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Planetary flow in a circular basin
Deep-SeaResearch, 1974, Vol. 21, pp. 69 to 77. PergamonPress. Printed in Great Britain.
SHORTER CONTRIBUTION Planetary flow in a circular basin SHmo IMAW~* and IO~zo T~dCANOt (Received 12 December 1972; in revised form 5 July 1973; accepted 16 July 1973)
Abstract--An analytical solution of the vorticity equation is obtained for a stationary, linear, planetary flow in a circular basin rotating about its center. Thc~'e is an inflow and an outflow on the cirmmfference. The inflow deflects leftward (westward in the case of the Northern Hemisphm~) immediately after entm~ng the basin and divides into two branches connecting the inlet to the outlet; one is a cyclonic flow and the other is an anticyclonic flow along the lateral wall. The results have been confirmed by a simple model experiment on a turntable. INTRODUCTION MOST OF the studies, either analytical or experimental, on circulation are concerned with the ocean in low and/or middle latitudes where the variation of the Coriolis parameter with latitude is of essential importance, as are the meridional boundaries for producing the principal features of the actual circulation. However, relatively little is known about the dynamics in oceans in high latitudes, such as the Arctic or the Antarctic oceans, which may differ somewhat from that in the other oceans, because the Coriolis parameter is large but its variation with latitude is very small and there are no long meridional boundaries. Our fundamental knowledge on the dynamic, thermal or thermodynamic processes, peculiar to these oceans is still sparse, though the interest and the importance of the polar oceans are increasing, particularly in relation to sea-air interaction and climatic problems. The present short note deals with steady, linear, two-dimensional fluid motions in a circular basin induced by an inflow and an outflow on the circumference. Our objective was to gain insight into some of the basic properties of the currents in a circular basin free from the meridional boundary, as a preliminary step in solving more substantial problems including thermohaline processes, ice phases and lateral and lower boundaries of irregular shape. First an analytical solution is presented, followed by experimental results on a turntable. Not a few studies (FALL~, 1960; IBnETSONand PHILLIPS, 1967; PEDLOSKYand G~NSPA~, 1967; HIDE, 1968; BEARDSLEY,1969; BAKER, 1971 ; HOLTON, 1971 ; BEARDSLL~,1973) were done on the flow in a circular basin. However, to the authors' knowledge, the present problem has not yet been studied. ANALYTICAL SOLUTION In a schematic representation of the basin considered (Fig. 1), an inlet and an outlet of arbitrary width are located on the circumference. Since only the steady state is studied, the total inflow Vt is made equal to the total outflow Vo. Both V~and Vo are constant in time. The bottom is horizontal. The surface is free from wind stress. The flow is sufficiently slow that the nonlinear effect (momentum advection) can be neglected. The equation of horizontal motion with spherical coordinates is written as f k r X v = -- 1/pgradp + f.
(1)
Here, r is the radial distance from the center of the Earth, 0 the colatitude and ~ the longitude (see Fig. 1), f the Coriolis parameter which equals 2¢o cos 0, oJ being the angular velocity of the Earth's rotation, kr the radial unit vector, v the horizontal velocity, p the density, p the pressure and f the friction due to eddy viscosity. Vertical integration of equation (1) from the bottom to the surface gives f k r x V = -- 1/pgradP + F + ~ B ,
*Geophysical Institute, Kyoto University, Kyoto 606, Japan. tRikagaku Kenkyusho, Wako-shi, Saitama 351, Japan. 69
(2)
70
Shorter Contribution
..... 9z
Vi Fig. 1. Schematic representation of the coordinate system with principal notation.
where V, P and F are the vertical integrals of horizontal velocity, pressure and lateral friction, respectively, and xB is the bottom friction. The two components Fo, Feo of the lateral friction F are given by 2cosO 0Vq~] 2cos00Vo]
F~=A~[AV~--R-~(I+si~)Vq~+R~sin,
O~
I,
where l
R~sinO O0 ~,
00/
R2sin20~2"
A~ denotes the coefficient of horizontal eddy viscosity, Vo, V~ the two components of volume transport V, and R the radius of the Earth. According to Ekman's theory, the bottom friction ~a is approximately written in terms of V as =
_
~B
~_/lsAo~ ~v + k, × v), HA/~
2 /
C3)
where A~ is the coefficient of vertical eddy viscosity and H the depth of the basin. Since the coefficient of vertical eddy viscosity A~ is much smaller than the coefficient of horizontal eddy viscosity A~, the importance of the bottom friction is much reduced. In fact it is justified a posteriori that the bottom friction is neglected. Equation of continuity is div V = 0. (4) Taking the curl of both sides of equation (2) to eliminate P and using equation (4), we have the vorticity equation • 2~sinO Vo = 0. curb F --tHere, 2o,sinO/R is usually written as 8.
(5)
Shorter Contribution
71
I f the volume transport stream function ¢ is defined as V =kr
× grade,
equation (5) is transformed into
Osb-~-~+ 0 3 b ~ +
~-~+01~+
0 4 - - +Os
+06
~ +07b~-
~
÷=0,
(6)
with 01_2cos0,0~= sin0 04
2 , ( 1 ) si--~'0 0a = -- 2 + si-~--0
2cos0,05 cos a0 0 -1 sin30 = si--~-0' e sin 4 0 07-
(7)
1 + 3 c o s ~0 2o~R~ sin 4 0 ' ~ = A---~-
The boundary conditions are as follows. Both meridional and zonal transports are zero at the wall (0 = 0~) except at the inlet and outlet. The meridional transport is prescribed across the inlet and outlet, where the zonal transport is assumed to be zero. The meridional and zonal transports are zero at the center (0 = 0). These conditions are written as (Fig. 2)
~ 1 0 - 0t =
~o ~/gx
for 0 _< ~? _< ~i
¢o
for ~x ~ ~ ~ ~
¢o -- 2¢0 ~ -- ~e ¢pa -- 92
for ~ _< ~ < ~
-- ~o
for ~a _~ ~ _~ ~4
- ~0 + ~0 ~9--,--~ 4~
forcp4_<
~0~0[o. o. = 0 ~¢ -- ~¢ -- 0 09 o0
I I t I
_<2~r
forO < ~ < 2~r for 0 = O,
I I I I
/
0
Fig. 2.
Stream function p r e ~ r i b e d o n the circumference.
)
72
Shorter Contribution
where 9, (i : 1, ..., 4) are constants specifying the locations of the inlet and outlet. The last conditions for 0 = 0 have no dynamic effect on the fluid motion. They are dummy conditions. The stream function prescribed on the circumference is expanded in Fourier series as $(Ob, ~) - ao + ~
(ancosn9 + b,,sinng),
n=l
where
L (2~
ao = 2 r r j ° ~b(0b, ~) d~ =
$0[(,2
-
,1)
-
(,4
-
,8)1/2,~
= ! [ "2~ an *rJo ~0~, 9)cosngd~
'~.]o The solution is assumed to be expanded in Fourier series as
4,(0, 9) = Ao(O) + ~. [A,(0)cosn~ + B,(0)sinn9].
(8)
The term A0(0) gives generally a zonal flow independent of longitude which satisfies the boundary conditions at 0 = Oband 0 = 0. Since only the flow caused by a n inflow and an outflow specified on the circumference is dealt with, the term Ao(0) must be a constant independent of O, which is equal to ao. It is noteworthy that the value of the stream function becomes a0 at the center (4[0-0 = ao), because the second term in the fight-hand side of(8) must vanish at the center (0 = 0) irrespectively of longitude 9. In other words, the transports through the left-hand side and the right-hand side of the center are immediately known from the locations of the inlet and outlet, without any long computation. The latitude 0 in (7) is assumed to be a constant 0c. It will be shown later that this simplification does not affect the result ser/ously. The function A~(0) and Bn(O) are, in turn, assumed to be of the form: 4
An(O) = ~ aknexp(t:O) k=l 4
Bn(O) = ~. b*nexp(t~nO), k=l where ak'* and b , '~ are coefficients to be determined by the boundary conditions. The solution (8) is substituted together with the above A~(0) and B,,(0) into the vorticity equation (6) to get t,n(k = 1, •.., 4; n = 1, ..., oo) as roots of the following quartic equation: t 4 + 01t a + (--n2OS + 03)t 2 + (--n204 + 05)t + naOe - - nSOv - - Kni = O.
In the case where the coefficient of horizontal eddy viscosity Ah is less than or equal to l0 s cmS/sec, the term 04~b/a04in equation (6) almost balances with the term - , ¢ ~ / 0 9 , the coefficient of which does not include 0. Hence no serious error should be made by replacement of 0 by 0c in Odi = I, ..., 7). For sufficiently large n and 0 _< 0 < 0~, the terms An(O) and Bn(0) are A,(O) = anGn(O) B . ( o) = b . G . ( O), where
Gn(O) - n(O.,~__O)exp [ sm oc
n(O,--O)] ~
.1"
Since the series ~ (a,,cosn9 + bnsinn~) converges and the magnitude of G,,(O) decreases faster than n=l
1/n for n--~o% the series ~ [A~(O)cosn~ + B,,(O)sinng] converges uniformly and can be differentiated term by term with respect to 0. This proves the validity of the above mathematical procedure.
Shorter Contribution
73
NUMERICAL EXAMPLE
A numerical computation was carried out for the following values; ~ = l 0 °, ~ = 170°, ~a = 190°, cp4 = 3 5 0 °,
Vo = 2~b0 = 10 i s c m S / s e c , 0~ = 2 0 °, 0c = l 0 ° a n d A~ = l 0 s c m 2 / s e c . T h e first 101 t e r m s
(n < 101) are retained in solution (8). The error introduced by neglect of higher terms is very small. The result is shown in Fig. 3.
180°
2
90 °
0o Fig. 3.
Stream lines in the case of a non-slip boundary. Contour interval 1018 cma/sec.
The main features of the flow are: (1) the inflow deflects leftward (westward in the case of the Northern Hemisphere) immediately after entering the basin, (2) then, it divides into two branches connecting the inlet to the outlet; one is a cyclonic flow and the other is a n anticyclonic flow along the lateral wall, and (3) these do not diffuse uniformly but are concentrated near the boundary. When the inlet and outlet are distributed asymmetrically with respect to the center, the flow pattern becomes somewhat more complicated, but the fundamental features (1) tO (3) are to be observed without any significant change. The flow pattern is not much changed in the case of a free-slip lateral boundary. The stream lines are slightly more concentrated toward the lateral wall (Fig. 4). Figure 5 shows the solution for a non-slip lateral boundary for the case of ~ ---- 0. Comparison of Fig. 5 with Figs. 3 and 4 clearly indicates the importance of the/3-effect. MODEL E X P E R I M E N T A model experiment on a small turntable (Fig. 6) was undertaken to confirm the existence of these flow patterns. A plastic vessel (a) of ! 9-2 cm diameter is placed on a turntable (b). It is covered by a plastic lid so that the fluid surface will not be disturbed by the ambient air motion relative to the water. The diameters of the cylindrical inlet (c) and outlet (d) are 2.5 cm. The mean water depth H is 2.3 cm. The turntable rotates clockwise with a speed of 33~ rev/min. Water is poured through the pipe (e) fastened o n a fixed stand into a reservoir ( 0 rotating together with the vessel. The amount of water
74
Shorter Contribution 180°
2
7
0
~
90*
~45
3 1 5 " ~
"
hhT~/ oo
Fig. 4. Stream linesin the case of a flee-slipboundary.Contourinterval 10t2 cruZ/see.
i80 ~
2
7
:515
0
"
~
90*
45*
0° Fig. 5. Streamlinesin the case of non-rotation or uniformrotation 03 : 0). Contour interval 10TM cm3lsec.
Shorter Contribution
75
passing through the tube (g) is controlled by a pet cock (h) and by adjustment of the water level in the reservoir. Water flows into the vessel through the inlet (c) and flows out through the outlet (d). The water is discharged through a circular conduit (i). Water in the reservoir is dyed with methylene blue when a steady state is reached. Before presenting the results we will consider the principal dynamic processes governing this experiment. Water density can be considered as constant. The horizontal pressure gradient is, therefore, replaced by the surface slope. The cylindrical coordinate system (r, 9, z) is used with the conventional notation. In the case of no water motion relative to the vessel, the free surface height Ho is given by tgHo - - co2r
~r
(9)
g '
since the resultant of gravity and the centrifugal force is perpendicular to the free surface. Here, cois the angular velocity of the turntable, which equals -- 3 '49 rad/sec, and g is the acceleration due to gravity. The equation of motion is given by fk~
×
V
=
--
gHgrad~ + F + lIP%B,
(10)
wherefequals 2oJ, kz denotes the vertical unit vector, ~ the surface displacement from the equilibriunt state, and H the surface height which equals H0 + ~. The bottom friction tn is given by relation (3), provided that A~ and kr in (3) are replaced by the coefficient of viscosity A and k~, respectively. Equation (10) and the equation of continuity lead to curlzF + l curl~n + 2~ar Vr : 0,
(I l)
Vr denoting the radial component of the volume transport V. Here, the relation (9) and the geostrophic relation 2~ Vr = --gH/r 0~/~ were used to eliminate the radial gradient for the free surface OHo/Orand the surface slope ~ / ~ , respectively. The relation (1 l) does not hold very well near the lateral boundary where the lateral friction is more or less important. This should, however, not be crucial for the present analysis. It might be parenthetically noted that 2ojar/gHis of importance as an alternate for ~. The second term is approximately given by _1curlz~n p
~/(oJA) [curlzV _ g-H<°~r(Vr + V~)j_], H
where V~ denotes the azimuthal component of the volume transport. While the bottom friction could be ignored in the analytical solution (8), this is not the ease with the laboratory experiment, as is reported in the studies cited above and others (for instance, Vm~or~ts, 1967). The method for solving equation (11) is similar to that for equation (5). Figure 7 shows the analytical solution. For comparison, the solution without bottom friction is shown in Fig. 8. Both results are not qualitatively very different, although water flows closer to the lateral wall in the ease of no bottom friction. Therefore, the ratio A/L a to/3 should be considered as a rough measure of the similarity between the model and the prototype, where A is the coefficient of viscosity, L is a characteristic length which could be taken to be the radius of the basin, ~ is 2oJsinOdR for the prototype and 2oard(g/-/) for the model, r, being L/2. The coefficient A in the model can be taken as that of the molecular viscosity of pure water (1.1 × 10-z cm2/sec) because the Reynolds number is small (about 5) enough for the flow to be laminar. The ratio is 3 × I0 -4 for the prototype and 6 x 10-5 for the model. Since these two values do not differ from each other in order of magnitude, the experimental result is expected to be comparable with the analytical solution of (5). Figure 9 illustrates the spreading pattern of the dyed water with time, calculated from the velocity field shown in Fig. 7, on the assumption that only advection governs the spreading of the dye. Figures 10a, b, c and d illustrate a series of photographs taken successively 6, 12, 18 and 24 minutes after the dyed water flowed into the vessel. The inflow rate is 16 cmS/min. The spreading pattern in the vessel closely resembles that in Fig. 9, especially in the following aspects: (1) dyed water divides into two branches immediately after flowing into the vessel; the right branch flows more rapidly than the left branch, (2) the tip of the right branch is roundish, whereas that of the left branch is sharp, (3) the right branch overshoots the outlet, and (4) both right and left branches reach the outlet simultaneously.
76
Shorter Contribution
~8o*
y Fig, 7. Stream lines with the bottom friction for the model experiment. Contour interval 0"03 cmS/sec.
leo ~
90*
o" Fig. 8.
Stream lines without the bottom friction for the model oxperimont. Contour interval 0.03 craB/see.
Im ~ o1
~D
c~
3
o
J~
lid
!i!¸
Fig. 10 (a).
Fig. 10 (b). Fig. 10. Experimental result. Spreading patterns of dyed water at 6(a), 12(b), 18(¢) and 24(d) minutes alter the injection of methylene blue.
Fig. 10 (c).
Fig. 10 (d).
SHORTER CONTRIBUTION Planetary flow in a circular basin SHmo IMAW~* and IO~zo T~dCANOt (Received 12 December 1972; in revised form 5 July 1973; accepted 16 July 1973)
Abstract--An analytical solution of the vorticity equation is obtained for a stationary, linear, planetary flow in a circular basin rotating about its center. Thc~'e is an inflow and an outflow on the cirmmfference. The inflow deflects leftward (westward in the case of the Northern Hemisphm~) immediately after entm~ng the basin and divides into two branches connecting the inlet to the outlet; one is a cyclonic flow and the other is an anticyclonic flow along the lateral wall. The results have been confirmed by a simple model experiment on a turntable. INTRODUCTION MOST OF the studies, either analytical or experimental, on circulation are concerned with the ocean in low and/or middle latitudes where the variation of the Coriolis parameter with latitude is of essential importance, as are the meridional boundaries for producing the principal features of the actual circulation. However, relatively little is known about the dynamics in oceans in high latitudes, such as the Arctic or the Antarctic oceans, which may differ somewhat from that in the other oceans, because the Coriolis parameter is large but its variation with latitude is very small and there are no long meridional boundaries. Our fundamental knowledge on the dynamic, thermal or thermodynamic processes, peculiar to these oceans is still sparse, though the interest and the importance of the polar oceans are increasing, particularly in relation to sea-air interaction and climatic problems. The present short note deals with steady, linear, two-dimensional fluid motions in a circular basin induced by an inflow and an outflow on the circumference. Our objective was to gain insight into some of the basic properties of the currents in a circular basin free from the meridional boundary, as a preliminary step in solving more substantial problems including thermohaline processes, ice phases and lateral and lower boundaries of irregular shape. First an analytical solution is presented, followed by experimental results on a turntable. Not a few studies (FALL~, 1960; IBnETSONand PHILLIPS, 1967; PEDLOSKYand G~NSPA~, 1967; HIDE, 1968; BEARDSLEY,1969; BAKER, 1971 ; HOLTON, 1971 ; BEARDSLL~,1973) were done on the flow in a circular basin. However, to the authors' knowledge, the present problem has not yet been studied. ANALYTICAL SOLUTION In a schematic representation of the basin considered (Fig. 1), an inlet and an outlet of arbitrary width are located on the circumference. Since only the steady state is studied, the total inflow Vt is made equal to the total outflow Vo. Both V~and Vo are constant in time. The bottom is horizontal. The surface is free from wind stress. The flow is sufficiently slow that the nonlinear effect (momentum advection) can be neglected. The equation of horizontal motion with spherical coordinates is written as f k r X v = -- 1/pgradp + f.
(1)
Here, r is the radial distance from the center of the Earth, 0 the colatitude and ~ the longitude (see Fig. 1), f the Coriolis parameter which equals 2¢o cos 0, oJ being the angular velocity of the Earth's rotation, kr the radial unit vector, v the horizontal velocity, p the density, p the pressure and f the friction due to eddy viscosity. Vertical integration of equation (1) from the bottom to the surface gives f k r x V = -- 1/pgradP + F + ~ B ,
*Geophysical Institute, Kyoto University, Kyoto 606, Japan. tRikagaku Kenkyusho, Wako-shi, Saitama 351, Japan. 69
(2)
70
Shorter Contribution
..... 9z
Vi Fig. 1. Schematic representation of the coordinate system with principal notation.
where V, P and F are the vertical integrals of horizontal velocity, pressure and lateral friction, respectively, and xB is the bottom friction. The two components Fo, Feo of the lateral friction F are given by 2cosO 0Vq~] 2cos00Vo]
F~=A~[AV~--R-~(I+si~)Vq~+R~sin,
O~
I,
where l
R~sinO O0 ~,
00/
R2sin20~2"
A~ denotes the coefficient of horizontal eddy viscosity, Vo, V~ the two components of volume transport V, and R the radius of the Earth. According to Ekman's theory, the bottom friction ~a is approximately written in terms of V as =
_
~B
~_/lsAo~ ~v + k, × v), HA/~
2 /
C3)
where A~ is the coefficient of vertical eddy viscosity and H the depth of the basin. Since the coefficient of vertical eddy viscosity A~ is much smaller than the coefficient of horizontal eddy viscosity A~, the importance of the bottom friction is much reduced. In fact it is justified a posteriori that the bottom friction is neglected. Equation of continuity is div V = 0. (4) Taking the curl of both sides of equation (2) to eliminate P and using equation (4), we have the vorticity equation • 2~sinO Vo = 0. curb F --tHere, 2o,sinO/R is usually written as 8.
(5)
Shorter Contribution
71
I f the volume transport stream function ¢ is defined as V =kr
× grade,
equation (5) is transformed into
Osb-~-~+ 0 3 b ~ +
~-~+01~+
0 4 - - +Os
+06
~ +07b~-
~
÷=0,
(6)
with 01_2cos0,0~= sin0 04
2 , ( 1 ) si--~'0 0a = -- 2 + si-~--0
2cos0,05 cos a0 0 -1 sin30 = si--~-0' e sin 4 0 07-
(7)
1 + 3 c o s ~0 2o~R~ sin 4 0 ' ~ = A---~-
The boundary conditions are as follows. Both meridional and zonal transports are zero at the wall (0 = 0~) except at the inlet and outlet. The meridional transport is prescribed across the inlet and outlet, where the zonal transport is assumed to be zero. The meridional and zonal transports are zero at the center (0 = 0). These conditions are written as (Fig. 2)
~ 1 0 - 0t =
~o ~/gx
for 0 _< ~? _< ~i
¢o
for ~x ~ ~ ~ ~
¢o -- 2¢0 ~ -- ~e ¢pa -- 92
for ~ _< ~ < ~
-- ~o
for ~a _~ ~ _~ ~4
- ~0 + ~0 ~9--,--~ 4~
forcp4_<
~0~0[o. o. = 0 ~¢ -- ~¢ -- 0 09 o0
I I t I
_<2~r
forO < ~ < 2~r for 0 = O,
I I I I
/
0
Fig. 2.
Stream function p r e ~ r i b e d o n the circumference.
)
72
Shorter Contribution
where 9, (i : 1, ..., 4) are constants specifying the locations of the inlet and outlet. The last conditions for 0 = 0 have no dynamic effect on the fluid motion. They are dummy conditions. The stream function prescribed on the circumference is expanded in Fourier series as $(Ob, ~) - ao + ~
(ancosn9 + b,,sinng),
n=l
where
L (2~
ao = 2 r r j ° ~b(0b, ~) d~ =
$0[(,2
-
,1)
-
(,4
-
,8)1/2,~
= ! [ "2~ an *rJo ~0~, 9)cosngd~
'~.]o The solution is assumed to be expanded in Fourier series as
4,(0, 9) = Ao(O) + ~. [A,(0)cosn~ + B,(0)sinn9].
(8)
The term A0(0) gives generally a zonal flow independent of longitude which satisfies the boundary conditions at 0 = Oband 0 = 0. Since only the flow caused by a n inflow and an outflow specified on the circumference is dealt with, the term Ao(0) must be a constant independent of O, which is equal to ao. It is noteworthy that the value of the stream function becomes a0 at the center (4[0-0 = ao), because the second term in the fight-hand side of(8) must vanish at the center (0 = 0) irrespectively of longitude 9. In other words, the transports through the left-hand side and the right-hand side of the center are immediately known from the locations of the inlet and outlet, without any long computation. The latitude 0 in (7) is assumed to be a constant 0c. It will be shown later that this simplification does not affect the result ser/ously. The function A~(0) and Bn(O) are, in turn, assumed to be of the form: 4
An(O) = ~ aknexp(t:O) k=l 4
Bn(O) = ~. b*nexp(t~nO), k=l where ak'* and b , '~ are coefficients to be determined by the boundary conditions. The solution (8) is substituted together with the above A~(0) and B,,(0) into the vorticity equation (6) to get t,n(k = 1, •.., 4; n = 1, ..., oo) as roots of the following quartic equation: t 4 + 01t a + (--n2OS + 03)t 2 + (--n204 + 05)t + naOe - - nSOv - - Kni = O.
In the case where the coefficient of horizontal eddy viscosity Ah is less than or equal to l0 s cmS/sec, the term 04~b/a04in equation (6) almost balances with the term - , ¢ ~ / 0 9 , the coefficient of which does not include 0. Hence no serious error should be made by replacement of 0 by 0c in Odi = I, ..., 7). For sufficiently large n and 0 _< 0 < 0~, the terms An(O) and Bn(0) are A,(O) = anGn(O) B . ( o) = b . G . ( O), where
Gn(O) - n(O.,~__O)exp [ sm oc
n(O,--O)] ~
.1"
Since the series ~ (a,,cosn9 + bnsinn~) converges and the magnitude of G,,(O) decreases faster than n=l
1/n for n--~o% the series ~ [A~(O)cosn~ + B,,(O)sinng] converges uniformly and can be differentiated term by term with respect to 0. This proves the validity of the above mathematical procedure.
Shorter Contribution
73
NUMERICAL EXAMPLE
A numerical computation was carried out for the following values; ~ = l 0 °, ~ = 170°, ~a = 190°, cp4 = 3 5 0 °,
Vo = 2~b0 = 10 i s c m S / s e c , 0~ = 2 0 °, 0c = l 0 ° a n d A~ = l 0 s c m 2 / s e c . T h e first 101 t e r m s
(n < 101) are retained in solution (8). The error introduced by neglect of higher terms is very small. The result is shown in Fig. 3.
180°
2
90 °
0o Fig. 3.
Stream lines in the case of a non-slip boundary. Contour interval 1018 cma/sec.
The main features of the flow are: (1) the inflow deflects leftward (westward in the case of the Northern Hemisphere) immediately after entering the basin, (2) then, it divides into two branches connecting the inlet to the outlet; one is a cyclonic flow and the other is a n anticyclonic flow along the lateral wall, and (3) these do not diffuse uniformly but are concentrated near the boundary. When the inlet and outlet are distributed asymmetrically with respect to the center, the flow pattern becomes somewhat more complicated, but the fundamental features (1) tO (3) are to be observed without any significant change. The flow pattern is not much changed in the case of a free-slip lateral boundary. The stream lines are slightly more concentrated toward the lateral wall (Fig. 4). Figure 5 shows the solution for a non-slip lateral boundary for the case of ~ ---- 0. Comparison of Fig. 5 with Figs. 3 and 4 clearly indicates the importance of the/3-effect. MODEL E X P E R I M E N T A model experiment on a small turntable (Fig. 6) was undertaken to confirm the existence of these flow patterns. A plastic vessel (a) of ! 9-2 cm diameter is placed on a turntable (b). It is covered by a plastic lid so that the fluid surface will not be disturbed by the ambient air motion relative to the water. The diameters of the cylindrical inlet (c) and outlet (d) are 2.5 cm. The mean water depth H is 2.3 cm. The turntable rotates clockwise with a speed of 33~ rev/min. Water is poured through the pipe (e) fastened o n a fixed stand into a reservoir ( 0 rotating together with the vessel. The amount of water
74
Shorter Contribution 180°
2
7
0
~
90*
~45
3 1 5 " ~
"
hhT~/ oo
Fig. 4. Stream linesin the case of a flee-slipboundary.Contourinterval 10t2 cruZ/see.
i80 ~
2
7
:515
0
"
~
90*
45*
0° Fig. 5. Streamlinesin the case of non-rotation or uniformrotation 03 : 0). Contour interval 10TM cm3lsec.
Shorter Contribution
75
passing through the tube (g) is controlled by a pet cock (h) and by adjustment of the water level in the reservoir. Water flows into the vessel through the inlet (c) and flows out through the outlet (d). The water is discharged through a circular conduit (i). Water in the reservoir is dyed with methylene blue when a steady state is reached. Before presenting the results we will consider the principal dynamic processes governing this experiment. Water density can be considered as constant. The horizontal pressure gradient is, therefore, replaced by the surface slope. The cylindrical coordinate system (r, 9, z) is used with the conventional notation. In the case of no water motion relative to the vessel, the free surface height Ho is given by tgHo - - co2r
~r
(9)
g '
since the resultant of gravity and the centrifugal force is perpendicular to the free surface. Here, cois the angular velocity of the turntable, which equals -- 3 '49 rad/sec, and g is the acceleration due to gravity. The equation of motion is given by fk~
×
V
=
--
gHgrad~ + F + lIP%B,
(10)
wherefequals 2oJ, kz denotes the vertical unit vector, ~ the surface displacement from the equilibriunt state, and H the surface height which equals H0 + ~. The bottom friction tn is given by relation (3), provided that A~ and kr in (3) are replaced by the coefficient of viscosity A and k~, respectively. Equation (10) and the equation of continuity lead to curlzF + l curl~n + 2~ar Vr : 0,
(I l)
Vr denoting the radial component of the volume transport V. Here, the relation (9) and the geostrophic relation 2~ Vr = --gH/r 0~/~ were used to eliminate the radial gradient for the free surface OHo/Orand the surface slope ~ / ~ , respectively. The relation (1 l) does not hold very well near the lateral boundary where the lateral friction is more or less important. This should, however, not be crucial for the present analysis. It might be parenthetically noted that 2ojar/gHis of importance as an alternate for ~. The second term is approximately given by _1curlz~n p
~/(oJA) [curlzV _ g-H<°~r(Vr + V~)j_], H
where V~ denotes the azimuthal component of the volume transport. While the bottom friction could be ignored in the analytical solution (8), this is not the ease with the laboratory experiment, as is reported in the studies cited above and others (for instance, Vm~or~ts, 1967). The method for solving equation (11) is similar to that for equation (5). Figure 7 shows the analytical solution. For comparison, the solution without bottom friction is shown in Fig. 8. Both results are not qualitatively very different, although water flows closer to the lateral wall in the ease of no bottom friction. Therefore, the ratio A/L a to/3 should be considered as a rough measure of the similarity between the model and the prototype, where A is the coefficient of viscosity, L is a characteristic length which could be taken to be the radius of the basin, ~ is 2oJsinOdR for the prototype and 2oard(g/-/) for the model, r, being L/2. The coefficient A in the model can be taken as that of the molecular viscosity of pure water (1.1 × 10-z cm2/sec) because the Reynolds number is small (about 5) enough for the flow to be laminar. The ratio is 3 × I0 -4 for the prototype and 6 x 10-5 for the model. Since these two values do not differ from each other in order of magnitude, the experimental result is expected to be comparable with the analytical solution of (5). Figure 9 illustrates the spreading pattern of the dyed water with time, calculated from the velocity field shown in Fig. 7, on the assumption that only advection governs the spreading of the dye. Figures 10a, b, c and d illustrate a series of photographs taken successively 6, 12, 18 and 24 minutes after the dyed water flowed into the vessel. The inflow rate is 16 cmS/min. The spreading pattern in the vessel closely resembles that in Fig. 9, especially in the following aspects: (1) dyed water divides into two branches immediately after flowing into the vessel; the right branch flows more rapidly than the left branch, (2) the tip of the right branch is roundish, whereas that of the left branch is sharp, (3) the right branch overshoots the outlet, and (4) both right and left branches reach the outlet simultaneously.
76
Shorter Contribution
~8o*
y Fig, 7. Stream lines with the bottom friction for the model experiment. Contour interval 0"03 cmS/sec.
leo ~
90*
o" Fig. 8.
Stream lines without the bottom friction for the model oxperimont. Contour interval 0.03 craB/see.
Im ~ o1
~D
c~
3
o
J~
lid
!i!¸
Fig. 10 (a).
Fig. 10 (b). Fig. 10. Experimental result. Spreading patterns of dyed water at 6(a), 12(b), 18(¢) and 24(d) minutes alter the injection of methylene blue.
Fig. 10 (c).
Fig. 10 (d).