Environmental Fluid Mechanics

, Volume 12, Issue 4, pp 361–378

# Oil spreading in instantaneous and continuous spills on rotating earth

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

## 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

Cw

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

FIV

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

FCV

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

FCT

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

FIR

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

FCR

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

h0

Characteristic oil slick thickness

k

Notation of discrete time t k

$${\ell}$$

Distance from the leading edge of slick

L0

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)

Np

Number of particles

q

Constant in source term

R

Horizontal scale of particle

r

Coordinate in symmetric case

rmax

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*

$${U/{\eta}_{max}}$$

uo = (uo, vo)

The layer averaged oil velocity

uw = (uw, vw)

Surface current velocity

V

Self-similar circumferential velocity

V*

$${V/{\eta}_{max}}$$

W

Volume of oil slick

W0

Characteristic volume of oil slick

Wi

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

δE

Thickness of the Ekman boundary layer

δw

Thickness of the boundary layer

δu = (δu, δv)

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

$${\sqrt{\nu_w f/2}}$$

μo

Oil viscosity

μw

Water viscosity

νw

Kinematic water viscosity

$${\xi}$$

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

ρo

Oil density

ρw

Water density

τb

Shear stress on the oil-water interface

τa

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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## 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