International Journal of Marine Science and Ocean Technology (IJMO)    IJMO-04-001e

Quantifying Anomalous Coastal Processes with Fractional Calculus-Based Models



Ninghu Su*

James Cook University, Cairns, Queensland, Australia.

*Corresponding Author

Ninghu Su,
James Cook University,
Cairns, Queensland 4870, Australia.
E-mail: ninghu.su@jcu.edu.au

Received: February 01, 2017; Published: February 04, 2017

Citation: Ninghu Su (2017) Quantifying Anomalous Coastal Processes with Fractional Calculus - Based Models. Int J Marine Sci Ocean Technol. 4(1e), 1-2.

Copyright: Ninghu Su© 2017. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.


Keywords

Coastal Regions; Oceans; Tides; Waves; Beaches; Seawater; Freshwater; Surface Water; Groundwater; Biogeochemical Cycles; Coastal Management; Mathematical Modelling; Fractional Calculus.

The emerging mathematical methods developed from fractional calculus are worth immediate attention in marine and coastal sciences.

The term “anomalous diffusion” refers to the mechanisms described by a fractional differential equation (FDE) or fractional partial differential equation (FPDE) compared to the processes described by the classic diffusion equation which is regarded as the “normal” diffusion [10, 16]. The recent advances in many fields have demonstrated that the anomalous process is universal, which implies that coastal processes are also anomalous.

The oceans are vital to human societies, life and the environment on the planet in many aspects by creating continuous hydrological cycles to provide water for sustaining life, environmental health and activities ensuring life’s survival. The United Nations Environment Programme (UNEP)12] estimates that “Half the world's population lives within 60 km of the sea, and three-quarters of all large cities are located on the coast. However, the seas and oceans are under increasing pressure from pollution. Much of this pollution comes from urban centres, and it creates environmental problems which threaten the viability of the cities themselves”. According to the United Nations Development Programme (UNDP)11], “Globally, the market value of marine and coastal resources and industries is estimated at $3 trillion per year or about 5% of global GDP, and an estimated 63% of global ‘ecosystems services’ are provided by marine and coastal systems. As much as 40% of the world oceans are considered as ‘heavily affected’ by human activities, including pollution, depleted fisheries, loss of coastal habitats such as coral reefs, mangroves and seagrasses, and by aquatic invasive species”.

The oceans of the world function as an irreplaceable source for water to replenish the otherwise dry and dead continents while embracing the influxes of various forms of pollutants, contaminants, pathogens, radioactive and other destructive materials. The data from the UN Water Assessment Programme 13] (2009, p. 138) indicates that “In at least 8 of the United Nations Environment Programme’s 13 Regional Seas Programme regions, over 50% of the wastewater discharged into freshwater and coastal areas is untreated, rising to over 80% in 5 regions”. These facts depict a grim situation for the health of the oceans.

The importance of the oceans and coastal regions to humans, the existing problems and challenges in coastal regions require consistent efforts and remedies to reduce the burdens on oceans and coastal regions, which require knowledge and tools for the quantification of water and solute exchanges between fresh and sea waters. With the inclusion of rainfall, physical evaporation and transpiration by plants, quantification of the exchanges between seawater and freshwater and solutes entrained within them pose challenges for numerical simulation and measurements for model parameters. While some of the existing models, particularly some numerical codes such as SUTRA [14] enjoyed success, our understanding of the physical, chemical, biological and other processes along coastal regions with limited data and simulated results is insufficient to inform various kinds of users such as irrigators on coastal farms, marine engineers, coastal conservationists and coastal region managers etc.

Recent advances in applying fractional calculus are encouraging [3, 4, 9]. In hydrology, very generic fractional partial differential equations (FPDEs) for solute movement in aquifers [2, 16], water movement in soils [6, 7] and groundwater flow in aquifers [8] have also been presented. Now the question is how the interactions between seawater and freshwaters in the coastal regions can be modelled using the FPDEs in order for the mathematical analyses across the different fields to be compatible.

The fractional calculus-based models and their parameters provide much more information about the stochastic processes of water flow and solute movement in porous media and fluid mechanics in the framework of the continuous-time random walk (CRTW) theory, which also connects the fractal geometry and the FPDEs [15].

Modelling coastal hydrological processes using FPDEs will certainly require more effort for parameterising the models compared to that for models based on integer calculus and integer partial differential equations (PDEs) such as in Bear (2005) [1], Liu et al. (2006) [5], and Voss and Provost (2010) [14]. The emergency of FPDEs in hydrological modelling is an important issue for consideration in that the extra information in the FPDE-based techniques underscores the advantages over their traditional counterparts. The processes, which can be modelled using these FPDE-based approaches, include coastal and marine processes and phenomena such as ocean waves, near-shore tidal waves, beach erosion processes, water movement and solute (salts and microbes etc.) transport in soils and aquifers, surface water and groundwater interactions, exchange between fresh water and seawater and biogeochemical cycles etc.


References

  1. Bear J (2005) Sea water intrusion into coastal aquifers. Encyclopedia of Hydrological Sciences, Wiley, Devon.
  2. Benson DA, Meerschaert MM, Revielle J (2013) Fractional calculus in hydrologic modelling: A numerical perspective. Adv Water Resour. 51: 479-497.
  3. Debnath L (2003) Recent applications of fractional calculus to science and engineering. Internl J Math Math Sci. 54: 3413-3442.
  4. Gorenflo R, Mainardi F (2009) Some recent advances in theory and simulation of fractional diffusion processes. J Comp Appl Math. 229: 400-415.
  5. Liu F, Anh V, Turner I, Bajracharya K, Huxley W, Su N (2006) A finite volume simulation model for saturated-unsaturated flow and application to Gooburrum, Bundaberg, Queensland, Australia. Applied Math. Modelling. 30(4), 352 – 366.
  6. Su N (2014) Mass-time and space-time fractional partial differential equations of water flow in soils: Theoretical framework and application to infiltration. J Hydrology. 519: 1792-1803.
  7. Su N (2017)a Exact and approximate solutions of fractional partial differential equations for water movement in soils. Hydrology, Special Issue: Groundwater, 4(1), 8: 1-13. doi:10.3390/hydrology4010008.
  8. Su N (2017)b The fractional Boussinesq equation of groundwater flow and its applications. J Hydrology. doi: 10.1016/j.jhydrol.2017.01.015.
  9. Tarasov VE (2013) Review of some promising fractional physical models. Internl J Modern Phys B. 27(9): 1330005.
  10. Tsallis C, Bukman DJ (1996) Anomalous diffusion in the presence of external forces: Exact time-dependent solutions and their thermostatistical basis. Phys Rev E. 54(3): R2197-R2198.
  11. UNDP. Water and Coastal Area Governance, Jan 2017.
  12. UNEP. Cities and Coastal Areas, Jan 17.
  13. UNESCO (2009) Water in a Changing World. The United Nations World Water Development Report 3, UNESCO Publishing, Paris.
  14. Voss CI, Provost AM (2010) SUTRA: A model for saturated-unsaturated, variable-density, ground-water flow with solute or energy transport. USGS Water-Resources Investigation Report- 02-4231, Reston, Virginia.
  15. Zaslavsky GM (2002) Chaos, fractional kinetics, and anomalous transport. Phys Reports. 371(6): 461-580.
  16. Zhang Y, Benson DA, Reeves DM (2009) Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications. Adv. Water Resour. 32: 561-581.

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