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mixing processes
Deep-Sea Research, 1974, Vol. 21, pp. 347 to 358. Pergamon Press. Printed in Great Britain.
The maintenance of the Pacific North Equatorial Countercurrent by thermal/mixing* processes WARRE~ B, W m ~ t (Received 15 May 1973; in revisedform 19 October 1973; accepted21 November 1973) Abstract--The strength and latitudinal position of the principal transport in the Pacific North Equatorial Countercurrent has been developed theoretically by Sv~u3ttup (1947) and by REID (1948) to be the result of a mass redistribution in response to Ekman divergence on the ~-plane. More recently, the principal transport of the Countercurrent was found in the surface layer above the thermocline (WYRT~ and I~NDALL, 1967) and a good approximation for the Countercurrent transport in the upper layer (< 150 m) can be obtained by considering only the slope of the thermocline in a fashion after the Margules equation. Independently, MmDPDL'SKIY(1970) developed a two-layer thermocline model from considerations of the conservation of heat and turbulent kinetic energy that prescribes the thickness of the upper layer in terms of the net heat flux and the flux of momentum across the sea surface. Therefore, it is possible that meridional variations in these fluxes could produce a meridional slope in the thermocline (a redistribution of mass induced by heating and mixing) that would be in geostrophic equilibrium with the flow in the upper layer. To determine whether this thermal/mixing process could maintain the Pacific North Equatorial Countercurrent, the long-term mean meridional profile of the heat and momentum fluxes at the sea surface were used to calculate the 'thermal/mixing' transport from 2.5°N to 10°N along 122.5°W. The direction of the 'thermal/mixing' transport is calculated to be eastward between 5 and 10°N in agreement with the long-term mean averaged direction of the North Equatorial Countercurrent (KENDALL, 1970), and in agreement with the results of Sverdrup's theory. In addition, the magnitude of the 'thermal/mixing' transport is comparable to the observed geostrophic transport of the Countercurrent, corresponding to a maximum vertically averaged eastward speed of 35 cm s- t near 7.5°N. 1.
INTRODUCTION
ONE PossmL~ m e c h a n i s m for the m a i n t e n a n c e o f the Pacific N o r t h E q u a t o r i a l C o u n t e r c u r r e n t has been shown b y SVEZDRUP (1947) a n d b y REID (1948) to b e the a c t i o n o f E k m a n divergence o n the t - p l a n e (in response to the t o r q u e o f the w i n d stress) t h a t leads to r e d i s t r i b u t i o n o f mass in the u p p e r ocean. T h e effect o f this is to generate m e r i d i o n a l g e o s t r o p h i c t r a n s p o r t to c o m p l y with the c o n s e r v a t i o n o f vorticity associated with the fluid. By continuity, this leads to the d e v e l o p m e n t o f z o n a l g e o s t r o p h i c t r a n s p o r t . H o w e v e r , there are o t h e r m e c h a n i s m s t h a t can induce a r e d i s t r i b u t i o n o f m a s s ; those o f t h e r m a l / m i x i n g processes which are p r e c l u d e d in the w o r k o f SVERDRUP (1947) a n d I ~ I D (1948). Recently, MIROPOL'SKIY (1970) developed a two-layer t h e r m o d i n e m o d e l for the ocean t h a t can a c c o u n t for this in response to the fluxes o f h e a t a n d m o m e n t u m across the sea surface. T h e r e d i s t r i b u t i o n o f mass is b r o u g h t a b o u t b y t u r b u l e n t exchange processes in the presence o f d i a b a t i c heating a n d manifests itself in the rise a n d fall o f the t h e r m o c l i n e in a s s o c i a t i o n with the change in the available p o t e n t i a l energy o f the u p p e r layer. Therefore, it m a y be expected t h a t differential spatial a c t i o n (in the m e r i d i o n a l direction) o f t h e r m a l / m i x i n g processes m i g h t induce *Thermal/mixing in this paper denotes thermal processes together with mixing processes. 3'Scripps Institution of Oceanography, University of California, San Diego, P.O. Box 1529, La Jolla, California 92037, U.S.A. 347
348
WASRE~ B. W m ~
slopes in the thermocline which would be in geostrophic equilibrium with the flow in the upper layer. This idea is not new; having been hypothesized by FREEMAN(1954). To test this hypothesis utilizing the Miropol'skiy thermocline model, it must be established that the equatorial North Pacific acts as a two-layer system. WYRTKI and KENDALL(1967) have calculated the zonal volume transport per unit width of the Countercurrent from hydrographic data along 79 meridional sections in the North Pacific and found that in the surface mixed layer it could be approximated by the 'thermoclinic' transport per unit width, a calculation based upon the meridional slope of the thermocline. This result supports the approximate two-layer notion of the equatorial oceans, associated with relatively swift surface flows above the thermocline overlying more sluggish deeper motions below the thermocline. Having established that the Miropol'skiy model may be applicable to the equatorial oceans, an equation similar to that of MIROPOL'SKIV(1970) to determine the depth of the thermocline is developed and then introduced into the 'thermoclinic' equation of W~rRa'ra and K~NDALL(1967). In this formulation, the resulting 'thermal/mixing' transport per unit width is calculated from the mean meridional distribution of the heat flux and the wind stress between 2.5°N and 10°N along 122°W, yielding a zonal Countercurrent with a magnitude that is comparable to both that of the Sverdrup geostrophic transport and the observed transport. 2.
MASS R E D I S T R I B U T I O N BY T H E R M A L / M I X I N G PROCESSES
Before proceeding directly to the calculation of the 'thermal/mixing' transport it is important to gain an understanding of the thermal/mixing processes that may be responsible for the redistribution of mass in the upper ocean. The following development is based on the work of MIROPOL'SKIY(1970), but differs from his approach in two ways; (1) with the inclusion of the advective term which cannot be neglected in the equatorial North Pacific Ocean and (2) with a different formulation for the turbulent kinetic energy production in the upper layer. Consider the familiar form of the steady-state equation for the conservation of heat in the upper layer of a two-layer model ocean: V.•O -- w(OO/Oz) = --O/~z[(q + I)/pcv] + 2(BS[z]/pcv) + V .V'0',
(2.1)
where 0 is the potential temperature of the upper layer, V is the mean horizontal velocity vector and w is the mean vertical velocity component (positive downward) with the primes denoting perturbation quantities, cp is the specific heat of sea water, p is the density of sea water, q is the turbulent heat flux in the vertical direction, I is the short-wave radiation flux, B is the latent heat, back radiation and reflected radiation flux, and 8(z) is the Dirac delta function indicating that the flux processes represented by B are restricted to the sea surface. Vertically integrating (2.1) from the sea surface to the interface separating the lower layer from the upper layer yields the heat balance for the upper layer:
M'~70 + wh(AO)-- pc~
+ flv'v'°'dz'
(2.2)
where .40 represents the absolute difference in potential temperature between the upper and lower layers, Qo represents the total heat flux through the sea surface (qo -]- Io + B),
The maintenance of the Pacific North Equatorial Countercurrent by thermal/mixing processes 349 w'O'[~.h represents the turbulent flux of heat at the interface, M is the horizontal volume transport (.per unit width) in the upper layer, and wh is the mean vertical velocity at the interface with the rigid lid approximation being applied at the sea
surface. The verticaladvcction of heat in (2.2) needs further explanation. At the interface, a discontinuity exists in temperature, resulting in the verticalgradient of temperature there being of the form of a Dirac delta function. The integralof the verticaladvcction of heat throughout the column can therefore bc determined in the following way: fhw~O d z :
w~ ~h
J0 ~z
~0 -- dz :
--wh(AO/2),
(2.3)
Jh_,~z
where wa is considered continuous and slowly changing in the vicinity of z : h. This term represents the advection of heat from the interface ( z : h) into the upper layer (z ---- h -- c). It is noted that the form of (2.2) can be determined in an alternate way by vertically integrating the flux form of the heat equation and applying the vertically integrated continuity relation: V " M = --w~. (2.4) The balance of thermal energy in (2.2) is between the advective fluxes on the lefthand side and the turbulent fluxes on the right-hand side. Up to this point we have not made any simplifying assumptions other than requiring the system to be in a steady state. The restriction is now made that the horizontal advection of the heat in (2.2) balances the horizontal turbulent diffusion of heat. The basis for this assumption stems from a comparison of observational work and numerical simulation. MONTGOMERY (1939) assumed this heat balance to exist in the Atlantic Equatorial Countercurrent, calculating a horizontal eddy diffusivity coefficient necessary for the balance to exist of 4 × 107 cm s s -1. This value approximates that (i.e. 107 cm 2 s -x) utilized by BRYAN and Cox (1968) in a numerical simulation of the North Atlantic Ocean, which produced transports and width scales for both the Gulf Stream and the Equatorial Undercurrent consistent with available observations. Applying this assumption to (2.2) yields: wn(AOI2) = (Qo/gc~) - w'O'lz-a,
(2.5)
which expresses the balance in the upper layer between the vertical advection of heat, and the vertical turbulent diffusion and radiation of heat. In (2.5), Q0 can be determined from bulk formulae that relate the total exchange of heat across the sea surface to such parameters as cloud cover, sea level pressure, air temperature, sea surface temperature, dew point temperature and wind speed (WYRTKI, 1965). The vertical velocity (wh) in (2.5) can be determined from the steady-state linear equations of motion used by EKMAN(1905) and the continuity relation, yielding on the f-plane: Wh--
curlz (Xo -- xa)
(2.6)
f
where xo (---- -- w'V') is the stress of the wind on the sea surface and xn is the interfacial stress.
The determination of the turbulent diffusion of heat across the interface (i.e. w'O'Iz-h ) is more difficult than establishing the other flux terms in (2.5), requiring
350
W~eN B. WroTE
consideration of the conservation equation for turbulent kinetic energy in the upper layer: O/St (V'.V'/2) = w'V'.(0V/0z) -- O[Oz b'w' + ga w'O' -- e,
(2.7)
where the velocity (V,w) and potential temperature (0) are as defined previously, b" = 1/2[(V'.V') + w'Z], g is gravity, and a is the thermal expansion coefficient. The terms in (2.7) are: V'.V'/2, the turbulent kinetic energy; w'V'.OV]Ox, the production of turbulent kinetic energy through a reduction in the mean vertical shear; b'w', the flux of energy due to the breakdown of surface or internal gravity waves to small-scale turbulence; w'O', the dissipation of turbulent kinetic energy by buoyancy forces; and ~, the viscous dissipation of eddy motion. All of the horizontal transports have been neglected in (2.7). In keeping with the stationary conservation of the total thermal energy within the upper layer, the total turbulent kinetic energy can be conserved by integrating (2.7) from the surface to the interface, and then setting the local time derivative to zero. This procedure yields the steady-state balance between the production and dissipation of turbulent kinetic energy in the upper layer: ga
w° dz =
Jo\
"fizz] dz + b'wlz= h --b w 1,=o +
fl ,dz.
(2.8)
By neglecting those production and dissipation terms in (2.8) due to gravity waves and viscous processes, emphasis is given to the production of turbulent kinetic energy through a reduction in the vertical shear of the mean flow contained within the upper layer and the dissipation of turbulent kinetic energy through the effects of buoyancy. The vertical shear in the upper layer can be of two types; Ekman shear and geostrophic shear. From the observed data (Section 4), the mean geostrophic flow displays a relatively small amount of shear in the main body of the upper layer, since the horizontal density gradients there are small. However, the geostrophic shear at the thermocline is very large, and can be represented mathematically in the present model as a Dirac delta function at the interface. However, the influence of this shear on the production of kinetic energy in the upper layer is small if the additional assumption is made that w'V' _ 0 at the interface. This will be true according to general circulation theory only if the depth of the upper layer is deeper than the depth of the Ekman layer and if the eddy viscous coupling (i.e. drag) between the upper and lower layer is absent. The fact that most of the transport of the Countercurrent (Section 4) is confined to the region above the thermocline suggests that this latter assumption is reasonable. Therefore, as a consequence, only the production of turbulent kinetic energy due to the reduction in Ekman shear is possible. The dissipation of turbulent kinetic energy by the effects of buoyancy, gaw'O', also represents the growth of average available potential energy in the upper layer (see KRAtJS, 1972, p. 23). This amounts to a redistribution of mass in the column induced by the vertical turbulent flux of heat. Because of the steady-state requirement this mass adjustment will result through turbulent diffusion processes that arise to balance the growth of turbulence at the expense of the mean shear of the Ekman flow, i.e. the driving of Ekman flow in the upper layer by the wind stress at the surface has the effect of generating turbulent diffusion processes that are in association with a redistribution
The maintenance of the Pacific North Equatorial Cotmtercurrent by thermal/mixing processes
351
of mass in the upper layer. This differs fundamentally from the redistribution of mass according to the Sverdrup theory, where the redistribution is induced by Ekman divergence on the fl-plane. To determine the growth of available potential energy in the water column from Ekman shear flow, equation (2.8) is rewritten with the assumption made above and by approximating the vertical eddy stress by its gradient formulation:
g
ofhoW'O'dz =
Kjo \
(2.9)
where K is the kinematic verticaleddy viscositycoefficient.O n the basis of the Ekman theory, we can determine the right-hand side of (2.9),which leads to ~i
w'O' dz
__-- 1 ~o 2 g--~ (2Kf)-----~"
(2.10)
This relation specifies the redistribution of mass in the upper layer due to the surface wind stresses, with the eddy viscosity coefficient controlling the Ekman layer depth scale and the Ekman vertical shear in the upper layer. Following MIROPOL'SKIY(1970) the integral equation is obtained for the turbulent kinetic energy balance from consideration of (2.10), (2.6), and (2.5): [(Qo/pC~) -- (curl'0/f) (A0/2)] h ---- (1/ga) ,rog"](2Kf) i,
(2.11)
where the left-hand side expresses the rate of change of available potential energy in the water column by heating [via Q/9cl, - curl]ft A0/2] the upper layer of thickness h. Given this form of the heating together with the wind stress at the sea surface, the thickness of the upper layer can easily be determined from (2.11), where h=
(1/ga) [To~/(2Kf) t] [(Qo]gC~) - (curl ~'o/f) ,40/2]"
(2.12)
This relation is similar in principal to that deduced by MmOPOL'SK~ (1970) for the steady-state case. 3.
THE ' T H E R M O C L I N I C ' TRANSPORT DUE TO THERMAL/MIXING PROCESSES
The mathematical expression utilized by WYRTKIand KENDALL(1967) to determine the zonal 'thermocllnic' flow (~) follows from the Margules relation (SVn~DRUP, JOHNSONand FLEMING, 1942) for geostrophic flow in a two-layer ocean with the lower layer at rest: :~ = ( - - g ' / f ) (Oh/Oy),
(3.1)
wherefis the Coriolis parameter, g is reduced gravity, h is the thickness of the upper layer, and y is the meridional spatial coordinate positive in the northward direction. Equation (3.1) differs from the Margules relation in that h defines the thickness of a layer of constant temperature rather than of constant density. However, in the equatorial ocean changes in density are determined principally by those in temperature [i.e. p -- ! o (1 --a0)] so the approximation is a good one. To determine the 'thermoclinic'
352
WA~N B. WroTE
transport multiply (3.1) by h, yielding
Mtz = (--g'/2f) (Oh2/Oy),
(3.2)
where Mtz is the zonal 'thermoclinic' volume transport per unit width. The substitution of (2.12) into (3.2) yields the theoretical 'thermoclinic' transport (per unit width) induced by thermal mixing processes (hereafter called the 'thermal/ mixing' transport): g ~
--
f (_~_~+~_j~/Q i~ AO~L~y \(2Kf)t]
\
~y
~y~ 2 f l J '
(3.3)
having neglected as small the zonal gradient of the meridional wind stress in curlz x0 and the term containing the meridional derivative of the Coriolis parameter. The relationship between Mt~ and ~2rz/~yZ is similar to that between the Sverdrup geostrophic transport Msz and the zonal wind distribution: M,x
:
aY 2
where fl is the meridional derivative of the Coriolis parameter. This expression differs from (3.3), in which ~%~/Oy~"and M~x are related on the f-plane and do not require the spatial integration of Ozr~/Oy~ from the eastern boundary.
4. C O M P A R I S O N OF THE ' T H E R M A L / M I X I N G ' COUNTERCURRENT
F L O W OF T H E
WITH OBSERVED FLOW
The North Equatorial Countercurrent (Fig. 1) lies between 5 ° and 10°N with the bulk of the transport confined to the surface layer ( < 150 m) above the 20 ° isotherm, the latter defining the middle of the thermocline. Because the principal transport of the Countercurrent is restricted to the surface layer, its magnitude is governed principally by the slope of the thermocline such as that of the 20°C isotherm. This observation is in agreement with the quantitative results of WYRTKI and KENDALL(1967) for a much larger number of such comparisons. To test the present model, both the thickness of the upper layer and the averaged zonal flow speed were calculated from long-term averaged values of heat flux and wind stress along 122.5°W for comparison with the observed data (Fig. 1). The long-term annual mean values of surface heat flux, taken from WYR'rrd (1965), are interpolated at 2.5 ° intervals from 0 ° to 12.5°N along 122.5°W. The long-term annual mean values of surface wind stress are from HELLERMAN(1967) along 122-5°W at 2.5 ° intervals over this same latitudinal range. From the basic wind stress data, both "ro~/(2Kf)t and 15.:~/OyA0/2fare then calculated, the former with an eddy viscosity coefficient of 100 cm 9-s -1 and the latter with AO[2 = 5°C. This value for Klies within the range of values given by SxrmU3RtrP,JOHNSONand FLEMIN~ (1942) for the upper ocean and gives an Ekman depth scale that is somewhat less than the average thickness of the upper layer (Fig. 1). In these calculationsfis allowed to vary as a function of latitude. From these values of Qo/pcl,, "ro2/(2Kf)i, ~ / ~ y /lO/2f the value of the upper layer thickness is calculated from (2.12) and displayed as a function of latitude in Fig. 2.
The maintenance of the Pacific North Equatorial Countercurrent by thermal/mixing processes
/-
\\
353
\\
"
~" - - - . _ i ~
~j..i
LATITUOE
o~
,
¥
,
,'7
,
6,"
,
T
,
(
\j
-I0
'
12~'W
?
Fig. I. The annual average temperature section between 110" and 140°W from the equator
to 10°N (taken from I ~ i ~ L , 1970) (above). The annual average gei~llropliicvelocity section between 110" and 140°W from the equator to 10°N (taken from I~I~rDALL, 1970) (below).
To compare this with the observed values of thermocline depth we also plot the observed depth o f t b e 20 ° isotherm (Fig. 2). The theoretical value o f h is always within 20 m of the depth of the 20 ° isotherm from 2.5 ° to 10°N, with the slopes of both curves being similar, having a positive slope between 0 ° and 5°N and a negative slope 5 ° and 10°N. The negative slopes in both cases are associated with the eastward flowing North Equatorial Countercurrent. F r o m an inspection of the relative magnitude of the terms in (2.12) we find that the meridional variability in h from 5 ° to 10°N is dominated by the production of turbulent kinetic energy from the E k m a n shear in the upper layer, with the effect of net surface heat flux remaining fairly constant but dominating the vertical advection of heat in the denominator of (2.12). However, from 0 ° to 5°N the meridional variability in h
354
WARRENB. Wnrr~ I
I
.
.
- -
.
.
I
q
THEORETICAL h DEPTH OF 20=C
50
-1-
IIit
,,=,2 E3
I II
I00
V 150 0°
I 2,5*
I 5°
DEGREES L A T I T U D E
Fig. 2.
1 75*
ALONG
I I0"
122.5*W
A plot of the theoretical value of thermocline depth together with the depth of the 20°C isotherm taken from the upper panel of Fig. 1.
is dominated by the vertical advection of heat into the upper layer, this term becoming large due to t h e f -1 term which it contains. From the meridional gradient of the Hellerman wind stress data and the Wyrtki heat flux data (using centered differences), the average zonal velocity (~) in the upper layer is calculated from consideration of (3.3), where t1 = Mtx/h. Both ~ and Mt~ are calculated using g' ----4 cm s -z, this value being slightly smaller than that (g' = 5 cm 2 s-1) used by W~tTra and K~NDALL (1967) to calculate the 'thermoclinic' transport of the North Equatorial Countercurrent. To compare a with the observed values of geostrophic flow, we plot (Fig. 3) the former with those (Fig. 1) of the latter digitized at 2.5 ° intervals. Both the latitudinal position and strength of the theoretical Countercurrent (Fig. 3) are in good agreement with the position and strength of the observed Countercurrent. Both have eastward speeds of 25-35 cm s -x near 7.5°N, changing to westward flow south of 5°N and north of 10°N. 5.
C O M P A R I S O N OF THE ' T H E R M A L / M I X I N G ' T R A N S P O R T W I T H THE SVERDRUP GEOSTROPHIC TRANSPORT
It is of interest to see how the maintenance of the North Equatorial Countercurrent by thermai/mixing processes compares with the maintenance of the Countercurrent through the theory developed by SVmtDRUP (1947) and REID (1948). Using the annual mean wind stress data for the eastern North Pacific from HELLF_JtMAN(1967), the Sverdrup geostrophic zonal transport M,~ has been calculated from (3.4). Values of ~ and h (Figs. 2 and 3) are used to compute the 'thermal/mixing' transport.
The maintenance of the Pacific North Equatorial Countercurrent by thermal/mixing processes
355
The direction of the 'thermal/mixing' transport and the Sverdrup geostrophic transport correspond in a general way with each other (Fig. 4) and with the observed data (Fig. 1). However, the maximum magnitude of the 'thermal/mixing' transport of the Countercurrent at 7.5°N is nearly three times as large as the maximum Sverdrup geostrophic transport at the same latitude, with an even greater disparity in the two t
t
. . . .
I
I
THEORETICAL OBSERVED
U
50
-~ i
0
S o bJ
-50
z o N
-I00
/ /
I
I I d"
-150
i
0°
2 5°
DEGREES
Fig. 3.
I
I
5°
7.5 °
LATITUDE
I
ALONG
I0 ° [22.5=W
A plot of the theoretical value of 'thermal/mixing' flow t~ together with the average geostrophic flow taken from the lower panel of Fig. 1. I
I SVERDRUP
....
i TRANSPORT
THERMAL/MIXING TRANSPORT
56 m
I-
/
,,=, "E o..
z
o
I I
-50
/ I I I I I
-I00
-150
J ,
0o
C
I
2.5 =
DEGREES
Fig. 4.
I
5=
LATITUDE
I
7.5 =
I
I00
A L O N G 122.5=W
A p l o t o f the ' t h e r m a l / m i x i n g ' t r a n s p o r t per unit, Mr=, t o g e t h e r w i t h the S v e r d r u p g e o s t r o p h i ¢ t r a n s p o r t per u n i t w i d t h , M a x .
356
WAmZEN
B. WmrrE
measurements south of 4°N. However, it is clear from a comparison of Figs. 2 and 3 that the 'thermal/mixing' transport has a magnitude that agrees better with the observed data than does the Sverdrup transport. 6.
DISCUSSION
In the steady-state 'thermal/mixing' theory the interface forms at a depth where the production of turbulent kinetic energy, associated with the reduction in Ekman shear flow in the upper layer, exactly balances the production of available potential energy associated with heating the upper layer by radiation, turbulent heat exchange and advection of heat. In this model there are no restrictions on the vertical velocity; its role in the balance is to provide for vertical advection of heat and n o t to represent the time rate of change of the thickness of the upper layer as is often assumed in quasigeostrophic ocean circulation theory. This approach is quite different from the steady-state Sverdrup theory in a quasigeostrophic two-layer model ocean, where no mass exchange takes place between the layers and hence no vertical advection of heat can take place across the interface. The interface in the steady-state Sverdrup theory exists at a depth where a balance occurs between the planetary vorticity and the production of vorticity by the torque of the wind stress. This is not to say that in the steady-state Sverdrup flow, the torque of the wind cannot produce vertical velocities at the base of the upper layer. Rather, implicit in this vorticity balance is the requirement that the Ekman vertical velocity at the interface be balanced by the compensating geostrophic vertical velocity that arises from mass conservation in the upper layer. One principal drawback of the thermal/mixing theory, as opposed to Sverdrup's theory, is the dependence of the former upon the turbulent exchange coefficients absent in the latter. Although at present there is little recourse to developing a steadystate model that describes the redistribution of mass through turbulent mixing processes, it is lamentable because it gives us two free parameters, the coefficients of horizontal eddy diffusivity and of vertical eddy viscosity. In the application of this theory to the North Equatorial Countercurrent the horizontal eddy diffusivity coefficient was set so that the horizontal diffusion of heat balanced the horizontal advection of heat, and the vertical eddy viscosity coefficient (K) was chosen to give a realistic magnitude of the 'thermal mixing' transport. Both of these turbulent exchange coefficients are very reasonable when compared with previous efforts to evaluate them (see SVERDRUP,JOHNSONand FLEMING, 1942, p. 484). Furthermore, the model is not very sensitive to changes in K, since it is taken to the square root power in (2.12). Moreover, changes in this parameter simply alter the absolute thickness of the upper layer, having no effect upon the sign of its slope. At this point, it is important to note that as Kincreases, the thickness of the upper layer decreases as the inverse square root, opposite to the Ekman layer depth which depends directly upon the square root of K. 7.
CONCLUSION
From this study, the volume transport of the Pacific North Equatorial Countercurrent can be adequately explained by considering thermal/mixing processes precluded in the theoretical development given by SVERDRUP (1947) and REID (1948).
The maintenance of the Pacific North Equatorial Countercurrent by thermal/mixing processes
357
The principal transport of the N o r t h Equatorial Countercurrent is observed to be confined to the surface mixed layer and is shown to be in geostrophic equilibrium with the slope of the near-surface thermocline, the depth of which can be determined by thermal/mixing processes. In heuristic terms, the thermocline exists at a depth where the production of turbulent kinetic energy in the upper layer from a reduction in the E k m a n shear flow is balanced by the growth of available potential energy of the column. The meridional profile of thermocline depth (and the resulting geostrophic flow and transport) in the vicinity of the Countercurrent from 5° to 10°N is due principally to the meridional variations in the production of turbulent kinetic energy by Ekman shear flow. F r o m 0 ° to 5°N, the meridional variability in the production of available potential energy by the vertical advection of heat into the upper layer becomes important and acts in conjunction with the former in establishing the meridional variability in the thermocline depth. Comparing the theoretical thickness of the upper layer along 122.5°W to the observed depth of the thermocline, shows good agreement to within 20 m from 2.5 ° to 10°N. Similar good agreement exists between observed geostrophic flow and the 'thermal• mixing' flow along this section. In both cases the N o r t h Equatorial Countercurrent is observed to lie between 5 ° and 10°N with the zonal maximum speeds of between 25 and 35 cm s-1. In addition, the 'thermal/mixing' transport (per unit width) is in good agreement with the Sverdrup geostrophic transport of the Countercurrent. This suggests that the thermal/mixing processes are at least as important toward the maintenance of the N o r t h Equatorial Countercurrent as the effect of Ekman divergence on the E-plane. This result has important implications for the near-surface general circulation the world over, suggesting that thermal/mixing processes may be significant in maintaining and generating surface geostrophic flows.
Acknowledgements--Tb.Js research was sponsored by the National Science Foundation Office of the Decade for Ocean Exploration and the Office of Naval Research as part of the North Pacific Experiment under the Office of Naval Research contract No. N000-14-69-A-0200-6043, and the University of California a t San Diego, Scripps Institution of Oceanography through the Ocean Research Division.
REFERENCES
BRYAN K. and M. Cox (1968) A nonlinear model of an ocean driven by wind and differential heating: Part 1. Journal era tmospheric Sciences, 25(6), 945-967. EIO~AN V. W. (1905) On the influence of the earth's rotation on ocean currents. Arkiv for Matematik, Astronomi oct Fysick, 2(11), 1-52. FREEMANJ. C. (1954) Wind mixing currents. Journal of Marine Research, 13(2), 157-165. I-IEL~IO~ANS. (1967) An updated estimate of the wind stress on the world ocean. Monthly Weather Reviews, 95(9), 607-614. [Corrections listed in 96(1), 63-74.] KENDALL R. (1970) The Pacific Equatorial Countercurrent, Int. Centre for Environmental Research, Laguna Beach, Calif., 19 pp. (Unpublished manuscript.) KnAus E. B. (1972) Atmosphere-ocean interaction, Oxford Monographs on Meteorology, Clarendon Press, 275 pp. MIROPOL'SKrYY. Z. (1970) Nonstationary model of the wind-convection mixing layer in the ocean. Fizika Atmosferi i Okeana, Izvestiz Akademii Nauk SSSR, 6(12), 1284-1294. MowrGoM~Y R. (1939) Ein Versueh, den vertikalen und seitlichen Austauch in der Tief der Sprungschricht im ~quatorialen Atlantischen Ozean zu bestimmen. Annalen der Hydrographie und maritime Meteorologie, 67, 242. ReID R. O. (1948) The equatorial currents in the eastern Pacific as maintained by the stress of the wind. Journal of Marine Research, 7, 74-99.
358
WAR~N B. Win-rE
SVERDROP H. U. (1947) Wind driven currents in a baroclinic ocean; with application to the equatorial currents of the eastern Pacific. Proceedings of the National Academy of Sciences, 33, 318-326. SVERDROPH. U., M. W. JOrlNSONand R. H. FImMINa(1942) The oceans: the irphysics, chemistry and general biology, Prentice-Hall, 1067 pp. WVaTKI K. (1965) The average annual heat balance of the North Pacific Ocean and its relation to ocean circulation. Journal of Geophysical Research, 70(18), 4547--4559. WYRTKI K. and R. KENDALL (1967) Transports of the Pacific Equatorial Countercurrent. Journal of Geophysical Research, 72(8), 2073-2076.
The maintenance of the Pacific North Equatorial Countercurrent by thermal/mixing* processes WARRE~ B, W m ~ t (Received 15 May 1973; in revisedform 19 October 1973; accepted21 November 1973) Abstract--The strength and latitudinal position of the principal transport in the Pacific North Equatorial Countercurrent has been developed theoretically by Sv~u3ttup (1947) and by REID (1948) to be the result of a mass redistribution in response to Ekman divergence on the ~-plane. More recently, the principal transport of the Countercurrent was found in the surface layer above the thermocline (WYRT~ and I~NDALL, 1967) and a good approximation for the Countercurrent transport in the upper layer (< 150 m) can be obtained by considering only the slope of the thermocline in a fashion after the Margules equation. Independently, MmDPDL'SKIY(1970) developed a two-layer thermocline model from considerations of the conservation of heat and turbulent kinetic energy that prescribes the thickness of the upper layer in terms of the net heat flux and the flux of momentum across the sea surface. Therefore, it is possible that meridional variations in these fluxes could produce a meridional slope in the thermocline (a redistribution of mass induced by heating and mixing) that would be in geostrophic equilibrium with the flow in the upper layer. To determine whether this thermal/mixing process could maintain the Pacific North Equatorial Countercurrent, the long-term mean meridional profile of the heat and momentum fluxes at the sea surface were used to calculate the 'thermal/mixing' transport from 2.5°N to 10°N along 122.5°W. The direction of the 'thermal/mixing' transport is calculated to be eastward between 5 and 10°N in agreement with the long-term mean averaged direction of the North Equatorial Countercurrent (KENDALL, 1970), and in agreement with the results of Sverdrup's theory. In addition, the magnitude of the 'thermal/mixing' transport is comparable to the observed geostrophic transport of the Countercurrent, corresponding to a maximum vertically averaged eastward speed of 35 cm s- t near 7.5°N. 1.
INTRODUCTION
ONE PossmL~ m e c h a n i s m for the m a i n t e n a n c e o f the Pacific N o r t h E q u a t o r i a l C o u n t e r c u r r e n t has been shown b y SVEZDRUP (1947) a n d b y REID (1948) to b e the a c t i o n o f E k m a n divergence o n the t - p l a n e (in response to the t o r q u e o f the w i n d stress) t h a t leads to r e d i s t r i b u t i o n o f mass in the u p p e r ocean. T h e effect o f this is to generate m e r i d i o n a l g e o s t r o p h i c t r a n s p o r t to c o m p l y with the c o n s e r v a t i o n o f vorticity associated with the fluid. By continuity, this leads to the d e v e l o p m e n t o f z o n a l g e o s t r o p h i c t r a n s p o r t . H o w e v e r , there are o t h e r m e c h a n i s m s t h a t can induce a r e d i s t r i b u t i o n o f m a s s ; those o f t h e r m a l / m i x i n g processes which are p r e c l u d e d in the w o r k o f SVERDRUP (1947) a n d I ~ I D (1948). Recently, MIROPOL'SKIY (1970) developed a two-layer t h e r m o d i n e m o d e l for the ocean t h a t can a c c o u n t for this in response to the fluxes o f h e a t a n d m o m e n t u m across the sea surface. T h e r e d i s t r i b u t i o n o f mass is b r o u g h t a b o u t b y t u r b u l e n t exchange processes in the presence o f d i a b a t i c heating a n d manifests itself in the rise a n d fall o f the t h e r m o c l i n e in a s s o c i a t i o n with the change in the available p o t e n t i a l energy o f the u p p e r layer. Therefore, it m a y be expected t h a t differential spatial a c t i o n (in the m e r i d i o n a l direction) o f t h e r m a l / m i x i n g processes m i g h t induce *Thermal/mixing in this paper denotes thermal processes together with mixing processes. 3'Scripps Institution of Oceanography, University of California, San Diego, P.O. Box 1529, La Jolla, California 92037, U.S.A. 347
348
WASRE~ B. W m ~
slopes in the thermocline which would be in geostrophic equilibrium with the flow in the upper layer. This idea is not new; having been hypothesized by FREEMAN(1954). To test this hypothesis utilizing the Miropol'skiy thermocline model, it must be established that the equatorial North Pacific acts as a two-layer system. WYRTKI and KENDALL(1967) have calculated the zonal volume transport per unit width of the Countercurrent from hydrographic data along 79 meridional sections in the North Pacific and found that in the surface mixed layer it could be approximated by the 'thermoclinic' transport per unit width, a calculation based upon the meridional slope of the thermocline. This result supports the approximate two-layer notion of the equatorial oceans, associated with relatively swift surface flows above the thermocline overlying more sluggish deeper motions below the thermocline. Having established that the Miropol'skiy model may be applicable to the equatorial oceans, an equation similar to that of MIROPOL'SKIV(1970) to determine the depth of the thermocline is developed and then introduced into the 'thermoclinic' equation of W~rRa'ra and K~NDALL(1967). In this formulation, the resulting 'thermal/mixing' transport per unit width is calculated from the mean meridional distribution of the heat flux and the wind stress between 2.5°N and 10°N along 122°W, yielding a zonal Countercurrent with a magnitude that is comparable to both that of the Sverdrup geostrophic transport and the observed transport. 2.
MASS R E D I S T R I B U T I O N BY T H E R M A L / M I X I N G PROCESSES
Before proceeding directly to the calculation of the 'thermal/mixing' transport it is important to gain an understanding of the thermal/mixing processes that may be responsible for the redistribution of mass in the upper ocean. The following development is based on the work of MIROPOL'SKIY(1970), but differs from his approach in two ways; (1) with the inclusion of the advective term which cannot be neglected in the equatorial North Pacific Ocean and (2) with a different formulation for the turbulent kinetic energy production in the upper layer. Consider the familiar form of the steady-state equation for the conservation of heat in the upper layer of a two-layer model ocean: V.•O -- w(OO/Oz) = --O/~z[(q + I)/pcv] + 2(BS[z]/pcv) + V .V'0',
(2.1)
where 0 is the potential temperature of the upper layer, V is the mean horizontal velocity vector and w is the mean vertical velocity component (positive downward) with the primes denoting perturbation quantities, cp is the specific heat of sea water, p is the density of sea water, q is the turbulent heat flux in the vertical direction, I is the short-wave radiation flux, B is the latent heat, back radiation and reflected radiation flux, and 8(z) is the Dirac delta function indicating that the flux processes represented by B are restricted to the sea surface. Vertically integrating (2.1) from the sea surface to the interface separating the lower layer from the upper layer yields the heat balance for the upper layer:
M'~70 + wh(AO)-- pc~
+ flv'v'°'dz'
(2.2)
where .40 represents the absolute difference in potential temperature between the upper and lower layers, Qo represents the total heat flux through the sea surface (qo -]- Io + B),
The maintenance of the Pacific North Equatorial Countercurrent by thermal/mixing processes 349 w'O'[~.h represents the turbulent flux of heat at the interface, M is the horizontal volume transport (.per unit width) in the upper layer, and wh is the mean vertical velocity at the interface with the rigid lid approximation being applied at the sea
surface. The verticaladvcction of heat in (2.2) needs further explanation. At the interface, a discontinuity exists in temperature, resulting in the verticalgradient of temperature there being of the form of a Dirac delta function. The integralof the verticaladvcction of heat throughout the column can therefore bc determined in the following way: fhw~O d z :
w~ ~h
J0 ~z
~0 -- dz :
--wh(AO/2),
(2.3)
Jh_,~z
where wa is considered continuous and slowly changing in the vicinity of z : h. This term represents the advection of heat from the interface ( z : h) into the upper layer (z ---- h -- c). It is noted that the form of (2.2) can be determined in an alternate way by vertically integrating the flux form of the heat equation and applying the vertically integrated continuity relation: V " M = --w~. (2.4) The balance of thermal energy in (2.2) is between the advective fluxes on the lefthand side and the turbulent fluxes on the right-hand side. Up to this point we have not made any simplifying assumptions other than requiring the system to be in a steady state. The restriction is now made that the horizontal advection of the heat in (2.2) balances the horizontal turbulent diffusion of heat. The basis for this assumption stems from a comparison of observational work and numerical simulation. MONTGOMERY (1939) assumed this heat balance to exist in the Atlantic Equatorial Countercurrent, calculating a horizontal eddy diffusivity coefficient necessary for the balance to exist of 4 × 107 cm s s -1. This value approximates that (i.e. 107 cm 2 s -x) utilized by BRYAN and Cox (1968) in a numerical simulation of the North Atlantic Ocean, which produced transports and width scales for both the Gulf Stream and the Equatorial Undercurrent consistent with available observations. Applying this assumption to (2.2) yields: wn(AOI2) = (Qo/gc~) - w'O'lz-a,
(2.5)
which expresses the balance in the upper layer between the vertical advection of heat, and the vertical turbulent diffusion and radiation of heat. In (2.5), Q0 can be determined from bulk formulae that relate the total exchange of heat across the sea surface to such parameters as cloud cover, sea level pressure, air temperature, sea surface temperature, dew point temperature and wind speed (WYRTKI, 1965). The vertical velocity (wh) in (2.5) can be determined from the steady-state linear equations of motion used by EKMAN(1905) and the continuity relation, yielding on the f-plane: Wh--
curlz (Xo -- xa)
(2.6)
f
where xo (---- -- w'V') is the stress of the wind on the sea surface and xn is the interfacial stress.
The determination of the turbulent diffusion of heat across the interface (i.e. w'O'Iz-h ) is more difficult than establishing the other flux terms in (2.5), requiring
350
W~eN B. WroTE
consideration of the conservation equation for turbulent kinetic energy in the upper layer: O/St (V'.V'/2) = w'V'.(0V/0z) -- O[Oz b'w' + ga w'O' -- e,
(2.7)
where the velocity (V,w) and potential temperature (0) are as defined previously, b" = 1/2[(V'.V') + w'Z], g is gravity, and a is the thermal expansion coefficient. The terms in (2.7) are: V'.V'/2, the turbulent kinetic energy; w'V'.OV]Ox, the production of turbulent kinetic energy through a reduction in the mean vertical shear; b'w', the flux of energy due to the breakdown of surface or internal gravity waves to small-scale turbulence; w'O', the dissipation of turbulent kinetic energy by buoyancy forces; and ~, the viscous dissipation of eddy motion. All of the horizontal transports have been neglected in (2.7). In keeping with the stationary conservation of the total thermal energy within the upper layer, the total turbulent kinetic energy can be conserved by integrating (2.7) from the surface to the interface, and then setting the local time derivative to zero. This procedure yields the steady-state balance between the production and dissipation of turbulent kinetic energy in the upper layer: ga
w° dz =
Jo\
"fizz] dz + b'wlz= h --b w 1,=o +
fl ,dz.
(2.8)
By neglecting those production and dissipation terms in (2.8) due to gravity waves and viscous processes, emphasis is given to the production of turbulent kinetic energy through a reduction in the vertical shear of the mean flow contained within the upper layer and the dissipation of turbulent kinetic energy through the effects of buoyancy. The vertical shear in the upper layer can be of two types; Ekman shear and geostrophic shear. From the observed data (Section 4), the mean geostrophic flow displays a relatively small amount of shear in the main body of the upper layer, since the horizontal density gradients there are small. However, the geostrophic shear at the thermocline is very large, and can be represented mathematically in the present model as a Dirac delta function at the interface. However, the influence of this shear on the production of kinetic energy in the upper layer is small if the additional assumption is made that w'V' _ 0 at the interface. This will be true according to general circulation theory only if the depth of the upper layer is deeper than the depth of the Ekman layer and if the eddy viscous coupling (i.e. drag) between the upper and lower layer is absent. The fact that most of the transport of the Countercurrent (Section 4) is confined to the region above the thermocline suggests that this latter assumption is reasonable. Therefore, as a consequence, only the production of turbulent kinetic energy due to the reduction in Ekman shear is possible. The dissipation of turbulent kinetic energy by the effects of buoyancy, gaw'O', also represents the growth of average available potential energy in the upper layer (see KRAtJS, 1972, p. 23). This amounts to a redistribution of mass in the column induced by the vertical turbulent flux of heat. Because of the steady-state requirement this mass adjustment will result through turbulent diffusion processes that arise to balance the growth of turbulence at the expense of the mean shear of the Ekman flow, i.e. the driving of Ekman flow in the upper layer by the wind stress at the surface has the effect of generating turbulent diffusion processes that are in association with a redistribution
The maintenance of the Pacific North Equatorial Cotmtercurrent by thermal/mixing processes
351
of mass in the upper layer. This differs fundamentally from the redistribution of mass according to the Sverdrup theory, where the redistribution is induced by Ekman divergence on the fl-plane. To determine the growth of available potential energy in the water column from Ekman shear flow, equation (2.8) is rewritten with the assumption made above and by approximating the vertical eddy stress by its gradient formulation:
g
ofhoW'O'dz =
Kjo \
(2.9)
where K is the kinematic verticaleddy viscositycoefficient.O n the basis of the Ekman theory, we can determine the right-hand side of (2.9),which leads to ~i
w'O' dz
__-- 1 ~o 2 g--~ (2Kf)-----~"
(2.10)
This relation specifies the redistribution of mass in the upper layer due to the surface wind stresses, with the eddy viscosity coefficient controlling the Ekman layer depth scale and the Ekman vertical shear in the upper layer. Following MIROPOL'SKIY(1970) the integral equation is obtained for the turbulent kinetic energy balance from consideration of (2.10), (2.6), and (2.5): [(Qo/pC~) -- (curl'0/f) (A0/2)] h ---- (1/ga) ,rog"](2Kf) i,
(2.11)
where the left-hand side expresses the rate of change of available potential energy in the water column by heating [via Q/9cl, - curl]ft A0/2] the upper layer of thickness h. Given this form of the heating together with the wind stress at the sea surface, the thickness of the upper layer can easily be determined from (2.11), where h=
(1/ga) [To~/(2Kf) t] [(Qo]gC~) - (curl ~'o/f) ,40/2]"
(2.12)
This relation is similar in principal to that deduced by MmOPOL'SK~ (1970) for the steady-state case. 3.
THE ' T H E R M O C L I N I C ' TRANSPORT DUE TO THERMAL/MIXING PROCESSES
The mathematical expression utilized by WYRTKIand KENDALL(1967) to determine the zonal 'thermocllnic' flow (~) follows from the Margules relation (SVn~DRUP, JOHNSONand FLEMING, 1942) for geostrophic flow in a two-layer ocean with the lower layer at rest: :~ = ( - - g ' / f ) (Oh/Oy),
(3.1)
wherefis the Coriolis parameter, g is reduced gravity, h is the thickness of the upper layer, and y is the meridional spatial coordinate positive in the northward direction. Equation (3.1) differs from the Margules relation in that h defines the thickness of a layer of constant temperature rather than of constant density. However, in the equatorial ocean changes in density are determined principally by those in temperature [i.e. p -- ! o (1 --a0)] so the approximation is a good one. To determine the 'thermoclinic'
352
WA~N B. WroTE
transport multiply (3.1) by h, yielding
Mtz = (--g'/2f) (Oh2/Oy),
(3.2)
where Mtz is the zonal 'thermoclinic' volume transport per unit width. The substitution of (2.12) into (3.2) yields the theoretical 'thermoclinic' transport (per unit width) induced by thermal mixing processes (hereafter called the 'thermal/ mixing' transport): g ~
--
f (_~_~+~_j~/Q i~ AO~L~y \(2Kf)t]
\
~y
~y~ 2 f l J '
(3.3)
having neglected as small the zonal gradient of the meridional wind stress in curlz x0 and the term containing the meridional derivative of the Coriolis parameter. The relationship between Mt~ and ~2rz/~yZ is similar to that between the Sverdrup geostrophic transport Msz and the zonal wind distribution: M,x
:
aY 2
where fl is the meridional derivative of the Coriolis parameter. This expression differs from (3.3), in which ~%~/Oy~"and M~x are related on the f-plane and do not require the spatial integration of Ozr~/Oy~ from the eastern boundary.
4. C O M P A R I S O N OF THE ' T H E R M A L / M I X I N G ' COUNTERCURRENT
F L O W OF T H E
WITH OBSERVED FLOW
The North Equatorial Countercurrent (Fig. 1) lies between 5 ° and 10°N with the bulk of the transport confined to the surface layer ( < 150 m) above the 20 ° isotherm, the latter defining the middle of the thermocline. Because the principal transport of the Countercurrent is restricted to the surface layer, its magnitude is governed principally by the slope of the thermocline such as that of the 20°C isotherm. This observation is in agreement with the quantitative results of WYRTKI and KENDALL(1967) for a much larger number of such comparisons. To test the present model, both the thickness of the upper layer and the averaged zonal flow speed were calculated from long-term averaged values of heat flux and wind stress along 122.5°W for comparison with the observed data (Fig. 1). The long-term annual mean values of surface heat flux, taken from WYR'rrd (1965), are interpolated at 2.5 ° intervals from 0 ° to 12.5°N along 122.5°W. The long-term annual mean values of surface wind stress are from HELLERMAN(1967) along 122-5°W at 2.5 ° intervals over this same latitudinal range. From the basic wind stress data, both "ro~/(2Kf)t and 15.:~/OyA0/2fare then calculated, the former with an eddy viscosity coefficient of 100 cm 9-s -1 and the latter with AO[2 = 5°C. This value for Klies within the range of values given by SxrmU3RtrP,JOHNSONand FLEMIN~ (1942) for the upper ocean and gives an Ekman depth scale that is somewhat less than the average thickness of the upper layer (Fig. 1). In these calculationsfis allowed to vary as a function of latitude. From these values of Qo/pcl,, "ro2/(2Kf)i, ~ / ~ y /lO/2f the value of the upper layer thickness is calculated from (2.12) and displayed as a function of latitude in Fig. 2.
The maintenance of the Pacific North Equatorial Countercurrent by thermal/mixing processes
/-
\\
353
\\
"
~" - - - . _ i ~
~j..i
LATITUOE
o~
,
¥
,
,'7
,
6,"
,
T
,
(
\j
-I0
'
12~'W
?
Fig. I. The annual average temperature section between 110" and 140°W from the equator
to 10°N (taken from I ~ i ~ L , 1970) (above). The annual average gei~llropliicvelocity section between 110" and 140°W from the equator to 10°N (taken from I~I~rDALL, 1970) (below).
To compare this with the observed values of thermocline depth we also plot the observed depth o f t b e 20 ° isotherm (Fig. 2). The theoretical value o f h is always within 20 m of the depth of the 20 ° isotherm from 2.5 ° to 10°N, with the slopes of both curves being similar, having a positive slope between 0 ° and 5°N and a negative slope 5 ° and 10°N. The negative slopes in both cases are associated with the eastward flowing North Equatorial Countercurrent. F r o m an inspection of the relative magnitude of the terms in (2.12) we find that the meridional variability in h from 5 ° to 10°N is dominated by the production of turbulent kinetic energy from the E k m a n shear in the upper layer, with the effect of net surface heat flux remaining fairly constant but dominating the vertical advection of heat in the denominator of (2.12). However, from 0 ° to 5°N the meridional variability in h
354
WARRENB. Wnrr~ I
I
.
.
- -
.
.
I
q
THEORETICAL h DEPTH OF 20=C
50
-1-
IIit
,,=,2 E3
I II
I00
V 150 0°
I 2,5*
I 5°
DEGREES L A T I T U D E
Fig. 2.
1 75*
ALONG
I I0"
122.5*W
A plot of the theoretical value of thermocline depth together with the depth of the 20°C isotherm taken from the upper panel of Fig. 1.
is dominated by the vertical advection of heat into the upper layer, this term becoming large due to t h e f -1 term which it contains. From the meridional gradient of the Hellerman wind stress data and the Wyrtki heat flux data (using centered differences), the average zonal velocity (~) in the upper layer is calculated from consideration of (3.3), where t1 = Mtx/h. Both ~ and Mt~ are calculated using g' ----4 cm s -z, this value being slightly smaller than that (g' = 5 cm 2 s-1) used by W~tTra and K~NDALL (1967) to calculate the 'thermoclinic' transport of the North Equatorial Countercurrent. To compare a with the observed values of geostrophic flow, we plot (Fig. 3) the former with those (Fig. 1) of the latter digitized at 2.5 ° intervals. Both the latitudinal position and strength of the theoretical Countercurrent (Fig. 3) are in good agreement with the position and strength of the observed Countercurrent. Both have eastward speeds of 25-35 cm s -x near 7.5°N, changing to westward flow south of 5°N and north of 10°N. 5.
C O M P A R I S O N OF THE ' T H E R M A L / M I X I N G ' T R A N S P O R T W I T H THE SVERDRUP GEOSTROPHIC TRANSPORT
It is of interest to see how the maintenance of the North Equatorial Countercurrent by thermai/mixing processes compares with the maintenance of the Countercurrent through the theory developed by SVmtDRUP (1947) and REID (1948). Using the annual mean wind stress data for the eastern North Pacific from HELLF_JtMAN(1967), the Sverdrup geostrophic zonal transport M,~ has been calculated from (3.4). Values of ~ and h (Figs. 2 and 3) are used to compute the 'thermal/mixing' transport.
The maintenance of the Pacific North Equatorial Countercurrent by thermal/mixing processes
355
The direction of the 'thermal/mixing' transport and the Sverdrup geostrophic transport correspond in a general way with each other (Fig. 4) and with the observed data (Fig. 1). However, the maximum magnitude of the 'thermal/mixing' transport of the Countercurrent at 7.5°N is nearly three times as large as the maximum Sverdrup geostrophic transport at the same latitude, with an even greater disparity in the two t
t
. . . .
I
I
THEORETICAL OBSERVED
U
50
-~ i
0
S o bJ
-50
z o N
-I00
/ /
I
I I d"
-150
i
0°
2 5°
DEGREES
Fig. 3.
I
I
5°
7.5 °
LATITUDE
I
ALONG
I0 ° [22.5=W
A plot of the theoretical value of 'thermal/mixing' flow t~ together with the average geostrophic flow taken from the lower panel of Fig. 1. I
I SVERDRUP
....
i TRANSPORT
THERMAL/MIXING TRANSPORT
56 m
I-
/
,,=, "E o..
z
o
I I
-50
/ I I I I I
-I00
-150
J ,
0o
C
I
2.5 =
DEGREES
Fig. 4.
I
5=
LATITUDE
I
7.5 =
I
I00
A L O N G 122.5=W
A p l o t o f the ' t h e r m a l / m i x i n g ' t r a n s p o r t per unit, Mr=, t o g e t h e r w i t h the S v e r d r u p g e o s t r o p h i ¢ t r a n s p o r t per u n i t w i d t h , M a x .
356
WAmZEN
B. WmrrE
measurements south of 4°N. However, it is clear from a comparison of Figs. 2 and 3 that the 'thermal/mixing' transport has a magnitude that agrees better with the observed data than does the Sverdrup transport. 6.
DISCUSSION
In the steady-state 'thermal/mixing' theory the interface forms at a depth where the production of turbulent kinetic energy, associated with the reduction in Ekman shear flow in the upper layer, exactly balances the production of available potential energy associated with heating the upper layer by radiation, turbulent heat exchange and advection of heat. In this model there are no restrictions on the vertical velocity; its role in the balance is to provide for vertical advection of heat and n o t to represent the time rate of change of the thickness of the upper layer as is often assumed in quasigeostrophic ocean circulation theory. This approach is quite different from the steady-state Sverdrup theory in a quasigeostrophic two-layer model ocean, where no mass exchange takes place between the layers and hence no vertical advection of heat can take place across the interface. The interface in the steady-state Sverdrup theory exists at a depth where a balance occurs between the planetary vorticity and the production of vorticity by the torque of the wind stress. This is not to say that in the steady-state Sverdrup flow, the torque of the wind cannot produce vertical velocities at the base of the upper layer. Rather, implicit in this vorticity balance is the requirement that the Ekman vertical velocity at the interface be balanced by the compensating geostrophic vertical velocity that arises from mass conservation in the upper layer. One principal drawback of the thermal/mixing theory, as opposed to Sverdrup's theory, is the dependence of the former upon the turbulent exchange coefficients absent in the latter. Although at present there is little recourse to developing a steadystate model that describes the redistribution of mass through turbulent mixing processes, it is lamentable because it gives us two free parameters, the coefficients of horizontal eddy diffusivity and of vertical eddy viscosity. In the application of this theory to the North Equatorial Countercurrent the horizontal eddy diffusivity coefficient was set so that the horizontal diffusion of heat balanced the horizontal advection of heat, and the vertical eddy viscosity coefficient (K) was chosen to give a realistic magnitude of the 'thermal mixing' transport. Both of these turbulent exchange coefficients are very reasonable when compared with previous efforts to evaluate them (see SVERDRUP,JOHNSONand FLEMING, 1942, p. 484). Furthermore, the model is not very sensitive to changes in K, since it is taken to the square root power in (2.12). Moreover, changes in this parameter simply alter the absolute thickness of the upper layer, having no effect upon the sign of its slope. At this point, it is important to note that as Kincreases, the thickness of the upper layer decreases as the inverse square root, opposite to the Ekman layer depth which depends directly upon the square root of K. 7.
CONCLUSION
From this study, the volume transport of the Pacific North Equatorial Countercurrent can be adequately explained by considering thermal/mixing processes precluded in the theoretical development given by SVERDRUP (1947) and REID (1948).
The maintenance of the Pacific North Equatorial Countercurrent by thermal/mixing processes
357
The principal transport of the N o r t h Equatorial Countercurrent is observed to be confined to the surface mixed layer and is shown to be in geostrophic equilibrium with the slope of the near-surface thermocline, the depth of which can be determined by thermal/mixing processes. In heuristic terms, the thermocline exists at a depth where the production of turbulent kinetic energy in the upper layer from a reduction in the E k m a n shear flow is balanced by the growth of available potential energy of the column. The meridional profile of thermocline depth (and the resulting geostrophic flow and transport) in the vicinity of the Countercurrent from 5° to 10°N is due principally to the meridional variations in the production of turbulent kinetic energy by Ekman shear flow. F r o m 0 ° to 5°N, the meridional variability in the production of available potential energy by the vertical advection of heat into the upper layer becomes important and acts in conjunction with the former in establishing the meridional variability in the thermocline depth. Comparing the theoretical thickness of the upper layer along 122.5°W to the observed depth of the thermocline, shows good agreement to within 20 m from 2.5 ° to 10°N. Similar good agreement exists between observed geostrophic flow and the 'thermal• mixing' flow along this section. In both cases the N o r t h Equatorial Countercurrent is observed to lie between 5 ° and 10°N with the zonal maximum speeds of between 25 and 35 cm s-1. In addition, the 'thermal/mixing' transport (per unit width) is in good agreement with the Sverdrup geostrophic transport of the Countercurrent. This suggests that the thermal/mixing processes are at least as important toward the maintenance of the N o r t h Equatorial Countercurrent as the effect of Ekman divergence on the E-plane. This result has important implications for the near-surface general circulation the world over, suggesting that thermal/mixing processes may be significant in maintaining and generating surface geostrophic flows.
Acknowledgements--Tb.Js research was sponsored by the National Science Foundation Office of the Decade for Ocean Exploration and the Office of Naval Research as part of the North Pacific Experiment under the Office of Naval Research contract No. N000-14-69-A-0200-6043, and the University of California a t San Diego, Scripps Institution of Oceanography through the Ocean Research Division.
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