1. Introduction
In the past 50 years, the quasigeostrophic (QG) approximation has been widely used to describe the ocean [see Berloff et al. (2007) for a recent example]. This approximation reveals the large-scale characteristics of the ocean circulation via the use of potential vorticity (PV). In the QG formalism, sources and sinks of PV are explicitly written into the equation of evolution. For example, the classical double gyre circulation is obtained by applying wind stress curl to the surface layer (Berloff et al. 2007). This formalism is appropriate to describe the large-scale evolution of layers that do not outcrop.
It is now widely recognized that the impermeability theorem applies to PV in the primitive equations (Haynes and McIntyre 1987). This theorem states that PV cannot cross isopycnal surfaces and is exactly conserved in an isopycnal layer (unlike mass). The only locations where PV can enter or exit a layer are at contacts with boundaries like the sea surface or the topography.
The circulation of PV in the ocean interior is also dictated by the orientation of J. This vector is not necessarily parallel to an isopycnal surface; components of J normal to the layer imply that the surface will move so that PV does not cross that surface.
Ertel and QG PV differ in that in the primitive equations a range of density layers receive PV at their out-/incrops, whereas in the QG formalism only the upper layer is forced by the atmosphere. This picture is further complicated by the seasonal cycle, which forces a considerable annual migration of the outcrops. This migration allows layers to receive atmospheric forcing contributions from various physical locations.
To better understand the PV dynamics in the ocean, we ask the following questions for a representative subset of isopycnals participating in the wind-driven gyre: How much PV enters the ocean via air–sea interaction? What is the main pattern of PV circulation in the interior and how much PV is created or destroyed at the topography?
To address these questions, we consider the PV budget using the output of a numerical model and separate the PV circulation in the ocean interior from the PV flux at the boundaries. All the numerical computations are performed with a 
Such a resolution (
The paper is organized as follows. Section 2 is dedicated to the description of PV flux at the surface. We review several mechanisms that are responsible for generating/extracting PV at the surface. Section 3 is centered on the ocean interior PV circulation. Conclusions are given in section 4. Circulation and surface flux are combined in a companion paper (Deremble and Dewar 2013, hereafter D13) that addresses the dynamics of mode water.
2. Surface PV flux
a. General statements
b. Validation in a model
Our first step is to validate Eq. (2) and the impermeability theorem in a numerical model. We compute the time rate of change of the total PV inside an isopycnal layer and compare it to the PV flux at the surface. This first evaluation is done while neglecting the flux of PV at the topography (see also D13 for an analytical derivation). The PV fluxes at the solid boundaries are discussed in section 3.
Figure 1a is a time series of the integrated PV in the layer 26.0 ± 0.25 kg m−3. Henceforth, we omit the density units when labeling an isopycnal layer. According to Fig. 1a, the integrated PV Q26 decreases from September to March and increases during the rest of the year. This is consistent with the seasonal evolution of the outcropping. Indeed, from Eq. (7), we know that the integrated PV is a rough measure of the surface extension of the isopycnal layer scaled by the Coriolis parameter. Figure 1a largely reflects the northern migration (in spring) and southern retreat (in autumn) of the outcrop.
(a) Red: 1-yr time series with a 5-day sampling of the integrated PV in Q26. Blue: reconstructed time series using the time integral of −Jz. Black: as in blue, but for the time integral of 
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
(a) Red: 1-yr time series with a 5-day sampling of the integrated PV in Q26. Blue: reconstructed time series using the time integral of −Jz. Black: as in blue, but for the time integral of
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
(a) Red: 1-yr time series with a 5-day sampling of the integrated PV in Q26. Blue: reconstructed time series using the time integral of −Jz. Black: as in blue, but for the time integral of
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
Differentiating this time series gives an estimate of the rate of PV production/destruction. This time rate of change of the integrated PV 
The correlation coefficient between these two curves is 0.96. A perfect matching between these two curves does not occur partly due to the neglect of processes related to the topography and because of spatial and temporal discretization (see the discussion in the next paragraph). However, the agreement between the two strongly supports the validity of Jz as expressed in Eq. (3).
In Fig. 1a, we also reconstructed the time series of the integrated PV using −Jz (blue curve). The red and blue curves are consistent with the curves showing the differentiated time series: Jz closely reflects the variations of PV (correlation coefficient: 0.99). The curve of integrated Jz shows an anomalous trend of PV creation, which we discuss next.
c. Global budget and aliasing issue
We now examine more closely the different components of Jz. The three contributions to Jz in Eq. (3) are k · ωσt, k · (ut × ∇σ), and k · (∇B × ∇σ). If we look at a daily snapshot of Jz, the dominant term is k · ωσt (not shown). This term is in fact another way to look at the huge seasonal meridional migration of isopycnals (several thousands of kilometers in one year). But in a 1-yr period, we expect an isopycnal to return near to its initial position, so the average of this term should be small compared to the other terms. After a long average is taken, we expect that the remnant will be the correlation between the relative vorticity and the time derivative of the density. This statement is true at every location, even when an isopycnal window is used as conditional averaging.
To demonstrate this proposition, suppose that we have perfect knowledge of the density at the surface of the ocean at any instant in time t. Suppose also that the state of the ocean at the end of the year is identical to the state of the ocean at the beginning of the year. We want to measure the contribution of the term k · ωσt to the surface PV flux. To a first approximation, this term is equal to fσt (we replace the vorticity by the planetary vorticity). We consider an isopycnal layer with the interval [σ1, σ2]. At a given location (x, y) and a given time t, if the surface density σ(x, y, t) ∉ [σ1, σ2] then Jz(x, y, t) = 0. Suppose now the isopycnal layer sweeps by this location (e.g., during the spring migration). As long as σ ∈ [σ1, σ2], there is a component of Jz that is equal to fσt. The summation (or integral) of all the instants in which σ ∈ [σ1, σ2] returns a global contribution equal to 
We have computed the term fσt in various locations and over 10 years and we find that this term is not equal to zero, or even small. In view of the above, we interpret this as an error of the analysis and explain it primarily as being due to the 5-day sampling of the model output. The spring migration of the outcrop is faster than the autumn retreat (not shown). At a given point (x, y), it is possible that one day in spring σ(x, y, t) < σ2 and five days later σ(x, y, t) > σ1. In such case there will not be any contribution at this precise location during the spring migration, whereas the real contribution should be f(σ2 − σ1) (with perfect sampling). During the return phase, which is slower, there is more chance that at this same location there will be a day for which σ1 < σ(x, y, t) < σ2, which means that there will be a contribution on the order of f(σ1 − σ2). The global contribution will appear as negative from the analysis when it is in fact zero.
The numerical evaluation of k · ωσt is at leading order equal to fσt. So replacing the planetary vorticity by the total vorticity does not significantly change the result.
We compute the spatial- and temporal-mean PV flux [using Eq. (3)] on several outcropping windows using Lagrangian averaging. For example, the mean of the blue curve in Fig. 1b (extrapolated for the 10-yr time series) gives the total amount of PV extracted at the air–sea interface in isopycnal 26.0. We normalize the result by the size of the outcrop area: dσ = 0.5. The same operation is performed on several density classes from 24.0 to 27.0 and the result is plotted in Fig. 2.
Solid line with plus signs: mean dJz/dσ (m2 s−2) computed on outcropping window from 24.0 to 27.0. Dashed line with crosses: the 
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
Solid line with plus signs: mean dJz/dσ (m2 s−2) computed on outcropping window from 24.0 to 27.0. Dashed line with crosses: the
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
Solid line with plus signs: mean dJz/dσ (m2 s−2) computed on outcropping window from 24.0 to 27.0. Dashed line with crosses: the
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
In this figure, the solid line is computed with the full Eq. (3). The dotted line with squares is the component 
The dashed line with asterisks is 
Finally the dashed curve with crosses is 
Using the curve 
d. Discrepancy with the bulk formulation
This component models the entrainment of water at the base of the mixed layer. This component of the PV flux is always positive since the wind can only mix the upper layer and has been shown to be of the same order of magnitude as 
These scaling laws [Eqs. (9)–(11)] are convenient as they are readily evaluated from observations external to the ocean and need little in situ data (only mixed layer depth). On the other hand, their accuracy has not been thoroughly assessed. We use model outputs to address this question, with the conclusion that these scaling laws are appropriate for the fall and winter season. Inaccuracies due to poorly observed shallow summer mixed layers tend to overwhelm the estimates away from the winter, leaving annual averages suspicious.
These scaling laws are compared to the time rate of change of the integrated PV [as was done in the previous subsection with the formulation of Jz given in Eq. (3)]. In Fig. 1c, the curve 
The primary reason for the difference is that 
We conclude from this comparison that bulk formulations of PV flux can be useful but must be viewed appropriately. Wintertime PV flux estimates are reliably portrayed in this manner, but summertime processes are too heavily dependent on the variable mixed layer to be trusted. Olsina et al. (2013) obtained annual-average PV flux estimates for the mode waters that suggested on average that the mode waters were receiving PV at the air–sea interface. It is likely that this result was biased due to an overestimated summer contribution.
e. The spatial structure of surface PV forcing
Figure 3 shows estimates of the PV extraction/input by different processes. These plots are a 10-yr average of a given component of Jz in the layer σ = 26.0 ± 0.25. To compute this average, we compute Jz for each output of the model (5-day sampling) and we apply a mask taking values of one on the outcropping window and zero elsewhere. Each of these maps is summed and then divided by the number of maps. All the variables needed to compute Jz either come from the model output or from the forcing used in the model (Brodeau et al. 2010). These forcing terms (wind, Qnet) are sampled on the same 5-day basis.
Maps of mean values of the different components of the Jz vector on the 26.0 ± 0.25 outcropping window: (a) 
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
Maps of mean values of the different components of the Jz vector on the 26.0 ± 0.25 outcropping window: (a)
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
Maps of mean values of the different components of the Jz vector on the 26.0 ± 0.25 outcropping window: (a)
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
To understand these maps of PV extraction/input, we need to make a clear distinction between the daily snapshot of Jz (not shown) and the time average of Jz (Fig. 3).
As mentioned earlier, instantaneous PV flux can differ significantly from the time-averaged PV flux. However, it is the daily flux pattern that leads to the idea that PV destruction occurs in winter and PV production occurs in summer. This is not true if we do the entire “summer” average (the period during which the outcrop is north of a certain latitude; roughly from March to September). Much of the summer production is removed in the following winter as the outcrop sweeps south (see also Fig. 1).
The first panel (Fig. 3a) is the component that has been validated in the previous section 
We now analyze each individual contribution as given by the scaling laws and compare it to this reference map. Buoyant PV generation 
As mentioned by Maze et al. (2013), mechanical PV generation 
Note that 
We now compare the sum of these three terms 
The large-scale pattern is different. Whereas for
The shallow water zones are completely different. This is probably due to the interaction between the mixed layer and the bottom topography leading to a completely different scaling than the one proposed in Eqs. (9)–(11) (see, e.g., the Grand Banks region or the continental shelf where the water is less than 100 m deep).
More fine structures occur in Jz than
The overall magnitude of Jz is smaller than
A visual comparison between these maps shows that the Fig. 3a 
In this section, we validated Eq. (3) for the day-to-day PV flux and Eq. (8) for the mean PV flux. We also demonstrated that the use of scaling laws is inaccurate to measure precisely the surface PV flux.
3. Interior circulation
Once PV is injected at the surface, it either circulates in the ocean interior or is re-extracted during the same annual cycle (or perhaps some years later). We now examine the main pathway of PV circulation and the different processes associated with its circulation.
a. The isopycnal formalism
The impermeability theorem states that PV, defined as in Eq. (1), does not cross surfaces of constant σ. This leaves open many choices for σ. Historically, the most natural choice is potential density, to draw connections with theory and because of the dynamical importance of density. There are issues that arise with this choice, however, in a model employing a real equation of state. We adopt potential density in this paper to connect with the literature, but point out the consequences of this choice in the following analysis (see also Marshall et al. 2001).
The relation between the Bernoulli function and the mean circulation of PV has already been demonstrated by Schär (1993) and used by Maze and Marshall (2011). However, the formulation of Eq. (14) provides a useful framework to diagnose and understand the PV circulation.
Solving the elliptic Eq. (14) with the Neumann boundary conditions (∂B/∂x and ∂B/∂y) given in Eq. (13) yields the mean Bernoulli function (see the appendix for the method used to solve this equation). Of course, it is also possible to construct the Bernoulli function directly from the output of an OGCM. We perform this computation as well, but argue that using both procedures has an advantage. An issue with the Bernoulli is that it involves two enormous but opposite-signed inputs (the pressure and the quantity ρgz). The dynamical content in the Bernoulli is actually much smaller than either of these by a factor of 10 000 (McDougall 2003). Given the inherent errors in a numerical model, it is not obvious a priori that the Bernoulli constituted from the numerical data would be accurate. In contrast, the present approach based on the elliptic equation relates the Bernoulli solely to the dynamical state of the model. Of course, solving an elliptic equation is more difficult than simply creating the Bernoulli, so each technique offers some advantages. Last, we point out that the elliptic operator also allows us to compute the contributions to PV streamfunction from each of the contributing terms: we can explicitly compute the roles of the advection, diabatics, or equation of state [see Eq. (5)].
b. PV circulation map
To validate this technique for finding the Bernoulli function, we plot in Fig. 4 the mean Bernoulli function for 1997 projected on layer 26.0. The northern boundary of the computational domain is given by the southernmost position of the outcrop during 1997 (red line). The two sets of contours (thick solid and thin dashed lines) show the two different methods used to compute the Bernoulli function. The thick lines are the result of the elliptic equation [Eq. (14)] and the thin lines are obtained with direct calculation [Eq. (4)]. A constant is added to the solution of the elliptic equation to match the Bernoulli contours obtained with the direct calculation (this unknown is due to the Neumann boundary conditions). The two estimates of the Bernoulli function agree to within a few percent, which argues for our purposes that either estimate will suffice.
Mean Bernoulli function for year 1997 on 26.0. Solid lines are obtained with Eq. (14) and thin dashed lines are obtained with Eq. (4). The contour interval is 0.5 m2 s−2. The computational domain is bounded by the southernmost position of the outcrop (red line).
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
Mean Bernoulli function for year 1997 on 26.0. Solid lines are obtained with Eq. (14) and thin dashed lines are obtained with Eq. (4). The contour interval is 0.5 m2 s−2. The computational domain is bounded by the southernmost position of the outcrop (red line).
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
Mean Bernoulli function for year 1997 on 26.0. Solid lines are obtained with Eq. (14) and thin dashed lines are obtained with Eq. (4). The contour interval is 0.5 m2 s−2. The computational domain is bounded by the southernmost position of the outcrop (red line).
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
As for the shape of the Bernoulli contours, we recognize the main features of the subtropical gyre: the Gulf Stream at the western boundary separates at Cape Hatteras and the return flow occurs at all longitudes.
As mentioned earlier, an advantage of solving the elliptic equation is to quantify the impact on the streamfunction of each term of the total PV flux. It appears that this field is almost exclusively due to the advection term on this surface. The effect of diabatic heating [H in Eq. (13)] is very small (four orders of magnitude lower). We expect that the ocean interior is largely adiabatic, hence this result agrees with our intuition. The new information here is that the ocean interior, as computed within the NEMO framework, also behaves in a largely adiabatic manner with regard to this important dynamical variable.
We do not explicitly compute the impact of “friction.” Indeed, this effect is expected to be weak. The confirmation of this expectation appears in that the Bernoulli functions computed using our two different approaches are very nearly identical. In principle, friction is an input to the directly constructed Bernoulli function. The absence of friction in our elliptic inversion, coupled with its coincidence with the direct Bernoulli function, implies that friction is a weak effect.
c. Evolution of the circulation with depth
The projection of the Bernoulli contours on the consecutively deeper 26.0, 26.25, and 27.0 isopycnals appears in Fig. 5 and a surprisingly large impact of thermobaricity appears. For the relatively shallow layer (26.0), the mean circulation of PV is clockwise, in agreement with the global circulation pattern of the subtropical gyre. Using the decomposition of Eq. (13), we obtain that this pattern is mostly the result of the advection term. As we go deeper, the circulation weakens. For the mode water layer (26.25), net PV circulation is quite weak. Thermobaricity, however, is still weaker than advection (not shown). This result is further analyzed in D13. The last panel of Fig. 5 is quite different from the previous two. The circulation of PV is reversed (counterclockwise) and much more intense (the contour interval is the same in all panels). Again, using the decomposition of Eq. (13), this circulation results mostly from the thermobaric effect. In fact, because of the presence of mode water, the depth of this isopycnal surface varies from 700 (below the mode water) to 300 m elsewhere. This variable depth is responsible for a large in situ density gradient on this surface and creates a counterclockwise PV circulation on this density surface.
Bernoulli contours projected on three different isopycnals: (a) 26, (b) 26.25, and (c) 27. The arrows indicate the direction of the circulation. The contour interval is 0.75 m2 s−2.
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
Bernoulli contours projected on three different isopycnals: (a) 26, (b) 26.25, and (c) 27. The arrows indicate the direction of the circulation. The contour interval is 0.75 m2 s−2.
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
Bernoulli contours projected on three different isopycnals: (a) 26, (b) 26.25, and (c) 27. The arrows indicate the direction of the circulation. The contour interval is 0.75 m2 s−2.
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
As mentioned by McDougall (1989), these plots can be counterintuitive because of the thermobaric effect, and the nature of this term depends on the choice of the variable σ. If σ were chosen as ρ, the thermobaric term would vanish, but then the source term H in the thermodynamic equation would contain a contribution from the material derivative of the pressure. In a previous study, Marshall et al. (2001) also mentioned this increasing role played by thermobaricity for deep isopycnal surfaces.
d. Sinks and sources of PV
We use this to diagnose the topographic PV generation. Consider the contour in Fig. 6, consisting of the topographic boundaries of the 26.0 isopycnal, the equator, and the position of the southernmost outcrop. For our numerical evaluation, we consider a layer of thickness Δσ = 0.5. The net PV flux through this contour is zero since it is a closed contour. The values of the Bernoulli function on that contour are plotted in the second panel of Fig. 6 with the numbers matching the numbers of the first panel.
(a) Mean Bernoulli contours projected on the isopycnal 26.0. (b) Value of the Bernoulli function on the boundary. The red numbers refer to positions on the contour in (a).
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
(a) Mean Bernoulli contours projected on the isopycnal 26.0. (b) Value of the Bernoulli function on the boundary. The red numbers refer to positions on the contour in (a).
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
(a) Mean Bernoulli contours projected on the isopycnal 26.0. (b) Value of the Bernoulli function on the boundary. The red numbers refer to positions on the contour in (a).
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
Starting at point 3 and moving counterclockwise to point 7, an arc that involves topographic interactions with South America and the cross-equatorial PV flux, the Bernoulli function is seen to be almost constant. Equivalently, this is almost a mean PV flux streamline, implying that the PV on this density surface is isolated from the Southern Hemisphere. From point 7 to 12, the Bernoulli function increases considerably. The map indicates that this is a net input of PV due to the atmosphere, as these streamlines all emanate from the southernmost outcrop line. The Bernoulli function increase along this arc is consistent with a southwestward flux of PV, in agreement with the overall sense of general circulation. We could essentially drop a line from point 12 to 4, with the Bernoulli difference between the two of 4 m2 s−2 being the net westward PV flux of the subtropical gyre, with about half entering the Caribbean Sea and a significant amount of the rest following the Antilles Current to the east of the Bahamas. Moving counterclockwise from point 15 to 4 is a mild Bernoulli increase of about 1 m2 s−2. These are locations where the isopycnal grounds into the subsurface Gulf of Mexico, so we here see injection into the gyre of PV from topographic sources. The increase here is consistent with an eastward flux of positive PV.
However, clearly the major action in the figure occurs between points 14 and 15. The line between points 14 and 15 defines the East Coast of North America up to the point of Gulf Stream separation. One sees a growth of the Bernoulli function moving counterclockwise on the curve from Cape Hatteras to the tip of Florida that compares in size to that occurring over the interior of the open Atlantic. These Bernoullis, from about −1.5 to −6 m2 s−2, have no counterparts along the surface outcrop and are thus unambiguously identified as of topographic origin. Equivalently, the sharp Bernoulli decrease from point 13 to 14 consists of recycling of interior PV (from Bernoullis 4 to −1.5 m2 s−2) and the expulsion of topographically created PV (from Bernoullis −1.5 to −5.5 m2 s−2) in roughly equal parts.
What is the fate of the topographically generated PV? The Bernoulli contours show that the topography is connected directly to the outcrop. The northern topographic input, north from Cape Hatteras to Newfoundland, forms its own circuit, directly from the generation site to the outcrop and is seemingly disconnected from the larger subtropical circulation. The U.S. East Coast PV injection takes a fast path into an area that in the mean experiences atmospheric interaction. This appears in Fig. 7 where a detailed view of the regional Bernoulli function appears superimposed on a map of the atmospheric PV injection. North of the outcrop, the PV flux is, strictly speaking, no longer nondivergent. Equivalently, the time-mean flux is the combination of two inputs, one from a streamfunction represented by the mean Bernoulli function and one from a velocity potential determined by atmospheric exchange. It is our experience that the vector sense of the PV flux is well approximated by lines of constant Bernoulli, so the sense of Fig. 7 is that the topographically generated PV enters a region of net extraction dominated by convection. In this sense, the topographically generated PV takes a fast path, entering at subsurface isopycnal incrops, moving north in the Gulf Stream to an atmospheric interaction that draws it out of the fluid. Viewed in terms of PV dynamics, part of the large region of heat loss in the separated Gulf Stream exists to balance the PV flux input to the gyre by the topography.
Mean Bernoulli contours and Jz (colors) for layer 26.0.
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
Mean Bernoulli contours and Jz (colors) for layer 26.0.
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
Mean Bernoulli contours and Jz (colors) for layer 26.0.
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
The regional net loss of PV appears to be stronger than the topographic injection. Overall balance on this isopycnal layer involves interactions of the recirculating fluid with regions of both injection and extraction of PV. Specifically, the recirculating PV near the Gulf Stream enters a region of net PV injection associated with the late winter early spring period. This injection is largely balanced by extraction just to the north. The recirculating fluid continues moving anticyclonically around the region of atmospheric interaction through a broader, but weaker, zone of PV injection. The fluid then exits the region of outcropping, anticyclonically winds around the gyre and shoots back into the atmospherically dominated PV forcing region.
e. The role of eddies
The solution so obtained is not unique because the boundary conditions can always be modified by a function that introduces no curl [see the discussion in Peterson and Greatbatch (2001)]. For that reason, we refer to these components as pseudo-Bernoulli contributions. Even though this ambiguity exists, we remark that the curvature of the field we compute will be unaffected by the boundary conditions, and hence the locations of extrema are unique. We will confine our interpretation of the pseudo-Bernoulli functions focusing on the extrema. We have also verified that the sum of all pseudo-Bernoulli functions obtained by solving the elliptic equation forced by the components of the flux corresponds to the total Bernoulli field with a maximum error of 10%. The full Bernoulli field was almost entirely recreated by the advective pseudo-Bernoulli functions and the thermobaric component.
In Fig. 8a we plot the pseudo-Bernoulli function due to 
Pseudo-Bernoulli function: (a) mean component of the advective part of the Bernoulli function and (b) eddy component. All plots are for isopycnal 26.25. The contour intervals are 0.25 m2 s−2 for (a) and 0.05 m2 s−2 for (b).
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
Pseudo-Bernoulli function: (a) mean component of the advective part of the Bernoulli function and (b) eddy component. All plots are for isopycnal 26.25. The contour intervals are 0.25 m2 s−2 for (a) and 0.05 m2 s−2 for (b).
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
Pseudo-Bernoulli function: (a) mean component of the advective part of the Bernoulli function and (b) eddy component. All plots are for isopycnal 26.25. The contour intervals are 0.25 m2 s−2 for (a) and 0.05 m2 s−2 for (b).
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
This contribution of the eddies to the circulation of PV makes no statement about their flux divergence. This property is indeed a crucial component of the PV dynamics and is examined for the Eighteen Degree Water in D13.
Values of (a) κ and (b) ν (m2 s−1) computed using Eq. (21). A 1° boxcar filter has been applied to smooth the data.
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
Values of (a) κ and (b) ν (m2 s−1) computed using Eq. (21). A 1° boxcar filter has been applied to smooth the data.
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
Values of (a) κ and (b) ν (m2 s−1) computed using Eq. (21). A 1° boxcar filter has been applied to smooth the data.
Citation: Journal of Physical Oceanography 44, 1; 10.1175/JPO-D-13-087.1
The cross-gradient component is a sizable component of the total field 
This mean/eddy decomposition is used in D13 to explain the maintenance of subtropical mode water. It is shown there that while the mean flux of PV is directed outward of the mode water, it is balanced by an eddy flux of the same magnitude directed in the opposite way.
4. Conclusions
In this paper, we clarify the pathways of PV in the North Atlantic subtropical gyre. To address this topic we separate the in-/output flux from the interior circulation.
a. Surface fluxes
We show that evaluating the PV input via air–sea interaction is not a trivial computation and must be addressed with care. In section 2, we argue that the scaling laws for the vertical PV flux at the surface (Fig. 3) are not accurate. They demonstrate skill in the wintertime, but yield inaccurate estimates in the summertime because of their sensitivity to mixed layer depth (Fig. 1). The complete formulation of flux is also hard to compute even from model outputs due to its dependence on nonconservative effects. The alternate formulation of the flux proposed in Eq. (3) is considerably simpler to use and returns generally more reliable estimates for the PV flux. We do remark, however, that computing the time-dependent flux accurately requires high-frequency observations of sea surface density to avoid aliasing. We compute mean PV budgets for several density classes (Fig. 2), using Eq. (8) to compute the air–sea PV flux. We show that for the density classes of the subtropical wind-driven gyre (from 24.0 to 27.0), the ocean loses PV at the air–sea interface. The annual mean remains very small compared to the daily PV flux.
b. Ocean interior
The PV circulation in the interior is diagnosed using density coordinates, which reveals the importance of the Bernoulli function in describing the mean flux of PV [see Eqs. (13) and (14)]. This set of equations permits the computation of the Bernoulli function in an indirect way by solving an elliptic equation. We validate this computation by comparing the Bernoulli functions obtained using direct and indirect computation (Fig. 4). This novel technique allows us to directly connect the Bernoulli to its dynamical content and can independently quantify the amplitudes of the several contributions to PV flux.
The evolution of the Bernoulli contours with depth is plotted in Fig. 5. This figure reveals a surprising feature of the mean PV circulation in deep layers. We showed that the thermobaric effect is responsible for overwhelming the clockwise circulation produced by advection and setting up a strong counterclockwise PV circulation (Fig. 5). The mode water layer (26.25) appears to be a layer of very weak circulation. This point is further investigated in D13.
Figure 6 argues that the dominant behavior of the PV is to recycle in the interior. The PV route for this component of the circulation consists of atmospheric input in the central and eastern parts the North Atlantic and atmospheric extraction in the west after a circuit in the gyre. Farther west, against the U.S. East Coast, PV is input to the gyre from topographic sources, and then follows a short path to extraction where convective activity is driven by buoyancy exchange with the atmosphere (Fig. 7). The magnitude of the topographic production is comparable to that of the surface PV flux.
Finally, the role of eddies is investigated in terms of the proposition of Rhines and Young (1982) that they promote downgradient transport. We found mean values for the eddy PV diffusivity of κ = 1000 m2 s−1 (Fig. 9). The primary departure from downgradient flux occurred in the western boundary layer, in the zone affected by topographically generated PV flux. The cross-gradient component of the eddies is far from negligible in the Gulf Stream, although this contribution is not yet fully understood.
c. Discussion and implication for future work
A full exploitation of Eqs. (13) and (14) requires an accurate method for the Helmholtz decomposition to decompose each contribution into a rotational part and divergent part. This decomposition is not unique and we are now developing a method to find the physically relevant rotational part and divergent part of a vector field (see also Fox-Kemper et al. 2003).
The interaction with the topography and the process by which PV is created is not yet fully understood. We are now trying to address this topic using high-resolution models. However, in contrast to surface PV flux that can be estimated from data, it is not currently possible to compare theoretical (or model) values of the PV flux at the topography with observed PV fluxes.
A direct application of this work can be found in D13. The results obtained here are used to explain the mode water dynamics. We show that the vertical component of the J vector does not play a direct role in mode water formation. The creation of low PV water mass is instead more directly related to the surface buoyancy flux.
Acknowledgments
We thank the Drakkar team and especially A.-M. Treguier, J.-M. Molines, and T. Penduff for providing their dataset and the associated diagnostic tools. T. J. McDougall provided valuable input on the subtleties of the Bernoulli and two reviewers provided perceptive comments. The work reported here was sponsored by NSF Grant OCE-0960500.
APPENDIX
The Elliptic Equation Solver
The resolution of an elliptic equation in an irregular domain is almost as straightforward as it is for a regular (rectangular) domain. We tackle this problem using direct computation of the type used by Gibou et al. (2002) rather than the immersed boundary condition described by Pares-Sierra and Vallis (1989).
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