# Separating Mesoscale and Submesoscale Flows from Clustered Drifter Trajectories

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## Abstract

**:**

## 1. Introduction

## 2. Modelling Framework

#### 2.1. Local Taylor Expansion

- $\{{x}_{k}\left(t\right),{y}_{k}\left(t\right)\}$ are observations from drifter k at time t;
- $\{{u}^{\mathrm{bg}}\left(t\right),{v}^{\mathrm{bg}}\left(t\right)\}$ is the spatially homogeneous time-varying background flow;
- $\{{u}_{0},{v}_{0},{u}_{1},{v}_{1},{\sigma}_{n},{\sigma}_{s},\zeta ,\delta \}$ are the model parameters for the mesoscale flow;
- $\{{x}_{0},{y}_{0}\}$ is the expansion location and has no consequence to the model, other than redefining $\{{u}_{0},{v}_{0}\}$;
- $\{{u}_{k}^{\mathrm{sm}}\left(t\right)$${v}_{k}^{\mathrm{sm}}\left(t\right)\}$ are the residual ‘submesoscale’ velocities for each drifter, assumed to be zero-mean in time, but also zero-mean in space across drifters.

#### 2.2. Diffusivity

#### 2.3. Model Solutions

## 3. Estimation and Hierarchical Modelling

#### 3.1. Parameter Estimation

#### 3.2. Flow Decomposition

#### 3.3. Hierarchical Modelling

#### 3.4. Selecting between Hierarchies

#### 3.4.1. Fraction of Variance Unexplained (FVU)

#### 3.4.2. Fraction of Diffusivity Unexplained (FDU)

## 4. Uncertainty Quantification and Capturing Temporal Evolution

#### 4.1. Uncertainty Quantification

#### 4.2. Time-Evolving Parameters Using Rolling Windows

#### 4.3. Slowly-Evolving Parameters Using Splines

## 5. Application to the Latmix Experiment

#### 5.1. Fixed Mesoscale Parameter Estimates

#### 5.2. Time-Evolving Parameters Using Rolling Windows

#### 5.3. Slowly-Evolving Parameters Using Splines

## 6. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Sensitivity Analyses

**Figure A1.**Relative standard error of stain rate, strain angle and vorticity over 100 repeated simulations for a varying number of drifters K. The simulation setup is as in Figure 1 in the strain-dominated model with trajectories simulated for 1 day. The initial drifter positions are sampled isotropically with expected distance to centre-of-mass fixed over all experiments to be identical to Latmix Site 1. Relative standard error is computed by dividing the observed sample standard error by the true parameter value. Therefore in this experiment we require approximately three drifters in the cluster before the standard errors are approximately half the true parameter value (and hence significantly non-zero).

**Figure A2.**Different cluster morphologies (deployment configurations) we shall consider. In the left panel we consider nine drifters deployed as at Latmix Site 1 (blue dots), together with nine drifters deployed parallel and orthogonal to the strain angle (red and green dots respectively). In the right panel we again consider nine drifters deployed as at Latmix Site 1 (blue dots), but this time the red and green dots are the same morphologies but with the respective distances to the centre-of-mass either doubled or halved. In both panels the velocity field is as in the strain-only simulation of Figure 1, and the positions are given in centre-of-mass coordinates.

**Figure A3.**Required window lengths to obtain significant strain rate estimates for different drifter configurations. The lines in the left/right panels correspond to the drifter configurations considered in the left/right panels of Figure A2 respectively, with the colours matching the corresponding configurations. Each line corresponds to the level where the standard error of the strain rate estimate is approximately half the true strain rate value. These lines are found as in Figure 8 over 100 repeated simulations over a grid of true strain rates and window lengths.

## References

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**Figure 1.**Simulation of nine drifters from Equation (2) over 6.25 days, with starting positions, number of drifters, and experiment length taken to match LatMix Site 1. In each panel the submesoscale velocities $\{{u}_{k}^{\mathrm{sm}}\left(t\right),{v}_{k}^{\mathrm{sm}}\left(t\right)\}$ follow a Wiener increment process with diffusivity equal to 0.1 m${}^{2}$/s. The top row shows drifter positions, and the bottom row shows positions with respect to centre-of-mass at each time step. From left to right we include the following model components. left: diffusivity only. Centre left: strain and diffusivity. Centre right: strain, vorticity, and diffusivity (strain dominated). right: strain, vorticity, and diffusivity (vorticity dominated). In each plot where a parameter is present, it has been set as $\sigma =7\times {10}^{-6}$/s, $\theta ={30}^{\circ}$, $\zeta =6\times {10}^{-6}$/s (centre right), and $\zeta =8\times {10}^{-6}$/s (right). We have set ${u}_{0}={v}_{0}={u}_{1}={v}_{1}={u}^{\mathrm{bg}}={v}^{\mathrm{bg}}=0$. The trajectories are simulated using the Euler–Maruyama scheme [13] and we include quivers in all plots representing the underlying velocity field.

**Figure 2.**The one-sided frequency spectrum for a particle integrated with Equation (9) is shown in black. The particle is initially placed at $\left\{x\right(0),y(0\left)\right\}=\{1\mathrm{km},1\mathrm{km}\}$ and integrated for 5 days in a strain-only model with simulation parameters set to $\kappa =0.1$ m${}^{2}$/s and $\sigma =1\times {10}^{-5}/$s. The theoretical spectrum of the mesoscale process, Equation (17), is shown in blue, and the theoretical spectrum of the white noise process, Equation (10), is shown in red.

**Figure 3.**Hierarchy of mesoscale models using the second-moment fitting method where p indicates the number of parameters. A model with increased complexity is used only if it explains significantly more variance than the lower complexity model. Models with fewer parameters are favoured when a choice must be made.

**Figure 4.**FVU (left column) and FDU (right column) for candidate models fitted to trajectories generated from the four model scenarios from Figure 1. Each subplot here is for a different true model scenario (the y-axis), and each box and whisker within a subplot provides the spread of FVU/FDU values from a fitted candidate model (the x-axis). The final box and whisker in each subplot is using the true mesoscale parameter values. The spread of results is over 100 repeated simulations using nine drifters sampled every 30 min for one day. The estimated theoretical FVU, obtained from Equation (28), and the estimated theoretical FDU, obtained from Equation (30), are overlaid by a red horizontal line in each subplot. Parameters are estimated using the second-moment fitting method, where results using the first and second-moment fitting method yield near identical results as ${u}_{0}={v}_{0}={u}_{1}={v}_{1}={u}^{\mathrm{bg}}={v}^{\mathrm{bg}}=0$ in these simulations.

**Figure 5.**Histogram of bootstrap parameter estimates for strain rate, strain angle, and vorticity, over 100 repeated simulations where $B=100$ for each simulation, thus obtaining 10,000 total bootstrapped parameter values. The trajectories are generated as in Figure 1 in the strain-dominated model for 1 day, and the parameters are estimated using the second-moment fitting method. Any bootstrap estimates outside the range of the x-axis are placed in the limiting visible bar in the histogram on each side. The red vertical line is the true parameter value, and the blue vertical line is the average bootstrap estimate.

**Figure 6.**Simulation of nine drifters using the identical configuration of Figure 1 (strain only) except that the strain rate changes linearly across time from $\sigma =1\times {10}^{-5}$/s to $\sigma =1\times {10}^{-6}$/s and $\kappa =0.5$ m${}^{2}$/s. The left panel displays drifter positions. The right panel displays drifter positions with respect to their centre-of-mass. The quiver arrows indicate the velocity field at the beginning of the simulation.

**Figure 7.**The left panel shows rolling-time window estimates of the varying strain rate from the data presented in Figure 6 over three choices of window lengths using the second-moment fitting method. The right panel shows the standard error of these time-varying estimates over 100 repeated simulations, plotted against the true value of $\sigma /2$.

**Figure 8.**Estimated standard errors for the strain rate (in the units of the true strain rate) across a dense grid of fixed strain rate values $\sigma $ and window lengths W in a strain-only simulation mirroring the setup in Figure 1. In the left panel we have set $\kappa =0.1$ m${}^{2}$/s and in the right $\kappa =1$ m${}^{2}$/s. The strain rate estimates are obtained using the second-moment fitting method of a strain-only model, and the standard errors are obtained over 100 repeated simulations. The standard errors in the heatmap are upper-bounded by 0.9 for representation purposes. We draw a red line where the standard error is approximated to be half the true parameter value for each value of the strain rate.

**Figure 9.**Hierarchy of first and second-moment mesoscale models where p indicates the number of parameters. A model with increased complexity is used only if it explains significantly more variance than the lower complexity model. Models with fewer parameters are favoured when a choice must be made.

**Figure 10.**LatMix trajectories of Site 1 (nine drifters) and Site 2 (eight drifters). top row are the positions in $\{{x}_{k}\left(t\right),{y}_{k}\left(t\right)\}$, bottom row are relative to centre-of-mass $\{{\overline{x}}_{k}\left(t\right),{\overline{y}}_{k}\left(t\right)\}=\{{x}_{k}\left(t\right)-\frac{1}{K}{\sum}_{k=1}^{K}{x}_{k}\left(t\right),{y}_{k}\left(t\right)-\frac{1}{K}{\sum}_{k=1}^{K}{y}_{k}\left(t\right)\}$. The black and red star in the top row of plots indicate the respective starting and ending centre-of-mass positions. $\{0,0\}$ in the $\{x,y\}$ components corresponds to $\{-73.0234,31.7424\}$ degrees longitude-latitude for Site 1 and $\{-73.6776,32.2349\}$ degrees longitude-latitude for Site 2.

**Figure 11.**Fixed (red) and time-varying (blue) parameter estimates, where the latter are generated with a one-day rolling window using the second-moment fitting method. Top-Left: strain rate estimates with the strain-only model (Site 1). Top-Right: strain rate estimates with the strain-only model (Site 2). Bottom-Left: strain rate estimates with the strain-vorticity model (Site 2). Bottom-Right: vorticity estimates with the strain-vorticity model (Site 2). 100 bootstrapped time-varying trajectories are shown in grey in each subplot.

**Figure 12.**Distribution of strain rate parameters estimated for the first two days of the LatMix experiment at Site 1. Contours indicate the percentage of samples enclosed. The left panel shows estimated strain rate parameters using only the second-moment fitting method, where the right panel shows estimates using the first and second-moment fitting method.

**Figure 13.**Parameters of the spline based strain model fits to Site 1 (left panel) and strain-vorticity model fits to Site 2 (right panel) using the first and second-moment fitting method. The most likely solution is highlighted, with 90% and 68% most likely solutions shown in grey and dark grey, respectively. The models are fit using four degrees of freedom per parameter with the splines shown in the bottom row.

**Figure 14.**Decomposition of the flow at LatMix Site 1 using the strain-only model fitted with splines using the first and second-moment fitting method. The left panel shows the the mesoscale solution in the fixed coordinate reference frame (compare to the upper-left panel of Figure 10). The centre panel shows the same solution in the centre-of-mass frame (compare to the lower-left panel of Figure 10). The top-right and bottom-right panels show the path-integrated background and submesoscale flow, respectively.

**Figure 16.**The top and bottom panels show the power spectra of the decomposed flow for Sites 1 and 2, respectively. The spectra shown are the spatially homogeneous background flow ${\mathbf{u}}^{\mathrm{bg}}$ (black), the average of the mesoscale component of the flow ${\mathbf{u}}^{\mathrm{meso}}$ (blue), and the average of the submesoscale component ${\mathbf{u}}^{\mathrm{sm}}$ (magenta). Anticyclonic oscillations are indicated by negative frequencies and cyclonic oscillations by positive frequencies. The vertical lines indicate the semi-diurnal tidal frequency and the inertial frequency on the positive and negative side, respectively.

**Table 1.**Observed standard errors from simulation, and average bootstrap standard error estimates from Equation (31) (where $B=100$), over 100 repeated simulations, for both the strain-only and strain-dominated simulations of Figure 1 over 1 day. We also provide the standard deviation of bootstrap standard error estimates across the 100 simulations, as indicated after the ± symbol.

$\mathit{\sigma}$$\left({\mathit{s}}^{-1}\right)\times {10}^{6}$ | $\mathit{\theta}$${(}^{\circ})$ | $\mathit{\zeta}\left({\mathit{s}}^{-1}\right)\times {10}^{6}$ | |
---|---|---|---|

Strain-only Simulation | |||

Simulated Bootstrap | $1.17$ $1.32\pm 0.365$ | $6.68$ $6.29\pm 2.47$ | N/A N/A |

Strain-dominated Simulation | |||

Simulated Bootstrap | $1.22$ $1.55\pm 0.459$ | $6.78$ $8.08\pm 3.43$ | $1.61$ $1.94\pm 0.572$ |

**Table 2.**LatMix submesoscale diffusivity estimates and associated FVU and FDU, estimated over candidate models in the hierarchy at each site using either fixed, rolling window, or spline parameter estimates. For fixed estimates we also show the mesoscale parameter estimates (scaled by the inertial frequency, ${f}_{0}$). The fixed and rolling-window estimates use the second-moment fitting method, whereas the spline estimates uses the first and second-moment fitting method.

Fixed Estimates (Site 1) | |||||||

model | $\sigma $$\left({f}_{0}\right)$ | $\theta $${(}^{\circ})$ | $\zeta $$\left({f}_{0}\right)$ | $\delta $$\left({f}_{0}\right)$ | $\kappa $ (m${}^{2}$/s) | FVU | FDU |

$\left\{\zeta \right\}$ | 0 | 0 | −0.000137 | 0 | 0.974 | 1.000 | 1.001 |

$\left\{\delta \right\}$ | 0 | 0 | 0 | 0.0493 | 0.361 | 0.983 | 0.371 |

$\{\sigma ,\theta \}$ | 0.0591 | −27.8 | 0 | 0 | 0.188 | 0.976 | 0.193 |

$\{\sigma ,\theta ,\zeta \}$ | 0.0785 | −15.3 | −0.0443 | 0 | 0.229 | 0.971 | 0.235 |

$\{\sigma ,\theta ,\delta \}$ | 0.0489 | −25.6 | 0 | 0.0137 | 0.174 | 0.976 | 0.179 |

$\{\sigma ,\theta ,\zeta ,\delta \}$ | 0.0711 | −12.2 | −0.0443 | 0.0137 | 0.216 | 0.971 | 0.221 |

Fixed Estimates (Site 2) | |||||||

model | $\sigma $$\left({f}_{0}\right)$ | $\theta $${(}^{\circ})$ | $\zeta $$\left({f}_{0}\right)$ | $\delta $$\left({f}_{0}\right)$ | $\kappa $ (m${}^{2}$/s) | FVU | FDU |

$\left\{\zeta \right\}$ | 0 | 0 | 0.00613 | 0 | 4.011 | 0.999 | 1.000 |

$\left\{\delta \right\}$ | 0 | 0 | 0 | 0.0125 | 1.886 | 0.997 | 0.470 |

$\{\sigma ,\theta \}$ | 0.0131 | −67.0 | 0 | 0 | 1.906 | 0.996 | 0.475 |

$\{\sigma ,\theta ,\zeta \}$ | 0.0642 | 78.0 | 0.0650 | 0 | 1.950 | 0.985 | 0.486 |

$\{\sigma ,\theta ,\delta \}$ | 0.0107 | −67.9 | 0 | 0.00258 | 1.874 | 0.996 | 0.467 |

$\{\sigma ,\theta ,\zeta ,\delta \}$ | 0.0637 | 77.0 | 0.0650 | 0.00258 | 1.919 | 0.985 | 0.478 |

Rolling Estimates (Site 1) | Rolling Estimates (Site 2) | ||||||

model | $\kappa $ (m${}^{2}$/s) | FVU | FDU | $\kappa $ (m${}^{2}$/s) | FVU | FDU | |

$\left\{\zeta \right\}$ | 0.995 | 0.992 | 1.022 | 2.924 | 0.872 | 0.729 | |

$\left\{\delta \right\}$ | 0.325 | 0.974 | 0.334 | 2.341 | 0.838 | 0.584 | |

$\{\sigma ,\theta \}$ | 0.183 | 0.961 | 0.188 | 1.680 | 0.710 | 0.419 | |

$\{\sigma ,\theta ,\zeta \}$ | 0.282 | 0.937 | 0.290 | 0.825 | 0.675 | 0.206 | |

$\{\sigma ,\theta ,\delta \}$ | 0.147 | 0.966 | 0.151 | 1.753 | 0.704 | 0.437 | |

$\{\sigma ,\theta ,\zeta ,\delta \}$ | 0.248 | 0.941 | 0.255 | 0.722 | 0.669 | 0.180 | |

Spline Estimates (Site 1) | Spline Estimates (Site 2) | ||||||

model | $\kappa $ (m${}^{2}$/s) | FVU | FDU | $\kappa $ (m${}^{2}$/s) | FVU | FDU | |

$\left\{\zeta \right\}$ | 1.742 | 1.025 | 1.791 | 3.059 | 0.973 | 0.697 | |

$\left\{\delta \right\}$ | 0.342 | 0.983 | 0.352 | 3.438 | 0.831 | 0.783 | |

$\{\sigma ,\theta \}$ | 0.178 | 0.976 | 0.183 | 2.118 | 0.837 | 0.483 | |

$\{\sigma ,\theta ,\zeta \}$ | 1.433 | 0.997 | 1.473 | 1.041 | 0.808 | 0.237 | |

$\{\sigma ,\theta ,\delta \}$ | 0.159 | 0.974 | 0.163 | 2.501 | 0.783 | 0.570 | |

$\{\sigma ,\theta ,\zeta ,\delta \}$ | 1.446 | 0.996 | 1.487 | 1.466 | 0.770 | 0.334 |

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**MDPI and ACS Style**

Oscroft, S.; Sykulski, A.M.; Early, J.J.
Separating Mesoscale and Submesoscale Flows from Clustered Drifter Trajectories. *Fluids* **2021**, *6*, 14.
https://doi.org/10.3390/fluids6010014

**AMA Style**

Oscroft S, Sykulski AM, Early JJ.
Separating Mesoscale and Submesoscale Flows from Clustered Drifter Trajectories. *Fluids*. 2021; 6(1):14.
https://doi.org/10.3390/fluids6010014

**Chicago/Turabian Style**

Oscroft, Sarah, Adam M. Sykulski, and Jeffrey J. Early.
2021. "Separating Mesoscale and Submesoscale Flows from Clustered Drifter Trajectories" *Fluids* 6, no. 1: 14.
https://doi.org/10.3390/fluids6010014