Rayleigh–Kuo criterion

Key takeaways

Rayleigh–Kuo criterion belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
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Reproduce the core statement of Rayleigh–Kuo criterion from memory before moving on to harder problems.

Reference excerpt

The Rayleigh–Kuo criterion (sometimes called the Kuo criterion) is a stability condition for a fluid. 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. Note that this criterion is a necessary condition, so if it does not hold it is not possible for a barotropic instability to form. But it is not a sufficient condition, meaning that if the criterion is met, this does not automatically mean that the fluid is unstable. If the criterion is not met, it is certain that the flow is stable. This criterion was formulated by Hsiao-Lan Kuo and is based on Rayleigh's equation named after the Lord Rayleigh who first introduced this equation in fluid dynamics.

Barotropic instability

Vortices like eddies are created by instabilities in a flow. When there are instabilities within the mean flow, energy can be transferred from the mean flow to the small perturbations which can then grow. 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). This means that surfaces of constant density (isopycnals) are also surfaces of constant pressure (isobars). Barotropic instability can form in different ways. Two examples are; when there is an interaction between the fluid flow and the bathymetry or topography of the domain; when there are frontal instabilities (may also lead to baroclinic instabilities). These instabilities are not dependent on the density and might even occur when the density of the fluid is constant. Instead, most of the instabilities are caused by a shear on the flow as can be seen in Figure 1. This shear in the velocity field induces a vertical and horizontal vorticity within the flow. As a result, there is upwelling on the right of the flow and downwelling on the left. This situation might lead to a barotropic unstable 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). This leads to a transfer of kinetic energy (not potential energy) from the mean flow towards the small perturbations (the eddies). 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. This happens in the centre of the eddies, where the acceleration of the flow perpendicular to the flow changes direction.

Examples The presence of these instabilities in a rotating fluid have been observed in laboratory experiments. The settings of the experiment were based on the conditions in the Gulf Stream and showed that within the ocean currents such as the Gulf Stream, it is possible for barotropic instabilities to occur. 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. The formation of these rings is a manifestation of a barotropic instability.

Derivation 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. This derivation is repeated and simplified below. First, the assumptions made by Hsiao-Lan Kuo are discussed. Second, the Rayleigh equation is derived in order continue to derive the Rayleigh–Kuo criterion. By integrating this equation and filling in the boundary conditions, the Kuo criterion can be obtained.

Assumptions In order to derive the Rayleigh–Kuo criterion, some assumptions are made on the fluids properties. We consider a nondivergent, two-dimensional barotropic fluid. The fluid has a mean zonal flow direction which can vary in the meridional direction. On this mean flow, some small perturbations are imposed in both the zonal and meridional direction: u ( y , t ) = U ( y ) + u ∗ ( y , t ) {\displaystyle u(y,t)=U(y)+u^{*}(y,t)} and v = v ∗ {\displaystyle v=v^{*}} . The perturbations need to be small in order to linearize the vorticity equation. 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. The derivation of the Kuo criterion will be done within the domain L = [ 0 , y ] {\displaystyle L=[0,y]} . On the northern and southern boundary of this domain, the meridional fluid is zero.

Rayleigh Equation

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Illustrations

Rayleigh–Kuo criterion: 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). — 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).

Worked examples

Example 1 — a first encounter with Rayleigh–Kuo criterion

Start with the simplest possible case. Write down what Rayleigh–Kuo criterion claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Rayleigh–Kuo criterion before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Rayleigh–Kuo criterion ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Rayleigh–Kuo criterion

In research: Rayleigh–Kuo criterion appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Rayleigh–Kuo criterion in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Rayleigh–Kuo criterion is common in secondary-school and first-year university syllabi. It links to neighbouring topics Fluid dynamics, so understanding it makes those chapters shorter.
In everyday life: Look for Rayleigh–Kuo criterion outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Rayleigh–Kuo criterion in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Rayleigh–Kuo criterion means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
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Frequently asked questions

What is Rayleigh–Kuo criterion in simple terms?

The Rayleigh–Kuo criterion (sometimes called the Kuo criterion) is a stability condition for a fluid. This criterion determines whether or not a barotropic instability can occur, leading to the presence of vortices (like eddies and storms).

Why does Rayleigh–Kuo criterion matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Rayleigh–Kuo criterion?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Rayleigh–Kuo criterion.