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Atmospheric Local Energetics and Energy Interactions between Mean and Eddy Fields. Part I: Theory SHIGENORI MURAKAMI Climate Research Department, Meteorological Research Institute, Tsukuba, Japan (Manuscript received 8 September 2010, in final form 23 November 2010) ABSTRACT A new diagnostic scheme for the atmospheric local energetics is proposed. In contrast to conventional schemes, this scheme correctly represents the local features of the Lorenz energy cycle for time-mean and transient-eddy fields. The key point is that the energy equation is divided not into two but into three parts consisting of the mean, eddy, and interaction energy equations, when basic variables are divided into mean and eddy fields. The interaction energy itself vanishes when appropriate averaging is taken. However, the equation for interaction energy does not vanish and gives a relationship between the interaction energy flux and the two types of energy conversion terms. These three quantities give the complete information for the energy interactions between mean and eddy fields. The Lorenz energy diagram is reconstructed to include a representation of this relationship. A brief discussion about the relationship with wave activity analysis is also given.
1. Introduction Atmospheric energetics analysis has had a long history since Lorenz (1955) formulated the concept of available potential energy (APE). Lorenz also formulated a diagnostic scheme for the atmospheric general circulation by dividing basic variables into zonal-mean and eddy fields and considering the energy conversions among the zonal and eddy components of APE and kinetic energy (KE). His analysis is conventionally represented by a box diagram and is referred to as the Lorenz energy cycle. This type of analysis has been extended along several directions. One was proposed by Saltzman (1957) and involved expanding the eddy component into Fourier series and considering the energy conversions on the wavenumber domain. The normalmode energetics developed by Tanaka et al. (1986) also develops in this direction. Considering the energy cycle using isentropic coordinates constitutes another type of extension (e.g., Iwasaki 2001). These diagnostic schemes basically give a global view of the energy cycle. It is also possible to consider the local energy balance by dividing basic variables into time-mean
Corresponding author address: Shigenori Murakami, Meteorological Research Institute, 1–1 Nagamine, Tsukuba, 305-0052 Japan. E-mail:
[email protected] DOI: 10.1175/2010JAS3664.1 Ó 2011 American Meteorological Society
and transient-eddy components. However, even in this case, if we try to draw the Lorenz diagram for a local energy cycle, we are faced with a difficulty since an energy conversion term has two different local expressions. Holopainen (1978, p. 141) expresses the situation as follows: ‘‘there is some ambiguity in the interpretation of which terms in the energy equations for an open system represent an energy conversion from one form to another.’’ Plumb (1983) also expressed doubts about the usefulness of the energy cycle analysis and proposed a diagnostic scheme of the transformed Eulerian mean (TEM) energy cycle. Despite of his criticism and the movement of developing three-dimensional wave activity flux (i.e., extended Eliassen–Palm flux) in the 1980s, energetics analysis has continued to be used (e.g., Mak and Cai 1989; Orlanski and Katzfey 1991). One reason for this is that the wave activity analysis is, in many cases, constructed on the framework of quasigeostrophic (QG) theory and is based on the assumptions of large-scale (basic state) motion and small-amplitude eddies. On the other hand, the classical energetics analysis is basically free from the restriction of QG theory, and threedimensional distributions of energy conversion terms between mean and eddy fields are easily calculated from Eulerian variables even though there is some ambiguity in its interpretation. The basic issue raised by Holopainen (1978), however, does not seem to be solved. As for the
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concept of interaction energy, Orlanski and Katzfey (1991) paid some attention to it, but they do not seem to deepen their analysis in this direction. The aim of this paper is to reconstruct the classical energetics analysis based on the concept of interaction energy and its flux, and to give some diagnostic tools for the local energetics analysis. It also gives an answer to the question raised by Holopainen (1978). In sections 2 and 3, a set of energy balance equations for KE and APE in a form slightly different from conventional ones is derived. Based on these equations, in section 4 a new box diagram, which correctly represents the local energy balance, is proposed. The usefulness of the interaction energy fluxes is also discussed in that section. Finally, a brief discussion about the relationship to traditional energetic analysis and to wave-activity flux analysis is given in section 5.
2. Governing equations and KE equations We start from a set of primitive equations for a dry atmosphere in the spherical pressure coordinate system: Du tanu 1 ›F uy f y 5 1 F l, Dt a a cosu ›l
(1a)
Dy tanu 2 1 ›F 1 u 1 fu 5 1 F u, Dt a a ›u
(1b)
›F a, ›p
(1c)
k p Du Q 5 0 , Dt p Cp
(1d)
05
1 ›u ›y cosu ›v 1 1 5 0, a cosu ›l ›u ›p
(1e)
where l is longitude, u is latitude, p is pressure, t is time, u is eastward wind speed, y is northward wind speed, v is pressure velocity, F is geopotential height, a is the radius of the earth, f is the Coriolis parameter, F 5 (Fl , Fu , 0) is horizontal friction force, Q is diabatic heating, p0 is reference pressure (51000 hPa), Cp is atmospheric specific heat at constant pressure, k is the ratio of the gas constant and specific heat (5R/Cp), u is potential temperature [5(p0 /p)kT, where T is temperature], and a is specific volume (5RT/p). The Lagrangian differentiation in this coordinate system is given by
Equations (1e) and (2) can be expressed in terms of divergence and gradient operators in this coordinate system as divu 5 0, and D › 5 1 u grad, Dt ›t
(2)
(4)
where u 5 (u, y, v) is the three-dimensional wind velocity field. By taking the inner product of u and Eqs. (1a)–(1c), a balance equation for the KE is obtained as 2 D u2 1 y 2 › u2 1 y 2 u 1 y2 5 1 u grad Dt ›t 2 2 2 5 u gradF av 1 u F .
(5)
From a vector identity and Eq. (1e), we can rewrite Eq. (5) as › u2 1 y 2 u2 1 y 2 1 div u 1 Fu 5 av 1 u F . ›t 2 2 (6) In pressure coordinates, all terms in the equations are quantities per unit mass, so that the above is a balance equation for the KE density per unit mass. It is also notable that the terms of the Coriolis and curvature effects do not appear in the (total kinetic) energy equation since they are inertial forces depending on the choice of specific coordinates. Next, we divide the basic variables u, y, v, F, and u (and a) into two components (e.g., u 5 u 1 u9), where () denotes the time-mean operator and ()9 denotes the deviation from it. In this case, the KE density K is divided into three terms as K5 5
(u 1 u9)2 1 (y 1 y9)2 2 u2 1 y 2 u92 1 y92 1 1 (uu9 1 yy9). 2 2
We refer to these terms as the (time) mean, (transient) eddy, and interaction components of KE and denote by KM, KT, and KI, respectively. By taking the time average of the above relation, we can easily see that K 5 KM 1 KT 5 K M 1 KT . Similarly, the KE flux can be divided into six terms Ku 5 K M u 1 K T u 1 K I u 1 KM u9 1 KT u9 1 KI u9
D › u › y › › 5 1 1 1v . Dt ›t a cosu ›l a ›u ›p
(3)
and time-averaging yields
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2 › u2 1 y2 u 1 y2 1 u grad 1 u gradF ›t 2 2
Ku 5 KM u 1 KT u 1 KT u9 1 KI u9. It is not obvious how the kinetic energy equation (6) can be divided when the basic variables are divided into time-mean and transient-eddy components. To determine the decomposition of Eq. (6), we consider the balance equations for KM, KT, and KI. From the equations for time-mean component of wind velocity field, ›u tanu 1 u gradu 1 divu9u9 (u y 1 u9y9) f y ›t a 5
1 ›F 1 F l, a cosu ›l
(7a)
›y tanu 1 u grady 1 divy9u9 1 (uu 1 u9u9) 1 f u ›t a 5
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1 ›F 1 F u, a ›u
(7b)
›F a, ›p
(7c)
applying the inner product with u, a balance equation for KM is obtained as
5 av u divu9u9 y divy9u9 1
tanu (uu9y9 yu9u9) 1 u F . a
Similarly, from the equations for transient-eddy components ›u9 1 u gradu9 1 u9 gradu 1 u9 gradu9 divu9u9 ›t tanu (u9y 1 uy9 1 u9y9 u9y9) f y9 a 1 ›F9 1 F l9 , 5 (9a) a cosu ›l ›y9 1 u grady9 1 u9 grady 1 u9 grady9 divy9u9 ›t tanu (u9u 1 uu9 1 u9u9 u9u9) 1 f u9 1 a 1 ›F9 5 (9b) 1 F u9 , a ›u 05
›F9 a9, ›p
(10)
u9 [Eqs. (7a c)] 1 u [Eqs. (9a c)], a balance equation for KI is obtained as
2 ›(u9u 1 y9y) u 1 y2 tanu 1 div (u9u9y9 y9u9u9) u9 1 (u9u 1 y9y)u 1 (u9u 1 y9y)u9 1 F9u 1 Fu9 ›t a 2 tanu tanu (uu9y9 uu9y9) 1 (yu9u9 yu9u9) a a 5 a9v av9 1 u9u9 gradu 1 y9u9 grady 1 u div(u9u9) 1 y div(y9u9).
The sum of Eqs. (8), (10), and (11) exactly equals Eq. (6). Therefore, these three equations just give the decomposition for which we were looking. By taking the time average of these three equations, the time-mean versions of the energy balance equations for KM, KT, and KI are obtained as
(9c)
applying the inner product with u9, a balance equation for KT is obtained as
2 2 › u92 1 y92 u9 1 y92 u9 1 y92 1 u grad 1 u9 grad 1 u9 gradF9 2 u9divu9u9 y9divy9u9 ›t 2 2 2 tanu tanu (u9u9y9 y9u9u9) 5 2a9v9 u9u9 gradu y9u9 grady (uu9y9 yu9u9) 1 u9 F 9 1 a a It should be noted that Eqs. (9a)–(9c) are not linearized but are exact equations for eddies. In addition, by taking
(8)
›KM 1 div(KM u 1 Fu) 5 av C1 1 u F , ›t
(11)
(12a)
›KT 1 div(K T u 1 KT u9 1 F9u9) 5 a9v9 C2 1 u9 F 9, ›t (12b)
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div(K I u9) 5 C1 1 C2 ,
(12c)
where C1 5 u divu9u9 1 y divy9u9
tanu (uu9y9 yu9u9), a (13a)
C2 5 u9u9 gradu 1 y9u9 grady 1
tanu (uu9y9 yu9u9). a (13b)
Duplicate terms in the equations are denoted as C1 and C2 here. It should be noted that Eq. (11) degenerates into Eq. (12c) since the term ›KI/›t vanishes in the timeaveraged version. The sum of these three equations is, of course, exactly equal to the time-averaged version of Eq. (6) as ›(KM 1 KT ) 1 div(K M u 1 K T u 1 K T u9 1 K I u9) ›t 1 div(Fu 1 F9u9) 5 av a9v9 1 u F 1 u9 F 9. (14) Equations (12a)–(12c) and (14) indicate that the timeaveraged KE flux Ku should be divided into three parts: one is the mean KE flux K M u, one is the transient-eddy KE flux K T u 5 K T u 1 K T u9, and the last is KI u9. From Eqs. (11) and (12c), we can see that the last part corresponds to the flux term of the time-averaged interaction energy equation. Thus, we refer to it as the interaction energy flux of KE (or the interaction KE flux). Equation (12c) also indicates that the interaction KE flux KI u9 can be further divided into two parts C1 and C2. From the form of Eqs. (12a)–(12c), we interpret the quantities C1 and C2 as the energy conversion rates from KM to KI and from KT to KI, and denote them as C(KM, KI) and C(KT, KI), respectively. Equation (12c) turns into a simple relation C(KM, KI) 1 C(KT, KI) 5 0 when averaged over the entire atmosphere, and both conversion terms can be expressed simply as C(KM, KT) and 2C(KM, KT). The derivation here explains the origin of the traditional energy conversion term C(KM, KT).
for APE. By taking the time average of Eq. (1d), the time-averaged version of the thermodynamic equation is obtained as k p ›u Q . 1 u gradu 1 divu9u9 5 0 ›t p Cp
(15)
Global averaging over isobaric surfaces yields k p hQi ›hui ›hvui ›hv9u9i 1 1 5 0 , ›t ›p ›p p Cp
(16)
where hi denotes the global-average operator in this coordinate system. Subtracting Eq. (16) from Eq. (15) and multiplying by u hui, a balance equation › (u hui)2 (u hui)2 hui2 1 u grad (u hui) ›t 2 2 k p (u hui)(Q hQi) ›hvui ›hv9u9i 1 5 0 3 ›p ›p Cp p uv
›hui (u hui) divu9u9 ›p
(17)
is obtained. Here, we used the identities gradu 5 grad(u hui) 1 gradhui and u gradhui 5 v›hui/›p. Similarly, from the transient-eddy component of the thermodynamic equation, ›u9 1 u gradu9 1 u9 gradu 1 u9 gradu9 divu9u9 ›t k p Q9 5 0 , (18) p Cp multiplying by u9, a balance equation › u92 u92 u92 1 u grad 1 u9 grad u9divu9u9 ›t 2 2 2 k p u9Q9 ›hui u9v9 5 0 u9u9 grad(u hui) Cp ›p p
(19)
3. APE equations In this section, we consider the decomposition of the APE equation, and consider the interaction energy flux
is obtained. In addition, u9 3 [Eqs. (15) (16)] 1 (u hui) 3 [Eq. (18)] yields
›u9(u hui) 1 u9u gradu 1 (u hui)u9 gradu 1 (u hui)u gradu9 1 u9divu9u9 1 (u hui) divu9u9 ›t k k p0 u9(Q hQi) p (u9 hui)Q9 ›hvui ›hv9u9i 1 5 u9 1 0 . ›p ›p Cp Cp p p
(20)
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The sum of Eqs. (17), (19), and (20) gives › (u hui)2 1 (u hui)u gradu (u hui) ›t 2 k p (u hui)(Q hQi) ›hvui ›hv9u9i 1 5 0 3 . ›p ›p Cp p (21) Multiplying by CP( p/p0)2kg, we obtain the following equation: 2k ›A p (u hui)2 hui2 g div 1 Cp u R ›t p0 2 (22)
where
R 5 Cp
p p0
2k
›hvui ›hv9u9i 1 , g(u hui) ›p ›p
(23a)
›AM p 1 Cp p0 ›t
g div
(23b)
(23c)
(u hui)2 hui2 u R(AM ) 2
5 g(T hTi)(Q hQi) av C(AM , AI ),
(24a)
!
2k ›AT p u92 u92 1 Cp u9 g div u1 p0 ›t 2 2 5 gT9Q9 a9v9 C(AT , AI ),
where AM 5
AT 5
Cp
2 Cp
2
p p0 p p0
2k
2k
g(u hui)2 ,
gu92 5
Cp 2
gT92 ,
(25a)
(25b)
(25c)
2k p C(AM , AI ) 5 Cp g(u hui) divu9u9, p0
(25d)
2k p C(AT , AI ) 5 Cp gu9u9 grad(u hui), p0
(25e)
The quantity g is known as the static stability index of dry atmosphere. The quantity A is equal to the approximate expression of APE in pressure coordinates derived by Lorenz (1955), if the global average over the isobaric surface is taken. The original definition of APE by Lorenz (1955) is given as a global quantity and not a local quantity. We can, however, treat the local contributions for it as a density, taking care of extra terms, such as R, in the balance equations. We refer to this quantity simply as APE density and the term R as the residual term. It should be noted that the residual term R vanishes when the global average is taken. Taking the time mean of Eqs. (17), (19), and (20), and multiplying by CP( p/p0)2kg, we obtain a set of balance equations: 2k
(24c)
R(AM ) 5 Cp
p p0
2k
›hvui ›hv9u9i . g(u hui) 1 ›p ›p (25f)
1 k p0 k dhui g5 . p p dp
2k p Cp g div[(u hui)u9u9] 5 C(AM , AI ) 1 C(AT , AI ), p0
2k p AI 5 C p gu9(u hui), p0
5 g(T hTi)(Q hQi) av,
Cp p 2k Cp g(T hTi)2 , g(u hui)2 5 A5 2 p0 2
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(24b)
The quantities AM, AT, and AI are the time-mean, transient-eddy, and interaction components of APE density A. It should be noted that the term ›AI /›t in Eq. (24c) vanishes since AI is always zero. The sum of Eqs. (24a), (24b), and (24c) is, of course, equal to the time-averaged version of Eq. (22) as 2k ›(AM 1 AT ) p ~ u1A ~ u1A ~ u9 1 Cp g div A M T T p0 ›t ! 2 ~ u9 hui u R(A ) 1A I M 2 5 g(T hTi)(Q hQi) 1 gT9Q9 av a9v9, (26) ~ 5 (u hui)2 /2, A ~ 5 u92 /2, and A ~ 5 (u where A M T I hui)u9. Similarly to the case of KE equations, we refer to the terms C(AM, AI) and C(AT, AI) as the energy conversion rates from AM to AI and from AT to AI. Equation (24c) gives the relationship between the interaction energy flux of APE and two energy conversion terms. It turns into a simple relationship, C(AM, AI) 1 C(AT, AI) 5 0, when averaged over the entire atmosphere, and both conversion terms are expressed simply as C(AM, AT) and 2C(AM, AT) respectively.
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4. Box diagram of local energy cycle and interaction energy fluxes The key point in the previous sections is that the KE (or APE) equation is divided not into two but into three parts consisting of the mean, transient, and interaction energies, when the basic variables are divided into timemean and transient-eddy components. Taking this into account, we express the time-averaged version of the balance equations for APE density as ›AM 5 G(AM ) C(AM , KM ) C(AM , AI ) ›t B(AM ) 1 R(AM ),
(27a)
›AT 5 G(AT ) C(AT , K T ) C(AT , AI ) B(AT ), ›t (27b) 0 5 C(AM , AI ) 1 C(AT , AI ) F(AI ),
(27c)
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 the available potential energy (APE) and kinetic energy (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 for APE and KE.
where G(AM ) 5 g(T hTi)(Q hQi),
(28a)
G(AT ) 5 gT9Q9,
(28b)
C(AM , K M ) 5 v a,
(28c)
C(AT , KT ) 5 v9a9,
(28d)
2k p (u hui)2 hui2 g div u , (28e) B(AM ) 5 Cp p0 2 B(AT ) 5 Cp F(AI ) 5 Cp
p p0 p p0
2k
! u92 u92 u9 , g div u1 2 2
(28g)
and for KE density as ›KM 5 C(AM , KM ) C(K M , KI ) D(KM ) B(K M ), ›t (29a) ›KT 5 C(AT , K T ) C(K T , KI ) D(KT ) B(KT ), ›t (29b) 0 5 C(K M , KI ) 1 C(KT , KI ) F(KI ), where
(30a)
D(KT ) 5 u9 F 9,
(30b)
u2 1 y 2 1F u , (30c) 2 " ! # u92 1 y92 u92 1 y92 B(K T ) 5 div 1 F9 u9 , u 1 div 2 2 (30d)
B(KM ) 5 div
F(K I ) 5 div(uu9u9 1 y y9u9). (28f)
2k g div[(u hui)u9u9],
D(K M ) 5 u F ,
(29c)
(30e)
The symbols G(*), D(*), and B(*) denote generation, dissipation, and boundary flux terms and the symbol C(A*, K*) denotes a conversion term from A* to K*. Other terms are already defined in the previous sections. It should be noted that the terms AT and K T are simply expressed as AT and KT in this section. From these equations, we can express the local balance of these terms as Fig. 1. To represent Eqs. (27c) and (29c) on the diagram, two nodes and wavy arrows, which denote the interaction energies and their fluxes, are added to the conventional one. Reflecting the fact that the time mean of interaction energies is always zero, the quantities AI and KI are denoted by nodes (not by boxes). The term R(AM) is included in the term B(AM) in the figure for simplicity. The diagram turns into the conventional one when averaged over the entire atmosphere. The energy diagram shown in Fig. 1 suggests that the local feature of the energy interactions between mean
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and eddy fields can take quite different form from the global one. Theoretically, the following six types of local interactions are possible for KM and KT: KM
KM
Y Y
Y Y
. ;;;;;
,
KT
, ;;;;;
KM
KM
[ [
[ [
. ;;;;;
,
KT
KT KM
Y [
[ Y
,
KT
,
, ;;;;;
(31)
.
KT
Similar interaction patterns are also possible for AM and AT. Moreover, energy interactions between distant locations are possible through the interaction energy fluxes, such as KM
KM
Y [ KT
. ;;;;;
[ Y
We refer to a dominant energy flow pattern extracted from an exact energy diagram as the energy path and a diagram such as diagram (31) or (32) as the simplified energy diagram. Simplified energy diagrams are especially useful for representing energy interactions like diagram (32), but we should be careful not to overinterpret the expression because they do not show the exact energy balance differently than the exact local energy diagram indicated in Fig. 1.
5. Discussion
, ;;;;;
KM
. ;;;;;
,
KT
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(32)
KT .
The variety of energy interaction patterns is quite large. However, the complete information about the energy interactions between mean and eddy fields will be given by plotting interaction energy fluxes AI u9 and KI u9 together with spatial distributions of energy conversion terms C(AM, AI), C(AT, AI), C(KM, KI), and C(KT, KI). In addition, the box diagram shown in Fig. 1 quantifies the balance of all the terms in the energy equations for the specific locations. In this meaning, the energy interaction fluxes and box diagrams of the local energy cycle give complementary tools for the diagnosis of the energy interactions between mean and eddy fields. In Part II of this paper (Murakami et al. 2011, hereafter Part II), we will see some examples.
In Holopainen (1978), the terms C(KM, KI) and C(KI, KT) are expressed as C*(kM, kT) and C(kM, kT), and the question is raised as to ‘‘which one of the terms should be interpreted as representing an energy conversion from kM to kT.’’ From the diagram provided in the previous section, it is obvious that both terms are required to represent the local feature of the energy conversion between mean and eddy fields. It is also obvious that the local values (and spatial distributions) of these two terms are different unless the interaction energy flux vanishes. Moreover, these two conversion terms and the interaction energy flux give complete information about the energy interactions between mean and eddy fields. As an example, we consider an idealized case such that the time-averaged quadratic quantities of eddy (such as u9y9) are spatially uniform. In this case, the term C(KM, KI) obviously vanishes, but the term C(KT, KI) remains if the mean fields are not spatially uniform. It means that, as a time mean, the eddy energy can interact only with the interaction energy and the energy for mean field is not affected. This represents in essence a nonacceleration theorem. Using the simplified energy diagram of this paper, the situation above can be expressed as KM ... .. .
[
. ;;;;;
KT
KM ... .. .
Y
(33)
KT .
Similarly, an interaction pattern such as KM
KM
Y
[
. .. ;;;;; .. .. KT
.. .. .. KT
(34)
is also possible. In this case, as a time mean, the mean energy interacts only with interaction energy and does
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not affect the eddy energy. In other words, as a time mean, eddies can transport the mean energy KM from a place to another as interaction energy K I u9 with no effect on the eddy energy KT. It may be difficult to observe the pure case of such energy interaction patterns in the real atmosphere. However, in Part II, we will see that cases of such dominant interaction patterns as KM ! K I ,K I ! KM
or
KT ! KI ,KI ! KT ,
(35) (36)
are commonly observed in the real (or simulated) atmosphere. In most of the literature on atmospheric energetics analysis, only the term C(KT, KI) is treated and is referred to as ‘‘barotropic conversion.’’ From the viewpoint of this paper, however, it is the term C(KM, KI), rather than C(KI, KT), that represents the energy conversion from KM and should be referred to as barotropic conversion. The energy conversion terms C(KM, KI) and C(KT, KI) are also associated with the barotropic part of the (extended) Eliassen–Palm flux (wave-activity flux). Indeed, in the framework of QG theory, the horizontal part of C(KM, KI) is equivalent to u divh Ex 1 y divh Ey u gradh KT
(37)
and the horizontal part of conventional barotropic conversion term C(KI, KT) is equivalent to 1 ›u ›y ›y ›u u9y9 1 , Ex D 5 (y9y9 u9u9) 2 ›x ›y ›x ›y (38) where the subscript h for gradient and divergence operators means the horizontal part of those, and 1 Ex 5 (y9y9 u9u9), u9y9 , 2 1 Ey 5 u9y9, (y9y9 u9u9) , 2 ›u ›y ›y ›u D5 , 1 ›x ›y ›x ›y
The wave-activity analysis is useful for the study of wave propagation because it gives a local conservation low associated with wave propagation. On the other hand, the purpose of the analysis here is to grasp the local feature of the energy interaction (exchanges of energy). From the viewpoint of Hamiltonian fluid dynamics (e.g., Salmon 1998), the relationship between the waveactivity analysis and the analysis provided here can be understood as follows. In the case of time-mean and transient-eddy fields, the wave-activity is related to the pseudoenergy (e.g., Plumb 1985, 1986). The pseudoenergy in a Hamiltonian system is defined symbolically as H(z) H(z0 ) 1 C(z) C(z0 ),
(40)
where H, C, z0, and z are the Hamiltonian, the Casimir invariant, a basic state of the system, and an arbitrary state in the neighborhood of z0, respectively (e.g., Holm et al. 1985; Abarbanel et al. 1986; McIntyre and Shepherd 1987). The Hamiltonian usually represents the total energy of the system. The Casimir invariant represents the first integral associated to some symmetry of the system, and the quantity H 1 C is referred to as energy-Casimir [an analog of momentum-Casimir discussed by Haynes (1988) and Ran and Gao (2007)]. In the case of fluid dynamics, a Casimir is a functional of potential vorticity. As emphasized in this paper, the difference of total energy between an arbitrary state and a basic state is not the eddy energy but the sum of eddy and interaction energies. These facts clarify the reason why the terms C(AM, AI) and C(KM, KI) appear in the TEM energetics although only the terms C(AT, AI) and C(KT, KI) appear in the classical energetics analysis, and why the interaction energy fluxes are intrinsically included in the wave-activity flux.
6. Summary
(39a)
(39b)
(39c)
(e.g., Hoskins et al. 1983; Simmons et al. 1983; Trenberth 1986). It is also notable that the interaction energy flux is (implicitly) included in the wave activity flux. These facts suggest that the energetics analysis and the waveactivity analysis are intrinsically compatible. But there are some differences between these two types of analysis.
The decomposition of energy equations, corresponding to the division of basic variables into time-mean and transient-eddy components, is considered. The energy equation is divided into three parts consisting of the mean, eddy, and interaction energy equations, and not into two. The flux term of the time-averaged interaction energy equation (i.e., the interaction energy flux) is further divided into two parts. Those just define the two types of energy conversion term between mean and eddy fields. Therefore, the energy interaction (conversion) between mean and eddy fields is simply a part of the flux term of the total energy equation, and a set of the interaction energy flux and these two conversion terms gives the complete information about the energy interaction. The local
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balance of all the terms in the time-averaged mean, eddy, and interaction energy equations for APE and KE is correctly represented by a box diagram that includes nodes for interaction energies and arrows for the interaction energy fluxes. The box diagram of the local energetics and the interaction energy flux will be useful for diagnosis of the atmospheric general circulation. A brief discussion about the relationship to the classical energetics analysis and to the wave-activity analysis is also given. Acknowledgments. The author thanks three anonymous reviewers. In particular, one of the reviewers provided many detailed comments upon the manuscript that greatly improved the paper. The author also thanks Eiki Shindo of Meteorological Research Institute for his useful discussions. REFERENCES Abarbanel, H. D. I., D. D. Holm, J. E. Marsden, and T. S. Ratiu, 1986: Non-linear stability analysis of stratified fluid equilibria. Philos. Trans. Roy. Soc. London, 318A, 349–409. Haynes, P. H., 1988: Forced, dissipative generalizations of finiteamplitude wave-activity conservation relations for zonal and nonzonal basic flows. J. Atmos. Sci., 45, 2352–2362. Holm, D. D., J. E. Marsden, T. Ratiu, and A. Weinstein, 1985: Nonlinear stability of fluid and plasma equilibria. Phys. Rep., 123, 1–116. Holopainen, E. O., 1978: A diagnostic study on the kinetic energy balance of the long-term mean flow and the associated transient fluctuation in the atmosphere. Geophysica, 15, 125–145. Hoskins, B. J., I. N. James, and G. H. White, 1983: The shape, propagation and mean-flow interaction of large-scale weather systems. J. Atmos. Sci., 40, 1595–1612. Iwasaki, T., 2001: Atmospheric energy cycle viewed from wave– mean flow interaction and Lagrangian mean circulation. J. Atmos. Sci., 58, 3036–3052.
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Lorenz, E. N., 1955: Available potential energy of the maintenance of the general circulation. Tellus, 7, 157–167. Mak, M., and M. Cai, 1989: Local barotropic instability. J. Atmos. Sci., 46, 3289–3311. McIntyre, M. E., and T. G. Shepherd, 1987: An exact local conservation theorem for finite amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnol’d’s stability theorems. J. Fluid Mech., 181, 527–565. Murakami, S., R. Ohgaito, and A. Abe-Ouchi, 2011: Atmospheric local energetics and energy interactions between mean and eddy fields. Part II: An example for Last Glacial Maximum climate. J. Atmos. Sci., in press. Orlanski, I., and J. Katzfey, 1991: The life cycle of a cyclone wave in the Southern Hemisphere. Part I: Eddy energy budget. J. Atmos. Sci., 48, 1972–1998. Plumb, R. A., 1983: A new look at the energy cycle. J. Atmos. Sci., 40, 1669–1688. ——, 1985: An alternative form of Andrews’ conservation law for quasi-geostrophic waves on a steady, nonuniform flow. J. Atmos. Sci., 42, 298–300. ——, 1986: Three-dimensional propagation of transient quasigeostrophic eddies and its relationship with the eddy forcing of the time-mean flow. J. Atmos. Sci., 43, 1657–1678. Ran, L., and S. Gao, 2007: A three-dimensional wave-activity relation for pseudomomentum. J. Atmos. Sci., 64, 2126–2134. Salmon, R., 1998: Lectures on Geophysical Fluid Dynamics. Oxford University Press, 378 pp. Saltzman, B., 1957: Equations governing the energetics of the large scales of atmospheric turbulence in the domain of wavenumber. J. Meteor., 14, 513–523. Simmons, A. J., J. M. Wallace, and G. W. Branstator, 1983: Barotropic wave propagation and instability, and atmospheric teleconnection patterns. J. Atmos. Sci., 40, 1363–1392. Tanaka, H., E. Kung, and W. E. Baker, 1986: Energetics analysis of the observed and simulated general circulation using threedimensional normal mode expansions. Tellus, 38A, 412–428. Trenberth, K. E., 1986: An assessment of the impact of transient eddies on the zonal flow during a blocking episode using localized Eliassen–Palm flux diagnostics. J. Atmos. Sci., 43, 2070– 2087.
