Prognostic Residual Mean Flow in an Ocean General ... - AMS Journals

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Prognostic Residual Mean Flow in an Ocean General Circulation Model and its Relation to Prognostic Eulerian Mean Flow JUAN A. SAENZ, QINGSHAN CHEN,* AND TODD RINGLER Fluid Dynamics and Solid Mechanics, Los Alamos National Laboratory, Los Alamos, New Mexico (Manuscript received 2 February 2015, in final form 19 May 2015) ABSTRACT Recent work has shown that taking the thickness-weighted average (TWA) of the Boussinesq equations in buoyancy coordinates results in exact equations governing the prognostic residual mean flow where eddy–mean flow interactions appear in the horizontal momentum equations as the divergence of the Eliassen–Palm flux tensor (EPFT). It has been proposed that, given the mathematical tractability of the TWA equations, the physical interpretation of the EPFT, and its relation to potential vorticity fluxes, the TWA is an appropriate framework for modeling ocean circulation with parameterized eddies. The authors test the feasibility of this proposition and investigate the connections between the TWA framework and the conventional framework used in models, where Eulerian mean flow prognostic variables are solved for. Using the TWA framework as a starting point, this study explores the well-known connections between vertical transfer of horizontal momentum by eddy form drag and eddy overturning by the bolus velocity, used by Greatbatch and Lamb and Gent and McWilliams to parameterize eddies. After implementing the TWA framework in an ocean general circulation model, the analysis is verified by comparing the flows in an idealized Southern Ocean configuration simulated using the TWA and conventional frameworks with the same mesoscale eddy parameterization.

1. Introduction Mesoscale eddies have a leading-order effect on the momentum and tracer budgets of the ocean and therefore on the dynamics and climate of the ocean and Earth system. However, it is not feasible for climate models to routinely resolve mesoscale eddies, and we therefore need to resort to parameterizing them with simplified models. The conceptual approach used to represent unresolved mesoscale eddies in ocean general circulation models (OGCMs) is to average the governing equations over an ensemble of realizations. This results in a set of governing equations in which the prognostic variables

Denotes Open Access content.

* Current affiliation: Department of Mathematical Sciences, Clemson University, Clemson, South Carolina.

Corresponding author address: Juan A. Saenz, P.O. Box 1663, Los Alamos, NM 87545. E-mail: [email protected] DOI: 10.1175/JPO-D-15-0024.1 Ó 2015 American Meteorological Society

are averages of the prognostic variables in the original system. Importantly, the averaged governing equations contain statistics of unresolved processes that we are required to model through parameterization. The Boussinesq equations are commonly used to represent the ocean circulation in climate models. A common approach, based on the fact that to a large extent the flow in the ocean occurs along surfaces of constant buoyancy, is to average the equations in isopycnal coordinates (Andrews and McIntyre 1976; de Szoeke and Bennett 1993). The prognostic variables in the resulting governing equations obtained this way represent the residual mean flow. However, the details of the residual mean flow equations are subtle and complicated, some aspects of which have remained uncertain until recently. The three-dimensional residual mean flow equations were first derived by de Szoeke and Bennett (1993) by 1) averaging the Boussinesq equations over scales up to representative convective length scales such as to produce an averaged field that is statically stable; 2) transforming the microscopically averaged equations to isopycnal coordinates; and then 3) taking the ensemble average at constant density surfaces, weighted by isopycnal thickness per unit density, over macroscopic

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length scales up to representative planetary wave scales. This approach is referred to as the thickness-weighted average (TWA) framework. Recently, Young (2012) generalized the work of de Szoeke and Bennett (1993) by taking the average in step 3 in buoyancy coordinates and weighting by the buoyancy layer thickness per unit buoyancy. In the momentum residual mean flow equations that result, eddy correlations are represented as the divergence of the Eliassen–Palm vectors (Eliassen and Palm 1961). A critical result of the formulation in Young (2012) is that the residual mean flow equations obtained are exact. Maddison and Marshall (2013) then showed that the eddy correlations in the momentum equations are in fact the result of the divergence of the Eliassen–Palm flux tensor (EPFT) and, among other things, discussed the gauge term and the equivalence between previously derived forms of the EPFT within this gauge term. In the prognostic residual mean flow equations, eddy–mean flow interactions appear in the momentum and tracer equations but not in the thickness or the density equations. The EPFT tensor can be used to diagnose eddy–mean flow interactions in eddy-resolving simulations (e.g., in western boundary currents; J. A. Saenz et al. 2015, unpublished manuscript). The system of equations used to represent the statistics of unresolved eddying dynamics has traditionally been the momentum equations for the Eulerian mean flow and equations for tracer budgets in which advection is done by the residual mean velocity. This approach derives from the seminal work in Gent and McWilliams (1990, hereinafter GM) and Gent et al. (1995), which has had a very large impact on our ability to represent ocean circulation in climate system models (Danabasoglu et al. 1994). Alternative frameworks that solve for the prognostic residual mean flow in OGCMs have been proposed (Ferreira and Marshall 2006; Eden and Greatbatch 2008). However, these approaches are intrinsically limited by the approximations used to derive the governing equations in each framework. It has been argued though that the exact nature and mathematical tractability of the governing equations and the clear physical interpretation of the EPFT and its relation to potential vorticity make the TWA framework a natural choice with intrinsic advantages for simulating ocean circulation with parameterized mesoscale eddies (Young 2012; Marshall et al. 2012; Maddison and Marshall 2013). However, the feasibility of the TWA framework for this purpose has not been tested. Furthermore, even though some parallels have been made between the TWA framework and the more conventional and well-established Eulerian mean flow–based framework [for a summary, see, e.g., Maddison and Marshall (2013)], the connections between these two

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frameworks have not been thoroughly explored. We address these two issues in this paper. It should be noted that many parameterizations used in ocean general circulation models rely on the Eulerian mean state of the flow. This may constitute a challenge for the implementation of these parameterizations in ocean general circulation models that use the TWA framework. These and other subtleties are out of the scope of this paper but will have to be addressed before the TWA framework is a viable choice for coupled climate simulation. In section 2, we start by presenting the TWA equations for the residual mean flow, along with terms and quantities that will be used throughout the paper. Then, in section 3, we simplify and reduce the Eliassen–Palm stress tensor that appears in the horizontal momentum equations and show that, under certain assumptions and limitations, this tensor can be reduced to the parameterization proposed by Greatbatch and Lamb (1990, hereinafter GL) and, equivalently, to the expression used to parameterize the eddy fluxes of thickness height perturbations in GM. After this, in section 4, we summarize the governing equations that are now used in conventional (CNV) OGCMs to represent the ocean circulation with parameterized eddies. In section 5, we use Ertel potential vorticity of the residual mean flow that results from the governing equations of the TWA and CNV frameworks, respectively, to illustrate the connections between the two. Finally, we proceed to verify our analysis by illustrating the results in a quantitative manner. For this, in section 6, we implement both the CNV and TWA frameworks in an OGCM and use the same mesoscale eddy parameterization to solve for the flow in an idealized configuration that resembles the flow in the Southern Ocean. In section 7, we provide a discussion of the results and summarize our findings.

2. Governing equations for the residual mean prognostic variables The residual mean flow equations are obtained by averaging the Boussinesq equations over microstructural length scales to produce a stably stratified field in which buoyancy decreases monotonically with depth, converting the resulting equations to buoyancy coordinates and then taking the average on constant buoyancy surfaces, weighted by buoyancy layer thickness. After this the equations can be transformed back to depth-based ‘‘z coordinates.’’ Following the notation in de Szoeke and Bennett (1993) and in Young (2012), the residual mean flow equations, or TWA equations, in z coordinates are

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DY u^ ›pY 1 $ Eu 5 R^x , 2 f ^y 1 ›x Dt DY ^y ›pY 1 f u^ 1 1 $ Ey 5 R^y , Dt ›y

(2)

indicate derivatives with respect to x~ and y~, where the tilde symbol e is used to indicate quantities at constant buoyancy. The double prime represents deviations from TWAaveraged quantities, for example,

(3)

u00 5 u 2 u^ ,

(4)

and primed quantities represent deviations from the ensemble average, for example,

(1)

›pY 5 bY , ›z ›^ u ›^y ›wY 1 1 5 0, and ›x ›y ›z DY bY 5 0, Dt

(5)

where u, y, and w are the zonal, meridional, and vertical velocities; p is pressure; R^x and R^y contain dissipation and forcing terms for the zonal and meridional momentum equations; b is the buoyancy of a fluid parcel; and x, y, and z represent the zonal, meridional, and vertical (depth) coordinates, respectively. The caret or ‘‘hat’’ symbol is used to represent the thickness-weighted average of a quantity, for example, u^ 5

us , s

(6)

where s5

›bY ›z

21 (7)

is the layer thickness per unit buoyancy. The overline symbol in the TWA framework, as in s and us, represents an ensemble average at fixed x, y, bY, t, and the sharp symbol Y indicates quantities associated with the residual mean flow without being averages themselves. In this sense, bY is the buoyancy variable in the TWA framework, and the depth coordinate z represents the mean depth of buoyancy layer bY . The material derivative following the residual mean velocity in z coordinates is given by DY › › › › 5 1 u^ 1 ^y 1 wY . ›x ›y ›z Dt ›t

(8)

Maddison and Marshall (2013) show that the vectors E and Ey are the first two columns in the EPFT:

z0 5 z 2 z .

1 0C C C C 0C C, C C A 0

(9)

where z is the z coordinate of a given buoyancy surface, and m 5 p/r0 2 bz is the Montgomery potential. Subscripts

(11)

The individual terms of the EPFT shown in (9) represent eddy Reynolds stresses or horizontal transfer of horizontal momentum ( b u00 u00, b u00 y 00, and b y 00 y 00 ), eddy potential 21 02 energy [(2s) z ], and interfacial form drag or vertical transfer of horizontal momentum (s21 z0 m0 and x~ s21 z0 m0 ), which is proportional to the eddy buoyancy y~ fluxes. The expression given in (9) is only unique to within a rotational gauge term that is divergence free and does not affect the dynamics of the flow (Maddison and Marshall 2013). The divergence of the EPFT gives forces that result from interactions between eddies and the mean flow or the eddy potential vorticity flux. Thus, $ Eu and $ Ey in (1)–(2) are the horizontal forces in x and y, respectively, associated with eddy–mean flow interactions. Recall that the only assumption required for deriving the TWA equations for the residual mean flow is that the flow be stably stratified. This assumption is rather minor for the implementation of the TWA framework, as it is satisfied naturally as a result of averaging over microstructural scales (Young 2012). The significance of the TWA (1)–(5) for the residual mean flow is that eddy– mean flow interactions appear only in the momentum equation through the divergence of the EPFT. Decomposing s, u, and y in (6) into the ensemble average and fluctuations from the average as in (11), the thickness-weighted average velocity is expressed as the sum of the ensemble average velocity and eddy thickness fluxes normalized by the mean layer thickness, namely, u^ 5 u 1

s0 u0 , and s

(12)

^y 5 y 1

s0 y 0 . s

(13)

u

0 1 02 00 00 00 y 00 b b u Bu u 1 2s z B B B 00 y 00 00 y 00 1 1 z0 2 b E5B u yb B 2s B B @ 1 0 0 1 0 0 zm zm x~ y~ s s

(10)

Isopycnal thickness fluctuation fluxes in (12)–(13) are often interpreted as the eddy-induced velocities, or bolus velocities: s0 u0 , u* 5 s y* 5

s0 y 0 . s

and

(14) (15)

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Thus, the residual velocities can be expressed as u^ 5 u 1 u*,

and

^y 5 y 1 y * .

(16) (17)

Given the above decomposition, the TWA velocity is referred to as the residual mean velocity, and we will use this naming convention throughout this paper. Although this decomposition is not needed in the TWA framework, we will use it later to make connections between existing eddy parameterizations and between the Eulerian mean velocity and the residual mean velocity in different OGCMs. The thickness equation for the TWA framework in buoyancy coordinates (see, e.g., de Szoeke and Bennett 1993; Young 2012) takes the form ›s › › › ~ 5 0, us) 1 (^y s) 1 (-s) 1 (^ x ›~ y ›~t ›~ ›b~

In this section, we invoke assumptions applicable under somewhat restricted, idealized conditions and use them to obtain a simplified model of the EPFT in (9). In doing so, we obtain an expression that can be written in two ways, one of which is identical to the widely used parameterization proposed by GM, and the other is the parameterization proposed by GL. The results in this section are in agreement with the developments presented by Ferreira and Marshall (2006), where similar expressions are obtained by transforming the governing equations in the Eulerian mean framework to obtain a representation of the flow in terms of residual mean prognostic variables. First, we assume that the Reynolds stress terms in (9) are small and can be ignored, after which we are left with the interfacial form drag terms s21 z0 m0x and s21 z0 m0y . Parameterizing form drag stress as downgradient diffusion of horizontal velocity with vertical viscosity m, we obtain z

s

z0 m0y

5 2m

›^ u , ›z

›^y . 5 2m ›z s

and

! Y › 1 0 0 1 › 2 ry mN Y , and z mx 5 ›z s f ›z rz Y › 1 0 0 1 › 2r mN xY . z my 5 2 ›z s f ›z rz

(21) (22)

An alternative derivation is obtained as follows. Starting with (1)–(2) and assuming geostrophy, we produce ›pY › 1 0 0 1 z mx 5 0, and ›x ›z s ›pY › 1 0 0 f u^ 1 z my 5 0. 1 ›y ›z s

(18)

3. Parameterizing the EPFT

m0x

We then use thermal wind balance in the above equation and express the Brunt–Väisälä frequency 2 Y Y 21 as N 5 2gr21 0 r z 5 1/bz 5 s . The resulting force due to form drag, which is expressed as the vertical derivative of the stress, becomes

2f ^y 1

where - represents the vertical velocity in buoyancy coordinates and is associated with diabatic effects. We will use this relation in the analysis below.

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(23) (24)

Decomposing the residual mean velocity using (12)– (13) and using the definitions in (14)–(15), we obtain an expression relating the form drag to eddy thickness fluxes: › 1 0 0 s0 y 0 [ f y *, and z mx 5 f ›z s s › 1 0 0 s0 u0 [ 2fu* . z my 5 2f ›z s s

(25) (26)

The above equation relates the bolus velocities to the eddy force associated with the form drag, the vertical variation of a horizontal interfacial stress. A similar relation was derived in Greatbatch (1998). Combining (21)–(22) with (25)–(26), we get ! rYy s0 y 0 › k Y , 5 ›z s rz s0 u0 › rY k xY , 5 ›z s rz

and

(27) (28)

where

(19)

2

k5m

N . f2

(29)

(20)

Ferreira and Marshall (2006) use the above expressions to parameterize the eddy stress in simulations using their transformed Eulerian mean formulation.

Defining the isopycnal slope vector S as $ rY S 5 2 zY , rz

(30)

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where $z is the horizontal gradient operator in x, y, the horizontal bolus velocity vector in Cartesian coordinates u* 5 u*i 1 y *j can be written as › u* 5 2 (kS) . ›z

(31)

The expression relating eddy viscosity and eddy diffusivity in (29) is the same as that suggested in GL and Ferreira and Marshall (2006). The above analysis illustrates, from the viewpoint of the TWA framework, the well-known link between the parameterization of mesoscale eddies as vertical diffusion of horizontal momentum by GL and the one by GM whereby mesoscale eddies are parameterized as the flux of isopycnal layer thickness. In their approach, GL propose the use of vertical viscosity m 5 Af 2 N 22 , without providing a physical justification. A result of our analysis is that A in GL and k in GM are identical, that is, A 5 k, which suggests that the two parameterizations are equivalent, consistent with the discussion in Ferreira and Marshall (2006). Note that these derivations are independent of the tracer equations and are thus independent of the eddy–mean flow interactions that appear in the tracer equations. Thus, when viewed from the perspective of the TWA equations, both GM and GL can be interpreted as representations of eddy forcing in the horizontal momentum equations. GL proposed to combine the expressions in (21)– (22) with the TWA system (1)–(5). In this approach, the effects of eddies are represented by a vertical flux of horizontal momentum by form drag in the momentum equations. GM, on the other hand, combine the above parameterization with the governing equations for the Eulerian mean flow prognostic variables. As a result, the GM parameterization is often viewed as representing eddy fluxes of isopycnal thickness fluctuations as downgradient diffusion of mean isopycnal thickness. From the GM perspective, the influence of eddies appears as an advection of tracers by an eddy-induced velocity, in addition to the advection by the Eulerian mean velocity, and so the net tracer advection is by the residual mean velocity. We will refer to this framework as the CNV framework. We address the differences between these two approaches in the following section.

4. Governing equations in the conventional Eulerian mean framework The governing equations solved in the CNV framework are

Du z ›p 2 f y z 1 5 Rx , ›x Dt Dy z ›p 1 f uz 1 5 Ry , ›y Dt ›p 5 b, ›z ›(uz 1 uz ) ›(y z 1 y z ) ›(wz 1 wz ) * 1 * 1 * 5 0, ›x ›y ›z Db ›b ›b ›b 1 uz* 1 y z* 1 wz* 5 0, Dt ›x ›y ›y

(32) (33) (34) (35) (36)

where the material derivative following the Eulerian mean velocity is given by D › › › › 5 1 u z 1 y z 1 wz , Dt ›t ›x ›y ›z

(37)

and uz* , y z* is the bolus velocity calculated using the Eulerian mean flow variables z r › k xz , and uz* 5 2 ›z rz z ry › y z* 5 2 k z , ›z rz

(38) (39)

representing the effects of eddies in the Eulerian mean framework. The z represents an ensemble average at a fixed point x, y, z, t. In a steady state, when the mean depth x, y~, bY ) of a buoyancy surface bY is constant in z 5 z(~ time, the ensemble average of a quantity f at x, y, z 5 z(~ x, y~, bY ) is equal to an ensemble average of f at fixed x~, y~, bY , or, in other words, fz 5 f. For now, in this paper we assume that the TWA dissipation terms R^x and R^y in buoyancy coordinates can be represented as ensemble average dissipation terms Rx and Ry in z coordinates. In the conventional framework, inertial terms in the horizontal momentum equations only involve the Eulerian mean velocity. In other words, the Eulerian mean velocity is advected by the Eulerian mean velocity. Buoyancy is advected by the sum of Eulerian mean velocity uz , y z , wz and the bolus velocity uz , y z , wz . * * *

5. Comparison of Ertel potential vorticity in the residual and Eulerian mean prognostic equations At this point, we have two sets of prognostic equations that can be used to model flow with unresolved eddies. One is the TWA framework and the second is the conventional framework outlined in section 4. In this section, we contrast the two frameworks by comparing the Ertel potential vorticity of each one. Ertel potential vorticity (EPV) is a good choice for this

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comparison because it provides an integrated view of the dynamics by combining the momentum and thickness equations. We first introduce the EPV of the residual mean flow, as derived by Young (2012) for the TWA (1)–(5) and summarize some of its well-known properties. We then derive the EPV of the residual mean flow that results from transforming the conventional prognostic equations (32)–(36). We end this section with a discussion comparing the two.

a. Ertel potential vorticity of the residual mean flow in the TWA framework

Y

(40)

as derived in Young (2012), is given by Y

DY P 5 2$ FY 2 $ GY , Dt

(41)

$ Ey $ Eu e1 2 e2 s s

(42)

is the EPV eddy flux vector, and GY 5

1 s

›^y 1 ›^ u ^ Y e3 ^ 2 R^y~ e1 1 ^ 1 R^x~ e2 2 -P 2 s ›b~ ›b~ (43)

is the time tendency in the EPV due to diabatic terms. It can be shown that the EPV eddy flux vector FY can be obtained from the curl of the eddy forces in the momentum equation $ E (Maddison and Marshall 2013). Equation (41) is written in vector form and is thus valid in any coordinates system. Equations (42) and (43) are written in buoyancy coordinates, where e1 and e2 are basis vectors tangent to a buoyancy surface, and e3 is the vertical basis vector. These basis vectors are nonorthogonal and are related to the orthonormal basis vectors in Cartesian coordinates i, j, and k by e1 5 i 1 zx~k , e2 5 j 1 zy~k, e3 5 sk.

(44) and

(47)

where u* is given by the parameterization in (31). Recall that the EPV eddy flux vector FY is related to the eddy forces $ E in the momentum equation. In the above equation where eddies have been parameterized using GM, these forces are given by f y* and 2fu*. Similar relations are discussed in Treguier et al. (1997) and Greatbatch (1998).

b. Ertel potential vorticity of the residual mean flow in the transformed CNV framework

where FY 5

trajectories but are identically zero when summed over the entire domain. Thus, EPV is globally conserved within the TWA system even in the presence of diabatic and eddy forcing. For a more detailed discussion of these and other properties of EPV, see Young (2012) and the references therein. We now implement the eddy parameterization derived in section 3. Thus, we reduce the EPFT to the terms associated with interfacial form drag by ignoring all other terms and combine (25)–(26) with (14)–(15) in (42). This way, we can write (41) as DY P 2 5 $ ( f u*N ) , Dt

The evolution of EPV of the residual mean flow ›^y ›^ u f1 2 ›~ x ›~ y , PY 5 s

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(45) (46)

When the flow is adiabatic in the interior ($ GY 5 0) and eddy effects are zero ($ FY 5 0), EPV is conserved following the residual mean flow. Diabatic and eddy processes serve as sources/sinks of EPV along particle

We transform the Eulerian mean flow of the conventional framework [(32)–(36)] to a residual mean flow and calculate the EPV of the latter. Using the chain rule and assuming a steady state such that u z 5 u, and so on, horizontal momentum [(32)– (33)] can be written as ›u ›u › 1 1 m 1 u 2 1 y 2 5 Rx , and 1 - 2 s yP 1 ›~ x 2 2 ›~t ›b~ ›y ›y › 1 2 1 2 m 1 u 1 y 5 Ry , 1 - 1 s uP 1 ›~ y 2 2 ›~t ›b~

(48) (49)

where we have defined the EPV of the Eulerian mean flow as

P5

f1

›y ›u 2 ›~ x ›~ y . s

(50)

Cross-differentiating and combining (48) and (49), we obtain › › ›y u sP 1 - 2 Ry (sP) 1 ›~ x ›~t ›b~ › ›u y sP 2 - 1 Rx 5 0. 1 ›~ y ›b~

(51)

Using the decomposition in (16)–(17) and assuming a steady state, we transform the Eulerian mean flow u, y

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to a residual mean flow u^, ^y . With this transformation, we then write the EPV of the Eulerian mean flow in terms of the EPV of the residual mean flow as

P5

f1

›^y ›^ u ›y* ›u* 2 2 v ›~ x ›~ y ›~ x ›~ y 2 5 PY 2 * , s s s

(52)

where we have used (40) and where v* is the relative vorticity associated with bolus velocities u* and y *. Using the thickness [(18)], we simplify and use (52) to rewrite (51) in terms of PY : DY PY DY 5 (v s21) 1 $ (u*PY) 2 $ (u*v*s21) 2 $ G, Dt Dt * (53)

1 ›^y ›y* 2 Ry e1 2 s ›b~ ›b~ 1 ›^ u ›u* 1 Rx e 2 1 22 s ›b~ ›b~ v 2 - PY 2 * e3 . s

by the eddy form drag forces in the EPFT, that is, $ (u*PY ) ’ $ (f u*s21 ). Assuming that diabatic effects are small in the interior of the flow, away from the mixed layer, we ignore G. With these approximations and using 2 N 5 2g/r0 (›r/›z) 5 (›bY /›z) 5 1/s, we are left with DY PY 2 5 $ (f u*N ) , Dt

(55)

which is the same expression we obtained in (47). This relation is consistent with previous work (see, e.g., Treguier et al. 1997; Greatbatch 1998) and results from the relationships between eddy buoyancy fluxes, vertical eddy momentum fluxes, and eddy potential vorticity fluxes.

6. Numerical simulations: Idealized Southern Ocean

where G5

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(54)

Comparing the right-hand sides of (41) and (53), it can be seen that the time tendency of EPV of the residual mean flow induced by eddies and diabatic processes is different in the TWA and CNV frameworks. These differences arise because the velocities that advect momentum in the TWA and CNV frameworks are the residual mean and Eulerian mean velocities, respectively. Therefore, in the CNV framework there are additional EPV fluxes on the right-hand side of (53): the first term is the material derivative of the relative vorticity of the bolus velocity; the second term has eddy forces f u* analogous to those in (42) and an additional term involving the relative vorticity of the residual velocity; the third term contains eddy fluxes of v* by the bolus velocity; and the fourth term has additional diabatic fluxes associated with the bolus velocity. Under certain approximations, the time tendency of EPV in the two frameworks is equal. When the effects of the bolus velocity on the momentum equations are small, we are left with $ (u*PY ), the second term on the right-hand side of (53). Note that this approximation is also used by Ferreira and Marshall (2006) in the horizontal momentum equations to derive their transformed Eulerian mean formulation, as we discuss in the appendix. In regions where geostrophy dominates the flow, away from topography and in regions where Reynolds stresses are negligible, PY is dominated by the planetary vorticity term PY ’ f /s, and the EPV flux is dominated

In this section, we test the TWA framework for simulating ocean circulation and dynamics with parameterized mesoscale eddies using an idealized Southern Ocean configuration. We verify the analysis made in section 5 by comparing the flows simulated with the TWA framework and the CNV framework using the mesoscale eddy parameterization presented in section 3.

a. Solving for the prognostic residual mean flow in an ocean general circulation model We added three new capabilities to the ocean general circulation implementation of the Model for Prediction Across Scales-Ocean (MPAS-O; Ringler et al. 2013). The first new capability involves the calculation of the eddy-induced thickness fluxes or bolus velocity in (31) and the (Redi 1982) isopycnal diffusion for a general vertical coordinate system. To calculate the eddyinduced thickness fluxes, we use the streamfunction formulation with the boundary value tapering scheme from Ferrari et al. (2010). This parameterization is then used in the two frameworks that we have discussed so far: the TWA framework and the CNV framework. The second capability is the ability to solve for the flow using the CNV framework. To add this capability, we define the tracer advection velocity as the sum of the Eulerian mean velocity and the bolus velocity, as we discussed previously. For the third new capability, we implemented the TWA framework in MPAS-O. To allow a direct comparison with the CNV framework, we implement the parameterization in (25)–(26). However, (19)–(20) likely would have resulted in a better representation of eddy form drag as discussed by Ferreira et al. (2005) and Ferreira and Marshall (2006). Note that (1)–(5) for the residual mean prognostic variables are structurally identical to (32)–(36) for the

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Eulerian mean prognostic variables solved by conventional OGCMs, with the exception of an extra term for the divergence of the EPFT. Thus, the TWA framework can be implemented in conventional OGCMs simply by reinterpreting the prognostic variables as TWA residual mean variables and adding tendency terms to the horizontal momentum equations representing the parameterized divergence of the EPFT. Many parameterizations used in ocean general circulation models rely on the Eulerian mean state of the flow. These and other subtleties accompanying this reinterpretation will have to be addressed before the TWA framework is a viable choice in coupled climate simulation. However, the TWA approach itself is achieved not through algorithmic changes within OGCMs but rather through a conceptual redefinition of existing variables.

b. Idealized Southern Ocean We simulate the flow in an idealized Southern Ocean (ISO) configuration consisting of a 2.5-km-deep circumpolar channel on a rotating sphere with vertical walls on the north and south, fn 5 508S and fs 5 708S, respectively. The flow is forced with boundary conditions based on an idealized, zonally averaged annual climatology of the Southern Ocean, shown in Fig. 1. We apply a zonal wind stress that varies as a function of latitude f: " # f 2 fm 2 p f 2 fm , cos t(f) 5 t 0 exp 2 2 Df Df

(56)

where t 0 5 0.2 N m22, Df 5 108, and midchannel latitude fm 5 608S. Temperatures at the surface are restored to a slightly asymmetrical profile centered at the midchannel latitude with the form f 2 fm f 2 fm 1 Tb Tr (f) 5 Tm 1 Ta tanh 2 , Df Df

The initial temperature profile at time t 5 0 throughout the channel was set to T(x, y, z, t 5 0) 5 T0 (z) 5 T1 1 T2 tanh

z 1 mT z, h1 (59)

where we used constants T1 5 68C, T2 5 3.68C, h1 5 300 m, and mT 5 7.5 3 1025 8C m21. The fluid is initially at rest.

c. Numerics (57)

where Tm 5 38C, Ta 5 28C, and Tb 5 1.258C. The time scale for restoring surface temperature is 30 days. Along the northern wall, a sponge layer is applied by restoring interior temperatures to an exponentially decaying vertical profile of the form z Trn (f, z) 5 Ts exp , ze

FIG. 1. Forcing and initial conditions for the simulations in the idealized Southern Ocean configuration. (top) Surface wind stress t(f) (black line) and the relaxation temperature at the surface Tr(f) (gray line) as functions of latitude. (bottom) Interior restoring temperature profile used in the sponge layer at the north wall of the channel Trn(f 5 508S) (black line) and the initial temperature profile T0(z) (gray line) as functions of the vertical coordinate z.

(58)

where Ts 5 Tr(f 5 508S) and ze 5 1 km. Restoring in the sponge layer is done with a time scale of 10 days and an exponential decay away from the northern wall with an e-folding length scale of 120 km.

We integrate the governing equations in time using the standard numerical configuration for MPAS-O, summarized as follows [for full details, refer to Ringler et al. (2013)]: The horizontal mesh is a spherical centroidal Voronoi tessellation that is composed primarily of hexagonal cells with a uniform spacing of 120 km. MPAS-O uses an arbitrary Lagrangian–Eulerian (ALE) approach to discretize the vertical direction (Petersen et al. 2015). For these simulations, we configure the ALE system to mimic a z* vertical coordinate (Adcroft and Campin 2004) with 40 vertical layers where thickness varies between 4 m at the surface and 250 m at depth. The eddy-induced velocity is zero at the surface by virtue of the boundary value tapering scheme from Ferrari et al. (2010), and we used a split–explicit time stepping

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algorithm (Higdon 2005) with a time step of 3600 s. The GM parameter in (31) is k 5 1200 m2 s21. The simulations include the standard horizontal biharmonic viscosity of 2.5 3 1013 m4 s21 and bottom drag coefficient of 1.0 3 1022. To suppress eddy activity, we use a Laplacian horizontal viscosity of 2500 m2 s21. We use the Richardson number–based vertical mixing scheme by Pacanowski and Philander (1981) with background vertical viscosity and diffusivity of 1.0 3 1024 and 1.0 3 1025 m2 s21, respectively. Tracers are advected using a quasi-thirdorder monotone advection scheme (Skamarock and Gassmann 2011). The fluid has a linear equation of state with a reference temperature of 198C, linear thermal expansion coefficient a 5 0.255 kg m23 8C21, and reference density of 1025 kg m23. There are no salinity fluctuations.

d. Results Two simulations are discussed below, one using the TWA framework and one using the CNV framework. Both simulations are integrated for 300 yr, at which point a circumpolar current is developed with an associated overturning circulation driven by the wind and buoyancy forcing, resembling the flow in the Southern Ocean. Instabilities are entirely damped first by the eddy parameterization and second by the Laplacian and biharmonic viscosities, thus resulting in a flow that is characterized by the absence of mesoscale eddies. All results are presented in terms of zonally averaged quantities where we use the notation h i to denote the zonal average of a variable. The streamfunction cY (x, y, z) associated with the residual flow, for which $ uY 5 0, is defined as ›cY ^y 5 2 , ›z

›cY wY 5 . ›y

(60)

For a flow in a domain bounded by solid walls as in the ISO configuration, the zonally averaged residual overturning streamfunction associated with cY , computed in buoyancy coordinates, becomes ð ð b(x,y,z) CY (y, b) 5 2

^y (x, y, z)s db0 dx0 .

(61)

x b(x,y,z52H )

Since the flow is stably stratified and b(x, y, z) increases monotonically with increasing z, we are able to map CY (y, b) in buoyancy coordinates to CY [y, z(b)] in depth coordinates. The overturning streamfunctions CY (y, z) for the flows in the TWA and CNV frameworks are shown in the upper and lower panels of Fig. 2, respectively. The overturning circulation is characterized by two surface overturning cells that ventilate the interior, as expected given the forcing by the imposed boundary conditions

FIG. 2. Zonally averaged residual overturning circulation CY (y, z) as a function of latitude y 5 f and depth z, where a positive streamfunction indicates clockwise circulation (Sv; color). Solid lines are zonally averaged density contours hr(x, y, z)i, ranging from 1029 to 1027.6 kg m23 from high to low latitudes at 0.1 kg m23 intervals. Dashed lines show contours of zonally averaged, zonal residual mean velocity h^ u(x, y, z)i, ranging from 0 cm s21 at the walls to 22.5 cm s21 at the core of the jet, at 2.5 cm s21 intervals. Quantities are obtained from the simulations using the (top) TWA and (bottom) CNV frameworks.

(e.g., Marshall and Radko 2003). In the interior, away from the diabatic surface mixed layer and the diabatic sponge layer near the northern wall, the streamfunction and the isopycnals in Fig. 2 are aligned, indicating that the circulation is isopycnal, which is consistent with the assumption made in section 5. At t 5 300 yr, there is a third interior overturning cell that is slowly decaying in time. We will show later that this third cell occurs entirely within neutrally stratified fluid; therefore its existence does not affect our analysis. The zonal current has a surface-intensified jet that is slightly displaced northward, as represented by the

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FIG. 3. Zonally averaged planetary potential vorticity hf N 2 i, the dominating term in the Ertel potential vorticity of the residual mean flow PY (s23; color). Contour lines represent the zonally averaged meridional overturning circulation streamfunctions CY at 2-Sv intervals, with CY (y, z) 5 0 shown by the white lines. Clockwise (CY . 0) and counterclockwise (CY , 0) circulation are represented by solid and dashed lines, respectively. Quantities are obtained from the simulations using the (top) TWA and (bottom) CNV frameworks.

zonally averaged, zonal residual mean velocity h^ ui, plotted in Fig. 2. The zonal and meridional circulations produced by the TWA and CNV frameworks are in excellent agreement in the interior. The strength of the overturning cells is slightly higher in the CNV framework, particularly at the surface. The zonal flow h^ ui is slightly lower in the CNV framework. The circumpolar transport, calculated as the integral of h^ ui across the channel, at 300 yr is 620.1 and 619.0 Sverdrups (Sv; 1 Sv [ 106 m3 s21) in the TWA and CNV frameworks, respectively. The EPV of the flow is dominated by the planetary potential vorticity, consistent with the assumptions 2 made at the end of section 5b. The zonal average hf N i is shown in Fig. 3. Thus, the EPV is largely dictated by

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FIG. 4. Zonally averaged stratification, represented by the 2 Brunt–Väisälä frequency hN i (s22; color). Contour lines represent the zonally averaged meridional overturning circulation streamfunctions CY at 2-Sv intervals, with CY (y, z) 5 0 shown by the white lines. Clockwise (CY . 0) and counterclockwise (CY , 0) circulation are represented by solid and dashed lines, respectively. Quantities are obtained from the simulations using the (top) TWA and (bottom) CNV frameworks.

the stratification set by the boundary conditions at the surface and northern wall and by the dynamics in the interior. The variation of EPV with latitude and depth is 2 almost entirely dictated by the stratification N , plotted in Fig. 4. As mentioned earlier, the overturning cell that does not ventilate the interior is contained entirely within the unstratified region (Fig. 4). In the interior, the change in EPV of a parcel traveling along an isopycnal is given by the time tendency of EPV on the right-hand side of (41). Given the zonal symmetry of the flow, the zonal component of the EPV fluxes does not affect the EPV time tendency for this flow. In the upper and lower panels of Fig. 5, we plot the meridional 2 component of the EPV fluxes FY 5 2f u*N in (47) and

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FIG. 5. Zonally averaged Ertel potential vorticity fluxes from 2 2 parameterized mesoscale eddy forces, hf u*N i 5 2h f ›z (kS)N i 24 (m s ; color). Contour lines represent the zonally averaged meridional overturning circulation streamfunctions CY at 2-Sv intervals, with CY (y, z) 5 0 shown by the white lines. Clockwise (CY . 0) and counterclockwise (CY , 0) circulation are represented by solid and dashed lines, respectively. Quantities are obtained from the simulations using the (top) TWA and (bottom) CNV frameworks.

(55) diagnosed for the TWA and CNV frameworks, respectively. The EPV fluxes plotted in Fig. 5 are driven by the eddy forces in (26) that act on the zonal momentum equation. Note that in the interior, EPV fluxes are positive on the upwelling branch of the overturning circulation and negative where there is downwelling. The zonal average of the divergence of the EPV fluxes is shown in Fig. 6 (as indicated in the caption, the data shown have been filtered to smooth out noise introduced by the diagnostic calculation of the divergence). In the interior, this divergence is negative where the flow is upwelling and zero elsewhere. The term associated with the gradient of the stratification on the right-hand side of

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FIG. 6. Zonally averaged time tendency in the Ertel potential 2 vorticity from parameterized mesoscale eddy forces h$ (f u*N )i 5 2 24 2h$ f ›z (kS)N i (s ; color). A uniformly weighted eight-point Laplacian filter is used to smooth noise. Contour lines represent the zonally averaged meridional overturning circulation streamfunctions CY at 2-Sv intervals, with CY (y, z) 5 0 shown by the white lines. Clockwise (CY . 0) and counterclockwise (CY , 0) circulation are represented by solid and dashed lines, respectively. Quantities are obtained from the simulations using the (top) TWA and (bottom) CNV frameworks.

(55), which, given the zonal symmetry of the flow, can be 2 2 approximated to f u* $N ’ f y *›y N , dominates the time tendency in Fig. 6 (not shown). Consider the upwelling branches of the overturning cells that ventilate the interior. At their lowest depth, the EPV 2 f N of parcels on this branch is negative. Then, as the time tendency on the right-hand side of (55) becomes positive and increases in magnitude (Fig. 6), the EPV becomes more negative, increasing in magnitude, as parcels move upward and approach the surface. Thus, each surface overturning cell in the adiabatic interior is associated 2 with a peak in EPV hf N i underneath the surface mixed

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layer. As the flow enters the surface mixed layer, where the stratification is zero (Fig. 4), the EPV quickly goes to zero. To illustrate this quantitatively, we track the time tendency induced by the divergence of the EPV fluxes for parcels moving in the interior along isopycnals on the upwelling branches of the overturning cells that venti2 late the surface. At the northern wall, f is fixed and N is set by the sponge layer interior restoring of temperature. Thus, as parcels emerge from the sponge layer, enter the interior, and start flowing upward at time t1, the initial EPV PY1 is specified by the boundary conditions. Integrating the right-hand side of (47) and (55) following a water parcel from t1 to just before it enters the mixed layer at time t2, we obtain DPY 5

ðt

2

t1

2

$ ( f u*N ) dt .

(62)

Taking the zonal average and then using dt 5 dy/h^y i, we get DhPY i 5

ðy

2

y1

h$ ( f u*N )ih^y i21 dy . 2

(63)

We define 19 discrete isopycnal surfaces evenly distributed between 1028 and 1028.6 kg m23 (e.g., Fig. 2). We bound each isopycnal on the north by y1 5 558S and on the south by y2, where the isopycnal layer reaches a depth of z 5 2290 m. We then use (63) to diagnose the time tendency change of EPV by eddy forces along isopycnal surfaces for the flows obtained with the TWA and CNV frameworks and plot these calculations in Fig. 7. Despite the noise in this plot, there is a clear match between the two frameworks. There is excellent agreement between the diagnosed EPV fluxes (Fig. 5), their divergence (Fig. 6), and the EPV time tendency (Fig. 7) induced by eddy forces for the TWA and CNV frameworks. This illustrates how the role of eddy forces is the same in both frameworks despite the differences in the way eddies are parameterized, as we discussed in section 5.

7. Discussion and conclusions The thickness-weighted average equations are an exact representation of the statistics of an ensemble realization of flows represented by the Boussinesq equations (de Szoeke and Bennett 1993; Young 2012; Maddison and Marshall 2013). The effects of eddies are represented through eddy correlation terms in the EPFT, which, in turn, appears as a divergence of a stress in the averaged horizontal momentum equations.

FIG. 7. Diagnosed change in EPV induced by the divergence of the EPV fluxes calculated by integrating the right-hand side of (63) along several isopycnal layers that exist in the adiabatic interior of the upwelling region. Calculations are diagnosed using the flows obtained with the TWA and CNV frameworks, indicated by the black and gray lines, respectively.

Using the EPFT in the TWA equations as a starting point, we derive the GM and GL mesoscale eddy parameterizations. Starting from the TWA equations, following a rather straightforward process that invokes thermal wind balance, we derive a relation between the eddy thickness fluctuation fluxes, or bolus velocity, and horizontal eddy forces represented as the vertical derivative of the horizontal stresses associated with eddy form drag. A similar relation was derived in Greatbatch (1998). The GM parameterization represents the effects of eddies as the tracer advection by the bolus velocity, or eddy fluxes of layer thickness, in addition to advection by the Eulerian mean velocity field. The GL parameterization, on the other hand, represents the effects of mesoscale eddies as vertical diffusion of horizontal momentum, or vertical transfer of horizontal momentum by form drag, in the ‘‘thermocline’’ horizontal momentum equations, which, with the benefit of hindsight, resemble the TWA horizontal momentum equations. From the perspective of the TWA framework, using the EPFT, we illustrate how GM and GL represent the net force of mesoscale eddies on the averaged horizontal momentum equations, as shown previously elsewhere (e.g., Treguier et al. 1997; Greatbatch 1998; Ferreira and Marshall 2006). We verify the adequacy of the TWA framework for modeling ocean flow in an OGCM with parameterized eddies and the derivations made to compare this framework with the CNV framework by simulating the

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flow in an idealized Southern Ocean configuration. To accomplish this comparison, we modified MPAS-O to support the CNV and TWA frameworks. In the CNV framework, we implement GM as in conventional OGCMs, namely, by solving for Eulerian mean prognostic variables, or the Eulerian mean flow, and parameterizing eddies with an additional advection of tracers by the bolus velocity. In the second framework, we directly solve the TWA equations for the residual mean prognostic variables. We then proceed to verify the derivations made and illustrate the equivalence within the restrictions imposed by the idealized nature of the flow. Comparisons of residual mean overturning circulation, residual mean zonal velocity, stratification, and potential vorticity of the residual mean flow obtained from the TWA framework and the CNV framework show excellent agreement. There are differences in the overturning circulation in the mixed layer where the structure of the streamfunctions present small, qualitative differences, and the magnitude is slightly higher in the CNV framework. Potential vorticity and its tendency also show excellent agreement. The agreement between the EPV in the flows simulated by the TWA and CNV frameworks relies on the simplified nature of the idealized Southern Ocean configuration we simulated. In this flow the inertial terms associated with the bolus velocity in the horizontal momentum equations of the residual mean are small. We expect the differences between flows produced by the two frameworks to be larger under conditions where the eddies are strong and perhaps under unstable regimes, for example, in western boundary currents and in the presence of topography, and in flows that are not in a statistically steady state, among other flow conditions. A purpose of this paper is to implement the new TWA framework and verify its viability by comparing it to the CNV framework and to investigate the connections between the two frameworks. Investigating the conditions under which the differences between the two flows become larger is out of the scope of this paper. The physical picture that arises from our analysis using the TWA framework, which is in agreement with the literature discussing eddy–mean flow interactions from the point of view of residual mean theory, can be summarized as follows. Consider two contiguous layers in a surface-intensified zonal flow such as the Antarctic Circumpolar Current in the Southern Ocean. The flow through a buoyancy layer is decelerated by form drag from eddy thickness fluctuations at the interface between it and the layer below where the flow is slower. The lower layer, on the other hand, is accelerated by the form drag from above.

Thus, form drag acts to redistribute horizontal momentum down the vertical gradient of horizontal velocity, as expressed in (19)–(20). By thermal wind balance, the redistribution of horizontal momentum in the vertical must be accompanied by a redistribution of buoyancy in the horizontal. Thus, the eddy-induced processes that act to weaken the vertical gradient of horizontal momentum simultaneously act to weaken the horizontal gradient of density. Since the redistributions of momentum and buoyancy are directly linked to one another, parameterizing either is sufficient to induce the other. In the TWA framework, eddy processes are modeled as an additional force in the momentum equation leading to downgradient diffusion. In the conventional framework, eddy processes are modeled as an addition advection of buoyancy leading to a flattening of isopycnal. We discussed connections between the TWA and the CNV frameworks that could suggest possible steps that one might follow to further refine the eddy parameterizations in conventional ocean general circulations models. Acknowledgments. This work is part of the ‘‘Multiscale Methods for Accurate, Efficient, and ScaleAware Models of the Earth System’’ project, supported by the U.S. Department of Energy’s Office of Science program for Scientific Discovery through Advanced Computing (SciDAC). Code developments and simulations relied heavily on the work of the MPAS dynamical core development team at LANL and NCAR and in particular the contributions from the MPAS-O development team at LANL. We gratefully acknowledge D. Jacobsen, P. Jones, M. Maltrud and M. Petersen for their contributions to MPAS-O. Simulations were conducted using an institutional computing allocation at LANL. Q. C. acknowledges the support of the Simons Foundation through a travel grant. We thank H. Aiki, R. Tailleux, J. Marshall, and an anonymous reviewer for constructive comments that led to a significantly improved manuscript.

APPENDIX Comparison between the TWA and the Eulerian Mean Flow Equations To compare the Eulerian mean flow [(32)–(36)] with the [TWA (1)–(5)], we transform the former in order to represent the residual mean flow. Assuming steady state, we obtain u z 5 u, uz 5 u*, and u z 1 uz 5 u 1 u* 5 u^, * * and so on. Replacing u 5 u^ 1 u* and y 5 ^y 1 y* into (32)– (36), we obtain

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DY › › u 2 u*) 2 y * (^ u 2 u*) (^ u 2 u*) 2 u* (^ ›x ›y Dt ›p 2 f (^y 2 y *) 1 5 Rx , ›x Y D › › (^y 2 y *) 2 u* (^y 2 y*) 2 y * (^y 2 y*) ›x ›y Dt ›p 1 f (^ u 2 u*) 1 5 Rx , ›x ›p 5 b, ›z › › ›w (^ u) 1 (^y ) 1 5 0, and ›x ›y ›z DY b 5 0. Dt

(A1)

(A2) (A3) (A4) (A5)

A similar transformation is done in Ferreira et al. (2005) and Ferreira and Marshall (2006) to derive their transformed Eulerian mean equations using additional assumptions. We then subtract the transformed Eulerian mean equations from (1) to (5). Invoking geostrophic balance, and using (25)–(26), the Coriolis force associated with the bolus velocity is approximately balanced by the vertical divergence of E. After these manipulations, we obtain DY u* Dt

›u ›u 1 u* 1 y * 5 0, and ›x ›y

DY y *

›y ›y 1 u* 1 y * 5 0. ›x ›y Dt

(A6) (A7)

These are also the terms that are neglected in the transformed Eulerian mean formulation by Ferreira and Marshall (2006), which they justify using scaling arguments. REFERENCES Adcroft, A., and J.-M. Campin, 2004: Rescaled height coordinates for accurate representation of free-surface flows in ocean circulation models. Ocean Modell., 7, 269–284, doi:10.1016/ j.ocemod.2003.09.003. Andrews, D. G., and M. E. McIntyre, 1976: Planetary waves in horizontal and vertical shear: The generalized Eliassen-Palm relation and the mean zonal acceleration. J. Atmos. Sci., 33, 2031–2048, doi:10.1175/1520-0469(1976)033,2031:PWIHAV.2.0.CO;2. Danabasoglu, G., J. C. McWilliams, and P. R. Gent, 1994: The role of mesoscale tracer transports in the global ocean circulation. Science, 264, 1123–1126, doi:10.1126/science.264.5162.1123. de Szoeke, R. A., and A. F. Bennett, 1993: Microstructure fluxes across density surfaces. J. Phys. Oceanogr., 23, 2254–2264, doi:10.1175/1520-0485(1993)023,2254:MFADS.2.0.CO;2. Eden, C., and R. J. Greatbatch, 2008: Towards a mesoscale eddy closure. Ocean Modell., 20, 223–239, doi:10.1016/j.ocemod.2007.09.002. Eliassen, E. T., and E. Palm, 1961: On the transfer of energy in stationary mountain waves. Geofys. Publ., 22 (3), 1–23. Ferrari, R., S. M. Griffies, A. G. Nurser, and G. K. Vallis, 2010: A boundary-value problem for the parameterized mesoscale

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