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- Author or Editor: Huijun Yang x

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## Abstract

A concept of the δ-approxiniation for the earth's surface has been introduced. Using the Rossby wave packet approximation and the WKB method, the evolution of a single geostrophic synoptic disturbance system has been further studied on the δ-surface of the earth. The global behavior of the structural changes of the wave packet due to the zonal, the meridional and the asymmetric basic currents and the variety of the topography on the δ-surface of the earth have been discussed and compared with those an the earth's,β-plane by using the WKB phase plane, i.e., the wave packet's local wavenumber phase plane. The results show that the governing system on the earth's δ-surface may be dynamically different from that on the earth's β-plane. Moreover, the wave packet structural vacillation has been found on both the β-plane and the δ-surface. Wave packet structural vacillation is characterized by the time-periodic changes of the wave packet's structure. Both the tilt and the spatial scales of the packet will evolve periodic changes simultaneously. The wave packet structural vacillations are also characterized by the closed WKB trajectories on the WKB phase plane. The results show that in the presence of the asymmetric basic currents, the WKB trajectories on the WKB phase plane appear simply to be elliptical, e.g., in the case of a southwesterly jet, or hyperbolic, e.g., in the case of a southeasterly jet. The results suggest that it is possible for the packet structural vacillations to exist in the presence of some asymmetric basic currents, e.g., a southwesterly jet. The behaviors related to topography in various distributions have also been discussed. It has been demonstrated that the quadratic east-west oriented topography modifies only the δ-effect, and that with some topographies, e.g., convex topographies, the wave packet structural vacillation can also exist. In some cases, however, the behaviors of the evolution of a packet will be qualitatively different on the earth's δ-surface from those on the earth's β-plane. For example, in the meridional basic current or on the north-south oriented topographies. only on the δ-surface of the earth do there exist such wave packet structural vacillations. On the other hand, in some cases, the wave packet solutions have been obtained on both the β-plane and the δ-surface. The wave packet vacillation suggests a possible mechanism of vacillations observed in the atmosphere.

## Abstract

A concept of the δ-approxiniation for the earth's surface has been introduced. Using the Rossby wave packet approximation and the WKB method, the evolution of a single geostrophic synoptic disturbance system has been further studied on the δ-surface of the earth. The global behavior of the structural changes of the wave packet due to the zonal, the meridional and the asymmetric basic currents and the variety of the topography on the δ-surface of the earth have been discussed and compared with those an the earth's,β-plane by using the WKB phase plane, i.e., the wave packet's local wavenumber phase plane. The results show that the governing system on the earth's δ-surface may be dynamically different from that on the earth's β-plane. Moreover, the wave packet structural vacillation has been found on both the β-plane and the δ-surface. Wave packet structural vacillation is characterized by the time-periodic changes of the wave packet's structure. Both the tilt and the spatial scales of the packet will evolve periodic changes simultaneously. The wave packet structural vacillations are also characterized by the closed WKB trajectories on the WKB phase plane. The results show that in the presence of the asymmetric basic currents, the WKB trajectories on the WKB phase plane appear simply to be elliptical, e.g., in the case of a southwesterly jet, or hyperbolic, e.g., in the case of a southeasterly jet. The results suggest that it is possible for the packet structural vacillations to exist in the presence of some asymmetric basic currents, e.g., a southwesterly jet. The behaviors related to topography in various distributions have also been discussed. It has been demonstrated that the quadratic east-west oriented topography modifies only the δ-effect, and that with some topographies, e.g., convex topographies, the wave packet structural vacillation can also exist. In some cases, however, the behaviors of the evolution of a packet will be qualitatively different on the earth's δ-surface from those on the earth's β-plane. For example, in the meridional basic current or on the north-south oriented topographies. only on the δ-surface of the earth do there exist such wave packet structural vacillations. On the other hand, in some cases, the wave packet solutions have been obtained on both the β-plane and the δ-surface. The wave packet vacillation suggests a possible mechanism of vacillations observed in the atmosphere.

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## Abstract

Using the Rossby wave packet approximation and the WKB method, the evolution of a single geostrophic synoptic disturbance has been studied. The structural changes of the wave packet due to the variation of β with latitude, the asymmetric basic currents and the variety of topography are thoroughly discussed. However, there are different effects on the wave packet in the various asymmetric basic currents, depending on the different positions of the wave packet relative to the basic current and the variety of topography. The δ-effect always lengthens the Rossby wave packet's longitudinal scale and causes the westward-tilting trough line to lean toward the *Y*-axis, i.e., to the north. When the Rossby wave packet is located to the left (right) side of a southwesterly jet its longitudinal scale (latitudinal scale) will lengthen and its latitudinal scale (longitudinal scale) will shrink, while the westward-tilting trough line will become more westward (toward the *Y*-axis). Linearly sloping topographies will not affect the structure of the Rossby wave packet, but nonlinear topographies do affect the structure of the packet. The results suggest that the mountains, especially the Rocky Mountains, may decrease (increase) the *X*-propagating disturbance system when it is westward- (eastward) tilted. The effects of topography on distributions of east-west oriented, north-south oriented, convex and concave models have been discussed in detail. Two examples of the entire evolution of the Rossby wave packet are also presented.

## Abstract

Using the Rossby wave packet approximation and the WKB method, the evolution of a single geostrophic synoptic disturbance has been studied. The structural changes of the wave packet due to the variation of β with latitude, the asymmetric basic currents and the variety of topography are thoroughly discussed. However, there are different effects on the wave packet in the various asymmetric basic currents, depending on the different positions of the wave packet relative to the basic current and the variety of topography. The δ-effect always lengthens the Rossby wave packet's longitudinal scale and causes the westward-tilting trough line to lean toward the *Y*-axis, i.e., to the north. When the Rossby wave packet is located to the left (right) side of a southwesterly jet its longitudinal scale (latitudinal scale) will lengthen and its latitudinal scale (longitudinal scale) will shrink, while the westward-tilting trough line will become more westward (toward the *Y*-axis). Linearly sloping topographies will not affect the structure of the Rossby wave packet, but nonlinear topographies do affect the structure of the packet. The results suggest that the mountains, especially the Rocky Mountains, may decrease (increase) the *X*-propagating disturbance system when it is westward- (eastward) tilted. The effects of topography on distributions of east-west oriented, north-south oriented, convex and concave models have been discussed in detail. Two examples of the entire evolution of the Rossby wave packet are also presented.

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## Abstract

Based on previous theory, which was made by using the Rossby wave-packet approximation and the WKB method, the evolution of a single geostrophic synoptic disturbance system has been further investigated. In the presence of both the basic current and topography, bifurcation properties of the evolution of the wave packet due to symmetric topography and asymmetric topography have been studied analytically as the topography parameter changes, using bifurcation diagrams and the WKB phase space, i.e., the local wavenumber phase span.

The results show that both topography and the basic current play very important roles in the bifurcation properties of the dynamics of geophysical flows. The results also show that the topological structure of the evolution of a Rossby wave packet in various basic currents and topographies can vary accordingly. Both subcritical and supercritical bifurcations have been found analytically. Moreover, reverse supercritical and sub-critical bifurcations also have been found. The effect of a zonal basic current on the bifurcation is different from that of a meridional basic current. For example, on concave topography the bifurcation will be supercritical (subcritical) when the packet is located on the left-hand side (right-hand side) of a westerly jet. However, in a meridional basic current, the evolution on the concave topography exhibits a reverse supercritical (subcritical) bifurcation on the left- (right) hand side of the current. On symmetric topography there exist only two kinds of equilibria, i.e., the largest scale state and the two pure latitudinal-scale states or the pure longitudinal-scale states. On asymmetric topography mixed-scale equilibrium states exist. The existence of the mixed-scale state suggests that there is a different kind of evolution for a Rossby wave packet. The results suggest that the evolution of a Rossby wave packet on one side of a basic current could be different from that on the other side, which is expected in real geophysical flows. At the center of a basic current, the dynamical system of the Rossby wave packet is structurally unstable.

The “homoclinic orbits” and the ”heteroclinic orbits” also have been discussed, which are separatrices of the wave-packet vacillations in the WKB phase space. The structural vacillations in the Rossby wave packet and their implications in geophysical flows associated with the bifurcation properties of the evolution of a Rossby wave packet are investigated. The results suggest that the transitions between two kinds of wave-packet structural vacillations only can occur on one side of the basic current as the topography parameter is varied.

## Abstract

Based on previous theory, which was made by using the Rossby wave-packet approximation and the WKB method, the evolution of a single geostrophic synoptic disturbance system has been further investigated. In the presence of both the basic current and topography, bifurcation properties of the evolution of the wave packet due to symmetric topography and asymmetric topography have been studied analytically as the topography parameter changes, using bifurcation diagrams and the WKB phase space, i.e., the local wavenumber phase span.

The results show that both topography and the basic current play very important roles in the bifurcation properties of the dynamics of geophysical flows. The results also show that the topological structure of the evolution of a Rossby wave packet in various basic currents and topographies can vary accordingly. Both subcritical and supercritical bifurcations have been found analytically. Moreover, reverse supercritical and sub-critical bifurcations also have been found. The effect of a zonal basic current on the bifurcation is different from that of a meridional basic current. For example, on concave topography the bifurcation will be supercritical (subcritical) when the packet is located on the left-hand side (right-hand side) of a westerly jet. However, in a meridional basic current, the evolution on the concave topography exhibits a reverse supercritical (subcritical) bifurcation on the left- (right) hand side of the current. On symmetric topography there exist only two kinds of equilibria, i.e., the largest scale state and the two pure latitudinal-scale states or the pure longitudinal-scale states. On asymmetric topography mixed-scale equilibrium states exist. The existence of the mixed-scale state suggests that there is a different kind of evolution for a Rossby wave packet. The results suggest that the evolution of a Rossby wave packet on one side of a basic current could be different from that on the other side, which is expected in real geophysical flows. At the center of a basic current, the dynamical system of the Rossby wave packet is structurally unstable.

The “homoclinic orbits” and the ”heteroclinic orbits” also have been discussed, which are separatrices of the wave-packet vacillations in the WKB phase space. The structural vacillations in the Rossby wave packet and their implications in geophysical flows associated with the bifurcation properties of the evolution of a Rossby wave packet are investigated. The results suggest that the transitions between two kinds of wave-packet structural vacillations only can occur on one side of the basic current as the topography parameter is varied.

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## Abstract

Based on a simplified mathematical model (Yang), using the Rossby wave packet approximation and the WKB method, the bifurcation properties of the evolution of a single geostrophic synoptic disturbance system were further studied. This analytical investigation was in the presence of an asymmetric basic current and utilized general topography as the topography parameter changes, using bifurcation diagrams and the WKB phase space.

The results showed that both the basic current and topography play very important roles in the bifurcation properties of geophysical flows. The results suggest that with the same type of topography, topological structure on one side of an asymmetric basic current will be different from that on the other side. On one side of an asymmetric basic current there exists only the primary bifurcation with three equilibrium states. However, on the other side of the asymmetric basic current, there are two distinct bifurcations. As the topography parameter reaches the first critical value, the primary bifurcation with five equilibrium states occurs. The equilibrium states are the 1argest spatial-scale state, two pure longitudinal-scale status and two pure latitudinal-scale status. As the topography parameter is increased further to the second critical value, a secondary bifurcation with the mixed-scale equilibrium states occurs. The evolution of a Rossby wave packet could exhibit the supercritical and subcritical primary bifurcations and the reverse supercritical and subcritical primary bifurcations accordingly, as the topography parameter varies. The secondary bifurcation, however, is the transcritical bifurcation. The WKB trajectories in the phase space are discussed for the different topography parameters. Both homoclinic orbits and heteroclinic orbits exist, as the separatrices of the wave packet structural vacillations. It has been shown that for the stable mixed-scale domain of the WKB phase space. For the unstable mixed-scale states the trajectories of the evolution constitute a family hyperbolic curves in the WKB phase space.

The structural vacilliations in the Rossby wave packet and their implications in geophysical flows are investigated. The results suggest that on side of an asymmetric basic current, the transitions can occur among three kinds of wave structural vacillations, while on the other side of the asymmetric basic current, the transitions can occur among three kinds of wave packet structural vacillations, while on the other side of the asymmetric basic current the transitions can only occur between two kinds of wave packet structural vacillations, as the topography parameter is varied.

## Abstract

Based on a simplified mathematical model (Yang), using the Rossby wave packet approximation and the WKB method, the bifurcation properties of the evolution of a single geostrophic synoptic disturbance system were further studied. This analytical investigation was in the presence of an asymmetric basic current and utilized general topography as the topography parameter changes, using bifurcation diagrams and the WKB phase space.

The results showed that both the basic current and topography play very important roles in the bifurcation properties of geophysical flows. The results suggest that with the same type of topography, topological structure on one side of an asymmetric basic current will be different from that on the other side. On one side of an asymmetric basic current there exists only the primary bifurcation with three equilibrium states. However, on the other side of the asymmetric basic current, there are two distinct bifurcations. As the topography parameter reaches the first critical value, the primary bifurcation with five equilibrium states occurs. The equilibrium states are the 1argest spatial-scale state, two pure longitudinal-scale status and two pure latitudinal-scale status. As the topography parameter is increased further to the second critical value, a secondary bifurcation with the mixed-scale equilibrium states occurs. The evolution of a Rossby wave packet could exhibit the supercritical and subcritical primary bifurcations and the reverse supercritical and subcritical primary bifurcations accordingly, as the topography parameter varies. The secondary bifurcation, however, is the transcritical bifurcation. The WKB trajectories in the phase space are discussed for the different topography parameters. Both homoclinic orbits and heteroclinic orbits exist, as the separatrices of the wave packet structural vacillations. It has been shown that for the stable mixed-scale domain of the WKB phase space. For the unstable mixed-scale states the trajectories of the evolution constitute a family hyperbolic curves in the WKB phase space.

The structural vacilliations in the Rossby wave packet and their implications in geophysical flows are investigated. The results suggest that on side of an asymmetric basic current, the transitions can occur among three kinds of wave structural vacillations, while on the other side of the asymmetric basic current, the transitions can occur among three kinds of wave packet structural vacillations, while on the other side of the asymmetric basic current the transitions can only occur between two kinds of wave packet structural vacillations, as the topography parameter is varied.

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## Abstract

In this study the author investigates the 3D evolution of the Ertel potential vorticity (PV) and N_{2}O in Northern Hemisphere winter from 1 to 10 February of model year 1984 in the GFDL SKYHI model. The diagnosis was done on three isentropic surfaces—800, 450, and 320 K—by using output from the GFDL SKYHI model, which has 1.2° × 1° horizontal resolution, 40 layers in vertical, and 60-s time resolution. The data output was taken twice daily. The high resolution Lagrangian Field Advection Model (FAM), which has no diabatic heating and diffusion, is used to study the evolution of these two fields. The 1° × 1° resolution in FAM gives reasonable results. The following results are found. 1) In the stratosphere there are two kinds of barriers, that is, subtropical barrier and polar barrier for both PV and N_{2}O. The existence of the subtropical barrier is coincident with the subtropical jet. The subtropical barrier is more permeable than the polar barrier. Even though the tropopause acts like a barrier, there is substantial exchange between the troposphere and the stratosphere at 320 K. 2) The inside edge of the polar vortex is marked by a nearby high-PV ring, and the outside edge is indicated by the maximum gradient of N_{2}O at 800 K. 3) On all three isentropic surfaces, both PV and N_{2}O are conserved quite well. 4) Poleward transport from the Tropics to the high latitudes and equaterward transport from the high latitudes to the middle latitudes due to Rossby wave breaking takes place simultaneously and is captured remarkably well in FAM. 5) The polar vortex is kinematically isolated from outside both on the 800- and the 450-K surfaces, with a small amount of outside air entrainment into the edge of the polar vortex on the 450-K surface. The probability distribution function and the finite-time Lyapunov exponents are proven to be useful to characterize the structure and mixing properties of PV and N_{2}O. 6) The transport and mixing channels between the Tropics and the high latitudes have been identified by the positive high Lyapunov exponents. 7) Three clear separate peaks in the PDFs identify with three distinct regions of N_{2}O (tropical reservoir, surf zone, and polar vortex), bounded by the subtropical barrier and the polar barrier between them, suggesting that the mixing may occur separately in each region in the stratosphere. The implications of these findings are discussed briefly.

## Abstract

In this study the author investigates the 3D evolution of the Ertel potential vorticity (PV) and N_{2}O in Northern Hemisphere winter from 1 to 10 February of model year 1984 in the GFDL SKYHI model. The diagnosis was done on three isentropic surfaces—800, 450, and 320 K—by using output from the GFDL SKYHI model, which has 1.2° × 1° horizontal resolution, 40 layers in vertical, and 60-s time resolution. The data output was taken twice daily. The high resolution Lagrangian Field Advection Model (FAM), which has no diabatic heating and diffusion, is used to study the evolution of these two fields. The 1° × 1° resolution in FAM gives reasonable results. The following results are found. 1) In the stratosphere there are two kinds of barriers, that is, subtropical barrier and polar barrier for both PV and N_{2}O. The existence of the subtropical barrier is coincident with the subtropical jet. The subtropical barrier is more permeable than the polar barrier. Even though the tropopause acts like a barrier, there is substantial exchange between the troposphere and the stratosphere at 320 K. 2) The inside edge of the polar vortex is marked by a nearby high-PV ring, and the outside edge is indicated by the maximum gradient of N_{2}O at 800 K. 3) On all three isentropic surfaces, both PV and N_{2}O are conserved quite well. 4) Poleward transport from the Tropics to the high latitudes and equaterward transport from the high latitudes to the middle latitudes due to Rossby wave breaking takes place simultaneously and is captured remarkably well in FAM. 5) The polar vortex is kinematically isolated from outside both on the 800- and the 450-K surfaces, with a small amount of outside air entrainment into the edge of the polar vortex on the 450-K surface. The probability distribution function and the finite-time Lyapunov exponents are proven to be useful to characterize the structure and mixing properties of PV and N_{2}O. 6) The transport and mixing channels between the Tropics and the high latitudes have been identified by the positive high Lyapunov exponents. 7) Three clear separate peaks in the PDFs identify with three distinct regions of N_{2}O (tropical reservoir, surf zone, and polar vortex), bounded by the subtropical barrier and the polar barrier between them, suggesting that the mixing may occur separately in each region in the stratosphere. The implications of these findings are discussed briefly.

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## Abstract

The thermodynamical process of latent heat flux is added to an analogical delayed oscillator model of the El Niño–Southern Oscillation (ENSO) that mainly considers equatorial ocean dynamics and produces regular, non–phase-locked oscillations. Latent heat flux affects the model sea surface temperature (SST) variations by a positive feedback between the surface wind speed and SST operating through evaporation, which is called the wind speed–evaporation–SST feedback. The wind speed–evaporation–SST feedback in which the atmosphere interacts thermodynamically with the ocean through surface heat flux differs from the conventional zonal wind stress–SST feedback in which the atmsophere interacts dynamically with the ocean through momentum flux.

The combination of equatorial ocean dynamics and thermodynamics produces relatively more realistic model oscillations. When the annual cycle amplitude of the zonal wind in the wind speed–evaporation–SST feedback is gradually increased, the model solution undergoes a transition from periodic to chaotic and then to periodic oscillations for some ranges of the parameters, whereas for other ranges of the parameters the transition goes from periodic to quasiperiodic and then to periodic oscillations. The route to chaos is the intermittency route. Along with such irregularity, the nonlinear interactions between the annual and interannual cycles operating through the wind speed–evaporation–SST feedback also produce a phase-locking of ENSO to the seasonal cycle. The model ENSO onset and peak occur in the boreal winter and spring, respectively, consistent with the observed phase-locking of ENSO in the far eastern Pacific. It is shown that ENSO decadal or interdecadal variability may result from the nonlinear interactions between the annual and interannual cycles in the Tropics.

## Abstract

The thermodynamical process of latent heat flux is added to an analogical delayed oscillator model of the El Niño–Southern Oscillation (ENSO) that mainly considers equatorial ocean dynamics and produces regular, non–phase-locked oscillations. Latent heat flux affects the model sea surface temperature (SST) variations by a positive feedback between the surface wind speed and SST operating through evaporation, which is called the wind speed–evaporation–SST feedback. The wind speed–evaporation–SST feedback in which the atmosphere interacts thermodynamically with the ocean through surface heat flux differs from the conventional zonal wind stress–SST feedback in which the atmsophere interacts dynamically with the ocean through momentum flux.

The combination of equatorial ocean dynamics and thermodynamics produces relatively more realistic model oscillations. When the annual cycle amplitude of the zonal wind in the wind speed–evaporation–SST feedback is gradually increased, the model solution undergoes a transition from periodic to chaotic and then to periodic oscillations for some ranges of the parameters, whereas for other ranges of the parameters the transition goes from periodic to quasiperiodic and then to periodic oscillations. The route to chaos is the intermittency route. Along with such irregularity, the nonlinear interactions between the annual and interannual cycles operating through the wind speed–evaporation–SST feedback also produce a phase-locking of ENSO to the seasonal cycle. The model ENSO onset and peak occur in the boreal winter and spring, respectively, consistent with the observed phase-locking of ENSO in the far eastern Pacific. It is shown that ENSO decadal or interdecadal variability may result from the nonlinear interactions between the annual and interannual cycles in the Tropics.