Bounds for rotating Rayleigh–Bénard convection at large Prandtl number

Bounds are derived for rotating Rayleigh–Bénard convection with free slip boundaries as a function of the Rayleigh, Taylor and Prandtl numbers Ra , Ta and Pr . At inﬁnite Pr and Ta > 130, the Nusselt number Nu obeys Nu (cid:2) 736 (cid:2) 4 / π 2 (cid:3) 1 / 3 RaTa − 1 / 3 , whereas the kinetic energy density E kin obeys E kin (cid:2) ( 7 / 72 π ) ( 4 / π ) 1 / 3 Ra 2 Ta − 2 / 3 in the frame of reference in which the total momentum is zero, and E kin (cid:2) ( 1 / 2 π 2 )( Ra 2 / Ta )( Nu − 1 ) . These three bounds are derived from the momentum equation and the maximum principle for temperature and are extended to general Pr . The extension to ﬁnite Pr is based on the fact that the maximal velocity in rotating convection at inﬁnite Pr is bound by 1 . 23 RaTa − 1 / 3 .

which an observable, such as the heat flow, is bounded in terms of the control parameters and the kinetic energy, which is itself an observable or a result of the dynamics. Bounds in this spirit were already derived for flows in periodic domains (Childress, Kerswell & Gilbert 2001;Doering & Foias 2002;Rollin, Dubief & Doering 2011;Tilgner 2017b) or flows around an obstacle (Tilgner 2021).
The general strategy of the derivation is to split the velocity field into a sum u + v of two parts u and v, where v solves the momentum equation for infinite Pr. It is possible to find bounds on the heat advected by v and the kinetic energy of v in terms of the amplitude of the temperature fluctuations as a consequence of the momentum equation alone. Together with the maximum principle for temperature, this leads to the bounds valid at infinite Pr detailed in § 3. In addition, one can find bounds on the maximal magnitude of v and on the dissipation in the field u which allow one to derive bounds on the heat transport and the kinetic energy at any Pr, but which are expected to be strictest at large Pr. These bounds are obtained in § 4 and discussed in § 5.

Basic equations
The non-dimensional equations of evolution for temperature T(r, t), velocity v tot (r, t) and pressure divided by density p tot (r, t) given as functions of position r and time t can be put in the form 1 Pr with the Rayleigh, Taylor and Prandtl numbers Ra, Ta and Pr. The unit vector in the z-direction is denoted byẑ. Gravity points along −ẑ and the rotation vector points in the direction ofẑ. In a Cartesian system in which the x and y axes lie in the horizontal, the velocity is required to fulfil the free-slip boundary conditions v tot,z = ∂ z v tot,y = ∂ z v tot,x = 0 at z = 0 and z = 1, whereas the temperature is fixed at those boundaries to T(z = 0) = 1 and T(z = 1) = 0. Periodic boundary conditions are assumed in the lateral directions with arbitrary periodicity length. It will be convenient to decompose temperature as T(r, t) = θ(r, t) + 1 − z with θ(z = 0) = θ(z = 1) = 0. The equations of evolution then become 1 Pr (2.5) Two types of averages will be used: the time average of a function f (t) will be denoted by an overline (2.6) and the volume average over a periodicity volume V of a function g(r) by angular brackets g(r) = 1 V g(r) dV. (2.7) The temperature T is restricted on the boundaries to 0 T 1 by the boundary conditions and obeys the advection and diffusion equation (2.3) in the bulk. It follows from the maximum principle for parabolic equations that 0 T 1 everywhere provided the initial conditions obeyed 0 T 1 (Evans 2010;Choffrut et al. 2016). On an attractor or in a statistically stationary state reached from this type of initial condition, the maximum principle for T thus requires |T − 1 2 | 1 2 , or equivalently |θ| 1 2 + |z − 1 2 |, which integrates to θ 2 7 12 .
(2.8) Multiplication with θ followed by integration of (2.5) leads to (2.9) and the product of v tot with (2.4) followed by integration and time averaging leads to the energy budget with the Nusselt number Nu given by where the last equality follows from (2.9). The velocity field will be decomposed into the sum v tot = v + u of two solenoidal fields v and u, where v solves the momentum equation for infinite Pr (2.12) and the remainder u solves (2.13) so that the energy budget for u reads 14) with the notation |∇u| 2 = i,j |∂ j u i | 2 . One of the goals of the calculations below is to find bounds on the kinetic energy in the flow. With free-slip boundaries, this is only possible if one specifies a particular frame of reference. The plane layer is invariant under translation in horizontal directions and the horizontal velocity of the boundaries does not appear in the boundary conditions for free-slip boundaries. The equations of evolution together with free-slip boundaries are therefore valid in any inertial frame of reference in which the boundaries have zero vertical but arbitrary horizontal velocity. It is thus possible to find arbitrarily large kinetic energies in the flow simply by observing the flow from a suitable frame of reference. We will select by convention the inertial frame in which the total momentum is zero, or u + v = 0. The vertical component of momentum is zero by virtue of the no-penetration conditions at the boundaries, and the horizontal components become zero through the choice of the frame of reference. The stress free boundaries exert no horizontal stress on the fluid layer, so that total momentum stays zero in a given inertial frame if it was zero initially. The selection of the frame of reference can thus be replaced by the requirement that the initial conditions The freedom in the choice of reference also applies to the subproblem of finding v. The diagnostic equation (2.12) with free-slip boundaries does not have a unique solution since an arbitrary uniform horizontal translation velocity can be added to any solution. We select the unique solution with v = 0. It follows from u + v = 0 and v = 0 that u = 0.
Poincaré's inequality states in a basic form for the boundary conditions under consideration and a function f (r) that |∇f | 2 η f 2 where η is the smallest eigenvalue of the Helmholtz equation −∇ 2 g = ηg if the functions g and f obey the same boundary conditions. Dirichlet boundaries at z = 0 and 1 imply η = π 2 . Neumann boundary conditions at z = 0 and 1 for functions satisfying f = 0 lead to η = min{π 2 , (2π/L) 2 } where L = max{L x , L y } and L x and L y are the periodicity lengths in the x and y directions, respectively. We first choose f to be u z + v z and find |∇( If, on the other hand, we select one of the horizontal components of u + v to be f , we find that |∇(u x + v x )| 2 min{π 2 , (2π/L) 2 } (u x + v x ) 2 and similarly for the y component. Adding the results for the three Cartesian coordinates yields No-slip boundaries allow the L independent conclusion |∇(u + v)| 2 π 2 |u + v| 2 . Another useful tool will be the decomposition of v, valid for any solenoidal vector field periodic in x and y, of the form (Schmitt & von Wahl 1992) (2.16) in which φ and ψ are poloidal and toroidal scalars satisfying φ = ψ = 0, and v mf is a mean flow field which depends spatially only on z and whose z-component is zero. If we insert this decomposition into the diagnostic equation (2.12), take the dot product of (2.12) with v mf and integrate the result over volume, we find |∂ z v mf | 2 = 0 and hence ∂ z v mf = 0. The condition v = v mf = 0 then yields v mf = 0, so that the poloidal and toroidal scalars suffice to fully determine v.

Infinite Prandtl number
This section will derive several properties of the solution v of the momentum equation (2.12) for infinite Pr. The results are independent of whether u = 0 or not and can thus be invoked in later sections on general Pr. The temperature advection equation (2.5) only appears in as far as we require θ to be a solution of it for any velocity field satisfying the no-penetration boundary conditions on the horizontal boundaries. This velocity field does not have to be v. Let us decompose v into poloidal and toroidal scalars φ and ψ as The z-component of the curl and the z-component of the curl of the curl of (2.12) yield A. Tilgner which can be combined to with the differential operator D = ∇ 2 ∇ 2 ∇ 2 + Ta∂ 2 z . The boundary conditions require φ = ∂ 2 z φ = ∂ z ψ = θ = 0 on z = 0 and z = 1. It follows from (3.2) that ∂ 4 z φ = 0 on the horizontal boundaries, too. Since θ is a solution of (2.5) and θ, v tot,z and v tot · ∇θ are all zero on the horizontal boundaries, ∂ 2 z θ = 0 there as well. This section will solve a series of variational problems in which the average of a function Z (either heat transport or some energy) is maximized subject to the constraints imposed by the momentum equations (3.2), (3.3) and the maximum principle (2.8). This is done by constructing the Lagrangian L with the help of the Lagrange multipliers μ 1 (r), μ 2 (r) and λ: The variation δL becomes independent of the behaviour of the variations δθ, δψ, δφ at the boundaries if μ 1 (r) and μ 2 (r) obey the same boundary conditions as φ and ψ, respectively. All integrations by parts are then possible to find: The functional L is a symmetric quadratic form in the variables μ 1 , μ 2 , φ, ψ, θ, so that the Euler-Lagrange equations are linear and lead to an eigenvalue problem or more generally to linear homogeneous equations with homogeneous boundary conditions and λ as parameter. The largest λ for which a non-trivial solution exists, λ max , provides us with the inequality (see vol. 1, chap. 6, § 1 of Courant & Hilbert 1989) in which we may furthermore insert the upper bound obtained for θ 2 from the maximum principle (2.8).

Upper bound on heat transport
The last two equations combine to On the boundaries z = 0 and z = 1, we required μ 1 = ∂ 2 z μ 1 = ∂ z μ 2 = 0 and we already It follows from (3.10) that ∂ 4 z μ 1 = 0 and from (3.11) that ∂ 6 z μ 1 = 0, and applying ∂ 4 z to (3.8) leads to ∂ 4 z θ = 0. One can now go in loops and take two additional derivatives of (3.4), (3.11) and again (3.8) to conclude that all even derivatives of θ, φ and μ 1 are zero at z = 0 and z = 1.
We next eliminate from (3.2), (3.3), (3.8), (3.9), (3.10) all spatially dependent variables except one to obtain λ Ra The eigenfunction compatible with all boundary conditions is of the form Upon insertion into (3.12) and after the substitutions k 2 = k 2 x + k 2 y = n 2 π 2 ξ and τ = Ta/n 4 π 4 one finds that the eigenvalues have the form (3.14) To find λ 1 , the largest of these eigenvalues, we may first optimize the fraction over ξ . A necessary condition for a maximum is The largest root of this polynomial will obey ξ 1 for τ 1 so that asymptotically, ξ 3 = 2τ . Inserting this into the expression for λ/Ra shows that the largest λ is realized for n = 1 so that asymptotically, This asymptotic expression for λ 1 also serves as an upper bound for λ 1 for Ta not too small. Figure 1(a) plots the largest root of the polynomial in (3.15) as a function of τ . The asymptotic expression ceases to be an upper bound for λ 1 at Taylor numbers too small to be of practical interest. The situation is similar for the variational problems that follow and we simply agree to be interested in Ta 130 only. With this restriction, the bound on the heat advection is We may note that we could have obtained the same result from a Fourier technique. If we start from the outset with a mode decomposition of the form θ = n,k x ,k yθ n,k x ,k y sin(nπz) exp(i(k x x + k y y)), (3.18) and similarly for φ with coefficientsφ n,k x ,k y and insert the sum into (3.4), we find that the amplitudes are related bŷ φ n,k x ,k y = Ra n 2 π 2 + k 2 (n 2 π 2 + k 2 ) 3 + Ta n 2 π 2θ n,k x ,k y .
( 3.19) The advective heat transport is then given by with n,k x ,k y 1 2 |θ n,k x ,k y | 2 = 7 12 . It now is enough to find the mode n, k x , k y which optimizes v z θ for the available θ 2 = 7/12. This optimization is algebraically identical with the optimization of λ given by (3.14). The Fourier technique appears to be simpler than the variational problem for the optimization of the heat transport, but the advantage is less clear for the following problems, and the variational formulation promises to be more convenient for other boundary conditions and geometries. That is why we will stick to the variational calculus. However, because of the connection with the mode decomposition, we will not discuss in detail the boundary conditions for θ and φ that derive from the Euler-Lagrange equations of the upcoming variational problems and simply assume an eigenfunction with a sinusoidal dependence on z.
3.2. Bounds on the kinetic energy Poloidal and toroidal fields are orthogonal in the sense of (∇ for any two vector fields a(r) and b(r) satisfying periodic boundary conditions in the horizontal directions and ∇ × ∇ × φẑ = (ẑ × ∇) × ∇φ. This helps to derive The variations of the Lagrangian (3.5) for Z = |v| 2 with respect to θ, ψ and φ yield, respectively, We again eliminate from (3.2), (3.3), (3.24), (3.25), (3.26) all variables except θ to find Inserting the eigenfunction θ ∝ sin(nπz) exp(i(k x x + k y y)) and substituting k 2 = k 2 x + k 2 y = n 2 π 2 ξ and τ = Ta/n 4 π 4 yields λ (3.28) A necessary condition for a maximum in ξ is that ξ satisfies which for large ξ leads to the solution ξ 3 = τ/2. The maximal eigenvalue λ 2 is realized for n = 1 which leads to the asymptotic expression Another result arises if one bounds |v| 2 in terms of |∇θ| 2 . To this end, one has to replace in the Lagrangian (3.5) the last term −λθ 2 by −λ|∇θ| 2 . Since δ |∇θ| 2 dV = −2 ∇ 2 θδθ dV, the only change to the Euler-Lagrange equations (3.24), (3.25), (3.26) is that λθ needs to be replaced by −λ∇ 2 θ, so that the calculation ends with − λ in place of (3.27). The same ansatz for the eigenfunction and the same substitutions as before now lead to λ . (3.33) The necessary condition for a maximum in ξ becomes which has a root for large ξ at ξ 4 = τ/3, which leads after the selection n = 1 and with figure 1(c) to |v| 2 1 π 2 Ra 2 Ta |∇θ| 2 . (3.35) The inequalities in the abstract result from (3.31) and (3.35) and E kin = |v| 2 /2.

3.3.
Maximal velocity This section determines the largest possible velocity in rotating convection at infinite Pr. To this end, we compute the Green's function so that, for any temperature distribution θ(r), the velocity is given by Because of the maximum principle |θ | 1, |v| is bounded by |v(r)| Ra |v G (r, r )| d 3 r . (3.38) We now decompose v G into poloidal and toroidal scalars and introduce the pair of Fourier transforms with respect to the horizontal coordinates and similarly for ψ G andψ G ;φ G andψ G have to obey for a source placed at r = (0, 0, z ). The homogeneous equations, valid for z / = z , are solved by In both regions, z < z and z > z ,φ G may be written as a linear combination with six coefficients c j and where the α j are the six roots of the following polynomial in α: Because of (3.43),ψ G is given bŷ There are six coefficients c j for z < z and another six for z > z so that 12 conditions are necessary to determine them all. These are the six boundary conditionsφ G = ∂ 2 zφ G = ∂ zψG = 0 at z = 0 and z = 1 together with six conditions at z = z . If we denote with square brackets the jump of a variable at z = z , (3.42) and (3.43) require that [ zφ G ] = 1. Because of the translational and rotational invariance of the system, φ G and ψ G can only depend on z, z , and the horizontal distance s = (x − x ) 2 + ( y − y) 2 between the points at r and r . We therefore change the list of arguments to φ G = φ G (s, z, z ) andφ G = φ G (k, z, z ) and note that for an axisymmetric function

A. Tilgner
The response v G to a source located at (0, 0, z ) is then given in cylindrical coordinates (s, ϕ, z) by where v ∞ denotes the supremum of |v| taken over the entire volume. In a next step, s s (0) and s ϕ (0) are computed for different Ta. The result is shown in figure 3. At large Ta, s s (0) s ϕ (0) because s s (0) asymptotes to s s (0) = 9Ta −1/2 , whereas s ϕ (0) approaches s ϕ (0) = 1.23Ta −1/3 . The exponents are familiar from vertical shear layers in rotating flows (Stewartson 1957). As seen in the figure, these power laws are approached from below so that they can be used as upper bounds. We conclude that v ∞ 1.23RaTa −1/3 . (3.54) This bound is entirely obtained from numerical evaluation of the Green's function and is not supported by an analytical calculation in an asymptotic limit as the other results in this paper. An analytical treatment of (3.52) is arduous because of the absolute values in the integrand which are essential even at z = 0. This is due to an interesting point about the structure of v G shown for an example in figure 4. The source term in (3.42) corresponds to an upward pointing force. The upwelling driven by this force, in conjunction with the boundaries, generates a converging flow below the source and a diverging flow above it. The Coriolis force then gives rise to a cyclonic circulation below the source and an anticyclonic swirl above it. These are indeed the main features observed in figure 4. However, there is also an anticyclonic ring that develops at some distance from the source near z = 0. This anticyclonic ring weakens as the source moves away from the boundary and is clearly the result of the interaction with the boundary. This ring also causes a change of sign in v Gϕ on z = 0 as a function of s and makes the absolute value in (3.52) indispensable. Figure 4 plots the product sv G,ϕ rather than v G,ϕ alone to clearly display the change of sign.

General Prandtl number
4.1. The additional constraint The previous section derived bounds on the heat advection and the kinetic energy of the field v defined by (2.12). These are simultaneously bounds on Nusselt number and total kinetic energy for convection at infinite Pr. At finite Pr, the velocity field is v + u. A bound on u is provided by the time average of (2.14) was used. One readily bounds the last term from The term with the time derivative on the other hand gives rise to another variational problem; ∂ t v has a decomposition in poloidal and toroidal scalarsφ andψ with ∂ t v = ∇ × ∇ ×φẑ + ∇ ×ψẑ,φ = ∂ t φ,ψ = ∂ t ψ and φ and ψ are the scalars for the decomposition of v itself;φ andψ obey the same boundary conditions as φ and ψ. The relation holds if v obeys no-penetration boundary conditions. The transformation is independent of boundary conditions;n is the outward pointing normal vector to the surface bounding a periodicity volume. The contribution to the boundary integrals from the vertical faces of the periodicity volume cancel because of periodicity;n ∝ẑ on the remaining faces. The second boundary integral is zero forn ∝ẑ independently of the behaviour ofψ, and the first integral is zero becauseφ = 0 on z = 0 and z = 1. The Schwarz inequality motivates us to seek bounds for |∇ ×φẑ| 2 and ψ 2 . The time variation of φ and ψ derives from the time dependence of θ. Let us rewrite (2.5) as Ra∂ t θ = ∇ · q with q = Ra∇θ − Ra(u + v)(T − 1 2 ) and take time derivatives of (3.2) and (3.3) to find Before starting the next variational problem, we first note that the preceding equations may alternatively be written as a direct time derivative of (2.12) as Taking the scalar product of this equation withṽ and averaging over all space leads to |∇ṽ| 2 = ṽ z (∇ · q) = − (∇ṽ z ) · q |∇ṽ z | 2 |q| 2 |∇ṽ| 2 |q| 2 , (4.9) and hence |∇ṽ| 2 |q| 2 . Both |∇ ×φẑ| 2 and ψ 2 are bounded in terms of |∇ṽ| 2 because of the Poincaré inequality, which means that both quantities are smaller than some finite factor multiplied by |q| 2 . The existence of such finite factors being established, we now proceed to determine Ta dependent values for these factors from a variational problem.

Discussion
The main results of this paper are the bounds for Nu and the kinetic energy at infinite Pr in (3.17), (3.31), (3.35) together with the pointwise bound on the magnitude of the velocity at infinite Pr in (3.54). The first three bounds are extended to finite Pr in (4.27), (4.30), (4.31).
The bound for Nu in (4.27) is valid as long as Ta > 130, which was necessary for the estimation of the largest root of the polynomial in (3.15). This bound reduces to a simpler expression for infinite Prandtl number already given in (3.17). The critical Rayleigh number for the onset of convection, Ra crit , is proportional to Ta 2/3 (Chandrasekhar 1961). Ignoring numerical prefactors, the bound (3.17) may be rewritten as Nu − 1 Ra 1/2 (Ra/Ra crit ) 1/2 . This may be compared with the bound derived in  which is apart from a prefactor Nu − 1 Ra 1/2 (Ra/Ra crit ) 3/2 . Both of these bounds are inferior to the uniformly valid Nu − 1 Ra 1/2 as soon as Ra > Ra crit . For Ra < Ra crit , the result in  is the sharper bound, which is not surprising since the bound (3.17) was derived without explicit reference to the temperature equation. However, the bound (3.17) has the merit to be generalizable to finite Pr in (4.27).
The bounds for general Pr and free-slip boundaries depend on the periodicity length and diverge if this length tends to infinity. However, if one wants to model a geometry similar to a deep spherical shell, such as the Earth's outer core, the vertical extent is of the same order of magnitude as the lateral periodicity length and the dependence on the periodicity length is not a major concern.
As soon as a bound on Nu is available, the Poincaré inequality yields a bound at least on the poloidal kinetic energy through a Poincaré inequality. Previous numerical optimization (Tilgner 2017a) obtained a better prefactor but not better exponents of the control parameters than obtained from the Poincaré inequality, neither at infinite nor at finite Pr. The bound derived in the present paper is a novelty in this respect. Straightforward application of the Poincaré inequality to either (3.17) or the bound from  leads to |v| 2 Ra 2 Ta −1/3 or |v| 2 Ra 3 /Ta, both of which are inferior to (3.31) for Ra > Ra crit . In the same manner, inequality (3.35), which is the reduction of (4.31) to infinite Pr, says that |v| 2 (Nu − 1)Ra 2 /Ta, which is better than the result from the Poincaré inequality for Ra < Ta.
However, the bounds on neither Nu nor |(u + v)| 2 reproduce the fact observed in both experiments and simulations that Nu − 1 and |(u + v)| 2 approach zero at Ra = Ra crit .

Bounds for rotating Rayleigh-Bénard convection
The exponents in both bounds can be obtained by simply inserting (3.54) into v z θ v ∞ and |v| 2 v 2 ∞ . The complexity of the calculation is reduced in this paper by solving two problems sequentially. In a first step, the constraints imposed by the momentum equation and the maximum principle are used at infinite Pr. No use of the temperature equation is made. It is then simple to obtain in a second step rotation dependent bounds at arbitrary Pr by including the constraint (4.24). A more comprehensive approach would be to solve an optimization problem which simultaneously includes the momentum equation, the temperature equation, and the constraint (4.24). This set of constraints guarantees tighter bounds for Nu at infinite Pr and presumably also improves the bounds at general Pr. However, this more involved problem will likely need a numerical optimization for its solution and is left for future work.
Inequalities like (3.35) or (4.31) may look like weak statements from the mathematical point of view because they relate Nu to both the control parameters and the kinetic energy, which is itself a result of the dynamics. However, these relations are more immediately applicable to geophysical and astrophysical situations than the other bounds, because the Rayleigh number is frequently impossible to determine from observations. To motivate the derivation of relations of this type and to keep the discussion simple, let us look at how the form for infinite Prandtl number may become useful and use dimensional variables. If the convecting fluid has viscosity ν, thermal diffusivity κ, density ρ, heat capacity c p and expansion coefficient α, its velocity V must be related to the non-dimensional velocity v by V = vκ/d in order to obtain (2.4), where d is the depth of the layer. If the heat flux through the layer is Q and the temperature difference across the layer is T, the rotation rate of the frame of reference is Ω and the gravitational acceleration is g, the control parameters in (2.4) are given by Let us take convection in the Earth's core as an example and pretend that the above relation is applicable to a spherical shell in the presence of a magnetic field. We ignore these features for now because the relevant point to be made about this relation does not concern specific values of prefactors or exponents, but depends on which quantities are well known and which are not. The rotation rate, the depth of the layer, and the gravitational acceleration are well known, and there are reasonable estimates of the material properties. Temporal variations of the geomagnetic field provide us with some information about |V | 2 , and the heat flux through the Earth's surface constrains Q. On the other hand, T is essentially unknown. A relation of the form of (4.31) thus allows us to put a constraint on a quantity impossible to determine otherwise.
(5.4) This inequality puts a formal limit on the structure of a condensate or a large eddy as observed in numerical simulations (Guervilly, Hughes & Jones 2014). The simplification brought about by the infinite Prandtl number is that the time derivative and advection terms disappear from the momentum equation (2.4). At large Pr, these terms are small compared with the viscous term. However, the two terms can also be negligible in comparison with the Coriolis term. Many results in classical geophysical fluid dynamics are based on an approach in which one postulates that the advection term and possibly the time derivative term are small, one solves the thus linearized momentum equation, and one finds good agreement for many large scale phenomena (Vallis 2017). This is not a widespread approach in simulations of convection (but see Calkins, Julien & Tobias 2017), but the inertial terms are found to be negligible with hindsight in some simulations. An example are the simulations in Schmitz & Tilgner (2009) (and analogous results for no-slip boundary conditions in Schmitz & Tilgner 2010). It was found in these simulations that the flows are dominated by the Coriolis force if |v| 2 Ta −1/2 < 1, in which case the flow is organized into columnar structures, similar to those visualized in Stellmach & Hansen (2004), which vary slowly in time and whose Rossby number is small. The time derivative and advection terms in these flows are small even though the Prandtl number is not large. These flows are good candidates to be modelled by the momentum equation with these terms dropped. It therefore makes sense to compare the numerical results of Schmitz & Tilgner (2009) not only with the bound (4.31), but also with the bound (3.35) in the form |v| 2 Ta Ra 2 (Nu − 1)/π 2 which is done in figure 5. As one can see, the latter bound is within an order of magnitude of the actual results. It remains a challenge for the future to prove ab initio that the inertial terms are negligible at the parameters of this figure. The bound (4.31) lies several orders of magnitude above the numerical data. Declaration of interests. The authors report no conflict of interest.