Dust devils are upward spiralling vortices. They are common atmospheric phenomena on Earth and Mars and, due to the presence of dust, these vortices can be visible as miniature or small tornadoes. This phenomenon can have a major impact on weather and climate and serve as nuclei of dust storms on Mars.

The main characteristics of dust devils have been analysed and collected in a number of in situ measurements , as well as laboratory and numerical experiments e.g.. It is well known that this phenomenon arises on sunny days when the superadiabatic lapse rate occurs. The latter leads to convective instability in the planetary boundary layer. This results in generation of convective cells with finite poloidal (horizontal) vorticity e.g..

Numerous observations and laboratory experiments have revealed that dust devils can be generated only in a convectively unstable atmosphere with large-scale seed vertical vorticity. Sinclair e.g. has analysed dust devil generation in the lee of small mountains. Based on the analysis of observational results, he concluded that dust devils probably arise due to a vortex embedding in the background flow together with superadiabatic lapse rate.

In such structures the dust grains are in contact with each other and, therefore, via friction or triboelectric processes can generate electric charges and related electric fields . The electric forces near and within dust devils are coherent and fairly strong, but nevertheless they are, as a rule, still small in comparison with corresponding pressure gradients.

The linear theory of dust devil generation, i.e. by assuming that the excited small-scale vertical vorticity is smaller than external vorticity, has been developed in . One of the main features of dust devil vortex generation refers to a strong increase in the tangential speed and vertical vorticity within a few seconds or even a fraction of a second. In this paper we study the nonlinear stage of dust devil generation. We show that the vertical vorticity and the toroidal speed in an unstably stratified atmosphere with large-scale seed vorticity rapidly grows within a time of the order of the inverse convective instability growth rate. Note that, in the present analysis, the dissipative processes (i.e. viscosity, thermal conductivity, heat flow, etc.) are neglected. Moreover, we disregard the effects of the dust component assuming that it plays a passive role in vortex generation.

We consider the fluid motion in cylindrical geometry (r,ϕ,z) and the axially symmetric case ∂/∂ϕ=0. The most general divergence-free flow velocity v can be decomposed into its poloidal v⟂=(vr,0,vz) and toroidal vϕe^ϕ parts, v=v⟂+vϕe^ϕ, respectively. Here v⟂=∇×(ψ×∇ϕ)=(∇ψ×∇ϕ)/r, where ψt,r,z is the stream function: vr=-1r∂ψ∂z,vz=1r∂ψ∂r. According to the equation describing the nonlinear internal gravity waves in the approximation ∂/∂r≫∂/∂z can be written as ∂2∂t2+ωg2Δr∗ψ+1r∂∂tJψ,Δr∗ψ=0, where ωg is the Brunt–Väisälä or buoyancy frequency given by ωg2=gγa-1γaH+1TdTdz, where J(A,B)=(∂A/∂r)∂B/∂z-(∂A/∂z)∂B/∂r is the Jacobian, γa is the ratio of specific heats, H=g/cs2 is the reduced height of atmosphere, cs is the adiabatic sound velocity and Δr∗=r∂∂r1r∂∂r. If ωg2<0, Eq. () describes the dynamics of convective cells in an unstable atmosphere. An equation similar to Eq. () has been obtained previously by for interpretation of behaviour of acoustic gravity vortices.

In this work we constrain the model stream function as ψ=αr2z2exp⁡(γt-r2/r02), where r0 is the vortex radius, γ is the growth rate of the rising air, and α is a constant. By applying the differential operator Eq. () to the stream function Eq. () it is easy to obtain that Δr∗ψ=-8r02ψ+4r02r2r02ψ. Therefore, by taking into account Eq. () the nonlinear term in Eq. () is negligible in comparison with the linear terms if αγr2r02exp⁡(γt-r2/r02)≪1. In this approximation from Eq. (), it follows that γ=|ωg|. Thus, the function in Eq. () is a solution of Eq. () if the condition in Eq. () is satisfied.

Using Eq. () the radial and vertical velocities can be written as vr=-αr2exp⁡(γt-r2/r02) and vz=αz1-r2r02exp⁡(γt-r2/r02). From Eqs. () and (), it follows that the poloidal vorticity at r≪r0 is ωϕ=∂vr∂z-∂vz∂r=4αrzr02exp⁡(γt). Thus, the parameter α corresponds to horizontal (poloidal) vorticity for 4rz=r02 and t=0 in the internal vortex region. To investigate the evolution of the vertical (toroidal) vorticity ωz we make use of the following equation: ∂ωz∂t+vr∂ωz∂r+vr∂ωz∂z=ωz∂vz∂z, which describes the interaction of vertical vorticity with poloidal motion. We consider that in the vortex area there is a stationary large scale, with characteristic scale R≫r0, vortex vorticity germ Ωzh(r) when z<h and Ωz(r) when z>h. Here h is the vertical scale due to the interaction of vortex with the ground surface, Ωzh(r)=Ωzh1-exp⁡r2R2≃Ωzhr2R2, and Ωz(r)=Ω1-exp⁡r2R2≃Ωr2R2, where Ω is a constant. We assume that ωz=Ωzhf(r,t) and ωz=ΩzF(r,t) when z<h or z>h, respectively. By substituting expressions for ωz into Eq. (11) one can obtain that f(r,t)=exp⁡[(α/γ)exp⁡(γt-r2/r02)] and F(r,t)=exp⁡[2(α/γ)exp⁡(γt-r2/r02)]. It is seen that in the internal vortex region at r2/r02≪1 and when t=0, we have f(r,t)=F(r,t)=1. The vortex generation when F(r,t)-1<1 and therefore F(r,t)=1+2(α/γ)exp⁡(γt-r2/r02) corresponds to the linear approximation which has been investigated before by . In this article we study the vertical vortex generation at arbitrary values f(r,t) and F(r,t) and pay special attention to the nonlinear stage of the vertical vorticity ωz generation when in the internal vortex region f(r,t)≫1 and F(r,t)≫1. The vertical vorticity ωz is related to tangential velocity vϕ by relation ωz=1r∂∂rrvϕ. Thus the tangential velocity for z<h is as follows: vϕ=vϕh0exp⁡[(α/γ)exp⁡(γt-r2/r02)]. For z>h, vϕ=vϕ0exp[2(α/γ)exp⁡(γt-r2/r02)]. Here, vϕh0=(ΩR/4)(r/R)3(z/h) and vϕ0=(ΩR/4)(r/R)3 are seeds of tangential velocities. To illustrate our results we make use of observations from . We consider an air temperature lapse rate of 2 C m-1 that corresponds to characteristic temperature vertical scale LT=150 m and the growth rate γ=0.25 s-1. Figures  and  show the dependence of normalized radial and vertical components of the velocity within the time interval t≤4s and when α=γ in the radial domains r/r0≤0.2 or r/r0≥2.0. Note that according to Eq. () the vertical velocity is proportional to z. Figure  is plotted for z=2r0. Equations () and () show that the poloidal (horizontal) vorticity is proportional to the radial velocity vr(r,t); therefore its dependence on the radial coordinate, and time is similar to that in vr(r,t). In the intermediate domain 0.2<r/r0<2.0, where the condition in Eq. () fails, the stream function in Eq. () is not a solution of Eq. (). In this case while solving Eq. (), the nonlinearity associated with the Jacobian should be taken into account. Further investigation of this nonlinearity in the region 0.2<r/r0<2.0 will be carried out elsewhere. Comparing the left and right panels of Figs.  and , using interpolation behaviour of velocities in the regions 0.2<r/r0<2.0 and r/r0≥2.0, one can assume that for r≃r0 the absolute values of the radial velocity and the horizontal vorticity attain their maximum values. Then the vertical component of the velocity, for r≈r0, vanishes and then changes the sign.

Figure  shows the amplification factor F in the radial intervals the same as in Figs.  and . Note the strong, i.e. 10-times, increase of F from t=5.6 to 6.4 s in the interior of the dust devil vortex, while in the external region (right panel) the amplification factor F remains close to the unity. Therefore, the amplification of the vertical vorticity is concentrated in the internal part of the vortex. Figure  indicates the normalized toroidal velocity in the internal part. The example shows a factor-of-8 increase in the toroidal velocity at r=r0/5 in the time interval δt=6.4-5.6=0.8 s.

For smaller temperature gradient with scale LT=300 m we have γ=0.18s-1, and the behaviour of vr and vz remains the same. However, the former time interval t≤4s is enlarged to t≤9 s following the condition γt=const.

In Figs. – we show the generation of dust devils in the Earth atmosphere with its characteristic parameters. When applied to the Martian atmosphere, these parameters will have other values; as a result, the instability growth rate and spatial scales of generated dust devils will be different. Nevertheless, the behaviour of radial velocity (Vr), vertical velocity (Vz), vertical vorticity (ωz), and toroidal velocity (Vϕ) will remain same. Therefore, Figs. – can also be used to illustrate the mechanism of dust devil generation on Mars.

In summary, the new nonlinear analysis developed here shows how an unstably stratified atmosphere develops high-speed toroidal rotation from a large-scale seed vertical vorticity. It is shown that convective cells in such a case can convert into dust devil vortex structures extremely fast, more rapidly than simply exponentially. From formulae () and (), it follows that the vertical vorticity ωz/Ωz0 and toroidal velocity vϕ/vϕ0 grow as ∼exp⁡[2(α/γ)exp⁡(γt-r2/r02)]. It is seen that, in the inner vortex region, r/r0≪1, for moderate time domain t≳γ-1 the amplification factor shows explosive behaviour if α/γ≳1.