Climate Dynamics (1994) 9:391-409
l:limai¢ Uynumia © Springer-Verlag 1994
Parameter sensitivity study of the saline circulation Rui Xin Huang 1, Ru Ling Chou 2 1 Department of Physical Oceanography,Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA 2 Lamont-DohertyEarth Observatory, Palisades, NY 10964, USA Received: 18 September 1992/Accepted: 7 June 1993
Abstract. Saline circulation forced by fresh water alone is studied for a broad region of parameter space by varying the amplitude and profile of evaporation minus precipitation, the vertical and horizontal mixing of salt, vertical and horizontal dissipation of momentum, and the horizontal resolution. The model is a modified Bryan-Cox model with a freshwater flux as the natural boundary condition for the salinity balance. For a model forced by a linear freshwater flux profile, as the amplitude of freshwater flux is increased from 0.01 m year-1 to 1 m year-1 with other parameters fixed, the system evolves from a steady state of no oscillation to a state of periodic oscillation whose frequency increases almost linearly with the amplitude of freshwater flux. When the freswater flux is fixed and the vertical mixing coefficient is increased from 0.5 to 2.5 cm2s-% the system evolves from a steady state to a state o f singleperiod oscillation, chaotic, a single period, and finally to a chaotic state when the vertical mixing coefficient is larger than 2 cm2s -l. One set of numerical experiments forced by a cosine shape of freshwater flux clearly reveals the transition from a state of single period oscillation to period doubling, period quadrupling, and a state of chaotic oscillation. Simple scaling analysis and numerical experiments indicate that the strength of the meridional overturning increases with the square-root of the vertical mixing and the IA power of the freswater flux. The mean sea surface salinity (deviation from 35 psu) increases with the 3/~ power of the freshwater flux and decreases with the 1A power of the vertical salt mixing.
1 Introduction Freshwater flux is an important component of the buoyancy forces driving the thermohaline circulation, * Contribution No. 8191 from the Woods Hole Oceanographic Institution Correspondence to: RX Huang
and it is also the most poorly understood component of the circulation. Most existing general circulation models use a relaxation boundary condition for the salinity balance. A parameter study based on the relaxation conditions has been carried out by F. Bryan (1987). The so-called mixed boundary conditions, i.e. a relaxation condition for the temperature and a salt flux condition for the salinity, were introduced by Bryan (1986) in a study of the halocline catastrophe. During the past few years there have been some efforts devoted to numerical experiments based on the mixed boundary conditions, such as those by Marotzke and Willebrand (1991), Weaver and Sarachik (1991). These studies have been carried out for thermohaline circulation forced by wind-stress, thermal forcing and virtual salt flux, with special attention to the role of different patterns of virtual salt flux. However, the dynamic character unique to freshwater flux through the ocean-atmosphere coupled system has not been fully realized. First, the relaxation conditions and the so-called mixed conditions are not suitable for climate study or oceanic forecasting. The relaxation condition for salinity requires a reference salinity distribution on the upper surface. Although the climatological mean sea surface salinity can be used. for simulating the present climate, this reference state is unknown for studies of the past/future climate or oceanic forecasting. In addition, the relaxation coefficient is not well defined. Thus, the virtual salt flux diagnosed from models based on relaxation conditions involves some artifacts. For example, in many numerical experiments based on such conditions, negative salt flux (or equivalent freshwater flux) is required for some parts of the Gulf Stream where evaporation should prevail. Second, a virtual salt flux across the air-sea interface and going through the oceans and the atmosphere is required for these two types of boundary conditions. It is clear that such a virtual salt flux is a misinterpretation of the hydrological cycle through the atmosphere and the oceans. For example, in all existing models there is a huge poleward salt flux, which is at least ten times larger than the real meridional salt fluxes due to
392 horizontal advection or diffusion. It seems clear that such an unphysical flux should be avoided. In addition, this approach gives rise to a systematic error in the salinity balance. This error is due to the fact that there is a positive correlation between the evaporation and high salinity, so the global integration of (E-P)S, is positive, although the global integration of E - P is zero for the present climate. In order to avoid a continuous buildup of salinity, the virtual salt flux has to be modified to ( E - P ) ~ , where ~ is the mean sea surfae salinity averaged over the domain of the model. This formulation would introduce a systematic bias for global models. For example, salinity can be lower than 33 psu in the Pacific, and can be higher than 37 psu in the Atlantic. Accordingly, the model based on the virtual salt flux would overestimate the salt flux in the Pacific, but underestimate it in the Atlantic. Similarly, such a model would have severe problems near river mouths and ice edges where salinity is extremely low. Recently, a natural boundary condition of the salinity balance in a general circulation model has been introduced (Huang 1993b). The freshwater flux comes into play through the continuity equation and a nosalt-flux condition applies to the salt balance. Under the rigid-lid approximation, this is reduced to specification of a vertical velocity w0 = E - P on the top of the ocean, z =0. The rigid-lid approximation is valid for motions of spatial scales smaller than planetary scale and of temporal scales longer than decadal. Numerical experiments for the thermohaline circulation driven by evaporation minus precipitation reveal very interesting structure of the saline circulation driven by precipitation and evaporation. In the new model, the water cycle through the oceanatmosphere is reproduced correctly, so the so-called barotropic Goldsbrough-Stommel gyres (Goldsbrough 1933; Stommel 1957) are reproduced properly. Most importantly, the meridional freshwater flux and salt flux are simulated accurately. It was also pointed out that at an equilibrium state the total meridional salt slux should be zero. Therefore, the meridional advective salt flux should be balanced by the meridional diffusive salt flux. Since the meridional advective salt flux due to the equatorward motion of the fresh water is missing in all existing traditional models, a reference flux correction term must be subtracted before the meridional salt flux is used for any physical interpretation (Huang 1993b). The sensitivity of the thermohaline circulation to the salt flux boundary condition has been studied in conjunction with the wind-driven and thermally driven circulation in previous studies, such as Weaver and Sarachik (1991), Weaver et al. (1991, 1993). Because the haline forcing has a quite different physical nature compared to other forcings, it seems important to study the sensitivity of the haline circulation without the complication due to wind stress or heat flux. Broecker et al. (1990) proposed a conceptional model for the salt oscillation in the glacial Atlantic. Since the salt oscillation in the oceans involves many competing forcings, such as heat flux, wind stress, and
Huang and Chou: Parameter sensitivity study of saline circulation freshwater flux due to evaporation minus precipitation and ice melting, it is desirable to study the salt oscillation in the simplest form, i.e. a salt oscillator forced by evaporation minus precipitation. One of the remarkable features of the haline circulation is the strong baroclinic circulation driven by the small amount of freshwater flux going through the airsea interface. Given such big amplification factors, it is easy to understand that the model is very sensitive to any changes in the forcings and the parameters. In this study, the sensitivity of an oceanic general cirulation model based on the natural boundary conditions for salinity to changes in closure parameters and surface forcing is determined by examining a series of solutions that span a broad range of parameter space. The model used in this study, described in section 2, is a modification of the GFDL (Geophysical Fluid Dynamics Lab) model, Cox (1984). Under the rigid-lid approximation the upper boundary conditions for the salinity are: a vertical velocity condition on the upper surface representing the freshwater flux across the air-sea interface, and a no-salt-flux condition for the salinity balance. These boundary conditions are natural and more accurate than the old boundary conditions used in existing models, and these boundary conditions have been tested in a series of numerical experiments. Because of the fundamental difference in the nature of the upper boundary conditions for temperature and salinity, it is desirable to isolate the haline circulation component from the thermally forced circulation. The philosophy of this study is to use the simplest possible geometric setting, a square basin with no topography and no bottom drag. Such a model can be integrated to quasi-equilibrium rather quickly and is therefore suitable for a parameter study. The horizontal and vertical resolution and other parameters used in this study are typical of those used in world ocean circulation and coupled ocean-atmosphere models. A scaling analysis for the thermohaline circulation is presented in Section 3, where the difference between the relaxation conditions and the freshwater flux condition is discussed. Seven sets of numerical experiments have been carried out to examine the sensitivity of the model to the amplitude of the freshwater forcing (Section 4), the vertical salt mixing coefficient (Section 5), the vertical momentum dissipation (Section 6), the horizontal salt mixing coefficient (Section 7), the horizontal momentum dissipation coefficient (Section 8), the horizontal resolution (Section 9), and the different evaporation minus precipitation profiles (Section 10). Finally, we present concluding in Section 11. 2 Description of the model This section gives a concise description of the model, especially the upper boundary conditions used for salinity balance. For a comprehensive discussion, the reader is referred to Bryan (1969) and Hnang (1993b). The model is based on primitive equations in spherical coordinates (A, ~b, z), where A is longitude, ~b is la-
Huang and Chou: Parameter sensitivity study of saline circulation titude, and z is heigl~t relative to the earth's mean radius. The essential difference from the traditional G F D L model is that freshwater flux comes into play through upper boundary conditions for the vertical velocity w0 = e - p ,
a t z = 0,
(1)
there e - p is the evaporation minus precipitation rate, in cm/s. The upper boundary condition for salinity is a no-salt-flux condition Sy=O.
(2)
Paralleling Bryan (1969), the vertically integrated mass flux is represented in terms of a barotropic streamfunction 91 and a barotropic potential function q,
393 ure will be used in numerical experiments discussed at the end of this paper,) The corresponding depth-integrated barotropic velocity consists of two parts. The first part comes from the barotropic streamfunction (due to the freshwater pumping at the upper surface), and it gives rise to the Goldsbrough-Stommel gyres. The second part comes from the barotropic potential function q5 (the source-sink-driven flow), which is in simple analytic form for the present case
aWo
cos~4(1-~n).
Hv-
Notice that H v is negative everywhere, and it reaches a minimum at 'h = 4,,/2 Humi n __
0
a l poudz= - qG + m qO,,
(3)
H 0
a j povdz =m ~z,+ @~,
(4)
H
where a is the earth's radius, (u, v) is the horizontal velocity, m = sec ~b. By cross-differentiating and adding (3, 4), we obtain
4)~ + ~
@4,
=
-po(e-p).
(5)
This equation should be solved with the no flux bound0q~ ary condition along the edge of the basin, i.e. - - = 0
On
along the lateral boundary. A simple, "linear" profile of evaporation minus precipitation is chosen for a basin confined between the equator q5= 0 and a northern boundary & = &n w0 -
COS ~
1-
(6)
where Wo > 0 is the amplitude of freshwater flux. This profile is labeled A in Fig. 1. (Other profiles in this fig-
(7)
a¢n W0<0. 2 cos (~bJ2)
(8)
The model is a 60°x60 ° square basin, extending from the equator to 60°N, with a constant depth of 5700 m. In the following discussion the model used in most numerical experiments is a 4°x 4 ° low-resolution model with 15 levels vertically, forced by freshwater flux alone. The temperature is held constant at 12.5 ° C. Since the horizontal resolution is rather crude, the nonlinear terms in the horizontal m o m e n t u m equations have been neglected. A pivotal case, discussed in detail by Huang (1993b), has been run with the following parameters: the freshwater flux amplitude W0 = 1 m year-1, the horizontal eddy dissipation (for the velocity) A , , = 1 0 9 c m 2 s -1, the horizontal eddy diffusivity (for the salinity) A, = 107 cmZs-2, both the vertical dissipation and diffusion k m = k s = l c m 2 s -1, the time steps are 2.5 days for the salinity and 3 hours for the baroclinic velocity and barotropic streamfunction. The standard G F D L convective adjustment loop is iterated five times for each time step to make sure that the water column is very close to gravitationally stable.
3 Scaling analysis a) Thermohaline circulation under relaxation conditions
I
1.0
__
I
I
[
[
In the wind-driven circulation theory, the Ekman pumping velocity plays a central role in determining the circulation. In the thermal-driven circulation theory, the north-south temperature difference controls the strength of the circulation. Under a relaxation condition, the north-south reference temperature difference can be used as a basic scale. The continuity gives
0.8
B
0.6 0.4-
E o
if) I
g
0.2-
0,0
C. -o.2 -0.4
-
UD=WL,
kJ
--0.6
(9)
and the thermal wind relation gives
--0.8 -1,0
a p _ fo U L 1 I0
r
20
310
I 40
I 50
60
Lofifude Fig, 1, E v a p o r a t i o n
study
minus precipitation
profiles
u s e d in this
Po
(10)
gD
Assuming both W and A p are specified on the upper surface, the advective vertical scale of the motion is obtained by eliminating U from (9, 10):
Hnang and Chou: Parameter sensitivity study of saline circulation
394
(f°WL2) 1/2 6~ = \gAp/Po] .
(11)
On the other hand, if the horizontal advection is negligible, the diffusive vertical scale is determined by k 8a = - - . W
(12)
For typical cases, f o = 1 0 - 4 s -1, W~=10 -4 cms -I, L = 6 x l 0 s cm, gzXp/po=l cms -2, and k = l cm2s -1, SO, 8~=600 m,
6a= 100 m.
(13)
However, horizontal advection is not negligible in the oceans; thus, one should not simply assume that W in (12) is equal to We. In fact, W should be determined internally, i.e. one has to combine (12) with (9, 10) to eliminate W, and an intermediate vertical scale is obtained
( f ° k L a / 1/3= t~2/3~1d/3~-" 340m. 8 = \gAplpo) ~
(14)
Accordingly, the meridional velocity is
(g2 Ap2k~l'3 U=
~ \~poL
,
]
(15)
and the strength of the meridional overturning is (g~*~L411/~
M,~=UDL=\.
fopo
J
'
(16)
tude of evaporation minus precipitation.) The salinity conservation gives
(UDL + L 2 E) ( S - AS) = UDLS.
(19)
Neglecting a small term L2EAS, one obtains UD AS = LES.
(20)
From (9, 12, 18, 20) one obtains the following scales for the haline circulation. The scale depth of the halocline is
(f°L2k211'4= 8X2 8~ 2, g = \ g3xEg] where 8A = ~ gfl~ ]
(21)
is the advective vertical scale
based upon the vertical velocity on the upper surface imposed by evaporation minus precipitation, and k. ~D = ~ lS the diffusive vertical scale. For fo = 10-4s--1, L = 6 x 108 cm. k s = l cm2s-1, E = 3 x 10 .6 c m s - 1 , and S-=35psu, the advective and diffusive scales are 6A~23.4 m and 8D~3,333 m. Thus, these scales are widely separated. They are either too small or too big and therefore unsuitable for describing the haline circulation. However, the geometric mean of these scales gives an intermediate scale 8, ~ 279 m, which is a fair description of the shallow halocline characteristic of the haline circulation. The north-south salinity difference is
(joE' LT4 .
Notice that the thermal forcing itself generates an intrinsic vertical velocity
AS = \- g--~7~k,2 /
k W, = - . 8
In the following analysis we will use the mean surface salinity & (deviation from the basin mean salinity of 35 psu). It is easy to show that ~ has the same scale as kS. The scales of the meridional velocity and the strength of the meridional overturning can be obtained by substituting Ap=pofi, AS into (15) and (16)
(17)
For the parameters used above We=3 x 10-4 cm s - l , which is the same order of the Ekman pumping imposed on the upper surface. The scaling analysis above is well known and can be found in many textbooks and papers, e.g. Bryan (1987).
b) Haline circulation forced by fresh water
gfi~AS-
foUL D '
u = \ (g sE £ / l '2 , Mm = UDL = \
Compared with the previous case, the major difference is that the north-south surface salinity difference AS (equivalent to Ap in the previous case) is unknown. The thermal wind relation becomes
(18)
where fis-~0.771x10 -3 is the contraction coefficient for the salinity. The basic salinity scale AS is determined by a global salt balance. Notice that the total mass flux above the halocline is UDL, this is also the strength of the meridional overturning. However, the southward flow above the halocline must carry slightly more water than the return flow. The difference is L2E, the total amount of precipitation (E is the ampli-
(22)
(23) ~
,]
(24)
Another important diagnostic quantity is the meridional salt flux. In many previous models based on virtual salt flux, there is always a large poleward salt flux. As discussed by Huang (1993b), such a large poleward salt flux is an artifact of models based on virtual salt flux. With the new formulation of freshwater boundary condition on the upper surface, the model is capable of keeping track of the meridional salt flux exactly. The meridional salt flux consists of two terms s s F s =Sad+Say
(25)
where FSa = jjvSdxdz is the salt flux due to the mer-
OS oy
idional advection, and FS; = - JJA, XL-_.dxdz is the salt
Huang and Chou: Parameter sensitivity study of saline circulation
395 0.0
flux due to diffusion, which is a parameterization for eddies. The corresponding scale for the meridional diffusive salt flux is
k
P
I
I
1.0/ IY"
FSs = As -kS z -6s =
A s t1f ,°2E/ \|
(26)
When the model approaches a quasi-equilibrium, the total meridional salt flux should vanish; however, these two components of salt flux are non-zero and salt advection is balanced by salt diffusion. From the scaling analysis the basic scales depend on the 1½-power of the vertical mixing coefficient, except the meridional salt flux which is independent of the vertical mixing. It is important to notice that our scaling analysis is consistent with the classical scaling analysis for thermohaline circulation. For example, (15) and (16) are still valid if we use the Zip from the numerical model's output. However, AS is not specified as a boundary condition for the present model. In fact, as shown above, AS is inversely proportional to the square root of ks. As a result, the strength of the circulation is proportional to the square root of vertical mixing. It has, thus, a slightly weaker dependence on vertical mixing, compared with the %-power law for the case of relaxation conditions discussed by Bryan (1987). More detail about the % power law will be provided in Section 5. The haline circulation generates an internal vertical velocity Ws = --~k2 3.6 x 10 -5 cm s -1,
(27)
6s which is about ten times larger than the vertical velocity imposed on the upper surface by the evaporation minus precipitation. Thus, haline forcing induces an internal vertical velocity which is comparable with the Ekman pumping velocity, and this internal vertical velocity is responsible for the strong baroclinic structure of the haline circulation discussed by Huang (1993b). In addition, the strength of the meridional circulation depends on the 1A power of the precipitation amplitude and the basin mean salinity. If there were no salt difference between the rain and the sea water (either no salt in the ocean or the rain has the same salinity as the sea water), our scale analysis would not apply, and the barotropic Goldsbrough-Stommel gyres would be the only circulation driven by evaporation and precipitation, with no baroclinic return flow at all. In such a fictitious ocean the profile of the vertical velocity would be a linear one, decreasing from the maximum velocity imposed on the upper surface to zero at the bottom. The fact that the rain has no salt and sea water is salty changes the picture entirely. Within the northern basin, salt is moved upward and mixed into the southward moving water in the upper ocean; within the southern basin, salinity increases due to evaporation and the dense water sinks forming deep water. It is important to notice that there is not just one type of deep water being formed. In fact, sinking takes place within
E
P',
2.0 -
/I 3.o-
/
/ ,/'
I J
;/ //
~
~' r 0
-10
u t
,~ /
I
5,0
,
,;
/
4.0 -
,,)/,~
d
I
C3
/,
i 10
i 20
w (0.00001 0,0
r
i
p
i 30
i
,~.j~...
i
1.0
, ,
¢'
~
4.o
L
",.%
b
5_
40
cm/s)
/,' a),~.
/(
d)
/
I [,
/"
/
]
//
/
~',,, -1200-1000
-800
-600
-400
-200
0
200
400
600
w (0.00001 c m / s )
Fig. 2a, b. Vertical velocity profiles of a haline circulation, the location of the profile is in non-dimensional (x, y)-coordinates, a: in the interior, a (0.567, 0.233); b (0.300, 0.233); c (0.833, 0.567); d (0.300, 0.567); b: near the southern boundary, a (0,967, 0.033); b (0.700, 0.033);c (0.967, 0.100); d (0.700, 0.100)
the southeastern basin and water parcels with different densities sink to different depths. As a result, the total amount of upwelling increases upward and reaches the maximum at the base of the halocline, so the vertical velocity also reaches the maximum, as seen in Fig. 2a. Vertical velocity is not uniform basin-wide and the scaling analysis applies to the ocean interior only. Along the southern and eastern boundaries of the basin the vertical velocity is much stronger than that calculated from the scaling analysis, see Fig. 2b. The nonuniformity of the vertical velocity is consistent with the analytical study based on a two-level model (Huang 1993a).
4 D e p e n d e n c e on the amplitude of freshwater flux
In the first set of numerical experiments, all the parameters are chosen the same as those of the pivotal case discussed by Huang (1993b), except the freshwater flux. A summary of the parameters used in these numerical experiments is given in Table 1. Since a simple "linear" profile, e - p =
Wo (l__~_O/
COS ~0 \
~n /
is used, the
freshwater flux is determined by a single parameter
Huang and Chou: Parameter sensitivity study of saline circulation
396
Table 1. Sensitivity study on the amplitude of the freshwater flux Exp.
Wo
Period
Merid. cell
W-1 W-2 W-3 W-4 W-5 W-6 W-7 W-8 W-9 W-10 W-ll W-12 W-13 W-14
0.01 0.05 0.1 0.125 0.15 0.2 0.3 0.4 0.5 0.75 1.0 1.1 1.5 2.0
Steady Steady Steady Steady 127.7 104.6 67.0 45.6 36.6 25.8 18.87 17.80 Chaotic Chaotic
-
16.74 59.26 79.92 83.48 84.53 69.33 72.91 79.09 85.84 98.63 108.36 113.45 124.79 130.99
-
0.026 0.110 0.196 0.235 0.273 0.345 0.479 0.602 0.717 0.982 1.235 1.327 1.661 2.055
A m = 1 0 9 c m 2 s -1, A , = 1 0 7 c m 2 s -1, k m = k , = l c m 2 s - 1 ; W0 in m y e a r - Z ; p e r i o d in years; meridional cell flux in S v (a negative sign indicates sinking along the equator); and ~ in psu. Each exp e r i m e n t is run for 1370 years, except W-11, which is run for 1712 years. The time mean is taken for one period or between 1 0 0 0 1370 y e a r s ( f o r the chaotic cases)
W0, which is varied from 0.01m year -~ to 2 m year - a. For the first four cases with freshwater amplitude less or equal to 0.125 m year-~, the circulation reaches a steady state after spin-up from an initial state of a homogeneous ocean. When the freshwater amplitude is between 0.15 and 1.1 m year-~, the model reaches 0.06
I
I
I
I
103
I
"-d" 0.05 o
10 2 "~
0.04
an equilibrium state of periodic oscillation. The most interesting phenomenon is that the frequency of oscillation is almost a linear function of the freshwater amplitude, Fig. 3a. Apparently, the period of the salt oscillation depends on the time scale for water to move across the basin. Assuming a speed of 1 cms -a, it would take 20 years for water parcels to move across a meridional distance of 6000 kin. Thus, the meridional advective time scale is consistent with the decadal periodic or chaotic oscillations observed in many numerical experiments. Thermohaline oscillation on decadal or century time scales have been discussed in many studies. One of the basic mechanism is the so called Malkus-Howard loop oscillator. The main idea is that in a weakly forced and dissipated loop, perturbation would circulate repeatedly. This idea has been applied to the thermohaline circulation by Welander (1986) and other investigators, such as Weaver et al. (1991) and Winton and Sarachik (1993). To explore the basic physics involved in the so-called loop oscillator, we can use an idealized model which consists of a loop of salty water forced by a sinusoidal pattern of evaporation and precipitation, Esin 4,. The salt conservation law is s, + [ ( . + E c o s 4,)
= o,
(28)
where u=k~AS is the mean circulation velocity (k is the frictional parameter), Ecos4, is the accumulated freshwater flux. The steady state solution of this equation is
{~--1/4
101
......
,,,I
, , , ..... I
, , ,,,,,,I
3/4
/
I~ 1oO ~
//
j
003 0.02
"r-
10-t
10 ~
q)
L~
0.01 C
0,00
r a~a 0.0 0.2
t
0.4
10 °
I
0.6
0.5
1.0
.....
10-2
1.2
.....
10-1
,"i
e I 0 -2
.......
100
, , , , , , i , ,
10 3
i ,H,,I
........
I
i ........ i0-I
i0-2
10 I
. . . . HI
101 _,
. . . . . . . .
I
,
,,
.....
i
E
........ 101
.... 1/4
L
,,
,,ml
1/2.
/-"
Y, lO°
10 2
0.4
i 100
Wo (m/year)
1./2 0.5
t.d
........
Wo (m/year)
Wo (m/year)
0.7
,'i
.
>
0.3
101 0.2 0,1
0.0
-~ .<
O
/
b I
I
I
I
I
I
I
T
0.00.20.40.60.81.01.21.41.61.82.0
Wo (m/year)
I
'c" o)
d 10 0
[
10-2
...... i
10 - I
....
,,'q
10 0
Wo (m/year)
Fig. 3a-f. Dependence on the freshwater amplitude of a the frequency of the oscillation; b the mean kinetic energy, in erg c m -3; e the direct meridional cell; triangles, the time-mean; diamonds, the maximum; and squares, the m i n i m u m ; d the direct zonal cell; triangles, the time-mean; diamonds, the maximum; and squares
10 -1
10 .2
.......
10 I
........ 10-2
i
10-1
'
........
' ','"'i
10 o
101
Wa (m/year) the m i n i m u m ; e the mean sea surface salinity (deviation from 35) in psu; squares, steady solutions; triangles, oscillatory solutions; f the meridional advective salt flux at 28 ° N, i n 106 k g s -1; squares steady solutions; triangles, oscillatory solutions
Huang and Chou: Parameter sensitivitystudy of saline circulation (u + Ecos ~b)S(~b) = constant.
(29)
Since u,>E, a salinity perturbation satisfies a linear equation
S/'+uS'~=O.
(30)
This is a first-order partial differential equation, which describes the salt anomaly advection. Because there is no salt flux across the air-sea interface (the outer surface of the loop model), salt is conserved and the salinity change is due to freshening and concentration. In many traditional models, the exact conservation of salt is treated in terms of the so-called weak relaxation or the equivalent salt flux. Although these approaches can help to illustrate the dynamics, we think that the best way of describing the salt oscillation is to use the natural boundary condition. It is readily seen that the period of the oscillation is L --, where L is the length of the loop. According to the u sclaing, the meridional velocity is proportional to Em. Thus, the salt oscillation would have a period proportional to E -m. However, the results of numerical experiments indicate that the period of oscillation is proportional to E-1. The reason for this discrepancy is not clear. Notice the analysis above is based on a linearization about the basic state, so that the salinity perturbation would not affect the circulation. In the nonlinear case, salt perturbation can interact with the mean circulation. Walin (1985) discussed a positive feedback between the salt perturbation and a thermal-dominated meridional circulation. Walin's argument has been used for explaining decadal variability observed in many numerical studies on the thermohaline circulation. In a purely saline circulation salt perturbation can lead to a negative feedback between the sea surface salinity and the overturning cell. Suppose there is a positive salt anomaly at low latitudes, this would enhance the meridional pressure gradient and the direct meridional cell. As a result slightly more fresh water from high latitudes would flow to low latitudes reducing salinity at low latitudes. Because the system has inertia (a time lag), it overshoots and a negative salinity anomaly develops at low latitudes. (Notice that as this new fresh anomaly appears at the low latitude surface, the previous salt anomaly moves downstream.) The decline in surface salinity at the southern basin gives rise to a smaller meridional salinity difference, which would cause the meridional circulation to slow down. When the meridional circulation is slowed down, salinity at low latitudes starts to increase due to evaporation/precipitation, and the whole cycle repeats again. For freshwater forcing stronger than 1.1 m y e a r - l , the model reaches a chaotic state after a thousand years of spin-up. These two chaotic states have been excluded from the frequency profile in Fig. 2a. In the following analysis, the mean properties of these two chaotic solutions are calculated by time-averaging over last 370 years of calculation.
397 When the amplitude of freshwater flux is below the critical value Woc (about 0.13 m y e a r - l ) , the mean kinetic energy increases rapidly with the increase of freshwater flux. As the salt oscillation appears for slightly larger freshwater flux, the mean kinetic energy increases at a slightly slower rate, and there seems to be a linear relation between the freshwater flux amplitude and the mean kinetic energy averaged over one period. This is consistent with the velocity scaling of E v2" According to (24) the amplification factor of the system is
dMm _ cE-3/4. (31) dE Hence, for small E the freshwater flux amplification factor is extremely large. For example, when W0=0.01 m year -1, the total amount of freshwater flux is about 0.0036 Sv, M,z = 16.74 Sv, so the amplification factor is 5000; when W0=0.1m y e a r - l , M,~ = 80 Sv, which corresponds to an amplification factor of 2000. Given such a large amplification factor, the system must be very sensitive to small changes in the forcing and parameters. One of the most appealing conclusions from this study is that the freshwater flux associated with evaporation and precipitation must be treated as accurately as possible. Although the present technology may not provide such accurate measurements, the vital importance of simulating the haline component of the oceanic circulation accurately will certainly stimulate more careful study in this direction. The strength of the meridional overturning Mm increases as the freshwater flux increases. A simple scaling analysis in the previous section suggested a 1A-power law. The maximum flux of both the meridional cell and the time-mean meridional cell in the numerical experiments seems to fit this law well, except for a case of very small freshwater flux, Fig. 3c. The zonal overturning streamfunction increases when Wo is increased from 0.15m year -1 to l m year-1. According to our numerical experiments, the time-mean meridional cell fits the 1A-power law; however, the maximum strength of the cell does not fit the V4-power law; instead, the ~/i-power law seems fit the data better, Fig. 3d. It is not clear why the maximum zonal circulation responds to the increase of freshwater flux in such a way. In addition, it is not clear why the sensitivity is different for very weak freshwater forcing. Fresh water dilutes the upper ocean; thus, the magnitude of the mean surface salinity ~ (deviation from 35 psu) increases as the amplitude of freshwater flux is increased. According to the simple scaling analysis in the previous section, ~ should increase in proportion to the 3A-power of freshwater flux; this is consistent with the numerical experiments, Fig. 3e. As discussed above, even when the model reaches a quasi-equilibrium there is a non-zero meridional advective salt flux, which should be balanced by an opposite meridional diffusive salt flux, Simple scaling analy-
398
Huang and Chou: Parameter sensitivity study of saline circulation
sis predicts a ~A-power law, which is confirmed by the meridional salt flux across 28°N calculated f r o m the numerical model, as shown in Fig. 3f. Notice that the meridional salt flux calculated f r o m the present model is one order of magnitude smaller than the meridional salt flux calculated f r o m m a n y existing models based on a relaxation condition or a virtual salt flux condition for the salinity. F u r t h e r m o r e , these two fluxes m a y have different signs at high latitudes (Huang 1993b). Thus, caution must be taken when salt flux calculated f r o m the traditional models is used in explaining the physics.
5 D e p e n d e n c e on the vertical salt mixing
1 m
Exp.
ks
Time
Period
Merid. cell
~
FS,
ks-1 ks-2 ks-3 ks-4 ks-5 ks-6 ks-7 ks-8 ks-9 ks-9 ks-10 ks-ll
0.5 0.6 0.8 1.0 1.1 1.25 1.5 1.75 2.0 2.1 2.25 2.5
1369 1369 1369 1712 1369 2054 2739 2054 1369 1369 1369 1369
Steady Steady 18.85 18.87 22.84 Chaotic Chaotic Chaotic 30.67 33.87 Chaotic Chaotic
-75.00 -77.84 -103.10 -108.36 -118.69 -128.24 -138.79 -149.61 -167.94 -170.52 -175.49 -179.84
-1.734 -1.594 -1.375 -1.235 -1.158 -1.092 -0,987 -0.920 -0.831 -0,810 -0.794 -0.746
-0.8887 -0.9030 -0.8836 -0.9266 -0.9015 -0.9122 -0.9608 -0.9931 -0.9574 -0.9725 -0.9813 -0.9883
A,n =10 9 cm2s -1, A s = 1 0 7 cmZs -], k,~=l cm2 s-l; ks in cm2 s - l ;
The strong haline circulation driven by evaporation minus precipitation is very similar to the salt-mixing induced recirculation in salty estuaries. Fresh water input F f r o m river run-off is relatively small. If there were no mixing, the only motion would be confined to a thin layer of plain water on the top. However, there is mixing induced by tide and waves or wind-stress, so salt is entrained into the u p p e r layer. Assuming a steady state, the salt balance between the out-going flow and the returning flow gives
(F+R) So=RSb,
Table 2. Sensitivity study on the vertical salt mixing ks. Wo= year -t
(32)
time of integration in years; period in years; TKE, the mean kinetic energy in erg cm -~; meridional and zonal cell flux in Sv (a negative sign indicates sinking along the equator); ~ in psu; FoSa, the meridional advective salt flux at 28° N, in 10 6 kg s -~. The time mean is taken as one period or the last 500 years for the chaotic cases 40 20 0 o0 cD L)
-20 -40 -60 -80
where R is the returning flow and Sb is the saliity of the returning flow; thus, the return flow is
-100 - 120
R-
So
- -
Sb - So
-140
F.
(33)
Generally, So=Sb due to strong mixing, so R>F, i.e. the return flow is m a n y times larger than the river runoff. Therefore, the haline circulation is critically dep e n d e n t on the vertical mixing. One set of experiments has b e e n carried out to examine the model's sensitivity with respect to the vertical salt mixing, Table 2. In this set of experiments, only the vertical salt mixing is changed, all other p a r a m e t e r s are fixed. In the first experiment, k~=0.5 cmZs - t , the m o d e l was spun up f r o m a h o m o g e n e o u s initial state. During the spin-up process the meridional circulation oscillated, but this oscillation diminishes within about 600 years, Fig. 4a. After 1000 years, the model reached a steady state. Similar to the pivotal case discussed by H u a n g (1993b), the meridional circulation is characterized by a direct cell (sinking along the equator) and an indirect cell (upwelling along the equator). The direct cell occupies most of the basin with a mass flux of 75 Sv; the indirect cell is located in the equatorial deep ocean, with a mass flux of 5 Sv. Apparently, the existence of the indirect cell is independent of whether the system oscillates or not. It is noticeable that the vertical resolution of the model is rather low, especially for the deep ocean. W e a v e r and Sarachik (1990) pointed out that for such a low vertical resolution, a necessary condition for nu-
0
100 200 300 400 500 600 700 800 900 1000110012001300 Years
40 20 0 -20
--
-40
(,~
-60
'~
-80 -100 -120 -140 0
100 200 300 400 500 600 700 800 9 0 0 1 0 0 0 1 1 0 0 1 2 0 0 1 3 0 0 YeGrs
Fig. 4a, b. Mass fluxes of the direct and indirect meridional overturning cells during the process of spin-up, W0=lmyear -1, a ks=0.5 cm~s -1, b k,=0.6 cm~s -1
merical instability is satisfied; thus, the subsurface cell near the equator m a y be a numerical artifact. In their numerical experiments with increased vertical resolution, there was no indirect cell near the equator. However, recent numerical experiments with high horizontal and vertical resolutions indicate the existence of a subsurface cell near the equator (Wang 1993). Therefore, we present the solutions, including the indirect cell near the equator, and h o p e further numerical experiments with high horizontal and vertical
Huang and Chou: Parameter sensitivity study of saline circulation I
1,2
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100 200 ,300 400 500 600 700 800 900 1 0 0 0 1 1 0 0 1 2 0 0 1 3 0 0
Years
Fig. 5a-f. Time evolution of the mean kinetic energy during spin-up process, W 0 = l m year - t and the vertical diffusivity k , = a 0,5, b 1.0, c 1.25, d 1.5, e 2.0, f2.5 cm2s -1
resolutions will eventually resolve the deep circulation more accurately. When k, is increased to 0.6 cm2s-1, the oscillatory phase of the spin-up continues until year 1200 before it finally disappears, Fig. 4b. For ks larger, the model enters the oscillatory state. As ks is increased to 1.0 cm2s -1, the pivotal value for the model, there is an 18 year oscillation, as discussed by Huang (1993b). As ks is increased to 1.25, 1.5, and 1.75 cm2s-1, the circulation becomes chaotic. For these three runs, the model has been run longer to make sure it is actually in a state of chaotic oscillation. Further increase in ks brings the system back to a single period oscillation of T=30.67 years for ks = 2.0 cm2s -1 and T =33.87 years for ks =2.1 cmZs -1. However, for k, larger than 2.2 cm2s -1 the circulation returns to the chaotic state. The changes in the circula-
tion can be seen from the time evolution of the mean total kinetic energy, Fig. 5. The sensitivity of the model with respect t o t h e change in vertical diffusivity is shown in Fig. 6. Notice that the mean strength of the meridional and zonal overturning cells is proportional to the square root of ks, while the mean surface salinity (deviation from 35 psu) is inversely proportional to the square root of ks, which is consistent with the scaling analysis. As discussed in Section 3, the meridional (and zonal) overturning obey the l/2 power law for a model based on a flux condition, and it is consistent with the % power law if the north-south salinity difference is treated as given. Why did Bryan (1986) obtain a 1/5 power law for the meridional overturning in his numerical experiments? Wright and Stacker (1992) found that under relaxation conditions the meridional overturning of the At-
400
Huang and Chou: Parameter sensitivity study of saline circulation 10 3
0,8
i
i
i
,
ll~ll
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101
i 2,0
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, illiill
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k s (cm2s -1)
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i
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i,ll
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c-1 ..-'
[13
)
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Io °
Fig. 6a-d. Dependence on the vertical salt mixing of: a the mean kinetic energy; b meridional overturning streamfunction; triangles, the mean; diamonds, the maximum; and squares, the minimum; dashed line indicates the 1/2-power
"CO.)
b 101
........ 1 0 -1
I 10 0
1 0 -1
........
k s (crn2s -1)
101
........ 1 0 -1
t 10 0
........
ks (cm2s -I)
lantic in their zonally averaged global ocean circulation model scales like k~/a for cases with no wind stress, and k~/3 for cases with wind stress. However, they noticed that the meridional overturning in the Pacific and Indian oceans obey the 2/3 power law. They proposed enhanced mixing due to deep convection as a possible mechanism for the deviation from the 2A power law. Notice that Bryan (1986) has observed that the zonal overturning obeys the 2/; power. If the deep convection is responsible for the deviation, then why is the zonal overturning not affected by the same deep convection? Therefore, deep convection may not be the primary cause of this deviation. It is noticeable that the relaxation condition used in most models sets a constraint on the north-south density difference. As a result, the zonal overturning obeys the 2/; power law predicted by scaling analysis. However, the West-East density differences across the basin interior and the western boundary currents are not determined a priori. In fact, such density differences are established as part of the global balance. The establishment of the WestEast density (pressure) gradient for cases with relaxation condition or flux condition seems very similar. First, it should be noted that in both cases, the thermohaline circulation is driven by deep water formation in a small area. As required by mass conservation, there is a basin-wide abyssal upwelling, which drives a southward motion in the upper part of the interior ocean. Apparently, the differences between these two cases are the location of deep water formation and the resulting structure of the meridional cell. In the haline circulation discussed here, dense water sinks along the equator so the meridional flow in the ocean interior is part of the direct cell. T h e r e is an indirect cell which consists of western boundary currents. Apparently, the southward motion in the upper part of the interior ocean can transport a substantial part of
101
law; c zonal overturning streamfunction; d mean sea surface salinity (deviation from 35 psu); the dashed line is an inverse square root law
the mass required for deep water formation, so the indirect cell is relatively weak. In the case discussed by Bryan (1986) dense water sinks at the northeastern corner, so the meridional flow in the ocean interior goes against the zonal-mean meridional cell. To maintain the mass flux required by the meridional overturning, strong western boundary layers appear that carry water southward in the deep ocean and northward in the upper ocean. Since the interior flow is against the zonal-mean cell, the mass flux in the deep western boundary current is much stronger than the rate of deep water formation. The net meridional overturning is the difference between the direct cell in the western boundary region and the indirect cell in the ocean interior. The density difference between the eastern boundary and the outer edge of the western boundary current is unknown, so the assumption of fixed density difference does not apply. As a result, the indirect cell in the ocean interior does not obey the 2/; power law. Similarly, the density difference across the western boundary is not fixed, so the 2/; power law does not apply for the direct cell either. In fact, the West-East density difference within the western boundary region and the ocean interior is set up by the global dynamics, similar to the case with a flux condition. Thus, the same 1½ power law applies to the meridional overturning. On the other hand, it is readily seen that there are no strong northern or southern boundary layers, so the north-south temperature (or density) difference is fixed for the case of relaxation condition and the 2/3 power law applies. To verify this simple argument we have carried out one set of numerical experiments for the purely thermal-forced circulation. The model's surface temperature is relaxed back to a linear reference temperature of 25 ° C at the equator and 0 ° C at 60 ° N, with a relaxa-
Huang and Chou: Parameter sensitivity study of saline circulation
tion constant of 0.7 m d a y - l , which is equivalent to a relaxation time of 43 days for an upper layer of 30 m. Otherwise, the model configuration and parameters are the same as the pivotal case, i.e. A ~ = l . 0 x l 0 9 cm2s -1, A s = 1.0 × 107 cm2s -1, and k~ = 1 cm2s -1. To make sure that the model reaches a quasi-equilibrium at the end of each run, we allowed the numerical experiments to run for an excessively long time based on the diffusion time for a thermohaline circulation T = H2/kt .
10 2
~)
-7.5~
%---~ 101
/'
/-
"C. O)
10 o
........ 1 0 -1
2.0
?
WO
m year -[
I
Chaotic
L+
1.5
1.0 0.8
/18 + + ~_Y +23 ~-~ "-Y-+~+9 '+ SteadYPeriodic +-26
0.6
If kt is reduced from 1.0 cmZs-1 to 0.1 cm2s -1, the model would take 10 times longer to reach the equilibrium state. Thus, the integration times for our test runs have been chosen as kt (in cm2s-i) (integration time): 2.0 (2740 years); 1.5 (2740 years); 1.0 (2740 years); 0.5 (5479 years); 0.25 (9000 years). Our preliminary results indicate that a purely temperature-relaxation model obeys the 1/2 power law closely. By the way, we notice that in Bryan's experiments, the integration time for a case with k=0.1 cmZs -1 was only 800 years. Apparently, the model may not have reached the equilibrium state at all. As indicated in Fig. 7, if each experiment is run for a relatively short time of 1300 years, the model's results may give a faulty impression of a % power law. (The sensitivity of the model forced by both thermal relaxation and wind stress is left for further study.) As a final remark, the spin-up time of the thermal circulation seems much longer than that of the haline circulation for the same vertical mixing coefficient. It seems that a thermal circulation is characterized by very energetic overturning in the beginning due to the large amount of dense water formed, and it takes a long time for the system to adjust in order to reach the final equilibrium state. In contrast, the saline circulation is characterized by relaltively weak deep water formation, and it takes less time for the system to reach a quasi-equilibrium. The discussion above is a simplified view of the subtle differences between these two forcings; a close examination is left for further study. Finally, we notice that the meridional advective salt flux at 28 ° N is alsmost constant for all runs in this set
G" (/3
401
I 10 0
0.4 02t__ 0
i
0.5
Steady f
+
Chaotic
+
31 34 ÷+
+
+
Periodic Chaotic
+-37 +-46 +-67 +1105 +-128
1.0
kS
1',5
2'.0
2'.5
" 1 Cff/2S-
Fig. 8. Oscillatory nature of the model in the ks- W0 parameter space
of experiments, see the last column in Table 2. This is consistent with the scaling relation (26). Fig. 8 gives a summary of the results of the first two sets of numerical integrations. This figure provides an overview of the oscillatory nature of the model in the k s - W0 plane. When the freshwater flux is small or the salt mixing is weak, the system is in a steady state. For medium freshwater flux and salt mixing, the system is in a limit cycle. As the freshwater flux or the salt mixing becomes large, the system becomes chaotic. This kind of behavior has been found in many studies on the thermohaline variability, such as those by Weaver et al. (1991, 1993). Our numerical results also indicate that there can be some isolated "islands" of limit cycle behavior within the chaotic regime. Thus, the model's behavior resembles that observed in many highly truncated chaotic systems, such as the very impressive tabulation of the chaotic behavior of the finite-amplitude baroclinic waves by Pedlosky and Frenzen (1980). Since the present model has 3 x 15 x 15 x 15 degrees of freedom, the chaotic behavior of the system provides a link between the highly truncated chaotic systems which have just a few degrees of freedom and the chaotic behavior of the thermohaline circulation and climate in the real world. The saline circulation discussed here is probably the simplest possible and most important component of the thermohaline circulation. Our results can only provide a glimpse of the landscape; much detail of the picture remains to be found in the future.
6 Dependence on the vertical momentum dissipation
........ 101
ks (cm2s -1)
Fig. 7. Meridional overturning forced by temperature relaxation as a function of the vertical mixing coefficient. Triangles indicate solutions close to equilibrium after long time integration, stars indicate solutions obtained after 1370 years of integration, which are not in final equilibrium
The third set of numerical experiments is designed to test the sensitivity of the model with respect to the vertical momentum dissipation. In this set of experiments all the parameters are the same as the pivotal case, except the vertical momentum dissipation, which varies from 0.5 to 2.0 cm2s-1, Table 3. The results of the ex-
Huang and Chou: Parameter sensitivity study of saline circulation
402
Table 4. Sensitivity study on the horizontal salt mixing A~. W 0 = l m year -1
Table 3. Sensitivity study on the vertical momentum dissipation km
Exp.
km Period
km-1 0.5 19.13 km-2 1.0 18.87 km-3 1.5 18.95 kin-4 2.0 18.95
TKE
Merid. cell
Zonal cell
~
FSa
Exp. A,
Period
TKE
0.363 0.363 0.362 0.366
-108.70 -108.69 -109.22 -108.69
35.68 35.49 36.11 35.57
-1.232 -1.232 -1.232 -1.233
-0,9405 -0.9398 -0.9061 -0,9267
As-1 As-2 As-3 As-4
Steady 24.21 18.87 Chaotic
0 . 2 5 6 -87.06 0.286 -97.18 0.363 -108.69 0.359 -78.45
W o = l m year-l; A.~=109 cmZs -~, As=107 cm2s -1, k,= 1 cm2s-1; km in cm28-1; TKE, the mean kinetic energy in erg cm-3; meridional and zonal cell flux in Sv (a negative sign indicates sinking along the equator); ~ in psu; FoSd, the meridional advective salt flux at 28° N, in 106 kg s-~. Each experiment is run for 1370 years, and the time mean for one period
One set of numerical experiments has been carried out to test the model's sensitivity with respect to the horiI
l
l
l
l
l
l
l
l
l
l
l
~
FSa
42.57 41.18 35.49 30.31
-1.192 -1.217 -1.232 -1.208
-1.9175 -1.3813 -0.9927 -0.4994
zontal diffusivity; see Table 4 for the parameters used in these runs. Since isohalines intercept horizontal planes, especially above the halocline, horizontal diffusion tends to reduce the horizontal salinity gradient. As a result, horizontal diffusion works against the external forcing. Generally, large horizontal diffusion corresponds to weak meridional overturning. In fact, for As = 2.0 x 107 cm2s -1, the strength of the meridional overturning is only 87 Sv. Furthermore, there is no salt oscillation, as seen from Fig. 9a. Apparently, strong horizontal salt mixing slows down the circulation and makes it steady. As As is reduced to 1.5x107 cm2s -1, the system starts to oscillate. As A~ is reduced to 0.5 x 107 cm 2 s -1 or lower, the system becomes chaotic, see Fig. 9d.
7 Dependence on the horizontal salt mixing
40
Zonal cell
A,~=109 cm2s -1, k,,~=k~=l cmas-~; A~ in 107 cmZs-~; TKE, the mean kinetic energy in erg cm-3; meridional and zonal cell flux in Sv (a negative sign indicates sinking along the equator); in psu; FS~, the meridional advective salt flux at 28°N, in 106 kgs-1. Each experiment is run for 1370 years, and the time mean for one period or from 1000 to 1369 years for the chaotic case
periments indicate that the saline circulation is very insensitive to the vertical momentum dissipation kin. The insensitivity of the saline circulation to the vertical momentum dissipation is due to the fact that the saline circulation is solely driven by the horizontal pressure gradient created by vertical mixing of salt. On the other hand, a free slip boundary condition is applied to both the upper and the lower boundary, so the vertical momentum mixing has a rather passive role in the model. This is different from the wind-driven circulation where the vertical transport of momentum can play a vital role in setting up the circulation.
a
2.0 1.5 1.0 0.5
Merid. cell
l C
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20
g" = L~
0 -20 -40
)
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-8(]
-100
120
-120 -140
i
~
i
100 200 300
i
400
i
i
i
SO0 600 700
i
800
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900 1 0 0 0 1 1 0 0 1 2 0 0 1 3 0 0
d
,o 20-
' 600
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' 1200
1400
1600
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Yeors
Years
b
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0
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-20 -
-40
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.~ - 1 2 0
-80 -
-160
-100
-200
-120
-240
-140 0
100 200 300
400
500 600 700 Years
800
900
1000110012001300
Years
Fig. 9 a-d. Time evolution of the meridional overturning cells for different horizontal diffusion coefficient A~., in cm2s-1, a 2.0 x 109, b 1.5x109 , c 1.0x109 , d 0.5 x109
Huang and Chou: Parameter sensitivity study of saline circulation It is interesting to notice that the zonal overturning declines as As is reduced. However, the meridional overturning grows as As is reduced, except the last case, As-4, in which the meridional overturning is reduced. Similarly, Ss becomes more negative (surface water becomes fresher) as As is reduced, except the last case, As-4. The reason for such behavior is unclear. According to (26), the meridional advective salt flux is proportional to the horizontal salt mixing As. The results from the numerical experiments indicate that this is indeed the case, see the last column in Table 4.
8 D e p e n d e n c e on the horizontal m o m e n t u m dissipation
In most numerical model experiments with low resolution, a high horizontal eddy dissipation coefficient is used. The criterion of choosing the horizontal dissipation is based on the requirement of resolving the western boundary current in the model (Bryan et al 1975). Accordingly, the dissipation coefficient is chosen as
Am >-/3(A ][3/,rr) 3
(34)
w h e r e / 3 = 2 . 0 x 1013s-1 c m - a , A is the grid size. Since the 4 ° x 4 ° model has a grid size of about 400 kin, the corresponding horizontal dissipation coefficient is about 2.5 x 109 cm2s-1. In the present cases, however, the western boundary currents are relatively weak, and we have been able to run the model with A m = l x l 0 9 cmZs--1 Notice that in the present G F D L code, one way of solving the barotropic streamfunction is with a standard relacation scheme. However, the maximum absolute error of the residuals has been chosen as the criterion of convergence. Such a criterion may not be suitable for climate study where the system approaches a quasi-equilibrium. No matter how small the absolute error bound is, after a certain time the forcing term will be smaller than the error bound. Consequently, the relaxation loop will be used only once for each time step. Over many time steps large error (noise) may be accumulated. This problem was not apparent and large horizontal eddy dissipation was used by many modelers to obtain a smooth solution. In order to reduce the eddy-diffusivity as much as possible the use of the relative error as the convergence criterion is suggested. In most of our numerical experiments, the relative error bound has been chosen as e_<0.001. This is certainly quite a demanding error bound. However, our solutions seem quite smooth even for a horizontal dissipation coefficient which is several times smaller than the value used in many other numerical experiments. W h e t h e r using the relative errors as the convergence criterion can reduce the eddy-diffusion necessary for obtaining a smooth solution for the general cases with wind stress and b o t t o m topography remains to be determined. Several numerical experiments have been carried out to test the model's sensitivity with respect to the
403 Table 5. Sensitivity study on the horizontal momentum dissipation Am
Exp.
Am Period TKE Merid. cell
Zonal ~ cell
fsa
Am-1 Am-2 Am-3 Am-4
2.0 1.5 1.0 0.5
35.11 34.61 35.46 37.72
-0.8775 -0.9067 -0.9266 -0.9370
18.84 18.80 18.87 41.10
0.305 0.325 0.363 0.432
-106.83 -106.77 -108.36 -115.56
-1.192 -1.217 -1.232 -1.208
W0=l myear 1; As=10 7 cm2s-1, k . , = G = l cm2s-1; A,~ in 10 9 cmZ s-1; TKE, the mean kinetic energy in erg cm 3; meridional and zonal cell flux in Sv (a negative sign indicates sinking along the equator); ~ in psu; FSa, the meridional advective salt flux at 28° N, in 10 6 kg s-1 Each experiment is run for 1370 years, and the time-mean for one period
horizontal eddy dissipation; see Table 5. As A,~ is reduced from 2 , 0 x 10 9 cm2s-1, the circulation does not change much. However, when Am is reduced to 0 . 5 x 1 0 9 cm2s -1, the salt oscillation period doubles; see Fig. 10. Overall, the model is rather insensitive to the horizontal eddy dissipation within the range of our experiments; see Table 5. This insensitivity is consistent with the passive role of m o m e n t u m mixing in the pure haline circulation.
9 D e p e n d e n c e on the horizontal resolution
In all numerical experiments discussed above, a standard 4 ° x 4 ° low resolution is used. To test the model's sensitivity with respect to the horizontal resolution, we have carried out one set of experiments with horizontal resolution gradually increased from 6 ° x 6 ° to 5 ° x 5 °, 3 ° x 3 °, and 2 ° x2°; parameters used in these experiments are listed in Table 6. (The time step used in these experiments changes in proportion to the horizontal resolution). As discussed in the previous section, the horizontal dissipation required for resolving the western boundary currents is proportional to the cube of the grid size and the time step is linearly proportional to the grid size. Thus, we have chosen A m = 2 . 0 x 10 9 cm2s-1, based on the low-resolution case of 6 ° x 6 °, and use this value to run the cases with fine resolution. For the case of 6 ° x 6 ° the salt oscillation is chaotic. However, as the resolution is increased to 5 ° x 5 °, the system enters a limit cycle (Fig. 11a, b). As the resolution is increased from 6°x 6 ° to 5 ° x 5 ° and 4 ° x 4 °, the meridional overturning increased. Consequently, the mean sea surface salinity deviation is larger. As the resolution is increased to 3 ° x 3 °, however, the circulation becomes chaotic and the meridional overturning declines. For the high-resolution case of 2 ° x 2 °, the circulation becomes steady and the meridional overturning declines further. The most remarkable difference is that the site of bottom water formation is moved away from the southern boundary. As seen from Fig. 12b,
Huang and Chou: Parameter sensitivity study of saline circulation
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Fig. 10a-d. Time evolution of the meridional overturning cells for different horizontal dissipation coefficient Am, in cm2s -1. a 2.0 x 109, b 1.5 x 109, e 1.0 x 10 9, d 0.5 x 109
Table 6. Sensitivity study on the horizontal resolution. Am = 2 . 0 x 1 0 9 cm2s -1
Exp.
3A X 6
Period
TKE
Merid. cell
R-1 R-2 R-3 R-4 R-5
6° x 6° 5° x 5° 4° x 4° 3° x 3° 2°x2 °
Chaotic 23.39 18.87 Chaotic Steady
0.304 0.289 0.363 0.531 0.339
- 87.97 - 94.92 - 108.37 - 84.86 -39.27
- 1.221 - 1.224 - 1.235 - 1.236 -1.165
A,=l.0xl07 cm2s -1, km=k~=l.0 cm2s-1; period and integration length in years; TKE in erg cm-3; meridional cell flux in Sv (a negative sign indicates sinking along the equator); ~ in psu. Each experiment is run for 1370 years, and the time-mean for one period or from 1000 to 1370 years for the chaotic cases
the salinity maximum is located at 10 ° N near the eastern boundary. Apparently, sinking along the eastern b o u n d a r y becomes so strong that the vorticity stretching actually pulls water into a cyclonic gyre in the southern basin, as seen from the velocity diagram, Fig. 13. Notice that b o t t o m water is not formed at the place where evaporation is the strongest. Instead, b o t t o m water formation is associated with the location of very slow horizontal motion, so evaporation can work most efficiently in creating salty and dense water which can sink to the b o t t o m of the ocean. Notice that the zonally integrated meridional overturning streamfunction still shows the strongest sinking from the surface face along the southern boundary, see Fig. 12a. The solution of the high resolution case also implies that the haline circulation strongly depends on the model's horizontal resolution. Its implication for cli-
mate modeling may need to be studied more carefully.
10 Haline circulation driven by different freshwater flux profiles So far all our discussion has been concentrated on the sensitivity for the haline circulation forced by a particular "linear" profile, which implies precipitation at high latitude and evaporation at low latitude. It is of interest to examine the model's behavior forced by different freshwater flux profiles. A n o t h e r simple profile is a weighted cosine profile. w0-
cos~b 2 cos 2~r
.
(35)
This is the B profile (Fig. 1) with W 0 = l m Year -~ (The factor of ~- is introduced in the definition so that the maximum meridional freshwater flux in profiles A and B is about the same, roughly equal to 0.36 Sv when W0 = 1 m y e a r - 1 . ) This profile implies precipitation at the equator and high latitudes, and evaporation at middle latitudes, which resembles the present day evaporation minus precipitation field. A set of numerical experiments has been carried out based on this freshwater flux profile in order to test the sensitivity of the model with respect to the vertical mixing; see Table 7. The basic structure of the circulation can be seen from Fig. 14, which is an instantaneous picture of the circulation at the end of 1370 years in case B-4. Since the evaporation maximum is
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located at 30 ° N, the meridional circulation is divided into two cells, a northern cell and a southern cell both of them driven by dense salty water formed at middle latitudes, see Fig. 14a. The oscillatory nature of the solution follows a pattern similar to the cases driven by the A profile. For a very low mixing rate, ks = 0.1 cmZs -1, the system is in a steady state; see Fig. 15. As ks is increased to 0.25 cmas -1, the system enters a periodic state. The dependence on the vertical salt mixing of the period is shown in Fig. 16. The square near the origin indicates the steady state for ks=0.1 cm2s -1. As ks is increased, the system enters a limit cycle where the period declines slightly as k, is increased. However, period doubling and quadrupling appear near ks=4.25 cm2s -1 and ks=6.25 cm;s -1, as shown in Fig. 15c, 15d, and Fig. 16. As k, increased to 10 cm2s-1, the system becomes aperiodic; see Fig. 15e. However, as ks is increased further, the system moves into another region of periodic oscillation in the parameter space. Notice that the first critical value for ks, at which the system transits from limit cycle behavior to chaotic, is rather high. Such a high ks may not be very realistic. However, the substantial difference in the first critical value of ks for profile A and B indicates that the transition from the region of limit cycle to region of chaotic behavior is very sensitive to the freshwater flux distribution. Consequently, the freshwater flux must be measured and specified as accurately as possible in order to understand the variability of the thermohaline circulation. We believe that the transition from the region of limit cycle to the region of chaotic behavior, including period doubling and quadrupling, can be
Huang and Chou: Parameter sensitivity study of saline circulation
406 Table 7. Sensitivity study on the vertical salt mixing under different evaporation minus precipitation profiles. B series is forced by the B profile in Fig, 1, and C-1 is forced by the C profile
Exp.
ks
B-1 0.1 B-2 0.25 B-3 0.5 B-4 1.0 B-5 3.5 B-6 4.0 B-7 4.5 B-8 5.0 B-9 6.0 B-10 7.0 B-11 7.5 B-12 8.0 B-13 10.0 B-14 15.0 C-1
Time
Period
TKE
S. merid. cell
N. merid. cell
Zonal cell
0-1600 2740-4110 1370-2740 0-1370 1370-4110 4110-5480 4110-5480 4110-5480 4110-5480 4110-5480 0-2054 2054-3424 0-2740 0-1370
Steady 147.40 128.63 113.97 104.93 102.88 199.45 178.45 176.03 357.26 354.25 348.90 Chaotic 83.46
0.0805 0.1352 0.1856 0.2597 0.4957 0.5349 0.5685 0.6220 0.6882 0.7600 0.7946 0.8275 0.9626 1.2460
4.05 7.56 10.04 12.78 24.09 25.12 25.46 25.73 26.20 26.90 27.52 28.40 31.40 38.92
-5.40 -5.34 -7.71 -10.22 -16.70 -17.72 -18.77 -20.26 -22.76 -24.38 -25.33 -26.24 -28.65 - 34.54
4.81 6.33 8.64 11.44 19.83 21.23 22.76 25.36 28.23 31.16 32.34 33.48 37.70 41.84
-4.669 -3,356 -2.422 -1.729 -0.922 -0.858 -0.806 -0.764 -0.690 -0.634 -0.610 -0.588 -0.518 - 0.409
0-1370
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1.0
0.2071 16.16
0.0
W0=l m year-l; A~=l.0xl09 cmls -l, A,=107 cmZs 1, /% =ks=l cm2s -~. Integration time in years; notice that B-3 and B-5 are restarted from B-4, and B-2 is restarted from B-3; B-6, B-7, B-8, B-9, and B-10 are restarted from B-5, and B-12 is restarted from B-11. Period in years; TKE, the mean kinetic energy in erg cm-3; northern and southern meridional and zonal cell flux in Sv (a negative sign indicates an anticlockwise cell when looking westward); ~ in psu; most experiments are run for 1370 years, and the time-mean for one period
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found with realistic value of k, in models forced by more realistic freshwater flux pattern. The dependency of the model's solution on the vertical mixing is shown in Fig. 17. It is readily seen that the model's results fit the scaling analysis very well. In fact, both the northern and southern meridional cells and the zonal cell obey the 1½ power law, and the mean surface salinity (deviation from 35 psu) obeys the -1/2 power law. The fact that both the southern and northern meridional cells satisfy the 1/2 power law suggests that the simple scaling analysis in Section 3 is rather robust, regardless of the exact shape of the freshwater flux profile. It is to be noticed that for the same parameters, the meridional overturning is about 1/10 of that forced by profile A, and the zonal overturning is about 1/3 of that forced by profile A. Such a great change in the strength of the circulation indicates that the saline circulation is very sensitive to change in the freshwater flux profile. This is consistent with the sensitivity of thermohaline circulation to change in the virtual salt
flux profile discussed by Weaver and Sarachik (1991), and Weaver et al. (1991, 1993). Further study is needed in order to understand the subtle role of freshwater flux forcing in determining the saline circulation. As the final example, we present a haline circulation driven by the C profile in Fig. 1, which is the reverse of the A profile, With the same parameters as the pivotal case, i.e. W 0 = l m year -1, the system reaches a periodic oscillation with 1000 years of integration, Fig. 18. The meridional overturning is characterized by salty water sinking at the northern boundary and upwelling basewide, which is rather similar to the standard case of thermal circulation. The surface salinity reaches the maximum at the northeast corner, as seen in Fig. 19b. Although such a freshwater flux profile seems unrealistic, the oscillatory nature of the solution demonstrates that the oscillatory nature of the saline circulation is very robust regardless of the shape of the freshwater flux.
11 Discussion In this study we have carried out extensive numerical experiments to test the sensitivity of a model based on the natural boundary conditions for the salinity balance. Results from our study are consistent with many previous studies. In addition, our study has provides a unified view of the dynamical nature of the saline circulation which is useful for understanding the thermohaline circulation and climate. First, haline circulation forced by evaporation and precipitation is very sensitive to changes in the freshwater flux. A small freshwater flux can drive a huge baroclinic circulation in a basin, and a small change in
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the freshwater flux can cause a substantial change in the haline circulation. Second, haline circulation forced by evaporation and precipitation is especially sensitive to the vertical mixing. In a way, the vertical mixing is the primary
mechanism which gives rise to the meridional overturning. Simple scaling suggests a 1/2-power law dependence, which is also confirmed by numerical experiments. Third, haline circulation forced by steady evaporation and precipitation can appear in quite different modes, ranging from steady circulation to single period oscillation, period doubling and quadrupling, and chaotic behavior. In general, for high resolution and vertical mixing the system is more likely to be chaotic; however, there are islands of limit cycle behavior within the chaotic regimes. As discussed by Huang (1993b), the time-mean circulation of our model is rather similar to other models based on the virtual salt flux condition. Therefore, the results from the parameter sensitivity study of the present model also apply to other models, except the oscillatory behavior and the meridional salt flux. Since the haline circulation is very sensitive to the freshwater flux, a slight change in the upper boundary condition
408 ,-¢
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can lead to substantial change in the oscillatory behavior of the solution; see Huang (1993b). We speculate that models based on virtual salt flux may have similar oscillatory behavior; however, the boundary of different zones and the detail of the oscillation may be different from the present model. We want to emphasize that vertical mixing is extremely important for the haline circulation driven by evaporation and precipitation. Models based on different schemes of vertical mixing, with or without a mixed layer, may have quite different character. In this study we have probably raised more questions than provided answers. We look forward to seeing more careful study on evaporation/precipitation-driven circulation and its links to climate. R.X. Huang was partially supported by the States World Ocean Circulation Experiments and the NaScience Foundation through grant O C E 90-17158 to Hole Oceanographic Institution. R . L . Chou was supby the Environmental Protection Agency, A. Weaver's review of the manuscript helped to clarify the presenta-
Acknowledgements.
Fig. 18. Time evolution of the meridional cells driven by the C profile in Fig. 1
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Fig. 19a, b. Instantaneous structure of the circulation driven by the C profile in Fig. 1 with k , = l cm2s-~, a meridional overturning streamfunction, in Sv; b sea surface salinity (deviation from
35 psu)
Broecker WS, Bond G, Klas M, Bonani G, Wolfli W (1990) A salt oscillator in the glacial Atlantic? 1. The concept. Paleoceanography 5 : 469-477 Bryan F (1986) High-latitude salinity effects and interhemispheric thermohaline circulations• Nature 323:301-304 Bryan F (1987) Parameter sensitivity of primitive equation ocean general circulation models. J Phys Oceanogr 17:970-985 Bryan K (1969) A numerical method for the study of the circulation of the world ocean• J Comput Plays 4:347-376 Bryan K, Manabe S, Pacanowski R C (1975) A global oceanatmosphere climate model. Part II. The oceanic circulation. J Comput Phys 1 5 : 3 0 4 6
Huang and Chou: Parameter sensitivity study of saline circulation Cox MD (1984) A primitive equation, 3-dimensional model of the ocen. GFDL Ocean Group Tech Rep No 1, GFDL/Princeton University Goldsbrough GR (1933) Ocean currents produced by evaporation and precipitation. Proc R. Soc London A141:512-517 Huang RX (1993a) A two-level model for the wind and buoyancy forced circulation. J Phys Oceanogr 23:104-115 Huang RX (1993b) Real freshwater flux as a natural boundary condition for the salinity balance and thermohaline circulation forced by evaporation and precipitation. J Phys Oceanogr (in press) Marotzke J, Willebrand J (1991) Multiple equilibria of the global thermohaline circulation. J Phys Oceanogr 21:1372-1385 Pedlosky J, Frenzen C (1980) Chaotic and periodic behavior of finite-amplitude baroclinic waves. J Atmos Sci 37 : 1177-1196 Stommel HM (1957) A survey of ocean current theory. Deep-Sea Res 4:149-184 Walin G (1985) The thermohaline circulation and the control of ice ages. Palaogeogr Palaeoclimatol Palaeoecol 50:323-332 Wang D (1993) Modeling deep equatorial circulation. PhD thesis, University of Hawaii
409 Weaver AJ, Sarachik ES (1990) On the importance of vertical resolution in certain ocean general circulation models. J Phys Oceanogr 20: 600-609 Weaver AJ, Sarachik ES (1991) The role of mixed boundary conditions in numerical models of the ocean's climate. J Phys Oceanogr 21 : 1470-1493 Weaver AJ, Sarachik ES, Marotzke J (1991) Freshwater flux forcing of decadal and interdecadal oceanic variability. Nature 355 : 836-838 Weaver AJ, Marotzke J, Cummins PF, Sarachik ES (1993) Stability and variability of the thermohaline circulation. J Phys Oceanogr 23 : 39-60 Welander P. (1986) Thermohaline effects in the ocean circulation and related simple models. In: Willebrand J, Anderson DLT (eds) Large-scale transport processes in oceans and atmosphere. D Reidel, Dordrecht, pp 163-200 Winton M, Sarachik ES (1993) Thermohaline oscillations induced by strong steady salinity forcing of ocean general circulation models. J Phys Oceanogr (in press) Wright DG, Stocker TF (1992) Sensitivity of a zonally averaged global ocean circulation model. J Geophys Res 97:1270712730
