Interplay between geostrophic vortices and inertial waves in precession-driven turbulence F. Pizzi

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Interplay between geostrophic vortices and inertial waves in precession-driven

turbulence

F. Pizzi

Institute of Fluid Dynamics, Helmholtz-Zentrum Dresden-Rossendorf,

Bautzner Landstrasse 400, D-01328 Dresden, Germany and

Department of Aerodynamics and Fluid Mechanics,

Brandenburg University of Technology, Cottbus-Senftenberg, 03046 Cottbus, Germany∗

G. Mamatsashvili

Institute of Fluid Dynamics, Helmholtz-Zentrum Dresden-Rossendorf,

Bautzner Landstrasse 400, D-01328 Dresden, Germany and

E. Kharadze Georgian National Astrophysical Observatory, Abastumani 0301, Georgia

A. J. Barker

Department of Applied Mathematics, School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK

A. Giesecke and F. Stefani

Institute of Fluid Dynamics, Helmholtz-Zentrum Dresden-Rossendorf,

Bautzner Landstrasse 400, D-01328 Dresden, Germany

The properties of rotating turbulence driven by precession are studied using direct numerical

simulations and analysis of the underlying dynamical processes in Fourier space. The study is

carried out in the local rotating coordinate frame, where precession gives rise to a background

shear ﬂow, which becomes linearly unstable and breaks down into turbulence. We observe that

this precession-driven turbulence is in general characterized by coexisting two dimensional (2D)

columnar vortices and three dimensional (3D) inertial waves, whose relative energies depend on the

precession parameter P o. The vortices resemble the typical condensates of geostrophic turbulence,

are aligned along the rotation axis (with zero wavenumber in this direction, kz= 0) and are fed by

the 3D waves through nonlinear transfer of energy, while the waves (with kz6= 0) in turn are directly

fed by the precessional instability of the background ﬂow. The vortices themselves undergo inverse

cascade of energy and exhibit anisotropy in Fourier space. For small P o < 0.1 and suﬃciently high

Reynolds numbers, the typical regime for most geo-and astrophysical applications, the ﬂow exhibits

strongly oscillatory (bursty) evolution due to the alternation of vortices and small-scale waves. On

the other hand, at larger P o > 0.1 turbulence is quasi-steady with only mild ﬂuctuations, the

coexisting columnar vortices and waves in this state give rise to a split (simultaneous inverse and

forward) cascade. Increasing the precession magnitude causes a reinforcement of waves relative to

vortices with the energy spectra approaching the Kolmogorov scaling and, therefore, the precession

mechanism counteracts the eﬀects of the rotation.

I. INTRODUCTION

Rotating turbulence is an ubiquitous phenomenon in a

broad context ranging from astrophysical and geophys-

ical ﬂows [1–4] to industrial applications [5, 6]. Under-

standing the impact of rotation on the turbulence dy-

namics is far from trivial due to the complexity of the

nonlinear processes involved. In general, when a ﬂuid

is subjected to rotational motion, the nonlinear interac-

tions are aﬀected by the Coriolis force whose strength is

quantiﬁed by the Rossby number, Ro (the ratio of the

advection time-scale to the rotation time-scale) and the

Reynolds number, Re (the ratio of advective to viscous

time-scale). If the Coriolis force is strong enough the for-

mation of coherent columnar vortices occurs inside the

ﬂuid ﬂow. This phenomenon has been observed in ex-

∗f.pizzi@hzdr.de

perimental campaigns for several systems such as oscil-

lating grids [7], for decaying turbulence [8, 9], forced tur-

bulence [10–12], and turbulent convection [13, 14]. Also

numerical simulations have been instrumental to analyze

such tendency in a myriad of cases [15–21], making use of

large eddy simulations [22–24] and even turbulence mod-

els [25, 26].

The emergence of columnar vortices aligned along the

ﬂow rotation axis is accompanied by inertial waves which

are inherent to rotating ﬂuids. Their frequency mag-

nitude ranges between zero and twice the rotation rate

Ω of the objects [27]. The dynamics and mutual cou-

plings between these two basic types of modes largely

depend on Re and Ro. The results of the asymptotic

analysis at Ro 1 indicate that three-dimensional (3D)

inertial waves and two-dimensional (2D) vortices are es-

sentially decoupled and evolve independently: vortices

undergo inverse cascade, while the wave energy cas-

cades forward through resonant wave interactions in the

regime of weakly nonlinear inertial wave turbulence [28–

arXiv:2210.12536v1 [physics.flu-dyn] 22 Oct 2022

31]. However, this picture does not carry over to mod-

erate Rossby numbers Ro ∼0.1, where the situation is

much more complex, since 3D inertial waves (the so called

fast modes) and 2D vortices (also called slow modes)

can coexist and be dynamically coupled. In this case,

asymptotic analysis cannot be used and more complex

mathematical models have been proposed to explain the

geostrophic vortices-wave interaction, such as the quartic

instability [32] or near-resonant instability [33]. However

progress can be made mainly by numerical simulations.

Several works are devoted to the study of these two mani-

folds and their interactions for forced rotating turbulence

[29, 34–36] and also for convective and rotating turbu-

lence [37, 38].

Indeed other forcing mechanisms have been shown to

be characterized by this interplay of vortices and waves,

such as elliptical instabilities [39–43] and tidal forcing

[44–48]. In this respect, the precession-driven dynamics

represents a possible candidate for the development of

both 3D waves with embedded 2D vortices [36, 49] but

so far these studies does not investigate a wide range of

governing parameters. Other works were devoted to the

stability analysis of the precession ﬂows [50].

The modiﬁed local Cartesian model of a precession-

driven ﬂow was proposed by Mason and Kerswell [51] and

later used by Barker [45] to study its nonlinear evolution.

In the ﬁrst paper, rigid and stress-free axial boundaries in

the vertical direction were used , while in the second pa-

per an unbounded precessional ﬂow was considered in the

planetary context, employing the decomposition of per-

turbations into shearing waves. In this paper, we follow

primarily the approach of Barker [45], who analyzed the

occurrence of vortices, function of the precession param-

eter (Poincar´e number), including energy spectrum and

dissipation properties. The main advantage of the lo-

cal model is that it allows high-resolution study of linear

and nonlinear dynamical processes in precession-driven

ﬂows, which is much more challenging in global models.

Also, this model allows to focus only on the dynamics

of the bulk ﬂow itself avoiding the complications due to

boundary layers. This is important for gaining a deeper

understanding of perturbation evolution in unbounded

precessional ﬂows and then, comparing with the global

simulations, for pinning down speciﬁc eﬀects caused by

boundaries.

In this paper, we continue this path and investigate in

detail the underlying dynamical processes in the turbu-

lence of precessional ﬂow in the local model. We decom-

pose perturbations into 2D and 3D manifolds and anal-

yse their dynamics and interplay in Fourier space. Our

main goal is to address and clarify several key questions:

(i) how the presence and properties of columnar vortices

depend on the precession strength and Reynolds number

(here deﬁned as the inverse Ekman number), (ii) what are

the mechanisms for the formation of the columnar vor-

tices in precessing driven ﬂows, in particular, how their

dynamics are aﬀected by precessional instability of iner-

tial waves, that is, if there are eﬀective nonlinear trans-

fers (coupling) between vortices and the waves; (iii) what

are the dominant nonlinear processes (channels) in this

vortex-wave system, i.e., the interaction of 2D-3D modes

or 2D-2D modes (inverse cascade), (iv) in terms of to-

tal shell-average spectral analysis, what type of cascades

(inverse, forward) occur and what kind of spectra char-

acterize precessional ﬂows.

The paper is organized as follows: in Section II the

local model and governing equations in physical and

Fourier space are presented, and numerical methods in-

troduced. Section III presents general evolution of the

volume-averaged kinetic energy and dynamical terms as

well as ﬂow structure. In Section II B we investigate the

nonlinear dynamics of 2D vortices and 3D inertial waves

and nonlinear interaction between them in Fourier space.

In this section we also characterize turbulent dissipation

as a function of precession parameter P o. Discussions

and the future perspectives are presented in Section V.

II. MODEL AND EQUATIONS

We consider a precessional ﬂow in a local rotating

Cartesian coordinate frame (also referred to as the ‘man-

tle frame’ of a precessing planet) in which the mean total

angular velocity of ﬂuid rotation Ω= Ωezis directed

along the z-axis. In this frame, the equations of motion

for an incompressible viscous ﬂuid take the form (see a

detailed derivation in Refs. [45, 51]):

∂U

∂t +U·∇U+ 2Ω(ez+(t)) ×U=

−1

ρ∇P+ν∇2U+ 2zΩ2(t),(1)

∇·U= 0,(2)

where Uis the velocity in this frame, ρis the spatially

uniform density and Pis the modiﬁed pressure equal to

the sum of thermal pressure and the centrifugal potential.

The last two terms on the left-hand side in the brackets

are the Coriolis and the Poincar´e forces, respectively, and

(t) = P o(cos(Ωt),−sin(Ωt),0)Tis the precession vec-

tor with P o being the Poincar´e number characterizing

the strength of the precession force. The last term on

the right-hand side is the second part of the precession

force with vertical shear, which is the main cause of hy-

drodynamic instability in the system, refereed to as the

precessional instability [52]. νis the constant kinematic

viscosity.

The basic precessional shear ﬂow in this local frame

represents an unbounded horizontal ﬂow with a linear

shear along the vertical z-axis and oscillating in time t,

i.e., Ub= (Ubx(z, t), Uby (z, t),0) with the components

given by [45, 51]

Ub=−2Ω P o 



0 0 sin(Ωt)

0 0 cos(Ωt)

0 0 0









≡Mr,

FIG. 1. Sketch of the periodic cubic domain with length L

in each direction where the base ﬂow inside it is Ubwith

superimposed perturbation velocity u.

where r= (x, y, z) is the local position vector. Our local

model deals with perturbations to this basic ﬂow, u=

U−Ub, for which from Eq. (1) we obtain the governing

equation:

∂u

∂t +u·∇u=−1

ρ∇P+ν∇2u−2Ωez×u−

2Ωε(t)×u−Mu−Mr· ∇u,(3)

where the last three terms on the rhs are related to pre-

cession and proportional to P o. The ﬂow ﬁeld is conﬁned

in a cubic box with the same length Lin each direction,

Lx=Ly=Lz=L. In other words, both horizontal and

vertical aspect ratios of the box are chosen to be equal to

one in this paper. Varying these aspect ratios aﬀects the

linearly unstable modes that can be excited in the ﬂow

and the properties of the vortices [39].

Below we use the non-dimensionalization of the vari-

ables by taking Ω−1as the unit of time, box size Las the

unit of length, ΩLas the unit of velocity, ρL2Ω2as the

unit of pressure and perturbation kinetic energy density

E=ρu2/2. The key parameters governing a precession-

driven ﬂow are the Reynolds number (inverse Ekman)

deﬁned as

Re =ΩL2

and the Poincar´e number P o introduced above.

A. Governing equations in Fourier space

Our main goal is to perform the spectral analysis of

precession-driven turbulence in Fourier (wavenumber k-)

space in order to understand dynamical processes (energy

injection and nonlinear transfers) underlying its suste-

nance and evolution. To this end, following [39, 45], we

decompose the perturbations into spatial Fourier modes

(shearing waves) with time-dependent wavevectors k(t),

f(r, t) = X

f(k(t), t)eik(t)·r,(4)

where f≡(u, P ) and their Fourier transforms are

f≡(¯

u,¯

P). In the transformation (4), the wavevector

Re = 103.5

P o N hEi

0.01 64 -

0.025 64 -

0.05 64 -

0.075 64 -

0.1 64 -

0.125 64 -

0.15 64 -

0.175 64 -

0.2 64 -

0.225 64 -

0.25 64 -

0.3 64 6.09 ×10−5

Re = 104

P o N hEi

0.01 64 -

0.025 64 -

0.05 64 -

0.075 128 -

0.1 128 -

0.125 128 1.89 ×10−5

0.15 128 6.04 ×10−5

0.175 128 1.42 ×10−4

0.2 128 3.54 ×10−4

0.225 128 6.11 ×10−4

0.25 128 9.94 ×10−4

0.3 128 2.10 ×10−3

Re = 104.5

P o N hEi

0.01 128 -

0.025 128 -

0.05 128 -

0.075 128 3.82 ×10−5

0.1 128 1.13 ×10−4

0.125 128 4.93 ×10−4

0.15 256 1.30 ×10−3

0.175 256 2.30 ×10−3

0.2 256 5.40 ×10−3

0.225 256 6.20 ×10−3

0.25 256 6.40 ×10−3

0.3 256 9.00 ×10−3

0.5 256 1.25 ×10−2

Re = 105

P o N hEi

0.01 256 -

0.025 256 -

0.05 256 4.55 ×10−5

0.075 256 2.37 ×10−4

0.1 256 1.10 ×10−3

0.125 256 5.90 ×10−3

0.15 256 6.80 ×10−3

0.175 256 7.50 ×10−3

0.2 256 8.90 ×10−3

0.225 256 1.02 ×10−2

0.25 256 1.14 ×10−2

0.3 256 1.15 ×10−2

TABLE I. List of all simulations performed in the present

work. Each subtable corresponds to a speciﬁc Reynolds num-

ber and various Poincar´e numbers P o (ﬁrst column). The

second column shows numerical resolution N(before dealias-

ing), which is the same in each direction, Nx=Ny=Nz=N

(total number of point is N3). The third column shows the

time- and volume-averaged kinetic energy hEi. Runs marked

with hyphen are not sustained and quickly decay. Notice that

for Re = 104.5we have run also a simulation at very large

P o = 0.5.

of modes oscillates in time,

k(t)=(kx0, ky0, kz0+ 2P o(−kx0cos(t) + ky0sin(t)))T,

(5)

about its constant average value hk(t)i= (kx0, ky0, kz0)

due to the periodic time-variation of the basic preces-

sional ﬂow Ub. Substituting Eq. (4) into Eq. (3) and

taking into account the above non-dimensionalization, we

obtain the following equation governing the evolution of

velocity amplitude

d¯

dt =−ik(t)¯

P−k2

Re ¯

u−2ez×¯

−2ε(t)×¯

u−M(t)¯

u+Q,(6)

k(t)·¯

u= 0.(7)

Note that the wavevector k(t) as given by expression (5)

satisﬁes the ordinary diﬀerential equation

dt =−MTk(8)

and as a result the last term on the rhs of Eq. (3) related

to the basic ﬂow has disappeared when substituting (4)

into it. The term Q(k, t) on the rhs of Eq. (6) represents

the Fourier transform of the nonlinear advection term

u·∇u=∇ · (uu) in the original Eq. (3) and is given by

convolution [35, 53]

Qm(k, t) = −iX

kn¯um(k0, t)¯un(k−k0, t),(9)

where the indices (m, n) = (x, y, z). This term describes

the net eﬀect of nonlinear triadic interactions (transfers)

among a mode kwith two others k−k0and k0and thus

plays a key role in turbulence dynamics.

Multiplying both sides of Eq. (6) by the complex con-

jugate of spectral velocity ¯

u∗, the contribution from Cori-

olis and part of the Poincar´e force in the total kinetic en-

ergy of a mode cancel out, since they do not do any work

on the ﬂow, ¯

u∗·(2ez×¯

u−2ε(t)×¯

u) = 0, and as a result

we obtain the equation for the (non-dimensional) spec-

tral kinetic energy density E=|¯

u|2/2 in Fourier space

dt =1

2¯

u∗(M¯

u) + ¯

u(M¯

u)∗

| {z }

injection

2[¯

u∗Q+¯

uQ∗]

| {z }

nonlinear transfer

−2k2

Re E

| {z }

dissipation

(10)

The pressure term also cancels out since ¯

u∗·k(t)¯

P= 0.

Thus, the rhs of Eq. (10) contains three main terms:

•Injection

A≡1

2¯

u∗(M¯

u) + ¯

u(M¯

u)∗,

which is of linear origin, being determined by the

matrix M, i.e., by the precessing background ﬂow

and describes energy exchange between the pertur-

bations and that ﬂow. If A > 0, kinetic energy is

injected from the ﬂow into inertial wave modes and

hence they grow, which is basically due to preces-

sional instability [45, 51, 52, 54], whereas at A < 0

modes give energy to the ﬂow and decay.

•Nonlinear transfer

NL ≡1

2[¯

u∗Q+¯

uQ∗]

describes transfer (cascade) of spectral kinetic en-

ergy among modes with diﬀerent wavenumbers in

Fourier space due to nonlinearity. The net eﬀect

of this term in the spectral energy budget summed

over all wavenumbers is zero i.e.,

NL(k, t) = 0,

which follows from vanishing of the nonlinear ad-

vection term in the total kinetic energy equation

integrated in physical space. Thus, the main eﬀect

of the nonlinear term is only to redistribute energy

among modes that is injected from the basic ﬂow

due to A, while keeping the total spectral kinetic

energy summed over all wavenumbers unchanged.

Although the nonlinear transfers NL produce no

net energy for perturbations, they play a central

role in the turbulence dynamics together with the

injection term A. The latter is thus the only source

of new energy for perturbations drawn from the in-

ﬁnite reservoir of the background precessional ﬂow.

Due to this, below we focus on these two main

dynamical terms – linear injection and nonlinear

transfer functions, compute their spectra and anal-

yse how they operate in Fourier space in the pres-

ence of precession instability using the tools of Refs.

[53, 55].

•Viscous dissipation

D≡ −2k2

Re E

is negative deﬁnite and describes the dissipation of

kinetic energy due to viscosity.

B. 2D-3D decomposition

In the present section we follow a widely used ap-

proach in the theory of rotating anisotropic turbulence

[29, 31, 35–37, 39] and decompose the ﬂow ﬁeld into

2D and 3D modes in Fourier space to better character-

ize this anisotropy between horizontal and vertical mo-

tions. This choice is motivated by the observation of two

main types of perturbations: vortices, which are essen-

tially 2D structures, and 3D inertial waves in rotating

turbulent ﬂows with external forcing such as libration,

elliptical instability [39, 42], precession [36, 45] and other

artiﬁcial types of forcing concentrated at a particular

wavenumber [34, 35, 56]. The 2D vortical modes, also

called slow (geostrophic) modes, have dominant horizon-

tal velocity over the vertical one and are almost uniform,

or aligned along the z−axis, i.e., their wavenumber par-

allel to this axis is zero kz= 0. This slow manifold

is also referred to as 2D and three-component (2D3C)

ﬁeld in the literature, since it varies only in the hor-

izontal (x, y)-plane perpendicular to the rotation axis,

but still involves all three components of velocity with

the horizontal one being dominant. On the other hand,

3D inertial wave modes, called fast (with nonzero fre-

quency ω=±2Ωkz/k) modes, have comparable hori-

zontal and vertical velocities and vary along z-axis, i.e.,

parallel wavenumber is nonzero kz6= 0. In the present

case of the basic precessional ﬂow, the kz(t) wavenumber

of modes oscillates in time according to Eq. (5), so we

classify 2D and 3D modes as having hkz(t)i=kz0= 0

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Interplaybetweengeostrophicvorticesandinertialwavesinprecession-driventurbulenceF.PizziInstituteofFluidDynamics,Helmholtz-ZentrumDresden-Rossendorf,BautznerLandstrasse400,D-01328Dresden,GermanyandDepartmentofAerodynamicsandFluidMechanics,BrandenburgUniversityofTechnology,Cottbus-Senftenberg,03046Cot...

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Interplay between geostrophic vortices and inertial waves in precession-driven turbulence F. Pizzi

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