## 1. Introduction

The low-frequency response to changes in external forcing or other parameters for various components of the climate system is a central problem of contemporary climate change science. Examples include the response of the mean and variance of the low-frequency teleconnection patterns of the atmosphere in addition to more familiar functions such as the mean temperature response in a global warming scenario. Leith (1975) suggested that if the climate system satisfied a suitable fluctuation–dissipation theorem (FDT), then climate response to small external forcing or other parameter changes could be calculated by estimating suitable statistics in the present climate. For the general FDT, see Deker and Haake (1975), Risken (1989), and Majda et al. (2005, hereafter MAG05). There are important practical and conceptual advantages for climate change science when a skillful FDT algorithm can be established for suitable low-frequency variables in a climate model. The linear statistical response operator produced by FDT can be utilized directly for multiple climate change scenarios, multiple changes in forcing and other parameters, and inverse modeling directly without the need of running the complex climate model in each individual case, often a computational problem of overwhelming complexity. With these interesting possibilities for FDT, Leith’s suggestion inspired a first wave of systematic research (Bell 1980; Carnevale et al. 1991; North et al. 1993; Gritsun and Dymnikov 1999; Gritsun 2001; Gritsun et al. 2002; Gritsun and Branstator 2007; Gritsun et al. 2008) for various idealized climate models for the mean response to changes in external forcing. All of this work utilizes the quasi-Gaussian approximation (qG-FDT) suggested by Leith (1975) (Abramov and Majda 2009; MAG05). Recently, mathematical theory for FDT (Majda and Wang 2010; MAG05; B. Gershgorin and A. Majda 2009, manuscript submitted to *Physica D*) has supplied important generalizations and new ways to interpret FDT. These developments have lead to improved theoretical understanding of the qG-FDT algorithm, new applications of qG-FDT beyond the mean response with significant skill (Gritsun et al. 2008), and new computational algorithms beyond qG-FDT with improved high skill for both the mean and variance of low-frequency response, which have been tested in a variety of models (Abramov and Majda 2007, 2008, 2009). There is also recent mathematical theory for identifying the “most dangerous” perturbations in a given class (Majda and Wang 2010; MAG05) as well as generalizations of FDT to time-dependent ensembles including the seasonal cycle (Majda and Wang 2010). In particular, FDT has been demonstrated to have high skill for the mean and variance response in the upper troposphere for changes in tropical heating in a prototype atmospheric general circulation model (GCM) and can be utilized for complex multiple forcing and inverse modeling issues of interest in climate change science (Gritsun and Branstator 2007; Gritsun et al. 2008). On the other hand, the skill of qG-FDT for the prototype GCM can often deteriorate in the lower-middle troposphere (Gritsun and Branstator 2007) and the reasons for these discrepancies are unclear at the present time.

**u**belongs to ℝ

*with*

^{N}*N*≫ 1, typically in the thousands, millions, etc., so a direct implementation of FDT for

**u**is impossible. Instead, one partitions the state space into lower-dimensional low-frequency components

**u**

*∈ ℝ*

_{I}*so thatand applies the approximate version of FDT on the*

^{M}**u**

*variable alone (Gritsun and Branstator 2007; Gritsun et al. 2008; Abramov and Majda 2009). A central topic of the present paper is to provide new mathematical guidelines for applying suitable approximations to FDT on the variables*

_{I}**u**

*alone and to demonstrate the skill of these approximations on elementary prototype test problems with unambiguous features (see sections 2, 3, 4 below) mimicking those in more complicated models.*

_{I}*A*(

**u**

*), and call this the “FDT response operator,”The second adhoc-FDT procedure stems from a suggestion of Penland and Sardeshmukh (1995). A linear regression approximate stochastic model (Delsole 2004) is found for the variables*

_{I}**u**

*of the formwhere 𝗟 has all eigenvalues with negative real parts and*

_{I}**Ẇ**is white noise forcing. Since 𝗟 has decaying spectrum, the operator in (3) satisfies its own linear regression FDT (LR-FDT) (Gardiner 2004), so the proposal (Penland and Sardeshmukh 1995) is to use the LR-FDT for (3) as a surrogate FDT. The above references often report limited skill and success in utilizing the adhoc-FDT methods in (2) and (3).

A second important topic in the present paper is to demonstrate unambiguously that the limited skill for the “heuristic-FDT” procedures in (2) and (3) often stems from mathematical incompatibility between these heuristic-FDT formulas and the exact FDT response. In particular, as shown in section 3, while LR-FDT provides an interesting null-hypothesis for testing FDT, it has no skill for the response of the variances in a nonlinear system (unlike qG-FDT), and the skill for the mean response can deteriorate significantly because of various model errors in (3), including the tacit Gaussianity of the process as well as the realizability of (3) (Delsole 2000). As shown in section 2, even for systems with Gaussian climate, the heuristic interpretation in (2) remains valid for a scalar variable **u*** _{I}* only under extremely restrictive conditions: namely, if

**u**

*is uncorrelated with all the other variables and*

_{I}*A*(

**u**

*) is a constant multiple of*

_{I}**u**

*.*

_{I}The remainder of the paper has the following outline. In section 2, the basic theory for FDT is summarized (Majda and Wang 2010; MAG05) and new applications to the central issues in (1) and (2) are developed. Section 3 contains both mathematical theory and unambiguous demonstration in the 40-mode L-96 model (Lorenz 1995; Abramov and Majda 2007; MAG05) of both low skill and moderate skill of LR-FDT in various circumstances for the mean response and complete lack of skill for the variance response. Recent research has demonstrated that low-frequency regimes of GCMs exhibit subtle but systematic and significant departures from Gaussianity (Branstator and Berner 2005; Berner and Branstator 2007; Majda et al. 2006; Franzke et al. 2007). A central issue for applications of FDT to climate change science is the fashion in which such departures influence the low-frequency FDT response and how to account for them systematically. A second important practical issue is how to account for the indirect influence of the unresolved variables **u*** _{II}* from (1) on the FDT response of the resolved variables

**u**

*beyond the bare truncation formulas elucidated in section 2. All of these features and the skill of new systematic nonlinear stochastic FDT (NS-FDT) algorithms are elucidated in an unambiguous test model (Majda et al. 2009) capturing crucial features of GCMs in section 4. Finally, section 5 contains a summary of the results as well as an outlook on future developments.*

_{I}## 2. General properties of FDT and applying FDT on reduced subspaces

**u**∈ ℝ

*, where*

^{N}**is an**

*σ**N*×

*K*noise matrix and

**Ẇ**∈ ℝ

^{K}is a

*K*-dimensional white noise. We assume that (4) is written in the Ito sense (Gardiner 2004) so that the associated Fokker–Planck equation for the probability density

*p*(

**u**,

*t*) iswhere 𝗤 =

*σσ*^{T}. The climate state associated with (4) is the probability density

*p*

_{eq}(

**u**) that satisfies

*L*

_{FP}

*p*

_{eq}= 0 and the climate statistics of some functional

*A*(

**u**) are determined by

### a. FDT and the quasi-Gaussian approximation

*δ*

**w**(

**u**)

*f*(

*t*); that is, consider the perturbed equationCalculate perturbed climate statistics by utilizing the Fokker–Planck equation associated with (7) with initial data given by the unperturbed climate equilibrium. Then, FDT states that if

*δ*is small enough, the leading-order correction to the statistics in (6) becomeswhere 𝗥(

*t*) is the linear response operator, which is calculated through correlation functions in the unperturbed climate:The noise in (4) is not needed for FDT to be valid, but in this form the equilibrium measure needs to be smooth; there is an alternative formulation of FDT for

**≡ 0 in (4) that avoids this smoothness requirement or even explicit knowledge of**

*σ**p*

_{eq}(

**u**) and is useful for developing short-time FDT algorithms based on a linear tangent model (Abramov and Majda 2007, 2008, 2009; Abramov 2009). These are useful generalizations of FDT that apply to ensemble predictions of the time-dependent systems and even allow for noise perturbations to assess model error (Majda and Wang 2010). Note that FDT does not require any linearization of the underlying dynamics in (4). One major stumbling block in applying FDT directly in the form in (9) is that the equilibrium measure

*p*

_{eq}(

**u**) is not known exactly. In the quasi-Gaussian approximation, one utilizes the approximate equilibrium measurewhere the mean

**u**

*p*

_{eq}. One then calculatesand replaces

*B*(

**u**) by

*B*(

^{G}**u**) in the quasi-Gaussian FDT (Leith 1975; Gritsun et al. 2008; MAG05). The correlation in (9) with this approximation is calculated by integrating the original system in (4) over a long trajectory or an ensemble of trajectories covering the attractor for shorter times assuming mixing and ergodicity for (4). For the special case of changes in external forcing

**w**(

**u**)

*=*

_{i}**e**

*, 1 ≤*

_{i}*i*≤

*N*, the response operator for qG-FDT is given by the matrixThese are the approximations utilized for the qG-FDT algorithm in recent applications (Gritsun et al. 2008; Abramov and Majda 2009; MAG05). Next, we systematically address the central issue in (1) for GCMs in utilizing FDT.

### b. Application of FDT on reduced subspaces: General principles

**u**∈ ℝ

*with*

^{N}*N*≫ 1. Practically, one cannot envision applying FDT on the entire phase space because of practical limitations in calculating the covariance matrix, response operator, etc. Instead, one hopes to compute the response operator on a reduced subspace as in (1). What properties should the subspace defined by

**u**

*have and how can these properties be reconciled with the FDT summarized above? A natural condition is imposed on the small perturbation vector field,*

_{I}*δ*

**w**(

**u**)

*f*(

*t*) so thatHere we ask and answer the following basic question: when is FDT approximately true when restricted to the subspace

**u**

*alone? The general answer is easy by using Bayes’ theorem and conditional probability (Gardiner 2004). Write*

_{I}*p*

_{eq}(

**u**

*,*

_{I}**u**

*) as*

_{II}**u**

*given the value*

_{II}**u**

*withWe have the following consequence of the second formula in (9), (14), and (15).Substituting the factorization in (14) and (15) into (9) yields (16). With general fact 1, we immediately haveThis condition is satisfied automatically for a perturbation*

_{I}**w**(

**u**

*) with*

_{I}**w**

*(*

_{II}**u**

*) = 0 provided that*

_{I}**u**

*. It is worth mentioning here that many unforced undamped geophyscal systems (such as quasigeostrophic flow on the sphere with topography) have Gaussian invariant measures that automatically have such a factorization (Majda and Wang 2006; MAG05) where the reduced subspace is just a projection on any finite number of spherical harmonics or Fourier modes. As an important example, let us compute what the proposition means in the Gaussian case. By centering so the mean is zero, assume*

_{I}*p*

_{eq}=

*C*exp(−½

_{N}**u**𝗖

^{−1}

**u**) where the correlation matrix 𝗖 > 0 has the block decomposition in terms of

**u**

*and*

_{I}**u**

*:There is a well-known regression formula (Gardiner 2004) to calculate the marginal distribution*

_{II}**w**(

**u**

*) with*

_{I}**w**

*(*

_{II}**u**

*) = 0 provided that*

_{I}### c. Implications for FDT modeling on reduced subspaces

*R*(

*t*) in (9) by also assessing the contributions from the modes

**u**

*that are not in the subspace spanned by*

_{II}**u**

*even for functionals*

_{I}*A*(

**u**

*) and perturbations*

_{I}**w**(

**u**

*) with*

_{I}**w**

*(*

_{II}**u**

*) = 0. These improved results by the blended response algorithm apply to the mean and variance response of the L-96 model in weakly chaotic regimes (Abramov and Majda 2007; see Table 1 below) and for the mean and variance response of a T-21 truncation on the sphere with realistic orography (Abramov and Majda 2009).*

_{I}Finally, it is easy to critique the ad hoc approach in (2) for computing the FDT response of a variable *A*(**u*** _{I}*) to external forcing by combining (8), (9), fact 2, and (20). Thus, for example, the heuristic FDT response formula in (2) is valid for the scalar variable

**u**

*if and only if*

_{I}*A*(

**u**

*) is a constant multiple of*

_{I}**u**

*and the perturbation*

_{I}**w**(

**u**

*) satisfies*

_{I}**w**

*(*

_{II}**u**

*) = 0 as well as*

_{I}**u**

*is uncorrelated with the variables*

_{I}**u**

*—that is, if*

_{II}**u**

*is a multiple of an EOF in a given metric with*

_{I}**u**

*spanned by the other EOFs in that metric.*

_{II}## 3. The skill of linear regression strategies for the FDT response

*L*,

*σ*, and

**F**in (3) to match the climatological mean and covariance matrix for the nonlinear dynamical system in (4) (Penland and Sardeshmukh 1995; Delsole 2000, 2004); that is,The second requirement for fitting a linear regression model is to match a suitable functional of the lag covariance matrix (Delsole 2000). For applications of linear regression models to approximate long-term climate response, it is natural to try to fit the infinite time-mean response of the linear regression model from (3) in order to match the quasi-Gaussian mean response to external forcing (Gritsun and Branstator 2007). Thus, from (8)–(12) with

*A*(

**u**) = (

**u**−

**u**

**u**

^{R}=

**u**

*L*, in (3) with eigenvalues with negative real parts simultaneously satisfying (22) and (23) with a given regression strategy (Delsole 2000, 2004). Under these circumstances, in general additional dissipation and white noise forcing are added so that

*L*is guaranteed to have eigenvalues with negative real parts at the expense of model errors in the climatological variance in (22) (Delsole 2000, 2004). Below we propose a perfect regression strategy for solving (22) and (23) without such errors and compare its skill to the standard regression strategy utilized in stochastic modeling of shear turbulence (Delsole 2004). Naively, one might think that a linear regression strategy satisfying (22) and (23) would have the same skill for calculating the FDT response as the quasi-Gaussian approximation developed above in (10)–(12). There is an important difference since the quasi-Gaussian approximation uses the lag covariance in (12) computed from the nonlinear dynamics in (4) while the LRM in (3) automatically assumes linear dynamics with a Gaussian invariant measure as approximations for the dynamics. In fact, this distinction is important, as we show next.

### a. No skill for the variance response for any LRM

*A*(

**u**) such as the variances given bywhere 𝗤 is any constant matrix. With (3) for the linear regression dynamics, the linear response operator in (8) and (9) for

*A*(

**u**) in (24) to a change in external forcing for the LRM is given byThe LRM in (3) always generates a Gaussian random field in time and a key property of Gaussian distributions is that all odd-centered moments automatically vanish (Gardiner 2004). Thus,For many dynamical systems of interest here ranging from GCMs (Gritsun et al. 2008) to the L-96 model (Abramov and Majda 2007, 2009) to the Kruskal–Zabusky model (MAG05), there is a significant response in the variance to changes in external forcing and this is captured with skill by the quasi-Gaussian approximation described in (10)–(12). It is worth mentioning that the Kruskal–Zabusky model has significant variance response even though the climatology is Gaussian (see section 2.9 of MAG05) and the blended response algorithms (Abramov and Majda 2007, 2009) have even higher skill than the quasi-Gaussian approximation for the variances. On the other hand, (26) shows that any LRM automatically has no skill for the variance response.

### b. The L-96 model as a test bed for the skill of LRM for the mean response

*j*= 0, … ,

*J*− 1, with

*J*= 40, and

*F*is a forcing parameter. The model is designed to mimic baroclinic turbulence in the midlatitude atmosphere with the effects of energy-conserving nonlinear advection and dissipation represented by the first two terms in (27). The L-96 model has been utilized as a test bed for FDT algorithms (Abramov and Majda 2007, 2008; MAG05) among other applications since it exhibits features of weakly chaotic (

*F*= 5, 6), strongly chaotic (

*F*= 8), and turbulent (

*F*= 16) mixing dynamical systems as

*F*is varied. Let the mean state be

*u*

*u*

_{j}, where the bar denotes the long time average in the statistical steady state. The mean fluctuations of the energy around the energy of the mean state are given by

*F*. To have the same energy variations for various values of

*F*, a rescaled system is considered (MAG05) with the transformation

*u*=

_{j}*u*

*E*

_{p}*ũ*,

_{j}*t*=

*t̃*/

*E*

_{p}*F*. The dynamical equation in terms of the new variables becomesThe main interest here is the skill of LRM strategies for calculating the mean response to a change in external forcing over variety of times,

*T*= 5, 20, ∞ in the nondimensional units utilized here (Abramov and Majda 2007; MAG05). However, here we begin with an important set of more general remarks. The L-96 model in (27) is translation invariant so that Fourier modes are eigenvectors for both the EOFs and the infinite time integrated covariance matrix of the right-hand side of (23) for the exact L-96 operator. The eigenvectors that diagonalize the right-hand side of (23) are the optimal low-frequency basis for mean response (Gritsun and Dymnikov 1999) and information theory has been utilized (Majda and Wang 2010; MAG05) to interpret those eigenvectors with the largest eigenvalues as having the most significant additional information in perturbations beyond the unperturbed climate. Figure 1 shows the EOFs (Fourier modes) with the largest variance compared with the optimal low-frequency-basis eigenvectors and eigenvalues as

*F*varies. From these two graphs, we see that especially for

*F*= 5, 6, 8, the EOFs with largest variance are concentrated in wavenumbers 6 ≤ |

*k*| ≤ 10 while the largest amplitude for the low-frequency basis for the infinite time response is concentrated on the disjoint band of wavenumbers 11 ≤ |

*k*| ≤ 14. The reason for this is that the Fourier modes with the largest variance have highly oscillatory correlation functions in the L-96 model (MAG05). Thus, the L-96 model provides interesting examples that illustrate that the modes with the largest climate variance do not necessarily contribute the largest climate response, so truncation to the leading EOFs requires caution in applying FDT to a truncated response. On the other hand, in atmospheric GCMs and prototype models for low-frequency atmosphere dynamics (Franzke and Majda 2006; Berner and Branstator 2007; Abramov and Majda 2009), the EOFs with the largest variance also have the largest correlation times and thus the largest low-frequency response.

*Ẇ*is complex white noise. Now, the linear SDE for each mode

*υ̂*becomeswhere

_{k}*ω*

_{1}(

*k*) = {

*u*

*πk*/

*J*) − cos(4

*πk*/

*J*)] − 1}/

*E*

_{p}*ω*

_{2}(

*k*) =

*u*

*πk*/

*J*) + sin(4

*πk*/

*J*)]/

*E*

_{p}*d*,

_{k}*ω*, and

_{k}*σ*such that the Fourier modes

_{k}*υ̂*have the same statistical properties in (22) and (23) as the Fourier modes

_{k}*û*from the L-96 model. In particular, we fit the values of the variance and the integral of the time autocorrelation function for each mode. The variance of each mode

_{k}*û*of the original system can be computed as

_{k}*T*is a decorrelation time of the mode

_{k}*û*. Therefore, we have three parameters, Var

_{k}*,*

_{k}*T*, and

_{k}*θ*, which will be used to determine three unknowns,

_{k}*d*,

_{k}*ω*, and

_{k}*σ*. For the linear process

_{k}*υ̂*(

_{k}*t̃*) given by (31), the corresponding values of the variance and of the integral of the correlation function are given byTo find the unknown parameters, we solve the following equations:The solution is given byThese formulas, together with (31), determine the perfect regression strategy. We have just shown that it is always realizable with the constraints in (22) and (23).

*ω*

_{2}(

*k*), fixed at the climatological background operator but to change the damping and white noise forcing to satisfy (22) and (23). In this case, the linear SDE for each mode

*υ̂*becomesNow, we have two unknowns,

_{k}*d*, and

_{k}*σ*, and two parameters,

_{k}*T*, Var

_{k}*. To find them, provided we utilize the real part of the integrated correlation function in (33), the equations becomeThe first equation is quadratic in*

_{k}*d*and has the following solutions:To ensure that the damping is real and positive, the expression under the square root has to be positive. Because of this restriction, we have a constraint on the correlation times

_{k}*T*of each of the modes. If the correlation time

_{k}*T*that we obtain from the original nonlinear model does not satisfy this constraint, then this value of

_{k}*T*is unrealizable and we have to “tune” the dissipation. The second parameter,

_{k}*σ*

_{k}^{2}, is computed via

For the LRM defined by either the perfect or standard regression strategies described in the last two paragraphs, the linear response operator in physical space can be calculated directly from the same linear response operator for the decoupled stochastic equations in (31) or (35) through the inverse Fourier transform in (29). In Table 1, we show the skill of these linear regression strategies for *F* = 5, 6, 8, 16 in computing the mean response at rescaled times *T* = 5, 20, ∞. The pattern correlations of the approximate response operator and the ideal response operator in physical space are reported there (Abramov and Majda 2007; MAG05) for the two LRM strategies as well as the correlation skill of the quasi-Gaussian and blended response algorithms for comparison (Abramov and Majda 2007; MAG05). For all values of *F* ranging from weakly chaotic (*F* = 5, 6) to strongly chaotic (*F* = 8) to fully turbulent (*F* = 16) in the L-96 model, the perfect regression strategy has significantly more skill than the standard regression strategy at all three response times. Furthermore, this skill of the perfect LRM strategy is comparable to the skill of the quasi-Gaussian approximation for all values of *F* but significantly lower than the skill of the blended response algorithm (Abramov and Majda 2007), especially for *F* = 5, 6, 8. Figure 2 shows the mean response operator at infinite time for the perfect and standard LRM strategies compared with the ideal response operator. The relative skill of the two regression strategies is apparent in Fig. 2. With the requirements in (22) and (23) satisfied exactly for the perfect regression strategy, its skill for mean response comparable to the skill of the quasi-Gaussian approximation is not surprising at infinite times but not obvious for times *T* = 5, 20.

In summary, the main results here show that any LRM for FDT has no skill for the variances in contrast to the quasi-Gaussian approximation. On the other hand, a perfect LRM strategy for FDT has essentially the same skill as the qG-FDT for the mean response to external forcing for the L-96 model. However, the standard LRM strategy for FDT from shear turbulence theory (Delsole 2004) has significantly less skill for the mean response than either the perfect LRM strategy or the quasi-Gaussian approximation. Also for the L-96 model, the EOFs with the largest climate variance are disjoint from the Fourier modes with largest low-frequency mean climate response.

## 4. Reduced stochastic models for FDT

An important practical issue for GCMs is how to account for the indirect influence of the unresolved variables **u*** _{II}* from (1) on the FDT response of the resolved variables

**u**

*beyond the bare truncation formulas elucidated earlier in section 2. GCMs often have too little variability and one way to cope with this lack of variability is to add suitable stochastic parameterizations (Palmer 2001; von Storch 2004; Majda et al. 2008). These considerations provide one motivation for developing reduced stochastic models for the variables*

_{I}**u**

*alone with high skill for the low-frequency FDT response. A second motivating direction stems from the fact that recent research has demonstrated that low-frequency regimes of GCMs exhibit subtle but systematic departures from Gaussianity (Branstator and Berner 2005; Berner and Branstator 2007; Majda et al. 2006; Franzke et al. 2007). A central issue for applications of FDT to climate change science is the fashion in which such departures influence the low-frequency FDT response. In a related direction, there is a developed theory for deriving reduced stochastic models for low-frequency teleconnection patterns (Majda et al. 1999, 2003; Franzke and Majda 2006; Majda et al. 2008) and a central issue is the skill of such reduced stochastic models in approximating the low-frequency response. These issues are elucidated here by studying them unambiguously in a family of prototype simplified models. We begin with these issues in the context of linear stochastic models and then discuss the subtle issues involving small departures from Gaussianity in a prototype nonlinear stochastic model (Majda et al. 2009).*

_{I}### a. Skill of reduced FDT approximations for linear stochastic models

**u**= (

**u**

*,*

_{I}**u**

*)*

_{II}^{T}in (1) given bywhich we also write compactly asThe parameter ε > 0 in (38) can be large or small here. Below, we require that 𝗟

^{ε}has eigenvalues with a negative real part for all ε and in particularfor

**u**≠ 0. These requirements guarantee that 𝗟

^{ε}is invertible and the climate mean state is given by 〈

**u**〉 = (𝗟

^{ε})

^{−1}〈

**F**〉; this fact together with (39) and (40) implies in particular that the change in the first components of the climate mean state,

*δ*〈

**u**

*〉, in response to a change in forcing,*

_{I}*δ*

**F**

_{1}, is given exactly byStochastic mode reduction techniques (Majda et al. 1999; Gardiner 2004; Majda et al. 2008; MAG05) systematically produce a reduced stochastic model for the variables

**u**

*alone, which is a valid model in the limit ε → 0; such models often have significant skill for moderate values of ε (Majda et al. 2003; Franzke and Majda 2006). Here, we focus on their skill in producing the infinite time-mean response in (41) of the full dynamics from (38) independent of ε.*

_{I}**u**

*variable alone (Kubo et al. 1985) given byFor simplicity in exposition, zero initial data are assumed for*

_{I}**u**

*. As discussed in detail elsewhere (Majda et al. 1999; MAG05), the second and third terms in (42) simplify in the limit ε → 0 and yield reduced simplified local stochastic dynamics for*

_{II}**u**

*alone given byThis is an explicit example of stochastic mode reduction where the variables*

_{I}**u**

*have been eliminated and there is a reduced local stochastic equation for*

_{II}**ũ**

_{I}alone with explicit corrections that reflect the interaction with the unresolved variables. Here, we address the skill of the approximation in (43) in recovering the exact mean climate response in (41) independent of ε. Reasoning as discussed earlier in general below (40), the response of the climate mean in (43) to a change in forcing is given exactly byThis general result points to the surprisingly high skill of the FDT for the reduced stochastic model in calculating the mean climate response. The next issue to address is the subtle role of nonlinearity and non-Gaussianity in influencing the skill of FDT algorithms based on reduced stochastic modeling.

### b. The skill for reduced nonlinear stochastic models for FDT

*u*

_{1}as the resolved variable with the variables (

*u*

_{2},

*u*

_{3})

^{T}=

**u**

*, the group of unresolved variables in (1) or (38). The linear stochastic terms in (46) have the same structural form as (38) and satisfy (40). The crucial new feature in (46) is the energy-conserving nonlinear dyad interaction between*

_{II}*u*

_{1}and

*u*

_{2}for

*I*≠ 0 that is responsible for departures from Gaussianity. As emphasized recently (Majda et al. 2009), such dyad interactions arise typically in an empirical basis of EOFs and explain nonlinearity and significant departures from Gaussianity in data from a single low-frequency variable of a prototype GCM. Application of the same stochastic mode reduction procedure (Majda et al. 1999, 2009; MAG05) as described earlier in (42) and (43) for ε ≪ 1 results in the nonlinear reduced stochastic equation for the

*u*

_{1}variable alone,where the explicit values of the coefficients that depend on ε are listed in Table 2. From Table 2, one can see that the reduced stochastic model for (47) necessarily has correlated additive and multiplicative (CAM) noise (i.e.,

*B*≠ 0 with

*A*≠ 0) simultaneously with nonlinear cubic damping

*c*> 0 if and only if the nonlinear dyad interaction coefficient

*I*≠ 0; the corresponding probability density function (PDF) for (47) can be written down explicitly (Majda et al. 2009) and has Gaussian tails with significant realistic non-Gaussian skewness and flatness statistics as in actual PDFs for low-frequency variables in GCMs (Branstator and Berner 2005; Majda et al. 2006; Franzke et al. 2007) (see Figs. 5 and 6). The reduced stochastic models in (47) have been utilized as nonlinear regression models (Majda et al. 2009) to fit low-frequency data from a prototype GCM with a significant role for the cubic damping,

*c*≠ 0. All this work contrasts strongly with the ad hoc proposal (Sardeshmukh and Sura 2009) to use linear stochastic models with CAM noise and power-law tails for fitting low-frequency variables.

*δF*, with

*B*(

*ũ*) from (9) given explicitly byThe interest here is the skill for FDT of the reduced stochastic model from (47) in capturing the mean and variance response of

*ũ*

_{1}from the full system in (46) to a change in external forcing. Below, we also consider the skill of the quasi-Gaussian approximation for the one-dimensional reduced stochastic system in (47) as well as the skill of the quasi-Gaussian approximation implemented directly for the full three-component system in (46). Below, we find that as in the linear Gaussian case reported in (45), for

*I*≠ 0 and ε = 1 the reduced nonlinear FDT response approximation using (47) and (48) has high skill in capturing the mean and variance response; there is less but still significant skill for the FDT response with the quasi-Gaussian approximation of the reduced stochastic model in (47) and surprisingly poor skill for the FDT response utilizing the quasi-Gaussian approximation of the full three-component system in (46).

To demonstrate these points, we utilize the system in (46) with explicit values *σ*_{2} = 1.2, *σ*_{3} = 0.8, 𝗟_{11} = −2, 𝗟_{12} = 0.2, 𝗟_{13} = 0.1, *γ*_{2} = 0.6, *γ*_{3} = 0.4, and *F* = 0 and vary ε systematically from ε = 0.1 to 1. The value of the nonlinear interaction coefficient *I* = 1 is utilized below unless noted otherwise. The numerical algorithms for integrating (46) and (47) and calculating the ideal response operator to the change in external forcing of the first component in (46) as well as the predictions of the various FDT approximations follow those developed earlier (MAG05, chapters 2 and 3). These algorithms utilize a sufficiently long time series to compute both the ideal response and the lag correlations required in (9)–(12) accurately. Given the structure in (46), it is not surprising that the EOFs for the unperturbed climatology line up in directions that are small departures from the coordinate axes, with only weak dependence on ε, and that most of the variance (98%) is associated with the EOFs essentially aligned with *u*_{2} and *u*_{3} components; this is natural since the components *u*_{2}, *u*_{3} in (46) are surrogates for all the unresolved variables in the GCM whereas *u*_{1} represents a single low-frequency mode (Majda et al. 2008).

Figure 3 shows the PDF and autocorrelation of *u*_{1} as well as the finite and infinite time response operators for the Gaussian case with *I* = 0 but with large ε, ε = 1. As predicted by the theory in (45), the mean response operator of the reduced stochastic model closely approximates the ideal response operator not only at infinite time but also at finite response times despite the discrepancy in the time scale of the autocorrelation of the reduced model. Next, the skill of the FDT response of the reduced stochastic model is tested with strong coupling, *I* = 1, and ε varying from 0.1 to 1. The autocorrelations of the three components and representative snapshots of the time series for *u*_{1} for the full and reduced models are depicted in Fig. 4. As expected for ε = 0.1 in (46), the variable *u*_{1} has a much longer autocorrelation than *u*_{2}, *u*_{3}; however, for ε = 1, the longest autocorrelation is given by *u*_{3} while *u*_{1} and *u*_{2} have shorter autocorrelation times. The representative time series for *u*_{1} in both cases show clearly skewed bursts of regime behavior that are captured with high skill for ε = 0.1 by the reduced stochastic model from (47) and with less skill for ε = 1. These results in Fig. 4 indicate that the case *I* = 1, ε = 1 is an extremely stringent test case for the skill of the FDT response from the reduced stochastic model. Figure 5 shows the expected skewed non-Gaussian PDF for *u*_{1} and the autocorrelation of *u*_{1} for both the full triad model in (46) and the reduced stochastic model for *I* = 1, ε = 0.1. The finite time FDT response operators for both the mean and variance are also depicted in Fig. 5. Here all three approximations to the mean response—namely, FDT for the reduced system, qG-FDT for the reduced system, and qG-FDT for the full system—have comparable and very high skill for the mean response at all times. On the other hand, both qG-FDT approximations overshoot the variance response significantly whereas the nonlinear FDT for the reduced system retains high skill. Figure 6 shows the same quantities as in Fig. 5 for the stringent test case, *I* = 1, ε = 1. Here, the reduced stochastic model recovers the non-Gaussian skewed shape of the normalized PDF despite the affects of large ε in the reduced approximation as shown in Fig. 4. The non-Gaussian FDT for the reduced system has extremely high skill for the mean and significant skill for the variance response despite the large values of ε. In this case the mean response predicted by both quasi-Gaussian approximations is not accurate, with the largest errors for the quasi-Gaussian approximation for the full triad system. The underestimate of the mean response with the quasi-Gaussian approximation for the reduced system can be understood through the shorter autocorrelation time for the reduced model in Fig. 6 and the fact that the mean response with the quasi-Gaussian approximation for (47) is just the integral of this correlation function.

## 5. Concluding discussion

A new perspective on potential applications of FDT for low-frequency climate response has been developed here through a combination of mathematical theory and prototype idealized models. The practical issue in (1) regarding applying FDT on a reduced set of variables is the main focus here. Section 2 contains precise mathematical theory and guidelines for the validity of the bare truncation of FDT to a reduced subspace; as a minimum requirement, the variables in the subspace of reduced variables should be weakly correlated with the remaining variables for such approximations to remain valid. The theory supplies a constructive criticism of the “ad hoc” FDT recipe in (2), utilizing the integral of an autocorrelation function to represent the FDT response; in section 2, it is shown that this ad hoc recipe is only valid under the very restrictive condition that the variable **u*** _{I}* is multiple of an EOF in some metric and the nonlinear functional,

*A*(

**u**

*), is in fact a constant multiple of*

_{I}**u**

*. The other popular ad hoc FDT method is through linear regression modeling as described in (3); this is the topic of section 3. There, it is established that linear regression models always necessarily have no skill in the response for the variances. Furthermore, as demonstrated in the L-96 model, the skill in the mean response from LRM to a change in external forcing can have moderate or significantly lower skill depending on the details of the linear regression strategy. A new perfect regression strategy, introduced in section 3, has much higher skill on the L-96 test bed for the mean response than the standard regression strategy from stochastic modeling of shear turbulence (Delsole 2004). The hope here is that these constructive criticisms of ad hoc FDT methods will inspire the potential users of FDT to examine more systematic approximations of FDT with firm mathematical underpinnings and with much higher skill than the ad hoc methods. The developments in section 4 center on the practical issue of developing FDT approximations for the resolved variables,*

_{I}**u**

*, which go beyond the bare truncation described in section 2 and involve reduced stochastic models for the explicit interaction of the resolved and unresolved variables. This paradigm is useful both for direct applications in GCMs (Palmer 2001; von Storch 2004) and also for the development of reduced stochastic models that incorporate subtle departures from Gaussianity for low-frequency patterns with high skill in their FDT response. Both reduced stochastic models in the linear case and nonlinear reduced stochastic models were developed systematically and tested on a prototype model. For linear stochastic systems, the reduced stochastic model, remarkably, has perfect skill in the long time-mean response even for large values of the expansion parameter ε [see Fig. 3 and (45)]. As reported in section 4, for the prototype nonlinear system in (46), the reduced nonlinear FDT response approximation has remarkably high skill in capturing the mean and variance response in non-Gaussian situations where once again the expansion parameter is large. On the other hand, the quasi-Gaussian approximation of the full three-component system in (46) has poor skill for the mean and variance response (Fig. 6) in this stringent regime. These encouraging results provide new directions for improving FDT approximations, which we hope to address in the near future. In particular, additional subtle issues occur for FDT approximation with a seasonal cycle (Majda and Wang 2010) and statistically exactly solvable test models have been developed recently (B. Gershgorin and A. Majda 2009, manuscript submitted to*

_{I}*Physica D*) to elucidate these issues.

## Acknowledgments

We thank Rafail Abramov for providing us with the ideal response operators for L-96 model. The research of Andrew J. Majda is partially supported by National Science Foundation Grant DMS-0456713, Office of Naval Research Grant N00014-05-1-0164, and Defense Advanced Research Projects Agency Grant N0014-07-1-0750. Boris Gershgorin is supported as a postdoctoral fellow through the last two agencies and Yuan Yuan is supported as a graduate research assistant by the first agency.

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FDT response using perfect regression [(31), (34), (29)], standard regression [(35), (36), (37), (29)], quasi-Gaussian approximation, and the blended response algorithm (Abramov and Majda 2007) for different response times, *T*, and different forcings, *F*.

Coefficients of the reduced one-dimensional model (47).