On the Boussinesq equations Joint work with S. Spirito (L’Aquila) Luigi C. Berselli Dipartimento di Matematica Applicata “U. Dini” Universit` a degli Studi di Pisa berselli@dma.unipi.it Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.1/17
Setting of the problem In some problem of geophysics (ocean modeling) one has to consider incompressible fluids with variable density/temperature, at least in the Boussinesq approximation. One of the most relevant phenomena is mixing since it determines the transport of pollutants, sediments and biological species, as well as the details of the global thermohaline circulation. The lack in space and time resolution, allow most coastal and general circulation models to provide partial information about oceanic mixing processes. Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.2/17
Setting of the problem The equations of motion are the following ∂ t u + ( u · ∇ ) u − 1 Re ∆ u + ∇ π = − 1 Fr 2 ρ ′ e 3 , ∇ · u = 0 , (1) 1 ∂ t ρ ′ + ( u · ∇ ) ρ ′ − Re Pr ∆ ρ ′ = 0 . The unknowns ( u, p, ρ ′ ) are velocity, pressure, and “salin- ity perturbation,” respectively and e 3 = (0 , 0 , 1) . The non- dimensional parameters are the Reynolds number Re , the Prandtl number Pr , and the Froude number Fr . Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.2/17
Challenge One challenge is the large range of scales in the ocean. Using a characteristic speed scale U = 10 − 1 ms − 1 , horizontal length scale L = 10 5 m and kinematic viscosity ν = 10 − 6 m 2 s − 1 this gives for the Reynolds number Re = UL/ν Re = 10 10 . Then K41 theory predicts for the degrees of freedom (for homogeneous, isotropic turbulence scales Re 9 / 4 ) N = O (10 22 ) . Challenging (or even not possible) to compute the state of the oceanic velocity and tracer fields. Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.3/17
LES and Ocean modeling One “natural” idea: try to use for these stratified flows the approach of Large Eddy Simulation (LES) in order to reduce the needed degrees of freedom and to have a computable problem, which can be used to make predictions. Several experiments performed by Özgökmen, Iliescu, Fischer, Srinivasan, & Duan, Large eddy simulation of stratified mixing in two-dimensional dam-break problem in a rectangular enclosed domain , Ocean Modelling (2007). Comparison of several different LES models and especially the Smagoringsky model clipped (with Ri ) in an anisotropic way. Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.4/17
Numerical tests Some simulations heavy fluid injected Özgökmen, Fischer, Duan, Iliescu, J. Physical Oceanography , 2004 lock exchange Özgökmen, Iliescu, Fischer, Srinivasan, Duan, Ocean Modelling , 2007 Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.5/17
Numerical tests Density perturbation. DNS at: (a) t=0.8; (b) t=1.2; (c) t=3.0; (d) t=5.0; and (e) t=45.0. B., Özgökmen, Iliescu, Fischer, J. Sci. Comput. , 2011 Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.5/17
Numerical tests Density perturbation snapshots. (a) DNS ; (b) DNS ∗ ; (c) Clark- α horizontal; and (d) RLES horizontal. B., Özgökmen, Iliescu, Fischer, J. Sci. Comput. , 2011 Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.5/17
Numerical tests RPE † (Reference potential energy) curves for DNS , DNS ∗ , Clark- α horizon- tal, RLES horizontal, Clark- α , and RLES. B., Özgökmen, Iliescu, Fischer, J. Sci. Comput. , 2011. † is the minimum potential energy that can be obtained through an adiabatic redistribution of water masses Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.5/17
Computational method Since we have a code for LES methods based on wavenumber asymptotics (Gradient, Clark, Clark- α , Rational) we adapted the same ideas on horizontal filtering and for the moment we studied the “horizontal version” of the Rational-Clark- α method ∂ t w + ∇ · ( w ⊗ w ) − 1 Re ∆ w + ∇ · ( I − α 2 ∆ h ) − 1 α 2 ∇ h w ∇ h w T + ∇ π = f = − 1 Fr 2 ρ ′ e 3 , Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.6/17
Computational method Since we have a code for LES methods based on wavenumber asymptotics (Gradient, Clark, Clark- α , Rational) we adapted the same ideas on horizontal filtering and for the moment we studied the “horizontal version” of the Rational-Clark- α method ∂ t w + ∇ · ( w ⊗ w ) − 1 Re ∆ w + ∇ · ( I − α 2 ∆ h ) − 1 α 2 ∇ h w ∇ h w T + ∇ π = f = − 1 Fr 2 ρ ′ e 3 , coupled with 1 ∂ t ρ ′ + ( w · ∇ ) ρ ′ − Re Pr ∆ ρ ′ = 0 . Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.6/17
Computational method Since we have a code for LES methods based on wavenumber asymptotics (Gradient, Clark, Clark- α , Rational) we adapted the same ideas on horizontal filtering and for the moment we studied the “horizontal version” of the Rational-Clark- α method ∂ t w + ∇ · ( w ⊗ w ) − 1 Re ∆ w + ∇ · ( I − α 2 ∆ h ) − 1 α 2 ∇ h w ∇ h w T + ∇ π = f = − 1 Fr 2 ρ ′ e 3 , This is not the LES method for which we can prove the best theoretical results, nevertheless some stability proper- ties hold true. Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.6/17
On smoothing In order to detect which type of smoothing is needed to stabilize the problem, we are led to consider the following two issues 1. Is the smoothing in the density equation not necessary? 2. Are the usual regularity conditions known for the NSE also valid for the viscous Boussinesq equations? Similar results, with the perspective of understanding Voigt models has been studied in the 2D case by Larios, Lunasin, and Titi (2010). Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.7/17
On smoothing From the mathematical point of view there are some results, as for instance those in Fan and Zhou, Proc. Edinburgh (2010) for a class of α -Boussinesq equations (with smoothing in both equations). Criteria for the regularity have been proved by several authors, Chae and Nam, Proc. Edinburgh (1997), Ishimura and Morimoto, M3AS (1999) Chae, Kim, and Nam, Nagoya (1999), Fan and Ozawa, Nonlinearity (2009), Liu, Wang, and Zhang, JMFM (2010). We also point out that in recent years the interest for the 2D Boussinesq, as a model problem for 3D Euler, has been emphasized, see Constantin, Publ. Mat (2008). Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.7/17
On smoothing Our aim is twofold To collect the regularity criteria, in a sort of unified treatment. To prove results in a bounded domain, with various boundary conditions. Observe that many results are scattered through the litera- ture and most are proved for the Cauchy problem, avoiding the relevant treatment of the boundary terms. Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.7/17
The mathematical problem The equations We call “viscous system” that with ν > 0 and the diffusivity k = 0 in Ω × [0 , T ] , ∂ t u − ν ∆ u + ( u · ∇ ) u + ∇ π = − ρ e 3 in Ω × [0 , T ] , ∇· u = 0 , in Ω × [0 , T ] . ∂ t ρ + ( u · ∇ ) ρ = 0 Both systems have to be supplemented with initial data ( u 0 , ρ 0 ) such that ∇· u 0 = 0 . Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.8/17
The mathematical problem The boundary conditions On Γ = ∂ Ω we observe that a natural condition is for any choice of κ, ν ≥ 0 is the slip condition on Γ × ]0 , T [ . (1) u · n = 0 Since we consider the problem with non-zero viscosity we supplement the boundary condition on the u with either a) on Γ × ]0 , T [ (Dirichlet), u × n = 0 b) on Γ × ]0 , T [ (Navier’s type), ω × n = 0 where ω = ∇ × u is the curl of the velocity. Concerning the density ρ since u is tangential we have no boundary conditions when k = 0 . Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.8/17
The mathematical problem R 3 a bounded smooth, and By standard techniques one can show Theorem Let be given Ω ⊂ simply connected open set. Let u 0 , ρ 0 ∈ H 3 (Ω) , with ∇· u 0 = 0 and u 0 satisfying the boundary condition u 0 = 0 and the compatibility condition � � + P [ ρ 0 e 3 ] = 0 − νP ∆ u 0 + P ( u 0 · ∇ ) u 0 ) where P is the Leray L 2 -projection operator. Then, there exists T ∗ = T ∗ ( � u 0 � 3 , � ρ � 3 , ν ) > 0 and a unique solution for the B. system with Dirichlet boundary conditions such that u, ρ ∈ C (0 , T ∗ ; H 3 (Ω)) . The same result holds true also with the Navier’s type Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.8/17
Regularity criteria The main continuation result we prove is the following. Theorem. Let ( u, ρ ) be a solution of the system B. system with boundary conditions as above. If for some T > 0 one of the following conditions holds true: Z T 2 q + 3 � u ( s ) � q for p > 3 , p ds < ∞ , p = 1 , 0 Z T 2 q + 3 for p > 3 �∇ u ( s ) � q p ds < ∞ , p = 2 , 2 , 0 Z T ‚ π ( s ) + | u ( s ) | 2 q + 3 2 for p > 3 q ‚ ‚ p ds < ∞ , p = 2 , 2 , ‚ ‚ 2 ‚ 0 „ ω ( x, t ) | ω ( x, t ) | , ω ( y, t ) « ≤ c | x − y | 1 / 2 , ∀ x � = y ∈ Ω : ω � = 0 and a.e. t ∈ [0 , T ] , ∡ | ω ( y, t ) | then is possible to continue u and ρ as a regular solution in L ∞ (0 , T ; H 3 (Ω)) . Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.9/17
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