thermobaricity, cabbeling, and water-mass...

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 92, NO. C5, PAGES 5448-5464, MAY 15, 1987

Thermobaricity, Cabbeling, and Water-Mass Conversion

TREVOR J. McDOUGALL

Division of Oceanography, Commonwealth Scientific and Industrial Research Organization Hobart, Tasmania, Australia

The efficient mixing of heat and salt along neutral surfaces (by mesoscale eddies) is shown to lead to vertical advection through these neutral surfaces. This is due to the nonlinearities of the equation of state of seawater through terms like c•2p/c•Oc•p (thermobaric effect) and c•2p/c•O 2 (cabbeling). Cabbeling always causes a sinking or downwelling of fluid through neutral surfaces, whereas thermobaricity can lead to a vertical velocity (relative to neutral surfaces) of either sign. In this paper it is shown that for reasonable values of the lateral scalar diffusivity (especially below a depth of 1000 m), these two processes cause vertical velocities of the order of 10-7 m s-x through neutral surfaces (usually downward!) and cause water-mass conversion of a magnitude equal to that caused by a vertical diffusivity of 10 -'• m 2 s-1 (often equivalent to a negative diffusivity). Both thermobaricity and cabbeling can occur in the presence of any nonzero amount of small-scale turbulence and so will not be detected by microstructure measurements. The conservation equations for tracers are considered in a nonorthogonal coordinate frame that moves with neutral surfaces in the ocean. Since only mixing processes cause advection across neutral surfaces, it is useful to regard this vertical advection as a symptom of various mixing processes rather than as a separate physical process. It is possible to derive conservative equations for scalars that do not contain the vertical advective term explicity. In these conservation equations, the terms that represent mixing processes are substantially altered. It is argued that this form of the conservation equations is the most appropriate when considering water-mass transformation, and some examples are given of its application in the North Atlantic. It is shown that the variation of the vertical diffusivity with height does not cause water-mass transformation. Also, salt fingering is often 3-4 times more effective at changing the potential temperature of a water mass than would be implied by simply calculating the vertical derivative of the fingering heat flux.

1. INTRODUCTION

Tracers are mixed quite efficiently in the ocean along quasi- lateral neutral surfaces by mesoscale eddies. In contrast, vertical mixing processes cause very small vertical fluxes (perhaps one thousandth of the lateral fluxes). Despite this disparity in the magnitude of the lateral and vertical property fluxes, the two types of mixing processes can make similar contributions to the conservation equations for fluid properties because the conservation equations involve flux divergences, and whereas the lateral fluxes vary over scales of several hundred kilome- ters, the vertical fluxes vary over scales of several hundred meters. The question of how much lateral and vertical mixing processes each contribute to water-mass transformation in the ocean is still unresolved, despite having a long history dating back to Iselin ['1939]. To resolve this question, it is necessary first to define the surfaces along which oceanic mesoscale eddies do their efficient lateral mixing (along neutral surfaces) and then to derive conservation equations for tracers on these surfaces so that observations of tracers can be correctly inter- preted. Water-mass conversion is deemed to be synonymous with changes of potential temperature on neutral surfaces.

Lateral mixing in the ocean occurs along surfaces of neutral stability, not along surfaces of constant potential density. An in situ parcel can be moved small distances along a "neutral surface" without experiencing a buoyant restoring force. Po- tential density surfaces do not possess this property because the compressibility of seawater is a function of temperature. It is common practice therefore to choose several different reference pressures for potential density when contouring deep-

Copyright 1987 by the American Geophysical Union.

Paper number 6C0448. 0148-0227/87/006C-0448505.00

ocean data (reference levels of 0, 2000, and 4000 dbar are routinely chosen [Reid and Lynn, 1971; lvers, 1975]). By re- garding the equation of state of seawater as a function of S, 0, and p, rather than as a function of S, T, and p, McDougall [ 1984, also On neutral surfaces, submitted to Journal of Physi- cal Oceanography, 1987] (hereinafter referred to as McDougall (1987)) has shown how neutral surfaces can be defined and plotted independently of the concept of potential densities. At a pressure of 1500 dbar in the North Atlantic, the spatial gradients of potential temperature in the potential density surface (referenced to a pressure of 0 dbar) can be as much as 300% more than the physically relevant spatial gradient of potential temperature in the neutral surface (McDougall, 1987).

Having defined the neutral surfaces along which lateral fluxes occur in the ocean, it is then logical to derive the conservation equations for tracers on these surfaces. In other words, the neutral surface is the natural reference frame in

which to measure the relative strength of various mixing processes. In this paper the development of the conservation equations by McDougall [1984] is followed closely. This approach depends on the conservation equations of potential temperature and salinity being considered concurrently and realizing that the vertical fluid velocity through neutral surfaces occurs only as a consequence of various mixing processes. It is then possible to derive a form of the conservation equations that does not explicitly contain the vertical advective process through neutral surfaces. Mixing processes enter these new conservation equations in altered expressions; the main point of McDougall [1984] was that small-scale turbulence, as parameterized by a vertical diffusivity D, appears in the conservation equation in a term proportional to D(d2S/d02). In this paper, the term that represents the total effect of small-scale turbulent mixing is shown as a ratio of the

5448

McDOUGALL: THERMOBARICITY, CABBELING, AND WATER-MASS CONVERSION 5449

usually assumed vertical diffusive contribution, DOzz, for three neutral surfaces in the North Atlantic. The total effect of verti-

cal turbulent mixing includes the diffusive contribution, DOzz, as well as its concomitant vertical advective term. The North

Atlantic data show that the total effect of vertical turbulence

bears little resemblance to, and often has the opposite sign of, the simple diffusive DOzz term. It is also shown that the vertical derivative of the vertical diffusivity D does not enter the new conservation equations. This means that vertical variations of D do not have the ability to cause water-mass trans- formations.

Corresponding maps of the total effect of salt fingering on water-mass conversion, divided by the normally assumed diffusive contribution, are also shown on three neutral surfaces in

the North Atlantic. Because of the special way in which salt fingers flux heat in relation to salt, they are up to 4 times more efficient at changing the potential temperature of a water mass than is the simple vertical derivative of the flux of heat due to salt fingering. Again, this results from salt fingers causing a certain vertical advection through neutral surfaces. When this effect is included in the conservation equation for potential temperature on a neutral surface, the total effect of salt fingers is up to 4 times as large as expected from the straightforward flux divergence term.

Due to nonlinearities of the equation of state of seawater, lateral fluxes of heat and salt (caused by, for example, mesoscale energetic eddies) also cause vertical advection of fluid through neutral surfaces. This type of vertical advection is due to two quite separate physical processes. The fact that •//• is a function of potential temperature and salinity (• is the thermal expansion coefficient and /• is the saline contraction coefficient) causes the cabbeling process, while the variation of •//• with pressure (or equivalently, the variation of the compressibility of seawater with potential temperature and salinity) causes the thermobaric effect. The term "cabbeling" was first used by Witte [1902] to describe the fact that two water masses of equal density but different temperatures and salinities produce a more dense water mass. Foster [1972] has reviewed the history of the term cabbeling, and Garrett and Horne [1978] have investigated when cabbeling is strong enough to cause frontogenesis. Cabbeling is also known to be important in freshwater lakes, where potential density is deter- mined only by potential temperature, and it is often the cause of deep vertical convection when the water temperature is close to 4øC [Carmack, 1979]. Foster and Carmack [1976] have also used the term cabbeling to describe the production of a dense water mass as a result of vertical mixing. On the basis of a S-O diagram, a criterion is readily developed to describe when a mixture of two water masses can produce water that is more dense than either of the parent water masses [Foster and Carmack, 1976]. This criterion leads to a rather sharp cutoff condition for the existence of cabbeling, and when it occurs, vigorous vertical overturning is assumed to occur. In this paper however, the term cabbeling is used to describe a more continuous process (without a sharp "cutoff" criterion) that is caused by lateral mixing. Following McDou- gall [1984], it is shown that in a vertically stratified ocean, the lateral mixing of heat along a neutral surface causes a concomitant vertical advection of water through the neutral surface due to the cabbeling nonlinearity terms in the equation of state. There is no sharp cutoff criterion for the existence of this vertical motion relative to neutral surfaces. This cabbeling ver-

tical advection will be shown to cause significant subsurface water-mass conversion.

The thermobaric process has some similarities with the cabbeling process in that it is also due to the lateral fluxes of heat and salt along neutral surfaces; however, the thermobaric process occurs only if pressure varies on the neutral surface, that is, if the neutral surface is not flat. (The adjective, thermobaric, has been coined to describe processes that depend on the second derivative of the equation of state with respect to potential temperature and pressure.) The thermobaric effect also relies on a nonlinear term in the equation of state, namely, the second derivative of the equation of state with respect to potential temperature and pressure (c32p/30t3p, also c32p/3S3p). This nonlinear term can be approximately regarded as either the dependence of the thermal expansion coefficient on pressure or as the dependence of the compressibility on potential temperature. It is this same nonlinearity of the equation of state that causes the differences between neutral surfaces and

potential density surfaces. While cabbeling always produces a sinking of fluid through neutral surfaces, the thermobaric vertical velocity relative to neutral surfaces can be either upward or downward. J. Shepherd (unpublished manuscript, 1981), reasoning from the motion of water parcels, was the first to point out that the thermobaric nonlinearities in the equation of state could cause vertical advection through neutral surfaces, a process he regarded as being due to the "fuzziness of neutral surfaces" in the ocean. McDougall [1984], on the other hand, assumed the hydrographic fields have a well-defined mean state with superimposed fluctuations due to several physical processes. As neutral surfaces are then locally unam- biguous and the cabbeling and thermobaric vertical advection processes are then described in terms of the lateral diffusivity operating on the lateral gradients of scalars, McDougall's [1984] assumptions are adopted in the present paper. The small nonlocal path-dependent nature of neutral surfaces is discussed by McDougall (1987).

The Levitus [1982] data set is used in this paper to evaluate the contribution of both cabbeling and thermobaricity to (1) vertical upwelling and downwelling velocities and (2) water- mass conversion in the North Atlantic Ocean. It is shown that

both of these processes are as important as the other water- mass transformation processes that operate in the ocean (namely, lateral mixing, vertical mixing, and double-diffusive convection). In particular, it is not uncommon for cabbeling and thermobaricity to cause downwelling velocities through neutral surfaces of 10-7 m s-•. This is of the same magnitude, but of opposite sign, to the vertical velocity required throughout the world's oceans to upwell the bottom water to the base of the thermocline [Munk, 1966]. It is premature to deduce the importance of cabbeling and thermobaricity in causing water-mass conversion in the whole world's ocean, but it is shown that at least in the North Atlantic, these two new water-mass transformation processes cause as much water- mass transformation at a depth of 1500 m as a vertical diffusivity of the order of 10 -4 m 2 s- •

2. CONSERVATION EQUATIONS FOR SCALARS ON NEUTRAL SURFACES

The approach taken here to develop the relevant conservation equations is basically that of McDougall [1984] except that here the exact nature of the coordinate system and the approximations are described in detail. Following McDougall

5450 McDOUGALL: THERMOBARICITY, CABBELING, AND WATER-MASS CONVERSION

Fig. 1. Two points, a and b, are shown on a neutral surface a short horizontal distance apart, 6x. The lateral gradient of, say, 0 in

mixing along neutral surfaces, small-scale turbulent mixing across neutral surfaces, and double-diffusive convection). Very similar equations to (4) and (5) above were the starting point of McDougall [1984, equations (1) and (2)], where they were simply written without derivation. As this form of the conservation equations on a sloping and undulating reference frame (namely, the neutral surface) was not justified in the previous paper, it is appropriate to do so now. Readers willing to accept (4) and (5) at face value may skip to section 2.1.

As conservation equations (4) and (5) can both be derived in exactly the same fashion, only (4) is discussed here. The starting point is that the material derivative, following .a fluid parcel DO/Dr is equal to minus the divergence of the total flux of 0 due to mixing processes [e.g., Batchelor, 1967]. -The material derivative is usually written in cartesian coordinates as O, + uO,• + roy + wO z, where the velocity of the fluid parcel is (u, v, w). Similar expressions are often used in spherical polar

this coordinate frame is defined as •O/•xln = lim 6x--•0 [0(b) - coordinates. Sutcliffe [1947] introduced an alternative "pro- O(a)]/6x. In this way, V,0 = c•O/•xl, i + c•O/•yl,j, where i and j are the jected isobaric coordinate system" into meteorology that has unit vectors in the x and y directions. since proved very useful in that subject. Just like neutral sur-

[1984], neutral surfaces are defined such that if a water parcel is moved a small distance isentropically and adiabatically (i.e., at constant potential temperature 0 and salinity S) in the neutral surface, it will not experience any buoyant restoring forces. Note that small movements of an in situ water parcel in a potential density surface do require work to be done against gravity. Spatial and temporal gradients of 0 and S on a neutral surface Vn0 and VnS were shown to be related by

•xVnO -- •VnS •0 t -- •S t (1)

where cz and fl are the thermal expansion coefficient and saline contraction coefficient, defined in terms of the in situ density p as

p •0 -1 øP s,p[8ø ]- p c•T •--•s,p (2)

1 1 •p •0 P O,p T,p T,p

T being the in situ temperature. Equations (1) can be regarded as defining a neutral surface. The buoyancy frequency N is given exactly by the equation

•l- 1N 2 = gOz __ fiS z (3)

where z is the vertical coordinate (positive upward). From the work of Gregg [1984] it can be shown that, for all practical purposes, the first law of thermodynamics can be regarded as a conservation equation for potential temperature 0. The conservation equations for heat and salt, written in the reference frame of a neutral surface, are

0 t '31- vn'Vn 0 -1-- eO z = V n '(KVnO ) + (DO•) z -- Fz ø (4)

S t q- vn'Vn S q-- eSz = V n '(KVnS ) q- (OSz) z -- Fz S (5)

Here 0 t and S t are the temporal changes of 0 and S on the neutral surface, and e is the vertical velocity of fluid through the neutral surface. The left-hand sides of (4) and (5) are simply the total derivatives of 0 and $, following a fluid parcel, and the right-hand sides are the divergences of the fluxes of 0 and $ due to three separate mixing processes (namely, lateral

faces in the ocean, isobaric surfaces in the atmosphere (and also in the ocean) are not flat (i.e., do not coincide with geopo- tentials) and are bumpy. The isobaric coordinate system uses pressure instead of z as the vertical coordinate but measures lateral distances in the horizontal plane (as before), even though the x and y axes are now inclined to the horizontal and in fact lie at the intersection of the p surface with the plane y = const (for the x axis) and at the intersection of the p surface with the plane x = const (for the y axis). This coordinate system is nonorthogonal. In a very clear paper on this subject, Bleck [1978] has shown that the material derivative operator in isobaric coordinates does not contain any "metric" terms but takes the simple form

_ Dt •t p p p The velocities u and v are the usual east and north horizontal

velocities and the lateral derivatives •/•x Ip and c•/Oy I•, are the derivatives, in the horizonal plane, of the projection of variables measured in the isobaric surface. Dp/Dt is the rate of change of pressure following the fluid parcel as it moves through the isobaric surface, and c•/c•pl,,y,t is the derivative operator in the vertical direction with respect to p. The left- hand side of (4) is a straightforward reinterpretation of (6). The reference surface is now a neutral surface rather than an iso-

baric surface, 0• is simply the temporal derivative of 0 on this surface, and Vn. ¾n0 is u(c•O/c•x)In + v(c•O/c•y)In (that is, the dot product of the horizontal velocity and the horizontal gradients of the 0 values, measured on the neutral surface and projected

to the horizontal; see Figure 1). Here, (Dp/Dt)(c•/c•p)l,,y,t is due to the vertical movement of the water parcel through the reference surface; this term is expressed here as the vertical fluid velocity through the neutral surface e, multiplied by the vertical spatial gradient O z . Note that this coordinate system differs slightly from the orthogonal coordinate system used by Mc- Dougall [1984], where lateral distances were measured along the neutral surface and motion through neutral surfaces was measured in the direction normal to the surface. The present coordinate system is preferred because it is more convenient to use maps of properties projected from the neutral surface onto the horizontal than to use these same properties on the wrin- kled neutral surface.


Having established that the le•hand sides of (4) and (5) are the material derivatives of 0 and S, some comments are re-

quired on the right-hand sides of these equations. Mesoscale eddy motions (meandering of current systems, isolated eddies, two-dimensional turbulence)cause fluctuations of properties on large lateral spatial scales (1-1000 km) and on time scales of days to several months. Fluid parcels are moved about laterally by these eddying processes in such a way that no work is done against gravity, and so this activity takes place along neutral surfaces (rather than along surfaces of constant potential density). A typical eddy diffusivity that represents the mixing of scalars by these mesoscale eddying motions is of the order of 10 3 m 2 s-•, a value that is consistent with the product of an eddy velocity scale of 0.1 m s- • and a length scale of 10 km. Armi and Stommel [1983] found a value of 500 m 2 s- 1 from analyzing salinity variations in the "fl triangle" region of the North Atlantic at a depth of approximately 700 dbar. Thiele et al. [1986] estimated values of between 1700 and 2900 m 2 s- • in the North Atlantic from distributions of tran-

sient tracers (tritium, 3He, and freons) on the potential density horizons a o -- 26.5 and 26.8, both of which lie above 300 dbar in this region. McWilliams et al. [1983] used SOFAR float data to find eddy diffusivities ranging from 6500 m 2 s- 1 at 700 dbar to 1500 m 2 s -1 at 1300 dbar at the Local Dynamics Experiment site in the North Atlantic, a region of only moder- ate eddy activity. Haidvogel et al. [1983] have reviewed present knowledge of the turbulent motions that "tease out" the lateral gradients of properties until the lateral length scales are small enough for some other process, such as shear flow dispersion [Young et al., 1982], to do the final intimate mixing on the molecular scale. In this paper a value of 1000 m 2 s- 1 is taken for the lateral diffusivity due to these mesoscale eddy motions, even though this is considerably lower than most of the recent estimates for the upper 1500 dbar of the North Atlantic.

Lateral mixing by mesoscale process could, at least in prin- ciple, occur without any mixing across neutral surfaces; the diffusivity tensor in the ocean is therefore very anisotropic [Redi, 1982]. In (4) the lateral flux divergence is written as ¾n(KYn), although it is a moot point whether the lateral distances involved in these spatial derivative operations should be measured on horizontal surfaces (as in the chosen projected isobaric coordinate system) or along the neutral surfaces (as was assumed by McDougall [1984] in his expression ¾•.(K¾i0)). Since a typical slope of neutral surfaces in the ocean is 10-3, the difference between these two expressions is O(10 -6) and so can safely be ignored.

The small-scale turbulent mixing processes that do the final, intimate, mixing of properties in the ocean involve three- dimensional turbulence, which can be assumed to be spatially isotropic. The diffusivity tensor associated with small-scale mixing of scalars can then be assumed to be isotropic. How- ever, since typical values for this diffusivity are of the order of 10 -'• m 2 s- l, small-scale turbulent mixing will be less effective than mesoscale eddy activity (by a factor of 107) in producing fluxes of properties along neutral surfaces. Because of this, small-scale turbulent mixing enters the conservation equation of 0 through a term (DOd) d, where d is measured normal to neutral surfaces as in the work by McDougall [1984]. The difference between this parameterization and the (DOz) z expression in (4) is again proportional to the slope squared of

neutral surfaces (i.e., O(10-6)), and so the simpler vertical coordinate z is chosen. Similar arguments can be made for the double-diffusive flux divergence terms (Fz ø and F• s) in (4) and (5).

The fundamental conservation laws that lead to (4) and (5) are the laws of conservation of mass, salt, and enthalpy. Since enthalpy is not exactly a linear function of potential temperature, there are in fact additional terms in (4) due to (1) the variation of the specific heat of seawater with potential temperature, (2) the variation of the specific heat of seawater with salinity, and (3) the "heat of mixing" of seawater, which is related to the variation of the chemical potential of seawater with temperature and salinity. Fofonoff [1962] gives an expression for the temperature changes caused by these three nonlinear effects when two water masses are mixed together. It is readily shown that the nonlinear terms in the equation of state that lead to cabbeling and thermobaricity swamp these three small nonlinear thermodynamic terms in the potential temperature conservation equation by a factor of 100, and so (4) is accepted as sufficiently accurate for all foreseen oceanographic use.

2.1. Equation for the Vertical Velocity Throu•7h Neutral Surfaces

Multiplying (5) by fl and substracting • times (4) leads to

(e - Dz)•7 -iN 2 = D(•Ozz -- fiSzz ) + (fiFz s -- •Fz ø)

Lc0 IV"ø12 + v.o.v.p (7) where •/•0 and c•/c•p are in fact shorthand notations for

and

respectively (see McDougall [1984] for details of the derivation of (7)). The most important point in (7) is that the vertical velocity through neutral surfaces e is a direct result of various mixing processes. The first term on the right-hand side of (7) is due to small-scale turbulent mixing. Note that the term (•0= - ilS=) is not simply equal to g-•(N2)z but contains additional terms because of the nonlinearities of the equation of state of seawater (see (13) below). Apart from these nonlinear terms, vertical mixing contributes to e the simple e•- pression N-2(DN2)z, which implies that if the flux of buoyancy DN 2 is independent of height z, then the vertical velocity through neutral surfaces e is zero. This obviously requires D to vary as N-2, a parameterization for D that has been used by McCreary [1981] to uncouple various modes of motion in upwelling current systems in the presence of mixing of scalars and momentum. Since upwelling is believed to occur at the equator, it is clear that McCreary's [1981] parameterization of D cannot be correct.

Spatial variations of diffusivities are usually termed "pseu- dovelocities" since they appear in conservation equations as advective terms but they do not appear in the continuity equation. In our neutral surface reference frame, however, D• causes a vertical velocity through the neutral surface exactly


45ON . • ...... o ..... •5 N

o

7SoW 65 ø 55 ø 45 ø 35 ø 25 ø 15 ø 5 o 75Ow 65 ø 55 ø 4•o 3•o 2•o 1•o

4 s ø N

35 ø -1 • -0.5

• -1 .5 •0.5 • 1 575•we,5o 5,5o 4• ø •o %0 •o •• 75we5 5,5 4: •5 25 ø •5 5 ø 4o.

1:: • • 5 ø 75ow65 ø 55 ø 45 ø 35 ø 25 ø 1• o 5 o 75ow65 ø 55 ø 45 ø 35 ø 2'5 ø 1• o 5 o

Fig. 2. Data displayed on a neutral surface that has a potential density of 26.80 at the southwest corner of the square of data. (a) The pressure on the surface (decibars). (b) Parameter 0• (øC m-2). (c) Ratio gN-20z3fl(d2S/d02)/O•. This is the ratio of (1) the total effect of the vertical diffusivity D at producing changes of 0 on neutral surfaces to (2) the straightforward vertical diffusive term DO.,•. (d) Ratio gN-20z3•(d2S/d02)/S•. This is the ratio of (1) the total effect of the vertical diffusivity D at producing changes of S on neutral surfaces to (2) the straightforward vertical diffusive term DS•. (e) Ratio (R•/7œ - 1)/(R• - 1). This is the factor that multiplies the vertical divergence of the flux of 0 caused by salt fingering in (8) and (11). (f) The reciprocal of (•//•)•/(•//•) R•/7œ(R• - 1). This length scale ranges from 200 m to over 800 m and represents the importance of nonlinear terms in the equation of state on the way the salt fingering causes water-mass transformation (see text). Values greater than 800 m are not contoured, since at these large length scales, the nonlinear terms are not important.

equal to D z. One can think of this as (minus) the vertical velocity of the neutral surface through the fluid. The terms in (7) proportional to the lateral diffusivity K are the main concern of this paper, the first being cabbeling and the second thermobaricity. These terms are discussed in section 3.

2.2. Water-Mass Transformation Equation

Having realized that the vertical velocity through neutral surfaces e is not a separate physical process but only occurs in response to mixing processes, it is therefore sensible to elimi- nate this vertical advective term from (4) using (7), obtaining

0, + (v" - V.K). V.0

--KYn20+OgN-20z3fld28 (• )-•- •oF• • _ F• ø } (• - •) ?

2 f• • } -{- K •J N - Oz • •-. • IVn012 -{- •pp Vn O . V nP (8)

where R,, = •xO•/flS•. From (8) it is seen that small-scale turbulent mixing causes changes of 0 on neutral surfaces at a rate proportional to the diffusivity of this process, D, multiplied by

the second derivative of S with respect to 0, d2S/dO 2, evaluated from a vertical cast of S and 0. This is because this term in (8) includes both the diffusive contribution, (DO0•, and the vertical advective contribution,

- D=O= - DO zgN- 2(•z0•z -- flS•)

to the rate of change of 0 on neutral surfaces. It is this total combination of terms that appears in (8) and has the ability to cause water-mass transformation (e.g., to change 0 on a neutral surface). Notice that vertical variations of D do not enter (8). This is because D• causes an additional vertical velocity through neutral surfaces of exactly the same size (i.e., e in- creases by D z) and because these two effects cancel in the conservation equation (8). That is, vertical variations of the vertical diffusivity D z do not cause water-mass conversion and so cannot change property-property plots like the O-S diagram. The main concern of this paper is the cabbeling and thermobaric processes that appear as the last two terms in (8), but before discussing these processes, some data from the North Atlantic are presented to illustrate the effects of small- scale turbulent mixing and double-diffusive convection on the water-mass transformation equation (8).


i i i i I i i i I i I i

55ON (a} . , _•/ 55ON• (b} , , •1

1 o •?•oo _

45 ø 700

•o • • •o• •x,7 • ow o

o (c o • •5ON• (d) .... •{

•sø • • • •sø • • • •ø•ow•o •;o •o ,•o •o •ø•oow••7•o•• •o.q,•,.; •,• •, • ß , , ,o , .

800 • 0• •ø / ::•• •ø"• "• ••øø •• • 2 45ø

I I I

50øW 40 ø 30 ø 20 ø 10 ø 0 ø 50øW 40 ø 30 ø 20 ø 10 ø 0 ø

Fig. 3. Data displayed on a neutral surface that has a potential density of 27.30 at the southwest corner. The potential density on this neutral surface varies from 27.30 to 27.39. For captions describing each of the plots, see Figure 2.

2.3. Small-Scale Turbulent Mixing and Salt Fingering

The influences of the vertical diffusivity D and of salt fingering on water-mass conversion have been investigated on three neutral surfaces in the North Atlantic. These neutral surfaces

were found from the Levitus [1982] annual mean data set by

ensuring that cz60 = figs (see (1)) along lateral integration paths where 60 and 6S are the changes in 0 and S from one conductivity, temperature, and depth cast to the next. The pressure on these neutral surfaces is shown in Figures 2a, 3a, and 4a. The potential densities are not constant on neutral surfaces but vary from 26.77 to 26.80 (Figure 2), from 27.30 to

5454 McDOUGALL' THERMOBARICITY, CABBELING, AND WATER-MASS CONVERSION

27.39 (Figure 3), and from 27.73 to 27.83 (Figure 4). The Lev- itus [1982] data set has data at a fixed number of depths, and in order to calculate second derivatives in the vertical direc-

tion, quadratics were fitted to the potential temperatures and salinities over six data points on each cast, three values from above the neutral surface and three from below.

The total effect of vertical mixing on water-mass transformation is given by

d2S

DgN- 20z3l• dO 2

(see (8)). For computational purposes, this can be expressed as

DgN- 20z3l• d2S = (9)

which is shown in Figures 2c, 3c, and 4c as a fraction of the diffusive flux divergence term DO=. Since 0= can be zero, the magnitude of the ratio was limited to 2.5 before contouring. It is seen that water-mass transformation bears little relation to

DO•z but in some place occurs at a small fraction of DO=, while there are large regions where water-mass transformation occurs at a rate of --DO= or less. On the shallow surface (Figure 2), the change in the sign of water-mass transformation as a fraction of DO• occurs due to the change of the sign of 0=, while d2S/dO 2 (and hence the vertical diffusivity term in (8)) is positive throughout this surface. On the intermediate neutral surface, both d2S/dO 2 and 0= are positive everywhere (Figures 3b and 3c), while on the deep neutral surface, 0= is everywhere positive and d2S/dO 2 changes sign and is negative southward of about 35øN. Also shown in Figures 2, 3, and 4 is the ratio of the term that causes changes of S in the neutral surface DgN-20z3o•(d2S/d02), divided by the diffusive term DS_.:. This ratio is also quite different from unity, and there is a substantial region on the shallow neutral surface (Figure 2d) where this ratio is negative.

Turning to the double-diffusive term in (8), consider now the salt-finger mechanism. The buoyancy flux ratio for salt fingers,

o•F ø

3's - flF s (10) is known to be a weak function of the stability ratio of the

water column, R, = •O:/flS: [Schmitt, 1979; McDougall and Taylor, 1984], and here we shall assume that 7s is a constant and equal to 0.5. Under this assumption, the double-diffusive term on the right-hand side of (8) can be expressed as

- ø]

(n. - FO • • Fz 0 7f(R.- 1) (a/fl) •(R.- 1)

The contribution to the change of 0 on neutral surfaces from the vertical divergence of the salt-finger flux of 0 is simply --F• ø. When the vertical advection caused by the salt fingering is also included, the total effect of salt fingering on the conservation equation for 0 on neutral surfaces is given by (11), where the vertical divergence term is multiplied by the factor (Rp/7•--1)/(R o -- 1) and a term appears due to the nonlinearities of the equation of state. The multiplying term,

(R./7• -- 1)/(R. -- 1), is shown in Figures 2e, 3e, and 4e for the three neutral surfaces. A typical value for the upper and lower surfaces is about 3, while the intermediate surface has slightly lower values, closer to 2.5. These values mean that salt fingers are actually approximately 3 times more effective at causing water-mass transformation (changes of 0 on neutral surfaces) than would be inferred by simply considering the vertical flux divergence term --F• ø. Under the Mediterranean tongue (see Figure 4e), where R. is low (about 1.5), the multiplying factor rises to values greater than 4, making salt fingering a very effective agent for water-mass transformation in this region. Figures 2f, 3f, and 4fshow a vertical length scale Z defined by

(•//5)z R. (12) (•//•) 7•(•. - 1)

If this length scale is significantly less than the vertical length scale over which F ø varies (Fø/Fzø), then the nonlinear term in (11) will be large in comparison with --F•ø(Ro/ •. -- 1)/(R, -- 1). If one assumes that (Fø/Fz ø) is likely to be no more than 200 m in the ocean, then it is clear from the figures that the nonlinear term in (11) will not often be more important than the modified flux divergence term

--F•ø(R./7j.- 1)/(R.- 1). While considering this vertical length scale Z associated with nonlinear terms in the equation of state, it is noted that the vertical velocity through neutral surfaces caused by vertical turbulent mixing can be expressed as (from (7))

e = N-2(DN2)• - D (Ra _ 1) R• D

-- (13) -• N- 2(DN2)z 2Z

since fi varies little in the ocean compared with •. There are, of course, additional contributions to e from salt fingers, cabbeling, and thermobaricity, which are not shown in (13).

2.4. Discussion of Vertical Mixing Processes in Wunsch's Inverse Box Models

In a series of papers beginning in 1978, Wunsch has pio- neered the application of the mathematical inversion tech- nique to obtain estimates of reference-level velocities from hydrographic data. These models are now becoming progres- sively more sophisticated. Wunsch et al. [1983] attempted to interpret the residuals of the fitted equations in terms of vertical advection of fluid through isopycnals, while Wunsch [1984a, b] specifically includes the vertical advection velocities in the model equations. In this section of the paper, it is shown that it is inconsistent to include diapycnal advection of properties in these models without, at the same time, including the vertical diffusive fluxes. Until this is done, any conclusions about upwelling rates (e.g., at the equator) are very uncertain and may even have the wrong sign.

In terms of the conservation equations (4) and (5), Wunsch's [1984a] procedure is equivalent to including the vertical advective terms, eO z and eS z, on the left-hand sides of these equations but ignoring the right-hand sides of the equation. Since Wunsch's interpretation of the origin of the diapycnal velocity did not include lateral mixing or double-diffusive convection, an attempt is made here to reinterpret Wunsch's results assuming that small-scale turbulence is the only type of mixing


i I i I I i i i i i i

' / // 1800 SSøN

25 ø 25 ø •2x1• e

$o $OOW 40 ø 80 ø 20 ø 10 ø Oø $OøW 40 ø 80 ø 20 ø 10 ø Oø

I I I I I I I I I I I I

4• o 1 •••• •o •o

i i

50øW 40 ø 30 ø 20 ø 10 ø Oø 50øW 40 ø 30 ø 20 ø 10 ø Oø

55øN

45 ø

35 ø

25 ø

15 ø

5 ø

;5 ø

5 ø

5 ø

5 ø

_ (e) 210 ß 2. 3.0 .

4.0

.

i

' 0 o 0 ø 30 ø 20 ø I

(f) I i i

i

50øW 40 ø 50øW 40 ø 10 ø 0 ø

8OO

ß 800

i

30 ø 20 ø

Fig. 4. Data displayed on a neutral surface that has a potential density of 27.75 at the southwest corner. The potential density on this neutral surface ranges from 27.73 to 27.83. For descriptions of each of the plots, see Figure 2.

process. While this may not be a reasonable first guess of the relative importance of various mixing processes in the ocean, adding one physical process at a time to a model is certainly a common and acceptable procedure for the purpose of model

building and testing. The point that will be emphasized in this section of the paper is that small-scale turbulent diffusion in (4) and (5) contributes two terms to both equations; the diffusive terms (e.g., (DOz)z) and the advective terms eO z.

5456 McDOUGALL; THERMOBARICITY, CABBELING, AND WATER-MASS CONVERSION

• b

I

Fig. 5. Figure illustrating the thermobaric vertical advection process (see text for a discussion). Note that the thermobaric effect causes the migration of the two water parcels, a and b, off the neutral surface as they approach the central point, c. Intimate mixing of the two parcels at point c consolidates this thermobaric vertical motion and also causes a further sinking of the mixed parcel from c to d. This further sinking is due to the cabbeling process.

An estimate can be made for the relative magnitudes of the total effect of vertical mixing in (4) and (5) compared with that due to diapycnal advection above, based on the equatorial portions of Figures 2c, 2d, 3c, 3d, 4c, and 4d. Since e is now caused only by the vertical diffusivity D, these figures display the combination of terms

½o m (DOz z __ eOz)/(DOzz)

(Figures 2c, 3c, and 4c) and

½t s -- (DS.... --

(Figures 2d, 3d, and 4d). Wunsch's [1984a, b] equations con- tained only the diapycnal advective term. The ratio of the total effect of vertical mixing to the advective term can be exl:'ressed in terms of ½0 and ½s as

(DO:: -- e0..)/(e0:) = ½tø/(1 -- ½t ø)

and

(DS.... - eS..)/(eS..) = ½s/(1 - ½s)

for 0, and S, respectively. At a depth of approximately 250 m in the equatorial Atlantic (Figures 2c and 2d), ½tø• 0.5 and ½s • 0, implying that the advective term e0 z is equal to the correct combination (DO.... -- eO..) (since ½tø/(1 -- ½t ø) - 1) in the heat equation but that the advection term eS z seriously misre- presents the total effect of vertical mixing in the salt conservation equation, since (DSz•- eSz)- 0. At a depth of about 750 m (Figures 3c and 3d) in the equatorial Atlantic, the "total to advective" ratios, ½0/(1 - ½0) and ½0/(1 - ½s), are both very

large (since ½0 • ½s • 1), implying that the total effect of vertical mixing is much larger than Wunsch's assumed advective terms. At a depth of about 1500 m, the ratio ½ø/(1- ½0) is approximately -0.4, whereas ½s/(1- ½s) is very large (since ½0 • -0.7 and ½s • 1).

These simple examples at just three depths show that the diapycnal advection terms that Wunsch [1984a] used bear no useful relation to the correct, total effect of vertical turbulent diffusion in the conservation equations. Wunsch's analysis also included a conservation equation for bomb radiocarbon, and there is no indication of the relative magnitude of its diffusive flux term to its advective term. Bennett [1986] has found that the diapycnal velocities obtained by Wunsch et al. [1983] are inconsistent with either a vertical diffusivity or salt fingering.

Bennett [1986] has alluded to a different but equally worri- some problem with the deduced diapycnal velocities of Wunsch et al. [1983]. The diapycnal velocities were found from the continuity equation. Bennett has argued convincingly that the noise in the continuity equation that arises from taking lateral differences of the lateral velocity components (which are themselves model outputs) is too large to be able to deduce the diapycnal velocity. I believe the solution to this problem is to use the advective form of the conservation equations (like (4), (5), (7), and (8)) rather than the flux form that has been used in the inverse studies discussed above. T. Joyce and T. J. McDougall (manuscript in preparation, 1987) and E. Bauer (personal communication, 1986) have both performed overdetermined inverse studies based on this approach (i.e., using (8) and similar equations for other tracers). The reader is referred to Joyce and McDougall's paper for a self-consistent way of casting (8) in finite-difference form between pairs of neutral surfaces.

3. THERMOBARICITY AND CABBELING

3.1. Dianeutral Advection

The physical basis for the thermobaric vertical advection process can be understood by considering the lateral mixing of water parcels in Figure 5. Here a neutral surface is sketched along which pressure p and potential temperature 0 both vary so that V,O.V,p > 0 (the lower sketch in Figure 5 shows the variation of 0 on the neutral surface). Along the neutral surface, variations of in situ density gp are related to variations of pressure •Sp through the relation gp = p7cSp, where 7 is the adiabatic and isentropic compressibility defined by ?- p-•(c•p/•P)lo.s. Note, however, that the compressibility is not constant but is a function of position on the neutral surface through the variables S, 0, and p. Consider moving two water parcels, a and b, together so that they meet in the middle at point c and then mix together. Until these parcels meet at point c, assume that no mixing occurs, so that along the dashed paths a-c and b-c, the potential temperature and salinity of the water samples are preserved at their original values. Since 0 (and S) change along the neutral surface, at each point along the paths a-c and b-c of the water parcels, they have a different 0 and S to the values on the neutral surface at that

latitude and longitude. It follows that the compressibility 7 of the water parcel is different from that on the neutral surface at this location. Subsequent movements of these water parcels toward point c will be such that the changes in gp and gp will be related by the in situ compressibility of the water parcel, rather than by the compressibility of the fluid on the neutral


45ON •///_•. .... o,,, , ......

• •o • -o.5•-o. •••

- ? 1_oo I• •o.o5•••

• -0. -U.• • ' 5 ø ?////•///• , , ,

, , , , , , • • 75øW 65 ø 55 ø 45 ø 35 ø 25 ø 7 45 ø 3

45ø" •<&-o'.2 ' ' ' 45ø" •//•& 0'.05 '0.0• ' ' •

o

-o. 1 -o.• 25 ø [ -o.2-o. 1 • 5 • , ' o ' o ' o ' o 5 o z5øw65 ø 55 45 3s 25 •o •o 75ow65 o 55 ø 45 ø 35 ø 25 ø

45øN

35 ø

25 ø

15 ø

5 ø

-'0,2 .-o'.o5' ;o. ' •((e) •/'•_f• •(')• ({••

• -o.o5 .

_ . • o • ¸ 0.05

75øW 65 ø 55 ø 4• ø 3• ø 2• ø 1• ø 5 ø 75•W 6• ø 5• ø 4• ø 3• ø 2• ø

Fig. 6. Data on the shallow neutral surface (see Figure 2 for the depth of this surface). (a) The vertical velocity through the neutral surface due to cabbeling. (b) The vertical velocity through the neutral surface due to thermobaricity. (c) The effective vertical diffusivity, Dcff,e Cab, that produces the same vertical velocity through the neutral surface as cabbeling. (d) The effective vertical diffusivity, Dcff,e tb, that produces the same vertical velocity through the neutral surface as thermobaricity. (e) The effective diffusivity, Dell,0 Cab, that produces the same amount of water-mass conversion as cabbeling. (f) The effective diffusivity, Dcff,0 tb, that produces the same amount of water-mass conversion as themobaricity.

surface at that latitude and longitude. In this way the water parcels move off the neutral surface and arrive at point c. If these parcels were moved back to their original positions, they would retrace their paths, and no irreversible effects would have occurred. However, the mixing of the two parcels together at point c consolidates the vertical movement of the parcels off the neutral surface. The terms in the equation of state that cause this effect are the combinations

: -- 7 c•Oe• -- •'p 3"•-pJ (14) which can be regarded as either the dependence of the thermal expansion coefficient • on pressure or as the dependence of the compressibility 7 on potential temperature, together with the smaller terms (ot/fi)(c•fi/t•p) = (ot/fi)(t•7/t•S). The c32p/c30t3p term in (14) is 10 times as large as the (ot/fip)(t•2p/t•St•p) term, and so we call the physical process of water-mass conversion caused by the combination of nonlinear terms in (14), thermobaricity. A polynomial expression for the thermobaric parameter (14) is given in the appendix.

Once the two water parcels in Figure 5 arrive at point c and mix, the cabbeling process occurs and causes a further vertical

movement of the mixed water parcel from point c to point d. Whereas thermobaricity is due to the variation of o•/fl with pressure along the neutral surface, cabbeling is due to the variations of •/fi with both 0 and S along the neutral surface. The relevant combination of nonlinear terms of the equation of state is

A polynomial expression for this cabbeling parameter is given in the appendix.

Note the subtle distinction between the thermobaric and

cabbeling vertical advection processes. Both depend on variations of 0•/fi along neutral surfaces; thermobaricity is due to the fact that •/fi is a function of the pressure along the neutral surface, while cabbeling is due to the dependence of •/fl on the (compensating) changes of 0 and S along neutral surfaces. As a result, fluid parcels can move off neutral surfaces due to thermobaricity without requiring any mixing of fluid parcels, whereas cabbeling requires the 0 and S of fluid parcels to change and so requires mixing down to the molecular level.

The strengths of the cabbeling and thermobaric processes have been evaluated on the three neutral surfaces in the North

Atlantic by assuming a lateral diffusivity K of 1000 m 2 s-x.


The vertical velocities through neutral surfaces caused by cabbeling and thermobaricity are (from (7))

0• •0• 0• 2 e•) q- 2 •2 (15) e cab = --gN-2KVnO' VnO •-• and

etb=--gN -2 KV,O'V,p( These velocities are contoured in Figures 6a, 6b, 7a, 7b, 8a, and 8b for the three neutral surfaces. Notice that cabbeling always produces a negative velocity e cab, whereas the thermobaric effect produces a vertical velocity through neutral surfaces that depends on the sign of VnO' VnP and so can be either upward or downward. However, thermobaricity, at least in this region of the ocean, tends to produce more downwelling than upwelling. Also, the magnitude of the sinking of fluid through neutral surfaces due to cabbeling tends to be larger than that due to thermobaricity (by a factor of, say, 3).

The strengths of cabbeling and thermobaricity can be expressed in other ways. For example, it is natural to ask what value of the vertical diffusivity would cause the same vertical velocity through neutral surfaces as cabbeling or thermobaricity. These "effective diffusivities" are found (from (7)) to be

Deff,e cab = -- KVnO. VnO •-• q- 2 fl eS •2 • (0•0zz -- •Szz) (17)

Deff,e tb --- -- KV.O . V.p fi (•zOzz -- fiSzz ) (18) They are shown in Figures 6c, 6d, 7c, 7d, 8c, and 8d. Since (•Ozz - [•Szz ) can take either sign (although it is generally positive), these effective diffusivities can be either positive or negative. Before contouring, the magnitudes of these effective diffusivities were limited to 3 x 10 -4 m 2 s-1. Again it is seen that cabbeling tends to be more important than thermobaricity.

Another way of displaying the strength of these processes is to ask what value of the vertical diffusivity would cause the same amount of water-mass transformation (i.e., changes of 0 on a neutral surface) as cabbeling or as thermobaricity. These effective diffusivities are found from (8) to be

Der f 0 cab = q- KVnO'VnO • q- 2 fi2 • 0z2fl ' (19)

Deff'0tb = + KVnO' VnP •PP • 5 Oz2• dO2/] (20) They are shown in Figures 6e, 6f, 7e, 7f, 8e, and 8f. The fact that the "effective diffusivities for upwelling" (Deff,e cab and Deff,e tb) are not equal to the "effective diffusivities for water- mass conversion" (Dell.0 cab and Dell,0 tb) reflects the fact that cabbeling and thermobaricity are not vertical diffusive processes and cannot be parameterized by a simple vertical eddy diffusivity. Rather, these two processes are inherently vertical, advection processes. Consider, for example, the common case in the upper kilometer of the ocean where (cz0zz- •S•)> 0 and d2S/dO 2 > 0. In this case, the downwelling caused by cabbeling (15) produces a negative Dcff,e cab but a positive D•ff,0ca•; large regions where this occurs are shown in Figures 6, 7, and

8. If one wished to parameterize, say, cabbeling by an effective vertical diffusivity, one would need values of different sign to represent the effects of cabbeling on producing a vertical velocity through neutral surfaces and to represent the effect of cabbeling on water-mass conversion. This emphasizes that, as cabbeling and thermobaricity are advective by nature, they must be treated as such, even though the concept of an "effective vertical diffusivity" is useful when comparing the strength of these newly recognized processes with previously known mixing processes.

The magnitudes of the cabbeling and thermobaric processes are very striking in Figures 6, 7, and 8. On the intermediate and deep neutral surfaces, between which the pressure varies from, say, 600 to 1500 dbar, the combined effect of cabbeling and thermobaricity produces a downwelling velocity through neutral surfaces of -0.5 x 10 -7 m s -1 throughout much of the North Atlantic. On the three surfaces shown in Figures 6, 7, and 8, the average vertical velocity due to thermobaricity was negative and equal to about one sixth of the downwelling velocity due to cabbeling. The vertical velocity through neutral surfaces due to both cabbeling and thermobaricity has

average values of -0.20 x 10-7 m s- • on the shallow surface, -0.41 x 10 -7 m s -• on the intermediate surface, and -0.52 x 10 -7 m s -• on the deep neutral surface. On the intermediate and deep neutral surfaces, the average downwelling velocity has a magnitude equal to one half of the canonical upwelling velocity of + 10 -7 m s-•, which must return from the bottom of the ocean across mean neutral surfaces [Munk, 1966]. The sign of the vertical velocity due to cabbeling and thermobaricity is, however, opposite to that of the traditional upwelling value. Also cabbeling and thermobaricity are more important in water-mass transformation at 1000 dbar in a large portion of the North Atlantic than is a vertical diffusivity of the traditional deep value of 10 -4 m 2 s -1. It is a little harder to make such comparisons for the shallow neutral surface (Figure 6), since upwelling velocities and vertical diffusivities are very poorly known for the upper ocean. However, values of effective diffusivities of the order of 0.1 x 10 -4 m 2

s- • (Figure 6e) are of the same order as the modern measured values of diffusivity from microstructure instruments in this depth range.

In preparing Figures 6, 7, and 8, I have attempted to be conservative, and it is likely that the effects of cabbeling and thermobaricity are more important than shown in these figures. First, the value chosen for the lateral diffusivity, K, of 1000 m 2 s-• is probably low by at least a factor of 2 (see the discussion above following (6) and the recent references men- tioned there). Second, the Levitus [1982] data contain substantial smoothing in space over 7 ø of latitude and longitude. Since both thermobaricity and cabbeling depend on the square of spatial gradients (¾,O'¾np and IV,012, respectively), the total effect of these processes, when integrated over a given area of the ocean, decreases in proportion to the first power of the horizontal smoothing scale. For this reason, cabbeling and thermobaricity are likely to be underestimated by a factor of, say, 2 by this smoothing. It is then easy to make a case that cabbeling and thermobaricity may each be, say, 4 times as large as displayed in Figures 6, 7, and 8. It is already clear, however, that these new processes are too large to ignore, either from the point of view of understanding how bottom water is upwelled in the world's ocean or as water-mass transformation processes in the ocean interior. For example, if a


35ø O•vv••

!' ,

[ i i i I i i i i i i i

.-I (•)-o.• -0.• •x• •o.• ,••,_• -o.• •L •/ ••--o,• ) • •• -1.0

ø" ø 35 ø

/ • -0.125••• • o•• •_o.•• •o

• •ow •o •o •o ,•o •o •oow •o o •o o •o o ,o o o o i I i I / i I i i i

55øN . (c) x 0 •• (d) -0 4 • 45 ø -

• •• 0 0

50øW 40 ø 30 ø 20 ø 10 ø 0 ø

c

50øW 40 ø 30 ø 20 ø 10 ø 0 ø

I i i i I i

(e) 0 6 5

45 ø 4

35 ø 3

15 ø 1

i i i O o o 50øW 40"30 ø 20 ø I 0

i I i

o.2••.• c}øL_..:.•_ o

o•

ø ••, i

50øW 40 ø 30 ø 20 ø 10 ø 0 ø

Fig. 7. Data on the intermediate neutral surface (see Figure 3 for the depth of this surface). See caption to Figure 6 for a description of Figures 7a-7f.

study of cabbeling and thermobaricity in the whole world ocean shows an average downwelling due to these processes of, say, - 10- ? m s- •, then small-scale turbulence in the ocean interior would have to be sufficiently intense to cause (by

itself) an upwelling velocity of 2 x 10- ? m s-•, which is the velocity that would be required, first, to overcome the net downwelling velocity of cabbeling and thermobaricity and, second, to upwell bottom water through the ocean with a


55ON•., (a) • I I '•,•L ' " " ' ' 55ON .• (b) 0.4 ?./.& I

'•i -\o• •'-ø'6

50ow 40 ø 30 ø 20 ø 10 ø 0 o o

•o,. ,c,' ' ' '•L •o,-I ,•,' ' ' ',••

45 - 45 • 0 4

3s ø / 3sø '1 ,,.•• o; "--'•.•( i I i e o

i i i i I ! 0 50øW 40 ø 30 ø 20 ø 10 ø 0 ø 50øW 40 ø 30 ø 2 o 10 ø 0 o

(e) [ • [ • (f) ' I-.1 [ •o, ,• •;•\ •o, -o.• •

45 ø 45 ø i • I• J• 35 ø . 35 ø

15 ø 15 ø

5 ø , 5 ø i i i i i i 0 ø 50øW 40 ø 30 ø 20 ø 10 ø 0 ø 50øW 40 ø 30 ø 20 ø I o

Fig. 8. Data on the deep neutral surface (see Figure 4 for the depth of this surface). See the caption to Figure 6 for a description of Figures 8a-8f

velocity e of 10- ? m s-•. If small-scale turbulence is not observed to be this strong, the solution to this paradox must lie with intense boundary or near-boundary mixing.

McDougall (1987) has shown that in the North Atlantic on

the same deep neutral surface that we have considered here, the lateral gradient of potential temperature in a potential density surface is up to 4 times the lateral gradient of 0 in the neutral surface. This means that if the magnitude of the cab-

McDOUGALL' THERMOBARICITY, CABBELING, AND WATER-MAss CONVERSION 5461

beling process had been evaluated without having realized the subtle (but large) differences between neutral surfaces and potential density surfaces, the vertical downwelling velocity would have been overestimated by a factor of up to 4 2= 16, making the downwelling velocity approximately equal to -32 x 10-7 m s-• near the Mediterranean outflow in Figure 8a. The thermobaric nonlinear terms in the equation of state are responsible for the differences between neutral surfaces and potential density surfaces, and so it is not logically consistent to estimate the magnitude of thermobaricity in a potential density reference frame; however, if one did, it too would be overestimated by a factor such as 4 or 16.

3.2. Do Cabbelinq and Thermobaricity Cause Significant Vertical Diffusion ?

The method of quantifying cabbeling and thermobaricity described above is based on the assumption that lateral fluxes of properties can be expressed as a lateral diffusivity multiplied by a lateral gradient. This derivation can be made a little more general without changing the results, by dealing with the lateral fluxes themselves (equal to KV,O and KV,S) without assuming that they can be represented by a lateral diffusivity. The lateral property fluxes in the real ocean are due to turbulent correlations of fluctuating quantities, and the question arises as to whether the smoothly varying ocean that has been considered above has masked any of the properties of cabbeling and thermobaricity. In particular, it can be shown that when a fluid parcel with a potential temperature anomaly of O' mixes with the environment in equal porportions, the vertical excursion of the mixed water due to cabbeling is

• (0') 2 gz = qN -2 (21)

c•0 2

Variations of O' with time imply a variety of vertical excursions, gz, and so one would expect that the effect of the cabbeling process would not be just a vertical advection process but would have superimposed upon this advection a certain amount of vertical dispersion or diffusion. Here it is shown that while this is theoretically true, this vertical dispersion aspect of cabbeling (and thermobaricity) is, in practice, small.

As an example of the use of (21), taking N 2 = 10-5 s-2 and O' = 0.2øC leads to a vertical excursion due to cabbeling, gz, of 0.2 m. Taking a vertical velocity through neutral surfaces due to cabbeling of e c - -10-7 m s-•, the time z between mixing events for an individual fluid particle must be z = gz/eC= 2 x 10 6 s, assuming that every mixing event has 0'= 0.2øC. Of

course in practice, the ocean contains a range of O' and so will produce a range of different vertical excursions, gz, due to cabbeling. An estimate for the ambiguity of gz (i.e., vertical diffusion due to cabbeling) in time z is the magnitude of gz itself (0.2 m in our example). Let us compare this distance with the vertical diffusion length scale (D'r) •/2 over which small- scale turbulent mixing, as parameterized by a vertical diffusivity D, will diffuse scalars in the same time z. Taking D = 10 -4 m 2 s- • and z = 2 x 106 s (as above), this vertical diffusion length scale is 14 m. This means that the vertical diffusion achieved by small-scale turbulence smooths properties over a vertical length scale that is 70 times larger than that achieved by the vertical diffusion aspect of the cabbeling process. An- other way of expressing this result is that the vertical dispersion or diffusion caused by cabbeling is of the same order as that caused by a vertical diffusivity of gze c= 0.2 x 10 -7 m 2

s-• in our example. Since this is less than the molecular diffusivity of heat and is a factor of more than 1000 less than traditional values of the vertical diffusivity D of small-scale turbulent mixing in the ocean, it is concluded that cabbeling, while ineffective at causing vertical diffusion of properties, is a very effective vertical advective process. The same conclusion can be reached for thermobaricity.

4. INTERPRETATION OF MICROSTRUCTURE DISSIPATION DATA

Thermobaricity and cabbeling are processes of a vertical advective nature but are caused by lateral mixing along neutral surfaces. These lateral mixing processes have been parameterized in this paper by a lateral diffusivity K. It is traditional to assume that mesoscale eddy motions or the energetic range of two-dimensional turbulence cause this lateral mixing (or stirring), and yet it is obvious that small-scale processes of some kind must ultimately be responsible for intimate mixing to the molecular scale. At present we have only a very basic understanding of how scalar property gradient variance is transferred from the mesoscale (105 m) to the microscale (10 -3 m). A common thought experiment involves releasing red dye in the ocean and to take snapshots of its evolution. Initially the dye is deformed into long, thin streaks of red fluid, but at some stage [Garrett, 1983] the large spatial gradients of red- ness and the greatly increased length of these gradient regions dictate that the diffusion of red material will eventually be the dominant dispersion process. The lateral processes that cause this diffusion are legion, and we mention here just two: lateral interleaving and shear-flow dispersion. Both of these processes can flux scalar properties along neutral surfaces for any (van- ishingly small) amount of microscale turbulence. The argu- ment here is that the fluxes along neutral surfaces are gov- erned by the mesoscale field and that the smaller scale processes will diffuse the scalars at the rate set by the mesoscale. If the microscale turbulence is particularly weak in a certain area, then the stirring by intermediate-scale processes must proceed for a little longer before the spatial gradients are sufficiently strong for the weak microstructure to be effective.

It is clear from the above that thermobaricity and cabbeling do not have any detectable microstructure signal. The dissipation of mechanical energy e, as measured from modern microstructure instrumentation, can be used to estimate the vertical eddy diffusivity D but is of no use in estimating the flux of heat along neutral surfaces, which is the cause of thermobaricity and cabbeling. Thermobaricity and cabbeling do not have a signature in the dissipation of mechanical energy e, which may be the reason why these two processes have been overlooked to date. Nevertheless both processes produce significant vertical downwelling (or upwelling) and water-mass transformation in the ocean. In other words, microstructure e

measurements are not capable of detecting all the processes that cause water-mass transformation or upwelling, and we must, in the future, use information on lateral turbulence to infer some of the vertical processes.

McDougall (1987) has found a third process that achieves vertical advection in the ocean but does not have any signature in the microstructure velocity field. This process is a consequence of neutral surfaces not being, in fact, well defined; it can be shown that after a loop around an ocean basin on a neutral surface, one will not, in general, arrive back at the same height at which the lateral integration process began. In other words, the definition for a neutral surface is path depen-


dent. It is shown by McDougall (1987) that this can be inter- preted as a vertical advection process.

5. SUMMA}•¾

The results of this paper can be summarized as follows. 1. The vertical advection of fluid through neutral surfaces

is best regarded as a result of various mixing processes rather than as a separate physical process in its own right.

2. A conservation equation for potential temperature 0 on a neutral surface has been derived (equation (8)) that does not explicitly contain the vertical advection term. The price paid for eliminating vertical advection through neutral surfaces is that the conservation equation now requires information on both the S and the 0 spatial fields.

3. The total effect of small-scale turbulent mixing on water-mass transformation is equal to the diffusivity D of this process, multiplied by a term proportional to the curvature of the O-S curve of the water column. This bears little relation-

ship (and is often of a different sign) to the purely diffusive term DO zz (see Figure 4c).

4. Variations of the vertical diffusivity with height D z do not cause any water-mass transformation. Dz causes an extra vertical velocity through neutral surfaces, and the total effect of this extra vertical velocity, together with the D•O• term, results in zero net change of scalars on neutral surfaces.

5. If the vertical diffusivity D is proportional to N-2 (as assumed by McCreary [1981] in an equatorial model), the upwelling velocity e through neutral surfaces is zero (assuming a linear equation of state). Since upwelling is believed to occur near the equator, McCreary's [1981] parameterization of D cannot be correct.

6. Salt fingering causes water-mass transformation at a rate significantly higher than the simple vertical derivative of the flux of a property due to the salt fingers themselves. For example, salt fingers often cause the potential temperature of a water mass to change 3-4 times as fast as the simple vertical derivative of the double-diffusive flux of 0, -F• ø (see Figure 4e).

7. It is shown that inverse box models that do not care-

fully consider both the diffusive and advective effects of mixing processes cannot yield any information on the magnitude of either upwelling or water-mass conversion. I believe the key to such inverse studies in the future is to cast the conservation

equations in the advective form, so that the continuity equation is always exactly satisfied (cfi (7) and (8)). It may also turn out to be more accurate to take a second-order spatial derivative of an observed variable such as 0 and S than to take a

first-order spatial derivative of a derived quantity like a velocity component (which appears in the continuity equation).

8. Because the equation of state is nonlinear, lateral mixing along neutral surfaces causes associated vertical advection through these neutral surfaces. This vertical advection is due to two quite separate processes: thermobaricity and cabbeling. Thermobaricity is due to the dependence of cz/• (where y is the thermal expansion coefficient and fi is the saline contraction coefficient) on pressure along a neutral surface. Equivalently, one can regard thermobaricity as due to the dependence of the compressibility of seawater on 0 and S along a neutral surface. Cabbeling is due to the dependence of •/fi on 0 and S along a neutral surface.

9. The magnitudes of thermobaricity and cabbeling have

been estimated on three neutral surfaces in the North Atlantic.

In the depth range from 800 to 1500 m, they produce a downwelling of fluid through neutral surfaces of the order of -0.5 x 10-7 m s -•. This estimate is likely to be conservative by a factor of 4 or so. If bottom water is to rise to the base of the thermocline in the interior of the ocean, the vertical diffusivity must be significantly larger than previously imagined in order to counteract the combined downwelling effects of thermobaricity and cabbeling, and then to cause some positive motion upward past neutral surfaces of the required size of 10 -? m s -•.

10. It may prove convenient for some purposes to parameterize cabbeling and thermobaricity by a vertical diffusivity. By considering data from the North Atlantic, it is shown that cabbeling and thermobaricity may behave like a diffusivity of one sign as far as water-mass conversion is concerned (this sign could be either positive or negative) and may cause a vertical velocity through neutral surfaces as would a vertical diffusivity of the opposite sign. Such a diffusive parameterization must then be used with extreme caution and may in fact only be useful as a scaling exercise.

11. Neither of these two new water-mass transformation

processes, thermobaricity or cabbeling, can be quantified by microstructure dissipation measurements. Rather, information must be collected on lateral mixing processes before these vertical advection mechanisms can be reliably estimated. The lack of a microstructure signature is probably the reason why the influence of cabbeling and thermobaricity has escaped at- tention to date.

12. It is shown that while cabbeling and thermobaricity are very effective at producing downwelling (or upwelling) of water through neutral surfaces, they are surprisingly ineffective at producing vertical dispersion (or diffusion) of tracers.

APPENDIX: POLYNOMIAL EXPRESSIONS FOR

CABBELING AND THERMOBARIC PARAMETERS

The thermobaric and cabbeling parameters are due to certain nonlinearities of the equation of state when expressed as a function of S, 0, and p. The following polynomial expressions have been fitted over the oceanographic ranges of 0, S, and p. S was varied from 25 to 40 practical salinity units (psu), and p was varied from 0 to 10,000 dbar (for 0 = 0øC), from 0 to 4000 dbar (for 0 = 10øC), and from 0 to 1000 dbar (for 0 = 20øC, 30øC, and 40øC). In this way, 248 data points were fitted to find the following polynomials. The cabbeling parameter is given by

+ 2 fi •S fi2 = +0.136108 x 10 -4 - 0.325332 X 10-60 + 0.663168 x 10-802-- 0.732603 x 10-•ø03

+ (S - 35.0){-0.106105 x 10 -6 + 0.205719 x 10-80}

+ p{-0.891152 x 10-9+ 0.191471 x 10-•ø0}

+ p(S - 35.0)(+ 0.469446 x 10- • •)

+ p2( + 0.120888 x 10- x 3)

The rms error of this fit is 0.00632 x 10-5(øC)-2, and a check value is 0.8008 x 10-5(øC)-2 at S = 40.0 psu, 0 = 10.0øC, and p = 4000 dbar.

McDOUGALL' THERMOBARICITY, CABBELING, AND WATER-MASS CONVERSION 5463

;'-, 1.o

E 6000 2000 • •-----'"'•'-'• 8000

• 000 I I I I I 0.5

._=_

o o.o 0 10 20 30

O (øC)

Fig. 9a

2.0 --

1.0

o.o 0

I I I I I

10 20 30

O (øC)

Fig. 9b

Fig. 9. Graphs of the (a) cabbeling parameter and (b) the thermobaric parameter as a function of potential temperature 0 for values of the pressure p (ranging from 0 to 10,000 dbar) and with the salinity equal to 35.0 psu.

The thermobaric parameter is given by

fi = +0.299559 x 10 -? --0.875412 x 10-90 + 0.150321 x 10-1ø02- 0.139391 x 10-1203

+ (S- 35.0){-0.153332 x 10 -9 + 0.248555 x 10-110}

+ p{-0.556927 x 10-12 + 0.272974 x 10-130

- 0.631646 x 10-1502}

+ p2{--0.453055 x 10 -16}

The rms error of this fit is 0.0112 x 10-8(øC)-I (dbar)-1, and a check value is 1.9810 x 10-8(øC) - 1 (dbar)-i at S = 40.0 psu, 0 = 10.0øC, and p = 4000.0 dbar.

Both the cabbeling parameter and the thermobaric parameter are shown graphically in Figure 9.

NOTATION

vertical velocity of fluid through neutral surfaces. gravitational acceleration. in situ pressure (dbar). vertical (strictly isotropic) diffusivity of scalars (m 2 s- 1).

K lateral (i.e., along neutral surfaces) diffusivity of scalars (m2 s- •).

$ salinity (practical salinity units, psu). N buoyancy frequency, defined by g-iN 2 = o•0 z -- flS z. R, stability ratio of the water column defined by

R,= •O=/fiS=. 0 potential temperature referenced to 0 dbar (øC).

F ø, F s double-diffusive vertical fluxes of 0 and S. V,0 lateral spatial gradient of 0 in a neutral surface.

y thermal expansion coefficient of seawater, - (1/p)(c•p/c•O)ls,v.

fi saline contraction coefficient of seawater, -- ( • /•)(•/•S)1o,..

7 compressibility of seawater, = (1/p)(t•p/c•P)ls,o. 7s buoyancy flux ratio of salt fingers, 7œ - YFø/• Fs. c dissipation rate of kinetic energy by microscale

turbulence (W kg- •).

Acknowledgments. This research was performed during a sabbati- cal visit to the Woods Hole Oceanographic Institution (WHOI) in 1985. Partial support (especially for the computer time) came from the Centre for Analysis of Marine Systems (CAMS) at WHOI. The plotting of the North Atlantic data could not have been performed without the excellent CAMS-ATLAS software plotting package [Sgouros and Keffkr, 1983]. Nikki Pullen did an excellent job with the word processing, and Josephine Nunn prepared the figures for publication. Helpful comments on a draft of this manuscript from Vivienne Aplin- Mawson, Tom Osborn, and Stuart Godfrey are gratefully acknowl- edged.

REFERENCES

Armi, L., and H. Stommel, Four views of a portion of the North Atlantic Subtropical Gyre, J. Phys. Oceanogr., 13, 828-857, 1983.

Batchelor, G., An Introduction to Fluid Dynamics, Cambridge Univer- sity Press, New York, 1967.

Bennett, S. L., The relationship between vertical, diapycnal, and isopycnal velocity and mixing in the ocean general circulation, J. Phys. Oceanogr., 16, 167-174, 1986.

Bleck, R., Finite difference equations in generalized vertical coordinates, I, Total energy conservation, Contrib. Atmos. Phys., 51, 360- 372, 1978.

Carmack, E. C., Combined influence of inflow and lake temperature on spring circulation in a riverine lake, J. Phys. Oceanogr., 9, 422- 434, 1979.

Fofonoff, N. P., Physical properties of seawater, in The Sea, vol. 1, edited by M. N. Hill, pp. 3-30, Wiley-Interscience, New York, 1962.

Foster, T. D., An analysis of the cabbeling instability in seawater, J. Phys. Oceanogr., 2, 294-301, 1972.

Foster, T. D., and E. C. Carmack, Temperature and salinity structure in the Weddell Sea, d. Phys. Oceanogr., 6, 36-44, 1976.

Garrett, C., On the initial streakiness of a dispersing tracer in two- and three-dimensional turbulence, Dyn. Atmos. Oceans, 7, 265-277, 1983.

Garrett, C., and E. Horne, Frontal circulation due to cabbeling and double-diffusion, d. Geophys. Res., 83, 4651-4656, 1978.

Gregg, M. C., Entropy generation in the ocean by small-scale mixing, d. Phys. Oceanogr., 14, 688-711, 1984.

Haidvogel, D. B., A. R. Robinson, and C. G. H. Rooth, Eddy-induced dispersion and mixing, in Eddies in Marine Science, edited by A. R. Robinson, Springer-Verlag, New York, 1983.

Iselin, C. O., The influence of vertical and lateral turbulence on the characteristics of the waters at mid-depths, Eos Trans. AGU, 20, 414-417, 1939.

Ivers, W. D., The deep circulation in the northern North Atlantic, with especial reference to the Labrador Sea, Ph.D. thesis, Univ. of Calif., San Diego, 1975.

Levitus, S., Climatological atlas of the world ocean, Prof Pap. 13, Natl. Oceanic and Atmos. Admin., U.S. Dep. of Commer., Wash- ington, D.C., 1982.

McCreary, J.P., A linear stratified ocean model of the equatorial undercurrent, Philos. Trans. R. Soc. London, Ser. A., 298, 603-635, 1981.


McDougall, T. J., The relative roles of diapycnal and isopycnal mixing on subsurface water-mass conversion, d. Phys. Oceanogr., 14, 1577-1589, 1984.

McDougall, T. J., and J. R. Taylor, Flux measurements across a finger interface at low values of the stability ratio, d. Mar. Res., 42, 1-14, 1984.

McWilliams, J. C., et al., The local dynamics of eddies in the western North Atlantic, in Eddies in Marine Science, edited by A. R. Robin- son, Springer-Verlag, New York, 1983.

Munk, W. H., Abyssal recipes, Deep Sea Res., 13, 707-730, 1966. Redi, M. H., Oceanic isopycnal mixing by coordinate rotation, d.

Phys. Oceanogr., 12, 1154-1158, 1982. Reid, J. L., and R. J. Lynn, On the influence of the Norwegian-

Greenland and Weddell seas upon the bottom waters of the Indian and Pacific oceans, Deep Sea Res., 18, 1063-1088, 1971.

Schmitt, R. W., Flux measurements on salt fingers at an interface, d. Mar. Res., 37, 419-436, 1979.

Sgouros, T. A., and T. Keffer, The CAMS interactive Atlas package: A computerized program for the archiving and presentation of oceanographic data, Tech. Rep. WHOI 83-39, Woods Hole Ocean- ogr. Inst., Woods Hole, Mass., 1983.

Sutcliffe, R. C., A contribution to the problem of development, Q. d. R. Meteorol. Soc., 73, 370-383, 1947.

Thiele, G., W. Roether, P. Schlosser, R. Kuntz, G. Siedler, and L. Stramma, Baroclinic flow and transient-tracer fields in the Canary-Cape-Verde basin, d. Phys. Oceanogr., 16, 814-826, 1986.

Witte, E., Zur Theorie der Stromkabbelungen, Gaea Koln, 484-487, 1902.

Wunsch, C., An estimate of the upwelling rate in the equatorial Atlan- tic based on the distribution of bomb radiocarbon and quasi- geostrophic dynamics, d. Geophys. Res., 89, 7971-7978, 1984a.

Wunsch, C., An eclectic Atlantic Ocean circulation model, I, The meridional flux of heat, d. Phys. Oceanogr., 14, 1712-1732, 1984b.

Wunsch, C., D. Hu, and B. Grant, Mass, heat, salt and nutrient fluxes in the South Pacific Ocean, d. Phys. Oceanogr., 13, 725-753, 1983.

Young, W. R., P. B. Rhines, and C. J. R. Garrett, Shear-flow dispersion, internal waves and horizontal mixing in the ocean, d. Phys. Oceanogr., 12, 515-527, 1982.

T. J. McDougall, Division of Oceanography, Commonwealth Sci- entific and Industrial Research Organization, GPO Box 1538, Hobart, Tasmania 7001, Australia.

(Received December 23, 1985; accepted July 14, 1986.)

thermobaricity, cabbeling, and water-mass...

Documents