Vertical Structure of the Beaufort Gyre Halocline and the Crucial Role of the Depth-Dependent Eddy Diffusivity

a. The Beaufort Gyre circulation

The Beaufort Gyre in the Canadian Basin of the Arctic Ocean is driven by the anticyclonic winds associated with the Beaufort High sea level pressure center (Proshutinsky and Johnson 1997). Ekman convergence accumulates low-salinity water (e.g., from river discharge and sea ice loss) and deforms the isopycnals, inducing an anticyclonic circulation in the halocline (Fig. 1a; Proshutinsky et al. 2002). The Beaufort Gyre contains ~20 000 km³ of freshwater, about one-fifth of the Arctic Ocean’s total (Haine et al. 2015). When the atmospheric forcing relaxes (i.e., is anomalously cyclonic) for a sustained period, freshwater is thought to be released from storage and fluxed to the subarctic seas. The weakening of the Beaufort Gyre in an ocean–sea ice model has been found to coincide with the so-called “Great Salinity Anomaly” events of the 1970s–90s (Proshutinsky et al. 2015), in which freshwater pulses circulated through the North Atlantic subpolar gyre (Dickson et al. 1988; Belkin et al. 1998; Belkin 2004). Upper-ocean freshening inhibits deep convective mixing in the Labrador Sea and Nordic seas, a process by which air–sea heat fluxes remove sufficient buoyancy to destabilize the stratification (Gelderloos et al. 2012; Lauvset et al. 2018). Therefore, Beaufort Gyre dynamics have potentially important implications for the thermohaline circulation (Jackson and Vellinga 2013), which has far-reaching connections with the broader climate system through its role in transporting surface heat and carbon to depth (Buckley and Marshall 2016).

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(a) Annual-mean salinity from the MIMOC (Schmidtko et al. 2013) at 200 dbar showing the pronounced salinity minimum in the Beaufort Gyre (color map). Bathymetry contours are shown in intervals of 1000 m (solid black lines). The dashed black line is the location of the pressure–latitude transect at 150°W shown in (b). White dots are the locations of BGOS moorings A–D. (b) Latitude–pressure transect at 150°W, showing the annual mean potential density (referenced to 0 dbar) as calculated from the MIMOC temperature and salinity data using the TEOS-10 equations of state. Selected isopycnals are indicated by black lines.

Citation: Journal of Physical Oceanography 51, 3; 10.1175/JPO-D-20-0077.1

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The Beaufort Gyre undergoes significant interannual variability. Recently, anomalously anticyclonic conditions have prevailed, as indicated by the deepening of isopycnals in the halocline (Zhong et al. 2019) and the accumulation of ~500 km³ yr⁻¹ of freshwater during 2003–18 (Proshutinsky et al. 2019a,b). The volume of Pacific Winter Water (PWW) (i.e., the water mass bounded by the 1026–1027 kg m⁻³ isopycnals between ~100–200-m depth) alone increased by 5000 km³, or about 18%, during 2002–16 (Zhong et al. 2019). The significant loss of sea ice after 2007 and the coincident spinup of the gyre have resulted in enhanced mechanical energy input because of a combined impact from wind and ice stresses (Armitage et al. 2020). Understanding the key processes leading to these changes and predicting the overall evolution of the Beaufort Gyre remains a major challenge.

Among the factors that have been hypothesized to affect the halocline dynamics are the Ekman pumping, sea ice cover, mesoscale eddies, and diabatic mixing. Observations indicate that Ekman pumping, driven by both sea ice and wind-induced ocean stresses, plays a major role in the halocline deepening (Proshutinsky et al. 2019b; Meneghello et al. 2018a,b, 2020). However, observations also suggest that the halocline is baroclinically unstable as mesoscale eddies are ubiquitous in the Canada Basin (Zhao et al. 2016, 2018). In general, mesoscale eddies can redistribute isopycnal layer thicknesses laterally and as a result affect the halocline depth (Manucharyan and Spall 2016). In addition, on sufficiently long time scales the vertical mixing can lead to significant changes in the halocline structure (Spall 2013). At present, the basic dynamical balance of the Beaufort Gyre remains uncertain and multiple theories have been proposed that can be at odds with one another and with the observational evidence in certain respects. Below we summarize existing Beaufort Gyre models and discuss their limitations with respect to capturing these dynamics.

b. Theories of the Beaufort Gyre dynamics

Spall (2013) proposed that eddy boundary fluxes of buoyancy are balanced by diapycnal diffusion in the gyre interior. In this framework, eddies transport cold, fresh shelf water from the boundary current to the upper halocline and warm, salty Atlantic water to the lower halocline, restoring the stratification that is homogenized by diapycnal diffusion. In this idealized configuration, the ocean was forced with uniform winds and the Ekman convergence is unnecessary to explain the isopycnal profile and boundary currents. As a result, there is an unrealistically homogeneous distribution of the freshwater content in the interior of the Beaufort Gyre and the halocline depth variations are not linked to Ekman pumping as they are in observations (Proshutinsky et al. 2019b).

To explore the role of eddies in the Beaufort Gyre equilibration forced by Ekman pumping, Manucharyan and Spall (2016) and Manucharyan et al. (2016) have developed a single-layer model using the Transformed Eulerian Mean formalism (Andrews and McIntyre 1976, 1978; Boyd 1976; Vallis 2017). Specifically, Ekman pumping gives rise to a baroclinically unstable tilt of isopycnals, generating mesoscale eddies that reduce their slope via along-isopycnal (i.e., adiabatic) fluxes represented by the Gent–McWilliams (GM) parameterization (Gent and McWilliams 1990). In a steady state, the residual circulation, a sum of the Ekman- and eddy-induced circulations, directly balances buoyancy sources and sinks (Andrews and McIntyre 1976). In the gyre interior, the residual circulation is balanced by diapycnal mixing (an effective volume source/sink of buoyancy) and is thought to be an order of magnitude smaller than the Ekman- and eddy-induced circulations.

Through a phenomenon coined the “ice–ocean stress governor,” Meneghello et al. (2018a) have suggested that the presence of sea ice (i.e., ice–ocean stresses) could obviate the need for an eddy-induced circulation to equilibrate the gyre. In a steady state, the net Ekman pumping might be negligibly small with wind-driven downwelling balancing the ice-driven upwelling—a mechanism also pointed out in Dewey et al. (2018) and Zhong et al. (2018). However, the Beaufort Gyre is not fully ice covered, and observations suggest that the ice–ocean stress governor mechanism reduces the effective strength of the Ekman pumping to about 2–3 m yr⁻¹ (Meneghello et al. 2017); this remaining Ekman pumping needs to be counteracted by some process (presumably eddy activity) in order to achieve a steady state. Using a hierarchy of models, Doddridge et al. (2019) have demonstrated that the steady-state balance is determined by wind stress on the ice-free ocean, the ice–ocean governor mechanism, and mesoscale eddy fluxes. The ice–ocean governor mechanism is thought to dominate the transient evolution on interannual time scales, with eddy fluxes becoming important on longer time scales (Meneghello et al. 2020). Analyzing the interannual gyre variability, Armitage et al. (2020) inferred that the eddy dissipation by friction against the sea ice must be present in order to balance the gyre’s mechanical energy sources and sinks, particularly since 2007 when sea ice concentrations have significantly decreased. Thus, while the ice–ocean governor mechanism does not require eddies, they are nonetheless prominent in the Beaufort Gyre (Zhao et al. 2016, 2018) and their observationally constrained diffusivity can be of sufficient magnitude to counteract the Ekman pumping (Meneghello et al. 2018b) and the accumulation of potential energy (Armitage et al. 2020).

In this study we point out the critical role of eddies in controlling the vertical structure of the Beaufort Gyre halocline. Specifically, we focus on explaining the significant variations of isopycnal slopes with depth that are evident in the climatological state of the gyre (Figs. 1b and 2c). Prior models (e.g., Manucharyan et al. 2016; Doddridge et al. 2019; Meneghello et al. 2020) have represented the halocline as a single layer or as multiple layers with a constant eddy buoyancy diffusivity and cannot explain the vertical structure, as will be demonstrated. To explain the observed isopycnal slope profile, we develop a framework for a multilayer gyre model and use it to evaluate the relative role of Ekman pumping, vertical mixing, and eddy overturning in driving the thickness variability in the PWW layer.

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Time-mean stratification at mooring B: (a) salinity (here labeled “S”; blue) and potential temperature θ (red), (b) potential density ρ (blue) and buoyancy frequency N² (red), and (c) mean isopycnal slope S estimated at moorings A–D from the MIMOC.

Citation: Journal of Physical Oceanography 51, 3; 10.1175/JPO-D-20-0077.1

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c. Overview of the study

In section 2, we investigate observational data from multiple sources to characterize the mean state and variability of the isopycnal structure in the Beaufort Gyre, with a focus on the PWW layer. In section 3, we derive a multilayer model that generalizes those of Spall (2013), Manucharyan and Spall (2016), Manucharyan et al. (2016), and Meneghello et al. (2018a) and includes all major physical processes: Ekman pumping, mesoscale eddy activity, and diapycnal diffusion. In the subsequent sections, we quantify how each term in the model contributes to the observed isopycnal slope profile and layer thickness variability in the gyre interior. In section 4, we quantify the depth dependence of the Eulerian-mean vertical velocity and demonstrate that its contribution to the PWW layer thickening is negligible. In section 5, we exclude diapycnal mixing as the cause of the PWW layer thickening. In section 6, we perform modeling experiments to demonstrate that increasing thickness in the gyre interior can be explained by the activity of mesoscale eddies in the GM parameterization if the eddy buoyancy diffusivity coefficient varies with depth. Combining this theoretical framework with observational data, we then infer the vertical structure of the eddy buoyancy diffusivity coefficient. In addition, we discuss the inability of alternative eddy flux parameterizations, such as potential vorticity or thickness diffusion, to explain the layer thickness variability. In section 7, we summarize and discuss the implications of our findings.

a. Model description

The Eulerian-mean vertical velocity w^Ek and eddy-induced vertical velocity w* are given by the radial derivatives of the respective streamfunctions. Then the time evolution of the ith isopycnal depth (h_i > 0) is controlled by three dynamical processes (Fig. 4): Ekman pumping, mesoscale eddy activity (including boundary fluxes), and diapycnal diffusion. Specifically,

${\displaystyle \frac{\partial h_i}{\partial t}}=\underset{\text{Ekman}:\hspace{0.17em}{w}^{\mathrm{Ek}}}{\underbrace{{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(r{\displaystyle \frac{\overline{\tau }}{{\rho }_{0}f}}\right)}}-\underset{\text{eddies}:\hspace{0.17em}{w}^{*}}{\underbrace{{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(r{K}_i^b{\displaystyle \frac{\partial h_i}{\partial r}}\right)}}-\underset{\text{diabatic}:\hspace{0.17em}{w}^d}{\underbrace{{\displaystyle \frac{\partial }{\partial z}}\left({\kappa }^d{\displaystyle \frac{\partial \overline{b}}{\partial z}}\right)/{\displaystyle \frac{\partial \overline{b}}{\partial z}}}}=0.$(4)

An axisymmetric coordinate system is used with r the radial coordinate and z positive up; here w^Ek < 0 for Ekman pumping.

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Schematic of a cross section of the Beaufort Gyre, including the major processes involved in setting its vertical structure. The gyre is bounded by the shelf on the south (left) and is open to the north (right). Eulerian-mean vertical velocity w^Ek and eddy-induced vertical velocity w* are shown. Also shown is an idealized profile of the isopycnals defining the PWW layer in a steady state in which K^b is constant with depth (blue lines) and in a transient state in which the surface stress intensifies and K^b decreases with depth (red lines).

Citation: Journal of Physical Oceanography 51, 3; 10.1175/JPO-D-20-0077.1

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We separate w^Ek into time-dependent and space-dependent components: w^Ek = p^Ek(t)w₁(r), where the prefactor p^Ek(t) is unitless and w₁(r) is an idealized spatial profile. For Eq. (4) with a fixed boundary condition (see discussion of boundary conditions below), we choose a profile of w₁ that is constant in the gyre interior as in Manucharyan and Spall (2016) and Manucharyan et al. (2016) (Fig. 5d). For Eq. (4) with a flux boundary condition, we choose a spatial profile of w₁ corresponding to nearly constant Ekman pumping in the gyre interior, subject to dynamical constraints. Specifically, w₁ has a contribution from strong coastal upwelling near the gyre boundary as suggested by observations and ensures that the model with a flux boundary condition is volume conserving in the absence of boundary fluxes (Q = 0) and diapycnal mixing (w^d = 0); this is equivalent to the condition τ(R) = 0. Note that Eq. (4) neglects the vertical variation of the Eulerian-mean vertical velocity, which is demonstrated to be small in section 4.

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(a) EOF 1, (b) EOF 2, and (c) EOF 3 of the observed Ekman pumping of Meneghello et al. (2017) over the given region. EOFs are scaled to have mean 1 m yr⁻¹ between the gyre center and ~400 km for comparison with the idealized forcing. (d) Spatial component w₁(r) of the idealized Ekman pumping in the model experiments with the fixed and flux (including no flux) boundary conditions, and azimuthal mean of the EOFs in (a)–(c). (e) Time-dependent component p^Ek(t) of the idealized Ekman pumping velocity in the model experiments and principal components corresponding to the EOFs in (a)–(c). See the text for details.

Citation: Journal of Physical Oceanography 51, 3; 10.1175/JPO-D-20-0077.1

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This decomposition of the Ekman pumping is validated by an EOF analysis of the observational data of Meneghello et al. (2017) over the region shown in Figs. 5a–c. The first three EOFs explain about 27%, 21%, and 15% of the variance, respectively. The first and second EOFs contain most of the variance in the radially symmetric patterns, and are qualitatively similar to the idealized spatial patterns of the Ekman pumping (Figs. 5a–d). Last, the velocity induced by diapycnal mixing is parameterized by a diapycnal diffusivity κ^d (m² s⁻¹). For simplicity, we have used the weak-slope approximation and an assumption that the background stratification $\partial \overline{b}/\partial z$ does not substantially change in time or space, resulting in $b-\overline{b}=\partial \overline{b}/\partial z\left(h-\overline{h}\right)$ for any buoyancy perturbations from the background stratification; the assumption is akin to one commonly used in the quasigeostrophic approximation.

b. Steady-state balance

If the gyre were fully equilibrated, Eq. (8) would provide a relationship among the (observed) Ekman pumping, isopycnal slope, and (unknown) vertically varying eddy buoyancy diffusivity. Zhong et al. (2019) have used a steady-state argument to estimate the recent isopycnal deepening in the PWW layer due to Ekman pumping and mesoscale eddy activity. Yet during the period 2005–18, the Beaufort Gyre was not in an equilibrated state, and ∂h/∂t is at least as large as the Ekman- and eddy-induced velocities in Eq. (4) (Figs. 6a,b). Thus, we consider a model for the transient dynamics of the gyre [Eq. (4) with all terms retained; see Fig. 4].

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(a) Estimates of ∂h/∂t, i.e., change of isopycnal depth with time, at indicated BGOS moorings as estimated from the linear trend over the time period of the available data. (b) Ekman pumping velocity penetrating to indicated depth, as calculated from Eq. (12) using the time-mean stratification from BGOS mooring B and the indicated wavelength L (km) of the surface Ekman pumping velocity. (c) Estimates of w^d (i.e., vertical velocity from diapycnal mixing) based on the time-mean ρ(z) at mooring B.

Citation: Journal of Physical Oceanography 51, 3; 10.1175/JPO-D-20-0077.1

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According to this model, several phenomena could potentially account for the expansion of the PWW layer, such as depth variation of Ekman pumping, K^b, diabatic processes, or explicitly prescribed boundary fluxes of buoyancy [i.e., the flux boundary condition of Eq. (6) with ∂Q/∂z ≠ 0]. In section 6, we show that thickening of the PWW layer results as a transient response to increasing Ekman pumping in the presence of depth-dependent eddy diffusivity K^b, while the vertical variations of the Ekman pumping velocity (section 4) and diapycnal mixing (section 5) act to contract the layer. In particular, we infer the vertical structure of the eddy buoyancy diffusivity from the transient evolution of the isopycnal depth and layer thickness, and show that, consistent with the structure inferred from Eq. (8), it is decreasing with depth in the PWW layer.

a. Eddy flux parameterizations

In this section, we investigate the role of mesoscale eddy activity in the transient evolution of the Beaufort Gyre. Eddy flux parameterizations arose from the need for coarse-resolution ocean models to parameterize subgrid-scale baroclinic processes. The optimal parameterization of eddy fluxes is uncertain, and multiple alternatives have been proposed, leading to different dynamics. The development of the GM parameterization around 1990 allowed climate models to run stably without flux corrections for the first time by eliminating the “Veronis effect” (i.e., spurious vertical velocities that result from the then-commonplace horizontal diffusion; see Redi 1982; Gent 2011). The GM scheme is predicated upon the principle that eddy fluxes should extract available potential energy from the fluid by reducing the slope of isopycnals while conserving the volume between isopycnals (Gent et al. 1995). Despite initial comparisons to a diffusion operator, the GM scheme constitutes an along-isopycnal, advective flux of buoyancy by eddy-induced transport velocities (i.e., υ* and w*; Gent et al. 1995; Treguier et al. 1997; Abernathey et al. 2013; Griffies 2018). The horizontal eddy velocity satisfies υ* = ∂(K^bS)/∂z, where $S=-\nabla b/\left(\partial \overline{b}/\partial z\right)$ represents the slope of buoyancy surfaces. In addition to adiabatic buoyancy fluxes, tracers are diffused along isopycnals with a Redi diffusivity, which differs in general from K^b but is often assumed to be equivalent (Redi 1982; Gent et al. 1995).

Alternatively, mesoscale eddy activity can be represented as a diffusion of potential vorticity within isopycnal layers. In this case, it follows that υ* = −K^q∂S/∂z (neglecting β, the meridional variation of the Coriolis parameter; Gent et al. 1995; Treguier et al. 1997; Abernathey et al. 2013; Griffies 2018). A related variant is the diffusion of thickness between isopycnal interfaces, which is similar to a potential vorticity diffusion. By way of distinction, Treguier et al. (1997) point out that in isopycnal models, the GM parameterization bases eddy fluxes on the height of isopycnals rather than the thickness of isopycnal layers, leading to significant differences in dynamics when the eddy diffusivity coefficient varies vertically.

Observational data suggest that potential vorticity gradients amplified the PWW layer when the gyre circulation intensified, which primarily took place during 2007–10 (e.g., Fig. 8 of Zhong et al. 2019). But if the Ekman-driven Eulerian mean circulation is incapable of affecting the interior PV gradients, what explains the amplification of the interior PV gradients during the gyre spinup and their decrease during spindowns? The effect of diabatic mixing in the gyre interior, away from coastal boundaries and surface mixed layer, is negligibly small on time scales of a few years, implying that answer lies in the eddy dynamics. However, considering the two common eddy parameterizations, the downgradient PV or layer thickness diffusion and the GM parameterization, only one can provide a sensible explanation. A thickness diffusion scheme for the PWW layer would direct eddy thickness fluxes down the mean gradient, i.e., outward from the gyre interior toward the boundary, leading to a reduction in the spatial gradient of thickness. In the absence of diabatic sources of layer thickness at the center of the gyre, the eddies would drive the isopycnals toward a state with zero thickness gradients in which their slopes are parallel. Thus, for a more energetic gyre with presumably stronger eddy variability one would expect to see a reduction of any preexisting interior thickness gradients, in contrast to the observations. The GM eddy parameterization can generate interior thickness gradients even in the absence of a residual mean circulation when the eddy buoyancy diffusivity K^b is depth dependent. Specifically, assuming that K^b is lesser at depth explains not only the observed mean state with nonparallel isopycnals but also the amplification of the interior layer thickness gradients occurring during the gyre spinup.

Numerical simulations of the Antarctic Circumpolar Current suggest that eddy diffusivities of different variables can have vastly different vertical structures. In a nonlinear, quasigeostrophic model, K^q was intensified near the critical level (~1000 m) at which Rossby waves propagate with the same velocity as the mean flow (Smith and Marshall 2009). In a primitive equation model, the eddy diffusivity of quasigeostrophic potential vorticity, Ertel potential vorticity, and a passive tracer (but not buoyancy) had similar vertical structures below ~500 m (Abernathey et al. 2013). While these findings depend upon the location of critical levels (and hence the baroclinic modes) that likely differ between the Beaufort Gyre and the Antarctic Circumpolar Current, they suggest that the equivalence of K^q and K^λ should not be assumed. Therefore, Eq. (15) cannot be directly integrated to obtain K^b, demonstrating the need for alternative methods to infer the vertical structure of the eddy buoyancy diffusivity.

b. Vertical structure of the eddy buoyancy diffusivity: Modeling experiments

To illustrate how the eddy diffusivity profile influences the transient evolution of the gyre, a series of nine numerical experiments are performed (Table 1). Specifically, the gyre model of Eq. (4) is set up with two isopycnal interfaces with initial depths of 100 and 200 m. The model is spun up for 30 years with a constant Ekman pumping velocity of −3 m yr⁻¹ and then forced with the interannually varying Ekman pumping of Meneghello et al. (2017). Three idealized profiles of eddy diffusivity are constructed: a uniform profile with K^b = 400 m² s⁻¹ for both layers; a profile that is decreasing with depth ($K_1^b$ and $K_2^b$ = 400 and 100 m² s⁻¹ respectively), and a profile that is increasing with depth ($K_1^b$ and $K_2^b$ = 100 and 400 m² s⁻¹ respectively). In addition, the boundary term (fixed boundary, no-flux boundary, and flux boundary) is systematically varied. In the flux case, the approximate PWW volume increase estimated by Zhong et al. (2019) (about 6.7 × 10³ km³ yr⁻¹) is forced into the gyre as a specified lateral flux; there is no injection of volume between the surface and the upper isopycnal. (Note that their estimate is time varying; our results, however, are rather robust to the choice of Q of this order of magnitude.) No boundary forcing is prescribed during the spinup. The vertical variation of the Eulerian-mean velocity and the diapycnal term have been shown to be small and are neglected.

Table 1.

Summary of numerical experiments performed with the three-layer model. Indicated are experiment identifier (ID), upper- and lower-layer eddy buoyancy diffusivity $K_1^b$ and $K_2^b$ (m² s⁻¹), the data source for Ekman pumping, boundary condition (BC), and length of integration (yr), excluding the spinup period of 30 yr. For forcing, “observed” refers to Ekman pumping that is based on Meneghello et al. (2017) and T^Ek refers to the time scale (yr) of the synthetic red noise. For the flux boundary condition, fluxes are estimated from Zhong et al. (2019) (see the text for details).

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At the end of the spinup period, a steady state has been reached in which the isopycnal slope is inversely proportional to the eddy buoyancy diffusivity, consistent with Eq. (8). In particular, the slope is constant with depth if K^b is constant; see dash–dotted lines in Fig. 7. Next, we discuss the transient dynamics in the experiments with the fixed and no-flux boundary conditions.

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Various states of the three-layer model from six experiments (summarized in Table 1). Two different profiles of the eddy buoyancy diffusivity are used: uniform profile ($K_i^b$ = 400 m² s⁻¹ for both interfaces) and surface-intensified profile ($K_1^b$ = 400 m² s⁻¹ and $K_2^b$ = 100 m² s⁻¹ for the upper and lower interface, respectively). In addition, three different boundary conditions (fixed, no flux, and flux) are used. The 1026 and 1027 kg m⁻³ isopycnals are represented by the blue and red lines, respectively; the initial state of the model (dashed lines), equilibrated state at the end of the 30-yr spinup (dash–dotted lines), and transient state at a selected time step (solid lines) are given. The experiments shown are (a) uniform eddy diffusivity and fixed boundary condition, (b) surface-intensified eddy diffusivity and fixed boundary condition, (c) uniform eddy diffusivity and no-flux boundary condition, (d) surface-intensified eddy diffusivity and no-flux boundary condition, (e) uniform eddy diffusivity and flux boundary condition, and (f) surface-intensified eddy diffusivity and flux boundary condition.

Citation: Journal of Physical Oceanography 51, 3; 10.1175/JPO-D-20-0077.1

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When a constant value of the eddy buoyancy diffusivity is imposed, the transient solution is characterized by constant isopycnal layer thickness over time, i.e., all isopycnals evolve in parallel when the model is initialized from a state of parallel isopycnals. (As a consequence of Eq. (4), isopycnals that are not parallel approach a parallel state approximately exponentially if K^b is constant with depth, neglecting the vertical variation of the Ekman pumping and diapycnal mixing.) However, when the eddy diffusivity varies with depth, changes in the Ekman pumping give rise to changes in both the isopycnal depth and layer thickness (cf. Figs. 7a,b and Figs. 7c,d). The relationship between the isopycnal depth and layer thickness variability reflects the eddy buoyancy diffusivity profile. For instance, layer thickness variations in the gyre interior in the experiment with $K_1^b>K_2^b$ are nearly opposite to those of the experiment with $K_2^b>K_1^b$ (Figs. 8e,f). With the fixed boundary condition, Ekman pumping drives changes in both the isopycnal depth and total layer volume if K^b varies with depth, since boundary fluxes depend upon the isopycnal slope at the boundary and hence the eddy diffusivity (Fig. 7b). Even if the total layer volume is constrained to be constant over time (as in the case of the no-flux boundary condition), mesoscale eddy fluxes induce layer thickness changes at fixed locations in the gyre interior that are compensated near the boundary (Fig. 7c).

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Time series of (a)–(c) depth h of the 1026 (blue) and 1027 kg m⁻³ (red) isopycnals, and (d)–(f) thickness Δh between the isopycnals, from selected model experiments (summarized in Table 1), forced with the observed Ekman pumping of Fig. 5e, with indicated profiles of the eddy buoyancy diffusivity, at a spatial location about 300 km from the gyre center. Dash patterns indicate different boundary conditions: fixed (solid line) and no flux (dashed line).

Citation: Journal of Physical Oceanography 51, 3; 10.1175/JPO-D-20-0077.1

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With the flux boundary condition, the isopcynal slope can vary with depth in the transient state even when the eddy buoyancy diffusivity is constant (Fig. 7e). However, this gives rise to a physically unrealistic profile of the isopycnal slope since the isopycnal must be deformed near the boundary to permit a flux into the gyre (Figs. 7e,f). In addition, since the gyre is thought to be driven by Ekman pumping (i.e., lateral fluxes respond to Ekman pumping in the gyre interior) rather than lateral buoyancy fluxes (i.e., boundary fluxes are forced and independent of the interior Ekman pumping), our subsequent theory and discussion focus upon the fixed and no-flux conditions.

c. A diagnostic for the vertical variation of the GM eddy diffusivity coefficient

This equation, combined with the variability of PWW isopycnal layer depths and thicknesses (Fig. 3) from the mooring observations, suggests the approximate profile of the eddy buoyancy diffusivity. The slope of the linear regression of Δh and h (Fig. 9) is an estimate of the left-hand side of Eq. (18). Using the observational data, the regression suggests that the eddy buoyancy diffusivity is ~10%, 40%, and 30% greater at the upper interface than the lower interface of the PWW layer for moorings A, B, and D, respectively. In this estimate, apparent outlying data points (gray dots in Fig. 3) are removed. If all the data were included, the estimates would change to ~10%, 50%, and 20%, respectively. As will be subsequently quantified, these values likely underestimate the true ratio.

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Scatterplot of Δh for the two isopycnals bounding the PWW layer (1026 and 1027 kg m⁻³) and h (for the upper isopynal); the same data are plotted as a time series in Fig. 3. A linear least squares fit over all data (dashed black line) and with apparent outliers of Δh (gray points) manually identified and removed (solid black line) is given; the slope is an estimate of the left-hand side of Eq. (18). The coefficient of determination R² is given in the upper left of each panel.

Citation: Journal of Physical Oceanography 51, 3; 10.1175/JPO-D-20-0077.1

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Equation (18) assumes that the gyre evolves in a state of dynamical equilibrium, which would be approximately satisfied if the temporal variation of Ekman pumping were at a lower frequency than the eddy-driven gyre equilibration time scale. The presence of Ekman pumping variability at a higher frequency than the gyre equilibration time scale would potentially introduce noise into Eq. (18). While the Ekman pumping and gyre equilibration time scales are not fully constrained by observations, several estimates of relevant time scales do exist. Experiments with a barotropic ocean–sea ice model forced with observed atmospheric variables suggest that the ocean circulation alternates between anomalously anticyclonic and cyclonic regimes with a 10–14-yr period (Proshutinsky and Johnson 1997; Proshutinsky et al. 2002, 2015), suggesting that the atmospheric forcing has an interannual to decadal memory. It was also shown that the Beaufort Gyre freshwater content has a decadal memory of the sea level pressure field (Johnson et al. 2018). The equilibration time scale of ~5 years was estimated for a surface stress-driven gyre and shown to be inversely proportional to the mesoscale eddy diffusivity (Manucharyan and Spall 2016). In addition, numerical simulations reveal that the eddy field itself might require a significant equilibration time scale of 2–6 years because of the so-called eddy memory effect (Manucharyan et al. 2017).

To investigate how the estimates of K^b depend on the time scale of the Ekman pumping variability, we perform numerical experiments with various synthetic forcing, each constructed as a red noise process with a specified persistence time scale T^Ek. These time series are scaled to produce reasonable means and variances compared with observations. We integrate the three-layer model with T^Ek ≈ T^eq ≈ 6 yr (where T^eq is the mean of the two interfaces) for 500 years, treating the initial 30-yr period as a spinup (experiment 10; Fig. 10a). A distribution of $K_1^b/K_2^b$ is obtained by regressing Δh and h over all overlapping moving windows of length 50 years (for downsampled data; the spinup is excluded). Means and 90% confidence intervals are then constructed from the resulting distributions. An illustration of the method of estimating $K_1^b/K_2^b$ is presented in Figs. 10a and 10b.

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(a) Time series of h (blue) and Δh (anomalies from time mean; red) from experiment 10 (see Table 1). (b) Scatterplot of Δh vs h from experiment 10 about 300 km from the gyre center to the boundary during a selected 50-yr period. The slope of the dashed black line represents the left-hand side of Eq. (18); the slope of the solid black line represents that recovered from least squares fit to modeled data. (c) Scatterplot of $K_1^b/K_2^b$ vs T^Ek/T^eq from a regression of Δh and h. ODE data are from Eq. (19); PDE refers to experiment 10. The dashed line represents the specified ratio. (d) Scatterplot of recovered vs specified $K_1^b/K_2^b$ for T^Ek ≈ T^eq ≈ 6 yr. Deviation from the dashed line represents the error. In (c) and (d), boxes represent the mean of the distribution of regression coefficients over 50-yr moving windows of the data; error bars are the 5th and 95th percentiles.

Citation: Journal of Physical Oceanography 51, 3; 10.1175/JPO-D-20-0077.1

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The error in recovering the ratio of eddy diffusivities from Eq. (18) generally decreases as T^Ek increases above T^eq (Fig. 10c) and the method tends to underestimate $K_1^b/K_2^b$ when $K_1^b>K_2^b$. Similarly, the method tends to overestimate $K_1^b/K_2^b$ when $K_1^b<K_2^b$. The underestimation is scale dependent, such that as the magnitude of $K_1^b/K_2^b$ increases, the bias increases (Fig. 10d). (As the deviation of K^b between the two layers increases, the deviation of the eddy equilibration time scale also increases, such that variations in layer depth evolve less coherently with variations in layer thickness.) For representative choices of K^b and T^Ek ≈ T^eq ≈ 6 yr (experiment 10), the method recovers a ratio of $K_1^b/K_2^b$ that is about 90% of that specified (Fig. 10d). Thus, Eq. (18) provides a reasonable estimate of the eddy buoyancy diffusivity ratio.