# Dynamical Filtering Highlights the Seasonality of Surface-Balanced Motions at Diurnal Scales in the Eastern Boundary Currents

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Data and Methods

#### 2.1. LLC4320

#### 2.2. Software for Data Processing

- 1.
- Download hourly snapshots for each variable, area and season.
- 2.
- Combine hourly data into time series for each variable and season, and the merge all variables into a single dataset per season and area.
- 3.
- Calculate dynamical filter in 3D $\omega -{k}_{h}$ spectral space.
- 4.
- Apply dynamical filter to each variable of interest, namely horizontal components of the ocean surface speed, from which one will obtain the low-pass (balanced motions) and high-pass (internal gravity waves) components.
- 5.
- Compute derived quantities (i.e.,$\zeta $ and $\delta $) for further analyses.

#### 2.3. $\omega -{k}_{h}$ Spectrum

#### 2.4. Temporal Variability of the Vorticity and Divergence Fields

#### 2.5. A Dynamical Filter to Discriminate BM from IGW

_{2}in the cases we examined). The resulting filter is a function in the $\omega -{k}_{h}$ spectral space that obtains a cutoff frequency that depends on ${k}_{h}$ in the form

_{2}tides in this example) is represented by the black dashed line in Figure 2.

#### Coherence and Phase Difference between Average Intensities of Divergence and Vorticity

## 3. Results

#### 3.1. Comparing $\zeta $ and $\delta $ at Seasonal and Diurnal Time Frames

#### 3.2. Rotational-Divergent Ratio in the Frequency–Wavenumber Space

#### 3.3. Separating IGW and BM

_{2}(1/12.42 h) (Figure 3). The fields presented before applying the filter are hereinafter referred to as unfiltered fields. After applying the filter, we follow the convention described in Section 2.4: BM (IGW) for motions below (above) the dispersion relation of the aforementioned vertical normal mode.

_{2}internal tides, since the dispersion relation used is limited at M

_{2}. However, the JPDF associated with this leakage in the IGW component is three orders of magnitude smaller than the JPDF of BM.

#### 3.4. Seasonal and Diurnal Variability of BM and IGW

#### 3.5. Diurnal Lag between Divergence and Vorticity

## 4. Discussion

_{2}internal tides. In this paper, we used the model output of the LLC4320 simulation with a nominal horizontal resolution of ∼2 km ($1/48\xb0$). Our method mitigates the leakage of information from BM to IGW during winter. Nevertheless, the horizontal and vertical resolutions of the LLC4320 prevent the proliferation of small-scale motions with frequencies larger than M

_{2}internal tides. Nelson et al. [37] demonstrated the strengthening of the kinetic energy frequency spectrum at frequencies greater than M

_{2}internal tides when the horizontal and vertical resolutions increase. However, this scenario needs to be tested by analyzing the Joint-PDF of $\zeta $ and $\delta $, as we did in this study.

## 5. Conclusions and Perspectives

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EBC | Eastern Boundary Current |

BM | Balanced motions (regime) |

SBM | Submesoscale balanced motions (regime) |

IGW | Internal gravity waves (regime) |

KE | Kinetic energy |

SSH | Surface sea height |

FFT | Fast Fourier Transform |

LLC4320 | MITgcm general circulation model (MITgcm) on a 1/48${}^{\xb0}$ nominal Lat/Lon-Cap (LLC) |

numerical grid | |

MITgcm | Massachusetts Institute of Technology 55 general circulation model |

RV | Vertical component of the relative vorticity (also $\zeta $) |

DIV | Horizontal divergence (also $\delta $) |

ASO | August–September–October months |

JFM | January–February–March months |

NASA | National Aeronautics and Space Administration |

SWOT | Surface Water and Ocean Topography (satellite mission) |

S-MODE | Sub-Mesoscale Ocean Dynamics Experiment |

TTTW | Transient turbulent thermal wind balance |

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**Figure 1.**Study areas within each of the Eastern Boundary Currents: California (North Pacific, from 24.17 N to 50.26 N), Canary (North Atlantic, from 13.8 N to 33.76 N), Peru–Chile (South Pacific, from 42.55 S to 13.7 S), and Benguela (South Atlantic, from 29.1 S to 8.32 S). Each tile in the map represents a quasi-quadrangular area of ∼6° side. Specific (latitude, longitude) locations of each quadrangular area can be found on Table 1.

**Figure 2.**Power spectral density of the surface kinetic energy (KE) in the frequency–horizontal wavenumber ($\omega $-${k}_{h}$) domain for the area centered at 26.64° N within the Canary current during the winter (January, February, March) of 2012. The black dotted lines represent dispersion relations for modes 1, 2, 3, and 10 of the internal gravity waves. The black dashed line denotes the minimum frequency between the internal gravity waves (IGW) at mode 10 and the ${M}_{2}$ tide, whereas the white dashed lines mark the ${M}_{2}$ and ${K}_{1}$ tide frequencies for reference purposes. The solid dark pink line corresponds to the average Coriolis frequency f in that area.

**Figure 3.**Dynamical filter in the k-l-$\omega $ space (

**left panel**) and in the $\omega $-${k}_{h}$ space (

**right panel**). Internal gravity waves (balanced motions) are located inside (outside) the cone shape in the left panel and in the black (white) region in the right panel.

**Figure 4.**Snapshots of the total (unfiltered normalized) relative vorticity ($\zeta /f$: (

**a**,

**b**)) and divergence ($\delta /f$: (

**c**,

**d**)) fields, along with their corresponding $\zeta -\delta $ joint probability distributions (

**e**,

**f**) at Canary (26.64 N) for summer (

**a**,

**c**,

**e**) and winter (

**b**,

**d**,

**f**) at times when the sea surface temperature was maximal (at around 5:00 p.m. local time). $\zeta $ and $\delta $ were then normalized by their Coriolis frequencies f. Joint PDF colors are presented on a logarithmic scale.

**Figure 5.**Snapshots of the relative vorticity ($\zeta $: (

**a**,

**b**)), divergence ($\delta $: (

**c**,

**d**)), and instantaneous $\zeta $-$\delta $ joint probability distributions (

**e**,

**f**) at Canary (26.64 °N) for times where the sea surface temperature was maximal (at around 5:00 p.m. local time, left) and minimal (at around 5:00 a.m. local time, right) on an arbitrary day in winter (1 March 2012). Joint PDF colors are presented on a logarithmic scale.

**Figure 6.**The quotient of spectral densities $K{E}_{\zeta}/K{E}_{\delta}$ in the frequency–horizontal wavenumber domain by current and season at selected areas within the California ($26.64$ N: (

**a**,

**b**)), Canary ($26.64$ N: (

**c**,

**d**)), Peru ($21.61$ S: (

**e**,

**f**)), and Benguela ($26.64$ S: (

**g**,

**h**)) current systems. Green and orange highlight scales where either $K{E}_{\zeta}$ or $K{E}_{\delta}$ dominate, respectively.

**Figure 7.**Snapshots of the normalized relative vorticity ($\zeta /f$: (

**a**–

**c**)), divergence ($\delta /f$: (

**d**–

**f**)), and joint $\zeta -\delta $ PDF (

**g**–

**i**) for a snapshot on 30 September 2012 at Canary ($26.64\xb0$ N). Fields are shown in an unfiltered state (

**a**,

**d**,

**g**) as well as for the BM (

**b**,

**d**,

**h**) and IGW (

**c**,

**e**,

**i**) regimes. JPDF bin colors are presented on a logarithmic scale.

**Figure 8.**Joint probability distribution of $\zeta $ (x-axis) and $\delta $ (y-axis) at selected study areas within the California ($26.64\xb0$ N: (

**a**–

**d**)), Canary ($26.64\xb0$ N: (

**e**–

**h**)), Peru ($21.61\xb0$ S: (

**i**–

**l**)), and Benguela ($26.64\xb0$ S: (

**m**–

**p**)) current systems for both the balanced motion (BM) and internal gravity wave (IGW) regimes. Both vorticity ($\zeta $) and divergence ($\delta $) are normalized by f. Bin colors are presented on a logarithmic scale.

**Figure 9.**Time series of dynamical variables for the study area centered at $26.6\xb0$ N within the Canary current from 2 August to 30 October 2012 (

**a**,

**c**,

**e**,

**g**) and from 2 January to 30 March 2012 (

**b**,

**d**,

**f**,

**h**) seasons. First row (

**a**,

**b**): mean values for the wind stress ($\left|\tau \right|$, blue) and the KPP turbulent boundary layer depth ($KP{P}_{hbl}$, red). Second row (

**c**,

**d**): mean values for the sea surface temperature (T, blue) and ocean net heat flux (oceQnet, red). Third row (

**e**,

**f**): standard deviation of the normalized vorticity ($\zeta /f$, magenta) and divergence ($\delta /f$, green) fields in the internal gravity wave (IGW) regime. Fourth row (

**g**,

**h**): standard deviation of the normalized vorticity ($\zeta /f$, magenta) and divergence ($\delta /f$, green) fields in the balanced motion (BM) regime.

**Figure 10.**Time series of dynamical variables for the study area centered at $26.6\xb0$ S within the Benguela current from 2 January to 30 March 2012 (

**a**,

**c**,

**e**,

**g**) and from 2 August to 30 October 2012 (

**b**,

**d**,

**f**,

**h**) seasons. First row (

**a**,

**b**): mean values for the wind stress ($\left|\tau \right|$, blue) and the KPP turbulent boundary layer depth ($KP{P}_{hbl}$, red). Second row (

**c**,

**d**): mean values for the sea surface temperature (T, blue) and the ocean net heat flux (oceQnet, red). Third row (

**e**,

**f**): standard deviation of the normalized vorticity ($\zeta /f$, magenta) and divergence ($\delta /f$, green) fields in the internal gravity wave (IGW) regime. Fourth row (

**g**,

**h**): standard deviation of the normalized vorticity ($\zeta /f$, magenta) and divergence ($\delta /f$, green) fields in the balanced motion (BM) regime.

**Figure 11.**Time series of dynamical variables for the study area centered at $26.6\xb0$ N within the Canary current from 17 September to 24 September 2012 (

**a**,

**c**,

**e**,

**g**) and from 17 February to 24 February 2012 (

**b**,

**d**,

**f**,

**h**). First row (

**a**,

**b**): mean values for the wind stress ($\left|\tau \right|$, blue) and the KPP turbulent boundary layer depth ($KP{P}_{hbl}$, red). Second row (

**c**,

**d**): mean values for the sea surface temperature (T, blue) and ocean net heat flux (oceQnet, red). Third row (

**e**,

**f**): standard deviations of the normalized vorticity ($\zeta /f$, magenta) and divergence ($\delta /f$, green) fields in the internal gravity wave (IGW) regime. Fourth row (

**g**,

**h**): standard deviation of the normalized vorticity ($\zeta /f$, magenta) and divergence ($\delta /f$, green) fields in the balanced motion (BM) regime.

**Figure 12.**Lag between the divergence and vorticity fields for the four EBC in summer (

**left**) and winter (

**right**) as a function of the latitude (absolute value). Data points were taken from Table 1, and solid lines correspond to a first-order linear regression for each current, calculated by first excluding data points with a coherence below the $90\mathsf{\%}$ confidence interval (as per Table 1).

**Table 1.**Phase difference $\mathrm{\Delta}t$ (in hours) between normalized divergence $\left(\delta \right)$ and vorticity $\left(\zeta \right)$ by current, center (latitude, longitude), and season for each quadrangular area examined. The phase difference is the angle of the complex power spectral density, calculated with a 10-day window using Welch’s method [22]. All phase differences correspond to the diurnal (24 h) component. Positive values indicate that divergence occurs first and is then followed by the relative vorticity. Rows in bold mark the study areas compared in this paper. Values with an asterisk (*) correspond to cases when the coherence did not pass the F-test for the $90\mathsf{\%}$ confidence interval.

Summer | Winter | |||||
---|---|---|---|---|---|---|

Current | Latitude | Longitude | $\mathsf{\Delta}\mathit{t}$ [h] | ${\mathit{C}}_{\mathit{\zeta}\mathit{\delta}}$ | $\mathsf{\Delta}\mathit{t}$ [h] | ${\mathit{C}}_{\mathit{\zeta}\mathit{\delta}}$ |

California | 48.4° N | 137° W | 2.41 | 0.6 | 3.86 | 0.97 |

California | 44.5° N | 131° W | 3.14 | 0.88 | 3.81 | 0.97 |

California | 40.4° N | 131° W | 3.57 | 0.88 | 3.55 | 0.95 |

California | 36.05° N | 131° W | 3.22 | 0.93 | 3.21 | 0.97 |

California | 31.46° N | 125° W | 2.95 | 0.95 | 3.02 | 0.97 |

California | 26.64° N | 125° W | 2.75 | 0.99 | 3.02 | 0.99 |

Canary | 31.46° N | 23° W | 3.61 | 0.95 | 3.83 | 0.99 |

Canary | 26.64° N | 23° W | 3.33 | 0.96 | 3.57 | 0.99 |

Canary | 21.61° N | 23° W | 3.35 | 0.98 | 3.33 | 0.99 |

Canary | 16.40° N | 29° W | 2.75 | 0.97 | 3.13 | 0.99 |

Peru | 16.39° S | 83° W | 2.16 | 0.67 | 3.09 | 0.99 |

Peru | 21.61° S | 77° W | 3.42 | 0.65 | 3.17 | 0.99 |

Peru | 40.41° S | 83° W | 3.15 | 0.94 | 2.66 | 0.97 |

Benguela | 11.03° S | 7° E | 2.96 | 0.76 | 2.79 | 0.99 |

Benguela | 16.39° S | 7° E | 2.98 | 0.63 | 3.19 | 0.99 |

Benguela | 26.64° S | 7° E | 8.5 | 0.19 * | 3.13 | 0.99 |

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**MDPI and ACS Style**

Quintana, A.; Torres, H.S.; Gomez-Valdes, J.
Dynamical Filtering Highlights the Seasonality of Surface-Balanced Motions at Diurnal Scales in the Eastern Boundary Currents. *Fluids* **2022**, *7*, 271.
https://doi.org/10.3390/fluids7080271

**AMA Style**

Quintana A, Torres HS, Gomez-Valdes J.
Dynamical Filtering Highlights the Seasonality of Surface-Balanced Motions at Diurnal Scales in the Eastern Boundary Currents. *Fluids*. 2022; 7(8):271.
https://doi.org/10.3390/fluids7080271

**Chicago/Turabian Style**

Quintana, Antonio, Hector S. Torres, and Jose Gomez-Valdes.
2022. "Dynamical Filtering Highlights the Seasonality of Surface-Balanced Motions at Diurnal Scales in the Eastern Boundary Currents" *Fluids* 7, no. 8: 271.
https://doi.org/10.3390/fluids7080271