Theoretical studies of atmospheric perturbations caused by the gravity field inhomogeneities

In solving atmospheric dynamics problems, the influence of the gravity field inhomogeneities (GFI) is neglected. The traditional basis for this is that the amplitude of gravity variations even in highly anomalous regions do not exceed 10−3 ms−2 in order of magnitude, i.e. 4 orders of magnitude less than average gravity. But in the presence of gravity anomalies, inhomogeneous forces of the same order act in the directions tangent to the mean Earth ellipsoid. The atmosphere dynamics is very sensitive to the forces of such directions. In highly anomalous regions, they are comparable with the main horizontal forces acting in the atmosphere, in particular with the forces of the pressure gradient and Coriolis forces. Therefore, it seems appropriate to analyze the possible effect of spatial variations of gravity on the dynamics of the atmosphere, primarily for mesoscale atmospheric disturbances. In a number of the authors’ previous works, theoretical studies of the possible effect of GFI on the atmosphere dynamics are carried out. This report contains some new relevant findings. A three-dimensional analytical model of geostrophic wind disturbances under the influence of GFI is developed. An analogy of atmospheric disturbances caused by thermal inhomogeneities of the underlying surface and GFI is shown. Attention is drawn to the possibility of “accumulation” of atmospheric effects associated with the gravitational field inhomogeneities.


Introduction
In a number of the authors' previous works, theoretical studies of the possible effect of the gravity field inhomogeneities (GFI) on the atmosphere dynamics are carried out [1][2][3]. The present report contains some new relevant findings. A three-dimensional analytical model of geostrophic wind disturbances under the influence of GFI is developed. An analogy of atmospheric disturbances caused by thermal inhomogeneities of the underlying surface and GFI is shown. Attention is drawn to the possibility of "accumulation" of atmospheric effects associated with the gravitational field inhomogeneities.

Three-dimensional analytical model of geostrophic wind disturbances influenced by the gravity field inhomogeneities
The results of two-dimensional model of the authors [1] are generalized to the case when the GFI depends on all three coordinates. In the absence of GFI, a uniform geostrophic flow is specified along one of the horizontal axes x : (1) Here y is the second horizontal coordinate (in direction transverse to the flow), f is the Coriolis parameter ( f -plane approximation is used), p is the pressure. It is assumed that the background density and pressure distributions (indicated by a bar) depend not only on the height z , but also on one of the horizontal coordinates y . For example, for analysis it is convenient to use model const, , exp , exp where axis z is directed vertically upwards; the meaning of constants This specifying the background state in the simplest cases allows to reduce the problem to a system of equations with constant coefficients.
Disturbances introduced in this flow by gravity field inhomogeneities are studied in the linear approximation. The horizontal and vertical components of these "additional" accelerations are described respectively by the values The total gravity is the vector sum of these perturbations and the average gravity, that below is indicated by a constant g , as usual. If the potential of gravity is denoted by Φ , then the relations are , , (3) The linearized system of equations for the stationary disturbances of velocity, pressure and density in the ideal incompressible medium [1,3] for a three-dimensional problem has a form: are the disturbances of the horizontal and vertical components of velocity y and z respectively; the disturbances of other quantities are indicated by a prime. There is given a non-permeability condition at the lower boundary of the medium. At a solid horizontal surface it has a form This formulation of the problem generalizes the two-dimensional problem considered in the works of the authors [1,3].
By eliminating part of the unknowns from set (4), it is not difficult to derive a system of two equations with constant coefficients Therefore, the above-mentioned term can be confidently neglected and the second equation (5) can be written in the form  is close to the previously considered two-dimensional problem, and the reverse inequality corresponds to an even less interesting case of flow directed along inhomogeneities). Let us introduce the dimensionless variables , where L -is the characteristic spatial scale of gravity field inhomogeneities. The equation (7) can be written as . , The designations for dimensionless parameters are introduced here: the inverse Burger number [5] We consider the situations when spatial scales of disturbances L are larger or about 100 km. With such characteristic spatial scales and for the above-mentioned values of the remaining parameters, all dimensionless coefficients in the equation (8) are small. By neglecting the terms with small coefficients, we obtain: Its meaning is completely clear: it describes flowing of surfaces with equal gravitational potential. But this solution does not meet the boundary condition . The latter is also understandable: if the underlying surface does not coincide with any equipotential surface, then the non-permeability condition on the underlying surface is obviously incompatible with the ideal flow around the curved isosurfaces of the potential. Therefore, a relatively thin boundary layer [1] is formed near the underlying surface; in which the solution (9) is substantially violated. In deriving (9), the terms with small coefficients for the highest derivatives by z in the dimensionless equations were not taken into account. For a correct description of the boundary layer, it is necessary to take them into account.
Let us dwell on the important limiting case for the small Rossby numbers: (in addition to the above assumptions, in particular, ). In this case, the trapped disturbances are generated, and a scale analysis results in a significant simplification of the system (5): . v Let us estimate the ratio of the two terms in the right part of the first of these equations: Here z ∆ is the characteristic vertical scale on which the velocity components (the thickness of the above-mentioned boundary layer) vary significantly. If this scale is less or on the order of the characteristic thickness of atmosphere H , then ratio (11) is much less than unity, and the first term on the right-hand side of the first equation (10) can be neglected (the smallness of this term can be verified further a posteriori). Then system (10) is reduced to the single equation In order to obtain a solution in an explicit analytical form, we consider the simplest model with periodic on x and y inhomogeneity of the gravity field: where G is GFI amplitude. The solution for disturbances is also sought in the form of a horizontal harmonic; in particular, For the amplitude function W the approximate equation is derived This is a special case of the approximate solution (9). To meet the boundary condition on the surface, it is necessary to add the corresponding solution of the homogeneous equation. Its characteristic equation has the form Let, for example, (16) According to the boundary conditions, we choose a negative value is the dimensionless number, so that the solution for the vertical velocity has the form For the components of horizontal velocity an approximate solution is obtained: Qualitatively, the solution has similarities to the previously considered two-dimensional problem [1]. For the parameter values under consideration, the dimensionless parameter 300 ≈ S . This means that a boundary layer of thickness on the order ( ) 1 − kS that ensures the fulfillment of the boundary condition (5) is added to slowly dampening expressions of the type (9), (14). In the numerical example considered, the thickness of the boundary layer is of the order of 1 km.
In the three-dimensional problem, inhomogeneities of equipotential surfaces can be flowed not only from above, but also in horizontal direction. Therefore in continuity equation the terms A boundary layer appears on the underlying surface, due to the fact that the flow around curved equipotential surfaces is incompatible with the non-permeability condition (absence of vertical velocity components) at the lower horizontal boundary. In this boundary layer, noticeable perturbations of buoyancy, pressure, and horizontal velocity can occur. The characteristic amplitudes of the velocity deviations due to local anomalies of gravity are obtained on the order of the product of the deviations of the geoid and the Brunt-Väisälä frequency. For highly anomalous regions, these amplitudes can reach several tenths of meters per second.

An analogy of atmospheric disturbances caused by thermal inhomogeneities of the underlying surface and inhomogeneities of the gravitational field
A comparative analysis of linear stationary models of mesoscale flows associated with thermal inhomogeneities of the underlying surface and with inhomogeneities of the gravitational field is performed. For both types of problems, there are considered the linear perturbations in a stably stratified semi-bounded volume of a medium rotating around a vertical axis. The Boussinesq approximation is used; it is assumed that the density of the medium linearly depends on the perturbations of the potential temperature.
For the problem of perturbations introduced by thermal inhomogeneities of the underlying surface, the two-dimensional stationary system of equations for linear perturbations has the form v 0 Here P is the ratio of the pressure perturbation to the average background density of the medium ρ ; T is temperature deviation; α is the thermal coefficient of the medium expansion; κ ν , are the exchange coefficients; is two-dimensional Laplace operator. Temperature stratification is assumed to be stable, so there is a background gradient of potential temperature. 0 > γ . On the lower boundary, stationary two-dimensional periodical horizontally thermal inhomogeneities and conditions of adhesion and impermeability are defined: (20) Here c is medium heat capacity, the meaning of parameters Q и k is obvious.
We seek the periodical horizontally solutions in the form (21) The system of equations for amplitudes has the form Excluding from the last system all unknowns except W , we obtain the equation The buoyancy frequency N , dimensionless variable kz Z = are introduced here; the control dimensionless parameters 0 R > , 0 Ta > are some analogues of Rayleigh and Taylor numbers. (26) From here, the remaining unknowns are easy to express. It makes sense to limit ourselves to the atmospheric values of the parameters under consideration, for which the inequalities are fulfilled (27) In this case, three roots of the characteristic equation (25) are expressed as follows: at that 1 Ta The approximate solution has the form: In the problem on disturbances caused by inhomogeneities of gravity field, there is supposed investigation of linear two-dimensional disturbances related to the presence of gravity inhomogeneities when its acceleration is are small deviations, connected by the ratios (3). In this case, the first and third dynamic equations, compared to (18), are modified as follows: At the lower boundary, instead of the first condition (20), it is assumed that there are no temperature deviations: Here ϕ is the angle of inclination of the underlying surface to the horizon, u is the component of velocity along this surface, axis n is perpendicular to that, θ is a temperature deviation from background (for the air in atmosphere -deviation of the potential temperature), 0 θ is the given value of this deviation at the surface.
that also has meaning of temperature at the lower boundary 0 = n . Since the denominator of the last expression contains ϕ sin , at 0 → ϕ the amplitude of the velocity disturbance max u not only does not vanish, but, on the contrary, tends to infinity.
These paradoxical properties of the slope flows (intense response to small surface slopes) were analyzed in detail in [4]. The fact is that with a decrease in the tilt angles, the time for establishing stationary solutions increases. At small time intervals, the effects of weak slopes are negligible, but they accumulate at sufficiently large times. In particular, with stable stratification of the medium, the

Conclusion
The performed analysis shows that the influence of GFI leads to perturbations of the horizontal velocity of relatively small amplitude -up to several tenths of ms -1 even in highly anomalous regions. But it must be borne in mind that the solutions obtained imply the existence of stable, albeit slow, but ordered flows over large territories. Consequently, the effect of GFI has a quasi-systematic character, CLIMATE 2019 IOP Conf. Series: Earth and Environmental Science 606 (2020) 012020 IOP Publishing doi:10.1088/1755-1315/606/1/012020 10 manifested primarily in the systematic errors of mathematical modeling of the atmosphere. The obtained results do not pretend to the conclusion that the effects under consideration are significant, but it seems that in any case they indicate, which disturbances can be induced by the considered mechanism, and allow reasonably decide to whether or not to take into account the influence of GFI.