Project on Stochastic Turbulence Theory

2016
Nikolaidis, M. - A., Farrell, B. F., Ioannou, P. J., Gayme, D. F., Lozano-Duran, A., & Jimenez, J. (2016). A POD-based analysis of turbulence in the reduced nonlinear dynamics system. Journal of Physics: Conference Series , 708 (1), 012002. .pdf
2014
Constantinou, N. C., Farrell, B. F., & Ioannou, P. J. (2014). Emergence and equilibration of jets in beta-plane turbulence. J. Atmos. Sci. , 71 (5), 1818-1842.Abstract

Stochastic structural stability theory (S3T) provides analytical methods for understanding the emergence and equilibration of jets from the turbulence in planetary atmospheres based on the dynamics of the statistical mean state of the turbulence closed at second order. Predictions for formation and equilibration of turbulent jets made using S3T are critically compared with results of simulations made using the associated quasi-linear and nonlinear models. S3T predicts the observed bifurcation behavior associated with the emergence of jets, their equilibration, and their breakdown as a function of parameters. Quantitative differences in bifurcation parameter values be- tween predictions of S3T and results of nonlinear simulations are traced to modification of the eddy spectrum which results from two processes: nonlinear eddy-eddy interactions and formation of discrete nonzonal struc- tures. Remarkably, these nonzonal structures, which substantially modify the turbulence spectrum, are found to arise from S3T instability. Formation as linear instabilities and equilibration at finite amplitude of multiple equilibria for identical parameter values in the form of jets with distinct meridional wavenumbers is verified, as is the existence at equilibrium of finite-amplitude nonzonal structures in the form of nonlinearly modified Rossby waves. When zonal jets and nonlinearly modified Rossby waves coexist at finite amplitude, the jet structure is generally found to dominate even if it is linearly less unstable. The physical reality of the manifold of S3T jets and nonzonal structures is underscored by the existence in nonlinear simulations of jet structure at subcritical S3T parameter values that are identified with stable S3T jet modes excited by turbulent fluctuations.

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2007
Farrell, B. F., & Ioannou, P. J. (2007). Structure and Spacing of Jets in Barotropic Turbulence. J. Atmos. Sci. , 64, 3652-3665 . J. Atmos. Sci.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 convec-
tion 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 produc-
ing and maintaining these jets and, specifically, to predict the jet amplitude, structure, and spacing. These 
jets are shown to be maintained by the continuous spectrum of shear waves and to be organized into stable 
attracting states in the mutually adjusted mean flow and turbulence fields. The jet structure, amplitude, and 
spacing and the turbulence level required for emergence of jets can be inferred from these equilibria. For 
weak but supercritical turbulence levels the jet scale is determined by the most unstable mode of the SSST 
system and the amplitude of the jets at equilibrium is determined by the balance between eddy forcing and 
mean flow dissipation. At stronger turbulence levels the jet amplitude saturates with jet spacing and 
amplitude satisfying the Rayleigh-Kuo stability condition that implies the Rhines scale. Equilibrium jets 
obtained with the SSST system are in remarkable agreement with equilibrium jets obtained in simulations 
of fully developed -plane turbulence.

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2004
Farrell, B. F., & Ioannou, P. J. (2004). Sensitivity of Perturbation Variance and Fluxes in Turbulent Jets to Changes in the Mean Jet. J. Atmos , Sci. , 61, 2644–2652 . J. Atmos , Sci.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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2003
Farrell, B. F., & Ioannou, P. J. (2003). Structural Stability of Turbulent Jets. J. Atmos, Sci. , 60, 2101-2118 . J. Atmos, Sci.Abstract

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.

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1998
Farrell, B. F., & Ioannou, P. J. (1998). Perturbation structure and spectra in turbulent channel flow. In (Vol. 11, pp. 237-250) . Theor. Comput. Fluid Dynamics.Abstract

The strong mean shear in the vicinity of the boundaries in turbulent boundary layer flows prefer- entially amplifies a particular class of perturbations resulting in the appearance of coherent structures and in characteristic associated spatial and temporal velocity spectra. This enhanced response to certain perturba- tions can be traced to the nonnormality of the linearized dynamical operator through which transient growth arising in dynamical systems with asymptotically stable operators is expressed. This dynamical amplification process can be comprehensively probed by forcing the linearized operator associated with the boundary layer flow stochastically to obtain the statistically stationary response.

In this work the spatial wave-number/temporal frequency spectra obtained by stochastically forcing the linearized model boundary layer operator associated with wall-bounded shear flow at large Reynolds number are compared with observations of boundary layer turbulence. The verisimilitude of the stochastically excited synthetic turbulence supports the identification of the underlying dynamics maintaining the turbulence with nonnormal perturbation growth

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1996
DelSole, T. M., & Farrell, B. F. (1996). The quasilinear equilibration of a thermally maintained, stochastically excited jet in a quasigeostrophic model. In (Vol. 53, pp. 1781-1797) . J. Atmos. Sci.Abstract

A theory for quasigeostrophic turbulence in barclonic jets is examined in which interaction between the mean flow and the perturbations is explicitly modeled by the nonnormal operator obtained by linearization about the mean flow, while the eddy--eddy interactions are parameterized by a combination of stochastic excitation and effective dissipation. The quasi-linear equilibrium is the stationary state in dynamical balance between the mean flow forcing and eddy forcing produced by the linear stochastic model. The turbulence model depends on two parameters that specify the magnitude of the effective dissipation an stochastic excitation. The quasi-linear model produces heat fluxes (upgradient), momentum fluxes, and mean zonal winds, which are remarkably consistent with those produced by the nonlinear model over a wide range of parameter values despite energy and edstrophy imbalances associated with the parameterization for eddy-eddy interactions. The quasi-linear equilibrium also appears consistent with most aspects of the energy cycle, and with the negative correlation between transient eddy transport and other transports observed in the atmosphere. The model overestimated the equilibrium eddy kinetic energy in cases in which it achieves correct eddy fluxes and energy balance. Understanding the role of forcing orthogonal functions rationalized this behavior and provides the basis for addressing the role of transient eddies in climate.

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1995
Farrell, B. F., & Ioannou, P. J. (1995). Stochastic dynamics of the mid-latitude atmospheric jet. In (Vol. 52, pp. 1642-1656) . J. Atmos. Sci.Abstract

The innate tendency of the background straining field of the midlatitude atmospheric jet to preferentially amplify a subset of disturbances produces a characteristic response to stochastic perturbation whether the perturbations are internally generated by non linear processes or externally imposed. This physical property of enhances response to a subset of perturbations is expressed analytically through the nonnormality of t he linearized dynamical operator, which can be studied to determine the transient growth of particular disturbances over time through solution of the initial value problem or, alternatively, to determine the stationary response to continual excitation through solution of the related stochastic problem. Making use of the fact that the background flow dominates the strain rate field, a theory for the turbulent state can be constructed based on the nonormality of the dynamical operator linearized about the background flow. While the initial value problem provides an explanation for the individual cyclogenesis events, solution of the stochastic problem provides a theory for the statistics of the ensemble of all cyclones including structure, frequency, intensity, and resulting fluxes of heat and momentum, which together constitute the synoptic-scale influence on midlatitude climate. Moreover, the observed climate can be identified with the background thermal and velocity structure that is in self-consistent equilibrium with both its own induced fluxes and imposed large-scale thermal forcing. In order to approach the problem of determining the self-consistent statistical equilibrium of the midlatitude jet it is first necessary to solve the stochastic problem for the mixed baroclinic/barotropic jet because fluxes of both heat and momentum are involved in this balance.

In this work the response to stochastic forcing of a linearized nonseparable quasigeostrophic model of the midlatitude jet is solved. The observed distribution of transient eddy variance with frequency and wavenumber, the observed vertical structures, and the observed heat and momentum flux distributions are obtained. Associated energetics and implications for maintenance of the climatological jet are discussed.

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DelSole, T. M., & Farrell, B. F. (1995). A stochastically excited linear system as a model for quasigeostrophic turbulence: analytic results for one- and two-layer fluids. In (Vol. 52, pp. 2531-2547) . J. Atmos. Sci.Abstract

The authors explore the hypothesis that nonlinear eddy interactions in quasigeosptrophic turbulence can be parameterized as a stochastic excitation plus an augmented dissipation in a statistically stationary equilibrium. The focus primarily on models sufficiently simple to be solves analytically a In particular, closed form solutions are obtained for the linear response to stochastic excitation of horizontally uniform barotropic and two-layer baroclinic flow. The response of the barotrophoic model is very simple to understand because the governing equations are mathematically normal. In contrast, the two-layer model is non-normal in the presence of vertical shear and/or vertically asymmetric dissipation and yields rather complicated results. The space-time spectra of the streamfunction and the hear fluxes derived from the two lyer model are in quantative agreement  with the corresponding observed quantities at 50N. The velocity variance predicted from the parameterication is a weaker function of the temperature gradient than indicated by observations. For strong thermal forcing, the parameterized fluxes vary inversely with the difference between a critical temperature gradient and the ambient gradient. This parameterization yields behavior suggestive of baroclinic adjustment  but operates by mechanisms fundamentally different from those conventionally associated with instability theory.

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1994
Farrell, B. F., & Ioannou, P. J. (1994). Variance maintained by stochastic forcing of non-normal dynamical systems associated with linearly stable shear flows. In (Vol. 72, pp. 1188-1191) . Phys. Rev. Letters.Abstract

The level of variance maintained in a stochastically forced asympttically stable linear dynamical system with a non-normal dynamical operator cannot be fully characterized by the decay rate of its normal modes, unlike normal dynamical systems. The nonorthogonality of modes may lead to transient growth which supports variance far in excess of that anticipated from the decay rate given by the eigenvalues of the operator. As an example, the variance maintained by stochastic forcing in a canonical shear slow is found to increase with a power of the reynolds number between 1.5 and 3. This great amplification of variance suggests a fundamentally linear mechanism underlying shear flow turbulence.

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Farrell, B. F., & Ioannou, P. J. (1994). Tropical cyclone formation. In J. Atmos. Sci. (19th ed. Vol. 51, pp. 2685-2698) . J. Atmos. Sci.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 central goal of dynamic meteorology and climate studies. Recently, stochastic forcing of highly nonnormal dynamical systems, such as arise from analysis of these equations governing perturbations to the midlatitude westerly jet, had 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 the stochastic forcing is describes 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 with observed seasonal mean energetics.

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1993
Farrell, B. F., & Ioannou, P. J. (1993). Stochastic forcing of the linearized Navier-Stokes equations. In (5th ed. Vol. 5, pp. 2600-2609) . Phys. Fluids.Abstract

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 Couette and Poiseuille flow implies a mechanism for producing and sustaining high levels of variance in shear flows from relatively small intrinsic or extrinsic forcing disturbances.
I

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Farrell, B. F., & Ioannou, P. J. (1993). Stochastic dynamics of baroclinic waves. In J. Atmos. Sci. (24th ed. Vol. 50, pp. 4044-4057) . J. Atmo. Sci.Abstract

The maintenance of variance and attendant heat flux in linear, forced dissipatative baroclinic shear flow 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 from the mean flow and project this energy on a distinct subset of response functions (EOFs) that are in turn 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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Farrell, B. F., & Ioannou, P. J. (1993). Stochastic forcing of perturbation variance in unbounded shear and deformation flows. In J. Atmos. Sci. (50th ed. pp. 200-211) . J. Atmos. Sci.Abstract

The problem of growth of small perturbations in fluid flow and the related problem of maintenance of perturbation variance has traditionally been studies by appeal to exponential modal instability of the flow. in the event that a flow supports and exponentially growing modal solution, the initially unbounded growth of the mode is taken as more of less compelling evidence for eventual flow breakdown. However, atmospheric flows are characterized by large thermally forced background rates of strain and are subject to perturbations that are not infinitesimal in amplitude. Under these circumstances there is an alternative mechanism for growth and maintenance of perturbation variance: amplification is straining flow of stochastically forced perturbations in the absence of exponential instabilities. From this viewpoint the flow is regarded as a driven amplifier rather than as an unstable oscillator. We explore this mechanism using as examples unbounded constant shear and pure deformation flow for which closed-form solutions are available and neither of which supports a nonsingular mode. With diffusive dissipation we find that amplification of isotropic band-limited stochastic driving is unlinear velocity profile. A phenomenological model of the contribution of linear and nonlinear damped modes to the maintenance of variance results in variance levels increasing linearly with shear. We conclude that amplification of stochastic forcing in straining field can maintain a variance field substantially more energetic than that resulting from the same forcing in the absence of a background straining flow. our results further indicate that existence of linear and non linear damped modes is important in maintaining high levels of variance byt he mechanism of stochastic excitation.

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