Some cOllsideratiollS OD the usual derivation 

or the Pockels (or Helmholtz) equation in steady 

two dimensionaI fluid dYllamics 

I~. lJRLI.'OS."lO (*) - E. SAJ:t:STI (**) - F. 7.IltILJ.I (***) 

:-\U'l\IARL 'l'hl' simplest non·!inear motion of <1 lIui,1 i~ stmlh'd; 
i.e. the ~tea,ly two dimension<11 motio!l of a pedect flni,\. 'l'hes!' equation8 
h<1YC il l"emarkahle pradiC'al importanc(' hecause they ,leSlTihe the air-
1l1Otion over Il1011ntains ami a wake on an oeeallic (lunen"" In partieular 
the numbel' of physieal ~olutions is discusse,l in r('!atio!l to the known bonnd-
i1l'y ('OIHlitions. 

RL\.~I'CYru. - :-:'i ~tlHlia il caHO più semplice di l1Ioto non lineare di 
nn fluido: t'ome, ad esempio, il moto non viscoso stazionario hidillwn~io­
n;J!e di un lluido omogenco. QUl'sj(' ('(Illazioni hanno applkazioni pratielw 
notevoli, come l1Ioto sopra lc montag!le o scia ,li isole in conenti sta· 
zionari!'. ~i esaminano l' discutono, in partif'olarc, il nlllll('TO Ili soluzioni 
fisielli' in rapporto alle conllizioni al ('ontol'!lO f'lIllOSI'.iuÌ\'. 

III t1w sl.m]y of tlte Vl'oblem or t.he 8teally two tlÌmlmsÌonal air-
Ho\\" over a mOlllltaÌll or of 1.JlC wake generatetl by an islantl OlT a 
sl.muly-stal.e no\\', many ]leOvle IIÌsellssel1 t.he Ilel'ivat,Ìon of all eqllatiol! 
fol' the stream fllllr,t,iol! a.m] st.lltlietl the conespontling solutions. 

(*) Istituto di Fisica dell'Atmosfera del C.~.H. - Hom<1. 
(U) Istitut.o ,li FisÌl'a dl'll'l~niYcrsitil, 1.X.F.X. _ Homa. 
(***) Rockfellpl" tllliversity, X,l',C., lO021 XCIV l'ork, U.i-l.A. 



3 8 L . D E L L ' O S S O - E . S A L U S T I - F . Z I K I L L I 

One of t h e s e equations has been known as t h e Pockels (or Helm-
holtz) equation, it is an elliptic linear p a r t i a l differential equation 
for t h e s t r e a m f u n c t i o n . 

I n this p a p e r , we derive t h i s equation, as is usually done(3'6-7 , 8-9'1 0), 
and we p o i n t o u t some inaccuracies of t h e usual derivation. 

The steady motion of a non viscous t w o dimensional fluid in t h e 
plane x, z can be described b y t h e following e q u a t i o n s : 

7)u 

Ï)X 

tWC 
a « 

Dx 

¡¡u 

+ w 

+ w -

+ w 

Ml 

ôit> 

w 

Î>71 
ix 

= — + À 0 

= — .S'(z) w 

+ 
i>W 

= a w 

[1] 

[2] 

[3] 

[4] 

where , are p a r t i a l derivatives respect t o x and 2 respectively, 
DX <)Z 

m, w are t h e velocity components in t h e x and z directions respectively, 
/ - c o n s t a n t is t h e so called convection or buoyancy p a r a m e t e r , S(z) 
is a given f u n c t i o n , a = c o n s t a n t (the t e r m aw in t h e c o n t i n u i t y equa-
tion [4] allows a r e d u c t i o n of t h e fluid density w i t h t h e a l t i t u d e 
if t h e z-axis is directed u p w a r d ) , n is a q u a n t i t y connected with 
t h e pressure and 0 is t h e p o t e n t i a l t e m p e r a t u r e . 

F o r a complete derivation of equations [1], [2], [3], [4] see t h e 
very interesting book of G u t m a n (6). 

I n t r o d u c i n g t h e s t r e a m f u n c t i o n y> b y eq. [5] 

az 
~iz t>x 

[5] 

we derive f r o m eqs. [1], [2], [3], [4] an e q u a t i o n for rp. 

F i r s t of all we notice t h a t eq. [4] is always satisfied b y t h e 
choice of eq. [5]. 

Multiplying eq. [3] b y c~az we h a v e 

-az -az W 
c u + e w 

7>x î)z 
S(z) w 



S O M E C O N S I D E R A T I O N S ON T I I E U S U A I . D E R I V A T I O N E T C . 3 9 

t h a t means by eq. [5] 

i*. J® J® 8(z) ^ = o 

t h a t is 

5 
a <*, , ) ^ + I *<*> d S ' } 

= 0 

o 

t h e d e t e r m i n a n t of t h e J a c o b i a n is zero. 

Then we h a v e 

0 + J S(z') dz' = ft (y>) [6] 

where fi(y>) is an a r b i t r a r y function (some r e q u i r e m e n t of regularity 
of fi{y>) is needed). 

E l i m i n a t i n g n f r o m equations [1], [2] and m a k i n g use of eq. [5] 
we h a v e 

where 

5 (ip, L y>) I ^ aO 
5 (x, z) ¡| 1 t>x 

2az ( a-w y-w 7>w L w = e - ——h —1 H 0 —— r \ a#2 az2 Dz 

[7] 

Using eq. [6] 

so t h a t 

iO „ ~dw 
,r = ™ J |S| 

a (f> L yj) I1 cty; 
a (x, z) i dx 

t h a t is in J a c o b i a n f o r m 

a 
a(x z) L f — z) I = o 



40 L . D E L L ' O S S O - E . S A L U S T I - F . Z I R I L L I 

Then we h a v e 

hip = U(y) + /•/'i(V') « 

where /2 as /1 is an a r b i t r a r y f u n c t i o n . 
Thus t h e system of eqs. [1], [2], [3], [4], has been reduced t o eq. [9] 

for t h e stream f u n c t i o n ip t h a t depends 011 t h e a r b i t r a r y f u n c t i o n s /1, /2. 
I n order t o h a v e a "well posed" problem t h e domain where t h e 

equation lias t o b e verified, t h e b o u n d a r y conditions and t h e properties 
of t h e f u n c t i o n s /1, /2 h a v e t o be specified. 

T h e Pockels (or Helmholtz) e q u a t i o n is a p a r t i c u l a r form of eq. [9]. 
In p a r t i c u l a r t h e Pockels e q u a t i o n is commonly derived in t h i s 

way. 
The region where eq. [9] is studied is of t h e t y p e z > 8 (x) where 

8(®) is a regular f u n c t i o n of x (i.e. S(x) is t h e profile of a m o u n t a i n ) 
see Fig. 1. 

(in order t o simplify t h e problem we assume now S(z) = S = const. 
a = 0 ) 

The b o u n d a r y condition 011 t h e system of eqs. [1], [2], [3], [4] 
are t h e following [ G u t m a n (")]: 

X 

x = — 00 z ^ 8 ( — 0 0 ) 
[10] 

[ 1 1 ] 



S O M E C O N S I D E R A T I O N S ON T I I E U S U A I . D E R I V A T I O N E T C . 41 

Wo cam assume S (— oo) = 0. 

By eqs. [5] and [10] we h a v e 

xp = y>x = Vz + c x = — oo, z > 0 [12] 

Assuming c = 0 b y eqs. [5] [11] a n d [12] we h a v e 

M F ) = - Y R - Y> X = — 0 0 , 2 > 0 [13] 

and finally from eqs. [9], [12] a n d [13] we obtain 

J2(YJ) = Y) X = O O , 2 ^ 0 . [ 1 4 ] 

Assuming t h a t fi(y>), U(f) h a v e everywhere t h e f o r m given by 
eqs. [13] [14] we reach to t h e Pockels' equation: 

- r- + - + —- (w — V z) = 0 [15 
7>x2 ii/2 V2 r L J 

A r a t h e r similar discussion is usually done for t h e oceanographic 
case: t h e wake of an island on an oceanic stream. I t has t o be remarked 
t h a t in this problem t h e Coriolis force is t a k e n into account, as in 
m a n y other meteorological cases. 

F r o m our p o i n t of view t h e derivation of eq. [15] is non satis-
f a c t o r y for t w o different reason: 

1) The t r a n s l a t i o n of t h e b o u n d a r y conditions on u, 0 in t e r m s 
of tp is n o t completely correct. 

I n f a c t u = V where x = — oo 2 > 0 and u = —- does n o t 
1>z 

imply y> = Vt in a n y finite region. The meaning of u = V when 
x = — oo, z ^ 0 is lim u (x, z) = V, z >-• 0. 

X —> - OO 

So t h a t w h a t we can expect is t h a t lim ip (x, z) = = Vz if 
X -»- - oo 

2 ^ 0 t h a t means y> = Vz, x = — oo, 2 ^ 0 . 



42 L . D E L L ' O S S O - E . S A L U S T I - F . Z I K I L L I 

Moreover t h e b o u n d a r y condition on 0 a n d eq. [6] tell us t h a t 

l i m 0 (x, z) = l i m (S z — ft(ip (x, z))), z > 0 
X—> — x — CO 

t h a t is 

0=£z — fi(Vz), 0. 

So we h a v e t h a t /i is fixed only for p o s i t i v e a r g u m e n t s (z ^ 0) and 
n o t everywhere as people seems t o believe. 

Finally using eq. [9| in order t o derive t h e f o r m of /2 we use t h e 
following f a c t : 

lim A y> (x, z) = A lim ip (x, z) = A y>x = 0, (A = 5 + ) 
X - CO \ 0Ji 0 Z 1 

this is also incorrect if no special assumptions are done on t h e previous 
limit. However if we s t u d y t h e problem in a compact domain, as 
it is t h e case of a numerical c o m p u t a t i o n , t h e f a r u p s t r e a m p a r t of 
t h e domain t a k e t h e rule of I n this case t h e s i t u a t i o n is t h a t 
people, hope t o know f r o m t h e conditions in p a r t of t h e b o u n d a r y not 
only the solutions of one well determined elliptic non linear differential 
equation b u t also t h e explicit shape of t h e non linear p a r t ji(y>) and 

Mr)-
This appears to us as an o v e r s t a t e m e n t . 

2) The b o u n d a r y condition eqs. [10], [11] are not enough t o deter-
mine an u n i q u e solution of eqs. [1], [2], [3], [4] and so also t h e equation 
[15] has only t h e b o u n d a r y condition given b y eq. [12] which is not, 
enough t o determine a u n i q u e solution. 

Concluding t h e idea t h a t t h e b o u n d a r y conditions can d e t e r m i n e 
t h e form of /i, /2 in eq. [9] seems due t o other non rigorous reasons, 
p e r h a p s of historical origin. 

I t has t o be remarked, however, t h a t t h e above derivation of 
eq. [15] is now a classical method in geophysics and t h a t an enormous 
a m o u n t of practical work is done on it. 

But it has t o be said also t h a t an enormous m a t h e m a t i c a l litera-
t u r e exists on eq. [9] under various assumptions of ft, /2 [see for exam-
ple (7)]. The problem however of determining t h e physical f o r m of 
/1, /•> and so t h e physical solutions of eq. [9] is in our opinion essentially 
open, so t h e use of t h e p r i m i t i v e equations [1], [2], [3], [4] seems to 
us t h e most reasonable way t o h a n d l e these problems. 



S O M E C O N S I D E R A T I O N S O N T I I E U S U A I . D E R I V A T I O N E T C . 43 

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