Angular Dependence of Magnetosonic and Alfvén Wave Speeds in a Stratified Solar Atmosphere

 Corresponding author: Onengiyeofori A. Davies Copyright © 2025 Author(s) retain the copyright of this article. This article is published under the terms of the Creative Commons Attribution License 4.0. Angular Dependence of Magnetosonic and Alfvén Wave Speeds in a Stratified Solar Atmosphere Chukwudi F. Asara, Chigozie Israel-Cookey, Friday B. Sigalo, Iyeneomie Tamunobereton-ari, Onengiyeofori A. Davies * and Patrick Oki Department of Physics, Faculty of Science, Rivers State University, Nkpolu-Oroworukwo, Port Harcourt, Rivers State, Nigeria. World Journal of Advanced Research and Reviews, 2025, 28(02), 832-841 Publication history: Received on 26 September 2025; revised on 03 November 2025; accepted on 06 November 2025 Article DOI: https://doi.org/10.30574/wjarr.2025.28.2.3738 Abstract This study quantitatively examines the angular dependence of the squared phase speed ratios (𝑣𝑝ℎ 2/𝑣𝑎2 ) for fast and slow magnetosonic waves in a highly gravitationally stratified solar atmosphere. The methodology employed the fundamental MHD governing equations (continuity, momentum, and energy), where the complex, non-linear system was simplified through normalization and subsequent linearization to derive the characteristic dispersion relation, which was then solved numerically using Python. Results for the Fast Magnetosonic Wave (FMSW) show near-unity ratios (≈1.0 ) at parallel propagation in the lower, denser layers (photosphere/chromosphere), but a dramatic, multiorder suppression of the ratio in the upper atmosphere. This suppression is a direct result of the increasing Alfvén speed (𝑣𝑎) due to decreasing mass density, confirming that FMSW phase speed becomes negligible relative to the background magnetic dynamics in the corona. The most significant finding concerns the Slow Magnetosonic Wave (SMSW), which exhibited large-magnitude negative values for 𝑣𝑝ℎ,𝑆𝑀𝑆𝑊/𝑣𝑎2 across wide oblique angles in the photosphere and chromosphere (e.g., as low as −2060.0 at 𝜃=60∘). The negative 𝑣𝑝ℎ 2 confirms that SMSW is predominantly evanescent or strongly damped in the lower atmosphere, suggesting that gravitational stratification effectively prevents this mode from efficiently transporting acoustic energy into the corona. These findings reinforce the crucial role of Alfvén waves as the dominant energy carriers in the upper solar atmosphere and offer new, highly anisotropic diagnostic ratios for solar magneto-seismology, critical for modeling energy flux and improving geomagnetic storm forecasting. Keywords: Magnetosonic Wave; Alfven Wave; Gravitationally Stratified Solar Atmosphere; Group speed; Numerical Model; Heat transfer 1. Introduction The outer layers of the Sun, encompassing the chromosphere and corona, represent a plasma environment characterized by extreme thermal and magnetic conditions [1, 2]. Understanding the mechanisms responsible for the immense heating of the solar corona and the acceleration of the solar wind requires a comprehensive theoretical framework, with magneto-hydrodynamic (MHD) waves being a leading candidate for energy transport. MHD wave theory predicts three fundamental modes of propagation in a magnetized plasma: the Alfvén wave and two magnetoacoustic waves, often referred to as fast and slow magnetosonic waves [3, 4]. These modes are instrumental in solar physics, providing diagnostics for plasma conditions through solar magneto-seismology and serving as conduits for mechanical energy from the convective zone into the upper atmosphere. World Journal of Advanced Research and Reviews, 2025, 28(02), 832-841 833 A crucial challenge in modeling wave behavior in the Sun is accounting for the inherent non-uniformity of the solar atmosphere [5, 6]. Unlike idealized uniform plasma models, the Sun's atmosphere is fundamentally influenced by gravitational forces, leading to significant density and pressure stratification over small vertical scales [7]. This gravitational stratification profoundly affects the characteristics of wave propagation, including wave speeds, damping, and mode conversion. The relationship between the phase speed (𝑣𝑝ℎ) of the waves, particularly their squared ratio relative to the Alfvén speed (𝑣𝑎2), and the angle of propagation (𝜃) with respect to the background magnetic field is not trivial in such a complex, non-idealized medium [5, 8]. Despite extensive studies into MHD wave characteristics, a rigorous and systematic analysis of the angular dependence of the ratios of squared phase speeds, specifically for the fast and slow magnetosonic modes relative to the Alfvén speed, within a highly gravitationally stratified plasma remains necessary [5, 9]. These ratios act as sensitive indicators of local plasma-𝛽 and anisotropic pressures, offering a refined diagnostic tool beyond simple phase speed analysis. The primary objective of the present study is, therefore, to quantitatively examine how the ratio of 𝑣{𝑝ℎ,𝑓𝑎𝑠𝑡} 2 / 𝑣𝑎2 and 𝑣{𝑝ℎ,𝑠𝑙𝑜𝑤} 2 / 𝑣𝑎2 evolves across the propagation angle in the context of the gravitationally stratified solar atmosphere. To achieve this goal, we employ the fundamental MHD governing equations, including the equations for mass continuity, momentum, and energy. The complex, non-linear system is simplified through normalization and subsequent linearization, allowing us to derive the characteristic dispersion relation for the waves. This complex algebraic relation is then solved numerically using Python, enabling the systematic calculation of the wave speed ratios for all propagation angles from 0∘ to 180∘. By mapping the angular dependence of this ratio across a stratified height profile, we aim to provide insight into how wave modes might preferentially propagate or convert in different atmospheric layers, under different magnetic-field inclinations and density gradients. The results have implications for solar wave diagnostics, mode‐coupling analyses and potentially for atmospheric heating mechanisms in magnetised plasmas. 2. Material and methods 2.1. Ideal Magnetohydrodynamic (MHD) Equations Ideal MHD equations describe the motion of a perfectly conducting (ideal) fluid and its interaction with a magnetic field in a uniform atmosphere [10, 11]. They stem from the combination of Maxwell’s equations with the equations of gas dynamics. It is assumed that the typical plasma velocities are much less than the speed of light. The MHD equations are the continuity, momentum, energy and diffusion equations as showed in eqns. 1 to 4. 𝜕𝜌 𝜕𝑡+ 𝜌∇.𝑣=0 (1) 𝜕(𝜌𝑣) 𝜕𝑡 + ∇.[(𝑣𝜌𝑣−𝐵𝐵)]+ ∇[(𝛾−1)(𝑒−𝜌𝑣2 2)]= 𝜌𝑔 (2) 𝜕𝑒 𝜕𝑡+ ∇.(𝑣𝑒−𝐹)+ ∇𝑃= 𝜌𝑔.𝑣 (3) 𝜕𝐵 𝜕𝑡+ ∇.(𝑣𝐵−𝐵𝑣)= 0 (4) The force field is given by; 𝐹= 𝐵𝐵.𝑣 + 𝑣(𝛾−1)(𝑒−𝜌𝑣2 2) (5) While the pressure field is given by; 𝑃= (𝛾−1)(𝑒−𝜌𝑣2 2) (6) Where 𝜌 is the mass density, 𝑝 is the scalar pressure, 𝑣 is velocity, g is gravity, 𝐵 is the magnetic field strength, 𝐹 is the force field, 𝜇0 is the permeability of free space, 𝑒 is the energy density and 𝛾 is the ratio of specific heats. Owing to this description being of ideal MHD, several terms are absent, such as those encompassing the effects of viscosity, resistivity, hall effects, etc. In any case, these omitted non-ideal terms are considered negligible on the length scales considered within this study. World Journal of Advanced Research and Reviews, 2025, 28(02), 832-841 834 2.2. Linearized MHD Equations In order to utilize the MHD equations to study the propagation of magnetosonic or P-waves, we rewrite the system of equations (1) - (4) in terms of an initial time-independent background equilibrium state (𝑝0, 𝜌0, 𝐵0,𝑣0) and small perturbations (𝑝1, 𝜌1, 𝐵1,𝑣1) to that state, such that;. 𝑝= 𝑝0+ 𝑝1 (7) 𝜌= 𝜌0+ 𝜌1 (8) 𝐵= 𝐵0+ 𝐵1 (9) 𝑣= 𝑣0+ 𝑣1 (10) 𝑒= 𝑒0+𝑒1 (11) Under static initial conditions; 𝑣0=0, 𝜕 𝜕𝑡=0, inserting equations 7 – 11 into equations 1 – 4, gives a new set of equations, which are; 𝜕𝜌1 𝜕𝑡+𝜌0∇.𝑣1=0 (12) 𝜕(𝜌0𝑣1) 𝜕𝑡 + ∇.(2𝐵0𝐵1) + ∇[(𝛾−1)(𝑒1−𝜌0𝑣12 2)]= 𝜌1𝑔 (13) 𝜕(𝑒1) 𝜕𝑡 + ∇.[𝑣1𝑒0−2𝐵0𝐵1𝑣1+ 𝑣1(𝛾−1)(𝑒1−𝜌0𝑣12 2)] + ∇(𝛾−1)(𝑒1−𝜌0𝑣12 2)= 𝜌0𝑔.𝑣1 (14) 𝜕𝐵1 𝜕𝑡 +∇.(𝑣1𝐵0− 𝐵0𝑣1)=0 (15) Assuming wave like solutions of the form 𝑒𝑥𝑝 [𝑖(𝒌 · 𝒓 − 𝜔𝑡)], where 𝒌 is the wave vector, 𝒓 contains the spatial coordinates, 𝜔 is the angular frequency, and 𝑡 is time, linearized equations 12 – 15 reduces to ; 𝜔𝜌1−𝜌0(𝑘.𝑣1)=0 (16) 𝜌0[𝑖𝜔𝑣1+(𝛾−1)(𝑖𝑘𝑣1)]= 𝜌1[𝑖(𝛾−1)𝑘𝑒1 𝜌1−𝑔] (17) 𝑖𝜔𝑒1+𝑖𝑘𝑣1𝑒1+𝑖𝑘𝑣1𝑒0−𝑖𝑘4𝐵0𝐵1𝑣1+(𝛾−1)[2𝑖𝑘𝑣1𝑒1−𝑖3 2𝑘𝑣12𝜌0−1 2𝑖𝑘𝑣13𝜌1] +(𝛾−1)[𝑖𝑘𝑒1−𝑖𝑘𝜌0𝑣1−𝑖𝑘1 2𝑣12𝜌1]= 𝜌0𝑔.𝑣1 (18) −𝑖𝑤𝐵1= 𝐵0𝑖𝑘𝑣1− 𝐵0𝑖𝑘𝑣1 (19) Under the assumption that 𝜔 ≠0, equations (2.16) - (2.19) yield the solutions, 𝜌1= 𝜌0𝑘.𝑣1 𝜔 (20) 𝜌0 𝜌1= [𝑖(𝛾−1)𝑘𝑒1 𝜌1− 𝑔] [𝑖𝜔𝑣1+(𝛾−1)(𝑖𝑘𝑣1)] (21) 𝜌0[𝑖3 2𝑘𝑣12+𝑖𝑘𝑣1]= 𝑖𝑘4𝐵0𝐵1𝑣1− (𝑖𝜔𝑒1+𝑖𝑘𝑣1𝑒1+𝑖𝑘𝑣1𝑒0) (𝛾−1)−[1 2𝑖𝑘𝑣13𝜌1+𝑖𝑘1 2𝑣12𝜌1]+ [2𝑖𝑘𝑣1𝑒1+𝑖𝑘𝑒1]−𝑔.𝑣1 (𝛾−1) (22) −𝑖𝑤𝐵1= 0 (23) According to Goedbloed and Poedts [12], in the case of a non-zero magnetic field and without loss of generality, the equilibrium magnetic field 𝐵0 can be directed along the z-axis with the wave vector 𝒌 lying in the x − z plane. Representing the linearized equation in matrix form, we have; [𝜔2−𝑘2𝑣𝑎2−𝑘2𝑣𝑠2𝑠𝑖𝑛2𝜃 0 −𝑘2𝑣𝑠2𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃 0 𝜔2−𝑘2𝑣𝑎2𝑐𝑜𝑠2𝜃 0 −𝑘2𝑣𝑠2𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃 0 𝜔2−𝑘2𝑣𝑠2𝑐𝑜𝑠2𝜃][𝑣𝑥 𝑣𝑦 𝑣𝑧]=0 (24) Here, the subscripts for the perturbation velocity have been dropped and replaced with their respective cartesian coordinates, θ is defined as the angle between 𝐵0 and 𝑘, 𝑘 = |𝑘|. The Alfvén speed and sound speed can thus be expressed as; World Journal of Advanced Research and Reviews, 2025, 28(02), 832-841 835 𝑣𝑎= √𝐵02 𝜇0𝜌0 (25) 𝑣𝑠= √𝛾𝑝0 𝜌0 (26) Solutions to equation (24) exist only when the determinant of the left-hand square matrix is zero, which in turn provides the dispersion relation. (𝜔2−𝑘2𝑣𝑎2𝑐𝑜𝑠2𝜃)[𝜔4−𝜔2𝑘2(𝑣𝑎2+𝑣𝑠2)+𝑘4𝑣𝑎2𝑣𝑠2𝑐𝑜𝑠2𝜃]=0 (27) There are three independent real roots of the above dispersion relation, which correspond to the three different types of waves that can propagate through an MHD medium. The first of these roots describes the shear Alfvén wave and is expressed as; 𝜔=𝑘𝑣𝑎𝑐𝑜𝑠𝜃 (28) This wave involves plasma motion strictly perpendicular to the magnetic field, a fact which can be seen through equations (20) and (21), which reveal zero perturbation to the density or pressure. The remaining two roots of the dispersion relation are gotten by solving quadratically the bi-quadratic equation using the formula, [𝜔4−𝜔2𝑘2(𝑣𝑎2+𝑣𝑠2)+𝑘4𝑣𝑎2𝑣𝑠2𝑐𝑜𝑠2𝜃]=0 (29) 𝜔2= 𝑃 (30) 𝐶𝑚𝑠= ±√(𝑣𝑎2+𝑣𝑠2) (31) Where 𝐶𝑚𝑠 is the magnetosonic speed. Inserting equations (30) and (31) into equation (29), we have [(𝜔2)2−𝜔2𝑘2𝐶𝑚𝑠 2+𝑘4𝑣𝑎2𝑣𝑠2𝑐𝑜𝑠2𝜃]=0 (32) [(𝑃)2−𝑃𝑘2𝐶𝑚𝑠 2+𝑘4𝑣𝑎2𝑣𝑠2𝑐𝑜𝑠2𝜃]=0 (33) Recall that a quadratic equation is generally defined as 𝑎𝑃2+𝑏𝑃+𝑐=0. From equation (33), 𝑎=1, 𝑏=−𝑘2𝐶𝑚𝑠 2 and 𝑐=𝑘4𝑣𝑎2𝑣𝑠2𝑐𝑜𝑠2𝜃. According to the almighty formular, 𝑃= 2𝑐 −𝑏±√𝑏2+4𝑎𝑐 (34) Inserting equation (30) and values of 𝑎, 𝑏 and 𝑐 into equation (34), 𝜔2= 2𝑘4𝑣𝑎2𝑣𝑠2𝑐𝑜𝑠2𝜃 𝑘2𝐶𝑚𝑠 2+√𝑘4(𝐶𝑚𝑠 2)2+4𝑘4𝑣𝑎2𝑣𝑠2(1−𝑠𝑖𝑛2𝜃) (35) From equation (35), we obtain two dispersion relations which are, 𝜔2= 2𝑘4𝑣𝑎2𝑣𝑠2𝑐𝑜𝑠2𝜃 𝑘2𝐶𝑚𝑠 2+√𝑘4(𝐶𝑚𝑠 2)2+4𝑘4𝑣𝑎2𝑣𝑠2(1−𝑠𝑖𝑛2𝜃) (36) 𝜔2 𝑘2= 2𝑣𝑎2𝑣𝑠2(1−𝑠𝑖𝑛2𝜃) 𝐶𝑚𝑠 2±√(𝐶𝑚𝑠 2)2+4𝑣𝑎2𝑣𝑠2(1−𝑠𝑖𝑛2𝜃) (37) 𝑣𝑝ℎ 2= 2𝑣𝑎2𝑣𝑠2(1−𝑠𝑖𝑛2𝜃) 𝐶𝑚𝑠 2±√(𝐶𝑚𝑠 2)2+4𝑣𝑎2𝑣𝑠2(1−𝑠𝑖𝑛2𝜃) (38) 𝑣𝑃ℎ,𝐹𝑀𝑆𝑊 2= 2𝑣𝑎2𝑣𝑠2(1−𝑠𝑖𝑛2𝜃) 𝐶𝑚𝑠 2+√(𝐶𝑚𝑠 2)2+4𝑣𝑎2𝑣𝑠2(1−𝑠𝑖𝑛2𝜃) (39) 𝑣𝑝ℎ,𝑆𝑀𝑆𝑊 2= 2𝑣𝑎2𝑣𝑠2(1−𝑠𝑖𝑛2𝜃) 𝐶𝑚𝑠 2−√(𝐶𝑚𝑠 2)2+4𝑣𝑎2𝑣𝑠2(1−𝑠𝑖𝑛2𝜃) (40) World Journal of Advanced Research and Reviews, 2025, 28(02), 832-841 836 Where 𝑣𝑃ℎ is the phase speed, 𝐶𝑚𝑠 is the magnetosonic speed, 𝑣𝑎 is the Alfven speed, 𝑣𝑠 is the sound speed, fast magnetosonic wave (FMSW) and slow magnetosonic wave (SMSW). Here, the fast and slow magnetosonic wave mode corresponds to the positive and negative solutions, respectively, of equation (38). These waves have non-zero perturbations to the density and pressure and involve plasma motion both perpendicular and parallel to the magnetic field. 2.3. Astrophysical Constants Certain astrophysical constants were required to estimate 𝑣𝑎 and 𝑣𝑠. These constants are shown in table 1 below. Table 1 Selected Astrophysical Constants for Solar Atmosphere [13] Solar Atmosphere Stratified /Layers Magnetic Field (𝐁𝟎)(G) Mass Density (𝛒𝟎)(𝐊𝐠𝐦−𝟑) Photosphere Lower 500 - 2000 1.65×10−6 Mid 1.4×10−8 Upper 5.5×10−6 Chromosphere Lower 1 - 500 3.75×10−10 Mid 2.25×10−11 Upper 7.5×10−12 Corona Lower 0.1 - 10 3.0×10−13 Mid 1.25×10−13 Upper 3.0×10−15 3. Results The evaluated ratios of the square of phase speed of the fast and slow magnetosonic waves to Alfven speed with the propagation angle in a highly gravitationally stratified solar atmosphere are presented in tables (2) to (4), while a curve of the ratio of phase speed of fast and slow magnetosonic waves against angle of propagation is presented in fures (1) to (3): Table 2 Estimated Ratio of the Square of Phase Speed of the Fast and Slow Magnetosonic Waves to Alfven Speed with the Propagation Angle in a Highly Gravitationally Stratified Solar Photosphere Propagation Angle Lower Photosphere Mid Photosphere Upper Photosphere (𝒗𝒑𝒉,𝑭𝑴𝑺𝑾 𝒗𝒂)𝟐 (𝒗𝒑𝒉,𝑺𝑴𝑺𝑾 𝒗𝒂)𝟐 (𝒗𝒑𝒉,𝑭𝑴𝑺𝑾 𝒗𝒂)𝟐 (𝒗𝒑𝒉,𝑺𝑴𝑺𝑾 𝒗𝒂)𝟐 (𝒗𝒑𝒉,𝑭𝑴𝑺𝑾 𝒗𝒂)𝟐 (𝒗𝒑𝒉,𝑺𝑴𝑺𝑾 𝒗𝒂)𝟐 0 0.999 −0.205 0.431 −2.47 0.372 −2.17 45 0.707 −2050.0 0.234 −2.28 0.202 −1.99 60 0.250 −2060.0 0.123 −2.52 0.106 −0.98 135 0.707 −2050.0 0.234 −2.28 0.202 −1.99 180 0.999 −0.205 0.431 −2.47 0.372 −2.17 World Journal of Advanced Research and Reviews, 2025, 28(02), 832-841 837 Table 3 Estimated Ratio of the Square of Phase Speed of the Fast and Slow Magnetosonic Waves to Alfven Speed with the Propagation Angle in a Highly Gravitationally Stratified Solar Chromosphere Propagation Angle Lower Chromosphere Mid Chromosphere Upper Chromosphere (𝒗𝒑𝒉,𝑭𝑴𝑺𝑾 𝒗𝒂)𝟐 (𝒗𝒑𝒉,𝑺𝑴𝑺𝑾 𝒗𝒂)𝟐 (𝒗𝒑𝒉,𝑭𝑴𝑺𝑾 𝒗𝒂)𝟐 (𝒗𝒑𝒉,𝑺𝑴𝑺𝑾 𝒗𝒂)𝟐 (𝒗𝒑𝒉,𝑭𝑴𝑺𝑾 𝒗𝒂)𝟐 (𝒗𝒑𝒉,𝑺𝑴𝑺𝑾 𝒗𝒂)𝟐 0 0.992 −667.0 0.983 −149.0 3.02×10−7 −0.972 45 0.497 −666.0 0.493 −297.0 1.51×10−7 −0.971 60 0.248 −666.0 0.247 −134.0 75.0×10−7 −0.974 135 0.497 −666.0 0.493 −297.0 1.51×10−7 −0.971 180 0.992 −667.0 0.983 −149.0 3.02×10−7 −0.972 Table 4 Estimated Ratio of the Square of Phase Speed of the Fast and Slow Magnetosonic Waves to Alfven Speed with the Propagation Angle in a Highly Gravitationally Stratified Solar Corona Propagation Angle Lower Corona Mid Corona Upper Corona (𝒗𝒑𝒉,𝑭𝑴𝑺𝑾 𝒗𝒂)𝟐 (𝒗𝒑𝒉,𝑺𝑴𝑺𝑾 𝒗𝒂)𝟐 (𝒗𝒑𝒉,𝑭𝑴𝑺𝑾 𝒗𝒂)𝟐 (𝒗𝒑𝒉,𝑺𝑴𝑺𝑾 𝒗𝒂)𝟐 (𝒗𝒑𝒉,𝑭𝑴𝑺𝑾 𝒗𝒂)𝟐 (𝒗𝒑𝒉,𝑺𝑴𝑺𝑾 𝒗𝒂)𝟐 0 0.00466 −1.02 0.00289 −1.01 0.00260 −0.010 45 0.00233 −1.01 0.00145 −1.00 0.00130 −1.003 60 0.11700 −1.04 0.00073 −1.00 0.00065 −1.010 135 0.00233 −1.01 0.00145 −1.01 0.00130 −1.003 180 0.00466 −1.02 0.00289 −1.01 0.00260 −1.005 Figure 1 A curve of the ratio of the square of phase speed of (a) fast magnetosonic wave to Alfven wave speed against angle of propagation in lower solar atmosphere (b) slow magnetosonic wave to Alfven wave speed against angle of propagation in lower solar atmosphere World Journal of Advanced Research and Reviews, 2025, 28(02), 832-841 838 Figure 2 A curve of the ratio of the square of phase speed of (a) fast magnetosonic wave to Alfven wave speed against angle of propagation in mid solar atmosphere (b) slow magnetosonic wave to Alfven wave speed against angle of propagation in mid solar atmosphere Figure 3 A curve of the ratio of the square of phase speed of (a) fast magnetosonic wave to Alfven wave speed against angle of propagation in upper solar atmosphere (b) slow magnetosonic wave to Alfven wave speed against angle of propagation in upper solar atmosphere 4. Discussion The results presented in the preceding tables provide a detailed quantitative characterization of the angular dependence of wave speed ratios (𝑣𝑝ℎ 2/𝑣𝑎2) for fast magnetosonic waves (FMSW) and slow magnetosonic waves (SMSW) across various regions of the gravitationally stratified solar atmosphere, from the photosphere to the corona. These findings illuminate the profound influence of density and temperature gradients on energy propagation and mode characteristics. Analysis of the FMSW ratio, 𝑉𝑝ℎ,FMSW 2 /𝑣𝑎2, consistently demonstrates a maximal value at 𝜃 = 0∘ and 180∘ (parallel propagation) across all atmospheric layers. According to Nakariakov and Verwichte [14], this behavior is characteristic of the fast mode, where phase speed is highest when propagating along the magnetic field lines. In the lower atmosphere, specifically the lower photosphere, lower chromosphere, and mid chromosphere, the ratio approaches unity (e.g., 0.999, 0.992, and 0.983 at parallel propagation). This proximity to 1.0 suggests that in these denser, cooler regions, the thermal speed (𝐶𝑚𝑠) is relatively low compared to the Alfvén speed, causing the FMSW to predominantly behave like a non-compressive Alfvén wave when traveling parallel to the field [14, 15]. As the propagation angle increases to 𝜃=45∘, the FMSW ratio drops significantly, exhibiting the characteristic 𝑐𝑜𝑠2𝜃 dependency expected from standard MHD theory for the fast mode. However, a striking transition occurs upon moving to the upper chromosphere and corona. Here, the FMSW ratios are orders of magnitude smaller (e.g., maximum of 3.02× 10−7 in the upper chromosphere, and 0.00466 in the lower corona). This dramatic reduction is the direct consequence of the sharp drop in mass density and the subsequent increase in the Alfvén speed (𝑣𝑎), which is inversely proportional to the square root of density. Because the phase speed 𝑣𝑝ℎ is constrained by the local background conditions, the massive relative increase in 𝑣𝑎 due to stratification causes World Journal of Advanced Research and Reviews, 2025, 28(02), 832-841 839 the ratio 𝑣𝑝ℎ,FMSW 2 /𝑣𝑎2 to plummet. This indicates that while the FMSW exists in the upper atmosphere, its phase speed becomes less influential relative to the background Alfvén speed in the highly tenuous, magnetically dominated upper layers [16, 17]. The most notable and theoretically significant finding is the highly unusual, large-magnitude negative values obtained for the SMSW ratio, 𝑣𝑝ℎ,SMSW 2/𝑣𝑎2, particularly in the denser layers of the photosphere and chromosphere. A negative value for the squared phase speed indicates that the corresponding wave mode is evanescent or non-propagating in the direction examined, meaning the energy associated with the SMSW is not transferred via standard propagating oscillations but is instead exponentially damped or trapped [14, 18]. The extremely large negative magnitudes (e.g., −2060.0 at 𝜃=60∘ in the lower photosphere) highlight a highly constrained or resonant regime. This prevalence of large negative values in the lower atmosphere suggests that the combined effects of strong gravitational stratification and the large pressure gradients overwhelm the magnetic restoring force for the slow mode in certain directions, according to Yu et. al. [19]. Specifically, the sharp, angle-dependent spike to −2060.0 at 𝜃=60∘ in the lower photosphere suggests that waves entering the solar atmosphere obliquely are immediately damped or reflect strongly due to the stiffening effect of the density gradient. The large negative values persist up through the mid chromosphere (e.g., −297.0 at 45∘/135∘), confirming that the SMSW is predominantly non-propagating in these gravitationally stressed regions for a significant range of angles. In contrast to the complex behavior below, the SMSW ratio in the corona stabilizes around a much smaller magnitude, near −1.0 (e.g., − 1.01 to −1.02). While still indicating an evanescent wave, the magnitude is now far smaller and more predictable, suggesting that in the high-Alfvén speed, low𝛽 coronal plasma, the wave is damped more uniformly, unlike the complex trapping/damping seen in the lower atmosphere [20, 21]. The highly anisotropic nature of both FMSW and SMSW demonstrated by this angular analysis has critical implications for coronal heating and solar wind acceleration: the significant damping and evanescent behavior of the SMSW in the chromosphere suggests this mode is unlikely to be an efficient large-scale transporter of acoustic energy to power the solar corona, meaning its energy is likely dissipated rapidly (mode-converted to heat) within the transition region [22, 23]. Conversely, the extreme values of 𝑣𝑎 implied by the very low 𝑣𝑝ℎ,FMSW 2/𝑣𝑎2 ratios in the corona strongly support the hypothesis that Alfvén waves are the dominant and most efficient movers of energy in the upper solar atmosphere [24, 25]. Their incompressible nature and lack of dependence on sound speed make them robust against gravitational stratification, supporting the long-standing recommendation that their large group speed is crucial for optimizing energy transfer from the stellar interior into the corona, thereby driving the stellar wind. Furthermore, the highly angledependent and distinct values of 𝑣𝑝ℎ,FMSW 2/𝑣𝑎2 and 𝑣𝑝ℎ,SMSW 2/𝑣𝑎2 across the different atmospheric layers provide a potential new diagnostic tool for solar magneto-seismology, allowing for a correlation between precise wave measurements at the photosphere and the specific phase speed ratios derived here. This correlation could lead to more accurate modeling of energy flux emergence, which ultimately impacts the intensity and characteristics of interplanetary disturbances relevant to space weather and geomagnetic storm forecasting. 5. Conclusion This study successfully quantified the angular dependence of the squared phase speed ratios for fast and slow magnetosonic waves (𝑣𝑝ℎ 2/𝑣𝑎2) across the highly gravitationally stratified solar atmosphere, from the photosphere to the corona. Using the fundamental MHD equations for mass continuity, momentum, and energy, we developed a theoretical model in which the complex, non-linear system was simplified through normalization and linearization to obtain the characteristic dispersion relation. The results provide critical insight into how density and pressure gradients fundamentally alter the classical behavior of these plasma wave modes, offering refined diagnostics for solar energy transport mechanisms. Numerical analysis of this relation across photospheric, chromospheric, and coronal layers revealed several key results: • The fast magnetosonic to Alfvén speed ratio is largest for propagation parallel to the magnetic field and decreases markedly at oblique angles, showing that angular dependence strongly governs wave-mode interactions. • The slow magnetosonic wave ratio varies irregularly with angle and height, indicating increased sensitivity to local plasma β and density gradients. • Gravitational stratification enhances Alfvénic dominance in the upper atmosphere, as the Alfvén speed rises rapidly with decreasing density, while sound speed remains comparatively low. • The observed angular trends imply that wave coupling and energy redistribution are most efficient at intermediate propagation angles, supporting theories that link Alfvénic perturbations to coronal heating and solar wind acceleration. World Journal of Advanced Research and Reviews, 2025, 28(02), 832-841 840 Overall, the analysis demonstrates that the combined effects of propagation angle and gravitational stratification are key to understanding MHD wave behavior in the solar atmosphere. Future work should extend this approach to include dissipative effects, nonlinear coupling, and magnetic field divergence to better quantify energy transport from the photosphere to the corona. Compliance with ethical standards Acknowledgements The authors gratefully acknowledge the support of the Authorities of Rivers State University, Port Harcourt, Rivers State, Nigeria. Disclosure of conflict of interest All authors declare that there are no conflicts of interest regarding this paper publication. References [1] De Moortel, I. and P. Browning, Recent advances in coronal heating. Philosophical Transactions of the Royal Society A: Mathematical, Physical Engineering Sciences, 2015. 373(2042): p. 20140269. 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