Limitations of the background field method applied to Rayleigh-Bénard convection

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1 Limitations of the background field method applied to Rayleigh-Bénard convection Camilla Nobili, and Felix Otto Citation: Journal of Mathematical Physics 58, 93 7; View online: View Table of Contents: Published by the American Institute of Physics Articles you may be interested in Solution of the Fokker-Planck equation with a logarithmic potential and mixed eigenvalue spectrum Journal of Mathematical Physics 58, 933 7;.63/.5386 Renormalization of SU Yang Mills theory with flow equations Journal of Mathematical Physics 58, ;.63/.54 Evolution of superoscillations for Schrödinger equation in a uniform magnetic field Journal of Mathematical Physics 58, 93 7;.63/ The classical dynamic symmetry for the Sp-Kepler problems Journal of Mathematical Physics 58, 97 7;.63/.5688 Attracting subspaces in a hyper-spherical representation of autonomous dynamical systems Journal of Mathematical Physics 58, 97 7;.63/.589 A perturbation theory approach to the stability of the Pais-Uhlenbeck oscillator Journal of Mathematical Physics 58, 935 7;.63/.538

2 JOURNAL OF MATHEMATICAL PHYSICS 58, 93 7 Limitations of the background field method applied to Rayleigh-Bénard convection Camilla Nobili,a and Felix Otto Department of Mathematics and Computer Science, University of Basel, Spiegelgasse, 45 Basel, Switzerland Max-Planck Institute for Mathematics in the Sciences, Inseltraße, 43 Leipzig, Germany Received 7 August 6; accepted 3 August 7; published online 9 September 7 We consider Rayleigh-Bénard convection as modeled by the Boussinesq equations, in the case of infinite Prandtl numbers and with no-slip boundary condition. There is a broad interest in boun of the upwar heat flux, as given by the Nusselt number Nu, in terms of the forcing via the imposed temperature difference, as given by the Rayleigh number in the turbulent regime Ra. In several studies, the background field method applied to the temperature field has been used to provide upper boun on Nu in terms of Ra. In these applications, the background field method comes in the form of a variational problem where one optimizes a stratified temperature profile subject to a certain stability condition; the method is believed to capture the marginal stability of the boundary layer. The best available upper bound via this method is Nu Ra 3 ln Ra 5 ; it procee via the construction of a stable temperature background profile that increases logarithmically in the bulk. In this paper, we show that the background temperature field method cannot provide a tighter upper bound in terms of the power of the logarithm. However, by another method, one does obtain the tighter upper bound Nu Ra 3 ln ln Ra 3 so that the result of this paper implies that the background temperature field method is unphysical in the sense that it cannot provide the optimal bound. Published by AIP Publishing. [ I. INTRODUCTION In a d-dimensional container of height normalized to unity, we consider Rayleigh-Bénard convection as modeled by the Boussinesq equations, which we consider in their infinite-prandtl-number limit: t T + u T = T for < z <, a u + p = RaTe z for < z <, b u = for < z <, c u = for z {, }, d T = for z =, e T = for z =. f Here u R d denotes the fluid velocity, T R denotes its temperature, and p R denotes its pressure. We denote with z the vertical component of the d-dimensional position vector x = y, z and with e z the upward unit normal in the vertical direction. As a convenient proxy of the side-wall effect, the functions u, T, and p, which depend on the spatial variable x and the time variable t, are supposed to be periodic in the d -horizontal directions y with period L, where L is the horizontal period. In our treatment, the dimension d is arbitrary and we think of L as being large. The first equation a Author to whom correspondence should be addressed: camilla.nobili@unibas.ch -488/7/589/93/46/$3. 58, 93- Published by AIP Publishing.

3 93- C. Nobili and F. Otto J. Math. Phys. 58, 93 7 encodes the diffusion of the temperature, driven by the Dirichlet boundary conditions e and f, and its advection by the fluid velocity. The second equation, the Stokes equation, encodes the fact that the fluid parcels move as a reaction to the buoyancy force RaTe z hotter parcels expand and thus experience an upwar force under gravity and are slowed down by viscosity u in conjunction with the no-slip boundary condition d. The last equation expresses the incompressibility of the fluid and is balanced by the pressure term p acting as a Lagrangian multiplier in the previous equation. The parameter appearing in the Stokes equation, the Rayleigh number Ra, expresses the relative strength of the buoyancy force and is given by Ra = αgt b T t h 3, νκ where α is the thermal expansion coefficient, ν is the kinematic viscosity, κ is the thermal conductivity, T b T t is the temperature gap between the bottom and the top plate, and h is the height of the container before the non-dimensionalization. In, the inertia of the fluid has been neglected, which amounts to sending the Prandtl number Pr = ν κ to infinity. Therefore Ra, next to L, is the only non-dimensional parameter. The linear stability analysis identifies a critical value Ra c, the Rayleigh number at which the solution of bifurcates from the linear conduction profile T = z, u =, p = z z, see, for instance, Ref.. When Ra > Ra c, the buoyancy forces trigger the formation of convection rolls. Eventually, when Ra Ra c, these convection rolls break down. This regime features boundary layers at the top and bottom plates, with a high vertical temperature gradient, from which small fluid parcels of different temperatures than the ambient fluid detach and deform, the so-called plumes. In this paper, we are interested in this turbulent regime of Ra and in the experimentally observed enhancement of the heat transport over the pure conduction state. An appropriate measure to quantify the vertical heat flux is the Nusselt number, Nu = ut T e z dz, which represents the average heat flux ut T passing through an area element. The bracket denotes the time and horizontal space average, t f lim sup t t L d f t, ydydt, 3 [,L d and might be thought as a statistical average. In the fifties, Malkus, considering flui with very high viscosity, performed experiments in which he noticed sharp transitions in the slope of the Nu Ra relation and suggested the scaling Nu Ra 3 for very high Ra numbers based on the marginal stability argument, which we reproduce now. Since the main temperature drops near the boundary, we can assume that in a bottom boundary layer of thickness δ to be determined, the temperature drops from to its average. Thanks to the average, we may extract the Nusselt number from the boundary layer, where by the no-slip boundary condition we have ut Te z z T so that Nu δ. We think of the boundary layer as a pure conduction state in the interval z δ. Marginal stability refers to the assumption that this state is borderline stable, meaning that its Rayleigh number is critical, which in view of means Ra c = gαt b T t δh 3 νκ from which, because Ra c, we infer δ Ra 3. Inserting this in the scaling of the Nusselt number above, one fin Nu Ra 3. The same conclusion can be achieved by rescaling Eq. according to x = Ra 3 ˆx, t = Ra 3 ˆt, u = Ra 3 û, p = Ra 3 ˆp and thus Nu = Ra 3 Nu. 4,

4 93-3 C. Nobili and F. Otto J. Math. Phys. 58, 93 7 In this way, we end up with the parameter-free system ˆt T + û ˆ T = ˆ T, ˆ û + ˆ ˆp = Teẑ, ˆ û =, which naturally lives in the half space. Since for the latter system, it is natural to expect that the heat flux is universal, i.e., Nu, we also obtain Nu Ra 3. The scaling Nu Ra 3 has been confirmed by experiments at relatively high Prandtl numbers cf. Ref. 3, p. 3, for a list of experimental results. Rigorous analyses have produced upper boun that capture this scaling up to logarithms which we report now. In the sixties, Howard 4 obtained an upper bound that scales like Ra, optimizing over a field of test functions satisfying physical constraints coming from the Navier-Stokes equation cf. Ref. 4, Sec. 3, while neglecting the incompressibility constraint. Later Busse 5 developed the theoretical tool of multiple boundary layer solution multi-α solution in order to solve Howard s variational problem when the incompressibility constraint is taken into account cf. Ref. 5, Sec.. The multi-α solution theory inspired Chan 6 in the seventies. He elegantly applied it, deriving an upper bound on the Nusselt number that scales like Ra 3 when additional conditions in the asymptotic analysis are assumed. In the nineties, Constantin and Doering, inspired by the studies of Malkus, Howard, and Busse, introduced the background field method in order to bound the average dissipation rate in plane Couette flow. This method was already implicitly used by Hopf 7 in the forties in the construction of solutions to the Navier-Stokes equations in the sense of Leray with inhomogeneous boundary data. Later, Constantin and Doering applied it to the Rayleigh-Bénard convection in order to derive rigorous upper boun for the Nusselt number Nu. Although Howard s problem and the background field method constitute dual variational problems, 8 the second method has the advantage to use simple test functions and functional estimates. Indeed this method turned out to be very fruitful: It has been extensively used and it has produced meaningful boun in the theory of turbulence. In the context of the Rayleigh-Bénard convection, this method consists of decomposing the temperature field T into a steady background temperature field profile τ = τz with driving boundary conditions, τ = for z = and τ = for z =, and into temperature fluctuations θ. As we will see in detail in Subsection I B, the advantage of the background field method is to transform the problem of finding upper boun for the Nusselt number into a purely variational problem: Find profile test functions τ that satisfy a certain stability condition and then select the one with minimal Dirichlet energy. The solution of this variational method produces a new number Nu bf such that Nu Nu bf. 5 Experiments suggest to try a profile τ that displays a drop by in a boundary layer and it is constant in the bulk. Such a profile satisfies the stability condition only if the boundary layer size δ is chosen artificially small and gives only suboptimal boun see Ref. 9. Replacing the constant bulk by a linearly increasing profile at the expense of making the drop in the boundary layers deeper does not improve the situation. The idea that the bad boundary layers can be more efficiently compensated by a profile that increases fast near the boundary and slowly almost constant away from them brought Doering, Reznikoff, and the second author in 6 to investigate the stability of a background profile that grows logarithmically in the bulk. This Ansatz indeed proved to be successful, yielding the bound Nu bf Ra 3 ln Ra 3 and therefore reproducing the scaling proposed by Malkus up to a logarithmic correction. Seis and the second author in improved the last bound by reducing the logarithmic correction Nu bf Ra 3 ln Ra 5. 6 They used the same logarithmic construction as in Ref. with a logarithmically larger boundary layer thickness, which they could afford using an additional estimate on the vertical velocity component w = u e z in terms of θ. In the context of the Rayleigh-Bénard convection, the background field method has also been used to study the case of free-slip boundary condition for the velocity field, of an imposed heat flux at the boundary 3 and in the bulk, 4 of mixed thermal boundary conditions, 5 and of rough boundaries. 6

5 93-4 C. Nobili and F. Otto J. Math. Phys. 58, 93 7 This method has been fruitfully applied to a variety of other problems in fluid mechanics, namely, plane Couette flow, 7 pipe flow, and arbitrary Prandtl number convection. Nicolaenko, Scheurer, and Temam 8 applied the background field method to derive an upper bound for the long-time limit of the L -norm of the solution of the Kuramoto-Sivashinsky equation. In this paper, we address the question of the optimality of the background field method in two ways: What is the optimal bound [in terms of the two scaling exponents in Ra µ ln Ra ν ] in the background field method? The answer given in our main result is that the construction in Refs. and leading to 6 is indeed optimal. Does the background field method catch the optimal bound on Nu, in other wor, is 5 optimal at least in terms of both scaling exponents? The answer given by our main result in conjunction with the bound in 9 obtained in Ref. is no. The main result of this paper is stated in the following. Theorem. For Ra, we have Nu bf Ra 3 ln Ra 5. 7 We refer to Theorem for the full formulation and the explanation of the notation and. This lower bound on Nu bf, together with the upper bound 6, implies Nu bf Ra 3 ln Ra 5 and in particular shows that the background field method cannot produce any smaller logarithmic correction than ln Ra 5. However, a combination of the background field with another method improves the logarithmic correction in 6, as we shall explain now. This other method, which we refer to as the maximal principle method, was also introduced by Constantin and Doering. 9 Indeed it is easy to verify that the temperature equation satisfies the maximum principle, leading to T, 8 possibly neglecting an initial layer. This L bound together with a maximal regularity estimate for Stokes equation in L yiel the bound Nu Ra 3 ln Ra 3, where, this time, the logarithm is an expression of the failure of the L norm to be a Calderón- Zygmund norm. Recently, Seis and the second author combined the maximum principle method with the background field method developed in Ref., obtaining Nu Ra 3 ln ln Ra 3, 9 which, to our knowledge, is the best rigorous upper bound. We observe that the combination of all the previous results yiel Nu Nu bf 9 Ra 3 ln ln Ra 3 Ra 3 ln Ra 5 7 Nu bf. So indeed, the background field method is not able to capture the behavior of the Nusselt number even in terms of the two scaling exponents. Therefore, the optimal background temperature profile cannot carry much of a physical meaning. In 5, Plasting and Ierley considered piecewise linear profiles with τ in the bulk and solved the variational problem numerically, finding Nu Ra 7. In 6, inspired by Chan s multi α solution treatment, Ierley, Kerswell, and Plasting, with help of a mixture of numerical and analytical metho, improved the previous result finding Nu c Ra µ ln Ra ν,

6 93-5 C. Nobili and F. Otto J. Math. Phys. 58, 93 7 where µ = and ν =.35. Clearly, since.35 <. 6 = 5, our result although slightly underestimated has been anticipated ten years ago. Incidentally, the background field method suffers a similar fate in the context of the Kuramoto- Sivashinsky equation: Bronski and Gambill identified the optimal scaling of the upper bound that can be obtained by this method, and soon later, Giacomelli and the second author showed that a tighter upper bound can be obtained by an alternative method, which has subsequently been further improved in Refs. 3 and 4. A. Notations The spatial vector: x = y, z [, L d [, H], where H denotes the height of the container and L is the lateral horizontal cell-size. Vertical velocity component: Background profile: Long-time and horizontal average: w u e z where u = uy, z, t. τ : [, H] R such that τ = and τh =, τ = τz, f lim sup t t t ξ dτ dz. L d f t, ydydt. [,L d Gradient: Laplacian: y f = f. z f = y f + z f. Horizontal Fourier transform: Ff k, z = L d e ik y f y, zdy, [,L d where k π L Zd is the dual variable of y. Real part of an imaginary number: Re stan for the real part of a complex number. Complex conjugate: Fw and Fθ are the complex conjugates of the complex valued functions Fw and Fθ. Universal and specific constants: We call universal constant a constant C such that < C < and it only depen on d but not on H, on L, and on the initial data. Throughout the paper, A B means A CB with C a universal constant. Likewise a condition A B means that there exists a possibly large universal constant C such that A C B. We indicate specific constants with C, C, C,.... B. Temperature background field method and main result We start by rescaling 4 suggested by Malkus marginal stability argument, that is, x Ra 3 x, t Ra 3 t, u Ra 3 u, and p Ra 3 p,

7 93-6 C. Nobili and F. Otto J. Math. Phys. 58, 93 7 and setting H Ra 3, we rewrite as t T + u T = T for < z < H, a u + p = Te z for < z < H, b u = for < z < H, c u = for z {, H}, d T = for z =, e T = for z = H. f Notice that in this non-dimensionalization of the equation, the only parameter appearing is the height H of the container and Malkus scaling Nu Ra 3 correspon to Nu. We recall from the Introduction that the Nusselt number is defined as Nu = H ut T e z dz, and now derive some useful representations starting from the equation for the temperature in a: Applying to Eq. a and qualitatively using bound 8 on T given by the maximum principle, it is easy to show that the upward heat flux is constant in the vertical direction, Nu = T w z T for z, H. Testing the equation with T, appealing to incompressibility c, and using for z =, we obtain see Ref. 9 the alternative representation Nu = T dz. 3 For the convenience of the reader, we sketch the derivation of the background field method, see Ref. 5 for more details. The background field method consists of decomposing the temperature field T into a steady background temperature profile τ which depen only on the vertical variable z and satisfies the inhomogeneous driving boundary conditions, τ = for z = and τ = for z = H, and into temperature fluctuations θ, with homogeneous boundary conditions θ = for z {, H}. Inserting this decomposition T = τ + θ 4 in the equation for temperature a, we find that the fluctuations θ evolve according to t θ + u θ θ = d τ dz w dτ dz. From the incompressibility condition c, we obtain by testing with θ, θ dz = dτ dz zθ dz Together with 3, this yiel the final representation dτ H Nu = dz dz dτ θw + θ dz dτ θw dz. dz dz. 5 Applying the divergence to the Stokes equation b, we find that the pressure satisfies p = z T. Inserting p into the equation b e z, we find the direct relationship between θ and the vertical velocity component w u e z, w = y θ for < z < H, w = z w = for z {, H}. Representation 5 shows: Any τ = τz that satisfies the driving boundary conditions and is stable in the sense that the following quadratic form is non-negative dτ θw + θ dz, 7 dz 6

8 93-7 C. Nobili and F. Otto J. Math. Phys. 58, 93 7 for every θ satisfying homogeneous boundary conditions [and w defined through 6], yiel an upper bound for the Nusselt number, dτ Nu dz. dz Note that in 7, we may disregard the time variable, which is only a parameter in 6, so that in 7 reduces to the horizontal average. This motivates to define the Nusselt number associated with the background field method as Nu bf which in view of 5 satisfies inf τ:,h R, τ=,τh= dτ dz τ satisfies 7, 8 dz Nu Nu bf. Our objective is to derive an Ansatz-free lower bound for Nu bf, trying to extract local information on τ from the completely non-local stability condition 7. The full formulation of the main result, already stated in the Introduction, is contained in the following equation. Theorem. Suppose that τ =, τh =, 9 and τ satisfies 7 for all θ, w related by 6 in its Fourier transformed version 6. Then for H, dτ dz ln H 5. dz In particular for the Nusselt number associated with the background field method, we have the lower bound Nu bf ln H 5. Here and in the sequel, stan for C for some generic universal constant C <. Likewise, H means that there exists a universal constant C < such that the statement hol for H C. Besides implying the non-optimality of the background field method, this theorem offers some insights. Indeed the proof is based on a characterization of profiles that satisfy the stability condition 7 see Sec. II B. This characterization is motivated by the analysis of a reduced form of the stability condition 8 which indicates that the long-wave length stability implies approximate logarithmic growth of τ in z, while the short wave-length stability implies approximate monotonicity see Proposition in Sec. II. It is convenient to introduce the slope ξ dτ dz of the background temperature profile. With this convention, the stability condition 7 can be rewritten explicitly as follows: ξ wθ dz + y θ dz + z θ dz, for all functions θ [and w related to θ via the fourth-order boundary value problem 6] that vanish at z {, H}. A major advantage of the background field method is that it is amenable to horizontal Fourier transform: Indeed, denoting by k π L Zd, k, the horizontal wavenumber, 6 turns into whereas assumes the form k d dz Fw = k Fθ for < z < H, Fw = d dz Fw = for z {, H}, ξfwfθ dz + k Fθ dz + d dz Fθ dz, 3

9 93-8 C. Nobili and F. Otto J. Math. Phys. 58, 93 7 where the bar denotes complex conjugation. Using Eq., we can eliminate θ from the stability condition 3, obtaining ξfw d dz + k + k d d dz dz + Fw k Fw dz 4 dz + d dz + k Fw dz, which has to be satisfied for all k π L Zd \ {} and all complex valued functions Fwz satisfying the three boundary conditions Fw = d dz Fw = d dz + k Fw = for z {, H}. 5 We now introduce a further simplification by letting L so that 4 has to hold for all k R d. This strengthening of the stability condition has the additional advantage that it becomes independent of the dimension d: We will henceforth say that ξ satisfies the stability condition if ξw d dz + k w dz + k d d dz dz + w k dz + d dz + k w dz 6 hol true for all k R and all complex valued functions wz satisfying the three boundary conditions 5 with Fw replaced by w. The analysis of the stability condition 6 imposed for all values of k R which correspon to assume L amounts to considering profiles τ that are stable even under perturbation that have horizontal wave-length much larger than H. In Ref. 6, it is shown that at least Proposition still hol true if the lateral size L is of order H. The rest of the paper is organized as follow: In Sec. II A, we study a reduced stability condition obtained by retaining only the indefinite term in the stability condition and show that in this case a stable profile must be increasing and logarithmically growing. This result is obtained by exploring the limit for small and large wavelengths in the reduced stability condition, respectively. In Sec. II B, when working with the original stability condition, we can no longer pass to the limit for small/large wavenumbers k to infer the positivity for ξ = dτ dz and the logarithmic growth for τ. Nevertheless, by subtle averaging of the stability condition, we construct a non-negative convolution kernel φ with the help of which we can express the positivity on average approximately in the bulk see Lemma. Likewise we recover the logarithmic growth, at least approximately in the bulk, on the level of the construction of ξ, see Lemma. Finally in Lemma 3, we connect the bulk with the boundary layers. The main result Theorem is proved in Sec. III and it consists of combining all the results contained in the lemmas together with an estimate that connects ξ to ξ, and in particular to ξ dz = [see Lemma 4, estimate 36]. In the rest of the paper, we omit the constant factor in front of the indefinite term in 6, which is legitimate since we are interested in the scaling of the Nusselt number. II. CHARACTERIZATION OF STABLE PROFILES A. Reduced stability condition We note that the stability conditions 6 and 5 are invariant under the following transformation: z = Lẑ and thus k = L ˆk, H = LĤ and ξ = L 4 ˆξ. 7

10 93-9 C. Nobili and F. Otto J. Math. Phys. 58, 93 7 Hence in the bulk z and H z, we expect that the first term in 6 dominates. This motivates to consider the reduced stability condition ξw d dz + k w dz, 8 for all k R and all complex valued functions wz satisfying the three boundary conditions 5. The following proposition is independent of the main result Theorem but it serves as a preparation: The ideas developed in the proof cf. Sec. II C will be adapted to the more challenging full stability condition 6. Proposition. Let ξ = ξz be such that for all k R and for all wz satisfying 5, it satisfies the reduced stability condition 8. Then ξ and 9 ξdz ξ dz. 3 /e ln H We notice that while 9 means that τ is an increasing function, the second statement 3 correspon to a logarithmic growth of τ see Fig.. Hence somewhat surprisingly, monotonicity is not sufficient for stability. The two estimates above are obtained by taking the limit for very small and very large horizontal wavenumbers in the reduced stability condition 8: The bound 9 arises from letting k, thus considering very small horizontal length-scales, while 3 is a consequence of letting k, which amounts to considering large horizontal length-scales. These boun are proved in detail in Subsection II C. B. Original stability condition: Statements of lemmas In the following lemmas, we derive properties of those profiles τ that, in terms of their slope ξ = dτ dz, satisfy the original stability condition 6 and for the last lemma the driving boundary conditions 9. These four lemmas are the only ingredients of the main theorem. They all are formulated on the level of the logarithmic variables s = ln z and ˆξ = zξ = dτ, cf. 4. Lemma establishes approximate positivity of the slope ˆξ in the bulk and thus is the generalization of 9 in Proposition, replacing the stricter reduced stability condition 8 there by the original stability condition 6 here. It does so in terms of a suitable convolution ˆξ of ˆξ in the logarithmic variable s. Lemma establishes an approximate logarithmic growth of the profile in the bulk, again on the level of ˆξ, and amounts to the generalization of 3 in Proposition. Lemma 3 is the most subtle and shows that the convolved slope ˆξ cannot be too negative in the boundary layer s provided it is sufficiently small in the transition layer s. In Lemma 4, the condition ξ dz = for the slope ξ, which expresses the driving boundary conditions 9 on τ, is translated into a condition for its logarithmic-variable convolution ˆξ. FIG.. The boun in Proposition describe the qualitative behavior of the background profile τ in the bulk: it grows monotonically 9 and at least logarithmically 3.

11 93- C. Nobili and F. Otto J. Math. Phys. 58, 93 7 Lemma. There exists φ, which will play the role of a convolution kernel, with the properties φ s, φ s =, supp φ z 4, 3, φ 4 z = φ + z, 3 such that, for all s ln H, ˆξ s exp 3s, 3 where ˆξ s ˆξs + s φ s. 33 Lemma. For S, we have ˆξ S S ˆξ Lemma 3. For all S and ε, we have ˆξ ˆξ + S + ε ε + ˆξ + ε exp5s. 35 S S Lemma 4. Suppose that the slope ξ of the profile τ satisfies ξ dz = and ξ dz ln H 5. Then ln H ˆξ. 36 In order to infer the approximate non-negativity from the original stability condition, we are forced to work with k. Indeed here, as opposed to the case of the reduced stability condition [see argument for 9 below], we need to control the error term in 6. Therefore, as expected, the estimates in Lemma and Lemma are valid in the region of large horizontal length-scales thus in the bulk, while Lemma 3 is obtained for k <, which correspon to the region of small horizontal length-scales where the specific boundary condition plays a role. Moreover, it is important to observe that the analysis at each scale is required as the lower bound on Nu bf consists in the combination of all lemmas stated above. C. Proof of Proposition Argument for 9 is Letting k, 8 reduces to ξ w dz for all compactly supported w, from which we infer 9. Argument for 3, heuristic version, is Letting k, 8 reduces to for all functions wz satisfying the three boundary conditions ξw d4 wdz 37 dz4 w = dw dz = d4 w = for z {, H}. 38 dz4 In fact, besides Subsection IV C, we will work with w compactly supported in z, H so that the boundary condition 38 is trivially satisfied. Focusing on the lower half of the container, we make the following Ansatz: w = z ŵ,

12 93- C. Nobili and F. Otto J. Math. Phys. 58, 93 7 where ŵz is a real function with compact support in, H. The merit of this Ansatz is that in the new variable ŵ, the multiplier in 37 can be written in the scale-invariant form φ = w d4 d4 w = ŵz dz4 dz 4 z ŵ = ŵ z ddz z + ddz + z d z ddz dz ŵ. 39 Note that the fourth-order polynomial in z d dz appearing on the rhs of 39 may be inferred, without lengthy calculations, from the fact that z d4 z annihilates {, dz 4 z z,, z}. This suggests to introduce the new variables s = ln z and ξ = z ˆξ, 4 for which the stability condition turns into ln H where φ = ŵ ˆξ φ, 4 d d d d + + ŵ, for all functions ŵ with compact support in s, ln H. Here comes the heuristic argument for 3: For H, we may think of test functions ŵ that vary slowly in the logarithmic variable s. For these ŵ, we have d d d d φ = ŵ + + ŵ ŵ d ŵ = d ŵ, 4 which in particular implies ln H ln H ˆξ φ Since ŵ was arbitrary besides varying slowly in s, it follows ˆξ d ln H d ˆξ ŵ = ŵ. d ˆξ, approximately on large s-scales. We expect that this implies that for any S ln H, ˆξ S S ˆξ, 43 which in the original variables 4, for S turns into 3. We now establish rigorously that 37 and 9 imply 43. Argument for 43, rigorous version: We start by noticing that because of translation invariance in s, 4 can be reformulated as follows: For any function ŵs supported in s and any s ln H, we have ˆξs φs s = ˆξs + s φs, 44 where the multiplier φ is defined as in 4. We note that 43 follows from 44 once for given S, we construct a family F = {w s } s of smooth functions w s parameterized by s R and compactly supported in z, i.e., s, ] and a measure ρ = ρs supported in s, ln H] such that the corresponding convex combination of multipliers {φ s } s shifted by s, i.e., φ s φ s s s ρs, 45

13 93- C. Nobili and F. Otto J. Math. Phys. 58, 93 7 satisfies for s φ s C S for s S, 46 else for a possibly large universal constant C. Indeed, using in conjunction with positivity 9 of the profile ˆξ, we have ˆξφ ˆξ + C S ˆξ, S 47 which implies 43. We first address the form of the family F. The heuristic observation 4 motivates the change of variables s = λŝ with λ, 48 our logarithmic length-scale, to be chosen sufficiently large. We fix a smooth, real-valued and compactly supported mask ŵ ŝ; it will be convenient to restrict its support to ŝ, ], say ŵ > in, and ŵ = else, 49 and, in order to justify the language of mollification by convolution, we impose the normalization ŵ =. By mask we mean that in 4, we choose ŵλŝ = λ / ŵ ŝ 5 see Fig.. With this change of variables, the multiplier can be rewritten as follows: 4 d d d d φ λ = ŵ + + ŵ 48&5 d = λ ŵ λ + d λ + d d λ λ ŵ d 4 = ŵ λ d 3 λ 4 3 d λ 3 d λ ŵ, and reordering the terms we have φ λ = d λ ŵ ŵ λ 3 ŵ ŵ + λ 4 ŵ 3 ŵ + λ 5 ŵ 4 ŵ. 5 d d 3 d 4 FIG.. Construction of the family of smooth and compactly supported test functions. Once a mask ŵ satisfying 49 is fixed blue line, we construct the family { ˆω α } α via rescaling of ŵ, see 5.

14 93-3 C. Nobili and F. Otto J. Math. Phys. 58, 93 7 Heuristically, for λ, the multiplier φ λ can be approximated by the first term on the rhs, φ λ s d s λ ŵ. λ Inserting this approximation into definition 45 of φ, we have φ s = = φs s ρs = d φsρs s s λ ŵ ρs s λ s d ρ λ ŵ λ s s. 5 δ s, in For λ smaller than the characteristic scale on which ρ varies, we may think of λ ŵ view of our normalization. This yiel φ d ρ, 53 which in view of 46 suggests that ρ should have the form { s ρs + for s = s S for s S s λ }, 54 see Fig. 3. We will now argue that we have to modify the Ansatz for both φ λ and ρ. To this purpose, we go through the above heuristic argument heuristically assessing the error terms. Expanding ρ in a Taylor series around s we may write φ s 45 = ρs ρs s ρs d ρ s s + d ρ s s, φ λ sρs s φ λ d ρ s sφ λ + d ρ s s φ λ. We note that the first term in 5, i.e., dŵ ŵ λ = dŵ λ, gives the leading-order contribution to the first and the second moment sφ λ s dŵ λ 48 = ŝ dŵ = ŵ = FIG. 3. Construction of the measure ρ. The heuristic argument above suggests the relation between the multiplier φ and the measure ρ, i.e., φ dρ. The comparison of this condition with the desired bound in 47 determines the shape of the measure ρ: it must be first increasing in an interval of length one and then decreasing in an interval of length S.

15 93-4 C. Nobili and F. Otto J. Math. Phys. 58, 93 7 and s φ λ s dŵ λ 48 = λ ŝ dŵ = λ ŝŵ, while the second term in 5 gives the leading-order contribution to the zeroth moment of the multiplier φ, φ λ λ 3 ŵ Hence we obtain the following specification of 5: φ s = λ ρs φ λ sρs s d ŵ 48 = λ dŵ. dŵ d ρ s + λ d ρ s ŝŵ. 55 Our goal is to specify choice 54 of ρ such that 46 is satisfied. This shows a dilemma: On the one hand, in the plateau region s S, we would need λ S so that the first rhs term in 55 does not destroy the desired S behavior. On the other hand, in the foot region s [, ], we would need λ so that the last term does not destroy the effect of the middle term. This suggests that λ should be chosen to be small in the foot regions and large on the plateau region. Therefore it is natural to choose λ = s 56 so that φ s in 45 indeed acquires a dependency on s besides the translation. For our choice of 56, 5 assumes the form φ s = d s ŵ ŵ s 3 ŵ ŵ + s 4 ŵ 3 ŵ + s 5 ŵ 4 ŵ. 57 d Note that with the choice 56 and s = s s, 48 turns into the nonlinear change of variables ŝ = s s s = s s s = s + ŝ. 58 We consider this as a change of variables between s and ŝ with s as a parameter; thanks to the support restriction 49 on ŵ, it is invertible in the relevant range ŝ [, ]: d = s d +ŝ = s d s and = s. From 45 and 57, we thus get the first representation +ŝ φ s = s + s 3 dŵ ρ s + ŝ d 3 ŵ ŵ 3 ρ + s 4 d 3 d ŵ + ŝ ŵ ρ d 4 + ŝ 3 d 4 ŵ ŵ 4 ρ. An approximation argument in ŵ below necessitates a second representation that involves ŵ only up to second derivatives. For this purpose, we rewrite 57 in terms of the three quadratic quantities ŵ, dŵ, and d ŵ, φ s = s dŵ + s 3 + dŵ s 5 = d s + d dŵ dŵ d ŵ + s 4 3 d + d 4 ŵ 4 d s 3 + d 3 s d 4 s 5 4 dŵ + s 5 s 3 3 d s 4 d s 5 ŵ dŵ + d3 ŵ 3 dŵ. 59

16 93-5 C. Nobili and F. Otto J. Math. Phys. 58, 93 7 Now in this formula, using the change of variables 58, we want to substitute the derivations s m dn n by linear combinations of derivations of the form d k + ŝ m n k for k =,..., n. The reason s m k k why this can be done is explained in Appendix A, where also the linear combinations are explicitly computed. The formulas A A3 allow to rewrite 59 as follows: φ s = d s [ + d + ŝ d3 s s 4 s 5 ] dŵ d s 3 + ŝ 3 + ŝ + + ŝ s + ŝ5 5 d s ŝ 3 ŵ d s 4 + ŝ. 6 dŵ The advantage of this form is that integrations by part in s become easy so that we obtain φ = s ŵ d ρ d ρ + ŝ + d 3 ρ + ŝ 3 + d 4 ρ + ŝ s s 4 s 5 + ŝ 3 dŵ ρ 3 s 3 8 s 4 + ŝ dŵ d ρ dŵ d ρ s 3 + ŝ + dŵ s 5 + ŝ 5 ρ. Finally, using the substitution s = +ŝ, the last formula turns into the desired second representation φ = ŵ d ρ + ŝ d ρ + ŝ 3 + d 3 ρ + ŝ d 4 ρ + ŝ s + 6 s 3 dŵ s 4 + ŝ ρ 3 s 8 dŵ d ρ s 3 dŵ d ρ s + ŝ + dŵ s 4 + ŝ 3 ρ. 6 From this representation, we learn the following: If we assume that ρs varies on large length-scales only so that dρ, d ρ, ρ and d ρ, d3 ρ, dρ 3 then for s, we obtain to leading order from the above d ρ φ + ŝ ŵ + s + ŝ dŵ ρ. If ρs varies slowly even on a logarithmic scale so that, e.g., s dρ is negligible with respect to ρ, the above further reduces to

17 93-6 C. Nobili and F. Otto J. Math. Phys. 58, 93 7 φ d ρs + ŝ ŵ + ρs s + ŝ dŵ, 6 which should be compared with 55. We see that the first, negative, right-hand-side term of 6 dominates the second positive term provided d ρ s. This is satisfied if ρ is of the form ρs = S dρ = S S s S s s S for some S to be chosen later; indeed for s S s. Disregarding for a couple of pages the fact that ρ nee to be supported in s, ln H, which will be achieved by cutting off at scales s S, we define as our intermediate goal to construct a measure ρs with infinite support such that φ s φ s s s ρs = s S < s > S, 63 for some universal S, to be chosen later. The above considerations motivate the following Ansatz for ρ: We fix a smooth mask ρ ŝ such that ρ = for ŝ, d ρ > for < ŝ, ρ = ŝ for ŝ 64 and consider the rescaled version ρs ŝ + = ρ ŝ, i. e. the change of variables s = S ŝ + 65 with S to be fixed later see Fig. 4. It is convenient to rescale s accordingly s = S ŝ. 66 With this new rescaling and the choice of the measure ρ [see 64 and 65], 6 turns into FIG. 4. The function ρ of the variable s = S ŝ +.

18 93-7 C. Nobili and F. Otto J. Math. Phys. 58, 93 7 φ = S ŵ + ŝ d ρ ŵ S ŵ d ρ + ŝ 3 ŵ + d 3 ρ d 4 ρ S 3 + ŝ S 4 + ŝ S ŝ + 6 S 3 ŝ 3 S 4 ŝ dŵ 4 + ŝ ρ 3 S 4 ŝ 3 8 S 5 ŝ dŵ d ρ 4 dŵ d ρ S 5 ŝ 3 + ŝ + dŵ S 4 ŝ 4 + ŝ 3 ρ. 67 Since in the integrals in formula 6, the argument of ρ was given by s = s +ŝ, cf. 58, it follows from 65 and 66 that the argument of ρ is given by ŝ = ŝ ŝ Thus all the integrals in 67 just depend on ŝ and not on S. Hence 67 makes the dependence of φ on S explicit. We are now ready to show that the construction of the family w s [cf. 5 and 56] and of the measure ρ [cf. 64 and 65] yield the intermediate goal 63 when S. In order to establish 63, it is convenient to distinguish three regions [note that if s, S ], all the integrals in 67 vanish because the supports of ŵ and ρ do not intersect, see below]. The range of large s is s 3S or equivalently ŝ Note that because of our support condition 49 on ŵ, all integrals in 67 are supported in ŝ [, ]. Together with our range 69, this yiel for the argument ŝ 68 = ŝ +ŝ of ρ and its derivatives that ŝ. Because of d ρ = for ŝ, cf. our Ansatz 64, the first integral in 67 reduces to ŝ ŵ d ρ + ŝ = ŵ + ŝ ŝ +ŝ = ŵ ŝ + ŝ ŝ ŵ, 7 for ŝ, whereas all the other integrals in 67 are O or smaller in ŝ because at least ŝ one derivation falls on ρ or have pre-factors or smaller. Since only the term in 67 coming from ŝ integral 7 has pre-factor S while all the other terms have pre-factors or smaller for S S, the first term in 67 uniformly dominates all other terms for S, φ S In conclusion, we have The range of intermediate s is [ ] 3 s 4 S, 3S ŵ ŝ + ŝ uniformly in ŝ 3 for S. 7 φ S ŝ < in the range ŝ 3 for S. 7 or equivalently ŝ [ ] 3 4, 3. 73

19 93-8 C. Nobili and F. Otto J. Math. Phys. 58, 93 7 Again, we consider the first integral in 67. Now we use that is strictly positive in ŝ +ŝ,, cf. 49, and that d ρ is strictly positive in ŝ >, cf. 64, that is, in ŝ < ŝ, cf. 68. We note that the two ŝ intervals, and, ŝ intersect for ŝ >. Hence by continuity of the first integral in 67 in its parameter ŝ, there exists a universal constant C such that ŵ d ρ + ŝ [ ] 3 C for ŝ 4, 3. Hence also in this range, the first term in 67 dominates all other terms, φ ŵ [ ] d ρ 3 S + ŝ uniformly in ŝ 4, 3 and we may conclude that φ S < in the range s ŵ [ ] 3 4 S, 3S for S, 74 for S. 75 Note that the above discussion on supports also yiel that φ is supported in ŝ [,. The range of small s is s S, 3 4 S or equivalently ŝ, We would like φ to be strictly negative in this range for S. Here, we encounter the second difficulty: No matter how large λ = s in 5 is, the behavior of φ s near the left edge of its support [, ] and also at its right edge, but there we do not care is dominated by the d ŵ 4 ŵ λ 5 -term 4 and thus automatically is strictly positive. Taking the ρs -average of the shifted φ s s s does not alter this behavior as long as ρ is non-negative in [S,, cf. 64: φ is strictly positive near the left edge S of its support. The way out of this problem is to give up smoothness of ŵ near the left edge of its support [, ]. In fact, we shall first assume that ŵ satisfies in addition ŵ = ŝ + [ for ŝ ], This means that ŵ has a bounded but discontinuous second derivative i.e., ŵ H,. This is the main reason why in 67 we expressed φ only in terms of up to second derivatives of ŵ. We argue that the so-defined φ is, as desired, strictly negative on s S, 3 4 S ] for all S. Indeed, in view of 57, form 77 implies, in terms of s = s s, φ s = s s s + 3 s 3 s s + ] < for s s, s In view of 64 and 65, ρ is strictly positive for s S,. On the other hand, it follows from 78 that s φ s s s is strictly negative for s s s, s 4 ], that is, for s [ 4 3 s, s and supported in s [s, s ]. Hence, by 45, φ is strictly negative for S [ 4 3 s, s, that is, for s S, 3 4 S ], for any value of S >. Now we approximate ŵ with a sequence of smooth functions ŵ ν in H, and we call φ ν the associated multiplier. Since φ involves ŵ only up to second derivatives [cf. 67] then φ ν converges uniformly in ŝ to φ. We conclude that φ < for s S, 3 4 S and S >. 79 Finally we fix a sufficiently large but universal S so that 79 together with 7 and 75 implies our intermediate goal 63. The choice 64 and 65 of ρ is not admissible since ρ should be supported in, ln H, which will be achieved by cutting off at scales s S. Only this cutoff will ensure 46 in the range s S.

20 93-9 C. Nobili and F. Otto J. Math. Phys. 58, 93 7 FIG. 5. The measure ρ is constructed from ρ see Fig. 4 by cutting off at scale s S. More precisely, in 45, we choose ρ to be ρs η s S where ρ is defined in 65, η is a mask for a smooth cutoff function with ηŝ = ŝ ŝ see Fig. 5 for an illustration of ρ. We argue that φ s satisfies 46 for S. This will follow immediately from the three claims.. φ s = for s S,. φ s = φ s for s S 4, 3. φ φ S for S 4 s S. Claims,, and 3 together with bound 63 on φ imply φ s = for s S for S + s S + for S s S 4. 8 S for S 4 s S = for s S Note that with the help of the scale invariance 7, which turns into a shift invariance in the logarithmic variables 4, we may shift φ and its estimate 8 by S + to the left and redefine S S noting that the universal constant S was fixed in the previous step recovering the desired form 46. Let us now establish the claims,, and 3. We start by noting that by the change of variables 58 in conjunction with the support condition 49 on ŝ, we have the following relation between the argument s of ρ and the variable s of φ in the representation 6, s S 4 S 4 s S S s s S S 4 s S S s. 8 Hence claims and are immediate consequences of the defining properties of the cutoff function η. We now turn to claim 3 and appeal to the representation 6 for φ, which we also use for φ and

21 93- C. Nobili and F. Otto J. Math. Phys. 58, 93 7 thus obtain a representation of φ φ with ρ replaced by η ρ. Clearly, all the terms bearing a pre-factor of or smaller are of higher order in the range s S s 4. Likewise, all the terms where at least one of the derivatives on the product η ρ falls on the cutoff η, thereby producing a factor S, are of the desired order or smaller. We are left with the terms ŵ η d ρ + ŝ d ρ + ŝ 3 + d 3 ρ + ŝ d ρ + ŝ In our range S 4 s S, the integrand is supported in S 4 s S, cf. 8. For these arguments, we have by choice 64 and 65 of the averaging function ρs = s S S s. Hence also the terms 8 are at least of order and thus of higher order. This establishes claim 3 and thus 8. S III. PROOF OF THE MAIN THEOREM Proof of Theorem. We start by combining Lemmas and 3. We first note that by rearranging 34 in Lemma and then adding ˆξ to both sides, while using S, we obtain by rearranging S ˆξ S ˆξ +, C where we momentarily retain the value C of the universal constant. We now add this to 35 in Lemma 3 in the form of S ˆξ C ˆξ + S + ɛ ɛ + ˆξ + ɛ exp5s S and adjust ɛ = C C S so that the pre-factor of the transition term ˆξ vanishes; this choice of ɛ is consistent with ɛ because of S. We end up with S S ˆξ S + S + to which we add S ˆξ S ˆξ, obtaining S ˆξ S S ˆξ + S exp5s, ˆξ + S + S exp5s, 83 where we replaced S by S + at the expense of changing the multiplicative constant hidden in. In order to provide ourselves with an additional parameter S next to S, S to optimize in, we now appeal to a scaling argument: The change of variables 7 that leaves the stability condition 6 invariant assumes the form of a shift, s = S + ŝ, ˆξ = exp 3S ˆξ, and thus ˆξ = exp 3S ˆ ξ, on the level of the logarithmic variables 4. We apply 83, which only relied on the stability condition, to the rescaled variables ŝ and ξˆ with the upper integral boundaries Ŝ and Ŝ and then revert to the original variables, Ŝ +S Ŝ +S ˆξ ˆξ + exp 3S Ŝ + exp5ŝ. Ŝ This estimate hol provided Ŝ, Ŝ. In terms of the original integral boundaries S = Ŝ + S and S = Ŝ + S, this rea S S ˆξ ˆξ + exp 3S S S + exp5s + S, 84 S S

22 93- C. Nobili and F. Otto J. Math. Phys. 58, 93 7 which is valid provided S S = Ŝ, S + S = Ŝ, and S ln H. Now is the moment to optimize in S by choosing it such that the two last terms are balanced, which is achieved by S S = exp 5 S +S. In our regime S +S, we have exp 5 S +S exp 5 S +S S +S so that the above choice means S + S exp 5 S + S, which implies exp 3S exp3s S + S 6 5. Hence with our choice, the entire second term in 84 is exp3s S + S 5, S ˆξ S ˆξ + exp3s S + S This estimate is valid provided S ln H and S + S since the latter by S + S exp 5 S + S ensures S + S and by S S = exp 5 S + S then also S S. We now make use of Lemma and Lemma 4. Note that without loss of generality., we may assume that our background profile τ satisfies on the level of its slope ξ dz ln H 5 next to ξ dz = since if such a profile would not exist, the statement of Theorem would be trivially true. Hence we are in the position to apply Lemma 4. By Lemma, we have S ln H ˆξ exp 3S. Adding this to 85, we obtain ln H ˆξ so that by 36 in Lemma 4, we get S S ˆξ + exp3s S + S 5 + exp 3S ˆξ + exp3s S + S 5 + exp 3S. Clearly, the optimal choice of S is given by saturating the constraint in the form of S = ln H, which by H turns into S ˆξ + exp3s ln H 5 86 and hol provided S. We finally connect this to the Nusselt number Nu bf, which on the level of the slope ξ and the logarithmic variables turns into Nu bf dτ H ln H dz = ξ dz = exp s ˆξ. 87 dz This allows us to estimate the first rhs term in 86: By definition 33 of the convolution ˆξ and the support property 3 of the kernel φ, we have S S ˆξ ˆξ exp S S exp s ˆξ, 88 where we used the Cauchy-Schwarz inequality in the last step. Inserting 87 into 88 and then into 86, we obtain exp S Nu bf + exp 3S ln H 5. Finally choosing S such that exp3s is a small multiple of ln H 5 so that the last term can be absorbed into the lhs and to the effect of exp S ln H 3, the above turns into the desired ln H 3 Nu bf. IV. PROOFS OF LEMMAS In this section, we will give the proofs of the four lemmas stated in Subsection II B.

23 93- C. Nobili and F. Otto J. Math. Phys. 58, 93 7 A. Approximate positivity in the bulk: Proof of Lemma Much of the effort of this construction will consist in designing the kernel φ in such a way that it is both non-negative and compactly supported. Non-negativity of φ s and its fast decay for s and support in s will be crucial in Subsections IV B and IV C, where we will work with convolution 33. In order to infer non-negativity, we can no longer let k in 6 as in the proof of Proposition since the two last terms would blow up. To quantify this qualitative observation, we restrict to k > and introduce the change of variable z = ẑ 89 k so that the stability condition 6 turns into kh ẑ ˆξ w d k dẑ + w dẑ ẑ kh + k 3 d w d dẑ dẑ + kh dẑ + k 3 w d dẑ + dẑ. 9 We shall restrict ourselves to k with kh and real functions wẑ compactly supported in ẑ, ] so that the boundary conditions 5 are automatically satisfied. In particular, an integration by parts [based on d + 4 dz = d 8 4 d6 + 6 d4 4 d + ] in the two last terms of 9, yielding dz 8 dz 6 dz 4 dz kh d dẑ = d w dẑ + d 5 w dẑ kh dẑ + + d 4 w dẑ 4 d w dẑ + + d 3 w dẑ 3 dẑ d w dẑ + 5 dw + w dẑ, 9 dẑ shows that there are no fortuitous cancellations: Provided the multiplier φ w d + dẑ w of ˆξ is non-negligible in the sense of φ dẑ ẑ = O, the two last terms of 9 are at least of Ok3. Hence we are forced to work with k and thus, as expected, the conclusion is only effective for z = ẑ k. In anticipation of 33, we introduce the logarithmic variables s = ln ẑ and s = ln k 9 so that the first term in the stability condition 9 can be rewritten as follows: kh ẑ ˆξ w d k dẑ + w dẑ ẑ = ˆξs + s w d dẑ + w. In view of this and 9, the stability condition 9 turns into: For all s ln H, we have ˆξs + s w d dẑ + w exp 3s d 5 w dẑ w dẑ. 93 Let us consider the lhs of 93 in more detail. In order to derive a result of the type of 3, it would be convenient to have a smooth compactly supported w such that the multiplier φ = w d + w is dẑ non-negative. Although we do not have an argument, we believe that such a w does not exist. Instead, we will construct a family F of smooth functions w supported in ẑ, ] and a probability measure ρdw on F which is invariant under the symmetry transformation w ŵ defined through the change of variables ŵ + z = w z such that the convex combination of the multipliers φ ẑ φẑ ρdw, 94 F

24 93-3 C. Nobili and F. Otto J. Math. Phys. 58, 93 7 where φ w d dẑ + w is non-negative, supported in [ 4, 3 4 ] and non-trivial and thus satisfies 3 after normalization. Roughly speaking, the reason why this can be achieved is the following: For any non-trivial smooth, compactly supported w we have φ = w d dẑ + w is positive on average, φ dẑ = d w dẑ + dw + w dẑ. dẑ φ = w d4 w + + w is positive near the edge of the support of w incidentally this would not be dz 4 true for the positive second order operator d +. dẑ Before becoming much more specific, let us address the error term stemming from the rhs of 93 for our construction, that is, d 5 w F dẑ w dẑ ρdw. 95 The functions in our family F will be of the form w l,ẑ ẑ l 3 ẑ ẑ w, 96 l that is, translations and rescalings of a mask w. The mask w is some compactly supported smooth function that we fix now, say exp for ẑ, w ẑ C ẑ, 97 else and the normalization constant C chosen such that dw dẑ =, 98 provided l 8 dẑ and ẑ 3 8, 5, 99 8 then w l,ẑ is, as desired, supported in ẑ [ 4, 3 4 ]. If we choose the length-scale to be bounded away from zero, i.e., l C, then the error term 95 is clearly finite so that 3 follows from 93 via integration w. r. t. ρdw. It thus remains to construct a probability measure ρ in l and ẑ with 99 and such that 94 is non-negative and non-trivial. Note that w l,ẑ in 96 is scaled such that the corresponding multipliers satisfy φ l,ẑ ẑ = l w d 4 dẑ w d ẑ ẑ 4 lw dẑ w + l 3 w, l d and w is normalized in 98 in such a way that w 4 w dẑ 4 dẑ =. Hence for all ẑ, we have convergence as l, φ l,ẑ ẑ δẑ ẑ when tested against smooth functions of ẑ.