The accurate representation of geostrophic balance is an essential requirement for numerical modelling of geophysical flows. Significant effort is often put into the selection of accurate or optimal balance representation by the discretisation of the fundamental equations. The issue of accurate balance representation is particularly challenging when applying dynamic mesh adaptivity, where there is potential for additional imbalance injection when interpolating to new, optimised meshes.
In the context of shallow-water modelling, we present a new method for preservation of geostrophic balance when applying dynamic mesh adaptivity. This approach is based upon interpolation of the Helmholtz decomposition of the Coriolis acceleration. We apply this in combination with a discretisation for which states in geostrophic balance are exactly steady solutions of the linearised equations on an f-plane; this method guarantees that a balanced and steady flow on a donor mesh remains balanced and steady after interpolation onto an arbitrary target mesh, to within machine precision. We further demonstrate the utility of this interpolant for states close to geostrophic balance, and show that it prevents pollution of the resulting solutions by imbalanced perturbations introduced by the interpolation.
A single-layer, quasi-geostrophic (QG), large-scale ocean circulation model is developed in this paper to study available ocean current energy potentials harnessed by using the ocean current turbines. Power extraction is modeled by adding a parameterized Rayleigh friction term in the barotropic vorticity equation. Numerical assessments are performed by simulating a set of mid-latitude ocean basins in the beta plane, which are standard prototypes of more realistic ocean dynamics considering inter-decadal variability in turbulent equilibrium. A sensitivity analysis with respect to the turbine pa…
We consider a 2-layer quasi-geostrophic ocean model where the upper layer is forced by a steady Kolmogorov wind stress in a periodic channel domain, which allows to mathematically study the nonlinear development of the resulting flow. The model supports a steady parallel shear flow as a response to the wind stress. As the maximal velocity of the shear flow (equivalently the maximal amplitude of the wind forcing) exceeds a critical threshold, the zonal jet destabilizes due to baroclinic instability and we numerically demonstrate that a first transition occurs. We obtain reduced equations of the…
The framework of invariant parameterization is extended to higher-order closure schemes. We also define, for the first time, generalized invariant parameterization schemes, where symmetries of the corresponding original model are preserved as equivalence transformations of related classes of closed system of differential equations. As a particular problem, we consider invariant parameterization schemes for geostrophic eddies in a barotropic ocean. Here the initial model is the barotropic vorticity equation, which is equivalent to the system of incompressible inviscid two-dimensional Euler equa…
A physically plausible lower bound on the spatially varying geostrophic mesoscale eddy energy dissipation time-scale within the ocean, related to the geographical energy transfer rate out of the geostrophic mesoscales, is provided by means of a simple and computational inexpensive inverse calculation. Data diagnosed from a high resolution global configuration ocean simulation is supplied to a parameterized model of the geostrophic mesoscale eddy energy, from which the dissipation time-scale results as a solution to an optimization calculation. We find that the dissipation time-scale is shortes…
A general method for the derivation of asymptotic nonlinear shallow water and deep water models is presented. Starting from a general dimensionless version of the water-wave equations, we reduce the problem to a system of two equations on the surface elevation and the velocity potential at the free surface. These equations involve a Dirichlet-Neumann operator and we show that all the asymptotic models can be recovered by a simple asymptotic expansion of this operator, in function of the shallowness parameter (shallow water limit) or the steepness parameter (deep water limit). Based on this met…
A strictly geostrophic product of sea-surface velocities from the SWOT fast-sampling phase
Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Université Grenoble Alpes, CNRS, INRAE, IRD, Grenoble INP, Institut des Géosciences de l’Environnement（格勒诺布尔阿尔卑斯大学）；Department of Earth and Environmental Sciences, University of Michigan（密歇根大学地球与环境科学系）；Naval Research Laboratory（海军研究实验室）；Center for Ocean-Atmospheric Prediction Studies, Florida State University（佛罗里达州立大学海洋大气预测研究中心）
We present an extensive numerical comparison of a family of balance models appropriate to the semi-geostrophic limit of the rotating shallow water equations, and derived by variational asymptotics in Oliver (2006) for small Rossby numbers ${\mathrm{Ro}}$. This family of generalized large-scale semi-geostrophic (GLSG) models contains the $L_1$-model introduced by Salmon (1983) as a special case. We use these models to produce balanced initial states for the full shallow water equations. We then numerically investigate how well these models capture the dynamics of an initially balanced shallow w…
The daily cycle of heating and cooling of the near-surface ocean may be quite different in a shallow lagoon with a few meters deep seafloor that can be heated directly by the sun. If important, the solar radiation will affect the local benthic communities. To study the physical processes associated with the daily cycle of south-Pacific lagoon Bora Bora, a vertical string of five high-resolution temperature sensors was moored at a 2-m deep site for 3 weeks. Besides the standard ocean warming (approximately during daytime) and cooling (approximately nighttime), the sensors show relatively highes…
Adjoint-based online learning of two-layer quasi-geostrophic baroclinic turbulence
Hong Kong University of Science and Technology（香港科技大学）；Institut de Physique du Globe de Paris（巴黎地球物理研究所）；Université Grenoble-Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；National Oceanography Centre, Southampton（南安普顿国家海洋中心）；Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）
Salmon's nearly geostrophic model for rotating shallow-water flow is derived in full spherical geometry. The model, which results upon constraining the velocity field to the height field in Hamilton's principle for rotating shallow-water dynamics, constitutes an important prototype of Hamiltonian balanced models. Instead of Salmon's original approach, which consists in taking variations of particle paths at fixed Lagrangian labels and time, Holm's approach is considered here, namely variations are taken on Lagrangian particle labels at fixed Eulerian positions and time. Unlike the classical qu…
We analyze shallow water wind waves in Currituck Sound, North Carolina and experimentally confirm, for the first time, the presence of $soliton$ $turbulence$ in ocean waves. Soliton turbulence is an exotic form of nonlinear wave motion where low frequency energy may also be viewed as a $dense$ $soliton$ $gas$, described theoretically by the soliton limit of the Korteweg-deVries (KdV) equation, a $completely$ $integrable$ $soliton$ $system$: Hence the phrase "soliton turbulence" is synonymous with "integrable soliton turbulence." For periodic/quasiperiodic boundary conditions the $ergodic$ $sol…
A systematic method to derive the Hamiltonian and Nambu form for the shallow water equations, using the conservation for energy and potential enstrophy, is presented. Different mechanisms, such as vortical flows and emission of gravity waves, emerge from different conservation laws (CLs) for total energy and potential enstrophy. The equations are constructed using exterior differential forms and self-adjoint operators and result in the sum of two Nambu brackets, one for the vortical flow and one for the wave-mean flow interaction, and a Poisson bracket representing the interaction between dive…
HClimRep-Ocean: A Global Ocean Emulator on an Unstructured Mesh
Alfred Wegener Institute, Helmholtz Centre for Polar and Marine Research（阿尔弗雷德·魏格纳亥姆霍兹极地与海洋研究中心）；Forschungszentrum Jülich GmbH（于利希研究中心）；Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；European Center for Medium-Range Weather Forecasts（欧洲中期天气预报中心）；University of Cologne（科隆大学）；University of Bremen（不来梅大学）
Lagrangian fronts (LF) in the ocean delineate boundaries between surface waters with different Lagrangian properties. They can be accurately detected in a given velocity field by computing synoptic maps of the drift of synthetic tracers and other Lagrangian indicators. Using Russian ship's catch and location data for a number of commercial fishery seasons in the region of the northwest Pacific with one of the richest fishery in the world, it is shown statistically that the saury fishing grounds with maximal catches are not randomly distributed over the region but located mainly along those LFs…
In an effort to study the applicability of adaptive mesh refinement (AMR) techniques to atmospheric models an interpolation-based spectral element shallow water model on a cubed-sphere grid is compared to a block-structured finite volume method in latitude-longitude geometry. Both models utilize a non-conforming adaptation approach which doubles the resolution at fine-coarse mesh interfaces. The underlying AMR libraries are quad-tree based and ensure that neighboring regions can only differ by one refinement level. The models are compared via selected test cases from a standard test suite for …
The aim of this paper is to use large deviation theory in order to compute the entropy of macrostates for the microcanonical measure of the shallow water system. The main prediction of this full statistical mechanics computation is the energy partition between a large scale vortical flow and small scale fluctuations related to inertia-gravity waves. We introduce for that purpose a discretized model of the continuous shallow water system, and compute the corresponding statistical equilibria. We argue that microcanonical equilibrium states of the discretized model in the continuous limit are equ…
