Soy..7. Phys. Oceanogr., Vol.2, No. 5, pp. 341-348 (1991) @ vsP 1991.

Propagation of internal Airy and Fresnel waves in unsteady media* V. V. B U L A T O V and YU. V. V L A D I M I R O V Abstract - - A problem on the propagation of internal Airy and Fresaael waves in unsteady stratified media is solved by the 'r, nnlng wave' method. An eikonal equation is derived for the determination of the location of wave fronts. Conservationlaws are obtained which allowus to determine the evolution of the wave width and amplitude with time. A problem on the propagation of an Airy wave from a pointwise mass source moving in unsteady media is solved numerically. Under the conditions of a real ocean, the V~istil~i-Bmnt frequency N 2 = -gO in p/Oz, where 9 is the acceleration of gravity and p is the undisturbed density which determines the basic characteristics of the internal gravity waves (IWs) and depends not only on the spatial variables z, V and z, but also on time t. The most typical forms of the time variability of N 2 are manifested in the thermocline's shallowing and deepening, as well as in the variation of its breadth, etc. [1]. Several time scales of the hydrophysical field variability in seas and ocean exist: the small-scale one with periods up to 10 rain; mesoscale with periods of several days; and synoptic and global variabilities with periods from months to several years [1]. Further, we will consider the propagation of IWs in unsteady media. Their parameters have variation periods of a day or more. This allows application of the geometrical optics approximation (GO) [2], since the period of IWs is several tens of minutes or less. The propagation of IWs in unsteady horizontally inhomogeneous stratified media in the G O approximation was considered in ref. 2. A problem on the propagation of internal Airy and Fresnel waves in a stratified horizontally in_homogeneous medium was solved in ref. 3 by the 2unning wave' method, which is a generalization of the G O method. The present work contains a solution of ~he problem on the propagation of internal Airy and Fresnel waves in a vertically stratified unsteady medium using the method given in ref. 3. The Airy and Fresnel waves emerge in the problem on the propagation of the source in the layer of a stratified fluid where the distribution of N2(z) has a variety of forms independent of time [4, 5]. As is shown in ref. 4 for the steady distribution of N~(z) in a finite layer of fluid with rigid boundaries, the solution for the vertical component w of the IW velocity consists of separate modes, and each one is expressed through the derivative of the Airy function in the vicinity of the wave front. The argument of the latter depends on the first two coefficients of the Taylor's expansion of the dispersion curve in zero. In the case of a finite stratified layer of fluid with a large underlying homogeneous layer, the asymptotics of a separate mode near the wave front are expressed in terms of the Fresnel integrals [5]. When the undisturbed density p depends on the variables z and t, the set of linearized *Translated by Mikhail M. Trufanov. UDK 551.466,

342

V. F. Bulatov and Yu. V. l/ladimirov

hydrodynamic equations is reduced to one equation; for example, for the vertical velocity

+ where

o~ / ~

~ +~

+ ~z ODzl ~ = g-~-z ~ '

02 02 A = 0-~2+ Oy2.

If we neglect the term containing 0 In p/Oz, we o b t ~ t the foUowing equation in the Boussinesq approximation:

It seems natural to neglect also the term containing a In p/at; this would correspond to successive application of the Boussinesq hypothesis. However, the density which characterizes the inert mass of fluid can be treated as constant. Then

0,( 0,)

0~--~ A + ~

w + ~v'(z, ~)A~, _- 0.

(11

This equation differs from the common equation of the internal waves in a steady medium by the parametric entry of the time t into the V~iisNi-Brunt frequency. A problem on the possible use of the Boussinesq approximation over time is discussed in ref. 6. The solution of (1) is sought as a sum of modes, each of them propagating independently (adiabatic approximation). We will consider a single separate mode omitting its index. Further, only the area near the wave front will be of interest to us, L e. we will consider only the time t which is close to the time of the wave front arrival, designated henceforth in terms of r. This means that a weak-dispersion approximation is used. Let us consider the Airy wave propagating in the layer of a stratified fluid - H < z < 0 with the V/iis~il~-Brunt frequency N2(z, t). We will seek the solution (1) with boundary conditions w = 0, z = 0, - H in the following form:

w = [A(ex, ey, e,z) + OA(ez, sy,Or r , z ) ( e t - r) + ...] Fo(W)

F ' + , ( ~ ) = F,~(~), where p = 2/3; 7- = r(ez, sy); and Fo(qa) = A~(~) is the derivative of the Airy function with the argument ~ = a(sz, sy)(er - r ( e z , ey))e-P of the order of unity. The function r describes the wave front location; the function a the evolution of the A i ~ wave width; the minor parameter e characterizes 'slow variables'. As we are interested only in 'slow times' et which are close to the time of arrival of the wave front r, all the functions prior to the functions F,~ are represented as Taylor power series of st - r -.~ e p. We represent N2(z, et) in the following form:

Na(z'st) = N=(z'r) + aN~(~' Or ~') ( a - ~-) + o(s=~).

(3)

Internal Airy and Fresnel waves

343

Substituting expansions (2) and (3) into (1) and equating the terms if powers of e are identical, we obtain the following equations at e p 02A Oz-----r +

IVrl2g2(z, r)A

A=0,

= 0,

z=0,-H,

(4) (5)

where V = (O/Oz, O/Oy). It is convenient to represent the eigenfunction of problems (4)-(5) in the form A ( x , y, r, z) = k~(x, y ) f ( z , r), where f ( z , r) satisfies the normalization 0

condition f N 2 ( z , r ) f 2 ( z , r )

dz = 1.

-H

It is assumed that eigenfunctions f ( z , r) and eigenvalues A(r) _= [Vr I of problem (4)-(5) are known; then to determine r, we have the eikonal equation

+\0y)

-

+

=:(T).

(6)

The respective characteristic system for the eikonal equation (6) has the following form: dz rc2(r) ' d-~ =

dr d---~ =

d(r) c(r) r,

dy qc2(r) ' d---~ =

dq c'(r) d---~ = - c(r) q"

(7)

Here c(r) = A -1 (r) denotes the maximum velocity of the long waves. To determine the functions a and ~, we equate terms of the order e2p after the substitution of expansions (2) and (3) into equation (1). We obtain the following expressions: A'(~)~(T) + V ~ W + ~ - I ( ~ ) V ~ ( T ) W " = O,

v \ ~, ] vT + / w = 0,

(8)

0

a(r) = / f 2 ( z , r ) dz. -H

The following law of conservation along the characteristics of (7) is valid for the determination of 4: ~2c3R a4 - constant, (9) where geometric divergence of the rays R is connected with the Jakobian D which describes the transition from spatial variables x, y to the ray coordinates r and To through the relation D = Rc. With the characteristics of (7), equation (8) is reduced to the Bernoulli equation da 1 dA(r) dT + ~(~) dT '~ =--~'a(T),

344

V. V. Bulatov and Yu. V. Vladimirov

and its solution has the form

a(z,y) = c(r)

-

;'

]

--1/3

a(t)ca(t) dt

L ~'o(~*)

Now we consider the Fresnel wave propagating in the layer of a stratified fluid of thickness H with the V~iis/il~-Brunt frequency N2(z, t) which lies on a large homogeneous layer with N2(z,t) = 0. We will seek a solution, for example, for the elevation 7/ (w = Orl/Ot). We take r / = 0 at the surface z = 0 and lOw/Ozl = Iv,H at the boundary of the stratified layer z = - H as the boundary conditions. The last condition provides the exponental decay of the solution with depth. The solution for r/is sought in the form

0o) rl*

[A(ex,ey, r,z) +

OA(~z, ~y, r, z) Or (~t- r) +

.] Fo(~)

[ OB(~z, sv, r,z) J - r ) + ]FI(~) +i~ p B(ez,~y,r,z) + Or (~ "'" [ OC(e~:,~y,r,z) r)+ ] F_x(~) + 0 ( ~ ' ) , +ie" C(ez, ~y, r, ~) + Or (et ... exp -it~o -

F0(qa) =

dr,

0

where p = 1/2; all the remaining symbols are retained henceforth except those specified. Substituting expansions (3) and (10) into equation (1), we obtain the eikonal equation (6) for determining r, where A(r) is derived from equation (4) with the boundary conditions A = 0, z = 0, OA/Oz = 0, z = - H . We have the conservation law (~cR)/a 2 = constant at the characteristics of the eikonal equation. The function a will be the solution of the respective Bernoulli equation

a ( x , y ) = c(r)

b(t)ca(t)dt /

,

b(r) = f2(-H,r).

J Now we consider the Airy wave arising when a pointwise source of mass moves in the finite layer of a stratified fluid with rigid boundaries. Let us assume that the source moves in the positive z direction with the velosity V at a depth zo. As a parameter for the integration of system (7), we take eikonal r. A single-parametric family of functions x(r, to), y(r, ro), r(r, to), q(r, to) is the solution of system (7); the first two functions define the ray at the plane z,y; ro is the initial eikonal or, which is the same, the time when the ray leaves the source. Let the source be located in the point (X(ro),y(ro))=(Vro, O) at the moment of time r = to; then we obtain the following set of equations for the determination of functions r(ro), q(ro):

r2(ro) + q2(ro) = c-2(ro),

(Ii)

Internal Airy and Fresnel waves ~(~o

ay(,o),

345

,

Equation (11) is the eikonal equation (6) at the moment of time r = to; equation (12) is derived through differentiation of the initial eikonal to(Z, y) with respect to to. From (11) and (12) we derive the functions r(ro), q(ro); their relation q(ro)/r(ro) determines the tangent of the angle between the ray which leaves the point (Vro,O) at the moment of time r = ro and the axis z. Then the initial data for system (7) will be the following:

9 (~o) = y,o,

~(~o) = 1IV, (13)

y(,o) = 0,

q(~0) = x/~-~(~o) - v-~.

When solving system (7) with the initial data (13), we obtain the equations of rays

x(r, ro) = VTo + ~ - ~ ] c(t) dt, "to

(14)

v-~/c(t)

y(r, To) = c ( r o ) ~ / c - ' ( , o ) -

at.

~'o

It follows from (14) that the rays are straight lines and their slope depends on the time of the emerging ray to. When r is fixed, we have the wave front; when ro is fixed, the beam. Transforming equations (14), we obtain r = r(x, y), 7"0= to(x, y ). The value of the constant on the right-hand side of equation (9), further denoted in terms of P(ro), can be determined for rectilinear and uniform motion of the pointwise source of mass. The value of P(ro) is obtained from the problem in which the Vfds~l~i-Brunt frequency is constant [4]:

csCTo)

(ofCzo,~o)~ ~ OZo J "

(151

P(r~ = 4~/p~--- c2(r0) k

Geometrical divergence of the rays R has the following form for the rectilinearly and uniformly moving source

eko_)

[

c(t) d~ + ~/V~ - ~(~-o).

(16)

Using (15) and (16), we can set down the explicit form of the first term of the Airy wave when the pointwise source of mass moves in an unsteady medium:

cs/2(ro)a2(x,y)f(z,r) = 2~/~(~-)R,/~(~,y)(w

- c~(~o))~/~

(

Of(zo, ro)Ai, a(z,y)t O~o

+ O(e4/~). ~/~

]

Data on the V~is/il~i-Brunt frequency variability [1] were used for the numerical computations. Figure 1 shows the profiles of N(z, t) at intervals of 4 h for which the computations were made. The distribution of N(z), which is denoted by the dashed line, was used as

346

V. V. Bulatov and Yu. V. Vladimirov

A/~Z) 7S-I

0.0411""""-""

-SO

Z/71"

Figure 1. Variability of N(z, t) with time.

the stationary distribution. Figures 2 and 3 give the results for the first mode for the unstationary (solid line) and stationary cases (dashed line) when the values of parameters were as follows: V = 1 m/s, ro = 0, r = 5000 s, z = - 10 m, zo = - 2 0 m. Figure 2 depicts the results of the calculation of the rays and fronts. Figure 3 describes the calculation of the vertical velocity w in the fixed point of space T. Thus, the solution of the problem on the propagation of the internal Airy and Fresnel waves in a stratified unstationary medium and the numerical computations made using the solution of this problem show that the time variability of the VS.isfil~-Brunt frequency

347

Inten~al Airy and Fresnel waves

/000

0 I\\

-I000 ~

. . . . .

~

~\-

_ / - -

~

--"""

I ./-

x,m

~-~ ;"~-

Figure 2. Fronts and rays of the Airy wave first mode.

,,.%

0.5

I

,

f I I

I

I |

I I I

| !

/-N

I

I

~000 I \

"0.5

I /

!

I I

l t

I

I

!

|

a

I

I

!

I %/

Figure 3. Vertical velocity of the Airy wave.

can affect appreciably the character of the I W propagation. Therefore, in solving probIems on the propagation of IWs one should also take into account the time variability of the Vfiis~l~-Brunt frequency equally with vertical stratification and the d e p e n d e n c e of N 2 on the horizontal variables.

348

V. V. Bulatov and Yu. It:. l/ladimirov

REFERENCES

L Nelepo, B. A. (Ed.). Variability of Hydrophysical Fields in the Black Sea. Leningrad: Gidrometeoizdat (1984), 240 p. 2. Miropol'sky, Yu. Z. Dynamics of the bztemal Gravity Waves b~ the Ocean. Leningrad: Gidrometeoizdat (1981), 302 p. 3. Bulatov, V. V. and Vladimirov, Yu. V. Propagation of the internal Airy and Fresnel waves in a horizontally inhomogeneous medium. Mar. Hydrophyz. r. (1989) 6, 14-19. 4. Borovikov, V. A., Vladimirov, Yu. V. and Kel'bert, M. Ya. The field of the internal gravity waves generated by localized sources. Izv. Akad. Nauk SSSR, Fi~ Atmos. Okeana (1984) 20, 526-532. 5. Borovikov, V. A., Bulatov, V. V. and Kel'bert, M. Ya. On intermediate asymptotics of the remote field of internal waves in the layer of a stratified fluid lying on a homogeneous layer./zv. Akad. Nauk SSSP,, Liq. Gas Mech. (1988) 3, 158-162. 6. Vladimirov, Yu. V. and Frost, V. A. lntemal Waves u~der Unsteady Conditions. Moscow (1989), 38 p. (Preprint of the Institute of Mechanics Problems of the USSR Acad. Sci.).