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- Author or Editor: Ronald M. Errico x

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## 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.

## 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.

## 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.

## 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.

## Abstract

A global, spectral model developed at the National Center for Atmospheric Research is investigated. It is first demonstrated that some of the model's normal modes tend toward an approximate dynamical balance. This is shown by presenting time series of a kind of mean frequency for the various types of modes. For the analyzed data investigated, almost all inertial-gravitational waves initially present are dissipated within two weeks. Most are dissipated much more quickly.

The model is then used to determine which modes are balanced. Only the balance described byMachenhauer is investigated. The relative magnitudes of various diabatic and adiabatic forces (including advection as a "force"), as they act to drive each normal mode, are compared with the time tendency of each mode. A mode is considered balanced if the magnitude of its time tendency is significantly smaller than the magnitudes of some forces acting upon it, implying that those forces tend to cancel each other.

Gravitational modes whose natural (i.e., resonant) periods are less than 20 h appear to be balanced; this balanced set includes modes of all vertical and horizontal scales, although not all combinations of such scales. That these modes are balanced implies that their amplitudes satisfy an approximate diagnostic relationship, although they are actually prognostically determined. Gravitational modes with longer natural periods appear to behave as forced waves. As expected, rotational modes are mostly driven by adiabatic, quasi-rotational dynamics, and exhibit neither balanced nor wavelike behavior to any great degree.

The forces which are in balance include the inertial-gravitational force (expressed by linear terms in the model) and the forcing of the gravitational modes by the rotational modes (expressed by nonlinear terms). For shallow modes, surface drag also balances the inertial-gravitational force. For no modes does heating by any process appear to participate in a balance of forces. The force which includes the advection of gravitational modes by the rotational wind also participates in the balance of forces, although its participationis second-order. For the model investigated, initialization using Machenhauer's scheme seems most appropriate when applied only to modes whose natural periods are less than 20 h, and only to the adiabatic plus surface drag forces.

## Abstract

A global, spectral model developed at the National Center for Atmospheric Research is investigated. It is first demonstrated that some of the model's normal modes tend toward an approximate dynamical balance. This is shown by presenting time series of a kind of mean frequency for the various types of modes. For the analyzed data investigated, almost all inertial-gravitational waves initially present are dissipated within two weeks. Most are dissipated much more quickly.

The model is then used to determine which modes are balanced. Only the balance described byMachenhauer is investigated. The relative magnitudes of various diabatic and adiabatic forces (including advection as a "force"), as they act to drive each normal mode, are compared with the time tendency of each mode. A mode is considered balanced if the magnitude of its time tendency is significantly smaller than the magnitudes of some forces acting upon it, implying that those forces tend to cancel each other.

Gravitational modes whose natural (i.e., resonant) periods are less than 20 h appear to be balanced; this balanced set includes modes of all vertical and horizontal scales, although not all combinations of such scales. That these modes are balanced implies that their amplitudes satisfy an approximate diagnostic relationship, although they are actually prognostically determined. Gravitational modes with longer natural periods appear to behave as forced waves. As expected, rotational modes are mostly driven by adiabatic, quasi-rotational dynamics, and exhibit neither balanced nor wavelike behavior to any great degree.

The forces which are in balance include the inertial-gravitational force (expressed by linear terms in the model) and the forcing of the gravitational modes by the rotational modes (expressed by nonlinear terms). For shallow modes, surface drag also balances the inertial-gravitational force. For no modes does heating by any process appear to participate in a balance of forces. The force which includes the advection of gravitational modes by the rotational wind also participates in the balance of forces, although its participationis second-order. For the model investigated, initialization using Machenhauer's scheme seems most appropriate when applied only to modes whose natural periods are less than 20 h, and only to the adiabatic plus surface drag forces.

## Abstract

The convergence properties of Machenhauer’s nonlinar normal-mode initialization scheme are explored. Only adiabatic initialization is considered. Several models are used, including an *f*-plane model. a numerical weather prediction model, and simple linear models with analytic solutions. The last are used to estimate a radius of convergence for Machenhauer's scheme.

It is fist demonstrated with the *f*-plane model that Machenhaur's scheme may be approximated by one that is linear in gravity-mode coefficients. The components which diverge when the scheme is applied are shown to be linear combinations of gravity modes which interact due to advection. Those combinations which diverge fall into two categories those whom phase speed more than doubles as a result of advection, and those whose direction of propagation changes due to advection. These results agree with those of the simpler model of Ballish.

Consideration of Ballish's model suggests that the inclusion of under-relaxation in Machenhauer's scheme, as suggested by Kitade, improves the convergence of eastward waves, but not that of westward waves. Experiments with both the *f*-plane and numerical weather prediction model also yield this result. Therefore, improvement in convergence by using under-relaxation will depend strongly on which modes are initialized.

## Abstract

The convergence properties of Machenhauer’s nonlinar normal-mode initialization scheme are explored. Only adiabatic initialization is considered. Several models are used, including an *f*-plane model. a numerical weather prediction model, and simple linear models with analytic solutions. The last are used to estimate a radius of convergence for Machenhauer's scheme.

It is fist demonstrated with the *f*-plane model that Machenhaur's scheme may be approximated by one that is linear in gravity-mode coefficients. The components which diverge when the scheme is applied are shown to be linear combinations of gravity modes which interact due to advection. Those combinations which diverge fall into two categories those whom phase speed more than doubles as a result of advection, and those whose direction of propagation changes due to advection. These results agree with those of the simpler model of Ballish.

Consideration of Ballish's model suggests that the inclusion of under-relaxation in Machenhauer's scheme, as suggested by Kitade, improves the convergence of eastward waves, but not that of westward waves. Experiments with both the *f*-plane and numerical weather prediction model also yield this result. Therefore, improvement in convergence by using under-relaxation will depend strongly on which modes are initialized.

## 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.

## 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.

## 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.

## 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.

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.

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.

## 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.

## 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.

## 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.

## 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.

## 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 ε^{2}.

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.

## 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 ε^{2}.

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.