This paper presents an efficient algorithm to solve a constrained optimization problem with a quadratic object function, one quadratic constraint and (positivity) bounds on the variables. Against little computational cost, the algorithm allows for the inclusion of positivity of the solution as prior knowledge. This is very useful for the solution of those (linear) inverse problems where negative solutions are unphysical.The algorithm rewrites the solution as a function of the Lagrange multipliers, which is achieved with the help of the generalized eigenvectors, or equivalently, the generalized singular value decomposition. The next step is to find the Lagrange multipliers. The multiplier corresponding to the quadratic constraint, which is known to be active, is easy to find. The Lagrange multipliers corresponding to the positivity constraints are found with an iterative method that can be likened to the active set methods from quadratic programming.

Abstract. This paper presents a model-independent algorithm for tomography of the ionosphere. Prior knowledge consists of the following pieces of information: electron density cannot be negative, the ionosphere is basically smooth and stratified, and electron density is low at high (•>700 km) and low (•<100 km) altitudes. Tests based on simulated measurements show that the method recovers the latitudinal structure well, whereas the vertical structure is recovered with moderate success: the estimated height of the layer of maximum electron density may be as much as 90 km in error. Because of the imposed smoothness the method tends to underestimate the peak in electron density by as much as one third in unfavorable cases. IntroductionSince its conception in 1986 [Austen et al., 1986], radiotomography of the ionosphere has grown into a relatively inexpensive technique to image the electron density distribution in vertical cross sections of the ionosphere. In general, tomography is a method to reconstruct a distribution from its line integrals. In ionospheric tomography the line integrals of electron density (called total electron content (TEC)) are measured by the differential Doppler technique, where a receiver registers the differential Doppler shift of an orbiting beacon satellite [Leitinger et al., 1984]. An array of receivers placed parallel to the satellite's ground path should yield enough data to make the tomographic inversion from the measurements into an image of electron density. The surface of reconstruction is defined by the lines of sight (or rather phase paths) between satellite and receivers (see Figure 1).The first studies in ionospheric tomography were based on simulated TEC measurements using model ionospheres. Later, real experimental data were used on the basis of phase shift measurements from either the U.S. Navy Navigation Satellite System (NNSS) or A literature study reveals that every research group has its own reconstruction algorithm. Why should this paper add one more to the stack? Here we argue why our algorithm is a valuable contribution to the field.The main problem in ionospheric tomography is the absence of (near-)horizontal line integrals in the data set [Yeh and Raymund, 1991]. This absence is the result of a geometry where the lines of integration correspond to lines of sight between an orbiting satellite and ground-based receivers (Figure 1). Consequently, there are no lines of sight that remain at an approximately constant altitude. A set of such lines would contain the vertical profile of ionospheric electron density. This information is virtually missing from the data set, and tomography cannot be expected to provide a reconstruction where the vertical profile is rendered truthfully. A mathematician would say that the inverse problem is ill posed. In this jargon the inverse problem is the reconstruction of the "cause" from the "effect," the cause being the ionospheric electron density distribution and the effect being the result of the experiment (the TEC data). The problem ...

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SUMMARYThe propagation of inertia-gravity waves (IGWs) through a dynamical transport barrier, such as the Antarctic polar vortex edge is investigated using a linear wave model. The model is based on the linearized, inviscid hydrostatic equations on an f -plane. Typical values for the parameters that are appropriate to the Antarctic polar vortex are given. The background ow U is assumed to be barotropic and its horizontal shear is represented by a hyperbolic tangent background wind pro le. The wave equation that describes the latitudinal structure of a monochromatic disturbance contains two singularities. The rst corresponds to the occurrence of a critical level where the intrinsic wave frequency Ä D ! ¡ kU becomes zero. ! is the absolute wave frequency and k its longitudinal wave number in the direction of U . The second is an apparent singularity and does not give rise to singular wave behaviour. It becomes zero whenever the square of the intrinsic wave frequency Ä 2 D f .f ¡ U y /, f being the Coriolis frequency and U y the horizontal shear of the ow. The wave equation is solved numerically for different values of the angles of incidence of the wave upon the background ow, of the wave frequency, of the horizontal wave number and of the Rossby number. Re ection (jRj) and transmission (jT j) coef cients are determined as a function of these parameters. The results depend on whether the ow is inertially stable or not. They also depend on the presence and location of the turning levels, where the wave becomes evanescent, with respect to the location of the Ä-critical levels. For inertially stable ows, the wave totally re ects at the turning level and never reaches the critical level. If the background ow is inertially unstable, turning levels can disappear and the wave can now reach the critical level. Then over-re ection, over-transmission and absorption can occur.

The effect of Raman backscattering (RBS) on high-energy electron generation in laser-plasma interaction has been investigated for laser intensities well above the wave breaking and electron trapping threshold. One-dimensional particle-in-cell simulations show that suppression of RBS increases the high-energy electron yield in this regime. RBS-induced heating causes heavy beam loading and damping of the laser wake. Its suppression leads to higher wake amplitudes and higher particle energies. RBS suppression through direct stimulation of Raman forward scatter is demonstrated. The implications for high-energy electron production through laser-plasma interaction are discussed.Introduction. -Several recent experiments (see [1,2] and references therein) on the interaction of intense laser pulses with underdense plasmas have demonstrated the production of energetic electrons in the self-modulated regime of the laser wakefield accelerator (LWFA) with accelerating gradients in excess of 100 GV/m. The resulting electron bunches are characterized by high charge (up to 10 nC), sub-ps duration, and an exponential energy distribution which extends out to hundreds of MeV. Applications of these bunches include injectors to secondary accelerators [3], short-pulse radiation sources [4], short-lived radio-isotope production [5], and fast-ignition fusion [6].Raman forward (RFS) and backward (RBS) scattering are laser-plasma instabilities that govern the self-trapping and acceleration dynamics in the self-modulated LWFA. According to basic Raman scattering theory [7], RBS is a three-wave interaction, in which the incoming laser light (carrier frequency ω 0 , wave number k 0 , peak amplitude E 0 = (m e ω 0 c/e)a 0 ) decays into a backscattered electromagnetic (EM) wave (ω 0 − ω p , −(k 0 − k p )) and a slow Langmuir wave (ω p , 2k 0 − k p ). Here, ω p = n 0 e 2 /(ε 0 m e ), k p = ω p /c, and n 0 is the unperturbed plasma electron density. The Langmuir wave phase velocity is approximately ω p c/(2ω 0 ) c for an