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  • View in gallery

    Comparison of |ud| given in (2.5) with the data of Fu (2009) and Chelton et al. (2011) at 150° and 110°W. The data are reproduced satisfactorily. In all the figures, the model results were obtained from an average of the last 3 years of a simulation with the GISS ER stand-alone OGCM (appendix C), which was run for 300 years.

  • View in gallery

    Maps of , at the (top) surface and (bottom) at 1798-m depth.

  • View in gallery

    Vertical profiles of the x and y components of the relative velocity that enters the eddy-induced velocity [(1.2)].

  • View in gallery

    Vertical profiles of the isopycnal slopes with (solid) and without (dashed) the new term in the eddy-induced velocity [(1.2)].

  • View in gallery

    (top) Map of the depth defined in (3.7) in units of the MLD computed from the potential density criterion Δσρ = 0.03 kg m−3. In all regions ≥ MLD, indicating that below the mixed layer the flow is still diabatic. (bottom) Map of the isopycnal slopes at . The value of is no longer arbitrary as in tapering schemes; it is now given by the model. It is not a universal constant but a location-dependent variable.

  • View in gallery

    Vertical profiles of the vertical buoyancy flux in the ACC, Gulf Stream, Kuroshio, and Sea of Japan. In the D regime, the flux is given by (3.4), while in the A regime Fυ(b) = κMN2s ∙ ξ. The depth z is in units of . To convert the flux to watts per square meter, one multiplies Fυ(b) by (108 s3 m−2) W m−2, αT = a010−4 K−1, where α0 is location dependent.

  • View in gallery

    Tapering function T(z) defined in (3.9) for the California Current, ACC, Gulf Stream, and Sea of Japan.

  • View in gallery

    ACC z profiles of the A-regime diffusivity tensor components K12 and K21 contributed by E in (7.5). The magnitudes are much smaller than the diagonal components K11 and K22 from the first tensor in (7.5).

  • View in gallery

    The z profiles of the three components of D-regime diffusivities K31, K32, and K33 in (7.6) in the ACC and Gulf Stream. The normalization was chosen so that at the bottom of the D regime, the diffusivities match those of the A regime.

  • View in gallery

    Deacon cell. (top) The GM model. (bottom) The present model, which induces a considerable reduction of the intensity of the cell compared to the GM model.

  • View in gallery

    Vertical profile of the average over the whole ocean of the in situ temperature minus the surface value in the present model and the WOA05 data (Locarnini et al. 2006). Figure 2a of Kuhlbrodt and Gregory (2012) shows that most model results fall to the right of the data corresponding to a weak stratification and to a large capacity for downward heat transport that is not observed. The present model induces a stronger stratification.

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Parameterization of Mixed Layer and Deep-Ocean Mesoscales including Nonlinearity

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  • 1 NASA Goddard Institute for Space Studies, and Department of Applied Physics and Applied Mathematics,
  • | 2 Columbia University, New York, New York
  • | 3 NASA Goddard Institute for Space Studies, and Center for Climate Systems Research, Columbia University, New York, New York
  • | 4 NASA Goddard Institute for Space Studies, New York, and Department of Physics and Computer Science, Medgar Evers College, City University of New York, Brooklyn, New York
  • | 5 NASA Goddard Institute for Space Studies, and SciSpace, LLC, New York, New York
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Abstract

In 2011, Chelton et al. carried out a comprehensive census of mesoscales using altimetry data and reached the following conclusions: “essentially all of the observed mesoscale features are nonlinear” and “mesoscales do not move with the mean velocity but with their own drift velocity,” which is “the most germane of all the nonlinear metrics.” Accounting for these results in a mesoscale parameterization presents conceptual and practical challenges since linear analysis is no longer usable and one needs a model of nonlinearity. A mesoscale parameterization is presented that has the following features: 1) it is based on the solutions of the nonlinear mesoscale dynamical equations, 2) it describes arbitrary tracers, 3) it includes adiabatic (A) and diabatic (D) regimes, 4) the eddy-induced velocity is the sum of a Gent and McWilliams (GM) term plus a new term representing the difference between drift and mean velocities, 5) the new term lowers the transfer of mean potential energy to mesoscales, 6) the isopycnal slopes are not as flat as in the GM case, 7) deep-ocean stratification is enhanced compared to previous parameterizations where being more weakly stratified allowed a large heat uptake that is not observed, 8) the strength of the Deacon cell is reduced. The numerical results are from a stand-alone ocean code with Coordinated Ocean-Ice Reference Experiment I (CORE-I) normal-year forcing.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: V. M. Canuto, vmcanuto@gmail.com

Abstract

In 2011, Chelton et al. carried out a comprehensive census of mesoscales using altimetry data and reached the following conclusions: “essentially all of the observed mesoscale features are nonlinear” and “mesoscales do not move with the mean velocity but with their own drift velocity,” which is “the most germane of all the nonlinear metrics.” Accounting for these results in a mesoscale parameterization presents conceptual and practical challenges since linear analysis is no longer usable and one needs a model of nonlinearity. A mesoscale parameterization is presented that has the following features: 1) it is based on the solutions of the nonlinear mesoscale dynamical equations, 2) it describes arbitrary tracers, 3) it includes adiabatic (A) and diabatic (D) regimes, 4) the eddy-induced velocity is the sum of a Gent and McWilliams (GM) term plus a new term representing the difference between drift and mean velocities, 5) the new term lowers the transfer of mean potential energy to mesoscales, 6) the isopycnal slopes are not as flat as in the GM case, 7) deep-ocean stratification is enhanced compared to previous parameterizations where being more weakly stratified allowed a large heat uptake that is not observed, 8) the strength of the Deacon cell is reduced. The numerical results are from a stand-alone ocean code with Coordinated Ocean-Ice Reference Experiment I (CORE-I) normal-year forcing.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: V. M. Canuto, vmcanuto@gmail.com

1. Introduction

Mesoscales (10–100 km) are energetic, long-lived (several months), coherent oceanic structures that are numerically unresolved in coarse-resolution ocean global circulation models (OGCMs) and must therefore be parameterized. Since the ocean contains two distinct regimes, adiabatic (A; deep ocean) and diabatic (D; upper layers), two independent parameterizations are needed.

a. Adiabatic: A regime

In this regime, water parcels move primarily along surfaces of constant density with negligible across-isopycnals (diapycnal) fluxes. The first mesoscale parameterization for this regime was proposed by Gent and McWilliams (1990, hereinafter GM). The 3D buoyancy fluxes were separated into along- and across-isopycnal components: the first flux is usually parameterized using the form suggested by Redi (1982), while the second flux is represented by the eddy velocity The GM form of reflects the suggestion that mesoscales act as a sink of mean potential energy (MPE). Since the reduction of MPE entails a flattening of the slopes of the mean isopycnals, the GM model is represented as a downgradient of the mean isopycnal slopes. The improvements in the OGCMs results obtained with the GM + Redi parameterizations were reviewed by Gent (2011), whose Fig. 1c reproduces the results of Danabasoglu et al. (1994), who showed that the GM model made the Deacon cell (considered an artifact) almost disappear. On the other hand, 16 modern OGCMs (Farneti et al. 2015) exhibit a robust Deacon cell of 20–28 Sv (1 Sv ≡ 106 m3 s−1). In section 8, we revisit the problem and show that the present model induces a reduction of the cell compared to the GM model.

Fig. 1.
Fig. 1.

Comparison of |ud| given in (2.5) with the data of Fu (2009) and Chelton et al. (2011) at 150° and 110°W. The data are reproduced satisfactorily. In all the figures, the model results were obtained from an average of the last 3 years of a simulation with the GISS ER stand-alone OGCM (appendix C), which was run for 300 years.

Citation: Journal of Physical Oceanography 48, 3; 10.1175/JPO-D-16-0255.1

Recently, a comprehensive census of altimetry data (Chelton et al. 2011, hereafter C11) highlighted new mesoscale features that were not present in the GM model and that we now include in the present parameterizations. The main conclusions reached by C11 are as follows:

  1. essentially all of the observed mesoscale features are nonlinear;

  2. mesoscales do not move with the mean velocity but with their own drift velocity ud; and

  3. ud is the most germane of all the nonlinear metrics.

It follows that ud cannot be represented by the Rossby phase velocity cR1 since ud is a nonlinear feature, while cR is a result of linear analysis; in addition, cR was shown not to reproduce altimetry data (Klocker and Marshall 2014).

To quantify the effect of nonlinearity, we begin with Killworth’s (1997) linear model that yielded the following form of the eddy velocity [his (35)]:
e1.1
where κM is the mesoscale diffusivity. The first term has the form of the GM model even though the linear nature of the treatment did not allow a determination of its magnitude. The novelty of (1.1) is that it predicts the presence of the second term representing the Rossby phase velocity cR. However, since C11 found that mesoscales do not move with the linear cR but with the nonlinear drift velocity ud, the second term in (1.1) is interesting but incomplete. Five years before C11, a nonlinear mesoscale model (V. M. Canuto and M. S. Dubovikov 2015, unpublished manuscript; http://www.giss.nasa.gov/staff/vcanuto/canuto_nonlinearity_201507.pdf) yielded the following eddy-induced velocity [Canuto and Dubovikov 2006, hereinafter CD6, their (4a)–(4g)]:
e1.2
where 2 is the mean velocity, σσt(1 + σt)−1, and σt is the turbulent Prandtl number O(1) (we have taken σ to be 1). The new form (1.2) reproduces Killworth’s linear model [(1.1)], while a new effect of nonlinearity is represented by the term containing , which vanishes when mesoscales comove with the mean velocity (steering level); the CD6 model also yielded an expression for ud that is presented in (2.5).

An interesting question arises as to whether the last term in (1.2) affects the transfer of MPE to mesoscales. In appendix C of Canuto and Dubovikov (2005, hereinafter CD5), it was shown analytically that the last term in (1.2) contributes negatively to the transfer of MPE to mesoscales. A simpler derivation of the same result is presented here in (2.13). In (2.15), we quantify the reduction of the transfer of MPE to mesoscales with the implication that the isopycnal slopes are not as flat as in the GM case (see Fig. 4). Since we are dealing with the adiabatic regime, the reduced energy transfer cannot be interpreted as because of the dissipation by small scales since the latter is an irreversible process. It must represent energy that is put back to MPE and did not become available to mesoscales.

The new parameterization [(1.2)] did not receive the attention the authors had hoped for and two reasons were suggested: 1) the form of ud was not assessed against data and 2) no OGCM results were presented to assess the implications of the new term in (1.2). The reason why ud was not assessed is because in 2005–06 there were no data to do so; the data having since become available, the model of ud is now shown to reproduce the Ocean Topography Experiment (TOPEX)/Poseidon (T/P) data satisfactorily (Fig. 1). Second, the reason why no OGCM results were presented is because the CD5 and CD6 parameterizations were for a fully adiabatic ocean, but an ocean also contains a diabatic D regime whose parameterization was not available in 2005–06 as it became available in 2011.

b. Diabatic: D regime

The upper layers of the ocean are not adiabatic but diabatic. In the latter regime mixing is strong, diapycnal fluxes are large, and water parcels no longer move along isopycnal surfaces as in the A regime. The fact that the D regime cannot be described by the A-regime streamfunction ΨA can be seen as follows. The latter is given by (e.g., Ferrari et al. 2008)
e1.3
where b is the buoyancy, F(b), FH(b) are the 3D and horizontal buoyancy fluxes, N is the Brunt–Väisälä frequency, and the last relation corresponds to the small slopes approximation. Using (1.3), the vertical eddy velocity is given by (we recall that )
e1.4
which is an unphysical result since the vertical eddy velocity must vanish at the surface. Though several heuristic suggestions were proposed to avoid (1.4) (Held and Schneider 1999; Marshall and Radko 2003; Aiki et al. 2004; Gnanadesikan et al. 2007; Ferrari et al. 2008; Cerovecki et al. 2009; Mahadevan et al. 2010), no unique formulation has emerged.
The physical reason for the inapplicability of the A-regime formulation to the D regime is because the two regimes obey different conservation laws. As discussed in appendix A, a key ingredient of nonlinearity is the turbulent viscosity νt. In the 3D case, kinetic energy cascades from large to small scales, which is represented by a νt > 0; in 2D turbulence there are two types of cascades, a forward enstrophy cascade from large to small scales [see Lesieur (1987) for definitions and discussion] corresponding to νt > 0 and a backward energy cascade (from small to large scales) corresponding to νt < 0. In the A regime, flows occur along isopycnal surfaces, and potential vorticity (PV) q = h−1(f + ζ) is conserved ( is the relative vorticity, h = ∂z/∂ρ is the layer thickness, and ρ is the density); only inverse energy cascade can occur corresponding to νt < 0 (Held and Larichev 1996; Smith and Vallis 2002; Arbic and Flierl 2004; Thompson and Young 2006). By contrast, in the D regime, the nonlinear interactions conserve the relative vorticity ζ, which entails an enstrophy cascade (Lesieur 1987) corresponding to νt > 0. Thus, in the AD regimes one has the following relations:
e1.5
e1.6
A D-regime parameterization based on the above conservation laws was presented in Canuto and Dubovikov (2011, hereinafter CD11). It has the following features: 1) it was assessed by Luneva et al. (2015) using a numerical simulation; 2) the streamfunction ΨD(z) automatically satisfies the condition ΨD(0) = 0, which yields instead of (1.4); and 3) it avoids ad hoc tapering functions whose shortcomings were discussed by Gnanadesikan et al. (2007).

c. Present work

The A-regime parameterization presented in CD5 and CD6 and the one for the D regime presented in CD11 and section 3 were necessary but not sufficient to obtain a mesoscale parameterization usable in OGCMs. The goal of this work is to provide the following missing ingredients: First, since OGCMs solve the equations of both active tracers such as temperature and salinity and passive tracers such as mean concentrations, the present parameterization treats an arbitrary tracer. Second, in the A regime, the tracer equation used in today’s OGCMs equation [(5.1)] is based on thickness-weighted averages that are different from the Eulerian averages that must be used in the D regime (Killworth 1997, 2005). To ensure matching at the AD interface, one must first ensure that the two regimes employ the same averages. Since in the D regime water parcels no longer move along isopycnal surfaces as they do in the A regime, the only alternative is to transform the A-regime thickness-weighted averages’ tracer equations to Eulerian-averaged equations. This is because Eulerian averages are applicable in both AD regimes, while thickness-weighted averages can only be applied in the A regime, not in the D regime, where, among other considerations, the slope of the isopycnal can become extremely large because of its inverse proportionality to N2, which is very small in the well-mixed D regime.

The transformation was carried out by McDougall and McIntosh (2001, hereinafter MM), who showed that a new vector E appears that they considered “very difficult to parameterize.” No attempts were made to parameterize E, which created an impasse that lasted some 15 years. In appendix B, we present the parameterization of E that allows the AD parameterizations to be used in OGCMs.

The structure of the paper is as follows: In sections 2 and 3, we discuss the new AD parameterizations with implications and results; in section 4, we discuss the case of an arbitrary tracer that is needed in OGCMs and how the AD parameterizations match at their interface. In section 5, we discuss the dynamic tracer equations to be used in OGCMs and the origin of the vector E, which is discussed in section 6. In section 7, we present the new 3D tracer diffusivity tensor to be used in OGCMs. In section 8, we discuss the reduction of the Deacon cell in the present model versus the one of the GM model. In section 9, we discuss how the present model enhances the stratification of the deep ocean and the relevance to the heat uptake problem. In section 10, we present some conclusions.

2. A regime: Eddy and drift velocities

In the A regime, the horizontal buoyancy flux is related to the eddy velocity as follows:
e2.1
where Hb is the depth of the ocean, and the eddy velocity is given by (1.2), which we find convenient to rewrite in the concise form
e2.2
and where the streamfunction is given by ΨA = κMez×ξ. The GM model is recovered in the limit ξs, where the vector s is the isopycnal slope. The vertical flux is obtained using the condition that in the A regime the diapycnal flux is negligibly small:
e2.3
where Fd and Fυ are the diapycnal and vertical fluxes, respectively.

a. D regime

As we discussed in CD11, in the D regime the horizontal buoyancy flux is given by
e2.4

b. Drift velocity

The AD horizontal buoyancy fluxes must match at some depth . Substitution of from (1.2) into (2.1) and equating the result to (2.4) yields the following drift velocity:
e2.5
where the average <..> is defined as follows:
e2.6
Here, is the isopycnal slope at . In an adiabatic ocean , one recovers (4f) and (4e) of CD6.
Figure 1 shows that ud given by (2.5) reproduces satisfactorily the T/P data (Fu 2009; D. B. Chelton and M. G. Schlax 2013, personal communication), while Klocker and Marshall (2014) showed that if one keeps only the first term in (2.5), the altimetry data cannot be reproduced. The additional terms in (2.5) represent the contributions of nonlinearity, which is also responsible for the presence of κM in the averages of (2.6), which differ from a straight vertical average since κM is surface enhanced. In Fig. 2, we show at two different depths. Since ud is a barotropic variable,3 while the mean velocity depends on z, near the depth where they coincide the difference ud decreases significantly. Using (2.5) and (1.2) for , the function ξ is computed from the second relation (2.2). Figure 3 shows the vertical profile of the zonal and meridional components of ud in the ACC. Combining these results with (1.2) and neglecting the cR term, one concludes that
e2.7
As for the effect of the new term in (1.2) on the isopycnal slopes, in Fig. 4 we show the vertical profiles of the isopycnal slopes with and without the new term in (1.2). The larger slopes in the present model with ξs mean that the “flattening of the isopycnals” is not as strong as in the GM case ξ = s, which is further reflected in the fact that less energy is drawn from the MPE, as we quantify below.
Fig. 2.
Fig. 2.

Maps of , at the (top) surface and (bottom) at 1798-m depth.

Citation: Journal of Physical Oceanography 48, 3; 10.1175/JPO-D-16-0255.1

Fig. 3.
Fig. 3.

Vertical profiles of the x and y components of the relative velocity that enters the eddy-induced velocity [(1.2)].

Citation: Journal of Physical Oceanography 48, 3; 10.1175/JPO-D-16-0255.1

Fig. 4.
Fig. 4.

Vertical profiles of the isopycnal slopes with (solid) and without (dashed) the new term in the eddy-induced velocity [(1.2)].

Citation: Journal of Physical Oceanography 48, 3; 10.1175/JPO-D-16-0255.1

c. Thickness versus PV parameterizations

Using (2.5) in (1.2), the latter acquires the more symmetric form:
e2.8
Treguier (1999) suggested that should be a downgradient of potential vorticity q = h−1(f + ζ) rather than a downgradient of layer thickness . The substitution4 introduces the term βf−1, which diverges at the equator. The problem is no longer present in the present model since βf−1 does not depend on z and cancels out in the difference a − 〈a〉. The first term in (2.8) then becomes
e2.9
which shows that the two representations are actually identical, a conclusion consistent with Roberts and Marshall (2000), who found no significant differences in the results of the two types of parameterizations.

d. At the A–D interface

If one integrates (2.2) and (2.8) from −Hb to and equates the results, one obtains that Since the constant of integration in the second of (2.2) is zero.

e. Depth-integrated eddy velocity

Consider the depth integral of the eddy velocity:
e2.10
In the A regime, we use (2.8); the first two terms yield zero identically, while the last term using (2.6) yields . Once the D-regime parameterization is derived in section 3, we show that the D-regime integral yields , which cancels the contribution of the A regime, leading to the result that satisfies the relation
e2.11
The zero vertically integrated momentum (2.11) reflects the fact that mesoscales do not exert a mean stress on the ocean but only distribute the surface stresses.

f. MPE to mesoscale energy transfer

Consider the total mesoscale energy production rate and make use of (2.1). Using the geostrophic form of the mean velocity and inverting the order of integration, we obtain
e2.12
If we substitute the first two terms of (2.8) corresponding to the A regime, we obtain
e2.13
which shows that the last term in (2.13) is negative whose implication is considered next.

g. Mesoscales as sink of mean potential energy

Consider the dynamic equation of the total eddy energy (TEE) defined as the sum of kinetic and potential eddy energies [CD6, their (2d)]:
e2.14
We computed the volume integral of the right-hand side of (2.14) for the GM case ξ = s and the present model ξs with the following results (1 TW = 1012 Watts)5:
e2.15
The following comments may be useful. First, in both GM and present parameterizations, the production corresponds to the same part of the Lorenz energy diagram, namely, the two lower blocks from MPE to the eddy potential energy (EPE) shown in Fig. 1 of CD6. Second, the difference of 0.23 TW in (2.15) is solely due to the different parameterizations of the bolus velocity in GM and the present model. Specifically, the term responsible for the last negative contribution in (2.13) comes from the mean velocity term in (2.8), which is not present in GM. Third, the results imply that the present parameterization of the eddy velocity entails a less efficient transfer of energy from MPE to EPE.

3. D-regime fluxes: Buoyancy

While (2.1) shows that in the A regime the horizontal buoyancy flux is the suitable variable to express the mesoscale eddy-induced velocity, in CD11 it was shown that in the D regime the appropriate variable is the vertical buoyancy flux, which vanishes at the surface, which ensures the vanishing of the streamfunction and avoids the unphysical result of (1.4). The form of the vertical buoyancy flux presented in CD11 [(35)] reads as follows:
e3.1
where the dimensionless function ω(z) is given by (we recall that z < 0)6
e3.2
High-resolution numerical simulations by Luneva et al. (2015) showed that (3.1) and (3.2) work well within and somewhat below the ML but not near the bottom of the D region, which makes (3.1) unsuitable to determine . The reason is that in deriving (3.1) it was assumed that in the D regime the mixing is strong, which is valid in most of the D regime but not all the way to the bottom of it. It is therefore necessary to divide the D regime into two parts: a mixed layer (ML) proper, where mixing is strong, and a transition (T) regime, where mixing is no longer as strong as in the ML but not yet as small as in the A regime and where neither the ML nor the A-regime parameterizations apply. Regrettably, we do not have a parameterization for the T regime derived from the mesoscale dynamic equations as in the cases of the ML and A regimes; we therefore present a heuristic parameterization of the T regime that matches the ML- and A-regimes parameterizations in the appropriate limits. We extend (3.1) across the T regime toward the bottom of the D regime using the following criteria: 1) in the ML, (3.1) must be recovered; 2) the diapycnal buoyancy flux must vanish at the bottom of the D regime; and 3) going from the bottom of the D regime across the T regime, the terms in the new flux, other than the original (3.1), must be decreasing fast enough so that criterion 1 is satisfied. Let us begin with criterion 2. Using (2.4), it translates into:
e3.3
For convenience of notation, we write the formula we seek as follows:
e3.4a
where the vector Ω is found next. To satisfy criterion 1, we must keep the original term ω×ez in Ω. To satisfy criterion 2, that is, Ω(−) = s(−), we notice that cannot appear in Ω(−), since and thus can acquire arbitrary values at z = −. Finally, to satisfy criterion 3, we introduce a function Φ(z) that approaches unity at |z| ~ and becomes vanishingly small for |z| ≪ . Putting these considerations together, the simplest expression of Ω(z) has the form
e3.4b
where an asterisk means a variable at z = −. Furthermore, the function Φ(z) must satisfy the boundary conditions
e3.5
As for the form of Φ(z) we reasoned that since for small |z|, (3.1) need not be modified, Φ(z) must decrease faster than ω; furthermore, in the region of a strong mixing where , the correction term proportional to Φ(z) should be small. Thus, to satisfy these requirements we suggest the following form of Φ(z):
e3.6
Since (3.4) are valid for any , the latter cannot be determined using (3.4). We therefore suggest the following: on the one hand, should be equal to or larger than the mixed layer depth (MLD; as shown by the numerical simulations); on the other hand, it should be less than the depth of the thermocline since at that depth the stratification is too strong for the D regime to exist. A simple way is to express as the depth halfway between the MLD and the thermocline, which leads to the relation
e3.7
Figure 5 shows a map of in units of the MLD denoted by h. The regions where the simulations of Mensa et al. (2013) and Veneziani et al. (2014) found > h are reproduced by (3.7).
Fig. 5.
Fig. 5.

(top) Map of the depth defined in (3.7) in units of the MLD computed from the potential density criterion Δσρ = 0.03 kg m−3. In all regions ≥ MLD, indicating that below the mixed layer the flow is still diabatic. (bottom) Map of the isopycnal slopes at . The value of is no longer arbitrary as in tapering schemes; it is now given by the model. It is not a universal constant but a location-dependent variable.

Citation: Journal of Physical Oceanography 48, 3; 10.1175/JPO-D-16-0255.1

a. Stream function

In CD11, a model was presented for the D-regime streamfunction that no longer leads to (1.4). To achieve that goal, CD11 exploited the fact that the tracer equation expressed in terms of Ψ, Fres is invariant under the transformation where A is often referred to as a gauge function. The choice of the latter required external inputs that were chosen to be as follows: 1) since there is no reason to introduce modifications in the A-regime deep ocean, Ψ + A must be almost equal to the adiabatic Ψ and A must therefore vanish in the deep ocean, and 2) at the surface, the function A in Ψ + A must cancel Ψ so as to attain the surface boundary condition Ψ(0) = 0. The new streamfunction was then found to have the following form:
e3.8
which we purposely wrote in the form used by tapering models with the difference that in the latter, the tapering function T(z) was unknown while in the present model it is expressed in terms of the vertical buoyancy flux whose form has already been derived in the forms (3.1)(3.4); specifically, we have
e3.9

b. Surface values

Since the vertical buoyancy flux vanishes at the surface, from (3.9) it follows that T(0) = 0 and thus
e3.10
The second relation is the expected result that the vertical eddy velocity vanishes at the surface.

c. Bottom of the D regime

Substituting in (3.9), one obtains
e3.11
While tapering models had to impose the relation , here it follows from the model; second, the value of at the bottom of the D regime is no longer unknown, it now follows from relation (3.11) (see Fig. 5, lower panel).

d. Results7

In Fig. 5, upper panel, we show the depth given by (3.7) in units of the MLD (denoted by h) computed from the potential density criterion Δσ = 0.03 kg m−3; is deeper than the MLD, indicating that below the ML the flow is still diabatic. The results are in agreement with those from the numerical simulations by Mensa et al. (2013) for the Gulf Stream and Veneziani et al. (2014) for the tropical South Atlantic. At the same time, the model also predicts that the ratio is very location dependent. An important consequence of the latter concerns studies of subduction that takes place at the mixed layer depth, which may or may not coincide with . Specifically, in regions where ~ h, subduction takes places near the A regime, whereas when h, it occurs well within the D regime. This implies that one must employ different parameterizations and that the assumption = h (Sallée and Rintoul 2011) must be revisited. In the lower panel of Fig. 5, we show the values of the isopycnal slopes . This variable turned out to be a crucial ingredient in the models based on a tapering function; the values = 1/500 and 1/100 suggested by Gnanadesikan et al. (2007) are reproduced by the present model, but they are not universal since the results show that is location dependent. This may explain why assuming a global value of led previous authors to the difficulties described in their work. In Fig. 6, we plot the vertical profiles of the buoyancy flux (3.4) in four regions: ACC, Gulf Stream, Kuroshio, and Sea of Japan. A further remark may be of interest. Vertical buoyancy fluxes caused by small-scale turbulent mixing are of the form Fυ(b) = −KυN2, where Kυ is the vertical diffusivity. If one employs the characteristic values Kυ = 102 cm2 s−1 and N2 = 10−6 s−2, the resulting buoyancy flux is O(10−8) m2 s−3, which is of the same magnitude as the fluxes in Fig. 6. However, the important difference is that while the mesoscale vertical buoyancy fluxes are positive, the ones caused by small-scale turbulent mixing may be of either sign depending on whether temperature and salinity gradients lead to instabilities or not. Therefore, the two fluxes can add or subtract from one another depending on the type of stratification. In strongly unstable configurations leading to convection, for example, in winter in the Labrador Sea, the value of Kυ is quite large (as in convective adjustment models) so as to overwhelm the contribution from the fluxes in Fig. 6. In Fig. 7, we show the annually averaged T(z) from (3.9). Finally, using (3.8) and one obtains the following expression for the D-regime eddy velocity: The D-regime integral in (2.10) yields , which in turn leads to (2.11).

Fig. 6.
Fig. 6.

Vertical profiles of the vertical buoyancy flux in the ACC, Gulf Stream, Kuroshio, and Sea of Japan. In the D regime, the flux is given by (3.4), while in the A regime Fυ(b) = κMN2s ∙ ξ. The depth z is in units of . To convert the flux to watts per square meter, one multiplies Fυ(b) by (108 s3 m−2) W m−2, αT = a010−4 K−1, where α0 is location dependent.

Citation: Journal of Physical Oceanography 48, 3; 10.1175/JPO-D-16-0255.1

Fig. 7.
Fig. 7.

Tapering function T(z) defined in (3.9) for the California Current, ACC, Gulf Stream, and Sea of Japan.

Citation: Journal of Physical Oceanography 48, 3; 10.1175/JPO-D-16-0255.1

4. Arbitrary tracers

The vertical buoyancy flux (3.1) is a particular case of the vertical tracer flux:
e4.1
As in the case of (3.1), the flux (4.1) is valid in most of the D region where mixing is strong. In the vicinity of the AD interface, (4.1) must be extended to match the A-region vertical tracer flux:
e4.2
In analogy with the treatment of buoyancy, we suggest the following extension of (4.1):
e4.3
At the AD interface, (4.3) matches (4.2). In the buoyancy case where , (4.3) becomes , which is (3.4a). In summary, the mean tracer equation has the form
e4.4
where ; the vertical flux is given by (4.3), and the horizontal flux is given by . Though the AD tracer parameterizations match at their interface, their use in OGCMs requires the solution of an additional problem that we discuss in the next section.

5. The OGCM tracer equations

The transformed Eulerian mean (TEM; Treguier et al. 1997) formulation of the A regime expresses the mean buoyancy equation in terms of skew and residual fluxes. However, (6) of Treguier et al. (1997) also shows the presence of a diapycnal flux (called Σ in their paper) that should not be present in an adiabatic regime. Rather than simply ignoring it, several authors (de Szoeke and Bennett 1993; Gent et al. 1995; Aiki et al. 2004; Gent 2011) suggested that since in the A regime water parcels move along isopycnal surfaces, the correct representation is in terms of thickness-weighted (isopycnal) averages denoted by a caret ^ and defined as , where h = gz/∂b is the layer thickness. MM called the approach temporal residual mean (TRM) and derived the following tracer dynamic equation in which there are no diapycnal fluxes and which MM called “exactly the same as are solved in modern oceanic GCMs”:
e5.1
Here, is the 3D mean velocity, and Q represents external sources, submesoscales, and small-scale mixing. The skew, 3D eddy velocity, and Redi flux are defined as follows: , , and . In the GM model, and . If the ocean were fully adiabatic, the above equations would suffice, but in the real ocean one must account for the presence of the diabatic D regime. As Killworth (2005) showed, the latter must be described using Eulerian averages denoted by an overbar, and the corresponding mean tracer reads as follows:
e5.2
where the horizontal and vertical tracer fluxes are defined as and where primes denote mesoscale fields. The difficulty is that the different averages represented by a caret and an overbar may prevent the AD-regime parameterizations from matching at their interface, and one must make sure to employ the same averages in both AD regimes. The choice is actually limited. In fact, since thickness-weighted averages do not apply to the D regime where water parcels do not move along isopycnal surfaces, the only alternative is to employ Eulerian averages in both AD regimes. The transformation of (5.1) from thickness-weighted isopycnal averages to Eulerian averages was carried out by MM with the following result:
e5.3
The unparameterized form of the new vector was derived by MM (p. 1234), who stated it “would be very difficult to parameterize.” No further work was done on parameterizing E, and an impasse was created that lasted until now when we present a parameterization of E (appendix B).

6. Parameterization of E

MM expressed E in terms of second-order correlation functions as follows:
e6.1
In appendix B we present the parameterization E2 with the result given by (B.25). Equations (22a) and (22d) of CD6 derived the following form of E1:
e6.2
where is a diapycnal tracer flux whose form is derived in appendix B: (B.14), while the form of is given by (B.15). The vector E2 contributes two new components to the total tracer flux that is not present in the GM–Redi model. Specifically, the xy components of the tracer flux are not zero even when the gradients of the mean tracer in those directions is zero, see (7.4), which represents a nonlocal contribution.

7. OGCMs implementation

Since most OGCMs represent the 3D tracer flux in the tensor form
e7.1
we do likewise. In appendix B, we derive the horizontal, vertical, and E fluxes [(B.10), (B.13), (B.14), and (B.25)] with which we derive the following form of the diffusivity tensor :
e7.2
The first tensor represents the symmetric part given by the Redi (1982) term, the second tensor represents the antisymmetric part that accounts for the difference between the slopes s and ξ, and the third tensor is due to E where the components are given by
e7.3
where γ = (rd/K3/2)(∂K/∂z). Without the function E and with ξ = s, the first two tensors in (7.2) yield the GM + Redi model [Griffies 1998, his (23)], with K13 = K23 = 0, K31 = 2sx, K32 = 2sy, K33 = s2. The ex, ηy new terms in K11, K22 may be relatively small since they enter as corrections to unity; the terms ez and ηz in K13, K23 may also be small since they scale like the square of the isopycnal slopes; the terms ey, ηx are the most significant new contributions to the diffusivity tensor since they substitute the zero components K12 and K21 from the sum of the first two tensors. Thus, because of the vector E, the x and y components of the flux now acquire the two new components:
e7.4
Because of the nonzero gradients in the y direction, there is a nonzero flux in the x direction and analogously for the y direction. The two fluxes of (7.4) represent a nonlocal feature. For use in OGCMs, we suggest the following simplified form of (7.2):
e7.5
In Fig. 8, we show the ACC z profiles of the components K12 and K21 contributed by E in (7.5). The magnitudes are much smaller than the diagonal components K11 and K22 from the sum of the first two tensors in (7.5). In the D regime, use of (4.3) and yields the following diffusivity tensor :
eq1
e7.6
At the AD interface, (7.5) and (7.6) match since In Fig. 9, we show the z profiles of the three components in (7.6). The normalization was chosen so that at the bottom of the D regime, the diffusivities match those of the A regime.
Fig. 8.
Fig. 8.

ACC z profiles of the A-regime diffusivity tensor components K12 and K21 contributed by E in (7.5). The magnitudes are much smaller than the diagonal components K11 and K22 from the first tensor in (7.5).

Citation: Journal of Physical Oceanography 48, 3; 10.1175/JPO-D-16-0255.1

Fig. 9.
Fig. 9.

The z profiles of the three components of D-regime diffusivities K31, K32, and K33 in (7.6) in the ACC and Gulf Stream. The normalization was chosen so that at the bottom of the D regime, the diffusivities match those of the A regime.

Citation: Journal of Physical Oceanography 48, 3; 10.1175/JPO-D-16-0255.1

8. Deacon cell

The Deacon cell (40°–60°S) is considered an artifact that does not reflect a real flow across isopycnals. In an influential paper, Danabasoglu et al. (1994) showed that use of the GM model made the Deacon cell almost disappear, a result that favored the acceptance of the GM model. Figure 1c of the 2011 review article by Gent (Gent 2011) cites that result. On the other hand, Fig. 15 of Farneti et al. (2015) shows that 16 modern OGCMs exhibit again a robust Deacon cell of 20–28 Sv. Figures 10a and 10b show the Deacon cell with GM and the present model. The latter causes a decrease of the magnitude of the cell with respect to the GM case.

Fig. 10.
Fig. 10.

Deacon cell. (top) The GM model. (bottom) The present model, which induces a considerable reduction of the intensity of the cell compared to the GM model.

Citation: Journal of Physical Oceanography 48, 3; 10.1175/JPO-D-16-0255.1

9. Ocean heat uptake: Enhanced stratification

A recent analysis by Cheng et al. (2017, p. 6) concluded that there is “a robust ocean warming in the upper 700 m, but the warming does not penetrate to the deep ocean below 700 m.” By contrast, Fig. 2a of Kuhlbrodt and Gregory (2012, p. 3) shows the results of several climate models that predict a warm deep ocean and point out that “a model with a weak vertical temperature gradient [weak stratification] has a larger capacity for downward heat transport.” In their Fig. 2a, the temperature profiles (minus surface temperature) from several OGCMs are shown together with the data from the World Ocean Atlas 2005 (WOA05; Locarnini et al. 2006). Profiles to the right of the latter exhibit a weaker stratification than the data and transport heat too deeply. The results are from ocean–atmosphere coupled models.

In the stand-alone ocean model treated here, the global-mean vertical profiles of the in situ temperature minus surface temperature are presented in Fig. 11 from the present mesoscale model together with the same WOA05 data. The results are from the last 3 years of 300-yr runs with the present version of the GISS OGCM (see appendix C). Since the results correspond to an increased stratification, it will be the task of future research to find out whether in a coupled simulation the present mesoscale model yields more stratification than previous models and whether this brings the coupled model results closer to observations.

Fig. 11.
Fig. 11.

Vertical profile of the average over the whole ocean of the in situ temperature minus the surface value in the present model and the WOA05 data (Locarnini et al. 2006). Figure 2a of Kuhlbrodt and Gregory (2012) shows that most model results fall to the right of the data corresponding to a weak stratification and to a large capacity for downward heat transport that is not observed. The present model induces a stronger stratification.

Citation: Journal of Physical Oceanography 48, 3; 10.1175/JPO-D-16-0255.1

10. Conclusions

The goal of this work was to derive and assess mesoscale parameterizations of both AD regimes taking into account the nonlinear nature of mesoscales as concluded from altimetry data.

The parameterizations describe arbitrary tracers. We cite the following features: 1) altimetry data imply that in the A regime the eddy velocity acquires a new, nondowngradient term; 2) the effect of the latter is to lower the transfer of mean potential energy to mesoscales with the result that isopycnal slopes are less flat; 3) the D-regime parameterization was assessed with a mesoscale-resolving simulation; 4) an expression for the depth of the D regime is suggested whose results agree with those of numerical simulations; 5) there is a reduction of the strength of the Deacon cell compared to the GM case; and 6) there is an enhancement of the deep-ocean stratification. Previous models yielded a weakly stratified deep ocean that entailed a larger ocean heat uptake than observed. For ease of implementation in coarse-resolution OGCMs, the 3D tracer fluxes are expressed in terms of 3D diffusion tensors (as done in previous parameterizations).

Acknowledgments

The authors thank two anonymous referees whose valuable criticism helped us to achieve a more focused presentation. The authors thank the NASA Climate Program and the NASA Oceanography Program managed by T. Lee and E. Lindstrom for financial support. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Center for Climate Simulation (NCCS) at the Goddard Space Flight Center. VMC would like to dedicate his work to Aura Sofia Canuto.

APPENDIX A

Treatment of Nonlinearity

Since altimetry data analyzed by C11 have shown the nonlinear nature of mesoscales, the key features of the treatment of nonlinearity used in this work must be highlighted. The key feature was the substitution of the Navier–Stokes equations (NSE) for the mesoscale velocitiesA1 with the stochastic Langevin equations. While the NSE are nonlinear in the velocities but linear in the viscosity term, the Langevin equations are the opposite: they are linear in the velocity field and nonlinear in the viscosity term that is now represented by a turbulent viscosity (often referred to as eddy viscosity). The advantage of the linear Langevin equations is that they allow analytic solutions once a representation is found of the turbulent viscosity. As discussed in V. M. Canuto and M. S. Dubovikov (2015, unpublished manuscript; http://www.giss.nasa.gov/staff/vcanuto/canuto_nonlinearity_201507.pdf), the above procedure was pioneered by turbulence theorists who successfully applied it to special turbulent flows. The contribution of two of the present authors was to show that flows of geophysical interests such as those driven by shear, buoyancy, and so on could also be reproduced by such methodology. Here, we cite only the expression of the turbulent viscosity for an eddy of arbitrary wavenumber k:
eea_1_1
Here, ν is the molecular viscosity, and E(k) is the eddy kinetic energy spectrum whose integral over all wavenumbers k yields the eddy kinetic energy K. The sum νd(k) = ν + νt(k) is often referred to as dynamical viscosity. Relation (A.1) encompasses both low and high Reynolds number regimes: 1) for small eddies (large k), the integral is small and the turbulent viscosity is small; 2) for large eddies (small k), the integral is large, the molecular viscosity is negligible, and the turbulent viscosity is large; and 3) if one could obtain the kinetic energy spectrum E(k) from observations and/or numerical simulations, (A.1) could be used to evaluate the turbulent viscosity for eddies of different sizes, but present data (e.g., Scott and Wang 2005) do not cover the entire spectrum in (A.1). The mixing length theory is a particular case of (A.1). In fact, if one assumes that E(k) is dominated by a single large eddy k0, then E(k) = E0δ(kk0) and thus K = E0k0; it follows that with , which is the Rossby deformation radius. To further clarify the procedure used to construct the mesoscale fluxes, consider the vertical tracer flux , where primes indicate mesoscales and an overbar stands for an ensemble average. Once the dynamic equations for the mesoscale fields τ′, w′ are derived, one constructs the second-order correlation . Since the advective term in the original dynamic equations contains the 2D mean velocity , the latter becomes part of the vertical tracer flux [see (3.1)(3.2)]. The above treatment of nonlinearity applies to both AD regimes where νt enters with different signs as discussed in the text [see (1.5)(1.6)].

APPENDIX B

Horizontal, Vertical, and E Fluxes

a. Horizontal fluxes

Since by definition we have
eeb_1_1
it follows that [see (6.1)]
eeb_2_2
Next, we use the relations
eeb_3a_3a
eeb_3b_3b
Equation (B.2) then becomes (κM = 1)
eeb_4_4
Consider now relation (6.2):
eeb_5_5
where by definition
eeb_6_6
and
eeb_7_7
Using the last of (B.6) and (B.7), (B.5) becomes
eeb_8_8
Substituting (B.8) into (B.4), yields the relation
eeb_9_9
The simplest ansazt is
eeb_10_10
which in the buoyancy case gives
eeb_11_11
which reproduces the last relation in (B.6).

b. Vertical tracer flux

We substitute (B.9) into (B.8) and obtain
eeb_12_12
which we then substitute into (B.5) and obtain
eeb_13_13
To ensure the antisymmetric nature of the second tensor in (7.2), we must require that the last term in (B.13) vanishes, which gives the following form of the diapycnal flux:
eeb_14_14
Together with (B.5)(B.6), this implies that
eeb_15_15

c. The function

Because of the geostrophic relation , the last term in E2 in (6.1) vanishes and
eeb_16_16
Next, we use double/single primes and double/single overbars to denote Eulerian/isopycnal mesoscale fields and Eulerian/isopycnal averaging. Using the transformation laws (e.g., appendix A of CD6), , , and E2 becomes
eeb_17_17
To parameterize , we consider the mesoscale tracer equation in isopycnal coordinates and adiabatic approximation:
eeb_18_18
where is the nonlinear term. Next, we Fourier transform (B.18) with respect to time and horizontal coordinates and use the dispersion formula derived in CD5, (15a), where ud is the eddy drift velocity (C11) whose specific form is not needed in the present context. Following a suggestion by Killworth (1997, 2005), we employ the same symbols τ and u in both physical and Fourier spaces; the difference is that in Fourier space we use the independent variable k and ω. We then obtain
eeb_19_19
Using the mixing length model, in the vicinity of the maximum of the energy spectrum at k ~ k0 ~ [this relation was derived in CD5, their (13b)], the nonlinear term is given by
eeb_20_20
where K is the eddy kinetic energy. Substituting the first of (B.20) into (B.19) and solving for τ″,
eeb_21_21
To use this result in the second of (B.17), we employ the relation , where B″ is the mesoscale component of the Montgomery potential defined in (1c) of CD5: B = p + gρz. Since mesoscale fields are almost geostrophic, we use the geostrophic relation, which in Fourier space reads as follows: Substituting the above relations into the second half of (B.17), we obtain the relation
eeb_22_22
In the denominator, the ratio of the second to the first term is such that , and thus we neglect the former. Then, because of the axial symmetry of mesoscales, the function does not depend on the direction of k. We substitute (B.22) into the first of (B.17) and average <…> over the angles of k, which gives . Using this result and
eeb_23_23
from (B.22), we obtain
eeb_24_24
We recall that the above relations are valid in the vicinity of the maximum of the energy spectrum, that is, at . Under the assumption that the shape of the spectrum of is similar to the one of the energy spectrum, relation (B.24) is still valid if we substitute <...> with the functions and K, respectively. Thus, after integrating (B.24) over d2k and substituting into (B.17), we obtain
eeb_25_25

APPENDIX C

OGCM Code Structure

We employed the GISS ER model, which is the ocean component of the coupled NASA GISS model E (Russell et al. 1995; Russell et al. 2000; Liu et al. 2003). Here, an early version of the revised E2-R code was run in stand-alone mode (Danabasoglu et al. 2014). It employs a mass coordinate, which is approximately proportional to pressure with 32 vertical layers whose thicknesses range from ≈12 m at the surface to ≈200 m at bottom and a horizontal resolution of 1.25° (longitude) by 1° (latitude). It is a fully dynamic, non-Boussinesq, mass-conserving, free-surface ocean model using a quadratic upstream scheme for the horizontal advection of tracers and a centered difference scheme in the vertical. A 1800-s time step is used for tracer evolution. The fluxes of ocean variables through unresolved straits are represented by a subgrid-scale parameterization. Sea ice dynamics, thermodynamics, and ocean–sea ice coupling are represented as in the CMIP5 model E configuration (Schmidt et al. 2014), save that here ice is on the ocean model grid. To force the model we adopted the Coordinated Ocean-Ice Reference Experiment I (CORE-I) protocol (Griffies et al. 2009), which forces the ocean with fluxes obtained from bulk formulas the inputs to which are the ocean model surface state and atmospheric conditions obtained from a synthesis of observations that repeat the seasonal cycle of a “normal year.”

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1

The Rossby phase velocity is given by , where ez = (0, 0, 1), rd is the Rossby deformation radius, and f is the Coriolis parameter.

2

An overbar denotes an Eulerian mean. In general, in a stationary process, time and ensemble averages are assumed to be identical, which is the so-called ergodic hypothesis whose proof is a complicated matter but which has never been shown to be incorrect (Bradshaw 1976).

3

The reason is given in CD5, where it is shown that ud is part of the dispersion relation (15a) obtained as a solution of the eigenvalue problem for the mesoscale Bernoulli function (11a).

4

Since .

5

It can be checked that the integral of (2.14) and geostrophy yield (2.12).

6

The physical meaning of the integral in (3.2) is as follows: the contribution to ω(z) given by the first term is local since it is computed at the same z as ω(z), while the second term is the sum of the contributions from z to z = 0 and represents a nonlocal contribution as expected since nonlinearity is a nonlocal feature.

7

See appendix C for details on the OGCM used to obtain the results.

A1

An arbitrary field A is split into mean and fluctuating (submesoscale) components ; substituting into the original equation, averaging and subtracting the mean equation from the total, one obtains the equations for the fluctuating fields.

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