A new numerical scheme for obtaining the steady state form of an internal solitary wave of large amplitude is presented. A stratified inviscid two dimensional fluid under the Boussinesq approximation flowing between horizontal rigid boundaries is considered. The stratification is stable, and buoyancy is continuously differentiable throughout the domain of the flow. Solutions are obtained by tracing the buoyancy frequency along streamlines from the undisturbed far field. From this the vorticity field can be constructed and the streamfunction may then be obtained by inversion of Laplace's operator. The scheme is presented as an iterative solver where the inversion of Laplace's operator is performed spectrally. The solutions agree well with previous results for stratification in which the buoyancy frequency is a discontinuous function. The new numerical scheme allows significantly larger amplitude waves to be computed than have been presented before and it is shown that waves with Richardson numbers as low as 0.062 can be computed straightforwardly. The method is also extended to deal in a novel way with closed streamlines when they occur in the domain. The new solutions are tested in independent fully nonlinear time dependent simulations and are verified to be steady. Waves with regions of recirculation are also discussed.

Synthetic Aperture Radar has a unique potential for continuous forest mapping as it is not affected by cloud cover. While longer wavelengths, such as L-band, are commonly used for forest applications, in this paper we assess the aptitude of C-band Sentinel-1 data for this purpose, for which there is much interest due to its high temporal resolution (five days) and “free, full, and open” data policy. We tested its ability to distinguish forest from non-forest in six study sites, located in Alaska, Colombia, Finland, Florida, Indonesia, and the UK. Using the time series for a full year significantly increases the classification accuracy compared to a single scene (a mean of 85 % compared to 77 % across the study sites for the best classifier). Our results show that we can further improve the mean accuracy to 87 % when only considering the annual mean and standard deviation of co-polarized (VV) and cross-polarized (VH) backscatter. In this case, separation accuracies of up to 93 % (in Finland) are possible, though in the worst case (Alaska), the highest possible accuracy using these variables was 80 % . The best overall performance was observed when using a Support Vector Machine classifier, outperforming random forest, k-Nearest-Neighbors, and Quadratic Discriminant Analysis. We further show that the small information content we found in the phase data is an artifact of terrain slope orientation and has a negligible impact on classifier performance. We conclude that for the purposes of forest mapping the smaller file size and easier to process GRD products are sufficient, unless the SLC products are used to compute the temporal coherence which was not tested in this study.

It is also shown that the critical value of L x /λ required for instability, where L x is the length of the region in a wave in which Ri < 1/4 and λ is the half width of the wave, is sensitive to the ratio of the layer thicknesses. Similarly, a linear stability analysis reveals † Email address for correspondence: magda@mcs.st-and.ac.uk 2 M. Carr, S. E. King and D. G. Dritschel that,σ i T w , whereσ i is the growth rate of the instability averaged over T w , the period in which parcels of fluid are subjected to Ri < 1/4, is very sensitive to the transition between the undisturbed pycnocline and the homogeneous layers and, the amplitude of the wave. Therefore, the alternative tests for instability presented in Fructus et al. (2009) and Barad & Fringer (2010), respectively, namely L x /λ 0.86 andσ i T w > 5, are shown to be valid only for a limited parameter range. Breaking ISWs can result in vertical mixing in the environment in which they propagate. They are an important source of mixing, turbulence, re-distribution of potential energy in the water column and mass and momentum transfer. In physical oceanography, one of the most topical issues of debate is the role of internal waves on the overall mixing of coastal oceans -a process that, in turn, has implications for global ocean circulation, heat transport and hence climate modelling, see Munk Introduction

We describe the gravity-driven flow of a viscous fluid in a semi-infinite porous layer, $x>0$, from which fluid can drain freely at $x=0$. New experiments using a Hele-Shaw cell confirm that when the base of the layer is impermeable the motion of the current is self-similar and the dipole moment of the flow is conserved, as proposed theoretically by Barenblatt & Zel'dovich (1957). We extend the model to allow fluid to drain through the base of the porous layer into a thin horizontal layer of lower permeability. In this case we predict that the dipole moment of the current decays exponentially with time. At early times we find that the loss of fluid from the gravity current in the high-permeability layer is dominated by the draining at $x=0$, whereas at long times, the gravity-driven leakage into the underlying low-permeability layer is dominant. We successfully compare these analytic solutions for such draining currents with further laboratory experiments in which fluid drains from the end and through the base of a Hele-Shaw cell. We discuss the implications of these results for the dispersal of chemicals or pollutants injected into a layered porous rock.

New analytical models are introduced to describe the motion of a Herschel–Bulkley fluid slumping under gravity in a narrow fracture and in a porous medium. A useful self-similar solution can be derived for a fluid injection rate that scales as time $t$; an expansion technique is adopted for a generic injection rate that is power law in time. Experiments in a Hele-Shaw cell and in a narrow channel filled with glass ballotini confirm the theoretical model within the experimental uncertainty.