1. Introduction

Eddy buoyancy fluxes tend to show a large diapycnal projection even when only weak (or even no) irreversible mixing is associated with the process leading to the eddying motion. Mathematically, this is related to a large rotational component of the diapycnal eddy flux, with no net effect on the mean buoyancy. To identify the rotational component and to differentiate it from the residual divergent component in an objective manner, McDougall and McIntosh (1996, 2001) developed the temporal residual mean (TRM) framework, which was later expanded to the generalized TRM (TRM-G) by Eden et al. (2007a, 2009). In the TRM-G framework, the residual divergent diapycnal eddy buoyancy flux is only related to irreversible changes in buoyancy due to dissipation of buoyancy variance (and higher order moments) by microscale processes (e.g., given by molecular diffusion, radiative and frictional heating, radioactive decay, etc) or temporal changes of variance and higher order moments. The remainder (which can sometimes dominate the total flux) of the diapycnal eddy flux is given by a rotational flux, which does not change the mean buoyancy. When estimating a diapycnal diffusivity from measurements or a numerical simulation of turbulent fluxes using a flux-gradient relationship, it is important to account for that rotational part of the flux, as otherwise unphysical (often large and negative) diapycnal diffusivities are estimated. The presence of diapycnal rotational eddy fluxes in large-scale oceanic flow and how they can be effectively removed from the flux is demonstrated in this study using an exemplaric numerical experiment of an eddying channel model.

On the other hand, it is obvious that rotational eddy fluxes might show up not only in the diapycnal projection of the eddy flux but also in the perpendicular component: that is, in the part of the eddy flux flowing on the surface (isopycnal) given by the mean buoyancy. It was noted first by Marshall and Shutts (1981) how horizontal mesoscale eddy buoyancy fluxes can be approximately decomposed into rotational and divergent parts. This idea was combined with TRM-G by Eden et al. (2007a) in a model-based estimate of lateral diffusivities appropriate to the isopycnal thickness diffusivity of the Gent and McWilliams (1990) parameterization. Note that the thickness diffusivity is the counterpart of the diapycnal diffusivity for turbulent mixing of buoyancy in the isopycnal direction and that this part of the mixing can be written as an eddy-driven advective (bolus) term in the mean buoyancy budget (Gent et al. 1995). Eden et al. (2007b) found anisotropic isopycnal thickness mixing in large parts of the North Atlantic and Southern Ocean; that is, they found that the thickness diffusivity might be better represented by a tensor including a skew (antisymmetric) component, instead of a single scalar, indicating isotropic lateral mixing. This result is qualified here on the basis of exemplaric numerical experiments of an eddying channel and a closed basin, both featuring strong eddy-driven zonal jets. The experimental setting might be thought of as an extreme example of trying to introduce anisotropic isopycnal eddy mixing.

It will be discussed below that the TRM-G framework defines the rotational isopycnal fluxes in such a way that the residual isopycnal flux is only related to production of eddy variance on a reference isopycnal or, in other words, with removal of mean (available) potential energy by eddy activity. It turns out that, in both eddying models, large rotational eddy fluxes show up that are related to a large skew component, but the residual part of the flux is downgradient, that is, isotropic. In other words, the TRM-G framework reveals that (in the eddying models) the use of a scalar thickness diffusivity is sufficient and consistent with the classical picture of the eddy-driven isotropic thickness diffusion by Gent et al. (1995). This is the main result of this study.

In the following section, the configuration of the exemplaric numerical experiments is discussed. In section 3, the TRM-G framework is discussed and applied to the diapycnal eddy buoyancy fluxes simulated by the numerical models, whereas in section 4 the isopycnal eddy buoyancy fluxes within the jets are discussed using the TRM-G framework. The last section summarizes and discusses the results.

2. Numerical model

The impact of rotational eddy fluxes and anisotropic mixing is discussed using fluxes from a primitive equation model in Cartesian coordinates on a β plane, which was also used in Eden (2010). The numerical code of the model can be found online (available online at http://www.ifm.zmaw.de/~cpflame). The first setup of the model (CHANNEL) is configured as a channel forced at the sidewalls by restoring zones. These zones are 6 grid points wide, located at the northern and southern boundary, and relax the buoyancy b toward its initial condition, which is characterized by a state with specified linear meridional (b_y ≡ M₀²) and vertical (b_z ≡ N₀²) gradients and no zonal variations. Note that the actual simulated zonally averaged buoyancy deviates not very much from the initial state, despite strong eddy activity. Dissipation is given by biharmonic diffusion and friction (with viscosity and diffusivity of 1.25 × 10¹¹ m⁴ s⁻¹) and additionally by an interior linear drag with coefficient r = 10⁻⁷ s⁻¹. There is also (small) harmonic vertical diffusion and friction with vertical diffusivity of 10⁻⁴ m² s⁻¹ and vertical viscosity of 10⁻³ m² s⁻¹. The quadratic channel of width L = 4000 km is resolved by 256 × 256 grid points and is 2000 m deep with 40 levels. The horizontal (vertical) resolution is constant at approximately 16 km (50 m). Boundary conditions are free slip and zero buoyancy fluxes at top, bottom, and lateral boundaries. The biharmonic and vertical mixing and the interior drag represents the unresolved effects of dissipation by smaller-scale dynamics on the quasigeostrophic dynamics in the model, whereas the restoring near the sidewalls represents large-scale baroclinic forcing of the flow.

The CHANNEL setup is characterized by a basin Rossby number

View Expanded

Burger number

View Expanded

and an Ekman number Ek = r/f related to the interior drag. Resulting important parameters are the Rossby radius L_r = (N₀h)/( fπ) and the inverse Eady growth rate σ = N₀/M₀². Here we discuss an experiment with L_r = 96 km and σ = 4.8 days. Figure 1 shows a snapshot of pressure and horizontal velocity at 500 m. Strong mesoscale eddy activity fills the whole model domain, which is generated by baroclinic instability feeding from the mean available potential energy generated by the restoring zone. Part of the eddy energy also feeds back on the mean kinetic energy, forming four eastward zonal jets, which can be seen in a time average (Fig. 1b). All time averages shown in this study are taken from a 5-yr-long period taken from the model integration following a spinup phase of 3 years, in which the four zonal jets are almost steady, as seen in Fig. 1c. Note that after this period some of the jets start to merge.

The second setup of the model (BASIN) is identical to CHANNEL but with closed sidewalls. In particular, the same relaxation zones are located at the northern and southern boundary and there is no other forcing. To obtain a stable integration, however, the following modification have been found to be necessary: there is no interior drag in BASIN but there is linear bottom drag with a time scale of 4 × 10⁻⁴ s⁻¹, the rigid lid in CHANNEL is replaced by a free surface formulation in BASIN, and M₀ and the reference Coriolis parameter are reduced such that the Rossby radius is decreased to L_r = 33 km and the inverse Eady growth rate is increased to 9.7 days. Strong eddy activity due to the action of the restoring zones still generates zonal jets in BASIN (Fig. 1d), which show, however, much more fluctuations in their meridional position as in CHANNEL. To obtain a simulation with fixed jet positions in BASIN as well, topography was introduced in the form of four zonal ridges of about 200-m height. These ridges are able to fix the jets in their position. Results are shown below as 10-yr averages after a 10-yr spinup integration of BASIN.

3. Diapycnal rotational eddy fluxes

The time mean buoyancy budget is written in TRM-G as

View Expanded

Here Q indicates microscale diabatic processes such as molecular diffusion, radiative or frictional heating, etc. or, in the case of the numerical model discussed below, subgrid-scale processes that change buoyancy such as the parameterization for isotropic turbulent mixing. Note that standard Reynolds averaging was applied; for example, the buoyancy b has been split into time mean b and deviation b′ before taking the time average. The eddy buoyancy flux u′b′ (and fluxes of higher order moments) resulting from the Reynolds averaging, has been decomposed according to the TRM-G formalism¹ as

View Expanded

that is, into a diffusive [given by the first term on the rhs of Eq. (2)], advective (second term), and rotational flux component (third term). Note that the eddy-driven advection velocity u*, which advects together with the Eulerian mean velocity u the mean buoyancy b in the mean budget Eq. (1), is given by u* = ∇ × B and that the diapycnal projection of the eddy flux K takes the physical meaning of a turbulent diapycnal diffusivity in the mean budget Eq. (1). Note also that the rotational flux component, ∇ × θ, drops out in the mean budget equation [Eq. (1)]. It does show up, however, in the budget for the buoyancy variance and higher order moments, offering the possibility for a physically consistent definition for the vector potential for the rotational eddy flux θ. More details about the choice for θ and its consequences can be found in Eden et al. (2007a, 2009). Here, it is stressed that the choice guarantees that the residual divergent diapycnal eddy fluxes and, thus, the diapycnal diffusivity K can be entirely related to dissipation or temporal changes of buoyancy variance and higher order moments by the microscale processes indicated by Q.

The vector streamfunction B for the eddy-driven flow, the diapycnal diffusivity K, and the vector potential for the rotational eddy flux θ are given in TRM-G by an infinite series of along-isopycnal fluxes of variance and higher order moments:

View Expanded

with the statistical moments ϕ_n = b′ⁿ/n, the cross-isopycnal unit vector n = ∇b/|∇b|, and the derivative normal to an isopycnal ∂_n(···) = −n × [∇ × (···)]. Note that the vector streamfunction B and the vector potential for the rotational flux θ are directed along isopycnals: that is, B · n = θ · n = 0. Further, both vectors are pseudo vectors since they are defined as a sum of vector products of two axial vectors, whereas K is a scalar with the physical meaning of a (diapycnal) diffusivity.

While Eq. (3) holds in general, a strong vertical stratification is assumed in the following; that is, it is assumed that |s_x| ≪ 1 and |s_y| ≪ 1 with the (negative) isopycnal slopes s_x = ∂_xb/∂_zb and s_y = ∂_yb/∂_zb. Note that this assumption holds to a very good approximation in the interior of the ocean and also in the numerical experiments discussed below such that consequences of the small-slope approximation are very small for all results discussed in the present study. In regions of weak stratification, however, as for instance in the surface mixed layer, the small-slope assumption might break down. Note also that the isopycnal vectors θ and B become quasi horizontal—only horizontal fluxes are involved—in the small-slope approximation and are given by

View Expanded

with the slope vector s = (s_x, s_y). Note that the vector subscript denotes a clockwise rotation of the horizontal vector u_h = (u, υ) by 90°; that is, . Note also that the same applies to the operator ∇_h = (∂_x, ∂_y); that is, . The divergence-free rotational eddy flux and the eddy-driven velocity are given by the horizontal parts of θ and B and become in the small-slope approximation

View Expanded

Note that the Gent and McWilliams (1990) parameterization, which is often used in noneddy-resolving ocean general circulation models, employs a simple downgradient closure for the horizontal eddy buoyancy fluxes (while setting θ_h = 0) such that u*_h = ∂_z(κs), with the lateral (isopycnal thickness) diffusivity κ = O(1000 m² s⁻¹).

In the small-slope approximation, the diapycnal diffusivity K in Eq. (3) is given by

View Expanded

In steady state and using the budgets for the variance and higher order moments the diffusivity K can also be written in the small slope approximation as (Eden et al. 2007a, 2009)

View Expanded

Note that Eq. (7)—resembling in fact a generalized Osborn–Cox relation (Eden et al. 2009)—shows that K can entirely be related to dissipation, that is, to genuine mixing processes, or temporal changes of buoyancy variance and higher order moments related to microscale processes indicated by Q in Eq. (1). The numerical example, discussed next, demonstrates the need to account for diapycnal rotational eddy fluxes since otherwise very large diapycnal mixing would be associated to mesoscale eddies.

Figure 2 shows the diapycnal projections of eddy buoyancy fluxes and rotational components in terms of diapycnal diffusivities in experiment CHANNEL. Very large diffusivities show up when ignoring any rotational eddy flux, that is, when calculating the diapycnal diffusivity from K₁∂_zb = −b′w′ − s · u′_hb′. However, the leading term of the diapycnal rotational eddy flux (Fig. 2b) almost entirely cancels the diapycnal component of the eddy buoyancy fluxes, leaving behind a residual diapycnal diffusivity smaller than 1 cm² s⁻¹ in the southern part of the domain, which is in fact smaller than the prescribed vertical diffusivity in the model. In the northern part, however, the diapycnal diffusivity K related to the residual divergent part of the diapycnal eddy flux is increasing toward values of 10 cm² s⁻¹. Note that the Rossby radius—as a measure of the relevant eddy length scales—is decreasing toward the north and is getting closer to the grid resolution. There are suggestions (Lee et al. 2002) that eddying models with coarser horizontal grid resolutions closer to the eddy length scales feature larger numerically induced buoyancy mixing compared to models with finer grids, which would explain the increased diapycnal mixing in the northern part of the model.

It turns out that the horizontal component of the rotational eddy flux is much less important than the vertical component (not shown) for the near cancellation between diapycnal eddy flux and its rotational component. It also turns out that the leading order term in the expansion series of θ_h is the most important, that is, the divergence of the horizontal divergence of the fluxes of buoyancy variance (variations of ∂_zb are a minor contribution), whereas the second term is already much smaller (Fig. 2c). Furthermore, it is the term related to zonal advection of variance ∂_x(uϕ₂/∂_zb) that dominates the meridional advection term ∂_y(υϕ₂/∂_zb) in the leading-order term of (not shown). Since ∂_zb (and u) is zonally almost constant and ∂_x(uϕ₂) ≈ ∂_x(uϕ) ≈ u∂_xϕ, it is the (zonal) advection of variance in the horizontal plane, which yields the bulk of the diapycnal rotational buoyancy fluxes.

Figure 3 shows K₁ and the leading term of the diapycnal rotational eddy flux in experiment BASIN in the western part of the domain, highlighting a single zonal jet (the situation in the other jets in BASIN, not shown, is very similar). Here, it also becomes obvious that, whenever the mean flow (Fig. 3c) is strong and enters a region of increasing (decreasing) variance, a downgradient (upgradient) rotational diapycnal flux component develops. Note that this is also the case for the northward mean current at the western boundary in Fig. 3; that is, this feature of rotational eddy fluxes appears to be independent of the direction of the mean flow. Again, the residual diapycnal eddy flux (not shown) is much smaller than the rotational one and is only close to the western boundary related to a diffusivity on the order of the prescribed vertical diffusivity in the model.

A schematic (Fig. 9) summarizes the situation, which was discussed by McDougall and McIntosh (1996) and which also applies to both numerical simulations discussed in the present study: a flow into a region with increasing (decreasing) variance leads to downgradient (upgradient) rotational diapycnal eddy fluxes. This can be easily seen for the case of a zonal jet since the diapycnal diffusivity related to the leading-order rotational flux component is given approximately by K ∼ u∂_xϕ. Because the diapycnal rotational eddy flux component is dominating the divergent part, the upgradient rotational fluxes in the region with decreasing variance are in fact strong enough to lead to upgradient total eddy buoyancy fluxes, implying the negative diapycnal diffusivities of large magnitude in Figs. 2a and 3a when ignoring the rotational eddy fluxes. The residual divergent part of the diapycnal eddy flux is much smaller—but still nonzero—than the rotational component. It does not show pronounced lateral fluctuations anymore as the rotational component does, and is directed predominantly downgradient in the numerical experiment as indicated in Fig. 9.

While the diapycnal diffusivity K is much affected by rotational eddy fluxes, this is much less so for the streamfunction B (not shown). The horizontal part of the streamfunction, B_h, is predominantly meridionally oriented since the leading order contribution to B_h given by the rotated eddy buoyancy flux is dominated by the zonal eddy fluxes (as shown in Fig. 4a). Note that the classical picture of baroclinic instability flattening mean isopycnals (Gent et al. 1995) would imply meridional eddy-induced velocities and thus a zonally oriented streamfunction B_h instead of a meridionally oriented streamfunction as in the numerical example. However, it will turn out that large parts of the eddy streamfunction, in particular the meridional components, are related to isopycnal rotational eddy fluxes. The isopycnal part of the eddy flux has not been affected by applying the TRM-G so far since, by construction, it applies to the diapycnal part of the eddy flux only.

4. Isopycnal rotational eddy fluxes

As first noted by Marshall and Shutts (1981), the isopycnal eddy fluxes can also contain a large rotational part; they, in fact, do in the eddying channel model as discussed below. The isopycnal rotational eddy flux can be identified and removed in an objective way as outlined by Eden et al. (2007b): that is, by identifying the residual divergent part of the isopycnal eddy flux with production (or temporal changes) of buoyancy variance and higher-order moments on an isopycnal related to a reference state, which is in turn equivalent to removal of mean available potential energy by eddy activity. To do so, the total buoyancy b is first decomposed into a stationary background state b̃ and a perturbation b̂ from that background: that is,

View Expanded

The idea for the definition of the background is, as formulated below, that it describes the dominant part of the variations of buoyancy, which is in the ocean given by the strong stratification in the vertical. Note, however, that this stratification might change with lateral position. Note also that often, for instance in quasigeostrophic approximation, the background state is chosen as a function of the vertical coordinate only; that is, b̃ = b̃(z). Such a simplified background state was also used by Marshall and Shutts (1981) and Eden et al. (2007b); here, however, a more general form is chosen to allow for horizontal variations of the vertical background stratification. In any case, the background buoyancy state can also be used as reference for the definition of available potential energy and, after time averaging (see below), for eddy and mean available potential energy in the usual manner (Lorenz 1955).

The buoyancy budget becomes

View Expanded

A diapycnal unit vector related to the background state is defined as ñ = ∇b̃/|∇b̃|, and the total velocity u is now decomposed into a cross-isopycnal component given by ũ_D = (u · ñ)ñ and an along-isopycnal component ũ_I = u − ũ_D. Note that for the choice b̃ = b̃(z) the isopycnal (diapycnal) velocity becomes horizontal (vertical). It is clear that the background buoyancy is advected only by the diapycnal velocity ũ_D; that is,

View Expanded

with ũ_D = |ũ_D| and Ѳ = |∇b̃|. The following assumptions were made: on the rhs of Eq. (10), it was assumed that the diapycnal variations of the perturbation buoyancy are much smaller than the one given by the background; that is, |ñ · ∇b̃| ≫ |ñ · ∇b̂| holds. Furthermore, it was assumed that the magnitude of the diabatic term Q is also small compared to Ѳũ_D. Note that, in fact, all diapycnal advection terms of the perturbation buoyancy can be moved to the right-hand side, where they are assumed to be dominated by the Ѳ term. Thus, in this respect, the isopycnal velocity ũ_I is free of divergence (analogous to the zero-order horizontal velocity in quasigeostrophic approximation). It is also assumed that the background isopycnal slopes given by s̃ = (∇_hb̃)/∂_zb̃ are small; that is, |s̃| ≪ 1 as before for the slopes of the total buoyancy b.

It is convenient to introduce the (approximate) horizontal isopycnal gradient ^iso = ∇_h − s̃∂_z, for which ũ_I · ∇(···) ≈ u_h · ^iso(···) holds within the small slope approximation. After Reynolds averaging with b̂ = b̂′ + b̂ and u_h = u′_h + u_h, the buoyancy budget becomes

View Expanded

The advective eddy fluxes in Eq. (11) have become two dimensional and therefore the eddy buoyancy fluxes—using the two-dimensional form of TRM-G—can be decomposed as

View Expanded

Note that ^isob̂ = ^isob and u′_hb̂′ = u′_hb′ such that the reference state b̃ plays only a role in defining ^iso. Now using the TRM-G formalism in the two-dimensional case leads to the definition of the isopycnal rotational eddy fluxes,

View Expanded

with the derivative normal to isolines of b in the isopycnal plane related to the background buoyancy ∂_n̂(···) = n̂ · ^iso(···) and with the lateral unit vectors and n̂ = |^isob|^−1isob pointing along and across isolines of b in the isopycnal surface related to b̃, respectively. Note that u_hb̂′ⁿ/n = u_hϕ_n. The potential for the isopycnal rotational eddy flux, , can now be combined with the vector potential θ for the diapycnal rotational flux as outlined in the appendix.

For the choice b̃ = b̃(z), appropriate to quasigeostrophic approximation, ŝ = 0 and the isopycnal gradient becomes simply ^iso = ∇_h. Note that this choice is discussed in Eden et al. (2007b). It is clear that the unit vectors n̂ and ŝ will then become ill defined in the case of flat mean isopycnals (related to total buoyancy), corresponding to a state of zero mean available potential energy. Note also that the same is true for the more general case where ∇b = ∇b̃: that is, gradients of mean perturbation buoyancy on mean isopycnals related to the background buoyancy—that is, nonvanishing mean available potential energy is necessary for the concept of defining isopycnal rotational eddy fluxes (since, otherwise, eddy activity cannot feed from the mean potential energy). However, note that the actual background diffusivity b̃ does not need to be specified in Eq. (13), only its slopes given by s, which are allowed to vary laterally and vertically, are necessary.

To leading order, the isopycnal rotational flux potential is given by the flux of variance in the direction of the mean isolines of buoyancy in the isopycnal surface related to the background state. Note that Marshall and Shutts (1981) assumed b̃ = b̃(z) and that the flux of variance is entirely directed along ; using the TRM-G formalism, that part of the flux in the direction ∇_hb can be also taken into account. Assuming furthermore a quasigeostrophic flow given by a streamfunction, which was assumed to be a function of buoyancy (and z) only, and assuming that the flux of variance is dominated by the mean advection, Marshall and Shutts (1981) demonstrated that the leading order rotational eddy flux circulates along isolines of variance ϕ₂ in the isopycnal related to the reference state. Using TRM-G, all assumptions used by Marshall and Shutts (1981)—except for the approximations involved in Eq. (10)—can be relaxed, but it will turn out that, in the numerical example, the isopycnal rotational eddy flux indeed tend to circulate around regions of enhanced or decreased variance.

In steady state and using the budgets for the variance and higher order moments for the perturbation buoyancy b̂ [using the same approximation as in Eq. (10)], the lateral diffusivity κ can also be written as (Eden et al. 2007b)

View Expanded

with the operator D_iso(···) = ^iso · [n(···)/|^isob|]. Note that Eq. (14) shows that the definition of the isopycnal rotational flux guarantees that the lateral diffusivity κ can be entirely related to diapycnal eddy buoyancy fluxes and diapycnal fluxes of variance and higher order moments that is, to the production (or temporal changes, not shown) of variance (and higher order moments) of buoyancy in the isopycnal surface related to the reference state or, equivalently, to production of eddy available potential energy and removal of mean available potential energy by mesoscale eddy activity as in the classical picture of eddy-driven overturning by Gent et al. (1995). Note that this feature is in analogy to the definition of diapycnal rotational fluxes, as discussed in the previous section and expressed in Eq. (7), which guarantees that the residual divergent diapycnal eddy flux is related only to dissipation (and temporal changes) of total buoyancy variance and higher order moments by microscale processes [indicated by Q in Eq. (1)]. It was discussed by Eden et al. (2007b) that the lateral diffusivity κ can be identified with the isopycnal thickness diffusivity appropriate to the Gent and McWilliams (1990) parameterization, whereas ν acts as a skew diffusivity (or streamfunction) for eddy-driven isopycnal thickness diffusion (advection). Note that a nonzero value of ν indicates anisotropic isopycnal thickness mixing.

Figure 4 shows the horizontal eddy buoyancy fluxes and the components of the isopycnal rotational eddy flux for the southernmost jet in experiment CHANNEL; u′_hb′ is largest at the northern and southern flanks of the jet. At the northern flank of the jets, the eddy fluxes are directed predominantly to the east, while at the southern side the direction is reversed. Note that the classical picture of baroclinic instability flattening mean isopycnals (Gent et al. 1995) implies meridional eddy-induced velocities within the jets, thus a zonally oriented streamfunction B_h. Ignoring isopycnal rotational fluxes, the streamfunction is given by and the eddy-induced velocities would be therefore zonally oriented, contradicting the classical picture of eddy-driven overturning by baroclinic instability flattening the isopycnals within the jet (Gent et al. 1995).

However, it is clear that the dominant part of the eddy buoyancy flux is rotating along isolines of buoyancy variance ϕ₂, which is at maximum at the core of the jet. Note that the eddy flux also tends to follow the small zonal variations of ϕ₂ and that the situation in the other jets is similar to the one shown in Fig. 4. Within the approximations used by Marshall and Shutts (1981), it can be shown that the (isopycnal) rotational eddy fluxes circulates around isolines of ϕ₂; that is, the bulk of the eddy flux in Fig. 4a appears to be rotational. Note that interpreting the zonal eddy fluxes circulating around isolines of ϕ₂ as rotational recovers the classical picture of baroclinic instability flattening the isopycnals within the jets. It turns out that this interpretation is also valid, relaxing the assumptions by Marshall and Shutts (1981) using the TRM-G framework for isopycnal rotational eddy fluxes.

The leading order term of the isopycnal rotational flux component, (shown in Fig. 4b), is dominated by the zonal mean advection of ϕ₂, that is, , and is therefore large and positive (∂_yb is negative) within the eastward jets and small and slightly negative outside (because of the small westward mean flow between the eastward jets). The related leading-order rotational eddy flux, , shares similar direction and magnitude as the horizontal eddy fluxes, that is, eastward (westward) at the northern (southern) side of the jet. Thus, a large part of the zonal eddy buoyancy flux is indeed rotational.

In contrast to the diapycnal rotational eddy fluxes, however, the higher order terms play a larger role for the isopycnal rotational potential flux . Figure 4c shows the second term, , which is as large, or even larger, at some location as the leading term. The is again dominated by zonal advection; that is, . The third moment ϕ₃ (not shown) is large within the jet and positive (negative) at the northern (southern) flank of the jet. The meridional derivative of ϕ₃ thus yields a maximum within the center of the jet, accompanied by two relative minima at the southern and northern flanks of the jet. The resulting second-order isopycnal rotational eddy flux acts to sharpen the leading order contribution toward the center of the jet. The residual divergent isopycnal eddy flux , shown in Fig. 4d, is much smaller than the total eddy flux u′_hb′ in the jets and downgradient to a large extent. Note that this implies isotropic isopycnal thickness mixing as anticipated by Gent and McWilliams (1990), whereas anisotropic mixing effects remain small despite a strong anisotropic configuration related to the zonal jets.

Figure 5 shows the buoyancy variance ϕ₂ and the horizontal eddy fluxes for experiment BASIN in the same zonal jet as in Fig. 3 [the situation in the other jets (not shown) is similar]. It is obvious that the eddy flux u′_hb′ also rotates in BASIN along isolines of the buoyancy variance ϕ₂ to a large extent; that is, u′_hb′ is to a large extent zonally oriented and follows the flanks of the jet (Fig. 3c). Further, the leading-order term of the isopycnal rotational flux component (shown in Fig. 5b) indicates that most of u′_hb′ is rotational, in particular the flux directed along the flanks of the jet. The next-order term, (shown in Fig. 5c), tends to sharpen the leading order contribution as before in CHANNEL. However, since |^isob| becomes very small at some locations in BASIN (marked white in Fig. 5), the estimation of becomes problematic and numerical errors grow in magnitude.

Figure 6 shows the projections of the isopycnal eddy fluxes along and across the gradients of b on the isopycnal related to the reference state, given by κ and v in the flux decomposition Eq. (13) in experiment CHANNEL. It is obvious from Fig. 6, that the dominant part of the along-gradient projection of the eddy fluxes given by the skew diffusivity is related to the zonally oriented isopycnal rotational eddy fluxes at the flanks of the jets. The rotational part of the flux yields large positive (negative) v₁ at the northern (southern) flank of the jet, indicating large (unphysical) zonal eddy-driven velocities at the flanks in opposite direction. Note that such zonal eddy-driven velocities would have a large (unphysical) impact on tracers with zonal gradients in the jets.

The residual divergent part of the isopycnal eddy flux shows much smaller skew thickness diffusivities (Fig. 6d). It is difficult to say here whether the small residual values of v are due to an incomplete removal of isopycnal rotational fluxes (related to higher order terms that have not been computed) or due to errors by higher-order finite differencing, which is used in computing the rotational fluxes (this error is particularly large for the higher order terms). Another complication and source of error² is the approximative form of the definition of the isopycnal flux decomposition. Similar errors are involved when computing the thickness diffusivity κ. Figure 6 demonstrates, however, that the regions of upgradient isopycnal thickness fluxes within the zonal jets, which can be seen without accounting for the rotational fluxes (Fig. 6a), are much reduced when doing so (Fig. 6b). Overall, the zonal and meridional variations in κ appear to be reduced by using the definition for isopycnal rotational eddy fluxes.

Figure 7 compares the horizontal eddy fluxes, the first two leading terms of the potential for the isopycnal rotational flux in TRM-G in experiment CHANNEL, and the potential χ for the rotational flux from a standard Helmholtz decomposition given by

View Expanded

Boundary conditions for the Helmholtz decomposition are zero rotational flux through the northern and southern boundaries. It turns out that both potentials, and χ, are rather similar. It appears therefore that the rotational flux in TRM-G, in fact, represents the “full” rotational part of the flux in this experiment and that the arbitrary choice of the boundary conditions for χ is unproblematic. Note that an analogous similarity shows up when comparing and χ in BASIN (not shown). However, the Helmholtz potential χ shows a much smoother spatial structure than (more so in BASIN than in CHANNEL) by construction, since is defined using derivatives, whereas χ is defined by an inverted Laplacian. Furthermore, the other above discussed sources of errors in estimating might lead to a noisy spatial structure in . It is therefore not surprising that the projection of the resulting rotational fluxes given by κ and v in the direction and perpendicular to ^isob shows also a much smoother structure using χ than using , as shown in Fig. 8.

In experiment CHANNEL (Figs. 8a–d), all negative values of κ disappear when using χ as rotational flux potential.³ Furthermore, the large values of v₁ at the flanks of the jets are much reduced; that is, they are nearly vanishing. It appears therefore that the small residual values of v seen in Fig. 6 are, indeed, vanishing when a less noisy rotational flux is removed from u′_hb′ and that there is no anisotropic isopycnal thickness mixing in experiment CHANNEL. This result is less clear in BASIN (Figs. 8e–h), where still small regions of negative values of κ can be seen in the vicinity of regions with nearly vanishing mean gradients (indicated by a solid line). However, in the western part of the domain where the zonal jets are most energetic in BASIN, the large values of v₁ disappear when using χ as rotational flux potential, except for small regions in which the mean gradients vanish and in the northeastern corner of the domain where negative values for v show up. Whether these values are an artifact of the Helmholtz decomposition owing to the specific choice of the boundary conditions remains unclear at this point. However, also in the zonal jets in BASIN no (or only very weak) anisotropic mixing of isopycnal thickness can be found.

Figure 9 summarizes the isopycnal rotational eddy fluxes, which develop when a jet is entering a region of enhanced variance. The leading-order contribution of the rotational eddy fluxes is given by the mean advection of variance, simply given by in case of a zonal jet as in CHANNEL, which forms a streamfunction for the isopycnal rotational eddy fluxes such that a downgradient rotational isopycnal flux develops in the region of increasing ϕ₂, a rotational isopycnal flux oriented along the northern and southern flanks of the region of enhanced ϕ₂, and an upgradient rotational isopycnal flux in the region of decreasing ϕ₂. In addition to the isopycnal rotational fluxes, a downgradient (upgradient) diapycnal flux develops when the flow enters (leaves) the region of enhanced variance, as already discussed above. In the numerical examples, both isopycnal and diapycnal rotational eddy fluxes dominate the residual divergent parts of the eddy fluxes. Although the leading order contribution of the diapycnal rotational eddy flux dominates the higher order terms, the second-order contribution is also of importance for the isopycnal rotational eddy fluxes. Both isopycnal and diapycnal residual divergent parts of the eddy flux are downgradient, and the corresponding isopycnal thickness mixing therefore appears isotropic in the numerical experiments.

5. Summary and discussion

Using a numerical simulation of mesoscale eddy-driven zonal jets, diapycnal and isopycnal rotational eddy fluxes are identified and discussed in this study. The identification implies that (in steady state) the residual divergent diapycnal eddy fluxes are only related to irreversible diabatic effects on buoyancy summarized by the diabatic source term Q in Eq. (1), that is, to “real” buoyancy mixing, whereas the isopycnal residual divergent eddy fluxes are related to the production of variance (and higher order moments) on isopycnals given by a prescribed reference state or, equivalently, to production (removal) of eddy (mean) available potential energy by mesoscale eddy activity consistent with the classical picture of eddy-driven overturning (Gent et al. 1995).

The effects of the rotational eddy fluxes are summarized in schematic form in Fig. 9: when a mean flow or a jet enters a region of enhanced variance, the rotational flux rotates around isolines of variance on an isopycnal given by the reference state, and it develops both a downgradient (upgradient) isopycnal and diapycnal component when it enters (leaves) the region of enhanced variance. The residual diapycnal flux, however, is related to real buoyancy mixing, whereas the residual isopycnal flux is related to removal of mean available potential energy.

Note that large diapycnal rotational eddy fluxes, hampering the estimate of diapycnal diffusivities from a flux-gradient relationship, have also been found in a nonhydrostatic simulation by Eden et al. (2009). In that numerical experiment, a strong horizontal mean current also enters a region with increasing variance (generated by vertical shear instability), which then leads to large downgradient rotational fluxes. Since in the simulation of Eden et al. (2009) the scales of the flow are very different from the present simulation, the effect of diapycnal rotational eddy fluxes appears to be a general feature of turbulent flow.

It was found in the present simulations, featuring energetic eddy-driven jets, that the physically relevant isopycnal thickness mixing is downgradient, that is, flattening isopycnals at the front within the jets and thus consistent with the classical picture of baroclinic instability (Gent et al. 1995), while the skew component of the isopycnal thickness mixing nearly vanishes. Despite the anisotropic configuration, isotropic isopycnal thickness mixing was thus diagnosed in the numerical experiments, which therefore supports the use of isotropic thickness mixing in contrast to previous suggestions (Smith and Gent 2004; Eden et al. 2007a).

It is important to note that the definition of isopycnal rotational eddy fluxes is subject to the definition of the reference background state and certain assumptions. Although the TRM-G framework and the definition of diapycnal rotational eddy fluxes is exact (and does not depend on the small-slope or any other approximation) and can be applied to the general situation, the definition of isopycnal rotational eddy fluxes is only applicable to a situation in which the buoyancy is strongly dominated by the background state, which is given by strong vertical stratification in the interior of the ocean. It was assumed that advection of perturbation buoyancy (perpendicular to the isopycnal related to the background field) can be neglected compared to the advection of the background field, and it was further assumed that diabatic processes, given by Q, remain small relative to the effect of advection of the prescribed background field. The latter might not be the case for passive tracers or for regions where diabatic forcing becomes large.