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Thomas Dubos and Marine Tort

of the equations in such a coordinate system is well known ( Kasahara 1974 ; Laprise 1992 ; Staniforth and Wood 2003 ). If the vertical coordinate is time independent (i.e., z based), standard expressions of the curl and grad operators in curvilinear coordinates can be used to rewrite the curl form as well. However, in the general situation where the vertical coordinate is time dependent, such a systematic treatment of the curl form is lacking. Moreover, given the close relationship between

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Hann-Ming Henry Juang

1. Introduction It is common to use hybrid vertical coordinates in atmospheric and oceanic modeling ( Simmons and Burridge 1981 ; Zhu et al. 1992 ; Konor and Arakawa 1997 ; Johnson and Yuan 1998 ; Bleck 2002 ; Benjamin et al. 2004 ). With hybrid coordinates, the atmospheric model can be integrated along different types of coordinate surfaces. The coordinates near the surface and lower atmosphere still use terrain-following sigma coordinates, but over the upper atmosphere better results

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Stephen Eckermann

troposphere or lower stratosphere ( Fels et al. 1980 ; Simmons and Burridge 1981 ; Simmons and Strüfing 1983 ; Simmons et al. 1989 ). Simmons and Strüfing (1981) and Simmons and Burridge (1981) tested different functional forms for h ( p , p S ) that yielded so-called hybrid σ – p coordinates. The final function considered by Simmons and Strüfing (1981) took the implicit form where p 0 is some nominal sea level pressure, typically ∼1000 hPa, and η̃ is the corresponding vertical profile of

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Günther Zängl

height-based Gal-Chen and Somerville (1975) system by Schär et al. and extended to pressure-based σ coordinate systems by Zängl (2003) . Another type of hybrid coordinates that is becoming more and more popular is the so-called σ – θ system, switching from a σ coordinate at low levels to isentropic coordinates at higher levels. Because of enhanced vertical resolution in regions of high static stability, a σ – θ system improves the representation of fronts and the tropopause and reduces the

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Rainer Bleck, Stan Benjamin, Jin Lee, and Alexander E. MacDonald

1980s saw some progress in the related field of ocean circulation modeling with an entropy-related vertical coordinate. Specifically, Bleck and Boudra (1981) developed a coordinate system that is mainly isopycnic but allows coordinate layers to turn into constant-thickness layers near the sea surface to overcome the massless-layer problem associated with modeling baroclinic ocean states. This may have been the first time that arbitrary Lagrangian–Eulerian (ALE)-like coordinates ( Hirt et al. 1974

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Michael D. Toy

1. Introduction Atmospheric models based on isentropic and hybrid-isentropic vertical coordinates have been well documented (e.g., Bleck 1984 ; Hsu and Arakawa 1990 ; Konor and Arakawa 1997 ; Benjamin et al. 2004 ). Recently, these coordinates have been implemented in nonhydrostatic models (e.g., Skamarock 1998 ; He 2002 ; Zängl 2007 ; Toy and Randall 2009 ). An advantage of using the quasi-Lagrangian θ coordinate is that for adiabatic flow, the cross-coordinate vertical mass flux is

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Michael D. Toy

1. Introduction Nonhydrostatic atmospheric models that use hybrid isentropic-sigma vertical coordinates have recently been developed but not yet put into practical application (e.g., Skamarock 1998 ; He 2002 ; Zängl 2007 ). In hybrid-coordinate models, an Eulerian terrain-following vertical coordinate is used near the surface, and in the free atmosphere the coordinate transitions to the vertically quasi-Lagrangian potential temperature coordinate θ . Using θ coordinates in atmospheric

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Akio Arakawa and Max J. Suarez

MONTHLY WEATHER REVIEW VOLUME 111Vertical Differencing of the Primitive Equations in Sigma Coordinates A~o ARAICAWA AND MAX J. SUAREZDepartmera of Atmospheric Sciences, University of California, Los Angeles, 90024(Manuscript received 22 December 1981, in final form 8 October 1982)ABSTRACT A vertical finite-difference ~cheme for the ptimitive equations in sigma coordinates is obtained by requirin~that the ~ equations retain ~ome

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Michael D. Toy and David A. Randall

, by the overturning of isentropic surfaces and planetary boundary layer (PBL) processes, and the intersection of isentropic surfaces with the lower boundary. The latter issue has been overcome in the successful development of various quasi-static θ -coordinate models (e.g., Eliassen and Raustein 1968 ; Bleck 1984 ; Hsu and Arakawa 1990 ). Hybrid vertical coordinates have been developed to address the issues of lower-boundary coordinate intersection and the lack of vertical resolution in the

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Suk-Jin Choi and Joseph B. Klemp

-following coordinates. Finally, a summary and discussion are given in section 5 . 2. Description of the smoothed hybrid sigma-pressure vertical coordinate a. Formulation A hybrid representation of the traditional terrain-following (HYB) coordinate may be written in the form: (1) π ⁡ ( i , η , t ) = B ⁡ ( η ) [ π sfc ⁡ ( i , t )   − π top ] + [ η − B ⁡ ( η

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