On the Boussinesq equations Joint work with S. Spirito (LAquila) - - PowerPoint PPT Presentation

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.

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Setting of the problem

The equations of motion are the following

∂tu + (u · ∇)u − 1 Re∆u + ∇π = − 1 Fr2ρ′ e3, ∇ · u = 0, ∂tρ′ + (u · ∇)ρ′ − 1 Re Pr∆ρ′ = 0.

(1)

The unknowns (u, p, ρ′) are velocity, pressure, and “salin- ity perturbation,” respectively and e3 = (0, 0, 1). The non- dimensional parameters are the Reynolds number Re, the Prandtl number Pr, and the Froude number Fr.

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Challenge

One challenge is the large range of scales in the ocean. Using a characteristic speed scale U = 10−1ms−1, horizontal length scale L = 105m and kinematic viscosity

ν = 10−6m2s−1 this gives for the Reynolds number Re = UL/ν Re = 1010.

Then K41 theory predicts for the degrees of freedom (for homogeneous, isotropic turbulence scales Re9/4)

N = O(1022).

Challenging (or even not possible) to compute the state of the oceanic velocity and tracer fields.

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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

• btained 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

∂tw + ∇ · (w ⊗ w) − 1 Re∆w + ∇ · (I − α2∆h)−1α2∇hw∇hwT +∇π = f = − 1 Fr2ρ′e3,

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

∂tw + ∇ · (w ⊗ w) − 1 Re∆w + ∇ · (I − α2∆h)−1α2∇hw∇hwT +∇π = f = − 1 Fr2ρ′e3,

coupled with

∂tρ′ + (w · ∇)ρ′ − 1 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

∂tw + ∇ · (w ⊗ w) − 1 Re∆w + ∇ · (I − α2∆h)−1α2∇hw∇hwT +∇π = f = − 1 Fr2ρ′e3,

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 ∂tu − ν∆u + (u · ∇) u + ∇π = −ρ e3

in Ω × [0, T],

∇· u = 0,

in Ω × [0, T],

∂tρ + (u · ∇) ρ = 0

in Ω × [0, T]. Both systems have to be supplemented with initial data

(u0, ρ0) such that ∇· u0 = 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

u · n = 0

• n Γ×]0, T[.

(1)

Since we consider the problem with non-zero viscosity we supplement the boundary condition on the u with either a)

u × n = 0

• n Γ×]0, T[

(Dirichlet), b)

ω × n = 0

• n Γ×]0, T[

(Navier’s type), 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

By standard techniques one can show Theorem Let be given Ω ⊂

simply connected open set. Let u0, ρ0 ∈ H3(Ω), with

∇· u0 = 0 and u0 satisfying the boundary condition u0 = 0

and the compatibility condition

−νP∆u0 + P

• (u0 · ∇) u0)
• + P[ρ0 e3] = 0

where P is the Leray L2-projection operator. Then, there exists T ∗ = T ∗(u03, ρ3, ν) > 0 and a unique solution for the B. system with Dirichlet boundary conditions such that

u, ρ ∈ C(0, T ∗; H3(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 u(s)q

p ds < ∞,

2 q + 3 p = 1, for p > 3, Z T ∇u(s)q

p ds < ∞,

2 q + 3 p = 2, for p > 3 2 , Z T ‚ ‚ ‚π(s) + |u(s)|2 2 ‚ ‚ ‚

q p ds < ∞,

2 q + 3 p = 2, for p > 3 2 , ∡ „ ω(x, t) |ω(x, t)|, ω(y, t) |ω(y, t)| « ≤ c|x − y|1/2, ∀ x = y ∈ Ω : ω = 0 and a.e. t ∈ [0, T],

then is possible to continue u and ρ as a regular solution in

L∞(0, T; H3(Ω)).

Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.9/17

Regularity criteria

Of these criteria, the first one is the well-known Leray-Prodi-Serrin one The second is the extension to gradients Beirão da Veiga, Chinese Ann. (1995), B. Diff. Int. Equ. (2002) That concerning the Bernoulli’s pressure extends those proved by Chae and Lee, Nonlinear Anal. TMA (2001), B. and Galdi, Proc. AMS (2002) (The result in the present form is new also when ρ = const.) That on the angle, is the most interesting in stratified prob- lems, where most of the vorticity is on the plane X- Y . This extends previous results of Constantin and Fefferman, IUMJ (1993) and B. and Beirão da Veiga, Diff. Int Eq. (2002), JDE (2009).

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Selective Smagorinsky model

One basic simple-too simple? Large Eddy Simulation (LES) model is the Ladyžhenskaya-Smagorinsky model (a lot of activity for the analysis of this model in the last years Prague, Freiburg, Pisa, Lisbon....)

ut + (u · ∇) u − ν∆u + ∇ · (CSδ2|∇su|∇su) + ∇p = 0

Model based on the idea that small scales have in the mean a dissipative effect. (Generally too dissipative). This method is very stable, try to improve performances by Adaptive/Selective Smagorinsky

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Selective Smagorinsky model

Cottet, Jiroveanu, & Michaux M2AN 2003, proposed a variant in which the turbulence model is applied only in regions of intense vortex activity. This regions are detected by misalignment of the vorticity’ direction

ut + (u · ∇) u − ν∆u + ∇ · (Ψ(x, t)CSδ2|∇su|∇su) + ∇p = 0 Ψ(x, t) =      1,

if β0 ≤ βm ≤ π − β0

0,

• therwise,

where β0 is some threshold angle (a common value of β0 is

π/12) of the vorticity direction near x.

Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.10/17

Idea of the proof

The proof, apart using different tools for any choice of the criteria and of the boundary conditions, is obtained through two main steps First step: By standard energy method we have

u ∈ L∞(0, T; L2(Ω)) ∩ L2(0, T; H1(Ω)), ρ ∈ L∞(0, T; Lp(Ω)),

for any p ∈ [1, ∞]

Then using one of the different conditions, it is possible to show that

u ∈ L∞(0, T; H1(Ω)) ∩ L2(0, T; H2(Ω))

Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.11/17

Idea of the proof

Second step: Use a bootstrap argument and first show some regularity for ut by

d dtut2

2 + ∇ut2 2 ≤ C

• ∇u4

2ut2 2 + ρ4 4 + u2∇u3 2

• .

Then consider the stationary system (with various bc)

−∆u + ∇π = −(u · ∇) u − ut − ρ e3

in Ω × [0, T],

∇· u = 0

in Ω × [0, T], to show regularity of the D2u. Then iterate to reach D2u ∈ L6 implying ∇u ∈ L∞. Finally plug in to show full regularity of ρ and on u (since ρ is a smooth rhs.)

Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.11/17

Idea of the proof

The proof do not introduce new tool, but we combined them in an organic setting, emphasizing how the bound on

∇u ∈ L∞(0, T; L2) is the crucial one, as for the NSE.

This shows that the equation for the density, in the weak interacting Boussinesq equations, plays a secondary role. The situation is not so easy for the full density-dependent NSE and this will be the object of future work.

Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.11/17

Singular limits

If we go back to the equations motion are the following

∂tu + (u · ∇)u − 1 Re∆u + ∇π = − 1 Fr2ρ′ e3, ∇ · u = 0, ∂tρ′ + (u · ∇)ρ′ − 1 Re Pr∆ρ′ = 0.

We can observe that in many interesting cases the parame- ters are very large

Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.12/17

Singular limits

It will be interesting for the numerical simulations to understand which properties are “stable” in the parameters, hence makes relevant to study the Euler-Boussinesq limit problem

ut + (u · ∇) u + ∇π = −ρ e3 ρt + (u · ∇) ρ = 0

at least in the space periodic case and the convergence of solutions of

∂tu − ν∆u + (u · ∇) u + ∇π = −ρ e3 ∇· u = 0 ∂tρ + (u · ∇) ρ = k∆ρ

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Singular limits

By adapting the proof in Masmoudi, CMP 2007 we have the following result

• Theorem. Let T ∗ = T ∗(u03, ρ03) > 0 be the maximal

time of existence of the unique smooth solution of the E.-B. eq., then for any n there exists a unique smooth solution

(ukn,νn, ρkn,νn) s.t. (ukn,νn, ρkn,νn) ∈ C(0, T ∗; H3).

In addition, for any sub-sequence {kn}n, {νn}n converging to zero (note that one of them can also be identically zero) it follows

sup

t∈]0,T ∗[

ukn,νn(t) − u(t)3 + ρkn,νn(t) − ρ(t)3

n→∞

→ 0

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Singular limits and boundaries

When considering the problem in a bounded domain, such limits cannot exists, due to the presence of a boundary layer (at least with Dirichlet conditions) To understand the core pof the problem, let us consider the case ρ = const., that is the NSE.

∂tuν + (uν · ∇)uν − 1 Re∆uν + ∇πν = 0

and consider convergence to the Euler equations

∂tuE + (uE · ∇)uE + ∇πE = 0

Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.13/17

Singular limits and boundaries

If the Navier-Stokes equations are equipped with Dirichlet conditions uν = 0 and the Euler equations by the slip

uE · n = 0, this mismatch will cause lack of convergence, at

least near to the boundary Better results can be obtained by assuming other boundary conditions. For instance, L∞(L2) convergence is obtained in Iftimie and Planas, Nonlinearity (2006) for the Navier-Stokes with Navier’s (with friction) conditions

uν · n = 0

• n Γ × [0, T)

[D(uν) + αuν]tan = 0

• n Γ × [0, T)

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Singular limits and boundaries

In particular, it is proved that if uν is a Leray-Hopf weak solution and uE is a smooth solutions of the Euler equations starting from the same initial datum u0 ∈ H3, in some interval [0, T], then

sup uν − uE2 + ν T ∇(uν − uE)2 = O(ν).

We will elaborate on the other condition uν · n = 0 and

ων × n = 0 we considered before.

We observe that the two conditions are connected by the identity conditions is expressed by the following identity, valid for all tangential vectors τ on the boundary Γ,

t · τ = ν 2(ω × n) · τ − ν u · ∂n ∂τ ,

• n Γ,

Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.13/17

Singular limits and boundaries

Recently the interest for these conditions involving the vorticity increased due to the work of Xiao and Xin, CPAM (2007), Beirão da Veiga and Crispo, JMFM (2009-10-11). What is relevant is that under this conditions on the vorticity better convergence can be proved. We investigate the case of the energy space and we will find a significant difference with the Navier’s one, connected also with the geometry of the solutions. We consider then NSE with b.c.

u · n = 0 ω × n = 0.

Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.13/17

Limit I

We prove now a first result, the idea is to take the difference

• f the NSE and Euler and test with uν − uE and integrate by

parts. Since the solutions of the Navier-Stokes are weak, this is not possible in this way, we have to work with the integral formulation and the Energy Inequality (but nevertheless this is just a technicality). We get

u(t)2 2 + t

• Ω

∇uν∇u dxdτ ≤

• t
• Ω

(u · ∇) uEu dxdτ

• +ν
• t
• Γ

uν∇nu dSdτ

• .

Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.14/17

Limit I

With parallelogram equality we get

2 Z t Z

Ω

∇uν∇u dxdτ = Z t Z

Ω

|∇uν|2 dxdτ + Z t Z

Ω

|∇u|2 dxdτ − Z t Z

Ω

|∇uE|2 dxdτ.

and since uE is classical

• t
• Ω

(u · ∇) uE u dxdτ

• ≤ C

t

• Ω

|u|2 dxdτ.

Concerning the boundary term, by using trace theorem and Young inequality we get, due to the smoothness of Γ,

ν

• Γ

|uν| |∇n| |u| dS ≤ ν 2∇uν2 + ν 2∇u2 + Cνuνu.

Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.14/17

Limit I

By collecting all the estimates and by using energy inequality we get that

u(t)2

2 + ν

t

• Ω

∇u(τ)2 dτ ≤ C t u(τ)2 dτ + ν

• .

We can now end the proof by using Gronwall lemma, obtain- ing the same as with the Navier’s conditions.

Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.14/17

Limit II

We treat now the l.h.s. in a different way by writing

t

• Ω

∇uν∇u dxdτ = t

• Ω

|∇u|2 dxdτ + t

• Ω

∇uE∇u dxdτ.

Then, we have

u(t)2 2 + ν t

• Ω

|∇u|2 ≤ C t

• Ω

|u|2 dxdτ + ν

• t
• Γ

uν(∇n)Tu dSdτ

• + ν
• t
• Ω

∇uE∇u dxdτ

• .

Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.15/17

Limit II

We integrate by parts using the Gauss-Green formula: for u and φ be two smooth enough vector fields, tangential to Γ.

−

• Ω

∆u φ dx =

• Ω

∇u∇φ dx −

• Γ

(ω × n) φi dS +

• Γ

φ(∇n)T u dS, −

• Ω

∆u φ dx =

• Ω

ω(curl φ) dx +

• Γ

(ω × n) φ dS.

We obtain

ν t

• Ω

∇uE∇u dxdτ ≤ ν

• t
• Γ

(ωE × n) u dSdτ

• + ν
• t
• Γ

uν · (∇n)T · u dSdτ

• + ν
• t
• Ω

∆uEu dxdτ

• Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.15/17

Limit II

By trace inequalities we estimate the red term as

ν

• Γ

(ωE × n)u dS

• ≤ Cνu

1 2

2,Γ∇u

1 2

2,Γ.

Then we obtain the following inequality

u(t)2

2 + ν

t ∇u2

2 ≤ C

t u2

2 + ν2 + ν

3 2

• ,

hence by using Gronwall-Lemma the rate

uν − uEL∞(L2) = O(ν

3 4)

Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.15/17

Limit III

Is it possible to improve the rate? One way will be that of getting rid of the boundary term

ωE × n.

This is connected with solving the following Euler equations

∂tuE + (uE · ∇) uE + ∇pE = 0

in Ω × (0, T),

∇ · uE = 0

in Ω × (0, T),

uE · n = 0

• n Γ × (0, T),

ωE × n = 0

• n Γ × (0, T),

uE(x, 0) = u0

in Ω.

Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.16/17

Limit III

This is seemingly an overdetermined problem, since only u tangential can be imposed. To better understand the problem, let us note that is ω0 × n = 0 this does not imply that ω × n = 0 at positive times. Observe that, for Euler equations the vorticity is transported by the velocity uE and stretched by ∇uE, by the well-known representation formula

ωE(X(α, t), t) = ∇αX(α, t)ωE(α, 0)

where the streamlines X : Ω × [0, TE[→ Ω ⊂

ODE

• d

dtX(α, t) = uE(X(α, t), t),

X(α, 0) = α,

Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.16/17

Limit III

The rotation introduced by ∇αX makes not possible a priori the propagation of the initial condition

ω0 × n = 0.

Nevertheless, we observe that, with the same representation formula, we have the following result: If

ω0(x) = 0 for all x ∈ Γ, then ωE(X(α, t), t) = ∇αX(α, t)ωE(α, 0) = 0 ∀ α ∈ Γ

Since uE ·n = 0 on the boundary, the streamlines starting on the boundary remain on the boundary.

Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.16/17

Limit III

The effect on our computations is the following: if ω0 = 0 on

Γ (this class is non empty, take u0 ∈ C∞

0 and

divergence-free) then ωE × n = 0 for all positive times and the bad boundary integral vanishes. Hence, we have that

u(t)2

2 + ν

t ∇u2

2 ≤ C

t u2

2 + ν2

• ,

and then the improved convergence rate

uν − uEL∞(L2) = O(ν), ∇uν − ∇uEL2(L2) = O(ν

1 2)

Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.16/17

Final Remarks

Navier’s conditions involve the symmetric part of the gradient, D := [∇u+(∇u)T ]

2

with the evolution equation

DD Dt + D2 + O2 = −Hπ

where D

Dt is the derivative along streamlines,

O := [∇u−(∇u)T ]

2

is “the vorticity” (since Oh = 1

2ω × h for each

vector h) and Hπ is the Hessian of the pressure. It seems that the evolution of D cannot be handled, since the pressure does not disappear. Hence, the problem related with vanishing-viscosity under the Navier b.c. seems to require different tools

Nonlinear Hyperbolic PDEs, Trieste July 2011 – p.17/17

Final Remarks

For the Boussinesq system one can apply a similar approach and the basic point is a representation formula for the vorticity satisfying the equation

ωt + (u · ∇) ω = (ω · ∇) u − curl(ρ e3).

In order to use the same approach, we need

curl(ρ e3) = 0

• n Γ.

Since ρ is transported by uE, the same argument can be employed if we add in addition

∇ρ0 = 0

for allx ∈ Γ.

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