Edward N. Lorenz

The predictability of a flow which possesses many scales of motion

It is proposed that certain formally deterministic fluid systems which possess many scales of motion are observationally indistinguishable from indeterministic systems; specifically, that two states of the system differing initially by a small “observational error” will evolve into two states differing as greatly as randomly chosen states of the system within a finite time interval, which cannot be lengthened by reducing the amplitude of the initial error. The hypothesis is investigated with a simple mathematical model. An equation whose dependent variables are ensemble averages of the “error energy” in separate scales of motion is derived from the vorticity equation which governs two-dimensional incompressible flow. Solutions of the equation are determined by numerical integration, for cases where the horizontal extent and total energy of the system are comparable to those of the earths atomsphere. It is found that each scale of motion possesses an intrinsic finite range of predictability, provided that the total energy of the system does not fall off too rapidly with decreasing wave length. With the chosen values of the constants, “cumulus-scale” motions can be predicted about one hour, “synoptic-scale” motions a few days, and the largest scales a few weeks in advance. The applicability of the model to real physical systems, including the earths atmosphere, is considered. DOI: 10.1111/j.2153-3490.1969.tb00444.x

Optimal Sites for Supplementary Weather Observations: Simulation with a Small Model

Abstract Anticipating the opportunity to make supplementary observations at locations that can depend upon the current weather situation, the question is posed as to what strategy should be adopted to select the locations, if the greatest improvement in analyses and forecasts is to be realized. To seek a preliminary answer, the authors introduce a model consisting of 40 ordinary differential equations, with the dependent variables representing values of some atmospheric quantity at 40 sites spaced equally about a latitude circle. The equations contain quadratic, linear, and constant terms representing advection, dissipation, and external forcing. Numerical integration indicates that small errors (differences between solutions) tend to double in about 2 days. Localized errors tend to spread eastward as they grow, encircling the globe after about 14 days. In the experiments presented, 20 consecutive sites lie over the ocean and 20 over land. A particular solution is chosen as the true weather. Every 6 h obs...

The Mechanics of Vacillation

Abstract The equations governing a symmetrically heated rotating viscous fluid are reduced to a system of fourteen ordinary differential equations, by a succession of approximations. The equations contain two external parameters-an imposed thermal Rossby number and a Taylor number. Solutions where the blow is purely zonal, and solutions with superposed “steady” waves which progress without changing their shape, are obtained analytically. Additional solutions exhibiting vacillation, where the waves change shape in a regular periodic manner in addition to their progression, and solutions exhibiting irregular nonperiodic flow, are obtained by numerical integration. For a given imposed thermal Rossby number, the flow becomes more complicated as the Taylor number increases. Exceptions occur at very high Taylor numbers, where the equations become unrealistic because of truncation. For values of the external parameters where steady-wave solutions are found, solutions with purely zonal flow also exist, but are un...

Barotropic Instability of Rossby Wave Motion

Abstract Zonal flow resembling zonally averaged tropospheric motion in middle latitudes is usually barotropically stable, but zonal flow together with superposed neutral Rossby waves may be unstable with respect to further perturbations. Rossbys original solution of the barotropic vorticity equation is tested for stability, using beta-plane geometry. When the waves are sufficiently strong or the wavenumber is sufficiently high, the flow is found to be unstable, but if the flow is weak or the wavenumber is low, the beta effect may render the flow stable. The amplification rate of growing perturbations is comparable to the growth rate of errors deduced from large numerical models of the atmosphere. The Rossby wave motion together with amplifying perturbations possesses jet-like features not found in Rossby wave motion alone. It is suggested that barotropic instability is largely responsible for the unpredictability of the real atmosphere.

Attractor Sets and Quasi-Geostrophic Equilibrium

Abstract The attractor set of a forced dissipative dynamical system is for practical purposes the set of points in phase-space which continue to be encountered by an arbitrary orbit after an arbitrary long time. For a reasonably realistic atmospheric model the attractor should be a bounded set, and most of its points should represent states of approximate geostrophic equilibrium. A low-order primitive-equation (PE) model consisting of nine ordinary differential equations is derived from the shallow-water equations with bottom topography. A low-order quasi-geostrophic (QG) model with three equations is derived from the PE model by dropping the time derivatives in the divergence equations. For the chosen parameter values, gravity waves which are initially present in the PE model nearly disappear after a few weeks, while the quasi-geostrophic oscillations continue undiminished. The states which are free of gravity waves form a three-dimensional stable invariant manifold within the nine-dimensional phase spac...

Computational chaos-a prelude to computational instability

Abstract Chaotic behavior sometimes occurs when difference equations used as approximations to ordinary differential equations are solved numerically with an excessively large time increment τ. In two simple examples we find that, as τ increases, chaos first sets in when attractor A acquires two distinct points that map to the same point. This happens when A acquires slopes of the same sign, in a rectifying coordinate system, at two consecutive intersections with the critical curve. Chaotic and quasi-periodic behavior may then alternate within a range of τ before computational instability finally prevails. Bifurcations to and from chaos and transitions to computational instability are highly scheme-dependent, even among differencing schemes of the same order. Systems exhibiting computational chaos can serve as illustrative examples in more general studies of noninvertible mappings.

Designing Chaotic Models

Abstract After enumerating the properties of a simple model that has been used to simulate the behavior of a scalar atmospheric quantity at one level and one latitude, this paper describes the process of designing one modification to produce smoother variations from one longitude to the next and another to produce small-scale activity superposed on smooth large-scale waves. Use of the new models is illustrated by applying them to the problem of the growth of errors in weather prediction and, not surprisingly, they indicate that only limited improvement in prediction can be attained by improving the analysis but not the operational model, or vice versa. Additional applications and modifications are suggested.

SEASONAL AND IRREGULAR VARIATIONS OF THE NORTHERN HEMISPHERE SEA-LEVEL PRESSURE PROFILE

Abstract The variations of five-day mean sea-level pressure, averaged about selected latitude circles in the northern hemisphere, and the variations of differences between five-day mean pressures at selected pairs of latitudes are examined statistically. The northern hemisphere is found to contain two homogeneous zones, one in the polar regions and one in the subtropics, such that pressures in one zone tend to be correlated positively with other pressures in the same zone and negatively with pressures in the other zone. Considerable difference is found between the seasonal and the irregular pressure-variations which result from mass transport across the equator, but the seasonal and the irregular variations of pressure differences resemble each other closely, as do the seasonal and the irregular pressure-variations which result from rearrangements of mass within the northern hemisphere. The most important rearrangements appear to consist of shifts of mass from one homogeneous zone to the other. These shif...

Forced and Free Variations of Weather and Climate

Abstract Variations of weather and climate are termed “forced” or “free” according to whether or not they are produced by variations in external conditions. In many simple climate models, the poleward transport of sensible heat in the atmosphere has been treated as a diffusive process, and has been assumed to be proportional to the poleward temperature gradient. The validity of this assumption, for various space and time scales, is tested with 10 years of twice-daily upper level weather data. The space scales are defined by a spherical harmonic analysis, while the time scales are defined by a “poor mans spectral analysis.” The diffusive assumption is verified for the long-term average and the seasonal variations of the largest space scale, but it fails to hold for most of the remaining scales. It is shown that diffusive behavior can be expected only for forced scales. It is suggested that most of the scales resolved by the data are free.

Nondeterministic theories of climatic change

Abstract A basic assumption in some climatic theories is that, given the physical properties of the atmosphere and the underlying ocean and land, specified environmental parameters (amount of solar heating, etc.) would determine a unique climate and that climatic changes therefore result from changes in the environment. The possibility that no such unique climate exists and that nondeterministic factors are wholly or partly responsible for long-period fluctuations of the atmosphere-ocean-earth system, is considered. A simple difference equation is used to illustrate the phenomena of transitivity, intransitivity, and almost-intransitivity. Numerical models of moderate size suggest that almost-intransitivity might lead to persistence of atmospheric anomalies for a whole season. The effect of this persistence could be to allow substantial anomalies to build up in the underlying ocean or land, perhaps as abnormal temperatures or excessive snow or ice. These anomalies could subsequently influence the atmosphere, leading to long-period fluctuations. The implications of this possibility for the numerical modeling of climate, and for the interpretation of the output of numerical models, are discussed.