Spatial sampling of the thermospheric vertical wind field at auroral latitudes

Results are presented from two nights of bistatic Doppler measurements of neutral thermospheric winds using Fabry!Perot spectrometers at Mawson and Davis stations in Antarctica. A scanning Doppler imager (SDI) at Mawson and a narrow-field Fabry-Perot spectrometer (FPS) at Davis have been used to estimate the vertical wind at three locations along the great circle joining the two stations, in addition to the vertical wind routinely observed above each station. These data were obtained from observations of the 630.0 nm airglow line of atomic oxygen, at a nominal altitude of 240 km. Low!resolution all-sky images produced by the Mawson SDI have been used to relate disturbances in the measured vertical wind field to auroral activity and divergence in the horizontal wind field. Correlated vertical wind responses were observed on a range of horizontal scales from ~150 to 480 km. In general, the behavior of the vertical wind was in agreement with earlier studies, with strong upward winds observed poleward of the optical aurora and sustained, though weak, downward winds observed early in the night. The relation between vertical wind and horizontal divergence was seen to follow the general trend predicted by Burnside et al. (1981), whereby upward vertical winds were associated with positive divergence and vice versa; however, a scale height approximately 3–4 times greater than that modeled by NRLMSISE-00 was required to best fit the data using this relation

Further refinements of the Heinz inequality

The celebrated Heinz inequality asserts that $2|||A^{1/2}XB^{1/2}|||\leq |||A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}|||\leq |||AX+XB|||$ for $X \in \mathbb{B}(\mathscr{H})$, A,B\in \+, every unitarily invariant norm $|||\cdot|||$ and $\nu \in [0,1]$. In this paper, we present several improvement of the Heinz inequality by using the convexity of the function $F(\nu)=|||A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}|||$, some integration techniques and various refinements of the Hermite--Hadamard inequality. In the setting of matrices we prove that \begin{eqnarray*} &&\hspace{-0.5cm}\left|\left|\left|A^{\frac{\alpha+\beta}{2}}XB^{1-\frac{\alpha+\beta}{2}}+A^{1-\frac{\alpha+\beta}{2}}XB^{\frac{\alpha+\beta}{2}}\right|\right|\right|\leq\frac{1}{|\beta-\alpha|} \left|\left|\left|\int_{\alpha}^{\beta}\left(A^{\nu}XB^{1-\nu}+A^{1-\nu}XB^{\nu}\right)d\nu\right|\right|\right|\nonumber\\ &&\qquad\qquad\leq \frac{1}{2}\left|\left|\left|A^{\alpha}XB^{1-\alpha}+A^{1-\alpha}XB^{\alpha}+A^{\beta}XB^{1-\beta}+A^{1-\beta}XB^{\beta}\right|\right|\right|\,, \end{eqnarray*} for real numbers $\alpha, \beta$.Comment: 15 pages, to appear in Linear Algebra Appl. (LAA