# An Oil Fate Model for Shallow-Waters

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## Abstract

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## 1. Introduction

**Figure 1.**Advection and dispersion aspects of the model that are implemented in these modules are examined in detail in this study. Brackets indicate that a suitable empirical representation of the phenomenon would be used to capture these in the fully developed model. The ocean, atmosphere, and waves are captured by existing, supported circulation models. See Figure 2 and Figure 3.

## 2. Background

## 3. Oil Dynamics

**Figure 4.**Schematic of the nearshore environment, z increases above the quiescent level of the sea $z=0$, $\mathbf{x}:=(x,y)$, and t is the time variable. The sea elevation $z=\eta $, includes a component of the free surface associated with the currents is $z=\eta (\mathbf{x},t)$ and a component associated with waves. The bottom topography $z=-h\left(\mathbf{x}\right)$ is referenced to the quiescent sea level height, $z=0$.

#### 3.1. The Oil Slick Component

**V**is supplied by the wave/current interaction equations. Approximating the surface velocity $\mathbf{u}$ by $\mathbf{V}$ is predicated on the fact that the transverse velocity is qualitatively similar, at the large spatio-temporal scales we have in mind. The appearance of the Stokes drift velocity, which can be comparable to the Eulerian ocean current in the nearshore, antecedes the inclusion of dispersive corrections to the oil slick transport equation and is elaborated upon in Section 3.2. The wind stress provides more local corrections via the second term in the advection term. The wind stress related advection can dominate the ocean related advection if ${s}_{i}\left|\tau \right|/2{\mu}_{i}>{C}_{xs}\left|\mathbf{V}\right|$. We also note that the variability of the τ is usually much larger and changes much faster in time than other advective mechanisms.

#### 3.2. The Sub-Surface Oil Component

#### 3.3. Dispersion

#### 3.4. Transformation Mechanisms: Chemistry and Physics of Oil

#### 3.4.1. Mass Exchanges between the Subsurface Oil and the Slick

#### 3.4.2. Aging: A Consequence of Grouping Chemicals and Unresolved Physics

#### 3.4.3. Emulsification and Changes to the Density, Surface Tension, and Viscosity of the Slick

#### 3.4.4. Evaporation

^{2}), ${V}_{i}$ is the molar volume (m

^{3}/mol) and ${m}_{i}$ is the molar mass (Kg/mol). This is a simplified version of the model that appears in [39].

#### 3.4.5. Photolysis, Biodegradation, Sedimentation

## 4. Ocean Dynamics

## 5. Energy Conservation

## 6. Illustrative Dynamic Examples

#### 6.1. Nearshore Sticky Waters in Shores with Intense Breaking

**Figure 5.**Schematic cross-section of the model domain. A light, thin oil slick sits atop the ocean. The ocean’s mixed layer of thickness P is laden with oil droplets, accounted for as a concentration. The distance from the shore, at $x=0$, is denoted by x. The break zone extends to $x=L$. The ocean surface is at $z=0$ and bottom topography is fixed and described by $z=-h\left(x\right)$.

^{2}/s, $\mathcal{S}\left(x\right)=(1+\mathrm{exp}{[(x-L)/w)]}^{-1}$, where $w=20$ m is the transition width, $L=200$m. The turbulent eddy viscosity is constant: ${D}_{eddy}=0.05$ m

^{2}/s.

**Figure 8.**(

**a**) Bottom topography $h\left(x\right)$ and $P\left(x\right)$; (

**b**) wave amplitude $A\left(x\right)$; (

**c**) oil slick velocity ${u}^{st}$ and subsurface oil velocity ${u}_{C}$; (

**d**) dispersion $0.16K\left(x\right)$.

**Figure 9.**Effective subsurface velocity ${u}_{e}\left(x\right)$. The advection associated with the dispersion dominates. The crossover from positive to negative occurs approximately at $x=480$ m.

**Figure 10.**(

**a**) Evolution of contours of $S(x,t)$; (

**b**) at $t=1000$ h, the distribution of s (solid) and S (dashed).

#### 6.2. Shelf Dynamics Examples

**Figure 11.**(

**a**) Areal view, with the domain of the computation, highlighted. South Etast of Galveston, TX, USA; (

**b**) bathymetry of the region [57].

^{2}/s for $\kappa {\delta}_{ij}+\beta \Theta H$, and 0.16 m

^{2}/s for the dispersion associated with waves, $\beta {\Theta}^{st}/k$.

**Figure 12.**Lagrangian path of an ideal tracer induced by the Stokes drift from the wind data. The path would be identical in all cases in this section in which we are activate the wave-induced advection. The path direction in time is to the right.

**Figure 13.**Waves can localize a spill: (

**a**) the steady currents; (

**b**) The source location. (

**c**) Spill under the action of the shear flow; (

**d**) spill under the action of shear and Stokes drift advection.

**Figure 14.**Effect of wave-driven diffusion/dispersion: (

**a**) Initial conditions. Two Gaussians, the upper one has been multiplied by a uniformly random amplitude. Final time configurations: (

**b**) Constant shear velocity applied as shown in Figure 13a, Ψ does not include waves and the advection is due to the shear only. (

**c**) Shear advection and diffusion includes wave component; (

**d**) Shear and waves included in the diffusion and the advection.

^{2}norm of the mean square distance. The mean square distance associated with the case depicted in Figure 13 is shown in Figure 15.

**Figure 16.**Tracer at the final time due to a steady source, located in the middle of the domain. Winter conditions. (

**a**) Currents only; (

**b**) Currents and Stokes drift. (

**c**) Mean square distance corresponding to case (

**b**); (

**d**) Empirical variance corresponding to currents only (thick line), and currents and waves (thin line). The currents are overwhelming and thus the waves have a minor effect on the evolution of the spill, mostly at short times. Advection due to Stokes Drift velocity only. The time interval between points is 10 min.

**Figure 17.**Summer conditions. Final configuration of tracer due to a point source located in the center of the domain, after about 333 h. (

**a**) Steady current, no Stokes drift velocity; (

**b**) steady currents and Stokes drift.

**Figure 18.**Summer conditions, currents and waves. Evolution of tracer leading to Figure 17b. The current is steady at 0.18 m/s, directed in the North East direction. The wave-induced Stokes drift and the currents are comparable in magnitude.

## 7. Recapitulation

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

Name | Symbol | Units |

fast/slow time | t, T | s |

transverse position vector. Cross-shore, along-shore coordinate | $\mathbf{x}=(\mathbf{x},\mathbf{y})$ | m |

depth coordinate | z | m |

cross-shore, along-shore unit vectors | $\widehat{\mathbf{x}}$, $\widehat{\mathbf{y}}$ | - |

sea elevation | $\eta ={\zeta}^{w}+{\zeta}^{c}+S$ | m |

bottom topography, total water column | h, $H=h+\eta +S$ | m |

spatial gradient operator | $\nabla =({\nabla}_{\perp},{\partial}_{z})$ | 1/m |

wave and mean (current) sea elevation | ${\zeta}^{w}$, ${\zeta}^{c}=\zeta +\widehat{\zeta}$ | m |

density of water | ${\rho}_{w}$ | Kg/m^{3} |

oil slick total mass | ${M}_{s}$ | Kg |

thickness of i-th component of oil slick | $\tilde{{s}_{i}}$ | m |

density of i-th oil slick component | ${\rho}_{i}$ | Kg/m^{3} |

viscosity of i-th oil slick component | ${\mu}_{i}$ | Kg/ms |

surface tension of i-th oil slick component | ${\gamma}_{i}^{t}$ | Kg/ms^{2} |

velocity of i-th oil slick component | $\tilde{{\mathbf{u}}_{\mathbf{i}}}$ | 1/m^{2} |

depth averaged velocity of i-th oil slick component | ${\stackrel{\u02c7}{\mathbf{u}}}_{i}$ | m/s |

outward normal vector to ocean surface | $\widehat{\mathbf{n}}$ | - |

transverse component of wind stress | τ | Kg/ms^{2} |

Eulerian ocean velocity at surface | $\mathbf{u}=(\mathbf{u},\mathbf{v},\mathbf{w})$ | m/s |

slip velocity parameter | ${C}_{xs}$ | - |

pressure, ambient plus dynamic | ${\tilde{p}}^{\prime}={p}_{0}+\tilde{p}$ | Kg/ms^{2} |

wave frequency, peak wave frequency | σ, ${\sigma}_{0}$ | rad/s |

depth-averaged transport velocity | $\mathbf{V}(\mathbf{x},t)={\mathbf{v}}^{c}+{\mathbf{u}}^{st}$ | m/s |

depth-averaged Eulerian velocity | ${\mathbf{v}}^{c}(\mathbf{x},t)=({u}^{c},{v}^{c})$ | m/s |

depth-averaged Stokes drift velocity | ${\mathbf{u}}^{st}(\mathbf{x},t)=({u}^{st},{v}^{st})$ | m/s |

tracer dispersion tensor | $\Psi =\Sigma +\Xi \approx D$ | m^{2}/s |

turbulent Reynolds stress tensor | Σ | m^{2}/s |

dispersion caused by averaging | Ξ | m^{2}/s |

dispersion due to fluctuations respect to friction velocity | Θ | m^{2}/s |

dispersion due to fluctuations respect to Stokes drift | ${\Theta}^{st}$ | m^{2}/s |

friction velocity | ${\mathbf{v}}_{*}$ | m/s |

wind speed | ${v}_{wind}$ | m/s |

fractional water content | ${F}_{wc}$ | - |

fraction of evaporated oil from slick | ${F}_{evap}$ | - |

reaction, mass exchange, and other rates of oil slick | ${E}_{i}^{s},\phantom{\rule{0.166667em}{0ex}}{R}_{i},{G}_{i}^{s}$ | m/s |

total mass of subsurface oil | ${M}_{c}$ | Kg |

concentration of i-th species | ${C}_{i}$ | Kg/m^{3} |

generic tracer concentration | $B=\overline{B}+{B}^{\prime}$ | Kg/m^{3} |

ocean velocity | $\mathbf{U}(\mathbf{x},z,t)=(U,V,W)=\overline{\mathbf{U}}+{\mathbf{U}}^{\prime}$ | m/s |

tracer molecular diffusion | κ | m^{2}/s |

eddy flux tensor | $\mathbf{F}$ | Kg/s m^{2} |

parcel coordinate | $\mathbf{Z}$ | m |

Kronecker delta tensor | δ | - |

subsurface concentration associated with intermediate time scales | b | m^{3}/m^{3} |

wave covariance | ${e}^{2}$ | m^{2} |

mixing layer thickness, oil mixed layer depth | P, $\xi \approx \mathrm{min}\left(H\right(x),P)$ | m |

wave oil diffusion coefficient | ${D}_{w}$ | m^{2}/s |

indicator function of oil slick | ${\chi}_{S}$ | - |

absolute and relative group velocity | ${\mathbf{c}}_{\mathbf{G}}$, ${\mathbf{C}}_{\mathbf{G}}$ | m/s |

unidirectional wave spectrum | F | s ${m}^{2}$ |

Current forces: wind, breaking, bottom drag, lateral viscosity | $\mathbf{S}$, $\mathbf{B}$, $\mathbf{D}$, $\mathbf{N}$ | ${\mathrm{m}/\mathrm{s}}^{2}$ |

vortex force | $\mathbf{J}$ | ${\mathrm{m}/\mathrm{s}}^{2}$ |

vorticity, Coriolis constant | ω, $2\mathsf{\Omega}$ | 1/s |

wind drag parameter | ${C}_{D}$ | - |

Manning drag parameter | ${C}_{M}$ | - |

loss term in the action equation | ϵ | Kg/s |

energy density | $\mathcal{E}$ | ${\mathrm{Kg}/\mathrm{s}}^{2}$ |

seafloor slope | m | m/m |

surf zone extent | L | m |

eddy viscosity | K | ${m}^{2}$s |

subsurface velocity | ${u}_{C}\left(x\right)$ | m/s |

subsurface effective bulk oil velocity | ${u}_{e}\left(x\right)={u}_{C}\left(x\right)+D\left(x\right)\frac{1}{\xi \left(x\right)}\frac{d\xi \left(x\right)}{dx}$ | m/s |

effective thickness of submersed oil | S | m |

nearshore bottom drag parameter | ${d}_{b}$ | - |

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**MDPI and ACS Style**

Restrepo, J.M.; Ramírez, J.M.; Venkataramani, S.
An Oil Fate Model for Shallow-Waters. *J. Mar. Sci. Eng.* **2015**, *3*, 1504-1543.
https://doi.org/10.3390/jmse3041504

**AMA Style**

Restrepo JM, Ramírez JM, Venkataramani S.
An Oil Fate Model for Shallow-Waters. *Journal of Marine Science and Engineering*. 2015; 3(4):1504-1543.
https://doi.org/10.3390/jmse3041504

**Chicago/Turabian Style**

Restrepo, Juan M., Jorge M. Ramírez, and Shankar Venkataramani.
2015. "An Oil Fate Model for Shallow-Waters" *Journal of Marine Science and Engineering* 3, no. 4: 1504-1543.
https://doi.org/10.3390/jmse3041504