Advances in sequential data assimilation and numerical weather forecasting: an Ensemble Transform Kalman-Bucy Filter, a study on clustering in deterministic Ensemble Square Root Filters, and a test of a new time stepping scheme in an atmospheric model

This dissertation deals with aspects of sequential data assimilation (in particular ensemble Kalman filtering) and numerical weather forecasting. In the first part, the recently formulated Ensemble Kalman-Bucy (EnKBF) filter is revisited. It is shown that the previously used numerical integration scheme fails when the magnitude of the background error covariance grows beyond that of the observational error covariance in the forecast window. Therefore, we present a suitable integration scheme that handles the stiffening of the differential equations involved and doesn’t represent further computational expense. Moreover, a transform-based alternative to the EnKBF is developed: under this scheme, the operations are performed in the ensemble space instead of in the state space. Advantages of this formulation are explained. For the first time, the EnKBF is implemented in an atmospheric model. The second part of this work deals with ensemble clustering, a phenomenon that arises when performing data assimilation using of deterministic ensemble square root filters in highly nonlinear forecast models. Namely, an M-member ensemble detaches into an outlier and a cluster of M-1 members. Previous works may suggest that this issue represents a failure of EnSRFs; this work dispels that notion. It is shown that ensemble clustering can be reverted also due to nonlinear processes, in particular the alternation between nonlinear expansion and compression of the ensemble for different regions of the attractor. Some EnSRFs that use random rotations have been developed to overcome this issue; these formulations are analyzed and their advantages and disadvantages with respect to common EnSRFs are discussed. The third and last part contains the implementation of the Robert-Asselin-Williams (RAW) filter in an atmospheric model. The RAW filter is an improvement to the widely popular Robert-Asselin filter that successfully suppresses spurious computational waves while avoiding any distortion in the mean value of the function. Using statistical significance tests both at the local and field level, it is shown that the climatology of the SPEEDY model is not modified by the changed time stepping scheme; hence, no retuning of the parameterizations is required. It is found the accuracy of the medium-term forecasts is increased by using the RAW filter.

Cloud-resolving model simulations with one- and two-way couplings via the weak temperature gradient approximation

A cloud-resolving model is modified to implement the weak temperature gradient approximation in order to simulate the interactions between tropical convection and the large-scale tropical circulation. The instantaneous domain-mean potential temperature is relaxed toward a reference profile obtained from a radiative–convective equilibrium simulation of the cloud-resolving model. For homogeneous surface conditions, the model state at equilibrium is a large-scale circulation with its descending branch in the simulated column. This is similar to the equilibrium state found in some other studies, but not all. For this model, the development of such a circulation is insensitive to the relaxation profile and the initial conditions. Two columns of the cloud-resolving model are fully coupled by relaxing the instantaneous domain-mean potential temperature in both columns toward each other. This configuration is energetically closed in contrast to the reference-column configuration. No mean large-scale circulation develops over homogeneous surface conditions, regardless of the relative area of the two columns. The sensitivity to nonuniform surface conditions is similar to that obtained in the reference-column configuration if the two simulated columns have very different areas, but it is markedly weaker for columns of comparable area. The weaker sensitivity can be understood as being a consequence of a formulation for which the energy budget is closed. The reference-column configuration has been used to study the convection in a local region under the influence of a large-scale circulation. The extension to a two-column configuration is proposed as a methodology for studying the influence on local convection of changes in remote convection.

Numerical simulation of tephra transport and deposition of the 1982 El Chichón eruption and implications for hazard assessment

El Chichón volcano, Chiapas, Mexico, erupted explosively on March 29th, 1982, after a repose period of about 550 years. Amongst ten eruptive episodes documented between March 29th and April 4th, only the three that occurred on March 29th and April 4th produced significant pyroclastic tephra deposits. Here we use analytical (HAZMAP) and numerical (FALL3D) tephra transport models to reconstruct the deposits and the atmospheric plume dispersal associated with the three main fallout units of the 1982 eruption. On the basis of such a reconstruction, we produce hazard maps of tephra fallout associated to a Plinian eruption and discuss the implications of such a severe eruption scenario.

Simulation of a domestic ground source heat pump system using a three-dimensional numerical borehole heat exchanger model

Common approaches to the simulation of borehole heat exchangers (BHEs) assume heat transfer in circulating fluid and grout to be in a quasi-steady state and ignore fluctuations in fluid temperature due to transport of the fluid around the loop. However, in domestic ground source heat pump (GSHP) systems, the heat pump and circulating pumps switch on and off during a given hour; therefore, the effect of the thermal mass of the circulating fluid and the dynamics of fluid transport through the loop has important implications for system design. This may also be important in commercial systems that are used intermittently. This article presents transient simulation of a domestic GSHP system with a single BHE using a dynamic three-dimensional (3D) numerical BHE model. The results show that delayed response associated with the transit of fluid along the pipe loop is of some significance in moderating swings in temperature during heat pump operation. In addition, when 3D effects are considered, a lower heat transfer rate is predicted during steady operations. These effects could be important when considering heat exchanger design and system control. The results will be used to develop refined two-dimensional models.

On the momentum fluxes associated with mountain waves in directionally sheared flows

The direct impact of mountain waves on the atmospheric circulation is due to the deposition of wave momentum at critical levels, or levels where the waves break. The first process is treated analytically in this study within the framework of linear theory. The variation of the momentum flux with height is investigated for relatively large shears, extending the authors’ previous calculations of the surface gravity wave drag to the whole atmosphere. A Wentzel–Kramers–Brillouin (WKB) approximation is used to treat inviscid, steady, nonrotating, hydrostatic flow with directional shear over a circular mesoscale mountain, for generic wind profiles. This approximation must be extended to third order to obtain momentum flux expressions that are accurate to second order. Since the momentum flux only varies because of wave filtering by critical levels, the application of contour integration techniques enables it to be expressed in terms of simple 1D integrals. On the other hand, the momentum flux divergence (which corresponds to the force on the atmosphere that must be represented in gravity wave drag parameterizations) is given in closed analytical form. The momentum flux expressions are tested for idealized wind profiles, where they become a function of the Richardson number (Ri). These expressions tend, for high Ri, to results by previous authors, where wind profile effects on the surface drag were neglected and critical levels acted as perfect absorbers. The linear results are compared with linear and nonlinear numerical simulations, showing a considerable improvement upon corresponding results derived for higher Ri.

A linear model of gravity wave drag for hydrostatic sheared flow over elliptical mountains

An analytical model of orographic gravity wave drag due to sheared flow past elliptical mountains is developed. The model extends the domain of applicability of the well-known Phillips model to wind profiles that vary relatively slowly in the vertical, so that they may be treated using a WKB approximation. The model illustrates how linear processes associated with wind profile shear and curvature affect the drag force exerted by the airflow on mountains, and how it is crucial to extend the WKB approximation to second order in the small perturbation parameter for these effects to be taken into account. For the simplest wind profiles, the normalized drag depends only on the Richardson number, Ri, of the flow at the surface and on the aspect ratio, γ, of the mountain. For a linear wind profile, the drag decreases as Ri decreases, and this variation is faster when the wind is across the mountain than when it is along the mountain. For a wind that rotates with height maintaining its magnitude, the drag generally increases as Ri decreases, by an amount depending on γ and on the incidence angle. The results from WKB theory are compared with exact linear results and also with results from a non-hydrostatic nonlinear numerical model, showing in general encouraging agreement, down to values of Ri of order one.

An analytical model of mountain wave drag for wind profiles with shear and curvature

An analytical model is developed to predict the surface drag exerted by internal gravity waves on an isolated axisymmetric mountain over which there is a stratified flow with a velocity profile that varies relatively slowly with height. The model is linear with respect to the perturbations induced by the mountain, and solves the Taylor–Goldstein equation with variable coefficients using a Wentzel–Kramers–Brillouin (WKB) approximation, formally valid for high Richardson numbers, Ri. The WKB solution is extended to a higher order than in previous studies, enabling a rigorous treatment of the effects of shear and curvature of the wind profile on the surface drag. In the hydrostatic approximation, closed formulas for the drag are derived for generic wind profiles, where the relative magnitude of the corrections to the leading-order drag (valid for a constant wind profile) does not depend on the detailed shape of the orography. The drag is found to vary proportionally to Ri21, decreasing as Ri decreases for a wind that varies linearly with height, and increasing as Ri decreases for a wind that rotates with height maintaining its magnitude. In these two cases the surface drag is predicted to be aligned with the surface wind. When one of the wind components varies linearly with height and the other is constant, the surface drag is misaligned with the surface wind, especially for relatively small Ri. All these results are shown to be in fairly good agreement with numerical simulations of mesoscale nonhydrostatic models, for high and even moderate values of Ri.

Inverse problems in neural field theory

We study inverse problems in neural field theory, i.e., the construction of synaptic weight kernels yielding a prescribed neural field dynamics. We address the issues of existence, uniqueness, and stability of solutions to the inverse problem for the Amari neural field equation as a special case, and prove that these problems are generally ill-posed. In order to construct solutions to the inverse problem, we first recast the Amari equation into a linear perceptron equation in an infinite-dimensional Banach or Hilbert space. In a second step, we construct sets of biorthogonal function systems allowing the approximation of synaptic weight kernels by a generalized Hebbian learning rule. Numerically, this construction is implemented by the Moore–Penrose pseudoinverse method. We demonstrate the instability of these solutions and use the Tikhonov regularization method for stabilization and to prevent numerical overfitting. We illustrate the stable construction of kernels by means of three instructive examples.

Numerical study of a M-cycle cross-flow heat exchanger for indirect evaporative cooling

In this paper, numerical analyses of the thermal performance of an indirect evaporative air cooler incorporating a M-cycle cross-flow heat exchanger has been carried out. The numerical model was established from solving the coupled governing equations for heat and mass transfer between the product and working air, using the finite-element method. The model was developed using the EES (Engineering Equation Solver) environment and validated by published experimental data. Correlation between the cooling (wet-bulb) effectiveness, system COP and a number of air flow/exchanger parameters was developed. It is found that lower channel air velocity, lower inlet air relative humidity, and higher working-to-product air ratio yielded higher cooling effectiveness. The recommended average air velocities in dry and wet channels should not be greater than 1.77 m/s and 0.7 m/s, respectively. The optimum flow ratio of working-to-product air for this cooler is 50%. The channel geometric sizes, i.e. channel length and height, also impose significant impact to system performance. Longer channel length and smaller channel height contribute to increase of the system cooling effectiveness but lead to reduced system COP. The recommend channel height is 4 mm and the dimensionless channel length, i.e., ratio of the channel length to height, should be in the range 100 to 300. Numerical study results indicated that this new type of M-cycle heat and mass exchanger can achieve 16.7% higher cooling effectiveness compared with the conventional cross-flow heat and mass exchanger for the indirect evaporative cooler. The model of this kind is new and not yet reported in literatures. The results of the study help with design and performance analyses of such a new type of indirect evaporative air cooler, and in further, help increasing market rating of the technology within building air conditioning sector, which is currently dominated by the conventional compression refrigeration technology.

Numerical simulation of the Filchner overflow

The plume of Ice Shelf Water (ISW) flowing into the Weddell Sea over the Filchner sill contributes to the formation of Antarctic Bottom Water. The Filchner overflow is simulated using a hydrostatic, primitive equation three-dimensional ocean model with a 0.5–2 Sv ISW influx above the Filchner sill. The best fit to mooring temperature observations is found with influxes of 0.5 and 1 Sv, below a previous estimate of 1.6 ± 0.5 Sv based on sparse mooring velocities. The plume first moves north over the continental shelf, and then turns west, along slope of the continental shelf break where it breaks up into subplumes and domes, some of which then move downslope. Other subplumes run into the eastern submarine ridge and propagate along the ridge downslope in a chaotic manner. The next, western ridge is crossed by the plume through several paths. Despite a number of discrepancies with observational data, the model reproduces many attributes of the flow. In particular, we argue that the temporal variability shown by the observations can largely be attributed to the unstable structure of the flow, where the temperature fluctuations are determined by the motion of the domes past the moorings. Our sensitivity studies show that while thermobaricity plays a role, its effect is small for the flows considered. Smoothing the ridges out demonstrate that their presence strongly affects the plume shape around the ridges. An increase in the bottom drag or viscosity leads to slowing down, and hence thickening and widening of the plume

Standardized test to evaluate numerical weather prediction algorithms

In order to assist in comparing the computational techniques used in different models, the authors propose a standardized set of one-dimensional numerical experiments that could be completed for each model. The results of these experiments, with a simplified form of the computational representation for advection, diffusion, pressure gradient term, Coriolis term, and filter used in the models, should be reported in the peer-reviewed literature. Specific recommendations are described in this paper.

On the approximation of local and linear radiative damping in the middle atmosphere

The validity of approximating radiative heating rates in the middle atmosphere by a local linear relaxation to a reference temperature state (i.e., ‘‘Newtonian cooling’’) is investigated. Using radiative heating rate and temperature output from a chemistry–climate model with realistic spatiotemporal variability and realistic chemical and radiative parameterizations, it is found that a linear regressionmodel can capture more than 80% of the variance in longwave heating rates throughout most of the stratosphere and mesosphere, provided that the damping rate is allowed to vary with height, latitude, and season. The linear model describes departures from the climatological mean, not from radiative equilibrium. Photochemical damping rates in the upper stratosphere are similarly diagnosed. Threeimportant exceptions, however, are found.The approximation of linearity breaks down near the edges of the polar vortices in both hemispheres. This nonlinearity can be well captured by including a quadratic term. The use of a scale-independentdamping rate is not well justified in the lower tropical stratosphere because of the presence of a broad spectrum of vertical scales. The local assumption fails entirely during the breakup of the Antarctic vortex, where large fluctuations in temperature near the top of the vortex influence longwave heating rates within the quiescent region below. These results are relevant for mechanistic modeling studies of the middle atmosphere, particularly those investigating the final Antarctic warming.

Some numerical experiments on the effect of the variation of static stability in two-layer quasi-geostrophic models

A method to solve a quasi-geostrophic two-layer model including the variation of static stability is presented. The divergent part of the wind is incorporated by means of an iterative procedure. The procedure is rather fast and the time of computation is only 60–70% longer than for the usual two-layer model. The method of solution is justified by the conservation of the difference between the gross static stability and the kinetic energy. To eliminate the side-boundary conditions the experiments have been performed on a zonal channel model. The investigation falls mainly into three parts: The first part (section 5) contains a discussion of the significance of some physically inconsistent approximations. It is shown that physical inconsistencies are rather serious and for these inconsistent models which were studied the total kinetic energy increased faster than the gross static stability. In the next part (section 6) we are studying the effect of a Jacobian difference operator which conserves the total kinetic energy. The use of this operator in two-layer models will give a slight improvement but probably does not have any practical use in short periodic forecasts. It is also shown that the energy-conservative operator will change the wave-speed in an erroneous way if the wave-number or the grid-length is large in the meridional direction. In the final part (section 7) we investigate the behaviour of baroclinic waves for some different initial states and for two energy-consistent models, one with constant and one with variable static stability. According to the linear theory the waves adjust rather rapidly in such a way that the temperature wave will lag behind the pressure wave independent of the initial configuration. Thus, both models give rise to a baroclinic development even if the initial state is quasi-barotropic. The effect of the variation of static stability is very small, qualitative differences in the development are only observed during the first 12 hours. For an amplifying wave we will get a stabilization over the troughs and an instabilization over the ridges.

Numerical prediction of atmospheric blocking-A case study

A series of numerical models have been used to investigate the predictability of atmospheric blocking for an episode selected from FGGE Special Observing Period I. Level II-b FGGE data have been used in the experiment. The blocking took place over the North Atlantic region and is a very characteristic example of high winter blocking. It is found that the very high resolution models developed at ECMWF, in a remarkable way manage to predict the blocking event in great detail, even beyond 1 week. Although models with much less resolution manage to predict the blocking phenomenon as such, the actual evolution differs very much from the observed and consequently the practical value is substantially reduced. Wind observations from the geostationary satellites are shown to have a substantial impact on the forecast beyond 5 days, as well as an extension of the integration domain to the whole globe. Quasi-geostrophic baroclinic models and, even more, barotropic models, are totally inadequate to predict blocking except in its initial phase. The prediction experiment illustrates clearly that efforts which have gone into the improvement of numerical prediction models in the last decades have been worth while.

Numerical modelling of the energy and water cycle of the Baltic Sea

This paper will introduce the Baltex research programme and summarize associated numerical modelling work which has been undertaken during the last five years. The research has broadly managed to clarify the main mechanisms determining the water and energy cycle in the Baltic region, such as the strong dependence upon the large scale atmospheric circulation. It has further been shown that the Baltic Sea has a positive water balance, albeit with large interannual variations. The focus on the modelling studies has been the use of limited area models at ultra-high resolution driven by boundary conditions from global models or from reanalysis data sets. The programme has further initiated a comprehensive integration of atmospheric, land surface and hydrological modelling incorporating snow, sea ice and special lake models. Other aspects of the programme include process studies such as the role of deep convection, air sea interaction and the handling of land surface moisture. Studies have also been undertaken to investigate synoptic and sub-synoptic events over the Baltic region, thus exploring the role of transient weather systems for the hydrological cycle. A special aspect has been the strong interests and commitments of the meteorological and hydrological services because of the potentially large societal interests of operational applications of the research. As a result of this interests special attention has been put on data-assimilation aspects and the use of new types of data such as SSM/I, GPS-measurements and digital radar. A series of high resolution data sets are being produced. One of those, a 1/6 degree daily precipitation climatology for the years 1996–1999, is such a unique contribution. The specific research achievements to be presented in this volume of Meteorology and Atmospheric Physics is the result of a cooperative venture between 11 European research groups supported under the EU-Framework programmes.