# Vortex dynamics on a Möbius strip

Jacques Vanneste

(School of Mathematics and Maxwell Institute for Mathematical Sciences,
University of Edinburgh, King’s Buildings, Edinburgh EH9 3FD, United Kingdom)

###### Abstract

We consider the dynamics of a two-dimensional incompressible perfect fluid on
a Möbius strip embedded in $\mathbb{R}^{3}$. The vorticity–streamfunction
formulation of the Euler equations is derived from an exterior-calculus form
of the momentum equation. The non-orientability of the Möbius strip and the
distinction between forms and pseudo-forms this introduces lead to unusual
properties: a boundary condition is provided by the conservation of
circulation along the single boundary of the strip, and there is no integral
conservation for the vorticity or for any odd function thereof. A finite-
difference numerical implementation is used to illustrate the Möbius-strip
realisation of familiar phenomena: translation of vortices along boundaries,
shear instability, and decaying turbulence.

##  1 Introduction

In this paper, we examine the dynamics of a two-dimensional incompressible
inviscid fluid confined to a Möbius strip. The main motivation is curiosity
about the impact of the non-orientability of the Möbius strip on the
phenomenology of vortex dynamics. A secondary, more frivolous motivation is
the wish to produce interesting fluid-dynamical pictures. The system is
physically realisable: soap films, for example, can be used as proxys for two-
dimensional fluids (e.g. Couder & Basdevant, 1986; Couder et al., 1989); with
a suitable wire frame they take the topology of a Möbius strip (e.g. Courant,
1940; Goldstein et al., 2010). The experiments in this paper are however
exclusively numerical.

The Möbius strip is a topological object but fluid dynamics requires a
geometry, which we need to choose. The simplest geometry is that of a flat
strip, imposing antisymmetry conditions across a ‘seam’ to achieve the
necessary twist. However, the flat Möbius strip cannot be embedded in
$\mathbb{R}^{3}$, a necessary condition for physical realisability. We
therefore have chosen another widely used model, the ruled surface constructed
by rotating the midpoint of a segment along a horizontal circle while rotating
the segment in the vertical at half the rate. This surface, which we denote by
$\mathcal{M}$, is parameterised as

$\displaystyle x$ $\displaystyle=(1+\zeta\cos\theta)\cos(2\theta),$ (1a)
$\displaystyle y$ $\displaystyle=(1+\zeta\cos\theta)\sin(2\theta),$ (1b)
$\displaystyle z$ $\displaystyle=\zeta\sin\theta,$ (1c)

with $-a\leq\zeta\leq a$ and $0\leq\theta\leq\pi$ (or rather $\theta$
interpreted as taken modulo $\pi$). The property
$\boldsymbol{x}(\zeta,\theta+\pi)=\boldsymbol{x}(-\zeta,\theta)$ reflects the
twist of the Möbius-strip topology. The parameterisation and shape of
$\mathcal{M}$ are illustrated in Fig. 1. The mean curvature of $\mathcal{M}$
does not vanish, so it is not a minimal surface as formed by a soap film. Its
Gaussian curvature does not vanish either, so it is not developable and cannot
be made of an inextensible material. Closed-form expressions are available for
Möbius strips in $\mathbb{R}^{3}$ that are either minimal (Odehnal, 2016) or
developable (see Schwarz, 1990) and could replace (1) to gain realism. The
equilibrium shape taken by an inextensible material twisted as a Möbius strip
can also be obtained, but this requires solving an energy-minimisation problem
numerically (Starostin & van der Heijden, 2007).

Figure 1: Möbius strip in $\mathbb{R}^{3}$ and its parameter space for
$a=0.25$. The Möbius strip is coloured according to the value of $|g|^{1/2}$,
with $|g|$ the metric determinant in (4), which ranges from about $1.5$ (dark
blue) to $2.5$ (light yellow) and whose variations are associated with
curvature effects.

The non-orientability of $\mathcal{M}$ is manifested in the fact that the
basis $(\partial_{\zeta}\boldsymbol{x},\partial_{\theta}\boldsymbol{x})$ of
its tangent space has opposite orientations for $\theta=0$ and $\theta=\pi$.
Equivalently, if a normal vector $\boldsymbol{N}$ is added to
$\partial_{\zeta}\boldsymbol{x}$ and $\partial_{\theta}\boldsymbol{x}$ to make
up a right-handed basis of $\mathbb{R}^{3}$, it satisfies
$\boldsymbol{N}(\theta=\pi)=-\boldsymbol{N}(\theta=0)$. Non-orientability is a
global property, not discernible at the level of a local patch of the strip;
incompressible fluid dynamics is also global, through the pressure field.
Nonetheless, for narrow strips such as the one in Fig. 1, and with the scale
of fluid structures set by the strip’s width, we can expect the non-
orientability to have a limited impact on the dynamics, which differs from
that in a flat cylinder (channel) primarily because of the strip’s curvature.
The main effect of non-orientability arises from the fact that the sense of
rotation of vortices is reversed when they are transported once along the
entire length of the strip, that is, when $\theta$ increases by $\pi$. We will
illustrate this by considering the dynamics of a single vortex propagating
along the strip’s boundary. A mathematical consequence is that vorticity
cannot be meaningfully integrated over the Möbius strip; as a result, the
familiar connection between circulation and vorticity integral is lost, and
the conservation of integrals of functions of the vorticity (Casimirs) is
restricted to even functions.

The paper is structured as follows. In §2, we derive the
vorticity–streamfunction formulation of the Euler equations for an
incompressible perfect fluid on $\mathcal{M}$ in terms of the coordinates
$(\zeta,\theta)$. We start with a convenient formulation of the Euler
equations that highlights the dual role of the vector and 1-form fields
associated with velocity, and treats vorticity as a 2-form. We carefully
distinguish between forms (such as the vorticity 2-form) and pseudo-forms
(such as the area 2-form, the streamfunction and a pseudo-scalar version of
the vorticity which we term ‘vorticity density’). The distinction is crucial
for a non-orientable surface such as $\mathcal{M}$. We derive the boundary
conditions for the vorticity–streamfunction formulation; these include a
condition involving the circulation along the (single) boundary of
$\mathcal{M}$. In §3 we describe a numerical implementation of the equations
which we employ to examine the propagation of a vortex along the strip’s
boundary, shear instability and decaying turbulence. Section 4 summarises the
main features of vortex dynamics on a non-orientable surface. Two appendices
provide mathematical details.

##  2 Euler equations on the Möbius strip

A coordinate expression for the Euler equations governing the dynamics of an
incompressible perfect fluid on an arbitrary surface was derived by Zermelo
(1902) (see Zermelo (2013) for a commented English translation). Modern
formulations with a focus on point vortices and closed surfaces can be found
in, e.g., Kimura (1999), Boatto & Koiller (2015), Dritschel & Boatto (2015),
Ragazzo & de Barros Viglioni (2017) or Gustafsson (2019). While Zermelo’s
expressions can be readily particularised to the Möbius strip, it is
instructive to start from a general, coordinate-free version of the Euler
equations. A convenient form is

$\displaystyle(\partial_{t}+\mathcal{L}_{\boldsymbol{u}})\nu$
$\displaystyle=-d\left(p-\tfrac{1}{2}|\boldsymbol{u}|^{2}\right),$ (2a)
$\displaystyle\quad\mathop{\mathrm{div}}\nolimits\boldsymbol{u}$
$\displaystyle=0.$ (2b)

Here $\boldsymbol{u}$ is the velocity (vector) field, $\nu$ is the momentum
1-form, obtained from $\boldsymbol{u}$ and the metric $g$ as
$\nu=g(\boldsymbol{u},\cdot)$, $\mathcal{L}_{\boldsymbol{u}}$ is the Lie
derivative and $d$ the exterior derivative (see e.g. Schutz, 1980; Arnold &
Khesin, 1998; Besse & Frisch, 2017; Gilbert & Vanneste, 2018).

The metric on the Möbius strip $\mathcal{M}$ is found by pulling back the
Euclidean metric $dx\otimes dx+dy\otimes dy+dz\otimes dz$ of $\mathbb{R}^{3}$,
that is, by using (1) to express $dx$, $dy$ and $dz$ in terms of $d\zeta$ and
$d\theta$ to find

$g=d\zeta\otimes
d\zeta+\left(4+8\cos\theta\,\zeta+(3+2\cos(2\theta))\zeta^{2}\right)d\theta\otimes
d\theta.$ (3)

This is a diagonal tensor, indicating that the coordinates $(\zeta,\theta)$
are orthogonal (i.e. the basis vectors $\partial_{\zeta}$ and
$\partial_{\theta}$ are orthogonal) with determinant

$|g|=4+8\cos\theta\,\zeta+\left(3+2\cos(2\theta)\right)\zeta^{2}.$ (4)

It follows from (3) that the vectors $\partial_{\zeta}$ and
$\partial_{\theta}$ have norms $|\partial_{\zeta}|=1$ and
$|\partial_{\theta}|=|g|^{1/2}$. Writing the velocity field in components as

$\boldsymbol{u}=u^{\zeta}\partial_{\zeta}+u^{\theta}\partial_{\theta},$ (5)

the corresponding momentum 1-form is found from $\nu=g(\boldsymbol{u},\cdot)$
as

$\nu=\nu_{\zeta}\,d\zeta+\nu_{\theta}\,d\theta=u^{\zeta}\,d\zeta+|g|u^{\theta}\,d\theta.$
(6)

Introducing (6) into (2a) with
$\mathcal{L}_{\boldsymbol{u}}=u^{\zeta}\partial_{\zeta}+u^{\theta}\partial_{\theta}$
and using the commutation
$\mathcal{L}_{\boldsymbol{u}}d=d\mathcal{L}_{\boldsymbol{u}}$ readily gives
the momentum equation in terms of $u^{\zeta}$ and $u^{\theta}$. This is worked
out in Appendix A.

The divergence in (2b) is defined by the relation
$\mathop{\mathrm{div}}\nolimits\boldsymbol{u}\,\mu=d(\boldsymbol{u}\lrcorner\mu)=\mathcal{L}_{\boldsymbol{u}}\mu$,
where $\mu$ is the volume form associated with the metric $g$ (area form since
$\mathcal{M}$ is two-dimensional) and $\lrcorner$ denotes the interior product
or contraction. The area form is given by

$\mu=\pm|g|^{1/2}d\zeta\wedge d\theta.$ (7)

This is a pseudo-2-form, which changes sign when the orientation of the basis
$(\partial_{\zeta}\boldsymbol{x},\partial_{\theta}\boldsymbol{x})$ changes, as
indicated by the notation $\pm$. Since $\mathcal{M}$ is non-orientable, a
change in the orientation of the basis is unavoidable; it occurs across
$\theta=0\mod\pi$ in the parameterisation (1). It can be verified that (7) is
the projection of the Euclidean volume $dx\wedge dy\wedge dz$ on the normal
$\boldsymbol{N}=(N^{1},N^{2},N^{3})$ to the Möbius strip, that is,
$\mu={\boldsymbol{N}}\lrcorner\,dx\wedge dy\wedge dz=N^{1}dy\wedge
dz+N^{2}dz\wedge\mathrm{d}x+N^{3}\mathrm{d}x\wedge dy$. With (7) we compute

$\boldsymbol{u}\lrcorner\mu=\pm|g|^{1/2}(u^{\zeta}\,d\theta-u^{\theta}\,d\zeta),$
(8)

and apply $d$ to obtain the incompressibility condition in the explicit form

$\mathop{\mathrm{div}}\nolimits\boldsymbol{u}={|g|^{-1/2}}\left(\partial_{\zeta}(|g|^{1/2}u^{\zeta})+\partial_{\theta}(|g|^{1/2}u^{\theta})\right)=0.$
(9)

The momentum and incompressibility equations (35) and (9) can be conveniently
replaced by the vorticity equation. This is best derived directly from (2a):
applying $d$ and using that $d$ and $\mathcal{L}_{\boldsymbol{u}}$ commute and
that $d^{2}=0$ gives

$(\partial_{t}+\mathcal{L}_{\boldsymbol{u}})d\nu=0,$ (10)

showing that the vorticity 2-form $d\nu$ is transported by the flow. A
vorticity density $\omega$ can be defined by

$\omega\mu=d\nu.$ (11)

This is a pseudo-scalar, which, like $\mu$, changes sign with the orientation
of the basis on $\mathcal{M}$. Applying $d$ to (5) and using (7) gives the
explicit expression

$\omega=\pm|g|^{-1/2}\left(\partial_{\zeta}(|g|u^{\theta})-\partial_{\theta}u^{\zeta}\right).$
(12)

Incompressibility ensures that
$(\partial_{t}+\mathcal{L}_{\boldsymbol{u}})\mu=0$, hence $\omega$ is also
transported by the flow:

$(\partial_{t}+\mathcal{L}_{\boldsymbol{u}})\omega=0.$ (13)

The velocity $\boldsymbol{u}$ can be related to the vorticity by means of a
streamfunction $\psi$, with

$u^{\zeta}=-|g|^{-1/2}\partial_{\theta}\psi\quad\textrm{and}\quad
u^{\theta}=|g|^{-1/2}\partial_{\zeta}\psi,$ (14)

so as to satisfy the incompressibility condition (9). Equivalently, this can
be written as

$d\psi=-\boldsymbol{u}\lrcorner\mu,$ (15)

where $\lrcorner$ denotes the interior product. With (14), (13) becomes

$\partial_{t}\omega+|g|^{-1/2}\partial(\psi,\omega)=0,$ (16)

where
$\partial(\psi,\omega)=\partial_{\zeta}\psi\partial_{\theta}\omega-\partial_{\theta}\psi\partial_{\zeta}\omega$
is the Jacobian operator. Substituting (14) into (12) then gives the
vorticity–streamfunction relation

$\omega=\Updelta\psi={|g|^{-1/2}}\left(\partial_{\zeta}\left(|g|^{1/2}\partial_{\zeta}\psi\right)+\partial_{\theta}\left(|g|^{-1/2}\partial_{\theta}\psi\right)\right),$
(17)

where we have identified the Laplacian $\Updelta$. A completely geometric
derivation of (14) and (17), which avoids coordinates but requires the
introduction of the Hodge * operator, is given in Appendix B.

Eqs. (16)–(17), together with the form (4) of the metric determinant $|g|$
constitute the vorticity–streamfunction formulation of the Euler equations on
the Möbius strip. In the familiar way, the vorticity density $\omega$ is the
dynamical variable from which the streamfunction $\psi$ is derived by
inverting the Poisson equation (17). This requires boundary conditions.
Because $\omega$ and $\psi$ are pseudo-scalars, they satisfy

$\omega(\zeta,\theta+\pi,t)=-\omega(-\zeta,\theta,t)\quad\textrm{and}\quad\psi(\zeta,\theta+\pi,t)=-\psi(-\zeta,\theta,t).$
(18)

In particular,

$\psi(\zeta,\pi,t)=-\psi(-\zeta,0,t),$ (19)

which provides a first boundary condition for (17). The no-normal-flow
condition

$u^{\zeta}(\zeta=\pm a,\theta,t)=0$ (20)

and (14) imply that $\psi$ depends only on $t$ along the boundary of the
strip. Because of the pseudo-scalar nature of $\psi$, this constancy
translates into

$\psi(\zeta=\pm a,\theta,t)=\pm C(t).$ (21)

This provides the second boundary condition for the Poisson equation (17). To
determine $C(t)$, we need to revert to the momentum formulation (2). It
implies conservation of the circulation along the (single) boundary, given by

$\displaystyle\Gamma$
$\displaystyle=\int_{\partial\mathcal{M}}\nu=\int_{0}^{\pi}\left(\nu_{\theta}(a,\theta,t)+\nu_{\theta}(-a,\theta,t)\right)\,d\theta$
$\displaystyle=\int_{0}^{\pi}\left({|g(a,\theta)|^{1/2}}\partial_{\zeta}\psi(a,\theta,t)+{|g(-a,\theta)|^{1/2}}\partial_{\zeta}\psi(-a,\theta,t)\right)\,d\theta.$
(22)

The equation

$\frac{\mathrm{d}\Gamma}{\mathrm{d}t}=0$ (23)

completes the formulation. The description of our numerical implementation in
§3 makes it clear how (23) determines $C(t)$ for use in (21).

The existence of infinitely many conserved integrals is a key feature of two-
dimensional inviscid incompressible fluids. Here the non-orientablity of the
Möbius strip leads to important differences compared with the familiar,
orientable case. This is because, while pseudo-2-forms can be integrated over
non-orientable surfaces, proper 2-forms cannot – the result of such an
integration would not be coordinate independent (e.g. Frankel, 2004). Thus
while the area of the Möbius strip $\int_{\mathcal{M}}\mu$ is well defined,
the integral $\int_{\mathcal{M}}d\nu$ of the vorticity 2-form is not (and
cannot be related to the circulation $\Gamma$). As a result, conserved
Casimirs functions

$\mathcal{C}_{f}=\int_{\mathcal{M}}f(\omega)\mu=\int_{-a}^{a}\int_{0}^{\pi}f(\omega)|g|^{1/2}\,d\zeta
d\theta,$ (24)

can only be defined for arbitrary even functions $f$. The evenness of $f$ is
required to ensure that the change of sign of $\omega$ which accompanies a
change of basis orientation has no effect on $f(\omega)$, so that
$f(\omega)\mu$ is a pseudo-2-form. The energy

$\mathcal{E}=-\frac{1}{2}\int_{M}\psi\omega\mu=-\frac{1}{2}\int_{-a}^{a}\int_{0}^{\pi}\psi\omega|g|^{1/2}\,d\zeta
d\theta$ (25)

is also well defined and conserved.

##  3 Numerical simulations

###  3.1 Numerical implementation

We solve the vorticity–streamfunction equations (16)–(17) numerically. To
control the generation of fine scales that inevitably accompanies non-trivial
flows, we modify (16), adding a small-scale dissipation to obtain

$\partial_{t}\omega+|g|^{-1/2}\partial(\psi,\omega)=\varepsilon\Updelta\omega,$
(26)

with $\varepsilon>0$ small. This requires an additional boundary condition,
taken as $\omega=0$ on the strip boundary $\partial\mathcal{M}$, corresponding
to a no-stress condition. With dissipation, the circulation $\Gamma$ along
$\partial\mathcal{M}$ changes. We derive the form of
$\mathrm{d}\Gamma/\mathrm{d}t$ in Appendix B where it is given as (41). Note
that the dissipation model chosen is not standard Newtonian friction: the
complete Navier–Stokes vorticity equation involves geometric terms omitted
from the right-hand side of (26) (see Appendix B, Gilbert et al., 2014;
Gilbert & Vanneste, 2021). In the simulations of §3.2, $\varepsilon$ is taken
small enough the results are representative of the limit $\varepsilon\to 0$
except for scales close to the grid size.

Figure 2: Visualisation of the non-zero entries in the matrix discretising the
Laplacian operator on the Möbius strip. The unusual anti-diagonal bands in the
top-right and bottom-left corners result from the twist of the Möbius strip.

To solve (17) and (26) numerically, we discretise $\omega$ and $\psi$ on a
regular grid in $(\zeta,\theta)$ coordinates. The Laplacian $\Updelta$ is
discretised using a straightforward 5-point stencil, accounting for the
condition (19) at the seam $\theta=0$ and assuming homogeneous conditions for
$\zeta=\pm a$ corresponding to the strip’s boundary $\partial\mathcal{M}$. The
resulting matrix is sparse and has an interesting structure displayed in
figure 2, with the standard band structure modified by anti-diagonal blocks in
the top-right and bottom-left corners. These result from the twist of
condition (19). The boundary condition (21) is implemented as follows. We
first compute a numerical approximation to the (time-independent) harmonic
function $\psi_{\mathrm{har}}$ solving

$\Updelta\psi_{\mathrm{har}}=0\quad\textrm{with}\ \
\psi_{\mathrm{har}}(\zeta=\pm a,\theta)=\pm 1,$ (27)

then, at each time step, solve the Poisson equation (17) with homogeneous
boundary condition

$\Updelta\psi_{\mathrm{hom}}=\omega\quad\textrm{with}\ \
\psi_{\mathrm{hom}}(\zeta=\pm a,\theta,t)=0.$ (28)

We compute the circulations $\Gamma_{\mathrm{har}}$ and
$\Gamma_{\mathrm{hom}}(t)$ associated with each solution by direct evaluation
of (22), and we obtain the total circulation $\Gamma(t)$ from the differential
equation (41). The solution of the Poisson equation satisfying the boundary
condition (21) is then calculated as

$\psi=\psi_{\mathrm{hom}}+C(t)\psi_{\mathrm{har}},$ (29)

with $C(t)$ given by

$C(t)=\left(\Gamma(t)-\Gamma_{\mathrm{hom}}(t)\right)/\Gamma_{\mathrm{har}}$
(30)

to ensure that the circulation associated with $\psi$ is $\Gamma(t)$ and (21)
holds.

We use a splitting scheme to advance (26) in time, with a three-step
Adams–Bashforth scheme for the nonlinear advection relying on the Arakawa
(1966) discretisation of the Jacobian, and a backward Euler scheme for the
viscous diffusion. For the simulations reported below, the half-width of the
Möbius strip is set to $a=0.25$; the spatial discretisation uses a $200\times
1000$ grid in $(\zeta,\theta)$; the time step is $0.2$ and the dissipation
parameter is $\epsilon=4\times 10^{-7}$. With this small value, energy is
approximately conserved over the duration of the simulations presented next.

###  3.2 Simulations

Figure 3: Propagation of a vortex along the boundary of the Möbius strip: the
vortex travels counterclockwise along the entire boundary, with a vorticity
vector field $\omega\boldsymbol{N}$ (shown by the arrow in the centre of the
vortex) reversing direction as the along-strip coordinate $\theta$ increases
by $\pi$ (see also movie 1).

We start by simulating the dynamics of a single vortex patch initialised near
the boundary of the Möbius strip. The vortex is expected to travel along the
boundary, forming a dipole with its notional image as is familiar in the
planar case. The Möbius strip case is intriguing in that the strip has a
single boundary, and the direction of rotation of the fluid can be reversed by
transporting the vortex once along the strip. The motion of the vortex is in
fact straightforward: the vortex travels continuously along the entire
boundary. When the along-strip coordinate $\theta$ of the vortex has increased
by $\pi$, so that the vortex returns to the same segment
$\theta=\mathrm{const}$, the sign of $\omega$ is reversed but the vortex
initially near the boundary at $\zeta=a$ is then near $\zeta=-a$ and the
propagation direction is unchanged. This is illustrated in the snapshots shown
in figure 3 and (better) in the accompanying movie 1. These visualise the
vorticity 2-form $d\nu$ which, unlike $\omega$, is a true geometric object,
independent of the choice of coordinate basis. The field $d\nu$ can be
represented as the (true) vector field $\omega\boldsymbol{N}$, normal to
$\mathcal{M}$ and defined intrinsically as dual to $d\nu$ via the
$\mathbb{R}^{3}$ volume-form $dx\wedge dy\wedge dz$. The visualisation in
figure 3 and movie 1 shows $\omega\boldsymbol{N}$ where $|\omega|$ is maximum,
that is, at the centre of the vortex, as well as the scalar field $|\omega|$
on the strip. The propagation of a vortex along the edge of the Möbius strip
makes the single-sidedness of the strip manifest and can be viewed as a fluid-
dynamical equivalent of Escher’s famous print Möbius Strip II showing ants
crawling along the strip.

Figure 4: Shear instability on a Möbius strip: the (pseudo-scalar) vorticity
density $\omega$ is shown at six successive time intervals, starting with a
perturbation of the field (31) (see also movie 2).

We next examine a version of shear instability by simulating the evolution of
the initial vorticity density

$\omega=\zeta(a^{2}-\zeta^{2}).$ (31)

This is not a stationary solution of the Euler equations because the
associated streamfunction depends on $\theta$ through the metric terms that
appear in the Poisson equation (17). Its evolution is nonetheless very slow
compared to that which occurs when a small $\theta$-dependent perturbation is
added – we use a perturbation of the form
$\omega^{\prime}\propto(a^{2}-\zeta^{2})\sin(8\theta)$. The rapid evolution in
the presence of a perturbation can be interpreted as resulting from shear
instability. Note the relative complexity of (31) compared with the quadratic
vorticity profile that is often employed to demonstrate shear instability in
planar flows (a quadratic vorticity has a gradient vanishing at a single
point, the inflection point of the velocity profile). This is imposed by the
topological constraint that 2-forms, here $d\nu=\omega dx\wedge dy$, must
vanish somewhere on non-orientable surfaces. Figure 4 shows a series of
snapshots of the pseudo-scalar vorticity density $\omega$ at equal time
intervals. For distributed vorticity distributions, this field is easier to
visualise than the vector field $\omega\boldsymbol{N}$ shown in figure 3, but
it has the drawback of an abrupt sign change across the seam $\theta=0$,
visible near the top right in the representation of the Möbius strip in figure
4. We emphasise that this discontinuity is artificial and that the physical
fields $\nu$ and $d\nu$ (or equivalently $\omega\boldsymbol{N}$) and the
direction of rotation of the fluid are continuous. The snapshots of figure 4
and the accompanying movie 2 show that shear instability on the Möbius strip
evolves very much as in planar channel, with the amplification of the initial
wavy perturbation of the constant-vorticity lines leading to the formation of
rows of counter-rotating vortices.

Figure 5: Decaying turbulence on a Möbius strip: the (pseudo-scalar) vorticity
density $\omega$ is shown at six successive equal time intervals (see also
movie 3).

Our final simulation, with results displayed in figure 5 and movie 3,
illustrates the dynamics of decaying turbulence on the Möbius strip. We build
a small-scale, random initial vorticity density $\omega_{0}(\zeta,\theta)$,
enforcing its pseudo-scalar property by first constructing a random function
$f(\zeta,\theta)$ that is $4a$-periodic in $\zeta$ and $2\pi$-periodic in
$\theta$ using a standard Fourier series method. The initial vorticity density
is then taken as

$\omega_{0}(\zeta,\theta)=f(\zeta,\theta)-f(2a-\zeta,\theta)+f(2a+\zeta,\theta+\pi)-f(-\zeta,\theta+\pi).$
(32)

The periodicity of $f$ ensures that $\omega_{0}$ vanishes at the boundary and
satisfies condition (18) at the seam $\theta=\pi$, as required for a pseudo-
scalar. The choice of peak wavenumbers $8\pi/a$ and $80\pi$ for $f$ leads to
the reasonably isotropic, small-scale $\omega_{0}$ shown in the first panel of
figure 5. In addition to an initial vorticity density, we also impose an non-
zero (weak) initial circulation which leads to an overall drift of the
vortices best seen in movie 3. The evolution shown in figure 5 and movie 3 is
characteristic of an inverse energy cascade, with vortex mergers that are
local and hence unaffected by the topology of the strip. Over time scales much
larger than those explored in our simulation, the scale of the vorticity field
may become such that the evolution is more strongly influenced by the topology
of $\mathcal{M}$.

##  4 Summary

This paper examines how the Euler equations governing the dynamics of two-
dimensional inviscid fluids can be formulated on a non-orientable surface,
using a Möbius strip embedded in $\mathbb{R}^{3}$ as a concrete example. The
velocity vector field $\boldsymbol{u}$, the corresponding momentum 1-form
$\nu$ and the pressure field $p$ are all geometrically intrinsic objects,
which are independent of the choice of coordinate basis regardless of whether
the surface is orientable or not. Pseudo-fields, which depend on coordinates
through the orientation of the basis, only appear when the Euler equations are
simplified to their vorticity–streamfunction formulation. While the vorticity
$d\nu$ is a proper 2-form, the more convenient $\omega$ is a pseudo-scalar
because its definition via $d\nu=\omega\mu$ involves the area pseudo-2-form
$\mu$. Similarly, the streamfunction $\psi$ is defined via the relation
$d\psi=-\boldsymbol{u}\lrcorner\mu$ and is a pseudo-scalar. With our choice of
coordinates $(\zeta,\theta)$, the difference between pseudo-scalars and
scalars is simply that the former satisfy the twist condition (19) across
$\theta=0\mod\pi$ instead of continuity. In the vorticity–streamfunction
formulation of the Euler equations, the condition that $\psi$ depends on time
only along each connected component of the boundary must be supplemented by
additional conditions tracing back to the original momentum formulation. On
the Möbius strip, the boundary is simply connected and the single function of
time $C(t)$ that needs to be determined is found straighforwardly from
conservation of the circulation along the single boundary. The main dynamical
effect of non-orientability is that the sign of $\omega$ can be changed by
transport: a vortex patch rotating clockwise transported once along the strip
returns rotating counterclockwise, as demonstrated in the numerical experiment
of figure 3. This is why odd functions of the vorticity density are not
conserved in an integral sense. Loosely speaking, the number of integral
conservation laws on the Möbius strip is half what it is on an orientable
surface. It would be interesting to examine the consequences of this for the
long-time dynamics of turbulence, e.g. as described by statistical mechanics.
It would also be interesting to study purely non-dissipative dynamics on a
Möbius strip by considering point vortices or contour dynamics.

Acknowledgments. JV thanks Stefan Llewellyn Smith for pointing out useful
references.

Declaration of interests. The author reports no conflict of interest.

##  Appendix A Momentum equation

We derive the momentum equation in coordinates $(\zeta,\theta)$. The
derivation illustrates the convenience of working directly with forms rather
than their components, as advocated by Frankel (2004). Starting from the
coordinate-free momentum equation (2a) we compute

$\displaystyle(\partial_{t}+\mathcal{L}_{\boldsymbol{u}})\nu$
$\displaystyle=(\partial_{t}+\mathcal{L}_{\boldsymbol{u}})(u^{\zeta}\,d\zeta+|g|u^{\theta}\,d\theta)$
$\displaystyle=(\partial_{t}+\mathcal{L}_{\boldsymbol{u}})(u^{\zeta})\,d\zeta+u^{\zeta}\,d(\mathcal{L}_{\boldsymbol{u}}\zeta)+(\partial_{t}+\mathcal{L}_{\boldsymbol{u}})(|g|u^{\theta})\,d\theta+|g|u^{\theta}\,d(\mathcal{L}_{\boldsymbol{u}}\theta)$
$\displaystyle=(\partial_{t}+\mathcal{L}_{\boldsymbol{u}})(u^{\zeta})\,d\zeta+u^{\zeta}\,du^{\zeta}+(\partial_{t}+\mathcal{L}_{\boldsymbol{u}})(|g|u^{\theta})\,d\theta+|g|u^{\theta}\,du^{\theta}$
(33)

and note that

$d(p-\tfrac{1}{2}|\boldsymbol{u}|^{2})=d(p-\tfrac{1}{2}(u^{\zeta})^{2}-\tfrac{1}{2}|g|(u^{\theta})^{2})=dp-u^{\zeta}\,du^{\zeta}-|g|u^{\theta}\,du^{\theta}-\tfrac{1}{2}(u^{\theta})^{2}(\partial_{\zeta}|g|\,d\zeta+\partial_{\theta}|g|\,d\theta)$
(34)

to find the two components

$\displaystyle(\partial_{t}+\mathcal{L}_{\boldsymbol{u}})u^{\zeta}-\tfrac{1}{2}(u^{\theta})^{2}\partial_{\zeta}|g|$
$\displaystyle=-\partial_{\zeta}p,$ (35a)
$\displaystyle(\partial_{t}+\mathcal{L}_{\boldsymbol{u}})(|g|u^{\theta})-\tfrac{1}{2}(u^{\theta})^{2}\partial_{\theta}|g|$
$\displaystyle=-\partial_{\theta}p,$ (35b)

where
$\mathcal{L}_{\boldsymbol{u}}=u^{\zeta}\partial_{\zeta}+u^{\theta}\partial_{\theta}$
since it acts on scalars.

##  Appendix B Geometric derivation

The vorticity–streamfunction formulation can be derived without resorting to
coordinates. Following Kimura (1999) and Ragazzo & de Barros Viglioni (2017),
this is best carried out using the Hodge * operator which, in two dimensions,
is determined by the following:

$*1=\mu,\quad*\alpha=\alpha^{\sharp}\lrcorner\mu\quad\textrm{and}\quad*\mu=1,$
(36)

where $\alpha$ is a 1-form and $\alpha^{\sharp}$ its dual vector, with
$\alpha=g(\alpha^{\sharp},\cdot)$. In coordinates, we have

$*(\alpha_{\zeta}\,d\zeta+\alpha_{\theta}\,d\theta)=|g|^{1/2}\alpha_{\zeta}d\theta-|g|^{-1/2}\alpha_{\theta}d\zeta.$
(37)

Noting that $**\alpha=-\alpha$ (e.g. Frankel, 2004), we can rewrite (15) and
(17) as

$\nu=*d\psi\quad\textrm{and}\quad\omega=*d\nu=*d*d\psi,$ (38)

which identifies the Laplacian acting on (pseudo-)scalars as $\Updelta=*d*d$.

This form of the Laplacian is also useful to compute the change in the
circulation along the boundary that is introduced by dissipation. For
simplicity, we take the our dissipative model to be

$(\partial_{t}+\mathcal{L}_{u})\nu=\varepsilon\Updelta\nu,$ (39)

where $\Updelta=d*d*+*d*d$ is the Laplace–de Rham operator. We emphasise that
this is not the standard (Navier–Stokes) viscous dissipation which, instead,
involves a different Laplacian defined via the divergence of the stress tensor
(Gilbert et al., 2014; Gilbert & Vanneste, 2021). Integrating over the
(closed) boundary $\partial\mathcal{M}$ and noting that the outer exact
differential in $\Updelta$ integrates to zero gives

$\frac{\mathrm{d}\Gamma}{\mathrm{d}t}=\varepsilon\int_{\partial\mathcal{M}}*d*d\nu=\varepsilon\int_{\partial\mathcal{M}}*d\omega,$
(40)

leading to the coordinate expression

$\frac{\mathrm{d}\Gamma}{\mathrm{d}t}=\varepsilon\int_{0}^{\pi}\left(|g|^{1/2}(d,\theta)\partial_{\zeta}\omega(d,\theta,t)+|g|^{1/2}(-d,\theta)\partial_{\zeta}\omega(-d,\theta,t)\right)\,d\theta$
(41)

on using (37) and that $\omega=0$ on $\partial\mathcal{M}$.

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# Flat Teams Drive Scientific Innovation

Fengli Xu
Knowledge Lab, Department of Sociology
University of Chicago
Chicago, IL
<EMAIL_ADDRESS>
Lingfei Wu
School of Computing and Information
University of Pittsburgh
Pittsburgh, PA
<EMAIL_ADDRESS>
James A. Evans
Knowledge Lab, Department of Sociology
University of Chicago
Chicago, IL
<EMAIL_ADDRESS>

###### Abstract

With teams growing in all areas of scientific and scholarly research, we
explore the relationship between team structure and the character of knowledge
they produce. Drawing on 89,575 self-reports of team member research activity
underlying scientific publications, we show how individual activities cohere
into broad roles of (1) leadership through the direction and presentation of
research and (2) support through data collection, analysis and discussion. The
hidden hierarchy of a scientific team is characterized by its lead (or
L)-ratio of members playing leadership roles to total team size. The L-ratio
is validated through correlation with imputed contributions to the specific
paper and to science as a whole, which we use to effectively extrapolate the
L-ratio for 16,397,750 papers where roles are not explicit. We find that
relative to flat, egalitarian teams, tall, hierarchical teams produce less
novelty and more often develop existing ideas; increase productivity for those
on top and decrease it for those beneath; increase short-term citations but
decrease long-term influence. These effects hold within-person – the same
person on the same-sized team produces science much more likely to
disruptively innovate if they work on a flat, high L-ratio team. These results
suggest the critical role flat teams play for sustainable scientific advance
and the training and advancement of scientists.

_Keywords_ Teams $\cdot$ Innovation $\cdot$ Science of science $\cdot$
Productivity $\cdot$ Novelty $\cdot$ Disruption

##  1 Introduction

Teams are the engines of modern science, having grown in prevalence and size
across all areas of scientific and scholarly investigation Wuchty et al.
(2007). Despite the known importance of team structure in many domains of
economy and society, little is known about how team structure in science
relates to innovation and discovery outcomes from a lack of consistent, large
scale data. Previous experimental and observational studies of emergent team
structure reveal that flatter teams with more balanced Woolley et al. (2010)
or synchronous communication between members Mayo and Woolley (2021); Saavedra
et al. (2011) achieve higher performance in problem solving Woolley et al.
(2010), sales Mayo and Woolley (2021), trading Saavedra et al. (2011) and
healthcare Kreps and Kunimoto (1994) settings, partially because coordinated
attention facilitates the adaptability needed to respond to uncertainty,
complexity and change Uitdewilligen et al. (2010). Hierarchy, by contrast,
accelerates rapid top-down communication and efficiency Anicich et al. (2015),
but necessarily reduces symmetric coordination Hackman (1987) and yields
greater inequality for team member benefits, ranging from higher deaths in
mountaineering expeditions Anicich et al. (2015) to uneven sacrifices in
market contexts Edge and Remus (1984).

Calls for more transparent, honest and equitable credit from the open science
movement have inspired increased mandatory reporting for individual research
contribution on published papers in most high-profile journals. Reporting has
become increasingly standardized to more accurately reflect researcher
contribution and signal contributor skills. In this paper, we use contributor-
level information to explore the relationship between the hierarchy of
individual team contributions and the character of the team’s contribution to
unfolding scientific advance. Recent studies analyzed the division of labor
across stated scientific contributions Larivière et al. (2016); Haeussler and
Sauermann (2020), but did not explore the hierarchical research roles that
emerge from the inequality of contributions (e.g., “core”/“lead” versus
“extended”/“supporting” team members Milojević (2014); Milojević et al.
(2018)).

Here, we demonstrate how specific scientist contributions cohere into
hierarchical roles that lead or follow in support of research publication and
yield a simple lead (or L)-ratio associated with each paper of $n$ authors
ranging from $1/n$ for maximum hierarchy to 1.0 for flatness. Teams with
higher L-ratios broadly share leadership opportunities in fulsome
collaboration, while those with low L-ratios segregate leading from supporting
contributions. We validate these patterns with the position of authors in
paper bylines, the imputed ideas and prior knowledge each scientist
contributes to each paper, and the history of contributions scientists have
made to science as a whole. These signals are available for all papers and
enable robust extrapolation from papers with self-reports to all papers in
science. These patterns reveal how team hierarchy may emerge as teams grow.
When it does, contributions for science differ dramatically.

Teams with higher L-ratios manifest greater novelty in the atypical
combination of ideas Uzzi et al. (2013), while those with lower L-ratios
engage in the development of established research directions Wu et al. (2019).
Teams with higher L-ratios facilitate greater productivity for the average
author, while those with low L-ratios amplify the productivity of just those
on top. Finally, teams with higher L-ratios are associated with the potential
for long-term scientific influence, while those with lower L-ratios contribute
to ensured short-term attention.

##  2 Results

Figure 1: The hidden hierarchy of scientific teams. (a) The co-occurrence of
research activities within individual authors across 89,575 contribution
statements. Three clusters including “Lead (red),” “Direct Support (blue),”
and “Indirect Support (light blue)” are identified. Arrows imply the direction
of influence. We verify L-ratio by demonstrating the distinct contributions of
lead and support authors to specific papers and science as a whole (b). Our
machine learning model classifies Lead and Support authors (precision 0.79,
recall 0.793) and predicts L-ratio (Pearson correlation coefficient 0.66)
(c-d). The composition of team roles (e) and the distribution of L-ratio (f)
changes with team size.

Drawing on 89,575 self-reports of team member research activity underlying
scientific papers published in PNAS, Nature, Science, and PLOS One from 2003
to 2020, we cluster the 25 most common research activities as a function of
their co-contribution by authoring scientists. These activities cluster into
broad roles of (1) leadership through the direction and presentation of
research and (2) direct or indirect research support through data collection,
analysis and discussion (see Fig. 1a). Specifically leadership involves the
activities: “conceive,” “design,” “lead,” “supervise,” “coordinate,”
“interpret,” and “write.” Direct and indirect support coherently separate into
their own clusters. Direct support involves the activities: “help,” “assist,”
“prepare,” “develop,” “collect,” “generate,” “purify,” “carry,” “do,”
“perform,” “conduct,” and “analyze.” Indirect support activities occur before
the research begins and after it is complete, including: “participate,”
“provide,” “contribute,” “comment,” “discuss,” and “edit.” The cynical
observer might reduce these roles to “brain,” “muscle,” and “fat,” the
essential anatomy of modern research teams. By contrast, we demonstrate that
when more members of the team are integrated into leading roles, the character
of research changes and comes to influence the unfolding of scientific advance
in strikingly different ways. The “L-ratio” quantifies the hierarchy of
scientific teams, defined as the fraction of authors playing lead roles among
all team members.

Leading authors make measurably distinct contributions from those playing only
supporting roles. Lead authors are 10% to 20% more likely than average to
introduce references, direct topics, initiate research as first author, and
manage communication as corresponding author. In contrast, support authors are
5% to 10% less likely than average to contribute to these tasks (Fig. 1b). We
find a comparable distinction between lead and support roles when analyzing
scientists’ cumulative contribution to science measured in career age,
citation impact, total number of topics studied, and total number of
previously published papers. These characteristics allowed us to build a
machine learning model to classify authors into “Lead” and “Support” roles
(Fig. 1c), with precision of 0.79 and recall of 0.793, and to robustly predict
the L-ratio of scientific teams (predicted and empirical L-ratios correlate at
0.66) (Fig. 1d). Using these models, we scale our measures of team L-ratios to
the complete sample of 16,397,750 papers published during 1950-2020 where
roles are not reported.

The composition of team roles changes with team size. The proportion of lead
authors peak in teams of size two, authors exclusively playing direct support
roles summit in teams of size 11, and those only in indirect support roles max
at teams of 25 members (Fig. 1e). While L-ratio is clearly associated with
team size such that smaller teams tend to have a higher L-ratio than larger
teams (Fig. 1f), substantial variance in L-ratio for teams of the same size
allows us to disentangle the relative effects of team hierarchy from size.

Figure 2: Tall vs. flat teams and the characters of research output.
Probability of writing a top 10% novel paper (red) increases with L-ratio,
whereas the percentile of development index (blue) decreases with it (A). Lead
authors are less productive in teams with a higher L-ratio (red), whereas
support authors experience productivity gains (blue) (B). Scientific
publications from high L-ratio teams receive more long-term citations after
twenty years (red) but fewer short-term citations within ten (blue) (C).
Bootstrapped 95% confidence intervals are shown as the shaded envelope for all
curves.

Hierarchy is deeply related to characteristics of the resulting research and
its recognition by others in science. The probability of writing a novel paper
(top 10% atypicality) increases with the team’s L-ratio, while the likelihood
that a team will be recognized by others as having incrementally developed
rather than radically disrupted prior ideas, measured by the development index
(the inverse of disruption score Wu et al. (2019)), decreases with it (Fig.
2a). We also find that lead authors are more productive in hierarchical teams
with a lower L-ratio, but support authors experience greater productivity on
flatter teams (Fig. 2b). Scientific publications from low L-ratio teams
receive more short-term citations, while those from high L-ratio teams
experience greater influence over the long term (Fig. 2c). We perform author
fixed-effects regressions and find that the L-ratio continues to play a
consistent and substantial role after controlling for team size and the career
age variance of team members in predicting all outcome variables (see SI for
details). This is important as changing the size or altering the age structure
of a team for optimal innovative performance involves hiring or firing
members, but our findings suggest the possibility that by simply re-organizing
tasks, the same goal may be achieved: leading roles can be extended across
large teams and involve younger scientists to maximize innovative potential.

##  3 Discussion

Tall teams provide obvious benefits for scientists who lead them, but do not
necessarily maximize the productivity and innovation potential of those who
support. They enable greater lead author productivity, maximize immediate
citations and so hedge against the risk that their work will not be credited.
Under some circumstances, hierarchies may represent the most efficient
allocation of effort for their research purposes, but they may impose costs on
supporting scientists who do not play leadership roles and produce fewer
papers. While the causality of these patterns remains unclear, our author
fixed-effect models suggest that as the same researchers shift from teams with
lower to higher L-ratios, their opportunities for leadership and productivity
expand, corroborating prior research on the distribution of member risk
associated with team hierarchy Anicich et al. (2015); Edge and Remus (1984).

Building on prior research about team member specialization Larivière et al.
(2016); Haeussler and Sauermann (2020), we uncover the hierarchy of research
roles and compare the lasting contributions of tall versus flat teams. Tall
teams produce less novel, more developmental, and shorter-lived contributions
to science, suggesting that scaling teams to increase innovation poses a
paradox, especially as sponsored science increasingly shifts from “little” to
“big” in the name of accelerating advance. In addition to steady growth in
team size Wuchty et al. (2007), team hierarchy has also markedly increased
over the past half century. If we define tall teams as those with an L-ratio
below .5 – where less than half of team members contribute to the conceptual
work, these increase from 50% in 1950 to 70% in 2015. Concerns over scientific
stagnation have arisen from apparent diminishing returns to scientific
investment, inferred from accelerated growth in publications and aging
scientists Blau and Weinberg (2017) but slowed expansion in new ideas Chu and
Evans (2021). Here we reveal the place of team hierarchy in the landscape of
innovation, and the critical role flat teams play in advancing supporting
scientists to grow next-generation scientific workforce for sustainable, long-
term scientific advance.

##  4 Methods

###  4.1 Datasets

We link and analyze two data sets, Microsoft Academic Graph (MAG) and author
contribution statements from four journals. Our MAG data include 16,397,750
journal articles written by two or more authors from 184 countries during
1950-2015. These papers contain 7.4 MAG topic keywords and 30.5 references on
average, and the average number of citations to them is 41.2. The average team
size (number of authors) is 4.8. We collect the author contribution statements
from four journals, including PNAS, Nature, Science, and PLOS ONE. Our data
cover a substantial body of all available contribution statements from the
time they were required, covering 13 years of PNAS (18,354 from 2003 to 2015),
15 years of Nature (9,364 from 2006 to 2020), 3 years of Science (1,176 from
2018 to 2020), and 9 years of PLOS ONE (60,681 from 2006 to 2014). We apply
NLP algorithms to extract the _subject_ and _predicate_ of each sentence and
identify the research activities associated with authors. The top 25
activities (Fig. 1a) cover 94.6% data and each paper lists 5.2 unique
activities on average. The “L-ratio,” the fraction of Lead authors in the
team, is 0.38 for Nature, 0.38 for Science, 0.45 for PNAS, and 0.43 for PLOS
ONE.

###  4.2 Predicting Lead vs. Support Roles and L-ratio

We apply Modularity-based algorithm Newman (2006) and identify three clusters
from the co-occurred research activities, including “Lead”, “Direct Support”
and “Indirect Support”. We analyze the distinct characteristics of Lead and
Support authors (Fig. 1b) and build a neural network to predict author roles
in 16,397,750 papers without explicit author contribution statements with
precision of 0.79 and recall of 0.793. We extend this model by including team
size and the unevenness of contributions and successfully predict L-ratio with
a Pearson correlation coefficient equaling 0.66 between predicted and
empirical values.

###  4.3 Quantifying Team Performances: Novelty, Development, Citations

Novelty (Fig.2a) quantifies to what extent an paper draws upon and combines
atypical ideas Uzzi et al. (2013). We applied a skip-gram word2vec model to
learn the vector representation of MAG topic keywords from their co-occurrence
within papers. The typicality of a pair of topic keywords is calculated as the
inner product of their embedding vectors and novelty is defined as negative
typicality. The developmental index (Fig.2a) is defined as the inverse of
disruption Wu et al. (2019). The Developmental index measures to what extend a
paper incrementally refines previous work rather than radically challenging
it. Productivity (Fig.2b) is measured as the total number of papers published
by an author in the same year when they contributed to the specific paper
under analysis. We separately analyzed the productivity of lead authors and
support authors for each paper (Fig. 2b). We measure short-term impact as the
number of citations a paper receives within ten years and long-term impact as
citations received after twenty years.

###  4.4 Evaluating the Impact of L-ratio on Team Performances

We evaluate the impact of L-ratio on different team performances by regressing
novelty, developmental index, productivity of lead authors, productivity of
support authors, short-term citation impact and long-term citation impact on
L-ratio, respectively. For each model, we also control for team size, mean
career age, and the standard deviation of career ages by including them
together with L-ratio as independent variables. Finally, we perform fixed-
effect regression to examine the impact of L-ratio for the same person moving
between different teams.

##  5 Acknowledgment

We are grateful for support from AFOSR (#FA9550-19-1-0354), NSF (#1829366 and
#1800956), and DARPA (#HR00111820006). L. W. acknowledges the support of Pitt
Cyber and Richard King Mellon Foundation.

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