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Bounds for rotating convection at infinite Prandtl number from semidefinite programs

Bounds for the poloidal and toroidal kinetic energies and the heat transport are computed numerically for rotating convection at infinite Prandtl number with both no slip and stress free boundaries. The constraints invoked in this computation are linear or quadratic in the problem variables and lead to the formulation of a semidefinite program. The bounds behave as a function of Rayleigh number at fixed Taylor number qualitatively in the same way as the quantities being bounded. The bounds are zero for Rayleigh numbers smaller than the critical Rayleigh number for the onset of convection, they increase rapidly with Rayleigh number for Rayleigh numbers just above onset, and increase more slowly at large Rayleigh numbers. If the dependencies on Rayleigh number are approximated by power laws, one obtains larger exponents from bounds on the Nusselt number for Rayleigh numbers just above onset than from the actual Nusselt number dependence known for large but finite Prandtl number. The wavelength of the linearly unstable mode at the onset of convection appears as a relevant length scale in the bounds.


I. INTRODUCTION
The optimum theory of turbulence attempts to replace the exact computation of time averaged global quantities, such as the heat transfer across a convecting layer, by a simpler calculation at the price of obtaining only upper bounds to the quantities of interest. The method most frequently used for this purpose is the background field method [1][2][3][4] , which is more convenient to use but which ultimately produces the same results as another method based on refs. 5-7, at least for the convection problem 8 . Both classes of methods do not make use of the full Navier-Stokes equation but use relations derived from integrals of this equation, such as the energy budget. As a consequence, a major disadvantage of these methods applied to convection flows is that they are insensitive to the presence of the Coriolis term in the momentum equation. Rotational effects appear in the bounds derived from straightforward optimum theory of convection only for reduced sets of equations 9,10 , the simplest representative of which is the model of the infinite Prandtl number.
Rotating convection is a problem with many applications in geophysics and astrophysics.
In this context, convection at large Prandtl numbers in particular also plays a role. Solutal convection in the oceans or in planetary cores 11 is characterized by Prandtl numbers around However, these applications motivate the study of convection at infinite Prandtl number as a simple limiting case. In addition, it was recently shown 12 how bounds derived for infinite Prandtl number convection can be used as a starting point for the computation of bounds for arbitrary Prandtl number. This calculation can be summarized as a background field method with a background field for velocity which is not fixed in advance and time independent, but which instead is the solution of the momentum equation for infinite Prandtl number.
The bounds derived in this manner for general Prandtl number cannot be lower than the bounds for infinite Prandtl number. It is therefore interesting to explore numerically which are the best possible bounds derivable at infinite Prandtl number from the constraints used in optimum theory of turbulence as we presently know it.
The known bounds derived by analytical means [12][13][14] for the heat transfer in rotating infinite Prandtl number convection are too loose to be satisfactory. While these bounds qualitatively reproduce a reduction of the heat transport by rotation, they do not restrict the advective heat transfer to zero for Rayleigh numbers below the critical Rayleigh number computed from linear stability analysis.
A numerical computation of upper bounds for heat transfer in rotating convection at infinite Prandtl number with stress free boundaries previously appeared in ref. 15. The parameter range covered in that study is too small to allow us to really judge the performance of the bounds by comparing them with experimental data. We also know from the non rotating problem that the optimum theory for infinite Prandtl number behaves very differently for stress free and no slip boundaries 16,17 , and the boundary conditions relevant for experiments are no slip rather than stress free boundaries. Finally, ref. 15 did not compute bounds on energies.
There are thus several motivations to revisit the problem of determining numerically the optimal bounds for rotating convection at infinite Prandtl number. The computation presented here makes use of the same constraints as previous work. These constraints only include linear and quadratic terms. This problem can be cast in the form of a semidefinite program (SDP). Numerical techniques of SDP have already been applied to convection problems 18,19 and they promise to be more generally applicable to bounding hydrodynamic quantities 4,20 . The SDP implemented for rotating convection at infinite Prandtl number is explained in the next section, and the results are presented in section 3.

II. THE SEMIDEFINITE PROGRAM
We consider a plane layer infinitely extended in the x and y directions and of thickness h with bounding planes perpendicular to the z axis in a frame of reference rotating with angular velocity Ω about the z axis. The gravitational acceleration acts in the direction of negative z. The layer is filled with fluid of density ρ, kinematic viscosity ν, thermal diffusivity κ, and thermal expansion coefficient α. Top and bottom boundaries are held at the fixed temperatures T top and T top + ∆T , respectively. Within the Boussinesq approximation, there are in general three control parameters: the Prandtl, Rayleigh and Taylor numbers Pr, Ra and Ta defined as Here, we immediately specialize to the limit of infinite Pr in which case the equations of evolution may be written as follows in terms of the non dimensional fields of velocity v(r, t), pressure p(r, t) and temperature deviation from the conductive temperature profile θ(r, t) where θ(r, t) = T (r, t) − 1 + z with T (r, t) being the temperature: z denotes the unit vector in z direction. The conditions at the boundaries z = 0 and z = 1 on the temperature imply that θ = 0 there. Both stress free and no slip conditions will be investigated. At stress free boundaries, with ∆ 2 = ∂ 2 x + ∂ 2 y . The boundary conditions become φ = ∂ 2 z φ = ∂ z ψ = 0 for stress free boundaries and φ = ∂ z φ = ψ = 0 for no slip boundaries.
The eqs. (5) and (6) determine φ and ψ, and hence v, in dependence of θ. The equations of evolution (2-4) therefore reduce to the single equation (3) in which v ultimately is some function of θ. Since there is no closed expression for this function, we keep the variables φ, ψ and v in subsequent formulae bearing in mind that they are nothing but functions of θ.
We next define averages over space of a function f (r) and over time of a function g(t) with the symbols where V is the volume of a periodicity volume. The product of eq. (3) with θ followed by a volume average leads in this notation to The quantities for which we will seek bounds in this paper are the time and volume averaged heat transport v z θ , the poloidal energy and the toroidal energy It only makes sense to ask for bounds on energy in convection with stress free boundaries if one specifies a certain frame of reference because any solution to the equations of evolution may be transformed to another solution with stress free boundary conditions simply by changing to another frame of reference moving at an arbitrarily large horizontal translation velocity. We adopt the same convention as in Ref. 12 and select the frame of reference in which total momentum is zero.
The method to find these bounds is exactly the same as in Ref. 18. The only difference in the numerical implementation is that in the non rotating problem of Ref. 18, the toroidal scalar ψ was always zero, whereas here, both φ and ψ have to be computed in terms of θ taking into account a Ta different from zero. The principle of the method is summarized here to make the paper self contained.
We select test functions ϕ n (z), n = 1...N which depend on z only and project onto them the temperature equation (3): We now build a functional G(λ 1 ...λ N , λ R , θ) as a linear combination of the right hand sides of the equations (11) and (8): Let us call Z the function for whose time average we want to find a bound. For example, Z = v z θ if we want to compute a bound for the heat transport. Z is a quadratic functional of θ for the problems of bounding heat transport, E pol and E tor . Let us suppose we know a number λ 0 and Lagrange multipliers λ 1 ...λ N , λ R such that all fields θ(r) obeying the boundary conditions for θ satisfy the following inequality: If we take the time average of this inequality keeping in mind that G is a linear combination of expressions equal to a time derivative, we find Z ≤ λ 0 .
The argument may equally well be recast in the form of a background method or in terms of an auxillary function 4 . If we call ϕ(z) = N n=1 λ n ϕ n (z), we may write G as so that the background field τ (z) used within the background field method applied to convection 3 is identical with 18 The best possible bound on Z is obtained by minimizing λ 0 over the λ 1 ...λ N , λ R subject to the constraint (13). This optimization problem turns into an SDP after discretization of The result is an SDP which yields the desired optimal bound, but it is not completely straightforward to compare the optimal coefficients λ n to the optimal ϕ(z) or τ (z) one might obtain from an analytical calculation. The reson for this is that the optimization problem includes constraints containing the integral of two functions f (z) and g(z), one of which (for example f ) is replaced in the course of the discretization by a sum of Dirac δ-functions as N n=1 f n δ(z − z n ) where the collocation points z n are given by The integral f (z)g(z)dz then becomes the sum N n=1 f n g(z n ). There is no immediate relationship between f (z) and the f n if the f n are chosen such that the sum N n=1 f n g(z n ) approximates as closely as possible the integral f (z)g(z)dz. For instance, these f n depend on the resolution N and the collocation points z n . The SDP as implemented returns a set of coefficients a n which appear in a representation of the first derivative of ϕ(z) as ϕ ′ (z) = N n=1 a n δ(z − z n ). The magnitude of these a n is not directly connected to optimal background fields obtained from analytical calculations, but the general dependence on space must be comparable.
Analytical treatment of infinite Prandtl number convection with the background field method 13,14,16,21 used functions τ (z) or ϕ(z) whose first derivative is either constant or piecewise constant, with a certain value for this derivative in the central region of the layer and another value in two boundary layers adjacent to the top and bottom boundaries. More sophisticated background fields were also used in the non rotating problem 22,23 . The thickness of the boundary layers is imposed by the Rayleigh number. Fig. 1 shows for a few examples the a n , each a n plotted at the location z n . The a n are approximately constant for the z n near the center of the layer, but there is a rapid variation of the a n within apparent boundary layers. More than one extremum appears for no slip boundaries at large Ta. The thickness of the boundary layer for no slip boundaries, defined from the location of the minimal a n , is compatible with the scaling in Ta −1/4 known from Ekman layers, whereas the layers remain thicker for free slip boundary conditions. The demand on resolution is therefore higher for no slip boundaries.

III. RESULTS
The bound H on the advective heat transport is obtained from the selection Z = v z θ .  from Z = 1 2 |∇ ×∇ ×(φẑ)| 2 and Z = 1 2 |∇ ×(ψẑ)| 2 , respectively. At any Ta, these bounds show again two regions of Ra in which rotation is either suppressing convection or not. To better identify the interval of Ra dominated by rotation, it helps to make a connection with the bounds on the heat transport. The z component of velocity is zero at the boundaries so that the bound H implies by virtue of the Poincaré inequality a bound on the poloidal energy 18 as However, one expects a stricter bound to be valid. Taking the dot product of the momentum equation (2) with v and integrating over space leads to If λ c is a characteristic length scale of the flow, one expects ij (∂ j v i ) 2 ∼ (E pol + E tor )/λ 2 c and therefore E pol ≤ RaH/λ 2 c . At Ra just above critical, the typical length scale of rotation is the wavelength of the most unstable mode in linear stability analysis, which leads to 24 λ 2 c ∝ Ta −1/3 . Replacing H by its power law fit deduced from fig. 2 and using Ra/Ra crit as variable, one expects the existence of a bound E p of the form This functional form, with a prefactor equal to 1, fits well the dependence of E p on Ra for rotating convection with both types of boundary conditions and Ra close to Ra crit . The bound E p obtained from the SDP is therefore sensitive to the variation of the characteristic length scale of convection with Ta, in agreement with analytically computed bounds 12 .
Eq. (19) fails to be a good fit if Ra is large enough, which marks the upper edge of the interval of Ra in which the bound E p is controlled by the Coriolis force. Fig. 3 shows that the bound on E pol is indicative of rapidly rotating convection only if E p Ta −0.6 ≤ 0.1 for no slip and E p Ta −0.8 ≤ 5 × 10 −3 for stress free boundaries. Fig. 4 repeats all the previous considerations for the toroidal energy. For no slip boundaries, the same function as in eq. (19) approximates also the bounds for the toroidal energy in the rapidly rotating regime, which implies near equipartition between the poloidal and toroidal energies in as far as bounds are concerned. In the case of stress free boundaries, the bound on toroidal energy E t exceeds the bound on poloidal energy. For both boundary conditions, E t exceeds E p in the high Rayleigh number regime even though the toroidal energy is zero in the non rotating case. At finite Pr, simulations 31 show that the fraction of toroidal to total energy decreases with Ra at large Ra. This behavior is not reproduced by the bound E t , possibly because eq. (6) prevents the fraction of toroidal to total energy from decreasing to zero which would mean that in this particular respect, rotating convection at infinite and finite Pr are fundamentally different from each other. Fig. 4 also shows that for no slip boundaries, the condition E t Ta −0.6 ≤ const. reasonably delimits the region in parameter space in which the bound E t is determined by the length scale of rapidly rotating convection near onset, λ c . Combined with the bounds for poloidal energy, this leads to (E p + E t )Ta −0.6 ≤ const.
as a criterion for the bounds being mainly controlled by the Coriolis term, which contains a different exponent than eq. (16).

IV. SUMMARY
Predictions of heat transport in turbulent convection necessarily rely on empirically motivated assumptions. An early approach assumed that the thermal boundary layers are marginally stable which leads in the non rotating case to 32,33  with β > 2 in fig. 2.
The SDP bounds the advective heat flux to zero for Ra < Ra crit . The SDP bounds also reflect several qualitative effects known from experiments and numerical simulations, such as a characteristic length scale varying as Ta −1/6 in the rotation dominated regime. In the case of no slip boundaries, the heat transport in rapidly rotating convection and in some interval of Rayleigh numbers exceeds the heat transport of non rotating convection at the same Rayleigh numbers. The exponent β in Nu (Ra/Ra crit ) β in the bounds for no slip boundary conditions increases with Ta (see fig. 2), just as it does in fits to experimentally and numerically determined Nu.
Rotating convection at infinite Prandtl number thus is a system in which significant improvement over existing analytical results is possible within the background method. It is also a system which is a promising testing ground for bounding methods involving additional constraints. It is possible to formulate an SDP which takes into account more constraints 20 , but at the cost of a significantly larger computational burden. A method of this type was recently demonstrated in the context of nonlinear stability analysis 36 . Rotating convection at infinite Prandtl number is a non trivial fluid dynamic system, which is relatively simple in the sense that it is governed by an equation for a single scalar field, and for which the known bounds do not scale with the control parameters in the same way as the results of numerical computations, so that this system promises to be particularly rewarding for any improved bounding method.