Chapter 2
1D Models of Ekman Layers
Abstract This chapter introduces the reader to one-dimensional water-column
models using fixed vertical levels. Such a model is applied to study the dynamics of
surface and bottom Ekman layers in the ocean.
2.1 Useful Background Knowledge
2.1.1 Inertial Oscillations
Before exploring the Ekman-layer dynamics, it is useful to revisit features inherent
with inertial oscillations. Flows under the sole influence of the Coriolis force are
described by the momentum equations:
∂u
∂t
+ f v = 0
(2.1)
∂v
∂t
− f u = 0
(2.2)
where f is the Coriolis parameter, given by f = 2Ω sin(ϕ), where Ω = 7.27 ×
10
−5 s
−1 is the rotation frequency of Earth, and ϕ is geographical latitude in radians.
For an initial flow in the x-direction of speed u o , the solution of the latter equations
is given by:
u(t) = +u o cos ( f t)
v(t) = −u o sin ( f t)
The resultant flow trajectories are circles of a radius of u o / | f |, called inertial
radius. The period of one complete cycle is 2π/ f , called inertial period. The inertial
period is 12 hrs at the poles and infinite directly at the equator where the Coriolis
force vanishes. Figure 2.1 shows flow paths associated with inertial oscillations for
u o = 0.1 m/s and f = 1 × 10
−4 s
−1 with and without ambient uniform flow.
J. K¨ ampf, Advanced Ocean Modelling, DOI 10.1007/978-3-642-10610-1 2,
C
Springer-Verlag Berlin Heidelberg 2010
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