An idealized model of Weddell Gyre export variability Zhan Su ∗ Andrew L. Stewart and Andrew F ...

50 Pages·2013·1.2 MB·English

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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
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Recent observations indicate that the export of Antarctic Bottom Water (AABW) from the
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Weddell Sea has a seasonal cycle in its temperature and salinity that is correlated with an-
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nualwind stress variations. This variability has been attributed to annual vertical excursions
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of the isopycnals in the Weddell Gyre, modifying the water properties at the depth of the
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Orkney Passage. Baroclinic adjustment of the wind-driven gyre has previously been invoked
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to explain these variations, but such adjustment typically requires several years due to the
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slow propagation speed of baroclinic Rossby waves. Here we explore an alternative mecha-
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nism in which the isopycnals respond directly to surface Ekman pumping and counteracting
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mesoscale eddy buoyancy fluxes.
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A conceptual model of the boundary that separates Weddell Sea Deep Water from Cir-
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cumpolar Deep Water is described in which the bounding isopycnal responds to a seasonal
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oscillation in the surface wind stress. Different parametrizations of the mesoscale eddy dif-
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fusivity are tested. The model accurately predicts the observed phases of the temperature
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and salinity variability in relationship to the surface wind stress. The model, despite its
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heavy idealization, also accounts for more than 50% of the observed oscillation amplitude,
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which depends on the strength of the seasonal wind variability and the parametrized eddy
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diffusivity. These results highlight the importance of mesoscale eddies in modulating the
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export of AABW in narrow boundary layers around the Antarctic margins.
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1
1. Introduction
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Observations indicate that the export of AABW from the Weddell Sea has a seasonal
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cycle in temperature and salinity (Gordon et al. 2010; McKee et al. 2011). Evidence of a
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link to surface wind forcing is given by Jullion et al. (2010) who found that Weddell Sea
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Deep Water (WSDW) properties in the Scotia Sea correlate with wind stress variations over
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the Weddell Gyre, with a phase lag of five months. Meredith et al. (2011) similarly find
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that temperature anomalies of WSDW in the Scotia Sea, and at the entrance of the Orkney
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Passage, lag the surface wind stress by 2 to 4 months. A leading candidate for this variability
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proposes that baroclinic variability of the gyre circulation, induced by changes in the surface
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wind stress, modify the outcrop position of isopycnals at the northern boundary of the gyre
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(e.g. Meredith et al. 2008; Coles et al. 1996). The dynamics responsible for this observed
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time-lag have not been explored.
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Here we focus on the transit of WSDW through the the Weddell Gyre (Figure 1), an
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important choke point of the global circulation. A major component of Antarctic Bottom
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Water (AABW), which ventilates the deep ocean, originates as WSDW, which we define
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as having a neutral density greater than γn 28.26kgm−3. WSDW circulates cyclonically
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∼
around the Weddell Gyre (Figure 1b) (Deacon 1979) with the strongest velocities found
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within narrow boundary currents (Figure 1d). As this boundary current intersects the South
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Scotia Ridge, WSDW may flow through deep passages and enter the Scotia Sea (Locarnini
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et al. 1993). Naveira Garabato et al. (2002) measure a WSDW transport of 6.7 1.7Sv
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±
through the South Scotia Ridge. The majority of this outflow, around 4 to 6Sv of WSDW
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colder than 0◦C, traverses the Orkney Passage and merges with the Antarctic Circumpolar
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Current (ACC) (Meredith et al. 2008; Naveira Garabato et al. 2002).
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An intuitive explanation for the correlation between surface winds and WSDW export is
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baroclinic adjustment of the Weddell Gyre. At large scales, the circulation of the Weddell
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Gyre is consistent with Sverdrup balance: the negative surface wind stress curl leads to a
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southward transport in the gyre interior that is balanced by a northward return flow in a
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2
western boundary current (Gordon et al. 1981; Muench and Gordon 1995). Radiation of
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Rossby waves is the principle mechanism via which a gyre-like flow responds to changes in
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wind forcing (Anderson and Gill 1975). The barotropic component of the gyre circulation
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can adjust over a time scale of a few days, but the baroclinic component requires several
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years because the barotropic and baroclinic Rossby wave speeds differ by 3–4 orders of
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magnitude. Thus baroclinic adjustment of the gyre occurs much too slowly to explain the
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annual variations of WSDW outflow properties.
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Here we explore an alternative mechanism that is motivated by recent work indicating
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that mesoscale eddies make a leading contribution to exchanges of mass, heat and salt across
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the Antarctic shelf break (Nøst et al. 2011; Dinniman et al. 2011; Stewart and Thompson
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2013). Anon-zerowindstresscurlovertheWeddellGyregeneratesverticalvelocitiesthrough
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divergence and convergence of surface Ekman transport that leads to vertical excursions of
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the interior isopycnals. Mesoscale eddies develop through baroclinic instability in order to
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extractthepotentialenergystoredinthetiltedisopycnals. Totestthishypothesiswedevelop
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aconceptualmodeloftheisopycnalseparatingWSDWfromtheoverlyingCircumpolarDeep
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Water (CDW) in the Weddell Sea. Furthermore, we cast this balance between wind-induced
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and eddy-induced circulation in the framework of residual-mean theory (RMT) (Andrews
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et al. 1987; Plumb and Ferrari 2005).
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RMT has been an important tool in understanding the principal balances in the ACC’s
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upper overturning cell (Marshall and Radko 2003), and has recently been extended to flows
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around the Antarctic margins (Stewart and Thompson 2013). This model moves away
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fromthetraditionalpicturethatcross-shelfexchangerequireslarge-scalealong-shelfpressure
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gradients (Ou 2007). Instead, we consider along-stream, or tangentially- averaged properties
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along the boundary of the Weddell Gyre. RMT is used to describe the evolution of the mean
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isopycnals in response to annual variations of the wind stress. This approach is motivated
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in part by the striking isopycnal tilt at the gyre boundary (Meredith et al. 2008).
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In 2 we describe our idealized domain and forcing and in 3 we derive a residual-mean
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§ §
3
model for the isopycnal bounding WSDW in the Weddell Gyre. In 4 we solve the evolution
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§
equation for the bounding isopycnal and discuss its sensitivity to wind stress and eddy
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diffusivity. In 5 we extend the model to include a representation of WSDW inflow to and
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§
outflow from the gyre. In 6, we compare our model predictions with observations and
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§
discuss the limitations and implications of our model. We draw conclusions in 7.
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§
2. An idealized Weddell Gyre
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Our approach adopts an idealized version of the Weddell Gyre that captures key aspects
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of the physics controlling isopycnal variability. Figure 2 shows a schematic of our concep-
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tual model. The gyre is assumed to be circular and azimuthally uniform with an applied
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azimuthally-uniform surface wind stress. This geometry motivates a description in terms of
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cylindrical coordinate (r,θ,z), where r = 0 at the gyre center, r = R = 680km at the gyre
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edge, and θ is anticyclonic.
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To explore the properties of WSDW exported from the edge of the Weddell Gyre, the
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model focuses on the evolution of an isopycnal that represents the division between WSDW
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and CDW (Figure 1). The model solves for the isopycnal’s azimuthal-mean position z =
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η(r,t) as a function of radius r and time t. The position at which this isopycnal outcrops
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from the bathymetry is denoted as r = r (t).
93 b
a. Bathymetry
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The idealized bathymetry is derived from the NOAA ETOPO1 data (Amante and Eakins
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2009) shown in Figure 1(b). The 1 km depth contour (magenta curve) defines the southern
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and western boundary of the Weddell Gyre. At the northern edge of the gyre the 1 km
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isobath is discontinuous and we use a straight line to approximate the boundary.
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We construct the model bathymetry as an average of 75 evenly-spaced sections that
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extend perpendicularly from the shelf break into the gyre interior (not shown). The sections
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4
are chosen to be 680km long so that they meet approximately in the gyre center. This
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procedure produces an accurate representation of the bathymetry, especially around the
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gyre edge (Figure 3). A simple polynomial approximation is also provided by the solid curve
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in Figure 3:
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η (r) = 4.54 103m+(1.85 10−26m−4)r5. (1)
b
− × ×
b. Azimuthal winds
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Wind stress data from the CORE.2 Global Air-Sea Flux Dataset (Large and Yeager
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2009), available from 1949 to 2006 with a monthly frequency and 1◦ resolution, is used to
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compute mean zonal and meridional wind stress along with the mean wind stress curl in
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Figure 4. The wind stress curl is almost uniformly negative over the gyre, and is particularly
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strong ( 2 10−7 Nm−3) close to the boundary.
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∼ ×
Thecomponentofthesurfacewindstressperpendiculartothe75sectionsdescribedin 2a
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§
are averaged to produce the mean wind stress tangential to the gyre boundary. Figure 4(d)
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shows the amplitudes of each Fourier mode of the azimuthal-mean wind stress at the shelf
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break τ(r = R,t). Negative values correspond to cyclonic wind stress. With the exception of
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the steady mode, whose amplitude is 0.073Nm−2, only the annual and semiannual modes
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−
are pronounced at the shelf break, having amplitudes of 0.029Nm−2 and 0.011Nm−2
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− −
respectively. Therelativeamplitudesofthemodesaresimilaratallradiifromthegyrecenter.
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For simplicity we include only the annual mode in our conceptual model, and neglect all
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other modes. Numerical experiments show that the semiannual mode can modify isopycnal
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excursions at the gyre edge by 10%–20%; this is discussed further in 6a.
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§
Figure 4(e) shows the radial variation in the amplitudes of the steady and annual az-
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imuthal wind stress modes. Both modes strengthen linearly from the gyre interior to the
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gyre boundary. The conceptual model uses linear fits to both the steady mode τ(r) = τ0r/R
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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.
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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
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zero, so for simplicity we approximate φ 300◦ as a constant. Thus our expression for the
129 12
≡
azimuthal wind stress is
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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
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with maximum amplitude at t = 5months, i.e., at the end of May. This wind pattern is
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consistent with previous observations (see e.g. Figure 1(d) of Wang et al. 2012).
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3. Residual-mean dynamics
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We now derive an evolution equation for the isopycnal z = η(r,t) using RMT. Our
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formulation is similar to that of Marshall and Radko (2003), except it is cast in terms of an
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azimuthal average of the buoyancy around our idealized Weddell Gyre. Our model describes
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the evolution of a single isopycnal surface γn = 28.26kgm−3, though it may be applied to
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any isopycnal in the Weddell Gyre.
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FollowingMarshallandRadko(2003),theazimuthally-averagedbuoyancymaybewritten
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as
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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
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h•i •
steamfunction ψ† describes the advectingHtwo-dimensional velocity field in the (r,z) plane,
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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
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h i
an eddy component ψ⋆,
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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
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ψ isrelatedtothesurfacewindstressusingtheazimuthally-averagedazimuthalmomentum
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h i
equation in the limit of small Rossby number. The same physical reasoning applies to both
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the ACC and the Weddell Gyre: in the zonal (azimuthal) mean the ACC (Weddell Gyre)
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cannot support a net zonal (azimuthal) pressure gradient, and thus no mean geostrophic
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meridional (radial) flow in the interior, so mean Ekman pumping driven by zonal (cyclonic)
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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.
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The eddy streamfunction arises from a downgradient eddy buoyancy flux closure (Gent and
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McWilliams 1990), where κ is the eddy buoyancy diffusivity and s = b / b is the
159 b −h ir h iz
isopycnal slope.
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As the azimuthal-mean isopycnal z = η(r,t) is a material surface in the sense of the
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residual advective derivative D†/Dt ∂ +u† , it may be shown (see Appendix A) that
162 t
≡ ·∇
it evolves according to
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∂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.
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Inserting (5) into (6) and noting s = ∂η/∂r, we obtain a forced-diffusive evolution equation
166 b
for the isopycnal height η,
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∂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.
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We impose no-flux boundary conditions (ψ† = 0) at the gyre center r = 0 and at the
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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
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∂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.
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As we have not prescribed any inflow nor outflow of WSDW at this point, our model
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conserves the total mass M beneath the isopycnal z = η(r,t),
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dM rb(t)
= 0, M = 2πrρ (η η )dr. (11)
0 b
dt −
Z0
8
4. The model solution
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a. Analytical solution in cylindrical basin
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In this section we solve the model evolution equation (7) analytically in a simplified case.
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The solution serves as a scaling for the isopycnal’s response to the annually-varying surface
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wind stress, and provides an intuitive interpretation of our later results.
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Figure 1(c) shows that the WSDW isopycnal outcrops at the steepest part of the bathy-
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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′.
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Taking the time average of (7) yields
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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