A technique for creating linearly stable localized atmospheric features with an application to nonlinear cyclogenesis

Abstract

A technique is described to construct persistent, three-dimensionally localized features. Localized is used here to signify a structure that is nonzero only in a small region of a larger domain. These features remain nearly coherent and stable in a linear calculation by constructing the local feature from neutral eigenmodes (sometimes called “continuum modes”). The limiting factors are how much the phase speed varies between the modal constituents of the localized feature and how good the match is between eigenvalue and initial-value versions of the governing equations. The governing equations used here are the quasi-geostrophic (QG) potential vorticity (PV) tendency equation. The linear form of the model specifies a zonal mean flow that can have both vertical and meridional shear. The prescribed, mean state has no variation along the flow. The construction of a localized state can be accomplished using fewer modes when meridional shear is present in the zonal mean flow. The localized features are tested as initial conditions (ICs) in linear calculations and then applied to study a problem in nonlinear extratropical cyclogenesis. The nonlinear simulations are not exhaustive. The scope is limited to considering whether nonlinear advection favors nonmodal growth (NG) or normal mode baroclinic instability (NMBI). The applications use several mean flows and IC amplitudes and structures representing conditions prior to observed cyclogenesis. Even though the scope of the application is limited, that scope is better served by using structures that approximate observed traveling, but not developing, localized troughs. The localization includes a procedure that removes linearly unstable normal modes from the IC. Removing the unstable normal modes allows tracking of how quickly growing structures are created by nonlinear advection. Results for selected ICs and basic flows find little NG. Adjacent to the original trough, unstable normal mode-like structures appear soon into the integration. Their properties are more consistent with normal mode growth than NG. Projecting the solution onto eigenmodes finds strong initial amplification of unstable normal modes by the nonlinear terms. The eddies in nonlinear integrations evolve towards a horizontal size that is greater than the linearly most unstable normal mode. Larger initial amplitude leads to faster breakdown of the localization. Emphasis is on ICs with mid or upper tropospheric isolated troughs and sufficient amplitude such that the sum of the IC and mean flow is either an “open wave” or a “closed contour”. The ICs considered develop a leading upper high and trailing lower high and the IC trough develops upstream tilt; both developments are similar to observed cyclogenesis.