1. Introduction

The low-frequency response to changes in external forcing for various components of the climate system is a central problem of contemporary climate change science. In particular, in the atmosphere the response of the low-frequency teleconnection patterns such as the Arctic Oscillation (AO), the North Atlantic Oscillation (NAO), and the Pacific–North America pattern (PNA), which steer the storm tracks, is a fundamental issue. Leith (1975) suggested that if the climate system satisfied a suitable fluctuation–dissipation theorem (FDT), then climate response to small external forcing 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). The topic of this paper is the use of a new algorithm (Abramov and Majda 2007, 2008, hereafter AM07, AM08) for calculating the FDT response to address the above issues in a prototype setting for the atmosphere.

In practice, Leith (1975) proposed that the response of the global climate due to changes in external parameters could be computed through an approximation, the quasi-Gaussian fluctuation–dissipation theorem (qG-FDT) for appropriate variables in the climate system. To predict global climate response in this setting, one observes the climatology of a model for a sufficiently long time under the tacit assumption that the dynamics is close to its statistical equilibrium and then, under the assumption of a Gaussian statistical equilibrium state, applies the quasi-Gaussian fluctuation–dissipation formula to predict mean climate response to small changes of the physical parameters of the dynamics; these estimates are modeled without actually simulating an appropriate scenario of climate development for those changes of parameters, which usually poses a computational problem of substantial complexity. Leith’s suggestion has inspired others such as Bell (1980), Carnevale et al. (1991), North et al. (1993), Gritsun and Dymnikov (1999), Gritsun (2001), Gritsun et al. (2002), Gritsun and Branstator (2007), and Gritsun et al. (2008) to apply the qG-FDT for idealized climate models with various approximations and numerical procedures. Bell (1980) considered a special truncation of the barotropic vorticity equation and studied the response of this truncation to small kick of trajectory at initial time, reporting that the predictions of classical FDT for the response of the system hold extremely well. Carnevale et al. (1991) generalize the quasi-Gaussian FDT formula to a non-Gaussian but smooth equilibrium state, also documenting reasonably high precision of the classical Gaussian FDT formula for a model suggested by Orszag and McLaughlin (1980) but its failure for the classical three-dimensional “Lorenz butterfly” model (Lorenz 1963). North et al. (1993) studied the fluctuation–dissipation of a general circulation model in response to a step-change in solar forcing. Gritsun and Dymnikov (1999), Gritsun (2001), and Gritsun et al. (2002) developed a systematic procedure to compute the complete response operator in matrix form for different response times; they also presented a robust algorithm of verification for the FDT response prediction through a series of direct numerical simulations with the actual perturbed system. Gritsun and Branstator (2007) and Gritsun et al. (2008) have applied qG-FDT to a comprehensive general circulation model for computing the response of the mean state and various quadratic functionals, such as variance and meridional eddy flux, with interesting results. In particular, the response to tropical heating is considered in these papers; qG-FDT typically has high correlation skill, above 0.8 in the upper troposphere response for linear and 0.7 for quadratic functionals, with amplitude errors in the 25%–50% range. In fact, this skill is sufficiently high for several successful applications of linear response modeling based on qG-FDT. On the other hand, the skill of qG-FDT in this setting can often deteriorate significantly in the lower-middle troposphere (Gritsun and Branstator 2007), and the reasons for these discrepancies are unclear at the present time.

The response of the forced-dissipative 40-mode Lorenz 96 (L96) model (Lorenz 1995; Lorenz and Emanuel 1998; Abramov and Majda 2004) has also been studied by MAG05 within the framework of the quasi-Gaussian FDT. It has been found in MAG05 that although the quasi-Gaussian FDT works reasonably well for the forced-dissipative L96 model in strongly chaotic dynamical regimes, for weakly chaotic regimes the predictions of the classical FDT become much worse. Although it is suitable for dynamical systems with a nearly canonical Gaussian equilibrium state, the fluctuation–dissipation theorem in its classical formulation is only partially successful for nonlinear dynamical systems with forcing and dissipation. The major difficulty in this situation is that the statistical equilibrium in the limit as time approaches infinity in this case is typically described by a Sinai–Ruelle–Bowen (SRB) probability measure, which is usually not smooth (Eckmann and Ruelle 1985; Ruelle 1997, 1998; Young 2002). In this context, AM07 and AM08 developed and tested novel computational algorithms for predicting the mean response of functionals of a chaotic dynamical system to small changes in external forcing via the fluctuation–dissipation theorem; the new algorithms include the short-time FDT (ST-FDT) and the blended short-time/quasi-Gaussian (ST/qG-FDT) algorithm. Unlike the earlier work in developing FDT-type computational strategies for chaotic nonlinear systems with forcing and dissipation, the new FDT methods developed in AM07 and AM08 are based on the theory of Sinai–Ruelle–Bowen probability measures, which commonly describe the equilibrium state of such dynamical systems. The ST-FDT algorithm takes into account the fact that the dynamics of chaotic nonlinear forced-dissipative systems often reside on chaotic fractal attractors, where the classical quasi-Gaussian formula of the fluctuation–dissipation theorem often fails to produce satisfactory response prediction, especially in dynamical regimes with weak and moderate chaos and slower mixing. It has been discovered in AM07 and AM08 that the ST-FDT algorithm is an extremely precise response approximation for short response times but is numerically unstable for longer response times. On the other hand, the classical qG-FDT method is numerically stable for all times; however, for a non-Gaussian climate it can produce significant errors at intermediate response times despite its guaranteed accuracy at extremely short times (MAG05). The blended ST/qG-FDT method employs the ST-FDT method at short response times and the qG-FDT method at longer response times, thus combining the advantages of both methods. The new response algorithms were tested in AM07 for the 40-mode L96 model in a wide range of dynamical regimes varying from weakly to strongly chaotic and to fully turbulent for both linear and quadratic functionals. The results in AM07 demonstrated a remarkably high level of precision for the blended response algorithms in all regimes for the L96 test bed for both linear and quadratic functionals. Furthermore, the ST/qG-FDT algorithm utilizes a linear tangent model at short times to calculate the response. All of these facts from AM07 suggest that the ST/qG-FDT algorithm is potentially useful in operational long-term climate change predictions for both linear and nonlinear response functions.

Here we test the new blended ST/qG-FDT response algorithm on the T21 barotropic model on the sphere with realistic topography and two different constant forcing fields, which are chosen to mimic the dynamical properties of the actual airflow at 500- and 300-hPa geopotential height. The same model and regimes were used by Abramov et al. (2005) to study the predictability for climate low-frequency variability. Here we address the response of the low-frequency teleconnection patterns in these models to external forcing. The 300-hPa constant forcing is developed by Franzke et al. (2005) and has a dynamically robust regime with a nearly Gaussian equilibrium state and rapidly mixing time autocorrelation functions. On the other hand, the 500-hPa constant forcing, developed originally by Selten (1995), has a strongly non-Gaussian equilibrium state and slowly decaying time autocorrelation functions. The dynamical system behavior in these two regimes of the model here roughly corresponds to the two extremes of behavior at low frequencies in more complex contemporary climate models. We test both the new blended ST/qG-FDT and classical qG-FDT climate response algorithms in these two different dynamical regimes and verify their predictions against the directly measured ideal response (Gritsun and Dymnikov 1999; MAG05; AM07; AM08), demonstrating the high skill and general superiority of the blended ST/qG-FDT algorithm in predicting the climate response in both situations.

Among other observations, a remarkable result here is the apparent insensitivity of the directly measured ideal low-frequency response to the structural changes of the equilibrium statistical state under changed external forcing. The T21 barotropic model, being a complex chaotic nonlinear dynamical system, is obviously structurally unstable in a geometric sense, which means that the modifications of external forcing parameters, no matter how small, cause structural changes in the topology of the statistical equilibrium state. It is impossible to discover these structural changes “remotely” through the tangent approximation, offered by the ST/qG-FDT because such bifurcations can only be captured by “running into them” (i.e., through the directly perturbed ideal response method). Yet, as presented below, the ideal response generally quite closely follows the ST/qG-FDT prediction. This behavior suggests that the climate response of simple linear and nonlinear response functions is not sensitive to the structural changes in a complex nonlinear climate state and can be accurately described by an FDT-type response approximation. The results presented here and in AM07 with the new blended response algorithm very strongly contrast with the pessimistic viewpoint advocated recently by McWilliams (2007) regarding predictive skill due to structural instability of climate models.

2. Climate response algorithms

Here we proceed under the general assumption that the climate model is numerically represented as a chaotic nonlinear dynamical system of ordinary differential equations for a state vector x ∈ ℝ^N, given by

where x in our case is a vector of principal components (PCs) corresponding to the kinetic energy EOFs. Let A(x) be a linear or nonlinear response function that is a part of the model climatology (in the climate response tests below, A(x) is set to represent the response of the climate mean state or its variance). The average value 〈A〉 of A(x) for the dynamics in (1) is computed as a time average along a single long-term trajectory of (1):

We assume that the dynamical system in (1) is perturbed by small external forcing as

In our framework, the external forcing represents the collection of altered geophysical parameters (such as solar radiative forcing) that cause climate change. Apparently, the solution y(t) will cause 〈A〉 to deviate from its original value, which represents the climate response to change in the external parameters:

The formula in (4) provides a straightforward estimate for the climate response, based on the distance between two different numerically simulated long-term trajectories of the perturbed and unperturbed models, respectively. However, it requires separate long-term numerical simulation with the climate model for each perturbation, which can be numerically expensive in the case of multiple climate change scenarios. Also, it computes the climate response at infinite time, and provides no information about transient time-dependent behavior of climate response when the changes in external forcing are applied to the dynamics.

Provide that the change in external forcing is small, the fluctuation–dissipation theorem offers a more versatile way to construct an approximation for δ〈A〉:

where t = 0 is the initial time when the forcing change is introduced into the dynamical system. Here, R(t) is the response operator, which directly relates the changes in external parameters δf to the climate response δ〈A〉. Note that R(t) does not depend on the magnitude and direction of δf and, once computed, provides the climate response for a wide range of magnitudes and directions δf. Also, one can study the singular directions of R to determine which changes in δf will result in a most catastrophic (or otherwise, most insignificant) response of δ〈A〉 (Gritsun and Dymnikov 1999; MAG05). Additionally, the formula in (5) provides the time-dependent climate response for finite values of t, which allows the study of transient climate change effects. In the situation when the forcing δf is a constant (which is “turned on” at time t = 0) and does not depend on space and time, a simplified version of (5) is used:

In the numerical experiments throughout the paper, a constant forcing δf is used.

Evaluating the response operator R(t) or its constant forcing counterpart R(t) is a nontrivial problem. Below we outline several mathematical and numerical methods to compute suitable approximations for R in a variety of dynamical regimes.

a. Quasi-Gaussian FDT method

Under the assumption of nearly Gaussian probability distribution with mean state x and covariance matrix σ² for the long-term climatological equilibrium, the response operator R(t) is evaluated as a time autocorrelation function (Deker and Haake 1975; Risken 1989; MAG05; AM07; AM08; Gritsun et al. 2008):

The formula in (7) is the quasi-Gaussian FDT response operator. When x is a vector of PCs, its climatology has zero mean state and identity covariance matrix, whereas the response of the climate mean state A(x) = x has a simple form:

The response formula in (8) was originally suggested by Leith (1975).

The main advantage of the quasi-Gaussian formula in (7) is its relative simplicity and low computational cost for a simple response function A(x), such as the response of the mean state or its variance. However, it is efficient only for dynamical regimes with a nearly Gaussian equilibrium state and robust mixing dynamics (MAG05; AM07; AM08). For a strongly non-Gaussian climatology with slow decay of correlation functions, the qG-FDT response algorithm rapidly loses its precision.

b. Short-time FDT method

A wide variety of practical geophysical models are complex nonlinear chaotic forced-dissipative dynamical systems with statistical attractors that are not smooth. The equilibrium states on such attractors are typically Sinai–Ruelle–Bowen probability measures (for details, see Eckmann and Ruelle 1985; Ruelle 1997, 1998; Young 2002). AM07 and AM08 reworked the FDT formula in such a way that an approximation to the equilibrium state is not required to compute the response, thus adapting the response formula to an arbitrary equilibrium state. This method is exact for a general dynamical system and is called short-time FDT because of its inherent numerical instability at longer times for chaotic dynamical systems. With this method, the time autocorrelation function is given by

where is a tangent map at x to t, which is computed by solving the equation

The ST-FDT response formula in (9) provides the precise response approximation for an arbitrary climatological equilibrium state, regardless of the dynamical robustness and mixing. The main drawback of the ST-FDT method is that the unstable Lyapunov directions of the tangent map in (10) grow exponentially in time and create numerical instability in (9) for long response times. Also, the ST-FDT method is computationally more expensive than the qG-FDT method. The detailed description of the ST-FDT method and its numerical implementation for computing (9) and (10) are given in AM07 and AM08 with illustrative detailed applications.

c. Blended ST/qG-FDT method

In the previous two sections we observed that the ST-FDT response operator and the qG-FDT approximation have somewhat opposite advantages and drawbacks. Whereas for shorter response times the ST-FDT approximation is superior to the qG-FDT response operator because of its precise geometric interpretation of the response formula in (9), it eventually blows up at longer times because of exponential growth in positive Lyapunov directions. On the other hand, the qG-FDT response operator is free of numerical instabilities, but it produces larger errors for weakly mixing non-Gaussian regimes. However, there is a simple way to avoid the blowup and increase the overall accuracy by blending the ST-FDT response with the qG-FDT response at later times before the numerical blowup occurs, thus combining the best properties of both responses through the following formula:

where ξ = ξ(t) is a blending function, chosen so that it is zero for small values of t, and thus the response at early times is computed through the highly accurate R_ST operator; however, ξ(t) assumes the value of 1 shortly before the numerical instability manifests itself in R_ST. The simplest and most straightforward choice of the blending function is the Heaviside step function

where T_cutoff is the cutoff time chosen just before the numerical instability occurs in R_ST. The blended response operator R_ST/qG combines the advantages of the ST-FDT for shorter times and qG-FDT for longer times. The Heaviside step-function (12) is used as a blending function for (11) throughout the paper and is developed and tested extensively in AM07. In particular, the universal empirical strategy T_cutoff = 3T_lyap, where T_lyap is the largest Lyapunov exponent, is utilized in AM07 for the L96 model with widely varying external forcing parameters for linear and quadratic functionals and yields excellent universal response performance. Note, however, that the choice of the blending function ξ(t) is not limited to the Heaviside function, and in fact can be chosen systematically to minimize errors between the ideal and predicted response. Detailed strategies for (12) are discussed in the application below for the T21 model.

3. Climatology of the T21 barotropic model

The model equations are

The physical units for the model length and time are meters and days, respectively. The model variables are as follows: ϕ, θ, and t are the longitudinal, latitudinal, and time coordinates, respectively; ψ = ψ(ϕ, θ, t) is the streamfunction; ζ = ζ(ϕ, θ, t) = Δψ is the relative vorticity; q = q (ϕ, θ, t) is the potential vorticity; and h = h(ϕ, θ) is the realistic earth topography (shown in Fig. 1) scaled by the height of the troposphere (H = 10 000 m). The physical parameters are the following: τ = 15 days is the Ekman friction time scale, Ω = 2π radian day⁻¹ is the angular velocity of the earth’s rotation, and the scale-selective damping coefficient K = 3.38 × 10⁻⁹ m⁶ day⁻¹ is chosen so that the damping time scale for wavenumber 21 is 3 h.

The model equations in (13) are projected onto the triangular-shaped array of spherical harmonics with maximum wavenumber 21 (standard T21 barotropic truncation). The T21 truncation is further restricted to the Northern Hemisphere with no flow across the equator; therefore, the truncated flow is completely described by 231 spectral coefficients. The standard fourth-order Runge–Kutta method is used to integrate the model in time.

The function ζ* = ζ*(ϕ, θ) is the time-independent forcing, which is chosen to roughly match the properties of model climatology with observations. In this work, two different types of steady forcing are used in the barotropic model to mimic a realistic climatology at 500- and 300-hPa heights in the troposphere.

For the quadratically nonlinear T21 barotropic model, the tangent map equation needed in (10) for the blended response algorithm is the linearized model truncation

for which the numerical implementation is readily available through a straightforward modification of (13). In (14), the operators ∇T, ΔT, ∇^⊥ΔT, and Δ³T are to be understood in the sense of their finite spectral representations applied to the columns of the tangent map T. In our setting, the tangent map Eq. (14) is solved using the fourth-order Runge–Kutta method with the same time step as for the T21 barotropic model itself.

a. Barotropic model climate for 500-hPa height

This type of climate is proposed by Selten (1995). Here, ψ_CL denotes the climatological mean field computed from 10 winters from European Centre for Medium-Range Weather Forecasts (ECMWF) daily analysis of 300-hPa vorticity fields rescaled to 500 hPa via multiplication by a factor of 0.6. The first guess for the forcing is determined through the formula proposed by Roads (1987):

where ψ′_CL and ζ′_CL are the deviations of the 10-day running mean from the climatological mean state and the angle brackets denote the time average. Then, model runs for 2000 days are performed, and each subsequent iteration of forcing is obtained by replacing the previous transient eddy forcing with the one from the current run:

The third iteration of (16) is implemented in the model and shown in Fig. 1.

b. Barotropic model climate for 300-hPa height

This type of climate is proposed by Franzke et al. (2005). Here the same observational data is used as in 500-hPa climate, but no additional rescaling is performed. The first guess for the forcing is determined through the same formula as in (15). Model runs for 3600 days are performed so that each subsequent iteration of forcing is obtained by replacing the model climatological mean state and transient eddy forcing from the previous run with the observed climatological mean state and transient eddy forcing:

The forcing is implemented after 30 iterations and shown in Fig. 1.

4. Low-frequency climate response for the mean state

Here we present the response of the climate mean state to changes in external forcing, which corresponds to the response function A(x) = x, where x is the vector of the principal components, corresponding to the kinetic energy EOFs. Both the qG-FDT and blended ST/qG-FDT response operators are computed along a single 1000-yr-long solution of the T21 barotropic model (13). The time discretization step used in the Runge–Kutta routine is 4 h for the 500-hPa regime and 3 h for the 300-hPa regime. The qG-FDT and ST/qG-FDT climate response operators are verified against the ideal response operator, which is computed by performing a series of runs with a 10 000-member ensemble solution that is directly perturbed by small changes in forcing. Forcing magnitudes with a strength that is 2% of the climate forcing are utilized to calculated this ideal response by direct numerical experiment with the barotropic model. These experiments are supplemented with others involving weaker and stronger forcings of strengths of 1% and 5% as checks for statistical accuracy. A detailed description of the computational algorithm for the ideal response operator and its performance is given in Gritsun and Dymnikov (1999), MAG05, AM07, and AM08. Both the forcing and response are considered in the kinetic energy EOF space; that is, the forcing is applied to a certain PC, and then the mean response is measured in terms of average PC deviations. Although both the qG-FDT and ST/qG-FDT response operators are automatically computed for the complete EOF space, here we demonstrate only parts of the FDT response operators that correspond to the low-frequency subspace of EOFs 1–4, which are the main focus here in a climate change scenario. The relative errors in figures and tables are computed as the ratio of the L₂-error norm between the ideal and the corresponding FDT operator to the norm of the ideal response operator itself. There is the usual caveat in implementing errors in the table that when the ideal response in a particular EOF is itself small, then such relative errors are less meaningful and high skill of the FDT response only requires a similar small magnitude and trend.

5. Low-frequency climate response for the variance

Here we present the response of the climate variance to changes in external forcing, which corresponds to the response function A_i(x) = x_i², where x is the vector of the PCs corresponding to the kinetic energy EOFs. Both the qG-FDT and blended ST/qG-FDT response operators as well as the ideal response operator are computed in exactly the same fashion as described at the beginning of section 4. Similar to section 4 for the mean state response, here the results are shown for PCs 1–4. The relative errors in figures and tables are computed as the ratio of the L₂ error norm between the ideal and corresponding FDT operator to the norm of the ideal response operator itself. Here, however, the actual response of the variances in the barotropic model is much smaller than the mean state response in both climates, probably because of the lack of baroclinic instability in the basic model; thus, a large relative error does not necessarily indicate a lack of skill in the variance response.

6. Conclusions

In two recent papers (AM07; AM08) the authors developed a new blended ST/qG-FDT response algorithm for predicting the climate response of a nonlinear chaotic forced-dissipative system to small external forcing. This new algorithm combines the geometrically exact general ST-FDT response formula at short response times using integration of a linear tangent model and the classical qG-FDT response algorithm at longer response times. This blended algorithm overcomes the inherent numerical instability in the ST-FDT formula arising because of positive Lyapunov exponents at longer times while delivering consistently better response prediction in the observed response time range. Section 2 describes the ST/qG-FDT and qG-FDT climate response algorithms.

The goal here is to develop a skillful algorithm to study the low-frequency climate response to external forcing through FDT algorithms in a prototype idealized atmospheric model. In the work here we tested the new blended algorithm on the T21 barotropic truncation on the sphere with realistic topography in two dynamical regimes corresponding to the mean climate behavior at 300- and 500-hPa geopotential height. It is shown in section 3 that these two regimes have somewhat opposite statistical dynamical properties: whereas the 300-hPa regime has robust strongly mixing behavior with nearly Gaussian equilibrium state distribution, the 500-hPa regime is weakly chaotic with slowly decaying time autocorrelation functions and strongly non-Gaussian climatology. The choice of these two different dynamical regimes (which both mimic realistic climate dynamics at different tropospheric heights) allows for an extensive test of both ST/qG-FDT and qG-FDT methods. In sections 4 and 5 the ST/qG-FDT and qG-FDT response algorithms are tested for the T21 barotropic model with realistic topography in the 300- and 500-hPa climate regimes. It is found that the blended ST/qG-FDT algorithm is highly skillful and generally superior to the classical qG-FDT algorithm for the observed time range of the response. Unlike the qG-FDT, the accuracy of the ST/qG-FDT blended algorithm does not deteriorate for the response of linear and quadratic functions for both dynamical regimes of weakly and strongly chaotic behavior because the ST/qG-FDT is represented by a much more precise geometric response formula for early response times, when significant response errors can occur. Furthermore, the response of the T21 barotropic model for the studied response functions does not seem to be affected by the model’s structural instability in a precise geometric sense; that is, the directly perturbed ideal response generally follows the tangent ST-FDT prediction at early response times for the climate mean state and variance response. At later times, the errors between the ideal and FDT response cannot be clearly attributed to the structural instability because of the presence of other error-producing factors, such as the developing nonlinearity of the ideal response and errors in finite time sampling of numerical integration of the FDT response operators. The main additional overhead of the ST/qG-FDT computation is the tangent map, which puts the blended ST/qG-FDT formula between the classical qG-FDT response and the full ideal response in terms of computational expense. The results here demonstrate significant skill for the blended algorithm on two different types of large-dimensional dynamical systems that sit at the extremes of statistical behavior in low-frequency regimes for contemporary climate models. Thus, these results suggest the use of the ST/qG-FDT response algorithm for predicting the low-frequency climate response of complex nonlinear geophysical models and for assessing their structural stability at low frequencies. Furthermore, the predicted response of the mean and covariance can be used to recalibrate the EOFs for nonequilibrium perturbed dynamics.

Future research in this direction will be centered on developing systematic adaptive algorithms for choosing optimal blending functions for the climate response in a variety of practical settings, as well as testing the blended response algorithms for more sophisticated climate models with more realistic large-scale features of the real atmosphere, ocean, and coupled atmosphere–ocean. This is possible because the new blended algorithms utilize integration of a linear tangent model for the short time response in a precise fashion (AM07; AM08) and such numerical algorithms are already available in many climate models. It remains to be seen whether less precise algorithms for the short time response introduce significant numerical errors that will swamp the skill demonstrated here on the barotropic model. To apply the ST/qG-FDT formula for the present climate models, an important extension of the FDT needs to be made to incorporate nonautonomous seasonal variations; such an extension has been developed recently by Majda and Wang (2008, manuscript submitted to Comm. Math. Sci.) In addition, climate models have nondifferentiable features due to moist convection processes, which usually occur on much shorter time scales. Thus, there is a possibility of applying blended response algorithms to reduced stochastic models for low-frequency components of the system.