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A mathematical model, developed earlier in the Polar Geophysical Institute, is applied to investigate the transformation of global gas flows in the Earth’s atmosphere over the course of a year. The model is based on the numerical solution of the system of gas dynamic equations. The mathematical model produces three-dimensional distributions of the gas dynamic parameters of the atmosphere in the height range from 0 to 126 km over the Earth’s surface. To investigate the seasonal transformation of the global circulation of the lower and middle atmosphere, simulations are performed for conditions corresponding to twelve dates, which belong to twelve different months. Results of simulations indicate that the variations of the solar illumination of the Earth’s atmosphere, conditioned by different positions of the Earth along its trajectory around the Sun, influence considerably the transformation of the planetary circulation of the lower and middle atmosphere over the course of a year.

Fluid dynamics may be successfully applied for solving problems of environmental sciences. To investigate gas flows in the Earth’s atmosphere with the help of the fluid dynamics, not only the theoretical and experimental but also computational studies may be applied. Several general circulation models of the lower and middle atmosphere have been developed during the last three decades (for example, see [

Not long ago, a non-hydrostatic mathematical model of the global neutral wind system of the Earth’s atmosphere has been developed in the Polar Geophysical Institute (PGI) [

This non-hydrostatic mathematical model has been utilized in order to investigate numerically how the horizontal non-uniformity of the atmospheric gas temperature affects the formation of the middle atmosphere global circulation [

The purpose of the present work is to continue these studies with the help of the non-hydrostatic model of the global neutral wind system, developed earlier in the Polar Geophysical Institute, and to investigate numerically the seasonal variations of the global circulation of the Earth’s lower and middle atmosphere which take place over the course of a year.

The utilized model differs from the bulk of existing global circulation models of the atmosphere, on principle. Firstly, the model does not include the pressure coordinate equations of atmospheric dynamic meteorology, in particular, the hydrostatic equation. Instead, the vertical component of the neutral wind velocity is obtained by means of a numerical solution of the appropriate momentum equation, with whatever simplifications of this equation being absent. Thus, three components of the neutral wind velocity are obtained by means of a numerical solution of the generalized Navier-Stokes equation.

Secondly, the model does not include the internal energy equation for the neutral gas. Instead, the global temperature field is assumed to be a given distribution. This peculiarity proceeds from complexity and uncertainty in various chemical-radiational heating and cooling rates, resulting in a discrepancy between the simulated and observed distributions of the atmospheric temperature. On the other hand, over the last years empirical models of the global atmospheric temperature field have been successfully developed. In the present study, we take the global temperature distribution from the NRLMSISE-00 empirical model [

The utilized model produces three-dimensional global distributions of the zonal, meridional, and vertical components of the neutral wind velocity and neutral gas density at the levels of the troposphere, stratosphere, mesosphere, and lower thermosphere. The model has the potential to describe the global neutral wind system under disturbed conditions when the vertical component of the neutral wind velocity at the levels of the lower thermosphere can be as large as several tens of meters per second [

The mathematical model, utilized in the present study, is based on the numerical solution of the system of equations containing the dynamical equation and continuity equation for the neutral gas. For solving the system of equations, the finite-difference method is applied. The dynamical equation for the neutral gas in vectorial form can be written as

where

where

where

In the mathematical model, utilized in the present study, we use a spherical coordinate system

where _{,}

The former tensor,

where

where

The utilized mathematical model contains the continuity equation of the neutral gas which can be written as

Besides, the state equation for an ideal gas is used in the utilized mathematical model for obtaining the con- nection between the pressure and temperature of the neutral gas.

The system of equations containing the momentum equations for the vertical, meridional, and zonal components of the neutral gas velocity, Equations (4)-(6), as well as the continuity equation, Equation (10), is nu- merically solved in a layer surrounding the Earth globally. The lower boundary of this layer is the Earth’s surface, which is assumed to be an oblate spheroid whose radius at the equator is more than that at the pole. The upper boundary of this layer is the sphere lain at the altitude of 126 km at the equator.

At the lower boundary, the velocity vector is determined from the no-slip conditions on the ground. Simulation results to be presented in this paper are obtained using the following condition at the upper boundary:

The utilized mathematical model allows us to calculate three-dimensional global distributions of the vertical, meridional, and zonal components of the neutral wind and neutral gas density. As pointed out previously, the finite-difference method is applied for solving the system of equations. Complete details of the utilized finite-difference method and numerical schemes have been presented in the paper of Mingalev et al. [

The utilized mathematical model of the global neutral wind system can be used for different solar illumination and geomagnetic conditions. To investigate the seasonal transformation of the global circulation of the lower and middle atmosphere, simulations are performed for conditions corresponding to twelve dates, which belong to twelve different months. These dates are 16 January, 16 February, 16 March, 16 April, 16 May, 16 June, 16 July, 16 August, 16 September, 16 October, 16 November, and 16 December. For each enumerated day, calculations were performed for identical solar cycle and geomagnetic conditions, namely, for moderate solar activity (F_{10.7} = 101) and low geomagnetic activity (Kp = 1). The variations of the atmospheric parameters with time were calculated until they become stationary. The steady-state distributions of the atmospheric parameters were obtained for twelve considered dates on condition that inputs to the model are time-independent and correspondent to the identical moment (10.30 UT) for each day. The temperature distributions, corresponding to this moment, were taken for each day from the NRLMSISE-00 empirical model [

Firstly, we made model calculations for the period from winter to summer in the northern hemisphere (from January to June). Secondly, the transformation of the global circulation of the lower and middle atmosphere was calculated for the period from summer to winter in the northern hemisphere (from July to December). Since the calculated transformations turned out different, it is convenient to present them separately.

In the present subsection, the numerical model is applied to simulate the global distributions of the atmospheric parameters in the lower and middle atmosphere for conditions corresponding to six dates, which belong to six different months beginning from January. The steady-state distributions of the atmospheric parameters were calculated for six considered dates on condition that inputs to the model correspond to 10.30 UT for each day. The obtained simulation results are shown in Figures 1-8. The given temperature distributions, obtained for the altitude of 50 km , are shown on the top panels of Figures 1-6. It is seen that the planetary distributions of the atmospheric temperature are non-uniform and distinct in different months. Their distinctions are conditioned by different conditions of solar illumination of the Earth’s atmosphere. It is seen that the distinctions between temperatures, obtained for different considered months, can achieve a few tens of degrees at identical points of the globe.

Let us consider global distributions of the horizontal and vertical wind for conditions corresponding to six dates, which belong to six different months (Figures 1-6). Three-dimensional global distributions of the horizontal and vertical components of the neutral wind velocity, calculated with the help of the mathematical model, illustrate both common characteristic features and distinctions caused by various conditions of solar illumination.

The calculated global atmospheric circulations possess the following common features. The horizontal and vertical components of the wind velocity are changeable functions not only of latitude and longitude but also of altitude. Maximal absolute values of the horizontal and vertical components of the wind velocity are larger at higher altitudes. Horizontal domains exist where the steep gradients in the horizontal velocity field take place. The horizontal wind velocity can have various directions which may be opposite at the near points, displaced for a distance of a few steps of finite-difference approximation. Moreover, horizontal domains exist in which the vertical neutral wind component has opposite directions. As a rule, the horizontal domains, where the vertical neutral wind component is upward, are significantly extended in both latitude and longitude. On the contrary, horizontal domains having a downward vertical neutral wind component are extended in longitude but narrow in latitude. Usually, the latter domains have a configuration like a long narrow band and coincide, as a rule, with the regions, where the steep gradients in the horizontal velocity field take place. Maximal absolute values of the upward vertical wind component are less than the maximal module of the downward vertical wind component. The vertical wind velocity can achieve values of a few m/s at levels of the lower thermosphere in the horizontal domains having a configuration like a limited narrow band. At levels of the middle atmosphere, the horizontal wind velocity can achieve values of more than 100 m/s.

Let us consider simulation results, obtained for different months, and their distinctions. From

It is know that the global atmospheric circulation can contain sometimes so called circumpolar vortices that are the largest scale inhomogeneities in the global neutral wind system. Their extent can be very large, sometimes reaching the latitudes close to the equator. It is well known from numerous observations that circumpolar vortices are formed at heights of the stratosphere and mesosphere in the periods close to summer and winter solstices, when there is no rebuilding of the atmosphere. The circumpolar anticyclone arises in the northern hemisphere under summer conditions, while the circumpolar cyclone arises in the southern hemisphere under winter conditions. On the contrary, the circumpolar cyclone arises in the northern hemisphere under winter conditions, while the circumpolar anticyclone arises in the southern hemisphere under summer conditions. Let us compare these experimental data with the simulation results, obtained for January and June conditions.

From

From

Incidentally, the qualitative agreement between January and June circulations of the middle atmosphere, obtained numerically and experimentally, manifests the adequacy of the distributions of atmospheric temperature, taken from the NRLMSISE-00 empirical model [

From

In the present subsection, the numerical model is applied to simulate the global distributions of the atmospheric parameters in the lower and middle atmosphere for the period from summer to winter in the northern hemisphere (from July to December). Calculations were made for conditions corresponding to six dates, which belong to six different months beginning from July. In Figures 9-16, simulation results, obtained for six different months on condition that the inputs to the model and boundary conditions are time-independent and correspondent to 10.30 UT for each day, are shown. Three-dimensional global distributions of the horizontal and vertical components of the neutral wind velocity, calculated with the help of the mathematical model, illustrate both common characteristic features and distinctions caused by various conditions of solar illumination.

In essence, the common characteristic features are the same as in the previous subsection, describing the results for the period from January to June. In particular, the distinctions between given temperatures, obtained for different considered months, can achieve a few tens of degrees at identical points of the globe. It is seen that horizontal non-uniformity of the atmospheric temperature, which is distinct in different months, influences considerably on the transformation of global circulation of the lower and middle atmosphere during the period from July to December.

Let us consider simulation results, obtained for different months, and their distinctions. From simulation results, we can see that the global distributions of the neutral wind, calculated for July conditions, in particular, the large-scale circumpolar vortices, are qualitatively similar to those, calculated for June conditions and discussed in the previous subsection. It can be seen that the global distributions of the neutral wind, calculated for December conditions, in particular, the large-scale circumpolar vortices, are qualitatively similar to those, calculated for January conditions and discussed in the previous subsection. It may be recalled that these large-scale circumpolar vortices correspond qualitatively to those, obtained from observations.

Let us consider the process of transformation of the global circulation of the lower and middle atmosphere during the period from July to December. The global atmospheric circulation, computed for August conditions, is similar to that, calculated for July conditions, with maximal absolute values of the horizontal component of the wind velocity being less in August than in July.

In the northern hemisphere, in September, at the levels of stratosphere and mesosphere, the module of the neutral gas velocity is reduced but the direction of the flow remains just the same (circumpolar anticyclone). Unlike, at the levels close to the stratopause (approximately 50 km ), the direction of the flow becomes contrary

(circumpolar cyclone). In October, in the northern hemisphere, at the levels close to the stratopause, the circumpolar cyclone increases; at the levels of stratosphere and mesosphere the direction of the flow becomes opposite, with the circumpolar cyclone arising. Thus, during September, in the northern hemisphere, at the levels of stratosphere and mesosphere, circumpolar vortices of the atmosphere changes significantly, with the direction of the flow becoming opposite.

In the southern hemisphere, the period of rebuilding of circumpolar vortices begins in September and lasts during October, with the process starting in the mesosphere and continuing in the stratosphere.

In November, in both hemispheres, the circumpolar vortices increase and, to December, they become analogous to those observed usually in the period close to the winter solstice.

Thus, during August and September, at the levels of stratosphere and mesosphere, circumpolar vortices of the atmosphere transform significantly, with the direction of the flow becoming opposite. This transformation is illustrated in

During the period from July to December, at the levels of 60 km , maximal absolute value of the horizontal component of the wind velocity in both hemispheres ought to be biggest in July, and that ought to be least in August.

To investigate the transformations of the global circulation of the lower and middle atmosphere, conditioned by changing solar illumination of the Earth’s atmosphere over the course of a year, the mathematical non-hydrostatic model of the global neutral wind system of the atmosphere, developed earlier in the Polar Geophysical Institute, was utilized. One of the peculiarities of the utilized model consists in using the global temperature field as a given distribution, i.e., the input parameter of the numerical model. The global temperature field is taken from the NRLMSISE-00 empirical model [

By means of the mathematical model, the steady-state distributions were obtained of the atmospheric parameters corresponding to 16 January, 16 February, 16 March, 16 April, 16 May, 16 June, 16 July, 16 August, 16 September, 16 October, 16 November, and 16 December (all at 10.30 UT). For each enumerated day, calculations were performed for identical solar cycle and geomagnetic conditions, namely, for moderate solar activity and low geomagnetic activity.

Simulation results indicate that, for all months, the horizontal and vertical components of the wind velocity are changeable functions not only of latitude and longitude but also of altitude. Horizontal domains exist where the steep gradients in the horizontal velocity field take place. Moreover, horizontal domains exist in which the vertical neutral wind component has opposite directions. As a rule, the horizontal domains, where the vertical neutral wind component is upward, are significantly extended in both latitude and longitude. On the contrary, horizontal domains having a downward vertical neutral wind component are extended in longitude but narrow in latitude.

The results of simulation indicated that the process of transformation of the global circulation of the lower and middle atmosphere during the period from January to June possesses the following peculiarities. In the northern hemisphere, the circulation, characteristic for period close to winter solstice, ought to exist up to April. During April this circulation is weakened and a circulation, characteristic for period close to summer solstice, arises. From May to June, the arisen circulations ought to increase. An analogous transformation takes place in the southern hemisphere during the past half of March and April. The process of the transformation of the global atmospheric circulation arises in the mesosphere and then propagates downward.

The results of simulation indicated that the process of transformation of the global circulation of the lower and middle atmosphere during the period from July to December possesses the following peculiarities. In the northern hemisphere, the circulation, characteristic for period close to summer solstice, ought to exist up to September. During September this circulation is weakened and a circulation, characteristic for period close to winter solstice, arises. This transformation ought to begin from the level of the stratopause and then to propagate upward and downward. An analogous transformation takes place in the southern hemisphere during the past half of September and October. This transformation ought to begin from the level of the mesosphere and then to propagate downward. From November to December, the arisen circulations ought to increase.

To understand the cause of the significant transformation of global gas flows in the Earth’s atmosphere over the course of a year, obtained numerically in the present study, it is useful to take into account the following circumstances. It is obvious that the horizontal non-uniformity of the neutral gas temperature, which is distinct in different months, influences considerably on the transformation of global circulation of the lower and middle atmosphere. It is clear that the global temperature field depends on changing solar illumination of the Earth’s atmosphere over the course of a year. It is well known that the Earth has an elliptical orbit, with the rotational axis of the Earth having a tilt of about 67˚ to the orbit plane (

When the northern pole is tilted toward the Sun the day lasts longer. This results in warmer average temperatures from the increase in solar radiation reaching the surface. When the northern pole is tilted away from the Sun, the reverse is true and the climate is generally cooler. Above the Arctic Circle, an extreme case is reached where there is no daylight at all for part of the year. This effect is called a polar night. The variations in solar radiation reaching the surface (because of the direction of the Earth’s axial tilt) results in the seasons.

The ellipticity of the Earth’s orbit leads to changes of the Sun-Earth distance over the course of a year. The changing Sun-Earth distance results in an increase of about 7% in solar energy reaching the Earth at perihelion relative to aphelion. Earth’s perihelion occurs around 3 January, and the aphelion around 4 July. Since the southern hemisphere is tilted toward the Sun at about the same time that the Earth reaches the closest approach to the Sun, the southern hemisphere receives slightly more energy from the Sun than does the northern over the course of a year.

Thus, different positions of the Earth along its trajectory around the Sun, in particular the changing Sun-Earth distance, ought to affect the formation of the global atmospheric temperature field in the lower and middle atmosphere. It may be recalled that, in the utilized mathematical model, we take the global temperature distribution from the NRLMSISE-00 empirical model [

the date and universal time (UT) are the input parameters of the latter empirical model.

As pointed out previously, the circumpolar vortices of the northern and southern hemispheres, obtained using the applied mathematical model at levels of the lower and middle atmosphere, are consistent with existing observational data, in particular, for winter and summer periods. Therefore, this fact manifests the adequacy of the distributions of the atmospheric temperature, calculated with the help of the NRLMSISE-00 empirical model [

It can be noted that modern scientific facility does not allow somebody to measure global three-dimensional fields of gas dynamical parameters, in particular global neutral wind system, of the lower and middle atmosphere at the same time, unfortunately. On the other hand, scientific researches, prediction applications and other environmental activities are in need of the planetary neutral wind systems for various geophysical conditions. The global neutral wind systems, obtained with the help of computational studies, in particular the simulation results, obtained in the present study for twelve different months, may be useful for these activities.

This work was partly supported by Grant No. 13-01-00063 from the Russian Foundation for Basic Research.