Formation of Jets by Baroclinic Turbulence

VOLUME 65 JOURNAL OF THE ATMOSPHERIC SCIENCES NOVEMBER 2008

BRIAN F. FARRELL Department of Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts

PETROS J. IOANNOU Department of Physics, National and Capodistrian University of Athens, Athens, Greece

(Manuscript received 4 September 2007, in final form 7 March 2008)

ABSTRACT

Turbulent fluids are frequently observed to spontaneously self-organize into large spatial-scale jets; geophysical examples of this phenomenon include the Jovian banded winds and the earth’s polar-front jet. These relatively steady large-scale jets arise from and are maintained by the smaller spatial- and temporal- scale turbulence with which they coexist. Frequently these jets are found to be adjusted into marginally stable states that support large transient growth. In this work, a comprehensive theory for the interaction of jets with turbulence, stochastic structural stability theory (SSST), is applied to the two-layer baroclinic model with the object of elucidating the physical mechanism producing and maintaining baroclinic jets, understanding how jet amplitude, structure, and spacing is controlled, understanding the role of parameters such as the temperature gradient and static stability in determining jet structure, understanding the phenomenon of abrupt reorganization of jet structure as a function of parameter change, and understanding the general mechanism by which turbulent jets adjust to marginally stable states supporting large transient growth. When the mean thermal forcing is weak so that the mean jet is stable in the absence of turbulence, jets emerge as an instability of the coupled system consisting of the mean jet dynamics and the ensemble mean eddy dynamics. Destabilization of this SSST coupled system occurs as a critical turbulence level is exceeded. At supercritical turbulence levels the unstable jet grows, at first exponentially, but eventually equilibrates nonlinearly into stable states of mutual adjustment between the mean flow and turbulence. The jet structure, amplitude, and spacing can be inferred from these equilibria. With weak mean thermal forcing and weak but supercritical turbulence levels, the equilibrium jet structure is nearly barotropic. Under strong mean thermal forcing, so that the mean jet is unstable in the absence of turbulence, marginally stable highly nonnormal equilibria emerge that support high transient growth and produce power-law relations between, for example, heat flux and temperature gradient. The origin of this power-law behavior can be traced to the nonnormality of the adjusted states. As the stochastic excitation, mean baroclinic forcing, or the static stability are changed, meridionally confined jets that are in equilibrium at a given meridional wavenumber abruptly reorganize to another meridional wavenumber at critical values of these parameters. The equilibrium jets obtained with this theory are in remarkable agreement with equilibrium jets obtained in simulations of baroclinic turbulence, and the phenomenon of discontinuous reorganization of confined jets has important implications for storm-track reorganization and abrupt climate change.

1. Introduction gaseous planets (Ingersoll 1990; Vasavada and Show- man 2005; Sánchez-Lavega et al. 2008). In the earth’s Coherent jets that are not forced at the jet scale are midlatitude troposphere, the polar-front jets are eddy often observed in turbulent flows, with a familiar geo- driven (Jeffreys 1933; Lee and Kim 2003). The earth’s physical-scale example being the banded winds of the equatorial stratosphere is characterized by the eddy- driven quasi-biennial oscillation (QBO) jet (Baldwin et al. 2001); the equatorial ocean supports eddy-driven Corresponding author address: Brian Farrell, Harvard Univer- sity, Department of Earth and Planetary Sciences, Geological Mu- equatorial deep jets (Muench and Kunze 1994). The seum 452, 24 Oxford Street, Cambridge, MA 02138. midlatitude ocean also supports alternating eddy- E-mail: [email protected] driven zonal jets, which are seen both in observations

DOI: 10.1175/2008JAS2611.1

JAS2611 3354 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65

(Maximenko et al. 2005) and in models (Nakano and unstable modes themselves (Stone 1978; Held 1978; Hasumi 2005; Richards et al. 2006). The phenomenon Lindzen and Farrell 1980; Gutowski 1985; Gutowski et of spontaneous jet organization in turbulence has been al. 1989; Stone and Branscome 1992; Lindzen 1993; extensively investigated in observational and in theo- Welch and Tung 1998; Zurita-Gotor and Lindzen retical studies (Rhines 1975; Williams 1979, 2003; 2004a,b; Schneider and Walker 2006) and scaling argu- Panetta 1993; Nozawa and Yoden 1997; Huang and ments for turbulent transport (Green 1970; Stone 1972, Robinson 1998; Manfroi and Young 1999; Vallis and 1974; Held 1978; Held and Larichev 1996; Held 1999; Maltrud 1993; Cho and Polvani 1996; Galperin et al. Lapeyre and Held 2003; Zurita-Gotor 2007). Baroclinic 2004; Lee 2005; Kaspi and Flierl 2007) as well as in turbulence is additionally characterized by power-law laboratory experiments (Krishnamurti and Howard behavior of fluxes as a function of variables controlling 1981; Read et al. 2004, 2007). The mechanism by which stability such as the temperature gradient (Held and eddies maintain jets against surface drag is upgradient Larichev 1996; Barry et al. 2002; Zurita-Gotor 2007) momentum flux produced by the continuous spectrum and a theory for baroclinic turbulence should account of shear waves as distinct from the discrete set of jet also for these power-law relations. modes (Huang and Robinson 1998; Panetta 1993; Vallis A model commonly employed for theoretical studies and Maltrud 1993). This upgradient momentum trans- of baroclinic turbulence is the two-layer model in a fer mechanism maintains jets in barotropic flows doubly periodic domain (Haidvogel and Held 1980; (Huang and Robinson 1998) and is also an important Panetta 1993). This model maintains a constant mean component of the eddy forcing maintaining baroclinic shear. Consequently, the Phillips (1954) two-layer jets (Panetta 1993; Williams 2003). However, baroclinic model stability criterion, if it is calculated using the jet structure is additionally influenced by eddy heat mean shear, cannot be changed by fluxes in this model. fluxes and secondary circulations as well as by exter- However, it does not follow that baroclinic adjustment nally imposed mean thermal forcing. does not occur. In the two-layer model, adjustment to A characteristic property of mean jets in strong syn- marginal stability can result from a combination of dis- optic-scale turbulence is marginal stability coexisting sipation, quantization by boundaries, and meridional with robust transient growth (Farrell 1985; Hall and confinement associated with the spontaneous emer- Sardeshmukh 1998; Sardeshmukh and Sura 2007). In gence of alternating jets. the case of the midlatitude jet, this marginally stable We show how turbulent atmospheric jets are ad- state is traditionally referred to as a baroclinically adjusted to stable but highly amplifying mean states. justed state after its original description by Stone Analogous behavior often arises when a feedback con- (1978), who drew attention to the proximity of mean jet troller is imposed to stabilize multidimensional me- states to the classical two-layer model stability bound- chanical or electronic systems and attempts to suppress ary. It is now recognized that observed and simulated the nonnormal growth in such systems gave rise to ro- marginally stable jets frequently exceed this necessary bust control theory (Zhou and Doyle 1998). It is com- condition for baroclinic instability that would apply to monly held that stable but fragile equilibria arise pri- the stability of a dissipationless and meridionally con- marily as a consequence of a design process (Bobba et stant shear. This is no contradiction of marginal jet sta- al. 2002). We find that the state of high nonnormality bility because a correct assessment of jet stability re- together with marginal stability is inherent to strongly quires an eigenanalysis in which proper account is turbulent eddy-driven baroclinic jets because turbu- taken of dissipation, quantization by boundaries, and lence maintains flow stability with a naturally occurring jet structure, particularly the stabilization effect of me- feedback controller. A fundamental advantage of our ridional shear (Ioannou and Lindzen 1986; James 1987; theory is that it clearly reveals this feedback stabiliza- Roe and Lindzen 1996; DelSole and Farrell 1996; Hall tion process and how it places the system in a stable but and Sardeshmukh 1998). highly amplifying state. The recognition, in both observations and simula- Excitation of the turbulent eddies can be traced ei- tions, of adjustment to marginal stability in baroclinic ther to exogenous processes such as convection, as in turbulence resulted in application of a variety of theo- the case of the Jovian jets, or to endogenous sources retical approaches to understanding this phenomenon. such as wave–wave interaction, as in the case of the This effort was motivated in part by the fact that baro- earth’s polar-front jet. These processes have short tem- clinic adjustment is a fundamental problem in the poral and spatial correlation scales compared with the theory of baroclinic turbulence but also by its implica- jet temporal and spatial scales, and their effect in forc- tions for flux parameterization in climate models. ing the synoptic-scale wave field can be approximated These theoretical approaches include adjustment by the as stochastic (Sardeshmukh and Sura 2007). The central

Unauthenticated | Downloaded 09/26/21 10:46 PM UTC NOVEMBER 2008 FARRELL AND IOANNOU 3355 component of a theory for the dynamics of jets in baro- excitation and no mean thermal forcing, the zonal jets clinic turbulence—namely, the method to obtain the that emerge spontaneously are nearly barotropic. If the structure of the turbulence and the associated fluxes stochastic excitation is not too strong, these jets equili- given the jet—is provided by stochastic turbulence brate nonlinearly with a nearly sinusoidal barotropic modeling (Farrell and Ioannou 1993c, 1994, 1995, 1996; structure. However, as the stochastic excitation is in- DelSole and Farrell 1995, 1996; DelSole 1996; Newman creased the nonlinearly balanced jets increase in ampli- et al. 1997; Whitaker and Sardeshmukh 1998; Zhang tude and encroach on eddy stability boundaries. This and Held 1999; DelSole 2004b). Once these fluxes are results in modification of the jet structure as SSST known, the equilibrium states of balance among the fixed-point equilibria enforce stability of the jet to eddy large-scale thermal forcing, the dissipation, and the en- perturbations because eddy fluxes diverge at stability semble mean turbulent flux divergences can be deter- boundaries.2 To avoid eddy instability, the jets become mined; this is the method of SSST.1 (Farrell and Ioan- increasingly east–west asymmetric, with the westward nou 2003, 2007). The interaction between the large- jet equilibrating primarily barotropically at the Ray- scale jet structure and the field of eddy turbulence is leigh–Kuo scale, which results in emergence of merid- nonlinear and results in a nonlinear trajectory for the ional jet spacing with the ␤ scale L␤. jet and the ensemble mean eddy field associated with it, When mean thermal forcing is imposed, baroclinic which often tends to an equilibrium and sometimes to a equilibria are found. Under strong stochastic excitation, limit cycle, but under extraordinary conditions is chaotic. the eastward part of these jet equilibria become sub- There are three length scales in the baroclinic dy- stantially baroclinic while at the same time becoming namics of the two-layer model on a ␤ plane: the scale sufficiently narrow meridionally so that perturbation ϭ imposed by the boundaries; the Rossby radius LR eddy stability is maintained by meridional confinement

NH/f0 (in which f0 is the Coriolis parameter, N the (Ioannou and Lindzen 1986; James 1987; Roe and buoyancy frequency, and H the depth of the fluid); and Lindzen 1996). The westward parts of these jets equili- the ␤ scale, L␤ ϭ ͌U/␤, where U is a characteristic brate near the Rayleigh–Kuo boundary with more velocity and ␤ is the meridional gradient of the Coriolis nearly barotropic structure. parameter. This U may be taken as the characteristic Although jet equilibria with marginal stability have velocity of the jet or of the eddies; when the rms eddy been recognized as characteristic of baroclinic turbu- velocity is taken for U, this length is proportional to the lence and meridional confinement has been implicated Rhines radius. When U is the jet velocity, it is the length in their stabilization, our theory is far more specific scale for which planetary and shear vorticity gradients than a diagnostic in that it provides a method for pre- are comparable. dicting the precise structure of the equilibria. These We find in the presence of turbulence, and even in equilibria are preferred states in the sense of being pre- the absence of an imposed mean thermal forcing, that cisely known marginally stable structures maintained there is a linear instability of the coupled turbulence– by the turbulence. However, their structure varies ei- mean flow system giving rise to zonal jets. This jet- ther continuously or discontinuously as a function of forming instability is an example of a new class of in- parameters influencing stability, such as the static sta- stability in fluid dynamics; it is an emergent instability bility or mean thermal forcing, so that no single pre- that arises from the interaction between the mean flow ferred structure can be said to characterize the adjusted and the turbulence. The modal jet perturbation growth state. rate is exponential because the jet modes organize the If the jets are quantized by meridional boundaries, as turbulence field to produce fluxes proportional in mag- would be the case for a planet of finite radius, a dis- nitude and consistent in structure with the jet mode. continuous reorganization of structure is induced at It is instructive to consider this jet forming instability threshold parameter values for which stable equilib- as a function of the strength of the stochastic excitation, the mean thermal forcing, the static stability, and dissipation. 2 It is logically necessary that exponential eddy instabilities be With sufficiently energetic homogeneous stochastic equilibrated in some manner. The mechanism of equilibration in our model, and we believe the primary mechanism in atmospheric jets generally, is through modification of the mean jet structure by the eddy fluxes. In our quasigeostrophic model this modification 1 The term “stochastic structural stability theory” was chosen to is necessarily confined to changes in jet velocity and attendant connote analysis of both the equilibria emergent from the coop- horizontal temperature gradients, but more generally modifica- erative interaction between turbulence and mean flows and the tion of static stability must be involved in some manner that re- structural stability of these equilibria with parameter change. mains to be investigated using a nongeostrophic model.

Unauthenticated | Downloaded 09/26/21 10:46 PM UTC 3356 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65 rium states with the extant marginally stable wavenum- obey the following evolution equations (DelSole and ber cease to exist. This provides an instructive example Farrell 1996; Vallis 2006): of the general phenomenon of eddy-driven jets appear- ϩ r2 ϩ Ϫ ϩ ing and disappearing discontinuously as a function of U ϭϪ ͑U Ϫ U ͒ ϩ F and ͑2͒ t 2 slow parameter change (Lee 1997; Farrell and Ioannou r 2003; Robinson 2006). Understanding the physical ͑ 2 Ϫ ␭2͒ Ϫ ϭ 2 2͑ ϩ Ϫ Ϫ͒ Ϫ ␭2͑ Ϫ Ϫ Ϫ͒ D 2 Ut D U U 2rR UR U mechanism determining the structure the eddy-driven 2 jet is a fundamental geophysical fluid dynamics prob- Ϫ ϩ D2F , ͑3͒ lem with important climate implications because discontinuous reorganization of eddy-driven jets can alter in which the term proportional to r2 represents linear both climate statistics, by altering storm tracks and the damping of the mean flow at the lower layer to a state source regions associated with isotopic signatures (Al- of rest (we assume no Rayleigh damping in the upper ley et al. 1993; Fuhrer et al. 1999; Wunsch 2003), and layer); the term proportional to rR represents linear relaxation of the baroclinic flow to the imposed baro- the climate itself, by changing the latitude of surface Ϫ ϩ Ϫ stresses and associated oceanic upwelling (Toggweiler clinic flow UR; and the terms F and F (explicit forms et al. 2006). This mechanism for abrupt change of cli- given later) represent the forcing of the zonal flow by mate and climate statistics arising from storm-track re- the eddies. The nondimensional Rossby radius of de- ϭ ␭ ϭ organization joins the short list of reorganization of formation is LR 1/ NH/fL, where L is the hori- thermohaline ocean circulations (Weaver et al. 1991), zontal scale, H the height of each layer, N the static ice sheet instability (MacAyeal 1993), and sea ice stability, and f the Coriolis parameters. The operator 2 ץ 2ץϵ 2 switches (Gildor and Tziperman 2003; Li et al. 2005) as D / y . possible mechanisms for explaining the observed The equations for the mean flow are written in the record of abrupt climate change. compact form In this work we study the structure and dynamics of dU ϭ GU ϩ H, ͑4͒ jets in baroclinic turbulence by joining the equation dt governing ensemble mean eddy statistics with the equa- in which the U is the full mean state velocity vector tion for the zonal mean to form a coupled wave–mean ϩ flow evolution system. This nonlinear coupled equation U U ϵ ͩ ͪ; ͑5͒ system is the basic tool of SSST analysis and we have UϪ applied it previously to the study of barotropic jets G is the linear dissipation operator (Farrell and Ioannou 2003, 2007). G11 G12 G ϭ ͩ ͪ, ͑6͒ G G 2. Dynamics of the zonal average velocity in 21 22 turbulent flows with components

r2 a. Formulation G ϭϪ , ͑7a͒ 11 2 A theory for jet dynamics in turbulent flow was de- r2 veloped in Farrell and Ioannou (2003) and applied to G ϭϩ , ͑7b͒ 12 2 the problem of the formation of jets in barotropic turbulence in Farrell and Ioannou (2007). Here we pro- Ϫ r2 G ϭ ͑D2 Ϫ 2␭2͒ 1ͩ D2ͪ, and ͑7c͒ vide a brief review of the salient ideas of this theory in 21 2 the context of baroclinic jets, which is the focus of this r2 G ϭ ͑D2 Ϫ 2␭2͒Ϫ1ͩϪ D2 ϩ 2r ␭2ͪ; ͑7d͒ work. 22 2 R Consider a turbulent rotating two-layer baroclinic and the mean flow forcing is composed of the eddy fluid on a ␤ plane, and let U (y, t) be the latitudinally H 1,2 forcing and relaxation toward the radiative equilibrium (y) and time (t) dependent mean velocities of the upper thermal wind UϪ; that is, (denoted as 1) and lower (denoted as 2) layers, the R ϩ mean being taken in the zonal (x) direction. The baro- F H ϵ ͫ ͬ. ͑8͒ tropic (denoted ϩ) and baroclinic (denoted Ϫ) mean 2 2 Ϫ1 2 Ϫ 2 Ϫ ͑D Ϫ 2␭ ͒ ͑D F Ϫ 2r ␭ U ͒ flows, defined as R R The eddy forcing of the mean flow is expressed in terms ϩ Ϫ ϩ U1 U2 Ϫ U1 U2 of the barotropic and baroclinic eddy streamfunction U ϭ and U ϭ , ͑1͒ 2 2 (DelSole and Farrell 1996; Vallis 2006) as

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ϩ ϭ ␺ϩ␺ϩ ϩ ␺Ϫ␺Ϫ ͑ ͒ eral boundary conditions have been incorporated. In F x yy x yy and 9a writing (13) we have included the linear operator pa- Ϫ ϭ ␺ϩ␺Ϫ ϩ ␺Ϫ␺ϩ Ϫ ␭2␺ϩ␺Ϫ ͑ ͒ F x yy x yy 2 x . 9b rameterization of the nonlinear terms in the complete The overbar denotes a zonal average and the perturba- perturbation equation as stochastic excitation and tion velocities are related to the streamfunctions ␺Ϯ as added dissipation in the form of linear damping of the streamfunction at rate reff (Farrell and Ioannou 1993b, Ϯ ϭϪ␺Ϯ ␷Ϯ ϭ ␺Ϯ ͑ ͒ u y and x . 10 1996; DelSole and Farrell 1996; Newman et al. 1997; Ϯ The perturbation streamfunctions are expanded in a DelSole 2004b). The coefficients r are the linear Fourier sum of zonal harmonics: damping rates of the barotropic and baroclinic streamfunction, respectively. ␺Ϯ͑ ͒ ϭ ␺Ϯ͑ ͒ ikx ͑ ͒ x, y, t ͚ k y, t e . 11 The stochastic excitation is given by k ⌬Ϫ1 ϩ͑ ͒ The zonal average ab of the product of two sinusoidally 0 W y 0 P ϭ ␣ ͫ ͬͫ ͬ. ikx ˆ ilx ˆ ␦ k k 2 Ϫ1 Ϫ varying fields aˆe and be is Re(aˆb*) kl /2 (* denotes 0 ͑⌬ Ϫ 2␭ ͒ 0 W ͑y͒ complex conjugate). Using this property, the eddy forc- ͑ ͒ ings [(9a), (9b)] are expressed in terms of the eddy 16 streamfunction Fourier amplitudes as The stochastic excitation of the potential vorticity rep- k resented by (16) is ⌬ correlated in time and spatially Fϩ ϭϪ͚ Im͑␺ϩD2␺ϩ* ϩ ␺ϪD2␺Ϫ*͒ and ͑12a͒ Ϯ ␣ 2 k k k k correlated by W (y). The scalar coefficient k is chosen k so that each perturbation wavenumber is forced with k equal energy. Ϫ ϭϪ ͑␺ϩ 2␺Ϫ ϩ ␺Ϫ 2␺ϩ Ϫ ␭2␺ϩ␺Ϫ ͒ F ͚ Im k D k * k D k * 2 k k * . k 2 The continuous operators are discretized and the dynamical operator is approximated by a finite dimen- ͑ ͒ Ϯ 12b sional matrix. The state ␺ (k) is represented by a col- Each Fourier component of the perturbation stream- umn vector with entries for the complex value of the function evolves according to the stochastically excited streamfunction at collocation points. In this approxima- linear equation tion, W represents the spatial (y) correlation in the stochastic excitation. Care must be taken that the struc- ␺ϩ ␺ϩ ␰͑ ͒ϩ d k k t ture of excitation matrix W does not bias the response. ͩ ͪ ϭ A ͑U͒ͩ ͪ ϩ P ͩ ͪ. ͑13͒ dt ␺Ϫ k ␺Ϫ k ␰͑ ͒Ϫ k k t We select W matrices producing Gaussian autocorrelation function about the level of excitation y propor- In (13), the second term on the rhs represents stochastic i tional to exp[Ϫ(y Ϫ y )2/␦2] so that ␦ controls the cor- excitation and in the first term the autonomous linear i relation distance in y. This forcing matrix is the same operator of the perturbation dynamics is for all zonal wavenumbers. ϩ͑ ϩ͒ ϩ͑ Ϫ͒ Ak U Ak U If the layers are excited equally, the stochastic exci- A ͑ ͒ ϭ ͫ ͬ ͑ ͒ ϩ k U Ϫ ϩ Ϫ Ϫ , 14 tation of the time dependence of the barotropic ␰ (t) A ͑U ͒ A ͑U ͒ Ϫ k k and baroclinic ␰ (t) components is equal and statisti- in which the individual linear operators are cally independent so that

ϩ ϩ Ϫ1 ϩ 2 ϩ ϩ ͑ ͒ ϭ ⌬ ͓Ϫ ⌬ Ϫ ͑␤ Ϫ ͔͒ Ϫ ϩ Ϫ† Ak U ikU ik D U r ͗␰ ͑t͒␰ ͑t͒͘ ϭ 0, ͑17͒ Ϫ ͑ ͒ reff, 15a where the angled brackets indicate the ensemble aver- ϩ͑ Ϫ͒ ϭ ⌬Ϫ1͑Ϫ Ϫ⌬ ϩ 2 Ϫ͒ Ϫ Ϫ ͑ ͒ age (i.e., an average over realizations of the forcing). Ak U ikU ikD U r , 15b We choose each of the excitations to be delta correlated Ϫ͑ ϩ͒ ϭ ͑⌬ Ϫ ␭2͒Ϫ1͓Ϫ Ϫ͑⌬ ϩ ␭2͒ ϩ 2 Ϫ Ak U 2 ikU 2 ikD U with zero mean and unit variance; thus, Ϫ Ϫ ⌬͔ ͑ ͒ ϩ Ϫ r , and 15c ͗␰ ͑t͒͘ ϭ ͗␰ ͑t͒͘ ϭ 0, and Ϫ͑ Ϫ͒ ϭ ͑⌬ Ϫ ␭2͒Ϫ1͓Ϫ ϩ͑⌬ Ϫ ␭2͒ ͗␰ϩ͑ ͒␰ϩ†͑ ͒͘ ϭ ͗␰Ϫ͑ ͒␰Ϫ†͑ ͒͘ ϭ ␦͑ Ϫ ͒ ͑ ͒ Ak U 2 ikU 2 t s t s I t s , 18 ϩ ͑␤ Ϫ 2 ϩ͒ Ϫ ϩ⌬ ϩ ␭2͔ Ϫ D U r 2rR reff, where I is the identity matrix. If only the upper layer is forced, then the barotropic and baroclinic forcing are ͑ ͒ 15d taken equal to ␰ϩ(t). where ⌬ϵD2 Ϫ k2 is the Laplacian and ⌬Ϫ1 is the The system consisting of (4) and (13) describes the inverse of the Laplacian in which the appropriate lat- dynamics of a single realization of a stochastically ex-

Unauthenticated | Downloaded 09/26/21 10:46 PM UTC 3358 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65 cited wave field interacting with the jet. This system ϩ Ϯ Ck Ck C ϭ ͫ ͬ ͑ ͒ results in strong fluctuations in jet structure. However, k Ϯ Ϫ , 19 ͑C ͒† C there are usually many independently excited waves k k simultaneously interacting with the jet, so the fluctua- ϩ ϭ͗␺ϩ␺ϩ†͘ Ϫ ϭ͗␺Ϫ␺Ϫ†͘ Ϯ ϭ͗␺ϩ␺Ϫ†͘ tions in jet structure are suppressed by the ensemble with Ck k k , Ck k k , and Ck k k , averaging of these independently excited wave fields. obeys the deterministic equation With this assumption of ensemble average wave forcing, the time evolution of the ensemble average covari- dCk ϭ A ͑U͒C ϩ C A†͑U͒ ϩ Q , with ͑20͒ ance matrix of the perturbation field dt k k k k k

Ϫ ϩ ϩ Ϫ ⌬ 1W ͑W ͒†⌬ 1† 0 Q ϭ |␣ |2ͫ ͬ ͑ ͒ k k Ϫ Ϫ Ϫ Ϫ , 21 0 ͑⌬ Ϫ 2␭2͒ 1W ͑W ͒†͑⌬ Ϫ 2␭2͒ 1† the spatial covariance matrix for equal stochastic exci- This ideal limit in which the ensemble average eddy tation of both layers (in the case of excitation limited to forcing is taken is of particular interest because in this the upper layer, Qk is modified appropriately). For our limit, although the effect of the ensemble average tur- purposes, Qk includes all the relevant characteristics of bulent fluxes is retained in the solution, the fluctuations the stochastic excitation; the meridional distribution of associated with the turbulent eddies are suppressed by the forcing is given by the diagonal elements of this the averaging and the coupled jet–turbulence dynamics matrix and the autocorrelation function by the rows. becomes deterministic. In this limit the mean zonal flow The ensemble mean baroclinic and barotropic eddy equilibria emerge with great clarity. This ideal limit is forcings [(12a), (12b)] are expressed in terms of the approached when the autocorrelation scale of the per- covariances as turbation field l is much smaller than the zonal extent L of the domain. In that case, taking the zonal average is ϩ k ϩ Ϫ ͗F ͘ ϭ ͚ Ϫ diag͓Im͑C ϩ C ͒D2†͔, ͑22a͒ equivalent to averaging N ϭ O(L/l) statistically inde- 2 k k k pendent realizations of the forcing, which converges to k the ensemble average as N → ϱ (Farrell and Ioannou Ϫ ϭ Ϫ ͓͑ Ϯ ϩ Ϯ† ͒ 2† Ϫ ␭2 Ϯ͔ ͗F ͘ ͚ diag͕ Im Ck Ck D 2 Ck ͖, k 2 2003). Examples of physical systems in which this limit applies include the Jovian upper atmosphere and the ͑ ͒ 22b solar convection zone, both forced by relatively small- and these ensemble means of zonal average forcings are scale convection. An experiment in which a jet emerged introduced in place of the single realization of zonal from rather coarse-grained turbulent convection was average forcing in (9a) and (9b). reported by Krishnamurti and Howard (1981). When In this limit, the evolution equation for the covari- this large ensemble limit is not approached sufficiently ance [(20)], the equation for the ensemble mean eddy closely, the system exhibits stochastic fluctuations forcing [(22a), (22b)] and the equation for the barotro- about the ideal equilibrium. A physical system that ex- pic and baroclinic mean zonal flow [(4)] define a closed hibits such behavior is the earth’s polar jet, in which the autonomous nonlinear system for the evolution of the number of independent cyclones around a latitude mean flow under the influence of its consistent field of circle is at most order 10 so that, while the underlying turbulent eddies. This system is energetically consistent order is revealed by the ideal dynamics, the observed and globally stable. For typical earth values of mean zonal jet exhibits stochastic fluctuations about the ideal baroclinic forcing, there is a small energy input by the jet (Farrell and Ioannou 2003). stochastic forcing, which can be accounted for in the A limitation of our analysis is the assumption that the energy balance by a slight increase in dissipation, stochastic excitation is independent of the mean and whereas the dominant balance is between energy ex- eddy fields. Although for the Jovian atmosphere this is tracted from the mean by baroclinic processes and en- probably a good assumption, because the stochastic ex- ergy lost to frictional dissipation. Typically the equilib- citation is thought to arise from internally generated rium solutions of this system are steady mean flows convection, it is a crude assumption for the earth’s polar maintained with constant structure by eddy forcing, al- jet, in which the excitation is influenced by the jet struc- though limit cycle and chaotic trajectories are found for ture and eddy amplitude itself. Obtaining the stochastic some parameter values (Farrell and Ioannou 2003). excitation consistent with the turbulence supported by

Unauthenticated | Downloaded 09/26/21 10:46 PM UTC NOVEMBER 2008 FARRELL AND IOANNOU 3359 a jet is equivalent to obtaining a closure of the turbulent where ⌬T is the temperature difference across 104 km, ϭ Ϫ4 Ϫ1 ϭ system, and progress on this problem has been made f0 10 s is the Coriolis parameter, and R 287 (DelSole and Farrell 1996; DelSole 1999). Although an J (kg K)Ϫ1 is the gas constant. This constant shear flow ⌬ Ͼ⌬ attractive avenue for future study, such a closure is not becomes baroclinically unstable when T Tc, which ⌬ ϭ necessary for understanding the basic dynamics under- with the above dissipation parameters is Tc 28.3 K lying emergence of zonal jets in turbulence; in the in- (104 km)Ϫ1. An important parameter is the criticality terest both of simplicity and of clarity of exposition, we defined as for the present retain a spatially uniform stochastic ex- ␰ ϭ ͑ Ϫ ͒ր͑␤␭2͒ ͑ ͒ citation. 4 U1 U2 . 25 Supercriticality (i.e., ␰ Ͼ 1) implies exponential insta- b. Scaling of the equations, boundary conditions, bility of the inviscid two-layer model for meridionally and parameters constant flows according to the Phillips (1954) criterion. The equations are nondimensionalized, using the The criticality is not a criterion for instability of realistic ϭ ϭ 6 flows, which are generally both meridionally varying earth day for the time scale, td 1 day, and L 10 m ϭ and dissipative. Because of the presence of dissipation for the length scale so that the velocity scale is L/td 11.6 m sϪ1. The calculations are performed in a doubly in the example above, the critical temperature gradient ⌬ ϭ 4 Ϫ1 periodic domain with width L ϭ 20 units in the me- for instability Tc 28.3 K (10 km) corresponds to y ␰ ϭ ridional direction. This doubly periodic model was in- criticality 2.03. troduced by Haidvogel and Held (1980) and used in the work of Panetta (1993) and Held and Larichev (1996). 3. Stability analysis of the ensemble mean coupled The size of the domain has been selected to be wide system enough to accommodate a number of jets; however, quantization effects do become important, as will be Recall that in the ensemble mean limit the eddy/ shown below. The calculations use typically 64-point mean jet system is governed by the coupled equations discretization in the meridional direction and 14 waves dCk in the zonal direction, consisting of global zonal wave- ϭ A ͑U͒C ϩ C A ͑U͒† ϩ Q and ͑26a͒ dt k k k k k numbers 1 to 14 in a periodic zonal domain of nondimensional length L ϭ 40. dU H. ͑26b͒؀x ϭ GU The strength of the stochastic excitation Qk is mea- dt ϭ sured by the forcing density F trace(MQk)/n, where M is the energy metric The equilibria of this set of equations consist of an equilibrium velocity UE and an associated equilibrium E 1 ⌬ 0 perturbation covariance C satisfying ϭϪ ͫ ͬ ͑ ͒ M 2 , 23 2n 0 ͑⌬ Ϫ 2␭ ͒ ͑ E͒ E ϩ E ͑ E͒† ϭϪ ͑ ͒ Ak U Ck Ck Ak U Qk, 27a ␺ϩ ␺Ϫ ␺ϩ ␺Ϫ † defined so that ( k , k )M( k , k ) is the total energy GUE ϩ H ϭ 0. ͑27b͒ ␺ϩ ␺Ϫ per unit mass of state ( k , k ) at each zonal wavenumber k, and n is the number of discretization points in the If stable, these equilibria may be found by forward meridional direction. For convenience, the stochastic integration of the coupled Eqs. (26a) and (26b); other- excitation will at times be expressed in dimensional wise, a root finder must be employed. units (mW kgϪ1). Two distinct concepts of jet stability arise in connec- Unless otherwise specified, friction parameters for tion with these equilibria. The first is the familiar eddy ϭ ϭ the perturbation dynamics are rR 1/15, r2 1/5, and perturbation stability determined by analysis of the op- ϭ ϭ E E reff 1/20. For the mean dynamics, r2 1/5 and vis- erator A . An equilibrium mean flow U is by necessity cosity ␯ ϭ (␦y)2, where ␦y is the grid size. The Froude stable in this sense because otherwise the eddy variance ␭2 ϭ 2 ϭ number in most of the calculations is 1/LR 1. would diverge. The converse, however, is not true; the For simplicity and to facilitate interpretation, we im- stability of AE does not imply the stability of the equi- pose meridionally homogeneous stochastic excitation librium (UE, CE), and there is a new stability concept and meridionally constant radiative equilibrium forcing arising from the interaction of the jet with its consistent with shear: eddy flux divergences. The stability of the coupled system (26a) and (26b) is determined by analysis of the R⌬T t Ϫ ϭ Ϫ ϭ d ͑ ͒ operator that governs the perturbation dynamics of its U1 U2 2UR 2 , 24 f0 20L equilibria, as discussed in Farrell and Ioannou (2003,

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Ϫ1 FIG. 1. The critical value of stochastic excitation Fc (m W kg ) required to structurally destabilize the baroclinic jet in thermal wind balance with mean thermal forcing ⌬T [K (104 Ϫ1 km) ]. The different curves correspond to the critical stochastic excitation Fc required for the emergence of a zonal jet with meridional wavenumber n ϭ 1,...,4.Theeddyfieldcomprises global zonal wavenumbers 1–14. Relevant physical parameters: ␤ ϭ 1.6 ϫ 10Ϫ11 mϪ1 sϪ1, ␭2 ϭ ϭ Ϫ1 ϭ Ϫ1 Froude number 1, eddy damping rates r2 1/5 day and rR 1/15 day , damping rate ϭ Ϫ1 ϭ Ϫ1 of the mean r2 1/5 day , and mean cooling rR 1/15 day . With these parameters insta- ⌬ ϭ 4 Ϫ1 bility of the baroclinic jet occurs at Tc 28.3 K (10 km) .

Ϫ 2007). This stability of the ensemble mean eddy–mean radiative equilibrium flow UR is an unstable fixed point jet equilibrium state depends on the strength of the of the coupled system (26a) and (26b). We wish to ex- stochastic excitation Q, the thermal forcing measured amine the coupled eddy–mean jet stability of these ra- ⌬ ⌬ Ͻ⌬ by the temperature difference T, the Froude number diative equilibrium fixed points. When T Tc, the ␭2 Ϫ , and the dissipation. We primarily examine the sta- radiative equilibrium UR becomes unstable only when bility of equilibrium states as a function of the stochas- the stochastic excitation F exceeds a critical value F c. ϭ ⌬ ⌬ Ͼ⌬ Ϫ tic excitation F trace(MQ)/n and T. When T Tc, the radiative equilibrium UR is un- It can be shown by integration of (27b) in y and stable for all F. Ϫ1 imposition of boundary conditions on the eddy motions The critical value of stochastic forcing Fc (mW kg ) that the meridional mean of an equilibrium UE has the is plotted as a function of ⌬T in Fig. 1. Because of the property meridional homogeneity of the perturbation equations, the perturbation eigenmodes are harmonic functions. Ly Ly ͵ Ϫ ϭ ͵ Ϫ ͑ ͒ Unstable jet perturbations of harmonic form in the me- U dy UR dy, 28 0 0 ridional direction emerge when F exceeds F c. Figure 1 shows the critical forcings required to destabilize zonal which implies that equilibration cannot result from flows with meridional wavenumbers n ϭ 1,...,4. change in the mean meridional shear or the mean tem- When the radiative equilibrium is unstable to the for- perature gradient. This is also true of the simulations of mations of jets, the growing zonal jets ultimately equili- Panetta (1993) and Held and Larichev (1996). brate as finite-amplitude extensions of the unstable ⌬ Ͻ⌬ Ϫ For T Tc, the radiative equilibrium flow U R is a eigenmodes of the coupled SSST wave–mean jet sys- fixed point of the coupled system (26a) and (26b) for tem. These equilibria may be stable or unstable fixed any excitation F because the ensemble mean eddy points. When unstable, the jets may go to a new stable ⌬ Ͼ⌬ fluxes are meridionally constant. For T Tc, the equilibrium or settle in a limit cycle having periodic

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FIG. 2. (a), (b) Meridional structure of the equilibrium upper-layer (continuous line) and lower-layer (dashed line) mean flows maintained by stochastically modeled turbulence in a doubly periodic domain with no mean thermal forcing. Equilibria for two levels of stochastic excitation F are shown. The stochastic excitation is (a) F ϭ Ϫ Ϫ 5.83 mW kg 1 and (b) F ϭ 58.3 mW kg 1 per zonal wavenumber. (c), (d) The corresponding mean potential vorticity gradients of the upper- and lower-layer flows. The nonnormality of the equilibrium flows has increased from 53.9 for the flows in (a) to 155 for the flows in (b). The eddy field comprises global zonal wavenumbers 1–14. ϭ Ϫ1 ϭ Ϫ1 ϭ The eddy damping parameters are r2 1/5 day and rR 1/15 day ; the damping rate of the mean is r2 1/5 Ϫ1 ϭ Ϫ1 day and the mean cooling is rR 1/15 day .

behavior; also, chaotic trajectories are possible (Farrell Examples of meridional wavenumber-2 jet equilibria and Ioannou 2003, 2007). However, over a significant are shown in Fig. 2 for two values of stochastic excita- parameter range stable fixed points are found, and tion. In the first case, the stochastic excitation is F ϭ 5.8 these are the subject of our study in this paper. mW kgϪ1 per wavenumber (Fig. 2a); in the second, the Ϫ stochastic excitation is F ϭ 55.8 mW kg 1 per wave- 4. Examples of jet equilibria number (Fig. 2b). The first case is stochastically excited with sufficient energy input rate to clearly show depar- a. Jet equilibria with no mean thermal forcing ture from the nearly harmonic equilibrium form ob- The equilibria found in the absence of mean thermal tained for slightly supercritical stochastic excitation. forcing are, for a wide choice of realistic parameter The second state is excited at a rate that places the values, nearly barotropic and similar to those discussed equilibrium state close to the stability boundary just in Farrell and Ioannou (2007), although with appropri- before the bifurcation to meridional wavenumber-1 ate parameter choice and/or vertical asymmetry of sto- equilibria as a function of stochastic excitation, and chastic excitation substantial baroclinicity can be main- therefore the second equilibrium state departs maxi- tained. mally from the harmonic equilibrium state that results For sufficiently strong stochastic excitation, multiple for slightly supercritical stochastic excitation. equilibria differing in jet meridional wavenumber exist In these examples the eddy field comprises 14 zonal at ⌬T ϭ 0 as indicated in Fig. 1. wavenumbers. Although the lower layer is Ekman

Unauthenticated | Downloaded 09/26/21 10:46 PM UTC 3362 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65 damped, the equilibria are nearly barotropic and equilibrate close to the Rayleigh–Kuo stability boundary so their meridional spacing scales as L␤ ϭ ͌U/␤, and there is pronounced asymmetry between the westward and eastward jets (Farrell and Ioannou 2007). The mean potential vorticity gradient is positive almost ev- erywhere, as shown in Fig. 2. Laboratory experiments using a stratified fluid in a rotating channel with a slop- ing bottom verify the formation of such stable persistent mean jets when there is no mean thermal forcing (Read et al. 2004, 2007). Although stable, these equilibrated flows are highly nonnormal, with nonnormality increasing3 from 53.9 to 155 as the stability boundary is approached. As an example of a nearly barotropic equilibrium, a calculation was done for the Jovian 23°N jet for which accurate velocity data is available (Sánchez-Lavega et FIG. 3. (a) Simulation of the 23°N jet on Jupiter (solid) com- al. 2008). Comparison between this accurately observed pared with observations (circles) adapted from Sánchez-Lavega et al. (2008). The equilibrium jet structure (solid) is barotropic and is jet and the corresponding equilibrium obtained using barotropically stable. (b) The associated mean vorticity gradient representative Jovian parameters is shown in Fig. 3. ϭ ϭ normalized by beta. Parameters are eddy damping rup rdown Noteworthy are the nearly constant shears and cusplike 5.6 ϫ sϪ1; mean damping 2.7 ϫ 10Ϫ9 sϪ1, as suggested by Conrath maximum of the eastward jet and the rounded profile et al. (1990) and used by Yamazaki et al. (2004); stochastic forcing Ϫ2 of the westward jet. The physics of these characteristic 0.34 W m per zonal wavenumber in a total Jovian layer depth of 5 bar; and ␤ value corresponding to Jovian planetographic lati- features of barotropic equilibria in strong turbulence is tude 23°N. The domain is doubly periodic with meridional width discussed in detail in Farrell and Ioannou (2007). equivalent to 28° Jovian latitude. The calculation was performed ␲ ϭ using the 14 zonal wavenumbers 2 n/Ly, with n 1,...,14. b. Baroclinic jet equilibria with barotropically unstable upper-level jets Q ϭ ␤ Ϫ UЉ ϩ ␭2͑U Ϫ U ͒, In the absence of mean thermal forcing the equilibria 1y 1 1 2 ϭ ␤ Ϫ Љ Ϫ ␭2͑ Ϫ ͒ ͑ ͒ are nearly barotropic. However, when the symmetry Q2y U2 U1 U2 , 29 between the layers is sufficiently disrupted, baroclinic are shown in Fig. 4. Also shown in the same figure are ␤ Ϫ Љ equilibria can be maintained with strong jets despite the the barotropic vorticity gradients U i (thick and absence of any mean thermal forcing. We find that thin dashed lines). This stable equilibrium jet is baro- strong jets satisfying Rayleigh–Kuo necessary condi- clinic, with the upper-level flow violating Rayleigh– tions for instability form spontaneously as equilibria in Kuo stability condition by 10␤. the presence of sufficient stochastic excitation and dissipation. As an example, consider the two-layer model c. Jet equilibria with mean thermal forcing ␤ with the following parameters: equal to one-fifth of Consider a stochastically excited baroclinic flow with ␤ ϭ ϭ the terrestrial , symmetric eddy damping r1 r2 1 mean thermal forcing. As the mean thermal forcing is Ϫ1 ϭ Ϫ1 day , eddy cooling of rR 1/15 day , and dissipation increased, jets equilibrate that are marginally stable ϭ ϭ Ϫ1 of the mean r1 r2 1/250 day with stochastic ex- and highly nonnormal. Examples of such baroclinically citation of only the upper layer at level F ϭ 0.972 mW Ϫ1 adjusted jets are shown in Fig. 5a for two different lev- kg . (Throughout this paper, day is abbreviated d.) Ϫ1 els of stochastic excitation, F ϭ 0.48 mW kg and F ϭ Equilibrium jet profiles together with the upper and 1.34 mW kgϪ1, with mean thermal forcing ⌬T ϭ 30 lower layer mean potential vorticity gradients 4 Ϫ1 ⌬ ϭ K(10 km) [recall that the stability boundary is Tc 28.3K (104km)Ϫ1]; this figure shows meridional wavenumber-3 jet equilibria. The second equilibrium in Fig. 3 The measure of nonnormality we use is the ratio of the vari- 5b is close to the structural stability boundary just be- ance maintained by stochastic excitation of this system to the fore the bifurcation to meridional wavenumber-2 equi- variance maintained in the equivalent normal system that has the same eigenvalues but orthogonal eigenvectors when both are libria as a function of forcing. Note that both equilibria ␰ ϭ Ϫ ␤␭2 ϭ forced with identical stochastic excitation Q ϭ I (Ioannou 1995; have a mean supercriticality parameter 8UR/( ) Farrell and Ioannou 1996). 2.15. The equilibrated flows in Fig. 5 and most of the

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FIG. 4. (top) Meridional structure of the equilibrium upper-layer (thick line) and lower-layer (thin line) mean flows maintained by stochastically modeled turbulence in a doubly periodic domain with no mean thermal forcing. The equilibrated flows violate the Rayleigh–Kuo barotropic stability condition but the flows are stable. The layers are equally damped at rate ϭ ϭ Ϫ1 ϭ Ϫ1 r1 r2 1 day , the eddy cooling rate is rR 1/15 day , and the dissipation of the mean ϭ ϭ Ϫ1 ϭ occurs at rate r1 r2 1/250 day . Stochastic excitation of only the upper layer at level F 0.972 mW kgϪ1 is used. The other parameters are ␭2 ϭ 1 and ␤ ϭ 1/5 of the terrestrial value in order to simulate Jovian conditions. (bottom) The corresponding mean potential vorticity gradients measured in units of ␤ of the upper-layer (solid thick line) and lower-layer (dashed thick line) flows. Also shown are the corresponding barotropic potential vorticity gradients (solid and dashed lines, respectively). examples shown in this paper have mean supercritical- subject of much study (Haidvogel and Held 1980; ity ␰ Ͼ 1 and yet are manifestly stable. Panetta and Held 1988; Held and Larichev 1996; Pavan The equilibrated eastward jets can be highly baro- and Held 1996; Lapeyre and Held 2003). We investigate clinic because baroclinicity of the eastward jet increases the role of emergent jets in this process by modeling a the already positive potential vorticity gradient—unlike turbulent supercritical flow in which equilibrated jets the case of the westward jets, for which the baroclinic have enforced stability. shear is suppressed at equilibrium to preserve stability, To address the role of multiple jets (each of which as can be seen in Figs. 5c,d. The presence of dissipation has been baroclinically adjusted to be near a stability allows the potential vorticity gradient of the equili- boundary) in regulating the heat flux across a wide brated jets to change sign, substantially violating the channel, we perform an experiment in which a super- Charney–Stern stability condition, while retaining sta- critical radiative equilibrium constant shear state is per- bility. These baroclinically adjusted states are primarily turbed by a small random initial jet structure and al- equilibrated by meridional confinement interacting lowed to equilibrate with zero initial eddy covariance. with dissipation (Ioannou and Lindzen 1986; James The temperature difference of the radiative equilibrium 1987; Roe and Lindzen 1996). is 35 K over a distance of 104 km, corresponding to a supercriticality of ␰ ϭ 2.51. The equilibration occurs in a characteristic progression in which the eddy field d. Mechanism controlling turbulent heat flux grows rapidly to reach a high overshoot eddy variance Regulation of the mean heat flux in a doubly periodic and heat flux, at which time the eddy field organizes jets two-layer baroclinic turbulence model has been the that grow and ultimately equilibrate the combined eddy

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FIG. 5. (a), (b) Meridional structure of the equilibrium upper-layer (continuous line) and lower-layer (dashed line) mean flows maintained by stochastically modeled turbulence in a doubly periodic domain with mean thermal Ϫ Ϫ forcing ⌬T ϭ 30 K (104 km) 1. Equilibria for two levels of stochastic excitation F are shown: (a) F ϭ 0.48 mW kg 1 Ϫ and (b) F ϭ 1.34 mW kg 1 per zonal wavenumber. (c), (d) The corresponding mean potential vorticity gradients of the upper- and lower-layer flows. Both equilibrium flows are supercritical with the same supercriticality parameter ␰ ϭ 2.18 but are exponentially stable because of the meridional variation of the flows and the dissipation. The nonnormality of the equilibrium flows has increased from (a) 26.7 to (b) 53. The eddy field comprises global ϭ Ϫ1 ϭ Ϫ1 ϭ Ϫ1 zonal wavenumbers 1–14. The eddy damping parameters are r2 1/5 day , rR 1/15 day , and reff 1/20 day . ϭ Ϫ1 ϭ Ϫ1 The damping rate of the mean is r2 1/5 day ; the mean cooling rate is rR 1/15 day . and mean jet system at a lower equilibrium total eddy energy and the eddy heat flux decrease when the relax- ϭ Ϫ1 ϭ energy. The time development of the flow and the as- ation is decreased to rR 1/15 day at t 60 day, sociated fluxes are shown in Figs. 6 and 7. This progres- allowing the jet to increase in amplitude, which in- sion is commonly observed when an eddy–mean jet sys- creases jet stability. This behavior is analogous to the tem is established or perturbed (Panetta and Held 1988; midwinter suppression of eddy variance that is ob- Panetta 1993; Chen et al. 2007). To better understand served in the Pacific storm track in winter (Nakamura control of the heat flux by the jets, an experiment was 1992; Zhang and Held 1999), a phenomenon that has performed in which for the first 60 days the coefficient also been simulated for conditions during the last gla- of Newtonian cooling of the mean baroclinic flow was cial maximum (Li and Battisti 2008). The common ϭ Ϫ1 ϭ set to rR 1/3 day before being switched to rR 1/15 thread in the two-layer model simulation and the ob- dayϪ1 at day 60. By t ϭ 60 the flow has been stabilized servations is that the westerlies become sharper and (Fig. 7c) and both the jets and the eddies have equili- stronger while the eddy activity is reduced. The time brated as indicated by the constant value of the maxi- development of the fluxes and variances show that at mum wind in layer 1 and the maximum shear shown in first both the mean flow and the eddy fluxes and vari- Fig. 7a, the latitudinally averaged heat flux in Fig. 7b, ances overshoot their equilibrium values before jets de- and the latitudinally averaged total eddy energy in Fig. velop, which then suppress both the eddy heat flux and 7d. Under strong mean thermal relaxation the jets are the variance, demonstrating the crucial role of the jets suppressed, and as a result the eddy equilibrium kinetic in controlling the mean equilibrium state. We conclude

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FIG. 6. Contours of the zonal mean velocity in the upper layer as a function of time and meridional distance. The flow starts at t ϭ 0 with a random perturbation about the radiative equilibrium uniform flow that maintains a temperature difference of 35° K (104 km)Ϫ1. The eddy field comprises global zonal wavenumbers 1–14. The eddy ϭ Ϫ1 ϭ Ϫ1 ϭ Ϫ1 ϭ damping parameters are r2 1/5 day , rR 1/15 day , reff 1/20 day . The damping rate of the mean is r2 Ϫ1 ϭ Ϫ1 ϭ Ϫ1 1/5 day ; the mean cooling rate is rR 1/3 day for the first 60 days and then changes to rR 1/15 day at Ϫ t ϭ 60 days. The stochastic excitation is F ϭ 0.48 mW kg 1 per zonal wavenumber. that regions containing a number of jets baroclinically gous to the dangerous inputs that render feedback- equilibrate each individual jet while the mean shear stabilized mechanical and electronic systems fragile remains supecritical and that this baroclinic equilibra- (Zhou and Doyle 1998), and it is excitation of these tion of the individual jets then determines the equilib- amplifying perturbations by the nonlinear wave–wave rium heat flux. interactions, here parameterized by stochastic excitation, that maintains the turbulent variance (Farrell and 5. Scaling, baroclinic adjustment, and the role of Ioannou 1993c,d; DelSole 2007). nonnormality In addition to explaining the stability, high nonnormality, and the centrality of transient growth in jet dy- Equilibrium jets that form in strong turbulence are namics, we argue that these equilibria provide an ex- adjusted to marginal stability. Although stable, these planation for the scaling laws of fluxes as a function of highly sheared flows are also highly nonnormal, sup- parameters influencing jet stability found in baroclinic porting large transient perturbation amplification. One turbulence simulations (Zhou and Stone 1993; Held consequence of this stability and nonnormality in the and Larichev 1996; Barry et al. 2002; Zurita-Gotor earth’s midlatitude atmosphere is the association of cy- 2007). clone formation with the chance occurrence in the tur- To introduce this last idea, consider the linear dy- bulence of optimal or near-optimal initial conditions namics of streamwise rolls in an unbounded shear flow (Farrell 1982, 1989; Farrell and Ioannou 1993c; DelSole with rotation in the spanwise direction. This system is 2004a, 2007). These optimal perturbations are analo- governed by the Reynolds matrix, which provides a

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FIG. 7. (a) Nondimensional maximum upper-level velocity (solid) and maximum shear (dashed line) as a function of time (unit velocity is 11 m sϪ1). (b) Mean heat flux as a function of time. (c) Maximum growth rate of the instantaneous flow as a function of time (dayϪ1). (d) Total eddy energy (solid) and ratio of eddy barotropic to eddy baroclinic energy (dashed) as a function of time. The parameters are as in Fig. 6. convenient example of a dynamical system in which rotation rate ⍀ destabilizes the Reynolds matrix by stability and nonnormality interact: coupling the streamwise and spanwise velocity components. With this addition, the Reynolds matrix 2 2 Ϫ␯͑l ϩ m ͒ Ϫ͑␣ Ϫ 2⍀͒ provides a versatile model of the interplay between sta- A ϭ ͫ ͬ. ͑30͒ Ϫ2⍀l 2ր͑l 2 ϩ m2͒ Ϫ␯͑l 2 ϩ m2͒ bility and nonnormality in the vicinity of a stability boundary such as that to which the coupled eddy–mean In (30), ⍀ is a rotation rate, ␣ the constant shear of the jet system generally adjusts jets. We now use this matrix mean flow, ␯ the coefficient of viscosity, l the spanwise to understand how power-law behavior arises in wavenumber, and m the vertical wavenumber (see ap- strongly turbulent equilibria such as the midlatitude jet. pendix). With ⍀ϭ0, this matrix models the dynamical When ␣ is increased in the Reynolds system and the system of a streamwise roll perturbation in an un- dynamical matrix A is forced with constant-variance bounded constant shear boundary layer (Farrell and white noise in each variable, the variance increases pro- Ioannou 1993a). In addition, with ⍀ϭ0 this dynamical portional to ␣2. In contrast, the variance in the equiva- system is stable, with decay rate Ϫ␯(l 2 ϩ m2), but with lent normal system, in which the change in the eigen- nonnormality increasing with ␣, so that it supports op- values is retained but the eigenvectors are assumed or- timal transient growth increasing proportional to ␣2 thogonal, exhibits no power-law behavior but only the ␯ 2 ϩ 2 Ϫ1 ␣ Ϫ ␣ Ϫ1 and occurring at time [ (l m )] , implying that the familiar resonance-induced ( c) divergence in variance grows proportional to ␣2 for constant viscosity. the immediate neighborhood of the stability boundary With the modification of including a spanwise oriented as shown in Fig. 8. The momentum flux also increases rotation rate ⍀, the decay of the least damped mode proportional to ␣, deviating from this power only in the ␣ ␣ ϭ ⍀ϩ also decreases as increases so that when c 2 immediate neighborhood of the critical shear. ␯ 2(l 2 ϩ m2)3/(2⍀l 2), the matrix becomes unstable. The Power-law behavior is similarly generic in strongly

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FIG. 8. For the Reynolds example system: the ensemble mean energy ͗E͘ as a function of the shear ␣ for rotation rate ⍀ϭ10Ϫ5, ␯ ϭ 0.1 and perturbation wavenumbers l ϭ m ϭ 1. The ␣ ϭ ␣2 critical shear at which the matrix becomes unstable is c 4000. The variance increases as , ␣ deviating from this power law behavior only for shears very close to c . The dashed–dotted curve shows the momentum flux ͗uw͘, which can be seen to vary as ␣. The dashed curve shows ͗ ͘ the variance En maintained by the equivalent normal system with the same decay rate as A; ␣ this variance is almost constant, increasing only for shears close to c. turbulent jet equilibria, but the specific power depends scaling of the flux–gradient relationship, which has on the stability parameter involved, the variable, and been interpreted as higher-order thermal diffusion with the growth mechanism. The origin of power-law scaling diffusivity increasing as the third power of the tempera- behavior can be understood by considering the growth ture gradient (Held and Larichev 1996; Zurita-Gotor of optimal perturbations, which dominate the mainte- 2007). The heat flux (at the center of the jet and its nance of variance in highly nonnormal systems. Figure mean value) increase with criticality as ␰3. At high criti- 9 shows the increase of the mean baroclinic and baro- cality, departures from the power-law behavior shown tropic energy generation rates in Fig. 10 result if the meridional average shear is used to calculate the criticality rather than the shear at the 1 ϭ ͵ Ϫ ␭2 Ϫ͗␺ϩ␺Ϫ ͘ ͑ ͒ jet axis. The reason is that the shear becomes meridi- Pem 2 U x dy and 31a 0 onally concentrated at high criticality. Figure 10 also

1 shows the heat flux as a function of the criticality taken ϭ ͵ ͓ ϩ͑͗␺ϩ␺ϩ ϩ ␺Ϫ␺Ϫ ͒͘ as the meridional average shear. Kem U x yy x yy 0 ϩ Ϫ͑͗␺ϩ␺Ϫ ϩ ␺Ϫ␺ϩ ͔͒͘ ͑ ͒ U x yy x yy dy, 31b 6. Quantization and abrupt jet reorganization and of the quantity As stochastic excitation is increased or mean shear is ␷T ϭ ͗␺ϩ␺Ϫ ϩ ␺Ϫ␺ϩ ͘, ͑32͒ increased or static stability decreased, equilibrium jets x yy x yy strengthen and adjust in structure to maintain a stable which is proportional to the heat flux, with criticality of equilibrium. However, if the jet number is quantized, as the equilibrated flow evaluated at the jet maxima ␰ ϭ is the case for the earth, eventually no stable jet equi- Ϫ ␤␭2 4(U1 U2)/( ). The energy generation rates increase librium is possible at the extant wavenumber and the as ␰ 4 in agreement with simulations showing power-law eddy–mean jet system undergoes a bifurcation in which

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Ϫ FIG. 9. The mean eddy baroclinic energy generation rate Pem (solid line), the mean eddy ␷ barotropic energy generation rate Kem (dashed line), and the heat flux T at the center of the ␰ ϭ Ϫ ␤ jet (dashed–dotted line) as a function of the criticality parameter [4(U1 U2)/ L2] evaluated at the jet maximum, together with the mean heat flux as a function of the mean criticality ⌶ (circles). The eddy generation rates increase as ␰ 4 (indicated with the upper dotted line) and the heat flux increases as ␰3 (indicated with the lower dotted line).The mean heat flux also has the same power-law behavior. The heat flux does not scale with the mean criticality ⌶ because as criticality increases it becomes increasingly localized near the jet Ϫ center. The stochastic excitation is F ϭ 0.24 mW kg 1 per mode. The eddy field comprises ϭ Ϫ1 ϭ global zonal wavenumbers 1–14. The eddy damping parameters are r2 1/5 day , rR 1/15 Ϫ1 ϭ Ϫ1 ϭ Ϫ1 day , and reff 1/20 day ; the damping rate of the mean is r2 1/5 day ; the mean cooling ϭ Ϫ1 rate is rR 1/15 day .

a new equilibrium is established, typically at the next world, troposphere static stability is expected to in- lowest allowed wavenumber. This reorganization oc- crease as a result of decrease in the saturated adiabatic curs abruptly as a parameter-controlling jet amplitude lapse rate. This effect is generally accepted for the trop- is changed. Similar behavior has been observed to oc- ics and subtropics (Held 1982; Sarachik 1985; Xu and cur in model systems (Lee 1997, 2005; Robinson 2006) Emanuel 1989), and although the extent of the increase and has been inferred to occur in the climate record in the extratropics is less well established, it is expected (Fuhrer et al. 1999; Wunsch 2003; Alley et al. 1993). It to also apply in midlatitudes (Juckes 2000; Frierson provides a mechanism for producing abrupt changes in 2006). The opposite tendency is expected to have char- climate statistics by changing the source region of cli- acterized ice age climates. mate markers such as isotopes that are recorded in ice The growth rate and structure of baroclinic waves are cores (Wunsch 2003). More fundamentally, because the strongly influenced by static stability, with the growth axis of the polar jet is also an axis of concentration of rate and the penetration depth scaling inversely with surface stress, the abrupt displacement of the polar jet the buoyancy frequency (Vallis 2006), so that it is of could change the climate itself by influencing the ven- interest to add static stability to stochastic excitation tilation rate of the intermediate depths in the ocean and thermal gradient as parameters in our analysis. (Toggweiler et al. 2006). Lower static stability implies a higher Froude num- A parameter of the baroclinic system that may vary ber as ␭2 ϭ (fL)2/(NH)2. We find that static stability with climate change is the static stability; in a warmer serves as a bifurcation parameter analogous to the

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͗␺ϩ␺Ϫ͘ FIG. 10. For the equilibrium jets, the maximum heat flux x as a function of Froude number ␭2.At␭2 ϭ 1.2 there is an abrupt transition from a meridional wavenumber-3 jet to a meridional wavenumber-2 jet. The thermal forcing is ⌬T ϭ 30 K (104 km)Ϫ1 and the Ϫ stochastic excitation is F ϭ 0.49 mW kg 1 per mode. The eddy field comprises global zonal ϭ Ϫ1 ϭ Ϫ1 wavenumbers 1–14. The eddy damping parameters are r2 1/5 day , rR 1/15 day , and ϭ Ϫ1 ϭ Ϫ1 reff 1/20 day ; the damping rate of the mean is r2 1/5 day ; the mean cooling rate is ϭ Ϫ1 rR 1/15 day . mean shear in its influence on jet equilibria and that tical shear (Lindzen 1993). The ensemble mean eddy– small changes in static stability can induce abrupt reor- mean jet equilibria provide insight into the mechanism ganization of jet equilibria as shown in Fig. 10. In this underlying the barotropic governor by showing that a example, bifurcation from a wavenumber-3 to a wave- barotropic governor stabilized state is an attracting number-2 jet occurs as shown in Fig. 11. At ␭2 ϭ 1.2, equilibrium state of the ensemble mean eddy–jet there is an abrupt transition from a meridional wave- coupled system. This mechanism of stabilization by number-3 jet to a meridional wavenumber-2 jet for confinement due to horizontal shear, which we have thermal forcing ⌬T ϭ 30 K (104 km)Ϫ1 and stochastic seen in the two-layer model, was analyzed by Ioannou Ϫ excitation of F ϭ 0.49 mW kg 1 per mode. and Lindzen (1986) and Roe and Lindzen (1996).

b. Relation between SSST equilibria and baroclinic 7. Discussion adjustment a. Relation between SSST equilibria and the Baroclinic adjustment was originally invoked to ex- barotropic governor plain the observed near-baroclinic neutrality of the It was noticed by James (1987) that baroclinic jets in midlatitude jets (Stone 1978; Stone and Miller 1980). GCM simulations tend to equilibrate by modifying the This adjustment to marginal stability has been con- barotropic shear and that substantially greater baro- firmed in observed flows (Hall and Sardeshmukh 1998; clinic shear can be maintained in these horizontally Sardeshmukh and Sura 2007) and in two-layer model sheared flows than would be compatible with stability turbulence (Cehelsky and Tung 1991; DelSole and Far- in a horizontally homogeneous flow. This phenomenon, rell 1996; Roe and Lindzen 1996). Baroclinic adjust- called the barotropic governor, was confirmed by sta- ment is important as a concept because of the insight it bility analyses, and it implies that the baroclinic adjust- offers into the nature of baroclinic turbulence. It is also ment process is in part independent of changes in ver- important as the theoretical underpinning for the con-

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FIG. 11. Structure of the equilibrium upper-layer (continuous line) and lower-layer (dashed line) mean flows maintained in a doubly periodic domain with mean thermal forcing ⌬T ϭ 30 Ϫ Ϫ K (104 km) 1 and stochastic excitation of F ϭ 0.49 mW kg 1 per mode. (a) The meridional wavenumber-2 jet that bifurcates from (b) the meridional wavenumber-3 jet. The parameters are as in Fig. 11. cept of compensation, which has implications for cli- in the following sense: the stability boundary is impor- mate dynamics. The idea of baroclinic compensation is tant but not, as in the original baroclinic adjustment that because baroclinic turbulence maintains a rela- interpretations, because growth occurs when this tively constant temperature gradient in mid and high boundary is exceeded; rather, it is because turbulent latitudes, the influence of any change in alternative fluxes produced by the nonnormal interaction among a heat transport mechanisms such as ocean heat transport larger set of waves increase rapidly as the stability is diminished: the baroclinic transport adjusts up or boundary is approached, thereby producing equilib- down to compensate for any variation in these trans- rium states of the coupled mean flow–turbulence sys- port mechanisms so their variation cannot substantially tem that occur near stability boundaries when the tur- change the climate. Baroclinic adjustment has been ex- bulence is strong. Turbulent transport is important be- tensively studied and two main ideas have emerged: (i) cause it is the spectrum of waves, not just the adjustment due to modal instabilities themselves pro- instabilities, that produces the transport; however, the ducing the heat flux when the stability boundary is equilibrium flux–gradient relationship cannot be under- crossed, with these modal instabilities then adjusting stood from homogeneous turbulence closure arguments the system to a stability boundary (Stone 1978; Cehel- because the transport at equilibrium is essentially con- sky and Tung 1991; Gutowski et al. 1989; Lindzen and trolled by the inhomogeneity of the equilibrium states Farrell 1980; Lindzen 1993); and (ii) high-order turbu- and particularly by the organization of the mean flow lent diffusion due to baroclinic eddies producing rapid into jets. Although a baroclinically adjusted state can increase in heat transport coincident with but not caus- result from a wide variety of jet structure modifications ally related to unstable growth itself (Pavan and Held (including reduction in the temperature gradient and 1996; Held and Larichev 1996). We have seen that en- meridional confinement), an advantage of the ensemble mean eddy–mean jet equilibria approach a sta- semble eddy–mean jet equilibria calculation we have bility boundary when the turbulence is sufficiently done is that it provides a prediction of the particular strong; thus, these equilibria provide a natural interpre- structure changes involved in producing the baroclini- tation for the phenomenon of baroclinic adjustment. cally equilibrated state so that it is a predictive and not This interpretation unites the previous interpretations simply a diagnostic theory for jet equilibria.

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8. Conclusions fied upgradient eddy momentum flux and dissipation. But as the stochastic excitation (or another stability- Large-scale zonal jets emerge spontaneously from related parameter such as the static stability) changes baroclinic turbulence in the absence of jet-scale forcing; so as to tend to destabilize the jet, its structure evolves examples include the Jovian banded winds and the in a characteristic progression that contrives to main- earth’s polar-front jet. The primary physical mechanism tain the jet equilibria near a modal eddy stability maintaining these jets is turbulent eddy fluxes that have boundary despite the increase in jet amplitude. The been systematically organized by the jet to support the eastward jet contracts and becomes increasingly baro- jet structure. This phenomenon arises from an ubiqui- clinic, whereas the westward jet expands, maintaining tous instability of turbulent fluids to jet formation, less baroclinicity. Contraction of meridional scale in the which occurs because turbulent eddies are always avail- eastward jet augments the effective beta, as does an able to be organized by an appropriately configured arbitrary vertical shear, so the eastward jet shear may perturbation zonal jet to produce a flux proportional to become quite large. This mechanism for stabilizing the jet amplitude. Because this local upgradient flux is pro- baroclinic mode in eastward shear may be viewed as a portional to jet amplitude, it results at first in an expo- manifestation of the barotropic governor (Ioannou and nential jet growth rate. This is a new mechanism for Lindzen 1986; James 1987; Lindzen 1993). No such sta- destabilizing turbulent flows that is essentially emer- bilization of the westward jet is possible, so the vertical gent from the interaction between the mean state and shear remains small in the westward jet to prevent its turbulence. In this work, we have provided a de- baroclinic eddy mode destabilization while the meridi- tailed dynamical explanation for the systematic mutual onal scale expands to approach the Rayleigh–Kuo con- organization of the jet and the turbulent eddy fluxes dition for barotropic modal stability. Consequently, the required to produce this instability. A key ingredient of equilibrium jet structure exhibits pronounced asymme- this theory is stochastic turbulence modeling, which try, with the eastward jet being sharper than the west- provides an analytic solution for the eddy covariance in ward jet; and of the three space scales in the jet struc- statistical equilibrium with the jet so that the feedback ture problem, the beta scale L␤ϭ͌U /␤ emerges as between the jet and the turbulence can be analyzed in max the primary meridional structure scale. detail. The primary analytical tool of SSST is the Dissipation and quantization of both meridional and coupled dynamical system consisting of the evolution zonal wavenumbers often results in stable states that in equation for the eddy fluxes obtained from a statistical some degree satisfy either or both of the Charney– equilibrium stochastic turbulence model together with the evolution equation for the zonal jet. This nonlinear Stern and Rayleigh–Kuo conditions. This is no contra- coupled set of equations exhibits robust and relatively diction because these conditions are necessary and not simple behavior. Using this model, we find that the sufficient conditions and in any case apply strictly only state of thermal wind balance with a constant tempera- to flows without dissipation. It is important to keep in ture gradient on a doubly periodic beta plane is a sta- mind that the fact that an observed jet satisfies one or tionary state of the coupled system for baroclinically both of these necessary conditions need not imply that stable shear; however, this stationary state is exponen- the jet is unstable. tially unstable to zonal jet perturbations if the turbu- Adjustment of jets to marginal stability can take lence is sufficiently strong, and these zonal perturba- place in many ways, including reducing the vertical tions evolve into jets that grow and adjust in structure shear, increasing the horizontal shear, and, in the primi- until reaching finite-amplitude equilibrium. If a baro- tive equations, modifying the static stability. Because clinically unstable mean shear is thermally forced, then all of these mechanisms are available, the mere knowl- the equilibration is again a function of both the thermal edge that the system is adjusted to a state of marginal gradient and the stochastic excitation, and equilibria stability, although useful as a diagnostic, does not con- are also found. stitute a predictive theory for jet structure. A particular With weak excitation, the equilibrated jets are sinu- advantage of the SSST is that it transcends this ambi- soidal and nearly barotropic. There may be a number of guity by the crucial additional requirement of equilib- unstable meridional jet wavenumbers each leading to rium between the turbulent fluxes and the jet, which an equilibrated state. When evolved from an unbiased identifies in the space of all marginal equilibria the con- initial condition, the wavenumber of a weakly forced jet sistent and therefore the physically relevant one. is that of the most unstable eigenmode of the linearized Adjustment to stable but highly amplifying states SSST system, and the amplitude at equilibrium results with power-law behavior for flux–gradient relations is from the balance between the weakly nonlinearly modi- characteristic of jet equilibria. Analogous behavior of-

Unauthenticated | Downloaded 09/26/21 10:46 PM UTC 3372 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 65 ten arises when a feedback controller is imposed to ported by an ELKE grant by the University of Athens stabilize a multidimensional unstable mechanical or and by the European Social Fund and National Re- electronic system (Zhou and Doyle 1998). We have sources–(EPEAEK II)–PYTHAGORAS. shown that the state of high nonnormality together with marginal stability is inherent because turbulence main- APPENDIX tains flow stability with a naturally occurring feedback controller. The Reynolds Matrix In the presence of meridional quantization, such as that enforced by finite planetary radius, an abrupt re- The perturbation equations governing the evolution organization of jet structure occurs as stochastic forcing of the (x, y, z) perturbation velocities (u, ␷, w) and or another parameter controlling jet amplitude in- pressure (p) of the harmonic form creases. This reorganization occurs as the increasing ͑ ϩ ͒ westward jet amplitude forces an incipient instability ͑u, ␷, w, p͒ ϭ ͓uˆ͑t͒, ␷ˆ ͑t͒, wˆ ͑t͒, pˆ͑t͔͒ei ly mz ͑A1͒ for the particular dissipation and spatial confinement of ␣ the model. At this point, the jet reorganizes to the next in an unbounded constant shear flow z i in a frame Ϫ⍀ lower allowed meridional wavenumber and the process rotating in the j direction are continues. duˆ This abrupt reorganization of jet structure as a func- ϭϪ͑␣ Ϫ 2⍀͒wˆ Ϫ ␯͑l2 ϩ m2͒uˆ, ͑A2a͒ dt tion of parameter change is an example of a fundamental process in jet dynamics with important climate con- d␷ˆ nections. Discontinuous reorganization of eddy-driven ϭϪilpˆ Ϫ ␯͑l2 ϩ m2͒␷ˆ, and ͑A2b͒ dt jets can alter both climate statistics (Fuhrer et al. 1999; Wunsch 2003) and the climate itself (Toggweiler et al. dwˆ ϭϪimpˆ Ϫ 2⍀uˆ Ϫ ␯͑l2 ϩ m2͒wˆ . ͑A2c͒ 2006). Abrupt change of climate and climate statistics is dt common in the climate record (Alley et al. 1993), but mechanisms for producing abrupt transitions have been These equations describe the evolution of x indepen- difficult to find. Storm-track reorganization is an alter- dent perturbations. These perturbations excite roll cir- native and perhaps complementary mechanism to reor- culations in the (y, z) plane associated with zonal (x) ganization of thermohaline circulations (Weaver et al. streaks (Farrell and Ioannou 1993a). 1991; Kaspi et al. 2004) and sea ice switches (Gildor and Using the continuity equation, we obtain Tziperman 2003) for explaining the record of abrupt climate change. 2im⍀ pˆ ϭ uˆ, ͑A3͒ We find that baroclinic adjustment in a wide channel l2 ϩ m2 occurs by the formation of multiple narrow jets, each of which while stable supports a substantial fraction of the and by eliminating the pressure from the vertical (z) mean thermal gradient. Moreover, the heat flux across velocity equation, we obtain the dynamical system the channel is controlled by these narrow jets. duˆ This work provides a detailed physical theory for the ϭϪ␯͑l2 ϩ m2͒uˆ Ϫ ͑␣ Ϫ 2⍀͒wˆ and ͑A4a͒ emergence of jet structure and structural transition in dt baroclinic turbulence. The equilibrium state obtained 2⍀ constitutes a theory for the general circulation of the dwˆ 2l 2 2 ϭϪ uˆ Ϫ ␯͑l ϩ m ͒wˆ . ͑A4b͒ midlatitude atmosphere. dt l2 ϩ m2 For reasons of theoretical clarity, we have chosen to concentrate on the case of jets in meridionally homo- For constant parameters, the perturbation dynamics geneous domains. However, the meridional structure of are governed by the matrix the earth’s subtropical jet is forced and this jet would Ϫ␯͑ 2 ϩ 2͒ Ϫ͑␣ Ϫ ⍀ exist even in the absence of eddies (Held and Hou l m 2 A ϭ ͫ ͬ. ͑A5͒ 1980). Extension of this work to address jet structure Ϫ2⍀l 2ր͑l2 ϩ m2͒ Ϫ␯͑l 2 ϩ m2͒ and transition when a large-scale meridional structure is imposed will be the subject of future work. The dynamics are stable if the shear of the mean flow is ␣ ϭ ⍀ϩ␯2 2 ϩ 2 3 ⍀ 2 smaller than c 2 (l m ) /(2 l ). Acknowledgments. This work was supported by NSF When the dynamics are stable, the ensemble mean ATM-0736022. Petros Ioannou has been partially sup- energy density for unit mass density ͗E͘ϭ͗uˆ 2ϩ␷ˆ2ϩwˆ 2͘/4

Unauthenticated | Downloaded 09/26/21 10:46 PM UTC NOVEMBER 2008 FARRELL AND IOANNOU 3373 maintained under white stochastic excitation with unit where the perturbation covariance matrix C ϭ covariance is ͗(uˆ,wˆ )(uˆ,wˆ )†͘ solves the Lyapunov equation

† l 2 ϩ m2 AC ؉ CA ϭϪI. ͑A7͒ 1 ͗E͘ ϭ ͩC ϩ C ͪ, ͑A6͒ 4 11 l 2 22 The solution is

2␯2͑l 2 ϩ m2͒3 Ϫ 2fl2͑␣ Ϫ 2⍀͒ ϩ ͑␣ Ϫ 2⍀͒2͑l 2 ϩ m2͒ C ϭ E , ͑A8a͒ 11 8␯3͑l 2 ϩ m2͒4 n

␯2͑l 2 ϩ m2͒4 Ϫ ⍀l 2͑␣ Ϫ 2⍀͒͑l 2 ϩ m2͒ ϩ ⍀2l 4 C ϭ E , and ͑A8b͒ 22 4␯3͑l 2 ϩ m2͒5 n

␣͑l 2 ϩ m2͒ ϩ 2⍀l 2 C ϭ C ϭϪ E , where ͑A8c͒ 12 21 8␯2͑l 2 ϩ m2͒3 n

␯͑l2 ϩ m2͒2 E ϭ ͑A9͒ n ␯2͑l 2 ϩ m2͒3 Ϫ 2⍀l 2͑␣ Ϫ 2f ͒ is the total variance maintained by the equivalent normal system

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