Diagnosing vertical motion in the Equatorial Atlantic

Abstract

Estimating the vertical velocity (w) in the oceanic upper-layers is a key issue for understanding the cold tongue development in the Eastern Equatorial Atlantic. In this methodological paper, we develop an expanded and general formulation of the vertical velocity equation based on the primitive equation (PE) system, in order to gain new insight into the physical processes responsible for the Equatorial and Angola upwellings. This approach is more accurate for describing the real ocean than simpler considerations based on just the wind-driven patterns of surface layer divergence. The w-sources/forcings are derived from the PE w-equation and diagnosed from a realistic ocean simulation of the Equatorial Atlantic. Sources of w are numerous and express the high complexity of terms related to the turbulent momentum flux, to the circulation and to the mass fields, some of them depending explicitly on w and others not. The equatorial upwelling is found to be mainly induced by the (i) the zonal turbulent momentum flux, (ii) the curl of turbulent momentum flux and (iii) the imbalance between the circulation and the pressure fields. The Angola upwelling in the eastern part of the basin is controlled by strong curl of turbulent momentum flux. A strong cross-regulation is evidenced between the w-forcings independent of w and dependent on w, which suggests an equatorial balanced-dynamics. The w-forcing depending on w represents the negative feedback of the ocean to the w-forcing independent of w: in the equatorial band, this adjustment is led by non-linear processes and by vortex stretching outside.

Appendix: The Ekman theory

The use of the Ekman assumptions in the primitive equation system leads us to consider the forcings involving the turbulent momentum flux and the Coriolis parameter only, and to neglect the non-linear forcings in Eq. 8. In consequence, system (9) reduces to five terms, out of 11, which are $F_{\tau_x}$, F _rotτ , F _divτ involving the turbulent momentum flux, the advection of planetary vorticity F _v regarding the external forcings and the linear form of the stretching term $F_{stretch\zeta}=\left(f^2 \frac{\partial w}{\partial z}\right)$ regarding the internal forcing. Therefore the Ekman hypotheses simplifies the vertical velocity Eq. 8 to the following expression:

$$ \begin{aligned} & \frac{\partial w(-z)}{\partial t} +f^2 \int_0^t w(-z) dt \\ &{\kern6pt}=\int_0^t\int_{-z}^0 \left(-2\beta {fv} -\beta \frac{\partial \tau_x}{\partial z} + f{\bf k}{\boldmath\nabla}\times\left(\frac{\partial {\boldmath\tau}}{\partial z}\right)\right)dzdt\\ &{\kern12pt}+\int_{-z}^0\left(\frac{\partial {\boldmath\nabla} {\boldmath\tau}}{\partial z}\right)dz \label{dwdt_ekman_eq} \end{aligned} $$

(14)

The vertical velocity at depth “z” and at current time “t” is obtained after the vertical (z) and temporal (t) integration of Eq. 14. Zebiak and Cane (1987) developed an expression for the equatorial Ekman pumping by introducing a Rayleigh friction in their shallow water model to obtain a steady state solution over a period of 2 days called T _adj here. Under this condition, Eq. 14 is integrated over the period T _adj to provide the similar stationarized expression of the Ekman pumping which is written as follows:

$$ \left\{ \begin{array}{@{}l} w(-z)=\dfrac{1}{\left( F_{adj}^2+f^2 \right)} \left(\int_{-z}^0 \left(-2\beta {fv} \right) dz -\beta \tau_{xs} \right. \\ \qquad\qquad\qquad\qquad\qquad\left.+f{\bf k}{\boldmath\nabla}\times {\boldmath\tau}_s +F_{adj} {\boldmath\nabla} {\boldmath\tau}_s \vphantom{\int_{-z}^0}\right)\!\!\!\!\!\\ \\ v=\dfrac{F_{adj} \tau_{ys}}{h\left(F_{adj}^2+f^2\right)} \mbox{where}\;\; h\;\; \mbox{is the Ekman depth}\!\!\!\!\\ \qquad \mbox{and} \boldmath\tau_s=(\tau_{xs},\tau_{ys}) \mbox{is the surface wind-stress}\!\!\!\! \end{array} \right. \label{w_ekman_eq} $$

(15)

It is important to note that Eq. 15 assumes that the Ekman pumping reaches a stationary regime at the time scale T _adj , which is viewed as an equatorial adjustment frequency (F _adj  = 1/T _adj ) while the more general expressions (8) and (14) are free from this assumption.

If we assume now that the external forcing reduces to the stress curl $\left(F_{ext}= f{\bf k}{\boldmath\nabla}\times\left(\frac{\partial {\boldmath\tau}} {\partial z}\right) \right)$ and exactly balances $\left(\frac{\partial w}{\partial t}=0\right)$, at all times, the internal forcing reduced to the stretching term $F_{stretch \zeta}=\left(f^2\frac{\partial w}{\partial z}\right)$ (because f > > ζ), then we obtain the well-known, classic, Ekman pumping of midlatitudes $\left(\frac{\partial w}{\partial z}=-\frac{1}{f}{\boldmath\nabla}\times \left(\frac{\partial {\boldmath\tau}}{\partial z}\right)\right)$. In fact, F _int (w) is a negative feedback to F _ext which gave birth to F _int (w) and thus cancels the tendency term $\left(\frac{\partial w(-z)}{\partial t}\right)$.

Finally, this demonstration shows that the expanded version of the vertical velocity Eq. 8 includes the Ekman theory.

Starting again from the expanded Eq. 8 for the vertical velocity, it is possible to propose a more general expression than Eq. 15 for the stationarized Ekman pumping. Now the Ekman assumptions are relaxed by taking into account all the 11 forcings listed in system (9) except the non-linear terms Tilt(ζ), Adv(D) and Def ₂(D) in the forcing F _NL . On the other hand, the assumption of stationarity at the frequency F _adj is kept. Under these conditions, the following extended formulation of the stationarized Ekman pumping is derived:

$$ \left\{ \begin{array}{@{}l} w(-z)\\ {\kern12pt} =\frac{1}{\left( F_{adj}^2+f(\zeta+\Delta \zeta+f)-\beta \Delta u \right)}\\ {\kern18pt}\times\displaystyle\int_{-z}^0 \left(-2\beta {fv} +\frac{\beta}{\rho}\frac{\partial P}{\partial x} +f Baro(\zeta) \right) dz+ F_{adj}\\ {\kern18pt}\times\!\displaystyle\int_{-z}^0 \!\!\left(\!\!\mathit{Baro}(\!D)\!-\!Def_1(\!D) \!+\!\left(f\zeta\!-\!\beta u\right)\!-\!\frac{1}{\rho}\nabla^2\! P \!\right)\!dz\\ {\kern18pt}-\beta (\tau_{xs}-\tau_{xb}) +f{\bf k}{\boldmath\nabla}\times ({\boldmath\tau}_s-{\boldmath\tau}_b) \\ {\kern18pt}+F_{adj} {\boldmath\nabla}({\boldmath\tau}_s-{\boldmath\tau}_b) \end{array} \right. \label{w_ekman_eq_ext} $$

(16)

where Δζ, Δu and τ _b are the vorticity and zonal current jumps and the turbulent momentum flux at depth “z”, respectively.

Since the right-hand-side terms are available from basic model outputs or gridded observations, the formulation (16) provides access to an estimation of the Ekman pumping but now in presence of a moving and heterogeneous ocean. This is not the case of Eq. 15, which assumes that the oceanic circulation derives from the surface wind-stress and the Coriolis force only. Note that expressions (15) and (16) also differ by their denominators (Γ) which can be seen as response-functions of the ocean for vertical motion. In expression (15), Γ depends only on the adjustment frequencies F _adj and f while, in the extended expression (16), Γ also depends on the dynamics through the vorticity and zonal current.