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- Author or Editor: A. WIIN-NIELSEN x

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## Abstract

The quasi-geostrophic, baroclinic stability problem is solved for an arbitrary zonal wind profile *U = U*(*p*) and for an adiabatic lapse rate. It is shown that the phase speed of the waves in this case depends on the vertical integrals of *U* and *U*
^{2}. Due to the assumption of an adiabatic stratification there is no short wave cutoff, but the effect of the variation of the Coriolis parameter will in all cases give stability for sufficiently long waves.

A number of numerical examples show that the region of instability, in a coordinate system with wavelength as abscissa and wind shear as ordinate, is the largest when the wind maximum is situated in the upper part of the atmosphere, and when the curvature of the zonal wind profile at the wind maximum has an intermediate value.

## Abstract

The quasi-geostrophic, baroclinic stability problem is solved for an arbitrary zonal wind profile *U = U*(*p*) and for an adiabatic lapse rate. It is shown that the phase speed of the waves in this case depends on the vertical integrals of *U* and *U*
^{2}. Due to the assumption of an adiabatic stratification there is no short wave cutoff, but the effect of the variation of the Coriolis parameter will in all cases give stability for sufficiently long waves.

A number of numerical examples show that the region of instability, in a coordinate system with wavelength as abscissa and wind shear as ordinate, is the largest when the wind maximum is situated in the upper part of the atmosphere, and when the curvature of the zonal wind profile at the wind maximum has an intermediate value.

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## Abstract

The kinetic energy of the horizontal, hydrostatic flow is divided into the kinetic energies of the vertically integrated flow and the deviation from this flow, the so-called shear flow. The energy transformation between the two types of flow is found in the general case of the primitive equations and also for the most simple quasi-non-divergent model. The two transformations are discussed, and the energy transformation in the quasi-non-divergent model in the two-parameter case is discussed as a function of wave number using linear theory. The energy conversion has been computed on a daily basis for the month of January 1959, and compared with earlier results of computations of transformations between available potential energy and shear flow kinetic energy. It is shown that the latter conversion changes the kinetic energy of the shear flow and not that of the mean flow. The residence time is estimated for the shear flow as well as the mean flow.

The energy transformation between the vertical shear flow and mean flow due to the non-divergent and divergent flow has been computed in the wave-number regime for the first 10 zonal wave numbers for each day in January 1959. It is found that the energy conversion between shear flow and mean flow is about 30 percent of the conversion between the available potential energy and the shear flow kinetic energy.

A further result is that the energy conversion between the shear flow and the mean flow due to the divergent part of the flow is estimated to be negative and ahout 10 percent of the conversion due to the non-divergent part of the flow.

The energy conversion as a function of wave number shows a maximum for the most unstable baroclinic waves.

## Abstract

The kinetic energy of the horizontal, hydrostatic flow is divided into the kinetic energies of the vertically integrated flow and the deviation from this flow, the so-called shear flow. The energy transformation between the two types of flow is found in the general case of the primitive equations and also for the most simple quasi-non-divergent model. The two transformations are discussed, and the energy transformation in the quasi-non-divergent model in the two-parameter case is discussed as a function of wave number using linear theory. The energy conversion has been computed on a daily basis for the month of January 1959, and compared with earlier results of computations of transformations between available potential energy and shear flow kinetic energy. It is shown that the latter conversion changes the kinetic energy of the shear flow and not that of the mean flow. The residence time is estimated for the shear flow as well as the mean flow.

The energy transformation between the vertical shear flow and mean flow due to the non-divergent and divergent flow has been computed in the wave-number regime for the first 10 zonal wave numbers for each day in January 1959. It is found that the energy conversion between shear flow and mean flow is about 30 percent of the conversion between the available potential energy and the shear flow kinetic energy.

A further result is that the energy conversion between the shear flow and the mean flow due to the divergent part of the flow is estimated to be negative and ahout 10 percent of the conversion due to the non-divergent part of the flow.

The energy conversion as a function of wave number shows a maximum for the most unstable baroclinic waves.

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## Abstract

Energy conversion between potential and kinetic energy is considered. Section 2 contains the derivations which are necessary to compute the energy conversion for a large region on the basis of vertical velocities and relative topography as obtained from a two-parameter model presently used by the Joint Numerical Weather Prediction Unit. The energy conversion is divided into three parts: (1) energy conversion due to a mean vertical velocity over the region, (2) energy conversion in meridional planes, and (3) energy conversion in the zonal planes.

Section 3 contains a discussion of the results obtained for the months January and April 1959. The energy conversion is positive for each day in both months, but the conversion in the meridional planes has a different sign in the two months, being positive in January and negative in April. The pattern of the mean meridional circulation is discussed and the frictional dissipation estimated.

Section 4 describes a procedure for an evaluation of the energy conversion for the different wave numbers, and discusses the results for the same two months. The same section contains a comparison with results obtained from a linear, adiabatic theory.

Section 5 contains a discussion of the modifications to the results in section 4 caused by the diabatic heating of the atmosphere It is made plausible that the maximum conversion found for the small wave numbers by an adiabatic computation is greatly altered due to the effects of the heating.

## Abstract

Energy conversion between potential and kinetic energy is considered. Section 2 contains the derivations which are necessary to compute the energy conversion for a large region on the basis of vertical velocities and relative topography as obtained from a two-parameter model presently used by the Joint Numerical Weather Prediction Unit. The energy conversion is divided into three parts: (1) energy conversion due to a mean vertical velocity over the region, (2) energy conversion in meridional planes, and (3) energy conversion in the zonal planes.

Section 3 contains a discussion of the results obtained for the months January and April 1959. The energy conversion is positive for each day in both months, but the conversion in the meridional planes has a different sign in the two months, being positive in January and negative in April. The pattern of the mean meridional circulation is discussed and the frictional dissipation estimated.

Section 4 describes a procedure for an evaluation of the energy conversion for the different wave numbers, and discusses the results for the same two months. The same section contains a comparison with results obtained from a linear, adiabatic theory.

Section 5 contains a discussion of the modifications to the results in section 4 caused by the diabatic heating of the atmosphere It is made plausible that the maximum conversion found for the small wave numbers by an adiabatic computation is greatly altered due to the effects of the heating.

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## Abstract

Different factors influencing the changes in the zonally averaged wind are investigated. It is evident from recent investigations of errors in the zonally averaged winds in the non-divergent, one-parameter model that the convergence of the meridional transport of zonal momentum concentrates too much momentum in the middle latitudes and predicts too small amounts of momentum in the low and high latitudes.

The other factors influencing the changes of zonal momentum, i.e., mean meridional circulations, friction, and vertical transport of momentum, are investigated on an averaged basis in the, following sections: In section 3 it is shown that the divergent, one-parameter model will reduce the errors in zonal momentum predicted by the non-divergent model, due to an implied mean meridional circulation. The corrections to the predictions of changes in zonal momentum caused by mean meridional circulations in a two-parameter model are investigated in section 4 by the aid of operationally computed initial values of the vertical velocities. It, is shown that reductions in the errors of the non-divergent model with respect to zonal momentum can be expected with a careful arrangement of the information levels in a two-parameter model.

Section 5 contains a similar investigation of the averaged contribution of vertical advection t o changes in zonal momentum. It is found that this contribution is smaller than the one resulting from mean meridional circulations, and further that the contribution from the vertical advection of momentum is not likely to reduce the errors found in the non-divergent predictions.

The main conclusion from the study is that the contributions from mean meridional circulations and surface friction are the most important for the reductions of errors in the prediction of zonal momentum in the non-divergent model. Some reduction of the errors can he expected in the divergent, one-parameter model or in a two-parameter model with a proper arrangement of the information levels. In order t o incorporate surface friction in a realistic way, and further in order to avoid the artificial constraint of a non-divergent level appearing in a two-parameter model, it is most likely that more than two parameters ale needed for accurate forecasts of zonal momentum.

## Abstract

Different factors influencing the changes in the zonally averaged wind are investigated. It is evident from recent investigations of errors in the zonally averaged winds in the non-divergent, one-parameter model that the convergence of the meridional transport of zonal momentum concentrates too much momentum in the middle latitudes and predicts too small amounts of momentum in the low and high latitudes.

The other factors influencing the changes of zonal momentum, i.e., mean meridional circulations, friction, and vertical transport of momentum, are investigated on an averaged basis in the, following sections: In section 3 it is shown that the divergent, one-parameter model will reduce the errors in zonal momentum predicted by the non-divergent model, due to an implied mean meridional circulation. The corrections to the predictions of changes in zonal momentum caused by mean meridional circulations in a two-parameter model are investigated in section 4 by the aid of operationally computed initial values of the vertical velocities. It, is shown that reductions in the errors of the non-divergent model with respect to zonal momentum can be expected with a careful arrangement of the information levels in a two-parameter model.

Section 5 contains a similar investigation of the averaged contribution of vertical advection t o changes in zonal momentum. It is found that this contribution is smaller than the one resulting from mean meridional circulations, and further that the contribution from the vertical advection of momentum is not likely to reduce the errors found in the non-divergent predictions.

The main conclusion from the study is that the contributions from mean meridional circulations and surface friction are the most important for the reductions of errors in the prediction of zonal momentum in the non-divergent model. Some reduction of the errors can he expected in the divergent, one-parameter model or in a two-parameter model with a proper arrangement of the information levels. In order t o incorporate surface friction in a realistic way, and further in order to avoid the artificial constraint of a non-divergent level appearing in a two-parameter model, it is most likely that more than two parameters ale needed for accurate forecasts of zonal momentum.

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## Abstract

A comparison is made between the stability properties of the non-divergent and divergent one-parameter models. It is show that the introduction of divergence in the one-parameter model reduces the rate of growth of unstable disturbances arid confines the instability to a more narrow band of wavelengths.

Changes in zonal momentum, momentum transports, and energy conversions between mean flow kinetic energy and eddy kinetic energy are investigated in the linear case, as well as by extended integrations of the spectral form of the prognostic equations allowing only a few wave numbers.

Long-term variations in barotropic flow, where the flow is initially stable, are investigated using the spectral formulation, but allowing only as many wave numbers as are needed to investigate variations in the profile of the zonal wind.

## Abstract

A comparison is made between the stability properties of the non-divergent and divergent one-parameter models. It is show that the introduction of divergence in the one-parameter model reduces the rate of growth of unstable disturbances arid confines the instability to a more narrow band of wavelengths.

Changes in zonal momentum, momentum transports, and energy conversions between mean flow kinetic energy and eddy kinetic energy are investigated in the linear case, as well as by extended integrations of the spectral form of the prognostic equations allowing only a few wave numbers.

Long-term variations in barotropic flow, where the flow is initially stable, are investigated using the spectral formulation, but allowing only as many wave numbers as are needed to investigate variations in the profile of the zonal wind.

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## Abstract

The equation for the vertical velocity in a quasi-non-divergent, three-parameter model has been solved for a certain simple flow pattern. The importance of vertical variation of static stability and of the horizontal wind is discussed from the solutions. Examples showing the distribution of divergence relative to the synoptic systems are computed and compared with those obtained by other investigators. A general discussion of the factors influencing the mid-tropospheric divergence follows in section 4, and section 5 contains finally some remarks on the divergence in very long waves.

## Abstract

The equation for the vertical velocity in a quasi-non-divergent, three-parameter model has been solved for a certain simple flow pattern. The importance of vertical variation of static stability and of the horizontal wind is discussed from the solutions. Examples showing the distribution of divergence relative to the synoptic systems are computed and compared with those obtained by other investigators. A general discussion of the factors influencing the mid-tropospheric divergence follows in section 4, and section 5 contains finally some remarks on the divergence in very long waves.

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## Abstract

It is shown that calculations of the energy conversion from available potential energy to kinetic energy based on vertical velocities obtained by the so-called adiabatic method may lead to erroneous results. An analysis of the method shows that it measures the difference between the energy conversion from available potential energy to kinetic energy, and the generation of available potential by diabatic heating. This result holds for both zonal and eddy avai1able potential energy.

The main conclusion is tested using numerical results from several energy conversion and energy generation studies.

The adiabatic method is compared with other methods to estimate vertical velocities.

## Abstract

It is shown that calculations of the energy conversion from available potential energy to kinetic energy based on vertical velocities obtained by the so-called adiabatic method may lead to erroneous results. An analysis of the method shows that it measures the difference between the energy conversion from available potential energy to kinetic energy, and the generation of available potential by diabatic heating. This result holds for both zonal and eddy avai1able potential energy.

The main conclusion is tested using numerical results from several energy conversion and energy generation studies.

The adiabatic method is compared with other methods to estimate vertical velocities.

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## Abstract

The problem of control of the ultra-long waves in numerical prediction is considered. Sections 2â€“4 contain a discussion of the one-level forecasts. It is shown that the vertical variation of the horizontal wind and the static stability are the main factors in determining the value of divergence a t 500 mb. An independent estimate of the size of the term governing the ultra-long waves in the atmosphere agrees well with the one determined by Cressman on an empirical basis.

Section 5 points out that any two-parameter model has to contain an effect similar to the one contained in the one-parameter model controlling the ultra-long waves. A modification of a two-parameter model is made in such a way that the ultra-long wares are controlled.

Section 6 describes a perturbation analysis of the model developed in Section 5 in order to investigate the effect of the modification also on the shorter waves. It is found that a certain stabilization of the shorter waves is produced. Baroclinic instability and growth rate are investigated.

Sections 7 and 8 contain a justification of certain approximations used in the earlier sections regarding the vertical variation of static stability and the profile of vertical velocity.

## Abstract

The problem of control of the ultra-long waves in numerical prediction is considered. Sections 2â€“4 contain a discussion of the one-level forecasts. It is shown that the vertical variation of the horizontal wind and the static stability are the main factors in determining the value of divergence a t 500 mb. An independent estimate of the size of the term governing the ultra-long waves in the atmosphere agrees well with the one determined by Cressman on an empirical basis.

Section 5 points out that any two-parameter model has to contain an effect similar to the one contained in the one-parameter model controlling the ultra-long waves. A modification of a two-parameter model is made in such a way that the ultra-long wares are controlled.

Section 6 describes a perturbation analysis of the model developed in Section 5 in order to investigate the effect of the modification also on the shorter waves. It is found that a certain stabilization of the shorter waves is produced. Baroclinic instability and growth rate are investigated.

Sections 7 and 8 contain a justification of certain approximations used in the earlier sections regarding the vertical variation of static stability and the profile of vertical velocity.

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## Abstract

The behavior of very long waves in a two-parameter model with no divergence in the mean flow has been investigated. It is found that the temperature field and the pressure field move almost independently of each other. The pressure field will retrograde with a speed comparable to the Rossby speed for non-divergent waves, while the temperature field progresses slowly. As a result, the latter field will precede the pressure field after a while, verifying an earlier observation in these forecasts. Introducing a divergence in the vertical mean flow not only greatly reduces the retrogression, but a stronger coupling then exists between the temperature and pressure fields.

## Abstract

The behavior of very long waves in a two-parameter model with no divergence in the mean flow has been investigated. It is found that the temperature field and the pressure field move almost independently of each other. The pressure field will retrograde with a speed comparable to the Rossby speed for non-divergent waves, while the temperature field progresses slowly. As a result, the latter field will precede the pressure field after a while, verifying an earlier observation in these forecasts. Introducing a divergence in the vertical mean flow not only greatly reduces the retrogression, but a stronger coupling then exists between the temperature and pressure fields.

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## Abstract

The normal mode initialization procedure is investigated. It is shown that a balance exists between the wind field and the mass field when the gravity modes have been removed from the initial fields. Adopting a representation in the spectral domain on the sphere it is shown that the vectors consisting of all amplitudes of the streamfunction and the velocity potential, respectively, are related to the vector consisting of all amplitudes of the geopotential by a square matrix which depends entirely on the eigenvalues and eigenvectors of the truncated systems.

The balance which exists after normal mode initialization is compared with the quasi-geostrophic balance when this procedure is applied to the adjusted initial fields which are obtained when the contribution from the gravity waves has been removed. It turns out that the balance from the normal mode procedure is virtually identical to the quasi-geostrophic balance *except* on the largest scales. The difference on the largest scale between the Rossby or rotational modes obtained by the two procedures is in the linear case entirely due to the sphericity of the earth since the modes would be identical if the Coriolis parameter were constant.

The modifications to the initial state created by the normal mode procedure are investigated in Section 5, and Section 6 contains an analysis of the first baroclinic mode analogous to the basic barotropic mode considered in the main body of the paper.

## Abstract

The normal mode initialization procedure is investigated. It is shown that a balance exists between the wind field and the mass field when the gravity modes have been removed from the initial fields. Adopting a representation in the spectral domain on the sphere it is shown that the vectors consisting of all amplitudes of the streamfunction and the velocity potential, respectively, are related to the vector consisting of all amplitudes of the geopotential by a square matrix which depends entirely on the eigenvalues and eigenvectors of the truncated systems.

The balance which exists after normal mode initialization is compared with the quasi-geostrophic balance when this procedure is applied to the adjusted initial fields which are obtained when the contribution from the gravity waves has been removed. It turns out that the balance from the normal mode procedure is virtually identical to the quasi-geostrophic balance *except* on the largest scales. The difference on the largest scale between the Rossby or rotational modes obtained by the two procedures is in the linear case entirely due to the sphericity of the earth since the modes would be identical if the Coriolis parameter were constant.

The modifications to the initial state created by the normal mode procedure are investigated in Section 5, and Section 6 contains an analysis of the first baroclinic mode analogous to the basic barotropic mode considered in the main body of the paper.