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In this paper, we find a new large scale instability displayed by a stratified rotating flow in forced turbulence. The tur bulence is generated by a small scale external force at low Reynolds number. The theory is built on the rigorous as ymptotic method of multi-scale development. There is no other special constraint concerning the force. In previous pa pers, the force was either helical or violating parity invariance. The nonlinear equations for the instability are obtained at the third order of the perturbation theory. In this article, we explain a detailed study of the linear stage of the instabil ity.

Large scale instabilities are very important in fluid dynamics. They generate vortices which play a fundamental role in turbulence and in transport processes. The characteristic dimensions of the large scale structures are greater than the typical scale of the turbulence. The turbulence is often simulated using a small scale external force. In this case, the large scale vortices are much greater than the scale of the external force. Large scale vortices are well observed in planetary atmospheres [1, 2], in numerical simulations, and in laboratory experiments [3-10]. The generation process of large scale instabilities has been studied in several papers [11-19]. In these papers, the turbulence which generates these coherent large scale structures cannot be homogenous, isotropic, or mirror invariant. A series of papers have shown that the essential mechanism which leads to the generation of large scale vortices is the lack of reflection invariance. This mechanism was called the hydrodynamic α-effect by analogy with the similar mechanism of generation of large scale magnetic fields.

Turbulence lacking reflection invariance is helical and a pseudo-scalar appears. Nevertheless, the helicity of turbulence by itself can not generate large scale vortices. Other factors which lack reflection invariance are necessary, such as, for instance, compressibility [16,19] or temperature gradients [17,18]. Large scale instability can also appear if the turbulence lacks parity invariance (AKA effect) [

Large scale instabilities in a stratified rotating flow were studied in [21,22]. In [

The occurrence of large scale instability in helical stratified turbulence was confirmed by the multi-scale development method in [

Direct numerical simulation of the Boussinesq Equation confirmed the existence of large scale vortex generation in stratified and rotating flows [24,25]. Sometimes the appearance of large scale vortex structures is accompanied by an inverse cascade of energy both in the threedimensional case (AKA-effect [

Let us consider the equations for the motion of an incompressible fluid with a constant temperature gradient in the Boussinesq approximation:

Here, , is the thermal expansion coefficient, is the constant equilibrium gradient of the temperature, , and. The external force has zero divergence. Let be, respectively, the characteristic scale, time, amplitude of the external force, and velocity of our system. We choose the dimensionless variables

Then,

where where R and

are respectively the Reynolds number and the Taylor number on scale. represents the Prandtl number. We introduce the dimensionless temperature

, and obtain the system of equations

Here, is the Rayleigh number on the scale. Furthermore, for the purpose of simplification, we will consider the case. We pass to the new temperature, and obtain

We will consider as a small parameter of an asymptotic development the Reynolds number

on the scale. Concerning the parameters and, we do not choose any range of values for the moment. Let us examine the following formulation of the problem. We consider the external force as being small and of high frequency. This force leads to small scale fluctuations in velocity and temperature against a background of equilibrium. After averaging, these quickly oscillating fluctuations vanish. Nevertheless, due to small nonlinear interactions in some orders of perturbation theory, nonzero 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 a large scale asymptotic development. So the purpose of this paper is to find and study the solvability equations, i.e., the equations for large scale perturbations. Let us denote the small scale variables by, and the large scale ones by. The small scale partial derivative operation, and the large scale ones are written, respectively, as

and. To construct a multi-scale asymptotic development we follow the method which is proposed in [

Let us search for the solution to Equations (4) and (5) in the following form:

Let us introduce the following equalities: and which lead to the expression for the space and time derivatives:

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

Substituting these expressions into the initial equations (4) and (5) and then gathering together the terms of the same order, we obtain the equations of the multiscale asymptotic development and write down the obtained equations up to order inclusive. In the order there is only the Equation

In order we have the equation

In order we get a system of equations:

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

which corresponds to a geostrophic equilibrum Equation, and

In zero order, 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:

In the second order, 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. 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 essential Equation for finding the nonlinear alpha-effect is Equation (36). In order to obtain these equations in closed form, we need to calculate the Reynolds stresses. First of all we have to calculate the fields of zero approximation. From the asymptotic development in zero order we have

Let us introduce the operator:

Using, we rewrite Equations (39) and (40):

Eliminating the temperature and pressure from Equation (42), we obtain

Here, is the projection operator

Dividing this equation by, we can write it in the form

where is the operator given by

We must now determine the inverse operator :

After some calculation, we find

Here,

and

Consequently, the expression for the velocity takes the form

In order to use these formulas, we have to specify in explicit form the external force. Let us specify it by

where

or

One can check that and

Formulas (50) and (52) allow us to easily make intermediate calculations, but in the final formulas we obviously shall take and as equal to unity, since the external force is dimensionless and depends only on the dimensionless arguments of space and time. The force (50) is physically simple and can be realized in laboratory experiments and in numerical simulations.

The force (50) can be written in complex form:

where and have the forms

The effect of the operator on the proper function has obviously the form

where is

From this it follows that

From Formulas (49) and (53), it follows that the field is composed of four terms:

where

Finally, we introduce the notation

where. Taking into account these formulas, we can write down the velocities in the form

where

and

We can now calculate the Reynolds stresses:

which can be decomposed into two components:

where and can be expressed as follows:

Taking into account Formulas (64) and (65), we obtain

We can write down the components and, which are the ones of interest:

Finally, using the following relations (we have similar formulas for after replacing with):

We can then express, , and:

where

Let us write down in the explicit form the equations for nonlinear instability:

where the components, , and of the Reynolds stress tensor are as defined in the previous section.

One can see that for small values of the variables and, Equations (83) and (84) are reduced to linear equations and describe the linear stage of instability:

where the coeficients, , and can be written as

with

and

which are the explicit forms of the quite bulky coefficients. However, these coefficients can be expressed using the internal helicity of the velocity field, calculated in Appendix B.

.

Therefore, we can write the constant coefficients and with respect to:

where.

Equations (85) and (86) can then be rewritten:

These formulas show that despite the zero helicity of the driving force, inside the system, an internal helicity is generated as a result of the joint impact of the Coriolis and buoyancy forces. This helicity plays an important role in the dynamics of the perturbations.

In order to find instabilities, we choose the velocity in the form:

Injecting these solutions into (85), we obtain the simple system of equations:

Evidently we get a quadratic equation for:

which allows us to obtain the dispersion equations for the different modes.

This equation is obtained by searching for solutions of (96) for which the discriminant is negative, namely,. We show Figures 1 and 2 representing the area (in gray) of the plane for which the discriminant is negative, this means that an instability can appear.

Finally, we get

where

(97) is the growth rate of the instability. We note that it is proportionnal to the square of the helicity.

This Equation is obtained by searching for solutions of (96) for which the discriminant is positive, namely,.

We obtain in this case two oscillatory modes, and, which are, respectively, a slow and a fast mode:

It appears that both slow and fast oscillatory frequencies are proportional to the square of the helicity as well.

In the same way as before, we get the system

We can then get a new quadratic Equation for:

The discriminant of this Equation is the same as in the nonviscous case, so the dispersion equation for the unstable mode has the same condition, namely , which leads to:

where

It is to be noted that the growth rate is maximal for, which can be considered as the characteristic scale of the generated vortex structures. Below is

It can be noted that if the discrimant is positive, we get an oscillation with an exponentially decreasing amplitude.

With increasing amplitude, the instability becomes nonlinear and stabilizes. As a result, nonlinear vortex structures appear. The nonlinear stage of this instability and the results of numerical simulations will be presented in a future paper.

In this paper, we showed that a large scale instability can appear in a rotating stratified fluid which is under the impact of a simple small scale external force (turbulence). The scale of this instability is much larger than the scale of the external force or turbulence. It is important to emphasize that, unlike previous papers about large scale instabilities, in the present paper, there are no special constraints imposed on the external force. It has a zero helicity and its parity needs not be violated; this means that this is a general force. Nevertheless, the small scale turbulence under the impact of the Coriolis force and the buoyancy force becomes helical. This helicity, finally, is responsible for the generation of large scale instabilities because the growth rate is proportional to. The instability itself is oscillating while its frequency and have, in principle, the same order. This means that the instability in the general case is aperiodic. The frequency of both the stable and unstable oscillations is also proportional to. So we can say

that the oscillation modes are inertial oscillations of the rotating fluid strongly modified by the helicity. There are two oscillating modes: one slow and one fast. These oscillations decay when the viscosity is taken into account and in the case of instability, the maximal growth rate is reached at a characteristic scale of. Thereby this scale is typical for vortex structures like Beltrami’s runaways. In this paper, the theory of a large scale instability was constructed using the method of multi-scale developments, which was proposed in the work of Frisch, She and Sulem [

In order to calculate the Reynolds stress, we begin with the general expression

with

and

Hence,

and has a similar expression.

Taking into account that only the components and of the external force are nonzero, and after some factorizations, we can write the two contribution of the Reynolds stress tensor in the following form:

The same calculation for the contribution gives us

Calculation of the HelicityThe driving force has no helicity, but the joint action of the external force, Coriolis force, and the buoyancy give the internal helicity.

The general helicity of the velocity field is expressed by

where we choose and such that

and

and in indicial notation:

We must calculate with

is calculated in the same way, by replacing with.

We finally obtain

After linearization:

where we recall that.

One can note that for small perturbations, the helicity approaches the constant:

which can be considered as the internal helicity of the field when there are no perturbations.