Joseph Pedlosky Geophysical Fluid Dynamics - CHAPTER 7 Instability Theory
Solar heating is the ultimate energy source for the motion of both the atmosphere and the oceans with the exception of the lunar forcing of the tides. The radiant energy emitted by the sun may vary somewhat over very long periods, but a sensible idealization for most meteorological and oceanographic purposes consists in considering the solar source strength as fixed. Temporal variations in the incident radiation (and its spatial distribution) are then fixed by the astronomical relation between the positions of the earth and sun, e.g., by the seasonal progress of the earth in its solar orbit. Quite clearly, though, the motions of both the atmosphere and the oceans exhibit fluctuations whose time scales are not directly related to the astronomical periodicities of the earth—sun system. The phenomenon of weather in the atmosphere is in fact nothing more than the existence of large-scale wavelike fluctuations in the circulation of the atmosphere whose occurrence cannot be predicted, as the tides can be, by a simple almanac of assured recurrence based on past experience. Observations of oceanic motions have also revealed fluctuations at periods which bear no evident relationship with the astronomical periods which characterize the externally imposed forces. Not only do the observed oceanic and meteorological fluctuations occur on time scales which do not match the periods of the external forcing, but in addition, any particular observation of the fluctuations in the circulation shows them to occur erratically if not randomly distributed in time.
It is possible, though, to imagine the atmosphere and the ocean in a dynamical state which would be consistent with the external forcing and boundary conditions in which all change would be predictable with the appropriate astronomical period, in which each season is identical to its predecessor and such that an almanac of the past would serve as an accurate predictor of the future. Such a physical system might be consistent with every physical principle, but it is not the state realized in nature. Happily, mankind instead experiences a rich variety of motions in the atmosphere and oceans which depart dramatically from a simple, repetitive recurrence.
The existence of fluctuations in the circulations of the atmosphere and oceans can be attributed to the instability of the dynamical state without fluctuations to very small wavelike disturbances. Such small disturbances are inevitably present in any real system, but their effect on stable systems is ephemeral. If a state of flow, however, is unstable with respect to small fluctuations, the fluctuations will grow in amplitude with time and space scales determined by the dynamics of the interaction of the initial perturbation and the structure of the original flow state. This at once leads to a natural explanation, conceptually, for the inevitable presence of fluctuation energy at nonastronomical periods. This hypothesis requires for its validation, however, two quite formidable problem elements whose sequence forms a program of investigation and whose relationship is crucial. First, if the existence of fluctuations is to be demonstrated as due to the instability of the circulation which would occur in the absence of the fluctuations, it is first necessary to know what that fluctuation-free state would be. This first task is ordinarily very difficult. In some classical problems in hydrodynamic instability, such as the instability of the conductive temperature gradient of a fluid layer heated uniformly from below, the calculation of the basic state is sufficiently simple that attention instinctively and immediately focuses on the second element of the program, i.e., the exposure of the initial state to small perturbations and their subsequent evolution. For the study of the stability of atmospheric and oceanic flows, the calculation of the physically and mathematically possible flows in the absence of fluctuations is itself so difficult that it is rarely possible to carry through even this first part of the program completely. It is then natural to ask whether there are alternatives to the detailed calculation of the fluctuation-free state of flow. Of course it is possible to observe the actual flow pattern and construct averages, in time say, and observationally produce a pattern which has filtered out the velocity and temperature variations associated with the fluctuations. This defines an observed mean flow state. It is crucial to realize that this state cannot, in general, be used as the flow whose stability or instability will determine whether the observed fluctuations can be attributed to an instability process. The structure of the observed mean flow will inevitably be affected by presence of the very fluctuations we seek to predict, since in general the fluctuations will give rise by nonlinear processes to fluxes of heat and momentum with nonzero time averages. The convergence of these must be balanced by dissipation or counterbalancing fluxes of the mean flow quantities if a time-averaged state is to exist. Consequently the structure of the observed mean flow already implicitly assumes the existence of fluctuations, and it is a generally misleading fiction to suppose that the stability of the averaged state accurately portrays the stability of the fluctuation-free states, since in most cases the nature of the fluctuations alters the fluctuation- free state in the direction of stability. That is, the time-averaged state, if considered as the initial state, is frequently found to be considerably more stable than the relevant initial state we should be examining.
The unknown nature of the precise fluctuation-free state required for the stability analysis may in fact be turned to advantage. Instead of precisely calculating the mean state, we may arbitrarily prescribe an initial state. For example, let us imagine a planet with no imposed longitudinal variations of flow produced by heating and topography. Any steady zonal flow, i.e., a flow independent of longitude, will then satisfy (6.5.21). We can imagine such a flow initially determined by the balance between friction and externally imposed heating, since these forces, while negligible for flows varying in x, become determining when the terms retained in (6.5.2 1) identically vanish, as they do for x-independent flows. Each imagined initial state will correspond to a particular distribution of heat sources and frictional forces which may be specified after the fact in order that the hypothesized flow may be a solution of the equations of motion. This allows the consideration of classes of initial states, each corresponding to a different constellation of forces, and the stability of each of these initial states may then be directly examined to see which feature of the initial states the instability can be attributed to. if the feature responsible for the instability is sufficiently general and robustly persistent in a variety of circumstances, and if the resulting instability can be identified as geophysically relevant, then the consideration of an imagined class of initial states, rather than a precisely calculated single example, serves to actually deepen our understanding of the nature of the instability process and the criteria for instability.
This in turn assumes that it is possible to judge whether the predicted mode of instability, i.e., the response of the initial state to a small perturbation, is indeed physically relevant to the ocean or the atmosphere. It is not evident a priori that the nature of the instability process will be clearly evident in the mature, finite-amplitude fluctuations that are realized in the ocean and atmosphere. A truly major contribution of the early pioneer workers in the field of atmospheric instability, such as Charney (1947) and Eady (1949), was their demonstration that the mode of instability of conceptually “reasonable” initial states possessed time and space scales and a physical structure remarkably close to the observed weather waves in the atmosphere. The notion that the observed fluctuations in the atmosphere could be explained in terms of the small-amplitude stability analysis of a highly idealized flow is not an obvious one, and its subsequent verification is a tribute to the profound physical insight of the early investigators.
The purpose of this chapter is to discuss the fundamentals of quasi-geostrophic instability theory. Although the unperturbed state of the atmosphere should in fact be characterized by a longitudinally varying flow as a consequence of both continent—ocean variations of surface temperature and topographic forcing, most stability theories idealize the initial state as zonally uniform, i.e., longitudinally invariant. The study of instability of such idealized initial states reveals in the most straightforward way the mechanisms which give rise to the instability process, and is capable of predicting the general character of the observed fluctuations.