## 1. Introduction

The warming of Earth’s surface and lower atmosphere due to increases in greenhouse gases (GHG) is associated with enhanced middle-atmosphere cooling and a possible strengthening of the Brewer–Dobson circulation through radiative–dynamical coupling. Because both the air density and the optical depths of major radiatively active species decrease with altitude, the physical state of the middle atmosphere as represented by various parameters such as temperature and winds is quite sensitive to climate forcing and is thus an important part of the signature of GHG-driven climate change (e.g., Akmaev et al. 2006). Hence, a more accurate quantification of the middle-atmosphere response to solar variability and anthropogenic changes in trace species is necessary to improve predictions of climate change.

Complex global circulation models are often used to study the combined climate response to specified changes in carbon dioxide, ozone, aerosols, or the solar forcing. However, the problem of identifying the individual contribution to the total predicted climate response in temperature and wind patterns by a specified change remains. A sensitivity investigation, where only one change at a time is specified in a numerical model, can give some information but neglects potentially important interactions between the different forcing perturbations. Moreover, examination of the climate evolution documented by reanalysis products and observations needs to include all historic forcing changes. It will be useful to be able to interpret the total climate change as a sum of perturbations with each contribution to the sum associated with specific forcing perturbations. One approach to identify forcing contributions is to use a radiation algorithm to linearly map the energy perturbations associated with forcing changes at all altitudes to small-amplitude temperature perturbations. An example of this approach is the climate feedback–response analysis method (CFRAM) that separates and estimates temperature responses due to external forcing and various climate feedbacks in the coupled troposphere–surface system based on data or model outputs that have already included all the resultant changes (Lu and Cai 2009, hereafter LC09; Cai and Lu 2009, hereafter CL09).

CFRAM is formulated based on the atmosphere–surface energy equation, and it explicitly decomposes the directly observable temperature change into partial temperature changes due to individual external forcing and feedback processes (LC09; CL09).^{1} The unique feature of CFRAM is that this decomposition into partial temperature changes is linearly additive, such that the sum of the partial temperature changes gives a total temperature change that can be compared directly to the observed temperature change at every point in space. From a modeling perspective, the so-called external forcing and its variation are akin to independent variables or parameters that would be prescribed as input values in the model. On the other hand, the feedback or internal processes of a system are similar to dependent variables or parameters that often constitute a set of model output values.

In this paper, CFRAM is extended to the middle atmosphere based on three physical features of this region: (i) the air density varies with altitude by several orders of magnitude and the energy deposition per unit mass generally slowly varies with altitude or log pressure, (ii) radiative energy exchange that can be directly evaluated from satellite observations plays a major role in the energy budget, and (iii) the energy flux associated with Earth’s surface can be considered as external forcing to the layered middle atmosphere. As a result, the middle-atmosphere climate feedback–response analysis method (MCFRAM) is formulated using an energy equation per unit mass in the commonly used units of kelvins per day for the middle atmosphere. It can be applied to both satellite observations and output fields of three-dimensional (3D) chemistry–climate models (CCM) to derive various partial temperature changes.

Because of gravity and planetary wave forcing, the middle atmosphere is often far from radiative equilibrium. Dynamical effects make important contributions to the middle-atmosphere energy budget, either through the eddy heat flux divergence or through adiabatic heating due to vertical motions. Therefore, changes to the middle-atmosphere climate due to dynamics need to be considered in addition to radiative forcing. With the radiative contributions being explicitly calculated in MCFRAM, the circulation changes can either be evaluated as residual terms in the energy budget equation or calculated directly if the winds are available. The newly developed MCFRAM method allows us for the first time to attribute temperature changes or anomalies in the middle atmosphere derived from observations and from numerical models to individual processes such as the solar cycle, anthropogenic greenhouse gas increases, changes in ozone, and changes in the Brewer–Dobson circulation. In this study, we illustrate the utility of MCFRAM by applying it to observations made by the Sounding of the Atmosphere using Broadband Emission Radiometer (SABER), a broadband radiometer on board the *Thermosphere, Ionosphere, Mesosphere, Energetics and Dynamics* (*TIMED*) satellite, and to the output of the Goddard Earth Observing System chemistry–climate model [GEOSCCM; Pawson et al. (2008) and references therein] over one solar cycle. The objective of the analysis is to ascertain the main factors that are responsible for the solar cycle variations in the middle atmosphere recorded in the SABER observations. The analysis of the GEOSCCM solar cycle climate simulations allows us to gain further insight into the role of the eddy-driven residual circulation in the middle atmosphere in response to solar cycle forcing.

In section 2, we extend CFRAM to the middle atmosphere. Then, we perform an eigenmode analysis of the generalized damping matrix derived from MCFRAM. The middle-atmosphere temperature and ozone fields needed in the analysis are derived from the *TIMED*/SABER instrument. Section 3 applies MCFRAM to the SABER observations while section 4 performs a set of more detailed MCFRAM analyses of the GEOSCCM. Section 5 summarizes the results.

## 2. Review and extension of the coupled feedback–response analysis method

This section presents the derivation of the MCFRAM equations. After a brief review of the CFRAM assumptions, the column energy balance equation for the middle atmosphere is used as a starting point for formulating the new MCFRAM equations.

### a. Formulation of the middle-atmosphere CFRAM

**R*** and

**S*** are column vectors of the infrared and solar flux differences corresponding to the vertical profiles of total radiative cooling and heating in each layer, respectively;

**Q*** is the nonradiative energy flux difference in each layer;

**T**is temperature profile; (

*r*,

*s*, …) are the mixing ratios of radiatively active species such as CO

_{2}, O

_{3}, H

_{2}O, and clouds; and (

*α*,

*β*, …) are parameters such as the solar irradiance at the top of the atmosphere (TOA), surface albedo, and solar declination angle. The terms in Eq. (1) for CFRAM have the units of energy flux (W m

^{−2}), which corresponds to the integrated heating or cooling rate per unit volume in a given layer. There are several advantages of adopting the flux form with units of watts per square meter in the conventional CFRAM: (i) the energy flux of the atmosphere can be seamlessly merged with the surface energy flux, (ii) the TOA version of CFRAM can be directly compared to a TOA-based climate feedback analysis such as the partial radiative perturbation method (Wetherald and Manabe 1988), and (iii) the layer thickness of the tropospheric CCMs is usually slowly varying in mass so the heating or cooling rate perturbations per unit space of different layers also vary slowly with altitude.

*ρ*, and Δ

*z*being the specific heat at constant pressure, air density, and layer thickness, respectively,

**R**and

**S**are the infrared radiative cooling and solar flux heating rates, respectively, and

**Q**is the nonradiative heating rate excluding the molecular thermal conductivity

**T**(Banks and Kockarts 1973). The units of all terms in Eq. (2) are kelvins per day. We now consider two statistical equilibrium states, 1 and 2, with two different sets of corresponding atmospheric parameters, each set satisfying the energy balance equation [Eq. (2)]. In practice, these two states can be two ensemble, time, or spatially averaged states. The difference of the energy balance equations between these two states is

**R**and

**T**

_{2}. Substituting Eq. (4) into Eq. (3), we obtain

^{−1}) calculated from the temperature dependence of infrared radiative cooling rate and molecular diffusion.

In the original CFRAM, where the surface and the atmosphere are in direct contact and thus strongly coupled dynamically and radiatively, the discretization of the energy equation [Eq. (1)] and the derivation of the “Planck feedback matrix” ^{−2}). The middle atmosphere is not in direct contact with Earth’s surface. It can therefore be discretized solely based on a layered atmosphere in an energy equation per unit mass [Eq. (2)]. The effect of the energy flux emergent from the lower boundary on the middle atmosphere is primarily the radiative flux that can be considered as an external forcing. For example, the effect of the solar radiative flux can often be parameterized by an effective albedo of the surface and lower atmosphere (^{−1}) due to a unit change in temperature (*z* (horizontal row).

_{2}, O

_{3}, and H

_{2}O (Zhu 1994). Therefore, the term

_{3}and O

_{2}mixing ratios, respectively. Likewise, we also let

_{2}and H

_{2}O, respectively. The quantity

^{−22}W m

^{−2}Hz

^{−1}), which is a parameter representing the solar flux variations, and

**T**and

_{2}O and clouds in the troposphere that can both radiatively heat and cool the atmosphere.

*y–z*meridional plane. Under such circumstances, we may choose

**Q**in Eq. (2) is given by (e.g., Dunkerton 1978; Andrews et al. 1987)

*a*is Earth’s radius,

*ϕ*is the latitude,

*z*is the geopotential height,

The physical meanings of all **S**, leading to a well-behaved additive relation [Eq. (10)]. The components of **S** induced by **R** −**Q**_{mol})/∂**T** term to form a revised generalized damping matrix. One important reason that

Partial temperature changes [

The additive relation (10) for the temperature changes is an alternative expression of the energy equation [Eq. (3)] that is also additive. A linear transformation that isolates out the temperature variation from the energy difference on the left-hand side of Eq. (3) leads to the partial temperature differences as shown in Eq. (11) and allows us to derive this alternative relationship. The principal advantage of the additive relation (10) for temperature over the additive relation (3) for energy is that

### b. Eigenmodes of the generalized damping matrix and illustration of MCFRAM

*N*is the total number of vertical layers. Equation (18) indicates that the eigenvalues of

Infrared radiative heat exchange by CO_{2} and O_{3} makes a major contribution, whereas infrared cooling by H_{2}O makes a minor contribution to the radiative cooling rate in the middle atmosphere (Zhu 1994). Here, we use the **T** and *TIMED* satellite to derive **T** and *TIMED*/SABER averaged over a 54°S–54°N latitudinal range and the 12-yr period 2002–13. The SABER observations ranging from 20 to 110 km in the middle atmosphere are provided on a 0.7-km grid and are merged with the U.S. Standard Atmosphere 1976 (COESA 1976) in the troposphere. The radiative heating and cooling rate calculations based on the JHU/APL radiation algorithm are performed in the entire vertical domain of 157 layers, whereas the MCFRAM is applied to the top 129 layers (*N* = 129) that corresponds to the middle atmosphere, ranging from 10 to 110 km. The matrix **T** and

There are two distinct features shown in Fig. 2. First, there exists a strong scale dependence of the eigenvectors for the generalized damping matrix

The effect of the vertical structure of _{10.7} is increased from 60 to 260. The vertical structure of the temperature differences (Fig. 4b) is smoother than and significantly different from that of the heating rate variations (Fig. 4a). This is mainly due to the scale dependence of the generalized damping rate (

In summary, the MCFRAM equations allow for the expansion of local temperature changes in terms of changes in forcing and dynamics. The availability of more input parameters on the climate change results in more complete information on the individual contributions to the observable total change in temperature to be extracted from the MCFRAM equations. To derive the generalized damping matrix

## 3. Application of MCFRAM to *TIMED*/SABER observations

Application of MCFRAM is straightforward as shown in Eqs. (5) and (9) together with Table 1 once the input fields of various parameter variations such as CO_{2}, O_{3}, winds, and solar cycle forcing are available. While climate models (such as GEOSCCM) can provide all the needed input fields distributed globally and uniformly, satellite observations often provide only some of the needed fields to derive the balanced additive relation (10). In general, the role of observations is to provide a verification to model’s credibility in simulating the actual physical processes. The original CFRAM has only been applied to model output fields because the major energy sources of sensible and latent heating rates in the troposphere can only be derived reliably from models. On the other hand, radiative heat exchange is the major energy source in the middle atmosphere, and it can be evaluated accurately based on the observed temperature and species distributions. In this section, we show MCFRAM analyzed results by using SABER-observed *T* and O_{3} fields (Russell et al. 1999). We use the V1.07 SABER data available to the public from the *TIMED* mission data center (http://www.timed.jhuapl.edu).

Figures 5a and 5b show the zonal-mean *T* and O_{3} fields in the middle atmosphere derived from SABER observations in the low and midlatitudes averaged over the 12-yr period of 2002–13. Shown in Figs. 5c and 5d are the *T* and O_{3} differences between two time-mean states covering the periods of 2002–03 and 2008–09, respectively. Though the temperature change in the middle atmosphere exhibits a noticeable decrease from the 2002–03 period near solar maximum to the 2008–09 period near solar minimum over most regions, there also exist regions where temperature increases between the same two periods when the solar energy input decreases. We note that the observed temperature difference represents the sum of effects contributed by various processes including the solar flux changes due to solar cycle and man-made variations in

We now apply MCFRAM to the SABER-observed *T* and O_{3} difference between the two periods: 2002–03 and 2008–09. The corresponding mean *F*_{10.7} used in MCFRAM analysis for these two periods are [*F*_{10.7} ~ 167.1] and [*F*_{10.7} ~ 68.1], respectively. There are six yaw cycles in each year, with each yaw cycle covering about 60 days. With both ascending and descending measurements, it is possible to have complete local time coverage in one yaw cycle, such that averaging over even one yaw cycle should remove any undersampled diurnal tidal signals. The corresponding local time and latitudinal coverage in two yaw cycles separated by 6 yr are nearly identical. Care has also been taken to remove quasi-biennial signals. The temperature difference shown in Fig. 5c represents the observed temperature difference _{10.7} varying with separate yaw cycles. *T*, O_{3}, and *F*_{10.7} variations and using the JHU/APL middle-atmosphere radiation algorithm, the first three components of _{2}O and solar flux heating by CO_{2} near 4.3 and 2.7 *μ*m only make minor contributions to the radiative cooling and heating rate in the middle atmosphere (López-Puertas and Taylor 2001), we expect the sum of the above three terms to be approximately the partial temperature change due to radiative transfer

We note that the middle-atmosphere cooling rate by the CO_{2} 15-*μ*m band is mainly contributed from its cool-to-space component with its escape probability slowly varying with altitude in the middle atmosphere (Zhu et al. 1992). A uniform change in *μ*m-band infrared cooling. Since there are both positive and negative ozone variations between the periods 2002–03 and 2008–09 (Fig. 5d), the induced response _{3} 9.6-*μ*m-band, cool-to-space variations in a more transparent atmosphere. Here, we note that the middle-atmosphere climate responses to the cooling rate changes induced by CO_{2} and O_{3} variations are different.

The overall spatial pattern and magnitude of

Though

It is worth pointing out that the results shown in Fig. 7 also verify both the SABER observations of *T* and O_{3} and the accuracy of the JHU/APL radiation algorithm for the middle atmosphere. One common way of verifying measurements and testing radiation algorithms is to evaluate the global radiative balance (Kiehl and Solomon 1986; Olaguer et al. 1992). A good radiation algorithm requires the globally averaged net radiative heating rate to be much smaller than the typical values of the localized net radiative heating rate. A more stringent requirement for a good algorithm is that the heating or cooling rate respond sensitively to variations in radiation parameters while still preserving the property of its globally averaged net radiative heating rate close to zero in range of altitude where the effect of eddy transport is negligible. In other words, the condition of a vanishing globally averaged net radiative heating is not artificially imposed but naturally derived. Note that _{2}, O_{3}, and *F*_{10.7} and yet the globally averaged *T* and O_{3} fields are accurate as well.

Using only SABER-observed *T* and O_{3} fields MCFRAM has the capability to identify the specific sources of middle-atmosphere climate change. The MCFRAM analysis shows that CO_{2}-induced temperature perturbations peak at the equatorial stratopause, while O_{3}-induced temperature perturbations have a more complex structure and are sensitive to the specific O_{3} change pattern. In addition, the solar forcing response was shown to increase with increasing altitude. Large changes due to dynamics are also evident based on the MCFRAM equations.

## 4. Application of MCFRAM to GEOSCCM output fields

We now apply the MCFRAM to 2D zonal-mean fields derived from the GEOSCCM where several dynamical variables are also available from the model output. The 3D GEOSCCM uses the GEOS-5 atmospheric general circulation model (Rienecker et al. 2008) in its forecast model component, coupled with the stratospheric chemical solver developed as a part of the GSFC 3D chemical transport model (Douglass et al. 1996; Pawson et al. 2008). With respect to Rienecker et al. (2008) this version of GEOSCCM also includes a treatment of stratospheric aerosol (Aquila et al. 2012, 2013), a mechanism to generate the QBO using a gravity wave drag parameterization (Molod et al. 2012), and variable solar irradiance forcing (Swartz et al. 2012).

In general, the usual output fields of GEOSCCM or any other CCMs are not specifically designed for directly performing a full MCFRAM analysis. Additional processing of some of the output fields is needed in order to produce a set of appropriate input fields for MCFRAM analysis. One potentially important input parameter as shown in Eq. (7) or (9) is the effective albedo of the surface and the lower atmosphere _{2} variation is negligible in the energy budget, we will neglect

In this paper, we choose the same output periods of 2002–03 (near solar maximum) and 2008–09 (solar minimum) from one GEOSCCM simulation as those for SABER observations used in the last section to perform the MCFRAM analysis. In Fig. 8, we show the variation in effective albedo of the surface and lower atmosphere scaled by the diurnally averaged solar radiation as a function of month and latitude over the 24-month period. Also shown in the figure are the corresponding partial temperature change ^{−2} that is about 2% of the globally averaged solar flux and is about one order of magnitude greater than the variation in the solar constant over the 11-yr solar cycle (Lean 1991). Significant geographic and transient variations are present with peak values appearing near equatorial and summer polar areas, where the maximum mean solar fluxes are deposited. The corresponding

We now examine

_{2}cooling in the high-latitude regions in the Southern Hemisphere mesosphere where the coldest temperature often occurs near the summer mesopause (Lübken 1999; Lübken et al. 1999). This is caused by heat exchange between the warmer stratopause and colder mesopause when the CO

_{2}15-

*μ*m-band transmission behaves transparently and the summer mesopause receives net radiative heating from the stratopause (Zhu et al. 1992). An increase in

_{2}15-

*μ*m cooling rate near the winter polar mesopause as a result of the combination of local thermodynamic equilibrium conditions, a near-uniform temperature, and a near transparent emission to space (Zhu 1994). This too contributes to the strengthening in

_{3}and its variability are very sensitive to a strong nonlinear coupling between photochemistry and dynamics. For example, the largely off-set peaks in

The partial temperature change _{2}O may increase with time as a result of increasing CH_{4} in the troposphere (Zhu et al. 1999). Its decadal change could also be well correlated with the equatorial sea surface temperature and the associated cooling of the cold point to limit the direct entry of H_{2}O into the stratosphere (Solomon et al. 2010). The existence of large regions of both positive and negative

Figures 10g and 10h show the partial temperature changes, _{2} 15-*μ*m band becomes transparent above the stratopause among different layers of atmosphere, whereas the O_{3} 9.6-*μ*m-band emission is largely transparent in the entire atmosphere (Zhu et al. 1991, 1992; Zhu 1994). Figure 9i shows the partial temperature change due to the subgrid-scale eddies

In Fig. 11, we show the global average of

Using the more complete global model output fields, MCFRAM is able to make further breakdown in contributions by various climate forcing and feedback processes. The CO_{2} and solar forcing response patterns are similar to those found in the SABER observations, whereas the patterns in the more sensitive O_{3} response differ from the SABER results. Latitudinal changes in the H_{2}O perturbation response are also found. Additional partitioning of the dynamical term enabled us to reconfirm that the adiabatic heating induced by vertical motion makes the largest dynamical contribution.

## 5. Summary

In this study, we have extended the climate feedback–response analysis method (CFRAM) for the coupled troposphere–surface system to the middle atmosphere. The middle-atmosphere CFRAM (MCFRAM) is built upon the atmospheric energy equation per unit mass, with radiative heating and cooling rates as its major thermal energy sources. In addition, molecular thermal conduction is added to the energy equation when the upper boundary is extended beyond the mesopause. MCFRAM preserves the unique additive property of the original CFRAM in that, under linear approximation, the sum of all the partial temperature changes [

The newly developed MCFRAM is applied to two sets of zonally averaged data. One is the middle-atmosphere zonal-mean fields of *T* and O_{3} derived from SABER observations in low and midlatitudes averaged over 60-day periods. The other comprises the zonal-mean fields of *T* and O_{3} plus several dynamical variables saved from GEOSCCM simulations. Results show that MCFRAM can complement both observations and climate model experiments. It is found that the spatial structures of the response, _{2}, O_{3}, and solar flux are different. *μ*m-band cooling by O_{3} are strongly influenced by the O_{3} distribution, and they contribute to _{3} heating and cooling rates,

Because not all of the required parameters are present in the input datasets, especially those datasets derived directly from observations, the partial temperature change due to nonradiative processes (

## Acknowledgments

This research was supported by NASA Living With a Star Program under Grant NNX13AF91G and NASA Geospace Science Program under Grant NNX13AE33G to The Johns Hopkins University Applied Physics Laboratory. Comments on the manuscripts by three anonymous reviewers and by Jae N. Lee are also appreciated.

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^{1}

Given an individual energy flux perturbation, CFRAM allows us to evaluate, under a linear approximation, what the required temperature change is such that its induced infrared radiation alone would exactly balance the energy flux perturbation under consideration. For this reason, we refer to such a required temperature change throughout this paper as the “partial temperature change” associated with a given energy flux perturbation.