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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 a time-marching numerical scheme. Furthermore
108 additional ad hoc arguments related to the conservation of energy had to be invoked in order
109 to ensure a unique solution to the elliptic problem. Nevertheless normal-mode analysis of
110 the unified system for small deviations from an isothermal state of rest revealed its excellent
111 accuracy from large to small horizontal scales.
112
113 Historically, the vast majority of approximate sets of equations used in geophysical fluid
114 dynamics have been derived heuristically, using a combination of scale analysis, intuition and
4 115 ingenuity to achieve both accuracy and dynamical consistency, defined as the existence of
116 conservation principles for suitably defined energy, potential vorticity and momentum. How-
117 ever it is now well-established that Hamilton’s principle of least action provides a systematic
118 procedure to derive dynamically consistent equations of motion, provided all approximations
119 are made directly in the expression of the action integral prior to invoking its stationarity
120 (Salmon 1983; Morrison 1998; Holm et al. 2002). Nevertheless, this approach remains the
121 exception (Tort and Dubos 2014a) rather than the rule, although most useful equations of
122 atmospheric motion can be shown to derive from a variational principle (Tort and Dubos
123 2014b). Especially sound-proof equations are obtained through the introduction of a La-
124 grange multiplier enforcing the constraint satisfied by density Cotter and Holm (2013); Tort
125 and Dubos (2014b).
126 Assuming that the AK09 equations conserve energy, potential vorticity and momentum,
127 the question emerges naturally as to whether they can be obtained from a variational princi-
128 ple. Such a formulation would provide additional insight into their structure, and may help
129 designing numerical schemes with desirable properties such as discrete conservation of energy
130 or self-adjointness of the discrete elliptic problem. In Tort and Dubos (2014b) a Lagrangian
131 description of the flow is adopted (Salmon 1983; Morrison 1998). This makes the variational
132 calculus simple, but would complicate the expression of the hydrostatic constraint, because it
133 involves a vertical derivative which is not straightforwardly expressed with Lagrangian vari-
134 ables. The most suitable framework is therefore a Eulerian description of the flow and the
135 relevant variational calculus (Holm et al. 2002; Cotter and Holm 2013). The work presented
136 hereafter started as an attempt at derive the AK09 equations from a Eulerian variational
137 principle. However it turned out that a slightly different set of equations was obtained, which
138 called for a physical interpretation of the additional terms and a normal-mode analysis to
139 check its accuracy.
140 The paper is organized as follows. Section 2 recalls the various forms of the inviscid,
141 compressible Euler equations for an arbitrary equation of state, as well as the AK09 uni-
142 fied equations. Section 3 applies the physical idea underpinning AK09 within an Eulerian
143 variational framework, introducing a Lagrange multiplier λ to impose hydrostatic balance.
144 The equations of motion are directly obtained in vector-invariant form. Comparison with
5 145 AK09 shows that an additional term has appeared, which suggests to interpret λ as a non-
146 hydrostatic vertical displacement from which all other non-hydrostatic perturbations derive.
147 In section 4 the conservation laws, which necessarily hold due to the variational approach, are
148 explicitly derived and confirm the physical interpretation of λ. The elliptic problem satisfied
149 by λ is obtained as well, invoking only the equations of motion and rigid-boundary bound-
150 ary conditions. The normal-mode analysis performed in section 5 shows that the equations
151 of motion filter acoustic waves and support accurate inertia-gravity waves and undistorted
152 long Rossby waves, with some caveats with respect to deep internal gravity waves. Section
153 6 summarizes and concludes.
154 2. The Euler and AK09 unified equations
155 a. Inviscid fully compressible Euler equations
156 For adiabatic motion of a compressible fluid, the mass and entropy budgets are : ∂ρ Ds + divρu =0, =0 (1) ∂t Dt
157 where ρ, v =1/ρ and s are density, specific volume and specific entropy and u = Dx/Dt is
158 velocity where x is position in the three-dimensional Cartesian coordinates x,y,z and D/Dt
159 is the material (Lagrangian) time derivative. The momentum budget can the be obtained
160 from Hamilton’s principle of least action, using either the Lagrangian or the Eulerian de-
161 scription of the flow. If the Lagrangian description is used (Salmon 1983; Morrison 1998)
162 then the equations of motion are obtained in Euler-Lagrange form : Du 1 + ∇p + ge =0. (2) Dt ρ z
163 where e(v,s) is the specific internal energy and p(ρ,s) = −∂e/∂v is pressure. The local
164 conservation law for total energy (kinetic, internal and potential) u · u E = ρ + e(1/ρ,s)+ gz , (3a) 2 ∂tE + div [(E + p) u]=0, (3b) 165
6 166 follows quite straightforwardly from (2).
167 On the other hand a Eulerian variational principle can be used (Newcomb 1962; Holm
168 et al. 2002) :
δ Ldt =0, L[ρ, u,s]= l(ρ, u,s, x)dx. (4) Z Z 169 In Hamilton’s principle (4), L[ρ, u,s] is a functional of the 3D, instantaneous Eulerian fields
170 ρ, u, s while l = lE(ρ, u,s, x) is an ordinary function (of 8 scalar variables) : u · u l (ρ, u,s, x)= ρ − e(1/ρ, s) − gz . (5) E 2 171 From (4) the equations of motion are naturally obtained in the curl form or vector-invariant
172 form (see Holm et al. (2002) and section 3) :
u · u p ∂ u +(curl u) × u + ∇ + e + − Ts + gz + s∇T =0 (6) t 2 ρ 173 The Lagrangian conservation of Ertel’s potential vorticity
1 q = ∇s · curl u, Dq/Dt =0 (7) ρ
174 follows relatively easily from (6).
175 Finally the conservation law for momentum is apparent in the flux-form of the equations
176 :
∂t (ρu)+ div (ρu ⊗ u)+ ∇p + ρgez =0. (8)
177 This form is easy to obtain from the Euler-Lagrange form (2). It does not directly follow,
178 but can be obtained, from a Eulerian variational principle if the Lagrangian is invariant with
179 respect to translations. As (2), (8) permits to identify unambiguously the forces acting on
180 the fluid.
−R/cp 181 Assuming an ideal perfect gas, vdp = θdπ where θ = θ(s) = T (p/p0) is potential
R/cp 182 temperature and π = ∂e/∂θ = cp (p/p0) is the Exner function (R, cp, p0 being the
183 constant pv/T , the specific heat capacity at constant pressure and an arbitrary reference
184 pressure). A form often encountered in the literature is then :
Du + θ∇π + ge =0. (9) Dt z
7 185 It must be stressed that this form is neither an Euler-Lagrange form, a curl-form, or a flux-
186 form. It is a mix between the Euler-Lagrange form (5) and the curl-form (6). Furthermore
187 for a general equation of state, although one can still define a potential temperature θ(s)=
188 T (s, p0) and an Exner function π = ∂e/∂θ = T ds/dθ, vdp − θdπ = d (e + pv − θπ) =6 0 so
189 that the contribution from thermodynamics ∇ (e + pv − θπ) does not generally vanish, as in
190 (6).
191 b. AK09 unified equations
192 Arakawa and Konor (2009) have derived unified equations based on the idea that the den-
193 sity field satisfies hydrostatic balance while velocity does not. Their approximation strategy
194 uses (9) as a starting point, and they expand the Exner function π into a quasi-static con-
195 tribution πqs determined by hydrostatic balance and the equation of state :
∂zpqs + ρg =0, (10a) 196 p κ p = p(ρ,θ), π = π(ρ,θ)= c qs , (10b) qs qs p p 0 ′ 197 and a non-hydrostatic contribution π to be determined by an elliptic equation deduced from
198 the mass budget :
∂tρ + ∇· (ρu) = 0 (11a) Du + θ∇π + ge = −θ∇π’. (11b) Dt qs z
199 Arakawa and Konor (2009) do not explicitly derive a flux-form for (11b). Such a form should
200 be obtained by multiplying (11b) by the density ρ obeying the mass budget (11a). Then
201 ρθ∇πqs = ∇pqs due to (10b) but it is not obvious how ρθ∇π’ can also be transformed into
202 a pure divergence.
203 3. Constrained variational principle
204 In this section the physical idea underpinning AK09 is implemented using a different
205 strategy : instead of manipulating the equations of motion, the Lagrangian is augmented
8 206 using a Lagrange multiplier λ in order to enforce (10a). The equations of motion follow,
207 which turn out to differ from (11b). A physical interpretation of additional terms is given,
208 and confirmed in the next section by analyzing the momentum budget in flux-form.
209 a. Lagrange multiplier
210 (10a) is an additional relationship that we wish to impose. It will restrict the motions
211 accessible to the flow, and is therefore a constraint. A constraint can be introduced into a
212 variational principle though a Lagrange multiplier λ provided the constraint acts only on
213 positions, not velocities. Such a constraint is termed holonomic (see Bernardet 1995 for a
214 discussion in the context of the anelastic approximation). From a fluid-parcel point of view,
215 ρ and s can be deduced from positions (Salmon 1983; Morrison 1998) so the hydrostatic
216 constraint is indeed holonomic. Since the constraint holds at every time and position, a field
217 λ(x,t) is necessary to enforce it. We therefore add to the Lagrangian the contribution ∂p ∂λ L [ρ,s,λ]= − λ qs + ρg d3x = p − ρgλ d3x (12) λ ∂z qs ∂z Z Z 218 where λ has the dimension of a length and boundary conditions will be discussed later. As
219 in AK09, we note pqs = p(ρ,s).
220 The constrained variational principle is therefore
δ Ldt =0, L[ρ, u,s,λ]= l(ρ, u,s, x λ,λz)dx (13) Z Z 221 where l is now a function of 10 variables including λ and λz = ∂λ/∂z and L is a functional
222 of the Eulerian fields ρ, u,s,λ :
l(ρ, u,s, x, λ,λz) = lE + lλ (14a)
lλ(ρ,s, λ,λz) = pqsλz − ρgλ (14b)
223 Vanishing variations with respect to λ impose : ∂l ∂l ∂l ∂ ∂l 0= δL = δλ + δλ dx = − δλdx (15a) ∂λ ∂λ z ∂λ ∂z ∂λ Z z Z z 224 hence ∂l ∂l λ = ∂ λ , (15b) ∂λ z ∂λ z 225 i.e. hydrostatic balance (10a)
9 226 b. Eulerian variational principle
227 We now obtain the equations of motion from Hamilton’s principle of least action. What
228 follows is essentially an application of the general Euler-Poincaré theory found in Holm et al.
229 (2002), exposed here using standard differential and vector calculus instead of the language
230 of differential geometry.
231 The key point is that variations of ρ, u, s are not mutually independent because ρ, u, s
232 must satisfy the purely kinematic transport equations :
∂ρ ∂s + divρu =0, + u ·∇s =0. (16) ∂t ∂t
233 Therefore δρ, δu, δs must satisfy :
∂ ∂ δρ + div (uδρ + ρδu)=0, δs + δu ·∇s + u ·∇δs =0. (17) ∂t ∂t
234 The general solution obtained by Newcomb (1962) is :
∂ξ δρ = −div (ρξ) , δu = + ∇ξ · u −∇u · ξ, δs = −ξ ·∇s (18) ∂t
235 where the virtual displacement ξ is arbitrary in the interior of the space-time domain (New-
236 comb 1962; Holm et al. 1998, 2002). We use standard tensor notation, i.e. ∇u · v and v ·∇u
237 are the vectors with components j (∂jui) vj and j vj (∂iuj), respectively. For a fixed spa-
238 tial domain, boundary conditionsP are that ξ = 0 Pat t = t1,t2 and ξ must be tangential at
239 the boundaries of the domain. Hamilton’s principle (4) holds for variations of the form (18)
240 only, not for arbitrary, independent variations of ρ,s, u.
241 Letting 1 v = ∂l/∂u (19) ρ
242 and taking into account (18), the Lagrangian density varies as
∂l ∂ξ ∂l δl = − divρξ + ρv · + ∇ξ · u −∇u · ξ − ξ ·∇s ∂ρ ∂t ∂s ∂l ∂ = −div ρ ξ + (ρv · ξ)+ div [(ρv · ξ) u] ∂ρ ∂t ∂ ∂l ∂l − (ρv) · ξ + ρξ ·∇ − ξ · div (v ⊗ ρu) − ρv ·∇u · ξ − ξ ·∇s. (20a) ∂t ∂ρ ∂s
10 243 The identities
div (v ⊗ ρu)= div (ρu) v+∇v·ρu, ∇ (u · v)= v·∇u+u·∇v, ∇v·u−u·∇v =(∇× v)×u (20b)
244 yield
∂ ∂l δl = (ρv · ξ)+ div (ρv · ξ) u − ρ ξ ∂t ∂ρ ∂v 1 ∂l − +(∇× v) × u + ∇B + ∇s · ρξ (20c) ∂t ρ ∂s
245 where B = u · v − ∂l/∂ρ is the Bernoulli function. Hamilton’s principle of least action then
246 yields the evolution equation for v in the curl form :
∂v 1 ∂l δ Ldt = − +(∇× v) × u + ∇B + ∇s · ρξ dxdt (21a) ∂t ρ ∂s Z Z 247 ∂v 1 ∂l δ Ldt =0 ⇔ +(∇× v) × u + ∇B + ∇s =0. (21b) ∂t ρ ∂s Z 248 Notice that with the Lagrangians (5,14a) considered in this work one finds simply v = u.
249 In a more general setting the Lagrangian would include a contribution generating the Coriolis
250 force, resulting in v being absolute instead of relative velocity. The Lagrangian (5) yields the
251 Euler equations (6) and we focus here on the additional terms resulting from the constraint
252 enforced by lλ. Using
∂lλ 2 ∂λ ∂lλ ∂p ∂λ ∂lλ ∂lλ = c − gλ, = , = −ρg, = pqs, (22) ∂ρ ∂z ∂s ∂s ∂z ∂λ ∂λz
253 and introducing the speed of sound c = ∂p/∂ρ, we find:
u · u p ∂l 1 ∂l ∂ u + curl u × u + ∇ + e + pv + gz − T ∇s = ∇ λ − λ ∇s (23) t 2 ∂ρ ρ ∂s ∂λ 1 ∂p ∂λ = ∇ c2 − gλ − ∇s. ∂z ρ ∂s ∂z
254 c. Comparison with AK09
255 To facilitate the comparison of (23) with AK09 we consider, in this subsection only, an
256 ideal perfect gas and replace s by θ. As a consequence T = ∂e/∂s becomes πqs = ∂e/∂θ in
11 2 2 257 (23). Noticing that ∂p/∂θ = −∂ e/∂θ∂v = −∂π/∂v = ρ ∂π/∂ρ , the r.h.s. of (23) can be
258 rewritten as : ∂λ 1 ∂p ∂λ p′ ∇ c2 − gλ − ∇θ = −g∇λ −∇ + π′∇θ (24) ∂z ρ ∂θ ∂z ρ 259 where we have defined
∂λ ∂p ∂λ ∂π 1 ∂p ∂λ ρ′ = −ρ , p′ = ρ′ = −ρc2 , π′ = ρ′ = − . (25) ∂z ∂ρ ∂z ∂ρ ρ ∂θ ∂z
′ ′ 260 Finally, vdp = θdπ implies p /ρ = θπ hence :
Du + θ∇π + ge = −θ∇π′ − g∇λ (26) Dt qs z
261 Clearly the variational approach yields equations of motion (26) that differ from those of
262 AK09 (11b).
2 ′ ′ 263 Assuming a characteristic vertical scale Lz, (25) yields ρc λ ∼ Lzp hence gλ ∼ (Lz/H) θπ 2 264 where H = c /g ≃ 10 km is the scale height. This suggests that the extra term g∇λ is gen-
265 erally small compared to the non-hydrostatic contributions already taken into account by
266 AK09, except for motions whose vertical scale is comparable to or larger than the scale
267 height.
268 d. Physical interpretation and boundary conditions for λ
269 The additional term −g∇λ found in (26) conveys at this stage no obvious physical in- ′ ′ 270 terpretation. All that can be said is that it can not be re-expressed in terms of p or π
271 because they depend on ∂zλ and do not carry information about the horizontal variations of
272 λ. However if we rewrite (26) as :
Du + ∇ [g(z + λ)] + θ∇ (π + π′)=0 (27) Dt qs
273 then a possible physical interpretation emerges : the true height of the air parcel may in ′ 274 fact not be z but Z = z + λ. This is consistent with π = πqs + π and the full density being ′ ′ 275 not ρ but µ = ρ + ρ = ρ (1 − ∂zλ). Indeed assuming ∂zλ ≪ 1 or, equivalently ρ ≪ ρ, −1 −1 276 µ ≃ ρ (1 + ∂zλ) = ρ (∂zZ) . If we consider a single atmospheric column with density and
277 specific entropy profiles µ(Z), s(Z) close to hydrostatic balance, label each air parcel by its
12 278 initial position Z and fictitiously and adiabatically displace each parcel to another altitude −1 279 z(Z) = Z − λ, the density profile becomes µ (dz/dZ) = ρ. Hence ρ is the density of the
280 atmospheric column after all parcels have been displaced from their true height Z to the
281 height z.
282 Now the internal and potential energy of the atmospheric column depends on the final
283 position z(Z) of air parcels as :
∂ z P [z] = e Z , s(Z) + gz µdZ (28a) µ Z ∂ z δP = −p Z , s(Z) ∂ δz + µgδz dZ µ Z Z ∂ z = ∂ p Z , s(Z) + µg δzdZ (28b) Z µ Z 284 where a rigid-lid is assumed so that δz =0 at the top and bottom of the atmospheric column.
285 Hence hydrostatic balance ∂zpqs + ρg =0 is satisfied precisely if the altitudes z are such that
286 P is minimized. The hypothetical process moving the air parcels from their true altitude Z
287 to the altitude z is therefore an (adiabatic) hydrostatic adjustment. z is the altitude an air
288 parcel would have after its atmospheric column has undergone a hydrostatic adjustment, and
289 λ is the difference between the true height Z and the adjusted height z, assumed small. This
290 interpretation provides the boundary conditions for λ. For rigid-lid boundary conditions one
291 should take λ =0 at the top and bottom.
292 The Lagrangian (14a) then appears as the result of an asymptotic expansion :
u · u (Dz/Dt + Dλ/Dt)2 1+ ∂ λ L = ρ H H + − e z , s − g (z + λ) dzdx 2 2 ρ Z ! u · u (Dz/Dt)2 1 p ≃ ρ H H + − e , s + ∂ λ − g (z + λ) dzdx (29) 2 2 ρ ρ z Z !
293 In addition to the assumption ∂zλ ≪ 1, the validity of approximation (29) requires that
294 Dλ/Dt ≪ Dz/Dt. The latter assumption will appear in section 6 as the main limitation of
295 our semi-hydrostatic theory.
13 296 4. Conservation laws and diagnostic equation for λ
297 Because the equations of motion (26) derive from a variational principle, it is guaranteed
298 that they possess conservation laws for momentum, potential vorticity and energy. Here we
299 obtain these conservation laws explicitly. The momentum budget will be especially helpful
300 to unambiguously identify forces, and to confirm the physical interpretation of λ. We start
301 from the curl-form (23) and focus on the contributions of the right-hand-side.
302 a. Potential vorticity
303 To see that the terms on the r.h.s of (23) do not contribute to the potential vorticity ∂p 304 ∂λ budget, take the curl, leaving only the contribution ∇s ×∇ ∂s ∂z in the vorticity budget. 305 Since this term is orthogonal to ∇s it does not contribute the evolution equation for (curl u)·
306 ∇s . Hence Ertel’s potential vorticity (1/ρ)(curl u) ·∇s is conserved as usual.
307 b. Momentum budget
308 Conservation of momentum results from the invariance of the Lagrangian with respect
309 to translations, which is expressed in our case by :
∂lλ ∂lλ ∂lλ ∂lλ ∇lλ = ∇ρ + ∇s + ∇λ + ∇λz (30) ∂ρ ∂s ∂λ ∂λz
310 After multiplying (23) by ρ we can transform the terms due to λ as follows :
∂lλ ∂lλ ∂lλ ∂lλ ∂lλ ∂lλ ρ∇ − ∇s = ρ∇ + ∇ρ + ∇λ + ∇λz −∇lλ ∂ρ ∂s ∂ρ ∂ρ ∂λ ∂λz ∂l ∂ ∂l = ∇ ρ λ − l + ∇λ λ (31) ∂ρ λ ∂z ∂λ z
311 where we have used the constraint (15b). Replacing lλ and its derivatives by their expressions
312 (22) as a function of ρ,s,λ yields :
∂λ ∂λ ∂ ∂ (ρu)+ div (ρu ⊗ u)+ ∇p + ρge = ∇ ρc2 −∇ p + (p ∇λ) (32a) t qs z ∂z qs ∂z ∂z qs
313 This shows that (26), contrary to (11b), clearly admits a flux-form momentum budget. This
314 budget is to be interpreted as follows :
14 315 • the vertical budget is Dw ∂p′ ρ = − . (32b) Dt ∂z
316 The weight of an air parcel is gdm, where dm = ρdxdydz = (ρ+ρ’) dxdyd (z + λ) is
317 the true elementary mass. The vertical gradient of hydrostatic pressure dpqs exactly
318 cancels, by design, this weight on the left-hand-side. Hence the vertical acceleration is ′ 319 solely due to non-hydrostatic pressure p defined as the extra pressure that arises if air
320 parcels undergo an adiabatic vertical displacement from their hydrostatically-adjusted
321 height z to their true height z +λ. Indeed as discussed previously if the parcels initially
322 at altitudes z and z + dz move to z + λ and z + dz + ∂λ/∂z dz, the volume they occupy ′ ′ 323 ∂λ is multiplied by 1+ ∂z hence the expressions (25) for ρ and p .
324 • the horizontal budget is :
Du ∂λ ∂ ρ H + ∇ p = −∇ p′ −∇ p + (p ∇ λ) (32c) Dt H qs H H qs ∂z ∂z qs H ′ 325 where uH and ∇H are the horizontal velocity and gradient. Apart from ∇H p already
326 discussed, two additional terms appear. They make sense if this momentum budget
327 is interpreted as a momentum budget over a vertically displaced infinitesimal domain,
328 located between x and x + dx, y and y + dy and z + λ and z + dz + λ + ∂λ/∂z dz. The
329 upper and lower boundaries of this domain are actually sloping, with a slope ∇H λ.
330 The pressure force exerted by this upper surface has a horizontal component pqs∇H λ.
331 ∂ The upper and lower contributions sum up to ∂z (pqs∇H λ). Furthermore the lateral
332 boundaries have height (1 + ∂λ/∂z)dz. The extra pressure forces exerted on these
333 ∂λ boundaries sum up to −∇H pqs ∂z . 334 Therefore the physical interpretation of λ as the vertical displacement between the true parcel
335 position and its position after the atmospheric column has undergone hydrostatic adjustment
336 is confirmed by the analysis of the momentum budget. In this budget, two categories of new ′ 337 terms arise. The first term ∇p is a true additional force, due to the fact that the actual ′ ′ 338 density and pressure of the air parcel are not ρ and pqs but ρ + ρ and pqs + p . The other
339 terms are pseudo-forces that arise when taking into account the slightly curvilinear character
340 of the coordinate system (x,y,z).
15 341 c. Energy
342 Regarding the energy budget, the new terms contribute : ∂l 1 ∂l ∂l ∂l ∂l ρu · ∇ λ − λ ∇s = div ρu λ + λ ∂ ρ + λ ∂ s (33a) ∂ρ ρ ∂s ∂ρ ∂ρ t ∂s t 343 Now invariance of the Lagrangian with respect to time yields :
∂lλ ∂lλ ∂lλ ∂lλ ∂tlλ = ∂tρ + ∂ts + ∂tλ + ∂tλz ∂ρ ∂s ∂λ ∂λz ∂l ∂l = λ ∂ ρ + λ ∂ s + ∂ (p ∂ λ) (33b) ∂ρ t ∂s t z qs t
344 Let us note ′ E = −lλ = ρgλ − pqs∂zλ (34a)
345 the additional internal and potential energy induced by the displacement λ. Notice that ′ ′ 346 hydrostatic balance (15b) implies that E dx = 0. Now using ρ∂lλ/∂ρ = −ρgλ − p = ′ ′ 347 −E − p − pqs∂zλ and collecting the standardR terms and the new terms, the full energy
348 budget reads
′ ′ ′ ∂t (E + E )+ div [(E + E + pqs + p ) u]+ div [pqs (u∂zλ + ∂tλez)]=0. (34b)
349 Boundary conditions of no-normal velocity and λ = 0 imply that no energy crosses the
350 boundaries of the domain, so that the domain-integrated energy, equal to the usual energy
351 Edx, is constant.
352 R The last term of this energy budget makes sense if we remember that the air parcels are ′ ′ 353 u u u Dλ e actually at altitude z + λ. Their actual velocity is + with = Dt z. Furthermore the
354 slope of the surfaces at altitude z + λ and the variation in “thickness” between surfaces at
355 altitude z+λ and z+λ+dz+dλ should be taken into account. These effects are already taken ′ 356 into account in ρ hence in E but not in pqs. They sum up to pqsu − pqsu ·∇λ + pqsu∂zλ = ′ 357 pqs (u∂zλ + ∂tλez). Hence the energy fluxes found in the local budget of E + E can all be ′ 358 interpreted either as the work of the pressure pqs + p or as due to the displacement λ.
359 d. Diagnostic equation for the non-hydrostatic displacement
360 The non-hydrostatic displacement λ must be diagnosed in order to close the semi-
361 hydrostatic system of equations. The hydrostatic constraint (15b) does not involve λ and
16 362 cannot be used to diagnose it. Instead it constrains ρ which is slaved to s. Additional hidden
363 constraints, obtained by time-differentiating hydrostatic balance, must exist. Because ∂ρ/∂t
364 and ∂s/∂t do not depend on λ, the first hidden constraint will not be solved for λ but for w.
365 Only after a second time differentiation does an equation for λ appear. These calculations
366 are done below assuming a flat bottom, but similar calculations can be done in the presence
367 of orography.
368 One time-differentiation of hydrostatic balance yields :
2 ∂p 2 ∂p ∂ c ∂ (ρw)+ w∂ s + gw + ∂ c ∂x · (ρu )+ u · ∂xs + g∂x (ρu )=0 (35) z z ∂s z z H ∂s H H 2 ∂p 2 ∂p 369 x x x Using −ρg = ∂zpqs = c ∂zρ + ∂s ∂zs, ∂ pqs = c ∂ ρ + ∂s ∂ s one finds the elliptic equation for
370 w :
A· w + B· uH =0, (36a) 371
2 where A : w 7→ ∂z ρc ∂zw (36b)
2 B : uH 7→ ∂z ρc ∂x · uH + uH · ∂xpqs + g∂x · (ρuH ) (36c)
∗ 2 B : λ 7→ ∂x ρc ∂zλ − ∂zλ∂xp − ρg∂ xλ (36d) 372 Assuming a flat bottom, w =0 at top and bottom, A is self-adjoint, positive definite while B ∗ 373 has adjoint B (with horizontally-periodic BCs and λ =0 at top and bottom). Equation (36a)
374 is Ooyama’s “neater form” of Richardson’s equation that arises when solving the primitive
375 equations in Eulerian coordinates (Richardson 1922; Ooyama 1990). It is also a particular
376 case of the form derived by Dubos and Tort (2014) for general quasi-hydrostatic systems.
377 Notice that this form differs from Richardson’s original formulation, which suffers from the
378 near-cancellation of two large terms (Arakawa and Konor 2009), unlike the “neater form”
379 (Ooyama 1990).
380 Now the momentum balance (32a) is precisely of the form 1 ∂ w − A· λ = r.h.s. (37a) t ρ 1 ∂ u − B∗ · λ = r.h.s. (37b) t H ρ
381 where r.h.s collects terms that do not involve λ. Time-differentiating (36a) and eliminating
382 ∂tuH yields therefore the self-adjoint problem :
17 B 1 B∗ A λ ρ = r.h.s (37c) A −ρ ∂tw 383 where λ and ∂tw satisfy homogeneous Dirichlet boundary conditions. Eliminating ∂tw yields
384 a single, positive definite problem for λ :
1 1 A A + B B∗ λ = r.h.s. (37d) ρ ρ
385 with inhomogeonous Dirichlet boundary conditions for Aλ. One sees that the “strange”
386 terms in (32a) are crucial to obtain an elliptic problem for λ.
387 e. Computational procedure
388 At this point, it is possible to outline a computational procedure to solve the semi-
389 hydrostatic equations. Using the form (26) and hydrostatic height z as a vertical coordinate,
390 possible prognostic variables of the system are uH , θ, ρ (the computational procedure will be
391 similar if the flux- or vector-invariant form of the system is considered instead). The major
392 diagnostic variables are pqs, λ, w. Prognostic variables are assumed to be known at time
393 level n. An explicit time-stepping organization for advancing from the current time level n,
394 to the next one n +1 can be sketched as follows:
(n) (n) (n) (n) 395 i. Hydrostatic quantities pqs , πqs are diagnosed from θ and ρ using the equation of
396 state (10b).
(n) 397 ii. Vertical wind w is diagnosed from Richardson’s equation (36a) with adequate bound-
398 ary conditions.
(n) 399 u (n) (n+1) iii. The full wind field H , w being now available, thermodynamic variables θ and (n+1) 400 ρ are prognosed using (16).
(n) 401 iv. Diagnostic equation (37d) for the non-hydrostatic displacement λ is solved at current
402 time level with upper and lower rigid-wall boundary conditions for λ. Collaterally, the ′ 403 non-hydrostatic departure π at time level n is obtained via (25).
18 (n+1) 404 u v. Horizontal wind H is pronostically obtained from (26), adding the extra tendency (n) 405 associated to λ to the right-hand side term of the horizontal momentum equation.
406 Notice that density can be predicted by the mass continuity equation without altering the
407 sound-filtering ability of the system provided that the hydrostatic constraint is initially
408 satisfied, and later maintained via the resolution of Richardson’s equation (36a). AK09
409 have devised a computational procedure that avoids solving Richardson’s equation. This
410 procedure may be applicable to the present system, but we leave the investigation of this
411 point for future work.
412 5. Normal-mode analysis
413 a. Resting isothermal base flow
414 Having explicitly verified the dynamical consistency of (23), we now proceed to assess
415 their accuracy. For this we reintroduce the Coriolis force, left aside for brevity in the pre-
416 vious section, and examine the dispersion relationship of infinitesimal perturbations to an
417 isothermal, resting atmosphere. We assume an ideal perfect gas and use potential tempar-
418 ture θ instead of entropy s. Including the Coriolis force, the full set of equations of motion
419 is :
Du 1 ∂λ = −fe × u − 1+ ∇p Dt z ρ ∂z qs 1 ∂λ + ∇ ρc2 − g∇λ − ge , (38a) ρ ∂z z Dθ = 0, (38b) Dt Dρ = −ρ ∇· u, (38c) Dt
∂zpqs = −ρg (38d)
κ 1−κ 420 where pqs(ρ,θ) satisfies ρRθ = p00/pqs with κ = R/Cp and p00 a reference pressure. For the ∗ 421 purpose of linearization of this system, the atmospheric basic-state X is chosen stationary, ∗ ∗ ∗ 422 resting, horizontally homogeneous, and hydrostatically balanced. Hence, θ = θ (z), pqs = ∗ ∗ ∗ ∗ ∗ 423 pqs(z), and ρ = ρ (z), linked by the hydrostatic balance dpqs/dz = −ρ g. In addition, the
19 ∗ ∗ ∗ 424 temperature of the basic state T = pqs/Rρ is assumed to be constant in order to be able
425 to find analytic solutions. The corresponding profiles are :
∗ ∗ −z/H∗ pqs(z) = pqs(0)e , (39a)
∗ ∗ ∗ ∗ −z/H ∗ −z/H∗ ρ (z) = (pqs(0)/RT )e ≡ ρse , (39b)
∗ ∗ ∗ ∗ −κ κz/H ∗ κz/H∗ θ (z) = T (pqs(0)/p00) e ≡ θs e (39c)
∗ 426 where H∗ = RT /g denotes the characteristic height-scale of the standard atmosphere, and ∗ 427 the basic surface pressure pqs(0) is an arbitrary constant. All other variables of the basic ∗ ∗ ∗ ∗ 428 state are set to zero, u = v = w = λ = 0 where u,v,w are the velocity components. ′ ∗ 429 The analysis is performed for small perturbation around the basic-state X = X − X , The ′ 430 perturbations X are assumed to remain small; thus, only the first order perturbed quanti-
431 ties is considered, (i.e any product of primed perturbation terms is neglected). Moreover,
432 the domain is restricted to a vertical plane along (x,z) directions for clarity. Finally, the
433 linearized system then writes
′ ∂u ′ 1 ′ 2 ′ = fv − ∂ p + c∗∂ − g ∂ λ (40a) ∂t ρ∗ x qs z x ∂v′ = −fu′ (40b) ∂t ′ ∂w 1 ∗ 2 ′ = ∂ ρ c∗∂ λ (40c) ∂t ρ∗ z z ′ 2 ∂b N∗ = − w′ (40d) ∂t g ∂ρ′ = −∂ (ρ∗u′) − ∂ (ρ∗w′) (40e) ∂t x z
∂zpqs = −ρg (40f) ′ ′ pqs ρ ′ (1 − κ) ∗ = ∗ + b (40g) pqs ρ
′ ′ ∗ 2 ∗ 434 with b = θ /θ , c∗ =(Cp/Cv)RT = gH∗/(1 − κ) the square of the acoustic phase speed and 2 2 ∗ 435 N∗ = g /(CpT )= κg/H∗ the square of Brunt-Vaisala frequency. From (40a) one sees again
436 that the extra term g∇λ is significant only for vertically-large motion ∂z ≤ 1/H∗.
437 For an isothermal profile c∗, N∗ and H∗ are constants and further progress can be made
438 analytically. In what follows, the normal mode analysis will be along the lines of Davies et al.
439 (2003) (hereafter DSWT03). Especially rigid lower and upper boundaries at respectively,
20 440 z = 0 and z = zT are assumed. However, assuming zT ≫ H∗, it is unlikely that a different
441 choice of upper boundary condition would materially change the conclusions, see Dukowicz
442 (2013). We first investigate slow modes on f- and β−, then external and internal modes on
443 the f−plane.
444 b. Slow modes
445 For constant f, (40a-40e) admit stationary solutions (degenerate Rossby modes). ∂/∂t = ∗ 446 0 leads to wˆ =u ˆ =0, corresponding to a non-divergent flow. Furthermore, ∂(ρ ∂λ/∂z)/∂z =
447 0 together with the rigid-wall boundary conditions λ(0) = λ(zT )=0 implies that λ = 0.
448 Therefore, only the geostrophic and the hydrostatic balances hold true for the perturbed ′ ′ ′ 449 hydrostatic pressure pqs, the normal wind component v and the density ρ , as expected.
450 We now sketch the solution of (40a-40e) on a midlatitude β-plane., i.e. f = f∗ + βy.
451 The issue is to assess the accuracy of the dispersion relationship for Rossby modes, espe-
452 cially long external Rossby modes since these modes, which are meteorologically meaningful
453 (Madden 2007), are strongly distorted in the pseudo-incompressible and anelastic/pseudo-
454 incompressible systems (Davies et al. 2003). Therefore the regime of interest is the quasi-
455 geostrophic regime, characterized by ∂/∂x = O(f∗/c∗) and ∂/∂t = O(βc∗/f∗) = O(εf∗),
2 456 where ε = βc∗/f∗ ≪ 1. Expanding (40a-40g) in powers of ε yields at 0−order the hydro-
457 static and geostrophic balances, with zero non-hydrostatic displacement. Non-zero λ appears
458 only at order ε but λ is not involved in the expression of potential vorticity, whose transport
459 by the 0−order geostrophic velocity determines the dispersion relationship. Therefore one
460 ends up, as in AK09, with the same dispersion relationship as with hydrostatic primitive
461 equations, i.e. Rossby modes are undistorted.
462 c. External modes on an f-plane
463 We are now interested in non-stationary modes for constant f = f∗. All coefficients in
464 (40a-40e) being horizontally homogeneous and time-independent, solutions are sought in the ′ 465 normal-mode form X = Xˆ(z)exp i (kx − iσt) for X = u, v, w, b, ρ, pqs, λ, where k is a
466 prescribed horizontal wave number and σ =06 is an unknown pulsation. Inserting this form
21 467 into (40a-40e) yields dp/ˆ dz = −ρgˆ and :
2 2 pˆ 2 d g f − σ uˆ = −σk − c∗ − λˆ , (41a) ρ∗ dz c2 ∗ f vˆ = − u,ˆ (41b) σ 2 c∗ d 1 dλˆ wˆ = − , (41c) σ dz H∗ dz 2 N∗ ˆb = w,ˆ (41d) σg ρˆ 1 d 1 ∗ = kuˆ + − wˆ , (41e) ρ σ dz H∗ pˆ 2 ρˆ = c∗ + ˆb , (41f) ρ∗ ρ∗ 468 Before focusing on modes that satisfy wˆ =6 0, which correspond to the internal modes, we
469 first consider modes without vertical motion, wˆ = 0, corresponding to the external modes
470 of the sytem. Together with rigid upper and lower boundary condition on λ, this leads to
2 2 2 2 471 ˆb =w ˆ = λˆ = 0. Eliminating u,ˆ v,ˆ ρˆ leaves (σ − f − c∗k )ˆp = 0. Assuming pˆ =06 , the
472 pulsation σ is determined by : 2 2 2 2 σ − f − c∗k =0, (42)
2 473 while hydrostatic balance (d/dz − g/c∗)ˆp =0 determines the vertical structure of the mode
474475 :
′ i(kx−σt) pqs =p ˆ0 exp[−(1 − κ)z/H∗] e (43a)
′ pˆ0 i(kx−σt) ρ = (1 − κ) exp[−(1 − κ)z/H∗] e , (43b) RT ∗ ′ σk pˆ0 i(kx−σt) ∗ u = − 2 2 ∗ exp(κz/H ) e , (43c) f − σ ρ0 ′ fk pˆ0 i(kx−σt) ∗ v = 2 2 ∗ exp(κz/H ) e (43d) f − σ ρ0
476 where pˆ0 denotes the value of the perturbed hydrostatic presure at surface (z = 0). Note
477 that the vertical modal structure (43a-43d) is also valid for the external degenerate external
478 Rossby mode (i.e with σ =0). Consequently, the semi-hydrostatic system correctly captures
479 the frequency and vertical structure of the external modes, for Lamb-modes as well as de-
480 generate Rossby modes. The external normal-mode are exactly identical to those found for
22 481 fully compressible equations, as with the AK09 unified equations, and unlike anelastic and
482 pseudo-incompressible systems, see DSWT03 and AK09.
483 d. Internal modes on an f−plane
484 We now consider the internal modes (wˆ =06 ). By combining Eqs (41d), (41e) and (41f)
485 with the hydrostatic balance, we obtain a first constraint (44a), equivalent to Richardson’s
486 equation (36a). By eliminating λˆ and pˆ from Eqs (41a) using respectively (41c),(41f), one
488487 obtains a second constraint (44b) that links uˆ to wˆ :
2 d N∗ d 1 dwˆ − kuˆ = − − (44a) dz g dz H∗ dz 2 2 2 2 d 1 duˆ 2 d g f − σ + c∗k − = c∗k − dz H∗ dz dz c2 ∗ σ2 d 1 d × − − wˆ (44b) c2 dz H∗ dz ∗
490489 Hence, eliminating uˆ in favor of wˆ provides the vertical structure equation :
2 2 2 ∗2 2 2 2 ∗ N∗ 2 σ − f ∆ + k N∗ − σ ∆ − σ wˆ = 0, (44c) z z c2 ∗ ∗ d d where ∆ = − 1/H∗ . (44d) z dz dz 491 The eigenmodes of the left-hand-side differential operator of (44c) are the same as those of
∗ 2 2 492 ∆z = (d/dz − 1/2H∗) − 1/4H∗ . The latter are of the form wˆ(z)=w ˆ0 sin mz exp(z/2H∗)
493 where the vertical wavenumber m is quantified as m=nπ/zT (n ∈ N) due to the rigid-wall ∗ 2 2 494 boundary conditions. Replacing ∆z by its eigenvalue −m − 1/4H∗ in (44c) yields :
2 1 2 2 1 2 m + +Γ m + − Γ− =0, (45a) 4H2 + 4H2 ∗ ∗ 495 with 1 2 2 2 2 2 2 2 2 2 2 2 k N∗ − σ 4N∗ σ k N∗ − σ Γ± = + ∓ , (45b) 2 σ2 − f 2 c2k2 (σ2 − f 2) 2 σ2 − f 2 " ∗ ∗ ∗ # ∗ 2 2 2 2 496 where Γ+/− are real positive values. Here we assume that N∗ >f and σ >f . Since m is
497 real due to boundary conditions and Γ+ > Γ−, only the second factor in (45a) can vanish.
498 Hence boundary conditions restrict the permissible frequencies σ to those that satisfy the
2 2 499 condition Γ− = m +1/4H∗ > 1/2H∗. Some more algebra yields the dispersion relation : p 23 2 2 2 2 2 k (N∗ − δmf ) σ = f + 2 2 2 (45c) δmk + m +Γ0 500 with
2 2 κ(1 − κ) m +Γ1 1 δm =1 − 2 = 2 2 , Γ0 = , Γ1 = (1 − 2κ)Γ0. (45d) (mH∗) +1/4 m +Γ0 2H∗
501 (45c) and other dispersion relationships are obtained by appropriately setting the switches
502 (δV ,δA,δU ) in the more general dispersion relationship :
2 2 2 2 2 k (N∗ − δV f ) σ = f + 2 2 . (46) 2 2 2 N∗ σ δV k + m +Γ1 + δA 2 + δV δAδU 2 c∗ c∗
503 (δV ,δA,δU ) = (1, 1, 1) corresponds to the fully compressible system, (δV ,δA) = (0, 1) to the
504 hydrostatic primitive equations and (δV ,δA) = (1, 0) to the pseudo-incompressible system
505 (Durran 1989; Davies et al. 2003). (δV ,δA,δU ) = (1, 1, 0) yields the AK09 unified system 2 2 2 2 506 while, using N∗ /c∗ = Γ0 − Γ1, (45c) corresponds to (δV ,δA,δU )=(δm, 1, 0). Since (45c)
507 has no quartic terms in σ, it provides only two solutions σ(k) where (46) with (δV ,δA,δU )=
508 (1, 1, 1) would provide four. Whenever δm ≃ 1, (45c) yields a frequency close to the one
509 arising in the AK09 soundproof system. Therefore the modes filtered by the semi-hydrostatic
510 equations are indeed acoustic.
511 Notice first that for mH∗ ≫ 1, 1 − δm ≪ 1. In this regime, assuming in addition
512 f ≪ N∗, the values of σ given by (46) with (δV ,δA,δU ) = (1, 1, 1), (δV ,δA,δU ) = (1, 1, 0)
513 and (δV ,δA,δU )=(1, δm, 0) all simplify to the common limit :
2 2 2 k (N∗ − f ) σ2 = f 2 + (47a) k2 + m2
514 Furthermore κ(1 − κ) < 0.25 so that 1 − δm is about 2% as soon as mH∗ is a on the order
515 of 3. With H∗ = 10km, mH∗ = 3 corresponds to a vertical wavelenght of 20km. Therefore
516 δm ≪ 1 actually covers a large number of real situations.
517 The situation is more complex when mH∗ = O(1). Interesting limits to the dispersion
518 relation (45c) are then :
519 • the steep gravity-wave regime k ≫ m. Since δm multiplies the horizontal wavenumber
520 k at the denominator of the right-hand-side of (45c), the effect of δm is most significant
24 521 in this regime : 2 N∗ σ2 = . (47b) δm
522 • the deep gravity-wave regime where k is O(Γ). In this regime σ = O(N∗) and
2 2 k 2 σ = 2 2 2 N∗ (47c) δmk + m +Γ
523 • the shallow gravity-wave regime where f/c∗ ≪ k ≪ Γ (note that f ≪ N∗ implies
524 Γ ≫ f/c∗). In this regime f ≪ σ ≪ N∗ :
2 2 2 k N∗ σ ≃ 2 2 (47d) m +Γ0
525 • the inertia-gravity wave regime where k ≤ O(f/c∗). In this regime σ = O(f) :
2 2 2 2 k N∗ σ ≃ f + 2 2 . (47e) m +Γ0
526 (46) with (δV ,δA,δU )=(1, 1, 1) coincides with the above limits only if k ≪ Γ, in the inertia
527 gravity-wave and shallow gravity-wave regimes. Conversely (46) with (δV ,δA,δU )=(1, 1, 1)
528 and (45c) disagree in the the steep and deep gravity-wave regimes. In fact in these regimes
−2 2 2 2 2 529 vertical momentum balance implies H∗ λ ∼ (σ/c∗)w hence ∂tλ/w ∼ σ H∗ /c∗ = O(1) which
530 violates the assumption Dλ/Dt ≪ Dz/Dt, required for the accuracy of approximation (29).
531 Notice that in the deep wavenumber region k ∼ m ∼ Γ, acoustic modes and internal
532 modes both have σ = O(N∗). Therefore accurately separating acoustic from gravity modes
533 for k ∼ m ∼ Γ may present a fundamental difficulty. Furthermore within the anelastic /
534 pseudo-incompressible approximations Γ0 is replaced by Γ1, which distorts the dispersion
535 relationship at scales larger than H∗ (Davies et al. 2003). This points are discussed more
536 in-depth in the conclusion (section 6).
537 Overall, except when k ∼ m ∼ Γ0 one can expect that (45c) differs very little from the
538 exact dispersion relationship at all scales in the geophysically relevant case f ≪ N∗. As
539 an illustration, Fig. 1 presents σ computed via (45c) and compared to the exact frequency ∗ 540 with basic state temperature T =250 K and upper rigid lid at zT =80 km as in DSWT03.
541 As expected, it is seen that the major discrepancy is most noticeable where kH∗ ≥ 1 and
542 mH∗ < 3. Note that the frequencies of the shallow internal gravity-modes such as mH∗ ≥5
543 are almost exactly captured by the system for any horizontal scale.
25 544 We finally examine the full mode structure, obtained by back-substituting the modal form
545 into (44a-44b) then (41a-41f). In order to compare with DSWT03, it is important to realize ′ 546 that the perturbations obtained so far are at altitude z + λ . Furthermore the total pres- ′ 547 sure perturbation is the sum of the hydrostatic and non-hydrostatic perturbations pqs and 2 ′ 548 −ρ∗c∗∂λ /∂z. Hence the perturbation of Exner pressure at fixed z is deduced from pˆqs and
549 λˆ as : ˆ ∗ pˆtot 2 ∂λ ∂pqs 2 d πˆtot = , pˆtot =p ˆqs − ρ∗c∗ − λˆ =p ˆqs + ρ∗ g − c∗ λˆ ρ∗θ∗ ∂z ∂z dz 550 which yields :
′ z i(kx−σt) λ =λˆ0 sin(mz) exp e (48a) 2H∗ ′ z i(kx−σt) w =ˆw0 sin(mz) exp e (48b) 2H∗ ′ 1 z i(kx−σt) θ = θˆ0 sin(mz) exp + κ e (48c) 2 H∗ ′ z i(kx−σt) u =ˆu0 (Γ1 sin(mz) − m cos mz) exp e (48d) 2H∗ ′ z i(kx−σt) pqs =ˆp0 (Γ1 sin(mz) − m cos mz) exp − e (48e) 2H∗ ′ 1 z i(kx−σt) πtot =ˆπ0 (Γ1 sin(mz) − m cos mz)exp − κ e (48f) 2 H∗
551 with
2 ∗ ˆ δmσ θs πˆ0 λ0 = − 2 2 2 , (49a) N∗ − δmσ c∗ 2 2 σ(m +Γ1) ∗ wˆ0 = i 2 2 θs πˆ0, (49b) N∗ − δmσ 2 2 ∗ ˆ g(m +Γ1) θs πˆ0 θ0 = 2 2 ∗ , (49c) N∗ − δmσ πs σk uˆ = θ∗πˆ , (49d) 0 σ2 − f 2 s 0 2 N∗ ∗ ∗ pˆ0 = 2 2 ρsθs πˆ0, (49e) N∗ − δmσ
∗ ∗ ∗ 552 where πs = cpT /θs , the amplitude πˆ0 is considered given, and σ is given either by the
553 dispersion relation (45c) for the internal gravity-modes or σ =0 for the degenerate internal
554 Rossby-modes.
26 555 The above modal structure is to be compared to that of the fully compressible system, i.e. Eqs
556 (5.15)-(5.24) of DSWT03, setting all the switches to one (notice that the DSWT03 definition
557 of Exner function π and ours differ by a factor cp). Examination of (48a)-(48f) shows that the
558 height scale of the internal modes and the location of modal zeros, determined by the vertical
559 phase angle ϕm = arctan(m/Γ1), are both correctly captured. The amplitudes (49a)-(49e)
560 also coincide with (5.21)-(5.24) of DSWT03 when σ =0, i.e for internal degenerate Rossby
561 modes. However this is not the case when σ =06 due to δm =16 . Consequently, there is some
562 energy redistribution for internal gravity modes. Nevertheless (49a)-(49e) do coincide with
563 (5.21)-(5.24) of DSWT03 if either σ ≪ N∗ or δm ≃ 1. Hence energy redistribution is small
564 in the wavenumber region where the dispersion relationship (45c) is accurate.
565 e. Diagnostic equation for λ
566 We finally perform a similar normal-mode analysis of the diagnostic equation (37c) for
567 λ,∂tw. (37c) is already a linear problem, so that linearizing about a background flow is not
568 necessary. We consider only the special case where the flow is horizontally homogeneous,
569 isothermal and at rest, so that it is characterized by a speed of sound c and a scale height
570 H.
571 Using first horizontal homogeneity ∂xpqs =0, ∂xρ =0 :
1 ∗ 2 B λ = c ∂ − g ∂xλ (50) ρ z
1 ∗ 2 2 B B λ = − g + c ∂ ρ g − c ∂ ∆ λ where ∆ λ = ∂x · (∂xλ) . (51) ρ z z H H 572 Using now the isothermal structure :
RT gH H = = cst, c2 = = cst, ∂ ρ = −ρ/H, (52) g 1 − κ z
573 so that for any field Λ :
α ∂ (ραΛ) = ραD Λ, g + c2∂ (ρΛ) = ρc2D Λ where D = ∂ − . (53) z α z κ α z H 574 Hence 2 1 ∗ 4 A = ρc D D , B B = ρc D D − ∆ . (54) 1 0 ρ κ 1 κ H
27 − − 575 2 1/2 1/2 Using DαD1−α = D1/2 − (1/4 − α(1 − α)) and D1/2 ρ Λ = ρ ∂zΛ, (37c) becomes : 2 1 1 1/2 c ∂zz − (1/4 − κ(1 − κ)) 2 ∆H ∂zz − 2 ρ λ H 4H = r.h.s. (55) 1 2 1/2 ∂zz − 4H2 −1/c ρ ∂tw 1/2 1/2 576 Projecting ρ λ, ρ ∂tw onto the eigenmodes of ∂zz with Dirichlet boundary conditions :
z (λ,∂ w) = exp λˆ , aˆ (x)sin mz (56) t 2H m m m X
577 where m is quantized by m = nπ/zT , n ∈ N yields a series of independant Helmholtz
578 problems for λˆm :
2 2 1 2 1 c m + (1/4 − κ(1 − κ)) 2 ∆H m + 2 λˆ H 4H = r.h.s. (57) 2 1 2 m + 4H2 1/c aˆ 579 2 2 1 ˆ Indeed eliminating aˆ = −c m + 4H2 λ + r.h.s leaves :
2 1 2 m + 2 4κ(1 − κ) ∆ − L−2 λˆ = r.h.s where L−2 = 4H ≥ (58) H m m 2 1 2 m + (1/4 − κ(1 − κ)) 2 H H 580 The significant point in (58) is that Lm ≤ O(H). Although the response of the flow to a
581 Dirac forcing r.h.s ∼ δ(x−x0) has presumably infinite support, so that information formally
582 propagates instantaneously to infinity, (58) implies that this response decays exponentially
583 fast beyond a region of extent ∼ H. Therefore in practice information does not significantly
584 propagate to arbitrarily distant places of the fluid. Numerically, this limits the stiffness of
585 the linear problem (37d).
586 6. Conclusion
587 Starting from Hamilton’s least action principle, compressible non-hydrostatic equations of
588 motion with density diagnosed from entropy (or potential temperature) through hydrostatic
589 balance have been derived for an arbitrary equation of state. They possess a vector-invariant
590 form (23,26) as well as flux-form (32a). The analysis of linear normal modes and dispersion
591 relationship for small departures from an isothermal state of rest on f− and β− planes shows
592 excellent accuracy from hydrostatic to non-hydrostatic scales, with the exception of the deep
28 593 gravity-wave regime, where both horizontal and vertical wave lengths are comparable to the
594 scale height. Especially the Lamb wave and long Rossby waves are not distorted, unlike with
595 anelastic or pseudo-incompressible systems.
596 The advantage of the variational approach is the guarantee to obtain dynamically consis-
597 tent equations. The difficulty lies in the physical interpretation of the Lagrange multiplier.
598 To this end the existence of a local momentum budget is of great help. Nevertheless this
599 budget includes, in addition to the gradient of a non-hydrostatic pressure perturbation,
600 unexpected terms with a non-straightforward interpretation. Providing this interpretation
601 requires the notion of hydrostatic height, defined as the height to which air parcels in a given
602 atmospheric column should move adiabatically in order to restore hydrostatic balance. This
603 notion is, as far as we are aware, new and we expect that it can be useful in other contexts
604 than the present work due to the preeminence of hydrostatic balance in geophysical flows.
605 The unexpected terms are then found to be an apparent force resulting from the vertical
606 coordinate not being the actual height of an air parcel, but its hydrostatic height. Conse-
607 quently, the Lagrange multiplier λ is identified as the non-hydrostatic vertical displacement,
608 i.e. the difference between the actual height of an air parcel and its hydrostatic height. By
609 comparing the volume of air parcels before and after an hypothetical hydrostatic adjust-
610 ment, the non-hydrostatic perturbations of density, pressure, and Exner pressure are found
611 proportional to the vertical derivative of λ, assumed small. In addition to ∂λ/∂z ≪ 1, the
612 condition Dλ/Dt ≪ w must be met for the Lagrangian (29) to be a good approximation
613 of the exact one. It is this condition that fails in the deep gravity-wave regime. Finally λ
614 is naturally subject to Dirichlet boundary conditions for the rigid boundaries considered here.
615
616 Enforcing one holonomic constraint supresses two degrees of freedom and results in the
617 associated Lagrange multiplier satisfying a self-adjoint diagnostic equation. Here, slaving
618 density to potential temperature suppresses the degrees of freedom supporting the propa-
619 gation of acoustic waves and results in a sound-proof system. As with hydrostatic prim-
620 itive equations, vertical velocity is diagnosed through Richardson’s equation (36a). The
621 semi-hydrostatic system has therefore precisely the same degrees of freedom as the hydro-
622 static primitive equations. This suggests that the difference between hydrostatic and non-
29 623 hydrostatic motion is essentially quantitative and not qualitative, i.e. any non-hydrostatic
624 phenomenon should have a hydrostatic counterpart, albeit probably with quantitatively dif-
625 ferent amplitude, dispersion relationship, etc. This conclusion is already suggested by the
626 success of the anelastic and pseudo-incompressible approximations which, apart from those
627 of the Lamb wave, have the same degrees of freedom as the hydrostatic equations.
628 In AK09, z is interpreted as the full non-hydrostatic height (Konor 2013). Consequently
629 the terms due to the difference between the actual height of an air parcel and its hydrostatic
630 height are omitted. In our case, these terms are crucial for the transparent derivation of
631 a well-posed self-adjoint problem (37d) for λ, from which non-hydrostatic pressure derives.
632 Generally speaking, the diagnostic equation satisfied by a Lagrange multiplier reflects the
633 structure of the constraint that it imposes (e.g. Bernardet 1995). Our self-adjoint problem
634 is vertically fourth-order with four boundary conditions for λ, reflecting the fact that the
635 hydrostatic constraint satisfied by density involves a vertical derivative, unlike the pointwise
636 constraint at the core of the anelastic and pseudo-incompressible approximations. Especially
637 it seems impossible to formulate a self-adjoint problem and boundary conditions directly for
638 non-hydrostatic pressure, as attempted by AK09. If following AK09, the term g∇λ is omitted
639 from horizontal momentum balance (26), then a linear problem similar to (37c) is obtained,
640 except that it loses its self-adjoint character.
641
642 It is natural to ask whether other sound-proof approximations fare better than ours in
643 the deep gravity-wave regime k ∼ m ∼ Γ. Separating accurately gravity waves from acous-
644 tic waves seems problematic in this regime on physical grounds, since both branches have
645 comparable frequencies O(N∗). To discuss this issue we let f = 0 for simplicity. Then, in
646 the deep gravity-wave regime, the dispersion relationships (46) with (δV ,δA,δU ) =6 (1, 1, 1)
647 differ at first order from the exact value given by (δV ,δA,δU ) = (1, 1, 1). This is all but
648 unexpected since the general dispersion relationship is of the form σ/N∗ = Σ(kH∗, mH∗, κ).
649 If both kH∗ and mH∗ are O(1) and κ is not small, there is no small adimensional parameter
650 left for an asymptotic expansion. Any expression for σ/N∗ is then either exact or inaccu-
2 651 rate. Furthermore for any sound-proof system σ must be the ratio of two polynomials in
2 652 kH∗, mH∗. The expression for σ solving (46) with (δV ,δA,δU ) =6 (1, 1, 1) involves a square
30 653 root and cannot exactly coincide with a polynomial fraction. Hence it seems impossible for a
654 sound-proof system to be accurate (in an asymptotic sense) in the deep gravity-wave regime.
655 In order to evaluate quantitatively the seriousness of this issue, we plot in Fig. 2 the
656 relative difference between the exact and approximate value of σ, for f = 0, κ = 2/7, as a
−2 657 function of kH∗, mH∗. The shaded region indicates where the relative error is > 10 . In
658 that sense, the pseudo-incompressible system is inaccurate for shallow gravity waves. Our
659 approximate system is inaccurate for steep gravity waves, as discussed in section 6, but
660 corrects much of the errors of the hydrostatic system. Especially the relative error never
661 exceeds 2κ/(1 − 2κ), yielding a maximum relative error of 4/3 (133%). Although large,
662 this is far from the complete breakdown of the hydrostatic equations for which relative
663 frequency error is unbounded at small-scale. The AK09 unified system has the smallest
664 possible inaccurate domain, for deep gravity waves only, as discussed above. Hence the term
665 g∇λ appearing in the horizontal momentum balance (26) restores desirable properties of the
666 equations such as unambiguous local budgets of energy and momentum as well as an explicit
667 self-adjoint diagnostic problem for λ at the price of some accuracy. Nevertheless the unified,
668 pseudo-incompressible and semi-hydrostatic systems are quite accurate as soon as mH∗ is
669 of order 1 − 3 (see Fig. 3) . The main difference is in the propation of long Rossby waves,
670 which is incorrect for the pseudo-incompressible equations.
671 It is not clear at this point whether numerically integrating the semi-hydrostatic equa-
672 tions is more efficient than integrating the full Euler equations with a proper treatment of
673 acoustic waves (Skamarock and Klemp 2008; Weller et al. 2013; Smolarkiewicz et al. 2014).
674 The answer will depend primarily on the possibility to efficiently solve the self-adjoint prob-
675 lem for λ. Whatever the eventual outcome, one can think of several other applications.
676 From a theoretical point of view, the semi-hydrostatic system provides an explicit separa-
677 tion between the acoustic and, broadly speaking, vortical motion, be it hydrostatic or non-
678 hydrostatic. Diagnosing the vertical velocity and non-hydrostatic displacement amounts to
679 filtering out the acoustic part of motion. Variational data assimilation systems may ben-
680 efit from such an accurate elimination of acoustic waves, reducing the optimization space.
681 Similarly, it could be useful for initialization purposes, avoiding the transient generation of
682 acoustic waves from an initial flow. It may also be helpful to occasionnally perform this
31 683 projection while integrating the fully compressible Euler equations, especially after physical
684 parameterizations have acted, potentially triggering the transient emission of acoustic waves.
685
686 The present semi-hydrostatic theory suggests a number of follow-up work. Dubos and
687 Tort (2014) provide the necessary formalism to obtain its Hamiltonian formulation in Eule-
688 rian and non-Eulerian coordinates, especially terrain-following coordinates (Kasahara 1974;
689 Laprise 1992), which can be useful for numerical implementation as well as for theoretical
690 purposes. Generalization to deep-atmosphere or spheroidal geometries is also straightfor-
691 ward using Tort and Dubos (2014b). Finally, the notion that air parcels are close to their
692 hydrostatic height could be used to revisit and possibly simplify quasi-hydrostatic systems
693 of equations as well (White and Bromley 1995; Tort and Dubos 2014a).
694 Acknowledgements
695 We would like to thank C.S Konor and R. Klein for their constructive reviews which
696 helped improve the accuracy and clarity the manuscript.
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35 768 List of Figures
769 1 Normalized frequencies σ/N* of internal gravity modes on an f-plane as func-
770 tions of normalized horizontal wavenumber kH∗, and for various normalized
771 vertical wavenumber mH∗ =1, 2,..., 10, 15, 25,..., 105. σ decreases as mH∗
772 increases for given kH∗. Solid : fully compressible system ; dashed : semi-
773 hydrostatic system. 37
774 2 Decimal logarithm of relative error ∆ = |σexact − σapprox| /σexact of the fre-
775 quency of internal normal modes of a non-rotating isothermal atmosphere
776 with κ = 2/7, as a function of normalized horizontal and vertical wavenum-
777 bers kH∗, mH∗, for the present approximation (semi-hydrostatic), AK09, hy-
778 drostatic and pseudo-incompressible approximations. The “inaccurate” region
−2 779 where ∆ ≥ 10 is shaded. 38
780 3 Normalized frequencies σ/N∗ of internal inertia-gravity mode for mH∗ =
781 0.33 (left) and mH∗ = 1 (right) as a function of the normalized horizontal
782 wavenumber kH∗ for the various systems : fully compressible (green dia-
783 monds), AK09 unified (pink square), semi- hydrostatic (blue circle) (lond-
784 dashed), pseudo-incompressible (green triangle), hydrostatic (blue triangle). 39
36 1.2
1.0
0.8 * N 0.6 Σ
0.4
0.2
0.0 0.01 0.1 1 10 100 1000
kH*
Fig. 1. Normalized frequencies σ/N* of internal gravity modes on an f-plane as functions of normalized horizontal wavenumber kH∗, and for various normalized vertical wavenumber mH∗ = 1, 2,..., 10, 15, 25,..., 105. σ decreases as mH∗ increases for given kH∗. Solid : fully compressible system ; dashed : semi-hydrostatic system.
37 semi−hydrostatic unified 3 3 10 −8 10 −8 −9 −11 −10 −7 −7 −7 −6 −7 −8 −12 −9 −11 −10 −12 2 −6 2 −5 −6 10 −5 10 −5 −4 −4
* 1 * 1 10 −3 −3 10 mH mH −2 −2 0 0 10 −1 −1 10
−1 0 0 −1 10 10 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 10 10 10 10 10 10 10 10 10 10 10 10 10 10 kH kH * * hydrostatic pseudo−incompressible 3 3 −7 10 −1 10
−6 −5 −4 −3 −2 −12 0 −11 −6 −8
−10
−9
−8
−7 2 0 2 −7 10 −1 1 10 −5 −9
* 1 * 1 1 −8 10 2 10
mH mH −11 −9 2 −12 0 0 −10 10 3 10
−1 −1 10 10 −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 10 10 10 10 10 10 10 10 10 10 10 10 10 10 kH kH * *
Fig. 2. Decimal logarithm of relative error ∆= |σexact − σapprox| /σexact of the frequency of internal normal modes of a non-rotating isothermal atmosphere with κ =2/7, as a function of normalized horizontal and vertical wavenumbers kH∗, mH∗, for the present approxima- tion (semi-hydrostatic), AK09, hydrostatic and pseudo-incompressible approximations. The “inaccurate” region where ∆ ≥ 10−2 is shaded.
38 mH* = 0.33 mH* = 1. 1.6 æ 1.2 æ
ô ææææææææææææææææææææææææææææ ææææææææææææææææææææææææææ ææææææææææ ææææææ ææææ æææ ô ææ ææ 1.1 æ æææææææææææææææææææ 1.4 æ æææææææææææææææææææææ æ æææææææææææ æ ô æææææææ æ æææææ æ ææææ æææ æ ææ æ ææ ææ ô æ 1.0 æ òòòòòòòòò æ òòòòòàòàòàòàòàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàìàìàìàìàìàìàìàìà æ òòòòòàòàòàòìàòìàòìàòìàòìàòìàìàìàìàìàììììì æ æ òòòòàòàòàòìàìàìàìàìàììì 1.2 ô òòòòàòàìàìàìàììì æ òòòààìàìàìì æ æ òòòààìàìì * * æ òòààìàìì òòàìì N N æ òààì òàì æ æ òàìì Σ Σ òàì ô 0.9 æ òàì òàì æ àì ô æòàì òàì 1.0 òòòòòòòòòòòòòàòàòàòàòàòàòàòàòàòàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàòìàò æ òòòòòòòòààààààìàìàìàìàìàìàìàìììììììììì òàì æ òòòò àààààìììììì òò àààìììì æòàì ò ààììì ò ààìì òàì ò à ì 0.8 ô àìì æì æò àì òà ò àì ô ì ì æòà àì ì 0.8 ò àì æòà æ ì òàì 0.7 à ô òì ì ôæ à ææææ æææà 2 4 6 8 10 2 4 6 8 10
kH* kH*
Fig. 3. Normalized frequencies σ/N∗ of internal inertia-gravity mode for mH∗ =0.33 (left) and mH∗ = 1 (right) as a function of the normalized horizontal wavenumber kH∗ for the various systems : fully compressible (green diamonds), AK09 unified (pink square), semi- hydrostatic (blue circle) (lond-dashed), pseudo-incompressible (green triangle), hydrostatic (blue triangle).
39