A numerical investigation of dilatation dynamics in stratified deep water under modified gravity

Nicholas N. Topmana; G. C. E. Mbah

doi:10.61298/pnspsc.2026.3.259

Authors

Nicholas N. Topmana
[email protected]

Department of Mathematics, Enugu State University of Science and Technology (ESUT), Nigeria
G. C. E. Mbah
Department of Mathematics, University of Nigeria Nsukka (UNN), Nigeria

Keywords:

Stratified deep-water dynamics, Modified gravity, Geophysical fluid dynamics, Finite-difference modelling, Dilatation

Abstract

This study presents a numerical investigation of dilatation dynamics in a vertically stratified deep-water column under modified gravitational forcing. The governing momentum equations were solved using a finite-difference scheme, and dilatation was evaluated as the divergence of the velocity field. The results show that initial perturbations decay rapidly because of strong stratification, whereas gravitational scaling mainly affects the transient response. The system evolves toward a quasi-incompressible state characterized by negligible volumetric deformation.

Dimensions

REFERENCES

[1] M. Roch, P. Brandt & S. Schmidtko, ``Recent large-scale mixed layer and vertical stratification maxima changes'', Frontiers in Marine Science 10 (2023) 1277316. https://doi.org/10.3389/fmars.2023.1277316.

[2] F. Paladini de Mendoza, K. Schroeder, L. Langone, J. Chiggiato, M. Borghini, P. Giordano & S. Miserocchi, ``Deep-water dynamics along the 2012--2020 observations on the continental margin of the Southern Adriatic Sea (Mediterranean Sea)'', Journal of Marine Science and Engineering 11 (2023) 1364. https://doi.org/10.3390/jmse11071364.

[3] T. M. Joyce, D. W. Waugh & R. M. R. Jensen, ``The impact of climate change on deep water masses and ocean circulation: A review'', Climate Dynamics 55 (2020) 735. https://doi.org/10.1007/s00382-020-05291-2.

[4] A. J. Chorin, ``Numerical solution of the Navier--Stokes equations'', Mathematics of Computation 22 (1968) 745. http://dx.doi.org/10.1090/S0025-5710242392-2.

[5] D. Chae, ``Note on the Liouville-type problem for the stationary Navier--Stokes equations in $R^3$'', Journal of Differential Equations 268 (2020) 1043. https://doi.org/10.1016/j.jde.2019.08.027.

[6] D. R. Durran, Numerical methods for wave equations in geophysical fluid dynamics, Springer, New York, USA, 2010. https://link.springer.com/book/10.1007/978-1-4419-6412-0.

[7] E. G. Tulapurkara, ``Hundred years of boundary layer -- Some aspects'', Sadhana 30 (2005) 499. https://doi.org/10.1007/BF02703275.

[8] A. Constantin, ``On the deep water wave motion'', Journal of Physics A: Mathematical and General 34 (2001) 1405. https://doi.org/10.1088/0305-4470/34/7/313.

[9] A. Constantin & E. Kartashova, ``Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves'', Europhysics Letters 86 (2009) 29001. https://doi.org/10.1209/0295-5075/86/29001.

[10] R. M. Chen, L. Fan, S. Walsh & M. H. Wheeler, ``Rigidity of three-dimensional internal waves with constant vorticity'', Journal of Mathematical Fluid Mechanics 25 (2023) 71. https://doi.org/10.1007/s00021-023-00816-5.

[11] C. I. Martin, ``Some explicit solutions to the three-dimensional water wave problem'', Journal of Mathematical Fluid Mechanics 23 (2021) 33. https://doi.org/10.1007/s00021-021-00564-4.

[12] C. I. Martin, ``On flow simplification occurring in three-dimensional water flows with non-vanishing constant vorticity'', Applied Mathematics Letters 124 (2022) 107690. https://doi.org/10.1016/j.aml.2021.107690.

[13] C. I. Martin, ``Liouville-type results for the time-dependent three-dimensional (inviscid and viscous) water wave problem with an interface'', Journal of Differential Equations 362 (2023) 88. https://doi.org/10.1016/j.jde.2023.03.002.

[14] G. C. E. Mbah & C. I. Udogu, ``Open channel flow over a permeable river bed'', Open Access Library Journal 2 (2015) 1. https://doi.org/10.4236/oalib.1101475.

[15] M. Liu, J. Park & J. C. Santamarina, ``Stratified water columns: Homogenization and interface evolution'', Scientific Reports 14 (2024) 11453. https://doi.org/10.1038/s41598-024-62035-w.

[16] C. Zhou, X. Xiao, W. Zhao, J. Yang, X. Huang, S. Guan, Z. Zhang & J. Tian, ``Increasing deep-water overflow from the Pacific into the South China Sea revealed by mooring observations'', Nature Communications 14 (2023) 2013. https://doi.org/10.1038/s41467-023-37767-4.

[17] N. T. Nnamani & G. C. E. Mbah, ``Mathematical modelling of geophysical fluid flow: The condition for deep water stratification'', Mathematical Modelling of Engineering Problems 11 (2024) 3509. https://doi.org/10.18280/mmep.111229.

[18] N. N. Topman & G. C. E. Mbah, ``The eigenspace of stratified deep water under modified gravity and Coriolis effect'', International Journal of Development Mathematics 2 (2025). https://doi.org/10.62054/ijdm/0203.20.

[19] N. T. Nnamani & G. C. E. Mbah, ``Mathematical model of stratified deep water flow under modified gravity using perturbation method analysis'', Mathematical Modelling of Engineering Problems 12 (2025) 1443. https://doi.org/10.18280/mmep.120434.

[20] N. N. Topman, G. C. E. Mbah, O. C. Collins & B. C. Agbata, ``An application of homotopy perturbation method (HPM) in a population model'', International Journal of Dynamics and Mathematical Sciences 6 (2023) 133. https://www.researchgate.net/publication/375279394_An_application_of_Homotopy_Perturbation_Method_HPM_for_solving_Influenza_virus_model_in_a_population.

[21] N. N. Topman & G. C. E. Mbah, ``Mathematical model on dimensional analysis of stratified deep water equations under modified gravity and Coriolis effect to obtain Reynolds number'', Journal of the Nigerian Association of Mathematical Physics 71 (2025) 199. https://doi.org/10.60787/jnamp.vol71no.627.

[22] N. N. Topman & G. C. E. Mbah, ``Mathematical model on impact of velocities for stratified deep water under modified gravity and Coriolis effect with simulation'', International Journal of Mathematical Analysis and Modelling 8 (2025) 437. https://www.researchgate.net/publication/399394365.

[23] L. P. Bailey, M. A. Clare, J. E. Hunt, I. A. Kane, E. Miramontes, M. Fonnesu, R. Argiolas, G. Malgesini & R. Wallerand, ``Highly variable deep-sea currents over tidal and seasonal timescales'', Nature Geoscience 17 (2024) 787. https://doi.org/10.1038/s41561-024-01494-2.

[24] J. M. Steinberg, C. G. Piecuch, B. D. Hamlington, P. R. Thompson & S. Coats, ``Influence of deep-ocean warming on coastal sea-level decadal trends in the Gulf of Mexico'', Journal of Geophysical Research: Oceans 129 (2024) e2023JC019681. https://doi.org/10.1029/2023JC019681.