Introduction

Despite that burning of fossil fuel results in global temperature rise and strong ambitions for renewable energy transition, searches for new profitable oil fields are still going strong1.Exploration includes increasingly higher latitudes, paradoxically made available by retreating sea ice because of anthropogenic warming. Moving into regions with lower temperatures, the biodegradation of oil is slower and the viscosity higher. Also, high latitude environments have more limited response possibilities due to the remote locations and harsh conditions2. Thus, the exploration of these new environments underscores the critical necessity of understanding impacts of accidental oil spills for improved decision making and preparedness.

Fish early life stages (ELS; eggs and larvae) are considered particularly vulnerable to accidental oil spills3,4. Impacts can occur either as acute mortality or delayed effects that eventually result in death because competition is high and only the fittest will survive5. Exposure to oil occurs because both fish ELS and oil disperse near-surface from the location where they are initiated and can intersect6. Oil will be present both as oil droplets and dissolved oil. Most often, only dissolved oil is considered in modelling studies addressing impacts of oil spill scenarios on fish ELS7 However, studies have shown that oil droplets can adhere to the eggs, leading to prolonged and continuous exposure8 even if the trajectories of oil and fish ELS diverge after intersection contrary to the corresponding scenario for dissolved oil.

The feature of sticky eggs, that enable oil droplets to adhere to the egg surface, varies among fish species9. Fish eggs with an adhesive chorion surface are typically associated with demersal non-buoyant eggs laid by species that attach their eggs to plants or substrates10,11,12,13. However, certain species with pelagic eggs also possess an adhesive chorion8,14,15. Atlantic cod and Atlantic haddock eggs have similar size and morphology, exhibiting comparable embryo development16,17,18. Notably, a key distinction is that haddock eggs have an additional adhesive membrane covering the primary egg envelope17,19. Unlike demersal species, where adhesivity results from polar interaction mechanisms10, the adhesive material in haddock eggs appears to be lipophilic and interacts with hydrophobic oil droplets as seen in Fig. 18,14,20,21. Other species with pelagic eggs, like hake (Merluccis merluccis), also possess an adhesive outer membrane15, suggesting they may have similar sensitivity to droplets of crude oil. Interestingly, the use of dispersant chemicals reduces the oil droplet adhesion to eggs14, but the overall effect of using dispersants is not entirely clear as this would also increase the number of droplets and therefore the contact rates between oil droplets and eggs.

Fig. 1: Oil droplets on fish eggs.
figure 1

Oil droplet fouling on the chorion of Atlantic haddock eggs 5.5 days post fertilization. Eggs were placed in submerged cages during exposure to prevent them from being affected by any surface oil slick. Arrows indicate examples of oil droplets on the chorion surface. A Unexposed. B Exposed to 300 μg oil/L for 72 h.

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The effect of adhesive eggs is thoroughly investigated in laboratory experiments8 and shown to reduce the oil concentration threshold for effects in eggs by a factor of 1022. Therefore, to obtain more ecologically realistic models for oil exposure scenarios, it is necessary to consider the toxic contribution from oil droplets in addition to dissolved oil. We know from both measurements23 and models24 that dissolved oil is present in the upper tens of meters, which is also where pelagic fish eggs are located25. Oil droplets and pelagic eggs behave similarly in the water column with a vertical distribution exponentially decaying from a maximum near the sea surface as a function of buoyancy and turbulence26.

Here, we develop a framework for assessing oil droplet interactions with fish eggs. Specifically, we present an algorithm to model the accumulation of oil droplets from crude oil on individual fish eggs due to turbulent mixing and differences in rise velocities between eggs and droplets. In the algorithm, oil droplets are represented by a given set of characteristic sizes corresponding to disjoint size classes (Tables 1 and 2). For each of these characteristic sizes, a normalized collision rate for describing the physical processes of turbulent mixing and buoyant rise is calculated for each egg based on the oceanographic conditions at its 3D position. Finally, this rate is multiplied with the concentration of oil droplets at the position to quantify the number of oil droplets colliding with each egg. It is assumed that all oil droplets colliding with an egg will adhere to the egg and stay until hatching, but pending updated knowledge a stickiness probability can be applied to differentiate between droplet collisions and droplet accumulation. For a more detailed description of the algorithm, see the section Materials and methods.

Table 1 Distribution of droplet size classes
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Table 2 Characteristics of droplet size classes
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To demonstrate the use of our algorithm, we consider an example scenario with an oil release at the shelf southwest of the Lofoten islands (67.700°N, 10.841°E, Fig. 2). This is close to the spawning grounds of Northeast Arctic (NEA) haddock, which is the largest haddock stock in the world with an estimated spawning stock biomass of 150 kt in 202427. The region is also subject of heated debates on whether to allow exploitation of identified petroleum resources. Previous studies have used the same oil release location28,29,30, which is located such that the Norwegian Coastal Current moves both the oil and the eggs northwards along a narrow shelf. Oil is released for 90 days using a surface release rate of 4500 m3 per day with the Balder oil blend 201031 and environmental conditions modelled to recapture March 1st until May 30th, 2001. The chosen release period is concurrent with the spawning season of NEA haddock and is previously shown to be the 90-day period resulting in the highest impact on early life stages of NEA haddock (including larval stage) for this release position29. Specifically, we want to address the following research questions for the example scenario:

  1. 1.

    What is the relative contribution to oil spill exposure of fish eggs from oil droplets and dissolved oil?

  2. 2.

    Is the impact region of oil droplets larger than that of dissolved oil?

  3. 3.

    Do oil droplets adhering to eggs lead to prolonged exposure durations as compared to dissolved oil?

Fig. 2: Study area.
figure 2

Overview of the example scenario. Oil is released from the white square at 10.841°E, 67.700°N with dynamic 3D oil concentrations being calculated within the oil domain (dashed line). Haddock eggs are released from the whole spawning area (orange). Both oil and eggs drift and develop according to physical forcing from an ocean model covering a domain extending far beyond the oil domain. Main ocean currents are indicated by red (Atlantic water) and green (Coastal water) arrows73. The inset shows the location of the region within Europe.

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To run the example simulation, we implement the droplet algorithm in the impact assessment model Symbioses described in Carroll et al.28,29. This is a modular model system with an ocean model, a nutrient-phytoplankton-zooplankton-detritus model (not considered here), a fish ELS individual-based model (IBM), and an oil spill and fate model. Haddock spawning is modelled by releasing 9000 eggs every third day from March 15th until May 14th, giving a total of 189,000. Eggs are released at 30 m depth and have an individual horizontal release position sampled uniformly from the spawning area. Vertical positioning of eggs is updated dynamically according to individual buoyancies resulting in an exponential depth distribution32. Hatching ages are calculated based on experienced temperatures, with all eggs hatching before the oil spill ends such that there is an ongoing spill during the entire egg stage of all eggs. Altogether, the model produces a high number of individual haddock egg exposure histories to dissolved oil and oil droplet collisions that will be used to address the research questions above.

Results

The model produces 3D concentration fields of both dissolved oil and droplet oil. Sampling these along the individual egg trajectories and applying the collision algorithm allows the development of egg exposure histories. In the following, we first present details of the dissolved and oil droplet concentration fields, and then the resulting oil exposure of the eggs.

Spatial oil concentrations

For a given concentration threshold, the oil droplets cover a larger area than the dissolved oil, and their concentration is about two orders of magnitude greater close to the release site. This can be seen in Fig. 3 of the time averaged concentrations for April, which is in the middle of the release period when the concentration levels have stabilized. Most of the oil follows the main currents downstream to the north while some of the droplet oil drifts off the shelf across the deep ocean and some even drifts upstream into the large bay to the east. Looking at the vertical distribution of the oil (insets), we see that both the dissolved oil and oil droplets are located mainly in the upper 50 m of the water column where also most of the eggs are found (Fig. S6).

Fig. 3: Oil concentrations.
figure 3

Time averaged oil concentrations for April, showing dissolved oil (left) and oil in droplet form (right). The maps show the vertical maximum after averaging, while the insets show the latitudinal maximum according to depth after time averaging. The large region indicated by a dashed line is the oil spill model domain, while the small region is the plotting region for the example egg in Fig. 4 and the white square is the release site.

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Oil exposure of eggs

The exposure history for the egg stage of an example egg is shown in Fig. 4 with a temporal resolution of 12 min. The egg is spawned south of the oil release site on Mar 21st and drifts northwards passing the release site until it hatches after 11.5 days. The egg experiences some oil the first 7.5 days of its life, but the exposure is several orders of magnitude greater during the last 4.0 days of the egg stage when the egg is close to or downstream of the oil release site. Experienced concentrations are rapidly fluctuating with maximums of \(120\, \upmu{{\rm{g}}}\cdot {{{\rm{L}}}}^{-1}\) dissolved oil and \(5100\, \upmu{{\rm{g}}}\cdot {{{\rm{L}}}}^{-1}\) droplet oil. Accumulated droplet mass stays below \(1.0\,\upmu {{\rm{g}}}\) until the egg is 7.6 days, but thereafter it increases to 270 µg at hatching due to the elevated concentrations of oil droplets experienced in the last four days of the egg stage.

Fig. 4: Example of life history.
figure 4

Exposure history and drift trajectory for the 11.5 days long egg stage of an egg in the model. Concentrations of dissolved (upper) and droplet (middle) oil are the concentrations at the time-varying position of the egg (right panels). The accumulated droplet mass (lower) is given by the collision model and increases when droplets attach to the egg. Note the use of log scale in all the left panels. In the trajectory plot, the black square is the spawning position where the egg is released, the white square is the oil release site, and the arrow is the hatching position. The map region is shown on the larger map in Fig. 3.

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In total, 57.7% of the released eggs are exposed to oil, with droplets having higher exposure concentrations than dissolved oil (Fig. 5). Of the exposed eggs, the median maximum dissolved concentration is \(0.097\, \upmu{{\rm{g}}}\cdot {{{\rm{L}}}}^{-1}\). Equivalently, the median maximum droplet concentration is about three orders of magnitude larger at \(109\, \upmu{{\rm{g}}}\cdot {{{\rm{L}}}}^{-1}\). For the accumulated droplet mass (at hatching), the median for the exposed eggs is \(6.3\, \upmu{{\rm{g}}}\).

Fig. 5: Exposure statistics.
figure 5

Fraction of the haddock eggs (y-axis) that experience at least the value indicated at the x-axis. The left panel shows maximum concentrations of dissolved oil (blue) and droplet oil (orange). The right panel shows the accumulated droplet mass (evaluated right before hatch). The left y-axes show the fraction with respect to all released eggs, and the right y-axes show the fraction with respect to only eggs that are exposed to oil. Percentiles and medians are calculated with respect to the exposed eggs, i.e., using the right y-axes. Values on the top x-axes are the median values given by the dashed lines.

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To study exposure durations, we apply the median values in Fig. 5 as thresholds and calculate exposure durations for the highest exposed half of the eggs. Most importantly, this reveals that exposure duration increases substantially when considering accumulation of oil droplets instead of droplet concentration directly (orange vs. yellow in Fig. 6, left panel). In fact, the mean duration for droplet oil is shorter than for dissolved oil (Fig. 6, left panel, orange), while the mean duration for accumulated mass (Fig. 6, left panel, yellow) is longer. A dissolved exposure duration of 8 days, for instance, corresponds to a shorter mean droplet exposure time of 4 days but a longer mean accumulated mass exposure duration of 10 days. For interpretation, remember that exposure duration is limited by hatching age (median of 12.7 days, Figure S8).

Fig. 6: Exposure durations.
figure 6

Left: Exposure durations for accumulated droplet mass (yellow) and droplet oil (orange) for a given exposure duration of dissolved oil, showing that droplet exposure duration increases when considering accumulation. Eggs considered when calculating the durations are those in the highest exposed half of all exposed eggs, obtained by applying the medians in Fig. 5 as exposure thresholds. Eggs are categorized into bins of 0.5 days according to their dissolved exposure duration, with the mean and standard deviation calculated for each of these bins. The dashed line shows an exposure duration equal to that of dissolved oil. Right: Histogram of exposure durations for accumulated droplet mass when considering the highest exposed half, showing that the most common duration is 10 days. Note that the exposure duration for accumulated mass is the time from the mass surpasses the threshold of 6.3 µg (median in Fig. 5) and until hatching.

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In addition to the accumulated exposure duration being longer than the droplet duration, it is also less dependent on the dissolved duration (Fig. 6, left panel). Eggs exposed to dissolved oil for less than one day, for example, are still continuously exposed to oil at elevated levels for about 7 days due to their accumulated droplet mass. In accordance with this, the distribution of exposure duration for accumulated mass (Fig. 6, right panel) peaks at around 10 days, showing that eggs experiencing a high accumulated mass do so for most of their egg stage.

Discussion

Oil exploitation introduces pollutants to the marine environment through the controlled release of produced water and accidental oil spill events. This may occur at the drill site or during transportation to refineries. Regulations require documented planning and execution of well development and production including environmental impact assessment and preparedness for spill events.

Numerical models for oil spill and fate quantify the development of oil released to the environment across different states in a 3D grid covering the relevant area. However, when quantifying exposure of vulnerable marine species and life stages to oil for impact assessment, typically, only the dissolved oil is considered. Oil droplets are merely considered a moving source of dissolved oil that transforms into dissolved oil according to oil characteristics and environmental conditions. However, many species that are sensitive to dissolved oil may also react to oil droplets, especially during certain life stages. Here, we have considered fish species with adhesive eggs, where oil droplets remain attached to the egg after collision and lead to prolonged exposure.

To address the impact of an oil spill on adhesive eggs, we have developed a framework for quantifying the collision frequency between fish eggs and oil droplets. For this, we consider the oceanographic conditions and oil droplet concentrations at the individual egg positions, in addition to the size and density of the eggs and oil droplets. By integrating across all eggs, we can assess the relative importance of dissolved and droplet oil during the egg stage for the entire year-class of recruits about to enter the fish stock. The framework is generic and may be applied to other species and geographical regions by entering scenario-specific information. It may also be integrated with additional features such as indirect exposure through fish larvae feeding on zooplankton28 that may ingest oil droplets33.

Two processes are considered key to collision between eggs and oil droplets; i) random turbulent mixing causing 3D trajectories to intersect, and ii) different rise velocities due to buoyancy differences. For the scenario described here, with an oil spill in the spawning habitat of the cold-water gadoid NEA haddock, the difference in rise velocities is the dominant process for collision between eggs and oil droplets. In a scenario elsewhere, this may be different, but the generic framework presented here could still be applied.

Our modeling results indicate that oil droplets play a larger role in exposure than dissolved oil, in having both greater mass concentrations and covering larger areas (Fig. 3). For species that engage with oil droplets in addition to dissolved oil, e.g. through adhesive surface8 or ingestion33 this causes a large increase in impacts if oil droplets interfere with vital organs or development processes similarly to dissolved oil. Furthermore, oil droplets permanently connected to sensitive marine species result in prolonged exposure not disrupted by diverging dispersal trajectories (Fig. 6). If not including this feature, short-term exposures to dissolved oil would lower the chance of severe impacts given that uptake kinetics slow down the alignment of external and internal toxin concentrations. Hence, impact assessments not taking oil droplets into account may drastically underestimate the impact of oil spills on the marine environment. The results displayed here also call for more experimental work on impacts of oil droplets on marine species, in the past dominated by dissolved oil. However, while one may easily manipulate concentrations of dissolved oil in tanks for experimental exposure studies, the dynamics of oil droplet exposure is more complex to replicate in an experiment even though there exist some studies like Sørhus et al.8 and Hansen et al.14.

A notable and distinct difference between dissolved oil and oil droplets is the oil composition. Dissolved oil includes higher proportions of low molecular weight and polar compounds, while oil droplets include larger and more lipid-soluble compounds14. Heavier compounds in general have higher octanol-water partition coefficient (Kow), associated with slow uptake rates but lower no-effect concentration and faster killing rate. For short exposures, the slow kinetics serves as a buffering capacity. For long exposures, such as a droplet adhered to an egg, the slow kinetics are dominated by the enhanced toxicity of these compounds. This shows that even if the oil composition had been the same for dissolved and droplet oil, one still needs to consider their different exposure dynamics. Related to this, several studies point out the total polyaromatic hydrocarbons (TPAH) of dissolved oil as the components resulting in impacts34,35,36. Recently, however, studies indicate that the TPAH may only act as a proxy for components that cause ecosystem impacts21, suggesting that there are unknown toxicity dynamics that potentially could be better explained by including oil droplets in assessments.

Oil droplets may coat the eggs and mechanically suffocate them by hindering oxygen uptake. Also, oil coating would make the eggs more buoyant if everything else is fixed. Here, the buoyancy of eggs is assumed unchanged in the model, which is justified by the fact that natural variance of buoyancy within the egg stage could be larger than \(1\,{{\rm{psu}}}\)37\(,\) equivalent to a density change of \(\sim0.8\,{{\rm{kg}}}\cdot {{{\rm{m}}}}^{-3}\). In comparison, to decrease the density of a \(1.6\,{{\rm{mg}}}\) egg with \(0.8\,{{\rm{kg}}}\cdot {{{\rm{m}}}}^{-3}\) by attaching oil droplets of \({\rho }_{{{\rm{o}}}}=864\,{{\rm{kg}}}\cdot {{{\rm{m}}}}^{-3}\), the adhered mass must be \(8\,\upmu{{\rm{g}}}\). This is larger than the median value of 6.3 µg accumulated mass reported in the simulations here. Also, this is ~40 times higher than the adhered oil reported by Hansen et al.14 in laboratory experiments. Hence, the effect of adhered oil on the buoyancy can be expected to be within the range of natural variability of eggs for the droplet sizes considered here. However, if larger droplets are present and adhere to the egg such that its density decreases subtantially, eggs would rest higher in the water column and potentially be exposed to more UV radiation which enhances toxicity21. Eggs reaching the ocean surface may also be exposed to and potentially get stuck within oil slicks.

Dispersants are one of several mitigation measures used by authorities and industry to break up oil slicks and speed up biodegradation. However, changing the number and sizes of droplets could affect the mass and size distribution of oil accumulated on eggs38. While this may be captured in standard oil spill dispersal and fate models that typically resolve droplet sizes in discrete bins, thereby allowing for calculations with the framework presented here, it is not clear whether smaller droplets more easily stick to an egg if colliding. Despite previous reports that dispersants have limited effects on exposure rates of fish early life stages to oil when considering dissolved oil6 it is not clear if this is true also when addressing oil droplets as studies are limited and typically address copepods or daphnids39,40. In addition, there may be other processes that alter the distribution of oil droplet size such as zooplankton manipulation41. For the calculation of accumulated mass, however, the exact size distribution and buoyancy of droplets do not matter. This is because collision rates are dominated by the rise velocity of the egg and thus almost constant across droplet sizes (Section 1 in the SI).

Knowledge about droplet adhesion and accumulation is scarce, but Hansen et al.14 exposed cod and haddock eggs to mechanical or chemical dispersed oil for 24 hours to study this. To do so, droplets were filtered to sizes with diameter below 100 µm with a mean droplet size in the order of 10 µm. Exposure concentrations of oil droplets for both cases were reported to be ~400 \(\upmu {{\rm{g}}}\cdot {{{\rm{L}}}}^{-1}\), which less than 5% of the eggs in our simulation experience (8% of the exposed). Resulting droplet oil adhered to the haddock eggs (Fig. 2 in Hansen et al.14) was reported to be ~100 µg oil per g egg for chemical dispersed oil and ~200 µg oil per g egg for mechanical dispersed oil. As an egg is roughly 1.6 mg this corresponds to ~0.2 µg oil. Comparing the amount of adhered mass directly, about 51% of all eggs (89% of exposed) in our example simulation accumulate at least 0.2 µg droplet mass. Hence, oil droplets adhere to the eggs much more effectively in the simulation than what is indicated from the experiment. By instead considering the percentile corresponding to the exposure level in Hansen et al.14, i.e. the top 5% exposed of all eggs in the simulation here, the percentile value for accumulated mass in the simulation is ~40 µg suggesting that ~0.5% of the collided droplet mass in the simulation should adhere to obtain the experimental value of ~0.2 µg. Remember, however, that turbulence existing in the field is notoriously difficult to replicate in experimental tanks and numerical experiments. Hence, the number of droplets adhered to an egg surface as reported in literature is likely biased as compared to what happens in the field. Similarly, numerical models typically have coarse resolution relative to turbulent length scales and are therefore biased to the effect of turbulence on eggs and oil droplets contact rates. Also, whether an oil droplet adheres if encountering marine life depends on various aspects such as the speed and angle at encounter, shape and size of contact area, adhesivity of marine life in question, and the density of objects colliding. Based on our findings, we strongly recommend experimental investigations of these mechanisms and oil droplet impacts on various species and stages to improve realisms in oil spill impact assessment models. Additionally, one needs to address uptake rates of oil droplets with various oil component compositions adhered to the egg surface experimentally and numerically.

Methods

In this section, we present the theoretical framework and the example scenario in more details. During the theoretical description, we use a standard environment of 7.0 °C and 35.0 psu for illustration as this is typical for the example region during spring42,43. In the model, however, the time-varying ambient temperature and salinity of each egg is used.

Theoretical framework

The collision process between eggs and droplets depends on their size and buoyancy. For simplicity, we assume that eggs are completely spherical with a fixed diameter \({D}_{{{\rm{e}}}}\), which is a common assumption37. Each egg has an individual buoyancy in psu which is converted to a regular density \({\rho }_{{{\rm{e}}}}\) in \({{\rm{kg}}}\cdot {{{\rm{m}}}}^{-3}\) based on the water temperature at the egg position.

Oil droplets have a continuous size distribution, but we discretize all droplets into a finite number of size classes. Each class \(i\) is defined by a diameter range with a characteristic diameter \({D}_{i}\) given by the geometric mean of the range limits, and has a number concentration \({n}_{i}\) at any point (x,y,z) where \(\left[{n}_{i}\right]={{{\rm{m}}}}^{-3}\). To convert between number and mass concentrations, droplets are assumed to have a fixed density \({\rho }_{{{\rm{o}}}}\), implying that the composition of the oil droplets is not considered.

To simplify later calculations, we define a characteristic length scale for collisions between an egg and a droplet of size class \(i\) as the sum \({L}_{i}=\frac{{D}_{{{\rm{e}}}}+{D}_{i}}{2}\) of egg and droplet radiuses.

Accumulation of oil droplets on fish eggs

Accumulation of oil droplets on a fish egg is modelled by quantifying the collision frequency between the egg and oil droplets. To do this, we consider the hydrographic conditions and oil droplet concentration at the position of the egg, in addition to the size and density of the eggs and oil droplets. The concentration of eggs is relatively low compared to the concentration of oil droplets. For a reference egg concentration, we consider NEA cod which is assumed to have some of the highest concentrations of eggs. This is due to cod having high fecundity44 and NEA cod being the largest cod stock in the world with spawning concentrated in time due to the high latitude spring bloom45. A year class of NEA cod is in the order of \({10}^{14}\) spawned eggs46 with reported peak concentrations in the order of \({10}^{4}\,{{\rm{eggs}}}\cdot {{{\rm{m}}}}^{-2}\)47\(.\) Assuming a vertical distribution of the eggs as in Sundby26 this gives a peak concentration in the order of 1 egg per L for NEA cod. For the oil droplet spectrum used here, 1 µg of oil corresponds to between 10 and \({10}^{2}\) droplets of various sizes (Table 2). Peak concentrations of oil droplets in our simulations are greater than \({10}^{4}\, \upmu{{\rm{g}}}\cdot {{{\rm{L}}}}^{-1}\), giving droplet concentrations in the order of 106 droplets per L. Based on this, we assume that concentrations of oil droplets are not affected by collisions with fish eggs and consider the accumulation of oil droplets on individual eggs without altering the oil droplet concentrations for the remaining eggs.

Now, let \({N}_{{{\rm{i}}}}\) be the number of droplets in size class \(i\) adhered to an individual egg. Using a similar notation to Bandara et al.48, the accumulation of droplets of this size is given as

$$\frac{d{N}_{i}}{{dt}}={\Gamma }_{i}\cdot {n}_{i}\quad{\rm{where}} \quad\left[\frac{d{N}_{i}}{{dt}}\right]={{{\rm{s}}}}^{-1}$$
(1)

and \({\Gamma }_{i}\) with units \(\left[{\Gamma }_{i}\right]=\frac{{{{\rm{m}}}}^{3}}{{{\rm{s}}}}\) is a rate describing the collision probability of the egg and a droplet from the size class under the hydrographic conditions at the position of the egg. Here,

$${\Gamma }_{i}={\alpha }_{i}\cdot \left({\theta }_{i}^{{{\rm{rise}}}}+{\theta }_{i}^{{{\rm{mix}}}}\right)$$
(2)

where the two collision contributions \({\theta }_{i}^{{{\rm{rise}}}}\) and \({\theta }_{i}^{{{\rm{mix}}}}\) are due to buoyant rise and mixing, respectively. For these, we use rate functions based on Bandara et al.48 with more details in the following subsection. \({\alpha }_{i}\) is the collision efficiency of size class \(i\), which can be interpreted as a stickiness parameter. The value of \({\alpha }_{i}\) would depend on collision angle, droplet size etc. For simplicity, however, we initially set \({\alpha }_{i}=\alpha =1\) implying that all droplets colliding with an egg will adhere. As stated towards the end of the Discussion, this parameter is species and stage specific and need to be addressed experimentally to determine the correct values. Using a numerical model, we assume \({\Gamma }_{i}\) and \({n}_{i}\) to be fixed within time steps of duration \(\Delta t\) such that the accumulation becomes discrete in time with

$$\Delta {N}_{i}=\Delta t\cdot {\Gamma }_{i}\cdot {n}_{i}$$
(3)

new droplets of the size class stuck within the time interval. Note that we don’t restrict \(\Delta {N}_{i}\) to integer values as we don’t model each actual collision, but instead consider \({N}_{i}\) a continuous variable. Having the numbers \({N}_{i}\) of stuck droplets for each size, the total droplet mass stuck on an egg is \(M={\sum }_{i}{N}_{i}\cdot {m}_{i}\) where \({m}_{i}=\frac{\pi }{6}{D}_{i}^{3}\cdot {\rho }_{{{\rm{o}}}}\) is the characteristic mass of a droplet from size class \(i\). With this formulation, droplets accumulate during the egg stage and stay until hatching.

Collisions rates

First, consider the collision contribution \({\theta }_{i}^{{{\rm{rise}}}}\) due to differences in rise velocity between the egg and oil droplets of size class \(i\). This is modelled as in Bandara et al.48 by

$${\theta }_{i}^{{{\rm{rise}}}}=\frac{\pi }{4}{\left({D}_{{{\rm{e}}}}+{D}_{i}\right)}^{2}\cdot \left|{w}_{{{\rm{e}}}}-{w}_{i}\right|=\pi {{\cdot L}_{i}}^{2}\cdot \left|{w}_{{{\rm{e}}}}-{w}_{i}\right|$$
(4)

where \(\pi \cdot {{L}_{i}}^{2}\) is the area that the centers of the egg and droplets must be within to collide. \({w}_{{{\rm{e}}}}\) is the rise velocity of the egg and \({w}_{i}\) the rise velocity of droplet size \(i\), both in m per s and calculated using Stokes’ law (see SI for more details). At salinity 35 psu (Figure S1), a haddock egg (\({w}_{{{\rm{e}}}} \sim {10}^{-3}\,{{\rm{m}}}\cdot {{{\rm{s}}}}^{-1}\)) rises faster than the droplets (\({w}_{i} \sim {10}^{-6}-{10}^{-4}\,{{\rm{m}}}\cdot {{{\rm{s}}}}^{-1}\)), as the larger size of the egg dominates the lower density of the oil droplets.

Next, consider the collision contribution \({\theta }_{i}^{{{\rm{mix}}}}\) for size class \(i\) due to mixing. In general, assuming that \({\theta }_{i}^{{{\rm{mix}}}}\) (units \({{{\rm{m}}}}^{3}\cdot {{{\rm{s}}}}^{-1}\)) is a function of the dissipation rate \(\epsilon\) (units \({{{\rm{m}}}}^{2}\cdot {{{\rm{s}}}}^{-3}\)), kinematic viscosity \(\nu\) (units \({{{\rm{m}}}}^{2}\cdot {{{\rm{s}}}}^{-1}\)) and lengt \({L}_{i}\) (unit m), dimensional analysis reveals that the possible functional relationships are on the form

$${\theta }_{i}^{{{\rm{mix}}}}\propto {\epsilon }^{\frac{1}{3}}\cdot {L}_{i}^{\frac{7}{3}}\cdot {\left({\epsilon }^{\frac{1}{3}}\cdot {\nu }^{-1}\cdot {L}_{i}^{\frac{4}{3}}\right)}^{\beta }$$
(5)

where \(\beta\) is any real number. Setting \(\beta =0\) gives a turbulent relationship without the viscosity \(\nu\), while \(\beta =0.5\) gives a viscous relationship with a volume factor \({L}_{i}^{3}\). Note that this dimensional argument does not require a spherical volume. According to Pecseli et al.49, the change between the two regimes happens at the modified Kolmogorov length scale

$${\eta }_{0}={\left(15{C}_{{{\rm{K}}}}\right)}^{\frac{3}{4}}\cdot \eta ={\left(15{C}_{{{\rm{K}}}}\right)}^{\frac{3}{4}}\cdot {\left({\nu }^{3}\cdot {\epsilon }^{-1}\right)}^{\frac{1}{4}}=13.3\cdot {\left({\nu }^{3}\cdot {\epsilon }^{-1}\right)}^{\frac{1}{4}}$$
(6)

where \(\eta ={\left({\nu }^{3}\cdot {\epsilon }^{-1}\right)}^{\frac{1}{4}}\) is the regular Kolmogorov length scale and \({C}_{{{\rm{K}}}}=2.1\) is a Kolmogorov constant. Assuming \(\epsilon ={10}^{-4}\,{{{\rm{m}}}}^{2}\cdot {{{\rm{s}}}}^{-3}\), which is a greater value than typically observed50 the numerical value becomes \({\eta }_{0}\approx 5.7\,{{\rm{mm}}}\). For the more realistic dissipation rate \(\epsilon ={10}^{-6}\,{{{\rm{m}}}}^{2}\cdot {{{\rm{s}}}}^{-3}\)51,52 it becomes \({\eta }_{0}\approx 18\,{{\rm{mm}}}\), fitting well with Mann and Lazier53 that report that the smallest turbulent length scale in the ocean is between 6 and 20 mm for typical conditions. Here, we compare the modified length scale \({\eta }_{0}\) to \(2{L}_{i}={D}_{{{\rm{e}}}}+{D}_{i}\approx 1.5\,{{\rm{mm}}}\) to decide which regime to use, placing us solely in the viscous regime. Fixing the length, the dissipation rate giving regime change is \({\epsilon \sim 2\cdot 10}^{-2}\,{{{\rm{m}}}}^{2}\cdot {{{\rm{s}}}}^{-3}\) as seen in Figure S4 and S5.

It could be argued that one should also consider the distance between oil droplets as turbulent mixing may play a role if the average distance between droplets is greater than the smallest turbulent length scale. For the droplet spectrum assumed here, a mass concentration of \(10\, \upmu{{\rm{g}}}\cdot {{{\rm{L}}}}^{-1}\) gives a droplet number concentration in the order of \({10}^{3}\,{{{\rm{L}}}}^{-1}={10}^{6}\,{{{\rm{m}}}}^{-3}\). Then, each droplet occupies an average volume of \({10}^{-6}\,{{{\rm{m}}}}^{3}\), giving a typical distance of \(10\,{{\rm{mm}}}\) between droplets for this concentration. Based on this, turbulence might move some droplets within collision range, but the collision process itself is still within the viscous regime giving

$${\theta }_{i}^{{{\rm{mix}}}}={\theta }_{i}^{{{\rm{visc}}}}={C}_{{{\rm{V}}}}\cdot \sqrt{\frac{\epsilon }{\nu }}\cdot {L}_{i}^{3}$$
(7)

where \({C}_{{{\rm{V}}}}\) is a proportionality constant. Bandara et al.48 have \({C}_{{{\rm{V}}}}=\sqrt{\frac{8\pi }{15}}=\approx 1.3\), which we will use. Pecseli et al.49 as a comparison suggest \({C}_{{{\rm{V}}}}\approx 1.1\). For completeness, the expression for the turbulent regime can be found in the Supplementary Information.

The collision rate due to difference in rise velocity is not a function of dissipation rate, while the collision rate due to mixing is increasing with dissipation rate. Assuming that the mixing is in the viscous regime, the contribution from rise and mixing is equal when

$${\theta }_{i}^{{{\rm{rise}}}}={\theta }_{i}^{{{\rm{mix}}}}\,\Longleftrightarrow \,{\theta }_{i}^{{{\rm{rise}}}}={\theta }_{i}^{{{\rm{visc}}}}\,\Longleftrightarrow \pi {{\cdot L}_{i}}^{2}\cdot \left|w-{w}_{i}\right|={C}_{{{\rm{V}}}}\cdot \sqrt{\frac{\epsilon }{\nu }}\cdot {L}_{i}^{3}$$
(8)

and by solving for \(\epsilon\) we get equal collision contribution of mixing and rise when

$$\epsilon =\frac{{\pi }^{2}\cdot \nu }{{C}_{{{\rm{V}}}}^{2}\cdot {L}_{i}^{2}}\cdot {{|w}-{w}_{i}|}^{2}=\frac{15\pi \cdot \nu }{8\cdot {L}_{i}^{2}}\cdot {{|w}-{w}_{i}|}^{2}\, .$$
(9)

For the given egg and droplet characteristics under the assumed standard conditions of 7 °C and 35 psu, the value giving equal contribution is \(\epsilon \sim {10}^{-4}\, {{\rm{m}}}^{2}\cdot{{\rm{s}}}^{-3}\) and still well within the viscous assumption as seen in Figure S5.

Example scenario

In the numerical model, we use the values for NEA haddock described in Carroll et al.29 with egg characteristics originally based on the KILO report54. The assumed diameter is \({D}_{e}=1.45\,{{\rm{mm}}}\), which is within the range of other reported egg sizes for haddock55,56,57. Egg buoyancies are drawn from a uniform distribution between 29.78 and 31.58 psu giving a mean of 30.68 psu. This is within the range of pelagic eggs25 and more specifically Atlantic cod58 whose values previously have been used for NEA haddock59 as both species are cold-water gadoids inhabiting the same areas45,60. Note that the buoyancy of an individual egg typically changes in the order of 1 psu during the development of the egg37,61. However, lacking haddock specific parametrizations for this process, each egg here has a fixed buoyancy during the whole egg stage. With the assumed properties, the eggs have a mass of \(\sim 1.6\,{{\rm{mg}}}\). Hatching day is calculated as

$${Da}{y}_{{{\rm{hatch}}}}=-15.22+\frac{55.72}{{\mathrm{ln}}T\,}$$
(10)

from Geffen et al.62 where \(T\) is the average temperature [°C] experienced by an egg.

Numerical model details

The ocean model is a primitive Navier-Stokes equation model with 4 km horizontal resolution63,64,65,66. Oil is modelled using a multi-phase particle based model67,68,69,70 that uses forcing from the ocean model and include environmental weathering process of the oil. Exposure concentrations are calculated by interpolating the oil particles on a grid with 1.5 km horizontal resolution and 20 vertical layers. The haddock IBM uses the same implementation as in Carroll et al.29 originally based on Vikebø et al.71 for cod. Horizontal release positions of the 189,000 eggs are sampled uniformly from the whole spawning area, and all eggs are released at 30 m depth. Every 12 min, data is saved for all eggs giving the full histories of position, oil exposure, experienced environment and accumulation of oil droplets. Additional model details are given in the Supplementary Information, including the calculation of dissipation rate.

Droplet size classes

We use the four droplet size classes shown in Table 1 which are based on data from nine simulations with the same oil release as in our simulation. The classes are chosen such that each class covers 25% of the droplets (in numbers), except the largest size class that covers 20%. The remaining 5% of the droplets (69.8% of the mass) have a diameter larger than \(100\, \upmu{{\rm{m}}}\) and are not considered for collisions. Having a comparable number of droplets in each size class, the mass distribution is dominated by the one with largest droplet size.

During the simulation, we need to convert from the total oil droplet mass to droplet number concentrations as we don’t have a live size spectrum available. To do this, we use the information in Table 2. First, mass is distributed across size classes using the mass distribution of Table 1 which results in the conversion factors in the right-most column in Table 2. Further, the mass of each size class is converted to a number concentration using a characteristic mass for a droplet from the size class. Altogether, this gives the conversion factors from total oil droplet mass to size class number concentrations (second column from the right in Table 2), which are the ones used in the simulation. The characteristic diameter of a size class is calculated as the geometric mean of the diameter limits, and the characteristic mass is then given by the fixed oil density \({\rho }_{{{\rm{o}}}}=864\,{{\rm{kg}}}\cdot {{{\rm{m}}}}^{-3}\) of the Balder blend 201031.

Based on the choice of size limits, the number concentration in Table 2 should be equal in the three first classes with the last one being 80% of this value as in Table 1. However, this is not the case as using the geometric mean of the diameter limits does not conserve both number and mass concentration. It is possible to conserve both quantities, but the characteristic size of the size classes should then be calculated based on these two quantities instead of only using the limits. Here, we have chosen to conserve mass and use the limits to calculate a characteristic size, resulting in seemingly inconsistent droplet number concentrations between the classes. The collision formulation, however, is linear in both the droplet number concentration and the characteristic droplet mass. This implies that the amount of accumulated mass for a class is robust with regard to the choice of characteristic droplet diameter for the size class.