Physics of the Eddy Memory Kernel of a Baroclinic Midlatitude Atmosphere

Elian Vanderborght aInstitute for Marine and Atmospheric Research Utrecht, Department of Physics, Utrecht University, Utrecht, Netherlands

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https://orcid.org/0009-0003-5500-2839
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Jonathan Demaeyer bRoyal Meteorological Institute, Brussels, Belgium

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Georgy Manucharyan cSchool of Oceanography, University of Washington, Seattle, Washington

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Woosok Moon dDepartment of Environmental Atmospheric Sciences, Pukyong National University, Pusan, South Korea

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Henk A. Dijkstra aInstitute for Marine and Atmospheric Research Utrecht, Department of Physics, Utrecht University, Utrecht, Netherlands
eCentre for Complex Systems Studies, Utrecht University, Utrecht, Netherlands

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Abstract

In recent theory trying to explain the origin of baroclinic low-frequency atmospheric variability, the concept of eddy memory has been proposed. In this theory, the effect of synoptic-scale heat fluxes on the planetary-scale mean flow depends on the history of the mean meridional temperature gradient. Mathematically, this involves the convolution of a memory kernel with the mean meridional temperature gradient over past times. However, the precise shape of the memory kernel and its connection to baroclinic wave dynamics remains to be explained. In this study we use linear and proxy response theory to determine the shape of the memory kernel of a truncated two-layer quasigeostrophic atmospheric model. We find a memory kernel that relates the eddy heat flux to the zonal mean meridional temperature gradient on time scales greater than 2 days. Although the shape of the memory kernel is complex, we show that it may be well approximated as an exponential, particularly when reproducing baroclinic low-frequency intraseasonal modes of variability. By computing the terms in the Lorenz energy cycle, we find that the shape of the memory kernel can be linked to the finite time that growing baroclinic instabilities require to adapt their growth properties to the local zonal mean atmospheric flow stability. Regarding the explanation for observed baroclinic annular modes in the Southern Hemisphere, our results suggest that it is physical for these modes to be derived directly from the thermodynamic equation by considering an exponentially decaying memory kernel, provided accurate estimates of the necessary parameters are incorporated.

Significance Statement

The goal of this study was to derive the memory of the zonal mean temperature field contained in eddy heat fluxes. To do this we used recent developments in a theory stemming from statistical mechanics, called proxy response theory. This theory facilitated direct numerical computations of the parameterization that links eddy heat fluxes to the zonal mean temperature field. Notably, this parameterization incorporates a crucial memory component, which we demonstrated to be essential in explaining the periodicity of low-frequency modes of variability, specifically the baroclinic annular mode (BAM). Understanding the role of memory as a driver of this variability holds great significance, as the BAM constitutes a dominant pattern of large annular variability within the Southern Hemisphere circulation. Enhanced comprehension of this driver, which is memory, can lead to improved understanding and predictive capabilities concerning observed annular weather patterns.

© 2024 American Meteorological Society. This published article is licensed under the terms of a Creative Commons Attribution 4.0 International (CC BY 4.0) License .

Corresponding author: Elian Vanderborght, e.y.p.vanderborght@uu.nl

Abstract

In recent theory trying to explain the origin of baroclinic low-frequency atmospheric variability, the concept of eddy memory has been proposed. In this theory, the effect of synoptic-scale heat fluxes on the planetary-scale mean flow depends on the history of the mean meridional temperature gradient. Mathematically, this involves the convolution of a memory kernel with the mean meridional temperature gradient over past times. However, the precise shape of the memory kernel and its connection to baroclinic wave dynamics remains to be explained. In this study we use linear and proxy response theory to determine the shape of the memory kernel of a truncated two-layer quasigeostrophic atmospheric model. We find a memory kernel that relates the eddy heat flux to the zonal mean meridional temperature gradient on time scales greater than 2 days. Although the shape of the memory kernel is complex, we show that it may be well approximated as an exponential, particularly when reproducing baroclinic low-frequency intraseasonal modes of variability. By computing the terms in the Lorenz energy cycle, we find that the shape of the memory kernel can be linked to the finite time that growing baroclinic instabilities require to adapt their growth properties to the local zonal mean atmospheric flow stability. Regarding the explanation for observed baroclinic annular modes in the Southern Hemisphere, our results suggest that it is physical for these modes to be derived directly from the thermodynamic equation by considering an exponentially decaying memory kernel, provided accurate estimates of the necessary parameters are incorporated.

Significance Statement

The goal of this study was to derive the memory of the zonal mean temperature field contained in eddy heat fluxes. To do this we used recent developments in a theory stemming from statistical mechanics, called proxy response theory. This theory facilitated direct numerical computations of the parameterization that links eddy heat fluxes to the zonal mean temperature field. Notably, this parameterization incorporates a crucial memory component, which we demonstrated to be essential in explaining the periodicity of low-frequency modes of variability, specifically the baroclinic annular mode (BAM). Understanding the role of memory as a driver of this variability holds great significance, as the BAM constitutes a dominant pattern of large annular variability within the Southern Hemisphere circulation. Enhanced comprehension of this driver, which is memory, can lead to improved understanding and predictive capabilities concerning observed annular weather patterns.

© 2024 American Meteorological Society. This published article is licensed under the terms of a Creative Commons Attribution 4.0 International (CC BY 4.0) License .

Corresponding author: Elian Vanderborght, e.y.p.vanderborght@uu.nl

1. Introduction

Midlatitude atmospheric flows display variability on weather time scales (synoptic variability) and on much longer time scales (low-frequency variability). Examples of such low-frequency-variability phenomena are blocking events in the Northern Hemisphere (Woollings et al. 2010) and the dominant intraseasonal mode in the Southern Hemisphere, the so-called baroclinic annular mode (BAM) (Thompson and Barnes 2014). Typically, these phenomena occur on planetary scales and involve variations of the midlatitude jets. The planetary-scale low-frequency variability is often related to the interaction between planetary and synoptic dynamics (Hartmann 1974; Benzi et al. 1986; Hansen and Sutera 1986; Lorenz and Hartmann 2001; Gerber and Vallis 2007). In the classical view of the Eliassen–Palm wave flux perspective, synoptic instabilities modify the zonal mean state through momentum and heat fluxes (Edmon et al. 1980). These instabilities ensure that the meridional temperature gradient does not significantly exceed criticality (Stone 1978; Stone and Miller 1980), resulting in a continuous exchange of energy between the internally generated eddies and the externally forced zonal mean state on time scales not exceeding a few months (Thompson and Woodworth 2014). Recent studies have suggested that the influence of eddy heat and momentum fluxes on the zonal mean state may decouple on sufficiently large time scales. Specifically on time scales > 10 days, variability in the eddy kinetic energy is only driven by eddy heat fluxes, and is therefore referred to as a baroclinic mode of variability (Thompson and Woodworth 2014; Boljka and Shepherd 2018; Boljka et al. 2018). The BAM was identified as such a quasi-periodic (∼20–30 days) low-frequency baroclinic mode (Thompson and Barnes 2014).

It is of interest to reproduce these baroclinic annular modes directly from the thermodynamic equations. This requires a parameterization of the eddy heat fluxes in terms of the planetary-scale zonal mean temperature field. However, when representing the effect of synoptic-scale eddy heat fluxes as a downgradient diffusion process, no such baroclinic modes are found (Moon and Cho 2020).

Problems with gradient-diffusion-type parameterizations become apparent when the mean-field changes occur on similar time scales as the turbulent correlation time scales. The elliptic nature of the diffusion equation requires information to propagate through the system at infinite speeds (Chen and Lin 1994), violating locality. A simple closure that avoids this problem is called the “minimal tau approximation” (MTA) (Blackman and Field 2002, 2003; Brandenburg et al. 2004). Here, using the arguments of isotropic turbulence, triple correlations are parameterized as a damping term, with time scale τ, in the tendency equation of the turbulent flux. This is equivalent to representing second-order correlations as a convolution integral of an exponentially decaying integral kernel and the gradient of the mean field (Brandenburg and Subramanian 2005a). It transforms the elliptic diffusion equation into a hyperbolic equation, where the propagation speed of information is finite. The validity of the MTA has been verified in several numerical studies (Brandenburg et al. 2004; Brandenburg and Subramanian 2005b). From MTA, it is expected that the memory kernel will behave as a decaying exponential, with a typical decay time scale τ. Santos Gutiérrez et al. (2021) showed that exponentially decaying memory kernels appear naturally in any reduced-order model for coupled dynamical systems. Specifically, they show that the memory kernel can be expressed as an eigenvalue decomposition of the generator of the uncoupled dynamics. As a consequence, the leading-order eigenvalues of the generator determine the memory time scale.

An exponentially decaying memory kernel has recently been used to explain low-frequency phenomena in the atmosphere and ocean. Manucharyan et al. (2017) used a finite memory kernel to explain the oscillatory exponentially decaying equilibration, observed in the spinup simulation of the Beaufort Gyre. They attributed this behavior to so-called eddy memory where the eddy-thickness flux is dependent on the history of the mean state. In the atmosphere, Moon et al. (2021) derived a generalized Langevin equation, where eddy heat fluxes are parameterized as the convolution product of an exponential memory kernel and the zonal mean planetary-scale meridional temperature gradient. This led a novel explanation for the baroclinic annular mode. In a follow-up study, Moon et al. (2022) showed that the inclusion of a finite memory kernel allowed for a new type of planetary waves, which may be relevant in the explanation of midlatitude blocking events.

In a weakly nonlinear context, within a two-layer baroclinic model, one can show explicitly that the memory kernel is a finite-time, exponentially decaying, kernel with a memory time scale τ, which is determined by the growth rate of the synoptic baroclinic instabilities (Dijkstra et al. 2022). In this case, baroclinic instability of the background state leads to a modification of the background flow, which takes a finite time to equilibrate. Such finite time equilibration can be described by a Ginzburg–Landau equation whose solution leads to an exponential decaying memory kernel. However, atmospheric flows are far above their linear stability thresholds and hence apart from eddy–mean flow interactions also eddy–eddy interactions may affect the eddy memory.

In this paper, we attempt to explicitly compute the memory kernel in a two-layer quasigeostrophic spectral model (QGS) (Demaeyer et al. 2020; De Cruz et al. 2016). This model is sophisticated enough to capture chaotic midlatitude atmospheric flows due to baroclinic instabilities, but simple enough to perform the extensive computations needed to identify the eddy-memory kernel.

To determine this kernel, we rely on recent advancements in linear response and proxy response theory. Linear response theory (LRT) enables one to predict the average first-order (linear) response of sufficiently smooth observables in axiom A dynamical systems when subjected to small perturbations (Ruelle 2009; Ragone et al. 2016; Lucarini et al. 2017). This framework can be extended into proxy response theory (Tomasini and Lucarini 2021; Lucarini 2018), where one is interested in using the linear response of one observable as a predictor for the linear response of another. Specifically, we here aim to predict the eddy heat flux response based on information of the zonal mean meridional temperature gradient response. In doing so, our study would be the first in deriving an eddy memory kernel without any prior assumptions. This then allows us to answer questions of physical relevance, being: What is shape of the memory kernel? Why does it have this shape? And can it be used as an explanation for baroclinic annular modes in the extratropics?

The paper has the following structure. Section 2a provides a concise overview of the essential aspects of the QG model. In section 2b, we present a summary of linear and proxy response theory, establishing the theoretical framework for our analysis. The derivation of the memory kernel is performed by comparing a forced and reference average response, and the results are presented in section 3. Subsequently, in section 4, we analyze the physics of the obtained memory kernel. We derive a reduced thermodynamic equation that incorporates an approximate memory kernel and compute the terms involved in the Lorenz energy cycle. Low-frequency modes of variability associated with specific memory kernels are derived and compared to observed low-frequency modes. By employing the Lorenz energy cycle, we can identify the underlying physical processes associated with these modes and establish connections with the shape of the kernel.

2. Formulation

In this section, we shortly summarize the main elements of the quasigeostrophic model (section 2a) and provide a short overview of the linear response theory approach being used (section 2b).

a. Model

We use a two-layer quasigeostrophic model, as implemented in the QGS framework (De Cruz et al. 2016), for a zonal channel of meridional width Ly = πL (y ∈ [0, πL]), and zonal width Lx = (2π/A)L (x ∈ [0, (2π/A)L]), where A is the aspect ratio 2Ly/Lx. The governing equations describe the evolution of the 750 and 250 hPa streamfunction fields (ψ3 and ψ1, respectively) and the atmospheric potential temperature Ta evaluated at 500 hPa. The model equations are given by
t2ψ1+J(ψ1,2ψ1)+βxψ1=kd2(ψ1ψ3)+f0ΔpW,
t2ψ3+J(ψ3,2ψ3)+J(ψ3,f0h/Ha)+βxψ3=kd2(ψ1ψ3)kd2ψ3f0ΔpW,
tTa+J(ψa,Ta)=σΔpRW+hd(T*Ta).
Here, ∇2 is the Laplacian operator, J is the Jacobian operator, β is the beta parameter, kd is an internal friction parameter, f0 is the Coriolis parameter, and Δp is the pressure difference between layer 3 and layer 1. Furthermore, Ha is the typical depth scale, h is a prescribed orography, kd is the bottom friction parameter, W is the vertical velocity in pressure coordinates, σ is the static stability parameter, R is the universal gas constant, and hd is the Newtonian cooling coefficient. The field ψa appearing on the right-hand side of Eq. (1c) is the barotropic streamfunction ψa = (ψ1 + ψ3)/2. The last term on the right-hand side of Eq. (1c) acts as a forcing of the meridional temperature profile at 500 hPa toward an equilibrium profile defined by T*(x,y). If the meridional temperature gradient, partially determined by T*, exceeds a critical value, the zonal flow defined by the thermal wind balance becomes baroclinically unstable to synoptic-scale perturbations (Eady 1949; Phillips 1951). Under strongly nonlinear conditions, the heat transport [represented by J(ψa, Ta)] and temperature profile relaxation [represented by hd(T*Ta)], cause the system to relax to a state of statistical equilibrium.
Equations (1) are rewritten in terms of the barotropic ψa and baroclinic θa = (ψ1ψ3)/2 streamfunction at 500 hPa. Using the hydrostatic and ideal gas relation, the temperature Ta at 500 hPa can be expressed in term of the baroclinic streamfunction θa, i.e., Ta = 2f0θa/R (De Cruz et al. 2016). Then, upon replacing Ta by θa and T* by θ*, and eliminating the vertical velocity W, the equations for ψa and θa describe the full dynamics of the model. These equations are projected on a set of orthogonal Fourier basis functions given by
FPA(x,y)=2cosPyL,FH,PK(x,y)=2cosHAxLsinPyL,FH,PL(x,y)=2sinHAxLsinPyL,
for H = 1, …, Hmax and P = 1, …, Pmax. The projection results in a set of 2na nonlinear ordinary differential equations for the projection coefficients, with the total number of basis functions used being na = Pmax(2Hmax + 1). For example, with Hmax = Pmax = 2, na = 10 the basis functions are F1A,F1,1K,F1,1L,F2A,F1,2K,F1,2L,F2,1K,F2,1L,F2,2K,andF2,2L. The resulting projection coefficients are ordered into the state vector η=(ψa,1,,ψa,na,θa,1,,θa,na)T. The orography h(x, y) and the equilibrium temperature profile T*=2f0θ*/R are also projected onto the same Fourier basis functions.
By projecting the evolution equations for ψa and θa on this set of orthogonal basis functions, the nonlinear partial differential Eqs. (1) defined on the spatial domain is represented by nonlinear ordinary differential equations in the spectral domain. The full evolution of a trajectory through 2na-dimensional phase space can then be described by
dηidt=j,k=02naTi,j,kηjηk=Fi(η).
Here Ti,j,k is the tensorial contraction containing all interaction coefficients. For a more elaborate description of how this model is implemented in the QGS framework (De Cruz et al. 2016), the reader is referred to the QGS documentation (https://github.com/Climdyn/qgs).

b. Linear response theory and the proxy Green’s function

The use of LRT has become increasingly popular in climate sciences, (Gritsun and Branstator 2007; Gritsun and Lucarini 2017; Ragone et al. 2016; Lucarini et al. 2017; Lembo et al. 2020; van Zalinge et al. 2017). Its theoretical foundation have been laid by Ruelle [see Ruelle (2009) for a review], who showed that linear response theory holds for axiom A (or uniformly hyperbolic) dynamical systems. These dynamical systems possess an invariant measure [called a Sinai–Ruelle–Bowen (SRB) measure; Young 2002] with which averages can be computed.

On the other hand, almost all the dynamical systems encountered in climate sciences are not uniformly hyperbolic. However, they are nonequilibrium, dissipative dynamical systems, evolving on a compact manifold MRn. When these dynamical systems are chaotic and possess many degrees of freedom, then the Gallavotti–Cohen hypothesis (Gallavotti and Cohen 1995) states that they can be considered as if they were uniformly hyperbolic. In practice, one can then apply linear response theory to these systems and compare the result with numerical experiments (Ruelle 1998).

In the following, we shall thus consider that an invariant measure exists for the dynamical system at hand (and verify it with numerical experiments). Consequently, as in Ruelle’s response theory (Ruelle 1998, 1999; Gallavotti and Cohen 1995; Ruelle 2009; Gritsun and Lucarini 2017), the change of this invariant measure due a weak forcing can be expressed in terms of the unperturbed (or background) invariant measure ρ0(dη). Suppose that we force the model by changing θ* and write (Lucarini et al. 2017)
dηdt=F(η)+ϵg(t,η),
where F(η) represents the tendencies of the unperturbed dynamical system, and suppose that we can write the perturbation as g(t, η) = Θ(t)X(η), where Θ(t) describes the temporal dependence of the perturbation. Due to the way the state variable η is defined, X represents both the field being forced (ψa or θa) and the spatial pattern (expressed in the spectral domain) of the small ϵ perturbation. The average response of a sufficiently smooth observable Ξ to this perturbation can then be formally evaluated with respect to the unperturbed measure ρ0 as (Ruelle 1998)
Ξηg(t)=Ξ(η)ρt(dη)=Ξη,0+j=1ϵjΞη,0g,(j)(t),
where the subscript 0 indicates that the averages are taken with respect to the unperturbed measure, i.e.,
Ξη,0=Ξ(η)ρ0(dη).
The first-order linear perturbation of the average can be written as (Ragone et al. 2016; Ghil and Lucarini 2020)
Ξη,0g,1(t)=tGΞX(s)Θ(ts)ds,
where GΞX(t) is the first-order Green’s function of the observable Ξ associated to the spatial pattern X. The Green’s function is causal, i.e., it is only defined for t > 0, and therefore obeys the Kramers–Kronig relation (Lucarini and Sarno 2011; Lucarini et al. 2005).
In practice, the Green’s function can be estimated from Eq. (7) by considering a particular time dependence for the forcing ϵΘ(t)=aH(t), where H is the Heaviside function, i.e., by considering the situation obtained when the forcing is the reference case multiplied by a factor (1 + a)X at t = 0. In this case, by differentiating (7) with respect to t, one can show that
GΞX(t)=1addtΞη,0g,1(t).
Hence, from a Heaviside-type perturbation, the Green’s function GΞX(t) for any observable Ξ and spatial pattern X can be determined. This enables us to determine the average linear response of Ξ under an arbitrary temporal forcing pattern, provided the perturbation is sufficiently small.

In this study, the response of the smooth observables υaTa¯ (from now indicated by Φ) and yTa¯ (from now indicated by ϕ) due to a modification (an increase) of the meridional gradient of the equilibrium temperature T* is considered at various latitudes y. The overbar represents the zonal mean operator, primed quantities represent the perturbations with respect to the zonal mean, and υa is the meridional velocity evaluated at 500 hPa.

As previously mentioned, the time scale separation between ϕ and Φ is insufficient to assume a Markovian parameterization. To address this issue, the Mori–Zwanzig formalism provides a framework that allows for the reduction of the thermodynamic evolution Eq. (1) of the slow observable by disentangling the slow dynamics from the memory effects induced by the fast dynamics (Zwanzig 1961; Wouters and Lucarini 2013). Applying the Mori–Zwanzig formalism, we gain insights into the representation of synoptic-scale eddy heat fluxes. These fluxes can be expressed as a convolution product between the planetary mean meridional temperature gradient ϕ and an integral kernel. This formulation naturally captures the memory of ϕ within the synoptic-scale eddy heat fluxes, highlighting the impact of past values on the present dynamics.

To compute this kernel, we first consider at any latitude the Green functions GΦX(y,t) and GϕX(y,t) providing the average response of 〈Φ〉 and 〈ϕ〉 to a corresponding perturbation g of the tendencies:
ΔΦηg(y,t)=tGΦX(y,s)Θ(ts)ds,Δϕηg(y,t)=tGϕX(y,s)Θ(ts)ds,
where Δ denotes the difference between the perturbed and reference state average of the considered observable. Because the Green’s functions GϕX and GΦX are causal, their Fourier transform (ΓϕX and ΓΦX, respectively), referred to as the susceptibility, is analytic in the upper-complex ω plane (Nussenzveig 1972; Lucarini et al. 2005). Hence, Eq. (9) implicitly requires higher-order terms in Eq. (5) are negligible compared to the linear response.
The concept of proxy response (Lucarini 2018; Tomasini and Lucarini 2021), i.e., the estimate of the response of a given observable with respect to another one, can then be applied. Indeed, taking the Fourier transform of both identities in Eq. (9), the convolution integral becomes a product of Fourier transforms. From this, the Fourier transform of the forcing Θ(t) can be eliminated and after inverse Fourier transformation, we can relate ΔΦηg(y,t) to Δϕηg(y,t) by
ΔΦηg(y,t)=tF1{ΓΦXΓϕX}(y,s)Δϕηg(y,ts)ds,
where F1{f} indicates the inverse Fourier transform of the function f to time. Hence, the response Δ〈Φ〉 can be written as a convolution integral of the zonal mean temperature gradient and the proxy Green’s function Gϕ,ΦX, whose Fourier transform or proxy susceptibility is given by ΓΦX/ΓϕX=Γϕ,ΦX(y,ω).
From (10), it follows that
ΔΦ˜ηg(y,ω)=Γϕ,ΦX(y,ω)Δϕ˜ηg(y,ω),
where we denote the Fourier transform of function f(t) as f˜(ω).
We shortly address two important aspects that require further clarification; for a more elaborate description on these aspects we refer to Lucarini (2018). First, the proxy Green’s function can have a singular part in t = 0, Sϕ,ΦX. The origin of the singular part is related to asymptotic decay of the susceptibilities, i.e.,
limωΓϕ,ΦX(y,ω)=limωΓΦX(y,ω)ΓϕX(y,ω)ωNM.
Here N and M describe the asymptotic decay of ΓϕX and ΓΦX, respectively. If NM, there will be a nonzero singular component in Gϕ,ΦX. This transforms Eq. (10) to
ΔΦηg(y,t)=sϕ,ΦX(y)Δϕηg(y,t)+tKϕ,ΦX(y,s)Δϕηg(y,ts)ds,
where we used Sϕ,ΦX(t,y)=sϕ,ΦX(y)δ(t). The singular and nonsingular [Kϕ,ΦX(t,y)] component of the proxy Green’s function, represent the local-in-time and nonlocal response of ΔΦηg in terms of Δϕηg, respectively.
Second, the proxy Green’s function is predictive if and only it is defined only for t > 0. This translates in the condition that ΓϕX(y,ω) has no complex zeros in the upper-complex ω plane, implying that all frequencies in the response of 〈Φ〉 can be linked to frequencies in the response of 〈ϕ〉. If this condition is not met, the proxy Green’s function will have a noncausal part, i.e., it may be nonzero for t < 0, and 〈ϕ〉 cannot explain all frequencies present in the response of 〈Φ〉. The causal and noncausal part are obtained by
Kϕ,ΦX,c(t,y)=H(t)Kϕ,ΦX(t,y),Kϕ,ΦX,nc(t,y)=H(t)Kϕ,ΦX(t,y).
Here the superscript c (nc) refers to the causal (noncausal) part of the nonsingular proxy Green’s function.
Parameterizing eddy heat fluxes in terms of a zonal mean meridional temperature gradient assumes that the latter is predictive for the former. Hence, in this study, we are only interested in the predictive, i.e., causal, part of the proxy Green’s function, which reads
Gϕ,ΦX,c(t,y)=Kϕ,ΦX,c(t,y)+sϕ,ΦX(y)δ(t).

When the nonsingular causal part of the proxy Green’s function is nonzero, the eddy heat fluxes will depend on the history of the zonal mean temperature gradient. As stated in the introduction, other studies have hypothesized Kϕ,ΦX,c to be an exponentially decaying memory kernel (Moon et al. 2021; Dijkstra et al. 2022).

The predictive power of observable 〈ϕ〉 for observable 〈Φ〉 is evaluated by computing the causality index (Tomasini and Lucarini 2021), defined as
CI(Gϕ,ΦX)=1Kϕ,ΦX,ncL1Kϕ,ΦX,cL1+sϕ,ΦX(y)L1,
where L1 is the L1 norm. If CI = 1 the proxy Green’s function is causal and only defined for t > 0. If CI is small, 〈ϕ〉 is a poor predictor for 〈Φ〉.

3. Results

The results are presented in three subsections. First, a reference equilibrium case is presented in section 3a. Next, the step-forced case, needed for computing the Green’s functions in the linear response theory is presented in section 3b. In section 3c, we then present the results on the memory kernel.

a. Reference case

As a reference case, we choose a configuration with standard parameters kd = 0.9 day−1, kd=0.09day1, hd = 0.4 day−1, Pmax = Hmax = 4, Ly = π L = 5000 km, A = 1.3, and σ = 2.2 × 10−6 m2 s−2 Pa−2. Similar values were used in the study of Reinhold and Pierrehumbert (1982), where they used Pmax = Hmax = 2 and hence projected the dynamics on 10 Fourier basis functions. However, in our study the number of Fourier basis functions is increased to 36 and the corresponding number of degrees of freedom is 72. Using only 10 components, the resulting dynamics are dominated by large-scale propagating waves. However, the real midlatitude atmospheric state more accurately resembles a state of geostrophic turbulence (Charney 1971; Rhines 1979), which is better represented when increasing the number of components to 36 (Pmax = Hmax = 4).

The orography and forcing are prescribed as
h(x,y)=h0F1,1K(x,y)=2h0cosAx/Lsiny/L,
θref*(x,y)=θ0F1A(x,y)=2θ0cosy/L,
with h0 = 1700 m and θ0 = 9.4 K. We notice that the forcing θ* is only varying along the meridional coordinate y, representing a zonally uniform equilibrium temperature profile.

We start from a random initial condition and first integrate 100 000 time steps of 0.01 days to find an initial point on the attractor. Next, we determine a trajectory on the attractor by integrating 200 000 time steps of 0.01 days starting from the previous end point.

Figure 1a shows a projection of the attractor in the (ψa,2, ψa,3) plane. Considering that (ψa,2, ψa,3) describe the planetary-scale structure of the atmosphere, Reinhold and Pierrehumbert (1982) defined different, well-defined, planetary-scale weather regimes based on the position of the trajectory in the (ψa,2, ψa,3) plane. For example, a zonally symmetric planetary-scale atmosphere is characterized by ψa,2ψa,3 ≈ 0. The transition from one weather regime to another is facilitated through instabilities (mainly baroclinic), the fussiness of the attractor in the (ψa,2, ψa,3) plane indicates that different planetary-scale weather regimes are occasionally visited in the model. However, the attractor is more densely populated in the lower-right quadrant of Fig. 1a, corresponding to a blocked weather regime. Figure 1b illustrates that 10 Lyapunov exponents are positive. The dynamics of the QGS model may therefore be regarded as chaotic and highly sensitive on the specified initial conditions.

Fig. 1.
Fig. 1.

(a) Projection of attractor in the nondimensional-(ψa,2, ψ3) plane. (b) Lyapunov exponents as determined from the reference trajectory.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0146.1

Figures 2a and 2b show a snapshot of the vorticity and geopotential height field at 500 hPa in the reference case, respectively. The abundance of synoptic-scale (∼900–1200 km) isolated concentrations of vorticity, is suggestive for the fluid being in a state of geostrophic turbulence (McWilliams 1990). The geopotential height field shows a dominant zonal mean component, on which wavelike perturbations are superposed.

Fig. 2.
Fig. 2.

(a) Vorticity field and (b) and geopotential height field (contours) at 500 hPa and the orography height (colored) in reference run when t = 60 days.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0146.1

Next, we randomly select 180 000 initial conditions on the attractor, from which we integrate a trajectory of 40 000 time steps of 0.01 days. To obtain a reference mean-state, we compute the average of the observable of interest over all 180 000 trajectories. The mean zonal average (MZA) temperature gradient and MZA eddy heat flux at a fixed value of y/L = 1.675 are plotted as blue curves in Figs. 3a and 3b, respectively. As expected, temperatures increase for lower latitudes, and on average heat is fluxed poleward.

Fig. 3.
Fig. 3.

Mean and standard deviation of both ensembles: reference (blue) and step forced (red). (a) Eddy heat flux and (b) meridional temperature gradient. All shown profiles are taken at y/L = 1.645. Insets show the latitudinal dependence of the peak value (indicated with a black dot on the red curve) of the considered variable.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0146.1

b. Step function forcing

Perturbing the equilibrium temperature profile θref*(x,y) with a Heaviside step function, H(t), the step-forced equilibrium temperature profile is formulated as
θ*(y)=(1+a)θref*(y).
For the forcing to be sufficiently weak, we choose a = 0.1. In the previous section’s formalism, it corresponds thus to a perturbation hdaθ0H(t)X of the tendencies with X11 = 1 and Xi = 0 for i ≠ 11, corresponding to a spatial pattern given by the mode F1A(x,y)=2θ0cosy/L of the model.

Next, we compute the mean response as in the reference case, i.e., a 400-day integration of 180 000 random initial conditions. The red curves in Figs. 3a and 3b show the MZA eddy heat flux and the MZA meridional temperature gradient, respectively, for the step-forced ensemble. The eddy heat flux becomes more positive than in the reference case, while the mean meridional temperature gradient becomes more negative. This agrees with the general notion of a downgradient flux of heat, which is proportional to the magnitude of the mean gradient. However, both the heat fluxes and the meridional temperature gradient surpass their statistical equilibrium value. Moreover, the Δ〈Φ〉 maximum lags the Δ〈ϕ〉 minimum by 4.4 days. Both observations do not follow from classical Fickian diffusion, which would describe an exponentially decaying equilibration (Manucharyan et al. 2017), and suggest that the nonlocal (nonsingular) link between Δ〈Φ〉 and Δ〈ϕ〉 significantly contributes in shaping the response.

The insets of Figs. 3a and 3b show the minimal (maximal) value of the changed MZA temperature gradient (eddy heat fluxes) as a function of latitude. The change of the eddy heat fluxes is maximal in the middle of the domain. This maximum does not coincide with the minimum of the MZA temperature gradient change, implying that the causal relation between Δ〈ϕ〉 and Δ〈Φ〉 is y dependent. In the following section we explore this in more detail.

Using Eq. (8), we compute the Green’s function of the MZA meridional temperature gradient and the MZA eddy heat fluxes for all latitudes. The Green’s function is computed for the full response (cf. Fig. 3) and a spline-smoothed response, the latter effectively removes all variability on time scale < 2 days. We refer to these Green’s functions as the filtered and unfiltered Green’s functions, respectively.

The resulting Green’s functions for 〈ϕ〉 and 〈Φ〉 are shown in Fig. 4a. Both Green’s functions display oscillatory exponentially decaying profile. The decay time scale of the observable response functions is similar and equals ∼3–4 days for all latitudes. Comparing this decay time scale to the dominant dissipative energy pathway time scale, being thermal cooling/heating hd12days (see section 2a), an opposing process must increase the relaxation time scale. The oscillation frequency of both response functions, shown in Fig. 4a, roughly equals 0.045–0.06 day−1, for all latitudes.

Fig. 4.
Fig. 4.

(a) Green’s function of MZA eddy heat flux and MZA meridional temperature gradient vs the forcing. (b) Normalized power spectrum of GΦX and GϕX. All shown profiles are taken at y/L = 1.645. The dotted and thick line show the Green’s function of the unfiltered and smoothed response, respectively.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0146.1

The phase of GΦX is shifted 90° to the left, with respect GϕX. As a consequence, GΦX practically vanishes for t → 0+. This makes sense, because ϕ is the variable to which the forcing is initially applied, and this forcing requires a finite time to affect Φ. Physically, this relates to the finite time required for the baroclinic eddies to modify accordingly, i.e., toward reaching a new equilibrium, to the changing mean state.

Figure 4b shows the power spectrum of GΦX and GϕX. Both spectra are right skewed, asymptotically decaying toward 0 for ν and are maximal at an intraseasonal time scale of 0.04–0.06 day−1. The power spectrum of GϕX is shifted toward slightly higher frequencies compared to |ΓΦX|2, which can be attributed to the phase difference between GϕX and GΦX (cf. Fig. 4a). Clearly M > N in Eq. (12), meaning that the perturbation’s effect is felt more rapidly in 〈ϕ〉 than on 〈Φ〉 (Tomasini and Lucarini 2021), which should be natural when using ϕ as a predictor for Φ. A similar conclusion follows from GΦXt0+O(t) (cf. Fig. 4a), suggesting that the singular part of the proxy Green’s function equals zero. This is remarkable because it contradicts the time locality assumed in standard Fickian diffusion, whereas it agrees with the Mori–Zwanzig formalism when the time scale separation between “fast” and “slow” variables is insufficient.

We can check whether the Green’s function response is indeed a good approximation of the response of the observables to an arbitrary forcing Θ(t). Here, we take a transient forcing of form
Θ(t)=tanh(5ttm),
where tm = 400 days. To obtain the statistics of the observables, under the forcing given in Eq. (18), another 180 000 trajectory mean, is computed. The LRT predicted response is then computed by using Eq. (9), where we integrate over the unfiltered Green’s functions (cf. Fig. 4a). The LRT predicted response of the MZA eddy heat flux (thick black line in Fig. 5), closely matches the numerically computed response (thick green line in Fig. 5), with an RMSE of 0.11 K m s−1.
Fig. 5.
Fig. 5.

Numerical and LRT computed response for the (a) eddy heat flux and (b) meridional temperature gradient to the forcing, (18). Profiles shown are located at y/L = 1.645. The insets in (a) and (b) show the RMSE of the LRT estimated response as function of latitude.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0146.1

Using the response profile of the standard deviation, shown in Figs. 3a and 3b, we can also compute the Green’s function of the standard deviations. The LRT computed standard deviation (black dashed curves) also approximates that of the forced ensemble very well. The inset of Fig. 5a shows the RMSE as a function of latitude. On average the RMSE equals 0.08 K m s−1, verifying that LRT accurately reproduces the response for all latitudes

In a similar fashion, we can integrate filtered GϕX (cf. Fig. 4a) over the forcing Θ(t) to obtain the LRT predicted computed MZA meridional temperature gradient response to the transient forcing. Figure 5b shows LRT computed response (thick black line) which agrees well with the numerical MZA temperature gradient (thick green line), with an RMSE of 0.038 K rad−1. The observed response is well approximated over the entire domain, with an average RMSE of 0.025 K rad−1 (cf. inset Fig. 5b). The LRT computed standard deviation for the MZA meridional temperature gradient closely resembles the observed standard deviation.

The remarkably low RMSE values confirm that we are operating within the linear response regime of the dynamical system (3), as anticipated. Nevertheless, it is evident from the insets of Fig. 5 that the RMSE exhibits a distinct dependence on latitude. Specifically, it reaches its maximum in the middle of the domain and gradually decreases toward the edges. This latitudinal variability can be attributed to the increased noise magnitude observed in the middle latitudes, primarily arising from the lower zonal flow stability and associated baroclinic fluxes present in that region. It is important to note that the linear approximation does not account for noise; thus, the observed correlation between RMSE and latitude comes as no surprise.

c. Memory kernel

The predictive power of GΦX and GϕX was verified in Figs. 5a and 5b. Linear response theory is now extended by using Δ〈ϕ〉 as a predictor for Δ〈Φ〉. Their relation is encapsulated in the proxy Green’s function which is computed as described in section 3b.

The computation of the proxy Green’s function involves both the filtered and unfiltered linear Green’s functions, as depicted in Fig. 4a. The combined result is presented in Fig. 6a. Notably, the nonsingular causal component of the proxy Green’s function remains nonzero for t > 0, indicating that eddies retain a finite memory of the past zonal mean meridional temperature gradient. Additionally, findings from the previous section suggest that the singular part of the proxy Green’s function can be disregarded, as illustrated in Fig. 4. By eliminating this singularity, the relationship between Δ〈ϕ〉 and Δ〈Φ〉 becomes nonsingular, resulting in a delayed response of Δ〈Φ〉 compared to Δ〈ϕ〉. This delayed response gives rise to the observed low-frequency mode (see, e.g., Fig. 3), which can be appropriately termed as a memory-induced low-frequency mode (Manucharyan et al. 2017; Moon et al. 2021). In section 4a we provide further arguments of this result.

Fig. 6.
Fig. 6.

(a) Proxy Green’s function relating the eddy heat flux to the meridional temperature gradient, illustrating the eddy memory for y/L = 1.645. Thick (dotted) lines represent obtained proxy Green’s function when filtered (unfiltered) Green’s functions are used in computation (cf. Fig. 5a). (b) Causality index, computed from Eq. (15), as a function of latitude.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0146.1

The effectiveness of the proxy Green’s function in making predictions is assessed using the CI index, which is calculated for all latitudes according to Eq. (15). The results are presented in Fig. 6b. When considering the unfiltered proxy Green’s function, the CI index exhibits values notably different from 1 across all latitudes, approximately around 0.8. This outcome can be attributed to the significant contribution of noncausal components to the overall proxy response function. The noncausal portion primarily consists of frequencies ranging ν ∈ [0.4–0.5] day−1.

For frequencies ν > 0.4 day−1 a significant part of the information required for predicting Δ〈Φ〉 is contained in the noncausal proxy response function. Therefore, Δ〈ϕ〉 may not be regarded as a good predictor for Δ〈Φ〉 on those time scales. Indeed, computing the predictability of the filtered Green’s functions (cf. Fig. 4a), where the nonpredictive frequencies are filtered out, we find a CI ∼ 1 (cf. Figs. 6a,b).

Physically, this is because on sufficiently large time scales (>2–3 days), Δ〈ϕ〉 and Δ〈Φ〉 are coupled through baroclinic instability. The growth of baroclinic eddies is partially determined by the zonal mean atmospheric flow stability, proportional to 〈ϕ〉 (Eady 1949). Hence, on these synoptic time scales, Δ〈ϕ〉 precedes Δ〈Φ〉, i.e., Δ〈ϕ〉 may be regarded as a cause for Δ〈Φ〉. Variability in Δ〈Φ〉 and Δ〈ϕ〉 on smaller time scales is likely due to noise resulting from the limited number of trajectories used to compute averages of observables.

The filtered proxy Green’s function at y/L = 1.645 (cf. Fig. 6a) exhibits an exponentially decaying behavior modulated with low-frequency oscillations (0.017 day−1). These oscillations become significant after around 50 days. We can obtain an estimate of the dominant decay time scale by fitting Γϕ,ΦX,c with
F{H(t)αexp(t/τ)}(y,ω)=α(y)τ(y)1+iω,
where α may be regarded as a feedback amplitude between Φ and ϕ and τ is the memory decay time scale (Brandenburg et al. 2004).

The fraction of explained variance, by the fit defined in Eq. (19), is shown in Fig. 7a. For all latitudes, the fit can explain at least 90% of the observed variance, within the frequency range of [0–0.5] day−1, verifying that the proxy Green’s function indeed closely resembles an analytic exponentially decaying function. However, the percentage of explained variance significantly decreases in the middle of the domain (i.e., y ∈ [1.3–2]).

Fig. 7.
Fig. 7.

(a) Fraction of explained variance when fitting the Γϕ,ΦX,c with Eq. (19) and (b) the resulting decay time scale and feedback amplitude.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0146.1

The inset of Fig. 7a shows the resulting fit from Eq. (19) at y/L = 1.645. Although describing Γϕ,ΦX well, the fit is unable to explain a spectrum peak in I{Γϕ,ΦX} located at ν ∼ 0.017 day−1, present for all y ∈ [1.3–2]; this peak results in a small amplitude oscillation modulation in the standard exponential decay, visible in Fig. 6a. We note that the decay rate and feedback amplitude were found to be slightly frequency dependent for all latitudes, especially close to the domain edges, explaining the decrease in explained variance also toward the edges of the domain.

As expected from the insets of Fig. 3, the shape defining parameters of the proxy Green’s function depend on latitude. Specifically, Fig. 7b shows the decay time scale τ and feedback amplitude α, as a function of latitude. On average, the decay time scale equals ∼8.5 days. In section 4b it is argued that this time scale relates to the relaxation time scale of growing baroclinic waves. The decay time scale is found to be maximal in the middle of the domain, and generally decreases toward the edges. The feedback amplitude on average equals 0.15 day−2 and has a similar latitudinal dependence.

We can evaluate the predictability of the filtered and unfiltered proxy Green’s function by integrating Eq. (10) over the temperature gradient response to both step and transient forcing scenarios [Eqs. (4) and (18), respectively]. The results are presented in Fig. 8. For both the filtered and unfiltered proxy Green’s function, the computed response closely approximates the mean response of 〈Φ〉, as anticipated from the high CI index.

Fig. 8.
Fig. 8.

Check of the linear response theory solution for (a) the forcing (4) and (b) the forcing (18) using the filtered and unfiltered proxy Green’s function (cf. Fig. 6b). The full (dotted) black line is the LRT computed for the (proxy) standard Green’s function. The inset shows the RMSE for filtered proxy Green’s function LRT computed response.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0146.1

The insets of Fig. 8 display the RMSE of the LRT computed response as a function of latitude. The average RMSE over all latitudes equals 0.05 and 0.1 K m s−1 for the transient forcing and step-forced forcing scenario, respectively. The primary contributor to the RMSE, in both forcing scenarios, is the poor predictive power on short time scales (cf. Fig. 6b). The RMSE of the filtered Green’s function LRT computed response is slightly higher in both forcing scenarios (∼0.2 K m s−1). However, the general shape of the mean response is approximated well.

The reduced accuracy of the filtered Green’s function is not surprising since the filtering process eliminates high-frequency information that could contribute to the prediction of Δ〈Φ〉. However, despite this loss of high-frequency detail, both the filtered and unfiltered Green’s functions successfully capture the essential low-frequency features of the response, including the delayed response of Δ〈Φ〉 and the introduced memory mode of variability (overshoot) (cf. Fig. 8a). The success can be attributed to the finding that proxy response function is causal for sufficiently small ν. Therefore, using Δ〈ϕ〉 to predict variability in Δ〈Φ〉 in the linear regime on sufficiently long time scale may be considered accurate, even when high-frequency behavior in Δ〈ϕ〉 is not considered (i.e., using the filtered proxy response function).

In a final numerical experiment, we examined the effect of removing the oscillatory component from the filtered proxy Green’s function on the predictions made using LRT. The results, illustrated by the bold blue line in Fig. 8, demonstrate that the removal of the oscillatory part has minimal impact on the predictions during the initial 50-day period. This outcome aligns with our expectations, considering the absence of oscillations in Fig. 6a within this time frame. Therefore, the memory-induced low-frequency mode continues to be accurately reproduced. Consequently, we argue that when it comes to predicting the overall response within a time range of up to 50 days, the exclusion of the oscillatory component does not significantly alter the findings compared to utilizing the complete filtered proxy Green’s function with oscillations.

Important here is that the elimination of the oscillatory component introduces a new memory mode of variability, characterized by a similar time scale as the oscillations depicted in Fig. 6a. Consequently, when utilizing the nonoscillatory proxy response function to reproduce low-frequency modes on time scales exceeding 50 days, caution is required.

4. Analysis

In the previous section, we showed that the zonal mean temperature gradient may be used as a causal predictor for the eddy heat fluxes, considering sufficiently large time scales (>2 days). The nonsingular nature of the proxy response function may allow for a novel explanation for low-frequency modes on a time scale spanning from 10 to 50 days. In this section, we provide further analysis of this result by comparing our results with an analytical model (section 4a) and computing terms in the Lorenz energy cycle (section 4b).

a. Analytic model

Because Δ〈ϕ〉 serves as a causal predictor for Δ〈Φ〉, within a sufficiently large time scale (cf. Fig. 6b), Δ〈Φ〉 may be parameterized in terms of Δ〈ϕ〉, allowing for a closure of the thermodynamic Eq. (1c). Figure 6a illustrated that the causal proxy Green’s function is nonsingular; hence, the parameterization involves a convolution product of Δ〈ϕ〉 and Kϕ,ΦX,c. Although some information is lost, particularly for Kϕ,ΦX,c in y ∈ [1.3, 2], Figs. 7a and 8 suggested that Kϕ,ΦX,c may be approximated as an exponentially decaying function, if the goal is to reproduce low-frequency memory induced modes on time scales < 50 days.

In this section we will explore to what extent an analytic exponential memory kernel, as assumed in Moon et al. (2021, 2022) and Manucharyan et al. (2017), can reproduce the numerically computed average response. Moreover, working in a simplified analytical model may be useful for interpreting the origin of the found decay time scale τ and feedback amplitude α (Fig. 7b).

In the case of an exponentially decaying causal nonsingular proxy response function, we may write the closure as (Brandenburg and Subramanian 2005b)
(t+1τ)ΔΦ=αΔϕ.

Here third moment correlations are represented as a damping force in the tendency equation of second moment correlations over a time scale τ, the memory time scale (Brandenburg and Sokoloff 2002; Brandenburg et al. 2004; Brandenburg and Subramanian 2005b). Dropping the time derivative in Eq. (20), recovers the classical Fickian diffusion closure scheme. We note that Eq. (20) is similar to the parameterization proposed in Thompson and Barnes (2014) to derive the dominant baroclinic mode in the extratropical Southern Hemisphere. However, a damping term in the tendency equation for Δ〈Φ〉 was not considered, i.e., τ. Therefore, synoptic eddies were assumed to have infinite memory of the zonal mean background temperature state.

Moon et al. (2021, 2022) used an exponentially decaying kernel to close the atmospheric planetary-scale heat equation. In the appendix we repeat this analysis in the framework of the QG equations. The analysis results in a second-order damped harmonic oscillator equation, given by
[2t2+(hd+1τ)t+(hdτ+λi)]ΔTa¯i=1τahdTi*+σΔpR(t+1τ)ΔW¯i.
Here λi are the eigenvalues of the one-dimensional diffusion operator (see appendix) and
ΔTa¯=iAnaΔTa¯iFiA(y).
The definition of the basis functions is given in Eq. (2). In the 36 component model, there are 4 modes iA.

It is crucial to highlight that the presence of the second-order derivative in Eq. (21) arises due to the inclusion of finite memory in the parameterization (20). By neglecting this memory, i.e., omitting the time derivative in Eq. (20), the thermodynamic equation reverts to a classical heat equation. Consequently, the solutions in the time domain exhibit only exponential decay, without any low-frequency modes. Therefore, it is appropriate to refer to the low-frequency modes of variability arising from Eq. (13) as “memory-induced low-frequency modes” (Manucharyan et al. 2017; Moon et al. 2021), a term previously mentioned in section 3c.

For simplicity we neglect contributions of ΔW¯i, which allows Eq. (21) to be solved easily using standard methods. In the next section we show that this approximation is reasonable.

Because λ1 is the eigenvalue of the least damped eigenmode of the diffusion operator (see appendix), it will dominate the variability in the temperature field. By projecting Eq. (21) on mode i = 1, and deriving with respect to y, we obtain a homogeneous solution by computing the roots of the differential operator’s characteristic polynomial. For realistic values of hd, τ, and α, the roots r1 and r2 will be complex conjugate. Hence, the solution of the temperature gradient reads
Δϕ(t,y)=2sin(y){Δϕ¯1p+exp(r1Rt)[C1cos(r1It)+C2sin(r1It)]},
where the superscripts I and R refer to the real and imaginary part, respectively, and the constants C1 and C2 are determined based on the conditions [Δ〈ϕ〉(0, y) = 0] and (Δϕ)/t(0,y)=ahdyT*(y). The second initial condition follows directly from the behavior of the thermodynamic equation for t → 0+.
The particular solution reads
Δϕ1p=ahdT1*hd+τλ1.
By using the kernel characteristics, τ and α (cf. Fig. 7b) we obtain an estimate for the roots of the characteristic polynomial. As such we can compute the response of Δ〈ϕ〉 and, using Eq. (20), Δ〈Φ〉. The result is shown in Fig. 9.
Fig. 9.
Fig. 9.

Analytically computed response of MZA (a) eddy heat fluxes and (b) meridional temperature gradient, to the step forcing, (18). The inset of (b) shows the proxy Green’s functions for which we analytically estimated the response; αR was set to 0.22 day−2.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0146.1

The correspondence between the analytic estimate and numerically found response is good (Fig. 9a), especially for τ = 12 days. For τ equal to 4, 12, and 20 days, the RMSE equals 0.2, 0.06, and 0.1 K rad−1, respectively. The higher RMSE for τ = 4 (or τ = 20) days, is mainly due to the overestimation (underestimation) of Δϕ1p. For all values of τ, the initial overshoot of Δ〈ϕ〉 is overestimated. We can relate this overestimation to a nonnegligible contribution of ΔW¯.

Note that for all values of τ, the frequency of the predicted response signal remains relatively similar. Consequently, the initial overshoot in Δ〈ϕ〉 (dashed black line) is correctly timed for all values of τ. This suggests that the resulting frequency of the response signal is dominated by the choice of the feedback amplitude α. Indeed, Fig. 9a shows that when the reference value αR is increased by a factor of 2, the frequency of oscillation is increased significantly, and the initial overshoot is estimated to be located at 2 days instead of 2.6 days. This can be explained from Eq. (20), since, for a higher value of α the initial growth rate of Δ〈Φ〉 increases and the heat flux magnitude will more quickly dominate over Newtonian cooling.

The results presented in Fig. 9b illustrate the convolution-type response of the analytically computed variation in Δ〈ϕ〉, when subject to an exponential integral kernel. Regardless of the chosen τ or α value, an initial overshoot of Δ〈Φ〉 (dashed black curve) in response to a step forcing is consistently overestimated. For α = αR, this overestimation can partially be attributed to a similar overestimation in 〈ϕ〉 (cf. Fig. 9a). For α = 2αR, the overestimation follows from the overestimated α value.

Notably, the degree of overestimation increases with larger τ values because the cumulative effect of the historical baroclinicity is larger. For all values of τ and α, the equilibration value of 〈Φ〉 is over- or underestimated. This outcome suggests that the more complex shape of the actual proxy response function (cf. Fig. 6a) must be taken into account for a more accurate prediction of the Δ〈Φ〉 equilibrium value.

It is interesting to note that, according to Fig. 9b, the behavior of limt0+(/t)ΔΦ is independent of the chosen value of τ and seems to depend only on the value of α. Specifically, increasing the value of α increases the tendency limt0+(/t)ΔΦ. This could have been anticipated from Eq. (20). Expanding Δ〈Φ〉 and Δ〈ϕ〉 for t → 0+, and using the findings from Figs. 3 and 4, we can approximate the growth rate tendency of the eddy heat flux for t → 0+ as (/t)GΦXαGϕX. Hence, the value of α quantifies the growth rate tendency of a baroclinic heat flux disturbance, with respect to the growth rate of the meridional temperature gradient perturbation, which for t → 0+ equals ahd(/y)T*(y). Therefore,
α(y)=1ahdyT*(y)limt0+[2t2ΔΦ(t,y)].
Inserting the values in Eq. (24), we indeed find an α = 0.21 day−2 for y/L = 1.645 and reproduce the observed latitudinal dependence for α (cf. Fig. 7b). Hence, the latitudinal dependence of α follows from the nonlinear relation between the growth rate tendency of eddy heat fluxes and the initial growth rate of the radiatively forced meridional temperature gradient. As the stability is lowest in the middle of the domain, we anticipate a disproportionately higher growth rate tendency of Δ〈Φ〉, which requires a larger α.

For α = αR, the analytic estimate of Δ〈Φ〉 demonstrates that the root-mean-square error (RMSE) of the analytically computed signal is minimized at τ = 4 days (equivalent to 2.47 K m s−1), while it reaches its maximum at τ = 20 days (equivalent to 4.6 K m s−1). Although the RMSE metric suggests that selecting τ = 4 days provides the most optimal results for predicting the outcome of Δ〈Φ〉, it underestimates an important aspect of the low-frequency variability, namely, the time difference between the minimum of Δ〈ϕ〉 and the maximum of Δ〈Φ〉 (cf. black dashed lines in Figs. 9a,b). Conversely, the time lag is overestimated for τ = 20 days, whereas it appears to be accurately estimated for τ = 12 days.

We can see how this time lag is related to the decay time scale by reconsidering Eq. (11). Writing Γϕ,ΦX as the product of the modulus and phase |Γϕ,ΦX|exp(iϕk), one can show that the heat flux response, at a monochromatic frequency ω, is related to the temperature gradient as (Hubbard and Brandenburg 2009)
ΔΦω(y,t)=|Γϕ,ΦX|(y,ω)Δϕ(y,tΔt).
Hence, the response of the heat flux at time t, for a monochromatic frequency ω, depends on the modulus of the proxy Green’s and the MZA temperature gradient at time t − Δt. When the proxy Green’s function is an analytic exponentially decaying function, with decay time scale τ, we can write this time lag as
Δt=1ωarctan(ωτ).
From Fig. 3a, we conclude that the Δ〈Φ〉 overshoot consist of a dominant mode, with ν = 0.05 day−1. Inserting this frequency in Eq. (26) and using τ = 12 days (cf. Fig. 7b), we find Δt = 4.2 days, close to the 4.3 days found from Figs. 3a and 3b. For τ = 4 days, we find Δt = 2.8 days, explaining the underestimation found in Fig. 9b. We also note that for τ, Δt = T/4, where T is the period, i.e., 〈ϕ〉 and 〈Φ〉 have a quadrature phase shift. This clearly does not follow from Fig. 3, therefore necessitating τ to be finite.

Using the fitted kernel decay time scale presented in Fig. 7b the latitudinal dependence of Δt was well approximated by Eq. (26). Because the dominant mode frequency remained relatively constant for all latitudes (∼0.05–0.055 day−1), the dependence of Δt translates into the dependence of τ on the latitude (cf. Fig. 7b).

The analysis demonstrates that the parameter values of α and τ, obtained by fitting Γϕ,ΦX to an equation of the form (19), are capable of explaining the initial growth rate tendency of Δ〈Φ〉 as well as the observed time lag between Δ〈Φ〉ω and Δ〈ϕ〉. Hence, utilizing these parameter values enables a relatively accurate reproduction of the mode of variability introduced by the finite memory in Δ〈Φ〉. This then also enables us to interpret τ as a damping time scale resulting from higher-order interactions within the growing disturbance [cf. Eq. (20)].

Although the memory induced low-frequency variability of Δ〈Φ〉 is nicely reproduced when describing the proxy Green’s function as a simple analytic decaying exponential with parameter values presented in Fig. 7b, the discrepancy between the analytical and LRT estimate is significant (cf. Fig. 9b). Much of this error can be attributed to the frequency dependence of both α and τ, resulting in a slight time dependence of these parameters. For example, the inset of Fig. 9b nicely illustrates the time dependence of τ. For the first 6 days, the decay time scale is closer to τ = 4 days, while increasing toward τ = 12 days afterward. Additional components of the error are due to neglecting the oscillatory component, contributing significantly to the LRT computed response > 40 days (cf. Fig. 8), and the assumption of τ and α being independent of y in the derivation of Eq. (21).

b. Lorenz energy cycle

Our findings demonstrate that the nonsingular nature of the proxy response function plays a crucial role in explaining low-frequency variability on intraseasonal time scales. However, while we have identified and characterized this behavior, further investigation is needed to understand the underlying physical mechanisms responsible for the observed intraseasonal memory-mode variability. As such, we aim to link the shape of the kernel to the physical mechanism it parameterizes. Unraveling the origin of low-frequency modes is perhaps most easily answered by looking at the Lorenz energy cycle (LEC; Lorenz 1955; see, e.g., Jüling et al. 2018) of the QGS model for the reference state and the step-forced perturbed average.

In the LEC framework, energy is stored in four reservoirs: zonal mean kinetic energy (K¯), zonal mean potential energy (P¯), eddy kinetic energy (K′), and eddy potential energy (P′), where the prime denotes deviations from the zonal mean. Energy stored in a particular reservoir X can change due to energy sinks (S), energy sources (Q), or conversions to reservoir Y (CX,Y) and vice versa (CY,X=CX,Y). A tendency equation for each reservoir is constructed by decomposing the dynamical Eqs. (1) into a zonal-mean part and an eddy part and computing the volume average (Phillips 1956; Schubert and Lucarini 2015). For the QGS model the equations become
K¯t=CP¯,K¯+CK,K¯SMIFSMGFTME,
P¯t=CP¯,K¯CP¯,P+QM,
Kt=CK,K¯+CP,KSEIFSEGF+TME,
Pt=CP,K+CP¯,P+QE.
The definition and meaning of all energy reservoirs and transfer terms are summarized in Table 1.
Table 1.

Definition of all conversion or source/sink terms in Eq. (27). S is the modified stability parameter, computed as σΔp2/2f02. ZM indicates zonal mean and ZA zonally asymmetric. The domain average is written as D.

Table 1.

In Fig. 10, the LEC for the equilibrated reference and step-forced case are shown. The main input of energy is provided through the zonal mean heating of the atmosphere, QM, increasing the baroclinicity. Through baroclinic instability, warm (cold) air is transferred poleward (equatorward) and upward (downward); hence, CP¯,P>0 and CP,K>0. Because both conversion process are linked to the growth of the same baroclinic wave, they occur almost simultaneously.

Fig. 10.
Fig. 10.

LEC for output of statistically stationary QGS model; red numbers represent the reservoir sizes and conversion/sink/source terms for the step-forced case. All conversion units are in J s−1; the reservoir units are in 106 J.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0146.1

The mean flow kinetic energy, K¯, is driven by CP¯,K¯ and CK,K¯. The former represents the zonal mean sinking (rising) of cold (warm) air, i.e., the resulting zonal mean flow of a thermally driven cell. The latter represents the barotropic decay stage of a nonlinear baroclinic wave in which eddy momentum is transferred to larger zonally symmetric jets (Vallis and Maltrud 1993; Vallis 2017). However, compared to other studies, investigating the global circulation energetics by working in a similar two-layer QG model, the barotropic conversion is small (Phillips 1956; Schubert and Lucarini 2015). For this reason, we argue that internally generated annular modes of low-frequency variability (see, e.g., Figs. 3a,b), are driven by growing baroclinic waves, i.e., eddy heat fluxes (CP¯,P and CP,K), and are approximately unaffected by eddy momentum fluxes (CK,K¯). These characteristics of the QGS model provides an ideal platform for examining baroclinic decoupled modes, the origin of which we aim to explain by finite memory effects.

It is known that the specific type of spectral model used in this study overestimates the importance of baroclinic dynamics (Cehelsky and Tung 1987; Charney and DeVore 1979). The underestimation of barotropic dynamics can be attributed to the absence of a momentum source, which in the real midlatitude atmosphere is provided by a northward flux of tropical westerly momentum (Vallis 2017).

Figures 11a and 11b show the response of the energy reservoirs and conversion rates, respectively, due to the step forcing. Initially, the atmosphere is heated by QM, causing P¯ to increase. The P¯ reservoir overshoots its statistical equilibrium value, introducing a low-frequency mode with time scale ∼20 days. Because QM(t)hd[P*P¯(t)], the overshoot can only be explained by CP¯,P. Indeed, we find that CP¯,P lags 3.5 days behind the response of P¯. Equation (25) shows that a delayed response at a monochromatic frequency may be expected if the proxy Green’s function has a nonsingular part, i.e., the conversion rate CP¯,P carries memory of P¯. The time lag between P¯ and CP¯,P is slightly smaller than, but comparable, to the time lag found in Figs. 3a and 3b. Since the time lag < T/4, the memory must again be finite.

Fig. 11.
Fig. 11.

(a) Energy reservoirs and (b) energy transfer rates response to step forcing in the QGS model.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0146.1

Based on Figs. 11a and 11b, we recognize the following energy pathway, QMP¯CP¯,PPCP,KK. This energy pathway is typical during the growth stage of a baroclinic wave (Simmons and Hoskins 1978). Whereas the dominant pathway during the decay stage (CK,K¯) of the baroclinic wave is negligible, as anticipated from Fig. 11b. Therefore, we interpret low-frequency variability in the response shown in Figs. 11a and 11b as being driven by the development of baroclinic waves and their interactions with other waves and the zonal mean state.

The origin of the oscillatory mode can be understood by considering that the growth rate of an instability is proportional to the local zonal mean flow instability, which we take proportional to ΔP¯. During the growth of an instability (CP¯,P and CP,K) it draws energy from the zonal mean available potential energy, thereby increasing the zonal mean flow stability. However, because the growth of the instability is not synchronized with the local temporal stability ΔP¯(t), but rather with the stability at a previous time ΔP¯(tΔt), the instability continues to grow even after reaching statistical equilibrium. Eventually, the growth of the instability diminishes to the point where Newtonian cooling dominates, causing ΔP¯, and the zonal mean flow instability, to increase again. This cycle repeats and leads to damped oscillations toward equilibrium.

The damping rate of these oscillations depends on two factors. First, it is influenced by the value of hd, which determines the point at which the zonal mean heating dominates over the growth of instabilities. Second, it is influenced by the time required for the instability to adapt its growth properties to the local zonal mean flow stability, which can be associated with the memory time scale τ. These relationships are consistent with the behavior described by Eq. (13).

This mechanism of baroclinic low-frequency oscillations is not new and has been reproduced from weakly nonlinear analysis and laboratory experiments (Lorenz 1963; Pedlosky 1970, 1971; Wang and Barcilon 1986; Dijkstra et al. 2022; Hart 1972; Früh and Read 1997; Harlander et al. 2011). Moreover, the mechanism of baroclinic wave vacillation has been used to explain baroclinic annular low-frequency variability (∼20–25 days) in the large-scale Northern and Southern Hemisphere circulation (Thompson and Barnes 2014; Boljka and Shepherd 2018; Ambaum and Novak 2014), where it was referred to as the “baroclinic oscillator.”

It is noteworthy that a secondary, smaller energy pathway exists, specifically P¯CP¯,K¯K¯. This energy pathway holds particular significance within the initial 4 days following the step forcing (cf. Fig. 11b). The response of the zonal mean temperature field during this period may be significantly influenced by this pathway. Notably, the presence of vertical velocity introduces an additional damping term in Eq. (13), as the thermally driven “Hadley” cell counteracts the growth of the first baroclinic Fourier basis function. In the previous section, we neglected the impact of vertical velocity, which remains valid for t > 4 days. However, within the first 4 days, disregarding the vertical velocity, this mode becomes underdamped, thereby accounting for the overestimation of the Δ〈ϕ〉 in Fig. 9a.

Dijkstra et al. (2022) derived an exponentially decaying memory kernel in a weakly nonlinear context, by considering the interaction of a baroclinic wave packet with the zonal mean state. This analysis excluded wave–wave interactions. We can assess the effect of wave–wave interactions by setting wave–wave interaction coefficients of the spectral QG model to zero, following Reinhold and Pierrehumbert (1982).

Figure 12 shows the response of the energy reservoirs and conversion rates, to a step forcing, when wave–wave interactions are removed (thick lines, referred to as the reduced model), compared to the response of the general model where wave–wave interactions are enabled (thin lines, as seen in Figs. 11a,b). Initially, the response of the reduced model is nearly identical to that of the general model. This is suggestive for the initial growth of baroclinic disturbances being approximately linear.

Fig. 12.
Fig. 12.

Response of energy reservoirs and conversion rates when wave–wave interactions are disabled (thick lines). For reference, the response including wave–wave interactions is also shown (thin lines). Note that the legend shown in Figs. 11a and 11b still applies.

Citation: Journal of the Atmospheric Sciences 81, 3; 10.1175/JAS-D-23-0146.1

However, for sufficiently large amplitude waves, nonlinear interactions become important and the reduced model starts deviating from the general model results. Specifically, the reduced model requires more time to reach equilibrium. The growth rate of CP¯,P is slower, and there is a significant increase in the time lag between P¯ and CP¯,P (5.8 days). These discrepancies suggest that wave–wave interactions have the effect increasing the value of α [cf. Eq. (24)] and decreasing the value of τ [cf. Eq. (25)].

It is important to highlight that even in the model where wave–wave interactions are disabled, the low-frequency variability is still driven by the same internal baroclinic oscillator mechanism discussed earlier. As a result, we expect the shape of the memory kernel to remain similar, characterized by an exponential decay.

5. Summary and discussion

The goal of this study was, using a simple 36-component QG model aimed to represent the midlatitude atmospheric dynamics, to determine the causal relation between the MZA eddy heat fluxes and MZA meridional temperature gradient. This causal relation is encapsulated in the causal proxy Green’s function. In a series of other studies, the causal proxy Green’s function has been hypothesized to be an exponentially decaying function (Moon et al. 2021, 2022; Manucharyan et al. 2017). Moreover, in a weakly nonlinear analysis describing the interaction between a baroclinic wave packet and the zonal mean background state, Dijkstra et al. (2022) showed that an exponentially decaying proxy response kernel naturally follows from the fact that a baroclinic wave packet requires a finite amount of time to adapt to the changing stability of the zonal-mean background state.

Using recent developments in proxy response theory (Lucarini 2018; Tomasini and Lucarini 2021), we determined the proxy Green’s function, relating the MZA eddy heat fluxes to the MZA meridional temperature gradient, in a simple 36-component two-layer QG model. We verified that the MZA meridional temperature gradient may be used as a good predictor for the MZA eddy heat fluxes, when making predictions on time scales longer than 2 days. Physically this makes sense, since on larger time scales, eddy heat fluxes are driven by synoptic instabilities in the atmosphere. The growth rate of these instabilities is proportional to the local flow stability, where stability can be characterized by the meridional temperature gradient. Therefore, the meridional temperature gradient precedes the eddy heat fluxes and consequently serves as a good predictor for sufficiently large time scales. Our study therefore confirms that proxy response theory is a suitable framework to derive a parameterization relating turbulence thickness flux to a mean-state variable if there exists a physical relation between both variables. This supports the idea that has been put forward by earlier work (Lucarini et al. 2017; Tomasini and Lucarini 2021).

One possible approach to address the lack of predictive power on short time scales is to employ multiple observables as proxies for the MZA eddy heat fluxes (Lucarini et al. 2017). By considering a forcing pattern characterized by N nonzero spatial modes, it becomes necessary to utilize N proxy variables for predicting 〈Φ〉. This expanded set of variables offers additional information that can possibly explain the tendencies of 〈Φ〉 on short time scales.

However, in this study, our primary focus was to identify the correct parameterization between the eddy heat fluxes and meridional temperature gradient, which would be able to explain observed low-frequency baroclinic modes in the atmosphere. Since the proxy response function is causal for those time scales using 〈ϕ〉 as predictor, we demonstrated the accurate reproduction of low-frequency variability in 〈Φ〉. Hence, the noncausality on short time scales is not of concern when explaining these modes with the determined proxy response function.

The relation between the MZA eddy heat fluxes and MZA meridional temperature gradient was found to be nonsingular for all latitudes. This implies that the heat fluxes require a finite amount of time to respond to changes in the temperature gradient, challenging the assumption of locality in standard Fickian diffusion. Physically this seems reasonable, because the considered forcing pattern directly perturbs the zonal mean temperature field. This perturbation requires a finite time to be transferred to zonally asymmetric waves. The nonsingular proxy Green’s function also emphasizes that synoptic-scale heat fluxes are influenced by the historical evolution of the zonal mean meridional temperature gradient, as hypothesized by Thompson and Barnes (2014) and Moon et al. (2021). This finding aligns with the hypothesis put forth by Zwanzig (1961) regarding variables that exhibit a lack of sufficient time scale separation, indicating that their dependence cannot be solely attributed to local-in-time interactions.

The nonsingular part of the proxy Green’s function was found to decay over a finite time scale, with small amplitude oscillations superimposed on it. These oscillations were primarily significant in the middle of the domain, and their inclusion in the memory kernel was shown to have a negligible effect in reproducing the characteristics of low-frequency (memory-induced) mode of intraseasonal time scale (cf. Fig. 8a). Therefore, low-frequency memory induced variability can be adequately captured by considering an exponentially decaying integral kernel.

The exponential finite decay of the integral kernel implies that the growth rate of the MZA eddy heat fluxes toward its equilibrium value is proportional to its distance from its equilibrium value (i.e., meridional temperature gradient at criticality). Equivalently, the adaptation rate of instabilities with respect to the zonal mean thermodynamic state is proportional to how well the instabilities are adapted to the zonal mean thermodynamic state. This is in agreement with Stone (1978) theory of baroclinic adjustment in the midlatitudes.

Due to the slight frequency dependence of the decay time scale and feedback amplitude (α), the memory kernel could not be described by a single decay time scale or feedback amplitude. Nonetheless, a significant portion of the variance in the spectral representation of the proxy response function was explained by a decay time scale of τ ∼ 6–12 days and feedback amplitude α ∼ 0.1–0.2 day−2. In comparison to the study by Moon et al. (2021), which achieved a good fit for the observed (reanalysis) low-frequency BAM power spectra with τ = 4 days and α = 0.1 day−2, our parameter values are relatively large. Recognizing the inherent limitations of the QGS model and leveraging insights from past studies estimating similar parameters (Thompson et al. 2017; Moon et al. 2021), we posit that both α and τ will exhibit lower values when applied to the actual extratropical atmosphere. This hypothesis is strongly supported by the overestimation of baroclinic dynamics in the QGS model (Cehelsky and Tung 1987), manifested as enhanced growth of baroclinic waves (corresponding to a larger α and smaller τ). This assertion can be readily verified by examining the LEC in section 4b.

To assess the validity of the decay time scale and feedback amplitude obtained, an analytical model was employed, incorporating an exponentially decaying memory kernel (Moon et al. 2021). By utilizing the identified parameter values, the model successfully reproduced the low-frequency mode of variability induced by the memory component, highlighting the legitimacy of the obtained decay time scale and feedback amplitude. Although the low-frequency memory induced mode was reproduced accurately, discrepancies in the overall response of Δ〈Φ〉 were significant and can be attributed mostly to the use of a time-independent decay rate and feedback amplitude.

From a Lorenz energy cycle analysis, our findings revealed that the zonally symmetric internal variability in the QG model primarily stemmed from the interaction of growing baroclinic waves with the zonal mean thermodynamic state, implying that annular low-frequency variability must be driven by eddy heat fluxes. Notably, the forcing mechanism of this variability is not adjusted to the local in-time stability, proportional to P¯, as would be expected under the assumption of regular Fickian diffusion. Due to this lack of adaptation between baroclinic wave growth and local in-time stability, the Newtonian cooling/heating component led to an overshoot of the dynamic equilibrium stability. As a consequence, the delayed response of baroclinic conversion also exhibits an overshoot beyond its dynamic equilibrium. This behavior ultimately contributed to the model’s progression toward statistically stationary dynamics in a decaying oscillatory manner. A behavior that has been extensively studied in weakly nonlinear analysis of growing baroclinic waves interacting with and adjusting to the zonal mean background state (Pedlosky 1970, 1971, 1972). These theories laid the groundwork for Stone’s (1978) theory of baroclinic adjustment and associated variability on intraseasonal time scales, which, we now verify, can be captured by a parameterization that includes memory in growing baroclinic waves or equivalently Δ〈Φ〉, as hypothesized in Dijkstra et al. (2022).

The theories on baroclinic wave interaction and adjustment are consistent with the recently found decoupled baroclinic annular modes, such as the BAM (Boljka and Shepherd 2018; Boljka et al. 2018; Thompson and Barnes 2014). Therefore, we argue that memory effects are of relevance in explaining baroclinic decoupled modes in the real atmosphere. Thompson et al. (2017) explained this decoupled baroclinic mode by the baroclinic oscillator mechanism, which relied on an infinite memory parameterization of the eddy heat fluxes [cf. Eq. (20)]. Our study now suggests that these decoupled baroclinic modes may indeed be explained as a result of memory in the eddy heat fluxes; however, we suggest that the memory is finite, as hypothesized by Moon et al. (2021) and Dijkstra et al. (2022).

Although the value of α may be associated with the initial linear growth rate tendency of the instability in response to a radiative forcing [as shown in Eq. (24)], the precise physical mechanism underlying the observed memory time scale τ remains elusive. However, from dynamical systems theory (Santos Gutiérrez et al. 2021; Wouters and Lucarini 2013), we expect that this time scale is determined leading-order eigenvalues of the Koopman operator generator associated with the uncoupled dynamics (i.e., independent of the meridional temperature gradient) of the eddy heat fluxes. It is therefore sensible to hypothesize that this time scale is determined by the relaxation time scale of a growing baroclinic disturbance, facilitated through the interaction of growing and decaying modes. This hypothesis also conforms with the assumptions of MTA. A comprehensive understanding of the suggested mechanism would necessitate a detailed examination of the equilibration process of instabilities in relation to the background state, which would likely involve a thorough analysis of energy transfer among various wave scales.

Two important issues regarding the used model remain to be discussed. First, the specific type of spectral model used in this study has been known to overestimate the importance of baroclinic dynamics (Cehelsky and Tung 1987). In section 1 we shortly addressed that baroclinic and barotropic annular modes may decouple on sufficiently large time scales, as shown in Boljka and Shepherd (2018) and Boljka et al. (2018). Since the QGS model was able to reproduce the mechanism behind decoupled annular baroclinic modes, i.e., the baroclinic oscillator, we argue that the proxy response function encompasses the necessary mechanisms to accurately reproduce the low-frequency characteristics of the baroclinic oscillator. Therefore, when working with a model that includes well-resolved barotropic dynamics, the proxy response function describing the interaction between baroclinicity and eddy heat fluxes on time scales exceeding 10 days is not expected to undergo significant changes in its shape.

Second, the chosen model configuration only has one stable weather regime. The absence of multiple stable weather regimes may affect the resulting Green’s function, since the feedback of synoptic eddies on the mean flow might differ in different weather regimes. To test this, we performed a similar analysis in a QGS model configuration, in which multiple weather regimes are stable (Reinhold and Pierrehumbert 1982). This model includes only 10 different spatial Fourier modes, such that the dynamics is highly simplified and governed by large-scale wave dynamics. A similar energy analysis revealed that the observed low-frequency annular modes were again baroclinic. The causal proxy response function computed in this model is similarly best described as an exponentially decaying function modulated with low-frequency oscillations. For this reason, we argue that the underlying physics that the causal proxy Green’s function contains also holds in the case of multiple stable weather regimes.

Future work could focus on determining the memory kernel in Earth system models (ESMs) of intermediate complexity, or more advanced state-of-the-art global climate models, where the influence of barotropic dynamics on the zonal mean global circulation in the extratropics are represented in more detail and in which a realistic representation of weather regimes is simulated. However, one must be reminded that in order to apply proxy response theory a large ensemble of trajectories is required. Ragone et al. (2016) showed that noisy climate variables, e.g., the eddy heat fluxes, require a larger number of ensemble members for statistical convergence of the Green’s functions. Based on Ragone et al. (2016) we therefore anticipate that, in a more advanced climate model, at least O(102) trajectories are required for statistical convergence.

Another interesting extension would include explicitly determining the kernel from reanalysis data. However, because the temporal and spatial forcing patterns are not exactly known, proxy response theory cannot be used to determine the kernel properties. Other possible methods include a linear inverse modeling approach and cross-spectral analysis. These methods might lead to feasible results if Eq. (20) holds for a certain range of time scales within the atmosphere.

Acknowledgments.

The authors gratefully acknowledge Stéphane Vannitsem and Francesco Ragone for their valuable discussions on the Lorenz energy cycle and proxy response theory, respectively. This project was funded by the European Research Council (Project TAOC, PI: Dijkstra H.; Grant Number: 101055096).

Data availability statement.

Results can be reproduced using QGS, which is freely available and can be downloaded at https://qgs.readthedocs.io/en/latest/. We further provide open access to the Python code used to obtain the results in this study and data at Vanderborght (2023).

APPENDIX

Derivation of Eq. (13)

Starting from the thermodynamic Eq. (1c), decomposing all variables into a zonal mean and perturbation part, assuming that the meridional derivative of the streamfunction is negligible, taking the zonal mean of the resulting equation and finally performing the ensemble mean, we end up with
tTa¯+yυaTa¯=hd(T*Ta¯)σΔpRW¯,
where t is short for /∂t and similarly for spatial derivatives. Setting up Eq. (A1) for the step forced and reference case, subtracting the resulting thermodynamic equation, deriving equation to time, and inserting parameterization (20) gives
ttΔTa¯y(αΔϕ+1τΔΦ)=hdtΔTa¯σΔpRtΔW¯.
Using Eq. (A1) to eliminate Δ〈Φ〉 in Eq. (A2) and assuming τ to be independent of y, we find
ttΔTa¯+(hd+1τ)tΔTa¯+hdτΔTa¯αyΔϕ=1τahdT*+σΔpR(t+1τ)ΔW¯.
We finalize Eq. (A3), by projecting
ΔTa¯(t,y)=iAnaΔTa¯i(t)FiA(y).
As such, the diffusion operator becomes
αΔTa¯i2y2FiA(y)=λiΔTa¯iFiA(y),
where λi=αPi2 is the ith eigenvalue of the one-dimensional diffusion operator.
Using orthogonality, the resulting differential equation reads
[tt+(hd+1τ)t+(hdτ+λi)]ΔTa¯i=1τahdTi*+σΔpR(t+1τ)ΔW¯i.

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