We study the scaling laws of stratified shear flows by performing high-resolution numerical simulations of inviscid compressible turbulence induced by Kelvin–Helmholtz instability.

We perform five sets of numerical experiments to examine the evolution processes of mode-2 internal solitary waves (ISWs) modulated by background shear currents. Three distinctly different shear-induced waves were identified as forward-propagating long waves, oscillating tails and amplitude-modulated wave packets. The background shear currents are found to play an important role for the short-lived nature and energy decay process of mode-2 ISWs observed previously by Shroyer et al. (2010).

This paper deals with the effects of nonlinearity in a state-of-the-art atmospheric global data assimilation system. It is shown that these effects have become increasingly important over the years due to increased model resolution and use of nonlinear observations. The ability to deal with nonlinearities has thus become a crucial asset of data assimilation algorithms. At ECMWF this is done in a perturbative fashion. Advantages and limitations of this technique are discussed.

We investigate the modeling of the effects of the unresolved scales on the large scales of the coupled ocean–atmosphere model MAOOAM. Two different physically based stochastic methods are considered and compared, in various configurations of the model. Both methods show remarkable performances and are able to model fundamental changes in the model dynamics. Ways to improve the parameterizations' implementation are also proposed.

We consider the nonlinear model well known in geophysics for deep water surface gravity waves – the envelope version of the Dyachenko–Zakharov equation. This model predicts that waves can propagate as a stable localized groups – solitons. We study numerically in detail the soliton collisions and find fundamentally different effects when compared to the previously known results. We demonstrate the formation of extreme amplitude waves that may cause serious damage appearing in seas and oceans.

Constructing good parametrizations is key when studying multi-scale systems. We consider a low-order model and derive a parametrization via a recently developed statistical mechanical approach. We show how the method allows for seamlessly treating the case when the unresolved dynamics is both faster and slower than the resolved one. We test the skill of the parametrization by using the formalism of the Wasserstein distance, which allows for measuring how different two probability measures are.

The problem on internal waves in a weakly stratified two-layered fluid is studied semi-analytically. We discuss the 2.5-layer fluid flows with exponential stratification of both layers. The long-wave model describing travelling waves is constructed by means of a scaling procedure with a small Boussinesq parameter. It is demonstrated that solitary-wave regimes can be affected by the Kelvin–Helmholtz instability arising due to interfacial velocity shear in upstream flow.

Laboratory and numerical experiments are conducted to investigate stem waves due to incidence of monochromatic waves. For larger-amplitude waves with smaller angle of incidence, the measured data clearly show stem waves. The resonant interactions between the incident and reflected waves predicted for solitary waves are not observed for the periodic Stokes waves. The existence and the properties of stem waves found theoretically via simulations are favorably supported by the physical experiments.

The forecast of weather prediction uncertainty is a real challenge and is crucial for risk management. However, uncertainty prediction is beyond the capacity of supercomputers, and improvements of the technology may not solve this issue. A new uncertainty prediction method is introduced which takes advantage of fluid equations to predict simple quantities which approximate real uncertainty but at a low numerical cost. A proof of concept is shown by an academic model derived from fluid dynamics.

The predictability of weather models is limited largely by the initial state error growth or decay rates. We have computed these rates for PUMA, a global model for the atmosphere, and MAOOAM, a more simplified, coupled model which includes the ocean. MAOOAM has processes at distinct timescales, whereas PUMA surprisingly does not. We propose a new programme to compute the natural directions along the flow that correspond to the growth or decay rates, to learn which components play a role.

The technique of microseismic source location is realized by calculating the time difference of arrival estimation of each mocroseismic signal. The time difference of arrival estimation directly affects the accuracy of microseismic source location. Microseismic signal with noises is proved to be closer to the α-stable distribution in this paper, and an improved time difference of arrival estimation approach microseismic signals is proposed.

The Utsu productivity law, one of the main relationships in seismicity statistics, gives the average number of aftershocks produced by a mainshock of a given magnitude. I demonstrate that the law can be formulated in the solid seismicity theory, where it is parameterized in terms of aftershock density within a geometrical solid, constrained by the mainshock size. This suggests that aftershocks can be studied by applying simple rules of analytic geometry on a static stress field.

Fractals are self-similar objects (which look the same at all scales), whose dimensions can be noninteger. They are mathematical concepts, useful to describe various physical systems, as the fractal dimension is a measure of their complexity. In this paper we study how these concepts can be applied to some problems in space plasmas, such as the activity of the Earth's magnetosphere, simulations of plasma turbulence, or identification of magnetic structures ejected from the Sun.

Accuracy of numerical weather prediction forecasts is strongly related to the quality of initial conditions employed. To improve them, it seems advantageous to use radar reflectivity observations because of their high spatial and temporal resolution. This is tested in a high-resolution model whose domain covers Italy. Results show that the employment of reflectivity observations improves precipitation forecast accuracy, but the positive impact is lost after a few hours of forecast.

This study presents, to our knowledge, the first data assimilation experiments in the Sea of Marmara. We propose a FerryBox network for monitoring the state of the sea and show that assimilation of the temperature and salinity improves the forecasts in the basin. The flow of the Bosphorus helps to propagate the error reduction. The study can be taken as a step towards a marine forecasting system in the Sea of Marmara that will help to improve the forecasts in the adjacent Black and Aegean seas.

We present a nonlinear analytic solution for barotropic flow relevant to the oceanographic slope region. A similarity approach is adopted and the solution takes the form of a Lambert W-function. A more general class of linear solutions is also discussed which take the form of error functions. The equations solved are similar to the heat equation and thus the results may be of interest beyond the geophysical community.

We develop a general framework for the frequency analysis of irregularly sampled time series. We also design a test of significance against a general background noise which encompasses the Gaussian white or red noise. Our results generalize and unify methods developed in the fields of geosciences, engineering, astronomy and astrophysics. All the analysis tools presented in this paper are available to the reader in the Python package WAVEPAL.

There is so far no general framework for handling the continuous wavelet transform when the time sampling is irregular. Here we provide such a framework with the Morlet wavelet, based on the results of part I of this study. We also design a test of significance against a general background noise which encompasses the Gaussian white or red noise. All the analysis tools presented in this article are available to the reader in the Python package WAVEPAL.

Data assimilation looks for an optimal way to learn from observations of a dynamical system to improve the quality of its predictions. The goal is to filter out the noise (both observation and model noise) to retrieve the true signal. Among all possible methods, particle filters are promising; the method is fast and elegant, and it allows for a Bayesian analysis. In this review paper, we discuss implementation techniques for (local) particle filters in high-dimensional systems.

Accurate estimation of subsurface geological parameters is essential for the oil industry. This is done by combining observations with an estimation from a model. Ensemble Kalman filter is a widely used method for inverse modeling, while ensemble transform particle filtering is a recently developed method that has been applied to estimate only a small number of parameters and in fluids. We show that for a high-dimensional inverse problem it is superior to an ensemble Kalman filter.

We propose a novel approach to solve conditional nonlinear optimal perturbation for identifying sensitive areas for tropical cyclone adaptive observations. This method is free of adjoint models and overcomes two obstacles, not having adjoint models and having dimensions higher than the problem space. All experimental results prove that it is a meaningful and effective method for solving CNOP and provides a new way for such research. This work aims to solve CNOP and identify sensitive areas.

A four-dimensional nonlinear spectral ocean model is used to study the transition to chaos induced by periodic forcing in systems that are nonchaotic in the autonomous limit. The analysis makes use of ensemble simulations and of the system's pullback attractors. A new diagnostic method characterizes the transition to chaos: this is found to occur abruptly at a critical value and begins with the intermittent emergence of periodic oscillations with distinct phases.

Using the framework Lyapunov vectors, we analyze the asymptotic properties of ensemble based Kalman filters and how these are influenced by dynamical chaos, especially in the context of random model errors and small ensemble sizes. Particularly, we show a novel derivation of the evolution of forecast uncertainty for ensemble-based Kalman filters with weakly-nonlinear error growth, and discuss its impact for filter design in geophysical models.

Ordinarily the climatic impact of systematic variations of parameters arising from anthropogenic effects is addressed on the basis of large numerical models, where parameters are set to a prescribed level and the system is subsequently left to relax. We have revisited the problem from a nonlinear dynamics perspective in which the time variation of parameters is fully incorporated into the evolution laws. Some universal trends of the response have been identified.

EnsVAR is fundamentally successful in that, even in conditions where Bayesianity cannot be expected, it produces ensembles which possess a high degree of statistical reliability. In non-linear strong-constraint cases, EnsVAR has been successful here only through the use of quasi-static variational assimilation. In the weak-constraint case, without QSVA, EnsVAR provided new evidence as to the favourable effect.

Ensemble variational assimilation (EnsVAR) has been implemented on two small-dimension non-linear chaotic toy models, as well as on a linearized version of those models. In the linear case, EnsVAR is exactly Bayesian and produced highly reliable ensembles. In the non-linear case, EnsVAR, implemented on temporal windows on the order of magnitude of the predictability time of the systems, shows as good performance as in the exactly linear case. EnsVar is as good an estimator as EnKF and PF.

Strong internal waves have a significant influence on underwater marine environment and off-shore engineering facilities. They induce noticeable currents and take part in the processes of mixing of water layers, suspension and transport of sediment particles, and formation of sea bottom relief. We consider probability of emergence of large-amplitude internal waves on the basis of instrumental measurements of internal wave fields in five different regions of the World Ocean.

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to find unknown parameters of the model. The observation data, and hence the optimal solution, may contain uncertainties. A response function is considered as a functional of the optimal solution after assimilation. The sensitivity of the response function to the observation data is studied. The results are relevant for monitoring and prediction of sea and ocean states.

Many applications in science require that computational models and data be combined. In a Bayesian framework, this is usually done by defining likelihoods based on the mismatch of model outputs and data. However, matching model outputs and data in this way can be unnecessary or impossible. This issue can be addressed by selecting features of the data, and defining likelihoods based on the features, rather than by the usual mismatch of model output and data.

Hurricanes are complex systems that carry large amounts of energy. Its impact produces, most of the time, natural disasters involving the loss of human lives and of materials and infrastructure that is accounted for in billions of US dollars. Not everything is negative as hurricanes are the main source of rainwater for the regions where they develop. In this study we make a nonlinear analysis of the time series obtained from 1749 to 2012 of the hurricane occurrence in the Gulf of Mexico.

This study generalizes a paper by Pires et al. (1996) to state-of-the-art data assimilation techniques, such as the iterative ensemble Kalman smoother (IEnKS). We show that the longer the time window over which observations are assimilated, the better the accuracy of the IEnKS. Beyond a critical time length that we estimate, we show that this accuracy finally degrades. We show that the use of the quasi-static minimizations but generalized to the IEnKS yields a significantly improved accuracy.

In this paper we investigate the spatiotemporal properties of waves in the Lorenz-96 model. In particular, we explain how these properties are related to the presence of Hopf and pitchfork bifurcations. We also explain bifurcation scenarios by which multiple stable waves can coexist for the same parameter values.

The use of groundwater as a low-enthalpy geothermal resource for heating and cooling of buildings and for agricultural and industrial processes is growing. The study has increased the understanding of heat transfer processes in the subsurface, encouraging the investigation into how further parameters affect the amount of energy that can be stored and dissipated. This is important to maximize the efficiency and minimize the environmental impact of the geothermal installations in groundwater.

Trajectory encounter volume – the volume of fluid that passes close to a reference fluid parcel over some time interval – has been recently introduced as a measure of mixing potential of a flow. We derived the analytical relationship between the encounter volume and diffusivity, which is the most commonly used characteristic of turbulent eddy diffusion. When applied to the altimetric velocities in the Gulf Stream region, the method illuminated transport properties of the Gulf Stream rings.

We devise a model to investigate the interplay between two common phenomena affecting the evolution of a seismogenic fault: the occurrence of earthquakes on neighbouring faults and the partial degree of anelasticity of rocks in the upper mantle, a feature that is often manifested by a post-seismic process known as viscoelastic relaxation. We show how this process dramatically complicates the way an earthquake may alter the magnitude and timing of a seismic event on the perturbed fault.

We have used numerical simulations to investigate the impact that rotation has on large waves existing internally in the ocean. In coastal regions these waves become trapped along the coast because of rotation. We have found that this trapping results in an adjustment of the form of the waves. The adjustment leads to heightened mixing along the coast, which has implications for nutrient and chemical distribution.

The ensemble forecast sensitivity to observation (EFSO) method can efficiently clarify under what conditions observations are beneficial or detrimental for assimilation. Based on EFSO, an offline assimilation method is proposed to accelerate the development of data selection strategies for new observing systems. The usefulness of this method is demonstrated with the assimilation of global satellite precipitation data.