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Peter Jan van Leeuwen

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Peter Jan van Leeuwen

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The propagation velocity and propagation mechanism for vortices on a β plane are determined for a reduced-gravity model by integrating the momentum equations over the β plane. Isolated vortices, vortices in a background current, and initial vortex propagation from rest are studied. The propagation mechanism for isolated anticyclones as well as cyclones, which has been lacking up to now, is presented. It is shown that, to first order, the vortex moves to generate a Coriolis force on the mass anomaly of the vortex to compensate for the force on the vortex due to the variation of the Coriolis parameter. Only the mass anomaly of the vortex is of importance, because the Coriolis force due to the motion of the bulk of the layer moving with the vortex is almost fully compensated by the Coriolis force on the motion of the exterior flow. Because the mass anomaly of a cyclone is negative the force and acceleration have opposite sign. The role of dipolar structures in steadily moving vortices is discussed, and it is shown that their overall structure is fixed by the steady westward motion of the mass anomaly. Furthermore, it is shown that reduced-gravity vortices are not advected with a background flow. The reason for this behavior is that the background flow changes the ambient vorticity gradient such that the vortex obtains an extra self-propagation term that exactly cancels the advection by the background flow. Last, it is shown that a vortex initially at rest will accelerate equatorward first, after which a westward motion is generated. This result is independent of the sign of the vortex.

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Peter Jan van Leeuwen

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A smoother introduced earlier by van Leeuwen and Evensen is applied to a problem in which real observations are used in an area with strongly nonlinear dynamics. The derivation is new, but it resembles an earlier derivation by van Leeuwen and Evensen. Again a Bayesian view is taken in which the prior probability density of the model and the probability density of the observations are combined to form a posterior density. The mean and the covariance of this density give the variance-minimizing model evolution and its errors. The assumption is made that the prior probability density is a Gaussian, leading to a linear update equation. Critical evaluation shows when the assumption is justified. This also sheds light on why Kalman filters, in which the same approximation is made, work for nonlinear models. By reference to the derivation, the impact of model and observational biases on the equations is discussed, and it is shown that Bayes’s formulation can still be used. A practical advantage of the ensemble smoother is that no adjoint equations have to be integrated and that error estimates are easily obtained. The present application shows that for process studies a smoother will give superior results compared to a filter, not only owing to the smooth transitions at observation points, but also because the origin of features can be followed back in time. Also its preference over a strong-constraint method is highlighted. Furthermore, it is argued that the proposed smoother is more efficient than gradient descent methods or than the representer method when error estimates are taken into account.

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Peter Jan van Leeuwen

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The application of particle filters in geophysical systems is reviewed. Some background on Bayesian filtering is provided, and the existing methods are discussed. The emphasis is on the methodology, and not so much on the applications themselves. It is shown that direct application of the basic particle filter (i.e., importance sampling using the prior as the importance density) does not work in high-dimensional systems, but several variants are shown to have potential. Approximations to the full problem that try to keep some aspects of the particle filter beyond the Gaussian approximation are also presented and discussed.

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Erik van Sebille and Peter Jan van Leeuwen

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The adiabatic transit time of wave energy radiated by an Agulhas ring released in the South Atlantic Ocean to the North Atlantic Ocean is investigated in a two-layer ocean model. Of particular interest is the arrival time of baroclinic energy in the northern part of the Atlantic, because it is related to variations in the meridional overturning circulation. The influence of the Mid-Atlantic Ridge is also studied, because it allows for the conversion from barotropic to baroclinic wave energy and the generation of topographic waves. Barotropic energy from the ring is present in the northern part of the model basin within 10 days. From that time, the barotropic energy keeps rising to attain a maximum 500 days after initiation. This is independent of the presence or absence of a ridge in the model basin. Without a ridge in the model, the travel time of the baroclinic signal is 1300 days. This time is similar to the transit time of the ring from the eastern to the western coast of the model basin. In the presence of the ridge, the baroclinic signal arrives in the northern part of the model basin after approximately 10 days, which is the same time scale as that of the barotropic signal. It is apparent that the ridge can facilitate the energy conversion from barotropic to baroclinic waves and the slow baroclinic adjustment can be bypassed. The meridional overturning circulation, parameterized in two ways as either a purely barotropic or a purely baroclinic phenomenon, also responds after 1300 days. The ring temporarily increases the overturning strength. The presence of the ridge does not alter the time scales.

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Geir Evensen and Peter Jan van Leeuwen

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It is formally proved that the general smoother for nonlinear dynamics can be formulated as a sequential method, that is, observations can be assimilated sequentially during a forward integration. The general filter can be derived from the smoother and it is shown that the general smoother and filter solutions at the final time become identical, as is expected from linear theory. Then, a new smoother algorithm based on ensemble statistics is presented and examined in an example with the Lorenz equations. The new smoother can be computed as a sequential algorithm using only forward-in-time model integrations. It bears a strong resemblance with the ensemble Kalman filter. The difference is that every time a new dataset is available during the forward integration, an analysis is computed for all previous times up to this time. Thus, the first guess for the smoother is the ensemble Kalman filter solution, and the smoother estimate provides an improvement of this, as one would expect a smoother to do. The method is demonstrated in this paper in an intercomparison with the ensemble Kalman filter and the ensemble smoother introduced by van Leeuwen and Evensen, and it is shown to be superior in an application with the Lorenz equations. Finally, a discussion is given regarding the properties of the analysis schemes when strongly non-Gaussian distributions are used. It is shown that in these cases more sophisticated analysis schemes based on Bayesian statistics must be used.

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Femke C. Vossepoel and Peter Jan van Leeuwen

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This paper presents a first attempt to estimate mixing parameters from sea level observations using a particle method based on importance sampling. The method is applied to an ensemble of 128 members of model simulations with a global ocean general circulation model of high complexity. Idealized twin experiments demonstrate that the method is able to accurately reconstruct mixing parameters from an observed mean sea level field when mixing is assumed to be spatially homogeneous. An experiment with inhomogeneous eddy coefficients fails because of the limited ensemble size. This is overcome by the introduction of local weighting, which is able to capture spatial variations in mixing qualitatively. As the sensitivity of sea level for variations in mixing is higher for low values of mixing coefficients, the method works relatively well in regions of low eddy activity.

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Melanie Ades and Peter Jan van Leeuwen

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The disadvantage of the majority of data assimilation schemes is the assumption that the conditional probability density function of the state of the system given the observations [posterior probability density function (PDF)] is distributed either locally or globally as a Gaussian. The advantage, however, is that through various different mechanisms they ensure initial conditions that are predominantly in linear balance and therefore spurious gravity wave generation is suppressed.

The equivalent-weights particle filter is a data assimilation scheme that allows for a representation of a potentially multimodal posterior PDF. It does this via proposal densities that lead to extra terms being added to the model equations and means the advantage of the traditional data assimilation schemes, in generating predominantly balanced initial conditions, is no longer guaranteed.

This paper looks in detail at the impact the equivalent-weights particle filter has on dynamical balance and gravity wave generation in a primitive equation model. The primary conclusions are that (i) provided the model error covariance matrix imposes geostrophic balance, then each additional term required by the equivalent-weights particle filter is also geostrophically balanced; (ii) the relaxation term required to ensure the particles are in the locality of the observations has little effect on gravity waves and actually induces a reduction in gravity wave energy if sufficiently large; and (iii) the equivalent-weights term, which leads to the particles having equivalent significance in the posterior PDF, produces a change in gravity wave energy comparable to the stochastic model error. Thus, the scheme does not produce significant spurious gravity wave energy and so has potential for application in real high-dimensional geophysical applications.

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Geir Evensen and Peter Jan van Leeuwen

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The ring-shedding process in the Agulhas Current is studied using the ensemble Kalman filter to assimilate Geosat altimeter data into a two-layer quasigeostrophic ocean model. The properties of the ensemble Kalman filter are further explored with focus on the analysis scheme and the use of gridded data. The Geosat data consist of 10 fields of gridded sea surface height anomalies separated 10 days apart that are added to a climatic mean field. This corresponds to a huge number of data values, and a data reduction scheme must be applied to increase the efficiency of the analysis procedure. Further, it is illustrated how one can resolve the rank problem occurring when a too large dataset or a small ensemble is used.

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Peter Jan van Leeuwen and Geir Evensen

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The weak constraint inverse for nonlinear dynamical models is discussed and derived in term of a probabilistic formulation. The well-known result that for Gaussian error statistics the minimum of the weak constraint inverse is equal to the maximum-likelihood estimate is rederived. Then several methods based on ensemble statistics that can be used to find the smoother (as opposed to the filter) solution are introduced and compared to traditional methods. A strong point of the new methods is that they avoid the integration of adjoint equations, which is a complex task for real oceanographic or atmospheric applications. They also avoid iterative searches in a Hilbert space, and error estimates can be obtained without much additional computational effort. The feasibility of the new methods is illustrated in a two-layer quasigeostrophic ocean model.

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