Surface Quasi-geostrophic Model Research Articles

The nonlinear evolution of two surface quasi-geostrophic vortices

We investigate numerically the evolution of a baroclinic vortex in a two-level surface quasi-geostrophic model. The vortex is composed of two circular patches of uniform buoyancy, one located at each level. We vary the vortex radii, the magnitude of buoyancy, and the vertical distance between the two levels. We also study different radial profiles of buoyancy for each vortex. This article considers two main situations: firstly, initially columnar vortices with like-signed buoyancies. These vortices are contra-rotating, are linearly unstable and may break. Secondly, we consider initially tilted vortices with opposite-signed buoyancies, which may align vertically. Numerical experiments show that (1) identical contra-rotating vortices break into hetons when initially perturbed by low azimuthal modes; (2) unstable, vertically asymmetric, contra-rotating vortices can stabilise nonlinearly more often than vertically symmetric ones, and can form quasi-steady baroclinic tripoles; (3) co-rotating vortices can align when the two levels are close to each other vertically, and when the vortices are initially horizontally distant from each other by less than three radii; (4) for initially more distant vortices, two such vortices rotate around the plane center; and(5) in all cases, the vortex boundaries are disturbed by Rossby waves. These results compare favorably to earlier results with internal quasi-geostrophic vortices. Further modelling efforts may extend the present study to fully three-dimensional ocean dynamics.

Particle dispersion and clustering in surface ocean turbulence with ageostrophic dynamics

Upper-ocean turbulent flows at horizontal length scales smaller than the deformation radius depart from geostrophic equilibrium and develop important vertical velocities, which are key to marine ecology and climatic processes. Due to their small size and fast temporal evolution, these fine scales are difficult to measure during oceanographic campaigns. Instruments such as Lagrangian drifters have provided another way to characterize these scales through the analysis of pair-dispersion evolution and have pointed out striking particle convergence events. By means of numerical simulations, we investigate such processes in a model of surface-ocean turbulence that includes ageostrophic motions. This model originates from a Rossby-number expansion of the primitive equations and reduces to the surface quasi-geostrophic model, a paradigm of submesoscale dynamics, in the limit of vanishing Rossby number. We focus on the effect of the ageostrophic dynamics on the pair-dispersion and clustering properties of Lagrangian tracer particles at the ocean surface. Our results indicate that while over long times the pair separation process is barely affected by the ageostrophic component of the velocity field, the latter is responsible for the formation of temporary particle aggregates, and the intensity of this phenomenon increases with the Rossby number. We further show that Lagrangian tracers preferentially accumulate in cyclonic frontal regions, which is in agreement with observations and other more realistic modeling studies. These findings appear interesting to improve the understanding of the turbulent transport by ocean fine scales and in light of upcoming, new high-resolution satellite data of surface velocity fields.

Basin-Scale to Submesoscale Variability of the East Mediterranean Sea Upper Circulation

Abstract The East Mediterranean Sea (EMS) circulation has previously been characterized as dominated by gyres, mesoscale eddies, and disjoint boundary currents. We develop nested high-resolution numerical simulations in the EMS to examine the circulation variability with an emphasis on the yet unexplored regional submesoscale currents. Rather than several disjoint currents, a continuous cyclonic boundary current (BC) encircling the Levantine basin is identified in both model solution and altimetry data. This EMS BC advects eddy chains downstream and is identified as a principal source of regional mesoscale and submesoscale current variability. During the seasonal fall to winter mixed layer deepening, energetic submesoscale [O(10) km] eddies, fronts, and filaments emerge throughout the basin, characterized by O(1) Rossby numbers. A submesoscale time scale range of ≈1–5 days is identified using spatiotemporal analysis of the numerical solutions and confirmed through mooring data. The submesoscale kinetic energy (KE) wavenumber (k) spectral slope is found to be k−2, shallower than the quasigeostrophic-like ∼k−3 slope diagnosed in summer. The shallowness of the winter spectral slope is shown to be due to divergent subinertial motions, consistent with the Boyd theoretical model, rather than with the surface quasigeostrophic model. Using a coarse-graining approach, we diagnose a seasonal inverse (forward) KE cascade above (below) 30-km scales due to rotational (divergent) motions and show that these commence after completion of the fall submesoscale energization. We also show that at scales larger than several hundred kilometers, the spectral density becomes near constant and a weak forward cascade occurs, from gyre scales to mesoscales.

Oceanic dipoles in a surface quasi-geostrophic model

Analysis of satellite altimetry and Argo float data leads Ni et al. (J. Geophys. Res., 125, 2020, e2020JC016479) to argue that mesoscale dipoles are widespread features of the global ocean having a relatively uniform three-structure that can lead to strong vertical exchanges. Almost all the features of the composite dipole they construct can be derived from a model for multipoles in the surface quasi-geostrophic equations for which we present a straightforward novel solution in terms of an explicit linear algebraic eigenvalue problem, allowing simple evaluation of the higher radial modes that appear to be present in the observations and suggesting that mass conservation may explain the observed frontogenetic velocities.

Baroclinic instability of spatially-periodic flows in a discrete surface quasi-geostrophic model with two levels

Within the framework of a discrete surface quasi-geostrophic (SQG) model with two levels, the hydrodynamic stability of a spatially periodic flow with sinusoidal velocity profile (Kolmogorov flow) is investigated. Such spatially quasi-periodic flows are often observed in the atmospheres of the giant planets and in the Southern Ocean. To describe the dynamics of perturbations, the Galerkin method with three basis trigonometric functions was used. It provides a tool to study both the linear and nonlinear perturbation dynamics. Analytical expressions are obtained for the growth rates of perturbations in the linear theory of stability, and it is shown that the fastest growing perturbations have a wavelength on the order of the spatial period of the main flow. Using the Galerkin method, it is also shown that the exponential growth of perturbations at the linear stage of development is replaced by the stage of stable nonlinear oscillations. Such oscillations are analogous to the nonlinear fluctuations, or vacillations, found in R. Hide's laboratory experiments. In terms of elliptic integrals, an analytical expression for the period of nonlinear oscillations is obtained, and it is shown that for typical values of the parameters relevant to the Earth’s atmosphere, the period of oscillations is about one month. The paper also proposes a long-wavelength version of the discrete SQG model, which describes the dynamics of perturbations with a horizontal scale much larger than the baroclinic Rossby radius of deformation. It is shown that the use of this version simplifies the study of stability and leads to asymptotically correct results.

Discrete SQG models with two boundaries and baroclinic instability of jet flows

In this paper, new vertically discrete versions of the surface quasigeostrophic (SQG) model with two boundaries are formulated. For any number of partition levels, the equations of the discrete model are written in the form of conservation laws for two Lagrangian invariants, which have the meaning of buoyancy distributions at the horizontal boundaries of the fluid layer. The values of the invariants are expressed in terms of the values of the stream function at two internal levels and contain higher order elliptic operators. The use of discrete models greatly simplifies the solution of problems of the linear theory of hydrodynamic stability and provides high accuracy even with a small number of vertical discrete levels. Using the two-level version of the SQG model, which is similar to the classical two–layer Phillips model, we investigated the linear stability of jet flows induced by piecewise constant boundary distributions of buoyancy. For these flows, analytical expressions for the growth rate of perturbations have been obtained and it is shown that the most unstable perturbation has a wavelength of the order of the Rossby baroclinic radius of deformation. Flows with vertical shear induced by smooth and slowly varying boundary buoyancy distributions are also considered. The instability of these flows is found to be absolute, that is, independent of the velocity profile horizontal structure.

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.

Long-wave instabilities in the SQG model with two boundaries

ABSTRACT Surface quasi-geostrophic (SQG) flows with a much larger horizontal scale than the Rossby radius of deformation are considered. A new version of the SQG model with two boundaries, which is reduced to a nonlinear system of partial differential equations, is proposed to describe the dynamics of such flows. This system describes the interaction between the barotropic and baroclinic components of the stream function and generalises the two-dimensional Euler equations for flows with a vertical velocity shear. The laws of conservation of both total and surface potential energies, which follow from this system, have been formulated. The solutions of a number of problems in the theory of baroclinic instability, which are in agreement with already known solutions, have been obtained within the framework of this system. It is shown that vertical shear flows are absolutely unstable, i.e. their instability is independent of the horizontal velocity profile structure. A generalised system of the SQG model equations, which additionally takes into account the β-effect and the Ekman bottom friction, has also been proposed. The transformation of jet flows due to the bottom friction and the influence of the β-effect on the stability of shear flows have been studied based on this system.

Instability of Surface Quasigeostrophic Spatially Periodic Flows

AbstractThe surface quasigeostrophic (SQG) model is developed to describe the dynamics of flows with zero potential vorticity in the presence of one or two horizontal boundaries (Earth surface and tropopause). Within the framework of this model, the problems of linear and nonlinear stability of zonal spatially periodic flows are considered. To study the linear stability of flows with one boundary, two approaches are used. In the first approach, the solution is sought by decomposing into a trigonometric series, and the growth rate of the perturbations is found from the characteristic equation containing an infinite continued fraction. In the second approach, few-mode Galerkin approximations of the solution are constructed. It is shown that both approaches lead to the same dependence of the growth increment on the wavenumber of perturbations. The existence of instability with a preferred horizontal scale on the order of the wavelength of the main flow follows from this dependence. A similar result is obtained within the framework of the SQG model with two horizontal boundaries. The Galerkin method with three basis trigonometric functions is also used to study the nonlinear dynamics of perturbations, described by a system of three nonlinear differential equations similar to that describing the motion of a symmetric top in classical mechanics. An analysis of the solutions of this system shows that the exponential growth of disturbances at the linear stage is replaced by a stage of stable nonlinear oscillations (vacillations). The results of numerical integration of full nonlinear SQG equations confirm this analysis.

Confinement of a hot temperature patch in the modified SQG model

In this paper we study the time evolution of a temperature patch in $\mathbb{R}^2$ according to the modified Surface Quasi-Geostrophic (SQG) patch equation. In particular we give a temporal estimate on the growth of the support, providing a rigorous proof of the confinement of a hot patch of temperature in absence of external forcing, under the quasi-geostrophic approximation.

Relative dispersion in generalized two-dimensional turbulence

The statistical properties of turbulent fluids depend on how local the energy transfers among scales are, i.e. whether the energy transfer at some given scale is due to the eddies at that particular scale, or to eddies at larger (non-local) scale. This locality in the energy transfers may have consequences for the relative dispersion of passive particles. In this paper, we consider a class of generalized two-dimensional flows (produced by the so-called $\unicode[STIX]{x1D6FC}$-turbulence models), theoretically possessing different properties in terms of locality of energy transfers. It encompasses the standard barotropic quasi-geostrophic (QG) and the surface quasi-geostrophic (SQG) models as limiting cases. The relative dispersion statistics are examined, both as a function of time and as a function of scale, and compared to predictions based on phenomenological arguments assuming the locality of the cascade. We find that the dispersion statistics follow the predicted values from local theories, as long as the parameter $\unicode[STIX]{x1D6FC}$ is small enough (dynamics close to that of the SQG model), for sufficiently small initial pair separations. In contrast, non-local dispersion is observed for the QG model, a robust result when looking at relative displacement probability distributions. However, we point out that spectral energy transfers do have a non-local contribution for models with different values of $\unicode[STIX]{x1D6FC}$, including the SQG case. This indicates that locality/non-locality of the turbulent cascade may not always imply locality/non-locality in the relative dispersion of particles and that the self-similar nature of the turbulent cascade is more appropriate for determining the relative dispersion locality.

Geophysical flows under location uncertainty, Part III SQG and frontal dynamics under strong turbulence conditions

Models under location uncertainty are derived assuming that a component of the velocity is uncorrelated in time. The material derivative is accordingly modified to include an advection correction, inhomogeneous and anisotropic diffusion terms and a multiplicative noise contribution. This change can be consistently applied to all fluid dynamics evolution laws. This paper continues to explore benefits of this framework and consequences of specific scaling assumptions. Starting from a Boussinesq model under location uncertainty, a model is developed to describe a mesoscale flow subject to a strong underlying submesoscale activity. Specifically, turbulent diffusion and rotation effects have similar orders of magnitude. As obtained, the geostrophic balance is modified and the Quasi-Geostrophic assumptions remarkably lead to a zero Potential Vorticity. The ensuing Surface Quasi-Geostrophic model provides a simple diagnosis of warm frontolysis and cold frontogenesis.

Surface Quasi-Geostrophy

Oceanic and atmospheric dynamics are often interpreted through potential vorticity, as this quantity is conserved along the geostrophic flow. However, in addition to potential vorticity, surface buoyancy is a conserved quantity, and this also affects the dynamics. Buoyancy at the ocean surface or at the atmospheric tropopause plays the same role of an active tracer as potential vorticity does since the velocity field can be deduced from these quantities. The surface quasi-geostrophic model has been proposed to explain the dynamics associated with surface buoyancy conservation and seems appealing for both the ocean and the atmosphere. In this review, we present its main characteristics in terms of coherent structures, instabilities and turbulent cascades. Furthermore, this model is mathematically studied for the possible formation of singularities, as it presents some analogies with three-dimensional Euler equations. Finally, we discuss its relevance for the ocean and the atmosphere.

Vortex merger in surface quasi-geostrophy

The merger of two identical surface temperature vortices is studied in the surface quasi-geostrophic model. The motivation for this study is the observation of the merger of submesoscale vortices in the ocean. Firstly, the interaction between two point vortices, in the absence or in the presence of an external deformation field, is investigated. The rotation rate of the vortices, their stationary positions and the stability of these positions are determined. Then, a numerical model provides the steady states of two finite-area, constant-temperature, vortices. Such states are less deformed than their counterparts in two-dimensional incompressible flows. Finally, numerical simulations of the nonlinear surface quasi-geostrophic equations are used to investigate the finite-time evolution of initially identical and symmetric, constant temperature vortices. The critical merger distance is obtained and the deformation of the vortices before or after merger is determined. The addition of external deformation is shown to favor or to oppose merger depending on the orientation of the vortex pair with respect to the strain axes. An explanation for this observation is proposed. Conclusions are drawn towards an application of this study to oceanic vortices.

The role of mixed-layer instabilities in submesoscale turbulence

Upper-ocean turbulence at scales smaller than the mesoscale is believed to exchange surface and thermocline waters, which plays an important role in both physical and biogeochemical budgets. But what energizes this submesoscale turbulence remains a topic of debate. Two mechanisms have been proposed: mesoscale-driven surface frontogenesis and baroclinic mixed-layer instabilities. The goal here is to understand the differences between the dynamics of these two mechanisms, using a simple quasi-geostrophic model. The essence of mesoscale-driven surface frontogenesis is captured by the well-known surface quasi-geostrophic model, which describes the sharpening of surface buoyancy gradients and the subsequent breakup in secondary roll-up instabilities. We formulate a similarly archetypical Eady-like model of submesoscale turbulence induced by mixed-layer instabilities. The model captures the scale and structure of this baroclinic instability in the mixed layer. A wide range of scales are energized through a turbulent inverse cascade of kinetic energy that is fuelled by the submesoscale mixed-layer instability. Major differences to mesoscale-driven surface frontogenesis are that mixed-layer instabilities energize the entire depth of the mixed layer and produce larger vertical velocities. The distribution of energy across scales and in the vertical produced by our simple model of mixed-layer instabilities compares favourably to observations of energetic wintertime submesoscale flows, suggesting that it captures the leading-order balanced dynamics of these flows. The dynamics described here in an oceanographic context have potential applications to other geophysical fluids with layers of different stratifications.

Realizing Surface-Driven Flows in the Primitive Equations

AbstractThe surface quasigeostrophic (SQG) model describes flows with surface buoyancy perturbations with no interior quasigeostrophic potential vorticity at small Rossby number Ro and O(1) Burger number, where quasigeostrophic dynamics are expected to hold. Numerical simulations of SQG dynamics have shown that vortices are frequently generated at small scales, which may have O(1) Rossby numbers and therefore may be beyond the limits of SQG. This paper examines the dynamics of an initially geostrophically balanced elliptical surface buoyancy perturbation in both the SQG model and the nonhydrostatic Boussinesq primitive equations (PE). In the case of very small Rossby number, it is confirmed that both models agree, as expected. For larger Ro, non-SQG effects emerge and as a result the solution of the PE deviates significantly from that of SQG. In particular, an increase in the Rossby number has the following effects: (i) the buoyancy filaments at the surface are stabilized in that they generate fewer secondary vortices; (ii) the core of the vortex experiences inertial instability, which results in a uniform buoyancy profile in its interior; (iii) the divergent part of the energy spectrum increases in magnitude; (iv) the PE model has significantly more gravity waves that are radiated from the vortex; (v) the magnitude of the vertical velocity increases; and (vi) in the mature stages of evolution, there are gravitational instabilities that develop because of the complicated dynamics inside the vortex. It is demonstrated that significant non-SQG effects are evident when the large-scale Rossby number of the initial flow is about 0.05 and the local Rossby number is O(1).

Spontaneous inertia-gravity-wave generation by surface-intensified turbulence

AbstractThe spontaneous generation of inertia-gravity waves (IGWs) by surface-intensified, nearly balanced motion is examined using a high-resolution simulation of the primitive equations in an idealized oceanic configuration. At large scale and mesoscale, the dynamics, which is driven by baroclinic instability near the surface, is balanced and qualitatively well described by the surface quasi-geostrophic model. This however predicts an increase of the Rossby number with decreasing spatial scales and, hence, a breakdown of balance at small scale; the generation of IGWs is a consequence of this breakdown. The wave field is analysed away from the surface, at depths where the associated vertical velocities are of the same order as those associated with the balanced motion. Quasi-geostrophic relations, the omega equation in particular, prove sufficient to separate the IGWs from the balanced contribution to the motion. A spectral analysis indicates that the wave energy is localized around dispersion relation for free IGWs, and decays only slowly as the frequency and horizontal wavenumber increase. The IGW generation is highly intermittent in time and space: localized wavepackets are emitted when thin filaments in the surface density are formed by straining, leading to large vertical vorticity and correspondingly large Rossby numbers. At depth, the IGW field is the result of a number of generation events; away from the generation sites it takes the form of a relatively homogeneous, apparently random wave field. The energy of the IGW field generated spontaneously is estimated and found to be several orders of magnitude smaller than the typical IGW energy in the ocean.

Dynamic growth estimates of maximum vorticity for 3D incompressible Euler equations and the SQG model

By performing estimates on the integral of the absolute value ofvorticity along a local vortex line segment, we establish a relativelysharp dynamic growth estimate of maximum vorticity under someassumptions on the local geometric regularity of the vorticity vector.Our analysis applies to both the 3D incompressible Euler equationsand the surface quasi-geostrophic model (SQG).As an application of our vorticity growth estimate, we apply ourresult to the 3D Euler equation with the two anti-parallel vortex tubesinitial data considered by Hou-Li [12]. Under someadditional assumption on the vorticity field, which seems to beconsistent with the computational results of [12], we showthat the maximum vorticity can not grow faster than double exponentialin time. Our analysis extends the earlier results byCordoba-Fefferman [6, 7] and Deng-Hou-Yu [8, 9].

Effects of surface quasi-geostrophic turbulence on phytoplankton competition and coexistence

This paper aims at studying numerically the competition between two mutually exclusive phytoplankton species in a fully-turbulent field consisting of interacting mesoscale and submesoscale structures. A simple NPPZD ecosystem model is embedded in a Surface Quasi-Geostrophic model which is able to reproduce frontogenesis and the associated nutrient vertical pump. The two simulated phytoplankton species differ by their size and their affinity for nutrients. In this study, we rationalize the role played by eddies and filaments in the distribution of the two phytoplankton species. We show that the SQG dynamics are responsible for the coexistence of the two phytoplankton species on a single limiting resource at statistical steady state. In addition, we show that as a result of strong vertical injections, filaments contain 64% of the phytoplankton biomass. The two phytoplankton species coexist in filaments but the large phytoplankton is predominant. By contrast, this latter is completely excluded from eddy cores where only the small phytoplankton develops. Since eddies are coherent structures (unlike filaments) and since their edges are almost impermeable to horizontal transport, the large phytoplankton can barely enter eddies. Therefore, eddies are ecological niches which shelter the small phytoplankton. Finally we show that interactions between eddies such as eddy merger can favor the survival of phytoplankton species within eddies on long time scales in the ocean.

Two Comments on the Surface Quasigeostrophic Model for the Atmospheric Energy Spectrum

Abstract The horizontal wavenumber spectra of wind and temperature in the upper troposphere and lower stratosphere display a narrow k−3 range at scales on the order of 1000 km and a broad k−5/3 range at mesoscales on the order of 1 to 500 km. Recently, Tulloch and Smith suggested that a surface quasigeostrophic (SQG) turbulence model can explain the observed spectra. Here, it is first argued that the mesoscale spectra are not likely to be explained by any quasigeostrophic model because the Rossby number corresponding to the mesoscale dynamics is on the order of unity or larger. Then it is argued that the SQG model in particular cannot explain the observations because its mesoscale spectrum displays a k−5/3 dependence only in a very thin layer just below the tropopause. The thickness of this layer can be estimated to be of the order of 10 m, whereas aircraft measurements are typically performed several hundred meters away from the tropopause.