Quasi-geostrophic Research Articles

Numerical investigation of the Danilov inequality for two-layer quasi-geostrophic systems

In the statistical steady state of forced-dissipative turbulence governed by the 2D incompressible Navier–Stokes (NS) equation, the difference between the energy flux ΠE(k) and the enstrophy flux ΠZ(k), k2ΠE(k) − ΠZ(k), where k is the wavenumber, has been proved to be negative outside the forcing wavenumber range. This is often referred to as the Danilov inequality and is important for determining the directions of the energy and enstrophy fluxes in the inertial ranges for 2D NS turbulence. We investigate numerically whether or not this inequality holds for two-layer quasi-geostrophic (QG) potential vorticity equations, which are a generalization of the vorticity equation for the 2D NS system. In two-layer QG systems forced by a horizontally homogeneous, baroclinically unstable basic flow, the corresponding difference between the total energy and total potential enstrophy fluxes is mathematically sign-indefinite due to the presence of the internal forcing term. Moreover, even if we concentrate on outside of the internal forcing range, the flux difference is still sign-indefinite when the dissipation mechanisms are asymmetric between the upper and lower layers. However, in our numerical experiments adopting both the internal forcing and asymmetric dissipations, the flux difference is negative across the whole wavenumber range. That is, the Danilov inequality continues to hold for two-layer QG systems, even though this cannot be proved mathematically.

Relative particle dispersion in two-dimensional and quasi-geostrophic turbulence

In this paper, phenomenological developments are used to explore relative particle dispersion (RPD) in two-dimensional (2D) and quasi-geostrophic (QG) fully-developed turbulence (FDT). The role played by the 2D and QG FDT cascade physics underlying this process is given special attention. Prevalence of spatial intermittency effects in 2D FDT enstrophy cascade, however small, is shown to lead to a structural change in the RPD growth law, more specifically the development of power-law scaling of RPD; this corroborates the difficulty in observing Lin’s (1972) exponential scaling law in laboratory experiments (Jullien (2003)). QG effects are found to lead to an enhanced RPD in the baroclinic regime of the energy cascade, which seems to be traceable to a negative eddy-viscosity characterizing the latter regime. They are also found to lead to particle clumping in the baroclinic regime of the enstrophy cascade (this aspect appears to be associated with the tendency of divorticity sheets to occur near the vortex nulls where particle clumping is known, as per numerical simulations and laboratory experiments, to be favored to occur). These results are developed from the established scaling relations for 2D and QG FDT and are validated further via alternative dimensional/scaling developments for 2D and QG FDT similar to the one given for 3D FDT by Batchelor and Townsend (1956). The feasibility of spatial intermittency effects is underscored via the nonlinear scaling dependence of RPD on the enstrophy (or energy) dissipation rate.

Numerical Development and Evaluation of an Energy Conserving Conceptual Stochastic Climate Model

AbstractIn this study we aim to present the successful development of an energy conserving conceptual stochastic climate model based on the inviscid 2-layer Quasi-Geostrophic (QG) equations. The stochastic terms have been systematically derived and introduced in such away that the total energy is conserved. In this proof of concept studywe give particular emphasis to the numerical aspects of energy conservation in a highdimensional complex stochastic system andwe analyzewhat kind of assumptions regarding the noise should be considered in order to obtain physical meaningful results. Our results show that the stochastic model conserves energy to an accuracy of about 0.5% of the total energy; this level of accuracy is not affected by the introduction of the noise, but is mainly due to the level of accuracy of the deterministic discretization of the QG model. Furthermore, our results demonstrate that spatially correlated noise is necessary for the conservation of energy and the preservation of important statistical properties, while using spatially uncorrelated noise violates energy conservation and gives unphysical results. A dynamically consistent spatial covariance structure is determined through Empirical Orthogonal Functions (EOFs). We find that only a small number of EOFs is needed to get good results with respect to energy conservation, autocorrelation functions, PDFs and eddy length scale when comparing a deterministic control simulation on a 512 × 512 grid to a stochastic simulation on a 128 × 128 grid. Our stochastic approach has the potential to seamlessly be implemented in comprehensive weather and climate prediction models.

Approaching Earth’s core conditions in high-resolution geodynamo simulations

SUMMARY The geodynamo features a broad separation between the large scale at which Earth’s magnetic field is sustained against ohmic dissipation and the small scales of the turbulent and electrically conducting underlying fluid flow in the outer core. Here, the properties of this scale separation are analysed using high-resolution numerical simulations that approach closer to Earth’s core conditions than earlier models. The new simulations are obtained by increasing the resolution and gradually relaxing the hyperdiffusive approximation of previously published low-resolution cases. This upsizing process does not perturb the previously obtained large-scale, leading-order quasi-geostrophic (QG) and first-order magneto-Archimedes-Coriolis (MAC) force balances. As a result, upsizing causes only weak transients typically lasting a fraction of a convective overturn time, thereby demonstrating the efficiency of this approach to reach extreme conditions at reduced computational cost. As Earth’s core conditions are approached in the upsized simulations, Ohmic losses dissipate up to 97 per cent of the injected convective power. Kinetic energy spectra feature a gradually broadening self-similar, power-law spectral range extending over more than a decade in length scale. In this range, the spectral energy density profile of vorticity is shown to be approximately flat between the large scale at which the magnetic field draws its energy from convection through the QG-MAC force balance and the small scale at which this energy is dissipated. The resulting velocity and density anomaly planforms in the physical space consist in large-scale columnar sheets and plumes, respectively, co-existing with small-scale vorticity filaments and density anomaly ramifications. In contrast, magnetic field planforms keep their large-scale structure after upsizing. The small-scale vorticity filaments are aligned with the large-scale magnetic field lines, thereby minimizing the dynamical influence of the Lorentz force. The diagnostic outputs of the upsized simulations are more consistent with the asymptotic QG-MAC theory than those of the low-resolution cases that they originate from, but still feature small residual deviations that may call for further theoretical refinements to account for the structuring constraints of the magnetic field on the flow.

Force balance in numerical geodynamo simulations: a systematic study

SUMMARY Dynamo action in the Earth’s outer core is expected to be controlled by a balance between pressure, Coriolis, buoyancy and Lorentz forces, with marginal contributions from inertia and viscous forces. Current numerical simulations of the geodynamo, however, operate at much larger inertia and viscosity because of computational limitations. This casts some doubt on the physical relevance of these models. Our work aims at finding dynamo models in a moderate computational regime which reproduce the leading-order force balance of the Earth. By performing a systematic parameter space survey with Ekman numbers in the range 10−6 ≤ E ≤ 10−4, we study the variations of the force balance when changing the forcing (Rayleigh number, Ra) and the ratio between viscous and magnetic diffusivities (magnetic Prandtl number, Pm). For dipole-dominated dynamos, we observe that the force balance is structurally robust throughout the investigated parameter space, exhibiting a quasi-geostrophic (QG) balance (balance between Coriolis and pressure forces) at zeroth order, followed by a first-order Magneto-Archimedean-Coriolis (MAC) balance between the ageostrophic Coriolis, buoyancy and Lorentz forces. At second order, this balance is disturbed by contributions from inertia and viscous forces. Dynamos with a different sequence of the forces, where inertia and/or viscosity replace the Lorentz force in the first-order force balance, can only be found close to the onset of dynamo action and in the multipolar regime. To assess the agreement of the model force balance with that expected in the Earth’s core, we introduce a parameter quantifying the distance between the first- and second-order forces. Analysis of this parameter shows that the strongest-field dynamos can be obtained close to the onset of convection (Ra close to critical) and in situations of reduced magnetic diffusivity (high Pm). Decreasing the Ekman number gradually expands this regime towards higher supercriticalities and lower values of Pm. Our study illustrates that most classical numerical dynamos are controlled by a QG-MAC balance, while cases where viscosity and inertia play a dominant role are the exception rather than the norm.

Exact solutions of asymmetric baroclinic quasi-geostrophic dipoles with distributed potential vorticity

An exact solution of a baroclinic three-dimensional vortex dipole in geophysical flows with constant background rotation and constant background stratification is provided under the quasi-geostrophic (QG) approximation. The motion of the dipole is unsteady but the potential vorticity contours move rigidly. The vortex comprises three potential vorticity anomaly modes, with a radial dependence given by the spherical Bessel functions and with azimuthal and polar dependences given by the spherical harmonics. The first mode, or spherical mode, accounts for the horizontal asymmetry of the vortex dipole and curvature of the dipole’s horizontal trajectory. The second mode, or dipolar mode, accounts for the speed of displacement of the vortex dipole. A third mode, or vertical tilting mode, accounts for the dipole’s vertical asymmetry. The QG vertical velocity field has two contributions: the first one is octupolar and depends entirely on the dipolar mode, and the second one is dipolar and depends on the nonlinear interaction between dipolar and vertical tilting modes.

Quasigeostrophic Controls on Precipitating Ascent in Monsoon Depressions

Abstract South Asian monsoon depressions are convectively coupled cyclonic vortices that form and intensify in a region of easterly vertical shear of the horizontal wind. Observations of maximum precipitation downshear of the cyclonic center have led to prior theories of quasigeostrophic (QG) control of moist convection in these storms. This study examines the interaction between adiabatic QG lifting and moist convection in monsoon depressions using an atmospheric reanalysis and idealized model. Inversion of the QG omega equation in the reanalysis shows that in the downshear, heavily precipitating region, adiabatic QG ascent, due to advection of vorticity and temperature, is comparable to diabatic ascent in the lower troposphere, while diabatic ascent dominates in the middle and upper troposphere. The causal influence of adiabatic QG lifting on precipitating ascent in monsoon depressions is then examined in the column QG modeling framework, where moist convection evolves in the presence of vorticity and temperature advection. The heavy observed precipitation rates are only simulated when moist convective heating amplifies QG ascent, with this interaction accounting for roughly 40% of the increase in precipitation relative to the basic state. Another 40% of this increase is produced by enhanced surface wind speed in the surface enthalpy flux parameterization, which represents the effect of cyclonic winds in the monsoon depression. Horizontal advection of the mean-state poleward moisture gradient accounts for the remaining 20% of the precipitation increase. In the upshear region, adiabatic QG subsidence and horizontal moisture advection both suppress precipitation, and are opposed by wind-enhanced surface enthalpy fluxes.

Hopf bifurcation and transition of three-dimensional wind-driven ocean circulation problem

The purpose of this article is three-fold. The first is to investigate the linear stability of the basic flow governed by the three-dimensional (3D) quasi-geostrophic (QG) equation with a steady wind force. We show that the basic flow will become unstable as the Reynolds number crosses a critical value. The second is to study the dynamic transition associated with the instability of the basic flow at the critical Reynolds number. It is proved that the type of the phase transition is determined by a parameter called transition number which is mainly determined by the Reynolds number and the aspect ratio. More precisely, the negative (positive) sign of the real part of the transition number corresponds to the continuous (catastrophic) transition. The last objective is to apply numerical experiments to examine the specific transition type for specified control parameters. We show that the real part of the transition number is always negative, leading to only the continuous transition.

Linear and nonlinear stability of a quasigeostrophic mixing layer subject to a uniform background shear

The aim of this work is to shed light by revisiting - through the kernel-wave (KW) perspective - the breakdown of a quasi-geostrophic (QG) mixing layer (or vortex strip/filament) in atmosphere under the influence of a background shear. The QG mixing layer is modelled with a family of quasi-Rayleigh velocity profiles in which the potential vorticity (PV) is constant in patches. In the KW perspective a counter-propagating Rossby wave (CRW) is created at each PV edge, i.e. the edge where a PV jump is located. The important parameters of our study are (i) the vorticity of the uniform shear m and (ii) the Rossby deformation radius Ld, which indicates how far the pressure perturbations can vertically propagate. While an adverse shear (m < 0) stabilizes the system, a favorable shear (m > 0) strengthens the instability. This is due to how the background shear affects the two uncoupled CRWs by shifting the optimal phase difference towards large (small) wavenumber when m < 0 (m > 0). As the QG environment is introduced a general weakening of the instability is noticed, particularly for m > 0...

Optimal Perturbations of an Oceanic Vortex Lens

The stability properties of a vortex lens are studied in the quasi geostrophic (QG) framework using the generalized stability theory. Optimal perturbations are obtained using a tangent linear QG model and its adjoint. Their fine-scale spatial structures are studied in details. Growth rates of optimal perturbations are shown to be extremely sensitive to the time interval of optimization: The most unstable perturbations are found for time intervals of about 3 days, while the growth rates continuously decrease towards the most unstable normal mode, which is reached after about 170 days. The horizontal structure of the optimal perturbations consists of an intense counter-shear spiralling. It is also extremely sensitive to time interval: for short time intervals, the optimal perturbations are made of a broad spectrum of high azimuthal wave numbers. As the time interval increases, only low azimuthal wave numbers are found. The vertical structures of optimal perturbations exhibit strong layering associated with high vertical wave numbers whatever the time interval. However, the latter parameter plays an important role in the width of the vertical spectrum of the perturbation: short time interval perturbations have a narrow vertical spectrum while long time interval perturbations show a broad range of vertical scales. Optimal perturbations were set as initial perturbations of the vortex lens in a fully non linear QG model. It appears that for short time intervals, the perturbations decay after an initial transient growth, while for longer time intervals, the optimal perturbation keeps on growing, quickly leading to a non-linear regime or exciting lower azimuthal modes, consistent with normal mode instability. Very long time intervals simply behave like the most unstable normal mode. The possible impact of optimal perturbations on layering is also discussed.

Analysis of the initiation of an extreme North Pacific jet retraction using piecewise tendency diagnosis

An unusually prolonged retraction of the tropopause‐level jet transformed the circulation in the North Pacific for weeks in mid‐February and March 2006 and preceded the development of a long‐lived negative Pacific–North American pattern. The initiation of this jet retraction, associated with a series of anticyclonic/LC1 wave‐breaking events at high latitude, is investigated through the lens of quasi‐geostrophic (QG) piecewise tendency analysis. Two key anticyclonic anomalies divert and retract the jet through serial LC1 wave‐breaking events, mostly confined to the 315–330 K isentropic layer and occurring in regions of strong deformation. The rapid development of the first anomaly coinciding with the beginning of retraction is diagnosed in a QG framework, in order to quantify contributions to QG height tendencies from various processes. The resultant analysis reveals that growth was induced by a deep potential vorticity intrusion that perturbed the jet, while the remainder of the ridge's life cycle was largely governed by upper‐level deformation in the jet exit region. Deformation facilitated both growth and decay, as the phasing between the ridge and deformation changed. Baroclinic development also contributed to growth but, unlike prior published analyses of anticyclones in the North Pacific, was not the dominant term.

Cluster Sampling Filters for Non-Gaussian Data Assimilation

This paper presents a fully non-Gaussian filter for sequential data assimilation. The filter is named the “cluster sampling filter”, and works by directly sampling the posterior distribution following a Markov Chain Monte-Carlo (MCMC) approach, while the prior distribution is approximated using a Gaussian Mixture Model (GMM). Specifically, a clustering step is introduced after the forecast phase of the filter, and the prior density function is estimated by fitting a GMM to the prior ensemble. Using the data likelihood function, the posterior density is then formulated as a mixture density, and is sampled following an MCMC approach. Four versions of the proposed filter, namely C ℓ MCMC , C ℓ HMC , MC- C ℓ HMC , and MC- C ℓ HMC are presented. C ℓ MCMC uses a Gaussian proposal density to sample the posterior, and C ℓ HMC is an extension to the Hamiltonian Monte-Carlo (HMC) sampling filter. MC- C ℓ MCMC and MC- C ℓ HMC are multi-chain versions of the cluster sampling filters C ℓ MCMC and C ℓ HMC respectively. The multi-chain versions are proposed to guarantee that samples are taken from the vicinities of all probability modes of the formulated posterior. The new methodologies are tested using a simple one-dimensional example, and a quasi-geostrophic (QG) model with double-gyre wind forcing and bi-harmonic friction. Numerical results demonstrate the usefulness of using GMMs to relax the Gaussian prior assumption especially in the HMC filtering paradigm.

Two Modes of Vertical Velocity in Subsurface Mesoscale Eddies

AbstractTwo fundamental modes of vertical velocity (w) in mesoscale subsurface eddies are described using the quasigeostrophic (QG) approximation and nonhydrostatic numerical modeling. The first mode of w (the spheroidal mode) arises when a spheroidal upright subsurface eddy acquires horizontal eccentricity and becomes an ellipsoid, still upright, vertically symmetric vortex. In this case, the vertical displacement of isopycnals vanishes at the middepth z = 0. Conservation of potential vorticity anomaly (PVA) on elliptical concave/convex isopycnals entails a three‐dimensional octupolar pattern of w which also vanishes at z = 0. The second mode of w (the tilted mode) arises when the eddy remains spheroidal but its vertical axis tilts relative to the vertical direction. In this case, the displacement of isopycnals is largest at the middepth z = 0 and has a dipolar distribution. The associated w is largest at the middepth and develops also a dipolar pattern. In both spheroidal and tilted modes, the vertical velocity pattern may be inferred from the fast advection of PVA conserving fluid particles on slower translating concave/convex or tilted isopycnals. This implies that the vertical velocity of both modes is approximately QG and may be correctly inferred from the QG omega equation as long as the Rossby number remains small. Under more general circumstances, the vortex is both spheroidal and tilted. In this case, both spheroidal and tilted modes coexist but remain, to a large extent, uncoupled, rotating with different and, at least at a first order of approximation, constant phase speeds.

Comparing a Multivariate Global Ocean State Estimate With High-Resolution in Situ Data: An Anticyclonic Intrathermocline Eddy Near the Canary Islands

The provision of high-resolution \textit{in situ} oceanographic data is key for the ongoing verification, validation and assessment of operational products, such as those provided by the Copernicus Marine Core Service (CMEMS). Here we analyze the ability of ARMOR3D - a multivariate global ocean state estimate that is available from CMEMS - to reconstruct a mesoscale anticyclonic intrathermocline eddy that was previously sampled with high-resolution independent \textit{in situ} observations. ARMOR3D is constructed by merging remote sensing observations with \textit{in situ} vertical profiles of temperature and salinity obtained primarily from the Argo network. \textit{In situ} data from CTDs and an Acoustic Doppler Current Profiler were obtained during an oceanographic cruise near the Canary Islands (Atlantic ocean). The analysis of the ARMOR3D product using the \textit{in situ} data is done over (i) a high-resolution meridional transect crossing the eddy center and (ii) a three-dimensional grid centered on the eddy center. An evaluation of the hydrographic eddy signature and derived dynamical variables, namely geostrophic velocity, vertical vorticity and quasi-geostrophic (QG) vertical velocity, demonstrates that the ARMOR3D product is able to reproduce the vertical hydrographic structure of the independently sampled eddy below the seasonal pycnocline, with the caveat that the flow is surface intensified and the seasonal pycnocline remains flat. Maps of ARMOR3D density show the signature of the eddy, and agreement with the elliptical eddy shape seen in the \textit{in situ} data. The major eddy axes are oriented NW-SE in both data sets. The estimated radius for the \textit{in situ} eddy is $\sim$ 46 km; the ARMOR3D radius is significantly larger at $\sim$ 92 km and is considered an overestimation that is inherited from an across-track altimetry sampling issue. The ARMOR3D geostrophic flow is underestimated by a factor of 2, with maxima of 0.11 (-0.19) m s$^{-1}$ at the surface, which implies an underestimation of the local Rossby number by a factor of 3. Both the \textit{in situ} and ARMOR3D eddies have decelerating flows at their northern edges. The ARMOR3D QG vertical velocity distribution has upwelling/downwelling cells located along the eddy periphery and similar magnitudes to the \textit{in situ}-derived QG vertical velocity.

On the Selection of Localization Radius in Ensemble Filtering for Multiscale Quasigeostrophic Dynamics

Covariance localization remedies sampling errors due to limited ensemble size in ensemble data assimilation. Previous studies suggest that the optimal localization radius depends on ensemble size, observation density and accuracy, as well as the correlation length scale determined by model dynamics. A comprehensive localization theory for multiscale dynamical systems with varying observation density remains an active area of research. Using a two-layer quasigeostrophic (QG) model, this study systematically evaluates the sensitivity of the best Gaspari–Cohn localization radius to changes in model resolution, ensemble size, and observing networks. Numerical experiment results show that the best localization radius is smaller for smaller-scale components of a QG flow, indicating its scale dependency. The best localization radius is rather insensitive to changes in model resolution, as long as the key dynamical processes are reasonably well represented by the low-resolution model with inflation methods that account for representation errors. As ensemble size decreases, the best localization radius shifts to smaller values. However, for nonlocal correlations between an observation and state variables that peak at a certain distance, decreasing localization radii further within this distance does not reduce analysis errors. Increasing the density of an observing network has two effects that both reduce the best localization radius. First, the reduced observation error spectral variance further constrains prior ensembles at large scales. Less large-scale contribution results in a shorter overall correlation length, which favors a smaller localization radius. Second, a denser network provides more independent pieces of information, thus a smaller localization radius still allows the same number of observations to constrain each state variable.

A New Interpretation of Vortex-Split Sudden Stratospheric Warmings in Terms of Equilibrium Statistical Mechanics

Abstract Vortex-split sudden stratospheric warmings (S-SSWs) are investigated by using the Japanese 55-year Reanalysis, a spherical barotropic quasigeostrophic (QG) model, and equilibrium statistical mechanics. The statistical mechanics theory predicts a large-scale steady state as the most probable outcome of turbulent stirring, and such a state can be computed without describing all the details of the dynamics. The theory is applied to a disk domain that is modeled on the polar cap north of 45°N in the stratosphere. The equilibrium state is obtained by computing the maximum of an entropy functional. In the range of parameters relevant to the winter stratosphere, this state is anticyclonic. By contrast, cyclonic states are quasi-stationary states corresponding to saddle points of the entropy functional. These results indicate that the mean state of the stratosphere associated with the polar vortex is not close to an equilibrium state but to a quasi-stationary state. The theoretical calculations are compared with the results of a quasi-static experiment in which a wavenumber-2 topographic amplitude is increased linearly and slowly with time. The results suggest that the S-SSW can be qualitatively interpreted as the transition from the cyclonic quasi-stationary state toward the anticyclonic equilibrium state. The polar vortex splits during the transition toward the equilibrium state.

Solutions of quasi-geostrophic turbulence in multi-layered configurations

We consider quasi-geostrophic (QG) models in two- and three-layers that are useful in theoretical studies of planetary atmospheres and oceans. In these models, the streamfunctions are given by (1+2) partial differential systems of evolution equations. A two-layer QG model, in a simplified version, is dependent exclusively on the Rossby radius of deformation. However, the f-plane QG point vortex model contains factors such as the density, thickness of each layer, the Coriolis parameter, and the constant of gravitational acceleration, and this two-layered model admits a lesser number of Lie point symmetries, as compared to the simplified model. Finally, we study a three-layer oceanography QG model of special interest, which includes asymmetric wind curl forcing or Ekman pumping, that drives double-gyre ocean circulation. In three-layers, we obtain solutions pertaining to the wind-driven doublegyre ocean flow for a range of physically relevant features, such as lateral friction and the analogue parameters of the f-plane QG model. Zero-order invariants are used to reduce the partial differential systems to ordinary differential systems. We determine conservation laws for these QG systems via multiplier methods.

The 23–26 September 2012 U.K. Floods: Using PV Surgery to Quantify Sensitivity to Upper-Level Forcing

Major river flooding affected the United Kingdom in late September 2012 as a slow-moving extratropical cyclone brought over 150 mm of rain to parts of northern England and north Wales. The cyclone deepened over the United Kingdom on 24–26 September as a potential vorticity (PV) anomaly approached from the northwest, elongated into a PV streamer, and wrapped around the cyclone. The strength and position of the PV anomaly is modified in the initial conditions of Weather Research and Forecasting Model simulations, using PV surgery, to examine whether different upper-level forcing, or different phasing between the PV anomaly and cyclone, could have produced an even more extreme event. These simulations reveal that quasigeostrophic (QG) forcing for ascent ahead of the anomaly contributed to the persistence of the rainfall over the United Kingdom. Moreover, weakening the anomaly resulted in lower rainfall accumulations across the United Kingdom, suggesting that the impact of the event might be proportional to the strength of the upper-level QG forcing. However, when the anomaly was strengthened, it rotated cyclonically around a large-scale trough over Iceland rather than moving eastward as in the verifying analysis, with strongly reduced accumulated rainfall across the United Kingdom. A similar evolution developed when the anomaly was moved farther away from the cyclone. Conversely, moving the anomaly nearer to the cyclone produced a similar solution to the verifying analysis, with slightly increased rainfall totals. These counterintuitive results suggest that the verifying analysis represented almost the highest-impact scenario possible for this flooding event when accounting for sensitivity to the initial position and strength of the PV anomaly.

Precipitating Quasigeostrophic Equations and Potential Vorticity Inversion with Phase Changes

Abstract Precipitating versions of the quasigeostrophic (QG) equations are derived systematically, starting from the equations of a cloud-resolving model. The presence of phase changes of water from vapor to liquid and vice versa leads to important differences from the dry QG case. The precipitating QG (PQG) equations, in their simplest form, have two variables to describe the full system: a potential vorticity (PV) variable and a variable M including moisture effects. A PV-and-M inversion allows the determination of all other variables, and it involves an elliptic partial differential equation (PDE) that is nonlinear because of phase changes between saturated and unsaturated regions. An example PV-and-M inversion is provided for an idealized cold-core cyclone with two vertical levels. A key point illustrated by this example is that the phase interface location is unknown a priori from PV and M, and it is discovered as part of the inversion process. Several choices of a moist PV variable are discussed, including subtleties that arise because of phase changes. Boussinesq and anelastic versions of the PQG equations are described, as well as moderate and asymptotically large rainfall speeds. An energy conservation principle suggests that the model has firm physical and mathematical underpinnings. Finally, an asymptotic analysis provides a systematic derivation of the PQG equations, which arise as the limiting dynamics of a moist atmosphere with phase changes, in the limit of rapid rotation and strong stratification in terms of both potential temperature and equivalent potential temperature.

On the vertical structure and stability of the Lofoten vortex in the Norwegian Sea

The Lofoten Vortex (LV), a quasi-permanent anticyclonic eddy in the Lofoten Basin of the Norwegian Sea, is investigated with an eddy-permitting primitive equation model nested into the ECCO2 ocean state estimate. The LV, as simulated by the model, extends from the sea surface to the ocean bottom at about 3000m and has the subsurface core between 50m and 1100m depths. Above and below the vortex core the relative vorticity signal decreases in amplitude while the radius increases by as much as 25–30% relative to the values in the core. Analyzing the model run, we show that the vertical structure of the LV can be casted into four standard configurations, each of which forms a distinct cluster in the parameter space of potential vorticity anomalies in and above the LV core. The stability of the LV for each of the configurations is then studied with three-layer and a two-layer (in winter) quasi-geostrophic (QG) models over a flat bottom as well as over a realistic topography. The QG results show a number of common features with those of the primitive equation model. Thus, among the azimuthal modes dominating the LV instability, both the QG model and the primitive equation model show a major role the 2nd and 3rd modes. In the QG model simulations the LV is the subject of a rather strong dynamic instability, penetrating deep into the core. The results predict 50–95% volume loss from the vortex within 4–5 months. Such a drastic effect is not observed in the primitive equation model, where, for the same intensity of perturbations, only 10–30% volume loss during the same period is detected. Taking into account the gently sloping topography of the central part of the Lofoten basin and the mean flow in the QG model, brings the rate of development of instability close to that in the primitive equation model. Some remaining differences in the two models are discussed. Overall, the LV decay rate obtained in the models is slow enough for eddy mergers and convection to restore the thermodynamic properties of the LV, primarily re-building its potential energy anomaly. This justifies the quasi-permanent presence of the LV in the Lofoten Basin.