Classifying general nonlinear force laws in cell-based models via the continuum limit

though discrete cell-based frameworks are now commonly used to simulate a whole range of biological phenomena, it is typically not obvious how the numerous different types of model are related to one another, nor which one is most appropriate in a given context. Here we demonstrate how individual cell movement on the discrete scale modeled using nonlinear force laws can be described by nonlinear diffusion coefficients on the continuum scale. A general relationship between nonlinear force laws and their respective diffusion coefficients is derived in one spatial dimension and, subsequently, a range of particular examples is considered. For each case excellent agreement is observed between numerical solutions of the discrete and corresponding continuum models. Three case studies are considered in which we demonstrate how the derived nonlinear diffusion coefficients can be used to (a) relate different discrete models of cell behavior; (b) derive discrete, intercell force laws from previously posed diffusion coefficients, and (c) describe aggregative behavior in discrete simulations.

The water vapour continuum: brief history and recent developments

The water vapour continuum is characterised by absorption that varies smoothly with wavelength, from the visible to the microwave. It is present within the rotational and vibrational–rotational bands of water vapour, which consist of large numbers of narrow spectral lines, and in the many ‘windows’ between these bands. The continuum absorption in the window regions is of particular importance for the Earth’s radiation budget and for remote-sensing techniques that exploit these windows. Historically, most attention has focused on the 8–12 μm (mid-infrared) atmospheric window, where the continuum is relatively well-characterised, but there have been many fewer measurements within bands and in other window regions. In addition, the causes of the continuum remain a subject of controversy. This paper provides a brief historical overview of the development of understanding of the continuum and then reviews recent developments, with a focus on the near-infrared spectral region. Recent laboratory measurements in near-infrared windows, which reveal absorption typically an order of magnitude stronger than in widely used continuum models, are shown to have important consequences for remote-sensing techniques that use these windows for retrieving cloud properties.

A continuum model of a fishing net

A working report for the Department of Agriculture and Fisheries, Scotland, Marine Laboratory Aberdeen

Absolute high spectral resolution measurements of surface solar radiation for detection of water vapour continuum absorption

Solar-pointing Fourier transform infrared (FTIR) spectroscopy offers the capability to measure both the fine scale and broadband spectral structure of atmospheric transmission simultaneously across wide spectral regions. It is therefore suited to the study of both water vapour monomer and continuum absorption behaviours. However, in order to properly address this issue, it is necessary to radiatively calibrate the FTIR instrument response. A solar-pointing high-resolution FTIR spectrometer was deployed as part of the ‘Continuum Absorption by Visible and Infrared radiation and its Atmospheric Relevance’ (CAVIAR) consortium project. This paper describes the radiative calibration process using an ultra-high-temperature blackbody and the consideration of the related influence factors. The result is a radiatively calibrated measurement of the solar irradiation at the ground across the IR region from 2000 to 10 000 cm−1 with an uncertainty of between 3.3 and 5.9 per cent. This measurement is shown to be in good general agreement with a radiative-transfer model. The results from the CAVIAR field measurements are being used in ongoing studies of atmospheric absorbers, in particular the water vapour continuum.

Water vapour foreign-continuum absorption in near-infrared windows from laboratory measurements

For a long time, it has been believed that atmospheric absorption of radiation within wavelength regions of relatively high infrared transmittance (so-called ‘windows’) was dominated by the water vapour self-continuum, that is, spectrally smooth absorption caused by H2O−H2O pair interaction. Absorption due to the foreign continuum (i.e. caused mostly by H2O−N2 bimolecular absorption in the Earth's atmosphere) was considered to be negligible in the windows. We report new retrievals of the water vapour foreign continuum from high-resolution laboratory measurements at temperatures between 350 and 430 K in four near-infrared windows between 1.1 and 5 μm (9000–2000 cm−1). Our results indicate that the foreign continuum in these windows has a very weak temperature dependence and is typically between one and two orders of magnitude stronger than that given in representations of the continuum currently used in many climate and weather prediction models. This indicates that absorption owing to the foreign continuum may be comparable to the self-continuum under atmospheric conditions in the investigated windows. The calculated global-average clear-sky atmospheric absorption of solar radiation is increased by approximately 0.46 W m−2 (or 0.6% of the total clear-sky absorption) by using these new measurements when compared with calculations applying the widely used MTCKD (Mlawer–Tobin–Clough–Kneizys–Davies) foreign-continuum model.

Climate change and coupling of macronutrient cycles along the atmospheric, terrestrial, freshwater and estuarine continuum

This paper provides an introduction to the Special Issue on “Climate Change and Coupling of Macronutrient Cycles along the Atmospheric, Terrestrial, Freshwater and Estuarine Continuum”, dedicated to Colin Neal on his retirement. It is not intended to be a review of this vast subject, but an attempt to synthesize some of the major findings from the 22 contributions to the Special Issue in the context of what is already known. The major research challenges involved in understanding coupled macronutrient cycles in these environmental media are highlighted, and the difficulties of making credible predictions of the effects of climate change are discussed. Of particular concern is the possibility of interactions which will enhance greenhouse gas concentrations and provide positive feedback to global warming.

Near-infrared water vapour self-continuum at close to room temperature

The gaseous absorption of solar radiation within near-infrared atmospheric windows in the Earth's atmosphere is dominated by the water vapour continuum. Recent measurements by Baranov et al. (2011) [17] in 2500 cm−1 (4 μm) window and by Ptashnik et al. (2011) [18] in a few near-infrared windows revealed that the self-continuum absorption is typically an order of magnitude stronger than given by the MT_CKD continuum model prior to version 2.5. Most of these measurements, however, were made at elevated temperatures, which makes their application to atmospheric conditions difficult. Here we report new laboratory measurements of the self-continuum absorption at 289 and 318 K in the near-infrared spectral region 1300–8000 cm−1, using a multipass 30 m base cell with total optical path 612 m. Our results confirm the main conclusions of the previous measurements both within bands and in windows. Of particular note is that we present what we believe to be the first near-room temperature measurement using Fourier Transform Spectrometry of the self-continuum in the 6200 cm−1 (1.6 μm) window, which provides tentative evidence that, at such temperatures, the water vapour continuum absorption may be as strong as it is in 2.1 μm and 4 μm windows and up to 2 orders of magnitude stronger than the MT_CKD-2.5 continuum. We note that alternative methods of measuring the continuum in this window have yielded widely differing assessment of its strength, which emphasises the need for further measurements.

A continuum model of melt pond evolution on Arctic sea ice

[1] During the Northern Hemisphere summer, absorbed solar radiation melts snow and the upper surface of Arctic sea ice to generate meltwater that accumulates in ponds. The melt ponds reduce the albedo of the sea ice cover during the melting season, with a significant impact on the heat and mass budget of the sea ice and the upper ocean. We have developed a model, designed to be suitable for inclusion into a global circulation model (GCM), which simulates the formation and evolution of the melt pond cover. In order to be compatible with existing GCM sea ice models, our melt pond model builds upon the existing theory of the evolution of the sea ice thickness distribution. Since this theory does not describe the topography of the ice cover, which is crucial to determining the location, extent, and depth of individual ponds, we have needed to introduce some assumptions. We describe our model, present calculations and a sensitivity analysis, and discuss our results.

Continuum sea ice rheology determined from subcontinuum mechanics

[1] A method is presented to calculate the continuum-scale sea ice stress as an imposed, continuum-scale strain-rate is varied. The continuum-scale stress is calculated as the area-average of the stresses within the floes and leads in a region (the continuum element). The continuum-scale stress depends upon: the imposed strain rate; the subcontinuum scale, material rheology of sea ice; the chosen configuration of sea ice floes and leads; and a prescribed rule for determining the motion of the floes in response to the continuum-scale strain-rate. We calculated plastic yield curves and flow rules associated with subcontinuum scale, material sea ice rheologies with elliptic, linear and modified Coulombic elliptic plastic yield curves, and with square, diamond and irregular, convex polygon-shaped floes. For the case of a tiling of square floes, only for particular orientations of the leads have the principal axes of strain rate and calculated continuum-scale sea ice stress aligned, and these have been investigated analytically. The ensemble average of calculated sea ice stress for square floes with uniform orientation with respect to the principal axes of strain rate yielded alignment of average stress and strain-rate principal axes and an isotropic, continuum-scale sea ice rheology. We present a lemon-shaped yield curve with normal flow rule, derived from ensemble averages of sea ice stress, suitable for direct inclusion into the current generation of sea ice models. This continuum-scale sea ice rheology directly relates the size (strength) of the continuum-scale yield curve to the material compressive strength.

A continuum anisotropic model of sea-ice dynamics

We develop the essential ingredients of a new, continuum and anisotropic model of sea-ice dynamics designed for eventual use in climate simulation. These ingredients are a constitutive law for sea-ice stress, relating stress to the material properties of sea ice and to internal variables describing the sea-ice state, and equations describing the evolution of these variables. The sea-ice cover is treated as a densely flawed two-dimensional continuum consisting of a uniform field of thick ice that is uniformly permeated with narrow linear regions of thinner ice called leads. Lead orientation, thickness and width distributions are described by second-rank tensor internal variables: the structure, thickness and width tensors, whose dynamics are governed by corresponding evolution equations accounting for processes such as new lead generation and rotation as the ice cover deforms. These evolution equations contain contractions of higher-order tensor expressions that require closures. We develop a sea-ice stress constitutive law that relates sea-ice stress to the structure tensor, thickness tensor and strain rate. For the special case of empty leads (containing no ice), linear closures are adopted and we present calculations for simple shear, convergence and divergence.

Global radiative and climate effect of the water vapour continuum at visible and near-infrared wavelengths

Recent laboratory measurements show that absorption by the water vapour continuum in near-infrared windows may be about an order of magnitude higher than assumed in many radiation codes. The radiative impact of the continuum at visible and near-infrared wavelengths is examined for the present day and for a possible future warmer climate (with a global-mean total column water increase of 33%). The calculations use a continuum model frequently used in climate models (‘CKD’) and a continuum model where absorption is enhanced at wavelengths greater than 1 µm based on recent measurements (‘CAVIAR’). The continuum predominantly changes the partitioning between solar radiation absorbed by the surface and the atmosphere; changes in top-of-atmosphere net irradiances are smaller. The global-mean clear-sky atmospheric absorption is enhanced by 1.5 W m−2 (about 2%) and 2.8 W m−2 (about 3.5%) for CKD and CAVIAR respectively, relative to a hypothetical no-continuum case, with all-sky enhancements about 80% of these values. The continuum is, in relative terms, more important for radiation budget changes between the present day and a possible future climate. Relative to the no-continuum case, the increase in global-mean clear-sky absorption is 8% higher using CKD and almost 20% higher using CAVIAR; all-sky enhancements are about half these values. The effect of the continuum is estimated for the solar component of the water vapour feedback, the reduction in downward surface irradiance and precipitation change in a warmer world. For CKD and CAVIAR respectively, and relative to the no-continuum case, the solar component of the water vapour feedback is enhanced by about 4 and 9%, the change in clear-sky downward surface irradiance is 7 and 18% more negative, and the global-mean precipitation response decreases by 1 and 4%. There is a continued need for improved continuum measurements, especially at atmospheric temperatures and at wavelengths below 2 µm.

The water vapour continuum in near-infrared windows – current understanding and prospects for its inclusion in spectroscopic databases

Spectroscopic catalogues, such as GEISA and HITRAN, do not yet include information on the water vapour continuum that pervades visible, infrared and microwave spectral regions. This is partly because, in some spectral regions, there are rather few laboratory measurements in conditions close to those in the Earth’s atmosphere; hence understanding of the characteristics of the continuum absorption is still emerging. This is particularly so in the near-infrared and visible, where there has been renewed interest and activity in recent years. In this paper we present a critical review focusing on recent laboratory measurements in two near-infrared window regions (centred on 4700 and 6300 cm−1) and include reference to the window centred on 2600 cm−1 where more measurements have been reported. The rather few available measurements, have used Fourier transform spectroscopy (FTS), cavity ring down spectroscopy, optical-feedback – cavity enhanced laser spectroscopy and, in very narrow regions, calorimetric interferometry. These systems have different advantages and disadvantages. Fourier Transform Spectroscopy can measure the continuum across both these and neighbouring windows; by contrast, the cavity laser techniques are limited to fewer wavenumbers, but have a much higher inherent sensitivity. The available results present a diverse view of the characteristics of continuum absorption, with differences in continuum strength exceeding a factor of 10 in the cores of these windows. In individual windows, the temperature dependence of the water vapour self-continuum differs significantly in the few sets of measurements that allow an analysis. The available data also indicate that the temperature dependence differs significantly between different near-infrared windows. These pioneering measurements provide an impetus for further measurements. Improvements and/or extensions in existing techniques would aid progress to a full characterisation of the continuum – as an example, we report pilot measurements of the water vapour self-continuum using a supercontinuum laser source coupled to an FTS. Such improvements, as well as additional measurements and analyses in other laboratories, would enable the inclusion of the water vapour continuum in future spectroscopic databases, and therefore allow for a more reliable forward modelling of the radiative properties of the atmosphere. It would also allow a more confident assessment of different theoretical descriptions of the underlying cause or causes of continuum absorption.