Benchmarking 2D hydraulic models for urban flooding

This paper describes benchmark testing of six two-dimensional (2D) hydraulic models (DIVAST, DIVASTTVD, TUFLOW, JFLOW, TRENT and LISFLOOD-FP) in terms of their ability to simulate surface flows in a densely urbanised area. The models are applied to a 1·0 km × 0·4 km urban catchment within the city of Glasgow, Scotland, UK, and are used to simulate a flood event that occurred at this site on 30 July 2002. An identical numerical grid describing the underlying topography is constructed for each model, using a combination of airborne laser altimetry (LiDAR) fused with digital map data, and used to run a benchmark simulation. Two numerical experiments were then conducted to test the response of each model to topographic error and uncertainty over friction parameterisation. While all the models tested produce plausible results, subtle differences between particular groups of codes give considerable insight into both the practice and science of urban hydraulic modelling. In particular, the results show that the terrain data available from modern LiDAR systems are sufficiently accurate and resolved for simulating urban flows, but such data need to be fused with digital map data of building topology and land use to gain maximum benefit from the information contained therein. When such terrain data are available, uncertainty in friction parameters becomes a more dominant factor than topographic error for typical problems. The simulations also show that flows in urban environments are characterised by numerous transitions to supercritical flow and numerical shocks. However, the effects of these are localised and they do not appear to affect overall wave propagation. In contrast, inertia terms are shown to be important in this particular case, but the specific characteristics of the test site may mean that this does not hold more generally.

Variation in dynamic elastic shear modulus of sandstone upon fluid saturation and substitution

Experimental acoustic measurements on sandstone rocks at both sonic and ultrasonic frequencies show that fluid saturation can cause a noticeable change in both the dynamic bulk and shear elastic moduli of sandstones. We observed that the change in dynamic shear modulus upon fluid saturation is highly dependent on the type of saturant, its viscosity, rock microstructure, and applied pressures. Frequency dispersion has some influence on dynamic elastic moduli too, but its effect is limited to the ultrasonic frequency ranges and above. We propose that viscous coupling, reduction in free surface energy, and, to a limited extent, frequency dispersion due to both local and global flow are the main mechanisms responsible for the change in dynamic shear elastic modulus upon fluid saturation and substitution, and we quantify influences.

Approximations to the scattering of water waves by steep topography

A new method is developed for approximating the scattering of linear surface gravity waves on water of varying quiescent depth in two dimensions. A conformal mapping of the fluid domain onto a uniform rectangular strip transforms steep and discontinuous bed profiles into relatively slowly varying, smooth functions in the transformed free-surface condition. By analogy with the mild-slope approach used extensively in unmapped domains, an approximate solution of the transformed problem is sought in the form of a modulated propagating wave which is determined by solving a second-order ordinary differential equation. This can be achieved numerically, but an analytic solution in the form of a rapidly convergent infinite series is also derived and provides simple explicit formulae for the scattered wave amplitudes. Small-amplitude and slow variations in the bedform that are excluded from the mapping procedure are incorporated in the approximation by a straightforward extension of the theory. The error incurred in using the method is established by means of a rigorous numerical investigation and it is found that remarkably accurate estimates of the scattered wave amplitudes are given for a wide range of bedforms and frequencies.

Steady periodic water waves with constant vorticity: regularity and local bifurcation

This paper studies periodic traveling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed. Using conformal mappings the free-boundary problem is transformed into a quasilinear pseudodifferential equation for a periodic function of one variable. The new formulation leads to a regularity result and, by use of bifurcation theory, to the existence of waves of small amplitude even in the presence of stagnation points in the flow.

The Stokes conjecture for waves with vorticity

We study stagnation points of two-dimensional steady gravity free-surface water waves with vorticity. We obtain for example that, in the case where the free surface is an injective curve, the asymptotics at any stagnation point is given either by the “Stokes corner flow” where the free surface has a corner of 120°, or the free surface ends in a horizontal cusp, or the free surface is horizontally flat at the stagnation point. The cusp case is a new feature in the case with vorticity, and it is not possible in the absence of vorticity. In a second main result we exclude horizontally flat singularities in the case that the vorticity is 0 on the free surface. Here the vorticity may have infinitely many sign changes accumulating at the free surface, which makes this case particularly difficult and explains why it has been almost untouched by research so far. Our results are based on calculations in the original variables and do not rely on structural assumptions needed in previous results such as isolated singularities, symmetry and monotonicity.

Numerical conformal mapping and mesh generation for polygonal and multiply-connected regions

Details are given of a boundary-fitted mesh generation method for use in modelling free surface flow and water quality. A numerical method has been developed for generating conformal meshes for curvilinear polygonal and multiply-connected regions. The method is based on the Cauchy-Riemann conditions for the analytic function and is able to map a curvilinear polygonal region directly onto a regular polygonal region, with horizontal and vertical sides. A set of equations have been derived for determining the lengths of these sides and the least-squares method has been used in solving the equations. Several numerical examples are presented to illustrate the method.

Multiple stationary solutions of an irradiated slab

A mathematical model describing the heat budget of an irradiated medium is introduced. The one-dimensional form of the equations and boundary conditions are presented and analysed. Heat transport at one face of the slab occurs by absorption (and reflection) of an incoming beam of short-wave radiation with a fraction of this radiation penetrating into the body of the slab, a diffusive heat flux in the slab and a prescribed incoming heat flux term. The other face of the slab is immersed in its own melt and is considered to be a free surface. Here, temperature continuity is prescribed and evolution of the surface is determined by a Stefan condition. These boundary conditions are flexible enough to describe a range of situations such as a laser shining on an opaque medium, or the natural environment of polar sea ice or lake ice. A two-stream radiation model is used which replaces the simple Beer’s law of radiation attenuation frequently used for semi-infinite domains. The stationary solutions of the governing equations are sought and it is found that there exists two possible stationary solutions for a given set of boundary conditions and a range of parameter choices. It is found that the existence of two stationary solutions is a direct result of the model of radiation absorption, due to its effect on the albedo of the medium. A linear stability analysis and numerical calculations indicate that where two stationary solutions exist, the solution corresponding to a larger thickness is always stable and the solution corresponding to a smaller thickness is unstable. Numerical simulations reveal that when there are two solutions, if the slab is thinner than the smaller stationary thickness it will melt completely, whereas if the slab is thicker than the smaller stationary thickness it will evolve toward the larger stationary thickness. These results indicate that other mechanisms (e.g. wave-induced agglomeration of crystals) are necessary to grow a slab from zero initial thickness in the parameter regime that yields two stationary solutions.

Stability of an ice sheet on an elastic bed

The stability of stationary flow of a two-dimensional ice sheet is studied when the ice obeys a power flow law (Glen's flow law). The mass accumulation rate at the top is assumed to depend on elevation and span and the bed supporting the ice sheet consists of an elastic layer lying on a rigid surface. The normal perturbation of the free surface of the ice sheet is a singular eigenvalue problem. The singularity of the perturbation at the front of the ice sheet is considered using matched asymptotic expansions, and the eigenvalue problem is seen to reduce to that with fixed ice front. Numerical solution of the perturbation eigenvalue problem shows that the dependence of accumulation rate on elevation permits the existence of unstable solutions when the equilibrium line is higher than the bed at the ice divide. Alternatively, when the equilibrium line is lower than the bed, there are only stable solutions. Softening of the bed, expressed through a decrease of its elastic modulus, has a stabilising effect on the ice sheet.

Sea ice–ocean coupling using a rescaled vertical coordinate z*

Realistic representation of sea ice in ocean models involves the use of a non-linear free-surface, a real freshwater flux and observance of requisite conservation laws. We show here that these properties can be achieved in practice through use of a rescaled vertical coordinate ‘‘z*” in z-coordinate models that allows one to follow undulations in the free-surface under sea ice loading. In particular, the adoption of "z*" avoids the difficult issue of vanishing levels under thick ice. Details of the implementation within MITgcm are provided. A high resolution global ocean sea ice simulation illustrates the robustness of the z* formulation and reveals a source of oceanic variability associated with sea ice dynamics and ice-loading effects. The use of the z* coordinate allows one to achieve perfect conservation of fresh water, heat and salt, as shown in extended integration of coupled ocean sea ice atmospheric model.

The impact of a seasonally ice free Arctic Ocean on the temperature, precipitation and surface mass balance of Svalbard

The observed decline in summer sea ice extent since the 1970s is predicted to continue until the Arctic Ocean is seasonally ice free during the 21st Century. This will lead to a much perturbed Arctic climate with large changes in ocean surface energy flux. Svalbard, located on the present day sea ice edge, contains many low lying ice caps and glaciers and is expected to experience rapid warming over the 21st Century. The total sea level rise if all the land ice on Svalbard were to melt completely is 0.02 m. The purpose of this study is to quantify the impact of climate change on Svalbard’s surface mass balance (SMB) and to determine, in particular, what proportion of the projected changes in precipitation and SMB are a result of changes to the Arctic sea ice cover. To investigate this a regional climate model was forced with monthly mean climatologies of sea surface temperature (SST) and sea ice concentration for the periods 1961–1990 and 2061–2090 under two emission scenarios. In a novel forcing experiment, 20th Century SSTs and 21st Century sea ice were used to force one simulation to investigate the role of sea ice forcing. This experiment results in a 3.5 m water equivalent increase in Svalbard’s SMB compared to the present day. This is because over 50 % of the projected increase in winter precipitation over Svalbard under the A1B emissions scenario is due to an increase in lower atmosphere moisture content associated with evaporation from the ice free ocean. These results indicate that increases in precipitation due to sea ice decline may act to moderate mass loss from Svalbard’s glaciers due to future Arctic warming.

The Greenland Ice Sheet's surface mass balance in a seasonally sea-ice free Arctic

General circulation models predict a rapid decrease in sea ice extent with concurrent increases in near surface air temperature and precipitation in the Arctic over the 21st century. This has led to suggestions that some Arctic land ice masses may experience an increase in accumulation due to enhanced evaporation from a seasonally sea ice free Arctic Ocean. To investigate the impact of this phenomenon on Greenland ice sheet climate and surface mass balance (SMB) a regional climate model, HadRM3, was used to force an insolation-temperature melt SMB model. A set of experiments designed to investigate the role of sea ice independently from sea surface temperature (SST) forcing are described. In the warmer and wetter SI + SST simulation Greenland experiences a 23% increase in winter SMB but 65% reduced summer SMB, resulting in a net decrease in the annual value. This study shows that sea ice decline contributes to the increased winter balance, causing 25% of the increase in winter accumulation; this is largest in eastern Greenland as the result of increased evaporation in the Greenland Sea. These results indicate that the seasonal cycle of Greenland's SMB will increase dramatically as global temperatures increase, with the largest changes in temperature and precipitation occurring in winter. This demonstrates that the accurate prediction of changes in sea ice cover is important for predicting Greenland SMB and ice sheet evolution.

Seasonality of submesoscale flows in the ocean surface boundary layer

A signature of submesoscale flows in the upper ocean is skewness in the distribution of relative vorticity. Expected to result for high Rossby-number flows, such skewness has implications for mixing, dissipation and stratification within the upper ocean. An array of moorings deployed in the Northeast Atlantic for one year as part of the OSMOSIS experiment reveals that relative vorticity is positively skewed during winter even though the scale of the Rossby number is less than 0.5. Furthermore, this skewness is reduced to zero during spring and autumn. There is also evidence of modest seasonal variations in the gradient Rossby number. The proposed mechanism by which relative vorticity is skewed is that the ratio of lateral to vertical buoyancy gradients, as summarized by the inverse gradient Richardson number, restricts its range during winter but less so at other times of the year. These results support recent observations and model simulations suggesting the upper ocean is host to a seasonal cycle in submesoscale turbulence.

Estimating surface CO2 fluxes from space-borne dry air mole fraction observations using an Ensemble Kalman filter

We have developed an ensemble Kalman Filter (EnKF) to estimate 8-day regional surface fluxes of CO2 from space-borne CO2 dry-air mole fraction observations (XCO2) and evaluate the approach using a series of synthetic experiments, in preparation for data from the NASA Orbiting Carbon Observatory (OCO). The 32-day duty cycle of OCO alternates every 16 days between nadir and glint measurements of backscattered solar radiation at short-wave infrared wavelengths. The EnKF uses an ensemble of states to represent the error covariances to estimate 8-day CO2 surface fluxes over 144 geographical regions. We use a 12×8-day lag window, recognising that XCO2 measurements include surface flux information from prior time windows. The observation operator that relates surface CO2 fluxes to atmospheric distributions of XCO2 includes: a) the GEOS-Chem transport model that relates surface fluxes to global 3-D distributions of CO2 concentrations, which are sampled at the time and location of OCO measurements that are cloud-free and have aerosol optical depths <0.3; and b) scene-dependent averaging kernels that relate the CO2 profiles to XCO2, accounting for differences between nadir and glint measurements, and the associated scene-dependent observation errors. We show that OCO XCO2 measurements significantly reduce the uncertainties of surface CO2 flux estimates. Glint measurements are generally better at constraining ocean CO2 flux estimates. Nadir XCO2 measurements over the terrestrial tropics are sparse throughout the year because of either clouds or smoke. Glint measurements provide the most effective constraint for estimating tropical terrestrial CO2 fluxes by accurately sampling fresh continental outflow over neighbouring oceans. We also present results from sensitivity experiments that investigate how flux estimates change with 1) bias and unbiased errors, 2) alternative duty cycles, 3) measurement density and correlations, 4) the spatial resolution of estimated flux estimates, and 5) reducing the length of the lag window and the size of the ensemble. At the revision stage of this manuscript, the OCO instrument failed to reach its orbit after it was launched on 24 February 2009. The EnKF formulation presented here is also applicable to GOSAT measurements of CO2 and CH4.

A review of the potential impacts of climate change on surface water quality

It is now accepted that some human-induced climate change is unavoidable. Potential impacts on water supply have received much attention, but relatively little is known about the concomitant changes in water quality. Projected changes in air temperature and rainfall could affect river flows and, hence, the mobility and dilution of contaminants. Increased water temperatures will affect chemical reaction kinetics and, combined with deteriorations in quality, freshwater ecological status. With increased flows there will be changes in stream power and, hence, sediment loads with the potential to alter the morphology of rivers and the transfer of sediments to lakes, thereby impacting freshwater habitats in both lake and stream systems. This paper reviews such impacts through the lens of UK surface water quality. Widely accepted climate change scenarios suggest more frequent droughts in summer, as well as flash-flooding, leading to uncontrolled discharges from urban areas to receiving water courses and estuaries. Invasion by alien species is highly likely, as is migration of species within the UK adapting to changing temperatures and flow regimes. Lower flows, reduced velocities and, hence, higher water residence times in rivers and lakes will enhance the potential for toxic algal blooms and reduce dissolved oxygen levels. Upland streams could experience increased dissolved organic carbon and colour levels, requiring action at water treatment plants to prevent toxic by-products entering public water supplies. Storms that terminate drought periods will flush nutrients from urban and rural areas or generate acid pulses in acidified upland catchments. Policy responses to climate change, such as the growth of bio-fuels or emission controls, will further impact freshwater quality.

Surface determination of the air-earth electrical current density using co-located sensors of different geometry

A vertical conduction current flows in the atmosphere as a result of the global atmospheric electric circuit. The current at the surface consists of the conduction current and a locally generated displacement current, which are often approximately equal in magnitude. A method of separating the two currents using two collectors of different geometry is investigated. The picoammeters connected to the collectors have a RC time constant of approximately 3 s, permitting the investigation of higher frequency air-earth current changes than previously achieved. The displacement current component of the air-earth current derived from the instrument agrees with calculations using simultaneous data from a co-located fast response electric field mill. The mean value of the nondisplacement current measured over 9 h was 1.76 +/- 0.002 pA m(-2). (c) 2006 American Institute of Physics.