A demonstration of cycled 4D‐Var in the presence of model errorCullen, M. J. P.
doi: 10.1002/qj.653pmid: N/A
The justification for the standard four‐dimensional variational data assimilation (4D‐Var) method used at several major operational centres assumes a perfect forecast model, which is clearly unrealistic. However, the method has been very successful in practice. We investigate the reasons for this using a toy model with fast and slow time‐scales and with non‐random model error. The model error is chosen so that the solution remains predictable on both time‐scales. The fast modes are much less well observed than the slow modes. We show that poorly observed modes can be best forecast by using a regularization matrix in place of the background‐error covariance matrix, and using it to give a much stronger constraint than that implied by the true background error for these modes. The effect is that use can be made of observations over a longer time period. This allows the resulting forecast‐error growth to be reduced to much less than that of random perturbations generated using the analysis‐error covariance matrix and even less than the model error growth given sufficiently accurate observations. © Crown Copyright 2010. Published by John Wiley & Sons, Ltd.
Can 4D‐Var use dynamical information from targeted observations of a baroclinic structure?Irvine, E. A.; Gray, S. L.; Methven, J.
doi: 10.1002/qj.673pmid: N/A
Targeted observations are generally taken in regions of high baroclinicity, but often show little impact. One plausible explanation is that important dynamical information, such as upshear tilt, is not extracted from the targeted observations by the data assimilation scheme and used to correct initial condition error. This is investigated by generating pseudo targeted observations which contain a singular vector (SV) structure that is not present in the background field or routine observations, i.e. assuming that the background has an initial condition error with tilted growing structure. Experiments were performed for a single case‐study with varying numbers of pseudo targeted observations. These were assimilated by the Met Office four‐dimensional variational (4D‐Var) data assimilation scheme, which uses a 6 h window for observations and background‐error covariances calculated using the National Meteorological Centre (NMC) method. The forecasts were run using the operational Met Office Unified Model on a 24 km grid. The results presented clearly demonstrate that a 6 h window 4D‐Var system is capable of extracting baroclinic information from a limited set of observations and using it to correct initial condition error. To capture the SV structure well (projection of 0.72 in total energy), 50 sondes over an area of 1×106 km2 were required. When the SV was represented by only eight sondes along an example targeting flight track covering a smaller area, the projection onto the SV structure was lower; the resulting forecast perturbations showed an SV structure with increased tilt and reduced initial energy. The total energy contained in the perturbations decreased as the SV structure was less well described by the set of observations (i.e. as fewer pseudo observations were assimilated). The assimilated perturbation had lower energy than the SV unless the pseudo observations were assimilated with the dropsonde observation errors halved from operational values. Copyright © 2010 Royal Meteorological Society
Diagnosis and formulation of heterogeneous background‐error covariances at the mesoscaleMontmerle, Thibaut; Berre, Loïk
doi: 10.1002/qj.655pmid: N/A
This study focuses on diagnosing variations of background‐error covariances between precipitating and non‐precipitating areas, and on presenting a heterogeneous covariance formulation to represent these variations in a variational framework. The context of this work is the assimilation of observations linked to precipitation (radar data especially) in the AROME model, which has been running operationally at Météo‐France since December 2008 over French territory with a 2.5 km horizontal resolution. This system uses multivariate background‐error covariances deduced from an ensemble‐based method. At first, such statistics have been computed for 17 precipitating cases using an ensemble of AROME forecasts coupled with an ALADIN ensemble assimilation. Results, obtained from 3 h forecast differences performed separately for non‐precipitating and precipitating columns, display large discrepancies in error variances, correlation lengths and the correlations between humidity, temperature and divergence errors. These results argue in favour of including heterogeneous background‐error covariances in AROME incremental 3D‐Var, allowing different covariances to be used in regions with different meteorological patterns. Such a method enables us to get increments more adequately structured in those regions, and thus potentially to make better use of observations in a data assimilation system. The implementation consists of expressing the analysis increment as the sum of two terms, one for precipitating areas and the other for non‐precipitating areas, making use of a mask that defines rainy regions. This implies a doubling in the size of the control variable and of the gradient of the cost function. The feasibility of this method is shown through experiments with four isolated observations. Copyright © 2010 Royal Meteorological Society
Representation of correlation functions in variational assimilation using an implicit diffusion operatorMirouze, I.; Weaver, A. T.
doi: 10.1002/qj.643pmid: N/A
Correlation models are required in data assimilation to characterize the error structures of variables defined on a numerical grid. Previous studies have shown that the diffusion equation can provide a flexible and computationally efficient framework for representing grid‐point correlation functions for problems of large dimension such as those encountered in atmospheric or ocean variational data assimilation. In this article, an implicit formulation of the diffusion‐based correlation model is presented as an alternative to the traditional explicit formulation. The implicit formulation is analyzed in detail for the one‐dimensional (1D) problem and shown to be closely related to the first‐order recursive filter. Integrating a 1D implicit diffusion equation, with constant coefficient, over M steps is shown to be equivalent to convolving the initial condition with an Mth order auto‐regressive (AR) function. Expressions for both the length‐scale of the AR function and the normalization factor required to generate unit‐amplitude (correlation) functions are given in terms of M and the diffusion coefficient. For a fixed length‐scale the Gaussian function, which is the only function that can be represented using an explicit formulation of the constant‐coefficient diffusion equation, is the limiting case as M → ∞ of the AR functions generated by the implicit diffusion equation. Generalizations of the diffusion model are discussed to allow for different shapes in the correlation function and spatial variations in the length‐scale. An important consequence of employing spatially varying length‐scales is that the normalization factors are no longer constant. Approximate expressions for the normalization factors are evaluated in terms of their effectiveness to provide viable alternatives to estimates produced using expensive algorithms such as randomization. Boundary conditions can distort the correlation functions near the boundaries and significantly degrade the accuracy of the analytical expressions for the normalization factors. These problems can be avoided through a straightforward extension of the diffusion model that makes the boundaries effectively transparent, although the solution comes at the expense of an extra application of the diffusion equation. Extensions of the method to construct two‐ and three‐dimensional correlation models are discussed. Copyright © 2010 Royal Meteorological Society
Development and evaluation of an ensemble forecasting system for coastal storm surgesFlowerdew, Jonathan; Horsburgh, Kevin; Wilson, Chris; Mylne, Ken
doi: 10.1002/qj.648pmid: N/A
The overtopping of flood defences by coastal storm surges is a significant threat to life and property. Like all forecasts, storm surge predictions have an associated uncertainty, but this has not been directly predicted by previous operational systems. The Met Office Global and Regional Ensemble Prediction System (MOGREPS), which has recently become operational, provides an explicit sample of the range of atmospheric evolutions consistent with the latest observations. The storm surge ensemble derives a storm surge forecast for each MOGREPS ensemble member, giving an explicit estimate of the risk of reaching significant water levels. The system has now been running for over two years in trial mode, with positive feedback from both Met Office and Environment Agency forecasters. For the surge on 9 November 2007, which received significant media coverage, it produced a clear signal of an abnormal event at the full 54‐hour lead time, with a useful indication of the range of possible water levels. Statistical verification covering two winters demonstrates that the ensemble spread is indeed a reliable indicator of the significantly increased uncertainty associated with large surge events, although the error in more normal situations is dominated by inaccuracies in the separate harmonic tide prediction which is used to convert surge to water level. Probabilistic verification shows some advantage over climatological error distributions, particularly for larger thresholds and longer lead times. Following this successful trial, the system was made operational in December 2009. Copyright © 2010 Royal Meteorological Society and Crown Copyright.
On the limiting effect of the Earth's rotation on the depth of a stably stratified boundary layerMironov, Dmitrii; Fedorovich, Evgeni
doi: 10.1002/qj.631pmid: N/A
Two alternative depth scales have been proposed for the case of a stably stratified boundary layer (SBL) where static stability is due to the surface buoyancy flux Bs. Kitaigorodskii in 1960 assumed that the Earth's rotation is no longer important as static stability becomes strong and that the SBL depth scales with the Obukhov length \documentclass{article}\usepackage{amsmath}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{amsfonts}\pagestyle{empty}\begin{document}$L=-u_*^3/B_{\rm s}$ \end{document}, u* being the surface‐friction velocity. Zilitinkevich in 1972 proposed an alternative scale, (u*L/|f|)1/2, which depends on the Coriolis parameter f no matter how strong the static stability. Similarly, two alternative depth scales have been proposed for the case of a SBL dominated by static stability at its outer edge with buoyancy frequency N. The depth scale u*/N introduced by Kitaigorodskii and Joffre in 1988 does not depend on the Coriolis parameter, whereas the Pollard, Rhines and Thompson scale u*/|Nf|1/2 introduced in 1973 does. In the present article, the above formulations for the SBL depth are shown to be consistent with the budgets of momentum and of turbulence kinetic energy in the SBL. Furthermore, it is demonstrated that in the case of sufficiently strong static stability the alternative depth‐scale formulations represent particular cases of more general power‐law expressions. For a SBL dominated by the surface buoyancy flux, the generalized depth scale is given by L(|f|L/u*)−γ. For a SBL dominated by outer‐edge static stability, the generalized scale is (u*/N)(|f|/N)−δ. The exponents γ and δ lie in the range from 0 to 1/2. With γ = 1/2 and δ = 1/2, these expressions yield the Zilitinkevich scale and the Pollard et al. scale, respectively. In the limits γ = 0 and δ = 0, the SBL depth scales cease to depend on the Coriolis parameter in their explicit form and the formulations proposed by Kitaigorodskii and by Kitaigorodskii and Joffre, respectively, are recovered. Simple dimensionality arguments are not sufficient to determine γ and δ. To do this would require an exact solution to equations governing the structure of mean fields and turbulence in the SBL. Since such a solution is not known, the exponents should be evaluated from experimental data. Available data from observations and from large‐eddy simulations are uncertain. They do not make it possible to evaluate γ and δ to adequate accuracy and to decide conclusively between the alternative formulations for the SBL depth. As regards practical applications, previously proposed multi‐limit formulations based on the above depth scales with γ and δ in the range from 0 to 1/2 are expected to give similar results for stability conditions typical of the atmospheric and oceanic SBLs, provided the disposable dimensionless coefficients in the multi‐limit formulations are appropriately tuned. Copyright © 2010 Royal Meteorological Society
A new stable boundary‐layer mixing scheme and its impact on the simulated East Asian summer monsoonHong, Song‐You
doi: 10.1002/qj.665pmid: N/A
This paper investigates the impact of stable boundary‐layer (SBL) mixing in a vertical diffusion package on the simulated climatology in a regional model. In contrast to previous studies, we focus on the sensitivity of the simulated climatology to the representation of SBL processes in the modelled atmosphere, paying particular attention to precipitation and associated large‐scale patterns. The new SBL scheme, based on the bulk Richardson number between the surface layer and the top of the boundary layer and implemented in the Yonsei University (YSU) boundary‐layer scheme, was evaluated against the local scheme in which the mixing coefficient is a function of the local Richardson number at a given model level. A statistical evaluation of a series of short‐range forecast confirms that the boundary‐layer structure is closer to the radiosonde observation when the new SBL scheme is used. In a regional climate framework, the results with the new SBL scheme in July 2006 demonstrate that modulating the subcloud structure with enhanced vertical mixing improves the simulated monsoon climatology by displacing the monsoonal precipitation southwards. Together with the local effects of the enhanced SBL mixing that warms and dries the boundary layer, the dynamical feedbacks accompanying strengthened moisture convergence results in enhanced precipitation towards what was observed. A ten‐member ensemble of three‐month June–July–August simulations for 1999–2008 shows that the revised SBL scheme improves the temperature and moisture profiles in the lower troposphere as well as the precipitation climatology. The interannual variation of seasonal precipitation is more realistic over both land and oceans. Copyright © 2010 Royal Meteorological Society
Boundary‐layer simulations for the Mars Phoenix lander siteSavijärvi, Hannu; Määttänen, Anni
doi: 10.1002/qj.650pmid: N/A
Diurnal simulations and sensitivity studies were made with a moist column model at the Mars Phoenix lander site (68°N) for the summer solstice (solar day 30) and for a later date (sol 99), when the LIDAR on board Phoenix detected fog, dust, ice clouds and even snowfall from cloud. The sol 30 simulation reproduces the observed repetitive diurnal 2 m temperature cycle quite well, displaying a well‐mixed boundary layer up to 4 km in the afternoon and a strong surface inversion to 500 m each night. Weak frost formation peaks at midnight and a very thin radiation fog appears during the coldest hour. The near‐surface water vapour pressure is underestimated during daytime but is close to the thermal and electrical conductivity probe observations during the night. The Prandtl slope wind mechanism produces veering winds in the model as observed by the ‘telltale’ device while coupled dust evolution implies well‐mixed dust to 4 km throughout the sol as observed by the LIDAR. The colder diurnal conditions around sol 99 are also simulated rather well. In these, the morning fog grows up to 800 m height and a water ice cloud forms at 4 km height at about 0300 local time, as observed. The cloud marks the radiatively cooled top of the moist residual boundary layer. Strong ground frost formation peaks in the evening, having a visible impact on the temperatures. The fog and cloud display weak feedbacks to the modelled radiative fluxes. Copyright © 2010 Royal Meteorological Society
Instability of surface‐temperature filaments in strain and shearHarvey, B. J.; Ambaum, M. H. P.
doi: 10.1002/qj.651pmid: N/A
The effects of uniform straining and shearing on the stability of a surface quasi‐geostrophic temperature filament are investigated. Straining is shown to stabilize perturbations for wide filaments but only for a finite time until the filament thins to a critical width, after which some perturbations can grow. No filament can be stabilized in practice, since there are perturbations that can grow large for any strain rate. The optimally growing perturbations, defined as solutions that reach a certain threshold amplitude first, are found numerically for a wide range of parameter values. The radii of the vortices formed through nonlinear roll‐up are found to be proportional to θ/s, where θ is the temperature anomaly of the filament and s the strain rate, and are not dependent on the initial size of the filament. Shearing is shown to reduce the normal‐mode growth rates, but it cannot stabilize them completely when there are temperature discontinuities in the basic state; smooth filaments can be stabilized completely by shearing and a simple scaling argument provides the shear rate required. Copyright © 2010 Royal Meteorological Society