## 1. Introduction

The attribution of climate change to a particular forcing agent (e.g., greenhouse gases, aerosols, volcanic activity, land use) is a problem of great significance in climate science. The most established systematic methodology of climate change detection and attribution is the “fingerprint” method (Hasselmann 1993; Hegerl et al. 1996; Hasselmann 1997), which is a statistical method of relating the observed climate response to model responses. In this approach, one needs the responses of many climate model experiments with different forcing functions corresponding to known drivers of climate change. A difficulty with this approach is that one needs to specify a priori the candidate forcing functions responsible for climate change when conducting the model experiments. To overcome this difficulty, we focus instead in this study on inverse methods. In an inverse method, the dynamics of the model enables one to directly calculate the forcing function responsible for a given climatic response. Given a sufficiently comprehensive model, the attribution problem is then reduced to identifying the agents responsible for a given forcing-function pattern and to quantify their relative contributions to the pattern.

In a typical (forward projection) climate model one simulates the response of the climate to a perturbation in known external forcing agents. In contrast, in an inverse model one infers the perturbation in external forcing from the observed climatic anomaly. This problem is complicated by the two-way interaction of the large-scale circulation and weather systems through heat and momentum fluxes, or potential vorticity fluxes, resulting in vacillation cycles of growth and decay (Frederiksen 1981). At steady state the mean circulation is balanced not only by the external forcing functions, but also by the eddy fluxes or second-order moments. Formally, the problem is equivalent to the statistical dynamical closure problem of turbulence where determination of the moment of a given order requires knowledge of moments of higher order. This implies that it is not sufficient to know the climate anomaly to determine the anomalous mean forcing function, but information about the climate variability is also needed. In other words, the full time series dataset is required, and this leads to a computationally expensive task.

*δ*〈

**f**〉 which implies thatwhere

*δ*〈

**x**〉 is a small change in the climate variable 〈

**x**〉. For a temporally constant perturbation, the sensitivity matrix can be written aswhere

*t*is the time and

*t*,

*s*) is the response function matrix, which describes the response of the system at time

*t*to a unit impulse at time

*s*. If the response function matrix can be estimated, then it becomes possible to efficiently calculate the forcing functions for potentially large numbers of possible climate perturbations—something that is impractical with numerical simulations of the circulation. The fluctuation–dissipation theorem (FDT) provides one approach to estimating the response function matrix. In this approach, suggested by Leith (1975),where

The fluctuation dissipation theorem has a long history in both classical and quantum physics (Kubo 1966). For many quantum systems the theory involves normal, often unitary, operators and the FDT is exact. In contrast, classical systems, which involve forcing and dissipation, are often described by operators that are not self-adjoint (Martin et al. 1973) and the FDT only holds approximately. Kraichnan (1959a) showed that the FDT is satisfied, in thermodynamical equilibrium, by general classical nonlinear dynamical systems. Further applications and extensions of his work were made by Leith (1975), Deker and Haake (1975), Bell (1980), Carnevale and Frederiksen (1983), and Carnevale et al. (1991). Carnevale and Frederiksen (1983) noted that Kraichnan's (1959b) direct interaction approximation (DIA) closure satisfies the FDT only at canonical equilibrium while it applies to simple linear systems in general. Herring's (1965) self-consistent field theory (SCFT) and McComb's (1974, 1990) local energy transfer (LET) have subsequently been shown to differ from the DIA only in how a FDT is employed (Frederiksen et al. 1994; Kiyani and McComb 2004). Interestingly, McComb's LET closure for homogeneous turbulence expresses the response function in terms of the two-time and single-time covariances in the same way as in Leith's approach for calculating the forcing function responsible for an anomalous climate state. The studies of McComb and Shanmugasundaram (1984) and McComb (1990) for three-dimensional turbulence and Frederiksen et al. (1994) and Frederiksen and Davies (2000) for two-dimensional turbulence assess the validity of the FDT through comparisons of the LET closure with the DIA, which does not employ the FDT, and with direct numerical simulations.

Recently, FDT-based methods of the type employed by Leith (1975) and Bell (1980) have been applied with some success to studies of atmospheric flows by Gritsun (2001), Gritsun and Branstator (2007), and Gritsun et al. (2008) although the flows are not strictly Gaussian. FDT-based methods that do not invoke the Gaussian assumption have also been constructed and successfully employed in atmospheric (Abramov and Majda 2008, 2009) and oceanic flows (Abramov and Majda 2012). Both the Gaussian and more general FDT-based methods have been extended so as to apply to variance response problems as well as mean response problems (Gritsun et al. 2008; Abramov and Majda 2009, 2012). Recently, the method has also been extended so as to apply to flows with a seasonal cycle (Majda and Wang 2010; Gritsun 2010).

In this study, in contrast to FDT-based methods, we present a method that closes the statistical equations for the climate by expressing the second-order moments in terms of the mean-field circulation. We are guided by the quasi-diagonal direct interaction approximation (QDIA) closure (Frederiksen 1999, 2012a) theory for the statistical dynamics of inhomogeneous flows. The QDIA may be derived from the more complex self-energy (SE) closure theory (Frederiksen 2012b) and has the property that the off-diagonal elements of the covariance and response functions in spectral space are expressed in terms of the diagonal elements and the mean field and topography. This makes the QDIA closure computationally tractable. It also results in expressions for the nonlinear damping in the mean-field equation being homogeneous in the horizontal. As reviewed by Frederiksen et al. (2012), the QDIA closure has been extensively tested and successfully applied to problems in dynamics, predictability, data assimilation, and subgrid modeling. In particular, it has been applied to strongly turbulent flows (O'Kane and Frederiksen 2004) and other far from equilibrium processes like Rossby wave dispersion in a turbulent environment (Frederiksen and O'Kane 2005). It should be mentioned that Jin et al. (2006a,b) have also proposed and applied a representation of vorticity transport by transient synoptic eddies that is linear in the mean field (but without the eddy-topographic force contribution) although this was not based on statistical dynamical closure theory.

In this paper we are guided in our approach by the structure of the QDIA equations. However, rather than solving the complex closure equations directly, here we close the equation for the mean climate by using the direct numerical simulation (DNS) statistics to express the second-order moments in terms of the mean field. As in the QDIA, the second-order moments are expressed as linear functions of the mean field but here the coefficients in the linear fit are obtained by least squares regression (in the QDIA the coefficients are functions of the transient covariances). This amounts to calculating a sensitivity matrix not for the mean forcing function 〈**f**〉, but for the second-moment terms 〈**T**〉. This sensitivity matrix, unlike in the fluctuation–dissipation theorem, is not related to lagged covariances but is instead directly calculated from the response of the system to numerous small climatic perturbations. Our approach is general and relatively simple to implement compared with solving the computationally demanding QDIA closure equations. Furthermore, unlike the FDT, which is a statistical linearization of the nonlinear equations and therefore assumes a linear response of climatic perturbations to forcing perturbations, it retains nonlinearity through the mean-field Jacobian terms.

This paper is structured as follows. In section 2 we outline our inverse modeling methodologies—namely, the iterative and closure methods—in their most general form. In section 3, we introduce the two-level quasigeostrophic (QG) model that is the basis of all the numerical experiments in this study. In section 4, we present some results employing perturbations of the July 1949–68 basic climate state. These perturbations consist of conical thermal sources in the Northern Hemisphere. We reconstruct the thermal sources from model climatology in spectral space using the methods outlined in section 2. In section 5, we experiment with multiple conical sources at different geographical locations. Finally, in section 6 we summarize the results and present our conclusions.

## 2. Methodologies for inverting climate models

*x*(

^{j}*λ*,

*μ*,

*t*) represent a prognostic field variable on the sphere, where

*λ*is the longitude,

*μ*is the sine of the latitude,

*t*represents time, and

*j*= 1, 2, 3, … is an index that represents vertically discretized fields such as vorticity, divergence, and temperature. The model that describes the evolution of the prognostic fields may also be horizontally discretized by expanding the fields in terms of spherical harmonics:where the double index

*mn*under the summation sign is a shorthand notation for summation over

*m*and

*n*. The gridpoint fields

*x*(

^{j}*λ*,

*μ*,

*t*) are then transformed into the spectral fields

*m*and total wavenumber

*n*describing horizontal degrees of freedom instead of the latitude and longitude;

*φ*(

^{j}*λ*,

*μ*) (for example, topography),

^{jkl}(

*m*,

*n*;

*p*,

*q*;

*r*,

*s*) and

^{jkl}(

*m*,

*n*;

*p*,

*q*;

*r*,

*s*) are interaction coefficients,

*p*and

*r*label the zonal wavenumbers while the indices

*q*and

*s*label the total wavenumbers in spectral space. The indices

*k*and

*l*label the distinct fields in the model at discrete vertical levels. The coefficients

*J*×

*J*matrix at each wavenumber, where

*J*is the number of distinct fields in the model.

### a. Iterative method

*κ*. That is,where

*γ*denotes an iteration parameter. The initial field

*κ*).

### b. Closure-based method

In this method, the contribution of the transient fields to the mean nonlinear Jacobian term is expressed in terms of the mean field, leading to a closed system at the mean-field level as shown below. Generally, we shall be using this method to calculate forcing functions corresponding to perturbations of a basic state. Hencewith, we shall work with the perturbations of a general variable *X* with respect to a basic-state variable *X*_{0}, denoted by *δX* = *X* − *X*_{0}.

*t*indicating that the generalized dissipation acting on the mean field and the mean forcing function in the model are modified by the transients) that we determine from the statistics of DNS. Note that in the QDIA closure the response function is inhomogeneous, but the associated nonlinear damping in the mean-field equation has the remarkable property of being homogeneous in the horizontal (Frederiksen 1999, 2012a).

We propose to calculate the coefficients from DNS as follows. Suppose we perform a (forward model) simulation using the mean forcing functions *s* = 1, 2, 3, … denotes distinct perturbations of the original climate state. Linear regression may then be used to determine

*r*indicates that the coefficient is renormalized (or redefined) and 〈

*x*(0)〉 corresponds to the basic-state climatology. Unlike Eq. (10), Eq. (15) is statistically closed in the sense that only the mean field is required to calculate the mean forcing and not higher-order moments. We examine the performance of Eq. (15) in relation to Eq. (10) to see how well the nonlinear mean-field tendencies are represented by the linear closure model represented by Eq. (14). We would expect that as long as the climate state in question is sufficiently close to the climate state used to calculate the linear regression coefficients

## 3. The two-level quasigeostrophic equations

*j*= 1 is the upper level and

*j*= 2 is the lower level. The (reduced) potential vorticity at level

*j*is defined aswhere

*ψ*are level-dependent streamfunctions,

^{j}*ζ*= ∇

^{j}^{2}

*ψ*are relative vorticities, and

^{j}*F*is the layer-coupling parameter. The latter is related to the internal Rossby radius of deformation

_{L}*r*

_{int}= (2

*F*)

_{L}^{−1/2}. The model formally includes drag at both levels specified by

*α*(in practice, however, the lower-level drag is substantially greater than the upper-level drag) and prescribed viscosities

^{j}*ρ*is a positive integer that describes the order of the Laplacian operator. The coefficient

*B*represents the beta effect. The equations are nondimensional, with Earth's radius as length scale and the inverse of Earth's angular velocity as a time scale; with these scalings

*B*= 2. The scaled topography is denoted by

*h*; it is given by

^{j}*h*= 2

^{j}*μH*/

^{j}*H*

_{0}, where

*H*is the topographic height and

^{j}*H*

_{0}is a scale height. We also allow for a simple representation of the effects of heating through a specification of the field

*κ*that controls how rapidly the flow is being relaxed toward this field. The parameterized heating creates a mean vertical shear in the flow that results in the formation of energetic eddies that are model representations of midlatitude weather systems.

*ψ*,

^{j}*ζ*,

^{j}*q*, and

^{j}*h*in terms of spherical harmonics as in Eq. (4):Here,In Eqs. (21) and (22), the interaction coefficient vanishes unless the selection rules

^{j}*m*+

*p*+

*r*= 0,

*n*+

*q*+

*s*equals an odd integer, and |

*q*−

*s*| <

*n*<

*q*+

*s*are satisfied. The real part of the complex operator

*x*with

^{j}*q*and

^{j}*φ*with

^{j}*h*and limiting the indices to

^{j}*j*,

*k*,

*l*=1, 2 in Eq. (5). Additionally,Here,

*δ*is the Kronecker delta function, which is unity if

^{jl}*l*=

*j*and otherwise vanishes, andMoreover, the matrix dissipation operator is given byand the bare forcing by

*c*= 1 + 2

_{n}*F*[

_{L}*n*(

*n*+ 1)]

^{−1}and

*ψ*

^{±}(

*λ*,

*μ*), using Eq. (4), and then using the definition of the zonal windThe spectral components of

*u*

^{±}(

*λ*,

*μ*)—namely,

*T*, the Jacobian term in potential vorticity space, in the same way as the flow field terms. The linear parameterization of the Jacobian term then still has the same generic form, as in Eq. (14), in all the three spaces.

^{j}## 4. A comparative study with a localized perturbation source

We choose the January 1949–68 climatology as a basic state for our studies. The iterative method described in section 2 is used to relax the simulation climatology toward this basic state. The parameters used in the quasigeostrophic models are as follows. The drag [*α*^{1}]^{−1} = 20 days and [*α*^{2}]^{−1} = 5 days; *ν*^{1} = *ν*^{2} = 1.55 × 10^{16} m^{4} s^{−1} and *ρ* = 2; *κ*^{−1} = 10 days; *F _{L}* = 3.125 × 10

^{−12}m

^{−2}. The model resolution is T31 and the global topography is closely similar to that shown in Frederiksen and Frederiksen (1993) with

*H*

_{0}= 8 km.

Figure 1 shows the resulting simulation climatology at statistical steady state, which is close to the observed January 1949–68 climatology (not shown). Having established a basic state approximately equal to an observed climate state we then introduce a perturbation in the forcing function. The systematic reconstruction of this anomalous forcing from the resulting anomalous climatology using the variety of techniques outlined in section 2 will occupy us for most of this study.

We perturb the simulated climate by specifying a thermal forcing perturbation of magnitude *p*; we shall experiment with different values of *p* in what follows. The spatial distribution of this thermal forcing, for *p* = 1, is shown in Fig. 2a. It can be seen to be an approximately conical cooling source that is centered at 40°N, 165°W with a central value of about −4.7 K day^{−1}. This thermal forcing is equivalent to a baroclinic potential vorticity forcing that follows from the thermal wind relation. The thermal forcing results in a thermal response, which is shown in Fig. 2b. This thermal response has a cooling of about −4.7 K near the center of the perturbation and is spread out over a larger region than the forcing function. Figure 3 shows the climatic zonal wind perturbation resulting from this perturbation. It leads to an increase of baroclinicity, and hence transient activity, southward of the center of the perturbation and a decrease of baroclinicity northward. Presumably, the increase of baroclinicity southward is also associated with the shift of the position of the barotropic jet due to nonlinear processes.

Having generated a perturbed climate state, we shall now focus on the techniques for reconstructing the perturbed forcing function associated with this perturbed climate state.

### a. Iterative method

The iterative method can be used to construct a simulation with approximately the same statistics as the simulation with a specified perturbation forcing function described above. We have experimented with a variety of integration times and number of iterations. Figure 4 shows the variation of the zonal wind forcing-function pattern correlation (with respect to the specified forcing function) *r* and its dependence on the number of iterations for both a 200- and a 2000-day integration per iteration step. In both cases, after a rapid initial growth in the pattern correlation with increase in the number of iterations, there is a saturation point at which the growth of the pattern correlation flattens out. This occurs after about 30–50 iterations. The short-integration-time case saturates at around the *r* = 0.9 level while the longer-integration-time case saturates at around the *r* = 0.95 level. Presumably, even longer integration times would yield higher saturation levels but the computational effort to achieve this would be considerable. Figure 5 shows the spatial patterns of the zonal wind forcing functions for both cases after 50 iterations. It demonstrates that the spatial pattern of the forcing-function perturbation is well captured, even for the short integration-time case.

### b. Closure-based method

Having established the performance of the iterative method with respect to the isolated source problem we now employ the closure-based method outlined in section 2. As discussed in the introduction, in this method the response of the mean field to the forcing is not completely linear because it retains the nonlinear mean-field Jacobian term in Eq. (15) and in this respect differs from the FDT method.

To calculate the model response to anomalous forcing, we generate an ensemble of 10 perturbed model climates by perturbing the relaxation fields with different values of the strength of the forcing perturbation *p*. We choose values of *p* in the range 0.525–0.75. For each member of this ensemble, we calculate the nonlinear eddy transfer terms *m* = 0) modes, which are much smaller in magnitude compared to the other modes.

Having obtained the aforementioned coefficients by sampling the ensemble of perturbations (with a limited range of the value *p*), we consider these coefficients to be quasi universal for small perturbations. Therefore they can be used to estimate the model response to perturbations with other values of *p*. If this is true then we have at our disposal a powerful method for estimating model responses to arbitrary perturbations. Figures 7 and 8 show how well this calculation of the model response performs when used to calculate the transient–transient Jacobian term for a perturbation of strength *p* = 0.5 directly from the perturbation mean field using these quasi-universal coefficients. We notice that the large-scale structures are well captured by this parameterization, consistent with the results of Fig. 6. Figure 9 compares the reconstructed forcing function, calculated from Eq. (15), with the actual forcing function used in the simulation; the pattern correlation is 0.93. Again, the larger-scale structures are well captured albeit with an underestimation of the peak magnitudes.

Figure 10 shows the result of the calculation when applied to climates with forcing perturbation of strength *p* = 1.0. The pattern correlation in this case is 0.91. The large-scale structures of the forcing function are again well captured albeit with a further underestimation of the magnitudes. We expect that for larger values of *p* the small change approximation implied by the form of Eq. (14) and the assumption of universal coefficients would cease to be valid. However, we see that even for *p* = 1, the method gives comparable results to those obtained with the iterative method as far as pattern correlations are concerned.

## 5. Experiments with test sources at arbitrary locations

### a. Single-location sampling

In this section we look at the performance of the closure-based method, initially again using the linear regression model coefficients calculated in the previous section. However, here we attempt to reconstruct test source functions placed at locations different from 40°N, 165°W, the location at which the ensemble of perturbations used to calculate the coefficients is located. This is a further test of the robustness of the method.

First, we consider a test source function placed at 40°S, 165°W. Figures 11 and 12 show, respectively, the barotropic and baroclinic components of the nonlinear transient tendency term, both actual and reconstructed. The pattern correlations between the actual and reconstructed terms are quite low (0.25 for the barotropic term and 0.44 for the baroclinic term). However, for the baroclinic term in particular, it is clear that the method is able to approximately replicate the main features of the pattern. It should be recalled that the original ensemble from which the coefficients have been calculated was formed from sources placed at a single location in the Northern Hemisphere, yet the reconstructed response is able to place the center of the response at the correct location in the Southern Hemisphere.

The results for the forcing function are shown in Fig. 13. The pattern correlation between the actual forcing function and the reconstructed one is 0.66. Even though the overall pattern correlation is inferior to pattern correlations obtained when the test source is at the same location as the ensemble of sources, this is still an impressive result given that the original ensemble used for sampling is in the Northern Hemisphere. The reason that the pattern correlation for the forcing function is greater than that for the nonlinear transient–transient tendency is that the other component of the nonlinear tendency—namely, the mean–mean Jacobian tendency term—is precisely calculated using the mean fields, and so the result is only partially dependent on the transient–transient term parameterization. This illustrates the advantage of a method that is partially nonlinear compared with a linear method such as the FDT method.

To explore this idea further we have placed test source functions at numerous other locations. Figure 14 shows the result for a source placed at 25°N, 30°E. The pattern correlation in this case is 0.59. The source pattern is still well replicated but there are weaker “false” source patterns at other locations, notably at the location where the sampling ensemble was placed. Figure 15 shows the result for a source placed at 50°N, 30°W. The reconstruction is much better in this case, giving a pattern correlation of 0.73. The pattern correlation is poorest at 0°, 180° (0.50), as shown in Fig. 16. Finally, Fig. 17 shows the result for two sources placed at 45°N, 90°E and 45°N, 90°W; the pattern correlation between the actual and reconstructed sources is 0.65.

As a final test of the universality of the coefficients calculated from an ensemble of sources at 40°N, 165°W, we construct a perturbation from National Centers for Environmental Prediction (NCEP) reanalysis data by calculating the difference of the climatological January data for the periods 1949–68 and 1975–94. We then use the iterative method described in section 2a to relax the baroclinic part of the flow toward this climatological perturbation, shown in Fig. 18. The correlation between these patterns is 0.93. The barotropic part of the flow is not forced so as to be consistent with the sampling method, which consists of purely baroclinic forcing functions.

The baroclinic climatic perturbation function as obtained by iteration is shown in Fig. 19a. The reconstructed version, using the closure method with coefficients calculated from the ensemble at 40°N, 165°W, is shown in Fig. 19b; the pattern correlation between these two figures is 0.66. This is quite an impressive result given that the sampling is done using an ensemble of isolated sources at a single location.

### b. Multiple-location sampling

In this subsection we attempt to improve the sampling by increasing the size of our ensemble. In particular, we look at the effect of sources placed at different locations as well having different strengths within the ensemble. We place thermal sources of the same basic form as described in section 4 at three different geographical locations (all in the Northern Hemisphere): 25°N, 30°E; 37.5°N, 180°; and 50°N, 30°W. The locations chosen are within a midlatitude band 25° wide and span over a wide range of longitudes. All of the potential vorticity sources are perturbed with 10 values of *p* in the range 0.975–0.525, bringing the total number of ensemble members to 30.

Figure 20 shows how the transient–transient perturbation Jacobian terms *m*, *n*) = (0, 2) (Fig. 6b, second row). Sampling a wider range has revealed that the trend has a negative slope rather than a positive one. The better-sampled coefficients

Figures 21a and 21b show the forcing functions of strengths *p* = 0.5 and *p* = 1.0, both located at 40°N, 165°W, which have been obtained in this manner. Note that in section 4 we also calculated the forcing functions corresponding to these perturbations but we used a 10-member ensemble obtained by perturbing the strength of a source at the same location (see Figs. 9 and 10). The range of the strengths of the perturbations was also narrower (0.75 ≥ *p* ≥ 0.525). If we compare the results of Fig. 21 and those of Figs. 9 and 10, we see that the former compare quite well with the latter. The pattern correlations are slightly lower, being 0.82 and 0.83, respectively. However, this is despite the fact that no information about the 40°N, 165°W source is included in the 30-member ensemble. This is an impressive result, for it is further evidence that the method yields good results for sources placed at arbitrary locations.

Figure 22a shows the reconstructed forcing function for the perturbation centered at 40°S, 165°W (Southern Hemisphere), this time with the 30-member ensemble. The correlation between this pattern and the actual forcing function (Fig. 13a) is 0.73. This is an improvement over the same calculation performed with model coefficients calculated with the 10-member single-location ensemble (Fig. 13b), which yielded a pattern correlation of 0.66. Similar or greater improvements in pattern correlations are observed for the remaining perturbations. The perturbation centered at 0°, 180° (Fig. 22b) has a pattern correlation of 0.67 with respect to the actual forcing function (Fig. 16a)—an improvement over the previous calculation, which yielded a correlation of only 0.5 (Fig. 16b). Additionally, the reconstruction of simultaneous sources placed at 45°N, 90°E and 45°N, 90°W (Fig. 22c) yields an improved pattern correlation of 0.74, up from the 0.65 obtained with the single-location samples (Fig. 17). Finally, the January (1975–94) minus (1949–68) climatic perturbation (Fig. 22d) now yields a pattern correlation of 0.76 with respect to the actual forcing function (Fig. 19a) as opposed to the 0.66 obtained previously (Fig. 19b). For the perturbations centered at 25°N, 30°E and 50°N, 30°W, pattern correlations of 0.90 and 0.94 were obtained, which are large improvements to the values of 0.59 and 0.73 that were obtained with the 10-member ensemble. Presumably, this is a result of sources at the same locations (but with different magnitudes) also being included in the sampling ensemble.

## 6. Summary and conclusions

In this study we have explored techniques for estimating the forcing functions responsible for climate perturbations. In general, this can be accomplished by a straightforward inversion procedure provided the full time series dataset of the perturbation is available. However, in many applications this is inconvenient or indeed impossible. For example, one might be interested to know what forcing is required to achieve some desired future climate state, in which case a dataset is not available.

It is with such issues in mind that we have investigated inverse models that do not require full datasets, but only require the mean (climate) perturbation state to estimate the forcing functions. Previous studies have attempted to construct such inverse models from the fluctuation–dissipation theorem (FDT), but we have chosen a different approach to the problem in the hope that some of the deficiencies observed when FDT-based methods are used might be overcome.

The most straightforward and reliable of these methods is the iterative method. This method employs a series of simulations in which the forcing function is continuously adjusted by the use of a term proportional to the difference between the target climate perturbation and the simulated climate perturbation. We found that we could successfully reconstruct the target perturbation at the 95% correlation level by using about 30–50 iterations, each employing about 5 years of simulated data. The difficulty with the iterative method is that this has to be repeated for each forcing function, corresponding to a given perturbation, that we wish to calculate. It is for this reason that we explore the other technique presented in this study—namely, the closure method.

The closure method has a similar philosophy to the FDT method in that it attempts to calculate the forcing function through the use of universal coefficients that only need to be calculated once for every conceivable small perturbation to the basic-state flow. It differs from the FDT in the manner by which these universal coefficients are calculated. In the closure method, an ensemble of climatic perturbations is sampled, while in the FDT-based methods, only the base climate is sampled. It should be noted, however, that in the FDT-based methods, the full covariance matrix in space and time needs to be calculated while in the closure method this is not required. Additionally, the mean-field response to the forcing retains nonlinearity through the mean–mean Jacobian term. The other component of the Jacobian term—namely, the transient–transient Jacobian term—is linearly parameterized in terms of the mean fields. The coefficients are calculated by sampling the ensemble of perturbations in which the mean fields and resulting Jacobian terms are varied. This enables us to calculate the coefficients of the linear model by linear regression.

The results obtained with the closure method are summarized in Table 1. We found that using a sampling ensemble consisting of isolated source function perturbations of different strength and at different locations yielded good results in calculating the forcing functions at arbitrary locations (at the 0.7–0.9 correlation levels with 30 samples), albeit with an underestimation of the peak magnitudes. This worked quite well (pattern correlation of 0.76) even when employing a perturbation calculated from NCEP data for the Northern Hemisphere winter, whose pattern resembles a superposition of isolated sources.

Pattern correlations obtained for various test perturbations using the closure-based method. Shown are results for the nonlinear tendency terms and forcing function. Additionally, sampling ensembles consisting of perturbation sources at a single location (10 members) and multiple locations (30 members) are considered.

The pattern correlations for the forcing functions are better than the pattern correlations for the nonlinear tendency terms in nearly all of the cases examined. This highlights the advantage of using a method in which the statistical linearization is only applied to part of the dynamical terms—namely, the transient–transient Jacobian terms. Even in cases where the parameterization performs poorly, as for the 40°S, 165°W source, the forcing function can still be estimated quite well. We notice that if sources at the same location as the test source, but not necessarily of the same magnitude, are included in the sampling ensemble, then the results are dramatically improved (up to the 0.9 correlation level for the 25°N, 30°E and 50°N, 30°W test sources). This suggests that with enough samples to cover the whole of the globe, forcing functions at arbitrary locations could be estimated extremely well. The systematic underestimation of the magnitudes of the forcing functions is presumably a result of the statistical linearization and is also noted in FDT-based results, as for example in Gritsun and Branstator (2007).

The closure-based method may be used as a low-cost alternative to the computationally costly but more accurate iterative method, and it is in this sense that we have discussed it in this paper. However, it can also be used in the context of the iterative method to give an improved initial estimate of the forcing function. This is demonstrated in Fig. 23, which shows how the correlation varies with number of iterations for calculations initiated from the basic state and from the closure-based calculation as applied to the NCEP data perturbation. After 10 iterations (and using 10 000 time steps per iteration) the calculation initiated from the forcing function as calculated by the closure-based method is 0.92 while the calculation initiated from the basic-state forcing function is only 0.7. In general it takes about 30 iterations to reach the 0.9 correlation level when the calculation is initiated from the basic-state forcing function. This further demonstrates the utility of the closure-based method.

Unlike the FDT-based methods discussed in the introduction, our method has been specifically formulated for solving the inverse modeling problem rather than the forward modeling problem. It is possible in principle to solve for the climate state in Eqs. (15) and (16) given the forcing function, which would constitute a solution to the forward problem. More difficult forward modeling problems such as estimating the variance response or applications to flows with a seasonal cycle go beyond the scope of the present paper. In future studies we hope to investigate these issues further and apply the current inverse modeling formalism to more complex circulation models.

The authors acknowledge support by the West Australian Department of Environment and Conservation under the Indian Ocean Climate Initiative Stage 3 and the Australian Climate Change Science Program of the Australian Department of Climate Change and Energy Efficiency.

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