A study of cross-shore maximum upwelling intensity along the Northwest Africa coast

Abstract

Satellite images of sea surface temperature (SST) show that the location of cross-shore SST minimum (LCSM) stretches along the isobaths in the Northwest Africa Upwelling System. To understand and interpret these observations better, we set up a two-dimensional analytical model that takes into account the surface and bottom Ekman transport and the alongshore geostrophic current, as well as bottom friction and variations in bottom topography. The structure of vertical velocity with a realistic topography clearly illustrates the variations of SST drop in a sample cross-shore section. Some idealized theoretical model experiments are carried out to examine the effects of eddy viscosity, Coriolis force, and cross-shore wind on the location of the cross-shore maximum upwelling intensity. The results show that the cross-shore wind largely impacts on the location where the coldest water outcrops to the surface through an adjustment of the cross-shore pressure gradient. This is also verified by the remotely sensed data, which indicate that the maximum correlation coefficient between cross-shore wind stress and the depth of LCSM is −0.65 with a lag of approximately 1 day.

Appendix: Derivation of the horizontal velocity

Welander (1957) extended the Ekman theory to varying bathymetry. Kamenkovich (1977) added a geostrophic component into the equation associated with the pressure gradient term. Estrade et al. (2008) finally obtained the geostrophic velocity and solved the exact solution using the assumption that cross-shore Ekman transport equals zero. Jiang et al. (2010) used a one-dimensional numerical model to solve the momentum equation. Following their work, in a steady, unstratified, and homogeneous ocean, we assume a constant density ρ = 1025 kg m⁻³ and vertical eddy viscosity A _z , and ignore the horizontal advection terms. The momentum equation is

$$ f\vec{k} \times \vec{V} = - \frac{\nabla P}{\rho } + A_{z} \frac{{\partial^{2} \vec{V}}}{{\partial z^{2} }}, $$

(2)

where P is the pressure, f is the Coriolis parameter, and $\vec{k}$ is a vertical unit vector. The solution of the equation can be obtained by separating the horizontal velocity components into an Ekman part and a geostrophic part $\vec{V} = \vec{u}_{\text{e}} + \vec{u}_{\text{g}} ,$ and by introducing the complex variables $\vec{u}_{\text{e}} = u_{\text{e}} + iv_{\text{e}}$, $\vec{u}_{\text{g}} = u_{\text{g}} + iv_{\text{g}},$ and wind stress $\vec{\tau } = \tau_{x} + i\tau_{y}$. Equation (2) can thus be written as

$$ \begin{aligned} f\vec{k} \times \vec{u}_{\text{e}} & = A_{z} \frac{{\partial^{2} \vec{u}_{\text{e}} }}{{\partial z^{2} }} \\ f\vec{k} \times \vec{u}_{\text{g}} & = - \frac{\nabla P}{\rho }. \\ \end{aligned} $$

(3)

Here, we take a right-handed coordinate system, in which x is in cross-shore direction, positive eastward; positive y is 90° counterclockwise from the positive x-direction; and z is the vertical direction, positive upward. The surface and bottom boundary conditions are

$$ \begin{aligned} z & = 0:\rho A_{z} \frac{{\partial \vec{u}_{\text{e}} }}{\partial z} = \vec{\tau } \\ z & = - h:\vec{u}_{\text{g}} + \vec{u}_{\text{e}} = 0. \\ \end{aligned} $$

(4)

The solution of Eq. (3) with boundary condition (4) is

$$ \vec{u}_{\text{e}} = \frac{{\vec{\tau }}}{{\rho A_{z} j}}\frac{\sinh [j(h + z)]}{\cosh [jh]} - \vec{u}_{\text{g}} \frac{\cosh [jz]}{\cosh [jh]}, $$

(5)

in which $j = (1 + i)\sqrt {f/2A_{z} } .$ The depth-integrated transport from z = −h to the surface is

$$ M = \int\limits_{ - h}^{0} {\vec{V}} {\text{d}}z = \frac{{\vec{\tau }}}{{\rho A_{z} j^{2} }}\left( {1 - \frac{1}{\cosh [jh]}} \right) + \vec{u}_{\text{g}} \left( {h - \frac{1}{j}\tanh [jh]} \right). $$

(6)

As Brink (1983) and Mitchum and Clarke (1986) did in their previous studies, we introduce Ekman layer structure functions which simplify the form of the solutions and therefore allow for an intuitive explanation of the results. These functions are defined as

$$ \begin{aligned} \delta & = \left\{ {\cos \left( \frac{h}{D} \right)\cosh \left[ \frac{h}{D} \right]} \right\}^{2} + \left\{ {\sin \left( \frac{h}{D} \right)\sinh \left[ \frac{h}{D} \right]} \right\}^{2} \\ S_{1} & = \frac{1}{\delta }\cos \left( \frac{h}{D} \right)\cosh \left[ \frac{h}{D} \right] \\ S_{2} & = \frac{1}{\delta }\sin \left( \frac{h}{D} \right)\sinh \left[ \frac{h}{D} \right] \\ T_{1} & = \frac{1}{\delta }\sinh \left[ \frac{h}{D} \right]\cosh \left[ \frac{h}{D} \right] \\ T_{2} & = \frac{1}{\delta }\sin \left( \frac{h}{D} \right)\cos \left( \frac{h}{D} \right), \\ \end{aligned} $$

(7)

where $D = \sqrt {2A_{z} /f}$ is the Ekman depth. Assuming that wind forcing is uniform and alongshore variation of terrain is small, the ∂P/∂y term becomes negligible so that the cross-shore geostrophic velocity u _g is zero. We then have the cross-shore Ekman transport

$$ M_{x} = \frac{{\tau_{y} }}{\rho f}(1 - S_{1} ) + \frac{{\tau_{x} }}{\rho f}S_{2} - \frac{{v_{\text{g}} D}}{2}(T_{1} - T_{2} ). $$

(8)

The no-flow penetration condition at the shore requires that the cross-shore transport M _x is equal to zero at the coast and must be equal to zero in the entire water column. The geostrophic velocity becomes

$$ \vec{u}_{g} = iv_{\text{g}} = \frac{2i}{\rho fD}\left[ {\frac{{\tau_{y} (1 - S_{1} )}}{{T_{1} - T_{2} }} + \frac{{\tau_{x} S_{2} }}{{T_{1} - T_{2} }}} \right]. $$

(9)

Substituting expression (9) into Eq. (5), we obtain the horizontal velocity $\vec{V}$ as

$$ \begin{aligned} \vec{V} & = \vec{u}_{\text{e}} + \vec{u}_{\text{g}} = \frac{{\tau_{x} + i\tau_{y} }}{{\rho A_{z} j}}\frac{\sinh [j(h + z)]}{\cosh [jh]} \\ & \quad + \frac{2i}{\rho fD}\left[ {\frac{{\tau_{y} (1 - S_{1} )}}{{T_{1} - T_{2} }} + \frac{{\tau_{x} S_{2} }}{{T_{1} - T_{2} }}} \right]\left( {1 - \frac{\cosh [jz]}{\cosh [jh]}} \right). \\ \end{aligned} $$

(10)