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## Abstract

The normal modes of a linearized discrete-time model have been determined. The model is spectral and global. Although the model actually uses the primitive equations, only the equivalent systems of shallow water equations are considered. The discrete-time scheme is semi-implicit. Modes of the corresponding continuous-time model are also determined. Structures of the corresponding modes of the two models are slightly different, especially those of the mixed Rossby-gravity or gravitational modes.

The set of modes which is linearly independent in the continuous-time model is not, in general, linearly independent in the discrete-time model. For the nonlinear versions of the models, nonlinear normal mode initialization for one is not equivalent to that for the other. Errors due to ignoring thew differences when initializing or investigating model balances are described. It is shown that the errors are typically small and may be neglected in most instances.

## Abstract

The normal modes of a linearized discrete-time model have been determined. The model is spectral and global. Although the model actually uses the primitive equations, only the equivalent systems of shallow water equations are considered. The discrete-time scheme is semi-implicit. Modes of the corresponding continuous-time model are also determined. Structures of the corresponding modes of the two models are slightly different, especially those of the mixed Rossby-gravity or gravitational modes.

The set of modes which is linearly independent in the continuous-time model is not, in general, linearly independent in the discrete-time model. For the nonlinear versions of the models, nonlinear normal mode initialization for one is not equivalent to that for the other. Errors due to ignoring thew differences when initializing or investigating model balances are described. It is shown that the errors are typically small and may be neglected in most instances.

## Abstract

Some basic aspects related to the problem of incorporating moist convective processes in a variational data assimilation framework are considered. The methodology is based on inverse problem theory and is formulated in its simplest context where the adjustment of temperature and humidity fields take place only in the vertical. In contrast to previous studies on the subject, the impact of error statistics from prior information and data sources of information is clarified. The accuracy of linearization of convection operators and the resulting impact in a minimization procedure are examined. The former was investigated using Monte Carlo methods. Versions of two schemes are examined: the Kuoâ€“Anthes scheme and the relaxed Arakawaâ€“Schubert scheme (RAS).

It is found, in general, that for nonpathological convective points (i.e., points where convection *always* remains active during the minimization process), a significant adjustment of convection (and precipitation rate) is realizable within the range of realistic background temperature and specific humidity errors and precipitation rate observation errors. Typically, three to five iterations are sufficient for convergence of the variational algorithm for both convective schemes. The degree of nonlinearity of both schemes appears comparable. The vertical correlation length for the background error temperature field is shown to produce a strong interaction with the RAS scheme in the minimization process where significantly different vertical structures of analysis increments for the temperature field are generated in the vicinity of a critical value of the correlation length.

## Abstract

Some basic aspects related to the problem of incorporating moist convective processes in a variational data assimilation framework are considered. The methodology is based on inverse problem theory and is formulated in its simplest context where the adjustment of temperature and humidity fields take place only in the vertical. In contrast to previous studies on the subject, the impact of error statistics from prior information and data sources of information is clarified. The accuracy of linearization of convection operators and the resulting impact in a minimization procedure are examined. The former was investigated using Monte Carlo methods. Versions of two schemes are examined: the Kuoâ€“Anthes scheme and the relaxed Arakawaâ€“Schubert scheme (RAS).

It is found, in general, that for nonpathological convective points (i.e., points where convection *always* remains active during the minimization process), a significant adjustment of convection (and precipitation rate) is realizable within the range of realistic background temperature and specific humidity errors and precipitation rate observation errors. Typically, three to five iterations are sufficient for convergence of the variational algorithm for both convective schemes. The degree of nonlinearity of both schemes appears comparable. The vertical correlation length for the background error temperature field is shown to produce a strong interaction with the RAS scheme in the minimization process where significantly different vertical structures of analysis increments for the temperature field are generated in the vicinity of a critical value of the correlation length.

## 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

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.

## 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.

## 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.

## 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

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 Machenhauerâ€™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 Machenhauerâ€™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 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

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.