Geophysical Research Letters

Is horizontal convection really “non-turbulent?”

Authors


Abstract

[1] Within the geophysical community Horizontal Convection (HC) has been considered irrelevant or nearly so in driving large scale overturning flows, based primarily on an inference based on a century old experiment by Sandström (1908), and on a theoretical argument that would prevent HC to sustain a true turbulent flow, the latter deemed necessary to achieve mixing. We revisit Paparella and Young's (2002) argument with the aid of DNS of HC at Rayleigh number up to 1010. We argue that the criterion used by these authors is overly restrictive. On the contrary, geometrical statistics show that HC possesses the characteristic of turbulent flows. The surprising result is that HC can transport very large quantities of heat and sustain large amounts of diapycnal mixing with a surprisingly small amount of dissipation. Values of diapycnal mixing and dissipation in the ocean are shown to be consistent with a HC driven ocean provided the effect of wind-forcing are included.

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

equation image

where κ is the diffusivity of the stratifying agent, H the depth of the domain and bmax 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 equation image and H to nondimensionalize the problem, Paparella and Young's result can be recast as (we assume fixed Prandtl number)

equation image

where D(KE) = ∫ εdV is the volume-integrated non-dimensional dissipation rate and Ra = bmaxH3/νκ 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

equation image

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 ∼ Ra1/3 up to the highest values of Ra attained (1017) [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.

2. Numerical Experiments

[8] We solve numerically the non-dimensional 3D Navier-Stokes equations in the Boussinesq approximation

equation image
equation image
equation image

where b is the buoyancy, signed so that for a stable configuration b increases with depth and Pr is the Prandtl number. The equations are solved on a staggered grid, using a standard projection method for time stepping (see Scotti [2008, and references therein] for details on the code). The scheme is overall second order in space and time. Verzicco and Camussi [2003] used the same discretization technique to simulate RBC. The domain is a three-dimensional channel of dimension Lx × Ly × 1. For computational efficiency, periodic conditions are assumed along the spanwise direction (y). Along the remaining directions arbitrary conditions can be applied.

2.1. Results

[9] We consider the geometry used by PY but with the third dimension added. The non-dimensional domain is 4 × 2 × 1 discretized uniformly with 512 × 256 × 128 points. All walls but the upper are adiabatic and free-slip. At the upper boundary free-slip conditions are imposed on the velocity, and a Dirichlet BC in the form of a sinusoidal buoyancy distribution is applied so that dense water with buoyancy equal to one is formed in the center of the channel and warming occurs at either ends (Figure 1) where buoyancy is set to zero. The location of maximum buoyancy is slightly modulated in the spanwise direction to force the flow to be three dimensional. Analysis of power spectra indicates that the resolution is adequate, as evidenced by the lack of accumulation of variance at the highest wave numbers. In all experiments, the fluid in the domain is initially quiescent, and has a uniform buoyancy of 1/2. Sensitivity runs started from a uniformly stratified fluid reached the same steady state. In the numerical experiments, we keep Pr fixed at 1, and consider several values of Ra ranging between 108 and 1010. Here we report results for Ra = 1010. In violation of Sandröm's inference, but in agreement with all the available experiments reviewed in the preceding sections, all three cases reach a statistically steady state characterized by a deep overturning circulation, affecting the entire channel, characterized by a narrow sinking plume and a broad upwelling, exactly as envisioned by Stommel [1962] in his “gedanken” HC model (Figure 1), and in agreement with Mullarney et al. [2004]. The buoyancy frequency profile at steady state shows the existence of a strong pycnocline overlying a weakly stratified interior. The stratification in the interior is two orders of magnitude smaller than the stratification in the pycnocline, as opposed to the real ocean, where the former is only one order of magnitude smaller.

Figure 1.

(left) The stream function of the spanwise averaged flow at Ra = 1010 at t = 15500. Sinking occurs in a narrow region near x = 0, while upwelling is diffuse, as in the Stommel [1962] model. (right) Buoyancy frequency profile.

2.2. Turbulence Criterion

[10] While there is still not a consensus within the turbulence community on what is turbulence (see Tsinober [2001] for a long list of proposed definitions), the last decade has seen a shift from emphasizing scaling and/or independence on the order parameter toward recognizing certain statistical properties of the velocity gradient tensor which (i) are common to all known turbulent flows and (ii) already appear at intermediate values of the order parameter. Interestingly, these properties are quite specific, and reflect underlying properties of the dynamics of strain and vorticity [Tsinober, 1996; Elsinga and Marusic, 2010].

2.2.1. Invariants of the Velocity Tensor and Geometrical Statistics

[11] The velocity gradient tensor Aij has three Cayley-Hamilton invariants. One, the trace, is trivially zero in incompressible flows. The other two are known as Q ≡ −1/2AijAji = (∥ω2 − 2SijSji)/4 and RAimAmnAni/3 = −(SijSjkSki + equation imageωiωkSij)/3, where ω is the vorticity and Sij the rate of strain tensor. Note that Q measures the relative size of strain over enstrophy, and R is a measure of production of strain versus production of enstrophy. A property that appears to be universal among turbulent flows is the particular shape that occurs when Q and R are plotted in a joint PDF map (the reader is referred to the recent review by Meneveau [2011] for a detailed discussion). This unmistakable tear-drop pattern appears in the R-Q PDF (Figure 2 shows the PDF from data at Ra = 1010), with the tail accumulating on the Viellefosse tail Q = −(3/2)2/3R2/3. This pattern is common to widely different turbulent flows [see, e.g. Tsinober, 2001, Figure 9.1]. Another geometrical statistical property common among turbulent flows is the PDF of (ω · W)/∥ω∥∥W∥, where WiωjSji is the vortex stretching vector. The PDF calculated from our experiments agrees very well with the PDF of grid turbulence at Reλ = 75 reported by Tsinober [1998] (Figure 3). Interestingly, these properties are established very quickly after the boundary condition is turned on, well before the flow reaches equilibrium, remain constant for the duration of the simulation and are substantially independent on the Rayleigh number. The overarching result is that from the point of view of geometrical statistics HC is turbulent.

Figure 2.

Q-R joint PDF at Ra = 1010. The black line shows the Viellefosse tail Q = −(3/2)2/3R2/3.

Figure 3.

PDF of the cosine of the angle between vorticity and the vorticity stretching vector: (blue) Ra = 1010; (black) Ra = 2.5 × 109; (red) Ra = 108; circles refer to grid turbulence at Reλ = 75.

3. The Energy Argument Revisited

[12] Paparella and Young's argument is formulated in terms of the sum of kinetic and potential energy. Let us consider instead the balance of volume integrated kinetic and available potential energy (APE) [Scotti et al., 2006] at steady state, which symbolically can be written as

equation image

where D(KE) and D(APE) are the dissipation rates of kinetic energy and APE. The forcing term in HC (with no external mechanical forcing, e.g., surface winds) is expressed as

equation image

where Q is the buoyancy flux at the surface, H the depth of the domain, z*(b) the equilibrium depth of a parcel with buoyancy b (we assume that the bottom is at z = 0, and the equilibrium depth is defined as in work by Winters et al. [1995]). At steady state ϕ must be balanced by dissipation of kinetic and available potential energy. An inspection of the equation for the integrated background potential energy ∫ (bzAPE)dV shows further that at steady state D(APE) is exactly balanced by ϕAPE [Hughes et al., 2009]. Hence, we have that D(KE) must be equal to the second term ϕKE, recovering Paparella and Young's [2002] result. This result, which Tailleux [2009] shows to hold beyond Bousinnesq, has profound consequences when applied to the overall mixing efficiency, defined here as

equation image

with the last equality holding at steady state. Thus, at steady state γ reflects the relative strength of ϕAPE versus ϕKE. The former is dominated by the cooling regions, as in the warming regions Hz*(b(x, y, z = H, t) is very small, and crucially, depends on the surface buoyancy flux. Let us assume that the Nusselt number in HC scales as Raα. Theoretical arguments show that α ≤ 1/3 [Winters and Young, 2009]. Earlier estimates by Rossby [1998] indicate that α = 1/5, a value that our numerical experiments confirm (though the limited range of Ra considered here prevents an unambiguous determination). In any case, it is reasonable to assume that α > 0. Then, just as for RBC,

equation image

which combined with Paparella and Young bound (2) shows that asymptotically

equation image

In our experiments, γ starts at about 0.5, a value observed in flows driven by an excess of APE [Dalziel et al., 2008], and, by the time steady state is reached, reaches values in excess of 0.85 (Figure 4), in agreement with (11).

Figure 4.

Time development of D(APE) (solid) and D(KE) (dashed) at Ra = 1010. Symbols show the mixing efficiency.

[13] The estimate in (11) is also in agreement with estimates given by Siggers et al. [2004], thus providing further confirmation for the paradoxical aspect of HC: It can very efficiently transport heat, when compared to conduction alone, with an extremely small amount of dissipation.

4. Is HC Relevant in the Ocean?

[14] Oceanic estimates for the terms in (9) have been provided by several authors. Oort et al. [1994] estimate ϕAPE from compiled data at 1.2 ± 0.7 TW, while Tailleux [2009] suggests that the lower bound of 0.5 TW may be closer to reality. In any case, these values are consistent with the diapycnal mixing rate inferred by Munk and Wunsch [1998]. Typical estimates of ε in the open ocean away from topography give 10−10 Wkg−1 [Toole et al., 1994]. Thus the observed γ is close to 0.9. Since the Rayleigh number for the ocean readily exceeds 1020, (11) would give a value much closer to one, even assuming for α the low-end value of 1/5. This is reflected in the estimate that Wang and Huang [2005] give for ϕKE ≃ 1.5GW based on (2), or an equivalent value for ε ≃ 10−12 Wkg−1.

[15] Thus, even though our results show that HC cannot be excluded on a qualitative basis, it fails this quantitative test. Obviously, the ocean is not a pure HC system, and indeed a host of processes mechanically force the ocean [Wunsch and Ferrari, 2004]. Following the hugely influential work of Munk and Wunsch [1998], oceanographers have focused on mechanical sources of deep mixing. The much larger forcing exerted by the winds at the surface has received much less attention. A steady meridional wind added to HC can provide a sink or a source of mechanical energy (depending on which direction the wind blows relative to the imposed buoyancy gradient) [Tailleux and Rouleau, 2010]. In a rotating ocean, zonal winds can produce an Ekman flux which crosses density lines, and has the same effect. A second effect of wind-forcing at the surface is to greatly enhance fluxes at the surface relative to a windless ocean. This obviously increases ϕAPE, since the latter depends on the buoyancy flux Q. Indeed, Oort et al. [1994] estimate for ϕAPE is based on measured values of the buoyancy flux across the actual, windy ocean surface.

[16] But it also increases ϕKE, as can be easily seen by repeating PY's analysis with a depth-varying eddy viscosity set to the molecular value below the pycnocline and increasing to a maximum value at the surface. But even if the use of an eddy viscosity may be not appropriate in this context, it is undeniable that wind-induced waves increase the actual surface of the ocean, both directly, as well as indirectly through the injection of bubbles, and thus in (2) a prefactor proportional to the ratio of the actual to the geometric surface should be included. In either case, it is clear that (11) cannot be applied senso strictu to a windy HC ocean. Yet, the fact that γ in the ocean is close to one should be considered as an indication that a wind-modified HC cannot be ruled out as a possible explanation for what drives the overturning circulation in the ocean, very likely together with other wind-mediated driving mechanisms [Gnanadesikan et al., 2005].

5. Conclusions

[17] The main result presented in this letter is that a flow driven by Horizontal Convection exhibits the geometrical statistics of a true turbulent flow. We argue that the criterion used by Paparella and Young [2002] to deny the turbulent nature of this flow is too restrictive, and we use conventional Rayleigh-Bènard convection as an example of a flow that as far as experimental evidence is concerned would fail such a test. Our three-dimensional simulations agree with earlier experiments and simulations in showing the occurrence of an overturning circulation that affects the entire water column, generating a healthy unsteady eddy field. Further, we show theoretically and confirm with our simulations that the mixing efficiency of Horizontal Convection flows approaches 100%. Application of the results above to the ocean suggests that, based on an energetics argument alone, it is not possible to dismiss HC as a potential co-driver of the overturning circulation provided the effects of surface wind-forcing on enhancing the overall thermodynamic forcing at the surface are included. The large amount of energy that the winds input to the ocean has been usually disregarded relative to deep-ocean mechanical stirring. Our results indicate that it may be appropriate to reconsider the relative balance of surface to deep-ocean stirring.

[18] Finally, it is important to bear in mind that our simulations do not include the effect of rotation, and are performed in a domain with an O(1) aspect ratio. While including rotation is in itself straightforward, to have a meaningful scale separation between quasi-geostrophic scales and the mixing scales, the aspect ratio ought to be increased at least a hundred-fold, which pushes the limit of what is computationally feasible, at least for a DNS.

Acknowledgments

[19] The authors wish to thank R. Tailleux, F. Paparella, W. R. Young and G. Hughes and two anonymous reviewers for useful comments on an earlier version of the manuscript. Financial support was provided by NSF under grants OCE-0726475. A.S. wishes to dedicate this letter to the memory of his late father, G. Scotti (1932–2011).

[20] The Editor thanks Anand Gnanadesikan and an anonymous reviewer for their assistance in evaluating this paper.

Ancillary