Ekman Spiral in a Horizontally Inhomogeneous Ocean with Varying Eddy Viscosity

Abstract

The classical Ekman spiral is generated by surface wind stress with constant eddy viscosity in a homogeneous ocean. In real oceans, the eddy viscosity varies due to turbulent mixing caused by surface wind and buoyancy forcing. Horizontally inhomogeneous density produces vertical geostrophic shear which contributes to current shear that also affects the Ekman spiral. Based on similar theoretical framework as the classical Ekman spiral, the baroclinic components of the Ekman spiral caused by the horizontally inhomogeneous density are obtained analytically with the varying eddy viscosity calculated from surface wind and buoyancy forcing using the K-profile parameterization (KPP). Along with the three existing types of eddy viscosity due to pure wind forcing (zero surface buoyancy flux), such an effect is evaluated using the climatological monthly mean data of surface wind stress, buoyancy flux, ocean temperature and salinity, and mixed layer depth.

Appendices

Appendix 1: General Solutions of Eq. (22)

Let the Ekman currents (u _E, v _E) be represented by a complex variable ψ,

$$\psi = u_{\text{E}} + iv_{\text{E}} , \, i = \sqrt { - 1} .$$

(52)

Substitution of (52) into (22) leads to

$$K\frac{{d^{2} \psi }}{{d\sigma^{2} }} + \frac{dK}{d\sigma }\frac{d\psi }{d\sigma } - if_{0} \psi = \phi ,$$

(53)

where

$$f_{0} = \frac{fh}{{\kappa u_{*} }}, \, \phi = \frac{{h^{2} }}{{f_{0} \kappa u_{*} V_{E} }}\frac{\partial }{\partial \sigma }\left[ {Ks_{y} } \right] - i\frac{{h^{2} }}{{f_{0} \kappa u_{*} V_{E} }}\frac{\partial }{\partial \sigma }\left[ {Ks_{x} } \right] .$$

(54)

The function $\phi$ represents the baroclinic effect. The second-order differential equation (53) needs two boundary conditions. The surface boundary condition (14) becomes

$$\frac{d\psi }{d\sigma }|_{\sigma = 0} = - \frac{{u_{*} }}{{\kappa K(0)V_{E} }}\left( {\cos \theta + i\sin \theta } \right) + \frac{{h^{2} }}{{f_{0} \kappa u_{*} V_{E} }}[s_{y} (0) - is_{x} (0)]. \,$$

(55)

The lower boundary condition of equation (53) is given by

$$\left| \psi \right|{\text{ finite as }}\sigma \to \infty .$$

(56)

to guarantee a physically meaningful solution, i.e., $\left| \psi \right| \,$ cannot be infinity as $\sigma \to \infty$. Eq. (53) is a linear inhomogeneous ordinary differential equation with the depth-varying coefficient K (z). Following Berger and Grisogono (1998), studying the Ekman atmospheric boundary layer, an approximate solution to the inhomogeneous problem (53) can be found with the variation of parameters technique, provided that an approximate solution of the homogeneous problem of (53),

$$K\frac{{d^{2} \psi }}{{d\sigma^{2} }} + \frac{dK}{d\sigma }\frac{d\psi }{d\sigma } - i \, f_{0} \psi = 0 ,$$

(57)

exists. If two independent approximate solutions to homogeneous problem of Eq. (57) are given by $\psi_{1} (\sigma )$ and $\psi_{2} (\sigma )$, the general solution of Eq. (53) is given by

$$\psi = c_{1} \psi_{1} (\sigma ) + c_{2} \psi_{2} (\sigma ) + \hat{c}_{1} (\sigma )\psi_{1} (\sigma ) + \hat{c}_{2} (\sigma )\psi_{2} (\sigma ) ,$$

(58)

where

$$\hat{c}_{1} (\sigma ) = - \int\limits_{\sigma }^{0} {\frac{{\psi_{2} \phi }}{{K(\zeta )[\psi_{1} (\zeta )d\psi_{2} (\zeta )/d\zeta - \psi_{2} (\zeta )d\psi_{1} (\zeta )/d\zeta ]}}d\zeta } ,$$

(59)

$$\hat{c}_{2} (\sigma ) = \int\limits_{\sigma }^{\infty } {\frac{{\psi_{1} \phi }}{{K(\zeta )[\psi_{1} (\zeta )d\psi_{2} (\zeta )/d\zeta - \psi_{2} (\zeta )d\psi_{1} (\zeta )/d\zeta ]}}d\zeta } .$$

(60)

Appendix 2: The WKB Method for Solving Eq. (57)

The WKB method can be used to obtain a good approximate solution of Eq. (57) if the vertical variation of K (σ) is slower than that of $\psi (\sigma )$ (Grisogono 1995),

$$\psi \propto \exp \left[ {\frac{{(S_{0} + S_{1} \varepsilon + S_{2} \varepsilon^{2} + \ldots )}}{\varepsilon }} \right] ,$$

(61)

where $\varepsilon$ is a presumably small parameter. Substitution of (61) into (57) leads to a set of equations in terms of powers of $\varepsilon$. If K (σ) does not vary too quickly with depth, we have

$$\frac{{\left| {S_{n + 1} (\sigma )} \right|}}{{\left| {S_{n} (\sigma )} \right|}} \ll 1, \, n = 0, \, 1, \, 2, \ldots$$

(62)

Solving for the first two terms S ₀ and S ₁ yields

$$S_{0} = \pm (1 + i)\sqrt {\frac{{\left| {f_{0} } \right|}}{2}} \int\limits_{0}^{\sigma } {\frac{d\zeta }{{\sqrt {K(\zeta )} }}} ,$$

(63)

$$S_{1} = \frac{1}{4}\ln \left[ {\frac{K(0)}{K(\sigma )}} \right] .$$

(64)

Thus, the two approximate solutions of the homogeneous equation (55) are

$$\psi_{1} (\sigma ) = A(\sigma )\exp [(1 + i)F(\sigma )], \, \psi_{2} (\sigma ) = A(\sigma )\exp [ - (1 + i)F(\sigma )] ,$$

(65)

where

$$A(\sigma ) = \left[ {\frac{K(0)}{K(\sigma )}} \right]^{1/4} , \, F(\sigma ) = - \sqrt {\frac{{\left| {f_{0} } \right|}}{2}} \int\limits_{0}^{\sigma } {\frac{d\zeta }{{\sqrt {K(\zeta )} }}} .$$

(66)

Substitution of (55) and (66) into (59) and (60) gives

$$\hat{c}_{1} (\sigma ) = - \frac{(1 - i)}{{2\sqrt {2\left| {f_{0} } \right|K(0)} }}\int\limits_{0}^{\sigma } {A(\zeta )\phi (\zeta )\exp \left[ { - (1 + i)F(\zeta )} \right]d\zeta } ,$$

(67)

$$\hat{c}_{2} (\sigma ) = - \frac{(1 - i)}{{2\sqrt {2\left| {f_{0} } \right|K(0)} }}\int\limits_{\sigma }^{\infty } {A(\zeta )\phi (\zeta )\exp \left[ {(1 + i)F(\zeta )} \right]d\zeta } .$$

(68)

Substitution of (65) into (58) gives

$$\psi = [c_{1} + \hat{c}_{1} (\sigma )]A(\sigma )\exp [(1 + i)F(\sigma )] + [c_{2} + \hat{c}_{2} (\sigma )]A(\sigma )\exp [ - (1 + i)F(\sigma )] .$$

(69)

It is noted that F (σ) < 0 leads to

$$\psi_{2} \to \infty {\text{ as }}\sigma \to \infty .$$

(70)

This leads to

$$c_{2} = 0$$

(71)

Substitution of (69) into the surface boundary condition (55) gives

$$c_{1} = \hat{c}_{2} (0) + (1 - i)\left\{ {\frac{{u_{*} (\cos \theta + i\sin \theta )}}{{\kappa \sqrt {2\left| {f_{0} } \right|K(0)} }} - \sqrt {\frac{K(0)}{{\left| {2f_{0} } \right|}}} \frac{h}{f}[s_{y} (0) - is_{x} (0)]} \right\}.$$

$$= \hat{c}_{2} (0) + V^{ + } - iV^{ - } + \sqrt {\frac{K(0)}{{\left| {2f_{0} } \right|}}} \frac{h}{f}\left\{ {\left[ {s_{x} (0) - s_{y} (0)} \right] + i\left[ {s_{x} (0) + s_{y} (0)} \right]} \right\}.$$

(72)