A coupled ocean-atmosphere box model of the Atlantic Ocean: A bimodal climate response

Journal of Marine Systems, 1 (1990) 197-208

197

Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

A coupled ocean-atmosphere box model of the Atlantic Ocean: a bimodal climate response G. Edward Birchfield, Matthew W y a n t * and Huaxiao Wang Department of Geological Sciences, Northwestern University, Evanston, IL 60208 (U.S.A.) Received September 20, 1989; revised version accepted October, 1989

ABSTRACT Birchfield, G.E., Wyant, M. and Wang, H., 1990. A coupled ocean-atmosphere box model of the Atlantic Ocean: a bimodal climate response. J. Mar. Syst., 1: 197-208. A coupled ocean-atmosphere box model is presented which, although extremely simplified, permits study of the time domain response of a model of the Atlantic Ocean coupled to an atmosphere. In particular the model can be used to systematically examine questions of multiple equilibria and steady state solutions of the earth's climate. The ocean model includes differential surface heating and evaporation, horizontal and vertical exchange of heat and salt between boxes, and a imply parameterized thermohaline circulation. Surface heat fluxes and evaporation are determined through the coupled ocean and energy-balance atmosphere models which treat fluxes of long and short wave radiation and sensible and latent heat. in this paper only circulations symmetric about the equator are presented. Two parameters represent the most important physics: t.t controls the magnitude of the thermohaline circulation; ¢ controls the strength of the hydrological cycle. For fixed p, two regimes are distinguished. One, associated with relatively small values of ~, has weak latitudinal water vapor transport in the atmosphere, a strong thermohaline circulation with sinking in high latitudes, upwelling in low latitudes and strong latitudinal transport of heat by the ocean. The second regime is characterized by strong latitudinal water vapor transport which, by reducing th surface salinity in high latitudes, nearly shuts down the thermohaline circulation and has relatively low latitudinal heat transport by the ocean. Although many important physical processes are not present in the model, the strong circulation regime corresponds roughly to a relatively warm interglacial climate and the weak circulation regime is associated with reduced ocean and atmosphere temperatures in high latitudes, i.e. with such events as the Younger Dryas or the last glacial maximum.

Introduction

This paper is concerned with the question of whether the ocean plays a dynamic or a passive role in the major climate changes seen in deepsea-sediment 6180 records and in ice core records from the polar ice caps (see for example: Ruddiman and Mclntyre, 1977; Dansgaard et al., 1984; Broecker et al., 1985; Broecker and Denton, 1989). Suggestions of the existence of multiple equilibria

* Present address: Department of Atmospheric Sciences, University of Washington, Seattle, WA (U.S.A.) 0924-7963/90/$03.50 © 1990 - Elsevier Science Publishers B.V.

for the ocean circulation have come from box model studies of Stommel (1961), Rooth (1982) and Welander (1986); Bryan (1986) and Manabe and Stouffer (1988) present results from experiments with general circulation models (GCM) which display multiple equilibria in ocean circulation. Evidence that the Younger Dryas climatic event between 11 and 10 ka was accompanied by a major change in circulation of the North Atlantic has been presented by Boyle and Keigwin (1987). They and Duplessy et al. (1988) reach similar conclusions for the glacial Atlantic. Rind et al.

G.E. BIRCHFIELDET AL.

198

(1986) and Manabe and Stouffer (1988) present numerical model results which suggest that changes in the circulation of the North Atlantic, in particular changes in the production rate of North Atlantic Deep Water (NADW), result in significant changes in the climate of the northern hemisphere. With a coupled o c e a n - a t m o s p h e r e box model for the northern hemisphere very much simpler than the G C M yet more explicit than the earlier box models, Birchfield (1989), henceforth referred to as BI, predicts a bimodal response in which one state has a strong thermohaline circulation and the other has a greatly reduced or possibly reversed circulation. A generalisation of BI is presented which includes a closed hydrological cycle and a representation for the entire Atlantic Ocean. In this paper only circulations symmetric about the equator are discussed. Weyl (1968) and Broecker and Denton (1989) speculate on the importance of water vapor transport on the thermohaline circulation and on climate change. The latter, in particular, assert that changes in latitudinal water vapor transport can affect the magnitude of the thermohaline circulation; an increase in transport may reduce and possibly shut down the circulation by reduction of surface salinity in high latitudes, while a decrease in transport may increase the strength of the circulation. With reduced strength of the thermohaline circulation, the important ocean heat transport to higher latitudes is reduced with significant effect on climate in the northern hemisphere. In this paper it is demonstrated that for symmetric circulations in the simple 4-box model of the Atlantic Ocean atmospheric water vapor transport is closely correlated with the magnitude of the thermohaline circulation. In the following section the ocean box model is described followed by a discussion of the atmospheric component. Solution of model equations is presented in the fourth section and results and discussion appear in the last section.

in Fig. 1. The low latitude surface box extends from 45°S to 45° N, the high latitude boxes to 70 ° N and S. Mean ocean depth is 4 kin; surface boxes are 800 m deep. Volumes and surface areas are given in Table 1. In the present version, the ocean is isolated from the rest of the world ocean; that is, there is no flow through the lateral boundaries of the ocean. The ocean model is forced by latitudinal differential heating at the surface of the ocean; that is, q2 watts flow into the surface of box 2 and qt and q4 watts flow out of boxes 1 and 4, respectively. The ocean is also forced by differential evaporation at the sea surface. Evaporation increases surface salinity and lowers the temperature through latent heat flux to the atmosphere. In each hemisphere a net amount of water vapor men or rues (m3s - t ) is transported from the low latitude atmosphere to the high latitude atmosphere; after precipitating it is returned via ocean circulation to the low latitude ocean box. The evaporation rate and the associated latent heat fluxes are determined by a complex set of processes associated with the hydrological cycle, described in the next section. The ocean water in individual boxes is not assumed to be well mixed. Properties of the water

M o d e l physics and equations: T h e ocean c o m p o -

Fig. 1. A schematic cross-section of the coupled ocean-atmosphere box model. Stipled region represents the ocean, ql, q2 and q4 denote surface heat fluxes; me, and rncs denote differential net surface evaporation, mort and rnos denote the magnitude of the deep water production in the northern and southern hemisphere high latitudes, respectively.

i4

i

I ATMOSPHERE I li

ql meal

,,l/J 1 /

The Atlantic Ocean is represented by a deep ocean box and three surface ocean boxes as shown

qz

men

D~On

lllOs

..... i- ............. -T-...... -t- ....... 3

m'ea

0,4

1 ,,/1

12

70~N nent

men

i

:--t--

ATL~TIC OCEAN EQ

70°S

COUPLED OCEAN-ATMOSPHERE BOX MODEL OF THE ATLANTIC TABLE 1 Coupled ocean-atmosphere model constants Constant

Value

v I (m 3) 02 (m 3) o3 (m 3) v4 (m 3) Aol (m 2) AO2 (m 2) Ao4 (m 2)

8.32.1015 5.184.1016 2.513.1017 10.93.1015 1.04.1013 6.48-1013 1.37.1013

ALl (m 2)

4.89.10 ~3 29.76.1013 5.23- 1013 8.297.106 6.488-105 2.021- 106 2.021 - 106 2.021.106 4.186.106 1027.84 9.622"10 -5 1027.84 9.622"10 -5 7.755" 10 -4 35.0 1.431 • 102 0.3854 0.0961 1.399" 102 0.4970 0.03086 0.1 0.1 107.65 170.40 107.65 2618.0 11.927 1.920 2554.8 9.6813 0.6163

AL2 (m 2) ALA (m 2) k21 (m3s - 1) k31 (m3s- 1) k23 (m3s-1) k34 (m3s -1) k24 (m3s- 1) co (J kg -1 K -1) Po (kg m -3) a (K -1) Po (kg m -3) a (K -1 ) fl So (permil) A 1 (K) B1 C1 A 2 (K) B2 C2 otL ao Sol ( w m -2) So2 (w m - 2) So4 (w m -2) Rio (w m -2) Ri1 (w m -2 K -1) R12 (w m -2 K -1) R2o (w m -2) R21 ( w m -2 K - l ) R22 (w m -2 K -1)

199

assumed equally effective in transport of heat and salt. In northern and southern high latitude boxes a thermohaline circulation m0, and m0s (m3s-1), respectively, exists either as net sinking in high latitudes (m 0 > 0) and upwelling in low latitudes, or as net sinking in low latitudes, upwelling in high latitudes (m 0 < 0). With vertical stratification in the model, static stability becomes an important consideration. Conservation properties of the model are water mass, heat and salt. The volumes of water in each box are assumed to remain fixed in time. The thermohaline circulation, m o, and the water mass return flow, m e, each imply salinity and heat fluxes; however, the heat flux by m e is neglected since usually rn e << m 0. In computing transport of heat and salt by volume flux of water between boxes, the temperature and salinity on the boundary between two boxes is approximated by taking the arithmetic mean of the mean temperature and salinity, respectively, of the two boxes. This approximation is justified below and in the Appendix. Statements of heat conservation for the three surface boxes and deep ocean are thus d

v , - ~ Tl = - 3 , + k21(T2- T1) + k 3 , ( T 3 - T,) m°n (T +~,-2--

d

v2-dT T2= + q 2 - k 2 , ( T 2 -

7"1) - k23(T z - T3)

- k z , ( T 2 - T4) m0n

+~

( T3 - T, ) + -~-~ ( T3 - T4 ), (2.1)

d o 3 ~ r~ = + k 2 3 ( T 2 in each box are represented by their mean values over the volume of the box. Linear eddy mixing of heat and salt occurs between surface boxes. The only manifestation of wind-driven circulation in the model is horizontal eddy mixing between the surface boxes. Vertical eddy mixing across the thermocline occurs in the model, i.e., between the surface and deep ocean boxes. Eddy mixing is

T3)

7"3) - k 3 , ( T 3 -

7",)

- k3.(T3 - 7"4)

+-~(T,d ~4~ T. =

-,h

T2) + - ~ - ~ ( T 4 - T2),

+ k2.(T~ - T.) + k3.(T~ - T.) m0s

+ - T ( ~ - ~),

G.E. BIRCHFIELD ET AL.

200

where 0 = q/poCo, Oo is a reference density and c o is the specific heat of sea water. Tj is the mean temperature in degrees Kelvin of box j. Vj is the volume of box j. k,j are the constant eddy mixing coefficients, where numerical values are specified in Table 1. Salt conservation requires, where Sj is the mean salinity of box j, d

UI-a'TS 1 -~- + k 2 1 ( S

2-

3 1 ) -4- k 3 1 ( S 3 -- S 1 )

+ - - ~ - ( S 2 - 83) - menSo,

d °2 T i

= -k2,(s2

-

s,)

- k23(S2

- s3)

- k=,(s: - s,) + -~(s3

- s,)

+-m-~-( S3-- S4) + (men + mes)SO, (2.2) d v3-yis

=

-

+

- &)

-- k34(T3 -- T4)

-.[- - ~ - ( 8 1 - 82 ) -.[- - . ~ ( 84 - 82 ) ,

d v,-.-~S 4 =

--}-k2,(S 2 - 8 4 ) -{- k 3 4 ( 8 3 - 8 4 ) +

(

-

m°n

Since the volumes of the surface boxes remain constant in time, the surface salinity flux is further simplified by replacing it in the above time rueSo. The equation of state for the ocean is taken to be linear in temperature and salinity. To approximately allow for the effects of ocean compressibility on static stability, a correction in temperature is added, 8j. (2.3)

where j = l . . . . . 4; 8 1 = 3 2 = 6 4 = 0 ; 3 3 = 0 . 2 ° K ; pj is the mean density of box j; a, fl are the thermal expansion coefficient and salinity coefficient, respectively, with both assumed constant; Po is a reference ocean density at To, So. Static stability is assumed to exist when 8Pl = P 3 - - P l > 0 ,

3PZ = P3 -- P2 > 0, 804 = P 3 - - P4 > 0.

P0 Pl -- P2),

/G m0s = ~ 0 ( P 4 - P2),

) - m o So.

p,=Po[1-a(Tj- To-3:)+fl(S,-So)],

The magnitude of the thermohaline circulation is determined by the following argument. In the high latitude surface box, it is assumed that, in addition to latitudinal differential heating, there is a local region in the box where production of water dense enough to sink to the deep ocean is occurring. Even though it may be a very local source region, the magnitude of the dense water production affects the mean density of the surface box such that the greater the production, the greater the mean box density. Therefore the magnitude of northern hemisphere high latitude sinking water flux, mow is proportional to Pl < P3- Likewise, dense water is being produced somewhere in the low latitude surface box due to the latitudinal net evaporation gradient. This dense water, which may be quite locally produced, tends to increase the mean density of the low latitude box the greater its production rate. We assume that the magnitude of northern hemisphere low latitude sinking of dense water, - m o w is proportional to Pz < P3. The water flux associated with the thermohaline circulation can be expressed, after using a similar argument for the southern hemisphere:

(2.4)

(2.5) (2.6)

where/~,, #s are constants of proportionality with dimensions m3s- 1. A specific effort to incorporate sub-box scale physics of deep water formation is made by choosing centered averages to approximate heat and salt fluxes between boxes. A consequence of this approximation is that the mean temperature a n d / o r salinity in the deep ocean box may be less than or greater than mean values in the surface ocean boxes. While in general, centered averages for numerical schemes of advection have the disadvantage that they allow local extrema to occur, here they permit, albeit in the simplest way, the important effects of local production of dense water to be incorporated into the model. A further discussion is presented in the Appendix. In BI it was estimated for the modern ocean that an appropriate order of magnitude for rn0n = 10.0-10 6 m 3 s - ] and for the heating, taking into

COUPLED OCEAN-ATMOSPHERE BOX MODEL OF THE ATLANTIC

20]

account box 2 includes both hemispheres, q2 = 1014 w; the hydrological mass flux was estimated to be men = 10 5 m3s -a. The magnitudes of the eddy mixing coefficients given in Table 1 were estimated with vertical coefficients based on an eddy diffusion coefficient of 1 cm s -a. Since the only manifestation of the wind-driven circulation in the model is through horizontal mixing, the horizontal coefficient based on a large lateral eddy diffusion coefficient of 108 cm s - I has been used. These values of the eddy mixing coefficients in Table 1 are used as estimates about which perturbations can be made to determine the sensitivity of the model circulation to the parameterization of the mixing processes.

ture, Ts. A important further simplification made here is the elimination of the temperature-albedo feedback; i.e., the surface albedos for land and for ocean, a L and ao, respectively, are fixed. The energy conservation equation is, for each box:

Model physics and equations: The atmospheric component With the focus on ocean physics forced by surface fluxes of heat and salinity, the atmospheric component, represented by an energy balance model (EBM), is simplified to the extreme. The atmosphere is divided into a high latitude box for each hemisphere and a single low latitude box, each extending from the surface to the top of the atmosphere. The high latitude box extends from latitude 45 ° to the pole (see Fig. 1). Each box of the atmosphere overlies a land fraction and an ocean fraction. Horizontal exchange of sensible heat in the atmosphere takes place across the boundary at latitude 45 o in each hemisphere and is modeled with an exchange coefficient. By treating the vertical energy fluxes in the atmospheric EBM in some detail, the role of the various fluxes at the ocean surface and the solar constant, Sc, itself become explicit. This allows a coupling of the hydrological cycle to the latent heat flux. The starting point for the approximation of the vertical fluxes is the two-layer seasonal energy balance model of Birchfield et al. (1982), henceforth refered to as BWL. The model is also similar in m a n y respects to the model of North et al. (1983). The BWL model is reduced to a single-layer mean annual model, assuming that each of the three boxes in the present model have a constant lapse rate; that is, the atmospheric temperature is uniquely determined by the surface air tempera-

A o ( Q T o - - QBo) + AL(QTL-- QSL) + FH = O, (3.1) where the subscripts O and L refer to atmosphere over the ocean and land sectors of the box, respectively, and subscripts T, B refer to the fluxes into the top and out the bottom of the atmosphere, respectively. These fluxes are multiplied by the ocean and land surface areas, A o, A L. F H is the eddy flux of heat from box 2 to box 1 or 4. The top and bottom fluxes may be represented as follows: QTO -----STO - LTO

(3.2)

QBo = SBo + LBo - QBBo -- QSHO -- QLHO

(3.3)

QTL ----"STL -- L T L

(3.4)

QBL = a B E "F LB L -- QBBL -- QSHL -- QLHL -----0

(3.5) where Svo, STL are the net incoming short wave radiation at the top of the atmosphere over ocean and land; LTO, LTL are the long wave radiative fluxes to space from over the ocean and land; QsHo, QSHL are the sensible heat fluxes into the atmosphere from the ocean and land surfaces; QLHO, QLHL represent the heating of the atmosphere due to the latent heat flux into the atmosphere from the ocean and land surfaces; L BO, LBL are the downward long wave fluxes to the surface of the ocean and land from the atmosphere; SBo, SBL are the short wave fluxes absorbed at the ocean and land surfaces; and QBB is the upward long wave flux from the ocean or land surface into the atmosphere. It is assumed that the total heat flux into the land surface vanishes exactly; that is, the land surface has no heat capacity. (a) Short wave fluxes Insolation entering the top of the atmosphere is averaged over the latitude range of each of the

G.E. BIRCHFIELD ET AL.

202 TABLE 2

These are f o u n d from:

Constants used in the atmospheric EBM

aAL = 0.281 + 0 . 5 0 a L,

Constant

OcAO= 0.281 + 0 . 5 0 a o,

Value

sI

0.700

s2

1.108

s4

0.700

S¢ (w m -2) e ca (kg m - 2 sec- 1) cp (J kg -l K -l) Lc (J kg- I )

1367.0 0.0 6.0185.10 - 3 1.0045"103 2.4904.106

O (W m - 2 K - a )

0.55697"10 -7

/~0 (w m - 2 )

455.77

/~I (wm-2 K-~) /~2 (w m -2 K -1) (w m -2) BI (w m -2 K -1) LT1 (w m -2) LT2 (w m -2 K -I) LT3 (w m -2 K -1) Lal (wm -2)

8.55 6.84 1.06"103

5.03 625.8 0.727 2.205 1755.0

LB2 ( w m - 2 K - I )

7.087

which implies that 50% of the incoming short wave radiation reaches the surface of the earth.

(b) Sensible heat flux The sensible heat flux from the surface to the atmosphere is, for land or ocean: Osn = CdCp(T* -- Ts) = F1(T*, Ts), where Cp is the specific heat of air, T, the surface temperature and Ts the surface air temperature.

(c) Long wave fluxes The black b o d y radiative flux from the surface is defined for land or ocean:

QBB = oT4 = Fz( T , ) ,

boxes and over the year. The resulting flux of short wave radiation entering the top of the atmosphere over each box is:

(3.6)

where o is the S t e f a n - B o l t z m a n constant. The long wave fluxes L T, L a, assuming fixed humidity distribution and cloudiness, depend only on the temperatures T,, Ts and are given in tabular form in Suarez and Held (1979).

sc sJ' = 4

d-2p-_ sJ,

(d) The latent heat flux and the hydrological cycle

where j = l , 2, 4 and sj is a function of the latitude range of the box and polar axis obliquity; e is the eccentricity; and Sc is the solar constant. sj values, together with other constants, are presented in Table 2. Temperature dependence of the short wave radiative parameters included in B W L is simplified here by using mean values. The net short wave flux entering the top of the atmosphere and that absorbed by the surface of the earth are then expressed, for each of the boxes: STL =

S'(1

STO =

S'(1

0tAL),

-

-

~tAO),

SaL = 0 . 5 0 S ' ( 1

-

SBo = 0.50S'(1

- ao)

aL) ,

- So,

where aALS', aAoS' represent the sum total of all reflected short wave radiation by processes in the atmosphere over land and ocean, respectively.

The latent heat flux to the atmosphere (w m - 2 ) from the surface of the earth is, for land or ocean:

O _ L H = C d L ¢ [ R ( T , ) - r R ( T s ) ] = F1(T,, Ts, r ) , (3.7) where c a is a drag coefficient, L c the heat of vaporization, R the saturation mixing ratio, and r the relative humidity at the top of the atmospheric b o u n d a r y layer. A fraction 0 < % < 1 of one half of the total water vapor flux to the atmosphere from the ocean in the low latitude box is transported in the atmosphere to the n o r t h e r n high latitude box before its latent heat is released; that is, we assume equal a m o u n t s of evaporation occur in each hemisphere of the low latitude box. Of the water vapor transported into the high latitude box, a fraction 0 < ~n < 1 is released in the atmosphere over the

COUPLED OCEAN-ATMOSPHEREBOX MODEL OF THE ATLANTIC

203

ocean in the high latitude box, the remainder, 1 - fn, over the land in box 1. After applying similar arguments to the southern hemisphere, the heating of the atmosphere over ocean by latent heat flux in boxes 2, 1 and 4, respectively, are:

tions about appropriate mean temperatures yield:

QLOH,2 =(1

'"

'_~^o

2 ] QLH,2,

2-

o

^o

en¢

o

^o

es

0L. =

+

QBB = --J~0 +/31T*, L T = --LTa + LT2T,

(3.9) +

LT3Ts,

L B = - L m + LB2T~,

Ao2 ~o

QLH,1 = QLH,1 + T~° AL-----~~LH.2, Ao2 ^o

QLH.4 = QLH,4 + TfS--~L4 QLH.2, where `4oj, A L j a r e the areas of ocean and land in box j. The latent flux heating of the atmosphere over land in the low latitude box is taken to be: L ^L QLH,2 = QLH,2,

(3.10)

where the coefficients are presented in Table 2. Further, since QaL = 0, the land surface temperatures 7", L can be expressed in terms of Ts and hence eliminated from eqn. (3.8) over the land. Equations (3.2)-(3.5) are then substituted into the energy conservation eqn. (3.1) for each atmospheric box. The resulting equations are then in terms of only T, o and Ts for each of the boxes. Solving these equations for Tsl, Ts2 and Ts4, it is found that:

while for box 1 and 4: Tsl = A 1 + B1T, ol + CiT, o2 Ao2 ~ o Q L . , , = QLH,, + -~ (1 -- ~,) ~ ~LH,2, L

^ L

(3.11a)

f'n

Ts2 = ,42 + B2T,02 + Ts4 =

L ^L Cs ~ ~ ,402 ^ O QLH.4 = QLH,4 q- -~-(1 - s , A L 1 QLH.2-

The water vapor flux from box 2 to 1, men, is in terms of the latent heat flux from the low-latitude ocean surface to the atmosphere,

~n Ao2 ^o

men = -~- #0Lc QLH,2"

(3.8a)

It is assumed that the fraction (1 - f~)men falling on land in box 1 is returned to the ocean in the same box. The hydrological cycle is completed in the ocean with the return flux men from box 1 to box 2. In a similar manner water vapor transport in the southern hemisphere is found to be % Ao2 ^o r u e s - 2 PoLC QLH.2"

(3.8b)

The next step is to determine each of the flux terms on the right hand side of eqns. (3.2)-(3.5) by its approximation above. These approximations depend on the solar constant, S¢; the surface ocean and land temperatures, T. o, T.L; the surface air temperature, Ts; the relative humidity in the atmospheric boundary layer, for the surface boxes and ~ and c for each hemisphere. Equations (3.6), (3.7) and those for L T and L B are nonlinear in temperature. Linear approxima-

A 4 +

C2T,01

B4T, o4 + C 4 T , o2

(3.11b)

(3.11c)

where T,o 1, T,m, T , o 4 a r e the ocean surface temperatures for boxes 1, 2 and 4, respectively. The above coefficients are expressed in terms of the coefficients appearing in the simplified representations of the individual fluxes and are evaluated in Table 1 for % - - G = 0.2. For the remaining discussion ~, = ~s = 0.5. The heat flux into the ocean, QBo, is also expressible in terms of the surface air and ocean temperatures using the above representations. With eqns. (3.11), the surface heat flux can then be expressed solely in terms of the ocean surface temperatures. In a similar fashion, men and mes can be expressed in terms of the surface ocean temperatures using eqns. (3.8), (3.9) and (3.11). The surface ocean temperatures can be expressed as the sum of the corresponding box mean temperature plus an estimated positive correction taken to be:

~T, = 3 . 5 ° C , 3 T 2 = 7.8° C, 3T4 = 3.5°C. With these corrections ql, q2 and q4 may be expressed in terms of only the mean temperatures

204

G.E. BIRCHFIELD ET AL.

of the surface ocean boxes and % and %, the hydrological parameters, men and rues can be similarly treated.

tures are: T, = T4 = (X + a3R2,)dl/D + (X + a3R,2)62/D r2 = ( x +

+

(x +

(4.1)

Coupled ocean-atmosphere model symmetric solutions

T3 = [)t + (k3, + rn)R2, + (k23 - re)R22 ] 6 , / D + [X + (k23 - m)R,l + (ks, + m)R,2 ] 62/0,

The coupled o c e a n - a t m o s p h e r e model is heated by solar radiation and cooled by long wave flux to space from the atmosphere and ocean surface. Because of net loss of energy at the top of the atmosphere in high latitudes and net gain in low latitudes, the motions of the atmosphere generated by this differential heating effect a transfer of heat from lower to higher latitudes. The latitudinal transport in the atmosphere is accomplished by eddy mixing and by the latent heat transport associated with the hydrological cycle and in the ocean by a combination of eddy mixing and transport by the thermohaline circulation. The fundamental parameters in the model are the solar constant S¢, % and %, which control the magnitude of the hydrological cycle in each hemisphere, /1. and /G, which control the magnitude of the thermohaline circulation in each hemisphere and the eddy mixing coefficients in atmosphere and ocean. In this paper we consider a simplification of the model: we look only for symmetric circulations about the equator. This is done by setting the southern hemisphere parameters equal to their northern hemisphere value, that is, % = % = e , /.t s = / . t n = p,, k34 = k31, etc. It follows that mo~ = mon =

m 0.

From BI it is known that # must be of the order of 10 l° m3s -1 in order to achieve thermohaline circulations of the order of 10 Sv. For /~ > 3.9 10 l° m3s - ~ the model is absolutely gravitationally stable in the same manner as in BI. Estimation of the hydrological parameter c appropriate for the modern climate is made using data from Baumgartner and Reichel (1975), Oort and Vonder Haar (1976) and Lorenz (1967). Estimates by more than one procedure lie between 0.1 and 0.25. The steady state form of eqns. (2.1) can be solved for the temperatures in terms of m 0 and the surface heat flux parameters. The tempera-

where rn = rno/2 and 61 = - - ( S o l

-~- R i o ) ;

62 = --(502

-~ R2o);

~k = O~1 + m 2 ,

D = d I + d:m 2, d 1 = a i d 2 + Or3( R 1 2 R 2 2 - R I 1 R 2 1 ) ; d 2 = ( - R n + R12 - R21 + R22 ) • The salinities are expressed from eqns. (2.2) in terms of m 0 and me:

S1 =34=Sm S2=Sm+--

a I + m2 meS° (

ot 1 + m

2

k31+m-

v3)

(4.2)

0-

meSom [ S 3 = S m - oq + rn2[1 - - ~ ) " Once the temperatures are found, rn e is known and the salinities may be expressed in terms of m 0. Note that while the mean salinity is arbitrary, the mean temperature of the ocean is not. As in BI, eqn. (2.6) is now used with the above to obtain a fifth degree polynomial for m: 2d2 ms + 2 ( d z x 1 + da)m 3

+ [x2(hod 2 + h 2 ) - x , l m 2 + 2dxalm + ( - x l a I +x2hod I + X2hl) = 0 , where x 1 = ~ a a a [ 6 1 ( R 2 2 - R21 ) + C2(Rll - gl2)] ,

x2 = ~Ba3So, h o = HHo; h 2 =

H(6, + 6:)(H,

+ H2),

h 1 = H{ H, [ 6 , ( a , + a3R21 ) + C2(a I "at- 013]~,2)] + H 2 1 6 , ( a , + a 3 R z z ) + C 2 ( a , + a3R,,)] }, H-

eAo2

/-COo "

COUPLED

OCEAN-ATMOSPHERE

1 root

BOX MODEL

OF THE

ATLANTIC

S

205

lower branch is stable. These results parallel those found in BI, where the bifurcation parameter was m e itself instead of c. The significance of the bimodal region in the present model for time

! I

me (m3/s)

!

!

1.60 105

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It is found that, as in BI, there exist at most three real roots; that is, the simplified coupled o c e a n atmosphere model with symmetric circulation about the equator admits either one or three possible equilibrium states for fixed /,, ~ and So. In the following section we discuss physical implications of the model solution as a function of the hydrological parameter c for fixed g. The dynamic stability of the solution determined for the magnitude of the equilibrium thermohaline circulation, m 0, shown in Fig. 3b, can be described as a function of c. For small c, a single and stable equilibrium exists; this is the upper branch in Fig. 3b. For large, ¢, a single and stable equilibrium exists, i.e., the lower branch. For intermediate c delineated by vertical dashed lines in Fig. 3b, three equilibria exist. For smaller ¢ in this region, the upper and lower branches are stable while the middle branch is unstable. For large ¢ in this region of multiple solutions only the

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G.E. B I R C H F I E L D ET AL.

evolution of the ocean circulation is discussed in the next section. Results and Discussion The solution is illustrated for variable ~ and fixed /~ = 4.0- 10 ~° m3s- 1. In Figure 2 this corresponds to a vertical line on the/~-c plane near the vertex of the three root region. Figure 3 illustrates the variation of latitudinal water vapor flux, m¢, thermohaline circulation, m 0 and ql = q2 = q4, the poleward laltitudinal heat transport by the ocean in each hemisphere. Three regions can be distinguished. For small c there is relatively low water vapor transport, a strong thermohaline circulation and high latitudinal ocean heat transport; for relatively large ~ there is high water vapor transport, low thermohaline circulation and low poleward ocean heat transported. For intermediate values of c a bimodal solution exists. For increasing c passing through this region there is a jump from the strong circulation to the weak circulation regime. The strong thermohaline circulation regime appears less stable than the weak regime; that is, once the thermohaline circulation has been shut down, with a fixed ~, there is no process in the model to bring it back to the other regime. The role of surface salinity in this is discussed below. In Figure 4 temperature and salinity variations with c for each box are shown for circulations symmetric about the equator, that is, with 7"1 = T4

and S 1 = $4. Low latitude temperature changes little with the hydrological parameter. The relatively small e regime, with strong thermohaline circulation, has relatively warmer high latitude surface ocean than does the weak thermohaline circulation regime; that is, consistent with the reduced poleward heat flux by the ocean, in the weak circulation regime, high latitudes are colder. The deep ocean temperature change with increasing e is striking. In the strong thermohaline regime, the deep ocean is cold as a consequence of the production of a large volume of deep water in the high latitude surface ocean; in the low thermohaline regime, however, this flux of cold water is shut down. Since the circulation is symmetric about the equator, there is no source of cold dense water and the deep ocean warms to a temperature intermediate to that of the high and low latitude surface ocean. High latitude surface salinity is relatively high for the small c strong thermohaline circulation regime and the deep ocean is relatively fresh. With a transition to the weak thermohaline circulation regime, surface salinity drops due to the increased water vapor transport. Again the deep ocean salinity takes on a value intermediate to that of low and high latitude surface ocean. For a strong circulation regime, the thermohaline transport dominates in the salinity balance in the high latitude surface ocean, whereas in the weak circulation regime, transport of salt by the thermohaline circulation is overshadowed by eddy mixing fluxes.

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COUPLED

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207

BOX MODEL OF THE ATLANTIC

The above response is consistent with data for the modern ocean and with several, but not all features postulated from proxy paleoclimatic data for the Younger Dryas event and for the last glacial maximum if the strong thermohaline circulation regime (small c) is associated with the modem ocean and the weak thermohaline circulation regime (larger c) is associated with the cold climates. The conclusions of Boyle and Keigwin (1987) and Duplessy et al. (1988), that production of N A D W was greatly reduced in the Younger Dryas, is support for the weak circulation regime in the model. However, they also predict a colder deep ocean due to increased transport of Antarctic Bottom Water into the North Atlantic. This clearly is not consistent with the model. The model, however, has only symmetric circulations about the equator; turning off deep water production occurs in both hemispheres in the model in the weak circulation regime. More general experiments with the two hemisphere model are presently under way.

Acknowledgements This research was made possible through support of the Climate Dynamics Section of the National Science Foundation under Grant ATM8705436. Valuable conversations were had with W. Broecker, K. Bryan and with J. Marotzke. Appreciation is extended to Cheril Cherverton for editing the text.

Appendix We wish to incorporate sub-box scale physics of deep water production into the model in the simplest way. Since individual boxes are not well mixed, water transported from a surface box to the deep ocean box may be warmer or colder than the mean temperature of the surface box. If the deep ocean box is colder than the mean temperature of the large surface box, e.g. by choice of initial conditions, and the temperature of the transported water is estimated as the average of the mean temperatures of the surface box and deep ocean box, the advected water will be colder than the surface box. We use this advection scheme

as a crude way to incorporate sub-box scale processes which in the prototype produce isolated extrema (in the mean) of heat and salinity in the deep ocean. Further support for this scheme makes use of the simple plume model of Morton et al. (1956). This consists of an axially symmetric plume with Boussinesq approximation and an arbitrary vertical stratification, with a radial Gaussian distribution of buoyancy flux, velocity, etc. The equations reduce to a set of ordinary differential equations which can be solved numerically. In particular, if the temperature is also Gaussian in r, the heat transport, say for the high latitude surface box to the deep ocean, can be reduced to transport = heat capacity × mass transport × ( T e + Tp)/2, where Te is the temperature of the environment, Tp that at the plume center. Since the buoyancy is produced at the surface of the surface box, we set T~ equal to the mean temperature of the surface box. In general, Tp << To. Taking as a crude estimate of Tp the mean temperature of the deep ocean box, the temperature of the transported water is then the mean of the surface and deep ocean boxes. While this argument applies specifically to production of deep water and not necessarily to return flow, it lends support to the the qualitative argument above. It also moves the domain of discussion from the question of difference equations approximation to differential equations to questions of parameterization of sub-box scale processes. It is also possible from the plume model to give a scale argument for the proportionality of the magnitude of the thermohaline circulation to the surface density gradient, employed in eqn. (2.6).

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