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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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Chris Snyder and Gregory J. Hakim

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Singular vectors (SVs) have been applied to cyclogenesis, to initializing ensemble forecasts, and in predictability studies. Ideally, the calculation of the SVs would employ the analysis error covariance norm at the initial time or, in the case of cyclogenesis, a norm based on the statistics of initial perturbations, but the energy norm is often used as a more practical substitute.

To illustrate the roles of the choice of norm and the vertical structure of initial perturbations, an upper-level wave with no potential vorticity perturbation in the troposphere is considered as a typical cyclogenetic perturbation or analysis error, and this perturbation is then decomposed by its projection onto each energy SV. All calculations are made, for simplicity, in the context of the quasigeostrophic Eady model (i.e., for a background flow with constant vertical shear and horizontal temperature gradient). Viewed in terms of the energy SVs, the smooth vertical structure of the typical perturbation, as well as its evolution, results from strong cancellation between the growing and decaying SVs, most of which are highly structured and tilted in the vertical.

A simpler picture, involving less cancellation, follows from decomposition of the typical perturbation into SVs using an alternative initial norm, which is based on the relation between initial norms and the statistics of initial perturbations together with the empirical assumption that the initial perturbations are not dominated by interior potential vorticity. Differences between the energy SVs and those based on the alternative initial norm can be understood by noting that the energy norm implicitly assumes initial perturbations with second-order statistics given by the covariance matrix whose inverse defines the energy norm. Unlike the “typical” perturbation, perturbations with those statistics have large variance of potential vorticity in the troposphere and fine vertical structure.

Finally, a brief assessment is presented of the extent to which the upper wave, and more generally the alternative initial norm, is representative of cyclogenetic perturbations and analysis errors. There is substantial evidence supporting deep perturbations with little vertical structure as frequent precursors to cyclogenesis, but surrogates for analysis errors are less conclusive: operational midlatitude analysis differences have vertical structure similar to that of the perturbations implied by the energy norm, while short-range forecast errors and analysis errors from assimilation experiments with simulated observations are more consistent with the alternative norm.

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Keith J. Harding and Peter K. Snyder

Abstract

The rapid expansion of irrigation in the Great Plains since World War II has resulted in significant water table declines, threatening the long-term sustainability of the Ogallala Aquifer. As discussed in Part I of this paper, the Weather Research and Forecasting Model (WRF) was modified to simulate the effects of irrigation at subgrid scales. Simulations of nine April–October periods (three drought, three normal, and three pluvial) over the Great Plains were completed to assess the full impact of irrigation on the water budget. Averaged over all simulated years, irrigation over the Great Plains contributes to May–September evapotranspiration increases of approximately 4% and precipitation increases of 1%, with localized increases of up to 20%. Results from these WRF simulations are used along with a backward trajectory analysis to identify where evapotranspiration from irrigated fields falls as precipitation (i.e., irrigation-induced precipitation) and how irrigation impacts precipitation recycling. On average, only 15.8% of evapotranspiration from irrigated fields falls as precipitation over the Great Plains, resulting in 5.11 mm of May–September irrigation-induced precipitation and contributing to 6.71 mm of recycled precipitation. Reductions in nonrecycled precipitation suggest that irrigation reduces precipitation of moisture advected into the region. The heaviest irrigation-induced precipitation is coincident with simulated and observed precipitation increases, suggesting that observed precipitation increases in north-central Nebraska are strongly related to evapotranspiration of irrigated water. Water losses due to evapotranspiration are much larger than irrigation-induced precipitation and recycled precipitation increases, confirming that irrigation results in net water loss over the Great Plains.

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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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Riwal Plougonven, David J. Muraki, and Chris Snyder
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Stewart W. Borland and John J. Snyder

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The market price of land intended for agricultural use is closely and directly related to its long-term productivity which depends, inter alia, upon regional climatology and the individual farm's soil quality. Low-quality soils, consistently low precipitation, and/or high frequency of damaging hail are therefore negatively capitalized into land values. A market-price function for cropland was formulated in terms of indices for hail frequency and severity, precipitation and temperature during the growing season, along with a number of non-weather variables including soil quality, extent of irrigation, degree of urbanization, and effective tax rates. Detailed information for 577 land transactions during the years 1968 through 1973 was obtained from courthouse files and detailed soil surveys covering 12 counties in Colorado and Nebraska. Values of all of the variables in the price function were then calculated for each transaction. Multiple regression techniques were used to estimate the extent to which variations in the deflated value per acre could be attributed to differences in these variables. Although the use of temperature variables did not appear helpful in the estimation equation, their influences will be re-examined when appropriate evapo-transpiration data have been developed for the sample regions. Results indicate that late-season precpitation is a net disbenefit in terms of its effect on the value of cropland, with the turning point occurring near the end of July. The estimated value of early season precipitation is consistent with yield effects previously obtained in field plot experiments. Previous estimates of hail damage, which have relied on the records of crop-hail insurance claims, appear to understate the actual losses. The direct benefits of a hail suppression program could be greatly enhanced (or more than offset) by concomitant effects on total precipitation since the value of a 20% decrease in hail is roughly equivalent to that of an 8% increase in early season rainfall. The positive price effects of the combined weather variables over the range of the sample are estimated to be about three times as large as those associated with observed differences in soil quality.

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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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J. H. Filloux and R. L. Snyder

Abstract

This paper is the first in a series reporting the results of a study of tides, setup and bottom friction in the Bight of Abaco, Bahamas. The paper describes three month-long field experiments. employing 15 tide gages and four weather stations distributed throughout the Bight.

The amplitude and phase of five principal tidal constituents and the M4 and M6 overtides are estimated for all stations and errors computed from a generalization/hybridization of the algorithm of Munk and Hasselman (1964) for tidal doublets. The resulting tidal distributions constitute an unusually complete data base against which to optimize the numerical models reported in Parts II and III of the series.

The relatively small amplitude of the override constituents along the western margin of the Bight suggests that these constituents are locally generated. Residual fluctuations are highly coherent with the wind field. Significant differential setup effects are apparent.

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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

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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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