Spherical Atmosphere Research Articles

Consistent quasi‐shallow models of the global atmosphere in non‐spherical geopotential coordinates with complete Coriolis force

A pair of dynamically consistent quasi‐shallow equation sets has recently been developed for global spherical atmospheres and oceans in spherical polar coordinates, using the standard spherical geopotential approximation. This pair is of intermediate complexity between two standard known pairs (one shallow, one deep). A surprising feature of this new pair is that the fluid is assumed to be of essentially shallow depth, yet the Coriolis force is completely represented. This is in contradistinction to the traditional shallow pair, where the Coriolis force is incompletely represented in order to achieve dynamical consistency.However, the Earth is better represented by a spheroid than by a sphere. It is shown herein that the aforementioned work can be generalised by making the weaker spheroidal geopotential approximation to obtain an analogous pair of quasi‐shallow equation sets using axisymmetric curvilinear orthogonal coordinates. This pair is derived in two different, but complementary, ways: via the traditional Eulerian approach, and via Hamilton's principle of least action.

Effect of the Kinematic Lower Boundary Condition on the Spectral and Autocorrelation Structure of Annular Variability in the Troposphere

Abstract The dynamical origin of the spectral and autocorrelation structure of annular variability in the troposphere is investigated by a deductive approach. Specifically, the structure of the power spectrum and autocorrelation function of the zonal-mean geopotential is analyzed for the case of a quasigeostrophic spherical atmosphere subject to a white noise mechanical forcing applied in a single Hough mode and concentrated at a particular level in the vertical, with vertically uniform Newtonian cooling and Rayleigh drag concentrated at a rigid lower boundary. Analytic expressions for the power spectrum are presented together with expressions for an approximate red noise (i.e., a Lorentzian-shaped) power spectrum. It is found that for an infinitely deep atmosphere the power spectrum can be well approximated by a red noise process for the first few Hough modes (associated with large Rossby heights), provided the distance from the forcing is not larger than about one Rossby height. When a frictional rigid lower boundary is included, however, the approximation is generally bad. The high-frequency part of the power spectrum exhibits near-exponential behavior and the autocorrelation function shows a transition from a rapid decay at short lags to a much slower decay at longer lags, if the thermal and mechanical damping time scales are sufficiently well separated. Since observed annular variability exhibits the same characteristics, the above results lead to the hypothesis that these characteristics may, to some extent, be intrinsic to the linear zonal-mean response problem—although the need for an additional contribution from eddy feedbacks is also implied by the results.

Nabro volcano aerosol in the stratosphere over Georgia, South Caucasus from ground-based spectrometry of twilight sky brightness

Abstract. Ground-based spectral measurements of twilight sky brightness were carried out between September 2009 and August 2011 in Georgia, South Caucasus. The algorithm which allowed to retrieve the lower stratospheric and upper tropospheric aerosol extinction profiles was developed. The Monte-Carlo technique was used to correctly represent multiple scattering in a spherical atmosphere. The estimated stratospheric aerosol optical depths at a wavelength of 780 nm were: 6 × 10−3 ± 2 × 10−3 (31 August 2009–29 November 2009), 2.5 × 10−3 ± 7 × 10−4 (20 March 2010–15 January 2011) and 8 × 10−3 ± 3 × 10−3 (18 July 2011–3 August 2011). The optical depth values correspond to the moderately elevated stratospheric aerosol level after the Sarychev eruption in 2009, background stratospheric aerosol layer, and the volcanically disturbed stratospheric aerosol layer after the Nabro eruption in June 2011.

Finite Element Modeling and Design of Rectangular Patch Antenna with Different Feeding Techniques

The finite element modeling of three dimensional structures is important for researchers especially in the field of antennas and other domains of electromagnetic waves. This paper presents a finite element calculations and numerical analysis for the microstrip patch antennas. In this paper, two different designs have been modelled and analyzed and both designs are based on the rectangular patches. The feeding point of one design is inside the patch while the other design contains feeding point outside the patch is T shaped. The computational analysis showed some interesting results for radiation pattern and far field domain. For these designs, the characteristic impedance taken is 50 Ω and the operating frequency domain is 1.4 to 1.7 GHz. The microstrip patch antennas are encapsulated in the inert spherical atmosphere of 20 mm thickness containing air inside it.

A hybrid method for modeling polarized radiative transfer in a spherical-shell planetary atmosphere

The Markov chain method is developed for polarized radiative transfer in a pseudo-spherical atmosphere with solar illumination. This solution is then used as an initial guess of the radiation field for a spherical atmosphere. By use of the short characteristic method, a convergent radiation field throughout the atmosphere is achieved after a few Picard iterations. We verified this hybrid method by comparing numerical results to those obtained by a backward Monte Carlo calculation. We carried out a demonstration calculation by simulating the Titan haze reflected intensity I and Stokes parameter Q, and degree of linear polarization at 934.8nm wavelength. Comparison of the I and Q images to those measured by the Imaging Science Subsystem instrument on the Cassini spacecraft shows the hybrid method to be useful for radiative transfer analyses for (both optically and physically) thick spherical atmospheres.

Compton broadening effect on spectral line formation

We have investigated the effects of Compton broadening due to electron scattering in an expanding stellar atmospheres. The line transfer equation is solved by including the second approximation of Edmonds (Astrophys. J. 119:58, 1954) which is due to Compton broadening and obtained the line profiles in (1) plane parallel (PP) (2) spherical (SS) atmospheres. The effect on spectral line formation is studied for different parameters like (a) optical depths (b) densities (c) frequencies (d) temperatures (e) thickness of the atmosphere and (f) expanding velocities. Various combination of the above parameters are used in computing the line profiles observed at infinity.

Hot Stars with Winds: The CMFGEN Code

Abstractcmfgen is an atmosphere code developed to model the spectra of a variety of objects – O stars, Wolf-Rayet stars, luminous blue variables, A and B stars, central stars of planetary nebula, and supernovae. The principal computational aim of cmfgen is to determine the temperature and ionization structure of the atmosphere, and the atomic level populations. Toward this end, we have developed several different radiative transfer modules that (a) solve the transfer equation for spherical geometry in the comoving frame, (b) solve the static transfer equation in the plane parallel approximation without, or with, a vertical velocity field, (c) solve the static transfer code for a spherical atmosphere allowing for all relativistic terms, (d) solve the time-dependent spherical transfer equation to first order in v/c for a homologous expansion, and (e) solve the time-dependent spherical transfer equation allowing for all relativistic terms. To achieve consistency between the radiation field, temperature structure, and level populations we use a linearization technique. Line blanketing is accurately treated while complex photoionization cross-sections, containing numerous resonances, can also be handled. Spectra, for comparison with observation, are computed using cmf_flux. Several other auxiliary programs have also been developed – these include diagnostic tools as well as programs that allow the effect on spectra of rotation and departures from spherical geometry to be investigated. In this presentation we will briefly describe cmfgen and auxiliary codes.

Statistical modeling of the point spread function in the spherical atmosphere and a criterion for detecting image isoplanarity zones

An algorithm is proposed to simulate a point spread function by the Monte Carlo method in order to represent external channel formation in the spherical model system of the “atmosphere-earth surface.” A new way of defining isoplanarity zones of optically homogeneous extended objects is described. We demonstrate that image restoration with a specified error for these objects when they are observed on the earth’s surface from space by an ideal optical system can determine a definite point spread function set.

Fundamental parameters of RR Lyrae stars from multicolour photometry and Kurucz atmospheric models - I. Theory and practical implementation

A photometric calibration of Kurucz static model atmospheres is used to obtain the following parameters of RR Lyrae stars: variation of stellar angular radius ϑ, effective temperature Te, gravity ge as a function of phase, interstellar reddening E(B−V) towards the star and atmospheric metallicity M. Photometric and hydrodynamic conditions are given to find the phases of pulsation when the quasi-static atmosphere approximation (QSAA) can be applied. The QSAA is generalized to a non-uniformly moving spherical atmosphere, and the distance d, mass and atmospheric motion are derived from the laws of mass and momentum conservation. To demonstrate the efficiency of the method, the UBV(RI)C photometry of SU Dra was used to derive the following parameters: [M] =−1.60 ± 0.10 dex, ⁠, equilibrium luminosity Leq= 45.9 ± 9.3 L⊙ and Teq= 6813 ± 20 K.

Picard iteration methods for a spherical atmosphere

Several methods for solving the radiative transfer equation in a spherical atmosphere are presented. These methods include the long characteristic Picard iteration, and the conventional and the accelerated short characteristic Picard iteration. Approximate methods as for instance the Picard iteration methods with open boundaries are also discussed. The analysis is focussed on the computational efficiency and accuracy of the proposed method and their applicability to atmospheric remote sensing.

Adjoint problem of radiative transfer for a pseudo-spherical atmosphere and general viewing geometries

We formulate the adjoint radiative transfer for a pseudo-spherical atmosphere and various retrieval scenarios. The single scattering radiance is computed in a spherical atmosphere by using the source integration technique, while for the multiple scattering radiance we formulate an one-dimensional adjoint radiative transfer equation in a plane-parallel atmosphere. The adjoint solution of the radiative transfer equation is obtained by employing the discrete ordinate method with matrix exponential. We provide an abbreviated derivation of our formalism as well as a discussion of the numerical implementation of the theory.

A generalized thermal wind equation and some non‐separable exact solutions of the flow equations for three‐dimensional spherical atmospheres

AbstractSteady axisymmetric flows in geostrophic and cyclostrophic balance under gravity are considered for the three‐dimensional Euler equations (with the spherical geopotential approximation but without the shallow‐atmosphere and ‘traditional’ approximations). The key to analytical specification of these flows is a compatibility condition involving the zonal flow and temperature fields. This condition is a generalization of the thermal wind equation for balanced zonal flow governed by the hydrostatic primitive equations. Two examples are presented in which the temperature field is specified as non‐separable two‐parameter functions of latitude and height, and corresponding zonal flows are derived analytically. Such flows extend the class of known analytical solutions of the governing equations, and their usefulness in numerical model development and testing is the focus of this study. © Crown Copyright 2008. Reproduced with the permission of the Controller of HMSO. Published by John Wiley & Sons, Ltd.

Unsteady exact solutions of the flow equations for three‐dimensional spherical atmospheres

AbstractTime‐dependent, closed‐form solutions of the 3D Euler equations describing motion relative to a uniformly rotating coordinate frame are derived. The spherical geopotential approximation is applied but not the shallow‐atmosphere and hydrostatic approximations. The solutions correspond to cyclostrophically and hydrostatically balanced vortices that are steady in inertial space and whose symmetry axes do not coincide with the rotation axis of the coordinate frame. The inertial‐frame flow velocities are readily transformed to a precisely spherical rotating coordinate system in which the 3D Euler equations contain centrifugal as well as Coriolis terms. In this form the solutions may be used to test numerical models formulated in spherical coordinates under the spherical geopotential approximation, so long as the centrifugal terms are explicitly included as forcing terms. The development is repeated for the hydrostatic and non‐hydrostatic primitive equations (with the shallow‐atmosphere approximation) and for the shallow‐water equations. In the latter case, the required explicit centrifugal force may be provided by a zonally symmetric addition to the free surface height, with an identically equal orographic elevation to ensure conservation of mass. The solutions are then identical to the unsteady shallow‐water solutions of Läuter et al. that inspired this study. ©Crown Copyright 2008. Reproduced with the permission of the Controller of HMSO and the Queen's Printer for Scotland. Published by John Wiley & Sons, Ltd.

SASKTRAN: A spherical geometry radiative transfer code for efficient estimation of limb scattered sunlight

The inversion of satellite-based observations of limb scattered sunlight for the retrieval of constituent species requires an efficient and accurate modelling of the measurement. We present the development of the SASKTRAN radiative transfer model for the prediction of limb scatter measurements at optical wavelengths by method of successive orders along rays traced in a spherical atmosphere. The component of the signal due to the first two scattering events of the solar beam is accounted for directly along rays traced in the three-dimensional geometry. Simplifying assumptions in successive scattering orders provide computational optimizations without severely compromising the accuracy of the solution. SASKTRAN is designed for the analysis of measurements from the OSIRIS instrument and the implementation of the algorithm is efficient such that the code is suitable for the inversion of OSIRIS profiles on desktop computers. SASKTRAN total limb radiance profiles generally compare better with Monte-Carlo reference models over a large range of solar conditions than the approximate spherical and plane-parallel models typically used for inversions.

McSCIA: application of the Equivalence Theorem in a Monte Carlo radiative transfer model for spherical shell atmospheres

Abstract. A new multiple-scattering Monte Carlo 3-D radiative transfer model named McSCIA (Monte Carlo for SCIAmachy) is presented. The backward technique is used to efficiently simulate narrow field of view instruments. The McSCIA algorithm has been formulated as a function of the Earth's radius, and can thus perform simulations for both plane-parallel and spherical atmospheres. The latter geometry is essential for the interpretation of limb satellite measurements, as performed by SCIAMACHY on board of ESA's Envisat. The model can simulate UV-vis-NIR radiation. First the ray-tracing algorithm is presented in detail, and then successfully validated against literature references, both in plane-parallel and in spherical geometry. A simple 1-D model is used to explain two different ways of treating absorption. One method uses the single scattering albedo while the other uses the equivalence theorem. The equivalence theorem is based on a separation of absorption and scattering. It is shown that both methods give, in a statistical way, identical results for a wide variety of scenarios. Both absorption methods are included in McSCIA, and it is shown that also for a 3-D case both formulations give identical results. McSCIA limb profiles for atmospheres with and without absorption compare well with the one of the state of the art Monte Carlo radiative transfer model MCC++. A simplification of the photon statistics may lead to very fast calculations of absorption features in the atmosphere. However, these simplifications potentially introduce biases in the results. McSCIA does not use simplifications and is therefore a relatively slow implementation of the equivalence theorem.

Remote sensing of three-dimensional cirrus clouds from satellites: application to continuous-wave laser atmospheric transmission and backscattering

A satellite remote sensing methodology has been developed to retrieve 3D ice water content (IWC) and mean effective ice crystal size of cirrus clouds from satellite data on the basis of a combination of the conventional retrieval of cloud optical depth and particle size in a horizontal plane and a parameterization of the vertical cloud profile involving temperature from sounding and/or analysis. The inferred 3D cloud fields of IWC and mean effective ice crystal size associated with two impressive cirrus clouds that occurred in the vicinity of northern Oklahoma on 18 April 1997 and 9 March 2000, obtained from the Department of Energy's Atmospheric Radiation Measurement Program, have been validated against the ice crystal size distributions that were collected independently from collocated and coincident aircraft optical probe measurements. The 3D cloud results determined from satellite data have been applied to the simulation of cw laser energy propagation, and we show the significance of 3D cloud geometry and inhomogeneity and spherical atmosphere on the transmitted and backscattered laser powers. Finally, we demonstrate that the 3D cloud fields derived from satellite remote sensing can be used for the 3D laser transmission and backscattering model for tactical application.

Recent developments in the line-by-line modeling of outgoing longwave radiation

High frequency resolution radiative transfer model calculations with the Atmospheric Radiative Transfer Simulator (ARTS) were used to simulate the clear-sky outgoing longwave radiative flux (OLR) at the top of the atmosphere. Compared to earlier calculations by Clough and coworkers the model used a spherical atmosphere instead of a plane parallel atmosphere, updated spectroscopic parameters from HITRAN, and updated continuum parameterizations from Mlawer and coworkers. These modifications lead to a reduction in simulated OLR by approximately 4.1%, the largest part, approximately 2.5%, being due to the absence of the plane parallel approximation. As a simple application of the new model, the sensitivity of OLR to changes in humidity, carbon dioxide concentration, and temperature were investigated for different cloud-free atmospheric scenarios. It was found that for the tropical scenario a 20% change in humidity has a larger impact than a doubling of the carbon dioxide concentration. The sensitive altitude region for temperature and humidity changes is the entire free troposphere, including the upper troposphere where humidity data quality is poor.

SCIATRAN 2.0 – A new radiative transfer model for geophysical applications in the 175–2400 nm spectral region

A successor version of the SCIATRAN radiative transfer model (RTM) has been developed to perform radiative transfer modeling in any observation geometry appropriate to measurements of the scattered solar radiation in the Earth’s atmosphere. The model is designed to be used as a forward model in the retrieval of atmospheric constituents from measurements of scattered solar light by satellite, ground-based, or airborne instruments in UV–Vis–NIR spectral region. Furthermore, it can be used to calculate air mass factors or fluxes. The new generation of the SCIATRAN model comprises all features of the latest SCIATRAN 1.2 RTM supporting additionally radiative transfer calculations in a spherical atmosphere. The program is written in FORTRAN 95 and suitable for parallel execution using the OpenMP standard. The wavelength range covered by the radiative transfer model is extended to 175–2380 nm including Schuman-Runge and Herzberg absorption bands of oxygen. The SCIATRAN 2.0 model exhibits the following new capabilities: (i) modeling of the scattered solar radiation in limb viewing geometry as well as any kind of measurements of the scattered radiation within the atmosphere, (ii) corresponding quasi-analytical calculation of weighting functions of atmospheric parameters, (iii) airmass factor calculations for ground-based, space and airborne measurements including off-axis geometry, (v) accounting for photochemically active species, i.e., radiative transfer calculations can be performed using solar zenith angle dependent vertical distributions of atmospheric species, (iv) height resolved radiation fluxes, including actinic fluxes for photolysis rate calculations, (vi) inelastic rotational Raman scattering in any supported viewing geometry, (vii) new effective approximations for radiative transfer modeling in presence of clouds. The SCIATRAN model is freely available via the world wide web for non-commercial scientific applications.

A polarized discrete ordinate scattering model for simulations of limb and nadir long‐wave measurements in 1‐D/3‐D spherical atmospheres

This article describes one of the scattering algorithms of the three‐dimensional polarized radiative transfer model ARTS (Atmospheric Radiative Transfer Simulator) which has been implemented to study for example the influence of cirrus clouds on microwave limb sounding. The model uses the DOIT (Discrete Ordinate Iterative) method to solve the vector radiative transfer equation. The implementation of a discrete ordinate method is challenging due to the spherical geometry of the model atmosphere which is required for the simulation of limb radiances. The involved numerical issues, which are grid optimization and interpolation methods, are discussed in this paper. Scattering simulations are presented for limb‐ and down‐looking geometries, for one‐dimensional and three‐dimensional spherical atmospheres. They show the impact of cloud particle size, shape, and orientation on the brightness temperatures and on the polarization of microwave radiation in the atmosphere. The cloud effect is much larger for limb radiances than for nadir radiances. Particle size is a very important parameter in all simulations. The polarization signal is negligible for simulations with completely randomly oriented particles, whereas for horizontally aligned particles with random azimuthal orientation the polarization signal is significant. Moreover, the effect of particle shape is only relevant for oriented cloud particles. The simulations show that it is essential to use a three‐dimensional scattering model for inhomogeneous cloud layers.

Linearized vector radiative transfer model MCC++ for a spherical atmosphere

Application of radiative transfer models has shown that optical remote sensing requires extra characteristics of radiance field in addition to the radiance intensity itself. Simulation of spectral measurements, analysis of retrieval errors and development of retrieval algorithms are in need of derivatives of radiance with respect to atmospheric constituents under investigation. The presented vector spherical radiative transfer model MCC++ was linearized, which allows the calculation of derivatives of all elements of the Stokes vector with respect to the volume absorption coefficient simultaneously with radiance calculation. The model MCC++ employs Monte Carlo algorithm for radiative transfer simulation and takes into account aerosol and molecular scattering, gas and aerosol absorption, and Lambertian surface albedo. The model treats a spherically symmetrical atmosphere. Relation of the estimated derivatives with other forms of radiance derivatives: the weighting functions used in gas retrieval and the air mass factors used in the DOAS retrieval algorithms, is obtained. Validation of the model against other radiative models is overviewed. The computing time of the intensity for the MCC++ model is about that for radiative models treating sphericity of the atmosphere approximately and is significantly shorter than that for the full spherical models used in the comparisons. The simultaneous calculation of all derivatives (i.e. with respect to absorption in all model atmosphere layers) and the intensity is only 1.2–2 times longer than the calculation of the intensity only.