Editorial Type:
Article
Article Type:
Research Article

Available Potential Energy and Irreversible Mixing in the Meridional Overturning Circulation

Graham O. Hughes The Australian National University, Canberra, Australia

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Andrew Mc C. Hogg The Australian National University, Canberra, Australia

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Ross W. Griffiths The Australian National University, Canberra, Australia

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Print Publication:
01 Dec 2009
DOI:
https://doi.org/10.1175/2009JPO4162.1
Page(s):
3130–3146
Full access

Abstract

The overturning circulation of the global oceans is examined from an energetics viewpoint. A general framework for stratified turbulence is used for this purpose; first, it highlights the importance of available potential energy in facilitating the transfer of kinetic energy to the background potential energy (defined as the adiabatically rearranged state with no motion). Next, it is shown that it is the rate of transfer between different energy reservoirs that is important for the maintenance of the ocean overturning, rather than the total amount of potential or kinetic energy. A series of numerical experiments is used to assess which energy transfers are significant in the overturning circulation. In the steady state, the rate of irreversible diapycnal mixing is necessarily balanced by the production of available potential energy sourced from surface buoyancy fluxes. Thus, the external inputs of available potential energy from surface buoyancy forcing and of kinetic energy from other sources (such as surface winds and tides, and leading to turbulent mixing) are both necessary to maintain the overturning circulation.

Corresponding author address: Graham O. Hughes, Research School of Earth Sciences, The Australian National University, Canberra, ACT 0200, Australia. Email: Graham.Hughes@anu.edu.au

Abstract

The overturning circulation of the global oceans is examined from an energetics viewpoint. A general framework for stratified turbulence is used for this purpose; first, it highlights the importance of available potential energy in facilitating the transfer of kinetic energy to the background potential energy (defined as the adiabatically rearranged state with no motion). Next, it is shown that it is the rate of transfer between different energy reservoirs that is important for the maintenance of the ocean overturning, rather than the total amount of potential or kinetic energy. A series of numerical experiments is used to assess which energy transfers are significant in the overturning circulation. In the steady state, the rate of irreversible diapycnal mixing is necessarily balanced by the production of available potential energy sourced from surface buoyancy fluxes. Thus, the external inputs of available potential energy from surface buoyancy forcing and of kinetic energy from other sources (such as surface winds and tides, and leading to turbulent mixing) are both necessary to maintain the overturning circulation.

Corresponding author address: Graham O. Hughes, Research School of Earth Sciences, The Australian National University, Canberra, ACT 0200, Australia. Email: Graham.Hughes@anu.edu.au

Keywords: Meridional overturning circulation; Diapycnal mixing; Kinetic energy; Ocean circulation

1. Introduction

The energy budget associated with the overturning circulation of the oceans provides insights into the governing dynamics. However, the budget currently contains many unknowns (e.g., see Huang 1998, 1999; Wunsch and Ferrari 2004). The resolution of these unknowns is essential in addressing problems such as the response of the overturning circulation to changes of forcing and in highlighting processes that need to be addressed in the development of general circulation models and climate models.

One area of debate motivating this paper concerns the extent to which the overturning circulation is powered by energy inputs from the surface winds and tides. Numerous energy analyses (e.g., Wunsch and Ferrari 2004) point out that the expansion and contraction of surface waters owing to surface buoyancy forcing can result in a small net change in potential energy. A current popular interpretation is that this mechanical energy input is unimportant for powering the circulation (Huang 1999; Wunsch and Ferrari 2004; Kuhlbrodt et al. 2007). However, a simple example illustrates the flaws in this reasoning (Fig. 1). The potential energy and total mass in the box are identical in Figs. 1a and 1b, but a flow will occur in Fig. 1b and not in 1a. The reason is that the fluid in Fig. 1b is not at equilibrium and therefore possesses available potential energy, which is released by driving a flow that acts to restore equilibrium. The rate of such conversion of available potential energy was estimated by Griffiths and Hughes (2004) for the (steady state) oceans as the product of the rate at which buoyancy is removed at the surface and the height through which the dense water falls. Based on the observed meridional heat transport (approximately 2 PW) and the assumption that dense surface waters sink to the bottom (at approximately 4-km depth), this conversion rate evaluated to 0.5 TW and therefore constitutes a significant contribution to that required for the maintenance of the density structure—thought to be of order 2 TW for the abyssal oceans (e.g., Munk and Wunsch 1998; Wunsch and Ferrari 2004), and probably significantly less if wind-induced upwelling (Toggweiler and Samuels 1998; Webb and Suginohara 2001; Gnanadesikan et al. 2005) and entrainment into sinking regions (Hughes and Griffiths 2006) are taken into account. It is the role of available potential energy in the ocean overturning circulation that we examine specifically in this paper.

Overlooking the role of available potential energy has led to ill-founded conclusions. The notion that the potential energy is little changed by balanced heating and cooling at the surface has been taken as confirmation of “Sandström’s theorem” (Huang 1999; Wunsch and Ferrari 2004; Kuhlbrodt et al. 2007), and to mean that “a closed steady circulation can be maintained in the ocean only if the heating source is situated at a lower level than the cooling source” (Defant 1961). It follows from these arguments that an overturning circulation ought not to be sustained in the simplified case of horizontal convection, where a closed volume is subject to heating and cooling along one horizontal boundary. However, this prediction is at odds with both laboratory experiments and numerical simulations of horizontal convection (Rossby 1965; Mullarney et al. 2004; Wang and Huang 2005). Indeed, this discrepancy motivated Coman et al. (2006) to re-create Sandström’s original experiments, finding his observations to be in error. In any real fluid, heat conduction has been shown (Jeffreys 1925; Coman et al. 2006) to render Sandström’s conclusions irrelevant (by effectively allowing heating to occur at levels below that of cooling). This is the case whether or not turbulent stirring is present in the flow.

Given the importance of available potential energy in horizontal convection forced purely by surface buoyancy fluxes, its role in the energy budget of the ocean overturning circulation needs to be carefully considered. Our aim here is to investigate how pathways associated with available potential energy modify the energy budget of the turbulent ocean. In section 2 we construct an energetics framework to describe the energy pathways and fluxes, and consider this in section 3 within the context of the ocean circulation. The approach we take in the remainder of the paper is to use a nonhydrostatic general circulation model to simulate an idealized flow subject only to surface buoyancy forcing. In doing so, we evaluate the effects on the energetics of common parameterizations typically used in numerical models for the ocean circulation. The results serve to clarify the significant energy pathways in the ocean overturning circulation, which are summarized in section 5.

2. Energetic framework

In this section we write down the equations describing the energy budget of the oceans. Similar equations have been derived previously (e.g., Winters et al. 1995; Huang 1998; Kuhlbrodt et al. 2007). We follow Winters et al. (1995) in delineating available and background potential energy, while also separating the mean and turbulent components of the kinetic energy.

Consider the Boussinesq equations of motion, written in tensor notation with summation implied:
i1520-0485-39-12-3130-e1
i1520-0485-39-12-3130-e2
i1520-0485-39-12-3130-e3
Here, ui is the velocity vector, xi the position vector, and t time. The subscript i = 3 corresponds to the vertical direction, which we define to be positive upward with the origin at the base of the ocean. For simplicity we assume a linear equation of state; the density is given by ρ and the constant reference density is ρ0. Molecular viscosity (ν) and diffusion (κ) coefficients are used.
The flow may be divided into large-scale (or “mean”) and small-scale (termed turbulent or fluctuating) flow components. The mean flow is obtained using an unspecified averaging filter that eliminates the fluctuations while still permitting slow variations of the mean flow. That is,
i1520-0485-39-12-3130-eq1
where the overbar indicates the mean flow and primed quantities denote fluctuating components. We define the averaging filter so that = 0.
Similar decompositions of the pressure and density fields allow us to rewrite (1)–(3) as
i1520-0485-39-12-3130-e4
i1520-0485-39-12-3130-e5
i1520-0485-39-12-3130-e6
where the averaging filter has been applied after decomposition to eliminate single perturbation terms.

This equation set can be used to determine the energy sources, sinks, and reservoirs of a turbulent ocean. For simplicity we assume that the ocean is rectangular, with no volume flux through the boundaries, and buoyancy flux only through the upper boundary. We now proceed to derive the kinetic energy and then the potential energy budgets for this flow. As we seek a description of the meridional overturning circulation (MOC) in terms of mechanical energy, we do not attempt to also close the internal energy budget [the reader is instead referred to recent discussions of internal energy considerations by Kuhlbrodt et al. (2007) and Tailleux (2009)]. In the current paper, internal energy is invoked only insofar as transfers to and from mechanical energy are required.

a. Kinetic energy

The kinetic energy of the mean flow is
i1520-0485-39-12-3130-e7
where V is the volume of the rectangular basin. Multiplying the momentum Eq. (4) by , integrating over the volume of the domain, and using boundary conditions of no normal flow gives
i1520-0485-39-12-3130-e8
where w = u3 is the vertical velocity and S is the surface bounding the volume V with unit normal nj.
Following Winters et al. (1995), we make several definitions to allow a shorthand notation for the above equation. The first term in (8) is the rate of production of the fluctuation kinetic energy from the mean shear (Tennekes and Lumley 1972) and is defined as
i1520-0485-39-12-3130-eq2
Second, we define the reversible buoyancy flux (Winters et al. 1995) of the mean flow to be
i1520-0485-39-12-3130-eq3
The third term in (8) is the rate of working by the boundary stress applied at the surface of the domain. For the oceans, we expect this term will constitute a net input of energy from the wind stress and thus write it as
i1520-0485-39-12-3130-eq4
although we note that boundary friction will provide a sink of energy here. In addition, we have not included explicitly terms to account for the energy input from the tides. Instead, we note that Φτ can be interpreted more generally to consist of the rate of energy input from external sources, which include the tides and surface wind stress (e.g., see Kuhlbrodt et al. 2007). Finally, we have the internal dissipation rate from the mean flow:
i1520-0485-39-12-3130-e9
which is a positive definite sink of kinetic energy. Thus, (8) simplifies to
i1520-0485-39-12-3130-e10
If the ocean is turbulent, then the kinetic energy in the fluctuation field will also play a significant role in the energy budget. To derive the appropriate energy equation, the full momentum Eq. (1) is multiplied by ui:
i1520-0485-39-12-3130-e11
Equation (11) can be decomposed into fluctuating and mean flow components, and averaging filters applied to eliminate single perturbation terms. This gives an equation incorporating both mean and turbulent kinetic energy; the mean component is subtracted so that integration over the domain yields
i1520-0485-39-12-3130-e12
where
i1520-0485-39-12-3130-e13
is the kinetic energy associated with the fluctuating component. The terms in Eq. (12) are, respectively, the rate of production of the turbulent kinetic energy (balancing the rate of loss from the mean kinetic energy, ), and the fluctuating component analogs of the buoyancy flux, wind stress working, and dissipation from (10). Equation (12) is written in shorthand form as
i1520-0485-39-12-3130-e14

b. Potential energy

The gravitational potential energy of the system is given by
i1520-0485-39-12-3130-e15
Note that there is no net potential energy associated with the fluctuating component because = 0. The potential energy can be evaluated by substituting (6) into (15), and integrating
i1520-0485-39-12-3130-e16
which is written in shorthand as
i1520-0485-39-12-3130-e17
The first two terms in (17) are the reversible buoyancy flux terms and describe the exchange of energy with the mean and turbulent kinetic energy [Eqs. (10) and (14), respectively]. The third term,
i1520-0485-39-12-3130-eq5
is the rate of energy supply owing to heat input through the surface of the domain. An important point in this analysis, and one alluded to by many previous authors, is that no net heat input occurs through the ocean surface (z = H) in a steady state. Thus, Φb1 = 0 when the surface heating and cooling are in balance.

The final term in (17), dubbed Φi[= −κgA(ρtop − ρbottom)], is the rate of conversion of internal to potential energy (Winters et al. 1995) and will be analyzed more completely in the following sections.

Background potential energy is the amount of potential energy remaining if the entire domain is adiabatically resorted into its minimum potential energy state. It can be written analytically following Winters et al. (1995):
i1520-0485-39-12-3130-e18
where
i1520-0485-39-12-3130-eq6
and the Heaviside step function is defined to be
i1520-0485-39-12-3130-eq7
The physical interpretation of z* is that it is the height to which a fluid element will move if the entire domain is adiabatically resorted to give a stable stratification. A key point about z* is that it depends upon the densities everywhere in the box. For this reason, a decomposition of z* into terms associated with the mean and fluctuating components in the flow is intractable, and we are forced to consider the total background potential energy.
The evolution of the background potential energy is derived by Winters et al. (1995) and can be found by substituting (3) into (18) to give
i1520-0485-39-12-3130-e19
We write z*(∂ρ/∂xi) = (∂ψ/∂xi), where ψ = ∫z* dρ, and apply boundary conditions of zero normal flow to give
i1520-0485-39-12-3130-e20
The first term here, Φb2, is similar to Φb1 from (17), except that it is the rate of supply of energy by buoyancy into the background potential field at level z*:
i1520-0485-39-12-3130-e21
Thus, provided the surface buoyancy fluxes maintain a density distribution in a state that is not at rest, even zero net buoyancy input at the surface will lead to a continual transfer of energy. Over the ocean, it is reasonable to expect that Φb2 is positive, describing a transfer of energy away from the background potential energy (which, as we shall see, produces available potential energy). The second term in (20) is
i1520-0485-39-12-3130-e22
and is the rate of energy conversion owing to the irreversible mixing of density (Winters et al. 1995). Given that the gradient of z* with density is negative, the irreversible mixing term is positive definite and will in general exceed Φi (which equates to vertical transport through the depth by molecular diffusion) in a turbulent field.
Available potential energy is defined to be the difference between the potential energy and the background potential energy:
i1520-0485-39-12-3130-e23
This is the amount of potential energy available to be converted into kinetic energy and, we will argue, is more dynamically relevant than potential energy Ep. The evolution equation for the available potential energy is simply
i1520-0485-39-12-3130-e24

The set of Eqs. (10), (14), (20), and (24) is represented schematically in Fig. 2. The kinetic and potential energy reservoirs (toward the bottom and top, respectively) are each divided into two separate reservoirs (Ek and E′k, and Ea and Eb, respectively), with exchanges of energy depicted by arrows. External forcing is shown by incoming arrows. We have also included in this diagram a reservoir of internal energy, which absorbs energy dissipated from its kinetic form by viscosity and which contributes energy through Φi to the background potential energy field. However, internal energy is dominated by heat flux terms, which are orders of magnitude larger than the terms shown here. The energetics (mechanics) of motion is not directly related to those heat fluxes and so they are omitted from our budget. Accordingly, we use a dashed boundary in Fig. 2 to denote that the internal energy reservoir is not closed by the above formulation. For a full analysis of the internal energy budget, see Kuhlbrodt et al. (2007) and Tailleux (2009).

The term denoted Φb2 is the rate of generation of the available potential energy by the surface buoyancy fluxes. This term represents a direct conversion from background to available potential energy, but it is catalyzed by an external energy source—the flux of buoyancy (internal energy) through the surface. It is only via this term (21) that surface buoyancy fluxes create motion.

The rate of conversion of internal to potential energy, Φi, occurs explicitly in the available potential energy Eq. (24) but not in the background potential energy Eq. (20). This is surprising because the molecular diffusion of a stably stratified fluid will alter the height of the center of mass (as diffusion acts to force the stratification toward thermodynamic equilibrium). If we consider, for simplicity, a fluid with a constant coefficient of thermal expansion, diffusion will increase the background potential energy without altering the available potential energy. This confusion can be resolved by realizing that Φi is a component of the irreversible mixing, Φd. For this reason we have chosen in Fig. 2 to illustrate Φi as contributing directly to the background potential energy, Eb, and the conversion of available to background potential energy via irreversible mixing as (Φd − Φi).

Figure 2 provides a basis from which to analyze the energy budget of the oceans, with particular reference to existing theory.

3. Discussion

a. Available potential energy

Figure 2 shows the inputs and outputs of energy, as well as internal transfers. Previous authors (Huang 1998, 1999; Wunsch and Ferrari 2004) have noted that Φb2 does not alter the potential energy, or the energy budget of the system as a whole, and conclude that surface buoyancy flux is not an important part of the circulation. However, this does not imply that the generation of available potential energy is unimportant. The primary result of Fig. 2 is that, when the system is in steady state, the rate of generation of available potential energy is exactly balanced by the rate of dissipation by irreversible mixing:
i1520-0485-39-12-3130-eq8
This point is also made in a recent paper by Tailleux (2009).
The link between rates of available potential energy generation and dissipation can be probed by making the assumption (common in oceanographic circles) that the efficiency of mixing is constant. According to Peltier and Caulfield (2003), mixing efficiency is defined as
i1520-0485-39-12-3130-eq9
If we assume globally constant e = ⅕, and that Φi is small (see the discussion in section 3c), then this can be rewritten as
i1520-0485-39-12-3130-eq10
implying that the rate of input of available potential energy is a constant fraction of the rate of input of other mechanical energy to the system.

The above equation does not imply causality, but does imply that when the ocean circulation finds a steady state, it adjusts to both mechanical and surface buoyancy forcings so that the above terms balance. It is likely that this adjustment occurs through the density field. We can consider this statement in the light of two thought experiments (retaining the assumption of constant mixing efficiency). In the first, imagine that the rate of mechanical energy input, Φτ = + Φ′τ is increased. The propagation of energy through the system will be as follows (refer to Fig. 2). There is a net input of kinetic energy, with the partition between the mean and fluctuating flow components being dependent upon the external mechanical forcing. Balance will be reestablished by increasing the rate of generation of turbulent kinetic energy, together with a corresponding increase in the rate of dissipation ϵ′. This equilibration will involve an increase in the rate of production of E′k from the mean shear (ΦT), but may also involve production from mean kinetic energy via available potential energy, that is, and Φ′z. It is worth noting that the transfer pathway from the mean shear via the available potential energy can operate in either direction. Nevertheless, the outcome is the same. The rate at which the available potential energy is removed by irreversible mixing is a fraction of the rate at which it is generated. Thus, the additional mechanical working enhances the rate of mixing and alters the density field. Consequently, the rate of production of the available potential energy changes; the final steady state depends upon each of these terms balancing.

The second thought experiment involves an increase in the surface buoyancy forcing, without altering the mechanical energy input. Increased buoyancy fluxes will act to enhance the generation rate of the available potential energy and, once a steady state is reached, the rate at which it is removed by irreversible mixing will again be in balance. The increased rate of irreversible mixing (under the assumption of constant mixing efficiency) depends upon an increase in the rate of viscous dissipation; this connection is achieved via enhanced buoyancy fluxes transforming the available potential energy to kinetic energy. It is reasonable to expect that this process will modify the overturning circulation.

There is scant evidence that the assumption behind these thought experiments (that mixing efficiency is a global constant) is true (Ivey et al. 2008). Thus, our argument is based on the idea that some proportion of the available potential energy production is converted to background potential energy via irreversible mixing. The primary result of the energetic framework exposed here is that, while available potential energy production does not change the globally averaged amount of potential energy, it does contribute energy to the ocean at a rate that balances the total rate of irreversible mixing. Thus, we contend that, contrary to the ocean energy synthesis of Wunsch and Ferrari (2004), surface buoyancy forcing produces available potential energy and is likely to be a first-order term in the ocean energy budget.

b. Comparison with previous work

In this section we examine how our energetics framework relates to previous work. The above analysis has emphasized that for a (quasi-)steady-state circulation it is the energy conversion rates between reservoirs that are relevant to the energy budget. However, the absolute amounts of energy in the kinetic and potential reservoirs must also be accounted for when seasonal or hemispheric budgets are analyzed for the oceans (e.g., see Oort et al. 1994). For simplicity, we restrict our attention to an energy budget for a (quasi-)steady state that is equivalent to a global annual mean within the context of the oceans.

1) Steady circulation without mechanical energy input

As noted earlier, numerous authors have equated no net surface heating of the oceans to mean that the buoyancy forcing cannot supply energy to maintain an overturning circulation. However, the same physics is applicable to horizontal convection having no mechanical energy input, where surface buoyancy fluxes clearly generate available potential energy (Φb2 > 0) from background potential energy to maintain an overturning circulation. Irreversible mixing Φd returns the available potential energy to background potential energy at the same rate; hence, no net change occurs to the total potential energy. That is, the dynamics of flow involves energy conversions.

As pointed out by Paparella and Young (2002), in the absence of wind forcing (i.e., Φτ = Φ′τ = 0), a steady-state flow has Φi = ϵ(= −z − Φ′z). This result places a strong constraint on the total rate of viscous dissipation + ϵ′ from the flow. However, it is important to note that this result (for finite ν and κ) does not strongly constrain the mean circulation (which will tend to be characterized by relatively large length scales). It is also worthy of note that Paparella and Young (2002) go on to argue that their result implies the flow is not turbulent (according to the “zeroth” law of turbulence) because the viscous dissipation tends to zero in the inviscid limit as ν → 0. We point out that Paparella and Young (2002) make two choices in their analysis: to allow κ → 0 as ν → 0, and to consider only fixed temperature boundary conditions. In taking these limits, the consequence is that Φb2 → 0: no available potential energy is created, so there can be no dissipation of kinetic energy. On the other hand, generalized boundary conditions that prescribe a component of the heat flux would give rise to a paradox: available potential energy is then created and can only be returned to background potential energy (in a statistically steady flow) if irreversible mixing occurs (i.e., Φd > Φi). This implies the existence of density variations and motion on small length scales in the flow. We suggest that this paradox can only be resolved if the flow evolves to have large top-to-bottom density differences in the limit κ, ν → 0, thereby allowing Φi to become large enough to supply sufficient energy to maintain fluctuations in the flow.

2) Ocean overturning circulation with energy from winds and tides

The energetics framework shown schematically in Fig. 2 illustrates how the input of the external energy Φτ from winds and tides plays a role in the MOC through the generation of available potential energy. We consider two categories of external forcing. First, external forcing of sufficient strength and coherence could drive a mean flow that creates available potential energy (i.e., positive ), which we expect to be balanced by release at relatively small length scales (i.e., negative Φ′z). An example is a wind-induced Ekman pumping “cell” in which dense waters are upwelled, and subsequently interact and combine with less dense waters in the vicinity of fronts, as in the Southern Ocean (Toggweiler and Samuels 1998; Webb and Suginohara 2001; Gnanadesikan et al. 2005). This type of cell was termed thermally indirect by Nycander et al. (2007). Although the upwelling may primarily follow isopycnal surfaces and the combination of water masses is often categorized as lateral (rather than vertical) mixing, we point out that surface buoyancy fluxes and cross-isopycnal mixing are integral to maintaining the density structure and thus the cell in the overall circulation. Hence, Φb2 and Φd (which depends upon z*) must be important in the energetics.

Second, the external forcing may induce motion at relatively small length scales (from the mean flow via ΦT or directly via Φ′τ). For instance, the external actions of winds and tides lead, through internal wave dynamics and interactions with topography, to a cascade of energy to localized turbulence (e.g., see Wunsch and Ferrari 2004; St. Laurent and Simmons 2006). Available potential energy is created by small-scale motions working against the stratification (i.e., positive Φ′z), and viscous dissipation and irreversible mixing will occur. The resulting (“thermally direct”; Nycander et al. 2007) mean flow will tend to release available potential energy (i.e., negative ), and surface buoyancy fluxes must again correspond to an important energetic term for the maintenance of the density structure.

Many authors have pointed out that the overall energy budget of the MOC is significantly reduced by wind-driven processes in the Southern Ocean (e.g., Toggweiler and Samuels 1998; Webb and Suginohara 2001; Gnanadesikan et al. 2005; Kuhlbrodt et al. 2007) upon comparison with that required if the circulation (albeit for the smaller volume comprising the abyssal waters only) corresponded to a balance between the upwelling and downward diffusion of heat (Munk 1966; Munk and Wunsch 1998). We can now see the context within which this is so. Wind-induced upwelling over vertical distances significantly greater than the largest turbulent overturn scale, followed by a component of stirring parallel to ∇ρ, will bring fluid parcels with relatively larger density contrasts into close proximity. [It is also worth noting that the global effects of entrainment into high-latitude sinking regions, studied by Hughes and Griffiths (2006), lead to a reduced energy budget for the overturning circulation for similar reasons.] Thus, for a given density structure (i.e., the distribution of ρ as a function of z* is fixed) in the oceans, (22) suggests that the rate of energy conversion owing to irreversible mixing would be much greater than that expected for (turbulent) diffusion in the same elemental volume. Put another way, the same rate of irreversible mixing is expected to accompany a lesser rate of external energy supply to the available potential energy, if the external energy is supplied via the mean flow pathway.

Figure 2 shows that for the circulation to be in a steady state, there can be no net buoyancy flux Φz(= + Φ′z) from the kinetic to the potential energy reservoir (assuming in the oceans that Φi is small; see section 3c). Widespread recent discussion in the oceanographic literature as to whether the overturning circulation is “pulled” by mixing (which is reliant on the generation of buoyancy fluxes at small scales Φ′z) or “pushed” by buoyancy forces (associated mainly with ) therefore makes little sense; a balance must operate, as already pointed out by Hughes and Griffiths (2006), on mechanical grounds. It is also worth considering Fig. 2 in light of numerous studies that have considered only a single reservoir each for potential and kinetic energy. By not allowing explicitly for the available potential energy, and further neglecting the role of surface buoyancy forcing in the energetics (on the basis of reasoning discussed earlier), irreversible mixing is itself excluded from the analysis. Thus, the conclusion that mixing “drives” the circulation (e.g., see Kuhlbrodt et al. 2007) presents an inherent contradiction.

c. Approximate energy pathways

We first consider the active energy pathways for a completely steady overturning circulation (i.e., without any turbulent or fluctuating component) that might be termed a weakly diffusive laminar flow. This is the flow that would result from a surface buoyancy forcing that is sufficiently weak that molecular diffusion alone maintains the potential energy reservoir at a constant level. Pathways reliant on fluctuations are unavailable as there is no fluctuating component of the flow. In this limit, Fig. 2 shows that a single one-way pathway exists for the transfer of energy. The conversion of internal energy to mechanical (potential) energy occurs at a rate Φi as molecular diffusion acts to force the system toward thermodynamic equilibrium. For the system to be in a steady-state balance, differential heating at the surface acts to reduce the background potential energy at the rate Φb2 = Φi, thus leading to the generation of available potential energy at the same rate. The associated buoyancy forces lead to a laminar flow in which balances Φb2. Note that one characteristic of this laminar flow is that the isotherms are barely distorted from their equilibrium positions and Φd = Φi; that is, the available potential energy is not removed by molecular diffusion in the flow. Viscosity acts to dissipate kinetic energy at the rate (note that the turbulent kinetic energy reservoir is absent), which is in balance with the rate of generation (= Φb2 = Φi). External energy input, if present, would merely increase the rate of viscous dissipation, so that = + Φi (i.e., no net work is done against buoyancy).

In contrast to the “laminar” energetic pathway just described, Fig. 2 highlights the possibility of more complicated pathways associated with flows that contain fluctuations. Explicit consideration of the available potential energy and fluctuation kinetic energy reveals two “loops” of energy cycling in this diagram (note, however, that these loops are not regarded as closed and independent; see also Winters and Young 2009). One might be referred to as the available potential energy (APE) loop, where APE is created at the expense of background potential energy (with no change to the potential energy itself). The other loop we call the turbulent kinetic energy (TKE) loop and it involves conversion at a rate ΦT of mean flow kinetic energy to that associated with the fluctuating component of the flow. The fluctuations in the velocity field excite a net upward buoyancy flux (Φ′z > 0), which we suggest is balanced by the advection of the density field by the mean flow < 0.

There is widespread agreement that in the oceans the presence of turbulence is important for the overturning circulation, and hence the release of internal energy by molecular diffusion Φi constitutes a relatively small term in the energetics budget. It is convenient to look at the energetics of this circulation in the simplified or approximate balance illustrated in Fig. 2, where we consider that terms corresponding to gray pathways and arrows are likely to be small in the oceans, and are thus eliminated. We have assumed that, provided the flow is turbulent, then ϵ′ ≫ and Φd ≫ Φi. We have also assumed that Φ′τ ≪ and Φb1 = 0. The resulting energy balance is substantially simplified and serves as a good model for ocean energetics.

The rate of viscous dissipation in the simplified energy budget is equal to the rate of input of mechanical energy, and is not affected by the cycling of energy around each of the APE and TKE loops. Many authors have interpreted this to mean that the dynamics associated with the energy loops are insignificant. However, Fig. 2 shows that available potential energy is crucial in facilitating the overturning circulation. In particular, the neglect of the forcing associated with either Φb2 or would lead to an overturning circulation that cannot be maintained in a statistically steady state. Thus, we argue that the roles of the external energy input and surface buoyancy forcing are complementary. In section 4 we confirm the importance of the APE and TKE loops by demonstrating that their respective rates of energy cycling are consistent with the time scale for the energy reservoirs to reequilibrate following changes in the flow forcing.

d. The energetics of ocean models

Given that the equations of motion for a Boussinesq ocean are known, it should in principle be possible to solve Eqs. (1)–(3) numerically and thereby test the relative sources and sinks of energy shown in Fig. 2. In practice, however, ocean models cannot resolve the length scales associated with rapid fluctuations in the flow, so that quantities such as the turbulent buoyancy flux (and thus the evaluation of Φ′z) must be parameterized. In this section we construct a revised energy budget that applies to a simple ocean model, and proceed in the following section to use an ocean model to illustrate the role of available potential energy.

There are two processes that will usually be parameterized in a large-scale ocean model. The first is the turbulent transport (associated with the fluctuating flow component), usually written as an enhanced diffusion and viscosity. The second is convection.

Turbulent parameterizations can include a wide range of schemes of differing complexity; here we focus on the simplest possible parameterization by representing the turbulent transports as a diffusive process (with transport down the mean gradient) with a uniform coefficient. That is,
i1520-0485-39-12-3130-e25
i1520-0485-39-12-3130-e26
where Kν and Kρ are the coefficients for momentum and density transport, respectively. We assume here that the mean flow is modeled accurately by the ocean model, and that all turbulent processes are incorporated into the parameterized transports. Furthermore, it is inherently assumed that changing the coefficients (for a given mean gradient) alters the turbulent transport (which is parameterized) relative to that in the mean flow (which is simulated numerically). This is usually not a good assumption, but it is the best that can be done with the available computing resources. This approach does not alter our main qualitative conclusions regarding the energetics.
The parameterization scheme outlined above allows redefinition of the terms that make up the energy budget. The term representing the rate of conversion by the turbulent buoyancy fluxes becomes
i1520-0485-39-12-3130-e27
which resembles the internal to potential energy conversion term, Φi, from (14). We point out that the turbulent buoyancy flux term Φ′z is not synonymous with the rate of background potential energy generation by irreversible mixing. The rate of irreversible mixing is determined by the diffusion of density down the local density gradients and, in a steady circulation, is regulated by surface buoyancy forcing [following from the discussion in section 3b(2)]. On the other hand, contributions to the turbulent buoyancy flux term Φ′z arise from localized stirring, which generates pressure gradients and thus flow on larger (mean flow) length scales, such that = −Φ′z. Although stirring tends to increase the local density gradients, there is no reason to expect that −Φ′z = Φd in general.
The background potential energy equation needs to be rederived by substituting (6) into (18). In that equation, the surface buoyancy term is then
i1520-0485-39-12-3130-e28
and the irreversible mixing term becomes
i1520-0485-39-12-3130-e29
These terms retain the same form as the original formulation, but use as the diffusion coefficient a turbulent Kρ, rather than the molecular κ. The use of Kρ in the surface flux term (28) recognizes (crudely) that the air–sea heat transfer involves breaking waves and radiation, and therefore cannot be based on the product of the molecular diffusivity and the mean gradient. The conversion term between the mean and turbulent kinetic energy flow components becomes
i1520-0485-39-12-3130-eq11
which now mimics the form of the dissipation term [cf. in (9)]. Finally, since the model does not simulate the turbulent kinetic energy, we set E′k = ϵ′ = 0. The terms in the parameterization-based formulation give rise to the model energy budget drawn schematically in Fig. 3.
The model energy budget shows that the APE loop is not significantly altered. Both the surface buoyancy flux and irreversible mixing terms take the same form as the real physical case [see (21) and (22)], but are altered (in both z* and ρ) by the omission of turbulent fields and the larger coefficient for downgradient density transport. On the other hand, the TKE loop is substantially altered. The TKE reservoir is eliminated; the turbulent buoyancy flux term now resembles that based on a diffusive flux and ΦT is the equivalent of a dissipative term. Importantly, there is no direct pathway from TKE production to turbulent buoyancy flux. In the steady state, we can see that
i1520-0485-39-12-3130-eq12
That is, in the absence of external energy input, the parameterized turbulent buoyancy flux term Φ′z balances the parameterized rate of the “viscous” dissipation ΦT [a parameterized version of the constraint obtained by Paparella and Young (2002)].

The second primary parameterization used in ocean models is that of convection through convective adjustment (e.g., Huang 1998; Nycander et al. 2007). Convective adjustment schemes mix vertically to eliminate statically unstable density gradients immediately. This has two consequences. One is to provide a direct sink of available potential energy without conversion to kinetic energy. The second is that statically unstable surface density anomalies are rapidly diluted, altering the z* distribution, and thereby markedly reducing the available potential energy production from the surface buoyancy flux. Convective adjustment thus renders the APE loop irrelevant.

We contend that a more accurate representation of the energetics of convection is critical for correctly modeling the available potential energy input into the ocean overturning circulation. For this reason, in the following section we explicitly model convection as a mean flow buoyancy flux (as in the regions of mean sinking of the MOC) using a nonhydrostatic model. We can therefore estimate the rate of available potential energy generation by the surface buoyancy flux and examine the dependence of this rate on the intensity (as characterized by the transport coefficients) of parameterized turbulence. We also compare the results for one case with those obtained using a convective adjustment scheme and otherwise identical parameters.

4. Model results

a. Parameters and model description

Modeling the overturning circulation, with the inclusion of convection, requires very high resolution in the vertical and horizontal planes, as well as a fully nonhydrostatic model—both of which add significantly to the computational cost. To be able to model this system, some simplifications are necessary: we consider nonrotating Boussinesq flow in two dimensions (the y–z plane) within an idealized domain, the buoyancy flux at the free surface is specified as a heat flux, and density is given by a linear equation of state (with a thermal expansion coefficient of α = 2 × 10−4 °C−1).

We use the Massachusetts Institute of Technology’s General Circulation Model (MITgcm) in fully nonhydrostatic mode. The box is 4000 m deep (with 80 grid cells varying from 75 to 10 m in vertical spacing) and 4000 km long in the y direction (with horizontal resolution ranging from 7.5 km in nonconvecting regions to 750 m in the high-latitude sinking regions). Sloping endwalls (with a steep 1% gradient) are introduced into the rectangular box to mimic the formation of gravity currents. The domain has no-slip boundaries on the bottom and endwalls, with an implicit free surface. We assume a specific heat capacity of 3900 J kg−1 °C−1 and a reference density of 1000 kg m−3.

The surface buoyancy forcing is a prescribed distribution of the heat flux—a half-sinusoid antisymmetrical about the center of the domain. As a standard case, we apply the maximum heat flux Q0 = 200 W m−2 (heat input) at one end of the box. At the other end the surface flux is −200 W m−2 (cooling). Symmetry about the midpoint of the domain means that the initial temperature (T = 20°C) remains the average temperature for the duration of the simulation, substantially reducing the required spinup time (relative to a run with thermal relaxation boundary conditions, e.g., in which the bulk temperature of the whole domain has to adjust to reach equilibrium). The surface buoyancy forcing is varied to gauge the effects on the generation of the available potential energy.

Wind forcing and tides are set to zero (i.e., Φτ = 0), but a uniform level of turbulence is assumed and parameterized by the vertical turbulent diffusion coefficient Kz. For simplicity, the coefficients describing the vertical diffusion of density and momentum are chosen to be equal (i.e., Kρ = Kν) and set to Kz. This diffusivity is varied to examine the effects of turbulent mixing on the circulation. Horizontal diffusion and viscosity are uniform with a coefficient Kh = 25 m2 s−1 in all runs. To obtain accurate energy conversion rates, we recalculate the background density distribution (i.e., the z* field) at each time step in all simulations.

b. Steady states

We first consider in Fig. 4 the dependence of the two-dimensional overturning circulation on the vertical diffusion coefficient Kz, where time-averaged streamlines are shown as a function of Kz (note that the streamfunction contour interval varies between plots). The thermocline depth is indicated by the 20°C isotherm. The mean circulation in each case features a tightly confined sinking region at one end wall and a relatively steady component of shallow circulation through the thermocline. The maximum mean overturning decreases with Kz, but it is primarily the shallow thermocline circulation that decreases. The Kz = 10−4 m2 s−1 case is the only one in which the deep circulation exceeds the shallow circulation.

A surprising feature of these simulations is that they exhibit strongly time-dependent behavior, and that this time dependence increases as Kz is reduced. The time dependence takes the form of a series of strong internal seiches and bores with a period of roughly 200 days, and which interact with the boundary conditions so that bottom water production is intermittent. It could be argued that the seiching behavior is aphysical. However, the resulting intermittent bottom water formation is not inconsistent with seasonal cycles. Thus, our approach has been to diagnose the circulation and energy fluxes over many cycles, so that the energetic analysis will be unaffected.

The top-to-bottom difference in the horizontally averaged density Δρ within the domain is shown in the caption of Fig. 4 for each case. This difference is very small when the diffusion is large, but approaches realistic values when Kz = 10−4 m2 s−1. It is reasonable to expect that this trend would continue with smaller Kz; this result lends support to the conjecture outlined in section 3b(1), which stated that the results of Paparella and Young (2002), when applied to constant buoyancy flux boundary conditions, would result in Δρ → ∞ as Kz → 0.

The rates of energy conversion between reservoirs can be diagnosed (Fig. 5) by considering averages over time scales significantly longer than those characterizing the long-period fluctuations. As expected, the energy conversion rates increase with Kz, and at each Kz, it can be seen that Φb2 ≈ Φd and − ≈ Φ′z ≈ ΦT. Importantly, the concept of balanced conversions in the steady state around each of the APE and TKE loops in Fig. 3 is supported. The discrepancies in these balances in Fig. 5 are due to the effects of dissipation from the numerical diffusion of density and momentum, and are small. For these simulations, the conversion rate in the APE loop is significantly greater than that in the TKE loop. It can be expected that the ratio of conversion rates in these two loops is different in the real ocean, where the TKE loop is fully operational and mixing is more closely coupled to the mechanical energy input. Nonetheless, it is apparent that the amount of irreversible mixing must still be regulated by the available potential energy generation from the surface buoyancy fluxes.

Figure 6 shows the dependence of the circulation upon the surface buoyancy flux. These results were obtained by altering the maximum surface heat flux from a standard solution with Q0 = 200 W m−2 (and Kz = 10−3 m2 s−1; Fig. 6b) to either 150 or 250 W m−2. The circulations in Figs. 6a–c correspond to the subsequent mean steady states; discussion of the transient evolution is delayed until section 4c. The flow structure consists of a tightly confined sinking region and a coherent full-depth overturning cell. As might be anticipated, the strength of the overturning circulation is observed to increase with the surface buoyancy throughput.

Plotted in Fig. 7 are the energy conversion rates as a function of maximum surface heat flux for the simulations in Fig. 6. The rates increase with Q0 and balanced conversions in the APE and TKE loops are observed in the steady state, that is, at each Q0, Φb2 ≈ Φd and − ≈ Φ′z ≈ ΦT. As above, the conversion rates in the APE loop are significantly greater than those in the TKE loop.

Table 1 summarizes the energy states for Kz = 10−3 and 10−1 m2 s−1 and the range of surface buoyancy forcings examined. Increasing the surface buoyancy forcing or decreasing Kz decreases the background potential energy (and also the total potential energy Eb + Ea) because the range of temperatures in the basin must increase. A stronger circulation is expected to result from increased buoyancy throughput and/or increased vertical diffusion because this increases both the mean kinetic energy and the available potential energy in the basin. It is striking that energy is partitioned between the available potential and mean kinetic energy reservoirs in approximately equal amounts, independent of Kz and Q0. The reason for this partition is not known.

An additional run with Kz = 10−3 m2 s−1 and Q0 = 200 W m−2 was carried out using MITgcm with a convective adjustment scheme (using hydrostatic pressure) and a mesh identical to that used in the other simulations (see section 4a). The results are compared with the corresponding fully nonhydrostatic simulation in Fig. 8: 8a and 8b highlight the substantial reduction of the energy conversion rates in the APE loop (Φb2 and Φd) and a striking change in the deep mean flow circulation (cf. Fig. 4c), respectively, associated with the use of the convective adjustment scheme. However, the shallow overturning and top-to-bottom difference in the horizontally averaged density in Figs. 8b and 4c are almost identical. This result confirms our assertion of section 3d that convective adjustment parameterization obscures the dynamically important role of available potential energy.

c. Transient adjustment

We consider here the evolution of the energy conversion rates in response to a sudden change in surface buoyancy throughput (Fig. 9). An initially steady-state circulation is established corresponding to a maximum surface heat flux of Q0 = 200 W m−2. The surface buoyancy throughput is changed at time t = 2400 days to that corresponding to Q0 = 150 W m−2 (Fig. 9a) and Q0 = 250 W m−2 (Fig. 9b). Note that at all times there is no net heating of the basin because both the heating and cooling fluxes are changed simultaneously. A vertical diffusion coefficient of Kz = 10−1 m2 s−1 is used in these adjustment experiments to minimize the distraction of large-amplitude long-period fluctuations in the time series. However, the mean response at smaller Kz is similar.

In both adjustment cases examined, it can be seen that the conversion rates to and from the mean kinetic energy reservoir, and ΦT, respond relatively rapidly. The re-equilibration of the available potential energy reservoir takes much longer, as is apparent from the relatively slow adjustment of the conversion rates associated with the irreversible mixing and turbulent buoyancy fluxes, Φd and Φ′z. The time scale for the adjustment seems best characterized by considering the evolution of the background potential energy Eb. Unlike the available potential energy reservoir, there is a single sink (Φb2, which we control) and source (Φd) operating in the approximate balance for Eb. If ΔQ0 is the change in maximum heat input that produces a change in background potential energy ΔEb between the initial equilibrium and re-equilibrated states, the net rate at which energy is (initially) removed from the background potential energy reservoir will be
i1520-0485-39-12-3130-e30
where t* = 0 is the time at which the sudden change is applied. Thus, the time scale for the adjustment Ta can be estimated as
i1520-0485-39-12-3130-e31
Substituting the values in Table 1 into (31), we estimate Ta to be of O(100) days for the adjustment experiments with Kz = 10−1 m2 s−1 (actually 103 and 81 days for the ΔQ0 = −50 and 50 W m−2 cases, respectively). This time scale is consistent with the observed adjustment in Fig. 9 and therefore supports the notion that the coupling between the surface buoyancy fluxes and irreversible mixing is important in the overturning circulation. Were this not the case, we would expect the adjustment to occur on a time scale consistent with the imbalance in energy conversion rates in the TKE loop, which in this case would be the relatively short time that characterizes the response of the kinetic energy reservoir.

5. Conclusions

We have used an energetics analysis to demonstrate that available potential energy is central to the energy budget for the meridional overturning circulation of the oceans. In a steady circulation, irreversible mixing is found to dissipate the available potential energy at the same rate at which it is generated by surface buoyancy forcing, and one transfer cannot exist without the other. Available potential energy is also generated by buoyancy fluxes from kinetic energy, which is in large part sustained by inputs from external sources (such as the winds and tides). Thus, buoyancy fluxes determine the rate of irreversible mixing by drawing on the available potential energy from several sources. A series of numerical experiments suggest that caution is required in obtaining an energetics interpretation of the meridional overturning circulation on the basis of output from general circulation models, in which both convection and turbulent mixing are parameterized. Realistic diagnostics of the energy budget require the use of a nonhydrostatic model.

We conclude that no single energy source should be regarded as dominant in “driving” the overturning circulation. This is consistent with the dynamically based arguments (Hughes and Griffiths 2006) that the surface buoyancy forcing and irreversible mixing must be coupled. The energetics framework developed here provides a new means of estimating the size of important terms involving surface buoyancy forcing and irreversible mixing in the energy budget of the meridional overturning circulation.

Acknowledgments

We thank Trevor McDougall, Rémi Tailleux, Bill Young, Kraig Winters, Anand Gnanadesikan, and an anonymous referee for their insightful discussions and comments that have improved this manuscript. The work was partly funded by the Australian Research Council (DP0664115), and the numerical simulations were supported by an award under the Merit Allocation Scheme on the NCI National Facility at The Australian National University. Additional calculations were performed on the Terrawulf II cluster, a computational facility supported through the AuScope initiative. AuScope Ltd. is funded under the National Collaborative Research Infrastructure Strategy (NCRIS), an Australian Commonwealth Government Programme.

REFERENCES

  • Coman, M. A., R. W. Griffiths, and G. O. Hughes, 2006: Sandstrom’s experiments revisited. J. Mar. Res., 64 , 783–796.

  • Defant, A., 1961: Physical Oceanography. Vol. I, MacMillan, 718 pp.

  • Gnanadesikan, A., R. D. Slater, P. S. Swathi, and G. K. Vallis, 2005: The energetics of ocean heat transport. J. Climate, 18 , 2604–2616.

    • Search Google Scholar
    • Export Citation

  • Griffiths, R. W., and G. O. Hughes, 2004: The energetics of horizontal convection. Proc. Fifteenth Australian Fluid Mechanics Conf., Sydney, NSW, Australia, University of Sydney, AFMC00080. [Available online at http://www.aeromech.usyd.edu.au/15afmc/proceedings/papers/AFMC00065.pdf].

    • Search Google Scholar
    • Export Citation

  • Huang, R. X., 1998: Mixing and available potential energy in a Boussinesq ocean. J. Phys. Oceanogr., 28 , 669–678.

  • Huang, R. X., 1999: Mixing and energetics of the oceanic thermohaline circulation. J. Phys. Oceanogr., 29 , 727–746.

  • Hughes, G. O., and R. W. Griffiths, 2006: A simple convective model of the global overturning circulation, including the effects of entrainment into sinking regions. Ocean Modell., 12 , 46–79.

    • Search Google Scholar
    • Export Citation

  • Ivey, G. N., K. B. Winters, and J. R. Koseff, 2008: Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech., 40 , 169. doi:10.1146/annurev.fluid.39.050905.110314.

    • Search Google Scholar
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  • Jeffreys, H. T., 1925: On fluid motions produced by differences of temperature and humidity. Quart. J. Roy. Meteor. Soc., 51 , 347–356.

    • Search Google Scholar
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  • Kuhlbrodt, T., A. Griesel, M. Montoya, A. Levermann, M. Hofmann, and S. Rahmstorf, 2007: On the driving processes of the Atlantic meridional overturning circulation. Rev. Geophys., 45 , RG2001. doi:10.1029/2004RG000166.

    • Search Google Scholar
    • Export Citation

  • Mullarney, J. C., R. W. Griffiths, and G. O. Hughes, 2004: Convection driven by differential heating at a horizontal boundary. J. Fluid Mech., 516 , 181–209.

    • Search Google Scholar
    • Export Citation

  • Munk, W. H., 1966: Abyssal recipes. Deep-Sea Res., 13 , 707–730.

  • Munk, W. H., and C. Wunsch, 1998: Abyssal recipes II: Energetics of tidal and wind mixing. Deep-Sea Res., 45 , 1976–2009.

  • Nycander, J., J. Nilsson, K. Döös, and G. Broström, 2007: Thermodynamic analysis of ocean circulation. J. Phys. Oceanogr., 37 , 2038–2052.

    • Search Google Scholar
    • Export Citation

  • Oort, A. H., L. A. Anderson, and J. P. Peixoto, 1994: Estimates of the energy cycle of the oceans. J. Geophys. Res., 99 , 7665–7688.

  • Paparella, F., and W. R. Young, 2002: Horizontal convection is non-turbulent. J. Fluid Mech., 466 , 205–214.

  • Peltier, W., and C. Caulfield, 2003: Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech., 35 , 135–167.

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Fig. 1.
Fig. 1.

Two cases with the same potential energy: (a) a homogeneous ocean and (b) a case where surface forcing of the upper layer results in equal and opposite density anomalies. In the latter case there is no change of potential energy, but there is a finite amount of available potential energy.

Citation: Journal of Physical Oceanography 39, 12; 10.1175/2009JPO4162.1

Fig. 2.
Fig. 2.

Energy pathways in the ocean as defined by Eqs. (10), (14), (20), and (24). The transfers between reservoirs of mean kinetic energy , fluctuation kinetic energy E′k, available potential energy Ea, and background potential energy Eb include conversions by the mean flow and turbulent buoyancy fluxes, and Φ′z; conversion via mean shear ΦT; kinetic energy dissipation from the mean and fluctuating flow components, and ϵ′; external input of kinetic energy to the mean and fluctuating flow components, and Φ′τ; internal energy release Φi by molecular diffusion; available potential energy generation due to external buoyancy forcing resulting in a change of bulk density Φb1 and redistribution of mass Φb2 in the volume; and available potential energy dissipation by irreversible mixing Φd. The reservoir of internal energy is not closed by the energetic framework in this paper and is thus denoted by a dashed boundary. Conversions (or directions) shown in gray are assumed to be relatively minor, and their omission yields a schematic for the approximate energetic balance for the oceans. For conversions that can be bidirectional, the direction defined as positive is indicated.

Citation: Journal of Physical Oceanography 39, 12; 10.1175/2009JPO4162.1

Fig. 3.
Fig. 3.

Energetics of an ocean model.

Citation: Journal of Physical Oceanography 39, 12; 10.1175/2009JPO4162.1

Fig. 4.
Fig. 4.

Dependence of the time-averaged overturning circulation upon the vertical diffusion coefficient (surface buoyancy flux is fixed, with Q0 = 200 W m−2). The maximum streamfunction quoted is that for a two-dimensional flow in a basin of 1-m width, while the density range is Δρ = ρbottom − ρtop. The 20°C isotherm is shown in gray.

Citation: Journal of Physical Oceanography 39, 12; 10.1175/2009JPO4162.1

Fig. 5.
Fig. 5.

Dependence of the various energy conversion rates upon the vertical diffusion coefficient, for a fixed buoyancy forcing: Q0 = 200 W m−2. The energy conversions are normalized by the mass M0 of the water in the basin (which, in the numerical simulations, has a width of 1 m).

Citation: Journal of Physical Oceanography 39, 12; 10.1175/2009JPO4162.1

Fig. 6.
Fig. 6.

Dependence of the time-averaged overturning circulation upon the surface buoyancy throughput (Kz is held constant at 10−3 m2 s−1). The maximum streamfunction and density range are as defined in Fig. 4.

Citation: Journal of Physical Oceanography 39, 12; 10.1175/2009JPO4162.1

Fig. 7.
Fig. 7.

Dependence of the various energy conversion rates upon the surface buoyancy throughput. The vertical diffusion coefficient Kz is held constant at 10−3 m2 s−1. The energy conversions are normalized by the mass M0 of water in the basin (which, in the numerical simulations, has a width of 1 m).

Citation: Journal of Physical Oceanography 39, 12; 10.1175/2009JPO4162.1

Fig. 8.
Fig. 8.

Summary of the model results for the statistically steady-state circulation with Kz = 10−3 m2 s−1 and Q0 = 200 W m−2, obtained using a convective adjustment scheme: (a) various energy conversion rates (per unit mass) compared with the corresponding fully nonhydrostatic simulation and (b) time-averaged overturning circulation (contour intervals as in Fig. 4c). The maximum streamfunction and density range are as defined in Fig. 4, and the 20°C isotherm is shown in gray. The maximum streamfunction below the 20°C isotherm is 9.4 × 103 kg s−1, which compares with 30.8 × 103 kg s−1 in Fig. 4c.

Citation: Journal of Physical Oceanography 39, 12; 10.1175/2009JPO4162.1

Fig. 9.
Fig. 9.

Adjustment of the various energy conversion rates (per unit mass) from an initially steady-state circulation forced with Q0 = 200 W m−2. The vertical diffusion coefficient is Kz = 10−1 m2 s−1. At time t = 2400 days, the surface buoyancy throughout is changed suddenly to that corresponding to (a) Q0 = 150 and (b) Q0 = 250 W m−2. Note that at all times, there is no net heating of the basin.

Citation: Journal of Physical Oceanography 39, 12; 10.1175/2009JPO4162.1

Table 1.

Summary of the energy reservoirs and conversion rates for Kz = 10−3 and 10−1 m2 s−1 as a function of maximum surface heat flux Q0 (W m−2). The energy stored in the reservoirs has units of J kg−1 and the conversions between reservoirs have units of W kg−1 (following normalization by the mass M0 of water in a basin of 1-m width). The background potential energy Eb is calculated with reference to an isothermal basin at 20°C.

Table 1.
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Journal of Physical Oceanography

  • Coman, M. A., R. W. Griffiths, and G. O. Hughes, 2006: Sandstrom’s experiments revisited. J. Mar. Res., 64 , 783–796.

  • Defant, A., 1961: Physical Oceanography. Vol. I, MacMillan, 718 pp.

  • Gnanadesikan, A., R. D. Slater, P. S. Swathi, and G. K. Vallis, 2005: The energetics of ocean heat transport. J. Climate, 18 , 2604–2616.

    • Search Google Scholar
    • Export Citation

  • Griffiths, R. W., and G. O. Hughes, 2004: The energetics of horizontal convection. Proc. Fifteenth Australian Fluid Mechanics Conf., Sydney, NSW, Australia, University of Sydney, AFMC00080. [Available online at http://www.aeromech.usyd.edu.au/15afmc/proceedings/papers/AFMC00065.pdf].

    • Search Google Scholar
    • Export Citation

  • Huang, R. X., 1998: Mixing and available potential energy in a Boussinesq ocean. J. Phys. Oceanogr., 28 , 669–678.

  • Huang, R. X., 1999: Mixing and energetics of the oceanic thermohaline circulation. J. Phys. Oceanogr., 29 , 727–746.

  • Hughes, G. O., and R. W. Griffiths, 2006: A simple convective model of the global overturning circulation, including the effects of entrainment into sinking regions. Ocean Modell., 12 , 46–79.

    • Search Google Scholar
    • Export Citation

  • Ivey, G. N., K. B. Winters, and J. R. Koseff, 2008: Density stratification, turbulence, but how much mixing? Annu. Rev. Fluid Mech., 40 , 169. doi:10.1146/annurev.fluid.39.050905.110314.

    • Search Google Scholar
    • Export Citation

  • Jeffreys, H. T., 1925: On fluid motions produced by differences of temperature and humidity. Quart. J. Roy. Meteor. Soc., 51 , 347–356.

    • Search Google Scholar
    • Export Citation

  • Kuhlbrodt, T., A. Griesel, M. Montoya, A. Levermann, M. Hofmann, and S. Rahmstorf, 2007: On the driving processes of the Atlantic meridional overturning circulation. Rev. Geophys., 45 , RG2001. doi:10.1029/2004RG000166.

    • Search Google Scholar
    • Export Citation

  • Mullarney, J. C., R. W. Griffiths, and G. O. Hughes, 2004: Convection driven by differential heating at a horizontal boundary. J. Fluid Mech., 516 , 181–209.

    • Search Google Scholar
    • Export Citation

  • Munk, W. H., 1966: Abyssal recipes. Deep-Sea Res., 13 , 707–730.

  • Munk, W. H., and C. Wunsch, 1998: Abyssal recipes II: Energetics of tidal and wind mixing. Deep-Sea Res., 45 , 1976–2009.

  • Nycander, J., J. Nilsson, K. Döös, and G. Broström, 2007: Thermodynamic analysis of ocean circulation. J. Phys. Oceanogr., 37 , 2038–2052.

    • Search Google Scholar
    • Export Citation

  • Oort, A. H., L. A. Anderson, and J. P. Peixoto, 1994: Estimates of the energy cycle of the oceans. J. Geophys. Res., 99 , 7665–7688.

  • Paparella, F., and W. R. Young, 2002: Horizontal convection is non-turbulent. J. Fluid Mech., 466 , 205–214.

  • Peltier, W., and C. Caulfield, 2003: Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech., 35 , 135–167.

  • Rossby, H. T., 1965: On thermal convection driven by non-uniform heating from below: An experimental study. Deep-Sea Res., 12 , 9–16.

  • St. Laurent, L., and H. Simmons, 2006: Estimates of power consumed by mixing in the ocean interior. J. Climate, 19 , 4877–4890.

  • Tailleux, R., 2009: On the energetics of stratified turbulent mixing, irreversible thermodynamics, Boussinesq models, and the ocean heat engine controversy. J. Fluid Mech., 638 , 339–382.

    • Search Google Scholar
    • Export Citation

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  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Two cases with the same potential energy: (a) a homogeneous ocean and (b) a case where surface forcing of the upper layer results in equal and opposite density anomalies. In the latter case there is no change of potential energy, but there is a finite amount of available potential energy.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Energy pathways in the ocean as defined by Eqs. (10), (14), (20), and (24). The transfers between reservoirs of mean kinetic energy , fluctuation kinetic energy E′k, available potential energy Ea, and background potential energy Eb include conversions by the mean flow and turbulent buoyancy fluxes, and Φ′z; conversion via mean shear ΦT; kinetic energy dissipation from the mean and fluctuating flow components, and ϵ′; external input of kinetic energy to the mean and fluctuating flow components, and Φ′τ; internal energy release Φi by molecular diffusion; available potential energy generation due to external buoyancy forcing resulting in a change of bulk density Φb1 and redistribution of mass Φb2 in the volume; and available potential energy dissipation by irreversible mixing Φd. The reservoir of internal energy is not closed by the energetic framework in this paper and is thus denoted by a dashed boundary. Conversions (or directions) shown in gray are assumed to be relatively minor, and their omission yields a schematic for the approximate energetic balance for the oceans. For conversions that can be bidirectional, the direction defined as positive is indicated.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Energetics of an ocean model.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Dependence of the time-averaged overturning circulation upon the vertical diffusion coefficient (surface buoyancy flux is fixed, with Q0 = 200 W m−2). The maximum streamfunction quoted is that for a two-dimensional flow in a basin of 1-m width, while the density range is Δρ = ρbottom − ρtop. The 20°C isotherm is shown in gray.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Dependence of the various energy conversion rates upon the vertical diffusion coefficient, for a fixed buoyancy forcing: Q0 = 200 W m−2. The energy conversions are normalized by the mass M0 of the water in the basin (which, in the numerical simulations, has a width of 1 m).

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Dependence of the time-averaged overturning circulation upon the surface buoyancy throughput (Kz is held constant at 10−3 m2 s−1). The maximum streamfunction and density range are as defined in Fig. 4.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    Dependence of the various energy conversion rates upon the surface buoyancy throughput. The vertical diffusion coefficient Kz is held constant at 10−3 m2 s−1. The energy conversions are normalized by the mass M0 of water in the basin (which, in the numerical simulations, has a width of 1 m).

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    Summary of the model results for the statistically steady-state circulation with Kz = 10−3 m2 s−1 and Q0 = 200 W m−2, obtained using a convective adjustment scheme: (a) various energy conversion rates (per unit mass) compared with the corresponding fully nonhydrostatic simulation and (b) time-averaged overturning circulation (contour intervals as in Fig. 4c). The maximum streamfunction and density range are as defined in Fig. 4, and the 20°C isotherm is shown in gray. The maximum streamfunction below the 20°C isotherm is 9.4 × 103 kg s−1, which compares with 30.8 × 103 kg s−1 in Fig. 4c.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    Adjustment of the various energy conversion rates (per unit mass) from an initially steady-state circulation forced with Q0 = 200 W m−2. The vertical diffusion coefficient is Kz = 10−1 m2 s−1. At time t = 2400 days, the surface buoyancy throughout is changed suddenly to that corresponding to (a) Q0 = 150 and (b) Q0 = 250 W m−2. Note that at all times, there is no net heating of the basin.

Fig. 1.

Two cases with the same potential energy: (a) a homogeneous ocean and (b) a case where surface forcing of the upper layer results in equal and opposite density anomalies. In the latter case there is no change of potential energy, but there is a finite amount of available potential energy.
Fig. 2.

Energy pathways in the ocean as defined by Eqs. (10), (14), (20), and (24). The transfers between reservoirs of mean kinetic energy , fluctuation kinetic energy E′k, available potential energy Ea, and background potential energy Eb include conversions by the mean flow and turbulent buoyancy fluxes, and Φ′z; conversion via mean shear ΦT; kinetic energy dissipation from the mean and fluctuating flow components, and ϵ′; external input of kinetic energy to the mean and fluctuating flow components, and Φ′τ; internal energy release Φi by molecular diffusion; available potential energy generation due to external buoyancy forcing resulting in a change of bulk density Φb1 and redistribution of mass Φb2 in the volume; and available potential energy dissipation by irreversible mixing Φd. The reservoir of internal energy is not closed by the energetic framework in this paper and is thus denoted by a dashed boundary. Conversions (or directions) shown in gray are assumed to be relatively minor, and their omission yields a schematic for the approximate energetic balance for the oceans. For conversions that can be bidirectional, the direction defined as positive is indicated.
Fig. 3.

Energetics of an ocean model.
Fig. 4.

Dependence of the time-averaged overturning circulation upon the vertical diffusion coefficient (surface buoyancy flux is fixed, with Q0 = 200 W m−2). The maximum streamfunction quoted is that for a two-dimensional flow in a basin of 1-m width, while the density range is Δρ = ρbottom − ρtop. The 20°C isotherm is shown in gray.
Fig. 5.

Dependence of the various energy conversion rates upon the vertical diffusion coefficient, for a fixed buoyancy forcing: Q0 = 200 W m−2. The energy conversions are normalized by the mass M0 of the water in the basin (which, in the numerical simulations, has a width of 1 m).
Fig. 6.

Dependence of the time-averaged overturning circulation upon the surface buoyancy throughput (Kz is held constant at 10−3 m2 s−1). The maximum streamfunction and density range are as defined in Fig. 4.
Fig. 7.

Dependence of the various energy conversion rates upon the surface buoyancy throughput. The vertical diffusion coefficient Kz is held constant at 10−3 m2 s−1. The energy conversions are normalized by the mass M0 of water in the basin (which, in the numerical simulations, has a width of 1 m).
Fig. 8.

Summary of the model results for the statistically steady-state circulation with Kz = 10−3 m2 s−1 and Q0 = 200 W m−2, obtained using a convective adjustment scheme: (a) various energy conversion rates (per unit mass) compared with the corresponding fully nonhydrostatic simulation and (b) time-averaged overturning circulation (contour intervals as in Fig. 4c). The maximum streamfunction and density range are as defined in Fig. 4, and the 20°C isotherm is shown in gray. The maximum streamfunction below the 20°C isotherm is 9.4 × 103 kg s−1, which compares with 30.8 × 103 kg s−1 in Fig. 4c.
Fig. 9.

Adjustment of the various energy conversion rates (per unit mass) from an initially steady-state circulation forced with Q0 = 200 W m−2. The vertical diffusion coefficient is Kz = 10−1 m2 s−1. At time t = 2400 days, the surface buoyancy throughout is changed suddenly to that corresponding to (a) Q0 = 150 and (b) Q0 = 250 W m−2. Note that at all times, there is no net heating of the basin.
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