On the Formation of Clouds in the Dusty Ionosphere of Mars

Dusty plasma clouds observed at altitudes of about 100 km in the mesosphere of Mars have been considered. Features of the dusty ionosphere of Mars compared to the dusty ionosphere of the Earth have been listed. The equations of the model describing self-consistently dusty plasma structures in the ionosphere of Mars have been presented. This model involves features that are important for the ionosphere of Mars but are ignored when describing the dusty plasma system in the ionosphere of the Earth. In particular, the model for Mars involves effects of deceleration of dust particles because of the adhesion of condensed molecules to them. An altitude distribution of particles constituting mesospheric clouds on Mars has been calculated with the self-consistent model. It has been shown that an important factor for the formation of dusty plasma clouds in the ionosphere of Mars is the Rayleigh–Taylor instability, which limits both the maximum size of dust particles that can form dusty plasma clouds and the maximum thickness of the dusty plasma clouds.

An important feature of the mesosphere of Mars is the presence of clouds at altitudes of about 100 km, which consist of dry ice particles with a characteristic size of about 100 nm formed as a result of carbon dioxide condensation in the ionosphere of Mars [1]. These clouds were discovered by the SPICAM infrared spectrometer on board the Mars Express orbiter. In March 2021, the Mars Science Laboratory Curiosity rover took photographs of Martian clouds [2], which apparently consist of dry ice particles. These clouds were detected above 60 km at sunset when the Sun illuminated the surface of dust particles and clouds against the background of the dark sky. Such a behavior, as well as an important role of condensation processes in the formation of constituent particles of clouds, implies some analogy [1] between mesospheric clouds on Mars and noctilucent clouds [3, 4] observed in the Earth’s atmosphere, although the mechanisms of the formation of Martian clouds are still not completely clear.

It is reasonable to assume that, much like noctilucent clouds in the Earth’s atmosphere, mesospheric clouds on Mars are dusty plasma structures in the ionosphere of Mars [5, 6]. This assumption is consistent with the concept of dusty plasmas, according to which one of the main features distinguishing dusty plasmas from a conventional one (free of charged dust particles) is the possibility of self-organization, leading to the formation of macroscopic structures such as dusty plasma clouds, drops, and crystals [7–10].

Figure 1 illustrates conditions in the ionosphere of Mars important for the physics of mesospheric dusty plasma clouds such as the altitude profiles of the temperature of the neutral gas and the pressures of saturated CO₂ vapor and CO₂ vapor. Conditions for the growth (nucleation) of particles are reached in the altitude range of 92–112 km, where CO₂ vapor is supersaturated. We note that mesospheric dusty plasma clouds are observed just in this altitude range. According to [5], in the case of condensation of gaseous carbon dioxide in the ionosphere of Mars, particles initially located in the upper part of the condensation region (where CO₂ vapor is supersaturated) absorb a large part of carbon dioxide and sediment. Different layers initially located at different altitudes absorb different amounts of the gas; as a result, layers can be mixed and dusty plasma clouds can be formed. The size of dust particles in the condensation region can reach 100 nm or more. The characteristic sedimentation time of such clouds in the condensation region is about several minutes. Condensed carbon dioxide below the condensation region is evaporated. Consequently, the characteristic sedimentation time of layers through the condensation region determines the characteristic existence time of dusty plasma clouds similar to noctilucent clouds in the Earth’s atmosphere. Although processes responsible for the formation of dusty plasma clouds in the ionosphere of Mars are described in [5], some significant questions primarily concerning the altitude distribution of particles constituting dusty plasma clouds remain unanswered.

Fig. 1.

Qualitative altitude profiles of (solid line) the temperature of the gas in the ionosphere of Mars, (dashed line) the partial pressure of CO₂ vapor, and (dash-dotted line) the pressure of saturated CO₂ vapor; CO₂ vapor in the altitude range of 92–112 km is supersaturated.

The description in [5] is similar to the description of noctilucent clouds in the Earth’s atmosphere [4, 11]. However, conditions in the ionosphere of Mars are significantly different from those in the ionosphere of the Earth. Therefore, it is important to take into account features of dusty plasma structures in the ionosphere of Mars compared to those in the Earth’s ionosphere.

The purposes of this work are to update the model presented in [5] to describe the dusty plasma structures in the ionosphere of Mars and to determine the altitude distribution of particles constituting the mesospheric clouds on Mars using the updated model. We also discuss the results of calculations in comparison with existing data and in terms of processes that can affect the characteristic sizes of dust particles, in particular, the Rayleigh–Taylor instability, which can play an important role, as shown in this work.

In terms of the thermodynamics of gaseous carbon dioxide, the considered region of Mars’ atmosphere can be divided into three characteristic parts: the middle part with strongly supersaturated carbon dioxide vapor and two peripheral parts with unsaturated CO₂ vapor (see Fig. 1). In terms of the dynamics of dust particles, the upper part with unsaturated vapor can be called the sedimentation region with a constant mass, the middle part can be referred to as the condensation region, and the lower part can be called the sublimation region. In the condensation region, the desublimation of CO₂ vapor leads to the growth of dust seeds; as a result, the layered structure of a dust cloud can be formed in the ionosphere [5]. In the sublimation region, previously condensed gaseous carbon dioxide is evaporated from the surface of dust particle, which finally results in the disappearance of the dust cloud; thus, the lifetime of the dust cloud is determined by the sedimentation time of microparticles.

The atmosphere of Mars has the following features compared to the Earth’s atmosphere.

(i) The main gas component of the atmosphere of Mars is carbon dioxide (about 95%) [12, 13], and dry ice particles constitute mesospheric clouds on Mars. In turn, water vapor, which forms composite ice particles constituting noctilucent clouds in the Earth’s ionosphere, contains only 0.5% of the mass of the atmospheric gas.

(ii) The density of water vapor in the Earth’s atmosphere is negligible compared to the density of nitrogen and oxygen, so that viscous friction is the main inhibiting factor during the entire sedimentation time to the ground. In the case of the sedimentation of dust particles in the condensation region of Mars’ atmosphere, the inhibiting factor of a dust particle owing to the adhesion of condensate molecules to it (analogous to the reactive force) is significant because the number density of desublimating carbon dioxide is high and CO₂ molecules adhered to the surface of microparticles have a nonzero relative velocity. At the same time, the viscous friction force is caused by only 5% of gases admixed to carbon dioxide in the atmosphere of Mars. In the sublimation region, the entire gas of Mars’ atmosphere creates the viscous friction force because the relative velocity of evaporating carbon dioxide molecules is zero in this case. Physically, this means that the deceleration of CO₂ molecules evaporated from the dust particle is due to molecules of the atmosphere rather than to the acceleration of the particle.

(iii) The atmosphere of Mars, as well as the Earth’s atmosphere, does not transmit ultraviolet radiation with quite short wavelengths. In particular, the transmission coefficients of Mars’ atmosphere at an altitude of 100 km, which is calculated from the SPICAM spectrometer data [14, 15], is almost zero for wavelengths less than 165 nm (which corresponds to photon energies exceeding 7.5 eV) and is nearly unity for longer wavelengths. The work function of dry ice is approximately equal to 11.5 eV (see Fig. 3 in [16]). Thus, it can be assumed that the photoelectric effect on the charging of dry ice particles at the considered altitudes is negligible, and the charges of dry ice particles are negative because the electron mobility is higher than the ion one.

In the model of dusty plasma structures in the mesosphere of Mars, the evolution of the distribution function of dust particles f_d(h, a, $v$, t) at an altitude of h is described by the kinetic equation

$$\begin{gathered} \partial {{f}_{{\text{d}}}}{\text{/}}\partial t + \frac{{{{\alpha }_{{{\text{C}}{{{\text{O}}}_{2}}}}}{{m}_{{{\text{C}}{{{\text{O}}}_{2}}}}}v_{{{\text{C}}{{{\text{O}}}_{2}}}}^{{{\text{th}}}}\left( {{{n}_{{{\text{C}}{{{\text{O}}}_{2}}}}} - n_{{{\text{C}}{{{\text{O}}}_{2}}}}^{{\text{s}}}} \right)}}{{4{{\rho }_{{\text{d}}}}}}\partial {{f}_{{\text{d}}}}{\text{/}}\partial a + v\partial {{f}_{{\text{d}}}}{\text{/}}\partial h \\ + \;\left( {g - \frac{{\pi \rho {{c}_{{\text{s}}}}{{a}^{2}}{{F}_{{\text{d}}}}(v + {{v}_{{{\text{wind}}}}})}}{{{{m}_{{\text{d}}}}}} + \frac{{{{u}_{{{\text{rel}}}}}}}{{{{m}_{{\text{d}}}}}}\frac{{d{{m}_{{\text{d}}}}}}{{dt}}} \right)\partial {{f}_{{\text{d}}}}{\text{/}}\partial v = 0. \\ \end{gathered} $$

(1)

Here, a and m_d are the characteristic size and mass of a dust particle, respectively; ${{m}_{{{\text{C}}{{{\text{O}}}_{2}}}}}$ is the mass of the CO₂ molecule; ${{\alpha }_{{{\text{C}}{{{\text{O}}}_{2}}}}}$ is the coefficient of accommodation of CO₂ molecules colliding with the dust particle (${{\alpha }_{{{\text{C}}{{{\text{O}}}_{2}}}}}$ in strongly supersaturated vapor is usually ~1); $v_{{{\text{C}}{{{\text{O}}}_{2}}}}^{{{\text{th}}}}$ is the thermal velocity of CO₂ molecules; c_s is the local speed of sound; ρ and ρ_d are the densities of the atmospheric gas and dust particle, respectively; $n_{{{\text{C}}{{{\text{O}}}_{2}}}}^{{\text{s}}}$ and ${{n}_{{{\text{C}}{{{\text{O}}}_{2}}}}}$ are the number densities of saturated CO₂ vapor over the surface of the dust particle and CO₂ in the mesosphere of Mars, respectively; ${{v}_{{{\text{wind}}}}}$ and $v$ are the vertical velocities of wind and the dust particle, respectively; F_d ~ 1 is a coefficient representing the effect of the shape of the dust particle; g is the gravitational acceleration; and ${{u}_{{{\text{rel}}}}}$ is the velocity of adhering/evaporating molecules with respect to the dust particle. The second term on the left-hand side of Eq. (1) describes the growth of dust particles in the ambient supersaturated gaseous carbon dioxide, the fourth term describes the sedimentation (ascent) of the dust particle interacting with neutrals (neutral drag), and the term with dm_d/dt is due to the reactive force.

The number density of saturated CO₂ vapor $n_{{{\text{C}}{{{\text{O}}}_{2}}}}^{{\text{s}}}$ over the surface of the particle is calculated conventionally [17]. To this end, the thermodynamic potential Φ of the system consisting of the dust particle, on the surface of which CO₂ molecules are condensed, and a layer of these molecules adjacent to the particle is introduced. Since the dust particle is charged and is surrounded by neutral molecules, as well as by ions and electrons screening the field of the dust particle, it is necessary to take into account the electrostatic interaction. Thus, the dependence of $n_{{{\text{C}}{{{\text{O}}}_{2}}}}^{{\text{s}}}$ on the size a of dust particles and their charge q_d, which also depends on the size a of particles, is taken into account. The thermodynamic potential of the system $\Phi$ has the form

$$\Phi = {{m}_{{\text{d}}}}({{\tilde {f}}_{{\text{d}}}} + P{{v}_{{\text{d}}}}) + {{m}_{{\text{g}}}}({{\tilde {f}}_{{\text{g}}}} + P{{v}_{{\text{g}}}}) + \sigma S + {{\Psi }_{E}},$$

(2)

where P is the pressure; ${{\tilde {f}}_{{\text{d}}}}({{v}_{{\text{d}}}},T)$ and ${{\tilde {f}}_{{\text{g}}}}({{v}_{{\text{g}}}},T)$ are the specific free energies of the dust particle and gas, respectively; ${{v}_{{\text{d}}}}$ and ${{v}_{{\text{g}}}}$ are the specific volumes of the dust particle and gas, respectively; T is the temperature; m_g is the mass of the gas; σ is the surface tension of the material of dust particles; S is the surface area of the dust particle; and ${{\Psi }_{E}}$ is the electrostatic energy, which is the sum of the electric field energies in and out of the dust particle:

$${{\Psi }_{E}} = \int\limits_{{\text{in}}} \varepsilon {{E}^{2}}{\text{/}}8\pi dV + \int\limits_{{\text{out}}} {{{E}^{2}}} {\text{/}}8\pi dV.$$

(3)

Here, ε is the relative permittivity of the material of the dust particle, E is the electric field, and V is the volume.

Analyzing the extrema of Eq. (2) under the assumption that the dust particle is spherically symmetric, ε is constant, and the electric field of the dust particle is screened according to the Yukawa potential with the screening length λ, one can derive the following transcendental relation between the pressure P_s of saturated CO₂ vapor over the dust particle, which has the size a and surface charge q_d, and the pressure P₀ of saturated CO₂ vapor over the flat surface:

$$\begin{gathered} {{v}_{{\text{d}}}}\left( {{{P}_{{\text{s}}}} - {{P}_{0}}} \right) - \frac{{{{N}_{{\text{A}}}}T}}{{{{\mu }_{{\text{g}}}}}}\ln \left\{ {\frac{{{{P}_{{\text{s}}}}}}{{{{P}_{0}}}}} \right\} \\ + \;2\sigma {\text{/}}a + q_{{\text{d}}}^{2}{\text{/}}2{{a}^{2}}\left( {1{\text{/}}\epsilon - 1 + \nu (\lambda ,a)} \right) = 0, \\ \end{gathered} $$

(4)

where μ_g is the molar mass of the gas, N_A is the Avogadro number, and

$$\nu (\lambda ,a) = \int\limits_a^\infty {\frac{{{{a}^{2}}{{{(\lambda + r)}}^{2}}}}{{{{r}^{2}}}}} \frac{{2a\exp \left\{ {2(a - r){\text{/}}\lambda } \right\}}}{{\lambda {{{(\lambda + a)}}^{3}}}}dr.$$

(5)

Correspondingly, the number density $n_{{{\text{C}}{{{\text{O}}}_{2}}}}^{{\text{s}}}$ is expressed in terms of P_s by the well-known relation $n_{{{\text{C}}{{{\text{O}}}_{2}}}}^{{\text{s}}} = {{P}_{{\text{s}}}}{\text{/}}T$ for the ideal gas.

The dynamic equation for the number density ${{n}_{{{\text{C}}{{{\text{O}}}_{2}}}}}$ of CO₂ vapor has the form

$$\begin{gathered} \frac{{\partial {{n}_{{{\text{C}}{{{\text{O}}}_{2}}}}}}}{{\partial t}} + \frac{{\partial {{\Gamma }_{{{\text{C}}{{{\text{O}}}_{2}}}}}}}{{\partial h}} \\ = {{P}_{{{\text{C}}{{{\text{O}}}_{2}}}}} - {{n}_{{{\text{C}}{{{\text{O}}}_{2}}}}}{{L}_{{{\text{C}}{{{\text{O}}}_{2}}}}} - \pi {{\alpha }_{{{\text{C}}{{{\text{O}}}_{2}}}}}v_{{{\text{C}}{{{\text{O}}}_{2}}}}^{{{\text{th}}}}{{n}_{{{\text{C}}{{{\text{O}}}_{2}}}}}\langle {{a}^{2}}{{n}_{{\text{d}}}}\rangle , \\ \end{gathered} $$

(6)

where ${{\Gamma }_{{{\text{C}}{{{\text{O}}}_{2}}}}}$ is the vertical diffusion flow of CO₂ vapor; ${{P}_{{{\text{C}}{{{\text{O}}}_{2}}}}}$ and ${{L}_{{{\text{C}}{{{\text{O}}}_{2}}}}}$ are the photochemical sources and sinks of CO₂ vapor in the mesosphere of Mars, respectively; and the last term on the right-hand side describes the absorption of CO₂ molecules by dust particles.

The kinetic equations (1) and (6) are represented in the one-dimensional approximation (the only spatial variable is the particle location altitude h). This approximation is justified because the vertical dimension of mesospheric clouds (~1 km) is much smaller than the horizontal dimension (~10–100 km) and the horizontal transfer velocity of dust particles is lower than or about the vertical transfer velocity; consequently, the horizontal displacement of dust particles can be neglected at the considered times (minutes or hours [5]).

The system of equations describing the effect of dust particles on the dynamics of charged particles in the ionosphere of Mars is written in the local approximation because the characteristic charging times for dust particles in the ionosphere are much shorter than their characteristic transfer times. The system includes the continuity equations [4, 18] for the electron density ${{n}_{e}}$, ion density ${{n}_{i}}$, and the charge of the dust particle $Z_{{\text{d}}}^{a}$ with the radius a:

$$\partial {{n}_{e}}{\text{/}}\partial t = {{q}_{e}} - {{\alpha }_{{{\text{rec}}}}}{{n}_{e}}{{n}_{i}} - L_{{{\text{dust}}}}^{e},$$

(7)

$$\partial {{n}_{i}}{\text{/}}\partial t = {{q}_{e}} - {{\alpha }_{{{\text{rec}}}}}{{n}_{e}}{{n}_{i}} - L_{{{\text{dust}}}}^{i},$$

(8)

$$\frac{{\partial Z_{{\text{d}}}^{a}}}{{\partial t}} = {{\nu }_{i}} - {{\nu }_{e}}.$$

(9)

Here, the terms $L_{{{\text{dust}}}}^{j} = \int {{\nu }_{j}}d{{n}_{{\text{d}}}}$ (j = e, i) describe the absorption of electrons and ions on dust particles, where ${{v}_{e}}$ and ${{v}_{i}}$ are the charging rates caused by collisions of electrons and ions with dust particles, respectively; and dn_d = f_dd$v$da, where f_d is the microparticle size distribution function.

The coagulation of dust particles colliding with each other in the mesosphere can be neglected because the characteristic coagulation time τ_coag ~ (n_d$v$πa²)^–1 ≥ 10⁶ s is much longer than all other characteristic times in the system. The Brownian motion of particles can also be neglected. Indeed, the characteristic Brownian displacement of particles can be estimated by the formula $\langle {{x}^{2}}\rangle = 2TB\tau = 2T\tau {\text{/}}(\pi \rho {{c}_{{\text{s}}}}{{a}^{2}})$, where $B = 1{\text{/}}(\pi \rho {{c}_{{\text{s}}}}{{a}^{2}})$ is the mobility of particles in a rarefied gaseous medium. At the temperature of the medium about T = 100 K, particle size a = 20 nm, and sedimentation time τ = 300 s, which are typical of the ionosphere of Mars, the diffusion drift is about 10 m, i.e., less than 1% of the descent altitude.

The microscopic electron and ion currents to microparticles are calculated in the orbit-limited probe model [19, 20], where the cross sections for the interaction of ions and electrons with a charged particle are determined from the momentum and energy conservation laws. For negatively charged dust particles, the orbit-limited probe model gives the following expressions for the charging rates:

$${{\nu }_{e}} \approx \pi {{a}^{2}}{{\left( {\frac{{8{{T}_{e}}}}{{\pi {{m}_{e}}}}} \right)}^{{1/2}}}{{n}_{e}}\exp \left( {\frac{{eq_{{\text{d}}}^{a}}}{{a{{T}_{e}}}}} \right),$$

(10)

$${{\nu }_{i}} \approx \pi {{a}^{2}}{{\left( {\frac{{8{{T}_{i}}}}{{\pi {{m}_{i}}}}} \right)}^{{1/2}}}{{n}_{i}}\left( {1 - \frac{{eq_{{\text{d}}}^{a}}}{{a{{T}_{i}}}}} \right).$$

(11)

Here, $q_{{\text{d}}}^{a} = Z_{{\text{d}}}^{a}e$ is the charge of the dust particle with the radius a; e is the elementary charge; m_e and m_i are the masses of the electron and ion, respectively; and ${{T}_{e}}$ and ${{T}_{i}}$ are the electron and ion temperatures, respectively.

The above equations constitute the self-consistent model of dusty plasma structures in the ionosphere of Mars. This system of equations is simpler than that for the Earth. In particular, only CO$_{2}^{ + }$ ions are taken into account because gaseous carbon dioxide composes about 95% of Mars’ atmosphere. Moreover, information on Mars’ atmosphere is less detailed than that on the Earth’s atmosphere and this simplified description seems sufficient at this stage. At the same time, the above system of equations involves features that are important for the ionosphere of Mars but are disregarded when describing the dusty plasma system in the ionosphere of the Earth. For example, Eq. (1) includes the terms describing the deceleration of the dust particle owing to the adhesion of condensate molecules to it (analog of the reactive force).

The system of Eqs. (1)–(11) allows one to evaluate the altitude distribution of particles constituting mesospheric clouds on Mars. Figure 2 shows such an altitude distribution formed as a result of the sedimentation of a dust cloud of seeds, which initially constitutes a model rectangular density profile at altitudes of 110–120 km. The initial size of dust particles in the cloud is 10 nm. Entering the condensation region (where gaseous carbon dioxide is strongly supersaturated), dust particles begin to grow owing to the desublimation of CO₂ vapor. Upper layers initially located at the interface between regions with supersaturated and unsaturated vapor evolve more slowly than layers initially located in the condensation region. At a certain time individual for each layer, all particles reach characteristic sizes of ~0.3–3 μm. According to the calculations, the characteristic sedimentation time of the dust cloud is about several minutes. The charge ${\text{|}}Z_{{\text{d}}}^{a}{\text{|}} \sim 10{-} 100e$ of particles noticeably perturbs the charged component of the ionospheric plasma because the total charge of dust particles becomes comparable with the equilibrium total charges of electrons and ions.

Fig. 2.

Time dependence of the altitude distribution of dust particles in mesospheric clouds on Mars formed as a result of the sedimentation of a cloud of seeds, which initially constitutes a model rectangular profile of number densities at altitudes of 110–120 km. The initial size of dust particles in the cloud is 10 nm and their number density is 100 cm^–3. Entering the condensation region, dust particles begin to grow owing to the desublimation of CO₂ vapor. The sizes of particles are given above the lines in microns.

According to the calculations with the presented model, dust particles in the mesospheric clouds on Mars are much greater than particles constituting noctilucent clouds in the Earth’s atmosphere because the density of gaseous carbon dioxide in the mesosphere of Mars is much higher than the density of water vapor in the ionosphere of the Earth. However, the calculated sizes of dust particles in the mesosphere of Mars (~0.3–3 μm) exceed the observed sizes (~0.1 μm) [1]. This can be due to processes disregarded in the system of Eqs. (1)–(11), in particular, the natural development of the Rayleigh–Taylor instability [21]. Indeed, such an instability can appear at the interface in the gas + dust system in the gravitational field in the situation where the upper half-space is occupied by dust and gas, whereas the lower half-space is filled only with gas [22]. In turn, mesospheric clouds have a sharp lower edge and the density of dust particles below this edge is negligibly low compared to the density of dust above it. When the upper half-space is occupied by dust and gas, whereas the lower half-space is filled only with gas and, in addition, the gas + dust system is not limited in the horizontal direction, instability is developed with the growth rate $\tilde {\gamma }$, satisfying the dispersion equation

$$\begin{gathered} {{{\tilde {\gamma }}}^{3}} + {{{\tilde {\gamma }}}^{2}}\left[ {\alpha \left( {1 + \frac{{{{n}_{{\text{d}}}}{{m}_{{\text{d}}}}}}{{{{n}_{{{\text{C}}{{{\text{O}}}_{2}}}}}{{m}_{{{\text{C}}{{{\text{O}}}_{2}}}}}}}} \right) + \frac{{{\text{|}}\mathbf{k}{\text{|}}g}}{\alpha }} \right] \\ + \;\tilde {\gamma }{\text{|}}\mathbf{k}{\text{|}}g - \frac{{{{n}_{{\text{d}}}}{{m}_{{\text{d}}}}{\text{|}}\mathbf{k}{\text{|}}g\alpha }}{{2{{n}_{{{\text{C}}{{{\text{O}}}_{2}}}}}{{m}_{{{\text{C}}{{{\text{O}}}_{2}}}}}}} = 0, \\ \end{gathered} $$

(12)

where k is the wave vector, α = $2\sqrt {2{\text{/}}\pi } G{{n}_{{{\text{C}}{{{\text{O}}}_{2}}}}}T{\text{/}}{{\rho }_{{\text{d}}}}v_{{{\text{C}}{{{\text{O}}}_{2}}}}^{{{\text{th}}}}a$, and G ~ 1 is a coefficient determined by the specificity of the interaction of neutrals with the surface of the particle. In particular, G = 1 in the case of the complete absorption or mirror reflection of neutrals from the surface of the particles after collision and $G = 1 + \pi {\text{/}}8$ in the case of complete accommodation [8]. In contrast to the dispersion equation presented in [22], where the condition Kn ≪ 1 was assumed, the dispersion equation (12) is obtained under the condition Kn ≫ 1 corresponding to the ionosphere of Mars.

The instability growth rate $\tilde {\gamma }$ reaches a maximum γ at a certain value |k|. Figure 3 shows the dependences of the maximum growth rate γ on the altitude h for different densities and sizes of dust particles. Figure 4 presents the dependences of the characteristic development time of the Rayleigh–Taylor instability $\tau = {{\gamma }^{{ - 1}}}$ and the sedimentation time of dust particles on their sizes for different altitudes and densities of dust particles. The calculations were performed with the following parameters of Mars’ atmosphere: T = 130.4 K and ${{n}_{{{\text{C}}{{{\text{O}}}_{2}}}}} =$ 6.85 × 10¹³ cm^–3 at h = 80 km, T = 105.8 K and ${{n}_{{{\text{C}}{{{\text{O}}}_{2}}}}} = 2.05 \times {{10}^{{13}}}$ cm^–3 at $h = 90$ km, $T = 81.2$ K and ${{n}_{{{\text{C}}{{{\text{O}}}_{2}}}}} = 5.48 \times {{10}^{{12}}}$ cm^–3 at $h =$ 100 km, and T = 82.85 K and ${{n}_{{{\text{C}}{{{\text{O}}}_{2}}}}} = 6.85 \times {{10}^{{11}}}$ cm^–3 at $h = 110$ km. It is seen in Fig. 4 that the sedimentation time is shorter than $\tau = {{\gamma }^{{ - 1}}}$ only for rather small dust particles. In particular, this is the case for sizes $a \lesssim 0.5$ μm at ${{n}_{{\text{d}}}} \gtrsim 100$ cm^–3. Since the development of the Rayleigh–Taylor instability can destroy a dusty plasma structure, dusty plasma clouds can exist only at small (submicron) sizes a of dust particles in agreement with observations [1]. Furthermore, smaller dust particles are located at higher altitudes (see Fig. 2). Thus, the possible development of the Rayleigh–Taylor instability limits the maximum thickness of the dusty plasma cloud.

Fig. 3.

Altitude profiles of the maximum growth rate of the Rayleigh–Taylor instability for dust particles with a size of (a) 0.1 and (b) 1 μm and the number density n_d = (solid line) 1, (dashed line) 10, and (dash-dotted line) 100 cm^–3.

Fig. 4.

(Thick solid lines) Sedimentation times of dust particles and (thin solid, dashed, and dash-dotted lines) the characteristic development times of the Rayleigh–Taylor instability versus their size calculated for altitudes of (a) 80, (b) 90, (c) 100, and (d) 110 km for the number density of dust particles n_d = (thin solid lines) 1, (dashed lines) 10, and (dash-dotted lines) 100 cm^–3.

To summarize, the equations of the model describing self-consistently dusty plasma structures in the ionosphere of Mars have been presented. This model involves features that are important for the ionosphere of Mars but are ignored when describing the dusty plasma system in the ionosphere of the Earth. In particular, the model for Mars involves effects of deceleration of dust particles because of the adhesion of condensed molecules to them (analog of the reactive force). An altitude distribution of dust particles constituting mesospheric clouds on Mars has been calculated using these equations. It has been shown that an important factor for the formation of dusty plasma clouds in the ionosphere of Mars is the Rayleigh–Taylor instability, which limits both the maximum size of dust particles that can form dusty plasma clouds and the maximum thickness of the dusty plasma clouds.