Quasi-geostrophic Model Research Articles

Positive and Negative Eddy Feedbacks Acting on Midlatitude Jet Variability in a Three-Level Quasigeostrophic Model

Abstract The variability of midlatitude jets is investigated in a long-term integration of a dry three-level quasigeostrophic model on the sphere. As for most observed jets, the leading EOF of the zonal-mean wind corresponds to latitudinal shifts of the jet, and the second EOF to pulses of the jet speed. The first principal component (PC1) is also more persistent than the second one (PC2); this longer persistence arises from different eddy feedbacks both in the short term (i.e., within a few days following the peak of the PCs) and in the long term. The short-term eddy feedbacks come from two distinct mechanisms. First, a planetary waveguide effect acts as a negative feedback on both PCs. The positive phases of PC1 and PC2, which correspond to poleward-shifted and accelerated jets, respectively, are first driven then canceled by planetary waves reflecting on the equatorward flank of the jet. A similar process occurs for the negative phases when planetary waves reflect on the poleward flank of the jet. Second, synoptic waves also exert a short-term negative feedback on PC2: when the jet accelerates, the enhanced meridional wind shear increases the barotropic sink of eddy energy and depletes it very rapidly, therefore preventing synoptic eddies from maintaining the accelerated jet. Finally, at lags longer than their typical time scale, synoptic eddies drive a positive feedback on PC1 only. This feedback can be explained by a baroclinic mechanism in which the jet shift modifies the baroclinicity, causing, first, eddy heat flux anomalies and, then, momentum convergence anomalies. This feedback is absent for PC2, despite some changes in the baroclinicity.

Potential Vorticity Dynamics of Coastal Outflows

AbstractA simple quasigeostrophic model is used to examine the outflow from a river, estuary, or strait into a coastal ocean. As shown by Johnson et al., these quasigeostrophic outflows are accurately described by analytical long-wave solutions. This paper first uses these solutions and contour dynamics simulations to discuss the behavior of coastal outflows. Second, it extends the model and the long-wave theory to consider the effects of ambient currents, tides, winds, or a variable source flux. Third, consideration of the momentum flux at the source is used to understand the turning of the current, showing that steady solutions conserve momentum, hence resolving the momentum imbalance paradox of Pichevin and Nof. Finally, a new numerical scheme to compute steady outflow boundaries is developed. The model focuses on the key dynamics driven by the source velocity and the generation of vorticity as the buoyant fluid adjusts. The simplicity of the model, and insight given by the long-wave solutions, enables a full understanding of the dynamics. The outflows display a range of behaviors, including indefinitely growing near-source bulges, steady boundary profiles with varying offshore width, bidirectional currents, and rarefying or eddy-like leading heads, all of which can be understood with the long-wave theory. Despite the simplicity of the model, the results show good agreement in comparison with observations, experiments, and numerical models.

Meandering Subtropical Jet and Precipitation over Summertime East Asia and the Northwestern Pacific

Abstract It has been revealed that in summertime, precipitation is enhanced to the south of the upper-level tropopausal potential vorticity contours, which are accompanied by instantaneous jets, over the eastern coastal region of China to the northwestern Pacific. It is frequently exhibited as precipitation bands ranging in size from over a thousand to several thousands of kilometers long. In this study, an analysis was conducted to quantify the relationship depending on the phase of upper-level disturbances. With composite analysis, it is shown that the enhancement along the contours occurs at all phases; it occurs not only to the east but also to the west of the upper-level troughs, although it is weaker. The midtropospheric distributions of upwelling and the Q-vector convergence are collocated with the precipitation enhancement, suggesting the importance of dynamical induction by geostrophic flow at all phases. The effects of upper-level disturbances and low-level jets (LLJs) with a southerly component are investigated by using an idealized nondimensional quasigeostrophic model supporting latent heating. While upper-level waves induce upwelling and downwelling to the east and west, respectively, of the upper-level troughs, LLJs tend to offset the downwelling, enabling precipitation to the west too. Both in the observational composite and the idealized model with LLJ, confluence and diffluence contribute to the Q-vector convergence to induce upwelling along the subtropical jet irrespective of upper-level disturbance phases. This induction is explained as a general feature of a veered jet where geopotential isolines rotate clockwise with height without requiring wind variation along the jet.

A note on ‘Toward a stochastic parameterization of ocean mesoscale eddies’

Porta Mana and Zanna (2014) recently proposed a subgrid-scale parameterization for eddy-permitting quasigeostrophic models. In this model the large-scale fluid is represented as a non-Newtonian viscoelastic medium, with a subgrid-stress closure that involves the Lagrangian derivative of large-scale quantities. This note derives this parameterization, including the nondimensional proportionality coefficient, using only two statistical assumptions: that the subgrid-scale term is locally homogeneous and decorrelates rapidly in space. The parameterization is then verified by comparing against eddy-resolving quasigeostrophic simulations, independently reproducing the results of Porta Mana and Zanna in a simpler model.

Fluctuations, response, and resonances in a simple atmospheric model

We study the response of a simple quasi-geostrophic barotropic model of the atmosphere to various classes of perturbations affecting its forcing and its dissipation using the formalism of the Ruelle response theory. We investigate the geometry of such perturbations by constructing the covariant Lyapunov vectors of the unperturbed system and discover in one specific case–orographic forcing–a substantial projection of the forcing onto the stable directions of the flow. This results into a resonant response shaped as a Rossby-like wave that has no resemblance to the unforced variability in the same range of spatial and temporal scales. Such a climatic surprise corresponds to a violation of the fluctuation–dissipation theorem, in agreement with the basic tenets of nonequilibrium statistical mechanics. The resonance can be attributed to a specific group of rarely visited unstable periodic orbits of the unperturbed system. Our results reinforce the idea of using basic methods of nonequilibrium statistical mechanics and high-dimensional chaotic dynamical systems to approach the problem of understanding climate dynamics.

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.

On the roles of baroclinic modes in eddy-resolving midlatitude ocean dynamics

This work concerns how different baroclinic modes interact and influence solutions of the midlatitude ocean dynamics described by the eddy-resolving quasi-geostrophic model of wind-driven gyres. We developed multi-modal energetics analysis to illuminate dynamical roles of the vertical modes, carried out a systematic analysis of modal energetics and found that the eddy-resolving dynamics of the eastward jet extension of the western boundary currents, such as the Gulf Stream or Kuroshio, is dominated by the barotropic, and the first and second baroclinic modes, which become more energized with smaller eddy viscosity. In the absence of high baroclinic modes, the energy input from the wind is more efficiently focused onto the lower modes, therefore, the eddy backscatter maintaining the eastward jet and its adjacent recirculation zones is the strongest and overestimated with respect to cases including higher baroclinic modes. In the presence of high baroclinic modes, the eddy backscatter effect on the eastward jet is much weaker. Thus, the higher baroclinic modes play effectively the inhibiting role in the backscatter, which is opposite to what has been previously thought. The higher baroclinic modes are less energetic and have progressively decreasing effect on the flow dynamics; nevertheless, they still play important roles in inter-mode energy transfers (by injecting energy into the region of the most intensive eddy forcing, in the neighborhood of the eastward jet) that have to be taken into account for correct representation of the backscatter and, thus, for determining the eastward jet extension.

Scale-aware deterministic and stochastic parametrizations of eddy-mean flow interaction

The role of mesoscale eddies is crucial for the ocean circulation and its energy budget. The sub-grid scale eddy variability needs to be parametrized in ocean models, even at so-called eddy permitting resolutions. Porta Mana and Zanna (2014) propose an eddy parametrization based on a non-Newtonian stress which depends on the partially resolved scales and their variability. In the present study, we test two versions of the parametrization, one deterministic and one stochastic, at coarse and eddy-permitting resolutions in a double gyre quasi-geostrophic model. The parametrization leads to drastic improvements in the mean state and variability of the ocean state, namely in the jet rectification and the kinetic-energy spectra as a function of wavenumber and frequency for eddy permitting models. The parametrization also appears to have a stabilizing effect on the model, especially the stochastic version. The parametrization possesses attractive features for implementation in global models: very little computational cost, it is flow aware and uses the properties of the underlying flow. The deterministic coefficient is scale-aware, while the stochastic parameter is scale- and flow-aware with dependence on resolution, stratification and wind forcing.

Piecewise uniform potential vorticity pancake shielded vortices

Shielded vortices consist of a core of potential vorticity (PV) of a given sign surrounded (or shielded) by a layer of opposite-signed PV. Such vortices have specific properties and have been the focus of numerous studies, first in two dimensional geometries (where PV is just the vertical component of the vorticity vector) and in geophysical applications (mostly in layered models). The present paper focuses on three-dimensional, spheroidal shielded vortices. In particular, we focus on vortical structures whose overall volume-integrated PV is zero. We restrict attention to vortices of piecewise uniform PV in the present research. We first revisit the problem within the quasi-geostrophic model, then we extend the results to the non-hydrostatic regime. We show that the stability of the structure depends on the ratio of PV between the inner core and the outer shield. In particular it depends on the polarity of the core and of the wavenumber of the azimuthal mode perturbed.

The Modified Quasi-geostrophic Barotropic Models Based on Unsteady Topography

New models using scale analysis and perturbation methods were derivated starting from the shallow water equations based on barotropic fluids. In the paper, to discuss the irregular topography with different magnitudes, especially considering the condition of the vast terrain, some modified quasi-geostrophic barotropic models were obtained. The unsteady terrain is more suitable to describe the motion of the fluid state of the earth because of the change of global climate and environment, so the modified models are more rational potential vorticity equations. If we do not consider the influence of topography and other factors, the models degenerate to the general quasi-geostrophic barotropic equations in the previous studies. Modelos semigeostróficos barotrópicos modificados con base en topografía inestableResumenEste trabajo deduce nuevos modelos con el uso de los métodos de análisis a escala y de perturbación a partir de las ecuaciones de aguas poco profundas con base en fluidos barotrópicos. En este artículo se obtuvieron algunos modelos semigeostróficos barotrópicos para aplicar en zonas de topografía inestable con diferentes magnitudes y considerar especialmente la condición del extenso terreno. La topografía inestable es más propicia para describir el movimiento del estado fluido de la tierra debido al cambio del clima y ambiente, por lo tanto los modelos modificados son ecuaciones de vorticidad potenciales más razonables. Si no se considera la influencia de la topografía y otros factores, los modelos se reducirían a las ecuaciones generales semigeostróficas barotrópicas de estudios anteriores.

Spatially extended estimates of analysis and short-range forecast error variances

Accurate estimates of ‘true’ error variance between Numerical Weather Prediction (NWP) analyses and forecasts and the ‘reality’ interpolated to a NWP model grid (Analysis and true Forecast Error Variance, hereafter AFEV) are critical for successful data assimilation and ensemble forecasting applications. Peña and Toth (2014, PT14) introduced a Statistical Analysis and Forecast Error estimation (hereafter called SAFE) algorithm for the unbiased estimation of AFEV. The method uses variances between NWP forecasts and analyses (i.e. ‘perceived’ forecast errors) and assumptions about the time evolution of true error variances. PT14 successfully tested SAFE for the estimation of area mean error variances. In the present study, SAFE is extended by mitigating the effects of increased sampling noise and by accounting for the spatiotemporal evolution of forecast error variances, both critical for gridpoint-based applications. The enhanced method is evaluated in a Simulated Nature, Observations, Data Assimilation, and Prediction Environment using a quasi-geostrophic model and an ensemble Kalman Filter (EnKF). SAFE estimates of true analysis error variance are within 6% of the actual values, as compared to 24–55% deviations in EnKF estimates. The spatial correlation between estimated and actual true error variances was also found high (above 0.9) and comparable with EnKF estimates, but much higher than NMC method estimates (0.63–0.78). Estimates of the other two SAFE parameters, the growth rate and decorrelation of analysis and forecast error variances are within 3% of the corresponding actual values.

Convergence of Extreme Value Statistics in a Two-Layer Quasi-Geostrophic Atmospheric Model

We search for the signature of universal properties of extreme events, theoretically predicted for Axiom A flows, in a chaotic and high-dimensional dynamical system. We study the convergence of GEV (Generalized Extreme Value) and GP (Generalized Pareto) shape parameter estimates to the theoretical value, which is expressed in terms of the partial information dimensions of the attractor. We consider a two-layer quasi-geostrophic atmospheric model of the mid-latitudes, adopt two levels of forcing, and analyse the extremes of different types of physical observables (local energy, zonally averaged energy, and globally averaged energy). We find good agreement in the shape parameter estimates with the theory only in the case of more intense forcing, corresponding to a strong chaotic behaviour, for some observables (the local energy at every latitude). Due to the limited (though very large) data size and to the presence of serial correlations, it is difficult to obtain robust statistics of extremes in the case of the other observables. In the case of weak forcing, which leads to weaker chaotic conditions with regime behaviour, we find, unsurprisingly, worse agreement with the theory developed for Axiom A flows.

Arctic-Mid-Latitude Linkages in a Nonlinear Quasi-Geostrophic Atmospheric Model

A quasi-geostrophic three-level T63 model of the wintertime atmospheric circulation of the Northern Hemisphere has been applied to investigate the impact of Arctic amplification (increase in surface air temperatures and loss of Arctic sea ice during the last 15 years) on the mid-latitude large-scale atmospheric circulation. The model demonstrates a mid-latitude response to an Arctic diabatic heating anomaly. A clear shift towards a negative phase of the Arctic Oscillation (AO−) during low sea-ice-cover conditions occurs, connected with weakening of mid-latitude westerlies over the Atlantic and colder winters over Northern Eurasia. Compared to reanalysis data, there is no clear model response with respect to the Pacific Ocean and North America.

A data-driven kernel method assimilation technique for geophysical modelling

Incorporating the quantity and variety of observations in atmospheric and oceanographic assimilation and prediction models has become an increasingly complex task. Data assimilation allows for uneven spatial and temporal data distribution and redundancy to be addressed so that the models can ingest massive data sets. Traditional data assimilation methods introduce Kalman filters and variational approaches. This study introduces a family of algorithms, motivated by advances in machine learning. These algorithms provide an alternative approach to incorporating new observations into the analysis forecast cycle. The application of kernel methods to processing the states of a quasi-geostrophic numerical model is intended to demonstrate the feasibility of the method as a proof-of-concept. The speed, efficiency, accuracy and scalability in recovering unperturbed state trajectories establishes the viability of machine learning for data assimilation.

The Role of Eddy Diffusivity on a Poleward Jet Shift

Abstract The authors use a quasigeostrophic (QG) two-layer model to examine how eddies modify the meridional asymmetry of a zonal jet. The initial asymmetry is introduced in the model’s “radiative equilibrium state” and is intended to mimic a radiatively forced poleward jet shift simulated by climate models. The calculations show that the initial “poleward” jet shift in the two-layer model is amplified by eddy potential vorticity fluxes. This eddy-accentuation effect is greater as the baroclinicity of the equilibrium state is reduced, suggesting that seasonal variations in baroclinicity may help explain observed and modeled jet-shift sensitivity to season. The eddy-accentuated jet shift from the corresponding radiative equilibrium state is more clearly visible in the slowly varying, eddy-free reference state of Nakamura and Zhu. This reference state formally responds only to nonadvective, nonconservative processes, but ultimately arises from the advective eddy fluxes. The implication is that fast eddies are capable of driving a slowly varying jet shift, which may be balanced by nonconservative processes such as radiative heating/cooling.

Stability of Baroclinic Vortices on the β Plane and Implications for Transport

AbstractThis paper explores stability of and transport by baroclinic vortices on the β plane using a two-layer, quasigeostrophic model. The study adapts a wave–mean flow formalism and examines interactions between the axisymmetric flow (“the vortex”) and residuals (“the waves”). Unlike baroclinically unstable vortices on the f plane, such vortices on the β plane can be also vulnerable to barotropic instability as revealed by the globally integrated energy balance analysis. The spatial structure of energy fluxes shows the energy leakage inside the vortex core when its breakdown occurs. Mixing by stable and unstable vortical flows is quantified through the computation of finite-time Lyapunov exponent (FTLE) maps. Depending on the strength of wave radiation, the upper-layer FTLE maps of stable vortices show either an annulus or volute ring of vigorous mixing inside the vortex interior. This ring region is disrupted when the vortex becomes unstable. Both stable and unstable vortices show the wavy patterns of FTLE in the near and far fields. Despite the fact that the initial vortex resides in the top layer only, significant FTLE patterns are observed in the deep layer at later times. Lagrangian analysis of the vortex-induced change of large-scale tracer gradient demonstrates significant effects of vortex instability in the top layer and the importance of the wavelike anomalies in the bottom layer.

Subcritical convection of liquid metals in a rotating sphere using a quasi-geostrophic model

We study nonlinear convection in a rapidly rotating sphere with internal heating for values of the Prandtl number relevant for liquid metals ($Pr\in [10^{-2},10^{-1}]$). We use a numerical model based on the quasi-geostrophic approximation, in which variations of the axial vorticity along the rotation axis are neglected, whereas the temperature field is fully three-dimensional. We identify two separate branches of convection close to onset: (i) a well-known weak branch for Ekman numbers greater than $10^{-6}$, which is continuous at the onset (supercritical bifurcation) and consists of thermal Rossby waves and (ii) a novel strong branch at lower Ekman numbers, which is discontinuous at the onset. The strong branch becomes subcritical for Ekman numbers of the order of $10^{-8}$. On the strong branch, the Reynolds number of the flow is greater than $10^{3}$, and a strong zonal flow with multiple jets develops, even close to the nonlinear onset of convection. We find that the subcriticality is amplified by decreasing the Prandtl number. The two branches can co-exist for intermediate Ekman numbers, leading to hysteresis ($Ek=10^{-6}$, $Pr=10^{-2}$). Nonlinear oscillations are observed near the onset of convection for $Ek=10^{-7}$ and $Pr=10^{-1}$.

Canonical Transfer and Multiscale Energetics for Primitive and Quasigeostrophic Atmospheres

AbstractThe past years have seen the success of a novel and rigorous localized multiscale energetics formalism in a variety of ocean and engineering fluid applications. In a self-contained way, this study introduces it to the atmospheric dynamical diagnostics, with important theoretical updates and clarifications of some common misconceptions about multiscale energy. Multiscale equations are derived using a new analysis apparatus—namely, multiscale window transform—with respect to both the primitive equation and quasigeostrophic models. A reconstruction of the “atomic” energy fluxes on the multiple scale windows allows for a natural and unique separation of the in-scale transports and cross-scale transfers from the intertwined nonlinear processes. The resulting energy transfers bear a Lie bracket form, reminiscent of the Poisson bracket in Hamiltonian mechanics; hence, we would call them “canonical.” A canonical transfer process is a mere redistribution of energy among scale windows, without generating or destroying energy as a whole. By classification, a multiscale energetic cycle comprises available potential energy (APE) transport, kinetic energy (KE) transport, pressure work, buoyancy conversion, work done by external forcing and friction, and the cross-scale canonical transfers of APE and KE, which correspond respectively to the baroclinic and barotropic instabilities in geophysical fluid dynamics. A buoyancy conversion takes place in an individual window only, bridging the two types of energy, namely, KE and APE; it does not involve any processes among different scale windows and is hence basically not related to instabilities. This formalism is exemplified with a preliminary application to the study of the Madden–Julian oscillation.

A Short-Term Negative Eddy Feedback on Midlatitude Jet Variability due to Planetary Wave Reflection

AbstractA three-level quasigeostrophic model on the sphere is used to identify the physical nature of the negative planetary wave feedback on midlatitude jet variability. A first approach consists of studying the nonlinear evolution of normal-mode disturbances in a baroclinic westerly zonal jet. For a low-zonal-wavenumber disturbance, successive acceleration and deceleration of the jet occur as a result of reflection of the wave on either side of the jet. The planetary wave deposits momentum in opposite ways during its poleward or equatorward propagation. In contrast, a high-zonal-wavenumber disturbance is not reflected but absorbed within the subtropical critical layer. It thus only induces poleward momentum fluxes, which accelerate the jet and shift it slightly poleward. A long-term simulation forced by a relaxation toward a zonally symmetric temperature profile is then analyzed. Planetary waves are shown to be baroclinically excited. When they propagate equatorward, they induce an acceleration of the jet together with a slight poleward shift. About two-thirds of the planetary waves are absorbed by the subtropical critical layer, which allows the accelerated poleward-shifted jet to persist for a while. For the remaining third, the potential vorticity equatorward of the jet is so well homogenized that a reflection occurs. It is followed by an abrupt jet deceleration during the subsequent poleward propagation. The reflection of planetary waves on the poleward side of the jet is more systematic because of the quasi-permanent presence of a turning latitude there. This negative planetary wave feedback is shown to act more on pulses of the jet than on its latitudinal shifts.

Dynamic modeling of the horizontal eddy viscosity coefficient for quasigeostrophic ocean circulation problems

This paper puts forth a simplified dynamic modeling strategy for the eddy viscosity coefficient parameterized in space and time. The eddy viscosity coefficient is dynamically adjusted to the local structure of the flow using two different nonlinear eddy viscosity functional forms to capture anisotropic dissipation mechanism, namely, (i) the Smagorinsky model using the local strain rate field, and (ii) the Leith model using the gradient of the vorticity field. The proposed models are applied to the one-layer and two-layer wind-driven quasigeostrophic ocean circulation problems, which are standard prototypes of more realistic ocean dynamics. Results show that both models capture the quasi-stationary ocean dynamics and provide the physical level of eddy viscosity distribution without using any a priori estimation. However, it is found that slightly less dissipative results can be obtained by using the dynamic Leith model. Two-layer numerical experiments also reveal that the proposed dynamic models automatically parameterize the subgrid-scale stress terms in each active layer. Furthermore, the proposed scale-aware models dynamically provide higher values of the eddy viscosity for smaller resolutions taking into account the local resolved flow information, and addressing the intimate relationship between the eddy viscosity coefficients and the numerical resolution employed by the quasigeostrophic models.