Response to forcing: the governing equations and some basic solutions - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 71

(a)

z/D

2

v

(b)

u

1

v

u

u g

Figure 3.9 Ekman veering caused by bed friction acting on a geostrophic flow. (a) Vertical
profiles of the u and v components of current; (b) the hodograph of the current vector. Adapted
from (Kundu and Cohen, 2008 ), courtesy of Academic Press.

In the bottom layer, the x and y equations of motion for steady flow are just:

0 @

1 t x

@

0 ¼ fv

z

ð 3 : 45 Þ

0 @

1 t y

@ 0 @

1 p

0 @

1 t y

@

0 ¼ fu

z y ¼

fu

z þ

fu g

@

where we have used Equation (3.44) to substitute for the pressure gradient. Expressing
the stresses above in terms of the velocity shear and an eddy viscosity (Equation 3.40) ,
we have:

2 u

N z @

þ

z 2 ¼

fv 0

@

ð 3 : 46 Þ

2 v

N z @ f ð u u g Þþ

z 2 ¼

0 :

@

For boundary conditions we have: at the bottom boundary u ¼ v ¼ 0atz ¼ 0. Far

above the bottom, the motion is just the geostrophic flow u .

With these conditions, the solution to Equations (3.46) is (e.g. Kundu, 1990 , see p. 622):

¼ u g ; v ¼ 0atz ¼1

e z = D cos u g e z = D sin

u ¼ u g 1 ð z = D Þ ; v ¼ ð z = D Þ ð 3 : 47 Þ

which is in the form of the steady current plus a spiral similar to that which we saw
at the surface. The spiral rotates clockwise with increasing height above bottom
(northern hemisphere). A plot of the boundary currents (steady current

spiral)
from Equation (3.47) is shown in Fig. 3.9 . Close to the bed and continuing up to a

þ