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G. W. Kent Moore
and
W. R. Peltier

trast, the behavior of the expansion for 0~ used in Moore and Peltier (1987) vanished for these limits. This improved behavior allows for a much more ac curate representation of 0 ~ near the boundaries z = 0 and z = H. Applying the weighting schemes described above re sults in a set of algebraic equations which after some rather tedious algebra may be written as the following matrix eigenvalue problem: s U~(~ I 2 3 = MK~,x~ Ux

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Álvaro Viúdez

understanding of a theorem resides in its derivation in addition to its simple exposition, those theorems based on a large and complicated proof appear obscure, whereas those based on a short and simple proof appear clear. Specifically one desires to make the derivation in a simple way and in a general framework where a comparison with other developments in physics is possible. The PV conservation theorem has been derived in several different ways since Ertel’s original work in 1942. The derivation that

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John A. Dutton

statistical properties of the usual unforced quasi-geostrophic flow are considered in phase space, andit is shown that the Poincar~ recurrence theorem applies and that long-term averages along the trajectoriesexist even though the flows are not ergodic. The thermally-forced model is developed by adding a Newtonian heating term to the First Law and byadding a dissipative term to the vorticity equation. In this model every initial set is mapped into a set ofvanishing measure as t --* oo. Moreover, it is

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Yongsheng Chen
and
M. K. Yau

assumptions of simplified models using idealized initial conditions. Recently, Liu et al. (1997, hereafter LZY97) successfully simulated Hurricane Andrew (1992) with a 3D triply nested nonhydrostatic primitive equation model MM5. Their results ( Zhang et al. 2001 ; Yau et al. 1999 ) indicate vortex wavelike structures in the simulated Andrew. Since a detailed comparison of realistic model results with vortex Rossby wave theory has not yet been undertaken, it is the purpose of this paper to fill the gap

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James A. Hansen
and
Leonard A. Smith

–653 . Pecora, L. M., and T. L. Carroll, 1990: Synchronization in chaotic systems. Phys. Rev. Lett., 64, 821–824 . Smith, L. A., and L. Gilmour, 1998: Accountability and internal consistency in ensemble formation. Proc. ECMWF Seminar on Predictability, Reading, United Kingdom, ECMWF, 113–127 . Stephenson, D. B. 1997: Correlation of spatial climate/weather maps and the advantages of using the Mahalanobis metric in predictions. Tellus, 49A, 513–527 . Strang, G., 1988: Linear Algebra and its

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Brian F. Farrell
and
Petros J. Ioannou

temporal structure of the errors incurred in the approximation and a constructive a method of determining the effective dimension of the dynamical system. In the following, methods of model order reduction are discussed, the balanced truncation is introduced, a comparison is made among these methods of model reduction, and an example problem is examined. 2. Methods for reducing the order of a linear dynamical systems a. Optimal reduction of the order of a matrix and reduction of order for normal

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Joseph Pedlosky
and
Lorenzo M. Polvani

and dissipation. Second, there is, at this order, a correction tothe mean flow in each layer. Both of these parts arefamiliar from the analysis of the single wave. theory(e.g., Pedlosky, 1970). Now, however, the wave-waveinteraction between the two marginal waves introducesinto the solution, at this order, contributions with thesum and difference wave numbers and frequencies. Aconsiderable amount of algebra allows us to write thesolutions as-b~(l) =A~A2R~ei(-'+-2) sin2/y + - ' +A

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A. Marshak
,
Y. Knyazikhin
,
J. C. Chiu
, and
W. J. Wiscombe

1. Introduction Recently several papers reported the discovery of spectrally invariant behavior in some simple algebraic combinations, called “spectral invariants,” of single scattering albedo and solar radiation reflected from or transmitted through vegetation canopies ( Knyazikhin et al. 1998 , 2005 ; Huang et al. 2007 ). The spectral invariant phenomenon is clearly seen in radiative measurements and remote sensing data ( Panferov et al. 2001 ; Wang et al. 2003 ). The phenomenon was

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Ruth S. Lieberman
and
Paul B. Hays

individual wave. Avg. vertical Local time wavelengthHRDI observing periods maps (km)20 December 1991-28 January 199229 January 1992-28 February 199229 February 1992-30 March 199229 April 1992-2 June 199222 July 1992-26 August 199227 August 1992-1 October 19922 October 1992-6 November 19927 November 1992-12 December 199219 December 1992

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Ali Asaadi
,
Gilbert Brunet
, and
M. K. Yau

elongated closed total PV contour associated with a PV strip. This time is considered to be “day 0” in our time-lagged analyses, and time-lagged composite maps of easterly waves associated with named storms were produced for day 0 to day +5. The formation time of the cat’s-eye is defined as the time when a second locally closed total PV contour appears. It was found that the latitudinal location of the center of the second closed PV contour was close to the critical latitude in each of the 54 cases. We

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