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A Semi-Hydrostatic Theory of Gravity-Dominated Compressible Flow

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A Semi-Hydrostatic Theory of Gravity-Dominated Compressible Flow Generated using version 3.2 of the official AMS LATEX template 1 A semi-hydrostatic theory of gravity-dominated compressible flow 2 Thomas Dubos ∗ IPSL-Laboratoire de Météorologie Dynamique, Ecole Polytechnique, Palaiseau, France Fabrice voitus CNRM-Groupe d’étude de l’Atmosphere Metéorologique, Météo-France, Toulouse, France ∗Corresponding author address: Thomas Dubos, LMD, École Polytechnique, 91128 Palaiseau, France. E-mail: [email protected] 1 3 Abstract 4 From Hamilton’s least action principle, compressible equations of motion with density diag- 5 nosed from potential temperature through hydrostatic balance are derived. Slaving density 6 to potential temperature suppresses the degrees of freedom supporting the propagation of 7 acoustic waves and results in a sound-proof system. The linear normal modes and dispersion 8 relationship for an isothermal state of rest on f- and β- planes are accurate from hydrostatic 9 to non-hydrostatic scales, except for deep internal gravity waves. Especially the Lamb wave 10 and long Rossby waves are not distorted, unlike with anelastic or pseudo-incompressible 11 systems. 12 Compared to similar equations derived by Arakawa and Konor (2009), the semi-hydrostatic 13 system derived here possesses an additional term in the horizontal momentum budget. This 14 term is an apparent force resulting from the vertical coordinate not being the actual height 15 of an air parcel, but its hydrostatic height, i.e. the hypothetical height it would have after 16 the atmospheric column it belongs to has reached hydrostatic balance through adiabatic 17 vertical displacements of air parcels. The Lagrange multiplier λ introduced in Hamilton’s 18 principle to slave density to potential temperature is identified as the non-hydrostatic ver- 19 tical displacement, i.e. the difference between the actual and hydrostatic heights of an air 20 parcel. 21 The expression of non-hydrostatic pressure and apparent force from λ allow the derivation 22 of a well-defined linear symmetric positive definite problem for λ. As with hydrostatic equa- 23 tions, vertical velocity is diagnosed through Richardson’s equation. The semi-hydrostatic 24 system has therefore precisely the same degrees of freedom as the hydrostatic primitive 25 equations, while retaining much of the accuracy of the fully compressible Euler equations. 1 26 1. Introduction 27 So far most atmospheric numerical modeling systems have specialized as limited-area, 28 mesoscale models, global climate models and global weather prediction models, each typi- 29 cally solving different sets of equations of motion, either the Euler equations or approximate 30 sets derived from them using various assumptions (see e.g. Tort and Dubos 2014b). Cli- 31 mate models resolve horizontal scales at which the hydrostatic approximation is accurate 32 and cost-effective. Mesoscale models need equations that remain accurate at small horizon- 33 tal scales, and typically solve either the compressible Euler equations or anelastic/pseudo- 34 incompressible equations (in the sequel we refer as “anelastic/pseudo-incompressible” to 35 both anelastic equations (Ogura and Phillips 1962; Lipps and Hemler 1982) and pseudo- 36 incompressible equations (Durran 1989, 2008; Klein and Pauluis 2011)). Global weather 37 forecasting models need accuracy at both small and large scales, which may rule out the 38 anelastic equations which distort long Rossby waves (Davies et al. 2003; Arakawa and Konor 39 2009; Dukowicz 2013). 40 The Euler equations support the propagation of acoustic waves. Resolving these meteo- 41 rologically unimportant motions not only involves taking care of useless degrees of freedom, 42 but also creates numerical difficulties. Although much progress has been achieved towards 43 efficient and accurate solutions to these difficulties (Skamarock and Klemp 2008; Weller et al. 44 2013; Smolarkiewicz et al. 2014), it can still be desirable to identify “unified” equations of 45 motion that would not support acoustic waves while retaining accuracy at large and small 46 scales. This is especially desirable if the specialization of modeling systems is to be aban- 47 doned in favor of all-scale, unified modeling systems, a strong current trend. Even if such 48 equations are eventually not chosen as the basis of a numerical model, they may help iden- 49 tifying the independent degrees of freedom of the atmospheric flow to be modeled and how 50 the dependent fields are related to the independent fields. 51 52 A major step towards the identification of such a unified system of equations has been 53 achieved by Arakawa and Konor (2009), hereafter AK09 (see also Konor 2013). In order to 54 discuss their work, it is useful to recall how acoustic waves are suppressed from the hydro- 2 55 static and anelastic/pseudo-incompressible equations, what the remaining degrees of freedom 56 are, and how they relate to large-scale waves. The hydrostatic equations of motion neglect 57 vertical acceleration in the vertical momentum balance. A consequence is that vertical ve- 58 locity is not prognostic any more. Furthermore for a given profile of potential temperature, 59 adequate top and bottom boundary conditions (either imposed pressure or imposed altitude 60 at the top of the domain) and a given column-integrated mass, there exists a unique profile 61 of density that satisfies hydrostatic balance. This fact is particularly apparent if the flow is 62 described using a Lagrangian or mass-based vertical coordinate (e.g. Dubos and Tort 2014). 63 From this point of view hydrostatic balance is the result of a vertical hydrostatic adjustment, 64 i.e. adiabatic vertical displacement of air parcels which adjusts their specific volume until 65 the resulting pressure profile reaches hydrostatic balance. Therefore the remaining degrees of 66 freedom are horizontal velocity, potential temperature and column-integrated mass. Vertical 67 velocity can be diagnosed by time-differentiating hydrostatic balance, yielding Richardson’s 68 equation, a one-dimensional vertical elliptic equation (Richardson 1922). The two degrees 69 of freedom lost, vertical velocity and density, were precisely those that allowed the acoustic 70 waves to propagate. 71 On the other hand with the anelastic/pseudo-incompressible equations density is locally 72 prescribed or diagnosed from potential temperature (Ogura and Phillips 1962; Lipps and 73 Hemler 1982; Durran 1989, 2008; Klein and Pauluis 2011) through a horizontally-uniform 74 hydrostatic background atmospheric profile assumed time-independent (Ogura and Phillips 75 1962; Lipps and Hemler 1982; Durran 1989, 2008; Klein and Pauluis 2011) or allowed to vary 76 in time (Almgren 2000; O’Neill and Klein 2014). This approach relies on the hypothesis that 77 the actual atmosphere does not depart too much from this background profile. Due to the 78 anelastic/pseudo-incompressible constraint, only two of the three velocity components are 79 prognostic. Although other choices are possible (Jung and Arakawa 2008), one may consider 80 horizontal velocity components as prognostic, vertical velocity being then diagnosed from the 81 anelastic/pseudo-incompressible constraint. Although it involves vertical acceleration, the 82 vertical momentum budget is then not a prognostic equation. Instead, combined with the 83 horizontal momentum budget, it leads to a Poisson-like, three-dimensional elliptic problem 84 for the deviation of pressure from its reference value. Compared to the hydrostatic equa- 3 85 tions, the anelastic/pseudo-incompressible equations have lost two two-dimensional degrees 86 of freedom : the column-integrated mass and the divergence of the column-integrated mass 87 flux (or potential temperature flux). As a consequence the anelastic/pseudo-incompressible 88 equations do not support the Lamb wave (Davies et al. 2003). The Lamb wave is by it- 89 self not important meteorologically, but the ability of the barotropic flow to be divergent 90 is crucial to the correct propagation of long barotropic Rossby waves (Cressman 1958), 91 which are meteorologically important (Madden 2007). This damages the validity of the 92 anelastic/pseudo-incompressible equations at large scales, although the seriousness of this 93 issue remains a subject of debate (Smolarkiewicz et al. 2014). 94 Generally speaking, acoustic waves are suppressed if the feedback between velocity diver- 95 gence and pressure is suppressed, which the hydrostatic and anelastic/pseudo-incompressible 96 equations achieve by slaving density to potential temperature, although in different ways. 97 Based on the above discussion, AK09 concluded that accuracy at large scales requires that 98 density be slaved the hydrostatic way, i.e. deduced from potential temperature and column- 99 integrated mass, and not the anelastic/pseudo-incompressible way. They recognized that 100 this assumption does not by itself imply the neglect of vertical acceleration in the vertical 101 momentum budget, only the converse being true. This observation enabled them to derive 102 non-hydrostatic equations of motion where density satisfies the hydrostatic constraint. A 103 delicate part of their derivation is the formulation of an elliptic problem yielding the devi- 104 ation of pressure from its hydrostatic value. They obtained an elliptic problem similar to 105 that of pseudo-incompressible equations. However one source term proportional to the sec- 106 ond time derivative of density remained unspecified, and they suggested an numerical time 107 extrapolation procedure to evaluate it in
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