Showing papers in "Tellus A in 1993"

Computation of optimal unstable structures for a numerical weather prediction model

Abstract: Numerical experiments have been performed to compute the fastest growing perturbations in a finite time interval for a complex numerical weather prediction model. The models used are the tangent forward and adjoint versions of the adiabatic primitive-equation model of the Integrated Forecasting System developed at the European Centre for Medium-Range Weather Forecasts and Meteo France. These have been run with a horizontal truncation T21, with 19 vertical levels. The fastest growing perturbations are the singular vectors of the propagator of the forward tangent model with the largest singular values. An iterative Lanczos algorithm has been used for the numerical computation of the perturbations. Sensitivity of the calculations to different time intervals and to the norm used in the definition of the adjoint model have been analysed. The impact of normal mode initialization has also been studied. Two classes of fastest growing perturbations have been found; one is characterized by a maximum amplitude in the middle troposphere, while the other is confined to model layers close to the surface. It is shown that the latter is damped by the boundary layer physics in the full model. The linear evolution of the perturbations has been compared to the non-linear evolution when the perturbations are superimposed on a basic state in the T63, 19-level version of the ECMWF model. DOI: 10.1034/j.1600-0870.1993.t01-4-00005.x

Scaling regimes of composite rainfall time series

Abstract: The scaling behaviour of rainfall is analysed both for a range of scales in time and for a given scale in intensity using the statistics of the Fourier transform and the cumulative probability distribution. The analyses are applied to sets of long time series of daily rainfall (26 (8) files of 45 (90) years at 13 European stations) and sets of 5-min totals (13 single station summer seasons) thus covering a wide scaling range. The results of both analyses are interpreted in terms of their asymptotically hyperbolic (i.e. power law) behaviour: The ensemble averaged power spectra exhibit distinct scaling regimes with their associated power law behaviour, P ( f ) ∼ f –b : a regime of climatic variability (> 3 years: b ∼ 0.7), a spectral plateau (3 years to 1 month: b – 0) of general circulation fluctuations, a transition regime (1 month to 3 days: a dropping power spectrum without scaling), and a range governed by frontal systems (< 3 days: b ∼ 0.5). The transition region is interpreted as being generated by both of its neighbouring regimes whose scaling can therefore be expanded into the transition regime. Finally an apparent break in scaling (at 2.4 h) can possibly be attributed to the instruments inability to measure frequent weak signals. The tail-end of the hyperbolic distribution (characterizing the intermittency regime) is not approached smoothly but shows a break from the rest of the distribution. Finally, an outlook to multifractal scaling is given. DOI: 10.1034/j.1600-0870.1993.t01-3-00004.x

Topographic effects in stratified flows resolved by a spectral method

Abstract: The aim of the paper is to validate a particular model formulation in the study of wind-induced response in a wide, rotating channel with varying bottom topography The model consists of a continuous stratified upper layer joined to a homogeneous lower layer For this density profile, standard normal mode solution techniques can be extended to forced models with sloping bottoms in order to separate the vertical dependency The governing equations are then reduced to a time dependent problem by expanding across channel variations in Fourier series and applying periodic boundary conditions along the channel An initial-value problem is solved numerically, using an eigenmode solution technique starting from a state of rest, where a constant wind stress is suddenly imposed and then switched off after a certain period of time The eigenmode solution technique is found to be applicable in the examination of the coupling between vertical modes and the analysis has resulted in a simplified coupling pattern for this specific two layer model Modified baroclinic eigenmodes are introduced in a superposition of each baroclinic and the barotropic mode, according to the eigenvalue problem For these modes the interaction amongst baroclinic modes is found to be insignificant while the barotropic eigenmodes are not found to be affected by the stratification For varying bottom topography these modified baroclinic modes will reflect both stratification and topography Model simulations show that the flow pattern is dominated by long period barotropic oscillations as topographic Rossby waves The waves show a dominance in the second lateral mode, and they are less affected by the stratification The barotropic response to the wind is mirrored in internal displacements and shows long periodic oscillations corresponding to the topographic waves The use of a bottom Ekman layer has shown that bottom friction mainly influences on long-period barotropic modes The baroclinic modes are damped by internal friction DOI: 101034/j1600-08701993t01-1-00003x

Balancing the carbon budget. Implications for projections of future carbon dioxide concentration changes

Abstract: A carbon cycle model is described incorporating CO 2 fertilization feedback and a convolution ocean model that allows the atmosphere-to-ocean flux to be varied. The main parameters controlling the model's behaviour are a fertilization feedback parameter (r) and an ocean flux scaling factor (characterized by the mean carbon flux into the ocean over the 1980s, F (1980s)). Since the model's 1980s-mean net land-use-change flux ( D n (1980s)) is a unique function of r and F (1980s), the model's behaviour can also be characterized by specifying D n (1980s) (instead of r) and F (1980s). The history of past land-use fluxes, D n (t), is derived by inverse modelling for a range of values of F (1980s) (1.0–3.0 GtC/yr) and D n (1980s) (0.6–2.6 GtC/yr). Even with this flexibility, the resultant D n (t) differs markedly from the observationally-based record of Houghton, particularly before 1950. The inverse calculations are used to determine the history of the so-called “missing sink”, as implied directly by the model and by the observationally-based record of D n (t), for a range of ocean uptake efficiencies as defined by F (1980s). Projections of future CO 2 concentration changes are made for the 6 emissions scenarios recently produced by the Intergovernmental Panel on Climate Change (IS92a–f). The ability to specify F (1980s) and D n (1980s) allows one to account for the missing sink in a variety of ways, and to account for uncertainties in the amount of missing carbon. This leads to a range of projections and provides some insights into the uncertainties surrounding these projections. DOI: 10.1034/j.1600-0889.1993.t01-4-00002.x

Important literature on the use of adjoint, variational methods and the Kalman filter in meteorology

Abstract: The use of adjoint equations is proving to be invaluable in many areas of meteorological research. Unlike a forecast model which describes the evolution of meteorological fields forward in time, the adjoint equations describe the evolution of sensitivity (to initial, boundary and parametric conditions) backward in time. Essentially, by utilizing this sensitivity information, many types of problems can be solved more efficiently than in the past, including variational data assimilation, parameter fitting, optimal instability and sensitivity analysis in general. For this reason, the adjoints of various models and their applications have been appearing more and more frequently in meteorological research. This paper is a bibliography in chronological order of published works in meteorology dealing with adjoints which have appeared prior to this issue of Tellus. Also included are meteorological works regarding variational methods (even without adjoints) and Kalman filtering in data assimilation, plus some references outside meteorology. These additional works are included here because the main thrust for adjoint application within meteorology is currently concentrated in the development of next-generation data assimilation systems. DOI: 10.1034/j.1600-0870.1993.t01-4-00002.x

Automatic differentiation in Odyssée

Abstract: This paper describes the design of Odyssee , a system for fortran programs manipulations and its application to automatic differentiation. The Odyssee system manipulates fortran programs as symbolic objects. It is an open system built as a toolkit, written in a high-level programming language adapted to this purpose. The use of a variational method to perform data assimilation requires the computation of the gradient of a cost function represented by a large-size fortran program. The usual drawback in the reverse automatic differentiation method is the storage requirement. The Odyssee system allows one to implement storage/recomputation strategies in order to fit the needed compromizes. We present the implementation of the strategy used in the weather forecasting arpege/ifs project to produce the adjoint code from the code representing the numerical model. Odyssee produces the same code as the hand-written adjoint code for thearpege/ifs project. DOI: 10.1034/j.1600-0870.1993.00016.x

Variational data assimilation with moist threshold processes using the NMC spectral model

Abstract: This paper describes a detailed study of variational 4-D data assimilation including the physical processes of large-scale precipitation and deep cumulus convection. The length of the assimilation window is 6 h, and the data are NMC's operational analyses. A comparison of the minimization behaviour, the computational complexity, the quality of the retrieved initial state, with and without physical processes is presented. The results demonstrate the ability to perform 4-D variational data assimilation with discontinuous physical processes. The experiments are carried out with the NMC global spectral model in a resolution of 18 layers in the vertical and a 40 wave triangular truncation. DOI: 10.1034/j.1600-0870.1993.t01-4-00004.x

Physical initialization using SSM/I rain rates

Abstract: Following our recent study on physical initialization for tropical prediction using rain rates based on outgoing long-wave radiation, the present study demonstrates a major improvement from the use of microwave radiance based rain rates. A rain rate algorithm is used on the data from a special sensor microwave instrument (SSM/1). The initialization, as before, uses a reverse surface similarity theory, a reverse cumulus parameterization algorithm and a bisection method to minimize the difference between satellite-based and the model-based outgoing long-wave radiation. These are invoked within a preforecast Newtonian relaxation phase of the initialization. These tests are carried out with a high-resolution global spectral model. The impact of the initialization on forecast is tested for a complex triple typhoon scenario over the Western Pacific Ocean during September 1987. A major impact from the inclusion of the SSM/I is demonstrated. Also addressed are the spin-up issues related to the typhoon structure and the improved water budget from the physical initialization. DOI: 10.1034/j.1600-0870.1993.t01-3-00001.x

Examination of the accuracy of a tangent linear model

Abstract: The accuracy of a tangent linear version of a 3-dimensional mesoscale primitive equation model (the PSU/NCAR MM4) is investigated by comparing its results with those produced by identical perturbations introduced in nonlinear forecasts of the original model. Moist physical processes are not considered in this study. For perturbation magnitudes as large as typical current analysis errors, the perturbation tendencies are shown to be very accurately estimated by the tangent linear model (TLM), with greater relative error in a summer case than in a winter one. The evolutions of perturbations in forecasts out to 72 h are also accurately estimated, although the unperturbed lateral boundary conditions that act to sweep perturbations out of the domain are an artificial means of perturbation constraint. It is shown that for many cases it is sufficient to approximate a true TLM by using an infrequent update of the basic state, thereby reducing the amount of stored fields and input required by the TLM software. For perturbations not smaller than one-tenth the size of typical analysis errors, a 30-min update appears sufficient. The TLM accuracy is related to the accuracy of adjoint sensitivity calculations with regard to finite-amplitude perturbations. An example of an adjoint application is shown to have two-digit accuracy for a moderately sized perturbation. All these results indicate that our TLM and corresponding adjoint yield quantitatively accurate results for many important uses. DOI: 10.1034/j.1600-0870.1993.00010.x

Stagnation points and bifurcation in 3‐D mountain airflow

Abstract: The effect of non-linearity on the formation of mountain-wave induced stagnation points is examined using the scaling laws for ideal hydrostatic flow and a series of runs with decelerating winds in a numerical model. In the limit of small deceleration rate (i.e., near steady state) runs with a variety of mountain heights and widths give similar results; i.e., the speed extrema values in the 3-D wave fields collapse onto “universal curves”. For a Gaussian hill with circular contours, stagnation first occurs at a point above the lee slope. This result contradicts the result of linear theory that stagnation begins on the windward slope. The critical value of ĥfor stagnation above a Gaussian hill is ĥ crit = 1.1 ± 0.1. For a 3/2-power hill, the critical height is slightly higher, ĥ crit = 1.2 ± 0.2. These values are significantly larger than the value for a ridge (ĥ crit = 0.85), due to dispersion of wave energy aloft. The application of Sheppard's rule and the vorticity near the stagnation point are discussed. As expected from linear theory, the presence of positive windshear suppresses stagnation aloft. With Richardson number = 20 for example, stagnation first begins at the ground at a value of ĥ= 1.6 ± 0.2. When a stagnation point first forms aloft in the unsheared case, the flow field begins to evolve in the time domain and the scaling laws are violated. We interpret these events as a wave-breaking induced bifurcation which leads to stagnation on the windward slope and the formation of a wake. DOI: 10.1034/j.1600-0870.1993.00003.x

Deep water exchange in the Baltic Proper

Abstract: Deep-water exchange and mixing properties in the Baltic Proper were analysed on the basis of temperature and salinity data measured during the period 1970–90. The data were analysed applying basic model concepts as the conservation principles, the two-layer approach and the geostrophic flow assumption. The renewal of the deep water in the Baltic Proper consists of inflow from upstream basins. The inflowing dense water is diluted by surface water and on a 20-year average increased by a factor of 4, when entering from the Kattegat into the Landsort Deep. Three main mixing zones were localized. Firstly, the Belt Sea and the Sound, where the deep-water inflow increases by 79%; secondly, the Arkona Basin, where vertical mixing causes the increase of deep current volume flow by an average of 53%; and thirdly, in the Stolpe Channel, where the turbulent entrainment adds an average of 28% to the deep current. Applying the geostrophic flow model on salinity data, time series of deep current flow rates were calculated. The model was calibrated by 20-year mean flows calculated from conservation principles. The effective sill depths were introduced as calibration coefficients. It was found that the geostrophic flow model described deep-layer flows in the Bornholm Channel well, but the flow rate was underestimated in the Stolpe Channel and overestimated in the Faro Channel. In the Stolpe Channel and in the Faro Channel, the deep-layer flow showed seasonal variations with rapid increase during the autumn and winter seasons, respectively. DOI: 10.1034/j.1600-0870.1993.t01-1-00006.x

Linearization and adjoint of parameterized moist diabatic processes

Abstract: The problems related to the development and applications of an adjoint model for a nonlinear model that has discontinuities in the governing equations are discussed in this study. We examine the hypothesis that a useful tangent linear model and its associated adjoint model can be developed for such a nonlinear model by using the reference state to determine the regimes associated with the discontinuities. Our nonlinear and linear experiments for the Pennsylvania State University/National Center for Atmospheric Research regional forecast model show that linearization of moist diabatic parameterizations is feasible using this method, but significant errors may be expected in the regions where transitions are frequent and associated with large tendency changes due to the model's moist diabatic parameterizations. We conclude that the tangent linear model and its associated adjoint model would be more accurate for a nonlinear model that has more well-behaved regime transitions. We suggest that nonlinear tests similar to the analysis shown in this study be performed prior to developing these linear models. If these tests show high nonlinearity due to regime transitions as a consequence of model deficiencies, we recommend first improving the nonlinear model before performing the linearization. DOI: 10.1034/j.1600-0870.1993.0012.x

Optimal determination of nudging coefficients using the adjoint equations

Abstract: The adjoint equations of a numerical model can be used for model-parameter estimation. In this study, a general computational procedure is developed to determine the size and distribution of any internal model parameter. The procedure is then applied to a one-dimensional shallow-water model in the context of analysis-nudging four-dimensional data assimilation (FDDA): the weighting coefficients used by the Newtonian nudging technique are determined such that the model error during the assimilation period is optimally reduced subject to some constraints. The sensitivity of these nudging coefficients to the optimal objectives and constraints is investigated using this simple grid-point model in an Observing Systems Simulation Experiments (OSSE) mode. The results show that in principle, it is feasible to determine a set of nudging weights which minimize the model error over the period covered by the observations. It is demonstrated, however, that the magnitude and distribution of these “optimal” nudging weights are sensitive to the prescribed estimate of the nudging weights and the corresponding coefficient matrix which define a penalty term in the cost function. The penalty term is a weak constraint on the size and distribution of the optimal nudging weights while the model is the strong constraint. The fit of the model to the data is greater when this constraint on the nudging weights is weaker, but then the nudging weights may be too large or even negative. Thus the “optimal” solution for this model parameter is not unique because specification of this penalty term in the cost function introduces a new uncertainty into the nudging FDDA framework. Nevertheless, this optimal-nudging approach does show promise, but the sensitivity of the technique to the penalty term requires further investigation under more realistic conditions. DOI: 10.1034/j.1600-0870.1993.t01-4-00003.x

The effects of discontinuities in the Betts’Miller cumulus convection scheme on four-dimensional variational data assimilation

Abstract: A tangent linear and an adjoint of the large-scale precipitation and the cumulus convection processes in the National Meteorological Center's NMC/ETA regional forecast model are developed. The effects of discontinuities in the Betts–Miller cumulus convection scheme are examined and applicability of derivative minimization methods in four-dimensional variational (4D VAR) data assimilation is considered. It is demonstrated that discontinuities present in the control Betts–Miller cumulus convection scheme increase linearization errors to a large extent and have adverse effects on 4D VAR data assimilation. In the experiments performed, discontinuities in the cumulus convection scheme have the most serious effect in low layers. These problems can be reduced by modifying the scheme to make it more continuous in low layers. Positive effects of inclusion of cumulus convection in 4D VAR data assimilation are found in upper layers, especially in humidity fields. The “observations” used are optimal interpolation analyses of temperature, surface pressure, wind and specific humidity. By inclusion of other data, more closely related to the convective processes, such as precipitation and clouds, more benefits should be expected. Even with the difficulties caused by discontinuities, derivative minimization techniques appear to work for the data assimilation problems. In order to get more general conclusions, more experiments are needed with different synoptic situations. The inclusion of other important physical processes such as radiation, surface friction and turbulence in the forecast and the corresponding adjoint models could alter the results since they may reinforce the effects of discontinuities. DOI: 10.1034/j.1600-0870.1993.00013.x

Residual currents in tidally dominated, well‐mixed estuaries

Abstract: An analytical model is developed to investigate the residual circulation originating from the interaction of the density field, river flow, and the nonlinear rectification of the periodic tides, in tidally-dominated, well-mixed estuaries. These residual flows effect the density field, which in turn drives the estuarine gravitational circulation. Previous analytical studies have excluded the tidal dynamics and have either imposed the density structure, thereby decoupling the problem, or assumed similarity solutions valid for specific regions within the estuary. Here, a theoretical model is developed that calculates, via a perturbation analysis, the lowest-order density field established from tidal mixing, tidally-induced residual flow, river runoff, and diffusion. Two special cases are considered, a long tidal river and a finite length, variable-breadth estuary. Once the density field is determined for each case, the density driven flow is calculated and added to the rectified tidal and river induced flows for both the Eulerian and Lagrangian reference frames. DOI: 10.1034/j.1600-0870.1993.00007.x

Comparison of initial and lateral boundary condition sensitivity for a limited-area model

Abstract: The sensitivities of two forecast aspects with respect to perturbations of initial and lateral boundary conditions are determined using an adjoint of a limited-area model. Sensitivities are presented as gradient fields. Some characteristics of the sensitivity fields have simple dynamical explanations, such as their strong dependence on winds, and do not require an adjoint model to reveal them. Other characteristics would be difficult to ascertain without an adjoint model, although they, too, in retrospect, have simple dynamical explanations. Still others, although equally important and undoubtedly dependent on the dynamics, would be practically impossible to determine without an adjoint. The quantity and quality of information provided by the adjoint solutions render the adjoint model a tool that is indispensable for sensitivity analysis. DOI: 10.1034/j.1600-0870.1993.00015.x

The dynamics of error covariances in a barotropic model

Abstract: Under the hypothesis of linear error growth, the tangent linear model can be used to provide a simplified form of stochastic-dynamic prediction, in which the uncertainty on a model forecast is defined by the error covariances between the model state variables. We apply this method on a global vorticity equation model with a simple initial condition for the covariance matrix. The forecast error variances and correlations undergo rapid modifications in relation with the atmospheric flow pattern. This may have some implications for operational assimilation schemes in numerical weather prediction. A deeper understanding of the relevant processes is sought, using an adjoint form of the prediction equation, in order to assess the impact of the initial variances and correlations, and of the intrinsic dynamics of the model flow. The application on some simplified flows shows the relevance of diffusion, advection, Rossby wave dispersion and barotropic instability to forecast skill prediction. DOI: 10.1034/j.1600-0870.1993.t01-2-00006.x

Statistical properties of sea ice surface topography in the Baltic Sea

Abstract: The results from helicopter-borne laser profiling of the ice surface in the Baltic Sea for March 1988 are presented. The laser was a PRAM III system with a measurement frequency of 1–4 kHz, a footprint of approximately 20 cm at a flight elevation of 100 m and a single-shot vertical resolution of 12.5 cm. The laser profiles were calibrated to remove aircraft motion and averaged to 1-m intervals which combined to produce an elevation accuracy of 1.8 cm. A total of 51 profiles with varying lengths from 6 to 24 km, in all 662 km, are analysed. The data were stratified into 5 groups representing different ice fields. The standard deviation of the surface elevations ranged from 7 to 14 cm. The spectrum of surface roughness displayed a red noise form (slope about − 1.5) through the wave-lengths 4 to 100 m. Ice ridges were identified using the Rayleigh criterion with a cutoff height of 40 cm. The mean ridge height ranged from 52 to 59 cm, the standard deviation 12 to 20 cm, and the maximum ridge height sampled was 197 cm. The number of ridges was 1.4 to 9.5 km −1 for 4 of the groups and up to 17.4 km −1 in one group representing an intense shear zone. Ridge heights were exponentially distributed whereas the log-normal distribution provided the best fit for the ridge spacings. Mean ridge height and ridge density were not strongly correlated. The amount of ridged ice was estimated to account for 1 to 18 cm of equivalent ice thickness while based on a crude extrapolation method rubble fields summed up to 2 to 10 cm. Deformed ice constituted 15 to 40% of the total ice volume. DOI: 10.1034/j.1600-0870.1993.t01-1-00004.x

A preconditioning algorithm for large-scale minimization problems

Abstract: A new preconditioning algorithm is proposed, employing a Taylor series expansion of the cost-function, and the relation between the adjustment of the control variable and the computed gradient norm. The preconditioning matrix is a positive definite diagonal matrix, being a product of two positive definite diagonal matrices. One is the weight matrix related to the Hessian matrix definition in the case of identity model operator (“rough” scaling), and the other matrix is interpreted as a refined scaling of the control variable. The procedure is quite easy to implement, and the computer time and space requirements are negligible. The algorithm was tested in two cases of realistic four-dimensional variational data assimilation experiments, performed using an adiabatic version of the NMC's new regional forecast model and operationally obtained optimal interpolation analyses. Test results show a significant improvement in the decrease of the cost-function and the gradient norm when using the new preconditioning procedure. The preconditioning was applied to a memoryless quasi-Newton method, however, the technique should be applicable to other minimization algorithms. DOI: 10.1034/j.1600-0870.1993.00011.x

Conversion among North Atlantic surface water types

Abstract: Climatological data sets are used to estimate the amount of water changing temperature and salinity at the sea-surface owing to air-sea exchanges. Considering this conversion as a function of sea-surface temperature and salinity leads to the definition of a conversion vector, which represents mass sources and sinks for each water type. The vector representation shows graphically the way the shape of the temperature-salinity relation is driven at the surface by the air-sea heat and fresh water fluxes. DOI: 10.1034/j.1600-0870.1993.00006.x

Linear unsteady mountain waves

Abstract: The object of this paper is to describe the generation of mountain gravity waves when the incident wind is transient. The incident flow is a stably stratified fluid which velocity starts from zero at a given time and returns to zero after a finite time. For the mountain, both a single harmonic and a single ridge topography are considered. The nature of transient effects on the disturbance is discussed according to the value of the parameter, ɛ = U 0 t f / L , which measures the ratio between the time scale t f of the temporal variation of the incident wind and the advective time scale L / U 0 . L and U 0 characterize the horizontal extension of the mountain and the incident wind intensity, respectively. Situations for which ɛ » 1 are referenced as quasi-steady and the wavefield can be conveniently described using ray-tracing techniques. Situations for which ɛ « 1, are referenced as very unsteady, and essentially reduce to a ground vibrating disturbance generated in a fluid at rest. In this paper, we consider both limit configurations as well as intermediate ones. This allows us to describe various mechanisms of generation of unsteady mountain waves. The incidence of the unsteadiness on the amplitude of the gravity wave momentum flux and on the location of the mountain wave breaking level are also discussed. DOI: 10.1034/j.1600-0870.1993.t01-2-00004.x

A comparison between planetary‐wave flow regimes and blocking

Abstract: An investigation of the relative similarities in regime classification of an index designed to isolate cases of blocking based upon the kinematic properties of the regional flow, and one which describes the amplitude of the planetary-scale eddy field, is undertaken with observations from 28 Northern Hemisphere winters. The occurrence of blocking patterns persisting 5 days or more over the eastern Pacific is found to coincide in 94% of all cases with the hemispheric flow being characterized by an amplified planetary-scale wave flow regime. However, the occurrence of such an amplitude regime does not necessarily imply the existence of a blocking pattern in the classical sense. In the eastern Pacific, the split flow blocking developed simultaneously with or during an amplified wave regime episode in 30 of 33 cases. Retrogression of blocking patterns to west of the dateline is a frequent occurrence accompanying the decay of amplified planetary-wave episodes. 3/4 of the blocking episodes occurring west of the dateline formed as a part of amplified wave regime events and persisted into their aftermath. Most European blocking episodes (78%) are a part of an amplified-wave flow regime during some part of their life cycle. Blocking in this sector developed along with or during an amplified wave regime in 30 of 51 cases. DOI: 10.1034/j.1600-0870.1993.t01-3-00003.x

The effects of discontinuities in the Betts-Miller cumulus convection scheme on four-dimensional variational data assimilation

Abstract: A tangent linear and an adjoint of the large-scale precipitation and the cumulus convection processes in the National Meteorological Center's NMC/ETA regional forecast model are developed. The effects of discontinuities in the Betts–Miller cumulus convection scheme are examined and applicability of derivative minimization methods in four-dimensional variational (4D VAR) data assimilation is considered. It is demonstrated that discontinuities present in the control Betts–Miller cumulus convection scheme increase linearization errors to a large extent and have adverse effects on 4D VAR data assimilation. In the experiments performed, discontinuities in the cumulus convection scheme have the most serious effect in low layers. These problems can be reduced by modifying the scheme to make it more continuous in low layers. Positive effects of inclusion of cumulus convection in 4D VAR data assimilation are found in upper layers, especially in humidity fields. The “observations” used are optimal interpolation analyses of temperature, surface pressure, wind and specific humidity. By inclusion of other data, more closely related to the convective processes, such as precipitation and clouds, more benefits should be expected. Even with the difficulties caused by discontinuities, derivative minimization techniques appear to work for the data assimilation problems. In order to get more general conclusions, more experiments are needed with different synoptic situations. The inclusion of other important physical processes such as radiation, surface friction and turbulence in the forecast and the corresponding adjoint models could alter the results since they may reinforce the effects of discontinuities.

Validation of a skill prediction method

Abstract: A large experiment is performed to validate two predictors for the quality of ECMWF forecasts over Western Europe. One predictor yields the spread of the probability distribution for the error in the predicted 500 hPa geopotential height. It is determined by the trace of the covariance matrix for the geographically local forecast error. In addition the spread for the 500 hPa vorticity error at a location near the Netherlands is computed. The local covariance matrix, necessary for determining both predictors, is computed for 607 days, using the tangent linear and an adjoint version of a quasi-geostrophic 3-level model with truncation T21. We assume linear error growth and the absence of model errors. The forward reference orbit is obtained by interpolating actual ECMWF forecasts with the 3-level model. Small values of the predictor imply small error growth, and therefore accurate forecasts. Large values may or may not be associated with large actual forecast errors, depending on whether the initial error strongly projects on the fastest growing modes. How the uncertainty in the structure of the initial error influences the performance of the skill predictor is studied by considering three different covariance matrices for the initial error. Validation of the predicted variance with the 2-day and 3-day ECMWF forecast error shows that for all initial covariance matrices, both predictors provide significant information about the quality of the forecast. In case of small and large predicted variance, the probabilities for small and large prediction errors are 10% higher than the climatological probabilities. Projection of the observed forecast error onto the eigenvectors of the local covariance matrix indicates that a few eigenvectors already describe a large portion of the forecast error. DOI: 10.1034/j.1600-0870.1993.t01-1-00007.x

Dynamics of a two-dimensional ribbon of shallow water on an f-plane

Abstract: An analytical solution is derived for the motion of an infinitely long fluid ribbon of shallow water, with parabolic depth and linear velocity distribution. The solution is of intrinsic dynamical interest and also offers a “bench-mark” for testing numerical methods which handle the intersection of a free surface with a horizontal boundary. The development is expressed in terms of the translation and deformation of the hump. The former component is determined uniquely by the initial velocity at the symmetry axis and the externally imposed uniform pressure gradient. On the other hand, the deformation component depends on the initial shear and divergence, but is unaffected by a uniform external pressure force. Relative to its axis, the fluid ribbon alternately expands and contracts, with a pulsation frequency (≥ the inertial frequency f) that is a function of a Rossby, and a rotational Froude number, defined in terms of the initial velocity, and height field. DOI: 10.1034/j.1600-0870.1993.00004.x

On the formation of ice on deep weakly stratified water

Abstract: Making use of the simplest possible model, we analyse the formation of ice on top of a weakly stratified ocean. Our interest is concentrated on the consumption of freshwater associated with ice formation and the dynamics of the system when the cooling continues beyond the point where further ice formation would destroy the stability of the system. We find that after an initial stage of ice formation the system will not overturn but go into a stage of development which may be called “freeze melting”. This stage is characterized by increasing mixed layer depth, slowly decreasing ice thickness and small but finite stability. If the freeze melting continues for a sufficiently long time, considerably longer than required for the initial ice formation, the ice cover may be removed altogether, whereupon the stratification overturns and the fresh-water in the top layer gets lost. It is suggested that if this happens one year it will contribute to preconditioning the system for ice-free conditions the following year. An essential condition for the analyses, which may be put in question, is the presence of at least some wind generated turbulence and that competing mixing processes, e.g., associated with cabbeling do not become dominating. Observations from the Wedell sea give support to the conclusion that late winter conditions in this area may be well described in terms of a freeze melting stage of development. DOI: 10.1034/j.1600-0870.1993.t01-1-00005.x

Treatment of on/off switches in the adjoint method: FDDA experiments with a simple model

Abstract: The purpose of this study is to illustrate the feasibility of using the adjoint method in the data assimilation procedure, where the assimilation model involves on/off switches associated with physical processes. Although the assimilation model used in this study is very simple, it has the on/off switches that are typical in more sophisticated and realistic assimilation models. The calculus of variations is used to confirm that variational FDDA using the adjoint method allows the equations of the assimilation model to have a finite number of first-order discontinuous points. These points represent the on/off switches for which the Jacobian matrix of the model equation may not exist. In practice, when the on/off switches are involved, the gradient of the functional can be obtained by assuming that an infinitesimally small initial perturbation does not change the time interval within which the switching takes place. The time when the corresponding switching in the adjoint equations is performed is the same as that in the forward integration of the model equations, i.e., the switching time is determined by the basic state. Numerical experiments are then conducted using the aforementioned ideas in a simple one-dimensional convection model involving on/off switches associated with latent heat release. The implication of the numerical experiments is that there is no difficulty, in theory, in treating the on/off switches in the adjoint of the assimilation model. Rather, the success of the adjoint method is dependent on the numerical aspects of the implementation, such as the proper scaling of the variables.

Retrieval of balanced fields: an optimal control method

Abstract: The nonlinear balance equation links the stream function and the geopotential in a steady-state approximation of the atmosphere. It has been widely used to retrieve coherent wind and mass fields in limited areas under the assumptions that one of these fields was known without error within the domain and the other partially known on the boundary. The method which is proposed in this paper is based on optimal control techniques; it permits one to alleviate the above assumptions. In a first part of the paper, we describe an iterative algorithm, the geopotential and mass term of the balance equation are adjusted, and the boundary conditions are controlled in such a way that the discrepancy between the observations and a solution of the balance equation is minimized. The convergence of the method is discussed and a numerical example is carried out. The nonlinear balance equation uses the wind fields through the stream function. Of course, only the wind fields are observed, and the second part is devoted to an algorithm which permits one to compute the stream function and the potential velocity from wind observations in a limited area. The method is also based on the control of the boundary conditions. DOI: 10.1034/j.1600-0870.1993.00009.x

Diurnal variation of the outgoing long-wave radiation: as revealed by the E.R.B.E.

Abstract: The diurnal variation of the Outgoing Longwave Radiation (OLR) and its annual variation were studied using scanner products from the Earth Radiation Budget Experiment (ERBE). Data from the combination of sun synchronous polar orbiter NOAA-9 and mid-inclined precessing orbit Earth Radiation Budget Satellite (ERBS), which provide the best diurnal global coverage, were used for the period of March 1985 to February 1986. Harmonic analysis was performed on monthly mean hourly OLR for both average sky and clear sky conditions in order to obtain the amplitude and phase of the OLR diurnal cycle. Large diurnal amplitudes are noticed mostly over desert regions. Over oceans, convectively active regions like the Inter Tropical Convergence Zone (ITCZ) and the South Pacific Convergence Zone (SPCZ) show noticeable diurnal variation compared to the rest of the regions. Annual variation of the OLR diurnal amplitude was compared with annual variation of the various earth radiation budget parameters. A good relationship was found between the OLR diurnal amplitude and the monthly mean net radiation and incoming solar radiation.

Generation of ringlets

Abstract: A mechanism for the generation of ringlets (i.e., small cyclonic eddies (20–40 km) which have recently been observed on the periphery of warm-core rings) is proposed. The suggested process is examined analytically using a reduced gravity one-and-a-half-layer and two-and-a-half-layer model. The underlying hypothesis is that the ringlets are formed by the expulsion of fluid from the outer rim of the warm ring and that this expulsion is the result of an absorption of foreign water into the core of the ring. This nonlinear process is examined using the not-so-frequently used integrated angular momentum constraint as well as the familiar conservation of potential vorticity and mass. These constraints show that when the ring interacts with other bodies of water such as shelf water or the Gulf Stream, the shelf water is sucked into the ring in such a manner that the entire ring is capped. To conserve angular momentum, some other fluid must then be pushed out and it is argued that via instability of the ring's edge, this expulsion forms ringlets. DOI: 10.1034/j.1600-0870.1993.t01-3-00005.x