The relationship between double-diffusive intrusions and staircases in the Arctic Ocean

YANA BEBIEVA ∗ AND MARY-LOUISE TIMMERMANS †

Department of Geology and Geophysics, Yale University, 210 Whitney Ave, New Haven CT 06511, USA.

ABSTRACT

The origin of double-diffusive staircases in the Arctic Ocean is investigated for the particular backgroundsetting in which both temperature and salinity increase with depth. Motivated by observations that show theco-existence of thermohaline intrusions and double-diffusive staircases, a linear stability analysis is performedon the governing equations to determine the conditions under which staircases form. It is shown that a double-diffusive staircase can result from interleaving motions if the observed bulk vertical density ratio is below acritical vertical density ratio estimated for particular lateral and vertical background temperature and salin-ity gradients. Vertical temperature and salinity gradients dominate over horizontal gradients in determiningwhether staircases form. Examination of Arctic Ocean temperature and salinity measurements indicates thatobservations are consistent with the theory for reasonable choices of eddy diffusivity and viscosity.

1. Introduction

The Arctic Ocean has a strong halocline and deeper wa-ter layers that are warmer than those at the surface in con-tact with sea-ice cover (e.g., Aagaard et al. 1981). Under-standing the mechanisms and magnitude of upward fluxesof deep ocean heat is essential to predictions of Arctic seaice and climate (e.g., Maykut and Untersteiner 1971; Wet-tlaufer 1991; Perovich et al. 2008; Carmack et al. 2015;Timmermans 2015). Relatively cold and fresh surface wa-ters, originating from net precipitation, river runoff, in-flows from the Pacific Ocean and seasonal sea-ice melt,occupy the upper ∼150–200 m water column (e.g., Steeleet al. 2008; Timmermans et al. 2014). Below these up-per layers lie relatively warm and salty waters associatedwith Atlantic Water (AW) inflows, centered around 400 mdepth in the Arctic Ocean’s Canada Basin, with lateraltemperature gradients indicating a general cooling movingeast from the warm core of the AW layer on the westernside of the basin (Figure 1a).

The basic vertical stratification, in which temperatureand salinity both increase with depth, provides condi-tions amenable to double-diffusive instability, believedto be a key physical process generating thermohaline in-trusions and staircases in the Arctic Ocean. Through-out much of the central Arctic Ocean basins, heat trans-

∗Corresponding author address: Department of Geology and Geo-physics, Yale University, 210 Whitney Ave, New Haven CT 06511,USAE-mail: yana.bebieva@yale.edu†Department of Geology and Geophysics, Yale University, 210

Whitney Ave, New Haven CT 06511, USA

fer from the AW layer is via double-diffusive convec-tion (e.g., Melling et al. 1984; Padman and Dillon 1987,1988; Timmermans et al. 2008; Polyakov et al. 2012; Sire-vaag and Fer 2012; Guthrie et al. 2015). Two types ofdouble-diffusive convection can arise in a stably-stratifiedocean: the case when both temperature T and salinity Sincrease with depth is referred to as diffusive convection(the DC regime, an overview is given by Kelley et al.2003), while the case when both temperature and salin-ity decrease with depth is referred to as the salt-finger(SF) regime (for overviews see, e.g., Kunze 2003; Schmitt2003). These stratifications may be characterized by adensity ratio, which we define here as Rρ = βSz/αTz,where the subscript z denotes vertical gradient, and β =(1/ρ0)(∂ρ/∂S)T,p and α = −(1/ρ0)(∂ρ/∂T )S,p (whereρ0 is a reference density) are the saline contraction andthermal expansion coefficients, respectively. The over-lines indicate a bulk gradient, taken to be linear over somespecified depth range.

At the top boundary of the AW layer in the CanadaBasin, a prevalent DC staircase is characterized by a se-quence of mixed layers of thickness on the order of sev-eral meters separated by sharp gradients in temperatureand salinity (Neal et al. 1969; Padman and Dillon 1987;Timmermans et al. 2008) (Figure 1b,c). Thermohaline in-trusions, characterized by DC and SF regions alternatingin depth, are often found underlying the staircase (e.g.,Carmack et al. 1998; Merryfield 2002; Woodgate et al.2007). Intrusions are believed to be associated with lat-eral (in addition to vertical) gradients in temperature andsalinity, and are driven partly by vertical buoyancy flux di-

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vergences (overviews of thermohaline intrusions are givenby Ruddick and Kerr 2003; Ruddick and Richards 2003).

Understanding and predicting the observed verticaltemperature-salinity structure (whether staircases or intru-sions), and associated vertical and lateral heat fluxes fromthe AW layer, requires knowledge of the origins of thesefeatures and their relationship to each other (see Kelley2001). Extensive efforts have been made to explain stair-case and intrusion origins and evolution with respect to theSF configuration of double diffusion. There exist aroundsix theories for the origin of SF staircases, as reviewed byRadko (2013). One of these theories is that interleavingmotions can develop into a staircase (Merryfield 2000).The idea relies on the presence of lateral temperature andsalinity gradients and builds on previous studies that in-voke a standard parametric flux model (Walsh and Rud-dick 1995, 2000). This model was first introduced by Stern(1967), who delineated three separate scales of motion:“small” to describe double-diffusive processes on cen-timeter to meter scales, “medium” to describe the scalesof interleaving (order tens of meters vertically and 1 kmlaterally), and “large” to characterize the background state(the full thickness of the double-diffusive thermocline andbasin scales laterally). The main assumption here is thatmedium-scale dynamics are qualitatively similar to small-scale dynamics and that the effects of double diffusionon medium-scale vertical fluxes can be parameterized interms of effective diffusivities and large-scale vertical gra-dients. Merryfield (2000) applied this formalism in his cal-culations showing that interleaving motions evolve into SFstaircases when the density ratio of the background (SF-stratified) state is below a certain value (i.e., there exists acritical density ratio that delineates the boundary betweenstaircase and intrusion formation).

Very little has been done with respect to analysis ofstaircase origins in the DC-stratified setting. The purposeof this study is to address the relationships between andorigins of DC staircases and intrusions, with applicationto the Arctic Ocean. Motivated by observations whichshow staircases and intrusions co-existing and evolving,we assess the main factors that determine whether stair-cases or intrusions will be observed. Perhaps the majorlimitation on developing a mechanism for staircase for-mation in the Arctic Ocean has been considered to relateto the fact that the magnitude of the observed vertical den-sity ratio (Rρ ∼ 4, see e.g., Timmermans et al. 2008) fallsoutside the DC-unstable range (1 < Rρ < 1.1) derived bylinear stability analysis of the Boussinesq equations (Vero-nis 1965). However, this derivation considers only small-scale dynamics (i.e., molecular diffusivities are used in thecomputation of fluxes), and no horizontal background gra-dients in temperature and salinity. Here, we invoke theparametric flux model (as used by Merryfield (2000) forthe SF case) to determine a critical vertical density ratioRcr

ρ for the DC stratification at which a transition between

staircases and intrusions occurs for the observed verti-cal and lateral temperature gradients in the Arctic Ocean.In this formalism, instability does not require Rρ to bebounded by 1.1.

The paper is organized as follows. In the next section,we formulate the governing equations (section 2a) andthen determine a constraint on the growth rate for grow-ing perturbations to evolve towards a DC staircase (sec-tion 2b). Next, in section 2c, we perform a linear stabilityanalysis on the governing Boussinesq equations to deter-mine the fastest growing mode as a function of verticaland lateral temperature and salinity gradients. Togetherwith the result from section 2b, this allows us to definea critical vertical density ratio Rcr

ρ above which intrusionsform, and below which staircases form. In section 3, weapply the theory to Ice-Tethered Profiler (ITP; Krishfieldet al. (2008); Toole et al. (2011)) measurements of tem-perature and salinity through the Canada Basin double-diffusive thermocline, which features both a staircase andintrusions. We examine ITP temperature and salinity pro-files to show that the observations are consistent with thetheory for reasonable choices of appropriate parameters.Findings are summarized and discussed in section 4.

2. Theory

a. Formulation of the governing equations

We begin by formulating the governing equations ac-counting for three scales of motion as described in section1. The governing set of Boussinesq equations (2D: hori-zontal and vertical dimensions) is as follows

ut +uux +wuz =−px/ρ0 +ν∇2u

wt +uwx +wwz =−pz/ρ0−gρ/ρ0 +ν∇2w

ux +wz = 0 (1)

Tt +uTx +wTz = κT ∇2T

St +uSx +wSz = κS∇2S

ρ = ρ0(1−α(T −T0)+β (S−S0)),

where u and w are horizontal and vertical components ofvelocity, p is pressure, κT and κS are the molecular co-efficients of heat and salt diffusion respectively, ν is themolecular kinematic viscosity, and T0 and S0 are referencevalues of temperature and salinity (corresponding to den-sity ρ0). We neglect rotation because we are primarily in-terested in the fastest growth rate; with respect to the SFcase, Kerr and Holyer (1986) have shown that the structureof the fastest-growing mode of perturbation (i.e., verticaland horizontal wave numbers along with the growth rate)is unaffected by rotation. This set of equations consists oftwo momentum equations, the continuity equation, con-servation equations for T and S and the equation of state.

We proceed by decomposing variables into mean largescale values (basin-wide scales, denoted with overbars),

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medium scales (intrusions, denoted with tildes) and smallscales (double-diffusive mixing, denoted by primes), forexample, u = u+ u+u′, T = T + T +T ′ etc.; this is sim-ilar to the separation of scales under Reynolds averaging(Reynolds 1894). All equations are next averaged oversmall scales associated with double-diffusive mixing ona time period that is sufficiently long to smooth transientfluctuations, yet short enough to retain the slow evolutionof the interleaving motion (operating on medium scales).Further, we assume that the background state is motion-less (i.e. u = 0 and w = 0). Note also that the back-ground temperature and salinity structure does not changeon the timescales of medium motion, so that ∂T/∂ t = 0and ∂S/∂ t = 0; in section 3b, we show this to be an ap-propriate approximation for the setting considered here.After time averaging (denoted by angle brackets, and notethat time averaging of any medium or large scale variableis equal to that variable: e.g., < u >= u) the governingequations become

ut + uux + wuz =−px/ρ0 +ν∇2u

− (< u′u′ >)x− (< u′w′ >)z

wt + uwx + wwz =−pz/ρ0−gρ/ρ0 +ν∇2u

− (< u′w′ >)x− (< w′w′ >)z

ux + wz = 0 (2)

Tt + uTx + wTz + uT x + wT z = κT ∇2(T + T )

− (< u′T ′ >)x− (< w′T ′ >)z

St + uSx + wSz + uSx + wSz = κS∇2(S+ S)

− (< u′S′ >)x− (< w′S′ >)z.

The last two terms (Reynolds stresses) on the right handside of the two momentum, heat and salt equations rep-resent the effects of fluctuations (associated with double-diffusive mixing) on interleaving motions. Horizontal di-vergences can be neglected with respect to vertical diver-gences since interleaving motions have small aspect ratio.

Analogous to Reynolds averaging, we combineReynolds stresses and frictional terms to define verticaleddy viscosity (A) and vertical eddy diffusivities for heat(KT ) and salt (KS) as follows

ν uz−< u′w′ >= Auz (3)

νwz−< w′w′ >= Awz (4)

κT (T + T )z−< w′T ′ >= KT Tz (5)

κS(S+ S)z−< w′S′ >= KSSz. (6)

The resulting linearized system of equations describes mo-tions on the medium scale:

ut + px/ρ0 +Fuz = 0 (7)

wt + pz/ρ0 +gρ/ρ0 +Fwz = 0 (8)

ux + wz = 0 (9)

Tt + uT x + wT z +FTz = 0 (10)

St + uSx + wSz +FSz = 0, (11)

where vertical fluxes of horizontal (Fu) and vertical (Fw)momentum as well as vertical fluxes of heat (FT ) and salt(FS) are defined as

Fu =−Auz, Fw =−Awz, (12)

FT =−KT Tz, FS =−KSSz. (13)

The salt flux may be expressed via FT and the ratio of thedensity flux of salt to the density flux of heat (i.e., the fluxratio RF = βFS/αFT ) as

FS = RFα

βFT . (14)

For the DC-type of double-diffusive convection it hasbeen shown experimentally that RF ≈ 0.15 when Rρ &2 (Turner 1965). Here we take RF = 0.15 for themedium scale motions under the assumption that fluxes onmedium-scales can be parameterized as double-diffusivemixing on medium scales (see Stern 1967) and that the DCfluxes dominate (cf. Toole and Georgi (1981) and Walshand Ruddick (1995, 2000) who followed the same reason-ing for the SF case).

1) ESTIMATES OF KT AND A

Eddy diffusivity KT parameterizes the effects of double-diffusive mixing (equation 5) on medium scales. In thestaircase region KT may be estimated using a double-diffusive heat flux parameterization based on a 4/3-fluxlaw (Kelley 1990) and the typical background verticaltemperature gradient; Guthrie et al. (2015) show this 4/3-flux law to be a reasonable representation of the fluxes.This yields KT = O(10−6) m2s−1 for typical double-diffusive fluxes in the Canada Basin around 0.1 – 0.2W/m2 (Padman and Dillon 1987; Timmermans et al.2008). This estimate for KT is consistent with estimatesby Guthrie et al. (2013) in the Canada Basin in the re-gion of the staircase, where they find KT = 1×10−6−5×10−6 m2s−1.

Eddy viscosity must be formulated to represent momen-tum transfer on medium scales due to double-diffusiveprocesses. In the theoretical framework, we considergrowing perturbations in the absence of large-scale back-ground shear. As perturbations grow, however, small ve-locities arise associated with interleaving. We assume that

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interleaving motions achieve a buoyancy-friction balance(McDougall 1985) some time after the onset of perturba-tions in the absence of large-scale shear and pressure work(Tennekes and Lumley 1972). An effective viscosity canthen be estimated using the kinetic energy balance (assum-ing that horizontal shear is the dominant term in the rate ofviscous dissipation of turbulent kinetic energy) as follows

Auzz = gαFT (1−RF) ⇒ A =gαKT Tz(1−RF)H2

2U0,

where uzz is scaled as 2U0/H2, and typical interleaving ve-locity along a layer (of characteristic height H = O(1) m)is U0 = O(1) mm/s (Ruddick and Hebert 1988; Walshand Carmack 2003). Our estimated value for KT to-gether with the typical mean vertical temperature gradi-ent in our observational setting (Tz ≈ 0.01◦C/m) yieldsA = O(10−8) m2s−1. This estimate of eddy viscosity issmaller than molecular viscosity, which may not be sur-prising in the context of pure double-diffusive mixing. Forexample, for the SF case, Ruddick (1985) estimated eddyviscosities in the range 10−9 − 10−14 m2s−1, where heattributed these small values to the small lateral veloci-ties between mixed layers in an SF staircase. Here weconsider shear generated only due to interleaving motionsoriginating from a motionless background state (due toconvergences and divergences of double-diffusive fluxes).This is principally different from the case where momen-tum transport is considered through thermohaline steps inthe presence of large-scale background shear (e.g., Pad-man 1994). Finally, the estimates for A are consistent withthose found in a laboratory study on double-diffusive in-terleaving: Krishnamurti (2006) showed that the ratio ofReynolds stress momentum fluxes to viscous stress mo-mentum fluxes (i.e., < u′w′ > /ν uz in our notation) canvary from 0.1 to 1 within the field of interleaving motion.Following (3), this yields a broad range for A with varia-tions by an order of magnitude (10−8 to 10−7 m2s−1).

b. Constraint leading to a staircase

In order to derive a constraint on the growth rate thatmust be satisfied for a perturbation to evolve to a stair-case, we consider wave-like solutions of the system (7)–(11). We write temperature and salinity fields as the sumof linear background gradients plus deviations (resultingfrom interleaving motions) in the form of a plane-wave asfollows:

T (x,z, t) = Tzz+Txx+ T = T zz+T xx+Teikx+imz+λ t

(15)

S(x,z, t) = Szz+Sxx+ S = Szz+Sxx+Seikx+imz+λ t ,(16)

where k and m are horizontal and vertical wavenumbersrespectively, λ is growth rate, and T and S are the ampli-tudes of the temperature and salinity perturbations. Merry-field (2000) provides an effective explanation for the pos-sible outcomes of a perturbation given initial conditionsof an SF stratification; we follow the same reasoning forthe DC case. At a particular x = x0 the evolution of tem-perature and salinity can be illustrated on the (αTz,βSz)plane (Figure 2a). Initially, the amplitudes of the pertur-bations are small and the T and S fields are represented bya single point on the plane. A growing perturbation is aline segment with slope βS/αT. If this slope is smallerthan the vertical density ratio Rρ characterizing the back-ground stratification (i.e., large scale) from which a per-turbation begins to grow, then the value of β Sz/αTz in-creases in time on one end of the segment and decreaseson the other end (Figure 2a, line denoted I for intrusion).An increase of β Sz/αTz leads to a doubly-stable stratifica-tion (i.e., the vertical temperature gradient reverses sign todecreasing temperature with depth, while the salinity gra-dient remains stable), with further increases leading to anSF-unstable stratification when the salinity gradient alsoreverses sign. The other end of the line segment I rep-resents DC-favorable conditions with decreasing β Sz/αTztowards unity, where convective instability takes place anda mixed layer is formed. Therefore, the evolution alongline I corresponds to the development of intrusions con-sisting of alternating mixed layers, DC-gradients, stableregions, and SF-gradients. The second scenario (depictedby the line S for staircase in Figure 2a) arises when theevolution of T and S gradients are on a slope that islarger than Rρ . In this scenario, as the perturbation grows,β Sz/αTz decreases on one end of the segment (towardsβ Sz/αTz = 1 and the convectively unstable region) and in-creases on the other end of the segment in the DC-unstableregion of the (αTz,βSz) plane (Figure 2a, S). The grow-ing perturbations result in a series of convectively unstablemixed layers and DC gradients, i.e., a DC-type staircase.Whether a perturbation evolves to intrusions (along lineI) or to a DC-staircase (along line S) depends upon thegrowthrate and wavelengths of the perturbations, a func-tion of the initial background temperature and salinity gra-dients. The implicit assumption here is that the evolutionof temperature and salinity is in the linear phase (i.e., per-turbations develop and grow exponentially, and retain thesame plane-wave spatial dependence as they grow frominfinitesimally small perturbations).

We begin by examining the case in which perturbationsto the linear background gradient evolve into a staircase.That is,

βSαT

> Rρ . (17)

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Combining (15)–(17) and introducing a stream functionψ = Ψeikx+imz+λ t (where Ψ is the amplitude) such that

ψx = w, ψz =−u, (18)

we obtain the following criterion that the growth rate λ

must satisfy

λ <m2KT

Rρ −1

[1−RF −

sΓ(Rρ −RF)

], (19)

in order for a perturbation to evolve to a staircase (fromcriterion 17). Here, Γ = Tx/Tz and s = k/m (the slope ofthe growing perturbations). We have assumed that the as-pect ratio of the background state is always larger thanthe aspect ratio of the perturbations (i.e., Γ/(Γ− s) > 0),which is always satisfied for the observed values of Tx andTz (see section 2d).

We next perform a linear stability analysis (see e.g.,Toole and Georgi 1981; Walsh and Ruddick 1995, 2000)on the system (7)–(11) to compute the most unstable mode(i.e., maximal λ and corresponding k and m) for a speci-fied Tx, Tz and Rρ . The result is used together with thecondition (19) to derive a critical density ratio Rcr

ρ belowwhich staircases are expected to be the end result of aninterleaving perturbation.

c. Critical density ratio

To reduce the number of equations in the system, wecombine (7)–(9) to yield an evolution equation for vortic-ity ∇2ψ

∇2ψt +g(β Sx−αTx)+Fw

zx −Fuzz = 0. (20)

Horizontal background gradients in temperature and salin-ity are taken to be density compensating such that αTx =βSx. For application to the observations, we consideralong-isopycnal gradients to avoid the effects of isopycnalheaving (induced by mesoscale eddies and other dynam-ics) that do not lead to water mass transformation. In thisway, we are effectively imposing ρx = 0 and as a conse-quence αTx = βSx. Using this fact, and the definition ofRρ , we express background salinity gradients in terms ofbackground temperature gradients as

Sx = αTx/β (21)

Sz = Rρ αTz/β . (22)

Multiplying (10) by α and (11) by β , the final systemof equations can be expressed as

∇2ψt +g(β Sx−αTx)−A∇

2ψzz = 0 (23)

αTt − ψzαTx + ψxαTz−αKT Tzz = 0 (24)

β St − ψzαTx + ψxRρ αTz−RF αKT Tzz = 0. (25)

Again assuming plane-wave solutions for the mediumscale motions (ψ, T , S) = (Ψ,T,S)eikx+imz+λ t , the system(23) – (25) reduces to −(k2 +m2)(Am2 +λ ) −igk igk

iα(kTz−mTx) λ +KT m2 0iα(Rρ kTz−mTx) RF KT m2 λ

Ψ

αTβS

= 0.

A solution to this exists only if the determinant of the co-efficient matrix is equal to zero, which gives

λ3m2(s2 +1)+λ

2m4(A+KT )(s2 +1)+

λ [AKT m6(s2 +1)+αTzgs2m2(1−Rρ)]+

αKT gm4s[Tx(1−RF)−Tzs(Rρ −RF)] = 0.

(26)

The growth rate λmax of the fastest-growing mode (andcorresponding values smax and mmax) can be determinedfrom (26) by applying a constrained optimization tech-nique using the method of Lagrange multipliers (Bert-sekas 2014), for a given Tx and Tz. We wish to maximizethe growth rate λ , which may be expressed as a functionof three variables f (λ ,m,s) = λ subject to the constraintgiven by (26), which we denote as G(λ ,m,s) = 0. In themethod of Lagrange multipliers, we construct a system ofequations that satisfy ∇ f (λ ,m,s) = a∇G(λ ,m,s) (wherea is a constant) subject to G(λ ,m,s) = 0. This yields fourequations and four unknowns. Next, using solutions of thesystem (i.e., λmax, mmax, smax) we can obtain Rcr

ρ from thecriterion (19). That is, we solve the following equation

λmax

m2max

=KT

Rcrρ −1

[1−RF −

sΓ(Rcr

ρ −RF)]. (27)

d. Interpretation

It is instructive to explore how Rcrρ is influenced by Tx,

Tz and Rρ for the general ranges observed in the CanadaBasin, and A and KT values described in section 2a. Thelinear stability analysis together with (27) indicates Rcr

ρ isinsensitive to variations in Tx for some specified Tz and Sz(Figure 2b, red contours). This may be explained by thefact that the ratio of horizontal to vertical wavenumbersof the most unstable mode varies proportionally to the as-pect ratio of the background temperature gradients. Thatis, smax/Γ = const (≈ 0.1 for the range of Tx and Tz here).Furthermore, λmax/m2

max also remains constant with vary-ing Tx, implying that (27) has the same coefficients over arange of Tx, and as a consequence Rcr

ρ remains unchanged.While the growth rate is larger for larger Tx, this does notaffect whether a staircase or intrusions are the end resultof a perturbation to a linear background stratification. Themain influencing parameters in this regard are Tz and Sz(and Rρ ). When Sz is held fixed (e.g., Sz = −0.002m−1)for smaller magnitude Tz, Rρ increases faster than Rcr

ρ (Fig-ure 2b). For the weaker Tz, it is more likely that Rcr

ρ < Rρ ,

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implying intrusions are the end state of an interleaving per-turbation. In the next section, we explore these relation-ships with respect to observed staircases and intrusions inthe Arctic Ocean.

3. Context with ITP observations

As a consistency check, we aim to examine whether thepresence of intrusions or staircases in the Arctic Oceanwater column is commensurate with the linear theory de-scribed in the previous section. Water-column measure-ments are from Ice-Tethered Profilers (ITPs, Krishfieldet al. 2008; Toole et al. 2011) that drifted in the CanadaBasin (Figure 1a). ITPs are automated profiling instru-ments that provide measurements of temperature, salinityand depth from several meters depth beneath the sea ice,through the Atlantic Water layer to about 750 m depth.The final processed data for two ITP systems are analyzedhere: ITP 1, operating between August 2005 and Septem-ber 2006, and ITP 41, operating between October 2010and January 2012. These ITPs were deployed in the samegeneral location close to the Northwind Ridge, and driftedalong similar trajectories across the Canada Basin (Figure1a) over the same seasons, separated by five years. Mea-surements have a vertical resolution of about 25 cm, anda horizontal profile spacing of a few kilometers; full pro-cessing procedures are given by Krishfield et al. (2008).ITP data have been analyzed in several past studies of dou-ble diffusion in the Canada Basin (e.g., Timmermans et al.2008; Radko et al. 2014; Bebieva and Timmermans 2016).We will examine ITP profiles to compare the observed ver-tical density ratio Rρ in depth regions exhibiting staircasesand intrusions with the critical density ratio Rcr

ρ computedfor the observed vertical and horizontal background tem-perature gradients and Rρ .

a. Quantifying the temperature and salinity gradients

Before applying the theory to the observations, we re-quire some method to assess the bulk temperature andsalinity gradients (Tz, Sz and Tx, Sx) applicable for a givenITP profile.

Vertical gradients. To compute Tz and Sz (and thereforeRρ ), we consider the depth range between a shallow boundof the AW layer (in the range around ∼230–300 m depth,Figure 3, 4) to the depth of the AW temperature maxi-mum around ∼380–410 m. Vertical profiles of Tz are con-structed by first performing a cubic spline with a smooth-ing parameter chosen to smooth the smallest finestructurescales in the profiles. In this smoothed profile, maximain the 2nd vertical derivative of temperature (Tzz) are se-lected as boundaries that allow for segmentation of theprofile into linear gradients Tz. If the estimated Tz withina segment is smaller than a specified threshold (taken tobe 0.001◦C/m), this segment is merged with the adjacent

deeper segment, and Tz is recomputed. This process is re-peated until either Tz is greater than 0.001◦C/m or a depthsegment extends to the depth level of the AW tempera-ture maximum. The final distribution of segments overa profile is insensitive to the choice of threshold gradientfor thresholds in the range 0.001 – 0.003 ◦C/m. Giventhe tight relationship between temperature and salinity,the same depth segments can be used to estimate verti-cal salinity gradients Sz. In computing Rρ , α and β areaveraged over a given depth segment. This segmentationmethod is optimal for the explicit purpose of returning alinear gradient for each different region of the profile.

Horizontal gradients. Considering the set of profilessampled by the ITP across the Canada Basin, Tx is esti-mated along isopycnals lying within selected depth seg-ments (determined for the profile of interest). The lateraltemperature gradient Tx is computed using standard lin-ear regression of the mean temperature within the rangeof isopycnals (in a given depth segment) versus distancefrom the most western profile near the Northwind Ridgeand in close proximity to the warm Atlantic Water bound-ary current (the general spatial gradient is clear in Figure1a).

b. Application of the theory

A selection of representative profiles are chosen herethat allow us to compare and contrast a variety of scenar-ios. First, we consider a profile taken by ITP 1 on June17, 2006 (Figure 3a), and a profile taken by ITP 41 onJune 17, 2011 (Figure 3d), where both profiles are in closeproximity (black and grey circles, respectively in Figure1a). For each profile, Rρ and Rcr

ρ are computed using esti-mated Tz, Sz and Tx (further taking KT = 10−6 m2s−1 andA = 10−8 m2s−1). The ITP 1 profile shows a robust stair-case around 250–300 m that is disrupted (in depth) by twothick apparently “mixed” layers characterized by a weakpositive temperature gradient (temperature decreases withdepth). This is a characteristic feature of the SF portionof an intrusion; there is evidence for weak salinity gradi-ents also decreasing with depth, however salinity changesover the SF portion (about 0.01%) are much smaller thantemperature changes (about 5%). This structure is consis-tent with Rρ < Rcr

ρ except in the depth range ∼270–290 mwhere Rρ >Rcr

ρ , consistent with the presence of intrusions.In the deeper portions of the profile above the AW temper-ature maximum (∼300–380 m) relatively strong reversedtemperature gradients associated with intrusions are ob-served (Figure 3a inset). This is consistent with Rρ > Rcr

ρ

in this depth range. The portions showing intrusions arecharacterized by slightly higher Rρ compared to the stair-case regions. However Rcr

ρ also varies with depth, indicat-ing that there is not a unique value of Rcr

ρ that determines

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whether staircases or intrusions will form, rather Rcrρ de-

pends upon the background vertical and horizontal gradi-ents.

Detailed examination of the ITP 41 temperature profile(Figure 3d) illustrates that the upper part of the AW layerconsists only of a staircase, as does the deep portion in-dicating thicker mixed layers around ∼340–410 m whichdo not show the temperature gradient reversals character-istic of intrusions. The entire profile showing a staircasestructure is consistent with Rρ < Rcr

ρ throughout the depthrange ∼300–410 m (Figure 3f). The major differencesbetween the ITP 1 (2006) and ITP 41 (2011) profiles isthat the later year was characterized by a weaker verti-cal background salinity gradient and a stronger verticalbackground temperature gradient (both of which resultedin smaller Rρ in 2011 compared to 2006). Referring tothe general relationships derived in the previous section(see Figure 2b) suggests the 2011 profile is more likely toshow staircases. Tx estimated over the basin in 2010–2011is about 4 times weaker than Tx in 2005–2006 throughoutthe depth range of interest (Figure 3b,e). However, ourlinear stability analysis has shown that Tx only has a rolein setting the aspect ratio of the initial interleaving motion(probably not representative of the scales of the final stepstructure), rather than the final response of a system to aperturbation.

Although the profiles considered in Figure 3 are gen-erally representative of the Canada Basin during each of2005-06 and 2010-11, we find that in the western surveyregion in the former period only a staircase is observed insome profiles, while in the eastern survey region in the lat-ter period, both staircases and intrusions are observed insome profiles. We next compare and contrast these casesin each year.

A profile from the western part of the ITP 1 drift track(taken on August 31, 2005) exhibits predominantly stair-cases (Figure 4a), by contrast with the ITP 1 profile to theeast (Figure 3a). This is likely because Tz in this region islarger compared to Tz in the east in that same year, whileSz is does not change between the two regions. Thus, Rρ

is lower for the western profile, and the observed staircasehere is consistent with the smaller Rρ . This lateral distri-bution of the vertical gradients is consistent with a pulse ofanomalously warm AW into the basin in the early to mid2000s (e.g., see Figure 2 in Polyakov et al. 2011). Thatis, the warmest waters are located close to the NorthwindRidge providing a strong vertical temperature gradient (atthe top boundary of the AW layer) in the western part ofthe basin.

A profile from the eastern part of the ITP 41 drift track(taken on December 31, 2011) exhibits weak intrusions inthe deeper regions (Figure 4d), by contrast with the ITP 41profile to the west that showed only a staircase (Figure3d). At depths where intrusions are observed in the east-ern profile, the magnitude of Tz is lower by a factor of 3,

while Sz is lower by a factor of 1.5 compared to the val-ues for the ITP 41 profile to the west. The net effect ofthese salinity and temperature changes is an overall largerRρ in the deep regions, where Rρ ∼ 4 in the west in 2011and Rρ ∼ 9.5 in the east in 2011. Rcr

ρ is also larger for theeastern profile but remains below Rρ consistent with theobserved intrusions. The weaker Tz and Sz to the east isconsistent with a general cooling and salinity diffusion ofthe AW core boundary current during its cyclonic transitaround the basin.

The horizontal and vertical wavelengths of the mostunstable modes are of the order of 1000–10,000 km and15–50 m respectively in all regions surveyed and in bothtime periods (i.e., 2005-06 and 2010-11). Similar scalesof the most unstable modes leading to either a staircaseor intrusions suggest that these two end states are of thesame nature. Of course the linear theory cannot predictthe transient evolution of the linear profile (e.g., how in-terleaving motions evolve, and the associated scale adjust-ments such as layer splitting/merging (Radko et al. 2014));this requires a separate analysis (as performed, for exam-ple by Walsh and Ruddick 1998; Li and McDougall 2015,for the SF case). The time scale of the instability (deter-mined from the fastest growing mode) ranges from aboutone year for intrusions to several years for a staircase. Asmaller magnitude of the horizontal temperature gradientshallower in the water column (e.g., Figure 4b) leads toan approximately order of magnitude decrease in λ (i.e.,an order of magnitude increase in time for a linear pro-file to develop staircases). Furthermore, at these shallowerdepths, it may be that other mechanisms (possibly moreone-dimensional in nature) dominate staircase formationand evolution.

In the absence of a continuous time series of temper-ature and salinity at a given location, the evolution of aprofile towards a staircase or intrusions in response to achanging background state is difficult to infer. (As wehave shown here, detailed knowledge of the backgroundhorizontal gradients is less essential.) Nevertheless, theabove analysis serves as a consistency argument betweenthe predictions of linear theory and observations. Withinthis analysis is the implicit assumption that the develop-ment of intrusions or a staircase does not affect the mag-nitudes of the background gradient (i.e., that the back-ground gradients do not evolve on timescales faster thanthe fastest growing mode). An order-of-magnitude esti-mate for the timescale on which double-diffusive fluxesmodify the background temperature gradients can be esti-mated as L2/KT , where L is a characteristic length scalefor the background gradient. Taking L ∼ 30 m for a ver-tical scale (a reasonable estimate over which linear gradi-ents have been fit to the typical profiles), and effective dif-fusivity KT = 10−6 m2s−1, yields timescales of decadesfor modification of the vertical background temperature

8

gradient. Similarly, considering an effective vertical dif-fusivity for salinity of around 10−7 m2s−1 (e.g., Bebievaand Timmermans 2016), we find timescales for modifi-cation of the vertical background salinity gradient to bean order of magnitude longer than this. Following thesame reasoning for the evolution of the lateral gradients(over O(100) km horizontal scales), and taking isopyc-nal diffusivities in the range 5 - 50 m2s−1 (Hebert et al.1990; Walsh and Carmack 2003), also yields timescalesof decades for the lateral gradients to evolve. Compari-son of these timescale estimates to the maximal growthrates found here (i.e., one to several years) suggests back-ground gradients do not evolve on timescales shorter thanthose associated with the development of a perturbationtowards either a staircase or intrusions; thus in this anal-ysis, the background gradients may be approximated asindependent of time.

4. Summary and discussion

We have examined a scenario for the origin of double-diffusive staircases and intrusions that are observed to co-exist in the Arctic Ocean’s AW. A linear stability analysisof the governing equations was performed to determine themost unstable mode for a given horizontal and vertical lin-ear temperature and salinity stratification that would leadtowards either staircases or intrusions. Staircases are theend result of a perturbation if the observed vertical densityratio is below a critical vertical density ratio (Rρ < Rcr

ρ ),and intrusions are expected to form otherwise (Rρ > Rcr

ρ ).The poor constraints on estimates of KT and A precludeany predictive capacity for our formalism; the theoreticalresults cannot be used to make viable predictions based onthe relative values of Rρ and Rcr

ρ . Within the uncertaintieson A and KT , a wide range of Rcr

ρ is possible. For example,over possible ranges for A (10−8 to 10−7 m2s−1) and KT(10−6 to 10−5 m2s−1), Rcr

ρ changes by a factor of 3. Nev-ertheless, application of our theoretical formalism to ob-servations serves as a consistency assertion, and we haveshown that the linear theory is consistent with the observa-tions for some reasonable choice of A and KT . We do notimplement a thorough statistical analysis of all ITP mea-surements here, but rather provide observational examplesto demonstrate the possibility of staircase formation in aDC background stratification subject to interleaving per-turbations. Future analyses are required to explore the pa-rameter space further, either numerically or in a laboratorysetting.

We have shown that the dominant factors that determinethe presence of either a staircase or intrusions are Tz, Szand Rρ , with lateral temperature gradients having little in-fluence. In general, we expect staircases in regions of rel-atively strong Tz (and small Rρ ) and intrusions where Tzis weaker. Consider, for example, an influx of warm AWin the water column. This would give rise to a stronger

Tz at the top boundary of the AW, and a weaker Tz aboveand close to the AW temperature maximum (i.e., withinan approximately homogeneous core). Such a modifica-tion of the background state would result in the formationof staircases at the AW top boundary and intrusions in theunderlying portions. This stratification allows for verticalas well as lateral heat transfer via the intrusions. If overtime the background temperature gradient in the deeperportion increases, this new Tz may be susceptible to a stair-case end state. One potential mechanism for increased Tzcould be that intrusions distribute heat more effectivelydownward via SF fluxes, rather than upward via DC fluxes(e.g., see heat flux estimates in Bebieva and Timmermans2016), although the overall effect of intrusive fluxes on thebackground gradient is unclear. Formation of a staircase inplace of intrusions would lead to reduced lateral transportof heat, and vertical fluxes would dominate heat transfer.In this respect, the lateral and vertical transfer of AW heatin the Arctic Ocean is dictated by the interplay betweentimescales for intrusive fluxes to modify the backgroundgradients and the lateral supply of AW heat. Further in-vestigation is required to quantify these timescales.

For the SF configuration, Merryfield (2000) showed thatstaircases are the end result of a perturbation when the ver-tical density ratio (defined in the conventional way for theSF configuration, inverse to the definition given here) issmall, while intrusions are the end state when that densityratio takes larger values. We have used the same formal-ism here to demonstrate that an analogous result is appli-cable to the DC case. That is, a staircase is a possible endstate of an interleaving perturbation. Merryfield (2000)showed that a staircase is more likely for smaller valuesof the effective diffusivity for salt. By contrast, we haveshown that (for the DC configuration) a staircase is morelikely for larger values of effective eddy diffusivity for heatKT , although the physics of these relationships needs to beexplained.

While the linear stability analysis and separation ofscales framework described here is instructive for un-derstanding the general structure of a water-column pro-file (i.e., staircases or intrusions), the observations oftendemonstrate a detailed finestructure that is somewhat morecomplicated. This analysis cannot describe layers in theobservations that are observed between and within themain staircase mixed layer and intrusive structures (e.g.,the smallest-scale temperature finestructure in the insetsof Figure 4a). These features, however, may be key to thetransition between interleaving structures and staircases –a conjecture that will require further analysis.

Acknowledgments. The Ice-Tethered Profiler data werecollected and made available by the Ice-Tethered ProfilerProgram based at the Woods Hole Oceanographic Institu-tion (Krishfield et al. 2008; Toole et al. 2011); data areavailable at http://www.whoi.edu/itp/data. Funding was

9

provided by the National Science Foundation Division ofPolar Programs under award number 1350046.

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Potential Temperature [°C]

100

200

300

400

500

600

700

100

200

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700

-2 -1.5 -1 -0.5 0 0.5 1

Dep

th [m

]

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Salinity

ITP 1

ITP 41

ITP 1

ITP 41

310

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390

310

330

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-0.2 0 0.2 0.4

0.7 0.8

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34.4 34.6

34.76 34.82

b)c)

0.5

0.6

Rus

sia

AlaskaCanada

Greenland

North

wind

ridge

AW

Tem

pera

ture

max

imum

[°C

]

a)

0.7

0.8

0.9

1

ITP 1

Augu

st 1

6, 2

005

Sept

embe

r 26,

200

6

ITP 41

Octo

ber 3

, 201

0

Janu

ary

14, 2

012

FIG. 1: a) Map showing locations of ITP 1 and ITP 41 profiles over the course of their drift;colors indicate the AW potential temperature maximum (◦C). b) Potential temperature (◦C,referenced to the surface) and c) salinity profiles measured by ITP 1 (black lines, 17 June2006) and ITP 41 (grey lines, 17 June 2011); locations where both profiles were sampledare marked by circles with corresponding colors on the map in a). The expanded scalesshow the double-diffusive staircase and intrusions for the ITP 1 profile. Diamonds markthe locations of two other profiles examined here, Figure 4.

12

0

0

I

S

R ρ=1

Rρ→∞

DC

SF β

S zConvectively unstable

Stable

αTz

a)

Rρcr

λmax x10-7 s-1

Rρ

Sta

irca

seIn

trus

ions

b)

Tz

[°C

/m]

-0.002

-0.006

-0.01

-0.014

-0.018

108642

3.6

4.6

6.5

10.8

32.3

Tx x 10-7 [°C/m]

FIG. 2: a) Schematic showing evolution of temperature and salinity according to (15) and(16) (following Merryfield 2002). Dashed lines are constant Rρ contours, bounded by twosolid lines where Rρ = 1 and Rρ →∞. Blue lines show evolution of the initial perturbationfrom a background state (red dot, characterized by Rρ) towards either a staircase (S) orintrusions (I). There are four regimes shown on the diagram depending upon values of(αTz,βSz): DC-unstable, convectively unstable, SF-unstable and stable stratification. b)Solutions to (27) for λmax (black curves) and Rcr

ρ (red curves) as a function of Tx and Tz,where the range of temperature gradients approximately correspond to observed values.Values A and KT are as given in the text and the background salinity gradient is heldfixed for the calculations (Sz = −0.002m−1). The approximate delineation between whenstaircases are expected and when intrusions are expected (based on values of Rρ and Rcr

ρ )is shown by a wavy line.

13

RρRρcr

RρRρcr

0 0.2 0.4 0.6 0.8 3.5 4 4.5 5 5.5

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]D

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d) e) f)

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4100.65 0.75

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4102.5 3 3.5 4 4.5 5

Staircase

Staircase

Intrusions

Intrusions

Staircase

RρPotential Temperature [°C]

RρPotential Temperature [°C]

269

272

275

0.365 0.368

310

330

350

0.7 0.76 0.82

34.81434.811

0 2 4 6 8Tx x 10-7 [°C/m]

ITP 1

0 2 4 6 8Tx x 10-7 [°C/m]

ITP 41

FIG. 3: a) Potential temperature (◦C, referenced to the surface) profile measured by ITP 1in the Canada Basin at the location of the black circle in Figure 1a. Red lines overlainindicate a piece-wise linear approximation to the profile. Zoomed-in potential temperatureprofiles are shown in the insets, with salinity shown by a blue line (bottom inset) withthe scale above. b) Lateral background temperature gradient (10−7 ◦C/m) for each depthsegment shown in a). c) Corresponding profiles of Rcr

ρ and Rρ (see text). d), e) and f) asfor a), b) and c) respectively but for a profile taken by ITP 41 (grey circle in Figure 1a).

14D

epth

[m]

0.2 0.4 0.6 0.8

Potential Temperature [°C]

0 2 4 6 8 4 5 6 7 8 9 10

Rρ

RρRρcr

Sta

irca

seIn

trus

ions

260

280

300

320

340

360

310

330

350

370

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410

240

a) b) c)

d) e) f)

0.2 0.4 0.6 0.8 1 2.5 3 3.5 4 4.5 5

RρRρcr

Dep

th [m

]

0 2 4 6 8Potential Temperature [°C] Rρ

0.68 0.7

390

400

ITP 41

ITP 1

Sta

irca

se

310

330

350

370

0.8 0.9 1

Tx x 10-7 [°C/m]

Tx x 10-7 [°C/m]

FIG. 4: a) Potential temperature (◦C, referenced to the surface) profile measured by ITP 1in the Canada Basin at the location of the black diamond in Figure 1a. Red lines overlainindicate a piece-wise linear approximation to the profile. b) Lateral background tempera-ture gradient (10−7 ◦C/m) for each depth segment shown in a). c) Corresponding profilesof Rcr

ρ and Rρ . d), e) and f) as for a), b) and c) respectively but for a profile taken by ITP 41(grey diamond in Figure 1a).