577 



dS/dt ~ SN^t (4) 



For small N't we obtain by integration 



(S - S )/S ~ N^t^ (5) 



o o 



(Sq is the initial area of horizontal projection of 

 the spot) . Thus, at the first stage the character- 

 istic size of the plan form of the turbulent spot, 

 L, changes proportionally to the square of time 



(L - 



L )/L 

 o o 



dL/dt ~ L N^t 

 o 



(6) 



[for the wake, S ~ L, and the relation (6) follows 

 from (5) in an elementary way; for the spot of the 



circular plan form, S 



but at 



Lo) << L, 



L'^ - Lq'^ - 2 (L - Lq) Lq and (6) follows again from 

 (5)]. 



The relations of the type of (6) were obtained 

 by J. Wu (1969) from the experimental investigation 

 for a spot having the form of a cylinder with a 

 horizontal axis; they were confirmed by some nu- 

 merical investigations [see Kao (1976) ] . Actually 

 they were confirmed to be valid to Nt ~ 2.5. 



At the intermediate stage the motive force of 

 the intrusion is balanced by form drag and wave 

 drag, thus, the velocity of the propagation of the 

 intrusion tongue is governed by the parameter of 

 stratification - Brunt-Vaisala frequency N - to- 

 gether with the actual height of the tongue, h, 

 whence by dimensional considerations we obtain 



dL/dt ~ Nh 



(7) 



We see that at this stage the dependence of the 

 velocity of the extension of the intrusion tongue 

 is different for various geometries of the problem. 

 In fact, the volume of the turbulent spot V is 

 constant; for the cylindrical spot h ~ V/LH (H is 

 the longitudinal size of the spot) and h ~ V/L^ for 

 a spot of the circular plane form. Therefore, we 

 obtain for the cylindrical spot 



dL^/dt - NV/H 



/NV(t 



(8) 



(to is a conditional time moment of the beginning 

 of the second stage) , whereas for the spot of the 

 circular plane form 



dL^/dt - NV 



L ~ /NV(t - 



(9) 



The relations of the type (8) were obtained by 

 J. Wu (1969) from the experimental data for collapse 

 of a turbulent wake of initial circular cross- 



FIGURE 3. Elementary particle of the diffusion tongue. 



section. They were confirmed to be valid for 

 3 < Nt < 25. 



3. FINAL, VISCOUS STAGE OF THE INTRUSION 



Under accepted assumptions the equation of mass con- 

 servation for a mixed fluid takes the following form 

 in hydraulic approximation. 



3 h + div (hv) 







(10) 



Here h(x,y,t) is the height of the intrusion 

 tongue; x,y are the spatial horizontal coordinates, 

 t is the time, y is the velocity of fluid displace- 

 ment averaged through the height of the tongue. 



For the determination of the velocity, v, let us 

 consider the system of forces acting on the cylin- 

 drical particle of the intrusion tongue leaning upon 

 the area 6S (Figure 3) . The motive force of this 

 particle is caused by the action of the gradient of 

 redundant pressure, p, and spatial variation of the 

 height of the tongue of intrusion 



Fm 



grad(ph) 6S 



(11) 



Furthermore, the drag force per unit area of a 

 particle surface due to the viscous character of 

 the drag at the final stage of the intrusion under 

 consideration is governed by the velocity, v, of 

 the particle relative to ambient fluid, viscosity 

 of the fluid, v, and particle height, h. The di- 

 mensional considerations give the viscous drag 

 force per unit area of particle surface proportional 

 to yv/h. Therefore, the viscous drag force acting 

 on the particle leaning upon the area, 6S, is equal 

 to 



Fr = Cviv6S/h 



(12) 



where C is a constant, under given assumptions - a 

 universal one. For estimating the constant, C, the 

 well-known solution of the problem of viscous flow 

 between flat plates may be used. This solution 

 gives for the viscous drag the value 12uv6S/h, whence 

 C = 12. Equaling drag force to motive force (the 

 inertia force, as at the second stage, is supposed 

 to be a negligible one) we find 



hgrad(ph) /Cy 



(13) 



To complete the statement of the problem we have 

 to find the redundant pressure in the mixed fluid. 

 In stratified fluids the density varies linearly with 

 height. The intrusion tongue propagates symmetri- 

 cally, thus, the equilibrium plane divides the 

 height of the tongue in half. Let us denote by pj 

 and pj, correspondingly, the pressure and the density 

 in stratified fluid at the level, z = Zj. Then, 

 evidently, the pressure in the stratified fluid 

 varies with depth following the relation 



Pl - Pig(z - zj) 

 + Pin2(z - zi)2/2 



(14) 



Here, as before, N is the Brunt-Vaisala frequency 

 N = ag, g is the gravity acceleration, a = (dp/dz)p]^. 

 Thus the pressure at the upper and the lower points of 

 a vertical section of the tongue z = zj ± h/2 are 

 equal, respectively, to 



