5.3 Exercise 9: Wave Refraction
97
Fig. 5.5 Boundary points that need to be specifie in the numerical code. In (a) v is predicted from
j = 1 to j = ny, whereas in (b) the prediction loop includes j = 0
boundary values of u at the northern and southern open boundaries are not required
since the finite-di ference equations do not access these values.
In this model application, we include j = 0 in the prediction of v and assume
that the second spatial derivative of η normal to the open boundary vanishes; that is:
η
n
0,k = 2η
n
1,k − η
n
2,k
η
n
ny+1,k = 2η
n
ny,k − η
n
ny−1,k
This condition induces a bias of the phase speed of waves in vicinity of the
boundary and can lead to severe misrepresentation of the dynamics in certain applications. This condition works reasonably well for the configuratio of this exercise.
Advanced methods are available for the numerical treatment of waves approaching an open boundary such as radiation conditions firs suggested by Sommerfeld
(1949). The basis of radiation conditions is that the phase speed normal to an open
boundary is the carrier of sea-level gradients across this boundary. This can be formulated by an advection equation reading:
∂η
∂t
+ c y
∂η
∂ y
= 0
(5.5)
where c y is the phase speed normal to the boundary. For processes of variable phase
speed, Orlanski (1976) proposed to compute c y using a diagnostic version of (5.5);
that is;
c y = −
∂η/∂t
∂η/∂y