578 



P = Pi - Pigh/2 + PiN^h^/S 

 P = Pi + Pigh/2 + PiN^h^/S 



(15) 



because at the upper and the lower points the pres- 

 sure in the tongue coincides with the pressure in 

 ambient stratified fluid. Hence, the pressure 

 within the tongue is distributed according to the 

 hydrostatic law 



P = Pi - Pig{z - zi) + PiN^h^/S (16) 



The pressure averaged over the section of the tongue 

 is equal to 



Pai = Pi + PlN2h2/8 



(17) 



The pressure averaged in the same way in the strati- 

 fied fluid due to (14) is equal to 



P^ = Pi + PiN2h2/24 



(18) 



deliberately is axisymmetric at the viscous stage. 

 Hence, Eq. (23) may be applied for its description. 

 Thus, the condition of conservation of the volume 

 of a turbulent spot takes the form 



2tt 



/ 



rh(r,t)dr 



Const 



(24) 



The asymptotic stage of the spreading of the spot 

 is of primary interest when the plane size of the 

 intrusion exceeds the corresponding initial size 

 of the turbulent spot. At this stage the details 

 of the initial distribution h(r,0) cease to be 

 essential and for an asymptotic description of the 

 viscous stage of the intrusion the initial distribu- 

 tion may be represented in the form of an instantan- 

 eous point source 



h(r,ti) ; (r -p^ 0) , 2ti / rh(r,ti)dr = V (25) 



Thus, the redundant pressure entering the expres- 

 sion of motive force of the intrusion tongue at a 

 given vertical line is 



p = p^^ - p^ = PiN2h2/12 



(19) 



The relations (13) and (19) give 



p n2 P n2 



V = - n^hgrad(h3) = - -i-_ h^grad (h) (20) 



Here, tj is the conditional time moment of the be- 

 ginning of the viscous stage. 



The solutions of such type for non-linear heat 

 conductivity equations with the power-type non- 

 linearity to which Eqs. (22, 23) belong were con- 

 sidered in the papers of Ya. B. Zel'dovich, A. S. 

 Kompaneets, and one of the present authors [see 

 Barenblatt et al. (1972) ] . In our case the solu- 

 tion depends on the quantities t - tj, n, V, r. 

 The dimensional considerations show that it is a 

 self similar one: 



Putting this expression into the equation of 

 mass conservation of mixed fluid (10) we obtain 

 for h a non-linear equation of the heat' conductivity 

 type 



3 h - nAh^ = , n = p iN^/aOCy = n2/20Cv (21) 



h = 



2TTn(t - tj) 



1/5 



X = r[V^n(t - ti)/16Tr^ 



f(C) 

 -1/10 



(26) 



Here A is the Laplace operator, v the kinematic 

 viscosity of the fluid. In particular, for one- 

 dimensional motions Eq. (21) takes the form 



Putting (26) into Eq. (23) and integrating the 

 ordinary differential equation obtained for the 

 function, f (C) / we find 



(22) 



3 h - n(l/r)3 r3 h^ = 

 t r r 



(23) 



for the plane and the axisymmetrical cases, respec- 

 tively. Here x is the horizontal Cartesian co- 

 ordinate, r the horizontal polar radius. 



4. 



SELF-SIMILAR ASYMPTOTIC LAWS OF TURBULENT SPOT 

 EXTENSION AT THE VISCOUS STAGE 



We neglected turbulent entrainment and the erosion 

 of a turbulent region; therefore, the volume of the 

 turbulent mixed region is considered to be constant 

 and equal to the initial volume of the turbulent 

 spot. It stands to reason that this assumption at 

 the viscous stage is valid for sufficiently high 

 stratification only. If the characteristic dimen- 

 sions of the plane form of a turbulent spot are 

 nearly equal, it is natural to expect that the ex- 

 tension of the intrusion starts already to be axi- 

 symmetric at the end of the intermediate stage and 



10 



1/5 \ \/h 



^0^ 



1/4 



f(C) 



0, c > c = 10^/5/2 ; 2 



' < C f c. 



(27) 



Thus, at each moment of time the intrusion tongue 

 stretches for a finite distance: this is (cf. Sec- 

 tion 1) the peculiar feature of non-linearity dis- 

 tinguishing the equation of intrusion from the 

 linear equation of heat conductivity. The edge of 

 the intrusion propagates following the law 



r (t) 

 o 



2(V^n(t 



ti)/16.^'/^° 



(28) 



The form of the intrusion tongue represented by 

 the curve 1 in Figure 4 also is peculiar: the 

 thickness of the tongue changes slowly to the very 

 edge where it comes abruptly to naught. The maxi- 



h(o,t) , also changes 



very slowly with time 



