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David J. Muraki
and
Gregory J. Hakim

Abstract

Tropopause disturbances have long been recognized as important features for extratropical weather since they produce organized vertical motion in the troposphere. Observations of cyclonic tropopause disturbances show localized depressions of the tropopause with stratospheric values of potential vorticity extending to lower altitudes; anticyclonic disturbances are associated with comparatively smaller upward deflections of the tropopause. Analytical solutions for nonlinear interfacial wave motions are derived for an intermediate balanced dynamics based on small Rossby number asymptotics. Beyond quasigeostrophy, traveling edge-wave solutions reveal realistic asymmetries such that cyclones are associated with greater deflections of the interface, as well as larger anomalies in pressure and vertical motion compared to anticyclones.

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David J. Muraki
and
Chris Snyder

Abstract

A new class of exact vortex dipole solutions is derived for surface quasigeostrophic (sQG) models. The solutions extend the two-dimensional barotropic modon to fully three-dimensional, continuously stratified flow and are a simple model of localized jets on the tropopause. In addition to the basic sQG dipole, dipole structures exist for a layer of uniform potential vorticity between two rigid boundaries and for a dipole in the presence of uniform background vertical shear and horizontal potential temperature gradient. In the former case, the solution approaches the barotropic Lamb dipole in the limit of a layer that is shallow relative to the Rossby depth based on the dipole’s radius. In the latter case, dipoles that are bounded in the far field must propagate counter to the phase speed of the linear edge waves associated with the surface temperature gradient.

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David J. Muraki
and
Richard Rotunno

Abstract

This work is motivated by an unusual feature associated with the start-up of a moist nearly neutral atmospheric flow over a mountain ridge that was previously observed in a full-physics numerical model. In that study, the upstream propagation of a wave of subsidence precluded the establishment of upward-displaced and saturated flow that might be expected upstream of the topography. This phenomenon was hypothesized to be a consequence of the peculiar property of saturated moist neutral flow: an upward air parcel displacement produces zero buoyancy, while a downward displacement desaturates the air parcel and produces a positive buoyancy anomaly. In the present study, this hypothesis is confirmed within numerical solutions to a reduced system of equations that incorporates the saturated-atmosphere property in a particularly simple manner. The relatively uncomplicated nature of these solutions motivates the numerical solution of a further simplified initial-value problem for both nonhydrostatic and hydrostatic flow. Exact analytic solutions are developed for the latter hydrostatic case, which explains the upstream-propagating wave of subsidence as a shock phenomenon.

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Riwal Plougonven
,
David J. Muraki
, and
Chris Snyder

Abstract

Normal modes of a linear vertical shear (Eady shear) are studied within the linearized primitive equations for a rotating stratified fluid above a rigid lower boundary. The authors' interest is in modes having an inertial critical layer present at some height within the flow. Below this layer, the solutions can be closely approximated by balanced edge waves obtained through an asymptotic expansion in Rossby number. Above, the solutions behave as gravity waves. Hence these modes are an example of a spatial coupling of balanced motions to gravity waves.

The amplitude of the gravity waves relative to the balanced part of the solutions is obtained analytically and numerically as a function of parameters. It is shown that the waves are exponentially small in Rossby number. Moreover, their amplitude depends in a nontrivial way on the meridional wavenumber. For modes having a radiating upper boundary condition, the meridional wavenumber for which the gravity wave amplitude is maximal occurs when the tilts of the balanced edge wave and gravity waves agree.

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David J. Muraki
,
Chris Snyder
, and
Richard Rotunno

Abstract

Quasigeostrophic theory is an approximation of the primitive equations in which the dynamics of geostrophically balanced motions are described by the advection of potential vorticity. Quasigeostrophy also represents a leading-order theory in the sense that it is derivable from the full primitive equations in the asymptotic limit of zero Rossby number. Building upon quasigeostrophy, and the centrality of potential vorticity, a systematic asymptotic framework is developed from which balanced, next-order corrections in Rossby number are obtained. The simplicity of the approach is illustrated by explicit construction of the next-order corrections to a finite-amplitude Eady edge wave.

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Richard Rotunno
,
David J. Muraki
, and
Chris Snyder

Abstract

Quasigeostrophic theory is an approximation of the primitive equations in which the dynamics of geostrophically balanced motions are described by the advection of potential vorticity. Quasigeostrophic theory also represents a leading-order theory in the sense that it is derivable from the primitive equations in the asymptotic limit of zero Rossby number. Building upon quasigeostrophic theory, and the centrality of potential vorticity, the authors have recently developed a systematic asymptotic framework from which balanced, next-order corrections in Rossby number can be obtained. The approach is illustrated here through numerical solutions pertaining to unstable waves on baroclinic jets. The numerical solutions using the full primitive equations compare well with numerical solutions to our equations with accuracy one order beyond quasigeostrophic theory; in particular, the inherent asymmetry between cyclones and anticyclones is captured. Explanations of the latter and the associated asymmetry of the warm and cold fronts are given using simple extensions of quasigeostrophic– potential-vorticity thinking to next order.

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Chris Snyder
,
Riwal Plougonven
, and
David J. Muraki

Abstract

Previous simulations of dipole vortices propagating through rotating, stratified fluid have revealed small-scale inertia–gravity waves that are embedded within the dipole near its leading edge and are approximately stationary relative to the dipole. The mechanism by which these waves are generated is investigated, beginning from the observation that the dipole can be reasonably approximated by a balanced quasigeostrophic (QG) solution. The deviations from the QG solution (including the waves) then satisfy linear equations that come from linearization of the governing equations about the QG dipole and are forced by the residual tendency of the QG dipole (i.e., the difference between the time tendency of the QG solution and that of the full primitive equations initialized with the QG fields). The waves do not appear to be generated by an instability of the balanced dipole, as homogeneous solutions of the linear equations amplify little over the time scale for which the linear equations are valid. Linear solutions forced by the residual tendency capture the scale, location, and pattern of the inertia–gravity waves, although they overpredict the wave amplitude by a factor of 2. There is thus strong evidence that the waves are generated as a forced linear response to the balanced flow. The relation to and differences from other theories for wave generation by balanced flows, including those of Lighthill and Ford et al., are discussed.

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Riwal Plougonven
,
David J. Muraki
, and
Chris Snyder
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David J. Muraki
,
Richard Rotunno
, and
Hugh Morrison

Abstract

A holepunch cloud is a curious phenomenon where a disturbance in a thin cloud layer initiates an expanding circular hole of clear air. Usually triggered by the passage of aircraft, observations of these holes in clouds date back to the earliest days of aviation, but only recently has a holepunch cloud been simulated within a full-physics numerical model. These computations confirm that ice crystal growth through the Wegener–Bergeron–Findeisen process creates a small cloud-free region whose subsequent outward spread defines the holepunch. The mechanics behind this continued expansion, however, has yet to be definitively identified. In this article, the motion of the cloud edge is explained as a propagating gravity wave front. To support this idea, a hierarchy of three idealizations is analyzed: a full-physics numerical model, a fluid mechanical model with simplified moisture effects, and a conservation law analysis for front motion. The essence of the holepunch cloud is established to be a moist air layer that is unsaturated (clear) and weakly stratified within the hole but saturated (cloudy) and moist neutral outside of it. The cloud edge thus represents a barrier to the outward propagation of gravity waves within the clear air—the result of this collision is a wave front whose velocity determines the growth rate of the hole.

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Gregory J. Hakim
,
Chris Snyder
, and
David J. Muraki

Abstract

Cyclonic vortices on the tropopause are characterized by compact structure and larger pressure, wind, and temperature perturbations when compared to broader and weaker anticyclones. Neither the origin of these vortices nor the reasons for the preferred asymmetries are completely understood; quasigeostrophic dynamics, in particular, have cyclone–anticyclone symmetry.

In order to explore these and related problems, a novel small Rossby number approximation is introduced to the primitive equations applied to a simple model of the tropopause in continuously stratified fluid. This model resolves dynamics that give rise to vortical asymmetries, while retaining both the conceptual simplicity of quasigeostrophic dynamics and the computational economy of two-dimensional flows. The model contains no depth-independent (barotropic) flow, and thus may provide a useful comparison to two-dimensional flows dominated by this flow component.

Solutions for random initial conditions (i.e., freely decaying turbulence) exhibit vortical asymmetries typical of tropopause observations, with strong localized cyclones, and weaker diffuse anticyclones. Cyclones cluster around a distinct length scale at a given time, whereas anticyclones do not. These results differ significantly from previous studies of cyclone–anticyclone asymmetry in the shallow-water primitive equations and the periodic balance equations. An important source of asymmetry in the present solutions is divergent flow associated with frontogenesis and the forward cascade of tropopause potential temperature variance. This thermally direct flow changes the mean potential temperature of the tropopause, selectively maintains anticyclonic filaments relative to cyclonic filaments, and appears to promote the merger of anticyclones relative to cyclones.

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