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  • View in gallery

    Box diagram of the local energy cycle. The boxes indicated by AM, AT, KM, and KT represent the mean and transient-eddy components of APE and KE. The arrows indicated by G(*), D(*), and B(*) represent generation, dissipation, and boundary flux terms in the energy balance equations. The arrows indicated by C(*, *) represent the energy conversion terms, and the wavy arrows indicated by F(AI) and F(KI) represent the interaction energy fluxes of APE and KE.

  • View in gallery

    Vertically integrated (a) APE density, (b) KE density, and (c) generation rate of APE for (left) CTL and (right) LGM simulations. Contour intervals are 30 × 105 J m−2 for APE, 6 × 105 J m−2 for KE, and 6 W m−2 for APE generation. Shading in right panels indicates deviation from the CTL.

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    Latitudinal profiles of vertically integrated and zonally averaged (a) total and zonal APE, (b) T-eddy APE, (c) S-eddy APE, (d) total and zonal KE, (e) T-eddy KE, and (f) S-eddy KE density. Solid and dotted lines indicate the values in LGM and CTL, respectively. Unit is 105 J m−2.

  • View in gallery

    Vertically integrated (a) transient-eddy APE (AT), (b) transient-eddy KE (KT), and (c) energy conversion rate from AT to KT for (left) CTL and (right) LGM simulations. Contour intervals are 10, 3, and 3 × 105 J m−2, respectively. Shading in right panels indicates deviation from the CTL.

  • View in gallery

    Vertically integrated (a) C(AM, AI), (b) C(AI, AT), and (c) interaction flux of APE and its convergence in the CTL simulation. Units for the conversion rate and flux are W m−2 and J m kg−1 s−1, respectively.

  • View in gallery

    (a) Zonal mean, (b) vertical section along 45°N, and (c) vertical integration of C(AI, AT) and interaction flux of APE in the CTL simulation. Units for the conversion rate are (a),(b) 10−4 W kg−1 and (c) W m−2. Units for the horizontal and vertical components of flux vector are J m kg−1 s−1 and J Pa kg−1 s−1, respectively. The vertical component of the flux vector is 100-fold in (a) and (b). Thin green contours indicate T-eddy APE density with contour intervals of 100 J kg−1, 180 J kg−1, and 5 × 105 J m−2.

  • View in gallery

    As in Fig. 6, but for the difference between LGM and CTL simulations. Thin green contours indicate T-eddy APE density at LGM.

  • View in gallery

    Vertically integrated (a) C(KI, KM), (b) C(KI, KT), and (c) interaction flux of KE and its convergence in the CTL simulation. Units for the conversion rate and flux are W m−2 and m3 s−3, respectively.

  • View in gallery

    (a) Zonal mean, (b) vertical section along 40°N, and (c) vertical integration of C(KI, KM) and interaction flux of KE in the CTL simulation. Units for the conversion rate are (a),(b) 10−4 W kg−1 and (c) W m−2. Units for the horizontal and vertical components of flux vector are m3 s−3 and Pa m2 s−3, respectively. The vertical component of the flux vector is 100-fold in (a) and 1000-fold in (b). Thin green contours indicate mean KE density with contour intervals 100 J kg−1, 200 J kg−1, and 5 × 105 J m−2.

  • View in gallery

    As in Fig. 9, but for the difference between LGM and CTL simulation. Thin green contours indicate mean KE density at LGM.

  • View in gallery

    Box diagrams of the local energy cycle corresponding to several typical locations shown in Fig. 12. Numerals indicate the vertically integrated values of the variables corresponding to Fig. 1. Upper and lower numerals correspond to the values in LGM and CTL, respectively. The values of R(AM) are included in the value of B(AM). Units for the values in the boxes and beside arrays are 105 J m−2 and W m−2, respectively. See the text for details.

  • View in gallery

    Maps of the conversion and convergence peaks of interaction energy over the (a) Pacific in CTL, (b) Pacific in LGM, (c) Atlantic in CTL, and (d) Atlantic in LGM. Color lines indicate the contour around the peaks of each quantity denoted in (a). Unit of the contours is W m−2. Rectangles denoted by the letters B–H indicate the areas where the energy diagrams in Fig. 11 are calculated. See the text for details.

  • View in gallery

    A six-box diagram of the global mean energy cycle. Upper and lower black numerals indicate the values in LGM and CTL, respectively. In addition, the percentages of increments at LGM to the CTL are denoted by colored numerals. Units for the values in the boxes and beside arrays are 105 J m−2 and W m−2, respectively. See the text for details.

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Atmospheric Local Energetics and Energy Interactions between Mean and Eddy Fields. Part II: An Example for the Last Glacial Maximum Climate

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  • 1 Climate Research Department, Meteorological Research Institute, Tsukuba, Japan
  • | 2 Research Institute for Global Change, JAMSTEC, Yokohama, Japan
  • | 3 Atmosphere and Ocean Research Institute, University of Tokyo, Kashiwa, and RIGC, JAMSTEC, Yokohama, Japan
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Abstract

The atmospheric local energy cycle in the Last Glacial Maximum (LGM) climate simulated by an atmosphere–ocean GCM (AOGCM) is investigated using a new diagnostic scheme. In contrast to existing ones, this scheme can represent the local features of the Lorenz energy cycle correctly, and it provides the complete information about the three-dimensional structure of the energy interactions between mean and eddy fields. The diagnosis reveals a significant enhancement of the energy interactions through the barotropic processes in the Atlantic sector at the LGM. Energy interactions through the baroclinic processes are also enhanced in the Atlantic sector, although those in the Pacific sector are rather weakened. These LGM responses, however, are not evident in the global energy cycle except for an enhancement of the energy flow through the stationary eddies.

Corresponding author address: Shigenori Murakami, Meteorological Research Institute, 1-1 Nagamine, Tsukuba 305-0052, Japan. Email: shimurak@mri-jma.go.jp

Abstract

The atmospheric local energy cycle in the Last Glacial Maximum (LGM) climate simulated by an atmosphere–ocean GCM (AOGCM) is investigated using a new diagnostic scheme. In contrast to existing ones, this scheme can represent the local features of the Lorenz energy cycle correctly, and it provides the complete information about the three-dimensional structure of the energy interactions between mean and eddy fields. The diagnosis reveals a significant enhancement of the energy interactions through the barotropic processes in the Atlantic sector at the LGM. Energy interactions through the baroclinic processes are also enhanced in the Atlantic sector, although those in the Pacific sector are rather weakened. These LGM responses, however, are not evident in the global energy cycle except for an enhancement of the energy flow through the stationary eddies.

Corresponding author address: Shigenori Murakami, Meteorological Research Institute, 1-1 Nagamine, Tsukuba 305-0052, Japan. Email: shimurak@mri-jma.go.jp

1. Introduction

The ultimate energy source of the atmospheric general circulation is the solar radiation reaching the earth. Spatial and temporal contrasts of the solar heating under the earth’s gravitational field generate the free energy that is stored in the atmosphere as the available potential energy (APE). A part of the APE is converted to kinetic energy (KE) through the convection or baroclinic processes. These processes maintain the atmospheric general circulation, and the KE dissipates into internal energy of the atmosphere through the friction process. The internal energy is converted to radiative energy and finally is emitted out to space as longwave earth radiation. The atmospheric part of these processes is often described and quantified using a box diagram of the Lorenz energy cycle (Lorenz 1955), which reveals many properties of the atmospheric general circulation. For example, the tropospheric general circulation is mainly maintained by the baroclinic processes of the midlatitudes (e.g., Oort 1964), and the lower stratospheric circulation is characterized as an indirect circulation maintained by the kinetic energy flux from the troposphere (e.g., Dopplick 1971). However, the Lorenz energy diagram gives only a summary of global energy cycle and cannot describe its local features. If we try to draw a box diagram to describe local features of the Lorenz energy cycle, we are faced with some difficulties because the energy conversion term between mean and eddy fields has two different local expressions whose spatial distributions are different, although they give the same value when averaged over the entire atmosphere (e.g., Holopainen 1978; Plumb 1983).

In Part I of this paper (Murakami 2011, hereafter Part I) one of the authors developed a new diagnostic scheme for the atmospheric local energetics analysis. The key concept of this analysis is the interaction energy flux. Using this concept, he shows it is possible to represent the local feature of the Lorenz energy cycle in a form of box diagram for the division of basic variables into time-mean and transient-eddy components. Moreover, a set of the interaction energy flux and the two local expressions of energy conversion term mentioned above provides the complete information about the three-dimensional structure of the energy interactions between mean and eddy fields. Here, as a first example (and a test) of this type of analysis, we investigate the energy interactions between time-mean and transient-eddy fields for the Last Glacial Maximum (LGM; 21 000 yr before present) climate simulated by a coupled atmosphere–ocean general circulation model (AOGCM). This is also a continuation of Murakami et al. (2008). In Murakami et al. (2008), the global-scale meridional energy transport in the LGM simulation was investigated. The target of this paper is to investigate the local feature of the energy transport related to the energy conversions between mean and eddy fields.

Since GCMs were put to practical use, many LGM climate simulations have been conducted. In early stages of paleoclimate modeling studies, atmospheric GCMs were used with lower boundary conditions reconstructed from the paleoclimate proxy data (e.g., Williams et al. 1974; Gates 1976). In a second stage, atmospheric GCMs coupled with mixed layer ocean models were used for those studies (e.g., Manabe and Broccoli 1985; Dong and Valdes 1998). In recent studies, fully coupled AOGCMs are usual tools for such the studies (e.g., Hewitt et al. 2001; Kitoh et al. 2001; Kitoh and Murakami 2002; Otto-Bliesner et al. 2006). Particularly over the recent few decades, the Paleoclimate Modeling Intercomparison Project (PMIP) has been playing an important role in leading paleoclimate modeling studies (e.g., Joussaume and Taylor 2000; Braconnot et al. 2007). Many simulations and analyses have been performed for the LGM climate in the framework of this project or in relation to it. Several studies focused on the transient eddy activity or the storm tracks in the LGM climate (e.g., Hall et al. 1996; Kageyama et al. 1999; Laîné et al. 2009; Li and Battisti 2008). This paper also deals with the transient eddy activity in one of such the simulations. However, the main target of this paper is to investigate the local feature of energy interactions between mean and eddy fields and how they respond to the LGM boundary conditions, which conventional energetics analysis could not reveal.

In section 2, we briefly describe the diagnostic scheme developed in Part I. A brief description of the LGM boundary conditions and some basic results are given in section 3. We also give a quick description of the energy density fields themselves in that section. In section 4, three-dimensional distributions of energy conversion terms between mean and eddy fields and interaction energy fluxes are plotted for the LGM and control (CTL) climate simulations. In section 5, box diagrams of the local energy cycle are shown for several typical locations in the Pacific and Atlantic sectors. Section 6 contains discussions and section 7 provides a summary.

2. Diagnostic scheme for local energetics

The key point of Part I is that each energy equation for APE or KE is divided not into two but into three parts consisting of the (time) mean, (transient) eddy, and interaction energy equations, when basic variables are divided into time-mean and transient-eddy components. We denote corresponding energy components as AM, AT, and AI for APE and KM, KT, and KI for KE (see appendix A for details). Using this notation, the basic equations of local energetics analysis are given as
i1520-0469-68-3-533-e1a
i1520-0469-68-3-533-e1b
i1520-0469-68-3-533-e1c
and
i1520-0469-68-3-533-e2a
i1520-0469-68-3-533-e2b
i1520-0469-68-3-533-e2c
where the term G(A*) denotes the APE generation rate, C(A*, K*) the energy conversion rate from APE to KE, C(*M, *I) the conversion rate from mean energy to interaction energy, C(*T, *I) the conversion rate from eddy energy to interaction energy, D(K*) the dissipation rate of KE, R(AM) the residual term, B(*) the boundary flux term, and F(*I) the flux term of the interaction energy equation. It should be noted that the time derivative terms in the interaction energy equations (1c) and (2c) vanish since the time averages of AI and KI are always zero. Detailed mathematical expressions for the above terms are given in appendix B, and the derivation of the equations is given in Part I.

The main differences of the above equations from the usual ones are the two additional equations (1c) and (2c) and the presence of two types of energy conversion terms between mean and eddy fields. In place of conventional conversion terms such as C(KM, KT), two conversion terms, C(KM, KI) and C(KT, KI), appear. The relationship among these two types of conversion terms and the flux term of interaction energy (hereafter referred to as the interaction energy flux) is given by (1c) or (2c) (see Part I for details). As shown after, these three quantities provide useful information about the energy interactions between mean and eddy fields. The balance among all the terms in the above equations is represented by a box diagram shown in Fig. 1. It should be noted that the residual term R(AM) in (1a) is, for simplicity, included in the term B(AM) in the diagram.

3. Experiments and basic results

First of all, we briefly summarize the LGM experiment analyzed in this paper. The model used here is the Center for Climate System Research (CCSR)–National Institute for Environmental Studies (NIES)–Frontier Research Center for Global Change (FRCGC) coupled GCM called the Model for Interdisciplinary Research on Climate 3.2.2 (MIROC3.2.2). This is basically the same model that was used in the Intergovernmental Panel on Climate Change’s (IPCC’s) Fourth Assessment Report (AR4) by the CCSR–NIES–FRCGC model group, but a bug fix about the treatment of momentum and heat fluxes over the ice sheets was made. A detailed description of the model is given in Hasumi and Emori (2004).

Two simulation runs were conducted using LGM and preindustrial control (CTL) boundary conditions. Under the PMIP2 protocols, the LGM experimental conditions are summarized as follows: 1) reduced greenhouse gas (GHG) concentrations, 2) insolation change due to 21 000 yr before present orbital parameters, 3) surface albedo changes due to prescribed ice sheets, 4) orography changes due to prescribed ice sheets, and 5) changes in land–sea distribution and altitude due to LGM sea level drop (about 120 m). Detailed descriptions of the experimental design can be found on the PMIP2 Web site (available online at http:/pmip2.lsce.ipsl.fr/).

The above boundary conditions cause a large surface air temperature (SAT) cooling in the northern high latitudes in addition to the global mean SAT cooling (about 4.5°C) compared to the CTL climate. This situation increases the north–south SAT gradient in the midlatitudes and enhances the low-level baroclinicity (see, e.g., Fig. 1 of Murakami et al. 2008). Moreover, large ice sheets on the North American continent enhance the stationary waves around that region (even for the annual mean) and enhance the poleward dry static energy transport (by stationary eddies) in the mid- and high latitudes (see Murakami et al. 2008 for details). These results are consistent with previous GCM simulations (e.g., Manabe and Broccoli 1985) or the linearized model calculations by Cook and Held (1988). In the upper troposphere, however, the relatively large cooling in the tropics rather acts to weaken the north–south temperature gradient. The main interest of this paper is how the transient eddies respond to such situations and interact with mean fields. All calculations in this paper were performed on the 30-yr time series of 6-hourly snapshot output for eastward wind speed u, northward wind speed υ, pressure velocity ω, temperature T, and diabatic heating Q.

Before investigating energy interactions, we briefly examine the spatial distributions of each energy component for APE and KE, and their response to the LGM boundary conditions. In this section, the time-mean fields are further divided into zonal-mean and stationary-eddy components, indicated respectively by subscripts Z and S (e.g., AZ, AS). Figure 2a shows vertically integrated APE density in both climates. Thick lines (solid and dotted) in Fig. 3a indicate the corresponding zonal mean profiles. Since APE is defined as square of temperature deviation from the global mean, the APE density is large in high latitudes and the response to the LGM conditions is also large in the northern high latitudes. This is consistent with the response of SAT mentioned above.

Figure 2b shows a similar map for KE, and the thick lines in Fig. 3d are the zonal-mean profiles of total KE. In contrast to the APE case, KE distribution has peaks in midlatitudes. Particularly in the NH, KE peaks are located over the two major oceans corresponding to the two jet stream maxima. At the LGM, the peak over the Atlantic increases more than 100%, but the peak over the Pacific decreases, except downstream of its maximum.

Figure 2c shows the vertically integrated generation rate of APE for both climates. It shows that G(A) is larger at high latitudes, similar to the APE density case, and at the convective precipitation zones in the tropics. At the LGM, it increases over the NH polar region, tropical western Pacific, and the ice-covered regions. The regions of increased G(A) in the tropics are also characterized by an increase of precipitation.

Next, we examine the each component of APE and KE. The AM and KM have similar spatial distribution to total APE and KE, respectively, and their responses to the LGM conditions are also similar to those of total energies (not shown; only the zonal mean profiles are shown in Figs. 3a and 3d). As shown in Fig. 4a, AT has two broad peaks over the eastern coast of the Eurasian continent and over the North American continent. On the other hand, KT has peaks over the North Pacific and the North Atlantic (Fig. 4b). The peaks of the energy conversion rate C(AT, KT) shown in Fig. 4c are located between the peaks of AT and KT. At the LGM, they decrease over the Pacific and increase over the Atlantic. This is consistent with response of KT density itself shown on Fig. 4b. In addition, a significant increase of AS and KS is observed over the North American continent at the LGM (not shown; only the zonal profiles are given in Figs. 3c and 3f).

4. Energy interactions between mean and eddy fields, and interaction energy fluxes

In this section, we investigate details of the energy interactions between mean and eddy fields using the interaction energy fluxes defined in Part I.

Figure 5 just shows the relationship represented by (1c) for the CTL climate. Figures 5a and 5b show the vertically integrated energy conversion terms C(AM, AI) and C(AI, AT) = −C(AT, AI). As mentioned in Part I, these two quantities have different spatial distributions. The former is larger in high latitudes and the latter has peaks in midlatitudes. These features are related to the fact that the generation and storage of APE are large in the high latitudes, and the conversion from APE to KE and the KE density itself are large in the midlatitudes (see Figs. 2c and 4c). The difference between these two spatial distributions implies the existence of an energy flow from the high latitudes to midlatitudes. Figure 5c displays the vertical integration of the horizontal part of interaction energy flux
i1520-0469-68-3-533-e3
(arrows) and its convergence (color shades). The interaction energy flux AIu well captures the energy flow from the regions where the APE generation is active (in high latitudes) to the regions where the energy conversion from eddy APE to eddy KE is active (in midlatitudes). In addition, the convergence of interaction energy flux for APE just agrees with C(AI, AT) − C(AM, AI) = −{C(AM, AI) + C(AT, AI)}. From these figures and relation (1c), we can confirm the following facts: 1) the local value of C(AM, AI) is equal to the sum of in situ conversion C(AI, AT) and the divergence of AI flux; 2) the local value of C(AI, AT) is equal to the sum of in situ conversion C(AM, AI) and the convergence of AI flux; and 3) the interaction energy flux AIu transports the interaction energy from its divergence regions to its convergence regions. In contrast to the conventional energetics analysis dealing only with the conversion term C(AI, AT), the diagnosis using these three quantities gives the complete information about the energy conversions (interactions) between mean and eddy fields. It should be noted that the divergence of interaction energy flux vanishes when averaged over the entire atmosphere, and the two conversion terms give the same value.
Another important feature seen from Fig. 5 is that the quantities C(AM, AI) and C(AI, AT) are positive in most regions. This means that the energy conversion AMAT is dominant in the real atmosphere. This is just the fact that the classical energetics analysis revealed. Therefore, we can generally recognize the following energy flow pattern (or energy path, in terms of Part I):
i1520-0469-68-3-533-e4
The chain of these processes should be referred to as baroclinic conversion from the viewpoint of this paper.

Figure 6 shows the three-dimensional structure of the energy interactions for APE in the CTL climate. Figures 6a and 6b display the zonal mean and vertical section along the 45°N parallel of C(AI, AT) and the corresponding interaction energy flux. Figure 6c is the same as Fig. 5c, but the vertical integration of C(AI, AT) is displayed as color shading. Each panel also displays the transient eddy APE density AT by thin green lines. In the troposphere, the interaction energy flux for APE generally tends to go from upper to lower levels and from the high to low latitudes. This is related to the fact that the APE generation peaks appear in the upper troposphere in high latitudes (except in the tropics). Also, AI converges to the eastern coasts of two major continents in the NH midlatitudes where the generation and growth of baroclinic disturbances (baroclinic processes) are active.

Figure 7 shows the same diagnosis for the difference between LGM and CTL. Under the LGM boundary conditions, the AI flux convergence and C(AI, AT) are significantly enhanced in the North American region, but those on the Eurasian side are rather weakened. In addition, as shown in zonal mean profiles of Fig. 7a, the interaction energy fluxes of APE and C(AI, AT) are weakened in the upper troposphere and enhanced in the lower troposphere. These responses are consistent with the temperature response mentioned in section 3 (not shown). Similar responses are also seen in the SH (Fig. 7a).

Figure 8 shows the relationship represented by (2c) for the CTL climate. In this case, C(KI, KM) and C(KI, KT) are plotted in Figs. 8a and 8b, respectively. Similarly to the case of APE, the spatial distributions of the two quantities C(KI, KM) and C(KT, KI) = −C(KI, KT) are different. In addition, in contrast to the case of APE, the direction of energy conversion varies from place to place, and moreover, in many locations, the directions of C(KI, KM) and C(KT, KI) are opposite (this is evident from the same color shading in Figs. 8a and 8b for the same locations). This means that the energy interaction pattern
i1520-0469-68-3-533-e5
is commonly seen for KE in the real (or simulated) atmosphere, which is never recognized from the conventional energetics analysis. The sum C(KI, KM) + C(KI, KT) = −{C(KM, KI) + C(KT, KI)} just gives the convergence of the interaction energy flux for KE:
i1520-0469-68-3-533-e6
Similarly to the case of APE, we can confirm the following facts: 1) the local value of C(KI, KM) is equal to the sum of in situ conversion C(KT, KI) and the convergence of KI flux; 2) the local value of C(KI, KT) is equal to the sum of in situ conversion C(KM, KI) and the convergence of KI flux; and 3) the interaction energy flux KIu transports the interaction energy KI from its divergence regions to its convergence regions.

Figure 9 shows the three-dimensional structure of the energy interactions for KE. Figures 9a and 9b display the zonal mean and vertical section along the 40°N parallel of C(KI, KM) and the corresponding interaction energy flux; Fig. 9c displays the vertical integration of C(KI, KM) and interaction flux of KE. In addition, the mean KE density KM is indicated by thin green contours as an indicator of the climatological jet stream. The KE interaction flux stands out mainly in the upper troposphere along the jet stream. It diverges at the jet entrance regions and converges at the exit regions. In addition, some vertical transport of KI is also observed, which is associated with the growth and decay of the baroclinic turbulence.

Figure 10 shows the same diagnosis for the difference between LGM and CTL. Under the LGM conditions, C(KI, KM) and interaction energy flux are significantly enhanced in the Atlantic sector according to the enhancement of the jet stream, although those in the Pacific are generally weakened. This result may be related to the existence of the Laurentide Ice Sheet on the North American continent, but a sensitivity study will be needed to separate the orographic effect. In addition, as seen from Fig. 10a, the zonal mean poleward KI flux is weakened in both hemispheres.

5. Box diagrams of local and global energy cycle

a. Local energy cycle

In this subsection, we draw box diagrams of the local energy cycle for several specific locations. Some features described in this section can be already seen from the figures and discussions in the previous sections. However, the utility of the local box diagram is that the diagram correctly represents the balance of all terms in the energy equations and makes it easy to quantify the differences between the different locations or between the different climate states.

Figure 11 show such diagrams for both climates. The locations of the chosen areas are shown in Fig. 12 as rectangles denoted by capital letters B–H, except for the area corresponding to Fig. 11a. The letters B–H correspond to Figs. 12b–h. Numerals in each diagram indicate the vertically integrated values of the variables defined in Fig. 1. Upper and lower numerals correspond to the values in LGM and CTL, respectively. A negative value means that the energy flows against the arrow. The values in parentheses (dissipation terms) are obtained as residuals of the balance equations. The values in curly brackets—that is, values of C(AM, KM)—are not directly used for the calculation because those values are significantly different from the others. Instead, the values of (α − 〈α〉)ω and −u · gradhΦ are used, where gradh means the horizontal component of the gradient operator. As defined in Part I, we refer to an energy flow pattern recognized in the energy diagram as an energy path. Thick arrows in each diagram indicate the dominant energy paths at that place.

We choose the first area A as the region north of 80°N (i.e., Arctic region) because the generation and storage of APE is large there. As shown by the diagram displayed in Fig. 12a, the generated mean APE AM is converted to AI and flows out from this region as the AI flux. At the LGM, the generation and storage of APE increase, but the increment of G(AM) mainly flows out as the boundary flux of B(AM) and the interaction energy flux does not necessarily increase in this region. The main energy path in this area can be expressed by the following simplified energy diagram:
i1520-0469-68-3-533-e7
This type of energy path is typically seen at high latitudes.

Area B (Fig. 12c) is chosen over Labrador, where the Laurentide Ice Sheet covered the ground at the LGM, and the APE generation might be enhanced (see Fig. 2c). The main energy path in this area at LGM is similar to area A. In this region, however, the increment of G(AM) mainly flows out as the interaction energy flux of APE (Fig. 11b).

We choose area C on the eastern coast of the Eurasian continent where the growth of the baroclinic disturbances is active (Fig. 12a). The characteristic of this area is shown in Fig. 12 as an overlap of a divergence peak of AI flux (green line) and a conversion peak of C(AI, AT) (red line). The conversion peak of C(AT, KT) shown in Fig. 4c also has a peak there. Therefore, as shown in Fig. 11c, the main energy path can be expressed as
i1520-0469-68-3-533-e8
This energy path is a typical one of the baroclinic conversion. Under the LGM conditions, this energy path weakens by about 10%. We can recognize another energy path in the diagram described by
i1520-0469-68-3-533-e9
Since area C is located at the entrance region of the Pacific jet, the convergence of KM [i.e., −B(KM)] is very large (about 470 W m−2). Most of converging KM is converted to AM (about 450 W m−2), a part of the residual is dissipated (about 10 W m−2), and the rest is converted to KI and flow outs as the interaction energy flux of KE (KI flux; 4.9 W m−2). It should be noted that only the net effects are indicated for B(KM) and B(AM) in the energy diagrams of Fig. 11, and the effect of dissipation is omitted in the simplified diagram (9) for simplicity. The intensity of this energy path does not change so much at LGM. Another notable feature for this area is that a part of converted energy KI [from KM as C(KM, KI)] further converted to KT as C(KI, KT) even though the value is small (about 1 W m−2) compared to other energy paths. This point will be addressed again.

Area D over the Atlantic is analogous to area C of the Pacific sector (Fig. 11c). In the CTL climate simulation, the intensity of the energy path described by diagram (8) in this place is less than half that at C. However, at the LGM this energy path intensifies corresponding to the enhancement of the baroclinicity at that region as expected from Fig. 7. In addition, a significant enhancement of the energy path described by diagram (9) is observed. This enhancement is related to the fact that the split jet streams by the Laurentide Ice Sheet join together near that place in the LGM simulation. This change is also clear from Fig. 10.

We choose area E as the location to the south-southeast of area C (Fig. 12a). This is also an area where the baroclinic process is active, but several differences can be seen. At this place, in contrast to area C, the converted KT from AT further converts to the interaction energy KI and diverges as KI flux. In addition, the energy conversions described by diagram (9) also occur. Therefore, this area is characterized by a strong divergence of KI. Figure 8 clearly shows this characteristic. The main energy path in this area is expressed as
i1520-0469-68-3-533-e10
The difference between Figs. 11c and 11e (or the diagrams of Figs. 8 and 9 and Fig. 10) represents the difference of climatological features for these two areas. We refer to the upper and lower branches in energy path (10) as the barotropic part and baroclinic part, respectively. At the LGM, the baroclinic part of this energy path is enhanced by about 10% and the barotropic part is weakened about 10%.

Area F in the Atlantic sector is similar to area E of the Pacific sector (Fig. 12c). The main energy path at this place is similar to that of area E except that the main energy source at this point is the convergence of the mean APE flux B(AM). At the LGM, changes from CTL are small, similar to place E except for the enhancement of barotropic part.

We choose area G over the central Pacific, where a convergence peak of KI flux and a peak of C(KI, KM) overlap (Fig. 12a). The main energy source at this place is convergence of interaction flux of KE, and the energy path is characterized by
i1520-0469-68-3-533-e11
Converging KE is mainly converted to KM and mainly flows out as the mean boundary flux B(KM), and is partly converted to KT. Under the LGM conditions, the main peaks of KI flux convergence and C(KI, KM) shift to the eastern Pacific (Fig. 12b). Therefore, the energy path is weakened at area G and enhanced over the eastern Pacific. It is also notable that there are typically two types of regions over the convergence area of KI flux, which can be determined from the sign of C(KT, KI). The interaction patterns of KE in those regions are expressed as
i1520-0469-68-3-533-e12
From Fig. 8, we can see that the latter pattern of energy interaction is dominant in the eastern North Pacific. This interaction pattern may be associated with the decay process of baroclinic disturbances (also see Fig. 9b).

Figure 11h shows the energy cycle at a region located over the eastern Atlantic indicated by rectangle H in Fig. 12c. This region is basically an energy converging region of KI similar to the eastern Pacific. The main energy path is same as diagram (11) and the intensity increases by more than 100% at the LGM. However, C(KT, KI) is basically positive in this region.

b. Global energy cycle

Finally we show the globally averaged energy balance in both climates. By integrating over the entire atmosphere, the diagram of Fig. 1 turns into the conventional four-box diagram. Here, we divide the time-mean state further into zonal-mean and stationary-eddy components and draw a six-box diagram, following Lee and Chen (1986), to clearly see the role of stationary eddies. Figure 13 displays this diagram for both climates (for details, see Lee and Chen 1986). The values of boundary flux terms and residual terms basically vanish except the contributions from the lower boundary. Those are less than 0.3 W m−2 as a global mean and are omitted in this diagram. Similarly to Fig. 11, the upper and lower numerals in the boxes or beside the arrows respectively denote the values for the LGM and CTL. We also add the percentage of increment at LGM to the CTL as the colored numerals. The main energy path we can recognize from the diagram is expressed as
i1520-0469-68-3-533-e13
i1520-0469-68-3-533-e14
which implies the importance of baroclinic process in the atmosphere, consistent with previous classical studies. Figure 13 also shows the 100% enhancement of the energy path through the stationary eddies (colored arrows in Fig. 13). However, some local characteristics and those responses to the LGM conditions that we have seen in the previous sections are not evident in Fig. 13.

6. Discussion

In this paper, in connection with the analysis in Murakami et al. (2008) and as the first test of the new scheme, we treated the data with annual mean basis and no filtering has been applied. It is difficult to compare our results directly with previous studies that mainly analyze the wintertime statistics of eddies, but we can find some similarities and some differences. Li and Battisti (2008), for example, report an intensification of the Atlantic jet and an enhancement of low-level baroclinicity for the National Center for Atmospheric Research (NCAR) Community Climate System Model version 3 (CCSM3) that are similar to the results in this paper, but they also report a weakening of wintertime transient-eddy KE in contrast with our results. Laîné et al. (2009) also report a weakening of wintertime total-eddy energy (AT + KT) for the MIROC3.2 model (the model version is slightly different from the model used here; see section 3). It is difficult to extract some robust results from those about the response of transient-eddy activity itself. However, as shown in the previous sections, the main target of this paper is a local feature of the energy interactions, which conventional energetics analysis cannot reveal.

As mentioned in the introduction of Part I, there are some studies that deal with the local energetics analysis by dividing basic variables into time-mean and transient-eddy fields. In those studies, only the terms C(AI, AT) and C(KT, KI) (in terms of this paper) are treated and referred to as baroclinic conversion and barotropic conversion, respectively. Therefore, the results related to those terms, particularly the results for baroclinic conversions, can be obtained from the methods used in those studies, although the interaction energy flux of this study provides more complete information. From the viewpoint of the present paper, as mentioned in section 5b, the classical concept of baroclinic conversion should be represented by a long chain of energy conversions described by diagram (14). The final part of this diagram (KTKM) may be accomplished by the decay process of baroclinic disturbances that is dominant over the eastern North Pacific or the North Atlantic. However, the local values of those processes are relatively small compared to other processes. From Fig. 8, we can recognize more intense interactions in the midlatitudes that are expressed as
i1520-0469-68-3-533-e15
i1520-0469-68-3-533-e16
Only the interaction pattern (16) or the direct interaction patterns KMKT and KTKM are recognized from conventional “local” energetics analysis, and only the global average of those three interactions is recognized as “barotropic” conversion from the classical Lorenz diagram. As seen from Figs. 8a and 8b, the interaction pattern (15) is generally more intense than pattern (16), and the energy convergence pattern in Fig. 8c reflects mainly the former pattern. Figure 10 shows that this interaction pattern is significantly enhanced in the Atlantic sector at LGM. As mentioned in Part I, this type of energy interaction transports the mean KE from a place to another as the interaction energy KI and hardly affects the transient-eddy KE field as a time mean. We may have overlooked this process by treating only the term C(KT, KI) as the energy conversion between mean and eddy fields. From the viewpoint of this paper, as mentioned also in Part I, it is rather C(KM, KI) that represents the energy conversion from (or to) mean KE and that is appropriately referred to as barotropic conversion.
A recent paper by Hernandez-Deckers and von Storch (2010), published after the submission of this paper, deals with the Lorenz energy cycle in a CO2 doubling experiment. Although their diagnosis is typical of the classical energetics analysis based on the global Lorenz energy cycle, they reported an enhancement of the energy path
i1520-0469-68-3-533-eq1
in the upper troposphere and lower stratosphere, and weakening in the middle and lower troposphere by dividing the whole atmosphere into upper and lower parts, where AE = AT + AS and KE = KT + KS. Those responses are generally opposite to the LGM case reported in this paper or in Li and Battisti (2008). It will be interesting to investigate global warming experiments in the manner of this paper in order to reveal the local features of energy interactions between mean and eddy fields.

7. Summary

The local feature of the energy interactions between time-mean and transient-eddy fields in the LGM simulation was investigated, using the interaction energy fluxes and box diagrams for the local energy cycle. The response of energy interactions to the LGM boundary conditions is quite different between in the Atlantic and Pacific sectors. The baroclinic conversion from mean APE to eddy APE is enhanced in the Atlantic sector corresponding to the increase of the baroclinicity in that region. On the other hand, in the Pacific sector the baroclinic conversion is rather weakened except in the lower troposphere. The energy interactions between mean KE and interaction KE (barotropic conversion; in terms of this paper) is significantly enhanced in the Atlantic sector simultaneously with an intensification of the Atlantic jet stream, but slightly weakened in the Pacific sector. These responses, however, are not evident in the classical Lorenz energy cycle.

Acknowledgments

The author thanks three anonymous reviewers. Their constructive comments greatly improved the paper. The computation of LGM and CTL simulations that gave a basis of the analysis in this paper was performed on the Earth Simulator of the JAMSTEC. A part of this work was performed when author Shigenori Murakami was at the Frontier Research Center for Global Change of JAMSTEC.

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APPENDIX A

Definitions of APE and KE Densities and Those Divisions

In the pressure coordinate system, APE and KE densities per unit mass are expressed as
i1520-0469-68-3-533-ea1a
i1520-0469-68-3-533-ea1b
where u is the eastward wind speed, υ the northward wind speed, p the pressure, p0 the reference pressure (=1000 hPa), T the temperature, Cp the atmospheric specific heat at constant pressure, κ the ratio of gas constant and specific heat (=R/Cp), θ the potential temperature [=(p0/p)κT], and γ is an index for static stability of the dry atmosphere defined by
i1520-0469-68-3-533-ea2
In these expressions, angle brackets denote the global average operator over the constant pressure surface and the overbar denotes the time mean operator. Corresponding to the division of basic variables into the time mean and transient eddy, APE and KE densities are divided into three components as
i1520-0469-68-3-533-ea3a
i1520-0469-68-3-533-ea3b
where
i1520-0469-68-3-533-ea4a
i1520-0469-68-3-533-ea4b
i1520-0469-68-3-533-ea4c
i1520-0469-68-3-533-ea5a
i1520-0469-68-3-533-ea5b
i1520-0469-68-3-533-ea5c
It follows directly from above definitions that the time mean of AI and KI vanish, and A = AM + AT and K = KM + KT hold. Of course, relations AM = AM and KM = KM also hold. In the main text, we denote AM, AT, KM, and KT simply as AM, AT, KM, and KT, respectively.

APPENDIX B

Detailed Expressions of the Terms in Energy Balance Equations

Detailed mathematical expression of the terms of energy balance equations in the spherical pressure coordinate system appearing in section 2 are given asgeneration terms:
i1520-0469-68-3-533-eb1a
i1520-0469-68-3-533-eb1b
dissipation terms:
i1520-0469-68-3-533-eb2a
i1520-0469-68-3-533-eb2b
conversion terms from APE to KE:
i1520-0469-68-3-533-eb3a
i1520-0469-68-3-533-eb3b
conversion terms between eddies and mean fields (APE):
i1520-0469-68-3-533-eb4a
i1520-0469-68-3-533-eb4b
conversion terms between eddies and mean fields (KE):
i1520-0469-68-3-533-eb5a
i1520-0469-68-3-533-eb5b
boundary flux terms:
i1520-0469-68-3-533-eb6a
i1520-0469-68-3-533-eb6b
i1520-0469-68-3-533-eb6c
i1520-0469-68-3-533-eb6d
and residual terms:
i1520-0469-68-3-533-eb7a
Here, Q, F, ω, α, and a are diabatic heating, horizontal friction force [i.e., F = (λ, φ, 0)], pressure velocity, specific volume, and the earth’s radius, respectively. The gradient and divergence operators in the spherical pressure coordinate system are given as follows:
i1520-0469-68-3-533-eb8a
i1520-0469-68-3-533-eb8b
where Ψ and X = (X, Y, Z) are arbitrary scalar and vector fields.

Fig. 1.
Fig. 1.

Box diagram of the local energy cycle. The boxes indicated by AM, AT, KM, and KT represent the mean and transient-eddy components of APE and KE. The arrows indicated by G(*), D(*), and B(*) represent generation, dissipation, and boundary flux terms in the energy balance equations. The arrows indicated by C(*, *) represent the energy conversion terms, and the wavy arrows indicated by F(AI) and F(KI) represent the interaction energy fluxes of APE and KE.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3583.1

Fig. 2.
Fig. 2.

Vertically integrated (a) APE density, (b) KE density, and (c) generation rate of APE for (left) CTL and (right) LGM simulations. Contour intervals are 30 × 105 J m−2 for APE, 6 × 105 J m−2 for KE, and 6 W m−2 for APE generation. Shading in right panels indicates deviation from the CTL.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3583.1

Fig. 3.
Fig. 3.

Latitudinal profiles of vertically integrated and zonally averaged (a) total and zonal APE, (b) T-eddy APE, (c) S-eddy APE, (d) total and zonal KE, (e) T-eddy KE, and (f) S-eddy KE density. Solid and dotted lines indicate the values in LGM and CTL, respectively. Unit is 105 J m−2.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3583.1

Fig. 4.
Fig. 4.

Vertically integrated (a) transient-eddy APE (AT), (b) transient-eddy KE (KT), and (c) energy conversion rate from AT to KT for (left) CTL and (right) LGM simulations. Contour intervals are 10, 3, and 3 × 105 J m−2, respectively. Shading in right panels indicates deviation from the CTL.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3583.1

Fig. 5.
Fig. 5.

Vertically integrated (a) C(AM, AI), (b) C(AI, AT), and (c) interaction flux of APE and its convergence in the CTL simulation. Units for the conversion rate and flux are W m−2 and J m kg−1 s−1, respectively.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3583.1

Fig. 6.
Fig. 6.

(a) Zonal mean, (b) vertical section along 45°N, and (c) vertical integration of C(AI, AT) and interaction flux of APE in the CTL simulation. Units for the conversion rate are (a),(b) 10−4 W kg−1 and (c) W m−2. Units for the horizontal and vertical components of flux vector are J m kg−1 s−1 and J Pa kg−1 s−1, respectively. The vertical component of the flux vector is 100-fold in (a) and (b). Thin green contours indicate T-eddy APE density with contour intervals of 100 J kg−1, 180 J kg−1, and 5 × 105 J m−2.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3583.1

Fig. 7.
Fig. 7.

As in Fig. 6, but for the difference between LGM and CTL simulations. Thin green contours indicate T-eddy APE density at LGM.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3583.1

Fig. 8.
Fig. 8.

Vertically integrated (a) C(KI, KM), (b) C(KI, KT), and (c) interaction flux of KE and its convergence in the CTL simulation. Units for the conversion rate and flux are W m−2 and m3 s−3, respectively.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3583.1

Fig. 9.
Fig. 9.

(a) Zonal mean, (b) vertical section along 40°N, and (c) vertical integration of C(KI, KM) and interaction flux of KE in the CTL simulation. Units for the conversion rate are (a),(b) 10−4 W kg−1 and (c) W m−2. Units for the horizontal and vertical components of flux vector are m3 s−3 and Pa m2 s−3, respectively. The vertical component of the flux vector is 100-fold in (a) and 1000-fold in (b). Thin green contours indicate mean KE density with contour intervals 100 J kg−1, 200 J kg−1, and 5 × 105 J m−2.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3583.1

Fig. 10.
Fig. 10.

As in Fig. 9, but for the difference between LGM and CTL simulation. Thin green contours indicate mean KE density at LGM.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3583.1

Fig. 11.
Fig. 11.

Box diagrams of the local energy cycle corresponding to several typical locations shown in Fig. 12. Numerals indicate the vertically integrated values of the variables corresponding to Fig. 1. Upper and lower numerals correspond to the values in LGM and CTL, respectively. The values of R(AM) are included in the value of B(AM). Units for the values in the boxes and beside arrays are 105 J m−2 and W m−2, respectively. See the text for details.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3583.1

Fig. 12.
Fig. 12.

Maps of the conversion and convergence peaks of interaction energy over the (a) Pacific in CTL, (b) Pacific in LGM, (c) Atlantic in CTL, and (d) Atlantic in LGM. Color lines indicate the contour around the peaks of each quantity denoted in (a). Unit of the contours is W m−2. Rectangles denoted by the letters B–H indicate the areas where the energy diagrams in Fig. 11 are calculated. See the text for details.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3583.1

Fig. 13.
Fig. 13.

A six-box diagram of the global mean energy cycle. Upper and lower black numerals indicate the values in LGM and CTL, respectively. In addition, the percentages of increments at LGM to the CTL are denoted by colored numerals. Units for the values in the boxes and beside arrays are 105 J m−2 and W m−2, respectively. See the text for details.

Citation: Journal of the Atmospheric Sciences 68, 3; 10.1175/2010JAS3583.1

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