1. Introduction
The western boundary current of the North Pacific separates from the coast of Japan as a fast, energetic narrow jet known as the Kuroshio Extension (KE). The KE jet path is variable and often highly meandering as it flows eastward, crossing ridges of relatively shallow bathymetry at approximately 140° and 160°E (Fig. 1). Part of this meandering pattern is quasi stationary, with mean crests in the jet path around 143°–144° and 150°E and a trough near 146°E; this pattern is attributed to lee waves downstream of the Izu–Ogasawara Ridge (Mizuno and White 1983). The jet is flanked by recirculation gyres to the south (e.g., Niiler et al. 2003) and north (Qiu et al. 2008; Jayne et al. 2009), though the subsurface northern gyres are weaker and generally linked to troughs in the quasi-stationary meanders (Jayne et al. 2009; Tracey et al. 2012).
Bathymetry in the Kuroshio Extension region. The magenta lines indicate the mean (solid) and 10th/90th percentile (dashed) jet axis positions computed from POP for 1995–2007. The jet axis position for each 5-day period is defined as the 5-day mean SSH contour associated with the steepest gradients of SSH in the study region (white rectangle). The jet axis position is then extended outside the study region along the same SSH contour. For more details on how the jet axis is defined, see section 3a.
Citation: Journal of Physical Oceanography 45, 5; 10.1175/JPO-D-13-0259.1
The KE is also associated with the highest levels of mesoscale eddy activity in the North Pacific (Qiu and Chen 2010). In energetic western boundary current extensions, mesoscale eddies are thought to play an important role in cross-jet transport of tracers such as heat (e.g., Wunsch 1999; Qiu and Chen 2005; Bishop et al. 2013) and momentum (e.g., Hall 1991; Adamec 1998; Greatbatch et al. 2010; Waterman et al. 2011). Mesoscale eddy activity in the KE region is complex and takes a variety of forms. Meanders in the KE jet are steepened, likely from baroclinic instability [as shown by Shay et al. (1995) in the Gulf Stream] driven by vertical coupling between the surface meanders and deep pressure/current anomalies (Bishop and Bryan 2013). These meanders then frequently pinch off the jet as rings that essentially extend to the bottom of the water column. Moreover, deep topographically controlled eddies (whether generated from the downstream jet or elsewhere) propagate generally southwestward along f/H contours, with length scales (half wavelengths) of 175–350 km and periods of 30–60 days (Greene et al. 2012). These eddies may produce changes in the path and cross-frontal structure of the KE jet (Tracey et al. 2012; Greene et al. 2012) and drive large divergent heat fluxes across the jet (Bishop 2013). Smaller perturbations (with approximately 100–200-km length scales and 4–60-day periods) in the KE jet, often called frontal waves, propagate downstream along the jet and may interact with the topographically controlled eddies to amplify or damp their influence, depending on their relative phasing (Tracey et al. 2012).
Because of the highly variable path of the KE jet, time averages of velocity and state variables in the KE region often smooth or obscure the true cross-jet structure; this problem has been successfully dealt with in the KE by transforming these fields into a stream coordinate reference frame relative to the jet (e.g., Howe et al. 2009; Waterman et al. 2011). Waterman et al. (2011) used this approach to estimate the eddy–mean flow interaction from observations, suggesting that eddies were helping to drive the mean jet and recirculations near the eddy kinetic energy (EKE) maximum at 146°E. However, in situ observations of eddy momentum fluxes have generally been limited to either a small number of transects or an array spanning 5°–6° longitude [i.e., the Kuroshio Extension System Study (KESS) array; Donohue et al. 2008] and are also constrained to time periods ranging from synoptic snapshots (Howe et al. 2009) to sporadic 2-yr field campaigns (Waterman et al. 2011).
The extension of spatial and temporal coverage offered by ocean general circulation models (OGCMs) provides an opportunity to study the along-jet and cross-jet variations in eddy forcing. Eddy forcing likely varies with longitude along the KE jet axis, influenced by bathymetric ridges underlying the jet (Fig. 1) as well as position relative to the maximum in EKE at 146°E. Quasigeostrophic models of idealized western boundary current extensions (e.g., Jayne et al. 1996; Waterman and Jayne 2011; Waterman and Hoskins 2013) in particular suggest that the sign of eddy forcing may vary in the along-jet direction near the eastward jet’s EKE maximum. Previous studies using OGCMs (Qiu et al. 2008; Taguchi et al. 2010) have considered the effect of eddy potential vorticity (PV) fluxes on the KE northern recirculation gyre at a middepth level (~27.6σθ). Qiu et al. (2008) determined that the eddy PV flux convergence largely reinforces the mean circulation at middepths, helping to drive the northern recirculation gyre. Additional insights can be gained from an OGCM regarding the long-term mean effects of eddy forcing in the near-surface ocean.
In this study, we examine how transient mesoscale eddies redistribute vorticity along the near-surface KE jet. The central objective of this work is to clarify the long-term effect of eddies on jet velocities and cross-frontal gradients as well as on the recirculation gyres flanking the jet. An eddying ocean simulation, run using the Parallel Ocean Program (POP), with 13 yr of simulated KE variability is used to construct a vorticity budget in the vicinity of the narrow jet. In particular, our analysis employs a jet reference frame to preserve the jet’s sharp gradients and so clarify the forcing from eddy vorticity fluxes on the mean jet, and how this forcing varies with longitude as well as across the jet. The paper is organized as follows: Section 2 describes the multiyear ocean model simulation. Section 3 details the stream coordinate or jet reference frame used in our analysis, with a comparison of the jet characteristics and eddy activity as viewed in geographic (i.e., Eulerian) and jet-following reference frames. Section 4 considers the depth-averaged vorticity budget in the geographic and jet reference frame, isolating the eddy terms and detailing their contribution to the budget. Section 5 discusses patterns of eddy forcing that are identified from the jet frame vorticity budget results. In section 6, a brief study of the long-term mean baroclinic instability criteria is presented to offer some context for the results of the jet frame analyses; section 7 offers a short summary of our findings and some conclusions.
2. Model description
POP is an ocean general circulation model that solves the three-dimensional primitive equations for ocean dynamics (Smith and Gent 2002; Smith et al. 2010). The model was run in the global domain, with nominal 0.1° horizontal resolution (~8 km in the Kuroshio Extension region) on a tripole grid, with two northern poles in Canada and Russia. The grid was configured with 42 vertical levels and ~10-m vertical spacing near the surface and utilizes the K-profile parameterization (KPP; Large et al. 1994) scheme for finescale (~10 m) vertical mixing. Biharmonic viscosity and diffusivity are used, with equatorial values of ν0 = −9 × 109 m4 s−1 and κ0 = −3 × 109 m4 s−1 that decrease as a function of the grid spacing cubed at higher latitudes so that the viscous term can balance the nonlinear advection terms (Maltrud et al. 1998). Modest surface salinity restoring was incorporated to limit drift, as were partial bottom cells to improve the representation of flow over bottom topography, which is important for representing the interactions with the ridges that underlie the KE.
The model run was initialized from year 30 of an existing POP run that was configured on the same tripole grid [for more details see Maltrud et al. (2010)], forced with Co-ordinated Ocean Research Experiments (CORE) normal-year surface fluxes representing a repeating annual cycle in the atmosphere (Large and Yeager 2004), with added synoptic-scale variability averaged to monthly intervals. Our model run was then forced with the CORE version 2 (CORE2) surface fluxes representing synoptically and interannually varying atmospheric conditions during the years 1990–2007 (Large and Yeager 2009). Daily mean state variables, surface fluxes, and advective fluxes were archived from 1995 to 2007 (postadjustment to high-frequency atmospheric forcing) for most of the North Pacific, including horizontal fluxes of momentum 
3. The jet reference frame
a. Defining a jet reference frame
Daily averages of the KE flow (e.g., Figs. 2a,b) typically depict much sharper sea surface height (SSH) gradients across the jet than are evident from geographic means over longer time periods (Figs. 2c,d). This discrepancy results from large, short-period fluctuations in the KE jet path (e.g., Fig. 2 in Qiu and Chen 2010), associated with propagating eddies and frontal waves. Consequently, the path of the jet varies meridionally by over 300 km in some areas (Fig. 1)—much more than the typical KE jet width of 100–200 km (Figs. 2a,b; Fig. 7 of Waterman et al. 2011).
POP SSH daily snapshots on (a) 22 Jun 1997 and (b) 17 Jul 1998, with the thick black line indicating the 50-cm contour. POP SSH annual geographic means for (c) 1997 and (d) 1998, with the 50-cm contour indicated as in (a) and (b). POP SSH annual jet frame means for (e) 1997 and (f) 1998, as computed using the steepest gradient SSH method; the thick black line indicates the jet axis. The color scale is the same for (a)–(f) and is indicated by the color bars below (e) and (f). The contour interval for all panels is 10 cm.
Citation: Journal of Physical Oceanography 45, 5; 10.1175/JPO-D-13-0259.1
A more useful method of averaging KE jet features over long time periods (e.g., Bingham 1992; Waterman et al. 2011) is to transform data into a jet reference frame. Bingham (1992) used a jet-following coordinate frame with two horizontal dimensions: the x coordinate was the longitude of the nearest point on the jet axis, and the y coordinate was the distance from the jet axis. Our approach differs slightly in that we retain the longitude of the data point itself as the x coordinate so that the effects of bathymetry are as faithfully represented in long-term means as possible. Fields in the POP model are averaged in bins that correspond to the longitude of the grid points and their distance to the closest point on the jet axis.
To transform into the jet reference frame, it is first necessary to come up with a consistent objective method for identifying the jet axis (the zero y coordinate in the jet reference frame). For the upper ocean, a fixed contour of SSH or temperature (Jayne et al. 2009; Waterman et al. 2011) or identified maxima in velocity magnitude (Howe et al. 2009) may be used to define the jet axis. Other criteria used to define the jet axis may be based on velocity shear (as described in Meinen and Luther 2003) or gradients of SSH, temperature, or other properties that vary across the front. To define a jet path that follows the along-stream direction of the flow as closely as possible, we considered jet definitions using fixed contours of SSH (50 cm; Jayne et al. 2009) and temperature (12°C at 350-m depth; Waterman et al. 2011). In addition, we implemented a “steepest (SSH/temperature) gradient” method that identifies the SSH or temperature contours at each time interval collocated with the steepest gradients of SSH and 350-m temperature in a geographic range (30°–40°N, 140°–160°E) that corresponds to the KE (Fig. 1). Of all these methods, the SSH steepest gradient method most consistently tracked the maximum velocity jet core in POP during the 13-yr study period; hence, our study employs this technique as described below.
To define the jet axis for each time period that will serve as the zero coordinate in the cross-jet direction, SSH from the model output was first averaged in 5-day periods. The 5-day time average was chosen to minimize the rapid oscillations of the jet path that can occur as closed SSH contours (representing rings) pinch off from or reattach to the jet axis contour, while still averaging at a short enough time scale to follow the variations in the jet path due to most mesoscale features.
Next, the value of the jet axis SSH contour was computed for each 5-day period. Zonal and meridional derivatives of the 5-day mean SSH were computed in the model native grid, with the SSH interpolated before the derivatives were taken so that the zonal and meridional derivatives were computed on the same grid as the SSH values they are derived from. From the zonal and meridional SSH derivatives, the magnitude of the SSH gradient |∇(SSH)| was obtained. Then the top 5% of |∇(SSH)| values were binned according to the values of SSH at the same locations, creating a probability distribution function (PDF). A Gaussian smoothing function was applied to the PDF to reduce the sensitivity of the maximum in the function to isolated peaks (such as might be associated with rings) and sampling biases that might result from the position of SSH contours relative to the model grid. The value of SSH associated with the maximum in the Gaussian-smoothed PDF was the SSH contour that defined the jet axis for that 5-day period. This method allows the contour to vary with seasonal and interannual changes in steric height, rather than using the same SSH contour to represent the jet axis at all time periods.
As a final filter, the length of the jet axis SSH contour was computed for each 5-day time period; in our case, this was done for a larger domain (135°–170°E) to allow for some continuity of the defined jet axis with regions just outside of the study domain 140°–160°E. The SSH contours that had a length below a certain threshold (80% of the zonal distance between 135° and 170°E) were considered unreliable, as these contours likely encompass rings rather than the true KE jet axis; this can occur during instances when a large ring has gradients around its edge that are nearly uniformly as steep or steeper than those at the true KE jet axis. In our analysis, the unreliable SSH contours constituted about 5% of all the jet axis contours, and the 5-day time periods corresponding to them were not included in the final averages. The remaining viable jet axes (which account for 95% of the 5-day periods from the 1995–2007 model output) were used in our jet frame analyses.
b. Jet reference frame binning and time averaging
Once the jet axes have been defined for each 5-day time period, the model grid points at each time period can be assigned a distance from the nearest point on the jet axis d and a local jet orientation angle θ that is important for averaging vector quantities in the jet frame (for a more detailed description of how d and θ are computed see the appendix, section a). The time average relative to the jet is computed by first binning the model grid points according to their longitude ϕ and values of d. Given a scalar quantity A, the angle bracket notation 
c. Jet characteristics in the geographic and jet reference frames
Time averages of jet properties such as SSH, currents, and pressure (Jayne et al. 2009; Waterman et al. 2011) are notably different when averaged in geographic and jet reference frames. The near-axis jet frame velocity maximum is more than twice the magnitude of the geographic mean velocity maximum in observations (Waterman et al. 2011), with steeper velocity gradients evident on the flanks of the jet. Here, we consider 0–250-m, depth-averaged properties of the jet that have been time averaged in geographic and jet reference frames. The upper 250 m of the water column encompasses the fastest velocities in the jet axis core as identified from observations (e.g., Howe et al. 2009; Waterman et al. 2011) and POP. Most of the eddy kinetic energy in the region of interest as depicted by POP also occurs in the upper 250 m. Figure 3 shows that the jet frame velocity variance terms at 146°E [the longitude of the observations discussed in Waterman et al. (2011)] decay rapidly and fairly uniformly with depth, indicating that the 0–250-m layer is representative of the upper ocean; hence, we use this layer in subsequent analyses.
Eddy variance terms (a) 
Citation: Journal of Physical Oceanography 45, 5; 10.1175/JPO-D-13-0259.1
As with observations, the cross-jet velocity profile in POP at 146°E is much sharper in the jet frame mean 
The 1995–2007 mean cross-jet velocity profile (0–250-m depth average) at 146°E in POP, as computed in the geographic (black) and jet (blue) reference frames. In the geographic reference frame, eastward velocity 
Citation: Journal of Physical Oceanography 45, 5; 10.1175/JPO-D-13-0259.1
A different view of the 0–250-m, depth-averaged EKE field also emerges when eddy velocities are computed and averaged in the jet frame versus the geographic frame (Fig. 5). The region of elevated geographic mean EKE 
(a) 1995–2007 geographic mean geostrophic EKE 
Citation: Journal of Physical Oceanography 45, 5; 10.1175/JPO-D-13-0259.1
4. The vorticity budget
Our formulation of the vorticity budget considers a layer of constant depth, with two fixed levels as upper and lower depth bounds. This form is most compatible with a z-level model; that is, one that uses depth as its vertical coordinate. The depth-averaged terms of the budget can then be time averaged either in the geographic or jet reference frame. We first consider the geographic time averages in section 4a to identify any aspects of the KE jet’s structure that can be readily understood in an Eulerian coordinate system as well as to provide a comparison with the jet frame results. The jet frame averages are then computed and discussed in sections 4b (full vorticity budget) and 4c (eddy forcing).
Equation (4) is then depth averaged from z = −h to z = 0, with h = 250 m in the open ocean to be consistent with the analysis of jet velocity profiles and EKE (section 3b). In a grid cell that has (or is adjacent to) bathymetry less than 250 m deep, h is instead the depth of the shallowest bathymetry in that cell or any adjacent cell. By not including depths that are laterally adjacent to land, the vorticity equation excludes areas where the curl of the pressure gradient in the discrete model grid is nonzero (appendix C.2 in Yeager 2013) and retains stretching terms near sloping bathymetry that would otherwise be effectively negated by the boundary condition w = 0. Hence, the vorticity equation [(4)] takes the same form over shallower bathymetry, as it does in areas with bathymetry deeper than 250 m [see Bell (1999) for further discussion of this method and how it relates to other forms of the vorticity balance].
a. Geographic vorticity budget and eddy forcing
Figure 6 shows the geographic 1995–2007 time mean of the terms in (5) from POP. The tendency term 
1995–2007 geographic time mean of the terms in the full vorticity budget in (5), vertically averaged, 0–250 m, from POP. The terms are the depth- and time-averaged (a) 
Citation: Journal of Physical Oceanography 45, 5; 10.1175/JPO-D-13-0259.1
The remaining terms appear to be negligible away from the coast, including the relative vorticity stretching 
The sum of the two eddy forcing terms on the right-hand side of (7) is the eddy relative vorticity flux convergence, that is, 
(a)–(c) 1995–2007 geographic mean eddy forcing terms in (7), vertically averaged 0–250 m from POP. The terms are the depth- and time-averaged (a) 
Citation: Journal of Physical Oceanography 45, 5; 10.1175/JPO-D-13-0259.1
b. Vorticity budget in the jet reference frame
Figure 8 illustrates the terms in the 1995–2007 jet frame mean of the vorticity equation [(8)] for all bins within the ranges 140°E ≤ ϕ ≤ 160°E and −250 ≤ d ≤ 250 km. In the jet frame, 
1995–2007 jet frame time mean of the terms in the full vorticity budget in (8), vertically averaged 0–250 m from POP. The terms are the depth-averaged and jet frame time-averaged (a) 
Citation: Journal of Physical Oceanography 45, 5; 10.1175/JPO-D-13-0259.1
As in the geographic frame, the jet frame time-mean vorticity budget indicates a three-way dominant balance in which 
Outside the jet core (approximately 100–200 km from the jet axis) there is a cross-jet asymmetry in the relative vorticity advection (Fig. 8b) and vertical stretching (Fig. 8f) terms that is not readily apparent in the geographic budget (Figs. 6b,f). In these areas on either side of the jet core, the vertical stretching term suggests downwelling south of the jet and upwelling north of the jet. We also note that in the jet frame budget, the 
c. Eddy forcing in the jet reference frame
As in the geographic case, the jet frame eddy forcing terms in (13), computed along the KE jet (Fig. 9), exhibit small-scale noise, particularly within 50 km of the jet axis. Nonetheless, the jet frame eddy forcing distribution can be much more readily associated with plausible dynamical mechanisms than the geographic eddy forcing. To aid the interpretation of the eddy forcing, we compare the total eddy forcing with the vorticity budget terms that represent the mean flow (Fig. 10). A qualitative comparison of the patterns in Fig. 10 is supplemented with spatial correlations and projections of the mean terms onto the total eddy forcing (Table 1), as a first-order attempt to quantify how much of the mean circulation (as indicated by vorticity) is eddy driven. The correlations and projections in Table 1 are computed separately for the high velocity jet core and for the regions outside the jet core, as the vorticity balances in these areas are quite different. In the high velocity jet core, the eddy forcing is most highly correlated with the mean relative vorticity advection 
(a)–(c) 1995–2007 jet frame, mean eddy forcing terms in (13), vertically averaged 0–250 m from POP. The terms are the depth-averaged and jet frame time-averaged (a) 
Citation: Journal of Physical Oceanography 45, 5; 10.1175/JPO-D-13-0259.1
1995–2007 jet frame mean circulation terms in the vorticity budget in (13). The mean circulation terms are (a) 
Citation: Journal of Physical Oceanography 45, 5; 10.1175/JPO-D-13-0259.1
Correlations and projections of mean terms M in the jet frame vorticity budget with the total eddy forcing 
5. Eddy forcing patterns
The eddy forcing of the mean flow in the jet reference frame (Figs. 9, 10) may be largely explained as the superposition of four patterns, represented schematically in Fig. 11. The first three of these patterns only act within the KE jet itself and are mainly balanced by the mean 
Schematic of eddy forcing patterns on the mean flow in the KE jet region; subplots show (a) pattern 1 refers to jet core deceleration, (b) pattern 2 refers to meander reinforcement, (c) pattern 3 refers to intermittent jet core acceleration, and (d) pattern 4 refers to forcing of recirculation gyres. The black line indicates the mean path of the KE jet. Ellipses indicate areas of eddy vorticity convergence (red) and divergence (dark blue), with the implied direction of eddy vorticity fluxes given by light blue arrows. The thick gray arrows illustrate the effective directions of the eddy momentum forcing from each pattern. Gray text indicates the vertical deformation of the upper layer because of the influence of the eddy forcing pattern.
Citation: Journal of Physical Oceanography 45, 5; 10.1175/JPO-D-13-0259.1
Focusing first on the eddy horizontal advection forcing term 
Pattern 3 (Fig. 11c) originates from the 
The effect of pattern 3 may be further clarified by considering the change in along-jet velocity from west to east (Fig. 12). A general deceleration of the jet occurs east of a maximum at 142°E, which reflects the influence of both eddy forcing patterns 1 and 3 as well as frictional dissipation. However, at 144°–145°E and 150°–151°E, the opposite occurs: a brief acceleration of the jet toward the east. Both of these locations are immediately downstream of crests in the long-term mean jet (Fig. 9d), and both coincide with the eddy acceleration from the 
1995–2007 jet frame mean along-jet velocity 
Citation: Journal of Physical Oceanography 45, 5; 10.1175/JPO-D-13-0259.1
The along-jet transitions in eddy forcing represented in pattern 3 also resemble in some aspects the downstream changes identified in idealized quasigeostrophic (QG) studies of barotropic (e.g., Jayne et al. 1996; Waterman and Hoskins 2013) and baroclinic (e.g., Holland and Rhines 1980; Mizuta 2009; Waterman and Jayne 2011) zonal jets. In the idealized studies, eddies develop from unstable regions in the mean flow, with downgradient eddy vorticity fluxes acting to decelerate the mean eastward jet. However, downstream of the unstable regions, the QG jet stabilizes and resembles a wave radiator, with advected and radiating instabilities inducing net upgradient eddy vorticity fluxes (e.g., Holland and Rhines 1980; Waterman and Jayne 2011) that help to drive the mean eastward jet and its recirculation gyres. This would appear to offer an explanation for the along-jet variations in eddy acceleration of the jet but does not explain why the eddy-induced eastward accelerations only appear in the 
Pattern 4 (Fig. 11d) is the dominant eddy forcing more than 80 km from the jet axis, originating from the 
6. Jet instability characteristics
Transects (Figs. 13a–c) of the jet at 142°E (northward mean jet and longitude of regional jet frame EKE maximum in Fig. 5c), 145°E (southward mean jet), and 148°E (northward mean jet) illustrate a notable asymmetry of the KE jet’s mean background state. While the PV gradient Qy (Figs. 13d–f) in the upper 100 m reverses on both flanks of the jet, only the gradient reversal on the north flank extends down to the thermocline and beyond. On the south flank of the jet, a strong positive gradient in Ertel PV at 100–500 m exists between the low PV subtropical mode water south of the jet and the jet axis, consistent with observations of PV structure across the KE jet (Howe et al. 2009). The strong positive PV gradient stabilizes the southern flank of the jet—likely explaining the minima in jet frame EKE immediately south of the jet axis (Fig. 5c). Because of the stabilizing PV gradient in the isopycnal range ρθ = 1024.5–1026.0 kg m−3 (Figs. 13d–f), the reversals in Qy on the south flank of the jet are displaced further from the jet axis than on the north flank, well outside the region of high velocity and high shear. This may be related to the jet frame EKE asymmetry (Fig. 5c), as much of the jet frame EKE is confined to the north flank of the jet.
Transects of Ertel potential vorticity from POP, 1995–2007 mean, with isopycnals (black contours) of potential densities relative to the surface (labeled on right axis) at (a) 142°E, (b) 145°E, and (c) 148°E. The color scale is indicated below (c) and is approximately logarithmic. (d)–(f) As in (a)–(c), but colors indicate cross-jet gradient of Ertel potential vorticity Qy along isopycnals. Thick black contours indicate zero crossings of Qy, corresponding to reversals in the along-isopycnal Ertel PV gradient. The color scale is indicated below (f) and is approximately logarithmic for both negative and positive values of Qy.
Citation: Journal of Physical Oceanography 45, 5; 10.1175/JPO-D-13-0259.1
As the jet flows eastward from 142° (Fig. 13d) to 148°E (Fig. 13f), the zero crossing of Qy on the north flank moves further from the high velocity jet core and its associated shear. Thus, a gradual lessening of the positive and negative Qy gradients occurs north of the jet axis, consistent with the dampening magnitude of jet frame EKE maxima as the jet moves eastward. However, the most favorable conditions for baroclinic instability remain to the north of the jet in all transects, where the PV gradient reversal is still closer to the jet in the 100–500-m depth range. This does not explain why EKE is higher south of the jet at 145°E (Fig. 5c), though it must be noted that the synoptic stability characteristics of the jet vary with time, and episodic shifts in the jet’s asymmetric structure might explain a shift in EKE structure. Yet, the along-jet variation in jet frame EKE (and eddy forcing; i.e., patterns 2 and 3 discussed in section 5) does not appear to result from along-jet changes in the baroclinic instability criterion, suggesting that the jet frame EKE at 144°–145°E may not be generated by the mean background state of the jet.
While the PV gradient is inconclusive regarding the along-jet variations in jet frame EKE, a comparison of the geographic versus jet frame EKE provides more insight. The EKE maximum just south of the mean jet at 144°–145°E is particularly large in the geographic frame (Fig. 5b) compared to the jet frame (Fig. 5c), while the jet frame does not remove as much eddy variability from the EKE maximum at 142°–143°E. This suggests that the jet position has a more variable distribution at 144°–145°E. The jet frame EKE also has large, well-defined maxima approximately 200 km away from the jet in either direction, which suggests that 144°–146°E is a favored area for ring separation from the jet. Hence, the displacement of the near-jet EKE maximum south of the jet at 144°–146°E (Fig. 5c) may be related to the complicated dynamics of the jet as rings separate from it.
7. Conclusions
In this study, we computed a vorticity budget from the archived output of an ocean GCM using a jet-following reference frame to elucidate eddy–mean interactions that might be partially or even fully obscured in geographic time averages. With this high-resolution model simulation, we show vorticity signatures consistent with some previously observed and explained phenomena in the KE jet: the quasi-permanent meanders that are essentially standing lee waves forced by bathymetry (White and McCreary 1976; Mizuno and White 1983) and the eddies playing a role in driving the time-mean recirculations (e.g., Jayne et al. 1996; Waterman and Jayne 2011). In addition, the jet frame time mean illustrates a fundamental asymmetry of instability development in the KE. The EKE maxima in the jet frame (Fig. 5c) occur on the north side of the jet, opposite regions of nearly zero EKE on the south side. The asymmetry can be readily explained by baroclinic instability criteria in the jet derived from observations (Howe et al. 2009) as well as in our model (Fig. 13), yet this asymmetry is not at all obvious from long-term means of geographic EKE (Figs. 5a,b). The jet frame EKE asymmetry is also consistent with other GCM studies (Qiu et al. 2008; Taguchi et al. 2010) that indicate eddy dissipation processes are necessary to simulate a realistically weak northern recirculation gyre. Our budget also demonstrates eddy forcing behaviors near the KE jet axis that have previously been suggested by idealized models of zonal jets or by observations but have not been explicitly identified in GCMs. In particular, eddies were found to play a role in the acceleration of the high velocity jet core just downstream of quasi-permanent crests in the jet, in contrast to the general decelerating trend of the jet toward the east (Figs. 11c, 12).
It is important to note that the patterns of eddy vorticity forcing identified in this study may not include all of the effects of mesoscale eddies on the vorticity structure of the jet. Rather, the primary focus of this study is on the role of eddies in the forward acceleration/deceleration of the jet and the changes in the cross-jet gradient associated with these velocity changes. In the high velocity jet core surrounding the jet axis, these effects can largely be described in terms of varicose modes of variability, which affect the jet’s width and cross-jet structure. Fluctuations in the jet path, which correspond closely if not exactly to sinuous modes of variability, are regarded in the jet frame as part of the mean flow at weekly or longer time scales. (For more background on sinuous and varicose modes, a number of previous studies have considered their stability characteristics using analytical methods; e.g., Talley 1983a,b; Pratt et al. 1991; Hogg 1994). Thus, the jet frame EKE and eddy forcing can be attributed mostly to varicose modes; the effects of sinuous modes are manifested in the mean circulation terms. Both sinuous and varicose modes may radiate instabilities away from the jet (Talley 1983a,b), and the effects of sinuous modes may be quantified as eddy forcing outside the high velocity jet core. Therefore, the jet frame mean-eddy decomposition implemented in this study is most useful for considering the effects of time-variable motions associated with 1) locally growing barotropic and baroclinic instabilities that excite varicose modes of variability as well as sinuous modes that may radiate away from the jet; 2) frontal waves that propagate in the along-jet direction, which may involve varicose modes; and 3) entrant eddies that originate outside of the jet or leave the jet and then impinge on the jet again, such as deep topographically controlled eddies (Tracey et al. 2012; Greene et al. 2012) whose structure is largely independent of the surface jet. Synoptic snapshots suggest that all three types of variability may contribute to the eddy forcing patterns we have identified in the Kuroshio Extension, though substantially more analysis would be needed to quantify the relative impact of each phenomenon on the 13-yr averages of eddy forcing.
One limitation of using eddy vorticity forcing to understand eddy–mean flow interactions is that it is not always a straightforward task to infer the horizontal momentum forcing on the jet. For example, a gyre in near-solid body rotation has negligible relative vorticity gradients ∇ζ ~ 0 but can still experience nonlinear momentum advection from the wind or eddies spinning up the gyre. The effect of this forcing on the mean flow will appear in the vorticity budget [(7)] or in (13) in the mean stretching terms −fwz and −ζwz; baroclinic adjustment must then be assumed before this forcing has an effect on the horizontal velocity. This issue is of little consequence near the jet where relative vorticity is effectively the cross-jet gradient of along-jet velocity 
Acknowledgments
A. S. Delman (ASD) and J. L. McClean (JLM) were supported by NSF Grant OCE-0850463 and Office of Science (BER), U.S. Department of Energy, Grant DE-FG02-05ER64119. ASD and J. Sprintall were also supported by a NASA Earth and Space Science Fellowship (NESSF), Grant NNX13AM93H. JLM was also supported by U.S. DOE Office of Science grant entitled “Ultra-High Resolution Global Climate Simulation” via a Los Alamos National Laboratory subcontract. S. R. Jayne was supported by NSF Grant OCE-0849808. Computational resources for the model run were provided by NSF Resource Grants TG-OCE110013 and TG-OCE130010. Output from this simulation is available through the Extreme Science and Engineering Discovery Environment (XSEDE). The authors also acknowledge useful discussions with Larry Pratt and Paola Cessi during the course of this work and the editor, two anonymous reviewers, and an associate editor whose suggestions greatly improved this manuscript.
APPENDIX
A Longitude-Preserving Jet Reference Frame
a. Jet frame coordinates for each grid point
The value of d is computed for each point as follows: Distances are first computed between discrete points on the jet axis and the array of grid points in the domain. Each grid point then has a discrete point on the jet axis that is the closest to it. The calculation of the distance is refined further by computing the orientation angle θ of the line segments between each discrete jet axis point. The value of θ is then determined for each model grid point by interpolating the values of θ from the two line segments adjacent to the nearest jet axis point. Finally, the exact value of d for the point is computed from the distance between the point and the perpendicular distance to the closest of the two line segments (Fig. A1). If the point that adjoins the two line segments (i.e., the original discrete jet axis point identified as closest) is closer than any other point on the two segments, then d is taken to be just the distance between the grid point and the original discrete jet axis point.
Schematic illustrating how d and θ are computed for sample model grid points, relative to a defined jet axis (black line and dots).
Citation: Journal of Physical Oceanography 45, 5; 10.1175/JPO-D-13-0259.1
Note that for each model grid point and 5-day time period, the calculation just described yields a coordinate value d but also an orientation angle θ. The orientation angle of each point is important in the calculation of vector mean quantities in the jet frame (e.g., velocities and fluxes), and therefore it is also necessary for the jet frame mean-eddy decomposition (appendix, section b).
b. The jet frame mean-eddy decomposition
Binning and averaging scalar quantities in the jet reference frame allows for a more faithful representation of the jet’s synoptic structure in long-term time averages (e.g., Fig. 2). However, in order to use the jet frame’s advantages to quantify the contribution of eddies to the mean circulation, it is necessary to define jet frame means of vector quantities (viz., velocities). Then the eddy part of the circulation can be defined as the deviation of the flow field at each time coordinate (i.e., 5-day time period) from the jet frame mean circulation.
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