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Abstract
On the basis of five years of Northern Hemisphere isobaric height data, states of the atmosphere separated by 12 days or less are found on the average to resemble each other more closely than randomly selected states, even after adjustment for seasonal trend has been made. The existence of partial predictability of instantaneous weather patterns at least 12 days in advance is thereby confirmed.
Abstract
On the basis of five years of Northern Hemisphere isobaric height data, states of the atmosphere separated by 12 days or less are found on the average to resemble each other more closely than randomly selected states, even after adjustment for seasonal trend has been made. The existence of partial predictability of instantaneous weather patterns at least 12 days in advance is thereby confirmed.
Abstract
A generalized vorticity equation for a two-dimensional spherical earth is obtained by eliminating pressure from the equations of horizontal motion including friction. The generalized vorticity equation is satisfied by formal infinite series representing the density and wind fields. The first few terms of a particular series solution are obtained explicitly. The series appear to converge near the north pole, and determine a model of a polar air mass. Within the air mass, the coldest winds are northeasterly and the warmest are southwesterly, while the coldest air of all is at the north pole. Heating occurs in the northwesterly winds and cooling in the southeasterlies, while aside from the effect of friction the air mass as a whole is cooled. The energy balance of the air mass is investigated. It is suggested that an analogous distribution of heating and cooling may be instrumental in maintaining the general circulation of the atmosphere.
Abstract
A generalized vorticity equation for a two-dimensional spherical earth is obtained by eliminating pressure from the equations of horizontal motion including friction. The generalized vorticity equation is satisfied by formal infinite series representing the density and wind fields. The first few terms of a particular series solution are obtained explicitly. The series appear to converge near the north pole, and determine a model of a polar air mass. Within the air mass, the coldest winds are northeasterly and the warmest are southwesterly, while the coldest air of all is at the north pole. Heating occurs in the northwesterly winds and cooling in the southeasterlies, while aside from the effect of friction the air mass as a whole is cooled. The energy balance of the air mass is investigated. It is suggested that an analogous distribution of heating and cooling may be instrumental in maintaining the general circulation of the atmosphere.
Abstract
Dynamical systems possessing regimes are identified with those where the state space possesses two or more regions such that transitions of the state from either region to the other are rare. Systems with regimes are compared to those where transitions are impossible.
A simple one-dimensional system where a variable is defined at N equally spaced points about a latitude circle, once thought not to possess regimes, is found to exhibit them when the external forcing F slightly exceeds its critical value F* for the appearance of chaos. Regimes are detected by examining extended time series of quantities such as total energy. A chain of k* fairly regular waves develops if F < F*, and F* is found to depend mainly upon the wavelength L* = N/k*, being greatest when L* is closest to a preferred length L 0. A display of time series demonstrates how the existence and general properties of the regimes depend upon L*.
The barotropic vorticity equation, when applied to an elongated rectangular region, exhibits regimes much like those occurring with the one-dimensional system. A first-order piecewise-linear difference equation produces time series closely resembling some produced by the differential equations, and it permits explicit calculation of the expected duration time in either regime. Speculations as to the prevalence of regimes in dynamical systems in general, and to the applicability of the findings to atmospheric problems, are offered.
Abstract
Dynamical systems possessing regimes are identified with those where the state space possesses two or more regions such that transitions of the state from either region to the other are rare. Systems with regimes are compared to those where transitions are impossible.
A simple one-dimensional system where a variable is defined at N equally spaced points about a latitude circle, once thought not to possess regimes, is found to exhibit them when the external forcing F slightly exceeds its critical value F* for the appearance of chaos. Regimes are detected by examining extended time series of quantities such as total energy. A chain of k* fairly regular waves develops if F < F*, and F* is found to depend mainly upon the wavelength L* = N/k*, being greatest when L* is closest to a preferred length L 0. A display of time series demonstrates how the existence and general properties of the regimes depend upon L*.
The barotropic vorticity equation, when applied to an elongated rectangular region, exhibits regimes much like those occurring with the one-dimensional system. A first-order piecewise-linear difference equation produces time series closely resembling some produced by the differential equations, and it permits explicit calculation of the expected duration time in either regime. Speculations as to the prevalence of regimes in dynamical systems in general, and to the applicability of the findings to atmospheric problems, are offered.
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.
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.
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. Rossby's 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.
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. Rossby's 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.
Abstract
Two states of the atmosphere which are observed to resemble one another are termed analogues. Eitherstate of a pair of analogues may be regarded as equal to the other state plus a small superposed "error."From the behavior of the atmosphere following each state, the growth rate of the error may be determined.
Five years of twice-daily height values of the 200-, 500-, and 850-mb surfaces at a grid of 1003 pointsover the Northern Hemisphere are procured. A weighted root-mean-square height difference is used as ameasure of the difference between two states, or the error. For each pair of states occurring within onemonth of the same time of year, but in different years, the error is computed.
There are numerous mediocre analogues but no truly good ones. The smallest errors have an averagedoubling time of about 8 days. Larger errors grow less rapidly. Extrapolation with the aid of a quadratichypothesis indicates that truly small errors would double in about 2.5 days. These rates may be comparedwith a 5-day doubling time previously deduced from dynamical considerations.
The possibility that the computed growth rate is spurious, and results only from having superposedthe smaller errors on those particular states where errors grow most rapidly, is considered and rejected. Thelikelihood of encountering any truly good analogues by processing all existing upper-level data appearsto be small.
Abstract
Two states of the atmosphere which are observed to resemble one another are termed analogues. Eitherstate of a pair of analogues may be regarded as equal to the other state plus a small superposed "error."From the behavior of the atmosphere following each state, the growth rate of the error may be determined.
Five years of twice-daily height values of the 200-, 500-, and 850-mb surfaces at a grid of 1003 pointsover the Northern Hemisphere are procured. A weighted root-mean-square height difference is used as ameasure of the difference between two states, or the error. For each pair of states occurring within onemonth of the same time of year, but in different years, the error is computed.
There are numerous mediocre analogues but no truly good ones. The smallest errors have an averagedoubling time of about 8 days. Larger errors grow less rapidly. Extrapolation with the aid of a quadratichypothesis indicates that truly small errors would double in about 2.5 days. These rates may be comparedwith a 5-day doubling time previously deduced from dynamical considerations.
The possibility that the computed growth rate is spurious, and results only from having superposedthe smaller errors on those particular states where errors grow most rapidly, is considered and rejected. Thelikelihood of encountering any truly good analogues by processing all existing upper-level data appearsto be small.
Abstract
We identify the slow manifold of a primitive-equation system with the set of all solutions that are completely devoid of gravity-wave activity. We construct a five-variable model describing coupled Rossby waves and gravity waves. Successive-approximation schemes designed to determine the slow manifold fail to converge when applied to the model, although they sometimes appear to converge before finally diverging. A noniterative scheme which demands only that the fast variables be functions of the slow variables yields a “Slowest invariant manifold,” which, however, is not unequivocally slow. We question whether the complete absence of gravity waves can be logically defined, and we note that the existence or nonexistence of a slow manifold does not depend upon the convergence or nonconvergence of a power series or a succession of approximations.
Abstract
We identify the slow manifold of a primitive-equation system with the set of all solutions that are completely devoid of gravity-wave activity. We construct a five-variable model describing coupled Rossby waves and gravity waves. Successive-approximation schemes designed to determine the slow manifold fail to converge when applied to the model, although they sometimes appear to converge before finally diverging. A noniterative scheme which demands only that the fast variables be functions of the slow variables yields a “Slowest invariant manifold,” which, however, is not unequivocally slow. We question whether the complete absence of gravity waves can be logically defined, and we note that the existence or nonexistence of a slow manifold does not depend upon the convergence or nonconvergence of a power series or a succession of approximations.
Abstract
A two-layer quasi-geostrophic beta-plane model is converted into a moist general circulation model by including total water content as an additional prognostic variable. The water-vapor and liquid-water mixing ratios are determined diagnostically from the total-water mixing ratio and the saturation mixing ratio. The underlying surface is ocean, which exchanges water with the atmosphere through evaporation and precipitation. The circulation is driven by solar heating. Thermodynamic and radiative effects of water are included. The model is reduced to a low-order model by expressing each horizontal field in terms of seven orthogonal functions.
When horizontal variations of solar heating are suppressed, there are sometimes two stable steady states—a cold, rather cloudy state and a warm, nearly clear state. A cloud-albedo feedback process appears to be responsible for the multiple equilibria. With variable solar heating the model produces cyclones and anticyclones, with maxima of relative humidity and precipitation ahead of the cyclones, and minima ahead of the anticyclones.
Abstract
A two-layer quasi-geostrophic beta-plane model is converted into a moist general circulation model by including total water content as an additional prognostic variable. The water-vapor and liquid-water mixing ratios are determined diagnostically from the total-water mixing ratio and the saturation mixing ratio. The underlying surface is ocean, which exchanges water with the atmosphere through evaporation and precipitation. The circulation is driven by solar heating. Thermodynamic and radiative effects of water are included. The model is reduced to a low-order model by expressing each horizontal field in terms of seven orthogonal functions.
When horizontal variations of solar heating are suppressed, there are sometimes two stable steady states—a cold, rather cloudy state and a warm, nearly clear state. A cloud-albedo feedback process appears to be responsible for the multiple equilibria. With variable solar heating the model produces cyclones and anticyclones, with maxima of relative humidity and precipitation ahead of the cyclones, and minima ahead of the anticyclones.
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 man's 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.
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 man's 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.
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 space. Points on this manifold are readily found by an algorithm based on the separation of time scales. The attractor set consists of a complex of two-dimensional surfaces embedded in this manifold. The geostrophic equation is a good approximation on most of the attractor, while the balance equation is better. The attractors of the PE and QG models are qualitatively similar.
Some speculations regarding the invariant manifold and the attractor in a large global circulation model are offered.
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 space. Points on this manifold are readily found by an algorithm based on the separation of time scales. The attractor set consists of a complex of two-dimensional surfaces embedded in this manifold. The geostrophic equation is a good approximation on most of the attractor, while the balance equation is better. The attractors of the PE and QG models are qualitatively similar.
Some speculations regarding the invariant manifold and the attractor in a large global circulation model are offered.
