# Role of Surface-Layer Coherent Eddies in Potential Vorticity Transport in Quasigeostrophic Turbulence Driven by Eastward Shear

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Model Description

`pyqg`[41] to simulate two-layer QG turbulence. The governing equations of the model are the forced-dissipative PV evolution equations in the two layers:

#### 2.2. Lagrangian Particle Advection and Lagrangian Diffusivity

#### 2.3. Identification of Coherent Eddies

- Identify all the LAVD maxima with a minimum separation of 20 pixels (minimum separation suggested by Tarshish et al. [38]).
- Search from each maximum in LAVD for the outermost LAVD contour satisfying CI $>-0.75$ using the bisection method with an initial contour level of 0.36 times the LAVD value of the maximum.
- Remove eddies containing fewer than 200 particles (approximately equivalent to an area of 274 ${\mathrm{km}}^{2}$).

`floater`modified from Abernathey [45].

## 3. Eddy Statistics

#### 3.1. Occurrence Frequency of Coherent Eddies

#### 3.2. Eddy Radius Distribution

#### 3.3. Radial Structure

#### 3.4. Eddy Propagation

#### 3.4.1. Zonal Propagation

^{−1}and ${c}_{BC}=0.51$ cm s

^{−1}. Figure 9 shows the zonal propagation velocities of coherent eddies are systematically eastward and larger than ${c}_{BC}$ except for the very weak friction and large thickness ratio cases. This may be due to the tendency of the eddies to concentrate in the faster upper layer when friction is stronger and stratification is more surface-intensified [43]. When ${r}^{\ast}$ decreases or $\delta $ increases, the eddy propagation velocity decreases or even turns westward, which is due to the larger contribution of the barotropic mode as bottom friction becomes weaker and stratification is less surface-intensified [43]. By varying the planetary vorticity gradient beta, the eddy propagation velocity is always eastward and decreases in the similar rate as the linear decrease of ${c}_{BC}$ as beta increases.

#### 3.4.2. Meridional Propagation

## 4. PV Transport by Coherent Eddies

#### 4.1. Transport by Trapping

#### 4.2. Transport by Stirring

## 5. Tracer Transport by Coherent Eddies

## 6. Summary and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LAVD | Lagrangian-averaged vorticity deviation |

PV | potential vorticity |

RCLV | rotationally-coherent Lagrangian vortex |

QG | quasigeostrophic |

EKE | eddy kinetic energy |

CI | coherency Index |

PPVI | piecewise PV inversion |

NSF | National Science Foundation |

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**Figure 1.**Time series of upper layer eddy kinetic energy (EKE) of the three simulations. Blue, orange, and green curves indicate the weak friction, control, and strong friction cases, respectively.

**Figure 2.**(

**a**,

**c**): Lagrangian averaged vorticity deviation (LAVD) fields at the initial particle positions for two example eddies (shading). Contours give values relative to the LAVD maximum at the eddy center with the outer boundary obtained from the threshold Coherency Index (CI) $>-0.75$ indicated by a thick blue contour. (

**b**,

**d**): Radial distribution of the CI for the eddies shown in (

**a**,

**c**), respectively, as a function of contour. Dashed red lines indicate the threshold value for the CI.

**Figure 3.**An example set of 30-day eddies from the weak friction case. The background field is the potential vorticity (PV) anomaly at the initial time, red curves give the boundaries of the coherent eddies at the initial time, black lines are the trajectories of eddy centers, and colored dots are the positions after 30 days of particles initially inside the eddies.

**Figure 4.**Eddy turnover time in days as a function of the (

**a**) friction parameter, ${r}^{\ast}$, (

**b**) layer thickness ratio, $\delta $, and (

**c**) beta parameter, ${\beta}^{\ast}$.

**Figure 5.**Number of coherent eddies in each time interval as a function of (

**a**) ${r}^{\ast}$, (

**b**) $\delta $, and (

**c**) ${\beta}^{\ast}$. Blue, orange, and green curves correspond to the mean of the number of 30-, 60-, and 90-day eddies, respectively. Error bars give two times the standard error of the mean over 100 time intervals. Red and black stars in the left panel indicate the mean for 30-day eddies detected by varying the CI threshold to $-0.5$ and $-1.0$, respectively. Average number of coherent eddies with lifetimes normalized by the eddy turnover time in each case for experiments varying (

**d**) ${r}^{\ast}$, (

**e**) $\delta $, and (

**f**) ${\beta}^{\ast}$. In (

**d**) and (

**e**), the black dashed line is an exponential fit $N={N}_{0}{e}^{-\lambda \tau}$, and the green dash-dot line is a power law fit $N={N}_{1}{\tau}^{-\alpha}$, where N is the number of coherent eddies, and $\tau $ is the normalized lifetime. In (

**f**), different colored dots indicate the simulations with different ${\beta}^{\ast}$.

**Figure 6.**Coherent eddy decay rate, $\lambda $, in units of inverse turnover time as a function of ${\beta}^{\u2606}$ (blue). The orange line gives $\lambda =2{\beta}^{\ast}+0.08$. The decay rate is obtained from exponential fits to the number of 30-, 60-, and 90-day eddies for each value of ${\beta}^{\u2606}$.

**Figure 7.**Radius of coherent eddies as a function of (

**a**) ${r}^{\ast}$, (

**b**) $\delta $, and (

**c**) ${\beta}^{\ast}$. Blue, orange, and green curves correspond to the mean of the radius for 30-, 60-, and 90-day eddies, respectively. Error bars give two times the standard error of the mean. Red and black stars in the left panel indicate the mean for 30-day eddies detected by varying the CI threshold to $-0.5$ and $-1.0$, respectively.

**Figure 8.**Mean radial structure of PV in coherent eddies. Panels from top to bottom are from the strong, control and weak friction cases. The orange solid curve is the PV normalized by its maximum value in the eddy center and averaged over all the coherent eddies. The blue shading spans the 20th–80th percentiles of the normalized PV at each radial distance normalized by the e-folding scale of the PV distribution. The red dashed line is the theoretical radial structure given by (11) and is normalized by its own e-folding scale.

**Figure 9.**Zonal propagation velocity of coherent eddies as a function of (

**a**) ${r}^{\ast}$, (

**b**) $\delta $, and (

**c**) ${\beta}^{\ast}$. Blue, orange, and green curves correspond to the mean of the zonal propagation velocity for 30-, 60-, and 90-day eddies, respectively. Error bars give two times the standard error of the mean. Red and black stars in the left panel indicate the mean for 30-day eddies detected by varying the CI threshold to $-0.5$ and $-1.0$, respectively. The dashed cyan, black and red lines give the upper layer mean flow ${U}_{1}$, depth-averaged mean flow ${U}_{b}$, and baroclinic Rossby wave speed ${c}_{BC}$, respectively.

**Figure 10.**Meridional propagation velocity of coherent eddies as a function of (

**a**) ${r}^{\ast}$, (

**b**) $\delta $, and (

**c**) ${\beta}^{\ast}$. Blue, orange, and green curves with correspond to the mean of the meridional propagation velocity for 30-, 60-, and 90-day eddies, respectively, and dashed curves with triangles and solid curves with circles indicate cyclonic and anticyclonic eddies, respectively. Error bars give two times the standard error of the mean. Red and black stars in the left panel indicate the mean for 30-day eddies detected by varying the CI threshold to $-0.5$ and $-1.0$, respectively.

**Figure 11.**Net meridional PV flux (per coherent particle) due to coherent eddies as a function of (

**a**) ${r}^{\ast}$, (

**b**) $\delta $, and (

**c**) ${\beta}^{\ast}$. Blue, orange, and green curves correspond to the mean of the net PV flux for 30-, 60-, and 90-day eddies, respectively. Error bars give two times the standard error of the mean over 100 time intervals. The dash-dot red, purple, and grey lines indicate the averaged incoherent meridional PV flux over the 100 30-, 60-, and 90-day intervals, respectively. Red and black stars in the left panel indicate the mean PV flux for 30-day eddies detected by varying the CI threshold to $-0.5$ and $-1.0$, respectively.

**Figure 12.**Ratio of meridional PV transport due to trapping of coherent eddies to total PV transport as a function of (

**a**) ${r}^{\ast}$, (

**b**) $\delta $, and (

**c**) ${\beta}^{\ast}$. Blue, orange, and green curves correspond to the mean of the ratio for 30-, 60-, and 90-day eddies, respectively. Error bars give two times the standard error of the mean over 100 time intervals. Red and black stars in the left panel indicate the mean for 30-day eddies detected by varying the CI threshold to $-0.5$ and $-1.0$, respectively.

**Figure 14.**Ratio of meridional PV transport by the flow induced by coherent eddies to total PV transport as a function of (

**a**) ${r}^{\ast}$, (

**b**) $\delta $, and (

**c**) ${\beta}^{\ast}$. Blue, orange, and green curves correspond to the mean of the ratio for 30-, 60-, and 90-day eddies, respectively. Error bars give two times the standard error of the mean over 100 time intervals. Red and black stars in the left panel indicate the mean for 30-day eddies detected by varying the CI threshold to $-0.5$ and $-1.0$, respectively.

**Figure 15.**Upper level Lagrangian diffusivity calculated by (8) (red curve) with shaded error bars of 2 times standard error, and Eulerian diffusivity (green dashed curve).

**Figure 16.**Time series of coherent diffusivity (per coherent particle) in three simulations. Blue, orange, and green lines are 30-, 60-, and 90-day coherent diffusivities, respectively. Error bars give two times the standard error. Black lines indicate the Lagrangian diffusivity averaged over all the particles in the same time period. Red dashed line indicates the Eulerian diffusivity.

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**MDPI and ACS Style**

Zhang, W.; Wolfe, C.L.P.; Abernathey, R.
Role of Surface-Layer Coherent Eddies in Potential Vorticity Transport in Quasigeostrophic Turbulence Driven by Eastward Shear. *Fluids* **2020**, *5*, 2.
https://doi.org/10.3390/fluids5010002

**AMA Style**

Zhang W, Wolfe CLP, Abernathey R.
Role of Surface-Layer Coherent Eddies in Potential Vorticity Transport in Quasigeostrophic Turbulence Driven by Eastward Shear. *Fluids*. 2020; 5(1):2.
https://doi.org/10.3390/fluids5010002

**Chicago/Turabian Style**

Zhang, Wenda, Christopher L. P. Wolfe, and Ryan Abernathey.
2020. "Role of Surface-Layer Coherent Eddies in Potential Vorticity Transport in Quasigeostrophic Turbulence Driven by Eastward Shear" *Fluids* 5, no. 1: 2.
https://doi.org/10.3390/fluids5010002