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- Author or Editor: A. WIIN-NIELSEN x
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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.
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.
Abstract
The results of extended integrations of a two-level, quasi-geostrophic model with Newtonian heating and dissipation in terms of surface friction, internal friction, and lateral diffusion are described. The major emphasis is on an analysis of the integrations in wavenumber space, including the calculations of spectra for available potential energy, kinetic energy, enstrophy, energy generations, conversions and dissipation, as well as the nonlinear cascades of the first three quantities.
It is found that the fluxes of available potential energy and kinetic energy through the wavenumber domain are very small above planetary wavenumber n = 10, while the enstrophy flux is large and positive for 6 ≤ n ≤ 10, but decreases rapidly for n > 10. The available potential energy, the kinetic energy and the enstrophy as a function of wavenumber vary approximately as n −5, n −3 and n −1, for 10 ≤ n ≤ 20.
Dimensional considerations based on a balance between the convergence of the enstrophy flux and the dissipation of enstrophy in wavenumber space is used to describe the experimental spectra for n > 10. In this analysis it is assumed that the flux of enstrophy is proportional to the product of the wavenumber and the enstrophy divided by a time scale which is related to the flux of enstrophy coming from the baroclinically active region in the wavenumber domain. Theory and experiment are compared with generally good agreement.
Abstract
The results of extended integrations of a two-level, quasi-geostrophic model with Newtonian heating and dissipation in terms of surface friction, internal friction, and lateral diffusion are described. The major emphasis is on an analysis of the integrations in wavenumber space, including the calculations of spectra for available potential energy, kinetic energy, enstrophy, energy generations, conversions and dissipation, as well as the nonlinear cascades of the first three quantities.
It is found that the fluxes of available potential energy and kinetic energy through the wavenumber domain are very small above planetary wavenumber n = 10, while the enstrophy flux is large and positive for 6 ≤ n ≤ 10, but decreases rapidly for n > 10. The available potential energy, the kinetic energy and the enstrophy as a function of wavenumber vary approximately as n −5, n −3 and n −1, for 10 ≤ n ≤ 20.
Dimensional considerations based on a balance between the convergence of the enstrophy flux and the dissipation of enstrophy in wavenumber space is used to describe the experimental spectra for n > 10. In this analysis it is assumed that the flux of enstrophy is proportional to the product of the wavenumber and the enstrophy divided by a time scale which is related to the flux of enstrophy coming from the baroclinically active region in the wavenumber domain. Theory and experiment are compared with generally good agreement.
Abstract
The geostrophic adjustment problem is considered for the case of a homogeneous fluid in a rotating cylindrical container. The formal solution for an arbitrary initial disturbance in the axisymmetrical case is obtained in terms of a series expansion in Bessel functions of zero order. The solution shows that the motion will consist of a time-independent part in geostrophic balance and a time-dependent part which is oscillatory. The general properties of the solution including the energeties are investigated.
The degree to which the process is simulated by various finite-difference methods for the time derivatives is investigated using the leap-frog, a semi-implicit scheme, and a scheme which combines forward and backward time differences (mixed scheme). It is found that all schemes are acceptable provided the time step is sufficiently small, but in general the simulation of the process by the leap-frog and the mixed scheme is more realistic.
The present conclusions are of importance in any scheme employed for the purposes of assimilation of meteorological data. The analysis can be expanded to a more general case than the axisymmetrical one.
Abstract
The geostrophic adjustment problem is considered for the case of a homogeneous fluid in a rotating cylindrical container. The formal solution for an arbitrary initial disturbance in the axisymmetrical case is obtained in terms of a series expansion in Bessel functions of zero order. The solution shows that the motion will consist of a time-independent part in geostrophic balance and a time-dependent part which is oscillatory. The general properties of the solution including the energeties are investigated.
The degree to which the process is simulated by various finite-difference methods for the time derivatives is investigated using the leap-frog, a semi-implicit scheme, and a scheme which combines forward and backward time differences (mixed scheme). It is found that all schemes are acceptable provided the time step is sufficiently small, but in general the simulation of the process by the leap-frog and the mixed scheme is more realistic.
The present conclusions are of importance in any scheme employed for the purposes of assimilation of meteorological data. The analysis can be expanded to a more general case than the axisymmetrical one.
