Deep-Sea Research, 1972, Vol. 19, pp. 707 to 718. Pergamon Press. Printed in Great Britain.

On the abyssal circulation of the world oeeanmV.

The influence of bottom slope on the broadening of inertial boundary currents

HENRY STOMMEL* and ARNOLD B. ARONSt

(Received 14 December 1971 ; in revised form 24 May 1972; accepted 24 May 1972)

Abstract--A potential vorticity conserving model is offered to explain the remarkably great width of western boundary bottom currents, such as that which conveys the Antarctic Bottom Water northward in the western trough of the South Atlantic Ocean. Effects of varying bottom slope, latitude, and transport are explored. The essential element in this explanation is the small slope of the bottom along which the current flows.

1. INTRODUCTION

DEEP WESTERN boundary currents have been discovered and surveyed in most major ocean basins. There is the well-known Antarctic Bottom Water in the South Atlantic Ocean as observed by the Meteor Expedition (WOST, 1937, 1957) and later during the I.G.Y. (FuGLISTER, 1960; MILLER, 1960). A particularly good section has recently been published (AMos, GORDON and SCHNEIDER, 1971) of the deep current along the Blake-Bahama Outer Ridge in the North Atlantic.

In the Indian Ocean there is a corresponding current off Madagascar (WARREN, 1970). In the South Pacific there is yet another (REID, STOMMEL, STROUP and WARREN, 1968; WARREN, 1970). The position and intensity of these currents correspond to the schematic theory of the abyssal circulation advanced by us earlier in Part II of this series (STOMMEL and ARONS, 1960). However, the width of the observed currents greatly exceeds that expected from simple inertial theory. Surface western boundary currents (the Gulf Stream, Somali Current, and Kuroshio, for example) are approximately 30 kilometers wide, and this corresponds very well with the width expected on the basis of elementary inertial theory (STOMMEL, 1954; ROBINSON, 1963). This 30 km width is the so-called ' radius of deformation '. If we now turn to the deep water, where the density stratification is at least ten times weaker, the radius of deformation is less than 10 km (MORGAN, 1965, Fig. 2). But observation of these deep currents shows that they are very wide--of the order of 500 km. This great width needs explanation.

In this paper we show that the presence of a sloping bottom is capable of producing a very substantial broadening of an inertial potential-vorticity conserving boundary current.

2. GEOMETRY AND G O V E R N I N G EQUATIONS, CASE OF CONSTANT SLOPE

Let us place the origin of a rectangular coordinate system at a depth of the density

*Massachusetts Institute of Technology, Cambridge, Mass. 02139, U.S.A. tDepartment of Physics, University of Washington, Seattle, Washington 98105, U.S.A.

707

708 HENRY STOMMEL and ARNOLD B. ARONS

surface which defines the upper boundary of deep or bot tom water in the interior of the ocean. The x-axis points eastward, the y-axis northward, and the z-axis upward.

Consider now a deep current flowing equatorward and conserving potential vorticity. For convenience of calculation we shall assume the potential vorticity of the current, f /Hi , is relatively uniform. In this way we do not have to distinguish different potential vorticity at each value of the transport function, and we can write the conservation law in the following simple form

~v f -t- ~--~. .f

h(x) H

where h(x) is the thickness of the moving layer, v is the northward component of velocity, and because of the narrowness of the stream, the relative vorticity av/Ox--Su/ay is approximately Ov/dx.

We choose a bottom of the form b = - sx (Fig. 1)*. I f there were no current, the

Z,Z'

Fig. 1. Geometry and definitions of the bottom and interface height of bottom current regime on slope. Dimensionless quantities shown by primes, or Greek letters. Stagnant water regions

shown by shaded areas.

deep water would be stagnant, and the top boundary of the deep water would extend in the level surface z = 0 right up to the origin where it would intersect the bot tom at x = 0, z = 0. The thickness of this stagnant layer is simply sx, and of course this does not satisfy the above conservation law. We can expect the law to apply only to portions of the fluid which are flowing from the source of uniform potential vorticity. Elsewhere we must take the fluid to be at rest, without a recent history of vorticity conservation, or else introduce some additional sources. We choose the former alternative as being simpler. Inside the current the topography of the top of the layer is h + b, and hence the geostrophic velocity v is given by

v = ~ ( h + b)

*In section 5 we will consider a bottom which is part sloping, part horizontal.

On the abyssal circulation of the world occan~V 709

where 7 = g/ fAp/p, Ap being the density difference between the bottom layer and the lighter resting water above the interface. Combining these two equations one obtains the familiar expression

~ h m = o ~ 2 ( h _ _ H )

where

f ~

(X 2 ~ - -

Ap g - - H

P

It is convenient at this point to introduce a non-dimensional set of variables in the following way: (1) the vertical scale is reduced to units of H, the new thickness of the layer being h' = h/H; (2) the x-scale is changed by introducing i = 0~x, where 0~- 1 is the natural scale of an inertial current against a vertical wall. The dimensionless coordinates are also illustrated in Fig. 1.

The solution is therefore h' = pe -e + qe ~ + 1

where the quantities p and q must be determined in terms of boundary conditions and integral constraints.

The equation of the bottom in the new coordinates is

b' = b / H = -- ai where

$ ( 7 ~ " '

0~H

3. B O U N D A R Y C O N D I T I O N S A N D I N T E G R A L C O N S T R A I N T S

In framing the boundary conditions we must provide for the fact that the position of the western and eastern edges, iw and in, (iw < IE) are both unknown, as are p and q. Four conditions are therefore necessary to completely specify the stream.

Condition (la): At IE the thickness of the current should match that of the interior stagnant mass: thus h' (IE) = a i r or

p exp ( - i ~ ) + q exp (t~) + 1 = alE.

Condition (lb): It is possible that there is no resting mass to the east of the current, in which case the depth must vanish at in:

p e x p ( - i n ) + q e x p ( I E ) + 1 = 0 .

Condition (2): In the case governed by condition (la) above, the velocity of the stream should vanish at i = ie , i.e., ~/~x (h -+- b) = 0 and therefore

-- p exp ( - - in) + q exp (ie) = or.

Condition (3a): At the western edge, ire, there may be a vertical wall whose position is given arbitrarily by i*. In this ca~e we have simply

~w = 4*.

710 HENRY STOMMEL and ARNOLD B. ARONS

Condition (3b): The thickness o f the current on the western edge may actually vanish somewhere on the sloping bo t tom and this is a posoible fo rm o f boundary condit ion for the western edge to the current

p e x p ( - - ~ w ) + q e x p ( ~ w ) + 1 = 0 .

The velocity of the s t ream at the point ~w is simply

[Oh' )

In order that the velocity shall not change sign anywhere in the s t ream the terms inside the parentheses must remain of the same sign. In the smooth por t ion o f the s t ream the te rm Oh'/b¢ vanishes, and therefore the te rm bh'/O¢ cannot exceed o in magni tude even at the western edge where h' is varying rapidly. For a b road s t ream we have near ~w

h' = pe -~ q- 1

Oh' _ _ = - - p e - ~ 0 ~

and hence at h = O, ¢ = ~w, bh'/O¢ = 1. Thus i f a < 1 there is a reversal o f velocity near ~w but this is inconsistent with our

picture of the whole current having a single source with fixed potential vorticity. Thus we can apply this boundary condit ion only for regimes where a ~'7 1. In the case where a < 1 we must seek another physical boundary condition which can be used at the western edge of the current without involving a countercurrent.

Condition (3c): At ~w the s t ream is bounded by another stagnant region of constant level h'(~w) + b'(~w), which lies at a higher level than that o f the interior, and extends f rom posit ion Cw westward to an intersection with the bot tom. This condit ion is one o f zero velocity at ~ --- ~w

- - p exp ( - - ~w) q- q exp (~w) = ~.

It can be used only when c r < l .

The four th condit ion is an integral constraint upon the total t ranspor t T of the stream, defined positively equatorwards.

Condition (4): Xe Xe

T = -- vh dx = 7 h (h + b) dx.

Xw Xw

Rewriting this in non-dimensional form, and performing the indicated integration we obtain

r = - - ~ { p2 [e (-- z eE) _ e (-2 ~w)] -i- q2 e (e -~E) - -e <~ ;w)] + 2 p (1 + cr) [e <- ~E) __ e (- ~w)] + 2q (1 - - or) [e (eE) __ e (+w)]

- 2 ~ ( ~ - ~w) } where

On the abyssal circulation of the world ocean--V 711

f g' H ~

Using these four conditions the form of the interface can be computed (i.e. p, q, ~w and ~E determined) as a function of the two dimensionless parameters a and z.

Solutions of the analogous problem of a stream of uniform potential vorticity applicable to the moving top layer of a two-density layered ocean and meant to represent the Gulf Stream have been given before (STOMMEL, 1958, pp. 108-111; ROBINSON, 1963, pp. 154-158). In these examples the width of the current is simply the ' radius of deformation '. A region of uniform thickness (as measured from the nearly level free surface) has vanishing velocity. In the case of the bottom currents discussed here, however, a region of uniform thickness H and zero relative vorticity, lies upon a sloping bottom and therefore can have a non-vanishing velocity of low relative vorticity. The width of this uniform region is quite independent of t h e ' radius of deformation ', which in fact only enters into determining the detailed form of the edges of the current where it joins, smoothly, the resting layer which may be on both sides of the current.

It should be pointed out that we do not have a rigorous justification for choosing the boundary conditions which bring the velocity of the stream to zero where it joins a stagnant layer other than the intuitive choice of avoiding introduction of shear layers. A reviewer has suggested that there may be additional solutions. Further types of boundary condition are also possible: for example, the interface may intersect the sloping bottom at the downslope edge of the current instead of joining to a large mass of stagnant water in the interior. Indeed both the overflow from the Mediterranean and the Denmark Strait exhibit this form. These streams both undergo strong dilution from entrainment and require a more elaborate theoretical treatment than our present model.

4. D E T E R M I N A T I O N OF p , q , ~r, AND ~ w , FOR o" < 1

(SMALL SLOPE)

Let us combine conditions (2) and (3c), solving for p and q:

__ cr e (~E) t r e ( -~w)

P = 1 + e ~ ' q--- 1 + e ~

where A~ = ~E - ~w, and ~E and ~w are yet to be determined. Substituting the values o f p and q into condition (4), for ~- we obtain

A~. = A ~ - - 2 t a n h - -

2

I f we now make use of condition (la) we obtain

~g = _I + t a n h A ~ . 2

Elimination of the tanh function between these two gives

o (~:w + ~:~) = 2 - - ~-.

Thus we have a transcendental equation for determining A ~ = ~r - ~w and a simple

712 I-~NRY STOMMEL and ARNOLD B. ARONS

solution for ~ , + ~E. From these two determinations we get ~ and ~vv separately. The solution for h' is then

1 Jr- a [ - - e (~E--~) + e (~ -ew) ] . ]/' = 1 -q-e ze

Figures 2a and 2b show the form of this solution for small slopes a = 0.05 and 0.1.

1,1

i (r = C,.05 T=I

21

(0)

(b)

Z'

G= % T=!

(c)

l cr =0.1

((j) (~ Q 11 12

o"= 0 5

2-4 3'3

(e'~

,z,

2"1

{f )

0,24 - -

(g)

0-:00 T : I ho'=4

5

~;:i;[ ;;,, %': 4.o y// / / / / .

0"4

Fig. 2. Various configurations o f the current computed with different choices o f parameters.

On the abyssal circulation of the world ocean--V 713

The transport is taken a s , = 1. There is a broad central region in each case where h' = 1, and where the current is uniform. Only near the edges 4 = 4r and ~ = 4w does the depth deviate from this uniformity in such a way as to bring the velocity to zero. Beyond each edge is a stagnant region (shaded). The width of the current is many inertial widths, and decreases as the slopes increases. Evidently in western boundary currents only equatorward currents can be widened. Poleward currents would be extremely narrow.

5. THE SOLUTION FOR MODERATE AND LARGE SLOPE, 0 " ~ 1

When the slope is large several complications arise: first, as a increases past a = 1, the boundary condition (3c) must be replaced by (3b). The stagnant layer bounding the western edge of the stream disappears. This somewhat complicates the expression for transport r, but there is no clear purpose in reproducing the algebra here. An example of this limiting case is shown in Fig. 2c.

The second complication is introduced because as the slope is increased we do not want our model ocean depth for small positive 4 to increase without limit. In fact we want to fix a maximum depth z' -- - Ho or ho' = Ho/H1 for the deep portion of the bottom at large x (or ~). The position of this foot of the slope is ~o -- ho'/a.

I f ~E < 4o then the current lies entirely upon the slope and our previous analysis applies, as is shown in Fig. (2d) where ho' is chosen as 1.2. But if the stream lies on both sides of ~o then the solution must be written separately in two regions, each with different constants p and q, and two more boundary conditions (joining h' and v) at ~o must be introduced. The determination of the constants therefore involves more algebra than we will reproduce here. An example is shown in Fig. (2e) where ho' = 1.2.

The limiting case for infinite slope is the easiest of all. The wall is now vertical (a = oo) and since ~o ~ 0, there is only one region of constant depth Ho (h' = ho). Obviously ~w = 0, and we need only determine ~ under the boundary conditions that h ' ( ~ ) = ho', and v(~E) = 0. The solution is therefore

h ' = 1 + (ho' - 1)cosh (4 - 4E)

where ~ is determined by the transport condition

(2~- + h'o2) ½ - - 1 cosh ~E =

h'o -- 1

Figures 2f and 2g show the solutions for the vertical wall with two choices of ocean depth ho' = 1.2, and 4.0. In the first case the stream is twice inertial width, in the second about half inertial width.

This sequence of figures, with a range of slopes that extends from very small to infinity, shows how gentle slopes broaden the boundary current.

6. AN EXAMPLE FROM THE SOUTH ATLANTIC

Sampling of bottom water is generally fairly sparse. One of the best sections obtained in the South Atlantic is the 16°S section made during the IGY (FUGLISTER,

714 HENRY STOMMEL and ARNOLD B. ARONS

1960). The western port ion o f the section, with contours o f specific volume anomaly c5 in the bo t tom water, is presented in Fig. 3. The 6 = 35 contour exhibits distinct

Om . . . . T . . . . . . . . . r - , . . . . . . . . . . . ~ _ _ _ _

Fig. 3.

6000

40" 35" 30* 2~ °

Contours of specific volume anomaly in deep water (Antarctic Bottom Water) of western trough of South Atlantic as shown by R.V. Crawford section on 16°S.

parallelism to the bot tom over almost 10 ° o f longitude. The difference o f density across ~5 = 40 is roughly 10 -4, hence g ' = 10 -1 cm sec -2. The depth H is evidently about 8 x 104cm, f = 0.5 × 10-4sec -1, - 1 = 18kin. The slope o f the bot tom s is 1.4 x 10 -3, and hence tr = 0-03. I f the t ransport o f the stream Tis o f the order o f 12.8 x 10 6 cm 3 sec -1 , then r = 1, and f rom r/tr = A ¢ - 2 tanh A ~/2 we obtain the width o f the stream A ~ = 33 radii o f deformation, i.e. 600 km.

The large width o f most observed bot tom flowing western boundary currents can be explained in terms of this simple model, but the currents in the South Pacific afford an exception (REID, STOMMEL, STROUP and WARREN, 1968): they are wide without a strong bot tom slope.

REFERENCES

AMOS A. F., A. GORDON and E. SCHNEIDER (1971) Water masses and circulation patterns in the region of the Blake-Bahama Outer Ridge. Deep-Sea Res. 18, 145-167.

On the abyssal circulation of the world ocean--V 715

FUGLISTER F. C. (1960) Atlantic Ocean atlas of temperature and salinity profiles and data f rom the Internat ional Geophysical Year of 1957-1958. Woods Hole oceanogr, lnstn Atlas Set. 1, 209 pp.

MILLER A. R. (1960) Oceanographic data f rom Atlantis cruise 247. Woods Hole oceanogr. Instn Rep. No. 60-40. (Unpublished manuscript).

MORGAN G. W. (1956) On the wind-driven ocean circulation. Tellus 8, 301-320. REID J. L., JR., H. STOMMEL, E. D. STROUP and B. A. WARREN (1968) Detect ion of a deep

boundary current in the western South Pacific. Nature, Lond., 217, 937. ROBINSON A. R. (1963) Wind-driven ocean circulation. Blaisdell Publishing Co., 161 pp. STOMMEL H. (1958) The Gulf Stream. Universi ty o f California Press, 202 pp. STOMMEL n . and A. B. ARONS (1960) On the abyssal circulation o f the world oceans--I I .

An idealised model of the circulation pattern and ampli tude in oceanic basins. Deep-Sea Res. 6(3), 217-233.

WARREN B. A. (1970) General circulation o f the South Pacific. In: Scientific exploration o f the South Pacific, W. S. WOOSTER, editor, U.S. Nat . Acad. Sci., 33-49.

WARREN B. g . (1971) Evidence for a Deep Western Boundary Current in the South Indian Ocean. Nature, Lond., Phys. Sci., 229(1), 18-19.

WriST G. (1937) Atlas zur Schichtung und Zirkulation des Atlantischen Ozeans. Schnitte und Karten yon Temperatur , Salzgehalt, und Dichte. Wiss. Ergebn. dr. atlant. Exped. ' Meteor' 1925-1927, 6, Atlas.

Wris t G. (1957) Quanti tat ive Untersuchungen zur Statik und Dynamik des Atlantischen Ozeans. Stromgeschwindigkeiten und Strommengen in den Tiefen des Atlantischen Ozeans. Wiss. Ergebn. dr. atlant. Exped. ' Meteor' 1925-1927, 6(2), (6), 261-420.

APPENDIX Effect of non-uniform slopes

(a) A wavy slope. Let us suppose that we add a sinusoidal perturbation of amplitude B and wave- number I to the x-dependence of bottom slope:

b = -- s x + B sin lx.

Our differential equation then becomes

b~h 3x--- ~ - - B 18 sin lx = ~2 ( h - - H ) ,

the solution of which is B

h' = 1 + pe -dx + qe dx ct~ sin Ix. l + f f

When I > > ~ (that is, the scale of bottom topography is small compared to the radius of deformation) the slope of the current's interface is just the same as the mean bottom; the current is uniform. On the other hand when I < < ~t, these larger scale corrugations in the bottom slope do affect the inter- face, which now tends to parallel the actual bottom. It is clear then that in discussing the parallelism of the current to the bottom topography we ought to smooth out the bottom features whose scale is less than that of the radius of deformation, 1/~.

(b) A broken slope. As another simple example we choose a bottom made up of segments of constant slope or vertical walls. The question arises: can the stream exist if the slope is segmented ? It appears that as long as the slope is convex down, a continuous band of current is possible, but that if it is convex up the stream cannot be continuous - - in fact it may be split. Let us choose two regions of different slope

~ < 0 , o = o l ; ~:> 0, o = o~.

The current is supposed to be of uniform depth I at both the limits ~: ~ 4- ~o. We may join the two regions by asserting continuity of the interface and of velocity. The solutions for the two regions

< 0 and ~ > 0 are then

~ < 0 : h ' = l + o r 2 °2e~

> 0 : h ' 1 + ° l - - ° 2 e - ~ . 2

If ol > o2 then there is always a solution corresponding to a continuous stream. This corresponds to a

716 HENRY STOMMEL and ARNOLD B. ARONS

' curva ture ' that is convex down - - and is actually the case that is characteristic o f most continental slopes. I f on the other hand the ' curva ture ' is convex up the stream can be cont inuous only if crz - or1 < 2. When c,2 -- ~rl :> 2 the solution implies vanishing o f the thickness of the stream near the break in slope, in which case there is more freedom in setting the form of the stream. This case o f convex up is more rare than the other in deep water, but corresponds rather more to the transition f rom deep continental slope to shelf in shallow water. Examples of these different cases are shown in Fig. 4.

Fig. 4.

o-=t

o - = 0 2

i

3

Shape o f the interface in bot tom current flowing out o f the plane of the page, parallel to a break in slope.

(c) The slope is uniform, but varies from one section to another. The discussion of how such a simple current actually manages to satisfy Bernoulli 's Principle by adjustment of velocity and pressure field as it moves f rom one slope to another needs further development and a specific example may help.

For simplicity let the stream be very broad. Let us compute the t ransport o f the stream, t(f) , between the edge ~:E and some point on the slope ~:.

t ( O = j v h d ~ = --½o 2 + o - - o ( ~ e - - ~)

but since ~E = (1 + e)/c,, t (O = a ~: --(1 + ~2/2) on the slope. At another section [which we denote by a star (*)], where the slope is a* (Fig. 5), the t ransport

may be written as

t * ( ,*) -~ a* ,* -- (1, + - ~ ) .

On the abyssal circulation of the world ocean--V 717

h

D*

Z )

Fig. 5. Same current on two different slopes to illustrate Bernoulli's Principle in this problem.

If we set t ( 0 = t* (f*), then ~* is the point on the starred section which has the same transport function as ~ on the original section. The relation between ~: and f* is then

i f * 2 __ i f2 ~* ~* - ~ ~ - _ _

2

The change of height along a line of constant transport in the interior is

(h* + b*) ~=~* -- (h + b) ~=~.

In the region of uniform thickness this is simply

H ( - - a* ~* + a ~).

The change in ' head ' is 2g' times this,

2 g ' n ( - - a * ~* + a ~),

or by the above identity

g'H(a z -- a*2).

This expression must now be shown to be equal to the change ia velocity squared,

V*2 - - V 2,

which, in the region of uniform thickness is given by the simple expression

(~)2 (s' -- s*2),

which is also simply g ' H (a2 - a*2),

and hence Bcrnoulli's Principle is clearly obeyed. Evidently in going from the region of large slope s, to that of more gentle slope s*, the decrease of (velocity) kinetic energy is absorbed by a slight increase of head (the line AA* joins abscissae of equal transport on the two slopes). Off hand, it might seem that since the lines of constant transport are spaced horizontally in inverse proportion to the slope in the region of uniform slope (and thickness) that there could be no change in head. However, this change occurs because the transport at the foot of the slope is actually sensitive to slope (is less

718 HENRY STOMMEL and ARNOLD B. ARONS

in the a* section in Fig. 5) and hence causes the head to change at all points further up the slope. Figure 5 is drawn for cr = 1, a* = 1/2. Thus, for example, the value of transport at point A

(~: = - 1 ) is the same as that at A* (~:*=--2"75). One can see that in moving from one profile to another the current has experienced an adverse pressure gradient (and slowed down). Similar matching of transport lines is shown in D, D*. In each case the change in level was 3/8 in units of H; the velocity on the * section was reduced to 1/2 its initial value.

(d) Critical latitude. The value of a depends strongly on latitude through the Coriolis parameter. For constant slope, s, a increases rapidly toward the equator. In the South Atlantic ~r passes through unity at about 3°S, which we may call a critical latitude. Poleward of this latitude boundary condition (3c) applies, and we have a stagnant layer of constant level along the western edge of the stream. Equatorward of the critical latitude boundary condition (3b) applies, and the western edge is no longer anchored to the constant level mentioned above. At the equator itself the levels at the western and eastern edges are the same. Hence a major change in depth of the level of the western edge might be expected to occur between 3°S and the equator. Unfortunately, even in the relatively well-surveyed South Atlantic, the available data do not enable us to test this idea in a definitive manner.

At the equator the velocities reach high values, corresponding to the full Bernoulli head available from upstream. After crossing the equator the functions which describe the shape of the current become trigonometric, and the current could pursue the western flank of the Mid-Atlantic Ridge as indicated in the FUGLISTER (1960) atlas. We are uncertain as to the uniform validity of our approxi- mations across the equator, however, and prefer to leave this question for future study.