A Theory of the Wind-Driven Beaufort Gyre Variability

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J OURNALOF P HYSICAL O CEANOGRAPHY

A theory of the wind-driven Beaufort Gyre variability

GEORGY EMANUCHARYAN∗

Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA.

MICHAEL ASPALL Department of Physical Oceanography, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts, MA, 02540, USA.

ANDREW FTHOMPSON Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA.

ABSTRACT

The halocline of the Beaufort Gyre varies significantly on interannual to decadal time scales affecting the freshwater content (FWC) of the Arctic Ocean. A recent study (Manucharyan and Spall 2016) emphasized the role of mesoscale eddies in setting equilibrium halocline properties. Here, we explore the role of eddies in the Ekman-driven gyre variability. Following the Transformed Eulerian Mean framework, we develop a theory that links the FWC variability to the stability of the large scale gyre, defined as the inverse of its equilibration time. We verify our theory’s agreement with eddy-resolving numerical simulations and demonstrate that the gyre stability is explicitly controlled by the mesoscale eddy diffusivity. The model suggests that a correct representation of the halocline dynamics requires the eddy diffusivity of 300 ± 200 m2 s−1, which is lower than what is used in most low-resolution climate models. We demonstrate that on seasonal time scales, FWC variability is explicitly governed by the Ekman pumping, whereas on interannual and longer time scales eddies provide an equally important contribution. In addition, only large-scale Ekman pumping patterns can significantly alter the FWC, with spatially localized perturbations being an order of magnitude less efficient. Incorporating the eddy effects, we introduce a FWC tendency diagnostic – the Gyre Index – that can be conveniently calculated using observations of surface stress and halocline slope only along the gyre boundaries. We demonstrate its strong predictive capabilities in the eddy- resolving model forced by stochastic winds and speculate that this index would be of use in interpreting FWC evolution in observations as well as in numerical models.

1. Introduction al 2009). The surface stress results largely from the sea ice drag and quantification of the oceanic response is sub- The observed increase of the Arctic freshwater content over the past two decades has been related partially to ject to our uncertainties in the sea ice-ocean momentum a deepening of the halocline and partially to water-mass exchange (Martin et al 2014; Giles et al 2012). freshening (Rabe et al 2014). In particular, a significant Modeling and observational studies demonstrate a di- contribution to the overall Arctic freshening came from rect relation between the freshwater content and the Ek- the Beaufort Gyre; here the freshwater content (FWC) in- man pumping (Proshutinsky et al 2002, 2009; Stewart and creased by about 30% over the past decade (Haine et al Haine 2013; Timmermans et al. 2011, 2014). Thus, Stew- 2015). art and Haine (2013) show that a reduction in the strength Observational evidence suggests that the halocline dy- of anticyclonic winds can lead to a redistribution of FWC namics of the Beaufort Gyre are to a large extent governed within the Arctic Ocean and to changes in the exchange by the anticyclonic atmospheric winds that drive the large- between North Atlantic and Pacific Oceans. The basic scale gyre circulation – and to a lesser extent due to the gyre dynamics are conventionally described in the follow- availability of Arctic freshwater sources (Proshutinsky et ing way: the surface Ekman transport converges surface fresh waters and deepens the halocline, thus storing fresh-

∗Corresponding author address: Georgy E Manucharyan, MC 131- water in the gyre (e.g. Proshutinsky et al 2009). This argu- 24, California Institute of Technology, 1200 East California Boulevard, ment, however, does not explain how a steady state can be Pasadena, CA, 91125, USA. achieved as it does not specify a mechanism opposing the E-mail: [email protected]

Generated using v4.3.2 of the AMS LATEX template 1 2 J OURNALOF P HYSICAL O CEANOGRAPHY continuous halocline deepening due to the Ekman pump- Ekman pumping ing. The importance of the opposing mechanism can be Fresh Fresh H clearly illustrated from the point of view of the large-scale al line ocl w loc gyre stability in its integral sense. By stability, we imply ine e Ha that there exists a statistical equilibrium state of the gyre

Eddies Eddies U p

(averaged over small scale features, e.g. internal waves g

and eddies) and that any deviations from this equilibrium l

e l

w i n

w e

Salty Salty g would decay on a finite time scale. Since the gyre is a p persistent large-scale feature of the Arctic Ocean, it is rea- U sonable to assume that it is a stable, externally-driven system. Using basic concepts of dynamical systems theory (e.g. Tabor 1989), the linear stability assumption implies Bottom Ekman layer that near its equilibrium, perturbations of the gyre state, a, obey a basic equation of forced exponential decay FIG. 1. Schematic view of the Beaufort Gyre circulation depicting a balance between Ekman pumping and eddy induced vertical velocity. The Ekman vertical velocity penetrates through the entire water column. da a = − + w. (1) Surface convergence of the Ekman transport is balanced by the corre- dt T sponding bottom divergence with the mean circulation being closed via coastal upwelling. We will formally derive Eq. 1 in this manuscript (see Sec- tion 6), providing physical interpretation for its variables, but at this point one can conceptually think of a and w as are, how they depend on external forcing, and how they measures of the bulk halocline deepening (or FWC) and affect the gyre variability. Ekman pumping correspondingly. The gyre stability is While much of the scientific effort has been devoted defined as the inverse of its equilibration time scale, T, to the exploration of the impact of Ekman pumping (e.g. and here we assume that the large-scale gyre circulation Proshutinsky et al 2002; Yang 2009; McPhee 2012; Tim- is always stable (i.e. T > 0). The damping term (−a/T) mermans et al. 2014; Cole et al 2014), factors controlling represents a linearization of a process opposing the Ekman the gyre stability are poorly understood. A recently pro- pumping near its equilibrium. posed hypothesis points to mesoscale eddy transport be- Here it is important to distinguish the concepts of large- ing a sufficient mechanism to oppose the Ekman pump- scale gyre stability in terms of a statistically-averaged ing (Davis et al 2014; Marshall 2015; Lique et al 2015; equilibrium and this state’s hydrodynamic stability char- Manucharyan and Spall 2016; Yang et al 2016). The acteristics. By gyre stability we refer to the equilibra- eddies are generated via baroclinic instabilities of the tion (exponential decay) of large-scale anomalies of a large-scale gyre circulation that release available poten- statistically-averaged circulation that does not contain in- tial energy associated with the halocline deepening. The formation about individual small-scale eddies. In contrast, eddy buoyancy fluxes (or layer thickness fluxes) through- baroclinic instabilities lead to the exponential growth of out this process act to adiabatically flatten the halocline, individual small-scale perturbations. Thus, the gyre can be thus opposing the deepening due to the Ekman pumping stable in a statistically-averaged sense and at the same time (schematically shown in Fig. 1). hydrodynamically unstable at small scales. Our proposed The mesoscale eddies discussed here are associated hypothesis implies that the stability of the large-scale gyre with baroclinic instabilities of a large scale halocline slope depends on the cumulative action of small-scale eddies, and hence predominantly carry the energy of the first baro- individually generated through baroclinic instabilities. clinic mode. The majority of the observational analysis in Based on a basic dimensional analysis, the halocline the Arctic Ocean, however, has been devoted to intense lo- deepening (units of meters) should depend not only on Ek- calized vortices (Manley and Hunkins 1985; Timmermans man pumping (units of meters per second) but also on the et al. 2008; Dmitrenko et al. 2008; Watanabe 2011; Zhao adjustment time scale (units of seconds). For example, et al 2014) that can form at boundary currents (Watanabe according to (1), a = wT for a system in steady state or 2013; Spall et al. 2008; Spall 2013) and at surface ocean subject to a slowly-evolving forcing. Moreover, the ampli- fronts (Manucharyan and Timmermans 2013). Although tude of the gyre variability in response to time-dependent observational analysis for the role of halocline-origin ed- Ekman pumping would also be directly proportional to T dies in the interior salinity budget is currently lacking, this (i.e. reduced stability would imply a larger variance for the is likely due to a lack of data sufficient to calculate energy same forcing). Since the gyre stability is directly related conversion rates and eddy salt fluxes in the basin interior. to the nature of the processes that counteract the Ekman Taking into account the eddy transport mechanism, pumping, it is essential to determine what these processes Davis et al (2014) have used a low-resolution shallow wa- J OURNALOF P HYSICAL O CEANOGRAPHY 3 ter model (with eddies parameterized as horizontal dif- a. TEM framework fusion) to explore seasonal gyre dynamics. Yang et al The Reynolds averaged buoyancy equation is written in (2016) explored the potential vorticity budget of the gyre cylindrical coordinates as: and reached the conclusion that the eddies are necessary ¯ ¯ ¯ 0 0 0 0 ¯ to close it. Previous studies have suggested that there are bt + v¯br + w¯bz + (v br) + (w bz) = S, (2) links between the dynamics governing the Antarctic Cir- cumpolar Current (ACC) (Marshall and Radko 2003) and where b denotes the buoyancy, (v¯,w¯) are the Eulerian gyres with azimuthally-symmetric circulations: Su et al. mean radial and vertical velocities in the (r,z) coordinates, (2014) in the Weddell Gyre, Marshall et al (2002) in labo- and S represents buoyancy sources and sinks. The bar here ratory experiments, and Marshall (2015) in the Beaufort represents an average either in the azimuthal direction or Gyre. The stratification in these regions is determined in an ensemble mean sense; we only consider axisymmet- from a leading order balance between mesoscale eddy ric solutions here. Next, the eddy buoyancy fluxes are as- transport and Ekman pumping. sumed to be predominantly aligned with isopycnals in the Vallis A recent study by Manucharyan and Spall (2016) (from interior of the ocean – the so called adiabatic limit ( 2006). In this limit the flux divergences can be represented now on MS16) demonstrated that the instabilities associ- as an additional advection of mean gradients by the eddy ated with the observed halocline slope are indeed generat- stream function such that: ing sufficient mesoscale activity to cumulatively counter- ∗ ∗ act the Ekman pumping. They also developed a set of an- b¯t + (v¯+ v )b¯r + (w¯ + w )b¯z = S¯, (3) alytical scaling laws that predict the gyre adjustment time scale, halocline depth, and maximum freshwater flux out where the eddy advection velocities can be defined from ∗ of the gyre and explicitly relate these key gyre character- an eddy stream function ψ as: istics to the mesoscale eddy dynamics. w0b0 v0b0 In light of this newly developed understanding it is im- ∗ = − = , ψ ¯ ¯ (4) perative to explore the role of mesoscale eddies in deter- br bz mining the stability of the gyre and its transient dynamics. ∗ ∗ ∗ 1 ∗ u = −ψ , w = (rψ )r. (5) Here, we present a basic theory of the wind-driven Beau- z r fort Gyre variability. The manuscript is organized in the The adiabatic assumption breaks down near boundary lay- following way. In Section 2 we briefly review the trans- ers that have significant diabatic fluxes due to buoyancy formed Eulerian mean framework that is used through- forcing and/or enhanced mixing. Near such boundaries out our analysis. In Section 3 we describe an idealized the processes obey different dynamics that we do not at- Beaufort Gyre model that we use for our process stud- tempt to represent here. ies. In Section 4 we diagnose a key eddy field charac- Thus, the ensemble mean buoyancy is advected by a teristic, the eddy diffusivity, and assess its sensitivity to residual circulation, ψ˜ ≡ ψ¯ + ψ∗, that exists in order to forcing. In Section 5 we demonstrate how mesoscale ed- balance the buoyancy sources and sinks. This formalism dies affect the gyre stability and equilibration. In Section clarifies that in the interior of the ocean, where the diabatic 6 we quantify the FWC response to periodic and spatially- fluxes are small, a steady state implies that the residual inhomogeneous Ekman pumping. In Section 7 we intro- circulation has to vanish, i.e. ψ˜ = 0. duce a new Gyre Index that includes the effects of both Ek- At this point, in order to make analytical progress in un- man pumping and mesoscale eddies in order to rationalize derstanding the buoyancy variability, it is necessary to in- FWC variability. We summarize and discuss implications troduce a closure for the eddy-driven stream function and in Section 8. to determine the Eulerian mean stream function.

b. Eulerian mean stream function 2. Theoretical background The azimuthally-averaged azimuthal momentum equation can be simplified for the gyre dynamics. First, we We briefly describe the mathematical formulation of the assume that contributions from the Reynolds stresses are Transformed Eulerian Mean framework (TEM) (Andrews negligible for large-scale flows. Second, the inertial term and McIntyre 1976; Vallis 2006; Marshall and Radko can also be neglected for time scales sufficiently larger 2003) that is used in our analysis. Within this framework than f −1. In addition, the azimuthal pressure gradient term the eddy buoyancy fluxes can be represented via an ad- vanishes as a result azimuthal averaging. Taking this into ditional eddy-induced stream function. This allows one account, a simplified form of the momentum equation rep- to view the ensemble mean buoyancy as being advected resents the steady Ekman dynamics: by the residual between the Eulerian mean and the eddy- driven circulations. f v¯ = −τ¯z/ρ0, (6) 4 J OURNALOF P HYSICAL O CEANOGRAPHY

−3 where τ is the azimuthal stress and ρ0=1023 kg m is the Surface stress at a particular location sets only the halo- reference ocean density. In the ocean interior the vertical cline slope, not its depth. The halocline slope defines the shear in stress is negligible, allowing one to obtain an ex- baroclinically-unstable azimuthal currents (via the thermal pression for the Eulerian stream function by vertically in- wind relation) that generate eddies to locally oppose the tegrating the momentum equation from the surface to any Ekman pumping. depth below the surface Ekman layer: Note that the Beaufort Gyre surface stress is on average anticyclonic (τ < 0) and hence the halocline is deeper in τ ∗ ψ¯ = , (7) the interior, the isopycnal slope br/bz < 0, and ψ > 0. ρ0 f Given a surface stress profile, the halocline deepening 1 v¯ = −ψ¯ , w¯ = (rψ¯ ) , (8) across the gyre can be determined by integrating (11): z r r 1 Z R −τ(r) n From now on τ = τ(r,t) denotes the radially- and time- ∆h = dr. (12) dependent azimuthal surface stress – an external forcing 0 ρ0 f k for the gyre. Thus, the large-scale Eulerian mean flow is Eq. 12 is identical to the expression for the ACC depth in a entirely determined by the surface stress distribution and theory developed by Marshall and Radko (2003). Assum- does not depend on the buoyancy field or the mesoscale ing the power n = 2, and considering a special case of uni- eddy activity. Note that this relation can be significantly form Ekman pumping (corresponding to τ(r) = −τˆr/R) affected by the presence of topography. the halocline deepens by

1 c. Mesoscale eddy parameterization 2 τˆ 2 ∆h = R . (13) Mesoscale eddies emerge due to an instability of the 3 ρ0 f k baroclinic flow and their amplitude is related to the stability characteristics of the flow. Here, we implement 3. Idealized Beaufort Gyre model a Gent-McWilliams type mesoscale eddy parameteriza- We implement an idealized model of the Beaufort Gyre tion (Gent and McWilliams 1990) that assumes a down- as was developed in MS16. It consists of a cylindrical gradient nature for the eddy buoyancy fluxes (or layer ocean basin (diameter 1200 km, depth 800 m) driven by thickness fluxes): an anticyclonic surface-stress τ(r). The primitive equations are solved using the MIT General Circulation Model v0b0 = −Kb . (9) r (Marshall, Hill et al 1997; Marshall, Adcroft et al 1997; The eddy diffusivity K = K(s) can in general depend on Adcroft et al. 2015) in its three-dimensional, hydrostatic the magnitude of the isopycnal slope s = |b¯r/b¯z|, thus configuration, with a 4 km horizontal resolution that is varying in space and time. Past studies suggested power- sufficient to permit Rossby deformation scale eddies. The law dependencies (K ∼ sn−1). Powers n = 1 or n = 2 (Gent deformation radius is about 20 km in our simulations (see and McWilliams 1990; Visbeck et al. 1997) are commonly more details in Appendix A of MS16). For simplicity, the used in low-resolution ocean models while MS16 suggest present configuration uses a flat bottom basin; MS16 in- that n = 3 might be more appropriate for the Beaufort cluded a continental slope. Gyre. Since observations are not sufficient to differentiate Here we focus on how surface stress and mesoscale ed- between these parameterizations, we continue the analysis dies influence the isopycnal distribution in the Beaufort assuming a general power-law dependence: Gyre and hence the surface buoyancy forcing is neglected. Within our idealized gyre model we prescribe a fixed- ψ∗ = Ks¯ = ks¯n, (10) buoyancy boundary condition at the coastal boundaries (via fast restoring to a given profile) and no-flux condi- 2 −1 where now k (having units of m s ) is a constant that we tions at the surface and bottom. The boundary conditions will refer to as eddy efficiency to distinguish it from time- aim to represent the unresolved dynamics that occur over and space-dependent eddy diffusivity K. the shallow shelves as well as the water-mass exchanges with other basins. We note, however, that these coastal dy- d. Steady state namics can in general depend on the surface stress, which In the absence of surface buoyancy forcing or strong causes upwelling and boundary mixing (Woodgate et al vertical mixing the steady state residual ocean circulation 2005; Pickart et at 2013). has to vanish (Marshall and Radko 2003; Su et al. 2014) The implemented fixed-buoyancy boundary conditions implying that imply that there exists an infinite reservoir of surface fresh water and bottom salty waters that the gyre is allowed τ b n to draw upon. The infinite water-mass reservoir repre- ψ˜ = + k − r = 0. (11) ρ0 f bz sents dynamics only on sufficiently long time scales that J OURNALOF P HYSICAL O CEANOGRAPHY 5

4. Mesoscale eddy diffusivity #10 15

) 3 0.4 The theoretical predictions of the gyre state rely on the

)

! s

S relevance of the mesoscale eddy parameterization which

( m

n uses an a priori unknown parameter k that is directly re-

k c

( 2 0.2 lated to the mean state eddy diffusivity K0. To ensure con-

u f

g sistency, we diagnose the eddy diffusivity using several

e m

n different methods based on our theoretical predictions.

c t

i We start with a direct estimate of diffusivity using the s

t 1 0

e l

n eddy buoyancy ﬂuxes (see Eq. 9) of the equilibrated gyre

u k

d simulation for a reference run:

e d

R 0 0

E v b 0 -0.2 K∗(r,z) = − , (14) 0 5 10 15 20 25 30 0 b¯ years r where the overline represents both temporal and azimuthal FIG. 2. Time series of the gyre integrated eddy kinetic energy and averaging and primes are the deviations from this mean. a corresponding residual circulation. The residual circulation was cal- The eddy diffusivity for a reference Beaufort Gyre sim- culated using diagnosed thickness fluxes in buoyancy coordinates (see ulation that has a relatively weak forcing (τ0 ∼ 0.015 N Appendix B in MS16) and averaged radially between 450km and 550km −2 2 −1 over the halocline layer bounded by salinities 30 and 32.5. Note that the m ) ranges between 100 − 500 m s (Fig. 3a). With amplitude of ψ˜ significantly reduces as the gyre spins up. the exception of the near coastal boundary layer, the diffusivity increases towards the gyre edges where the halocline slope (and hence the baroclinicity of the currents) is higher, which is consistent with our eddy parameterization (Eq. 10). The halocline appears rather thick in the figure allow for the continental shelf water masses to be replen- because of the spatial and temporal averaging of the thick- ished. Our idealized study does not include the processes ness variations represented by the mesoscale eddies. The of water-mass formation and exchange at the boundaries, halocline is typically 50 m thick locally in space and time. instead focusing on the long-term wind-driven halocline Using analytical predictions (Eqn. 13) we can also in- dynamics in the interior of the gyre. Nevertheless, once fer the mesoscale eddy diffusivity necessary to support the the understanding of internal gyre dynamics is gained, the simulated halocline deepening as: use of simplifying boundary conditions in the numerical ∆h 2 τˆ R model (and in analytical analysis) can be relaxed to in- K0 (τˆ) = , (15) 3 ρ0 f ∆h(τˆ) clude more realistic boundary processes. Spin-up simulations were initialized with a where we assumed an eddy parameterization power n = 2 horizontally-uniform stratification (50 m initial halo- (i.e. K = ks) as in Visbeck et al. (1997) and a charac- s = cline depth) forced by a uniform constant in time teristic slope was taken near the edge of the gyre [τˆ/(ρ f k)]1/n = 1.5∆h/R. Both (14) and (15) produce Ekman pumping (corresponding to a linear surface 0 similar eddy diffusivity estimates (Fig. 3b); they also show ˆ stress profile τ0 = −τr/R). The mean state simulations a similar sensitivity to surface stress forcing. These in- were spun up for at least 50 years to ensure equilib- ferred eddy diffusivities should be thought of as bulk val- rium. The following wind stress amplitudes were used ues representative of the gyre as a whole; the instantaneous τˆ = {0.005; 0.01; 0.015; 0.03; 0.05; 0.075}N m−2; we values can differ depending on location and time. Note consider τˆ = 0.015N m−2 as a reference run represen- that because s ∼ ∆h/R ∼ τ1/2, the definition of K in Eq. tative of the present-day Beaufort Gyre. Perturbation 15 is consistent with our assumed parameterization K ∼ s. experiments that are described in the following sections According to Visbeck et al. (1997) the coefficient k = 2 use spatially-inhomogeneous and time-dependent Ekman cNl where c = 0.015 is an empirical constant, N is the pumping. stratification parameter (N ≈ 130 f in our simulations), and l is the width of the baroclinic zone. Taking estimates for Following the growth of mesoscale eddies, this ideal- the diffusivity K = 300 m2 s−1 and the slope s = 10−4, ized Beaufort Gyre model achieves a statistically-steady we find k = 3 × 106 m2 s−1, which implies that l ≈ 150 state that coincides with a vanishing residual circulation km, using hte same value for the empirical constant c as in (Fig. 2). Thus, mesoscale eddies provide a mechanism Visbeck et al. (1997). Note that the estimated size of the to arrest the deepening of the halocline. We argue below baroclinic zone l is less than the gyre radius. This is qual- that eddies are also key to the temporal response of the itatively consistent with the numerical experiments show- halocline to surface stress perturbations. ing intense eddy generation near the edges of the gyre and 6 J OURNALOF P HYSICAL O CEANOGRAPHY not near its center where the halocline slope is negligibly A small. 0 The eddy diffusivity K increases with the surface stress following a nearly linear relationship (Fig. 3b) with the 50 rate of increase of about 170 m2 s−1 per 0.01 N m−2 increase in τˆ. This linear increase implies that the eddy sat- 100

uration regime would be reached for strong forcing, i.e. ) m for a sufﬁciently strong surface stress the halocline slope (

h 150

would approach a critical value because Ks = τ/(ρ0 f ) and t p

K ∼ τ. A corresponding halocline deepening is saturated e D at about ∆h ≈ 150 m. The eddy saturation phenomenon 200 has been extensively discussed in the ACC, where the baroclinic transport is only weakly sensitive to changes in 100 200 300 400 500 250 the strength of the surface westerlies (e.g. Hallberg and 2 !1 Gnanadesikan 2001; Meredith et al 2004; Munday et al eddy di,usivity (m s ) 2013). In contrast, because of the relatively weak forcing 300 0 200 400 600 the Beaufort Gyre is far from the eddy saturation limit and r (km) is highly sensitive to the surface stress (MS16). Thus, sig- B 2.0

niﬁcant gyre variability should be expected in response to ) 1

! T "h $

transient forcing. s K0 K0 K0

2 m

0 1.5

5. Transient TEM equations 0

0 1

Now that we have developed a basic understanding of # (

the mesoscale eddy response to forcing and conﬁrmed the y 1.0

t i

appropriateness of the eddy parameterization, we proceed v

i s

to explore the implications for the transient gyre dynam- u , ics. Combining the expressions for the Eulerian and the i 0.5 2 d K0 = 2420 =^=(;0f) + 230m =s

parameterized eddy stream functions, the evolution equa- y d tion for buoyancy is: d E 0 1 0 0.02 0.04 0.06 0.08 b + (ψ˜ r) b + ψ˜ b = S¯, (16) !2 t r r z z r Surface stress =^ (N m ) τ b n ψ˜ = + k − r . (17) ρ0 f bz FIG. 3. a) Spatial distribution of the equilibrium eddy diffusivity ∗ 0 0 ¯ K0 = −v b /br (14) as diagnosed from the eddy-resolving model for the − This nonlinear equation is relevant to the interior of the reference run (τ = 0.015 N m 2). Contours show equally spaced mean gyre and is subject to the appropriate boundary and initial state isohalines with salinity increment of 0.25. In weakly stratified regions shown in white the calculation of eddy diffusivity is not appropri- conditions: ate a the buoyancy gradient is close to zero there. The isopycnal slope in the top 50 m is reversed due to a presence of vertical mixing. b) Depen- b|r=R = b0(z), br|r=0 = 0, (18) dence of various definitions of the eddy diffusivity K on the magnitude T ∆h of the surface-stress. K0 (black circles, Eqn. 29) and K0 (red dia- bz|z=0,H = 0, b|t=0 = b0. (19) monds, Eqn. 15) are inferred from the diagnosed gyre adjustment time ∗ and the mean halocline depth, respectively; K0 (blue triangles, Eqn. 14) a. Scaling analysis are diagnosed from the model (averaged between 400 km and 450 km within the halocline layer). A linear fit to all the data points corresponds 2 −1 It is insightful to consider a scaling analysis for the gyre to K = 2420[τˆ/(ρ0 f )] + 230 m s (black dashed line). dynamics. On one hand, the system approaches equilibrium on an eddy-diffusion time scale defined as T ∼ R2/K, where eddy diffusivity K ∼ k(h/R)n−1 depends on the unknown halocline depth. Diffusive scaling implies that a deeper halocline would have a larger eddy diffusivity and would thus equilibrate faster. On the other hand, the pumping of freshwater and its horizontal diffusion due to isopycnal depth h together with the Ekman pumping ve- eddy transport the two mechanisms have to operate on a locity define a vertical advection time scale T ∼ h/we, where we ∼ τˆ/(ρ0 f R) is the Ekman pumping. Since the similar time scale. Following MS16, the two time scales dominant balance is achieved between the vertical Ekman can be equated to obtain a scaling for the halocline depth J OURNALOF P HYSICAL O CEANOGRAPHY 7

6. Gyre equilibration and stability 6 150 In this section we derive analytical time-dependent solutions for the halocline depth.

4 100 a. Linear dynamics near equilibrium

We assume that for a given mean surface stress τ0(r) there exists a steady state corresponding to a long-term time average (i.e. over time scales much longer than the 2 50 gyre spin-up time). We then assume that Ekman pumping perturbations lead to sufficiently small buoyancy perturba- Halocline deepening (m) deepening Halocline tions for which a linearization of the full nonlinear equa- Adjustment time scale (years) scale time Adjustment tion set (Eq. 16) is appropriate. In other words we ex- 0 0 plore the transient gyre dynamics where isopycnal depth 0 0.02 0.04 0.06 0.08 Surface stress (N=m2) perturbations can be considered small compared to their time-averaged state. FIG. 4. Equilibrium halocline deepening across the gyre (red y-axis) The linearized equations for the evolution of halocline and its e-folding adjustment time scale diagnosed from the spin up sim- depth anomalies h (see derivation in Appendix A) result in ulations (black y-axis) plotted as functions of the mean surface stress a diffusion equation forced by Ekman pumping: forcing (linear surface stress profile was used τ(r) = −τˆr/R). Theoret- ical scaling predictions are shown in solid lines for the power n = 3 and 1 1 −τ in dashed lines for n = 2 (see Eqns. 20-21). Each symbol is diagnosed ht = nK0rhr + r . (22) from a numerical model run with different surface stress. r r r ρ0 f r | {z } | {z } Eddy diffusion Ekman pumping and the adjustment time scale: Here τ is the surface stress perturbation from its mean value τ0, and K0(r) is a background eddy diffusivity set 1 by the mean isopycnal slope as τˆ n ∆h ∼ R , (20) n−1 ρ0 f k n n−1 −τ0(r) 1−n K0 = ks0 = k . (23) R2 τˆ n ρ0 f k T ∼ . (21) k ρ0 f k The first term on the right hand side of Eq. 22 acts to diffuse the isopycnal depth perturbations with a space- The expressions above imply that stronger pumping would dependent diffusivity equal to nK0; the prefactor n ap- lead to deeper halocline with faster adjustment time scale pears for a linear problem because of the a power-law de- (a more stable gyre). In addition, both scaling laws ex- pendence of eddy diffusivity on the slope (see Appendix plicitly depend on the eddy efficiency: for smaller k the A). This representation of mesoscale eddies as thickness halocline would be deeper and the adjustment time scale diffusivity is analogous to the expression in Gent and longer (a less stable gyre). MS16 show that these scaling McWilliams (1990); Su et al. (2014, e.g.). The second laws hold for a wide range of surface stress forcing and we term is the perturbation in Ekman pumping we that acts confirm (20) and (21) for the case of no topography under as a forcing for the isopycnal depth: consideration here (Fig. 4). Depending on the choice of the power n there can be −τ 1 −τ we = curl = r . (24) major qualitative differences in the gyre sensitivity to forc- ρ0 f r ρ0 f r ing. Thus, n = 1, equivalent to a constant eddy diffusivity, results in an equilibration time scale that is independent Thus, consistent with the mean state dynamics, it is an im- of the surface forcing and a halocline depth that scales lin- balance between the eddy diffusion and Ekman pumping early with the surface stress (implying there is no eddy sat- that drives the halocline depth perturbations. uration). This provides a poor representation of the simu- b. Gyre adjustment time scale lated gyre dynamics. Differences between n = 2 and n = 3 would manifest only for a sufficiently wide range of forc- One of the most important quantities that describes the ing magnitudes. For the range of forcing relevant to the gyre dynamics is its stability, i.e. the adjustment time scale −2 Arctic Ocean (τ . 0.1 N m ) n = 2 and n = 3 produce associated with the exponential decay of perturbations. In similar results (Fig. 4). Note, that the power n is not a this section we explore the impact of eddies on this adjust- priori constrained to be an integer. ment. 8 J OURNALOF P HYSICAL O CEANOGRAPHY

Given any radial proﬁle of the anticyclonic mean sur- 20 2 face stress [τ0(r)] one can calculate the background diffu- 0.005N/m ) 0.01N/m2

∗ 3 sivity K0(r) (from Eq. 23) and the eigenfunctions hi with 2

m 0.015N/m −1 corresponding eigenvalues T > 0 for the eddy diffusion k 15 0.03N/m2 i 0 0 0.05N/m2 operator: 0 2 1 h∗ 1 0.075N/m ∗ i # rK (r)h = − . (25) ( r 0 ir T

r i C 10 W

Here, homogeneous boundary conditions h(R) = 0 and F Slope = "S=Sref hr(0) = 0 should be used (see Appendix A). Since this diffusion operator is self-adjoint (for an r-weighted norm 5 and K0(r) > 0) any halocline depth perturbation can be 40 60 80 100 120 140 3 decomposed into contributions from its orthogonal eigen- V (#1000km ) ∗ modes (h(r)i ) as FIG. 5. Freshwater volume and freshwater content plotted against ∞ ∗ each for the spinup time series of different numerical model experi- h = ∑ ai(t)hi , (26) ments. Dashed line shows a slope of ∆S/Sre f supporting Eq. 31. i=1 where we sort the eigenfunctions corresponding to their the scaling laws (Eq. 21), here we determined the multi- eigenvalues starting from the smallest (their indices cor- plicative prefactor (nλ)−1 ≈ 0.1 which turns out to be an respond to the number of zero crossings). Thus, the first order of magnitude smaller than 1. eigenfunction corresponds to a large-scale halocline deep- Using the gyre adjustment time scale diagnosed from ening, whereas higher eigenmodes are more oscillatory the spinup time series (Fig. 4 black circles) we can infer in space and correspond to higher eigenvalues, discussed an appropriate eddy diffusivity that can generate such a more below. time scale as In an unforced case (i.e. Ekman pumping perturbations 2 we = 0), the amplitudes ai(t) would evolve independently T 1 R K0 (τˆ) = , (29) from each other according to a simple exponential decay λn T0(τˆ) law: dai ai where n = 2 and a corresponding λ = 4.7 are used (see = − . (27) dt Ti Eq. 28). Using the diagnosed equilibration time scale T0 This result confirms our a priori assumption about the (Fig. 4) we calculate the corresponding eddy diffusivity gyre being a stable system (see Eq. 1 and related discus- based on Eq. (29) above and plot in Fig. 3b (black circles). sions). Eigenvalues represent the inverse of the decay time This diffusivity estimate is consistent with the one inferred from the bulk halocline deepening (Eq. 15) as well as with scales Ti for each eigenmode. For a general perturbation that may consist of many different modes, the gyre would the one directly diagnosed from the eddy buoyancy fluxes equilibrate on a time scale corresponding to the longest (Eq. 14). A good agreement between the three indepen- one amongst all the possible modes: dent methods (Fig. 3b) provides strong support for the validity of our theory that draws direct connections be- 1 R2 tween eddy diffusivity, halocline depth, and gyre stability T0 = . (28) or equilibration time scale. nλ K0(R) Here λ is a positive dimensionless constant that arises 7. FWC response to Ekman pumping as a solution of the discussed eigenvalue problem. Note Defined as a linear measure of column salinity with re- that the eigenvalues depend on boundary conditions, how- spect to a reference salinity S , the amount of freshwater ever for diffusion operators they grow rapidly (roughly re f (FW) can be approximated as quadratically) with the number of zero crossings of the 0 corresponding eigenfunction. This implies that the spin- Z S(z) − Sre f ∆S up time scale observed in the numerical model corre- FW ≡ − dz ≈ h, (30) −H Sre f Sre f sponds to a decay of the large-scale gyre mode, whereas small-scale spatial perturbations in the halocline depth where the integration is to a depth H of a reference isoha- would decay much faster. line. The FW is proportional to the halocline depth h and For a linear surface-stress profile used in our reference to the bulk vertical salinity difference ∆S and thus can be gyre simulation (τ0 ∼ r) we obtain λ = {5.7,4.7,4.3} for altered either via water-mass modification or via changes powers n = {1,2,3} respectively. While the expression in halocline depth. Note that the presence of vertical mix- for the gyre adjustment time (Eq. 28) is consistent with ing can not significantly affect FWC unless it diffuses salt J OURNALOF P HYSICAL O CEANOGRAPHY 9 up from below the reference salinity. Mixing can become A

important in cases with weak forcing when the halocline ) m

becomes thin (Spall 2013). ( 20

s n

Using the approximation (30) it can be deduced that the o i

t 10

FWC, deﬁned as an area-integrated amount of freshwater, c

n u is proportional to the volume V of water above the halo- f

n 0

cline: e

g i R e $

Z e

∆S -10 h1; 61 = 4:7; n

FWC ≈ V, where V = 2π rhdr. (31) i $ l h2; 62 = 20:4;

Sre f 0 c $ o

l h3; 63 = 47:3.

a -20

Building on the linear relationship between FWC and V, H veriﬁed in the numerical model (see Fig. 5), we continue 0 0.2 0.4 0.6 0.8 1 our exploration of the forced freshwater volume dynam- r/R ics. In diagnosing V from the numerical model we de- B

ﬁned the halocline depth h via the location of the 32.25 ) 3 1.2

isohaline; the relative top-to-bottom salinity difference, m

k 0

∆S/Sre f ≈ 5/34 is essentially prescribed through bound- 0 1 0

ary conditions. In general, the temporal evolution of FWC 1 #

( 0.8 would also depend on water mass modiﬁcations that affect FWC

n 1 o surface and mid-depth salinities. i FWC

t 0.6 2 a

b FWC

r 3

u 0.4

a. Spatially inhomogeneous Ekman pumping t

r e Here we address the following question: amongst all p 0.2 the possible Ekman pumping distributions, which one is C W 0 most efficient in changing the FWC? We thus consider the F 0 5 10 15 evolution of halocline volume, V, under spatially inhomo- years geneous Ekman pumping. th Projecting Eq. 22 onto the i eigenmode we obtain that FIG. 6. a) First three halocline depth eigenfunctions; the legend each amplitude ai(t) is forced by the Ekman pumping pro- shows their corresponding nondimensional eigenvalues which grow ∗ rapidly with the number of zero crossings. b) Response of freshwa- jection wi onto a corresponding eigenfunction h (plotted i ter content to the mentioned surface stress profiles as simulated by the in Fig. 6a) such that eddy resolving numerical model (blue, black, and red correspondingly). Note that all stress perturbation had the same magnitude of area aver- dai ∗ ai ∗ hi = − hi + wi. (32) aged Ekman pumping. Perturbations were made on a control run. dt Ti An area integral of (Eq. 32) results in an evolution equation for the corresponding halocline volume Vi: order of magnitude larger. Thus, the most dominant contribution to freshwater content would be due to the Ekman dVi Vi = − +WE , (33) pumping pattern of the gravest eigenmode (i.e. gyre scale dt Ti anticyclonic surface stress pattern as shown in Fig. 6a, where WE is the area integrated Ekman pumping, i.e. the red). Eddy diffusion is efficient in damping the response Ekman transport. The equation above demonstrates the to spatially inhomogeneous Ekman pumping. This occurs stability of the gyre in terms of the decay of its halocline because highly oscillatory eigenmodes induce large halo- volume anomalies which supports our a priori stability as- cline slopes (see Fig. 6a) which quickly produce strong sumption (see Eq. 1 and discussions thereof). mesoscale transport that acts to damp them. Since we are considering Ekman pumping perturbations We now test this theoretical prediction within the eddy that have the same Ekman transport, a steady state volume resolving gyre model. We have perturbed the gyre from its anomaly is directly proportional to the corresponding time equilibrium state by increasing the Ekman pumping with scale Vi = WE Ti (Eq. 33). In turn, the time scales Ti reduce patterns corresponding to the first three eigenmodes (ra- rapidly in magnitude with their index dial distributions as in Fig. 6a). Ekman transport pertur- T bation amplitudes are 25% compared to the equilibrated i << 1 for i ≥ 1. (34) state transport. Note that all three perturbations have the T 0 same value of the area averaged Ekman pumping and yet For example, for the eigenfunctions plotted in Fig. 6 the theory predicts that the corresponding FWC responses T1/T0 = 0.23, T2/T0 = 0.1, implying that V0 is at least an should be dramatically different. 10 J OURNALOF P HYSICAL O CEANOGRAPHY

The numerical model, in agreement with the theory, with respect to its mean value (the large-scale spatial pat- shows that the FWC increases substantially more for the tern does not change in time). The normalized freshwater large-scale Ekman pumping pattern (Fig. 6b, red) and vir- volume amplitude and its phase lag are diagnosed from tually does not increase for the spatially inhomogeneous the model and are shown in Fig 7 (crosses). The eddy re- pumping (Fig. 6b, blue and black). This confirms that solving model is in close agreement with our theoretical only large-scale Ekman pumping can efficiently contribute predictions based on the adjustment by mesoscale eddy to changes in FWC. diffusion. By moving beyond the steady state solutions consid- The time scale T0 represents a transition in the sys- ered earlier, the transient dynamics reveal which Ekman tem from a one-dimensional to a three dimensional re- pumping modes provide the most efficient forcing. Non- sponse. If the eddies were unimportant to the dynam- linearity (not accounted for in our transient theory) might ics (a limit of infinitely large adjustment time scale T0 be important in determining the halocline depth anoma- in Eqns. 37-38), the volume perturbations would have lies that are large in amplitude, especially near the center been inversely proportional to the frequency of the Ekman of the domain (Fig. 6a). However, changes near the cen- pumping (V = WE /ω) and the phase lag would have been ter of the domain do not substantially contribute to FWC a quarter of a period for all frequencies (φ = π/2). This as it is weighted by area, meaning that changes are more limit is relevant for high-frequency forcing, where the ed- important at the edges. dies have insufficient time to respond to changes in Ek- man pumping; the gyre response in this case is entirely b. Periodic Ekman pumping due to the Ekman advection. However, the freshwater volume amplitude is maximized for slowly oscillating forcing Here we explore the response of FWC to periodic Ek- where it is more in phase with the forcing (Fig 7). In this man pumping. Modes with smaller-scale spatial variabil- regime the eddy transport nearly compensates for the Ek- ity are damped more efficiently by eddies (Eq.33), such man pumping. The response amplitude diagnosed from that the overall halocline volume anomaly is largely deter- the numerical model is slightly larger than the analytical mined by the volume of the first eigenmode: prediction (Fig. 7a) due to a presence of internal modes of ∞ the gyre variability excited by the forcing. V = ∑Vi ≈ V0. (35) The time scale T0 ≈ 2.1 years that was used in the theory i=0 (Eqns. 38-37) was estimated from bulk gyre sensitivity to forcing as The volume anomalies thus obey a simple equation of a ∆V damped-driven system that approaches equilibrium with T0 = , (39) ∆WE an e-folding time scale T0: τre f where ∆V is an equilibrium change in freshwater vol- dV V = − +W sinωt, (36) ume corresponding to an increase of Ekman transport by dt T E 0 ∆WE . This time scale represents the gyre stability to small A similar equation and role of eddies was found for tran- perturbations around its equilibrium state. It is slightly sient dynamics of the thermohaline circulation by Spall smaller than the 3 year non-linear adjustment time scale (2015). Note that here we consider the gyre forced by that was diagnosed from the spin-up time series (Fig. 4). the most efficient large-scale Ekman pumping pattern (i.e. ∗ 8. Predicting FWC using the Gyre Index we = w0h0). In general, WE would be the transport associated only with the Ekman pumping projection onto the Here, we propose a novel FWC tendency diagnostic – first eigenfunction w0 as the portions of the transport asso- the Gyre Index – that predicts the freshwater content and ciated with larger eigenvalue projections would insignifi- includes both the effects of wind forcing and eddies. cantly contribute to changes in volume. Solving the equation above we find that the halocline a. The Gyre Index volume, after an initial adjustment, approaches a simple periodic solution lagged with respect to the forcing by a We take (22) for the halocline perturbation and integrate phase φ: it over the surface area of the gyre to obtain an evolution equation for the changes in the volume of water above the V 1 halocline: = sin(ωt − φ), (37) W T p 2 E 0 1 + (ωT0) dV Z R dh −τ = 2πr dr = 2πR nK s + , (40) φ = arctan(ωT ). (38) o 0 dt 0 dt ρ0 f r=R Using the equilibrated control simulation we apply os- where the halocline slope s and surface stress τ are per- cillating Ekman pumping forcing with a 25% amplitude turbations from the mean state. Here we made use of the J OURNALOF P HYSICAL O CEANOGRAPHY 11

eddy and Ekman transports are compensated resulting in A GI = 0, the limit of a vanishing residual circulation.

e 1 Model Theory Because both terms in the GI are evaluated at the gyre m

u boundary (Eq. 41), the FWC tendency depends explic- l

o 0.8

v itly only on the boundary processes. On one hand, the 1 d 9 e 0.6 V 2 mesoscale eddies can only redistribute the halocline thick-

z p1+(!T0) i l ness within the gyre without affecting the total FWC un- a 0.4

m less there are eddy thickness sources at the boundaries. On r o the other hand, due to Stokes theorem, the area-integrated N 0.2 Ekman pumping is proportional to the contour integral 0 0 2 4 6 8 of surface stress around the gyre boundaries. Thus, any spatially-localized anomaly in Ekman pumping can only affect the halocline depth locally and changes in FWC B would depend only on the existence of a surface stress

:/2 along the boundaries. ?

g b. Diagnostic power of the Gyre Index

l e s We now proceed to investigate the extent to which the a :/4

h ? = arctg(!T0)

P GI can approximate FWC tendency within the eddy resolving numerical model. We simulate the gyre variability by applying time-dependent spatially-homogeneous Ek- 0 man pumping. Its amplitude evolves according to a red 0 2 4 6 8 noise process with a memory parameter of 1 year to mimic Ekman pumping frequency, ! (rad/yr) observations that show enhanced variability on interannual to decadal time scales (Proshutinsky et al 2009); time se- FIG. 7. Amplitude-phase response of the gyre freshwater content to ries are shown in Fig. 8a. The surface stress perturbations periodic Ekman pumping. a) Normalized amplitude of volume oscilla- from the reference run have a variance of 25% with respect tions (see Eq.37). b) corresponding phase delay relative to the oscillat- to the mean value of τ = 0.015 N m−2 (the reference case ing Ekman pumping (Eq. 38). 0 that we use for the Beaufort Gyre). The corresponding FWC undergoes significant variations of about 3000 km3 Stokes theorem for the Ekman pumping term and integra- on decadal time scales (Fig. 8a, red curve). tion by parts for the eddy diffusion term. We then compute the Gyre Index by diagnosing the Taking into account that the top-to-bottom salinity dif- azimuthally-averaged halocline slope perturbation s(t,R) ference is constant throughout our simulations, we can de- and the generated surface stress τ(t,R) at the gyre bound- fine the Gyre Index (GI) that closely approximates FWC ary (Eq. 41). Because we use stochastic in time Ekman tendency as: pumping perturbations the time series have large variances at short time scales (Fig. 8a). To illustrate its differences ∆S −τ(R) with the purely Ekman pumping-based index we evaluate GI = 2πR nK0(R)s(R)+ , (41) Sre f ρ0 f the performance of the Gyre Index on various time scales. | {z } | {z } In Figure 8b we compare FWC tendency and the GI for Eddy Ekman transport transport which a 5-year running mean smoothing has been applied in order to explore the interannual trends characteristic where we have made use of Eq. 31 (s and τ are perturba- of the Beaufort High variability. Using only the Ekman tions from their mean values, while K0 is the eddy diffu- pumping as an index (i.e. excluding the eddy transport sivity of the mean state). The GI approach is analogous to term in the GI) does not give an accurate representation conventional freshwater budget calculations with an addi- of FWC tendency (note the large differences between red tional eddy transport term. Thus, GI > 0 implies that the and black curves in Fig 8b). In contrast, the GI provides gyre is gaining FWC and GI < 0 implies the gyre is losing a much better agreement with the numerically-simulated FWC. The magnitude of the GI should be compared to the FWC evolution (Fig 8b), further supporting our theory of maximum FWC flux out of the gyre FWC/T0 ∼ τ/(ρ0 f ) the gyre variability. that is proportional to the Ekman transport and is indepen- We quantify the performance of the GI by calculating dent of the eddy diffusivity (see Eqns. 21 and 20). We the root mean square error as a function of the length of can also interpret the GI as being a measure of how far the running mean window used to smooth the time series the gyre is from its equilibrium state. In equilibrium the (Fig 8c). We see that the Ekman only predictor of FWC

12 J OURNALOF P HYSICAL O CEANOGRAPHY

) s

= A 2

m 0.06 16

(

)

3 n

o 0.04 15.5

k a

0.02 15 0

u 1

t 0 14.5

e (

p -0.02 14

e r

-0.04 13.5 W

s F

d -0.06 13 n i 0 20 40 60 80 100 120 140 160 180 200 W years

B )

r 300

y =

3 200 d=dt(F W C) Ekman Ekman+Eddies

k (

y 100

n e

d 0

n e

t -100

C W

F -200 0 20 40 60 80 100 120 140 160 180 200 years

C D )

r 1.2

y = 200 0

3 STD of FWC tendency

m 0 k [Ekman] error

( 160 1 K

r [Ekman+Eddies] error

r e

120 e v

i 0.8

l e

80 e

n r

e 0.6

t r

40 r

E W

F 0 0.4 0 5 10 15 20 0 100 200 300 400 500 600 2 smoothing interval (years) Eddy di,usivity K0(m =s)

FIG. 8. a) Response of the FWC (red) to time-dependent surface stress that has a fixed spatial pattern (same as for control run) but its amplitude oscillates around its mean state of 0.015 N m−2 with a 25% variance (shown in blue). Surface stress perturbation amplitude was generated as a red noise with a 1 year damping parameter. b) FWC tendencies as directly estimated from the model (black), as inferred using the Ekman pumping only (red), and the Gyre Index (blue). A five year running mean filter was applied to these time series. c) Standard deviation of FWC tendency (black), error approximating it with Ekman only term (red), and error of the Gyre Index (blue) plotted as functions of the smoothing interval. d) 2 −1 The dependence of error on the choice of the mesoscale eddy diffusivity showing the optimal choice of K0 ≈ 280 m s that was used for the time series shown in b). tendency provides a good match only on short time scales A calculation of GI requires the background eddy dif- where eddy activity is less important. However, on in- fusivity evaluated at the boundary (Eq. 41). We show that terannual and longer time scales it is necessary to account there exists an optimal eddy diffusivity that minimizes er- for the eddy activity as the Ekman predictor error becomes ror of the GI making it a factor of 2 smaller than that of the Ekman index alone (see Fig 8d). The optimal value of larger than the standard deviation of FWC tendency. This K ≈ 300 m2 s−1 is in a range of values diagnosed earlier implies that at interannual and longer time scales assum- 0 using the three other independent methods (see Fig. 3b). ing no change in FWC gives a better prediction than only In Section 7.1 we demonstrated that only large scale taking into account the Ekman pumping. In contrast, the variations in Ekman pumping significantly affect FWC. GI has a strong predictive capability on all time scales with Yet the GI, which accurately predicts variability of FWC, its error persistently smaller than the standard deviation of does not contain information about the horizontal scale the FWC tendency (Fig 8c). of the Ekman pumping. Instead, it only uses the Ek- J OURNALOF P HYSICAL O CEANOGRAPHY 13 man transport, which is the area integrated Ekman pump- forcing (Fig. 3). Our theory has only one a priori un- ing. This implies that for highly inhomogeneous Ekman known parameter, the mesoscale eddy diffusivity, which pumping (higher eigenmodes) the GI ≈ 0 and hence there we inferred in four ways based on: the buoyancy flux (Eq. should be a strong compensation between the eddy trans- 14); the bulk halocline deepening (Eq. 15); the adjustment port at the boundary and the Ekman transport. time scale (Eq. 29); and volume transport variability (Fig. 8d). These independent methods are consistent between 9. Summary and discussions each other and thus provide support for the theory. For conditions akin to a present-day Beaufort Gyre we pro- We explored the transient dynamics of an idealized vide an estimate of the characteristic diffusivity of about Beaufort Gyre in an eddy resolving general circulation 300 m2 s−1 (τˆ = 0.015 N m−2) and predict its sensitivity to model with particular emphasis on the FWC variability. be 170 m2 s−1 per 0.01 N m−2 increase in the azimuthal We performed a series of experiments exploring the gyre’s surface wind stress. These parameters should be tested response to a time-dependent surface stress. The results against observations. were interpreted using Transformed Eulerian Mean theory Third, motivated by a strong variability in atmospheric that explicitly includes the effects of mesoscale eddies. winds over the Beaufort Gyre, we explored a gyre re- Using an eddy parameterization, we provide theoretical sponse to spatially-inhomogeneous and time-dependent predictions for the gyre’s stability (inverse of the equili- Ekman pumping. Our analytical and numerical solu- bration time) as well as transient solutions for halocline tions show that amongst all possible stress distributions and FWC evolution under time-dependent Ekman pump- that have the same area averaged Ekman pumping, the ing. FWC is largely affected only by the gyre-scale Ekman Our model and theory neglect several processes that pumping (Fig. 6b). The FWC response to temporally- may be important in the real Beaufort Gyre. The theory periodic Ekman pumping is closely approximated by a is adiabatic and the model was run with a vertical diffu- simple damped-driven dynamical system that approaches sion coefficient of 10−5 m2 s−1. Observations indicate equilibrium with a known adjustment time scale controlled that diapycnal mixing is spatially and temporally inhomo- by eddy dynamics (Eq. 28). High-frequency oscillations geneous but, on average, between 10−6 and 10−5 m2 s−1 in the pumping (e.g. seasonal cycle) have little effect on (Guthrie et al 2013). Scaling theory in MS16 suggests that halocline depth, whereas the strongest effect is achieved the impact of diapcynal mixing is small but not negligible, for low-frequency forcing (e.g. on decadal time scales) especially when the surface stress is weak. We also neglect eddy salt fluxes (or thickness fluxes) that originate (Fig. 7). from eddies shed from the boundary currents that encircle Fourth, we proposed the use of the Gyre Index (Eq. 41) the Arctic basin (e.g. Manley and Hunkins (1985); Spall for monitoring/interpreting FWC tendency in the Beaufort et al. (2008); Watanabe (2013)). Scaling in Spall (2013) Gyre. Its key advantage is that calculating the GI requires suggests that the freshwater flux carried by these eddies is knowledge of the halocline slope and magnitude of sur- of similar amplitude as that carried by the Ekman trans- face stress evaluated only at the gyre boundaries (not in its port. It is important to note, however, that the Eulerian interior). Using a numerical model we demonstrated its streamfunction associated with the eddies vanishes so they strong predictive capability that is due to the fact that it in- do not provide an equivalent to the Eulerian Ekman pump- corporates the competing effects of both Ekman pumping ing velocity produced by the surface stress. We also do and mesoscale eddy transport. For interannual and longer not represent geometric complexities such as the Eurasian time scales the GI is far superior to using only the strength Basin, continental slopes, shelf dynamics, and mid-ocean of Ekman pumping in evaluating the FWC tendency (Fig. ridges. We view the present model as representing, in a 8). compact and transparent way, the leading order dynami- The GI can be readily generalized to include effects of cal balances for the Beaufort Gyre from which additional more realistic features of the Beaufort Gyre dynamics such processes may be considered. as the azimuthal asymmetry in Ekman pumping and eddy We now highlight four key results from this study. First, diffusivities. This would require calculating a contour in- we demonstrated that the presence of mesoscale eddies di- tegral along the gyre boundaries in order to evaluate terms rectly affects the Ekman-driven gyre variability (Eqns. 33 in the GI. Observationally, calculation of the GI could be and 37). By defining the gyre equilibration time scale (Eq. achieved through a network of instruments located along 28), we provided a theoretical expression for the time scale the gyre perimeter, thus complementing the observational and emphasize its explicit dependence on mesoscale eddy efforts in the interior of the gyre. We envision the GI cal- diffusivity (Eq. 28). culation requiring observations of density and velocities Second, we presented several estimates of the charac- from a set of moorings spaced around the gyre. Mooring teristic mesoscale eddy diffusivity for the gyre, demon- observations of ocean velocity and density fields would strating that it grows nearly linearly with the surface stress allow for calculation of the eddy fluxes by estimating the 14 J OURNALOF P HYSICAL O CEANOGRAPHY vertical shear of the first baroclinic mode of the horizon- does not change (b0z = const) and can thus define a per- tal velocity that is directly related to horizontal halocline turbation isopycnal displacement h as slope. We speculate that the GI would be a useful tool b(l,t) for interpreting the component of FWC variability that is h(l,t) = . (A5) related to changes in the halocline depth in both observa- b0z tional and modeling studies of the Beaufort Gyre. Taking into account that ∂l() = ∂r()rl + ∂z()zl we obtain Acknowledgments. The authors acknowledge the high- the following system for variables following characteris- performance computing support from Yellowstone pro- tics: vided by NCARs CIS Laboratory, sponsored by the NSF. 1 τ GEM acknowledges the support from the Howland Post- ht = − (ψ˜ l)l, ψ˜ = − nK0hl, (A6) l ρ0 f doctoral Program Fund at WHOI and the Stanback Fel- lowship Fund at Caltech. MAS was supported by NSF where K0(l) is a background eddy diffusivity set by the Grants PLR-1415489 and OCE-1232389. AFT acknowl- isopycnal slope of the steady state buoyancy distribution: edges support from NASA award NNN12AA01C. The au- n−1 n thors would like to thank Prof. John Marshall and an n−1 −τ0(l) K0 = ks0 = k . (A7) anonymous reviewer for helpful comments and sugges- ρ0 f k tions. The manuscript also benefited from discussions at the annual Forum for Arctic Modeling and Observing Syn- Eliminating ψ˜ from the equations above we obtain an thesis (FAMOS) funded by the NSF OPP awards PLR- equation for the time evolution of isopycnal displacement: 1313614 and PLR-1203720. 1 1 −τ ht = nK0rhr + r (A8) r r r ρ0 f r APPENDIX A b.c. : h|r=R = 0, hr|r=0 = 0. (A9) Time evolution of halocline depth perturbations that is now written in terms of the radial coordinate taking The nonlinear equation set 16-17 for the buoyancy evo- onto account that rl = 1. The fixed-buoyancy boundary lution can be linearized in the vicinity of its mean state to conditions for the original equations translated into the ho- obtain : mogeneous boundary conditions for the linearized system. 1 b + (ψ˜ r) b − ψ˜ b = 0, (A1) References t r r 0z z 0r Andrews, D. G., and M. E. 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Johnson (2014) On the Link between Arctic Sea Ice Decline and the Freshwater Content of the Beaufort one writes it along the characteristics defined by the halo- Gyre: Insights from a Simple Process Model. J. Climate, 27, 8170- cline depth of the mean buoyancy distribution. Eq. 11 8184. may be written in characteristic coordinates [r(l),z(l)] as Dmitrenko, I., S. Kirillov, V. Ivanov, and R. Woodgate (2008), bl = brrl + bzzl = 0, where the homogeneous right hand Mesoscale Atlantic water eddy off the Laptev Sea continental slope side indicates that buoyancy is conserved along the char- carries the signature of upstream interaction, Journal of Geophysical acteristic trajectory. The characteristic velocities are then Research: Oceans (1978–2012), 113(C7).

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