Partially Coherent Wave Scattering and Radiative Transfer: An Integral Equation Approach

This thesis consists of two parts. The first one is "Scattering of Waves in a Random Medium," and the second one is "Radiative Transfer in a Sphere Illuminated by a Parallel Beam: An Integral Equation Approach." In the first part, a new formalism tor partially coherent wave scattering in a random medium is developed. In this formalism the coherent wave is the solution of a phenomenological wave equation, and the mutual coherence function of the wave field satisfies a simple integral equation. Using this formalism, the Peierls equation can be readily derived. Also, an improved version of the Peierls equation is derived in which the intensity of the wave field and the first order derivative of the mutual coherence function are calculated at the same time. A simple problem is solved to find the mutual coherence function produced by a laser beam in the atmosphere. The similarity between the mutual coherence function and the density matrix or quantum mechanics is explored and a measure of the randomness is defined for the partially coherent wave field. In the second part of this work, the problem of multiple scattering of non-polarized light in a planetary body of arbitrary shape illuminated by a parallel beam is formulated using the integral equation approach. There exists a simple functional whose stationarity condition is equivalent to solving the equation of radiative transfer and whose value at the stationary point is proportional to the differential cross section. Our analysis reveals a direct relation between the microscopic symmetry of the. Phase function for each scattering event and the macroscopic symmetry of the differential cross section for the entire planetary body, and the intimate connection between these symmetry relations and the variational principle. The case of a homogeneous sphere containing isotropic scatterers is investigated in detail. It is shown that the solution can be expanded in a multipole series such that the general spherical problem is reduced to solving a set of decoupled integral equations in one dimension. Computations have been performed for a range of parameters of interest, and illustrative examples of applications to planetary problems are provided.</p

Jovian Stratosphere as a Chemical Transport System: Benchmark Analytical Solutions

We systematically investigated the solvable analytical benchmark cases in both one- and two-dimensional (1D and 2D) chemical-advective-diffusive systems. We use the stratosphere of Jupiter as an example but the results can be applied to other planetary atmospheres and exoplanetary atmospheres. In the 1D system, we show that CH_4 and C_2H_6 are mainly in diffusive equilibrium, and the C_2H_2 profile can be approximated by modified Bessel functions. In the 2D system in the meridional plane, analytical solutions for two typical circulation patterns are derived. Simple tracer transport modeling demonstrates that the distribution of a short-lived species (such as C_2H_2) is dominated by the local chemical sources and sinks, while that of a long-lived species (such as C_2H_6) is significantly influenced by the circulation pattern. We find that an equator-to-pole circulation could qualitatively explain the Cassini observations, but a pure diffusive transport process could not. For slowly rotating planets like the close-in extrasolar planets, the interaction between the advection by the zonal wind and chemistry might cause a phase lag between the final tracer distribution and the original source distribution. The numerical simulation results from the 2D Caltech/JPL chemistry-transport model agree well with the analytical solutions for various cases

Modulation of the Period of the Quasi-Biennial Oscillation by the Solar Cycle

The authors examine the mechanism of solar cycle modulation of the Quasi-Biennial Oscillation (QBO) period using the Two-and-a-Half-Dimensional Interactive Isentropic Research (THINAIR) model. Previous model results (using 2D and 3D models of varying complexity) have not convincingly established the proposed link of longer QBO periods during solar minima. Observational evidence for such a modulation is also controversial because it is only found during the period from the 1960s to the early 1990s, which is contaminated by volcanic aerosols. In the model, 200- and 400-yr runs without volcano influence can be obtained, long enough to establish some statistical robustness. Both in model and observed data, there is a strong synchronization of the QBO period with integer multiples of the semiannual oscillation (SAO) in the upper stratosphere. Under the current level of wave forcing, the period of the QBO jumps from one multiple of SAO to another and back so that it averages to 28 months, never settling down to a constant period. The “decadal” variability in the QBO period takes the form of “quantum” jumps; these, however, do not appear to follow the level of the solar flux in either the observation or the model using realistic quasi-periodic solar cycle (SC) forcing. To understand the solar modulation of the QBO period, the authors perform model runs with a range of perpetual solar forcing, either lower or higher than the current level. At the current level of solar forcing, the model QBO period consists of a distribution of four and five SAO periods, similar to the observed distribution. This distribution changes as solar forcing changes. For lower (higher) solar forcing, the distribution shifts to more (less) four SAO periods than five SAO periods. The record-averaged QBO period increases with the solar forcing. However, because this effect is rather weak and is detectable only with exaggerated forcing, the authors suggest that the previous result of the anticorrelation of the QBO period with the SC seen in short observational records reflects only a chance behavior of the QBO period, which naturally jumps in a nonstationary manner even if the solar forcing is held constant, and the correlation can change as the record gets longer

Nonstationary Synchronization of Equatorial QBO with SAO in Observations and a Model

It has often been suggested that the period of the quasi-biennial oscillation (QBO) has a tendency to synchronize with the semiannual oscillation (SAO). Apparently the synchronization is better the higher up the observation extends. Using 45 yr of the 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-40) data of the equatorial stratosphere up to the stratopause, the authors confirm that this synchronization is not just a tendency but a robust phenomenon in the upper stratosphere. A QBO period starts when a westerly SAO (w-SAO) descends from the stratopause to 7 hPa and initiates the westerly phase of the QBO (w-QBO) below. It ends when another w-SAO, a few SAO periods later, descends again to 7 hPa to initiate the next w-QBO. The fact that it is the westerly but not the easterly SAO (e-SAO) that initiates the QBO is also explained by the general easterly bias of the angular momentum in the equatorial stratosphere so that the e-SAO does not create a zero-wind line, unlike the w-SAO. The currently observed average QBO period of 28 months, which is not an integer multiple of SAO periods, is a result of intermittent jumps of the QBO period from four SAO to five SAO periods. The same behavior is also found in the Two and a Half Dimensional Interactive Isentropic Research (THINAIR) model. It is found that the nonstationary behavior in both the observation and model is caused not by the 11-yr solar-cycle forcing but by the incompatibility of the QBO’s natural period (determined by its wave forcing) and the “quantized” period determined by the SAO. The wave forcing parameter for the QBO period in the current climate probably lies between four SAO and five SAO periods. If the wave forcing for the QBO is tuned so that its natural period is compatible with the SAO period above (e.g., at 24 or 30 months), nonstationary behavior disappears

Meridional Transport in the Stratosphere of Jupiter

The Cassini measurements of C$_2$H$_2$ and C$_2$H$_6$ at $\sim$5 mbar provide a constraint on meridional transport in the stratosphere of Jupiter. We performed a two-dimensional photochemical calculation coupled with mass transport due to vertical and meridional mixing. The modeled profile of C$_2$H$_2$ at latitudes less than 70$^\circ$ follows the latitude dependence of the solar insolation, while that of C$_2$H$_6$ shows little latitude dependence, consistent with the measurements. In general, our model study suggests that the meridional transport timescale above 5-10 mbar altitude level is $\gtrsim$1000 years and the time could be as short as 10 years below 10 mbar level, in order to fit the Cassini measurements. The derived meridional transport timescale above the 5 mbar level is a hundred times longer than that obtained from the spreading of gas-phase molecules deposited after the impact of Shoemaker-Levy 9 comet. There is no explanation at this time for this discrepancy.Comment: 11 pages, 3 figures, 1 table. ApJL in pres

X_(CO2) retrieval error over deserts near critical surface albedo

Large retrieval errors in column‐weighted CO_2 mixing ratio (X_(CO2)) over deserts are evident in the Orbiting Carbon Observatory 2 version 7 L2 products. We argue that these errors are caused by the surface albedo being close to a critical surface albedo (α_c). Over a surface with albedo close to α_c, increasing the aerosol optical depth (AOD) does not change the continuum radiance. The spectral signature caused by changing the AOD is identical to that caused by changing the absorbing gas column. The degeneracy in the retrievals of AOD and X_(CO2) results in a loss of degrees of freedom and information content. We employ a two‐stream‐exact single scattering radiative transfer model to study the physical mechanism of X_(CO2) retrieval error over a surface with albedo close to α_c. Based on retrieval tests over surfaces with different albedos, we conclude that over a surface with albedo close to α_c, the X_(CO2) retrieval suffers from a significant loss of accuracy. We recommend a bias correction approach that has significantly improved the X_(CO2) retrieval from the California Laboratory for Atmospheric Remote Sensing data in the presence of aerosol loading

Assessing accuracy and precision for space-based measurements of carbon dioxide: An associated statistical methodology revisited

Analyzing retrieval accuracy and precision is an important element of space‐based CO_2 retrievals. However, this error analysis is sometimes challenging to perform rigorously because of the subtlety of Multivariate Statistics. To help address this issue, we revisit some fundamentals of Multivariate Statistics that help reveal the statistical essence of the associated error analysis. We show that the related statistical methodology is useful for revealing the intrinsic discrepancy and relation between the retrieval error for a nonzero‐variate CO_2 state and that for a zero‐variate one. Our study suggests that the two scenarios essentially yield the same‐magnitude accuracy, while the latter scenario yields a better precision than the former. We also use this methodology to obtain a rigorous framework systematically and explore a broadly used approximate framework for analyzing CO_2 retrieval errors. The approximate framework introduces errors due to an essential, but often forgotten, fact that a priori climatology in reality is never equal to the true state. Due to the nature of the problem considered, realistic numerical simulations that produce synthetic spectra may be more appropriate than remote sensing data for our specific exploration. As highlighted in our retrieval simulations, utilizing the approximate framework may not be universally satisfactory in assessing the accuracy and precision of X_(co_2) retrievals (with errors up to 0.17–0.28 ppm and 1.4–1.7 ppm, respectively, at SNR = 400). In situ measurements of CO_2 are needed to further our understanding of this issue and related implications

Short-period solar cycle signals in the ionosphere observed by FORMOSAT-3/COSMIC

We analyze 2 years of the FORMOSAT-3/COSMIC GPS radio occultation data to study the response of the Earth's ionosphere to the solar rotation (27-day) induced solar flux variations. Here we report electron density variations in the ionosphere (∼100–500 km) associated with the 27-day solar cycle. The peak-to-peak variation in electron density at low latitudes in the F2 region is about ∼10^4–10^5 electrons cm^(−3) or 20–40%, and can be as high as 60% depending on altitude, latitude, and season. The half and double periods of the 27-day are also observed at an amplitude comparable to that of the 27-day. The results place useful constraints for modeling chemical and dynamical processes in the ionosphere

Two-dimensional atmospheric transport and chemistry model: Numerical experiments with a new advection algorithm

Extensive testing of the advective scheme, proposed by Prather (1986), has been carried out in support of the California Institute of Technology–Jet Propulsion Laboratory two-dimensional model of the middle atmosphere. We generalize the original scheme to include higher-order moments. In addition, we show how well the scheme works in the presence of chemistry as well as eddy diffusion. Six types of numerical experiments including simple clock motion and pure advection in two dimensions have been investigated in detail. By comparison with analytic solutions it is shown that the new algorithm can faithfully preserve concentration profiles, has essentially no numerical diffusion, and is superior to a typical fourth-order finite difference scheme