The large scale instability and nonlinear vortex structures in obliquely rotating fluid with small scale non spiral force
In this paper, we find a new large scale instability which appears in obliquely rotating flow with the small scale turbulence, generated by external force with small Reynolds number. The external force has no helicity. The theory is based on the rigorous method of multi scale asymptotic expansion. N...
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Kopp, M.I. Tur, A.V. Yanovsky, V.V. 2017-01-18T19:44:43Z 2017-01-18T19:44:43Z 2015 The large scale instability and nonlinear vortex structures in obliquely rotating fluid with small scale non spiral force / M.I. Kopp, A.V. Tur, V.V. Yanovsky // Вопросы атомной науки и техники. — 2015. — № 4. — С. 264-269. — Бібліогр.: 17 назв. — англ. 1562-6016 PACS: 47.32C-; 47.27.De; 47.27.em; 47.55.Hd https://nasplib.isofts.kiev.ua/handle/123456789/112219 In this paper, we find a new large scale instability which appears in obliquely rotating flow with the small scale turbulence, generated by external force with small Reynolds number. The external force has no helicity. The theory is based on the rigorous method of multi scale asymptotic expansion. Nonlinear equations for instability are obtained in third order of the perturbation theory. In this article, we explain in details the nonlinear stage of the instability and we find the nonlinear periodic vortices and the vortex kinks of Beltrami type. Знайдено великомасштабну нестійкість, яка виникає в рідині, що обертається під нахилом, у маломасштабній турбулентності. Турбулентність генерується маломасштабною зовнішньою силою з малим числом Рейнольдса. Зовнішня сила не має спіральності. Теорія побудована з використанням послідовного багатомасштабного асимптотичного методу. Нелінійні рівняння для нестійкості отримано в третьому порядку теорії збурень. Проведено детальне дослідження нелінійної стадії нестійкості та знайдено нелінійні періодичні вихори Бельтрамієвського типу та вихорові кінки. Найдена новая крупномасштабная неустойчивость, которая возникает в наклонно вращающейся жидкости с мелкомасштабной турбулентностью. Турбулентность генерируется мелкомасштабной внешней силой с малым числом Рейнольдса. Внешняя сила не имеет спиральности. Теория построена строгим методом многомасштабного асимптотического разложения. Нелинейные уравнения для неустойчивости получены в третьем порядке теории возмущений. Проведено детальное исследование нелинейной стадии неустойчивости и найдены нелинейные периодические вихри Бельтрамиевского типа и вихревые кинки. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Нелинейные процессы в плазменных средах The large scale instability and nonlinear vortex structures in obliquely rotating fluid with small scale non spiral force Великомасштабна нестійкість та нелінійні вихорові структури в рідині, що обертається під нахилом, з маломасштабною зовнішньою неспіральною силою Крупномасштабная неустойчивость и нелинейные вихревые структуры в наклонно вращающейся жидкости с мелкомасштабной внешней неспиральной силой Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
The large scale instability and nonlinear vortex structures in obliquely rotating fluid with small scale non spiral force |
| spellingShingle |
The large scale instability and nonlinear vortex structures in obliquely rotating fluid with small scale non spiral force Kopp, M.I. Tur, A.V. Yanovsky, V.V. Нелинейные процессы в плазменных средах |
| title_short |
The large scale instability and nonlinear vortex structures in obliquely rotating fluid with small scale non spiral force |
| title_full |
The large scale instability and nonlinear vortex structures in obliquely rotating fluid with small scale non spiral force |
| title_fullStr |
The large scale instability and nonlinear vortex structures in obliquely rotating fluid with small scale non spiral force |
| title_full_unstemmed |
The large scale instability and nonlinear vortex structures in obliquely rotating fluid with small scale non spiral force |
| title_sort |
large scale instability and nonlinear vortex structures in obliquely rotating fluid with small scale non spiral force |
| author |
Kopp, M.I. Tur, A.V. Yanovsky, V.V. |
| author_facet |
Kopp, M.I. Tur, A.V. Yanovsky, V.V. |
| topic |
Нелинейные процессы в плазменных средах |
| topic_facet |
Нелинейные процессы в плазменных средах |
| publishDate |
2015 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Великомасштабна нестійкість та нелінійні вихорові структури в рідині, що обертається під нахилом, з маломасштабною зовнішньою неспіральною силою Крупномасштабная неустойчивость и нелинейные вихревые структуры в наклонно вращающейся жидкости с мелкомасштабной внешней неспиральной силой |
| description |
In this paper, we find a new large scale instability which appears in obliquely rotating flow with the small scale turbulence, generated by external force with small Reynolds number. The external force has no helicity. The theory is based on the rigorous method of multi scale asymptotic expansion. Nonlinear equations for instability are obtained in third order of the perturbation theory. In this article, we explain in details the nonlinear stage of the instability and we find the nonlinear periodic vortices and the vortex kinks of Beltrami type.
Знайдено великомасштабну нестійкість, яка виникає в рідині, що обертається під нахилом, у маломасштабній турбулентності. Турбулентність генерується маломасштабною зовнішньою силою з малим числом Рейнольдса. Зовнішня сила не має спіральності. Теорія побудована з використанням послідовного багатомасштабного асимптотичного методу. Нелінійні рівняння для нестійкості отримано в третьому порядку теорії збурень. Проведено детальне дослідження нелінійної стадії нестійкості та знайдено нелінійні періодичні вихори Бельтрамієвського типу та вихорові кінки.
Найдена новая крупномасштабная неустойчивость, которая возникает в наклонно вращающейся жидкости с мелкомасштабной турбулентностью. Турбулентность генерируется мелкомасштабной внешней силой с малым числом Рейнольдса. Внешняя сила не имеет спиральности. Теория построена строгим методом многомасштабного асимптотического разложения. Нелинейные уравнения для неустойчивости получены в третьем порядке теории возмущений. Проведено детальное исследование нелинейной стадии неустойчивости и найдены нелинейные периодические вихри Бельтрамиевского типа и вихревые кинки.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/112219 |
| citation_txt |
The large scale instability and nonlinear vortex structures in obliquely rotating fluid with small scale non spiral force / M.I. Kopp, A.V. Tur, V.V. Yanovsky // Вопросы атомной науки и техники. — 2015. — № 4. — С. 264-269. — Бібліогр.: 17 назв. — англ. |
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ISSN 1562-6016. ВАНТ. 2015. №4(98) 264
THE LARGE SCALE INSTABILITY AND NONLINEAR VORTEX
STRUCTURES IN OBLIQUELY ROTATING FLUID WITH SMALL
SCALE NON SPIRAL FORCE
M.I. Kopp1,2, A.V. Tur3, V.V. Yanovsky1,2
1Institute for Single Crystals, National Academy of Science Ukraine, Kharkov, Ukraine;
2V.N. Karazin Kharkiv National University, Kharkov, Ukraine;
3Université de Toulouse [UPS], CNRS, Institut de Recherche en Astrophysique et Planétol-
ogie, 9 avenue du Colonel Roche, BP 44346, 31028 Toulouse Cedex 4, France
E-mail: yanovsky@isc.kharkov.ua
In this paper, we find a new large scale instability which appears in obliquely rotating flow with the small scale
turbulence, generated by external force with small Reynolds number. The external force has no helicity. The theory
is based on the rigorous method of multi scale asymptotic expansion. Nonlinear equations for instability are ob-
tained in third order of the perturbation theory. In this article, we explain in details the nonlinear stage of the insta-
bility and we find the nonlinear periodic vortices and the vortex kinks of Beltrami type.
PACS: 47.32C-; 47.27.De; 47.27.em; 47.55.Hd
INTRODUCTION
It is well known, that the rotating effects play an im-
portant role in many theoretical and practical applica-
tions for fluid mechanics [1] and are especially im-
portant for geophysics and astrophysics [2 - 4], when
one have to deal with rotating objects such as the Earth,
Jupiter, the Sun, etc. Rotating fluids could generate dif-
ferent wave and vortex motions, for example, gyroscop-
ic waves, Rossby waves, internal waves, located vorti-
ces and coherent vortex structures [4 - 7]. Among the
vortex structures, the most interesting are the large scale
ones, since they carry out the efficient transport of ener-
gy and impulse. The structures which have characteris-
tic scale much more than the scale of turbulence or the
scale of external force which generates this turbulence
are understood as large scale ones. In this paper we find
a new large scale instability in obliquely rotating flow
which is influenced by the small scale external force
with zero helicity. Its axis of rotation does not coincide
with the Z axis. This force supports small scale turbu-
lent fluctuations in fluid. The nonlinear large scale heli-
cal vortex structures such as Beltrami vortices or local-
ized kinks appear as a result of the development of this
instability in rotating fluid. This supposes that the exter-
nal small-scale force substitutes the action of small-
scale turbulence. Further we consider that the external
force acts in the plane (X, Y). Instability occurs only
when the vector of angular velocity of rotation Ω
is
inclined relatively to the plane (X, Y), as shown in
Fig.1. If the fluid is rotating around the axis Z strictly,
then instability does not occur. The helical 2D velocity
field ,x yW W turns around the axis Z, when Z changes
in the periodic wave (Fig. 2) and makes one turn in the
kink (Fig. 3). The found instability belongs to the class
of instabilities called hydrodynamic α-effects. For these
instabilities the positive feedback between velocity
components is typical:
0, T x x y yW W W
z
α ∂
∂ −∆ − =
∂
0,T y y x xW W W
z
α ∂
∂ −∆ + =
∂
and leads to the instability. α-effect origins from mag-
netic hydrodynamics, where it engenders the increase of
large scale magnetic fields (see for example [16]). Later
it was extended to ordinary hydrodynamics. Several
examples of hydrodynamics α-effect [8-15] are known
for today. From this point of view, in this study we
found a new example of the α-effect. The theory of this
instability is based on a rigorous method of multi-scale
development, which was proposed by Frisch, She and
Sulem for the theory of the AKA effect [13]. This
method allows to find the equations for large scale per-
turbations as the secular equations of the asymptotic
theory, to calculate the Reynolds stress tensor and to
find the instability. The small parameter of asymptotical
development is the number of Reynolds , 1.R R ≤ Our
paper is organized as follows: in Section 2 we formulate
the problem and the main equations in rotating system
coordinates; in Section 3 we discuss the concept of mul-
ti-scale development and we give the secular equations.
In Section 4 we calculate the velocity field of zero ap-
proximation. In Section 5 we describe the calculations
of the Reynolds stress and find the large scale instabil-
ity. In Section 6 we discuss the saturation of the insta-
bility and find the nonlinear stationary vortex structures.
The results obtained are discussed in the conclusions
given in Section 7.
1. THE MAIN EQUATIONS
AND FORMULATION OF THE PROBLEM
Let us examine the equations of motion for non-
compressible rotating fluid with the external force 0F
in rotating coordinates system:
( )
0
0
2
1 ,
V V V V
t
P V Fν
ρ
∂
+ ∇ + Ω× =
∂
= − ∇ + ∆ +
(1)
0.divV =
(2)
The external force 0F
is divergence-free. Here Ω
is angular velocity of fluid rotation, is viscosity, is
ISSN 1562-6016. ВАНТ. 2015. №4(98) 265
constant fluid density. Let us design characteristic am-
plitude of force f0, and its characteristic space and time
scale λ0 and t0 respectively. Then 0F
= f0 0F
0 0
,x t
tλ
.
We will design the characteristic amplitude of veloc-
ity, generated by external force as v0. We choose the
dimensionless variables ( , , )t x V
0
0
0 0 0 0 0 0
, , , , ,x t V F Px t V F P
t v f Pλ ρ
→ → → → →
2 2
0 0 0 0 0
0 0 0 02
0 0
= , = , = , = .v v ft P f vλ ν ν λ
ν λ λ ν
Then, in dimensionless variables the equation (1)
takes following forme:
0( ) = ,V R V V D V P V F
t
∂
+ ⋅∇ + × −∇ + ∆ +
∂
(3)
0 0= , =vR D Taλ
ν
.
0 0= , =vR D Taλ
ν
. Where R and
2 4
0
2
4=Ta λ
ν
Ω
are respectively the Reynolds number and the Taylor
number on scale 0λ . Further we will consider the
Reynolds number as small 1R ≤ and will construct on
this small parameter the asymptotical development.
Concerning the parameter D, we do not choose any
range of values for the moment. Let us examine the fol-
lowing formulation of the problem. We consider the
external force as being of small scale and of high fre-
quency. This force leads to small scale fluctuations in
velocity. After averaging, these rapidly oscillating fluc-
tuations vanish. Nevertheless, due to small nonlinear
interactions in some orders of perturbation theory, non-
zero terms can occur after averaging. This means that
they are not oscillatory, that is to say, they are large
scale. From a formal point of view, these terms are
secular, i.e., they create the conditions for the solvability
of large-scale asymptotic development. So, the purpose
of this paper is to find and study the solvability equa-
tions, i.e., the equations for the large scale perturbations.
Let us denote the small scale variables by
0 0 0= ( , )x x t
, and the large scale ones by
= ( , )X X T
. The small scale partial derivative opera-
tion
0 0
,ix t
∂ ∂
∂ ∂
, and the large scale ones ,
X T
∂ ∂
∂ ∂
are
written, respectively, as , ,i t i∂ ∂ ∇ and T∂ . To construct
a multi-scale asymptotic development we follow the
method which is proposed in [16].
2. THE MULTI-SCALE ASYMPTOTIC
DEVELOPMENT
Let us search the solution to equations (2) and (3) in
the following form:
1 0 0 1
1( , ) = ( ) ( )V x t W X v x Rv
R − + + +
2 3
2 3 ,R v R v+ + +
(4)
3 2 1 0 03 2
1 1 1( , ) = ( ) ( ) ( ) ( )P x t P X P X P X P x
R R R− − −+ + + +
2 3
11 2 3( ( )) .R P P X R P R P+ + + + + (5)
We introduce the slow variables 2
0=X R x
and
4
0=T R t which lead to the following expressions for
the spatial and temporal derivatives:
2= ,i ii R
x
∂
∂ + ∇
∂
(6)
4= ,t TR
t
∂
∂ + ∂
∂
(7)
2
2 4= 2 .jj j j jjj j R R
x x
∂
∂ + ∂ ∇ + ∂
∂ ∂
(8)
Using initial notation, the system of equations can be
written as:
4 2
2 2 4
0
( ) ( )( ) =
( ) ( 2 ) ,
i i j j k
t T j j ijk
i i
j j jj j j jj
R V R R V V D V
R P R R V F
ε∂ + ∂ + ∂ + ∇ +
= ∂ + ∇ + ∂ + ∂ ∇ + ∇ +
(9)
( )2 = 0.i
i iR V∂ + ∇ (10)
Substituting these expressions into the initial equa-
tions (2) and (3) and then gathering together the terms
of the same order, we obtain the equations of the multi-
scale asymptotic development and write down the ob-
tained equations up to order 3R inclusive. In the order
3R− there is only one the equation:
3 3 3= 0, = ( ).iP P P X− − −∂ ⇒ (11)
In order 2R− we have the equation:
2 2 2= 0, = ( ).iP P P X− − −∂ ⇒ (12)
In order 1R− we get a system of equations:
1 1 1 1 3 1 1= ( ) ,i i j k i j
t jj ijk i i jW W D W P P W Wε− − − − − − −∂ − ∂ + − ∂ +∇ − ∂ (13)
1 = 0.i
iW−∂ (14)
The system of equations (13) and (14) gives the sec-
ular terms
3 1= ,j k
i ijkP D Wε− −−∇ (15)
which corresponds to a geostrophic equilibrium equa-
tion. In zero order 0R , we have the following system of
equations:
0 0 1 0 0 1 0( ) =i i i j i j j k
t jj j ijkv v W v v W D vε− −∂ − ∂ + ∂ + +
0 2 0( ) ,i
i iP P F−= − ∂ +∇ + (16)
0 = 0.i
iv∂ (17)
These equations give the following secular equation:
2 2= 0, = .P P Const− −∇ ⇒ (18)
Let us consider the equations of the first approxima-
tion R :
1 1 1 1 1 1 1 0 0
1 1 1 1
( ) =
( ) ( ),
i i j k i j i j i j
t jj ijk j
i j
j i i
v v D v W v v W v v
W W P P
ε − −
− − −
∂ − ∂ + + ∂ + +
= −∇ − ∂ +∇
(19)
1 1 = 0.i i
i iV W−∂ + ∇ (20)
Secular equations follow from this system of equations:
1 = 0,i
iW−∇ (21)
1 1 1( ) = ,i j
j iW W P− − −∇ −∇ (22)
ISSN 1562-6016. ВАНТ. 2015. №4(98) 266
The secular equation (21) and (22) are satisfied by
choosing the following geometry for the velocity field
(Beltrami field):
1 1 1 1= ( ( ), ( ),0); = ( );x yW W Z W Z T T Z− − − −
(23)
1 1= 0, = .P P Const− −∇ ⇒
In the second order 2R , we obtain the equations:
2 2 0
1 2 2 1 0 1 1 0 2
2
( ) =
i i i
t jj j j
i j i j i j i j j k
j ijk
v v v
W v v W v v v v D vε− −
∂ − ∂ − ∂ ∇ +
+∂ + + + +
1 0 0 1 2 0= ( ) ( ),i j i j
j i iW v v W P P− −−∇ + − ∂ +∇ (24)
2 0 = 0.i iv v∂ +∇ (25)
It is easy to see that there are no secular terms in this
order. Let us come now to the most important order 3R .
In this order we obtain the equations:
3 1 3 1 1
1 1 1 1 0 0
( 2 )
( )
i i i i i
t T jj j j jj
i j i j i j
j
v W v v W
W v v W v v
− −
− −
∂ + ∂ − ∂ + ∂ ∇ +∇ +
+∇ + + +
1 3 3 1 0 2 2 0 1 1( )i j i j i j i j i j
j W v v W v v v v v v− −+∂ + + + + +
13 3= ( ),j k
ijk i iD v P Pε+ − ∂ +∇ (26)
3 1 = 0.i iv v∂ +∇
From this we get the main secular equation:
11 1 0 0( ) = ,i i k i
T k iW W v v P− −∂ − ∆ +∇ −∇ (27)
There is also an equation to find the pressure 3P− :
3 1= .j k
i ijkP D Wε− −−∇ (28)
3. THE VELOCITY FIELD IN ZERO
APPROXIMATION
It is clear that the most important is equation (27). In
order to obtain these equations in closed form, we need
to calculate the Reynolds stress 0 0( )k i
k v v∇ . First of all
we have to calculate the fields of the zero approxima-
tion 0
kv . From the asymptotic development in zero order
we have:
0 0 1 0 0 0 0= ,i i k i j k i
t jj k ijk iv v W v D v P Fε−∂ − ∂ + ∂ + −∂ + (29)
Let us introduce the operator 0D :
0 .k
t jj kD W≡ ∂ − ∂ + ∂ (30)
Using 0D , we rewrite equation (29) in the form:
0 0 0 0 0= ,i j k i
ijk iD v D v P Fε+ −∂ + (31)
Pressure P0 can be found from condition 0.divV =
0
0 2
i
i
D v
P
×∂ =
∂
(32)
Let us introduce designations for the operators:
2
i
ij j
D
P
×∂ = ∂
∂
(33)
and for velocities: 0 0 0 0 0 0, , .x y zv u v v v w= = = Then ex-
cluding pressure from (31), we obtain the system of
equations to find the velocity field of zero approxima-
tion:
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
,
,
.
x
xx yx z zx y
y
xy z yy zy x
z
xz y yz x zz
D P u P D v P D w F
P D u D P v P D w F
P D u P D v D P w F
+ + − + + =
+ + + + − =
− + + + + =
(34)
In order to solve this system of equations we have to
set the force in the explicit form. Let us choose now the
external force in the rotating system of coordinates in
the following form:
( )
( ) ( )
0 1 20 0 2 1 1 0 2 0
1 20 0
0, ; , ,
1,0,0 , 0,1,0 .
zF F f iCos jCos k x t k x t
k k k k
ϕ ϕ ϕ ω ϕ ω⊥= = + = − = −
= =
It is obvious that divergence and helicity of this
force us equal to zero: 0 0.F rotF =
Thus, the external
force is given in the plane (x, y), which is orthogonal to
the projection of angular velocity Ω
.
Fig. 1. In general, the angular velocity Ω is in-
clined relative to the plane (X, Y) in which there
is an external force 0F ⊥
The solution for equations system (34) can be found
easily in accordance with Cramer's Rule:
1 2 3
0 0 0, , .u v w∆ ∆ ∆
= = =
∆ ∆ ∆
(35)
Here ∆ − is the determinant of the system (34):
0
0
0
,
xx yx z zx y
xy z yy zy x
xz y yz x zz
D P P D P D
P D D P P D
P D P D D P
+ − +
∆ = + + −
− + +
(36)
0
1 0 0
0
,
0
x
yx z zx y
y
yy zy x
yz x zz
F P D P D
F D P P D
P D D P
− +
∆ = + −
+ +
(37)
0 0
2 0
0
,
0
x
xx zx y
y
xy z zy x
xz y zz
D P F P D
P D F P D
P D D P
+ +
∆ = + −
− +
(38)
0 0
3 0 0 .
0
x
xx yx z
y
xy z yy
xz y yz x
D P P D F
P D D P F
P D P D
+ −
∆ = + +
− +
(39)
Expanding the determinant, we obtain:
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
0 0 0 0
0 0
1
1 ,
x
yy zz yz x zy x
y
zx y yz x yx z zz
u D P D P P D P D F
P D P D P D D P F
= + + − + − + ∆
+ + + − − + ∆
(40)
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
0 0 0
0 0 0
1
1 ,
x
xz y zy x xy z zz
y
xx zz xz y zx y
v P D P D P D D P F
D P D P P D P D F
= − − − + + + ∆
+ + + − − + ∆
(41)
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
0 0 0
0 0
1
1 .
x
xy z yz x xz y yy
y
xz y yx z xx yz x
w P D P D P D D P F
P D P D D P P D F
= + + − − + + ∆
+ − − − + + ∆
(42)
ISSN 1562-6016. ВАНТ. 2015. №4(98) 267
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
0 0 0
0
0 .
xx yy zz yz x zy x
yx z xy z zz xz y zy x
zx y xy z yz x yy xz y
D P D P D P P D P D
P D P D D P P D P D
P D P D P D D P P D
∆ = + + + − + − −
− − + + − − − +
+ + + + − + −
(43)
In order to calculate the expressions (40) - (43) we
present the external force in complex form:
( ) ( )2 2 1 10 0
0 0, .
2 2
i i i ix yf fF e e F e eϕ ϕ ϕ ϕ− −= + = + (44)
Then all operators in formulae (40) - (42) act from
the left on their eigen function. In particular:
( ) ( )
( ) ( )
2 2 1 1
2 2 1 1
0 0 2 0 0 0 1 0
2 0 1 0
, , , ,
, , ,
i i i i
i i i i
D e e D k D e e D k
e e k e e k
ϕ ϕ ϕ ϕ
ϕ ϕ ϕ ϕ
ω ω
ω ω
= − = −
∆ = ∆ − ∆ = ∆ −
(45)
To simplify the formulae, let us choose
0 0 01, 1, 1.k fω= = =
We will designate:
( ) ( ) ( ) ( )0 2 0 0 1 0, 1 1 , , 1 1 .y y x xD k i w A D k i w Aω ω− = − − = − = − − = (46)
Before doing further calculations, we have to note
that some components of tensors ( )1ijP k and ( )2ijP k
vanish. Let us write the non-zero components only:
( ) ( ) ( ) ( )1 1 2 2, , , .yx zx xy zyz y z xP k D P k D P k D P k D= = − = − = (47)
Taking into account the formulae (45) - (47), we can
find the determinant:
( ) ( )3 2 3 2
1 2, .x x x y y yk A D A k A D A∆ = + ∆ = + (48)
In a similar way we find velocity field of zero ap-
proximation:
2
0 2 2
1 . .,
2
i
y
y y
e A
u C C
A D
ϕ
= +
+
(49)
1
0 2 2
1 . .,
2
i
x
x x
e Av C C
A D
ϕ
= +
+
(50)
2 1
0 2 2 2 2
1 1 . ..
2 2
i i
y x
y y x x
e D e Dw C C
A D A D
ϕ ϕ
= − +
+ +
(51)
We note that the angular velocity zD component
disappears from the expression for the velocity field of
zero approximation, which is a consequence of the
properties of an external force.
4. REYNOLDS STRESS AND LARGE SCALE
INSTABILITY
To close the equations (27) we have to calculate the
Reynolds stresses 0 0w u and 0 0w v . These terms are easi-
ly calculated with help of formulae (49) - (51). As a
result we obtain:
0 0 0 02 22 2 2 2
1 1, .
2 2
y x
y y x x
D Dw u w v
A D A D
= = −
+ +
(52)
Now equations (27) are closed and take form:
22 2
22 2
1 0,
2
1 0.
2
y
T x x
y y
x
T y y
x x
D
W W
z A D
DW W
z A D
∂
∂ − ∆ + =
∂ +
∂
∂ − ∆ − =
∂ +
(53)
We calculate the modules and write the equations
(53) in the explicit form:
( ) ( )
( ) ( )
22 2
22 2
1 0,
2 4 1 2
1 0
2 4 1
.
2
y
T x x
y y y y
x
T y y
x x x x
D
W W
z w D w w
DW W
z w D w w
∂
∂ − ∆ + =
∂ − + + −
∂
∂ − ∆ − =
∂ − + + −
(54)
With small ,x yW W we obtain the linear zed equa-
tions (54)
0,
0.
T x x y y
T y y x x
W W W
z
W W W
z
α
α
∂
∂ − ∆ − =
∂
∂
∂ − ∆ + =
∂
(55)
( )
( )
( )
( )
2 2
2 22 2
2 2
2 , 2 .
4 4
y y x x
y x
y x
D D D D
D D
α α
− −
= =
+ +
The system (55) describes the positive feedback be-
tween the components of velocity. We will look for the
solution of linear system (55) in the following form:
( ), exp .x yW W T ikZγ +
(56)
Substituting (56) in equation (55), we obtain the dis-
persion equation:
2.x y k kγ α α= ± − (57)
The dispersion equation (57) shows the existence at
0x yα α
the large scale instability with maximum
growth rate
max ,
4
x yα α
γ = at the wave vector
max
1 .
2 x yk α α= As a result of the development of insta-
bility the large scale helical Beltrami vortices are gener-
ated in the system. When 0x yα α
, damped oscilla-
tions with a frequency 0ω = x y kα α arise instead of
instability. In fact the behavior of γ depends on how is
located the external force 0 0,x yF F with respect to the
perpendicular projections of the angular velocity of ro-
tation and the values of components , .x yD D If one of
the component ,x yD D is zero or equal to 2 , then the
instability is absent. Instability exists in the following
cases:
2 2
2 2
2 2 2 2
2 2 2 2
1. 2, 2;
2. , 0, 2, 2;
3. 0, 0, 2, 2;
4. 0, 0, 2, 2;
5. 0, 0, 2, 2; 2, 2;
6. 0, 0, 2, 2; 2, 2;
x y
x y x y
x y x y
x y x y
x y y x y x
x y y x y x
D D
D D D D
D D D D
D D D D
D D D D orD D
D D D D orD D
In all other cases damped oscillations occur.
5. SATURATION OF INSTABILITY AND
NONLINEAR VORTEX STRUCTURES
It is clear that with increasing of amplitude nonlinear
terms decrease and instability becomes saturated. Con-
sequently stationary nonlinear vortex structures are
formed. To find these structures let us choose for equa-
tions (54) 0
T
∂
=
∂
and integrate equations one time over
Z. We obtain the system of equations:
ISSN 1562-6016. ВАНТ. 2015. №4(98) 268
( ) ( )
( ) ( )
122 2
222 2
1 ,
2 4 1 2
1
2 4
.
1 2
y
x
y y y y
x
y
x x x x
D
W C
w D w w
DW C
w D
d
dZ
d
dZ w w
= +
− + + −
= − +
− + + −
(58)
Let's take for this system new variables:
1 ,1 .x x y yw u w u− = − = Then we obtain:
( ) ( )
( ) ( )
122 2 2 4
222 2 2 4
1 ,
2 1 2 1
1 .
2 1 2 1
y
y y y y
x
x x x x
x
y
du
dZ
du
D
C
D D u u
D C
D D u udZ
= −
=
+
+ + − +
+
+ + − +
(59)
The system of equations (59) can be written in Ham-
iltonian form:
,
.
x
y
y
x
du H
dZ u
du H
dZ u
∂
= −
∂
∂
=
∂
Where Hamiltonian H has the form:
( ) ( ), , ,x x y yH h D u h D u= + (60)
with function ( ),h D u :
( )
( ) ( )22 2 2 4
, .
2 1 2 1
D duh D u Cu
D D u u
= +
+ + − +
∫ (61)
Integral in expression (61) is calculated in elemen-
tary functions [17]. Let us choose for simplicity
1.x yD D D= = = In this case, the function (61) is equal
[17]:
( )
2
2 2
1 2 2 2ln .
16 2 2 2
u u uh u arctg Cu
u u u
+ +
= + + − + −
(62)
The sum ( ) ( )x yh u h u+ can be write down as one
formula. Then Hamiltonian is equal:
( )
( )
( )
( )
( ) ( )
( )
2 2
2 2
2 2
1 22 2 2
2 2 2 21 ln
16 2 2 2 2
2 2 2 21
16 2 4
x x y y
x x y y
y x x y
x y
x y x y
u u u u
H
u u u u
u u u u
arctg C u C u
u u u u
+ + + +
= +
− + − +
− + −
+ + +
+ − −
(63)
It is easy to construct the phase portrait of Fig. 4. for
Hamiltonian (63) and specific values 1C = 0.1, 2C = 0.1.
Fig. 2. Nonlinear spiral wave Beltrami, which corre-
sponds to a closed trajectory in the phase plane
( 1C =0.1, 2C = 0.1). The spiral is oriented along Z-axis
and inclined relative to the axis of rotation
The phase portrait shows the presence of closed tra-
jectories in the phase plane around elliptic points and
separatrix that connect hyperbolic points. It is obvious
that the closed trajectories correspond to nonlinear peri-
odic solutions. The separatrix correspond to localized
solutions of kink type.
Fig. 3. Localized solution (kink), which corresponds to
the separatrix in the phase plane (C1=0.1, C2= 0.1)
Fig. 4. Phase plane for Hamiltonian (63) (C1=0.1,
C2= 0.1). We see the presence of closed trajectories
around the elliptic points and separatrix that connect
hyperbolic points. Phase portrait is typical
for Hamiltonian systems
CONCLUSIONS
In this work we found new large scale instability in
rotating fluid. It is supposed that the small scale vortex
external force in rotating coordinates system acts on
fluid which maintains the small velocity field fluctua-
tions (small-scale turbulence with low Reynolds number
, 1R R ). For the real applications this Reynolds num-
ber should be calculated with help of the turbulent vis-
cosity. The asymptotic development of motion equa-
tions by small Reynolds number allows obtaining mo-
tion equations for the large scale. These equations are of
the hydrodynamic α- effect type, in which velocity
components Wx, Wy are connected by the positive feed-
back. This may result in the appearance of the large
scale vortex instability. This instability is responsible
for the formation in rotating fluid with small scale ex-
ternal force of large scale Beltrami vortices. With fur-
ther increase of amplitude the instability stabilizes and
passes to a stationary mode. In this mode the nonlinear
stationary vortex structures are formed. The most inter-
esting structures belong to a variety of vortex kinks.
These kinks connect stationary hyperbolic points of the
dynamical system (58).
Note that in contrast to previous work on the hydro-
dynamic α- effect in rotating fluid, the method enables
us to construct an asymptotic development in a natural
way and to explore non-linear theory of nonlinear sta-
tionary vortex kinks.
ISSN 1562-6016. ВАНТ. 2015. №4(98) 269
REFERENCES
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Article received 09.06.2015
КРУПНОМАСШТАБНАЯ НЕУСТОЙЧИВОСТЬ И НЕЛИНЕЙНЫЕ ВИХРЕВЫЕ СТРУКТУРЫ
В НАКЛОННО ВРАЩАЮЩЕЙСЯ ЖИДКОСТИ С МЕЛКОМАСШТАБНОЙ ВНЕШНЕЙ
НЕСПИРАЛЬНОЙ СИЛОЙ
М.И. Копп, А.В. Тур, В.В. Яновский
Найдена новая крупномасштабная неустойчивость, которая возникает в наклонно вращающейся жидко-
сти с мелкомасштабной турбулентностью. Турбулентность генерируется мелкомасштабной внешней силой с
малым числом Рейнольдса. Внешняя сила не имеет спиральности. Теория построена строгим методом мно-
гомасштабного асимптотического разложения. Нелинейные уравнения для неустойчивости получены в тре-
тьем порядке теории возмущений. Проведено детальное исследование нелинейной стадии неустойчивости и
найдены нелинейные периодические вихри Бельтрамиевского типа и вихревые кинки.
ВЕЛИКОМАСШТАБНА НЕСТІЙКІСТЬ ТА НЕЛІНІЙНІ ВИХОРОВІ СТРУКТУРИ В РІДИНІ,
ЩО ОБЕРТАЄТЬСЯ ПІД НАХИЛОМ, З МАЛОМАСШТАБНОЮ ЗОВНІШНЬОЮ
НЕСПІРАЛЬНОЮ СИЛОЮ
М.І. Копп, А.В. Тур, В.В. Яновський
Знайдено великомасштабну нестійкість, яка виникає в рідині, що обертається під нахилом, у маломасш-
табній турбулентності. Турбулентність генерується маломасштабною зовнішньою силою з малим числом
Рейнольдса. Зовнішня сила не має спіральності. Теорія побудована з використанням послідовного багатома-
сштабного асимптотичного методу. Нелінійні рівняння для нестійкості отримано в третьому порядку теорії
збурень. Проведено детальне дослідження нелінійної стадії нестійкості та знайдено нелінійні періодичні
вихори Бельтрамієвського типу та вихорові кінки.
Introduction
1. The Main Equations and Formulation of the Problem
Великомасштабна нестійкість та нелінійні вихорові структури В рідині, що обертається під нахилом, з маломасштабною зовнішньою неспіральною силою
|