Generated using version 3.2 of the official AMS LATEX template An idealized model of Weddell Gyre export variability 1 Zhan Su 2 ∗ Geological and Planetary Sciences, California Institute of Technology, Pasadena, California Andrew L. Stewart and Andrew F. Thompson 3 Environmental Sciences and Engineering, California Institute of Technology, Pasadena, California ∗author address: Zhan Su, Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125 E-mail: [email protected] 1 ABSTRACT 4 Recent observations indicate that the export of Antarctic Bottom Water (AABW) from the 5 Weddell Sea has a seasonal cycle in its temperature and salinity that is correlated with an- 6 nualwind stress variations. This variability has been attributed to annual vertical excursions 7 of the isopycnals in the Weddell Gyre, modifying the water properties at the depth of the 8 Orkney Passage. Baroclinic adjustment of the wind-driven gyre has previously been invoked 9 to explain these variations, but such adjustment typically requires several years due to the 10 slow propagation speed of baroclinic Rossby waves. Here we explore an alternative mecha- 11 nism in which the isopycnals respond directly to surface Ekman pumping and counteracting 12 mesoscale eddy buoyancy fluxes. 13 A conceptual model of the boundary that separates Weddell Sea Deep Water from Cir- 14 cumpolar Deep Water is described in which the bounding isopycnal responds to a seasonal 15 oscillation in the surface wind stress. Different parametrizations of the mesoscale eddy dif- 16 fusivity are tested. The model accurately predicts the observed phases of the temperature 17 and salinity variability in relationship to the surface wind stress. The model, despite its 18 heavy idealization, also accounts for more than 50% of the observed oscillation amplitude, 19 which depends on the strength of the seasonal wind variability and the parametrized eddy 20 diffusivity. These results highlight the importance of mesoscale eddies in modulating the 21 export of AABW in narrow boundary layers around the Antarctic margins. 22 1 1. Introduction 23 Observations indicate that the export of AABW from the Weddell Sea has a seasonal 24 cycle in temperature and salinity (Gordon et al. 2010; McKee et al. 2011). Evidence of a 25 link to surface wind forcing is given by Jullion et al. (2010) who found that Weddell Sea 26 Deep Water (WSDW) properties in the Scotia Sea correlate with wind stress variations over 27 the Weddell Gyre, with a phase lag of five months. Meredith et al. (2011) similarly find 28 that temperature anomalies of WSDW in the Scotia Sea, and at the entrance of the Orkney 29 Passage, lag the surface wind stress by 2 to 4 months. A leading candidate for this variability 30 proposes that baroclinic variability of the gyre circulation, induced by changes in the surface 31 wind stress, modify the outcrop position of isopycnals at the northern boundary of the gyre 32 (e.g. Meredith et al. 2008; Coles et al. 1996). The dynamics responsible for this observed 33 time-lag have not been explored. 34 Here we focus on the transit of WSDW through the the Weddell Gyre (Figure 1), an 35 important choke point of the global circulation. A major component of Antarctic Bottom 36 Water (AABW), which ventilates the deep ocean, originates as WSDW, which we define 37 as having a neutral density greater than γn 28.26kgm−3. WSDW circulates cyclonically 38 ∼ around the Weddell Gyre (Figure 1b) (Deacon 1979) with the strongest velocities found 39 within narrow boundary currents (Figure 1d). As this boundary current intersects the South 40 Scotia Ridge, WSDW may flow through deep passages and enter the Scotia Sea (Locarnini 41 et al. 1993). Naveira Garabato et al. (2002) measure a WSDW transport of 6.7 1.7Sv 42 ± through the South Scotia Ridge. The majority of this outflow, around 4 to 6Sv of WSDW 43 colder than 0◦C, traverses the Orkney Passage and merges with the Antarctic Circumpolar 44 Current (ACC) (Meredith et al. 2008; Naveira Garabato et al. 2002). 45 An intuitive explanation for the correlation between surface winds and WSDW export is 46 baroclinic adjustment of the Weddell Gyre. At large scales, the circulation of the Weddell 47 Gyre is consistent with Sverdrup balance: the negative surface wind stress curl leads to a 48 southward transport in the gyre interior that is balanced by a northward return flow in a 49 2 western boundary current (Gordon et al. 1981; Muench and Gordon 1995). Radiation of 50 Rossby waves is the principle mechanism via which a gyre-like flow responds to changes in 51 wind forcing (Anderson and Gill 1975). The barotropic component of the gyre circulation 52 can adjust over a time scale of a few days, but the baroclinic component requires several 53 years because the barotropic and baroclinic Rossby wave speeds differ by 3–4 orders of 54 magnitude. Thus baroclinic adjustment of the gyre occurs much too slowly to explain the 55 annual variations of WSDW outflow properties. 56 Here we explore an alternative mechanism that is motivated by recent work indicating 57 that mesoscale eddies make a leading contribution to exchanges of mass, heat and salt across 58 the Antarctic shelf break (Nøst et al. 2011; Dinniman et al. 2011; Stewart and Thompson 59 2013). Anon-zerowindstresscurlovertheWeddellGyregeneratesverticalvelocitiesthrough 60 divergence and convergence of surface Ekman transport that leads to vertical excursions of 61 the interior isopycnals. Mesoscale eddies develop through baroclinic instability in order to 62 extractthepotentialenergystoredinthetiltedisopycnals. Totestthishypothesiswedevelop 63 aconceptualmodeloftheisopycnalseparatingWSDWfromtheoverlyingCircumpolarDeep 64 Water (CDW) in the Weddell Sea. Furthermore, we cast this balance between wind-induced 65 and eddy-induced circulation in the framework of residual-mean theory (RMT) (Andrews 66 et al. 1987; Plumb and Ferrari 2005). 67 RMT has been an important tool in understanding the principal balances in the ACC’s 68 upper overturning cell (Marshall and Radko 2003), and has recently been extended to flows 69 around the Antarctic margins (Stewart and Thompson 2013). This model moves away 70 fromthetraditionalpicturethatcross-shelfexchangerequireslarge-scalealong-shelfpressure 71 gradients (Ou 2007). Instead, we consider along-stream, or tangentially- averaged properties 72 along the boundary of the Weddell Gyre. RMT is used to describe the evolution of the mean 73 isopycnals in response to annual variations of the wind stress. This approach is motivated 74 in part by the striking isopycnal tilt at the gyre boundary (Meredith et al. 2008). 75 In 2 we describe our idealized domain and forcing and in 3 we derive a residual-mean 76 § § 3 model for the isopycnal bounding WSDW in the Weddell Gyre. In 4 we solve the evolution 77 § equation for the bounding isopycnal and discuss its sensitivity to wind stress and eddy 78 diffusivity. In 5 we extend the model to include a representation of WSDW inflow to and 79 § outflow from the gyre. In 6, we compare our model predictions with observations and 80 § discuss the limitations and implications of our model. We draw conclusions in 7. 81 § 2. An idealized Weddell Gyre 82 Our approach adopts an idealized version of the Weddell Gyre that captures key aspects 83 of the physics controlling isopycnal variability. Figure 2 shows a schematic of our concep- 84 tual model. The gyre is assumed to be circular and azimuthally uniform with an applied 85 azimuthally-uniform surface wind stress. This geometry motivates a description in terms of 86 cylindrical coordinate (r,θ,z), where r = 0 at the gyre center, r = R = 680km at the gyre 87 edge, and θ is anticyclonic. 88 To explore the properties of WSDW exported from the edge of the Weddell Gyre, the 89 model focuses on the evolution of an isopycnal that represents the division between WSDW 90 and CDW (Figure 1). The model solves for the isopycnal’s azimuthal-mean position z = 91 η(r,t) as a function of radius r and time t. The position at which this isopycnal outcrops 92 from the bathymetry is denoted as r = r (t). 93 b a. Bathymetry 94 The idealized bathymetry is derived from the NOAA ETOPO1 data (Amante and Eakins 95 2009) shown in Figure 1(b). The 1 km depth contour (magenta curve) defines the southern 96 and western boundary of the Weddell Gyre. At the northern edge of the gyre the 1 km 97 isobath is discontinuous and we use a straight line to approximate the boundary. 98 We construct the model bathymetry as an average of 75 evenly-spaced sections that 99 extend perpendicularly from the shelf break into the gyre interior (not shown). The sections 100 4 are chosen to be 680km long so that they meet approximately in the gyre center. This 101 procedure produces an accurate representation of the bathymetry, especially around the 102 gyre edge (Figure 3). A simple polynomial approximation is also provided by the solid curve 103 in Figure 3: 104 η (r) = 4.54 103m+(1.85 10−26m−4)r5. (1) b − × × b. Azimuthal winds 105 Wind stress data from the CORE.2 Global Air-Sea Flux Dataset (Large and Yeager 106 2009), available from 1949 to 2006 with a monthly frequency and 1◦ resolution, is used to 107 compute mean zonal and meridional wind stress along with the mean wind stress curl in 108 Figure 4. The wind stress curl is almost uniformly negative over the gyre, and is particularly 109 strong ( 2 10−7 Nm−3) close to the boundary. 110 ∼ × Thecomponentofthesurfacewindstressperpendiculartothe75sectionsdescribedin 2a 111 § are averaged to produce the mean wind stress tangential to the gyre boundary. Figure 4(d) 112 shows the amplitudes of each Fourier mode of the azimuthal-mean wind stress at the shelf 113 break τ(r = R,t). Negative values correspond to cyclonic wind stress. With the exception of 114 the steady mode, whose amplitude is 0.073Nm−2, only the annual and semiannual modes 115 − are pronounced at the shelf break, having amplitudes of 0.029Nm−2 and 0.011Nm−2 116 − − respectively. Therelativeamplitudesofthemodesaresimilaratallradiifromthegyrecenter. 117 For simplicity we include only the annual mode in our conceptual model, and neglect all 118 other modes. Numerical experiments show that the semiannual mode can modify isopycnal 119 excursions at the gyre edge by 10%–20%; this is discussed further in 6a. 120 § Figure 4(e) shows the radial variation in the amplitudes of the steady and annual az- 121 imuthal wind stress modes. Both modes strengthen linearly from the gyre interior to the 122 gyre boundary. The conceptual model uses linear fits to both the steady mode τ(r) = τ0r/R 123 and the annual mode τ (r) = τ0 r/R, where τ0 = 0.072Nm−2 and τ0 = 0.026Nm−2 are 124 12 12 − 12 − constants. In a circular basin, the azimuthal wind stress must vanish at r = 0 by symmetry. 125 5 Figure 4(f) shows the radial variation of the phase φ of the annual mode, where the annual 126 12 mode is expressed as τ (r) sin(ωt+φ (r)) and ω = 2πyr−1 . The phase φ varies by less 127 12 12 12 · than 35◦ for r > 100km, and for r < 100km the amplitude of the annual mode is close to 128 zero, so for simplicity we approximate φ 300◦ as a constant. Thus our expression for the 129 12 ≡ azimuthal wind stress is 130 τ(r,t) = τ(r)+τ (r)sin(ωt+5π/3), (2a) 12 r r τ(r) = τ0 , τ (r) = τ0 . (2b) R 12 12 R Here t = 0 corresponds to the start of January, so the model wind field has an annual cycle 131 with maximum amplitude at t = 5months, i.e., at the end of May. This wind pattern is 132 consistent with previous observations (see e.g. Figure 1(d) of Wang et al. 2012). 133 3. Residual-mean dynamics 134 We now derive an evolution equation for the isopycnal z = η(r,t) using RMT. Our 135 formulation is similar to that of Marshall and Radko (2003), except it is cast in terms of an 136 azimuthal average of the buoyancy around our idealized Weddell Gyre. Our model describes 137 the evolution of a single isopycnal surface γn = 28.26kgm−3, though it may be applied to 138 any isopycnal in the Weddell Gyre. 139 FollowingMarshallandRadko(2003),theazimuthally-averagedbuoyancymaybewritten 140 as 141 b +J(ψ†, b ) = 0, (3) h it h i where b is the buoyancy and = (2π)−1 dθ denotes the azimuthal average. The residual 142 h•i • steamfunction ψ† describes the advectingHtwo-dimensional velocity field in the (r,z) plane, 143 defined as u† = u†e +w†e = (ψ†e ) with 144 r z θ ∇× ∂ψ† 1∂rψ† u† = , w† = . (4) − ∂z r ∂r 6 A turbulent diapycnal mixing term (κ b ) has been neglected from the right-hand side of 145 vh iz z (3), where κ is the vertical diffusivity. This term may be shown to be dynamically negligible 146 v in our model; this is discussed in detail in 6b. 147 § The residual streamfunction ψ† is comprised of a mean (wind-driven) component ψ and 148 h i an eddy component ψ⋆, 149 τ ψ† = ψ +ψ⋆, ψ = , ψ⋆ = κs . (5) b h i h i ρ f 0 0 Following a procedure analogous to Marshall and Radko (2003), the mean streamfunction 150 ψ isrelatedtothesurfacewindstressusingtheazimuthally-averagedazimuthalmomentum 151 h i equation in the limit of small Rossby number. The same physical reasoning applies to both 152 the ACC and the Weddell Gyre: in the zonal (azimuthal) mean the ACC (Weddell Gyre) 153 cannot support a net zonal (azimuthal) pressure gradient, and thus no mean geostrophic 154 meridional (radial) flow in the interior, so mean Ekman pumping driven by zonal (cyclonic) 155 surface winds penetrates to depth. Here ρ is the reference density and f is the reference 156 0 0 Coriolis parameter, the latitudinally-varying component of which has been neglected here. 157 The eddy streamfunction arises from a downgradient eddy buoyancy flux closure (Gent and 158 McWilliams 1990), where κ is the eddy buoyancy diffusivity and s = b / b is the 159 b −h ir h iz isopycnal slope. 160 As the azimuthal-mean isopycnal z = η(r,t) is a material surface in the sense of the 161 residual advective derivative D†/Dt ∂ +u† , it may be shown (see Appendix A) that 162 t ≡ ·∇ it evolves according to 163 ∂η(r,t) 1 ∂ = rψ† , 0 < r < r (t). (6) ∂t r∂r z=η(r,t) b ³ ´ ¯ 164 Equation (6) shows that the evolution of¯the isopycnal must be balanced by the radial divergence of ψ†, i.e. the net radial transport between the isopycnal and the ocean bed. 165 Inserting (5) into (6) and noting s = ∂η/∂r, we obtain a forced-diffusive evolution equation 166 b for the isopycnal height η, 167 ∂η 1 ∂ τ ∂η = r +κ , (7) ∂t r∂r ρ f ∂r · 0 0 ¸ 7 for0 < r < r (t). Hereκ = κ(r,t)istheeddybuoyancydiffusivityevaluatedontheisopycnal 168 b z = η(r,t), and the ∂η/∂r is isopycnal slope. 169 We impose no-flux boundary conditions (ψ† = 0) at the gyre center r = 0 and at the 170 isopycnal outcrop r = r (t). The azimuthally-averaged wind stress τ must vanish at r = 0 171 b by symmetry, so from (5) the boundary condition is 172 ∂η ψ† = 0 = = 0. (8a) r=0 ⇒ ∂r ¯r=0 ¯ ¯ ¯ ¯ 173 Similarly, at the isopycnal outcrop r = rb(t) we obtain¯ ∂η τ ψ† = 0 = = . (8b) r=rb ⇒ ∂r¯r=rb − ρ0fκ¯r=rb ¯ ¯ ¯ ¯ ¯ ¯ 174 In addition to (8a) and (8b) we also require an e¯volution equati¯on for the outcrop r = rb(t). By definition, r (t) satisfies 175 b η(r (t),t) = η (r (t)). (9) b b b Taking the time derivative of (9) and rearranging, we obtain an evolution equation for r (t) 176 b ∂η dr b ∂t = . (10) dt dη ∂η¯ b ¯ ¯ dr − ∂r ¯r=rb ¯ ¯ 177 Note that (10) does not add information to the evolu¯tion equation (7) and Neumann bound- ary conditions (8a)–(8b), but in practice it must be solved separately to track the position 178 of the outcrop. 179 As we have not prescribed any inflow nor outflow of WSDW at this point, our model 180 conserves the total mass M beneath the isopycnal z = η(r,t), 181 dM rb(t) = 0, M = 2πrρ (η η )dr. (11) 0 b dt − Z0 8 4. The model solution 182 a. Analytical solution in cylindrical basin 183 In this section we solve the model evolution equation (7) analytically in a simplified case. 184 The solution serves as a scaling for the isopycnal’s response to the annually-varying surface 185 wind stress, and provides an intuitive interpretation of our later results. 186 Figure 1(c) shows that the WSDW isopycnal outcrops at the steepest part of the bathy- 187 metric slope. Our numerical solutions in 4b show that the bathymetric slope at the gyre 188 § edge is an order of magnitude larger than the typical isopycnal slope. We therefore ap- 189 proximate the basin as a cylinder with vertical walls at r R = 680km. Our numerical 190 b ≡ solutions in 4b verify that the change of r is typically very small (< 4km). We choose 191 b § κ = constant = 300m2/s because this yields a range of isopycnal heights that approximately 192 matches the observed range in Figure 1(c). In 4c we examine the model’s sensitivity to the 193 § value of κ. 194 To solve the isopycnal evolution equation (7) we separate η and τ into time-mean com- 195 ponents η and τ, and time-dependent components η′ and τ′, i.e., η = η+η′ and τ = τ +τ′. 196 Taking the time average of (7) yields 197 1 ∂ τ ∂η r +κ = 0, (12) r∂r ρ f ∂r µ 0 0 ¶ where τ = τ0r/R from (2a). This equation defines the time-mean isopycnal position up to a 198 constant, which we choose to be the the basin-averaged isopycnal depth η = 2 Rηrdr/R2. 199 0 0 Solving (12) we obtain the time-mean isopycnal profile, R 200 τ0 η(r) = η + R2 2r2 . (13) 0 4ρ f κR − 0 0 ¡ ¢ The change in isopycnal height ∆η = η(r = 0) η(r = R) = τ0R/(2ρ f κ), depends on 201 0 0 − the amplitude of the mean wind stress component τ0 and the eddy diffusivity κ. Physically, 202 the time-mean isopycnal shape ensures that the time-mean wind-driven and eddy vertical 203 velocities exactly cancel. For ρ = 1000kgm−3 and f = 10−4s−1 we obtain ∆η = 860m, 204 0 0 − 9