The observed atmospheric spectrum of exoplanets and brown dwarfs depends critically on the presence and distribution of atmospheric condensates. The Ackerman and Marley methodology for predicting the vertical distribution of condensate particles is widely used to study cloudy atmospheres and has recently been implemented in an open-source python package, Virga. The model relies upon input parameter f sed, the sedimentation efficiency, which until now has been held constant. The relative simplicity of this model renders it useful for retrieval studies due to its rapidly attainable solutions. However, comparisons with more complex microphysical models such as CARMA have highlighted inconsistencies between the two approaches, namely that the cloud parameters needed for radiative transfer produced by Virga are dissimilar to those produced by CARMA. To address these discrepancies, we have extended the original Ackerman and Marley methodology in Virga to allow for non-constant f sed values, in particular, those that vary with altitude. We discuss one such parameterization and compare the cloud mass mixing ratio produced by Virga with constant and variable f sed profiles to that produced by CARMA. We find that the variable f sed formulation better captures the profile produced by CARMA with heterogeneous nucleation, yet performs comparatively to constant f sed for homogeneous nucleation. In general, Virga has the capacity to handle any f sed with an explicit anti-derivative, permitting a plethora of alternative cloud profiles that are otherwise unattainable by constant f sed values. The ensuing flexibility has the potential to better agree with increasingly complex models and observed data.

A significant challenge in radiative transfer theory for atmospheres of exoplanets and brown dwarfs is the derivation of computationally efficient methods that have adequate fidelity to more precise, numerically demanding solutions. In this work, we extend the capability of the first open-source radiative transfer model for computing the reflected light of exoplanets at any phase geometry, PICASO (Planetary Intensity Code for Atmospheric Spectroscopy Observations). Until now, PICASO has implemented two-stream approaches to the solving the radiative transfer equation for reflected light, in particular following the derivations of Toon et al. In order to improve the model accuracy, we have considered higher-order approximations of the phase functions; namely, we have increased the order of approximation from two to four, using spherical harmonics. The spherical harmonics approximation decouples spatial and directional dependencies by expanding the intensity and phase function into a series of spherical harmonics, or Legendre polynomials, allowing for analytical solutions for low-order approximations to optimize computational efficiency. We rigorously derive the spherical harmonics method for reflected light and benchmark the four-term method (SH4) against Toon et al. and two independent and higher-fidelity methods (CDISORT and doubling method). On average, the SH4 method provides an order-of-magnitude increase in accuracy, compared to Toon et al. Finally, we implement SH4 within PICASO and observe only a modest increase in computational time, compared to two-stream methods (20% increase).

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Heat transport in granular and porous media occurs through conduction in the solid and radiation through the voids. By exploiting the separation of length scales between the small typical particles or voids and the large size of whole region, the method of multiple scales can be applied. For a purely diffusive system, this yields a problem on the macroscale with an effective conductivity, deduced by solving a ‘cell problem’ on the microscale. Here, we apply the method when radiation and conduction are both present; however, care must be taken to correctly handle the integral nature of the radiative boundary condition. Again, an effective conductivity is found by solving a ‘cell problem’ which, because of the non-linearity of radiative transfer, to be solved for each temperature value. We also incorporate modifications to the basic theory of multiple scales in order to deal with the non-local nature of the radiative boundary condition. We derive the multiple scales formulation of the problem and report on numerical comparisons between the homogenised problem and direct solution of the problem. We also compare the effective conductivity to that derived using Maxwell models and effective medium theory.

Mathematical models are developed to explain the size distribution of particles in blenders to give insight into the behaviour of possible blender designs. The initial models consider idealized simplified situations, first with the chopping of long thin particles and then of spheres. The models are first presented using the idea of chopping at discrete places but then extended to account for chopping at any point via a continuous model. Some of the simple models can be solved analytically while others require numerical calculations. Comparisons of the predictions from the various models with experimental data at a fixed time are presented and show that the models account for much of the behaviour. The initial models do not however predict the large amount of extremely small particles (debris) that are observed at the end of the blending process. The models are thus modified using simple extensions to account for additional effects and numerical solutions of these models are compared with the observed data. The theory should provide a useful tool that eliminates the need to perform costly and time-consuming experiments when understanding how a particular food will be blended.