Waves, turbulent motions and mixing - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 91

Figure 4.1 Definition of parameters for
a surface wave.

z

h

c

z =0 x

l

w

u

z =- h

fact is expressed by the statement that the vorticity of the fluid is zero. Vorticity is
a vector quantity with components in x, y and z. For a wave travelling in the
x direction, for example, the y component of vorticity is @ u

@ z @ w

0. The significance
of the waves having zero vorticity is that the velocity field may then be expressed as
the derivative of a potential function, 1 i.e. we can write:

@ x ¼

Þ¼ @

f

x ; @

f

y ; @

f
@

u

¼ð

u ; v ; w ¼r f : ð 4 : 1 Þ

@ @ z

The analysis is then directed to find the form of f which satisfies the continuity
Equation (3.1) which, on substituting, becomes:

2 f 2 f 2 f

@

@

x 2 þ @ y 2 þ @

2 f

¼r ¼ 0 : ð 4 : 2 Þ

@ @ z 2

This is a well known equation in maths and physics called Laplace's equation.
We can set up the problem as shown in Fig. 4.1 . We choose the x axis to be the
direction of propagation of a plane wave which is uniform in the y direction so that
the

y terms are zero. As well as Laplace's equation, the velocity potential f must
also satisfy the boundary conditions:

@ / @

(i) at the bottom boundary (z ¼ h) flow cannot pass through the seabed:

¼ @

f

w

z ¼

0 ;

@

(ii) at the surface any vertical motion will be seen as changes in sea level:

¼ @ ¼ @

@

f

w

t ; @ z

1 A potential function is one from which a vector field (e.g. velocity) can be derived by taking the gradient
of the function. If such a representation of the vector field is possible, then there is a mathematical
requirement that vorticity or its equivalent is zero.