## Spring Weather

### Peter Lynch, Met Éireann.

#### To be published in the Irish Scientist Yearbook, 2001.

It is remarkable how the same ideas pop up in different contexts. Analogies between physical systems are a powerful means of gaining understanding of abstruse and complex phenomena from more familiar and simple systems. When the underlying equations are identical, more concrete conclusions can be reached. Recently, a mathematical equivalence between the interactions of large-scale waves in the atmosphere and the oscillations of a humble elastic pendulum was discovered, which has important consequences for weather forecasting.

### The Equivalence.

The large sinuous oscillations of the atmospheric flow are called Rossby waves, after their Swedish discoverer, Carl-Gustaf Rossby (1898-1957). They are dominant in determining the patterns of weather and climate in middle latitudes, in particular the changeable weather with which Ireland is blessed. You can see them any night on the TV weather forecast maps. These waves interact with each other in groups of three, known as resonant triads, and for small amplitude they are described by the three-wave equations. Their interactions are crucial for determining the distribution of energy in the atmosphere.

Recently, Holm and Lynch [1] discovered that these same equations also govern the dynamics of a simple mechanical system, the elastic pendulum, comprising a heavy mass suspended by a spring. Thus, the motion of a swinging spring gives us information about resonant triads. This equivalence allows us to deduce properties, not otherwise evident, of atmospheric flow from the behavior of the spring.

### Resonance and Precession of the Spring.

The swinging spring can oscillate up and down (springing motion) or from side to side (swinging motion). When the frequency of the springing is twice that of the swinging, the spring is said to be in resonance. It then behaves in a way which is fascinating to watch. When started with almost vertical springing motion, the movement gradually develops into an essentially horizontal swinging motion. This does not persist, but is soon replaced by springing oscillations similar to the initial motion. The alternation between swinging and springing continues indefinitely, and each time the horizontal swinging is in a different direction. The rotation of the swing angle is called precession.

Surprisingly, the characteristic stepwise precession of the swinging spring has been largely ignored by theoreticians, although it is immediately obvious upon observation of a physical pendulum in resonance. Indeed, it is almost impossible to suppress experimentally when the initial motion is close to vertical. The stepwise precession of the swinging spring is shown in Figure 1. In horizontal projection, a star-shaped pattern is traced out by the bob, each pair of spokes corresponding to a horizontal excursion in a different direction (see [2] for more details).

Figure 1. Resonant exchange and stepwise precession of the trajectory of a swinging spring. The horizontal projection of the bob is shown.

### Precession of Resonant Rossby Wave Triads.

Since the governing equations are identical, Rossby waves must behave in the same way as the spring. In Figure 2 we show a resonant triad solution: a star-shaped pattern appears once more (the sum of the amplitudes of the two lowest-frequency triad components is plotted against half the difference of their phases). The initial conditions are chosen to correspond to those for the spring, using the transformation which determines the equivalence between the systems. The similarity between the two figures is immediate, and clearly illustrates the precession for the resonant triad. This phenomenon has not been previously noted and is an example of the insight coming from the mathematical equivalence of the two systems (more details in [3]).

Figure 2. Resonant Rossby wave triad solution calculated using the barotropic potential vorticity equation (Charney equation).

### Predictability.

The precession has implications for the predictability of atmospheric motion. A flow dominated by a single Rossby wave is unstable, and will be rapidly distorted due to inevitable perturbations. Triad resonance is the primary mechanism for this breakdown. However, the resulting pattern is highly sensitive to details of tiny perturbations which are impossible to determine accurately. Thus, drastically different patterns can result from states which are initially very similar. And this divergence can happen within a matter of days. Since the three-wave equations are integrable, this sensitivity cannot be described in the usual terms of chaos (the solutions of these equations are regular). We therefore have a chaos-like phenomenon in an integrable system. The forecaster's task is even harder than we thought!

Of course, real atmospheric waves are not small. For larger amplitudes, both the spring and triad systems exhibit chaos and, since the nonlinearities are not identical, the equivalence between them no longer holds. Preliminary evidence indicates a period-doubling route from triad interaction to chaos, but sin scéal eile.

### Sources.

[1] Holm, Darryl D. and Peter Lynch, 2001: Stepwise Precession of the Resonant Swinging Spring. Submitted to SIAM Journal on Applied Dynamical Systems. http://arXiv.org/abs/nlin/0104038

[2] Lynch, Peter, 2001: Resonant Motions of the Three-dimensional Elastic Pendulum. To appear in Intl. J. Nonlin. Mech.

[3] Lynch, Peter, 2001: Resonant Rossby Waves and the Swinging Spring. In preparation for Bull. Amer. Met. Soc.

[4] Anyone interested in experimenting may find two MatLab programs, one for the spring and one for resonant triads, at: http://www.maths.tcd.ie/~plynch/Rossby_Wave_Triads

[5] For a Java Applet illustrating the motion of the swinging spring, go to: http://www.maths.tcd.ie/~plynch/SwingingSpring/SS_Home_Page.html