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134 6 Rotational Effects6.6.2 Instability to Long WavesOn the f -plane (β = 0), the shear-fl w profil shown in Fig. 6.7 satisfie the conditionsnecessary for instability to develop. It can be shown that initial disturbancesof 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 amplifiewith time. Disturbances of a shorter wavelength travel with the f ow without growth(Cushman-Roisin, 1994). Hence, the barotropic instability process discriminatesdisturbances according to their wavelength.6.7 Exercise 17: Barotropic Instability6.7.1 AimThe aim of this exercise is to simulate dynamic instabilities produced by horizontalshear fl ws.6.7.2 Model EquationsUnder the assumption of a steady zonal geostrophic background fl w, U geo ,theequations governing the problem can be written as:∂u∂t + (u + U geo) ∂u∂x + v ∂(u + U geo)− f v =−g ∂η∂y∂x(6.37)∂v∂t + (u + U geo) ∂v∂x + v ∂v∂η+ fu =−g∂y ∂y(6.38)∂η∂t + ∂(uh) ∂h+ U geo∂x ∂x + ∂(vh) = 0∂y(6.39)where u, v and η are f ow and sea level disturbances with respect to the ambientfl w. These equations are identical to those in Exercise 16, with the addition that thegeostrophic background fl w is allowed to vary in the y-direction. The reason whythe full equations rather than simplifie equations of the previous section are usedhere is that we want to be able to simulate the entire instability process and not onlyits initial phase.6.7.3 Task DescriptionThe model domain of this exercise is an open channel of 10 km in length and 5 kmin width, bounded by coasts along the northern and southern boundaries. Lateral