# Oil spreading in instantaneous and continuous spills on rotating earth

## Abstract

The effect of the Coriolis force on the oil spill spreading in the gravity-viscous regime is examined. A new shallow water model for the transport and spreading of oil slick of arbitrary shape is described in which the Coriolis force is included in the momentum equations and the oil–water friction is parameterized in a frame of the boundary layer theory including the Ekman friction. The numerical Lagrangian method based on smoothed particle dynamics is described. New similarity solutions of the model equations are obtained for unidirectional and axisymmetric spreading in gravity-viscous, gravity-turbulent and gravity-viscous-rotational regimes for instantaneous as well as continuous releases. The numerical simulation extends these results for the case of continuous release in the presence of currents. It was shown that Coriolis term in the momentum equation can be omitted if slick thickness is much less of the laminar Ekman layer thickness. However, the Ekman friction should be retained for slicks of any thickness for larger times. The Ekman friction results in the essential slowdown of the spreading as well as in the deflection of the oil spreading velocity at 45° from the direction of velocity in the non-rotation case. Numerical simulations of large-scale spills showed that after the 2 days the slick area with the Coriolis effect was approximately less than half of that without rotation. Therefore, the earth rotation can be also important in the oil weathering.

## Keywords

Oil slick Coriolis force Viscous stage Self-similarity## List of symbols

*a*Constant in the asymptotic solutions (41–42)

- \({C_w^\prime}\)
Constant in the Blasius solution

- \({C_{w}^{\prime\prime}}\)
Constant in the turbulent boundary layer solution

*C*_{w}Constant: \({C_{w} = C_w^\prime}\) at

*s*= 1/2 and \({C_{w}=C_w^{\prime\prime}}\) at*s*= 1/5*F*_{IV}Numeric factor in spreading law for instantaneous spill in gravity-viscous regime

*F*_{CV}Numeric factor in spreading law for continuous spill in gravity-viscous regime

*F*_{CT}Numeric factor in spreading law for continuous spill in gravity-turbulent regime

*F*_{IR}Numeric factor in spreading law for instantaneous spill in gravity-viscous-rotation regime

*F*_{CR}Numeric factor in spreading law for continuous spill in gravity-viscous-rotation regime

*f*Coriolis parameter

*g*Gravity

*g*′Reduced gravity

*H*Oil slick thickness in similarity solutions

*H*_{*}Normalised oil slick thickness in similarity solutions

*h*Oil slick thickness

*h*_{0}Characteristic oil slick thickness

*k*Notation of discrete time

*t*^{ k }- \({\ell}\)
Distance from the leading edge of slick

*L*_{0}Oil slick length scale

*m*Exponent in power law for source intensity

*n*Exponent (

*n*= 0 for unidirectional spreading and*n*= 1 for axisymmetric spreading)*N*_{p}Number of particles

*q*Constant in source term

*R*Horizontal scale of particle

*r*Coordinate in symmetric case

*r*_{max}Edge of slick in symmetric case

- \({{\rm Re}_\ell}\)
Reynolds number

- Ro
Rossby number

*s*Parameter:

*s*= 1/2 in laminar flow,*s*= 1/5 in turbulent flow*t*Time

*t*′Characteristic time scale for motions in the boundary layer

*U*Self-similar radial velocity

*U*_{*}\({U/{\eta}_{max}}\)

**u**_{o}= (*u*_{o},*v*_{o})The layer averaged oil velocity

**u**_{w}= (*u*_{w},*v*_{w})Surface current velocity

*V*Self-similar circumferential velocity

*V*_{*}\({V/{\eta}_{max}}\)

*W*Volume of oil slick

*W*_{0}Characteristic volume of oil slick

*W*_{i}Volume of particle

**x**= (*x*,*y*)The Cartesian coordinates

**z**The unit vector directed upward

- \({\tilde{z}}\)
Distance in water from surface

*α*Exponent in similarity solution

*β*Exponent in similarity solution

*γ*Exponent in similarity solution

*δ*_{E}Thickness of the Ekman boundary layer

*δ*_{w}Thickness of the boundary layer

*δ***u**= (*δu*,*δ**v*)Difference between oil and surface current velocities

- \({\Delta t}\)
Time step

- \({\epsilon}\)
Ratio of the slick thickness to the Ekman layer thickness

- \({\eta}\)
Similarity variable

- \({\eta_{max}}\)
Value of \({\eta}\) at

*r*=*r*_{ max }- \({\eta_{*}}\)
\({{\eta}/{\eta}_{max}}\)

*θ*Exponent in similarity solution

*λ*\({\sqrt{\nu_w f/2}}\)

*μ*_{o}Oil viscosity

*μ*_{w}Water viscosity

*ν*_{w}Kinematic water viscosity

- \({\xi}\)
\({\sqrt{f/(2\nu_w)}}\)

*ρ*_{o}Oil density

*ρ*_{w}Water density

**τ**_{b}Shear stress on the oil-water interface

**τ**_{a}Shear stress on the oil-air interface

- \({{\tau}_{w} = (\tau^x_{w}, \tau^y_{w})}\)
Shear stress caused by relative motion of slick and surface current

- \({\phi}\)
Exponent in similarity solution

*ω*Ratio of oil density to the water density

- \({\nabla}\)
Horizontal differential operator

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## References

- 1.ASCE (1996) State-of-art review of modeling transport and fate of oil spills. ASCE committee on modeling oil spills. J Hydraul Eng 122:594–609Google Scholar
- 2.Reed M, Johansen O, Brandvik PJ, Daling P, Lewis A, Fiocc R, Mackay D, Prentki R (1999) Oil spill modeling towards the close of the 20th century: overview of the state of art. Spill Sci Technol Bull 5: 3–16CrossRefGoogle Scholar
- 3.Fay JA (1969) The spread of oil slick on a calm sea. Oil on the sea. Plenum Press, NY, pp 53–63Google Scholar
- 4.Hoult DP (1972) Oil spreading on the sea. Ann Rev Fluid Mech 4: 341–348CrossRefGoogle Scholar
- 5.Fannelop TK, Waldman GD (1972) Dynamics of oil slicks. Am Inst Aeronaut Astronaut J 10: 506–510Google Scholar
- 6.Buckmaster J (1973) Viscous-gravitational spreading of an oil slick. J Fluid Mech 59: 481–491CrossRefGoogle Scholar
- 7.Foda M, Cox RG (1980) The spreading of thin liquid films on a water air interface. J Fluid Mech 101: 33–51CrossRefGoogle Scholar
- 8.Sundaram TR (1980) Spread of oil slicks on a natural body of water. J Hydronautics 14: 124–126CrossRefGoogle Scholar
- 9.Sundaram TR (1981) The spread of high-and low viscosity chemical on water. J Hydronautics 15: 100–102CrossRefGoogle Scholar
- 10.Camp DW, Berg JC (1983) The spreading of oil on water in the surface-tension regime. J Fluid Mech 184: 445–462CrossRefGoogle Scholar
- 11.Phillips WRC (1997) On the spreading radius of surface tension driven oil on deep water. Appl Sci Res 57: 67–80CrossRefGoogle Scholar
- 12.Chebbi R (2000) Inertia-gravity spreading of oil on water. Chem Eng Sci 55: 4953–4960CrossRefGoogle Scholar
- 13.Chebbi R (2001) Viscous-gravity spreading of oil on water. AIChE J 47: 288–294CrossRefGoogle Scholar
- 14.Nihoul JCL (1983) A non-linear mathematical model for the transport and spreading of oil slicks. Ecol Model 22: 325–339CrossRefGoogle Scholar
- 15.Brovchenko I, Maderich V (2002) Numerical Lagrangian method for the modelling of the surface oil slick. Appl Hydromech 4(76): 23–31Google Scholar
- 16.Brovchenko I, Kuschan A, Maderich V, Shliakhtun M, Koshebutsky V, Zheleznyak M (2003) Model for oil spill simulation in the Black Sea. In: Proceedings 3rd International Conference oil spills, oil pollution and remediation, 16–18 Sept 2003, Bogazici University, Istanbul, pp. 101–112Google Scholar
- 17.Brovchenko I, Kuschan A, Maderich V, Shliakhtun M, Yuschenko S, Zheleznyak M (2003) The modelling system for simulation of the oil spills in the Black Sea. In: Dahlin H, Flemming NC, Nittis K, Petersson SE (eds) Building the European capacity in operational oceanography. Proceedings third International Conference on EuroGOOS, 3–6 December 2002, Athens, Greece. Elsevier Oceanography Series, 69, Elsevier, pp 586–591Google Scholar
- 18.Tkalich P, Huda K, Gin K (2003) A multiphase oil spill model. J Hydraul Res 41(2): 1–11CrossRefGoogle Scholar
- 19.Kundu PK, Cohen IM (2004) Fluid mechanics (3 edn). Elsevier AP, San-DiegoGoogle Scholar
- 20.Schlichting H (1979) Boundary layer theory. McGraw-Hill, NYGoogle Scholar
- 21.Cushman-Roisin B (1994) Introduction to geophysical fluid dynamics. Prentice Hall Inc., Englewood CliffsGoogle Scholar
- 22.Lucy L (1977) Numerical approach to testing the fission hypothesis. Astron J 82: 1013–1024CrossRefGoogle Scholar
- 23.Hockney RW, Eastwood JW (1981) Computer simulation using particles. McGraw-Hill, New YorkGoogle Scholar
- 24.Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics: theory and application to non spherical stars. Mon Notices R Astron Soc 181: 375–389Google Scholar
- 25.Liu GR, Liu MB (2003) Smoothed particle hydrodynamics: a meshfree particle method. World Scientific Publishing Company, SingaporeCrossRefGoogle Scholar
- 26.Huppert HE (1982) The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J Fluid Mech 121: 43–58CrossRefGoogle Scholar
- 27.Chebbi R, Elrahman SA, Ahmed HK (2002) Experimental study of unidurectional viscous-gravity spreading of oil on water. J Chem Eng Jpn 35: 1330–1334CrossRefGoogle Scholar
- 28.Chebbi R, Abubakr AM, Jabbar AYAA, Al-Qatabri AM (2002) Experimental study of axisymmetric viscous-gravity spreading of oil on water. J Chem Eng Jpn 35: 304–308CrossRefGoogle Scholar