a. Connection between the eddy PV flux and eddy advection of enstrophy

The direction of the eddy PV flux is revealed by considering the scalar product of the eddy PV flux and PV gradient, u′Q′ · ∇Q, which can be expressed in terms of a time-mean contribution and eddy advection of enstrophy,

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These contributions can either reinforce or oppose each other. To understand the connection between the eddy PV flux and eddy advection of enstrophy, consider the following two limits.

1) Idealized perturbation during a spinup

During a spinup, there is a temporal change in the PV and enstrophy distributions, which is partly achieved by the eddy fluxes. Consider an instantaneous Q contour, which is perturbed along a zonal channel from time-mean zonal Q contours (full and dashed lines, respectively, in Fig. 1). The oscillation is drawn such that there is a downgradient eddy PV flux, υ′Q′ < 0, because υ′ > 0 and Q′ = Q − Q < 0 on the left-hand side and υ′ < 0 and Q′ > 0 on the right-hand side of Fig. 1.

There is a band of high enstrophy along the center of the channel, associated with the deviation of the instantaneous Q contour (shaded in Fig. 1), and there are zero values along the northern and southern boundaries. The velocity perturbation is drawn such that there is an eddy advection of enstrophy, υ′∂/∂y(Q′²/2) < 0, because υ′ and ∂/∂y(Q′²/2) have opposite signs over the channel [where ∂/∂y(Q′²/2) is evaluated where the velocity vector is marked between the central disturbance and the nearest boundary]. In this case, both υ′Q′∂Q/∂y and υ′∂/∂y(Q′²/2) reinforce each other, and, more general,

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Whether both terms are significant depends on the length scales over which Q and Q′ vary (which is explored further in section 3).

2) Statistically steady state

At a statistically steady state, the temporal change in the PV and enstrophy distributions becomes small. As a consequence, the eddy PV flux can become directed along instantaneous PV contours, rather than across them. Thus (2) reduces to u′Q′ · ∇Q ≃ 0, and

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These two limits are explored further in the subsequent eddy-resolving diagnostics in section 3.

a. Model formulation

The relationship between enstrophy distributions and PV fluxes is examined using eddy-resolving integrations of a five-layer, 1/16° isopycnic primitive equation model, a parallel version of the Miami Isopycnic Coordinate Ocean Model (Bleck and Smith 1990). The momentum and thickness diffusion have been modified to include biharmonic mixing in order to dissipate enstrophy near the grid scale, allowing large enstrophy gradients to emerge. The coefficients of thickness and momentum diffusivity are A_h = (Δx)³ × 5 × 10⁻³ m s⁻¹ ∼ 1.25 × 10⁹ m⁴ s⁻¹ and A_u ≥ (Δx)³ × 10⁻² m s⁻¹ ∼ 2.5 × 10⁹ m⁴ s⁻¹, respectively, the latter being in the form of Smagorinsky, deformation-dependent viscosity, where Δx is the grid spacing.

The model experiments are forced as a classical, wind-driven double gyre within a flat-bottom basin on a beta plane. The zonal wind stress is τ_x = −τ_o cos(2πy/L), where τ_o = 0.15 N m⁻² and L is the meridional basin scale. The domain extends from 14° to 36°N and spans 25° in longitude. There are vertical sidewalls with a free-slip boundary condition. Along the seafloor, there is a bottom drag, τ_b = −c_D(|u_bot| + 0.01)u_bot, where the drag coefficient c_D = 3 × 10⁻³. Initial layer interfaces are at depths of 500, 1250, 2000, and 2750 m, and a seafloor is at 4000 m.

The model does not include a mixed layer, although an artificial diapycnic transfer is included to prevent outcropping. The diapycnal transfer is only activated when the surface layer becomes less than 10 m in thickness; the layer thickness is locally relaxed back to 10 m and, to conserve volume of the surface layer, the layer thickness is uniformly decreased over the rest of the domain by an appropriate amount. The vorticity equation is formulated to conserve PV and enstrophy even in the limit of layers having vanishing thickness (Boudra and Chassignet 1988).

The model is integrated for 25 yr and diagnostics are conducted over the initial 3 yr of the spinup period or over the last 10 yr when the model is close to a statistically steady state. For simplicity we focus on the evolution of an intermediate layer, because there is no direct influence of wind forcing or bottom drag.

b. Time-mean state

The time-mean state is evaluated here from diagnostics extending from years 15 to 25 (with enstrophy now evaluated over this 10-yr period). The depth-integrated transport reveals a midbasin jet separating from the western boundary at 22°N with a maximum transport that reaches 100 Sv (1 Sv ≡ 10⁶ m³ s⁻¹; Fig. 2a, contours). The jet separates a relatively strong subtropical gyre and weak subpolar gyre. Tightly confined recirculations also emerge against the northern and southern boundaries.

The PV is evaluated using a layer definition, Q = (ζ + f)/h, where ζ is the relative vorticity, f is the planetary vorticity, and h is the layer thickness. All subsequent diagnostics are shown for an intermediate layer, layer 3, which is initially between interfaces at 1250 and 2000 m.

The time-mean PV reveals reduced PV contrasts over the gyre interior and enhanced gradients on the flanks of the combined subtropical and subpolar gyres (Fig. 2a). Along the gyre flanks, the flow is most unstable to baroclinic instability and instantaneous PV contours are, on average, preferentially deformed from their mean positions near the Rossby deformation scale. Thus, the time-mean enstrophy is relatively concentrated on the flanks of the combined subtropical and subpolar gyres (along 18° and 34°N), as well as along the western boundary (Fig. 2b). In the jet, the time-mean enstrophy is smaller because the horizontal shear deforms PV contours and enables the small-scale dissipation to remove enstrophy.

2) Parameterization of eddy thickness and PV fluxes

The implied diffusivities for the eddy PV flux and, for completeness, the eddy thickness flux are diagnosed using the time-mean diagnostics, for years 15–25 (shown in Fig. 2). The diffusivities are diagnosed using the following downgradient closures:

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where κ is the implied eddy diffusivity for PV or thickness, as denoted by the subscript Q or h, respectively.

The diagnosed κ show large positive and negative values for both PV and thickness when evaluated either in terms of the total flux or only in terms of the divergent flux component (Fig. 3). The κ generally reach their largest magnitude along the midbasin jet and have smaller values elsewhere. There is a smoother structure for κ_h when compared with κ_Q, which is probably a consequence of the smoother distribution of h when compared with PV. However, the κ for both thickness and PV show positive and negative values, reflecting the up- and downgradient direction of the eddy fluxes.

Given the changing sign in κ, we now consider why these patterns emerge and how they are controlled. Rather than duplicate the discussion, we focus on the control of the eddy PV flux, because PV is the dynamically conserved tracer.

c. PV and enstrophy evolution

The evolution of the PV and enstrophy distributions are revealed in snapshots shown in Fig. 4. There initially is a linear gradient in PV with zonal contours over the basin. Advection of PV along the western boundary leads to large contrasts in PV being formed across the meandering midbasin jet, and instability of the jet generates an active eddy field (Fig. 4a, left panel). As the spinup progresses, the eddy stirring leads to the interior PV distribution becoming more uniform, with PV contours expelled toward the gyre flanks (Figs. 4b,c). This result is evident in how two time-mean PV contours, marked by solid lines in Fig. 4, gradually acquire a wider spacing over the spinup; the time averaging for this figure is based upon a 600-day window centered on the time of each snapshot. The PV adjustment occurs mainly over the first 6 yr, and there is little subsequent evolution in the time-mean PV.

The enstrophy evolves according to the changes in PV (Fig. 4, right panel); the enstrophy is defined here by the PV deviation from the running time average applied over the 600-day window. There initially is concentration of enstrophy along and in the vicinity of the midbasin jet, consisting of a heterogeneous organization of deformation-scale patches (Fig. 4a, right panel). The enstrophy eventually becomes concentrated along the gyre flanks (Figs. 4b,c) as the PV gradients become expelled from the interior.

The subsequent detailed evolution of the enstrophy at a statistically steady state is revealed in Fig. 5. The sequence starts with a magnified snapshot as seen in Fig. 4c between 16° and 27°N. Over a 40-day period, a long, thin patch of high enstrophy in the jet (Fig. 5a) is stretched by the horizontal shear until it is thin enough to be diffused (Figs. 5b,c). A large patch of high enstrophy, created along the gyre flank, is advected simultaneously along the western boundary and midbasin jet into the gyre interior (Figs. 5a–d).

This cycle of creation of patches of high enstrophy near the deformation scale and diffusion at the smallest scales is observed throughout the basin. Animations of the enstrophy distribution suggest that there is preferential formation of enstrophy along the gyre flanks and destruction along the midbasin jet.

d. Time-mean local budget of enstrophy

The dominant balance in the enstrophy budget [(1b)] is now examined in more detail locally over two periods, during the early part of the spinup from years 0 to 3 and at a statistically steady state from years 15 to 25:

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where u is now taken as the velocity along the isopycnic layer.

For years 0–3, there is an oscillating pattern of up- and downgradient eddy PV fluxes, as revealed in the scalar product, −u′Q′·∇Q, over much of the domain (Fig. 6a). This reversing pattern is locally controlled by the advection of enstrophy, u·∇(Q′²/2) (Fig. 6b). The residual of these terms is relatively small and is concentrated over the western boundary where the numerical diffusion and dissipation of enstrophy are relatively large.

For years 15–25, there is a similar balance between the advection of enstrophy and the scalar product of the eddy PV flux and the mean PV gradient. The main difference is that a larger-scale pattern for the downgradient eddy PV flux emerges with regions of downgradient flux concentrated along the gyre flanks (along 18° and 34°N in the experiments).

Given how the enstrophy advection controls the direction of the eddy PV flux, rather than temporal change in the enstrophy or explicit forcing or dissipation of enstrophy, the separate contributions of the advection by the time-mean and eddy flows are considered (Fig. 7). The enstrophy advection by the time-mean and eddy flows contain opposing finescale structure, because the fine structure does not appear for the total advection of enstrophy (Fig. 6b). However, the larger-scale patterns in the enstrophy advection (Fig. 6) are controlled by the contribution from the eddy advection of enstrophy (Fig. 7a) rather than by enstrophy advection by the time-mean flow (Fig. 7b).

e. Interpretation of dominant enstrophy balance

The time-mean of the scalar product of the eddy PV flux and the PV gradient [(2)] is given by

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Thus, the limit when the enstrophy budget [(1b)] reduces to

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is equivalent to the eddy PV flux being aligned along the PV contours, u′Q′·∇Q = 0 [(3b)]. In our model diagnostics, this limit appears to apply, both at a statistically steady state for years 15–25 and in the earlier diagnostics at years 0–3, as depicted in Figs. 6 and 7.

We had expected a different balance during the spinup [(3a)] (see Fig. 1), when the eddy PV flux is expected to be directed downgradient and linked to an expelling of PV contours. However, our model diagnostics have not captured this signal, and it is probably difficult to achieve, because choosing an averaging period that is too short does not provide reliable diagnostics and choosing an averaging period that is too long leads to the initial signal being masked.

f. Time-mean flux form of the enstrophy budget

The enstrophy budget [(1b)] is now diagnosed in its flux form (see appendix) with a time average from years 15 to 25:

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The downgradient eddy PV flux is controlled by the divergence of the flux of enstrophy, with the dominant balance in (4) given by

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as revealed in Figs. 8a,b. In practice, diagnostics of (1/h)∇·[uh(Q′²/2)] are identical to those of (1/h)∇·[uh(Q′²/2)].

The large-scale patterns in the eddy PV flux show a downgradient flux along the gyre flanks and an upgradient flux over the midbasin jet. At a statistically steady state, these regions correspond to source and sink regions for enstrophy; enstrophy is created through baroclinic instability along the gyre flanks and is dissipated in a region of strong velocity shear along the midbasin jet. The flux form of the enstrophy budget emphasizes how these sources and sinks of enstrophy are connected by a thickness-weighted flux of enstrophy, uh(Q′²/2). However, although this interpretation might be plausible, the large rotational flux of enstrophy obscures this picture of a simple advective transfer between sources and sinks of enstrophy (Fig. 2d).

g. Scaling of advective terms in the enstrophy budget

Our model diagnostics suggest that, at a statistically steady state, the eddy PV flux is directed down- or upgradient according to the sign of the advection of enstrophy, with the eddy advection playing a dominant role. In contrast, previous quasigeostrophic studies, such as Holland and Rhines (1980) and Marshall and Shutts (1981), explained the eddy PV flux in terms of enstrophy dissipation and the advection by the time-mean flow. Given our contrasting results, we further examine the scaling necessary for our balances.

The ratio of the eddy advection of enstrophy to the scalar product of eddy PV flux and mean PV gradient is typically of order 1 in our model diagnostics (Fig. 9a). Applying scale analysis, this ratio is approximately related to a ratio of length scales

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where L_Q ∼ |(Q′²)^1/2/∇Q| and L_Q′ ∼ |(Q′²)^1/2/∇[(Q′²)^1/2]| reflect the length scales over which the enstrophy varies relative to the horizontal gradients in mean PV and perturbation PV (the latter defined from the square root of enstrophy). The ratio of these length scales is again typically of order 1 (Fig. 9b), suggesting that the scale over which the enstrophy varies is several 100 km (Fig. 9c); note that Figs. 9a and 9b do not simply reflect each other because of the scalar products in (5).

The advection of enstrophy by the mean and eddy flow is comparable on a finescale. However, on a larger regional scale, the advection by the mean flow does not make a significant contribution. Thus, the magnitude of the ratio of the mean advection of enstrophy to the scalar product of eddy PV flux and mean PV gradient is typically smaller than 1 in our model diagnostics (Figs. 6b and 7b). Applying scale analysis, this ratio is given by

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where |u′| ≡ (u′² + υ′²)^1/2. The relative smallness of this ratio is due to the magnitude of the mean velocity |u| being a factor of 2–4 smaller than the eddy velocity |u′| for this middepth layer (Fig. 9d). In other model studies, possibly different advective balances might emerge in the enstrophy budget through changes in these nondimensional ratios, L_Q/L_Q′ and |u|/|u′| (arising from different choices in the model domain, resolution, or forcing).