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Basile Gallet
Abstract

We consider the flow of a Newtonian fluid in a three-dimensional domain, rotating about a vertical axis and driven by a vertically invariant horizontal body-force. This system admits vertically invariant solutions that satisfy the 2D Navier-Stokes equation. At high Reynolds number and without global rotation, such solutions are usually unstable to three-dimensional perturbations. By contrast, for strong enough global rotation, we prove rigorously that the 2D (and possibly turbulent) solutions are stable to vertically dependent perturbations.

We first consider the 3D rotating Navier-Stokes equation linearized around a statistically steady 2D flow solution. We show that this base flow is linearly stable to vertically dependent perturbations when the global rotation is fast enough: under a Reynolds-number-dependent threshold value of the Rossby number, the flow becomes exactly 2D in the long-time limit, provided that the initial 3D perturbations are small. We call this property linear two-dimensionalization. We compute explicit lower bounds on and therefore determine regions of the parameter space where such exact two-dimensionalization takes place. We present similar results in terms of the forcing strength instead of the root-mean-square velocity: the global attractor of the 2D Navier-Stokes equation is linearly stable to vertically dependent perturbations when the forcing-based Rossby number is lower than a Grashof-number-dependent threshold value .

We then consider the fully nonlinear 3D rotating Navier-Stokes equation and prove absolute two-dimensionalization: we show that, below some threshold value of the forcing-based Rossby number, the flow becomes two-dimensional in the long-time limit, regardless of the initial condition (including initial 3D perturbations of arbitrarily large amplitude).

These results shed some light on several fundamental questions of rotating turbulence: for arbitrary Reynolds number and small enough Rossby number, the system is attracted towards purely 2D flow solutions, which display no energy dissipation anomaly and no cyclone-anticyclone asymmetry. Finally, these results challenge the applicability of wave turbulence theory to describe stationary rotating turbulence in bounded domains.

Two-dimensionalization of low- turbulence ]Exact two-dimensionalization of rapidly rotating large-Reynolds-number flows

1 Introduction

Global rotation is ubiquitous in geophysical, astrophysical and industrial flows. Uniform solid body rotation at angular frequency affects the fluid motion through the action of the Coriolis force, and allows for inertial waves: in an inviscid and incompressible fluid, an infinitesimal wave-like velocity disturbance obeys the inertial-wave dispersion relation,

(1.0)

where is the angular frequency, is the wave vector, , and is the component of the wave vector along the axis of global rotation (denoted as the vertical -axis by convention).

Both the linear and fully nonlinear behaviors of the flow are therefore affected by global rotation. For turbulent flows, the strength of global rotation can be characterized by the Rossby number , defined as the ratio of the global rotation period to the large-scale eddy turnover time. When the Rossby number is low, global rotation induces strong anisotropy: the flow tends to become two-dimensional, with flow structures weakly dependent on the coordinate along the rotation axis (Davidson 2013). This result is usually referred to as Taylor-Proudman theorem, which considers the asymptotic limit of vanishing Rossby number (infinite global rotation rate): fluid motion with characteristic time much larger than the rotation period is independent of the vertical.

Turbulent flows at large Reynolds number contain a broad range of spatial scales and temporal frequencies, including frequencies very large compared with the inverse large-scale eddy turnover time. While the large-scale and low-frequency structures of the flow become 2D for strong enough global rotation, the fate of small-scale high-frequency structures is less clear, and whether the latter become 2D as well for rapid global rotation is an open issue of rotating turbulence. This constitutes the central question of this study: are rotating flows more and more 2D as decreases, with a nonzero but decreasing fraction of the total energy contained in fully 3D flow structures, or do they become exactly two-dimensional under a critical value of the Rossby number, with no dependence at all along the vertical?

This central question is related to many of the fundamental questions addressed by experimental and numerical studies on rotating turbulence:

  • How much power per unit mass does a rotating turbulent flow dissipate? For stationary rotating turbulence with root-mean-square velocity and length scale , does display a dissipation anomaly, with , like in classical 3D turbulence (Frisch 1995; Doering & Foias 2002), or does it behave like 2D flows, with (Alexakis & Doering 2006)?

  • Why does rotating turbulence display less intermittency than its non-rotating counterpart (Baroud at al. 2003; Müller et al. 2007; Seiwert et al. 2008; Mininni et al. 2009)?

  • Global rotation induces an asymmetry of the vertical vorticity distribution. Such cyclone-anticyclone asymmetry is observed in experimental and numerical studies at moderately low values of the Rossby number (Bartello at al. 1994; Bourouiba & Bartello 2007; Smith & Waleffe 1999; Morize et al. 2005; Sreenivasan & Davidson 2008; Moisy et al. 2011; Deusebio et al. 2014; Gallet et al. 2014; Naso 2015). Does cyclone-anticyclone asymmetry persist for very low Rossby number, or is it a finite-Rossby-number effect?

  • Can low-Rossby-number rotating turbulence be described in the framework of weak turbulence of inertial waves (Galtier 2003; Cambon et al. 2004; Yarom & Sharon 2014; Scott 2015)?

Although these questions have been thoroughly addressed experimentally and numerically, exact mathematical results on this matter are scarce. Such exact results can be very valuable to test the various rotating turbulence models that have been proposed (see for instance Sagaut & Cambon (2008)): the model has to be compatible with the exact mathematical result in the range of parameters where the latter is valid.

In this paper we use rigorous analysis and estimates to answer the central question raised above: we consider the flow of a Newtonian fluid driven by a vertically invariant horizontal body force, and subject to steady global rotation about the vertical axis. We focus on domains that are periodic in the horizontal and bounded vertically by stress-free surfaces, although the results carry over to domains that are periodic in the three directions. Such boundary conditions are fashionable among numericists and more amenable to analysis than more realistic domains with no-slip boundary conditions. The system admits purely 2D (vertically invariant) solutions, either laminar or turbulent. In the absence of global rotation, such solutions are usually unstable to vertically dependent perturbations, so the flow is fully three-dimensional. By contrast, here we prove that, for strong enough global rotation, the 2D flow solutions are stable with respect to three-dimensional perturbations.

We first consider infinitesimal vertically dependent perturbations on a statistically-steady 2D base flow and prove linear two-dimensionalization: using a Reynolds number and a Rossby number based on the r.m.s. velocity (see section 2 for the exact definitions), we show that, for any given value of , there is a critical value of the Rossby number under which the – possibly turbulent – 2D flow is linearly stable to 3D perturbations. We compute some lower bounds on and we therefore determine a region of the parameter space where such exact two-dimensionalization takes place. For generic time-independent forcing the lower bound on is given by (6.0); it scales as , where is the vertical aspect ratio of the domain. This bound can be slightly improved if the forcing is of “single-mode” type, i.e., if it contains a single wavenumber (see equation (6.0)). We obtain similar results in terms of dimensionless numbers that involve the forcing strength instead of the r.m.s. velocity (exact definitions in section 2): when the forcing-based Rossby number is lower than a Grashof-number-dependent threshold value , the global attractor of the 2D Navier-Stokes equation is linearly stable to 3D perturbations. We determine regions of the parameter space where exact two-dimensionalization takes place by deriving a lower bound on , given by expression (6.0). It scales as .

We then consider the fully nonlinear rotating 3D Navier-Stokes equation, with arbitrarily large initial 3D velocity perturbations. Using a theorem from Babin et al. (2000) on the existence of a global attractor for the 3D rotating Navier-Stokes equation, we prove absolute two-dimensionalization: when the forcing-based Rossby number is lower than a threshold value , the flow becomes 2D in the long-time limit, regardless of the initial condition. This indicates that the global attractor of the 2D Navier-Stokes equation is the only attractor of the 3D rotating Navier-Stokes equation when .

The analysis consists in studying the stability of a (possibly turbulent) 2D base flow to vertically dependent 3D perturbations. The procedure is very different from a usual stability analysis, because we do not know the exact expression for this base flow, nor do we know its precise spatial and temporal dependence. The proofs therefore rely on rigorous upper bounds for several quantities associated with such 2D flows. These bounds provide sufficient information to determine regions of the parameter space where the 2D flow is stable to fully 3D perturbations.

A somewhat similar stability analysis of a possibly turbulent 2D base flow was performed in Gallet & Doering (2015), in the context of low-magnetic-Reynolds-number () magnetohydrodynamic (MHD) turbulence subject to a strong external magnetic field. However, we stress the fact that the proof of stability is very different in the two situations: in the low- MHD case, the proof relies on Ohmic dissipation strongly damping the 3D perturbations and therefore stabilizing the 2D flow. By contrast, for rotating flows the Coriolis force does not do work and global rotation does not appear directly in the energy budget. The essence of the proof is that global rotation strongly reduces the energy transfers from the 2D base-flow to the 3D perturbations: for strong enough rotation, these transfers are too weak to overcome viscous damping, and the 3D perturbations decay in the long-time limit. As a result, the 2D base-flow is stable.

The fact that global rotation reduces the transfers between the 2D and the vertically dependent 3D modes has been known since Greenspan (1990) and Waleffe (1993): in the asymptotic limit of low Rossby number, the 3D modes can be described in terms of weakly nonlinear inertial waves, and the dominant interaction between such waves consists of resonant triads. However, such resonant triads cannot transfer energy from the 2D modes to the 3D ones. This key result is obtained through a perturbative analysis and is valid to lowest order in Rossby number only. At higher order in Rossby number, near-resonant triads and four-wave interactions can transfer energy between the 2D and 3D modes (Smith & Waleffe 1999).

In the field of mathematical analysis, similar results were obtained by Babin et al. (1997, 2000), who translated into rigorous and exact analysis the concept of averaging over the fast rotation period, to study the regularity of solutions to the rotating Euler and Navier-Stokes equations. The proof of absolute two-dimensionalization that we present in section 7 makes extensive use of their theorem on the existence of bounded solutions to the rapidly rotating 3D Navier-Stokes equation (theorem 1 in Babin et al. (2000)).

Along the way to proving this theorem, they provide a decomposition of rotating flows on a time interval into three components: a 2D flow satisfying the 2D Navier-Stokes equation, some inertial waves that are advected and sheared by the 2D flow, and a small remainder. The wave part follows a reduced system of equations with coefficients depending on the 2D flow, and can be solved for exactly in some cases. The remainder decreases as but increases rapidly (typically exponentially) with the length of the time interval. This decomposition is useful when the remainder is indeed small, that is to say, in the limit of very fast rotation, for a given time interval. However, it cannot be used as is in the long-time limit () to answer the central question raised above. Here we therefore address stability to 3D perturbations head-on.

In the framework of geophysical fluid dynamics, the 2D base-flow corresponds to “balanced” fluid motion in the 2D slow manifold, while instability with respect to 3D perturbations corresponds to spontaneous wave generation (Vanneste 2013). The present study focuses on body forces that input energy directly into the 2D modes: we prove that the corresponding (possibly turbulent) flow settles exactly in the 2D slow manifold for low enough Rossby number, with no spontaneous wave generation.

We introduce the setup and notations in section 2, before describing the two-dimensional solutions to the three-dimensional problem. In sections 3 to 6, we consider the 3D rotating Navier-Stokes equation linearized about such a 2D base flow, and we prove the linear stability of these 2D solutions to 3D perturbations, for rapid global rotation: we derive sufficient criteria for such linear stability, either in terms of the Reynolds and Rossby numbers, or in terms of the Grashof and forcing-based Rossby numbers. The proof itself consists of 5 steps:

  • Write the evolution equation for the kinetic energy of the 3D perturbation.

  • Introduce a cut-off wavenumber , and control the small scales of the perturbation (wave numbers larger than ) with the viscous damping term (section 3).

  • Decompose the perturbation into helical modes (section 4).

  • Using an integration by parts in time, show that the transfer of energy from the 2D base flow to the large scales of the 3D perturbation (wave numbers smaller than ) is inversely proportional to the rotation rate (section 5).

  • For rapid global rotation, this transfer term is therefore weaker than the viscous damping of the 3D perturbation, hence the stability criterion (section 6).

In section 7, we consider arbitrary initial conditions for the velocity field, with arbitrarily large vertically-dependent perturbations. We repeat the steps listed above to prove absolute two-dimensionalization for fast enough global rotation.

On first reading, one may want to go directly from the end of section 2 to the concluding section 8, where we comment on the physical implications of these results.

2 Rotating turbulence in a periodic domain and 2D solutions

Figure 1: Flow of a Newtonian fluid in a frame rotating at angular velocity around the vertical axis. It is driven by a horizontal body-force f that is independent of the vertical. We assume periodic boundary-conditions in the horizontal, and stress-free boundaries at and .

2.1 Body-forced rotating turbulence

The setup is sketched in figure 1: an incompressible fluid of kinematic viscosity flows inside a domain with a Cartesian frame . The fluid is subject to background rotation at a rate around the axis, referred to as the vertical axis by convention. It is stirred by a steady divergence-free two-dimensional horizontal body-force that is periodic on a scale , an integer fraction of . That is, , where is periodic of period in each dimensionless variable, has vanishing spatial mean, and r.m.s. magnitude . We refer to as the amplitude, and as the shape of the force. We consider periodic boundary conditions in the horizontal directions, and stress-free boundary conditions in the vertical (although the proofs of the present study easily carry over to a 3D periodic domain). The velocity field follows the rotating Navier-Stokes equation,

(2.0)

together with the following boundary conditions at the top and and bottom boundaries,

(2.0)

We consider the solutions of equation (2.1) that have vanishing total momentum initially, and therefore at any subsequent time: the spatial average of the velocity field is zero at all time. From equation (2.1) we define the Reynolds number and the Rossby number based on the root-mean-square velocity of the flow, where the mean is performed over space and time:

(2.0)

We stress the fact that these dimensionless numbers combine the root-mean-square velocity of the solution with the forcing scale , instead of the typical scale of the velocity field (the integral scale). While the integral scale is probably close to for non-rotating 3D turbulence, it may increase very greatly and even reach the domain size for strong rotation, because of two-dimensionalization and enhanced inverse energy transfers. Nevertheless, the Reynolds and Rossby numbers defined in (2.1) are familiar to the theory of rotating turbulence, as well as to experimentalists: single-point velocity measurements usually lead to a good estimate of the root-mean square velocity and allow one to estimate and . Without loss of rigor, we will therefore present some results in terms of and defined in (2.1).

We also introduce dimensionless numbers based on the strength of the forcing. The Grashof number and the forcing Rossby number are

(2.0)

In contrast with and , and are control parameters: they do not require knowledge of the solution to be evaluated. For instance, and can be specified at the outset of a numerical simulation. In the following we present both results expressed in terms of and , which are useful for qualitative comparison with boundary-driven experiments, and results expressed with and , which are useful for comparison with body-forced numerical simulations or experiments.

In the following we use many inequalities. To alleviate the algebra somewhat, we make extensive use of the notation , where means that there is a dimensionless constant such that , where the constant is independent of the parameters of the problem: , , , , , , etc. This constant can depend only on the precise choice of the dimensionless shape function of the forcing. In the following, we denote as any such positive constant, and we sometimes use the same symbol to denote different constants in successive lines of algebra. Numbered constants ( in the appendix) keep the same value between different lines of algebra.

Finally, we consider only domains that are cubic or shallower than a cube, , and because we focus on the large-Reynolds-number behavior of the system, we restrict attention to and .

2.2 Two-dimensional solutions

Equation (2.1) with the boundary conditions (2.1) admits vertically invariant 2D solutions , where satisfies the two-dimensional Navier-Stokes equation,

(2.0)

Note that, in the 2D Navier-Stokes equation, the Coriolis force is a gradient that can be absorbed into the pressure term: disappears from the equation (and we keep using to denote the modified pressure). is a horizontal velocity field, with vanishing vertical component (for periodic boundary conditions in the vertical, the vertical component of satisfies a sourceless advection-diffusion equation and therefore vanishes in the long-time limit).

Rigorous bounds on the time-averaged enstrophy and enstrophy dissipation rate can be computed for solutions of the 2D Navier-Stokes equation (2.2). The derivation of these bounds is recalled in appendix A. In terms of the forcing amplitude , and denoting as the vertical vorticity of the 2D flow V, we obtain

(2.0)
(2.0)

where denotes time average, and is the standard norm in 3D:

(2.0)

Alternate bounds were obtained by Alexakis & Doering (2006) in terms of the r.m.s. velocity . Using our notations, their equations (23) and (19) translate into

(2.0)
(2.0)

where we restrict attention to .

These bounds for 2D flows can be further reduced if the forcing contains a single wavenumber in Fourier space, i.e., if it is such that is an eigenmode of the Laplacian operator (see Constantin et al. (1994) for a description of 2D turbulence driven by such forcing). These forcings are sometimes called “single-mode”, or Kolmogorov forcings. For such forcings the improved bounds on the enstrophy and enstrophy dissipation rate are

(2.0)
(2.0)

3 Linear perturbation to the 2D solution

Consider a 2D solution lying on the attractor of the 2D Navier-Stokes equation (2.2). Our goal is to prove linear two-dimensionalization: we wish to show that, for strong enough global rotation , this solution is stable with respect to infinitesimal 3D perturbations. We therefore consider the evolution of an infinitesimal perturbation to the two-dimensional flow . We write , where is infinitesimal, and consider the linearized evolution equation for :

(3.0)

where is the pressure perturbation. Because the base-flow is independent of the vertical, different vertical Fourier modes of evolve independently. We therefore consider a perturbation that has a single wavenumber in the vertical. More precisely, we consider the following Fourier decomposition of the perturbation,

(3.0)

where the wave vector takes the values , with . We denote this set of wave vectors as .

Dotting v into (3) and integrating over the domain leads to the evolution equation for the norm of the perturbation:

(3.0)

which we divide by to obtain

(3.0)

Our goal is to prove that, for large-enough , the time average of the right-hand-side of (3) is negative, and therefore decays to zero in the long-time limit.

3.1 Large versus small horizontal scales of the perturbation

From the spatial Fourier transform (3) of , we define a cut-off for the wavenumber , and we write , where contains all the Fourier modes with and contains all the Fourier modes with . Equation (3) becomes

We now bound all the terms on the right-hand side that involve . Using successively Hölder’s, the Cauchy-Schwarz, and Young’s inequalities together with an optimization,

(3.0)

where denotes the standard norm in space. From Poincaré’s inequality, and , hence

(3.0)

Inserting inequalities (3.0) and (3.0) in (3.1) results in

(3.0)

where . Notice that the triple velocity product in (3.0) does not involve anymore. In the following we choose the value

(3.0)

which leads to the following time-average of ,

(3.0)

which is sufficient control over small horizontal scales of the perturbation. The quantity is bounded from above in appendix A, which provides an upper bound on ,

(3.0)

where the right-hand-side involves the time-averaged enstrophy dissipation rate of the 2D base flow, which can be bounded in terms of its r.m.s. velocity using (2.2) or (2.2), or in terms of the forcing strength using (2.2).

4 Helical wave decomposition

We perform a standard helical wave decomposition of the Fourier amplitudes of the velocity perturbation (Cambon & Jacquin 1989; Waleffe 1993),

(4.0)

where is the frequency of a linear inertial wave with spatial structure . For non-vertical wave vectors, the latter is given by

(4.0)

with the unit vector along , and a sign coefficient. For vertical wave vectors, the structure of the helical modes is . These vectors are parallel to their curl, , and they are normalized, .

To obtain a similar decomposition for the -independent base-flow , we first write it as a Fourier series,

(4.0)

where the set contains all the horizontal wave vectors of the periodic domain: , with (recall that V has a vanishing average over the domain, see section 2.1). Because the frequency vanishes for horizontal wave vectors, the helical decomposition of each Fourier amplitude yields simply

(4.0)

and because is a horizontal flow, we get the additional relations

(4.0)
(4.0)

For brevity, in the following we often write , and only to designate respectively , and .

The oscillatory phases in the decomposition (4.0) absorb the Coriolis force: inserting the decompositions (3), (4.0), (4.0) and (4.0) into the curl of (3), we obtain

(4.0)

where the sum is over all and such that and over the two sign coefficients and . The following algebra involves many similar sums over wave vectors. In such sums, we omit to mention the sets in which the wave vectors are. In section 5, a sum over involving a base-flow component implies , while a sum over involving a perturbation component implies . We only mention the additional constraints under the sum symbol, e.g., . In section 7, we consider perturbations of arbitrary amplitude, and the sums over involving a perturbation component are over wave vectors , with , unless otherwise stated under the sum sign.

The coupling coefficient in (4.0) is

(4.0)

with

(4.0)

Using the decompositions (4.0) and (4.0), the triple velocity product in (3.0) reads

(4.0)

or, because and have symmetrical roles,

(4.0)

Because the sum is over , , and we can write

(4.0)

We see in this equation the result of Waleffe (1993) and Greenspan (1990): the coupling coefficient between the 2D base-flow and a couple of inertial waves is proportional to , and therefore it vanishes for resonant triads, i.e., for .

5 Control over the large scales of the perturbation

Integrate (3.0) from time to and divide by to obtain

Our goal is to show that the right-hand-side of this inequality is negative in the long-time- limit provided that is large enough: this ensures that decays. The key step is to prove that the contribution from the triple velocity product is small when is large. To wit, we insert the expression (4.0) of the triple velocity product, before performing an integration by parts in time, where we integrate the oscillatory exponential (and differentiate the rest of the integrand). This gives

(5.0)

where

(5.0)
(5.0)
(5.0)
(5.0)

To bound these terms, we use two inequalities on the coupling coefficients:

  • Because the helical vectors are normalized, . For wave vectors such that , and , we therefore have

    (5.0)
  • Using we also obtain

    (5.0)

Because the domain has a finite vertical extent , the vertical wavenumber satisfies for stress-free top and bottom boundaries (and for periodic boundary conditions). In either case, . Let us make use of (5.0) and the Cauchy-Schwarz inequality to bound :

Taking the limit ,

(5.0)

The time-averaged quantity appearing in the right-hand side is bounded in appendix A in terms of the time-averaged energy and enstrophy dissipation rates inside the 2D base flow (see A.5). The resulting bound on is

and after substituting the bound (3.0) on ,