Phases of the Isobaric Surface Shapes in the Geostrophic State of the Atmosphere and Connection to the Polar Vortices

This paper presents a theoretical study of the disturbed isobaric surface shape in the geostrophic state of the atmosphere. It has been shown that, depending on the overheat sign at the equator, the isobaric surface has the shape of an oblate or prolate geoid. If the geostrophic wind velocity is nonzero at the poles, the local pressure extrema (minima for oblate geoid and maxima for prolate geoid) appear at the poles in the geostrophic state. This result correlates with the well-known polar vortex phenomenon and possibly can refine our understanding and interpretation of the phenomenon. In other words, the existence of polar minima and maxima of the pressure field can be the peculiarity of the geostrophic state of the atmosphere. It has been found that air must be colder than the surrounding atmosphere for initiation of the zonal eastward transport. For warm air mass, only easterly winds will be observed

Phases of the Isobaric Surface Shapes in the Geostrophic State of the Atmosphere and Connection to the Polar Vortices

This paper presents a theoretical study of the disturbed isobaric surface shape in the geostrophic state of the atmosphere. It has been shown that, depending on the overheat sign at the equator, the isobaric surface has the shape of an oblate or prolate geoid. If the geostrophic wind velocity is nonzero at the poles, the local pressure extrema (minima for oblate geoid and maxima for prolate geoid) appear at the poles in the geostrophic state. This result correlates with the well-known polar vortex phenomenon and possibly can refine our understanding and interpretation of the phenomenon. In other words, the existence of polar minima and maxima of the pressure field can be the peculiarity of the geostrophic state of the atmosphere. It has been found that air must be colder than the surrounding atmosphere for initiation of the zonal eastward transport. For warm air mass, only easterly winds will be observed

Vortex Motion State of the Dry Atmosphere with Nonzero Velocity Divergence

In the present work an analytical model of the vortex motion elementary state of the dry atmosphere with nonzero air velocity divergence is constructed. It is shown that the air parcel moves along the open curve trajectory of spiral geometry. It is found that for the case of nonzero velocity divergence the atmospheric elementary state presents an unlimited sequence of vortex cells transiting from one to another. On the other hand, at zero divergence, the elementary state presents a pair of connected vortices, and the trajectory is a closed curve. If in some cells the air parcel moves upward then in the adjacent cells it will move downward and vice versa. At reaching the cell middle height the parcel reverses the direction of rotation. When parcel moves upward, the motion is of anticyclonic type in the lower part of the vortex cell and of the cyclonic type in the upper part. When parcel moves downward, the motion is of anticyclonic type in the upper part of the vortex cell and of the cyclonic type in the lower part

Convection of Moist Saturated Air: Analytical Study

In the present work, the steady-state stationary thermal convection of moist saturated air in a lower atmosphere has been studied theoretically. Thermal convection was considered without accounting for the Coriolis force, and with only the vertical temperature gradient. The analytical solution of geophysical fluid dynamics equations, which generalizes the formulation of the moist convection problem, is obtained in the two-dimensional case. The stream function is derived in the Boussinesq approximation with velocity divergence taken as zero. It has been shown that the stream function is asymmetrical in vertical direction contrary to the dry and moist unsaturated air convection. It has been demonstrated that the convection in moist atmosphere strongly depends on the vapor mass fraction gradient

Bäcklund Transformations for Liouville Equations with Exponential Nonlinearity

This work aims to obtain new transformations and auto-Bäcklund transformations for generalized Liouville equations with exponential nonlinearity having a factor depending on the first derivatives. This paper discusses the construction of Bäcklund transformations for nonlinear partial second-order derivatives of the soliton type with logarithmic nonlinearity and hyperbolic linear parts. The construction of transformations is based on the method proposed by Clairin for second-order equations of the Monge–Ampere type. For the equations studied in the article, using the Bäcklund transformations, new equations are found, which make it possible to find solutions to the original nonlinear equations and reveal the internal connections between various integrable equations

Bäcklund Transformations for Nonlinear Differential Equations and Systems

In this work, new Bäcklund transformations (BTs) for generalized Liouville equations were obtained. Special cases of Liouville equations with exponential nonlinearity that have a multiplier that depends on the independent variables and first-order derivatives from the function were considered. Two- and three-dimensional cases were considered. The BTs construction is based on the method proposed by Clairin. The solutions of the considered equations have been found using the BTs, with a unified algorithm. In addition, the work develops the Clairin’s method for the system of two third-order equations related to the integrable perturbation and complexification of the Korteweg-de Vries (KdV) equation. Among the constructed BTs an analog of the Miura transformations was found. The Miura transformations transfer the initial system to that of perturbed modified KdV (mKdV) equations. It could be shown on this way that, considering the system as a link between the real and imaginary parts of a complex function, it is possible to go to the complexified KdV (cKdV) and here the analog of the Miura transformations transforms it into the complexification of the mKdV

Influence of Atmosphere Near-Surface Layer Properties on Development of Cloud Convection

A two-dimensional mathematical model of moist air convection in the sub-cloud and cloud layers is proposed. A theoretical analysis of the influence of near ground atmospheric parameters on the development of sub-cloud and cloud convection is provided, and the criteria of convection development are considered. As a rule, this relationship is parameterized in general circulation, regional or mesoscale models of the atmosphere. Therefore, achieving a more complete and correct understanding of this relationship can lead to an improvement in the accuracy of weather forecasts. The mathematical model describes the system of the equations of motion, heat conductivity and the continuity equations for a two-dimensional vertical plane. The approximate analytical solution of the system of equations is obtained. Expressions for the estimation of the convection height and height of maximum velocity are derived for vertical and horizontal components of updraft wind and for vertical distribution of temperature. From the expressions obtained, the criterion of sub-cloud convection development is derived. The expressions for the convection parameters at a condensation level have also been formulated, from which the criterion of cloud development through convection is derived. It is established that the development of cloud convection depends on absolute values of the dew point deficit in a near-surface layer and, in a greater degree, on vertical gradients of water vapor mass fraction. It is shown that at certain critical values of a vertical gradient of water vapor mass fraction “explosive convective growth” is observed. The application of the obtained results to artificial stimulation of convection by means of air heating in the near-ground atmosphere has shown that the success of such an application and the required air heating-up depend on: (i) the vertical temperature gradient; (ii) the vertical dew-point gradient; and (iii) the value of the dew point deficit in the near-ground layer. The analysis performed has shown the possibility of successful stimulation of artificial convection under specific favorable atmospheric conditions