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Variational integrators for anelastic and pseudo-incompressible flows

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  • 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.

    Mathematics Subject Classification: Primary: 65P10, 76M60, 37N10; Secondary: 37K05, 37K65.

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  • Figure 5.1.  Notation and indexing conventions for the 2D simplicial mesh

    Figure 6.1.  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

    Figure 6.2.  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

    Figure 6.3.  Boussinesq scheme: snapshots of the wave propagation on the regular (left column) and the irregular (right column) mesh

    Figure 6.4.  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)

    Figure 6.5.  Boussinesq scheme: relative errors of total energy $ E(t) $ and mass $ M(t) $ for the regular (left column) and the irregular (right column) mesh

    Figure 6.6.  Anelastic scheme: relative errors of total energy $ E(t) $ and mass $ M(t) $ for the regular (left column) and the irregular (right column) mesh

    Figure 6.7.  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

    Figure 6.8.  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} $

    Figure 6.9.  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

    Figure 6.10.  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

    Table 5.1.  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)
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