Homoclinic Orbits In Slowly Varying Oscillators

We obtain existence and bifurcation theorems for homoclinic orbits in three-dimensional flows that are perturbations of families of planar Hamiltonian systems. The perturbations may or may not depend explicitly on time. We show how the results on periodic orbits of the preceding paper are related to the present homoclinic results, and apply them to a periodically forced Duffing equation with weak feedback

Constraining the evolution and origin of methane plumes on Mars

Future trace gas observations by the Nadir and Occultation for Mars Discovery (NOMAD) and Atmospheric Chemistry Suite (ACS) instruments on the ExoMars Trace Gas Orbiter (TGO) spacecraft will be the first instruments able to provide vertical profiles of multiple trace gas species, including methane. For interpretation and understanding of the retrieved methane vertical profiles, modelling studies are required to scrutinise between the different proposed mechanisms of methane release into the atmosphere, with global circulations models (GCMs) providing an invaluable tool to investigate the evolution of trace gas plumes and provide constraints on where the original source could be located, and potentially clues to its origin. This study investigates the vertical evolution of methane from multiple different source emission scenarios, using the state-of-the-art LMD-UK Mars GCM coupled to the Analysis Correction assimilation scheme. For the methane emission scenarios in this study, temperature retrievals from the Thermal Emission Spectrometer are assimilated. With the assimilation scheme ensuring the wind fields are consistent with the thermal data input to the model, the assimilation process ensures the optimal dynamical state of the atmosphere and subsequently the best constraint on the transport of tracers in the martian atmosphere. We show that at methane release rates constrained by previous observations and modelling studies, discriminating whether the methane source is a sustained or instantaneous surface emission requires at least ten sols of tracking the emission. A methane source must also be observed within five to ten sols of the initial emission to distinguish whether the emission occurs directly at the surface or within the atmosphere via destabilisation of metastable clathrates. The added constraint on global winds by the assimilation of thermal data is critical when attempting to backtrack the methane to its original source location

A General Framework for Updating Belief Distributions

We propose a framework for general Bayesian inference. We argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function, which is recovered under the special case of using self information loss. Modern application areas make it is increasingly challenging for Bayesians to attempt to model the true data generating mechanism. Moreover, when the object of interest is low dimensional, such as a mean or median, it is cumbersome to have to achieve this via a complete model for the whole data distribution. More importantly, there are settings where the parameter of interest does not directly index a family of density functions and thus the Bayesian approach to learning about such parameters is currently regarded as problematic. Our proposed framework uses loss-functions to connect information in the data to functionals of interest. The updating of beliefs then follows from a decision theoretic approach involving cumulative loss functions. Importantly, the procedure coincides with Bayesian updating when a true likelihood is known, yet provides coherent subjective inference in much more general settings. Connections to other inference frameworks are highlighted.Comment: This is the pre-peer reviewed version of the article "A General Framework for Updating Belief Distributions", which has been accepted for publication in the Journal of Statistical Society - Series B. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archivin

Assimilation of Mars Climate Sounder Dust Observations: Challenges and Ways Forward

Introduction: Atmospheric dust is ubiquitous on Mars, and as a result of its absorption and scattering of radiation, is the key driver of the martian circulation. Accurately representing the complex spatial and temporal distribution of dust is therefore crucial for understanding Mars’ atmospheric dynamics. In particular, the vertical representation of the dust distribution in Mars’ atmosphere has been shown to have a significant effect on results from modelling and assimilation [1,2,3]. With the goal of more accurately representing this distribution, the assimilation of dust vertical information is a valuable technique which is being increasingly explored [4,5]. However, it brings with it its own challenges and methodological questions to be explored. Model and assimilation details: We use the LMD-UK Mars Global Circulation Model (MGCM), which solves the meteorological primitive equations of fluid dynamics, radiative and other parameterised physics to calculate the state of the martian atmosphere [6,7]. The UK version of the MGCM possesses a spectral dynamical core and semi-Lagrangian advection scheme [8], and is a collaboration between the Laboratoire de Météorologie Dynamique, The Open University, the University of Oxford, and the Instituto de Astrofisica de Andalucia. The model was run using a range of spectral and vertical resolutions, the latter spaced logarithmically. The assimilation scheme used was a modified version of the Analysis Correction scheme developed at the Met Office [9], adapted for use on Mars [10]. This method has the advantage of being computationally in-expensive, and its use of repeated insertion, weighted over a time window of about six hours, helps counter the issue of relaxation of the atmospheric state – an especially significant problem given the low thermal inertia of Mars’ atmosphere. Retrievals: The retrievals used in this study are from the Mars Climate Sounder (MCS) instrument aboard the Mars Reconnaissance Orbiter (MRO) [11], which now has amassed over five full martian years’ worth of data. For this study, the assimilated MCS variables were temperature and dust profiles. Temperature profiles extend from the surface to approximately 100 km, and dust profiles from as low as 10 km above the surface up to a maximum height of approximately 50 km. Retrieval of dust profiles allows MCS to observe the complex vertical dust structure in the atmosphere. The retrieval version used is 5.2, a re-processing using updated 2D geometry [12]. This results in improved retrievals, especially in the polar regions. While not used in this study, the NOMAD instrument aboard ExoMars TGO will soon provide another high-volume source of dust profiles alongside MCS [13], and should return observations with an even higher vertical resolution. Discussion: The assimilation of MCS dust profiles poses unique technical challenges, but presents the opportunity of representing Mars’ vertical dust distribution with unprecedented spatial and temporal accuracy within a GCM. Some outstanding questions for further experimentation and discussion include: What are the optimal spatial and, in particular, vertical model resolutions for assimilation of this data? Can dust profile assimilation aid in forecasting? Previous indirect assimilation of vertical dust via its MCS temperature signature has yielded a forecast time of 10 sols [5]; how dependent is this on the assimilation scheme and the choice of assimilating variables? How should we approach the bimodal nature of MCS local times? Should we give higher weighting to nightside dust observations, which tend to have better vertical coverage due to reduced scattering? And how much can we validly infer from the high day-night variability seen in MCS dust profiles? What are the best heuristics for filtering spurious opacities which could disrupt the assimilation, for example due to CO2 ice or surface reflectance [16])? What are the optimal ways of dealing with spatial and temporal gaps in the dataset? How can we best represent the dust distribution beyond the range of MCS, especially in the lowest 5-10 km of the atmosphere? What are the advantages and disadvantages of directly assimilating the dust field vs indirectly up-dating the dust field via its temperature signature, as seen in Fig. 1? Dust profile assimilation has been used to track individual dust storm events [4]; what can this tell us about storm formation and evolution, and can it be used for storm forecasting? How can we best constrain and validate the column optical depths of MCS dust profiles? Some ways forward regarding these questions will be explored, including comparative reanalyses and validation against different orbital datasets. Comparisons against MCS and other retrievals (such as NOMAD) should provide insight into the advantages of various in-model representations of features such as the dust distribution as well as the possible advantages or disadvantages of pruning the assimilated dataset. Meanwhile, alternate orbital or even ground-based sources of column opacity (such as Mars Express and MSL) could help better con-strain the distribution of dust not seen by MCS and offer clues how best to proceed in periods when MCS data is missing or limited. Some results of intercomparisons will be presented with the aim of fostering a more general discussion on MCS assimilation techniques. References: [1] Lewis, S. R. et al., Icarus 192 (2), 327-347, 2007. [2] Rogberg, P. et al., QJRMS 136, 1614-1635, 2010. [3] Greybush, S. J. et al., JGR. 117, E11008, 2012. [4] Ruan, T., DPhil Thesis, 2015. [5] Navarro T. et al., Earth and Space Sci., 2017. [6] Forget, F. et al., JGR 104, 24155-24175, 1999. [7] Madeleine, J.-B. et al., JGR (Planets) 116, E11010, 2011. [8] Newman, C. E. et al., JGR 107, 5123, 2002. [9] Lorenc, A. C. et al., QJRMS 117, 59-89, 1991. [10] Lewis, S. R. et al., Icarus 192, 327-347, 2007. [11] McCleese, D. J. et al., J. Geophys. Res. 115, E12016, 2010. [12] Kleinböhl, A. et al., J. Quant. Spectrosc. Radiat. Transfer 187, 511-522, 2017. [13] Patel, M. R. et al., Appl. Opt. 56 (10), 2771-2782, 2017. [14] Navarro, T. et al., Geophys. Res. Lett. 41, 6620-6626, 2014. [15] Streeter, P. M. et al., 6th Intl. Workshop on the Mars Atmosphere, 2017. [16] Kleinböhl, A. et al., Icarus 261, 118-121, 2015

ASSIMILATION OF MARTIAN OZONE

Observations of atmospheric ozone on Mars can be used to develop the representation of trace gas transport, sources and sinks within global circulation models and constrain middle atmosphere wind speeds which are not observed directly. Ozone is also readily destroyed by OH which recycles CO2 to provide global stability of the atmosphere, a process still not fully understood. To make optimal use of information, observations and model information are combined by the process of data assimilation. Although data assimilation is now commonplace on Earth, it is a fairly new concept for other planetary systems, with Mars the only other current candidate. The satellites currently orbiting Mars, combined with the future planned satellite missions, create a great opportunity for the development of trace gas data assimilation techniques for extraterrestrial planets. For this project we use the LMD/UK Martian Global Circulation Model. The model uses a UK spectral dynamical core and transport scheme from a collaboration between the Open University and Oxford University along with physical parameterisations [6] primarily developed by the Laboratoire de Météorologie Dynamique and Instituto de Astrofísica de Andalucía. Combined with the LMD photochemical module and the UK Analysis Correction scheme tuned for Mars for assimilation of observations, we can investigate the evolution of ozone throughout a Martian year. Preliminary results are discussed from investigation of the adjusted model ozone abundance while testing a method of assimilating artificial ozone data. Once refined, the technique will then be used for the assimilation of real observations from the SPICAM and MARCI instruments which provide total ozone column abundance