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The preconditioning phase of MEDOC 1969—II. Topographic effects
Deep-Sea Research, 1973, Vol. 20, pp. 449 to 459. Pergamon Press. Printed in Great Britain.
The preconditioning phase of MEDOC 1969--H. Topographic effects NELSON G . H O G G *
(Received l September 1972; in revisedform 12 December 1972; accepted3 January 1973)
Abstract--In support of SWALLOWand CASTON'S(1973) conclusion that topography plays a significant role in determining the position of the dense water formation region in the northwest Mediterranean, a theoretical model conserving potential vorticity in flow past a conical bump abutted to the continental slope is studied. For parameters appropriate to this area it is shown that the flow falls in the weak stratification category of HOGG(1973), implying that the streamline pattern is quasi-two-dimensional. For typical values of the lateral shear and a topographic parameter, closed streamlines (i.e. a baroclinic Taylor column) are found over the bump in agreement with the observations. As the region is isolated and also possesses greater initial density and decreased stability, it is suggested that this is where dense water is most likely to be formed. INTRODUCTION
SWALLOW and CASTON (1973) (hereafter called 'Part I') in commenting upon observations taken during the preconditioning phase of bottom water formation in M E D O C , 1969, suggest that the location o f the dense water region in which bottom water later formed was influenced by the local bottom topography. Figure 5 of Part 1 is a chart of the northwest Mediterranean Sea giving both contours of bottom depth and of the surface dynamic height anomaly computed with respect to the 1800 dbar surface. Note that the bottom drops rather sharply from a continental shelf depth of 200 metres to an abyssal plain at roughly 2500 metres, except in a region off the south coast of France centred at 5°E 42°N known as the Rh6ne Deep Sea Fan. Here there exists a bulge in the bathymetric contours beginning at about 1500 metres, and extending to the 2500 metres contour which is displaced approximately 90 km offshore from its more usual position in the area. Note also the form of the dynamic height contours (dashed lines). Away from the bulge these contours are parallel to the coast. A trough or minimum in dynamic height exists from which the dynamic topography rises on either side and is consistent with the previously observed cyclonic surface circulation. With respect to 1800 dbar, at least, the surface flow is toward the southwest inshore of the trough and toward the northeast offshore. Over the bulge this pattern is profoundly altered. Contours inshore of the trough are deflected so as to cross the bathymetric contours and move further inshore while those offshore tend to avoid the bulge. Over the top a contour of even greater dynamic depth than is found in the trough occurs. Note that this -- 1100 dyn m contour, although drawn open, can be closed in a fashion consistent with the observations and that the *National Institute of Oceanography, Wormley, Godalming, Surrey, England. Present address: Department of Earth and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U,S.A. 449
450
NELSONG. HOGG
--1080 dyn m line can be extended to pass over the bulge. It was within this region over the Rh6ne Fan that bottom water was later observed to form. This surface pattern extends downward as illustrated by the contours of potential temperature in Figs. 12 and 13 (Part 1) at 1500 metres and 2000 metres, respectively. Although the surface general circulation is accepted to be cyclonic, little is known about the flow pattern at these depths. An analysis in Part I of the manner in which the bottom water formed and spread subsequent to the preconditioning period in question indicates that the bottom flow, offshore, was toward the southwest. If this was so and there existed an intermediate level of no motion then the bottom circulation would have been anticyclonic and opposed to the surface pattern. With this picture of general circulation in the northwest Mediterranean, away from the Rh(Sne Deep Sea Fan, that is directed along the bathymetry with lateral and vertical shears such that the flow is cyclonic at the surface and anticyclonic at the bottom, the effect of the bulge on such a circulation will be studied for an idealized model (see Fig. 1). Although this circulation is baroclinic it will be found that the appropriate nondimensional measure of the stratification is small, implying that the effect of the bulge on the horizontal streamline pattern is two-dimensional in accordance with the previous studies of HrDE (1971) and HOGG (1973). Consideration of the changes in vertical relative vorticity induced by the bulge on streamlines incident upon it gives rise to a tractable analytic problem. It will be shown that good qualitative agreement (Fig. 3) with Fig. 5, Part 1, can be achieved, and that
//
U~(E*)x(1- ~Y*/L)
~*=h °
Y
zO=ox * ~ /
r =L r*
Fig. 1. The theoretical model corresponding to Fig. 5, Part 1, giving the deYmition of the coordinate system, length scales and the undisturbed velocity profile,
The preconditioningphase of MEDOC 1969--II
451
the area over the bulge is topographically preconditioned to be the most favourable to bottom water formation processes. Arguments involving the generation of vorticity in flows over bottom topography in the ocean have been used with some success in the past (see e.g. DEFANT, 1961, p. 428). Of some relevance to the present problem is the laboratory and field study of flow over the Reykjanes Ridge by LANGSETHand BOYER(1972) who show that surface ship drift can be correlated with the disturbing effect of the ridge on an incident flow. THEORETICAL
MODEL
Shown ill Fig. 1 is an idealized representation of the geometry in Fig. 5, Part 1, and the assumed undisturbed current field. The continental slope is taken vertical and coincident with the x*-z* plane: it is horizontally infinite but bounded by fiat horizontal planes at the sea floor (z* = 0) and surface (z* = H). On the bottom, abutted to the coast and centred at the origin of the co-ordinate system, is an obstacle with half conical shape and maximum height h0. Away from this bump the current is directed along the coast and assumes the simple separable form a*(X*) ~ [u0*(z*) (1--ry*/L), O, 0] as r* ,~ 0%
(l)
where a*(2*) is the fluid velocity as a function of displacement 2" from the origin (an * is used to denote a dimensional quantity). At the coastline away from the topography the vertical current distribution is given by uo*(z*) and ~, is a measure of the lateral shear. The fluid is assumed to be incompressible and non-diffusive and the flow steady. Three parameters in addition to ~, are of particular importance to this study; the Rossby number defined as -~ V/fL, (2) where
V = { S~u°*2(z*)dz*H
}~
(3)
is a velocity scale; a stratification parameter
S -- gApH pf2L2'
(4)
where g is the acceleration due to gravity, zip is a density difference between bottom and surface, p an average density (Ap ,~ p) and f the Coriolis parameter; and lastly a topographic parameter /~ = h o . uo*(0)
~H
(5)
V
The importance of these parameters will be clarified in the ensuing analysis. It is possible to estimate the sizes of ~,, E, S and/3 from the observations reported in Part 1. The base of the conical bump is taken to be the 2500-metre isobath while the apex is at 1500 metres (see Fig. 4, Part 1) giving ho ~ 1000 metres, H ~ 2500 metres, and L ,.~ 90 kin. Equation (I) specifies a current field which vanishes at y*/L = 1[~. If the dynamic height computations are to be representative of this current distribution,
452
NELSONG. HOGG
then the offshore gradient of the dynamic height anomaly must vanish at y*/L = 1/V. From Fig. 1 this point occurs at approximately one-half the cone base diameter from shore making y ~ 2. In Fig. 6, Part 1, average vertical velocity profiles are given as computed dynamically with respect to 1800 dbar. Using equation O) on the innermost profile and allowing for the fact that the velocity applies to a point at about y*/L -=- 1/6 and should, therefore, be increased by about 50% to give the coastal value, one finds that V g 5-3 cm sec-L Becausef -~ 9.7 x 10-5 sec -1 then ~ ~ 7.3 7." 10-3 and the flow is geostrophic to a good approximation (provided that time variations are minimal). A small Rossby number indicates that the horizontal force balance is approximately between the Coriolis force and pressure gradient at right angles to the flow. Typical density differences between bottom and surface are small for the whole area and Ap/p ~ 10-4. Therefore S ~ 4.1 ~,~ 10 -2 from equation (4). Although this small value will be found to simplify the analysis by constraining the effect of the bump to be two-dimensional it does present some difficulties. It has been shown (see e.g. PEDLOSKY, 1970) that slow baroclinic flows may be unstable to infinitesimal disturbances provided that the stratification is weak enough. In fact, the theoretical analysis of flow between parallel vertical walls separated by a distance L indicates that instability occurs provided that S is less than some number of order 0"1 although significant viscosity can stabilize the flow. Clearly the value of S computed above from the observations falls below this critical value. The only justification for use of the following analysis which ignores the possibilities of time-dependent instabilities is that some agreement is achieved between theory and observation. Finally, the topographic parameter is computed to be/3 g 54u0*(0)/V, the ratio uo*(O)/V being observationally undetermined but, in all likelihood, small enough that /3 is not significantly greater than unity. From the observations, therefore, the following values of the important flow parameters have been decided upon: 7 = 2 = 0(1), E == 7.8 ~ 10 -`3 ,~ 1, S =- 4"1 :~ 10 -2 ~ 1 and/3 = 54uo*(O)/V:= 0(1).
(6)
Also indicated in (6) are the assumed orders of magnitude. FORMULATION
Let p*($*) denote pressure, p*(~*) density and a*(£*) == (u*, v*, w*) velocity. These variables can be nondimensionalized using the scales established in the preceding section by the following relations: (x*, y*, z*) = L(x, y, zH/L), (u*, v*, w*) = V(u, v, wH/L),
(7)
p* -= po -J- p~(z) + pUfLp(g), p* =: eo + ae{p,(z) +
,p(X)/s},
where po and p0 are the density and pressure to be found at z - 0 if the fluid is at rest and po + ps(z) is the pressure needed to balance the basic density stratification, po q- App,(z), hydrostatically. That is: 0
c3z ps(z)
==
--g[Po ~- aees(z)]H.
The preconditioning phase of M E D O C 1969--1I
453
The normalized density function re(z) is a negative function of z for static stability. With the Boussinesq assumption and the other assumptions mentioned previously the equations of motion take the form ~a" V u - v :
- p~,
(8)
~a" V v + u = - pu,
(9)
0 : (4s)a.
--pz - p
v,
(10)
+ ps'(z)w = o
uz + vu + wz : o.
(1 i)
(12)
Expanding all the variables in a power series for ~ of the form a(~; ,) =
~7 ,na(n)(£)
(13)
n=0
shows that the lowest order balance is geostrophic and hydrostatic. That is u(°) = --pu(O),
(14)
v(0) : p~(0),
(15)
0 = --pz(°) -- p(o).
(16)
Geostrophy implies that p(O) is equivalent to a stream function. Equations (14) and (15) when substituted into the continuity equation (12) give wz<°) = 0. In order to satisfy the vanishing constraint w(o) at the upper boundary w(°) ~ 0
(17)
everywhere and w is at most 0(~). Equation (11) becomes a(o) • V p (°) =
--a(o)
• ~ / ' p z (°)
= O,
(18)
[provided ps'(z) ~ O] after using (16). Writing (18) in the Jacobian form pu(O)pzx(o) pxtO)pzut o) = 0 and then integrating gives pz(O) = F(p(O)),
(19)
where F(p(°}) is, as yet, an arbitrary function ofp(O). It can be shown that equation (19) implies that the streamline pattern for the flow is depth independent. In particular, if we choose (1) for the undisturbed velocity field then p<0) ~ --uo(z)(1 -- ~,y/2)y as r ~ oo
(20)
by integrating (14) and (15) and settingpt0) -----0 on x : 0. Therefore p~
and
-uo'(z)y(1
-
~,y/2)
p¢O) -- uo'(z) oto) uo(z) : F(p(°)).
(21)
A further integration of (21) yields the important result that p(O) =_ uo(z)~(x, y),
(22)
with ~b(x,y) yet to be determined. As prO) is a streamfunction, equation (22) establishes
454
NELSON G. HOGG
the fact that the streamline pattern is independent of depth. In particular, if a closed streamline occurs at one depth then similar streamlines exist at all depths. Taking the 0(¢) versions of (8) and (9), cross differentiating to eliminate po), and then using (12) at 0(~) gives (23)
a~o>.v ~(o) = Wz~l>.
This equation states that 0(l) relative vertical vorticity, ~<0>=: vt0) _ uy
0
(24)
if tT(0)(x, y, z) = uo(z)~'(x, y) and wd) - (ho/~H)uo(O)~ • V h on the bottom [z = h o h ( x , y ) / H ] . Written in Jacobian form and integrated, equation (24) becomes \ / h 2 ¢ + flh = v
(25)
after using the far field form of ¢, namely, ~b ,,~ --y(l--?y/2) as r ,,~ oo
(26)
which is derived from (20). Finally, a condition is required that prohibits motion normal to the coastline at y =: 0. From (15) v~o) ---- px(°) = 0 at y =: 0. An integration and use of (20) and (22) establishes that ¢(x, y) -- 0 on y = 0. (27) SOLUTION
Equation (25) with conditions (26) and (27) pose a boundary value problem for a Poisson equation from which the effect of lateral shear is removed rather easily with the substitution ¢ = --y(1--~0.'/2) -t: flF(x, y).
(28)
This yields the simplified problem V h Z F + h - - O,
(29)
F~0asr~
(30)
F-
0%
0ony=0.
(31)
In the cylindrical co-ordinates (r, 0, z) defined in Fig. 1 the idealized obstacle shape can be written as { l--r,~
O, r >
.
The bump's finite dimensions allow the problem to be separated into the two regions r X 1. Solutions within each region can be found and then matched at the obstacle boundary to determine the constants of integration.
The preconditioning phase of MEDOC 1969--11
455
Using this technique and the separation of variables method it is possible to show that
F(r, O) : ,=l ~ Bn k n - 2 [ n + 2
n-3(n+3
2n
sin nO
n#3
+ Ba \ 5 and where
F(r, O) = n=l ~ B. [ 1 2~ -nSr2
36 +
1_ ]r_n sin nO, r > 1,
n+3J
B n = - 2 [ 1 - (-~l)n'] . L
sin 30, r < 1,
n
(33)
(34) (35)
_1
From this solution for ~(x, y) the pressure distribution is given by (22) ifu0(z) is known. Equations (14)-(16) then allow the horizontal velocity and density field to be computed at 0(1). DISCUSSION
The qualitative nature of the streamline solutions given by (28) with (33)-(35) depends crucially on the signs and magnitudes of the lateral shear parameter ~, and the topographic parameter/3. In particular, for certain regions of the (fl, ~,) space there occur stagnation points with associated closed streamlines. As was mentioned previously, the weak stratification present causes these closed streamline regions to be two-dimensional or columnar, an effect similar to that found in homogeneous rotating flow over topography where the region within the boundary closed streamline is known as a 'Taylor column'. Figure 2 defines the regions in 03, ~,) space where such Taylor column-like phenomena occur (see the appendix for the derivation of the region boundaries). Also sketched in a rough fashion are the characteristic forms assumed by the streamlines in each region. These patterns depend strongly on fl and ~, and Taylor columns appear in two regions. When fl < 0 and ~, > 0 the column is 'detached' and sits over the outer edge of the bump while iffl > ~r and y is less than some positive function offl the column hugs the coastline and is 'attached'. The best estimates from the observations of Part 1 led to the parametric values given in equation (6) of y = 2 and fl = 54u0(0)/1I. The picture of the general circulation illustrated in Fig. 1 has a bottom current at the coast which is directed along the negative x-axis so that u0(0) < 0. In all probability this bottom flow is small relative to the root mean square, 1I, and for lack of more complete knowledge this ratio uo(O)/V was chosen to be such that fl = --5 in the numerical computations. The corresponding contours of ~b(x, y) are presented in Fig. 3. There are three important properties of the solution illustrated in Fig. 3 which relate to the bottom water formation process. Firstly, from (16) and (22)
p~o~ = uo'(z)~(x, y).
(36)
The undisturbed geostrophic shear at the coast is positive [u0'(z) > 0] while ~(x, y) reaches its most negative value in the centre of the Taylor column. Therefore the
456
NwLsoN G. Ho~c
\ \',.\\'\ \ \
7"
\\"k
\\
",-15 ~ -
.X\'~ IO
,~\"
\
, \'\
,", ,,\-,
\,
.-,qi,"
5 ¸
,
\ \ '\
.
.
1'
.
\ ~\ \ \~ ' \~ ~
\
\
\\
\
",'\-,,5"
.
\
I
,
I
i
--
"
-5
\\
\
~
\
\-
\ \,\\
,'q-.i Fig. 2. The (fl, ~,) space showing regions in which a Taylor column is found to appear in the streamline pattern. Also sketched are characteristic forms of the streamlines in each region. In the shaded region a Taylor column does not occur.
j3 J
i J
!
m
-3
50
j 20
-2
-I
0
2
3
Fig. 3. The computed form of ¢(x, y) for/~ 5 and y -~ 2. The contours are to be compared with the dynamic height contours in Fig, 5, Part 1. Arrows show the direction of flow above the level of no motion: below this level the arrows are reversed. Away from the coast the plotted streamlines are not separated in magnitude by a constant increment.
density perturbation (36) attains its maximum negative value over the bulge. This resulting spatial variation in p(O)(x, y, z) is needed to balance the cyclonic motion at the surface within the Taylor column. It is interesting to note that ¢ goes from 0.0 at the coast to --0.2 along the northeast trough to --0.4 over the bulge while the surface density pattern shown in Fig. 3, Part 1, has roughly the same proportionate variation: =, changes from about 28.6 at the coast to 28.8 in the trough and 29.0 over the bulge. The source of this dense water presents difficulties of rationalization with the steady-state solution. For in the steady-state limit w ,-, ~ and, as density is conserved,
The preconditioning phase of MEDOC 1969--1I
457
this water cannot have been advected from upstream at the same level. A related problem arises from the existence of the closed streamlines which surround the dense water region. Within these streamlines the water is isolated and does not necessarily obey conditions derived from the undisturbed flow. In particular, equation (25) may not be valid within the Taylor column although the conclusion from (19) that the motion is two-dimensional remains valid. In our region of interest in the (//, y) space of Fig. 2 the closed streamline area actually extends to infinity. That is, ~b -------0.2 is the minimum value of ¢ at oo and intermediate values, - 0 . 2 > ~b > - 0 . 4 , have closed, roughly elliptical contours which become more elongated as ~b ,,, --0.2. It has not been possible to offer a rigorous justification of the solution given in Fig. 3 which treats the problem of what happens within the Taylor column. One could argue that in the transient development period there is sufficient vertical motion forced by the bulge (wt0) need not be zero during this time) that more dense water is upwelled into each horizontal section and is trapped by the Taylor column effect in the time-independent limit. This idea is consistent with the discussion in Part 1 of the observed surface properties where it was concluded that the dense surface water could not have been formed solely by cooling, evaporation, and mixing with deeper water but that intermediate water must have been closer to the surface over the bulge. In the end the best justifieation of this analysis is its good comparison with the observations of Part 1. Setting these difficulties aside the second property to note is the spatial dependence of the vertical stability as indicated by the theoretical density gradient. From (36) p~ = - uo"(z)~h(x, y).
(37)
Fig. 7, Part 1 shows the average geostrophic current profiles for the undisturbed area. Taking profile (a) to be representative of uo(z), one can see that both uo'(z) > 0 and uo"(z) > 0 so that pz > 0 within the Taylor column over the bulge. As the z-axis is vertical, antiparallel to gravity, pz > 0 means that the density gradient and the associated stability to convective processes is decreased within the column and is decreased the greatest amount in the centre. The final property of importance is that closed streamlines are found over the topography. It is possible, as was noted previously, to close the --110 dyn. m contour in Fig. 5 and to extend the --1080 dyn. m one over the inshore edge of the R h f n e Fan so as to give rough qualitative resemblance with Fig. 3. These three properties, the increased density, decreased stability, and Taylor column isolation, combine to make the area over the bump more susceptible to cooling and evaporative effects at the surface. For when the cold northwest winds begin blowing off the continent (the Mistral) the area directly over the bulge has been preconditioned by the topography to be initially more dense and less stable to convective overturning. In addition, although the Mistral is of a somewhat broader scale than the Rh6ne Fan, the water over the top is constrained to remain there while that further away can be advected out of the region of influence. Acknowledgements--I wish to express appreciation of the help received from Mrs. G. C,~STONin preparing the diagrams and Mrs. P. SEARSand Mrs. M. WILLIAMSin typing various forms of the manuscript. I am also grateful to Dr. J. SWALLOWfor discussing the observations with me and stimulating my thinking along the lines outlined above. This study was supported by the National Research Council of Canada in the form of a postdoctorate fellowship held at the National Institute of Oceanography.
458
NICLSON G. HOGG
REFERENCES DEI.ANT A. (1961)Physicaloceanography, Vol. I. P e r g a m o n , 729 pp. GRADSHTEYN I. S. a n d I. M. RVZH,K (1965) Table o f integrals, series a n d p r o d u c t s . A c a d e m i c Press, 4th ed., 1086 pp. H t D E R . (1971) O n geostrophic m o t i o n o f a n o n - h o m o g e n e o u s fluid. J. Fluid M e c h . , 49, 745-751. HoGG N. G. (1973) O n the stratified Taylor column. A c c e p t e d for publication in J. F l u i d M e c h . LANGSETH M. G. JR. a n d D. BOYER 0 9 7 2 ) The effect o f the Reykjanes R i d g e on the flow o f water a b o v e 2000 meters. In: S t u d i e s in p h y s i c a l o c e a n o g r a p h y , a t r i b u t e to G e o r g W i i s t on his 8 0 t h b i r t h d a y , A. L. GORDON, editor, G o r d o n & Breach, 2, 93-114. PEDLOS~:V J. (1970) Finite-amplitude baroclinic waves. J. M e t . , 27, 15-30. SWALLOW J. C. and G. F. CASTON (1973) The p r e c o n d i t i o n i n g phase o f M E D O C 1969--I. Observations. D e e p - S e a Res., 20, 429-448.
APPENDIX Closed streamlines (i.e. a Taylor column) occur when there is a local extremum in the stream function solution given by (28) and (33)-(35). It can be quite easily shown that ,~0 = 0 at 0 == rr/2 so that all extrema occur on 0 = ~r/2 as one might expect from the symmetrical nature of the solutions. On this line there is an extremum when 0
-~r]o
,~/2 "
-I
f yr l 3 F r .
(AI)
NOW (AI) specifies a family of lines in (#, y) space--a different line for each value of r. Another condition is needed to obtain the boundary of the Taylor column region. This condition is that stagnation occurs without closed streamlines and this happens when the extremum of (A 1) is an inflection point. That is 0 - q,,~ -= ~, r fltr,-.
{A2I
Frr 1 Y := -- F r ~ r F r - r ' ~ ~ }'r ~_2 }7~'r-r '
(A3)
Solving (A1) and (A2) for y and gives
or 7 and fl are determined in terms of the distance of the inflection point from the origin. It is possible to sum the infinite series of (33)-(35) for the derivatives Fr and Err at 0 = ~r/2 in analytic form with the help of series summation formulae given in GRADSHTEVN and RvzmK (1965). This being done the solution to (A3) is shown as the curved line which originates at Or, ~r) in the first quadrant of Fig. 2. The point (fl, 7) = (rr, ~r) occurs when the inflection point is at r = 0 or on the boundary. Here, condition (A2) is not necessary and (A1) (stating that the longshore velocity component vanishes) is a sufficient condition for the Taylor column to be on the verge of forming. It can be shown that Fr ~ I/~ as r ~ 0 so that (AI) gives fl -= rr,
(A4)
which is the vertical line in the first and fourth quadrants of Fig. 2. However, for this to be the boundary of a Taylor column region, this stagnation point must be a minimum in the f o r w a r d motion near the coastline (fl > 0 implies that the motion is along the positive x-axis near the shore). On 0 - ~r/2, u --: - ~ r so that ur - ~ r :- -- y q- fl F m At r -= 0 it can be shown that F~r ~ 1 so that -
-
ur = -- ~ "~" ft.
(A5)
If y < (/3 = 7r) then u~ > 0 and 13 ~ ~ is the boundary of a Taylor column region. If y > (/3 = ~r) then u~ < 0 and the forward motion reverses away from r = 0. A streamline pattern as sketched in the upper right portion of Fig. 2 is present. Provided that y > 0 there is a minimum offshore in the stream function (corresponding to the trough in the dynamic height anomaly). In the undisturbed flow this occurs at r = 1/y where 1 ( a t r = 1-,y>0).
(A6)
The preconditioning phase of M E D O C 1969--I1
459
The topography changes both the value and position of this minimum so that (~Frr)S+//F(at r = 1 _ tiff ,/3>0). (AT) 2~, Y v The second term on the fight-hand side, (flFrr)a/y, is always positive if y > 0 and therefore gives a negative contribution to ~min. It can be shown that F(r, ~/2) > 0 so that the contribution from the third term is also negative if fl < 0. In the second quadrant, therefore, the minimum value assumexi by the streamfunction over the bulge is less than its minimum undisturbed value and must be enclosed by closed streamlines. The second quadrant is, then, a Taylor column region. ~mtn
~
t~mtn und
--
The preconditioning phase of MEDOC 1969--H. Topographic effects NELSON G . H O G G *
(Received l September 1972; in revisedform 12 December 1972; accepted3 January 1973)
Abstract--In support of SWALLOWand CASTON'S(1973) conclusion that topography plays a significant role in determining the position of the dense water formation region in the northwest Mediterranean, a theoretical model conserving potential vorticity in flow past a conical bump abutted to the continental slope is studied. For parameters appropriate to this area it is shown that the flow falls in the weak stratification category of HOGG(1973), implying that the streamline pattern is quasi-two-dimensional. For typical values of the lateral shear and a topographic parameter, closed streamlines (i.e. a baroclinic Taylor column) are found over the bump in agreement with the observations. As the region is isolated and also possesses greater initial density and decreased stability, it is suggested that this is where dense water is most likely to be formed. INTRODUCTION
SWALLOW and CASTON (1973) (hereafter called 'Part I') in commenting upon observations taken during the preconditioning phase of bottom water formation in M E D O C , 1969, suggest that the location o f the dense water region in which bottom water later formed was influenced by the local bottom topography. Figure 5 of Part 1 is a chart of the northwest Mediterranean Sea giving both contours of bottom depth and of the surface dynamic height anomaly computed with respect to the 1800 dbar surface. Note that the bottom drops rather sharply from a continental shelf depth of 200 metres to an abyssal plain at roughly 2500 metres, except in a region off the south coast of France centred at 5°E 42°N known as the Rh6ne Deep Sea Fan. Here there exists a bulge in the bathymetric contours beginning at about 1500 metres, and extending to the 2500 metres contour which is displaced approximately 90 km offshore from its more usual position in the area. Note also the form of the dynamic height contours (dashed lines). Away from the bulge these contours are parallel to the coast. A trough or minimum in dynamic height exists from which the dynamic topography rises on either side and is consistent with the previously observed cyclonic surface circulation. With respect to 1800 dbar, at least, the surface flow is toward the southwest inshore of the trough and toward the northeast offshore. Over the bulge this pattern is profoundly altered. Contours inshore of the trough are deflected so as to cross the bathymetric contours and move further inshore while those offshore tend to avoid the bulge. Over the top a contour of even greater dynamic depth than is found in the trough occurs. Note that this -- 1100 dyn m contour, although drawn open, can be closed in a fashion consistent with the observations and that the *National Institute of Oceanography, Wormley, Godalming, Surrey, England. Present address: Department of Earth and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U,S.A. 449
450
NELSONG. HOGG
--1080 dyn m line can be extended to pass over the bulge. It was within this region over the Rh6ne Fan that bottom water was later observed to form. This surface pattern extends downward as illustrated by the contours of potential temperature in Figs. 12 and 13 (Part 1) at 1500 metres and 2000 metres, respectively. Although the surface general circulation is accepted to be cyclonic, little is known about the flow pattern at these depths. An analysis in Part I of the manner in which the bottom water formed and spread subsequent to the preconditioning period in question indicates that the bottom flow, offshore, was toward the southwest. If this was so and there existed an intermediate level of no motion then the bottom circulation would have been anticyclonic and opposed to the surface pattern. With this picture of general circulation in the northwest Mediterranean, away from the Rh(Sne Deep Sea Fan, that is directed along the bathymetry with lateral and vertical shears such that the flow is cyclonic at the surface and anticyclonic at the bottom, the effect of the bulge on such a circulation will be studied for an idealized model (see Fig. 1). Although this circulation is baroclinic it will be found that the appropriate nondimensional measure of the stratification is small, implying that the effect of the bulge on the horizontal streamline pattern is two-dimensional in accordance with the previous studies of HrDE (1971) and HOGG (1973). Consideration of the changes in vertical relative vorticity induced by the bulge on streamlines incident upon it gives rise to a tractable analytic problem. It will be shown that good qualitative agreement (Fig. 3) with Fig. 5, Part 1, can be achieved, and that
//
U~(E*)x(1- ~Y*/L)
~*=h °
Y
zO=ox * ~ /
r =L r*
Fig. 1. The theoretical model corresponding to Fig. 5, Part 1, giving the deYmition of the coordinate system, length scales and the undisturbed velocity profile,
The preconditioningphase of MEDOC 1969--II
451
the area over the bulge is topographically preconditioned to be the most favourable to bottom water formation processes. Arguments involving the generation of vorticity in flows over bottom topography in the ocean have been used with some success in the past (see e.g. DEFANT, 1961, p. 428). Of some relevance to the present problem is the laboratory and field study of flow over the Reykjanes Ridge by LANGSETHand BOYER(1972) who show that surface ship drift can be correlated with the disturbing effect of the ridge on an incident flow. THEORETICAL
MODEL
Shown ill Fig. 1 is an idealized representation of the geometry in Fig. 5, Part 1, and the assumed undisturbed current field. The continental slope is taken vertical and coincident with the x*-z* plane: it is horizontally infinite but bounded by fiat horizontal planes at the sea floor (z* = 0) and surface (z* = H). On the bottom, abutted to the coast and centred at the origin of the co-ordinate system, is an obstacle with half conical shape and maximum height h0. Away from this bump the current is directed along the coast and assumes the simple separable form a*(X*) ~ [u0*(z*) (1--ry*/L), O, 0] as r* ,~ 0%
(l)
where a*(2*) is the fluid velocity as a function of displacement 2" from the origin (an * is used to denote a dimensional quantity). At the coastline away from the topography the vertical current distribution is given by uo*(z*) and ~, is a measure of the lateral shear. The fluid is assumed to be incompressible and non-diffusive and the flow steady. Three parameters in addition to ~, are of particular importance to this study; the Rossby number defined as -~ V/fL, (2) where
V = { S~u°*2(z*)dz*H
}~
(3)
is a velocity scale; a stratification parameter
S -- gApH pf2L2'
(4)
where g is the acceleration due to gravity, zip is a density difference between bottom and surface, p an average density (Ap ,~ p) and f the Coriolis parameter; and lastly a topographic parameter /~ = h o . uo*(0)
~H
(5)
V
The importance of these parameters will be clarified in the ensuing analysis. It is possible to estimate the sizes of ~,, E, S and/3 from the observations reported in Part 1. The base of the conical bump is taken to be the 2500-metre isobath while the apex is at 1500 metres (see Fig. 4, Part 1) giving ho ~ 1000 metres, H ~ 2500 metres, and L ,.~ 90 kin. Equation (I) specifies a current field which vanishes at y*/L = 1[~. If the dynamic height computations are to be representative of this current distribution,
452
NELSONG. HOGG
then the offshore gradient of the dynamic height anomaly must vanish at y*/L = 1/V. From Fig. 1 this point occurs at approximately one-half the cone base diameter from shore making y ~ 2. In Fig. 6, Part 1, average vertical velocity profiles are given as computed dynamically with respect to 1800 dbar. Using equation O) on the innermost profile and allowing for the fact that the velocity applies to a point at about y*/L -=- 1/6 and should, therefore, be increased by about 50% to give the coastal value, one finds that V g 5-3 cm sec-L Becausef -~ 9.7 x 10-5 sec -1 then ~ ~ 7.3 7." 10-3 and the flow is geostrophic to a good approximation (provided that time variations are minimal). A small Rossby number indicates that the horizontal force balance is approximately between the Coriolis force and pressure gradient at right angles to the flow. Typical density differences between bottom and surface are small for the whole area and Ap/p ~ 10-4. Therefore S ~ 4.1 ~,~ 10 -2 from equation (4). Although this small value will be found to simplify the analysis by constraining the effect of the bump to be two-dimensional it does present some difficulties. It has been shown (see e.g. PEDLOSKY, 1970) that slow baroclinic flows may be unstable to infinitesimal disturbances provided that the stratification is weak enough. In fact, the theoretical analysis of flow between parallel vertical walls separated by a distance L indicates that instability occurs provided that S is less than some number of order 0"1 although significant viscosity can stabilize the flow. Clearly the value of S computed above from the observations falls below this critical value. The only justification for use of the following analysis which ignores the possibilities of time-dependent instabilities is that some agreement is achieved between theory and observation. Finally, the topographic parameter is computed to be/3 g 54u0*(0)/V, the ratio uo*(O)/V being observationally undetermined but, in all likelihood, small enough that /3 is not significantly greater than unity. From the observations, therefore, the following values of the important flow parameters have been decided upon: 7 = 2 = 0(1), E == 7.8 ~ 10 -`3 ,~ 1, S =- 4"1 :~ 10 -2 ~ 1 and/3 = 54uo*(O)/V:= 0(1).
(6)
Also indicated in (6) are the assumed orders of magnitude. FORMULATION
Let p*($*) denote pressure, p*(~*) density and a*(£*) == (u*, v*, w*) velocity. These variables can be nondimensionalized using the scales established in the preceding section by the following relations: (x*, y*, z*) = L(x, y, zH/L), (u*, v*, w*) = V(u, v, wH/L),
(7)
p* -= po -J- p~(z) + pUfLp(g), p* =: eo + ae{p,(z) +
,p(X)/s},
where po and p0 are the density and pressure to be found at z - 0 if the fluid is at rest and po + ps(z) is the pressure needed to balance the basic density stratification, po q- App,(z), hydrostatically. That is: 0
c3z ps(z)
==
--g[Po ~- aees(z)]H.
The preconditioning phase of M E D O C 1969--1I
453
The normalized density function re(z) is a negative function of z for static stability. With the Boussinesq assumption and the other assumptions mentioned previously the equations of motion take the form ~a" V u - v :
- p~,
(8)
~a" V v + u = - pu,
(9)
0 : (4s)a.
--pz - p
v,
(10)
+ ps'(z)w = o
uz + vu + wz : o.
(1 i)
(12)
Expanding all the variables in a power series for ~ of the form a(~; ,) =
~7 ,na(n)(£)
(13)
n=0
shows that the lowest order balance is geostrophic and hydrostatic. That is u(°) = --pu(O),
(14)
v(0) : p~(0),
(15)
0 = --pz(°) -- p(o).
(16)
Geostrophy implies that p(O) is equivalent to a stream function. Equations (14) and (15) when substituted into the continuity equation (12) give wz<°) = 0. In order to satisfy the vanishing constraint w(o) at the upper boundary w(°) ~ 0
(17)
everywhere and w is at most 0(~). Equation (11) becomes a(o) • V p (°) =
--a(o)
• ~ / ' p z (°)
= O,
(18)
[provided ps'(z) ~ O] after using (16). Writing (18) in the Jacobian form pu(O)pzx(o) pxtO)pzut o) = 0 and then integrating gives pz(O) = F(p(O)),
(19)
where F(p(°}) is, as yet, an arbitrary function ofp(O). It can be shown that equation (19) implies that the streamline pattern for the flow is depth independent. In particular, if we choose (1) for the undisturbed velocity field then p<0) ~ --uo(z)(1 -- ~,y/2)y as r ~ oo
(20)
by integrating (14) and (15) and settingpt0) -----0 on x : 0. Therefore p~
and
-uo'(z)y(1
-
~,y/2)
p¢O) -- uo'(z) oto) uo(z) : F(p(°)).
(21)
A further integration of (21) yields the important result that p(O) =_ uo(z)~(x, y),
(22)
with ~b(x,y) yet to be determined. As prO) is a streamfunction, equation (22) establishes
454
NELSON G. HOGG
the fact that the streamline pattern is independent of depth. In particular, if a closed streamline occurs at one depth then similar streamlines exist at all depths. Taking the 0(¢) versions of (8) and (9), cross differentiating to eliminate po), and then using (12) at 0(~) gives (23)
a~o>.v ~(o) = Wz~l>.
This equation states that 0(l) relative vertical vorticity, ~<0>=: vt0) _ uy
0
(24)
if tT(0)(x, y, z) = uo(z)~'(x, y) and wd) - (ho/~H)uo(O)~ • V h on the bottom [z = h o h ( x , y ) / H ] . Written in Jacobian form and integrated, equation (24) becomes \ / h 2 ¢ + flh = v
(25)
after using the far field form of ¢, namely, ~b ,,~ --y(l--?y/2) as r ,,~ oo
(26)
which is derived from (20). Finally, a condition is required that prohibits motion normal to the coastline at y =: 0. From (15) v~o) ---- px(°) = 0 at y =: 0. An integration and use of (20) and (22) establishes that ¢(x, y) -- 0 on y = 0. (27) SOLUTION
Equation (25) with conditions (26) and (27) pose a boundary value problem for a Poisson equation from which the effect of lateral shear is removed rather easily with the substitution ¢ = --y(1--~0.'/2) -t: flF(x, y).
(28)
This yields the simplified problem V h Z F + h - - O,
(29)
F~0asr~
(30)
F-
0%
0ony=0.
(31)
In the cylindrical co-ordinates (r, 0, z) defined in Fig. 1 the idealized obstacle shape can be written as { l--r,~
O, r >
.
The bump's finite dimensions allow the problem to be separated into the two regions r X 1. Solutions within each region can be found and then matched at the obstacle boundary to determine the constants of integration.
The preconditioning phase of MEDOC 1969--11
455
Using this technique and the separation of variables method it is possible to show that
F(r, O) : ,=l ~ Bn k n - 2 [ n + 2
n-3(n+3
2n
sin nO
n#3
+ Ba \ 5 and where
F(r, O) = n=l ~ B. [ 1 2~ -nSr2
36 +
1_ ]r_n sin nO, r > 1,
n+3J
B n = - 2 [ 1 - (-~l)n'] . L
sin 30, r < 1,
n
(33)
(34) (35)
_1
From this solution for ~(x, y) the pressure distribution is given by (22) ifu0(z) is known. Equations (14)-(16) then allow the horizontal velocity and density field to be computed at 0(1). DISCUSSION
The qualitative nature of the streamline solutions given by (28) with (33)-(35) depends crucially on the signs and magnitudes of the lateral shear parameter ~, and the topographic parameter/3. In particular, for certain regions of the (fl, ~,) space there occur stagnation points with associated closed streamlines. As was mentioned previously, the weak stratification present causes these closed streamline regions to be two-dimensional or columnar, an effect similar to that found in homogeneous rotating flow over topography where the region within the boundary closed streamline is known as a 'Taylor column'. Figure 2 defines the regions in 03, ~,) space where such Taylor column-like phenomena occur (see the appendix for the derivation of the region boundaries). Also sketched in a rough fashion are the characteristic forms assumed by the streamlines in each region. These patterns depend strongly on fl and ~, and Taylor columns appear in two regions. When fl < 0 and ~, > 0 the column is 'detached' and sits over the outer edge of the bump while iffl > ~r and y is less than some positive function offl the column hugs the coastline and is 'attached'. The best estimates from the observations of Part 1 led to the parametric values given in equation (6) of y = 2 and fl = 54u0(0)/1I. The picture of the general circulation illustrated in Fig. 1 has a bottom current at the coast which is directed along the negative x-axis so that u0(0) < 0. In all probability this bottom flow is small relative to the root mean square, 1I, and for lack of more complete knowledge this ratio uo(O)/V was chosen to be such that fl = --5 in the numerical computations. The corresponding contours of ~b(x, y) are presented in Fig. 3. There are three important properties of the solution illustrated in Fig. 3 which relate to the bottom water formation process. Firstly, from (16) and (22)
p~o~ = uo'(z)~(x, y).
(36)
The undisturbed geostrophic shear at the coast is positive [u0'(z) > 0] while ~(x, y) reaches its most negative value in the centre of the Taylor column. Therefore the
456
NwLsoN G. Ho~c
\ \',.\\'\ \ \
7"
\\"k
\\
",-15 ~ -
.X\'~ IO
,~\"
\
, \'\
,", ,,\-,
\,
.-,qi,"
5 ¸
,
\ \ '\
.
.
1'
.
\ ~\ \ \~ ' \~ ~
\
\
\\
\
",'\-,,5"
.
\
I
,
I
i
--
"
-5
\\
\
~
\
\-
\ \,\\
,'q-.i Fig. 2. The (fl, ~,) space showing regions in which a Taylor column is found to appear in the streamline pattern. Also sketched are characteristic forms of the streamlines in each region. In the shaded region a Taylor column does not occur.
j3 J
i J
!
m
-3
50
j 20
-2
-I
0
2
3
Fig. 3. The computed form of ¢(x, y) for/~ 5 and y -~ 2. The contours are to be compared with the dynamic height contours in Fig, 5, Part 1. Arrows show the direction of flow above the level of no motion: below this level the arrows are reversed. Away from the coast the plotted streamlines are not separated in magnitude by a constant increment.
density perturbation (36) attains its maximum negative value over the bulge. This resulting spatial variation in p(O)(x, y, z) is needed to balance the cyclonic motion at the surface within the Taylor column. It is interesting to note that ¢ goes from 0.0 at the coast to --0.2 along the northeast trough to --0.4 over the bulge while the surface density pattern shown in Fig. 3, Part 1, has roughly the same proportionate variation: =, changes from about 28.6 at the coast to 28.8 in the trough and 29.0 over the bulge. The source of this dense water presents difficulties of rationalization with the steady-state solution. For in the steady-state limit w ,-, ~ and, as density is conserved,
The preconditioning phase of MEDOC 1969--1I
457
this water cannot have been advected from upstream at the same level. A related problem arises from the existence of the closed streamlines which surround the dense water region. Within these streamlines the water is isolated and does not necessarily obey conditions derived from the undisturbed flow. In particular, equation (25) may not be valid within the Taylor column although the conclusion from (19) that the motion is two-dimensional remains valid. In our region of interest in the (//, y) space of Fig. 2 the closed streamline area actually extends to infinity. That is, ~b -------0.2 is the minimum value of ¢ at oo and intermediate values, - 0 . 2 > ~b > - 0 . 4 , have closed, roughly elliptical contours which become more elongated as ~b ,,, --0.2. It has not been possible to offer a rigorous justification of the solution given in Fig. 3 which treats the problem of what happens within the Taylor column. One could argue that in the transient development period there is sufficient vertical motion forced by the bulge (wt0) need not be zero during this time) that more dense water is upwelled into each horizontal section and is trapped by the Taylor column effect in the time-independent limit. This idea is consistent with the discussion in Part 1 of the observed surface properties where it was concluded that the dense surface water could not have been formed solely by cooling, evaporation, and mixing with deeper water but that intermediate water must have been closer to the surface over the bulge. In the end the best justifieation of this analysis is its good comparison with the observations of Part 1. Setting these difficulties aside the second property to note is the spatial dependence of the vertical stability as indicated by the theoretical density gradient. From (36) p~ = - uo"(z)~h(x, y).
(37)
Fig. 7, Part 1 shows the average geostrophic current profiles for the undisturbed area. Taking profile (a) to be representative of uo(z), one can see that both uo'(z) > 0 and uo"(z) > 0 so that pz > 0 within the Taylor column over the bulge. As the z-axis is vertical, antiparallel to gravity, pz > 0 means that the density gradient and the associated stability to convective processes is decreased within the column and is decreased the greatest amount in the centre. The final property of importance is that closed streamlines are found over the topography. It is possible, as was noted previously, to close the --110 dyn. m contour in Fig. 5 and to extend the --1080 dyn. m one over the inshore edge of the R h f n e Fan so as to give rough qualitative resemblance with Fig. 3. These three properties, the increased density, decreased stability, and Taylor column isolation, combine to make the area over the bump more susceptible to cooling and evaporative effects at the surface. For when the cold northwest winds begin blowing off the continent (the Mistral) the area directly over the bulge has been preconditioned by the topography to be initially more dense and less stable to convective overturning. In addition, although the Mistral is of a somewhat broader scale than the Rh6ne Fan, the water over the top is constrained to remain there while that further away can be advected out of the region of influence. Acknowledgements--I wish to express appreciation of the help received from Mrs. G. C,~STONin preparing the diagrams and Mrs. P. SEARSand Mrs. M. WILLIAMSin typing various forms of the manuscript. I am also grateful to Dr. J. SWALLOWfor discussing the observations with me and stimulating my thinking along the lines outlined above. This study was supported by the National Research Council of Canada in the form of a postdoctorate fellowship held at the National Institute of Oceanography.
458
NICLSON G. HOGG
REFERENCES DEI.ANT A. (1961)Physicaloceanography, Vol. I. P e r g a m o n , 729 pp. GRADSHTEYN I. S. a n d I. M. RVZH,K (1965) Table o f integrals, series a n d p r o d u c t s . A c a d e m i c Press, 4th ed., 1086 pp. H t D E R . (1971) O n geostrophic m o t i o n o f a n o n - h o m o g e n e o u s fluid. J. Fluid M e c h . , 49, 745-751. HoGG N. G. (1973) O n the stratified Taylor column. A c c e p t e d for publication in J. F l u i d M e c h . LANGSETH M. G. JR. a n d D. BOYER 0 9 7 2 ) The effect o f the Reykjanes R i d g e on the flow o f water a b o v e 2000 meters. In: S t u d i e s in p h y s i c a l o c e a n o g r a p h y , a t r i b u t e to G e o r g W i i s t on his 8 0 t h b i r t h d a y , A. L. GORDON, editor, G o r d o n & Breach, 2, 93-114. PEDLOS~:V J. (1970) Finite-amplitude baroclinic waves. J. M e t . , 27, 15-30. SWALLOW J. C. and G. F. CASTON (1973) The p r e c o n d i t i o n i n g phase o f M E D O C 1969--I. Observations. D e e p - S e a Res., 20, 429-448.
APPENDIX Closed streamlines (i.e. a Taylor column) occur when there is a local extremum in the stream function solution given by (28) and (33)-(35). It can be quite easily shown that ,~0 = 0 at 0 == rr/2 so that all extrema occur on 0 = ~r/2 as one might expect from the symmetrical nature of the solutions. On this line there is an extremum when 0
-~r]o
,~/2 "
-I
f yr l 3 F r .
(AI)
NOW (AI) specifies a family of lines in (#, y) space--a different line for each value of r. Another condition is needed to obtain the boundary of the Taylor column region. This condition is that stagnation occurs without closed streamlines and this happens when the extremum of (A 1) is an inflection point. That is 0 - q,,~ -= ~, r fltr,-.
{A2I
Frr 1 Y := -- F r ~ r F r - r ' ~ ~ }'r ~_2 }7~'r-r '
(A3)
Solving (A1) and (A2) for y and gives
or 7 and fl are determined in terms of the distance of the inflection point from the origin. It is possible to sum the infinite series of (33)-(35) for the derivatives Fr and Err at 0 = ~r/2 in analytic form with the help of series summation formulae given in GRADSHTEVN and RvzmK (1965). This being done the solution to (A3) is shown as the curved line which originates at Or, ~r) in the first quadrant of Fig. 2. The point (fl, 7) = (rr, ~r) occurs when the inflection point is at r = 0 or on the boundary. Here, condition (A2) is not necessary and (A1) (stating that the longshore velocity component vanishes) is a sufficient condition for the Taylor column to be on the verge of forming. It can be shown that Fr ~ I/~ as r ~ 0 so that (AI) gives fl -= rr,
(A4)
which is the vertical line in the first and fourth quadrants of Fig. 2. However, for this to be the boundary of a Taylor column region, this stagnation point must be a minimum in the f o r w a r d motion near the coastline (fl > 0 implies that the motion is along the positive x-axis near the shore). On 0 - ~r/2, u --: - ~ r so that ur - ~ r :- -- y q- fl F m At r -= 0 it can be shown that F~r ~ 1 so that -
-
ur = -- ~ "~" ft.
(A5)
If y < (/3 = 7r) then u~ > 0 and 13 ~ ~ is the boundary of a Taylor column region. If y > (/3 = ~r) then u~ < 0 and the forward motion reverses away from r = 0. A streamline pattern as sketched in the upper right portion of Fig. 2 is present. Provided that y > 0 there is a minimum offshore in the stream function (corresponding to the trough in the dynamic height anomaly). In the undisturbed flow this occurs at r = 1/y where 1 ( a t r = 1-,y>0).
(A6)
The preconditioning phase of M E D O C 1969--I1
459
The topography changes both the value and position of this minimum so that (~Frr)S+//F(at r = 1 _ tiff ,/3>0). (AT) 2~, Y v The second term on the fight-hand side, (flFrr)a/y, is always positive if y > 0 and therefore gives a negative contribution to ~min. It can be shown that F(r, ~/2) > 0 so that the contribution from the third term is also negative if fl < 0. In the second quadrant, therefore, the minimum value assumexi by the streamfunction over the bulge is less than its minimum undisturbed value and must be enclosed by closed streamlines. The second quadrant is, then, a Taylor column region. ~mtn
~
t~mtn und
--