϕ  ∂ v2 ∂b − − =− , ∂z ∂r r ∂r ∂z r ∂z r ∂r where ωϕ = r−1 (∂2 ψ/∂r2 − r−1 ∂ψ/∂r + ∂2 ψ/∂z2 ). For our vortex model, the thermal wind equation takes the form −(Q2 /2π2 )y

yr − 2Γ y yr = −∂b/∂r. We make use of y = ζ 2 , substitute (12) into the left-hand side of the last equation and anticipate the buoyancy field   2  2Q2  2 Γ 2 (h − z) 1 − r − (h − z)3 , r ≤ rm (z) 2 (z) (15) b(r, z) = a4 rm π 2 a8   0, r > rm (z) that satisfies the thermal wind equation at r < rm (z), matches to (13) at r = 0 and allows for b-discontinuity at r = rm (z). This solution (15) describes a warm-core vortex immediately below the critical level z = h, which by hydrostaticity arguments explains the emerging pressure deficit in the vortex center. In contrast, for lower heights, this pressure deficit is mainly explained by the ‘Bernoulli effect’, i.e. the pressure drops there owing to dramatic increase in the vortex flow speed, which would equally occur either for an updraft or a downdraft in the vortex core. To compensate this excessively strong pressure deficit, the vortex core is cold everywhere at low altitudes; see also (9). In our vortex model, the Ertel (1942) potential vorticity (PV) Π = ω·
(θ * /θ 0 ) is non-zero within the vortex core:

  r 2 (h − z)2 12Q2 Γ 4Γ 3 (h − z)2 1 ∂(rv) ∂b ∂v ∂b 1 − + − =− (h − z)4 , Π≡ g r∂r ∂z ∂z ∂r ga8 a4 gπ2 a12 (16) whereas outside the core, Π ≡ 0. This property makes our idealized vortex somewhat similar to larger-scale intense vortices, such as tropical cyclones and tornadoes, which have nonvanishing Ertel’s PV, and it is the radial PV-gradient that provides a vortex-Rossby-wave propagation mechanism (cf. Montgomery and Brunet, 2002). A case of non-helical vortices corresponds to setting Q = 0 in (11), and the simplest possible solution of (11), which is consistent with the assumption A2 , is given by ζ 2 = (b* /Γ 2 )(h − z), b(0, z) = b* = constant (b* > 0). It describes a cyclostrophically balanced warm-core vortex with everywhere vanishing Ertel’s PV. Recently, Kurgansky (2005) indicated a generalization of this simplest solution onto the case of a fully compressible

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atmosphere, without any Boussinesq approximation has been made. All other possible solutions to (11) of this sort (when Q = 0) inevitably have non-vanishing PV-field inside the vortex core. That particular solution, which corresponds to (12) in the limit case of Q → 0, shears the last property. Presumably, these non-helical, cyclostrophically-balanced solutions may have more in common with large-scale tropical cyclones and hurricanes than with smaller-scale dust devils and thermals. These non-helical vortices, with non-zero PV and whose radius grows with height, have buoyancy b decreasing with z and, therefore, do not satisfy the hydrostatic stability criterion. Nevertheless, they are likely to be almost everywhere stabilized (possibly, except of a thin layer adjacent to the critical level z = h) by the rapid fluid rotation about the vertical axis, in a full analogy to stability of a fast spinning Lagrangian symmetric top in the classical mechanics (Routh, 1960). For the vortex solution (12), non-zero Q-values further contribute its stability, as it follows from (15). Albeit these tentative arguments and analogies are believed to be applicable to our idealized vortex model, the question of its stability certainly needs a more rigorous analysis. The previous arguments use exclusively Eqs. (1) and (2) and their mathematical consequences, but not the thermodynamical Eq. (3). In fact, fluid flow within the vortex core is diabatically forced in our model, and it can be inferred from (3) and other formulas of this section—particularly (15)—that under neglect of the turbulent thermal diffusivity, kT = 0, the vortex flow needs non-zero diabatic forcing   Q g q= gΠ (17) cp T0 2πΓ for its thermodynamical support, where the potential vorticity Π is given by (16). In synthesis, this corresponds to diabatic cooling of the upper part of the vortex core and diabatic heating of its lower part, especially closer to the vortex core sidewalls. Formula (17) stands for a general property of steady helical flows, which becomes obvious for stationary helical Beltrami flows, with everywhere collinear velocity and vorticity vectors. Namely, based on the thermodynamic energy equation, it can be conjectured that steady helical flows of a baroclinic fluid have non-vanishing Ertel’s potential vorticity field if and only if they are diabatically driven.

3. Comparison with observational data Consider a realistic case, when the singular level (horizon) z = h is identified with the altitudinal position of a temperature inversion, which caps the convectively unstable boundary layer, whereas the earth’s surface coincides with the plane z = 0. In a meteorological context, the height h is order of 1 km. We consider the solution (12)–(14) only above a horizontal plane Σ (see Fig. 1), which truncates our idealized semi-infinite vortex. By the very construction this surface Σ is the level, z = δ, of the swirl velocity maximum, and it is further assumed that δ h. Above Σ, the swirl velocity decreases according to (14) and Γ -constancy; whereas below Σ, the turbulent friction reduces the velocity down to zero values at the earth’s surface. An inviscid solution (12)–(14) should smoothly match to a

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Fig. 1. Sketch of a dry convective Rankine-like vortex with the control surface Σ. Vortex cross-section in a vertical plane passing through r = 0 is depicted. Dotted lines represent the vortex core edge. Other notations are explained in the text. With regard to applications to atmospheric dust devils, the drawing is not in a proper scale, because in the real atmosphere rm /h ≈ δ/h ∼ 10−2 . Therefore, the upper part of the drawing, which is above Σ, must be stretched in vertical by a factor ∼10 and the sketch actually corresponds to a nearly columnar dust devil vortex in, at least, the lower half of the convective boundary layer; see also Section 4.

viscous flow within the surface-adjacent layer, which is beneath Σ, although we do not consider in detail this matching procedure but only assume that it is possible. For dust devils, it is optimal to construct Σ at a height of few meters. To maximum simplify arguments and in general accordance with the principal scheme of dust devil formation from Hess and Spillane (1990), see their Fig. 1(a) and Fig. 6(a) in Kanak et al. (2000), we attribute all the air mass radial convergence, necessary to explain the vertical mass flux Q within the vortex core, to this surface-adjacent layer of thickness δ. The notation rm is hereafter reserved for rm (δ), i.e. for the vortex core radius in the Σplane, and the notation vm is used for vm (δ). Now, from (12)–(14) and by using Γ = vm rm one gets that 2 = rm

2Γ 2 (1 − α2 ) , b(0, δ)(h − δ)

v2m =

b(0, δ)(h − δ) . 2(1 − α2 )

(18)

Here, α = Q/πrm Γ ≡ Q(h − δ)/πa2 Γ stands for the ratio of the vertical velocity w in the vortex to the maximum swirl velocity vm ; α-parameter is the reciprocal of the ‘swirl ratio’ S defined in Davies-Jones (1973) and it is π times the reciprocal of the ‘vortex parameter’ Ωe in Maxworthy (1982). Formulas (18) are valid for α2 < l; only under this condition the central warm part of the vortex core extends down to the surface Σ, see (15). The second formula (18) is reminiscent of the thermodynamically derived formula in Renn´o et

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al. (1998, 2000) for the wind speed around a dust devil. Also, Nikulin (1992) based on somewhat different arguments, had been applied to an idealized tornado-like vortex, and argued that v2m = 2b(0, δ)(h − δ), in our notations. More generally, the maximum wind speed, as shown in (18), may be referred to as the “thermodynamic speed limit” in intense atmospheric vortices, particularly in tornadoes (cf. Fiedler and Rotunno, 1986, and references therein), if to disregard for a while strong dependency of vm on the helical parameter α. In fact, formulas (18) clearly show that increasing of α leads to decreasing of the vortex size rm and simultaneously to enhancement of the maximum swirl velocity vm . Given the angular momentum Γ -value and other parameters in (18) were fixed, the vortex size rm is maximum and vm , respectively, is minimum when α2 = 0, i.e. the vortex is nonhelical. We shall now use formulas (18) for the purpose of comparison with observational data on dust devils in Australia, compiled by Hess and Spillane (1990). In the case of zero (or close to zero) environmental vorticity, one can conventionally set Γ ≈ u* r0 , where u* is the surface friction velocity and r0 the outer vortex radius (also see Fig. 1). In Hess and Spillane (1990), the convective boundary layer has an average height H ≈ 3200 m and the number density of dust devils is ≈4 per area of H2 . We deduce that r0 ≤ H/4 and use r0 = 800 as an upper estimate. This r0 -value is virtually related to horizontal dimension of Rayleigh–B´enard convective cells in the unstable boundary layer of height H, and the vertical dust-devil-like vortices are likely to form at or near the vertices of the developing convective elements (rings) (cf. Kanak et al., 2000). We deliberately assume that it is u* but not the convective velocity scale w∗ ≡ u∗ (−H/κL)1/3 proposed by Deardorff (1970), with κ as the von K´arm´an constant and L the Monin–Obukhov scale-length, which characterizes the natural background level of horizontal velocity fluctuations within the convective boundary layer at heights of 100 –101 m (i.e. order of −L) and may serve for a rough Γ -estimate. So, it is assumed that Γ = vm rm = u∗ r0 . According to the observational data in Hess and Spillane (1990), see their Table 1, the average dust devil diameter D is about 30 m; the friction velocity is estimated by u* ≈ 0.3 m s−1 from this table. Thus, we get Γ = u* r0 ≈ 240 m2 s−1 and vm ≈ 16 m s−1 if rm = D/2 is applied. This vm -value falls within the interval of experimentally measured velocities vm = 10–19 ms−1 reported in Hess and Spillane (1990). We further use the observational evidence by Hess and Spillane (1990) that a certain cluster of dust devils of apparently convective origin has average height h ≈ 0.51H. To make sure to apply our vortex model, we assume that this lesser height corresponds to a secondary thermal inversion within the main boundary layer bulk. It is known from observations (Sinclair, 1973) that dust devils have a warm core with typical temperature perturbations from 4 to 8 K above ambient desert values. So, we tentatively assume the temperature excess of .T ≈ 6 K in the center of a dust devil vortex. Now, for typical ambient air temperatures in the desert one has b(0, δ) ≈ 0.20 m s−2 and formulas (18) give that α2 ≈ 0.375, if the above indicated parameter values are used. If to accept the validity of our idealized vortex model, then it reveals √ essentially helical structure of the dust devil vortex with the updraft velocity w ≈ vm 0.375 ≈ 9.8 m s−1 . This value, α2 = 0.375, is slightly greater than the threshold value α2 = 1/3, starting with which one has diabatic heating in the vortex center near the earth’s surface; see (16) and (17) taken at r = 0 and z = δ. Now, formulas (15)–(18) show that the rate of diabatic heating in this particular point is given by q/cp ≈ 27.6 K h−1 . The

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obtained value matches to a hypothetical situation when a constant heat flux of 0.24 K m s−1 at the lower boundary (see Kanak et al., 2000) is applied to heat uniformly a 31 m thick surface-adjacent air layer.

4. Discussion and conclusions We have constructed an analytical asymptotic solution of Boussinesq equations for an inviscid helical Rankine-like vortex. The radius of the vortex core increases with height and becomes infinite at a certain critical level, z = h. In a theoretical viewpoint, the vortex motion spreads over the entire lower half-space, −∞ < z < h. We also considered a more realistic vortex model embedded in a convective boundary layer, with allowance for smallscale turbulent viscous friction within the surface-adjacent layer. The resulting approximate vortex solution has been applied to determine the maximum rotational wind speed and diameter of dry convective dust-devil-like vortices. One can draw two conclusions of some experimental value from formulas (14) and (18). Formula (14) reveals that our idealized vortex, with its core expanding with height and concentration of vertical vorticity in the vicinity of the earth’s surface, has a prominent mushroom-like shape, in accordance with many observed dust devils. In the classical meteorological literature, Humphreys (1920) mentioned this observational fact and wrote on ‘the mushroom capital’ of dust whirls. This ‘mushroom capital’ may also be characteristic for a larger thermal, at the bottom of which the dust devil usually originates, regardless of whether or not any dust devil vortex is actually embedded in the thermal. It should be emphasized that formula (14) does not contradict to often-reported moderate expansion with height of dust devil vortices (cf. Greely et al., 2003). In fact, only the lower part of the vortex, namely the dust column, is usually visible owing to the suspended rotating dust as a visualizer, whereas the upper part of the vortex remains invisible (cf. Renn´o et al., 1998). Tentatively assuming that the dust column corresponds to the lower half of the whole vortex, formula (14) gives rm (h/2)/rm ∼ = 2 and a small angle 2rm /(h − δ) ≈ 0.02 (see Section 3) stands for the dust column expansion with height. Certainly, in nature the dust whirl column may extend without noticeable expansion to greater altitudes—than our formula (14) predicts—before it will start rapid expansion with height and convert into the ‘mushroom capital’. In this particular sense, our model oversimplifies the reality but is believed to capture the overall vortex constitution. The first formula (18) allows one to examine the question of the possible causes of the considerable variability in observed dust devil size. In accordance with observations (Sinclair, 1966, 1973; Hess and Spillane, 1990) and simple theoretical arguments by Renn´o et al. (1998, 2000), it clearly shows that in environments of high wind speed and large horizontal wind shears, with presumably large initial Γ -values, dust devil diameters are biased toward larger values. Also, in strongly convectively unstable and otherwise calm, i.e. without any ambient wind shears imposed, atmospheric conditions addressed in Section 3, there is a lot of randomness in establishing a particular Γ -value (including its sign), the larger Γ -values being apparently far less probable than the smaller ones. By means of the first formula (18), it clearly hints on

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the predominance of smaller-scale dust-devil-like vortices if compared to the larger ones. A more detailed study of the statistical distribution of dust devils with respect to their size is planned be the topic of a forthcoming paper.

Acknowledgements I am grateful to two anonymous reviewers for their critical comments and constructive suggestions, which led to significant improvements in the manuscript. This work was partly supported by the Russian Foundation for Basic Research (Project No. 01-05-64232 and 04-05-64315).

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