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Petros J. Ioannou

Abstract

Recently, a new theoretical and conceptual model of quasigeostrophic turbulence has been advanced in which eddy variance is regarded as being maintained by transient growth of perturbations arising from sources including the nonlinear interactions among the eddies, but crucially without a direct contribution of unstable modal growth to the maintenance of variance. This theory is based on the finding that stochastic forcing of the subcritical atmospheric flow supports variance arising from induced transfer of energy from the background flow to the disturbance field that substantially exceeds the variance expected from the decay rate of the associated normal modes in an equivalent normal system. Herein the authors prove that such amplification of variance is a general property of the stochastic dynamics of systems governed by nonnormal evolution operators and that consequently the response of the atmosphere to unbiased forcing is always underestimated when consideration is limited to the response of the system's individual normal modes to stochastic excitation.

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Petros Ioannou and Richard S. Lindzen

Abstract

A formalism is developed for the calculation of baroclinic instability for barotropically stable jets. The formalism is applied to jet versions of both the Eady and Charney problems. It is found that jets act to confine instabilities meridionally, thus internally determining meridional wave scales. Once this internally determined meridional scale is taken into account, results correspond plausibly to classical results without a jet.

Consideration of the effect of such instabilities on the mean flow shows that they act to concentrate the jet barotropically while simultaneously reducing baroclinicity.

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Eli Tziperman and Petros J. Ioannou

Abstract

The physical mechanisms of transient amplification of initial perturbations to the thermohaline circulation (THC), and of the optimal stochastic forcing of THC variability, are discussed using a simple meridional box model. Two distinct mechanisms of transient amplification are found. One such mechanism, with a transient amplification timescale of a couple of years, involves an interaction between the THC induced by rapidly decaying sea surface temperature anomalies and the THC induced by the slower-decaying salinity mode. The second mechanism of transient amplification involves an interaction between different slowly decaying salinity modes and has a typical growth timescale of decades. The optimal stochastic atmospheric forcing of heat and freshwater fluxes are calculated as well. It is shown that the optimal forcing induces low-frequency THC variability by exciting the salinity-dominated variability modes of the THC.

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Brian F. Farrell and Petros J. Ioannou

Abstract

Temporally distributed deterministic and stochastic excitation of the tangent linear forecast system governing forecast error growth and the tangent linear observer system governing assimilation error growth is examined. The method used is to determine the optimal set of distributed deterministic and stochastic forcings of the forecast and observer systems over a chosen time interval. Distributed forcing of an unstable system addresses the effect of model error on forecast error in the presumably unstable forecast error system. Distributed forcing of a stable system addresses the effect on the assimilation of model error in the presumably stable data assimilation system viewed as a stable observer. In this study, model error refers both to extrinsic physical error forcing, such as that which arises from unresolved cumulus activity, and to intrinsic error sources arising from imperfections in the numerical model and in the physical parameterizations.

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Brian F. Farrell and Petros J. Ioannou

Abstract

Synoptic-scale eddy variance and fluxes of heat and momentum in midlatitude jets are sensitive to small changes in mean jet velocity, dissipation, and static stability. In this work the change in the jet producing the greatest increase in variance or flux is determined. Remarkably, a single jet structure change completely characterizes the sensitivity of a chosen quadratic statistical quantity to modification of the mean jet in the sense that an arbitrary change in the jet influences a chosen statistical quantity in proportion to the projection of the change on this single optimal structure. The method used extends previous work in which storm track statistics were obtained using a stochastic model of jet turbulence.

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Nikolaos A. Bakas and Petros J. Ioannou

Abstract

Zonal jets are commonly observed to spontaneously emerge in a β-plane channel from a background of turbulence that is sustained in a statistical steady state by homogeneous stochastic excitation and dissipation of vorticity. The mechanism for jet formation is examined in this work within the statistical wave–mean flow interaction framework of stochastic structural stability theory (SSST) that makes predictions for the emergence of zonal jets in β-plane turbulence. Using the coupled dynamical SSST system that governs the joint evolution of the second-order statistics and the mean flow, the structural stability of the spatially homogeneous statistical equilibrium with no mean zonal jets is studied. It is shown that close to the structural stability boundary, the eddy–mean flow dynamics can be split into two competing processes. The first, which is shearing of the eddies by the local shear described by Orr dynamics in a β plane, is shown in the limit of infinitesimal shear to lead to the formation of jets. The second, which is momentum flux divergence resulting from lateral wave propagation on the nonuniform local mean vorticity gradient, is shown to oppose jet formation. The upgradient momentum fluxes due to shearing of the eddies are shown to act exactly as negative viscosity for an anisotropic forcing and as negative hyperviscosity for isotropic forcing. The downgradient fluxes due to wave flux divergence are shown to act hyperdiffusively.

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Brian F. Farrell and Petros J. Ioannou

Abstract

Obtaining a physically based understanding of the variations with spatial scale of the amplitude and dispersive properties of midlatitude transient baroclinic waves and the heat flux associated with these waves is a central goal of dynamic meteorology and climate studies. Recently, stochastic forcing of highly nonnormal dynamical systems, such as arise from analysis of the equations governing perturbations to the midlatitude westerly jet, has been shown to induce large transfers of energy from the mean to the perturbation scale. In the case of a baroclinic atmospheric jet, this energy transfer to the synoptic scale produces dispersive properties, distributions of wave energy with wavenumber, and heat fluxes that are intrinsically associated with the nonnormal dynamics underlying baroclinic wave development.

In this work a method for calculating the spectrum and heat flux arising from stochastic forcing is described and predictions of this theory for a model atmosphere are compared with observations. The calculated energy spectrum is found to be in remarkable agreement with observations, in contrast with the predictions of modal instability theory. The calculated heat flux exhibits a realistic distribution with height and its associated energetic cycle agrees with observed seasonal mean energetics.

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Brian F. Farrell and Petros J. Ioannou

Abstract

The maintenance of variance and attendant heat flux in linear, forced, dissipative baroclinic shear flows subject to stochastic excitation is examined. The baroclinic problem is intrinsically nonnormal and its stochastic dynamics is found to differ significantly from the more familiar stochastic dynamics of normal systems. When the shear is sufficiently great in comparison to dissipative effects, stochastic excitation supports highly enhanced variance levels in these nonnormal systems compared to variance levels supported by the same forcing and dissipation in related normal systems. The eddy variance and associated heat flux are found to arise in response to transient amplification of a subset of forcing functions that obtain energy from the mean flow and project this energy on a distinct subset of response functions (E0Fs) that are in turn distinct from the set of normal modes of the system. A method for obtaining the dominant forcing and response functions as well as the distribution of heat flux for a given flow is described.

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Brian F. Farrell and Petros J. Ioannou

Abstract

Asymptotic linear stability of time-dependent flows is examined by extending to nonautonomous systems methods of nonnormal analysis that were recently developed for studying the stability of autonomous systems. In the case of either an autonomous or a nonautonomous operator, singular value decomposition (SVD) analysis of the propagator leads to identification of a complete set of optimal perturbations ordered according to the extent of growth over a chosen time interval as measured in a chosen inner product generated norm. The long-time asymptotic structure in the case of an autonomous operator is the norm-independent, most rapidly growing normal mode while in the case of the nonautonomous operator it is the first Lyapunov vector that grows at the norm independent mean rate of the first Lyapunov exponent. While information about the first normal mode such as its structure, energetics, vorticity budget, and growth rate are easily accessible through eigenanalysis of the dynamical operator, analogous information about the first Lyapunov vector is less easily obtained. In this work the stability of time-dependent deterministic and stochastic dynamical operators is examined in order to obtain a better understanding of the asymptotic stability of time-dependent systems and the nature of the first Lyapunov vector. Among the results are a mechanistic physical understanding of the time-dependent instability process, necessary conditions on the time dependence of an operator in order for destabilization to occur, understanding of why the Rayleigh theorem does not constrain the stability of time-dependent flows, the dependence of the first Lyapunov exponent on quantities characterizing the dynamical system, and identification of dynamical processes determining the time-dependent structure of the first Lyapunov vector.

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Nikolaos A. Bakas and Petros J. Ioannou

Abstract

In this paper, the emission of internal gravity waves from a local westerly shear layer is studied. Thermal and/or vorticity forcing of the shear layer with a wide range of frequencies and scales can lead to strong emission of gravity waves in the region exterior to the shear layer. The shear flow not only passively filters and refracts the emitted wave spectrum, but also actively participates in the gravity wave emission in conjunction with the distributed forcing. This interaction leads to enhanced radiated momentum fluxes but more importantly to enhanced gravity wave energy fluxes. This enhanced emission power can be traced to the nonnormal growth of the perturbations in the shear region, that is, to the transfer of the kinetic energy of the mean shear flow to the emitted gravity waves. The emitted wave energy flux increases with shear and can become as large as 30 times greater than the corresponding flux emitted in the absence of a localized shear region.

Waves that have horizontal wavelengths larger than the depth of the shear layer radiate easterly momentum away, whereas the shorter waves are trapped in the shear region and deposit their momentum at their critical levels. The observed spectrum, as well as the physical mechanisms influencing the spectrum such as wave interference and Doppler shifting effects, is discussed. While for large Richardson numbers there is equipartition of momentum among a wide range of frequencies, most of the energy is found to be carried by waves having vertical wavelengths in a narrow band around the value of twice the depth of the region. It is shown that the waves that are emitted from the shear region have vertical wavelengths of the size of the shear region.

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