Available potential energy and irreversible
mixing in the meridional overturning
circulation
Graham O. Hughes∗
Andrew McC. Hogg
Ross W. Griffiths
The Australian National University, Canberra, Australia
ABSTRACT
The overturning circulation of the global oceans is examined from an energetics
viewpoint. The general framework of Winters et al. (1995) for stratified turbulence
is used, first to highlight 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 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.
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
∗
Corresponding author address: Research School of Earth Sciences, The Australian National University,
Canberra ACT 0200, Australia
E-mail: [email protected]
(b)
(a)
ρ=ρ0−Δρ
ρ=ρ0
ρ=ρ0+Δρ
ρ=ρ0
Fig. 1. Two cases with the same potential energy: (a) homogeneous ocean; (b) 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 a finite amount of available potential energy.
the circulation (Huang 1999; Wunsch and Ferrari 2004; Kuhlbrodt et al. 2007). However, a
simple example illustrates the flaw in this reasoning (Figure 1). The potential energy and total
mass in the box is identical in (a) and (b), but a flow will occur in (b) and not in (a). The reason
is that the fluid in (b) 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) is 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 recreate 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.
2
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 §2 we construct
an energetics framework to describe the energy pathways and fluxes, and consider this in §3
in 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 effect 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 §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 kinetic energy.
Consider the Boussinesq equations of motion, written in tensor notation with summation
implied,
∂ui
∂p
∂ 2 ui
∂ui
,
(1)
+ ρ0 u j
=
− ρgδi3 + ρ0 ν
ρ0
∂t
∂xj
∂xi
∂xj 2
∂ui
= 0,
(2)
∂xi
∂2ρ
∂ρ
∂ρ
+ ui
=κ
.
(3)
∂t
∂xi
∂xi 2
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 upwards 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 which eliminates the fluctuations while still permitting slow variations of the mean flow.
That is,
ui (xi , t) = ui (xi , t) + u0i (xi , t),
where the overbar indicates the mean flow and primed quantities denote fluctuating components. We define the averaging filter so that u0i = 0.
Similar decompositions of the pressure and density fields allow us to rewrite (1)–(3) as
ρ0
∂u0i u0j
∂ui
∂ui
∂p
∂ 2 ui
+ ρ0 u j
+ ρ0
=
− ρgδi3 + ρ0 ν
,
∂t
∂xj
∂xj
∂xi
∂xj 2
∂ui
∂u0i
=
= 0,
∂xi
∂xi
(4)
(5)
∂ρ
∂ρ
∂u0 ρ0
∂2ρ
+ ui
+ i =κ
,
(6)
∂t
∂xi
∂xi
∂xi 2
where the averaging filter has been applied after decomposition to eliminate single perturbation
terms.
3
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 kinetic energy, and then the potential energy budgets for this flow. As we
seek a description of the meridional overturning circulation 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; Tailleux 2009). In the
current paper, internal energy is invoked only insofar as transfers to/from mechanical energy
are required.
a. Kinetic Energy
The kinetic energy of the mean flow is
ρ0
Ek =
2
Z
ui 2 dV,
(7)
V
where V is the volume of the rectangular basin. Multiplying the momentum equation (4) by
ui , integrating over the volume of the domain and using boundary conditions of no normal flow
gives
2
Z
Z
I
Z ∂ui
∂ui 0 0
dEk
∂ui
= ρ0
ui uj dV −
ρ wgdV + ρ0 ν ui
nj dS − ρ0 ν
dV,
(8)
dt
∂xj
∂xj
V
V ∂xj
V
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 for the
above equation. The first term in (8) is the rate of production of fluctuation kinetic energy
from the mean shear (Tennekes and Lumley 1972) and is defined as
Z
∂ui 0 0
ui uj dV.
ΦT = −ρ0
V ∂xj
Secondly, we define the reversible buoyancy flux (Winters et al. 1995) of the mean flow to be
Z
ρ wg dV.
Φz =
V
The third term in (8) is the rate of working by boundary stress applied at the surface of the
domain. For the oceans, we expect this term will constitute a net input of energy from wind
stress, and thus write it as
I
∂ui
Φ τ = ρ0 ν u i
nj dS,
∂xj
although we note that boundary friction will provide a sink of energy here. In addition, we
have not included explicitly terms to account for 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). Lastly
we have the internal dissipation rate from the mean flow
2
Z ∂ui
= ρ0 ν
dV,
(9)
∂xj
V
which is a positive definite sink of kinetic energy. Thus, (8) simplifies to
dEk
= −ΦT − Φz + Φτ − .
dt
4
(10)
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 equation (1) is multiplied by ui ,
ρ0 ∂uj u2i
∂ui p
∂
ρ0 ∂u2i
+
=
− ρwg + ρ0 ν
2 ∂t
2 ∂xj
∂xi
∂xj
∂ui
ui
∂xj
− ρ0 ν
∂ui
∂xj
2
.
(11)
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
dEk0
= −ρ0
dt
Z
V
∂ui 0 0
u u dV −
∂xj i j
Z
ρ0 w 0 g
I
dV + ρ0 ν
V
where
Ek0
ρ0
=
2
Z
∂u0
u0i i nj
∂xj
Z dS − ρ0 ν
u02
i dV
V
∂u0i
∂xj
2
dV,
(12)
(13)
V
is the kinetic energy associated with the fluctuating component. The terms in equation (12)
are respectively the rate of production of turbulent kinetic energy (balancing the rate of loss
from mean kinetic energy Ek ), and the fluctuating component analogues of buoyancy flux, wind
stress working and dissipation from (10). Equation (12) is written in shorthand form as
dEk0
= ΦT − Φ0z + Φ0τ − 0 .
dt
(14)
b. Potential energy
The gravitational potential energy of the system is given by
Z
Z
ρz dV.
ρz dV = g
Ep = g
(15)
V
V
Note that there is no net potential energy associated with the fluctuating component, since
ρ0 = 0. The potential energy can be evaluated by substituting (6) into (15), and integrating
Z
I
Z
dEp
∂ρ
=g
w ρ dV + g
w0 ρ0 dV + κg z
ni dS − κgA(ρtop − ρbottom ),
(16)
dt
∂xi
V
V
which is written in shorthand as
dEp
= Φz + Φ0z + Φb1 + Φi .
dt
(17)
The first two terms in (17) are the reversible buoyancy flux terms and describe the exchange of
energy with mean and turbulent kinetic energy (equations 10 and 14, respectively). The third
term,
I
∂ρ
Φb1 = κg z
ni dS
∂xi
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.
5
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 re-sorted into its minimum potential energy state. It can be written
analytically following Winters et al. (1995),
Z
Eb = g
ρz∗ dV,
(18)
V
where
1
z∗ (x, t) ≡
A
Z
H (ρ(x0 , t) − ρ(x, t)) dV 0 ,
V
and the Heaviside step function is defined to be
( 0 x < 0,
H(x) = 0.5 x = 0,
1 x > 0.
The physical interpretation of z∗ is that it is the height to which a fluid element will move if the
entire domain is adiabatically re-sorted 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
Z
Z
∂2ρ
∂ρ
dEb
= −g
ui z∗
dV + κg
z∗
dV.
(19)
dt
∂xi
∂xi 2
V
V
R
∂ψ
∂ρ
=
where
ψ
=
z∗ dρ and apply boundary conditions of zero normal flow to
We write z∗ ∂x
∂xi
i
give
dEb
= −Φb2 + Φd .
(20)
dt
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∗ ,
I
∂ρ
Φb2 = −κg z∗
ni dS.
(21)
∂xi
Thus, provided 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
Z
Φd = −κg
V
dz∗
dρ
∂ρ
∂xi
2
dV,
(22)
and is the rate of energy conversion owing to 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.
6
Available potential energy is defined to be the difference between the potential energy and
the background potential energy:
Z
Ea = Ep − Eb = g
ρ(z − z∗ )dV.
(23)
V
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
available potential energy is simply
dEa
= Φz + Φ0z + Φb1 + Φb2 − Φd + Φi .
dt
(24)
The set of equations (10), (14), (20) and (24) is represented schematically in figure 2. The
kinetic and potential energy reservoirs (towards the bottom and top, respectively) are each
divided into two separate reservoirs (Ek and Ek0 , 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 figure 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 available potential energy by the surface
buoyancy fluxes. This term represents a direct conversion from background to available potential energy, but it is catalysed 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 equation (24) but not the background potential energy equation (20). This is
surprising because the molecular diffusion of a stably stratified fluid will alter the height of the
centre of mass (as diffusion acts to force the stratification towards 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 available potential energy. This
confusion can be resolved by realising that Φi is a component of the irreversible mixing, Φd . For
this reason we have chosen in figure 2 to illustrate Φi as contributing directly to the background
potential energy, Eb , and 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; Nycander et al. 2007) 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 generation of available potential energy is unimportant. The primary
7
Φb1
Ea
Φb2
(Φd - Φi)
Φ z'
Φz
Eb
Φi
Internal
Energy
ε'
ε
Ek'
Ek
Φτ
ΦT
Φτ'
Fig. 2. Energy pathways in the ocean as defined by equations (10), (14), (20) and (24).
The transfers between reservoirs of mean kinetic energy Ek , fluctuation kinetic energy Ek0 ,
available potential energy Ea and background potential energy Eb include conversions by the
mean flow and turbulent buoyancy fluxes, Φz and Φ0z , conversion via mean shear ΦT , kinetic
energy dissipation from the mean and fluctuating flow components, and 0 , external input
of kinetic energy to the mean and fluctuating flow components, Φτ and Φ0τ , 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 grey 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.
result of Figure 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,
Φb2 = Φd .
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
e≡
Φ d − Φi
.
Φd − Φ i + 8
If we assume globally constant e = 1/5, and that Φi is small (see discussion in §3.c), then this
can be rewritten as
Φb2 = Φd = 0.25 = 0.25Φτ ,
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 forcing 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 = Φτ + Φ0τ is increased. The propagation of energy through the system will be as follows.
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 0 . This equilibration will involve an increase
in the rate of production of Ek0 from mean shear (ΦT ), but may also involve production from
mean kinetic energy via available potential energy, i.e. Φz and Φ0z . It is worth noting that the
transfer pathway from mean shear via available potential energy can operate in either direction.
Nevertheless, the outcome is the same. The rate at which 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 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 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 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 which 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 in the context of the oceans.
9
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 =
0
Φτ = 0) a steady state flow has Φi = (= Φz + Φ0z ). This result places a strong constraint on
the total rate of viscous dissipation + 0 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 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 Figure 2 illustrates how the input of
external energy Φτ from winds and tides plays a role in the meridional overturning circulation
(MOC) through the generation of available potential energy. We consider two categories of
external forcing. Firstly, external forcing of sufficient strength and coherence could drive a mean
flow that creates available potential energy (i.e. positive Φz ), which we expect to be balanced by
release at relatively small length scales (i.e. negative Φ0z ). 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.
Secondly, the external forcing may induce motion at relatively small length scales (from
the mean flow via ΦT or directly via Φ0τ ). For instance, the external action of winds and tides
leads, 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 Φ0z ), and viscous dissipation and irreversible mixing will occur. The resulting
10
(“thermally-direct”; Nycander et al. 2007) mean flow will tend to release available potential
energy (i.e. negative Φz ), and surface buoyancy fluxes must again correspond to an important
energetic term for 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 upwelling and downward diffusion of heat
(Munk 1966; Munk and Wunsch 1998). We can now see the context in 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 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 (= Φz + Φ0z ) from the kinetic to the potential energy reservoir (assuming in the oceans
that Φi is small; see §3.c). 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 Φ0z ) or “pushed” by buoyancy forces (associated mainly with
Φz ) 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 figure 2 in light of
numerous studies that have considered only a single reservoir each for potential and kinetic
energy. By not allowing explicitly for 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 which 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, Figure 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 towards 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 Φz balances Φb2 . Note that one characteristic of this laminar flow is that the isotherms
are barely distorted from their equilibrium positions and Φd = Φi , i.e. 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 Φz (= Φb2 = Φi ). External energy input, if present, would merely
11
increase the rate of viscous dissipation, so that = Φτ + Φi (i.e. no net work is done against
buoyancy).
In contrast to the “laminar” energetic pathway just described, figure 2 highlights the possibility of more complicated pathways associated with flows that contain fluctuations. Explicit
consideration of 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). 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 – 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 (Φ0z > 0) which we suggest is balanced by advection of the density field by the mean flow
Φz < 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 figure 2,
where we consider that terms corresponding to grey pathways and arrows are likely to be small
in the oceans, and are thus eliminated. We have assumed that, provided flow is turbulent, then
0 and Φd Φi . We have also assumed that Φ0τ Φτ 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, figure 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 Φτ would lead to an overturning circulation that cannot be maintained in a
statistically steady state. Thus, we argue that the roles of external energy input and surface
buoyancy forcing are complementary. In §4 we confirm the importance of the APE and TKE
loops by demonstrating that their respective rates of energy cycling are consistent with the
timescale for the energy reservoirs to re-equilibrate 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 equations (1)–(3) numerically, and thereby test the relative sources and
sinks of energy shown in Figure 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 evaluation of Φ0z ) must be parameterized. In this section we construct
a revised energy budget which 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 in the model.
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 turbulent transports as
a diffusive process (with transport down the mean gradient) with a uniform coefficient. That
is,
∂ui
−u0i u0j = Kν
,
(25)
∂xj
12
∂ρ
,
(26)
∂xi
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 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 which make up
the energy budget. The term representing the rate of conversion by turbulent buoyancy fluxes
becomes
Z
Z
∂ρ
0
0
0
ρ w g dV = −gKρ
dV = −gKρ A(ρtop − ρbottom ),
(27)
Φz ≡
V ∂z
V
−ρ0 u0i = Kρ
which resembles the internal to potential energy conversion term, Φi , from (14). We point out
that the turbulent buoyancy flux term Φ0z is not synonymous with the rate of background potential energy generation by irreversible mixing. The rate of irreversible mixing is determined
by diffusion of density down the local density gradients and, in a steady circulation, is regulated
by surface buoyancy forcing (following from the discussion in §3.b.2). On the other hand, contributions to the turbulent buoyancy flux term Φ0z arise from localized stirring, which generates
pressure gradients and thus flow on larger (mean-flow) length scales, such that Φz = −Φ0z .
Although stirring tends to increase the local density gradients, there is no reason to expect
that −Φ0z = Φd in general.
The background potential energy equation needs to be re-derived by substituting (6) into
(18). In that equation, the surface buoyancy term is then,
I
∂ρ
Φb2 = −Kρ g z∗
ni dS,
(28)
∂xi
and the irreversible mixing term becomes
Z
Φd = −gKρ
V
dz∗
dρ
∂ρ
∂xi
2
dV.
(29)
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 mean and turbulent kinetic energy flow components becomes
2
Z
Z ∂ui 0 0
∂ui
dV,
ΦT ≡ −ρ0
ui uj dV = ρ0 Kν
∂xj
V ∂xj
V
which now mimics the form of the dissipation term (cf. in (9)). Finally, since the model does
not simulate turbulent kinetic energy, we set Ek0 = 0 = 0. The terms in the parameterizationbased formulation give rise to the model energy budget drawn schematically in Figure 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 down-gradient density transport. On the other hand, the TKE loop
is substantially altered. The TKE reservoir is eliminated; the turbulent buoyancy flux term
13
Ea
Φb2
Φd
Φz
Φ z'
Eb
Ek
Φτ
ΦT
Fig. 3. Energetics of an ocean model
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
Φτ − ΦT + Φ0z = 0 (= Φτ − ΦT − Φz ).
That is, in the absence of external energy input, the parameterized turbulent buoyancy flux
term Φ0z balances the parameterized rate of “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 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
14
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 these add significantly to the computational cost. In order to be able to model this system,
some simplifications are necessary: we consider non-rotating 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 MITgcm in fully nonhydrostatic mode. The box is 4000 m deep (with 80 grid cells
varying from 75 m to 10 m vertical spacing) and 4000 km long in the y direction (with horizontal
resolution ranging from 7.5 km in non-convecting 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
end walls, 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 heat flux – a half-sinusoid antisymmetrical about the centre 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, for example, in which the bulk temperature of the whole domain has to
adjust to reach equilibrium). The surface buoyancy forcing is varied to gauge the effect on the
generation of 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 effect of turbulent
mixing on the circulation. Horizontal diffusion and viscosity are uniform with a coefficient
Kh = 25 m2 s−1 in all runs. In order 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 figure 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 endwall 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 which 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 behaviour, 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
15
which interact with the boundary conditions so that bottom water production is intermittent.
It could be argued that the seiching behaviour 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 horizontally-averaged density ∆ρ within the domain is
shown in the captions of figure 4 for each case. This difference is very small when 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 §3.b.1 which stated that the results of Paparella and Young (2002) when applied to constant
buoyancy flux boundary conditions would result in ∆ρ → ∞ as Kz → 0.
The rates of energy conversion between reservoirs can be diagnosed (figure 5) by considering
averages over timescales 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 −Φz ≈ Φ0z ≈ ΦT . Importantly, the concept of balanced conversions in the
steady state around each of the APE and TKE loops in figure 3 is supported. The discrepancies
in these balances in figure 5 are due to the effects of dissipation from 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 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 ; figure 6(b)) to either 150 W m−2 or 250 W m−2 . The
circulations in figure 6(a)–(c) correspond to the subsequent mean steady states; discussion of
the transient evolution is delayed until §4.c. 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 figure 7 are the energy conversion rates as a function of maximum surface heat
flux for the simulations in figure 6. The rates increase with Q0 and balanced conversions in
the APE and TKE loops are observed in the steady state, i.e. at each Q0 , Φb2 ≈ Φd and
−Φz ≈ Φ0z ≈ 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 forcing 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 §4.a). The results are compared with the corresponding fully non-hydrostatic simulation in figure 8: (a) and (b) highlight the substantial
reduction of energy conversion rates in the APE loop (Φb2 and Φd ) and a striking change in
the deep mean flow circulation (cf. figure 4(c)), respectively, associated with the use of the con16
(a) Kz = 0.1m2/s; ψ = 686×103 kg/s; ∆ρ = 0.02 kg/m3
Z (m)
−1000
−2000
−3000
500
1000
1500
2000
2500
3000
3500
(b) Kz = 0.01m2/s; ψ = 250×103 kg/s; ∆ρ = 0.08 kg/m3
Z (m)
−1000
−2000
−3000
500
1000
1500
2000
2500
3000
3500
(c) Kz = 0.001m2/s; ψ = 77×103 kg/s; ∆ρ = 0.29 kg/m3
Z (m)
−1000
−2000
−3000
500
1000
1500
2000
2500
3000
3500
(d) Kz = 0.0001m2/s; ψ = 28×103 kg/s; ∆ρ = 0.93 kg/m3
Z (m)
−1000
−2000
−3000
500
1000
1500
2000
Y (km)
2500
3000
3500
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 width 1 m, while the density range
∆ρ = ρbottom − ρtop . The 20◦ C isotherm is shown in grey.
vective adjustment scheme. However, the shallow overturning and top-to-bottom difference in
horizontally-averaged density in figures 8(b) and 4(c) are almost identical. This result confirms
our assertion of §3.d that convective adjustment parameterization obscures the dynamically
17
−8
1.8
x 10
Φ ′/M
z
1.6
Φ /M
T
1.4
0
−Φz/M0
0
Φb2/M0
Φd/M0
Power (W/kg)
1.2
1
0.8
0.6
0.4
0.2
0
−4
10
−3
−2
10
2
10
−1
10
Kz (m /s)
Fig. 5. Dependence of the various energy conversion rates upon the vertical diffusion coefficient,
for fixed buoyancy forcing Q0 = 200 W m−2 . 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.)
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
normalisation 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.
Kz
Q0
Eb /M0
Ea /M0
Ek /M0
Φb2 /M0
Φd /M0
−Φz /M0
Φ0z /M0
ΦT /M0
150
-0.268
0.00325
0.00348
0.370 × 10−8
0.258 × 10−8
0.142 × 10−8
0.116 × 10−8
0.133 × 10−8
10−3
200
-0.352
0.00366
0.00421
0.420 × 10−8
0.308 × 10−8
0.176 × 10−8
0.146 × 10−8
0.163 × 10−8
250
-0.390
0.00357
0.00411
0.456 × 10−8
0.324 × 10−8
0.203 × 10−8
0.179 × 10−8
0.187 × 10−8
important role of available potential energy.
18
150
-0.142
0.0190
0.0189
1.33 × 10−8
1.43 × 10−8
1.18 × 10−8
1.20 × 10−8
1.01 × 10−8
10−1
200
-0.179
0.0220
0.0217
1.66 × 10−8
1.70 × 10−8
1.50 × 10−8
1.52 × 10−8
1.28 × 10−8
250
-0.208
0.0243
0.0247
2.00 × 10−8
2.03 × 10−8
1.81 × 10−8
1.83 × 10−8
1.52 × 10−8
(a) Q0 = 150W/m2; ψ = 74×103 kg/s; ∆ρ = 0.22 kg/m3
Z (m)
−1000
−2000
−3000
500
1000
1500
2000
2500
3000
3500
(b) Q0 = 200W/m2; ψ = 77×103 kg/s; ∆ρ = 0.29 kg/m3
Z (m)
−1000
−2000
−3000
500
1000
1500
2000
2500
3000
3500
(c) Q0 = 250W/m2; ψ = 83×103 kg/s; ∆ρ = 0.34 kg/m3
Z (m)
−1000
−2000
−3000
500
1000
1500
2000
Y (km)
2500
3000
3500
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 is as defined in Figure 4.
c. Transient adjustment
We consider here the evolution of the energy conversion rates in response to a sudden change
in surface buoyancy throughput (figure 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
(figure 9a) and Q0 = 250 W m−2 (figure 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, Φz and ΦT , respond relatively rapidly. The re-equilibration of the
available potential energy reservoir takes much longer, as is apparent from the relatively slow
19
Φz′/M0
−9
5
x 10
−Φz/M0
ΦT/M0
Φ /M
4.5
b2
0
Φd/M0
4
Power (W/kg)
3.5
3
2.5
2
1.5
1
150
160
170
180
190
200
210
Q0 (W/m2)
220
230
240
250
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.)
adjustment of the conversion rates associated with irreversible mixing and turbulent buoyancy
fluxes, Φd and Φ0z . The timescale 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 ∆Q0 is the change in maximum heat input that produces a change in
background potential energy ∆Eb 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
∆Q0
dEb ≈
Φ
|
−
Φ
∼
Φb2 |t∗ <0 ,
(30)
−
b2
d
t
=0+
∗
dt t∗ =0+
Q0
where t∗ = 0 is the time at which the sudden change is applied. Thus the timescale for
adjustment Ta can be estimated as
Ta ∼ −
Q0 ∆Eb
.
∆Q0 Φb2 |t∗ <0
(31)
Substituting the values in Table 1 into (31), we estimate Ta of O(100) days for the adjustment
experiments with Kz = 10−1 m2 s−1 (actually 103 and 81 days for the ∆Q0 = −50 and 50 W m−2
cases, respectively). This timescale is consistent with the observed adjustment in figure 9 and
therefore supports the notion that the coupling between surface buoyancy fluxes and irreversible
20
−9
5
nonhydrostatic
conv. adj.
4
Power (W/kg)
(a)
x 10
3
2
1
0
Φz′/M0
−Φz/M0
ΦT/M0
Φb2/M0
Φd/M0
(b) Kz = 0.001 m2/s; ψ = 77×103 kg/s; ∆ρ = 0.30 kg/m3
−500
Z (m)
−1000
−1500
−2000
−2500
−3000
−3500
500
1000
1500
2000
Y (km)
2500
3000
3500
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 non-hydrostatic
simulation, and (b) time-averaged overturning circulation (contour intervals as for figure 4(c)).
The maximum streamfunction and density range is as defined in figure 4, and the 20◦ C isotherm
is shown in grey. 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 figure 4(c).
mixing is important in the overturning circulation. Were this not the case we would expect the
adjustment to occur on a timescale 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 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 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
21
−8
2.5
Φz′/M0
(a)
x 10
−Φ /M
z
0
Power (W/kg)
ΦT/M0
Φ /M
2
b2
0
Φ /M
d
0
1.5
1
2300
2400
2500
−8
Power (W/kg)
2.5
2600
2700
Time (days)
2800
2900
3000
2800
2900
3000
(b)
x 10
2
1.5
1
2300
2400
2500
2600
2700
Time (days)
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 W m−2 and (b) Q0 = 250 W m−2 . Note that
at all times, there is no net heating of the basin.
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.
22
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