## Abstract

The Kuroshio south of Japan shows bimodal path fluctuations between the large meander (LM) path and the nonlarge meander (NLM) path. The transition from the NLM path to the LM path is triggered by a small meander generated off southwestern Japan. The small meander first propagates eastward (downstream) along the Kuroshio and then rapidly amplifies over Koshu Seamount, located about 200 km south of Japan, leading to the formation of the LM path of the Kuroshio. Although Koshu Seamount is essential for the rapid amplification of the small meander, the underlying physical mechanism is not fully understood. In this study, the role of Koshu Seamount is revisited using a two-layer quasi-geostrophic model that takes into account the effects of bottom topography. Numerical experiments show that the transition from the NLM path to the LM path can be successfully reproduced only when bottom topography mimicking Koshu Seamount is incorporated. In this case, the upper-layer meander trough is rapidly amplified together with a lower-layer anticyclone by baroclinic instability during their passage over the northern slope of Koshu Seamount. A linear stability analysis shows that baroclinic instability over a seamount is caused by resonant coupling between the upper-layer Rossby wave in the eastward background flow and the lower-layer seamount-trapped wave during their eastward propagation over the northern slope of the seamount. The spatial scale and structure of this baroclinically unstable mode are close to those of the numerically reproduced small meander in its early amplification stage over the seamount.

## 1. Introduction

The Kuroshio is the western boundary current of the subtropical gyre in the North Pacific Ocean. It is well known that the Kuroshio path south of Japan shows a remarkable bimodal feature, that is, the nonlarge meander (NLM) path where the Kuroshio flows along the southern coast of Japan and the large meander (LM) path where the Kuroshio detours southward west of Cape Shiono-misaki and flows far away from the southern coast of Japan (e.g., Taft 1972; Shoji 1972; Nitani 1975; Kawabe 1985, 1995; Fig. 1). Each path usually persists for more than a year once it is formed, whereas the transition from one to the other occurs in a relatively short period of time (usually several months). Such a remarkable path variation is unique to the Kuroshio and is not found in the other western boundary currents including the Gulf Stream. Because the path variations of the Kuroshio exert a significant influence on fisheries, marine navigation, and regional climate along the southern coast of Japan (Nonaka and Xie 2003; Xu et al. 2010; Nakamura et al. 2012), this bimodal feature of the Kuroshio has received much attention (e.g., Chao 1984; Akitomo et al. 1991; Qiu and Miao 2000).

The transition from the NLM path to the LM path proceeds as follows: First, a small meander, also called a “trigger meander” (Solomon 1978), is generated off the southeastern coast of Kyushu through the interaction between the Kuroshio and an anticyclonic mesoscale eddy that approaches the Tokara Strait after propagating westward in the North Pacific or being advected downstream along the Kuroshio in the East China Sea (Mitsudera et al. 2001; Endoh and Hibiya 2001; Usui et al. 2008b; Miyazawa et al. 2008; Usui et al. 2013). The small meander then propagates eastward (downstream) along the Kuroshio up to Cape Shiono-misaki while being amplified through baroclinic interaction with an accompanying cyclone–anticyclone pair in the deep layer (Endoh and Hibiya 2000; Fujii et al. 2008; Usui et al. 2008a). After passing Cape Shiono-misaki, the small meander rapidly amplifies through the interaction with the accompanying abyssal anticyclone trapped over the local topographic feature, Koshu Seamount, located about 200 km south of Cape Shiono-misaki, leading to the formation of the LM path (Endoh and Hibiya 2001; Miyazawa et al. 2004; Tsujino et al. 2006).

Several observational and numerical studies showed the importance of Koshu Seamount for the rapid amplification of the small meander (Endoh and Hibiya 2001; Ambe et al. 2009; Endoh and Hibiya 2009; Endoh et al. 2011; Douglass et al. 2012). For example, using satellite altimeter data from 1993 through 2005, Ambe et al. (2009) showed that, although the generation and subsequent eastward propagation of a small meander were frequently observed throughout this period, the transition from the NLM path to the LM path occurred only when the amplitude of the small meander off Cape Shiono-misaki was large enough for the meander trough to reach over the northern slope of Koshu Seamount. Endoh and Hibiya (2001) were the first to numerically reproduce the transition from the NLM path to the LM path. They demonstrated that when Koshu Seamount was smoothed out in a numerical simulation, the small meander propagated away without being amplified. Furthermore, Endoh et al. (2011) carried out a linear stability analysis using a quasi-geostrophic (QG) equation for a continuously stratified ocean and showed that the Kuroshio became baroclinically most unstable when the water depth was reduced in the offshore direction. From these results, they concluded that enhanced baroclinic instability over the northern slope of Koshu Seamount should be a prerequisite to the formation of the LM path.

However, to simplify the linear stability analysis by allowing a plane wave solution for the QG equation, Endoh et al. (2011) assumed a uniform bottom slope throughout the analysis domain, which is quite different from the actual shape of Koshu Seamount. As a result, the underlying physical mechanism through which Koshu Seamount assists the development of baroclinic instability causing the rapid amplification of the small meander to form the LM path of the Kuroshio has not yet been fully understood.

In this study, the role of Koshu Seamount in enhancing baroclinic instability is investigated using a two-layer QG model, which is the simplest model that can take into account the effects of two-dimensional bottom topography on baroclinic waves. We first reproduce the transition from the NLM path to the LM path of the Kuroshio using a regional inflow–outflow numerical model incorporating idealized geometry and examine the evolution of baroclinic instability over Koshu Seamount causing the rapid amplification of the small meander. We then carry out a linear stability analysis of an idealized geostrophic flow over a seamount and provide a physical interpretation of the baroclinic instability over the seamount. Note that all the existing studies that examined the spatial and temporal characteristics of baroclinic instability (e.g., Smith 2007; Tulloch et al. 2011; Isachsen 2011; Chen and Kamenkovich 2013) ignored the effects of two-dimensional bottom topography as in Endoh et al. (2011), so this study is the first to examine the role of realistic bottom topography in promoting baroclinic instability.

## 2. Numerical experiments

The numerical model in this study is essentially the same as that used by Endoh and Hibiya (2000). The governing equations are the conservation of the QG potential vorticity in a two-layer ocean given by

and

where *t* is time; (*x*, *y*) are the horizontal coordinates; ∇^{2} = ∂^{2}/∂*x*^{2} + ∂^{2}/∂*y*^{2} is the horizontal Laplacian operator; *f* = *f*_{0} + *β*(*x* sin*θ* + *y* cos*θ*) is the Coriolis frequency with *f*_{0} = 7 × 10^{−5} s^{−1}, *β* = 2 × 10^{−11} m^{−1} s^{−1}, and *θ* = 20° (the inclination of the southern coastline of Japan to the zonal direction); *g*′ = 0.02 m s^{−2} is the reduced gravity; *H*_{1} = 500 m and *H*_{2} = 3500 m are the undisturbed upper- and lower-layer thicknesses, respectively; *A*_{H} = 500 m^{2} s^{−1} is the horizontal eddy viscosity coefficient; and *ψ*_{1} and *ψ*_{2} are the streamfunctions in the upper and lower layers, respectively, defined by

with (*u*_{i}, *υ*_{i}) as the horizontal current velocities in the (*x*, *y*) directions, respectively. The model geometry covers part of the western North Pacific south of Japan as well as part of the East China Sea (Fig. 2) with a horizontal grid resolution of 10 km. All the lateral boundaries are assumed to be no slip.

The most important difference between our configuration and that of Endoh and Hibiya (2000) is the inclusion of bottom topography denoted by *η*_{b}, which is assumed here to have the shape of an elliptical cone centered at (*x*, *y*) = (1225 km, 825 km) with the lengths of the major (oriented in the *x* direction) and minor axes of 240 and 160 km, respectively, and a height of 500 m (solid line in Fig. 2), all determined taking into account the actual shape of Koshu Seamount. Note that the relatively large major and minor axes of the seamount are based on the numerical results by Endoh et al. (2011), which showed that even the foot of Koshu Seamount played a crucial role in inducing rapid amplification of the small meander.

The Kuroshio is driven in the upper layer by prescribing outflow at the northeastern corner with a cross-sectional triangular velocity profile over a width of 100 km and inflow at the southwestern corner through boundary conditions given by

(see arrows in Fig. 2). As an initial state, we assume a narrow jet with the same profile as that at the outlet all the way from the inlet through the outlet (inside the dashed line in Fig. 2). After a quasi-steady state is attained through a few years of integration, a pulselike disturbance is superposed onto the Kuroshio in the Tokara Strait; that is, the value of *ψ*_{1} at the coast of the Nansei Islands as well as at the eastern and southern side boundaries is increased linearly from *ψ*_{1b} to *ψ*_{1b} + Δ*ψ*_{1} over 30 days and then decreased also linearly back to *ψ*_{1b} over 30 days to simulate the time variation of the Kuroshio volume transport actually observed in the Tokara Strait when an anticyclonic mesoscale eddy interacting with the Kuroshio generates a small meander off the southeastern coast of Kyushu (Endoh and Hibiya 2000).

To see the effects of Koshu Seamount, an additional numerical experiment is conducted in which the model configuration is kept identical but Koshu Seamount is removed. Although the results presented below are for the values of *ψ*_{1b} = 4.0 × 10^{4} m^{2} s^{−1} and Δ*ψ*_{1} = 0.8 × 10^{4} m^{2} s^{−1}, corresponding to the upper-layer volume transport of 20 and 4 Sv (1 Sv ≡ 10^{6} m^{3} s^{−1}), respectively, qualitatively similar results are obtained for a wide range of values of *ψ*_{1b} and Δ*ψ*_{1}.

## 3. Results

Figure 3 shows the time evolution of the upper- and lower-layer streamfunctions after the pulselike disturbance is superposed on the Kuroshio in the Tokara Strait in the numerical experiment with Koshu Seamount. Note that the streamfunctions are smoothed with a 12-day running mean to remove artificial basinwide oscillations in the lower layer. It can be seen that a small meander is generated off the southeastern coast of Kyushu in the upper layer (*t* ~ 90 days), and it propagates eastward along the Kuroshio while inducing a cyclone–anticyclone pair in the lower layer (*t* ~ 90–120 days). The small meander is gradually amplified together with the lower-layer cyclone–anticyclone pair (*t* ~ 120–180 days), as reproduced by Endoh and Hibiya (2000), who showed that baroclinic instability was responsible for this amplification off the southeastern coast of Kyushu.

The trough of the small meander is then deepened rapidly while passing over Koshu Seamount (*t* ~ 200–260 days). We can also find that, between the cyclone–anticyclone pair in the lower layer, a smaller-scale anticyclone is created over the northwestern slope of Koshu Seamount (*t* ~ 180–200 days). The abyssal anticyclone thus created is amplified simultaneously with the upper-layer meander trough advancing ahead during their eastward propagation over the northern slope of Koshu Seamount (*t* ~ 200–260 days), suggesting the occurrence of baroclinic instability. As the amplitude of the small meander becomes sufficiently large, its eastward propagation slows down and the LM path of the Kuroshio is finally achieved (*t* ~ 290 days, about half a year after the generation of the small meander), which stays nearly stationary just downstream of the seamount until the end of the simulation (*t* ~ 580 days). These transition processes from the NLM path to the LM path are consistent with those reproduced by Endoh and Hibiya (2001) and Endoh et al. (2011) using a three-dimensional numerical model with realistic bottom topography and density stratification. Note that, when the small meander cannot pass over the seamount because of its small initial amplitude (viz., the value of Δ*ψ*_{1}), the transition from the NLM path to the LM path does not occur (not shown), in agreement with the satellite-observed result (Ambe et al. 2009).

In contrast, such rapid amplification of the small meander does not occur when Koshu Seamount is removed (Fig. 4). Although the generation and propagation of the small meander are very similar to those obtained by Endoh and Hibiya (2000) until it reaches over the northern slope of Koshu Seamount (*t* ~ 180 days), the upper-layer meander trough and the lower-layer cyclone–anticyclone pair thereafter do not maintain the spatial structure favorable for baroclinic instability (i.e., the lower-layer cyclone is located just below the upper-layer meander trough; *t* ~ 200–220 days). Consequently, the small meander is not amplified enough to form the LM path of the Kuroshio (*t* ~ 240–290 days) and propagates away from the removed seamount while slowing down (*t* ~ 580 days).

Figure 5 shows the time series of each term of the upper- and lower-layer potential vorticity equations [(1) and (2)] averaged within the area including the cyclonic and anticyclonic disturbances over Koshu Seamount (1180 < *x* < 1400 km and 700 < *y* < 880 km for the upper layer and 1100 < *x* < 1340 km and 760 < *y* < 920 km for the lower layer), respectively, in the numerical experiment with Koshu Seamount. In the lower layer (Fig. 5b), anticyclonic relative vorticity is created at *t* ~ 200–260 days (the solid red line) mainly by the vertical vortex stretching (the black line). In the upper layer (Fig. 5a), on the other hand, small cyclonic relative vorticity is recognized to be created at *t* ~ 180–260 days (the solid red line), not only by the advection of relative vorticity (the dashed red line) but also by the vertical vortex stretching after *t* ~ 230 days (the black line). This joint evolution of the upper-layer cyclone and the lower-layer anticyclone through the vertical vortex stretching indicates that baroclinic interaction between the upper and lower layers is essential in this amplification process (Endoh and Hibiya 2001, 2009).

After *t* ~ 290 days, all the time-dependent terms become relatively small, and a quasi-steady balance is achieved by the advection of relative vorticity, the advection of planetary vorticity, and the vertical vortex stretching (the dashed red line, solid blue line, and black line in Fig. 5, respectively) in both the upper and lower layers. This balance is consistent with the results of previous studies that the LM path of the Kuroshio is maintained as a stationary Rossby wave under the effect of vertical vortex stretching (e.g., Tsujino et al. 2006; Usui et al. 2013; Kurogi et al. 2013).

The development of the small meander through baroclinic instability over Koshu Seamount is also confirmed by calculating the high-wavenumber components of the available potential energy (APE) defined by

where *ρ*_{0} = 1000 kg m^{−3} is the reference water density, and denotes a 200-km Gaussian high-pass filtered streamfunction. Figure 6 compares the time series of the tendency of the high-wavenumber APE integrated over the area 1000 < *x* < 1450 km and 600 < *y* < 1000 km obtained from the numerical experiments with and without Koshu Seamount. When Koshu Seamount exists, the high-wavenumber APE is significantly increased at *t* ~ 200–260 days, resulting in the accumulated high-wavenumber APE about 1.5 times that when Koshu Seamount is removed by *t* ~ 265 days. This difference indicates that the existence of Koshu Seamount enables efficient energy transfer from the large-scale background flow to the small-scale disturbances. Putting all these results together, we can conclude that the transition from the NLM path to the LM path of the Kuroshio, especially the rapid amplification of the small meander by enhanced baroclinic instability over Koshu Seamount, has been successfully reproduced using the two-layer QG model.

## 4. Linear stability analysis

We next carry out a linear stability analysis to provide a physical interpretation for the enhanced baroclinic instability over the seamount in the numerical experiment. The governing equations are the same as used in the numerical experiments but *A*_{H} and *β* are both assumed to be zero such that

and

Decomposing the streamfunction *ψ*_{i}(*x*, *y*, *t*) into the steady background component Ψ_{i}(*x*, *y*) and a time-dependent perturbation component as

and substituting it into Eqs. (6) and (7), we obtain the linear equations for the perturbation components:

and

where *U*_{1} = −∂Ψ_{1}/∂*y* is the background zonal velocity in the upper layer. For simplicity, the other components of the background velocity *U*_{2}, *V*_{1}, and *V*_{2} are all assumed to be zero.

Assuming a harmonic solution for the perturbation component

with *ω* and as the complex frequency and complex amplitude, respectively, and substituting it into Eqs. (9) and (10), we obtain

Equations (12) and (13), together with the appropriate boundary conditions, form an eigenvalue problem with *ω* and as the eigenvalue and corresponding eigenvector, respectively. A cyclic boundary condition is imposed at *x* = 0 and *x* = *L*_{x} where *L*_{x} is the alongshore length of the model domain, whereas a no-normal flow boundary condition is imposed at *y* = 0 and *y* = *L*_{y} where *L*_{y} = 350 km is the cross-shore width of the model domain assumed to be twice the distance between the southern coast of Japan and the center of Koshu Seamount (see Fig. 2). The values of *g*′, *f*_{0}, *H*_{1}, and *H*_{2} and the shape of the seamount are the same as those used in the numerical experiment, whereas *U*_{1} is fixed at a constant value of 0.4 m s^{−1} (the velocity of the Kuroshio averaged over its width of 100 km in the numerical experiment) across the model domain to rule out barotropic instability associated with the horizontal shear of the background flow.

To see if there exist unstable modes prescribed by the seamount, we numerically solve the eigenvalue problem for various values of *L*_{x}, which are found to give the dominant alongshore wavelengths of the obtained unstable modes. Figure 7 shows the growth rate and frequency of the most unstable mode as a function of *L*_{x} for the cases with and without the seamount, respectively. When the ocean bottom is flat (dashed line in Fig. 7), the growth rate, frequency, and spatial structure of the most unstable mode are all as expected from the classical two-layer baroclinic instability theory (e.g., Pedlosky 1987; Vallis 2006), where the maximum growth rate is found at *L*_{x} ~ 540 km (hereafter referred to as a classical unstable mode). The classical unstable mode has a sinusoidal amplitude distribution in the cross-shore direction and propagates eastward at a constant speed with the phase of the upper-layer wave lagging behind that of the lower-layer wave by ~60° (Fig. 8). The small meander amplified off the southeastern coast of Kyushu in the numerical experiment (*t* ~ 120–180 days; Figs. 3b–d) is close to this classical unstable mode.

When a seamount exists (solid line in Fig. 7), a distinctive unstable mode appears at *L*_{x} ~ 365 km (hereafter referred to as a seamount unstable mode), whose maximum growth rate is comparable to that of the classical unstable mode slightly modified by the existence of the seamount (*L*_{x} ~ 620 km; the spatial structure of this mode is very similar to that shown in Fig. 8). The spatial structure of the seamount unstable mode is shown in Fig. 9 when *L*_{x} = 365 km. The frequency and growth rate of this mode are 2.6 × 10^{−6} s^{−1} (a period of 28 days) and 6.1 × 10^{−7} s^{−1} (an *e*-folding time of 52 days), respectively. In the upper layer, the wave propagates eastward with the amplitude nearly uniform in the alongshore direction and sinusoidal in the cross-shore direction. In the lower layer, in contrast, the wave propagates clockwise around the seamount like a seamount-trapped wave but with a larger amplitude over the northern slope. The phase of the upper-layer wave lags behind that of the northern part of the lower-layer wave by ~60°, enabling the energy conversion from the background to the perturbation field through baroclinic instability.

The spatial structure and the temporal evolution of the seamount unstable mode are now compared with those of the small meander numerically reproduced over Koshu Seamount (Fig. 10). Although the cross-shore structure of the upper-layer component of the unstable mode is restricted by the width of the model domain used for the analysis, the alongshore scale of the upper-layer component of the unstable mode (~365 km) becomes nearly the same as that of the small meander obtained from the numerical experiment. Furthermore, the spatial structure of the lower-layer component of the unstable mode is largely determined by the shape of the seamount so that becomes comparable to that of the abyssal cyclone/anticyclone obtained from the numerical experiment. However, the time required for the amplification of the small meander is about 3 times that for the amplification of the seamount unstable mode (Figs. 10a–c vs Figs. 10e–g). In addition, the similarity in the spatial structure between the small meander and seamount unstable mode is limited to the time period of *t* ~ 200–240 days, namely, the early amplification stage (see also Figs. 5 and 6). In the later amplification stage (*t* ~ 240–260 days), the upper-layer meander trough no longer propagates eastward and the lower-layer anticyclone occupies the entire seamount (Fig. 10d). These discrepancies are probably caused by the fact that the small meander is amplified such that it cannot be treated just as a perturbation advected by the background flow, thus invalidating the application of the linear theory (e.g., Kubokawa 1989). Another possible explanation for these discrepancies is that the background flow assumed to be uniform in the linear stability analysis is different from that in the numerical experiment. In spite of these discrepancies, the similarity in the spatial scale and structure between the small meander and seamount unstable mode at the early amplification stage leads us to conclude that we have succeeded in identifying the baroclinic instability causing rapid amplification of the small meander (Endoh and Hibiya 2001, 2009; Endoh et al. 2011).

Finally, we examine the properties of the upper- and lower-layer waves, assuming that they are weakly coupled with each other. In this case, the characteristic features of the upper-layer (lower layer) wave in the original two-layer model can be examined using a 1.5-layer model with a finite upper layer (lower layer) above (below) an infinite lower layer (upper layer) (e.g., Iga 1993). The dispersion relation for the upper-layer Rossby waves in the background zonal flow is obtained by substituting a plane wave solution of into Eq. (12) with such that

where *k* and *l* = *π*/*L*_{y} are the wavenumbers in the alongshore and cross-shore directions, respectively. The spatial structure and propagation characteristics of the upper-layer component of the seamount unstable wave (Fig. 9a) are similar to those of this Rossby wave. In the lower layer, on the other hand, a topographically trapped wave propagating clockwise around the seamount (Fig. 11a) is modulated by the upper-layer eastward background flow (Fig. 11b). Note that these two modal structures are obtained by numerically solving Eq. (13) for *U*_{1} = 0 and 0.4 m s^{−1}, respectively, with the assumption that . The spatial structure and propagation characteristics of the lower-layer component of the seamount unstable wave (Fig. 9b) are similar to those of the seamount-trapped wave modulated by the upper-layer background flow (Fig. 11b).

The above result suggests that the seamount unstable wave is interpreted as a coupling between these two kinds of waves. In fact, for a fixed-shape seamount, the seamount unstable wave becomes most unstable when its frequency is close to that of the lower-layer seamount-trapped wave (Fig. 12a), and its dominant wavelength nearly satisfies the dispersion relation for the upper-layer Rossby wave (Fig. 12b). This suggests that the seamount-trapped wave prescribes the existence of the coupled unstable wave in terms of the temporal scale and not the spatial scale. Figures 10e–g further illustrate that the upper- and lower-layer waves grow together through resonant interactions during their eastward propagation over the northern slope of the seamount.

## 5. Summary and discussion

In this study, using a two-layer QG model, we have investigated the effects of Koshu Seamount on the development of baroclinic instability leading to the formation of the LM path of the Kuroshio. A linear stability analysis has shown that a distinctive unstable mode (a seamount unstable mode) grows as a result of coupling between the upper-layer Rossby wave propagating eastward in the background flow and the lower-layer topographically trapped wave propagating clockwise around the seamount. These two kinds of waves propagate in the same direction over the northern slope of the seamount while resonantly interacting with each other. Numerical experiments have shown that the transition from the NLM path to the LM path of the Kuroshio can be successfully reproduced only when bottom topography mimicking Koshu Seamount is incorporated. In this case, the upper-layer meander trough is rapidly amplified together with the lower-layer anticyclone during their passage over the northern slope of the seamount. The spatial scale and structure of the small meander during its early amplification stage are close to those of the seamount unstable mode. These results suggest that this seamount unstable mode is responsible for the baroclinic instability over Koshu Seamount pointed out by Endoh and Hibiya (2001, 2009) and Endoh et al. (2011) as an essential process for the rapid amplification of the small meander.

There are some limitations with the linear framework of this study. First, as described in the previous section, the time scale required for the small meander to grow while passing over the seamount is much longer than that required for the seamount unstable mode. In addition, the similarity in the spatial structure between the small meander and the seamount unstable mode is limited to the early stage of their amplification. To overcome these limitations, a nonlinear instability theory taking into account the effects of isolated bottom topography that allows the existence of bottom-trapped waves is indispensable.

Apart from these remaining issues, this study is the first to give a physical interpretation of the development of baroclinic instability enhanced over Koshu Seamount in terms of the existence of a seamount-trapped wave. We believe that this study provides a useful first step toward a general understanding of the role of variable bottom topography in controlling global ocean circulation through enhancing baroclinic instability.

## Acknowledgments

This work was supported in part by the Japan Society for the Promotion of Science through Grants-in-Aid for Scientific Research (Grant Numbers 25800260 and 17K14389). Figures were produced using the GFD-DENNOU Library.

## REFERENCES

*Geophysical Fluid Dynamics*. 2nd ed. Springer-Verlag, 710 pp.

*Kuroshio—Its Physical Aspects*, H. Stommel and K. Yoshida, Eds., University of Tokyo Press, 217–234.

*Kuroshio—Its Physical Aspects*, H. Stommel and K. Yoshida, Eds., University of Tokyo Press, 165–216.

*Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation.*Cambridge University Press, 745 pp.

## Footnotes

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).