Rayleigh–Kuo criterion

This criterion determines whether or not a barotropic instability can occur, leading to the presence of vortices (like eddies and storms).

The Kuo criterion states that for barotropic instability to occur, the gradient of the absolute vorticity must change its sign at some point within the boundaries of the current.

[1][2] Note that this criterion is a necessary condition, so if it does not hold it is not possible for a barotropic instability to form.

In a barotropic fluid the density is a function of only the pressure and not the temperature (in contrast to a baroclinic fluid, where the density is a function of both the pressure and temperature[3]).

This shear in the velocity field induces a vertical and horizontal vorticity within the flow.

The eddies that form alternatingly on both sides of the flow are part of this instability.

Another way to achieve this instability is to displace the Rossby waves in the horizontal direction (see Figure 2).

[5] The Rayleigh–Kuo criterion states that the gradient of the absolute vorticity should change sign within the domain.

In the example of the shear induced eddies on the right, this means that the second derivative of the flow in the cross-flow direction, should be zero somewhere.

The presence of these instabilities in a rotating fluid have been observed in laboratory experiments.

[6] But barotropic instabilities were also observed in other Western Boundary Currents (WBC).

In the Agulhas current, the barotropic instability leads to ring shedding.

The Agulhas current retroflects (turns back) near the coast of South Africa.

At this same location, some anti-cyclonic rings of warm water escape from the mean current and travel along the coast of Africa.

[7] The derivation of the Rayleigh–Kuo criterion was first written down by Hsiao-Lan Kuo in his paper called 'dynamic instability of two-dimensional nondivergent flow in a barotropic atmosphere' from 1949.

By integrating this equation and filling in the boundary conditions, the Kuo criterion can be obtained.

In order to derive the Rayleigh–Kuo criterion, some assumptions are made on the fluids properties.

On this mean flow, some small perturbations are imposed in both the zonal and meridional direction:

Vertical motion and divergence and convergence of the fluid are neglected.

When taking into account these factors, a similar result would have been obtained with only a small shift in the position of the criterion within the velocity profile.

On the northern and southern boundary of this domain, the meridional fluid is zero.

A zonal mean flow with small perturbations was assumed,

the small perturbations in the zonal and meridional components of the flow.

To find the solution to the linearized equation, a stream function was introduced by Lord Rayleigh for the perturbations of the flow velocity:

These new definitions of the stream function are used to rewrite the linearized barotropic vorticity equation.

To solve this equation for the stream function, a wave-like solution was proposed by Rayleigh which reads

It is therefore that Hsiao-Lan Kuo came up with a stability criterion for this problem without actually solving it.

Instead of solving Rayleigh's equation, Hsiao-Lan Kuo came up with a necessary stability condition which had to be met in order for the fluid to be able to get unstable.

To get to this criterion, Rayleigh's equation was rewritten and the boundary conditions of the flow field are used.

To get to the Kuo criterion, the imaginary part is integrated over the domain (

Figure 1: The induced horizontal and vertical circulation patterns induced by a horizontal shear flow at the surface of the fluid. The horizontal shear is indicated by the length and colour density of the arrows at the surface of the fluid. This shear in the velocity at the surface leads to both vertical and horizontal circulation patterns. Due to Ekman transport from the right side of the flow towards the left, there is divergence (convergence) on the right (left) side of the flow which leads to upwelling (downwelling) and together with the horizontal return flow, this is a full circulation pattern. In the horizontal there can be formation of eddies on either side of the flow which have a circular rotation pattern clockwise on the right side of the flow (blue) and anti-clockwise on the left side of the flow (red). The eddies form on alternating sides of the flow and move with the flow.
Figure 2: The shear stress in the flow of the fluid induces eddies. The upper left panel (a), shows the differences in speed with the length of the black arrows and the density of the colours (darker colours mean larger velocity). When this flow becomes unstable (b), there direction of the flow begins to change. This process is further enhanced in panel (c) until finally full eddies occur (d). These eddies form on alternating sides of the flow field with a clockwise motion on the right (blue circles) and an anti-clockwise circulation on the left side of the flow (red circles).