Environmental Fluid Mechanics

, Volume 12, Issue 4, pp 361–378 | Cite as

Oil spreading in instantaneous and continuous spills on rotating earth

  • Vladimir Maderich
  • Igor Brovchenko
  • Kyung Tae Jung
Original Article


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.


Oil slick Coriolis force Viscous stage Self-similarity 

List of symbols


Constant in the asymptotic solutions (41–42)


Constant in the Blasius solution


Constant in the turbulent boundary layer solution


Constant: \({C_{w} = C_w^\prime}\) at s = 1/2 and \({C_{w}=C_w^{\prime\prime}}\) at s = 1/5


Numeric factor in spreading law for instantaneous spill in gravity-viscous regime


Numeric factor in spreading law for continuous spill in gravity-viscous regime


Numeric factor in spreading law for continuous spill in gravity-turbulent regime


Numeric factor in spreading law for instantaneous spill in gravity-viscous-rotation regime


Numeric factor in spreading law for continuous spill in gravity-viscous-rotation regime


Coriolis parameter




Reduced gravity


Oil slick thickness in similarity solutions


Normalised oil slick thickness in similarity solutions


Oil slick thickness


Characteristic oil slick thickness


Notation of discrete time t k


Distance from the leading edge of slick


Oil slick length scale


Exponent in power law for source intensity


Exponent (n = 0 for unidirectional spreading and n = 1 for axisymmetric spreading)


Number of particles


Constant in source term


Horizontal scale of particle


Coordinate in symmetric case


Edge of slick in symmetric case

\({{\rm Re}_\ell}\)

Reynolds number


Rossby number


Parameter: s = 1/2 in laminar flow, s = 1/5 in turbulent flow




Characteristic time scale for motions in the boundary layer


Self-similar radial velocity



uo = (uo, vo)

The layer averaged oil velocity

uw = (uw, vw)

Surface current velocity


Self-similar circumferential velocity




Volume of oil slick


Characteristic volume of oil slick


Volume of particle

x = (x, y)

The Cartesian coordinates


The unit vector directed upward


Distance in water from surface


Exponent in similarity solution


Exponent in similarity solution


Exponent in similarity solution


Thickness of the Ekman boundary layer


Thickness of the boundary layer

δu = (δu, δv)

Difference between oil and surface current velocities

\({\Delta t}\)

Time step


Ratio of the slick thickness to the Ekman layer thickness


Similarity variable


Value of \({\eta}\) at r = r max




Exponent in similarity solution


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


Oil viscosity


Water viscosity


Kinematic water viscosity




Oil density


Water density


Shear stress on the oil-water interface


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


Exponent in similarity solution


Ratio of oil density to the water density


Horizontal differential operator


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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Vladimir Maderich
    • 1
  • Igor Brovchenko
    • 1
  • Kyung Tae Jung
    • 2
  1. 1.Institute of Mathematical Machine and System ProblemsKievUkraine
  2. 2.Korea Ocean Research and Development InstituteAnsanRepublic of Korea

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