Inertial waves in a rotating fluid

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ARCHIEF

Tech&sche Nogeschool OFFICE OF NAVAL RESEARCH

D UL

DEPARTMENT OF THE NAVY e

CONTRACT NONR-22O(35

INERTIAL WAVES IN A ROTATING FLUID

BY

GIULIO VENEZIAN

DIVISION OF ENGINEERING AND APPLIED SCIENCE

CALIFORNIA INSTITUTE OF TECHNOLOGY

PASADENA, CALIFORNIA

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Office of Naval Research Department of the Navy

Contract Nonr 220(35)

INERTIAL WAVES IN A ROTATING FLUID by

Giulio Vene zian

Reproduction in whole or in part is permitted

for any purpose of the United States Government

This document has been approved for public

release and sale; its distribution is unlimited.

Division of Engineering and Applied Science California Institute of Technology

Pasadena, California

Pre sent address: University of Hawaii

Report No. 97- 16 March 1969

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When the motion of a particle is referred to a rotating frame, two apparent forces act on the particle. These are the centrifugal force and

the Coriolis force. The centrifugal force is proportional to the distance

from the axis of rotation and it is directed away from that axis, while the Coriolis force is proportional to the velocity of the particle and is

directed at right angles to both the velocity and the axis of rotation.

The existence of these forces is due entirely to the non-inertial nature

of the frame of reference. Thus, if a particle is at rest in a rotating frame, it is actually moving in circles relative to an inertial frame and a centripetal force is required to keep it in its circular trajectory. Similarly, a particle at rest in an inertial frame will appear to be mov-ing in circles in a rotatmov-ing frame and thus, to an observer in a rotatmov-ing frame, the particle seems to be subjected to a force acting at right angles

to its velocity.

The motion of a fluid relative to a rotating frame is subject to the same two forces. _{The centrifugal force is a "potential" force, and for} this reason it is possible for a homogeneous fluid to remain in a state of uniform rotation, with the gradient of the centrifugal potential balanced by an equal and opposite pressure gradient. Any motion relative to the

rigid body rotation, however, is subject to the non-potential Coriolis force. _{Thus if through some imposed external conditions motion is} pro-duced in a rotating fluid, and then the external forces are removed, the fluid particles will be subjected to a force perpendicular to their velocity

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-2-and move in circular trajectories. These are inertial motions of the fluid, resulting in inertial currents and inertial waves.

Inertial waves can be produced in the laboratory, and they also occur in the ocean. Historically, they were first observed in experiments

performed by Lord Kelvin, who was also the first to describe their

dynamics. More recent experiments are those of Fultz3

and

To orn r e and

dridge4.

Inertial oscillations in the ocean were

ob-served by Defant5 in 1941.

Dynamics of rotating fluids

The equations of motion for a homogeneous fluid referred to a

rotating frame of reference are

+ ;.; +

= - Ç7p + V7V ₍₁₎

v=O

(2)

whe r e

P=pIp2Xr)2

In these expressions, y is the fluid velocity relative to the rotating

frame, 2

is the angular velocity of the frame, p is the pressure,

y

is the kinematic viscosity of the fluid, and p is the density. P denotes the reduced kinematic pressure, i.e. the kinematic pressure reduced by the centrifugal potential per unit mass. In the absence of free surfaces there is no need to distinguish between P and p,

so that P is often

called the pressure.

Equation (1) is the momentum equation, which states that the

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the Coriolis force and the viscous stresses. Equation (Z) is the con-tinuity equation, which for an incompressible fluid states that volume is conserved.

Inertial currents

The simplest type of inertial motion in a rotating fluid occurs

when P 0 and the velocity is transverse to the axis of rotation.

Neglecting the effect of viscosity, and quadratic terms in the velocity, the momentum equation oecomes

so that

and

With the viscosity and the non-linear effects neglected, the momentum equation is

+zx;=vP

(6)

The vorticity is defined by

(3)

(4)

:1 _z)

O (5)

It is clear from these equations that the fluid particles move with con-stant speed in circular paths with an angular frequency Z2. In terms of motion in the ocean, this corresponds to motion with the period of a

pendulum day. Such currents have been observed by Defant5 and others.

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-4-r, =VXv

(7)

The curl of Eq. (6) is

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and the divergence of that equation is

=v2p . (9)

Also, the scalar product of 2 with Eq. (6) is

so that

u

a t2

This equation is known as Poincar's

equation6,

although it had previous-ly been derived by

Kelvin2. Hough7

carried out an extensive

investiga-tion of flows inside ellipsoids, using this equainvestiga-tion. Equation (11) has plane wave solutions. Let

P = P cos(k. r_o - t) Then w2k2 - 4(.k)2 = O (13) Û r (10) (12) (14)

Thus wave motions are possible if i. e. for periods longer or equal to a pendulum day. The frequency of the wave motion does not

depend on the wavelength but on the angle between the direction of propaga-tion and the axis of rotapropaga-tion. The reason for this result is the following. Since the fluid is incompressible, the waves must be transverse, that is, the fluid moves in planes perpendicular to the direction of propaga-tion. _{Only the component of the} 2

parallel to k (perpendicular to y)

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enters into the Coriolis force, so that the fluid particles still move in circles but at a frequency corresponding to a smaller Coriolis force.

The wave character of the motion is due to the phase relationship be tween adjacent particles. The pressure is non-zero, but the gradient is parallel to k balancing the part of the Coriolis force which can be derived from

a potential.

Corresponding to the expression for P given in Eq. (12), the

velocity is

y =

Po Ç _{X (ÇX)cos (Ç.}

-t)+2ÇX

sin(Ç.-t)

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wk2

This expression represents a circularly polarized plane wave. A

some-what simpler form for P and y

is obtained by writing y in terms of a

vector U perpendicular to n, where n

is the unit vector in the direction of propagation. Then if

v=Ijcos(k.r -wt)±nX U sin(k.r -ut)

(16) P is given by

P=

[ff cos(k.r-t)±.(nXU)sin(k.r-t)]

(17) k. 2 with fore (18) Of particular interest is the energy flux associated with these waves.

From Eq. (6) it follows that

al

--v +7.(vP)=O

z

(8)

there-and

-6-;P=

ff+(U)xff

2k2

2k.Q

[(2U)U

+.(Xi)nX

]

Since nU is zero by definition, the energy flax is at right angles to

the direction of propagation. The same result is obtained from Eq. (18),

from which it follows that the group velocity is

-

2 2-k

=± k

k

k3

a vector which is clearly perpendicular to k. It is in fact oriented along

the projection of

normal to k.

This striking relation between the group and phase velocities will become even more apparent when we consider waves in a contained fluid.

Effect of viscosity on plane waves in an unbounded fluid

It was shown by Phillips8 that a single, damped, circularly

polarized plane wave is an exact solution of the Navier Stokes equations. This is because when only a single wave is considered, the non-linear term in the equation of motion is identically zero. The expressions for

y and P in this case are

k2t

-y = e V

[u

cos(k.r-.wt)± nXU sin(k.r-wt)]

z

-vt

P = [±2.ucos(k.r -wt) + 2.(nXU)sin(k.r -t)1 , (20) where, again w = ± (19)

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These expressions differ from those given previously only by the presence

z

-vkt

of the factor e . it is important to note that this damping factor has

a time constant proportional to v1 . As we shall see, this is not typical of a contained fluid, in which the damping constant is proportional to v2, usually a much shorter time.

Free inertial waves in a fluid layer

While an unbounded fluid would appear to be a suitable idealiza-tion of physical reality, this is not the case. The ocean, and even more so the atmosphere, seem unbounded to a human observer because the scales involved are much larger than those one normally experiences. When the ocean and atmosphere are regarded in a global view, however,

it is readily seen that the vertical dimensions are negligible compared

to the horizontal dimensions. _{Thus, a situation much closer to physical} reality is obtained by considering flows in a fluid contained between

infinite parallel planes. This is still a highly idealized model, but one that is a large step removed from the unbounded fluid model. As we

did before, we shall first consider the propagation of small amplitude

waves neglecting the effect of viscosity, and then take the effect of viscosity into account.

We consider now inertial waves in a fluid contained between the

planes _{z = O and z= h. For simplicity, let the axis of rotation be the}

z axis. _{The velocity components parallel to the x, y, z axes are u, y, w.} We seek solutions of Eq. (6) satisfying the requirement that w = O at

z O and z= h.

w itself satisfies Eq. (11), so that the entire problem

can be formulated in terms of w. One class of solutions will consist of plane waves, and for convenience we may take the x axis parallel to the direction of propagation. Then,

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-8-fliTZ

w = A s in c os (kx - wt)

represents a general travelling wave solution, provided that

(A)Z

k2 +

h2

The system thus acts as a waveguide, with a high frequency cut off.

This is in agreement with the previous results which required

Thus far, the constraint imposed by the presence of boundaries is to select a discrete set of wavenumbers associated with the z direction.

The wave represented by Eq. (21) may be interpreted as the superposition of two plane waves travelling in the directions which make

angles sìn'w/2 with the x-axis.The multiple reflection of these waves

gives rise to a standing wave in the z direction and a progressing wave in the x direction.

Two important points have to be noted. The direction of the waves depends only on the frequency, and not on the orientation of the boundaries. If the parallel planes had not been chosen normal to the axis of rotation, the two plane waves would have different wavenumbers

in the direction parallel to the planes, so the expression for would be

more complicated.

The second point has to do with the group velocity in this wave. The situation is appreciated best by sketching the group and phase velocities associated with each component. Figures 1(a) and (b) show the phase velocity (solid line) and the group velocity (dotted line) of plane waves travelling in the x-z plane in such a way that the phase has

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then (a) Figure 1 (b) (c) 8w _22knTr/h 3J2 (k2+n2rr2/h2)

so that w/k and 8w/8k have opposite signs. This result must be kept

in mind when one considers waves produced by _{a localized source, since} the radiation condition has to be imposed on the group velocity, not the phase velocity.

YAWAYAWAWAY

IÁWAWAYØÁYAWA

wavenumbers (k , O, k ) and (k , O, - k )

respectively. Figure 1(c)

1 3 1 3

represents a superposition of two such waves giving rise to a standing wave in the z direction, and a wave with a phase that progresses in the positive x direction, but for which the group velocity, and in fact the energy flux is actually in the opposite direction. Thus, even though w is proportional to cos(kx-wt), the energy is moving in the negative x direction.

This result can be obtained, of course, by direct differentiation. If

22n,r/h

w

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-lo-Effect of viscosity: the Ekman layers

If the viscosity is sufficiently small, the viscous stresses are

important only in the vicinity of the boundaries, and any other regions where sharp changes in the velocity may occur. The rest of the flow, which might be called the interior flow, is governed by the inviscid

equations of motion. In the more usual boundary layer problems, the interior flow provides the boundary conditions for the viscous boundary

layer flow. In the flow of a homogeneous rotating fluid, however, the

opposite is the case; the boundary layers impose severe constraints on

the interior flow, and though the local effects of viscosity are negligible, the entire fluid is affected by the viscous forces in the boundary layer.

We now turn our attention to the boundary layer near a wall in a rotating fluid. This problem was first analyzed by Ekman9, and boundary layers of this type are called Ekman layers. Figure 2 shows

the physical situation being considered, At some small distance from a

rigid horizontal wall, the interior flow has velocity components (U, V, W). At the wall, the no-slip condition requires that the velocity should be zero. In the boundary layer, which is the only region where viscous

stresses are important, the flow changes from the conditions at the

wall to the interior flow. This region is thin, so that only the normal

derivatives are of significance.

U,V,W

J

u, y, w

ô

z=O / / / / 7 / / / / / / / / /

/

J

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The boundary layer equations are correct in the formal limit y O.

It is convenient to define a stretched boundary layer coordinate

= z/E, where E = v/h2 is the Ekman number.

Since the viscosity,.

or more precisely the Ekman number, is supposed to be very small,

becomes very large a short distance away from the wall.

In the boundary layer approximation, the velocity components are split up as follows:

u = TJ(x,y,z_wall) +

with corresponding expressions for y, w,

and P.

The variables

u, y, w and P are non-zero only within the boundary layer, and go to

the tangential momentum equations become

In the case of waves, we are interested in flow fields which vary sinusoidally in time. Let the velocity components be defined as the real

part of complex variables which are multiplied by the factor e_1t. Then,

in terms of the linear combinations

u =u iv (28)

zero as . Then, since the interior variables satisfy the inviscid

equations, the tangential components of the momentum equation become

2 2h2

- = (25)

ar2

+

zT

= Ç2hZ ₍₂₆₎

at _8Z

and the continuity equation is

av ₍₂₇₎

(14)

or (l±i)(iTj3 )2/h u_±

=-Ue

± where

p =/22

From Eq. (27),

-12-(-ic± Zi2)u = c2h2a2u _lar:,2

00

=-E2 \ ud+

'a

_axj

(b00 _ay

- o

These integrals can be obtained from Eq. (29), and thus the Ekman layer

imposes a condition on the interior flow, whereby W cannot be specified independently of U and V.

Modification of the inertial waves by the Ekman layer

In this section, we shall carry out the steps outlined in the

pre-vious section to determine how the Ekman layers affect the interior flow.

This can be done by a systematic ordering procedure (see Greenspan0) of all the variables in powers of E2, or by matching the interior flow directly to the Ekman layers and then retaining only the significant terms in the expansion. We shall follow the second procedure here.

The boundary condition on w which was used in the inviscid case no longer holds. It is now replaced by the Ekman condition of Eq. (30).

Thus the expression for w given in Eq. (21) is no longer valid, and the and since + W W w=E O at = O C 0 _a 00 +

(+X-\d

_8y)

-E2)

( o

a)

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interior flow is now given by

w=Ae

ikP(1-Yz+Be ikp(lZ)2z

i

(31)

where A and B are as yet unspecified. The dependence on x and t

is to be understood throughout to be

e2

t) The corresponding expressions for u and y are:

i i = ik(l_Z)_2Z+Be_ik(l_Z)2Z) (32) u (-Ae J l-p2 and i -ik3(1 i ( ik(l-i2z Be

v=_1

Ae

The boundary conditions take _{on a simpler form if the position of the} plates is taken to be z = ±

h/2 instead of

_z = 0, h. With this shift in origin,

and conditions implied by Eq. (30) are

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B:

i

w(±h/2) = khE2

where, as before

U(z)

= U(x,y,z)± iV(x,y,z) (35)

Equations (31) - (35) result in a pair of homogeneous equations in A and

i

A [1 khE2 t(1+i)IT (lI)[iT

1

ikp(1-ph

i+p ,] e i ____ + B [1 F khE21(1+i)fi + (l-i)/T

i

4 _l-p _1+p = ° (1+i)IJ(±h/2)

(l-i)U(±h/2)

dl-p _I1+p (34) (36)

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k h

_{= mr +} Ji-.p2

-14-.

These equations have a solution only if

-i khE2 ±

e'

pZ)2h i + khE2 a where

(i+i)J (1-i)f i-p

a-Since E is very small, this dispersion equation can be solved to first

order in E2,

with the result

1EJi+p

i-p

To interpret Eq. (38) we may think of a wave of real wavenumber decaying in time (as for example when one is interested in the decay of an initial velocity field) or of a wave of real frequency and complex wavenumber (as in a problem involving a monochromatic source localized in space).

If 3

is regarded as real, then to first order in E2,

1 1

i +

E( i+p

i-p

i+p

i-p 1

k nir/i-.p2

L

i-p

₊ _J

= k ¿1+ (39)

i(kx-t)

Since the imaginary part of k is negative, a wave of the form e

increases in amplitude in the x direction. At this point it must be re-called that the direction of energy propagation is the -x direction, so that the amplitude decreases in the direction of energy travel.

If w is regarded as real, the corresponding result is

.

fi+p

(38)

fi-p

(17)

nw/h k2

khE[F(

)-iG( j k2 +

_i

k2 + 22 fl i h2 h2 where nu / h /kz + nz h2

is the value of 3 for the invis cid flow. Again, as in the case of k,

there is a small modification to the real part of 3 , plus the addition of

a negative imaginary part. In this case the imaginary part gives rise to

damping, with a time constant

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Ik

+ n22 /h2 ₍₄₁₎

kE2 G(130)

As it was indicated earlier, the damping introduced by the Ekman layers is much more significant than the viscous damping in the body of the fluid.

In the former case the relevant time constant is of order v2, while in

the latter case it is of order v1. The v2 decay time is characteristic

time for the decay of any free motion in a homogeneous rotating fluid, and

it will reappear in our discussion of spin-up.

Viscously excited waves and resonances

Thus far, we have only discussed free waves without reference to a source that might be producing them. In most cases of interest, one has

to deal with the decay of an existing flow after the forces which produced it have been removed (or equivalently on the approach to a steady state when steady forces are introduced). The response of the fluid to an

oscil-latory force is also of interest. A specific example of such a problem is

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-16-the following: a fluid is contained between two parallel planes which rotate with an angular velocity 2 + 2coswt. We want to find the resulting flow

field. A closely related problem is that of the spin-up of a fluid between

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parallel plates, which was discussed by Greenspan and Howard . Let

the fluid lie between the planes

z = h/2 and z = -h/2,

with the axis of

rotation parallel to the z axis. Let u, y, w be the velocity components in cylindrical coordinates. Since the flow is axially symmetric, the equation

of continuity is satisfied if u and w are derived from a stream function 4i(r, z, t) such that

u=- -

_8z

i a

(42)

w

- j (r4i)

Moreover, the boundary conditions are such that y and 4i are proportion-al to r The dependence is common to proportion-all problems involving rotating

flat plates, and was exploited by von Karman(1L) to analyze the non-linear flow over a rotating flat plate. Indeed, the solution of the exact non-linear equations could be found numerically in this case.

The linearized problem which we are discussing is such that the boundary layer equations are the same as the exact equations. However,

a considerable simplification results from using the boundary layer ap-proach, in that the boundary conditions are applied separately at each wall.

We shall make use immediately of the

r dependence of y and

4i, and write

y = rV(z)e-iwt

= r(z)e

-i(.)t (43)

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and

(vD2+i)V - 22D4 = O (44)

(vD2+1)D + Z2DV O , (45)

where D = d/dz. These are also the boundary layer equations. For the interior flow, Eqs. (44) and (45) simplify to

iwV - 22D = O , (46)

iwD + 22DV = O (47) Unless 47Z these equations require that

= DV = O . (48)

The symmetry of the problem is such that V is an even function of z,

and is an odd one. Therefore, i.n the interior,

= iAz

V=A

In the boundary layers,

(vD2+iw)2 D2 + 42ZD = O

For the boundary layer at z = -h/Z, let

= (z+h/2)/hE2

then z

t +ZiISj ---+4

d2

I dr d

d2

The boundary conditions are (to leading order)

=-iAh/2

,

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at Y, = O, and

as Ç-oo. Then

wh e r e and so that i - (pG -pC) = - A hE2 p = (1+i)Jl-p _, p = (l-i)Jl+13 i z Ah Ah C = - i13 C -'13

--i P2-P1 z 2 1 Now 2 d d hE2 -p -(pCd '

-pCe

2)

ii

z z hE2 At = O,

-18--0

- -p -p

=Ce

1

+Ce

i z

=A2-A

i p -p E2 2

Therefore, if neither p or p are of order E,

i

A E2

pp p₁₂

to leading order, and the interior flow is of an intensity proportional to

d2 ¿ so V i or A PP f..)2 1 2

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A2

and is given by Eqs. (49) and (51).

When Ç3 is close to ± 1, however, Eq. (50) gives

A=

3p p

1+ '

(p -p )E

2

To appreciate what is happening, suppose p is close to 1. Let

= i

- E, then to leading order,

A=

1+(1 -i)f

Thus in this case the interior flow is no longer of order E, but rather

of order 1. This is a considerable increase in the amplitude of the

interior flow. To explain it we must recall that the linear dependence

of y and q

on r corresponds to the case of zero horizontal wave-number. For this wavenumber, the free modes of oscillation for inertial

waves, according to Eq. (22) are only two: Ç3 = ± 1. Thus, when the

angular velocity of the parallel plates is modulated at or near one of these

frequencies, a resonance phenomenon occurs and the amplitude of the interior flow builds up to give a higher than usual response. Such

resonances have been observed

experimentally4.

A more precise analysis of the cases = ± 1 can be carried out

using Eqs. (44) and (45) to account for the fact that one of the terms in the boundary layer is not actually decaying, but the results do not differ

substantially from those outlined above.

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-ZU-Summary

The main points that have been brought out in this discussion are

the following. A rotating fluid can support transverse waves provided

42Z _{The energy is carried at right angles to the phase velocity.}

Ordinary dissipation leads to a decay time of order E, but in a

con-tamed fluid, reflections at the walls lead to a faster decay time, of order E2. Finally, in a contained fluid, discrete modes exist, and resonant responses are possible.

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References

i Lord Kelvin, Lectures (Macmillan and Co. , London, 1894), Vol. II,

pp. 129, 171, 183.

2

Lord Kelvin, bc. cit. p. 155.

3 Fuitz, D. IUTAM symposium on rotating fluid systems. La Jolla

1966. See:

Bretherton, F.P., Carrier, G.F., Longuet-Higgins,M.S.

J. Fluid Mech. 26, 393 (1966).

Aidridge, K. D. and Toomre A., unpublished. See Ref. 3.

Defant, A. Physical Oceanography (Pergamon Press, London, 1961).

Poincar, H.

, Acta Math. 7, 259 (1885).

Hough, S.S. Phil Trans. Roy. Soc. London 186, 469 (1895).

Phillips, O.M. , Phys. Fluids, 6, 513, 1963.

Ekman, V.W., Arkiv Math. Ast. Fys. Vol. 2, No. 11 (1905).

Greenspan, H. , J. Fluid Mech. 20, 673 (1964).

Greenspan, H., and Howard, L.N., J. Fluid Mech. 17, 385 (1963).

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D D FORM 4473_{NOV 65} _I (PAGE 1)

Unclassified

DOCUMENT CONTROL DATA - R & D

Security classification of tille, body of abstract and indexing annotation must be entered when lite overall report Is classified)

I ORIGINATING ACTIVITY (Corporale author)

Division of Engineering and Applied Science

28. REPORT SECURITY CLASSIFICATION

Unclassified

2h. GROUP

Not applicable

3 REPORT TITLE

INERTIAL WAVES IN A ROTATING FLUID

4 DESCRIPTIVE NOTES (Type of report and inclusive dales)

Technical Report

5. AU THORIS) (First name, middle initial, last name)

Venezian, Giulio

6 REPORT DATE

March 1969

78. TOTAL NO. OF PAGES

21

7h. NO. OF REFS 12 Sa. CONTRAC T OR GRANT NO.

Nonr 220(35)

b. PROJECT NO.

C.

d.

Sa. OR)GINATORS REPORT NUMBER(S)

Report No. 97-16

Sb. OTHER REPORT NO(S) (Any other numbers that may be assigned (his report)

tO. O(STRI BUTION STATEMENT

This document has been approved for public release and sale; its distribution is unlimited.

ti. SUPPLEMENTABY NOTES 2. SPONSORING MILITARY ACTIVITY

Office of Naval Research

t3. ABSTRACT

The problem of inertial waves in a rotating fluid, which is in a close analogy with the internal gravity waves in a stratified fluid, has

been further explored by Venezian. It has been shown that transverse

waves can exist in a rotating fluid provided their circular frequency is

less than twice the angular velocity of the rotating frame. The wave

energy is transported at right angles to the phase velocity. In a contained

fluid, discrete modes may exist, and resonance is possible.

(34)

Uncias sified Security Classificatìon D D 1N0V651473 (BACK)FORM I (PAGE 2) Uncias sified Security Classification 14

KEY WORDS LINK A LINK B LINK C

ROLE WT ROLE WT ROLlE WT

Rotating flow

Inertiai waves

Stratified flow