# Dynamically Consistent Parameterization of Mesoscale Eddies—Part II: Eddy Fluxes and Diffusivity from Transient Impulses

## Abstract

**:**

## 1. Introduction

## 2. Statement of the Problem

#### 2.1. Ocean Model

^{−3}is the upper layer density; $\beta =2\times {10}^{-11}$ m

^{−1}·s

^{−1}is the planetary vorticity gradient; $\nu =20$ m

^{2}·s

^{−1}is the eddy viscosity; $\gamma =4\times {10}^{-8}$ s

^{−1}is the bottom friction; and the stratification parameters ${S}_{1},{S}_{21},{S}_{22}$ and ${S}_{3}$ are chosen so that the first and second Rossby deformation radii are $R{d}_{1}=40$ km and $R{d}_{2}=20.6$ km, respectively. The prescribed steady Ekman pumping $W(x,y)$ is the only external forcing. The layer-wise model equations are augmented with the partial-slip lateral-boundary conditions and mass conservation constraints and solved by the high-resolution numerical algorithm [43] on the uniform ${513}^{2}$ grid with 7.5 km nominal resolution.

#### 2.2. Reference Solution and Eddy Backscatter

#### 2.3. Linearized Ocean Model and Transient Impulses

## 3. Transient-Impulse Solutions and Analyses

#### 3.1. Equivalent Eddy PV Flux

#### 3.2. Local Homogeneity Assumption

## 4. Discussion and Conclusions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Characteristics of the reference eddy-resolving solution: time mean. Time-mean transport streamfunction in the (

**a**) upper (Contour Interval (CI) $=3$ Sv); (

**b**) middle (CI $=6$ Sv); and (

**c**) deep layer (CI $=12$ Sv); (

**d**) upper-layer snapshot of the instantaneous transport streamfunction (CI $=3$ Sv); (

**e**) kinetic and (

**f**) potential energies of the upper-layer time-mean flow (all energies are normalized so that the basin average of the upper-layer time-mean eddy energy is unity, and the maximum and minimum values of the color scale are $\pm 10$); the upper-layer gradient of the time-mean Potential Vorticity (PV) anomaly is characterized by its (

**g**) zonal and (

**h**) meridional components; and (

**i**) absolute value (normalized so that the basin average of the absolute value is unity, and the maximum and minimum values of the color scale are $\pm 3$); the row of panels (

**j**–

**l**) shows the same as the above but for the middle layer (the maximum and minimum values of the color scale are $\pm 3$); (

**m**) absolute value of the upper-layer time-mean PV anomaly flux (normalized so that its basin average is unity, and the maximum and minimum values of the color scale are $\pm 5$).

**Figure 2.**Illustration of the eddy backscatter in action. (

**a**) instantaneous upper-ocean PV anomaly field is decomposed into (

**b**) large-scale and (

**c**) eddy components; The corresponding eddy forcing field is decomposed into the (

**d**) time-mean and (

**e**) transient parts, which are both positively correlated with the large-scale PV anomaly in the eastward jet region. On average, covariance of the transient eddy forcing with the large-scale PV anomaly is about ${10}^{4}$ times bigger than covariance of the time-mean eddy forcing, suggesting that the effect of the eddy backscatter completely dominates over the time-mean eddy stresses. Flow fields in the upper row of panels have the same but arbitrary units. Eddy forcing components are also shown with arbitrary units, but units in panel (

**e**) are $5\times {10}^{4}$ times bigger than in (

**d**), thus indicating that the transient component of the eddy forcing is much larger than the time-mean component.

**Figure 3.**Characteristics of the reference eddy-resolving solution: eddies. The upper-layer (

**a**) time-mean eddy energy and (

**b**) standard deviation of the eddy forcing (both fields are normalized so that their basin averages are unity, and the maximum and minimum values of the color scale are $\pm 6$); and (

**c**) ratio of these fields minus unity (the maximum and minimum values of the color scale are $\pm 1$). The lower row of panels shows the same quantities but for the middle layer. The time-mean transport streamfunctions are shown for convenience with CI $=6$ and 12 Sv.

**Figure 4.**Plunger-induced flows and footprints. The top row of panels show upper-layer velocity streamfunction anomalies induced at the end of the plunger impulse interval for three different locations of the plunger. Each plunger is located in the middle of the square outlining its surrounding region, and the lower row of panels shows the corresponding eddy forcing patterns in these squares. The time-mean transport streamfunction is shown for convenience with CI $=6$ Sv. The color scale units are chosen arbitrarily, since all solutions are linear but kept constant in each row of panels. Plungers are located in (

**a**,

**d**) westward return flow, (

**b**,

**e**) eastward jet, and (

**c**,

**f**) meridional interior-gyre flow.

**Figure 5.**Equivalent eddy fluxes and their divergences. (

**a**) zonal and (

**b**) meridional components of the upper-layer equivalent eddy flux; and (

**c**) the flux divergence; the same flux is represented by its (

**d**) ∇ and (

**e**) ${\nabla}_{\perp}$ components. The time-mean transport streamfunction is shown for convenience with CI $=6$ Sv. The color scale units are chosen so that MAX $=3$ for the fluxes, and MAX $=1$ for their divergences and divergence components.

**Figure 6.**Equivalent eddy fluxes and their divergences. The same as in Figure 5 but for the middle isopycnal layer: (

**a**) zonal and (

**b**) meridional component of the upper-layer equivalent eddy flux, and (

**c**) the flux divergence; the same flux is represented by its (

**d**) ∇ and (

**e**) ${\nabla}_{\perp}$ components. The time-mean transport streamfunction is shown for convenience with CI $=12$ Sv. The color scale units are chosen so that MAX $=2$ for the fluxes, and MAX $=0.2$ for the divergences.

**Figure 7.**Equivalent eddy diffusivity tensor. (

**a**) Upper-layer κ, (

**b**) upper-layer ${\kappa}_{\perp}$; (

**c**) middle-layer κ, (

**d**) middle-layer ${\kappa}_{\perp}$. The time-mean transport streamfunction is shown for convenience with CI $=6$ Sv (upper row) and CI $=12$ Sv (lower row). The color scale units are chosen arbitrarily but kept constant for all panels, and all fields were mildly smoothed by a running-average filter.

**Figure 8.**Effect of the local homogeneity assumption. (

**a**) Zonal and (

**b**) meridional components of the upper-layer equivalent eddy flux, and (

**c**) the flux divergence, all obtained with the local homogeneity assumption. These fields are to be compared with those shown in Figure 5a–c, and the corresponding differences are shown in the lower row of panels: differences in (

**d**) zonal and (

**e**) meridional component of the upper-layer equivalent eddy flux, and (

**f**) the flux divergence. The color scale units are as in Figure 5.

**Figure 9.**The eastward jet and its idealized approximation. (

**a**) upper-layer zonal velocity from the eddy-resolving reference solution; the color scale is saturated at 50 cm·s

^{−1}, thus indicating that the velocity in the western half of the jet exceeds this value; (

**b**) the meridional profiles of the zonal velocity component in all three isopycnal layers, zonally averaged over the western half of the basin; and (

**c**) the corresponding idealized zonal velocity profiles with $w=280$ km (curves with larger/smaller values correspond to the upper/deep isopycnal layers); (

**d**) upper- and (

**e**) middle-layer 2D velocity streamfunction fields constructed from the idealized velocity profile (the color scale is arbitrary).

**Figure 10.**Effect of a zonal jet on plunger-induced footprints and equivalent eddy fluxes. The equivalent across-jet eddy flux as function of the jet velocity amplitude U and width w (normalized by its value obtained for the zero-amplitude jet). The width of the jet shown in Figure 9c is indicated by the straight line. The zero value of the functions is indicated by the black curve; the flux changes sign for U about 0.5 m·s

^{−1}.

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**MDPI and ACS Style**

Berloff, P.
Dynamically Consistent Parameterization of Mesoscale Eddies—Part II: Eddy Fluxes and Diffusivity from Transient Impulses. *Fluids* **2016**, *1*, 22.
https://doi.org/10.3390/fluids1030022

**AMA Style**

Berloff P.
Dynamically Consistent Parameterization of Mesoscale Eddies—Part II: Eddy Fluxes and Diffusivity from Transient Impulses. *Fluids*. 2016; 1(3):22.
https://doi.org/10.3390/fluids1030022

**Chicago/Turabian Style**

Berloff, Pavel.
2016. "Dynamically Consistent Parameterization of Mesoscale Eddies—Part II: Eddy Fluxes and Diffusivity from Transient Impulses" *Fluids* 1, no. 3: 22.
https://doi.org/10.3390/fluids1030022