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Dynamical Relationship between the Phase of North Atlantic Oscillations and the Meridional Excursion of a Preexisting Jet: An Analytical Study DEHAI LUO
AND
TINGTING GONG
Physical Oceanography Laboratory, College of Physical and Environmental Oceanography, Ocean University of China, Qingdao, China
LINHAO ZHONG LACS, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing, China (Manuscript received 10 July 2007, in final form 6 November 2007) ABSTRACT In this paper, it is shown from an analytical solution that in the presence of a preexisting jet the interaction between the zonal jet and the topography of the land–sea contrast (LSC) in the Northern Hemisphere (NH) tends to induce a dipole component that depends crucially upon whether this zonal jet exhibits a north– south excursion. This phenomenon cannot be observed if the zonal jet has no north–south shift. When the preexisting jet is located more northward (southward), the induced dipole can have a low-over-high (highover-low) structure and thus can make the center of the stationary wave anomaly shift southward (northward), which can be regarded as an initial state or embryo of a positive (negative) phase North Atlantic Oscillation (NAO). This dipole component can be amplified into a typical NAO event under the forcing of synoptic-scale eddies. To some extent, this result provides an explanation for why the positive (negative) phase of the NAO can be controlled by the northward (southward) shift of the zonal jet prior to the NAO. In addition, the impact of the jet shift on the occurrence of NAO is examined in a weakly nonlinear NAO model if the initial state of an NAO is prespecified. It is found that the northward (southward) shift of a zonal jet favors the occurrence of the subsequent positive (negative) phase NAO event and then results in a northward (southward)-intensified jet relative to the preexisting jet. In addition, during the decaying of the positive phase NAO, a strong blocking activity is easily observed over Europe as the jet is moved to the north.
1. Introduction Recently it has been recognized that there is a possible link between the North Atlantic Oscillation (NAO) or Arctic Oscillation (AO), a dominant dipole mode in the Northern Hemisphere (NH), and the NH surface temperature, which has important implications in understanding the cause of global warming (Hurrell 1995, 1996; Thompson and Wallace 1998; Thompson et al. 2000). Thompson et al. (2000) noted that the observed positive trend in the NAO/AO index over the last three decades of the last century significantly contributed to the observed warming trend over Eurasia
Corresponding author address: Dr. Dehai Luo, College of Physical and Environmental Oceanography, Ocean University of China, Qingdao 266003, China. E-mail:
[email protected] DOI: 10.1175/2007JAS2560.1 © 2008 American Meteorological Society
and North America. Although the NAO is a natural mode of atmospheric variability (Feldstein 2003; Vallis et al. 2004), its close relationship with warming trends makes it difficult to distinguish whether the positive trend of the NAO at the end of the twentieth century is due to natural variability or due to anthropogenically forced warming. The NAO trend has been found to be related to ocean variability (Rodwell et al. 1999) or greenhouse gas concentrations (Shindell et al. 1999). However, recent observations show that the decadal trend of the NAO has reversed over the past several years and its link with global warming is not obvious (Cohen and Barlow 2005; Overland and Wang 2005). This means that the factor that is crucial for driving NAO trends is not clearly identified. Wittman et al. (2005) found in a simple stochastic model that a dipolar meridional structure of the NAO/AO is likely due to variations in the
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north–south position of the jet. However, why the phase of NAO depends on the meridional displacement of the jet is not sufficiently understood. The purpose of this paper is to provide a simple explanation for why the northward (southward) shift of a preexisting jet can excite the positive (negative) phase of the NAO by presenting an analytical solution of topographically forced stationary waves in a specified jet. It is also natural that anticyclonic (cyclonic) wave breaking is closely related to the northward (southward) displacement of a jet prior to the NAO. This paper is organized as follows: In section 2, the meridional structure of an observed jet prior to the NAO is described. In section 3 we present an analytical solution of topographically forced stationary waves in a specified jet, and find that the northward (southward) shift of the zonal jet tends to produce a dipole component with a low-over-high (high-over-low) anomaly over the North Atlantic through the interaction with the topography of the land–sea contrast (LSC). This dipole component can be regarded as the initial state of subsequent positive (negative) phase NAO events. In addition, a comparison with the numerical solutions is also discussed in this section. A weakly nonlinear NAO model established by Luo et al. (2007c) is extended in section 4 to include the role of a specified jet. In section 5 we discuss how the south–north displacement of a preexisting jet influences the life cycle of positive (negative) phase NAO events. Section 6 presents the relationship between the preexisting jet and subsequent jet. Conclusions and a discussion are summarized in section 7.
2. Meridional profile of Atlantic jet prior to the NAO The dataset used here is the daily mean, multilevel wind fields from the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR). To clearly see the meridional distribution of an Atlantic jet prior to the NAO, 20 negative and 10 positive phase events are chosen based upon the daily NAO index defined by Benedict et al. (2004). For each phase, the Atlantic jet prior to the NAO is defined as a time mean of zonal winds over the North Atlantic for 30 days before each NAO event begins, which is obtained through filtering out waves with periods from 2 to 21 days. Figure 1 shows that there is a northward (southward) shift of a jet over the North Atlantic prior to the positive (negative) phase of NAO in the troposphere from the upper layer to the surface. However, what causes the north–south excursion of the preexisting jet is unclear and is beyond the scope of this paper. As we will demonstrate in the next section, the north–south dis-
placement of the preexisting Atlantic jet is crucial for the phase of the subsequent NAO.
3. Asymptotic solution of linear topographically forced Rossby modes and a link to the meridional shift of a preexisting jet a. Asymptotic solution In a beta-plane channel with both a width of Ly and a barotropic background flow (u) having a horizontal shear, the nondimensional linearized barotropic vorticity equation that describes a stationary wave anomaly (A) forced by the large-scale topography in mid–high latitudes can be written as u
⭸ 2 ⭸h ⭸A ⵜ A ⫹ 共 ⫺ u⬙兲 ⫹u ⫽ 0, ⭸x ⭸x ⭸x
共1兲
where u⬙ ⫽ 2u/(y2), ⵜ2 ⫽ 2/(x2) ⫹ 2/(y2),  is the nondimensional meridional gradient of the Coriolis parameter, h is a nondimensional topography, and the boundary condition A/(x) ⫽ 0 at y ⫽ 0, Ly is satisfied. As a highly idealized case, the preexisting jet is assumed to be of the form u ⫽ u0 ⫺ ␣1 cos共my兲 ⫹ ␣2 cos
冉 冊
m y , 2
共2兲
where m ⫽ ⫾2/Ly, and u0 is a constant westerly wind; ␣1 is a constant and represents the strength of a barotropic jet, while ␣2 mainly denotes its meridional shift but in part reflects the jet strength. In the real atmosphere the zonal jet probably cannot be exactly represented by the mathematical expression given in (2), but can be decomposed into many meridional harmonic modes by using the Fourier expansion. In this case, this zonal jet can be reduced to (2) by neglecting modes higher than the first two modes. For example, for u0 ⫽ 0.85 and ␣1 ⫽ 0.24, u(y) is shown in Fig. 2 for ␣2 ⫽ 0, ␣2 ⫽ 0.24, and ␣2 ⫽ ⫺0.24, respectively. It is noted that this jet exhibits a west–east pulsing and no north–south shift for ␣2 ⫽ 0. However, a meridional shift of this jet is seen if ␣2 ⫽ 0. This zonal jet exhibits a southward shift for ␣2 ⫽ 0.24, but a northward displacement for ␣2 ⫽ ⫺0.24. Thus, the sign and magnitude of ␣2 can reflect the extent of the north– south wobbling of the jet, while the pulsing and bulging of the jet are mainly characterized by the value of ␣1. Although the jet profile shown in Fig. 2 cannot exactly correspond to the observed case, it can basically capture the main characteristics of an observed Atlantic jet prior to the NAO, as shown in Fig. 1. To obtain the analytical solution to (1), it is conve-
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FIG. 1. Meridional profile of the zonal jet at (a) 300 and (b) 850 mb and (c) the vertical mean over the North Atlantic between 300 and 850 mb for the 1-month mean prior to the positive (negative) phase of NAO. The solid and dashed curves denote the positive and negative phases, respectively.
nient to assume ␣i ⫽ ␣˜ i (i ⫽ 1, 2) for a small parameter satisfying 0 ⱕ K 1.0. Generally speaking, it is rather difficult to give the exact form of the topography of the LSC (hereafter LSC topography). However, the actual topography in the mid–high latitudes can be decomposed into zonal and meridional harmonic modes, in which the meridional distribution can be crudely represented by a sine curve (Hart 1979; Charney and DeVore 1979). In the zonal direction, zonal wavenumber 2
is dominant for this LSC topography. Thus, in the Northern Hemisphere the large-scale topography in mid–high latitudes that represents the LSC topography can be approximated as a topography with zonal wavenumber 2 and meridional monopole structure in a plane channel (Hart 1979; Charney and DeVore 1979) and in a spheric geometry (Legras and Ghil 1985). In this case, it is reasonable to assume h to be of the form of (Luo et al. 2007c)
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FIG. 2. Meridional profile of a specified jet, in which the dotted– dashed, solid, and dotted curves correspond in turn to ␣2 ⫽ 0, ␣2 ⫽ 0.24, and ␣2 ⫽ ⫺0.24.
h ⫽ h0 exp关ik共x ⫹ xT兲兴 sin共myⲐ2兲 ⫹ cc,
共3兲
where h0 is the topographic amplitude, xT is the position of the topographic trough relative to the NAO center if an NAO event is formed, k is the corresponding zonal wavenumber, and cc denotes the complex conjugate of the term preceding it. It is useful to expand A as
A ⫽ A0共x, y兲 ⫹ A1共x, y兲 ⫹ A2共x, y兲 ⫹ · · · . 2
共4兲 Substituting (3) and (4) into (1), it is easy to get the following approximate solution
A ⬇ A0 ⫹ A1 ⫽ A0 ⫹ A1C,
共5a兲
A0 ⫽ hAh0 exp关ik共x ⫹ xT兲兴 sin共myⲐ2兲 ⫹ cc, A1C ⫽ A11 ⫹ A12 ⫹ A13,
A11 ⫽ ⫺␣1
冋冉
2
⫺k2 ⫹
3m 4
冋 冉
2u0
冊
册 冊册
共5c兲
共5d兲
册
exp关ik共x ⫹ xT兲兴 sin共my兲 ⫹ cc,
⫺k2 ⫹
冊
册 冊册
3m2 hA ⫹ 1 4
冋 冉
9m2  ⫺ k2 ⫹ u0 4
h0
exp关ik共x ⫹ xT兲兴 sin共3myⲐ2兲 ⫹ cc,
h0
共⫺k2hA ⫹ 1兲 h0  2 2 2u0 ⫺ 共k ⫹ m 兲 u0
冋
A13 ⫽ ␣1
冋冉
2u0
exp关ik共x ⫹ xT兲兴 sin共myⲐ2兲 ⫹ cc,
A12 ⫽ ⫺␣2
共5b兲
hA ⫹ 1
m2  ⫺ k2 ⫹ u0 4
FIG. 3. (a) Distribution of the land–sea contrast topography, in which the solid and dashed curves represent the topographic ridge and trough, respectively, and topographically induced stationary wave anomaly for different uniform westerly winds (b) u0 ⫽ 0.85 and (c) u0 ⫽ 1.15, in which the solid and dashed curves represent positive and negative anomalies, respectively. The contour interval (CI) is 0.2.
共5e兲
共5f兲
where hA ⫽ ⫺1/[/u0 ⫺ (k2 ⫹ m2/4)]. Note that u0 is required to satisfy /u0 ⫺ (k2 ⫹ m2/4) ⫽ 0, /u0 ⫺ (k2 ⫹ m2) ⫽ 0, and /u0 ⫺ (k2 ⫹ 9m2/4) ⫽ 0 to ensure that the above analytical solutions presented in (5) are correct. It is found that in a jet with a horizontal shear the approximate solution for a planetary wave anomaly induced by the LSC topography exhibits a rather complicated spatial structure whose flow pattern depends
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FIG. 4. Anomaly fields A and A12 of topographically induced stationary wave components in a specified jet (CI ⫽ 0.1) in which the dashed and solid lines denote the negative and positive anomalies, respectively: (a) northward-shifting jet and (b) southward-shifting jet.
strongly on the amplitude of each component, as given in (5a)–(5f). An interesting point is that A12 can have a dipole structure as the preexisting jet exhibits a meridional displacement. This shows that the meridional excursion of the jet plays an important role in the excitation of the dipole mode through an interaction with the LSC topography in the NH.
b. Spatial structures of topographic stationary modes and meridional shift of a preexisting jet It is further shown that when the mean westerly wind u0 is weaker and is near the resonant westerly wind uc ⫽ /k2 ⫹ m2, A12 can become dominant compared to A11 and A13, but the solution of equations is incorrect if u0 ⫽ uc is satisfied. This situation will be avoided if u0 is somewhat far from the value of uc. For example, for u0 ⫽ 0.85, both A11 and A13 may be negligible because they are rather small, but A12 is significant. In a uniform westerly wind (␣i ⫽ 0, where i ⫽ 1, 2), the topographic planetary wave anomaly can only exhibit a monopole structure because there is only A ⫽ A0. A dominant dipole anomaly A12 tends to be excited through the interaction between the jet and LSC topography for a weak westerly wind u0 as the jet departs meridionally from its mean position (␣2 ⫽ 0). Nevertheless, both A11 and A13 can become more important as u0 increases further, because /u0 ⫺ (k2 ⫹ m2/4) and /u0 ⫺ (k2 ⫹ 9m2/4) become smaller. In any case, a dipole anomaly A12 can be excited by the LSC topography when the zonal jet undergoes a north–south excursion.
Because the LSC topography is dominated by zonal wavenumber 2 and because an NAO anomaly has a wavenumber 2 structure (Wallace 2000), we can choose k ⫽ 2/[6.371 cos(55°N)] in the whole paper (Luo et al. 2007a,b,c; Luo et al. 2008). In addition, xT ⫽ 0, h0 ⫽ 0.4, m ⫽ ⫺2/Ly, and Ly ⫽ 5 are fixed in this section, but u0 and ␣i (i ⫽ 1, 2) are allowed to vary to represent different cases of the zonal jet. Here, the LSC topography and associated topographically forced wave (A ⫽ A0) in a uniform westerly wind (␣i ⫽ 0, where i ⫽ 1, 2) are shown in Fig. 3 for u0 ⫽ 0.85 and u0 ⫽ 1.15. It is evident that when the background westerly wind is uniform, the LSC topography tends to induce a planetary wave with a meridional monopole structure as noted above, whose amplitude is dependent upon the setting of this uniform westerly wind. For u0 ⫽ 0.85 and ␣1 ⫽ 0.24, A and A12 are shown in Fig. 4 for ␣2 ⫽ 0.24 and ␣2 ⫽ ⫺0.24. It is noted that when the jet exhibits a northward (southward) shift, the positive anomaly center of A can be in the south (north) side of the ocean (North Atlantic), reflecting that A12 can have a low-over-high (high-overlow) dipole. Of interest is that A12, which has a dipole meridional structure, is dominant, in that both A11 and A13 are negligible for u0 ⫽ 0.85. The spatial structure of this dipole is found to depend upon the direction of the jet departing from its mean position. When the jet has a poleward (equatorward) displacement this dipole anomaly A12 exhibits a low-over-high (high-over-low) anomaly in the North Atlantic. Once this dipole anomaly interacts with synoptic-scale eddies upstream,
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FIG. 5. As in Fig. 4 but for anomaly fields A, A11, and A12.
it will be amplified into a positive or negative phase NAO (NAO⫹ or NAO⫺ hereafter) anomaly (Luo et al. 2007a). This can, to some extent, explain why the poleward (equatorward) displacement of the zonal mean jet is a precursor of NAO⫹ (NAO⫺) events, as noted in section 2. Of course, it is also important that the eddy fluxes from preexisting synoptic-scale eddies should have a matching spatial structure to provide a potential for the NAO occurrence (Luo et al. 2007a). Figure 5 shows the anomaly fields A, A11, and A12 of the topographically induced stationary wave for u0 ⫽ 1.15. For this case, A13 is not shown here because its
amplitude is rather weak. It is found that the amplitude of A for u0 ⫽ 1.15 is evidently larger than that for u0 ⫽ 0.85 as shown in Fig. 4, and at the same time A12 becomes weaker even though A11 can have a large amplitude. Thus, the setting of u0 does not affect the sign of the spatial structure of A12 unless it is particularly large. Certainly, the amplitude of A12 is also dominated by the extent of the north–south excursion of the jet. The results found in this section indicate that a dipole anomaly can be excited by the interaction between the jet and the LSC topography when the jet departs from
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FIG. 6. Anomaly field NAO of an eddy-driven positive phase NAO event (CI ⫽ 0.2): (a) symmetric jet (␣2 ⫽ 0), (b) northward-shifting jet (␣2 ⫽ ⫺0.18), and (c) southward-shifting jet (␣2 ⫽ 0.18).
its mean position. This dipole component due to the meridional wobbling of the jet can provide an initial state of the NAO anomaly although Wittman et al. (2005) noted in a stochastic model that the occurrence of a meridional dipole is a simple consequence of the north–south excursion of the jet center. More recently, Luo et al. (2008) found that anticyclonic wave breaking (AWB) or cyclonic wave breaking (CWB) tend to accompany the occurrence of an NAO⫹ or NAO⫺ event when an initial state with a low-over-high or high-over-
low dipole is prespecified. Thus, it is not difficult to conclude that the AWB (CWB) is more likely to be naturally linked with the preexisting northward (southward) shift of a zonal jet.
c. A comparison with numerical solutions in a beta plane and on sphere For the jet and topography given in (2) and (3), the numerical solution to (1) can be obtained by solving a
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FIG. 6. (Continued)
Poisson equation using a finite-difference method if A/(x) ⫽ Ax is defined in (1). It is found that for a northward (southward) jet prior to the NAO⫹ (NAO⫺), the center of the topographically induced stationary wave anomaly tends to be in the south (north) side of the ocean (not shown), which is consistent with both the analytical solution A presented in (5a) and the observational characteristics of stationary wave anomalies for two phases of the NAO obtained by Luo et al. (2007c, their Fig. 2). This result also holds for spherical geometry, the treatment for which is given in the appendix.
4. Generalized weakly nonlinear NAO model It needs to be pointed out that when the preexisting planetary wave is stationary, u0 ⫽ /(k2 ⫹ m2) will be satisfied so that the solution (5e) is invalid. This difficulty cannot be avoided unless some assumptions are made. To analytically treat the role of the jet shift in the NAO life cycle, we assume ␣i ⫽ 2␣i to avoid the above difficulty in treating the nonlinear evolution of NAO events. This is because the terms induced by the interaction between the jet shift and the topography
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FIG. 6. (Continued)
can only appear in the higher-order equation if ␣i ⫽ 2␣i is used. Such an assumption can allow us to obtain the weakly nonlinear asymptotic solutions of the NAO life cycle.
冉
As in Luo et al. (2007c), the nondimensional planetary () and synoptic (⬘) wave equations in a sheared basic flow u(y) can be obtained as
冊
⭸ ⭸h ⭸ ⭸ ⫹u 共ⵜ2 ⫺ F兲 ⫹ J共, ⵜ2 ⫹ h兲 ⫹ 共 ⫹ F u ⫺ u⬙兲 ⫹ u ⫽ ⫺J共⬘, ⵜ2⬘兲P, ⭸t ⭸x ⭸x ⭸x
冉
冊
⭸ ⭸ ⭸⬘ ⫹u ⫽ ⫺J共⬘, ⵜ2 ⫹ h兲 ⫺ J共, ⵜ2⬘兲 ⫹ ⵜ2* 共ⵜ2⬘ ⫺ F⬘兲 ⫹ 共 ⫹ F u ⫺ u⬙兲 S, ⭸t ⭸x ⭸x
共6a兲 共6b兲
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where and ⬘ are the planetary-scale anomaly and synoptic-scale field, respectively, *S is the synopticscale vorticity source, u⬙ ⫽ 2u/(y2), J is the Jacobian operator, and h is a nondimensional topographic variable; the other notation and boundary conditions used can be found in Luo et al. (2007a,b,c). Note that (6) reduces to the planetary- and synoptic-scale interaction equations derived by Luo et al. (2007c) if both u is a constant and u ⫽ u0 is allowed. Here h is chosen to be the same as (3). For ␣i ⫽ 2␣i in (2), following Luo et al. (2007c), the planetary- and synoptic-scale solutions of the NAO event and the generalized NAO equation can be obtained as
C ⫽ hAh0 exp关ik共x ⫹ xT兲兴 sin
冉 冊
m y ⫹ cc, 2
m ⫽ m1 ⫹ m2,
共7e兲
⬁
m1 ⫽ ⫺| B | 2
兺q g
n n
cos共n ⫹ 1Ⲑ2兲my,
冑
2 共Be⫺ikxT ⫹ B*eikxT兲 Ly
m2 ⫽ ⫺h0hA ⬁
兺 q˜ 共3a n
n
⫺ bn兲 cos共nmy兲,
⬘ ⬇ 3Ⲑ2共˜ ⬘0 ⫹ ˜ ⬘1兲 ⫽ ⬘1 ⫹ ⬘2,
w ⫽ ⫺u0 y ⫹ NAOA ⫹ C,
共7b兲
⬘1 ⫽ f0共x兲兵 exp关i共k˜1x ⫺ ˜ 1t兲兴
2 exp共ikx兲 sin共my兲 ⫹ cc, Ly
⬘2 ⫽ ⫺
m 4
⫺␣
冉 冊
⫹ ␣ exp关i共k˜2x ⫺ ˜ 2t兲兴其 sin
共7c兲
m y ⫹ cc, 2
共7h兲
共7i兲
冋 冉 冊 冉 冊册 冋 冉 冊 冉 冊册 冑 冋 冉 冊 冉 冊册 冑 冉 冊册
2 3m m Q Bf exp 兵i 关共k˜1 ⫹ k兲x ⫺ ˜ 1t兴其 p1 sin y ⫹ r1 sin y Ly 1 0 2 2
m 4
2 3m m Q2Bf0 exp 兵i 关共k˜2 ⫹ k兲 x ⫺ ˜ 2t兴其 p2 sin y ⫹ r2 sin y Ly 2 2
⫻ exp 兵i 关共 k˜1 ⫺ k兲x ⫺ ˜ 1t兴其 ⫻ s1 sin
冋 冉 冊
⫻ s2 sin
3m m y ⫹ h2 sin y 2 2
⫺
3m m y ⫹ h1 sin y 2 2
⫹␣
m 4
⫹
m 4
2 Q B*f Ly 1 0
2 Q B*f exp 兵i 关共k˜2 ⫺ k兲x ⫺ ˜ 2t兴其 Ly 2 0
m m f h exp 兵i 关共k˜1 ⫹ k兲 x ⫹ kxT ⫺ ˜ 1t兴其 sin共my兲 ⫺␣ f0 h02 4 0 0 1 4
⫻ exp 兵i 关共k˜2 ⫹ k兲x ⫹ kxT ⫺ ˜ 2t兴其 sin共my兲 ⫺
m m f h exp 兵i 关共k˜1 ⫺ k兲x ⫺ kxT ⫺ ˜ 1t兴其 sin共my兲 ⫺ ␣ f0 h02 4 0 0 1 4
⫻ exp 兵i 关共k˜2 ⫺ k兲 x ⫺ kxT ⫺ ˜ 2t兴其 sin共my兲 ⫹ cc,
共7j兲
⭸B ⭸B ⭸B ⫹ 2 ⫹ ␦ |B| 2B ⫹ ␣˜ h20共B ⫹ B*ei2kxT兲 ⫹ ␥␣2h0 exp共ikxT兲 ⫹ Gf 20 exp关⫺i共⌬kx ⫹ ⌬t兲兴 ⫽ 0, ⫹ Cg ⭸t ⭸x ⭸x
共7k兲
冉
i
共7g兲
n⫽1
共7a兲
冑 冑 冑
共7f兲
n⫽1
P ⬇ ⫺ u0 y ⫹ ⫹ m ⫽ w ⫹ m,
NAOA ⫽ B
共7d兲
冊
2
where hA ⫽ ⫺1/[/u0 ⫺ (k2 ⫹ m2/4)], ␥ ⫽ k(⫺k2hA ⫹ 1)/[2(k2 ⫹ m2 ⫹ F )]公Ly/2], P is the planetary-scale field, B is the amplitude of the NAO anomaly NAOA, 2 2 ␣ ⫽ ⫾ 1, m ⫽ ⫾ 2/Ly, f0 ⫽ a0e⫺ (x ⫹ x0) in which a0 is the eddy amplitude, ⬎ 0, and x0 is the position of the eddy activity center relative to the NAO; the other notation can be found in Luo et al. (2007c). Here h0 ⫽ ⫺0.4, m ⫽ 2/Ly and h0 ⫽ 0.4, m ⫽ ⫺2/Ly are allowed to represent a topographic trough (ocean) in that the positive and negative phases of the NAO require m ⫽ 2/Ly and m ⫽ ⫺2/Ly, respectively. Here T ⫽ P ⫹ ⬘ is defined as a total field of an NAO event. It is found that when the jet has no displacement from its mean position, that is to say, when ␣2 ⫽ 0,
(7k) becomes identical to the forced nonlinear Schrödinger (NLS) equation that governs the evolution of an NAO anomaly forced jointly by synopticscale eddies and LSC topography derived by Luo et al. (2007c). Thus, (7k) can be considered as an extension of the previously obtained forced NLS equation. If the initial states of the planetary and synoptic waves prior to the NAO are specified, one can predict the life cycle of an NAO event by solving (7k) when the preexisting jet undergoes a north–south shift. It seems that the solutions (7a)–(7k) are not directly related to the solution (5e). In fact, the dipole component described by solution (5e) is assumed to provide an initial state of the NAO represented by (7), but the
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FIG. 7. As in Fig. 6 but for the total field (CI ⫽ 0.3).
amplitude evolution of the subsequent NAO is still affected by the meridional shift of the zonal jet during its life cycle.
5. Life cycle of an NAO event and its link to the north–south shift of a jet In this section, ␣i is relaxed to be large although the strict assumption of ␣i ⫽ 2␣i may be violated. In addition, the parameters ␣1 ⫽ 0.18, a0 ⫽ 0.17, ⫽ 1.2, F ⫽ 1, and x0 ⫽ 2.87/2 are fixed, and without the loss of generality B(x, 0) ⫽ 0.4 is chosen as an initial amplitude
of the NAO anomaly. The other parameters of synoptic-scale eddies are chosen the same as in Luo et al. (2007c). Such a treatment can allow us to conveniently discuss the impact of the strong north–south excursion of the jet on the phase of the NAO.
a. NAO⫹ event and meridional shift of a preexisting jet The aim of this section is to look at the influence of the jet shift on the NAO occurrence if the initial state of the NAO anomaly is preexisting.
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FIG. 7. (Continued)
For three cases of ␣2 ⫽ 0, ␣2 ⫽ ⫺0.18, and ␣2 ⫽ 0.18, the anomaly field NAOA of an eddy-driven NAO⫹ event (␣ ⫽ 1, m ⫽ 2/Ly) is shown in Fig. 6 for xT ⫽ ⫺2 and h0 ⫽ ⫺0.4. It is found that for a symmetric jet an isolated NAO⫹ anomaly with a low-over-high dipole can be formed from a prespecified low-over-high anomaly under the forcing of synoptic-scale eddies (Fig. 6a). Once the zonal jet exhibits a meridional displacement, the northward shift of the jet will increase the amplitude of the NAO⫹ anomaly and strengthen its
downstream energy dispersion during the decaying process (Fig. 6b), but the southward shift of the jet will play a reverse role (Fig. 6c). Thus, the northward displacement of a preexisting jet is a favorable environment for the occurrence of the NAO⫹ anomaly and strong European blocking events. In a synoptic weather map the CWB is characterized by the poleward shift of warm air and the southward intrusion of cold air (Benedict et al. 2004), but a reverse movement of air characterizes the AWB. As seen in the total field in Fig. 7, the AWB
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FIG. 7. (Continued)
seems to accompany the occurrence of an NAO⫹ event, and the CWB occurs concurrently over Europe during the decaying phase (Fig. 7a). When the preexisting jet is located more northward, both the AWB in the Atlantic sector and subsequent CWB over the European continent are enhanced, and a strong European blocking is also observed (Fig. 7b). However, if the zonal jet is moved to the south, the NAO⫹ anomaly is strongly suppressed and no blocking event is seen over the European continent (Fig. 7c). Thus, the northward dis-
placement of a preexisting jet is conducive to the occurrence of NAO⫹ events, AWB, and European blocking events.
b. Impact of the meridional shift of a preexisting jet on the NAO⫺ event Observations show that the center of action of the NAO⫺ event is generally located more eastward than the region where the NAO⫹ event is (Benedict et al. 2004). As a result, we choose xT ⫽ ⫺1 and h0 ⫽ 0.4 to
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FIG. 8. Anomaly field NAOA of an eddy-driven negative phase NAO event (CI ⫽ 0.2): (a) symmetric jet (␣2 ⫽ 0), (b) southward-shifting jet (␣2 ⫽ 0.18), and (c) northward-shifting jet (␣2 ⫽ ⫺0.18).
discuss the evolution of an NAO⫺ event (␣ ⫽ ⫺1, m ⫽ ⫺2/Ly). Figure 8 shows the anomaly fields of NAOA for ␣2 ⫽ 0, ␣2 ⫽ 0.18, and ␣2 ⫽ ⫺0.18. It is found that for a symmetric jet an NAO⫺ event can be produced from an initial state with a high-over-low dipole through the forcing of synoptic-scale eddies (Fig. 8a). This NAO⫺ anomaly is enhanced as the zonal jet moves to the south (Fig. 8b), but weakened as the zonal jet moves to the north (Fig. 8c). At the same time, the decay of an NAO⫺ anomaly is also affected by the meridional shift of the preexisting jet. A prominent
CWB is also seen in the total field (Fig. 9). As indicated by Luo et al. (2008), the type of synoptic wave breaking is controlled by the spatial structure of the initial planetary wave that determines the phase of the NAO. Thus, the CWB is essentially linked with the negative phase of NAO. When the preexisting jet shows a southward displacement, the CWB is enhanced, thus naturally concluding that the equatorward shift of an Atlantic jet is a favorable environment for the CWB (Fig. 9b). In contrast, the CWB is strongly suppressed once the jet moves to the north (Fig. 9c).
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FIG. 8. (Continued)
6. Positive feedback between a preexisting jet and its subsequent evolution As noted in section 4, the poleward (equatorward) displacement of a zonal jet prior to the NAO implicates the more likely occurrence of subsequent NAO ⫹ (NAO⫺) events. More recently, Luo et al. (2007c) found that an intensified jet can be established and can move to the north (south) during the life cycle of an NAO⫹ (NAO⫺) event. Thus, it is not difficult to conclude that there is a positive feedback between a preexisting jet and the subsequent intensified jet during the
NAO life cycle. In other words, the northward (southward) shift of the zonal jet prior to the NAO implicates the likely emergence of a more northward (southward) intensified jet associated with an NAO⫹ (NAO⫺) event.
7. Conclusions and discussion In this paper, we have presented an analytical solution of topographically forced stationary modes in a specified jet to clarify why the poleward (equatorward)
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FIG. 8. (Continued)
shift of an Atlantic jet before an NAO event begins is more likely to excite positive (negative) phase NAO events. It is shown that a dipole component can arise from the interaction between a zonal jet and the LSC topography as the zonal jet has a north–south excursion. At the same time, the center of the topographically induced stationary wave anomaly shifts toward the north (south) as the preexisting jet is moved to the south (north). The spatial structure of the topographically induced dipole component depends strongly upon whether a preexisting jet moves to the north or the
south. When the jet undergoes a northward (southward) shift, the induced dipole anomaly can have a low-over-high (high-over-low) structure over the North Atlantic, which can be regarded as an initial state or embryo of positive (negative) phase NAO events. As indicated in Benedict et al. (2004) and Luo et al. (2008), anticyclonic (cyclonic) wave breaking always accompanies the occurrence of positive (negative) phase NAO events. Thus, it is possible that during the life cycle of a positive (negative) phase NAO event the anticyclonic (cyclonic) wave breaking is inevitably enhanced when
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FIG. 9. As in Fig. 8 but for the total field (CI ⫽ 0.3).
the preexisting jet exhibits a northward (southward) shift. On the other hand, in the present work a weakly nonlinear NAO model proposed by Luo et al. (2007c) has been extended to examine the impact of the north–south shift of a specified jet on the NAO life cycle if the initial state of the NAO is preexisting. It is shown that the poleward (equatorward) shift of the jet prior to the NAO is indeed favorable for the
occurrence of both the positive (negative) phase NAO event and anticyclonic (cyclonic) wave breaking, thus confirming previous observational results (Benedict et al. 2004; Franzke et al. 2004; Rivière and Orlanski 2007). Simmons and Hoskins (1980) and Thorncroft et al. (1993) have shown that the type of wave breaking is very sensitive to the meridional shear of the basic state zonal winds. Lee and Feldstein (1996)
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FIG. 9. (Continued)
found in an aquaplanet general circulation model that the meridional shear of the zonal wind in the upper troposphere, rather than the barotropic shear as in Thorncroft et al. (1993), is a crucial factor for determining the type of wave breaking. Orlanski (2005) recently noted that the type of wave breaking is in part controlled by SST anomalies created by ENSO because of their direct influence on the low-level baroclinicity. More recently, Woollings
et al. (2008) conjectured that the wave breaking arises from the interaction between planetary and synoptic waves. Even so, we find here that the north–south excursion of the zonal jet prior to the NAO seems a crucial factor determining the type of wave breaking. However, what causes the north–south excursion of the zonal jet prior to the NAO is a very interesting problem and deserves further study. Although the results here are gained in a beta-plane
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FIG. 9. (Continued)
channel, they are also concluded to hold in a spherical geometry from numerical studies in the beta-plane channel model and in a spherical model performed by Charney and DeVore (1979) and Legras and Ghil (1985). Of course, it is impossible to get the analytical solutions presented in this paper if a spherical model is used. However, using the barotropic vorticity equation in a beta-plane channel can make it easier to obtain the analytical solutions.
Acknowledgments. The authors acknowledge support from the National Outstanding Youth Natural Science Foundation of China under Grant 40325016, the National Natural Science Foundation of China (4057016), the CityU Strategic Research Grant 7002231, and Taishan scholar funding. The authors thank two anonymous reviewers for their useful suggestions in improving this paper.
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APPENDIX Linear Topographically Forced Stationary Wave Equation on a Sphere and Its Numerical Solution The linearized topographically forced stationary wave equation on a sphere can be rewritten as
1 cos
冦冤
⫹
⭸
冉
冊
sin u ⭸2u cos ⫺ 2 ⭸ ⭸
冥
⭸ⵜ2A ⭸A ⫹u ⭸ ⭸
⭸A sin ⭸h ⫹ u ⫽ 0, ⭸ cos ⭸
冧 共A1兲
where ⫽ U/(2⍀Re), Re ⫽ 6371 km is the radius of the earth with an annular velocity of ⍀, U ⫽ 10 m s⫺1 is the characteristic velocity of atmospheric horizontal motion, and are the latitude and longitude, respectively, and the other notation is the same as that in a beta plane. As in Legras and Ghil (1985), the topography on a sphere is chosen to be
冉
h ⫽ 2h0 cos共2兲 sin
冊
⫺ S , N ⫺ S
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obtained by integrating A with respect to using a Fourier expansion. For u0 ⫽ 1.15 and ␣1 ⫽ 0.24, the topographically induced stationary wave anomaly is shown in Fig. A2 for ␣2 ⫽ 0, ␣2 ⫽ ⫺0.24, and ␣2 ⫽ 0.24. It is found that for a symmetric jet (␣2 ⫽ 0), the center of the topographically induced stationary wave anomaly is not in the middle part of the latitude band from 30° to 80°N because of the effect of spherical geometry. Instead, it is in the south of this latitude band. This feature is different from the result obtained in a beta-plane channel. However, as we will note below, on a sphere the characteristics of a topographically induced stationary wave anomaly depending upon the meridional shift of a zonal jet are basically consistent with the analytical and numerical solutions obtained in a beta-plane channel. This point is clear in Fig. A2. When the zonal jet is shifted to the north, the positive center of topographically induced stationary wave anomaly is in the more south of the latitude band in the ocean (trough) as well as a negative center seen in the north of the latitude band. However, when this jet is displaced toward the south, the positive center over the ocean is shifted to
共A2兲
where 0 ⱕ ⱕ 360° and S ⱕ ⱕ N. The zonal jet is assumed to be of the form
冋
u ⫽ u0 ⫺ ␣1 cos
册
冋
册
2共 ⫺ S兲 共 ⫺ S兲 ⫹ ␣2 cos , N ⫺ S N ⫺ S 共A3兲
where the notation in (A3) is the same as that in (1). Here, we fix the parameters S ⫽ 30°N, N ⫽ 80°N, ␣1 ⫽ 0.24, and h0 ⫽ 0.4. The horizontal distribution of the topography with zonal wavenumber 2 on sphere can be seen in Fig. A1. For the topography given in Fig. A1 and the prescribed parameters, a finite-difference method is used to obtain A through solving a Poisson equation that is satisfied by A ⫽ A/(). On this basis, A(, ) is
FIG. A1. Horizontal distribution of wavenumber 2 topography on a sphere (CI ⫽ 0.1), in which the solid and dashed lines represent the ridge and trough, respectively.
FIG. A2. Anomalies of topographically induced stationary waves on sphere for u0 ⫽ 1.15 and different jets (CI ⫽ 0.5), in which the solid and dashed lines correspond to the positive and negative anomalies, respectively: (a) symmetric jet (␣2 ⫽ 0), (b) northward-shifting jet (␣2 ⫽ ⫺0.24), and (c) southward-shifting jet (␣2 ⫽ 0.24).
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the north. The meridional shift of the center of the forced stationary wave anomaly reflects that a dipole component can be excited by the interaction between the jet shift and topography, and can be regarded as the initial states of NAO events. This conclusion also holds for other values of u0 (not shown). In conclusion, the stationary waves on the sphere are found to be basically consistent with those in a beta plane, and they do not propagate in the meridional direction. This reflects that the stationary waves in the real atmosphere can, to large extent, be captured by the analytical and numerical solutions in a beta plane. REFERENCES Benedict, J. J., S. Lee, and S. B. Feldstein, 2004: Synoptic view of the North Atlantic Oscillation. J. Atmos. Sci., 61, 121–144. Charney, J. G., and J. G. DeVore, 1979: Multiple flow equilibria in the atmosphere and blocking. J. Atmos. Sci., 36, 1205–1216. Cohen, J., and M. Barlow, 2005: The NAO, the AO, and global warming: How closely related? J. Climate, 18, 4498–4513. Feldstein, S. B., 2003: The dynamics of NAO teleconnection pattern growth and decay. Quart. J. Roy. Meteor. Soc., 127, 901– 924. Franzke, C., S. Lee, and S. B. Feldstein, 2004: Is the North Atlantic Oscillation a breaking wave? J. Atmos. Sci., 61, 145– 160. Hart, J. E., 1979: Barotropic quasi-geostrophic flow over anisotropic mountains. J. Atmos. Sci., 36, 1736–1746. Hurrell, J. W., 1995: Decadal trends in the North Atlantic Oscillation: Regional temperatures and precipitation. Science, 269, 676–679. ——, 1996: Influence of variations in extratropical wintertime teleconnections on Northern Hemisphere temperature. Geophys. Res. Lett., 23, 665–668. Lee, S., and S. B. Feldstein, 1996: Two types of wave breaking in an aquaplanet GCM. J. Atmos. Sci., 53, 842–857. Legras, B., and M. Ghil, 1985: Persistent anomalies, blocking and variations in atmospheric predictability. J. Atmos. Sci., 42, 433–471. Luo, D., A. Lupo, and H. Wan, 2007a: Dynamics of eddy-driven low-frequency dipole modes. Part I: A simple model of North Atlantic Oscillations. J. Atmos. Sci., 64, 3–38. ——, T. Gong, and A. Lupo, 2007b: Dynamics of eddy-driven
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