The problem of symmetric stability is examined within the context of the direct Liapunov method. The sufficient conditions for stability derived by Fjørtoft are shown to imply finite-amplitude, normed stability. This finite-amplitude stability theorem is then used to obtain rigorous upper bounds on the saturation amplitude of disturbances to symmetrically unstable flows.By employing a virial functional, the necessary conditions for instability implied by the stability theorem are shown to be in fact sufficient for instability. The results of Ooyama are improved upon insofar as a tight two-sided (upper and lower) estimate is obtained of the growth rate of (modal or nonmodal) symmetric instabilities.The case of moist adiabatic systems is also considered

Grieger, N.

Schmitz, G.

Schnur, R.

von Storch, H.

The nonlinear evolution of perturbation superimposed on the barotropic, unstable observed mean easterly jet at 100 hPa is studied over the sphere. The nondivergent barotropic nonlinear global spectral model with rhomboidal truncation at zonal wavenumber 21 is integrated for 120 days for initial random and linear unstable perturbations. The model includes a Rayleigh friction and restoring mechanism for zonal wind to its initial distribution. Time variations of eddy and zonal kinetic energy, zonal-wave and wave-wave interactions, and eddy and zonal kinetic energy dissipations are examined. The growth of perturbation begins with exponential increase in its kinetic energy for a short period, followed by a linear increase. The perturbations undergo oscillations before approaching a steady state. During the initial exponential phase, the nonlinear interactions further destabilize the jet, and this effect is more pronounced for the random initial perturbation. It is found that oscillations in kinetic energy are due to the time lag between energy source (zonal-wave interaction) and sink (frictional dissipation). It is noticed that the random initial perturbation is more efficient than the linear unstable modes in reaching the steady state. Nonlinear interactions shift the preferred wave towards the lower wavenumber 6. Wave 6 accounts for more than 98% of eddy kinetic energy of steady state. The role of wave-wave interaction is secondary to zonal-wave interaction. Steady-state wave 6 has an inviscid growth rate of 0.104 × 10−5 s−1, a phase speed of −20.9° day−1, and a meridional scale comparable to the half-width of the easterly jet. The zonal scale of the wave is close to the Rossby radius of deformation. The maximum meridional velocity associated with the wave is 3.8 m s−1. Nonlinear interactions modify the jet to a more asymmetric distribution, which induces a southward shift in the wave- amplitude maximum location. Strong correlation between latitude of maxima for wave amplitude and meridional gradient of basic-state potential vorticity is noticed. A close agreement is seen between observation and nonlinear preferred wave 6. The kinetic energy cycle for steady-state wave 6 on the sphere and in the latitude belt 10°S–40°N is computed and discussed. The kinetic energy of wave 6 in the latitude belt agrees well with the observed value

Mishra, SK

Ministry of Earth Sciences, Government of India

Non linear barotropic instability of upper tropospheric tropical easterly jet on the sphere

The principal oscillation pattern (POP) analysis is a technique to empirically identify time-dependent spatial patterns in a multivariate time series of geophysical or other data. In order to investigate medium-scale and synoptic waves in the atmosphere it has been applied to tropospheric geopotential height fields of ECMWF analyses from 1984 to 1987. The data have been subjected to zonal Fourier decomposition and to time filtering so that variations with periods between 3 and 25 days were retained. Analyses have been performed separately for each zonal wavenumber 5-9 on the Northern Hemisphere in winter and on the Southern Hemisphere in summer (DJF). POPs can be seen as normal modes of a linear approximation to a more complex dynamical system. The system matrix is estimated from observations of nature. This concept is compared with conventional stability analysis where the system matrix of the linear system is derived from theoretical, in this case quasigeostrophic, reasoning. Only the mean basic flow depends on time- and space-averaged fields of observed wind and temperature from the ECMWF data. It turns out that the most significant POPs are very similar in time and spatial structure to the most unstable waves in the stability analysis. They describe the linear growth phase of baroclinic, unstable waves that propagate eastward with periods of 3-7 days. Since the POPs are purely derived from observations, the results indicate the appropriateness of the assumptions usually made in linear stability analysis of zonally symmetric flows to explain high-frequency atmospheric fluctuations. Moreover, the POP analysis reveals patterns that are not found in the linear stability analysis. These can possibly be attributed to the nonlinear decay phase of baroclinic waves. Eliassen-Palm cross sections help clarify the interpretation of the POPs in terms of the life cycle of nonlinear baroclinic waves

MPG.PuRe

English

Normal modes of the atmosphere as estimated by Prinicipal Oscillation Patterns and derived from quasi-geostrophic theory

The excitation and propagation of equatorial planetary waves and inertia-gravity waves were studied by comparing simulations from the comprehensive GFDL troposphere-stratosphere-mesosphere SKYHI general circulation model (GCM) and from a linear primitive equation model with the same domain and numerical resolution. The basic state of the linear model is time dependent and is derived from the mean zonal wind and temperature obtained from a simulation with the full SKYHI model. The latent and convective heating fields of this SKYHI integration are used as the forcing for the linear model in a parallel simulation. The wavelength and frequency characteristics of the prominent vertically propagating equatorial Kelvin and Rossby-gravity waves are remarkably similar in the linear model and in SKYHI. Amplitudes are also similar in the lower stratosphere, indicating that the latent and convective heating is the dominant mechanism producing equatorial wave activity in the GCM. The amplitude of these waves in the upper stratosphere and mesosphere is larger in the linear model than in SKYHI. Given that the linear and SKYHI models have comparable radiative damping and horizontal subgrid scale diffusion, it appears that the wave amplitudes in SKYHI are limited by some nonlinear saturation, possibly involving the subgrid-scale vertical mixing. At low latitudes the linear model reproduces the flux of upward-propagating inertia-gravity waves seen in the full model. The results also show that a significant fraction of the inertia-gravity wave activity found in the midlatitude mesosphere of the SKYHI model can be accounted for by tropical convective heating. The global-scale Rossby normal modes seen in observations were also identified in the analyses of westward-propagating planetary waves in both models. They are of realistic amplitude in the SKYHI simulation but are much weaker in the linear model. Thus, it appears that latent and convective heating is not the main source of excitation for the Rossby normal modes

Manzini, E.

HAMILTON, K.

Middle atmospheric traveling waves forced by latent and convective heating

Rigorous upper bounds are derived on the saturation amplitude of baroclinic instability in the two-layer model. The bounds apply to the eddy energy and are obtained by appealing to a finite amplitude conservation law for the disturbance pseudoenergy. These bounds are to be distinguished from those derived in Part I of this study, which employed a pseudomomentum conservation law and provided bounds on the eddy potential enstrophy. The bounds apply to conservative (inviscid, unforced) flow, as well as to forced-dissipative flow when the dissipation is proportional to the potential vorticity.



Bounds on the eddy energy are worked out for a general class of unstable westerly jets. In the special case of the Phillips model of baroclinic instability, and in the limit of infinitesimal initial eddy amplitude, the bound states that the eddy energy cannot exceed ϵβ2/6F where ϵ = (U − Ucrit)/Ucrit is the relative supercriticality. This bound captures the essential dynamical scalings (i.e., the dependence on ϵ, β, and F) of the saturation amplitudes predicted by weakly nonlinear theory, as well as exhibiting remarkable quantitative agreement with those predictions, and is also consistent with heuristic baroclinic adjustment estimates

Shepherd, Theodore G.

Central Archive at the University of Reading

Nonlinear saturation of baroclinic instability: part III: bounds on the energy Article Published Version Shepherd, T. G. (1993) Nonlinear saturation of baroclinic instability: part III: bounds on the energy. Journal of the Atmospheric Sciences, 50 (16). pp. 2697­2709. ISSN 0022­4928 doi: https://doi.org/10.1175/1520­0469(1993)050<2697:NSOBIP>2.0.CO;2 Available at http://centaur.reading.ac.uk/32904/ It is advisable to refer to the publisher’s version if you intend to cite from the work. Published version at: http://dx.doi.org/10.1175/1520­0469(1993)050<2697:NSOBIP>2.0.CO;2 To link to this article DOI: http://dx.doi.org/10.1175/1520­0469(1993)050<2697:NSOBIP>2.0.CO;2 All outputs in CentAUR are protected by Intellectual Property Rights law, including copyright law. Copyright and IPR is retained by the creators or other copyright holders. Terms and conditions for use of this material are defined in the End User Agreement . www.reading.ac.uk/centaur CentAUR Central Archive at the University of Reading Reading’s research outputs online

Nonlinear saturation of baroclinic instability: part III: bounds on the energy

Xie, Shang-Ping

Kubokawa, Atsushi

Hanawa, Kimio

Institutional Repositories DataBase (IRDB)

Evaporation-wind feedback and the organizing of tropical convection on the planetary scale. Part II: Nonlinear evolution

Cho, H-R.

Shepherd, T. G.

Vladimirov, V. A.

Application of the direct Liapunov method to the problem of symmetric stability in the  atmosphere Article Published Version Cho, H.­R., Shepherd, T. G. and Vladimirov, V. A. (1993) Application of the direct Liapunov method to the problem of symmetric stability in the atmosphere. Journal of the Atmospheric Sciences, 50 (6). pp. 822­836. ISSN 1520­0469 doi: https://doi.org/10.1175/1520­0469(1993)050<0822:AOTDLM>2.0.CO;2 Available at http://centaur.reading.ac.uk/32917/ It is advisable to refer to the publisher’s version if you intend to cite from the work. Published version at: http://dx.doi.org/10.1175/1520­0469(1993)050<0822:AOTDLM>2.0.CO;2 To link to this article DOI: http://dx.doi.org/10.1175/1520­0469(1993)050<0822:AOTDLM>2.0.CO;2 Publisher: American Meteorological Society All outputs in CentAUR are protected by Intellectual Property Rights law, including copyright law. Copyright and IPR is retained by the creators or other copyright holders. Terms and conditions for use of this material are defined in the End User Agreement . www.reading.ac.uk/centaur CentAUR Central Archive at the University of Reading Reading’s research outputs online