Also notice that in the component e

Also notice that in the component equations (and see Fig. 8.7 (a, b)) the
pressure gradient in the x-direction, dp/dx, is associated with v, while the
pressure gradient in the ^-direction, dp/dy, is associated with u. The x- and yequations
can be combined into a single one
GAMBAR HAL 91

2Qsin(/>FH = a - ^ (8.11)
dnH
where VH — magnitude of the vector sum of u and v
^tf + v2)1'2,
and dp/dnH = the horizontal pressure term perpendicular to the
direction of H (see Fig. 8.7(c)).
One way to remember the relative directions of the pressure force and the
velocity is to think of the sequence:
(1) the pressure gradient is initiated somehow,
(2) the fluid starts to move down the gradient,
(3) the fluid then experiences the Coriolis force to the right (in the northern
hemisphere) and therefore swings to the right,
(4) the fluid eventually moves along the isobars, i.e. along the slope, not
down it, with the pressure force down the slope balanced by the Coriolis
force up the slope.
The equivalent situation in the atmosphere was shown in Fig. 8.2 to aid in
defining the terms cyclonic and anticyclonic. It is left as a simple exercise for the
reader to verify that the circulations in this figure are consistent with
geostrophy.
Notice that an alternative procedure would be to start a fluid moving in some
direction, Coriolis force would then make it swing to the right (in the northern
hemisphere) and pile up there (slope up to the right) so developing a pressure
force to the left. Therefore the geostrophic equation simply tells us that the
pressure force balances the Coriolis force—it does not tell us which came first,
the pressure gradient or the motion.
Equation (8.11) is actually applicable no matter in which direction we take
the pressure derivatives. If nH is taken in an arbitrary direction then VH becomes
Vx, the component of the geostrophic velocity perpendicular to the direction
nH. In the northern hemisphere, taking nH to increase to the right the flow is
away from the observer if dp/dnH > 0 and toward the observer if dp/dnH < 0.
This is another way of stating that if the isobars slope up to the right (as in
Fig. 8.4) the flow is "into the paper".
How do we get from equations (8.10) and (8.11) to (8.9A) and (8.9B), the
practical forms of the geostrophic equation? The pressure derivatives in (8.10)
and (8.11) are taken on surfaces of constant z which are also surfaces of
constant Ö. The pressure derivatives in (8.10) and (8.11) are not directly
measurable, as already noted, so we must introduce the geopotential. Now
using the rule from differential calculus for implicit functions
( * ) . _ ( » ) /(*•)
/ y, z or Ö constant / y, p constant / P / x, y constant
CURRENTS WITHOUT FRICTION 81
and remembering that äÖ/äñ = — a = — 1/p (Section 8.3) we get dp/dx
= p (äÖ/ä÷) where this is the change in Ö as we go along an isobar in the xdirection.
Likewise dp/dy = p (äÖ/äã) and dp/dnH = ñ(äÖ/äçÇ). These relations
between p and Ö gradients can easily be obtained from first principles
instead of the calculus rule. Suppose that one moves a small distance bnH from
the point Ax in Fig. 8.4. Over this distance the height on the px isobar will
change by bz, the pressure on Ö÷ will increase by pg bz and Ö will increase by
g bz. Thus bp = ñbÖ and dividing both sides by bnH and taking the limit as
bnH -_ 0 gives the same relation for the derivatives as does the calculus rule.
Substituting the Ö term for the p terms in (8.10) and (8.11) gives an alternate
form for the geostrophic equations.
Now these Ö gradients cannot be measured either, but differences from one
level to another can be obtained from the density field. From equation (8.5) we
have
ö é =Ö2 + ÄÖ8ßá + ÄÖ
and ÄÖ81(1 is the same at every station, so its derivatives with respect to
horizontal coordinates are always zero. Consider the x-equation at levels 1 and
2 of Fig. 8.4,
^ • , fSp ßäÖË äÖ2 5(ÄÖ)
äÖ2 2Ù sin øö V2y = —— and the difference is dx
ä(ÁÖ)
2Ù sin ö (vl — v2 ) = —z—. Likewise
2Qsin0(ii1 - i i 2)= -^ô-1 and 2áþçö(í÷ - V2) = -—- (8.12)
cy cnH
where Vx, V2 represent the horizontal velocity components perpendicular to
the direction of nH, at levels 1 and 2.
These are differential forms of the geostrophic equation written in a way
which can be used with the kind of observations which we can make. Equations
(8.9), the practical equations, appear to be finite difference forms of equation
(8.12) but in fact they are integral forms. The average along a direction nH from
0 to L is, by definition,
^Jo
L
(quantity to be averaged) dnH.
Applying this to equation (8.12), using / f o r 2Qsin and an overbar to indicate
an average, gives
82 INTRODUCTORY DYNAMICAL OCEANOGRAPHY
<
d(A) . 1
Wi - V2) = T | - ^ d n H = Æ(ÄÖÂ-ÄÖ.). (8.13)
The only difference between equations (8.13) and (8.9) is that averaging is not
explicitly shown in the latter and we must assume that/( Vx - V2 ) = / ( Vx - V2 )
which will be a very good approximation since over the distances used in
practice fis nearly constant. In the example given in Table 8.2 with nH in the
southward direction (for which/variations with nH are a maximum),/changed
by only 1 % between A and B.
8.45 The "thermal wind" equations
These are another variation of the geostrophic equations originally derived
to show how temperature differences in the horizontal could lead to vertical
variations in the geostrophic wind velocity, hence the term thermal wind

2Qsin(/>FH = a - ^ (8.11)
dnH
where VH — magnitude of the vector sum of u and v
^tf + v2)1'2,
and dp/dnH = the horizontal pressure term perpendicular to the
direction of H (see Fig. 8.7(c)).
One way to remember the relative directions of the pressure force and the
velocity is to think of the sequence:
(1) the pressure gradient is initiated somehow,
(2) the fluid starts to move down the gradient,
(3) the fluid then experiences the Coriolis force to the right (in the northern
hemisphere) and therefore swings to the right,
(4) the fluid eventually moves along the isobars, i.e. along the slope, not
down it, with the pressure force down the slope balanced by the Coriolis
force up the slope.
The equivalent situation in the atmosphere was shown in Fig. 8.2 to aid in
defining the terms cyclonic and anticyclonic. It is left as a simple exercise for the
reader to verify that the circulations in this figure are consistent with
geostrophy.
Notice that an alternative procedure would be to start a fluid moving in some
direction, Coriolis force would then make it swing to the right (in the northern
hemisphere) and pile up there (slope up to the right) so developing a pressure
force to the left. Therefore the geostrophic equation simply tells us that the
pressure force balances the Coriolis force—it does not tell us which came first,
the pressure gradient or the motion.
Equation (8.11) is actually applicable no matter in which direction we take
the pressure derivatives. If nH is taken in an arbitrary direction then VH becomes
Vx, the component of the geostrophic velocity perpendicular to the direction
nH. In the northern hemisphere, taking nH to increase to the right the flow is
away from the observer if dp/dnH > 0 and toward the observer if dp/dnH < 0.
This is another way of stating that if the isobars slope up to the right (as in
Fig. 8.4) the flow is "into the paper".
How do we get from equations (8.10) and (8.11) to (8.9A) and (8.9B), the
practical forms of the geostrophic equation? The pressure derivatives in (8.10)
and (8.11) are taken on surfaces of constant z which are also surfaces of
constant Ö. The pressure derivatives in (8.10) and (8.11) are not directly
measurable, as already noted, so we must introduce the geopotential. Now
using the rule from differential calculus for implicit functions
( * ) . _ ( » ) /(*•)
 / y, z or Ö constant  / y, p constant /  P / x, y constant
CURRENTS WITHOUT FRICTION 81
and remembering that äÖ/äñ = — a = — 1/p (Section 8.3) we get dp/dx
= p (äÖ/ä÷) where this is the change in Ö as we go along an isobar in the xdirection.
Likewise dp/dy = p (äÖ/äã) and dp/dnH = ñ(äÖ/äçÇ). These relations
between p and Ö gradients can easily be obtained from first principles
instead of the calculus rule. Suppose that one moves a small distance bnH from
the point Ax in Fig. 8.4. Over this distance the height on the px isobar will
change by bz, the pressure on Ö÷ will increase by pg bz and Ö will increase by
g bz. Thus bp = ñbÖ and dividing both sides by bnH and taking the limit as
bnH -_ 0 gives the same relation for the derivatives as does the calculus rule.
Substituting the Ö term for the p terms in (8.10) and (8.11) gives an alternate
form for the geostrophic equations.
Now these Ö gradients cannot be measured either, but differences from one
level to another can be obtained from the density field. From equation (8.5) we
have
ö é =Ö2 + ÄÖ8ßá + ÄÖ
and ÄÖ81(1 is the same at every station, so its derivatives with respect to
horizontal coordinates are always zero. Consider the x-equation at levels 1 and
2 of Fig. 8.4,
^ • , fSp ßäÖË äÖ2 5(ÄÖ)
äÖ2 2Ù sin øö V2y = —— and the difference is dx
ä(ÁÖ)
2Ù sin ö (vl — v2 ) = —z—. Likewise
2Qsin0(ii1 - i i 2)= -^ô-1 and 2áþçö(í÷ - V2) = -—- (8.12)
cy cnH
where Vx, V2 represent the horizontal velocity components perpendicular to
the direction of nH, at levels 1 and 2.
These are differential forms of the geostrophic equation written in a way
which can be used with the kind of observations which we can make. Equations
(8.9), the practical equations, appear to be finite difference forms of equation
(8.12) but in fact they are integral forms. The average along a direction nH from
0 to L is, by definition,
^Jo
L
(quantity to be averaged) dnH.
Applying this to equation (8.12), using / f o r 2Qsin and an overbar to indicate
an average, gives
82 INTRODUCTORY DYNAMICAL OCEANOGRAPHY
<
d(A) . 1
Wi - V2) = T | - ^ d n H = Æ(ÄÖÂ-ÄÖ.). (8.13)
The only difference between equations (8.13) and (8.9) is that averaging is not
explicitly shown in the latter and we must assume that/( Vx - V2 ) = / ( Vx - V2 )
which will be a very good approximation since over the distances used in
practice fis nearly constant. In the example given in Table 8.2 with nH in the
southward direction (for which/variations with nH are a maximum),/changed
by only 1 % between A and B.
8.45 The "thermal wind" equations
These are another variation of the geostrophic equations originally derived
to show how temperature differences in the horizontal could lead to vertical
variations in the geostrophic wind velocity, hence the term thermal wind

0/5000

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Also notice that in the component equations (and see Fig. 8.7 (a, b)) thepressure gradient in the x-direction, dp/dx, is associated with v, while thepressure gradient in the ^-direction, dp/dy, is associated with u. The x- and yequationscan be combined into a single oneGAMBAR HAL 91 2Qsin(/>FH = a - ^ (8.11)dnHwhere VH — magnitude of the vector sum of u and v^tf + v2)1'2,and dp/dnH = the horizontal pressure term perpendicular to thedirection of H (see Fig. 8.7(c)).One way to remember the relative directions of the pressure force and thevelocity is to think of the sequence:(1) the pressure gradient is initiated somehow,(2) the fluid starts to move down the gradient,(3) the fluid then experiences the Coriolis force to the right (in the northernhemisphere) and therefore swings to the right,(4) the fluid eventually moves along the isobars, i.e. along the slope, notdown it, with the pressure force down the slope balanced by the Coriolisforce up the slope.The equivalent situation in the atmosphere was shown in Fig. 8.2 to aid indefining the terms cyclonic and anticyclonic. It is left as a simple exercise for thereader to verify that the circulations in this figure are consistent withgeostrophy.Notice that an alternative procedure would be to start a fluid moving in somedirection, Coriolis force would then make it swing to the right (in the northernhemisphere) and pile up there (slope up to the right) so developing a pressureforce to the left. Therefore the geostrophic equation simply tells us that thepressure force balances the Coriolis force—it does not tell us which came first,the pressure gradient or the motion.Equation (8.11) is actually applicable no matter in which direction we takethe pressure derivatives. If nH is taken in an arbitrary direction then VH becomesVx, the component of the geostrophic velocity perpendicular to the directionnH. In the northern hemisphere, taking nH to increase to the right the flow isaway from the observer if dp/dnH > 0 and toward the observer if dp/dnH < 0.This is another way of stating that if the isobars slope up to the right (as inFig. 8.4) the flow is "into the paper".How do we get from equations (8.10) and (8.11) to (8.9A) and (8.9B), thepractical forms of the geostrophic equation? The pressure derivatives in (8.10)and (8.11) are taken on surfaces of constant z which are also surfaces ofconstant Ö. The pressure derivatives in (8.10) and (8.11) are not directlymeasurable, as already noted, so we must introduce the geopotential. Nowusing the rule from differential calculus for implicit functions( * ) . _ ( » ) /(*•) / y, z or Ö constant / y, p constant / P / x, y constantCURRENTS WITHOUT FRICTION 81and remembering that äÖ/äñ = — a = — 1/p (Section 8.3) we get dp/dx= p (äÖ/ä÷) where this is the change in Ö as we go along an isobar in the xdirection.Likewise dp/dy = p (äÖ/äã) and dp/dnH = ñ(äÖ/äçÇ). These relationsbetween p and Ö gradients can easily be obtained from first principlesinstead of the calculus rule. Suppose that one moves a small distance bnH fromthe point Ax in Fig. 8.4. Over this distance the height on the px isobar willchange by bz, the pressure on Ö÷ will increase by pg bz and Ö will increase byg bz. Thus bp = ñbÖ and dividing both sides by bnH and taking the limit asbnH -_ 0 gives the same relation for the derivatives as does the calculus rule.Substituting the Ö term for the p terms in (8.10) and (8.11) gives an alternateform for the geostrophic equations.Now these Ö gradients cannot be measured either, but differences from onelevel to another can be obtained from the density field. From equation (8.5) wehaveö é =Ö2 + ÄÖ8ßá + ÄÖand ÄÖ81(1 is the same at every station, so its derivatives with respect tohorizontal coordinates are always zero. Consider the x-equation at levels 1 and2 of Fig. 8.4,^ • , fSp ßäÖË äÖ2 5(ÄÖ)äÖ2 2Ù sin øö V2y = —— and the difference is dxä(ÁÖ)2Ù sin ö (vl — v2 ) = —z—. Likewise2Qsin0(ii1 - i i 2)= -^ô-1 and 2áþçö(í÷ - V2) = -—- (8.12)cy cnHwhere Vx, V2 represent the horizontal velocity components perpendicular tothe direction of nH, at levels 1 and 2.These are differential forms of the geostrophic equation written in a waywhich can be used with the kind of observations which we can make. Equations(8.9), the practical equations, appear to be finite difference forms of equation(8.12) but in fact they are integral forms. The average along a direction nH from0 to L is, by definition,^JoL(quantity to be averaged) dnH.Applying this to equation (8.12), using / f o r 2Qsin and an overbar to indicatean average, gives82 INTRODUCTORY DYNAMICAL OCEANOGRAPHY<d(A) . 1Wi - V2) = T | - ^ d n H = Æ(ÄÖÂ-ÄÖ.). (8.13)The only difference between equations (8.13) and (8.9) is that averaging is notexplicitly shown in the latter and we must assume that/( Vx - V2 ) = / ( Vx - V2 )which will be a very good approximation since over the distances used inpractice fis nearly constant. In the example given in Table 8.2 with nH in thesouthward direction (for which/variations with nH are a maximum),/changedby only 1 % between A and B.8.45 The "thermal wind" equationsThese are another variation of the geostrophic equations originally derivedto show how temperature differences in the horizontal could lead to verticalvariations in the geostrophic wind velocity, hence the term thermal wind

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Juga perhatikan bahwa dalam persamaan komponen (dan lihat Gambar. 8.7 (a, b)) yang
gradien tekanan dalam arah x, dp / dx, terkait dengan v, sedangkan
gradien tekanan dalam ^ -direction, dp / dy , terkait dengan u. X dan yequations
dapat dikombinasikan menjadi satu satu
GAMBAR HAL 91 2Qsin (/> FH = a - ^ (8.11) DNH mana VH - besarnya penjumlahan vektor u dan v ^ tf + v2) 1'2, dan dp / DNH = istilah tekanan horisontal tegak lurus terhadap arah H (lihat Gambar 8.7 (c).). Salah satu cara untuk mengingat arah relatif dari kekuatan tekanan dan kecepatan adalah untuk memikirkan urutan: (1) gradien tekanan dimulai entah bagaimana, (2) dimulai fluida untuk bergerak ke bawah gradien, (3) cairan kemudian mengalami gaya Coriolis ke kanan (di utara belahan bumi) dan karena itu berayun ke kanan, (4) cairan akhirnya bergerak sepanjang isobar, yaitu di sepanjang lereng, tidak turun, dengan kekuatan tekanan menuruni lereng seimbang dengan Coriolis kekuatan atas lereng. Situasi setara di atmosfer ditunjukkan pada Gambar. 8,2 untuk membantu dalam mendefinisikan istilah siklon dan anticyclonic. Hal yang tersisa sebagai latihan sederhana untuk pembaca untuk memverifikasi bahwa sirkulasi dalam gambar ini konsisten dengan geostrophy. Perhatikan bahwa prosedur alternatif akan memulai cairan bergerak di beberapa arah, gaya Coriolis maka akan membuat ayunan ke kanan ( di utara belahan bumi) dan menumpuk di sana (lereng ke kanan) sehingga mengembangkan tekanan gaya ke kiri. Oleh karena itu persamaan geostropik hanya memberitahu kita bahwa kekuatan tekanan menyeimbangkan gaya Coriolis-ia tidak memberitahu kita yang datang pertama, gradien tekanan atau gerak. Persamaan (8.11) sebenarnya berlaku tidak peduli ke arah mana kita mengambil derivatif tekanan. Jika nH diambil dalam arah yang sewenang-wenang maka VH menjadi Vx, komponen kecepatan geostropik tegak lurus terhadap arah nH. Di belahan bumi utara, mengambil nH untuk meningkatkan ke kanan aliran ini jauh dari pengamat jika dp / DNH> 0 dan menuju pengamat jika dp / DNH <0. Ini adalah cara lain untuk menyatakan bahwa jika isobars kemiringan sampai dengan kanan (seperti dalam Gambar. 8.4) alirannya "ke dalam kertas". Bagaimana kita dapatkan dari persamaan (8.10) dan (8.11) ke (8.9A) dan (8.9B), yang bentuk praktis dari persamaan geostropik? Derivatif tekanan dalam (8.10) dan (8.11) diambil pada permukaan z konstan yang juga permukaan Ö konstan. Derivatif tekanan dalam (8.10) dan (8.11) tidak langsung terukur, sebagaimana telah dicatat, jadi kita harus memperkenalkan geopotensial tersebut. Sekarang menggunakan aturan dari diferensial kalkulus untuk fungsi implisit (*). _ (») / (* •) / y, z atau Ö konstan / y, p konstan / P / x, y konstan ARUS TANPA GESEKAN 81 dan mengingat bahwa ao / AN = - a = - 1 / p ( Bagian 8.3) kita mendapatkan dp / dx = p (AO / ä ÷) di mana ini adalah perubahan Ö karena kami pergi sepanjang isobar di xdirection tersebut. Demikian juga dp / dy = p (AO / AA) dan dp / DNH = ñ (AO / ACC). Hubungan ini antara p dan Ö gradien dapat dengan mudah diperoleh dari prinsip-prinsip pertama bukannya aturan kalkulus. Misalkan salah satu bergerak jarak BNH kecil dari titik Ax pada Gambar. 8.4. Selama jarak ini ketinggian pada isobar px akan berubah dengan bz, tekanan pada Ö ÷ akan meningkat pg bz dan Ö akan meningkat g bz. Jadi bp = NBO dan membagi kedua sisi dengan BNH dan mengambil batas sebagai BNH -_ 0 memberikan hubungan yang sama untuk derivatif seperti halnya aturan kalkulus. Mengganti istilah Ö untuk istilah p di (8.10) dan (8.11) memberikan alternatif bentuk untuk persamaan geostropik. Sekarang gradien Ö ini tidak dapat diukur baik, tetapi perbedaan dari satu tingkat ke yang lain dapat diperoleh dari bidang kepadatan. Dari persamaan (8.5) kita memiliki ö é = O2 + ÄÖ8ßá + ao dan ÄÖ81 (1 adalah sama di setiap stasiun, sehingga turunannya sehubungan dengan koordinat horisontal selalu nol. Pertimbangkan x-persamaan pada tingkat 1 dan 2 dari Gambar . 8.4, ^ •, FSP ßäÖË äÖ2 5 (AO) äÖ2 2U dosa oo V2y = - dan perbedaan adalah dx ä (AO) 2U dosa ö (vl - v2). = -z- Demikian juga 2Qsin0 (ii1 - ii 2) = - ^-ô 1 dan 2áþçö (í ÷ - V2) = - - (8.12) cy CNH mana Vx, V2 merupakan komponen kecepatan horisontal tegak lurus . arah nH, pada tingkat 1 dan 2 ini Bentuk diferensial dari persamaan geostropik ditulis dengan cara yang dapat digunakan dengan jenis observasi yang dapat kita buat. Persamaan (8.9), persamaan praktis, tampaknya bentuk beda hingga persamaan (8.12) tetapi sebenarnya mereka merupakan bagian integral bentuk. Rata-rata sepanjang arah nH dari 0 sampai L, menurut definisi, ^ Jo L (kuantitas dirata-ratakan) DNH. Menerapkan ini untuk persamaan (8.12), menggunakan / untuk 2Qsin dan overbar untuk menunjukkan rata-rata, memberikan 82 PENGANTAR Dynamical OCEANOGRAPHY < d (A

). 1
Wi - V2) = T | - ^ dn H = Æ (AOA-AO).. (8.13)
Satu-satunya perbedaan antara persamaan (8.13) dan (8.9) adalah rata-rata yang tidak
secara eksplisit ditampilkan dalam kedua dan kita harus mengasumsikan bahwa / (Vx - V2) = / (Vx - V2)
yang akan menjadi pendekatan yang sangat baik sejak lebih dari jarak yang digunakan dalam
praktek fis hampir konstan. Dalam contoh yang diberikan dalam Tabel 8.2 dengan nH di
arah selatan (yang / variasi dengan nH adalah maksimum), / berubah
oleh hanya 1% antara A dan B.
8.45 "Angin termal" Persamaan
ini variasi lain dari geostropik persamaan awalnya berasal
untuk menunjukkan bagaimana perbedaan suhu di horizontal dapat menyebabkan vertikal
variasi dalam kecepatan angin geostropik, maka angin termal jangka

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