^{1}

^{*}

^{2}

^{1}

In this paper, we find a new large scale instability in rotating flow forced turbulence. The turbulence is generated by a small scale external force at low Reynolds number. The theory is built on the rigorous asymptotic method of multi-scale development. The nonlinear equations for the instability are obtained at the third order of the perturbation theory. In this article, we explain the nonlinear stage of the instability and the generation vortex kinks.

It is well known, that the rotating effects play an important role in many practical and theoretical applications for fluid mechanics [_{x}, W_{y} turns around axis Z, when Z changes in the kink which links the hyperbolic point and the stable focus (_{x}, W_{y} is typical.

and leads to the instability. The α-effect is taking its origins from magnetic hydrodynamics, where it engenders the increase of large scale magnetic fields (see for example [

Let us examine the equations of motion for non-compressible rotating fluid with external force

The external force F_{0} is divergence-free. Here Ω-angular velocity of fluid rotation, _{0}, and its characteristic space and time scale

Then

Then, in dimensionless variables the Equation (1) takes forme:

ber on scale

Let us search for the solution to Equations (2) and (3) in the following form:

Let us introduce the following equalities:

Using indicial notation, the system of equation can be written as

Substituting these expressions into the initial Equations (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 obtained equations up to order R^{3} inclusive. In the order R^{−}^{3} there is only the equation

In order R^{−}^{2} we have the equation

In order R^{−}^{1} we get a system of equations:

The system of Equations (17) and (18) gives the secular terms

which corresponds to a geostrophic equilibrum equation.

In zero order R^{0}, we have the following system of equations:

These equations give one secular equation:

Let us consider the equations of the first approximation R:

From this system of equations there follows the secular equations:

The secular Equations (27) and (29) are satisfied by choosing the following geometry for the velocity field (Beltrami field):

In the second order R^{2}, we obtain the equations

It is easy to see that there are no secular terms in this order.

Let us come now to the most important order R^{3}. In this order we obtain the equations

From this we get the main secular equation:

There is also an equation to find the pressure

It is clear that the most important is Equation (36). In order to obtain these equations in closed form, we need to calculate the Reynolds stresses

Let us introduce the operator

Using

Pressure

Let us introduce designations for operatores:

and for velocities:

For simplicity, we choose the systeme of coordinates so that the axis Z coincides with the direction of angular velocity of rotation Ω. Then

It is obvious that divergence of this force us equal to zero. Thus, external force is given in plane (x, y), orthogonal to rotation axis.

The solution for equations system (34) can be found easily in accordance with Cramer’s Rule:

Here Δ is the determinant of the system (34):

After writing down the determinants in the explicit form, we obtain:

In order to calculate the expressions (40)-(43) we present the external force in complex form:

Then all operators in formulae (40)-(42) act from the left on their eigenfunctions. In particular:

To simplify the formulae, let us choose

We will designate

Before doing further calculations, we h ave to note that some components of tensors

Taking into account the formulae (45)-(47), we can find the determinant:

In a similar way we find velocity field of zero approximation:

To close the Equations (27) we have to calculate the Reynolds stresses

These terms are easily calculated with help of formulae (49)-(51). As a result we obtain:

Now Equations (27) are closed and take form:

We calculate the modules and write the Reynolds stresses (52) in the explicit form:

With small W_{x}, W_{y} Reynolds stresses (52) can be expanded in a series in the small parameters W_{x}, W_{y}. Taking into account the formula:

We obtain the linearized Equations (53):

We will search for the solution of linear system (55) in the form:

We substitute (56) in Equation (55) and obtain the dispersion equation:

The dispersion Equation (57) shows that equation system (55) has instable oscillatory solutions with oscillatory frequency

It is clear that with increasing of amplitude nonlinear terms decrease and instability becomes saturated. Consequently stationary nonlinear vortex structures are formed. To find these structures let us choose for Equations

(54)

From Equations (58) follows:

After integrating the system of Equations (59) we obtain:

Integrals in expression (60) are calculated in elementary functions (see [

Equations (58) can be easily ca lculated numerically using standard tools. In particular, this allows to construct phase portrait of the dynamical system (58) (

the stable knot and

In this work we find the 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 fluctuations (small scale turbulence with small Reynolds number R,_{x}, W_{y} are connected by the positive feedback. This may result in the appearance of the large scale vortex instability. The large scale vortices of Beltrami type are formed due to this instability in rotating fluid with small scale exterior force. With further increase of amplitude, the instability stabilizes and passes to stationary mode. In this mode the nonlinear stationary vortex structures form. Different vortex kinks belong to the most interesting structures. These kinks link stationary points of dynamical system (58). Kinks which link the hyperbolic point with the stable knot rotate around the stable knot as shown on

Let us note that unlike previous works about hydrodynamic α-effect in rotating fluid, the use of the asymptotic development allows constructing naturally the nonlinear theory and studying the stationary nonlinear vortex kinks.