Minimizing the stabbing number of matchings, trees, and triangulations

The (axis-parallel) stabbing number of a given set of line segments is the maximum number of segments that can be intersected by any one (axis-parallel) line. This paper deals with finding perfect matchings, spanning trees, or triangulations of minimum stabbing number for a given set of points. The complexity of these problems has been a long-standing open question; in fact, it is one of the original 30 outstanding open problems in computational geometry on the list by Demaine, Mitchell, and O'Rourke. The answer we provide is negative for a number of minimum stabbing problems by showing them NP-hard by means of a general proof technique. It implies non-trivial lower bounds on the approximability. On the positive side we propose a cut-based integer programming formulation for minimizing the stabbing number of matchings and spanning trees. We obtain lower bounds (in polynomial time) from the corresponding linear programming relaxations, and show that an optimal fractional solution always contains an edge of at least constant weight. This result constitutes a crucial step towards a constant-factor approximation via an iterated rounding scheme. In computational experiments we demonstrate that our approach allows for actually solving problems with up to several hundred points optimally or near-optimally.Comment: 25 pages, 12 figures, Latex. To appear in "Discrete and Computational Geometry". Previous version (extended abstract) appears in SODA 2004, pp. 430-43

Dependency of simulated tropical Atlantic current variability on the wind forcing

The upper wind-driven circulation in the tropical Atlantic Ocean plays a key role in the basin-wide distribution of water mass properties and affects the transport of heat, freshwater, and biogeochemical tracers such as oxygen or nutrients. It is crucial to improve our understanding of its long-term behaviour, which largely relies on model simulations and applied forcing due to sparse observational data coverage, especially before the mid-2000s. Here, we apply two different forcing products, the Coordinated Ocean-ice Reference Experiments (CORE) v2 and the Japanese 55-year Reanalysis (JRA55-do) surface dataset, to a high-resolution ocean model. Where possible, we compare the simulated results to long-term observations. We find large discrepancies between the two simulations regarding the wind and current field. In the CORE simulation, strong, large-scale wind stress curl amplitudes above the upwelling regions of the eastern tropical North Atlantic seem to cause an overestimation of the mean and seasonal variability in the eastward subsurface current just north of the Equator. The wind stress curl of JRA55-do forcing shows much finer structures, and the JRA55-do simulation is in better agreement with the mean and intraseasonal fluctuations in the subsurface current found in observations. The northern branch of the South Equatorial Current flows westward at the surface just north of the Equator. On interannual to decadal timescales, it shows a high correlation of R=0.9 with the zonal wind stress in the CORE simulation but only a weak correlation of R=0.35 in the JRA55-do simulation. We also identify similarities between the two simulations. The strength of the eastward-flowing North Equatorial Counter Current located between 3 and 10° N covaries with the strength of the meridional wind stress just north of the Equator on interannual to decadal timescales in the two simulations. Both simulations present a comparable mean, seasonal cycle and trend of the eastward off-equatorial subsurface current south of the Equator but underestimate the current strength by half compared to observations. In both simulations, the eastward-flowing Equatorial Undercurrent weakened between 1990 and 2009. In the JRA simulation, which covers the modern period of observations, the Equatorial Undercurrent strengthened again between 2008 to 2018, which agrees with observations, although the simulation underestimates the strengthening by over a third. We propose that long-term observations, once they have reached a critical length, need to be used to test the quality of wind-driven simulations. This study presents one step in this direction.</p

ENSO feedbacks and their relationships with the mean state in a flux adjusted ensemble

The El Niño Southern Oscillation (ENSO) is governed by a combination of amplifying and damping ocean–atmosphere feedbacks in the equatorial Pacific. Here we quantify these feedbacks in a flux adjusted HadCM3 perturbed physics ensemble under present day conditions and a future emissions scenario using the Bjerknes Stability Index (BJ index). Relationships between feedbacks and both the present day biases and responses under climate change of the mean equatorial Pacific climate are investigated. Despite minimised mean sea surface temperature biases through flux adjustment, the important dominant ENSO feedbacks still show biases with respect to observed feedbacks and inter-ensemble diversity. The dominant positive thermocline and zonal advective feedbacks are found to be weaker in ensemble members with stronger mean zonal advection. This is due to a weaker sensitivity of the thermocline slope and zonal surface ocean currents in the east Pacific to surface wind stress anomalies. A drier west Pacific is also found to be linked to weakened shortwave and latent heat flux damping, suggesting a link between ENSO characteristics and the hydrological cycle. In contrast to previous studies using the BJ index that find positive relationships between the index and ENSO amplitude, here they are weakly or negatively correlated, both for present day conditions and for projected differences. This is caused by strong thermodynamic damping which dominates over positive feedbacks, which alone approximate ENSO amplitude well. While the BJ index proves useful for individual linear feedback analysis, we urge caution in using the total linear BJ index alone to assess the reasons for ENSO amplitude biases and its future change in models

Variability and coherence of the Agulhas Undercurrent in a High-resolution Ocean General Circulation Model

The Agulhas Current system has been analyzed in a nested high-resolution ocean model and compared to observations. The model shows good performance in the western boundary current structure and the transports off the South African coast. This includes the simulation of the northward-flowing Agulhas Undercurrent. It is demonstrated that fluctuations of the Agulhas Current and Undercurrent around 50–70 days are due to Natal pulses and Mozambique eddies propagating downstream. A sensitivity experiment that excludes those upstream perturbations significantly reduces the variability as well as the mean transport of the undercurrent. Although the model simulates undercurrents in the Mozambique Channel and east of Madagascar, there is no direct connection between those and the Agulhas Undercurrent. Virtual float releases demonstrate that topography is effectively blocking the flow toward the north

Structural decomposition of decadal climate prediction errors: A Bayesian approach

Decadal climate predictions use initialized coupled model simulations that are typically affected by a drift toward a biased climatology determined by systematic model errors. Model drifts thus reflect a fundamental source of uncertainty in decadal climate predictions. However, their analysis has so far relied on ad-hoc assessments of empirical and subjective character. Here, we define the climate model drift as a dynamical process rather than a descriptive diagnostic. A unified statistical Bayesian framework is proposed where a state-space model is used to decompose systematic decadal climate prediction errors into an initial drift, seasonally varying climatological biases and additional effects of co-varying climate processes. An application to tropical and south Atlantic sea-surface temperatures illustrates how the method allows to evaluate and elucidate dynamic interdependencies between drift, biases, hindcast residuals and background climate. Our approach thus offers a methodology for objective, quantitative and explanatory error estimation in climate predictions