# Eddy Backscatter and Counter-Rotating Gyre Anomalies of Midlatitude Ocean Dynamics

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Double-Gyre Model

## 3. Analyses of the Double-Gyre Solutions

#### 3.1. Eddy Backscatter and Vertical Modes

**S**is the stratification matrix defined for $N\ge 2$ as

#### 3.2. Counter-Rotating Gyre Anomalies

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**A sequence of time-mean solutions for decreasing eddy viscosity ν and the constant wind forcing ${F}_{\mathrm{w}}$. The time-mean transport velocity streamfunction ${\overline{\psi}}_{1}$ for different models, grids G and viscosities $\nu \phantom{\rule{4pt}{0ex}}({\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1})$; contour interval is 0.5 Sv. Note that ${L}_{p}^{\left(3\right)}>{L}_{p}^{\left(6\right)}$, but ${L}_{p}^{\left(6\right)}\approx {L}_{p}^{\left(12\right)}$ for $\nu =\{50,100\}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}$.

**Figure 2.**Vertical structure of the potential vorticity q for the 3L, 6L, and 12L models at $\nu =100\phantom{\rule{4pt}{0ex}}{\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}$ and $G={257}^{2}$; contour interval is $0.05\phantom{\rule{4pt}{0ex}}\mathrm{Sv}$ normalized by the Coriolis parameter ${f}_{0}=0.83\times {10}^{-4}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{-1}$. Note that the upper layer is the most active, and activity rapidly decreases with depth. To compare the 3L, 6L and 12L solutions, we projected 6L and 12L solutions onto the 3 layers.

**Figure 3.**Flow response to the main eddy forcing components $\overline{\chi}$ (

**a**) and ${\chi}^{\prime}$ (

**b**) (used as forcing) in no-wind case (${F}_{\mathrm{w}}=0$); contour interval is 0.5 Sv.

**Figure 4.**The 3L and 6L model response to the forcing ${P}_{6\to 3}\left[\overline{\chi}\right]$ and ${P}_{3\to 6}\left[\overline{\chi}\right]$ (

**a**); ${P}_{6\to 3}\left[{\chi}^{\prime}\right]$ and ${P}_{3\to 6}\left[{\chi}^{\prime}\right]$ (

**b**) in no-wind case (${F}_{\mathrm{w}}=0$); contour interval is 0.5 Sv.

**Figure 5.**The nondimensional standard deviation of the eddy forcing fluctuating component $\sigma \left({\chi}^{\prime}\right)$ for (

**a**) 3L and (

**b**) 6L solutions.

**Figure 6.**The 3L and 6L model response to the force ${\widehat{\chi}}^{\prime \left(3\right)}$ (

**left**) and ${\widehat{\chi}}^{\prime \left(6\right)}$ (

**right**) in no-wind case (${F}_{\mathrm{w}}=0$); contour interval is 0.5 Sv.

**Figure 7.**The layerwise response of the 3L model to the selected-mode forcing ${\mathsf{\Theta}}_{m}^{-1}[{\tilde{\widehat{\chi}}}^{\prime \left(3\right)}],\phantom{\rule{4pt}{0ex}}m=0,1,2$; contour interval is 0.5 Sv. Note that the flow response is gradually decreasing with increasing m.

**Figure 8.**The layerwise response of the 6L model to the selected-mode forcing ${\mathsf{\Theta}}_{m}^{-1}[{\tilde{\widehat{\chi}}}^{\prime \left(6\right)}],\phantom{\rule{4pt}{0ex}}m=0,1,2$; contour interval is 0.5 Sv. Shown is only the response to the first three modes, since the higher modes induce a much weaker feedback.

**Figure 9.**The linear part of the time-mean transport velocity streamfunction: ${\overline{\psi}}_{1,\mathrm{lin}}^{\left(3\right)}$ (

**left**); ${\overline{\psi}}_{1,\mathrm{lin}}^{\left(6\right)}$ (

**middle**); and ${\overline{\psi}}_{1,\mathrm{lin}}^{\left(3\right)}-{\overline{\psi}}_{1,\mathrm{lin}}^{\left(6\right)}$ (

**right**) for $\nu =100\phantom{\rule{4pt}{0ex}}{\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}$; contour interval (CI) is 0.5 Sv; $\delta ({\overline{\psi}}_{1,\mathrm{lin}}^{\left(3\right)},{\overline{\psi}}_{1,\mathrm{lin}}^{\left(6\right)})=0.07$. Note that CI=0.1 Sv in the right figure.

**Figure 10.**Eddy-induced flow anomaly. The upper figure: The time-mean transport velocity streamfunction: ${\overline{\psi}}_{1,\oplus}^{\left(3\right)}$ (left column), ${\overline{\psi}}_{1,\oplus}^{\left(6\right)}$ (middle column), and ${\overline{\psi}}_{1,\oplus}^{\left(3\right)}-{\overline{\psi}}_{1,\oplus}^{\left(6\right)}$ (right column) at (

**a**) $\nu =100\phantom{\rule{4pt}{0ex}}{\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}$ and (

**b**) $\nu =50\phantom{\rule{4pt}{0ex}}{\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}$; contour interval is 0.5 Sv; $\delta ({\overline{\psi}}_{1,\oplus}^{\left(3\right)},{\overline{\psi}}_{1,\oplus}^{\left(6\right)})=0.91$ for $\nu =\{50,100\}\phantom{\rule{4pt}{0ex}}{\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}$. The lower figure: The time-mean PV anomaly: ${\overline{q}}_{1,\oplus}^{\left(3\right)}$ (left column), ${\overline{q}}_{1,\oplus}^{\left(6\right)}$ (middle column), and ${\overline{q}}_{1,\oplus}^{\left(3\right)}-{\overline{q}}_{1,\oplus}^{\left(6\right)}$ (right column) at (

**c**) $\nu =100\phantom{\rule{4pt}{0ex}}{\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}$ and (

**d**) $\nu =50\phantom{\rule{4pt}{0ex}}{\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}$; contour interval is $0.05\phantom{\rule{4pt}{0ex}}\mathrm{Sv}$ normalized by the Coriolis parameter ${f}_{0}=0.83\times {10}^{-4}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{-1}$.

**Figure 11.**The 3L and 6L model response to the forcing $-{\mathcal{F}}_{\overline{\chi}}$ in terms of the time-mean transport velocity streamfunction ${\overline{\psi}}_{1,\oplus}$ (

**a**) and ${\overline{\psi}}_{1}$ (

**b**) at $\nu =100\phantom{\rule{4pt}{0ex}}{\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}$; contour interval is 0.5 Sv; $\delta ({\overline{\psi}}_{1,\oplus}^{\left(3\right)},{\overline{\psi}}_{1,\oplus}^{\left(6\right)})=0.64$ and $\delta ({\overline{\psi}}_{1}^{\left(3\right)},{\overline{\psi}}_{1}^{\left(6\right)})=0.48$.

**Figure 12.**The 3L and 6L model response to the forcing ${\mathcal{F}}_{\overline{\chi}}$ in terms of the time-mean transport velocity streamfunction ${\overline{\psi}}_{1,\oplus}$ (

**a**) and ${\overline{\psi}}_{1}$ (

**b**), and their differences at $\nu =100\phantom{\rule{4pt}{0ex}}{\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}$; contour interval is 0.5 Sv; $\delta ({\overline{\psi}}_{1,\oplus}^{\left(3\right)},{\overline{\psi}}_{1,\oplus}^{\left(6\right)})=1.12$ and $\delta ({\overline{\psi}}_{1}^{\left(3\right)},{\overline{\psi}}_{1}^{\left(6\right)})=0.68$.

**Figure 13.**Upper-layer time-mean eddy forcing $\overline{\chi}$ for 3L and 6L solutions driven by the constant wind forcing ${F}_{\mathrm{w}}$, the time-mean $\overline{\chi}$ and fluctuating ${\chi}^{\prime}$ components of eddy forcing, respectively; contour interval is $5.0\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{-2}$.

Layers | ${\mathit{H}}_{\mathbf{1}}$ | ${\mathit{H}}_{\mathbf{2}}$ | ${\mathit{H}}_{\mathbf{3}}$ | ${\mathit{H}}_{\mathbf{4}}$ | ${\mathit{H}}_{\mathbf{5}}$ | ${\mathit{H}}_{\mathbf{6}}$ | ${\mathit{H}}_{\mathbf{7}}$ | ${\mathit{H}}_{\mathbf{8}}$ | ${\mathit{H}}_{\mathbf{9}}$ | ${\mathit{H}}_{\mathbf{10}}$ | ${\mathit{H}}_{\mathbf{11}}$ | ${\mathit{H}}_{\mathbf{12}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

3 | 0.25 | 0.75 | 3.0 | |||||||||

6 | 0.25 | 0.25 | 0.25 | 0.25 | 1.00 | 2.0 | ||||||

12 | 0.125 | 0.125 | 0.15 | 0.15 | 0.15 | 0.15 | 0.15 | 0.2 | 0.3 | 0.5 | 1.0 | 1.0 |

Layers | ${\mathit{R}\mathit{d}}_{\mathbf{1}}$ | ${\mathit{R}\mathit{d}}_{\mathbf{2}}$ | ${\mathit{R}\mathit{d}}_{\mathbf{3}}$ | ${\mathit{R}\mathit{d}}_{\mathbf{4}}$ | ${\mathit{R}\mathit{d}}_{\mathbf{5}}$ | ${\mathit{R}\mathit{d}}_{\mathbf{6}}$ | ${\mathit{R}\mathit{d}}_{\mathbf{7}}$ | ${\mathit{R}\mathit{d}}_{\mathbf{8}}$ | ${\mathit{R}\mathit{d}}_{\mathbf{9}}$ | ${\mathit{R}\mathit{d}}_{\mathbf{10}}$ | ${\mathit{R}\mathit{d}}_{\mathbf{11}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

3 | 40.0 | 23.0 | |||||||||

6 | 40.0 | 16.0 | 11.6 | 9.8 | 7.8 | ||||||

12 | 40.0 | 15.7 | 10.7 | 8.2 | 6.6 | 6.2 | 5.3 | 4.6 | 3.9 | 3.6 | 3.2 |

**Table 3.**Large-scale flow properties. The time-mean eastward jet penetration length ${L}_{p}\phantom{\rule{4pt}{0ex}}\left(\mathrm{km}\right)$, the total volume transport $Q\phantom{\rule{4pt}{0ex}}\left(\mathrm{Sv}\right)$, and the relative errors δ for different values of the eddy-viscosity $\nu \phantom{\rule{4pt}{0ex}}({\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1})$ in the 3L, 6L and 12L solutions.

ν | ${\mathit{L}}_{\mathit{p}}^{\mathbf{\left(}\mathbf{3}\mathbf{\right)}}$ | ${\mathit{L}}_{\mathit{p}}^{\mathbf{\left(}\mathbf{6}\mathbf{\right)}}$ | ${\mathit{L}}_{\mathit{p}}^{\mathbf{\left(}\mathbf{12}\mathbf{\right)}}$ | ${\mathit{Q}}^{\mathbf{\left(}\mathbf{3}\mathbf{\right)}}$ | ${\mathit{Q}}^{\mathbf{\left(}\mathbf{6}\mathbf{\right)}}$ | ${\mathit{Q}}^{\mathbf{\left(}\mathbf{12}\mathbf{\right)}}$ | $\mathit{\delta}\mathbf{(}{\overline{\mathit{\psi}}}_{\mathbf{1}}^{\mathbf{\left(}\mathbf{3}\mathbf{\right)}},{\overline{\mathit{\psi}}}_{\mathbf{1}}^{\mathbf{\left(}\mathbf{6}\mathbf{\right)}}\mathbf{)}$ | $\mathit{\delta}\mathbf{(}{\overline{\mathit{\psi}}}_{\mathbf{1}}^{\mathbf{\left(}\mathbf{6}\mathbf{\right)}},{\overline{\mathit{\psi}}}_{\mathbf{1}}^{\mathbf{\left(}\mathbf{12}\mathbf{\right)}}\mathbf{)}$ |
---|---|---|---|---|---|---|---|---|

100 | 2370 | 1740 | 1755 | 103 | 90 | 91 | 0.61 | 0.05 |

50 | 2865 | 2360 | 2302 | 123 | 119 | 131 | 0.57 | 0.05 |

**Table 4.**Transient eddy forcing. The basin-average integral of the standard deviation of the fluctuating component of the eddy forcing ${\chi}^{\prime}$.

Layer | ${\mathcal{I}}_{\mathbf{\Omega}}\mathbf{\left(}\mathit{\sigma}\mathbf{\left(}{\mathit{\chi}}^{\mathbf{\prime}\mathbf{\left(}\mathbf{3}\mathbf{\right)}}\mathbf{\right)}\mathbf{\right)}$ | ${\mathcal{I}}_{\mathbf{\Omega}}\mathbf{\left(}\mathit{\sigma}\mathbf{\left(}{\mathit{\chi}}^{\mathbf{\prime}\mathbf{\left(}\mathbf{6}\mathbf{\right)}}\mathbf{\right)}\mathbf{\right)}$ |
---|---|---|

1 | 18.85 | 15.55 |

2 | 2.55 | 5.77 |

3 | 0.78 | 2.43 |

4 | − | 1.19 |

5 | − | 0.54 |

6 | − | 0.52 |

**Table 5.**The efficiency of the modes $\mathcal{E}=\frac{\mathrm{Q}({\overline{\psi}}_{\mathcal{F}}^{\left(i\right)})}{\mathrm{Q}({\overline{\psi}}^{\left(i\right)})}$ for the selected-mode forcing $\mathcal{F}={\mathsf{\Theta}}_{m}^{-1}[\tilde{{\widehat{\chi}}^{\prime}}]$ in the 3L and 6L solutions.

m-th Mode | ${\mathcal{E}}^{\mathbf{\left(}\mathbf{3}\mathbf{\right)}}$ | ${\mathcal{E}}^{\mathbf{\left(}\mathbf{6}\mathbf{\right)}}$ |
---|---|---|

0 | 0.58 | 0.80 |

1 | 0.52 | 0.64 |

2 | 0.31 | 0.08 |

3 | − | 0.10 |

4 | − | 0.04 |

5 | − | 0.05 |

© 2016 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 license ( http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Shevchenko, I.; Berloff, P.
Eddy Backscatter and Counter-Rotating Gyre Anomalies of Midlatitude Ocean Dynamics. *Fluids* **2016**, *1*, 28.
https://doi.org/10.3390/fluids1030028

**AMA Style**

Shevchenko I, Berloff P.
Eddy Backscatter and Counter-Rotating Gyre Anomalies of Midlatitude Ocean Dynamics. *Fluids*. 2016; 1(3):28.
https://doi.org/10.3390/fluids1030028

**Chicago/Turabian Style**

Shevchenko, Igor, and Pavel Berloff.
2016. "Eddy Backscatter and Counter-Rotating Gyre Anomalies of Midlatitude Ocean Dynamics" *Fluids* 1, no. 3: 28.
https://doi.org/10.3390/fluids1030028