Two-layer Quasi-geostrophic Model Research Articles

Baroclinic vortex pulsars in unstable westward flows

We present a computational modeling study of geophysical coherent vortices embedded in horizontally homogeneous, baroclinically unstable, westward background flows with vertical shear. Within an idealized two-layer quasigeostrophic beta-plane model, we discovered two types of robust vortex-wave structures with distinct properties, which remain asymmetric and nonstationary in statistically-equilibrated turbulent flow regimes. The corresponding vortices, referred to as baroclinic vortex pulsars, are characterized by intense vorticity core coupled to the Rossby wave wake. The main conclusion — on the top of various analyses discussed in the paper — are that the vortex pulsars are fundamentally non-isolated coherent vortices, because they extract energy from the background circulation and expel excess potential vorticity, accumulating due to down-gradient material propagation, back into the environment. Both types may coexist as multiple statistically equilibrated states in some range of physical parameters, complicating any parameterization of eddy effects in climate-type models.

On a probabilistic evolutionary approach to ocean modelling: From Lorenz-63 to idealized ocean models

In this study we develop an alternative way to model the ocean reflecting the chaotic nature of ocean flows and uncertainty of ocean models — instead of making use of classical deterministic or stochastic differential equations we offer a probabilistic evolutionary approach (PEA) that capitalizes on the use of probabilistic dynamics in phase space. The main feature of the data-driven version of PEA proposed in this work is that it does not require to know the physics behind the flow dynamics to model it. Within the PEA framework we develop two probabilistic evolutionary methods, which are based on probabilistic evolutionary models using quasi time-invariant structures in phase space.The methods have been tested on complete and incomplete reference data sets generated by the Lorenz 63 system and by an idealized two-layer quasi-geostrophic model. The results show that both methods reproduce large- and small-scale features of the reference flow by keeping the probabilistic dynamics within the phase space of the reference flow. The proposed approach offers appealing benefits and a great flexibility to ocean modellers working with mathematical models and measurements. The most remarkable one is that it provides an alternative to the mainstream ocean parameterizations, requires no modification of existing ocean models, and is easy to implement. Moreover, it does not depend on the nature of input data, and therefore could work with both numerically-computed flows and real measurements from different sources (drifters, weather stations, etc.).

The quasigeostrophic two-layer model: Lax representations and conservation laws

We find two Lax representations for the quasigeostrophic two-layer model equations and use them to construct special Bäcklund transformations. Then we find all the conservation laws with the cosymmetries of order less or equal to 2 for the equations under the study.

On the dynamics of nonlinear barotropic–baroclinic interactions through a coupled Gardner hierarchies approach

The aim of this paper is on the propagations of barotropic–baroclinic coherent structures based on the two-layer quasi-geostrophic model (2LQG) through a Fourier spectrum compliant approach. First, by introducing the barotropic and baroclinic stream functions starting from the 2LQG model, a new coupled Gardner-type evolution equations, representing the interaction processes between the barotropic flow and baroclinic one, are obtained by combining the multi-scale method and the perturbation expansion method. Second, based on the obtained coupled model equations, the physical mechanisms of the nonlinear barotropic–baroclinic interaction are analyzed qualitatively. Within the range of parameters chosen in this paper, quantitative results show that the basic flow, the β effect, and the bottom topography are necessary factors to excite the nonlinear Rossby isolated waves. The results also declare that the dipole-like blockings are readily excited in the flow field and move slowly eastward in both barotropic and baroclinic flow fields.

Scaling for Saturated Moist Quasigeostrophic Turbulence

Abstract Much of our conceptual understanding of midlatitude atmospheric motion comes from two-layer quasigeostrophic (QG) models. Traditionally, these QG models do not include moisture, which accounts for an estimated 30%–60% of the available energy of the atmosphere. The atmospheric moisture content is expected to increase under global warming, and therefore, a theory for how moisture modifies atmospheric dynamics is crucial. We use a two-layer moist QG model with convective adjustment as a basis for analyzing how latent heat release and large-scale moisture gradients impact the scalings of a midlatitude system at the synoptic scale. In this model, the degree of saturation can be tuned independently of other moist parameters by enforcing a high rate of evaporation from the surface. This allows for study of the effects of latent heat release at saturation, without the intrinsic nonlinearity of precipitation. At saturation, this system is equivalent to the dry QG model under a rescaling of both length and time. This predicts that the most unstable mode shifts to smaller scales, the growth rates increase, and the inverse cascade extends to larger scales. We verify these results numerically and use them to verify a framework for the complete energetics of a moist system. We examine the spectral features of the energy transfer terms. This analysis shows that precipitation generates energy at small scales, while dry dynamics drive a significant broadening to larger scales. Cascades of energy are still observed in all terms, albeit without a clearly defined inertial range. Significance Statement The effect of moist processes, especially the impact of latent heating associated with condensation, on the size and strength of midlatitude storms is not well understood. Such insight is particularly needed in the context of global warming, as we expect moisture to play a more important role in a warmer world. In this study, we provide intuition into how including condensation can result in midlatitude storms that grow faster and have features on both larger and smaller scales than their dry counterparts. We provide a framework for quantifying these changes and verify it for the special case where it is raining everywhere. These findings can be extended to the more realistic situation where it is only raining locally.

Meridional Location and Profile of a Prematurely Separated WBC Extension Jet in a Two-Layer System

Abstract Western boundary currents (WBCs) under no-slip boundary conditions tend to separate from the coast prematurely (without reaching the intergyre boundary) and form eastward jets. This study theoretically considers the meridional structure and location of a prematurely separated WBC extension jet using a two-layer quasigeostrophic model. Assuming homogenized potential vorticity (PV) regions on both sides of and below the jet, we constructed a simple model for the meridional profiles of the zonal flows in the western subtropical gyre. This work clarifies that the meridional structure can be determined if two variables, such as the strength of the PV front (difference in PV across the jet) and the value of the streamfunction at the jet’s center, are given in addition to the meridional profile of the Sverdrup zonal flow and the vertical stratification. The zonal velocity profiles in both layers agreed well with those obtained by numerical experiments. When the jet is close to the intergyre boundary, the meridional location of the jet depends only on the front’s strength. When the northern recirculation gyre is detached from the intergyre boundary and the local wind effect on the jet is negligible, comparisons with the numerical experiments suggest that the jet’s central streamline connects to the central streamline of the eastward Sverdrup flow. We also found that a downward Ekman pumping velocity shifts the jet southward.

Dissipation Properties of Transport Noise in the Two-Layer Quasi-geostrophic Model

A stochastic version of the two-layer quasi-geostrophic model (2LQG) with multiplicative transport noise is analysed. This popular intermediate complexity model describes large scale atmosphere and ocean dynamics at the mid-latitudes. The transport noise, which acts on both layers, accounts for the unresolved small scales. After establishing the well-posedness of the perturbed equations, we show that, under a suitable scaling of the noise, the solutions converge to the deterministic 2LQG model with enhanced dissipation. Moreover, these solutions converge to the deterministic stationary ones on the long time horizon.

Evaluation of the sensitivity on mesoscale eddy associated with the sea surface height anomaly forecasting in the Kuroshio Extension

The sensitivity of the sea surface height anomaly (SSHA) forecasting on the accuracy of mesoscale eddies over the Kuroshio Extension region, which was determined by the conditional non-linear optimal perturbation (CNOP) method using a two-layer quasigeostrophic model, is evaluated by adopting multiply realistic marine datasets through an advanced particle filter assimilation method. It is shown that, if additional observations are preferentially assimilated to the sensitive area of mesoscale eddies identified by the CNOP, where the eddies present a clear high- to low-velocity gradient along the eddy rotation, the forecasting skill of the SSHA can be more significantly improved. It is also demonstrated that the forecasts of the SSHA in the region where the large-scale mean flow possesses much stronger barotropic and/or baroclinic instability tend to exhibit stronger sensitivity to the accuracy of the initial field in the sensitive area of mesoscale eddies. Therefore, more attention should be preferentially paid to the assimilation of the additional observations of the mesoscale eddies for the SSHA forecast in the region with a strong velocity shear of ocean circulation. The present study verifies the sensitivity on mesoscale eddies of SSHA forecasts derived by the two-layer quasigeostrophic model using multiply sets of realistic oceanic data, especially including observation and reanalysis data, which further additionally demonstrates the importance of targeted observations of mesoscale eddies to the SSHA forecast in the regions of strong velocity shear of ocean circulation and provides a more credible scientific basis for the field campaign of the targeted observations for mesoscale eddies associated with the SSHA forecasting.

Can We Accurately Quantify a Lateral Diffusivity from a Single Tracer Release?

Abstract Mixing along sloping isopycnals plays a key role in the transport and uptake of heat and carbon by the ocean. This mixing is quantified by a lateral diffusivity, which can be measured by tracking the lateral spreading of point release tracer patches. We present a definition for the area of a tracer patch, the time derivative of which provides the lateral diffusivity. To accurately estimate the diffusivity, an ensemble mean concentration field of many tracer release experiments is required. We use numerical experiments to quantify how accurately the “true” lateral diffusivity (obtained from the ensemble mean concentration field) can be estimated from a single tracer release experiment (one ensemble member). To simulate observational campaigns, we also estimate the diffusivity from a single tracer release that is spatially and/or temporally subsampled, quantifying how the error between the estimated diffusivity and the true diffusivity grows as this sampling resolution worsens. We perform these numerical experiments in a two-layer quasigeostrophic model of turbulent flow on a β plane, using an ensemble of 50 passive tracer release experiments, each initialized as a 2D Gaussian but with differing realizations of the turbulent flow. We find that the diffusivity estimates from the single tracer releases have a relative root-mean-square error (RMSE) of 1.43% from the true diffusivity. Subsampling a single tracer release experiment every 956 km increases the relative RMSE from the true diffusivity to 3.1%; also subsampling every 277 days raises this figure to 6.5%.

Exponential ergodicity for a stochastic two-layer quasi-geostrophic model

Ergodic properties of a stochastic medium complexity model for atmosphere and ocean dynamics are analyzed. More specifically, a two-layer quasi-geostrophic model for geophysical flows is studied, with the upper layer being perturbed by additive noise. This model is popular in the geosciences, for instance to study the effects of a stochastic wind forcing on the ocean. A rigorous mathematical analysis however meets with the challenge that in the model under study, the noise configuration is spatially degenerate as the stochastic forcing acts only on the top layer. Exponential convergence of solutions laws to the invariant measure is established, implying a spectral gap of the associated Markov semigroup on a space of Hölder continuous functions. The approach provides a general framework for generalized coupling techniques suitable for applications to dissipative SPDEs. In case of the two-layer quasi-geostrophic model, the results require the second layer to obey a certain passivity condition.

The Diabatic Rossby Vortex: Growth Rate, Length Scale, and the Wave–Vortex Transition

Abstract In idealized simulations of moist baroclinic instability on a sphere, the most unstable mode transitions from a periodic wave to an isolated vortex in sufficiently warm climates. The vortex mode is maintained through latent heating and shows the principal characteristics of a diabatic Rossby vortex (DRV) that has been found in a range of different simulations and observations of the current climate. Currently, there is no analytical theory for DRVs or understanding of the wave–vortex transition that has been found in warmer climates. Here, we introduce a minimal moist two-layer quasigeostrophic model with tilted boundaries capable of producing a DRV mode, and we derive growth rates and length scales for this DRV mode. In the limit of a convectively neutral stratification, the length scale of ascent of the DRV is the same as that of a periodic moist baroclinic wave, but the growth rate of the DRV is 54% faster. We explain the isolated structure of the DRV using a simple potential vorticity (PV) argument, and we create a phase diagram for when the most unstable solution is a periodic wave versus a DRV, with the DRV emerging when the moist static stability and meridional PV gradients are weak. Last, we compare the structure of the DRV mode with DRV storms found in reanalysis and with a DRV storm in a warm-climate simulation. Significance Statement Past research has identified a special class of midlatitude storm, dubbed the diabatic Rossby vortex (DRV), which derives its energy from the release of latent heat associated with condensation of water vapor and as such goes beyond the traditional understanding of midlatitude storm formation. DRVs have been implicated in extreme and poorly predicted forms of cyclogenesis along the east coast of the United States and the west coast of Europe with significant damage to property and human life. The purpose of this study is to develop a mathematical theory for the intensification rate and length scale of DRVs to gain a deeper understanding of the dynamics of these storms in current and future climates.

Interaction between an upper-layer point vortex and a bottom topography in a two-layer system

In this paper, the interaction between an upper-layer vortex and a bottom topography is investigated using an $f$ -plane two-layer quasi-geostrophic model with a point vortex and step-like topography. The contour dynamics method is used to formulate the model. A steadily propagating linear solution along the topography, known as the pseudoimage solution, is derived analytically for a weak point vortex, and the nonlinear solution is obtained numerically. Numerical experiments show that the nonlinear pseudoimage solution collapses with time. Saddle-node points in the velocity field are critical in this collapse. Even after the collapse, the point vortex propagates along the topography similarly to in the steadily propagating solution. Numerical experiments with various initial conditions show that the point vortex has two types of motion in this system: motion along the topography and motion away from the topography. In the latter case, the point vortex and lower-layer potential vorticity anomaly form a heton-like dipole structure. The motion classification results show that an anticyclonic (cyclonic) point vortex on the deeper (shallower) side is more likely to form a dipole structure than a cyclonic (anticyclonic) vortex on the deeper (shallower) side when their initial distance from the topography is the same.

Transitions of zonal flows in a two-layer quasi-geostrophic ocean model

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 system using the formalism of dynamic transition theory and establish two scenarios which completely describe this first transition. The generic scenario is that a conjugate pair of modes loses stability and a Hopf bifurcation occurs as a result. Under an appropriate set of parameters describing related midlatitude oceanic flows, we show that this first transition is continuous: a supercritical Hopf bifurcation occurs and a stable time periodic solution bifurcates. We also investigate the case of double Hopf bifurcations which occur when four modes of the linear stability problem simultaneously destabilize the zonal jet. In this case, we prove that, in the relevant parameter regime, the flow exhibits a continuous transition accompanied by a bifurcated attractor homeomorphic to \(S^3\). The topological structure of this attractor is analyzed in detail and is shown to depend on the system parameters. In particular, this attractor contains (stable or unstable) time-periodic solutions and a quasi-periodic solution.

How do discrete symmetries shape the stability of geophysical flows?

We examine the role discrete symmetries, time-reversal, and mirror symmetries play in the context of geophysical waves and instabilities. By looking at three special cases from the two-layer quasi-geostrophic model as well as developing a general framework for translating real-space transformations to Fourier space, we are able to (1) show that baroclinic instability is an example of spontaneous parity-time symmetry breaking; (2) show that pure parity symmetry for a fluid system is exactly analogous to charge-conjugation-parity symmetry in a condensed matter system; and (3) show that when a pure parity symmetry is broken, this is associated with the suppression of wave propagation. Furthermore, in the latter case, instability can arise without a corresponding symmetry breaking. This study highlights the role of symmetry breaking behind the dynamics of geophysical waves and instabilities.

Ensemble Riemannian data assimilation: towards large-scale dynamical systems

Abstract. This paper presents the results of the ensemble Riemannian data assimilation for relatively high-dimensional nonlinear dynamical systems, focusing on the chaotic Lorenz-96 model and a two-layer quasi-geostrophic (QG) model of atmospheric circulation. The analysis state in this approach is inferred from a joint distribution that optimally couples the background probability distribution and the likelihood function, enabling formal treatment of systematic biases without any Gaussian assumptions. Despite the risk of the curse of dimensionality in the computation of the coupling distribution, comparisons with the classic implementation of the particle filter and the stochastic ensemble Kalman filter demonstrate that, with the same ensemble size, the presented methodology could improve the predictability of dynamical systems. In particular, under systematic errors, the root mean squared error of the analysis state can be reduced by 20 % (30 %) in the Lorenz-96 (QG) model.

The most sensitive initial error of sea surface height anomaly forecasts and its implication for target observations of mesoscale eddies

Abstract We used the conditional nonlinear optimal perturbation (CNOP) approach to investigate the most sensitive initial error of sea surface height anomaly (SSHA) forecasts by using a two-layer quasigeostrophic model and revealed the importance of mesoscale eddies in initialization of the SSHA forecasts. Then, the CNOP-type initial errors for individual mesoscale eddies were calculated, revealing that the errors tend to occur in locations where the eddies present a clear high-to low-velocity gradient along the eddy rotation and the errors often have a shear SSHA structure present. Physically, we interpreted the rationality of the particular location and shear structure of the CNOP-type errors by barotropic instability from the perspective of the Lagrange expression of fluid motions. Numerically, we examined the sensitivity of the CNOP-type errors by using observing system simulation experiments (OSSEs). We concluded that if additional observations are preferentially implemented in the location where CNOP-type errors occur, especially with a particular array indicated by their shear structure, the forecast ability of the SSHA can be significantly improved. These results provide scientific guidance for the target observation of mesoscale eddies and therefore are very instructive for improving ocean state SSHA forecasts.

Moisture and the Persistence of Annular Modes

AbstractAnnular modes are the leading mode of variability in extratropical atmospheres, and a key source of predictability at midlatitudes. Previous studies of annular modes have primarily used dry atmospheric models, so that moisture’s role in annular mode dynamics is still unclear. In this study, a moist two-layer quasigeostrophic channel model is used to study the effects of moisture on annular mode persistence. Using a channel model allows moisture’s direct effects to be studied, rather than changes in persistence due to geometric effects associated with shifts in jet latitude on the sphere. Simulations are performed in which the strength of latent heat release is varied to investigate how annular mode persistence responds as precipitation becomes a leading term in the thermodynamic budget. At short lags (<20 model days, ≈4 Earth days), moisture increases annular mode persistence, reflecting weaker eddy activity that is less effective at disrupting zonal-mean wind anomalies. Comparisons to dry simulations with weaker mean flows demonstrate that moisture is particularly effective at damping high-frequency eddies, further enhancing short-lag persistence. At long lags (>20 model days), moisture weakly increases persistence, though it decreases the amplitudes of low-frequency annular mode anomalies. In the most realistic simulation, the greater short-lag persistence increases the e-folding time of the zonal index by 21 model days (≈4 Earth days). Moisture also causes a transition to propagating variability, though this does not seem to affect the leading mode’s persistence.

Nonlinear generation of long waves and the reversal of eddy momentum fluxes in a two-layer quasi-geostrophic model

AbstractAlthough classical theories of midlatitude momentum fluxes focus on the wave-mean flow interaction, wave-wave interactions may be important for generating long waves. It is shown in this study that this nonlinear generation has implications for eddy momentum fluxes in some regimes. Using a two-layer quasi-geostrophic model of a baroclinic jet on a β-plane, statistically steady states are explored in which the vertically integrated eddy momentum flux is divergent at the center of the jet, rather than convergent as in Earth-like climates. One moves towards this less familiar climate from more Earth-like settings by reducing either β, frictional drag, or the width of the baroclinic zone, or by increasing the upper bound of resolvable wavelengths by lengthening the zonal channel. Even in Earth-like settings, long waves diverge momentum from the jet, but they are too weak to compete with short unstable waves that converge momentum. We argue that long waves are generated by breaking of short unstable waves near their critical latitudes, where long waves converge momentum while diverging momentum at the center of the jet. Quasi-linear models with no wave-wave interaction can qualitatively capture the Earth-like regime but not the regime with momentum flux divergence at the center of the jet, because the nonlinear wave breaking and long wave generation processes are missing. Therefore, a more comprehensive theory of atmospheric eddy momentum fluxes should take into account the nonlinear dynamics of long waves.

Roles of vertical distributions of atmospheric transient eddy dynamical forcing and diabatic heating in midlatitude unstable air\u2013sea interaction

Atmospheric transient eddy dynamical forcing (TEDF)-driven midlatitude unstable air–sea interaction has recently been recognized as a crucial positive feedback for the maintenance of the extratropical decadal variabilities. Our recent theoretical work (Chen et al., Clim Dyn https://doi.org/10.1007/s00382-020-05405-0, 2020) has characterized such an interaction through building an analytical midlatitude barotropic atmospheric model coupled to a simplified upper oceanic model. This study extends the analytical model to including a two-layer quasi-geostrophic baroclinic atmospheric model and then identifies the roles of vertical distributions of atmospheric TEDF and diabatic heating in midlatitude unstable air–sea interaction. It is found that midlatitude air–sea coupling with more realistic vertical profiles of atmospheric TEDF and diabatic heating destabilizes oceanic Rossby wave modes over the entire range of zonal wavelengths, in which the most unstable coupled mode features an equivalent barotropic atmospheric low (high) pressure over a cold (warm) oceanic surface. Spatial structure and period of the most unstable mode are more consistent with the observation than those from in previous model. Although either TEDF or diabatic heating alone can lead to a destabilized coupled mode, the former makes a dominant contribution to the instability. The increase of low-layer TEDF stimulates the instability more effectively if the TEDF in upper layer is larger than in lower layer, while the TEDF in either high or low layers can individually cause the instability. The surface heating always destabilizes the air–sea interaction, while the mid-level heating always decays the coupled mode. The results of this study further confirm the TEDF-driven positive feedback mechanism in midlatitude air–sea interaction proposed by recent observational and numerical experiment studies.

Linear stability analysis for flows over sinusoidal bottom topography

Abstract