A theory of the wind-driven Beaufort Gyre variability · persistent large-scale feature of the Arctic Ocean, it is rea-sonable to assume that it is a stable, externally-driven sys-tem

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J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y

A theory of the wind-driven Beaufort Gyre variability

GEORGY E MANUCHARYAN∗

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

MICHAEL A SPALL

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

ANDREW F THOMPSON

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 thefreshwater content (FWC) of the Arctic Ocean. A recent study (Manucharyan and Spall 2016) emphasizedthe role of mesoscale eddies in setting equilibrium halocline properties. Here, we explore the role of eddiesin the Ekman-driven gyre variability. Following the Transformed Eulerian Mean framework, we develop atheory that links the FWC variability to the stability of the large scale gyre, defined as the inverse of its equi-libration time. We verify our theory’s agreement with eddy-resolving numerical simulations and demonstratethat the gyre stability is explicitly controlled by the mesoscale eddy diffusivity. The model suggests that acorrect representation of the halocline dynamics requires the eddy diffusivity of 300±200 m2 s−1, which islower 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 perturba-tions being an order of magnitude less efficient. Incorporating the eddy effects, we introduce a FWC tendencydiagnostic – the Gyre Index – that can be conveniently calculated using observations of surface stress andhalocline 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 FWCevolution in observations as well as in numerical models.

1. Introduction

The observed increase of the Arctic freshwater contentover the past two decades has been related partially toa deepening of the halocline and partially to water-massfreshening (Rabe et al 2014). In particular, a significantcontribution to the overall Arctic freshening came fromthe Beaufort Gyre; here the freshwater content (FWC) in-creased by about 30% over the past decade (Haine et al2015).

Observational evidence suggests that the halocline dy-namics of the Beaufort Gyre are to a large extent governedby the anticyclonic atmospheric winds that drive the large-scale gyre circulation – and to a lesser extent due to theavailability of Arctic freshwater sources (Proshutinsky et

∗Corresponding author address: Georgy E Manucharyan, MC 131-24, California Institute of Technology, 1200 East California Boulevard,Pasadena, CA, 91125, USA.E-mail: [email protected]

al 2009). The surface stress results largely from the seaice drag and quantification of the oceanic response is sub-ject to our uncertainties in the sea ice-ocean momentumexchange (Martin et al 2014; Giles et al 2012).

Modeling and observational studies demonstrate a di-rect relation between the freshwater content and the Ek-man pumping (Proshutinsky et al 2002, 2009; Stewart andHaine 2013; Timmermans et al. 2011, 2014). Thus, Stew-art and Haine (2013) show that a reduction in the strengthof anticyclonic winds can lead to a redistribution of FWCwithin the Arctic Ocean and to changes in the exchangebetween North Atlantic and Pacific Oceans. The basicgyre dynamics are conventionally described in the follow-ing way: the surface Ekman transport converges surfacefresh waters and deepens the halocline, thus storing fresh-water in the gyre (e.g. Proshutinsky et al 2009). This argu-ment, however, does not explain how a steady state can beachieved as it does not specify a mechanism opposing the

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2 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y

continuous halocline deepening due to the Ekman pump-ing.

The importance of the opposing mechanism can beclearly illustrated from the point of view of the large-scalegyre stability in its integral sense. By stability, we implythat there exists a statistical equilibrium state of the gyre(averaged over small scale features, e.g. internal wavesand eddies) and that any deviations from this equilibriumwould decay on a finite time scale. Since the gyre is apersistent large-scale feature of the Arctic Ocean, it is rea-sonable to assume that it is a stable, externally-driven sys-tem. Using basic concepts of dynamical systems theory(e.g. Tabor 1989), the linear stability assumption impliesthat near its equilibrium, perturbations of the gyre state, a,obey a basic equation of forced exponential decay

dadt

=− aT

+w. (1)

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 asmeasures of the bulk halocline deepening (or FWC) andEkman pumping correspondingly. The gyre stability isdefined as the inverse of its equilibration time scale, T ,and here we assume that the large-scale gyre circulationis always stable (i.e. T > 0). The damping term (−a/T )represents a linearization of a process opposing the Ekmanpumping near its equilibrium.

Here it is important to distinguish the concepts of large-scale gyre stability in terms of a statistically-averagedequilibrium and this state’s hydrodynamic stability char-acteristics. By gyre stability we refer to the equilibra-tion (exponential decay) of large-scale anomalies of astatistically-averaged circulation that does not contain in-formation about individual small-scale eddies. In contrast,baroclinic instabilities lead to the exponential growth ofindividual small-scale perturbations. Thus, the gyre can bestable in a statistically-averaged sense and at the same timehydrodynamically unstable at small scales. Our proposedhypothesis implies that the stability of the large-scale gyredepends on the cumulative action of small-scale eddies,individually generated through baroclinic instabilities.

Based on a basic dimensional analysis, the haloclinedeepening (units of meters) should depend not only on Ek-man pumping (units of meters per second) but also on theadjustment time scale (units of seconds). For example,according to (1), a = wT for a system in steady state orsubject to a slowly-evolving forcing. Moreover, the ampli-tude of the gyre variability in response to time-dependentEkman pumping would also be directly proportional to T(i.e. reduced stability would imply a larger variance for thesame forcing). Since the gyre stability is directly relatedto the nature of the processes that counteract the Ekmanpumping, it is essential to determine what these processes

Fresh Fresh

Ekman pumping

Bottom Ekman layer

Upw

ellin

g

Upw

ellingSalty

Eddies

Halocline

Salty

Eddies

Haloclinewe

we

FIG. 1. Schematic view of the Beaufort Gyre circulation depictinga balance between Ekman pumping and eddy induced vertical velocity.The Ekman vertical velocity penetrates through the entire water column.Surface convergence of the Ekman transport is balanced by the corre-sponding bottom divergence with the mean circulation being closed viacoastal upwelling.

are, how they depend on external forcing, and how theyaffect the gyre variability.

While much of the scientific effort has been devotedto the exploration of the impact of Ekman pumping (e.g.Proshutinsky et al 2002; Yang 2009; McPhee 2012; Tim-mermans et al. 2014; Cole et al 2014), factors controllingthe gyre stability are poorly understood. A recently pro-posed hypothesis points to mesoscale eddy transport be-ing a sufficient mechanism to oppose the Ekman pump-ing (Davis et al 2014; Marshall 2015; Lique et al 2015;Manucharyan and Spall 2016; Yang et al 2016). Theeddies are generated via baroclinic instabilities of thelarge-scale gyre circulation that release available poten-tial energy associated with the halocline deepening. Theeddy buoyancy fluxes (or layer thickness fluxes) through-out this process act to adiabatically flatten the halocline,thus opposing the deepening due to the Ekman pumping(schematically shown in Fig. 1).

The mesoscale eddies discussed here are associatedwith baroclinic instabilities of a large scale halocline slopeand hence predominantly carry the energy of the first baro-clinic mode. The majority of the observational analysis inthe Arctic Ocean, however, has been devoted to intense lo-calized vortices (Manley and Hunkins 1985; Timmermanset al. 2008; Dmitrenko et al. 2008; Watanabe 2011; Zhaoet al 2014) that can form at boundary currents (Watanabe2013; Spall et al. 2008; Spall 2013) and at surface oceanfronts (Manucharyan and Timmermans 2013). Althoughobservational analysis for the role of halocline-origin ed-dies in the interior salinity budget is currently lacking, thisis likely due to a lack of data sufficient to calculate energyconversion rates and eddy salt fluxes in the basin interior.

Taking into account the eddy transport mechanism,Davis et al (2014) have used a low-resolution shallow wa-

J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y 3

ter model (with eddies parameterized as horizontal dif-fusion) to explore seasonal gyre dynamics. Yang et al(2016) explored the potential vorticity budget of the gyreand reached the conclusion that the eddies are necessaryto close it. Previous studies have suggested that there arelinks between the dynamics governing the Antarctic Cir-cumpolar Current (ACC) (Marshall and Radko 2003) andgyres with azimuthally-symmetric circulations: Su et al.(2014) in the Weddell Gyre, Marshall et al (2002) in labo-ratory experiments, and Marshall (2015) in the BeaufortGyre. The stratification in these regions is determinedfrom a leading order balance between mesoscale eddytransport and Ekman pumping.

A recent study by Manucharyan and Spall (2016) (fromnow on MS16) demonstrated that the instabilities associ-ated with the observed halocline slope are indeed generat-ing sufficient mesoscale activity to cumulatively counter-act the Ekman pumping. They also developed a set of an-alytical scaling laws that predict the gyre adjustment timescale, halocline depth, and maximum freshwater flux outof the gyre and explicitly relate these key gyre character-istics to the mesoscale eddy dynamics.

In light of this newly developed understanding it is im-perative to explore the role of mesoscale eddies in deter-mining the stability of the gyre and its transient dynamics.Here, we present a basic theory of the wind-driven Beau-fort Gyre variability. The manuscript is organized in thefollowing way. In Section 2 we briefly review the trans-formed Eulerian mean framework that is used through-out our analysis. In Section 3 we describe an idealizedBeaufort Gyre model that we use for our process stud-ies. In Section 4 we diagnose a key eddy field charac-teristic, the eddy diffusivity, and assess its sensitivity toforcing. In Section 5 we demonstrate how mesoscale ed-dies affect the gyre stability and equilibration. In Section6 we quantify the FWC response to periodic and spatially-inhomogeneous Ekman pumping. In Section 7 we intro-duce a new Gyre Index that includes the effects of both Ek-man pumping and mesoscale eddies in order to rationalizeFWC variability. We summarize and discuss implicationsin Section 8.

2. Theoretical background

We briefly describe the mathematical formulation of theTransformed Eulerian Mean framework (TEM) (Andrewsand McIntyre 1976; Vallis 2006; Marshall and Radko2003) that is used in our analysis. Within this frameworkthe eddy buoyancy fluxes can be represented via an ad-ditional eddy-induced stream function. This allows oneto view the ensemble mean buoyancy as being advectedby the residual between the Eulerian mean and the eddy-driven circulations.

a. TEM framework

The Reynolds averaged buoyancy equation is written incylindrical coordinates as:

b̄t + v̄b̄r + w̄b̄z +(v′b′r)+(w′b′z) = S̄, (2)

where b denotes the buoyancy, (v̄, w̄) are the Eulerianmean radial and vertical velocities in the (r,z) coordinates,and S represents buoyancy sources and sinks. The bar hererepresents an average either in the azimuthal direction orin an ensemble mean sense; we only consider axisymmet-ric solutions here. Next, the eddy buoyancy fluxes are as-sumed to be predominantly aligned with isopycnals in theinterior of the ocean – the so called adiabatic limit (Vallis2006). In this limit the flux divergences can be representedas an additional advection of mean gradients by the eddystream function such that:

b̄t +(v̄+ v∗)b̄r +(w̄+w∗)b̄z = S̄, (3)

where the eddy advection velocities can be defined froman eddy stream function ψ∗ as:

ψ∗ =−w′b′

b̄r=

v′b′

b̄z, (4)

u∗ =−ψ∗z , w∗ =

1r(rψ

∗)r. (5)

The adiabatic assumption breaks down near boundary lay-ers that have significant diabatic fluxes due to buoyancyforcing and/or enhanced mixing. Near such boundariesthe processes obey different dynamics that we do not at-tempt to represent here.

Thus, the ensemble mean buoyancy is advected by aresidual circulation, ψ̃ ≡ ψ̄ + ψ∗, that exists in order tobalance the buoyancy sources and sinks. This formalismclarifies that in the interior of the ocean, where the diabaticfluxes are small, a steady state implies that the residualcirculation has to vanish, i.e. ψ̃ = 0.

At this point, in order to make analytical progress in un-derstanding the buoyancy variability, it is necessary to in-troduce a closure for the eddy-driven stream function andto determine the Eulerian mean stream function.

b. Eulerian mean stream function

The azimuthally-averaged azimuthal momentum equa-tion can be simplified for the gyre dynamics. First, weassume that contributions from the Reynolds stresses arenegligible for large-scale flows. Second, the inertial termcan also be neglected for time scales sufficiently largerthan f−1. In addition, the azimuthal pressure gradient termvanishes as a result azimuthal averaging. Taking this intoaccount, a simplified form of the momentum equation rep-resents the steady Ekman dynamics:

f v̄ =−τ̄z/ρ0, (6)


where τ is the azimuthal stress and ρ0=1023 kg m−3 is thereference ocean density. In the ocean interior the verticalshear in stress is negligible, allowing one to obtain an ex-pression for the Eulerian stream function by vertically in-tegrating the momentum equation from the surface to anydepth below the surface Ekman layer:

ψ̄ =τ

ρ0 f, (7)

v̄ =−ψ̄z, w̄ =1r(rψ̄)r, (8)

From now on τ = τ(r, t) denotes the radially- and time-dependent azimuthal surface stress – an external forcingfor the gyre. Thus, the large-scale Eulerian mean flow isentirely determined by the surface stress distribution anddoes not depend on the buoyancy field or the mesoscaleeddy activity. Note that this relation can be significantlyaffected by the presence of topography.

c. Mesoscale eddy parameterization

Mesoscale eddies emerge due to an instability of thebaroclinic flow and their amplitude is related to the sta-bility characteristics of the flow. Here, we implementa Gent-McWilliams type mesoscale eddy parameteriza-tion (Gent and McWilliams 1990) that assumes a down-gradient nature for the eddy buoyancy fluxes (or layerthickness fluxes):

v′b′ =−Kbr. (9)

The eddy diffusivity K = K(s) can in general depend onthe magnitude of the isopycnal slope s = |b̄r/b̄z|, thusvarying in space and time. Past studies suggested power-law dependencies (K∼ sn−1). Powers n = 1 or n = 2 (Gentand McWilliams 1990; Visbeck et al. 1997) are commonlyused in low-resolution ocean models while MS16 suggestthat n = 3 might be more appropriate for the BeaufortGyre. Since observations are not sufficient to differentiatebetween these parameterizations, we continue the analysisassuming a general power-law dependence:

ψ∗ = Ks̄ = ks̄n, (10)

where now k (having units of m2 s−1) is a constant that wewill refer to as eddy efficiency to distinguish it from time-and space-dependent eddy diffusivity K.

d. Steady state

In the absence of surface buoyancy forcing or strongvertical mixing the steady state residual ocean circulationhas to vanish (Marshall and Radko 2003; Su et al. 2014)implying that

ψ̃ =τ

ρ0 f+ k(−br

bz

)n

= 0. (11)

Surface stress at a particular location sets only the halo-cline slope, not its depth. The halocline slope defines thebaroclinically-unstable azimuthal currents (via the thermalwind relation) that generate eddies to locally oppose theEkman pumping.

Note that the Beaufort Gyre surface stress is on averageanticyclonic (τ < 0) and hence the halocline is deeper inthe interior, the isopycnal slope br/bz < 0, and ψ∗ > 0.Given a surface stress profile, the halocline deepeningacross the gyre can be determined by integrating (11):

∆h =∫ R

0

[−τ(r)ρ0 f k

] 1n

dr. (12)

Eq. 12 is identical to the expression for the ACC depth in atheory developed by Marshall and Radko (2003). Assum-ing the power n = 2, and considering a special case of uni-form Ekman pumping (corresponding to τ(r) = −τ̂r/R)the halocline deepens by

∆h =23

R(

τ̂

ρ0 f k

) 12. (13)

3. Idealized Beaufort Gyre model

We implement an idealized model of the Beaufort Gyreas was developed in MS16. It consists of a cylindricalocean basin (diameter 1200 km, depth 800 m) driven byan anticyclonic surface-stress τ(r). The primitive equa-tions are solved using the MIT General Circulation Model(Marshall, Hill et al 1997; Marshall, Adcroft et al 1997;Adcroft et al. 2015) in its three-dimensional, hydrostaticconfiguration, with a 4 km horizontal resolution that issufficient to permit Rossby deformation scale eddies. Thedeformation radius is about 20 km in our simulations (seemore details in Appendix A of MS16). For simplicity, thepresent configuration uses a flat bottom basin; MS16 in-cluded a continental slope.

Here we focus on how surface stress and mesoscale ed-dies influence the isopycnal distribution in the BeaufortGyre and hence the surface buoyancy forcing is neglected.Within our idealized gyre model we prescribe a fixed-buoyancy boundary condition at the coastal boundaries(via fast restoring to a given profile) and no-flux condi-tions at the surface and bottom. The boundary conditionsaim to represent the unresolved dynamics that occur overthe shallow shelves as well as the water-mass exchangeswith other basins. We note, however, that these coastal dy-namics can in general depend on the surface stress, whichcauses upwelling and boundary mixing (Woodgate et al2005; Pickart et at 2013).

The implemented fixed-buoyancy boundary conditionsimply that there exists an infinite reservoir of surface freshwater and bottom salty waters that the gyre is allowedto draw upon. The infinite water-mass reservoir repre-sents dynamics only on sufficiently long time scales that


years0 5 10 15 20 25 30

Eddy

kin

etic

ener

gy(k

gm

2s!

2) #10 15

0

1

2

3

Res

idual

stre

amfu

nct

ion

(Sv)

-0.2

0

0.2

0.4

FIG. 2. Time series of the gyre integrated eddy kinetic energy anda corresponding residual circulation. The residual circulation was cal-culated using diagnosed thickness fluxes in buoyancy coordinates (seeAppendix B in MS16) and averaged radially between 450km and 550kmover the halocline layer bounded by salinities 30 and 32.5. Note that theamplitude of ψ̃ significantly reduces as the gyre spins up.

allow for the continental shelf water masses to be replen-ished. Our idealized study does not include the processesof water-mass formation and exchange at the boundaries,instead focusing on the long-term wind-driven haloclinedynamics in the interior of the gyre. Nevertheless, oncethe understanding of internal gyre dynamics is gained, theuse of simplifying boundary conditions in the numericalmodel (and in analytical analysis) can be relaxed to in-clude more realistic boundary processes.

Spin-up simulations were initialized with ahorizontally-uniform stratification (50 m initial halo-cline depth) forced by a uniform constant in timeEkman pumping (corresponding to a linear surfacestress profile τ0 = −τ̂r/R). The mean state simulationswere spun up for at least 50 years to ensure equilib-rium. The following wind stress amplitudes were usedτ̂ = {0.005; 0.01; 0.015; 0.03; 0.05; 0.075}N m−2; weconsider τ̂ = 0.015N m−2 as a reference run represen-tative of the present-day Beaufort Gyre. Perturbationexperiments that are described in the following sectionsuse spatially-inhomogeneous and time-dependent Ekmanpumping.

Following the growth of mesoscale eddies, this ideal-ized Beaufort Gyre model achieves a statistically-steadystate that coincides with a vanishing residual circulation(Fig. 2). Thus, mesoscale eddies provide a mechanismto arrest the deepening of the halocline. We argue belowthat eddies are also key to the temporal response of thehalocline to surface stress perturbations.

4. Mesoscale eddy diffusivity

The theoretical predictions of the gyre state rely on therelevance of the mesoscale eddy parameterization whichuses an a priori unknown parameter k that is directly re-lated to the mean state eddy diffusivity K0. To ensure con-sistency, we diagnose the eddy diffusivity using severaldifferent methods based on our theoretical predictions.

We start with a direct estimate of diffusivity using theeddy buoyancy fluxes (see Eq. 9) of the equilibrated gyresimulation for a reference run:

K∗0 (r,z) =−v′b′

b̄r, (14)

where the overline represents both temporal and azimuthalaveraging and primes are the deviations from this mean.The eddy diffusivity for a reference Beaufort Gyre sim-ulation that has a relatively weak forcing (τ0 ∼ 0.015 Nm−2) ranges between 100− 500 m2 s−1 (Fig. 3a). Withthe exception of the near coastal boundary layer, the dif-fusivity increases towards the gyre edges where the halo-cline slope (and hence the baroclinicity of the currents) ishigher, which is consistent with our eddy parameterization(Eq. 10). The halocline appears rather thick in the figurebecause of the spatial and temporal averaging of the thick-ness variations represented by the mesoscale eddies. Thehalocline is typically 50 m thick locally in space and time.

Using analytical predictions (Eqn. 13) we can also in-fer the mesoscale eddy diffusivity necessary to support thesimulated halocline deepening as:

K∆h0 (τ̂) =

23

τ̂

ρ0 fR

∆h(τ̂), (15)

where we assumed an eddy parameterization power n = 2(i.e. K = ks) as in Visbeck et al. (1997) and a charac-teristic slope was taken near the edge of the gyre s =[τ̂/(ρ0 f k)]1/n = 1.5∆h/R. Both (14) and (15) producesimilar eddy diffusivity estimates (Fig. 3b); they also showa similar sensitivity to surface stress forcing. These in-ferred eddy diffusivities should be thought of as bulk val-ues representative of the gyre as a whole; the instantaneousvalues can differ depending on location and time. Notethat because s ∼ ∆h/R ∼ τ1/2, the definition of K in Eq.15 is consistent with our assumed parameterization K ∼ s.

According to Visbeck et al. (1997) the coefficient k =cNl2 where c = 0.015 is an empirical constant, N is thestratification parameter (N≈ 130 f in our simulations), andl is the width of the baroclinic zone. Taking estimates forthe diffusivity K = 300 m2 s−1 and the slope s = 10−4,we find k = 3× 106 m2 s−1, which implies that l ≈ 150km, using hte same value for the empirical constant c as inVisbeck et al. (1997). Note that the estimated size of thebaroclinic zone l is less than the gyre radius. This is qual-itatively consistent with the numerical experiments show-ing intense eddy generation near the edges of the gyre and


not near its center where the halocline slope is negligiblysmall.

The eddy diffusivity K increases with the surface stressfollowing a nearly linear relationship (Fig. 3b) with therate of increase of about 170 m2 s−1 per 0.01 N m−2 in-crease in τ̂ . This linear increase implies that the eddy sat-uration regime would be reached for strong forcing, i.e.for a sufficiently strong surface stress the halocline slopewould approach a critical value because Ks = τ/(ρ0 f ) andK ∼ τ . A corresponding halocline deepening is saturatedat about ∆h ≈ 150 m. The eddy saturation phenomenonhas been extensively discussed in the ACC, where thebaroclinic transport is only weakly sensitive to changes inthe strength of the surface westerlies (e.g. Hallberg andGnanadesikan 2001; Meredith et al 2004; Munday et al2013). In contrast, because of the relatively weak forcingthe Beaufort Gyre is far from the eddy saturation limit andis highly sensitive to the surface stress (MS16). Thus, sig-nificant gyre variability should be expected in response totransient forcing.

5. Transient TEM equations

Now that we have developed a basic understanding ofthe mesoscale eddy response to forcing and confirmed theappropriateness of the eddy parameterization, we proceedto explore the implications for the transient gyre dynam-ics. Combining the expressions for the Eulerian and theparameterized eddy stream functions, the evolution equa-tion for buoyancy is:

bt +1r(ψ̃r)rbz + ψ̃zbr = S̄, (16)

ψ̃ =τ

ρ0 f+ k(−br

bz

)n

. (17)

This nonlinear equation is relevant to the interior of thegyre and is subject to the appropriate boundary and initialconditions:

b|r=R = b0(z), br|r=0 = 0, (18)bz|z=0,H = 0, b|t=0 = b0. (19)

a. Scaling analysis

It is insightful to consider a scaling analysis for the gyredynamics. On one hand, the system approaches equilib-rium on an eddy-diffusion time scale defined as T ∼R2/K,where eddy diffusivity K ∼ k(h/R)n−1 depends on theunknown halocline depth. Diffusive scaling implies thata deeper halocline would have a larger eddy diffusivityand would thus equilibrate faster. On the other hand, theisopycnal depth h together with the Ekman pumping ve-locity define a vertical advection time scale T ∼ h/we,where we ∼ τ̂/(ρ0 f R) is the Ekman pumping. Since thedominant balance is achieved between the vertical Ekman

r (km)0 200 400 600

Dep

th(m

)

0

50

100

150

200

250

300

100 200 300 400 500

Surface stress =̂ (N m!2)0 0.02 0.04 0.06 0.08

Eddy

di,

usivity

(#10

00m

2s!

1)

0

0.5

1.0

1.5

2.0

KT0 K"h

0 K$0

eddy di,usivity (m2s!1)

A

B

K0 = 2420 =̂=(;0f ) + 230m2=s

FIG. 3. a) Spatial distribution of the equilibrium eddy diffusivityK∗0 =−v′b′/b̄r (14) as diagnosed from the eddy-resolving model for thereference run (τ = 0.015 N m−2). Contours show equally spaced meanstate isohalines with salinity increment of 0.25. In weakly stratified re-gions shown in white the calculation of eddy diffusivity is not appropri-ate a the buoyancy gradient is close to zero there. The isopycnal slope inthe top 50 m is reversed due to a presence of vertical mixing. b) Depen-dence of various definitions of the eddy diffusivity K on the magnitudeof the surface-stress. KT

0 (black circles, Eqn. 29) and K∆h0 (red dia-

monds, Eqn. 15) are inferred from the diagnosed gyre adjustment timeand the mean halocline depth, respectively; K∗0 (blue triangles, Eqn. 14)are diagnosed from the model (averaged between 400 km and 450 kmwithin the halocline layer). A linear fit to all the data points correspondsto K = 2420[τ̂/(ρ0 f )]+230 m2s−1 (black dashed line).

pumping of freshwater and its horizontal diffusion due toeddy transport the two mechanisms have to operate on asimilar time scale. Following MS16, the two time scalescan be equated to obtain a scaling for the halocline depth


Surface stress (N=m2)0 0.02 0.04 0.06 0.08

Adj

ustm

ent t

ime

scal

e (y

ears

)

0

2

4

6

Hal

oclin

e de

epen

ing

(m)

0

50

100

150

FIG. 4. Equilibrium halocline deepening across the gyre (red y-axis)and its e-folding adjustment time scale diagnosed from the spin up sim-ulations (black y-axis) plotted as functions of the mean surface stressforcing (linear surface stress profile was used τ(r) =−τ̂r/R). Theoret-ical scaling predictions are shown in solid lines for the power n = 3 andin dashed lines for n = 2 (see Eqns. 20-21). Each symbol is diagnosedfrom a numerical model run with different surface stress.

and the adjustment time scale:

∆h∼ R(

τ̂

ρ0 f k

) 1n

, (20)

T ∼ R2

k

(τ̂

ρ0 f k

) 1−nn

. (21)

The expressions above imply that stronger pumping wouldlead to deeper halocline with faster adjustment time scale(a more stable gyre). In addition, both scaling laws ex-plicitly depend on the eddy efficiency: for smaller k thehalocline would be deeper and the adjustment time scalelonger (a less stable gyre). MS16 show that these scalinglaws hold for a wide range of surface stress forcing and weconfirm (20) and (21) for the case of no topography underconsideration here (Fig. 4).

Depending on the choice of the power n there can bemajor qualitative differences in the gyre sensitivity to forc-ing. Thus, n = 1, equivalent to a constant eddy diffusivity,results in an equilibration time scale that is independentof the surface forcing and a halocline depth that scales lin-early with the surface stress (implying there is no eddy sat-uration). This provides a poor representation of the simu-lated gyre dynamics. Differences between n = 2 and n = 3would manifest only for a sufficiently wide range of forc-ing magnitudes. For the range of forcing relevant to theArctic Ocean (τ . 0.1 N m−2) n = 2 and n = 3 producesimilar results (Fig. 4). Note, that the power n is not apriori constrained to be an integer.

6. Gyre equilibration and stability

In this section we derive analytical time-dependent so-lutions for the halocline depth.

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-termtime average (i.e. over time scales much longer than thegyre spin-up time). We then assume that Ekman pumpingperturbations lead to sufficiently small buoyancy perturba-tions for which a linearization of the full nonlinear equa-tion set (Eq. 16) is appropriate. In other words we ex-plore the transient gyre dynamics where isopycnal depthperturbations can be considered small compared to theirtime-averaged state.

The linearized equations for the evolution of haloclinedepth anomalies h (see derivation in Appendix A) result ina diffusion equation forced by Ekman pumping:

ht =1r

(nK0rhr

)r︸︷︷︸

Eddy diffusion

+1r

(r−τ

ρ0 f

)r︸︷︷︸

Ekman pumping

. (22)

Here τ is the surface stress perturbation from its meanvalue τ0, and K0(r) is a background eddy diffusivity setby the mean isopycnal slope as

K0 = ksn−10 = k

[−τ0(r)ρ0 f k

] n−1n

. (23)

The first term on the right hand side of Eq. 22 acts todiffuse the isopycnal depth perturbations with a space-dependent diffusivity equal to nK0; the prefactor n ap-pears for a linear problem because of the a power-law de-pendence of eddy diffusivity on the slope (see AppendixA). This representation of mesoscale eddies as thicknessdiffusivity is analogous to the expression in Gent andMcWilliams (1990); Su et al. (2014, e.g.). The secondterm is the perturbation in Ekman pumping we that actsas a forcing for the isopycnal depth:

we = curl(−τ

ρ0 f

)=

1r

(r−τ

ρ0 f

)r. (24)

Thus, consistent with the mean state dynamics, it is an im-balance between the eddy diffusion and Ekman pumpingthat drives the halocline depth perturbations.

b. Gyre adjustment time scale

One of the most important quantities that describes thegyre dynamics is its stability, i.e. the adjustment time scaleassociated with the exponential decay of perturbations. Inthis section we explore the impact of eddies on this adjust-ment.


Given any radial profile of the anticyclonic mean sur-face stress [τ0(r)] one can calculate the background diffu-sivity K0(r) (from Eq. 23) and the eigenfunctions h∗i withcorresponding eigenvalues T−1

i > 0 for the eddy diffusionoperator:

1r

[rK0(r)h∗ir

]r=−h∗i

Ti. (25)

Here, homogeneous boundary conditions h(R) = 0 andhr(0) = 0 should be used (see Appendix A). Since thisdiffusion operator is self-adjoint (for an r-weighted normand K0(r) > 0) any halocline depth perturbation can bedecomposed into contributions from its orthogonal eigen-modes (h(r)∗i ) as

h =∞

∑i=1

ai(t)h∗i , (26)

where we sort the eigenfunctions corresponding to theireigenvalues starting from the smallest (their indices cor-respond to the number of zero crossings). Thus, the firsteigenfunction corresponds to a large-scale halocline deep-ening, whereas higher eigenmodes are more oscillatoryin space and correspond to higher eigenvalues, discussedmore below.

In an unforced case (i.e. Ekman pumping perturbationswe = 0), the amplitudes ai(t) would evolve independentlyfrom each other according to a simple exponential decaylaw:

dai

dt=−ai

Ti. (27)

This result confirms our a priori assumption about thegyre being a stable system (see Eq. 1 and related discus-sions). Eigenvalues represent the inverse of the decay timescales Ti for each eigenmode. For a general perturbationthat may consist of many different modes, the gyre wouldequilibrate on a time scale corresponding to the longestone amongst all the possible modes:

T0 =1

nλ

R2

K0(R). (28)

Here λ is a positive dimensionless constant that arisesas a solution of the discussed eigenvalue problem. Notethat the eigenvalues depend on boundary conditions, how-ever for diffusion operators they grow rapidly (roughlyquadratically) with the number of zero crossings of thecorresponding eigenfunction. This implies that the spin-up time scale observed in the numerical model corre-sponds to a decay of the large-scale gyre mode, whereassmall-scale spatial perturbations in the halocline depthwould decay much faster.

For a linear surface-stress profile used in our referencegyre simulation (τ0 ∼ r) we obtain λ = {5.7,4.7,4.3} forpowers n = {1,2,3} respectively. While the expressionfor the gyre adjustment time (Eq. 28) is consistent with

V (#1000km3)40 60 80 100 120 140

FW

C(#

1000

km

3)

5

10

15

20

0.005N/m2

0.01N/m2

0.015N/m2

0.03N/m2

0.05N/m2

0.075N/m2

Slope = "S=Sref

FIG. 5. Freshwater volume and freshwater content plotted againsteach for the spinup time series of different numerical model experi-ments. Dashed line shows a slope of ∆S/Sre f supporting Eq. 31.

the scaling laws (Eq. 21), here we determined the multi-plicative prefactor (nλ )−1 ≈ 0.1 which turns out to be anorder of magnitude smaller than 1.

Using the gyre adjustment time scale diagnosed fromthe spinup time series (Fig. 4 black circles) we can inferan appropriate eddy diffusivity that can generate such atime scale as

KT0 (τ̂) =

1λn

R2

T0(τ̂), (29)

where n = 2 and a corresponding λ = 4.7 are used (seeEq. 28). Using the diagnosed equilibration time scale T0(Fig. 4) we calculate the corresponding eddy diffusivitybased on Eq. (29) above and plot in Fig. 3b (black circles).This diffusivity estimate is consistent with the one inferredfrom the bulk halocline deepening (Eq. 15) as well as withthe one directly diagnosed from the eddy buoyancy fluxes(Eq. 14). A good agreement between the three indepen-dent methods (Fig. 3b) provides strong support for thevalidity of our theory that draws direct connections be-tween eddy diffusivity, halocline depth, and gyre stabilityor equilibration time scale.

7. FWC response to Ekman pumping

Defined as a linear measure of column salinity with re-spect to a reference salinity Sre f , the amount of freshwater(FW) can be approximated as

FW ≡∫ 0

−H−

S(z)−Sre f

Sre fdz≈ ∆S

Sre fh, (30)

where the integration is to a depth H of a reference isoha-line. The FW is proportional to the halocline depth h andto the bulk vertical salinity difference ∆S and thus can bealtered either via water-mass modification or via changesin halocline depth. Note that the presence of vertical mix-ing can not significantly affect FWC unless it diffuses salt


up from below the reference salinity. Mixing can becomeimportant in cases with weak forcing when the haloclinebecomes thin (Spall 2013).

Using the approximation (30) it can be deduced that theFWC, defined as an area-integrated amount of freshwater,is proportional to the volume V of water above the halo-cline:

FWC ≈ ∆SSre f

V, where V = 2π

∫ R

0rhdr. (31)

Building on the linear relationship between FWC and V ,verified in the numerical model (see Fig. 5), we continueour exploration of the forced freshwater volume dynam-ics. In diagnosing V from the numerical model we de-fined the halocline depth h via the location of the 32.25isohaline; the relative top-to-bottom salinity difference,∆S/Sre f ≈ 5/34 is essentially prescribed through bound-ary conditions. In general, the temporal evolution of FWCwould also depend on water mass modifications that affectsurface and mid-depth salinities.

a. Spatially inhomogeneous Ekman pumping

Here we address the following question: amongst allthe possible Ekman pumping distributions, which one ismost efficient in changing the FWC? We thus consider theevolution of halocline volume, V , under spatially inhomo-geneous Ekman pumping.

Projecting Eq. 22 onto the ith eigenmode we obtain thateach amplitude ai(t) is forced by the Ekman pumping pro-jection wi onto a corresponding eigenfunction h∗i (plottedin Fig. 6a) such that

dai

dth∗i =−ai

Tih∗i +wi. (32)

An area integral of (Eq. 32) results in an evolution equa-tion for the corresponding halocline volume Vi:

dVi

dt=−Vi

Ti+WE , (33)

where WE is the area integrated Ekman pumping, i.e. theEkman transport. The equation above demonstrates thestability of the gyre in terms of the decay of its haloclinevolume anomalies which supports our a priori stability as-sumption (see Eq. 1 and discussions thereof).

Since we are considering Ekman pumping perturbationsthat have the same Ekman transport, a steady state volumeanomaly is directly proportional to the corresponding timescale Vi = WETi (Eq. 33). In turn, the time scales Ti reducerapidly in magnitude with their index

Ti

T0<< 1 for i≥ 1. (34)

For example, for the eigenfunctions plotted in Fig. 6T1/T0 = 0.23, T2/T0 = 0.1, implying that V0 is at least an

r/R0 0.2 0.4 0.6 0.8 1

Haloclineeigenfunctions(m

)

-20

-10

0

10

20

h$1; 61 = 4:7;

h$2; 62 = 20:4;

h$3; 63 = 47:3.

years0 5 10 15

FWCperturbation(#

1000

km

3)

0

0.2

0.4

0.6

0.8

1

1.2

FWC1

FWC2

FWC3

A

B

FIG. 6. a) First three halocline depth eigenfunctions; the legendshows their corresponding nondimensional eigenvalues which growrapidly with the number of zero crossings. b) Response of freshwa-ter content to the mentioned surface stress profiles as simulated by theeddy resolving numerical model (blue, black, and red correspondingly).Note that all stress perturbation had the same magnitude of area aver-aged Ekman pumping. Perturbations were made on a control run.

order of magnitude larger. Thus, the most dominant con-tribution to freshwater content would be due to the Ekmanpumping pattern of the gravest eigenmode (i.e. gyre scaleanticyclonic surface stress pattern as shown in Fig. 6a,red). Eddy diffusion is efficient in damping the responseto spatially inhomogeneous Ekman pumping. This occursbecause highly oscillatory eigenmodes induce large halo-cline slopes (see Fig. 6a) which quickly produce strongmesoscale transport that acts to damp them.

We now test this theoretical prediction within the eddyresolving gyre model. We have perturbed the gyre from itsequilibrium state by increasing the Ekman pumping withpatterns corresponding to the first three eigenmodes (ra-dial distributions as in Fig. 6a). Ekman transport pertur-bation amplitudes are 25% compared to the equilibratedstate transport. Note that all three perturbations have thesame value of the area averaged Ekman pumping and yetthe theory predicts that the corresponding FWC responsesshould be dramatically different.


The numerical model, in agreement with the theory,shows that the FWC increases substantially more for thelarge-scale Ekman pumping pattern (Fig. 6b, red) and vir-tually does not increase for the spatially inhomogeneouspumping (Fig. 6b, blue and black). This confirms thatonly large-scale Ekman pumping can efficiently contributeto changes in FWC.

By moving beyond the steady state solutions consid-ered earlier, the transient dynamics reveal which Ekmanpumping modes provide the most efficient forcing. Non-linearity (not accounted for in our transient theory) mightbe important in determining the halocline depth anoma-lies that are large in amplitude, especially near the centerof the domain (Fig. 6a). However, changes near the cen-ter of the domain do not substantially contribute to FWCas it is weighted by area, meaning that changes are moreimportant at the edges.

b. Periodic Ekman pumping

Here we explore the response of FWC to periodic Ek-man pumping. Modes with smaller-scale spatial variabil-ity are damped more efficiently by eddies (Eq.33), suchthat the overall halocline volume anomaly is largely deter-mined by the volume of the first eigenmode:

V =∞

∑i=0

Vi ≈V0. (35)

The volume anomalies thus obey a simple equation of adamped-driven system that approaches equilibrium withan e-folding time scale T0:

dVdt

=−VT0

+WE sinωt, (36)

A similar equation and role of eddies was found for tran-sient dynamics of the thermohaline circulation by Spall(2015). Note that here we consider the gyre forced bythe most efficient large-scale Ekman pumping pattern (i.e.we = w0h∗0). In general, WE would be the transport asso-ciated only with the Ekman pumping projection onto thefirst eigenfunction w0 as the portions of the transport asso-ciated with larger eigenvalue projections would insignifi-cantly contribute to changes in volume.

Solving the equation above we find that the haloclinevolume, after an initial adjustment, approaches a simpleperiodic solution lagged with respect to the forcing by aphase φ :

VWET0

=1√

1+(ωT0)2sin(ωt−φ), (37)

φ = arctan(ωT0). (38)

Using the equilibrated control simulation we apply os-cillating Ekman pumping forcing with a 25% amplitude

with respect to its mean value (the large-scale spatial pat-tern does not change in time). The normalized freshwatervolume amplitude and its phase lag are diagnosed fromthe model and are shown in Fig 7 (crosses). The eddy re-solving model is in close agreement with our theoreticalpredictions based on the adjustment by mesoscale eddydiffusion.

The time scale T0 represents a transition in the sys-tem from a one-dimensional to a three dimensional re-sponse. If the eddies were unimportant to the dynam-ics (a limit of infinitely large adjustment time scale T0in Eqns. 37-38), the volume perturbations would havebeen inversely proportional to the frequency of the Ekmanpumping (V = WE/ω) and the phase lag would have beena quarter of a period for all frequencies (φ = π/2). Thislimit is relevant for high-frequency forcing, where the ed-dies have insufficient time to respond to changes in Ek-man pumping; the gyre response in this case is entirelydue to the Ekman advection. However, the freshwater vol-ume amplitude is maximized for slowly oscillating forcingwhere it is more in phase with the forcing (Fig 7). In thisregime the eddy transport nearly compensates for the Ek-man pumping. The response amplitude diagnosed fromthe numerical model is slightly larger than the analyticalprediction (Fig. 7a) due to a presence of internal modes ofthe gyre variability excited by the forcing.

The time scale T0≈ 2.1 years that was used in the theory(Eqns. 38-37) was estimated from bulk gyre sensitivity toforcing as

T0 =∆V

∆WE

∣∣∣∣τre f

, (39)

where ∆V is an equilibrium change in freshwater vol-ume corresponding to an increase of Ekman transport by∆WE . This time scale represents the gyre stability to smallperturbations around its equilibrium state. It is slightlysmaller than the 3 year non-linear adjustment time scalethat was diagnosed from the spin-up time series (Fig. 4).

8. Predicting FWC using the Gyre Index

Here, we propose a novel FWC tendency diagnostic –the Gyre Index – that predicts the freshwater content andincludes both the effects of wind forcing and eddies.

a. The Gyre Index

We take (22) for the halocline perturbation and integrateit over the surface area of the gyre to obtain an evolutionequation for the changes in the volume of water above thehalocline:

dVdt

=∫ R

02πr

dhdt

dr = 2πR(

nKos+−τ

ρ0 f

)∣∣∣∣r=R

, (40)

where the halocline slope s and surface stress τ are per-turbations from the mean state. Here we made use of the


0 2 4 6 8

Normalizedvolume

0

0.2

0.4

0.6

0.8

1 Model Theory

Ekman pumping frequency, ! (rad/yr)0 2 4 6 8

Phaselag?

0

:/4

:/2

V 9 1p1+(!T0)2

? = arctg(!T0)

A

B

FIG. 7. Amplitude-phase response of the gyre freshwater content toperiodic Ekman pumping. a) Normalized amplitude of volume oscilla-tions (see Eq.37). b) corresponding phase delay relative to the oscillat-ing Ekman pumping (Eq. 38).

Stokes theorem for the Ekman pumping term and integra-tion by parts for the eddy diffusion term.

Taking into account that the top-to-bottom salinity dif-ference is constant throughout our simulations, we can de-fine the Gyre Index (GI) that closely approximates FWCtendency as:

GI = 2πR∆SSre f

[nK0(R)s(R)︸︷︷︸

Eddytransport

+−τ(R)

ρ0 f

]︸︷︷︸

Ekmantransport

, (41)

where we have made use of Eq. 31 (s and τ are perturba-tions from their mean values, while K0 is the eddy diffu-sivity of the mean state). The GI approach is analogous toconventional freshwater budget calculations with an addi-tional eddy transport term. Thus, GI > 0 implies that thegyre is gaining FWC and GI < 0 implies the gyre is losingFWC. The magnitude of the GI should be compared to themaximum FWC flux out of the gyre FWC/T0 ∼ τ/(ρ0 f )that is proportional to the Ekman transport and is indepen-dent of the eddy diffusivity (see Eqns. 21 and 20). Wecan also interpret the GI as being a measure of how farthe gyre is from its equilibrium state. In equilibrium the

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

Because both terms in the GI are evaluated at the gyreboundary (Eq. 41), the FWC tendency depends explic-itly only on the boundary processes. On one hand, themesoscale eddies can only redistribute the halocline thick-ness within the gyre without affecting the total FWC un-less there are eddy thickness sources at the boundaries. Onthe other hand, due to Stokes theorem, the area-integratedEkman pumping is proportional to the contour integralof surface stress around the gyre boundaries. Thus, anyspatially-localized anomaly in Ekman pumping can onlyaffect the halocline depth locally and changes in FWCwould depend only on the existence of a surface stressalong the boundaries.

b. Diagnostic power of the Gyre Index

We now proceed to investigate the extent to which theGI can approximate FWC tendency within the eddy re-solving numerical model. We simulate the gyre variabilityby applying time-dependent spatially-homogeneous Ek-man pumping. Its amplitude evolves according to a rednoise process with a memory parameter of 1 year to mimicobservations that show enhanced variability on interannualto decadal time scales (Proshutinsky et al 2009); time se-ries are shown in Fig. 8a. The surface stress perturbationsfrom the reference run have a variance of 25% with respectto the mean value of τ0 = 0.015 N m−2 (the reference casethat we use for the Beaufort Gyre). The correspondingFWC undergoes significant variations of about 3000 km3

on decadal time scales (Fig. 8a, red curve).We then compute the Gyre Index by diagnosing the

azimuthally-averaged halocline slope perturbation s(t,R)and the generated surface stress τ(t,R) at the gyre bound-ary (Eq. 41). Because we use stochastic in time Ekmanpumping perturbations the time series have large variancesat short time scales (Fig. 8a). To illustrate its differenceswith the purely Ekman pumping-based index we evaluatethe performance of the Gyre Index on various time scales.In Figure 8b we compare FWC tendency and the GI forwhich a 5-year running mean smoothing has been appliedin order to explore the interannual trends characteristicof the Beaufort High variability. Using only the Ekmanpumping as an index (i.e. excluding the eddy transportterm in the GI) does not give an accurate representationof FWC tendency (note the large differences between redand black curves in Fig 8b). In contrast, the GI providesa much better agreement with the numerically-simulatedFWC evolution (Fig 8b), further supporting our theory ofthe gyre variability.

We quantify the performance of the GI by calculatingthe root mean square error as a function of the length ofthe running mean window used to smooth the time series(Fig 8c). We see that the Ekman only predictor of FWC


years0 20 40 60 80 100 120 140 160 180 200

FWCtendency(k

m3=yr)

-200

-100

0

100

200

300

d=dt(FWC) Ekman Ekman+Eddies

years0 20 40 60 80 100 120 140 160 180 200

Windstressperturbation(m

2=s)

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

FWC(#1000

km3)

13

13.5

14

14.5

15

15.5

16

smoothing interval (years)0 5 10 15 20

FWCtendencyerror(k

m3=yr)

0

40

80

120

160

200 STD of FWC tendency[Ekman] error[Ekman+Eddies] error

Eddy di,usivity K0(m2=s)

0 100 200 300 400 500 600

Errorrelativeto

K0=0

0.4

0.6

0.8

1

1.2

A

B

C D

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 amplitudeoscillates 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 rednoise with a 1 year damping parameter. b) FWC tendencies as directly estimated from the model (black), as inferred using the Ekman pumpingonly (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)The dependence of error on the choice of the mesoscale eddy diffusivity showing the optimal choice of K0 ≈ 280 m2 s−1 that was used for the timeseries shown in b).

tendency provides a good match only on short time scaleswhere eddy activity is less important. However, on in-terannual and longer time scales it is necessary to accountfor the eddy activity as the Ekman predictor error becomeslarger than the standard deviation of FWC tendency. Thisimplies that at interannual and longer time scales assum-ing no change in FWC gives a better prediction than onlytaking into account the Ekman pumping. In contrast, theGI has a strong predictive capability on all time scales withits error persistently smaller than the standard deviation ofthe FWC tendency (Fig 8c).

A calculation of GI requires the background eddy dif-fusivity evaluated at the boundary (Eq. 41). We show thatthere exists an optimal eddy diffusivity that minimizes er-ror of the GI making it a factor of 2 smaller than that ofthe Ekman index alone (see Fig 8d). The optimal value ofK0 ≈ 300 m2 s−1 is in a range of values diagnosed earlierusing the three other independent methods (see Fig. 3b).

In Section 7.1 we demonstrated that only large scalevariations in Ekman pumping significantly affect FWC.Yet the GI, which accurately predicts variability of FWC,does not contain information about the horizontal scaleof the Ekman pumping. Instead, it only uses the Ek-


man transport, which is the area integrated Ekman pump-ing. This implies that for highly inhomogeneous Ekmanpumping (higher eigenmodes) the GI ≈ 0 and hence thereshould be a strong compensation between the eddy trans-port at the boundary and the Ekman transport.

9. Summary and discussions

We explored the transient dynamics of an idealizedBeaufort Gyre in an eddy resolving general circulationmodel with particular emphasis on the FWC variability.We performed a series of experiments exploring the gyre’sresponse to a time-dependent surface stress. The resultswere interpreted using Transformed Eulerian Mean theorythat explicitly includes the effects of mesoscale eddies.Using an eddy parameterization, we provide theoreticalpredictions for the gyre’s stability (inverse of the equili-bration time) as well as transient solutions for haloclineand FWC evolution under time-dependent Ekman pump-ing.

Our model and theory neglect several processes thatmay be important in the real Beaufort Gyre. The theoryis adiabatic and the model was run with a vertical diffu-sion coefficient of 10−5 m2 s−1. Observations indicatethat diapycnal mixing is spatially and temporally inhomo-geneous but, on average, between 10−6 and 10−5 m2 s−1

(Guthrie et al 2013). Scaling theory in MS16 suggests thatthe impact of diapcynal mixing is small but not negligible,especially when the surface stress is weak. We also ne-glect eddy salt fluxes (or thickness fluxes) that originatefrom eddies shed from the boundary currents that encirclethe Arctic basin (e.g. Manley and Hunkins (1985); Spallet al. (2008); Watanabe (2013)). Scaling in Spall (2013)suggests that the freshwater flux carried by these eddies isof similar amplitude as that carried by the Ekman trans-port. It is important to note, however, that the Eulerianstreamfunction associated with the eddies vanishes so theydo not provide an equivalent to the Eulerian Ekman pump-ing velocity produced by the surface stress. We also donot represent geometric complexities such as the EurasianBasin, continental slopes, shelf dynamics, and mid-oceanridges. We view the present model as representing, in acompact and transparent way, the leading order dynami-cal balances for the Beaufort Gyre from which additionalprocesses may be considered.

We now highlight four key results from this study. First,we demonstrated that the presence of mesoscale eddies di-rectly affects the Ekman-driven gyre variability (Eqns. 33and 37). By defining the gyre equilibration time scale (Eq.28), we provided a theoretical expression for the time scaleand emphasize its explicit dependence on mesoscale eddydiffusivity (Eq. 28).

Second, we presented several estimates of the charac-teristic mesoscale eddy diffusivity for the gyre, demon-strating that it grows nearly linearly with the surface stress

forcing (Fig. 3). Our theory has only one a priori un-known parameter, the mesoscale eddy diffusivity, whichwe inferred in four ways based on: the buoyancy flux (Eq.14); the bulk halocline deepening (Eq. 15); the adjustmenttime scale (Eq. 29); and volume transport variability (Fig.8d). These independent methods are consistent betweeneach other and thus provide support for the theory. Forconditions akin to a present-day Beaufort Gyre we pro-vide an estimate of the characteristic diffusivity of about300 m2 s−1 (τ̂ = 0.015 N m−2) and predict its sensitivity tobe 170 m2 s−1 per 0.01 N m−2 increase in the azimuthalsurface wind stress. These parameters should be testedagainst observations.

Third, motivated by a strong variability in atmosphericwinds over the Beaufort Gyre, we explored a gyre re-sponse to spatially-inhomogeneous and time-dependentEkman pumping. Our analytical and numerical solu-tions show that amongst all possible stress distributionsthat have the same area averaged Ekman pumping, theFWC is largely affected only by the gyre-scale Ekmanpumping (Fig. 6b). The FWC response to temporally-periodic Ekman pumping is closely approximated by asimple damped-driven dynamical system that approachesequilibrium with a known adjustment time scale controlledby eddy dynamics (Eq. 28). High-frequency oscillationsin the pumping (e.g. seasonal cycle) have little effect onhalocline depth, whereas the strongest effect is achievedfor low-frequency forcing (e.g. on decadal time scales)(Fig. 7).

Fourth, we proposed the use of the Gyre Index (Eq. 41)for monitoring/interpreting FWC tendency in the BeaufortGyre. Its key advantage is that calculating the GI requiresknowledge of the halocline slope and magnitude of sur-face stress evaluated only at the gyre boundaries (not in itsinterior). Using a numerical model we demonstrated itsstrong predictive capability that is due to the fact that it in-corporates the competing effects of both Ekman pumpingand mesoscale eddy transport. For interannual and longertime scales the GI is far superior to using only the strengthof Ekman pumping in evaluating the FWC tendency (Fig.8).

The GI can be readily generalized to include effects ofmore realistic features of the Beaufort Gyre dynamics suchas the azimuthal asymmetry in Ekman pumping and eddydiffusivities. This would require calculating a contour in-tegral along the gyre boundaries in order to evaluate termsin the GI. Observationally, calculation of the GI could beachieved through a network of instruments located alongthe gyre perimeter, thus complementing the observationalefforts in the interior of the gyre. We envision the GI cal-culation requiring observations of density and velocitiesfrom a set of moorings spaced around the gyre. Mooringobservations of ocean velocity and density fields wouldallow for calculation of the eddy fluxes by estimating the


vertical shear of the first baroclinic mode of the horizon-tal velocity that is directly related to horizontal haloclineslope. We speculate that the GI would be a useful toolfor interpreting the component of FWC variability that isrelated to changes in the halocline depth in both observa-tional and modeling studies of the Beaufort Gyre.

Acknowledgments. The authors acknowledge the high-performance computing support from Yellowstone pro-vided by NCARs CIS Laboratory, sponsored by the NSF.GEM acknowledges the support from the Howland Post-doctoral Program Fund at WHOI and the Stanback Fel-lowship Fund at Caltech. MAS was supported by NSFGrants PLR-1415489 and OCE-1232389. AFT acknowl-edges support from NASA award NNN12AA01C. The au-thors would like to thank Prof. John Marshall and ananonymous reviewer for helpful comments and sugges-tions. The manuscript also benefited from discussions atthe annual Forum for Arctic Modeling and Observing Syn-thesis (FAMOS) funded by the NSF OPP awards PLR-1313614 and PLR-1203720.

APPENDIX A

Time evolution of halocline depth perturbations

The nonlinear equation set 16-17 for the buoyancy evo-lution can be linearized in the vicinity of its mean state toobtain :

bt +1r(ψ̃r)rb0z− ψ̃zb0r = 0, (A1)

ψ̃ =τ

ρ0 f−n

τ0

ρ0 fss0

, (A2)

ss0

=(

br

b0r− bz

b0z

), (A3)

where all dynamical variables are perturbations from theirmean state while 0-subscripts correspond to mean statevariables. The fixed-buoyancy boundary conditions trans-form into homogeneous conditions for the perturbationvariables (b = 0 at r = R and br=0 at r = 0).

The linearized equation set simplifies dramatically ifone writes it along the characteristics defined by the halo-cline depth of the mean buoyancy distribution. Eq. 11may be written in characteristic coordinates [r(l),z(l)] asbl = brrl + bzzl = 0, where the homogeneous right handside indicates that buoyancy is conserved along the char-acteristic trajectory. The characteristic velocities are then

dzdl

=−(−τ

ρ0 f k

) 1n

,drdl

= 1. (A4)

The mean vertical buoyancy gradient, following theisopycnals (characteristics) defined by the mean state,

does not change (b0z = const) and can thus define a per-turbation isopycnal displacement h as

h(l, t) =b(l, t)

b0z. (A5)

Taking into account that ∂l() = ∂r()rl + ∂z()zl we obtainthe following system for variables following characteris-tics:

ht =−1l(ψ̃l)l , ψ̃ =

τ

ρ0 f−nK0hl , (A6)

where K0(l) is a background eddy diffusivity set by theisopycnal slope of the steady state buoyancy distribution:

K0 = ksn−10 = k

[−τ0(l)ρ0 f k

] n−1n

. (A7)

Eliminating ψ̃ from the equations above we obtain anequation for the time evolution of isopycnal displacement:

ht =1r

(nK0rhr

)r+

1r

(r−τ

ρ0 f

)r

(A8)

b.c. : h|r=R = 0, hr|r=0 = 0. (A9)

that is now written in terms of the radial coordinate takingonto account that rl = 1. The fixed-buoyancy boundaryconditions for the original equations translated into the ho-mogeneous boundary conditions for the linearized system.

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A theory of the wind-driven Beaufort Gyre variability · persistent large-scale feature of the Arctic Ocean, it is rea-sonable to assume that it is a stable, externally-driven sys-tem