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.

Radar scattering by aggregate snowflakes

The radar scattering properties of realistic aggregate snowflakes have been calculated using the Rayleigh-Gans theory. We find that the effect of the snowflake geometry on the scattering may be described in terms of a single universal function, which depends only on the overall shape of the aggregate and not the geometry or size of the pristine ice crystals which compose the flake. This function is well approximated by a simple analytic expression at small sizes; for larger snowflakes we fit a curve to Our numerical data. We then demonstrate how this allows a characteristic snowflake radius to be derived from dual wavelength radar measurements without knowledge of the pristine crystal size or habit, while at the same time showing that this detail is crucial to using such data to estimate ice water content. We also show that the 'effective radius'. characterizing the ratio of particle volume to projected area, cannot be inferred from dual wavelength radar data for aggregates. Finally, we consider the errors involved in approximating snowflakes by 'air-ice spheres', and show that for small enough aggregates the predicted dual wavelength ratio typically agrees to within a few percent, provided some care is taken in choosing the radius of the sphere and the dielectric constant of the air-ice mixture; at larger sizes the radar becomes more sensitive to particle shape, and the errors associated with the sphere model are found to increase accordingly.

An estimate of the global impact of multiple scattering by clouds on outgoing long-wave radiation

Although the potential importance of scattering of long-wave radiation by clouds has been recognised, most studies have concentrated on the impact of high clouds and few estimates of the global impact of scattering have been presented. This study shows that scattering in low clouds has a significant impact on outgoing long-wave radiation (OLR) in regions of marine stratocumulus (-3.5 W m(-2) for overcast conditions) where the column water vapour is relatively low. This corresponds to an enhancement of the greenhouse effect of such clouds by 10%. The near-global impact of scattering on OLR is estimated to be -3.0 W m(-2), with low clouds contributing -0.9 W m(-2), mid-level cloud -0.7 W m(-2) and high clouds -1.4 W m(-2). Although this effect appears small compared to the global mean OLR of 240 W m(-2), it indicates that neglect of scattering will lead to an error in cloud long-wave forcing of about 10% and an error in net cloud forcing of about 20%.