6.6 Instability of Lateral Shear Flows
133
With addition of small perturbations; that is u = u + u
, v = v
, and η = η + η
, the
momentum equations can be written as (see Cushman-Roisin, 1994):
∂u
∂t
+ u
∂u
∂ x
+ v
∂u
∂ y
− f v
= −g
∂η
∂ x
∂v
∂t
+ u
∂v
∂ x
+ f u
= −g
∂η
∂ y
Based on scaling arguments and with a focus on the initial appearance of disturbances, the smallest terms such as u
∂u
/∂ x have been dropped in the above
equations. In addition to this, vertical velocity is assumed negligibly small, so that
the continuity equation can be expressed as:
∂u
∂ x
+
∂v
∂ y
= 0
Accordingly, sea surface pressure is not directly calculated from these equations,
but rather appears implicitly as a requirement to produce a horizontal fl w void of
lateral divergence/convergence. The f ow follows the contours of a certain streamfunction, called streamlines. Here, a streamfunction can be constructed from:
u
= −
∂ψ
∂ y
and v
= +
∂ψ
∂ x
which satisfie the previous continuity equation. By use of this streamfunction, the
linearised equations can be combined to yield a single equation for the streamfunction:
∂
∂t
+ u
∂
∂ x
∇
2
ψ +
β −
d
2
u
∂ y 2
∂ψ
∂ x
= 0
where β is the meridional variation of the Coriolis parameter. The solution of this
equation depends on the specifi form of u. It can be shown that a necessary condition for barotropic instability to occur is that the function:
β −
d
2
u
dy 2
(6.36)
vanishes at least once within the model domain. This result was f rst derived by Kuo
(1949).