Shallow water flow with cylindrical symmetry

A numerical scheme is presented tor the solution of the shallow water equations in a single radial coordinate. This can prove useful when testing codes for the two-dimensional shallow water equations. The scheme is applied with success to problems involving converging and diverging bores.

An algorithm for the shallow water equations with body fitted meshes

An efficient algorithm based on flux difference splitting is presented for the solution of the two-dimensional shallow water equations in a generalised coordinate system. The scheme is based on solving linearised Riemann problems approximately and in more than one dimension incorporates operator splitting. The scheme has good jump capturing properties and the advantage of using body-fitted meshes. Numerical results are shown for flow past a circular obstruction.

Measuring the impact of observations on the predictability of the Kuroshio Extension in a shallow-water model

In this paper sequential importance sampling is used to assess the impact of observations on a ensemble prediction for the decadal path transitions of the Kuroshio Extension (KE). This particle filtering approach gives access to the probability density of the state vector, which allows us to determine the predictive power — an entropy based measure — of the ensemble prediction. The proposed set-up makes use of an ensemble that, at each time, samples the climatological probability distribution. Then, in a post-processing step, the impact of different sets of observations is measured by the increase in predictive power of the ensemble over the climatological signal during one-year. The method is applied in an identical-twin experiment for the Kuroshio Extension using a reduced-gravity shallow water model. We investigate the impact of assimilating velocity observations from different locations during the elongated and the contracted meandering state of the KE. Optimal observations location correspond to regions with strong potential vorticity gradients. For the elongated state the optimal location is in the first meander of the KE. During the contracted state of the KE it is located south of Japan, where the Kuroshio separates from the coast.

Sediment fingerprinting as an environmental forensics tool explaining cyanobacteria blooms in lakes

Cyanobacteria (blue-green algae) blooms in water bodies present serious public health issues with attendant economic and ecological impacts. Llyn Tegid (Lake Bala) is an important conservation and amenity asset within Snowdonia National Park, Wales which since the mid-1990s has experienced multiple toxic cyanobacteria blooms threatening the ecology and tourism-dependent local economy. Multiple working hypotheses explain the emergence of this problem, including climate change, land management linked to increased nutrient flux, hydromorphological alterations or changing trophic structure - any of which may operate individually or cumulatively to impair lake function. This paper reports the findings of a sedimentfingerprinting study using dated lake cores to explore the linkages between catchment and lake management practices and the emergence of the algal blooms problem. Since 1900 AD lake bed sedimentation rates have varied from 0.06 to 1.07 g cm−2 yr−1, with a pronounced acceleration since the early 1980s. Geochemical analysis revealed increases in the concentrations of total phosphorus (TP), calcium and heavy metals such as zinc and lead consistent with eutrophication and a rising pollution burden, particularly since the late 1970s. An uncertainty-inclusive sedimentfingerprinting approach was used to apportion the relative fluxes from the major catchment land cover types of improved pasture, rough grazing, forestry and channel banks. This showed improved pasture and channel banks are the dominant diffuse sources of sediment in the catchment, though forestry sources were important historically. Conversion of rough grazing to improved grassland, coupled with intensified land management and year-round livestock grazing, is concluded to provide the principal source of rising TP levels. Lake Habitat Survey and particle size analysis of lake cores demonstrate the hydromorphological impact of the River Dee Regulation Scheme, which controls water level and periodically diverts flow into Llyn Tegid from the adjacent Afon Tryweryn catchment. This hydromorphological impact has also been most pronounced since the late 1970s. It is concluded that an integrated approach combining land management to reduce agricultural runoff allied to improved water level regulation enabling recovery of littoral macrophytes offers the greatest chance halting the on-going cyanobacteria issue in Llyn Tegid.

Comment on “A modified leapfrog scheme for shallow water equations” by Wen-Yih Sun and Oliver M.T. Sun

A recent paper published in this journal considers the numerical integration of the shallow-water equations using the leapfrog time-stepping scheme [Sun Wen-Yih, Sun Oliver MT. A modified leapfrog scheme for shallow water equations. Comput Fluids 2011;52:69–72]. The authors of that paper propose using the time-averaged height in the numerical calculation of the pressure-gradient force, instead of the instantaneous height at the middle time step. The authors show that this modification doubles the maximum Courant number (and hence the maximum time step) at which the integrations are stable, doubling the computational efficiency. Unfortunately, the pressure-averaging technique proposed by the authors is not original. It was devised and published by Shuman [5] and has been widely used in the atmosphere and ocean modelling community for over 40 years.

Interactions between shallow and deep convection under a finite departure from convective quasi–equilibrium

The present paper presents a simple theory for the transformation of non-precipitating, shallow convection into precipitating, deep convective clouds. In order to make the pertinent point a much idealized system is considered, consisting only of shallow and deep convection without large–scale forcing. The transformation is described by an explicit coupling between these two types of convection. Shallow convection moistens and cools the atmosphere, whereas deep convection dries and warms, leading to destabilization and stabilization respectively. Consequently, in their own stand–alone modes, shallow convection perpetually grows, whereas deep convection simply damps: the former never reaches equilibrium, and the latter is never spontaneously generated. Coupling the modes together is the only way to reconcile these undesirable separate tendencies so that the convective system as a whole can remain in a stable periodic state under this idealized setting. Such coupling is a key missing element in current global atmospheric models. The energy–cycle description as originally formulated by Arakawa and Schubert, and presented herein is suitable for direct implementation into models using a mass–flux parameterization, and would alleviate the current problems with the representation of these two types of convection in numerical models. The present theory also provides a pertinent framework for analyzing large–eddy simulations and cloud–resolving modelling.

Middle and Late Cenomanian anoxia in shallow and deeper shelf environments of western Morocco

The response of shallow-water sequences to oceanic anoxic event 2 and mid-Cenomanian events 1a and 1b was investigated along the west African margin of Morocco north of Agadir (Azazoul) and correlated with the deep-water sequence of the Tarfaya Basin (Mohammed Beach) based on biostratigraphy, mineralogy, phosphorus and stable isotopes. In the deeper Mohammed Beach section results show double peaks in δ13Corg for mid-Cenomanian events 1a and 1b (Rotalipora reicheli biozone, lower CC10a biozone), the characteristic oceanic anoxic event 2 δ13C excursion (Rotalipora cushmani extinction, top of CC10a biozone) and laminated (anoxic) black shale. In the shallow environment north of Agadir, a fluctuating sea-level associated with dysoxic, brackish and mesotrophic conditions prevailed during the middle to late Cenomanian, as indicated by oyster biostromes, nannofossils, planktonic and benthonic foraminiferal assemblages. Anoxic conditions characteristic of oceanic anoxic event 2 (for example, laminated black shales) did not reach into shallow-water environments until the maximum transgression of the early Turonian. Climate conditions decoupled along the western margin of Morocco between mid-Cenomanian event 1b and the Cenomanian–Turonian boundary, as also observed in eastern Tethys. North of Agadir alternating humid and dry seasonal conditions prevailed, whereas in the Tarfaya Basin the climate was dry and seasonal. This climatic decoupling can be attributed to variations in the Intertropical Convergence Zone and in the intensity of the north-east trade winds in tropical areas.

K-T transition in Deccan traps of central India marks major marine seaway across India.

Deccan intertrappean sediments in central India are generally considered as terrestrial deposits of Maastrichtian age, but the Cretaceous–Tertiary (K–T) position is still unknown. Here we report the discovery of the K–T transition, a marine incursion and environmental changes preserved within the intertrappean sediments at Jhilmili, Chhindwara District, Madhya Pradesh. Integrative biostratigraphic, sedimentologic, mineralogic and chemostratigraphic analyses reveal the basal Danian in the intertrappean sediments between lower and upper trap basalts that regionally correspond to C29r and the C29R/C29N transition, respectively. Intertrappean deposition occurred in predominantly terrestrial semi-humid to arid environments. But a short aquatic interval of fresh water ponds and lakes followed by shallow coastal marine conditions with brackish marine ostracods and early Danian zone P1a planktic foraminifera mark this interval very close to the K–T boundary. This marine incursion marks the existence of a nearby seaway, probably extending inland from the west through the Narmada and Tapti rift valleys. The Jhilmili results thus identify the K–T boundary near the end of the main phase of Deccan eruptions and indicate that a major seaway extended at least 800 km across India.

Desert dust deposition on Mt. Elbrus, Caucasus Mountains, Russia in 2009-2012 as recorded in snow and shallow ice core: high-resolution "provenancing", transport patterns, physical properties and soluble ionic composition

A record of dust deposition events between 2009 and 2012 on Mt. Elbrus, Caucasus Mountains derived from a snow pit and a shallow ice core is presented for the first time for this region. A combination of isotopic analysis, SEVIRI red-green-blue composite imagery, MODIS atmospheric optical depth fields derived using the Deep Blue algorithm, air mass trajectories derived using the HYSPLIT model and analysis of meteorological data enabled identification of dust source regions with high temporal (hours) and spatial (cf. 20–100 km) resolution. Seventeen dust deposition events were detected; fourteen occurred in March–June, one in February and two in October. Four events originated in the Sahara, predominantly in north-eastern Libya and eastern Algeria. Thirteen events originated in the Middle East, in the Syrian Desert and northern Mesopotamia, from a mixture of natural and anthropogenic sources. Dust transportation from Sahara was associated with vigorous Saharan depressions, strong surface winds in the source region and mid-tropospheric south-westerly flow with daily winds speeds of 20–30 m s−1 at 700 hPa level and, although these events were less frequent, they resulted in higher dust concentrations in snow. Dust transportation from the Middle East was associated with weaker depressions forming over the source region, high pressure centered over or extending towards the Caspian Sea and a weaker southerly or south-easterly flow towards the Caucasus Mountains with daily wind speeds of 12–18 m s−1 at 700 hPa level. Higher concentrations of nitrates and ammonium characterise dust from the Middle East deposited on Mt. Elbrus in 2009 indicating contribution of anthropogenic sources. The modal values of particle size distributions ranged between 1.98 μm and 4.16 μm. Most samples were characterised by modal values of 2.0–2.8 μm with an average of 2.6 μm and there was no significant difference between dust from the Sahara and the Middle East.

Data assimilation with correlated observation errors: experiments with a 1-D shallow water model

Remote sensing observations often have correlated errors, but the correlations are typically ignored in data assimilation for numerical weather prediction. The assumption of zero correlations is often used with data thinning methods, resulting in a loss of information. As operational centres move towards higher-resolution forecasting, there is a requirement to retain data providing detail on appropriate scales. Thus an alternative approach to dealing with observation error correlations is needed. In this article, we consider several approaches to approximating observation error correlation matrices: diagonal approximations, eigendecomposition approximations and Markov matrices. These approximations are applied in incremental variational assimilation experiments with a 1-D shallow water model using synthetic observations. Our experiments quantify analysis accuracy in comparison with a reference or ‘truth’ trajectory, as well as with analyses using the ‘true’ observation error covariance matrix. We show that it is often better to include an approximate correlation structure in the observation error covariance matrix than to incorrectly assume error independence. Furthermore, by choosing a suitable matrix approximation, it is feasible and computationally cheap to include error correlation structure in a variational data assimilation algorithm.

A Hamiltonian weak-wave model for shallow-water flow

A reduced dynamical model is derived which describes the interaction of weak inertia–gravity waves with nonlinear vortical motion in the context of rotating shallow–water flow. The formal scaling assumptions are (i) that there is a separation in timescales between the vortical motion and the inertia–gravity waves, and (ii) that the divergence is weak compared to the vorticity. The model is Hamiltonian, and possesses conservation laws analogous to those in the shallow–water equations. Unlike the shallow–water equations, the energy invariant is quadratic. Nonlinear stability theorems are derived for this system, and its linear eigenvalue properties are investigated in the context of some simple basic flows.

The energy-cycle analysis of the interactions between shallow and deep atmospheric convection

Interactions between different convection modes can be investigated using an energy–cycle description under a framework of mass–flux parameterization. The present paper systematically investigates this system by taking a limit of two modes: shallow and deep convection. Shallow convection destabilizes itself as well as the other convective modes by moistening and cooling the environment, whereas deep convection stabilizes itself as well as the other modes by drying and warming the environment. As a result, shallow convection leads to a runaway growth process in its stand–alone mode, whereas deep convection simply damps out. Interaction between these two convective modes becomes a rich problem, even when it is limited to the case with no large–scale forcing, because of these opposing tendencies. Only if the two modes are coupled at a proper level can a self–sustaining system arise, exhibiting a periodic cycle. The present study establishes the conditions for self–sustaining periodic solutions. It carefully documents the behaviour of the two mode system in order to facilitate the interpretation of global model behaviours when this energy–cycle is implemented as a closure into a convection parameterization in future.

Surface water temperature observations of large lakes by optimal estimation

Optimal estimation (OE) and probabilistic cloud screening were developed to provide lake surface water temperature (LSWT) estimates from the series of (advanced) along-track scanning radiometers (ATSRs). Variations in physical properties such as elevation, salinity, and atmospheric conditions are accounted for through the forward modelling of observed radiances. Therefore, the OE retrieval scheme developed is generic (i.e., applicable to all lakes). LSWTs were obtained for 258 of Earth's largest lakes from ATSR-2 and AATSR imagery from 1995 to 2009. Comparison to in situ observations from several lakes yields satellite in situ differences of −0.2 ± 0.7 K for daytime and −0.1 ± 0.5 K for nighttime observations (mean ± standard deviation). This compares with −0.05 ± 0.8 K for daytime and −0.1 ± 0.9 K for nighttime observations for previous methods based on operational sea surface temperature algorithms. The new approach also increases coverage (reducing misclassification of clear sky as cloud) and exhibits greater consistency between retrievals using different channel–view combinations. Empirical orthogonal function (EOF) techniques were applied to the LSWT retrievals (which contain gaps due to cloud cover) to reconstruct spatially and temporally complete time series of LSWT. The new LSWT observations and the EOF-based reconstructions offer benefits to numerical weather prediction, lake model validation, and improve our knowledge of the climatology of lakes globally. Both observations and reconstructions are publically available from http://hdl.handle.net/10283/88.