Journal of Modern Physics, 2013, 4, 505-516
http://dx.doi.org/10.4236/jmp.2013.44072 Published Online April 2013 (http://www.scirp.org/journal/jmp)
Postclassical Turbulence Mechanics
Jaak Heinloo
Marine Systems Institute, Tallinn University of Technology, Tallinn, Estonia
Email: jaak.heinloo@msi.ttu.ee
Received January 13, 2013; revised February 15, 2013; accepted February 26, 2013
Copyright © 2013 Jaak Heinloo. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
This paper surveys the formalism and applications of the postclassical turbulence mechanics (PCTM) grounded on the
characterization of turbulent flow field in infinitesimal surroundings of the flow field points besides the flow velocity at
these points also by the curvature of the velocity fluctuation streamlines passing these points. The PCTM applies this
step to found the turbulence split into the orientated and the non-orientated constituents. The split specifies the compe-
tence of the classical turbulence mechanics (CTM) to the description of the non-orientated turbulence constituent and
delegates the description of the orientated turbulence constituent (in the spirit of the theory of micropolar fluids) to the
equation of moment-of-momentum. The concurrent presence of the orientated (relatively large scale) and the non-ori-
entated (relatively small scale) turbulence constituents enables to compile the CTM and the conception of L. F.
Richardson and A. N. Kolmogorov about the cascading turbulence (RK conception) within a conjoint formalism. The
compilation solves the classical conflict between the CTM and the RK conception, though evinces a conflict of another
type characterized as paradigmatic.
Keywords: Fluid Mechanics; Turbulence; Micropolar Fluids
1. Introduction
According to the classical turbulence mechanics (CTM),
the flow state in the infinitesimal surrounding of each
flow-field point is uniquely characterized by the flow
velocity at this point. The postclassical turbulence me-
chanics (PCTM) modifies this statement by constituting a
flow state in the infinitesimal surrounding of each flow
point characterized in addition to the flow velocity at this
point also by the curvature of the velocity fluctuation
streamlines passing this point. The complementation is
an outcome of the analysis of the relation between the
CTM, the conception of L. F. Richardson [1] and A. N.
Kolmogorov [2] (RK conception) about the cascading
eddy structure of turbulence and the idea about the ap-
plicability of the theory of micropolar fluid (MF) [3-7] to
the description of turbulent flows [8-12]. The analysis
follows the general principles of the statistical physics
connecting the properties of the statistical ensembles to
the specific conditions of their formation formulated in
average terms [13,14]. The adjustment of these principles
within the context of the turbulence problem has been
explained in [15-21] and summarized in [22] (together
with the physical-historical background of the turbulence
problem) as the physical doctrine of turbulence (PDT).
The formalism of the PCTM (section 2) utilizes the
suggested complemented characterization of the flow-
field states in the infinitesimal surroundings of the flow-
field points as a precondition for definition of a kinema-
tical-dynamical pair of the Eulerian flow-field character-
istics reflecting local average effect of a prevailing ori-
entation of the large-scale turbulence constituent. The
turbulence property characterized by the defined quanti-
ties founds the decomposition of turbulence into its ori-
entated (relatively large scale) and non-orientated (rela-
tively small scale) constituents, delegates the description
of the orientated turbulence constituent (in the spirit of
the theory of MF) to the equation of moment-of-mo-
mentum, specifies the competence of the CTM to the
description of the non-orientated constituent of turbu-
lence and provides an opportunity to reflect the RK con-
ception (in two-scale approximation) in formulation of
turbulence mechanics (TM). Besides, the turbulence
properties reflected by the defined flow-field characteris-
tics introduce substantial particularization to the descrip-
tion of energetic and transport processes in turbulent me-
dia (A more general setup of the PCTM for the multi-
scale representation of turbulent flow-field can be found
in [19-21]).
Section 3 discusses a complementation of the ex-
C
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506
plained in the Section 2 all-purpose formalism of the
PCTM by the appropriate closure assumptions summing
up in the form of the theory of rotationally anisotropic
turbulence (the RAT theory) [15,16,23] (henceforth,
[23]). The applied closure constitutes the generalized
forces driving the motion linearly connected to the re-
spective generalized velocities, which ascribes the coef-
ficients introduced by the closure with a plain and unam-
biguous physical sense. The perspectives of application
of the RAT theory are exemplified in the Section 4 on
several examples.
Section 5 (Conclusion) comments the reception of the
PCTM (together with the RAT theory and its applica-
tions) by the scientific society engaged within the CTM.
The reception is characterized as evidencing a substantial
paradigmatic conflict between the expressed by the PDT
physical look to the turbulence problem and the look to
this problem kept by the CTM. The latter reduces the
turbulence problem to either a huge number of applied
tasks or to the problem of integration of equations of cla-
ssical fluid mechanics. The aim of the current paper is to
motivate physicists to determine their own position in
this conflict.
2. The Formalism of the PCTM
2.1. The Grounding Steps of Formulation of the
PCTM
The formalism of the PCTM begins from claiming the
momentary states of the turbulent flow-field fixed in in-
finitesimal surroundings of each flow-field point besides
the flow velocity v at this point by the curvature of the
velocity fluctuation streamline passing this point. The
claim is accompanied with the inclusion of the curvature
characteristics of the velocity fluctuation streamlines to
the arguments of the probability distribution of the me-
dium motion states at the flow-field points. In the fol-
lowing the PCTM applies this preliminary step as a nec-
essary precondition to determine the dynamical-kinema-
tical pair of the Eulerian flow-field characteristics
2
R R ee
Mv (1)
and
vk ee
, (2)
complementing the average flow field characteristics
introduced in the CTM. In (1) and (2) (and hereafter):
angular brackets denote statistical averaging,
vvu
(in which uv) denotes the fluctuating constituent of
the flow velocity; v
ev (in which v
v);
k
s
e (in which
s
is the length of the curve of
v
v
streamline passing a flow field point) is the curvature
vector of the streamline passing the flow field point;
2
kRk (in which kk) is the curvature radius
vector corresponding to k; RR; and the overdot de-
notes the full time derivative tv s
 .
The defined in (1) and (2) quantities characterize the
average state of motion of Lagrangian particles passing
the flow field points while M has the sense of the average
density (per unit mass) of the moment of fluctuating con-
stituent of momentum at the flow field points (hence-
forth—the moment-of-momentum) with R standing for the
arm of the moment and
has the sense of the average
angular velocity of rotation of the medium particles at the
flow field points in respect to the random curvature cen-
ters of the velocity fluctuation streamlines passing these
points. As the characteristics of a dynamical-kinematical
state of the flow-field at the flow-field points the defined
M and
determine the energy 1
2
KM specified
as a part of the total turbulence energy 2
1
2
K
v
0
rep-
resented as
KK
, (3)
where 01
2
K
M
MvRM
in which and
vk
. The energy split in (3) shows the me-
dium turbulence split into the orientated and the non-
orientated constituents characterized by the pair M,
and by 0
K
, respectively. Finally, proceeding from the
sense of M and
, it is natural to connect them by
MJ, (4)
J
. defining the tensor of effective moment of inertia
Let us emphasize the following:
1) The turbulence properties reflected by M and
relate the average turbulent continua to the class of MF
with their MF properties reflecting the local effect of the
prevailing orientation of eddy rotation;
2) Constituting M and
identically vanishing, the
CTM either confines its applicability to the situation with
the correlations expressed by (1) and (2) absent (which is
a physical assertion) or excludes axiomatically the cur-
vature of the velocity fluctuation streamlines from the set
of characteristics of the flow-field states in infinitesimal
surroundings of the flow-field points;
3) The split of turbulence into the orientated and the
non-orientated constituents reserves to the CTM the com-
petence of describing the non-orientated constituent of
turbulence;
4) The turbulence properties reflected by (1)-(3) pro-
vide a possibility to introduce the RK conception in an
explicit form (in the two-scale approximation) to the av-
erage description of turbulence.
2.2. The Balance Equations
The following particularizes the explained in 2.1 setup of
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turbulence description in terms of differential balance
equations for the average momentum, for the moment-of-
momentum, for different energy constituents and for the
concentration of scalar substance.
2.2.1. The Balance Equations for the Momentum and
Moment-of-Momentum
In the universal form the differential balance equations
for the momentum and the moment-of-momentum write
as [24]:

,ij j
σ
d
dt
uF
, (5)

,j
d
dkj
m
t
Mm
. (6)
In (5) and (6): ddtt u ;
is the medium
density; ij
are the components of the stress tensor; F
is the density (per unit mass) of the body force (hence-
forth—the body force); kj are the components of the
moment stress tensor describing the diffusive transport of
M;
m
is the dual vector to the antisymmetric constitu-
ent of the stress tensor coupling the fields of momentum
and the moment-of-momentum; m is the density (per unit
mass) of the body moment (henceforth—the body mo-
ment) acting on the medium; the index after the subscript
comma denotes differentiation along the respective space
coordinate while the Einstein summation is assumed and
the equivalent notation arbitrary tensor or vector quan-
tity {components of this quantity} is applied.
The characterization of the turbulent flow-field ex-
plained in 2.1 specifies u and M in (5) and (6) as the av-
erage flow velocity and the moment-of-momentum de-
fined in (1), respectively, while Equation (5) is the bal-
ance equation for the average momentum and Equation
(6) is the equation obtained from the averaged difference
of the balance equation of instantaneous momentum and
Equation (5) vector-multiplied by R from the right. The
derivation procedure provides all terms in (5) and (6)
with specific expressions via the momentary flow-field
characteristics [23]. In particular, it specifies the stress
tensor as mt
ij ijij

 m
, where ij
denote the compo-
nents of the molecular stress tensor and tvv

ij j i

12
denote the components of the turbulent stress tensor, and
the body moment m as
f
m mmm, (7)
where 1t
mvR

2
,
muM and
1

mfR
f
f
in which denotes the fluctuating constituent of the
body force acting on the medium.
Notice, that the asserted by Equations (5) and (6)
asymmetry of the turbulent stress tensor has the same
origin as the non-triviality of the turbulent flow-field
characteristics defined in (1) and (2).
2.2.2. The Energy Balance Equations
The full set of the energy balance equations of the PCTM
comprise the balance equations for 2
1
2
u
K
u (where
uu
) (derived as the scalar product of Equation (5)
and u), for ˆ
(deduced for
K
J
1J= constJ with
,
where denotes the unit tensor with the components
ˆ
1
ij
), for
, as the scalar product of Equation (6) and
0
K
(derived as the difference of equations of balance of
turbulence energy K and
K
) and for internal (thermal)
energy U, written as:
d
d
uu uuu
K
Aq
t

 h, (8)
ΩΩ ΩΩΩ
d
d
K
AB q
t

 h, (9)
00 00
d
d
u
K
Bq
t

 h, (10)
d
d
U
U
t
 h
0
,,,
uU
hhhh
(11)
In (8)-(11): denote diffusive flux vec-
tors of Ku,
, K0 and U, respectively;
K
A

(where 1
2
 u is the vorticity) denotes the work
realizing the energy exchange between Ku and
K
;
2
B
m
denotes the work realizing the energy ex-
change between
K
, 0
K
; (in which
 
,
ut
iji j
u

() 1
2
ttt
ijij ji



and

,,
,1
2ij ji
ij uuu) denotes the
work realizing the energy Ku scatter into the energy K0;
m

,1iji j
  m
denotes the work
resulting in the scatter of energy
K

into the energy
K0;

,0
um
iji j
u

0
00
, and reflect
the molecular dissipation of the energies Ku,
K
and
K0, respectively; 0u
 
u
q

;
F
u
q

,
f
0
qm
and
describe the effect of external
fields on the energies Ku, and K0.
K
A substantial implication of the energy balance situa-
tion represented by (8)-(11) is the specification of the
pairs of “generalized forces” and the respective “gener-
alized velocities” as
 
tu

,m
,iji j, ,iji j,
,
and
,
m1
. Each of the pairs determines an inde-
pendent physical process realizing the scatter of the en-
ergies Ku and/or
K
into the energy K0. Notice that,
unlike the positive u
and , the work A and the
work B may be either positive or negative. In particular,
positive A is related to the energy
K
feeding on the
energy Ku while the negative A declares the situation of
the eddy-to-mean energy conversion accompanied with
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508
the up-gradient momentum transfer. For the stationary
and homogeneous situation the latter possibility assumes
q
positive, i.e. external fields feeding the energy
K
. The other note relates to the energy K0 feeding on
the energies Ku and
K
c
, from which the RK conception
excludes the first energy source and the CTM excludes
the second. The concurrent inclusion of both situations in
set (8)-(11) suggests the compilation of the RK concep-
tion (in two-scale approximation) as well as the CTM
within one unique formalism.
2.2.3. The Balance Equation for the Scalar Substance
The turbulence properties reflected by the defined in (1)
and (2) quantities introduce a change not only into the
description setup of turbulent motions explained in 2.2.1
and 2.2.2 but also into the description of turbulent trans-
port processes. Indeed, denoting by , Cc
cc
and
the instantaneous, average and fluctuating
concentration of an arbitrary scalar substance (concentra-
tion of ingredients, temperature, etc.), respectively, and
using the identity
C
2
RvR


vR , the turbulent
flux vector of the substance, Cc

hv, in the balance
equation for C,
C
CQ h
0
CC C
hhh
d
dt, (12)
where Q denotes the body-source of C, becomes repre-
sented as
, (13)
In (13) 0
Cc

hR and C describe
the turbulent transport of C by the non-orientated and by
the orientated turbulence constituents, respectively.
c

hR
ˆ
3. The RAT Theory
3.1. The Closure Assumptions
The RAT theory [23] realizes the all-purpose formalism
of the PCTM explained above within a specific solution
of the closure problem of the balance Equations (5), (6)
and (12), formulated in three steps. The first step (already
applied while deriving Equation (9)) constitutes
J
1J=
0
, (14)
where . Notice that
J
J
determines the charac-
teristic length scale of eddies contributing to M and
.
Unlike , the length scale
R
J
is an average quantity.
In the second step the “generalized forces”, revealed in
the analysis of energy balance in 2.2.2, are set to linearly
depend on the respective “generalized velocities”, written
as

t
ij 
 
,
2
ij i j
pu

, (15)
0,1,2ijk kijij
m,ji
 

, (16)
4
,
and
14
(17)
m.
(15)-(18): p is the pressur
(18)
Ine; 0
is the coeffi-
y; ,, 0
cient of turbulence shear viscosit012

, are the
diffusion coefficients of M; 0
e coefficient of
turbulence rotational viscosity charhe shear
stresses in the relative rotatioe. for
is th
acterizing t
n, i.
; 0
interprets as the coefficient of decay of M due to the
cascading process. Relations (15)-(17) arear
respective relations within the MF theory whereas the
relation (18) reflects a fundamental difference in the pro-
perties of turbulent media and micropolar fluids—when
the turbulence structure requires incessant restoration
then the MF theory considers the media having a fixed
structure.
The third step constitutes 0
C
h and
similto the
cR, determin-
ing C
h in (13), expressed as 0
C0
kCh and
12
kC kC
c
R (with the latter derived as the
depeence of ndc
R on
linear in res
nd vanishing for 0C) re-
sulting in
CC
and C pect
to the both arguments a
hK. (19)
In (19)
K
is the turbulent tra
as
nsport tensor represented
s
as

K
KK, (20)
specifies the sym-
2
01
ˆˆ
skk11K
2
as k
in which
metric and

KE (wher
or)—the antisymmetric
e E is the Levi-
Civita tensconstituent of
K
.
Equations (19ain k0 and1 as the positive
coefficients characterizing normal (down-gradient) t-
bulent diffusion of C and k2—as the coefficient charac-
terizing the cross-gradient turbulent diffusion of C. For C
not contributing to the density field the sign of k2 speci-
fies as negative. Otherwise the sign of k2 becomes de-
pending on whether the C
) and (20) expl k
ur
constituent perpendicular to
the gravity acceleration amplifies of depresses
.
3.2. The Equations of the RAT Theory
The closure relations (14)-(19), turn the bal
tions (5), (6) and (12) (henceforth ,,,,
ance Equa-
, ,,
0120
J
k

 
,
and C
k1 and k2 are constituted to be constants) into the follow-
ing set of equations to determine u,

d2
dp
t
 
  
uuF
,
(21)


02 1
d4
d
4,
f
Jt
J
 
 


um
 

(22)
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
CC
Q 
K, (23)
where
d
dt
K
is specified in (20).
Let us accompany Equations (21)-(23
lowing comments.
closresulting in (21) and (22)
) with the fol-
1) Theure assumptions
specify also all terms in the energy balance Equations
(8)-(10) providing any solution of (21) and (22) with a
rigid physical sense expressed in energetic terms. In par-
ticular, the closure relation (17) specifies the work A in
(9) and (10) as

4A
 
which sets the prob-
lem of eddy-to-mean energy conversion into the dy-
namical context expressed by (21) and (22) avoiding the
application of negative viscosity [25] or the idealization
of the 2D turbulence.
2) Delegating the description of the orientated (rela-
tively large scale) turbulence constituent to the equation
of the moment-of-momentum, the formalism of the RAT
theory proves closer to the theory of MF than to the
CTM, which ignores this equation in its setup.
3) The constituted in the CTM statement of symmetry
of the turbulent stress tensor reads in terms of Equations
(21) and (22) as the condition

40

holding
either for 0
or for
which enlightens the
ambivalence of physical interpretation of the CTM. In
the first case, if 0
fmF and if is identically
zero at an initial time instant, then it appears vanishing
also for all following time instants. In the second case
Equation (22) should reduce to the equation for vorticity
following from Equation (21) which takes place if 0
and 1
J

.
4) Insofar as

as CCsK, in which
2
ks, Equation (23) can be rewritten also as

sCQ sK
explaining lent
transport similar to the advection by the velocity field
incomd

0 s.
urbulens under the
Influence of External Fields
unted
dCC
dt , (24)
the effect of the cross-gradient turbu
of
pressible flui
3.3. Turbulent Flows in Specific Conditions
3.3.1. Description of Tt Flow
The effect of external fields on turbulence is acco
for in (21) and (22) through the terms
F and
f
m. In
ses. the following the situation is particularized for two ca
The first case is related to the flows of electrically
conductive media under the influence of external mag-
netic field for small magnetic Reynolds number values,
where [26]
2
0000
ˆ
EB




FEBBB u1 (25)


2ˆ
1
f
JB

00 0
2
 mBB1 (26)
E
in which B0 denotes the induction of the ext
netic field,
ernal mag-
E
is the coefficient of electrical
ity and 01
conductiv-
denotes certain phenomenological
characteristic of the medium electric properti
n o
es. Expres-
sion (26) evinces the effect of external magnetic field
resulting for small magnetic Reynolds number values in
a suppressiof
. The situation changes for the me-
dium and/or large magnetic number values when the
magnetic field may prove acting as a source of energy of
the orientated constituent of turbulence or the medium
turbulence may prove acting as a course of generation of
magnetic field.
The second case is related to the flows under the influ-
ence of gravity force, where within the Boussinesq ap-
proximation [27] we have [26]

Fg
(27)
an
2,k
d

1ˆ
fk

 

g (28)
hich
1

mg
g
in w
is the characteristic constant density of
medium and
is the medium actual average density,
while
. Here, the integration of Equ
and (22) requires a specification of the equation
ations (21)
of state
expressing
through the characteristics of medium
ingredients contributing to
and Equations (21) and
(22) should be integrated together with the equations
formulated for all medium ingredients contributing to
. Notice, that expression (28) compiles in one single
formula the depression of the component of
perpen-
dicular to the gravity acceleration caused by the stable
stratification as well as its generation by the unstable
stratification, which substantially simplifies the descrip-
tion of gravitation-related processes involving the both
situations.
3.3.2. Description of Turbulent Flows in Rotating
Frames
For the description of motion in a frame rotating with a
constant angular velocity 0
, the flow velocity u is re-
0
placed by
or
t of
ur, where r denotes a radius-vect
from the arbitrary point on the rotation axes to a poin
low-field, 0
and
are replaced by the f
and 0
, and ddtu, ddtM are replaced by
0
ddt
uu
, 0
ddt
MM. The changes result in
the complementation of the right sides of (21) and (22),
respectively, by the Coriolis force term
0

2
C
Fu
, 29)
tiot term
(
and by the addinal body momen
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
0000
4JJ
 
 mu

(30)
s
K
by and in the replacement of the expression for

00
01
ˆˆ
skk
  

11K
20
. (31)
The expressions (29)-(31) evidence about a sub
in tat-
terthe
right side of (30) evinces the frame rotation pre
the anticyclonic (directed opposite ection of
stantial
difference between the turbulence properties the ro
ing and the non-rotating frames. So, the firstm on
ferring
0
to the dir
)
orientation of (gyration effect). The work done by
the moment 0
m may serve also as an additional cause
of eddy-to-mean energy conversion etc.
4. Examples of Application of the RAT
Theory
4.1. One-Dimensional Flows in Plain Channels,
Round Tubes, between Rotating Concentric
Fonsional flows in plain channels, round tubes,
-
Cylinders and Boundary Layers
r one-dime
between rotating concentric cylinders [23], and in boun
dary layers [28] the Equations (21) and (22) simplify to

2p
t
 
 
uu
F
, (32)

144
f
Jt

 
m
 (33)
where and u are orientated perpendicularly
pend on the coordinate perndicular to and
0
f
Fm the picted by (32) and (33) ve-
locity profiles were compared in [23] (for steady
hannels, in round tubes and between rotating con-
cy
inr os-
boundary
and de-
u.
pe
red
For
plain c
cil
flows in
centric linders) with data in [29-31] and (in case of
oscillating flow in round tube) with data [32]. Fo
lating layer generated by undulating free
flow the predicted by (32) and (33) velocity profiles were
compared in [28] with data in [33]. The predicted by (32)
and (33) velocity profiles for F and
f
m specified in (25)
and (26) were compared with data in [34]. In all cases the
predicted velocity profiles prove excellently matching
the actual velocity data. Notice, that for 0
Equa-
tions (32) and (33) coincide in written form with the re-
spective equations of the MF theory. However, the situa-
tions with 0
and 0
prove reflecting physi-
cally different situations. In particular, for steady flows
in round tubes and plain channels the solutio32) and
(33) predicts the flow velocity determined in the central
part of the flow region by the effective viscosity
ef
n of (


 whints to a substantial role of
the turbulence properties characterized by
h chi
and
in
this region with the property characterized by
play-
ing (in harmony with the RK conception) a marginal role.
In the central part of the flow, for 0
, the turbulence
epresented by the turbulence shear
viscosity only.
properties appear r
of concentration of the suspended
4.2. Vertical Distribution of Concentration of
Suspended Sediments in a River Estuary
In [35] Equations (21)-(23) were applied to describe the
vertical distribution
matter
C in a river estuary modeled as an open ch
nel with the fixed bottom slope angle

an-
and the tim
ing free surface angle


t

. Restricting t
e-
vary he
consideration with ,1
, the quasi-stationary flow
regime and with the concentrations small enough to not
influence density field, Equations (21)-(23) read as

e th

2
220
ut
z
z
 

g
 
 
, (34)

2
12420
u
z
z

 
 
, (35)

2
01 0
C
kk Q
zz






,
where
(36)
QwCz
, in which w is the settl
and z is the vertical coordinate directed upward. The term
ing velocity
g

in (34) expresses the summary effect of the
along-flow pressure gradient and of the gr
The determined from (34)-(36) vertical distributions of C
ompared wit
for differen
avity force.
were ch concentration of the resuspended
sediments observed in the Jiaojiang Estuary (China) [36]
t time instants of a spring tide cycle. The
comparison showed that the derived analytical formula
for C embraces two observed basic types of vertical dis-
tribution of concentration, one with a monotonic de-
crease of concentration gradient with distance from the
bottom and the other with a gradient maximum (luto-
cline) located at some distance from the bottom. (The
both types of vertical distribution of suspended sediments
were detected also in the bottom layer of natural water
body, studied in [37]).
4.3. Vertical Structure of the Upper Ocean
Consider now the situation in the upper ocean in Boussi-
nesq approximation specified by


,,0uzuzu,
xy
,,0zz 0
xy and
(the right-hand
Cartesian coordinate system

,,
x
yz with 0z
rected downward is assumed; hereafter 0
di-
is the angu-
lar velocity of the Earth rotation). From Equations (21)-
(23), where
F
is determined acco9) rding to (27) and (2
as 0
2

gu
,
f
mdetermined is according to (28)
as
1z
fkg

 m
and0 we with Qe hav
Copyright © 2013 SciRes. JMP
J. HEINLOO 511

20
222p
tz



uuu
(37)

2
11
4Jkg
tz
24
z
 
 


 
, (38)


pg


2
01
kk ztz
 



 


.
Equations (37)-(39) explain the stable stratificat
supressing the constituent perpendicular to the grav-
ity acceleration together with ancrease of vertical gra-
dient of density, and the unstable stratification in ampli-
fying the constituent perpendicular to t
eration together with a decrease of vertica
de
(39)
ion in
in
he gravity accel-
l gradient of
nsity.
For the constant
the solution of (37) and (38) [38,
39] for the velocity sums up from two addends reflecting
the Stokes drift effect [40] and the classical Ekman ver-
tical velocity profile [27] with the turbulence viscosity
replaced by the effective viscosity ef
. The solution
explains the Stokes drift effect in good agreement with
data in [41], in dimishing the angle between the flow
velocity and the shear stress. To demonstrate the stratifi-
cation effect Equations (37) and (38) were solved in [39]
for
in
constant everywhere instead of a density jump at
a certain depth modeling the assumed location of ther-
mocline in summer and winter. The calculated vertical
distribution of velocity was compared in both cases with
the velocity data in [42] showing a good agreement in the
dominating quality.
In [43,44], Equations (37)-(39) were applied to model
the reaction of the upper ocean to periodical cooling and
heating. Here zTzSz


, where T is
temperature, S is salinity,
is the coefficient of ther-
mal expansion and
characterizes the salinity contrac-
tion variance. It is shown that Equations (37)-(39) predict
the formation of a typical for the upper ocean vertical
density profile with the relatively uniform density distri-
bution in the layer next to the ocean surface separated
from lower layers by a stram of relatively abrupt den-
sity jump in a reasonable agreement with the observed
data.
4.4. Conjoint Effect of the Baroclinic Instability
and the Rotational Viscosity of Turbulence
In [45,46], Equations (21), together with
tu
F replaced
by C
F in (29), and (22), with
f
m specified in (28),
were applied to agphic correction ag
u to
geostrophically predicted net the Antarctic
-
tribu
calculateeostro
transport of
Circumpolar Current [47] from the observed spatial dis
tion of
. The idea lies in the determination of a
correction from vertical con-
he f
the balance condition of
stituent of the Coriolis force with the vertical constituent
of torce described by the term2
 in (21),
giving

ag
02 cos
cos
ur
 

, (40)
where
is determined from the zonal projection of
(22) as

1
21
g
rz
14kg k


 

Equations (40) and (41) explain ag
u for
joint effect of the rotational viscosity of medium turbu-
. (41)
med as a con-
lence
and the baroclinic instability (characterized
by 20k). The both mentioned effects are excluded
within the CTM. A similar setup was applied in [48] to
explain the formation of zonal winds in plan
mosphere with the main interest to the formation of east-
er
etary at-
lies in the equatorial zone showing a reasonable agree-
ment with the observed velocity data in [49].
4.5. Gyration Effect
In sub-subsection 3.3.2 it was pointed out that the frame
rotation prefers the anticyclonic orientation of
. The
represented in [50,51] zonally averaged
0,0,

and
0,0,
M
M estimated from the global surface
drifter data sets [52] (the right-hand coordinate system
,,z

, where π
2
π
2
 is latitude,
is longitude
,5
tical aspects
e gyration effe
4.5.1. Th
and axis z of the coordinate system is directed upward is
applied) confirm this conclusion, called in [501] the
gyration effect. In the following some theore
of thct from the position of the RAT the-
ory are commented.
eoretical Evidenceof the Gyration Effect
First consider how the gyration effect agrees with Equa-
tions (22) and (30). Restricting the consideration with the
effects of diffusion of moment-of-momentum neglected
we have from (22) and (30)
0sin
h




0sin
z
u
Jz
0
44
sin
J
 
   (4


where h
u
2)
denotes horizontal gradient operator. Insofar
as from the continuity equation 0 u follows that
hz
uz u Equation (42) rewrites also as


0
0
sin
44sin
h
J

 
 
 
u (43)
For 0
u Equation (43) gives
Copyright © 2013 SciRes. JMP
J. HEINLOO
512
0sin

 . (44)
Expression (44) explains the gyration effect
by the rotation of frame in balance of the shear
rotation and of the decrease of a prevailing orientation of
ed tation
4.5.2. Anomalous Turbulent Diffusi
Paper [53] exploits (44) within a model explaining the
ob
ulent diffusion. This
explanation follows from Equation (24) for C specified
generated
in relative
dy ro in cascading process.
ve Transport
served tongue-like structure of the salinity distribution
in the region of the Gibraltar Salinity Anomaly (GSA)
[54]. The tongue-like structure of the anomaly is explain-
ed as a result of cross-gradient turb
as salinity

S depending on
(longitude) and
(latitude) only, written as

2
0sinSkb S

 

s, (45)
where cosa
se (in which
e is the unit vector
directed to the east,

0
2
ak
) and


2
0
1
bk
. According to Equation (45) the
gyration effect stretches the salinty distribution out in
the east-west direction
2
k
i
0 and shifts the maxima
e latituistribution of salinity
the south with distance from the Gibraltar
ive agreemese
of thdinal d increasingly to
Strait in good
qualitatnt with the obrved situation repre-
sented in [54].
4.5.3. Eddy-to-Mean Energy Transfer in Geophysical
Jet Flows
For 0u and for 0sin
 conserved in a flow


0sin0
h

 u from (42) we have
0sin


 . (46)
Using (46), the work A 
performing the en-
exchange betw
ergy the average flow and the orien-
turbutituent, exp
een
tatedlence consresses as

0
4sinA



. (47)
Paper [55] employs (46) and (47) explaining the up-
grsfer and e
ception of 2D
ce [56,57]. The situation was particularized for
-sections of Gulf Stream at
nd at 35˚N (Onslow Bay) where, in harmony
y conver
additional result, the discussion evidences about
m.
adient momentum tranddy-to-mean energy
conversion

0A avoiding the negative viscosity
problem [25] or the application of the con
turbulen
the cross
Straits) a
wi
As an
26˚N (Florida
th (47) and the observational data in [58], the regions
with the up-gradient momentum transfer and eddy-to-
mean energsion were observed at the anticyc-
lonic sides

0
of the stream.
the insufficiency of the velocity covariance data for un-
ambiguous solution of the problem of turbulent stress
tensor properties. In particular, for antisymmetric stresses
the velocity covariance determines the stress tensor
components with the accuracy up to the sign, specified
from consideration of balance of internal moments acting
in the mediu
4.5.4. Topographically Generated Flows
Using the continuity equation in a shallow water region,
written as 1
z
uzHH
 u [27], from Equation (42)
it follows, that
00
sin sin
44J
H
HH


 
 u. (48)
0
H
For sin

 conserved in a shallow water
tant, and
region we have
0sin CH

 , (49)
where C is cons
0sin CH

 .
essing C through
(50)
Expr the depth cr
H
, where the en-
ergy
K
scatterergy 0
into en
K
obtains its minimum,
gives [59]
0
cr
1sin
H
C
and from (49) and (50) we have

0
cr
in1s
H
H


 


(51)
and
0
cr
1sin
H
H


 

 (52)
ith (44) for cr
.
H
H
Equations (51) and (52) agree w
suggesting identification of the actual depth H with cr
H
alongshore re- in the open part of the water-body. In the
gion (52) declares that 0
evidencing about a non-
ng velocity of flow in this region. In particular, in
the alongshore region of a closed water
ated flow velocity is directed anticyclon
islands cyclonically. For 0
vanishi
-body the gener-
ically and around
Equations (51) and (52)
re potentia
ped
. Unl
h the
duce to the condition of conservation of the l
vorticity predicting the motion in regions with slo
bottom, though leaving open the question about the
source of motion energyike the discussions in [60-
62] suggesting to overcome this shortcoming within
rather sophisticated theoretical constructions, the mecha-
nism suggested above explains the energy supply from
the energy associated witgyration effect converted
to the flow energy in shallow water regions.
Copyright © 2013 SciRes. JMP
J. HEINLOO 513
4.6. Summary to the Commented Applications of
the RAT Theory
According to the described above applications, the RAT
theory unites a broadened physical background of its
setup with a noteworthy simplification of discussion of
turbulence-related problems. The simplification follows
from the split of turbulence into the orientated (relatively
large scale) and the nonorientated (relatively small scale)
flowT theory, the CTM unites a
flows with
ex
M.
ts besides the
flow velocity at these points also by the curvature of the
streamlines passing these points. The
constituents with the orientated constituent of turbulence
dominating in the formation of properties of average
. Contrary to the RA
simplification of its setup by neglecting the orientated
turbulence constituent with complications in its applica-
tions following from the inconsistency between the ab-
sence of the orientated constituent of turbulence in the
CTM setup and its presence in actual flows.
Despite focusing on the orientated turbulence con-
stituent, the commented applications demonstrate con-
siderable perspectives of the RAT theory. So, the exam-
ples show the ability of the RAT theory to describe the
eddy-to-mean energy conversion avoiding the negative
viscosity or evading the actual 3D structure of turbu-
lence. The examples demonstrate also substantial per-
spectives of the RAT theory proceeding from the par-
ticularization of the interaction of turbulent
ternal fields and from the distinguishing the turbulence
properties in rotating and non-rotating frames. The latter
actualizes within the available ocean surface drifter data
providing a plain observational evidence to the prevailing
anticyclonic orientation of eddy rotation (gyration effect)
which has been excluded within the CTM. Due to the
inclusion of the gyration effect in its setup, the RAT the-
ory sets this effect into the dynamic context required to
explain the physical causes of the effect and its impact to
the dynamical, energetic and to some other (like trans-
port) processes in the upper ocean.
Despite the transparency of the applications formu-
lated as a direct inference from the same set of equations,
the focus of the applications on the orientated turbulence
constituent turns the formulated results incompatible
with the respective results formulated on the bases of the
CTM. The indicated discrepancy increases due to the
following from the RAT theory deficiency of contempo-
rary methods of experimental research of turbulence ad-
justed to the requirements of the CT
5. Conclusions
The PCTM is an implication of the physical-historical
point of view to the turbulence problem summarized as
the PDT [22]. It advances the TM realizing a small but
effective in its outcome modification at the very origin of
the setup of the TM. The modification stands in com-
plementation of characterization of the flow states in
infinitesimal surroundings of flow poin
velocity fluctuation
introduced modification was aimed to clarify the classi-
cal conflict between the CTM and the RK conception
actualized by the idea about applicability of the theory of
MF to the description of turbulent flows. The PCTM
accomplishes the task by compiling the CTM and the RK
conception in a single theoretical construction. The RAT
theory (complementing the universal formalism of the
PCTM by the appropriate closure assumptions) justifies
the applied modification from the pragmatic point of
view. It compiles a substantial enlargement of the com-
petence of the TM with a considerable simplification of
the discussion without losing the physical rigidity. Be-
sides grounding the RAT theory, the PCTM (especially if
complemented in its setup with the method of decompo-
sition of turbulent flow fields discussed and applied in
[19-21]) makes the turbulence problem an interesting
subject for theoretical discussions. As a mechanical out-
come of the PDT the PCTM esteems also the PDT as a
whole.
Unlike the PDT, the dominating up-to-date look on the
turbulence problem reduces it to a huge number of par-
ticular problems stressing rather on their particularities
than on their commonness, relates the RK conception to
the ideas of the past not worthy to be revived in modern
time and believes the fundamental aspects of the turbu-
lence problem belonging rather to mathematics than to
physics. The conflict emerged between the CTM and the
PCTM is enforced by the criticism of the PCTM in ad-
dress of the CTM. The conflict has all aspects archetypal
to the paradigmatic conflicts in science, always accom-
panied with the critics of the old paradigm from the point
of view of the novel paradigm and avoiding the discus-
sions which may insinuate doubts about the grounding
statements of the old paradigm.
In the end, any paradigmatic change in the science is
always preceded by the superfluous aplomb of the former
paradigm in its consummation, loss of adeptness for
self-criticism and relating the unresolved problems to the
solution nuances which cannot attaint the existing para-
digm as a whole. Though, the unresolved nuances may
incidentally actualize. Not finding answers within the
dominating paradigm they start looking for answers on a
wider scientific background embracing also neighboring
science fields. If succeeding, the new expanded point of
view may develop into an independent paradigm clarify-
ing its relation with the former paradigm through a para-
digmatic conflict. This is just the situation with the for-
mulation of the RAT theory, the PCTM and the PDT.
Started with formulation of the RAT theory initiated by
the discussed in 70-es idea about applicability of the the-
ory of MF to the description of turbulent flows, the for-
Copyright © 2013 SciRes. JMP
J. HEINLOO
514
mulation of this theory actualizes the RK conception as
well as evinces the incompatibility of the RK conception
with the CTM. The incompatibility raises several ques-
tions and the need to look for answers to these questions
within the frames of the general principles of statistical
physics collected together as the PDT. It motivates also
the formulation of the PCTM within the classical formal-
ism with an axiomatic change in the setup. This kind of
the setup turns the physical background of the PCTM
absolutely transparent and mandates the opponents either
to agree with the suggested change or to reject the
change by applying physical arguments. The fact that
neither of the possibilities has realized characterizes the
emerged paradigmatic conflict so deep that usually char-
acterizes breaking points in the respective science fields.
6. Acknowledgements
The preparation of the paper was supported by grant
ETF9381 of the Estonian Science Foundation. The au-
thor thanks Dr. Aleksander Toompuu for the idea to de-
termine the gyration effect from the ocean surface drifter
data providing the grounding statements of the RAT the-
ory with observational evidence.
REFERENCES
r Prediction by Numerical Pro-
bar, 1980.
[4] J. S. Dahler and
Continua,” Na , 1961, pp. 36-37.
[1] L. F. Richardson, “Weathe
cess,” Cambridge University Press, Cambridge, 1922.
[2] A. N. Kolmogorov, “The Local Structure of Turbulence
in Incompressible Viscous Fluids for Very Large Rey-
nolds Numbers,” Doklady Akademii Nauk SSSR, Vol. 30,
1941, pp. 376-387 (in Russian).
[3] A. C. Eringen, “Microcontinuum Field Theories II. Fluent
Media,” Krieger Pub. Co., Mala
L. F. Scriven, “Angular Momentum of
ture, Vol. 192, No. 4797
doi:10.1038/192036a0
[5] J. S. Dahler, “Transport Phenomena in a Fluid Composed
of Diatomic Molecules,” Journal of Chemical Physics,
Vol. 30, No. 6, 1959, pp. 1447-1475.
doi:10.1063/1.1730220
[6] A. C. Eringen, “Theory of Micropolar Fluids,” Journal of
Mathematics and Mechanics, Vol. 16, No. 1, 1966, pp. 1-
18.
[7] T. Ariman, M. A. Turk and D. O. Silvester, “Microcon-
tinuum Fluid Mechanics—A Review,” International Jour-
nal of Engineering Science, Vol. 11, No. 8, 1973, pp.
905-930. doi:10.1016/0020-7225(73)90038-4
70, pp. 1-8.
0.1016/0022-247X(72)90239-9
[8] A. C. Eringen and T. S. Chang, “Micropolar Description
of Hydrodynamic Turbulence,” Advances in Materials Sci-
ence and Engineering, Vol. 5, No. 1, 19
[9] A. C. Eringen, “Micromorphic Description of Turbulent
Channel Flow,” Journal of Mathematical Analysis and
Applications, Vol. 39, No. 1, 1972, pp. 253-266.
doi:1
072-9
[10] J. Peddieson, “An Application of the Micropolar Fluid
Model to Calculation of Turbulent Shear Flow,” Interna-
tional Journal of Engineering Science, Vol. 10, No. 1,
1972, pp. 23-32. doi:10.1016/0020-7225(72)90
Statistical Physics,”
.3.05
[11] V. N. Nikolajevskii, “Asymmetric Mechanics and the
Theory of Turbulence,” Archiwum Mechaniki Stosowanej,
Vol. 24, 1972, pp. 43-51.
[12] V. N. Nikolajevskiy, “Angular Momentum in Geophysi-
cal Turbulence: Continuum Spatial Averaging Method,”
Kluwer, Dordrecht, 2003.
[13] L. D. Landau and E. M. Lifshitz, “
Pergamon Press, Oxford, 1980.
[14] A. Ishiara, “Statistical Physics,” Academic Press, New
York-London, 1971.
[15] J. Heinloo, “Phenomenological Mechanics of Turbulent
Flows,” Valgus, Tallinn, 1984 (in Russian).
[16] J. Heinloo, “Turbulence Mechanics,” Estonian Academy
of Sciences, Tallinn, 1999 (in Russian).
[17] J. Heinloo, “On Description of Stochastic Systems,”
Proceedings of the Estonian Academy of Sciences, Phys-
ics and Mathematics, Vol. 53, No. 3, 2004, pp. 186-200.
[18] J. Heinloo, “A Setup of Systemic Description of Fluids
Motion,” Proceedings of the Estonian Academy of Sci-
ences, Vol. 58, No. 3, 2009, pp. 184-189.
doi:10.3176/proc.2009
, Vol. 8. No.
[19] J. Heinloo, “The Structure of Average Turbulent Flow
Field,” Central European Journal of Physics
1, 2010, pp. 17-24. doi:10.2478/s11534-009-0015-y
[20] J. Heinloo, “Setup of Turbulence Mechanics Accounted
for a Preferred Orientation of Eddy Rotation,” Concepts
of Physics, Vol. 5, No. 2, 2008, pp. 205-219.
doi:10.2478/v10005-007-0033-8
[21] J. Heinloo, “A Generalized Setup of the Turbulence De-
scription,” Advanced Studies in Theoretical Physics, Vol.
5, No. 10, 2011, pp. 477-483.
[22] J. Heinloo, “Physical Doctrine of Turbulence—A Re-
hy-
view,” International Journal of Research and Reviews in
Applied Sciences, Vol. 12, No. 2, 2012, pp. 214-221.
[23] J. Heinloo, “Formulation of Turbulence Mechanics,” P
sical Review E, Vol. 69, No. 5, 2004, Article ID: 056317.
doi:10.1103/PhysRevE.69.056317
[24] L. I. Sedov, “A Course in Continuum Mecha
ers-Noordhoff, Groningen, 1971.
nics,” Wolt-
[25] V. P. Starr, “Physics of Negative Viscosity Phenomena,”
McGraw-Hill, New York, 1968.
[26] J. Heinloo, “The Description of Externally Influenced Tur-
bulence Accounting for a Preferred Orientation of Eddy
Rotation,” European Physical Journal B, Vol. 62, No. 4,
2008, pp. 471-476. doi:10.1140/epjb/e2008-00187-8
[27] J. Pedlosky, “Geophysical Fluid Dynamics,” Springer,
New York, 1987.
[28] J. Heinloo and A. Toompuu, “A Model of Average Ve-
locity in Oscillating Turbulent Boundary Layers,” Jour-
nal of Hydraulic Research, Vol. 45, No. 5, 2009, pp. 676-
680. doi:10.3826/jhr.2009.3579
[29] J. Nikuradse, “Gesetzmässigkeiten der Turbulenten Strö-
Copyright © 2013 SciRes. JMP
J. HEINLOO 515
mung in Glatten Rohren,” VDI-Forschungsheft No 356,
1932, pp. 1-36.
[30] G. Comte-Bellot and A. Craya, “Écoulement Turbulent
entre Deux Parois Parallèles,” Fiche Détaillée, Paris, 1965.
nsfer in Round Channel with Inner Ro-
tal Study
d N. A. Carlsen, “Experimental and The-
[31] V. N. Zmeikov and B. P. Ustremenko, “Study of Ener-
getic and Heat Tra
tating Cylinder,” Problems of Thermoenergetics and Ap-
plied Thermophysics-1, Academy of Science of Kazakh-
stan SSR, 1964, p. 153 (in Russian).
[32] V. I. Bukreejev and V. M. Shakhin, “Experimen
of Unsteady Turbulent Flow in Round Tube,” Aerome-
hanika, Nauka, Moscow, 1976, p. 180 (in Russian).
[33] I. G. Jonsson an
oretical Investigations in an Oscillatory Turbulent Boun-
dary Layer,” Journal of Hydraulic Research, Vol. 14, No.
1, 1976, pp. 45-60. doi:10.1080/00221687609499687
[34] G. G. Branover and A. B. Tsinober, “Magnetic Hydro-
mechanics of Incompressible Fluids,” Nauka, Moscow,
1970 (in Russian).
[35] J. Heinloo and A. Toompuu, “A Model of Vertical Dis-
tribution of Suspended Matter in an Open Channel Flow,”
Environmental Fluid Mechanics, Vol. 11, No. 3, 2011, pp.
319-328. doi:10.1007/s10652-010-9180-1
[36] J. Jiang and A. J. Mehta, “Lutocline Behavior in High-Con-
centration Estuary,” Journal of Waterway, Port, Coastal,
and Ocean Engineering, Vol. 126, No. 6, 2000, pp. 324-
328. doi:10.1061/(ASCE)0733-950X(2000)126:6(324)
[37] J. Heinloo and A. Toompuu, “A Model of the Vertical
Distribution of Suspended Sediments in the Bottom Layer
of Natural Water Body,” Estonian Journal of Earth Sci-
ences, Vol. 59, No. 3, 2010, pp. 238-245.
doi:10.3176/earth.2010.3.05
[38] J. Heinloo and A. Toompuu, “A Modified Ekman Layer
Model,” Estonian Journal of Earth Sciences, Vol. 60, No.
2, 2011, pp. 123-129. doi:10.3176/earth.2011.2.06
[39] J. Heinloo and A. Toompuu, “A Modification of the Cla-
ssical Ekman Model Accounting for the Stokes Drift and
Stratification Effects,” Environmental Fluid Mechanics,
Vol. 12, No. 2, 2011, pp. 101-113.
doi:10.1007/s10652-011-9212-5
[40] O. M. Phillips, “Dynamics of Upper Ocean,” Cambridge
University Press, Cambridge, 1977.
[41] Y.-D. Lenn and T. K. Chereskin, “Observations of Ekman
Currents in the Southern Ocean,” Journal of Physical
Oceanography, Vol. 39, No. 3, 2009, pp. 768-779.
doi:10.1175/2008JPO3943.1
[42] R. R. Schudlich and J. F. Price, “Observations of Sea-
sonal Variation in the Ekman Layer,” Journal of Physical
Oceanography, Vol. 28, No. 6, 1998, pp. 1187-1204.
doi:10.1175/1520-0485(1998)028<1187:OOSVIT>2.0.C
O;2
[43] J. Heinloo and Ü. Võsumaa, “Rotationally Anisotropic
Turbulence in the Sea,” Annales Geophysicae, Vol
ol. 101, No. 11, 1996, pp.
25635-25646. doi:10.1029/96JC01988
. 10,
1992, pp. 708-715.
[44] Ü. Võsumaa and J. Heinloo, “Evolution Model of the
Vertical Structure of the Active Layer of the Sea,” Jour-
nal of Geophysical Research, V
ics,
1029/96JC01988
[45] J. Heinloo and A. Toompuu, “Antarctic Circumpolar
Current as a Density-Driven Flow,” Proceedings of the
Estonian Academy of Sciences, Physics and Mathemat
Vol. 53, No. 4, 2004, pp. 252-265.
doi:10.
hysicae, Vol. 24, No. 12, 2006, pp.
eanography, Vol.
[46] J. Heinloo and A. Toompuu, “Modeling a Turbulence
Effect in Formation of the Antarctic Circumpolar Cur-
rent,” Annales Geop
3191- 3196.
[47] T. Whitworth, W. D. Nowlin Jr. and S. J. Worley, “The
Net Transport of the Antarctic Circumpolar Current Trough
Drake Passage,” Journal of Physical Oc
12, No. 9, 1982, pp. 960-971.
doi:10.1175/1520-0485(1982)012<0960:TNTOTA>2.0.C
O;2
[48] J. Heinloo and A. Toompuu, “Mod
Effect in Formation of Zonal Winds,”
eling of Turbulence
The Open Atmos-
47-2
pheric Science Journal, Vol. 2, 2008, pp. 249-255.
[49] A. H. Oort, “Global Atmospheric Circulation Statistics,
1958-1973,” NOAA Professional Paper 14, Rockville,
1983
[50] J. Heinloo and A. Toompuu, “Gyration Effect of the
Large-Scale Turbulence in the Upper Ocean,” Environ-
mental Fluid Mechanics, Vol. 12, No. 5, 2012, pp. 429-
438. doi:10.1007/s10652-012-92
Global Surface Drifter Data in the Pacific Ocean,”
puu, J. Heinloo and T. Soomere, “Modelling of
preading at Ocean
[51] A. Toompuu and J. Heinloo, “Gyration Effect Estimated
from
IEEE/OES Baltic International Symposium, Klaipeda, 8-
10 May 2012, pp. 1-4.
[52] A. L. Sybrandy and P. P. Niiler, “WOCE/TOGA Lagran-
gian Drifter Construction Manual. WOCE Rep. 63,”
Scripps Institution of Oceanography, La Jolla, 1991.
[53] A. Toom
the Gibraltar Salinity Anomaly,” Oceanology, Vol. 29,
No. 6, 1989, pp. 698-702.
[54] R. W. Griffiths, E. J. Hopfinger, “The Structure of
Mesoscale Turbulence and Horizontal S
Fronts,” Deep Sea Research Part A. Oceanographic Re-
search Papers, Vol. 31, No. 3, 1984, pp. 245-269.
doi:10.1016/0198-0149(84)90104-3
[55] J. Heinloo and A. Toompuu, “Eddy-to-Mean Energy
No. 2-4, 1979, pp. 289-325.
Transfer in Geophysical Turbulent Jet Flows,” Proceed-
ings of the Estonian Academy of Sciences, Physics and
Mathematics, Vol. 56, No. 3, 2007, pp. 283-294.
[56] P. B. Rhines and W. R. Holland, “A Theoretical Discus-
sion of Eddy-Driven Mean Flows,” Dynamics of Atmos-
pheres and Oceans, Vol. 3,
doi:10.1016/0377-0265(79)90015-0
[57] J. R. Herring, “On the Statistical Theory of Two Dimen-
sional Topographic Turbulence,” Journal of the Atmos-
pheric Sciences, Vol. 34, No. 11, 1977, pp. 1731-1750.
1:OTSTOT>2.0.Cdoi:10.1175/1520-0469(1977)034<173
O;2
[58] F. Webster, “Measurements of Eddy Fluxes of Momen-
tum in the Surface Layer of the Gulf Stream,” Tellus, Vol.
17, No. 2, 1965, pp. 239-245.
[59] J. Heinloo, “Eddy-Driven Flows over Varying Bottom
Topography in Natural Water Bodies,” Proceedings of the
Copyright © 2013 SciRes. JMP
J. HEINLOO
Copyright © 2013 SciRes. JMP
516
o
0.1175/1520-0485(1986)016<2159:MCDBTD>2.0.
Estonian Academy of Sciences, Physics and Mathematics,
Vol. 55, No. 4, 2006, pp. 235-245.
[60] D. B. Haidvogel and D. H. Brink, “Mean Currents Driven
by Topographic Drag over the Continental Shelf and
Slope,” Journal of Physical Oceanography, Vol. 16, N
12, 1986, pp. 2159-2171.
.
doi:1
CO;2
[61] F. P. Bretherton and D. B. Haidvogel, “Two-Dimensional
Turbulence above Topography,” Journal of Fluid Me-
chanics, Vol. 78, No. 1, 1976, pp. 129-154.
doi:10.1017/S002211207600236X
ophic Eddies and the Mean Circulation over Large-
[62] S. T. Adcock and D.P. Marshall, “Interactions between
Geostr
Scale Bottom Topography,” Journal of Physical Ocean-
ography, Vol. 30, No. 12, 2000, pp. 3232-3238.