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Richard Grotjahn

Abstract

A new form of the linear, quasi-geostrophic model is derived on a sphere. The new feature is the use of empirically defined orthogonal basis functions (OBFs) to represent the vertical structure of the perturbation solutions. The prescribed basic state is expressed using a third vertical structure function. Spherical harmonics are used for the horizontal structure. The nonseparablc eigenvalue problem is derived. Solutions are presented using various vertical OBFs, basic flows (both zonally varying and zonally uniform) and horizontal truncations(both rhomboidal and triangular). One OBF (labeled "MOBF') is patterned after the structure found in the most unusable solution of a simpler problem. Another OBF (labeled "2-L") is intended to simulate a two-layer model.

Increasing the horizontal resolution from RI5 (rhomboidal truncation at zonal wavenumber 15) to R30 is found to decrease the growth rates in nearly all cas. (One exception is solid body rotation.) Surprisingly high resolution is needed to properly represent the instability of most of the basic flow jets studied herein. For some of these flows, R30 may not be high-enough resolution. In none of the flows examined did we conclude that RI 5 was adequate. The phase speeds in the "MOBF" cases are frequently much faster.than the most unstable modes in the "2-L" cases. In a few instances, the "2-U" version of the model obtains nearly stationary, rapidlygrowing modes whose counterpart is not found in the "MOBF" model.

Initially, our results suggested that much higher resolution may be needed than suggested by a previous researcher. This contradiction was seemingly resolved by our obse~vatlon that the convergence to the correct solution was faster when the basic jet was centered at a lower latitude. Some implications for low-resolution general circulaton models are made.

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Richard Grotjahn

Abstract

A preliminary investigation into the dynamical effects produced by the tropopause upon a mid-latitude wave cyclone is described. This article describes linear effects since the various structures of a fixed tropopause are examined. In general, the solutions are sensitive to changes in tropopause structure only when they have large amplitude in the tropopause vicinity or the forcing for the problem is significantly altered by the tropopause structure. The forcing is greatest at the bottom boundary and interior tropopause interface. The basic current contains an internal jet. Many characteristic properties of this jet were found in a less sophisticated antecedent study where the velocity maximum occurred at the top boundary. This research forms the basis for future inquiry into nonlinear tropopause dynamics.

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Richard Grotjahn

Abstract

The properties of wavelike eddies imbedded in zonal flows containing vertical and horizontal shear are examined via an analytical model of a midlatitude cyclone. The model combines and extends some work by several previous investigators. Perturbation methods are used to formulate and solve this model. A transformation to geostrophic coordinates is employed that includes some ageostrophic effects and additional ageostrophic terms are retained after scaling the primitive equations. The zonal flows are chosen to model conditions observed in the atmosphere during incipient wave-cyclone development. Solutions grow due to barotropic and (primarily) baroclinic instability of the zonal flow.

The stability, structure and energetics of some solutions are discussed. The lowest order solutions are in basic agreement with several previous studies. The effects of the intensity and vertical structure of the prescribed model thermal front are examined in a consistent fashion. As. the intensity of the front increases, the growth rate increases for most wavenumbers. As the meridional width of the east-west aligned frontal zone diminishes, 1) the most unstable wavelength shifts to shorter wavelengths, 2) the fastest moving wave shifts to longer wavelengths and 3) the meridional scale of the eddy decreases proportionally. The amplitude is increased in the vicinity of the front. The phase of the eddy pressure field is changed by the front in two ways: 1) barotropically unstable horizontal tilts are introduced and 2) the westward tilt with height is decreased in the upper region and increased in the lowest part by the horizontal shear. The energy conversions in these experiments reveal that the two instability mechanisms inhibit each other. This occurs because the two mechanisms are not independent. The ageostrophic terms 1) introduce meridional asymmetry into the solution, 2) reduce the growth rate and phase, speed and 3) tend to form a jet in a mean zonal flow that is initially only a function of height. Like the ageostrophic terms, the nonlinear distortion caused by the coordinate transformation improves the comparison between the model solutions and observed cyclones.

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Richard Grotjahn and Ching-Hua Wang

Abstract

The linear instability of a zonal flow passing over a large-scale mountain, having one of two orientations and two shapes, is considered via an eigenvalue/eigenvector problem using spherical coordinates in a quasi-geostrophic model. Topography enters only as slope flow in the bottom boundary condition. All variables are expressed using orthogonal functions in three dimensions. Realistic (variable) static stability is applied in the study.

The topography reduces the growth rates primarily by reducing the baroclinic energy conversion. For the two mountain orientations investigated here, when the ridge is oriented east–west the growth rates are reduced more than when the orientation is north-south. The highs and lows (at the surface) are deflected northward by the topography which places them where the basic flow vertical shear is less for a longer time when the ridge is oriented east–west. The deflection effects the eddy heat fluxes by increasing the meridional velocity on the southeastern side of each eddy. This increases the meridional heat fluxes, making the baroclinic conversion (from zonal mean to eddy available potential energy) largest on the upslope side. The eddy vertical velocities are also enhanced on the upslope (west) side of the ridge. This means that the conversion from eddy available potential to eddy kinetic energy is also larger there. On the downslope side the heat fluxes are usually reduced. In most cases the topography deflects the storm track more in the lower troposphere than in the upper troposphere. In a few cases, the topography causes the upper and lower level eddies to move at different rates and to be deflected in different directions; the phase relationship between temperature and pressure is altered such that negative baroclinic conversion occurs on the downslope side of the mountain.

Accurate solutions require even higher horizontal resolution than anticipated by earlier studies. But, much economy is gained by adopting a “parallelogramic” truncation, which uses more meridional than zonal wave-numbers.

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Richard Grotjahn, Roderick Pedersen, and Joseph Tribbia

Abstract

Normal-mode and nonmodal growth are investigated using initial value models. The initial value problems for the Eady and a generalized Eady model (the G model) are solved with no friction and with both Ekman and interior friction. The nonmodel growth is described as either a superposition of eigenmodes or as a transfer between the “thermal” and relative vorticity parts of quasigeostrophic potential vorticity. When all the eigen-modes are neutral, the growth rate (σH>) of enstrophy is zero, though the growth rate of energy (σE>) and amplitude (σL>) may be positive. For an initial condition having large upstream tilt and constant amplitude, a period of large initial growth in the energy and amplitude is followed by either oscillatory growth and decay (when all eigenmodes are neutral) or asymptotes to a rate given by the most unstable normal mode. In Part I, the authors show that interior friction strongly damps the continuum eigenmodes; however, nonmodal growth can still be significant even when interior friction is present in the Eady model. In the more realistic G model, less overlap between the eigenmodes is found and consequently the nonmodal growth by superposition is reduced compared to the Eady model studied previously by others.

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Richard Grotjahn, Min Chen, and Joseph Tribbia

Abstract

The eigenvalue problems for the original Eady model and a modified Eady model (the G model) are examined with no friction, Ekman friction only, and both Ekman and interior friction. When both Ekman and interior friction are included in the models, normal modes show little additional change when compared to the case with Ekman friction only, whereas the relevant “continuum modes” have large negative growth rates. Interior friction has a much greater effect on the continuum modes than on the normal modes because inviscid continuum modes have a delta-function vertical profile of potential vorticity q. In contrast, normal modes have much smoother profiles of q in the interior. Streamfunction profiles for the continuum modes are notably different in the two models. The continuum modes in the more realistic G model have sharp peak amplitudes that are not as broad in the vertical as in the Eady model.

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