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  • View in gallery

    Transient evolution of (a) the vertically averaged zonal-mean zonal wind and (b) the meridional power spectrum of the zonal-mean zonal wind for the control run (Wc = 10, β = 0.28 − 0.3). Note that β is changed at day 2500. Contour intervals are (a) 0.5 and (b) 10−3. Zero lines are omitted. Shading corresponds to values that are greater than (a) 1 and (b) 2 × 10−3.

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    The time-averaged meridional structures of the (top) zonal-mean zonal wind, (middle) upper-layer zonal-mean PV, and (bottom) meridional eddy PV flux for (a) PDJ (Wc = 10, β = 0.3), (b) IDJ (Wc = 10, β = 0.28), (c) sourceward migration (Wc = 10, β = 0.28), and (d) single jet (Wc = 5, β = 0.25). The time interval for the averaging is (a) days 4000–5000, (b) days 800–900, (c) days 1500–2500, and (d) days 1500–2500. In the top and bottom rows, the solid line is for the upper layer and the dashed line is for the lower layer. In the middle row, the solid line is for the time-mean upper-layer PV, and the dashed line is for the radiative equilibrium PV.

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    The meridional profiles of the zonal wind (thick, black solid curve) and PV (thick, black dashed curve), averaged following the center of the left jet for (a) PDJ and (b) IDJ, showing the (top) upper-layer and (bottom) lower-layer zonal wind and PV. For both PDJ and IDJ, 40 individual realizations of the zonal wind are shown in thin gray curves. For the PDJ, the mean of the 40 realizations is also shown (thin black curves).

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    Meridional profiles for the (top) eddy thickness flux, (middle) eddy momentum flux convergence, and (bottom) eddy PV flux for the (a) PDJ interjet mode, (b) PDJ conventional mode, and (c) IDJ interjet mode. The solid (dashed) line is for the upper (lower) layer.

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    The horizontal structure of the CRWs for the PDJ interjet mode. The CRWU (thick) and CRWL (thin) (a) PV and (b) streamfunction fields are superimposed onto the normal mode (shading). The contour interval is 0.2, except for the upper CRWψL in (b) (thin curve, top), whose contour interval is 0.05. The shading interval for the normal mode is 0.2. Dark shading denotes positive values and light shading negative values. Lower-layer critical latitudes (phase speed of the PDJ interjet mode is −0.35) are indicated by horizontal lines.

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    (a),(b) Schema of the CRWs for the PDJ interjet mode shown in Fig. 5: (a) CRWqU at x = x1 and (b) CRWqL at x = x2. The home base yr and the yc are indicated. Positive and negative signs are denoted. (c) The horizontal structure of the two upper-layer CRW-induced streamfunction fields; the sum of these two fields is equal to the normal mode. The dark shading denotes positive CRWq values and light shading negative values. The size of the plus and minus signs is roughly proportional to the amplitude of the CRWs.

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    As in Fig. 5, but for the IDJ interjet mode. The contour interval is 0.4 for CRWqU, CRWqL, CRWψU, and lower CRWψL and 0.2 for upper CRWψL in (b) (thin curve, top). The shading intervals in (b) are (top) 0.3 and (bottom) 0.2. Critical latitudes are indicated by four horizontal lines. The outer pair is for both layers and the inner pair is for the lower layer. The phase speed of the IDJ interjet mode is 0.35.

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    As in Figs. 5 and 7, but for the PDJ conventional mode. The contour interval is 0.4 for upper CRWqU, CRWqL, upper CRWψU, and lower CRWψL; in (b) it is 0.2 for lower CRWqU in (thick curve, bottom), lower CRWψU (thick curve, bottom), and upper CRWψL (thin curve, top). The shading interval is 0.3. Upper-layer critical latitudes are indicated by four horizontal lines, where the phase speed of the PDJ conventional mode is 0.75.

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    The lower-layer zonal-mean zonal wind tendency due to (a) the thickness flux, (b) the upper-layer eddy momentum flux, (c) the Ekman damping, and (d) the sum of (a),(b), and (c). (e) The actual tendency. The contour interval is 1 × 10−2, except in (a) where the contour interval is 5 × 10−3. Shading is for values greater than 1 × 10−2. In (b) and (d), critical latitudes for the fastest-growing normal modes are indicated (thick contours) by plotting the regions where the difference between the phase speed of the fastest-growing normal mode and the local zonal-mean zonal wind is less than 0.1.

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    Transient evolution of (a) the thickness flux, (b) the eddy momentum flux convergence, (c) the PV flux, and (d) the vertical shear of the zonal-mean zonal wind in the nonlinear life cycle experiment of the PDJ interjet mode. The contour intervals are (a),(c) 2 × 10−3, (b) 2 × 10−4, and (d) 0.1. Shading is for values (a) greater than 1 × 10−2, (b) less than −2 × 10−2, (c) greater than 1 × 10−2, and (d) greater than 0.9. The right panel in (c) shows the meridional profiles of the PV flux for the normal mode (thin curve) and at day 33 (thick curve).

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    (a) The time integration of the most unstable normal mode, (b) the upper-layer zonal-mean wind estimated by integrating the first unstable normal mode, and (c) the upper-layer zonal-mean zonal wind of the model. See the text for details. The contour intervals are (a) 0.1 and (b),(c) 0.8. Shaded values are greater than (a) 0.1 and (b),(c) 1.5.

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Persistent Multiple Jets and PV Staircase

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  • 1 Department of Meteorology, The Pennsylvania State University, University Park, Pennsylvania
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Abstract

The persistence of multiple jets is investigated with a quasigeostrophic, two-layer, β-plane channel model. Linearly unstable normal modes are found to be capable of qualitatively describing the eddy fluxes of the nonlinear model. For a persistent double jet (PDJ) state, the most unstable normal mode has its largest amplitude located between the two jets, with a downshear tilt that acts to keep the jets separated. The opposite tilt occurs for a double jet state that is intermittent. An analysis of these normal modes, which utilized the concept of counterpropagating Rossby waves (CRWs), suggests that the downshear tilt in the interjet region hinges on the presence of critical latitudes only in the lower layer. This conclusion in turn suggests that the initial generation of the persistent jets requires L/Cgy < r−1, where L is the distance between the wave source (jet) and sink (interjet), Cgy is the meridional group velocity, and r is the linear damping rate. Similar CRW analysis for a conventional normal mode, which has its largest amplitude at the jet centers, suggests that the downshear tilt adjacent to the jet maxima is associated with the presence of critical latitudes only in the upper layer. The PDJ is found to be accompanied by potential vorticity (PV) staircases in the upper layer, characterized by a strong PV gradient at the jet centers and a broad region of homogenized PV between the jets. This PV mixing is realized through baroclinic waves that propagate slowly westward in the interjet region. Nonlinear evolution of the most unstable normal mode of the PDJ shows that northward heat flux by these waves is crucial for broadening the interjet PV mixing zone necessary for producing the PV staircase.

Corresponding author address: Changhyun Yoo, Department of Meteorology, 503 Walker Building, The Pennsylvania State University, University Park, PA 16802. Email: cyoo@psu.edu

Abstract

The persistence of multiple jets is investigated with a quasigeostrophic, two-layer, β-plane channel model. Linearly unstable normal modes are found to be capable of qualitatively describing the eddy fluxes of the nonlinear model. For a persistent double jet (PDJ) state, the most unstable normal mode has its largest amplitude located between the two jets, with a downshear tilt that acts to keep the jets separated. The opposite tilt occurs for a double jet state that is intermittent. An analysis of these normal modes, which utilized the concept of counterpropagating Rossby waves (CRWs), suggests that the downshear tilt in the interjet region hinges on the presence of critical latitudes only in the lower layer. This conclusion in turn suggests that the initial generation of the persistent jets requires L/Cgy < r−1, where L is the distance between the wave source (jet) and sink (interjet), Cgy is the meridional group velocity, and r is the linear damping rate. Similar CRW analysis for a conventional normal mode, which has its largest amplitude at the jet centers, suggests that the downshear tilt adjacent to the jet maxima is associated with the presence of critical latitudes only in the upper layer. The PDJ is found to be accompanied by potential vorticity (PV) staircases in the upper layer, characterized by a strong PV gradient at the jet centers and a broad region of homogenized PV between the jets. This PV mixing is realized through baroclinic waves that propagate slowly westward in the interjet region. Nonlinear evolution of the most unstable normal mode of the PDJ shows that northward heat flux by these waves is crucial for broadening the interjet PV mixing zone necessary for producing the PV staircase.

Corresponding author address: Changhyun Yoo, Department of Meteorology, 503 Walker Building, The Pennsylvania State University, University Park, PA 16802. Email: cyoo@psu.edu

1. Introduction

Multiple zonal jets are ubiquitous not only in planetary atmospheres (e.g., Jupiter and Saturn), but also in the earth’s atmosphere and ocean. The satellite-derived total ozone data, analyzed by Hudson et al. (2003), show three distinctive regions separated by two boundaries in the earth’s atmosphere. One boundary closely coincides with the subtropical jet and the other with the polar-front jet. Recently, high-resolution ocean models (Nakano and Hasumi 2005; Richards et al. 2006) and satellite altimetry data have revealed strikingly coherent zonal jets in much of the World Ocean (Maximenko et al. 2005).

Multiple zonal jets have been successfully simulated by various numerical models, including barotropic (Williams 1978; Yoden and Yamada 1993; Huang and Robinson 1998; Huang et al. 1999), shallow-water (Cho and Polvani 1996), and baroclinic models on a β-plane channel (Williams 1979; Panetta and Held 1988; Vallis and Maltrud 1993; Panetta 1993; Lee 1997, hereafter L97), as well as on the sphere (Williams 1988; Lee 2005). Huang and Robinson (1998) found in their barotropic model that persistent multiple zonal jets are obtained only when the flow is stirred by eddies whose horizontal length scale is much smaller than the jet scale. In baroclinic models, in contrast, remarkably persistent multiple jets are spontaneously generated and maintained by baroclinic eddies whose length scale is comparable to the jet scale. While the scale of these simulated jets is reasonably well predicted (Panetta 1993; Lee 2005) by the turbulence arrest scales (Rhines 1975; Vallis and Maltrud 1993), relatively little attention has been paid to the question of why the multiple jets are so persistent. L97 found that slowly propagating baroclinic waves, developing between the jets, play an important role for the persistency because these waves take on a downshear tilt (an alignment of eddies that follows the background shear), which acts to divert westerly momentum from the interjet region. However, it is unclear both why the interjet waves take on the downshear tilt and, as L97 found, why the jets are more persistent with larger values of β.

In this study, we address the questions raised above by investigating the interjet baroclinic waves and the relationship between these waves and the zonal-mean flow. In addition, given the interest in what has become known as the potential vorticity (PV) staircase (Peltier and Stuhne 2002; Dunkerton and Scott 2008), we will also examine PV structure to discern the processes that are responsible for generating the PV staircase in a multiple-jet environment.

The numerical model is described in section 2. The results from the model experiments are presented in section 3. The results are interpreted in the context of a linear analysis and counterpropagating Rossby waves (section 4), critical latitudes (section 5), and interjet PV mixing (section 6). Concluding remarks and discussion follow in section 7.

2. Model

We employ a two-layer, quasigeostrophic (QG), β-plane channel model, with parameter settings similar to that used by L97. The model is governed by the dimensionless QG PV equations:
i1520-0469-67-7-2279-e1
where
i1520-0469-67-7-2279-e2
and ψj refers to the streamfunction. The subscripts j = 1 and 2 indicate the upper layer and lower layer, respectively. The horizontal length and time scales used for the nondimensionalization are the Rossby deformation radius λR and λR/Uo, respectively, where Uo is the horizontal velocity scale. The model includes Ekman damping in the lower layer and is also subjected to a relaxation toward a prescribed thickness field τe(y). Because thickness in the model is analogous to temperature in the continuous atmosphere, the thickness damping will be referred to as thermal damping. The values for the Ekman damping coefficient κM, the thermal damping coefficient κT, and the biharmonic diffusivity ν are set equal to 0.1, 30, and 6 × 10−3, respectively.
The profile of τe(y) is identical to that of L97. The corresponding vertical shear of the zonal-mean wind Ue(y) [=∂τe(y)/∂y] takes on the form
i1520-0469-67-7-2279-e3
where σ is set equal to 4. This wind shear profile provides a uniform baroclinic zone in the central region of the channel (The parameter 2Wc, which is dimensionally 2WcλR, is a measure of the width of the baroclinic zone.). The initial upper- and lower-layer zonal-mean zonal winds are set equal to Ue and zero, respectively.

The model employs finite differencing in the meridional direction and spectral transformation in the zonal direction. The channel width W is fixed at 90 (90λR in dimensional form) and is represented by 600 grid points. There are 20 zonal wavenumbers (0.1–2.0). For the control experiment, Wc is fixed at 10, and β (dimensionally βUo/λR2) is changed during the model integration from 0.28 to 0.3. These values are used to simulate a transition from a nonpersistent mostly single-jet state (β = 0.28) to a persistent double-jet state (β = 0.3). It is worth noting that the channel width (90λR) is much greater than the deformation radius, but the jet scale is about 5λR (see Figs. 1 and 2). While this scale is still greater than the jet scale in the atmosphere, which is close to being 1λR, we assume that the jet dynamics simulated by this idealized model is still relevant for the atmosphere. It may be that the atmospheric jets are relatively narrow because the jet (and eddy) size is constrained by the size of the earth, the radius of which is only about 2λR. In addition, it is important to remember that the subtropical jet is not driven by eddies but rather by an angular momentum–conserving overturning circulation. The presence of a subtropical jet, therefore, further acts to confine the eddies and the eddy-driven jet into an even smaller latitudinal zone. Without these constraints, the atmospheric eddy-driven jet may also take on a greater meridional scale. The calculations presented by Lee (2005) support this possibility.

The model is integrated for 2500 model days (1 “day” ≡ λR/Uo) for each value of β to ensure that a statistically steady-state solution is obtained.

3. Model results

Two distinct jet behaviors can be identified in the transient evolution of the zonal-mean flow in the control experiment (Fig. 1): meridionally migrating zonal jets prior to day 3000 and two persistent zonal jets afterward. These two jets emerged about 500 days after the value of β was increased to 0.3. We refer to this state as a persistent double jet (PDJ). Examining the migrating jets in more detail, it can be seen that the jets typically originate from either the northern or southern edge of the baroclinic zone and migrate toward the center of the channel. This type of zonal jet migration was observed in the atmosphere (Feldstein 1998; Riehl et al. 1950) wherein the migration occurs prevalently over a very broad range of latitudes from the tropics to about 70° latitude in both hemispheres. Similar poleward migration was also found in a barotropic flow where mountain-like orography acts as a wave source (Rhines 2007). The mechanism for this poleward migration of the zonal-mean flow was investigated using an idealized general circulation model (Lee et al. 2007). This study found that the poleward migration is caused by equatorward-propagating Rossby waves that break at successively higher latitudes, with most of the wave breaking taking place at critical latitudes (i.e., where the phase speed of the wave matches the zonal-mean zonal wind). Although not analyzed, we believe that a similar mechanism accounts for the zonal-mean flow migration observed in our model. Because of north–south symmetry in a channel model, the zonal flow anomaly migrates toward the channel center (Fig. 1a), the region of the wave activity source. For this reason, this zonal jet behavior will be referred to as sourceward migration.

Within this sourceward migration regime (i.e., t < day 2500) there are two zonal jets that intermittently coexist and remain persistent for a period as long as ∼200 days (Fig. 1). This can be seen, for example, between days 750 and 950. Figure 1b supports the presence of these multiple jets, as indicated by the presence of power at meridional wavenumbers corresponding to the scale of jets. The time period from days 800 to 900, showing two jets distinctly located on each side of the channel center, will be referred to as an intermittent double jet state (IDJ).

The time-averaged zonal-mean PV fields for the PDJ and the IDJ states are contrasted in Figs. 2a,b. The PDJ (Wc = 10, β = 0.3) is accompanied by a hint of a PV staircase (Fig. 2a, middle) in the upper layer. Following the terminology of Dunkerton and Scott (2008), which refers to the homogenized PV region as the “step” and the steep gradient as the “riser,” the PDJ PV structure can be described as steps and risers. The upper-layer PV flux that extends across the entire region between the two zonal jets is crucial for this PV structure (Fig. 2a, bottom). For the IDJ (Wc = 10, β = 0.28), in contrast, because the two jets are not as protruded as in the PDJ (Fig. 2b, top), the zonal-mean zonal wind is associated with narrower and weaker PV mixing within the interjet region (Fig. 2b, bottom), resulting in steps and risers that are barely noticeable (Fig. 2b, middle). If the averaging period is increased to include sourceward migration (Fig. 2c), the time mean PV is almost indistinguishable from the radiative equilibrium state because of the migrating mixing barriers. The zonal-mean zonal wind and the eddy PV flux profiles also show nearly uniform values throughout the region where the transitory jets are present (Fig. 2c, top and bottom).

A staircase-like PV profile emerges if the averaging is performed along the axis of the jet. The top panel of Fig. 3a shows the zonal wind and PV in the upper layer (i.e., U1 and Q1) for the PDJ averaged in these coordinates (thick black curves; the gray curves show individual realizations for each x and t). For the case shown here, the left jet is chosen as the reference jet. Because the jets meander in both x and t, this reference jet is much sharper than the zonally averaged jets (Fig. 2). The jet-coordinate averaged PV profile reveals a steep riser at the jet center and a quasi-flat step (thick, black dashed curve in Fig. 3a) between the jets. In these jet coordinates, even the IDJ is accompanied by a staircase-like PV profile (Fig. 3b), although the sample realizations (gray curves) show a large amount of scatter and the lower-layer zonal wind shows no hint of a double jet (compare the bottom row of Figs. 3a and 3b). This suggests that even transient jets are capable of being effective mixing barriers and the associated interjet region is accompanied by an intense PV mixing.

For completeness, several additional runs are performed to investigate the jet and PV characteristics in the Wcβ parameter space. As summarized in Table 1, the number of zonal jets increases either as Wc increases, or as the Rhines scale, which is inversely proportional to β1/2, decreases (Panetta 1993; L97; Lee 2005). The PV field for the single-jet case (Fig. 2d) shows a single riser at the center of the channel with two steps on each flank of the jet. Consistently, the upper-layer PV flux shows two peaks, representing strong PV mixing zones on each side of the jet. As was shown in Fig. 3, however, the double jets are associated with a broad PV step that spans the entire interjet region.

4. Eddy momentum flux and background flow

In this section, we explore why the jets are persistent in the PDJ but not in the IDJ. L97 suggested that downshear tilt of the interjet waves helps to maintain the multiple jets by transporting eastward momentum outward from the interjet region (see Fig. 5 in L97). L97 also showed that the most unstable normal mode can capture the key properties of the nonlinear interjet disturbances, including the sign of the eddy momentum flux. Following L97, this most unstable normal mode is referred to as the interjet mode. Figures 4a and 4c, respectively, show the interjet mode for the time-mean PDJ and IDJ state (Figs. 2a,b). The PDJ and the IDJ interjet modes differ from each other in terms of both relative vorticity and PV fluxes. At the center of the interjet region, while the PDJ interjet mode is characterized by a strong, narrow region of eddy momentum flux divergence in both layers (Fig. 4a, middle) the IDJ interjet mode reveals an eddy momentum flux convergence in the upper layer. To the extent that the most unstable normal mode is relevant for the nonlinear flow evolution, this momentum flux convergence can culminate in the termination of the double-jet state.1 In fact, eddy momentum flux convergence of the most unstable normal modes, calculated for instantaneous flows, are found to be reasonably successful at predicting the subsequent transient jet evolution in the nonlinear integration (see appendix A).

Given this capability of the normal mode, we exploit the normal mode as a tool to address the question of why the jets are persistent in the PDJ but not in the IDJ. Specifically, we ask what properties of the zonal-mean state are associated with the formation of the downshear (upshear) tilt of the PDJ (IDJ) interjet normal modes.2 To address this question, we follow Methven et al. (2005) and utilize the counterpropagating Rossby wave (CRW) perspective. The CRW was conceptualized and first used by Bretherton (1966) and later by Hoskins et al. (1985), wherein a pair of Rossby waves was introduced to explain baroclinic instability. The CRW concept was generalized for any zonal flow with dynamics described by PV conservation (Heifetz et al. 2004a) and then applied to the Charney model (Heifetz et al. 2004b). The CRW concept has also been used to explain the direction of the eddy momentum flux on a zonal jet (Methven et al. 2005). In this study, the three unstable normal modes, shown in Fig. 5, will be analyzed through this CRW perspective. Since our model has two layers, we adopt the “home base” method (Heifetz et al. 2004a) and thereby use the normal mode to construct the CRWs. The location of the home base was chosen so that it collocates with the value of y at which the streamfunction maximum of the normal mode occurs. For further description, the reader is referred to appendix B and also to Heifetz et al. (2004a).

The PV and streamfunction fields for each of the CRWs of the PDJ interjet mode are illustrated in Figs. 5a and 5b, respectively. To avoid confusion, we define a CRW that has its home base in the upper layer as CRWU (thick contours) and that with a home base in the lower layer as CRWL (thin contours). For this PDJ interjet mode, the home base of CRWU is chosen to be at the upper-layer channel center; similarly, the channel center in the lower layer was chosen for the home base of CRWL. In accordance with the definition of CRWs, it can be seen that the PV fields have no tilt in both the meridional and vertical directions, and that the PV of each CRW (hereafter CRWq) has its maximum amplitude at its home base and zero amplitude at the home base of the other CRW (Fig. 5a). That is, CRWqU (thick contours, Fig. 5a) has its maximum value at the upper-layer channel center and a zero value at the lower-layer channel center. Similarly, CRWqL (thin contours, Fig. 5a) has its maximum value at the lower-layer channel center and a zero value at the upper-layer channel center.

It turns out that CRWq structure is closely linked to critical latitudes. In the upper layer (Fig. 5a, top) CRWqU (thick contours) is confined mostly within the region bounded by the lower-layer critical latitudes indicated by the two horizontal lines. We do not yet have an explanation for this behavior. However, as will be shown throughout this section, the values of CRWq in both layers are closely tied to critical latitudes. In the lower layer (Fig. 5a, bottom) CRWqL (thin contours) is also confined to the central region bounded by the critical latitudes. The upper CRWqL (thin contours, Fig. 5a, top) has its maxima collocated with the critical latitudes. Similarly, in the lower layer, the CRWqL maxima (thick contours, Fig. 5a, bottom) also occur at the critical latitudes.

By comparing the streamfunction fields of the CRWs (hereafter CRWψ) with those of the normal mode, it can be seen that the difference in the meridional extension between the CRWψU and CRWψL can explain the meridional tilt of the normal mode.3 In Fig. 5b, CRWψU (thick contours) is induced by CRWqU (thick contours, Fig. 5a), and CRWψL (thin contours) is induced by CRWqL (thin contours, Fig. 5a). At each layer, the sum of these CRWψ fields is equal to the PDJ interjet normal mode streamfunction (shaded, Fig. 5b).

To help visualize the CRW structure, cross sections of these fields at two selected locations, x1 and x2 (the locations of x1 and x2 are shown in Fig. 5a), are shown schematically in Fig. 6. In this figure, CRWqU and CRWqL have the same value for y for the home base. It can be seen that for CRWqU there is vertical sign change with positive values at the center of the upper layer and negative values in the lower layer straddling the upper-layer positive region (Fig. 6a). Because of this sign change, the upper-layer CRWqU- and lower-layer CRWqU-induced streamfunction fields have the opposite sign near y = yc, where yc is the critical latitude. As a result of the cancellation between these two fields, CRWψU has a relatively small meridional scale (Fig. 6c). In contrast, the sign of the CRWqL does not change between the layers. As a result, CRWψL has a relatively large meridional scale. Moreover, because the phase relationship between the CRWψL and the CRWψL is such that the former leads the latter, as expected for growing normal modes, the above difference in the meridional scale generates the downshear tilt in the normal mode structure (Fig. 6c). Note that although small, this downshear tilt can be seen for the normal mode streamfunction field for 43 < y < 47 (shaded, Fig. 5b). This tilt produces the momentum flux divergence at the center of the interjet region.

The above interpretation of the relationship between the CRW and the background flow also holds for the IDJ interjet mode, although the CRW in this case takes on a somewhat more complex form because of the presence of the critical latitudes in both layers (Fig. 7) and of four, rather than two, critical latitudes in the lower layer. For these four lower-layer critical latitudes, the pair that is closer to (more distant from) the channel center will be referred to as the inner (outer) pair. It can be seen that there are four nodes in the upper CRWqU (thick contours, Fig. 7a, top). The central part of the upper CRWqU, bounded by the two inner nodes, is confined by the outer pair of critical latitudes (horizontal lines, Fig. 7). Similarly, the central part of the lower CRWqL (thin contours, Fig. 7, bottom) is confined by the inner pair. These features give rise to the CRWψU having a broader meridional scale than that of CRWψL, generating an upshear tilt between the two jets (Fig. 7b, top). As a result, the IDJ interjet mode acts to merge the two jets.4

Finally, for completeness, we consider the CRW for the second unstable normal mode of the PDJ, shown in Fig. 4b. This normal mode takes on the structure of a typical baroclinic wave associated with a westerly jet, with downshear eddy momentum flux on both sides of the jet maxima. For this reason, L97 referred to this mode as the conventional normal mode, and here we adopt the same terminology. In this case, where the critical latitudes are located in the upper layer, the CRWqU does not change sign between the two layers, while the CRWqL does have a sign change (Fig. 8). This behavior is opposite to that of the interjet CRW for the PDJ (Figs. 5 and 6). With the same reasoning used for explaining the interjet mode CRWs, it can then be inferred that CRWψU has a broader meridional scale than that of CRWψL. As a result, the conventional normal mode takes on a form that transports momentum toward the jet center.

Thus, it appears that the CRW that has its home base in the same layer as the critical latitude does not undergoes a sign change in its PV field, while the CRW that has its home base in the other layer undergoes a sign change. It is thus change in PV sign that shortens the meridional scale of the CRW streamfunction, which in turn accounts for the sign of the eddy momentum flux and thus the impact of the eddies on the zonal-mean flow.

5. Formation of the lower-layer critical latitude

The analysis presented in the previous section shows that the meridional scales of the CRWψ determine the direction of eddy momentum flux. The meridional scale of the CRWψ, in turn, is shown to be tied to the location of the critical latitudes. For the PDJ interjet mode, the downshear tilt is found to hinge on the presence of lower-layer critical latitudes. Therefore, we next ask how the lower-layer critical latitudes form. To address this question, we consider the transformed Eulerian mean (TEM) equations for the two-layer model (see Feldstein 1992):
i1520-0469-67-7-2279-e4
The lower-layer zonal-mean zonal wind due to each term on the right-hand side (rhs) of (4) is calculated by using the iterative tridiagonal solver. For example, the tendency induced by the thickness flux is obtained by performing the following calculation:
i1520-0469-67-7-2279-e5
where the subscript t denotes the time derivative and L−1 is the inverse of the linear operator on the left-hand side of (4). Figure 9 shows U2,t induced by a selection of terms on the rhs of (4). The tendencies due to the lower-layer eddy momentum flux, the thermal damping, and diffusion terms are relatively small and are therefore not displayed.

Figure 9 indicates that the lower-layer zonal-mean wind tendency is primarily driven by the upper-layer eddy momentum flux and is approximately balanced by the Ekman damping. In comparison, the contribution by the heat flux (Fig. 9a) is much weaker. The critical latitudes of the fastest-growing normal modes are superimposed in Figs. 9b and 9d. It can be seen that the lower-layer critical latitudes in the interjet region start to emerge near day 3000, at about the same time that the two distinct jets start to form. This evidence collectively suggests that the lower-layer critical latitudes initially form as a result of the deceleration of the lower-layer zonal-mean zonal wind, which is driven by the upper-layer momentum flux. Once this deceleration takes place, the jet remains significantly persistent via the processes described in section 4.

6. Interjet PV mixing

The findings in the previous section suggest that linear theory is capable of explaining the jet evolution and the direction of the eddy momentum flux. However, linear theory appears to be incapable of explaining the meridional scale of the interjet PV mixing, which is an important characteristic of the PV staircase for persistent multiple jets. As introduced in the previous section, to form a step–riser–step PV staircase, there must be a broad region of interjet PV mixing. A comparison between the upper-layer PV in Fig. 2a and the corresponding normal mode eddy PV flux in Fig. 4a indicates that the PV flux must broaden over time from its normal mode structure. In this section, therefore, a nonlinear life cycle experiment will be carried out to examine whether the PDJ interjet mode evolves in a manner that results in the broad and intense PV mixing shown in Fig. 2a. For this calculation, the basic state of the PDJ (Fig. 2a, top) is initialized with the PDJ interjet mode (Fig. 4a), and the nonlinear model is integrated only with a single zonal wavenumber (k = 0.6).

The nonlinear life cycle of the PDJ interjet mode shows that the meridional scale of both the eddy heat and PV flux broadens in time [Figs. 10a and 10c (including the right panel)]. Because the change in the structure of the eddy momentum flux (Fig. 10b) is minimal (note that contour interval for Fig. 10b is an order of magnitude smaller than that for Figs. 10a and 10c), the broadening in the PV flux is due almost entirely to the change in the heat flux. Snapshots at t = 0 days (thin) and t = 30 days (thick) illustrate this broadening (Fig. 10c, right). Because of this broadening, by day 30 the eddy PV flux attains a meridional e-folding scale comparable to that simulated by the fully nonlinear model (Fig. 2a). This nonlinear broadening in the PV flux is robust in that essentially the same behavior is obtained (not shown) in calculations that used different values of k (0.5 and 0.7). For calculations that included more than a single zonal wavenumber (e.g., k = 0.6, 1.2, and 1.8), the desired PV flux broadening was achieved within 20 model days (not shown), a shorter time period than that indicated in Fig. 10.

The above calculation indicates that a nonlinear process is essential for generating the required interjet PV mixing that produces the PV staircase. The workings of this nonlinear process can be gleaned by examining the time evolution of the vertical shear. Figure 10d shows that, in tandem with the eddy heat flux broadening, the vertical shear increases at both the southern and northern margins of the interjet region (i.e., at y = 39 and y = 51, respectively). Our interpretation of this property is as follows: because the eddy heat flux is a maximum at the center of the interjet region, there must be eddy heat flux convergence (divergence) to its immediate north (south). In response, the zonal-mean meridional thickness gradient must increase farther to the north (south) of the region of thickness-flux convergence (divergence). Because the eddy heat flux is always downgradient, this increase in the thickness-gradient of the northern and southern margins of the interjet region must conduct the eddy heat flux in these margins, whereby the eddy heat flux is broadened. As the heat flux develops in these margins, it decreases the mean thickness gradient (and thus barotropizes the mean wind) in these regions. At the same time, a zonal-mean thickness increase occurs even farther to the north and south, continuing to broaden the PV mixing zone. The barotropization process can be understood from the view point that the heat flux, in a QG flow, is equivalent to horizontal pressure gradient force between the two layers (i.e., a form drag) (e.g., Rhines 1979; Vallis 2006).

7. Discussion and concluding remarks

Using a two-layer QG model on β-plane channel, this study addressed questions on the persistence of multiple zonal jets and the associated PV staircase. The main question addressed by this study is what flow properties of the zonal-mean flow allow the multiple zonal jets to be so persistent. It was found for the persistent double-jet (PDJ) state that the eddy momentum flux diverges sharply at the center of the interjet region, while such a divergence is absent in the intermittent double-jet (IDJ) state. As was shown by L97, these key characteristics of the interjet disturbances are captured by the corresponding linearly unstable normal modes for both the PDJ and IDJ flows. Taking advantage of this capacity of the linear normal mode, counterpropagating Rossby waves (CRWs) are used to identify the property of the zonal-mean flow that enables the eddy momentum flux of the PDJ to diverge. Although it is unclear why the CRWs have this property, the CRW analysis suggests that eddy momentum flux divergence occurs in the interjet region if the critical latitude is present only in the lower layer. Similarly, for the conventional normal mode that has momentum flux convergence at the jet center, the critical latitude is present only in the upper layer.

In the introduction, we also raised the question as to why, everything else being equal, the multiple jets are more persistent as the value of β increases. This question may be addressed by turning to the above observation that the eddy momentum flux divergence in the interjet region is associated with the occurrence of critical latitudes only in the lower layer. If this can be treated as an axiom, it follows that the multiple jets can be persistent if the phase speed c of the interjet waves is less than U1 everywhere in the domain. For this to occur, there must be a sufficiently large deposit of westward momentum in the interjet region. To deduce the conditions under which the required deceleration can occur, we consider the following wave-mean flow relationship, which is applied heuristically to the upper layer of our model. Specifically, we ask what are the conditions that allow wave activity to propagate from its source region (i.e., the jet region) to the sink region (i.e., the interjet region) without being dissipated. Following the notation of Vallis (2006), for a steady, two-dimensional flow, the equations for time-mean, zonal mean, zonal wind and eddy enstrophy take the form of
i1520-0469-67-7-2279-e6
i1520-0469-67-7-2279-e7
where r is a linear damping coefficient for the zonal-mean flow U, γ = βUyy is the meridional potential vorticity gradient, and Fζ and Dζ are the source and sink, respectively, of perturbation vorticity. The angle bracket denotes the time mean. From (6) and (7), the time-averaged zonal-mean zonal wind can be expressed as
i1520-0469-67-7-2279-eq1
This equation tells us that for 〈U〉 in the wave activity sink region (the interjet region) to be smaller than that in the wave activity source region (the jets), the region of the wave activity sink, where 〈ζDζ〉 is large, should be separated from the region of the wave activity source, 〈ζFζ〉. Because wave activity propagates at the group velocity, assuming that the eddy damping time scale is also 1/r, this condition can be stated as
i1520-0469-67-7-2279-eq2
where L is the distance between the source and the sink regions, Cgy is the meridional group velocity, and k and l are, respectively, the zonal and meridional wavenumbers. Given k and l, this inequality is satisfied if β is sufficiently large and/or r is sufficiently small. This is consistent with the results of the numerical model calculations (Fig. 1; L97) and may offer an explanation for why zonal jets are so persistent in the fast-rotating, relatively inviscid Jovian atmosphere.

The analysis presented in this study also indicates that the upper-layer PV flux associated with the interjet waves is crucial for the formation of the PV staircase associated with the multiple jets. This interjet PV flux is dominated by the thickness flux, and while the linear normal mode captures both the PV and thickness fluxes, we found that nonlinear wave–mean flow interaction is necessary to produce the correct meridional scale of the PV flux.

Acknowledgments

This study is supported by the National Science Foundation through Grant ATM-0647776. We thank Dr. Steven Feldstein, who suggested the construction of Fig. 6, along with many other helpful comments; Dr. Peter Rhines for insightful comments, particularly for suggesting the calculation that led to Fig. 3; and two anonymous reviewers for their constructive comments and suggestions. We are grateful to Drs. Eyal Heifetz and Nili Harnik for helping us learn the CRW methodology.

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APPENDIX A

Prediction of Nonlinear Jet Evolution by the Most Unstable Normal Mode

The normal modes are obtained as solutions to the eigenvalue problem of a linearized version of (1), where the basic state is an appropriately chosen zonal-mean state. The eigenvalue problem was solved for k = 0.6. (Identical modes were found from additional calculations with different zonal wavenumbers.) In the wide-jet limit, defined as WcλR, the transformed Eulerian mean equation for the upper-layer zonal-mean zonal wind [(11) in Feldstein 1992] takes on the form
i1520-0469-67-7-2279-ea1
where an overbar and uppercase letters denote a zonal mean and a prime indicates the perturbation from the zonal mean. The subscript e denotes the radiative equilibrium state. A budget calculation of (A1) (not shown) indicates that the PV flux terms dominate the other terms by one order of magnitude. In addition, because the sum of the upper- and lower-layer heat fluxes is zero and the lower-layer vorticity flux is much smaller than the upper-layer vorticity flux, (A1) can be further reduced to
i1520-0469-67-7-2279-ea2
where ζ represents the relative vorticity.
We estimate U1, on day n, by performing the following integration:
i1520-0469-67-7-2279-ea3
In (A3), υ1 and ζ1 are the most unstable normal modes at day nj (i.e., the basic state is the zonal-mean state on day nj), and σnj is the corresponding growth rate. The normal mode is normalized by the maximum streamfunction amplitude in the upper layer on day nj. We base (A3) on the assumption that the evolving wave field is constantly adjusting to the form of the most unstable normal mode. The calculation in (A3) is performed over 15 model days to ensure that the number of days is longer than the wave decorrelation time scale (Leith 1973) and shorter than the model’s 30-day radiative relaxation time scale. Furthermore, the resulting U1 is found to be qualitatively insensitive to the choice of the time period unless it is shorter than ∼5 days or longer than ∼25 days. Finally, to facilitate comparison with the nonlinear model run, the second term on the right-hand side of (A3) is normalized by the maximum value of the upper-layer eddy vorticity flux obtained from the nonlinear model run.

The upper-layer zonal wind U1, constructed using (A3), is consistent with the nonlinear model run (cf. Figs. A1b,c). This is because the vorticity flux term [i.e., the second term on the right-hand side of (A3)], broadly resembles the zonal-mean zonal wind (Fig. A1a). Between days 2000 and 3000, the vorticity flux term captures much of the zonal jet meandering, including the sourceward migration. This vorticity flux is associated with the conventional mode (see Fig. 4b and the description in section 4). After day 3000, as the multiple jets become persistent, the interjet mode emerges as the most unstable normal mode (cf. Figs. A1a,c; see also Fig. 4a).

APPENDIX B

Construction of CRWs Using the Home Base Method

For given zonal wavenumber k, we consider a solution to the linearized form of (1) in terms of two neutral Rossby waves, that is,
i1520-0469-67-7-2279-eb1
where αU and αL are complex amplitudes and CjU(y, t = 0) and CjL(y, t = 0) are referred to as CRWqU and CRWqL, respectively. The subscripts j = 1 and 2 indicate the upper layer and lower layer, respectively. By construction, each CRW completely describes the PV at its home base. That is, at its home-base point y = y1, CRWqU = CjU(y1, 0) in the upper layer, and at y = y1 in the lower layer CRWqU = 0. Likewise, for CRWqL, at its home base, y = y2, CRWqL = CjL(y2, 0) in the lower layer and CRWqL = 0 in the upper layer. For the interjet mode, we choose the same home base, yb = y1 = y2, for both CRW1 and CRW2, where yb is the channel center, the location where the time-mean zonal-mean zonal wind is a minimum. For the conventional normal mode, yb is chosen to be the latitude where the jet maximum occurs.
Following Heifetz et al. (2004a), we construct CRWs using a growing mode and its decaying complex conjugate:
i1520-0469-67-7-2279-eb2
where Qj(y) = qj(y) exp[−j(y)]. Because the initial phase and amplitude can be expressed as αU(t = 0) = q1(yb)/|q1(yb)| and αL(t = 0) = q2(yb)/|q2(yb)|, in terms of the complex amplitude of the normal modes, (B2) can be rewritten as
i1520-0469-67-7-2279-eb3
where the asterisk indicates a complex conjugate. In the text, C1U is referred to as upper CRWqU, C2U as lower CRWqU, C1L as upper CRWqL, and C2L as lower CRWqL. Figure 6a displays a schematic for a y–z cross section of C1U and C2U for the PDJ interjet normal mode. Likewise, Fig. 6b illustrates a similar schematic for C1L and C2L By inverting these CRWq, the CRW streamfunction, CRWψ, can be obtained. Figure 6c displays a schematic for the CRWψ fields, corresponding to C1U and C2U.

Fig. 1.
Fig. 1.

Transient evolution of (a) the vertically averaged zonal-mean zonal wind and (b) the meridional power spectrum of the zonal-mean zonal wind for the control run (Wc = 10, β = 0.28 − 0.3). Note that β is changed at day 2500. Contour intervals are (a) 0.5 and (b) 10−3. Zero lines are omitted. Shading corresponds to values that are greater than (a) 1 and (b) 2 × 10−3.

Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3326.1

Fig. 2.
Fig. 2.

The time-averaged meridional structures of the (top) zonal-mean zonal wind, (middle) upper-layer zonal-mean PV, and (bottom) meridional eddy PV flux for (a) PDJ (Wc = 10, β = 0.3), (b) IDJ (Wc = 10, β = 0.28), (c) sourceward migration (Wc = 10, β = 0.28), and (d) single jet (Wc = 5, β = 0.25). The time interval for the averaging is (a) days 4000–5000, (b) days 800–900, (c) days 1500–2500, and (d) days 1500–2500. In the top and bottom rows, the solid line is for the upper layer and the dashed line is for the lower layer. In the middle row, the solid line is for the time-mean upper-layer PV, and the dashed line is for the radiative equilibrium PV.

Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3326.1

Fig. 3.
Fig. 3.

The meridional profiles of the zonal wind (thick, black solid curve) and PV (thick, black dashed curve), averaged following the center of the left jet for (a) PDJ and (b) IDJ, showing the (top) upper-layer and (bottom) lower-layer zonal wind and PV. For both PDJ and IDJ, 40 individual realizations of the zonal wind are shown in thin gray curves. For the PDJ, the mean of the 40 realizations is also shown (thin black curves).

Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3326.1

Fig. 4.
Fig. 4.

Meridional profiles for the (top) eddy thickness flux, (middle) eddy momentum flux convergence, and (bottom) eddy PV flux for the (a) PDJ interjet mode, (b) PDJ conventional mode, and (c) IDJ interjet mode. The solid (dashed) line is for the upper (lower) layer.

Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3326.1

Fig. 5.
Fig. 5.

The horizontal structure of the CRWs for the PDJ interjet mode. The CRWU (thick) and CRWL (thin) (a) PV and (b) streamfunction fields are superimposed onto the normal mode (shading). The contour interval is 0.2, except for the upper CRWψL in (b) (thin curve, top), whose contour interval is 0.05. The shading interval for the normal mode is 0.2. Dark shading denotes positive values and light shading negative values. Lower-layer critical latitudes (phase speed of the PDJ interjet mode is −0.35) are indicated by horizontal lines.

Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3326.1

Fig. 6.
Fig. 6.

(a),(b) Schema of the CRWs for the PDJ interjet mode shown in Fig. 5: (a) CRWqU at x = x1 and (b) CRWqL at x = x2. The home base yr and the yc are indicated. Positive and negative signs are denoted. (c) The horizontal structure of the two upper-layer CRW-induced streamfunction fields; the sum of these two fields is equal to the normal mode. The dark shading denotes positive CRWq values and light shading negative values. The size of the plus and minus signs is roughly proportional to the amplitude of the CRWs.

Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3326.1

Fig. 7.
Fig. 7.

As in Fig. 5, but for the IDJ interjet mode. The contour interval is 0.4 for CRWqU, CRWqL, CRWψU, and lower CRWψL and 0.2 for upper CRWψL in (b) (thin curve, top). The shading intervals in (b) are (top) 0.3 and (bottom) 0.2. Critical latitudes are indicated by four horizontal lines. The outer pair is for both layers and the inner pair is for the lower layer. The phase speed of the IDJ interjet mode is 0.35.

Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3326.1

Fig. 8.
Fig. 8.

As in Figs. 5 and 7, but for the PDJ conventional mode. The contour interval is 0.4 for upper CRWqU, CRWqL, upper CRWψU, and lower CRWψL; in (b) it is 0.2 for lower CRWqU in (thick curve, bottom), lower CRWψU (thick curve, bottom), and upper CRWψL (thin curve, top). The shading interval is 0.3. Upper-layer critical latitudes are indicated by four horizontal lines, where the phase speed of the PDJ conventional mode is 0.75.

Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3326.1

Fig. 9.
Fig. 9.

The lower-layer zonal-mean zonal wind tendency due to (a) the thickness flux, (b) the upper-layer eddy momentum flux, (c) the Ekman damping, and (d) the sum of (a),(b), and (c). (e) The actual tendency. The contour interval is 1 × 10−2, except in (a) where the contour interval is 5 × 10−3. Shading is for values greater than 1 × 10−2. In (b) and (d), critical latitudes for the fastest-growing normal modes are indicated (thick contours) by plotting the regions where the difference between the phase speed of the fastest-growing normal mode and the local zonal-mean zonal wind is less than 0.1.

Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3326.1

Fig. 10.
Fig. 10.

Transient evolution of (a) the thickness flux, (b) the eddy momentum flux convergence, (c) the PV flux, and (d) the vertical shear of the zonal-mean zonal wind in the nonlinear life cycle experiment of the PDJ interjet mode. The contour intervals are (a),(c) 2 × 10−3, (b) 2 × 10−4, and (d) 0.1. Shading is for values (a) greater than 1 × 10−2, (b) less than −2 × 10−2, (c) greater than 1 × 10−2, and (d) greater than 0.9. The right panel in (c) shows the meridional profiles of the PV flux for the normal mode (thin curve) and at day 33 (thick curve).

Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3326.1

Fig. A1.
Fig. A1.

(a) The time integration of the most unstable normal mode, (b) the upper-layer zonal-mean wind estimated by integrating the first unstable normal mode, and (c) the upper-layer zonal-mean zonal wind of the model. See the text for details. The contour intervals are (a) 0.1 and (b),(c) 0.8. Shaded values are greater than (a) 0.1 and (b),(c) 1.5.

Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3326.1

Table 1.

Dependency of jet characteristics on Wc and β. Here “single” indicates that one persistent jet exists, “sourceward” indicates nonpersistent jets that tend to migrate toward the center of the domain, “double” refers to two persistent jets, and “double–triple” is for the case where at any given time, either two or three jets are present.

Table 1.

1

An alternative way to consider this is that the interjet region of the PDJ may be thought of as a single westward (lower-layer) jet maintained by convergence of westward eddy momentum flux. On the contrary, for the IDJ, the double jets and the interjet region may be considered one broad eastward jet on which the eddy momentum flux converges.

2

In doing so, we are linking the linearly unstable normal modes to the nonlinear model solution by following the perspective of Simmons et al. (1983). That is, we are not claiming that the eddy fluxes are developing from small-amplitude perturbations. Instead, we consider that among the various growing and decaying disturbances, it is those disturbances that project onto the most unstable normal modes, during part of their evolution, that grow most rapidly and dominate the eddy field. In addition, nonlinear modeling studies find that the eddy momentum flux divergence of linear modes is often followed by wave breaking and PV mixing at the same location (Feldstein and Held 1989; Held and Phillips 1990).

3

The direction of the eddy momentum flux can be understood in terms of interactions between two horizontal circulations induced by each CRWq (see Fig. 13 in Methven et al. 2005). By construction, the sum of the two CRWψ fields in the upper layer is equal to the upper-layer normal mode (shaded, Fig. 5b, top). Similarly, in the lower layer, the sum of two CRWψ fields is equal to the lower-layer normal mode (shaded, Fig. 5b, bottom).

4

The narrow momentum flux divergence region in the lower layer (dashed, Fig. 4c, middle) can also be explained by CRW streamfunction fields because between y = 43 and y = 47, the CRWψU is narrower than the CRWψL (Fig. 7b, bottom). However, the resulting downshear tilt has negligible effect on the zonal-mean flow because it is weak and confined to a very narrow region.

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