Collusion in a Bertrand duopoly model with decreasing returns and product differentiation

We study cartel stability in a differentiated price-setting duopoly with returns to scale. We show that a cartel may be equally stable in the presence of lower differentiation, provided that the decreasing returns parameter is high. In addition we demonstrate that for a given factor of discount, there are technologies that can have decreasing returns to scale where the cartel always is stable independent of the differentiation degree.

Twin experiments with the equivalent weights particle filter and HadCM3

This paper investigates the use of a particle filter for data assimilation with a full scale coupled ocean–atmosphere general circulation model. Synthetic twin experiments are performed to assess the performance of the equivalent weights filter in such a high-dimensional system. Artificial 2-dimensional sea surface temperature fields are used as observational data every day. Results are presented for different values of the free parameters in the method. Measures of the performance of the filter are root mean square errors, trajectories of individual variables in the model and rank histograms. Filter degeneracy is not observed and the performance of the filter is shown to depend on the ability to keep maximum spread in the ensemble.

Aspects of particle filtering in high-dimensional spaces

Nonlinear data assimilation is high on the agenda in all fields of the geosciences as with ever increasing model resolution and inclusion of more physical (biological etc.) processes, and more complex observation operators the data-assimilation problem becomes more and more nonlinear. The suitability of particle filters to solve the nonlinear data assimilation problem in high-dimensional geophysical problems will be discussed. Several existing and new schemes will be presented and it is shown that at least one of them, the Equivalent-Weights Particle Filter, does indeed beat the curse of dimensionality and provides a way forward to solve the problem of nonlinear data assimilation in high-dimensional systems.

Reconstructing paleosalinity from δ18O: coupled model simulations of the Last Glacial Maximum, Last Interglacial and Late Holocene

Reconstructions of salinity are used to diagnose changes in the hydrological cycle and ocean circulation. A widely used method of determining past salinity uses oxygen isotope (δOw) residuals after the extraction of the global ice volume and temperature components. This method relies on a constant relationship between δOw and salinity throughout time. Here we use the isotope-enabled fully coupled General Circulation Model (GCM) HadCM3 to test the application of spatially and time-independent relationships in the reconstruction of past ocean salinity. Simulations of the Late Holocene (LH), Last Glacial Maximum (LGM), and Last Interglacial (LIG) climates are performed and benchmarked against existing compilations of stable oxygen isotopes in carbonates (δOc), which primarily reflect δOw and temperature. We find that HadCM3 produces an accurate representation of the surface ocean δOc distribution for the LH and LGM. Our simulations show considerable variability in spatial and temporal δOw-salinity relationships. Spatial gradients are generally shallower but within ∼50% of the actual simulated LH to LGM and LH to LIG temporal gradients and temporal gradients calculated from multi-decadal variability are generally shallower than both spatial and actual simulated gradients. The largest sources of uncertainty in salinity reconstructions are found to be caused by changes in regional freshwater budgets, ocean circulation, and sea ice regimes. These can cause errors in salinity estimates exceeding 4 psu. Our results suggest that paleosalinity reconstructions in the South Atlantic, Indian and Tropical Pacific Oceans should be most robust, since these regions exhibit relatively constant δOw-salinity relationships across spatial and temporal scales. Largest uncertainties will affect North Atlantic and high latitude paleosalinity reconstructions. Finally, the results show that it is difficult to generate reliable salinity estimates for regions of dynamic oceanography, such as the North Atlantic, without additional constraints.

Implicit equal-weights particle filter

Filter degeneracy is the main obstacle for the implementation of particle filter in non-linear high-dimensional models. A new scheme, the implicit equal-weights particle filter (IEWPF), is introduced. In this scheme samples are drawn implicitly from proposal densities with a different covariance for each particle, such that all particle weights are equal by construction. We test and explore the properties of the new scheme using a 1,000-dimensional simple linear model, and the 1,000-dimensional non-linear Lorenz96 model, and compare the performance of the scheme to a Local Ensemble Kalman Filter. The experiments show that the new scheme can easily be implemented in high-dimensional systems and is never degenerate, with good convergence properties in both systems.

What controls the vertical distribution of aerosol? Relationships between process sensitivity in HadGEM3–UKCA and inter-model variation from AeroCom Phase II

The vertical profile of aerosol is important for its radiative effects, but weakly constrained by observations on the global scale, and highly variable among different models. To investigate the controlling factors in one particular model, we investigate the effects of individual processes in HadGEM3–UKCA and compare the resulting diversity of aerosol vertical profiles with the inter-model diversity from the AeroCom Phase II control experiment. In this way we show that (in this model at least) the vertical profile is controlled by a relatively small number of processes, although these vary among aerosol components and particle sizes. We also show that sufficiently coarse variations in these processes can produce a similar diversity to that among different models in terms of the global-mean profile and, to a lesser extent, the zonal-mean vertical position. However, there are features of certain models' profiles that cannot be reproduced, suggesting the influence of further structural differences between models. In HadGEM3–UKCA, convective transport is found to be very important in controlling the vertical profile of all aerosol components by mass. In-cloud scavenging is very important for all except mineral dust. Growth by condensation is important for sulfate and carbonaceous aerosol (along with aqueous oxidation for the former and ageing by soluble material for the latter). The vertical extent of biomass-burning emissions into the free troposphere is also important for the profile of carbonaceous aerosol. Boundary-layer mixing plays a dominant role for sea salt and mineral dust, which are emitted only from the surface. Dry deposition and below-cloud scavenging are important for the profile of mineral dust only. In this model, the microphysical processes of nucleation, condensation and coagulation dominate the vertical profile of the smallest particles by number (e.g. total CN  >  3 nm), while the profiles of larger particles (e.g. CN  >  100 nm) are controlled by the same processes as the component mass profiles, plus the size distribution of primary emissions. We also show that the processes that affect the AOD-normalised radiative forcing in the model are predominantly those that affect the vertical mass distribution, in particular convective transport, in-cloud scavenging, aqueous oxidation, ageing and the vertical extent of biomass-burning emissions.

The impact of two coupled cirrus microphysics-radiation parameterizations on the temperature and specific humidity biases in the tropical tropopause layer in a climate model

The impact of two different coupled cirrus microphysics-radiation parameterizations on the zonally averaged temperature and humidity biases in the tropical tropopause layer (TTL) of a Met Office climate model configuration is assessed. One parameterization is based on a linear coupling between a model prognostic variable, the ice mass mixing ratio, qi, and the integral optical properties. The second is based on the integral optical properties being parameterized as functions of qi and temperature, Tc, where the mass coefficients (i.e. scattering and extinction) are parameterized as nonlinear functions of the ratio between qi and Tc. The cirrus microphysics parameterization is based on a moment estimation parameterization of the particle size distribution (PSD), which relates the mass moment (i.e. second moment if mass is proportional to size raised to the power of 2 ) of the PSD to all other PSD moments through the magnitude of the second moment and Tc. This same microphysics PSD parameterization is applied to calculate the integral optical properties used in both radiation parameterizations and, thus, ensures PSD and mass consistency between the cirrus microphysics and radiation schemes. In this paper, the temperature-non-dependent and temperature-dependent parameterizations are shown to increase and decrease the zonally averaged temperature biases in the TTL by about 1 K, respectively. The temperature-dependent radiation parameterization is further demonstrated to have a positive impact on the specific humidity biases in the TTL, as well as decreasing the shortwave and longwave biases in the cloudy radiative effect. The temperature-dependent radiation parameterization is shown to be more consistent with TTL and global radiation observations.