# American Institute of Mathematical Sciences

December  2019, 11(4): 511-537. doi: 10.3934/jgm.2019025

## Variational integrators for anelastic and pseudo-incompressible flows

 1 Imperial College London, Department of Mathematics, South Kensington Campus, London SW7 2AZ, UK, École Normale Supérieure, Laboratoire de Météorologie Dynamique, 24 Rue Lhomond, Paris, France 2 CNRS and École Normale Supérieure, Laboratoire de Météorologie Dynamique, 24 Rue Lhomond, Paris, France

To Darryl Holm, on the occasion of his 70th birthday

Received  November 2017 Revised  August 2019 Published  November 2019

The anelastic and pseudo-incompressible equations are two well-known soundproof approximations of compressible flows useful for both theoretical and numerical analysis in meteorology, atmospheric science, and ocean studies. In this paper, we derive and test structure-preserving numerical schemes for these two systems. The derivations are based on a discrete version of the Euler-Poincaré variational method. This approach relies on a finite dimensional approximation of the (Lie) group of diffeomorphisms that preserve weighted-volume forms. These weights describe the background stratification of the fluid and correspond to the weighted velocity fields for anelastic and pseudo-incompressible approximations. In particular, we identify to these discrete Lie group configurations the associated Lie algebras such that elements of the latter correspond to weighted velocity fields that satisfy the divergence-free conditions for both systems. Defining discrete Lagrangians in terms of these Lie algebras, the discrete equations follow by means of variational principles. Descending from variational principles, the schemes exhibit further a discrete version of Kelvin circulation theorem, are applicable to irregular meshes, and show excellent long term energy behavior. We illustrate the properties of the schemes by performing preliminary test cases.

Citation: Werner Bauer, François Gay-Balmaz. Variational integrators for anelastic and pseudo-incompressible flows. Journal of Geometric Mechanics, 2019, 11 (4) : 511-537. doi: 10.3934/jgm.2019025
##### References:
 [1] R. Abraham and J. E. Marsden, Foundations of Mechanics. II. Revised and Enlarged, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. [2] V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Applied Mathematical Sciences, 125, Springer-Verlag, New York, 1998. [3] W. Bauer, M. Baumann, L. Scheck, A. Gassmann, V. Heuveline and S. C. Jones, Simulation of tropical-cyclone-like vortices in shallow-water ICON-hex using goal-oriented r-adaptivity, Theoretical and Computational Fluid Dynamics, 28 (2014), 107-128.  doi: 10.1007/s00162-013-0303-4. [4] W. Bauer and F. Gay-Balmaz, Towards a variational discretization of compressible fluids: The rotating shallow water equations, Journal of Computational Dynamics, 6 (2019), 1–37, http://dx.doi.org/10.3934/jcd.2019001. doi: 10.3934/jcd.2019001. [5] C. J. Cotter and D. D. Holm, Variational formulations of sound-proof models, Quarterly Journal of the Royal Meteorological Society, 140 (2014), 1966-1973.  doi: 10.1002/qj.2260. [6] M. Desbrun, E. S. Gawlik, F. Gay-Balmaz and V. Zeitlin, Variational discretization for rotating stratified fluids, Discrete Continuous Dynamical Systems-A, 34 (2014), 477-509.  doi: 10.3934/dcds.2014.34.477. [7] D. R. Durran, Improving the anelastic approximation, J. Atmos. Sci, 46 (1989), 1453-1461.  doi: 10.1175/1520-0469(1989)046<1453:ITAA>2.0.CO;2. [8] D. R. Durran, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics, Texts in Applied Mathematics, 32. Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-3081-4. [9] E. S. Gawlik, P. Mullen, D. Pavlov, J. E. Marsden and M. Desbrun, Geometric, variational discretization of continuum theories, Physica D, 240 (2011), 1724-1760.  doi: 10.1016/j.physd.2011.07.011. [10] D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. in Math., 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721. [11] R. Klein, Asymptotics, structure, and integration of sound-proof atmospheric flow equations, Theor. Comput. Fluid Dyn., 23 (2009), 161-195.  doi: 10.1007/s00162-009-0104-y. [12] H. Lamb, Hydrodynamics, Ch. 309,310, Dover, 1932. [13] F. B. Lipps and R. S. Hemler, A scale analysis of deep moist convection and some related numerical calculations, J. Atmos. Sci., 29 (1982), 2192-2210.  doi: 10.1175/1520-0469(1982)039<2192:ASAODM>2.0.CO;2. [14] J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D: Nonlinear Phenomena, 7 (1983), 305-323.  doi: 10.1016/0167-2789(83)90134-3. [15] Y. Ogura and N. A. Phillips, Scale analysis for deep and shallow convection in the atmosphere, J. Atmos. Sci., 19 (1962), 173-179.  doi: 10.1175/1520-0469(1962)019<0173:SAODAS>2.0.CO;2. [16] D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. E. Marsden and M. Desbrun, Structure-preserving discretization of incompressible fluids, Physica D, 240 (2010), 443-458.  doi: 10.1016/j.physd.2010.10.012. [17] R. Wilhelmson and Y. Ogura, The pressure perturbation and the numerical modeling of a cloud, J. Atmos. Sci., 29 (1972), 1295-1307.  doi: 10.1175/1520-0469(1972)029<1295:TPPATN>2.0.CO;2.

show all references

To Darryl Holm, on the occasion of his 70th birthday

##### References:
 [1] R. Abraham and J. E. Marsden, Foundations of Mechanics. II. Revised and Enlarged, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978. [2] V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Applied Mathematical Sciences, 125, Springer-Verlag, New York, 1998. [3] W. Bauer, M. Baumann, L. Scheck, A. Gassmann, V. Heuveline and S. C. Jones, Simulation of tropical-cyclone-like vortices in shallow-water ICON-hex using goal-oriented r-adaptivity, Theoretical and Computational Fluid Dynamics, 28 (2014), 107-128.  doi: 10.1007/s00162-013-0303-4. [4] W. Bauer and F. Gay-Balmaz, Towards a variational discretization of compressible fluids: The rotating shallow water equations, Journal of Computational Dynamics, 6 (2019), 1–37, http://dx.doi.org/10.3934/jcd.2019001. doi: 10.3934/jcd.2019001. [5] C. J. Cotter and D. D. Holm, Variational formulations of sound-proof models, Quarterly Journal of the Royal Meteorological Society, 140 (2014), 1966-1973.  doi: 10.1002/qj.2260. [6] M. Desbrun, E. S. Gawlik, F. Gay-Balmaz and V. Zeitlin, Variational discretization for rotating stratified fluids, Discrete Continuous Dynamical Systems-A, 34 (2014), 477-509.  doi: 10.3934/dcds.2014.34.477. [7] D. R. Durran, Improving the anelastic approximation, J. Atmos. Sci, 46 (1989), 1453-1461.  doi: 10.1175/1520-0469(1989)046<1453:ITAA>2.0.CO;2. [8] D. R. Durran, Numerical Methods for Wave Equations in Geophysical Fluid Dynamics, Texts in Applied Mathematics, 32. Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-3081-4. [9] E. S. Gawlik, P. Mullen, D. Pavlov, J. E. Marsden and M. Desbrun, Geometric, variational discretization of continuum theories, Physica D, 240 (2011), 1724-1760.  doi: 10.1016/j.physd.2011.07.011. [10] D. D. Holm, J. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. in Math., 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721. [11] R. Klein, Asymptotics, structure, and integration of sound-proof atmospheric flow equations, Theor. Comput. Fluid Dyn., 23 (2009), 161-195.  doi: 10.1007/s00162-009-0104-y. [12] H. Lamb, Hydrodynamics, Ch. 309,310, Dover, 1932. [13] F. B. Lipps and R. S. Hemler, A scale analysis of deep moist convection and some related numerical calculations, J. Atmos. Sci., 29 (1982), 2192-2210.  doi: 10.1175/1520-0469(1982)039<2192:ASAODM>2.0.CO;2. [14] J. E. Marsden and A. Weinstein, Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids, Physica D: Nonlinear Phenomena, 7 (1983), 305-323.  doi: 10.1016/0167-2789(83)90134-3. [15] Y. Ogura and N. A. Phillips, Scale analysis for deep and shallow convection in the atmosphere, J. Atmos. Sci., 19 (1962), 173-179.  doi: 10.1175/1520-0469(1962)019<0173:SAODAS>2.0.CO;2. [16] D. Pavlov, P. Mullen, Y. Tong, E. Kanso, J. E. Marsden and M. Desbrun, Structure-preserving discretization of incompressible fluids, Physica D, 240 (2010), 443-458.  doi: 10.1016/j.physd.2010.10.012. [17] R. Wilhelmson and Y. Ogura, The pressure perturbation and the numerical modeling of a cloud, J. Atmos. Sci., 29 (1972), 1295-1307.  doi: 10.1175/1520-0469(1972)029<1295:TPPATN>2.0.CO;2.
Notation and indexing conventions for the 2D simplicial mesh
Section of central part of the irregular mesh with $\max_{{\bf x} \in \Omega}\Delta h({\bf x}) \approx 7$ for a resolution of $2\cdot 384 \times 20$ triangular cells
Initialization of the Boussinesq scheme by the buoyancy field $b(x,z,0)$, shown left. Initialization of the anelastic and pseudo-incompressible schemes by the potential temperature field $\theta(x,z,0)$, shown right
Boussinesq scheme: snapshots of the wave propagation on the regular (left column) and the irregular (right column) mesh
Anelastic scheme: snapshots of the wave propagation on the regular (left column) and the irregular (right column) mesh (snapshots for pseudo-incompressible scheme are very similar, hence not shown)
Boussinesq scheme: relative errors of total energy $E(t)$ and mass $M(t)$ for the regular (left column) and the irregular (right column) mesh
Anelastic scheme: relative errors of total energy $E(t)$ and mass $M(t)$ for the regular (left column) and the irregular (right column) mesh
Pseudo-incompressible scheme: relative errors of total energy $E(t)$ and mass $M(t)$ for the regular (left column) and the irregular (right column) mesh
Boussinesq scheme: frequency spectra for the regular (left block) and the irregular (right block) mesh determined on various points in the domain $\mathcal{D}$. The position in the panel indicates the corresponding position in $\mathcal{D}$, e.g. the upper left panel corresponds to a point at the upper left of $\mathcal{D}$
Anelastic scheme: frequency spectra for the regular (left block) and the irregular (right block) mesh determined on various points in the domain $\mathcal{D}$ similarly to Fig. 6.8
Pseudo-incompressible scheme: frequency spectra for the regular (left block) and the irregular (right block) mesh determined on various points in the domain $\mathcal{D}$ similarly to Fig. 6.8
Parallel between the continuous and discrete forms for the three models. Note that in the Euler-Poincaré form given in the sixth row of the first column, one has to compute the variational derivatives with respect to the three different weighted pairings in order to get the three models. The last row of the first column presents the continuous equations in a form that corresponds to the discrete forms obtained by variational discretization on 2D simplicial meshes. Note that these expressions are not the standard form of the models given in (2.4), (2.7), (2.8)
 Continuous diffeomorphisms Discrete diffeomorphisms Boussinesq: ${\rm Diff}_\mu (M)$ Boussinesq: ${\mathsf{D}}( \mathbb{M} )$ Anelastic: ${\rm Diff} _{\bar{ \rho }\mu}(M)$ Anelastic: ${\mathsf{D}}_{\bar \rho }( \mathbb{M} )$ Pseudo-incompressible: ${\rm Diff} _{\bar{ \rho }\bar\theta \mu} (M)$ Pseudo-incompressible: ${\mathsf{D}}_{\bar \rho \bar\theta }( \mathbb{M} )$ Lie algebras Discrete Lie algebras $\mathfrak{X}_ \mu (M),\;\; \mathfrak{X}_ {\bar{ \rho }\mu} (M), \;\;\mathfrak{X}_{ \bar{ \rho }\bar \theta \mu }(M)$ $\mathfrak{d} ( \mathbb{M} ), \;\;\mathfrak{d} _ {\bar{ \rho } } ( \mathbb{M} ), \;\;\mathfrak{d} _{ \bar{ \rho }\bar \theta }( \mathbb{M} )$ Euler-Poincaré form Discrete Euler-Poincaré form $\partial _t \frac{\delta \ell}{\delta {\bf{u}} }+\mathit{£} _{\bf{u}} \frac{\delta \ell}{\delta {\bf{u}} } + \frac{\delta \ell}{\delta \theta } {\bf{d}} \theta = - {\bf{d}} p$, Equation (4.12) Common form for the three models Common discrete form for the three models Form independent of the mesh Expression corresponding to the discrete form on 2D simplicial grids Discrete form on 2D simplicial grids Boussinesq: Discrete Boussinesq: $\partial _t {\bf{u}}^\flat + {\bf{i}} _{\bf{u}} {\bf{d}}{\bf{u}}^\flat =- z {\bf{d}} b - {\bf{d}} \tilde p$ Equation (5.2) Anelastic: Discrete Anelastic: $\partial _t {\bf{u}}^\flat + {\bf{i}} _{\bf{u}} {\bf{d}}{\bf{u}}^\flat =c_p \bar \pi {\bf{d}} \theta - {\bf{d}} \tilde p$ Equation (5.6) Pseudo-incompressible: Discrete Pseudo-incompressible: $\partial _t \Big( \frac{ {\bf{u}}^\flat}{ \theta } \Big) + \frac{1}{ \theta } {\bf{i}} _{\bf{u}} {\bf{d}}{\bf{u}}^\flat =-\Big(gz- \frac{1}{2} | {\bf{u}} | ^2 \Big)\frac{{\bf{d}} \theta}{\theta ^2 } - {\bf{d}} \tilde p$ Equation (5.8)
 Continuous diffeomorphisms Discrete diffeomorphisms Boussinesq: ${\rm Diff}_\mu (M)$ Boussinesq: ${\mathsf{D}}( \mathbb{M} )$ Anelastic: ${\rm Diff} _{\bar{ \rho }\mu}(M)$ Anelastic: ${\mathsf{D}}_{\bar \rho }( \mathbb{M} )$ Pseudo-incompressible: ${\rm Diff} _{\bar{ \rho }\bar\theta \mu} (M)$ Pseudo-incompressible: ${\mathsf{D}}_{\bar \rho \bar\theta }( \mathbb{M} )$ Lie algebras Discrete Lie algebras $\mathfrak{X}_ \mu (M),\;\; \mathfrak{X}_ {\bar{ \rho }\mu} (M), \;\;\mathfrak{X}_{ \bar{ \rho }\bar \theta \mu }(M)$ $\mathfrak{d} ( \mathbb{M} ), \;\;\mathfrak{d} _ {\bar{ \rho } } ( \mathbb{M} ), \;\;\mathfrak{d} _{ \bar{ \rho }\bar \theta }( \mathbb{M} )$ Euler-Poincaré form Discrete Euler-Poincaré form $\partial _t \frac{\delta \ell}{\delta {\bf{u}} }+\mathit{£} _{\bf{u}} \frac{\delta \ell}{\delta {\bf{u}} } + \frac{\delta \ell}{\delta \theta } {\bf{d}} \theta = - {\bf{d}} p$, Equation (4.12) Common form for the three models Common discrete form for the three models Form independent of the mesh Expression corresponding to the discrete form on 2D simplicial grids Discrete form on 2D simplicial grids Boussinesq: Discrete Boussinesq: $\partial _t {\bf{u}}^\flat + {\bf{i}} _{\bf{u}} {\bf{d}}{\bf{u}}^\flat =- z {\bf{d}} b - {\bf{d}} \tilde p$ Equation (5.2) Anelastic: Discrete Anelastic: $\partial _t {\bf{u}}^\flat + {\bf{i}} _{\bf{u}} {\bf{d}}{\bf{u}}^\flat =c_p \bar \pi {\bf{d}} \theta - {\bf{d}} \tilde p$ Equation (5.6) Pseudo-incompressible: Discrete Pseudo-incompressible: $\partial _t \Big( \frac{ {\bf{u}}^\flat}{ \theta } \Big) + \frac{1}{ \theta } {\bf{i}} _{\bf{u}} {\bf{d}}{\bf{u}}^\flat =-\Big(gz- \frac{1}{2} | {\bf{u}} | ^2 \Big)\frac{{\bf{d}} \theta}{\theta ^2 } - {\bf{d}} \tilde p$ Equation (5.8)
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