For modelling geophysical systems, large-scale processes are described through a set of coarse-grained dynamical equations while small-scale processes are represented via parameterizations. This work proposes a method for identifying the best possible stochastic parameterization from noisy data. State-of-the-art sequential estimation methods such as Kalman and particle filters do not achieve this goal successfully because both suffer from the collapse of the posterior distribution of the parameters. To overcome this intrinsic limitation, we propose two statistical learning methods. They are based on the combination of the ensemble Kalman filter (EnKF) with either the expectation–maximization (EM) or the Newton–Raphson (NR) used to maximize a likelihood associated to the parameters to be estimated. The EM and NR are applied primarily in the statistics and machine learning communities and are brought here in the context of data assimilation for the geosciences. The methods are derived using a Bayesian approach for a hidden Markov model and they are applied to infer deterministic and stochastic physical parameters from noisy observations in coarse-grained dynamical models. Numerical experiments are conducted using the Lorenz-96 dynamical system with one and two scales as a proof of concept. The imperfect coarse-grained model is modelled through a one-scale Lorenz-96 system in which a stochastic parameterization is incorporated to represent the small-scale dynamics. The algorithms are able to identify the optimal stochastic parameterization with good accuracy under moderate observational noise. The proposed EnKF-EM and EnKF-NR are promising efficient statistical learning methods for developing stochastic parameterizations in high-dimensional geophysical models.

Using a variational technique, middle atmosphere gravity wave drag (GWD) is estimated from Met Office middle atmosphere analyses for the year 2002. The technique employs an adjoint model of a middle atmosphere dynamical model to minimize a cost function that measures the differences between the model state and observations. The control variables are solely the horizontal components of GWD; therefore, the minimization determines the optimal estimate of the drag. For each month, Met Office analyses are taken as the initial condition for the first day of the month, and also as observations for each successive day. In this way a three-dimensional GWD field is obtained for the entire year with a temporal resolution of 1 day. GWD shows a pronounced seasonal cycle. During solstices, there are deceleration regions of the polar jet centered at about 63° latitude in the winter hemisphere, with a peak of 49 m s−1 day−1 at 0.24 hPa in the Southern Hemisphere; the summer hemisphere also shows a deceleration region but much weaker, with a peak of 24 m s−1 day−1 centered at 45° latitude and 0.6 hPa. During equinoxes GWD is weak and exhibits a smooth transition between the winter and summer situation. The height and latitude of the deceleration center in both winter and summer hemispheres appear to be constant. Important longitudinal dependencies in GWD are found that are related to planetary wave activity; GWD intensifies in the exit region of jet streaks. In the lower tropical stratosphere, the estimated GWD shows a westward GWD descending together with the westward phase of the quasi-biennial oscillation. Above, GWD exhibits a semiannual pattern that is approximately out of phase with the semiannual oscillation in the zonal wind. Furthermore, a descending GWD pattern is found at those heights, similar in magnitude and sign to that in the lower stratosphere.

Oceanic and atmospheric global numerical models represent explicitly the large-scale dynamics while the smaller-scale processes are not resolved, so that their effects in the large-scale dynamics are included through subgrid-scale parametrizations. These parametrizations represent small-scale effects as a function of the resolved variables. In this work, data assimilation principles are used not only to estimate the parameters of subgrid-scale parametrizations but also to uncover the functional dependencies of subgrid-scale processes as a function of large-scale variables. Two data assimilation methods based on the ensemble transform Kalman filter (ETKF) are evaluated in the two-scale Lorenz '96 system scenario. The first method is an online estimation which uses the ETKF with an augmented space state composed of the model large-scale variables and a set of unknown global parameters from the parametrization. The second method is an offline estimation which uses the ETKF to estimate an augmented space state composed of the large-scale variables and by a space-dependent model error term. Then a polynomial regression is used to fit the estimated model error as a function of the large-scale model variables in order to develop a parametrization of small-scale dynamics. The online estimation shows a good performance when the parameter-state relationship is assumed to be a quadratic polynomial function. The offline estimation captures better some of the highly nonlinear functional dependencies found in the subgrid-scale processes. The nonlinear and non-local dependence found in an experiment with shear-generated small-scale dynamics is also recovered by the offline estimation method. Therefore, the combination of these two methods could be a useful tool for the estimation of the functional form of subgrid-scale parametrizations.

An estimate of the three-dimensional gravity-wave drag field derived from middle-atmosphere analyses is presented. It is obtained using a novel technique based on variational data assimilation princi...