Brian F. Farrell

Three‐dimensional optimal perturbations in viscous shear flow

Transition to turbulence in plane channel flow occurs even for conditions under which modes of the linearized dynamical system associated with the flow are stable. In this paper an attempt is made to understand this phenomena by finding the linear three‐dimensional perturbations that gain the most energy in a given time period. A complete set of perturbations, ordered by energy growth, is found using variational methods. The optimal perturbations are not of modal form, and those which grow the most resemble streamwise vortices, which divert the mean flow energy into streaks of streamwise velocity and enable the energy of the perturbation to grow by as much as three orders of magnitude. It is suggested that excitation of these perturbations facilitates transition from laminar to turbulent flow. The variational method used to find the optimal perturbations in a shear flow also allows construction of tight bounds on growth rate and determination of regions of absolute stability in which no perturbation growt...

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...

Optimal Excitation of Baroclinic Waves

Abstract Development of perturbations in a baroclinic flow can arise both from exponential instability and from the transient growth of favorably configured disturbances that are not of normal mode form. The transient growth mechanism is able to account for development of neutral and damped waves as well as for an initial growth of perturbations asymptotically dominated by unstable modes at significantly greater than their asymptotic exponential rates. Unstable modes, which are the eigenfunctions of a structure equation, are discrete and typically few in number. In contrast, disturbances favorable for transient growth form a large subset of all perturbations. To assess the potential of transient growth to account for a particular phenomena it is useful to obtain from this subset the initial condition that gives the maximum development in a well-defined sense. These optimal perturbations have a role in the theory of transient development analogous to that of the normal modes in exponential instability theo...

A Simple Approximate Result for the Maximum Growth Rate of Baroclinic Instabilities

Abstract The Charney problem for baroclinic instability involves the quasi-geostrophic instability of a zonal flow on a β plane where the zonal flow is characterized by a constant vertical shear. The atmosphere is non-Boussinesq and continuous. The solution of this problem involves confluent hypergeometric functions, and the mathematical difficulty of the problem, for the most part, has precluded extracting simple results of some generality. In this note, it is shown that there does exist a very simple, powerful approximate result for the growth rate of the most rapidly growing instability, viz., that this growth rate is linearly proportional to the surface meridional temperature gradient. The coefficient of proportionality is also easily determined. Moreover, the result extends to substantially more general profiles than those in the Charney problem.

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 perturbations in viscous shear flow

Evidence, both theoretical and experimental, is accumulating to support a mechanism for transition to turbulence in shear flow based on the 3‐D secondary instability of finite 2‐D departures from plane parallelism. It is of central importance for using this mechanism to understand how the finite amplitude 2‐D disturbances arise. To be sure, it is possible that in many experiments the disturbance is produced by the intervention of a mechanism that directly injects the requisite disturbance energy without calling on the store of kinetic energy inherent in the shear flow. It is shown here that it is also possible to tap the mean shear energy using properly configured perturbations that develop into the required primary disturbance on time scales comparable to those associated with the secondary instabilities even though the shear flow is stable or supports, at most, weak exponential instability.

Tropical cyclone formation

Abstract The physics of tropical cyclone formation is not well understood, and more is known about the mature hurricane than the formative mechanisms that produce it. It is believed part of the reason for this can be traced to insufficient upper-level atmospheric data. Recent observations suggest that tropical cyclones are initiated by asymmetric interactions associated with migratory upper-level potential vorticity disturbances and low-level disturbances. Favored theories of cyclone formation, however, focus on internal processes associated with cumulus convection and/or air-sea interaction. This work focuses on external mechanisms of cyclone formation and, using both a two- and three-dimensional moist geostrophic momentum model, investigates the role of upper level potential vorticity disturbances on the formation process. A conceptual model of tropical cyclone formation is proposed, and implications of the theory are discussed.

Modal and Non-Modal Baroclinic Waves

Abstract Solution of the initial-value problem for the Eady model is presented. In the presence of boundaries, normal mode waves as well as non-modal waves exist. Energy extracted from the mean flow during the initial development of a perturbation is found to excite the persistent normal modes. It is suggested that this process may be important to cyclogenesis and in providing energy to neutral or near-neutral normal modes. In particular, the Petterssen criterion for cyclogenesis is clarified.

Mechanisms of Eastern Mediterranean Rainfall Variability

Abstract This paper presents a simple theory for the association between observed eastern Mediterranean (EM) rainfall anomalies and North Atlantic (NA) climate variability. Large-scale NA atmospheric mass rearrangements, primarily a modulation of the Icelandic low and the subtropical high pressure systems, tend to extend beyond the NA. A particularly strong such teleconnection exists between the northern NA and southern Europe and the Mediterranean Basin. Pressure anomalies over Greenland–Iceland are thus associated with reversed-polarity anomalies centered over the northern Adriatic, affecting the entire Mediterranean Basin; elevated Greenland pressure is accompanied by an anomalous cyclone over the Mediterranean, and a Mediterranean high pressure system is present when pressure over Greenland is reduced. In the EM, these anomalies result in anomalous southerlies during Greenland highs, and northerlies during Greenland lows. Eastern Mediterranean southerlies warm the EM, while northerlies cool locally. B...

Small Error Dynamics and the Predictability of Atmospheric Flows

Abstract Forecast reliability is known to be highly variable and this variability can be traced in part to differences in the innate predictability of atmospheric flow regimes. These differences in turn have traditionally been ascribed to variation in the growth rate of exponential instabilities supported by the flow. More recently, drawing on modern dynamical systems theory, the asymptotic divergence of trajectories in phase space of the nonlinear equations of motion has been cited to explain the observed loss of predictability. In this report it is shown that increase in error on synoptic forecast time scales is controlled by rapidly growing perturbations that are not of normal mode form. It is further noted that unpredictable regimes are not necessarily associated with larger exponential growth rates than are relatively more predictable regimes. Moreover, model problems illustrating baroclinic and barotropic dynamics suggest that asymptotic measures of divergence in phase space, while applicable in the...