Mechanics of Turbulence of Multicomponent Gases (Astrophysics and Space Science Library)


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The kinetic energy of turbulent motion of a system is generated by large energy carrying eddies due to a gradient in the hydrodynamic velocity V shear flow. The characteristic size of such eddies determine the external integral turbulence scale L. As an indication, eddies greater than L are nonisotropic, while eddies of smaller scales are approximately isotropic. The scale ranges from millimeters to thousands of kilometers, l 8 CHAPTER 1 cascade process occurs of energy transfer from large scale eddy motions to small scale ones. On the one hand, this quantity represents the transport rate of rotational kinetic energy from large atmospheric eddies to smaller ones.

The latter also follows from the dimensionality and similarity theory. The dependence of the effective vertical diffusivity v L on the local turbulence scale L z is shown in Figure 1. As it was mentioned above, for such large values of L , there is no generation or dissipation of kinetic energy but only its transfer to decreasingly smaller scales r down to the scale Decrease in scale fits the growth of the wave number in the curve of the energy spectral density of the velocity field see Figure 1.

At moderate Earth latitudes, Obviously, in this case the degeneration time of atmospheric kinetic energy due to turbulent viscosity matches the typical transformation time of the potential energy of synoptic processes i. Thus, the minimum scale of synoptic motions capable to overcome viscosity is estimated to be Monin, Such an approach agrees with the concept that there are universal relations between features of large- and small-scale fluctuations of turbulent flows, these relations being determined for by small-scale fluctuations rather than at large-scale velocity fluctuations.

The latter determines average values of all thermohydrodynamical quantities and, basically, are independent of Re. This embodies the Batchelor hypothesis about statistical independence of large- and small-scale motions. On the Spectrum of Atmospheric Processes.

In the theoretical study of turbulent atmospheric fluctuations of wind velocity or other thermohydrodynamical parameters fluctuating in a flow , the methods of mathematical statistics relevant to quasistationary stochastic processes, together with correlation and spectral analysis, are usually employed see Chapter 8.

The most commonly used statistical properties of a field of a random quantity A r,t involve its average value in what follows, the latter is designated with an overscribed bar and variance, various correlation and structural functions, spectral density functions, etc. Because motion systems of diverse spatialtemporal scales co-exist in the atmosphere, the thermohydrodynamical parameters describing the atmosphere must be averaged. In this approach it is assumed that individual realization of turbulent flows can be described with equations of multicomponent hydrodynamics for instantaneous motion.

To obtain representative statistical estimates of the parameters governing the fluctuating flow, the interval should be properly chosen. The larger the fluctuation scales, the longer this interval must be. The broad spectrum of fluctuations of the random variables mentioned above, whose periods range from a fraction of a second to thousands of years, is characteristic of the atmosphere see Monin, Within this spectrum, micrometeorological oscillations with periods from a fraction of a second to a few minutes deserve particular attention.

Specifically, these oscillations just arise in the atmospheric layer above the surface and represent small-scale isotropic turbulence serving as the most important mechanism for viscous dissipation. The maximum of the energy spectrum where is the spectral kinetic energy density of the flow fits a period minute. For an isotropic turbulent flow, the cascade process of kinetic energy transfer from large scale vortex motions to small scale eddies is convenient to analyze in the space of wave numbers using the spatial spectral energy density which is defined as Here is the Fourier transform of correlation function of the random velocity field where is the wave number is module, and summation is performed over repetitive indices.

Note that the kinetic energy of the fluctuating motion of a unit mass. Using the definition of it follows that turbulent energy equals Figure 1. It is important to emphasize that the redistribution of energy among spectral components occurs without a change in the total turbulent motion energy The second addend, describes the energy dissipation due to viscosity and characterizes the decrease in the kinetic energy of perturbation fluctuations with wave number being equal to the intensity of these perturbations multiplied by This means that, due to viscosity effects, the energy of long decreases much slower than the energy of wavelength perturbations with smaller short-wave perturbations.

This reflects the proportionality of friction force to the velocity gradient. The spectra with energy and energy dissipation are sketched in which determines the redistribution of Figure 1. As can be seen, negative values of function in the range of small which fit to the maximum of large-scale motion energy in the curve The latter correspond to the are replaced by positive values in the range of large This affirms the idea of the casmaximum energy dissipation in the curve cade energy transfer from large-scale components of motion to small-scale components.

Large local gradients are characteristic of the latter and, hence, it is here where viscosity plays an important role. Such an approach highlights the problem of the rate of energy redistribution over the spectrum proceeding from the basic spectral representation concept of the velocity field itself, which was developed by Batchelor Altogether, this brings additional support to the idea that the initial energy of large eddies, determining the dynamic and kinematic properties of a flow, is being expended on their fragmentation which is expressed by turbulent viscosity.

Ultimately, this energy determines the properties of viscous dissipation when the conditions of total energy balance are met. It states that the kinetic energy scale equals by order of magnitude the product of the energy supply rate to the system from outside in the form of insolation or heat from the planetary interior, or generated inside as or inhomogenities in the internal field and the least time scale pertinent to the system. This principle is intimately related with stochastics of events depending on their intensity and mode of energy transformation within the system.

Some Methods of Turbulence Simulation In addition to the statistical approach to turbulence modeling, the phenomenological semi-empirical approach and methods of direct numerical simulation of turbulence find an expanding application. They are based on solutions of special kinetic equations or non-stationary system of the three-dimensional Navier-Stokes equations, although only averaged properties of motion can be obtained because of the stochasticity of this phenomenon.

Nevertheless, sometimes this approach makes it possible to trace not only time evolution of the formation of various spatial structures but also to study the general dynamics and nature of turbulence development. This is in agreement with the known Kolmogorov-Obukhov law Gledser et al. It is interesting to note that examination of the stochastization process of dynamic systems and scenarios of passing to chaos in numerical turbulence simulations serves as an analog to the solution of incorrect ill-posed problems, which uses an averaging operator of the parametrical extension Tikhonov and Arsenin, With such an approach, the ordered structure of a turbulent current, which is determined as an attractor of an asymptotically steady solution for averaged magnitudes, represents its regularized description Belotserkovskii, However, it should be noticed that using the techniques of direct numerical turbulence simulation for solving practically important problems especially those related to calculations of turbulent heat and mass transport in multicomponent chemically active mixtures is often inconvenient or too bulky.

Therefore, such problems are more suitable to be solved with the use of less sophisticated semi-empirical theories. Thus, the equations obtained for the scale of averaged motion, due to nonlinearity of the initial Navier-Stokes equations, involve uncertain correlation terms such as turbulent diffusion vectors and heat, and turbulent Reynolds stress tensors and consequently they happen to be non-closed.

The closure of the hydrodynamic equations of a mixture, averaged according to Reynolds, is usually carried out with the help of some semi-empirical turbulence models this just being the subject of the present study. At the same time, it is important to indicate in advance a basic drawback of such an approach. The problem is that the Reynolds averaging is carried out over all scales of turbulence.

In other words, the simulations based on semi-empirical hypotheses of closure are simultaneously performed over the entire spectrum of multi-scale eddy structures. In contrast to a virtually universal spectrum of small-scale fluctuations for diverse patterns of currents , large-scale structures substantially differ from each other in different currents see Figure 1. Hence, it is obvious that creation of universal semi-empirical turbulence models suitable for description of the diverse turbulent mixture currents is hopeless and therefore, that the problem is focussed on the assessment of applicability limits for the most relevant turbulence model.

Turbulent Diffusion In the complex problems related to the theoretical examination of heat and mass transfer processes in a natural turbulent multicomponent medium, the transport modeling of minor admixtures is of considerable importance including air mass mixing regarding their chemical activity. In the atmosphere, in addition to gases, there also are aerosols of various types and sizes. They are partly involved in chemical transformations and phase transitions and include radioactive admixtures both of natural radon, thoron, and products of their decay and artificial due to production and testing of nuclear weapons, nuclear power station accidents, etc.

The transfer process of these admixtures and their mixing is conditioned by turbulent diffusion whose nature depends on the structure of the fluctuation velocity field and the turbulence energy distribution over fluctuations of different spatial scales. When describing diffusion processes in a turbulent atmosphere, it is possible to distinguish the average values of the admixture concentration see Chapter 2 and their fluctuative deviations as well as the average magnitude V and fluctuations V' of air motion velocity.

This makes it possible to depart from the diffusion equations for instantaneous concentration values to the diffusion equations for the CHAPTER 1 14 averaged motion scale using common averaging techniques see equation 3. Depending on the scale, the theory of turbulent admixture diffusion distinguishes gradienttype diffusion generated by rather small eddies, and non-gradient-type diffusion created by large eddies Monin and Yaglom, The phenomenological approach is applicable to the gradient-type turbulent diffusion see section.

As turbulence is a flow pattern property of a fluid rather than a property of the fluid itself, the mechanism of exchange of momentum as well as energy and substance implies only an implicit identity to molecular exchange of such a quantity. Nevertheless, there exists a definite analogy between mixture diffusion in the field of small-scale turbulence and molecular diffusion, which is also valid in the case of contamination of a medium by a fine-grain admixture, provided the turbulence scale is small compared to the scale of a contamination cloud.

Such analogy is based on an assumed proportionality between the turbulent flow of some diffusing substance and the gradient of its averaged concentration. Indeed, likewise a random molecular motion is characterized by some average velocity of the molecules and the free path length so that the molecular diffusivity bulent diffusivity locity fluctuations and ever, in contrast to random turbulent mixing can be described by the turwhere is the characteristic magnitude of turbulent veis local turbulence scale the so-called mixing length. Howand the parameters and are flow properties rather than fluid properties.

Accordingly, D and are addressed as constants of proportionality; D between the molecular diffusion flow of some substance and the gradient of its concentration of the substance in the fluctuating velocity field and between the turbulent flow and the gradient of its average concentration i. Monin, Note that since turbulent or eddy diffusion, unlike its molecular counterpart, usually is an anisotropic entity, in general, and must be treated as tensors.

In case of a conservative admixture, when individual properties of particles of a diffusing substance preserve while moving in a fluctuating flow, the Lagrangian concept see formula 3. The introduction of the Lagrangian turbulent fluctuation of the conservative property makes it possible to define a certain length roughly proportional to the average linear scale of the velocity fluctuations , in which the correlation between the initial and final velocities of the given Lagrangian particle disappears.

This allows us to simulate the turbulent diffusivity during the entire time period of the fluctuating motion Priestly, The possibility of some direct influence of molecular diffusion on eddy diffusion transport must not be ruled out, provided that the amount of the diffusing substance within a large-scale vortex noticeably change within a time frame, during which the correlation between the velocity values in the vortex remains fairly high.

Apparently, it also must be assumed that these multi-period diffusion processes have superposition features. This gives applicability limits to the parabolic diffusion equation see equation 3. In particular, under conditions present in the near-land layer of the atmosphere, the inequality is valid, which is conditioned by the fact that although is fair. In other words, the invoked analogy to the molecular diffusion is acceptable only for limited volumes of air parcels, as the velocity of admixture transport in the turbulent atmosphere is limited by the magnitude of the wind velocity fluctuations responsible for turbulent mixing.

Moreover, the analogy to molecular diffusion is wrong in the large-scale turbulence range, when the scale becomes commensurable to or exceeds the size L of contamination clouds and the distances between particles remain essentially unchanged as they are transported with large-scale motions, including large eddies. In this case, a more general statistical approach to large-scale turbulence discussed in section 1.

Two physical key quantities of a dissipative hydrodynamic system, that determine energy transfer from large eddies to smaller ones, and the turbuare used here: the viscous dissipation velocity of the turbulent energy lence scale L Monin and Yaglom, In this case, unlike in case of gradient diffusion, the specificity of a multicomponent medium is of no importance. When simulating the turbulent diffusivity for any possible atmospheric components including pollutants , it is necessary to account for factors most strongly affecting their mixing dispersion.

In particular, of primary importance is a priori evaluation of the mixing length scale of turbulence or its approximation. As it was already mentioned, in the general case of an anisotropic fluctuating velocity concentration field, the diffusion transport of minor components is characterized by the turbulent diffusion tensor or the viscosity tensor When currents feature a sharply determined direction of inhomogeneity as is, for example, the case for an atmosphere stratified in the gravitational field , the turbulent velocity fluctuations are superimposed on its average wind component in the horizontal direction, the latter being enhanced in the presence of shear.

For the currents of this kind, various simplified approximations for vertical turbulent viscosity such as the Prandtl mixing length model see formula 3. These approximations are fair only, however, if there is local equilibrium in a turbulent field see section 4. Nevertheless, generally speaking, such approaches are inapplicable to multicomponent chemically active mixtures because of the essential role played by chemical kinetics which breaks down the Lagrangian invariance of transferable substances.

Here again, we face the necessity to develop new methods to simulate turbulent mixture transfer factors that allows one to take into account the peculiarities of multicomponent media. The Dynamic Nature of Turbulence Let us now make some general remarks concerning the dynamic nature of turbulence in a non-linear dissipative gas-fluid system, that can exchange both energy and substance with surrounding bodies. By virtue of such exchanges, the formation of various spatialtemporal structures is possible whose succession makes up the process of selforganization.

In the presence of turbulence each individual particle in a medium moves randomly, so its coordinates and direction of motion vary in time by Markov's random process law. A complete statistical description of turbulent flow is reducible to the definition of a probability measure within its phase space r , p , involving all possible individual realizations of random thermodynamical fields which characterizes this space. Therefore, turbulence can be examined by means of the statistical mechanics of multiple particles see, for instance, Obukhov, Or it can be described with the kinetic equation, which is an analogue of the Boltzmann equation in a phase space for some conditional density function of the probability distribution serving as the basic statistical feature of pulsating motion Klimantovich, A characteristic property of an open system with a large number of independent dynamic variables r , p is its dynamic instability due to mixing that can be imagined as exponential divergence of initially close phase trajectories.

So any initial distribution of a probability density function in a phase space aims towards the limiting equilibrium distribution, that is, the most chaotic state with maximum entropy in the Boltzmann-Gibbs-Shannon sense. Turbulization of fluid or gas motion may be represented as a result of changing phase trajectory topologies, leading to rearrangement of the attractors and to qualitative modification bifurcation of the state of motion. Velocity correlation at any point of a flow is limited to small time intervals depending on the initial conditions. It is impossible to assess a causal dependence between velocity fields outside these intervals as well as correlation with an antecedent motion.

All this substantiates the idea of the stochastic nature of velocity fluctuations in a turbulent flow that arise due to loss of stability of laminar motion of a hydrodynamic system when changes in external controlling parameters occur for example, the number Re. From this point of view, turbulent motion is more random, than laminar motion, and turbulence is identified with chaos or noise.

The stochastic nature of turbulence is reflected by dense interlacing of phase trajectories with different asymptotic behavior topology and structure of attraction areas they enclose attractors. Such behavior of trajectories in phase space means that the system possesses ergodicity. In other words, for almost all realizations of a random field, its time average is equal to its statistical average, the system's time correlation functions rapidly decay, and the frequency spectra are continuous.

The ergodic property is, apparently, one of the characteristic features of a steady-state homogeneous small-scale turbulent field see, for example, Kampe de Feriet, On the other hand, in passing to the maximum shear turbulence in an open hydrodynamic system, new macro dependencies between individual areas set up due to collective interaction of the involved subsystems.

This enhances the internal ordering of the system as compared to arbitrary fluctuations at the molecular level. In this case, multiple spatial-temporal turbulence development scales correspond to the coherent behavior of a large number of particles. These structures, mentioned at the beginning of the present section, appear on the background of small scale turbulent motion.

Granulae in the solar photosphere are another example of the extensive family of coherent structures in turbulent flows Petrovay, All this supports the seemingly paradoxical inference that the state of mature turbulent motion, despite of its enormous complexity, is more ordered than the state of more symmetrical laminar motion. Interest in this problem, emerging in the most diverse natural phenomena and engineering areas, has recently grown.

Unfortunately, it should be noted, that although understanding of the synergetic nature of turbulence as a process of self-organizing in open hydrodynamic nonlinear systems is about thirty years old, up to now the concepts of emerging coherent spatial-temporal dissipative structures have not been embodied in usable engineering calculation methods for large-scale turbulence. For an outer gas-liquid giant planet, the notion of its atmosphere is synonymous to its exterior layers.

It may be considered as a free flowing fluid, for which gravity is furnished by the planet, and the energy to regularly lift this envelope and to ensure that gas flows is furnished by the Sun. As a rule, the Reynolds number Re exceeds in atmospheric gas flows and consequently, the currents are turbulent. Turbulization of atmospheric currents arises because of their deformation while flowing around irregularities of the underlying surface, or when a large-scale flow becomes unstable because of increased temperature and wind velocity gradients.

The loss of stability of internal gravity shear waves is the basic cause of turbulence generation in a free atmosphere. Instability of this kind is characteristic mainly of layers with strongly curved vertical temperature and wind velocity profiles. The presence of multiple components and chemical activity of atmospheric gases is one of the special features of a planetary atmosphere. Changes in density, temperature, and the mixture structures due to chemical processes can result in flow turbulization.


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The arising density gradients generate additional vorticity through interaction with surrounding gradients or fluctuations in the pressure field. It is related to the emerging of a source term equal to zero in barotropic media in the Friedman vorticity equation where is the absolute velocity vortex, is the vorticity vector, is the angular velocity vector of the planet's rotation; F is the sum of accelerations due to the viscosity force and external mass forces see, for example, Monin et al.

Thus, the occurrence of local mass density inhomogeneities gradients constitutes a major feature of reacting multicomponent currents which is usually neglected in classical turbulence models for homogeneous fluids. Local energy generated by chemical reactions is another complicating aspect of modeling multicomponent turbulence. Local heat release in gas flows accelerates the expansion of a medium and can induce Rayleigh-Taylor instabilities in currents stratified in a gravitational field in the presence of a buoyancy force , thus embodying a feedback to hydrodynamics.

In what follows we will consider some examples of turbulent natural multicomponent media. Planetary Atmospheres Properties of planetary atmospheres strongly differ from each other even within the relatively small region of the solar system occupied by the terrestrial planets. In addition to Earth, this group comprises Mercury, Venus, and Mars. Mercury, like the Moon, is virtually deprived of an atmosphere: the density of its gas envelope does not exceed at the surface, though a number of dynamic effects arise by interaction of solar wind plasma with the magnetosphere and the exosphere of Mercury.

Venus and Mars, as two extreme models of evolution of Earth-like planet, represent, from this viewpoint, the most interesting cases for comparative planetology in terms of specific features in their meteorology, climate formation and interior dynamics. Their atmospheres of secondary origin are oxidizing in character and mainly consist of carbon dioxide. At the same time, the atmospheres of the giant planets Jupiter, Saturn, Uranus, and Neptune , are reducing in their character and mainly consist of hydrogen, helium, and hydrogen-bearing compounds water, ammonia, methane and other hydrocarbons.

It is assumed that the primary atmosphere of these gas-fluid planets having no solid surfaces have essentially remained unchanged since the formation of the solar system Marov, Unlike stars, emitting radiation at the expense of nuclear energy generated in their interiors, planets are cold bodies reflecting and re-radiating absorbed energy from the Sun.

Winds in their atmospheres are principally driven by an atmospheric thermodynamic heat engine. From the viewpoint of energy exchange, both the underlying solid or liquid and gas layers represent a unified open thermodynamic system. It is controlled either by incoming solar radiation, or as in the case of Jupiter, Saturn and Neptune also by energy from the interior, which is two-three times greater than the energy these planets receive from the Sun. The energy absorbed by the atmosphere and transformed into internal and kinetic energy of gaseous media, eventually is radiated back to outer space as IR-radiation, ensuring total energy balance.

Through dissipation they thus exert an energy effect on the formation of large-scale weather processes. Condition 1. Dynamics of the Atmospheres of Earth and Venus In the atmosphere of the Earth, the forces conditioned by pressure gradients, are virtually balanced by the Coriolis forces the Rossby number Therefore, geostrophic wind is a typical synoptic feature and its zonal and meridional components can be assessed from the known pressure distribution.

Meanwhile, the Coriolis force effect is insignificant on the very slowly rotating Venus one revolution for terrestrial days , and the condition of cyclostrophic balance turns out to be justified Haltiner and Martin, ; Marov and Grinspoon, It is characteristic of the latter that the zonal latitudinal component, growing with altitude, is superimposed by a circulation cell the Hadley cell counterpart whose originates from the temperature gradient between the equator and pole.

As a result, the super-rotation of the atmosphere or the so-called carrousel circulation arises, such that on Venus the wind velocity grows from about 0. There, close to the tropopause, the upper cloud boundary is located. These winds are clearly traceable by the traveling of characteristic ultraviolet contrasts, supposedly being due to absorption of solar radiation by sulfur allotropes Figure 1. Although significant progress was accomplished in the study and the modeling of these specific patterns, the mechanism by which this vigorous circulation is maintained is poorly understood yet.

It is noteworthy that the static stability parameter of the atmosphere here is where is the potential temperature, is the standard pressure, is the adiabatic gradient and that the Richardson gradient number is Therefore, theoretically, generation of turbulence is impossible, because the necessary instability condition for small perturbations to exist in the inviscous shear current is Nevertheless it does occur as it also does in the terrestrial atmosphere with similar values of the Richardson gradient number, and agrees with temperature fluctuations of order 0.

This is verified by measurements of these temperature fluctuations at 45—50 km and above 60 km, derived from radio occultation experiments using spacecraft. Apparently, turbulence is also present in the underlying layers of the troposphere, at least in layers located higher than 15—20 km, where vertical velocity fluctuations of 0. A respective example is shown in Figure 1. Kerzhanovich and Marov, However, as can be seen in Figure 1. All arguments mentioned above, suggest that turbulence is an important element of the atmospheric dynamics on Venus. Indeed, its fraction of the total hydrogen content turned out almost two orders of magnitude larger than the terrestrial fraction.

This could be explained by the isotope separation process, caused by thermal dissipation of hydrogen from the atmosphere. In other words, deuterium could have been accumulated in the atmosphere as the water evaporated from the primary ocean and its molecules were dissociated by solar ultraviolet radiation. According to this mechanism, heavier atmospheric components are carried away jointly with hydrogen because of the high velocity of the hydrogen flux and hence no hydrogen-deuterium fractionation occurs.

One may assume that multicomponent turbulence significantly contributed to the implementation of such a process. Dynamics of the Martian Atmosphere The rarefied atmosphere of Mars possesses very small thermal inertia and its dynamics substantially differs from those of Earth and Venus.

The model of global circulation based on the geostrophic balance condition predicts similar a motion topology in the troposphere and in the stratosphere of Mars with dominance of eastward winds at high latitudes in winter and in the subtropics in summer, and with westward winds at other latitudes. At the same time, because of the very low temperature in the polar regions in the winter hemisphere, the carbon dioxide is partially freezing out off the atmosphere and forms ice deposits. Such seasonal exchange of carbon dioxide between the atmosphere and the polar caps is the main driving mechanism of gas transfer in the meridional direction.

This results in Hadley cells like configurations with ascending and descending flows and restructuring wind system near the surface and at high altitudes in the summer and winter hemispheres Zurek et al. The character of the circulation is strongly influenced by surface relief areography , on which both observable wind patterns and generation of horizontal waves of various spatial scales de- 24 CHAPTER 1 pend.

In turn, planetary waves, conditioned by baroclinic instabilities, and interior gravitational waves manifest themselves as irregularities in temperature and vertical motion profiles in the stratosphere. They also relate to observable wave motions in cloud structures at the lee side during flowing around an obstacle, which provides evidence for the occurrence of strong shear flows in the Martian atmosphere Briggs and Leovy, Apart from the characteristic seasonal mode due to at the polar caps, there are irregular variations at smaller temporal scales related to diurnal changes, local fluctuations of the atmospheric parameters, and wave processes.

The most pronounced variations occur in the records of the wind field, whose basic component is superimposed by effects of relief, surface roughness, and turbulence in the boundary layer. Proceeding from the available experimental data, it is possible to suppose that the Martian boundary layer is generally similar to that of Earth. Its structure is schematically shown in Figure 1. The layers B and C are characterized by the strongest turbulent convection, which the intensity substantially drops of outside the boundary layer i.

The convection compensates for high static instability of the Martian atmosphere, which is close to saturation even for the very low relative content of water vapor. At the same time, the thickness of the nocturnal boundary layer is much greater than that of the daytime layer and, like in the terrestrial atmosphere, the strongest winds night-time jet probably blow here without being decelerated by surface friction in the presence of the inversion layer Andre et al.

Thus, the entire near-surface atmosphere appears turbulent at the expense of shear currents in the night jet, even under conditions of steady stratification. Convection is also responsible for the large dust content constantly suspended in the Martian troposphere by virtue of which its optical depth usually does not happen to be less than This creates an additional dynamic effect superimposed on the global wind system, which is caused by positive feedback between the dust content and the heating of atmospheric gas.

The most dramatic phenomena, however, are periodically observed global dust storms when small-sized dust particles lifts higher than 20—30 km due to turbulent mixing. Owing to the high opacity of the troposphere an anti-greenhouse effect develops and strongly damps circulation transport. Local dust storms tornadoes, or dust devils can arise even more often in some regions of the planet.

Such local storms also are observed during the declining phase of a global dust storm lasting for several months.

Turbulence generated by a gas of electron acoustic solitons

Turbulence of a dispersed medium is intimately related to the processes responsible for triggering, maintenance, and decay of dust storms. The pattern of this kind of turbulence substantially depends on the dynamics and energy interaction of gaseous and dust phases, though details of these mechanisms are not quite clear. Admitting that the same approach is also applicable to the atmospheres of other planets, it is possible to reverse the problem and to estimate the respective profiles for the Martian boundary layer, these estimates being modified for the case of turbulent flows comprising heavy admixture Barenblatt, ; Golitsyn, It was shown that the average velocity profile for neutral stratification reduces to the form: where is the dynamic velocity friction velocity ; is the turbulent momentum flux; is the Karman constant; is the density of the gaseous phase.

The dimensionless parameter is determined by settling the velocity of dust particles , the ratio of coefficients of turbulent exchange for admixture and and the quantity Thus, it follows from 1. General Circulation of the Atmospheres of Giant Planets The structure and dynamics of the atmospheres of giant planets, whose typical representative is Jupiter, are distinguished by a number of specific features which differs them from the terrestrial planets. However, such a composition does not explain the presence of subtle color shades in discrete cloud systems, because simple condensation of gases and vapors in the jovian atmosphere would produce white clouds only.

One should therefore admit the existence of a specific atmospheric chemistry responsible for the formation of compounds with greater complexity, probably involving organic polymers. Their production is assumed to be driven by solar ultraviolet light, lightning discharges, and charged particles raining down from the magnetosphere in the polar regions. A similar cloud structure is assumed to exist in the atmosphere of Saturn, though the appearance of its globe is more subdued than that of Jupiter.

At the same time, on Uranus and Neptune, at lower effective temperatures, high cloud layers involve meth- 28 CHAPTER 1 ane, probably under which layers of condensed ammonia and sulfur-containing compounds are located. Intense absorption of the red part of the solar spectrum by atmospheric methane explains the characteristic aquamarine color of these planets.

The main property of the atmospheric circulation on Jupiter and Saturn is the presence of an ordered system of zones and belts at low and moderate latitudes and of strong jet streams traveling in the intrinsic rotation direction of the planets in the equatorial area Figures 1. The largest temperature gradients also are observed here, while temperature differences between equator and poles are hardly noticeable Figure 1. The higher located light zones are areas of ascending currents while the dark belts represent descending ones.

For these rapidly rotating planets, therefore, owing to the Coriolis interaction of meridional currents, strong zonal flows arise between zones and belts. At the same time, natural convection driven by the interior heat source, together with unordered motions and numerous coherent eddy structures convective cells observable at high latitudes, is the main mechanism of planetary dynamics. Obviously this mechanism is rather similar to the classical Rayleigh-Taylor hydrodynamic instability problem of a horizontal layer of fluid heated up from below, i.

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However, in the case under consideration, the convective interiors are in tight dynamic interaction with the upper gaseous layer where solar energy absorption occurs. Simultaneously, at the expense of declination of these cells due to rotation, a weak second-order mean flow arises consisting of differentially rotating coaxial cylinders, as shown in Figure 1. Such patterns of columns and cylinders, also obtained in experiments with barotropic fluids in rotated axisymmetrical containers, were associated with zones and belts in the atmospheres of Jupiter and Saturn.

It was shown that the occurrence of a secondary flow is caused by the nonlinear transformation of eddy energy into kinetic energy of the mean flow. Basically, the mean flow draws energy from convection cells by tilting them, causing a radial transfer of eastward momentum against the mean momentum gradient where and are the departures from the longitudinal means of eastward and radial velocities u and v , respectively, and r is the radial direction Ingersoll et al.

The idea that eddies are adding kinetic energy to the mean flow is confirmed by the same sign of and Unfortunately, this model yields no complete analogy to the observable currents. Also, there is no answer to the question concerning the layer thickness, within which zonal flows occur. In other words, it is unclear till how deep the temperature gradient remains close to adiabatic and how energy dissipation occurs that depends on the magnitude of the turbulent viscosity and the mean square velocity of vortex motions 30 CHAPTER 1 Ingersoll and Pollard have attempted to solve the problem, proceeding from the assumption that turbulent viscosity values equal thermal conductivity.

The latter was defined as the ratio between the turbulent heat flux and the potential temperature gradient Then the criterion for penetrating zonal currents into the deep of a gas-fluid planet may be where is the Brent-Vaisala fre- quency; g is the acceleration due to gravity; and T is the temperature. This idea agrees well with the theoretical prediction that inclined convective cells must occur in shear currents if the Richardson gradient number, Lipps, Besides, according to direct measurements of the atmospheric Galileo Jupiter Probe, that reached a pressure level of 21 bars, the wind velocity really increased during its descent Ingersoll et al.

On the contrary, in the upper troposphere and stratosphere wind velocities rapidly decrease with altitude. The Galileo probe also experienced small and essentially random accelerations during its descent measured by the Doppler frequency shift technique generally similar to that used in the above mentioned experiments with the Venera probes. These accelerations were caused jointly by turbulence along the descent path and buffeting an aerodynamic effect.

In the barotropic model, vorticity was introduced as a small-scale checkerboard pattern and was removed by eddy viscosity. It was found that an initially small mean flow was term to have the same sign as the meridional avamplified by the tendency of the erage velocity component Figure 1. In turn, in the baroclinic model, eddies arose spontaneously and were supported by instabilities that drew their energy from the meridional temperature gradient.

The pattern of eddy mean flow interactions was oscillatory due to the energy exchange between eddies and the mean flow, though initially it was similar to the pattern in the barotropic model. The positive correlation between and was confirmed on Jupiter by measurements on the Voyager spacecraft.

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However, despite significant distinction in inclination and energetics, both planets feature qualitatively identical meridional temperature and zonal wind profiles at cloud level, although on Uranus winds are approximately twice as weak as on Neptune. It is interesting to note that winds on Neptune and Uranus blow opposite to their rotational direction, which distinguishes these planets from Jupiter, Saturn, and Venus and also from the Sun and Titan , which feature equatorial superrotation, as follows from Figure 1.

When addressing Earth, one may conclude that, in contrast to Neptune, the terrestrial atmosphere has the highest dissipation level. This is mainly caused by processes related to hydrological cycles, as well as by small-scale convection and surface friction. Therefore, though Earth receives much more solar energy, the terrestrial wind velocities are almost an order of magnitude smaller than those on Neptune.

The comparison between the measured zonal wind velocity profile u on Neptune with the results of calculations in the shallow water approximation Allison and Lumetta, ; Ingersoll et al. A thin steady fluid layer of variable thickness h on a deep adiabatic sublayer was considered. The vertical was determined as a function of unrelative vorticity component der condition of potential vorticity preservation of the main current, where see condition 1.

The so-called deformation radius which describes eddy effects on rapidly rotating planets, was used is the ratio between the layer densities; all other desigas a free parameter. Here nations were given earlier. The best agreement with measurements was obtained under keeps constant not in the entire hemisphere but only within limthe condition that ited intervals, The parameter or in other words, the horizontal perturbation dimension whose kinetic energy, is comparable to the potential energy plays a key role in obtaining results from the modeling.

Obviously, when perturbations are smaller than and vise versa. Unfortunately, the results available do not allow to put specific constraints on this parameter. Particularly on giant planets having an internal heat source, numerous large-scale eddy formations stand out on the various cloud structure patterns see Figure 1. The GRS represents a giant anticyclone type vortex with an estimated lifetime of several thousand years.

It is located above the surrounding cloud layer due to ascending motions from beneath and the very complicated morphology of internal eddy currents. As yet, there is no satisfying explanation for the formation and maintenance of such steady structures in the atmospheres of Jupiter, Saturn, and Neptune on the background of chaotic small-scale activities in the form of relatively small clouds appearing and disappearing in a few hours. In addition, areas of descending motions where temperatures are higher than in surrounding clouds the so called 5micron hot spots, see Figure 1.

These areas are associated with some local changes in the chemical composition of the atmosphere. The observed longitudinal-latitudinal oscillations of spots, including GRS and GDS, resemble motion patterns of the upper parts of eddies in a steadily stratified shear flow. Like ordered zonal currents, it seems natural to treat them from the point of view of hydrological cycle formation in a stratified gas-fluid medium, taking into account its chemical composition, energetics, and its matching to the stability criterion considering the diverse relationships between the internal and solar energy sources.

A significant fraction of it is a characteristic example of a multicomponent turbulent medium.

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It is subjected to direct absorption of incident solar radiation and numerous chemical transformations together with the processes of heat and mass transport. Intense solar electromagnetic radiation is principally responsible for various photochemical processes such as photoionization, photodissociation, and excitation of inner degrees of freedom of atoms and molecules.

These processes are accompanied by inverse association reactions of atoms into molecules, ion recombinations, spontaneous photon emissions, and collisional deactivations. Properties of gas under influence of gravitational and electromagnetic fields are strongly dependent on the efficiency of molecular and eddy diffusion and heat transport at different height levels.

Due to temperature, concentration, and pressure gradients, the multi-scale turbulent hydrodynamic motions are developed from 36 CHAPTER 1 below up to the lower thermosphere. In addition, solar corpuscular radiation and some auxiliary energy sources such as tidal oscillations, viscous energy dissipation of internal gravity waves, etc. It is just in the upper part of the middle atmosphere and the overlying thermosphere defined as a region with a positive temperature gradient , where the main energy exchanges occur.

These exchanges are caused by direct absorption of solar extreme ultraviolet radiation EUV and soft X-rays approximately from and additionally by interactions with energetic particles originating from solar plasma after their acceleration in the terrestrial magnetosphere. Longer wavelength UV radiation in particular responsible for the formation of the ozonosphere, and solar particles with higher energies protons up to 15—30 MeV generated in solar flares, are absorbed in the lower regions of the middle atmosphere for details see Akasofu and Chapman, ; Marov and Kolesnichenko, The middle atmosphere itself hosts a complex interplay between radiative processes, chemistry, wave dissipation and turbulence, involving non-linear dynamics and electrodynamics.

The latter is mainly promoted by magnetospheric-ionospheric interaction processes which, in turn, exert the most pronounced effect on the distribution of thermohydrodynamical parameters of the high latitude thermosphere and ionosphere, due to energy dissipation of particles precipitated from the magnetosphere and excited current systems. These processes results in powerful heating of the medium accompanied by the growth of large local mass density gradients and dynamic transport. So, this level serves as a boundary between the homosphere and the heterosphere. It is also called the turbopause or homopause and characterizes the height at which turbulent mixing becomes inefficient and molecular diffusion begins to dominate, and the height distribution of atmospheric species is controlled by their particular scale heights.

However, despite a constant M in the middle atmosphere, its actual composition is subject to large variations caused by minor admixtures. This is due to the extreme complexity of chemical and dynamic processes occurring in the stratosphere and, of minor importance, in the mesosphere and the lower thermosphere. Nitrogen the main atmosphere component at the Earth's surface, dominates up to about km.

Above that level, in the thermosphere, atomic oxygen O , originating from dissociated starts to dominate, eventually being replaced by helium He and hydrogen at even higher levels Figure 1. Depending on the thermospheric temperature, mainly conditioned by the solar activity during the year 38 CHAPTER 1 cycle and the local time, the concentrations of He and O appear to be approximately equal to each other in the altitude range from to km, while the He and abundances are nearly the same between and km.

While and O concentrations grow with temperature, the content of on the contrary, drops because of its increasing dissociation rate. As far as He is concerned, its variations are even more complicated, because they exhibit a strong latitudinal dependence, related to the season and solar cycle phase. OH, In turn, hydrogen-, carbon-, and nitrogen-bearing compounds such as and play a noticeable role in the mechanisms of chemical transformations and radiative heat exchange.

The same is true when addressing some of their more complex derivatives including those of anthropogenic origin , and a number of other minor components, including metastable ones Figure 1. Various dynamic processes, wave motions included, are important features of the middle atmosphere and thermosphere. Dynamics, related to general circulation, causes redistribution of matter and energy at global scales. By means of mass, momentum, and heat exchange, this determines many aspects of the energy balance, implying an intimate coupling between the processes in near-planetary space.

Dynamic variations of the pressure field, first of all atmospheric tides, planetary waves, and IGWs, play an important role in this balance being observed as spatial-temporal variations of structural parameters in various areas. Wave-wave and wave-mean flow interactions affect minor species distributions and emission variations. Emergence of the above mentioned thermal tides is characteristic of Mars, while tidal effects in the dense Venus atmosphere probably facilitate its capture into resonance rotation with Earth see, e.

IGW dissipation is an important energy source in the upper atmosphere. Though there are many diverse sources of waves with various phase velocities, the presence of stationary irregularities at the base of an atmosphere creates a peak in the wave spectrum at zero horizontal velocity.

As wave absorption results in acceleration of a medium in the direction of the wave propagation, the total effect commonly displays itself as a braking of the atmosphere Fels and Lindzen, ; Lindzen, ; Holton, ; Andrews et al. Diverse perturbations serve as a source of IGWs. They are generated due to restructuring of meteorological processes, flow of air streams around ridges, wind shear instabilities, heating of auroral zones, etc.

In a stratified medium like an atmosphere, such waves usually propagate both in vertical and horizontal directions, though the horizontal component in an initially vertical perturbation may become predominant with increasing altitude. The heat release through energy dissipation of IGWs in the lower thermosphere turns out to be comparable to other energy sources related to the incident solar radiation at these altitudes Figure 1.

Turbulence, whose time and space morphology is not yet fully clarified, is an important factor in atmospheric dynamics and energetics, particularly in the lower thermosphere. Therefore, when analyzing thermohydrodynamic processes in the middle atmosphere, it is often necessary to simultaneously consider equations describing averaged concentration, temperature, and wind velocity fields, together with turbulent motion intensity features such as the eddy diffusion, The latter sometimes serves as a free parameter of the problem.

At the same altitudes, a fluctuation wind component, originating from wave motions, also occurs. Turbulent Diffusion in the Atmosphere of Terrestrial Planets The turbulent processes in the upper atmosphere of a planet are responsible for highaltitude redistribution of components due to eddy diffusion , variability in rate of chemical reactions due to turbulent mixing , and turbulent energy exchange heating through viscous dissipation of turbulent energy and cooling through turbulent thermal conductivity. Demokritov, Andrei N. Avram, Mikhail G. Nurushev, Mikhail F. Runtso, Mikhail N. Skulachev, Alexander V.

Bogachev, Felix O. Rubio, Yuri S. Nepomnyashchy, Alberto Ferrus, Eugene G. Giri, D. Goswami, A. Buildings Kartik N. Shinde, S. Dhoble, H. Binder, K. Dobrinets, Victor. Vins, Alexander M. Romero, Gabriela S. Korol, Andrey V. Solov'yov View Handbook of Polymer Nanocomposites. Kar, Jitendra K. Henn, Jan C. Kragh, James M. Hramov, Alexey A. Koronovskii, Valeri A.

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Mechanics of Turbulence of Multicomponent Gases (Astrophysics and Space Science Library) Mechanics of Turbulence of Multicomponent Gases (Astrophysics and Space Science Library)
Mechanics of Turbulence of Multicomponent Gases (Astrophysics and Space Science Library) Mechanics of Turbulence of Multicomponent Gases (Astrophysics and Space Science Library)
Mechanics of Turbulence of Multicomponent Gases (Astrophysics and Space Science Library) Mechanics of Turbulence of Multicomponent Gases (Astrophysics and Space Science Library)
Mechanics of Turbulence of Multicomponent Gases (Astrophysics and Space Science Library) Mechanics of Turbulence of Multicomponent Gases (Astrophysics and Space Science Library)
Mechanics of Turbulence of Multicomponent Gases (Astrophysics and Space Science Library) Mechanics of Turbulence of Multicomponent Gases (Astrophysics and Space Science Library)
Mechanics of Turbulence of Multicomponent Gases (Astrophysics and Space Science Library) Mechanics of Turbulence of Multicomponent Gases (Astrophysics and Space Science Library)
Mechanics of Turbulence of Multicomponent Gases (Astrophysics and Space Science Library) Mechanics of Turbulence of Multicomponent Gases (Astrophysics and Space Science Library)
Mechanics of Turbulence of Multicomponent Gases (Astrophysics and Space Science Library) Mechanics of Turbulence of Multicomponent Gases (Astrophysics and Space Science Library)

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