SEATTLE

UNIVERSITY OF WASHINGTON DEPARTMENT OF OCEANOGRAPHY(Formerly Oceanographic Laboratories)

Seattle, 'Washington

PROPAGATION AND DISSIPATION OF LONG INTERNAL WAVES

By

Maurice Rattray, Jr.

Technical Report No. 27

Office of Naval ResearchContract N8onr-520/II1

Project NR 083 012

Reference 54-11March 1954

ABSTRACT

LIST OF 51MBOLS

INTRODUCTION

MATHEMATICAL FORMULATION

DISCUSSION OF RESULTS

TABLE OF CONTENTS

Page

iii

iv

1

2

17

TABLE 10 Values of Wave Numbers and Damping CoefficientUnder Varying Conditions 19

FIGURE 1. Damping Coefficient Variation with Layer Depth 20

BIBLIOGRAPHY 21

-ii-

.~

ABSTRACT

The effect of friction on the propagation of long internal

waves in a rotating ocean is investigated theoretically for the simple

case of a two-layer system subjected toa constant eddy viscosity.

Waves of inertial period are excluded. Typical distances tor a 50%

decrease in amplitude are found to be 1,000 to 2,000 km•

-111-

Synibol

x, y, z:

t:

u, v, w:

t :J ·O·

g:

v :

CAl· t

PIflit

h:

o I

k:

y :

'\t

~2:

e :

R:

LIST OF SIMBOLS

Meaning

Cartesian coordinates of position, with z a 0 at themean interface level.

Time.

Components of water velocity.

Elevation of upper surface of layer.

Value of j at time t := 0 and position x =O.

Acceleration due to gravity.

Eddy viscosity.

Angular velocity of rotation of ocean area.

Density of sea water.

Density difference between two layers.

Mean layer depth.

Frequency of the internal wave.

Wave number of the internal wave.

Damping coefficient of the internal wave.

J (0 + 2 w}/2 v •

I (0 - 2 W}/2 v •I If

j 0/ Yo •g ~ ('Y+ ik}2 (hf+h fl )e

o 2 - 4w~ •

E>t

•

-iv-

Symbol Meaning

BI: (3 1 (h '+h lt ) •

B2 c [3 2(h t+hII ) 0

I (J).p • 12 - a. •

q: 1/2 + ~ •a

SO= (h l+h fl )2 •h'h"

SIt Real part ot !!. - 1 •So

Tlt Ima • RgJ.nary part of So - 1 •ij h fl

I q)O: h' +h" •~.

4'1 :.. e

Real part of "T::: - 1 0

0

'"'1'1:

e 1.Imaginary part ot ~ -0

P: p2B1 0

Q: q2B2 •

-v-

INTRODUCTION

Evaluation of oceanographio data is at best uncertain where the

influence of long internal waves cannot be determinedo This is clearly

shown by the California current where an apparent eddy structure is

found which is caused by internal oscillations of diurnal period

(Defant ~ 19'Oa). From the appa;rent wide distribution of such internal

waves it is reasonable to suppose that the possibility of error exists

in the interpretation of much similar data. A knowledge of the occurrence

and behavior of long internal waves in the ocean becomes then of prime

importance for the advance of oceanography.

Obtaining information on internal waves at sea is a difficult

process which requires long series of observations for statistically

valid results. It has been demonstrated that reported internal waves,

in many cases, may be only harmonic representations of short series of

random data (Haurwitz, 19,2, 19,3; Rudnick and Cochrane, 19,1). In

addition, in order to adequately describe the behavior of the internal

waves it is necessary to take measurements simultaneously at three

stations (Haurwitz, 19,3).

The essential reason for the difficulty in obtaining internal

wave data in the ocean is the lack of an adequate theory to guide the

investigations. Even the presence of tidal internal waves is not well

e~lained. The resonance theory (Haurwitz, 19,0; Defant, 19,0) is

inadequate because of the large frictional damping and narrow belt of

-1-

resonance for these waves. Since internal waves of tidal period have

not generally been found in the open ocean but mai.nly in offshore

regions, a mechanism of generation associated ~dth the ocean boundaries

is suggested (Rudnick and Cochrane, 1951). Model experiments have

shown the generation of internal waves by the passage of surface waves

over bottom irregularities (ZSilon, 1934). At that time it was felt

this did not offer an explanation for the observed tidal internal waves

since their amplitudes would be decreased rapidly by friction and

divergence 0 However, internal waves generated by the action of surface

tides over the continental shelves will not be subject to radial

divergence as from a point source, but will undergo convergence on

traveling seaward from a boundary source. It is proposed therefore to

derive expressions for the effect of friction. on internal, wave motion

for the case of a rotating earth.

MATHEMATICAL FOR1'lULATION

For mathematical simplicity the problem considered' is the

propagation of long internal waves through a two-layer fluid on a

rotating disk subject to constant eddy viscosity. A right-handed

coordinate system is used with the origin at the mean level of theI

i;nterface, the z-axis vertical, positive upwards, and the x-axis

positive in the direction of wave travel. A primed quantity refers

to the upper layer and a double-primed quantity to the lower layer.

The following assumptions are made r the vertical acoelerations are

~gligible permitting use of a hydrostatic pressure distribution;

-2-

all quantities are independent of position in the y direction; the water

is incompressible; and u, v, wand variations in p and p are sufficiently

small that terms of higher order can be neglected.

The equations of motion then become, for the upper layer:

av t

at

- 200 v t - V

+ 200u' - Y

.. -g~ '

= 0 ,

(1)

(2)

and for the lower layer:

CI _ g b~ t _ g ~p 0?;: tI

Ox P Ox(3)

'i. If ~2VIfQV + 2w uti - y ~ = 0 •~t dZ

The equation of continuity is

ht+ ~'

/" ~dz".J. i)x?;:"

for the upper layer, and

(4)

(5 )

nl';ltdz ~ - ~----

~t(6)

for the lower ~ayer.

The bounda~ conditions are:

at the free surface there is no tangential stress, or

at z g h', bv'-~z

cO; (1 )

at the interface, the velocities and tangential stresses are continuous, or

at z a 0, u' = un, vl::ll VII,

ourOZ

aUn

OZ , ~V"

,jz•, (8)

at the bottom there is no slippage, or

at z a -hn. , un = V" = O. (9)

Equations (1) through (9) are solved in a manner similar to that applied

to tidal waves on the Siberian shelf by Sverdrup (Sverdrup, 1926).

-4..

SOLUTION

A solution is found by assuming

y t ::I Y r ..y x i ( (1 t-kx)~ ~o e e ,

yll ::I yll -'YX i( a t-kx)~ ~O e e ,

(10)

(11)

• Itwhere ~ 0' ~0' r, 0, and k are all constants characteristic of the

motion. The general solutions of equations (1) through (4) are then

ig ~~ (r + 11<:)e..ty'X ei ( at-lac).. --------------02

- 4002

/',~ + Ar (1+i) A. z,) a 1 e ....J.

I

\.\

+ A~ e-(l+i)~lz - A3e(1+i)~2z - A4 e-(1+i)~2z (J

g ~~ ( Y + ik)e-Yx ei ( 0 t-kx)

v' = -02

- 4 002

(12)

(13)

(14)

ig (~b + ..MLp

~ It ) ("( + ik)e-Yx ei {a t-kx) (ull a _ 0 j 0 + Ai e (1+i ){31z

22\o .. 4 ell l~.

+ All -(1+i)(31Z All (1+i)(32z An -(1+i){32z 12

e - 3e

- 4 e )'

-5-

V" a

\

An -(1+i)f.l.1Z An (1+i)~2Z A" -(1+i)~2z;)+ 2 8 l-' + 3e + 4 8 ! •

•1

(15)

(31 and ~2 are constants given by I'a + 200 and)'a - 2CJl ,,2V \ 2v

respectively. Relations for the constant coefficients A, determined

from the boundary conditions, after some rearranging are:

a .. 2w I 6e (2 + '1. + A~) '" (e + -T)( (1; 2 W + Ai + Ali) J

eo (a + 200 , , ) ( 6p a + 200~ 2 - A3 - A4 :I e + p-)(2 - A3- Ai:) I

( ' '):1 (e +~) (AU .. An) ."e A1 ~ A2 P 1 2 '

-6-

(16)

(17)

(18)

(19)

(20)

(21)

(22 )

(23)

~twhere e III 4, a small quantity for internal waves. Solution of

~o

these equations yields the following expressions for the coefficients:I

(24)

-1-

2(e +~)____p_ (e..2(1+i)~h' + e2(1+i)~htt) Ai III 1:.~ (1 _ e-2(1+i)~h')a~2CJ) 2 p

- (e + ~)e(l+i)f1.h" J (27)p

2', e (e-2(1+i)~2h' + e2(1+i){32h") A4 a _ 1:.~ (1 + e2(1+i)132htt )a+2w 2 p

I

(30)

Thei equations of continuity can now be writteni

-8-

,

h'

f ~('f 2 + 2 ik'Y' - ~)r' (z)dz .. ia (~~ - ~~), (32)

o

and

where

and

r"(z) a ig (a+A~ e(l+i)'1.Z +A'2' e-(l+i)(31Z -A',3' e(l+i)(32Z

a 2- 400 2

Aft -(1+i)~2Z)- 4 e .•

Integration of the equations of continuity and substitution of the

values for the constants A, yields the following two expressions in

(r + ik) and e :

-9-

2(02 -4002) 0 (I-e) .. (0- 200) I:J.t sinh [ (1+i)131h'] {COSh [(1+i)131h"] + ~:I

g( "( + ik)2 (l+i)~l cosh [(1+1)~1 (h' + hit)]

( 0+ 200 ) -1- sinh [(1+i)~h'] {, cosh [(1+1)132h"] + I}

+(1+1)(32 cosh [(1+i)~2 (h t+ hit)]

[:

(0 .. 2(,J)sinh [(1+i)(31h'l (a+2CJ.»sinh [(l+i)~h' ')+e 2ah' + - '(,(34)

(1+1)(31 cosh [(1+1) '\ (h I + hit)] (1+1)(32 cosh [(l+i)~(h I + hlt)l)

2 22( a - 400 )oe

g( r+ik)2

It !J.nQ - 2 ah ..:::r_p

t(0- 200) -1J- {Sinh [(1+i)l3J.h '] + sinh [(1+i)131 (h'+ h"n)

(l+i)~l cosh [(1+1)(31 (h'+ hll

)]

( 0 + 200) -t- (Sinh [(1+1) 132h'] - sinh [(1+1>132 (h' + h" n)

, It

(1+1)(32 cosh [(1+i)(32 (h + h )]

t(O + 2(0) sinh [(1+i)(32 (h t + hit)]+e

(1+1)(32 cosh [(1+i)(32 (h'+ htl

)]

-10-

(35)

These equations in their present form are somewhat intractable. In order

to determine the relative importance of the various terms, the dimension-

less quantities

R =2 L.2

a - LW

, e = €

flp! p(36)

(37)

p = L.w/o2

, q = ~ +w!cr, (38)

are introduced. Equations 34 and 35 can now be written in the form

(1 - 7F-e) = p sinh [(1+1) fih'] {COSh [(1+1){31hll + 1}

R (l+i)}\ cosh [(1+i);31(h l +h IJ )]

+ q sinh [ (loL1){3~!J (COSh [ (1+1)f32h ll ] + 1}

(l+i) B2

cosh [(1+1)(3 (h' + hit) 1. 2

Gh l

+ e lith r +h II

P sw.h [(l+i )(31h 1 ] q si1"'.h (l+i )(32h I ] 1(1+1) Ii cosh [ (l+i}f31(h '+ h II)] - (1+i) P2 cosh m:+1 :f3 2 (h ' +h II )ij, (9)

-11-

R

= - h" P {sinh [(l+i)f\h l ] + sinh [ (l+i~l(h l + h ll)1h'+ hit (l+i)E]. .cosh [{l+i)~l(h

'+ hit)]

q sinh [{l+i )(32h r) - sitili [(l+i )(32 (h1+ h It )~

(l+i)11 cosh [(1+i)(32(h l +h ll )]

~• sinh [(I+i)(3~(hl+hll»)

+ e .r.

(l+i)~ cosh [(l+i:t3 2(h l +hll )]

These equations can be solved for any given values of a" w" v" h' ,

hn, and 6.p /p • However, a particular numerical solution will not indicate

the dependence of k, Y, ~ ~ , and ~~ on the above independent variables.

For the most important range of variables it is possible to approximate

the relations to give functional relationships between the above quantities.

The following ranges of variables will be considered, using

c.g.s. units:

11' ~ , x 103 ,

, x 10'~ 11 r: 4t Ie? ,l~V~la3,

gP ~ 10-3,00~ • 725 x 10-4 ,

0.7 x 10-4~ a ~ 3 x 10-4 ,

with the additional restriction· thatI

(40a)

-12-

The derived constants then have the following ranges:

(31 hi, (3 2 hi) 1.2 J

f3]. hit (32 h n ~ 24 J

Under, these conditions the following approximations are valid (to within

10 per cent):

and equations (39) and (40) can be approximated by:

6p

2:21(1 - -@) p q [PP = + - i +

R 2Bl 2B2 2B1

thi +~(cos ~ h ll - sin (3 hIt

@ 1 1 )+ hi + h ll 2Bl e131hh

(41)

q cos f32h ll - sin (32hn • p cos (31hll + sin (3 hIt '. q oo,s (3 hll+Sinf321r- - (. 132h it ) - ~ -2B ( (jlh II 1 ) + J. ....- (- ," ~2hif

2B2 e 1 e 2B2 e

-13-

6Pe h"

rp

-2~ ] - [2:1- 2:21p = + iR hI + hit _2Bl

+ef1_-l!... + ....9- + i -lL_~}(42)2B1 2B2 2Bl 2B2 •

The.unknown quantities are now split into real and imaginary parts:

e = CJ? + i 'l' 1:1 ~ 0 (1 + qi 1 + i'if 1) ,

(43 )

(44)

"l'

2where 8

0= _ (hI + h ll

) and ~ =_h Ihlt 0

h"hi + h"

, the limiting values for

the unknowns when v ~O. Upon substitution of these expressions into

equations (41) and (42) and replaoement of..E.- by P, ...9- by Q, the2Bl 2B2

result is:

4.Q. h lf

1 + i Ph' + h II (1 +CJ? 1 + i 'if 1) :: p + Q _ i (p + Q) _ h II (1 + CJ?1 + i'l' 1)_ (h I + h II ) 2 h I + h II

, h Ih tf (1 + 81 + i T1 )

~h I cos (31h" .. sin (3 h tl

+ P 1h-'-+-h-It e(31h Ii

oos A-h II. - sin r.t2h II_ Q ..; .. t"'"~ "'"----e""""!(3~2O;-h"n"h---

cos (3 2h II + sin (3 hII]0Q 2

+ J. J32hll

-14-

, (45)

h" h"so: _ + i(P - Q) - (p - Q) + - (1 +~ + i'.I' )

h '+h" 1 1h I+h lt

f+ P - Q - i (p - Q)]. (46)

These equations can be solved by approximation in a straightforward manner

if 51' Tl , ~l' and 'l'1 are all sufficiently small so that powers higher

than the first can be neglected, to give

: . 8P hit h' 0ht+hll)2 hI] [(hf +hll )2 h0(5 + iT ) = - - (1 - -) + - - - P + l- - + - Q

1 1 Ph'+hIt hi+h 11 . h th It hit h 'hIt hit

(47)

(eJ'?1 + i'I\) = ~ h'h" 2 + ~ (p _ Q) _ i~ (p _ Q) _ (p + Q) AP + i ~ (p + Q).P (h '+h") h ll hit P P

(48)

Now the following expressions can be written for the unlmownst

,

g 6p (h l + h")P: 2(y + ik) =-

2 1...2a - t.IW

and

h'h lt(49)

e6p/p

h"= - -- (1 + 911 + i'l' 1) •h l + hit

-lS-

(50)

In most cases of interest the terms in equations (47) and (48) conta~ning

2M will be negligibly small and!!..!. « (h' + h II) • Equations (47)p h" hlh"

and (48) then can be written:

-

h'= iiii (p - Q)(l - i) • (52)

Approximate solutions of these equations for the wave number, K, and the

damping coefficient, If, of the internal ·waves are:

.(h...'..,;.,+..;;;h~1I)~";...1_/2_[ (cr _ 'ail)3/2 + (cr+'ail )3/j14 [2 h 'h II cr( cr 2 - 4»2 )1/2 J,

(53)

11 3/2 U 2)f.cr - 2w ) 4: (cr + 2 w ) • (54)

The solution for e is then:

8 h ll

e=_M_-P h'+ h ll [

1/2 { 3/2 3/2~1 _ h!.. (I-i) (2,,) . (0' + 200) - (0 - 200) ] (55)

hit 40 (hl+ h U ) (02 _ 4002)1/2 •

-16-

DISCUSSION OF RESULTS

The wave number" k, is increased, thus wave length and velocity

are decreased by the effect of eddy viscosity. The correction term in

the expression for the wave number varies as the square root of the eddy

viscosity and inversely as the depth of the shallower layer. Its mag­

nitude is small unless the restriction of equation (4Oa) is violatedI

by approach to the inertial period.i

I

The damping coefficient, y, varies directly as the square root

at the eddy viscosity" and inversely as both the square root of theI

density difference and the three-halves power of the depth of the

shallower layer. It increases approximately as the square root of the

frequency but has only small dependence on latitude in the range under

consideration.I

The ratio of free surface to interface displacement is decreased

in magnitude by friction with a decrease in time lag for the free surface

mqtion. Again the correction terms vary directly as the square root

of the eddy viscosity and are small unless the inertial period is

approached.

Representative values for k and y under various typical con­

dftions are given in Table 1.

The distribution of internal waves will depend ve~ strongly

on, the value of the damping coefficient y. This quantity yaries fromI

a kaximum of 6.0 x 10-3 to a minimum of 0.045 x 10-3 for the conditions

considered. That is" wave amplitudes would be decreased by 50% in a

distance of 120 lon. and 15,000 lan., respectively. The actual distance

-11-

,

of wave travel will depend strongly on the prevailing oceanographic

conditions and the wave period but not on the latitude within the restric­

tions of equation (40a).

Let us consider a typical example. Since the depth of maximum

stability is that of maximum velocity shear, the effective eddy viscosity

is taken to be 1 cm. 2sec-1• An average value of 100 m. is used for the

upper layer depth and 10-.3 gms.,61J1.3 for the density difference across

the interface. Then at latitude l.S0 N, k and'Y will be, respectively,I

015 kIn-1• and 0.50 x 10-3 lan-l • for a 12 hr. period and .055 lan-l • andI

Ol32 x 10-3 lon-1• for a 24 hr. period. The 12 hr. and 24 hr. waves

Mve, respectively, wave lengths of 42 lone and no kIn. and distances

tor a 50% amplitude decrease ~f 1,400 lone and 2,100 lone Figure 1

shows the variation of y with the depths of the upper and lower laYers.

These results are the right order ot magnitude to give regions of

strong internal waves surrounding bottom irregularities and yet not

large enough to giva standing waves over the entire ocean.

-18-

:- TABLE 1

Values of wave number and damping coefficient under varying conditiona.

L T hi hit V 6p k V(degrees) (hra) (m) (m) (cm2sec-1) (gms.cm-3) (lon-1 ) (lan-I)

I

12 100 1000 100 10.3 .10 6.0 x 10-3

30 12 100 1000 100 10-.3 .14 5.4 x 10-3

15 12 100 1000 "lOa 10-3 .15 5.0 x 10-3

JS 24 100 1000 100 10-.3 .058 3.2 x 10-3

1, 24 100 1000 :10 10.3 .056 1.0 x 10-3

15 24 100 1000 1 10.3 .055 0.,32 x 10.3~

10.3 0.016 x 10.31IIl5 24 500 1000 1 .027

15 24 100 1000 1 2 x 10-3 .039 0.23 x 10-3".

0 12 100 1000 1 10-3 .15 0.49 x 10-3

0 24 100 1000 1 10-3 .076 0.35 x 10-3

,

-19-

800

Ta 12 hours

T a 24 hours

100

I,,\

1\

""I,"

"'I\II',\1\,,\1'II,III,\,I ,I ,,,\1\\\\\\\\\\\\\\\\,\\ \\ \\ \\ \\ \\ \\ \, \

\ \\ \\ ,, '"

" "'" .......... ........... ....." ............... "- __ --__ h D 1000 meters

- -:'!'- - ,.. ti 1& 5000 met.,s

0.0 '--__"'--__~___'____...r..___~___A_____'_____'__ ___'

o

'.3

0.1

1.1

I '

~O.9

1.2

200 300 400 500 600 700DEPTH OF UPPER LAYER. h' (Meters)

I·· PIOUBB 1. Damping Coefficient Variation with LQer Depth

(L II 1,°, 1"'= 1 cm.2 sec.-1, b. .p III J.()-3 11I8. em.-3).-B- "--20-

10:Io*08r'....0.7

H-ZUJ(3ii: 0.6IlLWo()

I

C> 0.5ZQ..:E~0.4

-i

,

,

BIBLIOGRAPHY

Defant, Albert1950'. On the origin of internal tide waves in the open sea. J.

Mar. Res., 9(2): 111-119.

1950a. Reality and illusion in oceanographic surveys. J. Mar. Res.,9(2): 120-138.

Haurwitz, Bernhardj1950. Internal waves of tidal oharacter. Trans. Amar. Geophys.

Un., 31(1)= 47-52.

1952. On the reality of internal lunar tidal waves in the ocean.Unpublished manuscript.

! 1953. Internal tidal waves in the ocean. Unpublished manusoript.

RUdnick, Philip and J. D. Coohrane11951. Diurnal fluctuations in bathythermograms. J. Mar. Res.,

10(3)= 257-262.

Sverdrup, H. U.1927. Dynamic of tides on the North Siberian shelf. Geofys. Publ.,

4(5): 1-75.

Zei10n, Nils ••1934. Experiments on boundary tides. Goteborg. VetenskSamh.

Handl., Fo1j 5, Sere B, Bd. 3, No. 10: 8 pp.

-21-

Department of OceanographyUniversity of Washington

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1 Office of Technical ServicesDepartment of CommerceWashington 25, D.C.