Two-level Quasi-geostrophic Model Research Articles

Regular and Chaotic Oscillations in a Geostrophic Flow with Vertical Shear

In the framework of a two-level quasi-geostrophic model, the stability of flow with a constant vertical shear is investigated. Analytical expressions for the increment of perturbation growth in linear stability theory were obtained. The Galerkin method with three basic Fourier harmonics was used to describe the nonlinear dynamics of perturbations. A nonlinear system of ordinary differential equations is formulated for amplitudes of Fourier harmonics. It is shown that in the absence of bottom friction all solutions of the system describe periodic mode of nonlinear oscillations or vascillations. The situation changes fundamentally in the model with bottom friction. In this case, for a wide range of parameter values, the system solutions exhibit complex chaotic behavior. Thus, chaos or turbulence emerges for large-scale motions.

Nonmodal growth of atmospheric disturbances relevant to the East Asian pressure surge in boreal winter

Significant events of East Asian pressure surge are identified based on sea level pressure for boreal winters of the period 1979–2016. These events typically grow from a high pressure anomaly over Northern Europe and the northwestern Russia and attain their peak over Siberia around ten days after the initiation. The structure of the pressure surge at its peak intensity mimics that of the semi-permanent Siberian High. With a two-level quasi-geostrophic (QG) model, a nonmodal instability analysis of the winter climatological flow reveals that one of the most rapidly amplifying “optimal disturbances” with an optimization time of 2 days resembles the structure of the East Asian pressure surge. This optimal disturbance originates from small-scale precursor disturbances over Northern Europe and the Barents Sea. Such initial disturbances grow in magnitude via Siberia towards East Asia, forming a low-level ridge extending to around 50° N. The optimal disturbance successfully captures the temporal and spatial scales and the overall structure of the circulation anomaly associated with the pressure surge, albeit with varying degrees of accuracy of the location of individual trough/ridge of the surge anomaly. Additional experiments suggest that (1) adding damping effect to the two-level QG model extends the low-level ridge further south to around 40° N and (2) the upper level wave train feature associated with pressure surge can be captured even in a barotropic model by optimal disturbances with an optimization time of 4 days.

A Nonmodal Instability Perspective of the Declining Northern Midlatitude Synoptic Variability in Boreal Summer

AbstractThe Pacific–North America–North Atlantic sector in general experienced a dryer and warmer climate in summer during the past 40 years. These changes are partly associated with declining midlatitude synoptic variability in boreal summer, especially over the two ocean basins. A nonmodal instability analysis of the boreal summer background flow is conducted for two periods, 1979–94 and 2000–15, to understand dynamical processes potentially responsible for the observed decline of synoptic variability. The synoptic variability associated with fast, nonmodal growth of atmospheric disturbances shows a decline over northern midlatitudes in the later period, in both a barotropic model and a two-level quasigeostrophic model. These results highlight the importance of the changing summer background flow in contributing to the observed changes in synoptic variability. Also discussed are factors likely associated with background flow changes including sea surface temperature and sea ice change.

Hybrid Levenberg–Marquardt and weak-constraint ensemble Kalman smoother method

Abstract. The ensemble Kalman smoother (EnKS) is used as a linear least-squares solver in the Gauss–Newton method for the large nonlinear least-squares system in incremental 4DVAR. The ensemble approach is naturally parallel over the ensemble members and no tangent or adjoint operators are needed. Furthermore, adding a regularization term results in replacing the Gauss–Newton method, which may diverge, by the Levenberg–Marquardt method, which is known to be convergent. The regularization is implemented efficiently as an additional observation in the EnKS. The method is illustrated on the Lorenz 63 model and a two-level quasi-geostrophic model.

Linear instability of broad baroclinic zones

Baroclinic energy conversions play a central role in mid-latitude synoptic-scale dynamics, drawing energy from the meridional temperature gradient and feeding it into the eddy dynamics. The physical mechanisms underlying these conversions can be investigated through the theory of small-amplitude instabilities growing on a steady basic state. In a recent study the author analysed the instability associated with a boundary temperature transition concentrated in a narrow front. This paper looks at the complementary problem of a broad baroclinic zone. The solution is applicable both to the Eady model and to the Phillips two-level quasi-geostrophic model. The reduction in the growth rate relative to an infinitely broad, homogeneous baroclinic mne is inversely proportional to the width of the baroclinic zone, as found in previous studies. The constant of proportionality is expressed in a simplified and more general form, related to the derivative with respect to zonal wave number of the complex phase speed of the homogeneous problem. If the flow is purely baroclinic the maximum growth rate is reduced compared with that of a homogeneous, infinitely broad frontal zone by a factor 1 - (2kmaxr)-1. where kmax is the wave number of the most unstable mode of the homogeneous problem and r is the width of the baroclonic zone. The width, r, is defined in terms of the second meridional derivative of the boundary temperature gradient evaluated at the maximum of the latter. Results in the literature give consistent, though less general, expressions for the growth rate, but various predictions for the structure of the waves. The numerical solutions show that as the jet width is reduced a new type of mode appears in which critical lines at the lid (tropopause) play an important role.

An attractor embedded in the atmosphere

A procedure for deriving a three-variable, one-level baroclinic model by truncating the familiar two-level quasigeostrophic model is described. The longitude, latitude and isobaric height of the low centre are introduced into the new model as alternative dependent variables. State space then becomes equivalent to geographical space, and the attractor becomes a structure within the atmosphere.

An attractor embedded in the atmosphere

A procedure for deriving a three-variable, one-level baroclinic model by truncating the familiar two-level quasigeostrophic model is described. The longitude, latitude and isobaric height of the low centre are introduced into the new model as alternative dependent variables. State space then becomes equivalent to geographical space, and the attractor becomes a structure within the atmosphere.

Thek−3andk−5/3Energy Spectrum of Atmospheric Turbulence: Quasigeostrophic Two-Level Model Simulation

The Nastrom‐Gage energy spectrum of atmospheric turbulence as a function of wavelength is simulated here with a two-level quasigeostrophic (QG) model. This simple model has no topography, no direct wave forcing, and no small-scale forcing, nor any kind of gravity wave generation. The two-level model does, however, allow for the simple mechanism of baroclinic energy injection at the large (synoptic) scales as the model atmosphere relaxes to a specified north‐south ‘‘radiative equilibrium’’ temperature gradient. It also has a small sink of energy at the small scales due to subgrid hyperdiffusion; this attempts to model the small-scale sink not resolved by the two-level QG model, in particular, enhanced viscous dissipation in atmospheric fronts. The magnitude and shape of the observed energy spectrum, with its characteristic k23 power-law behavior in the synoptic and subsynoptic scales (from several thousand to about eight hundred kilometers) and the characteristic k25/3 behavior in the mesoscales (less than about six hundred kilometers), are reproduced convincingly in the model. The picture that emerges for the energy spectrum of atmospheric turbulence from a few kilometers to tens of thousands of kilometers is actually quite simple. The potential energy of the mean flow, which is derived from solar heating with no scale dependence, is transferred selectively to the long synoptic scales of motion via the mechanism of (nonlinear) baroclinic instability. The injected energy moves both upscale, to the planetary waves where it is damped by Ekman damping, and also downscale, through the short synoptic waves, through the mesoscales, to the short mesoscales, where it can be damped by viscous dissipation. There is no need for dynamics other than QG to produce the spectrum. (However, the present work cannot be used to rule out other explanations, such as gravity wave generation, or a separate energy source at the small scales.)

Equilibration of a Simple Baroclinic Flow in aβChannel and on the Sphere

Abstract The statistical equilibration of baroclinic waves in a two-level quasigeostrophic model subject to forcing and dissipation is studied. The model employed may be formulated in either spherical or Cartesian geometry and is restricted to a midlatitude channel. Parameters are chosen so that only up to three waves can become supercritical (one planetary- and two synoptic-scale waves). It is found that both geometries exhibit essentially two equilibration regimes as the forcing temperature gradient varies. At low forcing, the planetary-scale wave is not excited while the two synoptic-scale waves equilibrate with steady finite amplitude. In this regime, the equilibrated temperature gradient is sensitive to forcing; the authors argue that this is due to the barotropic governor effect. At higher forcing, the planetary wave becomes active and the solution is aperiodic. In this regime, the planetary wave acts to reduce the barotropic shear spun up by the synoptic waves, thereby limiting the role of the baro...

Linear instability of broad baroclinic zones

Baroclinic energy conversions play a central role in mid-latitude synoptic-scale dynamics, drawing energy from the meridional temperature gradient and feeding it into the eddy dynamics. The physical mechanisms underlying these conversions can be investigated through the theory of small-amplitude instabilities growing on a steady basic state. In a recent study the author analysed the instability associated with a boundary temperature transition concentrated in a narrow front. This paper looks at the complementary problem of a broad baroclinic zone. The solution is applicable both to the Eady model and to the Phillips two-level quasi-geostrophic model. The reduction in the growth rate relative to an infinitely broad, homogeneous baroclinic mne is inversely proportional to the width of the baroclinic zone, as found in previous studies. The constant of proportionality is expressed in a simplified and more general form, related to the derivative with respect to zonal wave number of the complex phase speed of the homogeneous problem. If the flow is purely baroclinic the maximum growth rate is reduced compared with that of a homogeneous, infinitely broad frontal zone by a factor 1 - (2kmaxr)-1. where kmax is the wave number of the most unstable mode of the homogeneous problem and r is the width of the baroclonic zone. The width, r, is defined in terms of the second meridional derivative of the boundary temperature gradient evaluated at the maximum of the latter. Results in the literature give consistent, though less general, expressions for the growth rate, but various predictions for the structure of the waves. The numerical solutions show that as the jet width is reduced a new type of mode appears in which critical lines at the lid (tropopause) play an important role.

Nonlinear studies on the effects of topography on baroclinic zonal flows

We examine the generation of transient and stationary waves when baroclinic solid body rotation zonal currents flow over topography within an inviscid two-level nonlinear quasi-geostrophic model on a sphere. The equilibrium climate states resulting from different initial conditions are obtained for wide parameter ranges. The climates are obtained, not by direct integration and taking long time averages, but instead by using methods of equilibrium statistical mechanics. The effects of topography depend strongly on where, in the (U 1 ,U 3 ) phase space of solid body rotation components, the initial flow lies. Five regions of this phase space R 1 , R 2 , R T , R B1 and R B3 have been identified.

南北両半球における準地衡風渦位輸送と簡単な経度平均気候モデル

The meridional transport of quasi-geostrophic potential vorticity is calculated for the Northern and Southern Hemispheres. The results show that the transport of potential vorticity is negative over most of the atmosphere except over a thin layer near surface where it is positive. Exchange coefficients for the transport of potential vorticity are computed for both the hemispheres using the evaluations of the transport of quasi-geostrophic potential vorticity and calculating the meridional gradient of potential vorticity. The values of the exchange coefficients are used in a simple, two-level quasi-geostrophic numerical model for tonally averaged conditions. The model is applied to the Northern and Southern Hemispheres and the general characteristics of mean annual temperature and zonal wind variation are reproduced reasonably well.

Linear Model Predictions of Time Averages

Abstract Linear prediction models applicable to a basic nonlinear two-level quasi-geostrophic model and extended range forecasting are described. One prediction model is linearized around the nonlinear model baroclinic climatological state and solved via an expansion in normal modes. The skill of these predictions are superior to persistence forecasts of daily events for at least 20 days and time averages for at least 90 days. As might be expected, initial states that project strongly onto the linear baroclinic model slow modes provide skillful forecasts at long forecast lags (seasons), which thus provides a prediction of the quality of the prediction and a possible explanation as to why persistence and forecast skill have been found to be correlated at long lags. An equivalent method for partitioning extended range forecast quality is provided via an EOF expansion. Initial states strongly projecting onto the first and dominant EOF mode are predicted best by the linear baroclinic model. This dominant EOF ...

Numerical experiments on dynamic mechanism of blocking

By using a two-level quasi-geostrophic spectral model, the numerical experiments on blocking process affected by thermal forcing are conducted. The dynamic mechanisms of blocking, including the resonance between wave and forcing, the control effects of thermal forcing on the position and intensity of the highs and the instabilities of two kinds of equilibrium states are studied by controlled experiments.

A Numerical Investigation of the Dynamics of Blocking Waves in a Simple Two-Level Model

The atmospheric blocking problem is discussed within the framework of a spectral two-level quasi-geostrophic model with β-plane geometry. By further developing the basic ideas of Egger (1978, 1979) for the formation of blocking centers, a baroclinic mechanism is proposed which is almost as simple as the barotropic concept of Egger. In both concepts nonlinear interactions of forced and free slowly moving waves are important for the accumulation of eddy energy in only a few quasi-stationary waves. In this paper the possibility is studied that long waves get energy baroclinically from the zonal mean flow rather than from external sources. The baroclinic instability of the longest waves is not of linear nature but purely nonlinear. From a series of model runs some statistics of the dynamics of model blocking were extracted. Indeed, blocking dynamics are significantly different from the dynamics prevailing in blocking-free periods. The “nuclei” (Egger, 1978) of the model-generated-blockings are the 500 mb Rossby stationary pressure waves (together with the corresponding temperature modes). It was found that the proposed baroclinic type of block is well represented in the model simulations. One of the numerical experiments is studied in more detail in order to give more information on the spectral energetics of the flow.

Amplitude Vacillation on a β-Plane

Abstract Amplitude vacillation is studied using a spectral version of the two-level quasi-geostrophic model in a β-plane channel. The parameters are chosen so that only a single wave may be unstable to the undisturbed basic state. It is shown that the weakly nonlinear theories which have been proposed do not work well on the β-plane, because nonlinear interactions are not negligible even though only a single wave has any significant amplitude. The energetics and mechanism of amplitude vacillation on a β-plane are discussed for a case with moderate dissipation and contrasted with the mechanism on an f-plane.

Amplitude Vacillation on anf-Plane

Abstract Amplitude vacillation is studied using a spectral version of the two-level quasi-geostrophic model in a β-plane channel. The parameters are chosen so that only a single wave may be unstable to the undisturbed basic state. It is shown that the weakly nonlinear theories which have been proposed do not work well on the β-plane, because nonlinear interactions are not negligible even though only a single wave has any significant amplitude. The energetics and mechanism of amplitude vacillation on a β-plane are discussed for a case with moderate dissipation and contrasted with the mechanism on an f-plane.

Maintenance of Quasi-Stationary Waves in a Two-Level Quasi-Geostrophic Spectral Model with Topography

A study of the maintenance of the quasistationary waves forced by topography using a truncated two-level quasigeostrophic spectral model in a zonal channel on a beta-plane is presented. The model's motion contains wavenumbers 0, n, and 2n in the zonal direction, where n is the lowest eddy wavenumber and also the wavenumber of the topography. The study covered the two cases defined by n=2 and n=3; the spectral mode was integrated by initially perturbing the stationary solution of the equations governing the spectral coefficients, and a detailed energetics study was made of the quasiequilibrium state to study the maintenance of the quasistationary waves. The energy conversions required for maintaining these waves when n=3 imply that they are generated mainly by baroclinic stability of the forced waves; this type of baroclinic wave tends to become stationary to draw efficiently on the available energy of the forced wave.

Planetary Circulations: 3. The Terrestrial Quasi-Geostrophic Regime

Abstract The characteristics of the two-level quasi-geostrophic model are evaluated for a wide range of parameters in the terrestrial domain. Flow form is determined primarily by β (the Coriolis gradient) and by τD, (the time scale of the surface drag), acting through the influence of Rhines' transitional wave-number kβ = (β/2U)1/2 where U2 is the barotropic energy level. Two extreme types of circulation occur: jets when kβ is large, and gyres when wave propagation and drag are negligible. The present terrestrial circulation, in its quasi-geostrophic representation, is extremely efficient: the system can cope with increased heating rates without a significant rise in the pole-to-equator temperature differential. Although each hemisphere is, on occasion, near to transforming into a double-jet state, multi-jet circulations–corresponding to those in the Jovian regime–occur more readily at higher rotation rates. For the existing circulation to switch to a gyre form requires a large, unrealizable drop in surfa...

Planetary Circulations: 2. The Jovian Quasi-Geostrophic Regime

Abstract The characteristics of the two-level quasi-geostrophic model are evaluated for a wide range of parameter values in the Jovian domain. The results support the hypothesis that baroclinic instability energizes the circulation of Jupiter and Saturn and that the blocking effect of planetary wave propagation on quasi-geostrophic turbulent cascades determines the width and zonality of the bands—the degree of zonality being higher in the absence of surface drag. The model circulations consist of multiple westerly jets, separated by strong easterly flows—the result of momentum partitioning by the Kuo vortex separation process. There are no large-scale vertical motions. A cyclic variation occurs (with a time scale of several years) during which phases with intense, large-scale baroclinic activity alternate with longer, more quiescent phases involving weak, small-scale baroclinic instability and neutral baroclinic waves. These neutral waves, generated by quasi-two-dimensional cascades and propagating at spe...