The oil spill model OILTRANS and its application to the Celtic Sea ...

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1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465t413 k2 Vo(1)ok1wgwhere k 1 & k 2 are empirical coefficients (1.14 & 1.45 respectively, after Fay (1971)), V o is theoriginal volume of spilled oil (m 3 ), υ w is the kinematic viscosity of water (m 2 /s), g is gravitationalacceleration (m/s 2 ), and w is the relative density difference between water and oil given by:w oil (2)where ρ w is the density of water (g/cm 3 ) and ρ oil is the density of oil (g/cm 3 ). The area (m 3 ) overwhich the oil slick has spread at the end of this first spreading phase was determined by Fay (1971)to be:Ao4 5k 2Vog 2(3)k1 w2The second stage of slick spreading known as the gravity-viscous phase is governed by the balancebetween the viscosity of the oil and the oil-water interfacial tension. The viscous spreading phasecontinues to such a time that the slick gets so thin that surface tension forces alone play a role inspreading the slick, leading to the third, surface-tension, phase. This third phase is not modelled inOILTRANS as it occurs at much later stages in slick spreading, rather a terminal oil film thicknessis defined in OILTRANS to denote the end of the gravity viscous phase after which no furthermechanical spreading occurs. Generally, by this time significant weathering will have occurred andthe slick is allowed to disperse horizontally or break into smaller slicks due to surface currentshears.The general spreading formulae proposed by Fay (1971) only deal with an idealised spreading of aslick in a radial manner on calm seas. They take no account of wind conditions, which play animportant role in determining the shape and area of actual spills. Results of field experiments on thespreading of oil slicks have been reported by Elliott et al. (1986), Jeffrey (1973) and Lehr et al.(1984), all of which found that slicks developed elongated shapes, with the major axis orientated inthe direction of the wind. Lehr et al. (1984) devised a Fay-type formula to account for theelongation of the oil slick in the direction of the wind.Using the assumptions adopted by previous researchers (Al Rabeh et al., 2000; Chao et al., 2003;Gou and Wang, 2009; Inan and Balas, 2010, Nagheeby and Kolahdoozan, 2008), that the slick iselliptical in shape with the major axis orientated in the direction of the wind, the area, A (x10 3 m 2 ),of the oil slick once the gravity-viscous spreading phase has commenced, can be represented as:A 1 QR4(4)where Q and R are the lengths of the minor and major ellipse axis respectively, given by :
123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646514V 1 3Q 1.7t(5)o43wind34R Q 0.03Ut(6)where in this case, V o is the volume of the spill (bbl), t is the time after the oil slick commencesspreading (min) and U wind is the wind speed (kts). Following the method adopted by Al Rabeh et al.(1989, 1995, 2000), the slick can be conceptualised as a series of concentric ellipses in a localCartesian reference frame X * Y * with the centre of mass of the slick located at the local coordinatesof (x * , y * ) = (0,0), see Figure 2. The local X * axis of the slick is orientated in the direction of thewind. The elliptical spreading of the slick over time is assumed to be uniform, in that;rRqQr r (7)R Rq q (8)Q Qq r (9)Q Rwhere r and q are the lengths of the major and minor axis of an interior ellipse,rand q are theincreases in length of the major and minor axis of an interior ellipse in time-step Δt (min), andandt (min).Qare the increases in length of the major axis and minor axes of the outer ellipse in time-stepAny oil particle constituting the slick will therefore be subjected to a displacement proportional tothe increase in the length of the respective axes with time. Given the location of a particle on aninterior ellipse at coordinates, x * t, y * t, which can be rewritten as x * t = rcosθ and y * t = qsinθ, thedisplacement of the particle outwards by Δx * and Δy * by spreading has been calculated by Al Rabehet al. (2000) to be:xy*** x r rt* cos x Rr t R(10)* y r rt *sin y Q qt Q(11)where the values of1 3V 3 1 t . tQand Rare given by the formulae;4Q 1.7o(12)44 13R Q U 3 40.03windt . t4with units as those in equation (5) and (6). As the local Cartesian reference frame for the slick isorientated in the direction of the wind, the coordinates must be converted to the global cartesiancoordinate system used by the OILTRANS model by means of the following transformations; R(13)
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