## Introduction

Changes in the physical world are of two basic types: gradual and sudden. Gradual changes are the norm; sudden changes are less frequent, larger in magnitude, and easily recognizable. A physical system may experience gradual change over a long period of time, and then "all of a sudden," a rapid change may occur. For example, at a typical tropical island station in the Caribbean, the surface air temperature gradually rises after sunrise and continues to rise till late afternoon, but then suddenly drops owing to a thunderstorm. Often a sudden change occurs without any sudden rapid external stimulus, such that a slow gradual change in a system gives way to a sudden change. Water suddenly starting to boil in a heated pot is an example. At other times a sudden change is triggered by external events, but the amount of energy provided by the trigger is usually much less than that associated with the rapid change in the physical system initiated by the trigger. An example is the triggering of cumulus convection by air flowing over hills.

When the dependence of the equilibrium state of a dynamical system on an external parameter is nonlinear, multiple equilibrium states may occur. [The equilibrium state may be stationary (fixed point), time-periodic (limit cycle), or time-aperiodic (quasi-equilibrium, including strange attractor)]. Figure 1 illustrates such a conceptual picture. When the externally adjustable parameter e is ei < e < e2, two (quasi-)equilibria exist. If initially the system is on the lower branch when e is raised passing £2, the system jumps from the lower branch to the upper branch. The speed of the transition and the degree of overshooting before the system settles down on the new solution branch vary from system to system. Regardless, a transition such as this from one (quasi-)equilibrium solution to another when an external parameter passes a critical value, is called a catastrophe. Alternative to an external parameter passing a critical value, a catastrophe can be triggered by an external disturbance providing enough energy for the system to overcome the energy hump, which is of course larger when the external parameter is further away from the

Figure 1. Schematic diagram showing the state of a system as a function of an external parameter e. The system has two states between ei and e2.

critical value of e2. During the jump, the system may overshoot the new equilibrium and bounce back and oscillate about the new equilibrium before settling down. The overshooting may or may not occur, depending on the transient response of the system and the frictional force it experiences. A simple example to further illustrate a catastrophe is as follows (Fig. 2). A particle is held in a potential well as the structure of the well changes, the well eventually disappears, and the particle falls into a neighboring well. This fall takes place suddenly; it is called a spontaneous catastrophe. The fall can also take place without the well which the particle resides in disappearing, if the particle is pushed over the top of the well (energy barrier). This is called a triggered catastrophe.

The concept of a catastrophe (Thom, 1972; Poston and Stewart, 1978; Saunders, 1980; Iooss and Joseph, 1980) exists in many disciplines, and has been discovered by many researchers independently. Thus, it is not surprising that catastrophes have been given many other names.

Figure 2. Schematic diagram showing the state of a particle in a potential well. As an external parameter changes, the shape of the potential well changes such that the particle changes its position suddenly.

In many disciplines a catastrophe is called a "subcritical instability" (e.g. Drazin, 1992). The name "hysteresis" is also used when the emphasis is on the occurrence of multiple (quasi-) equilibria instead of the suddenness of the transitions. In engineering, it is called a "structural instability." In the physical sciences, it is often called a "critical phenomenon," owing to the fact that a parameter must exceed a critical value for the catastrophe to occur; or it may be called a "nonlinear instability," owing to the fact that, unlike in a linear instability, the growth rate is not constant. The term "nonlinear instability" includes subcritical and supercritical instabilities (Drazin, 1992). The forcing acting on a system to pull it away from an equilibrium that has just disappeared (in the case of a spontaneous catastrophe) increases as the system moves further away from the equilibrium. This is followed by a reduction of the forcing as the system approaches the other equilibrium. Thus the jump is under positive and then negative acceleration after the system overshoots the new equilibrium. Such accelerated growth lasts, of course, only for a finite time period and will eventually have to stop owing to the finite energy supply of the basic state; that is, when the system reaches the other quasi-equilibrium. The name "explosive instability" (Sturrock, 1966) is sometimes used, because the forcing pushing the system away from the initial (quasi-)equilibrium increases with time initially. Since it can be triggered by a finite-amplitude initial disturbance (a trigger), it is also called "finite-amplitude instability." In using the term "finite-amplitude instability," one should note that triggering is merely an alternative way to initiate the instability and is not necessary when an external parameter is changed to exceed a critical value. Nonetheless, in many cases triggering is the more likely way an instability gets started, owing to the frequent occurrence of perturbations in the atmosphere. Before a triggered catastrophe occurs, the system is stable, and this stability is called "metastability."

Figure 2. Schematic diagram showing the state of a particle in a potential well. As an external parameter changes, the shape of the potential well changes such that the particle changes its position suddenly.

The term "catastrophe," for a while, was considered an oversold concept and it incurred some connotation of a fraud when it was overly publicized in the 1970's. It has since regained its respectability after the initial commotion died down. It is now accepted as a standard mathematical term (Arnold, 1981) without the implication of something necessarily negative happening. We use it instead of terms equivalent to it — "explosive instability," "subcritical instability," "finite-amplitude instability," "critical phenomenon," and "structural instability" — for the sake of brevity.

In attempting to identify if a particular phenomenon is a catastrophe, one strives to ascertain if the common traits of catastrophes can be found. These common traits are bimo-dality (or multiple equilibria with at least two stable equilibria and an unstable one in between), suddenness of occurrence and spontaneous or triggered onset. Each catastrophe has its own cause for multiple equilibria, feedback processes in the transition, and triggering mechanisms (if they exist). The majority of the catastrophes can be easily identified through observation. One looks first for suddenness of the change. Suddenness means that the duration of the change is short, relative to the less eventful periods before and after the event. Bimodality — the next thing one looks for — means that the states before and after the change are clearly different, i.e. the change is sizable relative to the changes that take place before and after the transition. A phenomenon termed "onset" or "genesis" is very likely a catastrophe. Identification of the triggers of a catastrophe is not always very easy, since triggers do not always exist. Moreover, events concurrent with a catastrophe may be a result of that catastrophe rather than the triggers of it. Thus identifying triggers when they are not obvious can only be reliably accomplished through theoretical and modeling work, as in the case of a complete understanding of each catastrophe.

As far as understanding a catastrophe is concerned, the terms in the governing equation of a system that exhibits catastrophic behavior can be grouped (at least conceptually) into two competing sets (or two forcings), A and B. Thus, f —

In a multi-dimensional system, S is a judicially chosen single variable that represents the gross state of the system. Figure 3 is an example of Eq. (1). Initially, the state is at the stable equilibrium represented by point Si in Fig. 3.a Point S2 is an unstable (quasi-)equilibrium state and point S3 is another stable equilibrium state. As one or more parameters change such that curve A moves right ward while keeping its slope and/or the peak of curve B diminishes to such a degree that the point Si (quasi-)equilibrium can no longer be sustained, the state moves rapidly to (quasi-)equilibrium point S3. The movement

Figure 3. A and B as functions of S. The equilibrium states Si and S3 are stable, but not the one at S2.

aSince we have used S to denote a gross state, points 1 and 3 may be quasi-equilibria, rather than equilibria, in a multidimensional system.

is propelled by the difference between A and B in a "free fall" (more on this shortly). Figure 3 was used by Held (1983) to explain the topographically induced Rossby wave instability of Charney and DeVore (1979), which is a catastrophe. It is obvious that the existence of multiple quasi-equilibria alone is not sufficient for a catastrophe to occur. The relative movement of A and B induced by changes in external parameters or a trigger is necessary for a catastrophe to occur.

Figure 3 is a useful conceptual figure; however, it needs some clarification. According to Eq. (1), when the state reaches dS/dt = 0 and the state should stop changing and no overshooting should occur. But overshooting is often observed. The correct description is that A and B are functions of not only S but also dS/dt; what is depicted in Fig. 3 is A and B when dS/dt = 0, i.e. steady state forcings. Held (1983) pointed out that the curve B he used represents the drag exerted on the zonal flow by the steady forced waves generated by topography in the presence of dissipation. Thus the "free fall" is driven by a forcing greater than what appears in Fig. 3 as A-B.

Moreover, within the domain of the S-variable where one of the forcings varies highly nonlinearly, the other forcings usually do not vary highly nonlinearly and can thus be represented by a linear or a quasi-linear line; however, there can be exceptions.

Other examples of catastrophes are abundant. A simple example is the buckling found in flipping a wall switch or an earthquake. Another example is an explosion of any type. Other atmospheric examples are also abundant. In this article, three atmospheric catastrophes — tropical cyclogenesis, stratospheric warming, and monsoon onset — chosen based on the author's research interest are reviewed in some detail, to illustrate how one can study atmospheric catastrophes. Our understanding of these phenomena is still not complete; we will try to point out areas that require further study. Many more atmospheric catastrophes are enumerated below with varying degrees of commentary. In studying these catastrophes, one strives to identify the forces or forcings within that explain the associated multiple equilibria to arrive at a schematic diagram like Fig. 3. Also, one must support diagrams such as this by theoretical arguments and/or numerical experiments.

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