## 1. Introduction

[2] In geophysics we find many examples of fluids subject to a buoyancy gradient applied to an equigeopotential surface. For example, the buoyancy at the surface of the ocean is controlled by the exchange of heat with the atmosphere, by precipitation and by brine rejection during ice formation at high latitudes. Another example is the atmosphere of Venus, where most of the incoming solar radiation is completely absorbed at the upper edge [*Houghton*, 1977]. In both cases, a meridional gradient develops in the buoyancy field. A flow driven by such a gradient has been called Horizontal Convection (HC) [*Stern*, 1975]. Whether HC alone can drive a geophysically significant flow has been the object of an intense debate. Restricting our attention to the ocean, we observe that to date, most theoretical oceanographers dismiss HC as a potential player in the ocean [*Defant*, 1961; *Munk and Wunsch*, 1998; *Kuhlbrodt et al.*, 2007; *Wunsch and Ferrari*, 2004]. Such dismissal rests fundamentally on a century old inference by *Sandström* [1908], which states that to induce a flow, heating must occur lower than where cooling is applied. Applied to the ocean, this means that absent other mechanisms, aside from a weak residual flow within a thin surface layer, the latter would quickly become a stagnant cold and salty pool.

[3] Against Sandström inference there is a long list of laboratory [see, e.g., *Mullarney et al.*, 2004; *Coman et al.*, 2010; *Hughes and Griffiths*, 2008, and references therein] and numerical experiments [*Beardsley and Festa*, 1972; *Rossby*, 1998; *Paparella and Young*, 2002; *Siggers et al.*, 2004; *Coman et al.*, 2010] that show that HC generates a non trivial, unsteady, eddying overturning circulation that generally affects the entire vertical extension of the domain for sufficiently large values of the relevant Rayleigh number.

### 1.1. The “Anti-turbulence” Theorem

[4] Despite the experimental evidence to the contrary (even a replica of Sandström own experiments [*Coman et al.*, 2006] shows that HC indeed drives a significant overturning circulation), Sandström had a strong theoretical motive to back his inference. According to him, if we understand the HC as a thermodynamical engine no net energy can be obtained, since expansion and contraction occur at the same pressure. This argument was criticized for not taking into consideration heat conduction within the fluid. Indeed, *Jeffreys* [1925] showed that Sandrström argument really only implies that the path from the contracting to the expanding region must lie underneath the return path. The thermodynamic forcing was quantified much more recently by *Paparella and Young* [2002] (hereinafter referred to as PY), who showed that in a HC Boussinesq system forced at a geoequipotential surface by the imposition of a buoyancy boundary condition the mean kinetic energy dissipation rate per unit mass ε within the domain is bounded by

where κ is the diffusivity of the stratifying agent, *H* the depth of the domain and *b*_{max} is the largest buoyancy anomaly at the surface. The practical consequence of (1) is that HC may not be able to explain quantitatively the level of dissipation observed in the open ocean. We defer a discussion on this point to the end.

[5] However, this result, known as the “anti-turbulence” theorem, has been given a much stronger significance, because it has been used to imply that HC cannot drive a *bona fide* turbulent motion, and thus it cannot *qualitatively* explain the overturning circulation, since a non turbulent motion is deemed unable to provide the required mixing. This based on a “zeroth” law of turbulence [*Frisch*, 1995] which *requires* the rate of dissipation, when scaled in non-viscous units, to approach complete similarity [understood as in *Barenblatt*, 1996, p. 25] in the non-dimensional (maintaining fixed the diffusivity ratio). For the case at hand, using and *H* to nondimensionalize the problem, Paparella and Young's result can be recast as (we assume fixed Prandtl number)

where *D*(KE) = ∫ ε*dV* is the volume-integrated non-dimensional dissipation rate and Ra = *b*_{max}*H*^{3}/*ν*κ is the Rayleigh number which is the order parameter for convection (note that other authors have used the length of the box instead of the depth in the definition of Ra).

### 1.2. A Critique

[6] We do not dispute the scaling in (2), but we suggest that the “zeroth” law is too restrictive, since according to this strict definition, even canonical Rayleigh-Benàrd convection (RBC) would not be turbulent! Indeed, in the same units, we have for RBC

where Nu is the Nusselt number (the ratio of measured heat flux from the boundary to the heat flux in the absence of motion). All available experimental evidence shows that Nu ∼ Ra^{1/3} up to the highest values of Ra attained (10^{17}) [*Niemela et al.*, 2000; *Niemela and Sreenivasan*, 2006; *Verzicco and Camussi*, 2003; *Verzicco and Sreenivasan*, 2008; R. J. A. M. Stevens et al., Prandtl and Rayleigh number dependence of heat transport in high Rayleigh numberthermal convection, 2011, http://arxiv.org/abs/1102.2307, and references therein]. Hence, *D*(KE) ∼ Ra^{−1/6} achieves *incomplete* similarity and based on the strict “zeroth” law RBC ought not to be considered turbulent. Needless to say, this is not an opinion widely held. Rather, RBC is considered a “raw model for turbulence in general” [*Lohse and Xia*, 2009].

[7] Interesting, experimental and numerical evidence [*Paparella and Young*, 2002; *Mullarney et al.*, 2004; *Wang and Huang*, 2005] shows that as Ra increases, HC becomes more unsteady. To explore this apparent paradox, we performed numerical simulations of HC. Not willing to exclude *a priori* the possibility of turbulence, we solved the relevant equations in a three-dimensional domain. This distinguishes the present work from prior numerical simulations, which, to our knowledge, were all two-dimensional and thus intrinsically unable to address the turbulent versus laminar nature of HC. The results (see next section) show that HC generates flows whose geometrical statistics are unmistakably turbulent. Further, the energetics shows that the mixing efficiency is very high. HC has always being dismissed as an extremely inefficient thermal engine. However, what appears is that it is an extremely efficient heat mover, in agreement with recent theoretical estimates by *Siggers et al.* [2004]. Our analysis, in line with recent results by *Hughes et al.* [2009]; *Tailleux* [2009], demonstrates that the mixing efficiency of HC approaches 100%. In the last section, we apply our results to the real ocean, where we argue that the quantitative discrepancy between the dissipation rate implied by PY argument and the observed value can be attributed to the effect of wind forcing at the surface.