## 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

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