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## Abstract

Three years of twice-daily NMC global operational analyses were projected onto normal mode Rossby waves to produce a climatology of these waves. For zonal wavenumbers. 1 through 4, annual average geopotential amplitudes at 50 kPa are about 5 gpm for the gravest symmetric meridional mode, and 10 and 20 gpm for the next two meridional model although the amplitude for a given time and latitude can greatly exceed the average. Seasonal average amplitudes differ by less than ±25% from the annual average. The modes’ frequencies drift during the course of a year, but this variation is not correlated with season.

Autocorrelations of Rossby wave time series become negligible for lags greater than approximately ten days, which is of the order of the wave period.

For all ten modes examined, geopotential fluctuations exist in both Northern and Southern Hemispheres.

## Abstract

Three years of twice-daily NMC global operational analyses were projected onto normal mode Rossby waves to produce a climatology of these waves. For zonal wavenumbers. 1 through 4, annual average geopotential amplitudes at 50 kPa are about 5 gpm for the gravest symmetric meridional mode, and 10 and 20 gpm for the next two meridional model although the amplitude for a given time and latitude can greatly exceed the average. Seasonal average amplitudes differ by less than ±25% from the annual average. The modes’ frequencies drift during the course of a year, but this variation is not correlated with season.

Autocorrelations of Rossby wave time series become negligible for lags greater than approximately ten days, which is of the order of the wave period.

For all ten modes examined, geopotential fluctuations exist in both Northern and Southern Hemispheres.

## Abstract

This study addresses first the question of what normal-mode global Rossby waves might exist in the Earth's atmosphere. Then it identifies fourteen of these theoretically predicted waves in the NMC global tropospheric analyses.

Normal modes of linearized global primitive shallow water equations were found given a basic state of latitudinally dependent steady zonal flow. The solutions are free Rossby and gravity waves. Many of the waves' north-south structures are similar to Hough functions, which are the solutions of the simpler problem of free waves in an atmosphere at rest.

By projecting 1200 consecutive days of twice-daily NMC global tropospheric analyses of velocity and geopotential onto idealized three-dimensional, normal-mode Rossby wave structures, time series of wave amplitudes and phases were formed. Spectral analyses of these time series for zonal wavenumbers 1–4 revealed statistically significant peaks at eight out of 25 theoretical Rossby wave frequencies. Six additional woes may exist, but their significance could not be statistically supported because their spectral peaks fell into the red noise portion of the spectra. Periods for the fourteen waves lie between ∼2 and ∼30 days. Excluding the two weakest waves, average amplitudes at the surface range from 0.3 to 2 mb.

Prior to this study, only two normal-mode Rossby waves had been identified with confidence in the troposphere, a zonal wavenumber-1, 5-day wave and a zonal wavenumber-1, 16-day way. This study has identified up to ten more modes which have comparable amplitudes.

## Abstract

This study addresses first the question of what normal-mode global Rossby waves might exist in the Earth's atmosphere. Then it identifies fourteen of these theoretically predicted waves in the NMC global tropospheric analyses.

Normal modes of linearized global primitive shallow water equations were found given a basic state of latitudinally dependent steady zonal flow. The solutions are free Rossby and gravity waves. Many of the waves' north-south structures are similar to Hough functions, which are the solutions of the simpler problem of free waves in an atmosphere at rest.

By projecting 1200 consecutive days of twice-daily NMC global tropospheric analyses of velocity and geopotential onto idealized three-dimensional, normal-mode Rossby wave structures, time series of wave amplitudes and phases were formed. Spectral analyses of these time series for zonal wavenumbers 1–4 revealed statistically significant peaks at eight out of 25 theoretical Rossby wave frequencies. Six additional woes may exist, but their significance could not be statistically supported because their spectral peaks fell into the red noise portion of the spectra. Periods for the fourteen waves lie between ∼2 and ∼30 days. Excluding the two weakest waves, average amplitudes at the surface range from 0.3 to 2 mb.

Prior to this study, only two normal-mode Rossby waves had been identified with confidence in the troposphere, a zonal wavenumber-1, 5-day wave and a zonal wavenumber-1, 16-day way. This study has identified up to ten more modes which have comparable amplitudes.

Meteorologists need to be able to manipulate arbitrary dates in the past, present, and future. Here calendar rules for both the Julian (old) and Gregorian (modern) calendars are reviewed. The author describes free software available online to perform calendrical conversions involving (day, month, year), (day of the year, year), and Julian day number for both the Julian and Gregorian calendars.

Meteorologists need to be able to manipulate arbitrary dates in the past, present, and future. Here calendar rules for both the Julian (old) and Gregorian (modern) calendars are reviewed. The author describes free software available online to perform calendrical conversions involving (day, month, year), (day of the year, year), and Julian day number for both the Julian and Gregorian calendars.

## Abstract

The forward model solution and its functional (e.g., the cost function in 4DVAR) are discontinuous with respect to the model's control variables if the model contains discontinuous physical processes that occur during the assimilation window. In such a case, the tangent linear model (the first-order approximation of a finite perturbation) is unable to represent the sharp jumps of the nonlinear model solution. Also, the first-order approximation provided by the adjoint model is unable to represent a finite perturbation of the cost function when the introduced perturbation in the control variables crosses discontinuous points. Using an idealized simple model and the Arakawa–Schubert cumulus parameterization scheme, the authors examined the behavior of a cost function and its gradient obtained by the adjoint model with discontinuous model physics. Numerical results show that a cost function involving discontinuous physical processes is *zeroth-order* discontinuous, but piecewise differentiable. The maximum possible number of involved discontinuity points of a cost function increases exponentially as 2^{kn}, where *k* is the total number of thresholds associated with on–off switches, and *n* is the total number of time steps in the assimilation window. A backward adjoint model integration with the proper forcings added at various time steps, similar to the backward adjoint model integration that provides the gradient of the cost function at a continuous point, produces a one-sided gradient (called a subgradient and denoted as ∇^{s}
*J*) at a discontinuous point. An accuracy check of the gradient shows that the adjoint-calculated gradient is computed exactly on either side of a discontinuous surface. While a cost function evaluated using a small interval in the control variable space oscillates, the distribution of the gradient calculated at the same resolution not only shows a rather smooth variation, but also is consistent with the general convexity of the original cost function. The gradients of discontinuous cost functions are observed roughly smooth since the adjoint integration correctly computes the one-sided gradient at either side of discontinuous surface. This implies that, although (∇^{s}
*J*)^{T}
*δ*
**x** may not approximate *δJ* = *J*(**x** + *δ*
**x**) − *J*(**x**) well near the discontinuous surface, the subgradient calculated by the adjoint of discontinuous physics may still provide useful information for finding the search directions in a minimization procedure. While not eliminating the possible need for the use of a nondifferentiable optimization algorithm for 4DVAR with discontinuous physics, consistency between the computed gradient by adjoints and the convexity of the cost function may explain why a differentiable limited-memory quasi-Newton algorithm still worked well in many 4DVAR experiments that use a diabatic assimilation model with discontinuous physics.

## Abstract

The forward model solution and its functional (e.g., the cost function in 4DVAR) are discontinuous with respect to the model's control variables if the model contains discontinuous physical processes that occur during the assimilation window. In such a case, the tangent linear model (the first-order approximation of a finite perturbation) is unable to represent the sharp jumps of the nonlinear model solution. Also, the first-order approximation provided by the adjoint model is unable to represent a finite perturbation of the cost function when the introduced perturbation in the control variables crosses discontinuous points. Using an idealized simple model and the Arakawa–Schubert cumulus parameterization scheme, the authors examined the behavior of a cost function and its gradient obtained by the adjoint model with discontinuous model physics. Numerical results show that a cost function involving discontinuous physical processes is *zeroth-order* discontinuous, but piecewise differentiable. The maximum possible number of involved discontinuity points of a cost function increases exponentially as 2^{kn}, where *k* is the total number of thresholds associated with on–off switches, and *n* is the total number of time steps in the assimilation window. A backward adjoint model integration with the proper forcings added at various time steps, similar to the backward adjoint model integration that provides the gradient of the cost function at a continuous point, produces a one-sided gradient (called a subgradient and denoted as ∇^{s}
*J*) at a discontinuous point. An accuracy check of the gradient shows that the adjoint-calculated gradient is computed exactly on either side of a discontinuous surface. While a cost function evaluated using a small interval in the control variable space oscillates, the distribution of the gradient calculated at the same resolution not only shows a rather smooth variation, but also is consistent with the general convexity of the original cost function. The gradients of discontinuous cost functions are observed roughly smooth since the adjoint integration correctly computes the one-sided gradient at either side of discontinuous surface. This implies that, although (∇^{s}
*J*)^{T}
*δ*
**x** may not approximate *δJ* = *J*(**x** + *δ*
**x**) − *J*(**x**) well near the discontinuous surface, the subgradient calculated by the adjoint of discontinuous physics may still provide useful information for finding the search directions in a minimization procedure. While not eliminating the possible need for the use of a nondifferentiable optimization algorithm for 4DVAR with discontinuous physics, consistency between the computed gradient by adjoints and the convexity of the cost function may explain why a differentiable limited-memory quasi-Newton algorithm still worked well in many 4DVAR experiments that use a diabatic assimilation model with discontinuous physics.

## Abstract

An ensemble forecast is a collection (an ensemble) of forecasts that all verify at the same time. These forecasts are regarded as possible scenarios given the uncertainty associated with forecasting. With such an ensemble, one can address issues that go beyond simply estimating the best forecast. These include estimation of the probability of various events and estimation of the confidence that can be associated with a forecast.

Global ensemble forecasts out to 10 days have been computed at both the U.S. and European central forecasting centers since December 1992. Since 1995, the United States has computed experimental regional ensemble forecasts focusing on smaller-scale forecast uncertainties out to 2 days.

The authors address challenges associated with ensemble forecasting such as 1) formulating an ensemble, 2) choosing the number of forecasts in an ensemble, 3) extracting information from an ensemble of forecasts, 4) displaying information from an ensemble of forecasts, and 5) interpreting ensemble forecasts. Two synoptic- scale examples of ensemble forecasting from the winter of 1995/96 are also shown.

## Abstract

An ensemble forecast is a collection (an ensemble) of forecasts that all verify at the same time. These forecasts are regarded as possible scenarios given the uncertainty associated with forecasting. With such an ensemble, one can address issues that go beyond simply estimating the best forecast. These include estimation of the probability of various events and estimation of the confidence that can be associated with a forecast.

Global ensemble forecasts out to 10 days have been computed at both the U.S. and European central forecasting centers since December 1992. Since 1995, the United States has computed experimental regional ensemble forecasts focusing on smaller-scale forecast uncertainties out to 2 days.

The authors address challenges associated with ensemble forecasting such as 1) formulating an ensemble, 2) choosing the number of forecasts in an ensemble, 3) extracting information from an ensemble of forecasts, 4) displaying information from an ensemble of forecasts, and 5) interpreting ensemble forecasts. Two synoptic- scale examples of ensemble forecasting from the winter of 1995/96 are also shown.

## Abstract

This paper addresses the anomaly correlation of the 500-hPa geopotential heights from a suite of global multimodels and from a model-weighted ensemble mean called the superensemble. This procedure follows a number of current studies on weather and seasonal climate forecasting that are being pursued. This study includes a slightly different procedure from that used in other current experimental forecasts for other variables. Here a superensemble for the ∇^{2} of the geopotential based on the daily forecasts of the geopotential fields at the 500-hPa level is constructed. The geopotential of the superensemble is recovered from the solution of the Poisson equation. This procedure appears to improve the skill for those scales where the variance of the geopotential is large and contributes to a marked improvement in the skill of the anomaly correlation. Especially large improvements over the Southern Hemisphere are noted. Consistent day-6 forecast skill above 0.80 is achieved on a day to day basis. The superensemble skills are higher than those of the best model and the ensemble mean. For days 1–6 the percent improvement in anomaly correlations of the superensemble over the best model are 0.3, 0.8, 2.25, 4.75, 8.6, and 14.6, respectively, for the Northern Hemisphere. The corresponding numbers for the Southern Hemisphere are 1.12, 1.66, 2.69, 4.48, 7.11, and 12.17. Major improvement of anomaly correlation skills is realized by the superensemble at days 5 and 6 of forecasts. The collective regional strengths of the member models, which is reflected in the proposed superensemble, provide a useful consensus product that may be useful for future operational guidance.

## Abstract

This paper addresses the anomaly correlation of the 500-hPa geopotential heights from a suite of global multimodels and from a model-weighted ensemble mean called the superensemble. This procedure follows a number of current studies on weather and seasonal climate forecasting that are being pursued. This study includes a slightly different procedure from that used in other current experimental forecasts for other variables. Here a superensemble for the ∇^{2} of the geopotential based on the daily forecasts of the geopotential fields at the 500-hPa level is constructed. The geopotential of the superensemble is recovered from the solution of the Poisson equation. This procedure appears to improve the skill for those scales where the variance of the geopotential is large and contributes to a marked improvement in the skill of the anomaly correlation. Especially large improvements over the Southern Hemisphere are noted. Consistent day-6 forecast skill above 0.80 is achieved on a day to day basis. The superensemble skills are higher than those of the best model and the ensemble mean. For days 1–6 the percent improvement in anomaly correlations of the superensemble over the best model are 0.3, 0.8, 2.25, 4.75, 8.6, and 14.6, respectively, for the Northern Hemisphere. The corresponding numbers for the Southern Hemisphere are 1.12, 1.66, 2.69, 4.48, 7.11, and 12.17. Major improvement of anomaly correlation skills is realized by the superensemble at days 5 and 6 of forecasts. The collective regional strengths of the member models, which is reflected in the proposed superensemble, provide a useful consensus product that may be useful for future operational guidance.