Ekman transport
Definition of Ekman transport:
Ekman transport is the depth-integrated, wind-driven transport in the ocean surface boundary layer, directed approximately at right angles to the wind stress by the combined action of turbulent friction and the Coriolis acceleration.
This is the common definition for Ekman transport, other definitions can be discussed in the article
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Contents
Ekman spiral
In a rotating frame, turbulent friction and Coriolis acceleration together cause wind-driven currents to rotate with depth. Ocean currents are deflected to the right in the Northern Hemisphere and to the left in the Southern Hemisphere. This deflection results from the Coriolis acceleration associated with the Earth's rotation and explained in the article Coriolis acceleration. This means that currents driven by wind stress at the water surface do not follow the wind direction, but are deflected. The wind-driven water transport deflected by the Earth's rotation is called Ekman transport, named after the Swedish oceanographer Vagn Walfrid Ekman who first described this phenomenon[1]. The deflection angle of the current increases with depth, measured from the surface (Fig. 1). The veering of the current with depth depends on the intensity of the turbulent fluid motions in the water column, which can be captured in a so-called 'eddy viscosity' coefficient [math]K_z[/math]. Ekman transport is largely restricted to the ocean upper layer with thickness [math]D_E[/math], the so-called Ekman depth. The Ekman depth is often tens to about a hundred meters in the open ocean, but varies strongly with mixing intensity, stratification and latitude. The currents associated with Ekman transport are small, typically less than 0.1 m/s. However, because of their large horizontal extent, the total volume transport can be very large, comparable to the transport of major ocean currents such as the Gulf Stream or the Kuroshio[2].
When surface waters are exposed to wind stress, they drag the deeper layers with them. Each layer of water is moved by friction from the shallower layer and each deeper layer moves more slowly than the layer above it. The direction of flow in each layer is deflected by the Earth's rotation. The resulting veering of the current is called the Ekman spiral. In the classical Ekman model with constant eddy viscosity, the surface current is directed 45° to the right of the wind in the Northern Hemisphere. Observed angles often differ substantially. In the classical constant-eddy-viscosity solution, the depth at which the current has turned 180° relative to the surface current is sometimes called the Ekman depth. In observations, however, the effective Ekman layer thickness is usually diagnosed from the decay of wind-driven shear and may differ from this idealized value.[3] Field observations often show a flatter and more variable Ekman spiral than predicted by the classical constant-viscosity theory. The surface-current angle, decay scale and veering rate depend on stratification, turbulence, waves, time dependence and the history of wind forcing.[4].
Consequences of Ekman transport
Ekman transport generates vertical water motions (upwelling or downwelling) through several mechanisms. Vertical water motions are of great significance for the ocean ecosystem. In the case of upwelling, they stimulate primary production by bringing nutrient-rich water from the deep ocean to the surface. In the case of downwelling, Ekman transport can contribute to carbon sequestration by downward transport of dissolved inorganic and organic carbon into deep ocean layers[5].
Vertical water transport is generated by zonal wind stress close to the equator, resulting from the varying strength of Ekman transport with latitude and the associated convergence or divergence of ocean currents (see Appendix). Diverging Ekman transport produces upwelling, also called Ekman suction. Converging Ekman transport produces downwelling. The trade winds at low latitudes and the westerlies at mid-latitudes generate a large-scale anticyclonic curl (vortex) in the wind stress and corresponding large-scale anticyclonic gyres north and south of the equator, in both the Atlantic and the Pacific. Ekman transport converges in these gyres, generating large downwelling zones in the subtropical oceans[2] (see Ocean circulation).
Upwelling and downwelling are particularly important at the shelf sea boundaries with the ocean. Coast-parallel winds drive surface waters offshore (if the coast is to the left, in the Northern Hemisphere) or onshore (if the coast is to the right). In the former case (offshore driven surface waters), the water surface dips towards the coast. The associated hydrostatic pressure gradient generates onshore transport of deep ocean water (beneath the Ekman layer) that then upwells to the surface (Fig. 2). In the latter case (onshore driven surface waters), a rise in the water surface towards the coast generates a hydrostatic pressure gradient that causes the downward flow of shelf water that is then transported offshore to the deep ocean.
Sverdrup transport
The wind-stress curl and associated downwelling in the subtropical gyres drives a geostrophic flow in the ocean interior, from below the Ekman layer to about one kilometer depth. The momentum balance of this interior geostrophic flow depends on the latitudinal variation of the Coriolis parameter (see the Appendix). The interior geostrophic flow is therefore directed to the equator, generating so-called Sverdrup transport, on both the Northern and Southern Hemispheres. The compensation of these equatorward flows contributes to a poleward return flow along the western continental boundaries, the Kuroshio, Gulf Stream, Brazil current and Agulhas, the so-called western boundary currents, see also Ocean circulation. Observed transports agree well with predictions based on the wind field over large areas, primarily in the tropics and subtropics. Elsewhere, especially at higher latitudes and in boundary regions, Sverdrup balance does not accurately describe meridional geostrophic transports, possibly due to the increased importance of the baroclinic flow, nonlinear dynamics, and topographic influences[6].
Ekman bottom layer
In the open ocean, the Ekman surface layer does not extend to the bottom. The currents in the ocean interior are largely geostrophic, with very small vertical shear (see the article Geostrophic flow). However, close to the bottom, the flow experiences the friction of the seabed. In this bottom boundary layer, called Ekman bottom layer, the velocity is also deflected, and tends while rotating to zero at the seabed (Fig. 3). Velocity veering is due to the Earth's rotation, just as in the Ekman surface layer. In the Northern Hemisphere, the cross-flow component of the bottom Ekman transport is directed to the left of the geostrophic interior flow. This bottom transport causes convergence or divergence depending on the relative vorticity and topographic setting. The Ekman transport in the bottom layer contributes to shelf sea exchange with the ocean by crossing isobaths. The associated flow is therefore called 'ageostrophic'. Ekman transport in the bottom layer can also contribute to vertical water motion depending on the geostrophic flow pattern. Upwelling occurs where geostrophic flows form a cyclonic gyre and downwelling in the opposite case (see the Appendix).
Appendix: Mathematical derivations
Ekman transport
The ocean response to a surface wind stress [math]\vec{\tau}(z=0) = \vec{\tau}_w[/math] on a rotating Earth follows from the momentum balance equations (see Coriolis acceleration)
[math]\Large\frac{1}{\rho}\frac{\partial \tau^{(x)}}{\partial z}\normalsize = - f \, v \, \qquad (1)[/math]
[math] \Large\frac{1}{\rho}\frac{\partial \tau^{(y)}}{\partial z}\normalsize = f \, u . \qquad (2)[/math]
The Coriolis parameter [math]f[/math] depends on the Earth’s angular rotation rate [math]\Omega \approx 7.29211 \times 10^{−5} \, \text{rad s}^{−1}[/math] and on the latitude [math]\phi[/math] (expressed in radians, positive on the Northern Hemisphere, negative on the Southern Hemisphere). It is given by [math]f=2 \Omega \sin \phi[/math].
Meaning of the other symbols:
[math]x, \, y\,=[/math] spatial coordinates, along the horizontal Cartesian [math]x[/math] axis (taken here positive eastward) and [math]y[/math] axis (taken here positive northward)
[math]z=[/math] coordinate along the vertical axis (upwards positive, water surface at [math]z=0[/math] )
[math]u(z),\, v(z) =[/math] components of the current velocity [math]\vec{u}(z)[/math] along the [math]x[/math] and [math]y[/math] axis, respectively
[math]\rho=[/math] seawater density
[math]\tau^{(x)}(z), \, \tau^{(y)}(z) =[/math] components of the shear stress [math]\vec{\tau}[/math] along the [math]x[/math] and [math]y[/math] axis, respectively
The equations (1) and (2) are a simplified form of the momentum balance following the assumptions of (a) steady flow (no time dependence), (b) uniform seawater density and (c) no sea surface slope (i.e. no piling up of seawater against boundaries).
We assume that the wind stress acts in the [math]x[/math] direction, [math]\tau^{(x)}(z=0) = \tau_w \, , \, \tau^{(y)}(z=0)=0[/math]. The wind shear stress is related to the wind speed [math]V_{wind}[/math] (usually taken at 10 m above the water surface) and the roughness of the water surface. It can be estimated from the empirical formula [math]\tau_w = \rho_{air} C_D V_{wind}^2[/math], with [math]\rho_{air} \approx 1.2 \; kg \, m^{-3}[/math] and [math]C_D \approx (1-2) \, . \, 10^{-3}[/math].
Integration of Eq. (1) over the depth gives the net volume transports [math]M^{(x)} \, , \, M^{(y)}[/math] in the [math]x[/math] and [math]y[/math] directions respectively:
[math]M^{(x)} \approx \int_{-\infty}^0 u dz = 0 \, , \, M^{(y)} \approx \int_{-\infty}^0 v dz = - \Large\frac{\tau_w}{\rho f}\normalsize. \qquad (3)[/math]
This expression shows that wind stress along the [math]x[/math] direction produces a net volume transport, the so-called Ekman transport, to the right of the wind stress (in the negative [math]y[/math] direction, Northern hemisphere). In the ideal steady theory, the depth-integrated Ekman transport depends only on the applied wind stress and the Coriolis parameter, provided the stress vanishes below the Ekman layer. Stratification affects the vertical structure and thickness of the layer.
Ekman spiral
We will adopt a further simplification by assuming that shear stresses are related to the local velocity gradient with constant eddy viscosity [math]K_z[/math] (which is typically in the range 0.01-0.05 m2/s; it should be noted that a constant eddy viscosity is seldom observed in field situations). The corresponding formula is
[math]\vec{\tau}(z) = \rho \, K_z \, \Large\frac{\partial \vec{u}}{\partial z} \normalsize . \qquad (4)[/math]
The momentum balance equations (1, 2) then become
[math] K_z \Large\frac{\partial^2 u}{\partial z^2}\normalsize = - f \, v \, , \quad K_z \Large\frac{\partial^2 v}{\partial z^2}\normalsize = f \, u . \qquad (5)[/math]
With the assumption of great depth, the solution of these linear equations take the form
[math]u(z) = \Re [a \exp(\kappa z)] \, , \; v(z) = \Re [b \exp(\kappa z)] , \quad \Re[\kappa] \gt 0 , \qquad (6)[/math]
where [math]a, b, \kappa[/math] are complex numbers. Solving [math]\kappa[/math] by substitution of Eq. (6) in Eq. (5) gives
[math]a = \pm i b \, , \quad \kappa = \Large\frac{\pi \pm i \pi}{D_E}\normalsize , \qquad (7)[/math]
where [math]D_E[/math] is the Ekman depth given by [math]\; D_E = \pi \sqrt{\large\frac{2 K_z}{|f|}\normalsize} . \qquad (8)[/math]
The shear stress near the water surface [math]\rho K_z \Large\frac{d \vec{u}}{dz}\normalsize (z=0)[/math] is equal to the wind stress [math]\vec{\tau}_w[/math] along the [math]x-[/math]axis. This condition yields
[math]a = u_0 e^{-i \pi /4} \, , \quad u_0 = \Large\frac{\sqrt{2} \pi \tau_w}{\rho |f| D_E}\normalsize \qquad (9)[/math]
and in the Northern Hemisphere,
[math]u(z) = u_0 \cos \Big(-\large\frac{\pi}{4}\normalsize + \large\frac{\pi z}{D_E}\normalsize \Big) \exp\Big(\large\frac{\pi z}{D_E}\normalsize\Big) \, , \; v(z) = u_0 \sin \Big(-\large\frac{\pi}{4}\normalsize + \large\frac{\pi z}{D_E}\normalsize \Big) \exp\Big(\large\frac{\pi z}{D_E}\normalsize\Big) . \qquad (10)[/math]
This expression represents the Ekman spiral shown in Fig. 1.
Ekman pumping
Ekman pumping is a vertical water transport associated with horizontal Ekman transport driven by wind stress. The vertical velocity [math]w(z)[/math] is related to the horizontal flow components through the continuity equation
[math]\Large\frac{\partial w}{\partial z}\normalsize = - \Big( \Large\frac{\partial u}{\partial x}\normalsize + \Large\frac{\partial v}{\partial y}\normalsize \Big) . \qquad (11)[/math]
We assume that the Earth is a perfect sphere of radius [math]R[/math], with [math]\phi=[/math] latitude and [math]\theta=[/math] longitude (both expressed in radians). We use here the convention that [math]x=R \, \theta \, \cos \phi [/math] is the longitudinal coordinate and [math]y =R \, \phi[/math] the latitudinal coordinate. The longitudinal gradient of the Coriolis parameter [math]df/dx= 0[/math] and the latitudinal gradient [math]df/dy = \beta, \; \beta = 2 \Omega \cos \phi /R[/math].
Now we derive Eq. (2) with respect to [math]x[/math] and Eq. (1) with respect to [math]y[/math] and subtract the results. This yields
[math] \dfrac{1}{\rho}\dfrac{\partial}{\partial z} \Big( \dfrac{\partial \tau^{(y)}}{\partial x} - \dfrac{\partial \tau^{(x)}}{\partial y} \Big) - v \beta \equiv \dfrac{f}{\rho} \dfrac{\partial}{\partial z} \Bigg[ \vec{e_z} \Big( \vec{\nabla} \times \dfrac{\vec{\tau}}{f} \Big) \Bigg] = f \Big( \Large\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \normalsize \Big) = - f \Large\frac{\partial w}{\partial z}\normalsize , \qquad (12)[/math]
where [math]\vec{\nabla}[/math] is the vector [math](\partial/\partial x,\partial / \partial y, 0 )[/math] and [math]\vec{e_z}[/math] is the vector [math](0,0,1)[/math]. This equation (the 2nd and 4th members) can be integrated from the Ekman depth [math]-D_E[/math] to the surface. Neglecting shear stresses at the Ekman depth with respect to the surface shear stresses, the following estimate for the vertical velocity [math]w_E[/math] at the Ekman depth is obtained:
[math]w_E = \dfrac{1}{\rho} \vec{e_z} \Big( \vec{\nabla} \times \dfrac{\vec{\tau}}{f} \Big) \equiv \Large \frac{1}{\rho f} \Big( \frac{\partial}{\partial x}\normalsize \tau_w^{(y)} - \Large\frac{\partial}{\partial y}\normalsize \tau_w^{(x)} \Big) + \Large\frac{\beta}{\rho f^2}\normalsize \tau_w^{(x)} . \qquad (13)[/math]
The singularity of this expression at the equator (where [math]f=0[/math]) indicates that the mid-latitude Ekman approximation breaks down there. Equatorial upwelling is nevertheless strongly influenced by wind-driven divergence, but it must be described with equatorial dynamics rather than by directly applying the [math]f[/math]-plane Ekman formula. The subtropical trade winds which are oriented E-W ([math]\tau_w^{(x)} \lt 0[/math]) increase towards the equator ([math]f^{-1} \, \partial \tau_w^{(x)} / \partial y [/math] > 0). This produces downwelling currents ([math]w_E \lt 0[/math]) in the surface Ekman layer of the subtropical ocean. The transition zones from the low-latitude trade winds to the westerlies at higher latitudes can be viewed as large-scale anticyclonic gyres with corresponding downwelling currents[2], see Ocean circulation.
Sverdrup transport
Sverdrup transport is a direct consequence of the changing Coriolis acceleration with latitude. We assume that the flow in the interior of the ocean water column (below the Ekman layer) is geostrophic (see Geostrophic flow) with horizontal components [math]U, V[/math] along the [math]x, y[/math] axes,
[math]U = - \Large\frac{1}{\rho f}\frac{\partial p}{\partial y}\normalsize \, , \quad V = \Large\frac{1}{\rho f}\frac{\partial p}{\partial x}\normalsize \, . \qquad (14)[/math]
Substitution of [math]u=U[/math] and [math]v=V[/math] in the continuity equation (11) gives [math]\quad \Large\frac{\partial w}{\partial z}\normalsize = - \Large\frac{1}{\rho}\frac{\partial p}{\partial x}\frac{\partial}{\partial y}\big( \frac{1}{f} \big)\normalsize = \Large\frac{\beta}{f}\normalsize V . [/math]
We call [math]D[/math] the depth over which the flow is driven by the geostrophic balance (14), which is of the order of 1 km. By integrating this equation over the geostrophic interior of the ocean water column to the lower boundary of the Ekman layer (from [math]-D[/math] to [math]-D_E[/math]) and using the expression (13), we find the meridional transport in the geostrophic interior associated with the Ekman pumping velocity
[math]\int_{-D}^{-D_E} V \, dz = \dfrac{f}{\beta} w_E = \dfrac{1}{\rho \beta} \vec{e_z} \big( \vec{\nabla} \times \vec{\tau} \big) + \dfrac{1}{\rho f} \tau_w^{(x)} . \qquad (15)[/math]
The Sverdrup transport [math]M_{Sv}[/math] defined as [math]M_{Sv} = \dfrac{1}{\rho \beta} \vec{e_z} \big( \vec{\nabla} \times \vec{\tau} \big)[/math] is a meridional flow in the geostrophic upper part of the ocean produced by the large-scale curl of the wind stress. It is directed to the equator because [math]\partial \tau_w^{(x)} / \partial y [/math] is positive (negative) on the subtropical N (S) hemisphere. The additional last term in the r.h.s. of Eq. (15), [math]\tau_w^{(x)} \lt 0[/math], is related to the latitude variation of the Ekman transport itself.
Ekman bottom layer
We assume a geostrophic current [math]\vec{U} = (U, V, 0)[/math] which is uniform (no dependence on depth) in the ocean's interior, see Geostrophic flow. However, in a layer close to the bottom, the Ekman bottom layer, this current will experience the friction of the seabed. It will therefore be rectified by a current [math]\vec{u}' = (u', v', w')[/math] such that [math]\vec{u} = \vec{U} + \vec{u}' =0[/math] at the seabed. The [math]z-[/math]axis in now chosen such that [math]z=0[/math] at the seabed.
The equations of motion are
[math]\Large\frac{1}{\rho}\frac{\partial \tau^{(x)}}{\partial z}\normalsize = - f \, v' \, , \quad \Large\frac{1}{\rho}\frac{\partial \tau^{(y)}}{\partial z}\normalsize = f \, u' . \qquad (16)[/math]
We assume that in the Ekman bottom layer the shear stress can be represented by a constant eddy viscosity [math]K_z[/math], thus [math]\vec{\tau}(z) = \rho \, K_z \, \Large\frac{\partial \vec{u'}}{\partial z} \normalsize . [/math] Solving the equations (4) with the boundary condition [math]\vec{u}'=(-U,-V,0)[/math] at [math]z=0[/math] and [math]\vec{u}'=0[/math] for [math]z \rightarrow \infty[/math] gives (Northern Hemisphere, Fig. 3)
[math]u(z) = U + u'(z) = U \, \Bigl[1 - \cos \big( \large\frac{\pi z}{D_E}\normalsize \big) \exp\big(\large\frac{-\pi z}{D_E}\normalsize\big) \Bigr] - V \sin \big(\large\frac{\pi z}{D_E}\normalsize \big) \exp\Big(\large\frac{-\pi z}{D_E}\normalsize\big) \, , \; v(z) = V + v'(z) = U \sin \big(\large\frac{\pi z}{D_E}\normalsize \big) \exp\Big(\large\frac{-\pi z}{D_E}\normalsize\big) + V \, \Bigl[1 - \cos \big( \large\frac{\pi z}{D_E}\normalsize \big) \exp\big(\large\frac{-\pi z}{D_E}\normalsize\big) \Bigr], \qquad (17)[/math]
with the same formula (8) for the thickness [math]D_E[/math] of the Ekman bottom layer, [math]\; D_E = \pi \sqrt{\large\frac{2 K_z}{|f|}\normalsize} . [/math]
The velocity is veering in the Ekman bottom layer, as in the Ekman surface layer. The Ekman volume transport in the Ekman bottom layer is given by
[math]M^{(x)} = \int_0^{D_E} u'(z) dz \approx - \large\frac{D_E}{2 \pi}\normalsize (U+V) \, , \; M^{(y)} = \int_0^{D_E} v'(z) dz \approx \large\frac{D_E}{2 \pi}\normalsize (U-V) . \qquad (18)[/math]
The bottom Ekman transport is directed to the left of the geostrophic flow in the Northern Hemisphere. The vertical flow [math]w[/math] at [math]z=D_E[/math] can be derived from the continuity equation (11), using Eq. (18) and the geostrophic relation [math]\Large \frac{\partial U}{\partial x}\normalsize + \Large \frac{\partial V}{\partial y}\normalsize = 0[/math] ,
[math]w(z=D_E) = - \int_0^{D_E} \Big( \Large \frac{\partial u}{\partial x}\normalsize + \Large \frac{\partial v}{\partial y}\normalsize \Big) \, dz = - \int_0^{D_E} \Big( \Large \frac{\partial u'}{\partial x}\normalsize + \Large \frac{\partial v'}{\partial y}\normalsize \Big) \, dz = \large\frac{D_E}{2 \pi}\normalsize \Big( \Large \frac{\partial V}{\partial x}\normalsize - \Large \frac{\partial U}{\partial y}\normalsize \Big) . \qquad (19)[/math]
In the Northern Hemisphere, cyclonic vorticity of the geostrophic flow produces divergence of the bottom Ekman transport and therefore upward motion at the top of the bottom Ekman layer; the signs are reversed in the Southern Hemisphere.
Related articles
References
- ↑ Ekman, V.W. 1905. On the influence of the Earth’s rotation on ocean-currents. Arch. Math. Astron. Phys. 2: 1-52
- ↑ 2.0 2.1 2.2 Chereskin, T.K. and Price, J.F. 2001. Ekman transport and pumping. Encyclopedia of Ocean Science, vol. 2, p. 809-815. Elsevier
- ↑ Pinet P.R. 1998.Invitation to Oceanography. Jones and Barlett Publishers. p. 508
- ↑ Price, J.F. and Sundermeyer, M.A. 1999. Stratified Ekman layers. J. Geophys. Res. 104: 20.467-20.494
- ↑ Liao, E., Resplandy, L., Liu, J. and Bowman, K. W. 2020. Amplification of the ocean carbon sink during El Niños: Role of poleward Ekman transport and influence on atmospheric CO2. Global Biogeochemical Cycles 34, e2020GB006574
- ↑ Gray, A.R. and Riser, S.C. 2014. A Global Analysis of Sverdrup Balance Using Absolute Geostrophic Velocities from Argo. J. Phys. Oceanography 142: 1213-1229
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