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A Dynamical Model of “Quasi-stationary” States in Large-Scale Atmospheric Motions

H. MukougawaGeophysical Institute, Kyoto University, Kyoto, Japan

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Abstract

A theoretical study is made to establish the concept of “quasi-stationary (QS)” states in irregular, large-scale atmospheric motions by using a simple model. It is a truncated, two-layer, quasi-geostrophic model in a β-channel with surface topography, external thermal forcing (i.e., meridional differential heating) and dissipative processes. We define a QS state as a period when the magnitude of the time derivative of spectral components, |x˙(t)|, is smaller than a prescribed threshold value.

For a wide range of parameter values of the thermal forcing, each QS state is found to occur when the trajectory x(t) passes immediately next to a local minimum point (MP) of |x˙| in phase space; the MP is either a stationary solution or a nonstationary minimum point bifurcating from a limit point of a stationary solution. Moreover, we give the statistical significance to this relation between each QS state and an MP. The relation is almost independent of the threshold value defining the QS states. Therefore, the recurrence of the QS states is assured.

In irregular motions, some MPs are preferably selected to generate QS states independent of initial conditions. This selection does not depend on the local dynamical property of each MP, but is attributed to the inhomogeneous probability density distribution of the global trajectory in phase space.

The probability that each QS gate persists over n days is well expressed as exp(−n/τ), where τ is a characteristic time. This probability distribution is analytically deduced from a simple linearized system around an MP, We also obtain some examples of the linear growth of an unstable eigenmode of the nearest MP at the end of a QS state. These statistical and dynamical properties of the QS state support the relation between each QS state ,and an MP.

Abstract

A theoretical study is made to establish the concept of “quasi-stationary (QS)” states in irregular, large-scale atmospheric motions by using a simple model. It is a truncated, two-layer, quasi-geostrophic model in a β-channel with surface topography, external thermal forcing (i.e., meridional differential heating) and dissipative processes. We define a QS state as a period when the magnitude of the time derivative of spectral components, |x˙(t)|, is smaller than a prescribed threshold value.

For a wide range of parameter values of the thermal forcing, each QS state is found to occur when the trajectory x(t) passes immediately next to a local minimum point (MP) of |x˙| in phase space; the MP is either a stationary solution or a nonstationary minimum point bifurcating from a limit point of a stationary solution. Moreover, we give the statistical significance to this relation between each QS state and an MP. The relation is almost independent of the threshold value defining the QS states. Therefore, the recurrence of the QS states is assured.

In irregular motions, some MPs are preferably selected to generate QS states independent of initial conditions. This selection does not depend on the local dynamical property of each MP, but is attributed to the inhomogeneous probability density distribution of the global trajectory in phase space.

The probability that each QS gate persists over n days is well expressed as exp(−n/τ), where τ is a characteristic time. This probability distribution is analytically deduced from a simple linearized system around an MP, We also obtain some examples of the linear growth of an unstable eigenmode of the nearest MP at the end of a QS state. These statistical and dynamical properties of the QS state support the relation between each QS state ,and an MP.

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