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## Abstract

Recently, questions were raised concerning the ellipticity condition of the traditional balance equation, related to the application of nonlinear normal mode initialization for primitive equation prediction models. We investigate, in this paper, the occurrence of non-elliptic regions, using the FWE level IIIb analyses of the European Centre for Medium Range Weather Forecasts. We found that non-elliptic regions are ubiquitous in the tropics. For the traditional balance equation, non-elliptic regions are not realizable in the sense that balanced flows are not physically possible. To reconcile this dilemma, we postulate that the traditional balance equation lacks an additional term due, for example, to the effects of subgrid-scale motions. We evaluated this additional term as the residual by computing each term in the balance equation. This additional term modifies the ellipticity criterion of the balance equation. We call the modified ellipticity condition the realizability condition. This study concludes that the non-elliptic regions found in the tropics satisfy the realizability condition when this additional term in the balance equation is considered. This suggests that the apparent dilemma may be resolved by taking into account presently lacking physical processes in achieving the dynamical balance between the mass and wind fields, if the FGGE level IIIb analyses are supposed to be correct.

## Abstract

Recently, questions were raised concerning the ellipticity condition of the traditional balance equation, related to the application of nonlinear normal mode initialization for primitive equation prediction models. We investigate, in this paper, the occurrence of non-elliptic regions, using the FWE level IIIb analyses of the European Centre for Medium Range Weather Forecasts. We found that non-elliptic regions are ubiquitous in the tropics. For the traditional balance equation, non-elliptic regions are not realizable in the sense that balanced flows are not physically possible. To reconcile this dilemma, we postulate that the traditional balance equation lacks an additional term due, for example, to the effects of subgrid-scale motions. We evaluated this additional term as the residual by computing each term in the balance equation. This additional term modifies the ellipticity criterion of the balance equation. We call the modified ellipticity condition the realizability condition. This study concludes that the non-elliptic regions found in the tropics satisfy the realizability condition when this additional term in the balance equation is considered. This suggests that the apparent dilemma may be resolved by taking into account presently lacking physical processes in achieving the dynamical balance between the mass and wind fields, if the FGGE level IIIb analyses are supposed to be correct.

## Abstract

Spherical harmonics have been used to analyze global meteorological data, and because they are the solutions of a linearized nondivergent vorticity equation, it is appropriate to use them as orthogonal basis functions for analysis and prediction. However, for ultralong waves—geostrophic motions of the second type—horizontal divergence plays as essential a role as the vertical component of vorticity. Hence, it will be advantageous to use the solutions of linearized primitive equations over a sphere as basis functions. This will also permit identification of the characteristics of wave motions for the initialization of primitive equation models. Such solutions have been investigated in the past in conjunction with atmospheric tidal theories and the basic mathematical tools already available piecewise in the literature.

This paper reviews the mathematical development behind the construction of the eigensolutions (referred to as normal modes) of linearized primitive equations over a sphere. The basic state has no motion and the temperature is a function of height only. The solutions of both the vertical and horizontal structure equations are discussed.

The horizontal parts of such normal modes are called Hough harmonics Θ*
_{s}
^{l}
* exp (

*is*λ), where

*s*is zonal wavenumber, λ longitude and

*l*meridional mode index. Hough vector functions Θ

*consist of three components—zonal velocity Û, meridional velocity V̂ and geopotential height Ẑ, all of which are functions of latitude. There are three modes with distinct frequencies: eastward and westward propagating gravity waves, and westward propagating rotational waves of the Rossby/Haurwitz type. Hough harmonics are orthogonal and are conveniently used to decompose wind and mass fields simultaneously. Some examples are presented of global data decomposition in terms of Hough harmonics for studying ultralong waves in the atmosphere.*

_{s}^{l}## Abstract

Spherical harmonics have been used to analyze global meteorological data, and because they are the solutions of a linearized nondivergent vorticity equation, it is appropriate to use them as orthogonal basis functions for analysis and prediction. However, for ultralong waves—geostrophic motions of the second type—horizontal divergence plays as essential a role as the vertical component of vorticity. Hence, it will be advantageous to use the solutions of linearized primitive equations over a sphere as basis functions. This will also permit identification of the characteristics of wave motions for the initialization of primitive equation models. Such solutions have been investigated in the past in conjunction with atmospheric tidal theories and the basic mathematical tools already available piecewise in the literature.

This paper reviews the mathematical development behind the construction of the eigensolutions (referred to as normal modes) of linearized primitive equations over a sphere. The basic state has no motion and the temperature is a function of height only. The solutions of both the vertical and horizontal structure equations are discussed.

The horizontal parts of such normal modes are called Hough harmonics Θ*
_{s}
^{l}
* exp (

*is*λ), where

*s*is zonal wavenumber, λ longitude and

*l*meridional mode index. Hough vector functions Θ

*consist of three components—zonal velocity Û, meridional velocity V̂ and geopotential height Ẑ, all of which are functions of latitude. There are three modes with distinct frequencies: eastward and westward propagating gravity waves, and westward propagating rotational waves of the Rossby/Haurwitz type. Hough harmonics are orthogonal and are conveniently used to decompose wind and mass fields simultaneously. Some examples are presented of global data decomposition in terms of Hough harmonics for studying ultralong waves in the atmosphere.*

_{s}^{l}## Abstract

For numerical weather prediction with primitive equations (the Eulerian hydrodynamic equations modified by the assumption of hydrostatic equilibrium), various coordinate systems are used to represent the vertical structure of the atmosphere. In this paper, we review the essential features of prediction equations, satisfying the conservation of mass and total energy, in various vertical coordinate systems. We formulate the equations of horizontal motion, hydrostatic balance, mass continuity, and thermodynamics using a generalized vertical coordinate in which any variable that gives a single-valued monotonic relationship with a geometric height can be used as a vertical coordinate. Conditions to conserve total energy in a generalized vertical coordinate are investigated.

Various prediction schemes using pressure, height, and potential temperature as a vertical coordinate are derived from the set of basic equations in the generalized coordinate system. These three coordinate systems are unique in that the features of prediction equations in each system are all distinct. We place special emphasis on handling the earth's orography as the lower boundary condition. As an extension of the original idea of Phillips applied to the pressure-coordinate system, we propose transformed height and isentropic systems. In those systems, both the top of the model atmosphere and the earth's surface are always coordinate surfaces. It is hoped that these new schemes, as in the case of the Phillips' sigma-system, will enable us to handle the effect of the earth's orography in the prediction models without lengthy coding logic.

## Abstract

For numerical weather prediction with primitive equations (the Eulerian hydrodynamic equations modified by the assumption of hydrostatic equilibrium), various coordinate systems are used to represent the vertical structure of the atmosphere. In this paper, we review the essential features of prediction equations, satisfying the conservation of mass and total energy, in various vertical coordinate systems. We formulate the equations of horizontal motion, hydrostatic balance, mass continuity, and thermodynamics using a generalized vertical coordinate in which any variable that gives a single-valued monotonic relationship with a geometric height can be used as a vertical coordinate. Conditions to conserve total energy in a generalized vertical coordinate are investigated.

Various prediction schemes using pressure, height, and potential temperature as a vertical coordinate are derived from the set of basic equations in the generalized coordinate system. These three coordinate systems are unique in that the features of prediction equations in each system are all distinct. We place special emphasis on handling the earth's orography as the lower boundary condition. As an extension of the original idea of Phillips applied to the pressure-coordinate system, we propose transformed height and isentropic systems. In those systems, both the top of the model atmosphere and the earth's surface are always coordinate surfaces. It is hoped that these new schemes, as in the case of the Phillips' sigma-system, will enable us to handle the effect of the earth's orography in the prediction models without lengthy coding logic.

## Abstract

The linear response of model normal modes in a stratified atmosphere to tropical thermal forcing is investigated by using global primitive equations which are linearized with respect to a resting state and include a prescribed thermal forcing and momentum dissipation. By the method of separation of the variables, the basic equations are split up into vertical and horizontal equations. The homogeneous parts of these equations are solved spectrally to obtain the model normal modes. The forced problem is then solved by using a normal mode expansion.

For a parabolic form of heating in the vertical, it is shown that the internal modes corresponding to the equivalent height of a few hundred meters are favorably excited. This implies that the disturbances created by diabatic heating tend to have a typical baroclinic vertical structure. Numerical results are presented for the forced solutions generated by stationary and transient beat sources. For the case of *stationary* tropical heating, most of the excited energy goes into the rotational modes, but a significant portion also goes to the Kelvin modes, while other gravity wave modes play insignificant roles in general. For the case of *transient* tropical heating, the generation of gravity wave modes, except for the Kelvin modes, depends strongly on the time scale of heating, while the rotational modes and the Kelvin modes are dependent only weakly on the heating rate. The unique behavior of the Kelvin modes may be explained by the resemblance of the heating pattern to the horizontal structure of Kelvin modes and the closeness of their frequencies to those of rotational modes.

## Abstract

The linear response of model normal modes in a stratified atmosphere to tropical thermal forcing is investigated by using global primitive equations which are linearized with respect to a resting state and include a prescribed thermal forcing and momentum dissipation. By the method of separation of the variables, the basic equations are split up into vertical and horizontal equations. The homogeneous parts of these equations are solved spectrally to obtain the model normal modes. The forced problem is then solved by using a normal mode expansion.

For a parabolic form of heating in the vertical, it is shown that the internal modes corresponding to the equivalent height of a few hundred meters are favorably excited. This implies that the disturbances created by diabatic heating tend to have a typical baroclinic vertical structure. Numerical results are presented for the forced solutions generated by stationary and transient beat sources. For the case of *stationary* tropical heating, most of the excited energy goes into the rotational modes, but a significant portion also goes to the Kelvin modes, while other gravity wave modes play insignificant roles in general. For the case of *transient* tropical heating, the generation of gravity wave modes, except for the Kelvin modes, depends strongly on the time scale of heating, while the rotational modes and the Kelvin modes are dependent only weakly on the heating rate. The unique behavior of the Kelvin modes may be explained by the resemblance of the heating pattern to the horizontal structure of Kelvin modes and the closeness of their frequencies to those of rotational modes.

## Abstract

Solutions of the linearized global shallow-water equations (Laplace tidal equations) including the effect of a mean zonal flow are obtained by the Galerkin-transform method. Free oscillations of the first kind (gravity-inertia modes) are little affected by the zonal flow. Solutions of the second kind (rotational modes of the Rossby-Haurwitz type) are significantly affected by a zonal flow different from solid rotation. Only a few lowest rotational modes, whose angular phase velocities are less than the minimum velocity in the zonal flow, appear as discrete. The remaining angular phase velocities fall into a continuous spectrum which covers the interval between the minimum and maximum zonal velocities. An approximate, but accurate, frequency formula is obtained for the discrete modes of free oscillations under the effect of a mean zonal flow.

The frequencies and latitudinal structures of a few lowest rotational modes under the effect of a mean zonal flow are examined in detail and compared to observational evidence of westward propagating wavenumber 1 long-period oscillations in the atmosphere. The 5-day wavenumber 1 oscillation (the lowest symmetric rotational mode *I*
_{
R
}+1) is found insensitive to the presence of zonal flows. Other discrete modes are relatively sensitive to and their periods increased by the zonal flow effect. In particular, the period of the second symmetric rotational mode *I*
_{
R
}+3 (zonal wavenumber 1) increases to about 16–19 days in favor of the observations summarized by Madden (1978).

## Abstract

Solutions of the linearized global shallow-water equations (Laplace tidal equations) including the effect of a mean zonal flow are obtained by the Galerkin-transform method. Free oscillations of the first kind (gravity-inertia modes) are little affected by the zonal flow. Solutions of the second kind (rotational modes of the Rossby-Haurwitz type) are significantly affected by a zonal flow different from solid rotation. Only a few lowest rotational modes, whose angular phase velocities are less than the minimum velocity in the zonal flow, appear as discrete. The remaining angular phase velocities fall into a continuous spectrum which covers the interval between the minimum and maximum zonal velocities. An approximate, but accurate, frequency formula is obtained for the discrete modes of free oscillations under the effect of a mean zonal flow.

The frequencies and latitudinal structures of a few lowest rotational modes under the effect of a mean zonal flow are examined in detail and compared to observational evidence of westward propagating wavenumber 1 long-period oscillations in the atmosphere. The 5-day wavenumber 1 oscillation (the lowest symmetric rotational mode *I*
_{
R
}+1) is found insensitive to the presence of zonal flows. Other discrete modes are relatively sensitive to and their periods increased by the zonal flow effect. In particular, the period of the second symmetric rotational mode *I*
_{
R
}+3 (zonal wavenumber 1) increases to about 16–19 days in favor of the observations summarized by Madden (1978).

## Abstract

In an attempt to understand the dynamical influence of the earth's orography upon the large-scale motion of the atmosphere, the system of “shallow water” equations on the rotating earth is integrated numerically. The model consists of an incompressible, homogeneous, hydrostatic and inviscid fluid. The “beta-plane” approximation is used to simplify the model. The fluid is confined in a channel bounded on the north and south by two parallel “walls” extending in the cast-west direction. Periodicity is the boundary condition applied at the east and west boundaries to simulate the cyclic continuity of the zone with longitude. A circular obstacle of parabolic shape is placed at the bottom in the middle of the channel. The steady-state solutions in the absence of the obstacle are used as the initial conditions of the problem. Five different cases are investigated in detail. All computations were performed for an interval of 20 days (some cases were run longer) with a time step of 6 minutes.

The following main results were obtained: 1) Westerly flows past the obstacle produced a train of long waves on the lee side, which can be identified as “planetary” waves. On the other hand, easterly flows are little disturbed by the obstacle and long waves do not appear; 2) The number of waves produced in the westerly cases agrees with the number expected from the steady-state Rossby-Haurwitz wave formula for various intensities of zonal flow past the obstacle.

The results of the present calculations agree qualitatively with the data obtained in the early 1950's by Fultz, Long and Frenzen in laboratory experiments on the flow past a barrier in a rotating hemispherical shell. Finally, a theoretical consideration is given to explain the characteristic differences between westerly and easterly flows past the obstacle as observed in the numerical experiments.

## Abstract

In an attempt to understand the dynamical influence of the earth's orography upon the large-scale motion of the atmosphere, the system of “shallow water” equations on the rotating earth is integrated numerically. The model consists of an incompressible, homogeneous, hydrostatic and inviscid fluid. The “beta-plane” approximation is used to simplify the model. The fluid is confined in a channel bounded on the north and south by two parallel “walls” extending in the cast-west direction. Periodicity is the boundary condition applied at the east and west boundaries to simulate the cyclic continuity of the zone with longitude. A circular obstacle of parabolic shape is placed at the bottom in the middle of the channel. The steady-state solutions in the absence of the obstacle are used as the initial conditions of the problem. Five different cases are investigated in detail. All computations were performed for an interval of 20 days (some cases were run longer) with a time step of 6 minutes.

The following main results were obtained: 1) Westerly flows past the obstacle produced a train of long waves on the lee side, which can be identified as “planetary” waves. On the other hand, easterly flows are little disturbed by the obstacle and long waves do not appear; 2) The number of waves produced in the westerly cases agrees with the number expected from the steady-state Rossby-Haurwitz wave formula for various intensities of zonal flow past the obstacle.

The results of the present calculations agree qualitatively with the data obtained in the early 1950's by Fultz, Long and Frenzen in laboratory experiments on the flow past a barrier in a rotating hemispherical shell. Finally, a theoretical consideration is given to explain the characteristic differences between westerly and easterly flows past the obstacle as observed in the numerical experiments.

## Abstract

The steering flow of a hurricane is obtained by eliminating the vortex pattern from the total flow field. The evolution of the steering flow is predicted by solving the barotropic nondivergent vorticity equation. Based upon the steering-flow prediction, a forecast of the hurricane movement is obtained with the use of an equation which provides for interaction between the hurricane and the steering flow.

In the geostrophic model, the geostrophic-wind assumption is used in solving the vorticity equation. In the nongeostrophic model, the stream function governed by the ‘balance equation’ is adopted. With the above two prediction models, 45 pairs of predictions of the 24-hr and 48-hr movement of hurricanes Diane and Connie (August 1955) and Betsy (August 1956) at the 500- and 700-mb levels were prepared by the use of an electronic computer.

A detailed comparison between performances of the two prediction models is presented. Generally speaking, there is a remarkable similarity between the nongeostrophic and geostrophic forecasts of hurricane movement. The accuracy of the 700-mb hurricane forecasts appears to be comparable with that obtained from the 500-mb forecasts. However, for both prediction models, the ‘resultant’ forecast, which is the vector mean of the 700- and 500-mb predicted displacements, seems to provide a significant improvement over the accuracy of a single-level barotropic forecast. These analyses suggest that further improvements are possible by advancing the prediction model from the barotropic model to the equivalent barotropic or the baroclinic model.

## Abstract

The steering flow of a hurricane is obtained by eliminating the vortex pattern from the total flow field. The evolution of the steering flow is predicted by solving the barotropic nondivergent vorticity equation. Based upon the steering-flow prediction, a forecast of the hurricane movement is obtained with the use of an equation which provides for interaction between the hurricane and the steering flow.

In the geostrophic model, the geostrophic-wind assumption is used in solving the vorticity equation. In the nongeostrophic model, the stream function governed by the ‘balance equation’ is adopted. With the above two prediction models, 45 pairs of predictions of the 24-hr and 48-hr movement of hurricanes Diane and Connie (August 1955) and Betsy (August 1956) at the 500- and 700-mb levels were prepared by the use of an electronic computer.

A detailed comparison between performances of the two prediction models is presented. Generally speaking, there is a remarkable similarity between the nongeostrophic and geostrophic forecasts of hurricane movement. The accuracy of the 700-mb hurricane forecasts appears to be comparable with that obtained from the 500-mb forecasts. However, for both prediction models, the ‘resultant’ forecast, which is the vector mean of the 700- and 500-mb predicted displacements, seems to provide a significant improvement over the accuracy of a single-level barotropic forecast. These analyses suggest that further improvements are possible by advancing the prediction model from the barotropic model to the equivalent barotropic or the baroclinic model.

## Abstract

A steering method of predicting hurricane movement is formulated based upon a two-level baroclinic model. The upper steering-flow field is constructed from the pressure-weighted mean of the 200- and 500-mb steering-height fields, and the lower steering field is constructed from the pressure-weighted mean of the 700- and 1000-mb steering-height fields. Here, the steering flow of a hurricane is defined as the residual field after eliminating the vortex pattern from the total-flow field.

The evolutions of the upper and lower steering flows are predicted simultaneously by solving the two-level steering-flow vorticity equations. Based upon those steering-flow forecasts, the movement of a hurricane is predicted with the use of an equation which is derived from a solution of two vortex vorticity equations. A side condition is imposed that the upper and lower vortex patterns should move with the same velocity in the corresponding steering flows.

Ten cases of predicting the movement of hurricane “Betsy” (August 1956) up to 48 hr are presented. A preliminary comparison of the forecasts with those obtained from the barotropic model is also made.

## Abstract

A steering method of predicting hurricane movement is formulated based upon a two-level baroclinic model. The upper steering-flow field is constructed from the pressure-weighted mean of the 200- and 500-mb steering-height fields, and the lower steering field is constructed from the pressure-weighted mean of the 700- and 1000-mb steering-height fields. Here, the steering flow of a hurricane is defined as the residual field after eliminating the vortex pattern from the total-flow field.

The evolutions of the upper and lower steering flows are predicted simultaneously by solving the two-level steering-flow vorticity equations. Based upon those steering-flow forecasts, the movement of a hurricane is predicted with the use of an equation which is derived from a solution of two vortex vorticity equations. A side condition is imposed that the upper and lower vortex patterns should move with the same velocity in the corresponding steering flows.

Ten cases of predicting the movement of hurricane “Betsy” (August 1956) up to 48 hr are presented. A preliminary comparison of the forecasts with those obtained from the barotropic model is also made.

## Abstract

A model of a tropical cyclone is constructed which is based upon conservation of momentum, mass,water vapor and heat in the hydrostatic system. The horizontal and vertical eddy-exchange processes for momentum, moisture and heat are included in the equations in order to incorporate the planetary frictional (Ekman) layer into the model. The effects of the surface boundary (Prandtl) layer are simulated by the boundary conditions for the equations, which permit the evaluation of surface stress, the sensible heat transport and the evaporation of water vapor from the earth surface. The energy sources of the model are the latent heat of condensation released during the ascent of moist air and the sensible heat transported from the ocean surface.

The formulation of the finite-difference equations for the axially-symmetric case is presented, together with an examination of the computational stability. By means of a high-speed computer, two independent computations with and without the supply of latent heat were made from the same initial wind and temperature fields.

A comparison of the two cases reveals an important effect of latent heat of condensation upon the development of the tangential motion as well as its warm-core radial circulation. It is shown that the formation of cellar convective bands in the system is a manifestation of the gravitational instability which does not occur without the latent-heat supply.

## Abstract

A model of a tropical cyclone is constructed which is based upon conservation of momentum, mass,water vapor and heat in the hydrostatic system. The horizontal and vertical eddy-exchange processes for momentum, moisture and heat are included in the equations in order to incorporate the planetary frictional (Ekman) layer into the model. The effects of the surface boundary (Prandtl) layer are simulated by the boundary conditions for the equations, which permit the evaluation of surface stress, the sensible heat transport and the evaporation of water vapor from the earth surface. The energy sources of the model are the latent heat of condensation released during the ascent of moist air and the sensible heat transported from the ocean surface.

The formulation of the finite-difference equations for the axially-symmetric case is presented, together with an examination of the computational stability. By means of a high-speed computer, two independent computations with and without the supply of latent heat were made from the same initial wind and temperature fields.

A comparison of the two cases reveals an important effect of latent heat of condensation upon the development of the tangential motion as well as its warm-core radial circulation. It is shown that the formation of cellar convective bands in the system is a manifestation of the gravitational instability which does not occur without the latent-heat supply.

## Abstract

In order to increase the accuracy of prediction of hurricane movement beyond that which has been obtained with methods now in use, a numerical procedure based upon the barotropic model is presented here.

A vortex field is separated from the total flow to obtain the residual-steering flow field. By solving the equation for the steering field, which does not include any parameters related to the vortex, the prediction of the steering flow is executed in the ordinary manner. By solving the other equation, which includes the interaction terms between the hurricane and the steering flow, the movement of the vortex pattern is predicted. To make the latter problem more tractable, a velocity formula for the movement of the vortex center is derived on the basis of a few reasonable assumptions concerning the structure of the vortex and of the steering field. The variables in the formula are expressed only in terms of quantities determined by the steering field and a single characteristic parameter related to the scale of the hurricane. By this means, a forecast of the hurricane movement can be programmed as a subroutine which is executed in the course of the prediction of the steering flow.

Five cases of predicting the 24-hour and 48-hour movements of hurricane “Diane” and “Connie” (August 1955) at the 500-mb level are presented here. These results have been obtained by the use of a high-speed computer, from initial maps which were be-analyzed using all available data.

## Abstract

In order to increase the accuracy of prediction of hurricane movement beyond that which has been obtained with methods now in use, a numerical procedure based upon the barotropic model is presented here.

A vortex field is separated from the total flow to obtain the residual-steering flow field. By solving the equation for the steering field, which does not include any parameters related to the vortex, the prediction of the steering flow is executed in the ordinary manner. By solving the other equation, which includes the interaction terms between the hurricane and the steering flow, the movement of the vortex pattern is predicted. To make the latter problem more tractable, a velocity formula for the movement of the vortex center is derived on the basis of a few reasonable assumptions concerning the structure of the vortex and of the steering field. The variables in the formula are expressed only in terms of quantities determined by the steering field and a single characteristic parameter related to the scale of the hurricane. By this means, a forecast of the hurricane movement can be programmed as a subroutine which is executed in the course of the prediction of the steering flow.

Five cases of predicting the 24-hour and 48-hour movements of hurricane “Diane” and “Connie” (August 1955) at the 500-mb level are presented here. These results have been obtained by the use of a high-speed computer, from initial maps which were be-analyzed using all available data.