GAMBAR 100  the slopes in (b) are e

GAMBAR 100

 
the slopes in (b) are exaggerated! From the first thermal wind equation (8.14)
and with dv/dz = 0 we have/i? (dp/dz) = — g (dp/êx); horizontal gradients of
density and corresponding isopycnal slopes arise from vertical variations of
density associated with compression effects. In the Boussinesq approximation
we neglect density variations on the left-hand side and would get dp/dx
= dp/dy = 0 in the barotropic case; as we shall see these gradients and
corresponding isopycnal slopes are very small compared with those that occur
in baroclinic cases, so the approximation is reasonable. Alternatively, we could
use potential density, or at when it is a suitable approximation, which would be
constant in the barotropic case.
In the baroclinic case, there is no simple relation between the isobars and
isopycnals. From the geostrophic equation (8.6) the slopes of isobars are
proportional to the velocity. From the thermal wind equation (8.14) the
horizontal density gradients and corresponding isopycnal slopes are proportional
to d V/dz, the variation of velocity with depth change or the vertical
shear. Note that if density decreases to the right then the isopycnals will slope
down to the right.
Consider first the simplest idealization of baroclinic flow—a two-layer
system with the upper layer moving and the lower layer stationary. The upper
layer has constant potential density px and the lower layer has constant
potential density p2. In Section 9.14.2 we show that the slope of the interface
between the two layers (the isopycnal) = — pi/(p2 — Pi) times the slope of the
surface (an isobar). Thus in this simple case, the isopycnal slope is opposite in
sign to the isobaric slope. The magnitude of the isopycnal slope is about 1000
times that of the isobaric slope since (p2 — Pi) — 1 kgm"3 and pl
~ 1000 kgm"3 for a model of the ocean. This situation is illustrated in Fig.
8.9(c). Note again that the slopes are much exaggerated, with the isobaric slopes
and isopycnal slopes in the upper layer being much more exaggerated than the
interface slope. One can also consider case (c) to be a combination of cases (b)
for the upper layer and (a) for the lower layer. While this model is simplistic and
a bit unphysical, since both p and V have discontinuities at the interface, it does
illustrate that in baroclinic cases isopycnal slopes will be much larger than
isobaric slopes and of opposite sign, at least in part of the flow.
As a more realistic case consider the example of Section 8.43. By linear
interpolation the depth of the at = 27.0 surface is 130 m at station A and 280 m
at station B. Thus it descends 150 m from A to B while the isobars ascend about
0.13 m from A to B. The slopes of at surfaces will differ from the slopes of
isopycnals (density in situ) by the slopes of isobars which are negligible within
the observation errors, so we can use at slopes to represent isopycnal slopes.
The value at — 27.7 is found at 570 m at station A and at 750 m at station B for
a drop of about 180 m. In both cases the isopycnal slopes are opposite in sign
and about 1000 times the size of the isobaric slopes. However, at 100 m the
isopycnal is level within observation error and above 100 m the isopycnals
90 INTRODUCTORY DYNAMICAL OCEANOGRAPHY
slope the same way as the isobars because u decreases with height above 100 m.
For example, ot = 26.8 is at a depth of about 50 m at station A and must reach
the surface before station B, so the slope is greater than 50 m in 50 km. Overall,
the isopycnal slopes are opposite in sign and much larger in size than the
isobaric slopes.
We see from this example that the horizontal density gradients and
corresponding isopycnal slopes are large enough to be observed whereas the
pressure gradients and isobar slopes are too small to observe, except perhaps by
satellite altimetry for strong currents (since the expected accuracy of ± 0.1 m is
comparable to the true height variations over much of the ocean). Density
differences in the ocean are small and so isopycnal slopes are magnified from
isobaric slopes by the factor p/Ap. If we could measure density only to the same
accuracy as pressure and depth (about 1 in 103) we would not be able to detect
the isopycnal slopes either. Fortunately we can measure density changes,
through salinity and temperature, to higher accuracy (about ± 5 in 106) or
0.005 in óí If we could measure depth and pressure differences to 1 in 105 we
could use the pressure field to get the relative velocity field directly. Finally, if
we could establish the sea surface level to ± 1 cm in 100 km we could get the
absolute velocity field to ± 1 cm s"1 at mid-latitudes but this ability seems
unlikely in the foreseeable future.
Figure 8.10(d) shows a somewhat more realistic example similar to that of
Subsection 8.43 but without the decrease of V above 100 m. Note that unlike
the previous figures the exaggerations of the p and p slopes are different, i.e. p
by ^ 105 and p by 103, to illustrate that p slopes are greater although we cannot
show the 1000:1 ratio of the ocean. In this case, p slopes due to pressure effects
are too small to show compared with the slopes associated with the velocity
shear in the vertical. In the upper part of the flow, Vis large and independent of
z. The pressure surfaces are parallel and slope up to the right (slope in figure
~ 10" true slope ^ 10"6). The p surfaces are level (actual slopes with this
exaggeration would be 10" 3 and true slopes 10" 6). When the velocity begins to
decrease with depth, the isobar slopes gradually decrease to the level of no
motion. The isopycnal slopes are large at first and down to the right where
the shear is large; they gradually decrease to the level of no motion where
d V/dz = 0 as well as V = 0, and there is no discontinuity in either.
If one has a plot of the isopycnal surfaces only (dynamic heights not yet
having been computed) then one must integrate mentally starting from a level
where the motion is expected to be zero or at least small. In Fig. 8.10 (d), starting
at the level of no motion the isopycnal surfaces slope down to the right (light
water on the right) so Vrel must be increasing upward until the isopycnals level
out and V becomes uniform. Since V increases and is into the paper, the isobars
must slope up to the right (opposite to the isopycnals in the region where they
slope down).
Figure 8.11 shows a reasonably realistic and more complicated case. Again








GAMBAR 103

Hasil (Bahasa Indonesia) 1:

GAMBAR 100  lereng di (b) yang berlebihan! Dari persamaan termal angin pertama (8.14)dan dengan dv dz = 0 kami memiliki / saya? (dp/dz) =-g (dp/êx); horisontal gradien darikepadatan dan sesuai isopycnal lereng timbul dari variasi vertikalkepadatan yang terkait dengan efek kompresi. Dalam pendekatan Boussinesqkita mengabaikan kepadatan variasi di sebelah kiri dan akan mendapatkan dp dx= dp dy = 0 dalam kasus barotropic; seperti yang kita akan lihat gradien ini dansesuai isopycnal lereng sangat kecil dibandingkan dengan mereka yang terjadidalam kasus baroclinic, jadi perkiraan wajar. Selain itu, kita bisamenggunakan potensi kepadatan, atau ketika itu adalah pendekatan yang cocok, yang akankonstan dalam kasus barotropic.Dalam kasus baroclinic, ada tidak ada hubungan sederhana antara isobar danisopycnals. Dari persamaan geostrophic (8.6) dapat lereng isobarsebanding dengan kecepatan. Dari persamaan termal angin (8.14)horisontal kepadatan gradien dan sesuai isopycnal lereng yang proporsionald V/dz, variasi kecepatan dengan kedalaman perubahan atau vertikalgeser. Perhatikan bahwa jika kepadatan menurun ke kanan kemudian isopycnals akan lerengke kanan.Mempertimbangkan dulu idealization sederhana aliran baroclinic — dua lapisan-lapisansistem dengan lapisan atas bergerak dan lapisan bawah stasioner. Ataslapisan memiliki kepadatan potensi konstan px dan lapisan bawah memiliki konstanpotential density p2. In Section 9.14.2 we show that the slope of the interfacebetween the two layers (the isopycnal) = — pi/(p2 — Pi) times the slope of thesurface (an isobar). Thus in this simple case, the isopycnal slope is opposite insign to the isobaric slope. The magnitude of the isopycnal slope is about 1000times that of the isobaric slope since (p2 — Pi) — 1 kgm"3 and pl~ 1000 kgm"3 for a model of the ocean. This situation is illustrated in Fig.8.9(c). Note again that the slopes are much exaggerated, with the isobaric slopesand isopycnal slopes in the upper layer being much more exaggerated than theinterface slope. One can also consider case (c) to be a combination of cases (b)for the upper layer and (a) for the lower layer. While this model is simplistic anda bit unphysical, since both p and V have discontinuities at the interface, it doesillustrate that in baroclinic cases isopycnal slopes will be much larger thanisobaric slopes and of opposite sign, at least in part of the flow.As a more realistic case consider the example of Section 8.43. By linearinterpolation the depth of the at = 27.0 surface is 130 m at station A and 280 mat station B. Thus it descends 150 m from A to B while the isobars ascend about0.13 m from A to B. The slopes of at surfaces will differ from the slopes ofisopycnals (density in situ) by the slopes of isobars which are negligible withinthe observation errors, so we can use at slopes to represent isopycnal slopes.The value at — 27.7 is found at 570 m at station A and at 750 m at station B fora drop of about 180 m. In both cases the isopycnal slopes are opposite in signand about 1000 times the size of the isobaric slopes. However, at 100 m theisopycnal is level within observation error and above 100 m the isopycnals90 INTRODUCTORY DYNAMICAL OCEANOGRAPHYslope the same way as the isobars because u decreases with height above 100 m.For example, ot = 26.8 is at a depth of about 50 m at station A and must reachthe surface before station B, so the slope is greater than 50 m in 50 km. Overall,the isopycnal slopes are opposite in sign and much larger in size than theisobaric slopes.We see from this example that the horizontal density gradients andcorresponding isopycnal slopes are large enough to be observed whereas thepressure gradients and isobar slopes are too small to observe, except perhaps bysatellite altimetry for strong currents (since the expected accuracy of ± 0.1 m iscomparable to the true height variations over much of the ocean). Densitydifferences in the ocean are small and so isopycnal slopes are magnified fromisobaric slopes by the factor p/Ap. If we could measure density only to the sameaccuracy as pressure and depth (about 1 in 103) we would not be able to detectthe isopycnal slopes either. Fortunately we can measure density changes,through salinity and temperature, to higher accuracy (about ± 5 in 106) or0.005 in óí If we could measure depth and pressure differences to 1 in 105 wecould use the pressure field to get the relative velocity field directly. Finally, ifwe could establish the sea surface level to ± 1 cm in 100 km we could get theabsolute velocity field to ± 1 cm s"1 at mid-latitudes but this ability seemsunlikely in the foreseeable future.Figure 8.10(d) shows a somewhat more realistic example similar to that ofSubsection 8.43 but without the decrease of V above 100 m. Note that unlikethe previous figures the exaggerations of the p and p slopes are different, i.e. pby ^ 105 and p by 103, to illustrate that p slopes are greater although we cannotshow the 1000:1 ratio of the ocean. In this case, p slopes due to pressure effectsare too small to show compared with the slopes associated with the velocityshear in the vertical. In the upper part of the flow, Vis large and independent ofz. The pressure surfaces are parallel and slope up to the right (slope in figure~ 10" true slope ^ 10"6). The p surfaces are level (actual slopes with thisexaggeration would be 10" 3 and true slopes 10" 6). When the velocity begins todecrease with depth, the isobar slopes gradually decrease to the level of nomotion. The isopycnal slopes are large at first and down to the right wherethe shear is large; they gradually decrease to the level of no motion whered V/dz = 0 as well as V = 0, and there is no discontinuity in either.If one has a plot of the isopycnal surfaces only (dynamic heights not yethaving been computed) then one must integrate mentally starting from a levelwhere the motion is expected to be zero or at least small. In Fig. 8.10 (d), startingat the level of no motion the isopycnal surfaces slope down to the right (lightwater on the right) so Vrel must be increasing upward until the isopycnals levelout and V becomes uniform. Since V increases and is into the paper, the isobarsmust slope up to the right (opposite to the isopycnals in the region where theyslope down).Figure 8.11 shows a reasonably realistic and more complicated case. AgainGAMBAR 103