A New Mathematical Justification for the Hydrodynamic Equilibrium of Jupiter

In this paper, the case of Jupiter being found in hydrodynamic equilibrium is for the first time investigated solely by mathematical methods. With the help of the hydrodynamic method, formulas of energy balance for oval and vortex are found, which are summed as permanent kinetic energy and constantly provide equilibrium for the stable rotational movements of Jupiter. To find the total kinetic energy of the oval and vortex in turbulent mode, Green’s function methods with special definitions and flow functions that describe the movement of the vortex are applied. The results are expressed in lemmas and theorems. For the hydrodynamic equilibrium of Jupiter, the necessary and sufficient conditions for the preservation of the cyclone and the anticyclone are mentioned. The relationships between the angular velocity and the gradient of pressure and the Corolias parameter are also given. The Rossby number is given for steady rotational motion. These facts show the existence of necessary and sufficient conditions for maintaining the stability of rotational motion and prove the hydrodynamic equilibrium of Jupiter. In this case using stream function and constructing generalized Green’s function and accordance energy conservation laws, the hydrodynamic equilibrium of Jupiter is proved.


Introduction
It is known (see [1] [2] [3] [4] [5]) that the equatorial plane of the planet is close to the plane of its orbit (the inclination of the axis of rotation is 3.13˚ against 23.45˚ for the Earth, so there is no change of seasons on Jupiter). Jupiter orbits its axis faster than any other planet in the solar system. The rotational period at the equator is 9 h 50 min 30 s, and at mid-latitudes it is 9 h 55 min 40 s. Due to its rapid rotation, Jupiter's equatorial radius (71,492 km) is 6.49% larger than the polar radius (66,854 km); thus, the planet's contraction is 1: 51.4%. The model of the internal structure of Jupiter is contained under the clouds, there is a layer of a mixture of hydrogen and helium with a thickness of about 21 thousand km with a smooth transition from the gaseous to the liquid phase. This is followed by a layer of liquid and metallic hydrogen with a depth of 30 -50 thousand km. Inside Jupiter there may also be a solid core with a diameter of about 20 thousand km. Inside may be a solid core with a diameter of about 20 thousand km. The following model of Jupiter's inner structure has been recognized: The atmosphere is divided into three layers: the outer layer consisting of hydrogen; the middle layer consisting of hydrogen (90%) and helium (10%); and the bottom layer consists of hydrogen, helium and ammonia impurities, ammonium hydrosulfide and water, which form three layers of clouds: above, clouds of frozen ammonia (NH 3 ). It has a temperature of approximately 145˚C and a pressure of approximately 1 atm; below, a cloud of ammonium hydrosulfide (NH 4 HS); and at the bottom, water ice and possibly liquid water. The pressure in this layer is about 1 atm and the temperature is about 130˚C (143 K). Below this level, the planet is opaque. There is a layer of metallic hydrogen. The temperature of this layer varies from 6300 to 21,000 K and the pressure from 200 to 4000 GPa. The other part consists of the Stone Core. The question of the long-term existence of GRS and many processes are able to disperse atmospheric vortices similar to the Great Red Spot. Turbulence and atmospheric waves in the Red Spot region absorb the energy of its winds. The vortex loses energy by radiating heat. It should be noted that the absorption of smaller eddies by the Great Red Spot may be one of the mechanisms for maintaining its life and explains the long age of the largest atmospheric formation in the solar system. However, current models show that this is not enough. 3D models that take into account both horizontal and vertical gas flows show that when the slick loses energy, a temperature difference occurs, causing hot gas from the lower atmosphere to enter (vertically) into the GRS, which allows you to recover some of the lost energy. Thus, Jupiter's red spot is "fed" in (see [4] [5] [6]). As it turns out, vertical movement is the key to the "long life" of the Great Red Spot. The model also indicates the existence of a radial flow that "pulls" the wind from the high-speed currents and again directs them towards the center of the vortex. Therefore, it makes sense to build a global model to cover the above processes. Namely, our consideration studies this process in the energy sense and gives more clarity than previous studies. In the work, it is noted (see [7]) that hydrodynamic models are almost small applied to Jupiter because it is very difficult to construct hydrodynamic models for cyclones, anticyclones, turbulence, rotation, and the energetic budget of Jupiter. But last year, many investigators started studying mathematical and hydrodynamic models for GRS and Jupiter, which are being developed as new approaches. Our work (see [8] [9] [10]) is new approaches, and this presented work for the first time has the application of hydrodynamic models for ovals and vortex which summarize energy, finally helping to provide equilibrium in Jupiter.

Statement of Well-Posed Problems
Mathematical methods substantiate that Jupiter is in hydrodynamic equilibrium.
The main assumptions underlying it: 1) Jupiter is in hydrodynamic equilibrium; 2) Jupiter is in thermodynamic equilibrium. If you add the laws of conservation of mass and energy to these provisions, you get a system of basic equations. In addition, Jupiter's magnetic field circuit, like any field magnet, produces radio and X-ray radiation. Note that around Jupiter, as well as around most planets of the solar system, there is a magnetosphere, a region in which the behavior of charged particles, plasma, is determined by a magnetic field. In the case of Jupiter, the sources of such particles are the solar wind and its satellite Io The Coriolis force, or the deflecting force of rotation, appears in the equations of relative motion and is a fictitious force that describes the effect of the movement of the coordinate system associated with Jupiter: , if the x-axes is directed to the East, but y-to the North, z-vertically upwards and the wind speed component U,V,W along these axes. In this case w u  . The quantity where Ω , the rotation velocity of Jupiter, ϕ , along latitude. The ratio of the inertial force to the Coriolis force is called the Rossby number: , scales, ho-rizontal L, vertical H, (the atmosphere is anisotropic, and these scales differ significantly), the velocity scale U, the time scale for horizontal displacements 1 LU − for vertical HU ones, and the characteristic Coriolis parameter Here as a characteristic value of the inertial force equal to the acceleration of the particle, the characteristic value of the nonlinear term is taken, and this characteristic value is equal to is large, which means that the Corioli force can be neglected. As we can see, this depends both on the scale of motion (namely, the Coriolis force is negligible at small scales) and on the characteristic velocity: the larger it is, the larger Ro. At normal atmospheric velocities, the scale of the Coriolis force is not taken into account at mid-latitudes. Comparing the force of inertia with the force of friction, we find as a measure of their comparative significance the dimensionless Reynolds number for horizontal turbulent counter. The variation of pressure in the radial is given by The pressure at the axes of rotation is c P Therefore, the required pressure at the point r is

The Mathematical Justification for the Hydrodynamic Equilibrium of Jupiter
First, let's start with the fact that White Ovals, small vortices transfer their energies to a large vortex, including the GRS, as result of which is provided with constant kinetic energy. For the purpose we will try to build a visual description of this process by a mathematical formula, the justification of which is concrete and clear. Consider a number of isolated free vortices of force which is iindependents of time t, then the components of the i-th vortex ( 1, 2, 3, , i n =  ) (for example n = 100 ovals on Jupiter) have the following form: Here, in formulas (3.2), (3.3), the value in points is taken. Since the boundary on Jupiter is not solid, there is no stable flow in the region D. Therefore, in the hydrodynamic process, the flow of the function exists in the motion of i-th vortex and will have the following form: Now, starting from the system (3.4), we can determine the Green's function for a point in the region D. After that, it is easy to determine in the flow function by the definition of the Green's function, finding the harmonic function that expresses the system (3.2). The Green's function [16] must satisfy the condition where the G s * ∂ ∂ is taken as tangential derivative along the circle line. So, using the above, it seems possible to prove the symmetry property of this Green's function using standard methods. .Therefore, we summarize the results obtained in the following lemma.
This can be immediately seen by comparing the results obtained similarly to the results (3.2), (3.3) and (3.4). Note that the system of equation (3.12) is a Hamiltonian system of differential equations in the system of variables It is appropriate to note that in the work (see [8] [9] [10]) of the considering section "Mathematical description of the rotational details and motion process for the dynamic of the GRS on Jupiter" models for Jupiter were built on the basis of "spheroids" rotating differentially, whose semi axes are independent of each other: a problem that was solved using the law of rotation derived from a generalization of Bernoulli's theorem (for ideal gas where υ = V , P-pressure, ρ -density) also(see [9] [10]), which is valid only for axisymmetric masses. In this case, the term "quasi-potential" is additionally, introduced to pressure the rotation model. Each layer details with common boundaries of Jupiter rotate with its own angular velocity profile. The law of rotational has a simple dependence on the derivative the gravitational potential ( [8] [9] [10]). Despite the fact that no approved observational data has yet been found, that all layers have a common angular velocity profile which decreases from the pole to the equator, the angular velocities (the value of the angular velocities depending on this period of rotation changes) are clearly related by equality to pressures gradients and Carioles parameters. Therefore, the mathematical substantiation really allows finding out the laws of hydro dynamical properties of the equilibrium of the GRS and Jupiter. (See APPENDIX A).

Conclusions
The article first presents the facts and compares some results to create new con- • the movement of gas and liquid on the GRS is divided into three processes that combine laminar (or approximate, so-called quasi-laminar) and transitional flow along ovals with turbulent flow (See APPENDIX A); • in cyclones, the Coriolis force is directed from the center of the vortex, therefore, a decrease is formed in it, and in anticyclones, on the contrary, an increase in the gas density; • anticyclones are much longer-lived than cyclones, what is associated with the increased density inside them and, therefore, other things being equal, the total angular momentum of the anticyclone turns out to be higher than that of the cyclone, so it is more difficult for it to disintegrate; Open Journal of Applied Sciences • Rossby vortices slowly drift along the parallel to the west with a speed not exceeding dr where R V is the phase velocity of Rossby waves.
By means of stream function and Green's function constructed energy for one vortex motion and after summarized all ovals and energy for 1, 2, 3, , i n =  ( 100 n = , for Jupiter) (in particularly cases, ovals and vortex) vortexes motion. By the energy conservation laws, this summarized energy is constantan. It means that total motion of Jupiter rotation under indicated assumption always will be stability and therefore, the hydrodynamical equilibrium of Jupiter is proved..

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper. Open Journal of Applied Sciences    Figure A4. Illustration of scheme from laminar + quasi-laminar to transition turbulence regime (see [9] [10]). Figure A5. Zones belts and vortices on Jupiter. The wide equatorial zone is visible in the center surrounded by two dark equatorial belts (SEB and NEB). Figure A6. WHITE OVAL DE, JUPITER About 10 hours before closest approach to Jupiter, Voyager 1 acquired three 1 × 3 narrow angle green filtered mosaics of one of the three big, white ovals that were present in the South Temperate Zone at latitude 33˚S during the Voyager flybys. These ovals formed in 1939-1941 and had been shrinking since then.