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3637
The Structure and Maintenance of Stationary Waves in the
Winter Northern Hemisphere
TSING-CHANG CHEN
Atmospheric Science Program, Department of Geological and Atmospheric Sciences, Iowa State University, Ames, Iowa
(Manuscript received 9 June 2004, in final form 11 February 2005)
ABSTRACT
Previous studies of extratropical stationary waves in the winter Northern Hemisphere (NH) often focused
on effects of orography and land–ocean thermal contrast on the formation, structure, and maintenance of
these waves. In contrast, research attention to tropical stationary waves was attracted by the summer
monsoon circulations and the ENSO-related climate variability. Consequently, the structure and basic
dynamics of tropical stationary waves and the relationship of these waves with those in mid–high latitudes
have long been neglected. Thus, the following several distinct features of observed winter NH stationary
waves have not been explained: 1) an abrupt change in the longitudinal phase across 30°N; 2) a transition
from the vertical phase reversal of tropical stationary waves to the vertically westward tilt of extratropical
stationary waves; and 3) a longitudinally quarter-phase relationship between stationary waves and east–west
circulations, and a reversal of this relationship across 30°N. It is inferred from a spectral streamfunction
budget analysis with the NCEP–NCAR reanalyses that these wave features are caused by the transition of
wave dynamics from the Sverdrup regime in the Tropics to the Rossby regime in the mid–high latitudes.
Based on the simplified vorticity equations of these two dynamic regimes, analytic solutions obtained with
observed velocity potential fields (which were used to portray the global divergent circulation) confirm that
the aforementioned distinct features of stationary waves are attributed to the dynamics transition across
30°N. Since east–west circulations are part of the global divergent circulation, it is revealed from a diagnosis
of the velocity potential maintenance equation that this circulation component is maintained in the Tropics
primarily by diabatic heating and in the mid–high latitudes by both horizontal heat advection and diabatic
heating. Evidently, stationary waves are maintained by diabatic heating through the divergent circulation
and the dynamics transition of these waves from the Sverdrup regime to the Rossby regime is attributed to
strong midlatitude westerlies.
1. Introduction
It is perceivable from the Northern Hemispheric winter circulation depicted by streamlines and isotaches
(Fig. 1) that the upper-tropospheric flows (Fig. 1a) are
separated by jet streams at about 30°N into two different flow regimes: the mid–high latitudes and the Tropics. North of the jet streams exist three major troughs
(along the eastern seaboards of East Asia and North
America and central Europe—the Mediterranean Sea)
and three major ridges (along the Pacific coast of North
America, the eastern North Atlantic, and central Eur-
Corresponding author address: Tsing-Chang (Mike) Chen, Atmospheric Science Program, Department of Geological and Atmospheric Sciences, 3010 Agronomy Hall, Iowa State University,
Ames, IA 50011.
E-mail: tmchen@iastate.edu
© 2005 American Meteorological Society
JAS3566
asia). Corresponding to these asymmetric circulation
elements in mid–high latitudes, the tropical flow regime
south of the jet streams is characterized by tropical anticyclones/ridges (covering the tropical Africa–western
tropical Pacific region and the western tropical Atlantic), and tropical troughs (over the eastern Pacific and
Atlantic and the northern Arabian Sea). In the lower
troposphere (Fig. 1b), the mid–high-latitude flow regime exhibits three major low systems (the Aleutian
and Icelandic lows and the central European trough)
and three major high systems (the Alaska Pacific coast,
the western European coast, and the eastern Asian continent). These low-tropospheric asymmetric circulation
elements are coupled with their corresponding uppertropospheric troughs and ridges, respectively. For the
tropical flow regime, the salient lower-tropospheric circulation elements are overlaid by those with opposite
circulation characteristics in the upper troposphere.
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FIG. 1. Streamlines superimposed with isotaches (stippled areas) at (a) 200 and (b) 850 mb.
Dotted areas added to the 850-mb streamline chart are mountains with their surface pressure
less than 850 mb.
The asymmetric component of the atmospheric general circulation was often considered to be formed by
stationary waves. Their structure was portrayed in
terms of amplitudes and phases of zonal harmonic components (e.g., van Loon et al. 1973), while their dynamic role in the atmospheric circulation was illustrated by transport properties (e.g., Oort 1983) and energetics (e.g., Wiin-Nielsen and Chen 1993). To
understand their roles in the local balances of some
meteorological properties and their relationship with
the underlying orography and oceans, Lau (1979) described the three-dimensional structure of stationary
waves with horizontal maps and longitude–height cross
sections. Several interesting features revealed from the
spatial structure of these waves were
1) a westward tilt with height in mid–high latitudes,
2) an abrupt change in the upper-level longitude phase
at about 30°N, and
3) a transition from trapped waves at low latitudes to
vertically propagating waves at high latitudes.
The dynamics of stationary waves in explaining their
westward tilt in mid–high latitudes and vertical propagation at high latitudes have been well explored. Orography (Charney and Eliassen 1949; Bolin 1950) and diabatic heating (Smogorinsky 1953) play important roles
in the development of stationary waves. Realistic stationary waves in mid–high latitudes can be generated
by a combination of both forcings (Derome and Wiin-
Nielsen 1971). The sensible heat advection of these
waves is reflected in their westward vertical tilt (Holton
2004), while the vertical propagation is a result of the
energy leakage of ultralong waves from the troposphere to stratosphere through moderate polar westerlies (Charney and Drazin 1961). However, the introduction of the teleconnection theory of the Rossby
wave by Hoskins and Karoly (1981) reveals the impact
of the tropical heating on midlatitude stationary waves.
To understand this impact, the linear primitive equation stationary wave model on the sphere developed by
Eggar (1976) was adopted to simulate the Northern
Hemisphere wintertime stationary waves with the global diabatic heating (e.g., Nigam et al. 1986, 1988; Chen
and Trenberth 1988; Valdes and Hoskins 1989) and the
response of the extratropical stationary waves to tropical heating (e.g., Ting 1994). Transient eddy fluxes and
nonlinear interactions were also included by some studies (see a review by Held et al. 2002). Although tropical
stationary waves in summer and responses of the tropical flow to tropical diabatic heating exhibit a vertical
phase reversal, mechanism/dynamics responsible for
the vertical phases reversal of winter stationary waves
at low latitudes are still not well understood. Do these
waves behave like summer stationary waves, which are
characterized by a vertical phase reversal (White
1982)? What causes the longitudinal phase change of
stationary waves in the upper troposphere across 30°N?
The basic dynamics of stationary waves needed to an-
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swer those questions may not be revealed simply from
the forcing-response numerical simulation with the linear stationary wave models. Because of the limit of this
approach, a different method of exploring these questions is presented in this paper.
Accompanied by his depiction of the 200-mb divergent circulation in terms of velocity potential, Krishnamurti et al. (1973) presented the 200-mb large-scale
circulation portrayed with wind vectors and streamfunction. It was revealed from their presentation that
divergence centers coincide with continental highs and
convergence centers overlap with oceanic troughs. In
concert with Krishnamurti et al.’s depiction of largescale circulation, Streten and Zillman (1984) presented
a schematic diagram of the vertical structure of the
east–west circulations on the equatorial longtitude–
height cross section superimposed with upper- and
lower-level isobars. In the upper troposphere, the inphase relationships between divergence center and high
pressure and between convergence center and low pressure was clearly described by Streten and Zillman’s
cross section. In the lower troposphere, a convergent
center is overlaid by an upper-level divergence center
and vice versa. It is inferred from Streten and Zillman’s
schematic diagram that the structure of stationary
waves exhibits a vertical reversal associated with the
east–west circulations. It was observed by previous
studies (e.g., Kang and Held 1986; Chen et al. 1999;
Chen 2003) that in the summer hemisphere a spatially
quadrature relationship in the longitudinal direction exists between east–west circulations and tropical ultralong waves. Should there be an in-phase relationship
between the divergence/convergence center and the
high/low pressure center during the northern winter?
The 200-mb velocity potential used by Krishnamurti
et al. (1973) to portray the upper-level divergent circulation is dominated by the wave-1 component. The velocity potential of this wave component does not exhibit a longitudinal phase reversal across 30°N, like
geopotential height described by Lau (1979). Evidently,
relationships between divergent circulation and stationary waves in mid–high latitudes and the Tropics differ
from each other. What is the possible change in the
basic dynamics of stationary waves across 30°N reflected by this difference? Heat budget analyses of previous studies (e.g., Wei et al. 1983; Chen and Baker
1986; Kasahara et al. 1987; Rodwell and Hoskins 2001)
showed that the Tropics and mid–high latitudes belong
to the deep and slanted convection regimes, respectively. Since velocity potential is maintained by vertical
differentiations of sensible heat advection and diabatic
heating (Chen and Yen 1991a,b), is there any difference
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in maintaining velocity potential between the Tropics
and mid–high latitudes?
The pronounced anomalous circulation patterns of
climate variability, for example, the Pacific–North
America pattern (Wallace and Gutzler 1981; Horel and
Wallace 1981), the North Atlantic Oscillation (van
Loon and Rogers 1978; Rogers and van Loon 1979;
Meehl and van Loon 1979), and the North Pacific
ENSO short wave train (Chen 2002) are reflected by
interannual variations of stationary waves. Numerous
efforts have been made in the past decades to explore
the structure, basic dynamics, and causes of these
anomalous circulation patterns and their possible impacts on the global and regional climate/weather system. New research initiatives were planned and implemented under the Climate Variability and Prediction
Program (CLIVAR) to predict climate variability
[World Climate Research Programme (WCRP) 1997].
A better understanding of the structure and basic dynamics of stationary waves would be conducive to this
pursuit. The purpose of this study is to search for causes
of unexplained features of Northern Hemisphere wintertime stationary waves in terms of the relationship
between stationary waves and divergent circulation and
between diabatic heating and divergent circulation.
In view of the relationship between the east–west
circulation and large-scale circulation in the Tropics described by previous studies, a simplified streamfunction
budget analysis in the spectral domain (Holton and
Colton 1972) gives us an informative approach to explore the basic dynamics of stationary waves depicted
by the National Centers for Environmental Prediction–
National Center for Atmospheric Research (NCEP–
NCAR) reanalyses (Kalnay et al. 1996) for the period
of 1979–2002. The entire study is thus arranged in the
following manner. The relationship between stationary
waves and east–west circulations which was reexamined
with reanalyses is presented in section 2. The spatial
structure and maintenance mechanism of individual
stationary waves, which form the basis for the search of
the basic dynamics of these waves and causes of their
distinct features observed in section 2, are shown in
section 3. Based on the basic dynamics revealed from
the streamfunction budget, the analytic explanations of
stationary wave’s distinct features are illustrated in section 4. Some discussion and concluding remarks are
offered in section 5.
2. Relationship between stationary waves and
east–west circulations
As indicated by global maps of the National Oceanic
and Atmospheric Administration (NOAA) outgoing
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longwave radiation (e.g., Hartmann 1994), major tropical deep cumulus convection during northern winter
occurs over three tropical continents (the Maritime
Continent, tropical South America, and equatorial Africa). Krishnamurti et al.’s (1973, their Fig. 1) depiction
of the global divergent circulation at 200 mb showed
that three divergent centers over these tropical continents coincide with tropical convection centers and upper-level anticyclonic centers. Their finding was echoed
by Streten and Zillman’s (1984, their Fig. 87) equatorial
longitude–height schematic diagram of east–west circulations superimposed with upper- and lowertropospheric isobars: upward branches of the tropical
east–west circulations were coincident with upper
(lower) level anticyclones (cyclones), while downward
branches were coupled with the reversed circulation
condition. After the longitude–height cross sections of
stationary waves at different latitudes were presented
by Lau (1979), the vertical structure of these waves and
the relationship of these waves with the global divergent circulation were not explored further. In other
words, the in-phase relationship between the global divergent circulation and tropical stationary waves revealed from Krishnamurti et al.’s depiction of the east–
west circulations and Streten and Zillman’s schematic is
the current understanding of the tropical circulation
system.
Based on their energetics analysis, Krishnamurti et
al. (1973) argued that the three subtropical (East Asian,
North American, and North African) jets (Fig. 1a) associated with wintertime stationary waves were maintained by the global divergent circulation through the
overturning of the east–west circulation. In view of the
meridional extent of the global divergent circulation
from the Tropics to mid–high latitudes (which will be
illustrated later) and transitions in the vertical and horizontal structure of stationary waves at 30°N, what may
be the spatial relationship between stationary waves
and the east–west circulation north of this latitude? Numerous studies (e.g., Lau 1979) show that vertical motion associated with midlatitude stationary waves are
structured in the following manner: upward motion occurs ahead of major troughs, while downward motion
appears ahead of major ridges. According to the ␻
equation (Holton 2004), these vertical motions are primarily maintained by two processes: 1) positive vorticity advection ahead of troughs and negative vorticity
advection ahead of ridges in the upper troposphere,
and 2) warm-air advection east of major low centers
and cold-air advection east of major high systems in the
lower troposphere. It is inferred from this east–west
differentiation of vertical motion that midlatitude
troughs/ridges are spatially in quadrature with the as-
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sociated east–west circulations. This inference is supported by Blackmon et al.’s (1977) explanation for the
formation of subtropical jet streams, following Namias
and Clapp’s (1949) confluence theory. The jet is accelerated in its entrance by a thermally direct cross-jet
circulation driven by southward cold-air advection and
is decelerated in its exit by a thermally indirect cross-jet
circulation maintained by northward transient sensible
heat advection along the storm track.
Stationary waves in midlatitudes are often depicted
by the eddy component of geopotential height (ZE)
(e.g., Lau 1979), but the amplitude of geopotential
height perturbations in the Tropics is usually an order
of magnitude smaller (e.g., Holton 2004). In contrast,
the magnitudes of zonal winds in the Tropics and midlatitudes are the same order of magnitude. Because
zonal wind can be determined by the meridional gradient of streamfunction, magnitudes of eddy streamfunction (␺E) in the two latitudinal zones should be comparable. Thus, ␺E is preferred over ZE to portray stationary waves for both the Tropics and mid–high
latitudes. For this reason, the ␺E (200 mb) and ␺E (850
mb) fields are shown in Figs. 2a,d, respectively. Major
asymmetric components of the atmospheric circulation
depicted by streamlines in Fig. 1 are well represented
by ␺E anomalies in these two figures. The longitudinal
phase reversal of stationary waves at 30°N observed by
Lau (1979, his Fig. 2) is well depicted by ␺E (200 mb) in
Fig. 2a. A vertical westward tilt of stationary waves in
mid–high latitudes and a vertical phase reversal in the
Tropics are perceivable through the contrast between
␺E (200 mb) and ␺E (850 mb) (Fig. 2d). This transition
of the stationary waves’ vertical structure from mid–
high latitudes to the Tropics accompanied with the longitudinal phase reversal of stationary waves at 30°N is
further confirmed by longitudinal–height cross sections
of ␺E (50°N) (Fig. 2b) and ␺E (15°N) (Fig. 2c). The
monsoon circulation is characterized by a phase reversal in its vertical structure (Chen 2003). Since wintertime tropical stationary waves exhibit the same characteristic, the monsoon dynamics may be applicable to
these stationary waves. The transition in the spatial
structure of wintertime stationary waves is an indication of a change in their dynamics, but what is this
change and what may cause this change?
The east–west circulations in midlatitudes, (uD,
⫺␻)(50°N), and the Tropics, (uD, ⫺␻)(15°N), are superimposed on longitude–height cross sections of ␺E
(50°N) (Fig. 2b) and ␺E (15°N) (Fig. 2c), respectively.
As inferred from previous studies of planetary-scale
vertical motions (e.g., Lau 1979), upward (downward)
motion exists ahead of troughs (ridges) in midlatitudes.
The reversed relationship between ␺E (15°N) and
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FIG. 2. Eddy component of streamfunction (␺E) at (a) 200 mb, (b) 50°N superimposed with
the east–west circulation depicted by (uD, ⫺␻); (c) same as (b) expect for 15°N, and (d) 850
mb. Here uD and ␻ are the zonal component of divergent wind and p-vertical motion (⬅
dp/dt), respectively. Positive values of ␺E are stippled. Contour intervals of ␺E at (a) and
(b)–(d) are 2 ⫻ 106 m2 s⫺1 and 106 m2 s⫺1, respectively.
(uD, ⫺␻)(15°N) appears in the tropical upper troposphere. In other words, upward (downward) branches
appear east (west) of positive (negative) ␺E cells. As
observed in midlatitudes, ␺E (15°N) and (uD, ⫺␻)
(15°N) are in quadrature spatially. This quadrature relationship between the ␺E cells of stationary waves and
the east–west circulation revealed from Fig. 2c is dif-
ferent from that shown by Streten and Zillman’s (1984)
schematic diagram. Because of the vertical phase reversal of stationary waves in the Tropics, the phase relationship between ␺E (15°N) and (uD, ⫺␻)(15°N) in the
lower troposphere is also in quadrature, but with a longitudinal phase opposite to that in the upper troposphere. As shown in Fig. 2, the structure of stationary
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suggest that there is also a difference in the dynamics of
stationary waves between these two regions.
3. Maintenance of stationary waves
a. Structure of individual stationary waves
FIG. 3. A schematic diagram of the relationships between stationary waves (thick-solid sinusoidal lines) and the east–west circulation (thin-solid lines with shafts) in mid–high latitudes and the
Tropics.
waves (the vertical westward tilt in mid–high latitudes,
the vertical phase reversal in the Tropics, and the longitudinal phase reversal across 30°N) and their spatial
(quadrature) relationship with the east–west circulation
are summarized by a simple schematic diagram shown
in Fig. 3.
Krishnamurti et al. (1973) argued that the three subtropical jet streams associated with troughs of stationary waves in midlatitudes were energetically maintained by east–west circulations. However, the opposite
relationship between stationary waves and the east–
west circulation in the upper troposphere of midlatitudes and the Tropics strongly indicates that basic differences in the maintenance mechanism and dynamics
of stationary waves exist between these two latitudinal
zones. For example, it was shown by heat budget analyses in previous studies (e.g., Wei et al. 1983; Chen and
Baker 1986; Kasahara et al. 1987; Rodwell and Hoskins
1996) that planetary-scale vertical motion in the Tropics was maintained primarily by diabatic heating and by
both sensible heat advection and diabatic heating in
midlatitudes. The nonnegligible sensible heat advection
in midlatitudes distinguishes the midlatitude slanted
convection from the tropical deep convection. Apparently, the east–west circulations in midlatitudes and
Tropics are maintained by different mechanisms, and,
consequently, so are stationary waves. These arguments
Following Sanders (1984) and Kang and Held (1986),
the maintenance of winter stationary waves was explored by Chen and Chen (1990) in terms of a streamfunction budget analysis, but the distinct features of
these waves presented previously were not touched
upon by their study. To search for dynamical causes of
these features, the spectral decomposition of the
streamfunction budget introduced by Holton and Colton (1972) in examining the maintenance mechanism of
the Tibetan high is adopted in this study. The east–west
circulations are longitudinal–height cross sections of
the global divergent circulation, which is often depicted
in terms of velocity potential (␹). The relationship between stationary waves and east–west circulations may
be illustrated in terms of eddy components of streamfunction and velocity potential (␹). To match Holton
and Colton’s spectral streamfunction budget, both ␺E
and ␹E fields should be represented by their spectral
components. Two basic prerequisites are required by
this approach:
1) a proper representation of stationary waves by sufficient harmonic components, and
2) preservation of the distinct features of stationary
waves by selected harmonic components.
These two requirements can be satisfied by a simple
longitudinal harmonic (Fourier) analyses of both ␺E
and ␹E fields.
The power spectra of ␺E and ␹E in the upper and
lower troposphere at 50° and 15°N are displayed in Fig.
4. Except at 15°N, where only the first harmonic component of the ␺ n spectra is important, the first three
harmonic components stand out in the ␺ n spectra at 50°
and 15°N in both the upper and lower troposphere. In
contrast, the ␹ n spectra are always dominated by the
first harmonic component. According to the ␺n and ␹ n
spectra, stationary waves in the Tropics and mid–high
latitudes are well represented by the first three harmonic components. This representation of stationary
waves is supported by ratios of both the variances of ␺ n
and ␹ n in this long-wave regime and of the total asymmetric components (i.e., ␺E and ␹E) (Table 1). Variances of both variables represented by the first three
harmonic components, ␺1–3 and ␹1–3 are always larger
than 95% of their corresponding total eddies. Thus, we
shall focus our analysis on the long-wave regime. Fol-
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FIG. 4. Power spectra of ␺ n and ␹ n at 200 (stippled strips) and
850 mb (open strips): (a) ␺ n(50°N), (b) ␺ n(15°N), (c) ␹ n(50°N),
and (d) ␹ n(15°N). Scale on the left (right) side belongs to power
spectra at 200 (850) mb.
lowing van Loon et al. (1973), spatial structures of individual harmonic components are described in terms
of amplitude and phase in Fig. 5, instead of the spatial
distributions of ␺ n and ␹ n anomalies. With this approach, the distinct features of stationary waves observed by Lau (1979) and also those revealed from Fig.
2 can be measured quantitatively.
TABLE 1. Variance ratios of two variables [streamfunction and
velocity potential between their long-wave regime (1–3 superscript) and total eddy (E subscript)].
Latitude
Ratio of variance
50°N
15°N
Var(␺ )/Var(␺E) (200 mb)
Var(␺ 1–3)/Var(␺E) (850 mb)
Var(␹ 1–3)/Var(␹E) (200 mb)
Var(␹ 1–3)/Var(␹E) (850 mb)
96%
98%
99%
99%
98%
95%
99%
99%
1–3
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In the upper troposphere, ␺ n exhibit their maximum
amplitudes at about 50° and 20°N, and minimum amplitudes between 30° and 40°N where the westerly jet
streams exist and longitudinal phase changes of ␺E (200
mb) (Fig. 2a) occur. The ␹E maximum in the Tropics is
reflected by maximum amplitudes of ␹ n in Fig. 5a. In
addition, ␹ 2 and ␹ 3 exhibit their secondary maximum
amplitudes in midlatitudes where minimum amplitudes
and the longitudinal phase reversal of ␺ 2 and ␺ 3 (Fig.
5c) exist. In spite of the latitudinal phase reversal of ␺ n,
␹ n does not exhibit such a phase change. However, a
quadrature phase relationship appears between ␹ n and
␺ n in Fig. 5c. This ␹ n–␺ n phase relationship undergoes
a sign change across the latitude where the longitudinal
phase reversal in ␺ n takes place. That is, ␹ n is ahead of
␺ n north of 30°N, but behind ␺ n south of this latitude.
This change in the ␹ n–␺ n phase relationship is consistent with those between east–west circulations and the
associated ␺ n cells in midlatitudes and the Tropics.
In contrast to the ␺ n structure in the upper troposphere, a major maximum amplitude of ␺ n (850 mb)
(Fig. 5b) appears in midlatitudes and very minor maximum amplitudes of ␺ 2 and ␺ 3 emerge in the Tropics.
Actually, the continental high (e.g., Siberian high) and
oceanic lows (e.g., Aleutian and Icelandic lows) are reflected by the midlatitude ␺ n(850 mb) maximum amplitude. Here ␺ n(850 mb) does not show a longitudinal
phase reversal (Fig. 5d), but exhibits a spatial quadrature relationship with ␹ n(850 mb). Latitudinal distributions of ␹ n(850 mb) amplitude correspond to those of
␹ n(200 mb), except with an opposite phase between
them.
The vertical structure of individual wave components
may be inferred from the contrast between their amplitudes and phases at 200 and 850 mb. However, a
quantitative measurement of this vertical structure cannot be obtained without vertical distributions of the
amplitudes and phases. Shown in Fig. 6 are vertical
distributions of ␺ n and ␹ n amplitudes and phases at 50°
and 15°N. Because atmospheric flow is much stronger
in the upper troposphere, it is not surprising to see that
amplitudes of ␺ n are much larger in the upper troposphere of both the midlatitudes and the Tropics (Figs.
6a,b). Divergent winds are generally large in the upper
and lower troposphere and small in the midtroposphere, because of a nondivergent layer. Accordingly,
maximum amplitudes of ␹ n appear in the upper and
lower troposphere (Figs. 6a,b). As typical baroclinic
waves (e.g., Lau 1979), the vertically westward tilt of
stationary waves is revealed from the phase of
␺ n(50°N) in Fig. 6c. In the Tropics, a vertical phase
reversal (Fig. 6d) of stationary waves, which is a typical
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FIG. 5. Latitudinal distributions of amplitudes of (a) (␺ n, ␹ n) (200 mb), (b) (␺ n, ␹ n) (850 mb), and phases of (c)
(␺ n, ␹ n) (200 mb), and (d) (␺ n, ␹ n) (850 mb).
character of the monsoon circulation, is clearly shown
by the phase of ␺ n(15°N) (Fig. 6d).
Although ␺ n has a longitudinal phase change across
30°N, ␹ n covers the entire hemisphere without such a
change. This characteristic of ␹ n phase is reflected by
the similar vertical phase structure of ␹ n(50°N) and
␹ n(15°N). On the other hand, it is of interest to note
that there is a difference in the ␺ n–␹ n relationships between mid–high latitudes and the Tropics: ␺ n leads ␹ n a
quarter phase in the midlatitude upper troposphere,
but this ␺ n–␹ n relation is reversed in the Tropics. In the
lower troposphere, the ␺ n–␹ n relationships in the midlatitudes and the Tropics are more or less the same.
These ␺ n–␹ n relationships may be illustrated by the
contrast between east–west circulations and some circulation elements, for instance, the Pacific Northwest
ridge/the eastern tropical Pacific trough in the upper
troposphere (Fig. 2a) and the Pacific Northwest coast
FIG. 6. Same as in Fig. 5, except for vertical distributions.
OCTOBER 2005
anticyclone/the eastern tropical Pacific anticyclone in
the lower troposphere (Fig. 2d). The upper-level Pacific
Northwest ridge and the low-level West Coast anticyclone are coupled with a clockwise east–west circulation (Fig. 2b). In low latitudes, the upper-level eastern
Pacific trough and the eastern tropical Pacific anticyclone are also linked to a clockwise east–west circulation (Figs. 2b,c). The opposite ␺ n–␹ n relationship between midlatitudes and the Tropics is a reflection of the
difference in the dynamics of stationary waves in the
two latitudinal zones.
b. Maintenance of individual stationary waves
How do we illustrate the difference in the dynamics
of stationary waves in mid–high latitudes and the Tropics? Following Holton and Colton (1972), we express
the streamfunction budget equation simplified by Chen
and Chen (1990) in the longitudinal wavenumber domain to answer this question:
冉
0 ⫽ ⵜ⫺2 ⫺UZ
冊
⭸␨ n
⫹ ⵜ⫺2共⫺␯ n␤兲 ⫹ ⵜ⫺2共⫺f ⵱ ⭈ Vn兲,
⭸x
共1兲
n
␺A
1
n
␺A
2
␺ ␹n1
where the overbar signifies the seasonal mean value of
the variable. For convenience of discussion, an overbar
imposed on any variable is hereafter dropped. Equation (1) was simplified with the following conditions.
Any term in the complete streamfunction budget equation with its variance satisfying one of the two criteria is
neglected: Var( ␺ nA m )/Var( ␺ nA )ⱕ15% or Var( ␺ n␹ m )/
Var(␺ n␹)ⱕ15%, where Var represents variance averaged over the Northern Hemisphere between 0° and
60°N, and m ⫽ 1, 2, . . . , etc. The variance ratio of each
n
term in Eq. (1) with either ␺ nA (⫽兺 M
m⫽1 ␺ A m ) or
n
M⬘
n
␺ ␹ (⫽兺m⫽1␺ ␹m) is displayed in Table 2. Based on the
aforementioned criteria, other nonlinear and transient
terms in the complete streamfunction budget equation
are negligible in their contributions to streamfunction
tendency as shown by Chen and Chen (1990). Therefore, Eq. (1) is approximated by the following forms:
n
n
⫹ ␺A
⫹ ␺ ␹n1 ⯝ 0,
Mid–high latitudes: ␺ A
1
2
共2a兲
Tropics:
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n
␺A
⫹ ␺ ␹n1 ⯝ 0.
2
共2b兲
The vorticity equations corresponding to these two simplified streamfunction budget equations are
TABLE 2. Variance ratios of terms in the streamfunction budget
equation averaged over the Northern Hemisphere between the
equator and 60°N.
Wavenumber n
Wavenumber n
200 mb
850 mb
Ratio of variance
1
2
3
1
2
3
Var(␺ nA1)/Var(␺ nA)
Var(␺ nA2)/Var(␺ nA)
Var(␺ n␹1)/Var(␺ n␹)
Var(␺ n␹2)*/Var(␺ n␹)
60%
42%
89%
2%
62%
43%
88%
13%
51%
63%
91%
10%
8%
92%
90%
11%
9%
89%
93%
5%
9%
88%
94%
5%
* Here ␺ ␹2 ⫽ ⵜ⫺2(⫺␯D␤), which was considered to be significant
by Sardeshmukh and Hoskins (1985) in the vorticity budget analysis in the Tropics. However, this term, which does not noticeably
affect our analysis, is neglected to simplify discussion of results.
Mid–high latitudes UZ
⭸␨ n
⫹ ␯n␤ ⫽ ⫺f ⵱ ⭈ Vn,
⭸x
Rossby dynamics;
共3a兲
Tropics
␯ n␤ ⫽ ⫺f ⵱ ⭈ Vn, Sverdrup dynamics.
共3b兲
It is revealed from the comparison between Eqs. (3a)
and (3b) that the major difference in the dynamics of
stationary waves between mid–high latitudes and the
Tropics is the horizontal advection of relative vorticity
by strong midlatitude westerlies. The vorticity tendency
induced by this vorticity advection in mid–high latitudes enables wave perturbations to propagate eastward, while that induced by the meridional advection of
planetary vorticity works in an opposite manner. When
zonal westerlies reduce their magnitude in the lower
troposphere and reverse to easterlies in the Tropics, the
horizontal advection of relative vorticity may be negligible or function in a way similar to the meridional
advection of planetary vorticity. The Sverdrup dynamics therefore prevail in the lower troposphere and low
latitudes. As will be shown later, the basic characteristics of the three waves in the long-wave regime are
similar. Perhaps it is redundant to present the streamfunction budget analysis of each individual wave. The
Northern Hemisphere winter circulation is characterized by three midlatitude jet streams over North Africa,
East Asia, and North America (Fig. 1), and the power
spectra of ␺ 3(50°N) and ␺ 3(15°N) are larger than ␺ 2
and comparable to ␺ 1 (Fig. 4). To avoid the aforementioned redundancy, let us use the wavenumber-3 component as an example to elucidate the difference in the
dynamics of stationary waves between these two latitudinal zones. The detailed budget analysis of ␺ 3 is pre-
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FIG. 7. (a) Streamfunction anomalies of wavenumber-3 component, ␺ 3 (200 mb), and the streamfunction tendency at 200 mb induced by (a) relative vorticity advection, ␺ 3A1(200 mb), (b) planetary vorticity advection, ␺ 3A2(200 mb), and longitude–height cross sections of (c) (␺ 3A1, ␺ 3) (50°N), and (d) (␺ 3A2, ␺ 3)
(50°N). Positive (negative) values of ␺ 3A1 and ␺ 3A2 are contoured by solid (dashed) lines, while positive
(negative) values of ␺ 3 are stippled (dotted). The contour interval of ␺ 3A1 and ␺ 3A2 is 25 m2 s⫺2.
sented in Figs. 7–9. The major findings of this budget
analysis are presented as follows.
1) MID–HIGH
LATITUDES
␺ 3A1(200
Because
mb) (Fig. 7a) has a larger amplitude
than ␺ 3A2(200 mb) (Fig. 7b), a combination of these two
streamfunction tendencies, ␺ 3A12(200 mb) (Fig. 8a), induced by total vorticity advection is dominated by that
induced by relative vorticity advection. Evidently, be-
cause of strong westerlies in the upper troposphere of
the midlatitudes, ␺ 3A12(200 mb) is dominated by advection of relative vorticity. As shown in Fig. 8a, ␺ 3A12(200
mb) anomalies are longitudinally a quarter phase ahead
(east) of ␺ 3(200 mb) anomalies (positive values are
lightly stippled, while negative values are dotted).
Therefore, ␺ 3(200 mb) anomalies can be propagated
eastward by total vorticity advection. As revealed from
the contrast between Figs. 8a,b, the eastward propagation of ␺ 3(200 mb) anomalies in the midlatitudes by the
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FIG. 8. The streamfunction budget of ␺ 3 at 200 and 850 mb: (a) (␺ 3A12, ␺ 3) (200 mb), (b) (␺ 3␹1, ␺ 3) (200
mb), (c) (␺ 3A2, ␺ 3) (850 mb), and (b) (␺ 3␹1, ␺ 3) (850 mb). Contour intervals of all quantities in the
streamfunction budget are 25 m2 s⫺2. Positive (negative) values of ␺ 3 anomalies are stippled (dotted).
␺ 3A12(200 mb) tendency is counterbalanced by the
streamfunction tendency of ␺ 3␹1(200 mb) induced by the
vortex stretching associated with the east–west circulation in response to the eastward propagation of ␺ 3
anomalies. Consequently, the counterbalance between
␺ 3A12(200 mb) and ␺ 3␹1(200 mb) enables the ␺ 3(200 mb)
anomalies to become stationary.
The vertical structure of ␺ 3A1(50°N) and ␺ 3A2(50°N) is
portrayed in terms of longitude–height cross sections in
Figs. 7c,d, while the vertical structure of these two variables combined is shown in Fig. 9a. It becomes clear
that ␺ 3A1(50°N) ⬎ ␺ 3A2(50°N) in the upper troposphere
and ␺ 3A1(50°N) Ⰶ ␺ 3A2(50°N) in the lower troposphere
where midlatitude westerlies become weaker. Thus,
midlatitude ␺ 3 anomalies in the upper troposphere are
maintained by the counterbalance between ␺ 3A12 and
␺ 3␹1, and those in the lower troposphere by the counterbalance between ␺ 3A2 and ␺ 3␹1. This counterbalance is
further confirmed by these two variables at 850 mb
(Figs. 8c,d). Let us assume f ⫽ constant for discussion,
␺ 3␹1 ⫽ ⵱–2(–f ⵱·V3) ⫽ –f␹ 3. As inferred from the com-
parison of this simplified relationship and the longitude–height cross section of ␺ 3␹1 with ␺ 3 anomalies, a
counterclockwise east–west circulation is coupled with
a negative ␺ 3(50°N) cell at upper levels and the reversed east–west circulation is associated with a positive ␺ (50°N) cell at upper levels, as shown in Fig. 3.
2) TROPICS
In the Tropics, the upper-tropospheric flows are predominatly easterlies, which are weaker than midlatitude westerlies. Thus, it is not surprising to find that the
amplitude of ␺ 3A1(200 mb) (Fig. 7a) Ⰶ amplitude of
␺ 3A2(200 mb) (Fig. 7b) south of 30°N. Because ␺ 3(200
mb) anomalies undergo a longitudinal phase change
across 30°N, the ␺ 3A2(200 mb) streamfunction tendency
induced by the meridional planetary vorticity advection
is always a quarter phase behind (west of) ␺ 3 (200 mb)
anomalies. This phase relationship enables ␺ 3A2(200 mb)
to propagate ␺ 3(200 mb) anomalies westward. However, this dynamic process is counterbalanced by the
streamfunction tendency induced by vortex stretching,
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FIG. 9. Same as in Fig. 8, expect for longitude–height cross sections at 50° and 15°N.
␺ 3␹1(200 mb) (Fig. 8b), in such a way that ␺ 3(200 mb)
forms a stationary wave. It was revealed from Fig. 6
that ␺ 3 and ␹3 exhibit a vertical phase reversal in the
Tropics. How are tropical stationary waves maintained
in the tropical lower troposphere? The contrast between ␺ 3A2(15°N) (Fig. 9c) and ␺ 3␹1(15°N) (Fig. 9d) suggests that ␺ 3(15°N) in the upper and lower troposphere
are maintained by the same dynamical processes. Recall that ␺ 3␹1 ⯝ –f␹ 3, if we assume f ⫽ constant. It is
inferred from the counterbalance between ␺ 3␹1 and ␺ 3A2
in the Tropics that positive ␺ 3 anomalies are coupled
with a clockwise east–west circulation and negative ␺ 3
anomalies with counterclockwise east–west circulation,
as sketched by the schematic diagram shown in Fig. 3.
It was pointed out previously that the ␹n cell did not
change its phase latitudinally in the entire Northern
Hemisphere. This latitudinal phase structure of ␹3 may
be inferred from ␺ 3␹1 shown in Figs. 8b,d. Being part of
this divergent circulation cell, the east–west circulations
of the wavenumber-3 component in mid–high latitudes
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and the Tropics exhibit a similar structure (not shown).
Regardless of this similarity in the structure of the east–
west circulations in two latitudinal zones, the relative
vorticity advection by strong midlatitude westerlies enables east–west circulations of the same direction in
mid–high latitudes and the Tropics to couple with stationary waves of the opposite phase. It was revealed
from the simplified streamfunction budget analysis that
the major difference in the dynamics of stationary
waves between mid–high latitudes and the Tropics was
originated from the contrast between the longitudinal
relative vorticity advection [⫺UZ (⳵␨ n/⳵x)] and the meridional planetary vorticity advection (⫺ ␯ n ␤ );
⫺UZ (⳵␨ n/⳵x) ⬎ ⫺␯ n ␤ in mid–high latitudes, and
⫺UZ (⳵␨ n/⳵x) ⬍⬍ ⫺␯ n ␤ in the Tropics. The dynamics
of stationary waves inferred from diagnoses will be analytically substantiated in section 4.
4. Analytic solution
a. Formulation
It was revealed from spectral analyses of ␺E and ␹E
shown in Fig. 4 that stationary waves in both lower and
midlatitudes were basically formed by the long-wave
regime (waves 1–3). Let us represent harmonic components of ␺E and ␹E by the following forms:
␺ n ⫽ ⌿neikx,
␹ n ⫽ ⌾neikx,
Midlatitudes: ␺n ⫽ ⫺共1 ⫺ UZⲐC␤n 兲⫺1共n tan␸ csc␸兲␹ nei␲Ⲑ2,
共7兲
Tropics: ␺n ⫽ ⫺ 共n tan␸ csc␸兲␹nei␲Ⲑ2.
As indicated by Eqs. (7) and (8), the basic difference
of the (␺ n, ␹n) relationship between the Tropics and
midlatitudes is determined by the factor (1 ⫺ UZ/C n␤). It
may be inferred from Eqs. (1) and (2) that this factor
provides a measurement of the importance of the two
dynamical processes (zonal advection of relative vorticity and meridional advection of planetary vorticity) to
the dynamics of stationary waves. If UZ/C n␤ ⬎1 due to
strong westerlies in midlatitudes, the dynamics of stationary wave are dominated by the Rossby regime expressed by Eq. (7). On the other hand, if UZ/C n␤ ⬍ 1
because westerlies are not sufficiently strong in the
lower troposphere of the midlatitudes or yield to easterlies in the lower latitudes, the dynamics of stationary
waves belong to the Sverdrup regime depicted by Eq.
(8). To attain a clear perspective of the separation between these two dynamic regimes, the latitudinal distribution of UZ and UZ/C n␤ at 200 mb is displayed in Fig.
10. When UZ/Cn␤⫽1, both dynamic processes (UZ ⳵␨/⳵x
and ␯␤) are equally important to the dynamics of stationary waves. Based on this isoline, two different regimes of wave dynamics emerge in the upper troposphere:
共4兲
UZⲐC␤n Ⰶ 1 deep Tropics,
where k⫽n/a cos␸ and n and a are an integer and the
earth’s radius, respectively. Substituting Eq. (4) into
Eqs. (3a) and (3b), one can easily obtain the following
relationships between streamfunction and velocity potential:
UZⲐC␤n Ⰷ 1 midlatitudes.
Midlatitudes: ␺ n ⫽
Tropics:
␺n ⫽
冉 冊
冉 冊
CIn
UZ ⫺ C␤n
C In
⫺ C ␤n
␹nei␲Ⲑ2,
␹nei␲Ⲑ2,
共5兲
共6兲
where CnI ⫽ f/k, and C n␤ ⫽ ␤/k2. Note that f ⫽ 2⍀ sin␸,
␤ ⫽ 2⍀ cos␸/a, and k ⫽ 2␲/Lnx, where ⍀, ␸, a, Lnx(⫽2⍀a
cos␸/n) and n are earth’s rotational rate, latitude,
earth’s radius, wavelength, and wavenumber, respectively. With these parameters, the ratio CnI/Cn␤ may be
written as
CInⲐC␤n ⫽ n tan␸ csc␸.
Using this ratio, we may express Eqs. (5) and (6) in the
following forms:
共8兲
Another interesting feature of UZ/Cn␤ is that its maximum value of each wave component (indicated by a
light dashed line in Fig. 10b) appears slightly north of
30° where UZ (200 mb) (Fig. 10a) reaches its maximum
values. It was pointed out earlier that UZ became
weaker in the lower troposphere and easterly in the
tropical lower troposphere (Fig. 11a). Can the major
features of UZ/Cn␤ presented in Fig. 10 be maintained
over the entire troposphere? To answer this question,
latitude–height cross sections of UZ and UZ/Cn␤ for the
three long waves are shown in Fig. 11. For this longwave regime, UZ/Cn␤⬍⬍1 in the tropical upper troposphere and the lower troposphere. Thus, this argument
once again confirms that the dynamics of stationary
waves in these regions are dominated by the Sverdrup
regime.
b. Solutions
In this study, we are concerned with the following
distinct features of stationary waves in the winter
Northern Hemisphere:
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FIG. 10. Latitudinal distributions of (a) UZ (200 mb), and (b)UZ/Cn␤ (200 mb) for three
planetary waves n ⫽ 1, 2, and 3, which are marked by dots.
1) a longitudinal phase change at about 30°N in the
upper troposphere,
2) the westward tilt in the mid- and high latitudes,
3) a vertical phase reversal at low latitudes (south of
30°N), and
4) a quarter-phase relationship between the east–west
circulation and stationary waves and a phase reversal of this relationship across 30°N.
Can these special features of winter stationary waves be
illustrated/explained by solutions of Eqs. (7) and (8) in
the long-wave (1–3) regime?
It was revealed from the streamfunction budget
analysis in section 3 that the dynamics of stationary
waves in the upper troposphere at mid–high latitudes
belonged to the Rossby regime and those in the lower
troposphere and low latitudes belonged to the Sverdrup
FIG. 11. Latitudinal–height cross sections of (a) UZ, (b) UZ /C␤1 , (c) UZ /C ␤2 , and (d) UZ /C ␤3 . Positive values of UZ and values of
UZ /C n␤ ⱖ 1 are stippled. Contour intervals are labeled.
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FIG. 12. Latitudinal distributions of (a) UZ (200 mb), (b) UZ (850 mb), and amplitudes of (c)
␺ nAm(200 mb), and (d) AnAm(850 mb). Differences between the amplitudes of ␺ nA1 and ␺ nA2 are
heavily (lightly) stippled when the former is larger (smaller) than the latter.
regime. This division of wave dynamics is further supported by latitudinal distributions of ␺nA1 and ␺nA2 amplitudes at 200 and 850 mb for all three long waves (Fig.
12). In the upper troposphere, ␺nA1(200 mb) is much
larger than ␺nA2(200 mb) in mid–high latitudes and the
reverse is true in the Tropics. In contrast, ␺nA2(850 mb)
amplitudes are always larger than ␺nA1(850 mb). Since
amplitudes of ␺nA1(200 mb) and ␺nA2(200 mb) are equal
around 30°N, the regime transition of stationary wave
dynamics should occur across this latitude. Using
␹n(200 mb) portrayed in Figs. 5 and 6, we merge solutions of Eqs. (7) and (8) with an equal weight within the
latitudinal zone of 25°–35°N. Analytic solutions of the
streamfunction of every wave at 200 mb, ␺ nANA(200
mb), are displayed in Fig. 13. The representation of
observed streamfunction of every wave ␺ nobs(200 mb)
by the analytic solution may be measured by ratio of
Var[␺ nANA(200 mb)]/Var[␺ nobs(200 mb)] (Table 3) which
is always larger than 90%. Every ␺ nANA(200 mb) component exhibits a longitudinal phase reversal around
30°N. Based on the 200-mb analytic solutions of an
individual wave’s streamfunction, ␺ nANA (200 mb),
shown in Fig. 13, it becomes clear that the latitudinal
phase reversal of stationary waves across 30°N is a result of the dynamics transition from the Sverdrup regime in low latitudes to the Rossby regime in mid–high
latitudes. On the other hand, it was shown in Fig. 12b
that ␺nA1(850 mb) was always smaller in amplitude than
of ␺nA2(850 mb) in the long-wave regime. Evidently, the
dynamics of stationary waves in the lower troposphere
belong to the Sverdrup regime. The analytic solutions
of the streamfunction of every wave at 850 mb,
␺ nANA(850 mb), shown in Fig. 14 do not exhibit a longitudinal phase change in the midlatitudes. Ratios of
Var[␺ nANA(850 mb)]/Var[␺ nobs(850 mb)] (Table 3) for
three long waves are always larger than 90%. Based on
these analytic solutions, the dynamics transition of stationary waves across 30°N is reflected not only by the
longitudinal phase change across this latitude, but also
by the other three features of stationary waves in the
two latitudinal regions.
It was illustrated by Lau (1979) that, in mid–high
latitudes, positive total vorticity advection existed
ahead of an upper-level trough and negative total vorticity advection appears ahead of an upper-level ridge.
To maintain stationary waves, a balance between vorticity advection and vortex stretching is expected. That
is, significant vortex stretching occurs ahead of a trough
and vortex compression ahead of a ridge. Corresponding to these upper-level dynamic processes, there
should be low-level vortex compression ahead of a
trough and low-level vortex stretching ahead of a ridge.
Because the low-level basic flow is much weaker than
the upper-level flow, the horizontal advection of rela-
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FIG. 13. Analytic solutions of ␺ n at 200 mb for (a) the wave-1 component, ␺ 1ANA, (b) the
wave-2 component, ␺ 2ANA, and (c) the wave-3 component, ␺ 3ANA. Positive values of ␺ nANA are
stippled, while the contour interval of ␺ nANA is 2 ⫻ 106 m2 s⫺1.
tive vorticity becomes less important [as indicated by
the contrast between ␺ nA1(850 mb) and ␺ nA2(850 mb) amplitudes in Fig. 12b]. Thus, stationary waves in the
lower troposphere are primarily maintained by a balance between the meridional advection of planetary
vorticity and vortex stretching. The opposite phase of
vortex stretching/compression described here indicates
that upward vertical motion ahead of an upper-level
trough couples with a low-level low and downward vertical motion ahead of an upper-level ridge with a lowlevel high. To satisfy the mass continuity, these upward
and downward motions form east–west circulations
across positive/negative cells of stationary waves as
shown in Fig. 2b.
A low-level convergent center is overlaid by positive
total vorticity advection ahead of an upper-level
TABLE 3. Ratio of variance between ␺ nANA(analytic solution)
and ␺ nobs(observation).
Wavenumber
Ratio of variance
Var[␺
Var[␺
Var[␺
Var[␺
(200 mb)]/Var[␺ (200 mb)]
(850 mb)]/Var[␺ (850 mb)]
(50°N)]/Var[␺ nobs(50°N)]
(15°N)]/Var[␺ nobs(15°N)]
n
ANA
n
ANA
n
ANA
n
ANA
n
obs
n
obs
1
2
3
93%
94%
101%
96%
95%
99%
96%
98%
106%
96%
96%
98%
trough, while a low-level divergent center is overlaid by
negative total vorticity advection ahead of an upperlevel ridge. Therefore, a quarter-phase difference exists
between the upper-level trough (ridge) and the lowlevel (high) center, and a vertical westward tilt of stationary waves is expected. Following vertical distributions of ␺ nA1(50°N) and ␺ nA2(50°N) amplitudes in Fig.
15a, we merge solutions of Eqs. (7) and (8) to form
␺ nANA (50°N) in Fig. 16. Representations of ␺ nobs(50°N)
by these analytic solutions are tested by ratios
Var[␺ nANA(50°N)]/Var[␺ nobs(50°N)] ⱖ 95%. The vertical
westward tilt of stationary waves is a result of the dynamics transition from the Sverdrup regime in the
lower troposphere to the Rossby regime in upper troposphere.
The zonal-mean flows in low latitudes are either easterlies or weak westerlies. As revealed from vertical distributions of ␺ nA2 and ␺ nA1 in Fig. 15b, the former is
always larger than the latter. Thus, solutions of Eqs. (7)
and (8) in low latitudes should behave in the same manner. In other words, the dynamics of stationary waves in
the Tropics are prevailed by the Sverdrup regime. As
shown in Fig. 6, ␹nobs(15°N) exhibits a vertical phase
reversal. According to solution of Eq. (8), ␺ nANA(15°N)
should also undergo a vertical phase reversal. This expectation is confirmed by ␺ nANA(15°N) shown in Fig. 17.
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FIG. 14. Same as in Fig. 13, except at 850 mb and the contour interval of ␺ nANA is 2 ⫻
106 m2 s⫺1.
In addition to the vertical phase reversal of tropical
stationary waves, the Sverdrup dynamics [the solution
of Eq. (8)] require a quarter-phase difference between
␹n and ␺ n. Since Var[␺ nANA(15°N)]/Var[␺ nobs(15 oN)] ⱖ
95%, this quarter-phase difference is incorporated into
solutions of Eq. (8). It was depicted by Krishnamurti et
al. (1973) with the prior data from the First Global
Atmospheric Research Programme (GARP) Global
FIG. 15. Same as in Fig. 12, expect for vertical distributions at 50° and 15°N.
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FIG. 16. Same as Figs. 13 and 14, except for longitude–height cross sections at 50°N. The
contour interval of ␺ nANA(50°N) is 2 ⫻ 106 m2 s⫺1.
Experiment (FGGE) and schematically by Streten and
Zillman (1984) that the upper-level anticyclone (cyclonic) systems coincide with the upper-level divergent
(convergent) centers, while the reversed situation was
true in the lower troposphere. Solutions of Eq. (8) require this quarter phase between east–west circulations
and stationary waves as illustrated by the lower part of
Fig. 3.
It was shown in Figs. 2b,c that relationships between
␹n and the east–west circulation (unD, ⫺ ␻n) were opposite between mid–high latitudes and the Tropics. According to solutions (7) and (8), this difference is attributed to the factor of (1 ⫺ UZ/Cn␤). As shown in Fig.
10, strong westerlies in mid–high latitudes can make
this factor negative. In other words, the strong westerlies enable the relationship between ␺ n and ␹ n to be
opposite between mid–high latitudes and the Tropics.
This change in the ␺ n–␹ n relationship originates from
the difference between the Sverdrup and Rossby dynamics expressed in Eqs. (3a) and (3b). In mid–high
latitudes, the vorticity tendencies induced by the eastward relative vorticity advection associated with strong
westerlies and the meridional planetary vorticity advection oppose each other. When westerlies are sufficiently strong to make (1 ⫺ UZ/Cn␤)⬍ 0, the former
dynamic process overpowers the latter. It becomes
clear from the discussion here that the opposing relationship between ␺ n and the east–west circulation (uD,
⫺␻) in the two latitudinal zones is a result of the transition from the Sverdrup dynamics in the Tropics to the
Rossby dynamics in mid–high latitudes.
Up to this point, we explored only whether distinct
features of stationary waves born by individual waves in
the long-wave regime could be explained by solutions
of individual stationary waves. Although Var(␺ nANA)/
Var(␺ nobs) ⱖ 90% (Table 3), can the distinct features of
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FIG. 17. Same as in Fig. 16, except for 15°N. The contour interval of ␺ nANA is 2 ⫻ 106 m2 s⫺1.
␺E be explained by a combination of ␺ nANA in the long1⫺3
wave regime? Ratios of Var( ␺ 1⫺3
ANA )/Var( ␺ obs ),
1⫺3
1⫺3
Var(␺ obs )/Var(␺E), and Var(␺ ANA)/Var(␺E) are displayed in Table 4. Since all ratios are equal to/larger
than 90%, stationary waves (␺E) are well represented
by ␺ 1⫺3
ANA. Distinct features of stationary waves revealed
from Fig. 2 can be well explained by a combination of
solutions of individual stationary waves obtained from
Eqs. (7) and (8) in the long-wave regime.
5. Discussion and concluding remarks
a. Maintenance of global divergent circulation
In this study, stationary waves and divergent circulation are depicted in terms of streamfunction and velocity potential. It was observed that stationary waves
were spatially in quadrature with east–west circulations
(Fig. 2). Since the east–west circulations are part of the
global divergent circulation, relationships between sta-
1⫺3
TABLE 4. Ratio of variance between ␺ 1⫺3
ANA, ␺ obs , and ␺E.
Ratio of variance
Pressure level and latitude
200 mb
50°N
15°N
850 mb
Var(␺
)/Var(␺
1–3
ANA
96%
97%
98%
93%
n
obs
)
1–3
Var(␺ obs
)/Var(␺E)
Var(␺ 1–3
ANA)/Var(␺E)
97%
97%
94%
97%
93%
94%
92%
90%
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JOURNAL OF THE ATMOSPHERIC SCIENCES
tionary waves and the global divergent circulation in
mid–high latitudes and in the Tropics were expressed
by Eqs. (7) and (8), respectively. Using these ␺ n–␹ n
relationships, we are able to explain distinct features of
stationary waves found by Lau (1979) and the present
study. However, the observed ␹ n fields applied to illustrate these ␺ n–␹ n interactions are treated as forcings.
One may question how these ␹ n fields (e.g., global divergent circulations) are maintained. Perhaps the energetics relationship between atmospheric divergent and
rotational flows developed by Chen and Wiin-Nielsen
(1976) may constitute the dynamic basis to answer this
question. They demonstrated that the available potential energy generated by the north–south differential
heating is released through the global divergent circulation to maintain atmospheric rotational flow. It is inferred from Chen and Wiin-Nielsen’s (1976) energetics
cycle that the divergent circulation should be maintained by diabatic heating. Combining the continuity
and thermodynamic equations, Chen and Yen
(1991a,b) formulated the ␹-maintenance equation:
冋 冉
␹ ⯝ ⵜ⫺2
⭸ 1
V ⭈ ⵱T
⭸p ␴
冊册
␹H
冋 冉 冊册
⫹ ⵜ⫺2
⭸ ⫺1
Q̇
⭸p ␴Cp
˙
␹Q
,
共9兲
where ␴, V, T, Cp, and Q̇ are static stability, wind vector, temperature, specific heat with constant pressure,
and diabatic heating, respectively.
According to Eq. (9), velocity potential is maintained
by vertical differentiation of thermal advection and diabatic heating. Heat budget analyses of previous studies
(e.g., Wei et al. 1983; Chen and Baker 1986; Kasahara
et al. 1987; Rodwell and Hoskins 2001; and others)
showed that
0 ⯝ ⫺ V ⭈ ⵱T ⫹ ␴␻ ⫹
1
Q̇,
cp
Mid–high latitudes
共10a兲
1
0 ⯝ ␴␻ ⫹ Q̇.
cp
Tropics
共10b兲
This difference of atmospheric thermodynamics between these two latitudinal zones suggests that velocity
potential in the Tropics is maintained primarily by diabatic heating and in mid–high latitudes by both heat
advection and diabatic heating. To support this inference, eddy components of ␹, ␹Q̇, and ␹H at 200 mb are
shown in Fig. 18. Responses of the global divergent
circulation to different thermal forcings in mid–high
latitudes and the Tropics can be revealed from ratios
RQ̇ ⬅ Var[␹Q̇E (200mb)]/Var[␹E (200mb)] and RH ⬅
VOLUME 62
Var[␹HE (200mb)]/Var[␹E (200mb)]: RQ̇ ⫽ 90% in the
Tropics (south of 30°N), while RQ̇ ⫽ 51% and RH ⫽
49% in mid–high latitudes (30°–60°N). Although diabatic heating is the most important factor in maintaining the divergent circulation, heat advection cannot be
neglected in mid–high latitudes.
It has long been recognized that stationary waves in
mid–high latitudes are developed by orography and
land–sea contrast (Held et al. 2002). In the Tropics, it is
indicated by Streten and Zillman’s (1984) schematic
diagram of the Walker circulation at the equator superimposed with the upper- and lower-level isobars that
tropical stationary waves are maintained by diabatic
heating through the Walker circulations. This maintenance mechanism of tropical stationary waves is consistent with Krishnamurti et al.’s (1973) spectral energetics of tropical ultralong waves at 200 mb. In contrast,
the present study offers a different perspective of this
maintenance mechanism in terms of the following relationships:
1) the Q̇–␹ relationship: ␹-maintenance equation [Eq.
(9)], and
2) the ␹–␺ relationship: ␺-maintenance equation [Eqs.
(7) and (8)].
The approach adopted by this study does not provide a
direct response of stationary waves to orography and
land–ocean contrast, as the linear stationary wave models. Although studies using these types of models treat
individual forcings as linearly independent, it was
pointed out by Held et al. (2002) that diabatic heating
is nonlinearly affected by orography. As inferred from
the heat budget equation, diabatic heating Q̇ is modulated by orography and land–ocean contrast through
the surface ␻ field, transient heat flux along storm
tracks, and sensible heat advection channeled by these
two topographic factors. Actually, effects of orography
and land–ocean contrast are included in diabatic heating through these thermodynamic processes. In addition, the surface ␻ field induced by orography and
land–ocean contrast dynamically affects the atmospheric flow through vortex stretching. The goal of this
study is not to explore how stationary waves are generated by orography and diabatic heating. Instead, this
study applies the streamfunction budget analysis to obtain the basic dynamics of stationary waves in the Tropics and mid–high latitudes and explain three identified
features of stationary waves in section 2.
b. Major findings
Applying a harmonic analysis of the simplified budget equation, we explained four distinct features of stationary waves observed by Lau (1979) and the present
OCTOBER 2005
3657
CHEN
FIG. 18. Eddy components of all three terms in the ␹-maintenance equation: (a) [␹E (200 mb), P], (b)
(␹Q̇E, Q̇) (200 mb), and (c) ␹HE (200 mb). Here P and Q̇ are precipitation and diabatic heating,
respectively. The contour interval of (␹E, ␹Q̇E, ␹HE) (200 mb) is 2 ⫻ 105 m2 s⫺1.
study. Major results of this effort may be summarized as
follows.
1) A
QUARTER-PHASE RELATIONSHIP BETWEEN
STATIONARY WAVES AND EAST–WEST
CIRCULATIONS
As revealed from the upper-tropospheric circulation
(e.g., Schubert et al. 1990), easterlies in the Tropics
yield to strong westerlies in the midlatitudes. This
northward transition of atmospheric flow leads to a fundamental change in the dynamics of stationary waves.
In the Tropics, the easterly flow allows the dynamics of
stationary waves dictated by the Sverdrup vorticity balance. This balance results in a longitudinally quarter
phase of the east–west circulation coupled with ␹ n behind the stationary wave depicted by ␺ n. Because of
strong westerlies in midlatitudes, the relative vorticity
advection overpowers the effect of meridional planetary vorticity advection. Consequently, the dynamics
of stationary waves in mid–high latitudes belong to the
Rossby regime. This dynamics transition of stationary
waves accompanies a reversal of the relationship between the stationary waves and the east–west circulation. This relationship change is reflected by a longitudinally quarter phase of ␹ n ahead of the stationary wave
of ␺ n.
2) LONGITUDINAL
PHASE CHANGE
The meridional phase structure of ␹ n from the Tropics to mid–high latitudes does not show any dramatic
change. That is, the east–west circulation in the mid–
high-latitude region and the Tropics exhibit a more or
3658
JOURNAL OF THE ATMOSPHERIC SCIENCES
less uniform phase. However, the dynamics transition
of stationary waves from the tropical Sverdrup regime
to the mid–high-latitude Rossby regime results in a longitudinal phase change of stationary waves across 30°N
where the midlatitude jet cores exist.
3) A
VERTICAL PHASE REVERSAL IN THE
WESTWARD VERTICAL TILT IN MID–HIGH
LATITUDES
In addition to a transition from the tropical easterlies
to midlatitude westerlies, the atmospheric circulation in
midlatitudes also faces a transition from weak westerlies in the lower troposphere to strong westerlies in the
upper troposphere. These transitions of zonal winds in
the midlatitudes result in the dynamics transition of
stationary waves from the Sverdrup regime in the lower
troposphere to the Rossby regime in the upper troposphere. This transition is reflected by a vertical phase
reversal in the Tropics to a westward vertical tilt in the
mid–high latitudes, although this special feature of stationary waves has long been regarded as a result of
poleward sensible heat transport (Holton 2004).
c. Remarks
Relationships between the east–west circulations and
stationary waves in the mid–high latitudes and the
Tropics, and the transition of these relationships between two latitudinal zones were not fully understood
by previous studies. Thus, the structure and dynamics
of stationary waves in these two latitudinal regions
were often explored independently. Major findings of
this study enhance our understanding of the atmospheric general circulation and facilitate our search for
causes of weather and climate events related to stationary waves. Some examples are given below.
1) DEPICTION
ever, it was demonstrated in this study that the dynamical structure of the atmospheric circulation in both the
mid–high latitudes and the Tropics can be well illustrated by ␺ and ␹. In other words, the atmospheric
circulation can be alternatively portrayed by these two
variables.
TROPICS
The heat advection of large-scale motion in the Tropics is negligible in the tropical heat budget. This special
thermodynamic characteristic enables the tropical circulation to develop a deep convection regime within
the global circulation system. Consequently, velocity
potential ␹ n of this convection regime exhibits a vertical
phase reversal. Because the basic dynamics of tropical
stationary waves are depicted by the Sverdrup vorticity
balance, these waves therefore also undergo a vertical
phase reversal following velocity potential.
4) A
VOLUME 62
OF THE ATMOSPHERIC CIRCULATION
The atmospheric general circulation (e.g., Lorenz
1967) is generally depicted by dynamic (e.g., wind) and
thermodynamic variables (e.g., temperature). How-
2) TROPICAL–EXTRATROPICAL
INTERACTION
The tropical–extratropical interaction may be reflected by special weather events (e.g., cold surges in
East Asia; Lau and Chang 1987), and special climate
events (e.g., the impact of ENSO on the North America
weather) through a teleconnection wave train (Hoskins
and Karoly 1981). These interactions were illustrated
by previous studies primarily with changes of stationary
waves and divergent circulation in response to anomalous forcings without a clear perspective of the dynamics transition of stationary waves between the mid–high
latitudes and the Tropics. The disclosure of relationships between stationary waves and the east–west circulation and the dynamics transition of stationary
waves between two latitudinal zones will be of use to
search for a better understanding of the tropical–
extratropical interaction.
3) CLIMATE
SIMULATION
Global climate models are often used as a tool to test
various hypotheses for the possible impacts of anomalous forcings on the global climate system. As pointed
out above, the response of the global climate system to
these forcings may be reflected by changes in stationary
waves. For this reason, proper simulation of these distinct features of stationary waves would be crucial to
the climate variability prediction (WCRP 1997). In
spite of its cause, global climate modeling always faces
model bias. Distinct features of stationary waves are an
important means to validate global climate modeling.
Acknowledgments. This study is supported by the
NSF Grant ATM-0434798. Computational assistance
provided by Simon Wang and Paul Tsay were vital to
the success of this study. Typing support by Judy Huang
and editing assistance by Dave Flory are highly appreciated. Comments of two anonymous reviewers and Dr.
Peitao Peng were helpful in improving the presentation
of this paper.
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