The Efficiency of Upward Wave Propagation Near the Tropopause: importance of the form of the refractive index

The connection between the polar stratospheric vortex and the vertical component of the Eliassen-Palm flux in the lower stratosphere and upper troposphere is examined in model level data from the ERA-5 reanalysis. The particular focus of this work is on the conditions that lead to upward wave propagation between the tropopause and the bottom of the vortex near 100 hPa. The ability of four different versions of the index of refraction to capture this wave propagation are evaluated. The original Charney and Drazin index of refraction includes terms ignored by Matsuno that are shown to be critical for understanding upward wave propagation just above the tropopause in both the climatology and during extreme heat flux events. By adding these terms to the Matsuno index of refraction, it is possible to construct a useful tool that describes wave flux immediately above the tropopause and at the same time also describes the role of meridional variations within the stratosphere. It is shown that a stronger tropopause inversion layer tends to restrict upward wave propagation. It is also shown that while only 38% of extreme wave-1 Eliassen-Palm flux vertical component (Fz) at 100hPa events are preceded by extreme Fz at 300hPa, there are almost no extreme events at 100hPa in which the anomaly at 300hPa is of opposite sign or very weak. Overall, wave propagation near the tropopause is sensitive to vertical gradients in buoyancy frequency, and these vertical gradients may not be accurately captured in models or reanalysis products especially with lower vertical resolutions.

The existence and representation of the Drazin inverse of a 2 × 2 block matrix over a ring

In this paper, further results on the Drazin inverse are obtained in a ring. Several representations of the Drazin inverse of [Formula: see text] block matrices over an arbitrary ring are given under new conditions. Also, upper bounds for the Drazin index of block matrices are studied. Numerical examples are given to illustrate our results. Necessary and sufficient conditions for the existence as well as the expression of the group inverse of block matrices are obtained under certain conditions. In particular, some results of related papers which were considered for complex matrices, operator matrices and matrices over a skew field are extended to more general setting.

Bordering method to compute Core-EP inverse

Following the work of Kentaro Nomakuchi[10] and Manjunatha Prasad et.al., [7] which relate various generalized inverses of a given matrix with suitable bordering,we describe the explicit bordering required to obtain core-EP inverse, core-EP generalized inverse. The main result of the paper also leads to provide a characterization of Drazin index in terms of bordering.

A Note on the Drazin Indices of Square Matrices

For a square matrixA, the smallest nonnegative integerksuch that rank (Ak) = rank (Ak+1) is called the Drazin index ofA. In this paper, we give some results on the Drazin indices of sum and product of square matrices.

GENERALIZED INVERSES OF A SUM IN RINGS

We study properties of the Drazin index of regular elements in a ring with a unity 1. We give expressions for generalized inverses of 1−ba in terms of generalized inverses of 1−ab. In our development we prove that the Drazin index of 1−ba is equal to the Drazin index of 1−ab.

On the Drazin index of regular elements

It is known that the existence of the group inverse a # of a ring element a is equivalent to the invertibility of a 2 a − + 1 − aa −, independently of the choice of the von Neumann inverse a − of a. In this paper, we relate the Drazin index of a to the Drazin index of a 2 a − + 1 − aa −. We give an alternative characterization when considering matrices over an algebraically closed field. We close with some questions and remarks.