Algebraic Bounds on the Rayleigh-B\'enard attractor

The Rayleigh-B\'enard system with stress-free boundary conditions is shown to have a global attractor in each affine space where velocity has fixed spatial average. The physical problem is shown to be equivalent to one with periodic boundary conditions and certain symmetries. This enables a Gronwall estimate on enstrophy. That estimate is then used to bound the $L^2$ norm of the temperature gradient on the global attractor, which, in turn, is used to find a bounding region for the attractor in the enstrophy, palinstrophy-plane. All final bounds are algebraic in the viscosity and thermal diffusivity, a significant improvement over previously established estimates. The sharpness of the bounds are tested with numerical simulations.


Introduction
The long-time behavior of the Rayleigh-Bénard problem was analyzed in [10,17] for several types of boundary conditions. In that work the authors derived explicit estimates for enstrophy and the (L 2 -norm of the) temperature gradient on the global attractor for the case of no-slip boundary conditions in space dimension two. They also outlined the functional setting for the case of stress-free velocity boundary conditions (see (2.2a), (2.2b)), and mentioned that corresponding estimates can be carried out in a similar fashion. In this paper we revisit the 2D, stress-free boundary conditions case, and as in the case of rigorous bounds on the time averaged heat transport [18], we find estimates on the global attractor which are dramatically reduced from those in the no-slip boundary conditions case. We also derive estimates for the palinstrophy and H 2 -norm of the temperature.
One marked difference between no-slip and stress-free boundary conditions is that in the latter case, the system is not dissipative for general initial velocity data. This is due to the existence of steady states with arbitrarily large L 2 -norms, namely velocity of the form u(x, t) = (c, 0), (where c is a constant) with zero temperature θ(x, t) = 0 such as the shear-dominated flow investigated in [12]. Since, however, the spatial average is conserved for these flows, the system is dissipative within each invariant affine space of fixed horizontal velocity average. This wrinkle does not influence the estimates on the temperature or higher Sobolev norm estimates on the velocity.
The a priori estimates are carried out in Section 4. The key to finding sharper bounds in the stress-free case is to extend the physical domain, as done in [8], to one that is fully periodic and twice the height of the original. This makes the trilinear term vanish from the enstrophy balance, giving an easy bound that is O(ν −2 ) in terms of the kinematic viscosity ν. Though the trilinear term persists when estimating the temperature gradient, we are able to avoid the exponential bound that resulted from using a uniform Gronwall lemma in [10], by using the algebraic bound on the enstrophy. We find that on the global attractor the (L 2norm of the) temperature gradient satisfies a bound that is O(Ra 2 ), for Pr ∼ 1, where Ra is the Rayleigh number and Pr is the Prandtl number.
We then follow the approach in [5] for the Navier-Stokes equations (NSE) to obtain an estimate for the palinstrophy, with the temperature playing the role of the body force in the NSE. This leads to curves which bound the attractor in the enstrophy-palinstrophy plane, with an overall bound on palinstrophy that is O(Ra 3 ) for Pr ∼ 1. Using this palinstrophy bound, we then follow a similar procedure to find a bounding region for temperature θ in the ∇θ 2 In Section 5 we recall from [8] how all of these bounds impact the practicality of data assimilation by nudging with just the horizontal component of velocity of the stress-free Rayleigh-Bénard system. The sharpness of our rigorous bounds are tested with numerical simulations over a range of Rayleigh numbers in Section 6. Simulations are also presented there to demonstrate that the nudging algorithm works for data with much lower resolution than the analysis requires. This is actually what suggested we might improve on the exponential bounds in [10,17]. All the bounds here on the attractor are algebraic in the physical parameters.

Preliminaries
The Rayleigh-Bénard (RB) problem on the domain Ω 0 = (0, L) × (0, 1) can be written in dimensionless form as (see, e.g., [10]) where e 2 = (0, 1) and κ is the thermal diffusivity. In this paper, we consider the following set of boundary conditions that are stress-free on the velocity: in the x 2 -direction: u 2 , θ = 0 at x 2 = 0 and x 2 = 1, (2.2a) in the x 1 -direction: u, θ, p are of period L , where the indices 1 and 2 refer the horizontal and vertical components, respectively. Following [8], in the rest of this paper, we consider the equivalent formulation of problem (2.1) subject to the fully periodic boundary conditions on the extended domain Ω = (0, L) × (−1, 1) with the following special spatial symmetries: for (x 1 , x 2 ) ∈ Ω. As a result of this symmetry, we observe that smooth enough functions satisfy 2.1. Function spaces. We will use the same notation indiscriminately for both scalar and vector Lebesgue and Sobolev spaces, which should not be a source of confusion. We denote and |u| := (u, u) 1/2 , u := ((u, u)) 1/2 . Note that · is not a norm, but will form part of one in (2.4). We define function spaces corresponding to the relevant physical boundary conditions as in [8], where F 1 is the set of trigonometric polynomials in (x 1 , x 2 ), with period L in the x 1 -variable, that are even, with period 2 in the x 2 -variable, and F 2 is the set of trigonometric polynomials in (x 1 , x 2 ), with period L in the x 1 -variable, that are odd, with period 2 in the x 2 -variable. The space of smooth vector-valued functions which incorporates the divergencefree condition shall be denoted by We denote the closures of V and F 2 in L 2 (Ω) by H 0 and H 1 , respectively, which are endowed with the usual inner products (u, v) H0 := (u, v), (ψ, φ) H1 := (ψ, φ) and the associated norms u H0 := (u, u) 1/2 , ψ H1 := (ψ, ψ) 1/2 .
We define for k = 1, 2 Finally, we denote the closures of V and F 2 in H 1 per (Ω) by V 0 and V 1 respectively, endowed with the inner products and associated norms where |Ω| = 2L is the volume of Ω.

2.4.
Functional setting. Following [10], we have the functional form of the RB problem (2.1): where P σ denotes the Leray projector.

Statement of result
Theorem 3.1. The Rayleigh-Bénard problem (2.1) with stress-free boundary conditions (2.3) has a global attractor A α within the invariant affine space The elements in A α satisfy where the functions f , g are defined below in (4.28), (4.35), respectively, Pr is the Prandtl number ν/κ, Ra = 1/(νκ) is the Rayleigh number, and Q 1 Q 2 means Q 1 ≤ cQ 2 for a nondimensional universal constant c that is independent of the physical parameters.

A priori estimates
Global existence and uniqueness follows by the standard Galerkin procedure based on the trigonometric basis functions in the definitions of F 1 and F 2 . We thus proceed with a priori estimates. 4.1. L 2 bound on temperature. We have the following maximum principle from Lemma 2.1 in [10] where |Ω| is the volume of Ω, While the proof in [10] was done for no-slip boundary conditions, the only place the velocity u enters is the orthogonality property b 1 (u, θ, θ) = 0. The proof carries over verbatim to the stress-free case by (2.5). Consequently, we have (4.1) for each strong solution (u, θ) of (2.8).

L 2 bounds on velocity.
We denote the space average of the horizontal velocity over the extended domain by From (2.1) and the periodic boundary conditions on Ω, we find that the spatial average of the horizontal velocity is conserved, i.e., dα/dt = 0. It follows that Since u α has zero average, it satisfies the Poincaré inequality Note that since u 2 has zero mean, it satisfies a Poincaré inequality even though u 1 does not. Taking the scalar product with u α , and applying (2.5), the Cauchy-Schwarz and Young inequalities as well as (4.2), we get Applying (4.2) once again, together with (4.1) and Young's inequality, we have d dt and thus, 3. An enstrophy bound. We note that ∇u has zero average over Ω by the periodicity of u. As a consequence, we have the Poincaré inequality Taking the scalar product of (2.8a) with A 0 u, we have by the orthogonality property (2.6) 1 2 hence, by (4.1) and (4.6) we have d dt , and thanks to the Gronwall inequality we obtain Similar to the no-slip case analyzed in [10,17], if u(0) ≤ M 1 , θ(0) ≤ M 2 , and ε > 0 we have from (4.1), (4.5) and (4.8) that there exists t 0 = t 0 (M 1 , M 2 , ε) such that 4.4. Bound on the temperature gradient. We start by taking the scalar product of (2.8b) with A 1 θ = −∆θ, integrating by parts and applying the Cauchy-Schwarz and Young inequalities We apply integration by parts to rewrite the trilinear term as We then use the chain rule to rewrite the first sum, again apply integration by parts, and then incompressibility to find Applying the Hölder, Ladyzhenskaya and Young inequalities to each of the remaining four terms, we obtain (4.14) Now combine (4.12), (4.14) and the Poincaré inequality (4.3) so that We note that by the Cauchy-Schwarz inequality and (4.1), for ε as in (4.9)-(4.11). From (4.9), (4.11), (4.15) and Young's inequality, we have for all t ≥ t 0 To prove this we take ε > 0, as above, and consider two possibilities.
In either case we may now take ε → 0 + to conclude (4.16). Introducing the Rayleigh and Prandtl numbers in (4.16) and using the concavity of the square root function, we arrive at the bounding expression lim sup t→∞ θ(t) 2 ϑ max := |Ω|z max RaPr + |Ω| λ 1 z max RaPr is absorbing. This gives for each α the existence of a global attractor A α , within the invariant subspace of solutions (u, θ) where the spatial average of velocity is fixed at α. The global attractor is contained in B α (0).

Palinstrophy bound.
To estimate palinstrophy on A α we follow [5] almost verbatim except that the effect of time independent forcing of the Navier-Stokes equations is played by the bound θ 2 ≤ ϑ max . The other difference is that our velocity is not normalized as in [5]. For completeness, and in order to arrive at an overall bound in terms of ν, κ, we distill the essential argument here.
Returning to (4.7), we integrate by parts, and then apply the Cauchy-Schwarz inequality to get We denote Then whenever (4.20) Setting w = A 0 u in (2.7) and applying Agmon's inequality, we have . We next take the scalar product of (2.8a) with A 2 0 u, and integrate by parts to obtain 1 2

(4.24)
To close the system (eliminate ζ) we find that the maximum of ψ is achieved at We note that  A qualitative sketch of these three curves is shown in Figure 1. It is shown in [5] that the curve q = f 3 (z) does not intersect the curve q = 2 √ zϑ max /ν. Let (4.28) To prove (3.4), suppose there is an element in A α such that q(0) > f (z(0)). The solution through any element in A α exists for all negative time. If q(t) > f (z(t)) for all t < 0, since q(t) increases with negative time, as long as z(t) < z 2 , we have q(t) > min{q(0), q 0 }. By the upper bound in (4.20), z(t) would then exceed z 2 in finite negative time. Thus, we must have q(t) ≤ f (z(t)) at some t < 0. But forward in time, the region q ≤ f (z) is invariant, contradicting the assumption that the initial condition satisfied q(0) > f (z(0)).
We now find an overall bound on palinstrophy in A α . A straightforward calculation shows that substituting Similarly, using leads to 4.6. A bound on |A 1 θ|. From (4.12) and (4.14) we have Thus, if We next take the scalar product of the temperature equation with A 2 1 θ = ∆ 2 θ and using the fact that A 0 u = −∆u, write 1 2 We need to move two derivatives in the trilinear term in order to ultimately obtain a bound for it in which the highest order norm is |A 3/2 1 θ|. We integrate by parts to write We then integrate the first summation by parts and split the resulting first summation as Proceeding as in (4.13), we find that I a1 = 0. Integrating by parts again, we have Since the first sum is zero by incompressibility, we have by symmetry that I a2 = −I a2 , and thus I a2 = 0. Integrating by parts one more time, we have After gathering what remains, we use Agmon's and Ladyzhenskaya's inequalities to estimate the trilinear term as In terms of z 0 , our enstrophy bound on the attractor, (4.30) holds for Once again, by the Cauchy-Schwarz inequality, we have Thus for and solve dη dϑ We then find the intersection of this line with η = γϑ to be at (ϑ 1 , η 1 ), where , and η 1 = γϑ 1 . For η ≥ γϑ we combine As we argued in Section 4.5, if an element in the global attractor were to project in the ϑ-η plane above then by (4.33) the solution through it would, in finite negative time, have to enter the region below the curves in (4.35). Yet, this region is invariant. We conclude from (4.34) and (4.29) that we have an overall bound on the global attractor of A qualitative sketch of the region bounding the global attractor in this plane is shown in Figure 2.

Implications for data assimilation
Suppose reality is represented by a particular solution to an evolution equation where the initial data v(0) is not known. Instead continuous data of the form I h v(t) is known over an interval, t ∈ [t 1 , t 2 ], for a certain type of interpolating operator I h with spatial resolution h. The nudging approach to data assimilation amounts to solving the auxiliary system using any initial condition, e.g.,ṽ 0 = 0. It was shown in [2,3] that if µ > 0 is sufficiently large, and correspondingly, h sufficiently small, then v(t) −ṽ(t) → 0, in some norm, at an exponential rate, as t → ∞. In fact, computations indicate that this approach works with data that is much more coarse than suggested by rigorous estimates (see [1,6,7,11]). Flexibility in the choice of interpolant is one of the main advantages of injecting the observed data through a feedback nudging term, rather than into terms involving spatial derivatives [2,14]. Numerical errors are shown to be bounded uniformly in time for semi-discrete [15] and fully discrete schemes [13] for (5.1). Now consider this approach for the stress-free Rayleigh-Bénard system (2.8) using data from only the horizontal component of velocity. This means solving the auxiliary system dũ dt It was proved in [8] that if µh 2 ν and then u(t) −ũ(t) + |θ(t) −θ(t)| → 0 as t → ∞ at an exponential rate. Also shown there was that if then the stronger convergence holds at an exponential rate. The bounds in this paper on θ , |A 0 u| and |A 1 θ| are all algebraic, suggesting that data assimilation by nudging with just the horizontal velocity could be effective for the stress-free Rayleigh-Bénard system. We present computational evidence to this effect in the next section.

Computational Results
The computations presented below were done using Dedalus, an open-source package for solving partial differential equations using pseudo-spectral methods (see [4]). The time stepping is done by a four-stage third order Runge-Kutta method.
We solve (2.1) with L = 2 in the physical domain Ω 0 = (0, L) × (0, 1). The physical parameters of viscosity and thermal diffusivity are related to the Rayleigh and Prandtl numbers through We take Pr = 1 so that in our dimensionless variables Ra := (νκ) −1 = ν −2 and use n F Fourier modes in the x 1 -direction and n C Chebyshev modes in the x 2 -direction. The numbers of modes used are n F × n C = 256 × 128, 1024 × 512, and 2048 × 1024 for runs at Ra = 10 6 , 10 7 , 10 8 respectively.
6.1. Sharpness. Each plot in Figure 3 shows the projection of a solution after a transient phase in a plane spanned by the norms bounded by our analysis. The solutions are plotted over the time period 200 ≤ t ≤ 1000 for Ra = 10 6 , 10 7 and over 200 ≤ t ≤ 1485 for Ra = 10 8 (time units in the RB system (2.1)). The initial condition in each case is (u 0 , θ 0 ) = (0, 0) so the average α of the horizontal velocity is zero. It is not surprising that our rigorous overall bounds as well as the curves in Figures 1, 2 are orders of magnitude greater than the norms of these solutions. Plotting the bounds and curves together with the solutions is not revealing. Instead, to see a trend in sharpness, we plot in Figure 4 the ratios Using the numerical values for the z ratio, we gauge the (highest) power in (3.2) to be inflated (at least over this range of the Rayleigh number), by an addition of β, where 100 β = 3.14 × 10 6 7.05 × 10 4 , i.e., β = .824 . A similar calculation for the ϑ ratio gives β = 1.68. We note that the curves are bending favorably for the ratios for q = palinstrophy and η = |A 1 θ|.
where (x j 2 ) are the Chebyshev grid points in the x 2 -direction of the physical space. For n F × n C = 256 × 128, this means h(16) ≈ 0.196 and h(32) ≈ .0.393. The nudging parameter is fixed at µ = 1. Figure 5 shows that at h = 0.196 the solution to the data assimilation system converges to the reference solution at an exponential rate. At h = 0.393 the error appears to saturate around 10 −3 during rapid oscillations (see Figure 6). We found that at h = h(64) = 0.785 the nudged solution does not converge to the reference at all (not shown). This demonstrates a critical value of h.
Data assimilation by nudging works much more effectively than the rigorous analysis can guarantee. The value of µ and corresponding resolution h of the data suggested by the conditions (5.2) and (5.3) are based on compounded, conservative estimates derived using general inequalities which are not saturated by 2D convective flows. In addition, as demonstrated in (6.1), our algebraic rigorous estimates for θ , |A 0 u|, and |A 1 θ| in this case of stress-free boundary conditions, though much better than the exponential bounds previously found for the no-slip case in [10], are still somewhat artificially inflated. Numerical nudging tests in [7] for the Rayleigh-Bénard system with no-slip boundary conditions suggest that better bounds on the attractor might hold in that case as well. The key here in the stress-free case was extending the physical domain to be fully periodic, hence there is effectively, no boundary. Since, in the no-slip case one is unable to remove the physical boundary, one should have to resolve the boundary layer scales in order to determine the behavior of the solutions. This is even more pronounced in the estimates of the dimension of the global attractor of the 2D Navier-Stokes equations with no-slip boundary conditions in comparison to the case with periodic boundary conditions. Thus, improving the bounds in the no-slip case would require entirely different techniques.  Figure 6. Data assimilation at Ra = 10 6 and Pr = 1 with h ≈ 0.393.