# A Hybrid Lagrangian–Eulerian Particle Model for Ecosystem Simulation

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Property-Carrying Particle Model (PCPM)

- Read particle locations (x, y, z) and temperature. This step simply updates the location of each particle that is being used in the computation. Figure 2 is a conceptual representation of a PCPM computational cell, Particles (m
_{1}, m_{2}, m_{3},…) move in and out of the cell at each PCPM time step based on their trajectories as computed from the hydrodynamic model. The total number of particles for a particular computation is assumed to be fixed for the duration of the computation, although some particles may enter or leave the PCPM domain during the computation. Water temperature or other physical properties from the hydrodynamic calculation can be stored along with the pre-computed particle trajectories and can be included as one of the properties (P1, P2, P3, …) carried by the particle. - Determine the PCPM cell for each particle. In Figure 2, the PCPM cell is represented by the enclosing rectangle. The PCPM domain need not coincide with the domain that was used for the hydrodynamic simulation and computation of particle trajectories. It can be regular or irregular, as long as there is a prescribed method to calculate which PCPM cell contains a prescribed particle position (x, y, z). The PCPM cells are the volumes within which particle properties can interact, that is, during a single time step, all particles within a PCPM cell can influence the evolution of particle properties within that cell, but are independent of other cells.
- Apply boundary conditions to any particle-based properties that require them. If there is a property (e.g., concentration of a dissolved nutrient) that needs to be specified as a boundary condition, then particles within the cell where the boundary condition needs to be applied will have that property adjusted to meet the boundary condition. For example, in a cell that is associated with an inflow to the domain, the properties that are being carried into the domain through the inflow are adjusted to take account of the change in that property for particles within that cell. Alternatively, if particles from the hydrodynamic-based trajectory calculation are entering a PCPM cell, the values of the associated properties for each particle need to be specified.
- Calculate PCPM cell-based averages of each property. In this step, the averages of ${K}_{th}$ property for cell n are calculated as:$$\overline{P{K}_{n}}={\displaystyle \sum}_{j=1}^{L}P{K}_{{m}_{j}}/L$$
_{1}, m_{2},...${m}_{L}$) currently within cell n. L is the number of particles within that cell. If no particles are present in a particular cell, PCPM uses the values of $\overline{P{K}_{n}}$ from the previous time step. - Calculate the time evolution of the cell-based properties (and particle-based properties if necessary) using the process equation defined for that property. The process equations can incorporate terms which depend on either particle-based or cell-based properties, or both, i.e.,$$\overline{P{K}_{n}\left(t+\Delta t\right)}=FN\left(P{1}_{M}\left(t\right),P{2}_{M}\left(t\right),P{3}_{M}\left(t\right),\dots \overline{P{1}_{n}}\left(t\right),\overline{P{2}_{n}}\left(t\right),\overline{P{3}_{n}}\left(t\right),\dots \right)$$Note that M indicates m
_{1}, m_{2},...${m}_{L}$. The form of FN is completely general and depends on the problem being solved. For instance, in a NPZD model, the $Pi,(i=1,2,3,\dots $) would be N, P, Z, D, and water temperature, and the FN would be the process equations relating these properties.Since the cell-based averages have already been computed, the right-hand side of Equation (4) is independent of the left-hand side, so the computation of the evolution equations can be carried out in parallel. This is another key design feature of PCPM allowing it to take full advantage of multiprocessing computer environments, both symmetric multi-processing (SMP) and massively parallel processing (MPP). - Redistribute cell-based properties to particles within each cell by replacing the particle-based property with a weighted average of the cell-based property. After the evolution equations have been carried out (Step 5), particles within an individual cell most likely carry a range of different values of the various properties, which vary around the new cell-based average computed in Step 5, $\overline{P{K}_{n}\left(t+\Delta t\right)}$. PCPM provides an optional step to reduce the variance of the new particle-based properties within each cell. This optional step is applied as a ‘nudging’ term, i.e.,$$P{K}_{m}\left(t+\Delta t\right)=\left(1-{\alpha}_{i}\right)P{K}_{m}\left(t\right)+{\alpha}_{i}\overline{P{K}_{n}\left(t+\Delta t\right)}\text{}$$

#### 2.2. Idealized Case 1: Advection–Diffusion Plume

^{2}/s in this experiment), and $\Delta t$ is the time step for the particle trajectory calculation (1 s).

#### 2.3. Idealized Case 2: Vertical Settling

_{m}in vertical cell k looks like:

^{−4}m

^{2}s

^{−1}, and the redistribution parameter α = 0.1. Three runs were made with 5, 10, and 20 vertical cells respectively. The PCPM is integrated in time with $\Delta t$ = 1 h.

#### 2.4. Sandusky Bay Model

#### 2.5. Sandusky Bay Biological Model

## 3. Results

#### 3.1. Idealized Case 1: Advection–Diffusion Plume

#### 3.2. Idealized Case 2: Vertical Settling

#### 3.3. Application to Sandusky Bay

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Biological model formulation and parameters; the original value of the parameters is provided in ( ) if tuned value was used.

Parameters | Description | Units | Value Used | References |
---|---|---|---|---|

${k}_{s}$ | Half-Saturation constant | µmol N/L | 3 (0.6) | [17] |

$No$ | Nutrient threshold | µmol N/L | 0 (1.4) | [18] |

${\alpha}_{I}$ | Initial linear slope at low irradiances | $\frac{\mathrm{mgC}\xb7{\mathrm{m}}^{2}}{\mathrm{mgChl}\xb7\mathrm{Einst}}$ | 7 | [18] |

${\beta}_{I}$ | Negative slope at high irradiances | $\frac{\mathrm{mgC}\xb7{\mathrm{m}}^{2}}{\mathrm{mgChl}\xb7\mathrm{Einst}}$ | 0 | [43] |

${u}_{max}$ | Maximum potential growth rate | $\frac{\mathrm{mgC}}{\mathrm{mgChl}\xb7\mathrm{h}}$ | 2.4 | [43] |

${T}_{opt}$ | Optimum temperature | °C | 27.2 | [42] |

${T}_{min}$ | Minimum temperature | °C | 5.5 | [42] |

${u}_{p(max)}$ | Maximum growth rate for P | day^{−1} | 1.1 | |

${\gamma}_{p}$ | Phytoplankton respiration coefficient | day^{−1} | 0.01 | [17] |

${\gamma}_{T}$ | Exponential for Temperature forcing | dimensionless | 0.07 | [17] |

${\gamma}_{d}$ | Remineralization rate of detritus | day^{−1} | 0.015 | [17] |

${G}_{max}$ | Maximum P grazing rate by Z | day^{−1} | 0.4 | [17] |

${\sigma}_{P}$ | Preference coefficient of Z on P | ${\left({\mathrm{mmol}\text{}\mathrm{C}\text{}\mathrm{m}}^{-3}\right)}^{-1}$ | 0.5 | [17] |

${\sigma}_{D}$ | Preference coefficient of Z on D | ${\left({\mathrm{mmol}\text{}\mathrm{C}\text{}\mathrm{m}}^{-3}\right)}^{-1}$ | 0.1 | [17] |

${\u03f5}_{p}$ | Mortality rate of P | day^{−1} | 0.005 (0.01) | [17] |

${\u03f5}_{Z}$ | Mortality rate of Z | day^{−1} | 0.2 (0.01) | [17] |

${W}_{P}$ | Sink velocity of P | m/day | 0.6 | [17] |

${W}_{D}$ | Sink velocity of D | m/day | 0.6 | [17] |

${a}_{w}$ | Water attenuation coefficient | m^{−1} | 0.07 | [18] |

${a}_{p}$ | Phytoplankton attenuation coefficient | ${\mathrm{mgChl}}^{-1}{\mathrm{m}}^{2}$ | 0.03 | [18] |

${a}_{d}$ | Detritus attenuation coefficient | ${\mathrm{g}\text{}\mathrm{detritus}\text{}\mathrm{C}}^{-1}{\text{}\mathrm{m}}^{2}$ | 0.2 | [18] |

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**Figure 1.**Sandusky Bay is situated on Lake Erie’s south-west coast occupying a small portion of the Great Lake’s coastline. Sandusky Bay is relatively shallow bay with an average depth of ~2.6 m. The primary draining watershed to Sandusky Bay originates from the Sandusky River on the west end of the bay. Sampling stations ODNR1 and EC1163 are denoted with green dots.

**Figure 2.**Conceptual representation of a property-carrying particle model (PCPM) computational cell n and particles (m

_{1}, m

_{2}, m

_{3}, m

_{4}, m

_{5}…) within the cell n. PCPM cell-based average of each property ($\overline{P{1}_{n}}$, $\overline{P{2}_{n}}$, $\overline{P{3}_{n}}$,…) is determined by the property values carried by the particles that have entered in this cell. After time evolution of PCPM properties using process equations, the updated PCPM cell-based properties ($\overline{P{1}_{n}}$, $\overline{P{2}_{n}}$, $\overline{P{3}_{n}}$,…) are redistributed to particles with a weighted average. The particles then move around carrying the updated properties to different PCPM computational cells in the next cycle.

**Figure 3.**Finite volume community ocean model (FVCOM) model mesh for Lake Erie (

**upper**panel) and linked with a high-resolution mesh for Sandusky Bay (

**lower**panel). Only a portion of the Sandusky Bay mesh is displayed for a clear representation of the mesh’s resolution.

**Figure 4.**A schematic representation of the nutrient-phytoplankton-zooplankton-detritus (NPZD) model.

**Figure 5.**PCPM simulation of concentration plume in an idealized channel with four different values of the cell-based redistribution weight parameter (α = 0, 0.01, 0.1, 0.5). There are three panels for each value of α. The

**top**panel shows the locations of particles after 720 time steps (12 min). The

**second**panel shows the average concentration in each 10 m square cell with the same blue to red scale as the top panel, except cells with C = 0 are black. The

**third**panel compares concentration along the center line of the plume from the second panel to the analytical solution for a diffusive plume.

**Figure 6.**The PCPM simulation of vertical settling in comparison to the analytical solution at the end of 5000 time steps. Three runs were made with 5 (

**left**panel), 10 (

**middle**panel), and 20 (

**right**panel) vertical cells, respectively. The dots represent the locations of the particles on the vertical axis with their respective concentration on the horizontal axis. The thin line represents the cell average concentration and the thick line represents the analytical solution.

**Figure 7.**The time evolution of the root mean square difference (RMSD) between the cell averages and the analytical solution for the three cases presented in the Figure 3 (dark line for 5 cells, medium line for 10 cells, and light line for 20 cells).

**Figure 8.**River plumes at selected times (labelled in each panel) simulated with conventional soluble-tracer model (

**left**panels) and PCPM model (

**right**panels). The color scale represents the tracer concentration.

**Figure 9.**Observed and model simulated chlorophyll concentration at the sampling stations EC1163 (

**left**panels) and ODNR1 (

**right**panels). The upper panels are results from the PCPM-NPZD-NOADV (no advection) model simulations; the lower panel are the results from the realistic PCPM-NPZD model simulations where the three cases show nearly identical results, so only one result is plotted.

**Table 1.**The number of new particles released and the total water volume input from the Sandusky River in each month.

Month | Monthly Discharge (10^{8} m^{3}) | Particles (Number) |
---|---|---|

January | 0.364 | 10,405 |

February | 2.27 | 64,980 |

March | 7.19 | 205,367 |

April | 1.53 | 43,741 |

May | 0.443 | 12,640 |

June | 3.00 | 85,754 |

July | 1.97 | 56,347 |

August | 0.129 | 3680 |

September | 0.089 | 2544 |

October | 0.093 | 2654 |

November | 0.133 | 3798 |

December | 1.15 | 32,951 |

**Table 2.**The comparison of total run time using the method developed in this study (new method) and the grid-based Eulerian method (traditional method) in two scenarios. Scenario 1: conduct coupled biophysical model only once; Scenario 2: run ensemble simulation of the coupled biophysical model for 100 simulations with different biological parameterization.

Simulation Period (day) | Number of Central Processing Units (CPUs) | Time Per Run (Hour) | Scenario 1 | Scenario 2 | |||
---|---|---|---|---|---|---|---|

RequiredRuns | Total Time | RequiredRuns | Total Time | ||||

Hydrodynamic simulation (required for both methods) | 30 | 64 | 15 | 1 | 15 h | 1 | 15 h |

Biological simulation | |||||||

New Method | |||||||

Particle trajectory model | 30 | 1 | 1.5 | 1 | 1.6 hrs (1 CPU) | 1 | 11.5 hrs (1 CPU) |

PCPM-NPZD | 30 | 1 | 0.1 | 1 | 100 | ||

Traditional Method | |||||||

Eulerian, grid-based simulation | 30 | 32 | 5 | 1 | 5 hrs (32 CPUs) | 100 | 500 hrs (32 CPUs) |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Xue, P.; Schwab, D.J.; Zhou, X.; Huang, C.; Kibler, R.; Ye, X.
A Hybrid Lagrangian–Eulerian Particle Model for Ecosystem Simulation. *J. Mar. Sci. Eng.* **2018**, *6*, 109.
https://doi.org/10.3390/jmse6040109

**AMA Style**

Xue P, Schwab DJ, Zhou X, Huang C, Kibler R, Ye X.
A Hybrid Lagrangian–Eulerian Particle Model for Ecosystem Simulation. *Journal of Marine Science and Engineering*. 2018; 6(4):109.
https://doi.org/10.3390/jmse6040109

**Chicago/Turabian Style**

Xue, Pengfei, David J Schwab, Xing Zhou, Chenfu Huang, Ryan Kibler, and Xinyu Ye.
2018. "A Hybrid Lagrangian–Eulerian Particle Model for Ecosystem Simulation" *Journal of Marine Science and Engineering* 6, no. 4: 109.
https://doi.org/10.3390/jmse6040109