Local wave activity budgets of the wintertime Northern ...

Confidential manuscript submitted to Geophysical Research Letters

Local wave activity budgets of the wintertime Northern1

Hemisphere: Implication for the Pacific and Atlantic storm2

tracks3

Clare S. Y. Huang1∗, Noboru Nakamura14

1Department of the Geophysical Sciences, University of Chicago, Chicago, Illinois, USA.5

Key Points:6

• Seasonal mean budget of LWA reveals that the low-level meridional heat flux is the ma-7

jor source of LWA over both storm tracks.8

• On synoptic timescales the horizontal LWA flux convergence and diabatic heating play9

the leading roles in the LWA budget.10

• Convergence of the meridional eddy momentum flux describes the covariation of barotropic11

zonal wind and LWA over the Pacific but not the Atlantic.12

∗5734 S Ellis Ave, Chicago IL 60637

Corresponding author: Clare S.Y. Huang, [email protected]

–1–


Abstract13

A recently developed finite-amplitude local wave activity (LWA) diagnostic framework quan-14

tifies eddy-mean flow interaction on regional scales. Here we examine the column budgets of15

LWA for the winter Northern Hemisphere with the ECMWF ERA Interim product, with an16

eye toward quantifying the maintenance and variability of the Pacific and Atlantic storm tracks.17

The budget is governed by (i) low-level meridional eddy heat flux, (ii) horizontal convergence18

of the LWA flux, and (iii) nonconservative (diabatic) sources-sinks. In both regions the low-19

level meridional heat flux fuels LWA on seasonal timescales but the zonal LWA flux conver-20

gence and diabatic effects dominate the synoptic variability. Cospectral analysis shows that the21

interplay between barotropic zonal wind and column-averaged LWA through the meridional22

eddy momentum flux convergence is significant over the Pacific but not the Atlantic. A first23

attempt at partitioning LWA into stationary and transient eddy contributions is also discussed.24

1 Introduction25

Migratory weather systems populate the storm track regions of Earth’s midlatitudes and26

affect the lives of billions [Chang et al., 2002; Shaw et al., 2016]. In the Northern Hemisphere,27

storm tracks are localized over the Pacific and Atlantic sectors, whereas in the Southern Hemi-28

sphere, they are more zonally spread over the Southern Ocean [Hoskins and Hodges, 2002, 2005].29

Surface orography contributes significantly to the difference in the spatial structures of the storm30

tracks through stationary Rossby waves [Hoskins and Karoly, 1981; Held and Ting, 1990; Held31

et al., 2002; Wilson et al., 2009]. Storm tracks are generally more active in winter when the32

pole-to-equator and land-sea temperature gradients enhance baroclinicity, although the mid-33

winter suppression of the North Pacific storm activity is a notable exception [Nakamura, 1992].34

Commonly used metrics of storm track activities include the Eady growth rate [Lindzen35

and Farrell, 1980], variance in highpass sea level pressure and geopotential height [Nakamura,36

1992], transient eddy kinetic energy (EKE) [Orlanski, 1998; Deng and Mak, 2006] and aggre-37

gate potential vorticity (PV) anomalies contributed from individual storms [Hoskins and Hodges,38

2002, 2005]. To elucidate the underlying dynamics with meteorological data, metrics with known39

budget components are useful since they allow breakdown of the contributions from different40

physical processes. For example, Chang [2001] applies the budgets of EKE and wave activ-41

ity flux to the Southern Hemisphere storm tracks and show that the upstream generation of42

baroclinic wave activity maintains the downstream development of wave packets.43

The wave activity flux diagnostic [Plumb, 1985; Takaya and Nakamura, 2001] is widely44

used to describe the 3D propagation of Rossby wave packets. However, its derivation assumes45

that the wave amplitude is small. While the wave activity flux is readily calculable from data,46

the budget of wave activity cannot be closed with the small-amplitude assumption: it is dom-47

inated by triple products of eddy quantities at finite amplitude, and the (small-amplitude) wave48

activity itself becomes unreliable as the background PV gradient is reversed [Solomon and Naka-49

mura, 2012].50

Recently, Huang and Nakamura [2016] (hereafter HN16) introduced finite-amplitude lo-51

cal wave activity (LWA) to describe eddy-mean flow interaction on regional scales. LWA gen-52

eralizes the small-amplitude theory to eddies of arbitrary amplitude with a simple, closed bud-53

get. Both LWA and its eddy forcing terms are calculable from meteorological data, and the54

residual of the budget quantifies the net nonconservative processes. HN16 demonstrate that55

LWA captures the life cycle of anomalous wave events (see their Figs. 8 and 9). In this study,56

we analyze the column budget of LWA with the European Centre for Medium-range Weather57

Forecasts (ECMWF) ERA-Interim product [Dee et al., 2011] for the Northern Hemisphere win-58

ter to study the maintenance and variability of the Pacific and Atlantic storm tracks. A method59

will be introduced to decompose LWA into stationary and transient eddy contributions. As we60

will see, the transient component of LWA is consistent with, but better-behaved than, the small-61

amplitude transient wave activity defined by Plumb [1986]. We will also perform spectral anal-62

–2–


ysis of the forcing terms for synoptic to intraseasonal timescales over the two oceans to de-63

lineate the frequency dependence of the LWA budget.64

2 The local finite-amplitude wave activity (LWA)65

2.1 Definition of LWA66

LWA extends the finite-amplitude wave activity (FAWA) theory of Nakamura and Zhu67

[2010], formulated based on the conservation of quasigeostrophic PV (QGPV) on isobaric sur-68

face in a rotating, stratified atmosphere. FAWA measures the net ‘exchange’ of QGPV sub-69

stance across latitude circles by eddies. HN16 introduce LWA as the longitude-by-longitude70

contribution to FAWA. In spherical coordinates the interior LWA is defined as71

A∗(λ, φ, z, t) = −a

cos φ

∫ ∆φ

0qe(λ, φ, φ′, z, t) cos(φ + φ′)dφ′ z > 0, (1)72

where a is the radius of the planet, (λ, φ, z) defines longitude, latitude and pressure pseudo-73

height [z ≡ −H ln(p/p0) where p is pressure, p0 = 1000 hPa and H = 7 km is assumed].74

qe(λ, φ, φ′, z, t) ≡ q(λ, φ + φ′, z, t) − qREF(φ, z, t) is an ‘eddy’ component of the QGPV q, de-75

fined as the departure from a zonally symmetric, Lagrangian-mean reference state qREF(φ, z, t)76

at equivalent latitude φ [see Supporting Information (SI) sections 1-2 for details]. In the above77

φ′ is latitude relative to φ and used to describe the meridional displacement of QGPV from78

the reference state qREF(φ, z, t). ∆φ(λ, φ, z, t) is the meridional displacement of the QGPV con-79

tour q = qREF from the latitude circle at φ. (See HN16 Fig.1.) By definition, A∗ is positive80

definite and its zonal average recovers FAWA. Unlike previous studies with the FAWA formal-81

ism (e.g. Nakamura and Solomon [2010], Wang and Nakamura [2015], hereafter NS10 and WN15,82

respectively), the equivalent latitude and the reference state in this study are defined in a hemi-83

spheric domain, with the assumption that the effect of inter-hemispheric exchange of QGPV84

on the mid-latitude dynamics is insignificant.85

This paper focuses on the horizontal distribution of the density-weighted column aver-86

age of LWA and its fluxes, which we evaluate by the averaging operation87

〈(·)〉 ≡

∫ ∞0 e−z/H(·)dz

H. (2)

2.2 Comparison between instantaneous 〈 A∗〉 and 〈EKE〉88

LWA measures the meridional displacement of the QGPV substance. Unlike EKE, it is89

not a measure of how energetic eddies are locally: the two metrics quantify different aspects90

of eddies. To illustrate, Fig.1 compares daily snapshots of 〈A∗〉 and 〈EKE〉, together with the91

500hPa geopotential height (Z500) contours. During this period (December 14-17, 2010) there92

were a persistent anticyclone over the central Pacific (30−60◦N, 180◦) and a cyclone form-93

ing over the Gulf of Alaska (45−60◦N, 160◦W). 〈A∗〉 attains maxima at the centers of both94

features, where the zonal flows are most obstructed. This is consistent with the theoretical ex-95

pectation that LWA negatively covaries with the zonal flow (HN16, see also Fig.10 of SI). 〈EKE〉,96

in contrast, attains maxima around the features where the Z500 contours are densest (i.e. where97

the geostrophic flow is most energetic).98

2.3 Budget equations of LWA and zonal wind99

The column budget equations for the interior zonal wind u and LWA A∗ read:100

∂

∂t〈u〉 cos φ︸︷︷︸

zonal windtendency

≈ −1a∂Fu

∂λ︸︷︷︸zonal windzonal flux

convergence

−1

a cos φ∂

∂φ′

⟨ueve cos2(φ + φ′)

⟩︸︷︷︸

meridional eddymomentum

flux convergence

+ Gu︸︷︷︸ageostrophic

Coriolistorque

+ ˙〈u〉 cos φ︸︷︷︸residual

, (3)

–3–


∂

∂t

⟨A∗

⟩cos φ︸︷︷︸

LWA tendency

≈ −1a∂FA

∂λ︸︷︷︸zonal LWA

fluxconvergence

+1

a cos φ∂

∂φ′

⟨ueve cos2(φ + φ′)

⟩︸︷︷︸

meridional eddymomentum

flux divergence

+f cos φ

H

(veθe

∂θ/∂z

)z=0︸︷︷︸

low-level eddymeridional heat

flux

+˙⟨

A∗⟩

cos φ︸︷︷︸residual

,

(4)

where (u, v, θ) define the zonal and meridional wind velocities and potential temperature. The101

subscript e denotes the departure from the reference state (‘eddy’), Fu and FA are the column-102

averaged zonal fluxes of zonal wind and LWA, Gu is the Coriolis torque of the ageostrophic103

meridional wind (see Appendix A for the full expressions), f is the Coriolis parameter, θ(z, t)104

is the area-weighted average of potential temperature over the Northern Hemisphere, and u and105

˙A∗ represent non-conservative contributions. (See SI sections 3-5 for derivations and compu-106

tation.) The above equations generalize Eqns. (29) and (27) of HN16 for spherical coordinates.107

The zonal LWA flux FA includes advective fluxes [the first two RHS terms of (A.3)] that are108

cubic (or higher) in eddy products. The second RHS term of (4) represents the local transfer109

of barotropic momentum to and from the zonal wind. The third term is the upward wave ac-110

tivity input from the surface. The zonal convergence of the last term in (A.3), together with111

the second and third RHS terms of (4) make up the column average of the 3D Eliassen-Palm112

(EP) flux convergence. The last term of (4) represents nonconservative sources-sinks of LWA,113

including diabatic heating, dissipation through mixing, radiative and Ekman damping. The bud-114

get of surface wave activity associated with the meridional displacement of low-level poten-115

tial temperature will not be analyzed in this study. The main contribution to the column av-116

erage quantities over the oceans comes from the upper troposphere, where eddy amplitudes117

are greatest, even with the density-weighing that decreases with height (2) (HN16).118

The tendency terms in (3)-(4) are negligible upon time averaging over a season (December-119

February in this study). Denoting such time averaging by [(·)], (4) becomes:120

0 ≈ −1a∂[FA]∂λ

+1

a cos φ∂

∂φ′

[⟨ueve cos2(φ + φ′)

⟩]+

f cos φH

[(veθe

∂θ/∂z

)z=0

]+

[˙⟨

A∗⟩]

cos φ. (5)

This steady-state budget describes the balance between the flux convergence (the first three121

RHS terms) and sources-sinks of LWA (the last term), which will be evaluated in Fig.3 be-122

low.123

We also examine the frequency dependence of the budget by computing the cospectra124

(i.e. the in-phase signal of cross-spectra) between the LHS and each of the RHS terms of (3)125

and (4). We apply this spectral analysis after averaging each term in the equations over the126

respective domain (Fig.3). See Appendix B for the definition of the domains of averaging. The127

cospectrum of two quantities A and B will be denoted by Cosp(A,B) in the figure legends. The128

overall impact of nonconservative processes on LWA is estimated as the residual of the bud-129

get. No further breakdown of the nonconservative processes will be attempted. The residual130

also contains analysis errors (sampling errors, non-QG effects, truncation errors, etc.), which131

will not be quantified in this study. As we will see, the residual has nontrivial contributions132

to the LWA budget.133

3 Results134

3.1 Climatology of local wave activity in the boreal winter135

Figure 2a shows the December-January-February climatology (1979-2015) of the col-136

umn averaged LWA 〈A∗〉 cos φ (shades) and zonal wind 〈u〉 cos φ (contours). 〈A∗〉 cos φ is large137

at the poleward flanks of barotropic zonal jets over the storm track regions. Note that this seasonal-138

mean LWA includes contributions from both stationary and transient eddies. Decomposition139

of LWA and its fluxes into stationary and transient eddy contributions is not straightforward140

partly because the definition of LWA [(1)] requires a reference state based on the Lagrangian141

–4–


mean that does not commute with the Eulerian time mean. Nevertheless, having confirmed that142

the time mean of qREF does not deviate much from qREF computed from the Eulerian time mean143

of q at midlatitudes (See SI sections 6-7), we estimate the stationary eddy component of LWA144

by applying (1) to the time mean of QGPV field, i.e., 〈A∗([q])〉 (Fig.2b). The peaks at the east145

of the Mongolian Pleateau and Norwegian Sea arise largely from topography-induced merid-146

ional excursion of the QGPV contours at the lowest level (z = 1 km).147

The climatology of transient component of LWA is computed as the difference between148

the total and the estimated stationary component of LWA (Fig.2c), i.e. [〈A∗(q)〉]−〈A∗([q])〉.149

Its longitudinal extent over the North Pacific is consistent with other common metrics of storm150

tracks [e.g. Chang et al., 2002 Fig.2]. Over the Atlantic, where the barotropic jet is tilted north-151

eastward, the LWA maxima are found at both flanks of the jet, namely, over Quebec and west-152

ern Europe.153

One may question whether a zonally symmetric reference state is suitable to analyze wave154

dynamics over the northeast-tilted Atlantic storm track. As a comparison, we show in Fig.2d155

the climatology of transient wave activity proposed by Plumb [1986] [his Eq. (2.20)], which156

is based on a zonally-varying basic state for small-amplitude waves. Our estimated transient157

LWA (Fig.2c) has a spatial structure consistent with Plumb’s transient wave activity (Fig.2d),158

which implies that the obtained LWA structure, especially the relative minimum over the At-159

lantic, is not a consequence of a particular choice of the reference state. Rather, it is an in-160

trinsic property of wave activity that it is suppressed along the jet axis, as derived in HN16.161

This comparison also demonstrates the strength of the finite-amplitude LWA formalism: its162

magnitude is well-constrained by the wind field and much smoother, while the small-amplitude163

wave activity is plagued with spuriously large values in regions where the gradient of time-164

mean QGPV vanishes.165

3.2 Climatology of wave activity budgets166

As a first estimate of how the LWA budget is maintained, Fig.3 shows the climatology167

of the terms on the RHS of (5). The positive (poleward) low-level meridional heat flux (Fig.3c)168

is the major source of wave activity for both storm tracks. Over the North Pacific, its peaks169

are localized to the storm track entrance – from the Sea of Japan to the Kuroshio extension170

– and along the Alaska current. In contrast, large values of low-level poleward heat flux span171

over the majority of the Atlantic north of 40◦N. Much of this signal is due to the quasi-stationary172

zonal asymmetry in the low-level potential temperature forced by the underlying SST distri-173

bution and the associated meridional flow.174

Over the Atlantic, the low-level poleward heat flux is largely balanced by the zonal di-175

vergence of the LWA flux (blue in Fig.3a). Note that the region with large low-level heat flux176

(Fig.3c) is marked by relatively weak LWA (Fig.2a), indicating that LWA is moved away from177

the source region by the zonal flux into the downstream regions of convergence over Europe178

(red in Fig.3a). The meridional momentum flux convergence (blue in Fig.3b) also partially com-179

pensates the zonal flux convergence on the southern flank of the storm track, but the degree180

of this compensation is relatively small.181

Over the Pacific, there is considerable spatial variation in the balance of RHS terms of182

(5). In the source regions (e.g. the Sea of Japan and the Alaska current) a strong cancellation183

is still observed between the low-level poleward heat flux (Fig.3c) and the zonal divergence184

of the LWA flux (Fig.3a) as in the Atlantic. In the Central Pacific, where the low-level pole-185

ward heat flux is relatively weak, the zonal convergence of the LWA flux as a gain, as well186

as the meridional convergence of the eddy momentum flux associated with the equatorward187

radiation of LWA (Fig.3b) and the negative residual (Fig.3d) as losses, all contribute to the LWA188

budget. The magnitude of the residual is quite significant, being comparable to or greater than189

that of the momentum flux convergence. Assuming that the negative residual represents a lin-190

ear damping of LWA, the ratio of the area averaged residual to that of the column LWA gives191

a mean damping timescale of ∼ 12 days. Since this is much shorter than the typical radiative192

–5–


damping timescale in the troposphere (∼30 days), it may be related to Ekman damping or en-193

strophy dissipation (mixing) by wave breaking.194

The residual (Fig.3d) over the continental regions is largely negative, which suggests that195

friction over land surfaces promotes the demise of LWA. Exceptions are found on the lee side196

of mountains (e.g. the Mongolian Plateau and the Rockies). Over the oceans, the residual is197

generally weakly negative. However, there are pockets of weakly positive values over the At-198

lantic Ocean and the Norwegian Sea, hinting that the underlying warm ocean is providing ap-199

preciable diabatic sources of LWA (primarily through latent heat of condensation) that over-200

ride the effect of surface damping. We suspect that this partial cancellation causes the aver-201

age residual less negative over the Atlantic than over the Pacific.202

3.3 Synoptic to intraseasonal variability203

Much of the weather-related LWA variability occurs over synoptic to intraseasonal timescales,204

which is filtered out in the seasonal mean in the foregoing analysis. To delineate wave activ-205

ity budget on these timescales, we compute the cospectra of LWA tendency [the LHS of (4)]206

with each term on the RHS and compare it with the power spectrum of LWA tendency (Fig.4).207

Note that the sum of the cospectra resembles the power spectrum. The solid lines indicate the208

budget over a regular square box domain including both land and ocean grids, while the shad-209

ing indicates the budget change if the domain shrinks to oceanic regions only (Appendix B210

and SI section 8).211

Comparing Figs.4a and 4b, one sees that the variance in the Atlantic is more than twice212

larger than that in the Pacific but the spectral shapes are largely similar between the two re-213

gions. The power spectrum of LWA tendency maximizes around 0.15-0.25 cpd (4-7 days); over214

half of that is explained by the in-phase components of the zonal LWA flux convergence (blue)215

with a similar spectral shape. Cospectra with the meridional eddy momentum flux divergence216

(cyan) are an order of magnitude smaller than those with the zonal flux convergence (blue)217

for both regions, indicating that the LWA budget is dominated by the zonal passage of syn-218

optic weather systems in and out of the regions. Even though the low-level poleward heat flux219

dominates the LWA budget in the seasonal mean (Fig.2), it plays only a minor role in the LWA220

tendency except at low frequencies (< 0.05 cpd, red). The heat flux cospectrum over the At-221

lantic is broader than that over the Pacific, which is more right-shewed. Somewhat surprisingly,222

the residual (green) contributes to the LWA tendency much more than the heat flux over syn-223

optic timescales. Its contribution is comparable to the zonal advective flux convergence (blue)224

over the land-oceanic domain, while a bit smaller over oceanic domain. This suggests that there225

is significant diabatic forcing of wave activity in both regions.226

Given that the meridional eddy momentum flux divergence accounts for only a small frac-227

tion of LWA tendency, to what extent do LWA and the zonal wind covary through this term228

[(3) and (4)]? Figures 4c and 4d compare the cospectra of the meridional eddy momentum flux229

convergence with the LWA tendency (blue), with the zonal wind tendency (green), the cospec-230

tra of LWA tendency and the zonal wind deceleration (red), and the power spectrum of the zonal231

wind tendency (black) for the two storm track regions. The power spectrum of the zonal wind232

tendency peaks at around 0.05 cpd in both regions. Over the Pacific, the close alignment of233

blue, green, and black curves on synoptic timescales indicates that the convergence of the merid-234

ional momentum flux accomplishes barotropic conversion between 〈A∗〉 and 〈u〉 as suggested235

by (3) and (4), and it accounts for most of the zonal wind tendency. This is not the case with236

the Atlantic, where the meridional eddy momentum flux divergence has a negative contribu-237

tion to the LWA tendency and it accounts for much less fraction of zonal wind tendency. In-238

terestingly, the cospectra between the LWA tendency and the zonal wind deceleration (red) are239

positive throughout the frequency domain shown and deviate significantly from the other three240

curves at higher frequencies in both regions, particularly when the domain includes only oceanic241

regions: that is, the negative covariation of LWA and zonal wind is robust regardless of the242

contribution from the meridional eddy momentum flux divergence. Such covariation is espe-243

–6–


cially robust at the diffluent regions of the jets (See SI section 9). The implication is that the244

processes other than the meridional eddy momentum flux divergence, such as the zonal ad-245

vective flux convergence, maintain negative covariation of LWA and the zonal wind.246

Regression of LWA on various climate oscillation indices resembles the leading patterns247

of zonal wind distributions corresponding to the particular oscillation (See SI section 10). How-248

ever, the weak correlations suggest that exchange between LWA and the zonal winds is not249

the main mechanism governing such variability.250

4 Summary251

There has hitherto been no formal attempt to close the local angular momentum-wave252

activity budget for the midlatitude atmosphere. We have applied the column budget of finite-253

amplitude LWA, a density of angular pseudomomentum, to the winter storm tracks over the254

North Pacific and the North Atlantic using meteorological data. The regional budget of LWA255

is simpler than the budget of small-amplitude wave activity. The latter is often hard to close256

without large nonlinear terms and only the wave activity fluxes (but not the wave activity it-257

self) are used for diagnosis in this context [Plumb, 1985; Takaya and Nakamura, 2001; Chang,258

2001].259

Complementary to EKE, LWA maximizes where waves attain greatest cross-stream dis-260

placement of QGPV and weak zonal wind speeds (Fig.1). We have proposed an approximate261

partitioning of LWA into transient and stationary eddy contributions by decomposing the QGPV262

fields used in (1). The estimated transient eddy LWA climatology has spatial distribution con-263

sistent with that of Plumb’s small-amplitude transient wave activity [Plumb, 1986] with zonally-264

asymmetric basic state.265

The climatology of seasonal-mean LWA flux convergence gives a first estimate of how266

the LWA budget is maintained. In both storm track regions, the low-level poleward heat flux267

is a major source of LWA. The balancing mechanisms are nevertheless different. Over the At-268

lantic the loss is primarily through the zonal divergence. Over the Pacific, the LWA input by269

the heat flux is localized to the western and northeastern ends of the ocean basin, where it is270

largely balanced by the zonal LWA flux divergence. However, over the Central Pacific, the flux271

convergence is largely balanced by the loss through the residual (damping).272

On synoptic timescales, the area-averaged LWA tendency is closely associated with the273

convergence of the zonal LWA flux in both regions, corresponding to the transient passage of274

weather systems. However the residual term representing the net diabatic source has a com-275

parable magnitude whereas the low-level poleward heat flux plays a relatively small role. Whilst276

LWA and the zonal wind covary negatively at all scales, the barotropic conversion of zonal277

momentum plays only a limited role for this and is only significant over the Pacific.278

Future studies will compare interannual variability of LWA and fluxes over the two oceans.279

Also, since the net nonconservative sources and sinks of LWA are only inferred from the resid-280

ual of the budget in this study, more direct assessment of the diabatic sources of LWA in re-281

lation to the storm track maintenance [Hoskins and Valdes, 1990] will be performed with the282

aid of general circulation models. Nevertheless, this work provides a promising framework to283

delineate tendency of longitudinally localized wave activity with arbitrary amplitudes, which284

is potentially useful for comparing wave responses in models in climate change scenarios.285

A: Abbreviation in the flux equations313

The expressions for the column-averaged zonal fluxes in (3)-(4) are:314

Fu ≡⟨uREFue + u2

e

⟩(A.1)

315

Gu ≡ 〈 f ve cos φ〉 −1a∂ 〈ψ〉

∂λ(A.2)

–7–


120°E 150°E 180° 150°W 120°W

5.0

5.0

5.2

5.2

5.25.2

5.3

5.5

5.5 5.7

5.85.8

5.8

Shaded: <EKE> | Contours: Z500 | 2010/12/14

0

262

523

785

1047

1309

1570

1832

120°E 150°E 180° 150°W 120°W

5.0

5.0

5.0

5.2

5.2

5.2

5.25.35.5

5.55.7 5.8

5.8 5.8


0

262

523

785

1047

1309

1570

1832

120°E 150°E 180° 150°W 120°W

5.0

5.0

5.2

5.2

5.2

5.2

5.2 5.3

5.3

5.3

5.5

5.5

5.5

5.55.75.75.8

5.8


0

262

523

785

1047

1309

1570

1832

120°E 150°E 180° 150°W 120°W

5.0

5.05.25.2

5.2

5.3

5.3

5.3

5.5

5.5

5.5

5.75.8

5.8


0

262

523

785

1047

1309

1570

1832

30°N

40°N

50°N

60°N

70°N

120°E 150°E 180° 150°W 120°W

5.0

5.05.25.2

5.2

5.3

5.3

5.3

5.5

5.5

5.5

5.75.8

5.8

Shaded: <LWA>cosφ | Contours: Z500 | 2010/12/14

0

15

30

45

60

75

90

30°N

40°N

50°N

60°N

70°N

120°E 150°E 180° 150°W 120°W

5.0

5.05.25.2

5.2

5.3

5.3

5.3

5.5

5.5

5.5

5.75.8

5.8


0

15

30

45

60

75

90

30°N

40°N

50°N

60°N

70°N

120°E 150°E 180° 150°W 120°W

5.0

5.05.25.2

5.2

5.3

5.3

5.3

5.5

5.5

5.5

5.75.8

5.8


0

15

30

45

60

75

90

30°N

40°N

50°N

60°N

70°N

120°E 150°E 180° 150°W 120°W

5.0

5.05.25.2

5.2

5.3

5.3

5.3

5.5

5.5

5.5

5.75.8

5.8


0

15

30

45

60

75

90

Figure 1. Left column: Daily mean of 〈A∗〉 cos φ (shaded) and 500hPa geopotential height in km (contour)

over the Pacific. The four panels correspond to, from top down, December 14, 15, 16 and 17, 2010, respec-

tively. Right column: Same as left except that the shading indicates 〈EKE〉. Here EKE= 12 u′2 + v′2, where u

and v are the zonal and meridional wind velocities, whereas the overbar and prime denote the zonal mean and

departure from it, respectively. The red box indicates where a blocking event happens.

286

287

288

289

290

FA ≡⟨uREF A∗

⟩−

acos φ

⟨∫ ∆φ

0ueqe cos(φ + φ′)dφ′

⟩+

12

⟨v2

e − u2e −

RH

e−κz/Hθ2e

∂θ/∂z

⟩(A.3)

See SI sections 3-5 for further details.316

B: Computation of area-weighted average317

The land-ocean mask is retrieved from the ERA-Interim database as an invariant, with318

the same resolution (i.e. 1.5◦×1.5◦) as the other variables. The latitude range for area-weighted319

average for both oceans are [30◦, 75◦N]. The longitude range of Pacific is chosen to be [120◦E, 240◦E]320

and that of Atlantic is chosen to be [270◦W, 30◦E]. The results for land-ocean domain (solid321

lines in Fig.4) include all grid-points within the regions mentioned above. The results for oceanic322

domain only (the bound of shades in Fig.4) include only oceanic grid-points based on the land-323

sea mask. See SI section 8 for further sensitivity analysis.324

C: Treatment of data325

The budget terms in (3)-(5) are computed from 6-hourly temperature and wind fields on326

37 pressure levels from the European Centre for Medium-Range Weather Forecasts ERA-Interim327

datasets [Dee et al., 2011] for 1979-2015 with a horizontal resolution of 1.5◦×1.5◦. Data are328

–8–


30°N

40°N

50°N

60°N

70°N

30°N

40°N

50°N

60°N

70°N

60°E 60°E120°E 180° 120°W 60°W 0°60°E 60°E0

0

5 5

5

10

10

10

1015 15

15

20

2025

Climatology: DJF Seasonal mean <LWA> cosφ | Contour: cosφ

8

14

20

26

32

38

44

50

30°N

40°N

50°N

60°N

70°N

30°N

40°N

50°N

60°N

70°N

60°E 60°E120°E 180° 120°W 60°W 0°60°E 60°E0

0

5 5

5

10

10

10

1015 15

15

20

2025

Climatology: <LWA> of DJF Seasonal mean QGPV | Contour: cosφ

0

3

6

9

12

15

18

21

30°N

40°N

50°N

60°N

70°N

30°N

40°N

50°N

60°N

70°N

60°E 60°E120°E 180° 120°W 60°W 0°60°E 60°E0

0

5 5

5

10

10

10

1015 15

15

20

2025

Climatology: Estimated transient component of <LWA> cosφ | Contour: cosφ

4

9

13

18

22

27

31

30°N

40°N

50°N

60°N

70°N

30°N

40°N

50°N

60°N

70°N

60°E 60°E120°E 180° 120°W 60°W 0°60°E 60°E0

0

5 5

5

10

10

10

1015 15

15

20

2025

Climatology: Plumb (1986) transient wave activity for DJF | Contour: cosφ

8

14

20

26

32

38

44

50

Figure 2. (Shaded) Seasonal (DJF) climatology of (a) 〈A∗〉 cos φ, (b) estimated stationary-eddy component

of 〈A∗〉 cos φ in DJF, (c) estimated transient component of 〈A∗〉 cos φ taken as the difference of (a) and (b),

and (d) small-amplitude quasigeostrophic transient LWA computed with a zonally varying reference state

based on Plumb [1986]. In (d) a weak horizontal smoothing is applied to the seasonal-mean QGPV field

before computing the horizontal gradient. The color scale in (d) has been adjusted to be comparable to that

of (a)-(c) for comparison. Values exceeding the color range are indicated in brown. Contours indicate the

barotropic zonal wind 〈u〉. Both quantities have the unit [ms−1]. Regions masked by the gray stipples, where

the topography is higher than 1 km (i.e. zs > 1 km), have been excluded from analysis.

291

292

293

294

295

296

297

298

first interpolated vertically onto 49 equally spaced (1 km) pseudo-height levels as in NS10.329

The interpolated fields are then used to compute QGPV, wave activity and the fluxes at each330

pseudoheight level as functions of longitude and latitude. Evaluation of ue and θe involves com-331

putation of the reference state (uREF , θREF) from qREF . See SI section 4 for the details. Sim-332

ilar to WN15, the column average [(2)] in the discrete data is performed over all the ‘interior’333

(i.e. not at the boundaries) points. The lower boundary condition at z = 0 is replaced by the334

weighted mean values on the two lowest levels (z = 0 and 1 km), which will be referred to335

as ‘low level’ hereafter. See SI section 5 for the evaluation of the low-level meridional heat336

flux and the sensitivity to the choice of the lower boundary conditions.337

Acknowledgments338

This research is supported by NSF Grant AGS-1563307. The ECMWF ERA-Interim data [Dee339

et al., 2011] were downloaded from http://apps.ecmwf.int/datasets/data/interim-full-daily/levtype=sfc/.340

Helpful comments by two anonymous reviewers are gratefully acknowledged.341

–9–


30°N

40°N

50°N

60°N

70°N

30°N

40°N

50°N

60°N

70°N

60°E 60°E120°E 180° 120°W 60°W 0°60°E 60°E

(a) Zonal LWA Flux Convergence [x 10**{-5} m/s**2]

-21

-14

-7

0

7

14

21

30°N

40°N

50°N

60°N

70°N

30°N

40°N

50°N

60°N

70°N

60°E 60°E120°E 180° 120°W 60°W 0°60°E 60°E

(b) Meridional Eddy Momentum Flux Divergence [x 10**{-5} m/s**2]

-21

-14

-7

0

7

14

21

30°N

40°N

50°N

60°N

70°N

30°N

40°N

50°N

60°N

70°N

60°E 60°E120°E 180° 120°W 60°W 0°60°E 60°E

(c) Low Level Meridional Heat Flux [x 10**{-5} m/s**2]

-21

-14

-7

0

7

14

21

30°N

40°N

50°N

60°N

70°N

30°N

40°N

50°N

60°N

70°N

60°E 60°E120°E 180° 120°W 60°W 0°60°E 60°E

(d) Residual [x 10**{-5} m/s**2]

-21

-14

-7

0

7

14

21

Figure 3. The December-February climatology (ERA-Interim 1979-2015) of vertical column-average (with

cosine weighting) of (a) zonal LWA flux convergence, (b) meridional momentum flux divergence, (c) low-

level meridional heat flux, and (d) residual [see (5)]. The color scales for (b) to (f) are the same, with values

greater than the maximum given by the color bar is displayed in brown, while that less than the minimum

given by the color bar is displayed in green.

299

300

301

302

303

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cycle per day

(d)

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