A common task in oceanography is to model the vertical movement of particles such as microplastics, nanoparticles, mineral particles, gas bubbles, oil droplets, fish eggs, plankton, or algae. In some cases, the distribution of the vertical rise or settling velocities of the particles in question can span a wide range, covering several orders of magnitude, often due to a broad particle size distribution or differences in density. This requires numerical methods that are able to adequately resolve a wide and possibly multi-modal velocity distribution.

Lagrangian particle methods are commonly used for these applications. A strength of such methods is that each particle can have its own rise or settling speed, which makes it easy to achieve a good representation of a continuous distribution of speeds. An alternative approach is to use Eulerian methods, where the partial differential equations describing the transport problem are solved directly with numerical methods. In Eulerian methods, different rise or settling speeds must be represented as discrete classes, and in practice, only a limited number of classes can be included.

Here, we consider three different examples of applications for a
water column model: positively buoyant fish eggs, a mixture of positively
and negatively buoyant microplastics, and positively buoyant oil droplets
being entrained by waves. For each of the three cases, we formulate a model
for the vertical transport based on the advection–diffusion equation with
suitable boundary conditions and, in one case, a reaction term. We give a
detailed description of an Eulerian and a Lagrangian implementation of these
models, and we demonstrate that they give equivalent results for selected
example cases. We also pay special attention to the convergence of the model
results with an increasing number of classes in the Eulerian scheme and with the
number of particles in the Lagrangian scheme. For the Lagrangian scheme, we
see the

Studying the vertical transport of positively, negatively, or neutrally buoyant particles is a common task in oceanography. Examples include both anthropogenic and naturally occurring particles, such as microplastics, mineral particles, nanoparticles, aggregates, gas bubbles, oil droplets, fish eggs, or even particles with active swimming behaviour such as zooplankton. These particles may show a range of different behaviours, including rising, sinking, and interacting with the ocean surface and seafloor in different ways.

Vertical transport modelling may be applied at different scales. In a simple
one-dimensional water column model, the goal may be to investigate the timescale
of settling or surfacing for a specific type of particle. However, accurate
modelling of vertical transport is also key to predicting horizontal transport
at large scales, due for example to the vertical variability of horizontal
ocean currents

Commonly used transport models may be divided into two classes: Eulerian and
Lagrangian. Eulerian methods consist of solving the advection–diffusion–reaction
equation directly with numerical methods for partial differential equations.
Ocean circulation models, such as ROMS

A challenge of the Eulerian approach, when it comes to particles with a distribution of rise or settling speeds, is that one must use discrete speed classes and solve the advection–diffusion equation for each class, forming a large system of equations to solve simultaneously. Hence, it is of interest to know how many classes are needed to achieve the desired accuracy.

Lagrangian particle methods are quite a popular choice for modelling particle
transport in the ocean

One of the advantages of a Lagrangian approach to particle transport modelling is the ability to represent a wide range of properties or behaviours. By letting each numerical particle with its own properties move independently of the others, one can model physical particle distributions where the sizes, and hence the terminal velocities, can vary by several orders of magnitude.

In practice, Eulerian and Lagrangian models may offer complementary benefits,
and they are often used together. This can be as simple as forcing a Lagrangian
model with pre-calculated fields from an Eulerian model (offline Lagrangian
modelling; see e.g.

The purpose of this paper is to compare and discuss Eulerian and Lagrangian methods, with a focus on the numerical implementation of the models including different boundary conditions and a reaction term. We demonstrate that the two implementations give the same results, and we also address questions of efficiency and convergence. The methods are illustrated using three different one-dimensional cases as a basis for the discussion: fish eggs, microplastics, and oil droplets. The cases have been chosen because they represent simplified (but realistic) cases, and they highlight how particles can interact with the boundaries in different ways.

We consider the water column without background flow and investigate the
vertical transport of different types of particles. In Sect.

In Sect.

Some additional details are given in the Appendices:
Appendix

As our starting point, we assume that the movement of a collection of particles
with different rise or settling velocities in the water column may be described
by the advection–diffusion equation (see e.g.

A mathematically equivalent formulation of the advection–diffusion problem is to
consider an ensemble of numerical particles, whose positions develop in time
according to the stochastic differential equation (SDE):

The connection between Eqs. (

In many practical applications, one will have a size distribution of particles
obtained from measurements, a model, or other estimates. For the model we
consider here, however, the relevant property is not the size itself but the
terminal rise or settling speed of the particle, which is a function of size
(and other particle parameters such as shape and density, as well as the
viscosity of the ambient fluid). Note that we assume here that a particle will
immediately attain its terminal velocity and that particle motion can be
described as the combination of a constant terminal velocity and a series of
random displacements. This is a commonly used approximation, which holds well if
the timescale needed to reach terminal velocity is short compared to other
relevant timescales. As an example, consider a particle with some initial speed

In the example cases we consider in Sect.

Depending on the application, different boundary conditions may be used to
control the fluxes across the domain boundaries in a
water column model. We will deal separately with the diffusive flux,

For the boundary conditions for the diffusive flux, we first observe that
diffusion in our model represents the mixing that occurs due to the combination
of random turbulent fluctuations and molecular diffusion

Next, we consider the advective flux. In our model, the advection velocity

As another example, consider positively buoyant oil droplets. When oil droplets rise to the surface, they go from being individual droplets surrounded by water to forming small patches of floating oil at the surface or possibly to merging with a larger surface slick. When this happens, the oil droplets are no longer subject to random motion due to turbulent diffusion, and the oil will remain at the surface until some high-energy event like a breaking wave causes the surface slick to break up and form droplets.

To express the mechanism of oil droplets surfacing and leaving the water column,
we have two options: we can use a loss term, which removes oil droplets close to
the surface at some rate

The reasons we describe the surfacing of oil droplets through the boundary
condition instead of a loss term are that it is straightforward to express the
boundary condition in both the Eulerian and the Lagrangian implementations and
that we do not have to define what it means to be close to the surface for the
loss term. The idea behind allowing an advective flux while forcing the
diffusive flux to be zero is that buoyancy is the mechanism that leads to
surfacing. Additionally, if the diffusive flux out of the system is nonzero,
then increased diffusivity would lead to faster surfacing, which is contrary to
observations. For an additional discussion of this point, see

We note that similar reasoning to the above also applies to the boundary at the seafloor. For example, the settling out of negatively buoyant particles (microplastics, sediments, etc.) can be modelled as an advective flux leaving the model domain through the bottom boundary (i.e. settling on the seafloor or being incorporated into sediments). Note also that, here, advective flux means the flux due to the rise or settling speed of the particle and not that due to vertical currents.

An additional reaction term

In this study, we will only consider reaction terms that add mass. The reaction
term

For the Lagrangian implementation, adding particles is done in a stochastic
manner, designed to be consistent with the source term in the Eulerian
description. For a source term that adds mass at some rate, in some region of
the domain, we add a random number of particles at each time step, such that the
expected value of mass added at each time step is equal to the integrated
source term. The position of each added particle is drawn from a distribution
that corresponds to the spatial distribution of the source term. As this applies to only one of the three cases considered (entrainment of oil droplets), the
details are given in the description of that case in Sect.

When we solve an advection–diffusion–reaction problem by means of a stochastic
Lagrangian particle model; the position (and possibly other properties) of each
numerical particle represents a sample from an underlying distribution. When we
draw random samples from a distribution and calculate, for example, the mean of the
samples, there will be a random error in the sample mean relative to the true
(but usually unknown) mean of the distribution. In particular, if we draw

There is a wide range of numerical methods for PDEs to choose from for solving
the advection–diffusion–reaction equation (see for example

A detailed description of the discretisation, as well as of the boundary conditions
and the numerical solver, is given in Appendix

As mentioned in Sect.

With a finite-volume method, it is trivial to implement prescribed-flux boundary conditions by simply setting one or both of the advective and diffusive fluxes across the cell face at the end of the domain to the prescribed value. In our case, we implement no-flux boundary conditions by setting both fluxes to zero across the boundary and no-diffusive-flux boundary conditions by setting only the diffusive flux to zero, leaving the advective flux unchanged.

In this study, we are interested in the scenario where our particles have a distribution of rising and/or settling velocities. To investigate the convergence of our Eulerian method with an increasing number of velocity classes, we need an automated way to represent the velocity distribution as a given number of discrete classes. Depending on the application, different ways of dividing the velocity distribution into intervals may be preferable.

In our selected approach, we first choose lower and upper limits:

To set up the initial condition, the total mass must be divided among the
different velocity classes. The velocity distribution might be specified
directly (see e.g.

To set up a case with

With equal spacing on a logarithmic scale, the calculation of mass fractions is
similar, but the limits on the integral are different. For constant logarithmic
spacing

To compare our Eulerian and Lagrangian implementations, we will present direct
comparisons of the predicted concentration, while an investigation of the
convergence of our solutions as functions of the different numerical parameters
requires a quantitative measure of the error. We have chosen to use the first
moment (i.e. the centre of mass) of the distribution. The reason for
this choice is primarily due to numerical methods for SDEs usually having a
well-defined weak convergence in terms of the moments of the distribution

The first moment of a distribution

Numerically, we approximate the first moment for the Eulerian results by
calculating the integral in Eq. (

As we do not have analytical solutions for the case studies we consider in
Sect.

The starting point for our Lagrangian implementation for solving the
advection–diffusion equation is the SDE given by Eq. (

For the numerical solution, we use the Euler–Maruyama method

Here,

If a method convergences in the weak sense, this implies that the moments of the
modelled distribution converge to the moments of the true distribution as

As described in Sect.

We split our numerical scheme (Eq.

random displacement (

reflection at the surface (

rise due to buoyancy (

stop particles at the surface (

For no-flux boundary conditions at the seafloor for negatively buoyant particles, the steps are analogous but with reflection and stopping at the bottom boundary instead.

To implement a no-diffusive-flux boundary condition while allowing an advective
flux across the boundary at

random displacement (

reflection at the surface (

rise due to buoyancy (

remove particles above the surface (

Depending on the application, particles that have been removed from the simulation may be re-introduced with some probability to represent, for example, breaking waves entraining material from the water surface or strong currents resuspending material from the seafloor.

In a Lagrangian model, a distribution of terminal rising or sinking velocities
can be represented very naturally simply by allowing each particle to have its
own vertical velocity drawn from the desired distribution. Hence, any
distribution can be represented, and the quality of the representation will
depend on the number of particles. The requirement for assigning a random
velocity to each numerical particle is that we can draw samples from the
velocity probability distribution. Depending on the available information,
different approaches might be suitable (see e.g.

Mathematically, the link between the Lagrangian and the Eulerian methods is that
each solution of the SDE in the Lagrangian method represents a sample from the
distribution that evolves under the advection–diffusion equation in the Eulerian
method. However, in practical applications, we are often interested in the
distribution itself and not just samples. Numerous methods exist for
reconstructing the distribution from samples as this is a common problem not
just in applied geoscience but in statistics in general. For a review of a few
different approaches in the context of applied oceanography, see e.g.

Here, we have chosen to use the simple approach of constructing a histogram of particle positions. This works well when the number of particles is large relative to the number of cells, as is the case for our one-dimensional examples below. An additional benefit is that the only free parameter is the bin size of the histogram, and a very direct comparison between Lagrangian and Eulerian results is obtained by setting the bin size to be equal to the cell size of the Eulerian approach.

As described in Sect.

For the Lagrangian implementation, convergence with the number of particles

We present three different cases and simulate all three with both an Eulerian
and a Lagrangian implementation of our model. Then, we compare the results and
discuss convergence in terms of numerical parameters for both implementations.
The cases are described below, and an overview of some aspects of the cases is
presented in Table

A brief summary of the boundary conditions (BCs) and reaction terms used in the three different cases.

Numerical parameters used for convergence analysis. The same parameters are used in all three cases.

For case 1, we consider positively buoyant fish eggs with a distribution of rise speeds. Fish eggs will float to the surface but do not leave the water column; hence, we model these with a no-flux boundary condition at the surface (i.e. the particles cannot leave the water column).

For case 2, we consider microplastic particles with a velocity distribution that includes both rising and sinking particles, representing the diversity of densities associated with different polymer types. In this case, we again use a no-flux boundary at the surface, while the boundary at the seabed is zero flux in diffusion, though an advective flux is allowed to leave the domain. This represents negatively buoyant microplastic particles that are removed from the water column by settling onto the seabed.

For case 3, we consider oil which is entrained as droplets when a surface slick is broken up by waves. The droplets are all positively buoyant with the same density but have a size distribution which varies in time and which leads to a distribution of rising speeds. In this case, the boundary condition at the surface is zero diffusive flux, though an advective flux that represents droplets that merge with the surface slick is allowed. Additionally, we include the effect of entrainment of oil from the surface slick by breaking waves in our model.

A spatially variable diffusivity profile is used in all three cases. To
calculate eddy diffusivity as a function of depth, we have used the GOTM
turbulence model

To support re-gridding and differentiation, an analytical expression has been fitted
to the discrete diffusivity output from GOTM. In Fig.

Note that both implementations make use of the analytical diffusivity profile:
in the Eulerian implementation, the diffusivity is given by the value of
Eq. (

Eddy diffusivity output from a GOTM simulation, shown with the fitted analytical diffusivity profile used for the case studies.

For each of the three cases, our initial condition will be a submerged particle
distribution. The spatial distribution will be a Gaussian distribution centred
at

In all cases, we will present a direct comparison of the predicted total
water column concentrations at different times from the Eulerian and the
Lagrangian implementations. Total concentration in this case means that it is integrated
over the velocity distribution, showing the total suspended concentration
regardless of rising or settling velocity. Additionally, we show a numerical
convergence analysis separately for the two schemes. For the Eulerian
implementation, we consider the error to be a function of the number of velocity
classes. For the Lagrangian implementation, we present the error as a function of
the number of particles. As a measure of the error, we consider the first moment
(the centre of mass) of the spatial concentration distribution (see
Sect.

In case 1, we consider pelagic fish eggs with a distribution of rise velocities.
These positively buoyant particles will rise towards the surface but do not
form a surface slick. Rather, they rise to the surface in stagnant water but
stay submerged and can be mixed back down by the eddy diffusivity

Taking an example from

The results of our simulations for fish eggs are shown in
Fig.

In panel (b) of Fig.

Finally, in panel (c), we show the convergence of the Lagrangian
results as a function of the number of particles

Results for case 1: fish eggs. Panel

In this case, we consider a distribution of microplastic particles. As the only parameter describing the numerical particles in our model is the terminal rise and settling velocities, we have obtained a velocity distribution based on published descriptions of microplastics.

As our starting point, we consider particles with a distribution of densities
from 0.8 to 1.5 kg L

For the Eulerian simulations, discretisations of the velocity distribution into
different numbers of classes were obtained as histograms with different numbers
of bins of

In this case, particles are allowed to leave the water column via the seafloor at 50 m depth, where we enforce zero diffusive flux while allowing the advective flux to carry particles across the boundary. This represents particles that leave the water column (and thus the simulation) by settling onto the seafloor, where they are no longer able to be resuspended by the eddy diffusivity. Different resuspension mechanisms can of course be included in the model, but we have chosen to ignore that here.

The results of our simulations for microplastics are shown in
Fig.

In panel (b), we show the convergence of the first moment in the Eulerian
results as a function of the number of classes. The

In panel (c), we show the convergence with the number of particles of the
standard deviation of the first moment for the Lagrangian implementation. Again,
as expected from the discussion in Sect.

Results for case 2: microplastics. Panel

In contrast to case 1, here we have particles with both positive and negative
terminal velocities. As the particles are allowed to leave the water column, the
total mass in suspension (i.e. the integral of the concentration
profile) decreases with time. Hence, we also present the remaining suspended
mass as a function of time. Panel (a) of Fig.

In this case, we consider a simplified one-dimensional oil spill model, which includes entrainment of surface oil by breaking waves; rising of oil droplets due to buoyancy; turbulent diffusive mixing; and oil droplets rising to the surface, creating a surface slick. As the entrainment mechanism is unique to this case, it is described in some detail below.

To model the entrainment of surface oil by breaking waves, we need the
entrainment rate, the entrainment depth, and the size distribution of the
entrained droplets. We model entrainment as a first-order decay process

From Eq. (

Using uniform entrainment throughout the interval

In the Lagrangian implementation, we use the approximation that the analytical
solution of Eq. (

Surfacing of oil droplets is described as a boundary condition where an
advective flux is allowed to pass through the boundary at the surface while
enforcing zero diffusive flux, as described in Sect.

The results of our simulations for oil droplets are shown in
Fig.

In panel (b), we show the convergence of the Eulerian results as a
function of the number of classes. As in case 1, the error scales with

Results for case 3: oil droplets with entrainment. Panel

The total amount of submerged oil changes in time as oil droplets surface and
are re-entrained. Over time, the submerged size distribution shifts towards
smaller droplets as these take longer to reach the surface, causing the centre
of mass to shift downwards over time. As before, the two schemes give very
similar results. As for case 2, we also present here the remaining suspended
mass as a function of time. Panel (a) of Fig.

In this paper, we have conducted a comparison of an Eulerian and a Lagrangian implementation of a water column model for particles with different distributions of rising and sinking speeds. To highlight different choices of boundary conditions and a reaction term, we have chosen to use fish eggs, microplastics, and oil droplets as our example cases. The model can also easily be applied to other cases, such as mineral particles, nanoparticles, algae, and chemicals. More complex reaction terms, such as agglomeration, can also be added to the same modelling framework.

Boundary conditions are always an essential part of the problem for any model
based on PDEs, and indeed, the boundary conditions must be specified before the
problem can be said to be well posed

Transport modelling with stochastic particle methods, on the other hand, is
perhaps more of a niche endeavour, and there exists less general applied
literature on the topic of numerical methods for SDEs compared to PDEs. The
standard reference on numerical solution of SDEs,

More than 11 000 citations according to Google Scholar (as of June 2023).

, does not directly address the question of boundary conditions but mentions a few references in the section on bibliographical notesIn the mathematical literature, an Itô diffusion (of which Eq.

In the applied literature on atmospheric transport modelling with random flight
models, the issue of boundary conditions has received some attention (see
e.g.

We have provided a detailed description of our implementation of two
different types of boundary conditions in the Lagrangian scheme. While our
implementation may not have a rigorous foundation in the theory of SDEs, we do
present a comparison between our Lagrangian and Eulerian implementations, where
the Eulerian method uses a standard approach for FVMs. In cases 2 and 3
(microplastics and oil droplets), we show that the two implementations give very
similar predictions of the amount of mass which remains suspended in the water
column as a function of time, which of course indicates that the net effect of
the boundary conditions is the same in the two implementations. For an additional
discussion of this topic, see also

We have paid special attention to the representation of particles with a distribution of terminal rising and settling velocities. In a Lagrangian particle model, a distribution of terminal velocities may be straightforwardly represented since each numerical particle can have its own velocity. In an Eulerian model, we have to introduce discrete classes with different velocities to represent the true distribution. We have investigated how the error is reduced with an increasing number of particles and classes respectively.

As a measure of the error in the Lagrangian implementation, we have used the
standard deviation of the first moment of the position distribution,

For the Eulerian implementation, we have considered the error in the modelled
first moment

We note, however, that the result showing

If the integral is evaluated numerically, there will be a numerical
error, which will depend on the chosen approach. Our approach has been to divide
the underlying velocity distribution into equal-sized bins and to let each bin be
represented by its midpoint (on either linear or logarithmic scales, depending on
the case). This is equivalent to the midpoint quadrature method, which is known
to have an error proportional to

Numerous studies have discussed different aspects and applications of Lagrangian
and Eulerian methods and have compared the two approaches (see e.g.

First we consider case 1, in which we modelled positively buoyant fish eggs with a
Gaussian distribution of terminal rise velocities. For the Eulerian
implementation, the numerical error in the first moment due to a finite number
of velocity bins starts out between

For the Lagrangian implementation, on the other hand, we need to use

In case 2 with the microplastics, we have a more complicated velocity
distribution. The positive and negative parts of the distribution are shown
separately on log scales in Fig.

For the Lagrangian implementation, the results of case 2 look far more similar
to those of case 1, and we find that

In case 3 with the oil droplets, we consider a simplified one-dimensional oil
spill model. Lagrangian approaches have a long history in oil spill modelling,
both for horizontal transport

It is important to point out that the errors discussed above are of different
natures for the Eulerian and the Lagrangian implementations. In the Eulerian
approach, solving with a finite number of speed classes,

In this paper, we have implemented and compared two different versions of a water column model for particles that undergo diffusion and that rise or sink with different distributions of terminal velocities. Our Eulerian implementation used discrete velocity classes to represent the velocity distribution, while the Lagrangian implementation allowed each particle to have its own velocity.

We have studied the rate of convergence of the two different implementations
considering the centre of mass (i.e. the first moment of the
concentration profile) as a measure of the error. Our main interest has been to
show that we can implement different boundary conditions in an equivalent manner
in the two schemes and to demonstrate how the numerical error is reduced with
an increasing number of velocity classes and particles for the Eulerian and
Lagrangian implementations respectively. Convergence results for varying
time steps and (in the Eulerian case) spatial resolutions are also shown in
Appendix

Three different example cases were considered: positively buoyant fish eggs with
a Gaussian velocity distribution, microplastics with a distribution of positive
and negative velocities obtained by means of a Monte Carlo approach (see
Appendix

For the Eulerian implementation, we find that the error appears to scale as

Owing to current shear and density gradients, vertical behaviour is very important for correctly modelling horizontal transport in the ocean. While it might be hard to draw strict conclusions about three-dimensional simulations from one-dimensional examples, we would still argue that a one-dimensional model captures a large part of the complexity of modelling particle distributions: horizontal advection and diffusion affect all particles equally, and the differences in vertical behaviour are well represented in a water column model. Hence, it is clear from our results that the number of classes needed to get accurate results in an Eulerian simulation will depend strongly on the velocity distribution of the relevant particles. We observed that far fewer classes were needed for the fish egg case with a Gaussian velocity distribution than for the microplastic case, which used a wider, multi-modal distribution. Similarly, the time-dependent size distribution (and thus velocity distribution) seen in the oil case needed more classes to give accurate results than the fish egg case. Hence, our results would suggest that Lagrangian particle models have a particular advantage when wider, less normal velocity distributions are considered or when the size distribution changes with time.

The advection–diffusion–reaction (ADR) equation describes how the concentration

In finite-volume methods (FVMs), the PDE is converted to an integral form by
integration over control volumes or cells. As illustrated in
Fig.

The integral form of Eq. (

Illustration of the discretisation of the

Next, we make an approximation by assuming the concentration to be constant
within each control volume such that the cell averages

In Eq. (

For the advective fluxes, however, linear numerical schemes of second-order
accuracy and higher are known to yield numerical oscillations for
advection-dominated problems (i.e.

Here, positive and negative velocities are handled separately by letting

The flux limiting was done with the UMIST limiter function

We see that

As described in Sect.

First, we consider the cell adjacent to the surface boundary. Note that
the uppermost cell is indexed

We note that setting

To implement a zero-flux boundary condition, we force both the diffusive
and advective fluxes to be zero. This we can do by setting the diffusive flux to
zero as above and additionally setting

The same reasoning applies to the boundary conditions at the seafloor.

The scheme was discretised in time with a fixed time step, such that

With the chosen discretisation schemes in space and time, our numerical solver
can be rewritten into a matrix equation in the form

The matrices

Here,

In all the cases we consider below, the matrices describing the advection and
diffusion terms remain constant in time. The flux limiter correction and the
reaction terms, on the other hand, are themselves functions of the
concentration

Writing out Eq. (

This equation must be solved at every time step to find the concentration

We then refine our guess by letting

For the cases we consider,

Schematic illustration of Eq. (

Equation (

If there are no reaction terms, there is no interaction between the different classes of particles; hence, the advection–diffusion equation can be solved independently for each class, as described in the previous section. Nevertheless, for reasons of flexibility, it may be convenient to implement an approach where the advection–diffusion equation is solved simultaneously for all classes since this allows interacting reaction terms to be added.

When there are reaction terms that allow the classes to interact, i.e. if the
reaction term

Our chosen approach is first to set up the equation system for each individual
class. The concentration of class

As a practical matter, we note that since the matrices

To obtain a distribution of rising and settling velocities for microplastic
particles, we combine the distributions for density, size, and shape presented by

We employ a Monte Carlo approach to obtain a velocity distribution, where we
draw a sample from each of the distributions describing density, size, and shape,
which we can then map to a terminal velocity via the relations of

The size distribution found by

To draw random samples from the distribution with the probability density
function (PDF) given by Eq. (

For each of the shape categories, the distributions for width and height are
assumed to be symmetric, triangular distributions, as described in

Probability for a microplastic particle to belong to different
shape categories, as well as the parameters of the symmetric, triangular
distributions for the width and height for each shape category

Based on the above, our algorithm for the generation of a random CSF for a
microplastic particle is as follows:

Select a shape category, with the probabilities from
Table

Generate a random width

Calculate the CSF from Eq. (

Number distribution of terminal particle velocity, found by mapping

Particle properties used to calculate settling speed. All
parameters are generated randomly, except equivalent diameter and shape
factors, which are calculated from the random parameters as described. Note
that the ranges of width and height depend on length since we assume

Four different expressions are given for the drag coefficient, distinguishing
between particles that are lighter or denser than water, as well as
distinguishing fibres from other shapes (pellets and fragments). For sinking non-fibre particles, the
expression is

The drag coefficients are given in terms of the CSF, the Reynolds number, and in
one case, the Powers' roundness, with the Reynolds number given by

The Powers' roundness

Even though

Note that the expressions given by

Based on the description above, we generate random realisations of particle
properties, where each particle has the properties shown in
Table

For the Lagrangian implementation,

To get the discrete velocity classes, we generated

An example histogram showing the distribution of

Here, we present the convergence of the first moment of the distribution with respect to the time step for the Lagrangian implementation and with respect to the time step and grid cell size for the Eulerian implementation. These results were used to choose suitable numerical parameters for the investigation of convergence with the number of classes and number of particles. They also serve as a sanity check to demonstrate that the numerical schemes behave as expected.

The results are shown in Fig.

We also note that the error in the first moment of the particle distribution for
a Lagrangian stochastic method has two terms: a discretisation term due to the
time step and a stochastic term due to the finite number of samples

For the Eulerian implementation, the convergence with time step is shown in panel (b) of Figs.

Convergence of the first moment of the distribution in case 1 as a function of the
time step for the Lagrangian implementation

Convergence of the first moment of the distribution in case 2 as a function of the
time step for the Lagrangian implementation

Convergence of the first moment of the distribution in case 3 as a function of
the time step for the Lagrangian implementation

The code used to run the simulations and to create the plots
shown in this paper is available under a GPL-3.0 license and can be found at

TN wrote the Lagrangian implementation, ran all simulations, and wrote the first draft of the paper. RK wrote the Eulerian implementation. All the authors contributed significantly to the review and editing of the paper.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors would like to thank Svein Sundby for the helpful advice on the velocity distribution of pelagic fish eggs.

The work of Tor Nordam was supported in part by the Norwegian Research Council project INDORSE (grant no. 267793) and in part by the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 814426 – NanoInformaTIX. The work of Erik van Sebille was supported through funding from the Netherlands Organization for Scientific Research (NWO), Earth and Life Sciences (grant no. OCENW.KLEIN.085) and through funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 715386).

This paper was edited by Sylwester Arabas and reviewed by Alethea Mountford and one anonymous referee.