Eddy memory as an explanation of intra-seasonal periodic behavior in baroclinic eddies
EEddy memory as an explanation of intra-seasonal periodic behavior in barocliniceddies
Woosok Moon,
1, 2
Georgy E. Manucharyan, and Henk A. Dijkstra Department of Mathematics, Stockholm University 106 91 Stockholm, Sweden Nordita, Royal Institute of Technology and Stockholm University, SE-10691 Stockholm, Sweden ∗ School of Oceanography, University of Washington, Seattle, WA, USA Institute for Marine and Atmospheric Research Utrecht, Utrecht University, Utrecht, Netherlands (Dated: February 9, 2021)The baroclinic annular mode (BAM) is a leading-order mode of the eddy-kinetic energy in theSouthern Hemisphere exhibiting oscillatory behavior at intra-seasonal time scales. The oscillationmechanism has been linked to transient eddy-mean flow interactions that remain poorly understood.Here we demonstrate that the finite memory effect in eddy-heat flux dependence on the large-scaleflow can explain the origin of the BAM’s oscillatory behavior. We represent the eddy memory effectby a delayed integral kernel that leads to a generalized Langevin equation for the planetary-scaleheat equation. Using a mathematical framework for the interactions between planetary and synopticscale motions, we derive a reduced dynamical model of the BAM – a stochastically-forced oscillatorwith a period proportional to the geometric mean between the eddy-memory time scale and thediffusive eddy equilibration timescale. Our model provides a formal justification for the previouslyproposed phenomenological model of the BAM and could be used to explicitly diagnose the memorykernel and improve our understanding of transient eddy-mean flow interactions in the atmosphere.
I. INTRODUCTION
Large-scale atmospheric dynamics at mid-latitudes issignificantly influenced by the baroclinic wave life cy-cle initiated by shear instability [43]. Constrained bygeostrophic and hydrostatic balance, a meridional tem-perature gradient imposed by the heat imbalance be-tween tropics and high latitudes is proportional to thevertical shear of the mid-latitude jet, which acts as a keyparameter controlling the growth rate of baroclinic waves[8, 11, 41]. During the baroclinic growth of synopticwaves, poleward heat flux increases and the waves prop-agate upward, after which the waves break near criticallatitudes while propagating equatorward. The whole lifecycle of unstable baroclinic waves could be representedby energy exchange between zonal mean zonal wind andsynoptic eddies [12, 28, 33, 51]. This cycle is reflected indaily weather at mid-latitudes and understood as an en-gine to transfer residual energy from the tropics to highlatitudes. It determines many aspects of large-scale cir-culations in extra-tropical areas [19].The life cycle of baroclinic waves is an essential partof variability in mid-latitudes with its time-scale beingaround 3-4 days. However, the low-frequency intra-seasonal variability with a time scale longer than weatherbut shorter than seasonal [7, 16] is not entirely explainedby the traditional baroclinic wave life cycle [2, 44]. Forexample, the North Atlantic Oscillation (NAO) [22],which represents the fluctuation of the difference of at-mospheric pressure between the Icelandic Low and theAzores High, contains the intra-seasonal time-scale ofaround 10-days. Similarly, the Arctic Oscillation (AO) ∗ [email protected] [56] or the Southern Annular Mode (SAM) [25], repre-senting the zonally symmetric seesaw between sea levelpressures in polar and temperate latitudes in North-ern and Southern Hemisphere, respectively, also vary onintra-seasonal timescales.The variability of the mid-latitude westeries is com-monly defined using the zonal index [35] that was origi-nally defined by the pressure difference between 35N and55N, but there are many variants [24, 46]. The zonal in-dex, especially in the Southern Hemisphere, shows strongintraseasonal and interannual variability based on severalobservational data [17]. On intra-seasonal time-scales, itis hard to identify a major external forcing and hence themid-latitude jet variability could be controlled by inter-nal dynamics [7]. Indeed, the low-frequency variability ofthe zonal index has been identified in simple numericalmodels with zonally-symmetric thermal forcing [45], im-plying that variability can be caused by the interactionbetween synoptic eddies and zonal mean field.The eddy-mean flow interaction is a highly nonlin-ear process involving turbulent mixing of synoptic wavesleading to energy transfer among different scales thatare difficult to accurately parameterize. However, phe-nomenological models of eddy-mean flow interactionsexhibiting oscillatory behavior have been put forward.Thompson and Barnes [53] introduced a stochastic modelto explain the quasi-oscillatory behavior of the polewardheat flux in the Southern Hemisphere, also referred toas the Southern Hemisphere baroclinic annular mode(BAM) introuduced by [57]. Specifically, they suggesta two-dimensional stochastic model representing interac-tions between the poleward heat flux and the meridionaltemperature gradient to capture the quasi-oscillatoryvariability with a dominant time-period around 25 days.In this model, the increase of the poleward heat flux isproportional to the growth rate of eddies generated by a r X i v : . [ phy s i c s . a o - ph ] F e b baroclinic instability, which is proportional to the merid-ional temperature gradient. At the same time, the timeevolution of the anomalous meridional temperature gra-dient is controlled by the poleward heat flux with a damp-ing. This phenomenology results in a two-dimensionalstochastic dynamical system that contains oscillatory so-lutions depending on the choice of parameters. Withthis idealized description of the variability, Thompsonand Barnes [53] suggest that the quasi-oscillatory low-frequency variability can be caused by nonlinear eddy-mean flow interaction. Even though the model reflectsthe basic characteristics of the baroclinic wave life cycle,its direct connection to the primitive equations is notdescribed, which obscures the physical interpretation ofmodel parameters.The BAM is a companion index of the Southern Annu-lar mode (SAM) in the Southern Hemisphere represent-ing the characteristics of large-scale low frequencies inthe atmosphere. The SAM is defined by zonal mean ki-netic energy describing the variability of zonal mean windmainly influenced by the momentum flux. The BAM isconstructed by eddy kinetic energy mainly controlled bythe meridional heat flux in lower levels [57]. The twoprocesses, meridional heat flux in lower levels and mo-mentum flux in upper ones, show different characteris-tics in their variabilities [3, 4, 40, 60, 61]. The SAM iswell approximated by an autoregressive model of order1 (AR(1) process), having a red noise spectrum, but theBAM shows a distinct peak around 25 days in its powerspectrum. The stochastic oscillatory behavior shown inthe BAM is worthwhile to be investigated in detail dueto the expectation that it leads to significant progressin predictability in mid-latitudes on sub-seasonal time-scales. Furthermore, [55] argue that the BAM in theNorthern Hemisphere shows a dominant peak located ataround 20 days in its power spectrum. This implies thatthe oscillatory behavior in meridional heat flux or eddykinetic energy is an intrinsic feature of eddy-mean flowinteractions in large-scale atmospheric dynamics. There-fore, the major question is how the meridional heat fluxin lower levels can induce the oscillatory behavior in theinteraction with mean field.Generation of low-frequency oscillations in a rotatingfluid does not seem to be limited to the large-scale at-mosphere as similar internal oscillations were also foundin large-scale ocean currents. An eddy-resolving oceanmodel simulation with a repeated annual cycle forcingreveals an intrinsic mode in the Southern Ocean witha period of 40-50 year [23, 59]. An idealized modelof the surface-stress-driven Beaufort Gyre in the Arc-tic Ocean with mesoscale eddies as a key equilibrationprocess [29] generates an oscillation with a period of 50years [31]. To explain the oscillation Manucharyan et al. [31] introduce eddy memory into the relation betweenthe eddy buoyancy flux and the mean buoyancy gradi-ent, which leads to a modification of the commonly usedGent-McWilliams (GM) parameterization [15]. The in-clusion of eddy memory leads to a stochastic oscillation if the ratio between the memory timescale and the diffusiveequilibration timescale reaches a certain threshold [31].Generally, according to the Mori-Zwanzig formalism[62], the dynamic interactions between slow-evolvingmodes and fast ones are reflected as a delayed integralon the time evolution of the slowly-varying modes. Thememory effect, normally represented by a delayed termor integral, is not new in climate science. One of thesimplest models for the El-Nin˜o and Southern Oscilla-tion (ENSO) is a delayed ordinary differential equation[52]. Even with such one-dimensional model, it was foundthat the delayed model contains chaotic dynamics underseasonal forcing[58]. Recently, such delay equation cli-mate models were derived using the Mori-Zwanzig for-malism [13]. Considering the complex interactions influid-dynamical systems, the memory effect is an intrinsiccharacteristic leading to various internal variability.Our research focuses on the memory effect in the in-teraction between synoptic eddies and a zonal meanzonal wind and its relation to low frequency modes ontimescales longer than the weather. More specifically, wepropose a formalized explanation of the quasi-oscillatorybehavior of the BAM [53] by incorporating the eddymemory [31] into multi-scale equations of atmosphericdynamics [32]. We outline the multi-scale atmosphericmodel in Section II and we introduce the eddy memoryeffect onto the planetary -scale equations in Section 3.,where we formally derive the reduced model for the zonalmean flow, the stochastic oscillator. We conclude in Sec-tion 4. II. MULTISCALE MODEL OF ATMOSPHERICMOTIONS
The basic governing equation for Earth’s climate startsfrom the simplest heat flux balance between short-waveradiative flux and outgoing longwave radiative flux,which gives us a global average temperature as an equi-librium. This is possible when we consider the Earth as apoint object in the universe maintaining a thermal equi-librium state. If we magnify the size of the Earth from apoint to a sphere, we can see that there is an imbalance ofheat flux from tropics to higher latitudes, which requirestransferring heat from the tropics to higher latitudes. Asimple approximation of the poleward heat flux is a tur-bulent diffusion with a constant eddy diffusivity. In aneven more magnified view of the tropics and polar areas,the dominant physics shifts from energy flux balance topotential vorticity conservation, which acts as a theoreti-cal foundation for the generation of mid-latitude weathersystems. There are unclearly defined timescales betweenthe weather dominated by the vorticity dynamics and theseason controlled by external heat flux. These timescaleslie on around 10 to 30 days, where numerous variabil-ity has been detected in climate data. There should bean intermediate framework lying between the energy fluxbalance and the vorticity dynamics to investigate causesof the variability.The large-scale atmosphere is governed by the primi-tive equations which are comprised of three momentumbalances, the continuity equation and the heat budget.In Cartesian coordinates, the three momentum equationsare
DuDt − f v = − ρ ∂P∂xDvDt + f u = − ρ ∂P∂yDwDt = − ρ ∂P∂z − g, (1)where P and ρ are atmospheric pressure and density, re-spectively, and the velocities in the zonal, meridional, andvertical directions are u , v , and w . The Coriolis parame-ter f is approximated by f + βy , where f is a Coriolisparameter calculated at a reference latitude φ and βy isthe latitudinal variation of the Coriolis parameter where β = ∂f /∂y . The material derivative D/Dt is defined as
D/Dt ≡ ∂/∂t + u∂/∂x + v∂/∂y + w∂/∂z . Along with themomentum equations, the continuity equation is1 ρ DρDt + ∂u∂x + ∂v∂y + ∂w∂z = 0 . (2)Finally, to complete the equations, we need the ideal gaslaw P = ρRT and thermodynamic equation D Θ Dt = Θ C P T (cid:18) κρ ∇ T + Q (cid:19) , (3)where Θ is the potential temperature, i.e.,Θ ≡ T (cid:18) P P (cid:19) R/C P , (4)where P and T represent the pressure and the temper-ature, respectively, P is a reference pressure, typically1000mb, R is the gas constant for air, C P the specific heatat the constant pressure, and P is a reference pressure.The full equations are highly nonlinear and complicatedand to explain the BAM, we need to simplify the equa-tions using an appropriate scaling.The scale of atmospheric motions is distinguished bythe magnitude of the Rossby number, U/f L , where U isa horizontal velocity scale, and L is a horizontal lengthscale. When the Rossby number is asymptotically small,the geostrophic balance becomes a dominant force bal-ance in the horizontal momentum equations. The large-scale atmospheric flows satisfying the geostrophic and hy-drostatic balances are called geostrophic motions.There are two kinds of geostrophic motions dependingon the horizontal length scale [42]. If the length scale issimilar to the internal Rossby deformation radius ( L ∼ km ), the motion is called quasi-geostrophic and isdescribed by the conservation of potential vorticity,D H D t (cid:20) ∂ ψ∂x + ∂ ψ∂y + 1 ρ s ∂∂z (cid:18) ρ s S ∂ψ∂z (cid:19) + βy (cid:21) = 0 , (5) where D H D t = ∂∂t − ∂ψ∂y ∂∂x + ∂ψ∂x ∂∂y , (6) ψ is a leading-order pressure field acting as a stream func-tion, ρ s is a mean vertical density profile, S represents theaverage vertical stability, and ∂ ψ∂x + ∂ ψ∂y + ρ s ∂∂z (cid:16) ρ s S ∂ψ∂z (cid:17) + βy is the potential vorticity. All the variables are non-dimensionalized [39]. On the other hand, if the horizontallength-scale is close to the external Rossby deformationradius ( L D ∼ km ), the governing equations become u L = − ∂P L ∂y (7) v L = ∂P L ∂x (8) ρ L = − ρ s ∂∂z (cid:16) ρ s P L (cid:17) (9)1 ρ s ∂∂z (cid:16) ρ s w L (cid:17) − β L v L = 0 (10) ∂ Θ L ∂t + u L ∂ Θ L ∂x + v L ∂ Θ L ∂y + w L ∂ Θ L ∂z + w L (cid:15) Θ s dΘ s d z = Q L , (11)where P L is the planetary-scale pressure, u L , v L and w L are the zonal, the meridional and the vertical velocity, re-spectively, Θ s is the hemispheric average potential tem-perature, Θ L is an anomalous potential temperature, β L is the planetary-scale beta effect, and Q L represents ra-diative processes. The thermodynamic forcing Q L is aresidual of local radiative fluxes driving a local temper-ature change. The subscript L implies planetary-scalevariables. Eqs. (7)-(11) are usually referred to as plane-tary geostrophy, where the heat equation with advectionis constrained by the geostrophic and hydrostatic bal-ances, together with the Sverdrup relation [32]. Similarresults can be found in [9, 10], where the same leading-order equations are derived. They used a different smallparameter (cid:15) = ( a Ω /g ) instead of the Rossby number U/f L , where a is the earth’s radius and Ω is the earth’srotation frequency, thus the detailed derivations towardthe leading-order equations are different. On the otherhand, [32] derive the same results based on the Rossbynumber as a small parameter in the asymptotic analysisfollowing the historical development of theories of quasi-geostrophic motions in large-scale atmospheric dynamics[39].The two geostrophic motions coexist in the large-scaleatmosphere. Hence, the large-scale atmospheric dynam-ics should be represented by interactions between thetwo scales. Because these two scales are asymptoti-cally separate, we can use a multi-scale analysis in spa-tial and temporal domains. Based on the Rossby num-ber in the planetary scale (cid:15) = U/ ( f L D ), we can in-troduce the two scales, ( X, Y, ˜ t ) for the planetary scaleand ( x, y, t ) for the quasi-geostrophic (QG) one, where( x, y, t ) = (cid:15) / ( X, Y, ˜ t ). Thus, the time and spatialderivatives are scaled as ∂∂t → ∂∂ ˜ t + (cid:15) − / ∂∂t∂∂x → ∂∂X + (cid:15) − / ∂∂x∂∂y → ∂∂Y + (cid:15) − / ∂∂y . The two scales are separated by (cid:15) / which comes fromthe estimation that L/L D ∼ (cid:15) / in the large-scale at-mosphere in the earth. The internal Rossby deformationradius L is around 1000km in mid-latitudes and the ex-ternal (barotropic) one L D around 3000km.A regular perturbation analysis of the primitive equa-tions (1-4) [32] leads to u L = − ∂P L ∂Y , v L = ∂P L ∂X , ρ L = − ρ s ∂∂z ( ρ s P L ) (12) u = − ∂P ∂y , v = ∂P ∂x , ρ = − ρ s ∂∂z ( ρ s P ) (13) ∂w L ∂z − H w L = β L v L (14) ∂ Θ L ∂ ˜ t + u L ∂ Θ L ∂X + v L ∂ Θ L ∂Y + w L (cid:18) ∂ Θ L ∂z + S (cid:19) = − (cid:18) ∂∂t + u L ∂∂x + v L ∂∂y (cid:19) Θ − (cid:18) u ∂∂X + v ∂∂Y (cid:19) Θ L − ∂∂x (cid:16) u Θ (cid:17) − ∂∂y (cid:16) v Θ (cid:17) − w (cid:18) ∂ Θ L ∂z + S (cid:19) + Q L (15) ∂∂t ∇ P + ( u L + u ) ∂∂x ∇ P + ( v L + v ) ∂∂y ∇ P + βv = (cid:18) ∂∂z − H (cid:19) w , (16)where H = − ρ s dρ s ( z ) dz and S = F Θ s d Θ s ( z ) dz . The Burgernumber F is defined as L /L D where L is an inter-nal Rossby number and L D the external Rossby num-ber. These L and the numbers 0 and 1 are used asthe subscripts to represent planetary-scale and synoptic-scale variables, respectively. The number 0 (the num-ber 1) represents the leading-order (the next order)in the synoptic-scale. It is assumed that planetary-scale (synoptic-scale) variables are only dependent uponplanetary-scale (synoptic-scale) coordinates, X , Y , and˜ t ( x , y , and t ). The vertical coordinate z is used forthe both scales. The equations (12-16) show the dy-namics of the two scales and their mutual interactions.The detailed derivation and discussions are found in [32].The equations (12-14) describe that the two scales sat-isfy the basic balances, geostrophic and hydrostatic bal-ances. The continuity equation (14) in the planetaryscale includes the O (1) beta effect, which is known asthe Sverdrup relation. The two scales contribute to theenergy flux balance in the heat equation (15) with theirown temporal and spatial scales. The last equation (16) is the vorticity equation governing the quasi-geostrophicdynamics. Here the planetary geostrophic motion pro-vides a mean field for the development of vorticity on theRossby deformation scale. [10] also considered a similarmulti-scale analysis to represent the interactions betweenplanetary and synoptic scales, which leads to equivalentresults. However, their main focus lies on vorticity dy-namics at the two scales, and hence a planetary vortic-ity equation is derived from the combination of the heatequation and the Sverdrup relation along with the quasi-geostrophic potential vorticity equation. Our focus is inthe planetary-scale heat equation with thermal forcing Q L . Lying between the simple energy flux balance model[37] and the quasi-geostrophic dynamics, the planetaryheat equation together with basic dynamic balances con-nects the planetary thermal heat flux to fluid dynamicson planetary scales.The above equations become simpler when theplanetary-scale thermodynamic forcing Q L is zonally ho-mogeneous. The planetary scale preserves that symme-try, in which case all X -derivative terms vanish. Hence,the zonal means of the planetary meridional and verticalvelocities v L and w L become zero. This yields u L = − ∂P L ∂Y (17) ∂ Θ L ∂ ˜ t = − (cid:18) ∂∂t + u L ∂∂x (cid:19) Θ − ∂∂x (cid:16) u Θ (cid:17) − ∂∂y (cid:16) v Θ (cid:17) − w (cid:18) ∂ Θ L ∂z + S (cid:19) − v ∂ Θ L ∂Y + Q L (18) ∂∂t ∇ P + ( u L + u ) ∂∂x ∇ P + v ∂∂y ∇ P + βv = (cid:18) ∂∂z − H (cid:19) w . (19)We can introduce a planetary-scale time- and spatial-average to the QG variables under the assumption thatthe time- and spatial-average of a QG variable is closeto zero. This implies that the overall effect of synopticscales on the planetary-scale motions is represented bythe temporal and spatial average of QG-scale fluxes. Inparticular, it is important to consider a planetary-scalespatial-average over the terms containing the QG-scalespatial derivative such as ∂/∂x and ∂/∂y . Let’s considera large horizontal length l for the spatial-average in thesynoptic-scale and then the meridional average of ∂F∂y (cid:12)(cid:12)(cid:12) Y ,where Y represents a position in the planetary-scale co-ordinate, is then interpreted as ∂F∂y (cid:12)(cid:12)(cid:12) Y = 12 l (cid:90) Y + lY − l ∂∂y F dy (cid:39) lim ∆ Y → (cid:15) / Y ( F ( Y + ∆ Y ) − F ( Y − ∆ Y ))= (cid:15) ∂∂Y F. (20)Here l and ∆ Y are non-dimensional lengths in synoptic-and planetary-scale, respectively. The same length is ex-pressed in two length scales, i.e., l ∗ = ∆ Y ∗ where ∗ isused to represent dimensional quantities, which is sameas Ll = L D ∆ Y . Thus, l = ∆ Y /(cid:15) / where L/L D = (cid:15) / is used. Due to the asymptotic difference between thetwo scales, the length l in the synoptic-scale is approxi-mated by the limit of the planetary-scale length ∆ Y to-ward zero.The heat equation (18) and the QG vorticity equation(19), after taking the planetary-scale time-average ( · ),become ∂ Θ L ∂ ˜ t = Q L − ∂∂x u Θ − ∂∂y v Θ − w (cid:18) ∂ Θ L ∂z + S (cid:19) (21)1 ρ s ∂∂z ( ρ s w ) = ∂∂x u ∇ P + ∂∂y v ∇ P , (22)where the time average of linear terms in synoptic scalessuch as ∂ Θ /∂t and ∂ Θ /∂x is assumed to be zero andthe nonlinear terms representing heat and momentumfluxes are considered as non-zero average terms. Using ∇ P = ∂v /∂x − ∂u /∂y , we find that the equation (22)is equivalent to1 ρ s ∂∂z ( ρ s w ) = (cid:18) ∂ ∂x − ∂ ∂y (cid:19) u v − ∂ ∂x∂y u − v . (23)Now, we can take the planetary-scale spatial averageover the equations (21, 22), and combine the two by re-placing w by the horizontal vorticity convergence, whichleads to ∂ Θ L ∂ ˜ t = Q L − (cid:15) / (cid:18) ∂∂X u Θ + ∂∂Y v Θ (cid:19) + (cid:15) (cid:18) ∂ Θ L ∂z + S (cid:19) (cid:18) ∂ ∂X − ∂ ∂Y (cid:19) (cid:18) ρ s (cid:90) z ρ s u v d z (cid:48) (cid:19) (24) − (cid:15) (cid:18) ∂ Θ L ∂z + S (cid:19) ∂ ∂X∂Y (cid:18) ρ s (cid:90) z ρ s ( u − v ) d z (cid:48) (cid:19) . The equation (24), after taking the zonal-average (cid:104) ( · ) (cid:105) , becomes ∂ Θ L ∂ ˜ t = Q L − (cid:15) / ∂∂Y (cid:68) v Θ (cid:69) − (cid:15) (cid:18) ∂ Θ L ∂z + S (cid:19) ∂ ∂Y (cid:18) ρ s (cid:90) z ρ s (cid:68) u v (cid:69) d z (cid:48) (cid:19) , (25)where the dominant balance after ignoring the momen-tum flux contribution in the O ( (cid:15) ) order is ∂ Θ L ∂ ˜ t (cid:39) Q L − (cid:15) / ∂∂Y (cid:68) v Θ (cid:69) . (26)In the equation (26), we can consider an asymp-totic solution Θ L (cid:39) [Θ L ] + (cid:15) / η , in which case the radiative process Q L is represented approximately as Q L (˜ t, Θ L , Y ) (cid:39) Q L (˜ t, [Θ L ] , Y ) + (cid:15) / a ( Y, ˜ t ) η , where a ( Y, ˜ t ) = ∂Q L (˜ t, Θ L , Y ) /∂ Θ L | Θ L =[Θ L ] . The leading-order balance is ∂ [Θ L ] ∂ ˜ t (cid:39) [ Q L ] . (27)This is understood as a local radiative energy flux bal-ance, and [Θ L ] represents the seasonal mean of the po-tential temperature mainly determined by the seasonally-varying radiative flux balance [ Q L ]. The simplest rep-resentation of [ Q L ] is S (1 − α ) − σ [Θ L ] , where S isthe shortwave radiative flux, α is the local albedo, and σ [Θ L ] is the outgoing longwave radiative flux, but wehave to consider other heat fluxes such as incoming long-wave radiative flux and turbulent sensible and latent heatflux. In the planetary-scale heat equation, the leading-order governing physics is the local energy flux balancecontributing mainly to a seasonal cycle of the potentialtemperature.The O ( (cid:15) / ) order balance is ∂η∂ ˜ t = a ( Y, ˜ t ) η − ∂∂Y (cid:68) v Θ (cid:69) . (28)The fluctuation η around the seasonal cycle [Θ L ] is con-trolled by the seasonal sensitivity of the radiative pro-cesses a ( Y, ˜ t ) and the meridional heat flux convergenceinduced by synoptic eddies. [34] focus on the monthly-average variability of surface air temperature based ona periodic non-autonomous stochastic differential equa-tion ˙ η = a (˜ t ) η + N (˜ t ) ξ (˜ t ), where a (˜ t ) is equivalent to the a ( Y, ˜ t ) in (28), ξ (˜ t ) a white noise mimicking the overalleffect of weather-related processes and N (˜ t ) is a monthly-varying amplitude of noise. The noise forcing N (˜ t ) ξ (˜ t )can be understood as an approximation of the residualforcing R ( Y, ˜ t ) which can be considered as a contribu-tion of short-time processes in the equation (28). Themonthly statistics including variance and lagged correla-tions in a given monthly-averaged data such as surfacesea temperature or tropical climate indexes is regeneratedby the periodic non-autonomous stochastic model withan appropriate choice of the two periodic functions a (˜ t )and N (˜ t ). In particular, the positive sign of the a (˜ t ) im-plies existence of positive feedbacks which magnifies themagnitude of a given perturbation. For example, whenthe stochastic model is applied to the Nino 3.4 index,the a (˜ t ) is positive from July to November, which showsthe seasonality of the Bjerknes feedback. The construc-tion of the monthly sensitivity a (˜ t ) enables us to detecthow a positive feedback shapes the monthly statistics of aclimate variable. It is essential to understand how phaselocking and seasonal predictability barrier of climate phe-nomena are associated with the magnitude and timing ofpositive feedback.The equation (28) is an extension of the one-dimensional stochastic model considering meridionalvariation. Especially, it represents a strong influencefrom synoptic-scale eddies on the planetary-scale tem-perature. It is in contrast with [3] which provide a sim-ilar budget equation on the planetary scale based onthe multi-scale analysis suggested by [10]. In their bud-get equations, the planetary scale is independent fromthe synoptic scale, thus the two scales interact indi-rectly through source terms. On the other hand, theequation (28) tells that the variability of planetary-scaletemperature is mainly determined by the turbulent heatflux induced by synoptic eddies. One remaining step toclose the planetary-scale equations is to parameterize themeridional heat flux (cid:68) v Θ (cid:69) using the planetary-scaletemperature Θ L . III. EMERGENCE OF A GENERALIZEDLANGEVIN EQUATIONA. Fickian diffusion model
The meridional heat flux at mid-latitudes under thegrowth of synoptic waves is the major consequence ofbaroclinic instability. The baroclinic instability is initi-ated from an unstable mean state measured by the ver-tical shear of zonal mean wind ∂u L /∂z . The growthof baroclinic waves by a baroclinic instability startsfrom near surface, which induces a meridional heat flux.Hence, the meridional gradient of the planetary temper-ature ∂ Θ L /∂Y is strongly related to the meridional heatflux (cid:68) v Θ (cid:69) .The simplest representation of the mutual relationshipbetween ∂ Θ L /∂Y and (cid:104) v Θ (cid:105) is a turbulent flux gradi-ent parameterization, (cid:104) v Θ (cid:105) (cid:39) − K∂ Θ L /∂Y , where K is a constant [37]. If we use this parameterization in theequation (26), this leads to ∂ Θ L ∂ ˜ t (cid:39) (cid:15) / K ∂ Θ L ∂Y + Q L . (29)This simple energy flux balance model was first intro-duced by [50] and [36] to include the effect of large-scaleatmospheric dynamics upon the local energy flux bal-ance. The consequence of complicated large-scale atmo-spheric dynamics especially in mid-latitudes is to transferthe surplus of energy in low latitudes to high latitudes,which is simply approximated by a turbulent meridionaldiffusion.The temporal and spatial evolution of the perturbation η is represented by ∂η∂ ˜ t = K ∂ η∂Y + a ( Y, ˜ t ) η + R ( Y, ˜ t ) , (30)where we include the R ( Y, ˜ t ) for the contribution of short-time processes. We can consider two boundary conditionsin meridional coordinate, ∂η∂Y ( · , Y = 0) = ∂η∂Y ( · , Y = 1) =0 implying there is no meridional heat flux near the poleand the tropical areas. For the Y -direction diffusion operator, consider theeigenvalue problem, K ∂ ∂Y H n = − λ n H n (31)with ∂H n ∂Y ( Y = 0) = ∂H n ∂Y ( Y = 1) = 0. Here, H n = A n cos( nπY ) with λ n = Kn π and n = 0 , , , · · · .Hence, the temporal and spatial perturbation of the po-tential temperature anomaly η can be represented by aninfinite series of the eigenfunctions with time-varying co-efficients, η = ∞ (cid:88) n =0 x n (˜ t ) H n ( Y ) . (32)Consider first the case that a ( Y, ˜ t ) is a constant − γ and that the sum of short-time processes R ( Y, ˜ t ) is rep-resented as a sum of the same eigenfunctions, R = (cid:80) ∞ n =0 R n (˜ t ) H n ( Y ).The time-varying coefficient x n (˜ t ) then satisfies dx n (˜ t ) d ˜ t = − ( λ n + γ ) x n (˜ t ) + R n (˜ t ) . (33)If we consider the last term as a stochastic noise, in par-ticular, a Gaussian white noise, this becomes a Langevinequation with a decay time scale 1 / ( λ n + γ ). The timescale becomes shorter as n increases, which indicates thatthe first several modes representing large-scale motionsdominate in the overall fluctuations. When n is equal to0, the Langevin equation becomes ˙ x = − γx + R (˜ t ),where γ represents the seasonal sensitivity of local energyflux balance mainly dominated by radiative-convectiveequilibrium in land and ocean heat flux in ocean bound-ary layer. The formalism used in this argument wasfirst introduced by [30] to explain the internal variabil-ity caused by interactions between meso-scale eddies andmean fields under the framework of the Gent-McWilliamparameterization.The simple Langevin equation was introduced to cli-mate science by [18] to interpret climate variability interms of stochastic dynamics. The deterministic part isunderstood as a stabilizing process around a mean stateand the stochastic noise as the effect of weather. Climateis understood as a combination of the stability of slowly-evolving backgrounds and short-time processes approxi-mated as a noise. The interpretation was useful to ra-tionalize the ubiquitous emergence of red noise spectrafrom climate variables including sea surface temperature(SST) [14]. In a symmetric hemisphere, the meridionalvariation of mean near-surface temperature could be con-structed by the equation (29) and the fluctuation aroundthe mean be represented by the equation (33), whichmight be consistent with the usage of an AR1 processexplaining the zonal index variability [27]. B. Quasi-oscillatory behavior of the BAM [53] (TB) studied quasi-periodic variability of theSouthern Hemisphere baroclinic annular mode (BAM).Instead of the red noise spectrum originated from a sim-ple linear Langevin equation, the BAM shows a clear signof quasi-oscillation in its power spectrum. This indicatesthat the simple Langevin equation leading to a red-noisespectrum is not adequate to describe the quasi-periodicfluctuation.To explain a periodic behavior of meridional temper-ature gradient and poleward heat flux, TB introducemutual interaction and feedback between the meridionaltemperature gradient b ≡ ∂η/∂Y and the poleward heatflux (cid:104) v Θ (cid:105) . Baroclinic instability theory tells us thatthe growth rate of the baroclinic waves is proportionalto the meridional temperature gradient [26], which couldbe represented by ddt (cid:104) v Θ (cid:105) = − α (cid:104) b (cid:105) + (cid:15) ( t ) , (34)where α is a constant representing the amplitude of thefeedback between the baroclinicity and the meridionalheat flux [20]. Here, the noise forcing (cid:15) ( t ) is added toinclude the effect of chaotic weather. At the same time,TB consider a feedback of the poleward heat flux on themeridional temperature gradient, which simply assumesthat the meridional temperature gradient (cid:104) b (cid:105) increaseslinearly with respect to the poleward heat flux (cid:104) v Θ (cid:105) .Thus, the second equation in TB is dbdt = µ (cid:104) v Θ (cid:105) − br , (35)where µ is the degree of the feedback and r is a recover-ing time-scale of the meridional temperature gradient dueto diabatic processes and vertical motions. The combina-tion of (34-35) could generate an oscillation whose periodis determined by the relevant coefficients. While ignoringthe noise forcing in the equation (34), combining the twoequations [53] leads to d bdt + 1 r dbdt + αµb = G, (36)where the G is the time-derivative of the short-timescale forcing (cid:15) ( t ). Oscillatory behavior comes out when r − αµ <
0. TB used r (cid:39) αµ (cid:39) . − to generate a dominant peak of the power spectrum ofpoleward heat flux around 20 days. They suggest a mu-tual interaction between (cid:104) v Θ (cid:105) and ∂η/∂Y with thedamping represented by a Newtonian cooling. In themodel, it is hard to guess what determines the dampingtime-scale r and the feedback parameter αµ .By a simple comparison, the equation (35) is equiva-lent to the anomalous heat equation (28) and then theequation (34) provides a relationship between the pole-ward heat flux and the meridional temperature gradient.Even though the equation (35) is likely to be derived fromthe anomalous heat equation, the damping term − b/r isnot clearly understood physically. To understand the re-lationship between b and (cid:104) v Θ (cid:105) on intra-seasonal time-scales, it seems necessary what determines the dampingtime scale r . We can incorporate the equation (34) into the equation(28) after taking time-derivative on the equation (28).This results in ∂ η∂ ˜ t = α ∂ η∂Y − γ ∂η∂ ˜ t + ∂R∂ ˜ t , (37)where − γ is used instead of the a ( Y, ˜ t ) for simplicity. Theconstant γ implies the damping time scale of seasonal en-ergy flux balance. In land, the outgoing longwave radia-tive flux dominantly controls the time scale. Similarly,we can consider the eigenvalue problem for the diffusionoperator, α ∂ ∂Y H n = − λ n H n , (38)where η = (cid:80) ∞ n =0 x n (˜ t ) H n ( Y ). We obtain the time-evolution equation for x n (˜ t ), d x n d ˜ t + γ dx n d ˜ t + λ n x n = dR n d ˜ t , (39)where R = (cid:80) ∞ n =0 R n (˜ t ) H n ( Y ) is used. The equation for x n is similar to the equation (36), which suggests thatthe damping time scale is equivalent to 1 /γ . Physically,the γ is introduced as a sensitivity of the radiative en-ergy flux balance. In terms of time-scale, it could beinterpreted as a time-scale to return to a climatologicalseasonal cycle. [34] developed a time-series method toconstruct the γ from monthly-averaged surface air tem-perature. The time scale of the surface energy flux bal-ance in the Southern Hemisphere is around 1.5 months,which is much larger than 4 days. Therefore, the damp-ing time scale r does not come from diabatic processesrelated to radiative fluxes. C. Eddy memory and generalized Langevinequation
The damping time scale introduced to explain the oscil-latory behavior of the baroclinic mode in Southern Hemi-sphere is much shorter than that of the mean seasonalcycle of radiative processes. It is plausible that the timescale is related to synoptic eddies, rather than any ex-ternal influences which have longer time-scales. Synopticeddies generated from the baroclinic instability due toan unstable background undergoes a specific energy cy-cle with a zonal mean steady state. The baroclinic insta-bility enables the synoptic eddies to extract energy fromthe zonal mean state, from which the poleward heat fluxincreases near surface. The growing waves propagate up-ward and equatorward and meet critical latitudes wherephase speeds of waves are same as mean wind. The wavesbreak and give their energy back to the mean state bymomentum flux. This baroclinic wave life cycle is com-plete in a few days.The overall effect of the baroclinic wave life cycle on themeridional heat flux is represented by the planetary po-tential temperature anomaly η as (cid:104) v Θ (cid:105) = − K ∂η∂Y . ThisFickian diffusion approximation is well applied when thetime evolution of the planetary variable is much slowerthan the turn-over time-scale of synoptic eddies. On in-traseasonal time scales, however, the time scale for theevolution of planetary-scale variables is not clearly sep-arated from that for the baroclinic wave life cycle. Theadvective timescale for the synoptic waves is around 1.2days (where we use L = 1000km and U = 10m/s) andthe same timescale for the planetary waves around 3.5days (where we use L D = 3000km and U = 10m/s). Acomplete cycle of baroclinic wave lifecycle takes around3-4 days, hence the timescale for the planetary-scale mo-tion is comparable with that for the baroclinic wave lifecycles. It is questionable to apply the Fickian diffusionas a parameterization of the poleward heat flux in thesetime scales.Non-Fickian approximation for turbulent transport ordiffusion has been a central topic in turbulent closureproblems [38]. For a transient and chaotic turbulence,the Fickian approximation is not enough to capture non-local and non-Gaussian nature of turbulence. One of thesimplest approach is called the minimal τ approxima-tion, where the third-order momentum represented as aforcing for the time-evolution of second-order momentsis approximated as a damping term with the timescale τ [5, 6]. This is equivalent to apply an integral kernel in-stead of a constant diffusivity with a finite memory [21].The main idea is applied to the time-evolution of thepoleward heat flux of synoptic-scale waves, i.e., ∂∂t (cid:104) v Θ (cid:105) = (cid:104) ∂v ∂t Θ (cid:105) + (cid:104) v ∂ Θ ∂t (cid:105) . (40)The time-evolution equations for the synoptic-scalemeridional velocity v and potential temperature Θ [32]are ∂v ∂t = − ( u L + u ) ∂v ∂x − v ∂v ∂y − u − βy(cid:15) / u − ∂P ∂x∂ Θ ∂t = − ∂η∂ ˜ t − ( u L + u ) ∂ Θ ∂x − v ∂η∂Y − v ∂ Θ ∂y − w (cid:18) ∂ Θ L ∂z + S (cid:19) . (41)Hence, ∂∂t (cid:104) v Θ (cid:105) = −(cid:104) v (cid:105) ∂η∂Y + D, (42)where D contains the terms representing the contributionfrom synoptic eddies. After taking synoptic-time averageon both sides, we obtain ∂∂ ˜ t (cid:104) v Θ (cid:105) = −(cid:104) v (cid:105) ∂η∂Y + D. (43)The main assumption of the minimal τ approximation isthat the contribution of the synoptic eddies D is approx-imated by a damping of the (cid:104) v Θ (cid:105) with the designatedtime-scale r . Therefore, ∂∂ ˜ t (cid:104) v Θ (cid:105) = − K ∂η∂Y − (cid:104) v Θ (cid:105) r , (44) where K ≡ (cid:104) v (cid:105) suggesting how the eddy diffusivity K isrelated to the second-order statistics of synoptic eddies.The minimal τ approximation for the (cid:104) v Θ (cid:105) seems to beequivalent to the equation (34) in the TB. The feedbackparameter α could be understood as K . TB suggests therelationship based on the result from the baroclinic in-stability. On the other hand, we derived a similar oneby the simplest non-Fickian approximation as a closureof turbulent eddies. TB included a random forcing inthe relationship to include unresolved processes, but therandomness coming from short-time processes is consid-ered in the heat equation in our derivation. The aboveparameterization can be represented by an integral form,i.e., (cid:104) v Θ (cid:105) = − K ∂∂Y (cid:90) ˜ t −∞ η exp (cid:18) − ˜ t − ˜ t (cid:48) r (cid:19) d ˜ t (cid:48) . (45)The integral form originates from integrating (44) for the (cid:104) v Θ (cid:105) with respect to the time ˜ t . From this integralform, we can see that the poleward heat flux is the resultof accumulating the baroclinicity during a certain timeuntil present. It represents the memory effect caused bysynoptic eddies.The minimal τ approximation was tested in direct sim-ulations of isotropic 3D turbulence [5]. The statisticsof a passive scalar are compared between direct simu-lations and the parameterized equations, where decentmatches are obtained. In particular, the parameteriza-tion transforms the main parabolic tracer equation toa wave equation leading to an oscillatory behavior. Thesimulations show decayed oscillation with a certain choiceof the damping time-scale r . In an idealized BeaufortGyre numerical simulation, [31] introduced a finite mem-ory kernel instead of a constant diffusivity in the Gent-McWilliam parameterization, which generates the quasi-periodic variability in the eddy-mean interaction. Theminimal τ approximation can be understood as a finitememory effect represented by an integral kernel.Taking account of memory effect of eddies, we can in-troduce an integral kernel κ ( t − t (cid:48) ) on the parameteriza-tion of the meridional heat flux, (cid:104) v Θ (cid:105) = − K ∂∂Y (cid:90) t −∞ η ( t (cid:48) ) κ ( t − t (cid:48) ) dt (cid:48) , = − K ∂η ∗ ∂Y (46)which is a generalization of the minimal τ approximationin the equation (45) where η ∗ is an effective temperatureanomaly defined by η ∗ ≡ (cid:90) t −∞ η ( t (cid:48) ) κ ( t − t (cid:48) ) dt (cid:48) . (47)Thus, the equation (30) becomes ∂η∂ ˜ t = K ∂ η ∗ ∂Y − γη + R ( Y, η ) , (48)where we assume that a ( Y, ˜ t ) = − γ for simplicity. Fur-thermore, if we expand the η ∗ using the eigenfunctionsof the diffusion operator, η ∗ = (cid:80) ∞ n =0 x ∗ n (˜ t ) Q n ( Y ), andthe residual forcing, R = (cid:80) ∞ n =0 R n (˜ t ) Q n ( Y ), each time-dependent coefficient x n satisfies dx n d ˜ t = − λ n x ∗ n − γx n + R n = − λ n (cid:90) ˜ t −∞ x n (˜ t (cid:48) ) κ (˜ t − ˜ t (cid:48) ) d ˜ t − γx n + R n , (49)which is a generalized Langevin equation [63]. The gener-alized Langevin equation contains memory terms repre-senting the effect of past states, which is generally orig-inating from interactions with short-time scale compo-nents.The complicated chaotic processes to generate mem-ory are captured by the memory kernel κ (˜ t − ˜ t (cid:48) ). Thedynamics represented by a generalized Langevin equa-tion is dependent upon the choice of integral kernel. Wecan follow the theme of the τ approximation, using anintegral kernel with a finite-time memory. It has beensuggested that baroclinic eddies generated by baroclinicinstability are eventually organized to maintain a mid-latitude jet characterized by meridional temperature gra-dient [27, 47]. This implies that the effect of barocliniceddies upon the meridional heat flux is limited within afinite time-scale, which leads to as an approximation κ (˜ t − ˜ t (cid:48) ) = exp (cid:18) − ˜ t − ˜ t (cid:48) r (cid:19) . (50)Here r represent the finite memory of the baroclinic ed-dies, which is equivalent to dx ∗ n /d ˜ t = − x ∗ n /r + x n /r .Based on the specific memory kernel, the equation (49)becomes d x n d ˜ t + (cid:18) r + γ (cid:19) dx n d ˜ t + γ + λ n r x n = R n r + dR n d ˜ t . (51)There are two-damping time-scales in the above equa-tion defined by r and 1 /γ . The 1 /γ comes from theseasonal heat flux balance such that it is approximately1 . /r (cid:29) γ . The eigenvalues λ n come again from (31) with Q (cid:48) n (0) = Q (cid:48) n (1) = 0 implying no heat flux at both ends,which gives us λ n = Kn π and Q n ( Y ) = cos ( nπY ).The baroclinic annular mode (BAM) is defined by a dom-inant mode of the meridional heat flux maximized aroundthe center of mid-latitudes. This is similar to the modewith n = 1, Q ( Y ), whose derivative Q (cid:48) ( Y ), has its maxi-mum at the center. The time-dependent coefficient of themode, x , satisfies d x d ˜ t + 1 r dx d ˜ t + γ + Kπ r x = R r + dR d ˜ t , (52)which should be understood as an equivalent form to theequation (36). The damping time-scale r = 4 days comes from the memory effect of synoptic eddies on meridionalheat flux, which seems to be strongly related to the baro-clinic wave life cycle. Moreover, αµ in the equation (36)is same as ( γ + Kπ ) /r , which leads to K (cid:39) / π . cycles/day P o w e r r=4 daysr=3 daysr=2 daysr=1 daysBAM index (a) cycles/day P o w e r K /r=0.08 days -2 K /r=0.1 days -2 K /r=0.15 days -2 K /r=0.20 days -2 BAM index (b)
FIG. 1:
Power spectra of the stochastic oscillators representedby the equation (52) with several choices of the parameters r and K along with the power spectrum of the BAM index [57]. Withthe fixed K = 1 / π , four different r ’s are used for stochasticrealization (a). Similarly, four different K ’s are chosen forgenerating power spectra with a fixed r = 4 (b). Depending on the choice of the parameters r and K ,the equation (52) can show decayed oscillatory behav-ior or exponential decay. Under the assumption that R /r + dR /d ˜ t is approximately a Gaussian white noise,we can simulate the equation to generate a stochasticrealization. Figure 1 shows several power spectra ofstochastic realizations from the equation (52) with differ-ent parameters r and K along with the power spectrumof the BAM index [57]. With a fixed K , four differentmemory time-scales from 1 day to 4 days are used forstochastic realization. In figure 1(a), we see that longermemory (4 days) leads to oscillatory behavior, which hasthe similar power spectrum with that of the BAM in-dex, and shorter memory (1 day) to a red noise spec-trum (Fig. 1a). Similarly, we fix r = 4 days and vary K , which also shows the characteristics from stochasticoscillation to red noise process (Fig.1b). Similarly, when Kπ /r = 1 days − , the power spectrum is close to thatof the BAM index.The baroclinic wave life cycle results in recovering thevertical shear of the mean jet. This could be an ori-gin of the finite memory effect in meridional heat trans-fer in planetary-scale, which leads to oscillatory behav-ior of zonal mean potential temperature. Following theabove formalism, the fluctuation of potential tempera-ture anomaly η is represented as η = (cid:80) ∞ n =0 x n (˜ t ) Q n ( Y ),hence ∂η/∂Y = (cid:80) ∞ n =0 x n (˜ t ) dQ n /dY . By the thermalwind balance, ∂U/∂z ∝ (cid:80) ∞ n =0 x n (˜ t ) dQ n /dY , which im-plies that the vertical shear of the jet in lower levels couldshow quasi-oscillatory behavior depending on the time-scale of the eddy memory and relevant eigenvalues defin-ing the decay time-scale of a specific normal mode.The main focus lies on how to parameterize the merid-ional heat flux induced by synoptic eddies (cid:104) v Θ (cid:105) using0the planetary-scale potential temperature Θ L . Consider-ing that the time scale for synoptic eddies is not muchsmaller than that of planetary-scale motions, the merid-ional heat flux is not entirely determined by an instan-taneous potential temperature gradient; instead it is de-pendent upon the temporal history of the potential tem-perature gradient. Hence, the Fickian diffusion approxi-mation is not appropriate to parameterize the meridionalheat flux induced by synoptic eddies. The simplest wayconsidering the temporal history of the planetary-scalepotential temperature is to introduce a time integrationof the potential temperature with an appropriate mem-ory kernel, which enables to include the past influence ofplanetary-scale mean fields on the turbulent heat flux.As a result, fluctuations of potential temperature canbe described approximately by a generalized Langevinequation containing a memory term. The time-delayedeffect represented as an integral kernel in the generalizedLangevin equation can generate various types of vari-ability ranging from simple red-noise process to quasi-oscillations and chaos. Hence, an appropriate choice ofintegral kernel is crucial to define the statistical charac-teristics of planetary-scale variability. IV. CONCLUSION
The main goal of the study was to rationalize the quasi-oscillatory variability of the BAM [53] on intra-seasonaltime scales using the multi-scale representation of prim-itive equations of the large-scale atmosphere flow [32].Conceptually, the multi-scale approximation is a combi-nation of the energy flux balance in the heat equation andthe potential vorticity conservation. We found that thecritical issue under the framework is the specification ofan appropriate parameterization of the horizontal heatflux convergence from the synoptic and planetary-scaleeddy dynamics. If the heat fluxes are represented usingthe simplest diffusive (Fickian) parameterization with aconstant eddy diffusivity, the planetary-scale equationssimplify to an energy flux balance with a meridional tur-bulent diffusion, similarly to [36]. Other types of eddyparameterizations are also possible, of those most no-table is the residual mean theory of Andrews and Mcin-tyre [1] with a diffusive parameterization of isopycnaleddy fluxes of potential vorticity. While the residualmean formulation gives a different perspective on themean flow dynamics compared to the direct diffusive clo-sure of the heat fluxes [49], our study questions theirequilibrium-type Fickian assumption for the relation be-tween the eddy fluxes and gradients of the consideredtracer fields. Under the equilibrium-type eddy param-eterizations, the main equation in the planetary-scalesatmosphere is the heat equation with the basic balancesresulting in a parabolic partial differential equation thatdoes not support natural oscillations. The resulting tem-poral variability of observables obeys the Langevine equa-tion leading to a red-noise spectrum that does not ex- plain the quasi-oscillatory behavior of the BAM. Thusthe equilibrium-type eddy parameterizations are likelynot appropriate at the intraseasonal timescales and canonly be used for representing the relatively long-term av-erage of the planetary-scale dynamics.We find that using a non-equilibrium eddy parameter-ization with the eddy memory effect represented by thedelayed integral in the flux-gradient relation as was pro-posed for mesoscale ocean eddies [31] can explain the os-cillatory behavior of planetary-scale atmospheric flows.The rationalization for the eddy memory effect comesfrom the notion that the overall effect of synoptic eddiesis to maintain the vertical shear of a mid-latitude jet [47],implying that the eddy energy cycle starting from baro-clinic instability near the surface could be considered as afinite timescale process. Thus, we hypothesized that themid-latitude eddies have a finite memory and introducea simple exponentially decaying memory kernel into therelation between the poleward eddy heat flux and themeridional potential temperature gradient. The inclu-sion of the finite memory kernel directly converts the heatequation from parabolic to wave equation that allows dis-persive planetary-scale waves. As a result, instead of thered-noise model, the temporal variability of the anoma-lous potential temperature near the surface now obeys astochastic oscillator model, with spectral characteristicsthat are similar to the observables associated with theBAM.The key value of our theoretical study is that it high-lights the processes and clarifies the physical interpre-tation of the crucial parameters that may be governingthe spectral characteristics of the BAM. Specifically, ourstudy implies that the period of natural oscillations of theBAM is proportional to the geometric mean between theeddy memory and the diffusive equilibration timescales.The ratio of these timescales controls the existence of apronounced spectral peak while the variance is directlyproportional to the external noise variance in the heatequation. Validating our key hypothesis about the ex-ponential memory kernel and the finite eddy memorytimescale in the relation between the mean flow and eddyheat fluxes using atmospheric models would be the nextcrucial step towards validating our theoretical argumentsabout the BAM. Furthermore, understanding the natureof the eddy memory and how its timescale depends onthe mean flow or eddy characteristics would be necessaryto understand the climate conditions under which theBAM could exhibit oscillatory behaviour. Finally, ourresults emphasize the importance of the external noisein driving the variance of the BAM and the necessity tounderstand if it is dependent on the mean flow or acts asan external/independent source of energy.
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