zonal jets to close the circulation. In parallel,
Wyrtki (1961b) and Munk (1966) looked at the
scalar balance equations, and derived relationships
between vertical advection and diffusion. These
ideas that link mixing, the vertical velocity, and
the horizontal circulation have guided thinking
about the abyssal flow field for the past 40 years.
Munk’s (1966) seminal paper (revisited by
Munk and Wunsch, 1998) inferred the ocean mixing rate from a simple vertical advective–diffusive
balance for the main thermocline: w ⌰ z :(K ⌰ z ) z.
The model, fitted to observations of the typical
1-km e-folding scale for the potential temperature
profile (⌰(z)) and a guess at the average vertical
upwelling rate (10
97 m s
91
: equivalent to a global
bottom water formation rate of some 25 Sv uniformly upwelling into the thermocline), gives
Kϳ10
94 m
2 s
91
. Admittedly, this is a greatly simplified model; the ocean obviously has a fully
three-dimensional, time-varying circulation. Munk
certainly acknowledged this in his original paper,
suggesting that enhanced mixing in a (relatively
thin) layer above the (sloping) bottom followed by
lateral exchange with the interior might be the
actual mechanism sustaining the thermocline.
(This process has come to be known as boundary
mixing: see Armi, 1978; Garrett, 1991.) Comparably sized average diffusivity estimates have also
been derived from hydrographic box inverse
models (e.g. Wijffels, 1993; Ganachaud, 1999)
and extended beta-spiral calculations (e.g. Olbers
et al., 1985; Olbers and Wenzel, 1989). But as
Wunsch (1996) notes, small errors in the coefficient
matrices of these models ‘can easily produce spurious apparent mixing.’ (Indeed, an inverse model
developed by St Laurent (1999) that includes both
hydrographic and microstructure data finds that
skill in diagnosing the diapycnal velocity comes
largely from the latter.) Additionally, Davis (1994)
discusses how time dependence can introduce error
into K estimates inferred from a steady model.
Estimates of diathermal and diapycnal heat fluxes
in the abyssal ocean have also been derived from
heat budgets for semi-enclosed deep basins (e.g.
Hogg et al., 1982; Whitehead and Worthington,
1982; Saunders, 1987; Roemmich et al., 1996;
Morris et al., 1997). These analyses derive mass
and heat conservation statements for control volumes bounded by an interior isotherm (or isopycnal) and the bottom to relate estimates of the
mean temperature transport through an inlet strait
to the turbulent heat flux across the bounding
surfaces of the control volumes. When these turbulent fluxes are expressed in terms of an average
diathermal/diapycnal diffusivity, values of order
10
94 m
2 s
91 are again obtained. Interestingly,
related analyses for upper-ocean layers bounded
by isotherms (isopycnals) and the free surface find
similar need for significant diathermal fluxes (e.g.
Speer, 1997; Zhang and Talley, 1998). (Note that
the upper-ocean-layer analyses are far more complicated than their abyssal counterparts because of
the significant and time-variable exchanges of heat
and buoyancy at the air–sea interface (Garrett and
Tandon, 1997). In contrast, geothermal heat flux
generally makes negligible contribution to abyssallayer heat budgets.)
These indirect and budget calculations, while
valuable for setting bounds on the diapycnal mass
transports and property fluxes, do not shed light
on the possible mechanisms supporting the mixing,
or where within each control volume it occurs.
Indeed, the ocean mixing community has been
engaged in a 30-year effort to find processes capable of supporting diapycnal fluxes consistent with
O(10
94 m
2 s
91
) diffusivities. As discussed below,
some progress has been achieved in the last decade.
5.2.3 The Temporal-Residual-Mean
circulation
In this section we examine turbulence at the horizontal scale of tens of kilometres and we use our
basic knowledge of these mesoscale flows to
deduce something about how the fluxes achieved
by these motions should be parameterized in
coarse-resolution ocean models. Mesoscale eddies
cause significant lateral fluxes of density and
tracer, particularly in the Southern Ocean. Yet the
effects of mesoscale eddies still need to be parameterized in climate models because of the present
and foreseeable limitations of computer power that
constrain model resolution. The mixing achieved
by mesoscale eddies was traditionally parameterized as horizontal diffusion, but we have learnt in
the past decade what damage this horizontal mixing had been doing to the model results. Specifically, horizontal mixing implies an uncontrolled
amount of diapycnal property flux in regions of
sloping isopycnals, often dominating the parameterized diapycnal mixing in the model. What was
needed was a formulation for mesoscale mixing
SECTION 5 FORMATION AND TRANSPORT OF WATER MASSES
340
Wyrtki (1961b) and Munk (1966) looked at the
scalar balance equations, and derived relationships
between vertical advection and diffusion. These
ideas that link mixing, the vertical velocity, and
the horizontal circulation have guided thinking
about the abyssal flow field for the past 40 years.
Munk’s (1966) seminal paper (revisited by
Munk and Wunsch, 1998) inferred the ocean mixing rate from a simple vertical advective–diffusive
balance for the main thermocline: w ⌰ z :(K ⌰ z ) z.
The model, fitted to observations of the typical
1-km e-folding scale for the potential temperature
profile (⌰(z)) and a guess at the average vertical
upwelling rate (10
97 m s
91
: equivalent to a global
bottom water formation rate of some 25 Sv uniformly upwelling into the thermocline), gives
Kϳ10
94 m
2 s
91
. Admittedly, this is a greatly simplified model; the ocean obviously has a fully
three-dimensional, time-varying circulation. Munk
certainly acknowledged this in his original paper,
suggesting that enhanced mixing in a (relatively
thin) layer above the (sloping) bottom followed by
lateral exchange with the interior might be the
actual mechanism sustaining the thermocline.
(This process has come to be known as boundary
mixing: see Armi, 1978; Garrett, 1991.) Comparably sized average diffusivity estimates have also
been derived from hydrographic box inverse
models (e.g. Wijffels, 1993; Ganachaud, 1999)
and extended beta-spiral calculations (e.g. Olbers
et al., 1985; Olbers and Wenzel, 1989). But as
Wunsch (1996) notes, small errors in the coefficient
matrices of these models ‘can easily produce spurious apparent mixing.’ (Indeed, an inverse model
developed by St Laurent (1999) that includes both
hydrographic and microstructure data finds that
skill in diagnosing the diapycnal velocity comes
largely from the latter.) Additionally, Davis (1994)
discusses how time dependence can introduce error
into K estimates inferred from a steady model.
Estimates of diathermal and diapycnal heat fluxes
in the abyssal ocean have also been derived from
heat budgets for semi-enclosed deep basins (e.g.
Hogg et al., 1982; Whitehead and Worthington,
1982; Saunders, 1987; Roemmich et al., 1996;
Morris et al., 1997). These analyses derive mass
and heat conservation statements for control volumes bounded by an interior isotherm (or isopycnal) and the bottom to relate estimates of the
mean temperature transport through an inlet strait
to the turbulent heat flux across the bounding
surfaces of the control volumes. When these turbulent fluxes are expressed in terms of an average
diathermal/diapycnal diffusivity, values of order
10
94 m
2 s
91 are again obtained. Interestingly,
related analyses for upper-ocean layers bounded
by isotherms (isopycnals) and the free surface find
similar need for significant diathermal fluxes (e.g.
Speer, 1997; Zhang and Talley, 1998). (Note that
the upper-ocean-layer analyses are far more complicated than their abyssal counterparts because of
the significant and time-variable exchanges of heat
and buoyancy at the air–sea interface (Garrett and
Tandon, 1997). In contrast, geothermal heat flux
generally makes negligible contribution to abyssallayer heat budgets.)
These indirect and budget calculations, while
valuable for setting bounds on the diapycnal mass
transports and property fluxes, do not shed light
on the possible mechanisms supporting the mixing,
or where within each control volume it occurs.
Indeed, the ocean mixing community has been
engaged in a 30-year effort to find processes capable of supporting diapycnal fluxes consistent with
O(10
94 m
2 s
91
) diffusivities. As discussed below,
some progress has been achieved in the last decade.
5.2.3 The Temporal-Residual-Mean
circulation
In this section we examine turbulence at the horizontal scale of tens of kilometres and we use our
basic knowledge of these mesoscale flows to
deduce something about how the fluxes achieved
by these motions should be parameterized in
coarse-resolution ocean models. Mesoscale eddies
cause significant lateral fluxes of density and
tracer, particularly in the Southern Ocean. Yet the
effects of mesoscale eddies still need to be parameterized in climate models because of the present
and foreseeable limitations of computer power that
constrain model resolution. The mixing achieved
by mesoscale eddies was traditionally parameterized as horizontal diffusion, but we have learnt in
the past decade what damage this horizontal mixing had been doing to the model results. Specifically, horizontal mixing implies an uncontrolled
amount of diapycnal property flux in regions of
sloping isopycnals, often dominating the parameterized diapycnal mixing in the model. What was
needed was a formulation for mesoscale mixing
SECTION 5 FORMATION AND TRANSPORT OF WATER MASSES
340