Petros J. Ioannou

Generalized Stability Theory. Part II: Nonautonomous Operators

Abstract An extension of classical stability theory to address the stability of perturbations to time-dependent systems is described. Nonnormality is found to play a central role in determining the stability of systems governed by nonautonomous operators associated with time-dependent systems. This pivotal role of nonnormality provides a conceptual bridge by which the generalized stability theory developed for analysis of autonomous operators can be extended naturally to nonautonomous operators. It has been shown that nonnormality leads to transient growth in autonomous systems, and this result can be extended to show further that time-dependent nonnormality of nonautonomous operators is capable of sustaining this transient growth leading to asymptotic instability. This general destabilizing effect associated with the time dependence of the operator is explored by analysing parametric instability in periodic and aperiodic time-dependent operators. Simple dynamical systems are used as examples including th...

Stochastic forcing of the linearized Navier-Stokes equations

Transient amplification of a particular set of favorably configured forcing functions in the stochastically driven Navier–Stokes equations linearized about a mean shear flow is shown to produce high levels of variance concentrated in a distinct set of response functions. The dominant forcing functions are found as solutions of a Lyapunov equation and the response functions are found as the distinct solutions of a related Lyapunov equation. Neither the forcing nor the response functions can be identified with the normal modes of the linearized dynamical operator. High variance levels are sustained in these systems under stochastic forcing, largely by transfer of energy from the mean flow to the perturbation field, despite the exponential stability of all normal modes of the system. From the perspective of modal analysis the explanation for this amplification of variance can be traced to the non‐normality of the linearized dynamical operator. The great amplification of perturbation variance found for Couett...

Optimal excitation of three‐dimensional perturbations in viscous constant shear flow

The three‐dimensional perturbations to viscous constant shear flow that increase maximally in energy over a chosen time interval are obtained by optimizing over the complete set of analytic solutions. These optimal perturbations are intrinsically three dimensional, of restricted morphology, and exhibit large energy growth on the advective time scale, despite the absence of exponential normal modal instability in constant shear flow. The optimal structures can be interpreted as combinations of two fundamental types of motion associated with two distinguishable growth mechanisms: streamwise vortices growing by advection of mean streamwise velocity to form streamwise streaks, and upstream tilting waves growing by the down gradient Reynolds stress mechanism of two‐dimensional shear instability. The optimal excitation over a chosen interval of time comprises a combination of these two mechanisms, characteristically giving rise to tilted roll vortices with greatly amplified perturbation energy. It is suggested ...

Structural Stability of Turbulent Jets

Turbulence in fluids is commonly observed to coexist with relatively large spatial and temporal scale coherent jets. These jets may be steady, vacillate with a definite period, or be irregular. A comprehensive theory for this phenomenon is presented based on the mutual interaction between the coherent jet and the turbulent eddies. When a sufficient number of statistically independent realizations of the eddy field participate in organizing the jet a simplified asymptotic dynamics emerges with progression, as an order parameter such as the eddy forcing is increased, from a stable fixed point associated with a steady symmetric zonal jet through a pitchfork bifurcation to a stable asymmetric jet followed by a Hopf bifurcation to a stable limit cycle associated with a regularly vacillating jet and finally a transition to chaos. This underlying asymptotic dynamics emerges when a sufficient number of ensemble members is retained in the stochastic forcing of the jet but a qualitative different mean jet dynamics is found when a small number of ensemble members is retained as is appropriate for many physical systems. Example applications of this theory are presented including a model of midlatitude jet vacillation, emergence and maintenance of multiple jets in turbulent flow, a model of rapid reorganization of storm tracks as a threshold in radiative forcing is passed, and a model of the quasi-biennial oscillation. Because the statistically coupled wave‐mean flow system discussed is generally globally stable this system also forms the basis for a comprehensive theory for equilibration of unstable jets in turbulent shear flow.

State Estimation Using a Reduced-Order Kalman Filter

Abstract Minimizing forecast error requires accurately specifying the initial state from which the forecast is made by optimally using available observing resources to obtain the most accurate possible analysis. The Kalman filter accomplishes this for a wide class of linear systems, and experience shows that the extended Kalman filter also performs well in nonlinear systems. Unfortunately, the Kalman filter and the extended Kalman filter require computation of the time-dependent error covariance matrix, which presents a daunting computational burden. However, the dynamically relevant dimension of the forecast error system is generally far smaller than the full state dimension of the forecast model, which suggests the use of reduced-order error models to obtain near-optimal state estimators. A method is described and illustrated for implementing a Kalman filter on a reduced-order approximation of the forecast error system. This reduced-order system is obtained by balanced truncation of the Hankel operator ...

Structure and Spacing of Jets in Barotropic Turbulence

Abstract Turbulent flows are often observed to be organized into large-spatial-scale jets such as the familiar zonal jets in the upper levels of the Jovian atmosphere. These relatively steady large-scale jets are not forced coherently but are maintained by the much smaller spatial- and temporal-scale turbulence with which they coexist. The turbulence maintaining the jets may arise from exogenous sources such as small-scale convection or from endogenous sources such as eddy generation associated with baroclinic development processes within the jet itself. Recently a comprehensive theory for the interaction of jets with turbulence has been developed called stochastic structural stability theory (SSST). In this work SSST is used to study the formation of multiple jets in barotropic turbulence in order to understand the physical mechanism producing and maintaining these jets and, specifically, to predict the jet amplitude, structure, and spacing. These jets are shown to be maintained by the continuous spectru...

Accurate Low-Dimensional Approximation of the Linear Dynamics of Fluid Flow

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.

Transient Growth and Optimal Excitation of Thermohaline Variability

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.

Transient Development of Perturbations in Stratified Shear Flow

Abstract Transient development of perturbations in inviscid stratified shear flow is investigated. Use is made of closed form analytic solutions that allow concise identification of optimally growing plane-wave solutions for the case of an unbounded flow with constant shear and stratification. For the case of channel flow, variational techniques are employed to determine the optimally growing disturbances. The maximum energy growth attained over a specific time interval decreases continuously with increasing stratification, and no special significance attaches to Ri = 0.25. Indeed, transient growth can be substantial even for Ri = O(1). A general lower bound on the energy growth attained by an optimal perturbation in a stratified flow over a given time interval is the square root of the growth attained by the corresponding perturbation in unstratified flow. Enhanced perturbation persistence is found for mean-flow stratification lying in the range 0.1 < Ri < 0.3. Small but finite perturbations in mean flow...

Perturbation Growth and Structure in Time-Dependent Flows

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 longtime 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.