A New Analysis of Fractional-Order Equal-Width Equations via Novel Techniques

In this paper, the new iterative transform method and the homotopy perturbation transform method was used to solve fractional-order Equal-Width equations with the help of Caputo-Fabrizio. This method combines the Laplace transform with the new iterative transform method and the homotopy perturbation method. The approximate results are calculated in the series form with easily computable components. The fractional Equal-Width equations play an essential role in describe hydromagnetic waves in cold plasma. Our object is to study the nonlinear behaviour of the plasma system and highlight the critical points. The techniques are very reliable, effective, and efficient, which can solve a wide range of problems arising in engineering and sciences.

Numerical Investigation of Time-Fractional Equivalent Width Equations that Describe Hydromagnetic Waves

The present research article is related to the analytical investigation of some fractional-order equal-width equations. The homotopy perturbation technique along with Elzaki transformation is implemented to discuss the fractional view analysis of equal-width equations. For better understanding of the proposed procedure some examples related to equal-width equations are presented. The identical behavior of the derived and actual solutions is observed. The proposed technique can be modified to study the fractional view analysis of other problems in various areas of applied sciences.

On the Low Energy (< keV) O+ Ion Outflow directly into the Inner Magnetosphere

&lt;p&gt;The development of low energy (&lt; keV) O+ ions in the inner magnetosphere is a crucial issue for various aspects of magnetospheric dynamics: i) Recent studies have suggested that low energy O+ can be locally accelerated to few keV energies inside geosynchronous orbit, and thus can constitute a significant source of the storm-time ring current O+ that could dominate the energy density during storms, ii) Mass loaded densities are important for accurate location of the plasmapause, which, in turn, is necessary for meaningful calculation of the field line resonance radial frequency profiles of ULF hydromagnetic waves in plasmasphere, iii) since O+ is only of ionospheric origin, its outflow from ionosphere into the magnetosphere is a manifestation of fundamental processes concerning energy and mass flow within the coupled Magnetosphere &amp;#8211; Ionosphere system. Although a lot of progress has been made on O+ outflow at high latitudes and its subsequent transport and acceleration within the magnetotail and plasma sheet, the source of low-energy O+ within the inner magnetosphere remains a compelling open question. The Helium Oxygen Proton and Electron (HOPE) mass spectrometer instrument aboard Van Allen Probes, which move in highly elliptical, low inclination orbits with apogee of 5.8 RE, has repeatedly detected field aligned flux enhancements of eV to hundreds of eV O+ ions, which indicate O+ outflow directly into the inner magnetosphere. We systematically investigate, throughout the duration of the Van Allen Probes mission (2012 &amp;#8211; 2019), the occurrence of such events with respect to L and MLT, the dependence of their directionality (bi-directional or unidirectional) and the lowest and highest energies involved on L, MLT and MLAT. We categorize the outflow events with respect to plasmapause location (when its determination is possible) and identify whether there is enhancement of O+ density. This categorization is important because if the outflows occur close to the plasmapause location, and depending on the density enhancement they cause, they could be responsible for the formation of O+ torus, whose source has been under debate for years. Finally, in order to identify the physical processes that lead to the ionospheric outflow, we also examine whether there are dipolarizations and/or enhancements of the field-aligned poynting flux associated with these outflow events.&lt;/p&gt;

The interplay of fast waves and slow convection in geodynamo simulations nearing Earth’s core conditions

SUMMARY Ground observatory and satellite-based determinations of temporal variations in the geomagnetic field probe a decadal to annual timescale range where Earth’s core slow, inertialess convective motions and rapidly propagating, inertia-bearing hydromagnetic waves are in interplay. Here we numerically model and jointly investigate these two important features with the help of a geodynamo simulation that (to date) is the closest to the dynamical regime of Earth’s core. This model also considerably enlarges the scope of a previous asymptotic scaling analysis, which in turn strengthens the relevance of the approach to describe Earth’s core dynamics. Three classes of hydrodynamic and hydromagnetic waves are identified in the model output, all with propagation velocity largely exceeding that of convective advection: axisymmetric, geostrophic Alfvén torsional waves, and non-axisymmetric, quasi-geostrophic Alfvén and Rossby waves. The contribution of these waves to the geomagnetic acceleration amounts to an enrichment and flattening of its energy density spectral profile at decadal timescales, thereby providing a constraint on the extent of the $f^{-4}$ range observed in the geomagnetic frequency power spectrum. As the model approaches Earth’s core conditions, this spectral broadening arises because the decreasing inertia allows for waves at increasing frequencies. Through non-linear energy transfers with convection underlain by Lorentz stresses, these waves also extract an increasing amount of energy from the underlying convection as their key timescale decreases towards a realistic value. The flow and magnetic acceleration energies carried by waves both linearly increase with the ratio of the magnetic diffusion timescale to the Alfvén timescale, highlighting the dominance of Alfvén waves in the signal and the stabilizing control of magnetic dissipation at non-axisymmetric scales. Extrapolation of the results to Earth’s core conditions supports the detectability of Alfvén waves in geomagnetic observations, either as axisymmetric torsional oscillations or through the geomagnetic jerks caused by non-axisymmetric waves. In contrast, Rossby waves appear to be too fast and carry too little magnetic energy to be detectable in geomagnetic acceleration signals of limited spatio-temporal resolution.

Periodic mixed convection flow along the surface of a thermally and electrically conducting cone

The main aim of the present work is to highlight the significances of periodic mixed convection flow and heat transfer characteristics along the surface of magnetized cone by exerting magnetic field exact at the surface of the cone. The numerical simulations of coupled non-dimensional equations are computed in terms of velocity field, temperature and magnetic field concentration and then used to examine the periodic components of skin friction, ?w, heat transfer, qw, and current density, jw, for various governing parameters. A nice periodic behavior of heat transfer qw is concluded for each value of mixed convection parameter, ?, but maximum periodicity is sketched at ? = 50. It is also computed that the lower value of magnetic Prandtl number ? = 0.1 gets poor amplitude in current density but highest amplitude is sketched for higher ? = 0.5. The behavior of heat and fluid-flow in the pres?ence of aligned magnetic field is associated with the phase angle and amplitude of oscillation. It is also noted that due to the increase in magnetic force parameter, ?, there are wave like disturbances generate within the fluid layers. These disturbances are basically hydromagnetic waves which becomes more prominent as the strength of magnetic force parameter is increased.

Periodic mixed convection flow along the surface of a thermally and electrically conducting cone

The main aim of the present work is to highlight the significances of periodic mixed convection flow and heat transfer characteristics along the surface of magnetized cone by exerting magnetic field exact at the surface of the cone. The numerical simulations of coupled non-dimensional equations are computed in terms of velocity field, temperature and magnetic field concentration and then used to examine the periodic components of skin friction, ?w, heat transfer, qw, and current density, jw, for various governing parameters. A nice periodic behavior of heat transfer qw is concluded for each value of mixed convection parameter, ?, but maximum periodicity is sketched at ? = 50. It is also computed that the lower value of magnetic Prandtl number ? = 0.1 gets poor amplitude in current density but highest amplitude is sketched for higher ? = 0.5. The behavior of heat and fluid-flow in the pres?ence of aligned magnetic field is associated with the phase angle and amplitude of oscillation. It is also noted that due to the increase in magnetic force parameter, ?, there are wave like disturbances generate within the fluid layers. These disturbances are basically hydromagnetic waves which becomes more prominent as the strength of magnetic force parameter is increased.

Hydromagnetic waves in a compressed-dipole field via field-aligned Klein–Gordon equations

Hydromagnetic waves, especially those of frequencies in the range of a few millihertz to a few hertz observed in the Earth's magnetosphere, are categorized as ultra low-frequency (ULF) waves or pulsations. They have been extensively studied due to their importance in the interaction with radiation belt particles and in probing the structures of the magnetosphere. We developed an approach to examining the toroidal standing Aflvén waves in a background magnetic field by recasting the wave equation into a Klein–Gordon (KG) form along individual field lines. The eigenvalue solutions to the system are characteristic of a propagation type when the corresponding eigenfrequency is greater than a critical frequency and a decaying type otherwise. We apply the approach to a compressed-dipole magnetic field model of the inner magnetosphere and obtain the spatial profiles of relevant parameters and the spatial wave forms of harmonic oscillations. We further extend the approach to poloidal-mode standing Alfvén waves along field lines. In particular, we present a quantitative comparison with a recent spacecraft observation of a poloidal standing Alfvén wave in the Earth's magnetosphere. Our analysis based on the KG equation yields consistent results which agree with the spacecraft measurements of the wave period and the amplitude ratio between the magnetic field and electric field perturbations.