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 Dust devils are very common meteorological phenomena on the Earth as well as on Mars. They are an abbreviated wind-sand conveyance system. The moving particles in dust devils may become electrically charged, to the point of arcing to spacesuit or vehicle, and creating electromagnetic interference. In this paper a numerical model, which takes into consideration the effect of thermal flux from the surface to the atmospheric boundary layer, is employed to simulate a dust devil and to obtain its fine structure and its development. Then, on the basis of Coulomb's law, the electric field and its distribution in a dust devil are numerically simulated in this paper. The numerical results are consistent with theoretical models for dust devils. That is, the formation mechanisms of a dust devil can be explained with the theory of thermal convection. The numerical results also show that at the beginning stage of the evolution of a dust devil the electric field strengthens with time, but after 80 s the electric field changes little and the electric field has trended to a dynamic stabilization. The electric field in a dust devil has a maximum value at a certain height; the electric field will be increscent below this height and decrescent above this height at the interior of a dust devil.
 From 20 century more and more researchers are paying attention to the study of dust devils. Dust devils are significant meteorological phenomena and they contribute greatly to the global makeup of Mars, which is so dry and dusty that it is sometimes engulfed by global dust storms and scientists think dust devils are the seeds [Hess, 1973; Fernández, 1997]. Early studies of dust devils are field measurements [Sinclair, 1964, 1973; Hess and Spillane, 1990; Carroll and Ryan, 1970]. On the basis of some meteorological characteristics of dust devils obtained from field observations, Renno et al.  proposed a theoretical model to explain the formation mechanisms of a dust devil. In dust devils, grains in contact with each other and the surface are known to generate electric charge via frictional or triboelectric processes [Renno et al., 2004; Freier, 1960; Farrell et al., 2004]. Freier  and Crozier  observed that the electric field in a dust devil could reach to several hundred volt/m. In recent years the study of dust devils on Mars has indicated that the electric field is much stronger than that on Earth, and it can reach several thousand volt/m [Renno et al., 2004; Farrell et al., 2003, 2004]. For example, Renno et al.  measured an electric field of 10 kV/m in a large dust devil. The strong electric field of dust devils may be a possible nuisance or hazard to future human explorers on the surface of planets [Farrell et al., 2004], and therefore it is important to study the electric field in dust devils.
 Because it is difficult to get detailed information on the electric field in dust devils through measurement, the simulation becomes an effective way to study the electric field of field dust devils. But up to now there are few studies to simulate electric field of dust devils. Farrell et al.  presented an electrodynamic model of a dust devil to explain the field strengths measured in a dust devil and they concluded that lofting of the lighter preferentially charged grains leads to the development of substantial electric field in dust devils. Because of the shortage of detailed information on the dust devil, Farrell et al.  made several assumptions; for example, they assumed that there were only two species of particles: large dust grains (100 μm) in saltation and small grains (1 μm) driven by wind flow, and the horizontal velocities of grains were zero, which are apparently not in accord with the real situations in dust devils. Therefore this model cannot quantitatively depict the real electric field in dust devils.
 On the basis of the surface energy-balance equation, atmospheric movement equations, and Coulomb's law, the whole process of dust devil development and the electric field in a dust devil are numerically simulated in this paper, from the formation of thermal convection by heating of the local surface to the formation of a spinning vortex, and then charged sand grains are picked up and the dust devil and the electric field are formed accordingly. Then the profile of the electric field is quantitatively analyzed in detail.
2. Basic Equation
2.1. Surface Energy-Balance Equation
 A part of energy of solar radiation is absorbed by the atmosphere, the other part is absorbed by the land surface. In the meantime heat flux may be transmitted from the soil to the surface. Some of the energy at the surface will be transmitted or radiated to atmosphere. On the basis of the analyses of heating process of the land surface and quantities of heat absorbed and emanated from the surface, the surface energy-balance equation can be given as [Acquiemin and Noilhan, 1990]:
where TS is the surface temperature; RS is surface shortwave net radicalization; RL is surface long wave net radiation; ρLξ*ψ* is turbulent latent heat flux; ρCPξ*ϑ* is turbulent thermal flux; ω is the spin velocity of the earth; S is the average temperature of the surface; and CS, ρS and KS are specific heat, density and coefficient of heat exchange of the sand, respectively.
where S0 is solar constant, S0 = 1353 W m−2; Aa = 0.28/(1 + 6.43 cos z) is atmospheric albedo; AS is the surface albedo, AS = 0.2; ma is atmospheric moisture imbibition, ma = 5; and z is solar zenith angle and can be calculated as [Acquiemin and Noilhan, 1990]:
where ϕ is geographic latitude, ζ is solar declination, and h is hour angle of solar.
where Ta is lower atmosphere temperature; σ = 5.67 × 10−8 W m−2 K−4 is Stefan-Boltzmann constant; ɛa = 1.24(ea/Ta)1/7 is atmosphere radiant coefficient; ɛ is surface radiant coefficient; and ea is vapor pressure (hPa).
2.2. Equations of Atmospheric Movement
 To simulate small-scale atmospheric movement, basic equations of atmospheric movement with Boussinesq approximation can be expressed as [Stull, 1988]:
where θ is the virtual potential temperature and ρ is the air density. Km, KH and K are coefficients of diffusion; they can be obtained through the turbulent parameterized method of the boundary layer [Amada, 1983].
2.3. Sand Grain Movement Equations in a Dust Devil
 Usually in a dust devil the wind speed is bigger than 10 m/s; the drag force FD acting on the sand grains is much greater than other forces, such as gravitation and so on [Gu et al., 2006]. So we only consider the drag force and the movement equations of sand grain are as follows:
where m is the mass of a sand grain; Cd is the drag coefficient; D is the diameter of sand grain; and uD, vD, wD are velocity components of sand grain along x, y, z directions, respectively.
2.4. Electric Field Equations
 In a dust devil, as sand grains come in contact with the surface and with each other, they will have positive charge or negative charge and these charged sand grains generate electric field. According to Coulomb's law [Frankle, 1986], the electric field at a given point P(x, y, z) in a dust devil due to a grain with charge qi is given by
where k is electrostatic constant; qi is the charge quantity of the sand grain; and ri is vector distance between the charged sand grain and the point P(x, y, z). Then the total electric field at the point P(x, y, z) due to all charged sand particles in a dust devil may be written as
where n is the number of charged sand particles in the dust devil.
3. Algorithm for a Dust Devil and the Electric Field in It
 The calculation domain (Figure 1) is a sufficient large region (R0 = 100 m, r0 = 40 m, H = 200 m) to simulate a dust devil. On the bed the surface temperature TS = TS0 for r < 1 m, TS = TS0 − α1r for 1 m ≤ r < r0, TS = TS0 − 40α1 for r0 ≤ r ≤ R0 are initial conditions for equation (1).
 The boundary conditions and initial conditions for equations (5a)–(5f) are as follows:
 Boundary conditions:
At the top of the domain:
where α ∈ [0, 2π]; TS0 = 318 K; a1 = 2 K/m; Ta is lower atmosphere temperature; p0 = 1013 hPa; η = 0.0065 K/m; Cp is constant-pressure specific heat; and R is air constant.
 When wind speed exceeds the threshold velocity sand grains will lift off from bed. The lift off speed of sand grains is usually within the domain of [, 5u*] [Anderson and Haff, 1991]. u* is the friction velocity of wind. In this paper the initial conditions for equation (6) are given as follows: uD = vD = 0, wD is the random value between and 5u*.
 The actual calculation steps for simulating the developing process of a dust devil and the electric field in it proceed as follows:
 1. Input the initial values of u, v, w, p, T, TS0.
 2. Substitute TSi−1 (or TS0) and Tai−1 (or Ta0) into the surface energy-balance equation, and calculate the new surface temperature TSi. The superscript “i” is taken to express the results for the i-th iteration.
 3. Substitute the values of TSi−1 and TSi into equation (1) to calculate the new lower temperature of air Tai;
 4. A large-eddy simulation method is adopted to solve equations (5a)–(5f) and the wind velocity is calculated repeatedly until it is convergence. Then we obtain wind velocities, temperatures, and pressures of the moment for the i-th iteration in calculating domain.
 5. If the friction velocity is bigger the threshold friction velocity, sand grains will lift off the land surface with a random lift-off velocity between , and the motion of the sand grains are calculated by equation (6).
 6. Calculate the electric field of the moment for the i-th iteration with equations (7) and (8).
 7. Return to the second step and repeat the above procedures to calculate the wind velocity, the place of the moving sands till a dust devil is fully developed.
4. Numerical Results and Discussions
Figure 2 gives the isothermal distribution in the evolution of a dust devil. From Figure 2 it can be seen that hemispheric isothermal distribution is formed at the beginning of our simulation and isotherms deliver upward step by step. Then isotherms declutch at a height, and the high temperature delivers further upward. This indicates that high temperature of the local surface heats the lower atmosphere and the heated atmosphere will transfer upward heat. This process of heat diffusion causes unstable atmosphere state and generates intense turbulence and convective plumes, air converging into these plumes tends to conserve any initial angular momentum and a spinning vortex develops and the simulated results of flow field are presented in Figure 3.
Figure 3 shows the whole field (left) and section plane (right) of airflow at about 200 s from the beginning of our simulation. From Figure 3 it can be seen that a whirly wind field is formed in the calculating region. In the center and outside region of a dust devil the wind speed is small, but at the interior region between the center and outside region of a dust devil the wind speed is much greater and can reach a maximum value of 30 m/s.
 In our simulation, sand grains with three diameters (D = 0.15 mm, 0.2 mm, 0.25 mm) are stochastically paved on the bed. After about 200 s from the beginning of our simulation the wind speed exceeds the threshold velocity (Figure 3) and sand grains begin to be lifted from bed with a random lift-off velocity between and 5u*. The widely accepted expression of the threshold wind velocity by Bagnold  is used to determine the threshold velocity in this paper. Then more and more sand grains with different diameters are picked up into rotational wind field until the number of grains in air becomes to attain a dynamic constant. It needs about 90 s from the moment when a sand grain is lift off of the bed to a dust devil is fully developed. This process is presented in Figure 4. In Figure 4 red dots, green dots and blue dots denote sand grains with a diameter of D = 0.15 mm, D = 0.2 mm and D = 0.25 mm, respectively. It can be seen from Figure 4 that smallest sand grains (red dots, D = 0.15 mm) usually move higher and sand grains with biggest diameter (blue dots, D = 0.25 mm) move lower, and sand grains with D = 0.2 mm (green dots) move between smallest sand grains and biggest sand grains. This indicates that when sand grains with different diameters move in a dust devil, stratification will occur: small sand grains are usually above big sand grains.
 In dust devils, sand grains will have charge via frictional or triboelectric processes [Mills, 1977], and sand grains with different diameters may have different charges. Usually small sand grains have negative charge and big sand grains have positive charge and the charge quantity of sand grains are difficult to measure. We apply several charge-to-mass ratios by Huang and Zheng  as shown in Table 1 for sand grains with different diameters to calculate the electric field in a dust devil.
Table 1. Charge-to-Mass Ratios for Sand Grains With Different Diametersa
Charge-to-mass ratios are in μC/kg.
 On the basis of equations (7) and (8), the electric field is calculated through tracing every sand grain with different charge-to-mass ratios in a dust devil as shown in Figure 5. From Figure 5a it can be seen that the electric field of early measurements is only several hundred volt/m [Crozier, 1964; Freier, 1960], which is smaller than that of calculation results of Farrell et al.  (Figure 5b, Figure 5c and Figure 5d). It also can be seen that the calculated electric fields for sand grains with different charge-to-mass ratios differ greatly (Figure 5b, Figure 5c and Figure 5d). Comparing with the measurements of Farrell et al. , both maximum and the distribution of the calculated electric field at 20 m height for sand grains with charge-to-mass ratio of case (3) are more consistent than those of other cases. For example at 20 m height the calculated electric field for sand grains with charge-to-mass ratio of case (3) reaches 4.128 kV/m, which is only 10.2% smaller than 4.599 kV/m of the measurement. So it is reasonable we take the charge-to-mass ratios of sand grains with diameters 0.15 mm, 0.2 mm and 0.25 mm as −120 μC/kg, −60 μC/kg and 57 μC/kg, respectively, in a dust devil.
Figure 6 presents the calculated results of electric field distributions along the radius of a dust devil at 20 m height at different moments for sand grains with charge-to-mass ratios of case (3). From Figure 6 it can be seen that at the beginning of the evolution of a dust devil the electric field strengthens with time, but after 80 s the electric field changes little, which indicates that electric field of the dust devil has trended to stabilization. For instance it can be seen from Figure 6 that the curves of electric field at 80 s and 90 s are almost concurrent.
 The electric field at different height is described in Figure 7 when it trends to stabilization in a dust devil. From Figure 7 it can be seen that the electric field at different height reaches maximum at the center of the dust devil and minishes along the radial. In the meantime, the electric field increases to maximum and then decreases with the increment of height. Figure 8 shows the electric field distributions along height at different radial places inside the dust devil (r = 0 m, 2 m) and outside the dust devil (r = 20 m, 25 m). It can be seen that in the interior of a dust devil (r = 0 m, 2 m) the electric field increases to maximum and then decreases as height increases and the electric field attains maximum at about 6 m height. Outside the dust devil (r = 20 m, 25 m) the variational trend of electric field is similar to that of inside the dust devil, but there does not exist a certain height where the electric field attains maximum.
 In this paper a numerical model, which takes in to consideration of the effect of thermal flux from the surface to atmospheric boundary layer, is employed to simulate a dust devil and the numerical results show that the formation mechanisms of a dust devil can be explained with the theory of thermal convection, which is consistent with earlier theoretical models for dust devils [Renno et al., 1998, 2000]. The results also show that in dust devils sand grains with different diameter have stratification: smaller sand grains are usually above bigger sand grains. Then, on the basis of Coulomb's law, the electric field in a dust devil is numerically simulated in this paper to obtain its fine structure and its development. Six cases of charge-to-mass ratios with different diameters are employed to mathematically describe the electric field of a dust devil. When charge-to-mass ratios are −120 μC/kg, 60 μC/kg and 57 μC/kg for sand grain diameters of 0.15 mm, 0.2 mm and 0.25 mm, the simulated electric field is very close to the measurement. For electric field of a dust devil, it needs 80 s to attain to stabilization. And in the stable situation, the electric field will increases to maximum at a height and then decreases with height, the height where electric field reaches maximum is about 6 m inside dust devils.