Enseñar Economía: propuestas y métodos, Rafael Prieto-Puga

Since the horizontal component of∇small

N C is∇nCit would seem that the relative error

in neglecting the difference between | 1 +s2

∇_NC and ∇small

N C would scale as (using

Eq. (2.20)) |∇small N C− 1 +s2 ∇_NC| |∇_nC| = |(k· ∇_nz× ∇_nC)k× ∇_nz| |∇_nC| ∼s 2_{. (2.23)}

However, the relative error is actually significantly less than this scaling suggests because the magnitude of (k· ∇_nz× ∇_nC) in the ocean is usually much less thans|∇_nC|. Here we develop expressions for (k· ∇nz× ∇nC) and make the connection to neutral helicity.

Since ∇nC+∇zC+Cz∇nz, (k· ∇zC× ∇nC) is also equal to (k· ∇zC× ∇nC). Using

the expression (Eq. A.5) for the neutral tangent plane slope, we find

(k· ∇z× ∇nC) =−

g

N2k·(α∇zΘ−β∇zSA)× ∇zC. (2.24)

Consider now the case where the tracer C is Conservative Temperature Θ (we could equally well have chosen Absolute Salinity SA for this purpose since their epineutral

gradients are parallel), so that Eq. (2.24) becomes

(k· ∇_z× ∇_nΘ) = gβ

N2 (∇zSA× ∇zΘ)·k. (2.25)

To within the Boussinesq approximation we may take ∇zΘ and ∇zSA to be equal to

the corresponding gradients in an isobaric surface, ∇_pΘ and∇_pSA, so that

(k· ∇_z× ∇_nΘ) = gβ N2(∇zSA× ∇zΘ)·k (2.26) ≈ gβ N2(∇pSA× ∇pΘ)·k = − 1 gρ(∇nP × ∇nΘ)·k = − β ρN2∇P · ∇SA× ∇Θ = − H n ρTbN2 ,

where the last three parts of this equation have used the results of McDougall and Jackett (1988, 2007) and section 3.13 of IOC et al. (2010) that relate these various triple scalar products to neutral helicity, Hn, defined asHn =βTb∇P · ∇SA× ∇Θ where Tb is the

thermobaric parameter (McDougall, 1987b) that expresses the non-linear dependence of specific volume on both Conservative Temperature and pressure; a non-linear property that does not concern us in this section of the chapter.

The triple scalar product∇P· ∇SA× ∇Θ of neutral helicity has arisen in the context of

1. The ill-defined nature of neutral surfaces and the empty nature of ocean hydro- graphic data in SA−Θ−P space (McDougall and Jackett, 1988, 2007),

2. The mean vertical downwelling advection achieved by the helical nature of neutral trajectories (Klocker and McDougall, 2010),

3. The close connection between ∇P · ∇SA× ∇Θ and the spiraling of epineutral Θ

contours when the ocean is not motionless Zika et al. (2010a), and now,

4. The difference between the projected non-orthogonal epineutral gradient of Θ, ∇small

The quantification of the magnitude of neutral helicity in the ocean is far from complete, yet the above studies have shown that the angle between∇_nP and∇_nΘ is quite small in most of the ocean, even in the Southern Ocean where the influence of processes related to neutral helicity seem to be the largest (Klocker and McDougall, 2010). These results for neutral helicity were obtained in both a smooth ocean atlas and in an eddy-less ocean model, but neutral helicity has not yet been examined in the context of a high-resolution eddy-permitting ocean model. Nevertheless, if we take these results of McDougall and Jackett (2007) and Klocker and McDougall (2010) at face value then we would conclude that the relative magnitude of the error in the tracer flux due to using the small-slope approximation may be no more than a few percent of the magnitude estimated by the scale analysis of Eq. (2.23), that is, no more than a few percent of s2 =|∇nz|2. It must

be said that the reason why neutral helicity is as small as it is in the ocean is far from clear; see McDougall and Jackett (2007) and Klocker and McDougall (2010) regarding neutral helicity and the consequent requirement that the ocean is ‘thin’ in SA−Θ−P

space.

Specifically, if we letφbe the (small) angle between the two-dimensional gradients∇_nC

and∇nz, then from Eq. (2.23) we see that the relative error in neglecting the difference

between 1 +s2∇_NC and ∇small

N C is actually |∇small N C− 1 +s2 ∇NC| |∇_nC| =s 2_{sinφ. (2.27)}

The error made by neglecting this fraction of the epineutral tracer gradient can be com- pared with the very small errors that are made by taking Conservative Temperature Θ to be 100% conservative. From McDougall (2003) and Graham and McDougall (2013) we know that Conservative Temperature is approximately two orders of magnitude more conservative than is potential temperature (see Figures 2(a) and 4 of McDougall (2003) and Figure 8 of Graham and McDougall (2013)), while from Figure 11(a) of Graham and McDougall (2013) and Appendix A.14 of IOC et al. (2010) we see that the epineutral gradient of potential temperature, ∇nθ, is often about 1% different to that of Conser-

vative Temperature,∇nΘ. This 1% difference is due to the non-conservative nature of

potential temperature. We conclude that the relative error in using the epineutral gra- dient of Conservative Temperature is the product of these two factors of 10−2, namely 10−4. This can be compared with the relative error, Eq. (2.27), involved in fluxing a tracer in a direction in which there is no gradient. For an epineutral slope s of as large as 10−2 this relative error is s2sinφ= 10−4sinφwhich is smaller that the relative error in using Conservative Temperature by the factor sinφ, this factor representing the influence of neutral helicity. Since we expect sinφto be no larger than 0.05 (McDougall and Jackett, 2007), we conclude that this effect is absolutely tiny, even when compared

with the use of Conservative Temperature, which itself is an improvement by two orders of magnitude on oceanographic practice under EOS-80.

2.3.4 The component of the small-slope gradient in a direction in