Quasi-geostrophic Model Research Articles

Piecewise Potential Vorticity Inversion without Far-Field Response?

AbstractGiven a flow domain D with subdomains D1 and D2, piecewise potential vorticity inversion (PPVI) inverts a potential vorticity (PV) anomaly in D2 and assumes vanishing PV in D1 where boundary conditions must be taken into account. It is a widely held view that the PV anomaly exerts a far-field influence on D1, which is revealed by PPVI. Tests of this assertion are conducted using a simple quasigeostrophic model where an upper layer D2 contains a PV anomaly and D1 is the layer underneath. This anomaly is inverted. Any downward physical impact of PV in D2 must also be represented in the results of a downward piecewise density inversion (PDI) based on the hydrostatic relation and the density in D2 as following from PPVI. There is no doubt about the impact of the mass in D2 on the flow in the lower layer D1. Thus results of PPVI and PDI have to agree closely. First, PPVI is applied to a locally confined PV anomaly in D2. There is no far-field “response” in D1 if stationarity is imposed. Modifications of boundary conditions lead to “induced” flows in D1 but the results of PPVI and PDI differ widely. This leads to a simple proof that there is no physical far-field influence of PV anomalies in D2. Wave patterns of the streamfunction restricted to D2 are prescribed in a second series of tests. The related PV anomalies are obtained by differentiation and are also confined to D2 in this case. This approach illustrates the basic procedure to derive PV fields from observations which excludes a far-field response.

Lagrangian pair dispersion in upper-ocean turbulence in the presence of mixed-layer instabilities

Turbulence in the upper ocean in the submesoscale range (scales smaller than the deformation radius) plays an important role for the heat exchanges with the atmosphere and for oceanic biogeochemistry. Its dynamics should strongly depend on the seasonal cycle and the associated mixed-layer instabilities. The latter are particularly relevant in winter and are responsible for the formation of energetic small scales that extend over the whole depth of the mixed layer. The knowledge of the transport properties of oceanic flows at depth, which is essential to understand the coupling between surface and interior dynamics, however, is still limited. By means of numerical simulations, we explore the Lagrangian dispersion properties of turbulent flows in a quasi-geostrophic model system allowing for both thermocline and mixed-layer instabilities. The results indicate that, when mixed-layer instabilities are present, the dispersion regime is local from the surface down to depths comparable with that of the interface with the thermocline, while in their absence dispersion quickly becomes nonlocal with depth. We then identify the origin of such behavior in the existence of fine-scale energetic structures due to mixed-layer instabilities. We further discuss the effect of vertical shear on the Lagrangian particle spreading and address the correlation between the dispersion properties at the surface and at depth, which is relevant to assess the possibility of inferring the dynamical features of deeper flows from the more accessible surface ones.

A Multiscale Local Gain Form Ensemble Transform Kalman Filter (MLGETKF)

AbstractA new multiscale, ensemble-based data assimilation (DA) method, multiscale local gain form ensemble transform Kalman filter (MLGETKF), is introduced. MLGETKF allows simultaneous update of multiple scales for both the ensemble mean and perturbations through assimilating all observations at once. MLGETKF performs DA in independent local volumes, which lends the algorithm a high degree of computational scalability. The multiscale analysis is enabled through the rapid creation of many pseudoensemble perturbations via a multiscale ensemble modulation procedure. The Kalman gain that is used to update the raw background ensemble mean and perturbations is based on this modulated ensemble, which intrinsically includes multiscale model space localization. Experiments with a noncycled statistical model show that the full background covariance estimated by MLGETKF more accurately resembles the shape of the true covariance than a scale-unaware localization. The mean analysis from the best-performing MLGETKF is statistically significantly more accurate than the best-performing scale-unaware LGETKF. The accuracy of the MLGETKF analysis is more sensitive to small-scale band localization radius than large-scale band. MLGETKF is further examined in a cycling DA context with a surface quasigeostrophic model. The root-mean-square potential temperature analysis error of the best-performing MLGETKF is 17.2% lower than that of the best-performing LGETKF. MLGETKF reduces analysis errors measured in kinetic energy spectra space by 30%–80% relative to LGETKF with the largest improvement at large scales. MLGETKF deterministic and ensemble mean forecasts are more accurate than LGETKF for full and large scales up to 5–6-day lead time and for small scales up to 3–4-day lead time, gaining ~12 h–1 day of predictability.

Fast Quasi‐Geostrophic Magneto‐Coriolis Modes in the Earth's Core

AbstractFast changes of Earth's magnetic field could be explained by inviscid and diffusion‐less quasi‐geostrophic (QG) Magneto‐Coriolis modes. We present a hybrid QG model with columnar flows and three‐dimensional magnetic fields and find modes with periods of a few years at parameters relevant to Earth's core. For the simple poloidal magnetic field that we consider here they show a localization of kinetic and magnetic energy in the equatorial region. This concentration of energy near the equator and the high frequency make them a plausible mechanism to explain similar features observed in recent geomagnetic field observations. Our model potentially opens a way to probe the otherwise inaccessible magnetic field structure in the Earth's outer core.

Linear stability analysis for flows over sinusoidal bottom topography

Abstract

Can the Intermediate Western Boundary Current recirculation trigger the Vitória Eddy formation?

South of the Vitória-Trindade Ridge, a seamount chain off East Brazil, the Brazil Current (BC) meanders cyclonically within Tubarão Bight, occasionally forming the Vitória Eddy. It was recently found that the Intermediate Western Boundary Current (IWBC), which flows equatorward below the BC, cyclonically recirculate within Tubarão Bight. We present an analysis of AVISO observations that suggest that the Vitória Eddy formation is conditioned by the strength of the BC upstream of Tubarão Bight. A weak BC is prone to local meandering and eddy formation in the bight, while a strong BC suppresses eddy formation in the bight but triggers downstream meander growth. To study the effects of the IWBC recirculation on the BC meandering and the Vitória Eddy formation, we formulate a simple two-layer quasi-geostrophic model. In the model, the BC is represented by a meridional jet in the upper layer and the IWBC recirculation is a steady eddy in the lower layer. The lower-layer eddy effectively acts as a topographic bump, affecting the upper-layer jet via the stretching term \(\psi _{2}/{R_{d}^{2}}\), where ψ2 is the lower-layer streamfunction and Rd is the baroclinic deformation radius. Based on the AVISO sea-surface height data and previous observational studies, we define a stationary eddy and reference jet. We conduct a number of initial-value problem experiments varying the upper-layer jet speed. A weak upper-layer jet slowly meanders and develops a cyclone above the lower-layer eddy. As we increase the jet velocity, the meandering is faster and the cyclone is larger. But a too-strong jet has an opposite effect: the potential vorticity anomalies induced by the lower-layer eddy are quickly swept away, leading to explosive downstream meander growth; no cyclone is formed above the lower-layer eddy. In all cases, the initial meandering trigger is a linear process (the steering of the upper-layer jet by the lower-layer eddy). But even when the upper-layer jet is weak, nonlinearity quickly becomes important, dominating the dynamics after 10 days of simulation. The downstream meander growth is fully nonlinear. Our idealized QG model confirms that the IWBC recirculation can trigger the Vitória Eddy formation and elucidates the mechanisms involved in this process.

Global attractors of two layer baroclinic quasi-geostrophic model

<p style='text-indent:20px;'>We study the dynamics of a two-layer baroclinic quasi-geostrophic model. We prove that the semigroup <inline-formula><tex-math id="M1">\begin{document}$ \{S(t)\}_{t\geq 0} $\end{document}</tex-math></inline-formula> associated with the solutions of the model has a global attractor in both <inline-formula><tex-math id="M2">\begin{document}$ {{\dot H}_{p}}^1(\Omega) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ {{\dot H}_{p}}^2(\Omega) $\end{document}</tex-math></inline-formula>. Also we show that for any viscosity <inline-formula><tex-math id="M4">\begin{document}$ \mu>0 $\end{document}</tex-math></inline-formula>, there is an open and dense set of forcing <inline-formula><tex-math id="M5">\begin{document}$ \mathcal G\subset{{\dot H}_{p}}^0(\Omega) $\end{document}</tex-math></inline-formula> such that for each <inline-formula><tex-math id="M6">\begin{document}$ G = (g_1, g_2)\in \mathcal G $\end{document}</tex-math></inline-formula>, the set <inline-formula><tex-math id="M7">\begin{document}$ S(G, \mu) \subset {{\dot H}_{p}}^4(\Omega) $\end{document}</tex-math></inline-formula> of the steady state problem is non–empty and finite.</p>

A Deep Learning Approach to Spatiotemporal Sea Surface Height Interpolation and Estimation of Deep Currents in Geostrophic Ocean Turbulence

AbstractSatellite altimeters provide global observations of sea surface height (SSH) and present a unique data set for advancing our theoretical understanding of upper‐ocean dynamics and monitoring its variability. Considering that mesoscale SSH patterns can evolve on timescales comparable to or shorter than satellite return periods, it is challenging to accurately reconstruct the continuous SSH evolution as currently available altimetry observations are still spatially and temporally sparse. Here we explore the possibility of SSH interpolation via Deep Learning by using synthetic observations from an idealized quasigeostrophic model of baroclinic ocean turbulence. We demonstrate that Convolutional Neural Networks with Residual Learning are superior in SSH reconstruction to linear and recently developed dynamical interpolation techniques. Also, the deep neural networks can provide a skillful state estimate of unobserved deep ocean currents at mesoscales. These conspicuous results suggest that SSH patterns of eddies might contain substantial information about the underlying deep ocean currents that are necessary for SSH prediction. Our training data are focused on highly idealized physics and diversification of processes needs to be considered to more accurately represent the real ocean. In addition, methodological improvements such as transfer learning and implementation of dynamically aware loss functions might be necessary to consider before its ultimate use with real satellite observations. Nonetheless, by providing a proof of concept based on synthetic data, our results point to deep learning as a viable alternative to existing interpolation and, more generally, state estimation methods for satellite observations of eddying currents.

Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows

This study examines the Hopf (double Hopf) bifurcations and transitions of two dimensional quasi-geostrophic (QG) flows that model various large-scale oceanic and atmospheric circulations. Using the Kolmogorov function to represent an external forcing in the tropical region, it is shown that the equilibrium of the QG model loses its stability if the combination of the Rossby number, the Ekman number, and the eddy viscosity satisfies a specific condition. Further use of the center manifold technique reveals two different types of the dynamical transition from either a pair of simple complex eigenvalues or a double pair of complex conjugate eigenvalues. These dynamical transitions are confirmed in the numerical analyses of the QG dynamics at the equilibrium, which capture Hopf (double Hopf) bifurcations due to the existence of a nonzero imaginary part of the first eigenvalue. The transition from a pair of simple complex eigenvalues is of particular interest, because it indicates the existence of a stable periodic pattern that is similar to atmospheric easterly waves and related tropical cyclone formation in the tropical atmosphere.

The Parameter Dependence of Eddy Heat Flux in a Homogeneous Quasigeostrophic Two-Layer Model on a β Plane with Quadratic Friction

Abstract This study investigates the parameter dependence of eddy heat flux in a homogeneous quasigeostrophic two-layer model on a β plane with imposed environmental vertical wind shear and quadratic frictional drag. We examine the extent to which the results can be explained by a recently proposed diffusivity theory for passive tracers in two-dimensional turbulence. To account for the differences between two-layer and two-dimensional models, we modify the two-dimensional theory according to our two-layer f-plane analyses reported in an earlier study. Specifically, we replace the classic Kolmogorovian spectral slope, −5/3, assumed to predict eddy kinetic energy spectrum in the former with a larger slope, −7/3, suggested by a heuristic argument and fit to the model results in the latter. It is found that the modified theory provides a reasonable estimate within the regime where both and the strength of the frictional drag, , are much smaller than unity (here, c D is the nondimensional drag coefficient divided by the depth of the layer, k d is the wavenumber of deformation radius, and U is the imposed background vertical wind shear). For values of and that are closer to one, the theory works only if the full spectrum shape of the eddy kinetic energy is given. Despite the qualitative, fitting nature of this approach and its failure to explain the full parameter range, we believe its documentation here remains useful as a reference for the future attempt in pursuing a better theory.

Solitary magnetostrophic Rossby waves in spherical shells

Abstract

A Fuzzy-Logic-Based Covariance Localization Method in Data Assimilation

In ensemble data assimilation systems, the impracticalities of full sampling and systematic error often lead to spurious correlations between two variables with low actual correlations. To solve these problems, researchers have previously proposed a covariance localization (CL) method, which mainly involves the Schur product between a state error covariance matrix and a distance-based correlation matrix. Although this CL method can reduce spurious correlations to a certain extent, observational data remain difficult to be used effectively, which results in unreasonable assimilation. In this study, we develop a new CL method coupled with a fuzzy logic control algorithm, which we call the covariance fuzzy (CF) method. The proposed CF method is a distance-based localization method with “fuzzy” vanishing correlations in data assimilation (DA) systems. To verify the effectiveness of the new algorithm, we conducted a set of experiments using an ensemble Kalman filter (EnKF) that combines the nonlinear Lorenz-96 model or the quasi-geostrophic (QG) models. First, the performances of the CL and CF methods are discussed with respect to different strength forcings, ensemble sizes, and covariance inflation factors. The experimental results show that the proposed CF method can obtain a more effective observation weight than the CL method and can reduce the errors caused by spurious correlations. Additionally, using power spectral density (PSD) as a performance evaluation index, the robustness of the proposed fuzzy logic localization method is demonstrated. However, the application of the fuzzy logic-based localization methodology to a real atmospheric model remains to be tested.

Solitary waves of nonlinear barotropic–baroclinic coherent structures

This study describes the evolutionary mechanisms of nonlinear barotropic–baroclinic interactions, especially, on the excitations, propagations, and decreases of nonlinear coherent structures. Starting from the classical two-layer quasi-geostrophic potential vorticity conservation model equations, the barotropic and baroclinic model equations are derived from the classical work of Pedlosky and Thomson [J. Fluid Mech. 490, 189–215 (2003)]. By considering the effects of bottom topography and beta-plane approximation, the coupled nonlinear Korteweg–de Vries model equations for the evolutions of barotropic and baroclinic coherent structures are obtained by using the methods of multiple scales and perturbation expansions, respectively. Solitary wave solutions are given according to the method of elliptic function expansions, and the physical mechanisms for the evolutions of the nonlinear barotropic–baroclinic interactive coherent structures are analyzed based on the obtained solitary wave solutions. It will be potentially useful for further theoretical investigations on atmospheric blocking phenomena or wave–flow interactions.

N-symmetric interaction of N hetons. I. Analysis of the case N = 2

We examine the motion of N symmetric hetons (oppositely signed vertical dipoles) in a two-layer quasi-geostrophic model. We consider the special case of N-fold symmetry in which the original system of 4N ordinary differential equations reduces to just two equations for the so-called “equivalent” heton. We perform a qualitative analysis to classify the possible types of vortex motions for the case N = 2. We identify the regions of the parameter space corresponding to unbounded motion and to different types of bounded, or localized, motions. We focus on the properties of localized, in particular periodic, motion. We identify classes of absolute and relative “choreographies” first introduced by Simó [“New families of solutions to the N-body problems,” in Proceedings of the European 3rd Congress of Mathematics, Progress in Mathematics Vol. 201, edited by C. Casacuberta, R. M. Miró-Roig, J. Verdera, and S. Xambó-Descamps (Birkhäuser, Basel, Barcelona, 2000), pp. 101–115]. We also study the forms of vortex trajectories occurring for unbounded motion, which are of practical interest due to the associated transport of heat and mass over large distances.

Baroclinic Modes over Rough Bathymetry and the Surface Deformation Radius

AbstractThe deformation radius is widely used as an indication of the eddy length scale at different latitudes. The radius is usually calculated assuming a flat ocean bottom. However, bathymetry alters the baroclinic modes and hence their deformation radii. In a linear quasigeostrophic two-layer model with realistic parameters, the deep flow for a 100-km wave approaches zero with a bottom ridge roughly 10 m high, leaving a baroclinic mode that is mostly surface trapped. This is in line with published current meter studies showing a primary EOF that is surface intensified and has nearly zero flow at the bottom. The deformation radius associated with this “surface mode” is significantly larger than that of the flat bottom baroclinic mode. Using World Ocean Atlas data, the surface radius is found to be 20%–50% larger over much of the globe, and 100% larger in some regions. This in turn alters the long Rossby wave speed, which is shown to be 1.5–2 times faster than over a flat bottom. In addition, the larger deformation radius is easier to resolve in ocean models.

Mathematical Modeling of Vortex Interaction Using a Three-Layer Quasigeostrophic Model. Part 1: Point-Vortex Approach

The theory of point vortices is used to explain the interaction of a surface vortex with subsurface vortices in the framework of a three-layer quasigeostrophic model. Theory and numerical experiments are used to calculate the interaction between one surface and one subsurface vortex. Then, the configuration with one surface vortex and two subsurface vortices of equal and opposite vorticities (a subsurface vortex dipole) is considered. Numerical experiments show that the self-propelling dipole can either be captured by the surface vortex, move in its vicinity, or finally be completely ejected on an unbounded trajectory. Asymmetric dipoles make loop-like motions and remain in the vicinity of the surface vortex. This model can help interpret the motions of Lagrangian floats at various depths in the ocean.

Assimilating Observations with Spatially Correlated Errors Using a Serial Ensemble Filter with a Multiscale Approach

AbstractThe serial ensemble square root filter (EnSRF) typically assumes observation errors to be uncorrelated when assimilating the observations one at a time. This assumption causes the filter solution to be suboptimal when the observation errors are spatially correlated. Using the Lorenz-96 model, this study evaluates the suboptimality due to mischaracterization of observation error spatial correlations. Neglecting spatial correlations in observation errors results in mismatches between the specified and true observation error variances in spectral space, which cannot be resolved by inflating the overall observation error variance. As a remedy, a multiscale observation (MSO) method is proposed to decompose the observations into multiple scale components and assimilate each component with separately adjusted spectral error variance. Experimental results using the Lorenz-96 model show that the serial EnSRF, with the help from the MSO method, can produce solutions that approach the solution from the EnSRF with correctly specified observation error correlations as the number of scale components increases. The MSO method is further tested in a two-layer quasigeostrophic (QG) model framework. In this case, the MSO method is combined with the multiscale localization (MSL) method to allow the use of different localization radii when updating the model state at different scales. The combined method (MSOL) improves the serial EnSRF performance when assimilating observations with spatially correlated errors. With adjusted observation error spectral variances and localization radii, the combined MSOL method provides the best solution in terms of analysis accuracy and filter consistency. Prospects and challenges are also discussed for the implementation of the MSO method for more complex models and observing networks.

Data-adaptive harmonic analysis of oceanic waves and turbulent flows.

We introduce new features of data-adaptive harmonic decomposition (DAHD) that are showcased to characterize spatiotemporal variability in high-dimensional datasets of complex and mutsicale oceanic flows, offering new perspectives and novel insights. First, we present a didactic example with synthetic data for identification of coherent oceanic waves embedded in high amplitude noise. Then, DAHD is applied to analyze turbulent oceanic flows simulated by the Regional Oceanic Modeling System and an eddy-resolving three-layer quasigeostrophic ocean model, where resulting spectra exhibit a thin line capturing nearly all the energy at a given temporal frequency and showing well-defined scaling behavior across frequencies. DAHD thus permits sparse representation of complex, multiscale, and chaotic dynamics by a relatively few data-inferred spatial patterns evolving with simple temporal dynamics, namely, oscillating harmonically in time at a given single frequency. The detection of this low-rank behavior is facilitated by an eigendecomposition of the Hermitian cross-spectral matrix and resulting eigenvectors that represent an orthonormal set of global spatiotemporal modes associated with a specific temporal frequency, which in turn allows to rank these modes by their captured energy and across frequencies, and allow accurate space-time reconstruction. Furthermore, by using a correlogram estimator of the Hermitian cross-spectral density matrix, DAHD is both closely related and distinctly different from the spectral proper orthogonal decomposition that relies on Welch's periodogram as its estimator method.

Correcting for model changes in statistical postprocessing – an approach based on response theory

Abstract. For most statistical postprocessing schemes used to correct weather forecasts, changes to the forecast model induce a considerable reforecasting effort. We present a new approach based on response theory to cope with slight model changes. In this framework, the model change is seen as a perturbation of the original forecast model. The response theory allows us then to evaluate the variation induced on the parameters involved in the statistical postprocessing, provided that the magnitude of this perturbation is not too large. This approach is studied in the context of a simple Ornstein–Uhlenbeck model and then on a more realistic, yet simple, quasi-geostrophic model. The analytical results for the former case help to pose the problem, while the application to the latter provides a proof of concept and assesses the potential performance of response theory in a chaotic system. In both cases, the parameters of the statistical postprocessing used – the Error-in-Variables Model Output Statistics (EVMOS) method – are appropriately corrected when facing a model change. The potential application in an operational environment is also discussed.

Hybrid Gain Data Assimilation Using Variational Corrections in the Subspace Orthogonal to the Ensemble

AbstractThe viability of a parameterless hybrid data assimilation algorithm is investigated. As an alternative to the traditional hybrid covariance scheme, hybrid gain data assimilation (HGDA) was proposed to blend the gain matrix derived from the variational method and the ensemble-based Kalman filter (EnKF). A previously proposed HGDA algorithm uses a two-step process applying the EnKF with a variational update. The algorithm is modified here to limit the variational correction to the subspace orthogonal to the ensemble perturbation subspace without the use of a hybrid weighting parameter, as the optimization of such a parameter is nontrivial. The modified HGDA algorithm is investigated with a quasigeostrophic (QG) model. Results indicate that when the climatological background error covariance matrix B and the observation error covariance R are well estimated, state estimates from the parameterless HGDA are more accurate than the parameter-dependent HGDA. The parameterless HGDA not only has potential advantages over the standard HGDA as an online data assimilation algorithm but can also serve as a valuable diagnostic tool for tuning the B and R matrices. It is also found that in this QG model, the empirically best static B matrix for the stand-alone 3DVAR has high variance at larger spatial scales, which degrades the accuracy of the HGDA systems and may not be the best choice for hybrid methods in general. A comparison of defining the orthogonal subspace globally or locally demonstrates that global orthogonality is more advantageous for stabilizing the hybrid system and maintains large-scale balances.