O n l i n e a r i n t e r n a l w a v e s o n t h e s e a , 

s t r o n g l y v e r t i c a l l y t r a p p e d 

E . P U R I N I ( * ) - E . S A L T J S T I ( * * ) 

Received on A u g u s t 2nd, 1975 

Summary. — W e s t u d y some explicit cases of m a r i n e t h e r m o c l i n e . We 
f o c u s our a t t e n t i o n on t h e s t r o n g l y v e r t i c a l l y t r a p p e d i n t e r n a l waves, which 
in our cases allow an explicit dispersion r e l a t i o n a n d a simple b e h a v i o u r in 
t e r m s of e l e m e n t a r y f u n c t i o n s . T h e explicit f o r m of t h e V a i s a l a - B r u n t f r e -
q u e n c y N2{z) is p r o p o r t i o n a l t o 1 / \z—20| in one case a n d t o A2—B2(z—zD)2 

in t h e o t h e r . A c o m p a r i s o n w i t h some e x p e r i m e n t a l d a t a concerning t h e 
L i g u r i a n Sea is a c t u a l l y in course. 

R iassunto . — I n relazione a d e t e r m i n a t e condizioni di superfìcie, l a 
s t r u t t u r a v e r t i c a l e del m a r e si c a r a t t e r i z z a m e d i a n t e u n a b r u s c a v a r i a z i o n e 
nella d e n s i t à . Nel p r e s e n t e lavoro vengono s t u d i a t e le o n d e i n t e r n e che vi 
r i s u l t a n o f o r t e m e n t e i n t r a p p o l a t e , o t t e n e n d o relazione di dispersione, velocità 
di g r u p p o e correlazione in t e r m i n i di f u n z i o n i e l e m e n t a r i p e r d u e s i t u a z i o n i 
s p e r i m e n t a l i i n d i v i d u a b i l i a n a l i t i c a m e n t e a t t r a v e r s o la f r e q u e n z a di V à i s à l a -
B r u n t N2(z) p r o p o r z i o n a l e a I /\z—za \ in u n caso ed u g u a l e a A2-B2(z—z0)2 

n e l l ' a l t r o . È in corso u n c o n f r o n t o con i d a t i p r o v e n i e n t i d a c a m p a g n e di 
m i s u r a e f f e t t u a t e nel Mar L i g u r e . 

1 . - I N T R O D U C T I O N 

A n i n t e r e s t i n g p r o b l e m i n t h e e n e r g y b a l a n c e of a s e a , c o n c e r n s 
t h e i n t e r n a l w a v e s a n d t h e i r e n e r g i e s . T h e s e a r e w a v e s w h i c h p r o p a g a t e 
h o r i z o n t a l l y a n d t h e i r l a r g e s t a m p l i t u d e is r e l a t e d t o t h e v e r t i c a l 
v a r i a t i o n s of t h e d e n s i t y g(z). T h i s is d u e t o t h e v e r t i c a l v a r i a t i o n s 
of t e m p e r a t u r e a n d s a l i n i t y , w h i c h a r e o r i g i n a t e d b y t h e i n t e n s e e x -

(*) C . N . R . - I s t i t u t o Fisica d e l l ' A t m o s f e r a - R o m a , I t a l y . 
(**) I s t i t u t o di Fisica " G . M a r c o n i " , U n i v e r s i t à di R o m a , I t a l y -

I s t i t u t o di Fisica Nucleare - Sezione di R o m a 



2 4 2 11. P U K I N l - E . S A L U S T I 

change of m o m e n t and h e a t with t h e mowing a t m o s p h e r e . More in 
detail, one could first separate a " m i x e d l a y e r " which has a v e r t i c a l 
extension of some t e n s of m e t e r s under t h e air-sea surface. This layer 
is mixed b y t h e t u r b u l e n c e p r o p a g a t i n g downward f r o m t h e moving 
a t m o s p h e r e (this is one of t h e effects of winds, storms, a t m o s p h e r i c 
turbulence etc.). I t s energy p r o p a g a t e s downward freely a n d m a k e s 
t e m p e r a t u r e , salinity, density and motion-homogeneous in this " m i x e d 
l a y e r " . The surface u n d e r this m i x e d layer is t h e n a t u r a l d o m a i n of 
p r o p a g a t i o n f o r t h e m o s t intense i n t e r n a l waves. I t h a s however t o be 
r e m a r k e d t h a t it is n o t v e r y easy t o distinguish w h a t p e r c e n t a g e of 
atmospheric energy generates i n t e r n a l waves and w h a t p e r c e n t a g e of 
energy is used b y t h e system t o erode t h e u n d e r l y i n g stratified region. 
Practically, moreover, one can r e m a r k t h a t m a n y times t h e mixed layer 
has n o t a v e r y s h a r p division with t h e underlying stratified region, b u t 
has a vertical extension of 50 --500 meters. The resulting periodic 
p h e n o m e n a , t h e i n t e r n a l waves, are in this case related t o a smoother 
v a r i a t i o n of t h e density t h a n in t h e case of t h e s h a r p division between 
t h e mixed layer and t h e deeper stratified region. 

Tts stratification is a r a t h e r curious p h e n o m e n o n : one can experi-
mentally r e m a r k m a n y sheets of an horizontal extension of kilometers 
and this is a surprising contrast with t h e vertical extension of few 
centimeters. Practically, f o r g e t t i n g this " f i n e - s t r u c t u r e " , one could see 
it as a stable region of slowly v a r y i n g density g0(z). 

I n t h e lowest p a r t , g0(z) decreases with t h e d e p t h as a slow expo-
nential exp (— az), where a is a c o n s t a n t . 

I n this contest, we h a v e r e m a r k e d t h a t t h e surface b e t w e e n t h e 
mixed layer and t h e stratified thermocline is t h e d o m a i n of m a n y 
interesting and i m p o r t a n t p h e n o m e n a , related t o t h e i n t e r n a l waves 
d i s t r i b u t i o n of energy inside t h e fluid ("). I t can be shown, m o r e pre-
cisely, t h a t t h e knowledge of e x a c t s h a p e of g0(z), t h e s t a t i c d e n s i t y 
profile, could give essential informations concerning t h e i n t e r n a l waves 
s t r u c t u r e , their correlations and t h e i r energetics. 

More explicitly, it has t o be added t h a t t h e e x p e r i m e n t a l evidence 
stresses t h a t this is n o t really a s h a r p surface, b u t it appears r a t h e r as a 
vertical region of t r a n s i t i o n between t h e homogeneous mixed layer 
and t h e stratified thermocline. This appears interesting because t h e 
i n t e r n a l waves can be described b y a simple equation (if one assumes 
t h e linear waves and if t h e Boussinesq a p p r o x i m a t i o n is assumed v a l i d : 
see, for example, Phillips (°) and T h o r p e (7)) where t h e explicit shape 
of Q0{z) plays an explicit role. 



O N L I N E A R I N T E R N A L W A V E S O N T H E S E A E T C . 2 4 3 

Now, this equation has simple dispersion relation and solution in 
some cases r a t h e r well known in t h e literature. These are supplied 
when t h e Vaisala-Brunt f r e q u e n c y 

, , dp0 q 
N2 «) = r~ • — > 0 d Z Q O 

has a d(z) behaviour or JV-(z) = const, behaviour (Phillips (6)) and when 

N*(z) = 
0 for 0 <z<d ((j i s t ] i e d e p t h of t h e discontinuity 
e-az for z>d in t h e density) 

as in t h e classical analysis of Garret and Munk (3) and in few other 
cases studied b y T h o r p e (8). 

The realistic cases are r a t h e r different; one could easily solve t h e m 
numerically, b u t this would imply some loss of informations. F o r t h i s 
region we h a v e studied t w o r a t h e r realistic profiles 

N*(z) = a2l\z—z0\ and N2(z) — A2 — B2{z—z0)2 

These profiles can, in some cases, simulate correctly t h e e x p e r i m e n t a l 
situation and t h e y also allow an explicit calculation of t h e i n t e r n a l 
waves, their dispersion relation, their group and wave velocities, then-
correlations. 

An experimental verification is actually i n course (x). 

2 . L I N E A R T H E O R Y O F M A R I N E I N T E R N A L W A V E S 

The t h e o r y of internal waves is r a t h e r well known (Phillips (°), 
T h o r p e (8)). I n t h e following we will r e p e a t t h e essential results on 
t h e t i m e evolution of these waves. 

I n fact, in t h e case t h a t t h e Boussinesq a p p r o x i m a t i o n can be 
assumed and t h e e a r t h ' s r o t a t i o n can be disregarded, t h e velocity 
components satisfy t h e equations (6|7). 

1 i)« 
+ — - f - = 0 

Qo 

+ — ~ = 0 
(?o ty 

i q' *tw + — -J- + g = 0 
Q o "Z Qo 

10 



2 4 4 R . P U R 1 N I - E . S A L U S T I 

where p is t h e d e p a r t u r e f r o m t h e h y d r o s t a t i c pressure and one has 
assumed 

Q = QO(z) + Q' (X, y, z, T) 

with t h e density variation Q'<S:Q0. 
I n t h e same f r a m e of a p p r o x i m a t i o n , one also assumes t h a t t h e 

water is incomprensible: 

ixU + + izlV = 0 

d p V V <>g' 
at it ix 7>y iz 

Calculating first t h e t i m e derivative of t h e vorticity, one lias 

( i * - + — JL V = o 
\ iz ix J it Q O ÍX 

. iv iw \ i a io' 
Vu — — — + — - - — = 0 

iz iy J it go iy 

a n d , t a k i n g into account t h e <y~ = 0 relation, one h a s 
Cli 

i l g ig' g i i , g d g o iw 
~ì)t Qo ÌX Qo ÌX it ~ Qo dz iz 

i g io' g i 5 , g dp0 iio 
it go iy Qo iy it ^ ~ Qo d» iy 

Taking t h e n t h e horizontal divergence, one h a s t h e basic relation 

y-tt (Vxz + i\y + W (X, t) + A^X^+S2,,,,) W (X, z,t) = 0 [2.1] 

where N-(z) = — — - f - > 0 is t h e Vàisàla-Brunt frequency. 
QO d z 

Assuming a p l a n e progressive wave solution of t h e f o r m 

w (x, z, t) = 17 (z) exp i(Kxx + Kyy + Kzz — wt) 

one easily arrives t o t h e equation 

+ K2 — K2\ W(z) = 0 [2-2] 
d*W\z) , j N2(z) 

dz2 



ON L I N E A I ! I N T E R N A L W A V E S O N T I I E S E A E T C . 2 4 5 

w i t h 
17(0) = o a t t h e rigid f r e e surface (

9) 
W(—d) = 0 a t t h e b o t t o m 

The informations concerning t h e stratification are included in t h e 
p a r t i c u l a r shape of N2(z). This is r a t h e r c o n s t a n t in t h e mixed layer. 
I n t h e lower region it has m a n y sharp variations (the fine-structure). 

These variations give t h e classical behaviour of N2 (z) in t h e 
tliermocline when, in some sense, t h e y are averaged in 2. A t last N 2 —> 
const, value in t h e deepest regions can be f o u n d . 

As t h e various preceding cases h a v e been studied, we h a v e focused 
our a t t e n t i o n to two explicitly solvable cases: 

where a, A, B, are constants to be determined on experimental ground. 
The d e p t h z0 is t h a t corresponding t o t h e zone of highest v a r i a t i o n of 
t h e Y a i s a l a - B r u n t frequency N2(z). The cases seem to be general 
enough t o a p p r o x i m a t e realistic cases, particularly in t h e parabolic case. 

3 . T H E E X P L I C I T C A S E N2(Z) = a2/\z— z0\ 

W e are now going t o s t u d y t h e case above mentioned 
N2(z) = a21\z—z0\. The equation [2.2] now results 

W e assume t h a t t h e distance among zD, t h e b o t t o m z = — d and t h e 
surface 2 = 0 could be considered large. I n practice z0 is 20-50 m e t e r s 
for localized seas (*) and h u n d r e d m e t e r s for t h e Ocean. The d e p t h 
d of t h e b o t t o m is usually fixed t o be larger, in order t o avoid b o t t o m 
effects. F o r strongly t r a p p e d i n t e r n a l waves, t h e vertical region of 
i n t e r e s t is determined in order t o fix t h e vertical scale of motion. I t 
usually is of t h e order of m a g n i t u d e of few meters. Outside this region 
2 = z0, t h e Yaisala-Brunt f r e q u e n c y decays r a t h e r rapidly, as a power 
of 2. Much more quick, however, is t h e decay of t h e solution IF, 
which results in general a negative exponential. So one could also 
assume for these waves an idealized b o u n d a r y condition 

a) N2(z) = a211 z—z0 \ 
b) N2(z) = A2 — B2 (z—zo)2 

d2 W(z) 
dz2 

[3.1] 

W (± 00) = 0 



2 4 6 R . P U R 1 N I - E . S A L U S T I 

which simplifies t h e calculations. T h e n one can say t h a t our eigen-
value e q u a t i o n can a s s u m e a n infinity of eigensolutions, labelled 
b y 11 = 1, 2, 3 . . . 

T h e g e n e r a l solution is (see A p p e n d i x ) : 

Wfl = A^xo™o> L}^ {2K\z—z0\} [3.2] 

where A/t is a c o n s t a n t a n d t h e f u n c t i o n L)i_l (g) is called L a g u e r r e 
polynomial. A t z — za, t h e f u n c t i o n is c o m p l i c a t e d : t h e e q u a t i o n shows 
t h a t i t s second d e r i v a t i v e diverges. T h e n Wp results a c o n t i n u o u s 
f u n c t i o n s y m m e t r i c a r o u n d t h e p e a k z0 One c a n m o r e o v e r say (see 
A p p e n d i x ) t h a t t h e solution exist if a n d only if 

a2K 
- — = p =1,2, . . . 

I t ' s i n t e r e s t i n g t o n o t e t h a t t h e p r e c e d i n g dispersion r e l a t i o n for 
t h e s e i n t e r n a l waves is similar t o t h a t of t h e two-fluid s y s t e m . 

T h e n one can calculate t h e g r o u p v e l o c i t y cg 

da> 1 a 2 

°g = d K = Y 1W(2JL)
1'2 

One h a s also t o r e m a r k t h a t W¡i. is p r o p o r t i o n a l t o e x p - / t 12—z0\ 
so t h a t t h e b e h a v i o u r of t h e w a v e decays e x p o n e n t i a l l y outside t h e 
zone of s h a r p v a r i a t i o n of A2(2). 

This implies t h a t we m u s t consider only waves w i t h a large K 
v a l u e fixed b y t h e dimension of t h e physically i n t e r e s t i n g region t h r o u g h 
t h e r e l a t i o n K ~ 1 \L. One could enlarge t h e p r e c e d i n g t r e a t m e n t of 
t h e i n d i v i d u a l s t r u c t u r e of i n t e r n a l waves b y considering t h e correlation 
f u n c t i o n . This q u a n t i t y is 

+ 00 
B/tfl' (K) = l l > u y = I WM W%' dz' 

I t a p p e a r s i n t e r e s t i n g because it is a p o w e r f u l tool i n t h e comparison 
b e t w e e n t h e o r e t i c a l models a n d e x p e r i m e n t a l d a t a (4). I n our case 
i t r e s u l t s : 

LI 2 

B^ (K) = ^XKL j AfxA%' e - 2 ^ 0 ) Lji^ L\*_! d {2K\zr—z0\) 



ON L I N E A I ! I N T E R N A L W A V E S ON TIIE S E A E T C . 2 4 7 

The Laguerre polynomial are on orthogonal set, therefore one has 

1 0 for n ^ fi' 
BNi' ( K ) = i 1 — { l A p U / t - i y . ? 2C-i for ¡X = n ' 

this K ' 1 behaviour of t h e correlation f u n c t i o n is r a t h e r interesting 
and is in agreement with t h e results of Phillips (6) for a range of t h e 
s p e c t r u m of t h e lowest i n t e r n a l mode. If one w a n t s t o calculate also 
t h e cross-correlation WV, t h e c o n t i n u i t y equation 

i KyV + W = 0 

m u s t be used. This can give t h e m a t r i x element 

/ d W 1 ] W d = 0 
d z 

One could add t h a t t h e m a t r i x element 

dWn 
iz Wm &z' 

is n o t zero if (and only if) w = to ± 1. I n more detail 

d W. \ _ , , U 
W ^ dz' = 1 / J L 

dz J • "" ' 2 

and 

/ -dz ) " T i | 2 

To finish, also t h e case 

a2 B-
JV2(«) = — a, /? const. 

z z2 

could be exactly t r e a t e d . I t results, however, of different physical 
interest because it describes an instable case, whereas in this n o t e one 
studies stable phenomena only. 



2 4 8 R . PUR1NI - E . S A L U S T I 

4 . T I I E C A S E O F A P A R A B O L A A 2 (Z) = A2 — IP (Z—ZA)2 

B y repeating t h e preceding considerations, one can arrive to t h e 
e q u a t i o n : 

= (z-zo)2 1 F ( . ) - i ^ f ' - J D 1 * ( . ) ; ^ 
Az2 to2 [ u)2 J 

with t h e b o u n d a r y conditions (°), 

W ( 0 ) = 0 

W(—d) = 0 

I n t h i s case also t h e r e is an infinity of solutions (see A p p e n d i x ) : 

'<K , „ U K B V ' 2 
W, = At e"a j ,z'z°' Hq — (z—z0)\ q = 1, 2, . . . [4.1] 

Where A„ is a c o n s t a n t and Hq(o) is an H e r m i t e polynomial. F r o m 
this equation one can see t h a t t h e wave a m p l i t u d e decreases more 
rapidly t h a n in t h e preceding case (eq. [3.2]) as \z—z01 increases. 

There is also in this case a dispersion relation 

- - f 
I t has t o be r e m a r k e d t h a t <o decreases as q increases, t h a t is, for higher 
modes, t h e dimension L of t h e system fixes a K v a l u e K ~ 1 [L. F r o m 
t h e above equation, one can derive: 

— CB + (G2B2 + 4J£2A2)1/2 
<» = 2 K — 0 = g + l [4.2] 

and t h e n t h e group velocity can be deduced 

Cg = 
(G2B2 + h [ C B - ^ + 4 A 2 K ' ) v i 

I n this case also it is easy t o calculate t h e correlation f u n c t i o n 

L / 2 

B„' (K) = ~ f AqA*. e' K'1(z'-2a)1 Hq Hi dz' = 



ON L I N E A I ! I N T E R N A L W A V E S ON TIIE S E A E T C . 2 4 9 

LI 2 

I A.q A.q' Hqllq' 6 UJ d KB\ (z' Zo) 
o 

which, r e m e m b e r i n g t h e orthogonality of t h e H e r m i t e polynomials, 
becomes 

This result can be w r i t t e n in a more clear f o r m by using t h e disper-
sion relation. 

I n f a c t , f r o m eq. [4.2] one has t h a t 

roA'-i ~ JSC-2 

for which eq. [4.3] becomes 

I n this case, t h e behaviour of t h e correlation f u n c t i o n is different 
f r o m t h e other one. 

As in t h e preceding section, if one w a n t s t o calculate t h e correla-

t i o n WV, one m u s t use t h e c o n t i n u i t y equation, giving V as a func-

tion of ' ' ^ . Now, t h e resulting m a t r i x element 

can be found in t h e l i t e r a t u r e (5). 
The profile 

N2(z) = A2 — B2 (z—zo)2 + C (z—zo) + D ( z — z a ) 3 

can also be calculated. The result is exact for C ^ 0 and with some 
a p p r o x i m a t i o n for D ^ 0. 

To finish, we w a n t to note t h a t results obtained in this note for 
t h e i n t e r n a l waves in t h e presence of a strong stratification are used 
for a comparison with experimental d a t a concerning t h e Ligurian sea i1). 

[4.3] 

Bm' (K)ocK-v2 



2 5 0 R . P U R I N I - E . S A L U S T I 

A P P E N D I X 

a2 
a) The case. 

I Z—Z o \ 

This e q u a t i o n is r a t h e r well known. W e follow here a description 
of t h e solution which is widely used in q u a n t u m mechanics (2). I n 
adimensional form, it results 

d *~u ( I 1 \ 
T 7 + M == 0 

dp2 \q 4 ) 

Asimptotically t h e solution results 

u ~ e ± e/2 

and if one w a n t s u t o he finite, u ~ e-e/2 

Multiplication b y a polynomial d o e s n ' t change t h e a s i m p t o t i c 
value 

This gives 

= F (Q) e-e/2 

dg- dg g 

P u t t i n g now 

F = S AK G* 

K = 1 

inside t h e preceding equation, one has 

(A — 1 ) A , + 2 A , = 0 

K(K+1)Ak + 1 + ( A — K ) Ak = 0 

Now, if one assumes t h a t t h e series is infinite, t h e n 

A K + I _ A — A 1 _ 

Ak ~ E (E+l) K 

which correspond t o an exponential behaviour of F(g), is in contra-
diction with our hypothesis u ~ e~Si2. 



ON L I N E A R I N T E R N A L WAVES ON THE SEA ETC. 251 

T h e n t h e series cannot he infinite. 
The other only possibility is t h a t }. be an integer n. 
Then one has t h e "eigenvalue c o n d i t i o n " 

I = n 

AK = 0 Ii > ft 

The polynomials obtained in this way are related t o t h e associated 
L a g u e r r e polynomials: 

Ln(Q) = £ 0 ? " 

The first one are: 

L\ = 1 ; L\ = 4 — 2 g; L\ = 1 8 — 1 8 O — 3 <?-
L \ = 9 6 — 1 4 4 G + 4 8 O 2 — 4 Q 3 

b) The A2—B2(z—z0)2 case 

W e r e p e a t here t h e preceding t r e a t m e n t of t h e equation. I n adi-
mensional f o r m 

T ! + - « = 0 

The asimptotic value being 

u ~ e 

If one assumes 

u = H(Q)C'Q*'2 

t h e n obtains, for t h e basic equation 

^ - 2 , f + (A - 1) H = 0. [A.l] 
do2 do 

Now, writing for II 
N 

H = 2 an Q" 
H = 1 

t h e differential equation [A.l] gives 

« s + 2 ( 2 s + 1 ) — A 
(s + 2) (s + 1 ) 

s > 0 

10 



2 5 2 K. p u r i n i - E. SALUSTI 

Separating t h e even or add solutions, on t h e n has 

I = 2 n + 1 

The solutions are even or add polynomials, t h e H e r m i t e poly-
nomials 

d » o - G2 
H t { e ) = ( - D * • 

The first one are 

H o = 1 ; H , = 2 g ; H 2 = ±Q
2—2; H 3 = 8 Q

3 — 12 Q 

R E F E R E N C E S 

( 1 ) C o i . a c i n o 51., P u r i n i R . , R o v e l l i A . , S t o c c i i i n o C . , 1 9 7 0 . - Studio 
del bilancio termico diurno del ciclo di temperatura e delle onde interne 
in stazione fissa (Mar Ligure). N o t a i n t e r n a d e l l ' I s t i t u t o Idrografico 
della M a r i n a , G e n o v a . 

(2) D icke R., W i t t k e J . , 1900. - Introduction to Quantum Mechanics. " A d d i -
son W e s l e y , P u b l . Comp. I n c . " , L o n d o n . 

(3) G a r r e t C., Munk W . , 1972. - Space-Time Scale oj Internal Waves. "Geopli. 
F l u i d D y n . " , 2, p p . 225-264. 

(4) Monin A. S., Y aglom A. M., 1971. - Statistical Fluid Mechanics. " T h e 
Mit P r e s s " , I. 

(5) Morse P . , Feshbach H . , 1953. - Methods oj Theoretical Physics. "McGraw-
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(6) P h i l l i p s O. M., 1900. - The Dynamic of the TJpper Ocean. " C a m b r i d g e 
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(7) See t h e classical discussion in t h e book of Phillips, Reference (6). 
(8) Thorpe S. A . ; On the Shape of Progressive Internal Waves. " R o y a l Soc., 

L o n d o n , P h i l o s . T r a n s " . , V 2 0 3 A ; I I 4 5 ; p p . 5 0 3 - 0 1 4 . 
(°) Thorpe S. A. (and other articles), 1975. - " J . of Geopli. R e s e a r c h " , 80.