the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 28 Jan 2025
A Nonlinear Generalized Boussinesq Equation ((2+1)-D) for Rossby-Khantadze Waves
Abstract. In the following paper, we investigate nonlinear Rossby-Khantadze waves at a higher dimension, by taking the inhomogenities in the geomagnetic field and in angular velocity into account. Considering the system to be weakly nonlinear, we make use of perturbation theory to derive a new (2+1)–D general form of Boussineq equation, derived from the equation of potential vorticity. We evaluate the obtained equation by using the qualitative theory of ODEs, and bifurcation theory of dynamical systems. Through which we obtain the exact solution of the system in a co-moving frame of reference and for more information, we make use of dynamical analysis. Furthermore, we provide the exact numerical solutions. These results show that the aforementioned solutions of the traveling waves corresponds to Rossby-Khantadze solitons.
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RC1: 'Comment on egusphere-2025-123', Anonymous Referee #1, 02 Apr 2025
This paper addresses the role of sheared Rossby-Khantadze (RK) waves in global atmospheric circulation, particularly in the ionospheric E region. The study builds upon prior research (Kaladze et al., 2011-2014) and explores nonlinear processes associated with RK wave dynamics, including their self-organization into solitary dipolar vortices and their potential to generate sheared zonal flows. They are well studied by many scientists, in which fast and slow electromagnetic wave’s propagation features were presented in the shear flow driven ionosphere, fast ones exist in the upper layers of the ionosphere, where magnetic filed effect on the waves enhances, but the slow electromagnetic waves – at comparably lower altitudes due to Coriolis force (Aburjania et al, 2006, JGR). It seems that authors are not familiar of the previous works or deliberately ignore those works.
In this papers the waves are investigated with the novel approach - a system of equations for boussinesq model equation from the initial set of equations namely, momentum equation, continuity equation and Maxwell equation telling the nonlinear interaction of considered multiple-scale analysis and asymptotic expansion to derive the nonlinear Boussinesq equation, providing a systematic approach to understanding wave interactions in the ionospheric E-region, the results are verifying the works Khantadze et al, who was investigating these waves for aa long time, but I couldnot find the reference on his works. Can the authors explain this?
In overal, the method used in this paper and the results are valuable for investigation of these waves in the ionosphere. With improvements in writing, clarity, and interpretation, the paper could provide significant insights into ionospheric wave dynamics.
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RC2: 'Comment on egusphere-2025-123', Muhammad Fraz Bashir, 10 Apr 2025
The Authors investigate nonlinear Rossby-Khantadze waves, including the inhomogenities in the geomagnetic field and angular velocity. In my opinion, the use of perturbation theory is appropriate, and the paper's results are very interesting, with the potential of improving our understanding of the large-scale processes. The authors are well-qualified and have contributed to different aspects of this topic in the past. I would like to recommend the paper for publication after the authors address the following minor points in the revised manuscript:
1. In figure 1: Increase the font size of figure labels by at least the size of the main text. Also, provide information on normalization in the caption for the reader. Add a legend to specify the red and blue lines. The visualization can further be improved to fix the y-axis up to 1. This will capture all cases.
2. In figure 2: increase the font size for tick labels and also add axes labels.
3. The abstract can be revised to include key results, e.g., the findings of inhomogeneity effects, etc., to draw the reader's attention to key findings.
Citation: https://doi.org/10.5194/egusphere-2025-123-RC2
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