Self-energy flows in the two-dimensional repulsive Hubbard model

We study the two-dimensional repulsive Hubbard model by functional RG methods, using our recently proposed channel decomposition of the interaction vertex. The main technical advance of this work is that we calculate the full Matsubara frequency dependence of the self-energy and the interaction vertex in the whole frequency range without simplifying assumptions on its functional form, and that the effects of the self-energy are fully taken into account in the equations for the flow of the two-body vertex function. At Van Hove filling, we find that the Fermi surface deformations remain small at fixed particle density and have a minor impact on the structure of the interaction vertex. The frequency dependence of the self-energy, however, turns out to be important, especially at a transition from ferromagnetism to d-wave superconductivity. We determine non-Fermi-liquid exponents at this transition point.Comment: 48 pages, 18 figure

Functional renormalization and mean-field approach to multiband systems with spin-orbit coupling: Application to the Rashba model with attractive interaction

The functional renormalization group (RG) in combination with Fermi surface patching is a well-established method for studying Fermi liquid instabilities of correlated electron systems. In this article, we further develop this method and combine it with mean-field theory to approach multiband systems with spin-orbit coupling, and we apply this to a tight-binding Rashba model with an attractive, local interaction. The spin dependence of the interaction vertex is fully implemented in a RG flow without SU(2) symmetry, and its momentum dependence is approximated in a refined projection scheme. In particular, we discuss the necessity of including in the RG flow contributions from both bands of the model, even if they are not intersected by the Fermi level. As the leading instability of the Rashba model, we find a superconducting phase with a singlet-type interaction between electrons with opposite momenta. While the gap function has a singlet spin structure, the order parameter indicates an unconventional superconducting phase, with the ratio between singlet and triplet amplitudes being plus or minus one on the Fermi lines of the upper or lower band, respectively. We expect our combined functional RG and mean-field approach to be useful for an unbiased theoretical description of the low-temperature properties of spin-based materials.Comment: consistent with published version in Physical Review B (2016

A novel framework to harmonise satellite data series for climate applications

Fundamental and thematic climate data records derived from satellite observations provide unique information for climate monitoring and research. Since any satellite operates over a limited period of time only, creating a climate data record requires the combination of space-borne measurements from a series of several (often similar) satellite sensors. A simple combination of calibrated measurements from several sensors, however, can produce an inconsistent climate data record. This is particularly true of older, historic sensors whose behavior in space was often different from their behavior during pre-launch calibration in the laboratory. More scientific value can be derived from considering the series of historical and present satellites as a whole. Here we consider harmonisation as a process that obtains new calibration coefficients for revised sensor calibration models by comparing calibrated measurements over appropriate satellite-to-satellite match-ups, such as simultaneous nadir overpasses. When we perform a comparison of two sensors, however, we must consider that those sensors are not observing exactly the same Earth radiance. This is in part due to differences in exact location and time tolerated by the match-up process itself, but also due to differences in the spectral response functions of the two instruments, even when nominally observing the same spectral band. To derive a harmonised data set we do not aim to correct for spectral response function differences, but to reconcile the calibration of different sensors given their estimated spectral response function differences. Here we present the concept of a framework that establishes calibration coefficients and their uncertainty and error covariance for an arbitrary number of sensors in a metrologically-rigorous manner. We describe harmonisation and its mathematical formulation as an inverse problem. Solving this problem is challenging when some hundreds of millions of match-ups are involved and the errors of fundamental sensor measurements are correlated. We solve the harmonisation problem as marginalised errors in variables regression. The algorithm involves computation of first and second order partial derivatives, for which the corresponding computer source code is generated by Automatic Differentiation. Finally we present re-calibrated AVHRR radiances from a series of 10 sensors. It is shown that the new time series have much less match-up differences while being consistent with uncertainty statistics

Depth-resolved particle associated microbial respiration in the northeast Atlantic

Atmospheric levels of carbon dioxide are tightly linked to the depth at which sinking particulate organic carbon (POC) is remineralised in the ocean. Rapid attenuation of downward POC flux typically occurs in the upper mesopelagic (top few hundred metres of the water column), with much slower loss rates deeper in the ocean. Currently, we lack understanding of the processes that drive POC attenuation, resulting in large uncertainties in the mesopelagic carbon budget. Attempts to balance the POC supply to the mesopelagic with respiration by zooplankton and microbes rarely succeed. Where a balance has been found, depth-resolved estimates reveal large compensating imbalances in the upper and lower mesopelagic. In particular, it has been suggested that respiration by free-living microbes and zooplankton in the upper mesopelagic are too low to explain the observed flux attenuation of POC within this layer. We test the hypothesis that particle-associated microbes contribute significantly to community respiration in the mesopelagic, measuring particle-associated microbial respiration of POC in the northeast Atlantic through shipboard measurements on individual marine snow aggregates collected at depth (36–500 m). We find very low rates of both absolute and carbon-specific particle-associated microbial respiration (< 3 % d−1), suggesting that this term cannot solve imbalances in the upper mesopelagic POC budget. The relative importance of particle-associated microbial respiration increases with depth, accounting for up to 33 % of POC loss in the mid-mesopelagic (128–500 m). We suggest that POC attenuation in the upper mesopelagic (36–128 m) is driven by the transformation of large, fast-sinking particles to smaller, slow-sinking and suspended particles via processes such as zooplankton fragmentation and solubilisation, and that this shift to non-sinking POC may help to explain imbalances in the mesopelagic carbon budget

Particle flux in the oceans: Challenging the steady state assumption

Atmospheric carbon dioxide levels are strongly controlled by the depth at which the organic matter that sinks out of the surface ocean is remineralized. This depth is generally estimated from particle flux profiles measured using sediment traps. Inherent in this analysis is a steady state assumption; that export from the surface does not significantly change in the time it takes material to reach the deepest trap. However, recent observations suggest that a significant fraction of material in the mesopelagic zone sinks slowly enough to bring this into doubt. We use data from a study in the North Atlantic during July/August 2009 to challenge the steady state assumption. An increase in biogenic silica flux with depth was observed which we interpret, based on vertical profiles of diatom taxonomy, as representing the remnants of the spring diatom bloom sinking slowly (<40 m d-1). We were able to reproduce this behaviour using a simple model using satellite-derived export rates and literature-derived remineralization rates. We further provide a simple equation to estimate ‘additional’ (or ‘excess’) POC supply to the dark ocean during non-steady state conditions, which is not captured by traditional sediment trap deployments. In seasonal systems, mesopelagic net organic carbon supply could be wrong by as much as 25% when assuming steady state. We conclude that the steady state assumption leads to misinterpretation of particle flux profiles when input fluxes from the upper ocean vary on the order of weeks, such as in temperate and polar regions with strong seasonal cycles in export

Testing variational estimation of process parameters and initial conditions of an earth system model

We present a variational assimilation system around a coarse resolution Earth System Model (ESM) and apply it for estimating initial conditions and parameters of the model. The system is based on derivative information that is efficiently provided by the ESM's adjoint, which has been generated through automatic differentiation of the model's source code. In our variational approach, the length of the feasible assimilation window is limited by the size of the domain in control space over which the approximation by the derivative is valid. This validity domain is reduced by non-smooth process representations. We show that in this respect the ocean component is less critical than the atmospheric component. We demonstrate how the feasible assimilation window can be extended to several weeks by modifying the implementation of specific process representations and by switching off processes such as precipitation

Variational methods

International audienceThis contribution presents derivative-based methods for local sensitivity analysis, called Variational Sensitivity Analysis (VSA). If one defines an output called the response function, its sensitivity to inputs variations around a nominal value can be studied using derivative (gradient) information. The main issue of VSA is then to provide an efficient way of computing gradients. This contribution first presents the theoretical grounds of VSA: framework and problem statement, tangent and adjoint methods. Then it covers pratical means to compute derivatives, from naive to more sophisticated approaches, discussing their various 2 merits. Finally, applications of VSA are reviewed and some examples are presented, covering various applications fields: oceanography, glaciology, meteorology