OCEANOLOGICA ACTA 1979 -VOL. 2 - No 2
~
-----~-
Modelling
the oceanic upper layer
Oceanic upper layer
Air-sea interaction
Turbulence in stratified fluid
Turbulent boundary layers
Couche ·océanique superficielle
Interactions ocean atmosphère
Turbulence en milieu stratifié
Couche limite turbulente
S. S. Zilitinkevich a, D. V. Chalikov a, Yu. D. Resnyansky b
a P. P. Shirshov lnstitute of Oceanology, Academy of Sciences of the Ussr, Leningrad.
USSR
b Hydrometeorological Centre. Moscow, USSR.
Received 3/5/78, in revised form 2/9/78, accepted 12/9/78.
ABSTRACT
Empirical data on the vertical structure of the oceanic upper layer are given. Different
kinds of models based on prediction equations for the depth h and temperature T 0 of
the homogeneous upper layer are considered, together with more detailed models dealing
with equations for turbulent perturbation moments. Various approaches used in the
construction of these models are discussed in conjunction with examples of calculation
of h and T 0 , using real data assigned at the ocean surface and in application to the problem
of parameterizing the beat transfer processes in the active layer of the ocean, relevant
t.., the numerical models of large-scale circulations.
Oceanol. Acta, 1979, 2, 2, 219-240.
RÉSUMÉ
Modélisation de la couche océanique superficielle
On fournit en premier lieu des données empiriques sur la couche océanique superficielle.
Sont considérés ensuite différents types de modèles fondés sur les équations qui décrivent
l'évolution de la profondeur h et de la température T 0 , au même titre que des modèles
détaillés faisant appel à des équations pour les moments turbulents. Diverses approches
utilisées pour la mise au point d'un modèle sont discutées sur des exemples de calcul de h
et de T 0 • On introduit des données réelles fixées à la surfaQ< de l'océan et on les applique
à la défmition de paramètres représentant les échanges thermiques de la couche active
et adaptés aux modèles numériques de calcul à grande échelle.
Oceanol. Acta, 1979, 2, 2, 219-240.
wind-generated mixing and the sign and value of the
large-scale vertical velocity at the lower boundary of
the layer.
OBSERVED STRUCTURE OF THE UPPER LAYER
OF THE OCEAN
According to Filyushkin (1968); who formally defined
the lower boundary of the active layer as a level where
the vertical temperature gradient decreases to 0.1 K/ rn,
the depth of the active layer in the Pacifie varies from
100 rn in the western equatorial areas and near the
continents to 200-300 rn in the middJe latitudes of the
central regions of the oçean.
The input of solar radiation through the ocean surface
and thermodynamic interaction with the atmosphere
induce thermohaline changes in the density PlV of surface water, characterized by the vertical mass flux
M =p ~ w' (p ~ and w' being fluctuations of density
and vertical velocity). The main spatial and temporal
variations in density are localized in the upper active
layer, whose depth ha is determined by the amplitude
of annual changes in the mass flux, the intensity of
0399-1784/1979/219/$1.00/@Gauthier-Villars.
In high latitudes in winter, intensive cooling and increasing salinity due to ice formation may lead to the
development of deep convection, sometimes accompanied
219
S. S. ZILITINKEVICH, D. V. CHALIKOV, YU. D. RESNYANSKI
is the thermal expansion factor; and as = 8 x 10- 4 ( 0100 )- 1
is a similar factor for salinity. According to the given
expression for p'w, the Monin-Obukhov oceanic length
scale is expressed in the form
by penetration of the active layer down to the ocean
bottom. This phenomenon appears to play an important
rôle in the formation of cold deep water masses.
Except in the polar areas, where deep convection is
rather typical, the active layer usually has a standard
vertical structure. In its upper part, there exists a pronounced mixed and, to ali intents and purposes, vertically
homogeneous layer, whose depth h ranges from severa!
tens to one or two hundred metres, limited by a density
jump from below. In the lower part of the active layer
called the seasonal themwcline, density increases with
depth due to a decrease in temperature and increase
in salinity.
The homogeneous upper layer is induced by mixing as
a result of surface wave breaking, turbulence produced
by the velocity shear in drift currents, and convection
due to buoyancy produced by cooling of the ocean
·surface or increasing salinity, for example, in the course
of evaporation. As a result of rough measurements of
temperature and salinity vertical profiles, the layer in
question is often visualized as being relatively homogeneous (Francis, Stommel, 1953). More sensitive instruments reveal, however, the presence in this layer of a
microstructure with alternation of comparatively homogeneous sublayers separated by sharp jumps with a
temperature difference up to 0.1 K (Woods, 1968;
Monin, Fyodorov, 1973). Nevertheless, the vertical
changes in temperature and salinity are appreciably
smaller here than in the seasonal thermocline. I t is
therefore assumed that within the homogeneous upper
layer, density does not vary with depth and is coïncident
with surface value. It is usually supposed when constructing theoretical models of the oceanic upper layer that
both temperature T = T 0 and salinity s= s0 , are separa tel y
constant with depth.
It should, however, be remembered that the assumption
of approximately constant density in the case considered
does not testify to the absence of a vertical mass flux.
Indeed, because of the high values of beat capacity cw
and density Pw of the water, the almost imperceptible
vertical temperature differences here may be responsible
for a vertical beat flux of the same order of magnitude
as in the atmospheric surface layer above the sea. By
the order of magnitude, the ratio of the temperature
differences in the water and air assuring equal vertical
turbulent beat fluxes is
(1)
ltl
lu'w'l
1 2
1 2
/ =
Here, v.=
1 , Q=T'w' and S=s'w'
represent friction velocity, kinematic beat flux and
salt flux in the water near the surface; u' and w' are
fluctuations of the horizontal vector velocity (with
components u and v) and vertical velocity w the vertical
axis z is directed downward from the surface): p~-~ is
the vertical momentum flux in the water, assumed to
be equal to the wind stress at the surface.
According to (1), the magnitudes of the mass flux
M=10gm.cm- 2 .year- 1 (see empirical estimates by
Monin, 1970) and friction velocity in the water
v* = 1 cm. s- 1 (which corresponds to -a typical wind
stress t = 1 t 1= 1 dyn/ cm 2) yield L ~ 30 rn; which indicates
a practically neutra} stratification in a homogeneous
layer whose stratification .parameters Jl=h/L and
J..Lo =v* ( f L)- 1 appear to be of the order of severa!
units (fis the Coriolis parameter). This estimate, only
applies, however, to the mean annual values of the
mass flux. To evaluate possible variability in the conditions of static stability in the upper layer of the ocean,
we consider the ratio of atmospheric to oceanic MoninObukhov length scales which, for coïncident values of
heat fluxes in the water and air, equals
aT T 0 c p p1/2c-1
p-1/2 "'0 5
w
w
• •
If we also take into account the fact . that the depth of
the atmospheric boundary layer is one order of magnitude
greater than the depth of the homogeneous upper layer
of the ocean, it then follows that density stratification
in the ocean may be considered as neutra! not only
annually on an average, but also in the majority of
specifie situations.
Thus, in using the term "homogeneous layer" we must
bear in mind the presence of very small but finite, density
gradients which, as a result of vigorous mixing, assure
appreciable mass fluxes with practically neutra! strati'
fication.
Maximum d~nsity gradients are observed at the lower
boundary of the upper homogeneous layer, in the region
of the so-called main jump. Here, the temperature gradient
often reaches severa! degrees per metre.
Below the jump layer, a monotone increase of density
with depth is observed in the seasonal (upper) thermocline; here, however, one or more secondary jumps
may take place in specifie situations. A complicated
microstructure occurs against their background (much
more pronounced here than in the homogeneous layer),
so that the thermocline appears to be divided into a
multitude of homogeneous layers of depths ranging
from metres to centimetres, separated by thinner sublayers with density jumps (Woods, 1968; Monin, Fyo-
A temperature difference of 0.01 K in the upper, e. g.,
ten-metre, layer of the water is usually considered to
be negligibly small; from the point of view of beat
transfer, however; this is equivalent to a difference in
temperature of 1 K between the water surface and the
air at a height of 10 m.'
It is convenient to use the Monin-Obukhov length scale L
to describe density stratification in the upper layer
of the ocean. To define it, we use the equation of state
of sea water which, to a linear approximation, gives
P~= Pw (aps' -aT T') where p~, s'and T'are fluctuations
of density, salinity and temperature, aT=2 x w- 4 K - 1
220
MODELLING THE OCEANIC UPPER LAYER
zm3't.SO
0
35.00
38.00
35.50
36.50 S 0/oo
f',-
z/h
0.6
O.Lt
O.BKs
0
·~
x~
~
100
0.2
0
0.4
'
~~x
0.8
x/.7
/)
300
0.2
~
_,
1
O.'t-
/s
10
-2
0
15
20
0.4
0.6
1.0
0.8Kr
Figure 2
Vertical structure of the oceanic upper layer from measurements
on Ocean Station "Papa".
1. nondimensional temperature in the seasonal thermocline
KT=(T-T 0 )(T.-T0 )- 1 , after Miropolsky et al. (1970); 2. nondimensiona/ salinity in the active layer K,=(s -s 0 )(s. -s 0 )- 1 ,
after Reshetova and Chalikov (1977); the horizontal segments indicate
standard deviations.
and which is approximated quite weil by the fourth
power polinomial. The approximate self-similarity corres. ponding to Equation (2) for temperature and to a similar
equation for salinity was confirmed by subsequent
treatment of observations in the ocean (where
dorov, 1973; Fyodorov, 1976). An example of the vertical
temperature profile illustrating the presence of the
microstructure is given in Figure 1.
The microstructure-forming mechanism remains to be
fully clarified. There are few theoretical approaco:hes to
this problem. Long (1970) has pointed out that in the
neighbourhood of density profile point of inflection,
the effect of small-scale turbulence increases its gradient
and may lead to the formation of a jump growing with
time in the presence of a vertical mass flux. The most
promising results were obtained by Voronovich et al.
(1976), who found on the basis of the nonlinear theory
of internai waves that a stepped microstructure close
to that observed in the ocean may occur during the
passage of a wave envelope.
KT=8/3Ç-2Ç 2 + 1/3Ç4
see Figure 2) and by laboratory experiments (where
KT=2Ç-6/5Ç 2 + 1/5Ç4
see Figure 3).
Figure 3
Nondimensional density similar to the nondimensional temperature
KT as afunction of the nondimensional depth t;, in laboratory experiments (Linden, '1975).
Analysing smoothed temperature profiles in the seasonal
thermocline, Kitaigorodsky and Miropolsky (1970) have
found an apprQximate self-similarity of the nondimensional
temperature (T0 - T)/ (T0 - Ta) considered as a function
of the non-dimensio~al depth Ç= (z- h)/(ha- h) where Ta
is the temperature at the lower boundary of the active
layer, i.e. at z =ha. Th us, the following approximate
expression is valid in the active layer
{ T0
for z ;:;; h,
T 0 -(T0 -Ta)KT (Ç)
0.2
25 T°C
Figure 1
Examples of vertical temperature ana sanmty proj1les in the oceanic
upper layer, obtained by means of"Aist" sond: (1) and usual hydrological measurement series; (Il) (Fyodorov, 1976).
Sounding was performed for 20 minutes before the series on
13 June 1970 in the Tropical Atlantic: a, shown shifted by t•c;
b, shown shifted by 0. 2°1 00 .
T_
-
0.8~
--1
xJx
800
H
x--
500
f::s
~
..__
0.8 ~
1
-o-}11
-x-
)(
ltOO
1
for z > h,
(2)
where KT (Ç) is a nondimensional empirical function
which satisfies the conditions KT (0)=0 and KT (1)= 1
-0.8
221
-O.'t
0
O.'t
0.81.,
S. S. ZILITINKEVICH. D. V. CHALIKOV. YU. D. RESNYANSKI
There is more recent observational evidence, however,
that the "universal temperature profile" below the
mixed layer does not remain unchanged on the seasonal
time scale. lt rather appears that each weather station
has its own "universal profile" for each month. The mean
annual universal structure of seasonal thermocline may
therefore serve as a very rough approximation only.
Observed variations in the structure of the active layer
are illustrated in Figure 4, which shows a typical example
of the temperature seasonal course at various depths
in the North Pacifie. The horizontal distribution of the
homogeneous layer depth is shown in Figure 5.
(4)
vertical changes in its mean flux E=e' w'.
Using the approximate constancy of temperature with
depth in a mixed layer, we integrate Equation (3) vertically
w' is related to the horizontal
beat flux Q=T' w' by the expression M= -aTpwQ
where p~, T' and u;' are fluctuations of density, temperature and vertical velocity, and where P';;; and aT are
standard values of the density and the thermal expansion
factor of sea water.
We use the equation of heat transport
Figure 5
Observed distribution of depths h (metres) of the upper homogeneous
layer in the Pacifie Ocean (Bathen, 1972).
a, January; b, July.
(3)
and the equation of turbulent kinetic energy budget e
(which is understood to be a half-sum of the mean
squares of the fluctuation velocities)
Figure 4
Observed seasonal variations in temperature (degrees centigrade)
on Ocean Station N (Dorman, 1974).
zm
0
. 50
Il
oz
•x
The main features of the evolution of the oceanic active
layer can be satisfactorily simulated with the use of
simple one-dimensional models which do not take
account of advection, horizontal diffusion or largescale vertical motions. In presenting these models,
we assume for the sake of simplicity that water density
is dependent on the temperature alone, so that the
p~
oE
ot
'Here, G = -(t. ou/oz) is the rate of energy production
due to the mean current shear; u is the horizontal vector
of current velocity with components u and v along
the x- and y-axes: t = u' u-' is the vector of the vertical
momentum flux with components and ty, normalised
by the water density (the primes mean the fluctuations,
the overbar is used for averaging, and w' is the vertical
velocity fluctuation); z is the vertical coordinate directed
downward, t is the time; B=aT g is the buoyancy parameter (g is the gravity acceleration); F is the solar radiation flux normalised by cu; pu;; E is the energy dissipation
rate due to viscosity. The second term in Equation (4)
- ~ Q expresses generation or dissipation of turbulent
energy due to buoyancy forces (for Q > 0 i. e. for the
downward h~t flux, this term is negative); the third
term - oE/ oz describes the energy variations due to
NET BUDGET OF TURBULENT ENERGY IN
A MIXED LAYER
vertical mass flux M =
oe
-=G-~Q---E.
x
Xl/
222
MODELLING THE OCEANIC UPPER LAYER
from z = 0 to z =h-O. Designating
(Q+F)z=o=Qo,
formation of the turbulent kinetic energy produced in
a mixed layer into the potential energy of density stratification. According to sorne estimates (Pollard et al.,
1973; Mellor, Durbin, 1975) this value is approximately
one arder of magnitude smaller than the rate of mechanical production (or viscous dissipation) of turbulent
energy, i.e. Pis a small difference oflarge values. For this
reason, rough estima tes of turbulent energy budget components appear to be inadequate for determining P. Kra us
and Turner (1967), together with the majority of subsequent au thors, use the assumption of steady-state turbulent regime, having in mind a large difference brtween the
time scale of dissipation of turbulence te=e/ e and
heating-time tT=hôT/Q in the oceanic upper layer.
Indeed, for the temperature variation ôT = 1 K and
typical values of h=20m and Q=2x w- 3 K.cm.s- 1,
we obtain tT= 12 days, whereas the typical value of te
fore= 1 cm 2 • s- 2 ande= 10- 1 -10- 3 cm 2 • s- 3 does not
exceed a few tens of minutes (see Manin et al., 1974).
The effect of penetrative solar radiation, which contributes to the onset of convection in the uppermost layer
of the water is readily estimated if one considers the
exponential decay law F=F0 exp (-cu), where ex is
the absorption coefficient of the water, F 0 =F 1z=O·
In this case we obtain
Q lz =h-o=Qh
and assuming Flz=h=O we obtain
Ofo
h-=Qo-Qh.
ôt
(5)
Equation (5) is fundamental to ali simple models of
the homogeneous upper layer of the ocean, the depth
of the layer h and its temperature T 0 (coïncident with
the surface temperature) being used therein as unknown
variables. The specifie models differ firstly in their
methods of defining the heat flux Qh through the interface z=h, and secondly in the additional hypotheses
for estimating one term or another, in the equation of
turbulent kinetic energy budget (4).
Since Kraus and Turner (1967), the temperature profile
in a number of papers has been assigned in the form
of discontinuous function: T=T0 for O~z<h. T=Th
for z=h with 8T=T0 -Th>0. The beat exchange
between the homogeneous layer and seasonal thermocline in this case is determined by entrainment. The
corresponding heat flux Qh is defmed by the relation
which follows from (3) for the limiting transition from
the continuous to discontinuous distribution of temperature using the assumption Q 1;= =h+o = 0:
ôh
Qh=ôT ôt.
(6)
because usually h > ex- 1 =5-20 rn (see Kraus, Turner,
1967). The role of the term in question is small in corn pa-
Equation (6) can be reasonably interpreted only for
the case ôh/ ot ~ 0, when the water at temperature Th
penetrates the homogeneous upper layer and is mixed
there, th us cooling this layer. If oh/ ôt < 0, the application
of Equation (6) would mean that in the upper homogeneous layer there occurs a process which is the reverse
of stirring, in which water cooled by the value ôT penetrates into the lower layer. lt is supposed, in place of
this undoubtedly meaningless assumption, that for
oh/ ôt < 0 beat exchange is absent across the interface, i.e. Qh=O.
Integrating (4) on z from 0 to h and taking into account
the expression for Q which follows from (3) and (5)
we now obtain
rison, for example, with bulk dissipation
J" edz:
0
.
for the typical values of e, h, ex given above and the values
of F 0 =2x w- 3 K.cm.s- 1 and 13=10- 1 cm.s- 2 .K-t,
the ratio J3ex- 1 F 0 (e h)- 1 is as small as 5 x w- 4 -2 x 10- 1 •
Nevertheless, rough estimates of the main components
of the turbulent energy budget do not always permit
definition of the real significance of the effect of penetrative radiation as being negligibly small.
The most important terms on the right-hand side of
the second formula in Equation (7) in the Kraus and
Turner model were estimated in the following way
[ edz=O,
E0 +
J:
G dz oc
v~,
(8)
where v*= 1T 1112 is the friction velocity in the water
near the surface. ln a similar madel Elsberry et al. (1976),
and in sorne other modifications, the omitted nondimensional coefficient on the right-hand side of the
third formula in Equation (8) was assumed to be not
çonstant, but dependent on h [proportional to
exp (-hl h.), where h* is a dimensional constant chosen
for better agreement of the results of calculations with
observed data (see also Gill, Trefethen, 1977) ].
Hypotheses (8) permitted the qualitative simulation of
the process of deepening of a mixed layer in laboratory
experiments and a number of features relevant to the
Here, E 0 = E 1z =0 is the turbulent kinetic energy flux
from the atmosphere towards the ocean produced by
wave breaking; Eh=Eiz=h is the downward turbulent
energy flux at the lower boundary of the homogeneous
layer.
The right-hand side of the upper equation in (7), which
so far remains indefinite, describes the rate of trans-
223
S. S. ZILITINKEVICH, D. V. CHALIKOV, YU. D. RESNYANSKI
evolution of a mixed layer in the ocean throughout
the season (see Kraus, Turner, 1967; Turner, Kraus,
1967). These hypotheses, however, are known to be
approximate, and are· of limited application. The rough
estimate of the mechanical production ·of turbulent
energy, in particular, led to a lower rate of entrainment
oh/ ot in comparison with observations in the case of
wind-generated mixing, whereas neglect of dissipation
or, the assumption that it is proportional to mechanical
production (which, in fact, amounts to the same thing),
leads to an over-estimation of the rate of entrainment
in the case of convective mixing (see Gill, Turner, 1976).
A number of subsequent models were also based on
relations (5)-(7), differing only in the hypotheses relevant
to the terms in the expression for P. Sorne of these
hypotheses are summarized in Table 1, the last column
of which gives formulas for the limiting depth of the
homogeneous layer hm obtained in the given model
for t--+ oo, provided that the wind velocity V0 (or the
associated friction velocity P,J and downward (i. e.
positive) heat flux at the ocean surface remain unchanged
during an appreciable period of time [in Pollard et al.
(1973), the limiting value of h is reached during the
finite period t=n/ f].
ln addition to the above designations, the following
notation is used in the Table: U and V are the components
of the current velocity horizontal vector assumed to
be independent of depth in a mixed layer; À= v*/ j is
the Ekman Iength scale; j is the Coriolis parameter;
L=v; (~Q 0 )- 1 and L*=U6 (~Q 0 )- 1 are versions of .
the oceanic Monin-Obukhov (1954) length scale; Jlo= "A/L
is the Kazansky and Monin (1960) stratification parameter; N = (~ r) 112 is the VaisaHi-Brunt frequency corresponding to the vertical temperature gradient in the
seasonal thermocline r =-ô T /oz lz =h+o;~ e 0 and e0
are dimensional parameters in Alexander and Kim (1976)
and Kim (1976) which are, respectively, the background
value of the dissipation rate of turbulent energy and
typical value of turbulent energy in a mixed layer dependent on friction velocity (e 0 =4. 5 v'! for v*> 1 cm. s- 1 ,
e0 =4.5 cm 2 .s-_2 for v*~l cm/sec); e0 is also the
mean value of turbulent energy in a mixed layer in
Garwood (1977); À6 =v!/e 0 and ÀF=F 0 (CIQ 0 )- 1 are
the length scales related to dissipation and to penetrative radiation; a, a0 , a 1 , ••• , a6 are nondimensional
empirical constants; «P is the non-dimensional empirical
function oftwo arguments chosen by Resnyansky (1975).
Corrections of the Kraus and Turner mode! are reduced,
in the first place, to using more detailed expressions for
E0 +
[
G dz. and
J:
the inertial period 2n/ j, the integral [ G dz being
calculated by means of the nonstationary Ekman
equations, on the assumption that the velocity components in a mixed layer are constant with depth and are
equal to zero for z.> h. Neglecting the viscous dissipation
J:
e dz and the turbulent energy input from the atmo-
sphen~ E0 , and also the term '1 2 ~ h Q0 on the Ieft-hand
side of the first formula in Equation (7), the authors
obtained from this formula the relation
·
(it was assumed from additional considerations that
Ri= 1). By means of this mode!, Pollard et al. (1973) ·
calculated the adjustment of the homogeneous layer
structure to wind forcing which occurs instantaneously
and then remains constant (v* =constat t > 0). According
to these calculations, a mixed layer deepens rapidly
to the maximum value hm=2 314 v* {fN)- 1 ' 2 during
half the inertial period. During the process, about 40%
of energy input due . to wind stress at the surface is
expended on increasing the kinetic energy of mean
motion, whereas the rate of change of the potential
energy of density stratification appears to be three times
smaller. Th us, in the case of sharp changes in wind speed,
the velocity shear in non-steady drift currents appears to
be the predominant mechanism ofturbulence production
during the time of the order of the inertial period.
In real conditions, however, sharp wind changes are
only infrequently observed. Comparatively slow synoptic
variations with characteristic times of the order of
a day are more typical. In these cases, there is no reason
to neglect the energY. flux produced by surface wave
breaking. Moreover, in a stejldy regime, this flux can be
an order of magnitude larger than the energy flux
transferred by wind to drift currents (see Kitaigorodskii,
1970, Section 10. 2).
Taking this into account, Niiler (1975) elaborated the
previous mode!, taking account of the energy flux
E0 oc
The behaviour of the characteristics of the
homogeneous layer in this mode! at the very beginning
of deepening (t ~ 50N- 1) and at a finite stage (t :<; nj- 1)
is governed by the input of turbulence from the ocean
surface, whereas at intermediate times
v; ..
e. dz. We have already noted that
the main rote is played by the current velocity shear in
the jump layer-in exactly the same manner as in the
model of Pollard et al. (1973). The most rapid changes
in h take place during the initial period (t < nj- 1),
after which the depth of the mixed layer becomes approximitely equal to 2v* (JN)- 1 12. Subsequent deepening
occurs at a much lower rate, the Iimiting value of h being
equal to hm oc L as in the case of the Kraus and Turner
mode!.
We note that taking account of the additional source
the expression for E 0 used in Elsberry et al. (1976) was
specified by including an empirical coefficient dependent
on h on the right-hand side of the third formula in
Equation (8).
The role of the production term [ G dz was investigated
by Pollard et al. (1973). They considered the effect of
non-steady drift currents for time scales of the order of
224
MODELLING THE OCEANIC UPPER LAYER
Table 1
Closing assumptions for the bulk components of turbulent kinetic energy budget used in different models, and the corresponding estimates of the
maximum depths of the mixed layer heated from above.
-
I
oe
-dz
E0
o ot
Authors: Kraus, Turner (1967)
0
a.-t Fo
0
Author: Denman (1973)
Fo
0
- (1-exp (-cxh)]
0
ex
Gdz
[
- [ edz
h.,=limh
,_..,
0
aU~
Authors: Pollard et al. (1973)
0 .
0
0
0
0
2
ot
Author: Rusin (1973)
0
-a 0 U~-a 3 Ch
or
aU~
0
Y
2 [(a-a 0 )L.+ÀF]
Author: Garnich (1975)
0
aU~
0
2aL*
Author:. Niiler (1975)
0
0
u 2+v2 oh
a 1v ! + - 2
ot
0
-a 2v!
2(a 1 -a 2 )L
Author: Resnyansky (1975)
0
0
Authors: Kosnyrev et al. (1976)
0
h.,
0
Author: Kim (1976)
oh
F0
-e0 -[1+exp(-cxh)J
ot
ex
is a function of llo
2{(a 1 -a 2 )L+ÀF [1-exp(-cxh.,)J}
1 +2L/À'l
Authors: de Szoeke, Rhines (1976)
0
0
Authors: Gill, Turner (1976)
0
0
Author: Niiler (1977)
oh
2
-0.3v.0
ot
0
Authors: Niiler, Kraus (1977)
v; oh
0
2 ot
Author: Garwood (1977)
0
Not determined
h.,
0
225
is a function of llo
S. S. ZILITINKEVICH. D. V. CHALIKOV, YU. D. RESNYANSKI
of turbulence on the right -band side of the upper formula
in Equation (7) modifies the temporal behaviour of h
at the very beginning of the mixed layer development:
in the Niiler model, the asymptotic behaviour hoc t 113
occurs for t < N- 1, whereas the model of Poliard et al.
yields hoc t 112• Within the framework of such an
approach, consideration of the non-steady term
f:
The specifie formula on the right-hand side of Equation (9)
interpolates between two simple cases discussed below
in connection with Equation (10). In the case of heating
of the ocean surface (llo > 0), the following limiting
depth hm corresponds to the presentation of dissipation
given above
(10)
(oe/ ot) dz oc v; oh/ô t in the equation of turbulent
This expression appears to be in better agreement with
experimental data in comparison with earlier models
(see Fig. 6) and yields Rossby and Montgomery (1935)
formula hm oc À. for llo--+ 0 and Kitaigorodskii (1960)
formula hm oc L for J.lo--+ oo.
The first of the asymptotic relations mentioned above
is associated with the dependence of the bulk dissipation
rate on the Ekman length scale À.. Although the viscous
dissipation rate of turbulent energy is concentrated
on smallest-scale eddies, it is govemed by the cascade
energy transfer from larger to smaller eddies (see, for
example, Tennekes, Lumley, 1972). It is natural to
assume that the latter is inversely proportional to the
characteristic time scale of larger eddies, which for
h ~ L is of the order of j - 1• These considerations are
.taken as the basis ofheuristic derivation of the expression
energy budget leads to the same asymptotic behaviour
hoc t for small tas in the case of the mixed layer development in a homogeneous fluid (de Szoeke, Rhines, 1976;
Niiler, 1977).
The most questionable expressions in the above models
appear to be those concerning net viscous dissipation:
the latter is either assumed to be proportional to one
of the energy production terms (Niiler, 1975), oris entirely
neglected (Kraus, Turner, 1967; Pollard et al., 1973).
Underestimation of dissipation leads to continuai increase
in the potential energy of density stratification and
to a questionable unlimited deepening of a mixed layer
for Q 0 =0 (in the models suggested by Kraus and Turner
and by Niiler). The limiting value of the depth of this
layer given in the last column of Table 1 appears to be
proportional to the Monin-Obukhov length scale L,
whereas the depth of the real homogeneous layer in
the ocean is better estimated using the Ekman scale À.
(see Fig. 6).
for
Nondimen.~ional depth of the upper homogeneous layer hl L as a
function of the stratification parameter llo= À/ L.
Empirical points: monthly averaged data of Ocean Station "Papa" (1)
and "Tango'" (2), after Filyushkin (1968). Theoretical curves: (3) h ac L,
(4) h ac À., (5) h=hm according to Equation (10) for a 1 = 10, a4 =25,
a 5 =0.35.
ln(h/L)
1
·~
0
-----~...:
0
(19~7). The expression ob.tained
The field of application of Equation (9) is, however,
limited by the inequality h ~ (a tf a 4 ) À. ~ (0. 2-0.4) À..
The depth of the homogeneous layer in the convective
case usually exceeds this value and obliges us in consequence to choose more complicated empirical approximations to the function <D. Furthermore, the set of
governing parameters used in Equation (9) is uncomplete,
because turbulent characteristics in a mixed layer may
also be dependent, for example, on the temperature
jump () T at its Jower boundary (see, for instance, Gordeev,.
Kagan, 1976). ln this case, the nondimensional coefficients a 4 and a5 (and also a 1 in the expression for the
turbulent energy production, see Table 1) must be
functions of the Richardson muber Ri=~hôT/v;
where ve is a suitable velocity scale (see Niiler, Kraus,
1977; Kitaigorodskii, 1977 a, b). Laboratory experiments
testify, in particular, to a decrease in a 1 with growing Ri
(Kantha, Phillips, Azad, 1977). A detailed analysis of
the regimes of mixed layer deepening, taking into
account the dependence of the right-hand side of Equation (7) on Ri is given by Kitaigorodskii (1977 a, b)
and Garnich and Kitaigorodskii (1977).
Figure 6
'· ·,o
'
in Garwood
in this paper (see Table 1) ts analogous to Equation (9)
if we use e112 for the velocity scale, in place of v*' and
assume a 5 =0.
One may attempt to eliminate the above discrepancies
by specifying the expression for the viscous dissipation
of turbulent energy. Assuming the bulk dissipation to
be a function of v*, ~. Q 0 , f and h, Resnyansky (1975),
from dimensional considerations, evaluated the
expression
2
f: r.d~.
1
• 2
--3
-·-4
-s
Another empirical formula for defming
-1
J:
s dz was
suggested by Alexander and Kim (1976) and Kim (1976),
who take into consideration the background dissipation
r. 0 = 2 x 10- 8 rn 2 • s- 3 which presumably existed independently of local conditions at the .ocean surface. The
application of their model (see Table 1) is also limited:
in the convective case the hydrostatic instability may
occur at the lower boundary of the homogeneous layer
-2
226
MODELLING THE OCEANIC UPPER LAYER
(i.e. ôT becomes negative), so that Equations (5)-(7)
lose their physical sense. ln these cases, the model
provides for application of a convective adjustment
similar to that used in atmospheric models (see Manabe
et al., 1965; Zilitinkevich, Monin, 1971 ). Using the abovementioned expressions for separate components of
the net budget turbulent energy Equation (7), it is possible
to obtain different kinds of mixed-layer models having
one or another required property. The examples of
such a version and an analysis of the specifie regimes
of mixed-layer evolution are given in de Szoeke and
Rhines (1976), Niiler (1977) and Niiler and Kraus (1977).
Table 1 may be of use to forma general notion of existing
approaches in this field.
We must acknowledge that ail the works considered
in this section (and in subsequent sections) seem to
involve too many empirical hypotheses for definition
of the basic components of the bulk budget of turbulent
energy. As will be shown below, however, a suitable
choice of empirical coefficients permits the more or
Jess trustworthy simulation of the observed evolution
of the structure of the oceanic upper layer, by means of
calculations based on different model.s. This certainly
cannot verify whether the real physical processes occurring here are described correctly. Indeed, due to a large
number of goveming factors (wave and wind mixing,
convection, advective phenomena, upweliings or downwellings), and to complicated nature of their interaction, the approximate coïncidence of the simulated
and observed evolution of the depth and/ or temperature
of a mixed layer does not testify to adequate description
of ali of the above factors individualiy.
For this reason, laboratory experiments in which ali
the major mechanisms of interest can be realized and
studied separately are of special importance for understanding the ph ysical processes governing the entrainment
of a stratified fluid into a mixed layer. We should mention
in this connection the experiments by Kato and Phillips
(1969) and Kantha et al. (1977) involving the effect of
deepening of a mixed layer due to shear turbulence;
the experiments by Turner (1973), Linden (1975) and
Thompson and Turner (1975) studied the effect of
turbulent energy diffusion downwards from the surface
(turbulence production due to wave breaking was
simulated using oscillating grids); and the experiments
by Deardorff et al. (1969) and Wiliis and Deardorff
(1974) who investigated penetrative convection. Since
the present paper is devoted to oceanographie applications of mixed-layer models, it does not include a special
discussion of laboratory experiments. The latter are
treated in Turner (1973), Phillips (1977) and Kitaigorodskii (1977 a).
upper layer seems to be a promising issue, but so far
remains for future consideration.
TURBULENT ENERGY BUDGET AT THE OUTER
BOUNDARY OF A MIXED LAYER
Another approach to the investigation of the evolution
of a homogeneous mixed layer is based on evaluation
of the prediction equation for its depth h from consideration of turbulent kinetic energy budget at the level
z=h. Such models for cases of penetrative convection
and mechanicaliy forced deepening have been suggested
by Tennekes (1973, 1975) and Zilitinkevich (1975).
The papers quoted are devoted to the atmospheric
boundary layer, but may readily be paraphrased to
become relevant to the oceanic upper layer or to laboratory experiments.
As in the models treated in Section 2, the papers in
question consider the vertically homogeneous mixed
layer with temperature jump ôT at its lower boundary.
This leads to relations (5) and (6) which foliow from
the equation of beat transport (3) and which were
discussed earlier. Assuming the vertical temperature
gradient in the thermocline to be a known quantity
ôT
oz
=-r
for z > h,
(11)
(r may be an assigned function of depth), it is not difficult
to obtain from Equation (5) the foliowing expression
relating the changes in ôT and h
.
~ôT= Q0
at
h
+(r-
oT)oh.
h at
(12)
The equation of turbulent kinetic energy budget (4) in
a mixed layer ne ar the level z = h may be written in the
following approximate form
(13)
Here, the left-hand side is an estimate of the nonstationary term oe/ot; Cte 312 jh is an estimate [given
in the spirit of Kolmogorov (1942) approximate similarity hypotheses] of the input of turbulent energy
transferred by diffusion to the lower boundary of the
mixed layer; and c 1 and c3 are nondimensional empirical
constants. Shear production of turbulence and its viscous
dissipation are assumed to be insignificant for z = h - O.
Kitaigorodskii and Garnich (1978) and Kitaigorodskii
(1977 a, b) present a mode! of dynamically forced
deepening of a mixed layer based on Equations (5)-(7)
and on two empirical universal functions, which was
used to define the net rate of turbulent energy dissipation
in a mixed layer. The model offers a realistic simulation
of sorne laboratory experiments and incorporates as
specifie cases the entrainment mechanisms considered
by Kraus and Turner (1967) and Poliard et al. (1973).
The development of this model relevant to the oceanic
We specify Equation (13) relevant to the case of a pure
thermal penetrative convection (incidentally, the absence
of shear production in this case is self-evident and there
is sorne reason to neglect viscous dissipation, see Tennekes, Lumley, 1972). A regime close to free convection
is realized in the main part of the mixed layer, so it is
227
S. S. ZILITINKEVICH, O. V. CHALIKOV. YU. O. RESNYANSKI
The estimates of the entrainment rate oh/ ot based on
Equations (15) and (18) prove to be in satisfactory
agreement with corresponding laboratory experiments;
compared with the latter, they yield, in particular, the
above values of the empirical constants A 1 , A 2 , B 1
and B2 (see Tennekes, 1975; Zeman, Tennekes, 1976).
Neglecting the nonstationary term in Equation (13)
(i.e. taking c 3 =0 or A 2 =B 2 =0) the simplified versions
of the prediction equations for h considered above
may be combined by the following interpolation equation
natural to estimate the typical value of turbulent energy
by use of the well-known formula (see, e. g., Monin,
Yaglom, 1965, Chap. 4):
(14)
c
where 4 is another empirical nondimensional constant.
Using Equations (13), (14) and (6) we obtain the following
prediction equation for h
(15)
oh
at=
where B 1 =c 1 d 12 ~0.5 and B 2 =c 3 c 4 ~3.5. The set
of Equations (12) and (15) permits calculation of the
evolution of h and 15 T (and also ofT0 since the T profile
for z > h is assigned) for penetrative convection when
Qo<O.
Examples of solutions to these equations for specifie
conditions have been obtained by Burangulov (1977).
We will confme ourselves to the behaviour of the solution
at large t. Since h and ()T increase with time, the second
term in square brackets on the right-hand side of (15)
may be neglected if t is not very small. Then, for constant
rand Q 0 and zero initial conditions (h=O, ôT=O for
t=O), we will have the expressions
which, apparently, are the asymptotics of the solution
to the complete set of equations.
We now con si der another particular form ofEqua ti on ( 13)
corresponding to deepening of a neutrally stratified
layer in which mixing is due to breaking of wind-induced
surface waves, producing perturbations in the neacsurface layer of water and subsequent downward diffusion
of turbulent energy. In this case, there is no beat flux
at the upper boundary of the mixed layer (Q 0 =0) and a
typical value of turbulent kinetic energy is expressed, in
contrast with Equation (14), by the formula
c
where v* is the surface friction velocity,
is a nondimensional constant. Substitution of this relation into
Equation (13) and taking into account Equation (6)
gives
v;
oh
A1
ot- Ph8T+A2v;'
(18)
Analogous to Equation (16), the asymptotic solution
of Equations (12) and (18) for large t corresponding
to the zero initial conditions and constant r and v* will
r
have the form
rh
ôT=-·
2
(20)
which, as may readily be seen, coïncides, except for
the choice of the constants A 1 and B1 , with the prediction equation in the Kraus and Turner (1967) model
which follows directly from Equations (6)-(8) of the
preceding section.
Certainly, the simple regimes corresponding to Equations (15) or (18) do not include ali the possible mechanisms of mixed layer evolution. Suffice it to recall that
in the initial Equation (13) we do not take account of
the shear production of turbulence G near the surface
z=h, which, apparently, must be significant in the case
of sharp changes of drift currents (this mechanism was
considered in Section 2 in connection with the mode!
of Poliard et al., 1973), and also the viscous dissipation e
which in the presence of small-scale shear turbulence
may play a much more important role than in the case
of bombardment of the interface by buoyancy driven
convective plumes, when it is probably of little significance.
(16)
(17)
Atv!+BtPhiQol
phôT
Furthermore, the energy budget near the leve! Z=h
may be influenced by the energy flux divergence associated with radiation of internai waves into a stahly
stratified fluid below [the mechanism of energy transfer
from turbulence to internai waves is associated with
the pressure flux divergence which is not usually
accounted for and which, incidentally, is omitted in
Equation (4) ].
A detailed discussion of ail of these mechanisms following
the above approach is offered by Zeman and Tennekes
(1976). To defme the rates of viscous dissipation e and
shear production G, it is necessary, in the first place,
to be able to estimate the depth d of the transient zone
between the mixed homogeneous layer and the adjacent
unperturbed stratified fluid (this zone is called the
turbulent entrainment layer in laboratory experiments,
the jump layer in the ocean or the inversion errosion
layer in the atmosphere; usually, d ~ h which is the
reason for an approximate presentation of the temperature profile by the discontinuous function). If the
temperature jump ôT is not very large, the penetration
of eddies with typical velocities of order e1' 2 in the
· transient zone of depth d will be approximately expressed
by the apparent relation prd 2 oc e. This yields direct! y
the estimate
et/2
(19)
doc-,
N
228
(21)
MODELLING THE OCEANIC UPPER LAYER
where N=(PIJ 1 ' 2 is the Brunt-Vaisala frequency.
Assuming now that in the layer of turbulent entrainment
the length scale of turbulent fluctuations is of order d,
it is natural to define the value of e using the following
formula of the traditional type
or
e=csNe.
Here, c6 is a nondimensional constant; th ïs the vertical
momentum flux associated with entrainment which is
defined by the expression
(24)
similar to Equation (6) (see Deardorff, 1973); U is the
mixed-layer-averaged velocity of drift current which,
within the framework of the homogeneous layer mode!,
may naturally be assumed to be coïncident with the
absolute value of the velocity difference at the interface
1ou 1· The components of the latter may be evaluated
using the horizontal motion equations integrated over
the homogeneous layer depth (see Deardorff, 1973;
Kitaigorodskii, 1977 a, b). This gives the following
equation of the same type as Equation (12)
(22)
Except for the numerical constant, a similar expression
is obtained for the rate of energy loss due to the generation of internai waves (Linden, 1975). We may therefore
presume that with appropriate selection of the nondimensional constant cs, Equation (22) may be used to
estimate the rate of energy !osses due to the net effect
of viscous dissipation and radiation of internai waves.
The empirical value obtained by Zeman (1975) using
the data of laboratory · experiments on penetrative
convection (Willis, Deardorff, 1974) is cs ~ 0.023.
ô
-ôt (hou)=t Xo + fov,
(25)
where f is the Coriolis parameter, t x = t xl z = 0 and
are the normalised momentum flux
t Yo =t Y components near the surface (t 2Xo +t Yo2 =v!).
The consumption ei of turbulent energy generated in
a mixed layer for internai wave radiation from its lower
boundary was also investigated by Kantha (1977). ln
his paper, the theory of internai waves and dimensional
considerations were used to obtain an expression which
assures vanishing of ei in limiting cases of a two-layer
fluid (r -+ 0) and a very sharp temperature difference,
which appears to be necessary for simple physical
reasons. This expression has not yet been tested experimentally or used for oceanographie applications.
We also note that Gordeev and Kagan (1976), concurrently with Zeman and Tennekes (1976), proposed that
account be taken of the turbulent energy dissipation
rate e in the equation similar to (13) assuming the nondimensional ratio ehe- 3 ' 2 to be a linear function of
the Richardson number phoT1e. The corresponding
expression for e includes. two empirical constants.
Choice of the latter reasonably permitted [as in the case
of Equation (9) in Resnyansky, 1975] improvement on
the agreement between the simulated and observed
evolution of the oceanic upper layer in comparison
with calculations based directly on Equation (13).
In the presence of an appreciable velocity shear at the
leve! Z=h, for example in the case of a strong nonsteady
drift current, the turbulent energy equation similar
to Equation (13) must include a significant shear production term G. Assuming that the total difference in
the current velocity vector between the mixed layer and
the fluid below ori is localized in the layer of turbulent
entrainment of depth d, it is natural to define the shear
production G by the formula
or
ô
-ôt (hov)=t Yo -fou,
lz-o
We note that in using Equation (13) or its more detailed
versions, for example, including the viscous dissipation
e given by Equation (22) or shear production G given
by Equation (23), we need not confine ourselves to cases
of purely buoyant convection and purely mechanical
mixing (this was done earlier only for convenience of
the presentation). Intermediate regimes may be investigated in a similar manner taking for the turbulent kinetic
energy e the linear interpolation formula
(26)
connecting Equation (14) and Equation (17). Equation (20)
being the simplest example of the prediction equation
for h, may be derived immediately from such approach.
A more complicated method of obtaining similar interpolation expressions may be based on application of
the net turbulent energy budget Equation (7). Thus,
in Garwood (1977), in contrast to the models discussed
in Section 2, this equation is intended for direct estimation
of the mixed-Iayer-averaged turbulent energy, rather
than for obtaining the rate of entrainment [this purpose
is served by the equation at the lower boundary of a
mixed layer, similar to (13) ]. This leads to the construction
of the mode) which Iinked the approaches discussed
above.
The question remains concerning the degree· of realism
of the approach discussed here with regard to the evolution of the upper homogeneous layer in the ocean and,
in particular, to determine the extent to. which the
allowance for the viscous dissipation e and shear production G in the equation of turbulent energy budget is
necessary. Zeman and Tennekes (1976), however, have
presented data which verify the consistency of the
calculations based on the mode! given above with
observations of the development of a convective boundary
layer in the atmosphere (performed during the Wangara
experiment, see Clarke et al., 197D.
(23)
229
S. S. ZILITINKEVICH, D. V. CHALIKOV. YU. D. RESNYANSKI
Kato and Phillips' (1969) laboratory experiments mentioned earlier in this paper also provided interesting
data for testing this model. In these experiments, timeconstant stress was produced on the surface of a stratified ·
fluid, resulting irt an increasing drift current velocity,
whereas turbulence caused by the velocity shear led
to the appearance· of a homogeneous mixed layer and
its subsequent deepening (at a rate of oh/ ot). The paper
quoted gives an empirical graph showing the dependence of the nondimensional rate of entrainment
we=v- 1 oh/ot on the modified Richardson number
Ri= ~*hoT/ v (not to be confused with the different
version of Richardson number Ri designated similarly
previously). This graph is presented in Figure 7. Apart
from the empirical points, the figure presents three
theoretical relations: Kato and Phillips (1969) formula
z
tion (13) the viscous dissipation e defmed according
to Equation (22) for c 5 = 0. 023.
It may be seen in the figure that Equation (28) is in better
agreement with real data than other formulas, which,
· according to Zeman and Tennekes (1976), testifies to
a practically complete compensation for the effects
of shear production and viscous dissipation of turbulent
energy at the outer boundary of a mixed layer. If this
appears to be the case, then it follows that the field of
application of the prediction equations for h, similar
to Equations (15) and (18), is appreciably extended.
(27)
JOINT EVOLUTION OF A MIXED LAYER AND
SEASONAL THERMOCLINE
Calculations of the evolution of the depth and temperature of the upper homogeneous layer have been
combined in a number of papers with calculations of
the changes in the structure of the seasonal thermocline
(note that in the discussion of the mixed layer models
p. 222-227. the temperature profile in the thermocline was assumed constant and prescribed). The important additional difficulties in a problem of this kind are
associated with a very strong hydrostatic stability in
the seasonal thermocline, which leads to the suppression
of turbulence, its intermittence and to the development
of internai waves and specifie stepped microstructure
(see p. 219-222).
which neglects the nonstationary term, i. e. follows
from (13) for c 3 =0 or A 2 =0, on the assumption that
A 1 =2.5; Equation (18) with the above values of the
constants A 1 =3.4 and A2 = 13, which in the notation
considered takes the form
(28)
and a more detailed formula
(29)
Use of the traditional semi-empirical hypotheses relating
the turbulent fluxes to the mean gradients seem to be
obtained similarly to Equations (18) or (28) allowing
for Equation (12) and including in the energy Equa-
questionable below the jump layer, because the
mechanism of vertical transport is significantly different
here from that which is found in flows with developed
turbulence. Nevertheless, most calculations of the temperature profiles in the thermocline are based on this
approach. Sorne of these will be treated elsewhere in
this paper, but we note at once that even when the
calculated profiles appear to be close to observations,
due to a happy choice of turbulent exchange coefficients,
such an approach adds little to our understanding of
the physics of their formation.
Figure 7
Nondimensional rate of enrrainment w.=v;• iJh!iJt as a function of
the Richardson number Ri= Ph 0T 1v!.
Empirical points: from laboratory experiments (Kato and Phillips,
1969). Theoretical curves: 1, according to Equation (27); 2, Equation (28); 3, Equation (29).
A more convenient and natural approach (which, incidentally, does not clarify the physics of thermocline
formation), is based on the application of the universal
temperature profile in the upper layer of the ocean (see
Figs. 2, 3). According to Equation (2), the temperature
difference at the leve! z = h is not included, the layer of
turbulent entrainment being simulated by the sharp
portion of the temperature profile in the upper part of
the thermocline rather than the jump. Integration of
Equation (3) over the depth from z = h to z =ha allowing
for the approximation (2) in the absence of a beat flux
at the lower boundary of the upper active layer (z =ha)
and assuming the natural condition F = 0 in the region
h < z <ha, leads to the following relation o btained by
Kitaigorodskii and Miropolsky (1970)
We
0,2
0,02
--1
-·-2
-3
0
oh
OTo
Qh=Y (To- Ta) ot +(1-y) (ha-h) Tt'
20 JO 'tO 60 100
230
(30)
MODELLING THE OCEANIC UPPER LAYER
where y=
f
We tum now to a discussion of the realized calculations
based on Equation (30) and equations given p. 222-227.
The first of these was performed by Miropolsky (1970).
In his paper, the set of Equations (5)-(7), (30) closed by
means of hypotheses similar to Equation (8) (according
to which, however, the dissipation rate of turbulent
energy was assumed to be proportional to its production
rather than equal to zero), was used for calculating the
seasonal evolution of the upper layer of the ocean on
the basis of measurements of T 0 and v* on Ocean
Station "Papa" in the Pacifie Ocean. The calculated
and measured values of h appeared, on the whole, to be
in satisfactory agreement. Maximum errors occurred
during the winter period which, according to the discussion given p. 222-227. is due to underestimation of
the bulk dissipation in the case of convection (i. e. in
winter alone).
KT (Ç)dÇ= 11/15 is an empirical constant.
If now, in place of Equation (6), we use Equati?n (30)
together with Equations (5) and (7), then, followmg the
approach given p. 222. it is possible to obtain modified
versions of ail of the models (similar to Kraus, Turner,
1967) presented in that section, which will describe· the
joint evolution of the upper homogeneous layer and
seasonal thermocline and will differ from each other
depending on the chosen approximation to the righthand side of the upper equation in (7) (see Table 1).
Also, for y= 1, a continuous temperature .variati.on
in the seasonal thermocline is replaced by Jump-hke
variation such that Equation (30) takes the form of (6)
(certainly, if it is assumed that Ta=Th).
The models similar to Tennekes (1973, 1975) and Zilitinkevich ( 1975) described p. 227-230 may be modified in
a completely similar manner. In this case [using Equation (30) instead of Equation (6)] we obtain from Equation (5) and (30) the following equation for T 0 , in place
of (12)
To eliminate discrepancy in the framework of this
approach, Kamenkovich and Kharkov (1975) defined
[
Application of Equations (30) and (31) together with
the energy Equation (13) and with one or another
expression for the mean turbulent energy [Equations (14)
for convection, (17) for mechanical mixing or, finally,
(26) for a intermediate regime] permits the construction,
as p. 227-230. of prediction equations for the calculation
of h and T 0 allowing for the joint changes in the upper
homogeneous layer and seasonal thermocline. Not
only, however, were these equations not solved; they
were not even written out. It is so easy to obtain them
following the above approach that there is no point
in discussing ail the specifie cases; we thus confine
ourselves to one example of deepening of a neutrally
stratified mixed layer where Q 0 =0, e=c 2 v;. Equation (31) now is readily integrated and the set in question
takes the form
(T0 -
TJ
2]-1
yh
A
(1-y)ha+Yh + 2V*
'
using Resnyansky's formula (9). They also
allowed for changes in the upper layer depth ha and the
temperature at its lower boundary Ta evaluated by
integrating the beat transport equation in the deep
ocean. The problem was solved numerically for assigned
temporal evolution of the surface beat flux Q0 and
friction velocity v*. The response time (i.e. the period
of ajustment to the periodic regime) amounted to
150-200 years for the deep ocean and to 15-20 years
for the upper layer. The mean depth of the upper layer
obtained was close to 200 m. The effect of the variability
in ha and Ta on h and T 0 appeared to be more or less
insignificant.
(31)
X
E dz
The approximately universal behaviour of the vertical
temperature profile in the seasonal thermocline, as
found by Kitaigorodskii and Miroposky (1970), is very
convenient for simulation of the evolution of the upper
layer and, in particular, for its parameterization in
numerical models of large-scale oceanic circulation. At
the same time, however, the temperature distribution
in certain cases (more frequently in the polar regions),
appears to be inconsistent with Equation (2). To describe
such cases, upper layer models with continuous temperature proflle at depth z = h were constructed, using
hypotheses which relate' the beat flux Q 1z=h+o=Qh+o
to the temperature gradient in the seasonal thermocline
-oT/oziz=h+o=r. We considera few examples of
such models without attempting to oover ali the available
literature.
(
2)
3
[(1-y)ha+yh]=const.
It might ·be interesting to compare the calculations·
based on these equations (and similar relations for
penetrative convection) with observations in the ocean,
and also with data from laboratory experiments which
revealed the formation of the "thermocline" under a
mixed layer. Thus, Equation (32) seem to be just applicable to the Linden (1975) experiments quoted above,
which verified the universal structure of the laboratory
"thermocline" (see Fig. 3), but which in contrast with
the value y= 11/15 given earlier for the ocean thermocline, yielded y= 16/25.
Using Equation (3) written for the level z=h+O in the
absence of a temperature jump here and Equation (5)
and neglecting the solar radiation flux, we obtain
oh=-(OTo+oQI
Ôt
Ot
OZ z=h+O
)r-
1
(33)
231
S. S. ZILITINKEVICH, D. V. CHALIKOV, YU. D. RESNYANSKI
If we also neglect the turbulent heat flux for z ~ h, then
Equation (33) yields the following . simple equation
describing changes in h in the case of the so-called nonpenetrative convection
oh
-Qo
-=--·
ot rh
the dependence of sea water density on salinity and
empirically allowed for certain dynamic effects which
do not occur explicitly in Equation (33). With proper
choice of K 8 , the calculations of h and T 0 for periods
of severa} months appeared to be in satisfactory agreement
with observations for the Arctic basin. A similar model
was also used for calculating the température, salinity
and depth of the homogeneous layer in low latitudes
(Doronin et al., 1974).
(34)
considered by Zubov (1945) and later by many authors
(see, for instance, Bulgakov, 1975). The prediction
equation similar to Equation (34), in effect, is inferred
implicitly in convective adjustment schemes which are
widely used in numerical models of atmospheric and
oceanic general circulation (see Manabe et al., 1965;
Zilitinkevich, Monin, 1971; Zilitinkevich et al., 1978).
The possibility of applying Equation (37) to heat transport in a fluid with intermittent turbulence has been
discussed in Garnich (1975), where the value of K 8 was
assumed to be an effective conductivity due to the
evolution of turbulent sublayers. Equation (37) in this
paper was used to calculate the temperature field in
the thermocline, together with application of the model
similar to Kraus and Turner (1967) for description of
the upper homogeneous layer. Patching at the Ievel
Z=h involved the condition Qh+o=K8 r=O i.e. it was
assumed that the heat flux at the Iower boundary of a
mixed layer is due entirely to entrainment and is expressed
by Equation (6). A decay of temperature variations
was assigned for z--+ oo, as weil as in Doronin (1969).
This model exhibited an appreciable dependence of
the changes in the depth of a mixed layer after two
successive storms on the evolution of the temperature
profile in the seasonal thermocline: the difference in
the values of h calculated with and without allowance
for the variations of T in the thermocline reach~d 30%-
F.or values of h which are not too large in comparison
with the depth of a layer with purely wind-driven mixing,
the heat transfer at the lower boundary of the homogeneous layer in sorne cases cannot be assumed to be
negligibly small. Taking this into account, Kalatsky
(1973) used the traditional semi-empirical hypothesis
Qh+o=K8 r (where K 8 is the turbulent conductivity)
together with the relation
(au)-2
Rijz=h+O =f3r =Rie.
OZ z=h+O
(35)
A somewhat similar approach, which is also based on
a combination of the Kraus and Turner type model for
the upper homogeneous layer and the heat transfer
Equation (37) for the seasonal thermocline is applied
in Niiler (1977) and Kosnyrev et al. (1976). In the latter,
it involved, for oh/ ot ~ 0 the same patching conditions
for z = h as in Garnich (1975), and for the case of decreasing
depth of a mixed layer (oh/ ot < 0) it was assumed that
Qh=O, Qh+o= -'OT oh/ ot. This mode! could thus be
used for simulating the destruction of the temperature
jump below the mixed layer in the case of wind weakening.
In Arsenyev and Felzenbaum (1975), the Tennekesand-Zilitinkevich-type-model for the upper homogeneous
layer was combined once again with the equation of
heat transfer for the seasonal thermocline. It was used
here in place of assigning the poorly known and certainly
variable turbulent conductivity KH, the approximation
of the temperature profile by a polynomial to the second
power, with parameters to be defmed in terms of the
boundary conditions. The joint model took into account
the vertical beat flux below the mixed layer Qh+O (defmed
using the budget equation for temperature fluctuations),
whereas the heat flux at the lower boundary of the
homogeneous layer was treated in a more general
form than Equation (6)
Here, Ri is the Richardson number, Rie is its critical
value assumed to be constant, u is the drift current
velocity vector. The latter was determined from the
steady state Ekman equations using the eddy viscosity
being constant with depth but dependent on the wind
velocity. As in Zubov (1945) model, the heat flux divergence è QI è z 1 z=h+o was assumed to be zero. As a
results, Equation (33) reduced to the form
8[ f3Q0(au)- 2 J
at= h
l- RieKH oz z=h+O '
oh
K
(36)
where the value Ou/ oz 1z=h+o was expressed in the
usual manner in terms of the homogeneous layer depth h
and wind velocity. This equation does not take into
account the effect of temperature changes in the water
below on the variability of h.
A more detailed model follows from Equation (33) if
the time-dependent temperature profile in the seasonal
thermocline is to be chosen using the semi-empirical
heat transfer equation
a or
ot oz oz
or
- = - K8 -
for
Z>
h.
•(37)
(38)
Integration of Equations (37) and (33), provided that
T-+const for Z-+oo and Qh=Qh+ 0 =K8 r, was performed by Doronin (1969, 1975). He took into account
In subsequent papers by Arsenyev et al. (1976 a, b), this
model was used fu simulation of the resj:>onse of a stratified ocean to forcing by a typhoon with a ftxed centre.
232
MODELLING THE OCEANIC UPPER LAYER
formula, in place of Equation (41)
SIMULATION OF THE UPPER-LAYER EVOLUTION BASED ON SEMI-EMPIRICAL THEORY OF
TURBULENCE
1oc
Another approach applied to simulation of the formation
and growth of a mixed layer against the background
of a stahly stratified fluid is associated with local closure
hypotheses traditional in the semi-empirical theory of
turbulence (in contrast to bulk hypotheses conceming
the entire mixed layer in the models discussed in Sections 2 and 3). Thus, there were popular calculations
(~ainly in atmospheric problems) based on the following
sunple linear relations between turbulent fluxes and mean
gradients
au:
t=-KM-,
oz
oT
oz
oe
oz
Q=-KH-· E=-KE-·
(39)
A more detailed closure scheme including, apart from
the turbulent energy budget Equation (4), the prediction
equation for the energy dissipation rate E was used by
the group in the Computing Centre of the Siberian
Division of the Ussr Academy of Sciences (see Kochergin
et al., 1974, 1976; Marchuk et al., 1976). This madel
allowed for the difference between the eddy viscosity KM
and heat conductivity KH, the former being defined
by the equation KMoce 2 /& which follows from Equation (40) and the latter being determined using the
Munk and Andersen (1948) equation
e312
J '
(40)
where 1 is the mixing length or turbulence length scale.
The first example of applying the closure assumptions (39),
(40) in geophysical fluid dynamics was apparent! y offered
by Manin (1950) who proposed a madel of a neutrally
stratified planetary boundary layer of an atmosphere
where, in the light of Prandtle (1932) theory, it was
assumed that locz. In a stratified fluid, there is no reason
to consider 1as a linear function of z. Taking this fact into
account and considering, in the first place, atmospheric
applications, Zilitinkevich and Laikhtman (1965) suggested the following expression for l
loc'I'
O'P)-1,
(fu
e
'I'=1'
(42)
As in the case of Equations (39)-(41), ali the empirical
constants were defined through comparison of the
specifie solutions with reliable experimental data obtained
for the corresponding simple flows. Figure 8 illustrates
the Mellor and Durbin calculations of the evolution
of the temperature profile which was assumed to be
linear at the very beginning and to be forced by the
tangent stress Pw t=2dyn/cm 2 occurring instantly at
the water surface. The sequence of the temperature
profiles shawn here resembles to a great extent the
real development of a mixed layer in the ocean or in
laboratory.
(KM, KH and KE are, correspondingly, the turbulent
viscosity, heat conductivity and diffusivity for kinetic
energy perturbations) together with the turbulent energy
budget Equation (4) and Kolmogorov (1942) approximate
similarity hypotheses
EOC-
J~ ezdz(f~ edz r~.
KH=KM(1+10Ri) 1 1 2 (1- 1~Rir •
312
(43)
where Ri is the Richardson number [see Equation (35)].
Figure 8
Calculated evolution of the temperature profile T forced by the
turbulent stress on the ocean surface p r:=2dyn/cm 2 (Mel/or and
Durbin, 1975).
"'
The numbers at the curves indicate values of nondimensional time
2rt ft.
(41)
which. re~uces to the well-known von Karman (1930)
equation rn the case of neutral stratification.
Applications of the closure hypotheses (39)-(41) to the
ocean ~ay weil be illustrated by Laikhtman (1966, 1970)
calculabons of drift currents in a joint madel of the
atmospheric and oceanic boundary layers and his
unpublished calculations of the thermal structure of
the oceanic upper layer, and also in Romanov (1976 a, b)
who analysed the vertical structure of the boundary
layer under the ice cover.
Mellor and Durbin (1975) used a more or less similar
· version of closure of the equations of motion and heat
transport which, in contrast to the scheme mentioned
above, neglect the terms oe/ ot and - oE/ oz in Equation (4), using instead of Equation (40) for KM and KH .
more detailed expressions [according to which the nondimensional ratios KM(e 1 12 1)- 1 and KH(e11 2 1)- 1 are
assumed to be universal functions of the Richardson
number rather than constants], and applying a simpler
233
zm
0 .------r----
100 L----L---'-----_.1...------l
~
6
S. S. ZILITINKEVICH. D. V. CHALIKOV. YU. D. RESNYANSKI
This madel was used for performing a number of numerical experiments including calculation of the deepening
of the upper homogeneous layer during the passage
of storm (Marchuk et al., 1976), simulating the observations by Halpern (1974). The results given in Figure 9
appeared to be in good agreement with observations.
According to the calculations and also to the real data,
the deepening of a mixed layer began about six hours
later than wind strengthening (at t=O), continued for
about 24 hours and led to a growth of the homogeneous
layer depth from 18 to 25 m.
The most detailed calculations simulating the development of a mixed layer were performed on the basis
of the so-called "second-order closing" mainly applied,
however, to atmospheric boundary layers or laboratory experiments rather than to oceanic conditions.
One may consider as an atmospheric example of the
approximately homogeneous layer a convective boundary
layer occurring after sunrise and associated with the
entrainment of stahly stratified air above into the mixed
region through the upper boundary, which, in turn,
entails the growth of the mixed layer with time. This
phenomenon is completely similar to the deepening of
the homogeneous oceanic upper layer forced by the
downward mass flux at the surface (due to cooling or
increasing salinity at the surface).
The second-order closure hypotheses, together with the
usual statistically averaged equations of motion and
heat transport, involve a set of equations for ali the
single-point second-order moments of turbulent fluctuations, i. e. for dispersions and turbulent fluxes and
also for the dissipation rates of the kinetic energy of
turbulence e and of the r.m.s. temperature fluctuations eT (one of the first attempts in using such an approach
for the case of a stratified atmospheric boundary layer
is given by Monin, 1965). ln this way, there arises the
necessity of defming the third moments of turbulent
fluctuations appearing in the equations for second-order
values. To describe these third-order fluxes, Wyngaard
and Coté (1975), Lewellen and Tèske (1975), Yamada
and Mellor (1975) among others, used the simplest
flux-gradient relations [relevant to the turbulent energy
expressed by the third formula in Equation (39) ]. Zeman
and Lumley (1976) have discovered, however, that this
traditional idea appears to be not quite applicable to
the case of entrainment, and proposed a more detailed
mode! using the appropriate prediction equations
(Lumley, 1975 a, b; Lumley et al., 1978; Siess, 1977)
to define the third-order moments of the velocity field.
The fourth-order moments appearing in these equations
(except for the pressure terms) were described in terms
of second-order moments applying the Millionschikov
( 1941 a. h) hypothesis . concerning the vanishing of
the fourth-order semi-invariants.
The pressure terms were divided into two parts: the
return-to-isotropy part associated with internai interactions within the turbulent field and expressed by the
traditional approach (Rotta, 1951; Davydov, 1959;
Lumley, Khajeh-Nouri, 1974; Lumley, 1975 a, b), and
the "rapid" part which reflects the turbulence-mean
flow interactions, the latter being expressed according
to Zeman (1975). Equations fore and eT were 'constructed
by means of invariance reasons (Lumley, Khajeh-Noury,
1974; Lumley, 1975 a; Zeman, 1975).
Figure 10
Evolution of the vertical profile of the kinematic heat flux Q in the
case ofpenetrative convection calculated by Zeman and Lumley(1976)for
typical atmospheric values of external parameters.
The initial depth of the mixed layer is taken to be h 0 = 200 rn; the
kinematic heat flux through the surface Q0 =0.15 Km. s- 1 is
assumed to be constant with time; the potential temperature gradient
outside the mixed layer is taken to be equal to O. 01 KI m. The numbers
at the curves indicate values of nondimensional time t (~ Q0 h0 ) 1 i 3 h0 1 •
Nondimensional heat flux Q/Q 0 and nondimensional height z/h 0
are correspondingly plotted on the horizontal and vertical axes.
Figure 9
Development of the temperature profile during the passing of a storm,
from calculations by Marchuk et al. (1976).
Steady-state solution at t=O is marked by zero. Numbers 1, 2, 3, ...
indicate successive profiles after 6-hour time intervals. At the
moment "5", wind reached a maximum value of 15 m.s- 1 ; at the
moment "10", itfell to 5 m.s- 1
zjh 0
2
1
0
234
MODELLING THE OCEANIC UPPER LAYER
Using their mode!, Zeman and Lumley (1976) investigated the behaviour of a convective mixed layer at
a given value of its initial depth h0 and heat flux at
the surface Q0 varying other parameters (e. g. the temperature gradient below the mixed layer). The paper
also involved numerical simulation of Willis and Deardorff (1974) laboratory experiments on penetrative
convection. Typical values of the parameters of the
mode! were used here to simulate, in particular, qualitatively realistic vertical profiles of the turbulent heat
flux with pronounced negative values in the upper
part of the layer associated with the heat fluxes due to
entrainment (see Fig. 10).
1t is of fondamental importance that the vertical flux
of the vertical fluctuation energy in the calculations
appeared to be positive nearly ali over the mixed layer
(except for a very thin region near the surface), although
the r.m.s. vertical velocity attained maximum approximately in the middle of the layer. Such behaviour appears
to be in contradiction with the usual concept of gradient
transport (roughly speaking, in the lower half of the
convective boundary layer, the kinetic energy flux is
directed towards the region with higher energy), and
is consistent with available experimental data (see
Figs. 11 and 12 and also Townsend, 1956; Adrian, 1975).
This phenomenon could not be described in the light
of the gradient transport hypothesis, which led, in
previous models (Wyngaard, Coté, 1975; Lewellen,
Teske, 1975; Yamada, Mellor, 1975), to underestimation
of the heat fluxes due to entrainment at the outer boundary of a mixed layer.
Simulation of mixing and entrainment directly relevant
to the oceanic upper layer and based on the similar
second-order closing remains a matter of future concern.
Figure 11
Comparison with experimental data of the calculated vertical profile
Figure 12
Comparison of the calculated profile of the total turbulent energy
APPLICATION OF THE LOCAL MODELS TO
SIMULATION OF THE OBSERVED STRUCTURE
AND PARAMETERIZATION OF THE OCEANIC
UPPER LAYER
1t is convenient to test the models discussed above
by comparing appropriate calculations with the corresponding laboratory experiments. A review of experiments
which are of interest from this point of view may be
found in Long (1975), Turner (1973) and Phillips (1977).
It should not be forgotten, however, that the main
practical application of the models in question is to
simulate the variability of the oceanic upper layer
characteristics in specifie situations, in terms of empirical data at the water surface. Finally, sufficiently
simple and at the same time realistic models must
serve as the basis for parameterization of the active
layer in numerical models of the oceanic global circulation.
The first of the models discussed (Kraus, Turner, 1967)
appeared to be in satisfactory agreement with the
w; 3 with empirical data on the vertical
(e' w') w;.3 (Zeman, Lumley, 1976).
vertical flux (e' w' + p' w' / p)
of the vertical velocity fluctuations energy -;72"'normalised by the
convective scale w*=(~Q 0 h)tf3.
1, Zeman and Lumley (1976) theoretical curve; 2 and 3. cases S 1
and S 2, correspondingly, in Wil/is and Deardorff (1974) /ahoratorr
experiments; 4, 5, 6, atmospheric data by Lenschow (1970, 1974),
Lenschow and Johnson (1968), Telford and Warner (1964), correspondingly.
zjh
1.2
flux of kinetic energy
For the notation, see Figure 11. ln Zeman and Lumley (1976),
w'
e' + p' w' /p =3/5 e' w' so that for comparison with experimental
data, theoretical curve 1 should be stretched along the horizontal
axis with a ratio of 5/3.
_,
--2
z/h
Â-Â3
0
4
A
s
x 6_
0.8
235
S. S. ZILITINKEVICH, O. V. CHALIKOV, YU. O. RESNYANSKI
laboratory experiment by Turner and K.raus (1967),
in which the periodic entry of a light fluid through the
surface (corresponding to the periodic behaviour of the
beat flux) occurred at constant mixing rate. This mode!
described only qualitatively the real annual evolution
of T 0 and h in the ocean. A more detailed discussion
of a similar mode! (Denman, 1973) including, in particular, the specification of the empirical constants was
performed by Denman and Miyake (1973), who considered the synoptic scale variations of T 0 and h in
June 1970 on Ocean Station "Papa". In these calculations, it was even possible to simulate (against the
background of a structure formed earlier) the occurrence
of a secondary thinner homogeneous layer, which was
actually observed during periods of wind weakening
(Fig. 13). Satisfactory agreement of the calculations
with real data was apparently due to successful choice
of the initial state when the depth of the homogeneous
layer exceeded 25-30 m, so that the stage of rapid
deepening, which was poorly described by the model,
was excluded from the consideration (Niiler, 1975). In
addition, the boundary condition parameters (i.e. Q 0
and v*) during the period considered varied within
comparatively narrow limits; consequently the deviations from the linear dependence between the turbulent
energy production and dissipation rates used in the
mode! were not particularly noticeable.
A similar comparison of different homogeneous layer
models was performed by Thompson (1974, 1976) who
calculated seasonal changes in T 0 and h using the data
of Ocean Station N (in the Pacifie Ocean). The mean
annual correlation between T 0 (and h) calculated according to Pollard et al. (1973) and measured values of T 0 ·
(and h) was equal to 0.98 (and 0.8). Calculations based
on Denman (1973) and Warren (1972) models appeared
to be somewhat inferior. A qualitatively reasonable
behaviour of T 0 was obtained even in a very simple
model which provides constant depth of the mixed
layer (h= 15 m) and negligible beat flux at its lower
boundary. At the same time, none of the models could
Concerning the parameterization of the upper layer in
large-scale oceanic circulation modelling, we may refer
to Kamenkovich and Kharkov (1975) and Holland (1977)
quoted earlier and in application to the global air-sea
interaction, to the numerical experiments by the Geophysical Fluid Dynamics Laboratory (see, e. g. Bryan
et al., 1975) involving use of Kraus and Turner (1967)
mode!, and to the similar numerical experiments by the
Institute of Oceanology of the Ussr Academy of Sciences
(see Zilitinkevich et al., 1976, 1978), in which the
Kraus-and-Turner-type-model was used in conjunction
with Equation (30) for the heat flux due to entrainment.
Figure 14 taken from Zilitinkevich et al. (1978) shows
Figure 13
Variation with time in the ocean surface temperature T 0 (a) and
evolution of the vertical temperature profile T in the upper layer of
the ocean {b) after the data of Ocean Station "Papa" and calculations
by Denman and Miyake (1973).
Plotted on the horizontal axis in the upper graph is the time 13 through
24 June 1970. Dotted curves-experimental data; so/id curvescalculation.
be used to simulate satisfactorily high-frequency changes
in the water temperature. This may be explained both
in terms of inaccurate boundary data (highest errors
occur in the values of beat flux at the ocean surface Q0
calculated using rough empirical formulas) and also of
horizontal advection, together with certain other processes which were not taken into account.
Seasonal variations in the characteristics of the homogeneous oceanic upper layer have been calculed in
Kitaigorodskii and Miropolsky (1970), Miropolsky (1970),
Kamenkovich and Kharkov (1975) quoted above in this
paper and also in Rusin (1973), Lacombe (1974), Arsenyev
and Felzenbaum (1976), Kalatsky and Nesterov (1976),
Resnyansky (1976), and Kharkov (1977). The June issue
of Ocean Mode/ling ( 1977) is en ti rely devoted to these
calculations. In almost ali cases, the authors were
successful in reaching satisfactory agreement of the
variations in T 0 and h obtained with observations. It
should be remembered, however, that these calculations
are performed only for a few chosen points in the ocean
(mainly for Ocean Station "Papa"). Ali of the existing
models, therefore, should be estimated as being far
from completion, especially when account is taken of
the fact that empirical constants in the models were
often defined by means of the same data as those used
for comparison ofthe theoretical results with calculations.
-1
--2
T°C
8.5~
7.5t
ZM
0
a
-
~-~---'~~~----~--~----L---~--~----~---L----L----L--~
1'+
16
18
T°C
7 9
20
22
24
8
roc
7 9
'!l!!lll!JllllllllllJJJJ
236
MODELLING THE OCEANIC UPPER LAYER
Figure 14
Mean January field of the depth h of the upper homogeneous layer
of the ocean according to the numerical experiment with global
air-sea interaction mode! (Zilitinkevich et al., 1978).
Shading is used to indicate the continents represented roughly in
the mode/.
an example of calculated depth of the upper homogeneous layer in the World Ocean.
the depth of the homogeneous layer is dependent on
the cells (in particular, on their size) or whether on the
contrary, the cell properties are determined by the
vertical temperature profile, the latter being formed
as a result of turbulent mixing.
DISCUSSION
Finally, the assumption of horizontal homogeneity usual~y
applied is rare/y realized. Ocean response to atmosJ?henc
The formation and evolution of the oceanic upper layer
is determined by many physical phenomena. 7he most
effects occurs more often in the form of a comphcated
three-dimensional picture (for example, during the
passing of hurricane, see Fyodorov, 1972). Numerical
experiments performed with comparatively simple models
(Eisberry et al., 1976; Arsenyev et al., 1976 a, b) reveal
a significant role of largescale vertical motions (associated with the horizontal velocity divergence) in the
formation of the thermal structure observed in the
ocean. To allow for these motions, it is necessary to
consider not only the active layer, but also the largescale
circulation of the ocean in response to changes in the
externat conditions. Numerical simulation of oceanic
circulation for different scales with appropriate parameterization of air-sea interaction using the oceanic
active layer models is yet another practically interesting
problem for examination in the near future.
One fundamental problem which can hardly be advanced
by means of traditional semi-empirical approaches
important phenomenon appears to be turbulent mixing
due to local atmospheric forcing on the water surface.
The models discussed in our review are known to be
approximate, since they are based on semi-empirical
hypotheses with a limited field of application, which
require careful experimental testing. So far, there exists
no unified theory which would describe the formation
and development of the homogeneous layer together
with the seasonal thermocline forced by the limedependent conditions at the ocean surface. Moreover,
investigation of the oceanic upper layer in sorne respects
lags behind similar atmospheric research. Thus, we
have already noted p. 233-235 that no attempts to
simulate the evolution of the oceanic upper layer have
been made on the basis of detailed second-arder closures.
7hree-dimensional modelling of the oceanic upper layer
similar to Deardorff (1974) numerical experiments on
atmospheric boundary layer development is also lacking.
concerns the development of a theory explaining the
vertical structure and mechanism of vertical transport
in the seasonal thermocline. In such a stably stratified
This gap will apparently be filled in the near future.
A promising problem which has no atmospheric analo~e
fluid, a significant role may certainly be played by
internai waves which, together with the intermittent
turbulence which occurs here, are capable of realizing
momentum and energy transport (see, e. g. Wallace
and Kousky, 1968). It was shown in Voronovich et al.
(1976) quoted earlier in this paper that a microstructure
may develop against the background of the statically
stable mean density profile. No adequate description
is so far available of the mechanism of formation of
the density profile itself as a result of the contribution
of both turbulence and internai wave propagation to
beat and salt vertical fluxes.
and which is quite amenable to solution could constst
in numerical simulation of the mixed layer deepening
forced by the surface wave breaking, based on twodimensional non-steady equations similar to those used
in Chalikov (1976, 1978) numerical mode/ ofwind generation by turbulent wind.
The effect of Langmuir (1938) circulations on the structure
of the oceanic upper layer is not yet quite clear. According
to observations, the sizes of circulation cells in the
homogeneous upper layer are closely related to its
depth (Assaf et al., 1971). It remains to be seen whether
237
S. S. ZILITINKEVICH. D. V. CHALIKOV, YU. D. RESNYANSKI
The ever-growing interest in investigation of the oceanic
upper layer (revealed, in particular, in the sharp increase
in the number of relevant publications) leads us to
expect that many of the problems referred to here as
unsolved will be clarified in the near future.
Doronin Yu. P., 1975. Modelling of the active layer of freezing
seas as a consequence of their interaction with atmosphere, Tr.
Leningr. Gidrometeoro/. Inst .• 57, 5-16.
Doronin Yu. P., Karlin L. N., Balyasnikov S. B., 1974. Mathematical modelling of the formation of the thermocline as a consequence of air-sea interaction, Tr. Arkt. Antarkt. Nauchno-Jssled.
Inst., 316, 69-75.
Elsberry R. L., Fraim T. S., Trapnell R. N., Jr., 1976. A mixed
layer mode! of the oceanic thermal response to hurricanes, J. Geo·
phys. Res., 81, 6, ll53-ll62.
Filyushkin B. N., 1968. The thermal characteristics of the upper
layer of water in the North Pacifie Ocean, Okeanol. Jss/ed., 19,
Nauka, Moscow, 22-69.
Francis 1. K. D., Stommel H., 1953. How much does a gale mix
the surface layers of the ocean?, Q.J.R. Meteorol. Soc., 73, 342,
534-536.
Fyodorov K. N., 1972. The behaviour of the upper active layer
of the ocean un der the influence of tropical hurricanes and typhoons,
Okeanologiya, 12, 3, 387-393.
Fyodorov K. N., 1976. The Fine-Sca/e Thermohaline Structure of
the Ocean, Gidrometeoizdat, Leningrad, 184 p.
Garnich N. G., 1975. A mode! of continuous evolution of seasonal
thermocline, Okeanologiya, 15, 2, 233-238.
Gamich N. G., Kitaigorodskii S. A., 1977. On the rate of a mixed
layer deepening, Izv. Akad. Nauk Sssr, Fiz. Atmos. Okeana, 13,
12, 1287-1295.
Garwood R. W., 1977. An oceanic mixed layer mode! capable
of simulating cyclic states, J. Phys. Oceanogr., 7, 3, 456-468.
Gill A. E., Turner J. S., 1976. A comparison of seasonal thermocline models with observations, Deep-Sea Res., 23, 5, 391-401.
Gill A. E., Trefethen L. N., 1977. A mixed layer mode!, Ocean
Modelling, 5, 1-4.
Gordeyev R. G., Kagan B. A., 1976. The upper quasi-homogeneous
layer of the ocean at the initial stage of development of free convection, Meteorol. Gydrol., 10, 50-51.
Halpern D., 1974. Observation of the deepening of the windmixed layer in the Northeast Pacifie Ocean, J. Phys. Oceanogr.,
4, 3, 454-466.
Rolland W. R., 1977. The role of the upper ocean as a boundary
layer in models of the oceanic general circulation, in Modelling
and Prediction of the Upper Layers of the Ocean (Adv. Study
lnst., Urbino, ltaly, 1975), Pergamon Press, Oxford, 7-30.
Kalatzky V. 1., 1973. A two-layer mode! for calculating the depth
of the isothermallayer in the ocean, Meteorol. Gydrol., II, 60-68.
Kalatzky V. 1., Nesterov E. S., 1976. A calculation of the depth
of the homogeneous oceanic layer in the North Atlantic Ocean,
Tr. Gidrometsentra Sssr, 182, 37-42.
Kamenkovich V. M., Kharkov B. V., 1975. On seasonal changes
in the thermal structure of the oceanic upper layer, Oceanologiya,
15, 6, 978-987.
Kantha L. H., 1977. Note on the role of internai waves in thermocline erosion, in Mode/ling and Prediction of the Upper Layers
of the Ocean (Adv. Study lnst., Urbino, ltaly, 1975), Pergamon
Press, Oxford, 173-177.
Kantha L. H., Phillips O. M., Azad R. S., 1977. On turbulent entrainment at a stable density interface, J. Fluid Mech., 79, 4, 753-768.
von Karman Th., 1930. Mechanische Ahnlichkeit und Turbulenz,
Nad1r. Ges. Wiss, Güttingen, Math-Phys. KI, 5S-76.
Kato H., Phillips O. M., 1969. On the penetration of a turbulent
layer into a stratified fluid, J. Fluid Mech., 37, 4, 643-665.
Kazansky A. B., Monin A. S., 1960. On the turbulent regime
above the surface layer of air, lzv. Akad. Nauk Sssr, Ser. Geofiz.,
l, 165-168.
Kharkov B. V., 1977. On the structure of the oceanic upper layer,
Oceano/ogiya, 17, l, 37-43.
Kim J. W., 1976. A generalized bulk mode] of the oceanic mixed
layer, J. Phys. Oceanogr., 6, 5, 686-695.
Kitaigorodskii S. A., 1960. On calculation of the depth of the
wind-mixing layer in the ocean, Izv. Akad. Nauk Sssr, Ser. Geofyz.,
3, 425-431.
Kitaigorodskii S. A., 1970. The Physics of Air-Sea Interaction,
Gidrometeoizdat, Leningrad, 284 p.
Kitaigorodskii S. A., 1977 a. Oceanic surface boundary layer,
lnvited review paper at the Joel Scor Joint Study Conference on
General Circulation Models of the Ocean and their Relation to
C/imate, Helsinki, May 1977.
REFERENCES
Adrian J. R., 1975. Turbulent convection in water over ice,
J. Fluid Mech., 69, 4, 753-781.
Alexander R. C., Kim J. W., 1976. Diagnostic mode! study of
mixed-layer depths in the summer North Pacifie, J. Phys. Oceanogr.,
6, 293-298.
Arsenyev S. A., Felzenbaum A. 1., 1975. On a mode! of the quasihomogeneous layer and seasonal thermocline in the ocean, Dokl.
Akad. Nauk Sssr, 225, 1, 84-87.
Arsenyev S. A., Felzenbaum A. 1., 1976. On a simple method of
calculating thermal characteristics of the oceanic upper layer,
Dokl. Akad. Nauk Sssr, 227, 5, 1101-1103.
Arsenyev S. A., Sutyrin G. G., Felzenbaum A. 1., 1976 a. On the
response of a stratified ocean to a typhoon, Dokl. Akad. Nauk
Sssr, 231, 3, 567-570.
Arsenyev S. A., Sutyrin G. G., Felzenbaum A. 1., 1976 b. The thermal structure of the ocean in the region of a typhoon, Express
Information "Promyslovaya Okeanologiya", Ser. 9, Il, 1-7.
Assaf G., Gerald R., Gordon A. L., 1971. Sorne mechanisms of
oceanic mixing revealed in aerial photographs, J. Geophys. Res.,
76, 27, 6550-6572.
Bathen K. H., 1972. On the seasonal changes in the depth of the
mixed layer in the North Pacifie Ocean, J. Geophys. Res., 77,
36, 7138-7150.
Bryan K., Manabe S., Pacanowski R. C., 1975. A global oceanatmospheric climate model. Pt. II. The oceanic circulation.
J. Phys. Oceanogr., 5, 1, 30-46.
Bulgakov N. P., 1975. Convection in the Ocean, Nauka, Moscow,
272 p.
Burangulov N. 1., 1977. A mode! of penetrating convection with
applications to the atmosphere, ocean and laboratory experiments,
lzv. Akad. Nauk Sssr, Atmospheric and Oceanic Physics, 13, 12,
1248-1256.
Chalikov D. V., 1976. A mathematical mode! of wind-generated
waves, Dokl. Akad. Nauk Sssr, 229, 5, 1083-10S6.
Chalikov D. V., 1978. The numerical simulation of wind-wave
interaction, J. Fluid Mech., 87, 3, 561-582.
Clarke R. H., Dyer A. J., Brook R. R., Reid D. G., Troup A. J.,
1971. The Wangara Experiment: Boundary Layer Data, Division
of Meteorological Physics, Csiro Tech. Pap. No. 19, Melbourne,
Australia, 350 p.
Davydov B. 1., 1959. On statistical dynamics of a non-compressible
turbulent fluid. Dokl. Akad. Nauk Sssr, 127, 4, 768-771.
Deardorff J. W., 1973. An explanation of anomalously large
Reynolds stresses within the convective planetary boundary
layer, J. Atmos. Sei., 30, 6, 1070-1076.
Deardorff J. W., 1974. Three-dimensional numerical study of
turbulence in an entraining mixed layer, Boundary-Layer Meteorol.,
7, 2, 199-226.
Deardorff J. W., Willis G. E., Lilly D. K., 1969. Laboratory investigation of non-steady penetrative convection, J. Fluid Mech., 35,
1, 7-32.
Denman K. L., 1973. A time-dependent mode! of the upper ocean,
J. Phys. Oceanogr., 3, 2, 173-184.
Denman K. L., Miyake M., 1973. Upper layer modification of
Ocean Station "Papa·•: Observations and simulation, J. Phys.
Oceanogr., 3, 2, 185-196.
Dorman C.E., 1974. Analysis of Meteorological and Oceanographie
Data from Ocean Station Vesse! N (30"N, 140"W), Ph. D. Thesis,
Oregon State Univ., 136 p.
Doronin Yu. P., 1969. Thermal Interaction between the Atmosphere
and Hydrosphere in the Arctic, Gidrometeoizdat, Leningrad, 299 p.
238
MODELLING THE OCEANIC UPPER LAYER
Miropolsky Yu. Z., Filyushkin B. N., Chernyshov P. P., 1970.
On parameterized description of temperature profiles in the
active layer of the ocean, Oceanologiya, 10, 6, 1101-1106.
Monin A. S., 1950. Dynamic turbulence in the atmosphere, Izv.
Akad. Nauk Sssr, Ser. Geogr. Geofys., 14, 3, 232-254.
Monin A. S., 1965. On the properties of symmetry of turbulence
in the surface layer of the air, Izv. Akad. Nauk Sssr, Fiz. Atmos.
Okeana, 1, 1, 45-54.
Monin A. S., 1970. On turbulent mass fluxes in the oceans, Dok/.
Akad. Nauk. Sssr, 193, 5, 1038-1041.
Monin A. S., Obukhov A. M., 1954. Basic laws of turbulent mixing
in the surface layer of the atmosphere, Tr. Geofiz. Inst. Akad.
Nauk Sssr, 24 (151), 173-187.
Monin A. S., Yaglom A. M., 1965. Statistica/ Fluid Mn·hanics,
Part 1, Nauka, Moscow, 639 p.
Monin A. S., Fyodorov K. N., 1973. On the fine-scale structure
of the upper layer of the ocean, Izv. Akdd. Nauk Sssr, Fiz. Atmos.
Okeana, 9, 4, 442-444.
Monin A. S., Kamenkovich V. M., Kort V. G., 1974. Variability
in the World Ocean, Gidrometeoizdat, Leningrad, 262 p.
Monk W. H., Anderson E. K., 1948. Notes on the theory of the
thermocline, J. Mar. Res., 17, 3, 276-295.
Niiler P. P., 1975. Deepening of the wind-mixed layer, J. Mar.
Res., 33, 3, 405-422.
Niiler P. P., 1977. One-dimensional models of the seasonal thermocline, in The Sea: ldeas and Observations on Progress in the
Study of the Seas, Vol. 6, Wiley Interscience, New York, 97-115.
Niiler P. P., Kraus E. B., 1977. One-dimensional models of the
upper ocean, in Model/ing and Prediction of the Upper Layers
of the Ocean (Adv. Study Inst., Urbino, ltaly, 1975), Pergamon
Press, Oxford, 143-172.
Phillips O. M., 1977. Entrainment, in Model/ing and Prediction
of the Upper Layers of the Ocean (Adv. Study Inst., Urbino,
Italy, 1975), Pergamon Press, Oxford, 92-101.
Pollard R. T., Rhines P. B., Thompson R. O. R. Y., 1973. The deepening of the wind-mixed layer, Geophys. Fluid Dyn., 3, 4, 381-404.
Prandtle L., 1932. Meteorologische Anwendungen der Stromungslehre, Beitr. Phys. fr. Atmos., 19, 3, 188-202.
Reshetova O. V., Chalikov D. V., 1977. On the universal structure
of the active layer of the ocean, Oceanologiya, 17, 5, 774-777.
Resnyansky Yu. D., 1975. On parameterization of the bulk dissipation of turbulent energy in the upper quasi-homogeneous
layer of the ocean, Izv. Akad. Nauk Sssr, Fiz. Atmos. Okeana,
11, 7, 726-733.
Resnyansky Yu. D., 1976. On calculation of seasonal changes
in the depth and temperature of the upper quasi-homogeneous
layer of the ocean, Tr. Gidrometsentra Sssr, 182, 27-36.
Romanov V. F., 1976 a. Turbulent regime in a stratified sub-ice
boundary layer of the ocean, Oceanologiya, 16, 1, 32-40.
Romanov V. F., 1976 b. Pararnerization of the nonlinear smallscale air-sea interaction ta king into account the ice cover, Meteorol.
Gydrol., 12, 49-59.
Rossby C. G., Montgomery R. B., 1935. The layer of frictional
influence in wind and ocean currents, Pap. Phys. Ocean. Meteorol.,
3, 3, 1-101.
Rotta J. C., 1951. Statistische Theorie nicht-homogener Turbulenz,
Arch. Phys., 129, 547-572.
Rusio 1. N., 1973. On a method of calculating the depth of the
quasi-homogeneous layer of the ocean, Tr. Gl. Geojiz. Obs.,
315, 123-138.
Siess M. J., 1977. Un modèle numérique de pénétration par la
turbulence d'une couche stratifiée stable, Ann. Hydrogr., 5, 1,
746, 189-198.
de Szoeke R. A., Rhines P. S., 1976. Asymptotic regimes in mixedlayer deepening, J. Mar. Res., 34, 1, 111-116.
Telford J. L., Warner J., 1964. Fluxes of heat and vapor in the
lower atmosphere derived from aircraft observations, J. Atmos.
Sei., 21, 5, 539-548.
Tennekes H., 1973. A mode! for the dynamics of the inversion
above a convective boundary layer, J. Atmos. Sei., 30, 4, 558-567.
Tennekes 11., 1975. Reply, J. Atmos. Sei., 32, 5, 992-995.
Tennekes H., Lumley J. L., 1972. A First Course in Turbulence,
Cambridge, MIT Press, 300 p.
Thompson R., 1974. Prediction of the characteristics of the wellmixed layer, Woods Hole Oceanographie Institution, Tech. Rep.,
WHOI-74-82, Woods Hole, 76 p.
Kitaigorodskii S. A., 1977 b. The dynamics of the upper thermocline in the ocean, Itogi Nauki, Viniti, Ser. Oceano/ogiya, 4, 6-34.
Kitaigorodskii S. A., Miropolsky Yu. Z., 1970. On the theory
of the active layer of an open ocean, Izv. Akad. Nauk Sssr, Fiz.
Atmos. Okeana, 6, 2, 178-187.
Kitaigorodskii S. A., Garnich N. G., 1979. Asymptotic regimes
in mixed layer deepening, J. Phys. Oceanogr., in press.
Kochergin V. P., Sukhorukov V. A., Tsvetova E. A., 1974. Modelling
the processes of vertical turbulent diffusion in the ocean, in Numerical Methods in Calculating Oceanic Currents, Novosibirsk,
1:29-15:2.
Kochergin V. P., Klimok V. 1., Sukhorukov V. A., 1976. A turbulent
mode! of the Ekman layer of the ocean, Numerical Methods in
Mechanics of Continuous Medium, 7, 1, 72-84.
Kolmogorov A. N., 1942. Equations of turbulent motion of a noncompressible fluid, Izv. Akad. Nauk Sssr, Ser. Fiz., 6, 1-2, 56-58.
Kosnyrev V. K., Kuftarkov Yu. M., Felzenbaum A. 1., 1976. A onedimensional asymptotic mode! of the oceanic active layer, Dokl.
Akad. Nauk Sssr, 228, 1, 70-72.
Kraus E. B., Turner J. S., 1967. A one-dimensional mode! of
the seasonal thermocline: Pt. II. The general theory and its consequences, Tellus, 19, 1, 98-106.
Lacombe H., 1974. Modèles simples de prévision de l'état thermique de la mer et de l'immersion de la thermocline, Ann. Hydrograph., 2, N 741, 3-21.
Laikhtman D. L., 1966. The dynamics of boundary layers in the
atmosphere and the sea taking into account interaction and nonlinear effects, Izv. Akad. Nauk. Sssr, Fiz. Atmos. Okeana, 2, 10,
1017-1025.
Laikhtman D. L., 1970. The Physics of the Atmospheric Boundary
Layer, Gidrometeoizdat, 253 p.
Langmuir J., 1938. Surface motion of water induced by wind,
Science, 87, 119-123.
Lenschow D. H., 1970. Airplane measurements of planetary boundary layer structure, J. Appt. Meteorol., 9, 6, 874-884.
Lenschow D. H., 1974. Mode! of the height variation of the turbulent kinetic energy budget in the unstable planetary boundary
layer, J. Atmos. Sei., 31, 2, 465-474.
Lenschow D. H., Johnson W. B., Jr. 1968. Concurrent airplane
and balloon measurements of atmospheric boundary-layer structure over a forest, J. Appt. Meteoro/., 7, 1, 79-89.
Lewellen W. S., Teske M., 1975. Turbulence modeling and its
application to atmospheric diffusion: Pt. 1. Environmental Monitoring Series, TSI-EPA-600/4-75-016 a.
Linden P. F., 1975. The deepening of a mixed layer in a stratified
fluid, J. Fluid Mech., 71, 2, 385-405.
Long R., 1970. A theory of turbulence in stratified fluids, J. Fluid
Mech., 42, 2, 349-365.
Long R., 1975. The influence of shear on mixing across density
interfaces, J. Fluid Mech., 70, 2, 305-320.
Lumley J. L., 1975 a. Pressure-strain correlation, Phys. Fluids,
18, 6, 750.
Lumley J. L., 1975 b. Prediction methods for turbulent flows,
Notes, von Karman Institute, Rnode-Saint-Genèse, Belgium,
March 3-7, 1975.
Lumley J. L., Khajch-Nouri B., 1974. Computational modeling
of turbulent transport, in Turbul. Diffus. Environ. Po/lut. (Adv.
Geophys., Vol. 18 a). New York e. a., 169-192.
Lumley J. L., Zeman O., Siess J., 1978. The influence of buoyancy on turbulent transport, J. Fluid Mech., 84, 3, 581-597.
Manabe S., Smagorinsky J., Strickler R. F., 1965. Simulated
climatology of a general circulation mode! with a hydrologie
cycle, Mon. Weather Rev., 93, 12, 769-798.
Marchuk G. 1., Kochergin V. P., Klimok V. 1., Sukhorukov V. A.,
1976. The dynamics of the homogeneous layer in the ocean, Computing Centre, Siberian Division Acad. Sei. Ussr, Preprint, 11, 18 p.
Mellor G. L., Durbin P. A., 1975. The structure and dynamics
of the ocean surface mixed layer, J. Phys. Oceanogr., S, 4, 718-728.
Millionschikov M. D., 1941 a. On the theory of a homogeneous
isotropie turbulence, Dokl Akad Nauk Sssr, 32, 9, 611-614.
Millionscshikov M. D., 1941 b. On the theory of a homogeneous
isotropie turbulence, Izv. Akad. Nauk Sssr, Ser. Geogr. Geofys.,
s, 4-5, 433-446.
Miropolsky Yu. Z., 1970. A non-steady mode! of convective
and wind-generated mixing in the ocean, Izv. Akad. Nauk Sssr,
Fiz. Atmos. Okeana, 6, 12, 1284-1294.
239
S. S. ZILITINKEVICH. D. V. CHALIKOV. YU. D. RESNYANSKI
Thompson R., 1976. Climatological numerical models of the
surface mixed layer of the ocean, J. Phys. Oceanogr., 6, 4, 496-503.
Thomson S. M., Turner J. S., 1975. Mixing across an interface
due to turbulence generated by an oscillating grid, J. Fluid Mech.,
67, 2, 349-368.
Townsend A. A., 1956. The Structure of Turbulent Shear Flow,
Cambridge University Press, 315 p.
Turner J. S., 1973. Buoyancy Effects in Fluids, Cambridge University Press.
Turner J. S., Kraus E. B., 1967. A one-dimensional mode! of the
seasonal thermocline: Pt. 1. A laboratory experiment and its
interpretation, Tellus, 19, 1, 88-97.
Voronovich A. G., Leonov A. 1., Miropolsky Yu. Z., 1976. On the
theory of formation of the fine-scale structure of geophysical
fields in the ocean, Oceanologiya, 16, 5, 750-759.
Wallace J. M., Kousky V. E., 1968. Observational evidence of
Kelvin waves in the tropical stratosphere, J. Atmos. Sei., 25,
5, 900-907.
Warren B., 1972. Insensitivity of subtropical mode water characteristics to meteorological fluctuations, Deep-Sea Res., 19,
1, 1-19.
Willis G. E., Deardorff J. W., 1974. A laboratory mode! of the
unstable planetary boundary layer, J. Atmos. Sei., 31, 5, 1297-1307.
Woods J. D., 1968. An investigation of sorne physical processes
associated with the vertical flow of heat through the upper ocean,
Meteorol. Mag., 97, 1148, 65-72.
Wyngaard J. C., Coté O. R., 1975. The evolution of the convective planetary boundary layer. A higher-order closure model
study, Boundary-Layer Meteorol., 1, 3, 289-308.
Yamada T., Mellor G., 1975. A simulation of the Wangara atmospheric boundary layer data, J. Atmos. Sei., 32, 12, 2309-2329.
Zeman O., 1975. The Dynamics of Entrainment in the Planetary
Boundary Layer: A Study in Turbulence Modeling and Parameterization, Ph. D. thesis, Pennsylvania State Univ.
Zeman O., Lumley J. L., 1976. Modelling buoyancy driven mixed
layers. J. Atmos. Sei., 33, 10, 1974-1988.
Zeman O., Tennekes H., 1976. Parameterized energetics of boundary-Iayer growth in the atmosphere and oceans, Third Symposium
on Atmospheric Turbulence, Diffusion and Air Quality, October
26-29, 1976, Raleigh, N.C., American Meteorologica1 Society,
Boston, Massachusetts.
Zilitinkevich S. S., 1975. Comments on "A model for the dynamics
of the inversion above a convective boundary layer", J. Atmos. Sei.,
32, 5, 991-995.
Zilitinkevich S. S., Laikhtman D. L., 1965. Turbulent regime in
the surface layer of the atmosphere, Izv. Akad. Nauk Sssr, Atmospheric and Ocean Physics, 1, 2, 150-156.
Zilitinkevich S. S., Monin A. S., 1971. Turbulence in Dynamic
Models of the Atmosphere, Nauka, Leningrad, 44 p.
Zilitinkevich S. S., Monin A. S., Turikov V. G., Chalikov D. V.,
1976. Numerical simulation of the joint air-sea circulation, Dokl.
Akad. Nuuk Sssr, 230, 3, 556-559.
Zilitinkevich S. S., Monin A. S., Chalikov D. V., 1978. Air-sea
interaction, in Hydrophysics of the Ocean, Nauka, Moscow,
208-339.
Zubov N. N., 1945. The Arctic lee, Izdatelstvo Glavsevmorputi,
Moscow, 360 p.
240