Self-Similarity Concept in Marine System Modelling

Transcription

1 Self-Similarity Concept in Marine System Modelling Rein Tamsalu,, Pentti Mäi and Kai Myrerg Geopysica (997), 3 3(), 5-68 Finnis Institute of Marine Researc, P.O.Box 33, FIN-93, Helsini, Finland Estonian Marine Institute Paldisi Sir., EE, Tallinn, Estonia Astract Te self-similarity concept for sea temperature was firstly introduced y Kitaigorodsii and Miropolsi (97). Tey found tat a non-dimensional temperature is only dependent on a non-dimensional vertical coordinate. In our paper it is sown tat also te flux quantities of marine system variales (salinity, temperature, uoyancy, oxygen, nutrients etc.) ave a self-similar vertical structure. Te derivation of te equations for te fluxes allows us to present te fully 3-dimensional prolem in te sea y using te self-similarity concept. Te marine system variales mentioned aove seem to create self-similar structure in te sea: we suppose tat te currents destroy it, ecause no self-similarity profiles for currents ave een found. Key words: self-similarity, modelling, turulence, Baltic Sea. Introduction Te dynamics of te marine system is caracterized y processes covering a wide range of spatial and temporal time scales. Nioul and jenidi (987) presented tat te marine system can e descried y fairly well-defined "spectral windows" i.e. domains of lengt scales and time scales associated wit identified penomena. Te transfer of energy etween windows is effected y non-linear interactions. Te variaility in te sea is descried y tree main categories: small-scale, marine weater and long-term processes. Te small-scale processes are including microscale, mesialscale and mesoscale processes. Te microscale processes include 3-dimensional turulence and surface waves. Te time scale is from seconds to minutes. Te mesialscale processes are formed y internal waves and y microstructure "liny" turulence. Te time scale of tese processes is from minutes to ours. Te mesoscale processes include inertial oscillations, wic ave for example a time scale of 3-4 ours in te Baltic Sea, tides and storm surges. Te marine weater descries diurnal and synoptic variations. Tese processes ave a caracteristic time scale from a day up to some wees. Te long-term processes can e separated to seasonal and gloal variations. Te seasonal variations are formed y te seasonal canges of te atmosperic motion and te gloal variations are caracterized y climate cange processes. Te marine weater and te seasonal variaility form a two-layer vertical structure; a quasiomogeneous layer wit intense turulence and a stratified layer wit intermittent turulence. Te vertical structure of marine weater and long-term processes is caracterized y a so-called self-similarity profile. It means tat a self-similarity structure can e found from measurements of different marine system variales only if an averaging over small-scale oscillations is carried out. Te acground of self-similarity concept comes from nondimensional analysis. A classical scientist as von Karman as dealt wit suc an analysis in studies of flows in a pipe. A good review of nondimensional analysis and self-similarity concept in general as een given y Barrenlatt (975). Pulised y te Finnis Geopysical Society, Helsini

2 Te self-similarity of an unnown variale (for example temperature T in te pycnocline layer) in a two-dimensional coordinate system will e descried in a non-dimensional form: θ = Ts () t T(,) z t T() t T s H () y a non-dimensional coordinate: ξ = z () t H () t () So, θ = f(ξ) (3) T s (t)- temperature in te upper mixed layer, T(z,t) -temperature profile in te vertical direction, T H -temperature at te lower oundary of te ocean active layer, wic is approximated to e constant, z - te vertical coordinate, (t) - te ticness of te upper mixed layer, H - te dept of te ocean active layer, mime. In tis way a two-dimensional prolem ecomes as a one-dimensional prolem. So, y using te self-similarity approac we can reduce te dimension numer of te prolem. Te self-similarity in data analysis means tat te data measurements in nondimensional coordinates can e descried y a single curve.. Te irt and development of self-similarity concept More tan two decades ago Kitaigorodsii and Miropolsi (97) pulised te first paper, were te vertical structure of te ocean temperature was solved in terms of te self-similarity concept. By using tis concept tey were ale to calculate te seasonal variations of te ticness of quasi-omogeneous layer and of te vertical temperature profile in te seasonal termocline. Kitaigorodsii and Miropolsi (97) summarized teir new founding in te following way. Te dept of te ocean active layer is aout -5 m. Tere is at first te quasiomogeneous upper layer, ticness of wic as a great variaility in te function of time. Below te quasiomogenous layer, a termocline layer exists, were te temperature falls sarply. According to te aovementioned vertical structure Kitaigorodsii and Miropolsi (97) concluded: in te upper mixed layer te temperature cannot cange wit dept and it is equal to te surface temperature. Te vertical water temperature profile in te tennocline can e descried y a nondimensional temperature, wic only depends on a non-dimensional vertical coordinate ξ (see equation 4). Verification of tis ypotesis and determination of te function θ=θ(ξ) can e ased on measurements of te temperature profile over a long time interval. An approximate analytical expression for θ(ξ) can e found y using a metod similar to tat of Karman and Polausen in oundary layer teory. Kitaigorodsii and Miropolsi (97) got: θ(ξ) = 8/3ξ - ξ + /3ξ 4 (4) Tey studied te turulent excange troug te termocline in two separate cases: firstly, wen te termocline "locs" te eat flux coming from aove. In tat case te turulence in te termocline is intermittent from its origin and te principal source of turulence comes from te readown of internal

3 waves. Secondly, tey studied te case, were te eat flux is continuos across te oundary of te upper mixed layer and te termocline. By using tese two alternatives, different evolution equations forte ticness of te upper mixed layer were derived. Miropolsi et al. (97) found average montly dimensionless temperature profiles for two ocean stations "Papa" and "Tango". Tey concluded tat te internal consistency of te universal structures is mostdistinct in July-Septemer, wen te seasonal termocline is developed. Te great scatter of points in te profile in winter can e explained y te great ticness of te quasi-omogeneous layer and y te wea seasonal termocline. Karov (977) developed a parameterization for te two-layer structure of te upper ocean layer. He proposed a relationsip etween te omogeneous layer ticness and te termocline ticness on te asis of laoratory and oservational data on te ocean temperature field. Te study y Karov (977) confirmed te universal nature of te temperature distriution in te upper termocline. Kamenovic and Karov (975) studied te parameterization of te vertical eddy flux during te seasonal canges of te termal structure of te ocean. Tey used a tree-layer model wic consisted of te upper omogeneous layer, te seasonal tennocline and te main deep ocean. Te model was a one-dimensional one. Two separate cases were investigated: te mixed layer is increasing (entrainment) and te mixed layer is decreasing (detrainment). Te computations of temperature, turulent eat fluxes and te ticness of te upper mixed layer were compared wit measurements wit promising results. Arsenejev and Felsenautn (977) found a simple polynomial expression for self-similarity: θ = (-ξ) 3 (5) Resetova and Caliov (977) extended te self-similarity ypotesis for te first time to salinity. Tey calculated self-similarity profiles for salinity according to measurements carried out in te Pacific Ocean. According to Resetova and Caliov (977) te results indicate tat te dimensionless salinity and density ave a tendency to group along te universal profile wit a variance wic is appreciale less tan te caracteristic gradient witin te entire active ocean layer. However, te results were re-examined later on and it ecame out tat te scatter of points on te empirical curves was too large. Te idea of self-similarity ecame doutful. Linden (975) carried out laoratory investigations y using a rectangular tan. Tere was initially a two-layer vertical structure in te tan. Te upper layer was well-mixed and elow it tere was a layer wit a constant density gradient. Turulence was produced in te tan y oscillating a orizontal grid wit a stroe. After several experiments an average, non-dimensional vertical structure for density was found. Linden (975) found out tat it ad a very similar appearance as tose structures for temperature calculated y Kitaigorodsii and Miropolsi (97). Actually, te profile (4), wic was found y Kitaigorodsii and Miropolsi (97), is valid only for te case wen te mixed layer is increasing (entrainment). Te pysical conditions in te laoratory experiments carried out y Linden (975) presented not te case of entrainment. Te profile found y im as a resemlance to te new profile (7) found y Tamsalu (98) and y Mäli and Tamsalu (985) descried later in tis section. Te pysical acground of self-similarity as een investigated y several scientists. In 97s Barrenlatt (978) and Turner (978) made te first studies. Barrenlatt (978) concluded tat in te case of mixed layer increasing, te termocline is treated as a quasistationary termal and diffusion wave. According to Barrenlatt (978) and Turner (978) it is liely tat te energy needed to prevail te sarp gradient elow te surface layer in te upper termocline will e supplied y te reaing of te internal waves. Zilitinevic and Rumjantsev (99) concluded as follows. Effective eat conductivity K in termocline is muc iger tan te molecular conductivity and tat K increases wen -T/z increases (not te well-nown inversely dependence of K on -T/z. Te direct dependence etween K and -T/z can e explained in te following way: Te disturances at te lower oundary of mixed layer generate internal waves wic propagate downwards and terefore transfer inetic energy downwards. Te occurrence of reaing is more liely wen ig temperature gradients exist. In tis way, turulence "spots" are generated i.e. te waves expend a part of teir energy for te generation of intermittent turulence. Te teory

4 Self-Similarity Concept in Marine System Modelling 55 is similar to Turner's (978) ut Zilitinevic and Runtjantsev (99) expressed tat te role of uoyancy sould also e taen into account, so tat te mecanism would wor. Zilitinevic and Rutnjantsev (99) and Mironov et. al (99) pointed out tat processing of oceanic data (Miropolsi et al., 97, Resetova and Caliov, 977) revealed so great scatter of points on te empirical Curves θ(ξ) tat concept of self-similarity of te termocline ecame doutful. Tamsalu (98), Mäli and Tamsalu (985), y using te measured data of Nõmm (988), found tat te self-similarity profile strongly depends on te evolution of te mixed layer ticness. Tere are two different self-similarity structures: firstly, te case of entrainment wen te omogeneous layer is deepening (storm) and secondly, te case wen te mixed layer is decreasing (storm suside). Tus, te similarity function in te termocline (more generally in pycnocline) as te following forms: θ (ξ) = (-ξ) 3 (6) wen te mixed layer is increasing θ (ξ) = 4(-ξ) 3 + 3(-ξ) 4 (7) wen te mixed layer is decreasing (see Figure ). Fig.. Self-similarity structure for uoyancy θ as a function of ξ. Curve - mixed ayer dept is increasing; curve - mixed layer dept is decreasing. In te real situation tese two profiles are mixed, so te oservations situate etween tese two curves. We can suppose tat in tree-dimensional concept self-similarity of vertical fluxes is of primarily importance and tat a self-similarity profile for vertical fluxes can e found at sort time and space scales; i.e. in te turulent scale of motion. So, te self-similarity of different marine system variales is only a product of te flux -self-similarity. On te oter and, self-similarity profiles ave not een found for currents. Te role of currents seems to e to destroy self-similarity structure. Tat's proaly wy te self-similarity can e found for marine system variales only if an averaging over te small scale is carried out. Tus, te destroying effect of currents is smooted out. Te eat (uoyancy) flux self-similarity concept was proposed y. Leonov and Miropolsi (977) and y Tamsalu (98). In terms of traditional self-similarity concept and eat transfer equation Zilitinevic and Mironov (99) studied vertical fluxes troug te termocline. Tey developed a model of eat transfer in termocline from considerations of turulent energy udget and from expressions for effective eat conductivity, wic is ased on dimensional arguments using uoyancy parameter, temperature gradient and turulent lengt scale as governing parameters. Tis energy alance model is applicale to te mixed layer

5 deepening as well as for its steady state and collapse. In te next section, a new approac in self-similarity teory will e sown; te self-similarity profiles for vertical turulent fluxes are derived. 3. erivation of self-similarity profile for flux quantities Wit te linear version of te equation of state, te uoyancy can e written as follows. α u v w t z u v z w T I ' ' + ' ' + ' ' = c ρ z p (8) Te equation of continuity as te following form: u v w + + = (9) z ρ ρ u, v, z are velocity components in x, y and z-directions, = g - uoyancy, Hu I - ρ x-component of macroturulence, Hv''I - y-component of macroturulence, Hw''I - microturulence, α T - constant, ρ - mean density, I - penetrative component of solar radiation. Because te vertical structure of te sea as a clearly formed two-layer system, we write te equations (8) and (9) in a new coordinate system: z t = t', x = x', y = y', ξ = ; =, () z ; z H - sea-level deviation - mixed layer ticness H - sea dept (ticness of active or seasonal termocline) = ; = H - By using () we can write equations (8) and (9) as follows: t + u + v ω + + ξ δ u v q ' ' + ' ' + = ξ ξ t T I c ρ ξ p () ξ q = w' ' u' ' v' ' ξ ω = w u v ξ ξ

6 δ = +ξ =, For continuity we ave: u v ω + + = () ξ We will solve equation () y te split-up metod (Marcu, 975). In te first-order accuracy in time t i t t i+/ we will solve te equation: t + u + v ω y x u y v + + ' ' + ' ' = (3) ξ In te second order accuracy in time t i+/ t t i+ we will solve equation: t q δ + = ξ t ξ αt I c ρ ξ p (4) We can suppose tat equation (4) develops self-similarity structure and equation (3) destroys it. Tus, for determination of te flux structure we will concentrate our interest in to equation (4). We can write equation (4) for te upper mixed layer and for te pycnocline layer. For te mixed layer we ave: t q αt I + = ; (5) ξ c ρ ξ ξ p For te pycnocline layer we ave: δ q I + = ; (6) t t ξ ξ ξ We ave te following conditions: ξ = q =α T w T -β s w S =q ξ = and ξ = q l =q =q (7) ξ = q = 3 qdξ = mu * ; q = α( H) if t t > (8) q = qdξ = q _ if (9) t

7 were T is a fluctuation of temperature, S is a fluctuation of salinity, βs is a constant, u * is te friction velocity of wind and m is a experimental coefficient (m i =.6, Niiler and Kraus, 977). Buoyancy flux q will e written y using (7) as follows: [ ] q = q Q ( ξ) + Q ( ξ ) q () q = q( Q( ξ ) ), if t > ; q = qq ( ξ ) _ if < () t Wen te dept of te mixed layer is decreasing, te uoyancy flux etween te mixed layer and te pycnocline ecomes to zero and turulence in te pycnocline layer is formed y readown of internal waves. We propose ere tat te internal waves in te stratified layer are formed y te dynamics of te mixed layer: _ q = t () Te main prolem in te approximations ()-() is te nondimensional function Q (ξ ). In te upper quasiomogeneous layer Q (ξ ) is a linear function of coordinate (see for example Nifler, 975). δ Q( ξ) = = ( + ξ)/ (3) Te determination of Q (ξ ) is still an unnown function and tus it is a main topic of tis article. After integration of (5) and (6) in vertical direction (see appendix ) we get y using (7)-() t H = ; = εh ; = ε (4) t t H q R if q R t if = εq > ; _ = εq t R 3 mu I T q c e * α γ if = + + p > ρ γ t R αt I = q + c pρ if t g is te attenuation coefficient of solar radiation. In te calculation of te dept of te mixed layer te equation of turulent energy as een used in rater simple forms of te turulent energy equation as een developed (Garnic and Kitaigorodsii, 977, 978; Kitaigorodsii, 979) were e.g. te effects of te reaing of surface wind waves in te mixed layer as een taen into account. By sustituting (3) and (A.7) and (A.3) to () we find te following equation for q(ξ):

8 q q R = ϕ( ξ) if R < t > R = ϕ( ξ) if R > t (5) (6) φ (ξ )=/3(-ξ ) + /3(-ξ ) 4 if R < φ (ξ )=/ξ (-ξ ) 4 if R For calculation of te uoyancy we get in te second-order accuracy in time: t σ + = ϕ( ξ) (7) ξ σ = ξ 3 ϕ ξ = [ ξ ] ; ( ) ( ) if t H > 3 4 = ϕ ξ ξ ξ ξ = ( ) = 4 ( ) + 5 ( ) if t Sustituting (7) and (8) to (3) and (4) we get te following equation for determination of uoyancy witout te splitting-up metod; were te destroying effect of self-similarity y dynamics will e included: u v t x y x x y ω σ ξ µ ( + ) µ = q T I q + α c ρ ξ p (8) t + u + v y x x y ω σ ξ µ + ( + ) µ = α ϕ( ξ) R T I + (9) c ρ ξ p σ ξ ξ = ( ) 3 t H q R = if t > and q = if t Were te traditional way for parameterization of macroturulent mixing is used:

9 u' ' = µ ; v' ' = µ (3) were µ is te macroturulent coefficient. By te same way it is possile to descrie te equations for temperature, salinity and for te ecosystem components. Sustance equation for ecosystem component c, for example, as te following form: c t c + u + v c c y x c x y c ω σ ξ µ + ( + ) + µ = (3) ϕ ( c c ξ ) H H + G were G descries iocemical reactions. Te wole prolem will e calculated togeter wit equations of motion, equations of salinity and temperature, equation of state, equation of continuity and equation of te mixed layer ticness ( ). 4. Conclusions Te 5 years istory of self-similarity concept in marine science includes many important milestones. In te wor presented y Kitaigorodsii and Miropolsi (97) te existence of self-similarity of temperature in te Ocean was sown for te first time. However, some furter studies of self-similarity concept sowed tat te measured profiles and tose ones produced y using te self-similarity concept did not fit too well. Te self-similarity concept ecame doutful. Te wors y Tanisalu (98) and y Mäli and Tantsalu (985) sowed tat te self-similarity profiles depend on te time evolution of te upper mixed layer. In tis paper we ave derived a self-similarity profile for te vertical fluxes of turulence. So, y finding out te self-similarity structure also for te turulent fluxes, not only for single marine system variales, it ecomes possile to solve te 3-dimensional fields for marine system variales. Furter on, we can suppose tat te flux-self-similarity is actually of primarily importance and tat te self-similarity of different marine system variales is only a product of te flux - self-similarity in te turulent scale of motion. However, self-similarity structure as not een found for te vertical profiles of currents. It is most liely tat currents destroy te self-similarity structure. Tis is an explanation wy self-similarity profiles for marine system variales can e found only after averaging over te inertial period. Tus, te destroying effects of currents will e smooted out. Acnowledgements We would lie to tan r. Tilt Kullas for critical comments of te manuscript. 5. References Arsenejev, S. and A. Felsenaum, 977. Integral model of te ocean active layer. Izv., Atmosperic and Oceanic Pysics,Vol. 3, No., pp (englis edition). Barrenlatt, G., 975. Nondimensional analysis and self-similarity solutions. Pulications of te Academy of Sciences of te USSR, Moscow. First Issue. 54 pp. Barrenlatt, G, 978. Self-similarity of temperature and salinity distriutions in te upper termocline. Izv, Atmosperic and Oceanic Pysics, Vol. 4, No., 8-83 (englis edition).

10 Garnic, N. and S. Kitaigorodsii, 977. On te rate of deepening of te oceanic mixed layer. Izv., Atmosperic and Oceanic Pysics, Vol 3, No., pp Garnic, N. and S. Kitaigorodsii, 978. On te teory of deepening of te upper quasiomogeneous ocean layer owing te processes of purely wind-induced mixing. Izz, Atmosperic and Oceanic Pysics, Vol 4, No., pp Kamenovic, V. and B. Karov, 975. Seasonal variation of te termal structure of te upper ocean layer. Oceanology, Vol. 5, No. 6, pp (englis edition). Karov, B., 977. On te structure of te upper ocean layer. Oceanology, Vol. 7, No., pp. -3 (englis edition). Kitaigorodsii, S Review of te teories of wind-mixed layer deepening. - In: Marine forecasting, predictaility and modelling in ocean ydrodynamics. Edited y J. Nioul. Proceedings of te t international Liège colloquium on ocean ydrodynamics, Elsevier Oceanograpy series, 5, pp. -33, Amsterdam, Holland. Kitaigorodsii, S. and Y Miropolsi, 97. On te teory of te open-ocean active layer. Izv., Atmosperic and Oceanic Pysics, Vol. 6, No., pp (englis edition). Ejauss, W, 98. Te erosion of a termocline. J. Pys. Oceanogr.,, Leonov, A. and Y, Miropolsi, 977. Personal communication. Linden, R, 975. Te deepening of a mixed layer in a stratified fluid. J. Fluid. Mec., Vol. 7, part, pp Marcu, G., 975. Numerical metods in weater prediction. Academic Press, New Yor, 77 pp. Mironov,., S. Golosov, S.S., Zilitinevic., K. Kreiman and A. Terzevi, 99. Seasonal canges oftemperature and mixing conditions in a lae. In: Modeling air-lae interaction, ed. y S.S. Zilitinevic, pp Springer-Verlag, Berlin. Miropolsi, Y, B. Filyusin and P. Cernysov, 97. On te parametric description of temperature profiles in te active ocean layer. Oceanology, Vol., No. 6, pp (englis edition). MAli, P. and R. Tamsalu, 985. Pysical features of te Baltic Sea. Finnis Man Res. No. 5, pp., Helsini. Nioul, J. and S. jenidi, 987. Perspective in tree-dimensional modelling of te marine system. In: Tree-dimensional models of marine and estuarine dynamics edited y J. Nioul and B. Jantart, pp. -33, Elsevier Oceanograpy Series, Amsterdam, Holland. Nfiler, R, 975. eepening of te wind-mixed layer. J. Mar. Res, Vol. 33, pp Nfiler, P. and E. Kraus, 977. One-dimensional models of te upper ocean. In: Modelling and prediction of te upper layers of te ocean: edited y E. Kraus, Pergamon Press, Oxford, pp Nõmin, A., 988. Te investigation and simulation of te termoaline structure in te open part of te Gulf of Finland. P-tesis, Leningrad Hydrometeorological Institute, 4 pp. (in Russian). Resetova, O. and. Caliov, 977. Universal structure of te active layer in te ocean. Oceanology, Vol. 7, No. 5, 59-5 (englis edition). Tamsalu, R., 98. Paraineterization of eat flux in te sea. In: Second All-Union Oceanology Congress Papers, Vol., pp , Sevastopol, USSR. Turner, L, 978. Te temperature profile elow te surface mixed layer. Ocean Modelling, Vol., 6-8. Zilitinevic, S.S. and. Mironov, 99. Teoretical model of te termocline in a freswater asin. J. Pys. Oceanogr. Vol., No. 9, Zilitinevic, S.S. and V. Rurrijantsev, 99. A parameterized model of te seasonal temperature canges in laes. Environmental Software, Vol. 5, No., pp. -5.

11 APPENIX By using te self-similarity structure for uoyancy and for uoyancy flux equation (6) as te following form, if H is not a function of time t: d q dq H * ( ) + ( H )( ) θ θ t t + θ ξ dξ t = dξ (A.) were q* = q if t * > and q = _ q if t We ave te following oundary conditions: ξ = θ= Q= ξ = θ= Q = if t > (A.) ξ = θ= Q= ξ = θ= Q = if t (A.3) Integrating equation (A..) wit respect to ξ etween te limits and and y using te oundary conditions (A..) and (A..3) we otain: H ( ) + ( H) q κ κ t t + κ t = if t > (A.4) H ( ) + ( H ) κ κ t t + κ t = if t (A.5) oule-integration (A. ), first from to ξ, ten from to, yields: H ( / ) + ( H) mq κ κ t t + κ t = if t > (A.6) _ H ( / ) + ( H ) q κ κ t t + κ t = if t (A.7) ξ κ= θ d ξκ ; = θ d ξ d ξ ; m = Qd ξ From Mäli and Tamsalu (985) we get: κ =.75 ; κ =.3 and m =.6 if t > (A.8)

12 κ =.6; κ =. if (A.9) t By using condition (8) and equation () for te upper layer = (A.) t we ave te following equation for H, and q if > (R < ). t H t ε = R R ε ε ε t q R = H ; = ; = q (Al.) H κ κ / α κ m κ εh = ε κκ α ( κm κ) ; ( / ( )) κκ α ( κm κ) = α ε H Te proportional coefficient α is still unnown. Let's write Eq. (A.4) in an oter form. _ t _ α _ + ( ) = ( ) (A.) t κ t We got from Krauss (98) tat efore and after a storm, during wic te temperature decreased aout 4 degrees and te upper mixed layer deepened more tat meters, te mean temperature of te pycnocline layer T _ retained nearly as constant (see Figure ). Fig.. Temperature and salinity profiles: ()-efore a storm and ()-after a storm (from Krauss, 98).

13 _ t (A.3) It means, tat a = κ (A.4) By using (A.8) and (A.) we ave in te case tat te mixed layer is increasing t > : ε = /3 ε H = /3 ε q = / (A.5) Sustituting (A.) to (6) and y using (A.5) we ave te following equation for te determination of Q: dθ dq ( + εh) θ ε( ξ ) = εq (A.6) dξ dξ After integration of (A.6) wit respect to ξ, and y using condition (A.), we get: Q(ξ )=-/3(-ξ)-/3(-ξ) 4 (Al.7) In case, wen te mixed layer is decreasing, (/t and R > ) we propose tat: _ q R R = = α q (A.8) were α q is a coefficient of proportionality. Ten, y using conditions (9) and (A.), we get te following equations for H and : H t = εh ; = ε (Al.9) t H κ/ κ αqκ κ κ/ κκ + αqκ ε = ; εh = κκ κκ if ĸ =.6; ĸ =. (see Mäli and Tamsalu, 985) ten: ε = 5/6-5α q ; ε H = -/6 + 5α q (A.) In case of stale stratification and Ri numers aove critical, te principal source of turulence is te process of reaing of internal waves. Tis is called as te locing effect of turulence. Te internal waves ave a small amplitude in te ottom layer. Tus, te canges of te ottom temperature (uoyancy) are negligile. In te first approximation we tae, tat: e H = (A.) Ten we ave, tat:

14 α q = - /3 and ε m = /3 (A.) By using (A.) and (A.) we ave te following equation for Q, in te case wen te mixed layer is decreasing: Q = ξ (-ξ) 4 (A.3) Te distriution of non-dimensional turulent flux Q is presented in Figure 3 in cases of increasing (A.7) and decreasing (A.3) of te mixed layer dept. Fig. 3. Self-similarity structure for uoyancy flux Q as a function of ξ. Curve -mixed layer dept is increasing; curve - mixed layer dept is decreasing.