Open AccessArticle

Acoustic Drift: Generating Helicity and Transferring Energy

Andrey Morgulis

^1,2

I.I. Vorovich Institute for Mathematics, Mechanics and Computer Science, Southern Federal University, Rostov-na-Donu 344092, Russia

Southern Mathematical Institute—The Affiliate of Vladikavkaz Scientific Centre of Russian Academy of Sciences, Vladikavkaz 362025, Russia

Axioms 2024, 13(11), 767; https://doi.org/10.3390/axioms13110767

Submission received: 12 August 2024 / Revised: 24 October 2024 / Accepted: 26 October 2024 / Published: 4 November 2024

(This article belongs to the Special Issue Fluid Dynamics: Mathematics and Numerical Experiment)

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Figure 1
This figure illustrates the drift energy that arises from configuration (<a href="#FD26-axioms-13-00767" class="html-disp-formula">26</a>)–(<a href="#FD29-axioms-13-00767" class="html-disp-formula">29</a>) when the number of atoms tends to ∞ along a sequence of the doubled primes. The distribution of power spectrum over the atoms is the normal periodic one (right panel) or the uniform one (left panel). The graphs depict the total mean drift energy and its resonant or non-resonant fractions vs. the geometric parameter <math display="inline"><semantics> <mi>β</mi> </semantics></math>. Recall that the drift includes no uniform component for the uniform power spectrum distribution (see Remark 1). "> Figure 2
This figure displays samples of 20 level sets of the stream function (<a href="#FD59-axioms-13-00767" class="html-disp-formula">59</a>) for the drift field emergent from 22 atoms configuration defined by equalities (<a href="#FD26-axioms-13-00767" class="html-disp-formula">26</a>)–(<a href="#FD29-axioms-13-00767" class="html-disp-formula">29</a>) for equal amplitudes, <math display="inline"><semantics> <msub> <mi>φ</mi> <mi>j</mi> </msub> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>10</mn> </mrow> </semantics></math>. Upper-left, upper-right, lower-left, and lower-right frames display the samples taken for <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1.69</mn> <mo>,</mo> <mn>3.38</mn> <mo>,</mo> <mn>5.08</mn> </mrow> </semantics></math>, correspondingly. So, the sampling interval is approximately eight times smaller then the period. The next semi-period demonstrates the reverse changes. ">

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Abstract

This article studies the general properties of the Stokes drift field. This name is commonly used for the correction added to the mean Eulerian velocity for describing the averaged transport of the material particles by the oscillating fluid flows. Stokes drift is widely known mainly in connection with another feature of oscillating flows known as steady streaming, which has been and remains the focus of a multitude of studies. However, almost nothing is known about Stokes drift in general, e.g., about its energy or helicity (Hopf’s invariant). We address these quantities for acoustic drift driven by simple sound waves with finite discrete Fourier spectra. The results discover that the mean drift energy is partly localized on a certain resonant set, which we have described explicitly. Moreover, the mean drift helicity turns out to be completely localized on the same set. We also present several simple examples to discover the effect of the power spectrum and positioning of the spectral atoms. It is revealed that tuning them can drastically change both resonant and non-resonant energies, zero the helicity, or even increase it unboundedly.

Keywords:

oscillatory fluid flows; acoustic drift; Stokes’drift; helicity; Hopf’s invariant

MSC:

76M45; 76M50; 35Q30; 35C20

1. Introduction

In oscillating fluid flows, there is a difference between the averaged velocity of the material particles and the averaged Eulerian velocity. When referring to the vector field that describes the distribution of these corrections across the flow domain, we will be using the terms Stokes drift velocity, drift velocity, or simply drift. In the generic case, the Stokes drift velocity field in the leading approximation reads as

\begin{matrix} \bar{[ξ_{τ}, ξ]} / 2, \end{matrix}

(1)

where the notation

ξ

represents the field of ‘fast’ displacements while freezing the ‘slow’ variables, the overlining indicates the averaging over the ‘fast’ time, and the square brackets are the standard commutator of the vector fields in the slow variables. Here, the adjectives ‘slow’ and ‘fast’ indicate the phenomena occurring in essentially different time or space scales and the variables used for describing them. For instance, consider an oscillating rigid vessel filled with viscous incompressible fluid. Then, the oscillation period determines one time scale, while there is another one, namely,

L^{2} / ν

, where L is the characteristic size of the vessel, and

ν

is kinematic viscosity of the fluid. The case where the former is much smaller than the latter is the one to which Stokes drift and allied phenomena relate.

On average, the vibrations excite a slow flow, which is generically vortical and global, even if the vibrations are local. This feature is often called steady streaming (see review [1]). Regarding Stokes’ drift, the overall conclusion is that its long-term effect on the mean flow often is rather considerable. For some relations between the scales of the vibrational forcing and flow, there is a simple but mathematically consistent asymptotic theory [2,3,4]. It relies on a viscous incompressible approximation but allows an extension to the compressible flows where the fast displacements can propagate as the sound waves. In both cases, the leading approximation to the mean flow involves the Stokes drift.

Although the steady streaming remains rather popular research topic with numerous modern applications (e.g., [5,6]), this article addresses Stokes drift in a different respect. Namely, the Stokes drift vector fields often arise from commuting several basic fields that are set up equally and very simply. Depending on the compressibility, these can be irrotational fields, the potentials of which are plane sound waves or harmonic functions (see the articles cited above). However, the corresponding Stokes drifts are revealed to be divergence-free (always) and not irrotational (generically). As far as we are aware, the literature does not tell us anything else about the qualitative properties of Stokes drift in general, so there are a number of issues to raise. For example, to what extent are the invariants of the Stokes drift fields predictable? We particularly aim at Hopf’s invariant or helicity, which is common to see as a measure of the complexity of the phase trajectories of the corresponding vector field. The monograph [7] comprehensively exposes this topic—in particular, the relevance of helicity to the knotting and linking of the phase trajectories—and provides several key references on the subject. One more topic is the efficiency of Stokes drift upon transferring the energy from small scales to greater ones. In this article, we derive and discuss several simple examples that shed some light on these issues.

2. Results

2.1. List of Notation

$τ$ denotes fast time;
$\bar{(\cdot)}$ denotes the fast time averaging;
$ξ = ξ (x, τ)$ denotes the vector field of fast displacements due to the sound waves;
$φ = φ (x, t)$ denotes the potential of $ξ$ ;
$V$ denotes the vector field of Stokes drift velocity;
$I d$ denotes an identical mapping;
$c . c .$ denotes an abbreviation for ‘complex conjugate value’.
$Φ = Φ (x)$ denotes the amplitude of $φ$ , that is, $φ (x, t) = Φ (x) e^{- i τ} + c . c .$ ;
$\hat{Φ}$ denotes the Fourier transformation of $Φ$ ;
$S_{i}$ , $i = 1, 2$ denote the unit circumference and unit sphere;
$θ_{j} \in S_{2}$ , $φ_{j} \in C$ , $j = 1, 2, 3, \dots$ denote the positions and amplitudes of Dirac’s atoms, which constitute $\hat{Φ}$ ;
the term configuration means the set of positions of the aforementioned spectral atoms, that is, $A = {θ_{1}, θ_{2}, \dots} \subset S_{2}$ .
the term power spectrum means the set of amplitudes assigned to the spectral atoms—that is, ${φ_{1}, φ_{2}, \dots}$ ;
$〈 \cdot 〉$ denote the spatial averaging;
$R$ denotes the set of all resonant quartets of the atoms in a given configuration;
$I_{r e s}$ denotes the set of quartets of indices that correspond to the elements of $R$ ;
$I$ denotes the set of all quartets of indices such that the corresponding quartet of atoms contributes into the mean drift energy;
$I_{0} = I ∖ I_{r e s}$ ;
$D_{4}$ denotes the dihedral group;
$E_{r e s}$ , $E_{n r}$ denote the contributions to the mean energy of a drift field from the sets of resonant and non-resonant quartets of atoms, and ${\tilde{E}}_{n r} = E_{n r} - {〈 V 〉}^{2} / 2$ denotes non-resonant energy without taking into account the uniform drift component (if any).

2.2. Preliminaries

We consider the fast displacements due to the sound waves. Thus,

\begin{matrix} ξ_{τ} = \nabla φ, \bar{ξ} = 0, where φ_{τ τ} = Δ φ, \bar{φ} = 0, \end{matrix}

(2)

\begin{matrix} \bar{f} (x) = lim_{ℓ \to \infty} \frac{1}{2 ℓ} \int_{| τ | < ℓ} f (x, τ) d τ . \end{matrix}

(3)

Let

V

be the drift velocity determined by Formula (1) where the fast displacement field

ξ

is defined in (2). Then,

div V = 0,

as article [3] had reported. They omitted the proof. It is not very difficult, although it is not completely obvious as well. So, here is it. Since

[a, b] = (b, \nabla) a - (a, \nabla) b = curl (a \times b) + (div a) b - (div b) a,

the following identity holds true

2 V = curl (\bar{ξ_{τ} \times ξ}) + \bar{(div ξ_{τ}) ξ - (div ξ) ξ_{τ}} .

(4)

Hence,

\begin{matrix} div V = \bar{div (φ_{τ τ} ξ)} = - \bar{div (φ_{τ} \nabla φ)} = \\ - \bar{\nabla φ_{τ} \nabla φ + φ_{τ} Δ φ} = - \bar{{(φ_{τ}^{2} / 2 + {(\nabla φ)}^{2} / 2)}_{τ}} = 0 . \end{matrix}

Let us now consider

curl V

. Departing from (4) via straightforward calculations, we arrive at the expression

\begin{matrix} 2 curl V = - Δ (\bar{ξ_{τ} \times ξ}) + 2 \bar{ξ_{τ τ τ} \times ξ} & . \end{matrix}

(5)

To be more specific, let us restrict ourselves to within the waves propagating at the unit speed so that

φ (x, τ) = Φ (x) e^{- i τ} + c . c ., Δ Φ + Φ = 0,

(6)

where the abbreviation c.c. indicates the complex conjugate summand. Then,

φ_{τ τ} = - φ

, and

ξ_{τ τ τ} = - ξ_{τ}

. Hence,

curl V = - (Id + Δ / 2) (\bar{ξ_{τ} \times ξ}),

and it becomes clear that the case of

curl V = 0

represents a degeneration.

The wave amplitude,

Φ (x)

, reads as

\int_{S_{2}} e^{i x θ} \hat{Φ} (d S_{θ}),

where

\hat{Φ}

is a complex-valued Borel’s measure on the unit sphere

S_{2} = {| θ | = 1} \subset R^{3}

. Hence,

\begin{matrix} V = \int_{S_{2}^{2}} e^{i (θ_{1} - θ_{2}) x} (θ_{1} θ_{2}) (θ_{1} + θ_{2}) \hat{Φ} (d S_{θ_{1}}) {\hat{Φ}}^{*} (d S_{θ_{2}}), \end{matrix}

(7)

\begin{matrix} curl V = + 2 i \int_{S_{2}^{2}} (θ_{1} θ_{2}) e^{i (θ_{1} - θ_{2}) x} (θ_{1} \times θ_{2}) \hat{Φ} (d S_{θ_{1}}) {\hat{Φ}}^{*} (d S_{θ_{2}}), \end{matrix}

(8)

where (and everywhere in what follows) the expression in the form

(\cdot, \cdot)

denotes the canonical inner product in

R^{3}

, and superscript * indicates the complex conjugated value. At this point, we see that the irrotational Stokes drifts exist despite the fact that this is a degenerate case. For example, set

\hat{Φ} = φ_{1} δ (θ - e) + φ_{2} δ (θ + e), e \in S_{2} .

(9)

Then, Stokes drift reads as

V = 2 (| φ_{1} |^{2} - {| φ_{2} |}^{2}) e

One more uniform drift arises from a triplet of waves, the fronts of which are orthogonal, that is,

\hat{Φ} = φ_{1} δ (θ - e_{1}) + φ_{2} δ (θ - e_{2}) + φ_{3} δ (θ - e_{3}),

(10)

where the set

{e_{1}, e_{2}, e_{3}} \subset S_{2}

constitutes an orthogonal basis for

R^{3}

. Then,

V = 2 (| φ_{1} |^{2} e_{1} + | φ_{2} |^{2} e_{2} + {| φ_{3} |}^{2} e_{3}) .

For simplicity, we restrict ourselves from here on to within the class of amplitudes,

\hat{Φ}

, which are the purely atomic measures with finite numbers of atoms. Let positions of the spectral atoms constitute a set

A = {θ_{1}, θ_{2}, \dots, θ_{N}} \subset S_{2}

. Then,

\begin{matrix} Φ = \sum φ_{k} e^{i (θ_{k} x)} + c . c ., where φ_{k} \in C, θ_{k}^{2} = 1 \forall k = 1 \dots N, and θ_{k} \neq θ_{j} \forall k \neq j, \end{matrix}

(11)

\begin{matrix} ξ_{τ} = i \sum φ_{k} e^{i (θ_{k} x - τ)} θ_{k} + c . c ., ξ = - \sum φ_{k} e^{i (θ_{k} x - τ)} θ_{k} + c . c ., \end{matrix}

(12)

\begin{matrix} V = \sum_{j, k} (θ_{k} θ_{j}) φ_{j} {φ_{k}}^{*} e^{i x (θ_{j} - θ_{k})} (θ_{k} + θ_{j}) . \end{matrix}

(13)

For every field of (12) and every polynomial p on

R^{3}

, the function

f = p \circ ξ

possesses a well-defined temporal average (3) at every point

x \in R^{3}

and a well-defined spatial average at every point

τ \in R

. The latter reads as

〈 f 〉 = lim_{ℓ \to \infty} \frac{1}{{(2 ℓ)}^{3}} \int_{{(- ℓ, ℓ)}^{3}} f (x, τ) d x

(14)

The mappings

x \mapsto \bar{f} (x)

and

τ \mapsto 〈 f 〉 (τ)

define smooth and bounded functions on

R^{3}

and on

R

, and there exists the total average

〈 \bar{f} 〉 = \bar{〈 f 〉}

. In particular, averaging the drift velocity (13) gives us the uniform component of the drift, namely,

\begin{matrix} 〈V〉 = 2 \sum_{k} {| φ_{k} |}^{2} θ_{k} . \end{matrix}

(15)

Further, every displacement of (12) possesses the mean energy that takes the form

〈 \bar{{ξ_{τ}}^{2} / 2} 〉 = 〈 {| \nabla Φ |}^{2} 〉 = \sum {| φ_{k} |}^{2}

(16)

We normalize the displacements by setting the energy to unity so that

\sum | φ_{k} |^{2} = 1

(17)

in subsequent work. Averaging the drift velocity (13) gives us the uniform component of the drift, namely,

\begin{matrix} 〈V〉 = 2 \sum_{k} {| φ_{k} |}^{2} θ_{k} . \end{matrix}

(18)

For every drift velocity (13), the mean energy is also well defined and reads as

\begin{matrix} \frac{1}{2} 〈 V^{2} 〉 = \frac{1}{2} \sum_{(j, k, m, n) \in I} φ_{j} {φ_{k}}^{*} φ_{m} {φ_{n}}^{*} (θ_{k} θ_{j}) (θ_{m} θ_{n}) (θ_{j} + θ_{k}) (θ_{m} + θ_{n}) . \end{matrix}

(19)

\begin{matrix} I = {(j, k, m, n) : θ_{j} - θ_{k} + θ_{m} - θ_{n} = 0} . \end{matrix}

(20)

Set

I

includes a subset denoted as

I_{0}

below, that is trivial in the sense that it is not empty for every configuration of the spectral atoms. Namely,

I_{0} = {(j, k, m, n) : j = k = m = n or j = k, m = n, k \neq m or j = n, k = m, j \neq k} .

(21)

Definition 1.

A quartet of the spectral atoms

(e_{1}, e_{2}, e_{3}, e_{4}) \in A^{4}

is resonant if the following inequalities hold true:

e_{1} - e_{2} + e_{3} - e_{4} = 0, ({(e_{1} - e_{2})}^{2} + {(e_{3} - e_{4})}^{2}) ({(e_{1} - e_{4})}^{2} + {(e_{3} - e_{2})}^{2}) \neq 0

(22)

We denote the set of all resonant quartets emergent from configuration

A

R = R (A) \subset A^{4}

. The set of indices of the elements of set

R

I_{r e s} = I ∖ I_{0}

The atoms of a resonant quartet lay in some plane, and, moreover, they place themselves at the vertices of some rectangle. It follows by the definition of resonance since the atoms are at the unit sphere by construction.

Note that the resonant set,

R

, is invariant with respect to the following permutations:

\begin{matrix} T : (e, f, g, h) \mapsto (f, e, h, g), T_{1} : (e, f, g, h) \mapsto (g, f, e, h), T_{2} : (e, f, g, h) \mapsto (e, h, g, f) \end{matrix}

(23)

The set of resonant indices,

I_{r e s}

, possesses the same invariance. Permutations (23) form the following finite group with the composing operation:

\begin{matrix} {i d, T, T_{1}, T_{2}, T_{1} T_{2}, T_{1} T_{2} T, T_{1} T, T_{2} T} . \end{matrix}

(24)

The permutation subgroup (24) can be identified with the symmetry group of the square, also known as a dihedral group of order 8, which is usually represented by the notations

{Dih}_{4}

D_{4}

(see Appendix A).

2.3. Energy Transfer

To begin, let the resonant set (22) be empty, so that

I = I_{0}

, then the mean drift energy takes the form

\begin{matrix} \frac{1}{2} 〈 V^{2} 〉 = \frac{1}{2} {〈 V 〉}^{2} + \sum_{j \neq m} | φ_{m} |^{2} {| φ_{j} |}^{2} {(θ_{m} θ_{j})}^{2} (1 + (θ_{m} θ_{j})) \end{matrix}

(25)

where

〈 V 〉

is the uniform drift component that we have expressed with Formula (18). Clearly, the second summand on the right-hand side of (25) is non-negative as well, and one can see it as the mean non-resonant energy of the purely non-uniform component of a drift.

The mean non-resonant energies of the uniform and non-uniform components can (though not obliged, of course) be quite commensurable. For example, according to condition (17), the value of 2 is the upper bound for both mean non-resonant energies, and they reach this limit when the atoms come together at a given point on

S_{2}

Further, let all the atoms except for one gather around a given point, while the exceptional atom with squared amplitude equal to 1/2 places itself at the opposite point. Then, the mean energy of the non-uniform drift tends to

1 / 2

, and that of the uniform component tends to zero. Hence, the purely non-uniform drift itself can possess an energy commensurable with that of the purely uniform one.

Now, the objective is to determine how much energy is due to the resonances. First, we set up a suitable resonant configuration, namely,

\begin{matrix} A = {a, b, U a, U b, U^{2} a, U^{2} b, \dots, U^{M - 1} a, U^{M - 1} b} \overset{def}{=} A_{U} & , \end{matrix}

(26)

\begin{matrix} U \in S O (3) : U (a - b) = a - b, a \neq b, a, b \in S_{2} \end{matrix}

(27)

So, the notation of

U

stands for a rotation around the axis determined by

a - b

, and the atom positions read

\begin{matrix} θ_{2 p} = U^{p} a, θ_{2 p + 1} = U^{p} b, p = 0 \dots M - 1, \end{matrix}

(28)

Next, we assign the atom amplitudes as follows

φ_{2 p} = φ_{2 p + 1} = ψ_{p} \forall p = 0 \dots M - 1,

(29)

where numbers

ψ_{p} \in C

are free parameters obeying restriction

2 \sum_{j = 0 \dots M - 1} {| ψ_{j} |}^{2} = 1,

which is just another form of (17). Let

R_{U} \overset{def}{=} R (A_{U})

be the resonance subset in

A_{U}^{4}

. Since element

a - b

determines the axis of rotation

U

\begin{matrix} R_{U} \supset R_{U}^{0} \overset{def}{=} {(θ_{2 p}, θ_{2 p + 1}, θ_{2 q + 1}, θ_{2 q})}, \forall p, q = 0 \dots M - 1, θ_{ℓ} \in A_{U}, \forall ℓ = 0 \dots 2 M - 1 \end{matrix}

(30)

Consequently, by the dihedral invariance pointed out above, the resonant set also includes a subset

\begin{matrix} D_{4} R_{U}^{0} \overset{def}{=} ⋃_{θ \in R_{U}^{0}} D_{4} θ, \end{matrix}

(31)

where

D_{4} θ

stands for the orbit generated by the action of

D_{4}

-group on some quartet

θ \in R_{U}^{0}

For a while, let us assume that

D_{4} R_{U}^{0} = R_{U} .

Then, the contribution to the mean energy due to the resonances,

E_{r e s}

, taking into account partition (31), reads as

\begin{matrix} \sum_{p \neq q} | ψ_{p} |^{2} {| ψ_{q} |}^{2} ({(a, b)}^{2} (U^{p - q} (a + b), a + b) + (U^{p - q} a, a) (U^{p - q} b, b) (U^{p - q} b + b, U^{p - q} a + a)) \end{matrix}

For our purposes, it is convenient to put this using the semi-angle between elements a and b, which we denote as

β

. With this in mind, we arrive at the following expression:

\begin{matrix} E_{r e s} = \sum_{p \neq q} | ψ_{p} |^{2} {| ψ_{q} |}^{2} K (cos β, cos (α (p - q))) α = 2 π / M, p = 0 \dots M - 1, q = 0 \dots M - 1, \\ K (cos β, z) = K_{3} z^{3} + K_{2} z^{2} + K_{1} z + K_{0}, \\ K_{3} = K_{2} = 2 {cos}^{6} β, K_{1} = 2 (3 {cos}^{4} (β) - 1) {cos}^{2} β, K_{0} = 2 {sin}^{4} β ({cos}^{2} β - 2 {sin}^{2} β) \end{matrix}

Quite a simple transformation brings this into the form

\begin{matrix} E_{r e s} = (\sum_{ℓ = 0 \dots 3} c_{ℓ} {|F_{M}^{(ℓ)}|}^{2} + R_{M}), α = 2 π / M, \\ F_{M}^{(ℓ)} = \sum_{p = 0 \dots M - 1} | ψ_{p} |^{2} e^{i ℓ α p}, R_{M} = - 4 cos (2 β) (cos (2 β) {cos}^{2} β + 1) \sum_{p = 0 \dots M - 1} {| ψ_{p} |}^{4} \end{matrix}

(32)

\begin{matrix} c_{3} = \frac{K_{3}}{4} = \frac{{cos}^{6} β}{2}, c_{2} = \frac{K_{2}}{2} = {cos}^{6} β, c_{1} = \frac{3 K_{3} + 4 K_{1}}{4} = \frac{(15 {cos}^{4} β - 4) {cos}^{2} β}{2}, \end{matrix}

(33)

\begin{matrix} c_{0} = \frac{2 K_{0} + K_{2}}{2} = {cos}^{6} β + 2 {sin}^{4} β ({cos}^{2} β - 2 {sin}^{2} β) . \end{matrix}

(34)

Evaluating these sums becomes easier when the number of summands is great enough. Namely, let f be a continuous positive function on

S_{1}

and

\int_{S_{1}} f (s) d s = 1 .

Given constraint (17), put

2 | ψ_{p} |^{2} = \frac{f (α p)}{\sum_{p = 0 \dots M - 1} f (α p)} .

(35)

Then,

2 F_{M}^{(ℓ)} \to \hat{f} (ℓ) = \int_{S_{1}} f (p) e^{i ℓ p} d p, R_{M} \to + 0, E_{r e s} \to \frac{1}{4} \sum_{ℓ = 0 \dots 3} c_{ℓ} {| \hat{f} (ℓ) |}^{2}, M \to \infty .

where the coefficients,

c_{ℓ}

, have been specified by Formulae (33) and (34).

Next we calculate the non-resonant contribution into the mean energy of configuration

A_{U}

in the same manner but using Formulae (18) and (25) and arrive at equalities as follows

{〈 V 〉}^{2} / 2 = 2 {|\sum_{p = 0 \dots M - 1} {| ψ_{p} |}^{2} U^{p} (a + b)|}^{2} \to 2 {cos}^{2} β {| \hat{f} (1) |}^{2}

(36)

\begin{matrix} {\tilde{E}}_{n r} = 4 (\sum_{p, q = 0}^{M - 1} | ψ_{p} |^{2} | ψ_{q} |^{2} Q (cos β, cos (α (p - q)) - \sum_{p}^{M - 1} {| ψ_{p} |}^{4}), \\ Q = Q (cos β, z) = z^{3} {cos}^{6} β + z^{2} {cos}^{4} β + 3 z {sin}^{4} β {cos}^{2} β + {sin}^{4} β, \end{matrix}

Further, the limit of the mean non-resonance energy of the non-uniform flow reads

\begin{matrix} {\tilde{E}}_{n r} \to \sum_{ℓ = 0 \dots 3} C_{ℓ} {| \hat{f} (ℓ) |}^{2}, M \to \infty, \\ C_{0} = \frac{{cos}^{4} β + 2 {sin}^{4} β}{2}, C_{1} = \frac{3 ({cos}^{4} β + 4 {sin}^{4} β) {cos}^{2} β}{4}, C_{2} = \frac{{cos}^{4} β}{2}, C_{3} = \frac{{cos}^{6} β}{4}, \end{matrix}

(37)

Remark 1.

The uniform drift component vanishes for every configuration (26)–(29) such that the absolute values of all the amplitudes are equal, that is

| ψ_{0} | = | ψ_{1} | = \dots = | ψ_{M - 1} |

. This follows easily from expression (18). The left-hand side of the limit identity (36) is consistent with this conclusion, and the right-hand side also becomes consistent provided that

f \equiv const

, as it should be (see the discussion on concrete examples below).

Yet, it does not follow from any of the considerations above that set

D_{4} R_{U}^{0}

includes every resonant quartet of the atoms that configure set

A_{U}

. We have been assuming that it is so, and we believe that it holds true for every prime number M, or, at least, the exceptions are rare to such an extent that their contribution into the resonant energy becomes negligible when M tends to infinity. Below, we put a number of rationales for this conjecture.

There is an orthogonal decomposition

2 θ = \pm (a - b) + U^{p} (a + b) \forall θ \in A_{U}

, as element

a - b

determines the axis of rotation

U

by construction. Hence, every occurrence of a resonance entails an equality

(U^{p_{1}} - U^{p_{2}} + U^{p_{3}} - U^{p_{4}}) (a + b) = 0, M > p_{1} \geq p_{2} \geq p_{3} \geq p_{4} \geq 0, (p_{1}, p_{2}, p_{3}, p_{4}) \in N^{4},

where the quartet of exponents includes at least two different numbers. Since point

a + b

lays in the plane orthogonal to the axis of the rotation, this condition reduces to the following equation

u^{p_{1}} - u^{p_{2}} + u^{p_{3}} - u^{p_{4}} = 0, | u | = 1, u \in C, M > p_{1} \geq p_{2} \geq p_{3} \geq p_{4} \geq 1, (p_{1}, p_{2}, p_{3}, p_{4}) \in N^{4},

where

u = e^{i α}

, and

α

is the angle of revolving around the rotation axis. Evidently, this equation holds true for every

u \in C : | u | = 1

provided that an action of the

D_{4}

-group brings the quartet

(p_{1}, p_{2}, p_{3}, p_{4})

into the form

(p, p, q, q)

. Every such a resonance belongs to

D_{4} R_{U}^{0}

and vice versa. For any other resonances, one of the following equations must hold true for some

u : u^{M} = 1, u \neq 1

\begin{matrix} u^{p} - 1 = 0, M > p > 0 . \end{matrix}

(38)

\begin{matrix} 2 u^{p} - u^{q} - 1 = 0, M > p > q > 0, \end{matrix}

(39)

\begin{matrix} u^{p} - 2 u^{q} + 1 = 0, M > p > q > 0, \end{matrix}

(40)

\begin{matrix} u^{p} + u^{q} - 2 = 0, M > p > q > 0, \end{matrix}

(41)

\begin{matrix} u^{p} - u^{q} + u^{r} - 1 = 0, M > p > q > r > 0, \end{matrix}

(42)

\begin{matrix} u^{p} - u^{q} - u^{r} + 1 = 0, M > p > q > r > 0, \end{matrix}

(43)

\begin{matrix} u^{p} + u^{q} - u^{r} - 1 = 0, M > p > q > r > 0 . \end{matrix}

(44)

No equation holds in the first group provided that

u : u^{M} = 1, u \neq 1

, and M is a prime number. A direct numeric check using the Maple software (https://www.maplesoft.com/) does not reveal any equation that is held true in any of five other groups for at least the first hundred of the primes. As the one-hundredth prime number is as great as 541, the above limits deliver quite reasonable approximations of the resonant energy for the primes being comparable in magnitudes.

We have addressed the limits

M \to \infty

for two concrete configurations (26)–(29) with different power spectra. Figure 1 illustrates the results. One configuration corresponds to the uniform distribution of amplitudes, that is,

f \equiv const,

and another one corresponds to the periodization of the normal distribution density, so that

f (s) = \frac{1}{\sqrt{2 π}} \sum_{ℓ \in Z} exp (\frac{- {(2 π ℓ - s)}^{2}}{2}) .

(45)

For a finite number M and specific distribution density f, Equation (35) gives numbers

ψ_{p}

p = 0 \dots M - 1

, which, in turn, determine the power spectra by Equation (29). In particular, the uniform distribution leads to the power spectra that consists of equal real amplitudes, namely,

φ_{2 p} = φ_{2 p + 1} = ψ_{p} = {(2 M)}^{- 1 / 2}

p = 0 \dots M - 1

. We will proceed with discussing Figure 1 in Section 3.

Remark 2.

Every drift velocity (13) build on configuration (26)–(29) is periodic along the rotation axis with period

π / sin β

and its projection on this axis is constant along it. Additionally, this projection is equal to zero in the case of constant power spectrum, that is, when

ψ_{0} = ψ_{1} = \dots ψ_{M - 1}

. This follows easily by construction.

2.4. Generating Helicity

Let

T^{3}

denote a 3-torus with a planar Riemann’s metric. For a divergence-free vector field

w

, the common definition of the helicity reads as

\int_{T^{3}} (w, {curl}^{- 1} w) d x

where the round bracket represents the inner product in the tangent spaces determined by the Riemann’s metric, and inverting the

curl

-operator operates the vector fields obeying condition

〈 w 〉 = 0

by definition.

The helicity (also known as Hopf’s invariant) represents an invariant of action of the volume-preserving diffeomorphisms. We refer the reader to book [7] for more details, including a discussion on the links to the complexity of phase trajectories.

Analogous to the above, we restrict ourselves to within the drifts with a zero uniform component and define the helicity of the Stokes drift as follows

\begin{matrix} 〈 (V, {curl}^{- 1} V) 〉, 〈 V 〉 = 0 . \end{matrix}

(46)

Note that assuming no uniform drift means that all the summands with equal summation indices cancel themselves while calculating the drift velocity,

V

, by Formula (13). Hence, while doing so, it is correct to neglect the sum over coinciding indices. Then, inverting the operator of

curl

gives

\begin{matrix} {curl}^{- 1} V = 2 i \sum_{n \neq m} φ_{m} {φ_{n}}^{*} e^{i x (θ_{m} - θ_{n})} \frac{(θ_{n} θ_{m}) (θ_{m} \times θ_{n})}{{(θ_{m} - θ_{n})}^{2}} . \end{matrix}

Further, the mean helicity takes the form

\begin{matrix} 〈 (V, {curl}^{- 1} V) 〉 = 2 i \sum_{(j, k, m, n) \in I_{r e s}} φ_{j} {φ_{k}}^{*} φ_{m} {φ_{n}}^{*} \frac{(θ_{j} θ_{k}) (θ_{m} θ_{n}) (θ_{m} \times θ_{n}) (θ_{k} + θ_{j})}{{(θ_{m} - θ_{n})}^{2}}, \end{matrix}

(47)

where the notation of

I_{r e s}

stands for the set indices enumerating the set of resonant quartets,

R

(see (22)).

Now, we assert that there is a Stokes drift that obeys restriction (17), possesses as high of a helicity as one wishes, and such a drift can arise from a quintet of atoms. Let points

a, b

and rotation

U

be the same as those employed for constructing configuration

A_{U}

in Section 2.3, and let

β

be the semi-angle between a and b. Consider a configuration

a, b, U a, U b, c .

with the power spectrum

{φ_{1}, φ_{2}, φ_{3}, φ_{4}, φ_{5}},

where the order of writing defines the correspondence between the amplitudes and the atoms’ positions set above.

Quartet

a, b, U b, U a

is resonant, as are all those belonging to its

D_{4}

-orbit. This orbit produces a mean helicity that reads as

- 4 I m (φ_{1} φ_{2}^{*} φ_{4} φ_{3}^{*}) (\frac{{cos}^{2} 2 β {cos}^{2} β sin α}{sin β} - 2 sin β ({cos}^{2} β cos α + {sin}^{2} β) cot (\frac{α}{2})) .

(48)

In the derivation of this expression, the key point is the orthogonal decomposition using the rotation axis; however, we omit these quite routine details. Now, set

\begin{matrix} 0 < 2 β < π / 2, cos α = - {tan}^{2} β, \end{matrix}

(49)

Then, the expression (48) simplifies to

\begin{matrix} - 4 I m (φ_{1} {φ_{2}}^{*} φ_{4} φ_{3}^{*}) \frac{{cos}^{3 / 2} 2 β}{sin β} . \end{matrix}

(50)

Thus, the helicity produced by quartet

a, b, U b, U a

together with its

D_{4}

-orbit grows unboundedly when

β \to + 0

and the equality (49) holds true provided that the power spectrum obeys the following condition

I m (φ_{1} φ_{4} φ_{2}^{*} φ_{3}^{*}) > c_{0} > 0,

(51)

where

c_{0}

is an independent constant. Further, let us assume that another resonance has occurred in the same atoms’ configuration. Then,

c = \pm e_{1} \pm e_{2} \pm e_{3}, e_{i} \in {a, b, U a, U b},

and, moreover, the sign of one term on the right-hand side is always opposite to the other two signs. Hence,

(c, a - b) = \pm {(a - b)}^{2} / 2 = \pm 2 {sin}^{2} β .

Thus, the quartet

a, b, U b, U a

exhausts the resonant set emergent from the configuration

{a, b, U a, U b, c}

provided that

(a - b, c) = 0 .

(52)

(since it contradicts to the previous equality).

Given the condition (52), to eliminate the uniform component from the drift, the power spectrum of the configuration must meet the following constraints

\begin{matrix} | φ_{1} |^{2} - | φ_{2} |^{2} + | φ_{3} |^{2} - {| φ_{4} |}^{2} = 0 . \end{matrix}

(53)

\begin{matrix} 2 | φ_{5} |^{2} c + (| φ_{1} |^{2} + | φ_{2} |^{2}) (a + b) + (| φ_{3} |^{2} + | φ_{4} |^{2}) U (a + b) = 0, \end{matrix}

(54)

\begin{matrix} 2 | φ_{5} |^{2} = |(| φ_{1} |^{2} + | φ_{2} |^{2}) (a + b) + (| φ_{3} |^{2} + | φ_{4} |^{2}) U (a + b)|, \end{matrix}

(55)

Using equality (49), we obtain the following expression:

2 | φ_{5} |^{4} = {(| φ_{1} |^{2} + | φ_{2} |^{2} + | φ_{3} |^{2} + {| φ_{4} |}^{2})}^{2} cos 2 β + (| φ_{1} |^{2} + | φ_{2} |^{2} - | φ_{3} |^{2} - | φ_{4} {|^{2})}^{2} \neq 0, \forall β : β \in (0, π / 4) .

Hence, Equations (54) and (55) define point

c \in S_{2}

correctly. Further, constraints (53)–(55) are homogeneous in variables

φ_{1}, \dots, φ_{5}

. Therefore, if such a quintet meets them, then there is another one that meets them and restriction (17) at the same time.

Thus, let angles

α

and

β

obey the constraint (49), points

a, b, c

belong to

S_{2}

, and

(a, b) = cos (2 β)

. Let element

a - b

define the axis of rotation

U

and let

α

be the rotation angle. Finally, let elements

a, b, c

and the complex numbers

φ_{1}, φ_{2}, φ_{3}, φ_{4}, φ_{5}

obey the restrictions (53)–(55). Then, the configuration

{a, b, U a, U b, c}

with a power spectrum

{φ_{1}, φ_{2}, φ_{3}, φ_{4}, φ_{5}}

produces the drift without a uniform component. Hence, the helicity is well-defined for such a drift, and Formula (50) gives its value. Moreover, this value exceeds any bounds provided that angle

β

is small enough while the power spectrum obeys condition (51). Clearly, plenty of configurations meet all conditions listed here. For example,

φ_{1} = φ_{4} = \frac{1}{κ}, φ_{2} = φ_{3} = \frac{1 + i}{κ \sqrt{2}}, φ_{5} = \frac{\sqrt[4]{8 cos 2 β}}{κ}, κ = 2 \sqrt{1 + \sqrt{\frac{cos 2 β}{2}}} .

Slightly different examples result from the configurations (26)–(29). Let

M > 2

be a prime number and let the power spectrum read as

φ_{2 p} = \frac{e^{i α p}}{\sqrt{2 M}}, φ_{2 p + 1} = \frac{1}{\sqrt{2 M}}, p = 0 \dots M - 1 .

Since the absolute values of amplitudes do not vary within this spectrum, the corresponding configurations produce drifts with no uniform components. Hence, the helicity is well defined. Since it results from the resonant quartets only, let us follow the method employed for calculating the resonant energy in Section 2.3 but with the use of Formula (48) for the contribution of the individual

D_{4}

-orbits of a resonance from

R_{U}^{0}

. The sum of these contributions over set

D_{4} R_{U}^{0}

gives the helicity of the whole configuration, which reads

\begin{matrix} \frac{1}{M^{2}} \sum_{p = 0 \dots M - 1} \sum_{q = 0 \dots M - 1} (p_{0} (β) + (e^{i α (p - q)} P (cos β, e^{i α (p - q)}) + c . c .)), \end{matrix}

where

P (\cdot, z)

is a real-valued linear function in variable z. Next, we write this expression as follows

\begin{matrix} p_{0} + 2 p_{1} (cos β) {|\frac{1}{M} \sum_{p = 0 \dots M - 1} e^{i α p}|}^{2} + 2 p_{2} (cos β) {|\frac{1}{M} \sum_{p = 0 \dots M - 1} e^{2 i α p}|}^{2} \end{matrix}

using the transformation similar to that exploited in Section 2.3. Since the sums of exponents are equal to zero here, we do not specify the values of

p_{1}

and

p_{2}

. Hence, the total helicity of the aforementioned configuration is equal to

p_{0} (β)

and reads

- \frac{cos (6 β) + 19 cos (2 β) - 12}{8 sin (β)}

Thus, we have arrived at one more family of configurations that can produce drifts with arbitrarily high helicities. They can include from six to infinitely many atoms.

3. Discussion

From the graphs displayed in Figure 1, it follows that the resonances are capable of supplying or withdrawing a drift’s mean energy depending on the geometry of the configuration. Moreover, the resonant contribution is considerable and quite commensurable with the non-resonant one.

At the same time, the non-resonant quartets cannot produce the helicity, while generating it using resonances essentially depends on the phases of the power spectrum. Additionally, these phases can affect the energy gathering. Indeed, no helicity arises from the concrete configurations employed in Section 2.3 for drawing Figure 1. In contrast, the resonances in the quintets considered in Section 2.4 are capable of producing an arbitrarily great helicity with zero energy. Of course, the total energy is non-zero, but purely non-resonant.

The limits of great numbers of atoms addressed in Section 2.3 and Section 2.4 rely on assuming their number to be twice greater than a prime. This is an essential constraint. Indeed, the configuration (26)–(29) does not generate a drift at all for every even M and every value of angle

β

provided that all the spectral amplitudes are equal.

The well-known fact is that the leading approximation to the equation governing the averaged dynamics of material particles in an oscillating fluid flow takes the form

\dot{\bar{r}} = \bar{v} (\bar{r}, t) + V (\bar{r}, t)

(56)

where the notation of

\bar{r}

stands for the current particle’s position in

R^{3}

, the notations of

\bar{v}

and

V

are for the mean Eulerian flow velocity and for the Stokes drift, respectively. For the former, there is typically a closed equation which takes into account the vorticity transport due to the drift, e.g., Equation (35) in [3] (the unknown field denoted as

v_{1}

there is exactly what we have been denoting as

v

here). For a purely uniform drift,

V = 〈 V 〉

, (see configuration (9) or (10) in Section 2.2) this equation admits a simple solution:

\begin{matrix} \bar{v} = u (x + t 〈 V 〉), curl u = κ u . \end{matrix}

(57)

The solution (57) represents a travelling wave with a so-called ABC profile. There are examples of ABC vector fields that possess the complex topology of streamlines (see [7] for references). Hence, the travelling waves (57) can produce the chaotic dynamics of material particles inside uniformly drifting clouds.

It is worth noting that the uniform motion should not be neglected in our considerations, as we had fixed the speed of the sound waves already. Besides, the uniform flow velocity satisfies the aforementioned equation for the mean Eulerian velocity. Given these observations, it is of sense to regard Equation (56) with

\bar{v} \equiv const

. Then, the equation of the mean particle transport relative to a suitable moving frame of the Cartesian coordinates reads as

\dot{\bar{x}} = U (x, y, z + c t), \dot{\bar{y}} = V (x, y, z + c t) \dot{\bar{z}} = W (x, y, z + c t), V = (U, V, W), c = | \bar{v} | .

(58)

Here,

div V = U_{x} + V_{y} + W_{z} = 0

, as we have pointed out in Section 2.2. Consider now the ODE’ system (58) for drift flows produced by configurations (26)–(29). Then,

W_{z} = 0

(see Remark 2),

U_{x} + V_{y} = 0

. Hence, there exists a stream function

Ψ = Ψ (x, y, z)

such that

U (x, y, z) = Ψ_{y} (x, y, z), V (x, y, z) = - Ψ_{x} (x, y, z) .

(59)

Additionally, the field

V

\frac{π}{sin β}

-periodic in the z-coordinate, and the right-hand side in system (58) is time-periodic with the same period (see Remark 2 again). Consequently, system (58) is revealed to represent a kind of superstructure on the time-periodic Hamilton system produced by the Hamiltonian

Ψ

. Moreover, it is exactly a time-periodic Hamilton system provided that the amplitudes are equal within the power spectrum of the configuration. This follows from equality

W = 0

(see Remark 2 one more time)).

Figure 2 displays 20 level sets of Hamiltonian

Ψ

that arises from 22 atoms (

M = 11

) for equal real amplitudes and

β = π / 10

. The pictures of the first quarter-period discover rather rich reliefs. The most interesting feature is the numerous critical points organized in several rings. The closed level sets lie rather densely around critical points. This indicates the intensive eddies. The relief becomes much smoother during the second quarter-period. The rings of critical points undergo some rearrangements but persist overall. The next semi-period demonstrates the reverse changes. We conclude that the flow goes through phases of excitation and calming during each semi-period.

While looking at the pictures like these, the question arises as to whether the dynamics of the corresponding ODE system (58) is chaotic. We guess that the answer can be affirmative even for zero helicity. For example, the drift employed for drawing Figure 2 leads to the time-periodic system (58) written in a Hamilton form. Although it possesses only 1 degree of freedom, the chaotic dynamics are possible due to its time-periodicity, see [8]. We also expect occurrences of the chaotic dynamics for more general systems generated by configuration (26)–(29) given certain analogies with what has been treated in article [9].

4. Conclusions

Thus, we have addressed the Stokes drift driven by the trains of short sound waves dual to the finite linear combinations of Dirac’s atoms on the unit sphere in the frequency space (in the sense of a Fourier transform). We have been focused on generating the helicity of the drift and transferring the energy from the sound waves to it. The results discover that the mean drift energy includes a singular component which is localized on certain resonant set, which we have described explicitly. Moreover, the mean drift helicity turns out to be completely localized on the same set.

The power spectrum distribution across the atoms exerts a substantial effect on the resonant energy. In particular, there is special phase tuning that eliminates the resonant energy completely though generically its contribution is quite commensurable to the regular non-resonant component. The latter one depends on the absolute values of the spectral amplitudes only and tuning them changes the efficiency of the energy transfer essentially. We have presented examples of losing and amplifying the energy upon transferring from the sound waves to the drift due to altering the distribution of spectral amplitudes for one and the same configuration of the atoms. At the same time, the geometry of the atoms’ configuration also can exert a substantial effect on the magnitudes of both energy components when the power spectrum is fixed and even switch the resonances from supplying the energy to withdrawing it. Figure 1 illustrates all of what we have been saying in this paragraph.

The behavior of the helicity is a more subtle matter. Since the velocity field of a sound wave possesses zero helicity, we discuss not transferring but generating the helicity. It is a completely resonant feature; a non-resonant configuration generates nothing. A suitable spectral phase tuning can zero the helicity without any changes to the configuration of the atoms–that is, the wave fronts. At the same time, interactions of five acoustic waves can produce the drifts with arbitrarily high helicities while the total drift energy remains bounded. For every prime

M > 2

, tuning only the geometry

2 M

waves can cause the same. Here we mean manipulating only the wave fronts, with no altering the spectral amplitudes and phases. In all examples, pairing the waves with close to parallel fronts plays a crucial role.

Lacking the control of the helicity by the energy contrasts sharply with the common seeing of helicity as the quantity bounded on the energy level sets. Such a control, as well as the invariance of the helicity relative to the action of the volume preserving group and its relation to the knotting of and linking of the phase trajectories are studied well for the vector fields on compact manifolds only. So, the unbounded growth of the helicity reported above awaits the clarification of its meaning.

Yet another open question concerns the dynamics of the material particles (see the discussion around system (56)–(58) and the references therein), namely, to what extent does the correspondent dynamics depend on the configurations of atoms and the power spectrum? Can tuning them make it regular or chaotic?

Funding

This research received no external funding.

Data Availability Statement

The study did not report any data.

Acknowledgments

The author is grateful to the Southern Federal University and to the Southern Mathematical Institute of VSC RAS for the opportunity to conduct this research.

Conflicts of Interest

The author declares no conflicts of interest.

Appendix A

In general, a perfect n-gon possesses n symmetry axes, which generate n reflections. In turn, coupling them generates n rotations. Altogether, they constitute a dihedral group of order

2 n

. The rules which govern the corresponding multiplication table read as

\begin{matrix} S_{i} S_{j} = R_{i - j}, S_{i} R_{j} = S_{i - j}, R_{i} S_{j} = S_{i + j}, R_{i} R_{j} = R_{i + j}, R_{0} = i d, \end{matrix}

(A1)

where the notations

S_{j}

and

R_{i}

represent the reflections and rotations, respectively,

i, j = 0, \dots, n

, and calculation of the indexes must meet the rules of the modulo n arithmetic. A straightforward inspection shows that the permutation subgroup (24) meets the rules specified by the Formula (A1) upon setting

\begin{matrix} n = 4, S_{0} = T_{1} T_{2} T, S_{1} = T_{2}, S_{2} = T, S_{3} = T_{1} . \end{matrix}

(A2)

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Figure 1. This figure illustrates the drift energy that arises from configuration (26)–(29) when the number of atoms tends to ∞ along a sequence of the doubled primes. The distribution of power spectrum over the atoms is the normal periodic one (right panel) or the uniform one (left panel). The graphs depict the total mean drift energy and its resonant or non-resonant fractions vs. the geometric parameter

β

. Recall that the drift includes no uniform component for the uniform power spectrum distribution (see Remark 1).

β

. Recall that the drift includes no uniform component for the uniform power spectrum distribution (see Remark 1).

Figure 2. This figure displays samples of 20 level sets of the stream function (59) for the drift field emergent from 22 atoms configuration defined by equalities (26)–(29) for equal amplitudes,

φ_{j}

, and

β = π / 10

. Upper-left, upper-right, lower-left, and lower-right frames display the samples taken for

z = 0, 1.69, 3.38, 5.08

, correspondingly. So, the sampling interval is approximately eight times smaller then the period. The next semi-period demonstrates the reverse changes.

φ_{j}

, and

β = π / 10

. Upper-left, upper-right, lower-left, and lower-right frames display the samples taken for

z = 0, 1.69, 3.38, 5.08

, correspondingly. So, the sampling interval is approximately eight times smaller then the period. The next semi-period demonstrates the reverse changes.

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© 2024 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

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Morgulis, A. Acoustic Drift: Generating Helicity and Transferring Energy. Axioms 2024, 13, 767. https://doi.org/10.3390/axioms13110767

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Morgulis A. Acoustic Drift: Generating Helicity and Transferring Energy. Axioms. 2024; 13(11):767. https://doi.org/10.3390/axioms13110767

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Morgulis, A. (2024). Acoustic Drift: Generating Helicity and Transferring Energy. Axioms, 13(11), 767. https://doi.org/10.3390/axioms13110767

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Acoustic Drift: Generating Helicity and Transferring Energy

Abstract

1. Introduction

2. Results

2.1. List of Notation

2.2. Preliminaries

2.3. Energy Transfer

2.4. Generating Helicity

3. Discussion

4. Conclusions

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

Appendix A

References

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