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Chaotic behavior and controlling chaos in a fast-slow plankton-fish model

  • The interaction of different time scales in predator-prey models has become a common research topic. In the present article, we concentrated on the dynamics of interactions at two time scales in a plankton-fish system. To investigate the effects of the two time scales on plankton-fish dynamics, we constructed a new parameter with a corrected type that differs from the traditional slow parameter. In addition, zooplankton's refuge from the predator and phytoplankton mortality due to competition are incorporated into the model. Positivity and boundedness of solutions were proved. We then discussed feasibility and stability conditions of the equilibrium. We used a variety of means to support the existence of chaos in the system. Hopf bifurcation conditions were also obtained. Chaos control in the plankton-fish model is one of the main motivations for this study. In the slow-variable parameter case, we explored the control mechanism of gestation delay on chaotic systems, which are calmed by different periodic solutions. Moreover, under seasonal mechanisms, external driving forces can stabilize the system from chaos to periodic oscillations. Meanwhile, the sliding mode control (SMC) approach quickly calms chaotic oscillations and stabilizes it to an internal equilibrium state. The necessary numerical simulation experiments support the theoretical results.

    Citation: Guilin Tang, Ning Li. Chaotic behavior and controlling chaos in a fast-slow plankton-fish model[J]. AIMS Mathematics, 2024, 9(6): 14376-14404. doi: 10.3934/math.2024699

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  • The interaction of different time scales in predator-prey models has become a common research topic. In the present article, we concentrated on the dynamics of interactions at two time scales in a plankton-fish system. To investigate the effects of the two time scales on plankton-fish dynamics, we constructed a new parameter with a corrected type that differs from the traditional slow parameter. In addition, zooplankton's refuge from the predator and phytoplankton mortality due to competition are incorporated into the model. Positivity and boundedness of solutions were proved. We then discussed feasibility and stability conditions of the equilibrium. We used a variety of means to support the existence of chaos in the system. Hopf bifurcation conditions were also obtained. Chaos control in the plankton-fish model is one of the main motivations for this study. In the slow-variable parameter case, we explored the control mechanism of gestation delay on chaotic systems, which are calmed by different periodic solutions. Moreover, under seasonal mechanisms, external driving forces can stabilize the system from chaos to periodic oscillations. Meanwhile, the sliding mode control (SMC) approach quickly calms chaotic oscillations and stabilizes it to an internal equilibrium state. The necessary numerical simulation experiments support the theoretical results.



    The Rayleigh-Bénard convection model describes a buoyancy driven flow of a fluid heated from below and cooled from above. In a finite box with height h, the impermeable horizontal bottom an top plates are held at temperature Tb and Tt respectively, with Tt<Tb. Temperature differences trigger density differences within the fluid layers, which, in turn, generate convective motions. As the intensity of the buoyancy forces grows, the dynamics undergo a series of bifurcations: convection rolls are destabilized by plume-shaped structures forming at the boundaries and the motion becomes unpredictable, chaotic, turbulent. This model is a paradigm of nonlinear dynamics and pattern formation and has a myriad of applications in engineering of heat transfer [1]. The motion within the finite box is governed by an advection-diffusion equation for the temperature field T coupled with the Navier-Stokes equations for the velocity field u. Unless otherwise stated, we will focus on three-dimensional dynamics and denote by x=(x,z)=(x,y,z) a vector in R3 and by ez=(0,0,1)t the vertical normal vector. In dimensionless variables*, the velocity field u=(u,uz)=(ux,uy,uz):R3×(0,)R3 and the scalar temperature T:R3×(0,)R evolve in the domain Ω=[0,Lx]×[0,Ly]×[0,1] according to

    *see the Boussinesq equations in [3,Section 2.1] and their non-dimensionalization choosing length scales determined by the vertical gap height h, time scales defined by the thermal diffusion κ and temperature scales determined by the temperature gap TbTt.

    tT+uTΔT=0u=01Pr(tu+uu)+pΔu=RaTez. (1.1)

    This is also called Boussinesq system as, in its derivation, density variations generating the vertical buoyancy force appearing in the right-hand side of the velocity equation, are neglected elsewhere in the velocity and temperature equation (Boussinesq approximation). Two dimensionless parameters appear in the system after the non-dimensionalization:

    Ra:=αg(TbTt)h3κν

    is the (control parameter) Rayleigh number and

    Pr:=νκ

    is the (material parameter) Prandtl number. Here α is the thermal expansion coefficient, g is the acceleration of gravity, ν is the kinematic viscosity and κ is the thermal diffusivity. The pressure field p(x,t) appears as a Lagrange multiplier enforcing the incompressibility condition (u=0). In the non-dimensional varibles, the temperature boundary conditions are

    T=1 at z=0 and T=0 at z=1. (1.2)

    For technical convenience we will suppose all functions to be periodic in the horizontal variables x with period L=(Lx,Ly). We refer to [2] for details about the derivation of the system.

    One of the most important open problems in fluid dynamics is to determine the rate at which heat is transferred in turbulent convection. This theoretical problem finds uncountable many applications in meteorology, oceanography and industry [1]. The natural nondimensional quantity for gauging heat transfer in the upward direction is the Nusselt number Nu defined as the ratio between the convective to the conductive heat flux

    Nu=10(uTT)ezdz10Tezdz=10(uTT)ezdz, (1.3)

    where

    f=lim supt1tt01LxLyLx0Ly0f(,z)dxdydt.

    By similarity law, the Nusselt number Nu must be a function of the non-dimensional parameters appearing in the system, and thus obeys a functional relation of the type

    Nu=f(Ra,Pr).

    Theoretical predictions (Malkus [4], Kraichnan [5], Spiegel [6], Siggia [7]) and experiments (see [8] and references therein) suggest a power-law scaling of the form

    NuRaαPrβ for α,βR,

    where α and β depend on the characteristic of the flow. In Section 2 we will illustrate two physically motivated heuristic arguments predicting different scaling behaviors. The challenge for physicists, mathematicians, and engineers is to identify the value of the powers α and β, as Ra and Pr varies. As described by the stability theory, there exists a critical Rayleigh number, Rac, under which the pure conduction profile is stable. As Rayleigh number grows over this critical value, convection rolls appears and are eventually destabilized by plumes-like structures arsing from the boundary layers. At very high Rayleigh numbers the dynamics become turbulent. These three phases can be appreciated in Figure 1. As heat transport is enhanced in turbulent regimes, in the whole manuscript we will assume Ra1. The Prandtl number, instead will be mostly assumed to be large, because of limitation in our analysis. In fact, the smaller the Prandtl number, the stronger is the inertial term in the Navier-Stokes equations.

    Figure 1.  Three vertical slices in the xz-plane of three-dimensional, free-slip, simulation in [9] at Ra=3850, Ra=38501, and Ra=385014 and Pr=1. Each panel contains the instantaneous temperature field in the background with colors from blue (cold) to red (hot). The arrows project the instantaneous velocity onto the 2d-plane. The pictures describe three states as Ra increases: convection rolls, destabilization caused by plumes arising from the boundary layers and turbulent convection.

    In this paper we will review some of the major contributions concerning bounds for the Nusselt number in the Rayleigh-Bénard convection problem (1.1)–(1.2) in the last thirty years, distinguishing between no-slip, free-slip and Navier-slip type boundary conditions for the velocity field. Most numerical simulations and rigorous results assume no-slip boundary conditions in the horizontal plates and periodic sidewalls. Whether or not the no-slip conditions are the more suitable for this problem, remains a matter of debate. This is a general and very important question that has been posed by the founders of fluid dynamics and is still subject of ongoing research. Regarding the issue of detecting suitable boundary conditions for the Navier-Stokes equations, Serrin [10] writes "The conditions to be satisfied at a solid boundary are more difficult to state, and subject to some controversy. Stokes argued that the fluid must adhere to the solid, since the contrary assumption implies an infinitely greater resistance to the sliding of one portion of fluid past another than to the sliding of fluid over a solid. Another and perhaps stronger argument in favor of adherence, at least for the case of liquids at ordinary conditions, is found in experiments with tube viscometers where the fourth power of the diameter law is conclusively verified. Although these facts are quite convincing when moderate pressures and low surface stresses are involved, they do not apply in all cases; indeed in high altitude aerodynamics an adherence condition is no longer true". Many works took inspiration from Serrin's observation. Here we mention [11], where the authors addressed theoretical questions related to the choice of boundary conditions.

    In the last decade, free-slip boundary conditions for the Rayleigh-Bénard convection problem have been considered and rigorous upper bounds on the Nusselt number indicate the possibility of enhanced heat transport compared to the no-slip case. But which one of those boundary conditions better represent the physical picture? On the one hand, as remarked in [12], (especially) in "turbulent regimes" the boundary conditions might not be perfectly no-slip. On the other hand, the mathematically interesting free-slip boundary conditions are non-physical because of the absence of vorticity production at the boundary, as we will see later in Section 4. H. Navier in 1824 [13] suggested the boundary conditions

    2ν[Duˉn]ˉt+αuˉt=0 (1.4)

    where D is the rate of strain tensor, ˉt and ˉn are the tangential and normal vector to the boundary and α>0 is the friction coefficient. These conditions go by the name of Navier-slip boundary conditions and can be interpreted as an interpolation between no-slip and free-slip boundary conditions. Since, as observed in [14], very few surfaces are truly no-slip or free-slip, the Navier-slip boundary conditions appear reasonable and have been studied in a multitude of applied and theoretical problems, among which we want to mention the works on vanishing viscosity limits for the two-dimensional Navier-Stokes equations [15,16].

    In this manuscript we will focus on

    1) No-slip boundary conditions:

    uz=0,u=0 at z={0,1}. (1.5)

    2) Free-slip boundary conditions:

    uz=0zu=0 at z={0,1}. (1.6)

    3) Navier-slip boundary conditions:

    uz=0 at z={0,1},zu=1Lsu at z=1 and zu=1Lsu at z=0, (1.7)

    where Ls is the slip length.

    For these three types of boundary conditions we will critically review results concerning rigorous bounds on the Nusselt number and highlight differences in the analyses. As we will see later, while the model equipped with no-slip boundary conditions for u is most commonly investigated, the other two types of boundary conditions and their role have been less explored. The goal of this review is to show how the change of boundary conditions affects heat transport properties of the fluid.

    The review is organized as follows: In Section 2 we will go through two significant physical arguments predicting different scaling behavior for the Nusselt number. Moreover, starting from definition (1.3) and using a-priori estimates, we will obtain other identities involving the Nusselt number. Depending from the strategy adopted to produce bounds on the Nusselt number (introduced in the third subsection), each identity will later be used singularly or combined. Section 3 is divided in two parts: in Subsection 3.1 we will review some important contributions for the infinite Prandtl number system. In particular we will introduce the background field method and discuss its success and limitation. Then we will present an alternative argument proposed by Constantin and Doering in 1999 and show how this led Otto and Seis in 2011 to produce the optimal (so far) upper bound on the Nusselt number for no-slip Rayleigh-Bénard convection at infinite Prandtl number. In Subsection 3.2 we will go through three arguments to derive bounds for the Nusselt number in the finite Prandtl number case. Rigorous results for the system complemented with the other types of boundary conditions considered in this paper, namely free-slip and Navier-slip boundary conditions, are reviewed in Section 4. In all sections we will examine results in a (partial) chronological order and try to guide the reader through the reasoning behind development of new strategies and, eventually, optimal results.

    In the fifties Malkus [4], considering fluids with very high viscosity, performed experiments in which he noticed sharp transitions in the slope of the NuRa relation and suggested the scaling

    NuRa13 (2.1)

    for very high Rayleigh numbers by a marginal stability argument. This is based on the assumption that the heat flux is limited by transport across the (emergent) thermal boundary layer. We illustrate Malkus' argument here. Since the main temperature drop happens near the boundary, we can assume that in a bottom boundary layer of thickness δ (to be determined) the temperature drops from 1 to its average 12. Thanks to the average , we may extract the Nusselt number from the boundary layer, where by the no-slip boundary condition we have (uTT)ezzT so that Nu1δ. We think of the boundary layer as a pure conduction state in the interval 0zδ. Marginal stability refers to the assumption that this state is borderline stable, meaning that its Rayleigh number is critical, which in view of the definition of the Rayleigh number, means

    Rac=gα(TbTt)(δh)3νκ,

    from which, because Rac1, we infer δRa13. Inserting this in the scaling of Nusselt number above one finds

    NuRa13.

    The same conclusion can be achieved by rescaling the equations according to

    x=Ra13˜x,t=Ra23˜t,u=Ra13˜u,p=Ra23˜pandthusNu=Ra13~Nu. (2.2)

    In this way, neglecting the inertial term in the Navier-Stokes equations, we end up with the parameter-free system

    ˜tT+˜u˜T=˜ΔT˜Δ˜u+˜˜p=Te˜z˜˜u=0.

    Since for the latter system, it is natural to expect that the heat flux is universal, i.e., ~Nu1, we also obtain NuRa13. Most recently Malkus's theory was refereed to as classical theory [1].

    Kraichnan in [5] and Spiegel [6], instead, proposed the scaling

    NuPr12Ra12 (2.3)

    based on the assumption that the heat flux is limited by the transport of the fluid across the bulk. Recalling that, by definition, the convective and conductive heat flux are qconvρvcΔT and qcondρcκΔTh respectively and assuming

    vgαΔTh free fall velocity,

    then, the Nusselt number is

    NuqconvqcondρgαΔThcΔTρcκΔTh=(gαΔTh3κν)12(νκ)12=Ra12Nu12.

    We can obtain the same scaling by neglecting the diffusivity and the viscosity term in the model

    {tT+uT=01Pr(tu+(u)u)+p=RaTezu=0,

    and rescaling according to

    t=1(PrRa)12˜t,u=(PrRa)12˜u,p=Ra˜p and thus Nu=(PrRa)12~Nu.

    Then, imitating the previous argument, the Nusselt number for the parameter-free system

    {˜tT+˜uT=01Pr(˜t˜u+(˜u)˜u)+˜p=RaTez˜u=0

    is

    NuPr12Ra12.

    The Kreichnan-Spiegel theory was recently referred to as ultimate theory [1].

    In [6], Spiegel writes "What is suggested by these arguments is that at sufficiently high Ra, turbulent breakdown of the thermal boundary layer occurs and causes a transition from (2.1) to (2.3). No such transition has been detected experimentally, but this is presumably explained by the limitation of the experiments to what in stellar terms are modest Rayleigh numbers. The need to confirm (or deny) (2.3), which is intimately connected with basic ideas of stellar structure theory, poses a great challenge to the experimentalists.". Malkus's scaling NuRa13 has been been confirmed by experiments at (relatively) high Prandtl numbers (cf. [8] for a list of experimental results) while (up to today) it remains no indication of the Spiegel-Kraichnan scaling NuPr12Ra12.

    The Nusselt number defined in (1.3) admits other useful representations. As we will see later in the preliminaries, their employment depends on the method chosen to bound the Nusselt number. First, let us notice that if the initial temperature T(x,0)=T0(x) is chosen such that 0T0(x)1, then, by the standard maximum principle for parabolic equations, we have

    0T(x,t)1 for all t0. (2.4)

    The first representation can be obtained by taking the long-time and horizontal average of the temperature equation and using 12tT2=0 as a consequence of the maximum principle, we have

    zuzTzT=0.

    which implies that for all z(0,1)

    uzTzT=zT|z=0. (2.5)

    Recalling the definition of Nusselt number in (1.3), this identity yields

    Nu=uzTzT for all z(0,1). (2.6)

    Therefore the Nusselt number is independent of z, and it is the same for each horizontal layer of fluid. Another representation is derived by testing the temperature equation with T and integrating by parts. In fact, using the incompressibility together with the temperature boundary conditions and (2.5) we obtain

    Nu=10|T|2dz. (2.7)

    Instead, formally testing the Navier-Stokes equations with u and integrating by parts we have

    121Prddtu2L2=u2L2+RaΩTuz. (2.8)

    Note that this energy balance (formally) holds for all boundary conditions considered in this article (i.e., (1.5)–(1.7)) as it uses only that uz=0 at z={0,1} and the incompressibility condition.

    If the energy of the velocity remains bounded in time, averaging in (2.8) we obtain

    10|u|2dz=Ra10Tuzdz=Ra(Nu1). (2.9)

    This identity produces yet another useful representation of the Nusselt number

    Nu=1+1Ra10|u|2dz.

    Clearly this computation is rigorous in two dimensions, but only formal in three dimensions. In this last case, defining Leray-solutions we will obtain an energy inequality in (2.9).

    In this section we briefly introduce the reader to the most common techniques used to bound the Nusselt number. Some of these methods will be carefully explained and used in the next sections.

    The first rigorous result was proven by Howards [17] by transforming the problem of finding upper bounds on Nu into a variational problem. Howard found

    NuRa12

    under the hypothesis of "statistical stationarity", meaning that horizontal averages are time-independent. Later Busse extended Howard's result to solutions with multiple boundary-layer structure [18] obtaining the same bound with an improved constant prefactor.

    this holds for stationary flows.

    The background field method, proposed (in this context) by Doering and Constantin in [19], consists in decomposing the temperature field T as

    T(x,z,t)=τ(z)+θ(x,z,t), (2.10)

    where τ is a steady background profile satisfying the driven boundary conditions

    τ=1 at z=0 and τ=0 at z=1

    and θ represents the temperature fluctuations satisfying

    θ=0 at z{0,1}. (2.11)

    The fluctuations evolve according to

    tθ+uθ=τuz+Δθ+τ" (2.12)

    and the using the representation (2.7), the Nusselt number can be rewritten as

    Nu=10|τ|2dzQτ{θ,u}, (2.13)

    where, for fixed τ, Qτ is a quadratic functional in θ and u. In this way we can transform the problem of finding the optimal upper bound on the Nusselt number into a variational problem: find the background profile τ for which Qτ{θ,u}0 for all θ and minimizes the integral 10|τ|2dz. The background field method will be carefully described in Section 3.1.1 and we refer to [20] for a recent review on it.

    A generalization of the background field method was proposed in [21] to improve the available bounds. The method is based on the following observation: if V(t)=V{θ(,t),u(,t)} is a uniformly in time bounded and differentiable function with limtt0ddsV(s)ds=0 then, in order to prove

    Nu=10|T|2dzB, (2.14)

    it suffices to show

    ddtV+10|T|2dzHB0,

    where

    f(,z)H=1ALx0Ly0f(,z)dxdy.

    In fact, time-averaging this inequality we recover (2.14). It is easy to see that the auxiliary functional method and the background field method coincide when the auxiliary functional V is quadratic in u and θ [22]. The power of this method is to allow the functional V to be more general (than the quadratic form Qτ), to the expenses of an increasing analytical complexity which eventually requires computer-assisted investigation [23,24]. The auxiliary functional method has proven to be successful to produce sharp bounds for (system) of ordinary differential equations [25], but the application of this method for producing optimal bounds on the Nusselt number in the Rayleigh-Bénard convection can only be speculated.

    Another method to produce upper bounds based on the vertical localization of the Nusselt number was first used in [26] and then proposed by [27] as an alternative method to the background field method for a variety of fluid dynamics problems. This method is based on the observation that (2.6) can be averaged over any horizontal layer of fluid; in particular, averaging near the bottom boundary layer 0zδ we are led to the definition

    Nu=1δδ0uzTzTdz.

    In particular, because of the boundary conditions on the temperature and the fact that T0 for every x,t, by the maximum principle, we have

    Notice here that, if |t0uzTdxds|<g(z)L1, then the switch in order of the averages and the integral in z is justified by the Lebesgue dominated convergence theorem. The condition will be generally satisfied thanks to the a-priori (energy) estimates for u and T.

    Nu1δδ0uzTdz+1δ. (2.15)

    The goal is now to exploit the equations of motion to derive regularity bounds on uz and T in the strip [0,Lx]×[0,Ly]×[0,δ]. As we will see later in Section 3.2, identity (2.9) will be particularly useful, when adopting this method. Compared to the background field method, this approach has the advantage to be free from the rigid constraint imposed by the variational method. Nevertheless in Section 3.1.1 we explain that these two methods coincide for a particular choice of background profile.

    For some fluids the Prandtl number is so high (see [7], table 1) that the effect of inertia in the Navier-Stokes equations is almost negligible. This motivates the study of the infinite-Prandtl number model, obtained by setting Pr= in the momentum equation in (1.1)

    Δu+p=RaTezu=0. (3.1)
    Summary of best known results
    Boundary conditionsNo-slipNavier-SlipFree-slip
    2d, 3d
    Pr=
    Nu(lnlnRa)Ra13
    (Otto & Seis '15)
    NuRa512
    (Whitehead/Drivas, Nguyen, Nobili '21)
    NuRa512
    (Whitehead & Doering '11)
    2d, Pr< Nu{(lnRaRa)13PrRa13(lnRaRaPr)12PrRa13
    NuRa12
    (Choffrut, Nobili, Otto '16, Doering & Constantin'96)
    NuRa512+Ls2Ra12
    for Ls2Pr2Ra32
    (Drivas, Nguyen, Nobili '21)
    NuRa512
    (Whitehead & Doering '11)
    3d, Pr< same as 2D for Leray-weak solutions not known NuRa512+Gr2Ra14
    for Gr sufficiently small
    (Wang & Whitehead '12)

     | Show Table
    DownLoad: CSV

    Using the incompressibility condition it is easy to see that (3.1) can be rewritten as the fourth-order elliptic equation

    Δ2uz=RaΔHθuz=zuz=0, (3.2)

    where ΔH=2x+2y and the pressure is eliminated. Notice that by the incompressibility condition we infer that zuz=Hu=0 at z={0,1} since u=0 at z={0,1}. From relation (3.2), it is now clear that the temperature is "instantaneously" slaved to the velocity field. Xiaoming Wang in [28,29] reasonably asked whether this reduction is justified and whether the statistical properties of the flow in the Pr= model and in the finite (but large) Pr number model can be close in some sense. The author proved that, indeed, the global attractor of the Boussinesq system at large Prandtl number converges to that of the infinite-Prandtl number model, giving some indication on the possible closedness of the respective statistical properties.

    Introduction of the background field method: The first application of the background field method to find upper bounds on the Nusselt number for the Rayleigh-Bénard convection problem appears in the Doering and Constantin 1996 paper [19]. We reproduce their arguments, adapting them to the infinite Prandtl number case. We start by splitting T according to (2.10). Testing equation (2.12) with θ and integrating by parts in Ω we obtain

    12ddtθ2L2=Ωτuzθθ2L2+Ωτ"θ, (3.3)

    where we used incompressibility together with the condition uz=0 at z={0,1} to eliminate the term 12Ωuθ2; then, averaging we get

    10τzθdz=10|θ|2dz10τθuzdz.

    The combination of this identity with

    Nu(2.7)=10|T|2dz=10|τ|2dz+210τzθdz+10|θ|2dz, (3.4)

    yields

    Nu=10|τ|2dz10|θ|2dz210τuzθdz. (3.5)

    Thanks to this new representation of the Nusselt number we can formulate the following

    Criterion 3.1. If τ:[0,1]R satisfies

    Qτ{θ}=10|θ|2dz+210τuzθdz0, (3.6)

    for all fluctuations θ(x,z) satisfying (2.11) and related to uz through (3.2), then

    Nu10|τ|2dz. (3.7)

    Condition (3.6) is often referred to as "stability constraint" or "spectral constraint" of the background profile. The strategy just described goes by the name of background field (or background flow) method. Already introduced in 1992 by Constantin and Doering to bound the energy dissipation in shear driven turbulence [30], this method has been extensively used for estimating the Nusselt number. In fact, an upper bound is immediately obtained by computing the Dirichlet energy of a stable (in the sense specified above) background profile. Thus, each profile satisfying (3.6), produces an upper bound for the Nusselt number.

    The first example of a marginally stable profile is (a smooth approximation of)

    τ(z)={1zδ0zδ0δz1, (3.8)

    plotted in Figure 2, for which one easily shows NuRa12.

    Figure 2.  Example of background profile τ=τ(z) marginally stable for δRa12.

    We use this example to explain the adjective "marginal" in the stability condition. In analogy to the Malkus' argument reported in Section 2, this refers to the fact that the boundary layer [0,δ] adjusts itself to be "borderline" stable. We are going to see that the thickness of δ determines the validity of (3.6) and eventually the bound on the Nusselt number. The bigger δ is, the smaller the upper bound on the Nusselt number. Obviously there will be artificially thin boundary layers that will ensure the stability condition, but for optimality of the upper bound, we want to detect the thickest stable boundary layer. Any boundary layers thicker than this, will be unstable.

    Now we illustrate the idea behind the proof of the marginal stability of profile (3.8), as this will set the basis for future computations. Applying the Cauchy-Schwarz inequality we have

    Qτ{θ}=10|θ|2dz2δδ0uzθdz10|θ|2dz2δδ0|uz|2dz12δ0|θ|2dz12.

    Thanks to the boundary conditions for uz and θ, we can use the Poincaré estimates

    δ0|uz|2dzδδ0|zuz|2dz, (3.9)
    δ0|θ|2dzδδ0|zθ|2dz (3.10)

    together with (2.7) and (2.9) to get

    Nu(1δRa12)0.

    Optimizing on the boundary layers thickness, we select

    δRa12,

    yielding

    Nu10|τ|2dz1δRa12.

    Therefore we proved

    Theorem 3.2. For solutions of (1.1)–(1.2) with no-slip boundary conditions (1.5) the following upper bound holds:

    NuRa12forPr=.

    We remark that, in order to obtain this bound we could equivalently use the localization principle in (2.15), thus avoiding going through the background field method. In fact by the Cauchy-Schwarz inequality and estimates (3.9) and (3.10), we obtain

    Nu1δδ0uzTdz+1δδδ0|zuz|2dz12δ0|zT|2dz12+1δδ(Nu1)12Ra12Nu12+1δδNuRa12+1δ.

    Balancing, the optimal δ satisfies

    δ21NuRa12,

    leading to

    NuRa12.

    Therefore the localization method is equivalent to the background field method with the choice (3.8). More discussions and interesting observations about the relation between the background field method (and auxiliary functional method) and the localization method may be found in [22].

    Blooming of the background field method: In [31], Doering and Constantin choose (a smooth approximation of) the profile

    τ(z)={1z2δ0zδ12δz1δ1z2δ1δz1, (3.11)

    plotted in Figure 3, and improve Theorem 3.2 by employing finer estimates for the vertical velocity uz. Here we describe their argument.

    v(x,z)=jZ2ˆv(z)e2πiL(xj).
    Figure 3.  Plot of the background profile τ=τ(z) used in [31].

    Thanks to periodicity in the horizontal directions, passing to Fourier-series offers a big advantage. In fact, the unknown functions can be written as Fourier series with z dependent coefficients§

    §Here we assume, for simplicity L=Lx=Ly.

    Then the equation for (3.2) decomposes in

    (d2dz2+k2)2^uz=Rak2ˆθ, (3.12)

    where k=2πL|j| and the boundary conditions for uz are

    ^uz(0)=0=^uz(1) and z^uz(0)=0=z^uz(1). (3.13)

    Decomposing Qτ mode by mode in the translation invariant horizontal directions, it is clear that Qτ will be non-negative, if for each horizontal wavenumber k

    ˆQτ{ˆθ}=10(|zˆθ|2+k2|ˆθ|2+τ(ˆuzˆθ+ˆuzˆθ))dz, (3.14)

    is non-negative. Here the symbol denotes the complex conjugate. Because of the choice of the background profile in (3.11) we have

    10τ(^uzˆθ+^uzˆθ)dz1δ(δ0|^uz(z)||ˆθ(z)|dz+11δ|^uz(z)||ˆθ(z)|dz).

    The crucial element leading to the new bound is contained in the way the authors estimate the mixed term. To illustrate the argument, let us focus on the lower boundary layer. By Poincaré's inequality applied to uz, θ (see (3.9) and (3.10)) and zuz (thanks to the fact that zuz=0 at z={0,1}) we have

    1δδ0|^uz(z)||ˆθ(z)|dz1δ(δ0|^uz(z)|2dz)12(δ0|ˆθ(z)|2dz)121δδ2(δ0|2z^uz(z)|2dz)12δ(δ0|zˆθ(z)|2dz)12δ52supz|2z^uz(z)|(10|zˆθ(z)|2dz)12. (3.15)

    Again, due to the boundary conditions uz=0 and zuz=0 at z={0,1}, the L norm of uz can be bounded by using the following interpolation estimate

    Lemma 3.3. Let v(z):[0,1]R be a smooth function satisfying v=0 and zv=0 at z={0,1}, then

    2zv2L(0,1)24zvL2(0,1)2zvL2(0,1). (3.16)

    Applying this lemma to v=uz and using Young's inequality we obtain

    2z^uz2L(0,1)k22zuz2L2(0,1)+1k24zuz2L2(0,1).

    In order to conclude the argument, we now reconnect to the equation. In fact, thanks to the instantaneous slaving in (3.12) one can easily deduce

    k22z^uz2L2(0,1)+1k24z^uz2L2(0,1)1CRa2k2ˆθ2L2(0,1),

    for some numerical constant C>0. In fact, squaring (3.12) and integrating by parts we have

    Ra2k4ˆθ2L2(0,1)=4z^uz+k4^uz2k22z^uz2L2(0,1)4z^uz2L2(0,1)+k8^uz2L2(0,1)+4k42z^uz2L2(0,1)+2k4104z^uz^uzdz4k610^uz2z^uzdz4k210|4z^uz2z^uz|dz=4z^uz2L2(0,1)+k8^uz2L2(0,1)+6k42z^uz2L2(0,1)+4k6z^uz2L2(0,1)4k210|4z^uz||2z^uz|dz4z^uz2L2(0,1)+6k42z^uz2L2(0,1)a4z^uz2L2(0,1)4k4a2z^uz2L2(0,1).

    Choosing a=710, we obtain the desired bound. Finally going back to (3.15) we have

    1δδ0|^uz(z)||ˆθ(z)|dzδ52supz|2z^uz(z)|(10|zˆθ(z)|2dz)12δ52CRakˆθL2(0,1)zˆθL2(0,1)δ522CRa(k2ˆθ2L2(0,1)+zˆθ2L2(0,1)).

    Then, from (3.14)

    ˆQτ{ˆθ}(1δ522CRa)(zˆθ2L2+k2ˆθL2(0,1)).

    The choice δRa25 in (3.7) yields

    Theorem 3.4. For solutions of (1.1)–(1.2) with no-slip boundary conditions (1.5) the following upper bound holds:

    NuRa25forPr=.

    From the previous argument we see that the choice of bounding the mixed term with the Lnorm of the second derivative of uz "maximizes" the power of δ. In fact this last method gives us a δ52 prefactor against the δ2 prefactor of the previous section. Moreover notice that in this proof, the instantaneous slaving of uz to θ given by (3.2) was crucial. It is now natural to ask whether the profile chosen in (3.11) is optimal. Before jumping to the next two seminal results, let us comment on the background profile chosen in (3.11): The temperature field is supposed to be laminar in very thin boundary layers and this justifies the linear profile, decreasing from z=0 to z=δ and from z=1δ to z=1. In the bulk instead the temperature is not expected to vary wildly.

    Although the profile in (3.11) turned out to be marginally stable for some choice of very small δ, it does not reflect the idea that the fluid layers must be "stably stratified". This means that the lighter, warmer parcels of fluids are expected to be on the top of the cold, denser parcels. This observation leads us to the next result.

    Almost crowning of the background field method: In [32], Doering, Otto and Westdickenberg (neé Reznikoff) show stability of the logarithmic background profile

    τ(z)={1zδ0zδ12+λ(δ)ln(z1z)δz1δ(1z)δ1δz1, (3.17)

    with

    λ(δ)=12ln((1δ)/δ). (3.18)

    In this case, the derivative of the background profile τ produces weighted terms which are subtler to control, as we are going to describe now. With the choice profileserrin plotted in Figure 4, the quadratic form becomes

    Qτ{θ}=10|θ|2dz+2λ(δ)1δδ(1z+11z)θuzdz1δδ0θuzdz1δ11δuzθdz. (3.19)
    Figure 4.  Plot of the background profile τ=τ(z) chosen in [32].

    Adding and subtracting (λz+λ1z)uzθ in the boundary layers we rewrite

    Qτ{θ}=Qτ{θ}T+Qτ{θ}B,

    where

    QτB{θ}=120|θ|2dz+2λ(δ)10uzθzdzδ0(1δ+λ(δ)z+λ(δ)1z)uzθdz. (3.20)

    and

    QτT{θ}=11/2|θ|2dz+2λ(δ)10uzθ1zdz11δ(1δ+λ(δ)z+λ(δ)1z)uzθdz.

    Smuggling in the weight z2 and applying Hölder and Cauchy-Schwarz inequalities, the last term in QτB can be estimated as follows:

    δ0(1δ+λ(δ)z+λ(δ)1z)uzθdz=δ0(1δ+λ(δ)z+λ(δ)1z)z2|θ|212z12|uz|212z32dz(sup0<z<12|θ|212z12)(10|uz|2z3dz)12(δ0z4[1δ+λz+λ1z]2dz)12.

    By the fundamental theorem of calculus

    sup0zδ|θ|2z10|zθ|2dz10|θ|2dz,

    and, by a direct computation

    δ0z4(1δ+λz+λ1z)2dzδ3.

    The crucial estimate in this paper is contained in the following

    Lemma 3.5. For θL2(Ω) and w satisfying (3.12) and (3.13), the bound

    10wθzdz1Ra10|w|2z3dz. (3.21)

    holds.

    The combination of this lemma applied to w=uz with the estimates above yields

    QτB{θ}[2λRaδ3]10|uz|2z3dz.

    Arguing in the same way for QτT and imposing δ3ln((1δ)/δ)=12Ra, one finds that δ1RalnRa is optimal, and, since Nu10(τ)2dz1δ, this proves the following

    Theorem 3.6. For solutions of (1.1)–(1.2) with no-slip boundary conditions (1.5) the following upper bound holds:

    NuRa13(lnRa)13forPr=. (3.22)

    This result improves Theorem 3.4 and was the first one capturing (with an upper bound) Malkus' scaling Ra13 up to a logarithmic correction only using the background field method. The choice of the logarithmic background reveled itself to be "optimal" in a sense that we will specify at the end of this section. Roughly speaking, the logarithmic profile respects the stable stratification since it (monotonically) grows in the bulk. And the slow growth in the bulk maintains stability.

    Improving the (logarithmic) failure: The logarithmic correction in (3.22) was later improved by Otto and Seis in [26] using the following idea: going back to (3.19) and focusing on (3.20), the authors proved that, for (λRa)131 the upper bound

    sup0z(λRa)13|2zuz|2Ra53λ13[10|θ|2dz+λ10θuzz] (3.23)

    holds. Estimate (3.23) is crucial for reducing the power of λ in the bound: in fact, applying it in the estimate

    |δ0(1δ+λz+λ1z)θuzdz|sup0zδ|θ|212z12sup0zδ|uz|212z2δ0(1δ+λz+λ1z)z52dzδ52(δ0|zθ|2dz)12sup0zδ|2zuz|212δ52(δ0|zθ|2dz)12Ra56λ16[10|θ|2dz+λ10θuzz]12,

    where we used the Poincaré-type inequality

    sup0zδ|uz|2z4sup0zδ|2zuz|2,

    the authors then conclude

    QτB{θ}1210|θ|2dz+2λ10uzθzdz|δ0(1δ+λz+λ1z)θuzdz|1210|θ|2dz+2λ10uzθzdzδ52(δ0|θ|2dz)12Ra56λ162[1410|θ|2dz+λ410θuzzdz]121210|θ|2dz+2λ10θuzzdz2δ52Ra56λ16{ε10|θ|2dz+14ε[1410|θ|2dz+λ410θuzzdz]}.

    Choosing ε=14 and using 10θuzzdz0 by (3.21), we have

    QτB{θ}1210|θ|2dz+2λ10θuzzdzδ52Ra56λ16{1210|θ|2dz+2λ10θuzzdz}=(1δ52Ra56λ16){1210|θ|2dz+2λ10θuzzdz}.

    The optimal δ is

    δRa13(lnRa)115,

    and, as a result, we showed

    Theorem 3.7. For solutions of (1.1)–(1.2) with no-slip boundary conditions (1.5) the following upper bound holds:

    NuRa13(lnRa)115forPr=.

    The new estimate for the velocity in (3.23) improves the logarithmic correction substantially, but the bound still fails to reproduce the classical scaling Ra13. It is now natural to ask

    ● can we find a "better" profile which captures the Malkus' scaling without corrections? or

    ● is it possible to refine the regularity estimates to further reduce the logarithmic correction?

    We will (partially) answer these questions in the next sections.

    Limitations of the background field method: As we have seen in the previous section, the background field method transforms the problem of finding bounds on the Nusselt number into a variational problem. If we would find the minimizer of this problem, we would get the optimal upper bound that this method is able to produce. In particular, the shape of the optimal τ would give us precise information about the (statistical) behavior of the temperature.

    Let us rescale the infinite Prandtl number system according to the rescaling (2.2) suggested by Malkus' marginal stability argument, and set H=Ra13 to obtain

    tT+uT=ΔT0<z<HΔu+p=Tez0<z<Hu=00<z<Hu=0z{0,H}T=1z=0T=0z=H.

    Let us define the Nusselt number associated to the background flow method

    Nubf=infτ:[0,H]R:τ(0)=1,τ(H)=0{H0|τ(z)|2dzQτ{θ}} (3.24)

    with Qτ being

    Qτ{θ}=H0|θ|2dz+2H0τuzθdz, (3.25)

    where θ is instantaneously "slaved" to uz through

    Δ2uz=ΔHθ0<z<H,uz=zuz=0z{0,H}.

    Clearly,

    NuNubf,

    and the challenge is to understand how far the Nusselt number Nu is from the number Nubf produced by the background field method.

    In [33] the authors prove

    Theorem 3.8. For solutions of (1.1)–(1.2) with no-slip boundary conditions (1.5), Pr= and Ra1 the following upper bound holds:

    Nubf(lnRa)115Ra13. (3.26)

    This ansatz-free lower bound is proven by extracting local information on τ from the non-local stability condition (3.25), which written in Fourier-series reads:

    2H0τ^uz(d2dz2+|k|2)2^uzdz+H0|k|2|ddz(d2dz2+|k|2)2^uz|2dz+H0|(d2dz2+|k|2)2^uz|2dz0,

    with k2πLZd1{0} and

    ^uz=ddz^uz=(d2dz2+|k|2)2^uz=0 for z{0,H}. (3.27)

    Notice that in the quadratic form only the modulus of k appears, so that the stability condition is independent of the dimension. The authors initially show the result in a simplified setting. First, considering profiles τ that are stable even under perturbation that have horizontal wave-length much larger than H, letting the side-length L, implying kR. Second, reducing the stability condition to the second term in (3.25), which is the leading order term in the stability condition. Under these assumptions the authors characterize all stable profiles:

    Proposition 3.9. If dτdz satisfies

    H0τ^uz(d2dz2+|k|2)2^uzdz0 (3.28)

    for all kR and all ^uz satisfying (3.27), then

    τ0

    and

    11eτdz1lnHH1τdz.

    This means that, if τ satisfies the reduced stability condition (3.28), then it must be increasing and "logarithmically" growing in the bulk. We remark that, in [34] it is proved that the result above still holds true if the lateral size L is of order H. Extending this result to the full stability condition requires some careful analysis: by subtle averaging of the stability condition, the authors in [33] construct a non-negative convolution kernel with the help of which they can express the positivity on average approximately in the bulk. Also the logarithmic growth can be recovered at least approximately in the bulk. To finally achieve the lower bound, further estimates connecting the bulk with the boundary layers are provided. This result proves the optimality of the logarithmic profile.

    Interestingly, by a mixture of analytical and numerical analysis, Ierely, Kerswell and Plasting in 2006 [35] anticipated (although slightly underestimating) the result in [33], showing NuRa0.33173(lnRa)0.0325.

    In 1999 Constantin & Doering [36] caught the Malkus scaling Ra13 (up to a logarithm) without optimizing on the background profiles, but extracting finer information from the equation for the velocity. This paper marks the beginning of a new and powerful way of thinking about this problem. The authors choose τ to be (a smooth approximation of) the profile

    τ={1zδ0zδ0zδ, (3.29)

    in

    Nu=10|τ|2dz10|θ|2dz+210τuzθdz.

    An application of the maximum principle θL(Ω)1 to estimate the mixed term

    yields

    Nuδ2sup0z1|2zuz|dz+1δ, (3.30)

    where the bound |τ(z)|1δ was used.

    As we have seen in Section 3.1.1, the interpolation estimate (3.16) was the crucial ingredient for proving the bound NuRa25, but turned out to be sub-optimal. In this paper the authors proved the much finer estimate

    Proposition 3.10. Suppose w solves the problem

    Δ2w=ΔHfin[0,L]2×[0,1]w=zw=0atz={0,1}

    with horizontally periodic boundary conditions. Then, for any α(0,1) there exists a positive constant Cαsuch that the bound

    2zwL(Ω)CαfL(Ω)(1+log+fC0,α(Ω)))2, (3.31)

    holds, where

    fC0,α(Ω)=supxΩ|f(x,t)|+supxy|f(x,t)f(y,t)||xy|α.

    This maximal regularity estimate was obtained by decomposing the operator B=2z(Δ2)1ΔH into the sum of non translationally invariant operators, whose kernel is not explicit. Inserting the long-time average of (3.31) into (3.30) and using

    10|Δθ|2dzCRa2{1+10[|τ"(z)|2+z|τ(z)|2]dz}, (3.32)

    to estimate the C0,α norm in the logarithmic correction further, the authors obtain

    This estimate is simply obtained by testing (2.12) with Δθ, integrating by parts, and using the Cauchy-Schwartz and Youngs inequalities together with the interpolation inequality θ2L4(Ω)CθL(Ω)ΔθL2(Ω) and u2L2(Ω)CRa2, where the last inequality is derived by testing the Navier-Stokes equations, integrating by parts and using the maximum principle for the temperature.

    Nuδ2Ra(1+log(Ra))2+1δ

    and the choice δRa13(1+log+Ra)23 yields

    NuRa13(1+log+Ra)23.

    We notice that the logarithmic correction in this bound is bigger than the one produced in [26] using the logarithmic profile in the background field method. Nevertheless, we are now going to discuss that, by a refinement of this argument, Otto and Seis in [26] proved the best upper bound (up to today). We first observe that, by the maximum principle and Poincaré's inequality (applied twice in z) the bound

    Nu1δ10θuzdz+1δδ2supz|2zuz|12+1δ

    holds and is the same as (3.30) (which was obtained by (2.15)). The key of the result in [26] is the estimate on the hessian of uz

    supz(0,1)|2uz|2Ra2(Ra13Nu+ln2(Ra13Nuln12(Ra13))) (3.33)

    obtained by a clever combination of the maximal regularity estimate

    supz(0,1)|2uz|212Raln(10|θ|2dzsupxθ2+e)supxθ212 (3.34)

    and the weighted estimates in (3.21). Inserting (3.33) into the bound for the (localized) Nusselt number, one gets

    Nuδ2Ra[Ra16Nu12+ln(Ra13Nuln12(Ra13))]+1δ.

    Using Young's inequality we are led to

    Nuδ2Raln(NuRa13ln12(Ra13)+1δ.

    Choosing

    δ(Raln(NuRa13ln12(Ra13)))13,

    then

    NuRa13ln13(NuRa13ln12(Ra13)),

    yielding||

    ||Here one uses the fact that XYln13(XY)Y implies XYYln13(Y).

    Theorem 3.11. For solutions of (1.1)–(1.2) with no-slip boundary conditions (1.5) the following bound holds:

    NuRa13ln13(lnRa)forPr=.

    Besides the improvement of the logarithmic correction, the interesting aspect of this result is that the background field method has not been used. That is, this bound relies only on the maximum principle for the temperature and on regularity estimates derived from Stokes equation. We notice that the authors argue that it is sufficient to establish (3.34) for (3.2) with the free-slip boundary conditions

    uz=2zuz=0 for z{0,1},

    for details see Proof of Lemma 4 in [26]. Passing to this boundary conditions is fundamental in order to extend the problem to the whole space by periodicity and therefore being able to use Fourier series techniques in both (horizontal and vertical) variables**. In Fourier variables, the hessian of uz can be written as

    **this time the symbol ˆ denotes the Fourier transform in x=(x,z). With k and ξ we denote the horizontal and vertical Fourier variable, respectively.

    ^2uz=[kξ][kξ]|k|2(|k|2+ξ2)2ˆθ

    and, from this representation, mimicking a Littlewood-Paley decomposition, one can analyze the small-intermediate and large length scales separately. This "separation of scales" is crucial for obtaining the desired bound, since the Lnorm is critical for Calderón-Zygmund type estimates.

    In this section we review some results concerning upper bounds on the Nusselt number at finite Prandtl number. Differently from the previous section, in this case the temperature T and the velocity uz are no longer instantaneously slaved and the equation for uz can no longer be written as a fourth order boundary value problem as in the case Pr= (recall (3.2)). As a consequence, we will see that the application of the background field method becomes somewhat unnatural and new approaches have to be considered. In order to present the results concerning the three-dimensional case, we will sometimes implicitly assume regularity of the Navier-Stokes equations. Nevertheless the arguments can be made rigorous by working with Leray weak solutions, i.e., assuming

    uL((0,);L2(Ω))uL2((0,);L2(Ω)).

    Assuming regularity for the solution of the Navier-Stokes equations, adding the balances (2.8) and (3.3) we have

    12ddt(θ2L2+1Pru2L2)=Ωτuzθθ2L2+Ωτ"θu2L2+RaΩTuz, (3.35)

    Since u, T (and therefore θ) stay bounded in L2 for all time,

    we may take the long-time limit obtaining

    10τzθdz=10|θ|2dz10(τa)θuzdzaRa10|u|2dz,

    for some a>0. Here we used that

    10TuzdzH=10θuzdzH,

    since uzH=0 by the incompressibility condition. Combining this identity with (3.4) we obtain the new representation of the Nusselt number

    Nu=10|τ|2dz10|θ|2dz210(τa)uzθdz2aRa10|u|2dz.

    Clearly this coincides with (3.5) when a=0. If we can select a background profile τ with τ(0)=1,τ(1)=0 such that the quadratic form

    Qτ{θ,u}=10|θ|2dz+210(τa)uzθdz+2aRa10|u|2dz (3.36)

    is positive definite for all θ satisfying (2.12), then an upper bound is given by

    Nu10|τ|2dz.

    In [19], Doering and Constantin choose the following background profile

    τ(z)={1(1δ1)z0zδzδz1δ(1δ1)(1z)1δz1. (3.37)

    Inserting this choice of τ (see Figure 5) in the quadratic form with a=1, and focusing on the mixed term we get

    10(τ1)θuzdz=1δδ0θuzdz+11δθuzdz.
    Figure 5.  Plot of the background profile τ=τ(z) used in [19].

    By Cauchy-Schwarz, Poincaré and Young's inequality we obtain

    1δδ0θuzdzδδ0|zθ|2dz12δ0|zuz|2dz1214δ0|zθ|2dz+δ2δ0|zuz|2dz,

    and the same argument applies to the top boundary layer. Then the quadratic form is bounded from below by

    Qτ{θ,u}(2Ra4δ2)10|u|2dz.

    So, clearly, if δ is chosen to be

    δRa12,

    then

    Nu10(τ)2dz1δRa12.

    In particular, this shows

    Theorem 3.12. For solutions of (1.1)–(1.2) with no-slip boundary conditions (1.5) it holds

    NuRa12uniformly inPr.

    We notice that in this argument neither the incompressibility, nor the full set of boundary conditions was exploited. Therefore this result holds for all boundary conditions such that uz=0. In order to go beyond this result, the system have to be exploited extensively.

    In [29], Wang shows that the global attractors of the infinite-Prandtl and the large-Prandtl number convection model remain close, by viewing the Boussinesq system as a small perturbation of the infinite Prandtl number model. Subsequently he derived bounds on the Nusselt number at finite (but large) Prandtl number starting from the Constantin and Doering '99 result in [36]. Following up on his previous works [28,29,37], the central idea in Wang's result in [38] is to view the Navier-Stokes equations as a perturbation around the stationary Stokes equation,

    Au:=pΔu=fu=0,

    where f=RaTez1Pr(tu+uu) and apply the bound in [36] (reviewed in Section 3.1.2).

    Using (3.5) and inverting the Stokes operator we have

    Nu=10|τ|2dz10|θ|2+2RaτA1(θez)zθ21PrτA1(tu)zθ21PrτA1((u)u)zθdz. (3.38)

    Following the argument in Section 3.1.2 (applied to the Stokes equation), choosing the background profile (3.29) then

    1012|θ|2+2RaτA1(θez)zθdz0

    provided

    δRa13(lnRa)23. (3.39)

    A new argument is needed to estimate the last two terms in (3.38). In fact, for the first term, using the profile (3.29) together with the Cauchy-Schwarz inequality, Poincaré's estimate (twice, in z) and Stokes regularity one finds

    |2101PrτA1(tu)zθdz|2δ1Pr10|2zA1(tu)z|2dz1210|zθ|2dzcδ41Pr210|tu|2dz12+1410|θ|2dz,

    for some constant c=c(Ω)>0. Similarly, using the maximum principle, the bound

    |21Pr10τA1((u)u)zθdz|2δ1Pr10|(u)u|2dz12cδ1Pr10|u|2dz3410|Au|2dz14,

    holds, where the spatial L2norm of the nonlinear term was estimated by the Poincaré estimate and Agmon inequality in three dimensions, i.e.,

    uuL2(Ω)uL(Ω)uL2(Ω)cu12L2(Ω)Δu12L2uL2(Ω).

    The combination of these estimates in (3.38) yields

    Nu10|τ|2dz+δ41Pr210|tu|2dz12+δ1Pr10|u|2dz3410|Au|2dz14, (3.40)

    where we used again the Poincaré inequality. Finally, inserting the choice of δ in (3.39) and the (a-priori) upper bounds

    10|u|2dzRa3210|Au|2dzRa2 for PrRa10|tu|2dzRa72 for PrRa,

    in (3.40), Wang obtains

    Theorem 3.13. For solutions of (1.1)–(1.2) with no-slip boundary conditions (1.5) the following upper bound holds:

    NuRa13(lnRa)23forPrRa.

    We remark that the crucial (a-priori) upper bounds used to derive the final estimate on Nusselt number, were obtained by energy and enstrophy inequalities. In particular a Gronwall-type argument is needed to ensure the existence of an absorbing ball such that

    limtuL2R

    under the large-Prandtl number condition PrRa.

    The result in [39] improves Wang's result by perturbing around the nonstationary Stokes equation, i.e., considering

    1Prtu+pΔu=RaTez1Pr(uu)u=0,

    and proving a maximal regularity estimate of the type

    2uzRaTez1Pr(u)u,

    where the norm is strong enough to control the nonlinear term

    (u)uu2L2(0,1)(2.9)NuRa

    and sufficiently weak so that

    RaTezRaTL(0,1)(2.4)Ra.

    The interpolation norm, which is able to grant this controls, is

    f=f(0,1)=supf=f0+f1{sup0<z<1|f0|+10|f1|z(1z)dz}.

    In fact, the L1-weighted norm appears natural to bound the nonlinearity:

    10|(u)u|zdz10|u|2z2dz1210|u|2dz1210|u|2dz,

    where we used the Cauchy-Schwarz inequality together with Hardy's inequality and the dissipation bound in (2.9). Since the interpolation norm is critical for maximal regularity estimates (since the L and the L1weighted norm are both critical with this respect), the authors proved the following

    Proposition 3.14. Suppose u,p and f satisfy

    1Prtu+pΔu=finz(0,1)u=0inz(0,1)u=0atz={0,1}u=0att=0,

    and f is horizontally banded, i.e.,

    ˆf(k,z,t)=0unless1<R|k|<4.

    Then

    (t2z)u+tuz+u+2zuz+pf, (3.41)

    holds.

    As we can see, the authors do not obtain the full maximal regularity estimate due to cancellations that occurs as an effect of using the bandedness assumption. Nevertheless the crucial estimate

    2zuzf

    is not affected and allows to conclude the argument. In fact, if only horizontal intermediate wavelengths could be taken into account, then, inserting the maximal regularity estimate in the bound

    Nu(2.15)δ22zuz+1δ,

    and choosing δ((NuPr+1)Ra)13, we would be able to conclude

    Nu{(Ra)13Pr(Ra)13(RaPr)12Pr(Ra)13.

    This estimate is unfortunately not achieved as the high and low wavelength will weight in the final estimate. Nevertheless, by the incompressibility condition and the a-priori energy inequality (2.9), the authors show that the region of intermediate wavelengths (where estimate (3.41) holds) is large and that the correction from the desired bound is relatively small. In conclusion, we have

    Theorem 3.15. For solutions of (1.1)–(1.2) with no-slip boundary conditions (1.5), the following upper bound holds:

    Nu{(RalnRa)13Pr(RalnRa)13(RaPrlnRa)12Pr(RalnRa)13.

    In the previous sections, the no-slip boundary conditions

    uz=0 and zuz=0 at z={0,1}

    have been crucial to obtain upper bounds of the type Nu(logRa)γRa13. In fact by the application of Poincaré's estimate in z we bounded

    ††Recall from Section 3.1 that this is an equivalent way of writing the no-slip boundary conditions using incompressibility.

    uLp(0,δ)δ22zuLp(0,δ).

    With free-slip and Navier-slip boundary conditions this estimate is no longer available and it is therefore more difficult to appeal to maximal regularity estimates, which turned out to be crucial for producing optimal bounds in the no-slip setting. In the next section we see how to circumvent this problem and get sharp quantitative estimates on the flow. We want to notice here that, in terms of regularity estimates, the free-slip boundary conditions present a big advantage compared to the no-slip boundary conditions. In fact the free-slip boundary conditions (1.6) are equivalent to

    uz=0 and 2zuz=0 at z={0,1}

    since 2zuz=Hzu=0 thanks to the incompressibility condition. Now one can easily see that the equations can be naturally extended in the vertical direction by periodicity. This observation was used in [26] to derive a maximal regularity estimate and in [40] to show the existence of a global attractor in each affine space where the velocity has fixed spatial average. One major advantage is that the extension by periodicity allows to use Fourier-series techniques also in the vertical direction. In the no-slip case, this extension was not possible, and, in order to derive maximal regularity bounds we were forced to decompose the operator and exploit Fourier-series techniques only in the horizontal variables.

    Due to limitations in the analysis at finite Prandtl number, in the next sections we will often consider the two dimensional Rayleigh-Bénard system (1.1). In this case we will denote with x=(x,z) a vector in Ω=[0,L]×[0,1] and with u=(ux,uz) the velocity field.

    In his PhD thesis in 2002 [41], Jesse Otero reported numerical indication of the scaling NuRa512 for the two-dimensional Rayleigh-Bénard convection problem with free slip boundary conditions. Later in 2011, Doering and Whitehead in [42] proved rigorously the following result

    Theorem 4.1. For solutions of (1.1)–(1.2) and (1.6) in two-dimensions the following bound holds:

    NuRa512uniformly inPr. (4.1)

    The argument takes advantage of information on the scalar vorticity ω=xuzzux satisfying

    1Pr(tω+uω)Δω=RaxT in Ω=[0,L]×[0,1]ω=0 at z={0,1}. (4.2)

    The enstrophy balance is

    12Prddtω2L2(Ω)=ω2L2(Ω)+RaΩωxθdxdz. (4.3)

    from which one can show [40] that

    ωL2(Ω)CRa

    and obtain

    0=ω2L2+Ra10ωxθdz.

    Combining this, with the long-time averages of (3.3) and (2.8), the authors obtain the representation

    Nu=11b(10|τ(z)|2dzb)11bQτ

    where

    Qτ=10(|θ|2+aRa32|ω|2+bRa|ω|2+2τuzθ+aRa12ωxθ)dz, (4.4)

    with b(0,1) and a>0. By standard arguments it is not difficult to show that

    ^Qτzˆθ2L2+[ak2Ra32+1Ra(b2a24)]ˆω2L21δR{δ0^uzˆθdz+11δ^uzˆθdz},

    where R denotes the real part of a complex number. Since the first two terms of the right-hand side are already non-negative (for some choice of a and b), the authors need to argue for the (potentially negative) mixed term. We start with an observation: since

    10zˆuzdz=ˆuz(1)ˆuz(0)=0,

    there exists a z0(0,1) (no matters where it lies in (0,1)!) such that

    zˆuz(z0)=0

    Then the "missing" derivative (for applying Poincaré's estimate) is partially recovered by the estimate

    |zˆuz(z)|2=zz0z(zˆuz(z))2dz=2zz0zˆuz(z)2zˆuz(z)dz1k[2c1zˆuz2+k22c12zˆuz2],

    where we used the Youngs inequality. Then

    |ˆuz(z)|=|z0zˆuz(z)dz|z(10|zˆuz|2dz)12zk[2c1zˆuz2L2+k22c12zˆuz2L2]12

    Combining this estimate with

    |\hat\theta(z)|\lesssim z^{\frac 12}\|\partial_z \hat\theta(z)\|_2\,,

    which holds thanks to the assumptions on \theta , we showed

    Lemma 4.2. For w\in H^2(0, 1) with w = 0 at z = \{0, 1\} and \zeta\in H^1(0, 1) with \zeta = 0 at z = \{0, 1\} , the bound

    \begin{equation} \frac{1}{\delta}\int_0^{\delta}w(z)\zeta^{\ast}(z)\, dz \leq\frac{2}{25k}\delta^3\left[2c_1\|\partial_z^2 w\|_{L^2}^2+\frac{k^2}{2c_1}\|\partial_zw\|_{L^2}^2\right]+\frac 12\|\partial_z \zeta\|_{L^2}^2\,, \end{equation} (4.5)

    holds for some positive constants c_1 .

    We observe now that by using the (2d) relation \omega = \partial_z u^x-\partial_x u^z and the divergence-free condition in Fourier variables, one obtains

    ik \hat\omega = \partial_z^2\hat u^z-k^2\hat u^z\,,

    and easily find the bound

    \begin{equation} \frac{8}{9}\|\partial_z^2\hat u^z\|_{L^2}^2+\frac 83k^2\|\partial_z u^z\|_{L^2}^2\leq k^2\|\hat{\omega}\|_{L^2}^2\,. \end{equation} (4.6)

    In particular, (4.5) and (4.6) together with the choice c_1 = \frac{\sqrt{3}}{6} yield

    \begin{equation} \quad|\hat u^z(z)|\lesssim k^{\frac 12}z\|\hat\omega(z)\|_{L^2}\,. \end{equation} (4.7)

    Combining this a-priori bound with Lemma 4.2 applied to w = \hat{ u^z} and \zeta = \hat{\theta} , we find

    \begin{equation} \frac{1}{\delta}\int_0^{\delta}\hat{ u^z}(z)\hat{\theta}^{\ast}(z)\, dz \leq c \delta^3k\|\hat{\omega}\|_{L^2}^2+\frac 12\|\partial_z \hat\theta\|_{L^2}^2\,. \end{equation} (4.8)

    Inserting (4.8) in the lower bound for \hat{ \mathcal{Q}^{\tau}} , we immediately see that \hat{ \mathcal{Q}^{\tau}}\geq 0 if

    \frac{ak^2}{ {\rm{Ra}}^{\frac 32}}+\frac{1}{ {\rm{Ra}}}\left(b-\frac{a^2}{4}\right)-c\delta^3k\geq 0\,.

    Minimizing on \delta (over k ), choosing \delta\sim {\rm{Ra}}^{-\frac{5}{12}} where k\sim {\rm{Ra}}^{\frac 14} is the minimal wavenumber, we obtain the desired bound {\rm{Nu}}\lesssim {\rm{Ra}}^{\frac{5}{12}} .

    Recent numerical simulations [43] indicate the clear asymptotic scaling

    {\rm{Nu}}\sim {\rm{Ra}}^{\frac 13}\qquad\qquad \mbox{for} \quad {\rm{Ra}}\rightarrow \infty

    for steady 2d convection rolls between free-slip boundary conditions. It is an interesting problem to bridge this gap and see whether the result in [42] is optimal or steady 2d convection rolls (slightly) inhibit heat transport.

    In three dimensions, the vorticity is no longer a scalar and the enstrophy balance (4.3) does not hold. Then the variational problem to study is: find the optimal \tau that minimizes the right-hand side of

    {\rm{Nu}}\leq\frac{1}{1-b}\left(\int_0^1|\tau'|^2\, dz-b\right)

    constrained to

    \mathcal{Q}^{\tau}\{\theta\} = \langle\int_0^1|\nabla\theta|^2+\frac{b}{ {\rm{Ra}}}|\nabla \bf{u}|^2+2\tau' u^z\theta\, dz \rangle\geq 0\,.

    This is the same as (4.4) with a = 0 . In [44] and [35] the authors apply the background field method to the free-slip problem and their numerical simulations indicate

    {\rm{Nu}}\sim 0.1262 {\rm{Ra}}^{\frac{5}{12}}\,

    for stably stratified profiles in the bulk, with \tau' = p > 0 . Choosing this family of background profiles, Doering and Whitehead in [12] rigorously proved

    Theorem 4.3. For solutions of (1.1)–(1.2) and (1.6) in three dimensions the following upper bound holds:

    \begin{equation} {\rm{Nu}}\lesssim {\rm{Ra}}^{\frac{5}{12}} \qquad \mathit{\mbox{with}}\quad {\rm{Pr}} = \infty\,. \end{equation} (4.9)

    Confirming the optimality of the family of stably stratified profiles and the numerical result in [44]. The argument is almost identical to the one for Theorem 4.1, with the crucial bound (4.7) replaced by

    \begin{equation} |\hat{ u^z}(z)|\lesssim k^{\frac 12}z\|\hat{\zeta}\|_{L^2}\,, \end{equation} (4.10)

    where \zeta imitates the role of vorticity in two dimensions and is referred to as "pseudo-vorticity" (or "pseudo-enstrophy") and it is defined by‡‡

    ‡‡In real space the pseudo vorticity is defined through the relation

    \Delta u^z = |\nabla_{\rm H}|\zeta
    \begin{equation} (\partial_z^2-k^2)\hat{ u^z} = k\hat{\zeta}\,. \end{equation} (4.11)

    Recalling the elliptic fourth-order boundary value problem written in Fourier variables (3.12), the pseudo-vorticity satisfies

    \begin{eqnarray} (\partial_z^2-k^2)\hat{\zeta}& = & {\rm{Ra}} k\hat{T}\qquad \mbox{in } \Omega \end{eqnarray} (4.12)
    \begin{eqnarray} \hat{\zeta}& = &0 \qquad\qquad \mbox{at } z\in \{0,1\}\,. \end{eqnarray} (4.13)

    An interesting point discussed in [12] concerns the relation between the slope p of the profile \tau and the optimal constant c in (4.9). Fist, the authors clarify that, a stratified profile is the bulk or a balance parameter b with a non-stratified profile, produce the same scaling. In fact the authors demonstrate how setting p = 0 and optimizing in b produce a prefactor c that is almost indistinguishable from the case in which b = 0 and p is optimized. In particular, setting b = 0 and p = \frac{3}{29} , the authors find {\rm{Nu}}\lesssim0.2982 {\rm{Ra}}^{\frac{5}{12}} . As the slope parameter is nearly the same to p\sim 0.103 computed by Plasting and Ierley in 2005, the analysis of Whitehead and Doering is rather sharp. Second, the author showed that the best constant ( 0.28764 ) is achieved for p = -\frac{1}{17} and b = \frac{5}{12} . This in particular tells that, if b > 0 , the (optimal) bound is produced with a non-stably stratified profile (since the slope parameter is negative).

    Combining the methods in [38] and [12], Wang and Whitehead in [14] succeeded in treating the three-dimensional finite Prandtl number case as a perturbation of the infinite Prandtl number problem problem, i.e., studying

    \begin{equation*} \begin{array}{rll} A \bf{u}: = \nabla p-\Delta \bf{u}& = & {\rm{Ra}} T \bf{e_z}-\frac{1}{ {\rm{Pr}}} \bf{f}\\ \nabla \cdot \bf{u}& = &0\,, \end{array} \end{equation*}

    where \bf{f} = \partial_t \bf{u}+ \bf{u}\cdot \nabla \bf{u} and {\rm{Pr}} is supposed to be (very) large, allowing them to apply the argument in [12]. In particular, the authors eliminate the pressure and study the relation

    \Delta^2 u^z = - {\rm{Ra}}\Delta_{\rm H}T+\frac{1}{\Pr}\left(- \Delta_{{{\rm{H}}}} f^z+\partial^2_{xz} f^x+\partial^2_{yz} f^y\right)\,,

    and benefit from rewriting the equation in Fourier space

    k(-k^2+\partial_z^2)\hat{\zeta} = {\rm{Ra}} k^2\hat{T}+\frac{1}{ {\rm{Pr}}}(k^2\hat{ f^z}+ik_1\partial_z\hat{ f^x}+ik_2\partial_z\hat{ f^y})\,,

    where \zeta is the pseudo-vorticity defined in (4.10). Modulo the O(\frac{1}{ {\rm{Pr}}}) inertial terms, this relationship is identical to (4.12).

    Also here, as in Section 3.2.2, the idea is to use the background field method, splitting the quadratic form

    \mathcal{Q}^{\tau}\{\theta\} = \langle\int_0^1(|\nabla \theta|^2+2\tau' u^z\theta)\, dz \rangle\,.

    into a part (containing O(1) contributions already present at infinite Prandtl number) which is positive definite and a rest, of order o(\frac{1}{\Pr}) (and potentially negative) that needs to be controlled with other methods. They proved

    Theorem 4.4. Suppose that \bf{u}, T solve (1.1)–(1.2) and (1.6). Then there exists a non-dimensional constant c_0 such that if the Grashof number {\rm{Gr}} is sufficiently small, i.e.,

    {\rm{Gr}} = \frac{ {\rm{Ra}}}{ {\rm{Pr}}}\leq c_0\,,

    then

    {\rm{Nu}}\lesssim {\rm{Ra}}^{\frac{5}{12}}+ {\rm{Gr}}^2 {\rm{Ra}}^{\frac 14}\,.

    Here c_0 is a small number determined by ensuring the simultaneous validity of

    \langle\int_0^1 |A \bf{u}|^2\, dz \rangle\lesssim {\rm{Ra}}^2
    \langle\int_0^1 |\nabla \partial_t \bf{u}|^2\, dz \rangle\lesssim {\rm{Ra}}^{\frac 72}\, \qquad \mbox{and}
    \langle\int_0^1 |\nabla(( \bf{u}\cdot \nabla) \bf{u})|^2\, dz \rangle\lesssim\left( {\rm{Ra}}^{\frac{17}{4}}+\frac{1}{ {\rm{Pr}}^{\frac 32}} {\rm{Ra}}^5+\frac{1}{ {\rm{Pr}}^{6}} {\rm{Ra}}^{\frac{19}{2}}\right)\,.

    These are the fundamental bounds necessary to control the O(\frac{1}{\Pr}) negative contributions in the quadratic form.

    As observed before, while free-slip conditions are considered to be "non-physical" due to the absence of vorticity production at the boundary (i.e., \omega = 0 at z = \{0, 1\} ), the Navier-slip conditions are adapt to describe surfaces where some slip occurs, but there is some stress on the fluid. In fact, in realty most of the surfaces are neither perfectly free slip not no slip [12]. If we think of this boundary conditions as interpolating between no-slip and free-slip (as the slip length \rm{\mathcal{L}_s} increases from 0 to \infty ), we expect that the bounds on Nusselt number will interpolate as well, between the {\rm{Ra}}^{\frac 12} and the {\rm{Ra}}^{\frac{5}{12}} bounds. In two dimensions, the vorticity satisfies

    \begin{equation} \begin{array}{rrll} \frac{1}{ {\rm{Pr}}}\left(\partial_t\omega+ \bf{u}\cdot\nabla \omega\right)-\Delta \omega & = & {\rm{Ra}} \partial_x T & \mbox{ in } \Omega = [0,L]\times[0,1]\\ -\omega & = &\frac{1}{ \rm{\mathcal{L}_s}} u^x & \mbox{ at } z = 0\\ \omega & = &\frac{1}{ \rm{\mathcal{L}_s}} u^x & \mbox{ at } z = 1\,, \end{array} \end{equation} (4.14)

    and the enstrophy balance is

    \begin{eqnarray} &&\frac{1}{2 {\rm{Pr}}}\frac{d}{dt}\|\omega\|_{L^2(\Omega)}^2+\frac{1}{2 \rm{\mathcal{L}_s} {\rm{Pr}}}\frac{d}{dt}\left(\| u^x\|_{L^2(\{z = 1\})}^2-\| u^x\|_{L^2(\{z = 0\})}^2\right)+\|\nabla \omega\|_{L^2(\Omega)}^2\\ && = \frac{1}{ \rm{\mathcal{L}_s}}\left(\int_0^{L}p\partial_{x} u^x|_{z = 1}\, dx+\int_0^{L}p\partial_{x} u^x|_{z = 0}\, dx\right)+ {\rm{Ra}}\int_{\Omega}\omega \partial_xT\, dx\, dz\,. \end{eqnarray} (4.15)

    Notice that as \rm{\mathcal{L}_s}\rightarrow \infty , this balance reduces to (4.3). Following an idea in [16] applied to the two-dimensional unforced Navier-Stokes equations the authors in [45] prove

    Lemma 4.5. Let \omega satisfy (4.14), \rm{\mathcal{L}_s} \geq 1 and p \in [1, \infty) . Then

    \|\omega(t)\|_{L^p}\lesssim\|\omega_0\|_{L^p}+\frac{1}{ \rm{\mathcal{L}_s}}\|u_0\|_{L^2}+ {\rm{Ra}}\,.

    Thanks to this lemma, combining the averaged version on (4.15) with the balance

    \langle\int_0^1|\nabla u|^2\, dz \rangle+\frac{1}{ \rm{\mathcal{L}_s}}\left( \langle( u^x)^2|_{z = 1} \rangle+ \langle( u^x)^2|_{z = 0} \rangle\right)- {\rm{Ra}}( {\rm{Nu}}-1) = 0\,,

    we can write a new representation of the Nusselt number:

    (1-b) {\rm{Nu}}+b = \int_0^1|\tau'|\,dz+M {\rm{Ra}}^2- \mathcal{Q}^{\tau}\{\theta, \bf{u}\}

    where

    \begin{eqnarray*} \mathcal{Q}^{\tau}\{\theta, \bf{u}\} &: = &M {\rm{Ra}}^2+ \langle\int_0^1 |\nabla \theta|^2\, dz \rangle+\frac{b}{ {\rm{Ra}}} \langle\int_0^1|\omega|^2\, dz \rangle+a \langle\int_0^1|\nabla \omega|^2\, dz \rangle\\ &+&2 \langle\int_0^1\tau' u^z\theta\, dz \rangle+\frac{b}{ {\rm{Ra}} \rm{\mathcal{L}_s}}\left( \langle ( \bf{u}')^2|_{z = 1} \rangle+ \langle ( \bf{u}')^2|_{z = 0} \rangle\right)\\ &-&\frac{a}{ \rm{\mathcal{L}_s}}\left( \langle p\partial_{x} u^x|_{z = 1} \rangle+ \langle p \partial_{x} u^x|_{z = 0} \rangle\right) -a {\rm{Ra}} \langle\int_0^1\omega\partial_{x}\theta\, dz \rangle, \end{eqnarray*}

    with parameters a > 0 , b\in(0, 1) and M > 0 to be chosen at the end. The combination of the arguments in [42] (to bound the mixed term \langle\int_0^1\tau' u^z\theta\, dz \rangle ), Sobolev trace inequalities together with the regularity estimate on the pressure contained in

    Proposition 4.6. The pressure in (1.1) satisfies

    \begin{equation*} \begin{array}{rrll} \Delta p & = &-\frac{1}{ {\rm{Pr}}}\nabla \bf{u}^t:\nabla u+ {\rm{Ra}}\partial_zT & \mathit{\mbox{in}}\;\Omega\,,\\ -\partial_yp & = &\frac{1}{ \rm{\mathcal{L}_s}}\partial_x u^x- {\rm{Ra}} & \mathit{\mbox{at}}\;z = 0\,,\\ \partial_yp & = &\frac{1}{ \rm{\mathcal{L}_s}}\partial_x u^x & \mathit{\mbox{at}}\;z = 1\,, \end{array} \end{equation*}

    and for any r\in (2, \infty) the estimate

    \|p\|_{H^1}\lesssim \frac{1}{ \rm{\mathcal{L}_s}}\|\partial_x\omega\|_{L^2}+ {\rm{Ra}}\|T\|_{L^2}+\frac{1}{ {\rm{Pr}}}\|\omega\|_{L^2}\|\omega\|_{L^r}

    holds.

    yield the lower bound for \mathcal{Q}^{\tau}

    \begin{eqnarray*} \mathcal{Q}^{\tau}\{\theta,u\} &\geq& \frac 12 \langle\int_0^1|\nabla \theta|^2\, dz \rangle +a\left(\frac 14-\frac{C}{ \rm{\mathcal{L}_s}^2}\right) \langle\int_0^1|\nabla\omega|^2\, dz \rangle\\ &+&\left(\frac{b}{ {\rm{Ra}}}-\frac{a^2 {\rm{Ra}}^2}{2}-\frac{aC^2\|u_0\|_{W^{1,r}}}{2 \rm{\mathcal{L}_s}^2 {\rm{Pr}}^2}-\frac{aC^2 {\rm{Ra}}^2}{2 \rm{\mathcal{L}_s}^2 {\rm{Pr}}^2}-c\delta^6a^{-1}\right) \langle\int_0^1|\omega|^2\, dz \rangle. \end{eqnarray*}

    Optimizing in a, b, M and choosing \delta\sim {\rm{Ra}}^{-\frac{5}{12}} , Drivas, Nguyen and Nobili in [45] achieve the following result:

    Theorem 4.7. Let u, T solve (1.1)–(1.2) with Navier-slip boundary conditions (1.7). Then

    \begin{equation} {\rm{Nu}}\lesssim {\rm{Ra}}^{\frac{5}{12}}+ \rm{\mathcal{L}_s}^{-2} {\rm{Ra}}^{\frac 12} \qquad \mathit{\mbox{for}}\quad \rm{\mathcal{L}_s}^2 {\rm{Pr}}^2\geq {\rm{Ra}}^{\frac 32}\,. \end{equation} (4.16)

    Moreover, for the case {\rm{Pr}} = \infty in dimension d\geq 2 , the authors show that the bound

    {\rm{Nu}}\lesssim {\rm{Ra}}^{\frac{5}{12}}\,

    follows from the same arguments. Using similar techniques to the one in [12], this upper bound was independently (and previously) noticed by Whitehead in unpublished notes.

    In this paper we reviewed a selection of contributions in the field of quantitative bounds on the Nusselt number for the classical Rayleigh-Bénard convection problem between fixed impermeable horizontal boundaries. In the following chart we summarize the best upper bounds available, distinguishing between no-slip, Navier-slip and free-slip boundary conditions.

    There are various questions and open problems that remain open:

    Ultimate scaling: {\rm{Ra}}^{\frac 12} or {\rm{Ra}}^{\frac 13} ? Focusing on the no-slip boundary conditions, we see that, on the one hand, upper bounds of the type {\rm{Nu}}\lesssim {\rm{Ra}}^{\frac 13}(\log {\rm{Ra}})^{\gamma} hold only for {\rm{Pr}} = \infty or for very large Prandtl number (i.e., {\rm{Pr}}\gtrsim ({\rm{Ra}}\ln {\rm{Ra}})^{\frac 13} ). On the other hand, the upper bound {\rm{Nu}}\lesssim {\rm{Ra}}^{\frac 12} holds uniformly in {\rm{Pr}} and it improves the bound {\rm{Nu}}\lesssim ({\rm{Ra}}\ln {\rm{Ra}} {\rm{Pr}}^{-1})^{\frac 12} for low Prandtl numbers. At the same time, this bound is the same no matter what the boundary conditions are, as long as u^z = 0 at z = \{0, 1\} . Thus the analysis is not yet able to confirm or rule out any of the scaling theories. Also, direct numerical simulations see their limitations in power of prediction when {\rm{Ra}} becomes very big and often the data points reported are not sufficient to conclude a scaling behavior as argued in [46,47]. Nevertheless, in [48] the authors show that the three dimensional simulations-data fit {\rm{Nu}}\sim {\rm{Ra}}^{0.331} in a convection cell with small aspect ratio, \Pr = 1 and {\rm{Ra}}\in[10^{10}, 10^{15}] .

    Do boundary conditions matter in the ultimate regime? Whether the boundary conditions matter in the ultimate–turbulent–state, remains to be understood. Nevertheless the results in the chart seem to indicate that free-slip enhance vertical heat transport compared to no-slip boundary conditions. On the other hand, for the choice \rm{\mathcal{L}_s}\sim {\rm{Ra}}^{\alpha} and \Pr\geq {\rm{Ra}}^{\frac 34-\alpha} the bound in (4.16) reads

    {\rm{Nu}}\lesssim \begin{cases} {\rm{Ra}}^{\frac{5}{12}} & if \quad\alpha\geq \frac{1}{24}\\ {\rm{Ra}}^{\frac 12-2\alpha} & if \quad 0\leq\alpha\leq \frac{1}{24}\,. \end{cases}

    If we compare this bound for small slip-lengths with the upper bound for no-slip boundary conditions at relatively small Prandtl numbers ( {\rm{Nu}}\lesssim {\rm{Ra}}^{\frac 12} ) we observe that the Navier-slip conditions seem to slightly inhibit heat transport.

    Dimensions: can we say something more in 2d ? Again looking at the chart we can see that little difference can be appreciated between the results in two and three dimensions, especially for no-slip boundary conditions. While the analysis in three dimensions is limited by the complexity of the Navier-Stokes equations, the two dimensional dynamics are accessible and might reveal significant mechanism in heat transport. In fact, the two dimensional Rayleigh-Bénard problem displays many of the physical and turbulent transport feature of three dimensional convection and the structure and regularity of the flow allows finer estimates. In fact, as seen in [12,45] balances involving the vorticity (and its gradient) can be supplemented to the energy balance to discover important relations. The next challenge in mathematical analysis would be to improve the upper bounds in [39], as the arguments in this paper are blind towards dimensions. In particular, in two dimensions, can one extend the regime of validity of the bound {\rm{Nu}}\lesssim {\rm{Ra}}^{\frac 13} at finite Prandtl number? or can one prove a bound of the type {\rm{Nu}}\lesssim {\rm{Ra}}^{\beta} with \beta < \frac 13 in some (possibly low) Prandtl number regime?

    Are these results optimal? Let us notice that all rigorous results are in the form of upper bounds, which therefore give us indications about the "worst case" scenario. It remains to be seen whether these estimates are sharp. This is very challenging from a mathematical standpoint and, at the current time, no lower bounds for the Nusselt number have been derived. Constructing exact solutions that saturate the bounds is probably impossible, given the complexity of the system, even at infinite Prandtl number. Nevertheless, in this direction, we want to mention the works [49,50] in which Tobasco and Doering consider a passive tracer advected by an incompressible velocity field \bf{u} , satisfying \langle|\nabla \bf{u}|^2 \rangle\leq \rm{Pe}^2 , where \rm{Pe} is the Peclet number. By adopting optimal designing techniques in energy-driven pattern formation in material science, the authors succeeded in designing an incompressible flow satisfying the enstrophy constraint such that {\rm{Nu}}\gtrsim \rm{Pe}^{\frac 23}(\log\rm{Pe})^{-\frac 43} . Together with the upper bound {\rm{Nu}}\lesssim \rm{Pe}^{\frac 23} [51], this lower bound demonstrate {\rm{Nu}}\sim \rm{Pe}^{\frac 23} possibly up to a logarithm. In connection with the Rayleigh-Bénard convection problem, this result shows the existence of a flow which is not buoyancy driven but nevertheless realizes {\rm{Nu}}\sim {\rm{Ra}}^{\frac 12} .

    What are the bounds in case of rough boundaries? Another important problem to study is the role of rough boundaries in the bounds. Numerical studies [52] indicate that roughness can enhance heat transport. To the author's knowledge, the only rigorous result available is given by Goluskin and Doering in 2016. In their paper [53], they assume no-slip boundary conditions on a rough vertical walls. More precisely, the (vertical) boundaries are continuous (and piecewise differentiable functions) of the horizontal coordinates (that is z = h^T(x, y) and z = h^B(x, y) ) with finite L^2 gradients. By a (not standard) application of the background field method, the authors show

    {\rm{Nu}}\lesssim C(\|\nabla h\|_{L^2}^2)\, {\rm{Ra}}^{\frac 12}\,.

    The scaling exponent \frac 12 was observed in the experiments reported in [54] where all boundaries (including sidewalls) were rough and, most recently, in numerical simulations using sinusoidally rough upper and lower surfaces in two dimensions [55]. Nevertheless the bound derived by Goluskin and Doering depends on a constant of the L^2 -norm of the gradient of h . It is expected that this condition can be relaxed, allowing much weaker regularity. As remarked in the introduction, it is not clear whether the no-slip boundary conditions are the most suitable to describe the physical problem. Citing [56] "For many years it has been observed that there is no compelling argument to justify the standard "no-slip" boundary condition of textbook continuum hydrodynamics, which states that fluid at a solid surface has no relative velocity to it [57]. However, this assumption successfully describes much everyday experience. Indeed, Feynman [57] noted in his Lectures that the no-slip condition explains why large particles are easy to remove by blowing past a surface, but small particles are not. Similarly, readers who wash dishes have noticed that it is difficult to remove all the soap just by running water — a dishcloth is needed for effective cleaning. Why?". It is then reasonable to consider surfaces where some slip occurs but there is still some stress on the fluid [12,58]. This suggests to consider (in two dimensions) the Navier-slip conditions (1.4), where \alpha = \alpha(\bf{x}) is a given positive twice continuously differentiable function defined on \partial \Omega , which indicates the roughness of the surface. Future research with the aim of extending the results in [45] to this rough-setting is planned.

    The author thanks Charlie Doering for the uncountable explanations and discussions, for sharing his vision and passion on this fundamental problem. The author is grateful to Philippe Vieweg for providing the pictures in Figure 1, to Steffen Pottel for the many suggestions regarding the manuscript and to the anonymous reviewer for useful feedback. This work was partially supported by the DFG-grant GrK2583 and TRR181.

    The author declares no conflict of interest.



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