Surface Quasi-geostrophic Dynamics Research Articles

Eady edge waves, frontal wave-trains and type-B cyclogenesis

Eady edge waves (EEWs) propagate on surface baroclinic zones and they resemble synoptic and sub-synoptic scale frontal wave-trains. The wave-trains are frequently depicted on surface charts, and occur per se on a front with a local baroclinicity extremum. The possible linkage of EEWs to frontal waves is explored within the framework of so-called ‘surface quasi-geostrophic dynamics’ by establishing the properties of free and forced EEWs propagating on an ambient flow comprising a laterally confined baroclinic zone. It is shown that the dispersion properties of the free EEWs modes are influenced significantly both by the zone’s baroclinicity extremum and by key features that are akin to those of cold and warm fronts. In particular the larger-scale waves are nondispersive and a local wave-packet can retain coherency, the leading modes on a pseudo-cold front possess a phase speed related to the baroclinicity extremum, whereas waves on a pseudo-warm front propagate in the reverse direction. Also theoretical considerations demonstrate that forced modes can undergo resonant secular growth. It is further shown that transient or sustained forcing of a front by an upper-tropospheric potential vorticity anomaly (sic. short-wave trough) can serve to instigate features akin to frontal wave-trains and type-B cyclogenesis. More generally the study hints at the significance for NWP of a front’s structure and its dispersion properties, and that an element of predictability is introduced by the latter features due to dynamical scale selection.

A study of surface semi-geostrophic turbulence: freely decaying dynamics

In this study we give a characterization of semi-geostrophic turbulence by performing freely decaying simulations for the case of constant uniform potential vorticity, a set of equations known as the surface semi-geostrophic approximation. The equations are formulated as conservation laws for potential temperature and potential vorticity, with a nonlinear Monge–Ampère type inversion equation for the streamfunction, expressed in a transformed coordinate system that follows the geostrophic flow. We perform model studies of turbulent surface semi-geostrophic flows in a domain doubly periodic in the horizontal and limited in the vertical by two rigid lids, allowing for variations of potential temperature at one of the boundaries, and we compare the results with those obtained in the corresponding surface quasi-geostrophic case. The results show that, while the surface quasi-geostrophic dynamics is dominated by a symmetric population of cyclones and anticyclones, the surface semi-geostrophic dynamics features a more prominent role of fronts and filaments. The resulting distribution of potential temperature is strongly skewed and peaked at non-zero values at and close to the active boundary, while symmetry is restored in the interior of the domain, where small-scale frontal structures do not penetrate. In surface semi-geostrophic turbulence, energy spectra are less steep than in the surface quasi-geostrophic case, with more energy concentrated at small scales for increasing Rossby number. The energy related to frontal structures, the lateral strain rate and the vertical velocities are largest close to the active boundary. These results show that the semi-geostrophic model could be of interest for studying the lateral mixing of properties in geophysical flows.

Atmospheric Predictability: Why Butterflies Are Not of Practical Importance

Abstract The spectral turbulence model of Lorenz, as modified for surface quasigeostrophic dynamics by Rotunno and Snyder, is further modified to more smoothly approach nonlinear saturation. This model is used to investigate error growth starting from different distributions of the initial error. Consistent with an often overlooked finding by Lorenz, the loss of predictability generated by initial errors of small but fixed absolute magnitude is essentially independent of their spatial scale when the background saturation kinetic energy spectrum is proportional to the −5/3 power of the wavenumber. Thus, because the background kinetic energy increases with scale, very small relative errors at long wavelengths have similar impacts on perturbation error growth as large relative errors at short wavelengths. To the extent that this model applies to practical meteorological forecasts, the influence of initial perturbations generated by butterflies would be swamped by unavoidable tiny relative errors in the large scales. The rough applicability of the authors’ modified spectral turbulence model to the atmosphere over scales ranging between 10 and 1000 km is supported by the good estimate that it provides for the ensemble error growth in state-of-the-art ensemble mesoscale model simulations of two winter storms. The initial-error spectrum for the ensemble perturbations in these cases has maximum power at the longest wavelengths. The dominance of large-scale errors in the ensemble suggests that mesoscale weather forecasts may often be limited by errors arising from the large scales instead of being produced solely through an upscale cascade from the smallest scales.

Perturbed Rankine vortices in surface quasi-geostrophic dynamics

An analytical dispersion relation is derived for linear perturbations to a Rankine vortex governed by surface quasi-geostrophic dynamics. Such a Rankine vortex is a circular region of uniform anomalous surface temperature evolving under quasi-geostrophic dynamics with uniform interior potential vorticity. The dispersion relation is analysed in detail and compared to the more familiar dispersion relation for a perturbed Rankine vortex governed by the Euler equations. The results are successfully verified against numerical simulations of the full equations. The dispersion relation is relevant to problems including wave propagation on surface temperature fronts and the stability of vortices in quasi-geostrophic turbulence.

Surface kinetic energy transfer in surface quasi-geostrophic flows

The relevance of surface quasi-geostrophic dynamics (SQG) to the upper ocean and the atmospheric tropopause has been recently demonstrated in a wide range of conditions. Within this context, the properties of SQG in terms of kinetic energy (KE) transfers at the surface are revisited and further explored. Two well-known and important properties of SQG characterize the surface dynamics: (i) the identity between surface velocity and density spectra (when appropriately scaled) and (ii) the existence of a forward cascade for surface density variance. Here we show numerically and analytically that (i) and (ii) do not imply a forward cascade of surface KE (through the advection term in the KE budget). On the contrary, advection by the geostrophic flow primarily induces an inverse cascade of surface KE on a large range of scales. This spectral flux is locally compensated by a KE source that is related to surface frontogenesis. The subsequent spectral budget resembles those exhibited by more complex systems (primitive equations or Boussinesq models) and observations, which strengthens the relevance of SQG for the description of ocean/atmosphere dynamics near vertical boundaries. The main weakness of SQG however is in the small-scale range (scales smaller than 20–30 km in the ocean) where it poorly represents the forward KE cascade observed in non-QG numerical simulations.

The Response of a Uniform Horizontal Temperature Gradient to Heating

Abstract The response of a uniform horizontal temperature gradient to prescribed fixed heating is calculated in the context of an extended version of surface quasigeostrophic dynamics. It is found that for zero mean surface flow and weak cross-gradient structure the prescribed heating induces a mean temperature anomaly proportional to the spatial Hilbert transform of the heating. The interior potential vorticity generated by the heating enhances this surface response. The time-varying part is independent of the heating and satisfies the usual linearized surface quasigeostrophic dynamics. It is shown that the surface temperature tendency is a spatial Hilbert transform of the temperature anomaly itself. It then follows that the temperature anomaly is periodically modulated with a frequency proportional to the vertical wind shear. A strong local bound on wave energy is also found. Reanalysis diagnostics are presented that indicate consistency with key findings from this theory.

A New Surface Model for β-Plane Turbulence

Abstract This paper introduces a new numerical model for studying wave–turbulence interactions in a continuously stratified rotating flow, having a uniform potential vorticity and a rigid boundary. The meridional variation in the Coriolis parameter (β effect), a channel geometry, and the first-order nonlinear terms in a small Rossby number expansion are included into the surface quasigeostrophic dynamics. The model contains important dynamical characteristics of three-dimensional flows such as advection by ageostrophic winds, and stretching and tilting of relative vorticity. Nevertheless, it has the computational economy of two-dimensional flows. Long-term direct numerical simulations are performed for decaying turbulence arising from random initial conditions. In addition to the formation of steady zonal jets, frequently reported as a possible end state under the β effect, this model exhibits several other realistic physical effects that were lacking in the previously studied β-plane models for wave–turbulence interactions. There is a significant contrast in the spatial and time scales of the formed eddies, resulting in high meridional asymmetry of the flow. Mean surface cooling and persistent large-scale blocking eddies are observed as well. The surface potential temperature variance spectrum exhibits a well-resolved k−5/3 inertial range.

Local and nonlocal advection of a passive scalar

Passive and active scalar mixing is examined in a simple one-parameter family of two-dimensional flows based on quasi-geostrophic dynamics, in which the active scalar, the quasi-geostrophic potential vorticity, is confined to a single horizontal surface (so-called surface quasi-geostrophic dynamics) and in which a passive scalar field is also advected by the (horizontal, two-dimensional) velocity field at a finite distance from the surface. At large distances from the surface the flow is determined by the largest horizontal scales, the flow is spectrally nonlocal, and a chaotic advection-type regime dominates. At small distances, z, scaling arguments suggest a transition wavenumber kc∼1∕2z, where the slope of the passive scalar spectrum changes from k−5∕3, determined by local dynamics, to k−1, determined by nonlocal dynamics, analogous to the transition to a k−1 slope in the Batchelor regime in three-dimensional turbulence. Direct numerical simulations reproduce the qualitative aspects of this transition. Other characteristics of the simulated scalar fields, such as the relative dominance of coherent or filamentary structures, are also shown to depend strongly on the degree of locality.

Dynamics in the extratropical tropopause region: A case of transition between dynamically active and passive tracer advection?

It is argued that certain aspects of tracer patterns and related stirring by the flow in the extratropical tropopause region can be understood in terms of a transition between dynamically active and passive tracer advection, called ‘active-to-passive tracer transition’. In the framework of surface quasi-geostrophic dynamics, a specific initial-value problem is defined and investigated. It features a gradual transition between the two paradigms of tracer advection and allows a clear interpretation thanks to the idealized nature of the model setup. Physical reasoning and anecdotal evidence from previous studies suggest that, at least in specific cases, this interpretation is relevant for the real atmosphere. Copyright © 2005 Royal Meteorological Society