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Ronald M. Errico

Abstract

Several numerical weather prediction models now use nonlinear normal-mode initialization schemes. These schemes describe balanced states which act to limit the initial presence of high-frequency gravity waves and their subsequent growth by internal dynamics. It has been suggested that there may be states that are so balanced that these waves are never excited except through external forcing. These states have been termed “superbalanced” or “belonging to the slow manifold.”

The degrees to which various balance conditions describe solutions to primitive equations are determined using a sparse-spectral model. The degrees of balance are measured in terms of the portion of energy remaining in the unbalanced fields. Time scales of solutions are measured in terms of the power spectra of their normal linear modes. These measures are determined as a function of heating and dissipation rates and Rossby number.

The dynamics of imbalances is examined also. The tendency for an energy-conserving primitive-equation system to equipartition its energy among each independent mode is demonstrated. For non-adiabatic systems, the portion of the fields not described by superbalance conditions is shown to consist primarily of inertial- gravity waves, especially at the largest horizontal scales. These gravity waves occur intermittently and coincide with maxima in the Rossby number and with small-scale energy cascades. They are damped by eddy viscosity.

Mechanisms for generating imbalances are investigated by comparing various filtered versions of the model. Results indicate that high-frequency components of the quasi-geostrophic forcing terms are the, energy source. The gravity waves are amplified further by near-resonant ageostrophic interactions. However, the gravity modes act on the geostrophic field to increase the balance, probably by acting to damp high-frequency eddies. A true slow manifold does not exist in this model for time-mean Ro>0.1.

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Ronald M. Errico

Abstract

Ageostrophic effects on geostrophic motions are examined for f-plane flows characterized by small Rossby numbers. For small Rossby numbers, the ageostrophic field is determined by the geostrophic field through the familiar quasi-geostrophic relationships. Although the ageostrophic field is relatively weak, the two fields can interact nonlinearly to produce relatively large effects since the two are spatially well correlated. If the geostrophic field is very turbulent, energy exchanges by ageostrophic processes may significantly affect the geostrophic energy spectra. The ageostrophic effects examined here are those neglected by quasi-geostrophic theory.

It is hypothesized that time-mean statistics of some of these ageostrophic effects may be significant compared with similar quasi-geostrophic effects under atmospheric-like conditions. The hypothesis is tested with a low-order model which is described in terms of geostrophic and ageostrophic normal modes. Results suggest that energy exchanges between geostrophic modes due to ageostrophic processes can dominate at smaller synoptic scales for the reason suggested. The geostrophic energy at these scales is decreased if these mechanisms are excluded. Implications regarding studies of climate and geostrophic turbulence are presented.

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Ronald M. Errico

Abstract

We examine some properties of the extratropical atmosphere which act to maintain quasi-geostrophic balance. A nonlinear, f-plane, primitive equation, two-layer model is used. The momentum and temperature fields are described in terms of normal modes of the system given by the model's linear terms. These modes are classified as either geostrophic or ageostrophic depending on their associated eigenvalues. The original nonlinear equations are transformed into a system in which the modulations of the modal amplitudes by nonlinear effects are explicitly expressed in terms of the modal amplitudes themselves. This transformation facilitates a multiple-time-scale analysis.

The stability of simple finite-amplitude geostrophic solutions with respect to infinitesimal perturbations in other modes is investigated. Results are discussed for nondimensional unperturbed-state amplitudes of magnitude ε < 1. These geostrophic solutions may be unstable with respect to further geostrophic perturbations, with growth rates order ε. The ageostrophic modes in this case satisfy approximate quasi-geostrophic conditions. The geostrophic solutions may also be unstable with respect to ageostrophic perturbations, but only with growth rates of least order ε2.

The time-mean balance of ageostrophic energy is investigated using time-scale analysis and ordering arguments. Nearly resonant triad interactions between a geostrophic mode and a pair of inertial-gravity waves only weakly act to affect the geostrophic mode. This property and the separation between natural frequencies of geostrophic and ageostrophic modes suggests that energy is not readily exchanged between modes of the two types. Sufficiently strong dissipation by viscosity and diabatic effects is necessary to limit a slow accumulation of energy by inertial-gravity waves.

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Ronald M. Errico

Adjoint models are powerful tools for many studies that require an estimate of sensitivity of model output (e.g., a forecast) with respect to input. Actual fields of sensitivity are produced directly and efficiently, which can then be used in a variety of applications, including data assimilation, parameter estimation, stability analysis, and synoptic studies.

The use of adjoint models as tools for sensitivity analysis is described here using some simple mathematics. An example of sensitivity fields is presented along with a short description of adjoint applications. Limitations of the applications are discussed and some speculations about the future of adjoint models are offered.

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Ronald M. Errico

The Fifth International Workshop on the Applications of Adjoint Models in Dynamic Meteorology was convened in Mount Bethel, Pennsylvania, 21–26 April 2002. There were 62 participants from 12 countries. Topics included adjoint model development, sensitivity and stability analysis, ensemble forecasting, and several aspects of data assimilation.

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Ronald M. Errico

Abstract

The degrees to which mesoscale model simulations satisfy forms of the quasi-geostrophic omega-equation and nonlinear balance equation are determined for various vertical and horizontal scales. The forms of the equations are those consistent with the application of Bourke and McGregor's initialization scheme to the simulation model, and the vertical scales are those determined by the model's vertical modes.

Results indicate that the degree of balance is primarily a function of vertical (rather than horizontal) scale, with the larger vertical scales better balanced. The balance is observed simultaneously for the fields of velocity divergence and ageostrophic vorticity. Also, it is primarily an adiabatic balance, although at very small horizontal scales, diabatic processes (presumably model diffusion) are an important component of the balance. The degree of balance at any scale is apparently not strongly dependent on synoptic situation, although many significant exceptions are likely. In contrast, the initial interpolated analyses do not show similarly strong degrees of balance suggesting that those analyses require some form of nonlinear normal mode initialization.

Important conclusions are that both the quasi-geostrophic omega-equation and nonlinear balance equation are very applicable on the mesoscale if they are applied only to large vertical scales and if all significant nonlinear and diabatic processes are considered.

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Ronald M. Errico

Abstract

Time series of normal mode coefficients were determined by projection of data produced at every time step for 64 days of a long climate simulation with the NCAR Community Climate Model. From these, the coefficient tendencies, linear forcing, and nonlinear forcing were calculated. Given this nonlinear forcing, the error due to neglect of the time tendency term by Machenhauer's balance scheme was then determined along with errors for higher order schemes in which higher order tendencies were neglected.

Results indicate that the errors are typically less than 10% of the true mode amplitudes for most external and first-internal modes. As shallower modes are considered, typical errors are 1arger: for the fourth vertical mode a typical error is 30%, although greater for the Kelvin modes. Higher-order schemes (through sixth order) only improve the description of balance for some modes with resonant periods shorter than 10 hours. For other modes, the errors increase with the order of the scheme.

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Ronald M. Errico

Abstract

Time series of normal mode coefficients were determined by projection of data produced at every time step for 64 days of a long climate simulation with the NCAR Community Climate Model. Harmonic dials and power spectra for selected gravitational modes were examined. One result is that the greatest power for gravitational modes is typically at the longest periods examined, but noteworthy relative maxima also occur near a mode's resonant period and corresponding time-computational period. For most naturally fast modes, the power at long periods tends to be many times greater than the power near the resonant period, implying that the behavior of these modes may be characterized as approximately balanced. For naturally slow modes, such as Kelvin modes, however, the portion of power near the mode's resonant period is often nonnegligible, implying that these modes are characterized by quasi-linear, wavelike propagation, rather than by either diabatic or adiabatic balance behavior.

For the same simulated period, the forcing due to convective and stable-layer condensation was also projected onto the normal modes, and resulting harmonic dials and power spectra were examined. Typically, this power peaks at the longest periods and is approximately proportional to the period squared.

A response of each mode to its forcing by condensational heating was determined by assuming that linear damping acted on each mode with an e-folding period of 5 days and that this forcing was periodic. Results indicate this forcing has sufficient power at all periods to explain the near-resonant peaks in the observed power spectra of most gravitational model. In other words, the departure of the behavior of most modes from that of slow, nearly balanced motion may be explained as a consequence of the spatial and temporal characteristics of condensational heating, and the model's diabatic forcing destroys rather than creates balance. An important implication is that diabatic nonlinear normal mode initialization is a basically incorrect procedure for strengthening a tropical circulation otherwise weakened by applying an adiabatic initialization scheme.

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Ronald M. Errico

Abstract

The limited-domain, spectral analysis schemes or Errico and of Barnes are compared. They differ in their treatments of aperiodicity of the domain boundaries. The difference is shown to strongly affect treatment of the resolved small scales. Results indicate that the Barnes scheme has a smoother boundary trend field at the expense of greater misrepresentation of small interior scales after that trend is removed.

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Ronald M. Errico

Abstract

A method is presented for determining variance spectra of meteorological fields specified on limited-area grids. Spectra so obtained are compared with global spectra of the same data. An example of scale decomposition (i.e., filtering) using this method is also presented. The method is proposed as an analysis tool for data produced by limited-area models.

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