# “Explosively growing” vortices of unstably stratified atmosphere

## Authors

• The copyright line for this article was changed on 18 NOV 2016 after original online publication.

## Abstract

A new type of “explosively growing” vortex structure is investigated theoretically in the framework of ideal fluid hydrodynamics. It is shown that vortex structures may arise in convectively unstable atmospheric layers containing background vorticity. From an exact analytical vortex solution the vertical vorticity structure and toroidal speed are derived and analyzed. The assumption that vorticity is constant with height leads to a solution that grows explosively when the flow is inviscid. The results shown are in agreement with observations and laboratory experiments

## 1 Introduction

In this paper, we generalize the theory for finite r/r0 and show that these vortices exhibit “explosive growth.” The effect of amplitude saturation can be explained by energy dissipation, longitudinal temperature gradients, and the influence of dust on the instability growth rate. However, this analysis is out of scope of the present study and will be the subject of future research.

The paper is organized as follows. In the first part of section 2 we describe the nonlinear model which has been used for vortex structure analysis. In the second part of section 2 we obtain the expressions for radial and vertical velocity components of the generated convective motion. Using these expressions we investigate the generation of the vertical vorticity and toroidal velocity. Section 3 contains final comments and conclusions.

## 2 The Nonlinear Model

In cylindrical coordinates (r,ϕ,z) we only consider the axisymmetric case with /ϕ = 0. We also neglect the influence of dissipative processes such as viscosity, friction, thermal conductivity, and heat flows. The most general divergence-free flow velocity v can be decomposed into its poloidal v=(vr,0,vz) and toroidal parts, i.e., , where v=∇×(ψ×∇ϕ)=∇ψ×∇ϕ. Here is the stream function:

(1)

From Onishchenko et al. [2014] the ideal fluid equation describing nonlinear internal gravity waves is

(2)

where ωg stands for the Brunt-Väisälä or buoyancy frequency

(3)

and

(4)

Here J(A,B) = (A/r)B/z − (A/z)B/r is the Jacobian, γa is the ratio of specific heats, H is the reduced height of atmosphere, g is the gravity acceleration, and T is the temperature. Equation (2) can also be obtained from the system of two equations originally derived by Stenflo [1996] and has been discussed recently by Shukla and Stenflo [2012] and Onishchenko et al. [2013]. This equation describes the dynamics of convective motion in an unstable atmosphere, when [Onishchenko et al., 2014, 2015]. Using the approximation ωz/z = 0, where , the interaction of the growing convective motion with vertical vorticity ωz is described by the equation

(5)

Here the second term on the left-hand side and the term on the right-hand side give rise to a stretching of vortex lines which has the effect of intensifying vorticity. According to Kelvin's theorem, the total circulation of the vortex lines has to be constant, and thus, the axial stretching of vorticity lines increases their vorticity.

Nonlinear equations (2) and (5), with the use of equation (1), describe a generation of vortex structures in the stratified atmosphere. Here we are looking for an ωz solution with toroidal velocity vϕ bounded in the perpendicular direction. The stream function we use is

(6)

where R = r/r0, r0 is the vortex radius and γ stands for the constant growth rate. Assuming

(7)

we obtain:

(8)

By taking into account , in the linear approximation from equation (2) for unstable stratification we obtain γ=|ωg|. The radial and vertical velocity components are

(9)

and

(10)

where α > 0 is a constant value describing the strength of suction into the vortex.

The stream lines of the vortex are concentric circles given by dz/dr = vz/vr. Making use of equations (9) and (10), we obtain the following expressions for the particle trajectories in the internal and external regions zr3=const. The poloidal vorticity determined by ωϕ=−vz/r is

(11)

It is seen that ωϕ is always negative. Now we turn to the solution of equation (5) in the presence of an external background vertical vorticity which we represent in a simple one-parametric form of spatially localized vortex with a characteristic spatial scale a >> r0 and maximum vorticity Ω when r = a. We use this representation as an initial stage of the vertical vorticity , where Ω = const. After substituting equations (9) and (10) into equation (5), we find that

(12)

where  is the background vertical vorticity at r = r0. The double exponential term in equation (12) describes the explosively growing dependence of the ωz. The vertical vorticity attains the maximum value when R = 1/3, and signs of ωz and Ω coincide. Making use of equation (12) and relation rωz=(rvϕ)/r one obtains the toroidal velocity vϕ in the form

(13)

where is the background toroidal velocity at r = r0. The toroidal speed attains the maximum value at R = 1. The vertical vorticity and toroidal velocity shown in equations (12) and (13), respectively, are proportional to the double exponential function. This corresponds to the explosive growth of the vortex structure. Similar to the Burgers and Rankine vortices [see, e.g., Burgers, 1948; Balme and Greeley, 2006], the toroidal velocity vϕ is proportional to 1/r for large values of R.

## 3 Conclusions

In the framework of ideal hydrodynamics, we obtained a new exact solution which describes the explosive growth of vortex structures in the Earth, planetary and under some restrictions even in the solar atmosphere. Convective instabilities present in the atmosphere layer, along with background vorticity, are necessary conditions for this rapid vortex formation. Similar to Burgers's vortex model, the vertical and radial components of the flow (see equations (9) and (10)) depend on the parameter α, which characterizes the strength of suction into the vortex. The toroidal velocity distribution for r >> r0 decreases as an inverse function of radius in accordance with laboratory observations [see, e.g., Balme and Greeley, 2006; Mullen and Maxworthy, 1977], as well as in the models by Rankine and Burgers. From equations (12) and (13) it is seen that vertical vorticity ωz and toroidal velocity vϕ are independent of z. After a moderate time, i.e., t > γ−1, the amplification factor shows an anomalously strong character if . For a numerical calculation, we use the temperature lapse rate 2°C m−1 [see, e.g., Oke et al., 2007] which results in a convective instability growth rate of γ = 0.25 s−1. Figures 1 and 2 show the dependence of normalized vertical vorticity and toroidal velocities in a vortex within the time interval 5≤t≤9 s when α = 0.1γ.

Taking into account the radial dependance of the toroidal velocity vϕ for different times, one can calculate the pressure gradient , where p and ρ are the pressure and mass density, respectively. Figure 3 shows the time evolution of the normalized pressure gradient versus r/r0. It should be noted that this model of vortex generation is applicable to tornadoes or dust devil vortices with a single-celled structure [e.g., Balme and Greeley, 2006] and can also be used for the investigation of acoustic-gravity tornadoes [e.g., Shukla and Stenflo, 2012].

## Acknowledgments

This research is partially supported by the Program of the Russian Academy of Sciences No. 7 and by RFBR through grants 14-05-00850 and 15-05-07623. W.H. is supported by the U.S. Department of Energy Office of Fusion Energy Sciences under Award DE-FG02-04ER-54742 and by the Space and Geophysics Laboratory in the ARLUT at the University of Texas in Austin. V.F. would like to acknowledge STFC and the Royal Society for partial support received. The data used in this study can be obtained via e-mail request to V. Fedun (v.fedun@sheffield.ac.uk). The authors thank G. Verth and I. Giagkiozis for useful discussions and help with the preparation of the manuscript.