# Parametrization of Eddy Mass Transport in the Arctic Seas Based on the Sensitivity Analysis of Large-Scale Flows

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Model of the Kara Sea

#### 2.2. Parametrization of Cross-Isobathic Transport

#### 2.3. Large-Scale Representation of Mesoscale Characteristics

#### 2.4. Large-Scale Flow Characteristics

#### 2.5. Independence of Large-Scale Characteristics

- The depth of the ocean H, in situ depth z, vertical component of in situ density gradient $\partial \rho /\partial z$, and geographical latitude $\theta $ turned out to be strongly related to each other. Among them, the most prominent physical meaning has $\partial \rho /\partial z$ value as it defines the Brunt–Väisälä frequency of internal waves.
- The flow velocity in the direction of slope U, the component of the in situ density gradient in the direction of the slope $\partial \rho /\partial \overrightarrow{n}$ turned out to be strongly related to the value of this characteristic near the bottom ${\left.\partial \rho /\partial \overrightarrow{n}\right|}_{H}$. Among them, we chose the latter, because according to [30], it is an important characteristic of cascading.
- Similarly, the near-bottom density gradient component along the isobath ${\left.\partial \rho /\partial \overrightarrow{m}\right|}_{H}$ turned out to be strongly dependent on the in situ value of this gradient $\partial \rho /\partial \overrightarrow{m}$, so this value was neglected in favor of the second one.
- The flow velocity in the direction along the isobaths V turned out to be dependent on the down-slope component of the bottom density gradient ${\left.\partial \rho /\partial \overrightarrow{n}\right|}_{H}$, which is already chosen.
- The divergence of the velocity components $\partial U/\partial \overrightarrow{n}$ and $\partial V/\partial \overrightarrow{m}$ are related by the continuity equation, we chose the latter one of them.

#### 2.6. Clustering

- The center of the first cluster is determined randomly.
- The center of the second cluster is determined randomly with a probability proportional to the distance to the center of the first cluster.
- The center of the next i-th cluster is also determined randomly with a probability proportional to the minimum among the distances to the known cluster centers.

#### 2.7. Analysis of Dependencies on the Selected Parameters

## 3. Results

#### 3.1. Clustering Results

#### 3.2. Parametrization Dependencies—Cluster A

#### 3.3. Practical Use

#### 3.4. Diffusion Coefficient Test

#### 3.5. Counter-Gradient Tests

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Davies–Bouldin Index

#### Appendix A.2. Dunn Index

#### Appendix A.3. Silhouette Coefficients

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**Figure 2.**Geographical localization of clusters when the sample is split into 2, 3, 4, and 5 clusters. The cluster numbers in order of descending quantities of elements is to the right of the panels. Colors represent the horizontal distribution of the number of elements of each cluster corresponding to each model grid cell according to colorbar.

**Figure 3.**Localization of clusters in depth (vertical axis) depending on latitude (horizontal axis) when the sample is divided into 2, 3, 4, and 5 clusters: the labels are similar to Figure 2, the solid black line shows the maximum depth for different latitudes of the Kara Sea, the dotted red line shows the minimum depth. The position of each element on this section is marked with blue dot, thus the white area represents localities where there are no cluster elements.

**Figure 4.**Histograms of the distribution of the number of cluster elements depending on the values of parameters (

**a**) $\frac{\partial \rho}{\partial z}$, (

**b**) ${\left.\frac{\partial \rho}{\partial \overrightarrow{n}}\right|}_{H}$, and (

**c**) s relative to their averages. The horizontal intervals show the deviation from the mean in units of standard deviation, the vertical axis shows the number of sample elements in thousands. The red dotted line shows the shift of the mean relative to the zero value of the parameters. For comparison, the black solid line shows the corresponding normal distribution, and the red solid line shows the exponential distribution.

**Figure 5.**One-dimensional dependence of the value $log\frac{K}{{K}_{0}}$ on the parameter $\frac{\partial \rho}{\partial z}$ in cluster A: (

**a**) deviation of average values for each interval from the average for the cluster when divided into 50 intervals with an equal number of cluster elements (horizontal value in kg/m${}^{4}$); (

**b**) an enlarged view of the group of dots in a rectangle shown in (

**a**). Panels contain the closest curves for negative and positive values of the argument (red and blue solid lines). The vertical bars show the standard deviation for each interval.

**Figure 6.**Same as Figure 5, but depending on the derivative of the bottom density in the direction of the slope ${\left.\frac{\partial \rho}{\partial \overrightarrow{n}}\right|}_{H}$. Panels (

**a**,

**b**) contain the closest functional dependence curve.

**Figure 8.**One-dimensional dependence of $tanh\frac{q-\overline{q}}{\sigma \left(q\right)}$ on the parameter ${\left.\frac{\partial \rho}{\partial \overrightarrow{n}}\right|}_{H}$ (kg/m${}^{4}$) in cluster A: (

**a**) deviation of average values for each interval from the average for the cluster when divided into 50 intervals with an equal number of cluster elements; (

**b**) an enlarged view of the group of dots in a red box shown in (

**a**). The blue curve represents the closest functional relationship. The vertical bars show the standard deviation for each box.

**Figure 10.**The difference in mass (black curves), heat (red curves), and salt (magenta curves) content in terms of the increment in the mass of water in the Arctic (above 65 N): (

**a**) timeseries of the whole content, (

**b**) vertical distribution averaged over time in terms of incremental values in numerical experiment B to experiment A.

**Figure 11.**Annually averaged deviations of (

**a**) mass of water (kg/m${}^{2}$), (

**b**) its heat content (MJ/m${}^{2}$), and (

**c**) mass of salt (kg/m${}^{2}$) per square meter of the Arctic area (above 65 N), obtained as a result of vertical integration from the surface to the ocean floor in terms of incremental values in numerical experiment B to experiment A.

**Figure 12.**Timeseries of difference in mass (black curves), heat (red curves), and salt (magenta curves) content in terms of the increment in the mass of water in the Arctic (above 65 N): (

**a**) upper 30 m layer, (

**b**) 30–600 m layer, (

**c**) layer from 600 m to bottom in terms of incremental values in numerical experiment B to experiment A.

**Figure 13.**Deviations in (

**a**,

**d**) mass of water (kg/m${}^{2}$), (

**b**,

**e**) its heat content (MJ/m${}^{2}$), and (

**c**,

**f**) mass of salt (kg/m${}^{2}$) per square meter of the Arctic area (above 65 N), obtained as a result of vertical integration from the surface to the 30 m depth and averaged over April–June (

**a**–

**c**) and September–November (

**d**–

**f**) periods in terms of incremental values in numerical experiment B to experiment A.

**Figure 14.**Annually averaged deviations in (

**a**,

**d**) mass of water (kg/m${}^{2}$), (

**b**,

**e**) its heat content (MJ/m${}^{2}$), and (

**c**,

**f**) mass of salt (kg/m${}^{2}$) per square meter of the Arctic area (above 65 N), obtained as a result of vertical integration from 30 to 600 m depth (

**a**–

**c**) and from 600 m to the ocean floor in terms of incremental values in numerical experiment B to experiment A.

**Figure 15.**The difference in mass (black curves), heat (red curves), and salt (magenta curves) content in terms of the increment in the mass of water (kg) in the Arctic (above 65 N): (

**a**) time series of the whole content, (

**b**) vertical distribution averaged over time in terms of incremental values in numerical experiment C1 to experiment B.

**Figure 16.**Annually averaged deviations in (

**a**) mass of water (kg/m${}^{2}$), (

**b**) its heat content (MJ/m${}^{2}$), and (

**c**) mass of salt (kg/m${}^{2}$) per square meter of the Arctic area (above 65 N), obtained as a result of vertical integration from the surface to the ocean floor in terms of incremental values in numerical experiment C1 to experiment B.

**Figure 17.**The difference in mass (black curves), heat (red curves), and salt (magenta curves) content in terms of the increment in the mass of water (kg) in the Arctic (above 65 N): (

**a**) time series of the whole content, (

**b**) vertical distribution averaged over time in terms of incremental values in numerical experiment C2 to experiment B.

**Figure 18.**Annually averaged deviations in (

**a**) mass of water (kg/m${}^{2}$), (

**b**) its heat content (MJ/m${}^{2}$), and (

**c**) mass of salt (kg/m${}^{2}$) per square meter of the Arctic area (above 65 N), obtained as a result of vertical integration from the surface to the ocean floor in terms of incremental values in numerical experiment C2 to experiment B.

**Table 1.**Geographic and physical large-scale characteristics, used for analysis of eddy mass transport dependencies. Here $\overrightarrow{n}=\left({n}_{x},{n}_{y}\right)=\frac{\overrightarrow{\nabla}H}{\left|\overrightarrow{\nabla}H\right|}$ is the direction of the slope, and $\overrightarrow{m}=\left({n}_{y},-{n}_{x}\right)$ is the direction along the isobath. The first seven characteristics are selected after their dependency check.

No. | Symbolic Notation | Description |
---|---|---|

1 | s | The value of the bottom slope $\left|\overrightarrow{\nabla}H\right|$ |

2 | ${\left.\frac{\partial \rho}{\partial \overrightarrow{n}}\right|}_{H}$ | Component of the bottom density gradient in the direction of the slope |

3 | $\frac{\partial \rho}{\partial \overrightarrow{m}}$ | Component of the density gradient in the direction along the isobath |

4 | $\frac{\partial U}{\partial \overrightarrow{m}}$ | Shift of the velocity component U in the direction of the isobath |

5 | $\frac{\partial V}{\partial \overrightarrow{n}}$ | Shift of the velocity component V in the direction of the slope |

6 | $\frac{\partial V}{\partial \overrightarrow{m}}$ | Divergence of the velocity component V in the direction of the isobath |

7 | $\frac{\partial \rho}{\partial z}$ | Vertical component of the density gradient |

8 | H | Ocean depth |

9 | z | Local (in situ) depth |

10 | $\frac{\partial \rho}{\partial \overrightarrow{n}}$ | Component of the density gradient in the direction of the bottom slope |

11 | ${\left.\frac{\partial \rho}{\partial \overrightarrow{m}}\right|}_{H}$ | Component of the bottom density gradient in the direction along the isobath |

12 | U | Flow velocity component in the direction of the slope |

13 | V | Flow velocity component in the direction along the isobath |

14 | $\frac{\partial U}{\partial \overrightarrow{n}}$ | Divergence of the velocity component U in the direction of the slope |

15 | $\theta $ | Geographic latitude |

16 | $\lambda $ | Geographic longitude |

**Table 2.**The results of sample clustering of the selected large-scale parameters into sets of from 2 to 5 clusters. Cluster percentage represents the percentage of elements in each cluster relative to the sample volume. DB, D, ${\overline{S}}^{k}$, and ${S}^{*}\left(l\right)$ are the Davies–Bouldin index [37], Dunn index [38], clustering silhouette coefficient [39], and silhouette coefficient for each cluster [40], respectively (see description in Appendix A.1, Appendix A.2 and Appendix A.3). A, B, and C letters specify selected clusters referenced in text.

Number of Clusters | No. | Cluster Percentage | DB | D | ${\overline{\mathit{S}}}^{\mathit{k}}$ % | ${\mathit{S}}^{*}\left(\mathit{l}\right)$ % |
---|---|---|---|---|---|---|

2 | 1 | 91 | 1.62 | 0.953 | 78 | 95 |

2 | 9 | −89 | ||||

3 | 1 (A) | 88 | 1.37 | 0.958 | 74 | 96 |

2 (B) | 9 | −90 | ||||

3 (C) | 3 | −94 | ||||

4 | 1 | 81 | 1.70 | 0.526 | 57 | 91 |

2 | 13 | −84 | ||||

3 | 3 | −93 | ||||

4 | 3 | −95 | ||||

5 | 1 | 51 | 1.73 | 0.376 | 3 | 46 |

2 | 35 | −22 | ||||

3 | 9 | −81 | ||||

4 | 3 | −89 | ||||

5 | 3 | −93 |

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**MDPI and ACS Style**

Platov, G.; Iakshina, D.; Golubeva, E. Parametrization of Eddy Mass Transport in the Arctic Seas Based on the Sensitivity Analysis of Large-Scale Flows. *Water* **2023**, *15*, 472.
https://doi.org/10.3390/w15030472

**AMA Style**

Platov G, Iakshina D, Golubeva E. Parametrization of Eddy Mass Transport in the Arctic Seas Based on the Sensitivity Analysis of Large-Scale Flows. *Water*. 2023; 15(3):472.
https://doi.org/10.3390/w15030472

**Chicago/Turabian Style**

Platov, Gennady, Dina Iakshina, and Elena Golubeva. 2023. "Parametrization of Eddy Mass Transport in the Arctic Seas Based on the Sensitivity Analysis of Large-Scale Flows" *Water* 15, no. 3: 472.
https://doi.org/10.3390/w15030472