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Joseph J. Tribbia

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.

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Joseph J. Tribbia

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 methods—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.

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Joseph J. Tribbia

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.

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Joseph J. Tribbia

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.

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Roger Temam
and
Joseph Tribbia

Abstract

It was shown by Oliger and Sundström, 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.

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Arthur Mizzi
,
Joseph Tribbia
, and
James Curry

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 β 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 β 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.

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Tsing-Chang Chen
and
Joseph J. Tribbia

Abstract

Diagnostic computations of nonlinear cascades of enstrophy have been performed in the wavenumber domain for two northern summers. Attention is focused on the interactions among the waves, the interaction between the zonal flow and a given wave and the exchanges due to the beta effect. It is found that two wave ranges (low and intermediate wavenumbers) cascade enstrophy to two ranges of wavenumbers. Calculations are also performed to evaluate the contributions from the standing (92-day mean) and transient modes to the nonlinear enstrophy cascade.

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Richard Grotjahn
,
Min Chen
, and
Joseph Tribbia

Abstract

The eigenvalue problems for the original Eady model and a modified Eady model (the G model) are examined with no friction, Ekman friction only, and both Ekman and interior friction. When both Ekman and interior friction are included in the models, normal modes show little additional change when compared to the case with Ekman friction only, whereas the relevant “continuum modes” have large negative growth rates. Interior friction has a much greater effect on the continuum modes than on the normal modes because inviscid continuum modes have a delta-function vertical profile of potential vorticity q. In contrast, normal modes have much smoother profiles of q in the interior. Streamfunction profiles for the continuum modes are notably different in the two models. The continuum modes in the more realistic G model have sharp peak amplitudes that are not as broad in the vertical as in the Eady model.

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Richard Grotjahn
,
Roderick Pedersen
, and
Joseph Tribbia

Abstract

Normal-mode and nonmodal growth are investigated using initial value models. The initial value problems for the Eady and a generalized Eady model (the G model) are solved with no friction and with both Ekman and interior friction. The nonmodel growth is described as either a superposition of eigenmodes or as a transfer between the “thermal” and relative vorticity parts of quasigeostrophic potential vorticity. When all the eigen-modes are neutral, the growth rate (σ H>) of enstrophy is zero, though the growth rate of energy (σ E>) and amplitude (σ L>) may be positive. For an initial condition having large upstream tilt and constant amplitude, a period of large initial growth in the energy and amplitude is followed by either oscillatory growth and decay (when all eigenmodes are neutral) or asymptotes to a rate given by the most unstable normal mode. In Part I, the authors show that interior friction strongly damps the continuum eigenmodes; however, nonmodal growth can still be significant even when interior friction is present in the Eady model. In the more realistic G model, less overlap between the eigenmodes is found and consequently the nonmodal growth by superposition is reduced compared to the Eady model studied previously by others.

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Martin Ehrendorfer
and
Joseph J. Tribbia

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, SVs—constructed using covariance information valid at the initial time—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.

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