Due to their limited resolution, numerical ocean models need to be interpreted as representing filtered or averaged equations. How to interpret models in terms of formally averaged equations, however, is not always clear, particularly in the case of hybrid or generalized vertical coordinate models. We derive the averaged hydrostatic Boussinesq equations in generalized vertical coordinates for an arbitrary thickness weighted-average. We then consider various special cases and discuss the extent to which the averaged equations are consistent with existing model formulations. As previously discussed, the momentum equations in existing depth-coordinate models are best interpreted as representing Eulerian averages (i.e., averages taken at fixed depth), while the tracer equations can be interpreted as either Eulerian or thickness-weighted isopycnal averages. Instead we find that no averaging is fully consistent with existing formulations of the parameterizations in semi-Lagrangian discretizations of generalized vertical coordinate ocean models. Perhaps the most natural interpretation of generalized vertical coordinate models is to assume that the average follows the model’s coordinate surfaces. However, the existing model formulations are generally not consistent with coordinate-following averages, which would require “coordinate-aware” parameterizations that can account for the changing nature of the eddy terms as the coordinate changes. Alternatively, the model variables can be interpreted as representing either Eulerian or (thickness-weighted) isopycnal averages, independent of the model coordinate that is being used for the numerical discretization. Existing parameterizations in generalized vertical coordinate models, however, are usually not fully consistent with either of these interpretations. We discuss what changes are needed to achieve consistency.

Air-sea flux variability has contributions from both ocean and atmosphere at different spatio-temporal scales. Atmospheric synoptic scales and the air-sea turbulent heat flux that they drive are well represented in climate models, but ocean mesoscales and their associated variability are often not well resolved due to non-eddy-resolving spatial resolutions of current climate models. We deploy a physics-based stochastic subgrid-scale parameterization for ocean density, that reinforces the lateral density variations due to oceanic eddies, and examine its effect on air-sea heat flux variability in a comprehensive coupled climate model. The stochastic parameterization substantially modifies sea surface temperature (SST) and latent heat flux (LHF) variability and their co-variability, primarily at scales near the resolution of the ocean model grid. Enhancement in the SST-LHF anomaly covariance, and correlations, indicate that the ocean-intrinsic component of the air-sea heat flux variability improves with respect to high-resolution satellite observations, especially in Gulf Stream region.

Biased, incomplete models are often used for forecasting states of complex dynamical systems by mapping an estimate of a “true’ initial state into model phase space, making a forecast, and then mapping back to the “true’ space. While advances have been made to reduce errors associated with model initialization and model forecasts, we lack a general framework for discovering optimal mappings between the reference dynamical system and the model phase space. Here, we propose using a data-driven approach to infer these maps. Our approach consistently reduces errors in the Lorenz-96 system with an imperfect model constructed to produce significant model errors compared to a reference configuration. Optimal pre- and post-processing transforms leverage “shocks’ and “drifts’ in the imperfect model to make more skillful forecasts of the reference system. The implemented machine learning architecture using neural networks constructed with a custom analog-adjoint layer makes the approach generalizable to numerous applications.

A document by Nora Loose. Click on the document to view its contents.

Imperfect models are often used for forecasting and state estimation of complex dynamical systems, typically by mapping a reference initial state into model phase space, making a forecast, and then mapping back to the reference space. In many cases these mappings are implicit, and forecast errors thus reflect a combination of model forecast errors and mapping errors. Techniques to infer parameterizations and parameters to reduce model bias have been the subject of intense scrutiny; however, we lack a general framework for discovering optimal mappings between system and model attractors. Here we propose a novel Machine Learning paradigm for inferring cross-attractor transformations (CATs) that minimize forecast error. CATs are pairs of transformations from the phase space of a reference system to the phase space of a model and vice versa that serve as a bridge between the attractors of a true system and an imperfect model. A computationally efficient analog approximation to tangent linear and adjoint models is developed to enable efficient stochastic gradient descent algorithms to train CAT parameters. Neural networks constructed with a custom analog-adjoint layer permit specification of affine transformations as well as more general nonlinear transformations.

Air-sea flux variability has contributions from both ocean and atmosphere at different spatio-temporal scales. Atmospheric synoptic scales and the air-sea turbulent heat flux that they drive are well represented in climate models, but ocean mesoscales and their associated variability are often not well resolved due to non-eddy-resolving spatial resolutions of current climate models. We deploy a physics-based stochastic subgrid-scale parameterization for ocean density, that reinforces the lateral density variations due to oceanic eddies, and examine its effect on air-sea heat flux variability in a comprehensive coupled climate model. The stochastic parameterization substantially modifies sea surface temperature (SST) and latent heat flux (LHF) variability and their correlations, primarily at scales near the resolution of the ocean model grid. Changes in the SST-LHF anomaly correlations indicate that the ocean-intrinsic component of the air-sea heat flux variability improves with respect to the satellite observational product, especially in western boundary current extensions.

We describe a new way to apply a spatial filter to gridded data from models or observations, focusing on low-pass filters. The new method is analogous to smoothing via diffusion, and its implementation requires only a discrete Laplacian operator appropriate to the data. The new method can approximate arbitrary filter shapes, including Gaussian filters, and can be extended to spatially-varying and anisotropic filters. The new diffusion-based smoother's properties are illustrated with examples from ocean model data and ocean observational products. An open-source python package implementing this algorithm, called gcm-filters, is currently under development.

Energy exchanges between large-scale ocean currents and mesoscale eddies play an important role in setting the large-scale ocean circulation but are not fully captured in models. To better understand and quantify the ocean energy cycle, we apply along-isopycnal spatial filtering to output from an isopycnal 1/32$^\circ$ primitive equation model with idealized Atlantic and Southern Ocean geometry and topography. We diagnose the energy cycle in two frameworks: (1) a non-thickness-weighted framework, resulting in a Lorenz-like energy cycle, and (2) a thickness-weighted framework, resulting in the Bleck energy cycle. This paper shows that (2) is the more useful framework for studying energy pathways when an isopycnal average is used. Next, we investigate the Bleck cycle as a function of filter scale. Baroclinic conversion generates mesoscale eddy kinetic energy over a wide range of scales, and peaks near the deformation scale at high latitudes, but below the deformation scale at low latitudes. Away from topography, an inverse cascade transfers kinetic energy from the mesoscales to larger scales. The upscale energy transfer peaks near the energy-containing scale at high latitudes, but below the deformation scale at low latitudes. Regions downstream of topography are characterized by a downscale kinetic energy transfer, in which mesoscale eddies are generated through barotropic instability. The scale- and flow-dependent energy pathways diagnosed in this paper provide a basis for evaluating and developing scale- and flow-aware mesoscale eddy parameterizations.