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  • View in gallery

    The (left) strict surface-sensitivity analysis and (right) extended feedback loop. Vector gives the shape of the drifting atmospheric temperature profile.

  • View in gallery

    Fit of functions N(t) (green) and ΔTs(t) (blue): fit A (red) and B (black) (see text), and without the slowest exponential (thick black).

  • View in gallery

    Abrupt 4×CO2 regression: 3-yr average (red) and with fitted functions N(t) vs ΔTs(t) (green). Zoom indicated by arrow is on a hiatus period.

  • View in gallery

    Six atmospheric levels of the 12-month low-pass-filtered ratio ΔηkTs(t). From surface to stratosphere: level 17 (factor = 0.9; black), 14 (1.2; red), 11 (1.55; green), 8 (1.30; blue), 5 (0.2; cyan), and 2 (−1.9; magenta, shifted).

  • View in gallery

    The fitted step TLS model (red) applied to the 1pctCO2 experiment (blue). The asymptotic straight lines (green) and without the fast responses (magenta) are shown.

  • View in gallery

    Regression vs ΔTs with fitted functions (red), the influence of fast responses (green), and only the first ocean layer (blue).

  • View in gallery

    The inclusive water-cycle feedback system.

  • View in gallery

    Regression N (W m−2) vs ΔTs (K) with a 12-month low-pass filter (black lines) and monthly values (green line with points and blue crosses). Top-left curves are for Frcg and the bottom-right curves are for the W cycle.

  • View in gallery

    As in Fig. 8, but for evaporation (black) and total precipitation change (red) (mm day−1). The ΔPrecips/ΔTs ≃ 0.11 mm day−1 K−1. Blue line (6-month filtered) and symbols (monthly) are for evaporation minus precipitation.

  • View in gallery

    (top) Frcg: W-cycle (K K−1) and (bottom) atmospheric global temperature profiles for (right to left) 4, 8, 16, 32, 64, 128, and 256 days, and days 400 and 420 (enhanced symbol curves); Frcg begins at day 16. Temperature is scaled with surface global temperature change.

  • View in gallery

    Hovmöller diagram of monthly amount of deep-convection precipitation (mm day−1; contours) and its W-cycle response (shaded) in the intertropical zone (30°S–30°N). Contour interval is 1 mm day−1 and amounts range from 0 to 7 mm day−1.

  • View in gallery

    The W-cycle temperature IFC: year 12 (ΔTw; K; shaded) and forcing (ΔTfrcg; contours).

  • View in gallery

    The W-cycle response in net DC updrafts (shaded; kg m−2 h−1) and net downdrafts (contours).

  • View in gallery

    Monthly values (August, fourth year) of (bottom) ΔT and (top) Δq IFCs subcomponents of the W cycle. The forcing added to the three components— TRB, DC, and LSC—adds (Sum) to give the full water-cycle component (2×).

  • View in gallery

    Schematic implementation of the FGD double-call method.

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A Formal Analysis of the Feedback Concept in Climate Models. Part II: Tangent Linear Systems in GCMs

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  • 1 Laboratoire de Météorologie Dynamique, Paris, France
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Abstract

Climate analysis is greatly simplified in perturbation analysis when filtered anomalies show linear behavior. In the first part (part I) of this two-part analysis, the formal tangent linear system (TLS) that handles linear behavior was used to demonstrate the strict equivalence between feedback and sensitivity analysis but at the cost of reducing the generality of its application to GCMs. In this second part, the full feedback analysis is introduced from the application of the so-called regression method of Gregory et al. The authors give a complete example of its use in the global analysis of the phase 5 of the Coupled Model Intercomparison Project (CMIP5) abrupt 4×CO2 and ramp experiments. A simple 1D model with only two ocean layers is shown to be able to explain the slow climate warming of the next century. An extension of the formal results in part I allows a new perturbation method to be designed in GCMs to determine the TLS in models. A series of illustrations demonstrates the advantages of implementing such a method in GCMs.

Corresponding author address: Alain Lahellec, LMD/IPSL/Université Pierre et Marie Curie, Boite 99, 4 Place Jussieu, 75252 Paris CEDEX 05, France. E-mail: alain@lmd.jussieu.fr

Abstract

Climate analysis is greatly simplified in perturbation analysis when filtered anomalies show linear behavior. In the first part (part I) of this two-part analysis, the formal tangent linear system (TLS) that handles linear behavior was used to demonstrate the strict equivalence between feedback and sensitivity analysis but at the cost of reducing the generality of its application to GCMs. In this second part, the full feedback analysis is introduced from the application of the so-called regression method of Gregory et al. The authors give a complete example of its use in the global analysis of the phase 5 of the Coupled Model Intercomparison Project (CMIP5) abrupt 4×CO2 and ramp experiments. A simple 1D model with only two ocean layers is shown to be able to explain the slow climate warming of the next century. An extension of the formal results in part I allows a new perturbation method to be designed in GCMs to determine the TLS in models. A series of illustrations demonstrates the advantages of implementing such a method in GCMs.

Corresponding author address: Alain Lahellec, LMD/IPSL/Université Pierre et Marie Curie, Boite 99, 4 Place Jussieu, 75252 Paris CEDEX 05, France. E-mail: alain@lmd.jussieu.fr

1. Introduction

The ceaseless development of GCMs aiming to approach the complexity of climates has driven numerical models to become ever more complex themselves. The main difficulty stems from nonlinearity and the strong coupling between the elementary mechanisms represented in the models. Under these conditions, calibration of uncertain parameters, along with model verification and validation, has become a challenging task.

However, results of perturbation analysis bring some hope. In climate warming experiments, evidence has been found of linear behavior of the perturbed fields, suggesting that linear analysis can be applied to time- and scale-filtered variables. On the global scale, feedback analysis introduced in Hansen et al. (1984) and sensitivity factors from Wetherald and Manabe (1988) are based on linearity. Criticisms have nevertheless been voiced concerning the validity of these approaches, and some unfounded conclusions on climate sensitivity can indeed be found in the literature (e.g., Dessler 2010; Lindzen and Choi 2009, and comment articles). The methodological difficulty in separating climate forcing and rapid adjustment from feedback is also addressed in Stocker et al. (2014).1

It is mathematically difficult to justify a linear behavior from high-dimensional, nonlinear systems, because this calls for integro-differential functional techniques. In the first of these two papers (Lahellec and Dufresne 2013, hereafter Part I), we approached the problem from the opposite direction: we analyzed the tangent linear system (TLS) that represents the perturbed climate and proposed the assumption that, on large-enough time and spatial scales, a constant Jacobian matrix system can be used. We proposed some dynamical criteria to estimate the relevance of the TLS. The integration of this system should explain the transition from the unperturbed climate at equilibrium to the perturbed equilibrium state. When this is the case, we may conclude that the linear approach is valid.

The mathematical theory of linear systems provides numerous efficient methods. In particular, linearity allows climate sensitivity and feedbacks to be interpreted in a simple manner. At the scales where the TLS reveals itself, climate response can be decomposed into elementary perturbations, thus opening up the possibility of climate behavior becoming intelligible in terms of interactions between elementary mechanisms.

In Part I, we used the formal TLS to clarify a benchmark set of definitions and methods that are currently used to analyze climate feedbacks. We introduced the distinction between standard exclusive feedback analyses as obtained from an offline radiative computation technique and inclusive feedback as found in the standard suppression methods (e.g., Hall 2004; Hall and Manabe 1999; Hansen et al. 1997). The rigorous links between these methods were established with the severe limitation that the surface had to be at the origin of perturbations. We also stressed that speaking of feedback in loose terms can lead to difficulty in interpreting the results when the corresponding feedback loop is not clearly identified.

The dynamics of the TLS in the Laplace domain can be analyzed in a straightforward manner in terms of its characteristic times, and such an analysis allowed us to demonstrate that the regression method introduced by Gregory et al. (2004) gave correct sensitivity factors, the λ parameters, as soon as additional criteria were fulfilled. This can be checked in practice. In particular, we learned that when the correct filtering is applied, the atmosphere appears coupled to the drifting surface temperature; thus opening the way for a formal extension of our strict results of Part I. This second paper describes how we can extend feedback analysis to achieve the desired generality in GCMs.

After recalling the principal results of Part I in the next section, we discuss application in the GCM context in section 3. The formal extension of feedback decomposition necessitates a modification of the definition of feedback gains, as already noted by Lu and Cai (2009) in the CFRAM approach.

The main application concerns the phase 5 of the Coupled Model Intercomparison Project (CMIP5) idealized experiments; we investigate how our formal results can be used to derive unbiased feedback factors and how the conceptual linear model underlying these results could be made explicit. The abrupt 4×CO2 experiment is used to illustrate this issue and a comparison is made with the ramp forcing experiment.

So far, the formal TLS has been used to demonstrate the validity of classical methods. However, the limitations to filtered space and time scale using these methods call for more elaborate tools. For this purpose, we introduce a perturbation method based on the Gâteaux differential that enables the formally derived/ feedback characteristics to be obtained numerically in GCMs. A version of this method has been introduced in the Laboratoire de Météorologie Dynamique–Zoom, version 5 (LMDZ5), GCM and we give some of our first results in section 5. Finally, in the concluding section, we discuss the perspectives that our work has opened up and also explain the difficulties encountered in implementing the new method in LMDZ5.

2. Climate sensitivity and feedback

a. Tangent linear system

In Part I, the climate system was written as the general evolution equation
e1
where σ is the vector of the state variables. The diagonal matrix σ represents the inertia, and represents the external sources. Given initial conditions and sources, the (deterministic) system follows a given trajectory. In GCMs, σ classically represents temperature, specific humidity, wind field, etc. At this level of generality, the model can be full 3D or a single-column GCM as well as an mesoscale or global balance model. The tangent linear system of the climate model subjected to some forcing was introduced as
e2
where σ is the Jacobian matrix ∂σg. This system gives the deviation Δσ relative to the reference trajectory over time. On the integration time scale, σ and σ are dependent on the state, and while we qualify this TLS as circulating along the trajectory (CTLS), it is still a linear system because the Jacobian matrix depends on the reference trajectory only and not on the variables in Δσ. Section 5 will show how this CTLS can be implemented in GCM to analyze the climatic response Δσ(t) to .
In Part I, we simplified the problem thanks to low-pass filtering of the GCM results. When linearity results from such analysis at the global scale, a linear (1D) model can be inferred, which we can still write as Eq. (2), adding only that the matrices have constant entries—the CTLS is now the TLS. This hypothesis allows the Laplace transform of Eq. (2) to be used:
e3
where Δσ(μ) is the Laplace transform2 of Δσ(t). Now, the complexity of dealing with integro-differential systems is avoided and instead simple algebraic treatment can be used. We can reduce the dimension of the system by algebraically eliminating all other variables except temperature. This yields the energy system as
e4
The system obtained (the T system) incorporates all feedbacks from other variables in its reduced matrix (μ). In this system, is constant but was shown to vary because it inherits the inertia of all other mechanisms, such as from water vapor processes. The vector η represents temperatures.

Because climate analyses compare feedbacks with the specific Planck response,3 we separated matrix (μ) into and (μ), where represents longwave (LW) radiation exchange between layers, and represents the other climate mechanisms. Notice that , as representing instantaneous exchanges, does not depend on μ or on time.

b. Feedback loop

We introduced the feedback-loop structure in Eq. (4) to correspond to the radiative-feedback analysis hypothesis that “perturbations apply primarily to the surface.” This hypothesis was chosen because our purpose was to critically review the usual climate change analysis methods. As a consequence, an open-loop system representing the surface-Planck response is written as
e5
and the feedback-loop system equivalently written as
e6
This dual system is mathematically equivalent to Eq. (3) when Δη = Δηr + Δη0 and = . We also have a decomposition of matrix into , where , as the rightmost column of , represents all climate mechanisms that can immediately impact the troposphere from a surface forcing term. Hence represents all feedback mechanisms that are not directly affected by surface perturbations. The surface temperature response Δθs was defined as the global response from which the surface-Planck response Δθ0 is subtracted.

Let us describe how a feedback loop is implemented in the system. As sketched in Fig. 1 (left), the surface temperature response Δθs is extracted from the state vector as .4 It is easily verified that when Δθs is set to zero on the rhs of the first line of Eq. (6), it becomes equivalent to Eq. (5), the open-loop system. In Fig. 1, the surface forcing term Δθ0 propagates into the troposphere, which responds, and the perturbation provides feedback to surface, increasing Δθs. The feedback loop is closed when Δθs(t) is added to the forcing term. The loop is resumed in the expression of the gain in Eq. (8); reading the factors from right to left: propagates surface perturbations to atmosphere and provides feedbacks of the atmosphere response to surface processes.

Fig. 1.
Fig. 1.

The (left) strict surface-sensitivity analysis and (right) extended feedback loop. Vector gives the shape of the drifting atmospheric temperature profile.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0334.1

Equation (6) gives what is needed in climate sensitivity analyses: Δηr and Δθs are additional (feedback) responses to the Planck response (Δη0 and Δθ0). The solutions of Eq. (6) is
e7
where the feedback gain is given by
e8
e9
We called ρ the effective response function. In the Laplace domain, it is the response to the impulse function Δθ0δ(t).

c. Inclusive feedback components

The column matrix represents all the mechanisms linking surface to the atmosphere. It can be decomposed into different mechanisms—feedbacks— as partial elements of . Each of these mechanisms enters a feedback loop, the algebraic structure of which can be written in its forcing form as
e10
where represents the reaction of the system when the loop is opened—we dropped (μ) for clarity. In other words, i represents the whole system except that mechanism i, identified by , has no immediate influence on the response of the system to surface temperature perturbation. The solution reads
e11
where g*(μ) is the (inclusive) gain where one mechanism is associated with the feedback loop within the full system . Notice the difference with the elementary feedback gain from Eq. (8) that corresponds to a single active mechanism in the system, which we called exclusive gain, with the open-loop matrix . The solution to Eq. (6) gives an elementary warming component—that we called inclusive feedback component (IFC):
e12

Because all symbols represent functions of the Laplace variable μ, the preceding results are valid not only at equilibrium but also during transition, as soon as the time and spatial scales fit the relevance of the linear model. The analysis of the regression technique given below provides the criteria that have to be fulfilled.

d. Radiative sensitivity

Our second development in Part I served to link the feedback analysis with radiation sensitivity studies, including the partial radiative perturbation method (PRP). A simple decomposition of the Jacobian matrices allows the sensitivity factors λi to be introduced. We recall the notation
e13
which relates surface equations with the last row and column in matrices. With this notation, the feedback gain at equilibrium reads
e14
and the sensitivity factors
e15
Noting λ0 the surface-Planck sensitivity and the total leads to as the exclusive gains and as the inclusive responses. Thanks to the matrix decomposition, λi in Eq. (15) makes explicitly provides the feedback path from the atmosphere to the surface within a scalar expression: excitation of atmosphere from surface through , atmospheric response as the inverse matrix, and back to surface with closing the loop.

These are the main results we obtained showing that radiative sensitivity analysis and feedback analyses are consistent and share the same hypotheses. We finally justified the decomposition of temperature change into components (IFC) with the argument that the corresponding feedback loops can be built. This decomposition was already justified by Lu and Cai (2009) based on the fact that the top-of-the-atmosphere (TOA) budget can be decomposed into terms from different origins, in the same spirit as in Le Treut et al. (1994).

e. Regression

We finally analyzed the regression technique introduced by Gregory et al. (2004), which determines the global sensitivity factor λT from the relation we called (L2Obs):
e16
where f0 was shown to be the surface forcing term and the surface-layer thermal inertia. In addition to the linear hypothesis, Eq. (L2Obs) [Eq. (16)] relates the two global observables N—the incoming TOA radiation imbalance—and ΔTs over long time scales. We demonstrated that this relation was valid when the atmosphere was in quasi equilibrium with the surface. At these time scales, surface temperature drifts slowly and the atmosphere temperature follows: , where is the (constant) leading Lyapunov vector of the global TLS. In these conditions, the perturbed system evolution is known from the temperature at any given atmospheric level, which can be chosen as the surface ΔTs. Even though the forcing applies to the whole atmosphere, the surface forcing is still sufficient to dominate the feedback system. The ensuing extension of our formalism follows.

3. Application to analysis with GCMs

a. The attribution problem

So far, we have not treated in models, the explicit meaning of the forcing . The question is how the feedback mechanisms corresponding to can be identified (the attribution problem) and under what circumstances their effects can be added (the separation problem).

Clearly, the attribution problem is dependent on the mode and the way climate mechanisms are mathematically represented. We have found no better way to trace the feedback mechanisms in GCMs than that proposed by Lu and Cai (2009): vector Δη in our equations is the thermal flux budget of the GCM layers and is usually computed as a sum of separate parameterization contributions that give radiation budgets, turbulent and convective large-scale and mesoscale flows, their thermodynamic energy impacts, etc.

In such a sum of flux convergences: , each term formally corresponds to Δφi = iΔη, so that i, as part of the full matrix , can be associated with a mechanism, for example, a parameterization in the GCM, or a group of mechanisms following a feedback attribution sequence. Lu and Cai (2009) give further possible separations in 3D analysis. Usually, these parts in the sum can be identified with a sum of contributions to the evolution in temperature at each time step (“tendencies”) in the models, but this may be less obvious. In any case, the running sum ensures that the mechanisms are properly separated. The link with our formalism is now clear, because , it follows that is the way some specified mechanism immediately impacts the layer’s energy budget from a surface perturbation.

The preceding algebraic developments are based on the CTLS and can be numerically obtained using a model that actually computes system Jacobian matrices (e.g., Lahellec et al. 2008; Giering et al. 2003). GCMs in general do not compute these matrices explicitly. We will show that the CTLS can be formally obtained using the Gâteaux differential, with mathematical rational explained in Cacuci (1981) and Cacuci and Hall (1984). We just need a finite counterpart of this approach, which we call the finite Gâteaux differential (FGD) method, to numerically determine the feedback elementary factors from GCMs.

b. The method: Introducing finite Gâteaux differencing

Consider a model running along its reference trajectory, and in the same model we apply a perturbation Ts(t) + 1K only when calling a subprogram that takes a specific feedback mechanism into account. For instance, the LW radiation code is called with this perturbation, the rest of the code still seeing Ts(t). The result is a perturbation affecting the LW budget. The anomalies of this run are the solutions of a system analogous to the CTLS and should give the surface-Planck response Δθ0.

To be more specific, the perturbation amplitude should be large enough for its precision to overcome the numerical noise, but still small enough to preserve the linear relationship between perturbed variables. The linearity criterion is given mathematically in Cacuci (1981) as

  1. σ(t + dt) = [σ(t), Δx(t)] weakly Lipschitz (when close to equilibrium for instance);
  2. [σ(t) + ϵΔσ1 + ϵΔσ2] − [σ(t) + ϵΔσ1] − [σ + ϵΔσ2] + [σ(t)] = O(ϵ) (linearity).

Originally, is an operator on the nonlinear functions that compose the mathematical representation of the model—it is supposed to give the solution to Eq. (1). In numerical models such as GCMs, it represents the coded algorithms5 that compute the incremental evolution of the prognostic variables σ. The first criterion essentially concerns the uniqueness of the solution and its continuity when the initial conditions are moved. The linearity criterion allows anomalies to be determined and summed over different perturbation experiments. Neither of the criteria has any significant chance of being verified in the 3D fields. This is where the magic of averaging comes in: it allows a weak form of both criteria. In other words, if we apply the criteria to global averages of the anomalies, the linearity is known to apply, and can be checked. When the CTLS is integrated, that is, when the integration in time of the state variables ΔT, ΔQ for specific humidity, etc., are added to the standard calculation of the tendencies of T, Q, etc., the analysis of their global averaged trajectory should satisfy the required criteria, as long as climate bifurcation is not generated.

c. Extending sensitivity analysis in GCMs

In section 2, we used as the forcing term exciting the system from the surface, where is the rightmost column of matrix i. Perturbing the surface will immediately impact sensible and latent surface fluxes but, depending on the code, the impact is not obvious on other climatic mechanisms, such as deep convection or clouds. One can exemplify this facet by considering the vertical diffusion model in LMDZ, which represents all small-scale turbulent mechanisms: surface evaporation, surface-layer, boundary layer, and free-troposphere turbulent diffusion models (TRB; see section 5). The corresponding Jacobian matrix TRB is the three-diagonal diffusion matrix with turbulent conduction. The rightmost column of TRB, , has in addition the latent heat term from surface evaporation. It is actually the only Jacobian matrix that is filled as a matrix in this GCM; this is needed to apply the implicit time scheme for this model. So, represents the direct impact of the surface on the first atmospheric layer via sensible and latent fluxes. If full vertical turbulent exchanges are to be taken as a mechanism, the full matrix TRB must be included in the feedback. This is general in GCMs: a mechanism is represented by the full matrix i and we need to reformulate our results to take account of the direct impact of the forcing on the whole atmosphere, not only on the surface. Thanks to our Lyapunov argument concerning the slow climate response, after a transient evolution the troposphere is controlled by the drifting surface temperature. This provides an opportunity to extend our formalism to sensitivity analyses of climatic interest.

As an example, the full-Planck trajectory already computed in Part I [its Eq. (47)] corresponds to
e17
where the forcing Δx applies to the bulk of the atmosphere. Here, the open-loop matrix is the Planck matrix alone, contrary to Eq. (5) where it is b and hence includes the other tropospheric feedbacks—giving what we named the “surface Planck” response. We also need to constrain the tropospheric response to the surface response: , which states that, at the Lyapunov horizon, the tropospheric profiles only drift with the slow evolution of the surface temperature ( is constant6 for the slow Planck-response system).
With these assumptions, one can force the system with instead of . So instead of Eq. (10), the feedback system now reads
e18
Note that the Lyapunov horizon is also specified by writing the inertia matrix with the only remaining surface inertia . This is what matrix does.7 In this way, mechanisms that are not immediately sensitive to the surface are excited in the bulk of the troposphere. With the new decomposition of the system, a procedure analogous to the one applied to find Eq. (15) leads us to extend the sensitivity factors to include whole matrix i (see appendix A). The new expressions of the extended factors , given in appendix A, confirms what is intuitively obvious: the climate sensitivity to sensible and latent surface fluxes is at the very origin of most of the other climatic responses, such as deep and shallow convection and clouds, in addition to the radiative forcing. It also evidently conserves the global factor λT. This redistribution of sensitivity factors applies to the new response functions and consequently to the IFCs :
e19
that keeps the same form as in sections 2b and 2c.

With this extension, only valid for slow time scales, we have finally reached our goal of developing a numerical framework that is not as strict as our formal introduction but gives the possibility of analyzing the climate sensitivity through the mechanisms represented in the GCM codes. All the methods we have already reviewed are valid, from the PRP-equilibrium method to the regression technique. This constitutes the body of methods we shall illustrate in the next sections.

4. Application to analysis of CMIP5 experiments

We use the following analysis of the abrupt CMIP5 experiment as a test case to show how a rigorous procedure can ensure that a global analysis leads to relevant climate feedback parameters.

a. Dynamics

We analyze the LMDZ5A results of the abrupt 4×CO2 CMIP5 experiment (cf. Hourdin et al. 2013). Perturbations are computed as the difference between 140 yr of the perturbed trajectory and the average of the Pi-control trajectory with constant preindustrial CO2 concentration. Figure 2 gives the evolution of ΔTs and N (vertically shifted by +5.35) that are used in the L2Obs relation. It can be observed that one exponential function alone cannot fit the data. This is well known as far from equilibrium, some heat leaks to the deeper ocean—see Wigley and Schlesinger (1985), who introduced the “ocean heat uptake” to account for this feature. It is the reason why the TOA budget is that far from equilibrium after running 140 yr. We therefore need to complete our formal model with a deep ocean layer—note that this does not change the radiation budget part of our analysis. Supplementing our surface model [Eq. (16)] with this deep ocean layer means adding one equation to our model:
e20
where we have chosen κ as the opposite of the heat transfer coefficient for compatibility with the sign convention of the ocean heat uptake view (for brevity in this section, λ stands for our global factor λT).
Fig. 2.
Fig. 2.

Fit of functions N(t) (green) and ΔTs(t) (blue): fit A (red) and B (black) (see text), and without the slowest exponential (thick black).

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0334.1

The two e-folding times can be found by eliminating each variable in the other equation, which gives
e21
where .

We now have a modified e-folding time of for the mixed layer, with μτd ≫ 1. A longer e-folding time is introduced as .

We see that, as long as μτd ≫ 1, the surface temperature response tends to ; this gives the expression called “the transient climate response” (TCR) in Cubasch et al. (2001). The equilibrium response is (μτd → 0) ; on condition, however, that this two-ocean-layer system is valid up to equilibrium, which, as we shall see, is not the case. We still use ΔTeq in the following, meaning the asymptotic equilibrium temperature change is given by the two-layer model.

Details on the derivation of the solution ΔTs of Eq. (20) are given in appendix B—cf. Eq. (B2). It is found to be
e22
with the coefficients
e23
This verifies that b + c = 1, and hence .
Inversion to the time domain can be conceived as two operations: the first is the Laplace transform–time domain correspondence from symbolic to real space: ; then a first integration in time to obtain the response to the step forcing . We also apply a second integration to obtain the response to the ramp forcing for future use. The two responses are finally
e24
where S is the time for the ramp to reach the 4×CO2 forcing at 1% yr−1, and we have
e25
For the radiation budget
e26

The results of Fig. 2 will be shown to give a first ocean layer of ≃50 m and a deep layer of ≃600 m: the conditions are met for using the regression technique to determine the global sensitivity factor from the slope in Fig. 3. We find λT ≃ −0.85 W m−2 and f0 = 6.8 W m−2.

Fig. 3.
Fig. 3.

Abrupt 4×CO2 regression: 3-yr average (red) and with fitted functions N(t) vs ΔTs(t) (green). Zoom indicated by arrow is on a hiatus period.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0334.1

b. Regression

Coming to the evolution of the two variables, we have two exponential functions with e-folding times differing only by one order of magnitude, and we can apply the fit with different strategies. Consider the two functions to fit
e27
where N = −λTΔTsf0, and where we have introduced the two constants a and d to account for the fact that the L2Obs model might not verify ΔTs(0) = 0 because of “fast mechanisms” occurring before it reaches the Lyapunov horizon. The model could be forced to reach equilibrium by taking 1 − (d + e + f) = 0, which would lead to Eq. (26); equality could be imposed between the e-folding times, etc. Based on our argument concerning equality between the characteristic times in N(t) and ΔTs(t), we fit N + ΔTs (because |λ| ≃ 1) to determine the e-folding times and no other constraint. With these times, we fit the three other coefficients in each case (fit A). This provides the values seen in Table 1, where we also give the results for fit B, which, in addition, constrains the model to reach equilibrium [N(∞) = 0].
Table 1.

Parameters from the two fitting procedures [κ is taken from (see text); λ in fit B is from the regression].

Table 1.

The return to equilibrium in fit B adds 1.5 K to ΔT of fit A and artificially increases the second e-folding time. Note that, in both fits, we find e-folding times that are very comparable with those found by Jarvis and Li (2011) with a different method and with other atmosphere–ocean GCMs (AOGCMs).

We now discuss the more objective results of fit A. The value of κ is lower than the results of Raper et al. (2002) for nine AOGCMs in CMIP2, which ranged from 0.6 to 0.9 W m−2 K−1. Geoffroy et al. (2013) developed an identical model to ours and applied it to 16 models involved in the CMIP5 abrupt and ramp experiments. Their fitting procedure is, however, different; closer to our B fit but without using N. It still gives parameter values in the same range as ours: multimodel mean for f0 = 6.9 ± 0.9 W m−2 (their 6.8) and λ = −1.13 ± 0.3 W m−2 K−1 (their −0.85) (see their Table 4). The two e-folding times are also given: the fast one is 4 yr and the slow one is 249 yr, where we find 10 and 100 yr—these parameters are very sensitive to the fitting procedure. Notice that our model [L’Institut Pierre-Simon Laplace Coupled Model, version 5, coupled with NEMO, low resolution (IPSL-CM5A-LR) in the paper] does not need the heat-uptake efficiency factor introduced by the same authors.

Now, if the two-ocean-layer TLS model were adequate, the two periods where the first or the second exponential function is overwhelming would give the same sensitivity factor: e/b = f/c. This is not the case and, in the second period, N is less sensitive to ΔTs. By that time, ΔTs > 5 K and, because the specific humidity responds exponentially to temperature, saturation of water vapor absorption bands would suffice to explain this nonlinearity, but clouds might also have an influence. Such behavior is also noticeable in Forster and Taylor (2006) for some GCMs (see their Fig. 2). Andrews et al. (2012) give the regression results for the CMIP5 abrupt experiment in their Fig. 1, in terms of global annual averages, which are noisy; still it allows us to notice an analogous behavior for most models. In particular, there is evidence of two periods: (i) below about 10 yr, a linear regression is aligned with the fast-forcing value (red cross on their figure), and (ii) the remaining period with a different slope and an eventual saturation (curvature).

This nonlinear behavior is presently seen in Fig. 3, where the regression between the fitted functions is superimposed in dark green. We see that the corresponding regression deviates from the straight line at the end of the run. Figure 3 also shows that the rightmost green point, which looks like a stopping point, is still far from equilibrium. In these conditions, one cannot extrapolate toward N = 0 to determine an equilibrium surface temperature; this justifies our fitting method and we differ on this point with the cited authors.

After 140 yr, the ocean might react with a new structure incompatible with the two-ocean-layer model. This is what Gregory (2000) claims from the analysis of a ramp experiment with the Hadley Centre Coupled Model, version 2 (HadCM2) AOGCM, the coupled model of the Hadley Center: a change in deep water formation and upwelling. Gregory (2000) also uses a two-layer ocean model to show how the heat flux to the deep ocean can become to have little correlation with the surface warming. Li et al. (2013) show evidence of hemisphere and latitude contrasts in SST warming in the long-term abrupt experiment, where surface temperature only reached equilibrium after 1200 yr.

c. Checking the Lyapunov criteria

Following our methodology, we have to check that the troposphere temperature profile has reached the leading Lyapunov eigenvector in Eq. (A5). This is done by plotting the coefficients ΔηkTs for a few levels k as in Fig. 4. The asymptotic constancy is attained within a few years for the troposphere temperature but takes more than 5 yr in the stratosphere. Level 11 (400 hPa) in the mid–upper troposphere is the most sensitive level, 50% more than at the surface. This correctness criterion strongly constrains the regression method and, depending on the desired precision, excludes any attempt to link fast features with climate warming characteristics using this method; because fast features are incompatible with the constancy of . The figure also shows that data have to be filtered with a 36-month moving average at least. It should be emphasized that dealing with a global model artificially increases the inertia of the troposphere, because it includes inertia from horizontal transfers that are seen globally as vertical exchanges.

Fig. 4.
Fig. 4.

Six atmospheric levels of the 12-month low-pass-filtered ratio ΔηkTs(t). From surface to stratosphere: level 17 (factor = 0.9; black), 14 (1.2; red), 11 (1.55; green), 8 (1.30; blue), 5 (0.2; cyan), and 2 (−1.9; magenta, shifted).

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0334.1

d. CMIP5 ramp forcing experiment

CMIP5 defines the so-called 1pctCO2 experiment for which the CO2 concentration is increased by 1% every 1 January. Such an exponential increase induces a nearly linear forcing increase. Compared to the response to the step increase, a time integration allows us to write
e28
At t = S, the quadrupled value is reached (S = 140 yr =1680 months). The ramp TLS looks consistent with the fitted coefficients from the step experiment: in Fig. 5, where the step result from LMDZ and the fitted function are in red, the same function integrated over time (blue points) is consistent with the 1pctCO2 result (blue line). The same consistency between the step and ramp forcing is also found with the HadCM3 AOGCM by Good et al. (2011). Because the ramp forcing applies a (small) step each 1 January, it is possible to check that it shows the same surface temperature warming patterns as the abrupt forcing, and also the same precipitation and cloud change patterns, beginning with strong perturbations at high latitudes. The amplitudes, however, obviously increase faster in the abrupt case (not shown).
Fig. 5.
Fig. 5.

The fitted step TLS model (red) applied to the 1pctCO2 experiment (blue). The asymptotic straight lines (green) and without the fast responses (magenta) are shown.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0334.1

Now, looking at the 1pctCO2 regression curve in magenta in Fig. 6, we see that it does not follow a straight line. However, Gregory and Mitchell (1997) use this supposed linear regression to determine κ, whereas we proposed Eqs. (24) and (25) as an alternative. If we examine the fraction , recalling that b + c = 1 in the hypothesis that the model is valid up to equilibrium, we find
e29
We see that the regression could provide an approximation to κ or, in reality, to , that can be corrected according to b, τ1, and τ2, thanks to Eq. (25). It is the presence of τ1 in the numerator that limits the slope constancy and it can hardly be neglected for the fit with LMDZ5A. In addition, the criterion τ1tτ2 is hardly satisfied for the actual values of the two e-folding times. An approximate fit here gives κ ranging from 0.4 to 0.5, acceptable compared to the correct determination from λc/(b + a) = 0.5 W m−2 K−1. Another method adopted by Gregory and Mitchell (1997) takes κ = N(S)/ΔTs(S), giving 0.56; it can be improved with our correction, then giving 0.5 again.
Fig. 6.
Fig. 6.

Regression vs ΔTs with fitted functions (red), the influence of fast responses (green), and only the first ocean layer (blue).

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0334.1

We apply the regression method to the net TOA budget with the ramp forcing added, as in Forster and Taylor (2006), to determine the climate sensitivity factor (Fig. 6, thin blue line). The slope giving λT is found to be 0.82 (0.85 in the first half and 0.71 thereafter), close to the step results. Unlike the step characterization, the ramp determination of λT does not stand alone; it relies on the forcing, which, as we emphasized, has to be the y intercept of the step regression straight line. The straight line in red is obtained from the fitted functions of time for the step experiment.

e. On the fast responses

In the fitting procedures, we met two types of fast responses. When fitting the regression lines, the y intercept is currently interpreted as the forcing. In fact, it is a virtual forcing: it is the forcing that has to be applied to the L2Obs model, which, by its very essence, does not represent the fast responses as it does not include ΔTs = 0. As a second type of fast features, departure from the straight line is seen along the regression line.

1) Initial transient response

The N intercept cannot be the Planck forcing f0, because fast responses of, say, clouds and precipitation participate in these, as is the case for LMDZ5A (not shown). Depending on the authors, initial fast features are called atmospheric adjustments or semidirect effects (see, in particular, Colman and McAvaney 2011; Gregory and Webb 2008; Andrews and Forster 2008).

In the fitting of exponential evolution functions, coefficients like a in ΔTs(t) and d in N(t) [cf. Eq. (27)] also come from transient mechanisms, and depend on the smallest e-folding times of the TLS. They correspond to residuals of faster responses, when ΔTs(0) ≠ 0. The quantity N(0) again cannot be interpreted as the Planck forcing: considering the value of τ1 ≃ 10 yr, all mechanisms faster than, say, 3 yr should be considered fast. These residuals change the equations for the temperatures slightly: and , and also (see Table 1).

How these fast responses arise from our formal model can be seen in Part I [Eq. (39)]. The first order in μ forcing for the step function already leaves a constant residual adding to the integration of NL. Once the transient responses are damped during the 1% forcing, they follow a ramp function in time that becomes a new long-lasting forcing—their asymptotic influence. This is true for the fast mechanisms as well. In Fig. 6, the straight green line gives the influence of these fast mechanisms from the fitted functions. In blue, we have added the contribution of the first ocean layer. We see that the fast mechanisms explain a large part of the global sensitivity; see also Fig. 5 where the magenta line gives the 1pctCO2 temperature anomaly without considering the fast mechanisms.

2) Running fast response

Fast phenomena also arise along the path on the regression line. Consider, for instance, the first distinct spiral pattern in Fig. 3 at ΔTs = 5 K. It corresponds to a global cooling of about 0.3 K. Such a feature is called “a hiatus period” in Meehl et al. (2011), like the one observed during the decade 2000–09. This hiatus was associated with a TOA radiative imbalance of about 1 W m−2—very comparable to our 3-yr-average N swing of about 0.5 W m−2. We could indeed verify that, in the GCM results, it is associated with a strong La Niña event occurring during year 50–51 of the step experiment, the very reason invoked by Meehl et al. to explain the hiatus period. This imbalance nullifies the validity of the L2Obs method at short time scales, as we claimed. The difficulty is also acknowledged by Gregory and Forster (2008, p. 68), who note that “episodic volcanic forcing cannot be described by the same relationship and merits further investigation.”

An undesirable consequence of fast-feedback mechanisms is that forcing cannot be determined by evaluating only the radiative impacts of greenhouse gases and aerosols as in Forster and Gregory (2006). This difficulty unfortunately hinders the use of remote sensing short-range observations to determine the climate sensitivity with the L2Obs model. An evidence of this is found in Good et al. (2011), when they compare the (implicit) use of the L2Obs relation to fit the response to a 70-yr ramp-up followed by another 70-yr ramp-down forcing. Our interpretation is that because they use yearly values in their reconstruction (using a convolution product) fast mechanisms that particularly show up at the transition period cannot explain the HadCM3 GCM run. This is conspicuous for precipitation, which is predominantly a fast mechanism (see their Fig. 2).

To sum up, the consequence of the constraints imposed by the relevance of the L2Obs model is to limit the number of climate sensitivity factors that can be determined from regression. Take, for example, the lapse-rate factor in any of its definitions. It is necessarily a component of fast mechanisms because the tropospheric temperature anomaly has to keep the same profile (the leading eigenvector). More elaborate analyses, using our FGD perturbation method for example, have to be foreseen for a detailed description of the influence of climate mechanisms related to their characteristic times τ. However, if atmospheric adjustments are taken altogether, a very adequate method can be used to evaluate the response to forcing while keeping the fixed SST. This is the clever way to determine N for ΔTs = 0 in the L2Obs relation, as introduced by Hansen et al. (2005).

5. An implementation of feedback loops in LMDZ5

The preceding section has shown that a TLS at the global scale can be determined from a difference between two runs. On smaller regions and smaller time scales, this cannot work because climate trajectories are essentially stochastic. In contrast, if the CTLS [Eq. (2)] is integrated in time along a single trajectory, we have access to all spatial and time scales. We gave an example of such an implementation, concerning only radiation, in Part I. A version of the CTLS method has since been coded in the GCM and we report our first results in this section.

We take the analysis of the water-cycle (W cycle) feedback response as our test bed. In LMDZ5A, three models represent this mechanism: surface evaporation and vertical diffusion (TRB) (e.g., Deardorff 1974), deep convection (DC) from Emanuel (1991), and large-scale condensation (LSC; see Hourdin et al. 2006) in addition to the radiative shortwave (SW) and LW exchanges (RAD; cf. Fig. 7. These three mechanisms compose the total W-cycle feedback—evaporation, precipitation, and turbulent vertical exchanges. We recall that this feedback is not the PRP definition of the water vapor feedback (WV), which is only the greenhouse effect because of the added WV content, but here is the response of the mechanisms representing the water cycle in the GCM.

Fig. 7.
Fig. 7.

The inclusive water-cycle feedback system.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0334.1

Perturbations are not propagated through cloud models or orographic effects. Sea ice and land ice change are also excluded from the CTLS—as they are prescribed in the forced mode. What about dynamics? Perturbations should be advected by the unperturbed wind field, without dynamical feedback. In other words, (uη is taken into account, not (Δu)η.

The principal of applying the Gâteaux differential to obtain the CTLS is as follows. The variables to be perturbed are temperature η, and total water qT = q + ql (vapor + liquid). The trajectory is found by integrating the system
e30
Four mechanisms are involved in the perturbation: TRB, DC, LSC, and RAD, so , the sum over the four mechanisms, and where j covers TRB, DC, and LSC. TRB also treats the subsurfaces (ocean, land, land ice, and sea ice) with snow cover and runoff; we will not give the details here.8 The CTLS reads
e31
where we use dt as ∂t + V. Equation (31) is approximated by application of the FGD:
e32
The atmospheric liquid water is computed in LSC, so, in addition,
e33
and additional equations for the soil model, snow cover, etc., are included in this definition of the W-cycle feedback.

The differences in square brackets appearing in the equations are the mathematical representation of the double call to each of the mechanisms in the model (cf. appendix C). At the limit of small perturbations, Eqs. (32) and (33) should approximate the CTLS. However, we chose the forcing as 2×CO2 for convenience, but a smaller forcing can be used as long as round-off errors do not make the response too noisy. This system, which gives the inclusive W-cycle response to 2×CO2, can be schematized as in Fig. 7. The forcing (Frcg) is obtained from the integration of the Planck response as another CTLS—the one detailed in Part I.

Note that the system is structured as an exclusive feedback loop and some mechanisms can eventually be discarded to analyze stability. Another remark concerns the flexibility of building the CTLS: as our run is in the forced mode, SST is periodic from the data. The CTLS we built not only adds a perturbation to it, but also gives a certain inertia to the ocean surface—we chose the 50-m thickness found in the CMIP5 analysis, which applies to the Planck response as well. In this way, the CTLS includes a “slab ocean.”

The two CTLSs (Frcg and W cycle) are integrated in time along the trajectory following the same numerical method as the GCM for time-step integration except; of course, for the ocean surface. The perturbed fields (Δη, converted into potential temperature, and ΔqT) are given as passive tracers to the dynamical package. If advection were not taken into account, we would obtain for each column, the unstable response of the radiative–convective single-column model. With advection included, the so defined W-cycle response acts as forcing on the dynamic feedback.

a. Regression and forcing

The first series of results concerns the L2Obs regression curve. To be clear about our numerical experiment, the GCM is run with fixed periodic SSTs and preindustrial CO2 concentration. The “physics” driver has additional code to compute in parallel, at each time step, the anomalies responding to the 2×CO2 abrupt perturbation. Clouds and sea and land ice cover are not sensitive, but the snow cover is part of all processes in Fig. 7. A second CTLS representing the Planck response is also integrated as part of the process.

Figure 8 shows the resulting regression curves (N, ΔTs). Considering the monthly values (symbols and green line points), one can see that the warming buildup is not a trivial matter. The monthly values show the same complexity as in Fig. 3 with similar spiral and striation patterns but on a shorter time scale. However, there is no ENSO in our slab-ocean CTLS. Black lines are the 12-month averaged values that are both nearly straight lines. For the W-cycle curve (bottom right), the λ factor is found to be ~−1.64 W m−2 K−1. To compare with the PRP WV feedback, despite the difference in definition, we can combine the WV and lapse-rate factors with the forcing: the multimodel mean value in Dufresne and Bony (2008) gives an λp + λwv+lr of −2.2 ± 0.6 W m−2 K−1.

Fig. 8.
Fig. 8.

Regression N (W m−2) vs ΔTs (K) with a 12-month low-pass filter (black lines) and monthly values (green line with points and blue crosses). Top-left curves are for Frcg and the bottom-right curves are for the W cycle.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0334.1

So even linear modeling of the perturbation brings out the fact that the L2Obs regression is not a straight line. To complement what we said in section 4, the underlying unperturbed model is strongly nonlinear; clouds, etc. vary quickly with deep-convection events during the run, and the CTLS inherits these changing meteorological scenes with a fluctuating atmosphere. The top-left curves give the Planck response regression, with the slope giving λP = −2 W m−2 K−1, as already discussed in Part I.

We now examine the method of using the fixed SST perturbation to determine the initial forcing. We run the two CTLS with a quasi-infinite slab thickness, which gives the two red curves in the same figure—monthly values with blue crosses. This clearly shows that the method nearly determines that the integrated fast response is forcing the L2Obs straight line. However, this “initial” forcing of the L2Obs does not correspond to ΔTs = 0: this would only be true for an ocean planet. Continental surfaces warm up and the L2Obs forcing is to be taken for the global ΔTs obtained after the transient response of all surfaces other than the ocean. This is seen in Fig. 1 of Andrews et al. (2012) as well, where red crosses indicate the forcing found at fixed SST perturbation runs. The fixed SST method does not provide the exact fast-forcing constraint because of the slow-part response of continental surfaces. The fast response in the W-cycle surface temperature is found as 0.16 K, while it is 0.24 K for the Planck response; this gives a negative W-cycle effective response of −0.08 K and consequently, we can conclude that the fast W-cycle feedback is negative.

Figure 9 gives another look at that feature and shows the regression of total precipitation and evaporation with ΔTs for the first 10 yr. The transient lasts about 30 months when precipitation decreases more than evaporation, leading to a negative fixed SST yearly steady-state value of −0.04 mm day−1 (green symbols). What happens is that the radiative warming of the troposphere makes relative humidity decrease and so precipitation decreases. Specific humidity hence increases and evaporation also decreases in turn. This mechanism explains the negative fast feedback found in Hallegatte et al. (2006) with a simple 1D model. Colman and McAvaney (2011) suggest the same conclusion. It can also be seen that, after that transition, water is conserved in the W-cycle CTLS.

Fig. 9.
Fig. 9.

As in Fig. 8, but for evaporation (black) and total precipitation change (red) (mm day−1). The ΔPrecips/ΔTs ≃ 0.11 mm day−1 K−1. Blue line (6-month filtered) and symbols (monthly) are for evaporation minus precipitation.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0334.1

As we claimed, atmospheric lapse rates form part of the fast responses. Figure 10 shows that they need a year to approach steady-state. The Planck response (Fig. 10, top panel) shows the initial strong cooling of the stratosphere and heating of the low troposphere. The curves then become smoother, from right to left, giving an almost vertical profile. This gives a justification of a classical method of using the troposphere +1-K perturbation on the vertical as the Planck response profile (cf. Part I). The W-cycle temperature response shows a more complicated evolution, starting with an S shape (rightmost curve) similar to the initial Planck forcing, and later showing an enhanced warming in the mid- and upper troposphere, which can be attributed to enhanced convection in some regions. The final difference between the Planck profile and the W-cycle response (leftmost curves) is part of the PRP lapse-rate feedback. One can notice the similar profile obtained with a fixed SST perturbation run in Fig. 3 of Colman and McAvaney (2011).

Fig. 10.
Fig. 10.

(top) Frcg: W-cycle (K K−1) and (bottom) atmospheric global temperature profiles for (right to left) 4, 8, 16, 32, 64, 128, and 256 days, and days 400 and 420 (enhanced symbol curves); Frcg begins at day 16. Temperature is scaled with surface global temperature change.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0334.1

b. Decadal term

After about 3 yr of abrupt forcing, the CTLS structure follows the seasonal variation. Figure 11 is the Hovmöller diagram of deep-convection precipitation anomaly in the tropical region during 2 yr. Convection is almost always enhanced by the W-cycle feedback (by 10%). This is a classical result (e.g., Dufresne et al. 2013). The Maritime Continent and the western Pacific (140°–170°E) constitute an exception; where deep-convection decreases, large-scale condensation takes over. This region was the most convective one in this run (ω500 ≃ −50 hPa day−1). The W cycle brings more humidity, at higher temperature (about 2 K on the vertical), but convection does not increase as it should. This is clearly seen in Fig. 15 of Hourdin et al. (2006), where the Emanuel model begins to saturate when ω500 is lower than about −30 hPa day−1. In that case, large-scale condensation removes the surplus humidity. Resulting total precipitation change is positive almost everywhere over the region and amounts to about 15% (not shown).

Fig. 11.
Fig. 11.

Hovmöller diagram of monthly amount of deep-convection precipitation (mm day−1; contours) and its W-cycle response (shaded) in the intertropical zone (30°S–30°N). Contour interval is 1 mm day−1 and amounts range from 0 to 7 mm day−1.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0334.1

The atmospheric response to the Planck forcing (Frcg) and the induced W-cycle feedback response ΔTw are shown in Fig. 12. The Planck response to forcing (SW + LW) can be compared with the immediate forcing found in Taylor et al. (2013, Fig. 2c). Even though, the National Center for Atmospheric Research (NCAR) Community Climate System Model, version 4 (CCSM4), is a coupled model, we choose comparison with it because, using the Climate Feedback Response Analysis Method (CFRAM) for diagnostics, the analysis follows the same logic as ours (see appendix B). Temperature anomalies in our results have smaller amplitudes, because our Planck response is relaxed after 12 yr; also zonal patterns are different.

Fig. 12.
Fig. 12.

The W-cycle temperature IFC: year 12 (ΔTw; K; shaded) and forcing (ΔTfrcg; contours).

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0334.1

As for the W-cycle response, we have a different definition, since Taylor et al. (2013) use the classical WV-content radiative feedback. Compared to their Fig. 3c, our results show that the equatorial temperature change maximum is shifted up, because of rising deep convection, as shown in Fig. 13. In fact, our results are closer to their Fig. 1c, which gives the global response in temperature that includes all feedbacks (WV, clouds, surface albedo, and dynamics). We obtain, however, the same north–south asymmetry. Our maxima are of the same order but greater (4–5 K compared to 3 K) and the W-cycle ΔT pattern is more concentrated in the equatorial region. Comparison with a full coupled GCM is difficult to justify, but knowing the relative importance of the WV feedback, the comparison is still meaningful. Recall that our results in particular do not include dynamic responses and thus represent only a forcing to the large-scale flow. This might be among the reasons of the stronger amplitude change in our results: this forcing would very probably be relaxed by the dynamics. Our results are particularly close to the lower left graph in Fig. 4 of Hansen et al. (1984), giving the global warming in response to 2×CO2.

Fig. 13.
Fig. 13.

The W-cycle response in net DC updrafts (shaded; kg m−2 h−1) and net downdrafts (contours).

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0334.1

Figure 13 gives the change in the deep-convection-induced mass flux. In the model of Emanuel (1991), updrafts include the undiluted adiabatic core of the convective column and the mixing-induced updrafts, while downdrafts are composed of the unsaturated and saturated downwelling currents induced by precipitation. The net updrafts (shaded) show that DC influences the tropopause in the equatorial region while the net downdrafts (contours) display a systematic decrease in intensity. Deep convection is shifted upward all along the Hadley cell.

c. Decomposing the W cycle in mechanisms

When faced with results like the one obtained from the W-cycle response, developers are willing to analyze the effect of specific parameterizations on the global response. Because we included three mechanisms—TRB, DC, and LSC in that feedback—three CTLSs can be built. Each of these mechanisms is forced by the Planck response Δηp(t). The three feedback systems have the form
e34
where dt = (∂t + V · ).

So, for TRB-CTLS, for instance, the entries to the subroutine accounting for the mechanism are T + ΔT + ΔTfrcg, q + Δq, and similar entries account for surface and soil temperature and humidity (cf. Fig. C1). For the same CTLS, the other mechanisms depend on T + ΔT and q + Δq (no ΔTfrcg). The sum of these components and the forcing should, by linearity, give the full W-cycle CTLS. This can be mathematically checked by adding the four systems in Eq. (34), with i indicating TRB, DC, or LSC. This procedure was implemented using three runs.

The main test concerning the CTLS approach was to verify whether the sum of the three IFCs and the forcing gave the global response. We first explain the difficulty we met. During our numerous tests, we found that, below some threshold on the forcing amplitude, the response became independent of that amplitude; there was some pullback to an unexplained residual noise from DC that might be attributable to the triggering algorithm. This feature of the Emanuel model is obviously unpleasant and it can be expected that new triggering functions (using a statistical approach) could improve the model (cf. Rochetin et al. 2014). But the fact is that, meanwhile, we have to deal with this numerical problem. That problem, only detected when using the FGD method, shows the advantage this method may bring for testing parameterizations. We finally got rid of the problem by using a noise subtraction technique, which necessitates calling the DC code three times. The results are shown in Fig. 14 with the profiles of atmospheric temperature and specific humidity averaged over the tropical region—where the discrepancy between the sum of the IFCs and the full response is the most pronounced. Now, apart from a residual discrepancy in the boundary layer,9 both ΔT and ΔQ IFCs are additive. A probable origin of the discrepancy is the lack of a specific model for the boundary layer convection: without it, DC and TRB are used for this purpose, apparently in an excessive nonlinear behavior.

Fig. 14.
Fig. 14.

Monthly values (August, fourth year) of (bottom) ΔT and (top) Δq IFCs subcomponents of the W cycle. The forcing added to the three components— TRB, DC, and LSC—adds (Sum) to give the full water-cycle component (2×).

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0334.1

Concerning the IFCs obtained, it can be seen that DC has the opposite sign to Frcg in the lower and mid-atmosphere, unlike in the upper troposphere, and this is in agreement with the conclusion of Raymond and Herman (2011, p. 11) that “the convective quasi-equilibrium hypothesis appears to be valid in the lower troposphere but not in the upper troposphere.” Except in the boundary layer, LSC and TRB have similar responses, in both temperature and humidity,10 while DC is distinctive. The upward shift of deep convection is clearly seen with new heating near the tropopause.

We now conclude our illustration of the method: having a new technique in hand, some practice will be needed to learn how to use it properly. We hope other teams involved in GCM development will join us in investigating this new approach.

6. Concluding discussion

After having introduced a general formal framework in Part I and shown the equivalence between feedback and radiative sensitivity analyses in practice in the CMIP community, we have in this paper extended the formal analysis to render it more appropriate for numerical climate studies. The key point was to determine the rigorous dynamical conditions under which the regression technique provides relevant long-term climate sensitivity. These conditions are quite constraining, leading to time-scale low-pass filtering of the global climate evolution. Yet, at the same time, these constraints mitigate the limitations of the strict formal approach and extend the range of mechanisms that can be analyzed in GCMs with the same methodology. The same constraints also allow a regression of all available variables in the GCM with surface temperature anomaly because they are part of the same tangent linear system. This is of particular importance for CMIP analyses.

A major difference that we found with the standard analyses was that the lapse-rate feedback could not be chosen as a specific feedback and had to be distributed among the others, beginning with the Planck response, which already includes its own tropospheric lapse-rate response. We thus come to the same conclusion as Lu and Cai (2009) that the PRP lapse-rate feedback is irrelevant. Along the same line, Stocker et al. (2014) introduced the “effective radiative forcing” to account for that dynamical behavior.

Our main illustration concerned a global analysis of the CMIP5 quadrupling experiment with LMDZ. We were able to give details on the way fast mechanisms, not captured by the model underlying the regression method, explain much of the long-term evolution of the climate. The simplest dynamical model explaining the slow component of climate warming for the next century corresponds to the leading Lyapunov exponent of the 1D conceptual model underlying global mean analyses. It explains the PRP relation between the net TOA budget and the surface temperature anomalies in the regression method (the L2Obs relation). With only these two variables (N and ΔTs), we found that two layers of the ocean (down to about 600 m) are observable—in the sense of the control theory—from the global trajectory of the difference between the perturbed experiment and the control one.

Adding the dynamical analysis to the regression method as we did has a double advantage: it ensures that the Lyapunov constraint is attained and also determines the “ocean heat uptake” efficiency in a coherent way. It furthermore shows that the response to the abrupt quadrupling and to the 1% ramp forcings share the same tangent linear model, thus providing cross proof of the results through the use of both experiments.

The results of fitting this model to the LMDZ5A trajectory (and also to other models that we have already analyzed)11 show that this two-layer model cannot reach equilibrium, suggesting that the oceanic warming structure could change dramatically after 100 yr of the quadrupled CO2 experiment. Following our formal constraint, the two-layer model is, by definition, not appropriate for studying fast mechanisms, which we defined as the mechanisms that rule out equality between the TOA radiative and surface energy budgets, a simple criterion of the troposphere being in equilibrium with the surface. This does not mean that fast mechanisms cannot be analyzed with the tangent linear system; it only shows that the two variables N and ΔTs are not appropriate for this. But if the first time constant of about 10 yr found in section 4 is realistic, the forcing needs to include transient mechanisms that change the lapse rate, clouds, etc., in addition to the pure radiative forcing. This unfortunately makes it quite impossible to evaluate the real climate forcing needed in a remote sensing regression technique based on the L2Obs relation. Numerical models are essential for a detailed analysis of how the climate responds to perturbations, and this is why we need more efficient tools to meet the challenge of developing some understanding of how this complex natural system works.

We have proposed such a new method based on including the CTLS integration in GCMs. By introducing a finite version of the Gâteaux differential (FGD), we showed that each mechanism represented in a GCM can be differentially perturbed, the individual perturbation being or not propagated, through other chosen mechanisms, thus allowing the perturbed model behavior to be analyzed step by step from the exclusive to the inclusive approach. We used a first implementation of the method in LMDZ5 to analyze the water-cycle inclusive response to the CO2-doubling experiment. In contrast to perturbed runs, the CTLS method provides results at given times, region by region, opening up the possibility of analyzing fast responses as well. We gave a short illustration of such analysis for the year following the abrupt forcing and a few more in the decadal term.

Concerning the Planck response, now that we know how to determine it, we have to find an agreement on its definition. As it represents the no feedback response, it includes all mechanisms that are not considered feedbacks. For example, we deliberately included surface thermal inertia in it, but what about soil models and deep ocean processes? In any case, advection by the unperturbed wind fields has to be included if the forcing response and the IFCs are to give the total response.

The 2×CO2 forcing was chosen for our purpose, but obviously any other forcing can be used. The amplitude can possibly be adjusted to save the linearity of the response if necessary. The FGD method is very flexible and allows us to analyze changes in parameters and basic physical laws as well as sensitivity to various forcings. This should make it easier to develop and evaluate subgrid parameterization, adjust results to experiments, and calibrate models, without the need to build adjoint models. All this might find applications in numerical weather prediction methods. In any case, it should help coping with the important issue of evaluating model uncertainty.

The coding of the method gave us some difficulty, essentially because of the way the interfaces to the diverse mechanisms are written in LMDZ5. The first step of implementation is to integrate a CTLS with no forcing, which should result in a null response, and also without changing the reference trajectory. Some variables in the arguments of subroutines have to be saved and restored when the subroutines are called again, etc. These difficulties are thus strongly dependent on the GCM. A question remains concerning the dynamic feedback: is it necessary to use a numerical method for the atmospheric circulation feedback or is the two-run method enough? The answer will hopefully come from practice.

Acknowledgments

The authors gratefully acknowledge useful discussions with J. Y. Grandpeix and also with Q. Libois during his Master internship on CMIP5 analyses. Special thanks to Professor M. Ghil and to our editor M. Cai for their suggestions to improve our manuscript. Two anonymous reviewers also gave us valuable advice. We wish to thank Susan Becker for careful proofreading of our first English version.

APPENDIX A

Extended Sensitivity Analysis

The procedure (detailed in Part I) analogous to the one applied to find Eq. (15) leads to
ea1
and the Planck sensitivity factor to
ea2
(we use a superscript x to identify the extended factors). The new expressions come from the change in the forcing terms in Δθ0:
ea3
and their analogs for matrix . We keep Δθ0 as the forcing term, but its new value now includes impact of troposphere temperature change on the surface. Note that the global factor has not changed, because of the identity
ea4
the rhs term coming from the factor to .

In consequence, the new sensitivity factors are just a redistribution of the previous ones.

The preceding extension of sensitivity analysis has a drawback: the relation to feedback gains is partially lost. Instead of Eq. (6), the global feedback system with our assumptions would read
ea5
with the surface solution (1 − g′)ΔTs = Δθ0 introducing a new feedback gain
ea6
which looks fine: it corresponds to the new feedback loop in Fig. 1. The problem comes with the decomposition into partial gains, because we do not have the required equality . This contradicts the formal constraint that the feedback system must have the same factor (Δθs + Δθ0) on the rhs forcing term: see Eq. (10). With the new forms [Eq. (A5)], this is no longer possible; the normalized tropospheric profiles and have no reason to coincide—different responses have different lapse rates.A1 More explicitly, in the submatrix procedure, the troposphere responds with matrix only, and the surface solution may be written in the form
ea7
with another set of sensitivity factors:
ea8
which introduces “exclusive sensitivities”—for which the troposphere only responds to the Planck mechanism. What prevents the classical relation coming into play is the difference in λlr: it plays the same role as the PRP lapse-rate sensitivity, which is the reason why we used that symbol. Here, it is the difference between the true Planck profile and other temperature profiles, and the feedback loop cannot be opened without changing the forcing.
The difference in lapse rate of the IFC profiles is also discussed in Lu and Cai (2009). They propose replacing scalar feedback parameters by diagonal matrices. However, the 1D or 3D response matrix function—say i—corresponding to a feedback loop would read i = (−1i as the extension of Eq. (9), with no evidence of the gain matrix being diagonal. In fact, this is the case at the Lyapunov horizon. With our notation, CFRAM separates the energy divergence that balances the forcing into components Δφi, attached to the mechanisms (the same as our separation into i). The IFCs are obtained at equilibrium with matrix explicitly determined. In our framework, this reads
ea9
Noting the lth entry of vectors as (i), we then have
ea10
Hence we may write
ea11
where matrix (i)d is diagonal with entries:
ea12
We conclude that the CFRAM decomposition leads to proper IFCs, see Figs. 3, 5, and 7 in Lu and Cai (2010). This and Eq. (A10) give Lu and Cai’s (2010) Eqs. (23) and (26).A2 Accordingly, we share their conclusion that the PRP “lapse-rate feedback” is irrelevant.

These formal results also state that each IFC in CFRAM is associated with a feedback loop: this is what Cai and Lu verified numerically, successfully comparing the CFRAM decomposition and a special nonlinear suppression method (M. Cai and J. Lu 2012, personal communication), where the n inclusive feedback-loop responses were obtained from a difference between the unperturbed trajectory and n perturbed runs.

In summary, and because diagonal matrices commute in product, the extension of Eqs. (7), (9), and (12) are still valid at the cost of losing the simple scalar expressions: the (diagonal) exclusive gains become i = i/(+ ), etc.

APPENDIX B

The Two-Layer Ocean Model

With the two-layer ocean model, we encounter a situation where the surface temperature anomaly evolves with a more complex time function than in Eq. (16), with no impact on the radiation budget dependence on ΔTs and thus no change to the troposphere eigenvectors. Solving Eq. (20) for ΔTs gives
eb1
Dividing the numerator and denominator of the fraction by and factorizing yields
eb2
where and are the two e-folding times. Before inverting to the time domain, we need the poles—the zeroes of the denominator—which depend on the discriminant:
eb3
Using the smallness of the e-folding time fractions, we obtain an approximation of the poles as
eb4
Corresponding to the minus sign, we have the pole , the other being close to . The solution is classically found by identification with Eq. (B2), where τd is replaced in coherence with τ2:
eb5
finally yielding
eb6
with the coefficients
eb7

APPENDIX C

Implementing CTLS in GCMs

To explicit the logic of implementing the FGD scheme in a GCM, the diagram in Fig. C1 shows two columns. The left one corresponds to the main driver of physics in LMDZ. Without modifications (pink background), a subroutine SUB is called each time step with standard variables (Eta_seri) as input—which comprise atmospheric and surface temperature, humidity, and horizontal wind speed. The increment of these variables is output from SUB (d_Eta) and is used to update the standard variables.

Fig. C1.
Fig. C1.

Schematic implementation of the FGD double-call method.

Citation: Journal of the Atmospheric Sciences 71, 9; 10.1175/JAS-D-13-0334.1

The two rectangles contain the modifications, activated from the logical flag FGD. Basically, SUB is called with the perturbed variables as input: Eta_seri + DEta, where DEta are the CTLS-anomalies—Δσ in Eq. (2), which may include the forcing [see Eq. (32)]. The FORTRAN variable dDEta is initialized from SUB output. After the second call to SUB (second rectangle), dDEta is set to its increment and added to DEta (time integration). When SUB has F90 in–out variables, these have to be saved before the first call, so that at the (standard) second call they are reset to their standard values.

The right column gives details concerning subroutines that are calling other processes—this is the case in LMDZ for the subsurface treatment (ocean, land, sea ice, and land ice), where yStateVar is extracted from the global grid to the subsurface grids then recast into the global grid. In this case, a new flag (mod_pert) must be added in SUB input and passed to SubSub. It is used to save and restore internally modified variables, some of which are input to SubSub. During the first call, when mod_pert is True, the in–out variables are saved and internal standard variables are perturbed—StateVar + DSVar, where DSVar are the internal anomalies—notice that in LMDZ, SubSub unfortunately does not compute increments. After the call to SubSub, DSVar is initialized. At the second call, internal variables are restored to their standard values before calling SubSub. Returning from SubSub, the internal anomalies DSVar are computed.

Hence, the favorable case for FGD implementation is that (i) there are no in–out F90 variables, and (ii) algorithms determine incremental values of the variables. This is, anyway, good practice in general for GCMs from numerical concerns.

Globally, the CTLS state variables DEta and DSVar are thus added to the standard variables and output in the history files. Potentially all other diagnostic anomalies can also be computed and stored, hence doubling the number of history variables.

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1

Compare Stocker et al (2014): chapters 7.1.3, 9.7.2.1, and 12.5.3.

2

The Laplace variable μ has the dimension of per time.

3

The “no feedback” response, called the Planck response, is the hypothetical response to an increase of CO2 concentration in the climate system when only radiation is involved; it serves as a gauge for other climatic feedbacks, such as water vapor and surface albedo (cf. Part I).

4

If Δθs is the last entry of Δηr, the n-dimensional row matrix reads [0, 0, …, 1]. As in Part I, we note a single-column matrix as and a row matrix as , with the advantage of clearly recognizing scalars as form, as in Eq. (8).

5

FORTRAN code works more as an operator than a function, which is the reason why the Gâteaux differential has to be used; for instance, continuity in the variable is not required.

6

It is the eigenvector associated with the leading Lyapunov exponent.

7

The external product results in the null matrix with the exception of the last diagonal entry being unity.

8

These are, however, the main source of difficulty for implementation.

9

Only seen on oceanic surfaces (not shown).

10

While having, however, very different meridional structures.

11

Since this was written, Andrews et al. (2012) have given the regression results for 15 models involved in the CMIP5 abrupt experiment and proposed using a varying forcing or feedback for long term modeling.

A1

Example given in section 5.

A2

0 (=−1) in their Eq. (23) should be at left in the matrix product.

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