Funciones Ejecutivas

This section is a summary of a derivation by Groeskamp et al. (2014a) leading to an expression for the diathermohaline streamfunction, Ψdia_S

AΘ, which represent ocean cir- culation in Absolute Salinity SA and Conservative Temperature Θ coordinates. Here

Conservative Temperature is proportional to potential enthalpy (by the constant heat capacity factor c0_p), which represents the ‘heat content’ per unit mass of seawater (Mc- Dougall, 2003, Graham and McDougall, 2013). Absolute Salinity is measured on the Reference Composition Salinity Scale (Millero et al., 2008) and represents the mass fraction of dissolved material in seawater (in g kg−1_{), (IOC et al., 2010, McDougall}

Figure 4.1: The left shows the 2-dimensional (SA,Θ) (Thermohaline) version of

the Walin (1982) framework in Cartesian-coordinates. Advection through the pair of isotherms and isohalines enclosing ∆V (bold black lines), requires a change in the SA and Θ values of a part of ∆V. For such a change, the divergence of salt or heat

fluxes is required. These are provided by the thermohaline forcing terms, which are the diffusion of salt and heat through the isotherms and isohalines (MSA andMΘ, respec-

tively) and the surface fluxes of mass, salt and heat (FSA, FΘ and Fm, respectively).

The right shows the same ∆V represented in (SA,Θ) coordinates (bold black lines).

In these coordinates we have also defined the streamfunction, such that the diather- mohaline streamfunction difference is equal to the net flux through the isotherm or isohaline, which itself is provided by salt and heat fluxes gridded in the grid spanning

The superscript or prefix ‘dia’ indicates a transport through a surface. Hence we define

udia_C (x, t) as the velocity of a fluid parcel through a surface of constant conserved tracer

C = C(x, t), where x = (x, y, z). Hence udia_C (x, t) can be obtained from the material derivative of C (Griffies, 2004, Groeskamp et al., 2014a),

udia_C = 1 |∇C| DC Dt (4.1) = 1 |∇C| ∂C ∂t +u· ∇C |∇C| = 1 |∇C|(fC+mC).

Here u = u(x, t) = (u(x, t), v(x, t), w(x, t)) and the forcing terms fC = fC(x, t) and

mC = mC(x, t) (both in C s−1) are flux divergences of C due to boundary fluxes and

diffusive mixing processes, respectively. Eq. (4.1) shows that udia_C = udia_C (x, t) is the difference between the velocity of the fluid parcel in the direction normal to the surface of constantC[u·(∇C/|∇C|)] and the movement of the surface itself [(1/|∇C|)/(∂C/∂t)]. A trend in C over time period ∆t, leads to a net shift of the geographical position of the surface of constantC. This shift can be expressed as a net velocity of the surface of constantC over ∆t, given by,

utr_C = 1 |∇C| 1 ∆t Z t+∆t t ∂C ∂tdt. (4.2)

Here _∆1_tR_tt+∆t(...)dt= (...), is a time averaged (later denoted by an overbar) andutr_C =

utr_C(x, t).

Inserting C=SA(x, t) and C= Θ(x, t) in Eq. (4.1), we can define the diathermohaline

velocity vector, udia_S

AΘ = (u

dia

SA, u

dia

Θ ) as the velocity with which the fluid parcel crosses

isohalines and isotherms. Inserting C =SA(x, t) and C = Θ(x, t) in Eq. (4.2), we can

define the diathermohaline trend, utr_S

AΘ= (u

tr

SA, u

tr

Θ) as the net velocity of a shift of the

geographical position of the isohalines and isotherms, due to a trend in time in the local changes of SA and Θ. Following Groeskamp et al. (2014a), for a Boussinesq ocean in

which ∇ ·u= 0, the diathermohaline streamfunction Ψdia_S

AΘ is then given by,

Ψdia_S AΘ(SA,Θ) = Z t+∆t t Z Θ0_≤Θ|S A udia_S A−u tr SAdAdt (4.3) = − Z t+∆t t Z S_A0≤SA|Θ udia_Θ −utr_ΘdAdt. Here,R Θ0_≤Θ|

S_AdAis the area integral over all Θ

0_{, smaller than or equal to Θ on a surface}

of constantSAand

R

S0

A≤SA|ΘdAis the area integral over allS

0

SAon a surface of constant Θ. For a statistically steady ocean,utr_SAΘ= 0. Hence, Ψdia_SAΘ

represents the non-divergent component of the ocean circulation in (SA,Θ) coordinates,

while the diathermohaline trend represents the divergent component of this circulation.

From Eqs. (4.1) and (4.3) it is clear that Ψdia_S

AΘ can only be calculated if u(x, t) is known. An expression that provides Ψdia_S

AΘ from commonly observed variables (ocean hydrography and boundary salt and heat fluxes) is obtained when inserting the last line of Eq. (4.1) into Eq. (4.3) thereby directly relating Ψdia

SAΘ to thermohaline forcing, Ψdia_SAΘ(SA,Θ) = Z t+∆t t Z Θ0_≤Θ| S_A fSA+mSA |∇SA| −utr_SAdAdt = − Z t+∆t t Z S0 A≤SA|Θ fΘ+mΘ |∇Θ| −u tr ΘdAdt. (4.4)

However, this expression requires knowledge of the the ocean’s diffusive salt and heat fluxes everywhere. Unfortunately the ocean’s spatial and temporal varying diffusive salt and heat fluxes are not well known, requiring us to develop a different method to derive Ψdia_S

AΘ from observations, which is what we do in this chapter.

4.3

The Diathermohaline Volume Transport