Mathematical Model for Nth Dimensional Space and the Impact of Modified Gravity and Coriolis Effect on Stratified Deep Water Using Series Solution
DOI:
https://doi.org/10.62054/ijdm/0204.13Abstract
Stratified deep water refers to a layer of water that is deep enough that the effects of surface waves are negligible, and the water can be treated as a continuous, incompressible fluid. In such a system, the Coriolis effect and modified gravity can cause the water to stratify, or layer, with different densities and velocities in different regions. In an n-dimensional space, the Coriolis effect and modified gravity was shown to cause the water to stratify in complex ways, with different regions having different densities, velocities, and pressure gradients. The Coriolis effect, which is caused by the rotation of the Earth, causes the water to rotate and form vortices, while the modified gravity can cause the water to have different densities and pressures in different regions. A number of numerical simulations was carried out to understand the impact of modified gravity in deep water stratification and to predict the interactions between the internal waves within the layers. The model shows that stratification in deep water under the influence of the Coriolis effect and modified gravity have important implications for ocean circulation, climate, and marine ecosystems. Importantly, as there is no model presently which has successfully considered the impact of modified gravity and Coriolis effect together in the study of stratification in deep water. Therefore understanding the dynamics of stratified deep water in this context is undoubtedly an important area of research in oceanography and fluid dynamics in general.
References
Chae, D.: Note on the Liouville-type problem for the stationary Navier-Stokes equations in R3. J. Differ. Equ. 268, 1043–1049 (2020). https://doi.org/10.1016/j.jde.2019.08.027
Charney J. G. (1948), on that scale of Atmospheric Motion. Academic Press, New-York, 1982.
Chen, R.M., Fan, L., Walsh, S., Wheeler, M.H. (2023). Rigidity of three-Dimensional internal waves with constant vorticity. Journal of Mathematical Fluid Mechanics, 25(3): 71. https://doi.org/10.1007/s00021-023-00816-5
Chen, R.M., Fan, L., Walsh, S., Wheeler, M.H.: Rigidity of three-dimensional internal waves with constant vorticity. J. Math. Fluid Mech. (2023). https://doi.org/10.1007/s00021-023-00816-5 6.
Constantin, A., Kartashova, E.: Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves. Europhys. Lett. 86, 29001 (2009). https://doi.org/10.1209/0295-5075/86/29001 8. Constantin, A.: Nonlinear Water Waves with
Constantin, A.: Two-dimensionality of gravity water flows of constant non-zero vorticity beneath a surface wave train. Eur. J. Mech. B, Fluids 30, 12–16 (2011). https://doi.org/10.1016/j.euromechflu.2010.09.008 10. Constantin, A., Ivanov, R.I.: Equatorial wave-current interactions. Comm. Math. Phys. 370(1), 1–48 (2019).
Durran D. R. (2010). “Numerical Methods of Wave Equations in Geophysics Fluid Dynamics,” Springer-Verlag, New York.
Escher, J., Matioc, A.V., Matioc, B.V.: On stratified steady periodic water waves with linear density distribution and stagnation points. J. Differ. Equs. 251(10), 2932–2949 (2011). https://doi.org/10.1016/j. jde.2011.03.023
Gaspard-Gustave de Coriolis (1835): Surles equatios du movement relative des systemes de corps
Le Veque R. (2004). Finite Volume Methods for Hyperbolic Problems.
Liu, M., Park, J., Santamarina, J.C. (2024). Stratified water columns: Homogenization and interface evolution. Scientific Reports, 14(1): 11453. https://doi.org/10.1038/s41598-024-62035-w
Martin, C.I. (2021): Some explicit solutions to the three-dimensional water wave problem. J. Math. Fluid Mech. 23(2), 33 (2021). https://doi.org/10.1007/s00021-021-00564-4
Martin, C.I. (2022): On flow simplification occurring in three-dimensional water flows with non-vanishing constant vorticity. Appl. Math. Lett. 124, 107690 (2022).https://doi.org/10.1016/j.aml.2021.107690 39. Martin, C.I.: Liouville-type results for the time-dependent three-dimensional (inviscid and viscous) water
Martin, C.I. (2023). Liouville-Type results for the time-dependent three-dimensional (inviscid and viscous) water wave problem with an interface. Journal of Differential Equations, 362: 88-105. https://doi.org/10.1016/j.jde.2023.03.002
Mbah G. C.E . and Udogu, C. I. (2015) Open Channel Flow Over A Permeable River Bed Open Access Library Journal, 2, 1-7. https//doi.org/10.4236/oalib.1101475
Mengwei Liu, Junghee Park and J. Carlos Santamarina (2024). Stratified water columns: homogenization and interface evolution. https://doi.org/10.1038/s41598-024-62035-w
N.N Topman, G.C.E. Mbah, O.C. Collins and B.C. Agbata (2023). An application of Homotopy Perturbation Method (HPM) in a population. Volume 6, Issue 2 (Oct.-Nov.), 2023, ISSN(Online): 2682-5708
Nicholas N. Topman and Mbah G.C.E (2025). The Eigenspace of Stratified Deep Water Under Modified Gravity and Coriolis Effect. https://doi.org/10.62054/ijdm/0203.20
Nnamani Nicholas Topman and G.C.E Mbah (2024), Mathematical Modelling of Geophysical Fluid Flow: The Condition for Deep Water Stratification. https://doi.org/10.18280/mmep.111229
Nnamani Nicholas Topman and G.C.E Mbah (2025), Mathematical Model of Stratified Deep Water Flow Under Modified Gravity Using Perturbation Method Analysis. https://doi.org/10.18280/mmep.120434
Ponte R.M (2021). Deep water stratification and impacts on ocean currents and climate
Rippeth, T., Shen, S., Lincoln. B (2024). “The Deep Water Oxygen Deficit in Stratified Shallow Seas mediated by diapycnal mixing. Nat Commun 15, 3136
Roch, M., Brandt, P., Schmidtko, S. (2023). Recent large-scale mixed layer and vertical stratification maxima changes. Frontiers in Marine Science, 10: 1277316. https://doi.org/10.3389/fmars.2023.1277316
T. M. Joyce, D. W. Waugh, and R. M. R. Jensen (2020), "The impact of climate change on deep water masses and ocean circulation: A review". Climate Dynamics, 55(3-4), 735-754
Downloads
Published
Data Availability Statement
The useful that that aided this manuscript is readily available at doi.org/10.18280/mmep.111229 and doi.org/10.18280/mmep.120434
Issue
Section
License
Copyright (c) 2025 Nicholas N. Topman (Author); Arinze .L Ozioko (Translator); Emmanuel C. Duru (Author); Sunday I. S. Abang, Godwin C.E Mbah (Translator)
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors are solely responsible for obtaining permission to reproduce any copyrighted material contained in the manuscript as submitted. Any instance of possible prior publication in any form must be disclosed at the time the manuscript is submitted and a
copy or link to the publication must be provided.
The Journal articles are open access and are distributed under the terms of the Creative
Commons Attribution-NonCommercial-NoDerivs 4.0 IGO License, which permits use,
distribution, and reproduction in any medium, provided the original work is properly cited.
No modifications or commercial use of the articles are permitted.
