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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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Petros Ioannou and Richard S. Lindzen

Abstract

The baroclinic instability of a frontal mean state is investigated using the WKBJ approximation. The results are compared with numerical calculations performed on the same mean state. Excellent agreement (within 5%) is found for jets whose half-width is as small as a Rossby radius of deformation. For jets 20% broader, the agreement is almost perfect.

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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

Understanding the physical mechanism maintaining fluid turbulence remains a fundamental theoretical problem. The two-layer model is an analytically and computationally simple system in which the dynamics of turbulence can be conveniently studied; in this work, a maximally simplified model of the statistically steady turbulent state in this system is constructed to isolate and identify the essential mechanism of turbulence. In this minimally complex turbulence model the effects of nonlinearity are parameterized using an energetically consistent stochastic process that is white in both space and time, turbulent fluxes are obtained using a stochastic turbulence model (STM), and statistically steady turbulent states are identified using stochastic structural stability theory (SSST). These turbulent states are the fixed-point equilibria of the nonlinear SSST system. For parameter values typical of the midlatitude atmosphere, these equilibria predict the emergence of marginally stable eddy-driven baroclinic jets. The eddy variances and fluxes associated with these jets and the power-law scaling of eddy variances and fluxes are consistent with observations and simulations of baroclinic turbulence. This optimally simple model isolates the essential physics of baroclinic turbulence: maintenance of variance by transient perturbation growth, replenishment of the transiently growing subspace by nonlinear energetically conservative eddy–eddy scattering, and equilibration to a statistically steady state of marginal stability by a combination of nonlinear eddy-induced mean jet modification and eddy dissipation. These statistical equilibrium states provide a theory for the general circulation of baroclinically turbulent planetary atmospheres.

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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

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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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

Methods for approximating a stable linear autonomous dynamical system by a system of lower order are examined. Reducing the order of a dynamical system is useful theoretically in identifying the irreducible dimension of the dynamics and in isolating the dominant spatial structures supporting the dynamics, and practically in providing tractable lower-dimension statistical models for climate studies and error covariance models for forecast analysis and initialization. Optimal solution of the model order reduction problem requires simultaneous representation of both the growing structures in the system and the structures into which these evolve. For autonomous operators associated with fluid flows a nearly optimal solution of the model order reduction problem with prescribed error bounds is obtained by truncating the dynamics in its Hankel operator representation. Simple model examples including a reduced-order model of Couette flow are used to illustrate the theory. Practical methods for obtaining approximations to the optimal order reduction problem based on finite-time singular vector analysis of the propagator are discussed and the accuracy of the resulting reduced models evaluated.

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