Non-hydrostatic sound-proof equations of motion for gravity-dominated compressible fl ows. Thomas Dubos

Transcription

1 Non-hydrostatic sound-proof of for gravity-dominated compressible fl ows Thomas Dubos Laboratoire de Météorologie Dynamique/IPSL, École Polytechnique, Palaiseau, France Fabrice Voitus Centre National de Recherches Météorologiques, Météo-France, Toulouse, France I. Approximate of Hamilton's principle and asymptotics Soundproofing : anelastic vs hydrostatic II.A «semi-hydrostatic» approximation Derivation and interpretation «Elliptic» problem for non-hydrostatic pressure Accuracy : normal-mode analysis III.Conclusions

2 The atmosphere : a gravity-dominated, compressible flow Characteristic Velocity : Time : Length : small-scale 1 km Sound c ~ 340m/s Buoyancy oscillations N ~ g/c ~10-2 s-1 Scale height H=c2/g=10km mesoscale 10 km Wind U ~ 30m/s Coriolis f ~ 10-4 s-1 Rossby radius : R=c/f ~ 1000 km synoptic 100 km planetary 1000 km km Mach number : M=U/c <<1 Scale separation : f/n ~ H/R << 1 Small numbers => asymptotics => approximate But not at the expense of conservation : energy, momentum, potential vorticity 1/13

3 Can we make approximations while always maintaining conservation laws? HP for ideal fluid (Lagrangian description) Hamilton's least action principle (HP) Noether's theorems Maupertuis Euler, Lagrange Hamilton Noether Euler-Poincaré theory HP asymptotics HP for ideal fluid (Eulerian description) Relabelling symmetry => Kelvin's theorem & conservation of potential vorticity 1960 Eckart 1962 Newcomb 1996 Padhye & Morrison 2002 Holm, Marsden, Ratiu Reversible mechanical systems obey Hamilton's least action principle Conservation laws result from a symmetry of the action Hamilton's principle asymptotics : approximating the action instead of the of systematically produces approximate systems with all conservation laws 2/13

4 Least action principle in curvilinear coordinates (Tort & Dubos, accepted by J. Atmos. Sci.) Spherical geoid Shallow-atmosphere Compressible Sound-proof Traditional (Quasi-)Hydrostatic Spherical-geoid Quasi-hydrostatic (White & Wood, 2012) (Tort & Dubos, 2014b) Spherical-geoid Quasi-hydrostatic (White & Wood, 1995) Anelastic Pseudoincompressible Boussinesq Anelastic (Ogura & Phillips) Pseudoincompressible (Durran ; Pauluis) Non-traditional shallowatmosphere (Tort & Dubos, 2014a) Traditional shallowatmosphere (Phillips, 1966) NT shallow-atmosphere quasi-hydrostatic (Tort & Dubos, 2014a) Primitive (Richardson, 1922?) 3/13

5 Filtering acoustic waves : hydrostatic vs anelastic or Charybdis vs Scylla (waves over isothermal atmosphere : e.g. Davies et al., 2003 ; Arakawa & Konor, 2009) fast : >N N N O(N) Dz/Dt~Dx/Dt Non-Hydrostatic f f Anelastic Fully compressible Hydrostatic Dz/Dt<<Dx/Dt Hydrostatic kh =1 4/13

6 I. Approximate of Hamilton's principle and asymptotics Soundproofing : anelastic vs hydrostatic II.A «semi-hydrostatic» approximation Derivation and interpretation «Elliptic» problem for non-hydrostatic pressure Accuracy : normal-mode analysis III.Conclusions

7 Getting rid of the acoustic waves : the «unified» way Constraining (slaving) the density field breaks the pressure-density feedback loop key to suppressing acoustic waves «anelastic» slaving incorrect at large «hydrostatic» slaving is correct density pressure velocity acceleration Neglecting vertical acceleration implies hydrostatic balance but the converse is not true. Arakawa & Konor (2009) impose density through hydrostatic balance but retain vertical acceleration pressure is the sum of hydrostatic pressure and a non-hydrostatic deviation non-hydrostatic pressure determined from a Poisson-like problem Variational implementation of this physical idea? 5/13

8 Dubos & Voitus (submitted) hydrostatic balance is a holonomic constraint (velocity not involved) Let us introduce a Lagrange multiplier to impose it! 6/13

9 Dubos & Voitus (submitted) hydrostatic balance is a holonomic constraint (velocity not involved) Let us introduce a Lagrange multiplier to impose it! 6/13

10 Dubos & Voitus (submitted) hydrostatic balance is a holonomic constraint (velocity not involved) Let us introduce a Lagrange multiplier to impose it! Interpretation The true height of air parcels is z is their hydrostatic height λ is a vertical non-hydrostatic displacement Approximation is accurate if dλ/dz << 1 and either DZ/Dt<<Dx/Dt : hydrostatic or Dλ/Dt<<Dz/Dt : hydrostatic velocity is an accurate estimate of true vertical velocity 6/13

11 Dubos & Voitus (submitted) The true height of air parcels is z is their hydrostatic height λ is a vertical non-hydrostatic displacement coordinates (x,y,z) are slighty curvilinear ρ is the pseudo-density associated to (x,y,z) the true density and pressure are ρ+ρ' and p+p' : Advective form Usual terms Non-hydrostatic pressure Included in AK09 Pseudo-forces due to (x,y,z) being curvilinear neglected by AK09 7/13

12 Dubos & Voitus (submitted) The true height of air parcels is z is their hydrostatic height λ is a vertical non-hydrostatic displacement coordinates (x,y,z) are slighty curvilinear ρ is the pseudo-density associated to (x,y,z) the true density and pressure are ρ+ρ' and p+p' : Momentum budget Usual terms Non-hydrostatic pressure Pseudo-forces due to (x,y,z) being curvilinear 7/13

13 «Elliptic» problem for the non-hydrostatic displacement Assuming rigid boundaries, Dirichlet boundary conditions λ=0 Assuming flat boundaries, Dirichlet boundary conditions w=0 w=dz/dt obeys the same Richardson's equation as with hydrostatic (Richardson, 1922 ; see also Dubos & Tort, submitted to MWR) Now observe that the momentum budget is of the form One time-differentiation yields : Not Poisson-like : hydrostatic constraint involves vertical derivative of density Information propagates horizontally no farther than ~ scale height 8/13

14 Relative frequency error for waves in an isothermal atmosphere at rest Hydrostatic Accurate : Dz/Dt<<Dx/Dt Non-hydrostatic Hydrostatic Non-hydrostatic Accurate : Dλ/Dt<<Dz/Dt Inaccurate : Dz/Dt << Dx/Dt fails Accurate : DZ/Dt<<Dx/Dt Inaccurate : Dλ/Dt<<Dz/Dt fails The semi-hydrostatic approximation corrects much of the errors present in the hydrostatic approximation, except for vertically-long waves (several scale heights) 9/13

15 Relative frequency error for waves in an isothermal atmosphere at rest Hydrostatic Nonhydrostatic Inaccurate : Dλ/Dt<<Dz/Dt fails Hydrostatic Nonhydrostatic Hydrostatic Nonhydrostatic Inaccurate : Dz/Dt << Dx/Dt fails 11/13

16 Relative frequency error for waves in an isothermal atmosphere at rest 11/13

17 Relative frequency error for waves in an isothermal atmosphere at rest Acoustic frequency close to N No scale separation 11/13

18 Laws of (inviscid) atmospheric can be expressed concisely and safely from Hamilton's principle of least action Variational principles can be helpful for Classifying existing approximation : Tort & Dubos (J. Atmos.Sci., accepted) Deriving new, consistent approximations Tort & Dubos (QJRMS, 2014), Dubos & Voitus (J. Atmos. Sci., submitted) The semi-hydrostatic system : Conserves energy, momentum, potential vorticity Possesses a well-defined self-adjoint problem yielding NH pressure Is accurate from hydrostatic to NH Except horizontally short, vertically long gravity waves Has been derived for an arbitrary equation of state Can easily be extended to include : moisture, deep-atmosphere, etc. 12/13

19 Laws of (inviscid) atmospheric can be expressed concisely and safely from Hamilton's principle of least action Approximate laws of provide insight into Origin / nature of forces Actual, independant degrees of freedom 'Slaving' relationships between dependent and indepent DOFs For atmospherically-relevant flow regimes, including non-hydrostatic, the independent degrees of freedom of atmospheric are precisely those of the hydrostatic primitive 13/13