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Lateral mixing of water masses
Deep-Sea Research, 1967, Vol. 14, pp. 747 to 753. PergamonPress Ltd. Printed in Great Britain. /
Lateral mixing of water masses MELVIN E. STERN*
(Received 18 August 1967) Abstract--The vertical transport of heat and salt in the "salt-finger" convective regime can be enhanced by the vertical shear of a " m e d i u m " scale motion. A weak shear converts compensating temperature-salinity variations on isopycnal surfaces into strong vertical T-S gradients. The cumulative and selective modification of the density field amplifies the medium scale motion. The generation of geostrophic shear by this mechanism is discussed using a basic state with uniform horizontal and vertical mean T-S gradients. It is suggested that the main halocline of the oceaniccentral waters tends to a marginally stable and statistically steady state which has no T-S variations on isopycnal surfaces. Dimensional considerations lead to an order of magnitude estimate of the r.m.s, geostrophic velocity.
1. INTRODUCTION THE TERM "salt-fingers" refers to that convective regime whose ultimate source of mechanical energy is due to the sinking of salt through a stabilizing temperature field (STERN, 1960). Recent laboratory experiments by TURNER (1967) suggest that in an "evolved" state the non-dimensional ratio @) of the heat/salt flux is 3' = 0.56, over a wide range in the vertical gradients of temperature (T) and salinity (S). He utilized the idea that quasi-steady salt fingers will only exist in relatively thin discontinuity layers in the ocean. Large T - S gradients are to be found across and within this layer, which is bounded on either side by somewhat deeper mixed layers. The more familiar kind of convective turbulence is maintained in the latter region by the salt dripping through the discontinuity layer. The details of the stirring motion are supposed to be unimportant, but the intensity must be large enough to prevent the vertical extension of the salt chimneys. On this basis TURNERhas artificially produced a discontinuity layer by means of stirring grids, and then measured the subsequent fluxes with no stirring other than that which is produced by the free convection. Under certain conditions of similarity one can apply the empirical formula to observed T - S discontinuities to compute the convective salt flux. From one set of measurements beneath the Mediterranean outflow a salt flux of 10-7 g/cm2/sec was computed and a corresponding Austausch co-efficient of A = 5 cm2/sec. These figures, though tentative, must be scored against the salt fingers and we shall make use of them in the following discussion of effects expected on a larger scale. The ease with which organized salt fingers may be disrupted would seem to argue against their playing any significant role in the vertical mixing of a turbulent ocean. This is certainly true if there is another potent energy source maintaining small Richardson number in the main thermocline. One can hardly be sure that the windstress or the tide, acting through many non-linear transformations, can do the job. The purpose of this paper is to point out that if, on the other hand, the shear is weak *Graduate School of Oceanography, University of Rhode Island, Kingston, R.I., U.S.A.
747
748
MELVIN E. STERN
then it will lead to an enhancement of the salt-finger effect. Furthermore, and most important, there is a feedback which tends to amplify or maintain the horizontal shearing motion. The reasoning starts by postulating weak lateral T-S gradients in a region with no corresponding density or pressure gradients. It is well known that the advective effect of a horizontal current with vertical shear will be to tilt and stretch the T-S isolines. Eventually such a current will produce strong vertical T-S gradients, but still no laterial density gradients. STOMMELand FEDEROV (1967) have suggested that this is one way to account for the very thin laminae which they observed. Some of these will be favorable for intense fingering, and the resulting diabatic process will produce pressure gradients on a horizontal scale much larger than the thickness of the layers. The suggestion is made, and supported below, that the effect is such as to amplify the initial shearing motion. The horizontal T-S gradients which were postulated initially may either be relics of earlier convective activity or the transformation of very large scale baroclinic systems. In order to make any headway with this potentially vast maze of lateral interconnections we must now make some distinction between the various scales of motion, even though we recognize the probability that there will be no kinematically sharp separation. The term " small" scale will refer to centimeter wide salt fingers and laminae ranging up to ten meters in thickness. Medium scale (cf. HAMON, 1967) refers to vertical structure of 102 meters and at least one horizontal dimension as small as 101 kilometers. Here we also imply time scales large compared to one day so that the Coriolis force must be considered. " L a r g e " scale refers to vertical structure occupying the whole thermocline and horizontal structure of order 102 kilometers. A detached Gulf Stream eddy is probably "large," while the intrusions of shelf water found at the margin of the Gulf Stream by FORD, LONGARDand BANKS(1952) should be called " m e d i u m . " An example of a large scale T-S field which is in unstable equilibrium is shown in Fig. la. The broken lines represent the intersection of inclined surfaces of constant temperature and salinity with a horizontal plane. For the sake of simplicity the lateral T-S gradients will be taken as exactly compensating (fi~,x -- ~Tz = 0) so that the density surfaces are level and there is no mean geostrophic current. As an aid to visualization we have covered one half of the diagram with dots which move with the fluid. The vertical gradients ( e ~ >/3S~ > 0) are also constant and, following the idea of Tt~R~R (1967), we may imagine a concomitant homogeneous field of small scale convection whose salt flux is formally related to Sz by a constant Austausch (.4). A medium scale perturbation is now introduced as shown by the plan view in Fig. lb, and the two vertical sections in Fig. lc. The latter diagram shows our conception of the small scale structure, but the subsequent theory does not make explicit use of this picture. The small scale laminae are necessary to produce the strong vertical gradients across which the salt fingers form. It is suggested that these small scale layers originate in a way which is qualitatively similar to the formation of the medium scale tongues. II.
MEDIUM SCALE INSTABILITIES
Let (u', v', w') denote the (x, y, z) velocity components, respectively; and let S' (x,y, z, t) denote the salinity perturbation, so that the total salinity is S = S'+ S (x, z).
Lateral mixing of water masses
®
Cold and Fresh
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.
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, .
.
749
, .
.
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.
.
. . .
•
.
.
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.
-
,
.
.
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. •
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• -...i. :.;..,1. :.. .~~ E y ~
(~
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."
iI i J (:?
':\~'
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end
salt
fingers,
Fig. la. Top view of the isotherms and isohalines of the undisturbed large scale field. Figure lb shows the alteration in the isolines due to a " Medium" scale perturbation. The lines with arrowheads show the geostrophic streaming at a later stage of development. Figure lc shows two vertical cross sections; one through the center of the intruding fresh tongue and one through the saltier tongue. Note the increased intensity of the "small" scale convection below the center of the salt intrusion and above the center of the fresh intrusion. The thin lines extending downwardsfrom the laminae indicate the salt fingers. We assume that the infinitesimal change in salt flux is A 3S'/3z, and that the change in heat flux is 7Afl/~ 3S'/3z where 7 is a non-dimensional constant (TURNZR, 1967) and fl/¢ is the dimensional conversion factor obtained f r o m the linear equation o f state : p'/po = flS' -- ~T'. Notice that if a conventional Austausch representation were used then the perturbation would only depend on the stable density field and w o u l d not feel the horizontal T - S gradients. With T = T ' -k T denoting the total temperature field, the two t h e r m o d y n a m i c equations m a y be combined as : d
(~T -- rflS) = O.
(1)
MELVIN, E. STERN
750
The relevant equations for the medium scale perturbation are :
3u_.f'-- fv' = at
1 3p' po ~x
by-- + f u ' 3t
po
1 ~p'
by l ~P--~'-k g (~r' -- flS' )
0=--
po bz bu'
by'
bw' = 0
(2)
~x + ~ ; + bW ~-/~s' + w'/~g~ + u'/~g~ --- A~ ~- - S'
bt
bz2
b ~T' + w' ~Tz + u' ~ x 3t
7A/3b2 S' bz2
where f is the constant Coriolis parameter, g is gravity, p0 is average density, and p' is the perturbation pressure. Equations (2) have constant coefficients and solutions proportional to exp(•t -k ik (x -- ~z + ry)} where h is the growth rate of a mode whose half-wave lengths in the vertical and horizontal (y) directions are Lz : ,r [~k[-1 and Lu ~- 7r Irk]-1, respectively. When the harmonic eigenfunctions are substituted in (2) and the algebraic elimination carried through, one obtains the following equation for the growth rate :
O = ;t (h -k k2 a2 A) (~z -k f2 + ~ + (1-
g (~Tz -- [38z)} 7)k2(rA (gflSx(h + r f ) + hgflSz (1 ~ r 2 ) ) .
(3)
This quartic equation certainly has a real root (h > 0) if that term which does not contain h is negative. Consequently those disturbances are unstable for which ra has opposite sign to ~x. It can easily be shown that this implies that the fresh-cold intrusions are sinking (very slowly) while the hot-salty medium scale intrusion is rising. Two of the roots of the quartic are the high frequency inertia-gravity waves, and these may be filtered by noting that their magnitude is much greater than that of the remaining two roots. Use of this approximation reduces (3) to a quadratic. It may also be shown that optimum growth occurs when ~ is so large t h a t f 2 > gflSz (1 + r2)/az and by making this approximation one obtains the following reduced equation for the medium scale growth rate : k 2 ar A2 + Ak2 cr2 ~ + (1 )') gfiSx A 7 - = 0. (4) III.
THE G E O S T R O P H I C AND I S O P Y C N A L A P P R O X I M A T I O N
It is instructive to re-derive (4) by means of physical assertions made at the outset of the argument rather than by numerical considerations at the end. Because the basic isopycnal surfaces are horizontal and the growth of the perturbation is very slow, we shall assume that the latter is geostrophic in the strong sense of the term (w' = 0).
Lateral mixing of water masses
751
The combination of the geostrophic and hydrostatic equations gives the thermal wind equation (5)
fou'- " --_ g 3p__'.
bz
p0 by
The combination of (1) with the equation of state gives 1 dp po dt
d
(6)
at (fls - ~T) = (1 -- 7) fl aS.at
The acquisition of salt by the medium scale motion is proportional to -- 3Fs (x, y, z, t)[bz, where p0 F8 (g/cm~/sec) is the upwards flux of salt by the small scale turbulence. Equation (6) then becomes 1 dp _ pod t
(1 -- 7) fl bFs bz
(7)
Now dp/dt : bp'/bt + u' b#/bx -k w' b#/bz ~ 5p'/bt, because bp/bx : 0 and w' is asserted to be so small that the transport of heat normal to mean p surfaces (by the medium scale motion) is negligible. The elimination of the density from (5) give the relation f b U'
32 Fs
~ --
(8)
g (1 -- 7) fl by ~---~
which dearly exhibits the way y-variations in F~ increase the horizontal component of the vorticity vector. The equations for the salt balance are ~Fs ~z
or
-
5S' bt
+u'8~
/78 -~ -- A --bS bz ~2 Fs ~ Fs - bz 2 5t A
:
(9) if,x - -~u'
bz
which exhibits the way the vorticity modifies Fs. Eliminating F8 from (8) and (9) gives the following equation for the penetrative component of the velocity vector : 1 b2 u' A 3t 2
~3 u' gflS~ (1 -- 7) ~2 u' + = 0. 3t 3z9 f by bz
(10)
The harmonic solutions of (10) have a growth rate that is identical to that obtained from (4) and this justifies the approximation w' : 0 made in this section. An interesting relation is obtained from (10) by multiplying it by u' and integrating over a large volume (z), on whose boundaries u' is assumed to vanish : --
d'r ut uttt
This integral shows that if the mean square (horizontal) vorticity is to increase then the correlation between the two components of vorticity must have the same sign as 8x. IV.
ORDERS
OF
MAGNITUDE
From Equation (4) it is seen that for small values of A the growth rate increases as A~, and approaches the upper bound
752
MELVIN E. STERN L_
(1 -
for large A. A local gradient of/3Sx = 0"007%0 per km is neither so large as to be without observational precedent, nor is it so small as to be typical of the ocean. If one, again somewhat arbitrarily, chooses Lz = 100 m, Ly = 20 km, f = 10-4 sec-1, 9' = ½, A = 5 cm2/sec then the value of ;~ as computed from (4) is slightly less than ;I = 1 × 10-s sec-L It would require a frictional mechanism capable of absorbing 20 % of the geostrophic energy per day to inhibit this instability. A more provocative estimate of the effect of this process can be obtained from dimensional arguments stemming from (8). This equation says that the acceleration • of the disturbance is proportional to 3Fs/3y. The exponential increase of the incipient perturbation (Fig. 1) does not go on forever b u t " equilibrates " a t some characteristic velocity z~ in some characteristic time f. Let fie denote the characteristic salt flux differential, so that 3Fs/3y ~ ~ffs/L u. Subsequently all order unity factors, like ~r or/3 = 1 or 1 -- 9, = ½, will be ignored. Equation (8) suggests the order of magnitude relation
7 =7L
"
We now make the kinematical assumption z =Ly a implying that equilibration occurs in the time it takes a water parcel to cycle around a fraction of the perimeter of the eddy which is schematically indicated by the curved arrows in Fig. lb. From the two preceding equations one obtains : ~=
/(g~s).
(11)
Now consider those mid-latitude oceanic water masses where evaporation exceeds precipitation. Let an average of e (cm/sec) of pure water leave a surface of mean salinity So, so that po eSo g/cm2/sec of pure salt are released. A substantial fraction of this is transported downwards through the halocline, mixing laterally at the same time with the fresh Intermediate Water of Antarctic origin. If we assume that a finite fraction of the small scale flux occurs in conjunction with medium scale motion then -Fs = eS0
(12)
and (11) becomes
According to TURN~R'S (1967) single calculation beneath the Mediterranean outflow the value of Fs was 10-v cm/sec. It is an interesting coincidence that the value of eSo for an evaporation rate of 100 era/year and a surface salinity of 36%0 is also 10-~ cm/sec. Equation (13) is now put forward on purely dimensional grounds as the scaling velocity of the geostrophic fluctuations in those regions of relatively weak mean lateral T - S gradients (e.g. the halocline of the Central waters). Its value is about 1 cm/sec. The constant of proportionality relating it to the r.m.s, geostrophic velocity must depend on the thermal boundary conditions, among other things.
Lateral mixing of water masses V.
CONCLUDING
753
REMARKS
We have shown that in a water mass with large scale variations of temperature and salinity on isopycnal surfaces (and Sz > 0) instabilities develop which transport salt along the isopycnal surfaces. The energetics are primarily determined by the (laterally) available potential energy, and the increase in kinetic energy is accomplished quasistatically. Notice should be taken of the fact that the available potential energy in the mean ocean is very large compared to the kinetic energy of the mean motion, because the length of the ocean is large compared to Rossby's radius of deformation. A lot of transient motion can be maintained if just a little of the stored energy can be passed through t h e " escapement," especially if the latter happens to be locatedin the region of the spectrum where fluid motion is being dissipated into heat. Oceanographers interpret the mixing of water masses by means of a temperaturesalinity diagram. A large number of hydrographic stations, particularly in the North Atlantic Central Water, support the generalization that there are large regions where the temperature is a universal function of salinity (or density). This T - S correlation appears to be strongest in the main halocline of the Central waters. There appears to be considerably more scatter of individual soundings below this (viz. Antarctic Intermediate Water) and also in the surface" mixed" layer above the main thermocline. One also finds more scatter, although this is less documented, in other water masses, viz. Subarctic Pacific Water and parts of Antarctic Water. Several superficial explanations of the T - S correlation of the Central Water have been given; ours is that such a water mass (with ~Tz > fiSz > 0) tends to a state of marginal stability with respect to medium scale motion. Any large scale variation of T or S on isopycnal surfaces tends to be removed by instabilities, in the same way that super-adiabatic gradients in the free region of a heated fluid tend to be removed by vertical convection. There are undoubtedly many reasons for the lack of a strong correlation in those regions where Sz and 5~z are not positive. The precipitation region of the Subarctic Pacific is an interesting case because the long time averages are : S~ < 0 and Tz > 0, and there is no vertically available potential energy. None of the convective varieties that are due to the disparity of the two molecular diffusivities can occur, except in connection with the release of lateral potential energy of the adjacent Pacific Central Water or the fresh coastal water. Large amplitude baroclinic eddies could cause local reversals in the vertical gradients which lead to small scale salt convection. Such a process differs fundamentally from the model disscussed previously and there is no theoretical reason to expect a T - S correlation. REFERENCES
FORD W. L., J. R. LONGARDand R. E. BANKS(1952) On the nature, occurence, and origin of cold salinity water along the edge of the Gulf Stream. J. mar. Res., 11, 281-293. HAMON B. V. (1967) Medium scale temperature and salinity structure in the upper 1500 m in the Indian Ocean. Deep-Sea Res., 14 (2), 169-181. STERNM. E. (1960) T h e " Salt-fountain" and thermohaline convection. Tellus, 12, 172-175. STOMMELH. and K. N. FEDOROV(1967) Small scale structure in temperature and salinity. Tellus, 19, 306-325. TURNERJ. S. (1967) Salt fingers across a density interface. Deep-Sea Res., 14 (5), 599-611.
Lateral mixing of water masses MELVIN E. STERN*
(Received 18 August 1967) Abstract--The vertical transport of heat and salt in the "salt-finger" convective regime can be enhanced by the vertical shear of a " m e d i u m " scale motion. A weak shear converts compensating temperature-salinity variations on isopycnal surfaces into strong vertical T-S gradients. The cumulative and selective modification of the density field amplifies the medium scale motion. The generation of geostrophic shear by this mechanism is discussed using a basic state with uniform horizontal and vertical mean T-S gradients. It is suggested that the main halocline of the oceaniccentral waters tends to a marginally stable and statistically steady state which has no T-S variations on isopycnal surfaces. Dimensional considerations lead to an order of magnitude estimate of the r.m.s, geostrophic velocity.
1. INTRODUCTION THE TERM "salt-fingers" refers to that convective regime whose ultimate source of mechanical energy is due to the sinking of salt through a stabilizing temperature field (STERN, 1960). Recent laboratory experiments by TURNER (1967) suggest that in an "evolved" state the non-dimensional ratio @) of the heat/salt flux is 3' = 0.56, over a wide range in the vertical gradients of temperature (T) and salinity (S). He utilized the idea that quasi-steady salt fingers will only exist in relatively thin discontinuity layers in the ocean. Large T - S gradients are to be found across and within this layer, which is bounded on either side by somewhat deeper mixed layers. The more familiar kind of convective turbulence is maintained in the latter region by the salt dripping through the discontinuity layer. The details of the stirring motion are supposed to be unimportant, but the intensity must be large enough to prevent the vertical extension of the salt chimneys. On this basis TURNERhas artificially produced a discontinuity layer by means of stirring grids, and then measured the subsequent fluxes with no stirring other than that which is produced by the free convection. Under certain conditions of similarity one can apply the empirical formula to observed T - S discontinuities to compute the convective salt flux. From one set of measurements beneath the Mediterranean outflow a salt flux of 10-7 g/cm2/sec was computed and a corresponding Austausch co-efficient of A = 5 cm2/sec. These figures, though tentative, must be scored against the salt fingers and we shall make use of them in the following discussion of effects expected on a larger scale. The ease with which organized salt fingers may be disrupted would seem to argue against their playing any significant role in the vertical mixing of a turbulent ocean. This is certainly true if there is another potent energy source maintaining small Richardson number in the main thermocline. One can hardly be sure that the windstress or the tide, acting through many non-linear transformations, can do the job. The purpose of this paper is to point out that if, on the other hand, the shear is weak *Graduate School of Oceanography, University of Rhode Island, Kingston, R.I., U.S.A.
747
748
MELVIN E. STERN
then it will lead to an enhancement of the salt-finger effect. Furthermore, and most important, there is a feedback which tends to amplify or maintain the horizontal shearing motion. The reasoning starts by postulating weak lateral T-S gradients in a region with no corresponding density or pressure gradients. It is well known that the advective effect of a horizontal current with vertical shear will be to tilt and stretch the T-S isolines. Eventually such a current will produce strong vertical T-S gradients, but still no laterial density gradients. STOMMELand FEDEROV (1967) have suggested that this is one way to account for the very thin laminae which they observed. Some of these will be favorable for intense fingering, and the resulting diabatic process will produce pressure gradients on a horizontal scale much larger than the thickness of the layers. The suggestion is made, and supported below, that the effect is such as to amplify the initial shearing motion. The horizontal T-S gradients which were postulated initially may either be relics of earlier convective activity or the transformation of very large scale baroclinic systems. In order to make any headway with this potentially vast maze of lateral interconnections we must now make some distinction between the various scales of motion, even though we recognize the probability that there will be no kinematically sharp separation. The term " small" scale will refer to centimeter wide salt fingers and laminae ranging up to ten meters in thickness. Medium scale (cf. HAMON, 1967) refers to vertical structure of 102 meters and at least one horizontal dimension as small as 101 kilometers. Here we also imply time scales large compared to one day so that the Coriolis force must be considered. " L a r g e " scale refers to vertical structure occupying the whole thermocline and horizontal structure of order 102 kilometers. A detached Gulf Stream eddy is probably "large," while the intrusions of shelf water found at the margin of the Gulf Stream by FORD, LONGARDand BANKS(1952) should be called " m e d i u m . " An example of a large scale T-S field which is in unstable equilibrium is shown in Fig. la. The broken lines represent the intersection of inclined surfaces of constant temperature and salinity with a horizontal plane. For the sake of simplicity the lateral T-S gradients will be taken as exactly compensating (fi~,x -- ~Tz = 0) so that the density surfaces are level and there is no mean geostrophic current. As an aid to visualization we have covered one half of the diagram with dots which move with the fluid. The vertical gradients ( e ~ >/3S~ > 0) are also constant and, following the idea of Tt~R~R (1967), we may imagine a concomitant homogeneous field of small scale convection whose salt flux is formally related to Sz by a constant Austausch (.4). A medium scale perturbation is now introduced as shown by the plan view in Fig. lb, and the two vertical sections in Fig. lc. The latter diagram shows our conception of the small scale structure, but the subsequent theory does not make explicit use of this picture. The small scale laminae are necessary to produce the strong vertical gradients across which the salt fingers form. It is suggested that these small scale layers originate in a way which is qualitatively similar to the formation of the medium scale tongues. II.
MEDIUM SCALE INSTABILITIES
Let (u', v', w') denote the (x, y, z) velocity components, respectively; and let S' (x,y, z, t) denote the salinity perturbation, so that the total salinity is S = S'+ S (x, z).
Lateral mixing of water masses
®
Cold and Fresh
•. .. .,.. •. . . •
" •" .
.
y ,
-:-::-i.
, .
.
749
, .
.
".'Hot (]nd Salty: [:. •
.
.
. . .
•
.
.
.
•
.
-
,
.
.
•
. •
. -
• -...i. :.;..,1. :.. .~~ E y ~
(~
I
'.i
t
!!
."
iI i J (:?
':\~'
Z
J
"S alr'scole
end
salt
fingers,
Fig. la. Top view of the isotherms and isohalines of the undisturbed large scale field. Figure lb shows the alteration in the isolines due to a " Medium" scale perturbation. The lines with arrowheads show the geostrophic streaming at a later stage of development. Figure lc shows two vertical cross sections; one through the center of the intruding fresh tongue and one through the saltier tongue. Note the increased intensity of the "small" scale convection below the center of the salt intrusion and above the center of the fresh intrusion. The thin lines extending downwardsfrom the laminae indicate the salt fingers. We assume that the infinitesimal change in salt flux is A 3S'/3z, and that the change in heat flux is 7Afl/~ 3S'/3z where 7 is a non-dimensional constant (TURNZR, 1967) and fl/¢ is the dimensional conversion factor obtained f r o m the linear equation o f state : p'/po = flS' -- ~T'. Notice that if a conventional Austausch representation were used then the perturbation would only depend on the stable density field and w o u l d not feel the horizontal T - S gradients. With T = T ' -k T denoting the total temperature field, the two t h e r m o d y n a m i c equations m a y be combined as : d
(~T -- rflS) = O.
(1)
MELVIN, E. STERN
750
The relevant equations for the medium scale perturbation are :
3u_.f'-- fv' = at
1 3p' po ~x
by-- + f u ' 3t
po
1 ~p'
by l ~P--~'-k g (~r' -- flS' )
0=--
po bz bu'
by'
bw' = 0
(2)
~x + ~ ; + bW ~-/~s' + w'/~g~ + u'/~g~ --- A~ ~- - S'
bt
bz2
b ~T' + w' ~Tz + u' ~ x 3t
7A/3b2 S' bz2
where f is the constant Coriolis parameter, g is gravity, p0 is average density, and p' is the perturbation pressure. Equations (2) have constant coefficients and solutions proportional to exp(•t -k ik (x -- ~z + ry)} where h is the growth rate of a mode whose half-wave lengths in the vertical and horizontal (y) directions are Lz : ,r [~k[-1 and Lu ~- 7r Irk]-1, respectively. When the harmonic eigenfunctions are substituted in (2) and the algebraic elimination carried through, one obtains the following equation for the growth rate :
O = ;t (h -k k2 a2 A) (~z -k f2 + ~ + (1-
g (~Tz -- [38z)} 7)k2(rA (gflSx(h + r f ) + hgflSz (1 ~ r 2 ) ) .
(3)
This quartic equation certainly has a real root (h > 0) if that term which does not contain h is negative. Consequently those disturbances are unstable for which ra has opposite sign to ~x. It can easily be shown that this implies that the fresh-cold intrusions are sinking (very slowly) while the hot-salty medium scale intrusion is rising. Two of the roots of the quartic are the high frequency inertia-gravity waves, and these may be filtered by noting that their magnitude is much greater than that of the remaining two roots. Use of this approximation reduces (3) to a quadratic. It may also be shown that optimum growth occurs when ~ is so large t h a t f 2 > gflSz (1 + r2)/az and by making this approximation one obtains the following reduced equation for the medium scale growth rate : k 2 ar A2 + Ak2 cr2 ~ + (1 )') gfiSx A 7 - = 0. (4) III.
THE G E O S T R O P H I C AND I S O P Y C N A L A P P R O X I M A T I O N
It is instructive to re-derive (4) by means of physical assertions made at the outset of the argument rather than by numerical considerations at the end. Because the basic isopycnal surfaces are horizontal and the growth of the perturbation is very slow, we shall assume that the latter is geostrophic in the strong sense of the term (w' = 0).
Lateral mixing of water masses
751
The combination of the geostrophic and hydrostatic equations gives the thermal wind equation (5)
fou'- " --_ g 3p__'.
bz
p0 by
The combination of (1) with the equation of state gives 1 dp po dt
d
(6)
at (fls - ~T) = (1 -- 7) fl aS.at
The acquisition of salt by the medium scale motion is proportional to -- 3Fs (x, y, z, t)[bz, where p0 F8 (g/cm~/sec) is the upwards flux of salt by the small scale turbulence. Equation (6) then becomes 1 dp _ pod t
(1 -- 7) fl bFs bz
(7)
Now dp/dt : bp'/bt + u' b#/bx -k w' b#/bz ~ 5p'/bt, because bp/bx : 0 and w' is asserted to be so small that the transport of heat normal to mean p surfaces (by the medium scale motion) is negligible. The elimination of the density from (5) give the relation f b U'
32 Fs
~ --
(8)
g (1 -- 7) fl by ~---~
which dearly exhibits the way y-variations in F~ increase the horizontal component of the vorticity vector. The equations for the salt balance are ~Fs ~z
or
-
5S' bt
+u'8~
/78 -~ -- A --bS bz ~2 Fs ~ Fs - bz 2 5t A
:
(9) if,x - -~u'
bz
which exhibits the way the vorticity modifies Fs. Eliminating F8 from (8) and (9) gives the following equation for the penetrative component of the velocity vector : 1 b2 u' A 3t 2
~3 u' gflS~ (1 -- 7) ~2 u' + = 0. 3t 3z9 f by bz
(10)
The harmonic solutions of (10) have a growth rate that is identical to that obtained from (4) and this justifies the approximation w' : 0 made in this section. An interesting relation is obtained from (10) by multiplying it by u' and integrating over a large volume (z), on whose boundaries u' is assumed to vanish : --
d'r ut uttt
This integral shows that if the mean square (horizontal) vorticity is to increase then the correlation between the two components of vorticity must have the same sign as 8x. IV.
ORDERS
OF
MAGNITUDE
From Equation (4) it is seen that for small values of A the growth rate increases as A~, and approaches the upper bound
752
MELVIN E. STERN L_
(1 -
for large A. A local gradient of/3Sx = 0"007%0 per km is neither so large as to be without observational precedent, nor is it so small as to be typical of the ocean. If one, again somewhat arbitrarily, chooses Lz = 100 m, Ly = 20 km, f = 10-4 sec-1, 9' = ½, A = 5 cm2/sec then the value of ;~ as computed from (4) is slightly less than ;I = 1 × 10-s sec-L It would require a frictional mechanism capable of absorbing 20 % of the geostrophic energy per day to inhibit this instability. A more provocative estimate of the effect of this process can be obtained from dimensional arguments stemming from (8). This equation says that the acceleration • of the disturbance is proportional to 3Fs/3y. The exponential increase of the incipient perturbation (Fig. 1) does not go on forever b u t " equilibrates " a t some characteristic velocity z~ in some characteristic time f. Let fie denote the characteristic salt flux differential, so that 3Fs/3y ~ ~ffs/L u. Subsequently all order unity factors, like ~r or/3 = 1 or 1 -- 9, = ½, will be ignored. Equation (8) suggests the order of magnitude relation
7 =7L
"
We now make the kinematical assumption z =Ly a implying that equilibration occurs in the time it takes a water parcel to cycle around a fraction of the perimeter of the eddy which is schematically indicated by the curved arrows in Fig. lb. From the two preceding equations one obtains : ~=
/(g~s).
(11)
Now consider those mid-latitude oceanic water masses where evaporation exceeds precipitation. Let an average of e (cm/sec) of pure water leave a surface of mean salinity So, so that po eSo g/cm2/sec of pure salt are released. A substantial fraction of this is transported downwards through the halocline, mixing laterally at the same time with the fresh Intermediate Water of Antarctic origin. If we assume that a finite fraction of the small scale flux occurs in conjunction with medium scale motion then -Fs = eS0
(12)
and (11) becomes
According to TURN~R'S (1967) single calculation beneath the Mediterranean outflow the value of Fs was 10-v cm/sec. It is an interesting coincidence that the value of eSo for an evaporation rate of 100 era/year and a surface salinity of 36%0 is also 10-~ cm/sec. Equation (13) is now put forward on purely dimensional grounds as the scaling velocity of the geostrophic fluctuations in those regions of relatively weak mean lateral T - S gradients (e.g. the halocline of the Central waters). Its value is about 1 cm/sec. The constant of proportionality relating it to the r.m.s, geostrophic velocity must depend on the thermal boundary conditions, among other things.
Lateral mixing of water masses V.
CONCLUDING
753
REMARKS
We have shown that in a water mass with large scale variations of temperature and salinity on isopycnal surfaces (and Sz > 0) instabilities develop which transport salt along the isopycnal surfaces. The energetics are primarily determined by the (laterally) available potential energy, and the increase in kinetic energy is accomplished quasistatically. Notice should be taken of the fact that the available potential energy in the mean ocean is very large compared to the kinetic energy of the mean motion, because the length of the ocean is large compared to Rossby's radius of deformation. A lot of transient motion can be maintained if just a little of the stored energy can be passed through t h e " escapement," especially if the latter happens to be locatedin the region of the spectrum where fluid motion is being dissipated into heat. Oceanographers interpret the mixing of water masses by means of a temperaturesalinity diagram. A large number of hydrographic stations, particularly in the North Atlantic Central Water, support the generalization that there are large regions where the temperature is a universal function of salinity (or density). This T - S correlation appears to be strongest in the main halocline of the Central waters. There appears to be considerably more scatter of individual soundings below this (viz. Antarctic Intermediate Water) and also in the surface" mixed" layer above the main thermocline. One also finds more scatter, although this is less documented, in other water masses, viz. Subarctic Pacific Water and parts of Antarctic Water. Several superficial explanations of the T - S correlation of the Central Water have been given; ours is that such a water mass (with ~Tz > fiSz > 0) tends to a state of marginal stability with respect to medium scale motion. Any large scale variation of T or S on isopycnal surfaces tends to be removed by instabilities, in the same way that super-adiabatic gradients in the free region of a heated fluid tend to be removed by vertical convection. There are undoubtedly many reasons for the lack of a strong correlation in those regions where Sz and 5~z are not positive. The precipitation region of the Subarctic Pacific is an interesting case because the long time averages are : S~ < 0 and Tz > 0, and there is no vertically available potential energy. None of the convective varieties that are due to the disparity of the two molecular diffusivities can occur, except in connection with the release of lateral potential energy of the adjacent Pacific Central Water or the fresh coastal water. Large amplitude baroclinic eddies could cause local reversals in the vertical gradients which lead to small scale salt convection. Such a process differs fundamentally from the model disscussed previously and there is no theoretical reason to expect a T - S correlation. REFERENCES
FORD W. L., J. R. LONGARDand R. E. BANKS(1952) On the nature, occurence, and origin of cold salinity water along the edge of the Gulf Stream. J. mar. Res., 11, 281-293. HAMON B. V. (1967) Medium scale temperature and salinity structure in the upper 1500 m in the Indian Ocean. Deep-Sea Res., 14 (2), 169-181. STERNM. E. (1960) T h e " Salt-fountain" and thermohaline convection. Tellus, 12, 172-175. STOMMELH. and K. N. FEDOROV(1967) Small scale structure in temperature and salinity. Tellus, 19, 306-325. TURNERJ. S. (1967) Salt fingers across a density interface. Deep-Sea Res., 14 (5), 599-611.