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Oil spreading in instantaneous and continuous spills on rotating earth

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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.

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Abbreviations

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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Authors and Affiliations

  1. Institute of Mathematical Machine and System Problems, Kiev, Ukraine

    Vladimir Maderich & Igor Brovchenko

  2. Korea Ocean Research and Development Institute, Ansan, Republic of Korea

    Kyung Tae Jung

Authors
  1. Vladimir Maderich
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  2. Igor Brovchenko
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  3. Kyung Tae Jung
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Correspondence to Vladimir Maderich.

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Maderich, V., Brovchenko, I. & Jung, K.T. Oil spreading in instantaneous and continuous spills on rotating earth. Environ Fluid Mech 12, 361–378 (2012). https://doi.org/10.1007/s10652-012-9239-2

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  • DOI: https://doi.org/10.1007/s10652-012-9239-2

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