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  • View in gallery

    Four components of the leading mini-Clim Floquet vector throughout year for (K) and (mm). Thick lines are with SST feedback; thin lines are with forced SST. The tendency for the amplitude of the ocean temperature perturbation (blue) to be greater than that of the atmospheric temperatures (gray and red) suggests that FV1 is tied to oceanic mechanisms.

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    As in Fig. 1, but for the second FV. In contrast with FV1, the greater perturbation amplitude of the atmospheric temperature (red and gray) relative to the ocean temperature (blue) during the majority of the year and the larger amplitude in tropospheric water content (green) suggest that FV2 is tied to atmospheric mechanisms.

  • View in gallery

    Impact of the mini-Clim Floquet vectors on precipitation—in W m−2 as the energy release to the troposphere (a factor of 3.456/100 converts to mm day−1). The unperturbed precipitation amount along the trajectory (black curve; scaled) is also given for reference. Thick and thin lines are as in Figs. 1 and 2 for, respectively, free feedback from SST and forced SST.

  • View in gallery

    Regression curves between the OLR and perturbations of the two leading Floquet vectors. In accordance with the previous figures, thick lines (red and blue) are with SST feedback; thin lines (black and blue) are with forced SST. A comparison between their respective slopes and the Planck slope (black) highlights a qualitative difference between the feedbacks: the mean FV1 slope lies below the Planck slope, indicating a positive feedback, while the mean FV2 slope lies above, indicating a negative feedback.

  • View in gallery

    Effective W-cycle feedback responses for mini-Clim in different modes: the cycling response is in red, while the response from the CTLS at equilibrium is given by the dashed blue line, superimposed with the response retrieved from eigeninversion (see appendix C) for a 50-m ocean mixed layer. The remaining three responses are obtained from eigeninversion for different ocean thicknesses. (inset) The periodic trajectory reached by the cycling response after about 15 years is shown, along with the equilibrium response.

  • View in gallery

    FV1 oceanic temperature determined from two algorithms [Eq. (26), with ]: algorithm 1 in red and algorithm 2 in blue. The second method only is successful to produce a correct ocean temperature—compare to Fig. 1.

  • View in gallery

    IFC Floquet components of to a W-cycle perturbation with a forcing of 1 K. Total responses are also shown (black and light blue lines for OLR and Tτ, respectively). The first (slow) component clearly produces positive feedback (red and blue lines), while the second (fast) component produces negative feedback (green and yellow lines).

  • View in gallery

    As in Fig. 7, but for precipitation rate (mm day−1) and surface temperature (K). The slow components (black and red) again show a positive feedback, opposite to the fast (blue and orange) feedback.

  • View in gallery

    and ΔOLR of Katioucha (CTLS of LMDZ) with 1 K of Planck forcing and forced SST, hence the fast component. Dots represent the monthly average of the results between 30° and 40°N for the last 7 years of a 15-yr trajectory; thin line is a month-by-month average of the 7-yr results, giving a quasi-periodic response. As with mini-Clim, we see a negative feedback from this fast component.

  • View in gallery

    As in Fig. 9, but for total precipitation (black) and precipitation from deep convection only (blue). Both responses show a negative feedback, as in mini-Clim.

  • View in gallery

    Seasonal cycle of the fast (second) IFC components of mini-Clim (black and blue) compared with normalized and averaged forced-SST results from Katioucha (red and green) for and dRad, the TOA budget. The amplitudes of the results are the same order of magnitude, allowing further comparisons between the models.

  • View in gallery

    As in Fig. 11, but for the slow components. Slow components in Katioucha are obtained by subtracting the fast components of Fig. 11 from the total response (red and green).

  • View in gallery

    As in Fig. 11, but for total IFCs (dOLR in mini-Clim is the TOA budget perturbation; SW is unperturbed).

  • View in gallery

    Seasonal trajectory in the ΔTs–ΔOLR plane decomposed on the four mini-Clim FVs for 1 K of forcing. Katioucha results from a forced-SST run and a standard 50-m-thick ocean slab. This highlights that the qualitative feedback features of the two models are similar: negative feedbacks are associated with the fast components, and positive feedbacks are associated with the slow component. The total responses are, consequently, of smaller amplitude.

  • View in gallery

    As in Fig. 14, but for the change in precipitation rate. While the fast and slow responses of both models exhibit qualitatively similar responses, their total feedback has opposite sign. This discrepancy provides insight into the role of the lapse rate in the W-cycle feedback (see text).

  • View in gallery

    Changes in precipitation for three different mini-Clim lapse rate scenarios: the standard model (blue), the model with a modified feedback loop (red; see text), and the response after a reduction in the lapse rate by 2.5% (black). Reducing the lapse rate increases the response in total precipitation, from negative in the blue curve to positive in the black curve.

  • View in gallery

    Seasonal slow response in the lapse rate [contours of ] and DC updrafts (shaded; kg m−2 day−1). The warm-season decrease of the lapse rate (increased temperature at 200 hPa) coincides with more penetrative updrafts.

  • View in gallery

    Fast precipitation anomalies (mm day−1) from 24 responses (thin colored) and their mean (thick black) exhibiting exponential behavior from which the fast characteristic time of 30 days is drawn. Results are taken between 30° and 40°N, from 1 to 390 days with an interval of 10 days on an arbitrary date.

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A Formal Analysis of the Feedback Concept in Climate Models. Part III: Feedback Dynamics and the Seasonal Cycle in a Floquet Analysis

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  • 1 Laboratoire de Météorologie Dynamique, Paris, France
  • 2 Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada
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Abstract

This article introduces a new decomposition of climate feedback mechanisms based on their characteristic times. As the last of a series of three, it complements the first two parts by Lahellec and Dufresne to give a comprehensive review of climate feedbacks that will help to ensure consistency between practice and theory. In Parts I and II, analysis of the climate response to perturbations at the large spatial scales and time scales necessary to obtain linearity restricted the characterization to the slow components of the response. This part incorporates the fast mechanisms’ impact on the climate feedbacks, bringing the seasonal cycle into the analysis. Thanks to the Floquet theory, the authors could extend the formal framework of Parts I and II to incorporate the fast mechanisms. An illustration of the formal results with a simple 1D model highlights a clear distinction between the role of fast (intraseasonal) and slow (decadal) feedbacks, with an application to the water-cycle feedback of Part II. The same implementation of the Gâteaux difference in LMDZ, the GCM of LMD, as in Part II is used for comparing results with the authors’ toy model. This comparison essentially validates the Floquet decomposition obtained with the toy model: the fast component linked to precipitation is contributing negatively to the water-cycle feedback while the slow component is building up the positive feedback. The comparison also provides further insight on the sensitivity of models’ precipitation to climate warming.

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

This article introduces a new decomposition of climate feedback mechanisms based on their characteristic times. As the last of a series of three, it complements the first two parts by Lahellec and Dufresne to give a comprehensive review of climate feedbacks that will help to ensure consistency between practice and theory. In Parts I and II, analysis of the climate response to perturbations at the large spatial scales and time scales necessary to obtain linearity restricted the characterization to the slow components of the response. This part incorporates the fast mechanisms’ impact on the climate feedbacks, bringing the seasonal cycle into the analysis. Thanks to the Floquet theory, the authors could extend the formal framework of Parts I and II to incorporate the fast mechanisms. An illustration of the formal results with a simple 1D model highlights a clear distinction between the role of fast (intraseasonal) and slow (decadal) feedbacks, with an application to the water-cycle feedback of Part II. The same implementation of the Gâteaux difference in LMDZ, the GCM of LMD, as in Part II is used for comparing results with the authors’ toy model. This comparison essentially validates the Floquet decomposition obtained with the toy model: the fast component linked to precipitation is contributing negatively to the water-cycle feedback while the slow component is building up the positive feedback. The comparison also provides further insight on the sensitivity of models’ precipitation to climate warming.

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

Since its introduction into climate analysis, the feedback concept has been associated with characteristic times. Hansen et al. (1984) expressed that “Feedbacks modify the response-time since they come into play only gradually as the warming occurs” (p. 155). Yet, until recently, the dynamical aspect of feedbacks has seldom been brought up in the literature, and the main results of climate analysis have been discussed at equilibrium. A main objective of the three-part series ending with the present paper was to review current practice in the analysis of climate feedbacks. Therefore, Lahellec and Dufresne (2013, 2014, hereafter Part I and Part II, respectively) focused on the transition between two climate states at equilibrium.

The dynamics of feedback have occasionally been considered. Hall and Manabe (1999) asked “does the water-vapor feedback’s strength depend on the timescale of an internally generated temperature anomaly?” (p. 2328). The regression method in Gregory et al. (2004) indirectly reactivated this question because climate sensitivity is analyzed along GCM dynamical trajectories or even directly using satellite data. Part II gave conditions on the application of this method. In particular, it was shown that its application to GCM results (e.g., Gregory and Forster 2008; Andrews and Forster 2008) requires a low-pass filter of more than 3 years to deliver the slow (decadal) climate sensitivity from step experiments (CMIP51 abrupt 2 × [CO2], for instance). This procedure, by essence, can only determine slow feedbacks.

It was still unclear also whether the forcing determined from the regression method was a true forcing or if it included fast—or ultrafast—feedback mechanisms (currently called “adjustments”). The method applied to the ramp experiment (the CMIP5 1% every-year increase in the CO2 concentration) pointed out in Part II that about 30% of the warming was coming from the influence of fast mechanisms. These results led us to wonder about the explicit characteristic times attached to the mechanisms involved in feedbacks and what method could be developed to retrieve information on the fast feedbacks.

Another dynamical concern is the seasonal cycle. Part I analyzed the global climate in transition to the perturbed equilibrium. In such conditions, one can rigorously introduce feedback and sensitivity analyses on a formal basis. This approach was based on constant Jacobian matrices of the tangent linear system (TLS) that allowed analysis in the Laplace domain. However, the climate at equilibrium is following a seasonal cycle with amplitudes of variation exceeding the moderate forcing of even the 4 × [CO2] experiment. A new technique had thus to be utilized, and Part II introduced the circulating tangent linear system (CTLS) as a possible approach in GCMs.

Therefore, our objective with Part III was to analyze the impact of fast mechanisms on the perturbed climate, bringing the seasonal cycle into consideration. The seasonal cycle’s change under climate warming is, per se, of primary interest because it is known to play a major role in interannual climate oscillations and perturbations. For instance, the interannual frequency irregularity of El Niño–Southern Oscillation is linked to the nonlinear coupling between the characteristic time in an unforced system and the seasonal forcing (cf. Chang et al. 1995; Jin et al. 1994). Hence, considering the seasonal cycle as characterizing a periodic equilibrium climate, the Floquet theory seemed to be the proper tool for expressing feedback characteristics of such a dynamical system as well as for giving the Lyapunov exponents. Strong et al. (1995) used the Floquet theory on the CTLS built using a linearized version of a shallow-water-like model for the Northern Hemisphere. In their work, periodicity of the flow was obtained by extracting the mean seasonal cycle from a 100-yr-long run of the full model. In general, the sign of the real part of each Floquet exponent gives the stability along the cycle component—that is, whether a perturbation of initial conditions is amplified as it propagates from one year to the next. The interest of the Floquet analysis is to give intraseasonal activity of each Floquet component that can be associated with climate mechanisms. The leading perturbation pattern found in Strong et al. (1995) corresponds to an instability with a growing characteristic time of 54 days associated with the winter jet stream’s activity. The cited article is at the origin of our interest for the Floquet analysis.

Our hope beginning this work has been that the Floquet analysis, as decomposing a system response into elements with distinct dynamical characteristics (Floquet 1883), would provide results as simple as the fast-pole–slow-pole decomposition of Hallegatte et al. (2006, hereafter HLG): the water-cycle (W cycle) feedback was found to exhibit a positive component with a characteristic time of 4.8 years and a fast negative component prevailing in the global feedback sign even after the third year of application of the forcing. We will show that such a simple result can be retrieved for the periodic climate thanks to the Floquet theory. The Floquet theory is applied considering a perfectly periodic climate, but we will show that it might explain some remarkable features found in Part II with the step experiment.

It is important to address that although fast mechanisms are often associated with the clouds’ response (e.g., Gregory and Webb 2008), we have decided to keep the effect of clouds out of this analysis—not only because it is still uncertain among GCMs (e.g., Bony et al. 2006), but also to show that fast mechanisms occur independently of clouds.

In the next section, the formalism defining the CTLS already used in Part II is recalled in its generality. An original introduction of the Floquet theory in section 2b takes the particular point of view of the climate feedback issue. Section 3 gives the main characteristics of the toy model (mini-Clim) of the climate that, by providing numerical illustration, serves as support of the proposed methods. This allows us to illustrate how a wrong application of methods leads to false results. In section 4, the feedback-structured climate system is formally built, replacing the Laplace-transformed feedback system of Part I. The full dynamical aspects of the W-cycle effective response function are compared with the equilibrium ones. Finally, the Floquet theory is applied to the decomposition of the inclusive feedback components (IFCs) introduced in Part I to include their characteristic times.

Once the full set of elementary systems is established, it is integrated for the mini-Clim model and the results are analyzed in section 5. A comparison with the inverted Laplace effective response shows the analogy and the extensions brought by the Floquet analysis. Furthermore, the seasonal cycle of Floquet components allows an investigation into the way the global response builds up. The last section uses the methods developed in Part II to examine the fast, and subsequently slow, components of the W-cycle feedback in LMDZ, the GCM of LMD, and to make a comparison with our toy model. We end this analysis with a focus on the role of the lapse rate on the vertical extension of deep convection. General conclusions and perspectives for future work following this series of articles end this paper.

2. Floquet analysis

a. Circulating tangent linear system

Our application of the Floquet theory will be to the analysis of perturbations. We therefore first summarize how a tangent system is constructed from an original model in the context of climate systems. Let us consider the atmospheric-state-space model driven by an external forcing :
e1
where is the n-dimensional discretized state vector and g a nonlinear vector-valued function. When initial conditions are known, the (deterministic) system follows a trajectory.
Once a reference trajectory is followed, small perturbations to system (1) are solutions of the system obtained by a simple linearization of Eq. (1) around its trajectory:
e2
In this equation, is the deviation of the perturbed system from the reference trajectory at time t, caused by a perturbation starting at , and is the Jacobian matrix of the system.2 The initial forcing may have large-amplitude variation, so that is varying along the trajectory. We call Eq. (2) a circulating tangent linear system as in Part II, referencing the fact that, as the state circulates along , the Jacobian matrix is known and, in the linear assumption, is independent of the perturbation. This linear assumption, admittedly, allows us to analyze responses to small perturbations of the seasonal cycle. However, it opens up the possibility to make, as we shall see, useful decompositions. In our series of papers, it represents a second methodological step before addressing the full theory of both time-dependent, nonautonomous dynamical systems and of random dynamical systems (e.g., Ghil 2015).

When the system is slowly evolving as a result of a moderate forcing, can be taken as constant along the perturbed trajectory. This hypothesis has served in Part I to obtain the feedback form in the Laplace domain, as in Bode (1945). The method was used in Part II to analyze the slow response to perturbation in the CMIP5 experiments. In this case, linearity stemmed from low-pass filtering of the GCM results.

b. Introduction to the Floquet analysis

As we are considering here periodic climate models, Floquet theory, which applies to systems with periodic Jacobians, may be introduced as a tool for further understanding model sensitivity. The formal solution to system (2) is given by
e3
The state-transition matrix , or propagator, gives the perturbation at time t knowing it at time τ. It is the solution matrix of the following system:
e4
with the identity matrix (cf., e.g., Luenberger 1979).
Since the system is periodic, so is the Jacobian matrix , that is there is a period T for which . Moreover, as shown in appendix B, periodicity of translates to the propagator
e5
The periodicity relation of Eq. (5) along with well-known properties of the propagator renders the particular propagator from t to quite interesting, as is seen by the decomposition
e6
The last expression shows that the state-transition matrix on T-lag remains similar to along t and, hence, keeps the same eigenvalues. This essential property is at the foundation of the Floquet theory. We call this T-lag matrix 3. If the initial perturbation is taken to be an eigenvector of , the response after one period by a system with forcing is again aligned with , however with norm multiplied by the eigenvalue ( will be referred to as Floquet multipliers). Similarity of to for all t implies this remains true of the response after one period to a perturbation by any eigenvector of at time t. Moreover, periodicity of the propagator and thus implies has the same eigenvectors as . Therefore, continuing to advance the response to perturbation by an eigenvector of at time t, one finds that after n periods the response is aligned with with norm multiplied by .
By taking and decomposing it into its eigenelements, we extract the basic elements of the Floquet theory:
e7
where is the matrix of the right eigenvectors in columns and is diagonal, with coefficients the Floquet exponents. When is less than unity, the system is stable for initial perturbations along —in this case , which is possibly complex, is of negative real part.

c. Floquet vectors and exponents

The Floquet vectors (FVs) are defined to be the column vectors of
e8
Each column gives the perturbation trajectory from the corresponding but affected by the factor , which renders it periodic: as it is shown in appendix B. This can now be used to decompose the propagator into Floquet components: it is straightforward to verify that
e9
with as the matrix of the left Floquet vectors, cross normed with the right ones. With this decomposition of the propagator we can separate the response into components associated with the various characteristic times. This is made clear using the Schür sum decomposition of Eq. (9):
e10
As an illustration of the Floquet theory, let us consider a Poincaré section containing two FVs anywhere on the trajectory, say, at t. In this plane, a perturbation corresponds to a shifting of the origin—where the unperturbed trajectory crosses the plane. The corresponding CTLS will be given by . Taking the Floquet vectors as eigenvectors of now means that if the initial perturbation is taken along a (real) Floquet vector, it will remain aligned with that vector in the Poincaré section, with a distance to the origin multiplied by the Floquet multiplier at successive periods.

Theoretically, n Floquet vectors can be determined for a system of n state variables. In practice, only a few are numerically tractable, because each perturbation profile column in must be only partially damped by the end of the first period. Hence, a Floquet vector corresponds to a fundamental perturbation of the system that impacts the second period at least. In the climate analysis, it will inform on intraseasonal processes propagating their influence to the following years: in that sense, Floquet analysis is climatological and not meteorological. With a system of 231 state variables model of the Northern Hemisphere, Strong et al. (1995) determined that only three Floquet vectors are effective.

3. Illustration with mini-Clim

The Floquet decomposition is now demonstrated here with a 1D toy model (mini-Clim) derived from HLG. It has been rewritten in a new solver called mini_ker,4 which symbolically determines all partial derivatives and hence the Jacobian matrices of the system. The model has four state variables: three temperatures of a 50-m mixed ocean layer, of a tropospheric layer at constant lapse rate (0.006 K m−1), and of a stratospheric layer, also with a constant lapse rate (−0.0035 K m−1). The last state variable is the tropospheric water content. Three levels of clouds are fixed and constant for the intercept of shortwave (SW) and longwave (LW) radiation. The LW exchange model is a 62-layer narrow-band Malkmus model with two gases: CO2 and H2O (Cherkaoui et al. 1996). Sensible and latent heat fluxes are classically modeled between ocean surface and troposphere; the precipitation model condenses the water vapor when the relative humidity (RH) exceeds a given target (), with a time constant of 3 months. Compared to the HLG version, we added a static layer for the ocean-surface temperature whose associated equation is
e11
The fluxes are, respectively, the SW and LW surface budgets, the sensible transfers with the atmospheric boundary layer and the ocean mixed layer, and evaporation.

The model is forced with a seasonal solar flux cycle and is designed to give plausible dynamics of the climate between 30° and 40°N over periods of a few weeks—namely, discarding the diurnal cycle and fluctuations with time scales less than a few weeks. In these circumstances, the static surface layer might be considered to include about 1 m of the ocean surface, so that the absorbed SW is divided between this layer and the ocean slab. More details and parameter values are found in appendix A.

With mini-Clim, we find the four real Floquet multipliers. The first is close to 0.75 (characteristic time of 3.45 years) and the second is close to 0.038 (3.7 months). This means that an initial perturbation of the four state variables taken as the leading Floquet vector (FV1) is damped to 75% at the end of the first period, or to 3.8% if the initial perturbation is FV2. The last two multipliers are less than 10−13, showing that they are associated with rapidly damped perturbation vectors having characteristic times inferior to 11 days. It can be concluded that the propagator is quasi singular; while this feature is of no consequence to the analysis in the leading Floquet vectors, it is a source of difficulty, as we will see, for the analysis in Floquet feedback components (section 4e).

The two leading FVs are shown in Figs. 1 and 2 for their three temperature components (ocean , troposphere , and stratosphere ) and for the troposphere water content Q. The characteristic times reveal a slow mechanism associated to FV1, which suggests the inertia of the ocean is involved, and a fast one compatible with the troposphere inertia associated to FV2. Since climatologists classically differentiate global mechanisms by their characteristic times, in this sense the Floquet analysis selects climate mechanisms. This selection is, however, subtle: the instantaneous characteristic times at time t is given by the eigenvalues of the Jacobian matrix , which is varying along the trajectory, because of the nonlinearity of the model. What is selected by the Floquet analysis is the integrated influence of a mechanism on the period. This is clearly seen with Eq. (7) and will be discussed in section 4e.

Fig. 1.
Fig. 1.

Four components of the leading mini-Clim Floquet vector throughout year for (K) and (mm). Thick lines are with SST feedback; thin lines are with forced SST. The tendency for the amplitude of the ocean temperature perturbation (blue) to be greater than that of the atmospheric temperatures (gray and red) suggests that FV1 is tied to oceanic mechanisms.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0364.1

Fig. 2.
Fig. 2.

As in Fig. 1, but for the second FV. In contrast with FV1, the greater perturbation amplitude of the atmospheric temperature (red and gray) relative to the ocean temperature (blue) during the majority of the year and the larger amplitude in tropospheric water content (green) suggest that FV2 is tied to atmospheric mechanisms.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0364.1

Concerning the perturbed state variables, there are two interpretations of Floquet vectors. The first interpretation looks at the perturbed trajectory following the initial perturbation. In that case, the damping factor is to be applied to obtain the real perturbation trajectory: while FV1 is moderately damped, FV2 is rapidly decreasing.

Alternatively, if a vector is to be associated to some mechanism, the initial relative amplitudes5 of its components should select variables to perturb preferentially the considered mechanism. In this interpretation, if one compares the first two FVs, the initial FV1 points to the ocean temperature more than FV2. On the contrary, FV2 focuses more on the tropospheric water content, and is almost always well above the ocean temperature in FV2.

This interpretation is true all along the trajectory (at any time t) because it gives the initial perturbation that would have to be applied if the run were beginning at this time. Looking individually at the vectors, the FV1 ocean temperature perturbation tends to be greater than or equal to the atmosphere; it is the opposite for FV2, except during the warm season. These observations suggest that the ocean is driving the atmosphere for FV1, while for FV2, the atmosphere is a source of perturbation for the ocean, in accordance with its faster time characteristic—3 months, closely linked to parameter in Eq. (A7) for precipitation.

By following the variation in the initial perturbations to apply if the run were beginning at different times t over the year, that is looking at the periodic Floquet vectors, one is able to zoom in on seasonal features. For instance, we see that the initial warming applied to the two ocean temperatures, and , are slowly enhanced from March to September, while both tropospheric temperature components are cooling, especially for FV2: these atmospheric temperature changes even disappear mid-September. The same March–September period shows Q becoming the strongest driving perturbation. Figure 3 shows the precipitation energy released to the troposphere for the two FVs, which reveals these underlying features. During June and July, the amount of precipitation reaches a maximum, and perturbation of Q is left as the only variable that can still provide an increase of precipitation resulting from FV2. In this same period, we see in Fig. 1 that the slow component (FV1) is selected by a warming of the ocean in combination with a decrease of tropospheric temperature and an increase of Q. Figure 3 clearly shows that this causes an increase in precipitation coming from the slow component toward the end of the summer. The water content also has a greater amplitude in FV2, suggesting that precipitation is associated to this vector. Thus again one deciphers that FV1 is the perturbation giving the slow component of ocean-warming impact on the atmosphere, while FV2 is linked with the impact of precipitation on the atmospheric temperatures.

Fig. 3.
Fig. 3.

Impact of the mini-Clim Floquet vectors on precipitation—in W m−2 as the energy release to the troposphere (a factor of 3.456/100 converts to mm day−1). The unperturbed precipitation amount along the trajectory (black curve; scaled) is also given for reference. Thick and thin lines are as in Figs. 1 and 2 for, respectively, free feedback from SST and forced SST.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0364.1

While this application in mini-Clim has served to show how, using Floquet theory, a climate response to perturbation may be decomposed into components associated with various characteristic times, we have yet to discuss how further information on the sign and strength of climate feedbacks may be given by the Floquet analysis.

A first application to feedback analysis

In this direction we give now an application of the Floquet theory that sheds light on a controversy about the sign of atmospheric feedbacks from GCM experiments: Lindzen and Choi (2009, hereafter LC09) compare feedbacks determined from ERBE (1985–99) data and from the Atmospheric Model Intercomparison Projects (AMIP)6 of the Intergovernmental Panel on Climate Change (IPCC) AR4. Their analysis is made with forced SST and this has a strong influence on the model feedbacks since the SST feedbacks are no longer active. One can ask in this case if the model feedbacks remain climatologically relevant. We will only discuss here the model feedback issue—other critiques of the article can be found (e.g., Chung et al. 2010; Trenberth et al. 2009b). We have run mini-Clim in the forced-SST mode and recomputed the Floquet vectors, shown as thin lines in Figs. 13. The mixed-ocean-layer thermal capacitance has been multiplied by 106 to prevent the SST to respond to atmospheric forcing.

The consequence for FV1 is to be more concerned with the SST, which suggests this vector is closely related to the maximum inertial ocean. In coherence, forcing the SST leads to disconnect FV2 from the SST. Also, a part of the two atmospheric temperatures’ perturbation has moved from FV2 to FV1: because the fast feedback from SST is suppressed, part of precipitation also moves to FV1 (see Fig. 3). The consequence for exciting FV2 is left to Q, and indeed part of Q has moved from FV1 to FV2. The subtle link between precipitation and the two FVs shows that even if precipitation is still to be associated preferentially to FV2, the slow feedback also incorporates some part of this mechanism—one may guess it is from evaporation. The mathematical translation of this is that the two FVs are not orthogonal: the angle between them varies between 43° and 26°. We shall return to this feature in section 6 when we have the Floquet feedback decomposition and the GCM results available.

How we could use Floquet theory to characterize the strength of feedbacks is still an open question. One simple idea is to use the regression method as in LC09, Part I, and Part II. For instance, suppose we are estimating the feedback from any impulse radiative forcing that incites a response too fast to be considered by a model that is built to take account of slower mechanisms. In mini-Clim, which is built on a weekly time scale, this could be any response faster than a week. In this case the response may be rendered as immediate and our model would be perturbed by initial conditions —analogously to the forcing of the L2Obs regression in Part I.

Since the four Floquet vectors form a basis of the state space, can be decomposed on this basis. Each Floquet component of will give a trajectory following the periodic Floquet vector with decay . So and ΔOLR coming from a Floquet component have the same function of time and it follows that regression between them is meaningful if filtered to remove the modulation by the periodic functions.

The regression curves for the two leading FVs are seen in Fig. 4. The thick lines give the two trajectories running over five periods. We now clearly see the qualitative difference between the slow and fast feedbacks: while the mean slope of FV1 is about 2 W m−2 K−1, FV2 has a much higher slope. This is clearly discriminating between the signs of feedbacks associated with the two FVs: one is under the Planck slope, meaning a positive feedback, the other is above (negative feedback). The same is shown with the blocked ocean model (thin lines): FV1 is cycling with no damping,7 and FV2 is now entirely on the cooling side.

Fig. 4.
Fig. 4.

Regression curves between the OLR and perturbations of the two leading Floquet vectors. In accordance with the previous figures, thick lines (red and blue) are with SST feedback; thin lines (black and blue) are with forced SST. A comparison between their respective slopes and the Planck slope (black) highlights a qualitative difference between the feedbacks: the mean FV1 slope lies below the Planck slope, indicating a positive feedback, while the mean FV2 slope lies above, indicating a negative feedback.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0364.1

What is the proper interpretation of the forced-SST run? The conclusion of Trenberth et al. (2009b) on the LC09 procedure is “The use of models driven by observed SST to diagnose radiative feedbacks is questionable because feedbacks on the SSTs are not included” (p. 4). This is seen with the Floquet decomposition: FV2 only partially remains when the SST is forced. This conclusion that there is an absence of SST feedback in LC09 is, however, itself questionable because the forced climatological SST is indeed responding on a monthly basis to radiative forcing and to the real atmospheric forcings; if the atmospheric model is even partially correct, some of its forcing on the ocean should explain part of the forced SST. In this case, some part of the slow and positive feedback would remain in LC09 regression with the AMIP models, along with the monthly influence of the fastest feedbacks. Lindzen and Choi (2011) proposed a modified version of their previous method, responding to critiques of their methodology and extending their analysis to CERES (2000–09) data. The new results confirm our previous conclusion as will be discussed in our concluding section.

As a conclusion concerning our objectives, we have shown that the Floquet separation into slow and fast components of the climate response gives us some qualitative indications. Still, the mechanisms associated to FVs are not linked to some specific feedback. What is needed is a decomposition of the feedback effective response function—the IFCs of Part I and Part II—into Floquet components.

4. Floquet theory and feedback analysis

It was explained in Part I why a system requires a mathematical framework and, moreover, a feedback-loop structure in order to unambiguously define feedbacks associated to mechanisms. A method for bringing a generic atmospheric state-space model into such a form was demonstrated. In this context, each mechanism is understood to be represented by a part of the global system—for instance, in Part II, we considered deep convection as a mechanism represented by Emanuel’s scheme (1991). Here, precipitation takes account of the W cycle and corresponds to one equation only [Eq. (A7)].

A summary of the needed operations to go from system (2) to a feedback form of Part I, following Lahellec et al. (2008), are recalled here because it will guide our treatment of the dynamical problem:

  • to obtain a feedback-structured system, select a test variable, and plug it in to the state-advance system;
  • define a no-feedback system as following the base trajectory (the Planck trajectory in Part I);
  • subtract it from the forced system to get rid of the forcing vector ; and
  • decompose the effective response into feedback components.
We now apply this program to recast Eq. (2) in the classical feedback form.

a. General definition of dynamical feedback

Following the preceding procedure, the original tangent system (2) is modified to become
e12
Here is a general row matrix that defines a test variable from the state vector. The system above corresponds to the CTLS of the feedback-structured system:
e13
where the column and row matrices and formalize the structure of the feedback loop built with φ as the test variable. Notice that contrary to Part I, we cannot extract a subsystem as the temperature system: has to be in the full-dimensional state space, as in the first appendix in Part I. For system (12) to be mathematically equivalent to (2), the equality must hold. This is fundamental to keeping the same results as the original system.
We first define the base-response system, which is obtained by canceling in Eq. (12), hence giving the following open-loop system:
e14
We then subtract this last system from the preceding one:
e15
where we have in parallel applied a change in variables, the new ones representing the departure from the base trajectory:
e16
System (15) is in the feedback form that gives what we called in Part I “the effective response” to the prescribed perturbation . It is linear because although the Jacobian matrices depend on t along the reference trajectory, they do not depend on the unknowns (the variables). The general solution to the feedback system is
e17
Because we know how to decompose the propagator into Floquet components, a new Floquet feedback theory follows. This theory is introduced in section 4d.

b. Application to mini-Clim

To follow our previous analyzes, we first implement the W-cycle feedback in our toy model. Here, the mechanism controlling the atmospheric water-vapor content (WV) is a modified version of the Betts and Miller’s (1986) precipitation scheme; that is [see Eq. (A7)],
e18
with a function of a target RH and .8 This mechanism, as controlling the WV of the troposphere, is representing the W cycle of Part II. Following our directives, we have to make a copy of , the test variable, in order to have a “feedback variable” so that this mechanism can be perturbed. We choose this test variable as the troposphere temperature, instead of surface temperature, to follow HLG and for the necessity of section 6 where we shall also use the forced-SST mode in the GCM. Our forcing term is taken constant to focus on the model dynamics. Taking gives a proxy of the Planck response, as shown in Part II (Fig. 10), without introducing the interfering dynamics of the Planck response. We run the model in two modes: the first at equilibrium, as in HLG, with four ocean thicknesses, and the second with the seasonal cycling. Figure 5 gives the results. Notice that the effective response at equilibrium uses a novel algorithm to invert from the Laplace domain to the time domain, explained in appendix C. Two features stem from Fig. 5. The asymptotic limit of the response is independent of the ocean inertia, and, the deeper the ocean, the more negative the transient response. The figure shows how the cycling response follows the transient of the equilibrium response and finally reaches in about 15 years the periodic cycle—shown zoomed in in the inset. Our goal now is to decompose this periodic response into Floquet components in order to characterize fast feedbacks and link this with GCM experiments.
Fig. 5.
Fig. 5.

Effective W-cycle feedback responses for mini-Clim in different modes: the cycling response is in red, while the response from the CTLS at equilibrium is given by the dashed blue line, superimposed with the response retrieved from eigeninversion (see appendix C) for a 50-m ocean mixed layer. The remaining three responses are obtained from eigeninversion for different ocean thicknesses. (inset) The periodic trajectory reached by the cycling response after about 15 years is shown, along with the equilibrium response.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0364.1

c. The evolution of the effective feedback components

The application of Floquet theory to feedback analysis is not straightforward, mainly because the Floquet analysis is based on initial perturbations of the state variables: each ith column of , as an initial perturbation, results in a perturbed trajectory following the corresponding Floquet vector. After one period, the initial amplitudes are affected by the Floquet multiplier . Hence, for stable systems, the initial perturbation will be completely damped after a few periods. On the contrary, feedbacks respond to continuous forcing along the trajectory, and after a transient period, it can be foreseen that a limit cycle is reached (Fig. 5). In this context, we are willing to characterize the asymptotic limit of the effective feedback response:
e19
which is . The scalar function is any periodic forcing applied to , which, for instance, can be taken as the asymptotic Planck periodic response . Notice that the system is Eq. (17) with unperturbed initial conditions.
In the case of a stable system tends to the null matrix as n increases and exists:
e20
where we let —initial matrix in section 2b. This is derived in appendix B. Equation (20) shows that is integrated like only now with nonzero initial conditions. This keeps the numerical computation simple as the same algorithm may be used to determine both and . Essentially, we have determined which initial conditions render periodic. From Eq. (20) one sees these are .

Periodicity of is verified in appendix B; see Eq. (B5).

d. Floquet feedback components

Now using the Floquet decomposition of the propagator already established in Eq. (17), we may separate the effective feedback response into Floquet feedback components:
e21
and likewise their asymptotic limit:
e22
With Eq. (21), the Floquet feedback theory is based on a controlled system. Instead of perturbing the initial conditions to follow a Floquet vector, here the forcing that perturbs the feedback loop is projected on each FV to control the trajectory.
The total response is also of interest, because we shall meet numerical difficulty to determine the two last Floquet components. The sum on Floquet elements of initial conditions can easily be changed to depend only on the well-defined p first elements:
e23
with .
This general feedback system associated with the test variable is in consequence submit to a double decomposition:
  • a decomposition of in mechanisms (i) gives the IFCs (notation different from Part II9), and
  • the decomposition of propagator in its Floquet elements (k) gives the periodic elements of each IFC as .
The dynamical components are defined from this dual decomposition:
e24
For the ith feedback mechanism, the vector contains all the state-variable profiles of the limit cycle of perturbations. Each Floquet component adding a specific function of time associated to a Floquet vector—which hopefully can serve to aid further analysis of the dynamics of the feedback mechanisms. With this decomposition, one finally obtains dynamical components of each IFC with phases and amplitudes evolving through the season.

e. Numerical aspects of the system giving the periodic IFCs

With the large difference between Floquet multipliers that occurs (see section 3), the condition number of matrix is above 1012. Using 64-bit FORTRAN floating-point precision, this is not a problem for the computation of the right FVs from Eq. (8). However, we need to compute the left FVs as well and this cannot be done by inversion if the full matrix is not known. Alternatively, left FVs would need the adjoint model to be determined. Moreover, even once the left and right FVs are known, further integrations in time that are needed to compute , etc. rely on an initial monodromy matrix that already has bad conditioning. When integrating Eq. (8), for instance, the results diverge after t = 220 days, regardless of whichever method was tried.

Therefore, after many different algorithms were tried [one is described in Reimer (2012)], we chose finally to determine the FVs as eigenelements of because it allowed us to test different algorithms, just by checking that the eigenvalues remained constant. For instance, two contrasting time schemes can be used:
e25
In both cases the conditioning is bad because, while the first two eigenvalues are dominant in left factors, the two smaller are dominant on the right. Using the Schür sum on both sides, we get
e26
where are the eigenelements of Jacobian and those of . These forms point to how together the eigenelements build up the propagator, and it appears quite surprising that the resulting eigenvalues can remain constant. We eliminated the numerical problem using a physical argument: one can accept that the responses of longer characteristic times combine together to form the overall long-term response and, moreover, that the shortest responses should have no influence on them. Therefore, in both cases, we dropped the two fastest characteristic times to determine the first two eigenvector couples. With method 2, this has the consequence of keeping the first and second eigenvalues constant with, respectively, 0.1% and 4% precision (not shown). The response in the FV1 oceanic temperature is given for both methods in Fig. 6. The blue curve, obtained with the second method, coincides with the one obtained with the full set of eigenelements (cf. Fig. 1). We see, on the contrary, that with the first method (red curve), the two slowest time elements in the instantaneous Jacobian matrix are not sufficient to build up the longest response time. There is in consequence a loss in precision on the eigenvalue (not shown).
Fig. 6.
Fig. 6.

FV1 oceanic temperature determined from two algorithms [Eq. (26), with ]: algorithm 1 in red and algorithm 2 in blue. The second method only is successful to produce a correct ocean temperature—compare to Fig. 1.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0364.1

The converse method can be tried to determine the two last FVs—now keeping in Eq. (26). The result, not shown, is very noisy but gives a global accumulation of points along a curve that gives a proxy of FV3 and 4. An intriguing result is that eigenelements 3 and 4 also give a noisy signal compatible with FV3 and 4 when the two slowest e-folding times are kept. This contradicts our hypothesis and finally illuminates the subtlety of building up FVs in the model.10

5. Analysis of the W-cycle feedback with mini-Clim

a. Floquet components of the water-cycle feedback

Figure 7 shows the response in OLR and to the W-cycle perturbation obtained by resolution of system (24). Let us recall that the initial conditions are obtained from the response to the constant forcing of 1 K on the test variable, which is a copy of the tropospheric temperature [ in Eq. (A1)]. Both the total response and its decomposition in the first two Floquet components are given. The slow component (coming from FV1) clearly indicates a positive feedback, while, with the exception of summer when it leaves the slow-component alone, the fast component (coming from FV2) displays a negative one. The opposite cycling of these two components results in a total response which is one order of magnitude smaller than the components. Moreover, the components build up a total response that is almost constant along the seasonal cycle, with its slight oscillation coming from the fact that the seasonal variation is greater in the fast component. Because the climate is cycling at equilibrium, the yearly mean of the OLR perturbation cancels.

Fig. 7.
Fig. 7.

IFC Floquet components of to a W-cycle perturbation with a forcing of 1 K. Total responses are also shown (black and light blue lines for OLR and Tτ, respectively). The first (slow) component clearly produces positive feedback (red and blue lines), while the second (fast) component produces negative feedback (green and yellow lines).

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0364.1

Comparing now the precipitation cycles (Fig. 8), similar features occur: there is a fast negative component, a slow positive one, and the seasonal variation is greater for the fast component. The surface temperature cycle of the slow component, with its positive values and weakening in the summer, clearly confirms that FV1 is at the origin of a positive greenhouse feedback (the higher the specific humidity, the less sensitive the greenhouse effect to perturbation; see ΔOLR in Fig. 12). On the contrary, the fast response in temperature is small or negative, with the exception of summer. It shows, however, a strong negative impact on precipitation, 4–5 times the value of the slow component—this is about 10% of the yearly mean precipitation amount in mini-Clim for only 1 K of forcing. This mechanism can be explained by the decrease in relative humidity accompanying the forced atmosphere warming, which, in turn, results in a decline in precipitation as well as latent heat released to the atmosphere. We see that both components are equally important to explain the seasonal cycle of the perturbed climate. The summer features in all variables can be understood by the fact that when the surface is warmer than the ocean mixed layer, the inertia of the system is lower, so that the fast components of perturbation are privileged.

Fig. 8.
Fig. 8.

As in Fig. 7, but for precipitation rate (mm day−1) and surface temperature (K). The slow components (black and red) again show a positive feedback, opposite to the fast (blue and orange) feedback.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0364.1

b. Comparison with the Bode analysis

One may note that because perturbation of mini-Clim may be analyzed both from equilibrium and cycling runs, we may make a comparison of the Floquet analysis and the Bode analysis at Equilibrium in HLG. This comparison is summarized by the two functions
e27
e28
that can be obtained in the case of real exponents (cf. Fig. 5), with . The equilibrium response increases until it reaches the constant value , while the cycling increase is modulated by the , which is periodic, to finally follow the periodic function of time . The same form of decomposition is recovered when functions and are known on one period from the Floquet analysis.

6. Fast and slow components in the perturbed GCM step experiments

When the forced-SST runs of CMIP5 were analyzed in Part II, it was found that fast mechanisms led to a negative precipitation response, just as we find with the fast Floquet component here. This suggests that an interesting comparison can be attempted between the GCM responses to radiative forcing and mini-Clim. The GCM that we have used is again LMDZ5A (cf. Hourdin et al. 2013), with the implementation of the CTLS numerical method as described in Part II. Concerning this method, it suffices to say here that the CTLS results could be obtained via a two-run difference method, provided one could perfectly nudge the perturbed GCM to the unperturbed wind fields, meanwhile keeping the clouds unchanged. We note also that the CTLS incorporates a 50-m mixed ocean layer coupled with a periodic climatological SST. Thanks to this implementation, the CTLS response was shown to be essentially linear for a 2 × CO2 step forcing and therefore a comparison with a true CTLS simple model is meaningful. Because the parameters of mini-Clim were adjusted to mimic the GCM climate between the latitudes of 30° and 40°N, we have taken the average of the “Katioucha” results on this band [Katioucha is the name given to the CTLS code in LMDZ (its implementation is described in appendix C and section 5 of Part II)].

We begin our comparison by looking first at the broad qualitative behavior of the forced-SST response with Katioucha. Since forced-SST runs eliminate the thermal inertia of the ocean, only the fast components are left. Figure 9 displays the seasonal trajectory obtained during the last seven years of a 15-yr step experiment in the ΔTτ–ΔOLR plane and Fig. 10 shows the seasonal trajectory in the ΔTτ–Δprecips plane.11 The results from these 7 years are averaged month by month to give a quasi-periodic response. These figures show clearly that the fast components produce a negative feedback behavior, as in mini-Clim. Recall that Katioucha was built to give the response of the W cycle, which is essentially composed of the deep convection (DC) parameterization from Emanuel (1991) and a representation of the large-scale condensation process (LSC) (Hourdin et al. 2006). The cycle response of DC alone gives the blue curve in Fig. 10, which we see is noticeably less sensitive than the total. A remark concerning this feature and the feedback loop in mini-Clim is given later on.

Fig. 9.
Fig. 9.

and ΔOLR of Katioucha (CTLS of LMDZ) with 1 K of Planck forcing and forced SST, hence the fast component. Dots represent the monthly average of the results between 30° and 40°N for the last 7 years of a 15-yr trajectory; thin line is a month-by-month average of the 7-yr results, giving a quasi-periodic response. As with mini-Clim, we see a negative feedback from this fast component.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0364.1

Fig. 10.
Fig. 10.

As in Fig. 9, but for total precipitation (black) and precipitation from deep convection only (blue). Both responses show a negative feedback, as in mini-Clim.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0364.1

Returning to the total forced-SST results, we find that, if normalized through a division by the Planck response12 and averaged the same way as , they have amplitudes with order of magnitude comparable to the fast Floquet component. This allows the more precise comparison shown in Fig. 11. Moreover, our analysis of the characteristic times of the GCM-CTLS fast responses (see appendix D) shows that they nearly agree to a unique one:13 30–34 days for , , and the TOA radiation budget. We may therefore consider the forced-SST run experiment in the GCM-CTLS a proxy of the second Floquet feedback component in mini-Clim.14

Fig. 11.
Fig. 11.

Seasonal cycle of the fast (second) IFC components of mini-Clim (black and blue) compared with normalized and averaged forced-SST results from Katioucha (red and green) for and dRad, the TOA budget. The amplitudes of the results are the same order of magnitude, allowing further comparisons between the models.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0364.1

It follows naturally to ask “Can we compare the slow responses as well?” Because of the linearity found with Katioucha, it suffices to subtract the fast response from the total response to obtain the slow one. Figure 12 gives the resulting seasonal responses alongside the slow mini-Clim Floquet component. There are differences in the amplitudes, in particular with the mini-Clim jump in both parameters at day 250, but, again, not only is the same sign of feedback found, the seasonal variation also looks roughly similar.

Fig. 12.
Fig. 12.

As in Fig. 11, but for the slow components. Slow components in Katioucha are obtained by subtracting the fast components of Fig. 11 from the total response (red and green).

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0364.1

A comparison between Figs. 11 and 12 highlights the slow component’s connection to the greenhouse effect. In both models, the slow component points more to dRads (the change in TOA radiative budget) than the fast component—even if this is considerably less obvious in Katioucha. The connection is further confirmed by the reduced sensitivity, in both models, of the slow-radiative component in the summer. Dwyer et al. (2012) analysis of how variation in the seasonal cycle is affected by a warmer climate showed that a reduced seasonal amplitude in temperature is a robust features of CMIP3 models. Our results build on theirs, adding that it is not that the warming is smaller in summer, but that the slow and fast IFCs are weaker.

Finally, the results from Katioucha confirm that the W cycle can be thought of as having two main features: a slow (decadal) component that gives the WV greenhouse effect and fast (intraseasonal) features essentially due to the negative precipitation response. One can notice the close parallel of our use of the GCM step experiments with Colman and McAvaney’s (2011). There the authors also take the fast response from a forced-SST experiment with an abrupt 4 × [CO2] forcing, while the slow one is obtained from a standard CO2 concentration but with the perturbed SST. From the forced-SST results, the authors conclude that “Linear regression suggests a small but (just) significant negative [fast] response from WV to CO2 forcing” (p. 1656).

Following the success of the comparison between components, we now compare the total responses in Fig. 13. Because of the opposite feedback effects of the slow and fast components, the results here are of small amplitude along the entire seasonal cycle. One can conclude that without the fast mechanisms, the oscillation in temperature response would be larger by a factor of 2 and warmer by more than 0.4 K K−1 (i.e., per 1 K of forcing).

Fig. 13.
Fig. 13.

As in Fig. 11, but for total IFCs (dOLR in mini-Clim is the TOA budget perturbation; SW is unperturbed).

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0364.1

The overall results are summarized in Figs. 14 and 15—3 years are given, showing the quite perfect periodicity obtained from system (24). Compiled in the ΔT–ΔOLR plane are all our findings, which show that the qualitative feedback features of the two models are analogous: negative feedback is associated with the fast components, positive feedback is associated with the slow ones, and a total response with much smaller amplitude than the components. Contrary to mini-Clim, the yearly average of Katioucha’s change in OLR in the 30°–40°N band does not cancel. This is due to atmospheric transport of energy and moisture and may also affect the GCM components. Again the phase and amplitudes are different, but on the whole, considering the difference in models, it is surprising how well they compare.

Fig. 14.
Fig. 14.

Seasonal trajectory in the ΔTs–ΔOLR plane decomposed on the four mini-Clim FVs for 1 K of forcing. Katioucha results from a forced-SST run and a standard 50-m-thick ocean slab. This highlights that the qualitative feedback features of the two models are similar: negative feedbacks are associated with the fast components, and positive feedbacks are associated with the slow component. The total responses are, consequently, of smaller amplitude.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0364.1

Fig. 15.
Fig. 15.

As in Fig. 14, but for the change in precipitation rate. While the fast and slow responses of both models exhibit qualitatively similar responses, their total feedback has opposite sign. This discrepancy provides insight into the role of the lapse rate in the W-cycle feedback (see text).

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0364.1

We recall finally that these results are obtained for the perfectly periodic climate. When fluctuations arise, as is the case for a realistic forcing, the strong fast mechanisms may initiate large oscillations, breaking the balance between the slow and fast parts. In that sense the components might explain the smallest striations and spiral patterns found in the regression lines of Part II.15

Lapse rate and the W cycle

A comparison of the precipitation responses (Fig. 15) shows that the two models’ results are consistent for the components: again the slow and fast components have opposite sign. There is a discrepancy, however, between the total responses. By inquiring into the source of this discrepancy, further understanding of the slow-feedback component is found.

In Part II it was hypothesized that DC was less active in the warmer climate but penetrated farther into the upper troposphere, which finally explained the increase of DC precipitation in LMDZ. From this hypothesis, two questions arise here: Can we make the distinction between DC and LSC in mini-Clim? Could the constant lapse rate of our toy model be the reason for the difference in total precipitation between the two models?

Considering our feedback loop dedicated to the precipitation model, we see that two parts are entering the parameterization: evaporation and excess humidity, yet only excess sees the feedback-test variable. If however we consider evaporation as the recycling of precipitation on continental surfaces (e.g., Trenberth 1999), we may endow the feedback loop with the influence of this process—this is detailed in appendix A after Eq. (A7). Now, excess would represent LSC and DC would be evaporation. This renders the change in precipitation slightly less negative in mini-Clim—see the red curve in Fig. 16—but has only a small impact on the final seasonal cycle.16 If we now reduce the lapse rate by 2.5%, we obtain a final increase of precipitation rate (black curve), with only a short negative transient, which approaches the response of the GCM-CTLS (see also Part II’s Fig. 9). By decreasing the lapse rate, we mimic the vertical elongation of DC in the GCM and thus obtain a total precipitation response similar to that of the GCM, supporting our hypothesis.

Fig. 16.
Fig. 16.

Changes in precipitation for three different mini-Clim lapse rate scenarios: the standard model (blue), the model with a modified feedback loop (red; see text), and the response after a reduction in the lapse rate by 2.5% (black). Reducing the lapse rate increases the response in total precipitation, from negative in the blue curve to positive in the black curve.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0364.1

Figure 17 shows that Katioucha’s DC penetration in the upper troposphere (red band), particularly enhanced from June to October, is coupled with a decrease of the lapse rate: decreases belove the 500-hPa level and increases above.

Fig. 17.
Fig. 17.

Seasonal slow response in the lapse rate [contours of ] and DC updrafts (shaded; kg m−2 day−1). The warm-season decrease of the lapse rate (increased temperature at 200 hPa) coincides with more penetrative updrafts.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0364.1

Therefore, we see finally that the slow-feedback component is explained by a combined impact of the WV greenhouse effect, lapse rate modulation, and penetrative convection. The fact that the slow and fast responses both include precipitation in their mechanism finds a parallel in mini-Clim. As we have seen in section 3a, the correlation between the two leading FVs results in an angle of 43° in winter, decreasing to 26° in summer. This consolidates our comparison with the GCM but also shows that the change in the lapse rate is not the only factor explaining how the greenhouse effect in the slow component is combined with precipitation: the nonorthogonality just tells us that each leading mechanism does not play the game alone. Our conclusion of this analysis is that without the change in the lapse rate, which is strongly dependent on deep convection being more penetrative in the GCM, DC precipitation would decrease in a climate warming episode. This decrease would occur because of the fast mechanisms’ diminishing of the relative humidity. The Emanuel parameterization is responsible for this feature in LMDZ (cf. Part II), but the same occurs with the Tiedke scheme used in Colman and McAvaney (2011); see their Fig. 3.

7. Conclusions

In this final paper of the series, we have used Floquet theory to decompose feedback response according to characteristic times, in particular allowing us to analyze fast mechanisms, which were revealed to be important for the climate sensitivity to forcing in Part II. We have found Floquet theory to be very useful for climate analysis because it selects characteristic times that can be attributed to different mechanisms. Selection is made from the analysis of state-transition matrices propagating the system ahead by one period. This way, slow and fast mechanisms are characterized on a climatic time scale of 1 year. The simple model mini-Clim was used to illustrate our decomposition because Floquet analysis requires the Jacobian matrices to be determined. We found two leading Floquet vectors (FVs) among the possible four. The first FV is associated with an e-folding time of about 3 years, which points to the mixed-ocean-layer inertia. The second, faster FV (3 month) follows the decay time of the precipitation model and points to the decrease of relative humidity when atmospheric temperature increases as a result of internal variability. Comparison between the two initial FVs and their associated transfer variables is informative because the relative magnitude of components helps indicate a mechanism. The selection of mechanisms is not perfect, however, because the FVs form a nonorthogonal basis—the angle between FV1 and FV2 did not exceed in our toy model.

As a way to complete our introduction of the inclusive feedback components in the first two parts, we furthered our decomposition to discriminate between fast and slow components of a response using the Floquet theory. To our knowledge, this is a new result within the Floquet theory. The method applied to mini-Clim verified that the slow mechanisms are positively contributing to the climate feedbacks, while the fast mechanisms contribute negatively. This analysis, obtained within the linear-response hypothesis, might possibly be extended to high-amplitude perturbations and corresponding nonlinear responses using Chekroun et al.’s (2006) averaging methods to determine the periodic solution.

Comparison with a GCM could be made thanks to the implementation of the Gâteaux difference algorithms (Katioucha) documented in Part II. Two runs were used: the abrupt 2 × [CO2] experiment with a slab ocean and a forced-SST one. This comparison allowed us to evaluate the method of extracting fast mechanisms by forcing the SST. The same qualitative results as the Floquet decomposition were obtained for the components, and with surprisingly similar amplitudes despite the huge difference between the models. We found that, in a latitudinal band of ±50°, the W-cycle fast e-folding time is about 1 month and that the slow one depends on the slab thickness (3 years for 50 m). So the forced-SST experiments can be considered as providing a proxy of fast Floquet-component determination, at least in the case we analyzed—that is, with fixed clouds and frozen dynamics. An interesting method for determining feedbacks from observational data is proposed in Lindzen and Choi (2011). They restrict the regression between SST and OLR anomalies to short sequences (1–4 months), verifying from a lag analysis that the change in SST is at the origin of the change in OLR. After our analysis, it is clear that they are retrieving fast feedbacks. Indeed, they find a negative climate sensitivity, in coherence with our results. They wrongly conclude that their method retrieves the global climate feedback: their finding that the best correlation lag is a lag of 1 month, which is the same value as of our fast characteristic time found with Katioucha, confirms that it can only be a fast feedback. Nevertheless, their method provides a way to determine fast feedbacks from observational data, which can be compared with forced-SST models, as are the AMIP models used in their comparison.

The same decadal characteristic times were also found from coupled model experiments in Part II, with an additional centennial time scale of about 100 years. As an intermediate time scale, a realistic seasonal cycle in GCMs is crucial to elicit the links between fast and slower feedbacks. The possibility of analyzing separately fast seasonal and slow interannual responses to climate warming should help intercomparison between GCMs, in particular in deciphering models’ deficiencies in producing realistic tendencies at the decadal or seasonal time scale (e.g., Chiyuan et al. 2014; Grose et al. 2014).

The biggest difference between mini-Clim and the GCM-CTLS was found in the total precipitation response to warming: it was negative in mini-Clim, whereas it was slightly positive in the GCM. This difference was finally attributed to the change in lapse rate, which is fixed in mini-Clim. Therefore, completing our analysis of Part II (section 5), our conclusion is that, with a fixed lapse rate, the warming climate might be drier and that the increase of precipitation rate is closely linked to the more penetrative effect of deep convection—in other words, to more and longer violent storms.

This brings us to the end of this series of articles in which the concept of climate feedback, as put in practice in the CMIP community, has been reviewed. A specific formal framework has allowed us to clarify definitions of forcing and feedback. Quasi equivalence between Hansen’s feedback-gain approach and feedback factors in the partial radiative perturbation approach have been demonstrated. The difference between climate sensitivity and feedback was pointed out in that the first is based on field perturbations while the second is on mechanisms.

This review was done with the goal of attaining an overall consistency between methods and theory. Current methods for determining climate sensitivity have been discussed with consideration for the characteristic times involved. This enabled us to give criteria for the application of the regression methods in order to avoid their misusage. It was found that applying the regression method to the step CMIP5 experiment provides consistent results, but only on the decadal-to-more range, while the ramp forcing was seen to be 30% as a result of the fast mechanisms. This is why we took the Floquet theory to analyze fast features. The seasonal cycle of the response was revealed to be a balanced interplay between two feedback components with opposite cycles. One can be surprised by the determination of a decadal time scale from only one seasonal cycle. How to do this was reviewed by Trenberth et al. (2009b) and negatively answered “regressions of radiation with SSTs in the tropics may have nothing to say about climate sensitivity” (p. 1). Do our results give hope? Indeed, if one could benefit from a constant—or possibly periodic—forcing, ignore the noise introduced by clouds, and determine a propagator from weather forecast analysis, it would be possible to do so. It unfortunately looks out of range to implement Floquet analysis in GCM or NWP models yet, so methods like the forced-SST one are important if they are able to deliver the same important time scale information. Perhaps specific limit conditions, initial conditions, or forcings could also be looked at.

Part II also introduced a new analysis tool that applies the Gâteaux difference to determine specific sensitivity of the global models to, among other things, forcing, parameters, and the representation of mechanisms. It should permit one to disentangle causal connections between forcing and feedbacks. It allowed us in particular to define and determine an unambiguous Planck response. This innovation could prefigure new tools to complement the development of GCMs. There is still a lack of methods to cope with the complexity of both climate analysis and climate models (Ghil 2001).

The complexity involved with nonlinearity and stochasticity (e.g., Ghil 2015) is far from being correctly undertaken in models. Waiting for new paradigms, such as the stochastic dynamical systems approach, to give useful climate prediction, the development of analysis tools, including some embedded in GCMs codes, is in our opinion the condition for Charney’s (1979) approach to cope with the increasing complexity of our actual state-of-the-art models. For the time being, many attainable problems are still on the table: tuning, to begin with, but also rational evaluation of parameterization, estimation, and adjustment of parameters, evaluation of model uncertainty, and all of the analyses of specific climate phenomena on the global and regional scales.

Acknowledgments

The first author wishes to thank J. Y. Grandpeix for discussions on deep convection and its modeling. We are grateful to Prof. M. Ghil and an anonymous reviewer for many critiques, suggestions, and references that allowed us to clarify some parts of our manuscript and brought up stimulating questions. Katioucha’s runs were made possible thanks to Philippe Weill and his team, in charge of the development and managing of Ciclad, the IPSL farm of computers.

APPENDIX A

The mini-Clim Model

This appendix provides further details on the mini-Clim model introduced in section 3. Recall the model has four state variables: three temperature variables, one each associated with a mixed ocean layer, tropospheric layer, and stratospheric layer, and the tropospheric water content. The three temperature layers’ energy equations are
ea1
where denotes the ith layer’s capacitance and are transfers to be defined next. The tropospheric water content is given by
ea2
where is the water-vapor latent heat. The boundary layer temperature is derived from the tropospheric temperature taken to be at :
ea3
with lapse rate . The equation giving the surface temperature is static and associated with the following constraint:
ea4
with the weight retaining 70% of the SW in this layer and the rest going to the ocean cell.
There are 11 transfer equations. To begin, the sensible flux equations linking the surface temperature to the ocean and troposphere temperatures are
ea5
The sensible flux coefficient is a constant transfer coefficient of 2.8 W m−2 K−1. The one between the surface and the ocean is not constant: when , it takes a larger value than in the opposite situation, with a smooth variation between the extremes (0.56–20 W m−2 K−1) in a 2-K difference.
The evaporation energy flux is a function of the saturation at and :
ea6
where R is the relative humidity (rh) in the boundary layer and is the Tetens version of the Clausius–Clapeyron function of temperature. The coefficient is taken to be 0.7 × 10−2 s−1.
Precipitation (latent heat released to the troposphere) is modeled as the sum of evaporation and a term maintaining the target relative tropospheric humidity , taken to correspond to , with a time lag of , which is the characteristic time of return to equilibrium of the water content:
ea7
With this model, relative humidity varies from 0.4 to 1 throughout the year. In the basic feedback loop of the manuscript, the dependence of on is made depending on the test variable instead, on which a perturbation of 1 K is applied. In the modified feedback loop at the end of section 6 on the lapse rate influence, the variable in Eq. (A7) is replaced by its expression given by Eq. (A6) with the dependence on instead of .

The SW solar forcing is a sine function of mean 340 W m−2 with period 365 days and a semiamplitude of 85 W m−2; the date corresponds to the spring equinox. Three layers of clouds intercept the SW and the LW fluxes. The coefficients are given in HLG, along with all the other parameter values. The SW flux is thus constantly apportioned between the three atmospheric layers and the ocean: there is no cloud feedback in this model.

The four LW budgets for the surface, troposphere, stratosphere, and OLR are computed using the Malkmus narrow-band model from Cherkaoui et al. (1996) on a vertical grid of 62 layers for the CO2 and H2O greenhouse gases. Each function is locally linearized in order to determine the coefficients of the Jacobian matrices.

Detailed justifications of the assumptions are discussed in HLG.A1 To globally verify the retained values of the parameters, we compared the global energy budget at equilibrium with the values given by Trenberth et al. (2009a); see Table A1. This gave an acceptable agreement. The seasonal cycle introduced in mini-Clim is adjusted to mimic the LMDZ mean climatic variables in the latitude band of 30°–40°N—both ocean and land surfaces included. This relied in particular on the adjustment of the extreme values of .

Table A1.

Global energy budget (W m−2) on ocean compared with Trenberth et al. (2009a). LH is latent heat, S is sensible heat, and ASR is absorbed solar radiation.

Table A1.

The system programmed in mini_ker is not as simple as Eq. (15), where we omitted the numerous transfer equations for simplicity. The full system reads
ea8
where is the vector of all transfer variables, including the test variable, and are partial Jacobian matrices coupling to . A semi-implicit time scheme is applied, giving the matrix system
ea9
where and is the time increment of X from t to . The system is solved each time step by eliminating using the Schür complement method:
ea10
solving for , computing , and using the (eventually nonlinear) transfer equation to determine iteratively.
The same algorithm is used to determine the propagator . Determination of the propagator is also useful to find the initial conditions to attain periodicity of the trajectory: because the period is imposed by the forcing seasonal cycle, the minimization is applied to the residual
ea11
When initial conditions are changed , minimizing the changed residual
ea12
gives iteratively. When the fit is applied in the double-precision version, the final residual obtainedA2 is less than .

APPENDIX B

Properties of the Floquet Decomposition

a. Propagator and Floquet vectors

To see how periodicity of the Jacobian matrix applies to the propagator [Eq. (9)], let us consider a fundamental matrix of solution of the CTLS, such that . That such a matrix exists is a basic property of any state-transition matrix. We have
eb1
showing that is advancing as , so that the two solutions differ only by a constant matrix. For the propagator, it follows that
eb2
This property of the propagator is used to define periodic Floquet vectors. From its definition in Eq. (7), we have
eq1

b. Asymptotic Floquet effective response

Taking the effective response defined in system (19) further along its trajectory, we find that after one period
eb3
and, by iterating on n, that
eb4
where we let . When cancels for as for stable systems, Eq. (20) is obtained.
Periodicity of may be verified using Eq. (20):
eb5

APPENDIX C

Effective Response Function in the Perturbed Equilibrium Case

The expression of the effective response in the Laplace domain is, in HLG,
ec1
A Schür decomposition of matrix gives
ec2
with the left and right eigenvectors written and , respectively, and represents the eigenvalues. The perturbed equilibrium final response is obtained for . To analyze the response during the transition to the final equilibrium, one has to transform into the time domain. Equation (C2) is an addition of simple fractions that are trivial to inverse to the physical domain. The unit-source Laplace transformed is , and an integration is needed after inversionC1 of the simple fractions. For the single real pole , one obtains
ec3
Figure 5 shows that the inversion in the 50-m thickness case gives precisely the same result as running the CTLS at equilibrium (dark blue line superposed with blue points).
In the complex-conjugate pair case, we note , , and :
ec4
with
ec5

APPENDIX D

Fast Characteristic Time in the Forced-SST Experiment

Section 6 uses the forced-SST run of the CTLS method in LMDZ to characterize the fast-feedback components. What are the corresponding e-folding times of the fast response in that case? The seasonal cycle must be filtered out to give the response times. Katioucha was run for 26 years in 13-month periods, in which each period began with a new step with zeroed CTLS variables. Figure D1 shows each response in total precipitation amount shifted to the beginning of each initial period. The black line gives the average and reveals an integral–exponent behavior, giving by fit an e-folding time of 30 days.

Fig. D1.
Fig. D1.

Fast precipitation anomalies (mm day−1) from 24 responses (thin colored) and their mean (thick black) exhibiting exponential behavior from which the fast characteristic time of 30 days is drawn. Results are taken between 30° and 40°N, from 1 to 390 days with an interval of 10 days on an arbitrary date.

Citation: Journal of the Atmospheric Sciences 72, 9; 10.1175/JAS-D-14-0364.1

Analogous results are obtained for (34 days) and the TOA radiation budget change. With the low number of abrupt realizations for each month (only two for a 26-yr run), we conclude that the fast characteristic time of the W cycle is about 30 days.

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1

Coupled Model Intercomparison Project, phase 5; cf. Taylor et al. (2012).

2

is arbitrarily chosen, and in the case it comes from —say from a radiative forcing, for instance—it is .

3

G. Floquet named it “la matrice de monodromie,” the “monodromy matrix.”

5

Floquet vectors are of norm unity by convention. The sign is arbitrary; we chose the ocean warming option.

6

Models are run using climatological SST data entered on a monthly basis.

7

From Eq. (45) of Part I, the e-folding time of FV1 is . Thus, with infinite inertia, this e-folding time becomes ∞.

8

The true Betts and Miller scheme does not include the evaporation term .

9

Let us recall from Part II that if retains all influences of the test variable on the system, it can be decomposed into different mechanisms—say sensible heat flux and LW flux, for instance—corresponding to parts of . Here, we shall only retain a unique mechanism.

10

A remarkable effect was found during our numerous tests: deterioration of right eigenvectors—and consequently of the left eigenvectors—seems to compensate in products using : there is a kind of “push–pull” effect in the numerical determination of them. Mathematical investigation of this compensation might be of interest for computing sciences.

11

After about 6 years, the response was judged to be free from the transient response.

12

The Planck-CTLS is computed in parallel to the W-cycle one in Katioucha; see Part II.

13

This is true only in the band ±50° latitude.

14

The fast Floquet-component e-folding time of 3.7 months in mini-Clim is essentially controlled by the arbitrary parameter .

15

Not the “Hiatus period,” which requires an ocean dynamical model. With mini-Clim, to be more precise, the W-cycle feedback is at the origin of striations; we found that another feedback loop is at the origin of spiral patterns, involving the surface fluxes.

16

Notice this is comparable with Fig. 10, which shows the weaker sensitivity introduced by DC.

A1

The listing of the model in mini_ker is also available in http://web.lmd.jussieu.fr/ZOOM/doc/Mini-Clim.car.

A2

Computed with the LaPack driver DGELS.

C1

We use the sign “⊂ ” to symbolize this transformation.

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