Jet Formation and Evolution in Baroclinic Turbulence with Simple Topography

Transcription

1 Jet Formation and Evolution in Baroclinic Turbulence with Simple Topography Andrew F. Thompson Department of Applied Mathematics and Theoretical Physics University of Cambridge, Cambridge, U.K. Submitted to Journal of Physical Oceanography on January 13, 29 corresponding author address: Andrew F. Thompson Department of Applied Mathematics and Theoretical Physics Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 WA United Kingdom 1

2 Abstract Satellite altimetry and high-resolution ocean models indicate that the Southern Ocean is comprised of an intricate web of narrow, meandering jets that undergo spontaneous formation, merger and splitting events, and rapid latitude shifts over periods of weeks to months. The role of topography in controlling jet variability is explored using over 1 simulations from a doubly-periodic, forced-dissipative, two-layer quasigeostrophic model. The system is forced by a baroclinically-unstable, vertically-sheared mean flow in a domain that is large enough to accommodate multiple jets. The dependence of (i) meridional jet spacing, (ii) jet variability and (iii) large-scale, domainaveraged transport properties on changes in the length scale and steepness of simple sinusoidal topographical features is analyzed. The Rhines scale l β = 2π V e /β, where V e is an eddy velocity scale and β is the barotropic potential vorticity gradient, measures the meridional extent of eddy mixing by a single jet. The ratio l β /l T, where l T is the topographic length scale, governs jet behavior. Multiple, steady jets with fixed meridional spacing are observed when l β l T or when l β l T. When l β < l T, a pattern of perpetual jet formation and jet merger dominates the time evolution of the system. This unsteady structure reduces transport by a factor of two if the topography consists of zonally-invariant ridges, and increases transport by an order of magnitude or more if the topography consists of two-dimensional sinusoidal bumps. For certain parameters, bumpy topography gives rise to periodic oscillations in jet structure between purely-zonal and topographicallysteered states. In these cases, transport is dominated by bursts of mixing associated with the shifts between the two regimes. Unsteady jet behavior depends crucially on the feedback between changes in mean flow orientation, caused by topographic steering, and the conversion of potential energy to kinetic energy through baroclinic instability, as well as on asymmetric Reynolds stresses created by topographical modifications to the large-scale potential vorticity gradient. It is likely that these processes play a role in the dynamic nature of Southern Ocean jets. 2

3 1 Introduction Ocean flows are replete with coherent structures on scales ranging from the Rossby deformation radius λ, tens of kilometers, to the size of ocean basins, many thousands of kilometers. Ocean jets, defined as zonally-elongated flows typically exhibiting banded structure with alternating eastward and westward velocities, are an example of coherent structures observed throughout this range of scales. The near universal presence of jets in the ocean has been substantiated by observations of surface currents from satellite altimetry (Hughes and Ash 21, Maximenko et al. 25) as well as from finely resolved numerical simulations (Nakano and Hasumi 25). Jets that span roughly three degrees of latitude are latent features in the Pacific and Atlantic Ocean basin circulations (Richards et al. 26, Kamenkovich et al. 29). After sufficiently long averaging these features are found to fill the basins outside of equatorial regions. In marked contrast, smaller-scale jets appear explicitly in instantaneous images of Southern Ocean velocity fields obtained from both observations (Sokolov and Rintoul 27) and eddy-resolving numerical models (e.g. OCCAM 1/12 o model, Lee and Coward 23). These jets are distinct from ocean basin jets they are thin, ribbon-like features that undergo significant meandering. The location of these jets is largely fixed by topographical features, however over periods of weeks to months, jets may form and disappear, merge and split, and shift latitudes rapidly (Sokolov and Rintoul 27, Thorpe and Stevens unpublished manuscript). The existence of these jets relies on the unique properties of the Antarctic Circumpolar Current (ACC). The absence of continental boundaries across the latitudes spanning Drake Passage allows a stratification and circulation to develop that (a) has a larger zonal mean flow and (b) is more sensitive to mesoscale eddies and topographical features than ocean basins (see reviews by Rintoul et al. 21 and Olbers et al. 24). This produces an environment where zonal jets play a similar role to atmospheric storm tracks (Williams et al. 27). Still, the dynamics that set the jets horizontal length scales, time 3

4 scales of variability and velocity amplitudes are still being investigated. Not surprisingly, then, an understanding of how these jets influence large-scale transport of heat and tracers across the ACC is also underdeveloped. While the ACC s circumpolar flow is the primary mechanism for exchanging heat, chemicals, dissolved gases and other tracers between ocean basins, meridional transport across the ACC is also of vital importance for determining carbon dioxide distributions (Le Quéré et al. 27), the global heat balance (Hogg et al. 28, Gille 28) and the export of dense water formed around the margins of Antarctica (Orsi et al. 21). Thus, ocean circulation or coupled ocean-atmosphere climate models must accurately incorporate meridional transport processes in the ACC. This requires a representation (i.e. resolution or parameterization) of mesoscale eddies and jets, which can be as small as the deformation radius λ roughly 1 to 2 km in the ACC (Chelton et al. 1998). A complicating factor in modeling the ACC is the lack of consensus about the dynamicallyrelevant scales with regard to transport. At one extreme, Marshall et al. (26) calculate eddy diffusivities from tracer fields advected by observed surface velocities and conclude that the ACC acts as a single large-scale circumpolar transport barrier. This result differs from the traditional view that the ACC is comprised of three strong fronts that can be traced continuously around Antarctica following specific water mass properties (Orsi et al. 1995, Belkin and Gordon 1996). Recently, satellite altimetry has provided a more intricate view of the ACC s structure, with analysis of gradients in sea surface height or sea surface temperature indicating that the ACC is dominated by a complex web of filamentary structures (Hughes and Ash 21, Dong et al. 26, Sokolov and Rintoul 27). Sokolov and Rintoul (27) have shown that rather than three circumpolar jets, 6 to 1 individual narrow jets may be detected across the ACC at a given time and longitude. Eastward jets in the ACC are typically associated with barriers to meridional transport 1. Thus the distinction between circumpolar jets and more intricate webbed or braided 1 It has recently been shown that a jet s resilience to mixing may have complicated vertical variability 4

5 structures may be crucial for large-scale transport estimates. Figure 1 shows snapshots of layer-wise potential vorticity (PV) from two doubly-periodic quasi-geostrophic (QG) turbulence simulations on a β-plane, i.e. with a large-scale PV gradient one without and one with topography (upper and lower panels respectively). In the former case, regions of nearly homogeneous PV develop between strong eastward jets, which focus along sharp meridional gradients in PV (Rhines 1994, Dritschel and McIntyre 28). In the case with topography, a set of well-mixed PV regions is still apparent, although the boundaries separating these regions undergo large meridional excursions due to topographical steering. If these boundaries, or barriers, remain coherent, then topographical steering of the jets has little effect on large-scale meridional transport mixing across the entire domain remains limited. If, however, jet steering results in other processes, such as eddy shedding or a local breakdown of transport barriers, this may alter large-scale transport. Note that in the lower layer sharper boundaries develop than in the flat bottom case due to the enhancement of local PV gradients by topography. This study focuses on the role that simple topographies play in setting jet structure in a large domain with many jets. Specifically, a suite of over 1 simulations of equilibrated baroclinic turbulence in a two-layer, doubly-periodic QG model are used to address (a) the meridional scaling of the jets, (b) the spatial and temporal variability of the jet structure and (c) the resulting large-scale, domain-averaged transport characteristics. The simulations are performed in a domain that is large compared to λ so that, like the ACC, many jets may form. Following the results of Pavan and Held (1996), a doubly-periodic domain provides an accurate assessment of dynamics far from boundaries. QG models of the ACC with topography have been explored by Treguier and Hua (1988), Treguier and McWilliams (199) and Wolff et al. (1991) amongst others. In these studies, the emphasis was on quantifying energetics of the system, especially the role of bottom form stress. The studies of Vallis (Haynes and Greenslade 28, Smith and Marshall 28). In these studies jets are more effective barriers to transport near the surface. 5

6 and Maltrud (1993), Treguier and Panetta (1994), Sinha and Richards (1999) and Hogg and Blundell (26) consider how topography influences jet structure, although either due to the type of model used or because of the parameter regime explored, they were unable to capture the range of behaviors described here. Furthermore, the current study considers the implications of jet variability on large-scale transport properties. Section 2 briefly describes the model and two diagnostics used to analyze the flow: the Rhines scale l β and the effective diffusivity κ eff.. Section 3 considers jet structure in simulations with three types of topography: bottom slopes, zonally-invariant ridges and twodimensional arrays of sinusoidal bumps. A discussion of the mechanisms contributing to jet variability and the applicability of these mechanisms to Southern Ocean flows appears in section 4, while conclusions are given in section 5. Appendices A and B describe linear baroclinic instability over a topographic slope and the derivation of κ eff. respectively. 2 The model and diagnostics 2.1 Equations The choice of the two-layer Phillips model, described in Pedlosky (1987), is based on numerical simplicity, while allowing the system to be forced physically through baroclinic instability. The governing equations for the layer-wise PV are t Q i + J (Ψ i, Q i ) = κ δ i2 2 ψ 2 + ssd, (1) where J(a, b) = a x b y a y b x, δ ij is the Kronecker delta, Ψ i is the streamfunction and Q i is the potential vorticity of the upper (i = 1) and lower (i = 2) layers defined by Q i 2 Ψ i + f2 g H i ( 1) i (Ψ 1 Ψ 2 ) + β y + f H 2 δ i2 η. (2) Here g is the reduced gravity, H i are the mean depths of the layers and the Coriolis frequency is approximated by f = f + β y. The deformation radius λ is given by g H /f, where 6

7 H = H 1 H 2 /H. The ratio of the layer depths is given by δ = H 1 /H 2 and the total bottom depth is given by H = H + η. The Ekman damping coefficient is given by κ = f d E /H, where d E is the Ekman layer depth and ssd is small scale dissipation needed for numerical stability. Small scale dissipation is implemented using a wavenumber filter as described in the appendix of Smith et al. (22). The total streamfunctions are given by Ψ 1 = 2Uy +ψ 1 and Ψ 2 = ψ 2 so that the system is baroclinically unstable. The equations can be non-dimensionalized using λ and U which yields, dropping the ssd term: q 1t + G 1 ψ 1x + 2q 1x + J (ψ 1, q 1 ) =, (3) q 2t + G 2 ψ 2x + J (ψ 2, q 2 + h) = κ 2 ψ 2, (4) where h(x, y) f λη/uh 2 and κ is the non-dimensional parameter κ λ/u. The large-scale PV gradients are given by G i = β + 2δ i, where β = β λ 2 /U and perturbation PVs are q i = 2 ψ i + δ i (ψ 2 ψ 1 ). The weighting functions are defined by In a case with a bottom slope h = h y y, G 2 = β + 2δ 2 + h y. δ 1 = δ, δ 2 = δ 1 + δ. (5) The modal form of the equations are obtained using the relationships ψ = δψ 1 + ψ 2 δ (ψ1 ψ 2 ), τ =, (6) 1 + δ 1 + δ where ψ and τ are the perturbation streamfunctions of the barotropic and baroclinic modes respectively. The condition for linear instability in the case with κ = is that G 1 and G 2 have opposite signs (Charney 1971), although with the introduction of bottom friction, the system becomes unstable at values of β and h y beyond the frictionless critical point (Holopainen 1961). The energy balance is obtained by multiplying δ 2 ψ 1 to (3) and δ 1 ψ 2 to (4) and integrating over both turbulent fluctuations and the doubly periodic domain, indicated by, which gives 2δ 2 ψ 1 ψ 2x = κ ψ ssd. (7) 7

8 The contribution of the ssd term to the right hand side of (7) is never more than a few percent in these simulations. Examples of the topographies considered in this study are given in Figure 2. Panel (d) gives an example of how topographical features locally modify the barotropic PV and its gradient. For all simulations discussed here κ =.1, δ = 1, domain size is square with length L = 64λ and grid size is Diagnostics Effective diffusivity One diagnostic employed to analyze jet structure and transport in these simulations is the effective diffusivity κ eff. introduced by Nakamura (1996). Although primarily used in atmospheric contexts to indicate mixing regions and barriers to isentropic eddy transport, e.g. Haynes and Shuckburgh (2) and Allen and Nakamura (21), more recently, Marshall et al. (26) have used effective diffusivities to quantify eddy diffusivities in the Southern Ocean in a stream-wise framework. The values of κ eff. shown in the present study are not meant to imply quantitative similarity to eddy processes in the ocean or atmosphere. Instead, the emphasis is on quantifying how different topographies induce relative variations in transport properties. A careful derivation of the effective diffusivity calculation is given in Shuckburgh and Haynes (23) and is summarized in Appendix A. The magnitude of κ eff. is a measure of tracer contour complexity with the implicit assumption that stirring by eddies are responsible for increasing tracer complexity. Transport barriers are associated with simple tracer contours and therefore small values of κ eff., while regions of large κ eff. represent strong mixing. Throughout this study the effective diffusivity calculations are made using potential vorticity contours, making connections between the left hand side of (1) and advection of a passive tracer. Upper layer PV is advected by the upper layer streamfunction and is materially conserved, except for small-scale dissipation 8

9 which is small, therefore Q 1 mimics a passive tracer. Lower layer PV, on the other hand, is forced by bottom friction, and for this reason we focus on κ eff. distributions in the upper layer. It is shown in section 3 that regions of low κ eff. in the upper layer accurately track the position of the barotropic jets (see also Haynes et al. 27). The distribution of κ eff. for a reference flat bottom case with multiple jets is discussed in section Rhines scaling The Rhines scale l β is the crucial characteristic of the flow that determines its interaction with topography. Here we review the dependence of l β in simulations without topography to changes in the parameters β and κ. The data are taken from the simulations described in Thompson and Young (27), who re-visited the work of Panetta (1993) with the ability to significantly increase the domain size and horizontal resolution. Since the domain is doubly periodic, the simulations must select an integer number of jets; thus extending the domain size is an important check on a number of the features described in Panetta (1993). Thompson and Young (27) use a slightly different form of bottom friction than described in section 2.1, which does not significantly influence the behavior of the system. The jet spacing l J and wavenumber k J are estimated by counting the number of jets n J in the physical fields of the simulations. In most cases n J is unambiguous, although cases arise where n J is not clearly defined as it transitions from one integer to another. In this case we allow for half values, e.g. n J = 5.5. The jet wavenumber is then given by k J = 2πn J /L, where L is the domain size. Figures 3a and 3b provide a complete survey of the equilibrated value of n J and show that the dependence on κ and β is not a simple power law. For example, for the case β = 1/2, n J increases roughly like κ 1/4 (panel a), while for κ =.8, n J grows linearly with β (panel b). However, the panels taken together indicate that these simple scaling laws are special and do not apply except along these particular slices through parameter space, i.e. n J does not scale like βκ 1/4. 9

10 Although n J is too complicated to be described by a simple power law involving the external parameters β and κ, the hope is that n J can be simply related to other equilibrated statistics of the flow. Rhines scaling suggests that the wavenumber associated with the jet spacing k β should scale like β/v where V is a velocity scale. Panetta (1993) suggested that V should be associated with an eddy velocity scale V e such that where k β β V e, (8) V e ψ ψ 2 2. (9) The primes indicate that the zonal mean has been removed, i.e. ψ i = ψ i ψ i y. At small values of bottom friction, Panetta (1993) found good correlation between k J and k β (see his Figure 4). As further confirmation, Figure 3c shows contours of the ratio k J /k β over a range of κ and β values. The agreement between k β and k J breaks down as the jets become weaker, specifically in simulations with small β and large κ. In simulations where both β and κ are weak, as few as two jets occupy the domain, and quantization is likely responsible for the poorer agreement between k J and k β in this corner of parameter space. Rhines scaling was confirmed, using a similar definition of V, in a more realistic ocean model by Sinha and Richards (1999), although recent simulations by Berloff et al. (28) do not support Rhines scaling. We note that Rhines scaling does not work for other choices of V, such as the jet velocity scale u J = ψ 2 y or the meridional velocity scale V = ψx. 2 The main conclusion of this subsection is that Panetta s version of Rhines scaling is successful at condensing the results over a wide range of β and κ values, which is remarkable given the complex dependence of n J on κ and β summarized in Figure 3. Importantly, the agreement between l β and l J implies that l β represents the meridional extent of well-mixed PV regions in equilibrated baroclinic turbulence with a large-scale PV gradient (cf. Figure 1). 1

11 3 Results 3.1 Bottom slope In layer-wise QG models, topography modifies the PV gradient in a single, lower layer (although all modal PV gradients are affected). Another key layer-wise process is the forcing of the zonal mean flow by Reynolds stress correlations. This process, which generates upgradient momentum fluxes (Held and Andrews 1983) and non-local spectral transfers (Huang and Robinson 1998), was shown by Thompson and Young (27) to occur almost exclusively in the upper layer of a two-layer model. Since modification of PV gradients by topography and maintenance of the jets by Reynolds stresses occur in different layers, it is not immediately clear how topography might affect meridional jet scales. Thus we first consider how jet structure and meridional scaling responds to varying steepness of a uniform, meridional bottom slope h y. The role of topography in setting the meridional jet scale is examined by considering whether the Rhines scaling l β continues to predict the jet spacing. In this case β must be modified to account for the bottom slope. The best agreement was found by taking the modified β to represent the barotropic PV gradient: β BT = β + h y 2, (1) which is similar to the result found by Sinha and Richards (1999). Figure 4a shows a survey of the ratio k J /k β where k β = β BT /V e and k J = 2πn J /L where n J is obtained by counting the number of jets in the simulations. Each dot represents a simulation and its color gives the corresponding value of k J /k β. With the exception of the simulation with β = and h y = 1.5, k J /k β = 1 ±.15, which is comparable to the agreement in the flat bottom simulations (bottom friction is fixed at κ =.1). Panels (b) and (c) show how k J /k β and n J depend on β BT. Panel (c) indicates that while jet spacing generally decreases with increasing β BT, for a fixed β BT, different combinations of β and h y can produce various equilibrated 11

12 energy levels and jet spacings. The success of Rhines scaling with β BT indicates that although Reynolds stress forcing of the mean flow is a layer-wise process (Thompson and Young 27), the barotropic PV gradient ultimately determines the jet scale. This raises the question of how topographies that locally modify PV gradients, but do not alter the domain-averaged PV gradient, influence equilibrated jet structure. 3.2 Zonal Ridges The next level of topographic complexity considered is zonally-invariant (h x = everywhere), or zonal, ridges. Zonal ridges mark a departure from most studies of turbulence over topography which tend to include meridional ridges (e.g. Borowski et al. 22). This is because meridionally-oriented topography can support zonal pressure gradients, which are a key mechanism for removing momentum from channel-like flows, such as the ACC (Munk and Palmén 1951). The focus of this intermediate case between topographic slopes and twodimensional topographies is to understand how local PV modifications alter equilibrated jet structures, while restricting the mean flow to be zonal. All zonal ridge simulations were completed with base parameters β =.75 and κ =.1. In the absence of topography, these parameters produce a statistically equilibrated state with l β = 16λ. Therefore in a domain of size 64λ 2, four jets form, which are resolved accurately (four grid points per λ) with a computational grid that still permits sampling of a large region of parameter space. Holding β and κ fixed, two parameters are varied: the number of ridges n R = L/l T, where l T is the wavelength of a single ridge, and a measure of the topography steepness, h y. The ridges are described by h = h y k T cos (k T y), (11) where k T = 2π/l T. For β =.75, reversals in the sign of G 2 occur for h y >.25, while the barotropic PV gradient, (G 1 + G 2 )/2, only undergoes sign reversals for h y <

13 Figure 5 summarizes the behaviors of the jets by showing a series of Hovmöller (time/latitude) plots of upper layer κ eff. (left panels) and zonally-averaged zonal barotropic velocities (right panels) for simulations with varying n R and h y. Panels (a) show the structure of κ eff. for the n R =, h y = reference case. Four steady, eastward zonal jets form with weak westward flows occurring between these jets. The structure of κ eff. in the upper layer is a series of eight mixing regions and eight transport barriers. Strong mixing regions are found on the flanks of the eastward jets. Cores of the eastward jets are strong barriers to transport, while cores of the westward jets are weaker barriers to transport 2. Panels (b) show a simulation with n R = 3, h y = 1. and gives a typical example of the unsteady jet behavior that can occur. Multiple, coherent jets form, but now these jets migrate meridionally, unlike in the flat bottom case. This results in continuous jet formation and merger, giving rise to a braided jet structure reminiscent of the temporal variability of Southern Ocean jets (cf. Sokolov and Rintoul 27, Figure 1). The ratio l β /l T is given above each panel, where l β is defined by 2π V e /β, i.e. local PV modifications are neglected. For l β /l T 1, the system locks into a steady jet structure (panel c), however, the equilibrated jet scale (n J = 5) need not be the same as the flat bottom case (n J = 4). A pattern of 6 steady jets can also be obtained with l β /l T = The positions of the jets in panel (c) are fixed by the ridges with eastward jets located at the latitudes where the topographic slope most strongly enhances β, while the westward jets are located where the slope most strongly opposes β. As the scale of the topography is reduced, and l β /l T > 1, the system reverts to a steady pattern with n J = 4. Panels (e) show that the unsteady braided structure is a robust feature, even for relatively shallow topographies. Panels (f ) show that wide and steep topographies can support mixed behavior with weak unsteady braided behavior (e.g. along y e /λ = 32) occurring between strong steady jets. The ratio l β /l T is the key parameter determining the behavior of the jets. For l β /l T > 1, 2 Transport barriers coinciding with the westward jets gives support to recent work by Beron-Vera et al. (28), who have found similar barriers in westward jets in the atmosphere. 13

14 turbulent mixing occurs over horizontal scales greater than the scales of local PV modifications. The eddies essentially smear out the signature of the topography and the dynamics of the system and structure of the jets are observed to be similar to turbulence over a flat bottom. For l β /l T 1, the jets are steady and fixed by the scale of the topography. For l β /l T < 1 the system feels the local PV modifications imposed by the ridges. For.5 < l β /l T < 1, a braided structure is observed throughout the domain, while for l β /l T <.5, a combination of steady and braided jets is typically observed. Figure 6 describes how changes in jet structure, responding to variations in n R and h y, modify the domain-averaged energetics and transport of the system. Panel (a) gives the ratio l β /l T for all the simulations completed in the ridge configuration. Two parameter values, indicated by the dashed lines, are explored in detail: n R = 2 and h y = 1.. Panels (b) and (d) show time and domain-averaged values of eddy kinetic energy (EKE), zonal kinetic energy (ZKE) and potential energy (PE), where EKE = 1 ψ ψ 2 2, ZKE = 1 2 ψ 1 y 2 + ψ 2 2 y, PE = 1 (ψ1 ψ 2 ) 2. (12) 2 The values are normalized by the non-dimensional EKE, ZKE and PE values obtained from a simulation with n R = h y =, [8.95, 7.51, 5.75]. Panel (b) shows that for fixed h y = 1. kinetic energy levels are maximized when there are no ridges, and they decrease as n R increases up to 4. The n R = 4 simulation marks a minimum in both EKE and ZKE at about half the flat bottom value; the jets have a braided structure in this simulation. Energy levels rise rapidly for n R = 5 and 6 and remain largely unchanged for n R > 6, where l β /l T > 1. For small n R, PE jumps to over five times the flat bottom value, but for n R 4, the PE ratio falls to values less than 1, and for larger values of n R trends in PE follow EKE and ZKE. For n R 1, the energy ratios are approximately unity indicating that domain-averaged statistics are similar to flat bottom values for small scale topography. Similar behavior is observed in estimates of the domain-averaged transport characteristics (panel c). Two quantities are calculated to examine this, the eddy heat flux (HF) and the 14

15 domain averaged effective diffusivity κ eff., HF = 1 2 ψ 1ψ 2x, κ eff. = {κ eff. }, (13) where { } indicates a harmonic mean in the meridional direction and a temporal mean over turbulent fluctuations. The harmonic mean is the standard averaging technique for quantifying transport in the presence of transport barriers (cf. Nakamura 28). Similar to kinetic energy, topographical features with length scales greater than the equilibrated l β can reduce domain-averaged transport estimates by nearly a factor of two, but for topographical scales smaller than l β, transport properties revert back to flat-bottom levels. A key point, though, is that both energy and transport ratios have a non-monotonic dependence on n R. The data have three clear local minima at n R = 1, 4 and 8 that are related to changes in the jet structure as the wavelength of the topography is modified. Trends are clearer in the case where n R is held fixed (n R = 2) and h y is varied. As h y increases both EKE and ZKE decrease monotonically (panel d), as do the HF and κ eff. (panel e), (note that large h y and small n R pushes the limits of QG theory which requires changes in h remain small compared to the layer depth H 2 ). Conversely, PE rises smoothly as topography becomes steeper, indicating that the system becomes more stable and less energy is released through baroclinic instability. In simulations with large steepness, the flow is almost purely zonal as ZKE EKE. This monotonic reduction in transport as h y increases indicates that the stabilizing flank of a zonal ridge, where h y enhances β, regulates the domain-averaged transport. In other words, if larger, more energetic eddies develop in regions of reduced PV gradient, these are unable to break down the strong transport barriers that develop in regions of enhanced PV gradient. We conclude this subsection by considering how local PV modifications alter the Reynolds stress forcing of the mean flow. Figure 7 calculates the Reynolds stress (R) terms R i u iy (u i v i ) (14) 15

16 for three different simulations corresponding to Figures 5c, 5f and 5b. In panel (a) there are five steady zonal jets imposed by the topography since l β /l T 1. As in the flat bottom case, R terms are dominant in the upper layer (bold curve) and weak in the lower layer (thin curve); they occur on the flanks of the jet and are symmetric about the jet core. This occurs because the meridional scale of the jet is the same as the topographical scale, such that the flanks of the jet feel the same local PV gradient. Steeper topographies act to localize the R terms (panel b), creating a zonal velocity profile that is significantly different from the flat-bottom scenario. Finally, panel (c) shows a Hovmöller plot of R 1 in a simulation with a braided jet structure, where now only the zonal average is taken. This plot reveals that meridional jet migration is tied to the disagreement between l β and l T. Mixing regions on the flanks of the jets feel different local PV gradients resulting in asymmetric forcing of the mean flow. The jets are accelerated on one flank and decelerated on the other, which steers the jet across the mean PV gradient. This steering encourages jet merger and produces regions where new jets may form. 3.3 Bumps Topography is now allowed to vary zonally as well as meridionally. Local modifications of PV gradients will influence jet structure, however a new feature is the generation of nonzonal mean flows due to topographic steering. Steering is a result of PV conservation as fluid parcels act to minimize vortex stretching that occurs over topographical rises and depressions (Holloway 1987, Vallis and Maltrud 1993). These modified mean flows will then feed back on the baroclinic instability process. The results are based on a suite of simulations that keep the parameters β =.75, κ =.1 fixed, while varying the two parameters: n B = L/l T, where l T = 2π/k T is the wavelength of a bump, and bump steepness h y. We consider two 16

17 topographic forms: aligned and unaligned given by h aligned h unaligned = h y k T sin (k T y)cos (k T x), (15) = h y k T [sin (k T y) + cos (k T x)], (16) (see Figure 2). The aligned topography has zero zonal and meridional mean, while the unaligned topography retains zonally-averaged and meridionally-averaged structure. Jet behaviors are again summarized in a series of Hovmöller plots of upper layer κ eff. for the unaligned topographies (Figure 8, left panels). The behavior of the aligned topography cases is qualitatively similar and a separate plot is not included. The right panels again show Hovmöller plots of zonally-averaged zonal barotropic velocities; the correspondence with the left hand panels is weaker in some cases due to the effects of topographical steering. A number of behaviors observed in the zonal ridge configuration are repeated, including braided jets for l β /l T < 1 (panels b) and quasi-steady jets with spacing set by the topographic scale, i.e. l β l T (panels c) although now jet spacing is greater than in the flat bottom case. A new behavior is observed in panels (d), where at fixed latitudes, strong but transient transport barriers spontaneously form and then give way to strong mixing in a continuous process. The position of the strongest barriers are observed to jump throughout the domain, although they are constrained to one of three latitudes by topography (here n B = 3). As before, the jets regain their flat-bottom structure for large n B and h y 1. (panels e). However, for steeper topography with many bumps, the topographical scale appears in the upper layer jet structure, and the barriers associated with westward jets are lost (panels f ). Figures 9 and 1 show surveys of the domain-averaged statistics for the unaligned and aligned topographies. The statistics are the same as those defined for Figure 6, except EKE has been replaced by the total kinetic energy KE = 1 ψ ψ 2 2. Statistics are again normalized by the flat bottom values. The domain averaged statistics for the unaligned and aligned topographies have qualitatively the same behavior, although the unaligned topography has slightly larger variability. 17

18 For a fixed value of h y, KE, ZKE, PE, HF and κ eff. ratios 3 all have peaks at an intermediate value of n B that depends on the magnitude of h y. For steep topography (h y = 2) peak values occur for long topographical features n B = 2, while for shallower topography (h y = 1) the peak occurs for shorter topographical features (n B = 4). In general, PE is peaked at smaller values of n B than the kinetic energies. At large n B the statistical properties return to flat bottom values as in the ridge simulations. Energy levels and domain-averaged transport show a monotonic increase with increasing h y for a fixed value of n B. The trend is moderate for small h y, but increases rapidly as h y approaches 2β the value required to locally eliminate the barotropic PV gradient. For h y > 2β, the statistics plateau; the transition between low and high energy states is less abrupt in the aligned topography case. Heat flux ratios are larger than the κ eff. ratios in the high energy state, which likely reflects the additional contribution to the heat flux from the standing eddies. Unlike the ridge simulations, the introduction of topography uniformly increases statistical properties, with the exception of a decrease in ZKE for small n B. The inhibitive effect of topography locally enhancing β is no longer limiting because fluid can escape these regions, e.g. on the eastward and westward flanks of a bump. The uniform increase in energy and transport depends largely on the fact that baroclinic instability is more efficient at extracting PE from non-zonal mean flows (Spall 2, Arbic & Flierl 24, Smith 27). We now focus on jet behavior in two simulations shown in panels (b) and (d) of Figure 8. Figure 11a shows the time/latitude evolution of κ eff. as in Figure 8b. There is a persistent series of spontaneous jet formation and subsequent merger events. There are also distinct 3 Here the total heat flux is plotted, not the eddy heat flux. With a purely zonal topography, v = ψ x = therefore vτ = v τ. Correlations between the mean v and τ fields can develop in the case with topographic steering. Unfortunately it is not possible to save sufficient fields to accurately determine the mean field in each of the simulations shown here. Agreement between the HF and κ eff., which does not suffer from this problem, suggests that the behavior of the eddy heat flux is qualitatively similar. 18

19 periods when the strength of the transport barriers are minimized, e.g. tu/λ 55, 85 and 15. These periods are also apparent in the domain-averaged properties of the system as shown in panel (b), and correspond to peaks in domain-averaged KE and, to a certain extent, in κ eff. (here the time average has not been performed). Periods of high KE and κ eff. are typically preceded by a collapse or merger of transport barriers; the formation of new barriers reduces the energy and mixing rates again. These events occur in a quasi-periodic fashion with a period of roughly tu/λ = 3 in this simulation. The formation of new transport barriers can also lead to a shift in meridional position of enhanced mixing regions (e.g. around tu/λ = 85). More dominant periodic behavior may develop in simulations with steeper topography (Figure 12). Panel (a) shows the time evolution of the domain-averaged KE, ZKE and PE (corresponding to Figure 8d). There is a distinct quasi-periodic oscillation in these quantities with a period of approximately 45λ/U. All three quantities are out of phase with PE leading, followed by a rapid rise in KE and finally a peak in ZKE. Similar behavior has been observed in a study by Hogg and Blundell (26) using a three-layer QG model of the Southern Ocean. Panel (b) shows the corresponding zonally-averaged zonal barotropic velocity. Although n B = 3, the Rhines scale l β is comparable to the domain size during the high KE state. This results in a single strong jet whose position jumps from one topographical feature to another. Each of these shifts is preceded by a period of intense turbulent mixing, which allows the meridional structure to reorganize. There is no discernible trend for how the strong jet selects its latitude (cf. Figure 8d). Snapshots of the barotropic vorticity taken at three different times, indicated by the dashed lines, are shown in panels (c) through (e). During the low KE/high PE phase, PV is steered strongly by topography producing a non-zonal mean flow. At the peak KE stage, the eddy scale has grown dramatically and mixing is more vigorous. These large eddies effectively damp the signature of the topography and the flow becomes predominantly zonal. 19

20 As the KE decays, the eddies become smaller, topography begins to deflect the zonal flow again and the process repeats itself. This mechanism and its implication for transport in more realistic ocean flows are discussed in section 4. 4 Discussion 4.1 Variability mechanisms In simulations with zonal ridges, unsteady jet behavior involving persistent jet formation and merger, either throughout the domain or in regions between strong steady jets, occurs over a broad region of parameter space. The meridional extent over which PV becomes homogenized in β-plane, baroclinic turbulence is given by l β (Dritschel and McIntyre 28). Thus if two regions have roughly the same value of V e (9), but topography opposes β in one region and enhances β in the other, l β will be larger in the first region. In other words, one flank of a ridge generates larger eddies that mix over a wider region. Figure 13a shows time series of l β from the simulation in Figure 5b, estimated by dividing the domain into high PV gradient regions (β BT > 11/12) and low PV gradient regions (β BT < 7/12). Different estimates of l β are calculated from the mean values of β BT and V e in these separate regions. The eddy velocity scales V e are comparable in both high and low PV gradient regions, although V e experiences greater variability in regions of weak PV gradient (panel b). Thus local PV modifications caused by topography are the primarily means for developing the spatially variable l β values that lead to unsteady jet patterns. The variability mechanism in the case with bumpy topography can be very different as evidenced by the behavior of the energy diagnostics in Figure 12, which undergo quasi-periodic variability. In Figure 14 a large number of snapshots have been collected from the experiment shown in Figures 8d and 12 over a period of 12λ/U. The mean streamfunction is calculated from a running mean of 2 snapshots spanning a period 5λ/U. This allows an accurate cal- 2

21 culation of the EKE and thus l β. At the outset of a cycle (characterized by low KE and high PE values), l β is at a minimum and crucially, in these simulations, l β l T. Thus the jets feel the local PV modifications and are steered around the topographical features. During this stage, strong barriers to transport remain, despite the meandering. Non-zonal mean flows can generate eddy fields that can be more than 1 times more energetic than eddy fields arising from purely zonal mean flows (Arbic and Flierl 24). Thus the topographic steering initiates a burst in EKE, which allows eddy length scales to grow through an inverse cascade until l β exceeds l T. At this point, the controlling PV gradient is large-scale planetary β. Reynolds stress forcing of the mean flow acts preferentially in the zonal direction, and the flow becomes strongly zonal with l β > l T. As PE supplies are exhausted, V e decays and at some point l β l T. Topographic steering resumes, which builds up PE stores and initiates the cycle again. Panel (a) also shows a time series of κ eff., which indicates that the rapid growth of EKE is associated with bursts in turbulent mixing. This suggests that, in this regime at least, time and domain-averaged meridional transport may be dominated by intermittent events that may be difficult to capture observationally or to parameterize in numerical models. 4.2 Transport of passive tracers Effective diffusivity diagnostics calculated from upper layer PV give an accurate description of how a passive tracer, maintained by large-scale gradient, would behave in this layer. There are subtle differences between passive tracers and PV in the lower layer because of bottom friction effects. In particular, in flat bottom simulations κ eff. calculated in the lower layer from a passive tracer indicates mixing across eastward jets, while calculations using Q 2 indicate a barrier at the eastward jet, where bottom friction effects will be strongest. The details of this discrepancy are the focus of a separate study and are not discussed further here. The key point is that the transport barriers diagnosed in the upper layer are a good 21

22 representation of barriers associated with nearly barotropic jets (cf. Figures 5 and 8). The variability observed in these simulations complicates the standard view of eastward jets as strict barriers to transport. Two paradigms are summarized by the schematic in Figure 15. In the case where jets are zonally-oriented, steady, domain-wrapping features (panel a), we expect PV or tracers to become homogenized between the jets, with limited transport across the jets. If instead jets are continuously forming and merging, in either a temporal or streamwise sense (panel b), it becomes difficult to describe the meridional extent of a homogenized or well-mixed region. Furthermore, meridional jet migration offers a potential path for tracers to be carried through a latitude range greater than l β without ever crossing a sharp PV gradient. Thus while jets may be effective transport barriers, over large times tracers may exhibit a greater meridional spread than would be implied by local eddy diffusivities. The effective diffusivity is a useful diagnostic, but it only provides a zonally-averaged measure of mixing. Open questions remain as to whether there are preferred locations for meridional transport, and if so, how this relates to domain or circumpolar measures of transport. 4.3 Domain sensitivity Unsteady jet structure may also occur in β-plane, baroclinic QG turbulence in a doubly periodic domain without topography because of quantization problems, since the system must accommodate an integer number of jets (Panetta 1993, Thompson and Young 27). In this scenario, typically only an isolated region of the domain exhibits unsteady behavior. For example, if the system is varying between n and n+1 jets, the system typically has n 1 steady jets and another wider jet that develops continuous branches. In certain cases, this branching can lead to meridional movement of the jets, but more often the variability persists along a constant latitude. Thus, there are some clear distinctions between the variability observed in flat bottom simulations and topographically-induced variability. However, to 22

23 ensure that the variability described in this study is not attributable to domain constraints, especially when l β is comparable to L, a number of simulations were conducted in wider domains (128λ 2 ) keeping all other parameters fixed. In particular, the simulations shown in Figures 5b, 11 and 12 all exhibit the same behavior and have the same domain averaged statistics to within ±5%. Thus there is no evidence that dynamics in these simulations are controlled by domain size. 4.4 Applicability to the Southern Ocean The dynamics of the two-layer QG simulations represent a substantial simplification over processes occurring in the ocean. While the model captures the most relevant dynamics, e.g. topographic steering, baroclinic instability, and jet sharpening through Reynolds stresses, much detail has been sacrificed for numerical efficiency that has allowed a broad sweep of parameter space and the cataloguing of different behaviors. In particular, in a two-layer model topography almost necessarily influences the equilibrated dynamics of the system. Even a modest increase in vertical resolution to a three-layer model may provide new results since eddy generation through baroclinic instability could proceed independent of topographical influence in the upper two layers. This is an important aspect that should be tested in future work. Furthermore, incorporating layer depths more representative of ocean stratification, i.e. δ 1, could be useful, although typically this does not qualitatively change QG behavior (Arbic et al. 27). While the mean flow in this model may adjust, the domainaveraged velocity jump between layers is fixed. The vertically-sheared mean flow of the ACC will experience time dependence both in its amplitude and its orientation. This implies that regions of the ACC may flip between different steady jet spacings or alternatively between steady, unsteady, or quasi-periodic jet behaviors as baroclinic instability becomes more or less efficient at converting PE to KE. Despite these differences, it is worthwhile considering whether the behaviors documented 23

24 here have relevance to ocean flows, and in particular to jet structures observed in the ACC. Figure 16a shows the bathymetry of the Southern Ocean as given by Smith and Sandwell (1997). It is smoothed using a two-dimensional low-pass filter with a radius of roughly 1 km. Panel (b) shows the meridional gradient of the smoothed lower layer depth, where the depth of the lower layer is taken to be 8% of the total depth for simplicity. Finally the barotropic meridional PV gradient, as defined in (1) (in dimensional terms h y = f η y /H 2 ), is calculated and plotted in panel (c). Clearly, this is a crude estimate of local meridional PV gradients, however, it emphasizes the importance of topography on setting local PV gradients as a large portion of the domain experiences a gradient significantly different from the large-scale β. Panel (c) also indicates that topographical variations occur over a broad range of horizontal length scales that are available for interaction with the jet scales. For non-zonal topography we expect strong variability if l β l T ; achieving this regime in areas of the ACC seems not unlikely based on the results of Sokolov and Rintoul (27). Periodic behavior in QG simulations with topography has been observed in the work by Hogg and Blundell (26), which focused on large-scale circulation patterns that experienced variability on decadal time scales. Using typical Southern Ocean values, λ = 2 km and U = 2 cm s 1, the period 45λ/U in Figure 12 has a dimensional value of roughly 1 days. Thus on much shorter time scales, or scales comparable to the variability observed in the narrow meandering jets of the ACC, interaction between l β and l T is a potential mechanism for local jet re-organization. Although the model used here is simplified, the evidence that these re-organization events are associated with bursts in meridional transport (cf. Figure 8c), which may make a dominant contribution to the time-mean transports across the ACC, warrants further study into the role of small-scale jets on transport properties. 24

25 5 Conclusions The spontaneous formation and evolution of zonal jets on a doubly-periodic β-plane with simple topography, forced by a baroclinically-unstable vertical shear has been explored. The sensitivity of meridional jet spacing, jet pattern variability and domain-averaged transport properties to variations in topographical length scale and steepness have been addressed. The main conclusions are: 1. Jet spacing over a topographic slope in the lower layer is governed by the Rhines scale l β = 2π V e /β BT, where β BT = β + h y /2 is the barotropic PV gradient. 2. Zonal ridges locally modify the meridional PV gradients and can reduce domainaveraged meridional transport by up to a factor of 2. Minimum energy levels occur when the Rhines scale is comparable to the topographic scale l β l T. When l β < l T the system experiences continuous jet formation and merger events contributing to a braided jet structure. The dependence of domain-averaged transport on l T is nonmonotonic in this regime. Regions where topography enhances the meridional PV gradient regulate the domain-averaged statistics of the flow. 3. Two-dimensional topography both modifies the local PV gradients and induces a nonzonal mean flow through topographic steering. In contrast to the ridge simulations, energy levels and meridional transport increase with increasing slope steepness h y. For fixed h y energy levels and meridional transport peak when l β l T. Energy levels shift rapidly from a low energy state to a high energy state when h y is sufficiently large to induce local sign reversals in the meridional gradient of the barotropic PV. Transport barriers may exhibit steady, braided and quasi-periodic behavior. In the latter case, the ratio l β /l T oscillates between 1 and a value greater than 1 as more or less eddy kinetic energy is generated by baroclinic instability based on the orientation of the mean flow. Meridional transport is occurs in bursts in this case as jets reorganize 25

26 frequently. This study has provided a survey of mesoscale jet behavior over a range of simple topographies. Further work is required to fully understand the mixing and transport implications of some of the features described here, such as jet migration, jet merger and meridional jet shifts. The ultimate goal is to understand how similar jet variability influences meridional transport of heat, PV and tracers in the ACC. An important result is that topography does not simply steer jets in a passive manner. Through modification of the mean flow and generation of local PV gradients and their feedback on baroclinic instability, topography can generate a range of unsteady jet behaviors with significant consequences for large-scale transport properties. These processes are likely relevant to jet dynamics in the ACC. Acknowledgments The code used to calculate the effective diffusivities was provided by Peter Haynes. Insightful conversations with Pavel Berloff, Peter Haynes, Alberto Naveira Garabato and David Stevens that improved this manuscript are gratefully acknowledged. This work was supported by a Natural Environment Research Council (NERC) Postdoctoral Fellowship. 26

27 Appendix A: Derivation of effective diffusivity The effective diffusivity is calculated by considering the time evolution of a conservative tracer c(x, y, t) governed by the advection-diffusion equation c t + J (ψ, c) = k 2 c, (17) where ψ is a streamfunction with horizontal velocities u = ψ y, v = ψ x and k is a constant diffusivity. Irreversible mixing effects can be isolated from the effects of advection by rewriting (17) in the form of a diffusion-only equation by making a transforming to isotracer coordinates based on the evolving tracer itself. The coordinate system is also defined in terms of an equivalent latitude y e (C, t), which is the latitude the contour would have if it were remapped to be zonally symmetric, while retaining its internal area. In the new tracer-based coordinates, equation (17) becomes C (y e, t) t = ( ) C (y e, t) κ eff.(ye,t) y e y e (18) where κ eff. (y e, t) = k L2 eq. (y e, t). (19) L 2 min. Here L min. is the minimum contour length for a given y e, which in a simple square geometry is equal to L at all latitudes. The equivalent length L eq. measures the contour length after stirring and is given by L 2 eq. (y e ) = L 2 C (y e ) 1 c c L2 C (y e ), (2) where L C is the length of a tracer contour, L C (y e, t) = ds, and indicates a line average around the tracer contour corresponding to y e. Thus the equivalent length and effective diffusivities are essentially measures of tracer complexity, which accounts for the enhancement in diffusion that arises through the effects of eddy stirring and mixing. 27

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33 Williams, R. G., C. Wilson & C. W. Hughes, 27: Ocean and atmosphere storm tracks: The role of eddy vorticity forcing. J. Phys. Oceanogr. 37, Wolff, J.-O., E. Maier-Reimer & D. J. Olbers, 1991: Wind-driven flow over topography in a zonal β-plane channel: A quasigeostrophic model of the Antarctic Circumpolar Current. J. Phys. Oceanogr., 21,

34 Figure Captions Fig. 1. Snapshots of potential vorticity (PV) in the (a) upper and (b) lower layers of a quasigeostrophic simulation with a flat bottom (see section 2.1 for details). The parameters are βλ 2 /U =.75 and κλ/u =.1. Panels (c) and (d) show snapshots of PV in the upper and lower layers respectively of a simulation with bumpy bottom topography. The topography is defined by a series of sinusoidal bumps as in (16) with l T = 64λ/3 and h y = 1. (see also Figure 2). The topography causes the lower layer meridional PV gradient to vary between -1.25U/λ 2 and.75u/λ 2. All other parameters are the same as the upper panels. Fig. 2. Examples of bottom topographies considered in this study. They include (a) zonally-invariant sinusoidal ridges (11), (b) sinusoidal bumps with zero zonal/meridional means (15) and (c) sinusoidal bumps with non-zero zonal/meridional means (16). Solid and dashed curves indicate positive and negative contours respectively. Panel (d) gives an indication of how topography modifies the barotropic potential vorticity (PV) Q BT λ/u, given by the surface and contours, for a simulation with β =.75, h y = 1.. The colors indicate the magnitude of (Q BT λ/u). This is the same topography used in Figures 1c and 1d; here the meridional, barotropic PV gradient varies between.25u/λ 2 and 1.25U/λ 2. Fig. 3. Survey of the equilibrated number of jets n J in a square domain with sides of length 2π (25λ) as a function of non-dimensional (a) bottom friction κλ/u and (b) PV gradient βλ 2 /U for simulations with a flat bottom (data from Thompson and Young 27). (c) Contours of the ratio k J /k β for the same simulations. The wavenumber k β is defined in (8) and (9), and k J = 2πn J /L, where L is domain size and n J is determined from the observed number of jets. The crosses mark parameters where data is available. Fig. 4. Survey of the jet scale in simulations with a uniform slope h y in the lower layer. (a) The ratio k J /k β (given by the color of the circles) as a function of non-dimensional β and the bottom slope h y. (b) The ratio k J /k β as a function of the barotropic PV gradient 34

35 β BT = β + h y /2. (c) Number of observed jets n J in a 128λ 128λ domain as a function of β BT. Fig. 5. Time series of effective diffusivity κ eff. calculated from the upper layer potential vorticity field (left panels) and the zonally-averaged, zonal barotropic velocity (right panels) for parameters: (a) n R =, h y =, (b) n R = 3, h y = 1., (c) n R = 5, h y = 1., (d) n R = 8, h y = 1., (e) n R = 3, h y =.5 and (f ) n R = 3, h y = 2.. The ratio l β /l T is given for each set of panels. Fig. 6. Survey of the simulations with zonal ridges. (a) The ratio l β /l T (given by the color) as a function of ridge number n R = L/l T and ridge steepness h y (11). Data along the horizontal dashed line is shown in panels (b) and (c); data along the vertical dashed line is shown in panels (d) and (e) In all cases, βλ 2 /U =.75, κλ/u =.1 and L/λ = 64. (b) Time and domain-averaged values of eddy kinetic energy (EKE), zonal kinetic energy (ZKE) and potential energy (PE), defined in (12). (c) Time and domain averaged values of the eddy heat flux (HF) and effective diffusivity κ eff., defined in (13). Panels (d) and (e) show the same statistics as panels (b) and (c) respectively. Domain averaged statistics in panels (b)-(e) have been normalized by statistical properties in a simulation with n R = h y =. Fig. 7. Time and zonal averages of the Reynolds stress forcing of the zonal mean flow (14) by upper layer eddies (bold curve) and lower layer eddies (thin curve) in two simulations: (a) n R = 5, h y = 1. and (b) n R = 3, h y = 2. (Figures 5c and 5f ). The zonally-averaged zonal barotropic velocity is given by the dashed lines. (c) Time series of upper layer Reynolds stresses in a simulation with n R = 3, h y = 1. (Figure 5b). The values are smoothed with a running mean of duration 5λ/U. Fig. 8. Time series of effective diffusivity κ eff. calculated from the upper layer potential vorticity field (left panels) and the zonally-averaged, zonal barotropic velocity (right panels) for parameters: (a) n B =, h y =, (b) n B = 2, h y = 1., (c) n B = 3, h y = 1., (d) 35

36 n B = 3, h y = 2., (e) n B = 6, h y =.5 and (f ) n B = 6, h y = 1.. A time-averaged estimate of the ratio l β /l T is given in each set of panels. Fig. 9. Survey of the simulations with unaligned bumps. (a) The ratio l β /l T (given by the color) as a function of number of bumps n B = L/l T and bump steepness h y (16). Data along the horizontal dashed and dotted lines are shown in panels (b) and (c); data along the vertical dashed-dotted line is shown in panels (d) and (e). In all cases, βλ 2 /U =.75, κλ/u =.1, L/λ = 64. (b) Time and domain-averaged values of total kinetic energy (KE), zonal kinetic energy (ZKE) and potential energy (PE). (c) Time and domain averaged values of the total heat flux and the effective diffusivity κ eff. defined in (13). Panels (d) and (e) show the same statistics as panels (b) and (c) respectively. Domain averaged statistics in panels (b)-(e) have been normalized by statistical properties in a simulation with n R = h y =. Fig. 1. Same as Figure 9 but for simulations with aligned bumpy topography (15). Fig. 11. (a) Time series of the effective diffusivity κ eff. calculated from the upper layer potential vorticity field in a simulation with n B = 2, h y = 1.. (b) The corresponding time series of domain-averaged eddy kinetic energy EKE = KE-ZKE (solid line) and effective diffusivity κ eff. (dashed line); the gray shaded areas correspond to significant jet merger events in the upper layer. Fig. 12. (a) Time series of kinetic energy, potential energy and zonal kinetic energy for a simulation with n B = 3, h y = 2.. (b) Time series of zonally-averaged barotropic zonal velocity u J = ψ y in the same simulation. Panels (c)-(e) show snapshots of upper layer PV at times indicated by the dashed lines to the dashed lines in (a) and (b). Fig. 13. (a) Estimates of l β in a simulation with n R = 3, h y = 1.. The solid curve indicates l β based on domain-averaged values l β = 2π V e /β. The gray and black squares are values for l high β and l low β, where β BT and V e values are calculated by averaging over regions 36

37 with β BT > 11/12 and β BT < 7/12 respectively (β BT takes values between.25 and 1.25 over the domain). The dashed line gives the value of l T. The domain-averaged V e (solid line) and the values of V high e (gray s) and V low e (black s) are given in panel (b). Fig. 14. (a) Time series of effective diffusivity κ eff., eddy kinetic energy (EKE), mean kinetic energy (MKE) and zonal kinetic energy (ZKE) for a simulation with n B = 3, h y = 2 (Figure 8d). The mean fields are calculated from a 2 snapshot running mean spanning a period of 5λ/U. (b) Time series of the ratio l β /l T. Fig. 15. (a) Schematic showing the traditional zonal jet paradigm that strong eastward flows act as barriers to transport, while the region between the jets is well-mixed and becomes homogenized (see discussion of PV staircases in Dritschel and McIntyre (28) and references therein). (b) Schematic of the meandering and variable jet structure associated with the core of the Antarctic Circumpolar Current. The role of transport barriers in meridional mixing and property distributions is poorly understood. Fig. 16. (a) Bathymetry of the Southern Ocean as given by Smith and Sandwell (1997), smoothed with a two-dimensional low-pass filter. (b) Meridional gradient of H 2, where H 2 is taken to be.8h and H is ocean depth. (c) Estimate of the meridional gradient of barotropic PV, which in dimensional terms is given by G BT = β + f η y /1.25H 2. 37

38 64 (a) 64 (b) y/λ (c) (d) y/λ x/λ x/λ Figure 1: Snapshots of potential vorticity (PV) in the (a) upper and (b) lower layers of a quasi-geostrophic simulation with a flat bottom (see section 2.1 for details). The parameters are βλ 2 /U =.75 and κλ/u =.1. Panels (c) and (d) show snapshots of PV in the upper and lower layers respectively of a simulation with bumpy bottom topography. The topography is defined by a series of sinusoidal bumps as in (16) with l T = 64λ/3 and h y = 1. (see also Figure 2). The topography causes the lower layer meridional PV gradient to vary between -1.25U/λ 2 and.75u/λ 2. All other parameters are the same as the upper panels. 38

39 Figure 2: Examples of bottom topographies considered in this study. They include (a) zonally-invariant sinusoidal ridges (11), (b) sinusoidal bumps with zero zonal/meridional means (15) and (c) sinusoidal bumps with non-zero zonal/meridional means (16). Solid and dashed curves indicate positive and negative contours respectively. Panel (d) gives an indication of how topography modifies the barotropic potential vorticity (PV) Q BT λ/u, given by the surface and contours, for a simulation with β =.75, h y = 1.. The colors indicate the magnitude of (Q BT λ/u). This is the same topography used in Figures 1c and 1d; here the meridional, barotropic PV gradient varies between.25u/λ 2 and 1.25U/λ 2. 39

40 15 12 (a) (b) 8 8 n J 4 slope = 1/4 βλ 2 /U = 1/4 2 βλ 2 /U = 1/2 βλ 2 /U = 3/4 βλ 2 /U = κλ/u n J slope = 1 κλ/u =.2 κλ/u =.4 κλ/u =.8 κλ/u =.16 κλ/u =.32 κλ/u = βλ 2 /U 1.25 (c) βλ 2 /U κλ/u Figure 3: Survey of the equilibrated number of jets n J in a square domain with sides of length 2π (25λ) as a function of non-dimensional (a) bottom friction κλ/u and (b) PV gradient βλ 2 /U for simulations with a flat bottom (data from Thompson and Young 27). (c) Contours of the ratio k J /k β for the same simulations. The wavenumber k β is defined in (8) and (9), and k J = 2πn J /L, where L is domain size and n J is determined from the observed number of jets. The crosses mark parameters where data is available. 4

41 1.5 (a) (b) 1.5 k J /k β h y (c) β n J 1 5 k J /k β β BT Figure 4: Survey of the jet scale in simulations with a uniform slope h y in the lower layer. (a) The ratio k J /k β (given by the color of the circles) as a function of non-dimensional β and the bottom slope h y. (b) The ratio k J /k β as a function of the barotropic PV gradient β BT = β + h y /2. (c) Number of observed jets n J in a 128λ 128λ domain as a function of β BT. 41

42 64 (a) l β /l T = 1 4 y e /λ (b) l β /l T = y e /λ (c) l β /l T = y e /λ (d) l β /l T = y e /λ (e) l β /l T = y e /λ (f) l β /l T = y e /λ tu/λ tu/λ 5 Figure 5: Time series of effective diffusivity κ eff. calculated from the upper layer potential vorticity field (left panels) and the zonally-averaged, zonal barotropic velocity (right panels) for parameters: (a) n R =, h y =, (b) n R = 3, h y = 1., (c) n R = 5, h y = 1., (d) n R = 8, h y = 1., (e) n R = 3, h y =.5 and (f ) n R = 3, h y = 2.. The ratio l β /l T is given for each set of panels. 42

43 h y (a) Energy (b) n R Transport (c) n R n R l β /l T Energy (d) EKE.2 ZKE PE h y Transport (e) Eddy heat flux κ eff. (Upper layer) h y Figure 6: Survey of the simulations with zonal ridges. (a) The ratio l β /l T (given by the color) as a function of ridge number n R = L/l T and ridge steepness h y (11). Data along the horizontal dashed line is shown in panels (b) and (c); data along the vertical dashed line is shown in panels (d) and (e) In all cases, βλ 2 /U =.75, κλ/u =.1 and L/λ = 64. (b) Time and domain-averaged values of eddy kinetic energy (EKE), zonal kinetic energy (ZKE) and potential energy (PE), defined in (12). (c) Time and domain averaged values of the eddy heat flux (HF) and effective diffusivity κ eff., defined in (13). Panels (d) and (e) show the same statistics as panels (b) and (c) respectively. Domain averaged statistics in panels (b)-(e) have been normalized by statistical properties in a simulation with n R = h y =. 43

44 6 (a) 6 (b) 6 (c) y/λ tu/λ 1 Figure 7: Time and zonal averages of the Reynolds stress forcing of the zonal mean flow (14) by upper layer eddies (bold curve) and lower layer eddies (thin curve) in two simulations: (a) n R = 5, h y = 1. and (b) n R = 3, h y = 2. (Figures 5c and 5f ). The zonally-averaged zonal barotropic velocity is given by the dashed lines. (c) Time series of upper layer Reynolds stresses in a simulation with n R = 3, h y = 1. (Figure 5b). The values are smoothed with a running mean of duration 5λ/U. 44

45 64 (a) l β /l T = 1 4 y e /λ (b) l β /l T = y e /λ y e /λ (c) l β /l T =.94 (d) l β /l T = y e /λ (e) l β /l T = y e /λ y e /λ (f) l β /l T = tu/λ Figure 8: Time series of effective diffusivity κ eff. calculated from the upper layer potential vorticity field (left panels) and the zonally-averaged, zonal barotropic velocity (right panels) for parameters: (a) n B =, h y =, (b) n B = 2, h y = 1., (c) n B = 3, h y = 1., (d) n B = 3, h y = 2., (e) n B = 6, h y =.5 and (f ) n B = 6, h y = 1.. A time-averaged estimate of the ratio l β /l T is given in each set of panels. 45

46 (a) Energy (b) Transport (c) h y n B n B (d) 1 3 (e) n B Energy KE ZKE PE Transport Heat Flux κ eff. (Upper layer) l β /l T h y h y Figure 9: Survey of the simulations with unaligned bumps. (a) The ratio l β /l T (given by the color) as a function of number of bumps n B = L/l T and bump steepness h y (16). Data along the horizontal dashed and dotted lines are shown in panels (b) and (c); data along the vertical dashed-dotted line is shown in panels (d) and (e). In all cases, βλ 2 /U =.75, κλ/u =.1, L/λ = 64. (b) Time and domain-averaged values of total kinetic energy (KE), zonal kinetic energy (ZKE) and potential energy (PE). (c) Time and domain averaged values of the total heat flux and the effective diffusivity κ eff. defined in (13). Panels (d) and (e) show the same statistics as panels (b) and (c) respectively. Domain averaged statistics in panels (b)-(e) have been normalized by statistical properties in a simulation with n R = h y =. 46

47 (a) Energy (b) Transport (c) h y n B n B (d) 1 3 (e) n B Energy KE ZKE PE Transport Heat Flux κ eff. (Upper layer) l β /l T h y h y Figure 1: Same as Figure 9 but for simulations with aligned bumpy topography (15). 47

48 y e /λ (a) (b) Kinetic Energy Effective diffusivity Time (tu/λ) Figure 11: (a) Time series of the effective diffusivity κ eff. calculated from the upper layer potential vorticity field in a simulation with n B = 2, h y = 1.. (b) The corresponding time series of domain-averaged eddy kinetic energy EKE = KE-ZKE (solid line) and effective diffusivity κ eff. (dashed line); the gray shaded areas correspond to significant jet merger events in the upper layer. 48

49 6 4 (a) Kinetic Energy Potential Energy Zonal KE 2 y/λ (b) tu/λ (c) 6 (d) 6 (e) y/λ y/λ y/λ x/λ x/λ x/λ Figure 12: (a) Time series of kinetic energy, potential energy and zonal kinetic energy for a simulation with n B = 3, h y = 2.. (b) Time series of zonally-averaged barotropic zonal velocity u J = ψ y in the same simulation. Panels (c)-(e) show snapshots of upper layer PV at times indicated by the dashed lines to the dashed lines in (a) and (b). 49