The elliptic sine-Gordon equation: a nonlinear elliptic integrable PDE

In this paper, we summarise this recent progress to underline the features specific to this nonlinear elliptic case, and we give a new classification of boundary conditions on the semistrip that satisfy a necessary condition for yielding a boundary value problem can be effectively linearised. This classification is based on formulation the equation in terms of an alternative Lax pair.

Estimating Lorenz's reference state in an ocean with a nonlinear equation of state for seawater

The study of the mechanical energy budget of the oceans using Lorenz available potential energy (APE) theory is based on knowledge of the adiabatically re-arranged Lorenz reference state of minimum potential energy. The compressible and nonlinear character of the equation of state for seawater has been thought to cause the reference state to be ill-defined, casting doubt on the usefulness of APE theory for investigating ocean energetics under realistic conditions. Using a method based on the volume frequency distribution of parcels as a function of temperature and salinity in the context of the seawater Boussinesq approximation, which we illustrate using climatological data, we show that compressibility effects are in fact minor. The reference state can be regarded as a well defined one-dimensional function of depth, which forms a surface in temperature, salinity and density space between the surface and the bottom of the ocean. For a very small proportion of water masses, this surface can be multivalued and water parcels can have up to two statically stable levels in the reference density profile, of which the shallowest is energetically more accessible. Classifying parcels from the surface to the bottom gives a different reference density profile than classifying in the opposite direction. However, this difference is negligible. We show that the reference state obtained by standard sorting methods is equivalent, though computationally more expensive, to the volume frequency distribution approach. The approach we present can be applied systematically and in a computationally efficient manner to investigate the APE budget of the ocean circulation using models or climatological data.

A traceable physical calibration of the vertical advection-diffusion equation for modelling ocean heat uptake

The classic vertical advection-diffusion (VAD) balance is a central concept in studying the ocean heat budget, in particular in simple climate models (SCMs). Here we present a new framework to calibrate the parameters of the VAD equation to the vertical ocean heat balance of two fully-coupled climate models that is traceable to the models’ circulation as well as to vertical mixing and diffusion processes. Based on temperature diagnostics, we derive an effective vertical velocity w∗ and turbulent diffusivity k∗ for each individual physical process. In steady-state, we find that the residual vertical velocity and diffusivity change sign in mid-depth, highlighting the different regional contributions of isopycnal and diapycnal diffusion in balancing the models’ residual advection and vertical mixing. We quantify the impacts of the time-evolution of the effective quantities under a transient 1%CO2 simulation and make the link to the parameters of currently employed SCMs.

Infrared and millimetre-wave scintillometry in the suburban environment – Part 1: Structure parameters

Scintillometry, a form of ground-based remote sensing, provides the capability to estimate surface heat fluxes over scales of a few hundred metres to kilometres. Measurements are spatial averages, making this technique particularly valuable over areas with moderate heterogeneity such as mixed agricultural or urban environments. In this study, we present the structure parameters of temperature and humidity, which can be related to the sensible and latent heat fluxes through similarity theory, for a suburban area in the UK. The fluxes are provided in the second paper of this two-part series. A millimetre-wave scintillometer was combined with an infrared scintillometer along a 5.5 km path over northern Swindon. The pairing of these two wavelengths offers sensitivity to both temperature and humidity fluctuations, and the correlation between wavelengths is also used to retrieve the path-averaged temperature–humidity correlation. Comparison is made with structure parameters calculated from an eddy covariance station located close to the centre of the scintillometer path. The performance of the measurement techniques under different conditions is discussed. Similar behaviour is seen between the two data sets at sub-daily timescales. For the two summer-to-winter periods presented here, similar evolution is displayed across the seasons. A higher vegetation fraction within the scintillometer source area is consistent with the lower Bowen ratio observed (midday Bowen ratio < 1) compared with more built-up areas around the eddy covariance station. The energy partitioning is further explored in the companion paper.

Infrared and millimetre-wave scintillometry in the suburban environment – Part 2: Large-area sensible and latent heat fluxes

A millimetre-wave scintillometer was paired with an infrared scintillometer, enabling estimation of large-area evapotranspiration across northern Swindon, a suburban area in the UK. Both sensible and latent heat fluxes can be obtained using this "two-wavelength" technique, as it is able to provide both temperature and humidity structure parameters, offering a major advantage over conventional single-wavelength scintillometry. The first paper of this two-part series presented the measurement theory and structure parameters. In this second paper, heat fluxes are obtained and analysed. These fluxes, estimated using two-wavelength scintillometry over an urban area, are the first of their kind. Source area modelling suggests the scintillometric fluxes are representative of 5–10 km2. For comparison, local-scale (0.05–0.5 km2) fluxes were measured by an eddy covariance station. Similar responses to seasonal changes are evident at the different scales but the energy partitioning varies between source areas. The response to moisture availability is explored using data from 2 consecutive years with contrasting rainfall patterns (2011–2012). This extensive data set offers insight into urban surface-atmosphere interactions and demonstrates the potential for two-wavelength scintillometry to deliver fluxes over mixed land cover, typically representative of an area 1–2 orders of magnitude greater than for eddy covariance measurements. Fluxes at this scale are extremely valuable for hydro-meteorological model evaluation and assessment of satellite data products

Changes in significant and maximum wave heights in the Norwegian Sea

This paper analyses 10 years of in-situ measurements of significant wave height (Hs) and maximum wave height (Hmax) from the ocean weather ship Polarfront in the Norwegian Sea. The 30-minute Ship-Borne Wave Recorder measurements of Hmax and Hs are shown to be consistent with theoretical wave distributions. The linear regression between Hmax and Hs has a slope of 1.53. Neither Hs nor Hmax show a significant trend in the period 2000–2009. These data are combined with earlier observations. The long-term trend over the period 1980–2009 in annual Hs is 2.72 ± 0.88 cm/year. Mean Hs and Hmax are both correlated with the North Atlantic Oscillation (NAO) index during winter. The correlation with the NAO index is highest for the more frequently encountered (75th percentile) wave heights. The wave field variability associated with the NAO index is reconstructed using a 500-year NAO index record. Hs and Hmax are found to vary by up to 1.42 m and 3.10 m respectively over the 500-year period. Trends in all 30-year segments of the reconstructed wave field are lower than the trend in the observations during 1980–2009. The NAO index does not change significantly in 21st century projections from CMIP5 climate models under scenario RCP85, and thus no NAO-related changes are expected in the mean and extreme wave fields of the Norwegian Sea.

Wave height analysis from 10 years of observations in the Norwegian Sea

Large waves pose risks to ships, offshore structures, coastal infrastructure and ecosystems. This paper analyses 10 years of in-situ measurements of significant wave height (Hs) and maximum wave height (Hmax) from the ocean weather ship Polarfront in the Norwegian Sea. During the period 2000 to 2009, surface elevation was recorded every 0.59 s during sampling periods of 30 min. The Hmax observations scale linearly with Hs on average. A widely-used empirical Weibull distribution is found to estimate average values of Hmax/Hs and Hmax better than a Rayleigh distribution, but tends to underestimate both for all but the smallest waves. In this paper we propose a modified Rayleigh distribution which compensates for the heterogeneity of the observed dataset: the distribution is fitted to the whole dataset and improves the estimate of the largest waves. Over the 10-year period, the Weibull distribution approximates the observed Hs and Hmax well, and an exponential function can be used to predict the probability distribution function of the ratio Hmax/Hs. However, the Weibull distribution tends to underestimate the occurrence of extremely large values of Hs and Hmax. The persistence of Hs and Hmax in winter is also examined. Wave fields with Hs>12 m and Hmax>16 m do not last longer than 3 h. Low-to-moderate wave heights that persist for more than 12 h dominate the relationship of the wave field with the winter NAO index over 2000–2009. In contrast, the inter-annual variability of wave fields with Hs>5.5 m or Hmax>8.5 m and wave fields persisting over ~2.5 days is not associated with the winter NAO index.

Wave spectra assimilation in typhoon wave modeling for the East China Sea

The East China Sea is a hot area for typhoon waves to occur. A wave spectra assimilation model has been developed to predict the typhoon wave more accurately and operationally. This is the first time where wave data from Taiwan have been used to predict typhoon wave along the mainland China coast. The two-dimensional spectra observed in Taiwan northeast coast modify the wave field output by SWAN model through the technology of optimal interpolation (OI) scheme. The wind field correction is not involved as it contributes less than a quarter of the correction achieved by assimilation of waves. The initialization issue for assimilation is discussed. A linear evolution law for noise in the wave field is derived from the SWAN governing equations. A two-dimensional digital low-pass filter is used to obtain the initialized wave fields. The data assimilation model is optimized during the typhoon Sinlaku. During typhoons Krosa and Morakot, data assimilation significantly improves the low frequency wave energy and wave propagation direction in Taiwan coast. For the far-field region, the assimilation model shows an expected ability of improving typhoon wave forecast as well, as data assimilation enhances the low frequency wave energy. The proportion of positive assimilation indexes is over 81% for all the periods of comparison. The paper also finds that the impact of data assimilation on the far-field region depends on the state of the typhoon developing and the swell propagation direction.

Equation for the microwave backscatter cross section of aggregate snowflakes using the self-similar Rayleigh–Gans approximation

In this paper an equation is derived for the mean backscatter cross section of an ensemble of snowflakes at centimeter and millimeter wavelengths. It uses the Rayleigh–Gans approximation, which has previously been found to be applicable at these wavelengths due to the low density of snow aggregates. Although the internal structure of an individual snowflake is random and unpredictable, the authors find from simulations of the aggregation process that their structure is “self-similar” and can be described by a power law. This enables an analytic expression to be derived for the backscatter cross section of an ensemble of particles as a function of their maximum dimension in the direction of propagation of the radiation, the volume of ice they contain, a variable describing their mean shape, and two variables describing the shape of the power spectrum. The exponent of the power law is found to be −. In the case of 1-cm snowflakes observed by a 3.2-mm-wavelength radar, the backscatter is 40–100 times larger than that of a homogeneous ice–air spheroid with the same mass, size, and aspect ratio.

Discontinuous Galerkin methods for the p-biharmonic equation from a discrete variational perspective

We study discontinuous Galerkin approximations of the p-biharmonic equation for p∈(1,∞) from a variational perspective. We propose a discrete variational formulation of the problem based on an appropriate definition of a finite element Hessian and study convergence of the method (without rates) using a semicontinuity argument. We also present numerical experiments aimed at testing the robustness of the method.

Response of the North Atlantic wave climate to atmospheric modes of variability

This study investigates the relationship between the wind wave climate and the main climate modes of atmospheric variability in the North Atlantic Ocean. The modes considered are the North Atlantic Oscillation (NAO), the East Atlantic (EA) pattern, the East Atlantic Western Russian (EA/WR) pattern and the Scandinavian (SCAN) pattern. The wave dataset consists of buoys records, remote sensing altimetry observations and a numerical hindcast providing significant wave height (SWH), mean wave period (MWP) and mean wave direction (MWD) for the period 1989–2009. After evaluating the reliability of the hindcast, we focus on the impact of each mode on seasonal wave parameters and on the relative importance of wind-sea and swell components. Results demonstrate that the NAO and EA patterns are the most relevant, whereas EA/WR and SCAN patterns have a weaker impact on the North Atlantic wave climate variability. During their positive phases, both NAO and EA patterns are related to winter SWH at a rate that reaches 1 m per unit index along the Scottish coast (NAO) and Iberian coast (EA) patterns. In terms of winter MWD, the two modes induce a counterclockwise shift of up to 65° per negative NAO (positive EA) unit over west European coasts. They also increase the winter MWP in the North Sea and in the Bay of Biscay (up to 1 s per unit NAO) and along the western coasts of Europe and North Africa (1 s per unit EA). The impact of winter EA pattern on all wave parameters is mostly caused through the swell wave component.

Easterly wave disturbances over Northeast Brazil: an observational analysis

This paper aims to identify the circulation associated with Easterly Wave Disturbances (EWDs) that propagate toward the Eastern Northeast Brazil (ENEB) and their impact on the rainfall over ENEB during 2006 and 2007 rainy seasons (April–July). The EWDs identification and trajectory are analyzed using an automatic tracking technique (TracKH). The EWDs circulation patterns and their main features were obtained using the composite technique. To evaluate the TracKH efficiency, a validation was done by comparing the EWDs number tracked against observed cases obtained from an observational analysis. The mean characteristics of EWDs are 5.5-day period, propagation speed of ~9.5 m·s−1, and a 4500 km wavelength. A synoptic analysis shows that between days −2 d and 0 d, the low level winds presented cyclonic relative vorticity and convergence anomalies both in 2006 and 2007. The EWDs signals are strongest at low levels. The EWDs propagation is associated with relative humidity and precipitation positive anomalies and OLR and omega negative anomalies. The EWDs tracks are seen over all ENEB and their lysis occurs between the ENEB and marginally inside the continent. The tracking captured 71% of EWDs in all periods, indicating that an objective analysis is a promising method for EWDs detection.

Pseudopod growth and evolution during cell movement is controlled through SCAR/WAVE dephosphorylation

Regulated dephosphorylation of a fraction of the cellular SCAR pool is a key step in SCAR activation during pseudopod growth. Phosphorylation increases autoinhibition of the intact complex. Dephosphorylation weakens this interaction and facilitates SCAR activation but also destabilizes the protein. We show that SCAR is specifically dephosphorylated in pseudopods, increasing activation by Rac and lipids and supporting positive feedback of pseudopod growth.