576 



£ 



c cc c 



2=^xZ) ^ ^ Z^ 



c c cc 



^^ ^ z> 



> 



r 



FIGURE 2. The intrusion of a turbulent spot into con- 

 tinuously stratified fluid. 



(3) Final viscous stage. The motive force is 

 balanced at this stage mainly by viscous drag. 



Of course, between the first and second and the 

 second and third stages there exist intermediate 

 transitional periods. When the third stage comes 

 to the end the spot is mixed due to diffusion with 

 ambient fluid and disappears. 



The turbulent motion inside of the intrusion 

 tongue is supported by general shear stress together 

 with eddy motions inside of the intrusion due to the 

 difference of the velocities of the tongue and en- 

 vironmental non-turbulent fluid. The boundary of 

 turbulent and non- turbulent fluid is sharp and if 

 the thickness of the intrusion is not too small, 

 the shear required for supporting the turbulence 

 within the intrusion is not large. 



Indeed, let us consider the equation of the 

 balance of turbulent energy in a shear flow of 

 stratified fluid neglecting, as usually, the viscous 

 transfer term [Monin and Yaglom (1971)-] 



3^E 



+ 3 {w'E' + p'w'} 



pw g 



pe - p u w 



(1) 



Here t is the time, E the turbulent energy of 

 unit mass, e the dissipation rate per unit mass, 

 u the longitudinal and w the vertical velocity 

 components, p the pressure. The flow is considered, 

 for the estimates we need, as horizontally homogen- 

 eous and the Boussinesque approximation is accepted, 

 i.e., the density variation is taken into account 

 only if it is multiplied by very large factor - 

 gravity acceleration g. 



Let us accept for the terms of the equation of 

 balance of turbulent energy, the Kolmogorov approxi- 

 mations [Monin and Yaglom (1971)] 



the turbulent and the non-turbulent regions becomes 

 completely transparent from this equation. In fact, 

 Eq. (3) is a non-linear equation of heat conductivity 

 type with heat inflow where the coefficient of trans- 

 fer of turbulent energy equal to SI/B tends to zero 

 with turbulent energy itself. For such equations 

 under zero initial conditions the disturbed region, 

 in contrast to the linear heat conductivity equation, 

 is always finite; this explains (cf. below) mathe- 

 matically the existence of a sharp interface between 

 the turbulent and the non-turbulent regions. 



It is important that, due to mixing following the 

 generation of a turbulent spot, the losses of turbu- 

 lent energy for the work of suspending a stratified 

 fluid [the second term of the right-hand side of the 

 Eq. (3)] disappear because the density within the 

 spot becomes uniform. Furthermore, the first term 

 of the right-hand side of (3) governs the diffusional 

 transfer of turbulent energy within the mixed region 

 and does not influence the averaged, through the 

 spot, value of turbulent energy. Therefore, the 

 decay of turbulent energy within the spot is governed 

 by the balance of the two last terms of the right- 

 hand side of the equation (3) representing genera- 

 tion and dissipation of turbulent energy, 

 respectively. 



It seems natural to accept that the external scale 

 of turbulence I, within a factor of the order of 

 unity, coincides with the transverse size of the 

 tongue of intrusion h; the constant y by estimates 

 has a value of about 0.5. Thus, the shear B^u ~ 

 vg/h is sufficient to support the turbulence within 

 the spot at a steady level together with the state 

 of mixing within the spot. If h has the value of 

 tens of centimeters - one meter or more, then for 

 the value /B ~ 1 cm/sec, characteristic of oceanic 

 turbulence , the shear required for supporting steady 

 turbulence is small. In thin layers it is large; 

 therefore, the turbulence in thin layers decays 

 rather quickly and the spot of mixed fluid exists 

 during the time interval required only for the dif- 

 fusional mixing of the spot with the ambient strati- 

 fied fluid. 



Furthermore, available experimental data show 

 [J. Wu (1969)1 that turbulent entrainment and the 

 erosion of a turbulent spot may be neglected, start- 

 ing from a very early stage of the evolution till 

 rather late stages of this process. Therefore, we 

 shall take the volume of turbulent spot constant at 

 all stages of its collapse to be described. 



For simplicity we shall further suppose that the 

 initial form of a turbulent spot is symmetric in 

 respect to the equilibrium plane where the densities 

 of stratified fluid and mixed fluid coincide. 



w'E 



ps,/e 3 B 



J./g' 3 u, e = y'*B^/2/ji 



(2) 



INITIAL STAGES OF THE EVOLUTION OF THE SPOT OF 

 MIXED FLUID 



Here 6 = E/p is the mean turbulent energy per 

 unit mass, S, the external turbulent scale. Thus, 

 the equation of balance of turbulent energy takes 

 the form 



3^6 = 3 l/^ 3 6 - pw^g/p 

 t z z 



+ 2./B (3 u)2 - y'*6^/2/X. (3) 

 The mathematical nature of sharp interface between 



At the first stage, free fall (lifting from below) 

 of the particles of mixed fluid to the equilibriiom 

 plane takes place, followed by the spreading of 

 fluid particles along this plane. Therefore, the 

 rate of change of the area of horizontal projection, 

 S, of a turbulent spot is proportional at this stage 

 to the product of the actual area by the rate of 

 fluid influx to the equilibrium plane. The latter 

 quantity is equal to the product of the acceleration 

 of free fall proportional to N^ and time t. Thus, 

 we obtain for the initial stage 



