134
6 Rotational Effects
6.6.2 Instability to Long Waves
On the f -plane (β = 0), the shear-fl w profil shown in Fig. 6.7 satisfie the conditions necessary for instability to develop. It can be shown that initial disturbances
of a wavelength greater than 9.8 L, where L is the half-width of the shear zone,
are subtle to instability and grow rapidly. This wave does not travel but amplifie
with time. Disturbances of a shorter wavelength travel with the f ow without growth
(Cushman-Roisin, 1994). Hence, the barotropic instability process discriminates
disturbances according to their wavelength.
6.7 Exercise 17: Barotropic Instability
6.7.1 Aim
The aim of this exercise is to simulate dynamic instabilities produced by horizontal
shear fl ws.
6.7.2 Model Equations
Under the assumption of a steady zonal geostrophic background fl w, U geo , the
equations governing the problem can be written as:
∂u
∂t
+ (u + U geo )
∂u
∂ x
+ v
∂(u + U geo )
∂ y
− f v = −g
∂η
∂ x
(6.37)
∂v
∂t
+ (u + U geo )
∂v
∂ x
+ v
∂v
∂ y
+ f u = −g
∂η
∂ y
(6.38)
∂η
∂t
+
∂(uh)
∂ x
+ U geo
∂h
∂ x
+
∂(vh)
∂ y
= 0
(6.39)
where u, v and η are f ow and sea level disturbances with respect to the ambient
fl w. These equations are identical to those in Exercise 16, with the addition that the
geostrophic background fl w is allowed to vary in the y-direction. The reason why
the full equations rather than simplifie equations of the previous section are used
here is that we want to be able to simulate the entire instability process and not only
its initial phase.
6.7.3 Task Description
The model domain of this exercise is an open channel of 10 km in length and 5 km
in width, bounded by coasts along the northern and southern boundaries. Lateral