Lagrangian analysis is a powerful way to analyse the output of ocean circulation models and other ocean velocity data such as from altimetry. In the Lagrangian approach, large sets of virtual particles are integrated within the three-dimensional, time-evolving velocity fields. Over several decades, a variety of tools and methods for this purpose have emerged. Here, we review the state of the art in the field of Lagrangian analysis of ocean velocity data, starting from a fundamental kinematic framework and with a focus on large-scale open ocean applications. Beyond the use of explicit velocity fields, we consider the influence of unresolved physics and dynamics on particle trajectories. We comprehensively list and discuss the tools currently available for tracking virtual particles. We then showcase some of the innovative applications of trajectory data, and conclude with some open questions and an outlook. The overall goal of this review paper is to reconcile some of the different techniques and methods in Lagrangian ocean analysis, while recognising the rich diversity of codes that have and continue to emerge, and the challenges of the coming age of petascale computing

Oceanographic observations are limited by sampling rates, while ocean models are limited by finite resolution and high viscosity and diffusion coefficients. Therefore, both data from observations and ocean models lack information at small and fast scales. Methods are needed to either extract information, extrapolate, or upscale existing oceanographic data sets, to account for or represent unresolved physical processes. Here we use machine learning to leverage observations and model data by predicting unresolved turbulent processes and subsurface flow fields. As a proof of concept, we train convolutional neural networks on degraded data from a high‐resolution quasi‐geostrophic ocean model. We demonstrate that convolutional neural networks successfully replicate the spatiotemporal variability of the subgrid eddy momentum forcing, are capable of generalizing to a range of dynamical behaviors, and can be forced to respect global momentum conservation. The training data of our convolutional neural networks can be subsampled to 10–20% of the original size without a significant decrease in accuracy. We also show that the subsurface flow field can be predicted using only information at the surface (e.g., using only satellite altimetry data). Our results indicate that data‐driven approaches can be exploited to predict both subgrid and large‐scale processes, while respecting physical principles, even when data are limited to a particular region or external forcing. Our in‐depth study presents evidence for the successful design of ocean eddy parameterizations for implementation in coarse‐resolution climate models.

The resolution of climate models is limited by computational cost. Therefore, we must rely on parameterizations to represent processes occurring below the scale resolved by the models. Here, we focus on parameterizations of ocean mesoscale eddies and employ machine learning (ML), namely, relevance vector machines (RVMs) and convolutional neural networks (CNNs), to derive computationally efficient parameterizations from data, which are interpretable and/or encapsulate physics. In particular, we demonstrate the usefulness of the RVM algorithm to reveal closed‐form equations for eddy parameterizations with embedded conservation laws. When implemented in an idealized ocean model, all parameterizations improve the statistics of the coarse‐resolution simulation. The CNN is more stable than the RVM such that its skill in reproducing the high‐resolution simulation is higher than the other schemes; however, the RVM scheme is interpretable. This work shows the potential for new physics‐aware interpretable ML turbulence parameterizations for use in ocean climate models.

Significance: Since the 19th century, rising greenhouse gas concentrations have caused the ocean to absorb most of the Earth’s excess heat and warm up. Before the 1990s, most ocean temperature measurements were above 700 m and therefore, insufficient for an accurate global estimate of ocean warming. We present a method to reconstruct ocean temperature changes with global, full-depth ocean coverage, revealing warming of 436×1021J since 1871. Our reconstruction, which agrees with other estimates for the well-observed period, demonstrates that the ocean absorbed as much heat during 1921–1946 as during 1990–2015. Since the 1950s, up to one-half of excess heat in the Atlantic Ocean at midlatitudes has come from other regions via circulation-related changes in heat transport.

Coupled climate simulations that span several hundred years cannot be run at a high‐enough spatial resolution to resolve mesoscale ocean dynamics. Recently, several studies have considered Deep Learning to parameterize subgrid forcing within macroscale ocean equations using data from ocean‐only simulations with idealized geometry. We present a stochastic Deep Learning parameterization that is trained on data generated by CM2.6, a high‐resolution state‐of‐the‐art coupled climate model. We train a Convolutional Neural Network for the subgrid momentum forcing using macroscale surface velocities from a few selected subdomains with different dynamical regimes. At each location of the coarse grid, rather than predicting a single number for the subgrid momentum forcing, we predict both the mean and standard deviation of a Gaussian probability distribution. This approach requires training our neural network to minimize a negative log‐likelihood loss function rather than the Mean Square Error, which has been the standard in applications of Deep Learning to the problem of parameterizations. Each estimate of the conditional mean subgrid forcing is thus associated with an uncertainty estimate–the standard deviation—which will form the basis for a stochastic subgrid parameterization. Offline tests show that our parameterization generalizes well to the global oceans and a climate with increased levels without further training. We then implement our learned stochastic parameterization in an eddy‐permitting idealized shallow water model. The implementation is stable and improves some statistics of the flow. Our work demonstrates the potential of combining Deep Learning tools with a probabilistic approach in parameterizing unresolved ocean dynamics.