Complex dynamics in simple systems with periodic parameter oscillations

We study systems with periodically oscillating parameters that can give way to complex periodic or nonperiodic orbits. Performing the long time limit, we can define ergodic averages such as Lyapunov exponents, where a negative maximal Lyapunov exponent corresponds to a stable periodic orbit. By this, extremely complicated periodic orbits composed of contracting and expanding phases appear in a natural way. Employing the technique of ϵ-uncertain points, we find that values of the control parameters supporting such periodic motion are densely embedded in a set of values for which the motion is chaotic. When a tiny amount of noise is coupled to the system, dynamics with positive and with negative nontrivial Lyapunov exponents are indistinguishable. We discuss two physical systems, an oscillatory flow inside a duct and a dripping faucet with variable water supply, where such a mechanism seems to be responsible for a complicated alternation of laminar and turbulent phases.

Thermobaric control of gravitational potential energy generation by diapycnal mixing in the deep ocean

Sources and sinks of gravitational potential energy (GPE) play a rate-limiting role in the large scale ocean circulation. A key source is turbulent diapycnal mixing, whereby irre- versible mixing across isoneutral surfaces is enhanced by turbulent straining of these surfaces. This has motivated international observational efforts to map diapycnal mixing in the global ocean. However, in order to accurately relate the GPE supplied to the large scale circulation by diapycnal mixing to the mixing energy source, it is first necessary to determine the ratio, ξ , of the GPE generation rate to the available potential energy dissipation rate associated with turbulent mixing. Here, the link between GPE and hydro- static pressure is used to derive the GPE budget for a com- pressible ocean with a nonlinear equation of state. The role of diapycnal mixing is isolated and from this a global cli- matological distribution of ξ is calculated. It is shown that, for a given source of mixing energy, typically three times as much GPE is generated if the mixing takes place in bottom waters rather than in the pycnocline. This is due to GPE destruction by cabbelling in the pycnocline, as opposed to thermobaric enhancement of GPE generation by diapycnal mixing in the deep ocean.

Multi-season eddy covariance observations of energy, water and carbon fluxes over a suburban area in Swindon, UK

Eddy covariance measurements of the turbulent sensible heat, latent heat and carbon dioxide fluxes for 12 months (2011–2012) are reported for the first time for a suburban area in the UK. The results from Swindon are comparable to suburban studies of similar surface cover elsewhere but reveal large seasonal variability. Energy partitioning favours turbulent sensible heat during summer (midday Bowen ratio 1.4–1.6) and latent heat in winter (0.05–0.7). A significant proportion of energy is stored (and released) by the urban fabric and the estimated anthropogenic heat flux is small but non-negligible (0.5–0.9 MJ m−2 day−1). The sensible heat flux is negative at night and for much of winter daytimes, reflecting the suburban nature of the site (44% vegetation) and relatively low built fraction (16%). Latent heat fluxes appear to be water limited during a dry spring in both 2011 and 2012, when the response of the surface to moisture availability can be seen on a daily timescale. Energy and other factors are more relevant controls at other times; at night the wind speed is important. On average, surface conductance follows a smooth, asymmetrical diurnal course peaking at around 6–9 mm s−1, but values are larger and highly variable in wet conditions. The combination of natural (vegetative) and anthropogenic (emission) processes is most evident in the temporal variation of the carbon flux: significant photosynthetic uptake is seen during summer, whilst traffic and building emissions explain peak release in winter (9.5 g C m−2 day−1). The area is a net source of CO2 annually. Analysis by wind direction highlights the role of urban vegetation in promoting evapotranspiration and offsetting CO2 emissions, especially when contrasted against peak traffic emissions from sectors with more roads. Given the extent of suburban land use, these results have important implications for understanding urban energy, water and carbon dynamics.

Multi-site evaluation of an urban land-surface model: intra-urban heterogeneity, seasonality and parameter complexity requirements

An extensive off-line evaluation of the Noah/Single Layer Urban Canopy Model (Noah/SLUCM) urban land-surface model is presented using data from 15 sites to assess (1) the ability of the scheme to reproduce the surface energy balance observed in a range of urban environments, including seasonal changes, and (2) the impact of increasing complexity of input parameter information. Model performance is found to be most dependent on representation of vegetated surface area cover; refinement of other parameter values leads to smaller improvements. Model biases in net all-wave radiation and trade-offs between turbulent heat fluxes are highlighted using an optimization algorithm. Here we use the Urban Zones to characterize Energy partitioning (UZE) as the basis to assign default SLUCM parameter values. A methodology (FRAISE) to assign sites (or areas) to one of these categories based on surface characteristics is evaluated. Using three urban sites from the Basel Urban Boundary Layer Experiment (BUBBLE) dataset, an independent evaluation of the model performance with the parameter values representative of each class is performed. The scheme copes well with both seasonal changes in the surface characteristics and intra-urban heterogeneities in energy flux partitioning, with RMSE performance comparable to similar state-of-the-art models for all fluxes, sites and seasons. The potential of the methodology for high-resolution atmospheric modelling application using the Weather Research and Forecasting (WRF) model is highlighted. This analysis supports the recommendations that (1) three classes are appropriate to characterize the urban environment, and (2) that the parameter values identified should be adopted as default values in WRF.

Generation of a buoyancy-driven coastal current by an Antarctic Polynya

Descent and spreading of high salinity water generated by salt rejection during sea ice formation in an Antarctic coastal polynya is studied using a hydrostatic, primitive equation three-dimensional ocean model called the Proudman Oceanographic Laboratory Coastal Ocean Modeling System (POLCOMS). The shape of the polynya is assumed to be a rectangle 100 km long and 30 km wide, and the salinity flux into the polynya at its surface is constant. The model has been run at high horizontal spatial resolution (500 m), and numerical simulations reveal a buoyancy-driven coastal current. The coastal current is a robust feature and appears in a range of simulations designed to investigate the influence of a sloping bottom, variable bottom drag, variable vertical turbulent diffusivities, higher salinity flux, and an offshore position of the polynya. It is shown that bottom drag is the main factor determining the current width. This coastal current has not been produced with other numerical models of polynyas, which may be because these models were run at coarser resolutions. The coastal current becomes unstable upstream of its front when the polynya is adjacent to the coast. When the polynya is situated offshore, an unstable current is produced from its outset owing to the capture of cyclonic eddies. The effect of a coastal protrusion and a canyon on the current motion is investigated. In particular, due to the convex shape of the coastal protrusion, the current sheds a dipolar eddy.

Modelling of urban sensible heat flux at multiple spatial scales : a demonstration using airborne hyperspectral imagery of Shanghai and a temperature–emissivity separation approach

Urbanization related alterations to the surface energy balance impact urban warming (‘heat islands’), the growth of the boundary layer, and many other biophysical processes. Traditionally, in situ heat flux measures have been used to quantify such processes, but these typically represent only a small local-scale area within the heterogeneous urban environment. For this reason, remote sensing approaches are very attractive for elucidating more spatially representative information. Here we use hyperspectral imagery from a new airborne sensor, the Operative Modular Imaging Spectrometer (OMIS), along with a survey map and meteorological data, to derive the land cover information and surface parameters required to map spatial variations in turbulent sensible heat flux (QH). The results from two spatially-explicit flux retrieval methods which use contrasting approaches and, to a large degree, different input data are compared for a central urban area of Shanghai, China: (1) the Local-scale Urban Meteorological Parameterization Scheme (LUMPS) and (2) an Aerodynamic Resistance Method (ARM). Sensible heat fluxes are determined at the full 6 m spatial resolution of the OMIS sensor, and at lower resolutions via pixel aggregation and spatial averaging. At the 6 m spatial resolution, the sensible heat flux of rooftop dominated pixels exceeds that of roads, water and vegetated areas, with values peaking at ∼ 350 W m− 2, whilst the storage heat flux is greatest for road dominated pixels (peaking at around 420 W m− 2). We investigate the use of both OMIS-derived land surface temperatures made using a Temperature–Emissivity Separation (TES) approach, and land surface temperatures estimated from air temperature measures. Sensible heat flux differences from the two approaches over the entire 2 × 2 km study area are less than 30 W m− 2, suggesting that methods employing either strategy maybe practica1 when operated using low spatial resolution (e.g. 1 km) data. Due to the differing methodologies, direct comparisons between results obtained with the LUMPS and ARM methods are most sensibly made at reduced spatial scales. At 30 m spatial resolution, both approaches produce similar results, with the smallest difference being less than 15 W m− 2 in mean QH averaged over the entire study area. This is encouraging given the differing architecture and data requirements of the LUMPS and ARM methods. Furthermore, in terms of mean study QH, the results obtained by averaging the original 6 m spatial resolution LUMPS-derived QH values to 30 and 90 m spatial resolution are within ∼ 5 W m− 2 of those derived from averaging the original surface parameter maps prior to input into LUMPS, suggesting that that use of much lower spatial resolution spaceborne imagery data, for example from Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) is likely to be a practical solution for heat flux determination in urban areas.

On the Nusselt number for frazil ice growth—a correction to ”Frazil evolution in channels“ by Lars Hammar and Hung-Tao Shen

The growth (melt) rate of frazil ice is governed by heat transfer away from (towards) the ice crystal, which can be represented by the Nusselt number. We discuss choices for the Nusselt number and turbulent length scale appropriate for frazil ice and note an inaccuracy in the study ”Frazil evolution in channels“ by Lars Hammar and Hung-Tao Shen, which has also led to potentially significant errors in several other papers. We correct this error and suggest an appropriate strategy for determining the Nusselt number applicable to frazil ice growth and melting.

Influence of mass diffusion in sea ice dynamical models

The mixing of floes of different thickness caused by repeated deformation of the ice cover is modeled as diffusion, and the mass balance equation for sea ice accounting for mass diffusion is developed. The effect of deformational diffusion on the ice thickness balance is shown to reach 1% of the divergence effect, which describes ridging and lead formation. This means that with the same accuracy the mass balance equation can be written in terms of mean velocity rather than mean mass-weighted velocity, which one should correctly use for a multicomponent fluid such as sea ice with components identified by floe thickness. Mixing (diffusion) of sea ice also occurs because of turbulent variations in wind and ocean drags that are unresolved in models. Estimates of the importance of turbulent mass diffusion on the dynamic redistribution of ice thickness are determined using empirical data for the turbulent diffusivity. For long-time-scale prediction (≫5 days), where unresolved atmospheric motion may have a length scale on the order of the Arctic basin and the time scale is larger than the synoptic time scale of atmospheric events, turbulent mass diffusion can exceed 10% of the divergence effect. However, for short-time-scale prediction, for example, 5 days, the unresolved scales are on the order of 100 km, and turbulent diffusion is about 0.1% of the divergence effect. Because inertial effects are small in the dynamics of the sea ice pack, diffusive momentum transfer can be disregarded.

The influence of ocean flow on newly forming sea ice

The heat and mass balance of the Arctic Ocean is very sensitive to the growth and decay of sea ice and the interaction between the heat and salt fields in the oceanic boundary layer. The hydraulic roughness of sea ice controls the detailed nature of turbulent fluxes in the boundary layer and hence is an important ingredient in model parameterizations. We describe a novel mechanism for the generation of corrugations of the sea ice–ocean interface, present a mathematical analysis elucidating the mechanism, and present numerical calculations for geophysically relevant conditions. The mechanism relies on brine flows developing in the sea ice due to Bernoulli suction by flow of ocean past the interface. For oceanic shears at the ice interface of 0.2 s−1, we expect the corrugations to form with a wavelength dependent upon the permeability structure of the sea ice which is described herein. The mechanism should be particularly important during sea ice formation in wind-maintained coastal polynyas and in leads. This paper applies our earlier analyses of the fundamental instability to field conditions and extends it to take account of the anisotropic and heterogeneous permeability of sea ice.

The effect of a new drag-law parameterization on ice shelf water plume dynamics

A drag law accounting for Ekman rotation adjacent to a flat, horizontal bou ndary is proposed for use in a plume model that is written in terms of the depth-mean velocity. The drag l aw contains a variable turning angle between the mean velocity and the drag imposed by the turbulent bound ary layer. The effect of the variable turning angle in the drag law is studied for a plume of ice shelf wat er (ISW) ascending and turning beneath an Antarctic ice shelf with draft decreasing away from the groundi ng line. As the ISW plume ascends the sloping ice shelf–ocean boundary, it can melt the ice shelf, wh ich alters the buoyancy forcing driving the plume motion. Under these conditions, the typical turning ang le is of order 10° over most of the plume area for a range of drag coefficients (the minus sign arises for th e Southern Hemisphere). The rotation of the drag with respect to the mean velocity is found to be signifi cant if the drag coefficient exceeds 0.003; in this case the plume body propagates farther along and across the b ase of the ice shelf than a plume with the standard quadratic drag law with no turning angle.

Energy exchange in a dense urban environment – part I: temporal variability of long-term observations in central London

Though anthropogenic impacts on boundary layer climates are expected to be large in dense urban areas, to date very few studies of energy flux observations are available. We report on 3.5 years of measurements gathered in central London, UK. Radiometer and eddy covariance observations at two adjacent sites, at different heights, were analysed at various temporal scales and with respect to meteorological conditions, such as cloud cover. Although the evaporative flux is generally small due to low moisture availability and a predominately impervious surface, the enhancement following rainfall usually lasts for 12–18 h. As both the latent and sensible heat fluxes are larger in the afternoon, they maintain a relatively consistent Bowen ratio throughout the middle of the day. Strong storage and anthropogenic heat fluxes sustain high and persistently positive sensible heat fluxes. At the monthly time scale, the urban surface often loses more energy by this turbulent heat flux than is gained from net all-wave radiation. Auxiliary anthropogenic heat flux information suggest human activities in the study area are sufficient to provide this energy.

Energy exchange in a dense urban environment Part II: impact of spatial heterogeneity of the surface

The centre of cities, characterised by spatial and temporal complexity, are challenging environments for micrometeorological research. This paper considers the impact of sensor location and heterogeneity of the urban surface on flux observations in the dense city centre of London, UK. Data gathered at two sites in close vicinity, but with different measurement heights, were analysed to investigate the influence of source area characteristics on long-term radiation and turbulent heat fluxes. Combining consideration of diffuse radiation and effects of specular reflections, the non-Lambertian urban surface is found to impact the measurements of surface albedo. Comparisons of observations from the two sites reveal that turbulent heat fluxes are similar under some flow conditions. However, they mostly observe processes at different scales due to their differing measurement heights, highlighting the critical impact of siting sensors in urban areas. A detailed source area analysis is presented to investigate the surface controls influencing the energy exchanges at the different scales

Freshwater and heat transports from global ocean synthesis

[1] An eddy-permitting ¼° global ocean reanalysis based on the Operational Met Office FOAM data assimilation system has been run for 1989–2010 forced by ERA-Interim meteorology. Freshwater and heat transports are compared with published estimates globally and in each basin, with special focus on the Atlantic. The meridional transports agree with observations within errors at most locations, but where eddies are active the transports by the mean flow are nearly always in better agreement than the total transports. Eddy transports are down gradient and are enhanced relative to a free run. They may oppose or reinforce mean transports and provide 40–50% of the total transport near midlatitude fronts, where eddies with time scales <1 month provide up to 15%. Basin-scale freshwater convergences are calculated with the Arctic/Atlantic, Indian, and Pacific oceans north of 32°S, all implying net evaporation of 0.33 ± 0.04 Sv, 0.65 ± 0.07 Sv, and 0.09 ± 0.04 Sv, respectively, within the uncertainty of observations in the Atlantic and Pacific. The Indian is more evaporative and the Southern Ocean has more precipitation (1.07 Sv). Air-sea fluxes are modified by assimilation influencing turbulent heat fluxes and evaporation. Generally, surface and assimilation fluxes together match the meridional transports, indicating that the reanalysis is close to a steady state. Atlantic overturning and gyre transports are assessed with overturning freshwater transports southward at all latitudes. At 26°N eddy transports are negligible, overturning transport is 0.67 ± 0.19 Sv southward and gyre transport is 0.44 ± 0.17 Sv northward, with divergence between 26°N and the Bering Strait of 0.13 ± 0.23 Sv over 2004–2010.

Quantitative ventilation assessments of idealized urban canopy layers with various urban layouts and the same building packing density

This paper investigates urban canopy layers (UCL) ventilation under neutral atmospheric condition with the same building area density (λp=0.25) and frontal area density (λf=0.25) but various urban sizes, building height variations, overall urban forms and wind directions. Turbulent airflows are first predicted by CFD simulations with standard k-ε model evaluated by wind tunnel data. Then air change rates per hour (ACH) and canopy purging flow rate (PFR) are numerically analyzed to quantify the rate of air exchange and the net ventilation capacity induced by mean flows and turbulence. With a parallel approaching wind (θ=0o), the velocity ratio first decreases in the adjustment region, followed by the fully-developed region where the flow reaches a balance. Although the flow quantities macroscopically keep constant, however ACH decreases and overall UCL ventilation becomes worse if urban size rises from 390m to 5km. Theoretically if urban size is infinite, ACH may reach a minimum value depending on local roof ventilation, and it rises from 1.7 to 7.5 if the standard deviation of building height variations increases (0% to 83.3%). Overall UCL ventilation capacity (PFR) with a square overall urban form (Lx=Ly=390m) is better as θ=0o than oblique winds (θ=15o, 30o, 45o), and it exceeds that of a staggered urban form under all wind directions (θ=0o to 45o), but is less than that of a rectangular urban form (Lx=570m, Ly=270m) under most wind directions (θ=30o to 90o). Further investigations are still required to quantify the net ventilation efficiency induced by mean flows and turbulence.

Simulated climate change impact on summer dissolved organic carbon release from peat and surface vegetation: implications for drinking water treatment

Uncertainty regarding changes in dissolved organic carbon (DOC) quantity and quality has created interest in managing peatlands for their ecosystem services such as drinking water provision. The evidence base for such interventions is, however, sometimes contradictory. We performed a laboratory climate manipulation using a factorial design on two dominant peatland vegetation types (Calluna vulgaris and Sphagnum Spp.) and a peat soil collected from a drinking water catchment in Exmoor National Park, UK. Temperature and rainfall were set to represent baseline and future conditions under the UKCP09 2080s high emissions scenario for July and August. DOC leachate then underwent standard water treatment of coagulation/flocculation before chlorination. C. vulgaris leached more DOC than Sphagnum Spp. (7.17 versus 3.00 mg g−1) with higher specific ultraviolet (SUVA) values and a greater sensitivity to climate, leaching more DOC under simulated future conditions. The peat soil leached less DOC (0.37 mg g−1) than the vegetation and was less sensitive to climate. Differences in coagulation removal efficiency between the DOC sources appears to be driven by relative solubilisation of protein-like DOC, observed through the fluorescence peak C/T. Post-coagulation only differences between vegetation types were detected for the regulated disinfection by-products (DBPs), suggesting climate change influence at this scale can be removed via coagulation. Our results suggest current biodiversity restoration programmes to encourage Sphagnum Spp. will result in lower DOC concentrations and SUVA values, particularly with warmer and drier summers.