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## Abstract

The efficacy of the nonlinear initialization technique for use on global-scale numerical models is tested using a normal-mode, spectral model of the shallow-water equations on an equatorial beta-plane. Despite the nonexistence of strong, frequency separation for the ultralong, equatorially trapped modes, test integrations show that the nonlinear initialization scheme acts to smooth the most rapid oscillations in the system. Further integrations involving only spectral components associated with low-frequency, rotational modes show that the rotational mode trajectories are nearly unaffected by the presence of the balanced gravitational modes. The likely distortion of the divergence field obtained from a rotational-mode-only calculation makes this filtering-through-truncation technique appear unattractive, so an alternative scheme which uses both the truncation and nonlinear initialization schemes is proposed.

## Abstract

The efficacy of the nonlinear initialization technique for use on global-scale numerical models is tested using a normal-mode, spectral model of the shallow-water equations on an equatorial beta-plane. Despite the nonexistence of strong, frequency separation for the ultralong, equatorially trapped modes, test integrations show that the nonlinear initialization scheme acts to smooth the most rapid oscillations in the system. Further integrations involving only spectral components associated with low-frequency, rotational modes show that the rotational mode trajectories are nearly unaffected by the presence of the balanced gravitational modes. The likely distortion of the divergence field obtained from a rotational-mode-only calculation makes this filtering-through-truncation technique appear unattractive, so an alternative scheme which uses both the truncation and nonlinear initialization schemes is proposed.

## Abstract

An algorithm for obtaining high-order mode initialization of the type first proposed by Baer and Tribbia is developed which is free from the major difficulty of previous methodsâ€”the necessity of calculating Frechet derivatives of the nonlinear terms. This new method is shown to be a logical extension of the technique proposed by Machenhauer; thus the asymptotic equivalence of the Machenhauer and Baer-Tribbia initialization methods is accomplished. A comparison between the new algorithm and the older method of calculating second-order initialization demonstrates the accuracy and ease of implementation of the new technique.

## Abstract

An algorithm for obtaining high-order mode initialization of the type first proposed by Baer and Tribbia is developed which is free from the major difficulty of previous methodsâ€”the necessity of calculating Frechet derivatives of the nonlinear terms. This new method is shown to be a logical extension of the technique proposed by Machenhauer; thus the asymptotic equivalence of the Machenhauer and Baer-Tribbia initialization methods is accomplished. A comparison between the new algorithm and the older method of calculating second-order initialization demonstrates the accuracy and ease of implementation of the new technique.

## Abstract

Using a low-order, spectral, shallow-water model on an *f*-plane, the conditions under which height-constrained nonlinear normal mode initialization fails and the existence of realizable balancing wind fields are examined. The relationship of this nonrealizability condition and the ellipticity condition for the standard nonlinear balance equation is also examined. A conclusion from this analysis is that non-elliptic geopotential regions must be accompanied by transient gravity wave motion if there is no forcing mechanism.

The low-order results are extended through the use of a global shallow-water model. The relationship between the local *f*-plane results and the global results is analyzed and a strong correlation between the appearance of non-elliptic geopotential regions and the breakdown of the iteration scheme used in non-linear normal mode balancing is noted.

It is concluded that moderately weak anticyclonic disturbances in equatorial areas may act as regions of energy exchange between rotational and gravitational modes. Also, the climatological existence of these regions implies the necessity forcing to maintain them in the atmosphere and numerical forecast models.

## Abstract

Using a low-order, spectral, shallow-water model on an *f*-plane, the conditions under which height-constrained nonlinear normal mode initialization fails and the existence of realizable balancing wind fields are examined. The relationship of this nonrealizability condition and the ellipticity condition for the standard nonlinear balance equation is also examined. A conclusion from this analysis is that non-elliptic geopotential regions must be accompanied by transient gravity wave motion if there is no forcing mechanism.

The low-order results are extended through the use of a global shallow-water model. The relationship between the local *f*-plane results and the global results is analyzed and a strong correlation between the appearance of non-elliptic geopotential regions and the breakdown of the iteration scheme used in non-linear normal mode balancing is noted.

It is concluded that moderately weak anticyclonic disturbances in equatorial areas may act as regions of energy exchange between rotational and gravitational modes. Also, the climatological existence of these regions implies the necessity forcing to maintain them in the atmosphere and numerical forecast models.

## Abstract

The variational problem of initial data specification from observations with the strong constraint of the elimination of transient gravity waves through nonlinear normal mode balancing is reconsidered. The exact formulation of this problem is contrasted to the approximate formulation previously given by Daley.

Through the judicious use of model normal modes, an alternative algorithm is developed which allows the use of confidence weights which reflect the fidelity of observations in a more realistic manner than previously possible. In particular, longitudinally varying confidence weights can be utilized. Examples of the use of this technique using the Machenhauer and the second-order Baer-Tribbia initialization are given. An attempt to ascertain the validity of Daley's method demonstrates the accuracy of this approximation and justifies its continued use.

## Abstract

The variational problem of initial data specification from observations with the strong constraint of the elimination of transient gravity waves through nonlinear normal mode balancing is reconsidered. The exact formulation of this problem is contrasted to the approximate formulation previously given by Daley.

Through the judicious use of model normal modes, an alternative algorithm is developed which allows the use of confidence weights which reflect the fidelity of observations in a more realistic manner than previously possible. In particular, longitudinally varying confidence weights can be utilized. Examples of the use of this technique using the Machenhauer and the second-order Baer-Tribbia initialization are given. An attempt to ascertain the validity of Daley's method demonstrates the accuracy of this approximation and justifies its continued use.

## Abstract

It was shown by Oliger and SundstrĂ¶m, in 1978, that the initial boundary value problems for the hydrostatic primitive equations of meteorology and oceanography are ill posed if the boundaries are open and fixed in space. In this article it is shown, with theory and computation, that the same problems are well posed for a suitable set of local (pointwise applied) boundary conditions, if a mild vertical viscosity is added to the hydrostatic equation. Some indications on the behavior of the solutions, as the vertical viscosity parameter goes to zero, are also given. The Boussinesq equations are shown to be well posed in the same context of boundaries open and fixed in space. Finally, numerical simulations supporting the analysis are included.

## Abstract

It was shown by Oliger and SundstrĂ¶m, in 1978, that the initial boundary value problems for the hydrostatic primitive equations of meteorology and oceanography are ill posed if the boundaries are open and fixed in space. In this article it is shown, with theory and computation, that the same problems are well posed for a suitable set of local (pointwise applied) boundary conditions, if a mild vertical viscosity is added to the hydrostatic equation. Some indications on the behavior of the solutions, as the vertical viscosity parameter goes to zero, are also given. The Boussinesq equations are shown to be well posed in the same context of boundaries open and fixed in space. Finally, numerical simulations supporting the analysis are included.

## Abstract

Optimal perturbations, also referred to as singular vectors (SVs), currently constitute an important guideline for the generation of initial ensembles to be used for ensemble prediction. The optimality of these perturbations refers to their property of maximizing prespecified quadratic measures of error growth, given that tangent-linear error evolution is assumed. The goal of ensemble prediction is the accurate prediction of the uncertainty of forecasts made with dynamical numerical weather prediction models.

In the present paper the theoretical justification for the use of SVs in ensemble prediction systems is investigated. It is shown that, in a tangent-linear framework, SVsâ€”constructed using covariance information valid at the initial timeâ€”evolve into the eigenvectors of the forecast error covariance matrix valid for the end of the optimization interval. As such, SVs represent the most efficient means for predicting the forecast error covariance matrix, given a prespecified number of allowable (tangent-linear) model integrations. Such optimal prediction is of particular importance in light of the fact that the forecast error covariance matrix is summarizing important information about the probability density function of the model state at a given future time.

Based on the above result, optimal covariance prediction through appropriately determined SVs is demonstrated here for a three-dimensional Lorenz model, as well as for a barotropic model of intermediate dimensionality, both within a perfect-model framework. In the case of the barotropic model it is found that less than 15% of the SVs suffice to account for more than 95% of the total final error variance. Viewed differently, at least 80% of the final error variance is accounted for by retaining those SVs that are amplifying in terms of an enstrophy norm. In addition, variances and covariances predicted through SVs agree closely with independently obtained Monte Carlo estimates, as long as the tangent-linear approximation is sufficiently accurate.

Further, the problem of approximating the forecast error covariance matrix in the presence of a state-independent model-error representation is briefly considered. The paper is concluded with a summary of the results and a discussion of their possible implications on data assimilation procedures and on the further development of ensemble prediction systems.

## Abstract

Optimal perturbations, also referred to as singular vectors (SVs), currently constitute an important guideline for the generation of initial ensembles to be used for ensemble prediction. The optimality of these perturbations refers to their property of maximizing prespecified quadratic measures of error growth, given that tangent-linear error evolution is assumed. The goal of ensemble prediction is the accurate prediction of the uncertainty of forecasts made with dynamical numerical weather prediction models.

In the present paper the theoretical justification for the use of SVs in ensemble prediction systems is investigated. It is shown that, in a tangent-linear framework, SVsâ€”constructed using covariance information valid at the initial timeâ€”evolve into the eigenvectors of the forecast error covariance matrix valid for the end of the optimization interval. As such, SVs represent the most efficient means for predicting the forecast error covariance matrix, given a prespecified number of allowable (tangent-linear) model integrations. Such optimal prediction is of particular importance in light of the fact that the forecast error covariance matrix is summarizing important information about the probability density function of the model state at a given future time.

Based on the above result, optimal covariance prediction through appropriately determined SVs is demonstrated here for a three-dimensional Lorenz model, as well as for a barotropic model of intermediate dimensionality, both within a perfect-model framework. In the case of the barotropic model it is found that less than 15% of the SVs suffice to account for more than 95% of the total final error variance. Viewed differently, at least 80% of the final error variance is accounted for by retaining those SVs that are amplifying in terms of an enstrophy norm. In addition, variances and covariances predicted through SVs agree closely with independently obtained Monte Carlo estimates, as long as the tangent-linear approximation is sufficiently accurate.

Further, the problem of approximating the forecast error covariance matrix in the presence of a state-independent model-error representation is briefly considered. The paper is concluded with a summary of the results and a discussion of their possible implications on data assimilation procedures and on the further development of ensemble prediction systems.

## Abstract

A model of the tropical ocean and global atmosphere is described. It consists of an aqua-planet form of version one of the NCAR Community Climate Model coupled to a primitive equation model for the upper tropical ocean in a rectangular basin. A 24-year simulation is described that has almost no climate drift, a good simulation of the mean temperature gradient across the ocean, but smaller than observed annual and interannual variability. The coupled model is analyzed to see where it occurs on the schematic bifurcation diagram of Neelin. In years 9â€“16 of the simulation there is a dominant oscillation with a period of two years. The spatial pattern of this oscillation shows up clearly in the first empirical orthogonal function calculated from monthly averages of sea surface temperature anomalies. A series of 19 model-twin predictability experiments were carried out with the initial perturbation being a very small change in the ocean temperature field. The correlation coefficient of monthly sea surface temperature anomalies from these model-twin experiments decreases rapidly over the first 6 months and after that, more slowly, showing that there is some predictability out to a year. The predictability times are marginally increased if only the coefficient of the first empirical orthogonal function of monthly averaged sea surface temperature anomalies or NIN03 sea surface temperature is predicted. There is some evidence to indicate that it is easier to predict the onset of a model warm event than to predict the onset of a model cold event. More detailed analysis of the first model-twin experiment shows that the initial divergence in the integrations is a change at day 6 in the incoming solar radiation due to a change in the atmospheric model clouds. The dominant early change in sea surface temperature occurs by this change in radiative heat flux. If the cloud feedback is set to zero, then the first changes are delayed to day 12 and occur in the evaporative and sensible heat fluxes and in the atmospheric wind stress. In this case the dominant early change to sea surface temperature is by advection due to the changed wind stress.

## Abstract

A model of the tropical ocean and global atmosphere is described. It consists of an aqua-planet form of version one of the NCAR Community Climate Model coupled to a primitive equation model for the upper tropical ocean in a rectangular basin. A 24-year simulation is described that has almost no climate drift, a good simulation of the mean temperature gradient across the ocean, but smaller than observed annual and interannual variability. The coupled model is analyzed to see where it occurs on the schematic bifurcation diagram of Neelin. In years 9â€“16 of the simulation there is a dominant oscillation with a period of two years. The spatial pattern of this oscillation shows up clearly in the first empirical orthogonal function calculated from monthly averages of sea surface temperature anomalies. A series of 19 model-twin predictability experiments were carried out with the initial perturbation being a very small change in the ocean temperature field. The correlation coefficient of monthly sea surface temperature anomalies from these model-twin experiments decreases rapidly over the first 6 months and after that, more slowly, showing that there is some predictability out to a year. The predictability times are marginally increased if only the coefficient of the first empirical orthogonal function of monthly averaged sea surface temperature anomalies or NIN03 sea surface temperature is predicted. There is some evidence to indicate that it is easier to predict the onset of a model warm event than to predict the onset of a model cold event. More detailed analysis of the first model-twin experiment shows that the initial divergence in the integrations is a change at day 6 in the incoming solar radiation due to a change in the atmospheric model clouds. The dominant early change in sea surface temperature occurs by this change in radiative heat flux. If the cloud feedback is set to zero, then the first changes are delayed to day 12 and occur in the evaporative and sensible heat fluxes and in the atmospheric wind stress. In this case the dominant early change to sea surface temperature is by advection due to the changed wind stress.

## Abstract

Attempts to represent the vertical structure in primitive equation models of the atmosphere with the spectral method have been unsuccessful to date. The linear stability analysis of Francis showed that small time steps were required for computational stability near the upper boundary with a vertical spectral method using Laguerre polynomials. Machenhauer and Daley used Legendre polynomials in their vertical spectral representation and found it necessary to use an artificial constraint to force temperature to zero when pressure was zero to control the upper-level horizontal velocities. This ad hoc correction is undesirable, and an analysis that shows such a correction is unnecessary is presented.

By formulating the model in terms of velocity and geopotential and then using the hydrostatic equation to calculate temperature from geopotential, temperature is necessarily zero when pressure is zero. This strategy works provided the multiplicative inverse of the first vertical derivative of the vertical basis functions approaches zero more slowly than pressure.

The authors applied this technique to the dry-adiabatic primitive equations on the equatorial Î˛ and tropical *f* planes. Vertical and horizontal normal modes were used as the spectral basis functions. The vertical modes are based on the vertical normal modes of Staniforth et al., and the horizontal modes are normal modes for the primitive equations on a Î˛ or *f* plane.

The results show that the upper-level velocities do not necessarily increase, total energy is conserved, and kinetic energy is bounded. The authors found an upper-level temporal oscillation in the horizontal domain integral of the horizontal velocity components that is related to mass and velocity field imbalances in the initial conditions or introduced during the integration. Through nonlinear normal-mode initialization, the authors effectively removed the initial condition imbalance and reduced the amplitude of this oscillation. It is hypothesized that the vertical spectral representation makes the model more sensitive to initial condition imbalances, or it introduces imbalance during the integration through vertical spectral truncation.

It is also found slow spectral convergence properties for our vertical basis functions. It is concluded that a desirable vertical basis set should have the following properties: 1) nearly uniform distribution of zeros and rapid spectral convergence; 2) vertical structure functions that are bounded at the upper boundary and a multiplicative inverse of the first derivative that goes to zero more slowly than pressure, and 3) expansions for derivatives of the basis functions that need to converge quickly.

## Abstract

Attempts to represent the vertical structure in primitive equation models of the atmosphere with the spectral method have been unsuccessful to date. The linear stability analysis of Francis showed that small time steps were required for computational stability near the upper boundary with a vertical spectral method using Laguerre polynomials. Machenhauer and Daley used Legendre polynomials in their vertical spectral representation and found it necessary to use an artificial constraint to force temperature to zero when pressure was zero to control the upper-level horizontal velocities. This ad hoc correction is undesirable, and an analysis that shows such a correction is unnecessary is presented.

By formulating the model in terms of velocity and geopotential and then using the hydrostatic equation to calculate temperature from geopotential, temperature is necessarily zero when pressure is zero. This strategy works provided the multiplicative inverse of the first vertical derivative of the vertical basis functions approaches zero more slowly than pressure.

The authors applied this technique to the dry-adiabatic primitive equations on the equatorial Î˛ and tropical *f* planes. Vertical and horizontal normal modes were used as the spectral basis functions. The vertical modes are based on the vertical normal modes of Staniforth et al., and the horizontal modes are normal modes for the primitive equations on a Î˛ or *f* plane.

The results show that the upper-level velocities do not necessarily increase, total energy is conserved, and kinetic energy is bounded. The authors found an upper-level temporal oscillation in the horizontal domain integral of the horizontal velocity components that is related to mass and velocity field imbalances in the initial conditions or introduced during the integration. Through nonlinear normal-mode initialization, the authors effectively removed the initial condition imbalance and reduced the amplitude of this oscillation. It is hypothesized that the vertical spectral representation makes the model more sensitive to initial condition imbalances, or it introduces imbalance during the integration through vertical spectral truncation.

It is also found slow spectral convergence properties for our vertical basis functions. It is concluded that a desirable vertical basis set should have the following properties: 1) nearly uniform distribution of zeros and rapid spectral convergence; 2) vertical structure functions that are bounded at the upper boundary and a multiplicative inverse of the first derivative that goes to zero more slowly than pressure, and 3) expansions for derivatives of the basis functions that need to converge quickly.

## Abstract

Recent experimental results indicate that there are serious problems in forecasting planetary scales of motion. In contrast with predictability theory which suggests that the planetary scales are the most predictable, forecast experiments indicate that the long waves are predicted less accurately than the synoptic scales.

The present work suggests that one of the causes of model long wave error is the spurious excitation of transient external large-scale Rossby modes. It was found that these modes can be excited by the imposition of an equatorial wall, or by the use of unsuitable data in the tropics. Paradoxically, the imposition of a wall north of the equator may tend to suppress these spurious Rossby modes. The excited external Rossby modes are relatively fast-moving and can have a substantial negative impact on midlatitude forecast skill after only 24 h of integration.

## Abstract

Recent experimental results indicate that there are serious problems in forecasting planetary scales of motion. In contrast with predictability theory which suggests that the planetary scales are the most predictable, forecast experiments indicate that the long waves are predicted less accurately than the synoptic scales.

The present work suggests that one of the causes of model long wave error is the spurious excitation of transient external large-scale Rossby modes. It was found that these modes can be excited by the imposition of an equatorial wall, or by the use of unsuitable data in the tropics. Paradoxically, the imposition of a wall north of the equator may tend to suppress these spurious Rossby modes. The excited external Rossby modes are relatively fast-moving and can have a substantial negative impact on midlatitude forecast skill after only 24 h of integration.

## Abstract

The Maddenâ€“Julian oscillation (MJO), a planetary-scale eastward-propagating coherent structure with periods of 30â€“60 days, is a prominent manifestation of intraseasonal variability in the tropical atmosphere. It is widely presumed that small-scale moist cumulus convection is a critical part of its dynamics. However, the recent results from high-resolution modeling as well as data analysis suggest that the MJO may be understood by dry dynamics to a leading-order approximation. Simple, further theoretical considerations presented herein suggest that if it is to be understood by dry dynamics, the MJO is most likely a strongly nonlinear solitary Rossby wave. Under a global quasigeostrophic equivalent-barotropic formulation, modon theory provides such analytic solutions. Stability and the longevity of the modon solutions are investigated with a global shallow-water model. The preferred modon solutions with the greatest longevities compare well overall with the observed MJO in scale and phase velocity within the factors.

## Abstract

The Maddenâ€“Julian oscillation (MJO), a planetary-scale eastward-propagating coherent structure with periods of 30â€“60 days, is a prominent manifestation of intraseasonal variability in the tropical atmosphere. It is widely presumed that small-scale moist cumulus convection is a critical part of its dynamics. However, the recent results from high-resolution modeling as well as data analysis suggest that the MJO may be understood by dry dynamics to a leading-order approximation. Simple, further theoretical considerations presented herein suggest that if it is to be understood by dry dynamics, the MJO is most likely a strongly nonlinear solitary Rossby wave. Under a global quasigeostrophic equivalent-barotropic formulation, modon theory provides such analytic solutions. Stability and the longevity of the modon solutions are investigated with a global shallow-water model. The preferred modon solutions with the greatest longevities compare well overall with the observed MJO in scale and phase velocity within the factors.