Eddy-mean flow interaction in a multiple-scale ocean model

Transcription

1 1 Eddy-mean flow interaction in a multiple-scale ocean model 2 Bruno Deremble 3 Laboratoire de Météorologie Dynamique ENS, Paris, France 4 5 William K. Dewar Dept. of Earth, Ocean, and Atmospheric Science, Florida State University, Tallahassee, Fl, USA 6 7 Ian Grooms and Keith Julien Dept. of Applied Math, University of Colorado, Boulder, Co, USA 8 9 Roger M. Samelson College of Earth, Ocean, and Atmospheric Sciences, Oregon State University, Or, USA 1 11 Corresponding author address: 24 rue Lhomond, 755 Paris, France bruno.deremble@ens.fr Generated using v4.3.2 of the AMS LATEX template 1

2 ABSTRACT We investigate the eddy-mean flow interaction in the ocean with a multiple scale expansion of the primitive equations. In this framework large-scale dynamics driven by surface heat flux and wind is described by the planetary geostrophic (PG) formalism. Small-scale dynamics driven by the large-scale flow is described by the quasi-geostrophic (QG) formalism. Instabilities in the large-scale flow generate small-scale eddies, which then evolve according the non-linear dynamics. The average feedback of these oceanic eddies rectifies the large-scale flow by flattening isopycnal surfaces and by maintaining a sharp western boundary current. 2

3 21 1. Introduction In the extratropical Atlantic and Pacific basins, the combination of wind and surface heat fluxes maintain the subtropical and subpolar gyres. These gyres have a large-scale component, also known as the ventilated thermocline which is well described by the planetary geostrophic (PG) formalism (Luyten et al. 1983). In places where the large-scale flow is baroclinically unstable, small-scale eddies are generated with characteristic length scale the deformation radius R d, which is O(5 km) in mid-latitudes. The baroclinic instability and the eddy dynamics are well described by the quasi-geostrophic (QG) formalism. Such transfer of energy from large scales to small scales is a quasi universal feature of 3d turbulent flows, and is inevitably occurring in the ocean (McWilliams 216; Musacchio and Boffetta 217). But the ocean is not a 3d isotropic turbulent fluid: rotation and stratification damp vertical motion and maintain the turbulence in a QG regime rather than a fully 3d regime. Quasi-geostrophic turbulence driven by baroclinic instability is known to develop an inverse cascade of energy: conservation of energy and enstrophy force the energy to cascade to the gravest mode where it is ultimately dissipated by bottom friction (Charney 1971). In the ocean context, this means that small-scale eddies merge into bigger eddies to eventually rectify the large-scale circulation (e.g. Scott and Arbic 27). In the present article, we attempt to characterize these energy transfers in the oceanic context and answer the following questions: how and where does the large-scale flow generate baroclinic eddies? How do these eddies rectify the large-scale flow? A popular approach to study the eddy-mean flow interaction is to work with ocean models of increasing horizontal and vertical resolution and to compare the dynamics of eddy-resolving models with coarse resolution models. With this strategy, Penduff et al. (211) showed that the ocean is capable of generating an intrinsic chaotic variability and that the eddy rectification plays 3

4 an important role in the establishment of the large-scale dynamics. However, this methodology is limited by the fact that the small-scale dynamics is function of the slow evolution of the large-scale dynamics: in this hierarchy of models, it is not easy to separate eddies and mean flow because they evolve according to a single set of equations. We propose here to work with a multiple scale model originally derived by Pedlosky (1984) and confirmed and expanded by Grooms et al. (211). The derivation of this multiple scale model relies on a principle of scale separation between the large-scale dynamics and the small-scale turbulent flow. Using this separation of scales, one can decompose the Navier-Stokes equations into two equation sets: an equation set for the large-scale PG dynamics and one equation set for the small-scale QG dynamics. The QG model derived in this framework of multiple scale formalism is somewhat richer than the classic QG equations because the background stratification and the Coriolis parameters are functions of space (and thus, the deformation radius is no longer restricted to be spatially uniform). This contrasts with the classic QG formulation in which the only large scale effect is beta, and for which, mean state stratification is specified and assumed independent of horizontal position. In the derivation of the multiple scale model, Pedlosky (1984) and Grooms et al. (211) highlighted the paradox that there is no explicit feedback from small scales to large scales 1. The absence of feedback in the isotropic case is in contradiction with many examples in the literature who show the crucial role of eddies in the rectification of the large-scale flow. One famous example is the Gent and McWilliams (199) parameterization which states that eddies tap into the large-scale potential energy reservoir and have thus a tendency to flatten isopycnal surfaces. One 1 Grooms et al. (211) also used different scalings to derive a different model where the PG feels the eddies and the eddies do not feel the PG, and another model where there is two-way coupling when the PG scales are anisotropic (e.g. near boundaries). 4

5 65 66 goal of this article is to study this paradox and determine the conditions for which there could be an eddy feedback to the large-scale flow Multiple scale expansion From the primitive equations in the oceanic context, we derive a multiple-scale model as described in Pedlosky (1984) and Grooms et al. (211). We only recall here the main steps of the derivation, as the interested reader can refer to the full derivation in the aforementioned articles. We consider a stratified ocean in a closed domain that we describe with the hydrostatic incompressible Boussinesq primitive equations Du Dt f v = p x + F u + D u Dv Dt + f u = p y + F v + D v p z = b u x + v y + w z = Dρ Dt = F ρ + D ρ, (1a) (1b) (1c) (1d) (1e) with all notations standard, and D and F are unspecified dissipation and forcing. Two nondimen- sional parameters characterize the dynamics of the flow: the Rossby number Ro = u s f R d, (2) with u s a velocity scale, f the local value of the Coriolis parameter, and R d the local value of the deformation radius. The second important nondimensional number is the Froude number 5

6 Fr = u s NH, (3) 77 with N the buoyancy frequency and H the vertical length scale. We also define the aspect ratio a = H L (4) In the oceanic context, we consider that these three non dimensional numbers Ro and Fr, and a are O(ε), with ε a small parameter, and the ratio Ro/Fr O(1). Moreover we also consider that there is a good scale separation between the turbulent length scale R d and the planetary scale L such that their ratio δ = R d /L is another small parameter of the problem: δ O(ε). Because we assume there is a good scale separation, we introduce a system of coordinates (x,y ) to describe the small-scale dynamics and a system of coordinates (X,Y ) for the large-scale dynamics. To account for these two space coordinates, all horizontal derivatives are changed to x x + δ X, and y y + δ Y. (5) 85 Henceforth, we note the gradient on the small-scale coordinates ( ) x, y, (6) 86 and the gradient on the large-scale horizontal coordinates (with a bar) ( X, ). (7) Y In a similar way, we introduce a fast time scale t for the description of the eddy dynamics and a slow time scale T for the description of the gyre-scale dynamics, and we hypothesize that there is a good time scale separation, such that 6

7 t t + δ T. (8) 9 Each variable is now a function of all space and time dimensions as shown here for the variable 91 b b(x,y,z,t) b(x,y,z,t,x,y,t ), (9) where we recall that x and X are independent coordinates. We henceforth drop the primes to des- ignate the small-scale coordinate and the fast time scale. We also introduce an averaging operator (denoted with a bar) to filter the small scales and fast time dynamics as shown here for the variable 95 b b(x,y,z,t ) = 1 b(x,y,z,t,x,y,t )dxdydt, (1) τ f s f s f τ f where the integral is performed over an area of size s f large compared to eddy scale R d and for a time τ f long compared to to eddy time-scale. We expand each variable in a power series of ε, as shown here for the buoyancy b b = ε 1 b + b 1 + εb 2, (11) where the scaling in ε 1 is required to ensure that the small length scale is O(R d ) (cf. Eq. 3.3 in Pedlosky 1984). At order ε 1, we show that b and p are function of the large-scale and slow time coordinate only. At order, we obtain the multi-scale geostrophic balance f v 1 = p1 x + δ p ε X f u 1 = p1 y δ ε p Y. (12a) (12b) 7

8 Since p is function of the large-scale dimensions only, it is convenient to split u 1 and v 1 into two components: (u 1,v 1 ), function of the large-scale dimensions only, and (u 1,v 1 ) = (u 1,v 1 ) (u 1,v 1 ), function of both small- and large-scale coordinates such that (12) can be decomposed as f v 1 = δ p ε X p Y f u 1 = δ ε f v 1 p 1 = x f u 1 p 1 = y. (13a) (13b) (13c) (13d) Finally, at order 1, we obtain two sets of equations: the planetary geostrophic equation for the large-scale dynamics and the quasi-geostrophic equations for the small-scale dynamics that we both describe in the next two sections. 18 a. The planetary geostrophic (PG) equations 19 In dimensional variables, the planetary geostrophic model is f k U = Φ + D U Φ z = B U + w z = B + U B + w B T z = F B + D B + R B, (14a) (14b) (14c) (14d) where Φ is the large-scale pressure (p in the non dimensional notation), U is the large-scale horizontal velocity (u 1 and v 1 in the non dimensional notations), k is the unit vertical vector, and w is the vertical velocity. A dissipative term D U is added in (14a) to enforce the boundary 8

9 conditions (we provide more details on this point in section 3). B is the large-scale buoyancy. For simplicity we used a linear equation of state B = gαθ, (15) with Θ the large-scale temperature, α the thermal expansion coefficient, and g the acceleration due to gravity. The Coriolis parameter is a function of the large-scale latitude coordinate f = f (Y ) and for sake of simplicity we use the β-plane approximation f = f + βy, (16) where f is the Coriolis parameter in the middle of the domain and β is meridional gradient of f. Equation (14a) is the large-scale geostrophic balance (cf. Eq. 13a,b), equation (14b) is the hydrostatic balance, equation (14c) is the conservation of mass for an incompressible fluid, and finally, equation (14d) is the main prognostic equation for the time evolution of large-scale 122 buoyancy. The main terms in this last equation are: a full 3d advective term, a dissipation D B = Ah 2 B + Az 2 B/ z 2 ; with Ah, and Az the horizontal and vertical diffusivity coefficients respectively, and a forcing F B which is a combination of solar input and air-sea heat fluxes. The last term in Eq. (14d), R B corresponds to the eddy feedback to the large-scale flow. In principle, there is no explicit eddy feedback at this order in the multiple scale expansion, but there is a consensus in the ocean modeling community to add such term in low-resolution ocean model to mimic the fact that eddies flatten isopycnal surfaces (Treguier et al. 1997). Gent and McWilliams (199) proposed a parameterization for R B in the form of an additional advective term by the eddy induced velocity R B = u B, (17) 9

10 with u the non divergent eddy induced velocity, itself a function of the slope of the isopycnal surfaces (Grooms 216). Although this parameterization is well accepted it has rarely been validated by a systematic comparison between an eddy resolving and a coarse resolution model. The multiple-scale approach offers this possibility to explicitly compute the eddy field and the eddy feedback to the large-scale buoyancy field (cf. section 4). The main dynamic variable of the PG system is the large-scale buoyancy: the tilt of isopycnal surfaces is related to the strength of the circulation via the thermal wind equation (a quiescent ocean would have flat isopycnal surfaces). We characterize the magnitude of the large-scale oceanic circulation with the available potential energy (APE). The APE is a measure of the deflection of the isopycnal with respect to the horizontal (state of rest), and its formal definition is APE = 1 B(z z )dxdy dz, (18) SH X,Y,z where the integral is computed over the entire domain, normalized by the volume of the domain SH, S being the area of the large-scale domain, and H the depth of the ocean. In the integral, z is the height to which a fluid element would move if the entire domain were adiabatically leveled to give a horizontally uniform stable stratification (Winters et al. 1995; Hughes et al. 29). In an idealized square basin geometry without any topography, and with a linear equation of state, the computation of z is trivial. Note that in the PG formalism, the energy conservation law includes only potential energy, the sole contributor at scales large compared to the deformation radius (Colin de Verdière 1988). There is a large-scale kinetic energy, but it is a diagnostic equation slaved to the evolution of the large-scale buoyancy. In the absence of forcing/dissipation, the conserved quantity is the large-scale potential vorticity (PV) 1

11 Q = f B g z. (19) 152 b. Quasi-geostrophic (QG) equation The small-scale dynamics is governed by the quasi-geostrophic framework. The main variable is the quasi-geostrophic potential vorticity (QGPV) q = 1 f ( 2 ψ + Γψ ), (2) 155 with Γ the vertical stretching operator Γψ = z ( f 2 ) ψ N 2, (21) z and with ψ the pressure field (p 1 in non-dimensional variables). The main difference between this definition of PV and the usual definition found in textbooks (Vallis 26) is that the Coriolis parameter f and the buoyancy frequency N = B z, (22) are both functions of the large-scale coordinates (since the buoyancy B is a function of the large- scale coordinates only). The equation of evolution of the QGPV is q t + u q + U q + u Q = D q + F q, (23) 161 with D q the dissipative effects and bottom friction D q = A 2 f 4 ψ A 4 f 6 ψ r 2 ψ, (24) 11

12 with A 2 and A 4 the harmonic and bi-harmonic viscous coefficients, and r the bottom friction coefficient (non zero in the lower layer only) r = h e 2h n, (25) with h e the thickness of the Ekman layer and h n the thickness of the lowermost layer (cf. ap- pendix A). The last term in Eq. (23), F q is a filter added to ensure that q remains a small scale variable F q = (u q + U q + u Q), (26) This term is in theory not necessary because the secular growth of the small scale PV on the long time scale T is not permitted by the solvability condition. We will see in the numerical implementation of the model that without this term, the model tend to generate a solution with q. In (23), the small-scale velocity field is u = 1 f ( ψ y, ψ ), (27) x which is the dimensional form of (13c-d). The large-scale velocity field U is defined as in Eq. (14a) and the gradient of the large-scale vorticity field is Q = (ΓV,β ΓU). (28) Note that Q is not the exact gradient of Eq. (19). It is written here in a convenient compact form but it is not strictly speaking the gradient of the large-scale PV (see also the appendix of Grooms et al. 211). The small-scale buoyancy field b is related to the stream function via the hydrostatic balance 12

13 ψ z = b. (29) We expect that QG dynamics will draw its energy from the PG APE reservoir. formalism, an estimate of that APE reservoir per unit volume is In the QG E = SH X,Y,z B 2 dxdy dz. (3) N Rather than computing the energy of the QG solution (which is a perturbation to the large scale APE), we compute the total energy in the QG framework E = KE + APE qg = ssh X,Y x,y z ( ) 1 2 f ψ (B + b)2 + N 2 dzdxdydxdy, with s the area of the small-scale domain. the sign of E E will help us understand how the QG field will drain the PG APE reservoir. We find it more relevant to compute E E rather than the energy of the perturbation only (for which we would simply take B = in Eq. (31)) because the energy of the perturbation is positive definite and one cannot tell if there is an increase or decrease of total APE. To get the QG energy equation, one multiplies (23) by (ψ + Φ)/ f and integrates over both the small-scale and large-scale domains (31) de dt = 1 ssh = 1 ssh X,Y x,y z X,Y x,y ψ + Φ q f z (ψ + Φ) t t dzdxdydxdy ( 1 f 2 2 ψ + z ( ) 1 ψ N 2 + ( )) 1 Φ z z N 2 dzdxdydxdy, z (32) We added the last term in (32) because it is independent of the fast time scale (so its time deriva- tive is zero). We can do the usual integration by parts because the two coordinate systems are independent, and we get 13

14 de dt = d dt KE + d dt APE qg = 1 (D q + F q )dzdxdydxdy, (33) ssh X,Y x,y z and we used the fact that the small-scale gradient of the large-scale variable Φ is identically zero ( Φ = ). In a statistical steady state (de/dt = ), the energy dissipation at small-scale via the frictional and viscous operators D q must balance the energy injection at large-scale via the filtering operation F q. 195 c. Numerical formulation 196 There are several challenges related to the numerical implementation of the multiple scale 197 model. In the numerical modeling community, the traditional approach to implement a superparameterization-type model would be to implement the PG model on a coarse grid and subdivide this large-scale domain into a multitude of subdomains on which we can implement the small-scale QG dynamics (e.g. Campin et al. 211). This superparameterization approach has the advantage of being straightforward to implement as each QG domain corresponds to a classic QG (because all large-scale variables are then constant on each QG domain). The downside of this approach is that each domain behaves independently of its neighbors: there is no continuity in eddy dynamics from one QG domain to the other. This is problematic because we expect QG dynamics to exhibit an inverse cascade that we cannot capture with such representation. To overcome this limitation, we choose to implement the QG and PG equations in the same geographical domain. The QG grid is finer than the PG grid but there is only one QG domain that spans the entire PG domain. We interpolate all large-scale variables on the QG grid and we construct with care the spatial operators ( and ) to make sure we do not take spurious spatial derivative of large-scale fields. For example, when we compute the gradient of the small-scale PV (for the advective term in Eq. 2), we first compute the gradient of the stream function ψ and then reconstruct the gradient 14

15 212 of the PV field, as shown here for the x derivative q x 1 ( 2 ψ ) f x + Γ ψ. (34) x For the vertical discretization, one could in principle implement the QG and PG models with the same vertical grid because there is no scale separation in z. However, because the turbulent component is written in terms of PV dynamics, there are some important numerical constraints to ensure that the PV is well defined in the entire domain. An important requirement is that the large-scale stratification N 2 must be non zero to avoid a singularity in the stretching term of the quasi-geostrophic PV. For that reason, we will implement the QG model with a small number of vertical layers (to avoid the vertical discretization of the surface mixed layer and the deep ocean). More details about the construction of the numerical model are given in appendix A Planetary geostrophic dynamics The dynamical variable of the planetary geostrophic model is the buoyancy (cf. Eq. 14): from the buoyancy field, we reconstruct the pressure field with the hydrostatic equation. We compute the horizontal velocities with the gradient of the pressure field, and the vertical velocity is computed with the incompressibility equation. Once the 3d velocity field is known, the buoyancy field can be advected in time. There have been several numerical implementations of this model such as Colin de Verdière (1988), Salmon (199), Samelson and Vallis (1997), Huck et al. (1999), Grooms (216). The main subtlety of the numerical implementation is the treatment of the lateral boundaries: away from lateral boundaries, the planetary geostrophic system of equations is well posed and its numerical implementation is straightforward. But the presence of lateral boundaries adds serious limitations to the model: to satisfy the impermeability condition, the pressure must be uniform at the boundary, such that the buoyancy should also be uniform at the boundary. To 15

16 overcome this limitation, Samelson and Vallis (1997) added a frictional (drag) term in the momentum equation which alters the geostrophic balance. In such conditions the pressure is no longer required to be constant at the boundary. The combination of no normal flow and no flux boundary condition provides a 1d differential equation along the boundary, which is solved to get the buoyancy at the boundary. There are other strategies to handle the boundary condition such as the one proposed by Colin de Verdière (1988): he added a viscous term in the momentum equation to force a viscous-type boundary layer to connect the interior flow to the boundary. This issue of the boundary conditions reveals the limits of the multiple scale expansion near boundaries: the length scale associated with the boundary layer is much smaller than the planetary scale such that the multiple scale expansion takes a different flavor near the boundary (Deremble et al. 217). In the present study, we opt for Samelson and Vallis (1997) s implementation because the western boundary current is handled with a Rayleigh friction rather than with a viscous operator. The full description of the planetary geostrophic model can be found in Samelson and Vallis (1997) and we recall here the value of the main parameters used in the model. The model is meant to capture the large-scale dynamics of an extra-tropical basin such as the north Atlantic ocean. The equations are integrated in a square domain 5 km 5 km wide, and 5 m deep. We keep the same dimensional scaling as in Samelson and Vallis (1997) except for the vertical velocity: we choose W = m s 1 compared to their value of W = 1 6 m s 1. We made this modification to get a better agreement between the deformation radius computed in the model and the observed mid-latitude deformation radius (Chelton et al. 1998). The Coriolis parameter in the middle of the domain is f = s 1 and β = m 1 s 1. The model is forced at the surface with via a relaxation to a prescribed buoyancy profile with uniform meridional gradient of m s 2 km 1. The surface wind forcing drives a steady 16

17 256 barotropic response. The wind stress profile has a zonal component only τ X = ρ w e f L ( ) 2πY 2π sin, (35) L with w e = m s 1, the strength of the Ekman pumping. This wind profile was originally chosen to ensure that there is no Ekman flow at the northern and southern boundaries. We plot in Fig. 1a the large-scale pressure field averaged over the upper 2 m. This pressure field is the sum of a barotropic component that develops in response to the wind forcing, and of a baroclinic component which is driven by horizontal buoyancy gradients. With the chosen wind profile, the model captures the entire subtropical gyre and two smaller cyclonic gyres near the northern and southern boundaries. The flow in the western boundary is intensified even though the model is not capable of handling a realistic western boundary current. The strength of this circulation decreases with depth (not shown). The corresponding upper layer buoyancy field is plotted in Fig. 1b. This solution exhibits a large-scale north-south gradient of buoyancy, as expected from the atmospheric forcing. A warm core western boundary current is present and detaches from the western boundary near the 4 km tick. We compute the first deformation radius at each location as the smallest eigenvalue of the Sturm-Liouville equation (A2). We will discuss the consequences of the large-scale stratification more in details in section 4, but for now note only that the spatial variations of the deformation radius are mostly influenced by the variation of the Coriolis parameter. The vertical buoyancy profiles shown in Fig. 2 are taken along the dashed line in Fig. 1b. We only plot the upper 2 m as the deep-ocean buoyancy (below 2 m) is almost uniform. Unstratified deep ocean is a characteristic feature of closed basin models for which the deep circulation cannot be maintained by an inter-hemispheric water exchange (Nikurashin and Vallis 212). The thermocline which separates the deep ocean from the ventilated layers is visible in the north-south 17

18 section. The depth of this internal boundary layer is determined by a combination of the surface heat fluxes and the interior diapycnal diffusivity (Samelson and Vallis 1997). Near y = 25 km we see a pool of weakly stratified water that is reminiscent of the subtropical mode water (Deremble and Dewar 213). In this model, it is not clear whether this mode water is maintained by a surface buoyancy flux or the Ekman flow convergence at the surface (Dewar et al. 25). The western boundary current is clearly visible in the east-west section (strong tilt of the isopycnal surfaces and warm core at the surface). We measure the amplitude of the tilt of the isopycnal surfaces with the APE. For this particular PG solution, we get a value of APE = 12 m 2 s 2 (energy per unit volume). This potential energy will be the reservoir from which the quasi geostrophic dynamics will draw its energy. This solution was computed with R B =. To examine how the GM parameterization would modify this solution, we plot in Figure 1c the term R B averaged between 1 m and 325 m. This terms was calculated following the procedure described in Grooms (216) with a diffusivity coefficient κ = 2 m s 2. The effect of this term is to warm the subsurface layer near the northern edge of the domain. In the western boundary current area there is a dipole pattern which effect is to restore the isopycnal towards a state of rest. These two regions (northern and western boundaries) are indeed regions where isopycnal surfaces are almost vertical (cf. Fig. 2). Given the vertical structure of the PG solution, we can now choose the vertical layer decomposition in the QG model to ensure that N is non-zero everywhere: we avoid the vertical discretization of the surface mixed layer and capture the largely unstratified bottom of the ocean as a single layer. We implemented a 4-layer QG model with thicknesses equal to 2 m, 45 m, 75 m, and 3 m (from top to bottom) in order to have a good representation of the thermocline. All large-scale variables are averaged over the depth of each individual layer. After this vertical averaging, there are still places where N 2 is small especially in the northern part of the domain where 18

19 deep convection occurs. In such regions, we artificially increase the stratification by imposing a minimum reduced gravity g min = 1 3 m s 2. Non-hydrostatic dynamics might actually be more appropriate in the northern part of the domain (Julien et al. 26) Quasi-geostrophic dynamics The large-scale PG solution described in section 3 is a stationary flow for the fast time QG dynamics. In this section, we describe the small-scale QG turbulent flow driven by the large-scale PG flow. Instabilities in the QG flow will grow in places where the PG flow is baroclinically unstable: infinitely small perturbations will first grow exponentially according to the linear theory, they will then saturate and evolve according to the fully non-linear QG dynamics. To identify these most unstable modes, we first conduct a linear stability analysis of the multiple scale QG equation (Smith 27). 313 a. Linear dynamics: barotropic and baroclinic instabilities The large-scale buoyancy field (Fig. 1 and 2) is stable with respect to PG dynamics but this stratification may be unstable with respect to QG barotropic and baroclinic instabilities. To identify these instabilities, we consider the linear inviscid QG equation q t + U q + u Q =. (36) 317 We substitute the plane-wave solution of the form ψ = ˆψ(z)exp(iKx ωt) + c.c., (37) 318 with ˆψ(z) the complex vertical structure and c.c. the complex conjugate, such that (36) becomes 19

20 (KU ω)( K 2 + Γ) ˆψ + ˆψ(kQ Y lq X ) =, (38) and where we used the definition of q in (2). The gradient of large-scale PV Q (Eq. 28) is computed with the PG solution presented in section 3. At each point (X,Y ), this equation is an eigenvalue problem for ω when K is set. We solve this eigenvalue problem on an array of (k,l) values and look for the most unstable mode, which corresponds to the maximum of the real part of ω, noted ω m. If the system does not have any eigenvalues with positive real part, then the large scale flow is stable. In Fig. 3, we plot ω m in the (k,l) space in the middle of the physical domain (X = 25 km, Y = 25 km). The k and l axis are normalized by the local deformation radius (R d ) at this specific location. In the middle of domain, there is only one local maximum at kr d =.9 and lr d =.6 and the associated time scale of the most unstable mode is 7 days (label 1 in Fig. 3). The extension of the unstable region in the (k,l) space is reminiscent of the classic baroclinic instability: the 2 lobes are oriented in the direction of the mean flow, which is in the south-west direction in this example (cf. Fig. 1a). As in the canonical 2-layer Phillips model, the most unstable mode (label 1) projects both on the barotropic and first baroclinic components (the modal coefficients on the barotropic and first baroclinic modes are.25,.14); the projection on the second and third baroclinic modes is at least one order of magnitude smaller. We refer to label 1 as a baroclinic instability. The region near label 2 in Fig. 3 corresponds to a barotropic instability with a time scale of 13 days while near label 3 (smaller k and l), the unstable mode projects almost only on the first baroclinic (with a time scale again of 13 days). We do the same computation to get the most unstable mode at each physical location (X,Y ) of the PG domain, and we plot in Fig. 4 a map in the physical space of the time scale of the most unstable mode. We divide this map into 3 subdomains: region 1 where the growth time scale is faster than 25 days (dotted area), region 2 where the growth time scale is between 25 and 1 days (hatched area), and 2

21 region 3 where the growth time scale is higher than 1 days. The delimitation of these regions is mostly zonal with the exception of the western boundary and the mode water area. The structure of the unstable mode in the mode water area is still a baroclinic instability, but the longer time scale is set by the weak surface stratification in this area (not shown). In region 1, instabilities will grow at the fastest rate near the northern boundary, a region of intense return flow. The time scale of the instability in this area is order of one day and is clearly the dominant mode in the entire domain. We may question the physical relevance of these unstable modes as they are located in a region of strong upwelling in the PG model that is in fact due to the poor representation of the boundary flow in the PG system. We also emphasize that the linear instability analysis may not be valid near the western and southern boundaries, since the unstable modes are too big to fit into a local patch of ocean (Hristova et al. 28). Still in region 1, the region of the separated western boundary current is also subject to fast growing instabilities and we expect energetic eddies to be generated in this area. The region near the southern boundary does not exhibit any instability (the QG flow is stable there). This is due to the presence of the β effect which is known to stabilize zonal flows. The left panel in Fig. 5 is the length scale K of the most unstable mode. To facilitate the comparison between this figure and Fig. 4 we report the regions 1, 2, and 3 with the same convention as in Fig. 4. In first approximation, the modes with the shortest wave length also have the fastest growth rate. In region 1, the wavelength of the most unstable mode is between 5 and 1 km, whereas in regions 2 and 3, it falls between 1 and 5 km. There are zones of discontinuities (e.g. the long zonal band next to the northern boundary, or the diagonal in the middle of the mode water area). These discontinuities are present when there are two local maxima (most unstable modes) in the (k,l) space and when the most unstable mode switches from one maxima to the other (for example, in Fig. 3, that would be a transition between the absolute maximum (label 1) 21

22 and another local maximum as (e.g. label 3) for which the modulus of K is discontinuous between these two points). The ratio of the most unstable length scale and the deformation radius is plotted in the right panel in Fig. 5. Overall, the length scale is always bigger than the deformation radius 2. This ratio is the highest in the western boundary region where the instability grows on spatial scale order 1 times bigger than the deformation radius. The southern part of the gyre is also a region that favors the development of instability that are spatially large. There are places where the most unstable mode projects on a higher baroclinic mode (similar to the surface intensified modes shown in Smith (27). In our 4-layer configuration, the severe vertical mode truncation will naturally damp these modes. However, we show in appendix B that higher vertical resolution changes the picture dramatically with the appearance of surface intensified modes at high wave numbers (Smith 27). 377 b. From linear to non linear dynamics We now retain all terms in (23) except F q. The dissipative term D q is a combination of harmonic and bi-harmonic viscous operators that are meant to represent sub-grid scale phenomena (bottom friction will be added later). We choose A 2 = 1 m 2 s 2, and A 4 = 1 1 m 4 s 2, which are standard values in realistic modeling context (Chassignet and Garraffo 21). Such dissipative effects are a positive definite energy sink. Note that with these terms, the most unstable modes are slightly different than the one shown in the previous section. The non-linear model is integrated forward in time with an Adam-Bashforth scheme on a uniform horizontal grid (7.5 km grid spacing). 2 We chose to characterize the size of the instability with the wavelength of the most unstable mode. Another possibility would have been to choose half of a wave length. 22

23 Figure 6 is a time series of the kinetic and potential energy computed in the QG framework as stated in (31). We subtract to the potential energy the APE of the PG state projected on the 4 layer truncation: E 4l = m 2 s 2. Note that this value is 2 orders of magnitude smaller than the APE of the full PG state. Two reasons explain this discrepancy. First, when we compute the large- scale stratification N 2, we introduce a threshold such that the QGPV is well defined everywhere (cf. end of section 2c). This threshold introduces a spurious stratification and artificially decreases the APE. And second, the computation of the APE in Eq. (3) is done with the QG APE formula which is a first order Taylor expansion of the exact formula (cf. Pedlosky 1987). Such formula is not expected to be valid in regions of weak stratification. During the first 5 years of the integration, we notice a decrease of APE and an increase of KE. This corresponds to the classic scenario of the baroclinic instability: energy is drawn from the background PG flow and converted into smaller scale KE. The total energy is not constant because of the viscous sink which dissipates energy. To evaluate this sink and the validity of the energy budget, we plot in Fig. 6b the time derivative of the energy and the energy tendency due to the viscous terms (right hand side of Eq. (33) without F q ). There is a good agreement between the two curves for the first 2 years, and then, while the viscous terms remain an energy sink, de/dt tends to zero (statistical steady state). This discrepancy means that after 2 years, there is a spurious energy injection that is not physically relevant. To explain this discrepancy, we plot in Fig. 7 the mean upper layer stream function in the statistical steady state (between year 25 and 5). This map exhibits a large-scale circulation with opposite sign and comparable magnitude with respect to the PG solution. We conclude that since the QG solution contains a large-scale component, we are beyond the limit of validity of the scale separation. Indeed, the QG equation is only expected to be valid locally and for short time scale (which is not the case in Fig. 7). We attribute the origin of such large-scale mode in the QG model to the fact that QG dynamics has a natural tendency to 23

24 develop an inverse cascade of energy. This has been shown for idealized QG models of turbulence (e.g. Scott and Arbic 27; Arbic and Flierl 24; Venaille et al. 214), but this is also true for this QG model even if the equations are written with small-scale and large-scale variables. We can thus see Fig. 7 as a first estimate of the small-scale feedback (rectification) to the large-scale flow. In the next section we propose several ideas to remove this large-scale component from the QG flow and eventually work with a fully coupled PG-QG system of equations. 416 c. Toward a fully coupled PG-QG system The PG and QG equations are supposed to handle the large-scale and small-scale dynamics respectively. But as we saw in the previous section, it is not obvious how to untangle large- and small-scale dynamics in the QG system. To address the issue of the inverse cascade highlighted in the previous section, we need to design a filter to remove the stationary large-scale component of the QG solution (term F q in (23)) and add it to the PG solution (term R B in (14d)). These two steps correspond to the implementation of the missing coupling between the small-scale eddies and large-scale circulation. We expect that the effect the coupling will be to flatten isopycnal surfaces which corresponds to the classic view of eddies rectification (Gent and McWilliams 199). There are several ways to implement such a filter: according to the multiple scale expansion formalism, the small-scale PV should vary with the short time scale and on the small-scale grid. We first adopt a simple filtering strategy: every year, we remove the mean of the past year. The advantage of such filter is that the QG dynamics is not altered except at very specific moment where the solution is rescaled. This filter lacks realism in two ways: first it introduces a discontinuity in the solution. And second because the time window of 1 year may not be relevant for all regions of the domain: according to Fig. 4, we know that the time scale of the instabilities varies by a factor of 1 from the southern boundary of the domain to the northern boundary. In the northern 24

25 part of the domain, instabilities will saturate faster, and we expect that the system will reach a statistically steady state in less than one year. With all these deficiencies in mind, we analyze the new KE and APE time series in Fig. 8. The periodic rescaling of the solution appears clearly on the time series: every year, roughly 5% of the potential energy is removed from the QG solution. In a fully coupled model, this buoyancy anomaly would be transferred to the PG dynamics and would be damped by air-sea heat fluxes. The KE time series is barely affected by these periodic rescalings, which is consistent with the fact that there is little transfer of kinetic energy from the QG to the PG model. We also validate the energy budget by comparing the time derivative of the total energy (omitting the discontinuities) with the integral of the dissipative effects (Fig. 8b). The match between these two time series is still not perfect because of the aforementioned caveats in the choice of the filter, but the agreement is still better than without any filtering (Fig. 6b). To get a better overview of the effect of the filter, we plot in Fig. 9 the upper layer QG buoyancy b before and after the rescaling operation. We superimpose contours of the PG buoyancy B projected on the upper QG layer. As anticipated, the large-scale QG buoyancy is of opposite sign compared to the PG buoyancy. The magnitude of this rectification is not uniform in the domain: it is maximum in the northern part of the subtropical gyre and near the northern boundary. This corresponds to region 1 in Fig. 4 where the growth time scale of the instability is shorter than 25 days. The rescaling of the QG solution leaves most of the small-scale, fast dynamics unchanged, as can be seen in Fig. 9b. We plot in Fig. 9c the average buoyancy difference before and after the filtering, which corresponds to a buoyancy tendency over a period of one year (rescaling interval). This plot corresponds to the term R B in Eq. (14d). The pattern and the magnitude of this term are different than the GM estimate plotted in Fig. 1c. In both cases, the rectification term acts to flatten isopycnal surfaces but in the QG case, the bulk of the rectification occurs in the northern 25

26 part of the subtropical gyre. In the GM case, the rectification is maximum in the western boundary current and near the northern edge 458 d. Role of bottom friction We have not mentioned the realism of the QG solution yet. In fact, we have discussed the coupling mechanism in an idealized framework but we are missing several elements that will give more realistic pattern of turbulence without fundamentally affecting the aforementioned mechanism. One such element known to have a tremendous impact on the development of turbulence is bottom friction (see e.g. Arbic and Flierl 24; Venaille et al. 214). Bottom friction will indeed halt the inverse cascade and so in our framework, it may modify the QG PG transfer of energy. In the solution shown in Fig. 9 (without bottom friction), the flow settles in a barotropic state such that after the transient, nearly 1% of the eddy kinetic energy is in the barotropic mode. To halt the barotropization of the flow, we add a bottom Ekman layer of thickness h e = 1 m, which corresponds to a spindown time of 8 days (computed with an average value of the Coriolis parameter). In this configuration, the QG-PG coupling mechanism highlighted in the previous section is still present but the eddies remain in a baroclinic structure and their length scale is at least twice smaller than in the configuration without bottom friction. In the simulation with bottom friction, 65% of the kinetic energy is in the barotropic mode and 3% of the kinetic energy is in the first baroclinic mode. These numbers are in fair agreement with the study of Wunsch (1997) who estimated similar percentages with mooring data. In the North Atlantic map of Wunsch (1997), there is a clear increase of the fraction of kinetic energy in the first baroclinic mode over the mid-atlantic ridge reaching 6% of the total eddy kinetic energy whereas in the rest of the domain the fraction of the kinetic energy contained in the first baroclinic mode is in the range 2-3%, which is more in agreement with our estimate. Surprisingly the pattern and the magnitude of the rectification R B 26

27 computed in the cases with and without bottom friction are similar even though the eddy field is very different (not shown). 481 e. Implementation of two other filters Although the simulation with bottom friction and periodic rescaling reaches a statistically turbulent steady state, we wish to implement different filtering strategies to see the effect on R B. We first implement an exponential runing average of the PV field that we subtract every 2 days to the PV. We compute the average potential vorticity as q(t + dt) = αq(t) + (1 α)q(t), (39) where q(t + dt) and q(t) are the potential vorticity at time t + dt and t respectively and α the smoothing factor. The characteristic time scale associated to this filter is τ f = dt/α and for this experiment, we choose τ f = 2 days. To illustrate the strength of the eddy field in this solution compared to the large-scale flow, we plot a snapshot of the sum of the QG and PG upper level stream function in Fig. 1a. The vigorous eddy field modifies the large-scale flow (Fig. 1a) mostly in the region of the separated western boundary current. Note that the QG flow in this figure is similar to a snapshot of the QG flow obtained with the periodic rescaling strategy (not shown). We plot in Fig. 1b the rectification term computed as the average of the fields subtracted with the filter divided by the filtering frequency (2 days). The pattern is more consistent with the GM term (Fig. 1c): most of the rectification occurs in the western boundary current and its extension. The cooling pattern in this area is more pronounced and acts to flatten isopycnal surfaces in the mode water area (see Fig. 2 and the orange patch in Fig. 4). The magnitude of the three estimates of the rectification term (Figs. 1c, 9c and 1b) should be interpreted with care since the GM term 27

28 was computed with an arbitrary diffusivity coefficient (κ = 2 m s 2 ) and the magnitude of the patterns in Fig. 1c depends on the rescaling interval. Last, we also implemented a spatial filter to remove the large-scale component of the QG solution. To design an optimal spatial filter, the challenge is to find the best length scale cutoff l c for the filter to only let the turbulence develop and not the inverse cascade. In practice, we first implemented a Gaussian space filter with cutoff length scale l c that has the interesting property of commuting with the spatial derivatives if l c is uniform (Leonard 1974) q(x) = q(x ) g(x x )dx with g(x) = 1 ) 2πlc 2 exp ( x2 2lc 2. (4) We did two runs with l c = 75 km and l c = 15 km and we verified with an energy budget that both runs have little spurious energy injection by the advection operators (not shown). The downside of this filter with uniform l c is that the characteristic size of the eddies is uniform in the entire domain which is not realistic: we know from the linear stability analysis that eddies should be smaller in the northern part of the domain compared to the southern part of the domain. Guided by our results on the linear stability analysis, we define a cutoff length scale which is function of the large-scale coordinates l c (X,Y ). We set its minimum value to 75 km (based on the simulation with uniform l c ), and its maximum value to 45 km (an estimate of the scale beyond which the dynamics should be handled by the PG equations). Between these two extrema, we adjust the spatial pattern of l c to match the inverse time scale of the most unstable mode (Fig. 4). Another choice would have been to set the spatial pattern of l c to match the length scale of the most unstable mode (Fig. 5). These two maps are in fact very similar but the time scale map is smoother. Both the stream function snapshot and the rectification terms obtained with this filtering strategy (Fig. 11a,b) are similar to the one obtained with the running average strategy (dominant eddy activity in the northern part of the domain; unstable western boundary current). The main difference 28

29 is the magnitude of the rectification term which is weaker with the spatial filter. However, we do not wish to conclude that this spatial filtering strategy is better than the periodic time rescaling or the running average because we chose the filtering length scale l c (X,Y ) based on the linear stability analysis and not based on considerations on the non-linear dynamics Conclusion We implemented a prototype multiple-scale ocean model for which the large-scale component is described by PG dynamics and the small-scale component is described by QG dynamics. This model relies on the scale separation between the (small) eddy scale and the (large) thermocline 53 scale. The main conceptual difficulty with this model is that, while the eddies are driven by the large-scale flow, there is no explicit feedback from small to large scales (in the formalism). However, the numerical implementation revealed that the QG model, driven by baroclinic and barotropic instabilities exhibits an inverse cascade that rectifies the large-scale flow. The exact nature of this rectification depends on the nature of the filter implemented in the QG model. Indeed, to run the QG model for long time scales (which is normally not permitted by the scale separation), we need to filter the large-scale component that appears in the QG solution. This large-scale component extracted from the QG flow is in turn used to rectify the large-scale PG flow. The outcome of this sequence of operation is that the QG and PG systems are fully coupled. Interestingly, the coupling is strongest in the northern and western regions where the large-scale flow is anisotropic (an element already emphasized by Grooms et al. 211). Three different filtering strategies revealed that the role of the eddies is to flatten isopycnal surfaces in agreement with the classic picture of the eddy-mean flow interaction (Gent and McWilliams 199). The identification of the exact nature of this coupling is now a matter of finding the right filter which would have minimal effect on the small scale dynamics and such a filter remains to be determined. 29