## 1. Introduction

Interaction between mean flow and eddy perturbations is one of the key issues in physical oceanography, particularly in and around the western boundary currents (WBC) such as the Kuroshio and the Gulf Stream, where strong currents and large eddy activities are often observed (e.g., Chelton et al. 2011). While eddy perturbations are usually considered as a source of dissipation of the mean flow, they sometimes function instead as a driving force for the mean flow. There have been many attempts to investigate the dynamics of strong jets associated with the WBC and their relations to eddy kinetic energy (EKE) and eddy structures (e.g., Qiu et al. 2008; Waterman and Jayne 2011). These studies show, for example, that a recirculation gyre of the WBC is induced through divergence of the Reynolds stress due to eddy perturbations near strong WBC jets and that eddy vorticity forcing changes mean potential vorticity (PV) fields and generates recirculation gyres to the north and south of the WBC jets. Also, quasi-stationary meanders of the WBC jets are considered to be related to eddy–mean flow interactions (e.g., Qiu and Chen 2010).

The Lorentz diagram (Lorentz 1955) is a powerful diagnostic tool for studying such eddy–mean flow interactions and has been widely utilized to show energy budgets in the ocean in previous studies (e.g., Ogata and Masumoto 2011; Magalhães et al. 2017; Wang et al. 2017). The diagram focuses on energy conversion rates among mean kinetic energy, mean available potential energy, EKE, and eddy available potential energy, with quantitative assessment of barotropic and baroclinic instabilities. A remarkable advantage of this diagram and the associated energy estimate formulae is their simplicity, yielding straightforward implementation of the formulae for an analysis of outputs from ocean general circulation models (OGCMs).

The Lorentz diagram, however, is originally intended to be used for global mean condition. An application of this method to a regional analysis and/or discussions on spatial distribution of the energy conversion rates may create misunderstandings regarding the energy flow (Plumb 1983). One example is that the barotropic energy conversion rates in the Lorentz diagram could indicate a large value even without any local instability processes due to energy fluxes from other areas. Therefore, the energy conversion rates in the Lorentz diagram are inadequate in some cases for estimation of eddy–mean flow interactions in a limited region.

Alternative methods based on the wave activity or the pseudo-momentum have been proposed to overcome the above issue in the atmospheric literature (e.g., Andrews 1983; Plumb 1985a,b, 1986; Takaya and Nakamura 2001). However, the assumptions adopted to develop the alternative methods are inapplicable in many cases to a region near the oceanic WBC jets, in which background potential vorticity changes their magnitude abruptly along their stream lines (e.g., Waterman and Jayne 2011). Furthermore, estimation of the wave activity requires rather complicated calculations when using outputs from OGCMs or atmospheric general circulation models in some cases. For these reasons, the Lorentz diagram is widely used in the analysis of oceanic energetics.

Another approach proposed by Murakami (2011) and Chen et al. (2014, 2016) is to modify the energy conversion rates in the Lorentz diagram for a local estimation. This approach allows us to estimate locally induced energy transfer between mean flow and eddy perturbations explicitly, as well as the effects of “nonlocal” energy fluxes (Chen et al. 2014, 2016). Murakami (2011) further examines the relationship between these modified energy transfers and wave activity and indicates that the modified Lorentz diagram can capture eddy–mean flow interactions in a way that is consistent with other related methods. However, since physical interpretations of the nonlocal energy fluxes are not given explicitly and horizontal distributions of the nonlocal energy fluxes and their link to mean structures of flow fields in an oceanographic context have not been analyzed yet in detail, our understanding of energetics in a regional area is still limited, and further investigation is required.

There are several possible ways as described above to examine the transfer of energy between mean flow and eddy perturbations. In the present study, we first review energy conversion rates and related terms in the modified Lorentz diagram. We extend our arguments to potential enstrophy, which is another conservative quantity for a water parcel, to obtain a complementary view to the energy diagram. The Lorentz-type diagram in terms of the potential enstrophy and its relation to the nonlocality of eddy–mean flow interactions are documented for the first time in this study. Then, we apply the modified Lorentz diagram to a region in the Kuroshio Extension (KE) region. Our particular focus here is to discuss physical interpretations of the interaction energy flux and equivalent fluxes in related expressions and to show their distributions in the KE region. As an example of the advantage of combining several viewpoints of the energy diagram, we will briefly discuss the dynamics of the KE recirculation gyres.

In general, both the baroclinic and barotropic processes are important in the WBC jet regions. However, to highlight the above points in a clear and concise way, we only focus on barotropic processes in the energy analysis in the present study. More complete descriptions of a total view of energetics in the KE region is beyond the scope of this paper. Note that baroclinic processes can be treated in the same manner as that for barotropic processes demonstrated in this article, and baroclinic effects are also included in the case of potential enstrophy conversion rates.

This paper is organized as follows. Brief descriptions for each method are shown with their physical interpretations in section 2. Section 3 compares energy conversion rates between the classical Lorentz diagram and the modified diagram in the KE region as a key example. Also, the potential enstrophy conversion rates are discussed. Section 4 shows the analysis of energy and potential enstrophy for the KE northern recirculation gyre and demonstrates the nonlocality of the eddy–mean flow interactions. Finally, section 5 provides a summary of this paper.

## 2. Lorentz diagram and the modified energy and potential enstrophy transfer diagram

### a. Modified energy conversion rates

*x*that can be described by its time-mean value

*x*′ as a deviation from the mean, i.e.,

*ρ*

_{0}is the constant reference density;

**u**is the velocity, with

*u*and

*υ*as the zonal and meridional components; and ∇ is the three-dimensional gradient operator. This BTR is the product of the gradient of the mean flow and eddy Reynolds stresses, and a positive BTR value denotes energy gain in EKE through eddy–mean flow interactions. The BTR is considered to be the energy released from the mean kinetic energy in the classical Lorentz diagram.

*K*

_{E}is the EKE; and

*K*

_{M}(

*K*

_{E}) due to friction, wind stress, and bottom drag. Here,

**u**

_{h}is the horizontal velocity and ∇

_{h}is the horizontal gradient operator. Since the vertical terms in the energy conversion rates, such as

However, this is not the case for an energy diagram in a limited domain, which is frequently used in the oceanographic literature. The difference term, *K*_{I} vanishes in the time mean, its flux does not. Horizontal distribution of the kinetic energy is therefore affected by this energy flux, which is not evaluated explicitly in the classical Lorentz diagram.

The interaction energy flux can be considered as seeds of instability in some cases, as illustrated in Fig. 1. There are two regions located face-to-face with different background velocity fields (Fig. 1a); a uniform zonal current can be seen in region A, and zonal currents with meridional shear that allows barotropic instability in region B. Consider a case in which a small disturbance appears in region A. Since the necessary condition for barotropic instability is not satisfied in region A, the EKE there does not grow, i.e., BTR = 0. On the other hand, *K*_{I} transported from region A interacts with the mean shear flow to initiate barotropic energy conversion, which requires

### b. Enstrophy budget

*q*is defined by

*ζ*denotes the relative vorticity,

*f*is the Coriolis parameter,

*f*

_{0}is the Coriolis parameter averaged in the target region,

*σ*is the potential density, and ⟨

*σ*⟩ is the potential density averaged over time and space (Nakamura and Chao 2001; Qiu et al. 2008). This definition of QGPV was previously used to investigate the Kuroshio northern recirculation gyre generation mechanism by Qiu et al. (2008). By neglecting the external forcing and subgrid dissipation, the potential enstrophy equation for mean flow and eddy perturbation can be obtained as below:

*e*

_{E}through

A schematic diagram indicating the enstrophy transfer through eddy–mean flow interactions. Other elements such as advection are omitted. The mean flow loses its potential enstrophy corresponding to

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0124.1

A schematic diagram indicating the enstrophy transfer through eddy–mean flow interactions. Other elements such as advection are omitted. The mean flow loses its potential enstrophy corresponding to

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0124.1

A schematic diagram indicating the enstrophy transfer through eddy–mean flow interactions. Other elements such as advection are omitted. The mean flow loses its potential enstrophy corresponding to

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0124.1

*e*

_{M}. When

*q*is shifted by an arbitrary constant value

*f*

_{c}, the corresponding Eq. (11) should be modified as

## 3. Interpretations of energy and potential enstrophy budget in the Kuroshio Extension region

### a. Data and basic structures of the KE jet

To demonstrate the differences in the energy transfer terms in the classical and modified Lorentz diagrams and their interpretations, results from an eddy-resolving OGCM named OGCM for the Earth Simulator (OFES) (Masumoto et al. 2004), are used in the following analyses. OFES is based on the Modular Ocean Model version 3 (MOM3) developed at GFDL (Pacanowski and Griffies 2000) and optimized for the massively parallel computational architecture of the Earth Simulator. The horizontal grid spacing is 0.1° × 0.1°, and there are 54 vertical levels. The 3-day, NCEP-run snapshots from 1993 to 2012, which are forced by NCEP reanalysis products, are used [see Sasaki et al. (2006, 2008) for more detailed model settings]. It has been shown that OFES captures large-scale circulations as well as mesoscale eddies realistically (e.g., Masumoto et al. 2004; Sasaki et al. 2008; Masumoto 2010, and references therein), providing a reasonable platform to examine the eddy–mean flow interactions discussed in the previous section. We consider, for the following analyses, time averaged values to be background mean conditions, and deviations from them are defined as eddy perturbations.

Horizontal distributions of the mean EKE and sea surface elevation averaged from 1993 to 2012 calculated from the OFES results are shown for the KE region in Fig. 3. The KE is represented as the strong eastward meandering jet, whose volume transport per unit width integrated in the depth exceeds 200 m^{2} s^{−1}. In the region west of 150°E, the EKE has large values along the stationary meandering Kuroshio jet. After the EKE indicates its maximum around 153°E, the zonal jet is stabilized in the region east of 155°E. The jet broadens around 155°E, and northern and southern edges of the jet tend to turn back westward in a region north and south of the jet, respectively. The southern recirculation gyre (SRG), with the anticyclonic circulation and associated higher sea surface height (SSH), occupies the region 32°–34°N, 140°–155°E (Fig. 3b). The cyclonic northern recirculation gyre (NRG) can be seen in the region 35°–38°N, 145°–153°E, but the westward flow in the northern part is weaker and the lower SSH is less visible compared to the SRG (Qiu et al. 2008; Aoki et al. 2016). All these characteristics of the KE jet and the associated recirculation structures are consistent with previous results (e.g., Qiu et al. 2008; Qiu and Chen 2010, and references therein).

Horizontal distribution of (a) the EKE and (b) the mean sea surface elevation obtained from the OFES results for the period of 1993–2011. The EKE is integrated from the ocean surface to the ocean bottom. Black arrows represent horizontal volume transport per unit width integrated through the water column.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0124.1

Horizontal distribution of (a) the EKE and (b) the mean sea surface elevation obtained from the OFES results for the period of 1993–2011. The EKE is integrated from the ocean surface to the ocean bottom. Black arrows represent horizontal volume transport per unit width integrated through the water column.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0124.1

Horizontal distribution of (a) the EKE and (b) the mean sea surface elevation obtained from the OFES results for the period of 1993–2011. The EKE is integrated from the ocean surface to the ocean bottom. Black arrows represent horizontal volume transport per unit width integrated through the water column.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0124.1

### b. Energy conversion rates

Figure 4 shows spatial distributions of the three terms discussed in section 2 for the energy diagram, i.e.,

Horizontal distributions of (a)

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Horizontal distributions of (a)

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Horizontal distributions of (a)

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Both ^{9} W, suggesting the mean Kuroshio jet releases a large amount of its energy to the eddy field. However, the magnitude of ^{9} W, which is 54% of the

Area- and depth-integrated

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0124.1

Area- and depth-integrated

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Area- and depth-integrated

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On the other hand, the area integrated ^{9} W, while the area-integrated ^{9} W, which is positive. This indicates that the eddy perturbations cannot locally supply the kinetic energy for the mean flow, suggesting another energy source exists for

The above discrepancies between

In the upstream region between 145° and 150°E, the important role played by the interaction energy flux in the modified energy diagram can also be seen, but within relatively small regions. For example, the mean flow acceleration around 145°E is located mainly in the region of positive

The above examples clearly demonstrate that the arguments solely based on the barotropic energy conversion rates, as in discussions based on the classical diagram, may result in misinterpreting energy relations in a limited region. For a more accurate understanding of the eddy–mean flow interactions, we should consider

### c. Potential enstrophy conversion rates

The potential enstrophy conversion rates shown in Fig. 6 provide a complementary view for the energy analysis. While ^{3} s^{−3}, of which 88% is due to the local mean potential enstrophy loss through ^{3} s^{−3}, is supplied from the convergence of the interaction potential enstrophy flux. On the other hand, both

As in Fig. 4, but for (a)

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0124.1

As in Fig. 4, but for (a)

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As in Fig. 4, but for (a)

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0124.1

It should be noted here that the direction of the interaction potential enstrophy transport is opposite to that of the interaction energy flux. The interaction potential enstrophy flux integrated between 150° and 165°E (including both the upstream and downstream regions) is 0.01 m^{3} s^{−3}, suggesting that the total potential enstrophy is transported from the upstream/downstream region to the nearshore region. This may be of relevance to the wave maker mechanism suggested by previous studies (Rhines and Holland 1979; Waterman and Jayne 2011; Waterman and Hoskins 2013), in which the eddies radiate Rossby waves toward the upstream region. This process may be reflected in the direction of the interaction enstrophy flux and also related to the direction of the wave activity flux, which is parallel to the direction of wave propagation in the barotropic plane wave case.

## 4. Application to the northern recirculation gyre

The NRG may be one of the best phenomena for which arguments based on the energy and potential enstrophy conversion rates can be applied to better understand its dynamics. It has been shown that the NRG spans from east of the Japan Trench at around 145°E, to around 156°E west of the Shatsky Rise, and is confined to the north by the subarctic boundary along 40°N within the depth range from 600 m to around 1500 m (e.g., Qiu et al. 2008) (Fig. 7a). Qiu et al. (2008) suggested that the NRG was driven by the convergence of the Reynolds stress, which accelerates the westward mean flow north of the Kuroshio jet, particularly in the eastern part of the NRG. Their potential vorticity budget also indicated a possibility of the eddy-driven NRG. In addition, they investigated a model based on the turbulent Sverdrup equation by eddy PV forcing,

Horizontal distributions of (a) the EKE and (b) zonal component of the Reynolds stress at a depth of 1342 m. Positive values in (b) correspond to eastward forcing. The black arrows indicate horizontal velocities at a depth of 1342 m. The white contours in (a) indicate the 5000-m isobath. The region surrounded by this contour to the east of 155°E corresponds to the Shatsky Rise. The Japan Trench off the Japan archipelago is also represented by the 5000-m isobath. The zonal eddy forcing accelerates the mean flow in both region X and region Y.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0124.1

Horizontal distributions of (a) the EKE and (b) zonal component of the Reynolds stress at a depth of 1342 m. Positive values in (b) correspond to eastward forcing. The black arrows indicate horizontal velocities at a depth of 1342 m. The white contours in (a) indicate the 5000-m isobath. The region surrounded by this contour to the east of 155°E corresponds to the Shatsky Rise. The Japan Trench off the Japan archipelago is also represented by the 5000-m isobath. The zonal eddy forcing accelerates the mean flow in both region X and region Y.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0124.1

Horizontal distributions of (a) the EKE and (b) zonal component of the Reynolds stress at a depth of 1342 m. Positive values in (b) correspond to eastward forcing. The black arrows indicate horizontal velocities at a depth of 1342 m. The white contours in (a) indicate the 5000-m isobath. The region surrounded by this contour to the east of 155°E corresponds to the Shatsky Rise. The Japan Trench off the Japan archipelago is also represented by the 5000-m isobath. The zonal eddy forcing accelerates the mean flow in both region X and region Y.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0124.1

In this section, we revisit NRG dynamics from the viewpoints of energy conversion rates and potential enstrophy conversion rates. To highlight the NRG characteristics, we analyze horizontal distributions of various parameters at a depth of 1342 m. This choice of depth is arbitrary, as we have confirmed that the spatial structure for each conversion rate is essentially the same between depths of 1000 and 1500 m.

### a. Energy-based analysis

The horizontal distributions of the EKE magnitude and Reynolds stress near the recirculation gyres are shown in Fig. 7. Large EKE is confined in a latitude band of the meandering jet and the EKE maximum is located around the bifurcation point of the jet at 152°E. A relatively small secondary maximum can be seen at 37°N, 154°E, to the west of the Shatsky Rise. The zonal component of the Reynolds stress (Fig. 7b) shows that the Kuroshio jet is decelerated (accelerated) by eddy forcing in the nearshore (upstream) region. After a part of the Kuroshio jet bifurcates to the north in the downstream region around 152°E, the southern branch seems to be accelerated by a convergence of the zonal Reynolds stress (region X in Fig. 7b), while a divergence of the zonal Reynolds stress is observed in the region around 37.5°–39.5°N, 150°–155°E (region Y in Fig. 7b) for the northern branch of bifurcated jet, which turns westward within the region. The northern branch of the bifurcated jet joins the westward flow centered at 39°N and results in the formation of the eastern part of NRG. This is reflected in the

Horizontal distributions of (a)

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0124.1

Horizontal distributions of (a)

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0124.1

Horizontal distributions of (a)

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0124.1

The analysis of energy conversion rates also suggests the importance of the nonlocal eddy–mean flow interactions. In particular,

The horizontal distributions of

It is worth noting that the westward eddy forcing inside the NRG, indicated by the negative minimum of Reynolds stress convergence, has little effect on the kinetic energy (Figs. 7b and 8a). However, eddy variability also affects horizontal structure around the center of the NRG through the potential enstrophy transport, which will be discussed in the next subsection.

### b. Potential-enstrophy-based analysis

Horizontal distributions of mean potential enstrophy and convergence of the QGPV flux at 1342-m depth are shown in Fig. 9. The mean QGPV distribution indicated by the black contours in Fig. 9 well reflects the Kuroshio jet and the structures of the recirculation gyres. In the region west of 150°E, the QGPV front exists along the strong Kuroshio jet. The meridional gradients of the QGPV (not shown) change their sign at the flanks of both sides of the jet, satisfying the necessary conditions for an unstable jet. The QGPV contours broaden in the downstream (east of 150°E) region, and the jet becomes stable there as discussed in Waterman and Jayne (2011). On the southern (northern) flank of the jet, there is a positive (negative) maximum of QGPV associated with the southern (northern) recirculation gyre. Reflecting the closed QGPV structures, the mean potential enstrophy distribution has maxima located within the recirculation gyres. Influences of eddy perturbations on the mean flow through the eddy QGPV flux can be identified by comparing the mean QGPV distribution with the convergence of the QGPV fluxes (see Fig. 9b). In the nearshore region between 142° and 146°E, convergence (divergence) of the QGPV fluxes appears along the northern (southern) half of the Kuroshio jet, indicating that the eddy components tend to weaken the mean QGPV front by increasing (decreasing) the mean QGPV in the northern (southern) region, associated with the unstable nature of the jet. On the other hand, relatively large negative values of the eddy QGPV flux convergence are observed inside the NRG, with large minima located around (37°N, 148°E), (38°N, 154°E), and (36°N, 157°E), and a smaller one at (37°N, 146°E). The divergence of the eddy QGPV flux corresponds to enhancement of the mean potential enstrophy, which supports the suggestion by Qiu et al. (2008) that the NRG could be the eddy driven recirculation. To the south of the Kuroshio jet, large positive values of

Horizontal distributions of (a) the mean potential enstrophy and (b) the convergence of the QGPV flux at a depth of 1342 m in color shades. Black contours in (a) and (b) represent the QGPV with a contour interval of 0.5 × 10^{−5} s^{−1}. Dashed contours indicate negative QGPV values. Black arrows in (a) show horizontal velocities at 1342-m depth.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0124.1

Horizontal distributions of (a) the mean potential enstrophy and (b) the convergence of the QGPV flux at a depth of 1342 m in color shades. Black contours in (a) and (b) represent the QGPV with a contour interval of 0.5 × 10^{−5} s^{−1}. Dashed contours indicate negative QGPV values. Black arrows in (a) show horizontal velocities at 1342-m depth.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0124.1

Horizontal distributions of (a) the mean potential enstrophy and (b) the convergence of the QGPV flux at a depth of 1342 m in color shades. Black contours in (a) and (b) represent the QGPV with a contour interval of 0.5 × 10^{−5} s^{−1}. Dashed contours indicate negative QGPV values. Black arrows in (a) show horizontal velocities at 1342-m depth.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0124.1

The potential enstrophy conversion rates

Horizontal distributions of (a) ^{−5} s^{−1}. Dashed contours indicate negative QGPV values. Black arrows in (c) indicate the interaction enstrophy flux. The length of the black arrows is drawn to be proportional to the square root of the interaction potential enstrophy flux amplitudes to clearly illustrate weaker fluxes.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0124.1

Horizontal distributions of (a) ^{−5} s^{−1}. Dashed contours indicate negative QGPV values. Black arrows in (c) indicate the interaction enstrophy flux. The length of the black arrows is drawn to be proportional to the square root of the interaction potential enstrophy flux amplitudes to clearly illustrate weaker fluxes.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0124.1

Horizontal distributions of (a) ^{−5} s^{−1}. Dashed contours indicate negative QGPV values. Black arrows in (c) indicate the interaction enstrophy flux. The length of the black arrows is drawn to be proportional to the square root of the interaction potential enstrophy flux amplitudes to clearly illustrate weaker fluxes.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0124.1

Since the mean QGPV is highly homogenized inside the recirculation region, eddy perturbations do not easily grow locally. In fact, the meridional gradient of the mean QGPV does not change its sign in the regions outside of the Kuroshio jet (not shown). While

## 5. Summary and conclusions

We have shown the detailed interpretations of terms in the modified Lorentz diagram, especially focusing on the roles played by the interaction energy flux, as a possible diagnostic tool for eddy–mean flow interactions in a limited oceanic region. The barotropic conversion rate in the classical Lorentz diagram considers only a local eddy–mean flow interaction. As documented in Chen et al. (2014, 2016), however, there exist both local and nonlocal eddy–mean flow interactions in the ocean. The energy released from the mean flow is not completely used locally to enhance an eddy field through the barotropic instability process. The remaining energy, once stored in the form of interaction energy, is transported out from the domain in the form of interaction energy flux, and an eddy perturbation field can interact using this interaction energy with the background field in specific regions where the interaction energy flux converges. This suggests that the interaction energy can be considered as a reservoir of nonlocal eddy–mean flow interactions, and as a seed for eddy–mean flow interactions in other regions in a sense. Although the relation to a similar expression using the EP fluxes has already been mentioned in previous studies, we have shown for the first time the relation between the energy diagram and the potential enstrophy diagram, which play complementary roles in understanding eddy–mean flow interactions in the ocean.

The evaluation of each term in the modified energy flux diagram and the potential enstrophy diagram is conducted for the Kuroshio Extension region as an example featuring strong WBC jets. It is found that acceleration of mean flow in the downstream region is partly accomplished by the convergence of interaction energy fluxes. The barotropic conversion rate in the downstream region cannot supply all the kinetic energy gained by the mean flow through eddy–mean flow interaction, and this energy deficit is thus compensated by the interaction energy transported from the upstream region. It is this interaction energy flux that also explains the horizontal distribution of EKE.

The potential enstrophy budget also suggests the mean flow stabilization in the downstream region. One important difference between the energy diagram and the potential enstrophy diagram is the directions of the interaction fluxes. While the interaction energy flux converges in the downstream region, the interaction potential enstrophy flux radiates from the downstream region and converges in the nearshore region. Since the interaction energy flux is related to the Rossby propagation in the idealized case as shown in the appendix, this may be related to the Rossby wave radiation from the EKE maximum near the boundary between the upstream and downstream regions.

Finally, to highlight the advantage in combining different viewpoints for analysis of the eddy–mean flow interactions, the concepts of energy and potential enstrophy diagrams are applied to the NRG in the Kuroshio Extension region. These analyses provide further information about the roles of nonlocal eddy–mean flow interactions, adding to the previous understanding based on momentum and potential vorticity budget (Qiu et al. 2008). While the local energy conversion accelerates the northeastern part of the NRG with the aid of the relatively small nonlocal interaction energy flux convergence, the interaction potential enstrophy flux convergence is crucial around the center of the NRG.

Each method has its own advantages and limitations. They can provide insights from different viewpoints that complement each other. Therefore, the combined use of several methods is required to obtain a comprehensive view of eddy–mean flow interactions, particularly in a limited region.

## Acknowledgments

This work was supported by a JSPS Grant-in-Aid for Scientific Research(A) (17H01663) and a JSPS Grant-in-Aid for Transformative Research Areas(B) (20H05731).

## APPENDIX

### Other Interpretations of the Interaction Energy Flux

Since the interaction energy *K*_{I} vanishes in the time-averaged mean, it is not easy to give a physical meaning to the interaction energy flux in a definitive way. Here, the relationship between the energy conversion rate and the Eliassen–Palm fluxes are discussed. We also show two different interpretations of the interaction energy flux divergence *ρ*_{0} from the energy equations and associated conversion rates for simplicity.

#### a. Link to the Eliassen–Palm fluxes

Hereafter, we denote the first group of two terms in parentheses on the right-hand side, −*M*_{EP}, and the last term of the right-hand side, *M*_{EKE}. While there are several definitions for the EP flux (e.g., Plumb 1985a; Trenberth 1986; Thompson and Garabato 2014), the barotropic EP flux vectors in the form of (A1) and (A2) are helpful to interpret the energy conversion rate. This definition of EP flux corresponds to the anisotropic part of the Reynolds stress and *M*_{EP} can be considered as the work done by the EP flux. On the other hand, *M*_{EKE} is related to an isotropic part of the Reynolds stress. Since the EKE does not change the mean PV, the influence of the EKE term on the mean flow is often neglected as noneffective forcing in the transformed Eulerian mean (TEM) framework (e.g., Waterman and Jayne 2011), while the pressure forcing associated with the EKE contributes to the mean momentum balances (Aoki et al. 2016).

*D*

_{1}and

*D*

_{2}are values related to the deformation of two-dimensional mean velocity fields (Okubo 1970; Weiss 1991; Holton and Hakim 2013).

*D*

_{1}is a pure stretching term, by which a square fluid element deforms into a rhombus. Positive (negative)

*D*

_{1}indicates a flow field condition for which a fluid element tends to be stretched along the

*x*(

*y*) axis. On the other hand,

*R*

_{1}is related to the anisotropy of eddies (Hoskins et al. 1983; Marshall et al. 2012; Waterman and Hoskins 2013; Waterman and Lilly 2015). Positive

*R*

_{1}, in which eddies tend to have an elongated shape along the

*y*axis [see Waterman and Hoskins (2013) in detail], disturbs the mean flow and causes it to stretch more in the meridional direction than in the zonal direction. Therefore, a positive value of the product of

*D*

_{1}and

*R*

_{1}implies the extraction of the kinetic energy from the mean field by weakening the mean flow stretching. Likewise,

*R*

_{2}indicates a meridional tilting of eddies, and a positive value of

*R*

_{2}corresponds to an elongation in the forward direction (Waterman and Hoskins 2013). This term, together with the shear of the mean flow

*D*

_{2}, acquires energy from the mean flow through relaxation of the shear in the mean flow. Finally,

*γ*

_{m}was previously defined in Hoskins et al. (1983) as

*γ*

_{m}can be considered as a ratio of the anisotropic Reynolds stress that increases the anisotropy of mean flow to the EKE effect that increases isotropy of mean flow.

In the case of a two-dimensional barotropic flow, the EP flux term (

#### b. Wave activity conservation (pseudo-momentum)

As a different way to understand the interaction energy flux, we consider the wave activity conservation law for small-amplitude disturbances in the beta-plane quasigeostrophic (QG) system. Although the assumption of small disturbances may not be adequate for the Kuroshio Extension region, it is useful for obtaining the relationship between the interaction energy flux divergence

*ψ*=

*p*/

*f*, where

*p*is pressure and

*f = f*

_{0}+

*βy*is the Coriolis parameter, and the QG potential vorticity is written as

*N*is the buoyancy frequency and

*q*=

*Q*+

*q*′, where

*q*′ ≪

*Q*. Then, the linearized potential vorticity equation for QG flow becomes

**U**= (

*U*,

*V*, 0) is the background flow,

*Q*is the background potential vorticity, and

*S*′ is the nonconservative eddy source of potential vorticity. It is further assumed that the background flow is conservative, i.e.,

The relation (A11) means that the variation of background potential vorticity is relatively small compared to the enstrophy variation along the background mean flow. The assumptions (A12) and (A13) indicate that the direction of background potential vorticity gradient does not change abruptly.

*A*=

*e*/|∇

_{h}

*Q*| is the pseudo-momentum, i.e., the wave activity. The radiative wave activity flux is written as

*γ*= 1 (

*γ*= −1) if the background flow is pseudo-eastward (pseudo-westward). Here, pseudo-eastward (pseudo-westward) means the background mean flow is perpendicular to ∇

_{h}

*Q*, with the larger

*Q*value on its right (left) hand side, i.e.,

**M**

_{R}is connected to the group velocity theory. If the streamfunction has the wave form

**c**

_{g}= (∂

*ω*/∂

*k*, ∂

*ω*/∂

*l*, ∂

*ω*/∂

*m*) is the group velocity. Since

**M**

_{R}is everywhere parallel to the group velocity relative to the mean flow.

**M**

_{R}is connected to the interaction energy flux

**F**

^{(EP)}= −(

*U*

**E**

_{u}+

*V*

**E**

_{υ}) is the EP flux component of the interaction energy flux in the previous section [see Eq. (A7)]. From the relation (A20), the diabatic component of the interaction energy flux is parallel to the wave activity flux, and therefore the interaction energy flux corresponds to the momentum transport of the waves. Moreover, multiplying (A14) by |

**U**|, under the assumption of a slowly varying magnitude of the background velocity compared to that of the wave activity

*A*, gives the modified wave activity equation

*γ*

**F**

^{(EP)}is shown to be the radiative flux of the modified wave activity

Thus, the interaction energy flux neglecting noneffective eddy forcing represents the wave propagation and its momentum transport. If the barotropic eddy–mean flow interaction occurs nonlocally, i.e.,

#### c. Total energy conservation

**F**

_{eddy}, the pressure work

**F**

_{I}. The energy equation takes the form of a conservation equation, where the total (i.e., radiative plus advective) energy flux

**F**

_{I}term, except for the advection terms. The horizontal divergence of the interaction flux, ∇ ⋅

**F**

_{I}, corresponds to the interaction energy flux divergence

**F**

_{I}= 0, and therefore, eddy–mean flow interaction does not affect the total energy distribution. This indicates that the interaction energy flux divergence appearing in the modified Lorentz diagram represents the transport of total energy due to the nonlocality of eddy–mean flow interaction. In this sense, we may call

*energy flux divergence of the eddy–mean flow interaction*instead of the interaction energy flux divergence.

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