Quarterly Journal of the Royal Meteorological Society

The model is described and the results of three integrations presented. Its main novel features are the inclusion of non‐hydrostatic terms in a pressure co‐ordinate system, and the treatment of flow through the lateral boundaries. The results of the integrations, one with an ambient vertical wind shear and two without, indicate that the cloud dynamics is quite sensitive to assumptions made regarding the cloud microphysics, particularly at the stage when the downdraught is produced.

The non‐hydrostatic equations governing the inviscid, adiabatic motion of a perfect gas are formulated using pressure as vertical coordinate; and M. J. Miller's 1974 approximate quasi‐non‐hydrostatic pressure coordinate equations are derived by applying a systematic scale analysis and power series expansion. The derivation makes clear that these equations are the pressure coordinate counterparts of the anelastic height coordinate equations obtained by Y. Ogura and N. A. Phillips in 1962. The two sets cannot be interconverted by coordinate transformation and so they are not physically equivalent; but the differences are small at the order of validity of both sets. Consideration of a quasi‐hydrostatic approximation emphasizes the non‐hydrostatic character of Miller's equations. Sigma coordinate quasi‐non‐hydrostatic equations are obtained by direct transformation of the pressure coordinate forms, and consistent energy equations are derived for both sets. Convenient diagnostic partial differential equations for the geopotential field are obtained for both pressure and sigma coordinate forms. As shown by Miller, the quasi‐non‐hydrostatic formulation does not permit vertically propagating acoustic waves. Horizontally propagating acoustic waves (Lamb waves) are in general allowed, but can be removed from the pressure coordinate system by applying suitable boundary conditions. Some aspects of the treatment of the Lamb wave problem are corrected in this study. The quasi‐non‐hydrostatic sigma coordinate system permits Lamb waves, but it may still be considered suitable for convective and (especially) mesoscale modelling with or without orography. The possible use of the quasi‐non‐hydrostatic system in large‐scale theory and modelling is also discussed.

A study has been made of some aspects of frictionless stratified flow past three‐dimensional isolated mountains. the study uses a three‐dimensional non‐hydrostatic numerical model to investigate the behaviour of the flow as a function of the Froude number, and produces a picture of the dependence of the gravity‐wave drag on the Froude number for a wide range of that parameter. At the same time, the results of the numerical experiments clarify the behaviour of the flow in the transition from high to low Froude number, showing the relative importance of wave breaking and flow splitting in the transitional regime.

The frequency of various types of ‘present‐weather’ observations are analysed and a method is derived using these to obtain estimates of mean monthly precipitation. The method is applied to observations from ocean weather ships in the North Atlantic Ocean to obtain the mean monthly, seasonal and annual precipitation over the ocean during the five‐year period 1953 to 1957. The results are then compared with previous estimates.

A quantitative analysis of the properties of several Mt Ruapehu, New Zealand, ash plumes has been performed using multispectral satellite data from the AVHRR‐2 and ATSR‐2 instruments. The analysis includes: identification of the plume from background clouds using the ‘reverse’ absorption effect in the thermal channels: modelling and retrieval of particle sizes; determination of the plume height from cloud shadows, stereoscopy and meteorological data; and estimates of the mass of fine particles (radii less than 10 μm). A new spectral technique for identifying opaque, silica‐rich ash clouds is demonstrated by utilizing the near‐infrared (1.6 μm) and visible (0.67 μm) channels of the ATSR‐2, and the optical properties of a simple volcanic cloud are presented for use in radiative transfer studies. It is found that the Ruapehu eruption cloud contained silica‐rich ash particles with radii generally less than a few micrometres. The distribution of fine particles is monomodal with a dominant mode peak of about 3 μm radius. Mass loadings of fine particles are found to be in the range ≈︁1 to ≈︁7 mg m⁻³, and are consistent with estimates of mass loadings of volcanic clouds from eruptions of other volcanoes. The height of the plume top, derived from radiosonde data and plume‐top temperatures in the opaque regions, was found to be between 7.5 and 8.5 km, while the plume thickness was estimated to be between 1.5 and 3 km. Cloud height derived from ATSR‐2 stereoscopy on a different plume gave heights in the range 5 to 8 km.

The results of this study provide important information on the optical properties of nascent volcanic eruption plumes. This information may prove useful in determining the potential effects of volcanic clouds on local climate, and in assessing any hazard to aviation.

The paper presents the analysis of field measurements in the atmospheric surface layer over the floor of the Adige Valley, near the city of Bolzano in the Alps. Turbulence quantities, such as drag coefficient, displacement height and roughness length, appear similar to those reported in the literature concerning surface‐layer turbulence over flat uniform terrain. The analysis of the non‐dimensional standard deviations (σ_u, σ_v, σ_w) legitimates the adoption, for all the wind components, of the same Monin—Obukhov similarity relationship in the form $\sigma_{i}/u_{\ast}=\alpha_{i}\left(1+\beta_{i}\left|\zeta\right|\right)^{1/3}$, originally proposed only for flat uniform terrain under steady state conditions, and the extension of this expression to the case of winds over a valley floor in slowly varying situations. The coefficients α_i are very similar for along‐valley and cross‐valley winds, as are the β_i ones. Moreover the α_u values are only slightly larger than the α_v, contrary to what is reported in the literature. Conversely, σ_θ/θ_* shows distinctly different behaviour in the stable and unstable regimes. In particular, in the latter case a large scatter in the data is observed. However, since the most scattered data turn out to occur in transition periods (i.e. sunrise and sunset), after a specific data selection the best‐fit parameters are more accurately estimated. In addition, emphasis is laid on the relevance of an appropriate formulation and use of a suitable recursive filter to separate the low‐frequency unsteadiness of the mean flow from the turbulence signal. Copyright © 2009 Royal Meteorological Society

Following the a posteriori diagnosis approach proposed by some authors, a practical computation of the expectation of sub‐parts of the value of a cost function at the minimum is shown to be feasible by using a randomization technique based on a perturbation of observations or background fields. These computations allow the tuning of observation‐error weighting parameters by applying a simple iterative fixed‐point procedure. The procedure is first tested in a simplified variational scheme on a circular domain and then in a similar scheme but with the addition of the vertical coordinate. The relationship between the proposed approach and the Generalized Cross Validation is also shown. A test in the French Action de Recherche Petite Echelle Grande Echelle (ARPEGE) three‐dimensional variational framework with both simulated observations and background fields is finally performed. It shows that a complete description of observation‐error parameters can be retrieved with only a few iterations and, thus, at a reasonable cost.

The importance of prior error correlations in data assimilation has long been known; however, observation‐error correlations have typically been neglected. Recent progress has been made in estimating and accounting for observation‐error correlations, allowing for the optimal use of denser observations. Given this progress, it is now timely to ask how prior and observation‐error correlations interact and how this affects the value of the observations in the analysis. Addressing this question is essential to understanding the optimal design of future observation networks for high‐resolution numerical weather prediction. This article presents new results, which unify and advance upon previous studies on this topic.

The interaction of the prior and observation‐error correlations is illustrated with a series of two‐variable experiments in which the mapping between the state and observed variables (the observation operator) is allowed to vary. In an optimal system, the reduction in the analysis‐error variance and spread of information is shown to be greatest when the observation and prior errors have complementary statistics: for example, in the case of direct observations, when the correlations between the observation and prior errors have opposite signs. This can be explained in terms of the relative uncertainty of the observations and prior on different spatial scales. The results from these simple two‐variable experiments are used to inform the optimal observation density for observations of a circular domain (with 32 grid points). It is found that dense observations are most beneficial when they provide a more accurate estimate of the state at smaller scales than the prior estimate. In the case of second‐order auto‐regressive correlation functions, this is achieved when the length‐scales of the observation‐error correlations are greater than those of the prior estimate and the observations are direct measurements of the state variables.

In this paper, the optimal configurations of model resolution, observation resolution and observation density are investigated in a simple one‐dimensional framework. In this context, the representativeness error is formalized and estimated before being used in the analysis‐error formulation. Some optimal and suboptimal assimilation‐schemes, differing from different approximations of observation‐error covariance and observation operator, are compared. The optimal observation‐extent is determined as a function of model resolution. Increasing the observation density is usually beneficial, except for suboptimal schemes similar to the ones used in operational practice. The impact of thinning the observations with correlated error is also studied from a suboptimal viewpoint. Copyright © 2002 Royal Meteorological Society