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A. J. Simmons

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A. J. Simmons

Abstract

A stability analysis has been performed for an easterly lower tropospheric jet which approximates the observed flow over West Africa during summer. The most unstable disturbance has a growth rate of 0.27 day−1, a wavelength close to 4000 km and a phase speed of about 9 m s−1. Although gross features of its energetics are similar to those found by Rennick in a recent theoretical study, there are significant differences in horizontal structure. The present results appear to be in closer agreement with observation.

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A. J. Simmons

Abstract

Further numerical and analytical solutions are obtained for unstable disturbances in a two-layer, hydrostatic, quasi-geostrophic model with both vertical and meridional shear in the basic zonal wind, the meridional shear being confined to the upper layer. Contrary to a suggestion of Stone, the meridional scale of the most unstable wave is not, in general, the radius of deformation. As in previous studies, when the meridional scale of the zonal flow is of the order of the radius of deformation, the fastest-growing disturbance is found to take a similar meridional scale. However, for zonal flows of greater meridional extent, numerical solutions show the most unstable wave to have a meridional scale larger than the radius of deformation, although less than that of the zonal flow. In this case the meridional scale of the wave is shown analytically to be given, in general, by the geometric mean of the radius of deformation and a length scale determined by the curvature of the flow profile at the position of maximum upper-level wind. When this curvature vanishes, the disturbance takes a meridional scale closer to that of the zonal flow, with a weaker dependence an the radius of deformation.

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M. Hortal and A. J. Simmons

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Integrations of spectral models are presented in which the “Gaussian” grid of points at which the nonlinear terms are evaluated is reduced as the poles are approached. A maximum saving in excess of one-third the number of points covering the globe is obtained by requiring that the grid length in the zonal direction does not exceed the grid length at the equator, and that the number of points around a latitude circle enables the use of a fast Fourier transform. Tests are reported for Eulerian and semi-Lagrangian barotropic models, mostly at T106 resolution, and a summary is given of experiments based on the T106 primitive-equation model used for operational forecasting at ECMWF. The results show that such a reduced grid can be used for short- and medium-range prediction (and presumably also for climate studies) with no significant loss of accuracy compared with use of a conventional grid, which is uniform in longitude. The saying in computational time is between 20% and 25% for the T1O6 forecast model. There are also potential reductions in the memory requirement of the model and in the storage needed for the archiving of model results.

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A. J. Simmons and C. Temperton

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A study is made of the computational stability of semi-implicit treatments of gravity wave motion suitable for use with two-time-level advection schemes. The analysis is for horizontally uniform reference values of temperature and surface pressure, and for hybrid pressure-based vertical coordinates. Stability requires the use of reference temperatures that are warmer than those that can be used safely with the corresponding three-time-level scheme. The reference surface pressure should also be higher. When stable, the two-time-level scheme is damping, although the largest scales are damped less than by the three-time-level scheme if the latter uses a typical time filtering. The first-order decentered averaging of gravity wave tendencies used in a number of semi-Lagrangian models reduces the need for a relatively warm reference temperature profile but causes a quite substantial damping of otherwise well-represented low-wavenumber modes. The low-wavenumber damping can be avoided by using an alternative, second-order averaging involving a third (past) time level. For this alternative averaging, an economical spatial discretization is proposed that requires no additional departure point. Phase speeds show little sensitivity to these changes in formulation. All variants of the semi-implicit method substantially reduce the phase speeds of the fastest high-wavenumber modes when use is made of the large time steps possible with semi-Lagrangian advection.

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A. J. Simmons and B. J. Hoskins

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A. J. Simmons and B. J. Hoskins

Abstract

The authors' previous study of baroclinic instability using a primitive equation model with spherical geometry is extended to include more realistic initial distributions of zonal wind and temperature in the upper troposphere and lower stratosphere. Results show little difference in the low-level structure of normal modes, but generally larger upper-level amplitudes for wavelengths close to or longer than that giving maximum linear growth rate. Near the tropopause these disturbances may extend significantly toward the equator, a result shown to be consistent with the forced barotropic response of tropical regions. Their eddy momentum fluxes exhibit some of the variability noted previously, but transfer tends to be predominantly poleward at upper levels. The upper-level heat flux is stronger relative to the momentum flux than is found in general circulation statistics.

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A. J. Simmons and B. J. Hoskins

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The growth rate, phase speed, structure and transfer properties of normal modes of the primitive and quasi-geostrophic equations have been determined by applying an initial value technique to global nonlinear atmospheric models. Results are presented for three zonal flows that have the same vertical structure but quite different meridional variations. Use of a variety of vertical and horizontal resolutions gives important indications of truncation error.

Many properties of the unstable modes are much as found in simpler models of baroclinic instability, but spherical geometry has a significant effect on the location of the disturbances, particularly those of low zonal wavenumber, and on eddy momentum fluxes. The latter vary greatly from profile to profile, but mean meridional circulations are such as to give little net variation in the pattern of induced mean zonal surface winds. In fact, the change in vertical shear at the surface is shown to depend in the quasi-geostrophic limit only on the poleward eddy heat flux, which varies little, except in meridional position. Quasi-geostrophic solutions are generally similar to those of the primitive equations, although small differences are often of consistent sign. However, neglect of vertical eddy heat transfer, and to a lesser extent momentum transfer, is a poor approximation.

The present results are in some qualitative agreement with others obtained independently using two-level models, but such models are shown to be subject to severe quantitative error. More generally, vertical truncation error is found to give rise to spurious high-wavenumber growth.

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A. J. Simmons and B. J. Hoskins

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Baroclinic instability calculations published by Gall are repeated using the model developed by the authors for use in their own stability studies. Results indicate that some of the differences in the wavelength of maximum linear growth rate found previously may be accounted for by differences in flow profile, but there remains a discrepancy which is a likely consequence of either truncation error in Gall's calculations or a coding error in one or the other model (or both).

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A. J. Simmons and D. M. Burridge

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An energy and angular-momentum conserving vertical finite-difference scheme is introduced for a general terrain-following vertical coordinate which is a function of pressure and its surface value. A corresponding semi-implicit time scheme is also defined. These schemes am used to compare the usual sigma coordinate with the hybrid coordinate which reduces to pressure above a fixed level and with a modified hybrid coordinate which tends uniformly to pressure at upper levels. Error in the representation of the stratospheric pressure gradient over steep orography can be significantly reduced by use of the hybrid coordinate but the semi-implicit scheme is less stable. The modified hybrid coordinate offers a useful compromise.

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