- Email: [email protected]
Numerical Prediction of the Vertical Thermal Structure in the Bohai and Huanghai Seas—Two-Dimensional Numerical Prediction Model
277
NUMERICAL PREDICTION OF THE VERTICAL THERMAL STRUCTURE IN THE BOHAI A M ) HUANGHAI SEAS--TWO-DIMENSIONAL NUMERICAL PREDICTION MODEL Wang Zongshan, Xu Bochang, Zou Emei, Gong Bin, and Li Fanhua First Institute of Oceanography, SOA, Qingdao, China
ABSTRACT By using observed sea temperature data, we caIculated the non-dimensional depth q and its rele vant non-dimensional temperature OT to construct a similarity function OT= f(q ). Based on these, the thickness of the upper homogeneous layer h = h(x,y,t), surface temperature Ts= Ts(x,y,t), temperature in t h e thermocline T z = T,(x,y,z,t), bottom temperature T, = T,(x,y,t), the current velocity u = u(x,y,t), v = v(x,y,t) and the surface topography [ = [(x,y,t) were assumed to establish a numerical prediction model of the vertical thermal structure. The solution of this two dimensional model was obtained by using "ADT" and "HN" methods. The results of a trial prediction are satisfactory.
1 INTRODUCTION Until now, the studies of prediction on the vertical temperature structure have been focused on the characteristics of the upper ocean homogeneous layer (Kitaigorodskii, 1977; Nesterov, 1978; Resnanskii et al., 1980,1983,1986; Kraus et al., 1975,1977). Few papers are concerned with the prediction of the vertical temperature structure. Early mathematical models are divided into three categories: (1). Models for solving the closure equations including the momentum equation, continuity equation, equation of state, thermal equation and salinity equation (Kitaigorodskii, 1977; Kraus, 1977; Kalazkii, 1978, 1980; Murakami et al., 1985; Omsteds et al. 1983); (2). Forecasting models for describing the characteristics of the upper thermal structure according to the universal function derived by similarity theory (Wang et al., 1986); (3). Studies of the methods for calculating and simulating the vertical temperature structure in terms of the similarity function of the vertical temperature distribution (Xu et al., 1983, 1984) and thermal equation (Malkki et al., 1985). Category (1) is too difficult to put into effect because so many hydrographical and meterological factors have to be known for prediction. The coefficients in the equations are artificially taken as constants in most cases, which often produces errors in forecasting. As for category (2), though the model itself is simple, it describes only the mean characteristics of the vertical temperature structure. Category (3) need not some relevant coefficients in the equations, and the vertical temperature structure can be obtained somewhat objectively. Based on the category (3) and the available data of wind and air temperature field, a one-dimensional numerical prediction model for the vertical temperature structure in the Bohai and Huanghai Seas was developed (Wang et al., 1990). In the model, the effects of advection and lateral mixing are neglected, which holds good for the Bohai and Huanghai Seas. We could also verify it with the satisfactory trial prediction of the above model. However, with a strong cyclone passing, the simulated current velocities can reach 130 and 300 cm / s in some areas of the Bohai and Huanghai Seas, respectively (Zhang et al., 1983, 1988). Such huge wind-induced current will induce a large change in the three dimensional temperature structure. When a cyclone with wind speed 25-30111 / s passes over the deep sea at mid-latitudes, the upper layer temperature in a large area decreases by more than 2C and the upper homogeneous layer thickness increases by more than 6 m. The depth influenced by wind could reach lOOm (Japan Meteorol. Agency, 1985; Halpern, 1974; Nesterov,
.
278 1986). The upper homogeneous layer thickness in the Bohai and Huanghai Seas can increase by 5-8 m when heavy wind blows continuously over 24 hours (Ocean group of STC, PRC, 1964). The large. change of the thermal structure makes effective operation of acoustic instrument and fishing activities difficult. Therefore, the effect of the wind-induced current on the thermal structure redistribution needs to be understood. In this context, we developed a numerical prediction model for the vertical temperature structure as associated with wind-indud current and lateral mixing.
2 PHYSICAL MODEL 2.1 SIMILARITY FUNCTION OF THE VERTICAL TEMPERATUREPROFILE The vertical temperature distribution in the Bohai and Huanghai Seas in warm seasons is schematically shown in Fig. 1, which is divided into three layers: (1) the upper homogeneous layer, whose
r
I
I
I
I
h
2
Fig. 1. Schematic represemtation of the vertical temperature profie. thickness is h; (2) the thermocline; (3) the deep layer. Therefore, in the two-dimensional temperature structure, the non-dimensional depth q and its revelant non-dimensional temperature Or is written as: T -T Z-h OT=t q=(1) T,-T, H-h where Ts(x,y,t) denotes the surface temperature (or the tempcraturc in the upper homogeneous layer); TZ(x,y,z,t), the temperature which vanes with depth z; T,(x,y,t), the temperature in the bottom layer; H, the water depth. Dimensionless variable OT is a function of tj (Kitaigorodskii et al., 1970; Malkki et al., 1985; Xu et al., 1983, 1984). Based on the observed data of vertical temperature profile and meeting to boundary condictions: 'T={
=O =1
when
d
=O 1 I
'
a similarity function has been constructed by using the least square method( solid line in Fig. 2):
,,
where a a z, a 3, a ,
O,=a,tj+a,tja
+a,i'+a,v'
are empirical constants determined by 5053 historical temperature profile data
(2)
279
a4
a2
0
a6
a8
1.0
Fig. 2. The similarity profile of sea temperature in the Bohai and Huanghai Seas. obtained by Nansen casts in the Bohai and Huanghai Seas during May-October from 1958 to 1982. We by: denote K, and
zT
I
The average value of K, and K rfor the Bohai and Huanghai Seas are 0.75 and 0.30, respectively. According to the above formulae, the vertical temperature structure is expressed by following equa-
tions:
T , = T,(x,y,t),
OGzGh
T , = T,(x,y,t) - B,(W',(x,y,t)
-
T,,(x,y,t)l,
h szGH
(4)
2.2 GOVERNING EQUATIONS T o describe the two-dimensional current field, we use the following equations (Zhang et al., 1983, 1988):
aV at + u a-V+ ~ - +JV f i + ~ - - Ka(- = JX
Jy
ay
a2v 7 JY
---+--------1 aP 1 P , =Y
P,
Z a y -'by
H
in which u = u(x,y,t), v = v(x,y,t) are the velocity components, C = [(x,y,t) the sea level, K, the turbulent viscosity coefficient, f the Coriolis parameter, pw the water density (taken as l), H the water depth, P the pressure, g the gravitational acceleration, and 7b the surface and bottom frictional stress given by:
280
and
with pa (air density), C , (the drag coefficient), K,= 1 / M ?"(friction coefficient at the bottom), M (Manning coefficient), E(mean depth in each mesh). In Eqs. ( 5 ) and (a), the first term of right side can be ignored in comparision with the second term. Neglecting the relevant processes of the molecular diffusion, the heat conduction equation is:
"r + u-JT + v-JJyT + w-aJZT - A , V 1 T = Jf JX
-
J(
w'T+R) JZ
~
where T is the temperature, A,the horizontal turbulent heat conduction coefficient, W'T'the average vertical turbulent heat flux, R the penetrative component of the solar radiation divided by C,p,( Cpis specific heat of the sea water). Since u and v are independent ofz, w = O at z=O, and the solar energy is assumed to be entirely absorbed by the sea surface, integration of Eq. (10) over the upper homogeneous layer leads to:
- + u S +JT v L - A ,JT v JT, Jl
z T , = -Q s - Q H h
JY
JX
~
where Qs = QL/ (C,p,) is the surface heat flux, Qh= W'T'I h the mean turbulent heat flux at the lower boundary of the upper homogeneous layer, and QLtheheat budget at the sea surface. In order to derive an equation for h and T,, we substitute Eq. (4) into Eq. (10) and then integrate it with respect to z from h to H to get:
JT JT (l-O,)[L+u$ at
JT JT JT +v~-AA,v2T,]+0,[2+uL JY
Jl
JX
JT, +V--A,~~T,,]-(~ JY
Integration of Eq. (12)with respect to { from 0 to 1 with the boundary condition Q, = 0 gives:
JT
(1 - K,)[$
JT
aT
JT
+ u JX s + VS - R r V Z T , ]+ K,[-$ Jy
T -T H-h
4-K , I S - ' -
ah ah $ +u- +v- --Arvzh]-2A,[(g Har ax sy
JT
JT
+ U> J X + v HJY h
JT
JX
-A
J T , ah
--)- Jx
Jx
rv'T,I JT,
+(---)-I Jy
Jy
ah Jy
281
Double integration of Eq. (12)with respect to I,first from 0 to q, then from 0 to 1, yields: 1 -J T , aT, JT -JT JT JT ( s - K r ) [ y + U ~ + V ~ - - A ~ V ’ T ~ I + KJt, [ +- u’ A-J X+ v L - AJy JY -T , - T n ah ah ah 1 2 A , aT, a T ~ah 2K7 -{ (- + U- + V- - A ,.V h)l--I- OX JX H-h Jl JX Jy
=[(ax
+
T ~ - T ah~ 2 [(GI ( H - h)’
JT J T ~ ah +(2 --)-1-2A JY JY JY
-4
I
0
~
-
T s - T T , ah a h - 1 ) e 7 d ~ i A r[(GI ( H - h)’
u =(Q,
where
vlT 1
- i Q d q ) / Q,,
ah +(G) 11+[2K,-a, 1
+ (JhG z)I-= V Q r
=o
Q = (w”+ R). Simultanous equations (Eqs. (13) and (14)) to-
0
gether with Eq. (1 I ) give: JT,, JT, -+ UJt
ax
aT + YH - A,v’T,, ay =C
‘ h
,
z -C
+ C,A.=
QS-Qh
l
T - T ahi N), ( H - h)’
aha +(GI 1
h
(15)
and
C, = ( 2 K Y - K,K,C, = [2a,K+,
2K
1
- K ) / (KrK,) 2 7
rc-,
a K
+ 4K
I
J ( q- 116 ,dq] / (K 7 K T ) 0
C, = ( K , v - K ~ ) / ( K , K , )
c, = (-21 K 1 - K<)/ (K,K,) I
C , = [a , ( K r -);K
- 4K7 J(v - lYrdql/ ( K r K f ) 0
According to a processing method presented by Wang et al. (1990), Qs,Qh and y in Eqs. ( l l ) , (IS) and (16) are defined as follows. Qs is written by a formula (Wang, 1983):
282 where T, is the air temperature, and e,, e,empirical constants. It is difficult to get accurate value of the heat flux (QJ through the lower boundary of the upper homogeneous layer, because Qhis related to the entrainment of the lower cold water to the upper homogeneous layer rather than the simple change in h. Based on turbulent kinetic energy equation, Resnanskii (1980, 1983, 1986) showed that the vertical salinity variation in the mid- latitude areas has almost no effect on the buoyance flux, and entirely depends on the vertical turbulent heat flux W’T’. He also neglected the horizontal variations in the mean velocity, turbulent kinetic energy and their flux. Integration of the turbulent kinetic energy equation from 0 to h and use of Eqs. (10) and (11) gives: 2F Q = - Q -Bh Here B= ga,is the buoyance coefficient, a,the heat expansion coefficient of sea water, F the conversion rate of turbulent kinetic energy in the upper homogeneous layer to potential energy, which is subject to the following conditions: the predicted h is close to the measured h under the wind mixing, andQ,and entrainment velocity decrease with increasing h. Parameter F is written as (Resnanskii, 1983):
in which,b,, bz m are empirical constants taken as 2,5, and 2, respective1y.A = A(x) is the unit function of independent variable x= (U, -b,hlfl) withA = 1 when x>O and A = 0 when x < 0. U . = ( T /~ pw)”*the friction velocity of sea water. In order to determine u, Eqs. (13) and (14) are transformed into: d[l - K , + K , n + K , B
+ (2Kr -n,)a]=
1
(21)
Eqs. (21) and (22) give: 1 u = [- - KY(1- n - 2 8 2
I
201)- (a1 + 4 j ( q - l)Ordq)al/ [I - K,(I - n -8 0
- 2a)-
a,a].
By use of historical temperature profile data and meteorological data, and corresponding u, v,
(24)
283
Qs and Qh calculated from Eqs. (5)-(7), (IS), and (191, y can be determined with Eq. (24). The average value of y is 0.06 in the warming period and 0.53 in the cooling period in the Bohai and Huanghai Seas. Then the numerical prediction model is constructed by using Eqs. (5) to (7),(ll), (IS) and (16). At the closed boundary, V, = 0, Ts =TH,h = H, Q, = Qh = QH = 0. At the open boundary from the Changjiang river mouth to Chejudo island, au / aX = 0, aV / ay = 0. The sea level ( at the closed boundary is specified by:
c=
n
c ~ ( x , Y ) , C o 4 U -l t
I- I
(25)
R(X,Y),I
where i = 1,2, ... ... ,n denotes the serial number of tidal constituent, Hiand giare the harmonic constants of ith tidal constituent, ci is angular velocity. The initial conditions are given by:
u=v=O, C=O, h=h, Ts=T,. TH=Tho, whent=t,,. 3 NUMERICAL MODEL The parallel of 36 N is taken as x axis (positive eastward), and the meridian of 120 E as y axis (positive northward), the mean sea level a8 z = 0 (positive upward). 3.1 DIFFERENCESCHEME In order to get a finite difference analog of the prediction model in differential form, we chose Platman's alternative grid, and use alternating direction implicit method (ADI) (Leendertse, 1967; Zhang et al., 1983, 1988). The features of this method are: (1) the calculation is simple because an implicit differential form is alternatively used in one direction when estimating the variables in x and y directions. (2) an implicit and an explicit differential form are alternatively used in x and y directions that make the calculation stable and quick convergent. In the meantime a hydrodynamic numerical method with temporal forward difference and special central difference (HN)is used to disctetize the Eqs. (It), (15) and (16) describing the vertical temperature structure. We evaluate ( ,h. T , That a grid point (i. 2, u, w,at a grid point (i+l / 2, 3, v, w,at a grid point (i, jtl / 2) and H at a grid point (i+l / 2, j+l / 2). O
0 1 0 1 0 - + - + O
-
J
.
I
1 - 7 0 j-1
-
+ 1 t
-
i 4
O -
l
G
t
-
0 1 0 1 0 -
f
0
1
-
0
+ '
Fig. 3. Definition for computed grid points and variables.+-(.
0
h, Ts,TH; --u, w,;
I-v,~,;
0-H
284 3.2 DIFFERENCE FORM Variables U’, V’, Ckareobtained from Eqs. ( 5 ) to (7) at k A t intervals. The appendix gives the implicit differential form for C and u, and explicit differential form for v from k A t to (k+1/ 2)A t; the implicit differential form for and v, and explicit differential form for u from (k+l / 2) A t to (k+l)A t; and the direrentialform ofEqs. (ll), (15) and (16)from k A t to (k+l)At. 4 APPLICATION TO TIDAL PREDICTION The wind speed w and air temperature T,,are known. The grid spacing is 20 km. The time step is 900 seconds. The basic parameters are specificed as in Table 1. TABLE 1 The relevant parameters in the model (cgs)
I
I
PI
I
I .229 x lo-)
I
CD
c,
0.938
K,
a
0.25 x lo-)
K,
g
I
980
I
I
(0.8719+0.000704W~0)Xlo-’
I
10’
I
1 / ME’’‘ ,M = 0.016-0.018
* W,, means the wind speed at 10 m above the sea surface. In the forecasting procedure, first of all, we calculate the current field from the given wind field, then put the current field and the given air- temperature field into the governing equations to predict h, Tsand TH.Computation was made on IBM-4381 computer. 5 THE RESULTS AND DISCUSSION OF TRIAL PREDICTION Because of the limited observations in this area, we only take the wind, air temperature and sea surface temperature data during FGGE as known variables to forecast the vertical temperature structure. The period of validity is about 4 days. The surface temperature (Ts), thickness of the upper homogeneous layer (h), bottom temperature (TI(), meridional distribution of temperature along 123.5 ’ E and vertical temperature profiles on July 8, 1979 are presented in Figs. 4 to 8. These figures show that the sea surface temperature in the study area is 22-23C in most areas except that the SST in Haizhou Bay is higher than 25C and there is a cold eddy north of Cheng Shantou (Fig. 4). There are four areas where the thickness of the upper homogeneous layer is great (Fig. 5): the first one is located in the central Bohai Sea (greater than 5 m), the second one lies in the north ofthe northern Huanghai Sea Cold Water (greater than 15 m), the third one is at about 34.5 N, 123 ’ E (greater than 10 m) and the last one in the Huanghai Trough area has the greatest thickness (20 m-25 m). The thickness ofthe upper homogeneous layer reaches 10 m off the coast of northern Jiangsu province due to strong mixing and is about 5 m in the other areas. The temperature profile from the trial prediction (Fig. 8a) clearly shows the upper homogeneous layer, thermocline and deep layer. In order to verify the reliability of the trial prediction, we roughly compare the calculated results with observed data obtained from a simultaneous hydrographic survey (Fig. 8b). Comparison shows that: ( I ) the temperature profile of trial prediction coincides with the observation north of 36 N, but there is a discrepancy between them in the deep layer south of 36 ‘ N , which might be due to a large meridional distance and asynchronous observational data between stations. (2) the thermocline obtained from the trial prediction is thicker than the observed one and the predicted temperature is vertically homogeneous in the thermocline, wbicb may be due to the assumption that the current velocity is vertically homogeneous that induces the strong mixing between thermocline and its lower boundary. So three dimensional numerical prediction method is needed to put forward in the future.
285
30 *
Fig. 4. Sea surface temperature ( C ) from the trial prediction (July 8,1979 in the Bohar and Huanghai Seas.
Fig. 5. Thickness (m)of the upper homogeneous layer from the trial prediction (July 8, 1979) in the Bohai and Huanghai Seas.
30
Fig. 6 . Bottom temperature ( C , July 8,1979) in the Bohai and Huanghai Seas.
Fig. 7. The vertical temperature profile from trial prediction(Tp) and observation (T,) at some grid points in the Bohai and Huanghai Seas. a-j are the station numbers shown as in Table 2.
286
Fig. 8. Trial-predicted (a) and observed (b) rneridional temperature (C)at 123.5 ' E. Table 2 Characteristics of predicted (T,) and observed (T,) at 10 grid points in the Bohai Hunghai Seas
ACKNOWLEDGEMENTS The authors wish to thank Dr. Takano for his valuable suggestions and thanks go to Ms.Zhang Hongnuan for drawing the figures. This work was supported by the National Planing Committee, PRC.
287 Appendix: The implicit differential from For C and u ,and explicit differential from for v from k A t to &+1/ 2)At for Eqs . (5)-(7) are as follows:
where
where
288 the implicit differential form for ( and v, and explicit differential form for u from (k+l / 2)At to (k+l)Atare as follows :
where
where
289
vn
Velocity comonents F,T,Y, and (26)-(31) denote the average values at the calculated points, and d, is two times average depth of the point (i, j ). The differential forms of Eqs. (11),(15) and (16) from K A t to ( k + l ) A t are as follows :
290
REFERENCES Halpern, D., 1974, Observation of the deepening of the wind-mixed layer in the northeast Pacific Ocean. J. of Phys. Oceanogr., 4,454--466. Japan Meteorological Agency, 1985, The results of marine meteorological and oceanographical observations. No. 78. Kalazkii, V. E., 1978, Simulation of vertical thermal structure for the Ocean active layer. Leningrad (in Russian), 1978. Kalakzkii, V. E. and E. S. Nesterov. 1980, Numerical prediction of the ocean thermal structure with the influence of atmospheric process in weather scale. Tr. GMC SSSR (in Russian), No. 229; 37-44. Kitaigorodskii, S. A. and V. Z. Miropolskii, 1970, On the theory of the ocean active layer. LEV. Akad. Nauk SSSR, Ser. F A 0 (in Russian), No. 6,177--188. Kitaigorodskii, S. A. 1977, Dynamics of seasonal thermocline. Okeanologiya (in Russian), No. 4,6-34. Kraus, E. R. and J. S. Turner, 1975, A one-dimensional model of the seasonal thermocline II : the general theory and its consequences. Tellus, No. 19,98--106 Kraus, E. R., 1977, Modelling and prediction of the upper layers of the ocean. Pergmon Press. 1977. Leendertse, J. J., 1967, Aspects of a computional for long period water-wave propagation. Memorandum, RM--5394--PR, Rand Corporation, May, 1967. Malkki, P. and R. Tamsalu, 1985, Physical features of the Baltic sea. Finnish Marine Res., No. 252, 50--67. Murakami, M. et al., 1985, A numerical simulation of the distribution of water temperature and salinity in the Seto Inland Sea. J. of Oceanogr. SOC.of Japan, No.41,213--224. Nesterov, E. S. 1978, Numerical prediction for the thermal features of the upper layer in the north Atlantic Ocean. Tr. GMC SSSR (in Russian) ,No. 200,22-29. Nesterov, E. S., 1986, Response of upper ocean to the temperate zone cyclone. Tr. GMC SSSR (in Russian), No. 281.24-34. Ocean Group of the Science and Technology Commission, PRC, 1964, Comprehensive marine investigation report. No. 2 and 5 Omsteds, A. et al., 1983, Measured and numerically-simulated autumn cooling in the Bay of Bothnia. Tellus, No. 35a, 231-240. Resnanskii, U.D. and E. V. Trosnikov, 1980, Parametrization in the ocean active layer developed from the simulating zonal atmospheric circulation. Tr. GMC SSSR (in Russian), No. 229, 18--31. Resnanskii, U. D.. 1983, The influence of current on the evolution of features in the ocean active layer. Tr. GM C SSSR (in Russian), No.282,23--33. Resnanskii, U. D., 1986, Numerical experiment in the ocean active layer considering the term of space variation. Tr. GMC SSSR (in Russian) No. 291,3--14. Wang Zongshan, 1983, Asimplifed method of computing heat budget at the 8ea surface. Mar. S. Bull. (in Chinese), No., 2,22-25.
291
Wang Zongshan and Zou Emei, 1986, A parametric modle for thermal structure features of the ocean upper layer. Acta Oceanol. sinica, No. 5,16--21. Wang Zongshan, Xu Bochang, Zou Emei, Yang Keqi and Li Fanhua, 1990, A study on the numerical prediction model for the vertical thermal structure in the Bohai and huanghai seas ( I )---One-dimensional numerical prediction model. Acta Oceanol. Sinica (in Chinese), 14(2), in Press. Xu Bochang et al., 1983, Simulation of vertical distribution of sea-water density (ul) in the shallows. Mar. S. Bull. (in Chinese), No. 1,9-12. Xu Bochang et al., 1984, Simulation of vertical distribution of water temperature in the shallows. Mar. s. Bull. (in Chinese), No.3,1-6. Zbang Yangting and Wang Yijiao, 1983, Simulation of wind field and numerical computation of storm surge in the Bohai sea. Acta Oceanol. Sinica (in Chinese), 5(3), 261--272. Zhang Yangting and Wang Yijiao, 1988, The properties and the numerical modelling of the Typhoon surge of Huanghai sea. Preceedings of third science meeting of Chinese Oceanol. and Limnol. SOC. (in Chinese),57--63.
NUMERICAL PREDICTION OF THE VERTICAL THERMAL STRUCTURE IN THE BOHAI A M ) HUANGHAI SEAS--TWO-DIMENSIONAL NUMERICAL PREDICTION MODEL Wang Zongshan, Xu Bochang, Zou Emei, Gong Bin, and Li Fanhua First Institute of Oceanography, SOA, Qingdao, China
ABSTRACT By using observed sea temperature data, we caIculated the non-dimensional depth q and its rele vant non-dimensional temperature OT to construct a similarity function OT= f(q ). Based on these, the thickness of the upper homogeneous layer h = h(x,y,t), surface temperature Ts= Ts(x,y,t), temperature in t h e thermocline T z = T,(x,y,z,t), bottom temperature T, = T,(x,y,t), the current velocity u = u(x,y,t), v = v(x,y,t) and the surface topography [ = [(x,y,t) were assumed to establish a numerical prediction model of the vertical thermal structure. The solution of this two dimensional model was obtained by using "ADT" and "HN" methods. The results of a trial prediction are satisfactory.
1 INTRODUCTION Until now, the studies of prediction on the vertical temperature structure have been focused on the characteristics of the upper ocean homogeneous layer (Kitaigorodskii, 1977; Nesterov, 1978; Resnanskii et al., 1980,1983,1986; Kraus et al., 1975,1977). Few papers are concerned with the prediction of the vertical temperature structure. Early mathematical models are divided into three categories: (1). Models for solving the closure equations including the momentum equation, continuity equation, equation of state, thermal equation and salinity equation (Kitaigorodskii, 1977; Kraus, 1977; Kalazkii, 1978, 1980; Murakami et al., 1985; Omsteds et al. 1983); (2). Forecasting models for describing the characteristics of the upper thermal structure according to the universal function derived by similarity theory (Wang et al., 1986); (3). Studies of the methods for calculating and simulating the vertical temperature structure in terms of the similarity function of the vertical temperature distribution (Xu et al., 1983, 1984) and thermal equation (Malkki et al., 1985). Category (1) is too difficult to put into effect because so many hydrographical and meterological factors have to be known for prediction. The coefficients in the equations are artificially taken as constants in most cases, which often produces errors in forecasting. As for category (2), though the model itself is simple, it describes only the mean characteristics of the vertical temperature structure. Category (3) need not some relevant coefficients in the equations, and the vertical temperature structure can be obtained somewhat objectively. Based on the category (3) and the available data of wind and air temperature field, a one-dimensional numerical prediction model for the vertical temperature structure in the Bohai and Huanghai Seas was developed (Wang et al., 1990). In the model, the effects of advection and lateral mixing are neglected, which holds good for the Bohai and Huanghai Seas. We could also verify it with the satisfactory trial prediction of the above model. However, with a strong cyclone passing, the simulated current velocities can reach 130 and 300 cm / s in some areas of the Bohai and Huanghai Seas, respectively (Zhang et al., 1983, 1988). Such huge wind-induced current will induce a large change in the three dimensional temperature structure. When a cyclone with wind speed 25-30111 / s passes over the deep sea at mid-latitudes, the upper layer temperature in a large area decreases by more than 2C and the upper homogeneous layer thickness increases by more than 6 m. The depth influenced by wind could reach lOOm (Japan Meteorol. Agency, 1985; Halpern, 1974; Nesterov,
.
278 1986). The upper homogeneous layer thickness in the Bohai and Huanghai Seas can increase by 5-8 m when heavy wind blows continuously over 24 hours (Ocean group of STC, PRC, 1964). The large. change of the thermal structure makes effective operation of acoustic instrument and fishing activities difficult. Therefore, the effect of the wind-induced current on the thermal structure redistribution needs to be understood. In this context, we developed a numerical prediction model for the vertical temperature structure as associated with wind-indud current and lateral mixing.
2 PHYSICAL MODEL 2.1 SIMILARITY FUNCTION OF THE VERTICAL TEMPERATUREPROFILE The vertical temperature distribution in the Bohai and Huanghai Seas in warm seasons is schematically shown in Fig. 1, which is divided into three layers: (1) the upper homogeneous layer, whose
r
I
I
I
I
h
2
Fig. 1. Schematic represemtation of the vertical temperature profie. thickness is h; (2) the thermocline; (3) the deep layer. Therefore, in the two-dimensional temperature structure, the non-dimensional depth q and its revelant non-dimensional temperature Or is written as: T -T Z-h OT=t q=(1) T,-T, H-h where Ts(x,y,t) denotes the surface temperature (or the tempcraturc in the upper homogeneous layer); TZ(x,y,z,t), the temperature which vanes with depth z; T,(x,y,t), the temperature in the bottom layer; H, the water depth. Dimensionless variable OT is a function of tj (Kitaigorodskii et al., 1970; Malkki et al., 1985; Xu et al., 1983, 1984). Based on the observed data of vertical temperature profile and meeting to boundary condictions: 'T={
=O =1
when
d
=O 1 I
'
a similarity function has been constructed by using the least square method( solid line in Fig. 2):
,,
where a a z, a 3, a ,
O,=a,tj+a,tja
+a,i'+a,v'
are empirical constants determined by 5053 historical temperature profile data
(2)
279
a4
a2
0
a6
a8
1.0
Fig. 2. The similarity profile of sea temperature in the Bohai and Huanghai Seas. obtained by Nansen casts in the Bohai and Huanghai Seas during May-October from 1958 to 1982. We by: denote K, and
zT
I
The average value of K, and K rfor the Bohai and Huanghai Seas are 0.75 and 0.30, respectively. According to the above formulae, the vertical temperature structure is expressed by following equa-
tions:
T , = T,(x,y,t),
OGzGh
T , = T,(x,y,t) - B,(W',(x,y,t)
-
T,,(x,y,t)l,
h szGH
(4)
2.2 GOVERNING EQUATIONS T o describe the two-dimensional current field, we use the following equations (Zhang et al., 1983, 1988):
aV at + u a-V+ ~ - +JV f i + ~ - - Ka(- = JX
Jy
ay
a2v 7 JY
---+--------1 aP 1 P , =Y
P,
Z a y -'by
H
in which u = u(x,y,t), v = v(x,y,t) are the velocity components, C = [(x,y,t) the sea level, K, the turbulent viscosity coefficient, f the Coriolis parameter, pw the water density (taken as l), H the water depth, P the pressure, g the gravitational acceleration, and 7b the surface and bottom frictional stress given by:
280
and
with pa (air density), C , (the drag coefficient), K,= 1 / M ?"(friction coefficient at the bottom), M (Manning coefficient), E(mean depth in each mesh). In Eqs. ( 5 ) and (a), the first term of right side can be ignored in comparision with the second term. Neglecting the relevant processes of the molecular diffusion, the heat conduction equation is:
"r + u-JT + v-JJyT + w-aJZT - A , V 1 T = Jf JX
-
J(
w'T+R) JZ
~
where T is the temperature, A,the horizontal turbulent heat conduction coefficient, W'T'the average vertical turbulent heat flux, R the penetrative component of the solar radiation divided by C,p,( Cpis specific heat of the sea water). Since u and v are independent ofz, w = O at z=O, and the solar energy is assumed to be entirely absorbed by the sea surface, integration of Eq. (10) over the upper homogeneous layer leads to:
- + u S +JT v L - A ,JT v JT, Jl
z T , = -Q s - Q H h
JY
JX
~
where Qs = QL/ (C,p,) is the surface heat flux, Qh= W'T'I h the mean turbulent heat flux at the lower boundary of the upper homogeneous layer, and QLtheheat budget at the sea surface. In order to derive an equation for h and T,, we substitute Eq. (4) into Eq. (10) and then integrate it with respect to z from h to H to get:
JT JT (l-O,)[L+u$ at
JT JT JT +v~-AA,v2T,]+0,[2+uL JY
Jl
JX
JT, +V--A,~~T,,]-(~ JY
Integration of Eq. (12)with respect to { from 0 to 1 with the boundary condition Q, = 0 gives:
JT
(1 - K,)[$
JT
aT
JT
+ u JX s + VS - R r V Z T , ]+ K,[-$ Jy
T -T H-h
4-K , I S - ' -
ah ah $ +u- +v- --Arvzh]-2A,[(g Har ax sy
JT
JT
+ U> J X + v HJY h
JT
JX
-A
J T , ah
--)- Jx
Jx
rv'T,I JT,
+(---)-I Jy
Jy
ah Jy
281
Double integration of Eq. (12)with respect to I,first from 0 to q, then from 0 to 1, yields: 1 -J T , aT, JT -JT JT JT ( s - K r ) [ y + U ~ + V ~ - - A ~ V ’ T ~ I + KJt, [ +- u’ A-J X+ v L - AJy JY -T , - T n ah ah ah 1 2 A , aT, a T ~ah 2K7 -{ (- + U- + V- - A ,.V h)l--I- OX JX H-h Jl JX Jy
=[(ax
+
T ~ - T ah~ 2 [(GI ( H - h)’
JT J T ~ ah +(2 --)-1-2A JY JY JY
-4
I
0
~
-
T s - T T , ah a h - 1 ) e 7 d ~ i A r[(GI ( H - h)’
u =(Q,
where
vlT 1
- i Q d q ) / Q,,
ah +(G) 11+[2K,-a, 1
+ (JhG z)I-= V Q r
=o
Q = (w”+ R). Simultanous equations (Eqs. (13) and (14)) to-
0
gether with Eq. (1 I ) give: JT,, JT, -+ UJt
ax
aT + YH - A,v’T,, ay =C
‘ h
,
z -C
+ C,A.=
QS-Qh
l
T - T ahi N), ( H - h)’
aha +(GI 1
h
(15)
and
C, = ( 2 K Y - K,K,C, = [2a,K+,
2K
1
- K ) / (KrK,) 2 7
rc-,
a K
+ 4K
I
J ( q- 116 ,dq] / (K 7 K T ) 0
C, = ( K , v - K ~ ) / ( K , K , )
c, = (-21 K 1 - K<)/ (K,K,) I
C , = [a , ( K r -);K
- 4K7 J(v - lYrdql/ ( K r K f ) 0
According to a processing method presented by Wang et al. (1990), Qs,Qh and y in Eqs. ( l l ) , (IS) and (16) are defined as follows. Qs is written by a formula (Wang, 1983):
282 where T, is the air temperature, and e,, e,empirical constants. It is difficult to get accurate value of the heat flux (QJ through the lower boundary of the upper homogeneous layer, because Qhis related to the entrainment of the lower cold water to the upper homogeneous layer rather than the simple change in h. Based on turbulent kinetic energy equation, Resnanskii (1980, 1983, 1986) showed that the vertical salinity variation in the mid- latitude areas has almost no effect on the buoyance flux, and entirely depends on the vertical turbulent heat flux W’T’. He also neglected the horizontal variations in the mean velocity, turbulent kinetic energy and their flux. Integration of the turbulent kinetic energy equation from 0 to h and use of Eqs. (10) and (11) gives: 2F Q = - Q -Bh Here B= ga,is the buoyance coefficient, a,the heat expansion coefficient of sea water, F the conversion rate of turbulent kinetic energy in the upper homogeneous layer to potential energy, which is subject to the following conditions: the predicted h is close to the measured h under the wind mixing, andQ,and entrainment velocity decrease with increasing h. Parameter F is written as (Resnanskii, 1983):
in which,b,, bz m are empirical constants taken as 2,5, and 2, respective1y.A = A(x) is the unit function of independent variable x= (U, -b,hlfl) withA = 1 when x>O and A = 0 when x < 0. U . = ( T /~ pw)”*the friction velocity of sea water. In order to determine u, Eqs. (13) and (14) are transformed into: d[l - K , + K , n + K , B
+ (2Kr -n,)a]=
1
(21)
Eqs. (21) and (22) give: 1 u = [- - KY(1- n - 2 8 2
I
201)- (a1 + 4 j ( q - l)Ordq)al/ [I - K,(I - n -8 0
- 2a)-
a,a].
By use of historical temperature profile data and meteorological data, and corresponding u, v,
(24)
283
Qs and Qh calculated from Eqs. (5)-(7), (IS), and (191, y can be determined with Eq. (24). The average value of y is 0.06 in the warming period and 0.53 in the cooling period in the Bohai and Huanghai Seas. Then the numerical prediction model is constructed by using Eqs. (5) to (7),(ll), (IS) and (16). At the closed boundary, V, = 0, Ts =TH,h = H, Q, = Qh = QH = 0. At the open boundary from the Changjiang river mouth to Chejudo island, au / aX = 0, aV / ay = 0. The sea level ( at the closed boundary is specified by:
c=
n
c ~ ( x , Y ) , C o 4 U -l t
I- I
(25)
R(X,Y),I
where i = 1,2, ... ... ,n denotes the serial number of tidal constituent, Hiand giare the harmonic constants of ith tidal constituent, ci is angular velocity. The initial conditions are given by:
u=v=O, C=O, h=h, Ts=T,. TH=Tho, whent=t,,. 3 NUMERICAL MODEL The parallel of 36 N is taken as x axis (positive eastward), and the meridian of 120 E as y axis (positive northward), the mean sea level a8 z = 0 (positive upward). 3.1 DIFFERENCESCHEME In order to get a finite difference analog of the prediction model in differential form, we chose Platman's alternative grid, and use alternating direction implicit method (ADI) (Leendertse, 1967; Zhang et al., 1983, 1988). The features of this method are: (1) the calculation is simple because an implicit differential form is alternatively used in one direction when estimating the variables in x and y directions. (2) an implicit and an explicit differential form are alternatively used in x and y directions that make the calculation stable and quick convergent. In the meantime a hydrodynamic numerical method with temporal forward difference and special central difference (HN)is used to disctetize the Eqs. (It), (15) and (16) describing the vertical temperature structure. We evaluate ( ,h. T , That a grid point (i. 2, u, w,at a grid point (i+l / 2, 3, v, w,at a grid point (i, jtl / 2) and H at a grid point (i+l / 2, j+l / 2). O
0 1 0 1 0 - + - + O
-
J
.
I
1 - 7 0 j-1
-
+ 1 t
-
i 4
O -
l
G
t
-
0 1 0 1 0 -
f
0
1
-
0
+ '
Fig. 3. Definition for computed grid points and variables.+-(.
0
h, Ts,TH; --u, w,;
I-v,~,;
0-H
284 3.2 DIFFERENCE FORM Variables U’, V’, Ckareobtained from Eqs. ( 5 ) to (7) at k A t intervals. The appendix gives the implicit differential form for C and u, and explicit differential form for v from k A t to (k+1/ 2)A t; the implicit differential form for and v, and explicit differential form for u from (k+l / 2) A t to (k+l)A t; and the direrentialform ofEqs. (ll), (15) and (16)from k A t to (k+l)At. 4 APPLICATION TO TIDAL PREDICTION The wind speed w and air temperature T,,are known. The grid spacing is 20 km. The time step is 900 seconds. The basic parameters are specificed as in Table 1. TABLE 1 The relevant parameters in the model (cgs)
I
I
PI
I
I .229 x lo-)
I
CD
c,
0.938
K,
a
0.25 x lo-)
K,
g
I
980
I
I
(0.8719+0.000704W~0)Xlo-’
I
10’
I
1 / ME’’‘ ,M = 0.016-0.018
* W,, means the wind speed at 10 m above the sea surface. In the forecasting procedure, first of all, we calculate the current field from the given wind field, then put the current field and the given air- temperature field into the governing equations to predict h, Tsand TH.Computation was made on IBM-4381 computer. 5 THE RESULTS AND DISCUSSION OF TRIAL PREDICTION Because of the limited observations in this area, we only take the wind, air temperature and sea surface temperature data during FGGE as known variables to forecast the vertical temperature structure. The period of validity is about 4 days. The surface temperature (Ts), thickness of the upper homogeneous layer (h), bottom temperature (TI(), meridional distribution of temperature along 123.5 ’ E and vertical temperature profiles on July 8, 1979 are presented in Figs. 4 to 8. These figures show that the sea surface temperature in the study area is 22-23C in most areas except that the SST in Haizhou Bay is higher than 25C and there is a cold eddy north of Cheng Shantou (Fig. 4). There are four areas where the thickness of the upper homogeneous layer is great (Fig. 5): the first one is located in the central Bohai Sea (greater than 5 m), the second one lies in the north ofthe northern Huanghai Sea Cold Water (greater than 15 m), the third one is at about 34.5 N, 123 ’ E (greater than 10 m) and the last one in the Huanghai Trough area has the greatest thickness (20 m-25 m). The thickness ofthe upper homogeneous layer reaches 10 m off the coast of northern Jiangsu province due to strong mixing and is about 5 m in the other areas. The temperature profile from the trial prediction (Fig. 8a) clearly shows the upper homogeneous layer, thermocline and deep layer. In order to verify the reliability of the trial prediction, we roughly compare the calculated results with observed data obtained from a simultaneous hydrographic survey (Fig. 8b). Comparison shows that: ( I ) the temperature profile of trial prediction coincides with the observation north of 36 N, but there is a discrepancy between them in the deep layer south of 36 ‘ N , which might be due to a large meridional distance and asynchronous observational data between stations. (2) the thermocline obtained from the trial prediction is thicker than the observed one and the predicted temperature is vertically homogeneous in the thermocline, wbicb may be due to the assumption that the current velocity is vertically homogeneous that induces the strong mixing between thermocline and its lower boundary. So three dimensional numerical prediction method is needed to put forward in the future.
285
30 *
Fig. 4. Sea surface temperature ( C ) from the trial prediction (July 8,1979 in the Bohar and Huanghai Seas.
Fig. 5. Thickness (m)of the upper homogeneous layer from the trial prediction (July 8, 1979) in the Bohai and Huanghai Seas.
30
Fig. 6 . Bottom temperature ( C , July 8,1979) in the Bohai and Huanghai Seas.
Fig. 7. The vertical temperature profile from trial prediction(Tp) and observation (T,) at some grid points in the Bohai and Huanghai Seas. a-j are the station numbers shown as in Table 2.
286
Fig. 8. Trial-predicted (a) and observed (b) rneridional temperature (C)at 123.5 ' E. Table 2 Characteristics of predicted (T,) and observed (T,) at 10 grid points in the Bohai Hunghai Seas
ACKNOWLEDGEMENTS The authors wish to thank Dr. Takano for his valuable suggestions and thanks go to Ms.Zhang Hongnuan for drawing the figures. This work was supported by the National Planing Committee, PRC.
287 Appendix: The implicit differential from For C and u ,and explicit differential from for v from k A t to &+1/ 2)At for Eqs . (5)-(7) are as follows:
where
where
288 the implicit differential form for ( and v, and explicit differential form for u from (k+l / 2)At to (k+l)Atare as follows :
where
where
289
vn
Velocity comonents F,T,Y, and (26)-(31) denote the average values at the calculated points, and d, is two times average depth of the point (i, j ). The differential forms of Eqs. (11),(15) and (16) from K A t to ( k + l ) A t are as follows :
290
REFERENCES Halpern, D., 1974, Observation of the deepening of the wind-mixed layer in the northeast Pacific Ocean. J. of Phys. Oceanogr., 4,454--466. Japan Meteorological Agency, 1985, The results of marine meteorological and oceanographical observations. No. 78. Kalazkii, V. E., 1978, Simulation of vertical thermal structure for the Ocean active layer. Leningrad (in Russian), 1978. Kalakzkii, V. E. and E. S. Nesterov. 1980, Numerical prediction of the ocean thermal structure with the influence of atmospheric process in weather scale. Tr. GMC SSSR (in Russian), No. 229; 37-44. Kitaigorodskii, S. A. and V. Z. Miropolskii, 1970, On the theory of the ocean active layer. LEV. Akad. Nauk SSSR, Ser. F A 0 (in Russian), No. 6,177--188. Kitaigorodskii, S. A. 1977, Dynamics of seasonal thermocline. Okeanologiya (in Russian), No. 4,6-34. Kraus, E. R. and J. S. Turner, 1975, A one-dimensional model of the seasonal thermocline II : the general theory and its consequences. Tellus, No. 19,98--106 Kraus, E. R., 1977, Modelling and prediction of the upper layers of the ocean. Pergmon Press. 1977. Leendertse, J. J., 1967, Aspects of a computional for long period water-wave propagation. Memorandum, RM--5394--PR, Rand Corporation, May, 1967. Malkki, P. and R. Tamsalu, 1985, Physical features of the Baltic sea. Finnish Marine Res., No. 252, 50--67. Murakami, M. et al., 1985, A numerical simulation of the distribution of water temperature and salinity in the Seto Inland Sea. J. of Oceanogr. SOC.of Japan, No.41,213--224. Nesterov, E. S. 1978, Numerical prediction for the thermal features of the upper layer in the north Atlantic Ocean. Tr. GMC SSSR (in Russian) ,No. 200,22-29. Nesterov, E. S., 1986, Response of upper ocean to the temperate zone cyclone. Tr. GMC SSSR (in Russian), No. 281.24-34. Ocean Group of the Science and Technology Commission, PRC, 1964, Comprehensive marine investigation report. No. 2 and 5 Omsteds, A. et al., 1983, Measured and numerically-simulated autumn cooling in the Bay of Bothnia. Tellus, No. 35a, 231-240. Resnanskii, U.D. and E. V. Trosnikov, 1980, Parametrization in the ocean active layer developed from the simulating zonal atmospheric circulation. Tr. GMC SSSR (in Russian), No. 229, 18--31. Resnanskii, U. D.. 1983, The influence of current on the evolution of features in the ocean active layer. Tr. GM C SSSR (in Russian), No.282,23--33. Resnanskii, U. D., 1986, Numerical experiment in the ocean active layer considering the term of space variation. Tr. GMC SSSR (in Russian) No. 291,3--14. Wang Zongshan, 1983, Asimplifed method of computing heat budget at the 8ea surface. Mar. S. Bull. (in Chinese), No., 2,22-25.
291
Wang Zongshan and Zou Emei, 1986, A parametric modle for thermal structure features of the ocean upper layer. Acta Oceanol. sinica, No. 5,16--21. Wang Zongshan, Xu Bochang, Zou Emei, Yang Keqi and Li Fanhua, 1990, A study on the numerical prediction model for the vertical thermal structure in the Bohai and huanghai seas ( I )---One-dimensional numerical prediction model. Acta Oceanol. Sinica (in Chinese), 14(2), in Press. Xu Bochang et al., 1983, Simulation of vertical distribution of sea-water density (ul) in the shallows. Mar. S. Bull. (in Chinese), No. 1,9-12. Xu Bochang et al., 1984, Simulation of vertical distribution of water temperature in the shallows. Mar. s. Bull. (in Chinese), No.3,1-6. Zbang Yangting and Wang Yijiao, 1983, Simulation of wind field and numerical computation of storm surge in the Bohai sea. Acta Oceanol. Sinica (in Chinese), 5(3), 261--272. Zhang Yangting and Wang Yijiao, 1988, The properties and the numerical modelling of the Typhoon surge of Huanghai sea. Preceedings of third science meeting of Chinese Oceanol. and Limnol. SOC. (in Chinese),57--63.