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## Abstract

This paper describes further improvement in a new spectral model of the global barotropic primitive equations (Kasahara, 1977) which utilizes Hough harmonies as basis functions. A review is presented on a method of constructing Hough harmonics (normal modes of Laplace's tidal equations) with new results of the eigensolutions for the logitudinal wavenumber zero case.

Applying this complete set of orthonormal Hough harmonics, we formulate a spectral model of the nonlinear, barotropic primitive equations (shallow-water equations) over a sphere which eliminates separate treatment of the zonally averaged component equations in the previously proposed model by the author. An example of the model calculation with Haurwitz wavenumber 6 initial conditions is presented.

## Abstract

This paper describes further improvement in a new spectral model of the global barotropic primitive equations (Kasahara, 1977) which utilizes Hough harmonies as basis functions. A review is presented on a method of constructing Hough harmonics (normal modes of Laplace's tidal equations) with new results of the eigensolutions for the logitudinal wavenumber zero case.

Applying this complete set of orthonormal Hough harmonics, we formulate a spectral model of the nonlinear, barotropic primitive equations (shallow-water equations) over a sphere which eliminates separate treatment of the zonally averaged component equations in the previously proposed model by the author. An example of the model calculation with Haurwitz wavenumber 6 initial conditions is presented.

## Abstract

A new spectral model is formulated using Hough harmonics as basis functions to solve numerically the nonlinear barotropic primitive equations (shallow water equations) over a sphere. Hough harmonics are eigensolutions of free oscillations (normal modes) for linearized shallow water equations over a sphere about a basic state of rest and a prescribed equivalent height. Hough harmonics are expressed by Θ*
_{l}
^{s}
* exp(

*is*λ) with zonal wavenumber

*s*, longitude λ and meridional index

*l*. Hough vector functions Θ

*consist of three components-zonal velocity*

_{l}^{s}*U*, meridional velocity

*V*and geopotential height

*Z*, all functions of latitude. There are three modes with distinct frequencies for

*s*≥1: eastward and westward propagating gravity waves and westward propagating rotational waves of the Rossby/Haurwitz type.

The advantage of using Hough harmonies for a spectral barotropic global primitive equation model is that the prognostic variables are efficiently represented because Hough harmonics are normal modes of the prediction model. Initialization is no longer a separate procedure but is built into the forecasting scheme. The characteristics of wave motions are associated with the expansion functions, so that the filtering of unwanted wave components can be performed easily.

The nonlinear advection term in the spectral equation is calculated by the transform method–a combination of Fourier transform and Gaussian quadrature. The results of a test calculation with a balanced non-divergent initial state (a Haurwitz wave) compare favorably with those of an integration using a fourth-order finite-difference model.

## Abstract

A new spectral model is formulated using Hough harmonics as basis functions to solve numerically the nonlinear barotropic primitive equations (shallow water equations) over a sphere. Hough harmonics are eigensolutions of free oscillations (normal modes) for linearized shallow water equations over a sphere about a basic state of rest and a prescribed equivalent height. Hough harmonics are expressed by Θ*
_{l}
^{s}
* exp(

*is*λ) with zonal wavenumber

*s*, longitude λ and meridional index

*l*. Hough vector functions Θ

*consist of three components-zonal velocity*

_{l}^{s}*U*, meridional velocity

*V*and geopotential height

*Z*, all functions of latitude. There are three modes with distinct frequencies for

*s*≥1: eastward and westward propagating gravity waves and westward propagating rotational waves of the Rossby/Haurwitz type.

The advantage of using Hough harmonies for a spectral barotropic global primitive equation model is that the prognostic variables are efficiently represented because Hough harmonics are normal modes of the prediction model. Initialization is no longer a separate procedure but is built into the forecasting scheme. The characteristics of wave motions are associated with the expansion functions, so that the filtering of unwanted wave components can be performed easily.

The nonlinear advection term in the spectral equation is calculated by the transform method–a combination of Fourier transform and Gaussian quadrature. The results of a test calculation with a balanced non-divergent initial state (a Haurwitz wave) compare favorably with those of an integration using a fourth-order finite-difference model.

## Abstract

Roles of the horizontal component of the earth's rotation, which is neglected traditionally in atmospheric and oceanographic models, are studied through the normal mode analysis of a compressible and stratified model on a tangent plane in the domain that is periodic in the zonal and meridional directions but bounded at the top and bottom. As expected, there exist two distinct kinds of acoustic and buoyancy oscillations that are modified by the earth's rotation. When the cos(latitude) Coriolis terms are included, there exists another kind of wave oscillation whose frequencies are very close to the inertial frequency, 2Ω sin(latitude), where Ω is the earth's angular velocity.

The objective of this article is to clarify the circumstance in which a distinct kind of wave oscillation emerges whose frequencies are very close to the inertial frequency. Because this particular kind of normal mode appears only due to the presence of boundary conditions in the vertical, it may be appropriate to call these waves boundary-induced inertial (BII) modes as demonstrated through the normal mode analyses of a homogeneous and incompressible model and a Boussinesq model with thermal stratification. Thus, it can be understood that the BII modes can coexist with the acoustic and inertio-gravity modes when the effect of compressibility is added to the effects of buoyancy and complete Coriolis force in the compressible, stratified, and rotating model.

## Abstract

Roles of the horizontal component of the earth's rotation, which is neglected traditionally in atmospheric and oceanographic models, are studied through the normal mode analysis of a compressible and stratified model on a tangent plane in the domain that is periodic in the zonal and meridional directions but bounded at the top and bottom. As expected, there exist two distinct kinds of acoustic and buoyancy oscillations that are modified by the earth's rotation. When the cos(latitude) Coriolis terms are included, there exists another kind of wave oscillation whose frequencies are very close to the inertial frequency, 2Ω sin(latitude), where Ω is the earth's angular velocity.

The objective of this article is to clarify the circumstance in which a distinct kind of wave oscillation emerges whose frequencies are very close to the inertial frequency. Because this particular kind of normal mode appears only due to the presence of boundary conditions in the vertical, it may be appropriate to call these waves boundary-induced inertial (BII) modes as demonstrated through the normal mode analyses of a homogeneous and incompressible model and a Boussinesq model with thermal stratification. Thus, it can be understood that the BII modes can coexist with the acoustic and inertio-gravity modes when the effect of compressibility is added to the effects of buoyancy and complete Coriolis force in the compressible, stratified, and rotating model.

The future observing system of the global atmosphere which has been contemplated for the Global Atmospheric Research Program (GARP) is a combination of various observing subsystems including satellites, constant-level balloons, automatic ocean buoys, etc., as well as conventional upper-air and surface networks. All observing subsystems are neither perfect nor ideal. Numerical experiments (with global circulation models) are needed to evaluate the subsystems in terms of the accuracy, density, and frequency of observations. They are called Observing Systems Simulation Experiments (OSSE).

The purpose of this report is to describe the present activities in OSSE. Most useful up-to-date information was obtained from the oral presentation of papers at the International Symposium on Four-Dimensional Data Assimilation, Princeton, N. J., 19–22 April 1971.

Since a global circulation model is used as an integrator and analyzer of observed data in three-dimensional space and time taken from various observing subsystems, the question of error growth of the prediction model in relation to the accuracy of observation is discussed as the predictability experiment.

Another important property of the model atmosphere is the ability to adjust the model's variables for forced prescription of incomplete observed data. This adjustment property is used to assimilate observed data in four dimensions. The method of direct substitution is used to determine the basic data requirements for observation of wind, temperature, and surface pressure.

Various methods of four-dimensional data assimilation are reviewed for the purpose of optimum design for OSSE. Finally, questions concerning the reference level information are reviewed and results of some numerical experiments conducted at the National Center for Atmospheric Research are presented.

The future observing system of the global atmosphere which has been contemplated for the Global Atmospheric Research Program (GARP) is a combination of various observing subsystems including satellites, constant-level balloons, automatic ocean buoys, etc., as well as conventional upper-air and surface networks. All observing subsystems are neither perfect nor ideal. Numerical experiments (with global circulation models) are needed to evaluate the subsystems in terms of the accuracy, density, and frequency of observations. They are called Observing Systems Simulation Experiments (OSSE).

The purpose of this report is to describe the present activities in OSSE. Most useful up-to-date information was obtained from the oral presentation of papers at the International Symposium on Four-Dimensional Data Assimilation, Princeton, N. J., 19–22 April 1971.

Since a global circulation model is used as an integrator and analyzer of observed data in three-dimensional space and time taken from various observing subsystems, the question of error growth of the prediction model in relation to the accuracy of observation is discussed as the predictability experiment.

Another important property of the model atmosphere is the ability to adjust the model's variables for forced prescription of incomplete observed data. This adjustment property is used to assimilate observed data in four dimensions. The method of direct substitution is used to determine the basic data requirements for observation of wind, temperature, and surface pressure.

Various methods of four-dimensional data assimilation are reviewed for the purpose of optimum design for OSSE. Finally, questions concerning the reference level information are reviewed and results of some numerical experiments conducted at the National Center for Atmospheric Research are presented.

## Abstract

Two finite-difference methods for geophysical fluid problems are described, and stability conditions of these schemes are discussed. These two schemes are formulated based upon a similar procedure given by Lax and Wendroff in order to obtain a second-order accuracy in finite-difference equations. However, the two schemes show remarkable differences in their computational stability. One scheme is stable, as one might expect, under the usual stability conditions of Courant-Friedrichs-Lewy and Lax-Wendroff. However, the other scheme is conditionally stable only if the flow is supereritical (supersonic in the case of gas dynamics) and unconditionally unstable if the flow is suberitical (subsonic).

## Abstract

Two finite-difference methods for geophysical fluid problems are described, and stability conditions of these schemes are discussed. These two schemes are formulated based upon a similar procedure given by Lax and Wendroff in order to obtain a second-order accuracy in finite-difference equations. However, the two schemes show remarkable differences in their computational stability. One scheme is stable, as one might expect, under the usual stability conditions of Courant-Friedrichs-Lewy and Lax-Wendroff. However, the other scheme is conditionally stable only if the flow is supereritical (supersonic in the case of gas dynamics) and unconditionally unstable if the flow is suberitical (subsonic).

## Abstract

In the formulation of nonlinear normal mode initialization (NNMI), we apply the separation of variables to a primitive equation model which is linearized about a basic state at rest. This results in the vertical and horizontal structure equations from which normal mode functions are constructed as their solutions. This paper is concerned with the property of the vertical normal mode functions. They vertical structure functions constructed in the framework of a vertically staggered discretized atmospheric model may not, in general, be orthogonal. There is an obvious advantage in using an orthogonal set of expansion functions. We propose in this paper the construction of orthogonal vertical structure functions for the NNMI with an atmospheric prediction model for which non-orthogonal vertical structure functions have been adopted in the past. We discuss the difference between the new formulation which yields orthogonal vertical structure functions and the old formulation which does not, in general, yield orthogonality. We also present the results of a sensitivity test with a global model on 55 day forecasts starting from two sets of initial conditions in which two versions of vertical normal mode functions are used. It is shown that the differences between the two 5-day forecasts are negligibly small.

## Abstract

In the formulation of nonlinear normal mode initialization (NNMI), we apply the separation of variables to a primitive equation model which is linearized about a basic state at rest. This results in the vertical and horizontal structure equations from which normal mode functions are constructed as their solutions. This paper is concerned with the property of the vertical normal mode functions. They vertical structure functions constructed in the framework of a vertically staggered discretized atmospheric model may not, in general, be orthogonal. There is an obvious advantage in using an orthogonal set of expansion functions. We propose in this paper the construction of orthogonal vertical structure functions for the NNMI with an atmospheric prediction model for which non-orthogonal vertical structure functions have been adopted in the past. We discuss the difference between the new formulation which yields orthogonal vertical structure functions and the old formulation which does not, in general, yield orthogonality. We also present the results of a sensitivity test with a global model on 55 day forecasts starting from two sets of initial conditions in which two versions of vertical normal mode functions are used. It is shown that the differences between the two 5-day forecasts are negligibly small.

## Abstract

To represent atmospheric data spectrally in three indices (zonal wavenumber, and meridional and vertical modal indices), we propose to use three-dimensional normal mode functions (NMF's) to express the wind and mass fields simultaneously. The NMF's are constructed from the eigensolutions of a global primitive equation model and they are orthogonal functions. The vertical parts are obtained from the solutions of the vertical structure equation with the equivalent height as the eigenvalue. The vertical modal index is associated with a different value of the equivalent height. The horizontal parts of NMF's are Hough harmonics with zonal wavenumber and meridional modal index as two-dimensional scalings. The expansion of global data in terms of NMF's permits the partition of energy into two distinct kinds of motions-gravity-inertia modes and rotational modes of Rossby/Haurwitz type. Both kinds of motion are also partitioned into different vertical modes. Results of the spectral distribution of atmospheric energy, obtained by expanding in the NMF's hemispherical data of the National Meteorological Center, are presented. Information obtained will be useful to select proper horizontal and vertical computational resolutions for representation of atmospheric data.

## Abstract

To represent atmospheric data spectrally in three indices (zonal wavenumber, and meridional and vertical modal indices), we propose to use three-dimensional normal mode functions (NMF's) to express the wind and mass fields simultaneously. The NMF's are constructed from the eigensolutions of a global primitive equation model and they are orthogonal functions. The vertical parts are obtained from the solutions of the vertical structure equation with the equivalent height as the eigenvalue. The vertical modal index is associated with a different value of the equivalent height. The horizontal parts of NMF's are Hough harmonics with zonal wavenumber and meridional modal index as two-dimensional scalings. The expansion of global data in terms of NMF's permits the partition of energy into two distinct kinds of motions-gravity-inertia modes and rotational modes of Rossby/Haurwitz type. Both kinds of motion are also partitioned into different vertical modes. Results of the spectral distribution of atmospheric energy, obtained by expanding in the NMF's hemispherical data of the National Meteorological Center, are presented. Information obtained will be useful to select proper horizontal and vertical computational resolutions for representation of atmospheric data.

## Abstract

We discuss momentum balance and energetics in the stratosphere (18–36 km) based on the simulated mean January results described in Part I. The latitudinal distributions of component terms in the momentum budget equation clearly demonstrate the two-cell structure of the mean meridional circulation in the stratosphere as opposed to the three-cell structure in the troposphere. In contrast to the troposphere, the eddy and mean transports of momentum are equally important in the stratosphere in all latitudes with some tendency for the eddy transport to counteract the mean transport. The latitudinal distributions of component terms in the energy budget equations suggest the following mechanism for maintaining the zonal and eddy kinetic energies. The vertical flux of wave energy through the lower boundary of the stratosphere provides a major source of the eddy kinetic energy in the stratosphere. On the other hand, the zonal kinetic energy in the stratosphere appears to be maintained by energy conversion from the eddy kinetic energy against the energy loss by frictional dissipation. The present results of zonal and eddy kinetic energy budgets are compared with those based on real data of the Northern Hemisphere by various investigators and those of a numerical simulation experiment by Manabe and Hunt. The zonal internal energy budget shows that the zonal internal energy in the stratosphere is maintained by the supply of sensible heat from the troposphere and energy conversion from the eddy kinetic energy against the loss of energy by radiative cooling and the downward transport of internal energy by mean vertical motion. In addition, we find that interpretation of the energetics of the stratosphere depends upon how various energy terms are combined and energy conversion terms formulated. This special consideration is needed because the vertical transports of energy at the interface between the troposphere and the stratosphere play important roles in the energetics of the stratosphere.

## Abstract

We discuss momentum balance and energetics in the stratosphere (18–36 km) based on the simulated mean January results described in Part I. The latitudinal distributions of component terms in the momentum budget equation clearly demonstrate the two-cell structure of the mean meridional circulation in the stratosphere as opposed to the three-cell structure in the troposphere. In contrast to the troposphere, the eddy and mean transports of momentum are equally important in the stratosphere in all latitudes with some tendency for the eddy transport to counteract the mean transport. The latitudinal distributions of component terms in the energy budget equations suggest the following mechanism for maintaining the zonal and eddy kinetic energies. The vertical flux of wave energy through the lower boundary of the stratosphere provides a major source of the eddy kinetic energy in the stratosphere. On the other hand, the zonal kinetic energy in the stratosphere appears to be maintained by energy conversion from the eddy kinetic energy against the energy loss by frictional dissipation. The present results of zonal and eddy kinetic energy budgets are compared with those based on real data of the Northern Hemisphere by various investigators and those of a numerical simulation experiment by Manabe and Hunt. The zonal internal energy budget shows that the zonal internal energy in the stratosphere is maintained by the supply of sensible heat from the troposphere and energy conversion from the eddy kinetic energy against the loss of energy by radiative cooling and the downward transport of internal energy by mean vertical motion. In addition, we find that interpretation of the energetics of the stratosphere depends upon how various energy terms are combined and energy conversion terms formulated. This special consideration is needed because the vertical transports of energy at the interface between the troposphere and the stratosphere play important roles in the energetics of the stratosphere.

## Abstract

An attempt is made to investigate theoretically the controlling influence of compensating downward motions on the development of cumulus clouds and the size of the cloudless areas associated with them. The model consists of two circular concentric air columns, the inside column corresponding to the updraft (cloud) region and the outside concentric annular column to the downward motion region. The combined cell is surrounded by the atmospheric at rest. The governing equations of both the updraft and the compensating downward motion are derived from the conservation equations of momentum, heat, moisture and mass.

The differential equations are solved numerically to compute the vertical velocity, temperature, specific humidity and liquid water content in and out of the cloud as functions of height and time. Two experiments were performed with and without the effect of compensating downward motion. The main conclusions are the following: Without the effect of the compensating motion, the structure of the solitary updraft tends to a steady state. However, with the compensating motion, no tall cloud is maintained (unless there is a steady source of moisture at the cloud base), since the compensating downward motion acts as a “break.” Also, it was found that the most active cloud system develops when the ratio of the cloud area over the entire area (including the cloudless area associated with the updraft) is of the order of several per cent.

## Abstract

An attempt is made to investigate theoretically the controlling influence of compensating downward motions on the development of cumulus clouds and the size of the cloudless areas associated with them. The model consists of two circular concentric air columns, the inside column corresponding to the updraft (cloud) region and the outside concentric annular column to the downward motion region. The combined cell is surrounded by the atmospheric at rest. The governing equations of both the updraft and the compensating downward motion are derived from the conservation equations of momentum, heat, moisture and mass.

The differential equations are solved numerically to compute the vertical velocity, temperature, specific humidity and liquid water content in and out of the cloud as functions of height and time. Two experiments were performed with and without the effect of compensating downward motion. The main conclusions are the following: Without the effect of the compensating motion, the structure of the solitary updraft tends to a steady state. However, with the compensating motion, no tall cloud is maintained (unless there is a steady source of moisture at the cloud base), since the compensating downward motion acts as a “break.” Also, it was found that the most active cloud system develops when the ratio of the cloud area over the entire area (including the cloudless area associated with the updraft) is of the order of several per cent.