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## Abstract

Results from the European Centre for Medium-Range Weather Forecasts (ECMWF) high-resolution (T213L31) global analysis-forecast system are analysed as a function of scale. Three regimes of forecast skill are identified, only one of which exhibits classical predictability behavior in which error, initially concentrated at smaller scales, penetrates up the spectrum and saturates at values roughly twice the observed variance. Two of the regimes are seen at spectral ranges previously accessible in global analyses and forecasts (roughly for spherical harmonic wavenumbers *n* < 80). The large-scale low-wavenumber regime (*n* < 10) is dominated by stationary (largely zonal) structures that are relatively uncontaminated by error up to the 10-day forecast limit. These structures represent “climatological” aspects of the flow, and their simulation is not considered to represent useful forecast skill. The intermediate wavenumber range (10 < *n* < 80) exhibits classical predictability behavior. A third regime at high wavenumbers (*n* > 100) unexpectedly, and in contrast to the classical predictability regime, exhibits forecast skill up to the 10-day forecast limit. The apparent enhancement of predictability at these small scales is due to local topographic forcing in the ECMWF analysis-forecast system.

## Abstract

Results from the European Centre for Medium-Range Weather Forecasts (ECMWF) high-resolution (T213L31) global analysis-forecast system are analysed as a function of scale. Three regimes of forecast skill are identified, only one of which exhibits classical predictability behavior in which error, initially concentrated at smaller scales, penetrates up the spectrum and saturates at values roughly twice the observed variance. Two of the regimes are seen at spectral ranges previously accessible in global analyses and forecasts (roughly for spherical harmonic wavenumbers *n* < 80). The large-scale low-wavenumber regime (*n* < 10) is dominated by stationary (largely zonal) structures that are relatively uncontaminated by error up to the 10-day forecast limit. These structures represent “climatological” aspects of the flow, and their simulation is not considered to represent useful forecast skill. The intermediate wavenumber range (10 < *n* < 80) exhibits classical predictability behavior. A third regime at high wavenumbers (*n* > 100) unexpectedly, and in contrast to the classical predictability regime, exhibits forecast skill up to the 10-day forecast limit. The apparent enhancement of predictability at these small scales is due to local topographic forcing in the ECMWF analysis-forecast system.

## Abstract

The scale-dependent behavior of atmospheric flow on the sphere is investigated in terms of the spectra and spectral budgets of enstrophy and kinetic and available potential energy. The decomposition into both the *α* = (*n, m*) and the more usual *n*-spectral forms is considered. Several novel spectral results are obtained. First, a *direct* test of the extent to which large-scale atmospheric flow satisfies the necessary and sufficient conditions for homogeneous and isotropic two-dimensional turbulent behavior is carried out by calculating “spectral teleconnections” from data. The conditions are satisfied for the higher wavenumber transient component of the flow. Second, the spectral budget equations are extended to include the decomposition into time mean and transient components, a method that, although common and straightforward in real-space calculations, is novel in the spectral domain. The terms in these budgets are evaluated from data. Finally, the flow of energy and enstrophy through α-spectral space is displayed in terms of a novel “spectral potential function,” which is related to spectral fluxes.

The results give information about the scales and wavenumbers at which sources and sinks of the mean and transient components of energy and enstrophy are found and how these quantities are transferred between scales and converted from one form to another. The interaction between the high wavenumber homogeneous and isotropic transient components and the low wavenumber inhomogeneous and nonisotropic mean components is seen to be an essential aspect of the spectral budgets.

## Abstract

The scale-dependent behavior of atmospheric flow on the sphere is investigated in terms of the spectra and spectral budgets of enstrophy and kinetic and available potential energy. The decomposition into both the *α* = (*n, m*) and the more usual *n*-spectral forms is considered. Several novel spectral results are obtained. First, a *direct* test of the extent to which large-scale atmospheric flow satisfies the necessary and sufficient conditions for homogeneous and isotropic two-dimensional turbulent behavior is carried out by calculating “spectral teleconnections” from data. The conditions are satisfied for the higher wavenumber transient component of the flow. Second, the spectral budget equations are extended to include the decomposition into time mean and transient components, a method that, although common and straightforward in real-space calculations, is novel in the spectral domain. The terms in these budgets are evaluated from data. Finally, the flow of energy and enstrophy through α-spectral space is displayed in terms of a novel “spectral potential function,” which is related to spectral fluxes.

The results give information about the scales and wavenumbers at which sources and sinks of the mean and transient components of energy and enstrophy are found and how these quantities are transferred between scales and converted from one form to another. The interaction between the high wavenumber homogeneous and isotropic transient components and the low wavenumber inhomogeneous and nonisotropic mean components is seen to be an essential aspect of the spectral budgets.

## Abstract

The appropriateness of using simplified versions of the equations of motion to explain the response of the atmosphere to external forcing, such as that associated with the 1982/83 El Niño, is considered. In particular, the terms in the baroclinic equations linearized about a zonal basic state or about the three-dimensional mean climatic state are evaluated, and the appropriateness of the linearization is considered with reference to the results from both the observations and a simulation with a general circulation model. A similar analysis is undertaken of the terms of the barotropic vorticity equation at 200 mb. Considerations of the specification or parameterization of the source/sink term in the simplified equations are also discussed.

Generally it is found that the equations which arise when linearizing about a zonal basic state are clearly unsuitable in this case. The neglected terms are of the same order as those retained and the balances of energy and vorticity which occur in the general circulation model and the observations are not those of the linearized equations. The linearization about the three-dimensional mean state in the case of the baroclinic equations is considerably less in error. The linearization can be carried out successfully in the tropical region of direct response to the El Niño forcing but it is not successful in the case of the remote extratropical response to the tropical forcing. In the extratropics, the nonlinear transient terms cannot be neglected compared to the linear terms even when the linearizations are about the three-dimensional mean state. For the barotropic vorticity equation, linearization is generally unsuccessful and here the nonlinear mean as well as the non-linear transient terms cannot be neglected.

Finally, the correct specification of the sources and sinks for the simplified equations is an area of difficulty since, in the case of the response to the El Niño, both localized anomalous source and sink regions are found while the usual approach is to specify or parameterize a localized source but a distributed sink. Apparently an “anomaly” physics would be required in which both localized sources and sinks of energy and vorticity would be specified or parameterized.

## Abstract

The appropriateness of using simplified versions of the equations of motion to explain the response of the atmosphere to external forcing, such as that associated with the 1982/83 El Niño, is considered. In particular, the terms in the baroclinic equations linearized about a zonal basic state or about the three-dimensional mean climatic state are evaluated, and the appropriateness of the linearization is considered with reference to the results from both the observations and a simulation with a general circulation model. A similar analysis is undertaken of the terms of the barotropic vorticity equation at 200 mb. Considerations of the specification or parameterization of the source/sink term in the simplified equations are also discussed.

Generally it is found that the equations which arise when linearizing about a zonal basic state are clearly unsuitable in this case. The neglected terms are of the same order as those retained and the balances of energy and vorticity which occur in the general circulation model and the observations are not those of the linearized equations. The linearization about the three-dimensional mean state in the case of the baroclinic equations is considerably less in error. The linearization can be carried out successfully in the tropical region of direct response to the El Niño forcing but it is not successful in the case of the remote extratropical response to the tropical forcing. In the extratropics, the nonlinear transient terms cannot be neglected compared to the linear terms even when the linearizations are about the three-dimensional mean state. For the barotropic vorticity equation, linearization is generally unsuccessful and here the nonlinear mean as well as the non-linear transient terms cannot be neglected.

Finally, the correct specification of the sources and sinks for the simplified equations is an area of difficulty since, in the case of the response to the El Niño, both localized anomalous source and sink regions are found while the usual approach is to specify or parameterize a localized source but a distributed sink. Apparently an “anomaly” physics would be required in which both localized sources and sinks of energy and vorticity would be specified or parameterized.

## Abstract

Vertically integrated budgets of mass and energy for the global atmosphere, based on the ECMWF and GFDL FGGE III-b datasets for July 1979, have been obtained and compared with one another and with the results of a general circulation model. The rotational component of the mass and energy fluxes are reasonably well-measured and differences between the two FGGE analyses are an order of magnitude smaller than the values of the quantities themselves.

This is not the case for the divergent component of the mass and energy fluxes where the difference between analyses is of the same order of magnitude as the fields themselves. Since it is the divergence of this component of the energy flux that gives the distribution of energy sources and sinks, this quantity also differs markedly between analyses.

The difference in the calculated divergent energy fluxes, and the sources and sinks is primarily due to the difference in the time–mean component of the energy flux. The transient component of the flux is in comparatively good agreement.

The energy fluxes and source/sink distribution simulated by the CCC GCM agree, at least qualitatively, with those of the observations for the rotational component of the energy flux. For the divergent component, the model results agree most closely with those of the ECMWF analysis, which are weaker than those of the GFDL analysis.

If the differences in energy fluxes and energy sources and sinks between these two FGGE analyses are indicative of the uncertainties in these quantities, it would appear that it is not yet possible to infer oceanic transports from the knowledge of atmospheric transports and the radiative balance at the top of the atmosphere.

## Abstract

Vertically integrated budgets of mass and energy for the global atmosphere, based on the ECMWF and GFDL FGGE III-b datasets for July 1979, have been obtained and compared with one another and with the results of a general circulation model. The rotational component of the mass and energy fluxes are reasonably well-measured and differences between the two FGGE analyses are an order of magnitude smaller than the values of the quantities themselves.

This is not the case for the divergent component of the mass and energy fluxes where the difference between analyses is of the same order of magnitude as the fields themselves. Since it is the divergence of this component of the energy flux that gives the distribution of energy sources and sinks, this quantity also differs markedly between analyses.

The difference in the calculated divergent energy fluxes, and the sources and sinks is primarily due to the difference in the time–mean component of the energy flux. The transient component of the flux is in comparatively good agreement.

The energy fluxes and source/sink distribution simulated by the CCC GCM agree, at least qualitatively, with those of the observations for the rotational component of the energy flux. For the divergent component, the model results agree most closely with those of the ECMWF analysis, which are weaker than those of the GFDL analysis.

If the differences in energy fluxes and energy sources and sinks between these two FGGE analyses are indicative of the uncertainties in these quantities, it would appear that it is not yet possible to infer oceanic transports from the knowledge of atmospheric transports and the radiative balance at the top of the atmosphere.

## Abstract

A spectral analysis of error at 24, 48 and 72 hours in an operational forecast system is carded out for a winter and a summer month. The scale-dependent behavior of the error and the several factors governing error growth, namely the nonlinear production term, the nonlinear transfer of energy between scales and the model-related error source term are evaluated.

It is found that the largest errors occur at low wavenumbers where the amplitude of the flow is largest, while the largest relative errors occur at high wavenumbers when error is approaching saturation. Early in the forecast the growth of error is governed by the model error source term which supplies error at all scales. At later forecast times the nonlinear production and interaction terms become important. Error doubling times and the flux of error through wavenumber space are also calculated.

The predictability of the flow resides in the low wavenumber region. The behavior of the error and its transient and stationary parts markedly with season at these scales.

## Abstract

A spectral analysis of error at 24, 48 and 72 hours in an operational forecast system is carded out for a winter and a summer month. The scale-dependent behavior of the error and the several factors governing error growth, namely the nonlinear production term, the nonlinear transfer of energy between scales and the model-related error source term are evaluated.

It is found that the largest errors occur at low wavenumbers where the amplitude of the flow is largest, while the largest relative errors occur at high wavenumbers when error is approaching saturation. Early in the forecast the growth of error is governed by the model error source term which supplies error at all scales. At later forecast times the nonlinear production and interaction terms become important. Error doubling times and the flux of error through wavenumber space are also calculated.

The predictability of the flow resides in the low wavenumber region. The behavior of the error and its transient and stationary parts markedly with season at these scales.

## Abstract

The assumption that the streamfunction for two-dimensional nondivergent flow on the sphere is a homogeneous and isotropic random field is used to obtain a variety of results for the study of large-scale atmospheric turbulence. These results differ somewhat from those for Cartesian geometry.

It is shown that the necessary and sufficient condition that the turbulent flow be homogeneous and isotropic is the statistical independence of the spherical harmonic expansion coefficients of the streamfunction and the dependence of the variance of the expansion coefficients only an the wavenumber *n*. The homogeneity and isotropy of the vorticity field and of the velocities along and perpendicular to the geodesic connecting points on the sphere follows. The usual zonal and meridional velocity components on the sphere are an-isotropic although, at a point, these velocity components are uncorrelated and have equal amounts of kinetic energy.

Equations for the covariance function and the spectrum for flow on the sphere am obtained from the barotropic vorticity equation.

## Abstract

The assumption that the streamfunction for two-dimensional nondivergent flow on the sphere is a homogeneous and isotropic random field is used to obtain a variety of results for the study of large-scale atmospheric turbulence. These results differ somewhat from those for Cartesian geometry.

It is shown that the necessary and sufficient condition that the turbulent flow be homogeneous and isotropic is the statistical independence of the spherical harmonic expansion coefficients of the streamfunction and the dependence of the variance of the expansion coefficients only an the wavenumber *n*. The homogeneity and isotropy of the vorticity field and of the velocities along and perpendicular to the geodesic connecting points on the sphere follows. The usual zonal and meridional velocity components on the sphere are an-isotropic although, at a point, these velocity components are uncorrelated and have equal amounts of kinetic energy.

Equations for the covariance function and the spectrum for flow on the sphere am obtained from the barotropic vorticity equation.

## Abstract

Diagnostic studies of the general circulation are conducted, almost exclusively, using data and equations where pressure is the vertical coordinate. The complexity of the lower boundary condition in these isobaric coordinates may lead to some ambiguity in the manner of calculating and displaying the pertinent diagnostic statistics and budget terms.

A formalism is developed using the unit and delta functions which permits an unambiguous calculation of the appropriate diagnostic terms. Using this formalism, budget equations are derived which display a number of unfamiliar boundary terms. The more familiar budget equations are seen to be approximations to these more complete equations.

Examples of some basic general circulation statistics involving temperature are used to illustrate the application of the formalism.

## Abstract

Diagnostic studies of the general circulation are conducted, almost exclusively, using data and equations where pressure is the vertical coordinate. The complexity of the lower boundary condition in these isobaric coordinates may lead to some ambiguity in the manner of calculating and displaying the pertinent diagnostic statistics and budget terms.

A formalism is developed using the unit and delta functions which permits an unambiguous calculation of the appropriate diagnostic terms. Using this formalism, budget equations are derived which display a number of unfamiliar boundary terms. The more familiar budget equations are seen to be approximations to these more complete equations.

Examples of some basic general circulation statistics involving temperature are used to illustrate the application of the formalism.

## Abstract

A dynamical extended-range forecast consisting of a series of six monthly forecasts for each of the eight Januarys from 1979 to 1986 is carried out with the Canadian Climate Centre low-resolution general circulation model. Results, in terms of the 500-rnb height field, are presented for the mean January systematic and random error and for the skill of the January mean forecast. The evolution of error as a function of time and spatial scale is investigated and the equations governing the growth of systematic and random error are derived and evaluated.

The mean January systematic error in the 500-mb height field is modest and is largely in other than the zonal structures. Systematic-error variance is about an order of magnitude smaller than random-error variance and is concentrated in the larger scales of the flow. The January mean anomaly correlation indicates marginal forecast skill, as well as some connection between the spread of the forecasts and the skill of the average forecast.

The budget equation for random error indicates that the interaction between it and the systematic error is small so that, for this model at least, the removal of systematic error by subtraction is plausible. The nonbarotropic source-sink term dominates random-error growth early in the forecast, while nonlinear barotropic generation does so at later times. The growth of the (smaller) systematic-error variance is importantly affected by its interaction with random error at early forecast times and also by nonlinear barotropic generation at later times. The relative sizes of the nonlinear barotropic generation and baroclinic source-sink terms, together with the interactions between the two forms of error, may reveal differences between model behavior and suggest areas of improvement.

## Abstract

A dynamical extended-range forecast consisting of a series of six monthly forecasts for each of the eight Januarys from 1979 to 1986 is carried out with the Canadian Climate Centre low-resolution general circulation model. Results, in terms of the 500-rnb height field, are presented for the mean January systematic and random error and for the skill of the January mean forecast. The evolution of error as a function of time and spatial scale is investigated and the equations governing the growth of systematic and random error are derived and evaluated.

The mean January systematic error in the 500-mb height field is modest and is largely in other than the zonal structures. Systematic-error variance is about an order of magnitude smaller than random-error variance and is concentrated in the larger scales of the flow. The January mean anomaly correlation indicates marginal forecast skill, as well as some connection between the spread of the forecasts and the skill of the average forecast.

The budget equation for random error indicates that the interaction between it and the systematic error is small so that, for this model at least, the removal of systematic error by subtraction is plausible. The nonbarotropic source-sink term dominates random-error growth early in the forecast, while nonlinear barotropic generation does so at later times. The growth of the (smaller) systematic-error variance is importantly affected by its interaction with random error at early forecast times and also by nonlinear barotropic generation at later times. The relative sizes of the nonlinear barotropic generation and baroclinic source-sink terms, together with the interactions between the two forms of error, may reveal differences between model behavior and suggest areas of improvement.

## Abstract

The cyclonic vorticity budget of the hemispheres is investigated using the output of the CCC GCM and objective analyses for the FGGE period. Analysis of GCM output gives direct estimates of the “topographic” and “frictional” vorticity sources and sinks, which are generally otherwise unavailable. In a broad sense, vorticity flows from source regions associated with high mean sea level pressure to sink regions associated with low mean sea level pressure.

Direct estimates of the topographic source/sink term show a strong vorticity source associated with drainage flow from the elevated cold Antarctic plateau in Southern Hemisphere winter and, the reverse situation, a strong vorticity source associated with upward motion over the elevated warm Tibetan plateau in Northern Hemisphere summer.

Direct estimates of the frictional source/sink term show strongest values over oceans in winter and over portions of the hot continental land masses of the Northern Hemisphere in summer. In both cases the instability of the lower layers of the atmosphere means that there is a good frictional connection between the surface and the free atmosphere. The seasonality of this behavior is especially notable over the southern oceans.

From the zonally averaged point-of-view there is an interesting duality in the vorticity budget of the Southern Hemisphere. An almost complete cancellation occurs between the vorticity flux associated with the upper branches of the meridional circulation and that associated with the transient eddies. The vorticity transport is thus accomplished by the lower branches of the meridional circulation. When the budget terms are vertically integrated, however, the vorticity flux associated with the upper and lower branches of the meridional circulation tends to cancel so that the result is completely dominated by the transient eddy term.

## Abstract

The cyclonic vorticity budget of the hemispheres is investigated using the output of the CCC GCM and objective analyses for the FGGE period. Analysis of GCM output gives direct estimates of the “topographic” and “frictional” vorticity sources and sinks, which are generally otherwise unavailable. In a broad sense, vorticity flows from source regions associated with high mean sea level pressure to sink regions associated with low mean sea level pressure.

Direct estimates of the topographic source/sink term show a strong vorticity source associated with drainage flow from the elevated cold Antarctic plateau in Southern Hemisphere winter and, the reverse situation, a strong vorticity source associated with upward motion over the elevated warm Tibetan plateau in Northern Hemisphere summer.

Direct estimates of the frictional source/sink term show strongest values over oceans in winter and over portions of the hot continental land masses of the Northern Hemisphere in summer. In both cases the instability of the lower layers of the atmosphere means that there is a good frictional connection between the surface and the free atmosphere. The seasonality of this behavior is especially notable over the southern oceans.

From the zonally averaged point-of-view there is an interesting duality in the vorticity budget of the Southern Hemisphere. An almost complete cancellation occurs between the vorticity flux associated with the upper branches of the meridional circulation and that associated with the transient eddies. The vorticity transport is thus accomplished by the lower branches of the meridional circulation. When the budget terms are vertically integrated, however, the vorticity flux associated with the upper and lower branches of the meridional circulation tends to cancel so that the result is completely dominated by the transient eddy term.

## Abstract

Analyzed and forecast 200-mb velocity potential statistics from the ECMWF analysis-forecast cycle for the December–February and June–August seasons of the years 1986–91 are considered. The forecast skill of this representation of the tropical divergent flow is analyzed as is its connection with the skill of forecasting both the tropical and extratropical rotational flow. The mean structure of the velocity potential is concentrated in the tropical region as is its transient variability, which is dominated by the large-scale wavenumber 1 component. Hovmöller diagrams of the velocity potential averaged over the equatorial region show episodic propagating features identified as the Madden-Julian oscillation (MJO) in both analyses and forecasts.

Local deterministic forecasts of velocity potential show a comparatively rapid decrease of skill in the tropical region. There is a general loss of transient variance with forecast range. The forecast skill of tropical velocity potential decreases more rapidly with forecast range than does that of tropical or extratropical streamfunction. Tropical velocity potential forecasts are more skillful in December–February than in June–August and levels of skill differ from year to year. There is little evidence of a dependence of forecast skill on the state of the MJO or the E1 Niño/Southern Oscillation (ENSO). There is some evidence of a weak connection between the skill of the forecast of the tropical divergent component and that of rotational component in both the Tropics and extratropics. This connection is apparently independent of forecast range and also of the state of the MJO and ENSO.

## Abstract

Analyzed and forecast 200-mb velocity potential statistics from the ECMWF analysis-forecast cycle for the December–February and June–August seasons of the years 1986–91 are considered. The forecast skill of this representation of the tropical divergent flow is analyzed as is its connection with the skill of forecasting both the tropical and extratropical rotational flow. The mean structure of the velocity potential is concentrated in the tropical region as is its transient variability, which is dominated by the large-scale wavenumber 1 component. Hovmöller diagrams of the velocity potential averaged over the equatorial region show episodic propagating features identified as the Madden-Julian oscillation (MJO) in both analyses and forecasts.

Local deterministic forecasts of velocity potential show a comparatively rapid decrease of skill in the tropical region. There is a general loss of transient variance with forecast range. The forecast skill of tropical velocity potential decreases more rapidly with forecast range than does that of tropical or extratropical streamfunction. Tropical velocity potential forecasts are more skillful in December–February than in June–August and levels of skill differ from year to year. There is little evidence of a dependence of forecast skill on the state of the MJO or the E1 Niño/Southern Oscillation (ENSO). There is some evidence of a weak connection between the skill of the forecast of the tropical divergent component and that of rotational component in both the Tropics and extratropics. This connection is apparently independent of forecast range and also of the state of the MJO and ENSO.