Analysis of the Phase Velocities of (Pure, Slow and Fast) Alfvèn Waves in the E Region of the Ionosphere for Low Latitudes

In this paper, (pure, slow, and fast) Alfvèn waves for the accepted conditions in the Northern-hemisphere at the EF-region of ionospheric plasma were calculated with low latitudes using Eqs. (20, 25, 26) and the real geometry of the Earth's magnetic field, at hours 12.00 LT for the 1990 year when sunspot activity was at its peak. One of the most important findings of this research is that the phase velocities of magnitudes of "MHD modes = (pure, slow, and fast) Alfvèn waves" are analytically dependent not only on the angle between the wave propagation vector (k) and the magnetic field (B), but also on the declination (D = It is the angle value between the direction of the sun's rays and the equatorial plane) and magnetic dip angle (I = It is the angle between real north and magnetic north). According to the results, the behavior of the magnitudes of the squares of the phase velocities of all MHD modes is consistent with the behavior of the distribution of electron density with low geographic latitude, even when the magnetic field vector is perpendicular and parallel to the wave's propagation vector. In addition, the wave phase velocities are greater in the summer than in the winter. In this point, MHD wave behavior is critical in magnetosphere-ionosphere coupling. On the other hand The propagation velocities of the fast and slow MHD modes in the magnetic equatorial trough region have been determined to be very small at (= I), with almost no energy, but if = 90 + I, the energy increases with latitude and is approximately maximum at the low latitude limit. The minimum points are between 0 and 10 degrees latitude, where the wave energies are the smallest, and the maximum points are between 20 and 30 degrees latitude, where the wave energies are the greatest.


Introduction
Magnetohydrodynamics (MHD) is a field of research used to describe an electrically conductive fluid. In a medium with a magnetic field, low-frequency Alfvèn waves are very important in heating, transporting, and maintaining a stable structure. Besides, wave propagation in any medium such as magnetosphere and ionosphere plays an important role in order to gain information about the properties of the medium. The ionosphere may support both high-frequency (3-30 MHz) radio waves and very low-frequency Alfvén's waves [1][2][3][4][5][6][7][8][9][10][11]. Besides, low-frequency waves emitted in the ionosphere have a significant impact on the ionosphere-magnetosphere interaction. Including neutral dynamics in these processes means including the response of the medium to the low-frequency wave [12]. With this, it is aimed to make a self-consistent fluid definition in the partially ionized ionosphere plasma. These approaches allow us to use the usual MHD (manyeto-hydrodynamic). These approaches allow the effects of neutral dynamics in the ionosphere on wave propagation to be easily explained [4,[13][14][15][16]. The electrons, ions, and neutral particles constitute partially ionized ionospheric plasma [1,2,[5][6][7]. Their frequencies are very small and below a few Hertz (Hz). At lower frequencies below certain collision frequencies, the plasma behaves like a fluid [3,4,[9][10][11]. Generally, the magnetic field is present in all plasmas and is generally a good conductor. Therefore, the magnetic field is a quantity that is frozen in the fluid and moves with the fluid. In MHD equations, generally high and low beta [β = P/ (B 2 /2µ o )] distinction is made. P represents plasma pressure and (B 2 /2µ o ) magnetic pressure, respectively. If β >> 1, then plasma pressure is bigger than magnetic pressure and plasma drags magnetic field, otherwise magnetic field drags plasma. When plasma drags the magnetic field, the stresses occurring in the plasma propagate with the sound waves, these stresses propagate Alfven waves [3,4,[9][10][11][13][14][15][16]. The momentum changes in plasma particles caused by the collision between plasma particles and neutral particles facilitates the transfer of magnetic stress. At low frequencies, waves propagate, losing very little energy in the ionosphere if plasma inertia disappears. Due to the neutral charge inertia on the plasma, waves can be excited. If it has a very long wavelength, it can easily enter the magnetosphere. The ionosphere is a partially ionized natural plasma to occur strong collisions. The magnetosphere plasma above 400 km is collision-free in general approximation and is accepted as a magnetohydrodynamic (MHD) fluid. This leads to the conclusion that scientists who want to study the magnetosphere and ionosphere should take different approaches. While collision-free magnetospheric plasma is considered as the ideal magnetohydrodynamic, ionospheric dynamics is described using current (J) and electric field (E). Since the electric field in the ionosphere is considered to be of external origin, it cannot penetrate the quasi-neutral ionospheric plasma except in a very thin boundary layer of the order of the Debye length. The observations of the mid-latitude E-region of the ionosphere suggest the presence of the very low frequency (10 −4 s −1 long-wavelength 103 km) [1-7, 10, 11, 13-16]. Phase velocities of waves change from day to night in the E-region. Due to the strong day-night phase changes in the E-region, it is not possible to define these waves as conventional MHD waves [16]. On the other hand, it is worth stressing that there are some another studies on ionospheric plasma dynamics in the literature. For instance, the characteristics of Alfvénic wave structures associated with intermediate-scale plasma density were studied by. The equatorial plasma irregularities [17] and also characteristics of the plasma irregularities inside the mid-latitude ionospheric [18] can be seen in the literature. The distribution characteristics of the ionospheric plasma [18,19], and also Alfven waves related to atmospheric physics can be observed in the studies [20][21][22].
Until now, many scientists have done very valuable studies about many properties, physical structure, and chemical structure of the Earth's ionosphere. Since the ionosphere has a conductive structure, the behavior of the electromagnetic wave in this environment under various conditions and its response to the wave for high-frequency waves have been studied and continue to be studied. Although a lot of the studies on low-frequency waves have been done, it is observed that these studies are not enough in some respects [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16].
In fact, electron density is the driving force behind all physical changes in the ionosphere (electric field, conductivity, diffusion, etc.). All of the waves discussed in this article are directly affected by electron density. "Rishbeth 1967" claims that the electron density of 5 degrees south and about 15 degrees create troughs in the latitudes between the peaks at these latitudes at low latitudes [1]. The magnetic equatorial trough is the name given to this trough. The behavior of the magnitudes of these waves' phase velocities is similar to that of the electron density. The behavior of the magnitudes of the squares of the phase velocities of all MHD models is consistent with the behavior of the distribution of electron density with geographic latitude if the magnetic field is both perpendicular and parallel to the wave's propagation vector. Summer wave phase velocities are greater than winter wave phase velocities. MHD wave behavior is critical in magnetosphere-ionosphere coupling.

The MHD Equations for Ionospheric Plasma at Northern-Hemisphere
MHD equations have nonlinear equations with no continuous solutions. The solutions of such equations are extremely useful in studying the interaction with electric and magnetic fields in laboratory plasma, investigating steady states in astrophysical plasmas, observing the propagation of electric and magnetic waves in the atmosphere, studying symmetrical and asymmetrical explosions, and understanding many physical problems [12]. It is well known that MHD equations governing a compressible non-viscous conduction fluid immersed in a magnetic field are given by [3, 4, 9-11, 13, 16], If the Eqs. 1, 3, 4, 5 and 6 are substituted in Eq. 2 after mathematical manipulation, we obtain the momentum equation as follows: where all fields change as follows, Herein, u, m and B show the fluid's velocity, mass density and magnetic field, respectively. Under equilibrium conditions, the fluid is assumed to be spatially uniform with a constant density (ρ m0 ). Also, the equilibrium velocity is accepted zero and throughout the fluid, the magnetic induction B 0 is uniform and constant. When both electrical and fluid equations are used within each other, Eq. (7) is obtained as follows, Some expressions in this equation, respectively,ω: Wave frequency, k = Wave vector, Alfven velocity, and V s ; Adiabatic sound speed.
On the other hand, the magnetic field vector in Fig. 1 is as the following, where B x = B 0 CosI Sind, B y = B 0 CosI Cost, and B z = − B 0 SinI, with the magnetic dip I and the magnetic declination angles d, respectively. Wave vector is in the form, k = k y + k z = ksin(θ)a y + kcos(θ)a z . By using the geometry of magnetic field in Fig. 1 we can write the velocity V A with its components, as follows, Fig. 1 The geometry of Earth's magnetic field for the Northern hemisphere [23,24] Here, the components are as V Ax = V A Cosi Sind, V Ay = V A CosI Cosd and V Az = − V A SinI. However, adiabatic sound speed is given by V s = (γk b T/ρ m ) (1/2) , where γ is degrees of freedom, k b is Boltzmann constant and T shows Fluid temperature.
Using the real geometry of the Earth's magnetic field for Northern-hemisphere at ionospheric plasma, the velocity components of the Eq. The difference of the Eq. (20) from the other pure Alfven wave is the angle multiplier according to the geometry in Fig. 1. If Eqs. (16,17) are rearranged to the magnetic field and the accepted conditions, y, and z-components are obtained as follows.
The matrix form of Eqs. (21,22) can be written as follows: If the coefficient determinant of the Eq. (23) is taken the form, Equation (24) turns into a quadratic equation concerning (ω/k) 2 . Positive sign corresponds to fast Alfven mode; the negative sign corresponds to slow Alfven mode. From Eq. (24) we obtain the following equations,

Numerical Analysis and Results
In this study, MHD waves for the accepted conditions in Northern-hemisphere at E-region of ionosphere plasma were calculated with low latitudes by using Eqs. (20,25,26), at hours 12.00 LT for the year of 1990. We have studied special days 21 March (Northern and Southern Hemispheres. Sun rays fall at noon at perpendicular to the Equator. From this date, the sun's rays begin to fall perpendicular to the Northern Hemisphere. The nights begin to be longer than the days in the Southern Hemisphere) and 21 June (the Summer Solstice is the longest day of the year. The longest day affects not only the Northern hemisphere but also the Southern hemisphere. On this date, winter begins in the Southern Hemisphere. Longest night and shortest day in the Southern Hemisphere) for MHD wave modes. The ionosphere parameters (20, 25, 26) used for calculation were obtained by using the IRI model according to the accepted conditions. We investigated the seasonal change of Eqs. (20,25,26) concerning latitude for 12.00 LT.
As seen in the Fig. 2, when k is perpendicular to B according to the considered geometry on March 21, the square of the phase velocity of the Pure Alfven wave increases with latitude in the E-region. This wave is independent of the magnetic field, the propagation vector of the wave, and the magnetic dip. However, the square of the phase velocities of The fast MHD increases linearly with latitude and the magnitude of the phase velocity of the wave increases. The magnitude of the phase velocity of the slow MHD mode increases almost linearly concerning latitude, making a maximum at 20° North latitude, but decreasing again after this latitude. For the case of March 21, we can generally say that except for the magnitude of the phase velocity of the pure Alfven wave, the magnitude of the phase velocity of the other fast and slow MHD waves are perpendicular to the wave vector of the magnetic field; increases in parallel. In Fig. 3, the variation of the phase velocities of the MHD waves and the magnitudes of the squares with latitude on 21 June 1990 at noon is given. Accordingly, the trends of all modes are similar to Fig. 1. However, the squares of the phase velocities of all wave modes increased. The change in the magnitude of the squares of the phase velocities of the pure Alfven wave increases exponentially, the magnitude of the square of the phase velocity of the fast MHD decreases rapidly from South to North when the propagation vector of the wave and magnetic field vector is perpendicular to each other, passes through a minimum for each height at 5° North, and increases rapidly again. When the wave vector and the magnetic field vector are parallel to each other, the magnitude of the phase velocities increases from south to north and shows an almost linear behavior. For the slow MHD mode, the changing trend is the same as for fast MHD mode but crosses a maximum at 10° North for θ = I (k⊥B) and 20° North for θ = 90° + I (k//B).

Conclusion
In this study, MHD waves for the accepted conditions in Northern-hemisphere at E-region of ionospheric plasma were calculated with low latitudes by using Eqs. (20,25,26), at hours 12.00 LT for the year of 1990. When the results are evaluated in this article, the outstanding results are as follows: it has been shown that all modes of all MHD waves (Pure Alfven, fast and slow MHD) depend on the angle between the magnetic field and the wave propagation vector, as well as on the declination and magnetic dip angle. The values of NmF2 during the night as a function of the latitude exhibit a condition so-called "cavity" focused on the lowest magnetic point of the equator with "peaks" in 15 °N-20 °S latitudes in the northern hemispheres. Electromagnetic drift (⊥B) and diffusion (//B) combine and cause an upward increase in plasma motion like a "fountain". In this way, anomaly peaks are fed from the high regions on the Equator by diffusion. The production rate in this region is very low. However, plasma is drawn from lower levels around F2-peak, where the production rate is greater. According to these results, it is possible to say that when the wave vector is perpendicular to the magnetic field (θ = I) vector, the phase velocities of fast and slow MHD waves from troughs at different latitudes within two months (21 March and 21 June), but when the wave is parallel to the magnetic field according to Fig. 1, the phase velocities of fast and slow MHD waves become maximum at 20°-30° North. If the magnetic field is both perpendicular and parallel to the wave's propagation vector, the behavior of the magnitudes of the squares of the phase velocities of all MHD models is consistent with the behavior of the distribution of electron density with geographic latitude. The wave phase velocities are greater in the summer than in the winter. MHD wave behavior plays an important role in magnetosphere-ionosphere coupling.
Funding The author declares that no funds, Grants, or other support were received during the preparation of this manuscript.
Data availability Data sharing not applicable to this article as no datasets were generated or analysed during the current study.