Dynamics of Synoptic Eddy and Low-Frequency Flow Interaction.Part III: Baroclinic Model Results

L.-L. PAN* AND F.-F. JIN

Department of Meteorology, The Florida State University, Tallahassee, Florida, and Department of Meteorology, University ofHawaii at Manoa, Honolulu, Hawaii

M. WATANABE

Graduate School of Environmental Earth Science, Hokkaido University, Sapporo, Japan

(Manuscript received 18 April 2005, in final form 9 November 2005)

ABSTRACT

In this three-part study, a linear closure has been developed for the synoptic eddy and low-frequency flow(SELF) interaction and demonstrated that internal dynamics plays an important role in generating theleading low-frequency modes in the extratropical circulation anomalies during cold seasons.

In Part III, a new linearized primitive equation system is first derived for time-mean flow anomalies. Thedynamical operator of the system includes a traditional part depending on the observed climatological meanstate and an additional part from the SELF feedback closure utilizing the observed climatological propertiesof synoptic eddy activity. The latter part relates nonlocally all the anomalous eddy-forcing terms in equa-tions of momentum, temperature, and surface pressure to the time-mean flow anomalies. Using the obser-vational data, the closure was validated with reasonable success, and it was found that terms of the SELFfeedback in the momentum and pressure equations tend to reinforce the low-frequency modes, whereasthose in the thermodynamic equation tends to damp the temperature anomalies to make the leading modesequivalent barotropic. Through singular vector analysis of the linear dynamical operator, it is highlightedthat the leading modes of the system resemble the observed patterns of the Arctic Oscillation, AntarcticOscillation, and Pacific–North American pattern, in which the SELF feedback plays an essential role,consistent with the finding of the barotropic model study in Part II.

1. Introduction

In the extratropics, atmospheric variability is domi-nated by abundant transient synoptic eddies. Embed-ded in such a turbulent circulation, there are prominentrecurring patterns such as the Arctic Oscillation (AO)(Thompson and Wallace 1998, 2000) or North AtlanticOscillation (NAO) (Wallace and Gutzler 1981; Hurrell1995; Wallace 2000), Antarctic Oscillation (AAO)(Gong and Wang 1999; Thompson and Wallace 2000),

and Pacific–North American (PNA) pattern (Wallaceand Gutzler 1981), which are also known as telecon-nection patterns dominating in the monthly or seasonalmean anomaly fields. The AO and AAO are often re-ferred together to as annular modes (Thompson andWallace 2000).

Dynamical origins of these patterns of the low-frequency variability have been the central subject inmany studies (e.g., Qin and Robinson 1992; Branstator1992; Robinson 1991, 1994; Kimoto et al. 2001; Lorenzand Hartmann 2001; Feldstein 2002; Koo and Ghil2002; Koo et al. 2002; Kravtsov et al. 2003, 2005; Wa-tanabe and Jin 2004; Kondrashov et al. 2004), and arestill in debates (Shindell et al. 1999; Baldwin andDunkerton 1999; DeWeaver and Nigam 2000; Ambaumand Hoskins 2002). Given that earlier observationalstudies suggest that the transient eddy forcing associ-ated with changes in storm tracks is the major source ofthe persistent low-frequency anomalies (Egger andSchilling 1983; Hoskins et al. 1983; Lau 1988, 1991;

* Current affiliation: International Pacific Research Center,School of Ocean and Earth Science and Technology, University ofHawaii at Manoa, Honolulu, Hawaii.

Corresponding author address: Dr. Lin-Lin Pan, InterantionalPacific Research Center, School of Ocean and Earth Science andTechnology, University of Hawaii at Manoa, 2525 Correa Road,Honolulu, HI 96822.E-mail: lpan@hawaii.edu

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© 2006 American Meteorological Society

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Metz 1987; Karoly 1990; Cai and van den Dool 1991), avariety of stochastic dynamical models have been de-veloped to parameterize the transient eddy forcing(e.g., Penland and Ghil 1993; Penland and Matrosova1994; Newman et al. 1997; Whitaker and Sardeshmukh1998; Zhang and Held 1999; Majda et al. 1999, 2003;DelSole 2001; Winkler et al. 2001). Whether the tran-sient eddies serve merely as a stochastic forcing or apositive feedback to the mean flow was later ques-tioned (Feldstein and Lee 1998). However, the increas-ing evidence (Hartmann and Lo 1998; Limpasuvan andHartmann 2000; Pan and Jin 2005) indicates that tran-sient synoptic eddy momentum fluxes associated withthe low-frequency zonal flow anomaly tend to reinforcethe zonal wind variation: a two-way positive feedback.It is becoming clear that the internal dynamical pro-cesses, particularly the synoptic eddy and low-frequency flow (SELF) feedback, are indispensable forthe low-frequency variability (Pan 2003).

In Parts I and II of this three-part study (Jin et al.2006a,b, hereafter Part I and Part II), we have proposedan alternative dynamical approach for describing theSELF feedback, and derived a linear SELF closure in abarotropic model (see also section 2b). The anomaloustime–mean eddy forcing is linearly related to time-mean flow anomaly through the SELF closure. Basedon this closure, we established a linear barotropicframework with respect to a stochastic basic flow thatincludes not only basic mean flow but also climatologi-cal properties of the synoptic eddy statistics. The baro-tropic model study reveals that the scale-selectiveSELF feedback is of essential importance in the originof low-frequency modes such as the AO, AAO, andPNA, and it also can amplify the atmospheric responseto remote forcing.

In this Part III, we extended the previous barotropicmodel study to a primitive equation framework. Sec-tion 2 describes the linear primitive equation model[referred to as the linear baroclinic model (LBM)] andpresents the derivation of the general form of lineardynamic system with SELF feedback, based on ob-served climatological mean flow and climatologicalproperties of the three-dimensional synoptic eddy sta-tistics. Section 3 presents the construction of the three-dimensional stochastic flow. Section 4 gives validationof the linear SELF feedback closure using observa-tional data. Sections 5 and 6 elucidate the role of theSELF feedback in the formation of low-frequencymodes and regulating the atmospheric response to re-mote forcing by analyzing the leading modes of thelinear dynamic operator and forced solutions to ideal-ized forcing, respectively. The conclusions are given in

section 7, whereas the details of the model equationsare provided in the appendix.

2. A primitive equation framework with SELFfeedback

a. The primitive equation model

In this study, we develop the primitive equationframework from the LBM used in earlier studies (Wa-tanabe and Kimoto 2000, 2001; Watanabe and Jin2004), and the detailed equations are given in the ap-pendix. The model variables consist of vorticity (�), di-vergence (D), temperature (T), and logarithm of sur-face pressure (�). We choose a version with five verticallevels (� � 0.8987, 0.6983, 0.4439, 0.2220, 0.06224), anda T21 spectral resolution. Furthermore, we only con-sider zonal wavenumbers 0–15 to reduce the computa-tional burden even though the total degrees of freedomare 7056, still quite large. Three dissipation terms areincluded: a biharmonic horizontal diffusion with thedamping time scale of one day for the smallest wave, aweak vertical diffusion (damping time scale of 1000days), and the Newtonian damping and Rayleigh fric-tion. The latter is represented by a linear drag that hasa one-day damping at the lowest layer while the damp-ing rate is fifteen days elsewhere. This linear damping isadequate to suppress the unstable growth of the linearbaroclinic waves (Hall and Sardeshmukh 1998).

b. The closure for SELF feedback

The concepts and approaches described in Parts Iand II for deriving the SELF feedback closure in baro-tropic model can be naturally extended into a generalprimitive equation framework. For simplicity, we let xrepresent state vector of the variables in the baroclinicmodel, and separate it into three parts:

x � x c � xa � x�, �1�

where the superscripts a and c stand for anomaly andclimatology, respectively. The prime indicates high-frequency variability, and the overbar represents timemean such as monthly mean. Thus x c is the climatologi-cal mean state obtained through long-term average ofthe observations; x a denotes the time-mean flowanomaly, which we will loosely refer to as the low-frequency anomaly; and x� represents the high-frequency transient eddy field.

We write symbolically the linearized equation for thelow-frequency variability xa as follows:

�

�txa � L�x c�xa � A�x��x�

a� Q

a, �2�

1710 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 63

where the detailed equations are given in the appendix.Here, L is a linear operator depending on climatologi-cal basic state, Q

adenotes time-mean anomalies in the

external forcing. This equation extends the simplebarotropic model of Part I to a primitive equationmodel. The anomalous transient eddy forcing termA(x�)x�

ais defined as the difference between the time-

mean (such as monthly mean) eddy forcing A(x�)x� andits long-term climatological mean A(x�)x�

c. Without

A(x�)x�a, Eq. (2) reduces to the conventional LBM

(e.g., Hoskins and Karoly 1981).Similar to Part I, we assume that the anomalous time-

mean eddy forcing can be linearly related to time-meanflow anomaly through a SELF feedback closure. In theprimitive equation model, this closure can be written as

A�x��x�a Lf xa, �3�

where Lf is the closure operator for the SELF feedback.In the next section, we will derive an explicit, nonlocalform of the closure operator Lf in the primitive equa-tion model.

c. The derivation of the closure

We consider a hypothetical ensemble of the quasi-stationary stochastic state, X, in which the observedatmospheric circulation x is regarded as one particularrealization. Let angle brackets represent the ensemblemean. Then X can be decomposed as

X � �X� � X�, �4�

where X� represents the stochastic eddy field in whichx� is one particular realization as well. With an ergodicassumption (see section 3 of Part I), the ensemble-mean quantities can be substituted by the time meansof one realization as

�X� x � x c � xa. �5�

As in Part I, the stochastic eddy field X� is separatedinto two parts: a major but stochastic stationary com-ponent X�c that characterizes the climatological proper-ties of the observed storm track and a minor but quasi-stationary stochastic component X�a that represents thesynoptic eddies modulated by the low-frequencyanomaly. Thus X� can be expressed as

X� � X�c � X�a. �6�

Based on the ergodic assumption, the time mean ofeddy flux term, its climatology, and low-frequencyanomaly can be approximated by the corresponding en-semble means based on X�; namely,

A�x��x�c �A�X�c�X�c�, �7a�

A�x��x�a �A�X�a�X�c� � �A�X�c�X�a� � �LE�X�c�X�a�,

�7b�

where LE is a linear operator depending on X�c. In Eq.(7b) we have neglected the nonlinear term �A(X�a)X�a�by assuming that it is relatively small. Using Eq. (7b),Eq. (2) becomes

�

�txa � L�x c�xa � �LE�X�c�X�a� � Q

a. �8�

To obtain the closure for the term representing theensemble-mean anomalies of synoptic eddy forcing inEq. (8), we use the following linearized equation for X�a(refer to the appendix for the detailed derivation),

�

�tX�a � L�x c�X�a � LE�X�c�x

a � Q�a, �9�

where all nonlinear terms are neglected. This equationis an extension of the barotropic version of Eq. (8) inPart I. For simplicity, we also omit the high-order tem-poral terms in the transient flow equation. Equations(8) and (9) now form a coupled dynamic system thatdescribes the interaction between the low-frequencyanomaly xa and anomalous synoptic eddies X�a.

For a given x c, Eq. (9) is a linear model of anomalousstorm track activities. Our approach for modelinganomalous storm tracks using Eq. (9) avoids the timeintegration for the basic stochastic eddy–flow ensembleX�c. Instead, the basic stochastic eddy state is prescribedas a part of the climatological state, which enables us toexplicitly solve Eq. (9) to obtain a closure for Eq. (8).

Recall that X�c represents typical, synoptic travelingdisturbances, and the stochastic high-frequency statemay be expressed as

��c��, �, �, t� � n�1

Nc

�n2�n�t��n��, �, ��ei�nt � c.c., �10�

where c.c. stands for complex conjugate; �, �, and �represent longitude, latitude and vertical level, respec-tively. Each �n(�, �, �)ei�nt is a pair of complex em-pirical orthogonal function (CEOF) patterns associatedwith the angle phase speed �n and variance �2

n, all ob-tained from the observation. As in Part I, each �n(t) isa normalized and independent red noise process [d�n /dt� �n/�n � wn(t); wn(t) is a white noise] with the de-correlation time scale �n, which is estimated from thee-folding time of the autocorrelations of each CEOFtime series. In Eq. (10), Nc is the number of CEOFmodes used. The details of the CEOF will be given insection 3.

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With the definition of X�c given in Eq. (10), the solu-tion for the anomalous high-frequency ensemble hasthe following form:

X�a��, �, �, t� � n�1

Nc

X�n��, �, �, t�ei�nt � c.c. �11�

Then using Eqs. (10) and (11) and defining

Xn � ��n�t�X�n�, �12�

where the tilde means complex conjugate, we have

�LE�X�c�X�a� � n�1

Nc

�n2LE�Xn�Xn � c.c.. �13�

Multiplying �n(t) to both sides of Eq. (9) and taking theensemble average, we obtain

�n1 i�n � L�x c ��Xn � LE�Xn�x

a. �14�

Here we used the relation

� �n�t��X�n�t � �

�Xn

�t���n�t�X�n�

n��n�t�X�n�

n, �15�

where d�n/dt � �n/�n � wn(t), the definition of �n(t), isused. We assumed �Xn /�t 0 following the quasi-equilibrium assumption (see appendix A of Part I).

Combining Eqs. (8), (13), and (14), we obtain an ex-plicit form of the closed system for the low-frequencyanomalies:

�

�txa � L�x c�xa � Lfx

a � Qa, �16a�

Lfxa �

n�1

Nc

�n2LE�Xn���n

1 i�n

� L�x c��1LE�Xn��xa � c.c., �16b�

where the operator Lf represents a linear, nonlocal clo-sure for the SELF interaction. In Eq. (16), the dynami-cal impact of the climatological stationary waves on thelow-frequency variability is included in the conven-tional linear operator L. The impact of climatologicalproperties (such as its typical spatial pattern, variance,typical lifetime, and phase speed) of the synoptic eddieson the low-frequency variability is included in theSELF feedback closure operator Lf. The one and onlydifference between Eq. (16) and the traditional LBM isthe SELF feedback closure. With the SELF feedback,Eq. (16) becomes a new baroclinic framework for thelow-frequency variability. Since this is an anomalymodeling approach, the climatological properties of thestorm track is externally given as a part of the basicstate, but the small change in the storm track associatedwith the low-frequency anomaly is dynamically deter-mined in the system.

The system (16) is symbolically identical to the baro-tropic counterpart described in Part I, which in factmounts to a conceptual difference between the tradi-tional and this extended linear framework, as schemati-cally illustrated by Fig. 4 of Part I. In a traditional LBMframework, linear dynamics of the low-frequency vari-ability is controlled by the climatological mean flow andother sources for low-frequency variability, includingeddy forcing, are all considered as external forcing. Inthe extended LBM framework (16), the climatologicalproperties of synoptic eddies also become importantelements in generating low-frequency modes of vari-ability.

3. Properties of a three-dimensional stochasticsynoptic eddy field

a. Reconstruction of three-dimensional synopticeddy fields

To ensure that the stochastic basic state as expressedby Eq. (10) captures the basic nature of observed syn-optic eddies, we take an approach similar to Part I (seealso Pan 2003), but extend it to three-dimensional eddyfields. We adopt a CEOF decomposition method (Bar-nett 1983; Horel 1984) to derive the basic patterns andassociated parameters that characterize the climatologi-cal storm track. The primary dataset is the NationalCenters for Environmental Prediction–National Centerfor Atmospheric Research (NCEP–NCAR) reanalysis(Kalnay et al. 1996). It includes daily data of zonal wind(u), meridional wind (�), temperature (T), and surfacepressure (ps) during the cold season (November–Aprilfor the Northern Hemisphere and May–October for theSouthern Hemisphere) between the years of 1979 and1995. A bandpass filter retaining 2–8 days is applied tothese fields in order to extract transient baroclinic ed-dies. Each variable of the transient fields is weighted byits global-mean variance, horizontal area, and verticalmass distribution before performing the CEOF analy-sis.

As in the previous section, let x� represent the fil-tered fields (u�, ��, T �, p�s), then the CEOF provides thefollowing expansion,

x���, �, �, t� � n�1

N

Xn��, �, ��Yn�t� � c.c.

� n�1

N

Xn��, �, ���nein�t� � c.c., �17�

n�t� � arctan�ImYn�t�

ReYn�t��� �nt � �n�t�, �18�

�n2 � �Yn�t�Yn�t��, �19�

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where �n(�, �, �) and Yn(t) are the nth complex EOFpattern and time coefficient, respectively. From the realand imaginary parts of Yn(t), we obtain the angle phase�n(t), which is further decomposed into a linearly fittedphase propagation �nt and a small residual ��n(t). Theabsolute of the time coefficient, Yn(t)Yn(t), is used tocompute the variance �2

n and e-folding decay time (�n),the latter being estimated from the autocorrelationfunction. Through the above analysis, high-frequencyfields are approximately described in terms of variance�2

n, primary frequency �n, spatial structure �n(�, �, �),and the decorrelation time �n. Using the set of (Xn, �n,�n, �n; n � 1, Nc) to capture the climatological proper-ties of the observed synoptic eddy field, the stochasticsynoptic eddy field X�c in Eq. (10) is reconstructed. Itshould be noted that the constructed field of X�c is dif-ferent from in two senses: the CEOFs are truncated atNc in constructing X�c besides the phase, amplitude, anddecay time are fixed at constant for each CEOF. Con-sequently, the constructed X�c excludes an irregular, oranomalous, part of the high-frequency states, which willbe internally solved in the SELF closure.

Figure 1a presents the spatial structure for the realpart of the first CEOF as represented by the meridionalwind �� during the cold season of the Northern Hemi-sphere (NH). There are approximately 4–5 negative/positive centers in the �� field, located mainly over thePacific and the Atlantic regions. The first CEOF ac-counts for 10% of the total variance. The pattern forthe imaginary part is similar to the real part but isshifted eastward around 15° in longitude and the phasedifference is 90°. The results for other variables, such asu�, T �, and p�s, are similar [not shown here, for detailssee Pan (2003)]. The wavelike structure in the velocityfield tilts slightly to the west in the vertical cross section(Fig. 1b) and the T � field tilts slightly eastward (notshown).

The relative position between p�s, u�, ��, and T � at thelowest level (� � 0.8987) is shown in Fig. 1c. Warmtemperature anomalies are located on the southeastside of the low pressure center, where the southerlyanomaly dominates. This structure of synoptic eddies isconsistent with those of typical cyclones and anticy-clones. The second CEOF, which accounts about 8% ofthe total variance, shares many common features withthe first CEOF. Similar results are also obtained for theSouthern Hemisphere (SH; not shown).

b. Representation of the climatological eddy forcing

We now examine whether the constructed stochasticeddy states capture the climatological eddy forcingfields. On one hand, we calculate the climatologicaleddy forcing term A(x�)x�

cdirectly using the observed

data of the high-frequency fields. On the other hand,we can use the reconstructed stochastic synoptic fieldX�c to evaluate the ensemble mean of the climatologi-cal eddy forcing �A(X�c)X�c�. Then we will compare�A(X�c)X�c� with A(x�)x�

cto verify that X�c captures the

climatological properties of synoptic eddies. Using theorthogonality of the CEOF time coefficients in Eq.(17), we have A(x�)x�

c� N

n�1A(Xn)Xn � c.c.. Owingto the assumption of independence of each complexred noise processes in Eq. (10), �A(X�c)X�c� can be ex-pressed as

�A�X�c�X�c� � n�1

Nc

A�Xn�Xn � c.c., �20�

which includes the eddy-induced forcing term for thevorticity, divergence, temperature and surface pressure

FIG. 1. (a) Real part of the first CEOF as represented by themeridional wind �� at middle level (� � 0.4439) for the NH, (b)vertical cross section of �� at 40°N, and (c) anomalies of surfacepressure (thin line), temperature (thick line), and winds (arrow) atsurface layer (� � 0.8987). Units are hPa for pressure, K fortemperature, and m s1 for winds.

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fields. The details of these terms are given in Eqs. (A7),(A10), (A11), and (A12) in the appendix.

Thus the difference between A(x�)x�c

and �A(X�c)X�c�is mainly due to the truncation error, namely Nc � N.By setting Nc � 20, the majority of mean eddy varianceand eddy fluxes is well captured. The vorticity eddyforcing terms are expressed in terms of the streamfunc-tion tendency by operating an inverse Laplacian factor.For the NH, the climatological eddy vorticity forcing(Figs. 2a and 2b) is positive in the polar region at thelower level and negative at the upper level. The eddythermal forcing has a north–south dipole structure withpositive value in the north and negative value in thesouth, corresponding to a northward eddy heat trans-port (Figs. 2c and 2d). The results are consistent withprevious studies (e.g., Lau and Holopainen 1984). Simi-lar results are also obtained for surface pressure eddyforcing (Figs. 2e and 2f). All main centers of these eddyforcing fields (Figs. 2a–f) are located over the Pacificand Atlantic storm track regions, respectively. The re-sults for the SH are very similar except that the eddyforcing terms are more zonally symmetric. Thus with arelative small number of Nc, the stochastic representa-tion of the synoptic eddies captures well the climato-logical eddy forcing fields.

4. The validation of the SELF feedback closure

As expressed in Eq. (3), we can relate the anomalouseddy forcing directly to low-frequency flow anomaliesby the SELF feedback operator Lf, which is derivedthrough a series of assumptions and approximationsand depends only on the climatological mean state andclimatological measures of the synoptic eddies (Xn, �n,�n, �n; n � 1, Nc). It uses neither any information aboutobserved anomalous eddy forcing A(x�)x�

anor infor-

mation about observed low-frequency variability of xa,so the feedback operator Lf has no direct built-in an-swer about the observed anomalous eddy activity andlow-frequency anomalies. Therefore, we can validatethis dynamically derived SELF feedback operator byapplying Lf onto the observed xa to calculate param-eterized eddy forcing as Lf xa and then comparing itwith the observed eddy-forcing anomalies.

As in Part I, tests are conducted with the focus on theleading low-frequency modes and the associated eddyforcing patterns. We first regressed the anomalous tran-sient eddy forcing A(x�)x�

a, calculated directly from the

filtered data of x�, onto the monthly means of indicesfor low-frequency variability such as the AO and AAOduring the cold season. The definitions for the AO andAAO indices are the same as those defined by Thomp-son and Wallace (1998) and Gong and Wang (1999),

respectively. A similar regression is made to themonthly mean anomaly field xa, which represents thethree-dimensional structure of the AO and AAO fromthe observation. Then we calculate the parameterizededdy-forcing term (Lfx

a) by applying the SELF feed-back operator on the regressed low-frequency patterns.

The observed and parameterized transient eddy forc-ing fields for streamfunction, temperature, and thelogarithm of surface pressure associated with the AOresemble each other reasonably well (Figs. 3a–c). Bothparameterized and observed streamfunction forcing(Fig. 3a) have negative and positive values at low andmiddle latitudes, respectively, with the magnitudestronger in the upper troposphere. The forcing at low

FIG. 2. Climatological eddy forcing associated with (a), (b)streamfunction at � � 0.222, (c), (d) temperature at � � 0.8987,and (e), (f) surface pressure over the NH. The panels (a), (c), (e)are observed climatological mean while (b), (d), (f) are obtainedfrom the stochastic representation of the synoptic eddy field [Eq.(10)]. The contour intervals are 20 m2 s2, 3 � 106 K s1, and 4� 106 hPa s1 for streamfunction, temperature, and surface pres-sure forcing, respectively.

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levels of the polar region is slightly positive. The re-gressed streamfunction anomaly field associated withthe AO (Figs. 4a and 4b) correlates positively with thetendency fields induced by synoptic eddies. Given thatthe observed eddy-forcing pattern can be reproduced asthe response to the AO in the dynamical closure, thisstrongly suggests a positive SELF feedback in the vor-

ticity equation. The temperature forcing field is positiveat high-latitude regions, negative at middle-latitude re-gions, and slightly positive at low latitudes (Fig. 3b).The regressed temperature anomaly field associatedwith the AO (Fig. 4c) correlates negatively with thetemperature tendency field induced by synoptic eddies.Thus, the transient eddy forcing has a damping effecton the meridional temperature gradient associated withthe AO. This indicates that the eddy forcing in thetemperature equation tends to reduce the vertical shearof zonal flow. As the result, eddy forcing tends to make

FIG. 4. The regression pattern of the AO: (a) streamfunctionanomaly at � � 0.8987, (b) streamfunction anomaly at � � 0.222,(c) lower-level (� � 0.4439) temperature anomaly average, and(d) surface pressure anomaly. The units are 106 m2 s1 for stream-function, K for temperature, and hPa for surface pressureanomaly.

FIG. 3b. Eddy forcing of temperature anomaly associated withthe AO: (top) vertical average, and (bottom) north–south crosssection at 50°W. Left (right) panels are results from SELFclosure (observations). The contour interval is 0.5 and the unit is106 K s1.

FIG. 3a. Eddy forcing of streamfunction anomaly associatedwith the AO: (top) spatial structure at �� 0.4439, and (bottom)zonal mean cross section. Left (right) panels are results fromSELF closure (observations). The contour interval is 1 and theunit is m2 s2.

FIG. 3c. Eddy forcing of surface pressure anomaly associatedwith the AO: (left) model result, and (right) observational result.The contour interval is 0.5 and the unit is 106 hPa s1.

JULY 2006 P A N E T A L . 1715

the mode to become equivalent barotropic in its verti-cal structure. The eddy forcing for the surface pressurefield is positive at midlatitudes and negative at highlatitudes, which also correlates positively with the re-gression pattern for surface pressure anomaly corre-sponding to the AO (Fig. 4d). This also mounts to thepositive eddy feedback for the AO. Overall, the SELF-feedback operator captures the different roles of theeddy feedback for different components of the AOmode.

The results for the AAO (Figs. 5a–c) are essentiallysimilar. The observed eddy forcing is well captured bythe SELF feedback closure. The synoptic eddy-inducedstreamfunction tendency also shows a clear zonal meancomponent, with the positive tendency near the polarregion and a negative tendency in the middle latitude,serving as a positive eddy feedback to reinforce thestreamfunction field of the AAO (Fig. 5a). The eddyforcing in temperature equation again tends to reducethe north–south gradient (Fig. 5b). Compared with theNH, the transient eddy forcing for the SH is clearlymore zonally distributed, owing to the zonally uniformdistributions of the mean storm track. Similar valida-tion and analysis are also conducted for the PNA,which also indicate the SELF feedback closure capturesreasonably well the impact of the eddy feedback on thisdominating low-frequency flow pattern (not shown).

5. The leading modes in the linear dynamicalsystem

In this section, we further calculate the singular vec-tors of the linear operator, L(x c) � Lf , by means of the

singular value decomposition (SVD; Navarra 1993;Kimoto et al. 2001; Watanabe and Jin 2004), to examinethe leading low-frequency modes of the baroclinicmodel. We will examine separately the role of theSELF feedback as well as the stationary waves of thebasic state in the leading dynamical modes.

The leading singular mode of the linear matrix forthe NH cold season (Fig. 6), defined as a mode havingthe smallest singular value hence corresponding to thenear-neutral mode, is found to be of similar patterns tothe observed AO (Fig. 4). The patterns of streamfunc-tion anomalies associated with the leading mode at dif-ferent vertical levels (Figs. 6a and 6b) generally wellresemble those observed (Figs. 4a and 4b). In particu-lar, the dipole structure is locally pronounced over theAtlantic sector, as it is observed. This leading mode hasan equivalent barotropic structure in the vertical (notshown), which is also consistent with observations(Thompson and Wallace 2000). Both the simulated

FIG. 5a. As in Fig. 3a except for the AAO.

FIG. 5b. As in Fig. 3b except for the AAO. (upper) The contourinterval is 0.3 and (lower) the contour levels are 0.3, 0.1, 0.1,0.3, and 0.5.

FIG.5c. As in Fig. 3c except for the AAO.

1716 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 63

(Fig. 6c) and observed (Fig. 4c) temperature anomaliesdecrease with height. The surface pressure anomalypattern of the leading singular mode (Fig. 6d) alsoclosely resembles the observed pattern of the AO (Fig.4d), with a negative anomaly over the polar region, astrong positive center over the North Atlantic, andweak one at Pacific regions.

The second leading mode under the NH cold seasonbackground condition has a wavelike pattern (notshown), which does not appear to be closely related tothe observed leading low-frequency modes. However,third singular mode is reminiscent of the PNA pattern,having a wave-train type pattern located mainly overthe Pacific sector with four action centers over theNorth Pacific and North America (Fig. 7). A great partof the variability of the PNA is directly related to re-mote forcing from Tropics, nevertheless, it is alsoknown that even in the absence of external forcing,PNA-like pattern with significant amount of low-frequency variance is still simulated in atmosphericmodels (Straus and Shukla 2002). That the PNA-likepattern appears as one of the leading modes of thebaroclinic model supports the notion that internal dy-namics, particularly the SELF feedback, plays a signifi-cant role in the formation of the PNA pattern.

The leading singular mode of the linear matrix forthe SH cold season is also similar to the observed AAO(Figs. 8 and 9). The main features of AAO are wellreproduced by the leading mode of linear system with

the SELF feedback. For instance, both the leadingmode and observed AAO have equivalent barotropicstructures in the streamfunction field (Figs. 8a,b and9a,b). The observed and simulated patterns of tempera-ture fields have negative anomalies in the polar regionand positive anomalies in the extrapolar region (Figs.8c and 9c). The surface pressure anomaly of the leadingsingular mode is also similar to that of the observedAAO (Figs. 8d and 9d).

To isolate the effect of the SELF feedback, we set itto zero and recalculate the singular vectors of the con-ventional linear operator L(x c). The leading mode ofL(x c) from traditional LBM under the SH cold season

FIG. 8. As in Fig. 4 except for the AAO.

FIG. 6. As in Fig. 4 except for the solution of the leading SVDmode for the NH.

FIG. 7. As in Fig. 6a except for the third mode at � � 0.4439.

JULY 2006 P A N E T A L . 1717

background conditions bears no resemblance with theAAO (not shown), which is also consistent with thebarotropic study in Part II. The leading mode of underthe NH winter climate state does have dipolelike zonalmean flow anomalies and bears some similarity to theAO pattern, which is consistent with the other studies(Limpasuvan and Hartmann 2000; Kimoto et al. 2001;Watanabe and Jin 2004) who found that stationarywaves in the nonzonal basic flow may contribute to theformation of AO. Yet, the spatial pattern of this leadingmode (Fig. 10) without SELF feedback resembles theobserved AO mode to a significantly less degree thanFigs. 6a and 6b, suggesting the dominating impact of theSELF feedback on the formation of the AO mode.

To further demonstrate the effect of the SELF feed-back, we deliberately suppress the zonally asymmetricpart of the climatological mean state (i.e., stationarywaves) in the linear operator, but still retain the fullSELF feedback operator, which depends on the entirefull basic flow and basic eddy properties. In this case,we found that the AO- and AAO-like modes still ap-pear as the leading singular modes of the system (Figs.11–12) of the dynamical system. A PNA-like mode alsoexists in the system as the third mode (not shown). Thestrong regional characteristics of the AO in the Atlanticregion, for instance, are thus almost exclusively relatedto the storm track captured in Lf . In other words, theselocalized features of AO and AAO are generated bythe SELF feedback in the storm track region (see alsoPart II; Pan 2003).

In summary, our analysis indicates that the SELFfeedback is of essential importance in forming the AO-,AAO-, and PNA-like leading modes. The baroclinicmodel results show that the main conclusions drawn inthe barotropic study are valid. This is because SELFfeedback in the baroclinic framework tends to naturallyselect the leading modes with the equivalent barotropicstructure. One indication that SELF feedback favorsthe equivalent barotropic structure is the fact that eddyterms tends to damp temperature anomalies of theleading mode, as discussed in section 4. However, morein-depth study is needed to understand this effect of theSELF feedback.

6. Role of SELF feedback in teleconnections

Some of the teleconnection patterns in the extratro-pics are known to be excited by tropical forcing (Hosk-

FIG. 11. Streamfunction of the leading SVD mode at � � 0.4439for the NH with zonal symmetric basic state. The unit is 106

m2 s1.

FIG. 9. As in Fig. 8 except for the solution of the leading SVDmode.

FIG. 10. Streamfunction of the leading SVD mode at � � 0.4439for the NH without SELF feedback. The unit is 107 m2 s1.

1718 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 63

ins and Karoly 1981; Jin and Hoskins 1995). However,the eddy forcing also plays an important role in themiddle-latitude response to such remote forcing (e.g.,Held et al. 1989; Ting and Peng 1995; Peng et al. 2005).With the proposed SELF closure in the primitive equa-tion model, we revisit the role of SELF feedback inregulating the atmospheric teleconnection.

A forced experiment is conducted with Eq. (16) byputting a diabatic heating over the central Pacific (0°,140°W). The vertical heating distribution in the tropicalregion follows the observed result by Yanai and Tomita(1998), which has a top-heavy profile. The remote re-sponse in midlatitude (Fig. 13a) is modest without theeddy feedback (Lf � 0), but significantly enhancedwhen the eddy feedback is included (Fig. 13b), which isconsistent with previous numerical studies (e.g., Held etal. 1989). The difference between the responses withand without the SELF feedback, as shown in Fig. 13c,has some projections onto the PNA- and NAO-like pat-terns. This result is consistent with our findings in thebarotropic model (see Part I). Namely, the externalforcing excites Rossby wave trains that tend to perturbthe synoptic eddy activity in the stormy basic flow in anorganized manner to intensify the remote response.Therefore, the SELF feedback is an important part ofthe relay for the atmospheric teleconnections.

7. Summary and concluding remarks

In this three-part study, we proposed a linear dy-namical framework with respect to a stochastic basicflow and derived a linear SELF closure to investigatethe dynamics of the low-frequency variability of theatmospheric circulation. The approach relies on sepa-rating the quadratic, eddy–eddy interaction term into aclimatological component and an anomalous one; the

anomalous component presumably varies on a slowtime scale and slaved by the large-scale low-frequencyflow. The key idea is to prescribe the climatologicalmean flow and climatological properties of the synopticeddy flow and then to provide a linear closure between

FIG. 12. As in Fig. 11 except for the SH.

FIG. 13. The response of the streamfunction field to the forcingin the tropical Pacific at � � 0.4439 for the NH. (a) Without SELFfeedback, (b) with SELF feedback, and (c) difference between thesolutions in (b) and (a). The unit is 106 m2 s1.

JULY 2006 P A N E T A L . 1719

the anomalous synoptic eddy activity and the anoma-lous low-frequency flow.

We laid out in Part I, with a barotropic framework,the basic steps for establishing a linear dynamic closurefor the SELF feedback. After this linear closure isimplemented and tested in Part I using a barotropicmodel, we analyzed in Part II leading singular modes ofthis barotropic model with SELF feedback and theirrelevance to the observed low-frequency modes, andfurther provided a theoretical analysis of the SELFfeedback.

In Part III, we demonstrate that the barotropicmodel study of Parts I and II can be naturally extendedto the primitive equation model. We first derived thegeneral forms of the linear dynamic equations with theSELF feedback closure, which depends on observedclimatological mean state and observed climatologicalproperties of synoptic eddy activity. We used theCEOF decomposition to the observed bandpass-filtered data to attain the climatological properties ofthe synoptic eddies, such as the spatial structure, vari-ance, e-folding time scale, and propagation speeds.

We validated the nonlocal dynamical SELF feedbackclosure by using the observational data. The observedanomalous three-dimensional synoptic eddy-forcingfields associated with the AO and AAO are reasonablyreproduced by the SELF feedback closure. The ob-served positive feedbacks associated with the momen-tum and surface pressure eddy forcing and negativefeedback associated with eddy thermal forcing are suc-cessfully captured by the SELF feedback closure. Thefeedback associated with eddy–heat forcing tends tomainly weaken the baroclinic structure of the low-frequency modes, which may be responsible for theirequivalent barotropic structures.

We further performed SVD analysis using the linearbaroclinic model to investigate the role of the SELFfeedback in forming the leading low-frequency modes.Our analyses show that both AO- and AAO-like modesare the leading internal modes of the atmospheric low-frequency dynamic system in each hemisphere undercold season background conditions, whereas the PNA-like pattern emerges as the third SVD mode in the NH.With the full SELF feedback in the model, the AO-,AAO-, and PNA-like modes remain as the leadingmodes of the system even when the zonal asymmetricpart of the climatological mean state is removed. Thus,the SELF feedback is of essential importance in theformation of the leading low-frequency modes of theextratropical circulations. Moreover, all the leadingmodes are of equivalent barotropic structure, and thusthe main findings on the dynamical origin of the low-frequency modes from this baroclinic model study and

the barotropic model study in the Parts I and II areconsistent. Both barotropic and baroclinic model re-sults showed that the SELF feedback significantly am-plifies the middle-latitude atmospheric responses to thetropical forcing.

Our finding that the linear dynamic system withSELF feedback closure can produce basic troposphericfeatures of the AO and AAO suggests that the funda-mental patterns of these low-frequency modes arelargely controlled by internal dynamics within the tro-posphere. However, it has been noted that the annularmodes (or AO and AAO) are influenced by interactingwith the stratosphere (Baldwin and Dunkerton 1999,2001; Shindell et al. 1999; Ambaum and Hoskins 2002).Owing to the limited vertical resolution, our currentmodel framework is inadequate to address this issue;however, with higher resolutions, we will further inves-tigate this problem. The new linear framework devel-oped in this study may also be used to investigate thepossible role of the extratropical ocean–atmosphere in-teraction in the dynamics of the extratropical low-frequency variability. Moreover, nonlinear correctionsto the SELF feedback closure, other nonlinearities ofthe slow dynamics, feedbacks between dynamics andphysical processes, may also be incorporated into thisframework for more comprehensive studies of the low-frequency variability.

Acknowledgments. The research is supported byNOAA GC01-229, NSF ATM 0226141 and ATM0424799, and National Natural Science Foundation ofChina (NSFC) Grant 40528006. M. Watanabe is alsosupported by a Grant-in-Aid for Scientific Researchfrom MEXT, Japan. The authors are thankful for valu-able comments from Dr. M. Ghil and stimulating dis-cussions with Drs. A. I. Barcilon, S.-P. Xie, K. Hamil-ton, B. Qiu, B. Wang, W. Robinson, and M. Cai. Themanuscript was edited by D. Henderson.

APPENDIX

Model Equations

The primitive equation model employs � (�p/ps) co-ordinate and has four variables (�, �, D, T). The con-tinuity equation in terms of the logarithm of surfacepressure (� � lnps) is

��

�t� V · �� � � · V �

��

��� 0, �A1�

where � is vertical velocity, V � (u, �) the zonal andmeridional velocity.

1720 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 63

Each variable is separated into three terms as in Eq.(1). The continuity equation can be rewritten as

���� � � c � �a�

�t� �V� � V

a� V

c�

· ���� � �a � � c�

� · �V� � Va� V

c�

���� � �a � � c�

��. �A2�

By assuming that the basic flow term is much largerthan the anomaly term for each variable, we can get theclimatology equation after performing the long termaverage,

Vc

· �� c � � · Vc�

�� c

��� V� · ���

c. �A3�

Taking a time average such as monthly (or seasonal)average and removing the climatology part, we obtainthe linearized anomaly equations for the low-frequencyvariability:

��a

�t� V

a· �� c � V

c· ��a � � · V

a�

��a

��

� V� · ���a. �A4�

Based on the discussion in section 2, we use roman andbold italic variables below to distinguish the stochastic

eddy state such as V� from its one particular realizationV�. Let

V� � V�c � V�a, �� � ��c � ��a, �A5�

and consider the ergotic approximation, we have

V� · ���c �V�c · ���c�,

V� · ���a �V�c · ���a� � �V�a · ���c�, �A6�

where V�c, ��c, V�a, and ��a are the basic and anomalousstochastic states. Substituting Eq. (A6) into Eqs. (A3)and (A4), we have

Vc

· �� c � � · Vc�

�� c

��� �V�c · ���c�, �A7�

��a

�t� V

a· �� c � V

c· ��a � � · V

a�

��a

��

� �V�c · ���a� �V�a · ���c�. �A8�

Using Eqs. (A1), (A7), and (A8), we get an equationfor ��. Since V�, ��, and �� are all one particular real-ization of V�, ��, ��, and can be expanded according toEq. (A5), we obtain the linearized equation for ��a asfollows:

���a�t

� V�a · �� c � Vc

· ���a � � · V�a ����a��

� V�c · ��a Va

· ���c. �A9�

Here higher-order and nonlinear terms are neglected.Similarly, we can get the corresponding equations for

vorticity, divergence, and temperature. The climato-logical mean equations are

1

a cos�

�A�c

���

1a cos�

�

���A

uccos�� � �� c� �

1a cos�

��A ��c���

1

a cos�

�

����A �uc� cos��, �A10�

1

a cos�

�Auc

��

1a cos�

�

���A

�ccos�� � �2��

c� R�T�c� c �

uc2 � � c2

2 � � �Dc�

�1

a cos�

��A �uc���

�1

a cos�

�

����A ��c� cos�� �2��uc�

2� � �� c�2�

2 �, �A11�

1a cos�

�ucTc

���

1a

�

���� cT

c

cos�� Tc

Dc� �

c �Tc

�� �T

c��� c

�t� V

c· �� c �

�c

�� � �T

c�

�1

a cos�

��u�cT �c���

�1a

�

������cT �c� cos�� � �T �cD �c� ���c

�T �c��� � �� T �c����c

�t� V

c· ���c � V�c · �� c �

��c���

� �Tc�V�c · ���c�, �A12�

JULY 2006 P A N E T A L . 1721

where a represents the radius of the earth, � is lineardamping rate, � the geopotential height, R the atmo-spheric gas constant, Cp the specific heat at constantpressure, � � R/Cp, and

T � �T ���� � T, �A13�

Auc� ��

c� f �� c � c

�uc

��

RTc

a cos�

�� c

��, �A14�

A�c� ��

c� f �uc � c

�� c

��

RTc

a

�� c

��, �A15�

�A �uc� � ���c��c� ���c�u �c��� � RT �c

a cos�

���c��� , �A16�

�A ��c� � ���cu�c� ���c���c��� �RT �c

a

���c��� . �A17�

The equations for low-frequency anomalies are

��a

�t

1a cos�

�A�a

���

1a cos�

�

���A

uacos�� � ��

a� �

1a cos�

��A ��a���

1

a cos�

�

����A �ua� cos�� �A18�

�Da

�t

1a cos�

�Aua

��

1a cos�

�

���A

�acos�� � �2��

a� R�T �c�a � uauc � � c�a� � �D

a�

�1

a cos�

��A �ua���

�1

a cos�

�

����A ��a� cos�� �2��u�au�c� � ���c��a��, �A19�

�Ta

�t�

1a cos�

��uaTc

� ucTa

�

���

1a cos�

�

�����aT

c

� � cTa

� cos�� Ta

Dc T

c

Da� �a

�Tc

��� � c

�Ta

��

�Ta��� c

�t� V

c· �� c �

� c

�� �T

c���a

�t� V

a· �� c � V

c· ��a �

�a

�� � �T

a�

� 1

a cos�

���u �aT �c� � �u�cT �a����

1

a cos�

�

�������aT �c� � ���cT �a�� cos�� � �T �aD �c� � �T �cD �a� ���a

�T �c���

���c�T �a��� � ��T �a����c

�t� V

c· ���c � V �c · �� c �

��c���

� ��T �c����a�t

� V�a · �� c � Vc

· ���a ���a� �� � ��T

c�V�a · ���c � V�c · ���a��, �A20�

where

Aua� �

a� c � ��

c� f ��a �a

�uc

�� � c

�ua

��

RTc

a cos�

��a

��

RTa

a cos�

�� c

��, �A21�

A�a� �

auc � ��

c� f �ua �a

�� c

�� � c

��a

��

RTa

a

�� c

��

RTc

a

��a

��, �A22�

�A �ua� � ���a��c� � ���c��a� ���a�u�c��� ���c

�u�a��� � RT �c

a cos�

���a��� � RT �a

a cos�

���c��� , �A23�

�A �ua� � ���a��c� � ���c��a� ���a���c��� ���c

���a��� �RT �a

a

���c��� �RT �c

a

���a��� . �A24�

1722 J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S VOLUME 63

The equations for the ensemble eddy anomaly (or high-frequency anomaly) can be expressed as

���a�t

1

a cos�

�A ��a

���

1a cos�

�

���A �ua cos�� �

1a cos�

�A ��c

��

1a cos�

�

���A �uc cos�� ���a�, �A25�

�D�a�t

1

a cos�

�A �ua

��

1a cos�

�

���A ��a cos�� � �2���a � R�T �c��a � u �auc � ��a� c� � �D�a�

�1

a cos�

�A �uc

���

1a cos�

�

���A ��c cos�� �2�u�cu

a � ��c�a�, �A26�

�T �a�t

�1

a cos�

�u�aTc

� ucT �a��

�1

a cos�

�

�����aT

c

cos� � � c T �a cos�� Tc

D�a T �aDc� � c

�T �a��

� ��a�T

c

��

�T �a��� c

�t� V

c· �� c �

� c

�� �T

c����a�t

� Vc

· ���a � V�a · �� c ���a�� � �T �a�

� 1

a cos�

�u�cTa

� uaT �c��

1

a cos�

�

�����cT

a

cos� � �aT �c cos�� � Ta

D�c � T �cDa �a

�T �c��

��c�T

a

��

� �T �c���a

�t� V

a· �� c � V

c· ��a �

�a

�� � �T

a����c�t

� Vc

· ���c � V�c · �� c ���c��, �A27�

where

A �ua � ��a� c � ��c� f ���a ��a

�uc

�� � c

�u �a��

RT

c

a cos�

���a��

RT �a

a cos�

�� c

��, �A28�

A ��a � ��auc � ��c� f �u�a � c

���a��

��a�� c

��

RT �aa

�� c

��

RT

c

a

���a��

, �A29�

A �uc � �a��c � ��c�

a �a�u�c��

��c�ua

��

RT �ca cos�

��a

��

RT

a

a cos�

���c��

, �A30�

A ��c � �au�c � ��cu

a ��c��a

�� �a

���c��

RT

a

a

���c��

RT �c

a

��a

��, �A31�

using x to represent the variable group (�, D, T, �) andX to represent the corresponding stochastic state, wecan symbolically express the high-frequency and low-frequency anomaly equations as Eqs. (8) and (9):

�

�tX�a � L�x c�X�a � LE�X�c�x

a, �A32�

�

�txa � L�x c�xa � �LE�X�c�X�a�, �A33�

where L and LE are linear matrices, and LE may alsodepend on x c. The external forcing Q is set to zero here.

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